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+preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text +preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/README.md b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..d235c91e40f0da1225f2169ed15864b77ad368a6 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/README.md @@ -0,0 +1,17 @@ +# [One-sided negative sectional curvature and the holomorphic Liouville property](paper.pdf) + +OpenAI +September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026, + author = {{OpenAI}}, + title = {{One-sided negative sectional curvature and the holomorphic Liouville property}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf}{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/03-analytic.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/03-analytic.tex new file mode 100644 index 0000000000000000000000000000000000000000..f452d61bb5b8835515fddae5512e1ead87d5ca55 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/03-analytic.tex @@ -0,0 +1,760 @@ +\section{Polynomial tests for bounded holomorphic functions} +\label{sec:analytic} + +We will force the $w$-derivative of every bounded holomorphic function +$f$ on the tube to vanish. The elementary estimate comes from a holomorphic +disk through a point $(s_0,\zeta,w_0)$: +\[ + t\longmapsto(s_0,\zeta+t,w_0+i\alpha b t),\qquad |t|0,\quad |b|=1. +\] +If this disk lies in the tube and $|f|\le1$, Cauchy's inequality gives +\[ + |f_y(s_0,\zeta,w_0)+i\alpha b f_w(s_0,\zeta,w_0)|\le L^{-1}. +\] +We must prevent the two derivatives from canceling at many spatial +points. We arrange those points in root-of-unity cycles and hold $b$ +constant on each cycle. A finite Fourier average then keeps only Taylor +degrees in one residue class. The phases will separate the prescribed +lowest-degree part of $f_w$ from the first few surviving terms of $f_y$, +with one phase choice valid for every polynomial of that degree. + +The grids constructed below provide equidistributed projective directions +for polynomial sampling and cyclic affine lifts for this extraction. +Their additional separation and shell estimates will be used in +Section~\ref{sec:assembly} to localize the shear coefficients in disjoint +spatial neighborhoods when realizing the disks. After constructing the grids +and choosing the phases, we formulate the precise disk conditions and +prove that repeated tests force the holomorphic Liouville property. + +Throughout this section, $\sigma$ denotes the unitary-invariant +probability measure on $\mathbb P^{n-1}$. All norms on vectors and +derivatives are Euclidean unless indicated otherwise. Constants may +depend on the fixed dimension $n$; the exponents in the separation +estimate below do not. + +\subsection{Polynomial grids and their affine lifts} + +\begin{lemma}[Homogeneous polynomial grids] +\label{grid:polynomials} +Fix $n\ge2$. For every sufficiently large integer $N$ one can choose a +system of homogeneous polynomials of degree $N$, +\[ + Q_N=(H_N,p_N):\mathbb C^n\longrightarrow\mathbb C^{n-1}\times\mathbb C, +\] +with the following properties. These choices can be made along all +integers $N\to\infty$, or along any prescribed sequence of degrees +tending to infinity. +\begin{enumerate} +\item There is a constant $C_n$ such that +\begin{equation} + \label{grid:sphere-bounds} + N^{-C_n}\le |Q_N(u)|\le N\qquad (|u|=1). +\end{equation} +\item The projective zero set of $H_N$ consists of +$M_N=N^{n-1}$ transverse points. A subset $\mathcal G_N$ of these +points satisfies +\[ + |\mathcal G_N|=(1-o(1))M_N, + \qquad + \frac1{M_N}\sum_{\xi\in\mathcal G_N}\delta_\xi + \ \rightharpoonup\ \sigma. +\] +For every $\xi\in\mathcal G_N$ and every unit representative $u$ of +$\xi$, +\begin{equation} + \label{grid:goodness} + N^{-1}\le |p_N(u)|\le N, + \qquad + \operatorname{sing}_{\min} + \bigl(DH_N(u)|_{u^\perp}\bigr)\ge N^{-1}. +\end{equation} +\item Above each $\xi\in\mathcal G_N$, choose any lift $v_\xi$ such +that $p_N(v_\xi)=1$. All $N$ lifts +\[ + e^{2\pi i q/N}v_\xi,\qquad 0\le qN)\le C_n' N^{2n-1}e^{-c_n' N^2}\longrightarrow0. +\] +On the event $A\le N$, the Lipschitz constant on the sphere is at +most $CN^2$. Use a second net of mesh $cN^{-C_n-2}$. If the norm +somewhere on the sphere is less than $N^{-C_n}$, it is less than +$2N^{-C_n}$ at a point of this net. A standard complex Gaussian +vector in $\mathbb C^n$ has probability at most $C_n' r^{2n}$ of +lying in a ball of radius $r$. The union bound is therefore at most +\[ + C_n' N^{(C_n+2)(2n-1)-2nC_n} + =C_n' N^{-C_n+4n-2}. +\] +Choose, for example, $C_n=8n$. This proves both sphere bounds with +probability tending to one. + +Put $d=n-1$. Almost surely the projective zeros of the $d$-component +system $H_N$ are transverse. To see this directly, let $\mathcal A$ +be its finite-dimensional complex coefficient space. Evaluation at +any point of $\mathbb P^d$ maps $\mathcal A$ onto the fiber of +$\mathcal O(N)^{\oplus d}$. Hence the universal zero incidence +manifold is smooth, and a coefficient vector is a regular value of its +projection to $\mathcal A$ precisely when the corresponding section +has only transverse zeros. Sard's theorem gives a Lebesgue-null +exceptional set; the Gaussian law is absolutely continuous. +Transversality and compactness imply that the zero set is finite. +The projective B\'ezout theorem for a zero-dimensional intersection of +$d$ hypersurfaces of degrees $N,\ldots,N$ gives total intersection +number $N^d$; every transverse intersection has multiplicity one +\citep[Theorem~III-71]{EisenbudHarris2000GeometrySchemes}. Thus the number of zeros is the deterministic +quantity $M_N=N^d$ almost surely. + +We next justify the counting formulas needed for equidistribution. +They are the one- and two-point, zero-dimensional cases of the +correlation formula in \citet[Theorem~2.1]{BleherShiffmanZelditch2000}. +Our subsequent off-diagonal argument proves the particular convergence +needed here; the more general simultaneous-zero convergence theorem of +\citet[Corollary~1.3]{Shiffman2008Convergence} supplies a broader context. +In a holomorphic coordinate chart of $\mathbb P^d$, regard the system +as a holomorphic map $H_a(z)$ depending linearly on its coefficients. +Locally split those coefficients as $(a,b)$ so that +\[ + H_{a,b}(z)=E(z)a+R(z,b),\qquad \det E(z)\ne0. +\] +This is possible because evaluation is surjective. The equation +$H_{a,b}(z)=0$ is equivalent to +$a=A(z,b):=-E(z)^{-1}R(z,b)$. At a zero, with +$J=D_zH_{a,b}(z)$, +\[ + D_zA=-E(z)^{-1}J. +\] +For fixed $b$, the change-of-variables formula with multiplicity for +$z\mapsto A(z,b)$ thus introduces the real Jacobian +$|\det E|^{-2}|\det J|^2$. Integrating in $b$ against the smooth +Gaussian coefficient density yields the one-point zero density +\begin{equation} + \label{grid:one-point-density} + \rho_{1,N}(z)= + p_{H_N(z)}(0)\, + \mathbb E\bigl[|\det DH_N(z)|^2\mid H_N(z)=0\bigr]. +\end{equation} +The same argument works with a bounded measurable mark multiplying +the zero count. It first applies to nonnegative marks, and the +general bounded case follows by decomposition. + +For two distinct projective points, joint evaluation onto the two +fibers is surjective. Indeed, powers of linear forms that vanish at +one point and not the other give independent prescribed values for +each component. Solve now for $2d$ coefficient variables and apply +the same change of variables in the two point coordinates. The +point Jacobian is block diagonal, giving +\begin{equation} + \label{grid:two-point-density} + \begin{split} + \rho_{2,N}(z,z')={}&p_{(H_N(z),H_N(z'))}(0,0)\\ + &\quad\cdot + \mathbb E\bigl[ + |\det DH_N(z)|^2|\det DH_N(z')|^2 + \mid H_N(z)=H_N(z')=0\bigr]. + \end{split} +\end{equation} +Partitions of unity give these formulas on projective space and on +the complement of its diagonal, respectively. This derivation uses +only the coefficient density and ordinary change of variables; no +independence of zeros has been assumed. + +At a point represented by a unit vector $u$, take affine tangent +coordinates $u+\sum_{j=1}^d z_j e_j$ with $(e_j)$ orthonormal in +$u^\perp$. Differentiating the covariance shows that $H_N(u)$ is +independent of $N^{-1/2}DH_N(u)|_{u^\perp}$, and that the latter is +a $d\times d$ matrix of independent standard complex Gaussians. +At two distinct projective points, the cross-covariances of these +normalized value and first-derivative data tend to zero locally +uniformly off the diagonal. More explicitly, on a compact set of +pairs with $|u\cdot\bar v|\le\rho<1$, they are bounded by a fixed +power of $N$ times $\rho^{N-2}$. This follows by differentiating +$(u\cdot\bar v)^N$ at most once at each point. The joint Gaussian +covariance consequently converges uniformly there to the direct sum +of its one-point covariances. Conditional Gaussian covariances given +the values zero converge as well. Gaussian moments of the fixed +polynomials $|\det J|^2$ and their products therefore converge. +Equations \eqref{grid:one-point-density} and +\eqref{grid:two-point-density} imply that the two-point density, +divided by $M_N^2$, converges locally uniformly off the diagonal to +the product of the normalized one-point densities. + +The exact total number of zeros lets us pass from off-diagonal +decorrelation to weak convergence without estimating the two-point +density near the diagonal. Let +\[ + \mu_N=\frac1{M_N}\sum_{H_N(\xi)=0}\delta_\xi. +\] +Unitary invariance and the deterministic count imply +$\mathbb E\mu_N=\sigma$. The positive measure +$\mathbb E(\mu_N\otimes\mu_N)$ has total mass one. Off the +diagonal, its density converges to that of $\sigma\otimes\sigma$, +by the preceding calculation. This implies convergence on the +whole product: given $\varepsilon>0$, choose a compact subset of the +off-diagonal region having $(\sigma\otimes\sigma)$-mass greater +than $1-\varepsilon$. Its mass under +$\mathbb E(\mu_N\otimes\mu_N)$ is greater than $1-2\varepsilon$ +for all sufficiently large $N$. At most $2\varepsilon$ of the +total mass can remain elsewhere. Since the diagonal has zero +$(\sigma\otimes\sigma)$-mass, there is no additional limiting +mass there. Thus +\[ + \mathbb E(\mu_N\otimes\mu_N) + \rightharpoonup\sigma\otimes\sigma. +\] +For every continuous test function, its integral against $\mu_N$ +therefore has variance tending to zero. This proves weak +convergence to $\sigma$ in probability. + +It remains to discard the bad zeros. Use +\eqref{grid:one-point-density} with marks. If $G$ is a $d\times d$ +standard complex Gaussian matrix, then at a zero the derivative +condition in \eqref{grid:goodness} fails only when +$\operatorname{sing}_{\min}(G)N) + =1-e^{-N^{-2}}+e^{-N^2}\longrightarrow0. +\] +Markov's inequality shows that the total discarded fraction tends +to zero in probability. Removing that fraction does not alter weak +convergence. The modulus and singular-value conditions are +unchanged on multiplying a unit representative by a phase, so they +hold for every such representative. + +We may now select deterministic systems. Take a countable uniformly +dense family of continuous functions on projective space, and impose +successively accurate conditions on successively longer finite +subfamilies. Convergence in probability, the vanishing bad +fraction, and the probability-one transversality statement show that +these conditions and \eqref{grid:sphere-bounds} have a nonempty +intersection for every sufficiently large degree. Choose one system +from that intersection. A diagonal choice gives all the asserted +limits and works equally well along any prescribed sequence of +degrees. + +We have obtained the projective sampling properties. It remains to +provide disjoint neighborhoods in which Section~\ref{sec:assembly} can +localize its shear coefficients. Fix a good direction and a lift +$v$ with $p_N(v)=1$. +Homogeneity and \eqref{grid:goodness} give +$N^{-1/N}\le |v|\le N^{1/N}$. In the splitting of the domain into +the radial complex line and its Hermitian orthogonal complement, +$DH_N(v)$ vanishes on the radial line, while +$Dp_N(v)$ has radial derivative $N/|v|$. The transverse derivative +of $H_N$ has least singular value at least $N^{-4}$ for large $N$: +the scaling from a unit representative costs at most one additional +factor $N$, so this is a deliberately weaker bound than necessary. +Homogeneity, the sphere upper bound, and Cauchy's inequality at +scale $c/N$ give first- and second-derivative bounds $O_n(N^5)$ in +the stated neighborhood. At the lift itself, the transverse first +derivatives are $O_n(N^4)$. Block inversion therefore gives +\[ + \bigl\|D(H_N,p_N-1)(v)^{-1}\bigr\|\le N^9 +\] +for all sufficiently large $N$. Taylor's formula, uniformly for +$|z-v|\le4N^{-30}$, now yields +\[ + |(H_N(z),p_N(z)-1)| + \ge N^{-9}|z-v|-C_n' N^5|z-v|^2 + \ge\tfrac12 N^{-9}|z-v|. +\] +This proves \eqref{grid:shell-bound}. It also excludes every other +zero of $(H_N,p_N-1)$ within distance $4N^{-30}$, proving +disjointness of the balls about all good lifts. +\end{proof} + +\subsection{Choosing phases uniformly for a fixed Taylor order} + +The grids now supply many separated evaluation points. Their next role +is linear algebraic: we choose a phase at each projective point so that +no fixed-order polynomial jet can be hidden among the first few Taylor +degrees surviving a root-of-unity average. The independent-phase moment +calculation is the same quadratic-form identity used for random-phase +trace estimates in \citet{IitakaEbisuzaki2004}. +The additional rank and polynomial superlevel estimates give one phase +choice uniformly for an entire coefficient space. + +\begin{lemma}[Phases separate low-degree jets] +\label{grid:phases} +Fix positive integers $k,n$, with $n\ge2$, such that +\begin{equation} + \label{grid:dimension-choice} + \sum_{j=1}^{k-1}\frac{j^{n-1}}{(n-1)!}<\frac1{100}. +\end{equation} +Fix a nonnegative integer $h$. For sufficiently large polynomial +grids from Lemma~\ref{grid:polynomials}, choose one lift $v_\xi$ +above each good direction and let $V_N\subset\mathbb C^{\mathcal G_N}$ +be the space of evaluation vectors of sums of homogeneous +polynomials of degrees +\[ + h,\ h+N,\ \ldots,\ h+(k-1)N. +\] +Let $\Pi_N$ be orthogonal projection onto $V_N$ for the unweighted +Euclidean point norm. There are phases $b_\xi$, $|b_\xi|=1$, and a +constant $\delta_h>0$, independent of $N$ and of the chosen lift +phases, such that +\begin{equation} + \label{grid:phase-lower-bound} + M_N^{-1/2} + \left\|(1-\Pi_N) + \bigl(b_\xi P(v_\xi)\bigr)_{\xi\in\mathcal G_N}\right\| + \ge\delta_h\|P\| +\end{equation} +for every homogeneous polynomial $P$ of degree $h$. Here $\|P\|$ +is any fixed Euclidean norm on its coefficient space, chosen +independently of the grid. All required grid accuracies and degree +thresholds depend only on the fixed $n,k,h$ and this norm. +\end{lemma} + +\begin{proof} +Dimension gives +\[ + \frac{\operatorname{rank}\Pi_N}{M_N} + \le N^{1-n}\sum_{j=0}^{k-1} + \binom{h+jN+n-1}{n-1}<\frac2{100} +\] +for all sufficiently large $N$. Indeed its limit is the sum in +\eqref{grid:dimension-choice}. + +There is a number $t_h>0$ such that, on every sufficiently accurate +grid, every polynomial with $\|P\|=1$ satisfies +\begin{equation} + \label{grid:uniform-superlevel} + |P(v_\xi)|>t_h + \quad\hbox{at at least }0.8M_N\hbox{ good directions}. +\end{equation} +For completeness, a nonzero homogeneous polynomial has a +projective zero set of $\sigma$-measure zero. In affine charts this +is the elementary fact that the zero set of a nonzero complex +polynomial has Lebesgue measure zero, obtained by induction and +Fubini's theorem. Thus, for each norm-one polynomial $P$, some +positive threshold has a superlevel set of measure greater than +$0.9$. Uniform continuity in the coefficients makes the same +superlevel estimate, with a smaller threshold, valid in a +neighborhood of $P$. The coefficient unit sphere is compact, so +finitely many such neighborhoods suffice and their thresholds have +a positive common lower bound. Weak convergence of the grid +measures applied to these finitely many open superlevel sets gives +the estimate for unit representatives, with a fixed margin in both +threshold and counting proportion. Since +$|v_\xi|^h\to1$ uniformly, it gives +\eqref{grid:uniform-superlevel}. This proof requires only finitely +many fixed approximation conditions for the given $h$. + +The projection has $0\le(\Pi_N)_{\xi\xi}\le1$ and diagonal sum +less than $0.02M_N$. Consequently at most $0.04M_N$ diagonal +entries exceed $1/2$. Choose the $b_\xi$ temporarily as independent +uniform phases. Writing $Q_N'=1-\Pi_N$, independence gives +\[ + \begin{split} + &\mathbb E\left[ + M_N^{-1}\left\|Q_N' + \bigl(b_\xi P(v_\xi)\bigr)_\xi\right\|^2\right]\\ + &\hspace{25mm}= + M_N^{-1}\sum_\xi + \bigl(1-(\Pi_N)_{\xi\xi}\bigr)|P(v_\xi)|^2 + \ge0.38t_h^2 + \end{split} +\] +when $\|P\|=1$. The last inequality uses the intersection of the +$0.8M_N$ indices in \eqref{grid:uniform-superlevel} with the +complement of the at most $0.04M_N$ high-leverage indices. + +We need a single phase choice for all polynomials, rather than a +choice depending on $P$. Let $(e_a)$ be a fixed orthonormal basis +of the degree-$h$ coefficient space, and put $q_a(\xi)=e_a(v_\xi)$. +These evaluations have a common bound depending only on $n,h$ and +the chosen norm. The Gram matrix of the random linear map in +\eqref{grid:phase-lower-bound} has entries +\[ + G_{ab}=M_N^{-1}\sum_{\xi,\eta} + \overline{q_a(\xi)}\,\overline{b_\xi} + (Q_N')_{\xi\eta}b_\eta q_b(\eta). +\] +Its diagonal-index terms are deterministic. For distinct indices, +the products $\overline{b_\xi}b_\eta$ are orthonormal in +$L^2$ of the phase probability space: two such ordered products +have nonzero inner product only when their ordered pairs agree. +In particular, reversal of the pair gives expectation +$\mathbb E(\overline{b_\xi}^{\,2}b_\eta^2)=0$. +It follows that +\[ + \begin{split} + \mathbb E|G_{ab}-\mathbb EG_{ab}|^2 + &\le C_{n,h}M_N^{-2} + \sum_{\xi\ne\eta}|(\Pi_N)_{\xi\eta}|^2\\ + &\le C_{n,h}M_N^{-2}\operatorname{rank}\Pi_N + =O_{n,h}(M_N^{-1}). + \end{split} +\] +The Gram matrix has fixed finite size. Chebyshev's inequality +therefore implies that its operator-norm distance from its mean +tends to zero in probability. Its mean is bounded below by +$0.38t_h^2$ times the identity. A phase choice with least +eigenvalue at least $0.19t_h^2$ exists for every sufficiently large +grid. Taking $\delta_h=\sqrt{0.19}\,t_h$ proves the assertion. +All estimates used only absolute values of lift evaluations, so +the threshold and lower bound are uniform over their arbitrary +representative phases. +\end{proof} + +\begin{remark} +\label{grid:finite-dimension} +For every fixed $k$, condition \eqref{grid:dimension-choice} holds +in some sufficiently large finite dimension. Indeed, with +$r=n-1$, its left side is bounded by +$(k-1)^{r+1}/r!$, which tends to zero as $r\to\infty$. +The argument will use one such fixed dimension, not an assertion +in every dimension. +\end{remark} + +\subsection{From disk estimates to the holomorphic Liouville property} + +\begin{lemma}[Root-of-unity extraction] +\label{grid:frequency} +Let $0\le hp_1$. Choose an integer $k$ +and a finite dimension $n\ge2$ so that +\begin{equation} + \label{grid:exponent-choice} + 0.99k>p_0+p_1+1, + \qquad + \sum_{r=1}^{k-1}\frac{r^{n-1}}{(n-1)!}<0.01. +\end{equation} +Let $(\zeta_j,h_j)$ be a sequence of pairs from a countable dense +subset of an open disk in $\mathbb C$ and the nonnegative integers, +with every pair repeated infinitely often. At stage $j$, choose +numbers $R_j\ge1$, $c_j\ge1$, an integer $N_j>h_j$, and a grid +from Lemma~\ref{grid:polynomials} sufficiently large and accurate +for Lemma~\ref{grid:phases} at order $h_j$. Set +\[ + S_j=R_j e^{c_j/N_j},\qquad \log N_j\le c_j/100. +\] +Choose heights $D_j$ and positive numbers $\alpha_j$ such that +\[ + \alpha_j^{-1}\le e^{p_1c_j}, +\] +and, along the repetitions of every fixed pair, +\begin{equation} + \label{grid:height-rates} + c_j\longrightarrow\infty,\qquad + D_j\longrightarrow\infty,\qquad + \frac{D_j}{c_j}\longrightarrow0. +\end{equation} +There are phases $b_{j,\xi}$, one per good projective direction +and constant throughout its cycle of lifts, with the following +property. + +Let $\ell$ be a smooth real function on $\mathbb C^n\times\mathbb C$, +and let +\[ + \mathcal T=\{(s,y,w):\operatorname{Im}w>\ell(s,y)\}. +\] +Suppose every stage satisfies these conditions: +\begin{enumerate} +\item On $|s|\le S_j$, +\begin{equation} + \label{grid:height-gap} + \ell(s,\zeta_j)\le D_j-2. +\end{equation} +\item For every holomorphic $f$ on $\mathcal T$ with $|f|\le1$, +every $w$ with $\operatorname{Im}w=D_j$, and every $|s|=S_j$, +\begin{equation} + \label{grid:sphere-derivative} + |f_y(s,\zeta_j,w)|\le e^{p_0c_j}. +\end{equation} +\item For every good direction $\xi$, every lift +$v=e^{2\pi iq/N_j}v_\xi$, and every $w$ on that height line, the +holomorphic disk +\begin{equation} + \label{grid:sheared-disk} + t\longmapsto + \bigl(R_jv,\ \zeta_j+t,\ w+i\alpha_j b_{j,\xi}t\bigr), + \qquad |t|\ell(0,\zeta)$. At every point of this +half-plane a spatial neighborhood is available, so the derivative +is well-defined and holomorphic there. Choose once a small +closed spatial polydisk about zero and a number $t_0$ exceeding +the supremum of $\ell(s,\zeta)$ on that polydisk by more than two. +Cauchy's inequalities in $s$ and on a unit $w$-disk give +\[ + |g_\beta(w)|\le B_\beta + \qquad (\operatorname{Im}w\ge t_0), +\] +where $B_\beta$ is independent of every testing stage and of +$\operatorname{Re}w$. Along the stages assigned to $(\zeta,h)$, +\eqref{grid:jet-decay} gives +\[ + \sup_{\operatorname{Im}w=D_j}|g_\beta(w)| + \le C_h e^{-\varepsilon c_j}. +\] +For any fixed $t>t_0$, apply Hadamard's three-lines theorem +\cite[Theorem~7.13]{Tropp2022} to the horizontal strip between heights +$t_0$ and $D_j$, once $D_j>t$. +Writing $\theta_j=(t-t_0)/(D_j-t_0)$, it yields +\[ + \sup_{\operatorname{Im}w=t}|g_\beta(w)| + \le B_\beta^{\,1-\theta_j} + (C_h e^{-\varepsilon c_j})^{\theta_j}. +\] +If $B_\beta=0$ there is nothing to prove. Otherwise, the +logarithm of the right side is +\[ + \log B_\beta+ + \frac{t-t_0}{D_j-t_0} + \bigl(\log(C_h/B_\beta)-\varepsilon c_j\bigr) + \longrightarrow-\infty +\] +by \eqref{grid:height-rates}. Hence $g_\beta$ vanishes in the +upper half-plane and, by the identity theorem, throughout +$\operatorname{Im}w>\ell(0,\zeta)$. Repeating this argument for +each order proves that all spatial Taylor coefficients of $f_w$ +at zero vanish at every scheduled center. Dependence of the +constants and of the preliminary height $t_0$ on the order causes +no difficulty: each order has its own infinite sequence, and +analytic continuation gives the same full half-plane afterwards. + +To pass from these statements to a common open set, choose a +spatial ball about zero and an open subdisk of the center disk, +both with compact closures. Choose $L$ larger than the supremum +of $\ell$ on their product. On the product with +$\operatorname{Im}w>L$, the function $f_w$ is jointly holomorphic. +At each scheduled center in the subdisk, all of its spatial +Taylor coefficients vanish for every such $w$. Taylor uniqueness +and the identity theorem on the spatial ball give vanishing on +that ball. Continuity in $y$ and density of the scheduled centers +then give vanishing on the whole product. The tube is connected: +the coordinates +\[ + (s,y,\operatorname{Re}w, + \operatorname{Im}w-\ell(s,y)) +\] +identify it with +$\mathbb C^{n+1}\times\mathbb R\times(0,\infty)$. +The several-variable identity theorem therefore gives $f_w=0$ +everywhere on $\mathcal T$. + +For every fixed $(s,y)$ the $w$-fiber is a connected half-plane, +so $f$ is constant along that fiber. It descends to a bounded +function $F$ on $\mathbb C^{n+1}$. This descended function is +holomorphic: near any base point choose a single constant $w_0$ +whose imaginary part exceeds the supremum of $\ell$ on a small +base neighborhood, and write $F(s,y)=f(s,y,w_0)$ there. These +local holomorphic definitions agree on overlaps by fiber +constancy. Liouville's theorem, applied on complex lines in +$\mathbb C^{n+1}$, makes $F$, and therefore $f$, constant. +\end{proof} + +\begin{remark} +\label{grid:geometric-interface} +Condition \eqref{grid:sphere-derivative} follows immediately if, +at the stated sphere points and heights, the unsheared $y$-disk +of radius $e^{-p_0c_j}$ lies in the tube. Thus the criterion asks +the geometric construction for only three quantitative facts: a +height bound on a spatial ball, sufficiently wide unsheared disks +on its boundary, and much wider disks in the prescribed sheared +directions at its polynomial nodes. The phase choices are made +before any bounded holomorphic function is considered. +\end{remark} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/04-fiber.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/04-fiber.tex new file mode 100644 index 0000000000000000000000000000000000000000..ad22a78f5989b6688a9ff967fe9f7ad1b6af507c --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/04-fiber.tex @@ -0,0 +1,326 @@ +\section{The fiber Legendre transform and its curvature}\label{sec:fiber} + +The analytic criterion leaves us a geometric task: construct the base +potential while retaining explicit curvature margins for later changes. +We first derive the metric and its six curvature blocks for a fixed fiber +center. The partial Legendre transform uses, up to our fixed metric +normalization, the fiber momentum of the circle-invariant constructions +of Calabi and Hwang--Singer \citep{Calabi1979,HwangSinger2002}. +We allow the radial profile to vary over the spatial base, which adds +spatial-derivative terms to the curvature. We compute the full tensor +before estimating these additional terms. + +All spatial derivatives in this section are taken with the fiber momentum +\(\tau\) fixed, unless a different convention is stated explicitly. +The spatial variables are \(s=(s_1,\ldots,s_n)\), and the fiber variable is +\(y\in\mathbb C\). Its center \(\zeta\) is constant in this section. +Let \(q(x,s)>0\), where \(x=\log\tau\), and choose \(F\) by +\begin{equation}\label{fiber:F} + F_x=q^{-1},\qquad \lim_{x\to-\infty}(F(x,s)-x)=0. +\end{equation} +We assume that this normalization is possible, that \(F\to+\infty\) +as \(x\to+\infty\), and that \(q\), regarded as a function of \(\tau\), +is smooth at zero with \(q(0,s)=1\). All these statements are local +uniformly in \(s\). For smooth real functions \(B,\psi\), set +\begin{align} + U(s,\tau)&=B(s)+\tau\psi(s)-\int_0^\tau F(\log v,s)\,dv, + \label{fiber:U}\\ + \rho&=\log|y-\zeta|^2=F(\log\tau,s)-\psi(s), + \label{fiber:inverse}\\ + \ell_{\mathrm{rad}}&=U+\tau\rho + =B+\int_0^\tau q(\log v,s)^{-1}\,dv. + \label{fiber:potential} +\end{align} +The last identity follows by integration by parts; the boundary term +\(vF(\log v,s)\) tends to zero as \(v\downarrow0\). +Since \(F_x>0\) and its endpoint limits are infinite with the indicated +signs, \eqref{fiber:inverse} determines a unique \(\tau>0\) for every +\(y\ne\zeta\). + +Hereafter \(C=-R\) denotes the negative Hermitian curvature tensor: +\begin{equation}\label{fiber:curvature-convention} + C_{\alpha\bar\beta\gamma\bar\delta} + =\partial_\alpha\partial_{\bar\beta}g_{\gamma\bar\delta} + -g^{\mu\bar\nu} + (\partial_\alpha g_{\gamma\bar\nu}) + (\partial_{\bar\beta}g_{\mu\bar\delta}). +\end{equation} +The sectional-curvature sign associated with this convention is verified +below; in particular, positivity of the relevant contractions of \(C\) +means nonpositive Riemannian sectional curvature. + +\begin{lemma}[Metric and regularity of the Legendre transform] +\label{fiber:metric} +Write +\[ + P=\tau q,\qquad V=(B_{i\bar j}),\qquad + G=(\psi_{i\bar j}),\qquad h=(U_{i\bar j}),\qquad + \mathcal N=h_\tau. +\] +Where \(h>0\), the complex Hessian of \(\ell_{\mathrm{rad}}\) is +\begin{equation}\label{fiber:metric-form} + h_{i\bar j}\,ds_i\,d\bar s_j+ + P\left|d\log(y-\zeta)+U_{\tau i}\,ds_i\right|^2. +\end{equation} +It is positive off the axis. Its spatial Schur complement is exactly +\(h\). The potential and its complex Hessian extend smoothly across +\(y=\zeta\); on that axis the Hessian in the original splitting is +\(V\oplus e^\psi\), and is positive whenever \(V>0\). +\end{lemma} + +\begin{proof} +Use the holomorphic coordinate \(z_0=\log(y-\zeta)\), so that +\(\rho=z_0+\bar z_0\). Differentiating \(U_\tau+\rho=0\) gives +\[ + U_{\tau\tau}=-P^{-1},\qquad + \partial_{z_0}\tau=P,\qquad + \partial_{s_i}\tau=P U_{\tau i}. +\] +Differentiation of \(\ell=U+\tau\rho\) now gives +\[ + g_{0\bar0}=P,\qquad g_{i\bar0}=P U_{\tau i},\qquad + g_{i\bar j}=h_{i\bar j}+P U_{\tau i}U_{\tau\bar j}, +\] +which is \eqref{fiber:metric-form}. For regularity, write +\(F=\log\tau+f(\tau,s)\). The equation +\(f_\tau=(q^{-1}-1)/\tau\), with \(f(0,s)=0\), shows that \(f\) +is smooth. Equation \eqref{fiber:inverse} becomes +\(\tau e^{f(\tau,s)}=e^\psi|y-\zeta|^2\). Its derivative with respect +to \(\tau\) at zero is one, so the smooth implicit-function theorem +applies. In particular, +\(\tau=e^\psi|y-\zeta|^2+O(|y-\zeta|^4)\). +Substitution in \eqref{fiber:potential} proves the stated extension and +axis metric. +\end{proof} + +We use a local holomorphic change \(z_0\mapsto z_0+f(s)\) to make +\(U_{\tau i}=0\) at a specified point. The corresponding change of +\(U\) is linear in \(\tau\) with a pluriharmonic spatial coefficient; +it leaves \(h\) and \(\mathcal N\) unchanged. The quadratic jet of +\(f\) can also make \(U_{\tau ik}=0\) there. We call these the tilted +logarithmic coordinates. The coordinate changes used to evaluate a jet +are held fixed while that jet is differentiated. + +\begin{lemma}[The six curvature blocks]\label{fiber:blocks} +In tilted logarithmic coordinates, the tensor +\eqref{fiber:curvature-convention} has the following blocks: +\begin{align} + A:=C_{0\bar0 0\bar0} + &=P^2P_{\tau\tau}-|\partial P|_h^2, + \label{fiber:blockA}\\ + A_i:=C_{i\bar0 0\bar0} + &=P\bigl(\partial_iP_\tau- + (\mathcal N h^{-1}\partial P)_i\bigr), + \label{fiber:blockAi}\\ + W_{i\bar j}:=C_{i\bar j0\bar0} + &=P\bigl(\partial_i\partial_{\bar j}\log P + +P_\tau\mathcal N_{i\bar j} + -P(\mathcal N h^{-1}\mathcal N)_{i\bar j}\bigr), + \label{fiber:blockW}\\ + C_{i\bar0 k\bar0}&=(\nabla_h^{2,0}P)_{ik}, + \label{fiber:blockspin}\\ + K_{ik\bar j}:=C_{i\bar j k\bar0} + &=P\nabla_i^h\mathcal N_{k\bar j} + +P_i\mathcal N_{k\bar j}+P_k\mathcal N_{i\bar j}, + \label{fiber:blockK}\\ + L_{i\bar j k\bar l}:=C_{i\bar j k\bar l} + &=C^h_{i\bar j k\bar l} + +P(\mathcal N_{i\bar j}\mathcal N_{k\bar l} + +\mathcal N_{i\bar l}\mathcal N_{k\bar j}). + \label{fiber:blockL} +\end{align} +Here \(C^h\) and \(\nabla^h\) are the curvature with sign +\eqref{fiber:curvature-convention} and the K\"ahler connection of the +fixed-\(\tau\) spatial metric. Index positions in the products use the +indicated Hermitian metric, with the usual conjugations. +\end{lemma} + +\begin{proof} +In addition to the tilt, first choose spatial coordinates normal for +\(h\) at the point. On functions of \((s,\tau)\), physical coordinate +derivatives act as +\[ + D_i=\partial_i+P U_{\tau i}\partial_\tau, + \qquad D_0=P\partial_\tau, +\] +and their conjugates. At the point the complete list of cubic entries +needed below is +\begin{equation}\label{fiber:cubics} + \ell_{00\bar0}=PP_\tau,\quad + \ell_{i0\bar0}=P_i,\quad + \ell_{00\bar j}=P_{\bar j},\quad + \ell_{i0\bar j}=P\mathcal N_{i\bar j},\quad + \ell_{ik\bar0}=0,\quad \ell_{ik\bar j}=0. +\end{equation} +For example, the identity +\(P^2\partial_\tau U_{\tau i}=P_i\), obtained by spatial +differentiation of \(U_{\tau\tau}=-1/P\), and its conjugate give the +second and third entries. +Directly differentiating the metric entries before evaluating gives +\begin{align*} + D_0D_{\bar0}P&=P(P_\tau^2+PP_{\tau\tau}),\\ + D_iD_{\bar0}P&=P_iP_\tau+P\partial_iP_\tau,\\ + D_iD_{\bar j}P&=P_{i\bar j}+P\mathcal N_{i\bar j}P_\tau,\\ + D_iD_{\bar j}(PU_{\tau k}) + &=P_i\mathcal N_{k\bar j}+P_k\mathcal N_{i\bar j} + +P\partial_i\mathcal N_{k\bar j},\\ + D_iD_{\bar j}(h_{k\bar l}+PU_{\tau k}U_{\tau\bar l}) + &=\partial_i\partial_{\bar j}h_{k\bar l} + +P\mathcal N_{i\bar j}\mathcal N_{k\bar l} + +P\mathcal N_{i\bar l}\mathcal N_{k\bar j}. +\end{align*} +For completeness, the fourth derivative giving the spin block has a +cancellation which is useful later. Put \(a_i=U_{\tau i}\). Before +evaluation, +\[ + \ell_{ik\bar0}=Pa_{ik}+P_i a_k+P_k a_i+PP_\tau a_i a_k. +\] +At the point \(a_i=a_{ik}=0\), while +\[ + (a_i)_\tau=P_i/P^2,\qquad + (a_{ik})_\tau=P_{ik}/P^2-2P_iP_k/P^3. +\] +Applying \(D_{\bar0}=P\partial_\tau\) therefore gives \(P_{ik}\): +the two positive terms \(P_iP_k/P\) cancel the negative doubled term. + +Subtract the cubic contractions in +\eqref{fiber:curvature-convention}. For A the vertical contraction is +\(PP_\tau^2\) and the spatial one is \(|\partial P|_h^2\). +For \(A_i\) they are \(P_iP_\tau\) and +\(P(\mathcal Nh^{-1}\partial P)_i\). For \(W\) they are +\(P_iP_{\bar j}/P\) and \(P^2(\mathcal Nh^{-1}\mathcal N)_{i\bar j}\). +The first combines with \(P_{i\bar j}\) to give +\(P\partial_i\partial_{\bar j}\log P\). +For the purely spatial block the contraction is the one defining +\(C^h\). These calculations prove all formulas in normal spatial +coordinates. Without spatial normalization, the last entry of +\eqref{fiber:cubics} is \(\partial_i h_{k\bar j}\); its contractions +replace \(P_{ik}\) by \(\nabla^h_i\partial_kP\) and +\(\partial_i\mathcal N_{k\bar j}\) by +\(\nabla_i^h\mathcal N_{k\bar j}\). This also verifies the displayed +tensorial formulas in general spatial coordinates. +\end{proof} + +The six blocks give the complete tensor, but the desired sign concerns +real sectional curvature. The next lemma identifies the matrix tests +corresponding to real two-planes and records exactly how much of the +diagonal curvature can absorb each mixed block. + +We specify the norm convention needed to keep all subsequent constants +independent of \(n\). The norm of a covariant tensor is its +multilinear operator norm on unit vectors in the stated fixed frame. +The norm of a Hermitian test matrix is its Frobenius norm. Thus, for a +Hermitian matrix \(T\) of rank at most two and a covariant Hermitian form \(Q\), +\begin{equation}\label{fiber:ranknorm} + |Q:T|\le\sqrt2\|Q\|\,|T|. +\end{equation} +Indeed diagonalize \(T\); the sum of the absolute values of its at most two +nonzero eigenvalues is at most \(\sqrt2|T|\). +The same decomposition bounds a curvature pairing \(C(T,T)\) by +\(2\|C\||T|^2\). Contractions of cubic tensors through an inverse +metric can instead be regarded as inner products of two covectors. +Their operator bounds therefore involve no summation factor depending +on the dimension. + +\begin{lemma}[Real planes and rank-two absorption]\label{fiber:ranktwo} +If \(C(H,H)\ge0\) for every Hermitian contravariant matrix \(H\) of rank +at most two, the underlying real sectional curvatures are nonpositive. +Write the spatial, cross, and fiber diagonal blocks of \(H\) as +\(T,b_\times,t\). The contraction has the terms +\begin{equation}\label{fiber:contraction} + At^2+2t(W:T)+2W(b_\times,\bar b_\times)+L(T,T), +\end{equation} +and its other terms have absolute value at most +\begin{equation}\label{fiber:otherterms} + 2|\nabla_h^{2,0}P(b_\times,b_\times)| + +4|tA_i b_\times^i|+4|(K:T)b_\times|. +\end{equation} +In particular the following quantitative criterion suffices. Suppose +\(A>0\), \(W>0\), and a nonnegative quadratic form \(\mathcal R(T)\) +satisfies +\begin{align} + L(T,T)-\frac{|W:T|^2}{.98A}&\ge\mathcal R(T), + \label{fiber:reserve}\\ + |\nabla_h^{2,0}P(b,b)|&\le\tfrac14 W(b,\bar b), + \label{fiber:spinabs}\\ + |A_i b^i|&\le\eta\sqrt{A W(b,\bar b)}, + \label{fiber:Aiabs}\\ + |(K:T)b|&\le\eta\sqrt{\mathcal R(T)W(b,\bar b)}, + \label{fiber:Kabs} +\end{align} +for all such tests, where \(\eta\le1/100\). Then +\begin{equation}\label{fiber:retained} + C(H,H)\ge .01At^2+W(b_\times,\bar b_\times) + +\tfrac12\mathcal R(T). +\end{equation} +All constants are independent of dimension. +\end{lemma} + +\begin{proof} +For real vectors with \((1,0)\) parts \(v,w\), set +\(H=i(v\otimes\bar w-w\otimes\bar v)\). This is Hermitian of +rank at most two. K\"ahler symmetry gives +\[ + C(H,H)=2\{C(v,\bar v,w,\bar w) + -\operatorname{Re}C(v,\bar w,v,\bar w)\}. +\] +Expand \(R(v+\bar v,w+\bar w,w+\bar w,v+\bar v)\). The only four +nonzero terms are the two alternating type patterns and their +conjugates. In the convention in which a real vector with complex +components \(v\) has squared length \(g_{i\bar j}v^i\bar v^j\), the complex +bilinear extension of the real metric on a holomorphic and an +antiholomorphic vector is one half the Hermitian pairing. Accordingly +the real curvature numerator is \(-\tfrac12 C(H,H)\). This proves the +sign assertion. +Expanding a general block matrix \(H\) gives +\eqref{fiber:contraction}--\eqref{fiber:otherterms}; each mixed +conjugate pair accounts for its factor two or four. A principal block +\(T\) has rank at most two. + +For the quantitative assertion use +\(2t(W:T)\ge-.98At^2-|W:T|^2/(.98A)\). +The remaining diagonal coefficient is .02A. The spin term costs at +most \(W/2\). The other two terms obey +\[ + 4\eta |t|\sqrt{AW}\le .01At^2+400\eta^2W, + \qquad + 4\eta\sqrt{\mathcal R W}\le\tfrac12\mathcal R+8\eta^2W. +\] +Since \(1.5-408\eta^2\ge1\) when \(\eta\le1/100\), +\eqref{fiber:retained} follows. +\end{proof} + +\begin{lemma}[Exact perturbation identity]\label{fiber:perturbation} +In any fixed holomorphic coordinates let \(H\) denote the cubic jet viewed +as a map from two tangent inputs to a covector, and write \(H=\Gamma g\). +For a perturbation with \(g+\Delta g>0\), the change of the cubic +contraction \(Hg^{-1}H^*\) is exactly +\begin{equation}\label{fiber:perturbformula} + \Gamma\Delta H^*+\Delta H\Gamma^*-\Gamma\Delta g\Gamma^* + +(\Delta H-\Gamma\Delta g)(g+\Delta g)^{-1} + (\Delta H-\Gamma\Delta g)^*. +\end{equation} +Consequently, if \(\|g^{-1}\Delta g\|\) in metric-unit axes is at most +one half, the inverse of the perturbed metric is at most twice that +of \(g\), and +\begin{align}\label{fiber:perturbbound} + \|\Delta C\|\le {}&\|\Delta j^4\ell\| + +2\|\Gamma\|\|\Delta H\| + +\|\Gamma\|^2\|\Delta g\|\\ + &+2\|g^{-1}\| + (\|\Delta H\|+\|\Gamma\|\|\Delta g\|)^2. + \notag +\end{align} +The same bound, with an absolute factor, controls rank-two contractions. +\end{lemma} + +\begin{proof} +Set \(Q=\Delta H-\Gamma\Delta g\). Then +\(H+\Delta H=\Gamma(g+\Delta g)+Q\). Multiply out the corresponding +cubic contraction and subtract \(\Gamma g\Gamma^*\). The two cross +terms and \(\Gamma\Delta g\Gamma^*\) combine to give the first three +terms in \eqref{fiber:perturbformula}. Taking operator norms gives +\eqref{fiber:perturbbound}. The inverse comparison is the elementary +spectral estimate for \(I+g^{-1/2}\Delta g g^{-1/2}\); the final statement +uses \eqref{fiber:ranknorm} twice. +\end{proof} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/05-quiet.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/05-quiet.tex new file mode 100644 index 0000000000000000000000000000000000000000..fe2990b27888daca4f2091cfe36601f98d70a66a --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/05-quiet.tex @@ -0,0 +1,600 @@ +\section{A spatially varying quiet fiber profile}\label{sec:quiet} + +We now choose a radial fiber profile that tolerates spatial variation +of its parameters. Its steep part gives the curvature margins needed +near the testing disks; its logarithmic tail keeps the whole affine +fiber available. After establishing the scalar estimates, we use the +six-block formulas to prove a uniform curvature criterion and then +show that it survives localized shear terms. + +Fix \(b_0\ge1000\) and \(K_0\ge1\); increasing \(K_0\) later is +allowed. Define a positive function \(u\) of \(\tau>0\) by +\begin{equation}\label{quiet:u} + u=\tau\exp\left(\int_0^\tau + \frac{(1+v)^{-1/16}-1}{v}\,dv\right). +\end{equation} +It satisfies \(u/\tau\to1\) at zero and +\(d\log u/d\log\tau=(1+\tau)^{-1/16}\). It is increasing, is +comparable to \(\tau\) on \((0,1]\), and tends to a finite positive +limit at infinity: the last assertion follows by integrating +\((1+\tau)^{-1/16}d\tau/\tau\) from 1 to infinity. +More specifically, convexity gives +\((1+v)^{-1/16}\ge1-v/16\), whence +\(e^{-1/16}\tau\le u\le\tau\) for \(0<\tau\le1\). +For a real parameter \(E\), called the severity, set +\begin{equation}\label{quiet:q} + q=q_0(x,E)=(1+e^E u)^{b_0}+K_0^{-1}\log(1+\tau), + \qquad x=\log\tau. +\end{equation} +Constants in this section may depend on \(b_0,K_0\), but are independent +of the spatial dimension. A lower threshold for \(E\) will always be +chosen sufficiently large for these constants. The primitive \(F\) has +the normalization \eqref{fiber:F}. Because \(q-1=O_E(\tau)\) at zero, +this normalization exists. Because \(q\) grows only linearly in \(x\) at +the right end, \(F\to+\infty\) there. + +\begin{lemma}[Scalar estimates for the profile]\label{quiet:scalar} +Put +\[ + q_a=(1+e^Eu)^{b_0},\quad r=r_q=q_x/q,\quad + D=D_q=(q_x+q_{xx})/q=r(1+r)+r_x,\quad e_0=q_E/q. +\] +For all sufficiently large \(E\) and all \(\tau>0\), +\begin{gather} + q\ge1,\quad q_x>0,\quad + D\ge r(r+\tfrac12),\quad q_{xx}/q_x\ge-\tfrac1{16}, + \label{quiet:slopes}\\ + 00\). Taking the weighted mean with +weights \((q_a)_x,(q_b)_x\) gives the other slope bound. + +Every positive \(E\) derivative of \(q_a\) of order at most four is bounded +by \(Cq_a\min(1,z)\). Differentiating \(q^{-1}\) thus gives +\begin{equation}\label{quiet:reciprocal} + |\partial_E^j(q^{-1})| + \le C\frac{q_a}{q^2}\min(1,z),\qquad 1\le j\le4. +\end{equation} +Likewise \((q_a)_x\) and its \(E\) derivative have ratio bounded in +absolute value by a profile constant, so \(|r_E|\le Cr\). +Since \(a\ge2^{-1/16}\) for \(\tau\le1\), comparison with \(p\) gives +\(e_0,q_{EE}/q\le Cr\) there. The remaining bounds in +\eqref{quiet:derivatives} follow directly from the displayed formulas. + +Differentiate the normalized integral for \(F\). For \(x\le0\), use +\(d x=d\log z/a\), where \(a\ge2^{-1/16}\), in +\eqref{quiet:reciprocal}; the resulting absolute integral is at most +\[ + C\int_0^\infty\frac{\min(1,z)}{(1+z)^{b_0}}\,\frac{dz}{z}<\infty. +\] +For \(x\ge0\), put \(M=e^{b_0E}\). The positive lower and upper +bounds on \(u\) for \(\tau\ge1\) imply +\(cM\le q_a\le CM\) and +\(q\ge cM+x/K_0\). The bound in \eqref{quiet:reciprocal} is therefore +at most \(CM/(cM+x/K_0)^2\), whose integral over \([0,\infty)\) +is uniformly bounded. This proves \eqref{quiet:Fderivatives}, also +for the average by integration. + +Integration by parts gives, including for \(j=0\), +\begin{equation}\label{quiet:averageidentity} + \partial_E^j(F-\bar F) + =\int_0^1\partial_E^j(q^{-1})(\log\tau+\log t,E)\,dt. +\end{equation} +If \(\tau\ge e^{-E/4}\), split at \(t=e^{-E/4}\). On the first +part the integrand is uniformly bounded. On the second part +\(\tau t\ge e^{-E/2}\), and +\(u(v)\ge c\min(v,1)\) implies \(e^E u(\tau t)\ge c e^{E/2}\). +There the integrand, also for \(j=0\), is at most \(C e^{-b_0E/2}\). +This proves the first bound in \eqref{quiet:difference}. +For \(\tau\ge2\), split instead at \(t=\tau^{-1/2}\). +Monotonicity and the positive finite limit of \(u\), together with +\(\log(1+\sqrt\tau)\ge\tfrac12\log(1+\tau)\), give +\(q(\log\sqrt\tau,E)\ge c q(\log\tau,E)\). Thus the integral is +at most \(C(\tau^{-1/2}+q^{-1})\). +The estimates +\(q\le C(M+\log\tau)\) and +\(q_x\ge c(1+M\tau^{-1/16})\) show that +\(q\tau^{-1/2}\le Cq_x\) and \(1\le Cq_x\); hence this last bound is +at most \(Cr\). On \([e^{-E/4},2]\) the explicit formula for \(p\) gives a +positive lower bound on \(r\). This completes \eqref{quiet:difference}. + +The estimates \eqref{quiet:tail} follow from the same two summands of \(q\). +For \(1\le\tau\le e^{3b_0E}\), the logarithmic summand divided by +\(q_a\) tends uniformly to zero as the lower threshold on \(E\) increases; +also \(z/(1+z)\to1\) uniformly. This proves +\eqref{quiet:middle-slope}. On +\([e^{-E/4},e^E]\) one has +\(z\ge c e^{3E/4}\) and \(a\ge c e^{-E/16}\), so +\[ + q_x\ge(q_a)_x=b_0az(1+z)^{b_0-1} + \ge c\exp((3b_0/4-1/16)E)\ge e^E. +\] +Finally, on \((0,2e^{-E}]\), the quantity \(z\) is comparable to +\(e^E\tau\) and remains bounded by a profile constant. The formulas +for the first two \(x\) derivatives of \(q\) then give +\eqref{quiet:axis-slope}. Between \(2e^{-E}\) and 1 the same formulas +give \(q_x\ge c\); for \(\tau\ge1\), use \eqref{quiet:tail}. +These statements also prove the two final consequences. +\end{proof} + +The scalar estimates make two tasks compatible: the spatial Schur +metric remains positive, and the fiber curvature absorbs the trace +couplings in a real-plane test. The following criterion states the +spatial hypotheses needed for both tasks. The word \emph{quiet} refers +to the smallness of the severity derivatives in the metric of the twist. + +\begin{proposition}[Quiet profile criterion]\label{quiet:profile} +There exist positive constants \(\epsilon,\mu\) and a lower threshold +\(E_{\min}\), depending on the fixed profile constants but not on \(n\), +with the following property. Let \(E\) be a smooth spatial function and +use \(q=q_0(x,E(s))\) with constant center and no shear. At a spatial +point suppose, in some fixed holomorphic affine coordinates, that: +\begin{enumerate} +\item \(G\ge V>0\). The operator norms of \(V,G\), their inverses, +their first and second metric jets, and all derivatives of \(E\) of orders +one through four are at most \(e^{\epsilon E}\). +\item For every Hermitian contravariant matrix \(T\) of rank at most two, +\(C^V(T,T)\ge e^{-\epsilon E}|T|^2\), with the norm in those affine +coordinates. +\item In \(G\)-unit axes, +\[ + |\partial E|_G^2\le\mu,\qquad + \|\partial\bar\partial E\|_G\le\mu,\qquad + \|\nabla_h^{2,0}E\|_G\le\mu + \quad\hbox{for every }\tau>0. +\] +\end{enumerate} +If \(E\ge E_{\min}\), the resulting metric is positive for every \(y\) +over that spatial point and has nonpositive real sectional curvature +there. Uniformly in \(\tau\), +\begin{equation}\label{quiet:schur} + \mathcal N=(1+O(\mu))G,\qquad + h\asymp V+\tau G. +\end{equation} +More precisely, off the fiber axis the rank-two curvature calculation +retains fixed positive multiples of +\begin{equation}\label{quiet:margins} + At^2+W(b_\times,\bar b_\times)+P|T|_{\mathcal N}^2 + +\boldsymbol{1}_{\{\tau\le e^{-E/4}\}} + e^{-\epsilon E}|T|^2. +\end{equation} +Here the fiber blocks use the tilted logarithmic coordinates, +\(|T|_{\mathcal N}\) is its Frobenius norm in \(\mathcal N\)-unit +spatial axes, and \(\boldsymbol{1}\) denotes the indicated indicator. +The meaning of the final hypothesis is noncircular: positivity of \(h\) +follows from the preceding hypotheses and does not use +\(\nabla_h^{2,0}E\). +\end{proposition} + +\begin{proof} +Differentiating \(U_\tau=\psi-F\) gives +\begin{equation}\label{quiet:N} + \mathcal N=G-F_E\partial\bar\partial E + -F_{EE}\partial E\otimes\bar\partial E, + \qquad h=V+\tau\bar{\mathcal N}. +\end{equation} +The bar denotes averaging from 0 to \(\tau\). The bounded derivatives +in \eqref{quiet:Fderivatives} give +\((1-C\mu)G\le\mathcal N,\bar{\mathcal N}\le(1+C\mu)G\). +Choose \(\mu\) small enough that \(C\mu<1/10\). This proves +positivity and \eqref{quiet:schur}. + +We will repeatedly use finite-jet estimates of the following precise +kind. Differentiating \eqref{quiet:N} at most twice spatially gives +sums with at most four derivatives of \(E\) and at most four \(E\) derivatives +of \(F\) or \(\bar F\). Each summand is a product of a fixed number of +the assumed jets. The formulas for the inverse, connection, and +curvature add a fixed number of inverse metric factors. Consequently +there is an absolute finite constant \(C_*\), independent of dimension, +such that these estimates cost at most \(C e^{C_*\epsilon E}\), with +the factors \(1+\tau\) or \((1+\tau)^{-1}\) displayed explicitly +below. This conclusion follows by the product rule and +\(h^{-1}\le C e^{\epsilon E}(1+\tau)^{-1}I\); it uses operator norms, +not sums of tensor entries. The same observation applies after +normalizing spatial axes by \(\mathcal N\), because +\(e^{-\epsilon E}I/C\le\mathcal N\le C e^{\epsilon E}I\). +Enlarge \(C_*\) once to cover these finitely many products, and choose +\(\epsilon>0\) so small that \(C_*\epsilon<1/100\). +Any profile-dependent multiplicative constants are absorbed by raising +\(E_{\min}\). + +In \(\mathcal N\)-unit axes define +\begin{equation}\label{quiet:Y} + \mathbf Y=I-\tau\mathcal N^{1/2}h^{-1}\mathcal N^{1/2}. +\end{equation} +The displayed formula indicates the normalization; equivalently it is +\(I-\tau h^{-1}\) after \(\mathcal N=I\) has been imposed in the +fixed axes at the point. We claim +\begin{equation}\label{quiet:Ybounds} + \|\mathbf Y\|\le r/20\quad(\tau\ge2e^{-E}),\qquad + \mathbf Y=I+O(e^{-(1-C_*\epsilon)E}) + \quad(\tau\le2e^{-E}). +\end{equation} +Always \(\|\mathbf Y\|\le1+C\mu\), because +\(h\ge\tau\bar{\mathcal N}\) and +\(\bar{\mathcal N}\) is close to \(\mathcal N\). +On \([2e^{-E},1]\), the explicit slope \(p\) in the scalar lemma yields +\(r\ge .35b_0\) for all sufficiently large \(E\): indeed +\(u\ge e^{-1/16}\tau\), \(z\ge2e^{-1/16}\), +\(a\ge2^{-1/16}\), and +\(q_a/q\ge(1+\log2/K_0)^{-1}\ge(1+\log2)^{-1}\). +The product of these three lower bounds in +\(p(q_a/q)=b_0a[z/(1+z)](q_a/q)\) exceeds \(.35b_0\). +This proves the first bound on that interval. +For \(\tau\ge1\), \eqref{quiet:difference} gives +\(\|\bar{\mathcal N}-\mathcal N\|_{\mathcal N}\le C\mu r\). +Set \(h_0=V+\tau\mathcal N\). The inverse identity and +\(h,h_0\ge c\tau\mathcal N\) show that replacing \(h\) by \(h_0\) in +\eqref{quiet:Y} costs at most \(C\mu r\). In the normalized axes +\(V\le\gamma I\), where \(\gamma=(1-C\mu)^{-1}\), and hence +\[ + \|I-\tau h_0^{-1}\|\le\frac\gamma{\gamma+\tau}. +\] +This is at most \(r/40\) on \([1,e^{3b_0E}]\), by +\eqref{quiet:middle-slope} and \(b_0\ge1000\). +For \(\tau>e^{3b_0E}\), use \(r\ge c/q\) and +\(q/\tau\le C(e^{b_0E}+\log(1+\tau))/\tau\); the supremum of the +last expression on this range tends to zero with the lower threshold +on \(E\). Thus the same estimate holds there. Decreasing \(\mu\) pays +the inverse-identity error. Finally, if \(\tau\le2e^{-E}\), then +\(h\ge V\) and the raw size bounds give +\(\tau\|\mathcal N^{1/2}h^{-1}\mathcal N^{1/2}\| + \le C\tau e^{C_*\epsilon E}\), proving the second bound. + +Since \(P_\tau=q(1+r)\) and +\(P^2P_{\tau\tau}=Pq^2D\), the curvature blocks in +Lemma~\ref{fiber:blocks} become +\begin{align} + \frac W{Pq}&=rI+\mathbf Y+q^{-1}\partial\bar\partial\log q, + \label{quiet:Wformula}\\ + P^{-1}\nabla_h^{2,0}P + &=e_0\nabla_h^{2,0}E+(q_{EE}/q)\partial E\otimes\partial E. + \label{quiet:spinformula} +\end{align} +The final term in \eqref{quiet:Wformula} and the spin form divided by +\(Pq\) have norm at most \(C\mu r\) when \(\tau\ge2e^{-E}\), +and at most \(C\mu\) on the remaining interval. This follows from +\eqref{quiet:derivatives}, the three smallness assumptions on \(E\), and +\(q^{-1}\le Cr\) away from the exceptional interval. +After decreasing \(\mu\), +\begin{equation}\label{quiet:Wbounds} + W\asymp Pqr\,\mathcal N\quad(\tau\ge2e^{-E}),\qquad + W\asymp Pq(1+r)\,\mathcal N\quad(\tau\le2e^{-E}), +\end{equation} +and the spin inequality \eqref{fiber:spinabs} holds. + +For the fiber diagonal term, +\[ + \frac{|\partial P|_h^2}{Pq^2D} + \le C\mu\frac{e_0^2}{qD}\le C\mu. +\] +Here \(h\ge c\tau G\); the final scalar quotient is bounded using +\(e_0\le Cr\) for \(\tau\le1\), and +\(qD\ge cqr=cq_x\ge c\) for \(\tau\ge1\). +Thus +\begin{equation}\label{quiet:Abounds} + A=(1+O(\mu))Pq^2D>0. +\end{equation} +In the same normalized axes the cancellation in +\eqref{fiber:blockAi} gives +\[ + |A_{\cdot}|_{\mathcal N} + \le CPq\sqrt\mu\bigl(|e_0|(r+\|\mathbf Y\|)+|r_E|\bigr) + \le CPq\sqrt\mu\,r. +\] +The last inequality uses \(e_0\le Cr\) on the exceptional interval. +Combining with \eqref{quiet:Wbounds} and +\eqref{quiet:Abounds}, the squared ratio to the dual norm determined +by \(AW\) is at most \(C\mu\). Consequently +\eqref{fiber:Aiabs} holds with any fixed desired small \(\eta>0\) +after decreasing \(\mu\). + +We next pay the trace coupling. Outside the exceptional interval, +\eqref{quiet:Ybounds} and \eqref{quiet:Wformula} imply +\[ + |W:T|\le Pqr\left(|\operatorname{tr}_{\mathcal N}T| + +\frac{\sqrt2}{16}|T|_{\mathcal N}\right). +\] +By \(D\ge r^2\), its cost against .98A is at most +\begin{equation}\label{quiet:tracecost} + \frac P{.98(1-C\mu)} + \left(|\operatorname{tr}_{\mathcal N}T| + +\frac{\sqrt2}{16}|T|_{\mathcal N}\right)^2. +\end{equation} +For \(0\le s\le\sqrt2\), +\((s+\sqrt2/16)^2/(1+s^2)<.8\); this can also be checked by +differentiating, since its maximum on this interval occurs at the +right endpoint. Because \(T\) has rank at most two, decreasing \(\mu\) +therefore makes \eqref{quiet:tracecost} at most +\(.85P((\operatorname{tr}_{\mathcal N}T)^2+|T|_{\mathcal N}^2)\). +These are exactly the positive symmetrized terms in +\eqref{fiber:blockL}. +On \(\tau\le2e^{-E}\), the bounds +\(D\ge c e^E\tau\), \(q,r\le C\), and +\eqref{quiet:Wbounds} instead give a trace cost at most +\(Ce^{-E}|T|_{\mathcal N}^2\). + +To estimate the remaining \(C^h\), note from \eqref{quiet:N} that +the difference \(h-V=\tau\bar{\mathcal N}\) and its first two jets +are at most \(C\tau e^{C_*\epsilon E}\). +On \(\tau\le e^{-E/4}\), the curvature formula or +Lemma~\ref{fiber:perturbation} consequently gives +\[ + |C^h(T,T)-C^V(T,T)| + \le C\tau e^{C_*\epsilon E}|T|^2 + \le\tfrac12e^{-\epsilon E}|T|^2. +\] +In the complementary range, the inverse bound and the jet bounds give +\[ + |C^h(T,T)|\le C(1+\tau)e^{C_*\epsilon E}|T|^2. +\] +This is arbitrarily small compared with \(P|T|_{\mathcal N}^2\): +indeed, for \(\tau\ge e^{-E/4}\), +\(\tau q_a/(1+\tau)\ge e^{2E}\) after increasing the threshold on \(E\), +by the explicit power \(b_0\ge1000\); normalization of \(T\) costs only +\(e^{C_*\epsilon E}\). On the exceptional interval, its trace cost +\(Ce^{-E}|T|_{\mathcal N}^2\) is likewise negligible compared with +\(e^{-\epsilon E}|T|^2\). +We have thus obtained \eqref{fiber:reserve} with +\begin{equation}\label{quiet:Rreserve} + \mathcal R(T)\ge cP|T|_{\mathcal N}^2 + +c\boldsymbol{1}_{\{\tau\le e^{-E/4}\}} + e^{-\epsilon E}|T|^2, +\end{equation} +before paying the \(K\) coupling. + +For that coupling split \(K\) into \(P\nabla^h\mathcal N\) and the two +terms containing \(P_i=Pe_0E_i\). The latter have normalized size at +most \(CP|e_0|\sqrt\mu\); their squared dual \(W\) norm after pairing +with \(T\) is at most \(C\mu P|T|_{\mathcal N}^2\). To see this directly +outside the exceptional interval, the scalar quotient to bound is +\(e_0^2/(qr)\); it is bounded by \eqref{quiet:derivatives} and +\eqref{quiet:tail}. On the exceptional interval replace \(r\) in that +denominator by \(1+r\), which is easier. +For \(P\nabla^h\mathcal N\) use three ranges: +\begin{enumerate} +\item If \(\tau\le e^{-E/4}\), then \(W\ge cPq\mathcal N\), and +the raw jet bounds give squared dual \(W\) cost at most +\(C\tau e^{C_*\epsilon E}|T|^2\), negligible compared with the +second term of \eqref{quiet:Rreserve}. +\item If \(e^{-E/4}\le\tau\le e^E\), use +\(\|\nabla^h\mathcal N\|_{\mathcal N}\le Ce^{C_*\epsilon E}\). +The squared dual cost is at most +\(CPe^{C_*\epsilon E}(qr)^{-1}|T|_{\mathcal N}^2\). +By \eqref{quiet:intermediate} this is negligible compared with +the first term of \eqref{quiet:Rreserve}. +\item If \(\tau>e^E\), the identity \(\nabla^h h=0\) permits +\[ + \nabla^h\mathcal N + =\nabla^h(\mathcal N-\bar{\mathcal N}-V/\tau). +\] +Equation \eqref{quiet:difference}, differentiated spatially once, +and the raw bounds give a normalized norm at most +\(C e^{-E/4+C_*\epsilon E}\). In making this estimate the connection +has raw norm at most \(Ce^{C_*\epsilon E}\), because \(h\) has first +jets at most \(C(1+\tau)e^{C_*\epsilon E}\) and inverse norm at most +\(Ce^{C_*\epsilon E}/(1+\tau)\). Since \(qr=q_x\ge c\), the +squared dual \(W\) cost is again negligible compared with +\(P|T|_{\mathcal N}^2\). +\end{enumerate} +Fix \(\eta\le1/100\) in Lemma~\ref{fiber:ranktwo}. Choose \(\mu\) +small enough for the terms proportional to \(\mu\), and then +\(E_{\min}\) large enough for the exponentially small terms, so that +\eqref{fiber:Kabs} holds. That lemma proves positivity of every +rank-two contraction with the retained margins +\eqref{quiet:margins}, and hence nonpositive real sectional curvature. +Smoothness and positivity at the axis follow from +Lemma~\ref{fiber:metric}; curvature extends by continuity. +\end{proof} + +To realize the disks of Proposition~\ref{grid:criterion}, we need to +add a term linear in the fiber variable whose coefficient varies over +the spatial base. A locally constant coefficient contributes only a +pluriharmonic potential. The issue is therefore its derivative support, +where the estimates below must hold for arbitrarily large fiber radius. + +\begin{proposition}[Localized shear perturbations]\label{quiet:shear} +Under the hypotheses of Proposition~\ref{quiet:profile}, suppose \(E\) and +the center are constant on the spatial region under consideration. +Assume also that the first and pure second spatial derivatives of +\(\psi\), in the affine coordinates of that proposition, have norms +at most \(e^{\epsilon E}\). Add +\[ + \ell_{\mathrm{sh}}=\operatorname{Re}(c_{\mathrm{sh}}(s)(y-\zeta)). +\] +On the support of its coefficient's derivatives suppose +\(\psi\ge-J\), where \(J\ge0\), and +\begin{equation}\label{quiet:shear-small} + \|j^{\le4}c_{\mathrm{sh}}\|\le e^{-C E-J/2}. +\end{equation} +For a sufficiently large \(C\) depending only on the profile and the +constants in the quiet criterion, this perturbation preserves metric +positivity and nonpositive real sectional curvature for all \(y\). +Moreover its metric is at least one half the unperturbed metric, and +fixed fractions of the curvature margins remain. The choice of \(C\) is +independent of dimension. +\end{proposition} + +\begin{proof} +Where the coefficient is locally constant the added potential is +pluriharmonic, so it changes no metric. It remains to estimate the +derivative collar in \eqref{quiet:shear-small}. Since \(E\) is constant, +\(P_i=0\), \(\mathcal N=G\), and \(h=V+\tau G\). +The holomorphic tilt used in Lemma~\ref{fiber:blocks} has linear and +quadratic coefficients bounded by \(Ce^{\epsilon E}\), because +\(U_\tau=\psi-F\). Its constant term may and will be zero. Derivatives +through order four of the exponential of this fixed quadratic +polynomial at the point cost at most \(Ce^{C\epsilon E}\), by the +ordinary product rule. These are fixed-order multilinear estimates. + +First consider \(\tau\ge e^{-E}\) in tilted logarithmic coordinates. +The metric inverse has norm at most \(Ce^{C\epsilon E}/\tau\). +By \eqref{quiet:Abounds}, \eqref{quiet:Wbounds}, +\eqref{quiet:Rreserve}, and \(q_x\ge c\) on this range, the retained +curvature on a rank-two matrix \(H\) is at least +\begin{equation}\label{quiet:large-margin} + c\tau e^{-C_1E}|H|^2 +\end{equation} +in these fixed coordinates, for a fixed \(C_1\). +The cubic table \eqref{fiber:cubics} shows explicitly that the only +possibly nonzero connection blocks are +\[ + \Gamma^0_{00}=P_\tau,\qquad + \Gamma^j_{i0}=P(h^{-1}G)^j_i,\qquad + \Gamma^k_{ij}=(\Gamma^h)^k_{ij}. +\] +Consequently their operator norm is at most +\(Ce^{C_1E}q\), since \(r\) is bounded. Allowing a larger fixed power +of \(q\) in the following estimates would give the same result. + +The inequality \(q\ge1\) and the left normalization imply \(F\le x\). +Thus +\[ + |y-\zeta|=e^{(F-\psi)/2}\le e^{-\psi/2}\sqrt\tau. +\] +After the tilt, all shear jets needed for the metric and curvature, +of orders two, three, and four, are therefore at most +\begin{equation}\label{quiet:shear-large-jets} + C\sqrt\tau\exp((-C+C_2)E), +\end{equation} +where \(e^{-J/2}e^{-\psi/2}\le1\) was used. +For every fixed integer \(r_1\), the scalar estimate +\eqref{quiet:tail}, also valid as an upper bound for small \(\tau\), +gives +\begin{equation}\label{quiet:log-absorption} + \sup_{\tau\ge e^{-E}}\frac{q^{r_1}}{\sqrt\tau} + \le C_{r_1}e^{C_{r_1}E}. +\end{equation} +Indeed \(q^{r_1}\le C_{r_1}(e^{r_1b_0E} ++\log(1+\tau)^{r_1})\); on \([e^{-E},1]\) use +\(\tau^{-1/2}\le e^{E/2}\), and on \([1,\infty)\) the quotient +\(\log(1+\tau)^{r_1}/\sqrt\tau\) is bounded. + +For clarity, this bound controls the quadratic perturbation as well +as the linear terms. Write \(\delta=e^{-CE}\) and absorb all other +fixed \(E\) powers into \(e^{cE}\). The relative metric error is bounded +by \(\delta e^{cE}/\sqrt\tau\). Each linear curvature error from +\eqref{fiber:perturbbound}, divided by +\eqref{quiet:large-margin}, is bounded by +\(\delta e^{cE}q^{r_1}/\sqrt\tau\) for some fixed \(r_1\). +The quadratic contraction error before division by that margin is +at most \(\delta^2e^{cE}q^{2r_1}\); after division its bound is +\[ + \delta^2e^{c'E}\left(\frac{q^{r_1}}{\sqrt\tau}\right)^2. +\] +Equation \eqref{quiet:log-absorption} makes all these quantities +arbitrarily small by a single sufficiently large \(C\). It follows that +the metric is at least half its old value and that the curvature +errors are smaller than a fixed fraction of the retained margin, +uniformly on this whole fiber range. + +For \(0<\tau\le e^{-E}\), exponentiate the tilted logarithmic +coordinate and use the affine fiber coordinate with scale +\(e^{-(\psi(s_0)+E)/2}\) at the fixed spatial point \(s_0\). +Denote its value by \(z\). The zero constant term of the tilt and +\eqref{quiet:axis-slope} give +\begin{equation}\label{quiet:small-z} + |z|^2=e^{E+F}\asymp e^E\tau. +\end{equation} +For example, \(|F-x|\le Ce^E\tau\) follows by integration of +\((q^{-1}-1)d\tau/\tau\) on this range, and proves the comparison. +The metric blocks in affine coordinates are \(h\) and \(P/|z|^2\). +Both have inverse norm at most \(e^{C_3E}\). The curvature tensor +transforms tensorially: its fiber diagonal block divides by \(|z|^4\), +its \(W\) block by \(|z|^2\), and its spatial block is unchanged. +Since +\[ + A\ge c\tau^2e^E,\qquad W\ge c\tau G,\qquad + \mathcal R(T)\ge c e^{-\epsilon E}|T|^2, +\] +the retained rank-two curvature margin in affine coordinates is at +least \(e^{-C_3E}|H|^2\), after increasing \(C_3\). + +The connection in the new coordinates is also bounded by +\(e^{C_3E}\), without an apparent singularity at \(z=0\). +Indeed the logarithmic block \(\Gamma^0_{00}=P_\tau\) becomes +\((P_\tau-1)/z\), and +\(P_\tau-1=O(e^E\tau)\); thus it is bounded by +\(C\sqrt{e^E\tau}\). The mixed spatial block becomes +\(Ph^{-1}G/z\), bounded by \(e^{C_3E}\) using +\eqref{quiet:small-z}. The spatial-output, two-fiber-input block is +zero because \(P_i=0\), and the spatial connection is already bounded. +These checks cover all cubic entries. The same bounds extend to zero +by the smoothness proved in Lemma~\ref{fiber:metric}. + +In these affine coordinates the shear jets of orders two through four +are bounded by +\[ + e^{-CE-J/2}e^{-(\psi(s_0)+E)/2}Ce^{C_4\epsilon E} + \le C\exp(-(C+1/2-C_4\epsilon)E). +\] +The estimate includes derivatives of the tilt and uses that \(|z|\) is +bounded on this range. Apply Lemma~\ref{fiber:perturbation}, with the +inverse metric, connection, and reciprocal margin bounded by fixed +powers of \(e^{C_3E}\). Increasing the same \(C\) makes all relative +metric errors and all curvature errors smaller than one half their +respective retained bounds, including at the axis. The estimates +overlap at \(\tau=e^{-E}\), so they cover the entire fiber. + +Finally, all exponents just used arise from derivatives of order at +most four, a fixed finite number of products in +\eqref{fiber:perturbformula}, and operator norms of linear maps. +Rank-two contractions introduce only the absolute constants in +\eqref{fiber:ranknorm}. None of these operations counts spatial +indices. This proves the asserted independence from \(n\). +\end{proof} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/06-radial.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/06-radial.tex new file mode 100644 index 0000000000000000000000000000000000000000..30b73f9d872337d58dfafa3a3a6c23183a93eb2b --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/06-radial.tex @@ -0,0 +1,273 @@ +\section{Radial spatial geometry and aligned profiles}\label{sec:radial} + +The quiet criterion supplies the testing regions, where $G\ge V$. +A center change requires a different arrangement: $V$ must be close to +$\lambda G$ with a large scalar $\lambda$. This section constructs radial +spatial metrics that can pass between these arrangements and proves the +curvature estimate for the aligned regime. Here \emph{aligned} means +that their metric jets through order two are close to proportional, +with the scalar frozen at the evaluation point. + +All norms of covariant jets in this section are multilinear operator norms in the stated fixed affine coordinates. The norm of a Hermitian contravariant matrix is its Frobenius norm. Constants called absolute may depend on a fixed upper bound for the first two derivatives of $\kappa$ in $\log Y$, but not on the spatial complex dimension. This convention is essential when choosing the dimension only after the geometric constants. + +\subsection{The radial clocks} + +Write $t=|s|^2$, $\nu=\log t$, and let $B$ and $\Psi$ be radial potentials. Put +\begin{equation}\label{radial:definitions} + \begin{gathered} + a=B'(t),\qquad a_\psi=\Psi'(t)=a/\lambda,\qquad + u=\log a,\\ + Y=\frac{du}{d\nu},\qquad + H=\frac{Y}{t a_\psi},\qquad + p=\frac{d\log a_\psi}{du},\qquad + \kappa=\frac{dY/d\nu}{Y(1+Y)}. + \end{gathered} +\end{equation} +Here a prime on a radial potential means differentiation in $t$. We use $u$ for this radial variable in this section and $u_s$ in Section~\ref{sec:assembly}. We write $V=\partial\bar\partial B$ and $G_0=\partial\bar\partial\Psi$ for their Hermitian matrices. + +These quantities separate the controls needed in the construction. +The ratio $\lambda$ compares the transverse coefficients of $V$ and +$G_0$, while $H/\lambda=Y/(ta)$ is the common scale of the spatial +curvature components computed below. The evolution equations will give +$d\Psi/du=1/H$ and $d\nu/du=1/Y$: large $H$ limits the change of the +radial twist $\Psi$ during growth, and large $Y$ limits the spatial radius consumed. +We use the first feature to prepare a center change and the second to +keep each accelerated episode inside a finite radial interval. Capped +growth then lets the spatial radius continue between successive tests. + +\begin{lemma}[Radial evolution]\label{radial:odes} +Suppose $a,a_\psi,Y>0$, $\lambda\ge1$, and +\begin{equation}\label{radial:kappa-assumptions} + \frac1{1+Y}\le\kappa\le1.1,\qquad + \frac{d\kappa}{d\log Y}\ge-\kappa. +\end{equation} +Then +\begin{align}\label{radial:clock-equations} + \frac{d\nu}{du}&=\frac1Y,& + \frac{d\log Y}{du}&=\kappa(1+1/Y),& + \frac{d\log\lambda}{du}&=1-p,\\ + \frac{d\Psi}{du}&=\frac1H,& + \frac{d\log H}{du}&=\kappa(1+1/Y)-p-1/Y.\notag +\end{align} +If $Y\ge t$ initially, then $Y\ge t$ subsequently. With smooth bounded controls $p,\kappa$ on each finite $u$ interval, positive solutions have no blowup on that interval. A cap of the form $\kappa=C_\kappa/(1+Y)$, with fixed $C_\kappa\ge1$, has no blowup at finite $\nu$ for the prescribed bounded controls. + +If $p=1$, then +\begin{equation}\label{radial:H-logY} + \frac{d\log H}{d\log Y}=1-\frac1\kappa. +\end{equation} +In particular, $p=\kappa=1$ holds both $\lambda$ and $H$ constant. If $\kappa\ge1$ on an interval beginning at $u_0$, then the total increase of $\nu$ on that interval is at most $1/Y(u_0)$, even if its finite $u$ length is arbitrarily large. +\end{lemma} + +\begin{proof} +The first three identities follow by changing the independent variable from $\nu$ to $u$. Since $d\Psi/d\nu=t a_\psi$ and $H=Y/(t a_\psi)$, the fourth follows as well. Differentiating $\log H=\log Y-\nu-\log a_\psi$ gives the fifth. The lower bound on $\kappa$ implies +\[ + \frac{d\log(Y/t)}{d\nu}=\kappa(1+Y)-1\ge0. +\] +In $u$ time, $dY/du=\kappa(1+Y)\le1.1(1+Y)$, so $Y$ remains finite and positive on each finite interval. The remaining variables are obtained by integrating the displayed equations there. In a cap, $dY/d\nu=C_\kappa Y$, which gives the asserted global $\nu$ continuation. Dividing the last equation of \eqref{radial:clock-equations}, with $p=1$, by the second proves \eqref{radial:H-logY}. Finally, $\kappa\ge1$ gives $Y(u)\ge Y(u_0)e^{u-u_0}$, and integration of $d\nu/du=1/Y$ gives the last assertion. +\end{proof} + +The inequalities in \eqref{radial:kappa-assumptions} allow the smooth changes used later. For example, to decrease $\kappa$ between two fixed positive values, make the change over a sufficiently long fixed interval in $\log Y$; then $d\log\kappa/d\log Y\ge-1$. To pass from $\kappa=1$ to a cap, choose a smooth function $K(Y)$ which initially equals $1+Y$, has $0\le K'\le1$, and eventually is constant, and put $\kappa=K(Y)/(1+Y)$. The change can be made on a fixed increasing-factor interval of $Y$, so its first two $\log Y$ derivatives are bounded. Moreover $1\le K\le1+Y$ and +\[ + Y\kappa_Y+\kappa + =\frac{K+Y(1+Y)K'}{(1+Y)^2}\ge0. +\] +Thus all inequalities persist, and $H$ is nonincreasing during this change when $p=1$. + +\subsection{Curvature and coordinate bounds} + +\begin{lemma}[Radial curvature]\label{radial:curvature} +Under \eqref{radial:kappa-assumptions}, the negative Hermitian curvature tensor $C^V=-R^V$ satisfies +\begin{equation}\label{radial:rank-reserve} + C^V(T,T)\ge c\,\kappa\frac{H}{\lambda}|T|_V^2 + \qquad\text{if }T=T^*,\quad\operatorname{rank}T\le2, +\end{equation} +where $c>0$ is absolute. Here the subscript $V$ means that the matrix is expressed in $V$-unit axes. +\end{lemma} + +\begin{proof} +At $s=(\sqrt t,0,\ldots,0)$ the metric has entries +\[ + V_{i\bar j}=a\delta_{ij}+a'\bar s_i s_j, +\] +and its radial and transverse eigenvalues are $a(1+Y)$ and $a$. The unitary symmetry in the transverse variables and the K\"ahler symmetries leave only a transverse block, a radial--transverse bisectional block, and a pure radial entry. After removing the common factor $Y/(ta)=H/\lambda$, their respective values in $V$-unit axes are +\begin{equation}\label{radial:curvature-components} + \delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj},\qquad + \kappa\delta_{kl},\qquad + D=\kappa\bigl(1+\kappa+Y\kappa_Y\bigr). +\end{equation} +For completeness, the transverse coefficient is $a'/a^2=Y/(ta)$. The radial--transverse coefficient is +\[ + \frac{dY/d\nu}{t a(1+Y)}=\kappa\frac{Y}{ta}. +\] +For the pure radial entry, differentiate the logarithm of the radial metric along the radial complex line. Since +\[ + \frac{d}{d\nu}\log\bigl(a(1+Y)\bigr)=Y(1+\kappa), +\] +its second $\nu$ derivative divided by $ta(1+Y)$ is exactly the last expression in \eqref{radial:curvature-components} times $Y/(ta)$. These calculations also follow directly by subtracting the cubic contractions in the K\"ahler curvature formula. + +Write a Hermitian test matrix in radial/transverse blocks as +\[ + T=\begin{pmatrix}r&b^*\\ b&S\end{pmatrix}. +\] +The contraction after removing $Y/(ta)$ is +\[ + Dr^2+2\kappa r\operatorname{tr}S+2\kappa|b|^2 + +(\operatorname{tr}S)^2+|S|^2. +\] +The assumption on $\kappa_Y$ gives $D\ge\kappa$. Absorb the cross term with $0.9Dr^2$. Its cost is at most +\[ + \frac{\kappa^2}{0.9D}(\operatorname{tr}S)^2 + \le\frac{1.1}{0.9}(\operatorname{tr}S)^2. +\] +Since $\operatorname{rank}S\le2$, $|\operatorname{tr}S|\le\sqrt2|S|$. The transverse reserve is therefore at least $5|S|^2/9$, in addition to $0.1Dr^2$ and $2\kappa|b|^2$. As $\kappa\le1.1$, these bound an absolute positive multiple of $\kappa(r^2+2|b|^2+|S|^2)$. This proves \eqref{radial:rank-reserve}. +\end{proof} + +\begin{lemma}[Scaled affine jets]\label{radial:scaled} +Assume \eqref{radial:kappa-assumptions} and $\lambda\ge1$. +Suppose $Y\ge1$, $Y\ge t$, the first two derivatives of $\kappa$ in $\log Y$ are bounded by a fixed constant, and +\begin{equation}\label{radial:p-close} + |p-1|+\left|\frac{dp}{d\log Y}\right| + +\left|\frac{d^2p}{d(\log Y)^2}\right|\le\delta, + \qquad 0<\delta\le\tfrac1{10}. +\end{equation} +At a point use $G_0$-unit affine axes divided by $\sqrt H$. In these axes $G_0=H^{-1}I$ at the point. The transverse and radial Euclidean axis lengths are +\begin{equation}\label{radial:axis-lengths} + \sqrt{\frac tY},\qquad + \sqrt{\frac{t}{Y(1+pY)}}. +\end{equation} +In these fixed coordinates: +\begin{enumerate} +\item derivatives of $\Psi$ and $B$ in orders one through four are bounded by $C/H$ and $C\lambda/H$, respectively; +\item metric jets through order two of $V-\lambda G_0$, with $\lambda$ frozen at its value at the point, are bounded by $C\delta\lambda/H$; +\item derivatives of $u$ in orders one through four are bounded by $C$; +\item if a further potential has Euclidean derivatives in orders two through four bounded by $D_e$, its Hessian and first two metric jets here are bounded by $CD_e/(a_\psi H)$. +\end{enumerate} +All constants are independent of dimension. +\end{lemma} + +\begin{proof} +The eigenvalues of $G_0$ are $a_\psi$ transversely and $a_\psi(1+pY)$ radially, giving \eqref{radial:axis-lengths}. In these axes $dt$ and $d^2t$ have operator norms at most $Ct/Y$, and higher derivatives of $t$ vanish. The first bound includes the fact that the only nonzero first derivative is radial; thus its axis length includes the extra factor $(1+pY)^{-1/2}$. + +Set a dot to mean $d/d\nu$. We have +\[ + \dot Y=\kappa Y(1+Y)=O(Y^2),\qquad + \ddot Y=O(Y^3),\qquad \dddot Y=O(Y^4). +\] +The last two estimates use, respectively, the first and second indicated derivatives of $\kappa$. Also $\dot p=O(\delta Y)$ and $\ddot p=O(\delta Y^2)$. Direct differentiation gives +\begin{align*} + a'&=aY/t,\\ + a''&=a\bigl(Y^2-Y+\dot Y\bigr)/t^2,\\ + a'''&=a\bigl(Y^3-3Y^2+2Y+3(Y-1)\dot Y+\ddot Y\bigr)/t^3. +\end{align*} +Consequently $|a^{(j)}|\le Ca(Y/t)^j$ for $0\le j\le3$. The same formulas for $a_\psi$ use $pY$ in place of $Y$ and its derivatives. They imply the corresponding estimate for $a_\psi$, and comparison at the point, where $a=\lambda a_\psi$, improves their difference by the factor $\delta$. + +The chain rule for a radial potential in at most four spatial slots is a finite sum of products of its $t$ derivatives and copies of $dt,d^2t$. For a term containing its $l$th $t$ derivative, the product of these latter factors is bounded by $C(t/Y)^l$. Since $B^{(l)}=a^{(l-1)}$, every such product is at most $Cat/Y=C\lambda/H$, and the same argument gives $C/H$ for $\Psi$. The improved difference estimate proves the metric-jet assertion. For $u$, its $t$ derivatives through order four are $O((Y/t)^j)$, by the displayed bounds on $Y$ and its derivatives; the same chain rule proves the third assertion. All sums here have a number of terms bounded in terms of the derivative order, not the dimension. + +Finally, the largest axis length in \eqref{radial:axis-lengths} is $\sqrt{t/Y}\le1$. Applying two, three, or four such axes to the extra potential therefore gives at most $CD_e t/Y=CD_e/(a_\psi H)$. +\end{proof} + +\subsection{Constant-center aligned estimates} + +In the remainder of this section the fiber center and severity $E$ are constant, the fiber profile is $q=q_0(x,E)$ from Proposition~\ref{quiet:profile}, and no shear is added. We use the fiber notation +\[ + x=\log\tau,\quad P=\tau q,\quad r=q_x/q,\quad + D_q=(q_x+q_{xx})/q. +\] +The elementary profile estimates of Lemma~\ref{quiet:scalar} give $q\ge1$, $r>0$, $D_q\ge r^2$, and, for constants allowed to depend on the fixed profile and $E$, +\begin{equation}\label{radial:profile-bounds} + \begin{split} + &D_q\ge c_E\tau\quad(0<\tau\le1),\qquad q\le C_E\quad(0<\tau\le1),\\ + &q/D_q\le C q^2,\qquad q\le C_E(1+\log\tau)\quad(\tau\ge1). + \end{split} +\end{equation} +These bounds also follow directly from the positive logarithmic summand of $q_0$: for $\tau\ge1$ it supplies a positive lower bound for $q_x+q_{xx}$, whereas the other summand is bounded in $\tau$ at fixed $E$. + +\begin{proposition}[Aligned profile]\label{radial:aligned} +There is an absolute $\delta_0>0$ with the following property. Fix $\sigma>0$ and take the severity $E$ sufficiently large in terms of $\sigma$ and the fixed profile. Assume the hypotheses of Lemma~\ref{radial:scaled}, with $\delta\le\delta_0$, and also +\[ + \tfrac12\le\kappa\le1.1,\qquad H\ge\sigma, + \qquad G=G_0+G_e,\qquad + |\partial^{[j]}G_e|\le\delta/H\quad(0\le j\le2) +\] +in its scaled affine axes. Here $G=\partial\bar\partial\psi$ and $\partial^{[j]}$ denotes any real or complex metric jet of order $j$, measured as a multilinear operator. Suppose either +\begin{equation}\label{radial:aligned-alternatives} + \lambda=1,\qquad\text{or}\qquad + H\ge M_E(1+\log\lambda)^2, +\end{equation} +where $M_E$ is sufficiently large depending only on the fixed profile and $E$. Then the constant-center fiber potential defines a positive metric with nonpositive real sectional curvature. Its spatial Schur complement satisfies $h\ge cV$. + +More precisely, in scaled spatial axes and the tilted logarithmic fiber coordinate of Lemma~\ref{fiber:blocks}, the contraction of its negative curvature on each Hermitian rank-at-most-two matrix with blocks $(T,b,t_f)$ is bounded below by +\begin{equation}\label{radial:aligned-margin} + c\left(A t_f^2+W(b,\bar b) + +\frac{\lambda+\tau}{H}|T|^2 + +P|T|_G^2\right), +\end{equation} +where $c>0$ can be fixed independently of dimension and the stage parameters once the stated thresholds hold. The coefficient $A$ and the form $W$ are those of Lemma~\ref{fiber:blocks}. +\end{proposition} + +\begin{proof} +Since $q$ has no spatial dependence, $\mathcal N=G$ and $h=V+\tau G$. Lemma~\ref{radial:scaled} shows that, through two metric jets, $h$ differs from the constant multiple $(\lambda+\tau)V/\lambda$ by at most $C\delta(\lambda+\tau)/H$. In particular +\[ + h\asymp\frac{\lambda+\tau}{H}I,\qquad + \Gamma^h=O(1). +\] +Lemma~\ref{radial:curvature} gives a covariant curvature lower bound $c\lambda|T|^2/H$ for $V$ in these scaled axes. Multiplication of a metric by a positive constant multiplies its covariant curvature by that constant. The curvature difference is at most $C\delta(\lambda+\tau)|T|^2/H$: indeed its fourth-derivative term has this bound, and its cubic contractions have the same bound by the inverse estimate $|h^{-1}|\le CH/(\lambda+\tau)$ and the first-jet bounds. Lemma~\ref{fiber:perturbation} makes this calculation explicit. Reducing $\delta_0$ therefore gives +\begin{equation}\label{radial:h-margin} + C^h(T,T)\ge c\frac{\lambda+\tau}{H}|T|^2 + \quad(\operatorname{rank}T\le2). +\end{equation} + +Let $\theta=\lambda/(\lambda+\tau)$. The identity $\nabla^h(V+\tau G)=0$ implies +\[ + \nabla^h G=\theta\bigl(\nabla^hG-\lambda^{-1}\nabla^hV\bigr), + \qquad |\nabla^hG|\le C\delta\theta/H. +\] +The curvature table gives $A=Pq^2D_q$, vanishing spin and $A_i$ terms, and $K=P\nabla^hG$. In $G$-unit normalization it gives +\begin{equation}\label{radial:W-normalized} + \frac{W}{Pq}=rI+\mathcal Y, + \qquad \mathcal Y=V_G(V_G+\tau I)^{-1}, + \qquad \mathcal Y\asymp\theta I. +\end{equation} +Here $V_G$ denotes $V$ expressed in $G$-unit axes; its eigenvalues are $(1+O(\delta))\lambda$. In particular $W>0$. Spending a fixed small portion of $W$ on the $K$ coupling costs at most +\[ + C\delta^2\frac{P\theta^2}{qH(r+\theta)}|T|^2 + \le C\delta^2\frac{\tau\theta}{H}|T|^2, +\] +which is an arbitrarily small portion of \eqref{radial:h-margin} after reducing $\delta_0$. + +It remains to pay for $2t_f(W:T)$. Spending $0.98A t_f^2$ costs $(W:T)^2/(0.98A)$. First suppose the second alternative in \eqref{radial:aligned-alternatives} holds. Split the two terms in \eqref{radial:W-normalized} using $(a+b)^2\le1.1a^2+11b^2$. The $r$ term costs at most +\[ + \frac{1.1}{0.98}P(\operatorname{tr}_G T)^2. +\] +The horizontal curvature contains the positive term +\[ + P\bigl((\operatorname{tr}_G T)^2+|T|_G^2\bigr). +\] +As $|\operatorname{tr}_G T|\le\sqrt2|T|_G$, their difference retains more than $P|T|_G^2/2$. The remaining trace cost is at most +\begin{equation}\label{radial:trace-remainder} + C\frac{P\theta^2}{D_qH^2}|T|^2. +\end{equation} +For $\tau\le1$, \eqref{radial:profile-bounds} bounds $P/D_q$ by $C_E$. For $\tau\ge1$, it bounds \eqref{radial:trace-remainder} by +\[ + \frac{C_E}{H^2}\tau\theta^2(1+\log\tau)^2|T|^2 + \le \frac{C_E(1+\log\lambda)^2}{H^2} + (\lambda+\tau)|T|^2. +\] +For the last inequality, put $z=\tau/\lambda$ and use boundedness of +$z(1+z)^{-3}$ and $z(1+z)^{-3}(\log z)^2$ on $(0,\infty)$. Taking $M_E$ sufficiently large absorbs this cost into a small portion of \eqref{radial:h-margin}. + +Now suppose $\lambda=1$. The profile estimates give, for sufficiently large $E$, +\begin{equation}\label{radial:lambda-one-slope} + \|\mathcal Y\|\le r/18\qquad(\tau>2e^{-E}). +\end{equation} +To see the uniformity of this assertion, $\|\mathcal Y\|\le(1+C\delta)/(1+\tau)$. On $2e^{-E}\le\tau\le1$, the first summand of $q_0$ gives a logarithmic slope bounded below by a large fixed multiple of one. On $1\le\tau\le e^{3b_0E}$, it gives $r\ge0.9b_0(1+\tau)^{-1/16}$ after increasing $E$. Beyond this interval the logarithmic summand gives $r\ge c/q$, which exceeds $40/(1+\tau)$ by its linear-in-$\log\tau$ upper bound and the starting size $\tau\ge e^{3b_0E}$. These statements are unchanged by taking $\delta_0$ smaller. + +By \eqref{radial:lambda-one-slope}, the trace expenditure outside the exceptional interval is at most +\[ + \frac{P}{0.98} + \left(|\operatorname{tr}_GT|+\frac{\sqrt2}{18}|T|_G\right)^2. +\] +For $0\le s\le\sqrt2$, the ratio $(s+\sqrt2/18)^2/[0.98(1+s^2)]$ is strictly less than one, by an absolute amount. Thus a fixed portion of $P|T|_G^2$ remains. On $\tau\le2e^{-E}$, the explicit left-end profile gives $q\le C$, $r\le C$, and $D_q\ge c\tau e^E$, where constants may depend on the fixed profile but not $E$. Hence the trace cost is at most $Ce^{-E}|T|^2/H^2$. Since $H\ge\sigma$, this is a small portion of $(1+\tau)|T|^2/H$ for $E$ sufficiently large in terms of $\sigma$. + +All cases retain fixed portions of $A$, $W$, \eqref{radial:h-margin}, and the symmetrized $PG^2$ term, proving \eqref{radial:aligned-margin}. Lemma~\ref{fiber:ranktwo} gives nonpositive real sectional curvature off the fiber axis. The metric is positive there and $h\ge cV$. Smoothness of the original affine fiber potential and its positive limiting metric extend both conclusions across the axis. +\end{proof} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/07-centers.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/07-centers.tex new file mode 100644 index 0000000000000000000000000000000000000000..896c8506317c40dfaf3d4cce33820a54888864c5 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/07-centers.tex @@ -0,0 +1,345 @@ +\section{Changing the fiber center}\label{sec:centers} + +A moving center is not a holomorphic change of coordinates. We therefore first widen the fibers, then estimate the actual change of potential as a perturbation in fixed holomorphic coordinates. The widening and the metric scales must be chosen in the order specified below. + +Throughout this section $p=\kappa=1$ and $Y\ge\max\{1,t\}$, so $\lambda,H$ are constant by Lemma~\ref{radial:odes}, and $V=\lambda G_0$ as metrics, including their jets. We use the scaled affine spatial coordinates of Lemma~\ref{radial:scaled}. The severity $E$ is constant, and no shear term is present. The old twist has the form +\[ + \psi_{\rm old}=\Psi+\psi_e, + \qquad G_e=\partial\bar\partial\psi_e, + \qquad |\partial^{[j]}G_e|\le\delta/H\quad(0\le j\le2), +\] +with $\delta$ sufficiently small as in Proposition~\ref{radial:aligned}. Suppose also that, throughout the interval being configured, +\begin{equation}\label{center:old-data} + \psi_{\rm old}\le A_{\rm old},\qquad + |\partial^{[j]}\psi_{\rm old}|\le A_{\rm old} + \quad(1\le j\le4) +\end{equation} +in these fixed scaled axes at each point. The number $A_{\rm old}$ is known independently of the target sizes of $\lambda,H$. Additive constants already placed in the twist are included in this upper bound. + +\begin{proposition}[Center change]\label{center:change} +Choose the profile constants in the following order: first the fixed exponent $b_0$ of the quiet profile; then a sufficiently large absolute constant $d_0$; and then $K_0$ sufficiently large in terms of $d_0$ and absolute constants specified in the proof. Fix $E$ and old data as in \eqref{center:old-data}. There is a width parameter $L$, chosen using only these data and the fixed profile, such that the following construction is possible. + +After choosing $L$, choose a sufficiently large $\lambda$, and then a sufficiently large $H$; explicitly one can require +\begin{equation}\label{center:target-order} + \lambda\ge\max\{e^{3L},\Lambda_f\}, + \qquad H\ge M_f(1+\log\lambda)^2, +\end{equation} +where $\Lambda_f,M_f$ are finite constants determined by the fixed profile, $E,L$, the old data, and $\delta$, but not by $\lambda,H$. Over three fixed units of the variable $\Psi$, one can change from any prescribed constant center in the unit disk to any other such center, returning to the original profile $q_0(x,E)$ at the end. The final twist differs from the old twist by a nonnegative constant, possibly very large. + +During the change the potential is smooth, its metric is positive and has nonpositive real sectional curvature, and the metric bounds $cV$ from below on spatial projection, for a fixed $c>0$. There is no restriction on the cost of the final constant offset. All power exponents and smallness tolerances used before choosing the spatial dimension are dimension independent. +\end{proposition} + +\subsection{The profile deformation and its elementary bounds} + +Write $F^0_x=1/q_0$ and normalize $F^0(x)-x\to0$ at the left end. We record properties of the concrete quiet profile that will be used here. At fixed $E$, +\begin{equation}\label{center:q-properties} + \begin{split} + &q_0\ge1,\quad (q_0)_x>0,\quad + (q_0)_{xx}/(q_0)_x\ge-1/16,\\ + &|(q_0)_x|+|(q_0)_{xx}|\le C q_0,\qquad + q_0(x,E)\le C_E+C(1+\max(x,0)),\\ + &\inf_{x\ge-20}(q_0)_x>0. + \end{split} +\end{equation} +Here and below a subscript $E$ permits dependence on $E$ and the already fixed profile. The last infimum can be bounded below using just the derivative of $K_0^{-1}\log(1+e^x)$. The inequality for the second derivative follows by differentiating both summands in the quiet profile. In particular $(q_0)_x+(q_0)_{xx}\ge(15/16)(q_0)_x$. + +Two endpoint properties are also needed. The function $Q_0(v)=q_0(\log v,E)$ is smooth down to $v=0$, with +\begin{equation}\label{center:left-slope} + Q_0(0)=1,\qquad Q_0'(0)=b_0e^E+K_0^{-1}>0. +\end{equation} +At the other end the auxiliary function in the quiet profile tends to a positive finite limit with error $O(e^{-x/16})$, so +\[ + q_0(x,E)=c_E+x/K_0+O_E(e^{-x/16})\quad(x\longrightarrow\infty). +\] +Integration, using $1/q_0-K_0/x=O_E(x^{-2})$, gives +\begin{equation}\label{center:primitive-asymptotic} + F^0(x)=K_0\log x+O_E(1),\qquad + \int_0^L\frac{dx}{q_0(x,E)}=K_0\log L+O_E(1). +\end{equation} +No uniformity in $E$ is asserted or needed in these endpoint constants; $E$ has been fixed before $L$. + +Take $L$ large. Choose a smooth function $\chi$ with +\[ + 0\le\chi\le1,\quad \operatorname{supp}\chi\subset[-20,3L],\quad + \chi=1\text{ on }[0,L],\quad \int\chi=2L,\quad \chi'\ge-1/16, +\] +and with its derivatives through any fixed needed order bounded absolutely. Such a function is obtained from a fixed rising cutoff, an adjustable plateau, and a fixed descending cutoff of length greater than $16$; adjusting the plateau gives the integral exactly. Define the increasing inverse map $m=m(x)$ by +\begin{equation}\label{center:inverse-map} + x=m+L-\frac12\int_{-\infty}^m\chi(v)\,dv. +\end{equation} +Then +\begin{equation}\label{center:map-bounds} + 1\le m'\le2,\quad m\le x,\quad + m=x-L\ (x\le L-20),\quad m=x\ (x\ge X:=3L). +\end{equation} +Its derivatives of any fixed order are bounded absolutely. Set +\[ + q_1(x)=q_0(m(x),E)\le q_0(x,E). +\] +Since $m''=\chi'(m)(m')^3/2$, putting $a=m'\in[1,2]$ gives +\begin{align}\label{center:q1-positive} + (q_1)_x+(q_1)_{xx} + &\ge(q_0)_x(m)\bigl(a-a^2/16-a^3/32\bigr)\\ + &\ge\tfrac34(q_0)_x(m)a=\tfrac34(q_1)_x>0.\notag +\end{align} +Also $|(q_1)_x|+|(q_1)_{xx}|\le Cq_1$, with constants independent of $L$. These bounds follow from \eqref{center:q-properties} and the bounds on $m',m''$. + +\subsection{Offsets during profile interpolation} + +On the first unit of $\Psi$, choose a smooth nondecreasing flat-ended cutoff $\beta=\beta(\Psi)$ from $0$ to $1$, with fixed bounds on its derivatives through order four. Define +\begin{equation}\label{center:forward-profile} + q(x,\beta)=(1-\beta)q_0(x,E)+\beta q_1(x),\qquad + \mathfrak s=1/q, +\end{equation} +and let $F^\beta$ have derivative $F^\beta_x=\mathfrak s$ and left normalization $F^\beta-x\to0$. Every such primitive exists, tends to infinity on the right, and differs from $F^0$ by a constant for $x\ge X$. All spatial derivatives in the rest of this subsection are at fixed $\tau=e^x$. + +Use an offset $w^+(\beta)$ defined by $w^+(0)=0$ and +\begin{equation}\label{center:offset-definition} + (w^+)'(\beta)=\partial_\beta F^\beta(X) + +\frac{d_0}{1-\beta+1/L},\qquad + D(x,\beta)=(w^+)'(\beta)-\partial_\beta F^\beta(x). +\end{equation} +Here the prime on $w^+$ denotes differentiation in $\beta$. Since $\mathfrak s_\beta\ge0$ and vanishes for $x\ge X$, +\begin{equation}\label{center:D-domination} + D\ge\frac{d_0}{1-\beta+1/L}. +\end{equation} +This lower bound can dominate $\sup_x|\mathfrak s_\beta|$ by any required fixed factor if $d_0$ is sufficiently large. Indeed +\[ + 0\le\mathfrak s_\beta=(q_0-q_1)/q^2 + \le\min\{(1-\beta)^{-1},\sup_{x\le X}q_0(x,E)\}. +\] +By taking $L$ sufficiently large relative to $C_E$, the second entry is at most $CL$ with an absolute $C$. The elementary bound +$\min\{z^{-1},CL\}\le(C+1)/(z+1/L)$ proves the assertion, including $\beta=1$ by continuity. + +For later restoration use, on another unit of $\Psi$, the reverse mixture $q=(1-\beta)q_1+\beta q_0$ with a fresh increasing cutoff. Now $\mathfrak s_\beta\le0$ and $\partial_\beta F^\beta\le0$. Choose a new offset $w^-(\beta)$ with $w^-(0)=0$ and constant positive derivative greater than a sufficiently large fixed multiple of $1+\sup_{x,\beta}|\mathfrak s_\beta|$. With $D=(w^-)'-\partial_\beta F^\beta$, the same domination holds. No estimate on the size of $w^-(1)$ is required. + +Once $L$ is fixed, all the following constants are finite and may be included in a number $C_f$: upper bounds for the offsets and their derivatives through order four; for $D$ and its first three $\beta$ derivatives; for the profile and the finite-order derivatives used below on $x\le X$; for their left-end decay coefficients; and the reciprocal lower constants in the positive estimates on that interval. To check finiteness at the noncompact left endpoint, both $q_0-1$ and $q_1-1$, with their needed derivatives, are $O_f(e^x)$. Thus $\beta$ derivatives of $\mathfrak s$ are $O_f(e^x)$, and their integrals defining derivatives of $F^\beta$ converge. In particular $C_f$ may include $e^X$ and large functions of $E,L$. It is selected before $\lambda,H$. + +\begin{lemma}[Curvature during interpolation]\label{center:interpolation} +During either profile switch above, keep the center constant and set +$\psi=\psi_{\rm old}+w^\pm(\beta(\Psi))$, also including the already frozen offset when making the reverse switch. If $\lambda$ and then $H$ satisfy sufficiently large thresholds of the form \eqref{center:target-order}, the metric is positive, $h\ge cV$, and its negative curvature has a strictly positive rank-two reserve off the fiber axis. These conclusions extend across that axis by continuity. +\end{lemma} + +\begin{proof} +A prime on $\beta$ now means differentiation in $\Psi$. Direct differentiation of $\psi-F^\beta$ gives +\begin{equation}\label{center:N-formula} + \mathcal N=(1+D\beta')G_0+G_e + +\bigl(D\beta''+D_\beta(\beta')^2\bigr) + \partial\Psi\otimes\bar\partial\Psi. +\end{equation} +Here $D_\beta$ is the derivative at fixed $x$; thus all dependence of the primitive on the spatial cutoff is included. The last term and its first two spatial jets are $O(C_f/H^2)$, because all positive derivatives of $\Psi$ in scaled axes are $O(1/H)$. The derivatives of the scalar $D\beta'$ beyond its value are $O(C_f/H)$. With $d_f=1+D\beta'\ge1$, these facts show +\[ + \mathcal N\asymp d_fG_0>0, + \qquad h=V+\int_0^\tau\mathcal N(v)\,dv\ge cV. +\] +The constants in $\asymp$ can be absolute by making $H$ large enough; the value $d_f$ itself is bounded by $C_f$. + +First consider $x\le X$. Since $\tau\le e^X$, the first two metric jets of $h-V$ are $O(C_f/H)$. Hence +\begin{equation}\label{center:finite-x-h} + \Gamma^h=\Gamma^{G_0}+O(C_f/\lambda),\qquad + C^h(T,T)\ge c\lambda|T|^2/H + \quad(\operatorname{rank}T\le2). +\end{equation} +The second estimate follows from Lemma~\ref{radial:curvature}, the scaled jet estimates, and the curvature perturbation identity, taking $\lambda\gg C_f$. + +The mixed form in Lemma~\ref{fiber:blocks} is +\[ + \frac{W}{Pq}=(1+r)\mathcal N + -\tau\mathcal N h^{-1}\mathcal N + +q^{-1}\partial\bar\partial\log q. +\] +The last term equals +\[ + -\mathfrak s_\beta\beta'G_0+O(C_f/H^2). +\] +The spin form, divided by $Pq$, equals +\[ + -\mathfrak s_\beta\beta'\nabla_h^{2,0}\Psi+O(C_f/H^2). +\] +In fact $q_\beta/q^2=-\mathfrak s_\beta$; the remaining terms contain two first derivatives of $\Psi$. Lemma~\ref{radial:scaled} and \eqref{center:finite-x-h} give +$|\nabla_h^{2,0}\Psi|\le C/H$ with absolute $C$. Also +$|\tau\mathcal N h^{-1}\mathcal N|\le C_f/(\lambda H)$. Choose $d_0$ first so that the domination in \eqref{center:D-domination} is sufficiently strong in comparison with this absolute $C$, and then choose $\lambda,H$ large compared with $C_f$. It follows that +\begin{equation}\label{center:W-spin} + W\ge c\frac{Pq}{H}d_f I, + \qquad |\nabla_h^{2,0}P(b,b)|\le\tfrac14 W(b,\bar b). +\end{equation} +This argument also applies to the reverse mixture, using absolute values of $\mathfrak s_\beta$. + +By \eqref{center:q1-positive} and convexity of the mixture, +\[ + D_q=(q_x+q_{xx})/q\ge c_f\min(1,\tau)\quad(x\le X). +\] +Left-end decay and boundedness on the rest of this interval give +\[ + |\partial\log P|+|\partial P_\tau| + \le C_f\min(1,\tau)/H. +\] +It follows from the curvature table that +\begin{equation}\label{center:A-control} + A=(1+o(1))Pq^2D_q,\qquad + |A_i|\le C_f P\min(1,\tau)/H, +\end{equation} +where the relative error tends uniformly to zero as $\lambda,H$ increase. To verify uniformity at $\tau=0$, the subtracted term in $A$ is at most +$C_fP^2\min(1,\tau)^2/(\lambda H)$, while the leading term is $Pq^2D_q$. The ratio tends to zero uniformly under the stated choices. Similarly \eqref{center:W-spin} and \eqref{center:A-control} make the $A_i$ coupling arbitrarily small in units $(AW)^{1/2}$. + +Finally $|K/P|\le C_f/H$. Spending fixed portions of $A,W$ according to Lemma~\ref{fiber:ranktwo} costs at most +\[ + C_f\frac{P}{qH}|T|^2\le\frac{C_f}{H}|T|^2 + \quad\text{for the }K\text{ coupling},\qquad + C_f\frac{P}{D_qH^2}|T|^2\le\frac{C_f}{H^2}|T|^2 + \quad\text{for the trace coupling}. +\] +In these displayed bounds $C_f$ is enlarged to include $e^X$; also $P/D_q$ is bounded on $x\le X$, including the left endpoint. Both costs are small compared with \eqref{center:finite-x-h}. Thus fixed fractions of $A$, $W$, and $\lambda|T|^2/H$ remain. + +Now let $x>X$. Since the two profiles coincide exactly here, $F^\beta(x)-F^0(x)$ is independent of $x$. Consequently $\mathcal N=\mathcal N_\infty$ is independent of $\tau$, and +\begin{equation}\label{center:finite-tail} + h=V_f+\tau\mathcal N_\infty, + \qquad V_f=V+\int_0^{e^X} + \bigl(\mathcal N(v)-\mathcal N_\infty\bigr)\,dv. +\end{equation} +There is no infinite tail in this expression. Its first two spatial jets differ from those of $V$ by $O(C_f/H)$. Treating $d_f=1+D\beta'$ at the point as constant, the jets of $\mathcal N_\infty$ differ from those of $d_fG_0$ by at most $C\delta d_f/H$, after taking $H\gg C_f/\delta$; similarly those of $V_f$ differ from $\lambda G_0$ by a sufficiently small relative amount after taking $\lambda\gg C_f/\delta$. + +The aligned argument can now be applied with the two spatial scales $\lambda$ and $d_f$. More explicitly, let $\theta_f=\lambda/(\lambda+\tau d_f)$. The comparisons and the identity $\nabla^h(V_f+\tau\mathcal N_\infty)=0$ give +\[ + C^h(T,T)\ge c\frac{\lambda+\tau d_f}{H}|T|^2, + \qquad |\nabla^h\mathcal N_\infty| + \le C\delta d_f\theta_f/H. +\] +The $K$ cost is bounded by $C\delta^2(\lambda+\tau d_f)|T|^2/H$. The $rI$ part of the trace cost uses the same rank-two symmetrized $P\mathcal N_\infty^2$ reserve as before. The remaining cost is at most +\[ + C\frac{\tau qd_f^2\theta_f^2}{D_qH^2}|T|^2 + \le C_f\frac{(1+\log\lambda)^2(\lambda+\tau d_f)}{H^2}|T|^2. +\] +For the last step, use \eqref{radial:profile-bounds} and the scalar inequality from the proof of Proposition~\ref{radial:aligned} with $\lambda/d_f$ in place of $\lambda$. We have $1\le d_f\le C_f$ and may require $\lambda\ge C_f$, so the logarithm is bounded by $\log\lambda$. Taking $H\ge M_f(1+\log\lambda)^2$ with sufficiently large $M_f$ absorbs this last cost. Spin and $A_i$ vanish in this region because $q=q_0$ is spatially constant. + +We have proved positivity and nonpositive curvature off the axis, with fixed fractional reserves. The profiles are smooth in $\tau$ at zero, equal to one there, and have the prescribed left normalization. Thus the potential and metric are smooth in the original affine fiber coordinate, whose metric at the axis has entries $V,e^\psi$. Curvature extends by continuity. This proves the lemma. +\end{proof} + +\subsection{The width gained before moving the center} + +At the end of the first switch freeze $\beta=1$ and the offset, and write +$\psi=\psi_{\rm old}+w^+(1)$. Changing variables in the defining integral gives +\begin{equation}\label{center:F1-formula} + F^1(x)=L+F^0(m(x)) + -\frac12\int_{-\infty}^{m(x)}\frac{\chi(v)}{q_0(v,E)}\,dv. +\end{equation} +Both sides have derivative $1/q_1$ and the same left normalization. At $x=X$, $m(X)=X$, so integration of \eqref{center:offset-definition} yields +\begin{equation}\label{center:offset-total} + w^+(1)=L-\frac12\int_{\mathbb R}\frac{\chi(v)}{q_0(v,E)}\,dv + +d_0\log(1+L). +\end{equation} +In particular this offset is nonnegative: the integral is at most $\int\chi=2L$. + +Define the pointwise width scales +\[ + R_0=\exp\bigl((L-\psi)/2\bigr),\qquad + Z=\exp\bigl((F^1(x)-\psi)/2\bigr)=|y-\zeta|. +\] +Because $\chi=1$ on $[0,L]$, \eqref{center:primitive-asymptotic} and \eqref{center:offset-total} imply +\begin{equation}\label{center:R-growth} + R_0\ge c_{E,A_{\rm old}}L^{K_0/4-d_0/2}. +\end{equation} +Choose $K_0$ so large relative to $d_0$ that $K_0/4-d_0/2>K_0/9$. Then, for $L$ sufficiently large depending on old data, +\begin{equation}\label{center:width-bounds} + R_0\ge L^{K_0/9},\qquad + Z\ge x^{K_0/9}\quad(x\ge L-20). +\end{equation} +Here is the second bound in detail. At $x=L-20$, formula \eqref{center:F1-formula} gives $F^1=L+F^0(-20)$, so $Z$ is a fixed old-data multiple of $R_0$. Monotonicity of $F^1$ and \eqref{center:R-growth} prove the desired bound on $[L-20,3L]$, absorbing the fixed factor $3^{K_0/9}$ by increasing $L$. For $x\ge3L$, the integral in \eqref{center:F1-formula} has reached its total and cancels the integral in \eqref{center:offset-total}. Thus exactly +\[ + F^1(x)-\psi=F^0(x)-\psi_{\rm old}-d_0\log(1+L). +\] +The asymptotic \eqref{center:primitive-asymptotic}, together with $x\ge3L$ and the strict exponent inequality already chosen, gives the second bound on this remaining interval. These estimates are uniform over the spatial interval because only the upper bound in \eqref{center:old-data} was used. + +\subsection{The actual nonholomorphic center motion} + +The preceding calculation has produced width before any center is moved. +We now spend that width to control the metric and curvature errors of +moving the center. Two coordinate systems cover the fiber: logarithmic +coordinates away from the enlarged core and a scaled affine coordinate +on the core, including its axis. + +On the next unit of $\Psi$, hold $q=q_1$ and all offsets fixed, and let $\zeta$ be a smooth flat-ended convex-combination interpolation of the two prescribed centers. Its derivatives through order four as a function of $\Psi$ are bounded absolutely. By Lemma~\ref{radial:scaled}, and $H\ge1$, its spatial derivatives in scaled axes satisfy +\begin{equation}\label{center:zeta-jets} + |\partial^{[j]}\zeta|\le C/H\qquad(1\le j\le4). +\end{equation} +At a point $s_0$, compare the moving-center potential with the potential obtained by replacing $\zeta(s)$ by the constant $\zeta(s_0)$ in a neighborhood. This is a comparison in fixed holomorphic coordinates, not a claim that $y-\zeta(s)$ is a holomorphic coordinate. All constants in the following two comparisons may depend on old data, $E$, and the fixed profile, but are independent of $L,\lambda,H$ once the stated thresholds hold. + +First suppose $x\ge L-20$. For the frozen center use scaled spatial axes and the tilted logarithmic fiber coordinate of Lemma~\ref{fiber:blocks}. The previous interpolation proof with cutoff constant retains fractions of $A,W$ and the horizontal curvature reserve. On this region $m\ge-20$, so \eqref{center:q-properties} and \eqref{center:q1-positive} give +\[ + A\ge c\tau,\qquad W\ge c\tau H^{-1}I. +\] +The horizontal reserve is at least $c\tau/H$: on $x\le X$ use $\lambda\ge e^X\ge\tau$ in \eqref{center:finite-x-h}, and on $x>X$ use the aligned reserve. Consequently the full rank-two curvature contraction is bounded below by $c\tau/H$ times the squared Frobenius norm of the test. The metric has the same lower scale and its inverse has norm at most $CH/\tau$. These assertions also follow directly for the metric from $h=V+\tau G$ and $P=\tau q$. + +The polynomial holomorphic tilt has coefficients bounded by the first two old twist derivatives; the offset is constant. In these tilted coordinates the frozen connection has norm at most $Cq$. Indeed the possibly large entries in the cubic table are $P_\tau=q+q_x=O(q)$ and $P\mathcal Nh^{-1}=O(q)$, whereas $P_i=0$, the tilted pure holomorphic second spatial derivative vanishes, and $\Gamma^h=O(1)$. + +Let $\Phi$ be the one-variable fiber potential as a function of +$\rho+\psi=\log|y-\zeta|^2+\psi$. Its first derivative is $\tau$, and differentiation in this argument is $q\partial_x$. Thus its derivatives of orders one through four have size at most $C\tau q^3$. For example +\[ + \Phi''=\tau q,\quad + \Phi'''=\tau q(q+q_x),\quad + \Phi''''=\tau q\bigl((q+q_x)^2+q(q_x+q_{xx})\bigr). +\] +The fixed argument has bounded jets through order four in the chosen frame. The difference in its argument after moving the center is +\begin{equation}\label{center:log-argument-difference} + \log\left|1-\frac{\zeta(s)-\zeta(s_0)}{y-\zeta(s_0)}\right|^2. +\end{equation} +It vanishes at the evaluation point, and each positive jet contains a derivative of $\zeta$ and a denominator of size at least $Z$. By \eqref{center:zeta-jets}, bounded tilt coefficients, and $HZ\ge1$, its jets through order four are $O(1/(HZ))$. The chain rule consequently bounds every potential-jet difference through order four by +\begin{equation}\label{center:outer-jet-error} + C\frac{\tau q^3}{HZ}. +\end{equation} +In particular its relative metric error is at most $Cq^3/Z$. Once this is small, Lemma~\ref{fiber:perturbation}, the bound $\Gamma=O(q)$, and the inverse bound $O(H/\tau)$ show that the curvature error on unit rank-two tests is at most +\begin{equation}\label{center:outer-curvature-error} + C\frac{\tau}{H}\left(\frac{q^5}{Z} + +\frac{q^8}{Z^2}\right) + \le C\frac{\tau q^8}{HZ}. +\end{equation} +The first term accounts for all linear fourth-jet and cubic-contraction changes, and the second is the quadratic inverse-contraction term. Fixed-rank contraction costs only an absolute factor. + +For $x\ge L-20$, \eqref{center:q-properties} and $m\le x$ give $q=q_1\le Cx$ once $L$ is large relative to $C_E$. This constant is independent of $L,\lambda,H$. Choose, in addition to the earlier requirement, $K_0/9>10$. Then \eqref{center:width-bounds} makes the ratios in \eqref{center:outer-jet-error} and \eqref{center:outer-curvature-error} arbitrarily small compared with the metric and curvature reserves by increasing $L$. This choice precedes $\lambda,H$. + +It remains to include $x0$; positivity away from zero follows from \eqref{center:q-properties}. For $W$, the term $Pq\mathcal YG$ has a positive lower bound of this size because $\tau\le e^{L-20}$ and $\lambda\ge e^{3L}$. Divide the vertical and cross curvature blocks by $|z|^4$ and $|z|^2$, respectively, on changing from logarithmic to affine fiber coordinates. The retained vertical and cross reserves are then at least $c_Ee^L$ and $c_Ee^L/H$. The horizontal reserve $c\lambda/H$ also dominates $ce^L/H$. The metric has the same smallest scale, so its inverse is bounded by $C_EH/e^L$. + +Undoing the exponentiated tilt changes the affine fiber variable by $z\mapsto e^{\theta(s)}z$, where $\theta(s_0)=0$ and its coefficients are bounded by old twist derivatives. Since $|z|$ is bounded, the differential and its inverse, and the needed higher jets, have old-data bounds independent of $L,\lambda,H$. This preserves the indicated rank-two lower margin up to old-data constants. The connections in these coordinates are bounded: the logarithmic entry transforms to +\[ + (P_\tau-1)/z=O_E(|z|), +\] +and the spatial-fiber entry is bounded by +\[ + |P\mathcal Nh^{-1}|/|z| + \le C_E\tau/(\lambda|z|)\le C_E|z|, +\] +using $\lambda\ge e^L$. The pure vertical-to-spatial entry vanishes before the bounded tilt is undone. The remaining spatial entries are bounded by the radial jet estimates. + +In this affine frame the moving smooth argument, before applying the fixed smooth radial function, is +\begin{equation}\label{center:affine-argument} + \left|z-\frac{\zeta(s)-\zeta(s_0)}{R_0(s_0)}\right|^2 + e^{\psi(s)-\psi(s_0)}. +\end{equation} +Its difference from the frozen argument has jets through order four bounded by $C/(HR_0)$, including at $z=0$, by \eqref{center:zeta-jets}. Therefore the potential jets change by at most $Ce^L/(HR_0)$. The inverse metric and bounded connection estimates give a curvature error at most +\begin{equation}\label{center:inner-error} + C\frac{e^L}{H}\left(R_0^{-1}+R_0^{-2}\right). +\end{equation} +By \eqref{center:width-bounds}, this is a small fraction of the $e^L/H$ reserve for $L$ sufficiently large, independently of the later choices of $\lambda,H$. The metric error is small by the same estimate. Smoothness includes the axis, so no limiting logarithmic-coordinate assertion is being used at a singular coordinate point. + +We have proved that the moving-center metric remains, say, at least half the frozen-center metric and that its real sectional curvature is nonpositive. Its spatial projection bound is therefore inherited from the frozen Schur bound $h\ge cV$. + +\begin{proof}[Completion of Proposition~\ref{center:change}] +Choose $d_0$ to dominate the absolute spin constants in the interpolation estimate, and choose $K_0$ so that $K_0/4-d_0/2>K_0/9$ and $K_0/9>10$. The latter explicit exponent is larger than the exponent $8$ in \eqref{center:outer-curvature-error}. For fixed old data and $E$, choose $L$ sufficiently large for \eqref{center:width-bounds} and for the moving-center metric and curvature errors to be less than the retained reserves. Next compute the finite constants $C_f$ for both switches and choose $\lambda$ and then $H$ according to \eqref{center:target-order}, with the larger thresholds needed in Lemma~\ref{center:interpolation}. This is the required noncircular order of choices. + +Perform the forward switch with constant old center, the center interpolation with frozen profile $q_1$ and offset, and finally the reverse switch with constant new center. Lemma~\ref{center:interpolation} controls the two switches and the preceding argument controls the actual center motion. The cutoffs are flat-ended, so all joins are smooth. At the end the profile is $q_0$, the center is the prescribed new constant, and the added offset is $w^+(1)+w^-(1)\ge0$. All profiles are at least one, have smooth left endpoints with value one, and have normalized primitives tending to infinity on the right; thus they define smooth fiber potentials for all affine fiber values throughout. The spatial projection bound follows on each part, with a fixed fractional loss during the center motion. No bound on the final constant offset is required or asserted. +\end{proof} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/08-assembly.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/08-assembly.tex new file mode 100644 index 0000000000000000000000000000000000000000..15aa53bcfc518e66448f0fa3aa8110aae1f520f7 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/08-assembly.tex @@ -0,0 +1,1102 @@ +\section{Assembly of the complete base}\label{sec:assembly}\label{assembly:section} + +The local estimates are now available. We must realize infinitely many +analytic tests with one finite spatial dimension, preserve the earlier +tests when adding later wells, and obtain a complete global metric. +We choose the spatial data and join the local regimes. All tensor and +fixed-order derivative estimates in this section use operator or +multilinear norms in fixed affine coordinates. In particular, their +leading exponents do not include a trace over the spatial dimension. + +\begin{proposition}\label{assembly:base} +For some fixed finite integer $n$, there is a smooth strictly +plurisubharmonic potential $\ell$ on $\mathbb C^n\times\mathbb C$ +whose K\"ahler metric is complete and has nonpositive real sectional +curvature. The potential satisfies every hypothesis of +Proposition~\ref{grid:criterion}, for a countable dense set of centers +in the unit disk and every spatial Taylor order. Thus all bounded +holomorphic functions on its half-plane tube are constant. +\end{proposition} + +At each spatial point the fiber potential is recovered from +\[ + F(\log\tau,s)=\log|y-\zeta(s)|^2+\psi(s) +\] +with the left normalization of Section~\ref{sec:fiber}. The radial +function $B$ supplies the spatial metric. The twist $\psi$ starts with +a radial background $\Psi$: polynomial wells make selected fibers wide, +and additional radial plurisubharmonic terms control the derivatives +of the severity. A term +$\operatorname{Re}(c_{\rm sh}(s)(y-\zeta(s)))$ prescribes the sheared +disk directions. We choose these data in increasing spatial radius. +The polynomial wells are the only later additions that extend into +earlier regions, so their inward tails must be reserved in advance. + +The proof occupies this section. A cycle starts on a capped spatial +interval, raises the severity, performs one test, and raises severity +again on a post-test ramp. At constant severity it then accelerates the +spatial metric, enters the aligned regimes, changes center, and restores +the spatial parameters. Finally it increases severity while the +normalized spatial geometry is fixed, until the severity can pay for +the next capped interval. +\input{figures/stage-clock} + +We retain the radial notation of Lemma~\ref{radial:odes}: +\[ + \begin{gathered} + t=|s|^2,\quad \nu=\log t,\quad a=B'(t),\quad u_s=\log a,\\ + Y=\frac{du_s}{d\nu},\quad a_\psi=\Psi'(t)=a/\lambda,\quad + H=\frac{Y}{ta_\psi},\quad p=\frac{d\log a_\psi}{du_s}. + \end{gathered} +\] +Here $H$ is a scalar; the polynomial system below is denoted by +$\mathcal H$. The evolution equations are +\begin{align} + \frac{dY}{du_s}&=\kappa(1+Y),& + \frac{d\nu}{du_s}&=Y^{-1},& + \frac{d\log\lambda}{du_s}&=1-p,\label{assembly:clocks}\\ + \frac{d\Psi}{du_s}&=H^{-1},& + \frac{d\log H}{du_s} + &=\kappa(1+Y^{-1})-p-Y^{-1}. + \label{assembly:H-clock} +\end{align} +The admissibility and derivative conditions on $\kappa,p$ in the radial +lemmas will always be imposed. + +\subsection{Wells and the order of constants} + +Fix a sequence $(\zeta_j,h_j)$ in which every pair from a fixed countable +dense subset of $\{|y|<1\}$ and $\mathbb Z_{\geq0}$ occurs infinitely +often. The parameters of a test will be +\begin{equation}\label{assembly:test-parameters} + E_*=B_*c,\qquad \eta=e^{-c},\qquad + N=\lfloor e^{\gamma E_*}\rfloor,\qquad + l=\frac N2\log(t/R^2),\qquad S=Re^{c/N}. +\end{equation} +The constants $B_*,\gamma$ are fixed below. At each stage $R$ is fixed +before $c$. For a degree-$N$ system $(\mathcal H,p_N)$ from +Lemma~\ref{grid:polynomials}, define +\begin{equation}\label{assembly:well} + W_j(s)=d\log + \frac{|\mathcal H(s/R)|^2+|p_N(s/R)-1|^2+e^{-2c}} + {1+e^{-2c}}. +\end{equation} +The polynomial $p_N$ is unrelated to the radial control $p$. +The logarithmic sum-of-squares formula shows that $W_j$ is smooth +and plurisubharmonic. + +\begin{lemma}\label{assembly:well-bounds} +There is an absolute exponent $C_{\rm pol}$ such that, for $t\geq1$ +and $1\leq r\leq4$, +\begin{equation}\label{assembly:well-raw} + |\partial^{[r]}W_j| + \leq C(n,d)\exp\{C_{\rm pol}(\max(l,0)+c+\log N)\}. +\end{equation} +There is a constant $L_n<\infty$ such that, after increasing the degree +threshold, +\begin{equation}\label{assembly:well-tail} + |\partial^{[r]}W_j|\leq N^{L_n} + \qquad(1\leq r\leq4,\ l\geq L_n\log N). +\end{equation} +Both statements hold throughout the indicated outer ranges. +\end{lemma} + +\begin{proof} +Homogeneity and the spherical upper bound give +$|(\mathcal H,p_N)(s/R)|\leq Ne^l$. Cauchy estimates at scale +$c_0\sqrt t/N$ bound its derivatives of order at most four by this +quantity times a fixed power of $N$, since $t\geq1$. In a differentiated +logarithm there are only a fixed number of factors and denominator +powers; the denominator in \eqref{assembly:well} is at least $e^{-2c}$. +This proves \eqref{assembly:well-raw} with an absolute exponent; +fixed-dimensional constants and $d$ affect only its prefactor. + +The spherical lower bound also gives +$|(\mathcal H,p_N)(s/R)|\geq e^lN^{-C_n}$. Once $l$ exceeds a sufficiently +large multiple of $\log N$, subtraction of $(0,1)$ leaves norm at least +$e^lN^{-C_n}/2$. The factors $e^l$ in derivatives of the logarithm can +then be canceled. The remaining costs are fixed powers of $N$, with +exponents depending on the already fixed $n$. Increase $L_n$ to pay +them and the degree threshold to pay the prefactor. +\end{proof} + +For each stage we also introduce smooth nonnegative radial +plurisubharmonic functions $Z_{j,\rm in}$ and $Z_{j,\rm out}$. They +vanish before their stage and eventually become affine nondecreasing +functions of $\nu$. The center changes add offsets that vanish before +their change interval and are constant afterwards. We seek the final +twist in the form +\begin{equation}\label{assembly:twist} + \psi=\Psi+\sum_j W_j+\sum_j(Z_{j,\rm in}+Z_{j,\rm out}) + +\text{center-change offsets}. +\end{equation} +The center is constant except on its change intervals. There is one +fiber profile at each spatial point: it is $q_0(x,E(s))$ except during +the coordinated switches of Proposition~\ref{center:change}. +Shears occur only in the disjoint test collars. The recursion below +will make the well sum converge smoothly on compact spatial sets; +the other added terms are locally finite. + +The constants must provide the three outputs of +Proposition~\ref{grid:criterion}: a fixed-height spatial ball, small +unsheared disks on its boundary, and large prescribed sheared disks at +the nodes. Their exponents determine $k$ and hence the dimension. +Accordingly we first choose all geometric constants independently of +$n$, and only then use stagewise size choices to pay dimensional +prefactors. + +We give the full order of choices. First choose the profile constants, +including $d_0,K_0$ for the center change, then $\epsilon,\mu$ for +Proposition~\ref{quiet:profile}, the shear exponent in +Proposition~\ref{quiet:shear}, and the tolerance $\delta$ for +Proposition~\ref{radial:aligned}. These choices are independent of $n$. +Use control transitions whose first two derivatives in $\log Y$ have +absolute bounds. Let $C_{\rm rad}$ dominate the absolute exponents +needed for radial raw jets through order four. Choose +\begin{equation}\label{assembly:first-constants} + \epsilon_1\ll\epsilon/(1+C_{\rm rad}),\qquad + C_\uparrow\gg(1+C_{\rm rad})/\epsilon. +\end{equation} +Choose next +\begin{equation}\label{assembly:second-constants} + m_z\geq10(C_{\rm pol}+1),\qquad + B_*\geq\frac{C_*(1+m_z+C_{\rm pol}+C_{\rm rad})}{\epsilon_1}, +\end{equation} +where the absolute $C_*$ is large enough for the finitely many raw +exponent comparisons below. Choose $C_s$ larger than the quiet shear +severity exponent, and $C_w$ large enough to pay the shell loss and +four derivatives of its cutoffs; $C_w>5k_*+20$ suffices for these latter +costs after increasing the other fixed constants if necessary. Here +$k_*$ is the absolute separation exponent of the polynomial grids. +Choose +\begin{equation}\label{assembly:gamma-kb} + 0<\gamma\ll\epsilon_1,\qquad + \gamma B_*\leq\min\{.005,(8C_w)^{-1}\},\qquad + k_b>\max\{8/\gamma,100/\epsilon\}. +\end{equation} +Choose $M_b$ sufficiently large for the ratios in +Lemma~\ref{assembly:quiet-check}, and then $\sigma>0$ sufficiently small +for that lemma and Lemma~\ref{assembly:reset}. These are choices made +from fixed constants; in particular +$\sigma M_bk_b^2\ll1$, and $\sigma$ pays all quiet smallness requirements +involving $C_\uparrow$. Severity thresholds can depend on $\sigma$. +Finally take +\begin{equation}\label{assembly:depth} + d\geq8(M_bk_b+C_s+1)B_*. +\end{equation} +All constants so far are independent of dimension. + +Set +\begin{equation}\label{assembly:test-exponents} + m_0=d/2,\qquad p_0=2d,\qquad + p_1=C_sB_*+C_wd\gamma B_*. +\end{equation} +Then $p_1\leq d/4p_0+p_1+1$, and only now fix a finite $n$ sufficiently large that +\begin{equation}\label{assembly:dimension} + \sum_{r=1}^{k-1}\frac{r^{n-1}}{(n-1)!}<.01. +\end{equation} +Such a finite $n$ exists because $k$ is fixed. The constants $C_n,L_n$ +and all dimensional prefactors will be paid by the stagewise choice +of large $c$; they do not change \eqref{assembly:test-exponents}. + +\subsection{Recursive convention and capped intervals} + +Every estimate for an already configured region is made with an +additional allowance of one for the absolute value and spatial +jets through order four of all future wells. Let $\nu_j^{\rm sat}$ +denote the designated endpoint of severity saturation at stage $j$, +and let +\[ + K_j=\{s:|s|^2\leq e^{\nu_j^{\rm sat}}\} +\] +be the entire inner spatial ball ending there. The testing annulus lies +beyond this ball. At stage $j$ we impose +\begin{equation}\label{assembly:diagonal} + \sup_{s\in K_j}|\partial^{[r]}W_j(s)|\leq2^{-j}, + \qquad 0\leq r\leq j+4. +\end{equation} +Each new trigger begins beyond every completed stage, so every already +configured region lies in every later $K_j$. The omitted well sum +therefore has absolute value and spatial jets through order four +bounded by $\sum_j2^{-j}\leq1$ there. Its Hessian is also positive +semidefinite. Feasibility is proved below. This convention reserves the +entire future allowance before selecting any future coefficients. We apply +the local all-fiber lemmas to the final twist with these allowances; +we do not infer a uniform curvature statement from small perturbations +on compact subsets of the total space. + +Initially put +\begin{equation}\label{assembly:initial-data} + a=a_\psi=a_{\rm init}e^t,\quad Y=t,\quad + \lambda=p=1,\quad\kappa=(1+Y)^{-1}, +\end{equation} +with $a_{\rm init}$ so large that $H\leq\sigma$, and choose $B,\Psi$ +with equal initial constants. Up to a positive scale and an additive +constant, this initial spatial potential is the exponential radial +example of \citet[Section~3]{Seshadri2006}. Take center $\zeta_1$ and a constant +severity sufficiently large for the initial compact region and the +tail allowance. The radial curvature formula is strictly positive +on nonzero rank-two Hermitian tests at $t=0$, by its limit or by +direct differentiation of $a_{\rm init}e^t$. Thus the quiet criterion +also applies at the spatial origin. + +Before every trigger the radial data will have the form +\begin{equation}\label{assembly:cap} + \lambda=p=1,\qquad + \kappa=\frac{C_\kappa}{1+Y},\qquad + 1\leq C_\kappa\leq1+Y,\qquad H\leq\sigma, +\end{equation} +with $C_\kappa$ constant on that interval. Then +$dY/d\nu=C_\kappa Y$, so the cap can be continued through any +prescribed finite interval of $\nu$ without a pole. All previously +generated nonradial twist terms have bounded positive-order Euclidean +jets on the entire outer range: use \eqref{assembly:well-tail} and +the affine $Z$ tails. Hence a cap can also be continued until +$a=e^{u_s}$ dominates any prescribed bound for those old jets. + +Choose the next trigger start $\nu_B$ with +$Y_B=Y(\nu_B)\gg C_\kappa$, $e^{\nu_B}\geq1$, and severity constant +there at a value satisfying +\begin{equation}\label{assembly:trigger-initial} + E_0\geq C_\uparrow u_s(\nu_B). +\end{equation} +The reset below provides this inequality and any additional fixed +threshold. Put +\begin{equation}\label{assembly:trigger-radius} + \xi=Y_B(\nu-\nu_B),\qquad + \nu_*=\nu_B+Y_B^{-1},\qquad R=e^{\nu_*/2}. +\end{equation} +These data precede the choice of $c$. Retain the capped radial data +through +\begin{equation}\label{assembly:post-length} + l=T_*+4,\qquad T_*=10\{c+(1+L_n)\log N\}. +\end{equation} +For large $c$ this lies inside $\nu\leq\nu_B+2/Y_B$, since +$(T_*+4)/N\to0$. Throughout this fixed continuation +$Y\asymp Y_B$ and $u_s=u_s(\nu_B)+O(1)$, with constants allowed +to depend on the already fixed cap data. + +\begin{lemma}\label{assembly:height} +Before choosing $c$ at stage $j$, one can fix $D_j\geq j$ such that, +for every sufficiently large permitted $c$ and every final choice of +the reserved tails, +\begin{equation}\label{assembly:height-bound} + \ell(s,\zeta_j)\leq D_j-2\qquad(|s|\leq Re^{c/N}). +\end{equation} +\end{lemma} + +\begin{proof} +On the spatial region following the previous center-change interval, +the center is $\zeta_j$ and $\ell(s,\zeta_j)=B(s)$, including on shear +supports. The same identity holds on the initial cap at the first +stage. Thus the bound there follows from the fixed cap continuation +up to $\nu_B+2/Y_B$. +The earlier spatial region is a fixed compact set $K$ with already +chosen profiles and centers; future stages do not change $q(x,s)$ +on $K$. The reservation gives $|\psi-\psi_{\rm old}|\leq1$ there, +including the current and all later wells. Both centers are in the unit disk, so +$|\zeta_j-\zeta(s)|\leq2$. Choose +$A>\log4+\sup_K\psi_{\rm old}+1$. +For each $s\in K$, $F(x,s)\to\infty$ as $x\to\infty$. +Continuity, a finite cover, and monotonicity in $x$ give a common +finite $X$ with $F(X,s)>A$ on $K$. Inverting the fiber relation +therefore gives $\tau\leq e^X$. Since $q\geq1$, the radial potential +is bounded by $\sup_K B+e^X$; the fixed shear adds at most +$2\sup_K|c_{\rm sh}|$. At center coincidences this argument is +understood by the smooth limit $\tau=0$. These bounds are independent +of $c$ and of the future wells. Add two and increase to at least $j$. +\end{proof} + +We always require +\begin{equation}\label{assembly:height-rate} + c\geq j(D_j+1) +\end{equation} +in addition to the finitely many current thresholds. Thus $D_j\to\infty$ +and $D_j/c_j\to0$. + +\subsection{The severity trigger} + +Choose a smooth convex nondecreasing $h_b$ on $(-\infty,1)$, zero +for $\xi\leq0$, equal to $-\log(1-\xi)$ up to an affine function near +$1$. To construct it, cut on the nonnegative second derivative +$(1-\xi)^{-2}$ smoothly and integrate twice from zero. +Use +\begin{equation}\label{assembly:trigger} + E=E_0+k_bh_b(\xi),\qquad Z_{j,\rm in}=M_bk_b^2h_b(\xi) +\end{equation} +until severity is near $E_*$. Smoothly saturate $E$ at $E_*$ over +a unit interval of its argument, so that $E=E_*$ by +$k_bh_b=E_*-E_0+1$. Derivatives of the saturation function in that +argument are bounded. Keep the unsaturated convex $Z_{j,\rm in}$ +slightly longer, and once $k_bh_b\geq E_*-E_0+2$, cut off its +second derivative nonnegatively over half the remaining gap to +$\xi=1$. Continue it affinely in $\nu$ thereafter. This is a smooth +convex nondecreasing function. Such a function of $\log|s|^2$, +zero near the origin, is a smooth radial plurisubharmonic function: +its transverse and radial Hessian entries are its first and second +$\nu$ derivatives divided by $t$. + +Near this cutoff the gap $1-\xi$ is comparable, with fixed stagewise +factors, to $e^{-(E_*-E_0)/k_b}$. The final slope is bounded by +\begin{equation}\label{assembly:trigger-slope} + C M_bk_b^2Y_Be^{(E_*-E_0)/k_b}. +\end{equation} +The value there is $M_bk_b(E_*-E_0+O(1+k_b))$. Multiplication +of \eqref{assembly:trigger-slope} by the remaining gap to $\nu_*$ +costs only a fixed $O(M_bk_b^2)$, and its product with $4c/N$ +tends to zero. Thus for large $c$, +\begin{equation}\label{assembly:trigger-value} + Z_{j,\rm in}\leq2M_bk_bE_*\qquad(|l|\leq2c). +\end{equation} +Through saturation, its derivatives and those of $E$ through order +four have bounds given by fixed prefactors times +$(1+Y_B)^4 e^{4(E-E_0)/k_b}$. +The bound with $E_*$ also holds for positive-order derivatives of +the affine tail. + +The well satisfies \eqref{assembly:diagonal} after increasing $c$. +Indeed throughout severity saturation, +\begin{equation}\label{assembly:activation-gap} + l\leq-C(Y_B,E_0)N e^{-E_*/k_b}. +\end{equation} +The absolute value on the right grows exponentially in $E_*$, +because $\log N\sim\gamma E_*$ and $\gamma>4/k_b$. +Homogeneity makes the polynomial evaluations on the whole inner +ball exponentially small in this quantity. Cauchy on a collar +of scale $R/N$ gives the same conclusion for any fixed derivative +order, with only powers of $N$ as further costs. The logarithm in +\eqref{assembly:well} is then close to zero with all these +derivatives. We can therefore enforce the bound $2^{-j}$ for its +value and derivatives through order $j+4$. This decay pays all +fixed exponential-in-$c$ costs needed at stage $j$. + +\begin{lemma}\label{assembly:quiet-check} +The trigger satisfies the hypotheses of +Proposition~\ref{quiet:profile}, uniformly for every $\tau\geq0$. +After saturation the same criterion applies through $l=4c$, including +the initial part of the post-test term defined below. +\end{lemma} + +\begin{proof} +For raw estimates use Euclidean axes. Here $t\geq1$, $Y\asymp Y_B$, +and $\log Y_B\leq C u_s(\nu_B)$; the last estimate follows by +integrating $d\log Y/du_s=\kappa(1+Y^{-1})\leq2.2$ for $Y\geq1$. +Differentiation of $a'=aY/t$ bounds the required radial metric jets +by $e^{C_{\rm rad}u_s}$. Their inverses cost no more. The radial +curvature estimate gives +\begin{equation}\label{assembly:euclidean-curvature} + C^V(T,T)\geq c\kappa H|T|_V^2 + \geq c(a/t)|T|^2, +\end{equation} +since $V\geq aI$ and +$\kappa H=C_\kappa Y/((1+Y)ta)\geq c/(ta)$. +The initial severity pays these bounds, the old data and the future +allowance. The trigger cost $e^{4E/k_b}$ fits the quiet raw exponent. +The current well is small before saturation. Also $G\geq V$, since +all nonconstant extra twist terms in this interval are +plurisubharmonic. + +For the covariant condition, write +\[ + \mathcal B(\nu)=B(e^\nu),\qquad + \mathcal U(\nu)=\Psi(e^\nu)+Z_{j,\rm in}(e^\nu), +\] +and use primes for $\nu$ derivatives within this proof. On the +severity-varying interval, +\begin{align} + \frac{\mathcal U''}{\mathcal U'},\ + \frac{|\mathcal U'''|}{\mathcal U''},\ + \frac{\mathcal B''}{\mathcal B'},\ + \frac{|\mathcal B'''|}{\mathcal B''} + &\leq C\sqrt{\mathcal U''}, + \label{assembly:connection-ratios}\\ + \frac{|E'|}{\mathcal U'},\ + \frac{|E'|}{\sqrt{\mathcal U''}},\ + \frac{|E''|}{\mathcal U''},\ + \frac{|E'''|}{(\mathcal U'')^{3/2}} + &\leq\varepsilon_b, + \label{assembly:small-ratios} +\end{align} +where $\varepsilon_b$ can be made as small as required. +Away from the pole, $h_b$ and its relevant derivatives are bounded, +whereas +$\mathcal B'=Y/H$ and $\mathcal B''=Y(1+Y)/H$. +The choice $\sigma M_bk_b^2\ll1$ makes these background terms +dominate. Near the pole let $D_b=Y_B/(1-\xi)$. Then +\[ + Z_{j,\rm in}'\asymp M_bk_b^2D_b,\quad + Z_{j,\rm in}''\asymp M_bk_b^2D_b^2,\quad + |Z_{j,\rm in}'''|\leq C M_bk_b^2D_b^3, +\] +and $|E^{(r)}|\leq Ck_b^rD_b^r$ for $1\leq r\leq3$, including +saturation. These prove \eqref{assembly:connection-ratios} and +make \eqref{assembly:small-ratios} small by first enlarging $M_b$ +and then decreasing $\sigma$. The fixed intervening interval of +$\xi$ is covered by the first estimates. We also have +$\mathcal U''\geq1$. + +At fixed $\tau$, the radial part of the Schur metric is the Hessian of +\[ + J_h=\mathcal B+\tau(\mathcal U-\bar F(x,E)), +\] +where the bar denotes averaging from zero to $\tau$ in the $\tau$ +variable. Bounded severity derivatives of $\bar F$ and +\eqref{assembly:small-ratios} show that the first two derivatives +of $\mathcal U-\bar F$ are positively comparable with those of +$\mathcal U$ and its third derivative is bounded by +$C(\mathcal U'')^{3/2}$. +Use affine axes normalized by the Hessian of $\mathcal U$, with +radial radius $\sqrt{t/\mathcal U''}$ and transverse radii +$\sqrt{t/\mathcal U'}$. +At a radial point, the only nonzero holomorphic connection types +for a radial Hessian metric are radial--radial to radial and +radial--transverse to transverse. Differentiation gives the bounds +\begin{equation}\label{assembly:connection-exact} + \frac{|J_h'''/J_h''-1|}{\sqrt{\mathcal U''}}, + \qquad + \frac{|J_h''/J_h'-1|}{\sqrt{\mathcal U''}}. +\end{equation} +They are bounded by \eqref{assembly:connection-ratios}. No estimate +of a radial/transverse condition number is used. + +Write the remaining Hessian as $\tau G_e$, with $G_e\geq0$. +Its size and first jets in these axes are bounded: old Euclidean +bounds are dominated by the large initial $a$, and the current +and future wells have the reserved absolute bounds. The normalizing +radii are at most $a^{-1/2}$ because this background is at least +$V\geq aI$. If $h_r$ denotes the radial Schur metric, the exact +connection difference is, with the usual index placement, +\begin{equation}\label{assembly:connection-perturbation} + \Gamma^h-\Gamma^{h_r} + =\tau(\partial G_e-\Gamma^{h_r}G_e)(h_r+\tau G_e)^{-1}. +\end{equation} +Since $h_r\geq c\tau I$, $\tau h^{-1}$ is uniformly bounded for +all $\tau>0$. Thus the full connection is uniformly bounded as well. + +In these axes the first derivative of $E$ has norm +$|E'|/\sqrt{\mathcal U''}$; the complex Hessian entries are bounded +by $|E'|/\mathcal U'$ and $|E''|/\mathcal U''$, and the only pure +second derivative has size $|E''-E'|/\mathcal U''$. +Consequently subtracting the connection times the first derivative +gives a small $\nabla_h^{2,0}E$. Since the actual $G$ is at least the +normalizing background, the same smallness holds in $G$ units. +This proves all quiet smallness conditions uniformly in $\tau$. + +After saturation, severity derivatives vanish. Through $l=4c$ the +well's raw costs are bounded by $e^{C(c+\log N)}$; the post term +below has cost at most a fixed multiple of $N^4e^{2m_zc}$. +Both fit $\epsilon E_*$ by \eqref{assembly:second-constants}, while +the inverse and radial curvature estimates persist. +\end{proof} + +\subsection{Testing disks and the post ramp} + +Choose $c$ sufficiently large that the degree in +\eqref{assembly:test-parameters} meets the grid accuracy required for +$h_j$, and $N>h_j$. At every good lift $v$, use a smooth cutoff +supported in $|s-Rv|<2RN^{-k_*}$ and equal to one on +$|s-Rv|\leq RN^{-k_*}$. Assign on the inner ball +\begin{equation}\label{assembly:shear-amplitude} + c_{\rm sh}=\alpha b_v,\qquad + \alpha=\exp(-C_sE_*-C_wd\log N), +\end{equation} +and multiply by the cutoff in the collar. The phases have modulus +one, are independently assignable by projective direction, and +repeat on every lift of that direction. The outer balls are disjoint. +A cutoff $\chi(|s-Rv|^2/(RN^{-k_*})^2)$ has fixed-order multilinear +derivative bound $C_r(RN^{-k_*})^{-r}$ with absolute $C_r$. +At most one contributes at any point, so there is no factor for +the number of nodes or dimension. + +On the derivative supports, the shell estimate gives +\begin{equation}\label{assembly:shear-lower} + \psi\geq-2d(k_*+11)\log N-C_{\rm old}. +\end{equation} +Old values are bounded on the fixed cap continuation, future values +have the allowance, and the trigger term is nonnegative. +These balls have $|l|\leq\log N+1$ and lie beyond severity +saturation. First and pure second derivatives of $\psi$ also fit +the quiet raw bounds; the extra background first-derivative cost +is at most $a\sqrt t$. The constants $C_s,C_w$ pay respectively +the severity exponent of Proposition~\ref{quiet:shear} and half the +loss \eqref{assembly:shear-lower} together with the four cutoff +derivatives. Large $c$ pays fixed old factors. Thus that proposition +applies for all fiber radii with metric comparison loss at most +$1/2$. Where a shear coefficient is constant, its potential term +is pluriharmonic. + +At a node the new well is $-2dc+O(1)$; on $|s|=S$ it is at most +$2d(c+\log N)+O_d(1)$. All other values on the fixed continuation +are bounded independently of $c$, except the trigger term. +The post term is still zero. Hence +\begin{align} + \psi_{\rm node}&\leq-2dc+2M_bk_bB_*c+C_{\rm old}, + \label{assembly:node-twist}\\ + \psi_{\rm sphere}&\leq2d(c+\log N)+2M_bk_bB_*c+C_{\rm old,d}. + \label{assembly:sphere-twist} +\end{align} +For a small fixed profile-dependent $c_0>0$, the disk +\begin{equation}\label{assembly:fiber-disk} + |y-\zeta_j|0, +\] +so the large-$H$ aligned condition remains valid. Restore $p=1$. + +Finally reduce $H$ to $\sigma$. Transition $\kappa$ to $1/2$, +hold there for an adjustable interval, then return to one from +below. For $p=1$ the exact relation is +\begin{equation}\label{assembly:H-descent} + \frac{d\log H}{d\log Y}=1-\kappa^{-1}. +\end{equation} +Its value is $-1$ on the middle plateau. The endpoint transitions +have fixed finite effects on $\log H$; the initial $H$ can be +made as large as needed. Choose the plateau length so that the +final value is exactly $\sigma$. Throughout this descent +$H\geq\sigma$, $\lambda=1$, and $1/2\leq\kappa\leq\kappa_*$, +so the $\lambda=1$ aligned alternative applies. Choose the +downward transitions slowly enough to preserve the same +derivative inequality as above. + +All these intervals have finite $u_s$ or finite $\log Y$ length. +Equations~\eqref{assembly:clocks} show that $Y$ remains finite +on each finite $u_s$ interval and that the radius advances by +a finite positive amount. A putative pole of an indefinitely +continued accelerated equation is therefore never reached +during an episode. This proves the claim. +\end{proof} + +\begin{lemma}\label{assembly:reset} +From the endpoint in Lemma~\ref{assembly:change-order}, severity +can be increased and a new cap started so that the next trigger +has \eqref{assembly:trigger-initial}, while the metric continues +to satisfy the quiet criterion. The cap can be extended to +arbitrarily large spatial radii. +\end{lemma} + +\begin{proof} +The spatial scale $u_s$ may now be enormous, and $E_{\rm post}$ need +not dominate it. Proposition~\ref{quiet:profile} permits any fixed +holomorphic affine frame at the evaluation point. We first use +$V$-unit affine axes, where the curvature scale is the fixed $\sigma$ +and the relevant jets are bounded independently of $u_s$. This will +justify the initial severity increase before arranging the size +inequality needed for the next trigger. + +First hold $\kappa=p=\lambda=1$ and $H=\sigma$. +Let $E$ increase smoothly as a function of $u_s$, starting at +$E_{\rm post}$, with a fixed flat-ended transition to slope +$4C_\uparrow$. Its derivatives through order four in $u_s$ +are bounded by constants depending on the fixed parameters, +not on $\sigma$ or the possibly enormous $u_s$ at entry. +Continue long enough that +\begin{equation}\label{assembly:catchup} + E\geq2C_\uparrow u_s +\end{equation} +and any desired fixed threshold. The greater slope ensures +that this occurs after a finite interval. + +We verify the quiet criterion even before \eqref{assembly:catchup}. +Use $V$-unit affine axes. The scaled-jet radial estimates give +metric jets of orders $r=1,2$ bounded by $CH^{r/2}$ in these +axes, and scaled derivatives of $u_s$ through order four are +bounded by an absolute constant. Therefore +\[ + |\partial^{[r]}E|_{V}\leq C H^{r/2},\qquad 1\leq r\leq4. +\] +The extras in \eqref{assembly:aligned-extra} satisfy the same +bounds after this normalization, and $G\geq V$. +For $\sigma$ small, the severity terms in $\mathcal N$ are +small relative to $G$, so +$h\asymp(1+\tau)V$, with first derivatives bounded in these +axes by $C(1+\tau)\sqrt H$. +Its connection is consequently $O(\sqrt H)$ uniformly in +$\tau$. The first, complex second, and pure covariant second +severity derivatives are small for every $\tau$. + +The raw curvature margin on this hold is +$C^V(T,T)\geq c\sigma|T|^2$ in $V$-unit axes, because +$\kappa=1$ and $H=\sigma$. All other raw data are bounded there +by fixed constants. The previously chosen $E_{\rm post}$ +exceeds the fixed profile and $\sigma$ thresholds, so it +pays this margin even when it is much smaller than the current +$u_s$. Thus there is no gap before severity catches up. + +After \eqref{assembly:catchup}, keep the same slope while +turning $\kappa$ into a cap. Here is an explicit admissible +transition. If $Y_0$ is its starting clock value, set +\[ + A(Y)=1+Y_0+\int_{Y_0}^Y \eta_0(v/Y_0)\,dv,\qquad + \kappa(Y)=\frac{A(Y)}{1+Y}, +\] +where $\eta_0$ is smooth, equals one near $1$, decreases +nonnegatively to zero, and is zero for arguments at least two. +For $Y\leq Y_0$ use $A=1+Y$. +Then $1\leq A\leq1+Y$, so +$(1+Y)^{-1}\leq\kappa\leq1$. Moreover +\[ + Y\kappa_Y+\kappa + =\frac{YA'(Y)}{1+Y}+\frac{A(Y)}{(1+Y)^2}\geq0. +\] +The first two derivatives of $\kappa$ in $\log Y$ have absolute +bounds, because $Y/Y_0$ ranges over a fixed interval while +$\eta_0$ changes. For example, writing $b=Y/(1+Y)$ and +$D=Y\partial_Y$, one has +$D\kappa=b(\eta_0-\kappa)$ and +$D^2\kappa=b(1-2b)(\eta_0-\kappa)+bD\eta_0$. +At the end $A$ is the new constant $C_\kappa$. +Since $\kappa\leq1$ and $p=1$, $H$ is nonincreasing and +stays at most $\sigma$. + +The same normalized jet and connection estimates continue +through the transition and the cap. To check the raw curvature +margin there, note that $Y\geq t$: it holds initially, and +$d\log Y/d\nu=\kappa(1+Y)\geq1$ preserves it. +Also $u_s\geq t$ after choosing the initial constant large +enough, since $du_s/dt=Y/t\geq1$. +Hence +\[ + \kappa H\geq c/(ta)\geq c e^{-2u_s} + \quad(t\geq1), +\] +and \eqref{assembly:catchup} with large $C_\uparrow$ pays the +quiet lower bound $e^{-\epsilon E}$. The positive-order +extras stay bounded relative to $V$ on an arbitrarily long +cap. More explicitly, a fixed Euclidean jet bound $D_{\rm out}$ +costs at most +$C(D_{\rm out}/a)H^{r/2}$ in $V$-unit axes for Hessian jets +of order $r\leq2$, since $aH=Y/t\geq1$. +The ratio only decreases as $a$ grows. +Future wells are included in the fixed allowance until +their own activations. + +Coast as far in $\nu$ as desired. Shortly before the next +chosen trigger, turn off the severity slope smoothly over +a fixed-length $u_s$ interval. The margin in +\eqref{assembly:catchup} is sufficient to retain +$E_0\geq C_\uparrow u_s(\nu_B)$ after this bounded final +increment, by making the cap endpoint large enough. +The existing raw jet bounds can also be absorbed by making +that endpoint large. This produces +\eqref{assembly:trigger-initial} and the next cap data. +\end{proof} + +\subsection{Global realization and completeness} + +We now carry out the recursion. At each stage first extend the +cap so that its trigger starts beyond all previous stages and +with $\nu_B\geq j$. Fix its old-data bounds and testing height +by Lemma~\ref{assembly:height}. Then choose $c$ large enough +for \eqref{assembly:height-rate}, the polynomial approximation +for $h_j$, \eqref{assembly:diagonal}, and all the finite +thresholds in the trigger, test, post ramp, and aligned entry. +The later finite center-change targets are chosen in the +order proved in Lemma~\ref{assembly:change-order}. +Lemma~\ref{assembly:reset} completes the cycle. +Thus no clock or already configured profile is determined +by an unknown future well. + +For completeness, the future allowance is valid in every +kind of local regime. In a quiet interval, later wells have +positive-semidefinite Hessians and bounded absolute jets; +they preserve $G\geq V$, and the preceding connection proof +uses exactly these bounds. In an aligned or switch interval, +their scaled Hessian jets are bounded by +$C/(a_\psi H)$, which fits the reserved part of +$\delta/H$ at the large chosen $a_\psi$. In a center move, +their values change the radius estimates by at most a fixed +factor, and their scaled derivatives are within the old-data +allowance included before choosing the shift. On a shear +collar they lower the twist by at most one, already included +in the lower bound used for its coefficient. Consequently +the local lemmas apply to the final infinite twist for +every fiber radius, even though curvature margins need not +have a positive infimum over all fiber radii. + +The condition $\nu_B\geq j$ rules out finite accumulation of +stages. Any compact spatial set lies before the activation +of all sufficiently late wells. For a fixed derivative order +$r$, those later wells satisfy +$|\partial^{[r]}W_j|\leq2^{-j}$ once $j+4\geq r$. +The Weierstrass criterion in every derivative order proves +compact-smooth convergence of $\sum_jW_j$. +The $Z$ terms and nonconstant offsets are locally finite. +Flat-ended transitions, finite positive clock evolutions, +and the prescribed initial data therefore give global smooth +spatial data. On a compact spatial set the profile is smooth +to $\tau=0$ with $q(0,s)=1$; its normalized $F-x$ is smooth +there as well. The smooth inversion at the fiber center +therefore gives a global smooth potential +\[ + \ell(s,y)=\ell_{\rm rad}(s,y)+ + \operatorname{Re}\bigl(c_{\rm sh}(s)(y-\zeta(s))\bigr). +\] +The shears are absent where the center changes. +All local curvature and positivity conclusions apply, +including the fiber axes by their smooth limits. + +We spell out the metric comparison used for completeness. +In the constant-center regimes the Schur metric is bounded +below by a fixed positive multiple of $V$; during profile +switches, positivity of $\mathcal N$ for all $\tau$ also +gives $h=V+\int_0^\tau\mathcal N\,dv\geq V$. +The shear and moving-center comparisons can each be chosen +to lose at most a factor $1/2$. Their spatial supports are +separated, so these losses do not multiply over infinitely +many stages at a point. Hence a single $c_1>0$ satisfies +\begin{equation}\label{assembly:spatial-completeness} + g_\ell(\dot s,\dot y)\geq c_1 V(\dot s,\dot s) + \geq c_1a_{\rm init}|\dot s|^2 +\end{equation} +everywhere. + +A finite-length curve therefore has finite Euclidean +spatial variation and remains in a compact spatial set. +On such a compact set, for large $|y|$, the metric is also +bounded below by a fixed positive multiple of the frozen +log-block metric at the point. The tilt coefficients are +uniformly bounded in $\tau$ there. Indeed they are spatial +derivatives of $\psi-F$: on quiet intervals they use bounded +$F_E$; on switch intervals they use bounded $F_\beta$, which +is constant in $x$ past the fixed switch cutoff; and on a +center move the profile is spatially fixed apart from the +already bounded twist. There are only finitely many profile +episodes on the compact set, and the well sum is smooth. +Thus some compact-set constant $C_K$ bounds the tilt one-form. + +Also $q\geq1$ and $F\leq x$, so at sufficiently large $|y|$ +the fiber value satisfies $\tau\geq1$ and $P=\tau q\geq1$, +uniformly on the compact spatial set. +The frozen fiber differential is $dy/(y-\zeta(s))$; freezing +means that the center has its point value in this comparison. +The block bound and the triangle inequality imply along +the curve, when $|y|$ is large, +\[ + \frac{|dy|}{|y-\zeta(s)|} + \leq C_K\,ds_{g_\ell}+C_K|ds|. +\] +Both terms on the right have finite integral by +\eqref{assembly:spatial-completeness}. Since $|\zeta(s)|<1$, +this bounds the total variation of $\log|y|$ outside a +fixed disk and prevents fiber escape. Thus no divergent +curve has finite length. Equivalently, a Cauchy sequence +lies in a compact Euclidean coordinate set, where the +smooth positive metric gives convergence. The base metric +is complete. + +\begin{proof}[Proof of Proposition~\ref{assembly:base}] +The recursion gives a smooth strictly plurisubharmonic +potential on $\mathbb C^{n+1}$, with nonpositive real +sectional curvature by the local lemmas and completeness +by the preceding argument. Every required pair +$(\zeta,h)$ is tested infinitely often. The stages satisfy +$R\geq1$, $N\to\infty$, $c\to\infty$, +$\log N\leq.005cp_1$ and $c/(D_j+1)\to\infty$. +In particular the elementary holomorphic functions $e^{ikw}$, +$k>0$, are unbounded on the tube; they do not contradict its +bounded-function conclusion. +\end{remark} + +\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}] +Apply Lemma~\ref{tube:transfer} to the base just +constructed. Its tube is connected, complete, and +diffeomorphic to Euclidean space, hence simply connected. +Its real sectional curvatures are at most $-1$ by that lemma, +and its bounded holomorphic functions are constant by +Proposition~\ref{assembly:base}. The complex dimension is the fixed finite +$m=n+2$. + +There is no finite lower sectional-curvature bound. +At $s=0,y=\zeta_1$, the center and profile are constant +nearby and there is no shear. On the center axis the +spatial curvature tensor of the base equals that of $V$: +the added radial-fiber potential vanishes to second +order in $y-\zeta_1$, and its cubic derivatives with two +spatial indices and one fiber index vanish there. +The initial radial metric has a strictly negative +holomorphic-plane curvature in a spatial direction. +Corollary~\ref{tube:negative-plane} therefore shows that the tube's +sectional curvatures are unbounded below. +This proves every assertion of Theorem~\ref{thm:main}. +\end{proof} + + +\paragraph{The two-sided question.} +Neither this example nor the independent pinched threefold of +\citet[Theorem~1.1]{OpenAIPinchedThreefold2026} decides whether a +nonconstant bounded holomorphic function must exist under finite +two-sided negative real sectional pinching. The present curvature is +unbounded below, while the threefold retains nonconstant bounded +base-coordinate functions. diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/stage-clock.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/stage-clock.tex new file mode 100644 index 0000000000000000000000000000000000000000..c4083949d91bc10ef2e4f374c580a908ceae8098 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/stage-clock.tex @@ -0,0 +1,43 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[ + stage/.style={draw=black!55,rounded corners=2pt,fill=black!3, + text width=3.65cm,minimum height=1.27cm,align=center,font=\small}, + flow/.style={-{Latex[length=2mm]},thick,draw=black!65}, + note/.style={font=\footnotesize,align=center}] +\node[stage] (cap) at (0,0) {Cap and coast\\ + $\lambda=1$, $H\le\sigma$}; +\node[stage] (trigger) at (4.6,0) {Severity trigger\\ + raise $E$ to $E_*$}; +\node[stage] (test) at (9.2,0) {Polynomial test\\ + fixed center and severity}; +\node[stage] (post) at (9.2,-2.05) {Post-test ramp\\ + reach $E_{\rm post}$}; +\node[stage] (accelerate) at (4.6,-2.05) {Accelerate growth\\ + enter aligned regime}; +\node[stage] (grow) at (0,-2.05) {Grow $H$, then $\lambda$\\ + $E=E_{\rm post}$}; +\node[stage] (center) at (0,-4.1) {Widen, move, restore\\ + hold $\kappa=p=1$}; +\node[stage] (reset) at (4.6,-4.1) {Return $\lambda$ to $1$\\ + then lower $H$ to $\sigma$}; +\node[stage] (catchup) at (9.2,-4.1) {Severity catch-up\\ + before the next cap}; +\draw[flow] (cap)--(trigger); +\draw[flow] (trigger)--(test); +\draw[flow] (test)--(post); +\draw[flow] (post)--(accelerate); +\draw[flow] (accelerate)--(grow); +\draw[flow] (grow)--(center); +\draw[flow] (center)--(reset); +\draw[flow] (reset)--(catchup); +\draw[flow] (catchup.south)--++(0,-.48) + node[below,note]{Next stage: cap and coast}; +\end{tikzpicture} +\caption{One stage, in increasing spatial radius. The test is installed +while the spatial growth is capped. The intervening growth and center +change occupy finite clock intervals. After the reset, severity must +catch up with the accumulated spatial scale before the next capped +regime can begin. The arrows indicate order, not elapsed radius.} +\label{fig:stage-clock} +\end{figure} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/tube.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/tube.tex new file mode 100644 index 0000000000000000000000000000000000000000..8c658edee8f16ccae1b43913fb3447fb0277d7c6 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/tube.tex @@ -0,0 +1,18 @@ +\begin{tikzpicture}[x=1.15cm,y=.85cm] + \fill[black!5] + (-3,1.8) .. controls (-2.2,.45) and (-1.05,.55) .. (0,1.05) + .. controls (1.05,1.55) and (2.2,.65) .. (3,1.45) + -- (3,4.15) -- (-3,4.15) -- cycle; + \draw[->] (-3.45,0) -- (3.5,0) node[right] {$z$}; + \draw[->] (-3.35,-.1) -- (-3.35,4.4) node[above] {$v$}; + \draw[thick] + (-3,1.8) .. controls (-2.2,.45) and (-1.05,.55) .. (0,1.05) + .. controls (1.05,1.55) and (2.2,.65) .. (3,1.45); + \node[anchor=north] at (1.8,.82) {$v=\ell(z)$}; + \draw[dashed] (0,0) -- (0,1.05); + \draw[<->] (0,1.1) -- (0,3.05) + node[midway,right] {$b=v-\ell(z)$}; + \fill (0,3.12) circle (1.6pt); + \node[above right] at (0,3.12) {$(z,w)$}; + \node at (-1.75,3.55) {$\mathcal T_\ell:\ b>0$}; +\end{tikzpicture} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/main.tex b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..cd6f5bc9f79bd9bcd5da16699eba270d29777a15 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/main.tex @@ -0,0 +1,59 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs,array} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,fit,decorations.pathreplacing} +\usepackage[numbers,sort&compress]{natbib} +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{24,68,102} +\PassOptionsToPackage{hyphens}{url} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref} +\hypersetup{pdftitle={One-sided negative sectional curvature and the holomorphic Liouville property},pdfauthor={OpenAI}} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\setlist{topsep=5pt,itemsep=3pt,parsep=0pt} +\allowdisplaybreaks[2] +\emergencystretch=1.5em +\title{One-sided negative sectional curvature\\and the holomorphic Liouville property} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +We construct, in some sufficiently large fixed finite complex dimension +$m$, a domain in $\mathbb C^m$ diffeomorphic to $\mathbb R^{2m}$ that +admits a complete K\"ahler metric with real sectional curvature at most +$-1$ and has only constant bounded holomorphic functions. Its sectional +curvatures are unbounded below. The construction gives a negative answer +to the one-sided bounded-holomorphic-function question. +\end{abstract} +\clearpage +\begingroup +\small +\tableofcontents +\endgroup +\clearpage +\input{01-introduction} +\input{02-tube} +\input{03-analytic} +\input{04-fiber} +\input{05-quiet} +\input{06-radial} +\input{07-centers} +\input{08-assembly} +\bibliographystyle{plainnat} +\bibliography{refs} +\end{document} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/refs.bib b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/refs.bib new file mode 100644 index 0000000000000000000000000000000000000000..5ea2f9cc14f5f6ac7a02110c5608531e711eee2f --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/refs.bib @@ -0,0 +1,185 @@ +@article{WuYau2020, + author = {Wu, Damin and Yau, 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{Mathematical Research Letters}, + volume = {13}, + number = {3}, + pages = {495--500}, + year = {2006}, + doi = {10.4310/MRL.2006.v13.n3.a13}, + eprint = {math/0602289}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + url = {https://arxiv.org/abs/math/0602289} +} + +@misc{OpenAIPinchedThreefold2026, + author = {{OpenAI}}, + title = {{A negatively pinched K{\"a}hler threefold without bounded holomorphic coordinates}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf}{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026}}, + year = {2026} +} + +@misc{Tropp2022, + author = {Tropp, Joel A.}, + title = {{ACM} 204: Matrix Analysis}, + howpublished = {Caltech CMS Lecture Notes 2022-01}, + year = {2022}, + doi = {10.7907/nwsv-df59}, + url = {https://tropp.caltech.edu/notes/Tro22-Matrix-Analysis-LN.pdf}, + note = {\href{https://tropp.caltech.edu/notes/Tro22-Matrix-Analysis-LN.pdf}{Lecture notes}, Theorem~7.13} +} diff --git a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d77eebe4659a69d96895aa32fa8aa0aaded86616 --- /dev/null +++ b/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dedbbd2f24f8281c384e8927c78a54fd71d894113da70e5f702b8deb1f5bc046 +size 701177 diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/README.md b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..8279765792416947a02c6bb5d7205c393b76967f --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [Open Homomorphisms of Global Solvably Closed Galois Groups](open-homomorphisms-solvably-closed-galois-groups.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026, + author = {{OpenAI}}, + title = {{Open Homomorphisms of Global Solvably Closed Galois Groups}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf}{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/main.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..e75d090a4701fe69b90cf5d10153e7a7409cddc4 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/main.tex @@ -0,0 +1,50 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{tikz-cd} +\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref} +\hypersetup{pdftitle={Open Homomorphisms of Global Solvably Closed Galois Groups},pdfauthor={OpenAI}} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\C}{\mathbb C} +\newcommand{\F}{\mathbb F} +\newcommand{\X}[2]{X_{#1}(#2)} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Ind}{Ind} +\DeclareMathOperator{\Fr}{Fr} +\DeclareMathOperator{\cor}{cor} +\DeclareMathOperator{\res}{res} +\DeclareMathOperator{\Ad}{Ad} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Cl}{Cl} +\DeclareMathOperator{\Tr}{Tr} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\title{Open Homomorphisms of Global Solvably Closed Galois Groups} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required. +\end{abstract} +\input{sections/introduction} +\input{sections/preliminaries} +\input{sections/characters} +\input{sections/generation} +\input{sections/matching} +\input{sections/identification} +\input{references} +\end{document} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/references.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..68664ffded3265bd6b6dc5f198fa09b36b634427 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/references.tex @@ -0,0 +1,101 @@ +\begin{thebibliography}{99} + +\bibitem{bass1966} +Hyman Bass, +\emph{Generators and relations for cyclotomic units}, +Nagoya Math. J. \textbf{27} (1966), no.~2, 401--407. +\href{https://doi.org/10.1017/S0027763000026258}{doi:10.1017/S0027763000026258}. + +\bibitem{dasgupta2023} +Samit Dasgupta, +\emph{Ranks of matrices of logarithms of algebraic numbers. I: The theorems of Baker and Waldschmidt--Masser}, +Essential Number Theory \textbf{2} (2023), no.~1, 93--138. +\href{https://doi.org/10.2140/ent.2023.2.93}{doi:10.2140/ent.2023.2.93}. + +\bibitem{hoshi2025} +Yuichiro Hoshi, +\emph{Homomorphisms of global solvably closed Galois groups compatible with cyclotomic characters}, +Tohoku Math. J. (2) \textbf{77} (2025), no.~1, 17--32. +\href{https://doi.org/10.2748/tmj.20230414}{doi:10.2748/tmj.20230414}. + +\bibitem{hoshi2026} +Yuichiro Hoshi, +\emph{A note on open homomorphisms between global solvably closed Galois groups}, +Kodai Math. J. \textbf{49} (2026), no.~1, 63--79. +\href{https://doi.org/10.2996/kmj49104}{doi:10.2996/kmj49104}. + +\bibitem{laurent1991} +Michel Laurent, +\emph{Sur quelques r\'esultats r\'ecents de transcendance}, +Ast\'erisque \textbf{198--199--200} (1991), 209--230. +\href{https://doi.org/10.24033/ast.93}{doi:10.24033/ast.93}. + +\bibitem{maosaidi2025} +Yu Mao and Mohamed Sa\"idi, +\emph{Open homomorphisms between $m$-step solvable Galois groups compatible with the cyclotomic characters}, +preprint, 2025. +\href{https://arxiv.org/abs/2512.13312}{arXiv:2512.13312}. + +\bibitem{masser1981} +D.~W. Masser, +\emph{On polynomials and exponential polynomials in several complex variables}, +Invent. Math. \textbf{63} (1981), no.~1, 81--95. +\href{https://doi.org/10.1007/BF01389194}{doi:10.1007/BF01389194}. + +\bibitem{neukirch1969a} +J\"urgen Neukirch, +\emph{Kennzeichnung der $p$-adischen und der endlichen algebraischen Zahlk\"orper}, +Invent. Math. \textbf{6} (1969), 296--314. +\href{https://doi.org/10.1007/BF01425420}{doi:10.1007/BF01425420}. + +\bibitem{neukirch1969b} +J\"urgen Neukirch, +\emph{Kennzeichnung der endlich-algebraischen Zahlk\"orper durch die Galoisgruppe der maximal aufl\"osbaren Erweiterungen}, +J. reine angew. Math. \textbf{238} (1969), 135--147. +\href{https://doi.org/10.1515/crll.1969.238.135}{doi:10.1515/crll.1969.238.135}. + +\bibitem{neukirch1977} +J\"urgen Neukirch, +\emph{\"Uber die absoluten Galoisgruppen algebraischer Zahlk\"orper}, +Ast\'erisque \textbf{41--42} (1977), 67--79. +\url{https://www.numdam.org/item/AST_1977__41-42__67_0/}. + +\bibitem{neukirch1999} +J\"urgen Neukirch, +\emph{Algebraic Number Theory}, +translated by Norbert Schappacher, +Grundlehren der mathematischen Wissenschaften, vol.~322, +Springer-Verlag, Berlin, 1999. +\href{https://doi.org/10.1007/978-3-662-03983-0}{doi:10.1007/978-3-662-03983-0}. + +\bibitem{saiditamagawa2022} +Mohamed Sa\"idi and Akio Tamagawa, +\emph{The $m$-step solvable anabelian geometry of number fields}, +J. reine angew. Math. \textbf{789} (2022), 153--186. +\href{https://doi.org/10.1515/crelle-2022-0025}{doi:10.1515/crelle-2022-0025}. + +\bibitem{uchida1976} +K\^oji Uchida, +\emph{Isomorphisms of Galois groups}, +J. Math. Soc. Japan \textbf{28} (1976), no.~4, 617--620. +\href{https://doi.org/10.2969/jmsj/02840617}{doi:10.2969/jmsj/02840617}. + +\bibitem{uchida1979} +K\^oji Uchida, +\emph{Isomorphisms of Galois groups of solvably closed Galois extensions}, +Tohoku Math. J. (2) \textbf{31} (1979), no.~3, 359--362. +\href{https://doi.org/10.2748/tmj/1178229803}{doi:10.2748/tmj/1178229803}. + +\bibitem{uchida1981} +K\^oji Uchida, +\emph{Homomorphisms of Galois groups of solvably closed Galois extensions}, +J. Math. Soc. Japan \textbf{33} (1981), no.~4, 595--604. +\href{https://doi.org/10.2969/jmsj/03340595}{doi:10.2969/jmsj/03340595}. + +\bibitem{waldschmidt1981} +Michel Waldschmidt, +\emph{Transcendance et exponentielles en plusieurs variables}, +Invent. Math. \textbf{63} (1981), no.~1, 97--127. +\href{https://doi.org/10.1007/BF01389195}{doi:10.1007/BF01389195}. + +\end{thebibliography} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/characters.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/characters.tex new file mode 100644 index 0000000000000000000000000000000000000000..11f5b00a6d469e269c85c81c4a3c90f892848978 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/characters.tex @@ -0,0 +1,420 @@ +\section{Additive characters and Frobenius detection}\label{sec:characters} + +We will compare homomorphisms of Galois groups by evaluating additive +$\ell$-adic characters at Frobenius elements. The main fact needed for +this comparison is that infinitely many primes of absolute residue +degree one detect a character. We first prove the logarithm-rank +statement behind this fact, then establish the versions involving +averaging and corestriction. The section ends with a representation +contained in the character space that will supply enough characters for +the comparison. + +\subsection{A logarithm-rank lemma} + +For a finite extension $K/\Q_\ell$, normalize its absolute value by +$|\ell|_\ell=\ell^{-1}$. The $\ell$-adic logarithm on +$\mathcal O_K^\times$ is the continuous homomorphism that agrees with +the usual power series near $1$; its kernel is the finite group of +roots of unity in $K$. Logarithms of points in a torus are taken +coordinatewise. + +For a number field $M$, class field theory expresses an additive +character of $G_M$ as a linear form on $M\otimes_\Q\Q_\ell$. +A zero Frobenius value at a prime $v\nmid\ell$ forces this form to +annihilate $\log b$, where $(b)=v^h$ for some $h>0$. +For primes of residue degree one chosen over distinct rational primes +unramified in a Galois closure, the conjugate generators have disjoint +ideal supports; the following lemma turns their multiplicative +independence into full logarithmic rank. + +This special rank estimate admits an +interpolation-determinant proof. It is a special case of the +$\ell$-adic Waldschmidt--Masser rank theorem +\cite{waldschmidt1981,masser1981}. We retain a proof using the +translate estimates of that theory and Laurent's interpolation +determinants \cite{laurent1991}; see also +\cite[Theorems~5.1 and~5.8]{dasgupta2023}. + +\begin{lemma}[Logarithm rank]\label{lem:log-rank} +Let $n,m\geq1$, let $E$ be a number field embedded in a finite +extension $K/\Q_\ell$, and let +\[ + u_i=(u_{i1},\ldots,u_{in})\in(E^\times)^n, + \qquad 1\leq i\leq m, +\] +have algebraic-integer coordinates that are units in $K$. Suppose that +for every $a=(a_1,\ldots,a_m)\in\Z^m\setminus\{0\}$, the coordinates +of $u^a:=\prod_{i=1}^m u_i^{a_i}$ are multiplicatively independent +modulo roots of unity. If $m>n(n-1)$, then +\[ + \operatorname{span}_K\{\log u_1,\ldots,\log u_m\}=K^n. +\] +\end{lemma} + +\begin{proof} +The multiplicative hypothesis implies that the cyclic subgroup +generated by every $u^a$ with $a\ne0$ is Zariski dense in +$\mathbb G_m^n$. Indeed, distinct Laurent monomials have distinct +values at $u^a$. If a Laurent polynomial vanished at all nonnegative +powers of $u^a$, a Vandermonde matrix applied to its finitely many +monomials would force all its coefficients to vanish. + +Suppose that the logarithms span a subspace of dimension $r(nD)^n, + \qquad S=O(D^{n/m}). +\end{equation} +Consider the matrix with rows indexed by +$\alpha\in\{0,\ldots,D\}^n$, columns indexed by +$a\in\{0,\ldots,nS\}^m$, and entries $(u^a)^\alpha$. +We claim that its rank is $A$. + +Otherwise a nonzero polynomial $P$ of degree at most $D$ in each +variable would vanish at every point $u^a$ in this grid. Over an +algebraic closure of $E$, define closed subsets of the torus by +\[ + Z_j=\{x\in\mathbb G_m^n: + P(u^a x)=0\ \hbox{for all }a\in\{0,\ldots,jS\}^m\}, + \qquad 0\leq j\leq n. +\] +These sets are nested and nonempty, since $1\in Z_n$, whereas +$\dim Z_0\leq n-1$. Hence some adjacent pair $Z_j,Z_{j+1}$ has +the same dimension $d$. Let $V$ be a $d$-dimensional irreducible +component of $Z_{j+1}$. For every +$b\in\{0,\ldots,S\}^m$, the translate $u^bV$ is contained in +$Z_j$ and has dimension $d$, so it is an irreducible component of +$Z_j$. All these translates are distinct: an equality +$u^bV=u^{b'}V$ for $b\ne b'$ would make $V$ stable under every +power of $u^{b-b'}$. The density proved above would then force +$V=\mathbb G_m^n$, contrary to $V\subseteq Z_0$. + +For completeness, a common zero set of polynomials of total degree +at most $\delta\geq1$ in $\mathbb A^n$ has at most $\delta^n$ +irreducible components. One can obtain this form of the affine +B\'ezout bound by assigning a component $Y$ the weight +$\delta^{\dim Y}\deg Y$. Start with $\mathbb A^n$, of weight +$\delta^n$, and impose the equations successively. A component +contained in the next hypersurface retains its weight; a component +not contained in it is cut into components of dimension one less, +whose total degree is at most $\delta\deg Y$ by B\'ezout. Thus +the total weight does not increase, and removing redundant components +can only decrease it. The same component bound holds after +restriction to the torus. Applying it to $Z_j$, whose equations +have total degree at most $nD$, contradicts +\eqref{eq:log-grid-size}. This proves the rank claim. + +Choose a nonzero $A\times A$ minor $\Delta_D$ of the evaluation +matrix. It is an algebraic integer in the fixed field $E$. +Expand its rows using the convergent power series for $f_\alpha$. +A nonzero term must involve $A$ distinct monomials in the $r$ +variables $z_1,\ldots,z_r$, since repeated monomials give repeated +rows in the corresponding evaluation determinant. The sum of the +degrees of any $A$ distinct such monomials is at least +$c_rA^{1+1/r}$ for all sufficiently large $A$: the number of +monomials of degree at most $t$ is $\binom{t+r}{r}=O_r(t^r)$, +so at least half of the $A$ monomials have degree bounded below by +a positive multiple of $A^{1/r}$. Their evaluations have absolute +value at most $1$. The nonarchimedean triangle inequality and +\eqref{eq:log-taylor-bound}, first for truncated series and then by +convergence, therefore give a constant $c>0$ such that +\begin{equation}\label{eq:log-determinant-small} + \log|\Delta_D|_\ell\leq-cA^{1+1/r}. +\end{equation} + +At every archimedean embedding $\tau:E\hookrightarrow\C$, an +entry of this minor has absolute value at most $\exp(CDS)$, with +$C$ independent of $D$. Expanding the determinant thus gives +\begin{equation}\label{eq:log-determinant-large} + \log|\tau(\Delta_D)|\leq C'ADS+A\log A. +\end{equation} +Let $\lambda$ be the place of $E$ induced by $E\hookrightarrow K$. +Use absolute values at finite places extending the usual ones on +$\Q$, and sum over all embeddings $E\hookrightarrow\C$, so that +each complex place occurs twice. Writing $p_v$ for the rational +prime below a finite place $v$, the product formula reads +\[ + 0=\sum_{\tau:E\hookrightarrow\C}\log|\tau(\Delta_D)| + +\sum_{v\text{ finite}}[E_v:\Q_{p_v}] + \log|\Delta_D|_v. +\] +Every term in the second sum is nonpositive because $\Delta_D$ is +an algebraic integer. Retaining its $\lambda$-term and using +\eqref{eq:log-determinant-small}--\eqref{eq:log-determinant-large} +would imply +\[ + c[E_\lambda:\Q_\ell]A^{1+1/r} + \leq [E:\Q](C'ADS+A\log A). +\] +This is impossible as $D\to\infty$: the left side has order +$D^{n+n/r}$, whereas the right side is +$O(D^{n+1+n/m}+D^n\log D)$, and +\[ + \frac nr\geq\frac n{n-1}>1+\frac nm +\] +by $m>n(n-1)$. This contradiction proves the lemma. +\end{proof} + +\subsection{Detection, averaging, and corestriction} + +For a number field $M\subset\overline{\Q}$, write +\[ + \X{\ell}{M}:=\Hom_{\mathrm{cont}}(G_M,\Q_\ell), + \qquad G_M=\Gal(\overline{\Q}/M), +\] +where $\Q_\ell$ is an additive group with trivial Galois action. +Local class field theory shows that these characters are unramified +at every finite prime $v\nmid\ell$: the local unit group has a +pro-$p$ subgroup of finite index for $p\ne\ell$, and has no +nonzero continuous homomorphism to $\Q_\ell$. Thus +$x(\Fr_v)$ is defined for $x\in\X{\ell}{M}$ and $v\nmid\ell$. + +If $M/\Q$ is Galois, $R=\Gal(M/\Q)$ acts by +$(g x)(h)=x(\widetilde g^{-1}h\widetilde g)$, where +$\widetilde g\in G_\Q$ lifts $g$. The definition is independent +of the lift. For a subgroup $J\subseteq R$, write +\[ + E_J=\frac1{|J|}\sum_{g\in J}g +\] +for the projector onto the $J$-invariants. We use +$\res_{L/M}$ and $\cor_{L/M}$ for restriction and corestriction +on additive characters for a finite extension $L/M$. + +\begin{proposition}[Frobenius detection]\label{prop:character-detection} +Let $M$ be a number field, let $\ell$ be a rational prime, and let +$x\in\X{\ell}{M}$. +\begin{enumerate} +\item[(i)] If $x(\Fr_v)=0$ at infinitely many primes $v\nmid\ell$ +of absolute residue degree one, then $x=0$. +\item[(ii)] Suppose that $M/\Q$ is Galois, and fix +$\delta\in\Gal(M/\Q)$. If $x(\Fr_v)=0$ at infinitely many +primes $v\nmid\ell$ lying over rational primes unramified in $M$ +whose Frobenius element at $v$ in $\Gal(M/\Q)$ is $\delta$, then +$E_{\langle\delta\rangle}x=0$. +\item[(iii)] If $\cor_{M/\Q}x\ne0$, then +$x(\Fr_v)=0$ at only finitely many primes $v\nmid\ell$, with no +restriction on their residue degrees. +\end{enumerate} +\end{proposition} + +\begin{proof} +We first recall the class-field-theoretic description of the character +space. Let $V_M=M\otimes_\Q\Q_\ell$, and let $U_M\subseteq V_M$ +be the $\Q_\ell$-linear span of the logarithms of the global units, +with logarithms taken at all places above $\ell$. Then +\begin{equation}\label{eq:character-class-field} + \X{\ell}{M}\simeq + \{\lambda\in\Hom_{\Q_\ell}(V_M,\Q_\ell): + \lambda(U_M)=0\}. +\end{equation} +Indeed, pass to the inverse limit of the ray class sequences with +moduli supported above $\ell$. Local logarithms identify the +rationalized local unit groups with $V_M$; the global units impose +exactly the subspace $U_M$. The ordinary class group and the factors +at real places are finite, so contribute no additive +$\Q_\ell$-characters. This description uses the actual logarithmic +image of the units and makes no assertion about its dimension. + +For $v\nmid\ell$, choose $h_v>0$ and $b_v\in\mathcal O_M$ with +$(b_v)=v^{h_v}$. Such a choice is possible because the ideal class +group is finite. Principal reciprocity, with arithmetic Frobenius, +gives for the linear form corresponding to $x$ the identity +\begin{equation}\label{eq:character-principal-reciprocity} + \lambda(\log b_v)=-h_vx(\Fr_v). +\end{equation} +Here $b_v$ is a unit at every place above $\ell$; all other local +unit contributions are killed by $x$. + +For (i), set $n=[M:\Q]$ and choose $m>n(n-1)$ of the zero primes, +over distinct rational primes unramified in a Galois closure $E$ of +$M$. Let $b_i$ be the elements just chosen for these primes, and +let +\[ + u_i=(\sigma(b_i))_{\sigma:M\hookrightarrow E}\in(E^\times)^n. +\] +All coordinates are algebraic integers and are units at every place +of $E$ above $\ell$. We verify the multiplicative hypothesis of +Lemma~\ref{lem:log-rank} by valuations. + +Put $G=\Gal(E/\Q)$ and $H=\Gal(E/M)$. For a selected degree-one +prime $v_i$ and a prime $\mathfrak p_i$ of $E$ above it, the +decomposition group $D_i$ is contained in $H$. In fact, the rational +prime is unramified in $E$, and the degree-one condition says that +$[D_i:D_i\cap H]=1$. The support of $v_i\mathcal O_E$ is +indexed by $H/D_i$. Its translates corresponding to the embeddings +$M\hookrightarrow E$, indexed by $G/H$, have supports indexed by +the disjoint sets $gH/D_i$. Distinct indices $i$ involve distinct +rational primes. It follows that the $mn$ numbers +$\sigma(b_i)$ have pairwise disjoint nonempty supports of their +principal ideals. They are therefore multiplicatively independent +modulo roots of unity. In particular, for any nonzero +$a\in\Z^m$, the $n$ coordinates of $u^a$ are multiplicatively +independent modulo roots of unity. + +Embed $E$ in a finite extension $K/\Q_\ell$. Under the splitting +\[ + (M\otimes_\Q\Q_\ell)\otimes_{\Q_\ell}K + \simeq\prod_{\sigma:M\hookrightarrow E}K, +\] +the vector $\log b_i$ becomes $\log u_i$. +Lemma~\ref{lem:log-rank} says that these vectors span the right +side. Equations~\eqref{eq:character-class-field} and +\eqref{eq:character-principal-reciprocity} therefore force +$\lambda=0$, and hence $x=0$. + +For (ii), put $C=\langle\delta\rangle$, $F=M^C$, and +$y=E_Cx$. Since $\delta$ fixes every indicated prime $v$, averaging +preserves its Frobenius value, so $y(\Fr_v)=0$. The +restriction--corestriction identity gives +\[ + \res_{M/F}\cor_{M/F}y=\sum_{g\in C}g y=|C|y. +\] +Thus $y$ is the restriction of +$z=|C|^{-1}\cor_{M/F}y\in\X{\ell}{F}$. At an indicated prime, +the decomposition group of $M/\Q$ is exactly $C$. The prime +$u=v\cap F$ consequently has absolute residue degree one, and +\[ + 0=y(\Fr_v)=[\kappa(v):\kappa(u)]z(\Fr_u). +\] +Infinitely many distinct $u$ occur because $M/F$ is finite. +Part~(i) gives $z=0$, and therefore $y=0$. + +For (iii), suppose that infinitely many zero primes exist. Let +$E/\Q$ be a finite Galois closure of $M$, put +$R=\Gal(E/\Q)$, and let $y=\res_{E/M}x$. Discarding finitely +many primes, choose primes of $E$ above these zero primes whose +rational primes are unramified in $E$. Restriction multiplies a +Frobenius value by the relative residue degree, so they are zero +primes for $y$. An infinite subset has one fixed Frobenius element +$\delta\in R$. By (ii), +$E_{\langle\delta\rangle}y=0$, and hence $E_Ry=0$. It follows that +\[ + \res_{E/\Q}\cor_{E/\Q}y=|R|E_Ry=0. +\] +Restriction on additive characters across a finite extension is +injective: a character that vanishes on a subgroup of finite index +has finite image in the torsion-free group $\Q_\ell$, and so is +zero. On the other hand, transitivity and the +corestriction--restriction identity give +\[ + \cor_{E/\Q}y + =\cor_{M/\Q}\cor_{E/M}\res_{E/M}x + =[E:M]\cor_{M/\Q}x\ne0, +\] +a contradiction. +\end{proof} + +\subsection{A signed representation in the character space} + +The class field description also controls the supply of characters +when the number field varies. The regular representation gives upper +bounds for dimensions of subgroup invariants, while the signed +representation below gives lower bounds and ensures a faithful action +on the character space. + +\begin{proposition}\label{prop:signed-representation} +Let $M/\Q$ be a finite, totally imaginary Galois extension, put +$R=\Gal(M/\Q)$, and let $c\in R$ be a complex conjugation. +For every rational prime $\ell$, there are $R$-equivariant embeddings +\begin{equation}\label{eq:signed-representation} + W_M:=\Ind_{\langle c\rangle}^{R}(\mathrm{sign}) + \lhook\joinrel\longrightarrow\X{\ell}{M} + \lhook\joinrel\longrightarrow\Q_\ell[R], +\end{equation} +where $\mathrm{sign}(c)=-1$. In particular, the action of $R$ on +$\X{\ell}{M}$ is faithful. +\end{proposition} + +\begin{proof} +By the normal basis theorem, $V_M=M\otimes_\Q\Q_\ell$ is the +regular representation of $R$, as is its dual. The naturality of +\eqref{eq:character-class-field} gives the second embedding. + +The equivariant Dirichlet unit theorem +\cite[Section~3]{bass1966} identifies the representation +$\mathcal O_M^\times\otimes_\Z\Q_\ell$ with the permutation +representation on the archimedean places minus its trivial summand. +Since these places are indexed by $R/\langle c\rangle$, writing +$U^{\mathrm{alg}}=\mathcal O_M^\times\otimes_\Z\Q_\ell$ gives +\[ + \Q_\ell\oplus U^{\mathrm{alg}} + \simeq\Ind_{\langle c\rangle}^{R}\mathbf1. +\] +The character identity follows first over the real numbers from the +usual logarithmic unit embedding, and hence over $\Q_\ell$ by +semisimplicity in characteristic zero. In particular, +$U^{\mathrm{alg}}$ is self-dual. Its $\ell$-adic logarithmic image +$U_M$ is a quotient of $U^{\mathrm{alg}}$; no injectivity of the +logarithm on this tensor product is needed. + +The decomposition of the regular representation induced from the +two characters of $\langle c\rangle$ is +\[ + \Q_\ell[R] + \simeq\Ind_{\langle c\rangle}^{R}\mathbf1\oplus W_M + \simeq\Q_\ell\oplus U^{\mathrm{alg}}\oplus W_M. +\] +Since $U_M^\vee$ is a subrepresentation of +$(U^{\mathrm{alg}})^\vee\simeq U^{\mathrm{alg}}$, semisimplicity +and $\X{\ell}{M}\simeq(V_M/U_M)^\vee$ show that +$\X{\ell}{M}$ contains $\Q_\ell\oplus W_M$. This proves the +first embedding in \eqref{eq:signed-representation}. + +Finally, realize $W_M$ as the direct sum of the signed lines indexed +by the left cosets of $\langle c\rangle$. An element acting +trivially must fix the line of the identity coset, and therefore +belongs to $\langle c\rangle$. The nonidentity element $c$ acts +by $-1$ on that line. Thus only the identity acts trivially on +$W_M$, proving faithfulness. +\end{proof} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/generation.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/generation.tex new file mode 100644 index 0000000000000000000000000000000000000000..6952c84c60aefe0714ac362f3bfa557dfb87d844 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/generation.tex @@ -0,0 +1,425 @@ +\section{Normalizer extension and Frobenius generation}\label{sec:generation} + +Let $F$ be a number field and let +\[ + \alpha_0:G_F\longrightarrow P +\] +be a continuous open homomorphism. Our eventual aim is to identify this +map with ordinary restriction, up to conjugation in $P$. We first enlarge +an open part of its domain in a canonical way. For the enlarged map we +will prove surjectivity and equality of cyclotomic logarithms at an +infinite set of auxiliary primes. The main step is to show that images +of local Frobenius elements generate prescribed cyclic subgroups in +suitable finite quotients of $P$. + +Choose an open normal subgroup $U\trianglelefteq\mathcal G$ contained in +$G_F$, and put +\[ + A=\alpha_0(U),\qquad N=\ker(\alpha_0|_U),\qquad + T=N_{\mathcal G}(N). +\] +The subgroup $A$ is open in $P$. Since $U\subset T$, the normalizer $T$ +is open and closed in $\mathcal G$; write $T=G_k$. + +\begin{lemma}[Extension to the normalizer]\label{lem:normalizer-extension} +There is a unique continuous open homomorphism +\[ + \beta:T\longrightarrow P +\] +whose restriction to $U$ is $\alpha_0|_U$. +\end{lemma} + +\begin{proof} +Conjugation by $t\in T$ induces a continuous automorphism of $U/N=A$. +By Theorem~\ref{thm:uchida-isomorphism}, applied to +$\Q^s/(\Q^s)^A$, this automorphism is conjugation by a unique element +$\beta(t)$ of $P$. Indeed, the implementing field automorphism of +$\Q^s$ fixes $\Q$. Uniqueness shows both that $\beta$ is a homomorphism +and that $\beta(u)=\alpha_0(u)$ for $u\in U$. It also shows that any +extension of $\alpha_0|_U$ to $T$ must equal $\beta$: its values must +induce these same conjugations on $A$. + +Continuity on the open subgroup $U$ implies continuity on $T$. +Moreover, $\beta(T)$ contains $A$ and is therefore open in $P$. +A continuous homomorphism of profinite groups is open onto its image, +so $\beta$ is open. +\end{proof} + +We keep $U,A,N,T,k$, and $\beta$ fixed for the rest of the argument. +List the distinct conjugates of $N$ other than $N$ itself as +\[ + N_i=t_iNt_i^{-1}\quad(1\leq i\leq r),\qquad t_i\notin T, + \qquad H_i=\beta(N_i). +\] +There are finitely many because $U$ normalizes $N$. Normality of $U$ +in $\mathcal G$ implies that each $N_i$ lies in $U$ and is normal in +$U$, so $H_i$ is normal in $A$. +Each $H_i$ is nontrivial. Otherwise $t_iNt_i^{-1}\subset N$, and +iterating this inclusion would give a descending chain returning to +$N$: some power of $t_i$ lies in $U$ and hence normalizes $N$. +All inclusions would be equalities, contrary to $t_i\notin T$. +We allow $r=0$, with sums indexed by $i$ then equal to zero. + +\subsection{A finite quotient with enough characters} + +The next lemma constructs characters that distinguish a proper +subgroup of a cyclic group while avoiding the invariant spaces attached +to the $H_i$. The latter condition will prevent cancellation by +conjugates of $N$ when we average a pulled-back character on the source. +For a finite subgroup $J$ acting on a characteristic-zero vector space, +we use the averaging operator $E_J=|J|^{-1}\sum_{g\in J}g$. + +\begin{lemma}[Quotient amplification]\label{lem:quotient-amplification} +Fix an odd prime $\ell$ and a finite quotient $R_0$ of $P$. There is +a finite Galois extension $L/\Q$ in $\Q^s$, totally imaginary, such that +the quotient $R=\Gal(L/\Q)$ dominates $R_0$ and has the following +property. For every $\gamma\in R$, with $C=\langle\gamma\rangle$, and +every proper subgroup $D1$. As $H_{i,R}$ is normal in $A_R$, +\[ + (g_i-1)V=[h_i,V]\subset H_{i,R}. +\] +On $m$ regular modules the rank of $g_i-1$ is +$mn(1-1/d_i)\geq mn/2$. Consequently +\begin{equation}\label{eq:large-subgroups} + |H_{i,R}|\geq 2^{mn/2}. +\end{equation} +If $c\in R$ is a complex conjugation, its image in $R_1$ is a +nontrivial involution because $L_1$ is totally imaginary. +Conjugation by $V$ alone therefore gives at least $2^{mn/2}$ distinct +conjugates of $c$; thus +\begin{equation}\label{eq:large-conjugacy-class} + |c^R|\geq 2^{mn/2}. +\end{equation} + +Set $\eta=2^{-mn/2}$ and write $X=\X{\ell}{L}$. By +Proposition~\ref{prop:signed-representation}, $X$ embeds in the regular +$\Q_\ell[R]$-representation and contains +\[ + W=\Ind_{\langle c\rangle}^{R}(\operatorname{sign}). +\] +The regular-representation bound and \eqref{eq:large-subgroups} give +\begin{equation}\label{eq:forbidden-dimension} + \dim\left(\sum_{i=1}^rX^{H_i}\right) + \leq\sum_{i=1}^r\frac{|R|}{|H_{i,R}|} + \leq r\eta|R|. +\end{equation} +The character of $W$ takes the value $|R|/2$ at the identity, +the value $-|R|/(2|c^R|)$ on $c^R$, and zero elsewhere. +Hence, for every subgroup $J\subset R$, +\begin{equation}\label{eq:signed-invariants} + \left|\dim W^J-\frac{|R|}{2|J|}\right| + \leq\frac{\eta|R|}{2}. +\end{equation} +For $D1$ +in this range divides $m_0$, hence divides $\ell-1$. +Consequently $f=1$. + +Taking logarithms in \eqref{eq:generating-norm} gives +\[ + (\log\chi_\ell\circ\beta)(\sigma_v) + =\log p + =\log\chi_{k,\ell}(\sigma_v). +\] +The difference is an element of $\X{\ell}{k}$ vanishing at +infinitely many primes of absolute residue degree one. +Proposition~\ref{prop:character-detection}(i) proves +\eqref{eq:cyclotomic-logarithms}. +\end{proof} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/identification.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/identification.tex new file mode 100644 index 0000000000000000000000000000000000000000..2b37ecb2116306139499e06cb809226116553db9 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/identification.tex @@ -0,0 +1,102 @@ +\section{Identifying the homomorphism}\label{sec:identification} + +We now turn the prime correspondence into equality of group homomorphisms. The argument uses one fixed coefficient prime $r$, although the auxiliary prime witnessing activity is allowed to vary. This distinction is what makes the correspondences constructed in Section~\ref{sec:matching} sufficient. + +Retain the map $\beta:T=G_k\to P$, the infinite set $\mathcal L$, and the fields $B=\Q(i)$ and $K$ from Sections~\ref{sec:generation}--\ref{sec:matching}. Thus $G_K=\beta^{-1}(P_B)$, and Theorem~\ref{thm:normalized-map} gives +\begin{equation}\label{eq:fixed-r-log} + \log\chi_r\circ\beta=\log\chi_{k,r}\qquad(r\in\mathcal L). +\end{equation} +Write $\sigma:T\to P$ for ordinary restriction. + +\begin{proposition}\label{prop:identify-normalized} +There is an element $a\in P$ such that $\beta=\Ad(a)\circ\sigma$ on $T$. +\end{proposition} +\begin{proof} +Fix a finite Galois extension $L/\Q$ inside $\Q^s$ containing $B$, and put $R=\Gal(L/\Q)$. Choose a finite Galois extension $F'/\Q$ containing $K$ and $L$ such that $\beta(G_{F'})\subset P_L$. This is possible by taking the core in $\mathcal G$ of the appropriate open subgroup of $T$. Fix $r\in\mathcal L$. + +The maps $\beta$ and $\sigma$, restricted to $G_{F'}$, give two pullbacks +\begin{equation}\label{eq:two-pullbacks} + i_\beta,i_\sigma:\X rL\longrightarrow\X r{F'}. +\end{equation} +Both are injective: their underlying homomorphisms have open image, and an additive $\Q_r$-valued character vanishing on a finite-index subgroup is zero. For $a\in P$ let $\sigma_a=\Ad(a)\circ\sigma$ and let $i_a$ be its pullback. Since $L/\Q$ is Galois, $\sigma_a(G_{F'})\subset P_L$. The map $i_a$ depends only on $a$ modulo $P_L$, because inner automorphisms of $P_L$ act trivially on additive characters. + +\emph{Frobenius evaluations.} +Let $q\ne r$ range over rational primes split completely in $F'$. By Proposition~\ref{prop:same-characteristic}, for all but finitely many such $q$ there is a prime $v$ of $K$ above $q$ whose decomposition group maps into a decomposition group at a prime $w$ of $B$ above the same $q$. Choose a prime $v'$ of $F'$ above $v$ and compatible prolongations. Then +\[ + \beta(D_{v'})\subset D_u\subset P_L +\] +for a prime $u$ of $L$ above $w$. Every prime here has absolute residue degree one, because $q$ splits completely in $F'$. + +Choose a lift $h$ of arithmetic Frobenius in $D_{v'}$. Let $a_q\in\widehat{\Z}$ be the residue exponent of $\beta(h)$ at $u$. Equation~\eqref{eq:fixed-r-log} gives +\[ + (a_q)_r\log_r q=\log_r q. +\] +The number $\log_r q$ is nonzero: $q\in\Z_r^\times$ is a positive rational integer larger than one and is not a root of unity. Consequently $(a_q)_r=1$. Every $x\in\X rL$ is unramified at $u$, so +\begin{equation}\label{eq:matched-evaluation} + i_\beta(x)(\Fr_{v'})=x(\Fr_u). +\end{equation} +This calculation uses only decomposition containment from the active correspondence. It does not require that its auxiliary prime be $r$. + +Under ordinary restriction, the same Frobenius maps to Frobenius at another prime of $L$ above $q$. An element of $R$ carries this prime to $u$. Choose a lift $a\in P$ of that element. Then +\[ + i_\beta(x)(\Fr_{v'})=i_a(x)(\Fr_{v'}) + \qquad\text{for every }x\in\X rL. +\] +The choice of $a$ modulo $P_L$ may initially depend on $q$. Since $R$ is finite, an infinite set of distinct $q$ uses one fixed choice. For each $x$, the difference $i_\beta(x)-i_a(x)$ therefore vanishes at infinitely many degree-one primes of $F'$. Proposition~\ref{prop:character-detection} gives +\begin{equation}\label{eq:equal-pullbacks} + i_\beta=i_a. +\end{equation} + +\emph{From characters to the finite group.} +The field $F'/\Q$ is Galois, so $T$ acts on $\X r{F'}$ by conjugation. The two pullbacks intertwine this action with the respective actions on $\X rL$ induced by $\beta$ and $\sigma_a$. If $i$ denotes their common injective value in \eqref{eq:equal-pullbacks}, then for $t\in T$ and $x\in\X rL$, +\[ + i\bigl(\beta(t)\cdot x\bigr) + =t\cdot i(x) + =i\bigl(\sigma_a(t)\cdot x\bigr). +\] +The field $L$ is totally imaginary because it contains $B$. By Proposition~\ref{prop:signed-representation}, $R$ acts faithfully on $\X rL$. Hence +\[ + \beta(t)\bmod P_L=\sigma_a(t)\bmod P_L\qquad(t\in T). +\] + +\emph{One conjugation for all finite quotients.} +For each such $L$, let +\[ + C_L=\{a\in P:\ \beta(t)\equiv\Ad(a)(\sigma(t))\pmod{P_L} + \text{ for every }t\in T\}. +\] +We have proved that $C_L$ is nonempty. It is closed, and indeed clopen, since the defining condition depends only on $a$ modulo $P_L$. If $L$ contains finitely many fields $L_1,\ldots,L_s$, then $C_L\subset\bigcap_j C_{L_j}$. Composita thus give the finite-intersection property. Compactness of $P$ supplies an element in every $C_L$. The extensions $L$ containing $B$ are cofinal among finite Galois subextensions of $\Q^s/\Q$, so this element gives the asserted equality in $P$. +\end{proof} + +We remove the normalizer extension and return to the original source. +\begin{theorem}\label{thm:prosolvable-rigidity} +Let $F$ be a number field in $\overline{\Q}$. Every continuous open homomorphism $\varphi:G_F\to P$ is conjugate in $P$ to ordinary restriction. +\end{theorem} +\begin{proof} +As in Lemma~\ref{lem:normalizer-extension}, choose an open normal subgroup $U$ of $\mathcal G$ contained in $G_F$, and extend $\varphi|_U$ to the normalizer of its kernel. Proposition~\ref{prop:identify-normalized} shows that, after one conjugation of $\varphi$, it agrees on $U$ with ordinary restriction $\sigma:G_F\to P$. + +For $g\in G_F$ and $u\in U$, normality of $U$ gives +\[ + \varphi(g)\sigma(u)\varphi(g)^{-1} + =\varphi(gug^{-1}) + =\sigma(gug^{-1}) + =\sigma(g)\sigma(u)\sigma(g)^{-1}. +\] +Thus $\sigma(g)^{-1}\varphi(g)$ centralizes $\sigma(U)=\varphi(U)$, an open subgroup of $P$. Lemma~\ref{lem:centralizer} makes this element trivial. The equality holds for every $g\in G_F$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Choose copies of the given fields inside $\overline{\Q}$. By Lemma~\ref{lem:solvable-closure}, each $E_i$ contains $\Q^s$. Form the composite +\begin{equation}\label{eq:final-composite} + G_{F_1}\longrightarrow G_1\xrightarrow{\alpha}G_2 + \longrightarrow P, +\end{equation} +where the outer arrows are restrictions. The first is surjective. The last has open image, since its composite with $G_{F_2}\twoheadrightarrow G_2$ is ordinary restriction from an open subgroup of $\mathcal G$ to $P$. Continuous surjective homomorphisms between profinite groups are open, so \eqref{eq:final-composite} is open. + +By Theorem~\ref{thm:prosolvable-rigidity}, this composite is conjugate to ordinary restriction. Cyclotomic characters are unchanged by conjugation. All roots of unity lie in $\Q^s$, so the full cyclotomic characters factor through the groups in \eqref{eq:final-composite}. Surjectivity of the first arrow therefore yields +\[ + \chi_{E_2/F_2}\circ\alpha=\chi_{E_1/F_1} + \quad\text{in }\widehat{\Z}^{\times}. +\] +Theorem~\ref{thm:hoshi} gives an equivariant embedding $E_2\hookrightarrow E_1$. Its uniqueness is Uchida's theorem cited after Theorem~\ref{thm:hoshi}. +\end{proof} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/introduction.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..34168259f24ca9588a7956f5c4c24e7d906abc3e --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/introduction.tex @@ -0,0 +1,39 @@ +\section{Introduction}\label{sec:introduction} + +An algebraic extension of a number field is \emph{solvably closed} if it has no nontrivial abelian algebraic extension. Algebraic closures and maximal prosolvable extensions are examples. The isomorphism theorems of Neukirch and Uchida show that Galois groups of such fields retain remarkably precise arithmetic information. The corresponding question for open homomorphisms asks whether this reconstruction also recovers field embeddings. + +Let $E_i/F_i$ be Galois extensions of number fields, with each $E_i$ solvably closed, and write $G_i=\Gal(E_i/F_i)$. A field embedding $j:E_2\hookrightarrow E_1$ \emph{induces} a homomorphism $\alpha:G_1\to G_2$ if +\[ + g\circ j=j\circ\alpha(g)\qquad(g\in G_1). +\] +In particular, such an embedding sends $F_2$ into $F_1$. Uchida conjectured that every continuous homomorphism with open image arises in this way \cite[p.~595]{uchida1981}. We prove the conjecture in its full form. + +\begin{theorem}\label{thm:main} +Let $F_1,F_2$ be number fields, and let $E_i/F_i$ be possibly infinite solvably closed Galois extensions. Every continuous open homomorphism +\[ + \alpha:\Gal(E_1/F_1)\longrightarrow\Gal(E_2/F_2) +\] +is induced by a unique field embedding $j:E_2\hookrightarrow E_1$. +\end{theorem} + +The kernel is unrestricted. The proof establishes the necessary arithmetic compatibility from openness itself. + +\subsection{Historical context} +The reconstruction of number fields from their Galois groups begins with Neukirch's work on normal number fields and decomposition groups \cite{neukirch1969a,neukirch1969b}. The passage to arbitrary number fields and prescribed group isomorphisms involved Uchida, Iwasawa, and Ikeda; their contributions are recorded in the contemporary accounts \cite{uchida1976,neukirch1977}. Uchida subsequently established the isomorphism theorem for arbitrary solvably closed Galois extensions of number fields \cite{uchida1979}. + +In the paper formulating the homomorphism conjecture, Uchida proved it when the source number field is $\Q$, proved uniqueness in general, and established existence under a suitable local condition on decomposition groups \cite[Theorem~1, Proposition~2, and Theorem~2]{uchida1981}. His local analysis also associates primes to one another whenever a chosen pro-$\ell$ inertia group survives under the homomorphism, for an odd auxiliary prime $\ell$. For each fixed auxiliary prime $\ell$, this partial correspondence reaches all but finitely many target primes \cite[Proposition~1]{uchida1981}. + +A later reduction of Hoshi identifies an exact arithmetic condition for existence: an open homomorphism is induced by a field embedding if and only if it preserves the full cyclotomic character \cite[Theorem~3.4]{hoshi2025}. Hoshi's subsequent work reformulates the unrestricted conjecture in terms of radicals and normality of the field fixed by a kernel \cite[Theorem~B]{hoshi2026}. Related work on finite solvable quotients includes the isomorphism theorem of Sa\"idi and Tamagawa \cite{saiditamagawa2022} and the cyclotomic-compatible homomorphism theorem of Mao and Sa\"idi \cite{maosaidi2025}. Our proof uses the cyclotomic criterion at its endpoint. The main task is to derive that criterion from the partial local correspondences, even though their domains and exceptional sets initially depend on $\ell$. + +\subsection{The main ideas} +Two arguments used below are useful beyond the final reduction. First, an additive $\ell$-adic character of the absolute Galois group of a number field is determined by its values on any infinite set of primes of absolute residue degree one. More precisely, a character vanishing at all those primes is zero. The logarithmic-rank input is a special case of the $\ell$-adic Waldschmidt--Masser theorem \cite{waldschmidt1981,masser1981,dasgupta2023}. We give a complete proof using Laurent's interpolation-determinant method \cite{laurent1991} and a translate-and-degree zero estimate on a torus. Its variants for Frobenius averages and corestriction permit us to treat primes of arbitrary residue degree as well. These statements concern the actual unit-logarithm image and require no hypothesis on its rank. + +Second, we refine the Kummer argument in Uchida's proof of cofinite target coverage \cite[Proposition~1]{uchida1981}. After logarithmic cyclotomic compatibility has been established, the number of target primes missed by the partial correspondence can be bounded independently of $\ell$. A missed prime supplies a Kummer class. On killing a finite cyclotomic twist, these classes lie in one character component of an $S$-unit representation. The full extension degree may grow with $\ell$, but the multiplicity of that character is bounded by the number of orbits of $S$, hence by the degree of the fixed source field. This uniformity allows us to choose $\ell$ by simultaneous Kummer conditions and force enough corresponding primes to have the same rational residue characteristic. + +Here is the route from these ingredients to Theorem~\ref{thm:main}. Put $P=\Gal(\Q^s/\Q)$, where $\Q^s$ is the maximal prosolvable extension of $\Q$. For a number field $F$, let $G_F$ denote its absolute Galois group. It suffices to identify every continuous open homomorphism $G_F\to P$ with ordinary restriction up to conjugation. Indeed, every solvably closed field contains $\Q^s$, and this identification will give the full cyclotomic compatibility required by Hoshi's theorem. + +We first restrict to a normal open subgroup of $G_{\Q}$ and extend the map to the normalizer of its kernel. Write this extension as $\beta:G_k\to P$. Following the regular-module constructions used by Uchida \cite{uchida1979}, we enlarge finite quotients of $P$ by elementary abelian kernels consisting of many copies of a regular representation. Each quotient $R=\Gal(L/\Q)$ acts on the additive character space of $G_L$. Suitable characters distinguish a cyclic subgroup generated by a target Frobenius element from each of its proper subgroups. Comparing Frobenius averages proves that $\beta$ is surjective and preserves logarithmic cyclotomic characters for an infinite congruence class of auxiliary primes. + +Next, pass to the preimage of $P_{\Q(i)}=\Gal(\Q^s/\Q(i))$. A sign character shows that, for almost every split target prime, only finitely many source primes can correspond to it as $\ell$ varies. The uniform Kummer bound then forces a correspondence in the same residue characteristic for almost every such target prime. Finally, use rational primes split completely in suitable finite extensions. Their additive Frobenius evaluations identify the two character pullbacks, one from $\beta$ and one from ordinary restriction. A faithful representation in the character space identifies the group maps modulo every finite quotient; compactness supplies a single conjugation. + +Section~\ref{sec:preliminaries} states the anabelian inputs and proves the local norm identity. Section~\ref{sec:characters} develops the character tests. Section~\ref{sec:generation} obtains surjectivity and logarithmic cyclotomic compatibility. Section~\ref{sec:matching} proves the uniform bound and matches residue characteristics. Section~\ref{sec:identification} identifies the map to $P$ and proves Theorem~\ref{thm:main}. diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/matching.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/matching.tex new file mode 100644 index 0000000000000000000000000000000000000000..b572a1251fd54d1c2122d11ed3300f6363fcffd8 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/matching.tex @@ -0,0 +1,180 @@ +\section{Matching residue characteristics}\label{sec:matching} + +We now show that almost every prime of $\Q(i)$ above a split rational prime has an active partner of the same residue characteristic. The auxiliary prime defining activity may vary with the pair. Two finiteness statements make this possible: almost every such target prime has only finitely many possible partners, and the number of target primes missed at a fixed auxiliary prime admits a uniform bound. + +Keep the surjection $\beta:G_k\to P$ and the set $\mathcal L$ from Theorem~\ref{thm:normalized-map}, and put +\[ + B=\Q(i),\qquad G_K=\beta^{-1}(P_B),\qquad + \beta_B=\beta|_{G_K}:G_K\twoheadrightarrow P_B. +\] +Thus $K/k$ is quadratic. An \emph{active pair at $\ell$} in this section means a source prime $v$ of $K$ active at $\ell$ for $\beta_B$, together with its corresponding prime $w$ of $B$ from Theorem~\ref{thm:local-correspondence}. In particular, both primes lie away from $\ell$. For a prime $w$ of $B$, define +\[ + \mathcal V_w= + \{v:\text{$(v,w)$ is an active pair at some $\ell\in\mathcal L$}\}. +\] + +\subsection{Finite fibers and a uniform bound on missed primes} + +\begin{lemma}\label{lem:finite-fibers} +For all but finitely many primes $w$ of $B$ above rational primes split in $B$, the set $\mathcal V_w$ is finite. +\end{lemma} +\begin{proof} +Fix $r\in\mathcal L$. Proposition~\ref{prop:signed-representation} supplies a nonzero character $\psi\in\X{r}{B}$ on which $\Gal(B/\Q)$ acts by its nontrivial character. By Proposition~\ref{prop:character-detection}(i), $\psi(\Fr_w)$ is nonzero at all but finitely many degree-one primes $w\nmid r$. Fix such a $w$ above a split rational prime $q\ne r$, and put +\[ + x_0=(\log\chi_r)|_{P_B},\qquad + z=\frac{\psi(\Fr_w)}{\log_r q}. +\] +The denominator is nonzero: the kernel of the logarithm on $\Z_r^\times$ consists of roots of unity, and the positive rational integer $q$ is not a root of unity. Thus $z\ne0$, and $\psi-zx_0$ vanishes on the decomposition group at $w$, since it is unramified there and vanishes on Frobenius. + +Consider its pullback +\[ + y=\beta_B^*(\psi-zx_0)\in\X{r}{K}. +\] +The pullback of $\psi$ transforms by sign under $G_k/G_K$. Its corestriction to $k$ is therefore zero: restricting that corestriction back to $K$ gives the sum of the two translates, and restriction is injective for additive characters with characteristic-zero coefficients. Theorem~\ref{thm:normalized-map} identifies $\beta_B^*x_0$ with $\log\chi_{K,r}$. Consequently +\[ + \cor_{K/\Q}(y) + =-z[K:\Q]\log\chi_{\Q,r}\ne0. +\] +For every $v\in\mathcal V_w$ away from $r$, decomposition containment gives $y(\Fr_v)=0$, regardless of which auxiliary prime makes the pair active. Proposition~\ref{prop:character-detection}(iii) permits only finitely many such $v$. There are also only finitely many primes of $K$ above $r$, proving the assertion. +\end{proof} + +For $\ell\in\mathcal L$, let $\mathcal M_\ell$ be the set of primes of $B$ which occur in no active pair at $\ell$. This includes the primes above $\ell$. + +We seek a bound for $\#\mathcal M_\ell$ depending only on $[K:\Q]$. The proof will place independent classes attached to missed primes in one character space of an $S$-unit group. We first record the standard cohomological identification of that group, including the completion step; see also \cite{neukirch1999}. + +\begin{lemma}[Unramified Kummer classes]\label{lem:kummer-unramified} +Let $F$ be a number field, let $\ell$ be a rational prime, and let $S$ be a finite set of places of $F$ containing all places above $\ell$ and all archimedean places. Then Kummer theory induces a natural isomorphism +\begin{equation}\label{eq:unramified-kummer} + H^1_{\mathrm{cont}}(G_F,\Q_\ell(1)) + _{\text{unramified outside }S} + \simeq \mathcal O_{F,S}^{\times}\otimes_{\Z}\Q_\ell. +\end{equation} +It is equivariant for every group of field automorphisms of $F$ preserving $S$, with the natural action on cohomology combining conjugation on $G_F$ and the action on the Tate twist. +\end{lemma} +\begin{proof} +Integral Kummer theory gives +\[ + H^1_{\mathrm{cont}}(G_F,\Z_\ell(1)) + \simeq\varprojlim_n F^\times/(F^\times)^{\ell^n}. +\] +The inverse-limit assertion follows from the finite-level Kummer isomorphisms; the degree-zero groups are finite and satisfy the Mittag--Leffler condition. Every continuous rational cocycle has bounded image, since $G_F$ is compact, and hence becomes integral after multiplication by one power of $\ell$. The same observation for coboundaries gives rationalization of this integral cohomology. At a prime outside $S$, inertia acts trivially on $\Z_\ell(1)$, and its Kummer evaluation is the valuation times the tame $\ell$-adic character. These integral inertia homomorphisms are torsion-free. Thus unramifiedness after rationalization is exactly the vanishing of all integral valuation coordinates outside $S$. + +For completeness, put $U_S=\mathcal O_{F,S}^\times$ and let +\[ + H_n=\{[a]\in F^\times/(F^\times)^{\ell^n}: + \operatorname{ord}_v(a)\equiv0\pmod{\ell^n} + \text{ for every }v\notin S\}. +\] +Writing $\Cl_S(F)$ for the $S$-ideal class group, there is an exact sequence +\begin{equation}\label{eq:s-unit-completion} + 0\longrightarrow U_S/U_S^{\ell^n} + \longrightarrow H_n + \longrightarrow \Cl_S(F)[\ell^n] + \longrightarrow0. +\end{equation} +The last map sends $[a]$ to the ideal class of +$\operatorname{div}_S(a)/\ell^n$, where $\operatorname{div}_S$ records the valuations outside $S$. The transition from level $n+1$ to level $n$ induces multiplication by $\ell$ on these obstruction classes. Since $\Cl_S(F)$ is finite, their inverse limit is zero. It follows that the kernel of all outside-$S$ valuations in the completed multiplicative group is precisely $\varprojlim_n U_S/U_S^{\ell^n}$. The argument takes place in the full completed multiplicative group; it does not assume finite valuation support for its elements. Finally $U_S$ is finitely generated, so rationalizing its completion gives $U_S\otimes\Q_\ell$. This proves \eqref{eq:unramified-kummer}. All the constructions commute with field automorphisms preserving $S$, giving the asserted equivariance. +\end{proof} + +Uchida's proof of target-side finiteness associates Kummer classes to missed primes and bounds them by a unit rank \cite[Proposition~1]{uchida1981}. We refine that argument by keeping the classes in one character space of a cyclic extension. The multiplicity of this character has a bound independent of $\ell$, even when the extension degree grows. + +\begin{proposition}\label{prop:uniform-missed-primes} +For every $\ell\in\mathcal L$, +\[ + \#\{w\in\mathcal M_\ell:w\nmid\ell\}\le 2[K:\Q], + \qquad + \#\mathcal M_\ell\le 2[K:\Q]+2. +\] +\end{proposition} +\begin{proof} +Fix $\ell\in\mathcal L$ and any finite set $J\subset\mathcal M_\ell$ of primes away from $\ell$. For each $w\in J$, choose $a_w\in B^\times$ and $h_w>0$ with +\[ + (a_w)=w^{h_w}. +\] +Its compatible Kummer cocycles define a class +\[ + \kappa_w\in H^1_{\mathrm{cont}}(P_B,\Q_\ell(1)). +\] +They factor through $P_B$ because $\Q^s$ contains the roots of unity and all the required radicals. Valuations at the distinct primes $w$ show that these classes are linearly independent. Each is unramified at every finite prime other than $w$ and the primes above $\ell$. + +Write +\[ + \rho=\chi_\ell\circ\beta_B,\qquad + \epsilon=\rho\chi_{K,\ell}^{-1}. +\] +By Theorem~\ref{thm:normalized-map}, $\epsilon$ takes values in $\mu_{\ell-1}\subset\Q_\ell^\times$. Let $K'/K$ be the cyclic extension fixed by its kernel, and let $\Delta=\Gal(K'/K)$. Pullback along the surjection $\beta_B$ is injective on $H^1$, with pulled-back coefficient module $\Q_\ell(\rho)$. Restriction to $G_{K'}$ remains injective: its kernel is a first cohomology group of the finite group $\Delta$, which vanishes in characteristic zero. We therefore obtain independent classes +\[ + c_w\in H^1_{\mathrm{cont}}(G_{K'},\Q_\ell(1)) + \qquad(w\in J). +\] +Under the natural $\Delta$-action on this cohomology, they all lie in the $\epsilon^{-1}$ eigenspace. Indeed the original coefficient module is $\Q_\ell(1)\otimes\epsilon$, so invariance for its twisted action becomes the eigenvalue $\epsilon^{-1}$ after restriction. + +We claim that every $c_w$ is unramified away from $\ell$. Suppose that it is ramified at a prime $u\nmid\ell$ of $K'$, and let $v$ be the prime below it in $K$. The cyclotomic module is trivial on $I_u$, so the restricted cocycle is a nonzero continuous homomorphism from $I_u$ to the additive group $\Q_\ell(1)$. Choose a pro-$\ell$ Sylow subgroup $J_u\subset I_u$ on which it is nonzero, and a pro-$\ell$ Sylow subgroup $J_v\subset I_v$ containing $J_u$. Then $\beta_B(J_v)\ne1$, so $v$ is active at $\ell$. Theorem~\ref{thm:local-correspondence} sends $J_v$ into inertia at a corresponding prime $t\nmid\ell$ of $B$. Since $t$ occurs in an active pair, $t\notin J$, whereas $\kappa_w$ is unramified at $t$. Inertia at $t$ acts trivially on the coefficient module, so the Kummer cocycle itself vanishes on that inertia group. Its pullback therefore vanishes on $J_u$, a contradiction. This proves the claim. + +Let $S$ consist of the archimedean places of $K'$ and its primes above $\ell$. By Lemma~\ref{lem:kummer-unramified}, the classes $c_w$ give independent elements of the $\epsilon^{-1}$ eigenspace in $\mathcal O_{K',S}^{\times}\otimes\Q_\ell$. It remains to bound this eigenspace. The equivariant Dirichlet $S$-unit theorem \cite[Section~3]{bass1966} gives the character of the full $S$-unit module as +\[ + [\mathcal O_{K',S}^{\times}\otimes\Q_\ell] + =[\Q_\ell[S]]-[\Q_\ell]. +\] +Each $\Delta$-orbit in $S$ contributes at most one copy of the one-dimensional character $\epsilon^{-1}$ to the permutation representation: for a stabilizer $H\subset\Delta$, its multiplicity in $\Q_\ell[\Delta/H]$ is one if it is trivial on $H$, and zero otherwise. There are +\[ + \#(S/\Delta) + =r_1(K)+r_2(K)+\#\{v\text{ of }K:v\mid\ell\} + \le2[K:\Q] +\] +such orbits. Subtracting the trivial representation cannot increase this multiplicity. Hence the independent classes $c_w$ lie in a space of dimension at most $2[K:\Q]$, and $\#J\le2[K:\Q]$. Since $J$ was arbitrary, this bounds all missed primes away from $\ell$. There are at most two primes of $B$ above $\ell$, giving the second bound. +\end{proof} + +The bound depends on $K$, but neither on $\ell$ nor on the degree of the cyclic extension $K'/K$. We can therefore choose one auxiliary prime that excludes every possible partner of several specified target primes, and contradict this bound. The following elementary Kummer construction provides that choice. + +\subsection{Simultaneous separation by power residues} + +\begin{lemma}\label{lem:power-separation} +Let $m,d\ge1$, let $q_1,\ldots,q_n$ be distinct rational primes, and for each $j$ let $\mathcal P_j$ be a finite set of rational primes not containing $q_j$. Given any finite set of rational primes to avoid, there are infinitely many odd primes $\ell\equiv1\pmod m$ outside that set and all the displayed prime sets such that +\[ + p^f\bmod\ell\notin\langle q_j\bmod\ell\rangle + \quad + (1\le j\le n,\ p\in\mathcal P_j,\ 1\le f\le d). +\] +The sets $\mathcal P_j$ may overlap, and may contain $q_i$ for $i\ne j$. +\end{lemma} +\begin{proof} +Choose distinct auxiliary primes $s_j$ larger than $m$, $d$, and every prime in the lists, and put +\[ + C=\Q(\zeta_{m\prod_j s_j}). +\] +For each $j$, the classes of the distinct rational primes in $\{q_j\}\cup\mathcal P_j$ are independent in $C^\times/(C^\times)^{s_j}$. Indeed, at any prime in those lists, the ramification index in $C/\Q$ divides $[\Q(\zeta_m):\Q]$: the other cyclotomic conductors are prime to that prime. This index is prime to $s_j$, and valuations at the separate rational primes give the asserted independence. + +Kummer theory therefore identifies the Galois group over $C$ of +\[ + E_j=C\bigl(q_j^{1/s_j},\ p^{1/s_j}\ (p\in\mathcal P_j)\bigr) +\] +with an elementary abelian $s_j$-group having one independent coordinate for each listed prime. Choose $\sigma_j\in\Gal(E_j/C)$ which fixes $q_j^{1/s_j}$ and acts nontrivially on every $p^{1/s_j}$ with $p\in\mathcal P_j$. The extensions $E_j/C$ are linearly disjoint, since their degrees are powers of distinct primes. Thus these automorphisms combine to an element $\sigma\in\Gal(E/C)$, where $E$ is their compositum. + +The extension $E/\Q$ is Galois. Moreover, conjugation in $\Gal(E/\Q)$ preserves whether each of the specified radical coordinates of $\sigma$ is trivial: a conjugation only raises its root-of-unity multiplier to a power prime to the corresponding $s_j$. Chebotarev's theorem supplies infinitely many rational primes $\ell$ with Frobenius in the conjugacy class of $\sigma$, avoiding the prescribed finite set, all listed primes, and $2$. They split completely in $C$, so $\ell\equiv1\pmod m$ and $\ell\equiv1\pmod{s_j}$. At these primes, $q_j$ is an $s_j$th power modulo $\ell$ and every $p\in\mathcal P_j$ is not. + +The quotient of $\F_\ell^\times$ by its $s_j$th powers has prime order $s_j$. A nontrivial class stays nontrivial on raising it to any exponent $1\le f\le d2[K:\Q]+2, +\] +among them, above distinct split rational primes $q_1,\ldots,q_n$, with every $\mathcal V_{w_j}$ finite. Let $\mathcal P_j$ be the set of rational residue characteristics of the primes in $\mathcal V_{w_j}$. The failure of the asserted conclusion means $q_j\notin\mathcal P_j$. + +Apply Lemma~\ref{lem:power-separation} with $m=m_0$ and $d=[K:\Q]$. It gives $\ell\in\mathcal L$, away from all the primes involved, such that +\begin{equation}\label{eq:norm-separation} + Nv\bmod\ell\notin\langle q_j\bmod\ell\rangle + \qquad(v\in\mathcal V_{w_j},\ 1\le j\le n), +\end{equation} +because $Nv=p^f$ with $p\in\mathcal P_j$ and $1\le f\le[K:\Q]$. If $w_j$ occurred in an active pair at this $\ell$, its source prime would lie in $\mathcal V_{w_j}$. Applying Lemma~\ref{lem:norm-identity} to a lift of source Frobenius and reducing modulo $\ell$ would give +\[ + Nv\bmod\ell\in\langle Nw_j\bmod\ell\rangle + =\langle q_j\bmod\ell\rangle, +\] +contrary to \eqref{eq:norm-separation}. All $n$ primes $w_j$ therefore belong to $\mathcal M_\ell$, contradicting Proposition~\ref{prop:uniform-missed-primes}. +\end{proof} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/preliminaries.tex b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..570d1a7c37452c46a1ebda3d6c78a5f69a4bbe3a --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/preliminaries.tex @@ -0,0 +1,92 @@ +\section{Galois groups and partial local correspondences}\label{sec:preliminaries} + +We collect the anabelian inputs and the local calculation that will connect Frobenius elements with cyclotomic characters. The substantial inputs from Uchida and Hoshi are stated explicitly; the remaining arguments use standard global and local class field theory, Kummer theory, and Chebotarev's theorem. + +\subsection{Conventions and solvably closed fields} +Fix an algebraic closure $\overline{\Q}$ of $\Q$. For a number field $M\subset\overline{\Q}$ put $G_M=\Gal(\overline{\Q}/M)$, and put +\[ + \mathcal G=G_{\Q},\qquad + P=\Gal(\Q^s/\Q), +\] +where $\Q^s\subset\overline{\Q}$ is the maximal prosolvable extension of $\Q$. For a number field $M\subset\Q^s$ write $P_M=\Gal(\Q^s/M)$. All homomorphisms between profinite groups and all characters in this paper are continuous. An open homomorphism is a homomorphism with open image: a continuous surjection between profinite groups is a quotient map and is open. + +We use finite primes unless infinity is explicitly mentioned. For a finite prime $v$, its absolute norm is $Nv$, and its absolute residue degree is $[\kappa(v):\F_p]$, where $p$ is its rational residue characteristic. Frobenius is arithmetic. A chosen prolongation of $v$ determines a decomposition group $D_v$ and inertia group $I_v$; a different prolongation conjugates both. Expressions involving several such groups always use specified compatible prolongations. + +\begin{lemma}\label{lem:solvable-closure} +Every solvably closed algebraic extension $E$ of $\Q$ contains $\Q^s$. The field $\Q^s$ is solvably closed. For every number field $M\subset\Q^s$, the extension $\Q^s/M$ is the maximal prosolvable extension of $M$. +\end{lemma} +\begin{proof} +Since $E(\mu_n)/E$ is abelian, $E$ contains every root of unity. Adjoining an $n$th root of an element of $E$ then gives a cyclic extension, so $E$ is closed under radicals. Finite solvable Galois extensions can be built by towers of radical extensions after adjoining roots of unity; hence $\Q^s\subset E$. + +The same tower argument shows that $\Q^s$ contains all roots of unity and is closed under radicals. Any nontrivial finite abelian extension would have a cyclic subextension of prime degree, which Kummer theory excludes. Any nontrivial abelian algebraic extension would have a nontrivial finite abelian subextension. Thus $\Q^s$ is solvably closed. + +Finally, a finite solvable Galois extension of $M$ lies in a finite solvable Galois extension of $\Q$: first take a solvable normal closure of $M/\Q$ inside $\Q^s$, and then the compositum of the finitely many conjugate extensions. The group of this compositum over that normal closure embeds in a product of solvable groups. This proves the last assertion. +\end{proof} + +In particular, every additive $\ell$-adic character of $G_M$ factors through $P_M$ when $M\subset\Q^s$. For any solvably closed Galois extension $E/F$, the action on roots of unity defines its cyclotomic character +\[ + \chi_{E/F}:\Gal(E/F)\longrightarrow\widehat{\Z}^{\times}. +\] +We use $\chi_{M,\ell}$ for the usual $\ell$-adic character of $G_M$, and $\chi_\ell$ for that of $P$. The $\ell$-adic logarithm is normalized to vanish on the finite torsion subgroup of $\Z_\ell^\times$. + +\subsection{The anabelian inputs} +The first input permits a homomorphism to be extended from an open subgroup to the normalizer of its kernel. + +\begin{theorem}[Uchida]\label{thm:uchida-isomorphism} +Let $E_i/F_i$ be solvably closed Galois extensions of number fields. Every isomorphism +$\theta:\Gal(E_1/F_1)\xrightarrow{\sim}\Gal(E_2/F_2)$ +is induced by a unique field isomorphism $j:E_2\xrightarrow{\sim}E_1$, with +$g j=j\theta(g)$ for all $g$. +\end{theorem} +This is \cite[Theorem]{uchida1979}. One consequence we shall use twice is the following. +\begin{lemma}\label{lem:centralizer} +The centralizer in $P$ of any open subgroup of $P$ is trivial. +\end{lemma} +\begin{proof} +An open subgroup is $P_M$ for a number field $M\subset\Q^s$. An element centralizing $P_M$ is a field automorphism of $\Q^s$ implementing the identity automorphism of $P_M$. The identity field map also implements it, so uniqueness in Theorem~\ref{thm:uchida-isomorphism} gives the assertion. +\end{proof} + +Here is the local input, in the precise setting in which we need it. For a homomorphism $\varphi:G_M\to P_N$, an odd prime $\ell$, and a prime $v\nmid\ell$ of $M$, say that $v$ is \emph{active at $\ell$} if $\varphi$ is nontrivial on a pro-$\ell$ Sylow subgroup of $I_v$. The condition is independent of the chosen Sylow subgroup and prolongation, since such choices are conjugate. + +\begin{theorem}[Uchida's partial local correspondence]\label{thm:local-correspondence} +Let $M$ be a number field, let $N\subset\Q^s$ be a number field, and let $\varphi:G_M\to P_N$ be open. Fix an odd prime $\ell$. Every prime $v\nmid\ell$ active at $\ell$ determines a unique prime $w\nmid\ell$ of $N$ for which, with suitable prolongations, +\[ + \varphi(D_v)\subset D_w. +\] +Moreover, $\varphi$ is injective on a pro-$\ell$ Sylow subgroup of $I_v$, and sends this subgroup into $I_w$. All but finitely many primes of $N$ arise from such active primes of $M$. +\end{theorem} +This is the odd-prime case of the trichotomy in \cite[\S1, pp.~595--596]{uchida1981}, together with its Proposition~1. In that trichotomy, nontrivial pro-$\ell$ inertia is case~(ii); case~(i) kills that inertia, and the torsion case~(iii) requires $\ell=2$. The open-image hypothesis, rather than surjectivity onto $P_N$, suffices. We will apply the theorem after restricting to smaller open subgroups as well. The cofinite target coverage in this theorem is for a fixed $\ell$; its exceptional set is not asserted to be uniform in $\ell$. + +The uniqueness assertion concerns primes of the finite target field $N$. When restricting a map to smaller open subgroups, we will intersect a containing target decomposition group with $P_{N'}$ for a finite extension $N'/N$ inside $\Q^s$, and then use uniqueness at the $N'$-level. Proposition~\ref{prop:frobenius-generation} carries out this comparison with explicit source and target fields; no uniqueness of infinite prolongations is required. + +The final input converts cyclotomic compatibility into the required field embedding. +\begin{theorem}[Hoshi]\label{thm:hoshi} +Let $E_i/F_i$ be solvably closed Galois extensions of number fields, and let +$\alpha:\Gal(E_1/F_1)\to\Gal(E_2/F_2)$ be open. There exists an equivariant field embedding $E_2\hookrightarrow E_1$ if and only if +\[ + \chi_{E_2/F_2}\circ\alpha=\chi_{E_1/F_1} + \quad\text{in }\widehat{\Z}^{\times}. +\] +\end{theorem} +We use \cite[Theorem~3.4]{hoshi2025}. Uniqueness of the embedding, when it exists, is \cite[Proposition~2]{uchida1981}. + +\subsection{Tame inertia and norms} +Decomposition groups in $P_M$ are the full absolute Galois groups of the corresponding local fields. Here is a justification for using the usual local ramification structure. Fix a prime of $\overline{\Q}$ above a prime of $M$, and form inside a local algebraic closure the union of the completions of finite fields between $M$ and $\Q^s$. This union contains all roots of unity and is closed under radicals: for a fixed exponent, a nonzero local element can be approximated by a global element in the field defining that completion, and their ratio can be made a power. Thus the union has no cyclic extension. Every finite Galois group over a nonarchimedean local field is solvable, by its wild inertia, tame inertia, and unramified filtration. Every closed subgroup of its prosolvable absolute Galois group is prosolvable as well; a nontrivial such group has a nontrivial finite cyclic quotient after passing through a finite solvable quotient. The union is therefore the full local algebraic closure. The usual decomposition-group identification proves the claim. + +For an active pair $(v,w)$ from Theorem~\ref{thm:local-correspondence}, and $g\in D_v$, let $b(g)\in\widehat{\Z}$ be its source residue exponent and let $a(g)\in\widehat{\Z}$ be the target residue exponent of $\varphi(g)$. These are measured using $Nv$- and $Nw$-Frobenius, respectively. The target residue exponent is defined on $D_v$ by composition; we do not assume that the map sends all source inertia into target inertia. + +\begin{lemma}\label{lem:norm-identity} +For an active pair $(v,w)$ at an odd prime $\ell$, +\begin{equation}\label{eq:norm-identity} + (Nv)^{b(g)}=(Nw)^{a(g)}\quad\text{in }\Z_\ell^\times + \qquad(g\in D_v). +\end{equation} +Powers by profinite integers are interpreted by continuous powering in the compact group $\Z_\ell^\times$. +\end{lemma} +\begin{proof} +Let $p\ne\ell$ be the residue characteristic of $v$, and choose a topological generator $\tau$ of a pro-$\ell$ Sylow subgroup of $I_v$. Its image in source tame inertia generates the pro-$\ell$ factor. The tame conjugation relation shows that +\[ + e=g\tau g^{-1}\tau^{-(Nv)^{b(g)}} +\] +belongs to source wild inertia, a pro-$p$ group. In the target tame quotient, the image of $e$ also lies in the pro-$\ell$ factor of tame inertia: both $\varphi(\tau)$ and its conjugate do. Its image is therefore trivial, because it belongs to a pro-$p$ subgroup and to a pro-$\ell$ group. The nontrivial image of $\varphi(\tau)$ in the torsion-free target pro-$\ell$ tame factor then gives \eqref{eq:norm-identity} by the target tame conjugation relation. The target residue characteristic is different from $\ell$ by Theorem~\ref{thm:local-correspondence}. +\end{proof} diff --git a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b4c09c906fbe42c3450754430731b5a9e22b6147 --- /dev/null +++ b/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:36a58d5f408acc3d5430e1da1397aaaa7900321c3322a2001dcf7e6fc4de7fc0 +size 456051 diff --git a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/README.md b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..f036e810af16eb88c249c47b8242db8c72249397 --- /dev/null +++ b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/README.md @@ -0,0 +1,16 @@ +# [Optimal-order mixing of the Thorp shuffle](paper.pdf) + +OpenAI · September 26, 2026 + +## Citation + +```bibtex +@misc{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026, + author = {{OpenAI}}, + title = {{Optimal-order mixing of the Thorp shuffle}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf}{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/conditional-flow.tex b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/conditional-flow.tex new file mode 100644 index 0000000000000000000000000000000000000000..c469ed5bbdcd412b1fd48c9a8293c312f40a1791 --- /dev/null +++ b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/conditional-flow.tex @@ -0,0 +1,39 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[>=Stealth, every node/.style={font=\small}, + free/.style={circle,draw=linkblue,fill=white,minimum size=6mm}, + seen/.style={circle,fill=black!65,minimum size=3mm,inner sep=0pt}] + \node at (2,2.6) {Two free positions}; + \node[free] (a) at (0,1.7) {}; + \node[free] (b) at (0,0.3) {}; + \node[left=1mm of a] {$u$}; + \node[left=1mm of b] {$v$}; + \node[free] (c) at (4,1.7) {}; + \node[free] (d) at (4,0.3) {}; + \node[right=1mm of c] {$(u+v)/2$}; + \node[right=1mm of d] {$(u+v)/2$}; + \draw[->,linkblue] (a)--(c); + \draw[->,linkblue] (a)--(d); + \draw[->,linkblue] (b)--(c); + \draw[->,linkblue] (b)--(d); + \node at (2,-0.35) {Unrevealed fair switch}; + \begin{scope}[xshift=7.3cm] + \node at (1.5,2.6) {One free position}; + \node[free] (e) at (0,1.7) {}; + \node[seen] (f) at (0,0.3) {}; + \node[left=1mm of e] {$u$}; + \node[seen] (g) at (3,1.7) {}; + \node[free] (h) at (3,0.3) {}; + \node[right=1mm of h] {$u$}; + \draw[->,linkblue,thick] (e)--(h); + \draw[->,black!65,dashed] (f)--(g); + \node at (1.5,-0.35) {Switch fixed by the observed path}; + \end{scope} +\end{tikzpicture} +\caption{The conditional flow on one matching edge. Open circles are free +positions; filled circles contain an exposed card. A switch with two free +positions averages the centered masses $u,v$. With one free position, +the observed path fixes the switch and transports the centered mass $u$. +The latter case produces the martingale noise before the switch is revealed.} +\label{fig:conditional-flow} +\end{figure} diff --git a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/main.tex b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..4bfc10093e86ab9d132859e313598ed63f006b3b --- /dev/null +++ b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/main.tex @@ -0,0 +1,601 @@ +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{20,69,103} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref} +\hypersetup{pdftitle={Optimal-order mixing of the Thorp shuffle},pdfauthor={OpenAI},bookmarksopen=true,bookmarksnumbered=true} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\title{Optimal-order mixing of the Thorp shuffle} +\author{OpenAI} +\date{September 26, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that the Thorp shuffle on $2^d$ cards mixes in $\Theta(d)$ complete shuffles. The full permutation law after $1600d$ shuffles converges to uniform in total variation as $d\to\infty$, uniformly over the initial deck, while a support count gives a lower bound of $2d-O(1)$. +\end{abstract} + +\section{Introduction}\label{sec:introduction} + +A single card can become uniformly distributed long before a shuffled +deck is close to uniform. The Thorp shuffle makes this distinction +particularly clear. Split a deck into two equal halves, pair cards that +occupy corresponding positions, and choose the order of each pair by an +independent fair coin. Interleaving the pairs produces the new deck. +For a deck of size $n=2^d$, every individual card is uniform after $d$ +shuffles. The question is how much longer it takes to randomize the +joint order of all $n$ cards. + +We prove that a constant multiple of $d$ shuffles suffices. Let $q_d$ +denote the law of one complete shuffle as a permutation of the +positions, and let $U_n$ be uniform on the symmetric group $S_n$. +Convolution is the law of composition of independent permutations, and +\[ + \|\mu-\nu\|_{\mathrm{TV}} + =\frac12\sum_g|\mu(g)-\nu(g)|. +\] +Write $t_{\mathrm{mix}}(d)$ for the least nonnegative integer $t$ at which +$\|q_d^{*t}-U_n\|_{\mathrm{TV}}\le1/4$. + +\begin{theorem}[Optimal-order mixing]\label{thm:mixing} +As $d\to\infty$, +\[ + \|q_d^{*(1600d)}-U_n\|_{\mathrm{TV}}\longrightarrow0. +\] +For every integer $t\ge0$, +\[ + \|q_d^{*t}-U_n\|_{\mathrm{TV}} + \ge 1-\frac{2^{tn/2}}{n!}. +\] +Consequently the mixing time at distance $1/4$ satisfies +\[ + t_{\mathrm{mix}}(d)\ge + \left\lceil\frac2n\log_2\frac{3n!}{4}\right\rceil + =2d-O(1), + \qquad t_{\mathrm{mix}}(d)=\Theta(d). +\] +\end{theorem} + +One step in this theorem is one complete physical shuffle. Changing the +initial deck translates the permutation law, so its distance from +uniform is unchanged. The upper bound is asymptotic in $d$; the +constant $1600$ is an explicit choice in the proof, not a proposed +sharp constant. Determining a leading constant or a cutoff remains a +separate question. + +The full-law conclusion controls every statistic of the final deck and, +in particular, the joint positions of any deterministic selection of +cards, by contraction of total variation under a map. It also shows +that this shuffle produces an approximately uniform permutation using +$O(n\log n)$ independent fair bits: each of the $O(\log n)$ steps uses +$n/2$ bits. The support bound identifies the same order of randomness +as necessary for this construction. + +\subsection{History and related work} + +Thorp introduced the model in his 1973 study of the effects of +nonrandom shuffling on Faro \cite[Section~3.1]{thorp}. Its simple +pairwise rule conceals a difficult question about dependence across the +whole deck. For dyadic decks, Morris obtained the first bound +polynomial in $d$, namely $O(d^{44})$, using evolving sets and a +chameleon process that tracks conditional card probabilities +\cite{morris44}. Montenegro and Tetali sharpened this analysis to +$O(d^{29})$ by spectral-profile and evolving-set methods +\cite[Theorems~6.14--6.15]{montenegro-tetali2006}. + +A second line of attack used relative entropy. Morris related entropy +decay to pairwise card collisions and obtained $O((\log n)^4)$ +shuffles for every even $n$ \cite{morris4}. He then proved $O(d^3)$ +for $n=2^d$ by a stronger contraction over a block of $d$ shuffles +\cite{morris3}. Theorem~\ref{thm:mixing} attains the logarithmic order +discussed in Jonasson's notes \cite[Section~5]{jonasson} for dyadic +decks. The corresponding optimal-order question for arbitrary even +deck sizes is not addressed by the cube argument here. + +Card marginals have also been studied for their own sake and for +cryptography. Morris, Rogaway and Stegers analyzed the joint images of +ordered input lists in their construction of enciphering schemes on +small domains \cite[Theorem~1]{mrs}. Czumaj and V\"ocking proved that +$O((\log n)^2)$ Thorp shuffles mix any fixed fraction of the cards +strictly below one, using non-Markovian couplings and butterfly-network +estimates \cite{CzumajVocking2014}. Our first stage needs a different, +averaged guarantee: after $200d$ shuffles, the joint positions of a +uniformly sampled list of $7n/8$ labels are close to uniform on average +over the list. The passage from these averaged marginals to the full +law is a separate part of the proof. + +The conditional-flow method has several predecessors. In the original +Thorp analysis, Morris used a chameleon process to represent the law of a tagged card +conditional on the set-valued trajectory of a collection containing +that card; the conditional probabilities average across available pairs +and are transported across pairs with only one available position +\cite[Section~4, Lemma~3]{morris44}. Hoang, Morris and Rogaway later +used conditional squared energy and a sequential total-variation +comparison to analyze the swap-or-not shuffle +\cite[Section~3, Theorem~3 and Lemma~4]{hmr2014}. Their uniformly random +keys make the next partner uniform over the underlying group. +Here the directions are prescribed cyclically. The exact identity +retaining only one sweep of noise, together with independence of the +two opposite halves during the intervening updates, provides the needed +contraction. + +For the final passage to the group, we use complete reducibility and +Schur orthogonality, together with the tableau description and +hook-length formula for symmetric-group degrees +\cite{etingof,sagan}. The inverse-degree summability needed here is a +special case of a stronger theorem of Liebeck and Shalev +\cite[Theorem~2.6]{liebeck-shalev2004}; we give an elementary proof of +the required case. The final total-variation comparison is the +finite-group Fourier method of Diaconis and Shahshahani +\cite[Section~3, Lemma~14]{DiaconisShahshahani1981}. +The additional step here is to turn bounds on complementary card +marginals into operator bounds for matrix Fourier transforms, using a +common trimming and the span of one vector's orbit under a block +subgroup. + +\subsection{Proof overview} + +Label positions by binary strings of length $d$, the vertices of a +cube. Undoing a deterministic cyclic rotation turns successive shuffles +into switch layers along successive coordinate directions: the two +positions in one switch differ only in the active bit. A sweep through +all $d$ directions takes $d$ physical shuffles. +Section~\ref{sec:model} gives this change of frame and proves the +support lower bound. + +For the upper bound, first choose an ordered list of $7n/8$ labels +uniformly and independently of the shuffle. Expose these cards one at +a time. Conditional on the paths already exposed, the next card has a +probability distribution on the remaining positions. On a switch with +two remaining positions its masses are averaged; on a switch with +only one remaining position its mass follows the route fixed by the +exposed path. Subtracting the uniform distribution on the remaining +positions gives a centered flow. + +A full sweep of coordinate averages annihilates every mean-zero +vector. The actual conditional update differs from coordinate averaging +only by noise with conditional mean zero on switches with one available +position. +Consequently the expected squared norm at the present time is an exact +weighted sum of noise contributions from the preceding sweep, as +proved in Section~\ref{sec:flow}. To control those contributions, +Section~\ref{sec:contraction} looks back from the next active direction +through the $d-1$ other directions. These updates act independently +inside the two opposite halves. A flow value in one half is therefore +independent of the availability of its partner in the other half, +conditional on the history at the start of this interval. A random list +leaves a positive fraction of available positions in both halves with +high probability. This yields a scalar energy recurrence and then the +averaged marginal estimate of Proposition~\ref{prop:averaged-marginal} +at time $200d$. + +The second stage recovers the dependencies involving the omitted +cards. Partition the labels into eight equal blocks, choosing the +partition so that the eight complementary marginals have small total +error. Each marginal is a distribution on cosets of the subgroup that +permutes its omitted block. Removing a small amount of probability +makes all eight coset densities bounded under one common law. +Restrict an irreducible representation of $S_n$ to the product of these +eight subgroups. Each tensor constituent has at least one factor of +small degree. Even with arbitrary multiplicity, the orbit of a single +vector under the corresponding block subgroup spans a space of +controlled dimension. +Averaging projections onto its translates then bounds each Fourier +operator. Sections~\ref{sec:degrees} and~\ref{sec:lift} supply the degree +estimate and this transfer; Section~\ref{sec:completion} combines eight +independent time intervals of $200d$ shuffles and handles parity to complete the proof. + +The companion on random coordinate frames gives an independent +$32800d$ proof \cite[Theorem~1.1]{companion-frames}. The +conditional-information companion obtains bounds uniform over specified +input lists \cite[Theorem~1.1]{companion-conditional}, while the Fourier +companion develops general transfers from complementary coset bounds +\cite[Theorem~1.1]{companion-transfer}. The present proof uses none of +these results: its cyclic-coordinate estimate and eight-block transfer +are proved here. + +\section{The model and the support obstruction}\label{sec:model} + +Let $d\ge1$ be an integer. Identify the positions with \(V=\mathbb F_2^d\), write $e_1,\ldots,e_d$ for its standard basis, and label each card by its initial position. A permutation maps a label to its current position; composition means \((gh)(x)=g(h(x))\). With +\[ + R(x_1,\ldots,x_d)=(x_2,\ldots,x_d,x_1), +\] +one physical shuffle is \(RS\), where \(S\) independently fixes or swaps each pair \(\{x,x+e_1\}\). Thus its action is +\[ + (x_1,u)\longmapsto (u,x_1+\xi_u), +\] +with independent fair bits \(\xi_u\). + +Let $S_1,S_2,\ldots$ be independent copies of $S$. If \(Y_t=RS_tY_{t-1}\) is the physical permutation, started from $Y_0=\mathrm{id}$, and \(X_t=R^{-t}Y_t\), then +\[ + X_{t+1}=Q_tX_t,\qquad + Q_t=R^{-t}S_{t+1}R^t. +\] +The \(Q_t\) are independent matching switches in the directions +\(i_t=1+(t\bmod d)\), since $R^{-t}e_1=e_{i_t}$. A block whose length is a multiple of \(d\) ends in the physical frame. We analyze this cyclic-coordinate process, writing \(\Pi_t=X_t\). + +During one full sweep each coordinate of a given card is refreshed +exactly once by a coin fair conditional on the card's earlier path. +These refreshed bits form a uniform binary string. Restoring the +deterministic rotation preserves uniformity, proving the one-card +assertion in the introduction. + +Uniform measure is invariant under every position permutation. Starting from any fixed deck translates the law started at the identity and leaves its total-variation distance from uniform unchanged. Moreover, \(t\) shuffles have at most \(2^{tn/2}\) possible coin strings. Their support \(A\) therefore gives +\[ + \|q_d^{*t}-U_n\|_{\mathrm{TV}} + \ge q_d^{*t}(A)-U_n(A) + \ge1-\frac{2^{tn/2}}{n!}. +\] +Distance at most \(1/4\) forces \(2^{tn/2}\ge3n!/4\), proving the rounded lower bound. The inequalities \((n/e)^n\le n!\le n^n\) give its asymptotic form. + +For completeness, mixing exists in each fixed dimension. A full coordinate sweep has positive probability of being the identity, and of making any specified single cube-edge transposition. Edge transpositions on the connected cube generate \(S_n\). Every permutation can therefore be obtained in some number of sweeps; identity sweeps pad these representations to one common length. The kernel at that length assigns positive mass to every group element and hence dominates a positive multiple of \(U_n\). Iteration proves convergence. This finite-dimensional observation does not assert the numerical bound \(1600d\) for every \(d\). + +\section{A conditional flow with one sweep of memory}\label{sec:flow} + +Choose an ordered list \((c_1,\ldots,c_k)\) of distinct labels independently of the shuffle. We will ultimately average over uniform lists, but first fix the list. For $0\le\ell2d\), use induction in \eqref{eq:h-2}; all indices \(t-j\) are at least \(d\). The geometric contribution is at most \(\gamma^{t-2d}\), because +\[ + (1-\beta)\sum_{j=1}^{\infty}(2\gamma)^{-j} + =\frac{1-\beta}{2\gamma-1}=1. +\] +The constant contribution is at most +\((1-\beta)\eta/\beta+\eta=\eta/\beta\). +This proves \eqref{eq:h-6}. + +We next turn the energy estimate into a bound on the positions of an ordered list. We use the sequential comparison of Morris \cite[Section~5.3, Lemma~12]{morris-exclusion}, also given by Morris, Rogaway and Stegers \cite[Appendix~A, Lemma~2]{mrs}. It applies to any law on distinct-position tuples; $U_{\mathrm{inj}}$ denotes the uniform law on such tuples. +\begin{lemma}[Sequential total variation]\label{lem:sequential-tv} +For a random tuple $(Z_1,\ldots,Z_k)$ of distinct positions, +\begin{equation}\label{eq:h-7} + \|\mathcal L(Z_1,\ldots,Z_k)-U_{\mathrm{inj}}\|_{\mathrm{TV}} + \le\sum_{j=1}^k + \mathbb E\left\|\mathcal L(Z_j\mid Z_1,\ldots,Z_{j-1}) + -U_{V\setminus\{Z_1,\ldots,Z_{j-1}\}}\right\|_{\mathrm{TV}}. +\end{equation} +Every expectation is under the actual prefix law. Refining the conditioning in a summand can only increase its expected distance, provided the refining sigma-field contains the prefix. +\end{lemma} +\begin{proof} +At the $j$th step compare the actual joint $j$-tuple with the law that +samples its first $j-1$ coordinates from their actual distribution and +appends a uniform available position. Their distance is exactly the +$j$th summand. Applying this same uniform extension kernel to a uniform +prefix gives a uniform $j$-tuple and cannot increase total variation. +The triangle inequality and induction prove \eqref{eq:h-7}. The uniform comparison law is prefix-measurable, so the tower property and convexity of total variation prove the refinement assertion. +\end{proof} + +\begin{proposition}[Averaged ordered marginals]\label{prop:averaged-marginal} +Let \(n=2^d\), \(T=200d\), and \(1\le k\le7n/8\). Choose an ordered list \(C=(c_1,\ldots,c_k)\) uniformly among all lists of distinct labels, independently of the shuffle. Then +\begin{equation}\label{eq:h-8} + \begin{aligned} + \mathbb E_C + \left\|\mathcal L\bigl((\Pi_T(c_1),\ldots,\Pi_T(c_k))\mid C\bigr) + -U_{\mathrm{inj}}\right\|_{\mathrm{TV}} + &\le\varepsilon_n,\\ + \varepsilon_n:=\frac{n^{3/2}}2 + \sqrt{(31/32)^{198d}+32e^{-n/256}} + &=O(n^{-5/2}). + \end{aligned} +\end{equation} +\end{proposition} +\begin{proof} +First fix \(C\) and apply Lemma~\ref{lem:sequential-tv} to \(Z_j=\Pi_T(c_j)\). In its \(j\)th summand, condition further on the paths through time \(T\) of the first \(j-1\) cards. These paths determine the prefix of endpoints, so the lemma's refinement assertion applies. + +Let \(w_T^{(j)}\) be the resulting centered flow, with \(\ell=j-1\) and \(m_j=n-j+1\). Its conditional distance is \(\frac12\|w_T^{(j)}\|_1\). Averaging over both the list and the shuffle, Cauchy--Schwarz gives +\[ + \frac12\mathbb E_{C,\mathrm{shuffle}}\|w_T^{(j)}\|_1 + \le\frac12\sqrt{m_j\, + \mathbb E_{C,\mathrm{shuffle}}\|w_T^{(j)}\|_2^2}. +\] +Each stage leaves at least \(n/8\) free positions, so \eqref{eq:h-6} applies to the expectation under this joint law, for every \(j\le k\). Since \(m_j\le n\) and \(k\le n\), summing the last display proves the bound in \eqref{eq:h-8}. Finally, \(198\log_2(32/31)>8\) gives \((31/32)^{198d}\le n^{-8}\), which proves the stated asymptotic estimate. +\end{proof} +The bound averages over the input list. We will use that average to choose a single partition into eight blocks for which all eight complementary marginals have small total error. + +The remaining question concerns the full permutation, including dependencies involving the omitted cards. We now pass to representations of \(S_n\). Restricting a representation to the subgroups that permute labels within the eight blocks separates it into tensor factors. At least one factor is small enough that the span generated by a single vector remains controlled, even when that factor occurs with multiplicity. This converts the marginal estimates into bounds on the Fourier matrices of a slightly trimmed law. First we prove the dimension estimate needed to sum those bounds; then we carry out the trimming and orbit-span argument. + +\section{The representation degrees needed for the lift}\label{sec:degrees} + +The transfer needs two properties of symmetric-group representation degrees. A uniform bound on their reciprocal sum will control orbit dimensions for the label-block subgroups in Proposition~\ref{prop:orbit-lift}. Decay of the sum away from the trivial and sign representations will control the full-group Fourier error in Section~\ref{sec:completion}. We prove both properties here. + +Let \(\lambda\vdash s\) be a partition of \(s\), represented by its Young diagram, whose row lengths are the parts of \(\lambda\). Write \(f^\lambda\) for the degree of the corresponding irreducible representation of \(S_s\). A standard Young tableau fills the diagram with \(1,\ldots,s\), increasing along each row and column. We use the classical facts that \(f^\lambda\) counts these tableaux, is unchanged by transposing the diagram, and satisfies the hook-length formula +\[ + f^\lambda=\frac{s!}{\prod_{z\in\lambda}h(z)}, +\] +where $h(z)$ counts the box $z$ together with the boxes to its right +and below it. The row $(s)$ and column $(1^s)$ give the trivial and +sign representations, respectively \cite{sagan,etingof}. + +\begin{lemma}[Reciprocal representation degrees]\label{lem:reciprocal-degrees} +The reciprocal-degree sums satisfy +\begin{equation}\label{eq:h-9} + C_1:=\sup_{s\ge1}\sum_{\lambda\vdash s}(f^\lambda)^{-1}<\infty, + \qquad + \sum_{\substack{\lambda\vdash s\\\lambda\notin\{(s),(1^s)\}}}(f^\lambda)^{-1}\longrightarrow0\quad(s\to\infty). +\end{equation} +\end{lemma} +Stronger asymptotics for every fixed positive inverse-degree exponent +were proved by Liebeck and Shalev \cite[Theorem~2.6]{liebeck-shalev2004}. +The elementary proof below gives all the summability used here. +\begin{proof} + +First, let \(p(u)\) be the number of partitions of \(u\). The identity +\[ + \sum_{v\ge0}p(v)x^v=\prod_{j\ge1}(1-x^j)^{-1} +\] +gives \(p(u)\le e^{3\sqrt u}\) for \(u\ge1\): evaluate at \(x=e^{-1/\sqrt u}\), when the logarithm of the product is +\[ + \sum_{r\ge1}\frac1{r(e^{r/\sqrt u}-1)} + \le\sqrt u\sum_{r\ge1}r^{-2}\le2\sqrt u, +\] +and multiplying by \(x^{-u}\) bounds its \(u\)th coefficient as claimed. + +For a diagram of size \(s\), let \(L\) be the larger of its first row and column lengths. Transpose so its first row has length \(L\), and put \(u=s-L\). If \(1\le u\le s/4\), then +\[ + f^\lambda\ge \binom Lu\ge(L/u)^u\ge3^u. +\] +For the first inequality place labels \(1,\ldots,u\) in the first \(u\) top-row boxes; choose any \(u\) labels from the remaining \(L\) labels for the lower diagram, filling it in row-major order; and fill the remaining top row increasingly. Every lower column lies under one of the first \(u\) top boxes, so all column and row inequalities hold. There are at most \(2p(u)\) shapes with this tail size, including transposes. Their reciprocal contributions are bounded by the summable sequence \(2p(u)3^{-u}\); for each fixed positive \(u\), the binomial lower bound tends to infinity with \(s\). Dominated convergence makes the whole contribution tend to zero. + +If \(u>s/4\) and \(L\ge s/8\), retain the top row and the first \(b=\lfloor s/16\rfloor\) boxes below it, read row by row from top to bottom and left to right. The retained boxes form a Young diagram. Every standard tableau of the retained diagram extends to the full diagram by filling the remaining boxes in the same row order with the larger labels. In the retained diagram, fix \(1,\ldots,b\) in the first \(b\) top-row boxes and choose the \(b\) lower labels from the remaining \(L\) labels, as above. Since \(L\ge2b\), this gives +\[ + f^\lambda\ge\binom Lb\ge2^b +\] +for all sufficiently large \(s\). There are at most \(e^{3\sqrt s}\) diagrams, so their reciprocal contribution tends to zero. If \(L2|H_a|/|G|\), and let \(E\) be what remains after all eight removals. A bad coset satisfies +\(\mu(C)\le2(\mu(C)-|H_a|/|G|)\). +The sum of positive excesses is its marginal total-variation distance, so the union bound gives +\[ + M:=\mu(E)\ge1-2\delta_n. +\] +For all sufficiently large \(n\), we have \(M\ge1/2\) and may define the conditional probability law \(\nu=\mu(\cdot\mid E)\). Then +\begin{equation}\label{eq:h-11} + \|\nu-\mu\|_{\mathrm{TV}}=1-M=o(1), + \qquad + \nu(gH_a)\le\frac2M\frac{|H_a|}{|G|} + \le4\frac{|H_a|}{|G|}. +\end{equation} +A coset removed for that subgroup has no retained mass; every other coset had the required original cap. This proves the same cap for all eight systems under one common law. Consequently, for every nonnegative function that is constant on the cosets of any one \(H_a\), its expectation under \(\nu\) is at most four times its uniform expectation. The next proposition applies this comparison to projections onto orbit spans. + +We use finite-dimensional complex Hilbert spaces. The operator norm is +$\|B\|_{\mathrm{op}}$, and the Hilbert--Schmidt norm is +$\|B\|_{\mathrm{HS}}^2=\operatorname{tr}(B^*B)$ with ordinary, +unnormalized trace. Write $\widehat H_a$ for the set of equivalence classes of irreducible representations of $H_a$. For a probability measure \(\nu\) on \(G\) and a unitary representation \(\rho\), define +\(\widehat\nu(\rho)=\sum_g\nu(g)\rho(g)\). +\begin{proposition}[Eight-block Fourier bound]\label{prop:orbit-lift} +Let \(n\) be a positive multiple of eight, and partition the labels of \(G=S_n\) into sets \(M_1,\ldots,M_8\) of size \(n/8\). For each \(a\), let \(H_a\) be the subgroup fixing every label outside \(M_a\). Suppose a probability measure \(\nu\) on \(G\) satisfies +\[ + \nu(gH_a)\le4\frac{|H_a|}{|G|} + \qquad(g\in G,\ 1\le a\le8). +\] +Then every irreducible unitary representation \(\rho\) of \(G\), of degree \(D\), satisfies +\begin{equation}\label{eq:h-12} + \|\widehat\nu(\rho)\|_{\mathrm{op}} + \le A D^{-5/16},\qquad A=\sqrt{32C_1}. +\end{equation} +\end{proposition} +\begin{proof} +The disjoint block subgroups form \(H=H_1\times\cdots\times H_8\). The restriction of \(\rho\) to \(H\) decomposes orthogonally into spaces +\[ + (V_{\pi_1}\otimes\cdots\otimes V_{\pi_8}) + \otimes\mathcal A_{\boldsymbol\pi}, +\] +where the last space records arbitrary multiplicity and \(H\) acts trivially on it. Every occurring tensor type has product of factor degrees at most \(D\); assign its whole isotypic space to one factor whose degree is at most \(D^{1/8}\). This gives an orthogonal decomposition \(V_\rho=E_1\oplus\cdots\oplus E_8\). Each \(E_a\) is \(H\)-invariant, and all the \(H_a\)-irreducible types occurring in it have degree at most \(D^{1/8}\). + +Fix \(u_a\in E_a\), and let +\(W_a=\operatorname{span}\{\rho(h)u_a:h\in H_a\}\). +Let \(u_{a,\pi}\) be the projection of \(u_a\) onto the full \(H_a\)-isotypic component \(V_\pi\otimes\mathcal A_\pi\), with \(b=\dim\pi\). Its orbit lies in the image of +\[ + \operatorname{End}(V_\pi)\longrightarrow V_\pi\otimes\mathcal A_\pi, + \quad Q\longmapsto(Q\otimes I)u_{a,\pi}. +\] +Its dimension is at most \(b^2\), independently of multiplicity. Grouping all equivalent copies first, and counting each irreducible type once, \eqref{eq:h-9} gives +\begin{equation}\label{eq:h-13} + \dim W_a\le\sum_{\substack{\pi\in\widehat H_a\\\dim\pi\le D^{1/8}}} + (\dim\pi)^2 + \le D^{3/8}\sum_{\pi\in\widehat H_a}(\dim\pi)^{-1} + \le C_1D^{3/8}. +\end{equation} +The uniformity of \(C_1\) applies to every block size. + +The space \(W_a\) is \(H_a\)-invariant, so \(\rho(g)W_a\) depends only on \(gH_a\). If \(P_W\) denotes orthogonal projection, Schur's lemma and the trace give +\[ + \mathbb E_{g\sim U_G}P_{\rho(g)W_a} + =\frac{\dim W_a}{D}I. +\] +Indeed this average commutes with \(\rho(G)\), and its trace is \(\dim W_a\). Applying the assumed coset cap to the nonnegative coset-constant function \(\|P_{\rho(g)W_a}z\|^2\), we obtain +\[ + \begin{aligned} + |\langle z,\widehat\nu(\rho)u_a\rangle|^2 + &\le\mathbb E_{g\sim\nu}|\langle z,\rho(g)u_a\rangle|^2\\ + &\le\|u_a\|^2\mathbb E_{g\sim\nu} + \|P_{\rho(g)W_a}z\|^2\\ + &\le4\frac{\dim W_a}{D}\|u_a\|^2\|z\|^2 + \le4C_1D^{-5/8}\|u_a\|^2\|z\|^2. + \end{aligned} +\] +For \(u=\sum_a u_a\), sum these bounds after taking square roots, and use \(\sum_a\|u_a\|\le\sqrt8\|u\|\). This proves \eqref{eq:h-12}. +\end{proof} + +\section{Eight convolution factors and the final comparison}\label{sec:completion} + +The final comparison uses the finite-group Fourier method of +Diaconis and Shahshahani \cite[Section~3]{DiaconisShahshahani1981}. +The shuffle law is not constant on conjugacy classes, so we retain matrix norms throughout. +By \eqref{eq:h-11} and Proposition~\ref{prop:orbit-lift}, the nearby law +\(\nu\) has the required Fourier operator bound. We first show that +\(\nu^{*8}\) is close to uniform, then compare it with the law of eight +independent intervals of \(200d\) shuffles, +\(\mu^{*8}=q_d^{*(1600d)}\). + +\begin{proof}[Completion of the proof of Theorem~\ref{thm:mixing}] +With our Fourier normalization, every probability measure \(\sigma\) on \(G\) obeys +\begin{equation}\label{eq:h-14} + 4\|\sigma-U_G\|_{\mathrm{TV}}^2 + \le\sum_{\rho\ne\mathbf1}D_\rho + \|\widehat\sigma(\rho)\|_{\mathrm{HS}}^2. +\end{equation} +For clarity, Cauchy--Schwarz bounds the left side by +\(|G|\sum_g|\sigma(g)-|G|^{-1}|^2\). +The regular-character orthogonality identity gives, for every real mass function \(a\), +\[ + \sum_\rho D_\rho\left\|\sum_g a(g)\rho(g)\right\|_{\mathrm{HS}}^2 + =\sum_{g,h}a(g)a(h)\sum_\rho D_\rho + \operatorname{tr}(\rho(h)^*\rho(g)) + =|G|\sum_g a(g)^2. +\] +Apply this to \(a=\sigma-U_G\); its trivial coefficient is zero and uniform measure has zero transform in every nontrivial irreducible. This proves \eqref{eq:h-14}, with the ordinary, unnormalized Hilbert--Schmidt norm fixed in Section~\ref{sec:lift}. + +Convolution multiplies Fourier matrices. The operator norm is submultiplicative and \(\|B\|_{\mathrm{HS}}\le\sqrt D\|B\|_{\mathrm{op}}\), without any normality assumption. Therefore the nonsign, nontrivial contribution for \(\sigma=\nu^{*8}\) is at most +\[ + \begin{aligned} + \sum_{\rho\ne\mathbf1,\mathrm{sgn}}D_\rho + \|\widehat\nu(\rho)^8\|_{\mathrm{HS}}^2 + &\le A^{16}\sum_{\rho\ne\mathbf1,\mathrm{sgn}} + D_\rho^{2-16(5/16)}\\ + &=A^{16}\sum_{\rho\ne\mathbf1,\mathrm{sgn}}D_\rho^{-3} + \longrightarrow0, + \end{aligned} +\] +by \eqref{eq:h-9}, since \(D^{-3}\le D^{-1}\). + +The sign representation has degree one and requires a separate argument. In the final physical shuffle of a positive-time block, condition on every coin but one. Flipping that last coin changes the resulting permutation by a transposition, so the original block law \(\mu\) has zero expected sign. Since \(\nu=\mu\mathbf1_E/M\), +\[ + |\widehat\nu(\mathrm{sgn})| + =\frac1M\left|\sum_{g\notin E}\mu(g)\mathrm{sgn}(g)\right| + \le\frac{1-M}{M}\longrightarrow0. +\] +Its contribution in \eqref{eq:h-14} after eight factors also tends to zero. Thus \(\|\nu^{*8}-U_G\|_{\mathrm{TV}}\to0\). + +Finally, replacing independent convolution factors one at a time costs at most their individual total-variation distances. By \eqref{eq:h-11}, +\[ + \|\mu^{*8}-U_G\|_{\mathrm{TV}} + \le8(1-M)+\|\nu^{*8}-U_G\|_{\mathrm{TV}} + \longrightarrow0. +\] +Each convolution factor represents \(200d\) physical shuffles, so \(\mu^{*8}=q_d^{*(1600d)}\). Translation invariance gives the same bound from every deterministic initial deck. Together with the support inequality and fixed-dimensional convergence already proved, this completes the theorem. +\end{proof} + +\input{references} +\end{document} diff --git a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/references.tex b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..a2dd7db6514f89c4fb64748b719ab6c5c1f349bf --- /dev/null +++ b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/references.tex @@ -0,0 +1,152 @@ +\begin{thebibliography}{99} + +\bibitem{azuma1967} +K. Azuma, +\emph{Weighted sums of certain dependent random variables}, +T\^ohoku Mathematical Journal (2) 19 (1967), no.~3, 357--367, +\href{https://doi.org/10.2748/tmj/1178243286}{doi:10.2748/tmj/1178243286}. + +\bibitem{CzumajVocking2014} +A.~Czumaj and B.~V\"ocking, +\emph{Thorp shuffling, butterflies, and non-Markovian couplings}, +Automata, Languages, and Programming, +Lecture Notes in Computer Science 8572, Springer, 2014, 344--355. +\href{https://doi.org/10.1007/978-3-662-43948-7_29} +{doi:\nolinkurl{10.1007/978-3-662-43948-7_29}}. + +\bibitem{DiaconisShahshahani1981} +P.~Diaconis and M.~Shahshahani, +\emph{Generating a random permutation with random transpositions}, +Zeitschrift f\"ur Wahrscheinlichkeitstheorie und verwandte Gebiete +57 (1981), 159--179. +\href{https://doi.org/10.1007/BF00535487}{doi:10.1007/BF00535487}. + +\bibitem{etingof} +P.~Etingof, O.~Golberg, S.~Hensel, T.~Liu, A.~Schwendner, +D.~Vaintrob, and E.~Yudovina, +\emph{Introduction to Representation Theory}, +Student Mathematical Library, vol.~59, American Mathematical Society, +2011. +Author's lecture-note version, January 10, 2011: +\url{https://math.mit.edu/~etingof/replect.pdf}. + +\bibitem{hmr2014} +V. T. Hoang, B. Morris, and P. Rogaway, +\emph{An enciphering scheme based on a card shuffle}, +Advances in Cryptology---CRYPTO 2012, +Lecture Notes in Computer Science 7417, Springer, 2012, 1--13. +The cited Section 3, Theorem 3, and Lemma 4 refer to +\href{https://arxiv.org/abs/1208.1176v2}{arXiv:1208.1176v2}, +21 November 2014. + +\bibitem{hoeffding1963} +W. Hoeffding, +\emph{Probability inequalities for sums of bounded random variables}, +Journal of the American Statistical Association 58 (1963), no.~301, 13--30, +\href{https://doi.org/10.1080/01621459.1963.10500830}{doi:\nolinkurl{10.1080/01621459.1963.10500830}}. + +\bibitem{jonasson} +J.~Jonasson, +\emph{The Mathematics of Card Shuffling}, +course notes, April 2009, Section~5. +\url{https://www.math.chalmers.se/~jonasson/convrates.pdf}. + +\bibitem{liebeck-shalev2004} +M.~W.~Liebeck and A.~Shalev, +\emph{Fuchsian groups, coverings of Riemann surfaces, subgroup growth, +random quotients and random walks}, +Journal of Algebra 276 (2004), no.~2, 552--601. +\href{https://doi.org/10.1016/S0021-8693(03)00515-5} +{doi:\nolinkurl{10.1016/S0021-8693(03)00515-5}}. +Author version: \url{https://wwwf.imperial.ac.uk/~mwl/fuchs.pdf}. + +\bibitem{montenegro-tetali2006} +R. Montenegro and P. Tetali, +\emph{Mathematical aspects of mixing times in Markov chains}, +Foundations and Trends in Theoretical Computer Science 1 (2006), no.~3, +237--354, \href{https://doi.org/10.1561/0400000003}{doi:10.1561/0400000003}. + +\bibitem{morris44} +B.~Morris, +The mixing time of the Thorp shuffle, +\emph{SIAM Journal on Computing} \textbf{38} (2008), no.~2, 484--504. +\href{https://doi.org/10.1137/050636231}{doi:10.1137/050636231}. +Preprint: \url{https://arxiv.org/abs/math/0507307v1}, 15 July 2005. +The cited Section~4, Lemma~3 refers to this preprint version. + +\bibitem{morris-exclusion} +B.~Morris, +The mixing time for simple exclusion, +\emph{Annals of Applied Probability} \textbf{16} (2006), no.~2, 615--635. +\href{https://doi.org/10.1214/105051605000000728} +{doi:10.1214/105051605000000728}. +Electronic reprint: \url{https://arxiv.org/abs/math/0405157v2}, +12 July 2006. + +\bibitem{morris4} +B.~Morris, +Improved mixing time bounds for the Thorp shuffle and $L$-reversal chain, +\emph{Annals of Probability} \textbf{37} (2009), no.~2, 453--477. +\href{https://doi.org/10.1214/08-AOP409}{doi:10.1214/08-AOP409}. +Preprint: \url{https://arxiv.org/abs/0802.0339}. + +\bibitem{morris3} +B.~Morris, +Improved mixing time bounds for the Thorp shuffle, +\emph{Combinatorics, Probability and Computing} \textbf{22} (2013), +no.~1, 118--132. +\href{https://doi.org/10.1017/S0963548312000478} +{doi:10.1017/S0963548312000478}. +Preprint: \url{https://arxiv.org/abs/0912.2759}. + +\bibitem{mrs} +B. Morris, P. Rogaway, and T. Stegers, +\emph{How to encipher messages on a small domain: +deterministic encryption and the Thorp shuffle}, +Advances in Cryptology---CRYPTO 2009, +Lecture Notes in Computer Science 5677, Springer, 2009, 286--302, +\href{https://doi.org/10.1007/978-3-642-03356-8_17}{doi:10.1007/978-3-642-03356-8\_17}. +Theorem 1 and Appendix A, Lemma 2, are cited from the +\href{https://www.cs.ucdavis.edu/~rogaway/papers/thorp.pdf}{author's conference text}. +A later journal version is \emph{Deterministic encryption with the Thorp shuffle}, +Journal of Cryptology 31 (2018), no.~2, 521--536, +\href{https://doi.org/10.1007/s00145-017-9262-z}{doi:10.1007/s00145-017-9262-z}. + +\bibitem{companion-conditional} +OpenAI, +\emph{Conditional information under deterministic coordinate sweeps}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf}{OAI:Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026}, +2026. + +\bibitem{companion-transfer} +OpenAI, +\emph{From partial permutation information to Fourier bounds}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf}{OAI:From-partial-permutation-information-to-Fourier-bounds-September-26-2026}, +2026. + +\bibitem{companion-frames} +OpenAI, +\emph{Random coordinate frames and partial permutation laws}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf}{OAI:Random-coordinate-frames-and-partial-permutation-laws-September-26-2026}, +2026. + +\bibitem{sagan} +B.~E.~Sagan, +\emph{The Symmetric Group: Representations, Combinatorial Algorithms, +and Symmetric Functions}, second edition, +Graduate Texts in Mathematics, vol.~203, Springer, 2001. +\href{https://doi.org/10.1007/978-1-4757-6804-6} +{doi:10.1007/978-1-4757-6804-6}. + +\bibitem{thorp} +E.~O.~Thorp, +Nonrandom shuffling with applications to the game of Faro, +\emph{Journal of the American Statistical Association} \textbf{68} +(1973), no.~344, 842--847. +\href{https://doi.org/10.1080/01621459.1973.10481434} +{doi:10.1080/01621459.1973.10481434}. + +\end{thebibliography} diff --git a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3b0f96a517819a5fe2d92d6b38e76be906201319 --- /dev/null +++ b/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9b02c5b13a61d6c071b9a12849d0177a4ece970f3b182893a1c3d14b660c5984 +size 399392 diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/README.md b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..7955c68bf897cb8a4777a11c4deee3a73dfa5cd3 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/README.md @@ -0,0 +1,18 @@ +# [Orbifold and logarithmic Iitaka subadditivity](paper.pdf) + +**Author:** OpenAI + +**Date:** September 26, 2026 + +## Citation + +```bibtex +@misc{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026, + author = {{OpenAI}}, + title = {{Orbifold and logarithmic Iitaka subadditivity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/figures/two-maps.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/figures/two-maps.tex new file mode 100644 index 0000000000000000000000000000000000000000..3f6a0c372fe70c686eb01ca6cc3dc0fffec78434 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/figures/two-maps.tex @@ -0,0 +1,13 @@ +\begin{figure}[ht] +\centering +\begin{tikzcd}[column sep=large,row sep=large] + X \arrow[r,"g"] & W \arrow[r,"h"] \arrow[d,"p"'] & Y\\ + & S & +\end{tikzcd} +\caption{The relative Iitaka reduction factors the original fibration +through $W$. The map $h$ retains the original base and has +$k$-dimensional general fibers. The independent map $p$ expresses +$M$ as the pullback of a nef line $P$ on a projective base where +$K_S+aP$ is big. No map between $Y$ and $S$ is asserted.} +\label{fig:iitaka-period} +\end{figure} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/main.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..99cdd9f5360df1a8b9dde6017344590dbb0f0b27 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/main.tex @@ -0,0 +1,87 @@ +\documentclass[11pt]{amsart} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools} +\usepackage[margin=1.08in]{geometry} +\usepackage[expansion=false]{microtype} +\newcommand{\DoNotLoadEpstopdf}{} +\usepackage{enumitem,booktabs,longtable,array,needspace,tikz-cd} +\usepackage[hidelinks,pdfusetitle]{hyperref} +\usepackage[capitalise,noabbrev,nameinlink]{cleveref} +\hypersetup{pdftitle={Orbifold and logarithmic Iitaka subadditivity},pdfauthor={OpenAI},bookmarksdepth=2} +\newcommand{\C}{\mathbf C} +\newcommand{\Q}{\mathbf Q} +\newcommand{\R}{\mathbf R} +\newcommand{\Z}{\mathbf Z} +\newcommand{\N}{\mathbf N} +\newcommand{\cO}{\mathcal O} +\newcommand{\cV}{\mathcal V} +\newcommand{\cA}{\mathcal A} +\newcommand{\ii}{\sqrt{-1}} +\newcommand{\ddc}{dd^c} +\newcommand{\kbar}{\bar\kappa} +\newcommand{\red}{\mathrm{red}} +\newcommand{\hor}{\mathrm{hor}} +\DeclareMathOperator{\Gr}{Gr} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Supp}{Supp} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\trdeg}{trdeg} +\DeclareMathOperator{\Frac}{Frac} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Bir}{Bir} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\lct}{lct} +\DeclareMathOperator{\mult}{mult} +\DeclareMathOperator{\id}{id} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{construction}[theorem]{Construction} +\newtheorem{convention}[theorem]{Convention} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{example}[theorem]{Example} +\numberwithin{equation}{section} +\setcounter{tocdepth}{1} +\allowdisplaybreaks[2] +\emergencystretch=1em +\title[Orbifold and logarithmic Iitaka subadditivity]{Orbifold and logarithmic Iitaka subadditivity} +\author{OpenAI} +\date{September 26, 2026} +\begin{document} +\begin{abstract} +We prove Campana's orbifold Iitaka subadditivity conjecture for rational +simple normal crossing boundaries on compact manifolds in Fujiki class +$\mathcal C$, including coefficient one. Ordinary and logarithmic +subadditivity follow. +\end{abstract} +\maketitle +\tableofcontents +\input{sections/introduction} +\input{sections/models} +\input{sections/root-comparison} +\input{sections/adjoint-addition} +\input{sections/period-positivity} +\input{sections/general-type-addition} +\input{sections/applications} +\input{sections/reduced-root} +\appendix +\input{sections/normalization} +\raggedbottom +\input{references} +\end{document} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.bib b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..06a63da71b49f48060a279c623a1cfef63b762f6 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.bib @@ -0,0 +1,741 @@ +@misc{AdiprasitoLiuTemkinSemistable, + archiveprefix = {arXiv}, + author = {Adiprasito, Karim and 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{2023}, + pages = {163--228}, + doi = {10.1007/s00222-022-01166-1}, + note = {Theorem locators refer to arXiv:1811.12230v3, 23 October 2022}, + url = {https://arxiv.org/abs/1811.12230v3} +} + +@misc{BFMT, + archiveprefix = {arXiv}, + author = {Bakker, Benjamin and Filipazzi, Stefano and Mauri, Mirko and Tsimerman, Jacob}, + eprint = {2508.19215}, + note = {\href{https://arxiv.org/abs/2508.19215v2}{arXiv:2508.19215v2}, 18 December 2025}, + primaryclass = {math.AG}, + title = {{Baily--Borel} compactifications of period images and the b-semiampleness conjecture}, + url = {https://arxiv.org/abs/2508.19215v2}, + year = {2025} +} + +@article{BakkerTsimermanAxSchanuel, + author = {Bakker, Benjamin and Tsimerman, Jacob}, + doi = {10.1007/s00222-019-00863-8}, + journal = {Inventiones Mathematicae}, + note = {The cited locators refer to \href{https://arxiv.org/abs/1712.05088v1}{arXiv:1712.05088v1}}, + number = {1}, + pages = {77--94}, + title = {The {Ax--Schanuel} conjecture for variations of {Hodge} structures}, + url = {https://arxiv.org/abs/1712.05088v1}, + volume = {217}, + year = {2019} +} + +@article{BierstoneMilmanFunctorial, + author = {Bierstone, Edward and Milman, Pierre D.}, + doi = {10.2977/PRIMS/1210167338}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + number = {2}, + pages = {609--639}, + title = {Functoriality in resolution of singularities}, + url = {https://ems.press/journals/prims/articles/2154}, + volume = {44}, + year = {2008} +} + +@article{BirkarIitakaSix, + author = {Birkar, Caucher}, + doi = {10.1112/S0010437X09004187}, + journal = {Compositio Mathematica}, + note = {arXiv:0806.4413, first posted 27 June 2008}, + number = {6}, + pages = {1442--1446}, + title = {The {Iitaka} conjecture {$C_{n,m}$} in dimension six}, + url = {https://arxiv.org/abs/0806.4413}, + volume = {145}, + year = {2009} +} + +@article{BCHM, + author = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James}, + doi = {10.1090/S0894-0347-09-00649-3}, + journal = {Journal of the American Mathematical Society}, + note = {The cited locators refer to the 81-page \href{https://arxiv.org/abs/math/0610203v2}{arXiv:math/0610203v2}}, + number = {2}, + pages = {405--468}, + title = {Existence of minimal models for varieties of log general type}, + url = {https://arxiv.org/abs/math/0610203v2}, + volume = {23}, + year = {2010} +} + +@article{BoucksomVolume, + author = {Boucksom, S{\'e}bastien}, + doi = {10.1142/S0129167X02001575}, + journal = {International Journal of Mathematics}, + note = {Theorem locators refer to arXiv:math/0201031v1, 5 January 2002}, + number = {10}, + pages = {1043--1063}, + title = {On the volume of a line bundle}, + url = {https://arxiv.org/abs/math/0201031v1}, + volume = {13}, + year = {2002} +} + +@article{BEGZ, + author = {Boucksom, S{\'e}bastien and Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed}, + doi = {10.1007/s11511-010-0054-7}, + journal = {Acta Mathematica}, + pages = {199--262}, + title = {{Monge--Amp{\`e}re} equations in big cohomology classes}, + url = {https://www.math.univ-toulouse.fr/~guedj/fichierspdf/BEGZ10.pdf}, + volume = {205}, + year = {2010} +} + +@article{BrunebarbeCadorel, + author = {Brunebarbe, Yohan and Cadorel, Beno{\^i}t}, + doi = {10.1093/imrn/rny054}, + journal = {International Mathematics Research Notices}, + note = {The cited locators refer to \href{https://arxiv.org/abs/1707.01327v1}{arXiv:1707.01327v1}}, + number = {6}, + pages = {1601--1609}, + title = {Hyperbolicity of varieties supporting a variation of {Hodge} structure}, + url = {https://arxiv.org/abs/1707.01327v1}, + volume = {2020}, + year = {2020} +} + +@article{Campana2004, + author = {Campana, Fr{\'e}d{\'e}ric}, + doi = {10.5802/aif.2027}, + journal = {Annales de l'Institut Fourier}, + number = {3}, + pages = {499--630}, + title = {Orbifolds, special varieties and classification theory}, + url = {https://www.numdam.org/item/10.5802/aif.2027/}, + volume = {54}, + year = {2004} +} + +@article{CampanaOrbifolds, + author = {Campana, Fr{\'e}d{\'e}ric}, + doi = {10.1017/S1474748010000101}, + journal = {Journal of the Institute of Mathematics of Jussieu}, + note = {The cited locators refer to \href{https://arxiv.org/abs/0705.0737v8}{arXiv:0705.0737v8}}, + number = {4}, + pages = {809--934}, + title = {Orbifoldes g{\'e}om{\'e}triques sp{\'e}ciales et classification bim{\'e}romorphe des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes compactes}, + url = {https://arxiv.org/abs/0705.0737v8}, + volume = {10}, + year = {2011} +} + +@article{CaoSurfaces, + author = {Cao, Junyan}, + doi = {10.14231/AG-2018-020}, + journal = {Algebraic Geometry}, + number = {6}, + pages = {728--741}, + title = {{Kodaira} dimension of algebraic fiber spaces over surfaces}, + url = {https://content.algebraicgeometry.nl/2018-6/2018-6-020.pdf}, + volume = {5}, + year = {2018} +} + +@article{CaoPaunAbelian, + author = {Cao, Junyan and P{\u a}un, Mihai}, + doi = {10.1007/s00222-016-0672-6}, + journal = {Inventiones Mathematicae}, + note = {Theorem locators refer to arXiv:1504.01095v3, submitted 3 August 2016}, + number = {1}, + pages = {345--387}, + title = {{Kodaira} dimension of algebraic fiber spaces over abelian varieties}, + url = {https://arxiv.org/abs/1504.01095v3}, + volume = {207}, + year = {2017} +} + +@article{CKS, + author = {Cattani, Eduardo and Kaplan, Aroldo and Schmid, Wilfried}, + doi = {10.2307/1971333}, + journal = {Annals of Mathematics}, + number = {3}, + pages = {457--535}, + series = {2}, + title = {Degeneration of {Hodge} structures}, + url = {https://annals.math.princeton.edu/1986/123-3/p02}, + volume = {123}, + year = {1986} +} + +@misc{ChenLimits, + archiveprefix = {arXiv}, + author = {Chen, Qianyu}, + eprint = {2103.03983}, + note = {arXiv:2103.03983; the cited Lemma 8.1 is in the \href{https://qymath.github.io/Steenbrink.pdf}{61-page author manuscript}}, + primaryclass = {math.AG}, + title = {Limits of {Hodge} structures via holonomic {$D$}-modules}, + url = {https://qymath.github.io/Steenbrink.pdf}, + year = {2021} +} + +@article{DeligneHodgeII, + author = {Deligne, Pierre}, + title = {Th{\'e}orie de {Hodge}. {II}}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {40}, + year = {1971}, + pages = {5--57}, + doi = {10.1007/BF02684692}, + url = {https://www.numdam.org/item/PMIHES_1971__40__5_0.pdf} +} + +@article{DemaillyRegularization, + author = {Demailly, Jean-Pierre}, + journal = {Journal of Algebraic Geometry}, + note = {The cited locator refers to the \href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}{author manuscript}}, + number = {3}, + pages = {361--409}, + title = {Regularization of closed positive currents and intersection theory}, + url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}, + volume = {1}, + year = {1992} +} + +@article{DemaillyPaun2004, + author = {Demailly, 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\href{https://arxiv.org/abs/2404.05124v3}{arXiv:2404.05124v3}, 12 October 2024}, + pages = {7667--7688}, + title = {Semistable reduction for complex analytic spaces}, + url = {https://arxiv.org/abs/2404.05124v3}, + volume = {378}, + year = {2025} +} + +@article{FujikiDouady, + author = {Fujiki, Akira}, + doi = {10.2977/PRIMS/1195189279}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + number = {1}, + pages = {1--52}, + title = {Closedness of the {Douady} spaces of compact {K{\"a}hler} spaces}, + url = {https://ems.press/content/serial-article-files/41937}, + volume = {14}, + year = {1978} +} + +@incollection{FujinoWeakPositivity, + author = {Fujino, Osamu}, + booktitle = {Algebraic Varieties and Automorphism Groups}, + doi = {10.2969/aspm/07510073}, + note = {The cited locators refer to \href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{author version 0.54, 30 June 2015}}, + pages = {73--118}, + publisher = {Mathematical Society of Japan}, + series = {Advanced Studies in Pure Mathematics}, + title = {Notes on the weak positivity theorems}, + url = {https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}, + volume = {75}, + year = {2017} +} + +@article{FujinoAnalyticAdjunction, + author = {Fujino, Osamu}, + doi = {10.3792/PJAA.100.002}, + journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences}, + number = {2}, + pages = {7--11}, + title = {Log canonical inversion of adjunction}, + url = {https://www.math.kyoto-u.ac.jp/~fujino/on-inversion-of-adjunction-pja-final.pdf}, + volume = {100}, + year = {2024} +} + +@article{FujinoFujisawa, + author = {Fujino, Osamu and Fujisawa, Taro}, + doi = {10.4171/PRIMS/145}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + note = {\href{https://doi.org/10.4171/PRIMS/145}{doi:10.4171/PRIMS/145}}, + number = {4}, + pages = {589--661}, + title = {Variations of mixed {Hodge} structure and semipositivity theorems}, + url = {https://ems.press/content/serial-article-files/41240}, + volume = {50}, + year = {2014} +} + +@article{FujinoFujisawaVMHS, + author = {Fujino, Osamu and Fujisawa, Taro}, + journal = {International Journal of Mathematics}, + note = {Published article: 43 pp.; theorem locators refer to the 36-page \href{https://www.math.kyoto-u.ac.jp/~fujino/vmhs-applications8.pdf}{author version 0.55, 11 March 2025}}, + number = {9}, + pages = {2550020}, + title = {Variation of mixed {Hodge} structure and its applications}, + url = {https://www.math.kyoto-u.ac.jp/~fujino/vmhs-applications8.pdf}, + volume = {36}, + year = {2025} +} + +@article{FujinoGongyo, + author = {Fujino, Osamu and Gongyo, Yoshinori}, + doi = {10.5802/aif.2894}, + journal = {Annales de l'Institut Fourier}, + note = {\href{https://doi.org/10.5802/aif.2894}{doi:10.5802/aif.2894}}, + number = {4}, + pages = {1721--1735}, + title = {On the moduli b-divisors of lc-trivial fibrations}, + url = 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Math. IH{\'E}S 16 (1963), 35--36}, + pages = {5--64}, + title = {Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen}, + url = {https://www.numdam.org/item/PMIHES_1960__5__5_0/}, + volume = {5}, + year = {1960} +} + +@article{GrauertDirectImagesCorr, + author = {Grauert, Hans}, + doi = {10.1007/BF02684291}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + pages = {35--36}, + title = {Berichtigung: Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen}, + url = {https://www.numdam.org/articles/10.1007/BF02684291/}, + volume = {16}, + year = {1963} +} + +@article{GriffithsPeriodsIII, + author = {Griffiths, Phillip A.}, + doi = {10.1007/BF02684654}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + pages = {125--180}, + title = {Periods of integrals on algebraic manifolds, {III}: Some global differential-geometric properties of the period mapping}, + url = {https://www.numdam.org/item/PMIHES_1970__38__125_0/}, + volume = {38}, + year = {1970} +} + +@article{GuedjZeriahiEnergy, + author = {Guedj, Vincent and Zeriahi, Ahmed}, + doi = {10.1016/j.jfa.2007.04.018}, + journal = {Journal of Functional Analysis}, + note = {The cited locators refer to \href{https://arxiv.org/abs/math/0612630v2}{arXiv:math/0612630v2}}, + number = {2}, + pages = {442--482}, + title = {The weighted {Monge--Amp{\`e}re} energy of quasiplurisubharmonic functions}, + url = {https://arxiv.org/abs/math/0612630v2}, + volume = {250}, + year = {2007} +} + +@incollection{HaconPopaSchnellAbelian, + address = {Providence, RI}, + author = {Hacon, Christopher D. and Popa, Mihnea and Schnell, Christian}, + booktitle = {Local and Global Methods in Algebraic Geometry}, + doi = {10.1090/conm/712/14346}, + note = {Theorem locators refer to the author manuscript dated 9 May 2017}, + pages = {143--195}, + publisher = {American Mathematical Society}, + series = {Contemporary Mathematics}, + title = {Algebraic fiber spaces over abelian varieties: around a recent theorem by {Cao} and {P{\u a}un}}, + url = {https://people.math.harvard.edu/~mpopa/papers/HPS.pdf}, + volume = {712}, + year = {2018} +} + +@article{Hashizume, + author = {Hashizume, Kenta}, + doi = {10.3792/pjaa.96.017}, + journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences}, + note = {Theorem numbering is cited from arXiv:1902.10923v2}, + number = {10}, + pages = {87--92}, + title = {Log {Iitaka} conjecture for abundant log canonical fibrations}, + url = {https://arxiv.org/abs/1902.10923v2}, + volume = {96}, + year = {2020} +} + +@article{HironakaFlattening, + author = {Hironaka, Heisuke}, + doi = {10.2307/2373721}, + journal = {American Journal of Mathematics}, + number = {2}, + pages = {503--547}, + title = {Flattening theorem in complex-analytic geometry}, + url = {https://doi.org/10.2307/2373721}, + volume = {97}, + year = {1975} +} + +@article{IitakaOriginal, + author = {Iitaka, Shigeru}, + doi = {10.2969/jmsj/02320356}, + journal = {Journal of the Mathematical Society of Japan}, + number = {2}, + pages = {356--373}, + title = {On {$D$}-dimensions of algebraic varieties}, + url = {https://www.jstage.jst.go.jp/article/jmath1948/23/2/23_2_356/_article}, + volume = {23}, + year = {1971} +} + +@incollection{IitakaLog, + author = {Iitaka, Shigeru}, + booktitle = {Complex Analysis and Algebraic Geometry: A Collection of Papers Dedicated to {K. Kodaira}}, + doi = {10.1017/CBO9780511569197.014}, + editor = {Baily, Jr., Walter L. and Shioda, Tetsuji}, + note = {\href{https://doi.org/10.1017/CBO9780511569197.014}{doi:10.1017/CBO9780511569197.014}}, + publisher = {Iwanami Shoten and Cambridge University Press}, + title = {On logarithmic {Kodaira} dimension of algebraic varieties}, + url = {https://doi.org/10.1017/CBO9780511569197.014}, + year = {1977} +} + +@article{IitakaClassification, + author = {Iitaka, Shigeru}, + doi = {10.3792/pja/1195517881}, + journal = {Proceedings of the Japan Academy}, + number = {3}, + pages = {103--105}, + title = {Classification of Algebraic Varieties}, + url = {https://www.jstage.jst.go.jp/article/pjab1945/53/3/53_3_103/_pdf}, + volume = {53}, + year = {1977} +} + +@article{KawamataCurves, + author = {Kawamata, Yujiro}, + doi = {10.1007/BF01404756}, + journal = {Inventiones Mathematicae}, + number = {1}, + pages = {57--71}, + title = {{Kodaira} dimension of algebraic fiber spaces over curves}, + url = {https://link.springer.com/article/10.1007/BF01404756}, + volume = {66}, + year = {1982} +} + +@article{KawamataKodaira, + author = {Kawamata, Yujiro}, + doi = {10.1515/crll.1985.363.1}, + journal = {Journal f{\"u}r die reine und angewandte Mathematik}, + pages = {1--46}, + title = {Minimal models and the {Kodaira} dimension of algebraic fiber spaces}, + url = {https://doi.org/10.1515/crll.1985.363.1}, + volume = {363}, + year = {1985} +} + +@incollection{Kollar87, + address = {Tokyo and Amsterdam}, + author = {Koll{\'a}r, J{\'a}nos}, + booktitle = {Algebraic Geometry, Sendai, 1985}, + doi = {10.2969/aspm/01010361}, + pages = {361--398}, + publisher = {Kinokuniya and North-Holland}, + series = {Advanced Studies in Pure Mathematics}, + title = {Subadditivity of the {Kodaira} dimension: fibers of general type}, + url = {https://doi.org/10.2969/aspm/01010361}, + volume = {10}, + year = {1987} +} + +@article{KovacsPatakfalvi, + author = {Kov{\'a}cs, S{\'a}ndor J. and Patakfalvi, Zsolt}, + doi = {10.1090/jams/871}, + journal = {Journal of the American Mathematical Society}, + note = {The cited locators refer to the \href{https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf}{62-page author manuscript}}, + number = {4}, + pages = {959--1021}, + title = {Projectivity of the moduli space of stable log-varieties and subadditivity of log-{Kodaira} dimension}, + url = {https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf}, + volume = {30}, + year = {2017} +} + +@misc{MaeharaIitaka, + archiveprefix = {arXiv}, + author = {Maehara, Kazuhisa}, + eprint = {0910.0604}, + howpublished = {arXiv preprint}, + note = {Version 3, 16 May 2010; first posted 4 October 2009}, + title = {{Iitaka--Viehweg} conjectures {$C$} and {$C^{++}$}}, + url = {https://arxiv.org/abs/0910.0604v3}, + year = {2010} +} + +@book{MilneAlgebraicGroups, + author = {Milne, James S.}, + doi = {10.1017/9781316711736}, + note = {Cited from the corrected 2022 reprint}, + publisher = {Cambridge University Press}, + series = {Cambridge Studies in Advanced Mathematics}, + title = {Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field}, + url = {https://www.jmilne.org/math/Books/iAG2022.pdf}, + volume = {170}, + year = {2017} +} + +@article{NamikawaProjectivity, + archiveprefix = {arXiv}, + author = {Namikawa, Yoshinori}, + eprint = {math/0101019}, + journal = {International Journal of Mathematics}, + note = {Theorem locators refer to arXiv:math/0101019v6, 12 November 2001}, + number = {2}, + pages = {125--135}, + title = {Projectivity criterion of {Moishezon} spaces and density of projective symplectic varieties}, + url = {https://arxiv.org/abs/math/0101019v6}, + volume = {13}, + year = {2002} +} + +@misc{IitakaVariationCompanion, + author = {{OpenAI}}, + title = {{Logarithmic Kodaira dimension and whole-fiber variation}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf}{OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026}}, + year = {2026} +} + +@misc{IitakaProjectiveCompanion, + author = {{OpenAI}}, + title = {{Projective Hodge lines and ordinary Iitaka subadditivity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf}{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026}}, + year = {2026} +} + +@misc{IitakaAdditivityCompanion, + author = {{OpenAI}}, + title = {{The reverse logarithmic Kodaira inequality and additivity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf}{OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026}}, + year = {2026} +} + +@misc{OpenAILogAbundance2026, + author = {{OpenAI}}, + title = {{Log abundance in characteristic zero}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}}, + year = {2026} +} + +@article{PopaSchnellKahler, + author = {Pareschi, Giuseppe and Popa, Mihnea and Schnell, Christian}, + doi = {10.2140/gt.2017.21.2419}, + journal = {Geometry \& Topology}, + pages = {2419--2460}, + title = {{Hodge} modules on complex tori and generic vanishing for compact {K{\"a}hler} manifolds}, + url = {https://msp.org/gt/2017/21-4/gt-v21-n4-p12-s.pdf}, + volume = {21}, + year = {2017} +} + +@article{PatakfalviFibered, + author = {Patakfalvi, Zsolt}, + doi = {10.1090/tran/6386}, + journal = {Transactions of the American Mathematical Society}, + number = {3}, + pages = {1837--1869}, + title = {Fibered stable varieties}, + url = {https://arxiv.org/abs/1208.1787v4}, + volume = {368}, + year = {2016} +} + +@article{Schmid, + author = {Schmid, Wilfried}, + doi = {10.1007/BF01389674}, + journal = {Inventiones Mathematicae}, + number = {3--4}, + pages = {211--319}, + title = {Variation of {Hodge} structure: The singularities of the period mapping}, + url = {https://doi.org/10.1007/BF01389674}, + volume = {22}, + year = {1973} +} + +@article{SommesePeriodMapping, + author = {Sommese, Andrew John}, + title = {On the rationality of the period mapping}, + journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, + series = {4}, + volume = {5}, + number = {4}, + year = {1978}, + pages = {683--717}, + url = {https://www.numdam.org/item/ASNSP_1978_4_5_4_683_0/} +} + +@misc{TsujiDirectImages, + archiveprefix = {arXiv}, + author = {Tsuji, Hajime}, + eprint = {1012.0884}, + howpublished = {arXiv preprint}, + note = {Version 1, 4 December 2010}, + title = {Global generation of the direct images of relative pluricanonical systems}, + url = {https://arxiv.org/abs/1012.0884v1}, + year = {2010} +} + +@book{UenoClassification, + address = {Berlin}, + author = {Ueno, Kenji}, + doi = {10.1007/BFb0070570}, + publisher = {Springer}, + series = {Lecture Notes in Mathematics}, + title = {Classification Theory of Algebraic Varieties and Compact Complex Spaces}, + url = {https://doi.org/10.1007/BFb0070570}, + volume = {439}, + year = {1975} +} + +@incollection{ViehwegAdditivity, + address = {Tokyo and Amsterdam}, + author = {Viehweg, Eckart}, + booktitle = {Algebraic Varieties and Analytic Varieties (Tokyo, 1981)}, + doi = {10.2969/aspm/00110329}, + pages = {329--353}, + publisher = {Kinokuniya and North-Holland}, + series = {Advanced Studies in Pure Mathematics}, + title = {Weak positivity and the additivity of the {Kodaira} dimension for certain fibre spaces}, + url = {https://esaga.uni-due.de/eckart.viehweg/ecpubl/}, + volume = {1}, + year = {1983} +} + +@misc{VilladsenHodgeKahler, + archiveprefix = {arXiv}, + author = {Villadsen, Mads Bach}, + eprint = {2401.09544}, + note = {\href{https://arxiv.org/abs/2401.09544v1}{arXiv:2401.09544v1}, 17 January 2024}, + primaryclass = {math.AG}, + title = {{Hodge} modules and {K{\"a}hler} morphisms}, + url = {https://arxiv.org/abs/2401.09544v1}, + year = {2024} +} + +@article{WangKahlerIitaka, + author = {Wang, Juanyong}, + doi = {10.5802/afst.1690}, + journal = {Annales de la Facult{\'e} des Sciences de Toulouse. Math{\'e}matiques}, + number = {4}, + pages = {813--897}, + series = {6}, + title = {On the {Iitaka} Conjecture {$C_{n,m}$} for {K{\"a}hler} Fibre Spaces}, + url = {https://afst.centre-mersenne.org/item/10.5802/afst.1690.pdf}, + volume = {30}, + year = {2021} +} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..0a87e9fd29bb503168ee836ad5b5106d7ff83b6a --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.tex @@ -0,0 +1,406 @@ +\begin{thebibliography}{99} + +\bibitem{AdiprasitoLiuTemkinSemistable} +Karim Adiprasito, Gaku Liu and Michael Temkin. +\emph{\href{https://arxiv.org/abs/1810.03131v2}{Semistable reduction in characteristic {0}}}. +Preprint, 2019. +\href{https://arxiv.org/abs/1810.03131v2}{arXiv:1810.03131v2}, 17 June 2019; 22-page full preprint. + +\bibitem{AmbroBoundary} +Florin Ambro. +\emph{\href{https://arxiv.org/abs/math/0210271v1}{{Shokurov}'s boundary property}}. +Journal of Differential Geometry \textbf{67} (2004), no.~2, 229--255. +Theorem numbering is cited from arXiv:math/0210271v1. + +\bibitem{AmbroModuli} +Florin Ambro. +\emph{\href{https://doi.org/10.1112/S0010437X04001071}{The moduli b-divisor of an lc-trivial fibration}}. +Compositio Mathematica \textbf{141} (2005), no.~2, 385--403. +\href{https://doi.org/10.1112/S0010437X04001071}{doi:\nolinkurl{10.1112/S0010437X04001071}}. + +\bibitem{AndreMonodromy} +Yves Andr{\'e}. +\emph{\href{https://www.numdam.org/item/CM_1992__82_1_1_0/}{{Mumford--Tate} groups of mixed {Hodge} structures and the theorem of the fixed part}}. +Compositio Mathematica \textbf{82} (1992), no.~1, 1--24. + +\bibitem{BakkerMatsushita} +Benjamin Bakker. +\emph{\href{https://benjamin-bakker.github.io/matsushita.pdf}{A short proof of a conjecture of {Matsushita}}}. +Advances in Mathematics \textbf{482} (2025), 110554. +\href{https://doi.org/10.1016/j.aim.2025.110554}{doi:\nolinkurl{10.1016/j.aim.2025.110554}}. +Part A. + +\bibitem{BakkerBrunebarbeTsimerman} +Benjamin Bakker, Yohan Brunebarbe and Jacob Tsimerman. +\emph{\href{https://arxiv.org/abs/1811.12230v3}{o-minimal {GAGA} and a conjecture of {Griffiths}}}. +Inventiones Mathematicae \textbf{232} (2023), no.~1, 163--228. +\href{https://doi.org/10.1007/s00222-022-01166-1}{doi:\nolinkurl{10.1007/s00222-022-01166-1}}. +Theorem locators refer to arXiv:1811.12230v3, 23 October 2022. + +\bibitem{BFMT} +Benjamin Bakker, Stefano Filipazzi, Mirko Mauri and Jacob Tsimerman. +\emph{\href{https://arxiv.org/abs/2508.19215v2}{{Baily--Borel} compactifications of period images and the b-semiampleness conjecture}}. +Preprint, 2025. +\href{https://arxiv.org/abs/2508.19215v2}{arXiv:2508.19215v2}, 18 December 2025. + +\bibitem{BakkerTsimermanAxSchanuel} +Benjamin Bakker and Jacob Tsimerman. +\emph{\href{https://arxiv.org/abs/1712.05088v1}{The {Ax--Schanuel} conjecture for variations of {Hodge} structures}}. +Inventiones Mathematicae \textbf{217} (2019), no.~1, 77--94. +\href{https://doi.org/10.1007/s00222-019-00863-8}{doi:\nolinkurl{10.1007/s00222-019-00863-8}}. +The cited locators refer to \href{https://arxiv.org/abs/1712.05088v1}{arXiv:1712.05088v1}. + +\bibitem{BierstoneMilmanFunctorial} +Edward Bierstone and Pierre D. Milman. +\emph{\href{https://ems.press/journals/prims/articles/2154}{Functoriality in resolution of singularities}}. +Publications of the Research Institute for Mathematical Sciences \textbf{44} (2008), no.~2, 609--639. +\href{https://doi.org/10.2977/PRIMS/1210167338}{doi:\nolinkurl{10.2977/PRIMS/1210167338}}. + +\bibitem{BirkarIitakaSix} +Caucher Birkar. +\emph{\href{https://arxiv.org/abs/0806.4413}{The {Iitaka} conjecture {$C_{n,m}$} in dimension six}}. +Compositio Mathematica \textbf{145} (2009), no.~6, 1442--1446. +\href{https://doi.org/10.1112/S0010437X09004187}{doi:\nolinkurl{10.1112/S0010437X09004187}}. +arXiv:0806.4413, first posted 27 June 2008. + +\bibitem{BCHM} +Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan. +\emph{\href{https://arxiv.org/abs/math/0610203v2}{Existence of minimal models for varieties of log general type}}. +Journal of the American Mathematical Society \textbf{23} (2010), no.~2, 405--468. +\href{https://doi.org/10.1090/S0894-0347-09-00649-3}{doi:\nolinkurl{10.1090/S0894-0347-09-00649-3}}. +The cited locators refer to the 81-page \href{https://arxiv.org/abs/math/0610203v2}{arXiv:math/0610203v2}. + +\bibitem{BoucksomVolume} +S{\'e}bastien Boucksom. +\emph{\href{https://arxiv.org/abs/math/0201031v1}{On the volume of a line bundle}}. +International Journal of Mathematics \textbf{13} (2002), no.~10, 1043--1063. +\href{https://doi.org/10.1142/S0129167X02001575}{doi:\nolinkurl{10.1142/S0129167X02001575}}. +Theorem locators refer to arXiv:math/0201031v1, 5 January 2002. + +\bibitem{BEGZ} +S{\'e}bastien Boucksom, Philippe Eyssidieux, Vincent Guedj and Ahmed Zeriahi. +\emph{\href{https://www.math.univ-toulouse.fr/~guedj/fichierspdf/BEGZ10.pdf}{{Monge--Amp{\`e}re} equations in big cohomology classes}}. +Acta Mathematica \textbf{205} (2010), 199--262. +\href{https://doi.org/10.1007/s11511-010-0054-7}{doi:\nolinkurl{10.1007/s11511-010-0054-7}}. + +\bibitem{BrunebarbeCadorel} +Yohan Brunebarbe and Beno{\^i}t Cadorel. +\emph{\href{https://arxiv.org/abs/1707.01327v1}{Hyperbolicity of varieties supporting a variation of {Hodge} structure}}. +International Mathematics Research Notices \textbf{2020} (2020), no.~6, 1601--1609. +\href{https://doi.org/10.1093/imrn/rny054}{doi:\nolinkurl{10.1093/imrn/rny054}}. +The cited locators refer to \href{https://arxiv.org/abs/1707.01327v1}{arXiv:1707.01327v1}. + +\bibitem{Campana2004} +Fr{\'e}d{\'e}ric Campana. +\emph{\href{https://www.numdam.org/item/10.5802/aif.2027/}{Orbifolds, special varieties and classification theory}}. +Annales de l'Institut Fourier \textbf{54} (2004), no.~3, 499--630. +\href{https://doi.org/10.5802/aif.2027}{doi:\nolinkurl{10.5802/aif.2027}}. + +\bibitem{CampanaOrbifolds} +Fr{\'e}d{\'e}ric Campana. +\emph{\href{https://arxiv.org/abs/0705.0737v8}{Orbifoldes g{\'e}om{\'e}triques sp{\'e}ciales et classification bim{\'e}romorphe des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes compactes}}. +Journal of the Institute of Mathematics of Jussieu \textbf{10} (2011), no.~4, 809--934. +\href{https://doi.org/10.1017/S1474748010000101}{doi:\nolinkurl{10.1017/S1474748010000101}}. +The cited locators refer to \href{https://arxiv.org/abs/0705.0737v8}{arXiv:0705.0737v8}. + +\bibitem{CaoSurfaces} +Junyan Cao. +\emph{\href{https://content.algebraicgeometry.nl/2018-6/2018-6-020.pdf}{{Kodaira} dimension of algebraic fiber spaces over surfaces}}. +Algebraic Geometry \textbf{5} (2018), no.~6, 728--741. +\href{https://doi.org/10.14231/AG-2018-020}{doi:\nolinkurl{10.14231/AG-2018-020}}. + +\bibitem{CaoPaunAbelian} +Junyan Cao and Mihai P{\u a}un. +\emph{\href{https://arxiv.org/abs/1504.01095v3}{{Kodaira} dimension of algebraic fiber spaces over abelian varieties}}. +Inventiones Mathematicae \textbf{207} (2017), no.~1, 345--387. +\href{https://doi.org/10.1007/s00222-016-0672-6}{doi:\nolinkurl{10.1007/s00222-016-0672-6}}. +Theorem locators refer to arXiv:1504.01095v3, submitted 3 August 2016. + +\bibitem{CKS} +Eduardo Cattani, Aroldo Kaplan and Wilfried Schmid. +\emph{\href{https://annals.math.princeton.edu/1986/123-3/p02}{Degeneration of {Hodge} structures}}. +Annals of Mathematics (2) \textbf{123} (1986), no.~3, 457--535. +\href{https://doi.org/10.2307/1971333}{doi:\nolinkurl{10.2307/1971333}}. + +\bibitem{ChenLimits} +Qianyu Chen. +\emph{\href{https://qymath.github.io/Steenbrink.pdf}{Limits of {Hodge} structures via holonomic {$D$}-modules}}. +Preprint, 2021. +arXiv:2103.03983; the cited Lemma 8.1 is in the \href{https://qymath.github.io/Steenbrink.pdf}{61-page author manuscript}. + +\bibitem{DeligneHodgeII} +Pierre Deligne. +\emph{Th\'eorie de Hodge. II}. +Publications Math\'ematiques de l'IH\'ES \textbf{40} (1971), 5--57. +\href{https://doi.org/10.1007/BF02684692}{doi:\nolinkurl{10.1007/BF02684692}}. + +\bibitem{DemaillyRegularization} +Jean-Pierre Demailly. +\emph{\href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}{Regularization of closed positive currents and intersection theory}}. +Journal of Algebraic Geometry \textbf{1} (1992), no.~3, 361--409. +The cited locator refers to the \href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}{author manuscript}. + +\bibitem{DemaillyPaun2004} +Jean-Pierre Demailly and Mihai P{\u{a}}un. +\emph{\href{https://annals.math.princeton.edu/2004/159-3/p05}{Numerical characterization of the {K{\"a}hler} cone of a compact {K{\"a}hler} manifold}}. +Annals of Mathematics (2) \textbf{159} (2004), no.~3, 1247--1274. +\href{https://doi.org/10.4007/annals.2004.159.1247}{doi:\nolinkurl{10.4007/annals.2004.159.1247}}. + +\bibitem{Douady} +Adrien Douady. +\emph{\href{https://www.numdam.org/item/AIF_1966__16_1_1_0/}{Le probl{\`e}me des modules pour les sous-espaces analytiques compacts d'un espace analytique donn{\'e}}}. +Annales de l'Institut Fourier \textbf{16} (1966), no.~1, 1--95. +\href{https://doi.org/10.5802/aif.226}{doi:\nolinkurl{10.5802/aif.226}}. + +\bibitem{EnokizonoHashizumeSemistable} +Makoto Enokizono and Kenta Hashizume. +\emph{\href{https://arxiv.org/abs/2404.05124v3}{Semistable reduction for complex analytic spaces}}. +Transactions of the American Mathematical Society \textbf{378} (2025), 7667--7688. +\href{https://doi.org/10.1090/tran/9365}{doi:\nolinkurl{10.1090/tran/9365}}. +The cited locators refer to \href{https://arxiv.org/abs/2404.05124v3}{arXiv:2404.05124v3}, 12 October 2024. + +\bibitem{FujikiDouady} +Akira Fujiki. +\emph{\href{https://ems.press/content/serial-article-files/41937}{Closedness of the {Douady} spaces of compact {K{\"a}hler} spaces}}. +Publications of the Research Institute for Mathematical Sciences \textbf{14} (1978), no.~1, 1--52. +\href{https://doi.org/10.2977/PRIMS/1195189279}{doi:\nolinkurl{10.2977/PRIMS/1195189279}}. + +\bibitem{FujinoWeakPositivity} +Osamu Fujino. +\emph{\href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{Notes on the weak positivity theorems}}. +In \emph{Algebraic Varieties and Automorphism Groups}. +Advanced Studies in Pure Mathematics 75. +Mathematical Society of Japan, 2017, pp.~73--118. +\href{https://doi.org/10.2969/aspm/07510073}{doi:\nolinkurl{10.2969/aspm/07510073}}. +The cited locators refer to \href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{author version 0.54, 30 June 2015}. + +\bibitem{FujinoAnalyticAdjunction} +Osamu Fujino. +\emph{\href{https://www.math.kyoto-u.ac.jp/~fujino/on-inversion-of-adjunction-pja-final.pdf}{Log canonical inversion of adjunction}}. +Proceedings of the Japan Academy, Series A, Mathematical Sciences \textbf{100} (2024), no.~2, 7--11. +\href{https://doi.org/10.3792/PJAA.100.002}{doi:\nolinkurl{10.3792/PJAA.100.002}}. + +\bibitem{FujinoFujisawa} +Osamu Fujino and Taro Fujisawa. +\emph{\href{https://ems.press/content/serial-article-files/41240}{Variations of mixed {Hodge} structure and semipositivity theorems}}. +Publications of the Research Institute for Mathematical Sciences \textbf{50} (2014), no.~4, 589--661. +\href{https://doi.org/10.4171/PRIMS/145}{doi:\nolinkurl{10.4171/PRIMS/145}}. + +\bibitem{FujinoFujisawaVMHS} +Osamu Fujino and Taro Fujisawa. +\emph{\href{https://www.math.kyoto-u.ac.jp/~fujino/vmhs-applications8.pdf}{Variation of mixed {Hodge} structure and its applications}}. +International Journal of Mathematics \textbf{36} (2025), no.~9, 2550020. +Published article: 43 pp.; theorem locators refer to the 36-page \href{https://www.math.kyoto-u.ac.jp/~fujino/vmhs-applications8.pdf}{author version 0.55, 11 March 2025}. + +\bibitem{FujinoGongyo} +Osamu Fujino and Yoshinori Gongyo. +\emph{\href{https://www.numdam.org/item/10.5802/aif.2894.pdf}{On the moduli b-divisors of lc-trivial fibrations}}. +Annales de l'Institut Fourier \textbf{64} (2014), no.~4, 1721--1735. +\href{https://doi.org/10.5802/aif.2894}{doi:\nolinkurl{10.5802/aif.2894}}. + +\bibitem{FujinoMori} +Osamu Fujino and Shigefumi Mori. +\emph{\href{https://www.math.kyoto-u.ac.jp/~fujino/Canonical.pdf}{A canonical bundle formula}}. +Journal of Differential Geometry \textbf{56} (2000), no.~1, 167--188. + +\bibitem{GrauertDirectImages} +Hans Grauert. +\emph{\href{https://www.numdam.org/item/PMIHES_1960__5__5_0/}{Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen}}. +Publications Math{\'e}matiques de l'IH{\'E}S \textbf{5} (1960), 5--64. +\href{https://doi.org/10.1007/BF02684746}{doi:\nolinkurl{10.1007/BF02684746}}. +With corrigendum, Publ. Math. IH{\'E}S 16 (1963), 35--36. + +\bibitem{GrauertDirectImagesCorr} +Hans Grauert. +\emph{\href{https://www.numdam.org/articles/10.1007/BF02684291/}{Berichtigung: Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen}}. +Publications Math{\'e}matiques de l'IH{\'E}S \textbf{16} (1963), 35--36. +\href{https://doi.org/10.1007/BF02684291}{doi:\nolinkurl{10.1007/BF02684291}}. + +\bibitem{GriffithsPeriodsIII} +Phillip A. 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Milne. +\emph{\href{https://www.jmilne.org/math/Books/iAG2022.pdf}{Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field}}. +Cambridge Studies in Advanced Mathematics 170. +Cambridge University Press, 2017. +\href{https://doi.org/10.1017/9781316711736}{doi:\nolinkurl{10.1017/9781316711736}}. +Cited from the corrected 2022 reprint. + +\bibitem{NamikawaProjectivity} +Yoshinori Namikawa. +\emph{\href{https://arxiv.org/abs/math/0101019v6}{Projectivity criterion of {Moishezon} spaces and density of projective symplectic varieties}}. +International Journal of Mathematics \textbf{13} (2002), no.~2, 125--135. +Theorem locators refer to arXiv:math/0101019v6, 12 November 2001. + +\bibitem{IitakaVariationCompanion} +{\raggedright +{OpenAI}. +\emph{Logarithmic Kodaira dimension and whole-fiber variation}. +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf}{OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026}, 2026.\par} + +\bibitem{IitakaProjectiveCompanion} +{\raggedright +{OpenAI}. +\emph{Projective Hodge lines and ordinary Iitaka subadditivity}. +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf}{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026}, 2026.\par} + +\bibitem{IitakaAdditivityCompanion} +{\raggedright +{OpenAI}. +\emph{The reverse logarithmic Kodaira inequality and additivity}. +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf}{OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026}, 2026.\par} + +\bibitem{OpenAILogAbundance2026} +{\raggedright +{OpenAI}. +\emph{Log abundance in characteristic zero}. +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}, 2026.\par} + +\bibitem{PopaSchnellKahler} +Giuseppe Pareschi, Mihnea Popa and Christian Schnell. +\emph{\href{https://msp.org/gt/2017/21-4/gt-v21-n4-p12-s.pdf}{{Hodge} modules on complex tori and generic vanishing for compact {K{\"a}hler} manifolds}}. +Geometry \& Topology \textbf{21} (2017), 2419--2460. +\href{https://doi.org/10.2140/gt.2017.21.2419}{doi:\nolinkurl{10.2140/gt.2017.21.2419}}. + +\bibitem{PatakfalviFibered} +Zsolt Patakfalvi. +\emph{\href{https://arxiv.org/abs/1208.1787v4}{Fibered stable varieties}}. +Transactions of the American Mathematical Society \textbf{368} (2016), no.~3, 1837--1869. +\href{https://doi.org/10.1090/tran/6386}{doi:\nolinkurl{10.1090/tran/6386}}. + +\bibitem{Schmid} +Wilfried Schmid. +\emph{\href{https://doi.org/10.1007/BF01389674}{Variation of {Hodge} structure: The singularities of the period mapping}}. +Inventiones Mathematicae \textbf{22} (1973), no.~3--4, 211--319. +\href{https://doi.org/10.1007/BF01389674}{doi:\nolinkurl{10.1007/BF01389674}}. + +\bibitem{SommesePeriodMapping} +Andrew John Sommese. +\emph{\href{https://www.numdam.org/item/ASNSP_1978_4_5_4_683_0/}{On the rationality of the period mapping}}. +Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (4) +\textbf{5} (1978), no.~4, 683--717. + +\bibitem{TsujiDirectImages} +Hajime Tsuji. +\emph{\href{https://arxiv.org/abs/1012.0884v1}{Global generation of the direct images of relative pluricanonical systems}}. +arXiv preprint, 2010. +Version 1, 4 December 2010. + +\bibitem{UenoClassification} +Kenji Ueno. +\emph{\href{https://doi.org/10.1007/BFb0070570}{Classification Theory of Algebraic Varieties and Compact Complex Spaces}}. +Lecture Notes in Mathematics 439. +Springer, Berlin, 1975. +\href{https://doi.org/10.1007/BFb0070570}{doi:\nolinkurl{10.1007/BFb0070570}}. + +\bibitem{ViehwegAdditivity} +Eckart Viehweg. +\emph{\href{https://esaga.uni-due.de/eckart.viehweg/ecpubl/}{Weak positivity and the additivity of the {Kodaira} dimension for certain fibre spaces}}. +In \emph{Algebraic Varieties and Analytic Varieties (Tokyo, 1981)}. +Advanced Studies in Pure Mathematics 1. +Kinokuniya and North-Holland, Tokyo and Amsterdam, 1983, pp.~329--353. +\href{https://doi.org/10.2969/aspm/00110329}{doi:\nolinkurl{10.2969/aspm/00110329}}. + +\bibitem{VilladsenHodgeKahler} +Mads Bach Villadsen. +\emph{\href{https://arxiv.org/abs/2401.09544v1}{{Hodge} modules and {K{\"a}hler} morphisms}}. +Preprint, 2024. +\href{https://arxiv.org/abs/2401.09544v1}{arXiv:2401.09544v1}, 17 January 2024. + +\bibitem{WangKahlerIitaka} +Juanyong Wang. +\emph{\href{https://afst.centre-mersenne.org/item/10.5802/afst.1690.pdf}{On the {Iitaka} Conjecture {$C_{n,m}$} for {K{\"a}hler} Fibre Spaces}}. +Annales de la Facult{\'e} des Sciences de Toulouse. Math{\'e}matiques (6) \textbf{30} (2021), no.~4, 813--897. +\href{https://doi.org/10.5802/afst.1690}{doi:\nolinkurl{10.5802/afst.1690}}. + +\end{thebibliography} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/adjoint-addition.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/adjoint-addition.tex new file mode 100644 index 0000000000000000000000000000000000000000..eaacdfa11ae42481d37b26f057e6b08f41490734 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/adjoint-addition.tex @@ -0,0 +1,247 @@ +\section{The adjoint-addition principle}\label{addition:section} + +The section comparison reduces subadditivity to a statement about +$K_W+T+M$. We prove that statement here from the two positivity +inputs specified below. Its two maps have different purposes: +$h$ is the fibration whose fiber dimension is to be added, whereas +$p$ expresses the nef term as a pullback from a projective base. + +The first input is the precise adjoint-positivity theorem for the +line produced by the construction. We use the highest-step and +parabolic-extension conventions of Section~\ref{rankone:section}. + +\begin{theorem}[Adjoint-positivity input]\label{period:adjoint} +Let $B$ be a smooth compact connected manifold in class $\mathcal C$ +and $B^\circ\subset B$ a dense Zariski open set with SNC complement. +Let $\mathbb V$ be a pure real-polarizable variation of Hodge +structures on $B^\circ$ with an integral ambient lattice. Suppose +a complex direct summand of $\mathbb V_{\mathbf C}$ has a +rank-one highest nonzero Hodge step, with parabolic rational +extension $L$. There are a modification $\tau:B'\to B$, a +surjective morphism $p:B'\to S$ with connected fibers to a smooth +projective variety, and a nef rational line bundle $P$ on $S$ such that +\begin{equation}\label{period:conclusion} + \tau^*L=p^*P,\qquad K_S+aP\text{ is big for some }a\in\mathbf Q_{>0}. +\end{equation} +Positive rational rescaling of $L$ is allowed, and the line identity +persists under further modifications. If $S$ is a point, $\tau^*L$ +is rationally trivial. +\end{theorem} + +The full proof is in Section~\ref{period:section}. The lattice belongs to the ambient +variation: the complex summand need not itself be defined over +$\mathbf Q$, and the real polarization need not be rational. + +The second input supplies addition when the fiber already has a big +log canonical divisor. It will be proved in Section~\ref{generaltype:section}. + +\begin{proposition}\label{generaltype:addition} +Let \(h\colon V\to Y\) be a surjective holomorphic map with connected fibers +between smooth compact complex manifolds in Fujiki class \(\mathcal C\). +Let \(T\) and \(C\) be effective rational divisors with simple normal crossing +support and coefficients in \([0,1]\) on \(V\) and \(Y\), respectively. +Suppose that +\begin{enumerate}[label=\textup{(\roman*)}] +\item \(K_F+T_F\) is big for a very general smooth fiber \(F\) of \(h\); +\item \(C\leq B(h,T)\); +\item \(T_E=1\) for every prime divisor \(E\) of \(V\) whose image in \(Y\) +has codimension at least two. +\end{enumerate} +Then +\begin{equation}\label{generaltype:inequality} + \kappa(V,K_V+T)\ \geq\ + \dim F+\kappa(Y,K_Y+C). +\end{equation} +As usual, the assertion is automatic if the last term is \(-\infty\). +\end{proposition} + + +General-type addition will give sections after a positive ample +twist from $S$. To remove that twist, we will need one fixed section +with a negative ample twist. The next lemma produces it from a big +base divisor, once nonvanishing on the general $p$-fiber is known. +It uses weak positivity with exceptional poles and +descent to a fixed smooth reference, following the method of +\cite[Remark~6.5(3) and Proposition~6.6]{CampanaOrbifolds}. +We include the descent step explicitly. + +\begin{lemma}[Effectivity after a big base twist] +\label{application:weak-effectivity} +Let $p:(W,T)\to S$ be surjective, with $W$ smooth compact in +class $\mathcal C$, $S$ smooth projective, and $T$ a rational SNC +boundary. Suppose one fixed divisible relative log pluricanonical +system is nonzero on very general fibers. For every big rational +line bundle $H$ on $S$, the rational line bundle +$K_{W/S}+T+p^*H$ has a nonzero section in some positive degree. +\end{lemma} + +\begin{proof} +Choose a K\"ahler source modification, flatten over a smooth +projective modification $\pi:S'\to S$, and resolve the flat main +transform. Write $\mu:W'\to W$ and $p':W'\to S'$ for the +resulting maps. Give $W'$ the strict transform of $T$ plus the +reduced $\mu$-exceptional divisor, denoted $T'$. Then +\begin{equation}\label{application:source-discrepancy} + K_{W'}+T'=\mu^*(K_W+T)+E, + \qquad E\geq0\ \text{and $\mu$-exceptional}. +\end{equation} +Every source prime over a subset of codimension at least two in +$S'$ is exceptional over the fixed $W$: before resolving the +source, flatness bounds that inverse image by codimension at least +two. Put $R=K_{S'/S}\geq0$ and $H'=\pi^*H+R$. The exact identity +\begin{equation}\label{application:twist-cancellation} + K_{W'/S'}+T'+p'^*H' + =\mu^*(K_{W/S}+T+p^*H)+E +\end{equation} +will allow descent. + +Choose a divisible $m>0$ such that +$\mathcal E=p'_*\mathcal O_{W'}(m(K_{W'/S'}+T'))$ has positive +rank. Generic base change and the hypothesis provide such an $m$. +Fujino's weak positivity theorem applies to this log canonical pair +and its projective base \cite[Theorem~1.1]{FujinoWeakPositivity}. +Choose an ample Cartier divisor $A_0$ and an integer $\alpha>0$ +such that $H'-A_0/(m\alpha)$ is big. For some integer $\beta>0$, +\[ + (\operatorname{Sym}^{\alpha\beta}\mathcal E)^{**} + \otimes\mathcal O_{S'}(\beta A_0) +\] +is generated by global sections at the generic point. + +On the locally free locus of $\mathcal E$, multiplication of +relative sections gives an evaluation map to +\[ + \mathcal O_{W'}\bigl(m\alpha\beta(K_{W'/S'}+T') + +\beta p'^*A_0\bigr). +\] +It is nonzero generically, because a nonzero relative section has +nonzero powers. Generic generation supplies a global section whose +image is nonzero. The complement of the locally free locus has +codimension at least two. Multiply a sufficiently divisible power +of the evaluated section by a section of a sufficiently divisible +multiple of +$m\alpha\beta\,p'^*(H'-A_0/(m\alpha))$. +The latter exists by bigness. This produces a nonzero section of a +multiple of the left side of \eqref{application:twist-cancellation}, except possibly +at source primes above that codimension-two set. + +Locally, a meromorphic frame of $\mathcal E$ expresses the evaluated +section as a meromorphic combination of products of relative +sections. Thus the resulting form is meromorphic, and its only +possible divisorial poles lie above that omitted codimension-two set. +Those primes, and $E$, are exceptional over $W$. At every prime +of the original smooth $W$, the resulting meromorphic form therefore +has the required order for a multiple of $K_{W/S}+T+p^*H$. +Divisorial regularity and extension across codimension two give a +global section on $W$. If $S$ is a point, the same conclusion is +the assumed nonvanishing. +\end{proof} + +\begin{proposition}[Adjoint addition] +\label{addition:principle} +Let $h:W\to Y$ be a fibration between smooth compact manifolds in +class $\mathcal C$. Let $T,C$ be rational SNC boundaries with +$C\leq B(h,T)$, and suppose every prime of $W$ whose image has +codimension at least two in $Y$ has coefficient one in $T$. +Let $p:W\to S$ be a surjective morphism with connected fibers to +a smooth projective variety. Suppose $P$ is a nef rational line +bundle on $S$, $M=p^*P$, and $K_S+a_0P$ is big for some rational +$a_0>0$. If $K_W+T+M$ is big on very general $h$-fibers, then +\[ + \kappa(W,K_W+T+M) + \geq\dim(W/Y)+\kappa(Y,K_Y+C). +\] +\end{proposition} + +\begin{proof} +The assertion is automatic if the last term is $-\infty$. +Assume it is nonnegative, put $D=K_W+T$, and write +$k=\dim(W/Y)$. We may first replace $W$ by a K\"ahler +modification, adding the reduced exceptional divisor to its strict +boundary. The log section spaces, also after tensoring by the +pullback of $M$, are unchanged. Fiberwise bigness and the +inf-multiplicity boundary are preserved; newly exceptional primes +have coefficient one. Thus all hypotheses remain valid. + +If $S$ is a point, $M$ is rationally trivial and +\Cref{generaltype:addition} proves the proposition. Suppose $\dim S>0$, +and choose an ample Cartier divisor $A$ on $S$. A very general +$h$-fiber is K\"ahler and carries the big rational line +$(D+M)|_{W_y}$, so it is projective. Openness of its big cone +gives a rational $c$ with $00}). +\end{equation} + +The systems in \eqref{application:perturbed-bound} already have the required +image dimension. It remains to remove their ample term without +losing these ratios of sections. In particular one such divisor has +a nonzero section in a fixed +positive degree. Restriction of that section to a very general +$p$-fiber is nonzero; both pulled-back twists become trivial +there. Hence +\begin{equation}\label{application:p-fiber-nonvanishing} + \kappa(W_s,(K_W+T)|_{W_s})\geq0 + \quad\text{for very general }s\in S. +\end{equation} +One fixed relative log pluricanonical direct image for +$p:(W,T)\to S$ therefore has positive rank. + +Choose rational $a>\max\{1,a_0\}$. Since $P$ is nef, +$K_S+aP$ is big. For sufficiently small rational $\lambda>0$, +$H=K_S+aP-\lambda A$ is still big. +Lemma~\ref{application:weak-effectivity} now gives a nonzero section of a +positive multiple of +\begin{equation}\label{application:negative-twist} + E_-=D+aM-\lambda p^*A + =K_{W/S}+T+p^*H. +\end{equation} +Put +\[ + \theta=\frac{1-c}{a-c}, + \qquad \eta=\frac{(1-c)\lambda}{a-1}, + \qquad E_+=D+cM+\eta p^*A. +\] +Then $0<\theta<1$, $\eta>0$, and direct calculation gives +\begin{equation}\label{addition:cancel} + D+M=(1-\theta)E_++\theta E_-. +\end{equation} +Fix a nonzero section $e$ of $rE_-$ for one denominator-clearing +$r>0$. For sufficiently divisible $n$, multiplication by +$e^{n\theta/r}$ maps +\[ + H^0(W,n(1-\theta)E_+) + \lhook\joinrel\longrightarrow H^0(W,n(D+M)). +\] +The exponent is an integer, and the source degrees range through a +Veronese subsequence. On the complement of the zero divisor of $e$, +every ratio of two sections is unchanged. These systems therefore +have the same image dimensions before and after multiplication. +Equation~\eqref{application:perturbed-bound} for the displayed value of $\eta$ +proves the asserted lower bound. +\end{proof} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/applications.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/applications.tex new file mode 100644 index 0000000000000000000000000000000000000000..59dec4dd59104fae00a5a07af42f2e7bbede8e16 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/applications.tex @@ -0,0 +1,441 @@ +\section{Return to the original fibration and consequences} +\label{application:section} + +The previous sections supply the two positivity statements and the +exact section comparison. We now apply them to the original fibration, +keeping its base fixed while changing the intermediate Iitaka space. + +\begin{proof}[Proof of \Cref{main:orbifold}] +If either summand on the right is \(-\infty\), the asserted inequality +is automatic. Suppose henceforth that both are nonnegative, and put +\[ + k=\kappa(F,K_F+\Delta_F),\qquad b=\kappa(f\mid\Delta). +\] + +\smallskip\noindent +\emph{Fixing the base.} +Use Section~\ref{setup:section} and \Cref{setup:flattening} to resolve the diagram, +take K\"ahler models, and flatten before resolving the source. Use +strict transforms plus reduced exceptional divisors at every source +modification. Resolve the discriminant and boundary-stratum images +on the base, then fix this smooth K\"ahler base \(Y\). The source +and very general fiber log section rings are unchanged, and every +source prime mapped into codimension at least two in \(Y\) has +coefficient one. Put +\[ + C=B(f,\Delta) +\] +on this model. It has SNC support, and the definition of the invariant +gives +\begin{equation} + \kappa(Y,K_Y+C)\geq b. + \label{application:invariant-bound} +\end{equation} +\Cref{setup:simultaneous} preserves \(C=B(f,\Delta)\) and the +coefficient-one condition through all further changes over \(Y\). + +\smallskip\noindent +\emph{The relative Iitaka map and its Hodge line.} +Apply \Cref{application:relative-iitaka}, obtaining +\[ + X\xrightarrow{g}W\xrightarrow{h}Y, + \qquad \dim W_y=k, + \qquad \kappa(G,K_G+\Delta_G)=0. +\] +Make \(W\) K\"ahler, resolve the diagram, and fix the current smooth +source pair \((X_0,\Delta_0)\) as reference. Flatten over \(W\) +and resolve the generator and boundary data as required by +\Cref{rankone:comparison}. Thus every prime mapped by \(g\) into +codimension at least two is exceptional over \(X_0\), and on a dense +smooth log open the relative +systems have rank one in all sufficiently divisible degrees. + +Apply \Cref{rankone:comparison,period:adjoint}. Their further +modifications remain over \(Y\); \Cref{setup:simultaneous} preserves +the exceptional reference, while the variation and its parabolic line +pull back from the common open. The relative Iitaka property persists +by \Cref{application:relative-iitaka}. On the resulting common model +we obtain a rational line \(M\), a rational SNC boundary \(T\) in +\([0,1]\), and a surjective map to a smooth projective variety: +\begin{equation} + p:W\longrightarrow S, + \qquad M=p^*P, + \qquad P\text{ nef}, + \qquad K_S+a_0P\text{ big for some }a_0>0, + \label{application:period-data} +\end{equation} +such that +\begin{equation} + B(g,\Delta)\leq T\leq1. + \label{application:rankone-boundary} +\end{equation} +Compute \(T\) by the rank-one order formula on this final model. +Parabolic pullback and \Cref{rankone:comparison} identify the divisible +systems of \(K_W+T+M\) with those on \((X_0,\Delta_0)\), also +relatively over \(Y\), preserving all ratios. + +\smallskip\noindent +\emph{Relative bigness and the base boundary.} +Set \(D_W=K_W+T\). The systems defining the relative Iitaka image +have image dimension \(k\) on very general \(f\)-fibers. Push these systems +into \(D_W+M\) using \Cref{rankone:comparison}. Their ratios are +unchanged, so the resulting systems have image dimension \(k\) on +very general \(h\)-fibers, whose dimension is \(k\). Consequently +\begin{equation} + (D_W+M)|_{W_y}\text{ is big for very general }y. + \label{application:relative-big} +\end{equation} +Here the systems are obtained by generic base change over an open patch +of \(Y\) and extend on the smooth \(W_y\) by the relative section +comparison. Adjunction identifies the restricted adjoint bundles. + +By \Cref{application:composition}, +\begin{equation} + B(h,T)\geq C. + \label{application:composed-boundary} +\end{equation} +The same lemma gives coefficient one in \(T\) for every prime of +\(W\) mapped into codimension at least two in \(Y\). Thus both +boundary hypotheses of \Cref{generaltype:addition} hold. + +The hypotheses of \Cref{addition:principle} have now all been +verified. Its conclusion is +\begin{equation}\label{application:final-intermediate-bound} + \kappa(W,K_W+T+M)\geq k+\kappa(Y,K_Y+C). +\end{equation} +The proof of that proposition uses a single nonzero section to cancel +the ample perturbation and preserves the ratios defining this bound. + +Finally, \Cref{rankone:comparison} lifts these systems by the relative +generators, preserving their ratios and image dimensions. Untested +\(g\)-contracted primes are exceptional over \((X_0,\Delta_0)\), +so the forms descend and extend there. By +\Cref{setup:log-modification}, they then descend to the original source. +Combining \eqref{application:invariant-bound} and +\eqref{application:final-intermediate-bound} proves +\[ + \kappa(X,K_X+\Delta) + \geq \kappa(F,K_F+\Delta_F)+\kappa(f\mid\Delta). +\] + +Bigness on a point means Iitaka dimension zero, and a zero-dimensional +connected fiber has trivial relative line. If \(Y\) is a point the +inequality is an identity; if the relative Iitaka fiber is a point its +Hodge line is trivial. Together with the point-period-base case above +and the initial \(-\infty\) convention, these observations cover all +dimensions and values of the two summands. +\end{proof} + +\subsection{The logarithmic and ordinary inequalities} +\label{spec:section} + +We first compare the invariant orbifold base with a prescribed +logarithmic base. This makes the two classical specializations of +Theorem~\ref{main:orbifold} immediate, while retaining control of the +base under modification. + +\begin{lemma}[Comparison with a logarithmic base] +\label{spec:base-comparison} +Let $f:(X,\Delta)\to Y$ be a fibration of smooth compact complex +manifolds, where $\Delta$ has rational coefficients in $[0,1]$ and +simple normal crossing support. Let $D_Y$ be a reduced simple normal +crossing divisor on $Y$. Suppose that every prime component of +$f^*D_Y$ has coefficient one in $\Delta$. Then +\[ + \kappa(f\mid\Delta)\geq\kappa(Y,K_Y+D_Y). +\] +\end{lemma} + +\begin{proof} +Choose a neat model as in Lemma~\ref{setup:flattening}, with +$p:X'\to X$, $q:Y'\to Y$, $f':X'\to Y'$, and +$\Delta'=p_*^{-1}\Delta+\operatorname{Exc}(p)_{\mathrm{red}}$. +We may arrange that +$E_Y=(q^{-1}\operatorname{Supp}D_Y)_{\mathrm{red}}$ has simple +normal crossings. Every prime of $X'$ dominating a component of +$E_Y$ has coefficient one in $\Delta'$: an exceptional prime has +coefficient one by definition, and a nonexceptional prime is the strict +transform of a component of $f^*D_Y$. Thus its weighted multiplicity +is infinite, and the inf-multiplicity rule gives +\[ + B(f',\Delta')\geq E_Y. +\] +Pullback by $q$ takes logarithmic top forms with poles along $D_Y$ +to logarithmic top forms with poles along $E_Y$. In local coordinates +this follows by pulling back each factor $dz_i/z_i$; the same assertion +holds for all tensor powers. Consequently, pullback embeds every +plurilogarithmic system on $(Y,D_Y)$ into the corresponding system of +$K_{Y'}+B(f',\Delta')$, preserving all ratios of sections. +The neat-model identity~\eqref{setup:neat-formula} now proves the +claim. +\end{proof} + +\begin{corollary}[Logarithmic subadditivity] +\label{spec:logarithmic} +Let $f:X\to Y$ be a surjective morphism with connected fibers between +smooth connected projective complex varieties. Let $D_X,D_Y$ be +reduced simple normal crossing divisors, with the zero divisor allowed, +such that +\[ + \operatorname{Supp}(f^*D_Y)\subseteq\operatorname{Supp}(D_X). +\] +For a very general fiber $F$, put $D_F=D_X|_F$. Then +\begin{equation}\label{spec:log-inequality} + \kappa(X,K_X+D_X) + \geq\kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y). +\end{equation} +The same statement holds for compact manifolds in Fujiki class +$\mathcal C$ and holomorphic fibrations between them. +\end{corollary} + +\begin{proof} +Theorem~\ref{main:orbifold} applies to $(X,D_X)$, and +Lemma~\ref{spec:base-comparison} bounds its base term from below. +If either term on the right of~\eqref{spec:log-inequality} is +$-\infty$, the assertion follows from our convention for that value. +\end{proof} + +In the projective setting, this is the usual compactification form of +the logarithmic Iitaka conjecture. Indeed, for a smooth +quasi-projective variety $U$ with a smooth projective compactification +$X$ and reduced simple normal crossing boundary $D_X=X\setminus U$, +its logarithmic Kodaira dimension is +$\overline\kappa(U)=\kappa(X,K_X+D_X)$ +\cite{IitakaLog}. A dominant morphism $U\to V$ between smooth +connected quasi-projective varieties, with geometrically connected +general fiber, admits compatible such compactifications: +compactify the graph and resolve the graph and boundaries. The +Stein factor of the resulting morphism $X\to Y$ is finite +birational over the normal variety $Y$, hence equals $Y$. +Thus $f$ has connected fibers, and +$\operatorname{Supp}(f^*D_Y)\subseteq\operatorname{Supp}(D_X)$. +For very general $v\in V$, $(X_v,D_X|_{X_v})$ compactifies $U_v$. +Thus Corollary~\ref{spec:logarithmic} gives +$\overline\kappa(U)\geq\overline\kappa(U_v)+\overline\kappa(V)$. + +Ordinary subadditivity also enters the Albanese reduction of +\cite{OpenAILogAbundance2026}. That work combines +it with separate nonvanishing and good-model arguments to prove log +abundance in characteristic zero. + +\begin{corollary}[Ordinary subadditivity in characteristic zero] +\label{spec:ordinary} +Let $k$ be an algebraically closed field of characteristic zero, and +let $f:X\to Y$ be a surjective morphism with connected fibers between +smooth connected projective $k$-varieties. If $F$ is its geometric +generic fiber, then +\begin{equation}\label{spec:ordinary-inequality} + \kappa(X)\geq\kappa(F)+\kappa(Y). +\end{equation} +More generally, the logarithmic inequality of +Corollary~\ref{spec:logarithmic}, with the same boundary hypotheses, +holds over $k$, with the geometric generic fiber in place of a very +general complex fiber. +\end{corollary} + +\begin{proof} +Over $\mathbb C$, the ordinary assertion follows by taking +$D_X=D_Y=0$ in Corollary~\ref{spec:logarithmic}. In a projective +family over $\mathbb C$, the Iitaka dimension of the geometric +generic fiber equals that of a very general fiber. To see this, +apply cohomology and base change in each positive divisible +pluricanonical degree. After shrinking the base separately in each +degree, the corresponding relative rational map also has constant +fiberwise image dimension. Excluding the resulting countable +union of proper closed subsets identifies all these image +dimensions with those of the generic fiber. + +For the field extension, let $K\subseteq K'$ be fields and let $V$ +be a smooth projective geometrically integral $K$-variety with a +rational divisor $D$. For every $m$ for which $mD$ is integral, +flat base change gives +\[ + H^0(V,\mathcal O_V(mD))\otimes_K K' + \simeq H^0(V_{K'},\mathcal O_{V_{K'}}(mD_{K'})). +\] +The complete linear system on the right is the scalar extension of +the one on the left, so nonvanishing and image dimension are +unchanged. Hence Iitaka dimension is invariant under extension +of the ground field. Applying this observation to the generic +fiber over the function field of the base proves the corresponding +invariance for geometric generic fibers. + +Descend $f$, its projective embeddings, and its boundary divisors +to a finitely generated subfield $K\subset k$ over $\mathbb Q$. +The descended varieties are smooth and geometrically integral; +the boundary conditions descend as well. The equality +$f_*\mathcal O_X=\mathcal O_Y$ descends by proper flat base change +and faithful flatness, and therefore also holds after an embedding +$K\hookrightarrow\mathbb C$. Apply the complex logarithmic +inequality to this base change. The preceding invariance identifies +its three Iitaka dimensions with those over $k$, proving both +assertions. +\end{proof} + +\subsection{Kodaira dimension along the core} +\label{spec:core} + +A smooth compact orbifold pair $(X,\Delta)$ in class $\mathcal C$ is +\emph{special} if it has no meromorphic fibration onto a +positive-dimensional base of orbifold general type; here general +type means that the invariant base dimension equals the dimension +of the base. Campana constructed its \emph{core} +\[ + c_{(X,\Delta)}:(X,\Delta)\dashrightarrow C(X,\Delta), +\] +whose very general orbifold fiber is special and whose orbifold base is of general +type, unless it is a point +\cite[Theorem~9.1]{CampanaOrbifolds}. In particular, the core has +a point base exactly when $(X,\Delta)$ is special. Fiber properties +of the core and the tower below refer to Campana's stable orbifold +fibers: they are computed on the suitable equivalent holomorphic +representatives furnished by these constructions +\cite[Definition~2.49 and \S7.1]{CampanaOrbifolds}. +A neat representative $g:(Z,\Gamma)\to T$ in the sense of +Section~\ref{setup:section} is \emph{strictly neat} if its +inf-multiplicity boundary on the smooth base is SNC. It is +\emph{high} if $g$ is an orbifold morphism to this base: for every +base prime $D$ and source prime $E$ with +$a_E=\ord_E(g^*D)>0$, including primes contracted by $g$, +\[ + a_E m_\Gamma(E)\geq m_{B(g,\Gamma)}(D). +\] +These are the model conditions of +\cite[Definition~3.8 and Proposition~3.10]{CampanaOrbifolds}. +The high condition is not automatic: the inf-multiplicity base tests +divisors dominating base primes, whereas an orbifold morphism must +also control primes contracted into higher codimension +\cite[Remark~3.3(2)]{CampanaOrbifolds}. +For the core and its tower we choose compatible strictly neat and +high representatives supplied by Campana's constructions +\cite[Proposition~3.15 and Corollary~4.14]{CampanaOrbifolds}. +The section comparisons of Section~\ref{setup:section} do not +assert that arbitrary model changes preserve every stable-fiber +property. + +\begin{corollary}[Kodaira dimension along the core] +\label{spec:core-dimension} +Let $(X,\Delta)$ be a smooth compact orbifold pair in Fujiki class +$\mathcal C$, with rational simple normal crossing boundary +coefficients in $[0,1]$. Let $C=C(X,\Delta)$ be its core base and +$(G,\Delta_G)$ a very general orbifold fiber. Then +\begin{equation}\label{spec:core-equality} + \kappa(X,K_X+\Delta) + =\kappa(G,K_G+\Delta_G)+\dim C. +\end{equation} +If $\kappa(X,K_X+\Delta)\geq0$, then +\[ + 0\leq\dim C\leq\kappa(X,K_X+\Delta), +\] +and equality in the second inequality holds precisely when +$\kappa(G,K_G+\Delta_G)=0$. +In particular, $\kappa(X,K_X+\Delta)=0$ implies that +$(X,\Delta)$ is special. +\end{corollary} + +\begin{proof} +Resolve the core map, using the boundary convention of +Lemma~\ref{setup:log-modification}. The total and fiber Iitaka +dimensions are preserved. Theorem~\ref{main:orbifold} gives +the lower bound in~\eqref{spec:core-equality}, since the invariant +base dimension is $\dim C$. + +For completeness, the reverse bound is the elementary upper +addition inequality. For each divisible $m$, let $\phi_m$ be +the meromorphic map defined by $|m(K_X+\Delta)|$, when this +system is nonempty. The map $(c,\phi_m)$ has image dimension +at most $\dim C+\kappa(G,K_G+\Delta_G)$: on a very general +$c$-fiber its second component is given by a subspace of +$H^0(G,m(K_G+\Delta_G))$. Since projection from this image +onto the image of $\phi_m$ is surjective, the same upper bound +holds for $\dim\phi_m(X)$. Taking the maximum over $m$ proves +the desired inequality. If the fiber Iitaka dimension is +$-\infty$, restriction to a very general fiber shows that no +global pluricanonical section can be nonzero, so both sides of +\eqref{spec:core-equality} are $-\infty$. + +Finally, if the total Iitaka dimension is nonnegative, a nonzero +section restricts nontrivially to a very general core fiber. +Its Iitaka dimension is therefore nonnegative. The dimension +bound and its equality case follow from~\eqref{spec:core-equality}. +When the total dimension is zero, the core base is a point, +which is equivalent to specialness. +\end{proof} + +These statements recover, for the core, Campana's general-type-base addition +theorem and its consequences +\cite[Theorems~6.3, 6.7, and~6.8]{CampanaOrbifolds}. +The existence of the core and the implication from Kodaira +dimension zero to specialness are established results of his +theory. + +\subsection{The core tower} +\label{spec:core-tower-section} + +For a smooth orbifold pair $(Z,\Theta)$, the notation +$\kappa_+(Z,\Theta)=-\infty$ means that every meromorphic +fibration onto a positive-dimensional base has invariant +orbifold Kodaira dimension $-\infty$; the condition is vacuous +for a point. Equivalently, one may test all dominant meromorphic +maps: in Stein factorization, the finite orbifold canonical-divisor +inequality shows that passing to the connected-fiber factor cannot +decrease the invariant base dimension. Indeed, over a divisor +with multiplicity $m_D$, a prime with ramification index $e$ and +orbifold multiplicity $m_E$ satisfies $m_D\leq e m_E$; the +canonical-divisor difference has coefficient +$e/m_D-1/m_E\geq0$. +The \emph{$\kappa$-rational quotient} is a meromorphic fibration +$r:(Z,\Theta)\dashrightarrow R$ with $\kappa(r\mid\Theta)\geq0$ +whose very general stable orbifold fiber has $\kappa_+=-\infty$ +\cite[Definition~5.23 and Corollary~6.14]{CampanaOrbifolds}. +Its \emph{stable orbifold base} is the inf-multiplicity base on the +compatible representatives just described +\cite[Proposition~3.15, Corollary~4.14, and \S7.1]{CampanaOrbifolds}. + +\begin{corollary}[Campana's decomposition of the core] +\label{spec:core-tower} +Let $(X,\Delta)$ be as in Corollary~\ref{spec:core-dimension}, +and set $n=\dim X$. Starting from $(X,\Delta)$, successively +take the $\kappa$-rational quotient $r$ and the +Moishezon--Iitaka fibration $M$ of its stable orbifold base. +At each iteration choose such compatible representatives and equip +the new base with the stable inf-multiplicity orbifold structure +of the composite. +The composite after at most $n$ +iterations is the core, up to bimeromorphic equivalence of +fibrations: +\[ + c_{(X,\Delta)}=(M\circ r)^n. +\] +The positive-dimensional very general stable orbifold fibers of +the $r$-steps have $\kappa_+=-\infty$, and those of the +$M$-steps have Kodaira dimension zero. In particular, for +a special pair this tower ends at a point. +\end{corollary} + +\begin{proof} +Theorem~\ref{main:orbifold} establishes the hypothesis of +Campana's Corollary~6.14, which supplies the +$\kappa$-rational quotient. Its scope is exactly that used +here: smooth compact class-$\mathcal C$ pairs with rational +coefficients in $[0,1]$ +\cite[Conjecture~6.1 and Remark~6.2]{CampanaOrbifolds}. +Its stable base has nonnegative Kodaira dimension, so $M$ +is defined and has very general orbifold fiber of Kodaira +dimension zero. The new smooth base remains in class +$\mathcal C$, with rational boundary coefficients in $[0,1]$. +Thus the construction can be repeated. + +Campana's Lemma~10.1 states that each composite $M\circ r$ +has special general orbifold fiber and decreases dimension +unless its source pair is of general type. His +Theorems~10.2--10.3 identify the resulting stabilized +composite with the core. Their sole conjectural input is +the existence of the $\kappa$-rational quotient, now supplied +above. Strict dimension decrease allows at most $n$ +nontrivial iterations; identity steps give the displayed +formula. A point is fixed throughout. +These are precisely the decomposition and fiber assertions of +\cite[Lemma~10.1, Theorems~10.2--10.3, and Corollary~10.4] +{CampanaOrbifolds}. +\end{proof} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/general-type-addition.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/general-type-addition.tex new file mode 100644 index 0000000000000000000000000000000000000000..619114ac5b3aafb5c894d5f968c989a9bcb99ef9 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/general-type-addition.tex @@ -0,0 +1,685 @@ +\section{Addition with a log-general-type fiber}\label{generaltype:section} + +We prove subadditivity when the fiber is of log general type. The base +may be nonprojective: the projective positivity theorem will be applied +to a stable family over the image of its classifying map. + +We prove \Cref{generaltype:addition}. + +The proof constructs finitely many relative pluricanonical forms whose +ratios give a generically finite map on a general fiber. Their poles +must be controlled at every vertical divisor, including divisors whose +image on the original base has codimension at least two. Multiplying +these forms by base pluricanonical sections then retains both the fiber +coordinates and the base Iitaka coordinates. + +We first construct a projective stable family over the image of a +classifying map. Positivity is applied to that family. Its sections +are pulled back, descended through a finite cover, and tested at +vertical valuations. All degrees below clear the denominators of the +relevant divisors. + +\subsection{Comparison with a stable family} + +We use the stable-family convention of +\cite[Notation~3.7, Remark~3.8, and Definitions~3.9 and~3.11] +{KovacsPatakfalvi}. A stable pair has semi-log-canonical singularities +and an ample rational log canonical divisor. A stable family over a +normal base is flat, proper and surjective with connected stable +fibers; its boundary support avoids the generic points and the +codimension-one singular points of every fiber. Its relative log +canonical divisor is rational Cartier and relatively ample, so the +family is projective. Boundaries are pulled +back by divisorial restriction, and Cartier multiples of the relative +log canonical divisor commute with base change between normal bases. +Semi-log-canonicity +allows nonnormal fibers and tests log canonicity on the normalization +with the conductor boundary. The general fibers in our application +are additionally Kawamata log terminal (klt): they are normal and +all their log discrepancies are positive. This extra condition will +be checked before applying positivity. +A stable family has \emph{maximal variation} when its classifying map +to the moduli space of stable pairs is generically finite onto its image. + +Fiberwise finite generation gives an embedding degree separately on +each fiber. The next lemma selects one degree on an analytic base. +As throughout, \emph{very general} excludes a countable union of proper +closed analytic subsets. + +\begin{lemma}\label{generaltype:selection} +Let \(Y\) be an irreducible compact complex space. For each positive integer +\(j\), suppose that \(q_j\colon Q_j\to Y\) is a proper surjective map from an +irreducible compact complex space and that, over a dense analytic open +\(U_j\subset Y\), its fibers are irreducible of a fixed dimension \(r_j\). +Let +\[ + \phi_j\colon Q_j\dashrightarrow Z_j +\] +be a meromorphic map with irreducible projective image \(Z_j\). +Suppose that a dense open part of \(q_j^{-1}(U_j)\), called the genuine +frame locus, meets every fiber densely and that \(\phi_j\) is holomorphic +there. Let \(G_j\subset Z_j\) be a countable union of algebraically +constructible subsets. + +Let \(\Lambda_j\subset U_j\) be sets whose union contains a very general +subset of \(Y\), such that for \(y\in\Lambda_j\) all genuine frames above +\(y\) map into \(G_j\). Then some \(\Lambda_j\) is not contained in a +countable union of proper closed analytic subsets of \(Y\), and for +that index one of the constructible subsets making up \(G_j\) is dense +in \(Z_j\). In particular \(G_j\) contains a nonempty algebraic open +subset of \(Z_j\). +\end{lemma} + +\begin{proof} +Some \(\Lambda_j\) is not contained in a countable union of proper +closed analytic subsets; otherwise their union would be so contained. +Resolve the graph of \(\phi_j\) and shrink \(U_j\), if necessary. +This discards only a proper analytic subset of \(Y\) and preserves +the stated properties of the general fibers. +Write \(G_j=\bigcup_iG_{ji}\). If no \(G_{ji}\) is dense, the inverse +images +\[ + E_{ji}=\phi_j^{-1}(\overline{G_{ji}}) +\] +are proper closed analytic subsets of the resolved \(Q_j\). +For \(y\in\Lambda_j\), the genuine frame fiber is covered by these +subsets. Baire's theorem implies that one \(E_{ji}\) contains a +relatively open part of the fiber. Irreducibility then implies +that \(E_{ji}\) contains the whole compact fiber. + +The locus +\[ + A_{ji}=\{y\in U_j: + \dim(E_{ji}\cap q_j^{-1}(y))\geq r_j\} +\] +is analytic, by the fiber-dimension theorem for proper maps. It is +proper: otherwise \(E_{ji}\) would contain the general fibers and +hence all of \(Q_j\). The same theorem before restriction to \(U_j\) +shows that \(A_{ji}\), together with \(Y\setminus U_j\), is contained +in a proper closed analytic subset of \(Y\). +This would place \(\Lambda_j\) in countably many proper closed +analytic subsets, a contradiction. Finally, a dense constructible +subset of an irreducible algebraic variety contains a nonempty +open subset. +\end{proof} +\begin{lemma}[Analytic comparison with a stable family] +\label{generaltype:stable-comparison} +Let \(h\colon V\to Y\) be a surjective holomorphic map with connected +fibers, where \(V\) is a smooth compact Kähler manifold and \(Y\) is +a smooth compact manifold in class \(\mathcal C\). Let \(A\) be an +effective rational SNC divisor on \(V\), horizontal over \(Y\), with +coefficients strictly less than one. Suppose \(K_F+A_F\) is big on +very general fibers \(F\) of positive dimension. There are +\begin{enumerate}[label=\textup{(\roman*)}] +\item a proper generically finite map \(\tau\colon Y'\to Y\), with \(Y'\) +smooth, compact, and in class \(\mathcal C\); +\item a surjective holomorphic map \(\rho\colon Y'\to R\), with \(R\) +smooth and projective; +\item a projective stable family +\(\pi\colon(\mathcal V,\mathcal A)\to R\) of maximal variation, with klt +general fiber; +\end{enumerate} +such that the main component of \(V\times_Y Y'\) is bimeromorphic +over \(Y'\) to \(\mathcal V\times_RY'\). Over a dense analytic open, +this meromorphic comparison is the fiberwise log canonical model map +for \(K_F+A_F\), with the horizontal pushforward of \(A\) as boundary. +The space \(R\) may be a point. +\end{lemma} + +\begin{proof} +The construction has three outputs. Relative pluricanonical images +produce algebraic parameter data even though the base is analytic. +A classifying map then supplies a projective stable family after a +finite base cover. Finally, an isomorphism cover matches that family +with the relative image; fiberwise canonical models determine the +horizontal discrepancy signs. + +\emph{Relative images and frames.} +Set \(\mathcal L=\mathcal O_V(K_{V/Y}+A)\), understood as a rational +line bundle. For the factorial multiples \(m\) clearing its denominator, +the sheaves +\[ + \mathcal E_m=h_*\mathcal L^{\otimes m} +\] +are coherent. First restrict to a dense analytic open where \(h\) and +the horizontal boundary strata are smooth, so the proper family and +its line bundles are flat over the base. Generic constancy of the +fiber section dimension and cohomology and base change then identify +the fibers of \(\mathcal E_m\) with the complete systems, separately +for each \(m\); see \cite[\S7, nos.~4--6, S\"atze~3--5]{GrauertDirectImages} +and its corrigendum \cite{GrauertDirectImagesCorr}. +After a modification of \(Y\), +the strict transform of \(\mathcal E_m\), modulo torsion, is locally free. +The evaluation map gives a relative meromorphic map into its projective +bundle. Resolve this map and take its proper analytic image +\(\mathcal Z_m\). We also take the proper images of the horizontal +boundary components and retain their divisorial cycle parts. +Analytic flattening makes these embedded image and cycle data flat on a +dense analytic open, with fixed Hilbert and cycle polynomials. +These uses of coherent direct images and analytic flattening are +independent of any minimal-model theorem; see \cite{HironakaFlattening}. + +Write \(s_m=\rank\mathcal E_m\) on this open; indices with \(s_m=0\) +are omitted. Its projective frame bundle has a natural compactification +\[ + \overline Q_m= + \mathbb P\!\left(\Hom(\mathcal O_Y^{\,s_m},\mathcal E_m)\right). +\] +Here \(Y\) denotes the modification chosen for this degree; its map to +the original base remains proper and bimeromorphic. The locus of +invertible matrices modulo scalars is a principal +\(\operatorname{PGL}(s_m)\)-bundle. +A genuine frame embeds the corresponding fiber of \(\mathcal Z_m\) +in a fixed \(\mathbb P^{s_m-1}\). There is therefore a holomorphic +Hilbert--Chow parameter map on the genuine frame locus. +The Hilbert spaces here are the algebraic Hilbert schemes of +projective space, viewed analytically; equivalently one may use +their Douady realization \cite{Douady}. + +To extend this map meromorphically, use the existing compact relative +image and the projective linear action in local trivializations. +The graph of that action extends over the projective matrix +space: inversion is given by the adjugate matrix. Apply this construction +to the proper family \(\mathcal Z_m\) and its marked cycles, take proper +analytic images, and flatten by strict transform. One obtains a proper flat family +of embedded subspaces over a modification of \(\overline Q_m\). +Its Hilbert--Chow map is holomorphic. The local graph constructions +agree on the genuine frames and hence glue. Thus the original +parameter map is meromorphic. After resolving it, its image is +compact analytic in a fixed projective Hilbert--Chow space, and is +consequently algebraic. Denote the irreducible image by \(Z_m\). +This construction uses a frame bundle, not a meromorphic +trivialization of \(\mathcal E_m\). + +\emph{The good parameter locus.} +In \(Z_m\), call an embedded pair \((Z,D)\) good if it is klt and stable, +has the prescribed boundary weights, has a complete nondegenerate +embedding, and satisfies +\begin{equation}\label{generaltype:embedded-adjoint} + \mathcal O_Z(1)\simeq + \mathcal O_Z\bigl(m(K_Z+D)\bigr). +\end{equation} +One may also impose the generic Hilbert and marking data. +This is a countable union of algebraically constructible loci. +Here is the parameter-space justification. Stratify the universal +embedded families by their Hilbert polynomials, Cartier indices, +and base-change loci. On each such stratum, +\Cref{generaltype:embedded-adjoint} can be imposed by parameterizing +the map +\[ + \omega_Z^{[m]}\longrightarrow\mathcal O_Z(1) +\] +with its prescribed divisor. These parameter spaces are of finite +type. The stable-pair and klt conditions are constructible there, +and their images are constructible by Chevalley's theorem. +This is the embedded construction used for stable log-variety +moduli in \cite[Section~6.A, especially Propositions~6.8 and~6.11 +and Lemma~6.12]{KovacsPatakfalvi}. In particular no assertion +about analytic minimal models is hidden in the good locus. + +Every very general fiber is projective: bigness makes it Moishezon, +and it is Kähler. We use Moishezon's projectivity criterion, which +is also the smooth case of \cite[Theorem~6]{NamikawaProjectivity}. +The projective klt finite-generation theorem +\cite[Theorem~1.1 and its consequences]{BCHM} therefore gives its log +canonical model. Every sufficiently large divisible pluricanonical +degree embeds that model and satisfies +\Cref{generaltype:embedded-adjoint}. Exclude at once the countably +many generic base-change loci, relative-rank loci, and generic-degree +loci for the maps constructed above. Let \(\Lambda_m\) consist of +the remaining parameters for which the degree-\(m\) map actually +constructs the fiber's log canonical model. +Their union contains a very general subset of \(Y\), and all genuine +frames over a point of \(\Lambda_m\) give good data. +Apply \Cref{generaltype:selection}. We obtain one fixed \(m\) for which +\(\Lambda_m\) is not contained in a countable union of proper analytic +subsets and good data hold on an algebraic open of \(Z_m\). + +For this degree the relative image map is generically birational. +Indeed its generic finite degree agrees with its fiberwise degree +outside the generic-degree exceptional set already excluded. +Choose a point of \(\Lambda_m\) there; that degree is one. +Retain \(\Lambda_m\): its actual canonical models will determine the +horizontal discrepancy signs at the end of the construction. + +\emph{The invariant classifying map.} +Fix the dimension, the volume read from +\Cref{generaltype:embedded-adjoint}, and a finite coefficient set +containing the coefficients of \(A\) and closed under sums at most one. +The good algebraic parameter locus has a classifying map to the +projective coarse space \(M\) of stable pairs with these data. +If an additional parameterization of the map in +\Cref{generaltype:embedded-adjoint} was used, its moduli graph descends: +the embedded pair determines a unique isomorphism class, so the +closure of that graph is generically a singleton over the good +parameter locus. Restricting a dense algebraic open therefore gives +the same rational classifying map on \(Z_m\). +It is invariant under change of projective frame. + +Compose with the meromorphic frame parameter map. +The resulting map \(\overline Q_m\dashrightarrow M\) is constant +on general frame fibers. Its graph has a proper analytic image +in \(Y\times M\). This image is generically a singleton over \(Y\), +so its projection to \(Y\) is bimeromorphic. It is the graph of +a meromorphic map +\[ + \mu\colon Y\dashrightarrow M. +\] +Let \(M_0\) be the closure of its image; it is projective. + +\emph{A stable parameter cover and the matching cover.} +Stable moduli supplies a finite parameter cover +\(S\to M\) from a normal scheme carrying a stable family +\cite[Corollary~6.19]{KovacsPatakfalvi}. +The coarse space \(M\) is projective +\cite[Corollary~7.3]{KovacsPatakfalvi}, so its finite cover \(S\) +is projective as well. The moduli construction for the specified +coefficient set is given in +\cite[Definition~6.2, Proposition~6.4, and +Proposition~6.14]{KovacsPatakfalvi}. +Resolve \(\mu\), and normalize a component of the pullback of +\(S\to M\) dominating \(Y\). Let \(R_0\) be its image in \(S\). +The restricted stable family over \(R_0\) has maximal variation, +because \(R_0\to M_0\) is finite. + +At this point the projective family exists over \(R_0\). +Equality of coarse moduli points identifies its general fibers with +those of the original analytic image, but does not identify the +families. We construct a further generically finite cover carrying +such an identification. + +On a dense analytic open of the comparison base, embed the original +flat image by the complete system of +\(\mathcal O_{\mathcal Z_m}(k)\), and the pulled-back stable family +by its \(km\)-adjoint system, for a common sufficiently large and +divisible \(k\). The two sheaves of sections are coherent direct +images of compact families; generic cohomology and base change make +them locally free after shrinking. On a common compact modification +of the base, make their torsion-free strict transforms locally free, +resolve the evaluation maps, and flatten the embedded image families +and their weighted boundary cycles. This gives holomorphic relative +Hilbert--Chow data. The modification concerns the base itself, not a +space of choices of frames. + +On every general good fiber, \Cref{generaltype:embedded-adjoint} +identifies the two polarizations. Thus the projective linear maps +identifying the two embedded varieties and their weighted boundary +cycles form a nonempty Isom set. This set is finite, since stable +pairs have finite automorphism groups +\cite[Propositions~6.5 and~6.8]{KovacsPatakfalvi}; completeness and +nondegeneracy of the embeddings exclude additional ambient +stabilizers. At this stage we use only the embedded analytic image +and its polarization. Its \(\Q\)-Gorenstein family structure will +be transported from the stable family by the identification. + +To compactify these Isom sets, work in local trivializations of the +two section bundles. In the product of their projective parameter +spaces with the projective matrix space, take the algebraic closure +of the incidence relation for invertible matrices identifying the +embedded varieties and weighted boundary cycles, after clearing the +fixed coefficient denominators. Pull back this closure by the two +Hilbert--Chow maps. Equivariance under changes of both frames glues +the local incidence spaces into a proper analytic space in the +projective bundle of homomorphisms between the section bundles; no +meromorphic trivialization is used. Retain its compact irreducible +components that dominate the base and meet the invertible locus, +and normalize them. The invertible locus has the finite Isom fibers +just described, so the resulting cover is proper and generically +finite. Its tautological isomorphism identifies the generic families, +including the stable family's \(\Q\)-Gorenstein structure; its graph +therefore extends meromorphically over the compact base. + +Finally resolve \(R_0\) projectively and resolve the graph of the +map from the comparison cover to it. Denote the resulting spaces +by \(R\) and \(Y'\). Stable families and their divisible relative +adjoint line bundles commute with base change, with boundary understood +by divisorial restriction \cite[Remark~3.8]{KovacsPatakfalvi}. +Thus the pullback family over \(R\) remains stable; it has klt general +fibers and maximal variation. All covers and modifications used above stay +in class \(\mathcal C\). + +On a common resolution we can now compare the source adjoint +divisor with the pullback of the stable relative adjoint divisor, +which is rational Cartier. There are only finitely many horizontal +prime coefficients in their difference. On fibers above +\(\Lambda_m\), away from the finitely many generic comparison +exceptions, they are the log canonical model discrepancy coefficients. +They are therefore nonnegative and supported on divisors exceptional +over the stable model. The same statements hold horizontally in +the family. Remove the images of the remaining vertical discrepancy +divisors. On the resulting dense analytic open this is the log +canonical model comparison, with its divisorial pushforward boundary. +This proves the asserted meromorphic comparison without relative +analytic finite generation. If \(M_0\) is a point, the construction +uses the fixed stable pair and \(R\) is a point. +\end{proof} + +\subsection{Relative forms and vertical valuations} + +For a meromorphic section \(\sigma\) of +\(\mathcal O_V(bK_{V/Y})\), its horizontal pole bound \(bA\) +means that \(\ord_E(\sigma)\geq-bA_E\) at every horizontal prime, +using a local nonvanishing base volume to interpret relative forms. +At a vertical prime over a divisor \(D\subset Y\), write +\(\omega_D\) for a local nonvanishing logarithmic top form on the +base with a simple pole along \(D\), and no other pole at its generic +point. We say that \(\sigma\) has the relative logarithmic bound at +this prime if +\begin{equation}\label{generaltype:relative-log-bound} + \ord_E\bigl(\sigma\wedge h^*\omega_D^{\,b}\bigr)\geq-b. +\end{equation} +The notation \(\wedge\) denotes the relative-times-base identification +of the corresponding pluricanonical line bundles. + +\begin{lemma}\label{generaltype:stable-orders} +In the situation of \Cref{generaltype:stable-comparison}, pull back a +holomorphic section of a divisible multiple of +\(K_{\mathcal V/R}+\mathcal A\), and compare it with relative pluriforms +on the main component of +\(V\times_Y Y'\). +It satisfies the horizontal bound given by the pullback of \(A\). +Moreover, on a common resolution it satisfies +\Cref{generaltype:relative-log-bound} at every prime lying over a +divisor of any smooth modification of the comparison base. +\end{lemma} + +\begin{proof} +On general fibers, pullback from the log canonical model lies in the +original log pluricanonical system. The effective canonical-model +discrepancy on a common resolution therefore proves the horizontal bound. + +For the vertical assertion, first fix a smooth modification of the +comparison base and a prime divisor on it. The stable family and its +divisible relative adjoint line bundles commute with this base change. +Work transversely to a general point of that divisor. The resulting +one-parameter family is still stable. This order of operations is +what permits the same bound at divisors exposed only on a higher base +model. Its total pair with the reduced +central fiber added is log canonical. More explicitly, choose the +disk with klt fibers away from its origin. Its total space is normal: +flatness over the smooth disk and the \(S_2\) property of the fibers +give \(S_2\). A codimension-one point on the central fiber is a +generic point of one of its reduced components. Its local ring is +one-dimensional, and its quotient by the disk parameter is a field; +its maximal ideal is consequently generated by that parameter, so +this local ring is regular. Away from the central fiber the total +space is normal, because its fibers are klt. Serre's criterion now +gives normality. +Compatibility of the relative adjoint divisor with the stable fiber +identifies the adjunction different on the normalization of the central +fiber with its stable boundary and conductor. The resulting pair is +log canonical because the central fiber is stable; see +\cite[Lemma~2.10 and Corollary~2.11]{PatakfalviFibered} for this +normalization-and-conductor identity. +The analytic Cartier-divisor inversion theorem +\cite[Theorem~1.1]{FujinoAnalyticAdjunction} consequently applies. +The argument also applies after any finite ramified disk change. + +Let \(t\) be the disk parameter and let \(\xi\) denote the +pulled-back relative section in degree \(b\). Then +\(\xi\wedge(dt/t)^b\) is a section of the total log canonical +multiple with the central fiber added. On a log resolution the +log canonical discrepancy bound says that this form has pole +order at most \(b\) at every vertical prime. A further common +resolution with the original family preserves this bound. +This is \Cref{generaltype:relative-log-bound}. Smooth transverse +parameters do not change any of these orders. +\end{proof} + +\begin{lemma}\label{generaltype:relative-system} +Under the hypotheses of \Cref{generaltype:stable-comparison}, there +is a positive integer \(b_0\) and a finite-dimensional space +\[ + \mathcal S\subset + H^0_{\mathrm{mer}}\bigl(V,\mathcal O_V(b_0K_{V/Y})\bigr) +\] +whose associated meromorphic map has rank \(\dim F\) on general +\(h\)-fibers, with the following properties: +\begin{enumerate}[label=\textup{(\roman*)}] +\item every member has horizontal pole bound \(b_0A\); +\item every member satisfies the relative logarithmic bound +at primes over base divisors; +\item the same bound may be tested after any modification of +the base, at a divisor exposed on that modification. +\end{enumerate} +The same assertions hold for the systems of products of members +of \(\mathcal S\). +\end{lemma} + +\begin{proof} +Use \Cref{generaltype:stable-comparison}. If \(\dim R>0\), the +stable-family bigness theorem +\cite[Theorem~8.1 and Corollary~8.3]{KovacsPatakfalvi} applies: +the base is normal projective, the family is stable and has +maximal variation, and its general fiber is klt. +Thus \(K_{\mathcal V/R}+\mathcal A\) is big. +Choose finitely many sections of one sufficiently divisible +multiple whose map is generically finite. Its image has dimension +\(\dim F+\dim R\), and its restriction to a general \(\pi\)-fiber +has rank \(\dim F\). If \(R\) is a point, choose such +sections of the ample log canonical divisor of the fixed stable +pair instead. Pullback and the meromorphic comparison preserve these +image dimensions and give the pole bounds of +\Cref{generaltype:stable-orders}. + +Replace the comparison cover by a Galois cover and resolve the +diagrams equivariantly. To do this analytically, first take its Stein +factorization over \(Y\); its finite factor has the same cover over a +dense open set. Over the finite étale locus, the Galois closure is a +component of a sufficiently large fiber product of this finite factor. +Take the corresponding compact component, normalize, and then resolve +the graph of its map to the comparison cover. Equivariant resolution +gives the required diagram; write \(G\) for its finite group. +This construction uses finite analytic maps and does not require the +meromorphic function field of \(Y\) to have transcendence degree +\(\dim Y\). + +Let \(\mathcal B\) be the graded \(\C\)-algebra generated by the +chosen finite set of relative sections and all its \(G\)-conjugates. +Every homogeneous element has the required horizontal and vertical +linear pole bounds: products add the orders and sums cannot +decrease the lower bounds. The algebra is integral and finite over +\(\mathcal B^G\). Indeed each homogeneous generator satisfies its +orbit polynomial with invariant coefficients, and invariants of +a finite group acting on a finitely generated algebra are finitely +generated. The corresponding map +\(\operatorname{Proj}\mathcal B\to\operatorname{Proj}\mathcal B^G\) +is finite. Before taking invariants, the system contains the sections +pulled back from \(\mathcal V\), whose restriction to a general source +fiber has image dimension \(\dim F\). A finite map preserves the +dimension of the image of that fiber, even though \(G\) may permute +the fibers above one point of \(Y\). Choose a Veronese degree for +\(\mathcal B^G\) generated by finitely many elements of that single +degree. Their ratios therefore still have rank \(\dim F\) on general +fibers. Denote their pluricanonical degree by \(b_0\). +The same finite quotient also preserves the total image dimension, +which is at least \(\dim F+\dim R\). + +Use the natural identification of relative canonical bundles on the +common smooth locus for every base change or modification. On the +dense étale locus, invariant relative +tensors therefore descend to \(V\). They extend meromorphically +across the remaining locus by finite analytic descent. The +horizontal bounds descend because the comparison cover is +generically étale along each horizontal divisor. + +For completeness, the vertical order calculation is exact. +Let \(E'\) lie over \(E\), with ramification index \(e\). +For a degree-\(b_0\) top pluriform \(\alpha\), +\begin{equation}\label{generaltype:ramified-order} + \ord_{E'}(\alpha_{\mathrm{pullback}})+b_0 + =e\bigl(\ord_E(\alpha)+b_0\bigr). +\end{equation} +A logarithmic top form on the base pulls back to a nonvanishing +logarithmic top form at a prime dominating its divisor. +Apply \Cref{generaltype:ramified-order} to the relative form +wedged with this base logarithmic form. +The bound upstairs is therefore equivalent to +\Cref{generaltype:relative-log-bound} downstairs. +For (iii), pull the chosen Galois comparison cover and its original +sections to a common refinement over the modified base. +Lemma~\ref{generaltype:stable-orders} applies to these same pullbacks +at primes above the newly exposed base divisor; its bounds pass to +their conjugates, products, and invariant combinations. The +ramification calculation therefore gives the bound for the pullback +of each fixed descended member of $\mathcal S$. +Products add the order inequalities, and taking a full product system +is a Veronese operation on its image, so products preserve the relative +rank as well. +\end{proof} + +\begin{lemma}\label{generaltype:expose-valuation} +Let \(h\colon V\to Y\) be a proper surjective holomorphic map +of irreducible compact complex spaces, with \(Y\) smooth. +If a prime divisor \(E\) of a smooth \(V\) does not dominate \(Y\), +there is a modification of the base and a dominating main-component +model on which the strict transform of \(E\) maps onto a prime +divisor of the modified base. +\end{lemma} + +\begin{proof} +Apply analytic flattening to \(h\), by strict transform, and resolve +the modified base. Pullback preserves the flatness of the intermediate +family, whose fibers have the generic relative dimension \(d\). +This intermediate source is proper and bimeromorphic over \(V\). +On a resolution dominating it, the strict transform of \(E\) remains +a divisor, because a proper modification of the smooth \(V\) is an +isomorphism at the generic point of \(E\). Its image in the flat +intermediate source has dimension \(\dim V-1\): it still dominates +\(E\). The fiber-dimension bound consequently gives an image on the +new base of dimension at least +\(\dim V-1-d=\dim Y-1\). +That image is proper, since its projection to \(Y\) is contained in +\(h(E)\), and properness makes it a closed analytic set. It is +therefore a prime divisor. Further source resolutions retain the +same generic image and do not change the valuation of \(E\). +\end{proof} + +\subsection{Proof of the addition proposition} + +\begin{proof}[Proof of \Cref{generaltype:addition}] +We may suppose \(\kappa(Y,K_Y+C)\geq0\). +Choose a Kähler modification \(\nu\colon\widetilde V\to V\), +resolve the boundary, +and put +\[ + \widetilde T=\nu_*^{-1}T+\operatorname{Exc}(\nu)_{\mathrm{red}}. +\] +By \Cref{setup:log-modification,setup:modification-compatibility}, +pullback and descent identify the divisible log pluricanonical rings +on the total spaces and on their very general fibers. In particular, +fiberwise bigness is preserved, and +\(B(h\circ\nu,\widetilde T)=B(h,T)\). +Every prime over base codimension at least two still has coefficient +one: it is either a strict transform of such a prime or is newly +exceptional. Write \((V,T)\) for this Kähler model; its sections +descend to the original pair. + +If \(\dim F=0\), the map is bimeromorphic. Pull back sections +of \(K_Y+C\). At a prime dominating a base divisor the +multiplicity is one and the condition \(C\leq B(h,T)\) gives +the required pole bound. At an exceptional prime the +log canonical discrepancy bound for \((Y,C)\), together with +\(T_E=1\), gives the same conclusion. Thus pullback preserves +all the base systems and proves the assertion. If \(Y\) is a +point, the assertion is simply the assumed bigness. We may +therefore assume both dimensions are positive. + +Write \(T^{\mathrm h}\) for the horizontal part of \(T\). +Choose a rational \(\epsilon>0\) sufficiently small and set +\[ + A=(1-\epsilon)T^{\mathrm h}. +\] +Its coefficients are strictly less than one. +To make \(K_F+A_F\) big uniformly on very general fibers, consider +\(\epsilon=1/j\) and all denominator-clearing degrees. Choose a fiber +outside their countably many generic base-change and relative-rank +exceptional sets. +Openness of the big cone on that projective fiber gives one +choice of \(\epsilon\) and one degree with full image dimension. +The generic-rank choice makes the same degree have full rank +on general fibers. Notice that no vertical coefficient has +been lowered, and \(C\) has not been changed. + +Apply \Cref{generaltype:relative-system} to \(A\), and write +\(\sigma\) for a member of its system in degree \(b\), including +any product degree. Let +\[ + \eta\in H^0\bigl(Y,\mathcal O_Y(b(K_Y+C))\bigr). +\] +The relative-times-base product +\(\sigma\wedge h^*\eta\) is a meromorphic section of +\(\mathcal O_V(bK_V)\). We check that its poles are bounded by +\(bT\) at every prime. + +There is nothing further to check at horizontal primes, since +\(A\leq T^{\mathrm h}\). Suppose next that \(E\) maps onto a +prime divisor \(D\subset Y\), and put +\[ + a_E=\ord_E(h^*D). +\] +Relative to a local nonvanishing logarithmic base volume +\(\omega_D\), the section \(\eta\) vanishes to order at least +\(b(1-C_D)\). The relative logarithmic bound gives +\begin{equation}\label{generaltype:orbifold-order} + \ord_E(\sigma\wedge h^*\eta) + \geq -b+b\,a_E(1-C_D). +\end{equation} +The inf-multiplicity convention is precisely +\[ + B(h,T)_D + =1-\max_{E\mapsto D}\frac{1-T_E}{a_E}. +\] +Consequently \(C_D\leq B(h,T)_D\) implies +\[ + a_E(1-C_D)\geq1-T_E, +\] +and \Cref{generaltype:orbifold-order} is at least \(-bT_E\). +This uses no integrality condition on the boundary multiplicities. + +Finally, suppose \(\operatorname{codim}_Y h(E)\geq2\). +Expose a base divisor \(D'\) below the strict transform of \(E\) +by \Cref{generaltype:expose-valuation}, and resolve the diagrams. +The pullback of \(\eta\) has at most a logarithmic pole along +\(D'\): this is the divisorial discrepancy inequality for the +log canonical smooth pair \((Y,C)\). +Property (iii) of \Cref{generaltype:relative-system} therefore +gives order at least \(-b\) at the strict transform of \(E\). +The order is unchanged, since source modifications are isomorphisms +at the generic point of \(E\). +The assumption \(T_E=1\) supplies exactly this allowance. +We have proved, at all prime divisors, that +\begin{equation}\label{generaltype:product-sections} + \sigma\wedge h^*\eta + \in H^0\bigl(V,\mathcal O_V(b(K_V+T))\bigr). +\end{equation} +On a smooth space these divisorial bounds imply holomorphic +extension of the corresponding log pluricanonical section. + +Choose a degree in which the complete base system has image +dimension \(\kappa(Y,K_Y+C)\), and pass to a common multiple +with \(b_0\) in \Cref{generaltype:relative-system}. +The product system of relative sections still has image dimension +\(d=\dim F\) on a general fiber. The corresponding product of the +chosen base system has image dimension \(\kappa(Y,K_Y+C)\), since a +Veronese embedding preserves its image. In this common degree write +the nonzero relative generators as \(\sigma_i\) and the base +generators as \(\eta_j\). +The total system in \Cref{generaltype:product-sections} contains +all \(\sigma_i\wedge h^*\eta_j\). +On the open set where \(\sigma_0\) and \(\eta_0\) are nonzero, +ratios in this system include +\[ + \frac{\sigma_i}{\sigma_0}, + \qquad + h^*\!\left(\frac{\eta_j}{\eta_0}\right). +\] +Its image therefore maps onto the base Iitaka image, and over +a general point of that image it contains the \(d\)-dimensional +image of a general \(h\)-fiber. +The dimension-of-fibers inequality gives image dimension at +least \(d+\kappa(Y,K_Y+C)\). This proves +\Cref{generaltype:inequality} on the Kähler model. +The initial log pluricanonical descent then proves it on the +given smooth source. +\end{proof} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/introduction.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..89a7bd12f4411f963dfbc4ab00e2266b40e3d185 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/introduction.tex @@ -0,0 +1,312 @@ +\section{Introduction}\label{intro:section} + +The Kodaira dimension measures the growth of pluricanonical forms. +For a surjective morphism $f:X\to Y$ of smooth projective complex varieties +with connected fibers, Iitaka's subadditivity problem asks whether +\[ + \kappa(X)\geq\kappa(F)+\kappa(Y), +\] +where $F$ is a general fiber. The logarithmic version permits poles +along reduced boundary divisors and therefore applies to open +varieties. We prove a common generalization in which rational boundary +coefficients and multiple fibers both contribute to the inequality. + +Iitaka's divisor-dimension and logarithmic theories provide the +language for this question \cite{IitakaOriginal,IitakaLog}; an +explicit logarithmic subadditivity conjecture appears in +\cite[\S3]{IitakaClassification}. +The lower bound requires global pluricanonical forms that retain +the growth of both the fibers and the base. Direct-image positivity +established central cases: Kawamata treated curve bases and fibers +admitting good minimal models, Viehweg developed weak positivity +and addition over a base of general type, and Koll\'ar proved +addition for general-type fibers +\cite{KawamataCurves,KawamataKodaira,ViehwegAdditivity,Kollar87}. +Weak positivity controls symmetric powers after small positive +ample twists; a general-type base provides enough canonical +positivity to absorb them. + +Other advances exploit dimension or the geometry of the base: +Birkar treated total dimension at most six, Cao--P\u{a}un and +Hacon--Popa--Schnell treated abelian or maximal-Albanese-dimension +bases, and Cao treated projective klt pairs over surfaces +\cite{BirkarIitakaSix,CaoPaunAbelian,HaconPopaSchnellAbelian,CaoSurfaces}. +Thus the established cases draw positivity from different parts of +the fibration, rather than imposing one uniform condition on it. + +For logarithmic pairs, Kov\'acs--Patakfalvi proved the projective +log-general-type-fiber case \cite[Theorem~9.6]{KovacsPatakfalvi}, +while Hashizume proved reduced-SNC subadditivity for fibers with +abundant log canonical divisor \cite[Theorem~1.2]{Hashizume}. +Maehara had established logarithmic addition over a base of log +general type; Fujino gives a proof through twisted weak positivity +\cite[Theorem~1.9]{FujinoWeakPositivity}. +Campana proved addition over an orbifold base of general type, and +Wang proved klt K\"ahler subadditivity over complex tori, including +an inf-multiplicity refinement +\cite[Theorem~6.3]{CampanaOrbifolds} +\cite[Main Theorem~A and Theorem~E]{WangKahlerIitaka}. +The present argument uses general-type addition only after a relative +Iitaka reduction; the original fiber need not be of general type +or admit a good minimal model. + +Earlier preprints also state general ordinary inequalities. +Tsuji states ordinary subadditivity following a direct-image generation +theorem \cite[Theorems~1.9 and~1.13]{TsujiDirectImages}; his +Remark~1.14 distinguishes an unproved variation refinement. +Maehara states an ordinary determinant inequality involving variation +\cite[\S7, Theorem~8]{MaeharaIitaka}. This later preprint claim is +distinct from his logarithmic general-type-base theorem above. +Neither preprint statement is a proof input here. + +\subsection{The orbifold base and the main theorem} + +Multiple fibers carry information that the ordinary base omits. +Campana's orbifold base records it through inf multiplicities, and +his birational invariant gives the corresponding form of Iitaka's +conjecture \cite{Campana2004,CampanaOrbifolds}. We specify that +invariant before stating the theorem. + +\label{setup:orbifold-base} +A fibration is a proper surjective holomorphic map with connected +fibers. A rational SNC boundary on a smooth manifold is a finite +sum $\Delta=\sum_i\delta_i\Delta_i$, with +$\delta_i\in\mathbf Q\cap[0,1]$ and simple normal crossing support. +Write $\Delta_E$ for the coefficient of a source prime $E$. Set +\[ + m_\Delta(E)=\frac1{1-\Delta_E}, + \qquad \frac1{0}=\infty, +\] +where $\Delta_E=0$ off the boundary. If the base of +$f:(X,\Delta)\to Y$ is smooth, define +\begin{equation}\label{setup:inf-boundary} +\begin{gathered} + m(f,\Delta;D)=\min_{E\mapsto D} + \bigl\{\operatorname{ord}_E(f^*D)m_\Delta(E)\bigr\}, + \qquad + B(f,\Delta)=\sum_D\left(1-\frac1{m(f,\Delta;D)}\right)D. +\end{gathered} +\end{equation} +Here $D$ ranges over base prime divisors, $E$ over source primes +dominating $D$, and $1/\infty=0$. The minimum is nonempty and +only finitely many coefficients of $B(f,\Delta)$ are nonzero. +There is no divisibility condition on these multiplicities. + +For smooth sources and bases, an elementary change of model is a +commutative diagram of fibrations +\[ +\begin{tikzcd} + (X',\Delta') \arrow[r,"u"] \arrow[d,"f'"'] & + (X,\Delta) \arrow[d,"f"]\\ + Y' \arrow[r,"v"'] & Y +\end{tikzcd} +\] +where $u,v$ are proper bimeromorphic holomorphic maps, +$u_*\Delta'=\Delta$, and both boundaries are rational SNC +boundaries in $[0,1]$. We require, for every prime $D$ on $X$ +and prime $E$ on $X'$ with $t=\operatorname{ord}_E(u^*D)>0$, +\begin{equation}\label{setup:orbifold-morphism} + t\,m_{\Delta'}(E)\geq m_\Delta(D). +\end{equation} +We permit $\infty\geq\infty$ and impose no condition for $t=0$. +Write $f'\sim f$ for the equivalence relation generated by these +diagrams, allowing arrows in either direction. Define +\begin{equation}\label{setup:invariant-base} + \kappa(f\mid\Delta)= + \inf_{f':(X',\Delta')\to Y'\sim f} + \kappa\bigl(Y',K_{Y'}+B(f',\Delta')\bigr). +\end{equation} +For a normal singular base $Y$, resolve $Y$, resolve the main +component of the fiber product with $X$, and give the resulting +smooth source the strict transform of $\Delta$ plus the reduced +exceptional divisor over $X$. After resolving this boundary, apply +the preceding definition. It is independent of the chosen resolution +\cite[Definitions~2.1, 2.3, 3.2, 4.7, and~4.10; +Remark~4.13 and Corollary~4.14]{CampanaOrbifolds}. + +For a rational line bundle $L$, $\kappa(Z,L)$ is the maximal image +dimension of its complete linear systems in sufficiently divisible +positive degrees, and is $-\infty$ if all these systems are empty. +A point has Kodaira dimension zero, and +$(-\infty)+a=-\infty$, including $a=-\infty$. +Fujiki class $\mathcal C$ consists of compact complex spaces +bimeromorphic to compact K\"ahler spaces. + +\begin{theorem}[Orbifold subadditivity]\label{main:orbifold} +Let $X$ be a smooth compact connected complex manifold in Fujiki +class $\mathcal C$, let $\Delta$ be a rational SNC boundary, and +let $f:X\to Y$ be a surjective holomorphic map with connected +fibers onto a normal compact irreducible complex space. +For a very general smooth fiber $F$, put $\Delta_F=\Delta|_F$. +Then +\begin{equation}\label{main:inequality} + \kappa(X,K_X+\Delta) + \geq\kappa(F,K_F+\Delta_F)+\kappa(f\mid\Delta). +\end{equation} +\end{theorem} + +This is a positive resolution of Campana's orbifold Iitaka +conjecture in its rational SNC, Fujiki-class-$\mathcal C$ +formulation, including coefficient one +\cite[Conjecture~6.1 and Remark~6.2]{CampanaOrbifolds}. +Here a very general fiber is chosen outside a countable union of +proper closed analytic subsets, including the critical values and +the exceptional images of boundary strata. + +For reduced SNC divisors $D_X,D_Y$ satisfying +$\operatorname{Supp}(f^*D_Y)\subseteq\operatorname{Supp}(D_X)$, +the invariant base term is at least $\kappa(Y,K_Y+D_Y)$. +Theorem~\ref{main:orbifold} therefore gives +\[ + \kappa(X,K_X+D_X) + \geq\kappa(F,K_F+D_X|_F)+\kappa(Y,K_Y+D_Y) +\] +for fibrations between smooth compact class-$\mathcal C$ manifolds. +For a smooth quasi-projective variety $U$, its logarithmic Kodaira +dimension is $\bar\kappa(U)=\kappa(X,K_X+D_X)$ on a smooth +projective compactification with reduced SNC boundary +$D_X=X\setminus U$. Compatible compactifications give logarithmic +subadditivity for dominant morphisms of smooth quasi-projective +complex varieties with geometrically connected general fiber. +Empty boundaries give ordinary subadditivity. The projective +ordinary and logarithmic statements also hold over algebraically +closed fields of characteristic zero, using geometric generic fibers. + +\subsection{The reduction and the two positivity inputs} + +The proof first replaces the original fiber by a general-type +object without losing its complete pluricanonical systems. +Assume the two terms on the right of Theorem~\ref{main:orbifold} +are nonnegative, and put $k=\kappa(F,K_F+\Delta_F)$. +A relative Iitaka map gives, after the model changes of Section~\ref{setup:section}, +\[ + X\xrightarrow{g}W\xrightarrow{h}Y, + \qquad \dim W_y=k, + \qquad\kappa(G,K_G+\Delta_G)=0 +\] +for a very general $g$-fiber $G$. + +The canonical-bundle-formula strategy separates the contribution +of singular fibers from a moduli contribution on $W$. +Fujino--Mori developed a higher-dimensional logarithmic formula, +and Ambro organized its discriminant and moduli parts birationally +\cite{FujinoMori,AmbroBoundary,AmbroModuli}. +Our use of this strategy requires an exact comparison on the fixed +original source. Section~\ref{setup:section} obtains it from the orders of relative +pluriforms. A sufficiently divisible pluricanonical system on $G$ +is one-dimensional. Taking a root of its generator on a cyclic +cover gives a differential form, and the line it spans belongs to +the cohomology of that cover. Tensor descent and extension across +the boundary produce a rational line bundle $M$ on $W$. +The orders of the same form determine a boundary $T$ such that +\[ + B(g,\Delta)\leq T\leq1, + \qquad + H^0\bigl(X,m(K_X+\Delta)\bigr) + \simeq H^0\bigl(W,m(K_W+T+M)\bigr) +\] +in sufficiently divisible degrees. The comparison also holds +relatively over $Y$. Thus $K_W+T+M$ is big on the +$k$-dimensional fibers of $h$. + +Two inputs turn this fiberwise bigness into the required global +lower bound. The first concerns the line $M$: the period map, +which records how the Hodge structures of the covers vary, supplies +a second fibration $p:W\to S$, with $S$ smooth projective, and +a nef rational line bundle $P$ such that +\[ + M=p^*P,\qquad K_S+aP\ \text{is big} +\] +for some positive rational $a$. The maps $h$ and $p$ have different +jobs, as Figure~\ref{fig:iitaka-period} shows. The second input is +log-general-type addition: a log canonical divisor that is big on +the fibers of $h$ has Kodaira dimension at least their dimension +plus the orbifold contribution of $Y$. + +\input{figures/two-maps} + +Section~\ref{addition:section} states both inputs precisely and proves the intervening +addition principle. Write $D_W=K_W+T$. For a fixed rational $c<1$ +close to one, general-type addition gives sufficiently many ratios +of sections of $D_W+cM+\eta p^*A$, where $A$ is ample on $S$ +and $\eta>0$ is rational. Weak positivity then supplies one fixed +section of $D_W+aM-\lambda p^*A$ for suitable $a>1$ and +$\lambda>0$. A positive rational combination cancels the ample +terms and gives coefficient one on $M$. Multiplication by powers +of that fixed section preserves the ratios of the first system. +This is an exact section argument; it requires neither abundance +of $P$ nor a limiting assertion for Kodaira dimensions. + +The full proofs of the inputs follow this calculation. +Section~\ref{period:section} proves adjoint positivity of the Hodge line. It combines +the curvature and boundary theory of Griffiths, Schmid, and +Cattani--Kaplan--Schmid with a projective period quotient +\cite{GriffithsPeriodsIII,Schmid,CKS}. +The construction retains suitable rational adjoint factors of the +ambient integral variation, associated with its monodromy, from which +a positive power of the chosen line can be recovered. Their full +periods give the quotient, using Villadsen's class-$\mathcal C$ +compactification theorem and the period-quotient approach of +Bakker--Brunebarbe--Tsimerman and Sommese; curvature and a volume +argument establish its projectivity +\cite{BakkerBrunebarbeTsimerman,SommesePeriodMapping,VilladsenHodgeKahler}. + +On this quotient $S$, the retained full period map is generically +immersive, but the map remembering only the chosen line may have +smaller rank. Curvature identifies the latter rank with the numerical +dimension $\nu(P)$, measured by its nonzero intersection powers. +Let $D$ be the reduced boundary where the retained variation has +nonidentity local monodromy. Brunebarbe--Cadorel gives $K_S+D$ big +\cite{BrunebarbeCadorel}. Using Andr\'e's monodromy normality and +Bakker--Tsimerman's Ax--Schanuel theorem to analyze fibers of the +line map, the boundary argument proves +$\nu(P|_{D_i})<\nu(P)$ for every component $D_i$ of $D$, including +finite nonidentity monodromy +\cite{AndreMonodromy,BakkerTsimermanAxSchanuel}. +The boundary loss in the asymptotic section count consequently has +lower degree in the coefficient of $P$ than the available sections +on $S$. Lemma~\ref{period:subtract} turns this comparison into +bigness of $K_S+aP$. + +Section~\ref{generaltype:section} proves log-general-type addition in class $\mathcal C$. +Kov\'acs--Patakfalvi's stable-family positivity applies to projective +families. We construct such an auxiliary family and descend its +relative sections to the original class-$\mathcal C$ base. +Fujino's weak positivity supplies the other direct-image input +used in Section~\ref{addition:section} \cite{KovacsPatakfalvi,FujinoWeakPositivity}. +Section~\ref{application:section} then returns once to the original fibration, applies the +addition principle to $(W,T,M)$, and derives the ordinary, +logarithmic, characteristic-zero, and core consequences. The core +and the relevant specialness of Kodaira-zero fibers are Campana's +constructions and results \cite{Campana2004,CampanaOrbifolds}. + +\subsection{The reduced-boundary refinement} + +For a rational boundary, the root form above spans one character +subspace of the cyclic cover's highest Hodge space; the whole +highest space can be larger. For projective reduced-boundary pairs +of logarithmic Kodaira dimension zero, Section~\ref{cyc:section} proves more: +the minimal root cover has a one-dimensional entire logarithmic +top-form space. Mixed-Hodge extension and semipositivity in the +forms of Fujino--Fujisawa control the coefficient-one boundary +\cite{FujinoFujisawa,FujinoFujisawaVMHS}. +We identify the resulting line with the moduli divisor in the +projective subpair setting of Bakker--Filipazzi--Mauri--Tsimerman +\cite[Theorem~6.28]{BFMT}. This identification does not use their +separate semiampleness theorem, whose hypotheses include +effectiveness over the generic point \cite[Theorem~1.5]{BFMT}. +Section~\ref{cyc:section} also gives the exact divisorial normalization. +Appendix~\ref{p1logcbfalt:section} collects complementary projective +calculations by valuations and toroidal models, together with the +comparison on higher models. + +The logarithmic lower bound, adjoint positivity, and this +reduced-boundary refinement supply the stated inputs to the +whole-fiber-variation companion \cite{IitakaVariationCompanion}. The reverse-inequality companion +\cite{IitakaAdditivityCompanion} uses the lower bound only for additivity; its upper theorem has +additional smoothness hypotheses on all boundary strata. +Neither companion is used in the proof of Theorem~\ref{main:orbifold}. +The separate article on projective Hodge lines +\cite{IitakaProjectiveCompanion} gives a complete +ordinary proof through rationally polarized integral variations, +together with the distinct integral and analytic boundary arguments. diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/models.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/models.tex new file mode 100644 index 0000000000000000000000000000000000000000..2ee241bb07c96e5a8aa7a514097d4e1da12fe7ab --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/models.tex @@ -0,0 +1,364 @@ +\section{The relative Iitaka reduction and its Hodge line} +\label{setup:section} + +The first task is to replace the fibers by their Iitaka images while +retaining every sufficiently divisible pluricanonical system. We first +fix the birational models on which divisorial order tests suffice. The +root form of a Kodaira-zero fiber will then provide a Hodge line and an +exact comparison of sections on the intermediate base. + +We use additive notation for rational line bundles; equality of them +means equality in $\operatorname{Pic}\otimes\Q$. A rational line is +effective if a positive integral multiple has a nonzero section. +All degree assertions below are made after clearing denominators. +For a fibration $g:(Z,\Gamma)\to T$ and a very general smooth +fiber $F$, adjunction gives +\begin{equation}\label{setup:fiber-adjunction} + (K_Z+\Gamma)|_F=K_F+\Gamma|_F. +\end{equation} + +\subsection{The fixed smooth models} + +A smooth-base fibration $g:(Z,\Gamma)\to T$ is \emph{neat} if there is a +proper bimeromorphic orbifold morphism +$w:(Z,\Gamma)\to(Z_0,\Gamma_0)$ to a smooth pair, with +$w_*\Gamma=\Gamma_0$, such that every prime divisor sent by $g$ into +codimension at least two is also sent by $w$ into codimension at least +two. Campana's construction gives suitable neat models, model +independence, and the formula +\begin{equation}\label{setup:neat-formula} + \kappa(g\mid\Gamma)=\kappa\bigl(T,K_T+B(g,\Gamma)\bigr) + \quad\text{on a neat model} +\end{equation} +\cite[Definition~3.8, Proposition~3.10, Corollary~4.11, and +Remark~6.2]{CampanaOrbifolds}. +For the proof we only need the defining inequality, valid on every +allowed smooth model: +\begin{equation}\label{setup:base-comparison} + \kappa\bigl(T,K_T+B(g,\Gamma)\bigr) + \ \geq\ \kappa(g\mid\Gamma). +\end{equation} + +\subsection{Reduced exceptional boundaries} + +\begin{lemma}[Logarithmic modifications]\label{setup:log-modification} +Let $p:Z'\to Z$ be a proper bimeromorphic holomorphic map between smooth +compact manifolds. Suppose that $\Gamma$ has simple normal crossing +support and coefficients in $\Q\cap[0,1]$, and set +\[ + \Gamma'=p_*^{-1}\Gamma+\operatorname{Exc}(p)_{\mathrm{red}}. +\] +Assume, after further resolution if necessary, that this boundary has +simple normal crossing support. There is an effective $p$-exceptional +rational divisor $A$ such that +\begin{equation}\label{setup:log-discrepancy} + K_{Z'}+\Gamma'=p^*(K_Z+\Gamma)+A. +\end{equation} +For every sufficiently divisible positive integer $m$, the natural +identification of meromorphic pluricanonical forms induces +\begin{equation}\label{setup:section-invariance} + p_*\mathcal O_{Z'}\bigl(m(K_{Z'}+\Gamma')\bigr) + =\mathcal O_Z\bigl(m(K_Z+\Gamma)\bigr). +\end{equation} +These identifications respect multiplication, so the divisible section +rings and their Iitaka dimensions agree. Moreover, $p$ satisfies +\eqref{setup:orbifold-morphism}. +\end{lemma} + +\begin{proof} +The difference in \eqref{setup:log-discrepancy} is exceptional, since +$p$ is an isomorphism at the generic point of every target prime. +To compute its coefficient at an exceptional prime $E$, choose +compatible local canonical forms. +Near a general point of its center on $Z$, let $z_1,\ldots,z_n$ be +coordinates in which the boundary components containing that center are +$z_i=0$ for $1\leq i\leq r$, with coefficients $\gamma_i$. +Put $t_i=\ord_E(p^*z_i)$ and let $k_E$ be the order of the Jacobian of +$p$ along $E$. The pullback of +\[ + \frac{dz_1}{z_1}\wedge\cdots\wedge\frac{dz_r}{z_r} + \wedge dz_{r+1}\wedge\cdots\wedge dz_n +\] +has at most a simple pole along $E$. Indeed, at a general smooth point of +$E$, each logarithmic factor has the form $t_i\,du/u$ plus a holomorphic +one-form, and a nonzero wedge product contains at most one factor +$du/u$. Consequently $k_E+1\geq\sum_i t_i$. The coefficient in question +is therefore +\[ + k_E+1-\sum_i\gamma_i t_i + \ \geq\ \sum_i(1-\gamma_i)t_i\ \geq\ 0. +\] +For an effective integral exceptional divisor $mA$, normality of $Z$ +gives $p_*\mathcal O_{Z'}(mA)=\mathcal O_Z$: a meromorphic function +with poles only on $mA$ descends outside the codimension-two exceptional +image and extends by Hartogs' theorem. The projection formula proves +\eqref{setup:section-invariance}, compatibly with products and ratios. + +Finally, an exceptional prime has infinite boundary multiplicity. +A nonexceptional prime is a strict transform, with pullback order one +and unchanged coefficient. These two cases prove +\eqref{setup:orbifold-morphism}. +\end{proof} + +\begin{corollary}[Descent across exceptional centers] +\label{setup:exceptional-descent} +In the situation of \Cref{setup:log-modification}, a meromorphic +$m$-pluricanonical form on $Z'$ satisfying the boundary bounds of +$m\Gamma'$ at every nonexceptional prime descends to a section of +$m(K_Z+\Gamma)$ whenever $m\Gamma$ and $m\Gamma'$ are integral. +The assertion also holds after tensoring by the pullback of a line +bundle on $Z$. +\end{corollary} + +\begin{proof} +On the complement of the exceptional image the form is a holomorphic +section of the prescribed line bundle on $Z$: divisorial regularity +suffices on a smooth space. A local trivialization reduces extension +across the remaining codimension-two analytic subset to Hartogs' +extension theorem. The tensor-twisted assertion has the same proof. +\end{proof} + +\begin{corollary}[Fibers, composition, and a fixed base] +\label{setup:modification-compatibility} +Let $p:(Z',\Gamma')\to(Z,\Gamma)$ be a modification with the boundary +rule of \Cref{setup:log-modification}. Then: +\begin{enumerate}[label=\textup{(\roman*)}] +\item If $p$ is a modification over a fixed base, it induces a +logarithmic modification on a very general smooth fiber. The log +Kodaira dimensions of the corresponding fibers agree. +\item A finite composition of such modifications again has boundary +equal to the strict transform of the initial boundary plus the reduced +exceptional divisor of the composite. +\item If $g:(Z,\Gamma)\to T$ has smooth base and $g'=g\circ p$, then +\begin{equation}\label{setup:fixed-base-boundary} + B(g',\Gamma')=B(g,\Gamma). +\end{equation} +\end{enumerate} +\end{corollary} + +\begin{proof} +For (i), choose the parameter so that the relevant smoothness and +dimension statements hold for the finitely many exceptional components +and their images. A component dominating the base restricts to a +reduced divisor, and its exceptional image restricts to codimension at +least two. Components not dominating the base do not meet this fiber. +The restricted map has the same boundary rule, so +\Cref{setup:log-modification} applies on the fiber. For (ii), every +prime exceptional for the composite is either newly exceptional or the +transform of an earlier exceptional prime; in both cases its coefficient +is one. + +For (iii), fix a prime $D\subset T$. Every old prime dominating $D$ +has a strict transform, with the same pullback order and boundary +multiplicity. Every new prime in the minimum defining +$m(g',\Gamma';D)$ is exceptional for $p$, so its boundary multiplicity +is infinite. Adding these entries leaves the old minimum unchanged, +including when that minimum was already infinite. +\end{proof} + +For choice independence in the singular-base definition, take a common +resolution of two chosen diagrams. Their induced source boundaries +coincide: old primes retain their coefficients and all primes exceptional +over the initial $X$ have coefficient one. The comparison maps satisfy +\Cref{setup:log-modification}, so the smooth fibrations are equivalent +and their total-space and very general fiber log Kodaira dimensions agree. + +\subsection{Flattening relative to a fixed reference pair} + +To compare forms over generic points of base divisors, we need control of +source divisors over higher codimension. Proper analytic flattening gives +a base modification whose strict transform is flat +\cite{HironakaFlattening,FujikiDouady}. Here the strict transform is the +closure of the original family over the isomorphism locus, with base +torsion removed. Further base change preserves flatness. We use only +the resulting equidimensionality before resolving this intermediate +source. + +\begin{lemma}[Flattening and the exceptional reference] +\label{setup:flattening} +Let $g_0:(Z_0,\Gamma_0)\to T_0$ be a fibration from a smooth compact +pair with rational SNC boundary in $[0,1]$ to a smooth compact manifold. +There are a smooth-base modification +$q:T\to T_0$, a smooth-source modification $p:Z\to Z_0$, and a +fibration $g:Z\to T$ with $q\circ g=g_0\circ p$ such that, for +\[ + \Gamma=p_*^{-1}\Gamma_0+\operatorname{Exc}(p)_{\mathrm{red}}, +\] +the boundary has simple normal crossing support and every prime divisor +$E\subset Z$ satisfying $\operatorname{codim}_Tg(E)\geq2$ is +$p$-exceptional. In particular every such prime has coefficient one in +$\Gamma$, and the resulting fibration is neat with reference pair +$(Z_0,\Gamma_0)$. + +One may resolve prescribed proper analytic bad loci on the base during +this construction. After any finite sequence of further base +modifications, one can choose dominating smooth source models for which +the same exceptional assertion holds with respect to the original +reference $Z_0$. +\end{lemma} + +\begin{proof} +Flatten first, before resolving the source. After resolving the new base +by further base change, denote the equidimensional strict transform by +$\bar g:\bar Z\to T$, and its modification to $Z_0$ by $\bar p$. +Resolve $\bar Z$ and all boundary data, and write +\[ + Z\xrightarrow{r}\bar Z\xrightarrow{\bar p}Z_0, + \qquad p=\bar p\circ r, + \qquad g=\bar g\circ r. +\] +Let $N=\dim Z_0$, $b=\dim T$, and $d=N-b$. If a prime $E$ on $Z$ +has image contained in an analytic subset $C\subset T$ of codimension +at least two, equidimensionality gives +\[ + \dim\bar g^{-1}(C)\leq\dim C+d\leq N-2. +\] +It follows that $r(E)$, and therefore also $p(E)$, has dimension at most +$N-2$. Thus $E$ is exceptional over $Z_0$. The boundary and +orbifold-morphism assertions follow from \Cref{setup:log-modification}. +The new map has connected general fiber. +Its Stein factorization has a finite bimeromorphic map to the normal +base $T$, which is an isomorphism, so all its fibers are connected. + +For any further base modification $T'\to T$, pull back the flat +intermediate family and take a common resolution dominating the previous +smooth source. The same dimension estimate proves exceptionality over +$Z_0$. This includes principalizations of prescribed bad loci and their +SNC resolutions on the base. Iterating proves the persistence assertion; +\Cref{setup:modification-compatibility}\textup{(ii)} identifies the +stepwise boundary with that computed directly over $Z_0$. +\end{proof} + +The resolution need not be flat: the conclusion we retain is +exceptionality over $Z_0$, which permits +\Cref{setup:exceptional-descent}. + +All constructions remain available in Fujiki class $\mathcal C$. +A smooth compact member admits a compact K\"ahler modification +\cite[Theorems~0.7 and~3.4]{DemaillyPaun2004}. Starting with one for +$Z_0$, graph resolutions and principalizations by smooth-center blowups +give a K\"ahler dominating source. K\"ahlerness is preserved by these +projective modifications, and proper meromorphic images remain in +class $\mathcal C$ \cite[Lemmas~4.4 and~4.6\textup{(3)}]{FujikiDouady}. +Thus the bases also admit K\"ahler models. The preceding dimension +argument still refers to the original $Z_0$. + +\subsection{Keeping the original base fixed} + +\begin{corollary}[Simultaneous model control]\label{setup:simultaneous} +Suppose a smooth pair $(X_1,\Delta_1)$ has a fibration $f_1:X_1\to Y$ +to a fixed smooth base, and every prime divisor contracted by $f_1$ into +codimension at least two has coefficient one in $\Delta_1$. Put +$C=B(f_1,\Delta_1)$. Let a factorization through another smooth compact +space be given, after an allowed source modification if necessary, by +\[ + X_1\xrightarrow{g_1}W_1\xrightarrow{h_1}Y. +\] +Apply \Cref{setup:flattening} to $g_1$ and make any finite number of +subsequent modifications of $W_1$, always taking dominating smooth +source models with reduced added exceptional boundaries. The resulting +diagram $X\xrightarrow{g}W\xrightarrow{h}Y$, with source boundary +$\Delta$, can be chosen to have all the following properties: +\begin{enumerate}[label=\textup{(\roman*)}] +\item Every prime contracted by $g$ into codimension at least two is +exceptional over the chosen smooth reference source for $g_1$. +\item Every prime contracted by $f=h\circ g$ into codimension at least +two in $Y$ has coefficient one in $\Delta$. +\item $B(f,\Delta)=C$. In particular, for each prime $D\subset Y$ and +each prime $E\subset X$ dominating $D$, +\begin{equation}\label{setup:retained-multiplicity} + \ord_E(f^*D)\,m_\Delta(E)\geq m_C(D). +\end{equation} +\item The log Kodaira dimensions of the total source and of its very +general fibers over $Y$ are unchanged by these source modifications. +\end{enumerate} +The hypothesis on contracted divisors follows from +\Cref{setup:flattening}, applied to the original map to its resolved +base before fixing $Y$. +\end{corollary} + +\begin{proof} +Assertion (i) is \Cref{setup:flattening}. For (ii), a source prime +is either newly exceptional, with coefficient one, or the strict +transform of an old prime with unchanged image in $Y$. The hypothesis +handles the latter case. Assertions (iii) and (iv) are +\Cref{setup:modification-compatibility}; +\eqref{setup:retained-multiplicity} is the defining minimum for $C$. +Flattening the original map supplies the initial coefficient-one +condition by making its contracted primes exceptional. +\end{proof} + +In applications we include the critical values and images of boundary +strata among the base bad loci, so that $C$ has SNC support. We then +fix $Y$ and use \Cref{setup:simultaneous} for changes of intermediate +bases. + +\subsection{The relative Iitaka map} + +\begin{lemma}[Relative Iitaka construction] +\label{application:relative-iitaka} +Let \(f:(X,\Delta)\to Y\) be a fibration between smooth compact +manifolds in Fujiki class \(\mathcal C\), with \(\Delta\) a rational +SNC boundary in \([0,1]\). Suppose that +\[ + k=\kappa(F,K_F+\Delta_F)\geq 0 +\] +on very general smooth fibers. After modifications of the source, adding +the reduced exceptional divisor to the strict transform of the boundary, +there is a factorization +\[ + X\xrightarrow{g}W\xrightarrow{h}Y +\] +with smooth compact intermediate space in class \(\mathcal C\), both +maps proper with connected fibers, and +\[ + \dim W_y=k,\qquad + \kappa(G,K_G+\Delta_G)=0 +\] +for very general fibers of \(h\) and \(g\), respectively. The factorization +restricts to the log Iitaka fibration on very general \(f\)-fibers. +These properties persist on higher models obtained by the same rule for +the source boundary. +\end{lemma} + +\begin{proof} +Consider all degrees clearing the denominators of \(\Delta\). +Their relative adjoint direct images are coherent. First shrink to +the dense analytic open where the map and boundary strata are smooth. +There the family and its adjoint line bundles are flat over the base; +generic constancy of the fiber section dimension permits cohomology +and base change \cite[\S7, nos.~4--6, S\"atze~3--5]{GrauertDirectImages}, +with its corrigendum \cite{GrauertDirectImagesCorr}. +Together with the evaluation ranks this gives a dense analytic open +for each degree; +outside the resulting countable collection of exceptional sets, the +maximum image dimension is \(k\), attained in one fixed degree. + +Choose a sufficiently divisible degree giving the stabilized maps, +resolve the relative meromorphic map to the projective bundle of its +direct image, and take Stein factorization onto the image. The Iitaka +fibration theorem for line bundles on compact complex spaces identifies +the resulting map on very general fibers +\cite[Chapter~II, \S5; Chapter~III, \S7]{UenoClassification}. +No finite generation is required. The image has relative dimension \(k\); its +Stein factor is in class \(\mathcal C\), being a proper image of a space +in that class. Resolve it and the resulting source map. Connectedness +of the fibers of \(h\) follows from that of \(f\), since their images +under \(g\) are precisely the fibers of \(h\). + +The assertion that the restriction has Kodaira dimension zero concerns +the original line bundle. Its standard proof applies here: coherent +direct images on the projective Iitaka image acquire sections after an +ample twist. If the +restriction had positive Iitaka dimension, multiplying those sections +by powers of the defining system would enlarge the Iitaka image. +A nonzero defining section restricts nontrivially to a very general +Iitaka fiber, so the restriction has dimension exactly zero. + +Apply this to the resolved log adjoint on a very general \(f\)-fiber. +Its divisible section ring is unchanged by +\Cref{setup:log-modification}, and adjunction identifies the restriction +with \(K_G+\Delta_G\). Further source modifications preserve the same +section ring, hence the relative Iitaka factorization and its asserted +fiber dimensions. +\end{proof} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/normalization.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/normalization.tex new file mode 100644 index 0000000000000000000000000000000000000000..8b70fec2ec65e214918785fd27ef140e58274528 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/normalization.tex @@ -0,0 +1,428 @@ +\section{Supplementary projective calculations} +\label{p1logcbfalt:section} + +The reduced-boundary construction of Section~\ref{cyc:section} admits +complementary projective proofs and calculations. We first give +divisorial proofs of the section comparisons and recover the +fixed-source descent by extracting valuations on the base. The fixed +logarithmic zero divisor then gives another discrepancy-sheaf test, +and a toroidal calculation proves logarithmic extension of the root +form. The final subsection tracks the auxiliary subpair when the +minimum-rule divisor is recomputed on a higher base model. This last +operation is distinct from the prescribed boundary modification of +Lemma~\ref{lem:normalized-models}: its discriminant is recomputed, +rather than defined as the strict boundary plus the reduced exceptional +divisor. Throughout these calculations, modifications of the original +source carry strict boundary plus reduced exceptional boundary. + +\subsection{Divisorial sections and relative Iitaka systems} + +The following elementary comparisons fix the complete systems and the +original models used in the normalization. All varieties in this +subsection are projective over $\C$, all rational divisor degrees are +sufficiently divisible, and the very general fibers are taken integral. +The logarithmic modification identity, also with a pulled-back rational +twist, is \Cref{setup:log-modification}. + +\begin{lemma}[Divisorial section test]\label{p1logcbfalt:section-test} +Let \(V\) be normal and \(D\) an integral Weil divisor. A rational section +belongs to \(H^0(V,D)\) if and only if it satisfies the corresponding order +inequality at every prime divisor of \(V\). If \(\pi:V'\to V\) is finite +surjective with \(V'\) normal and \(D\) is Cartier, this test on \(V'\) needs +only primes lying over prime divisors of \(V\). If \(\pi\) is Galois, then +\[ +H^0(V',\pi^*D)^{\operatorname{Gal}(V'/V)}=H^0(V,D). +\] +\end{lemma} + +\begin{proof} +The first assertion is the description of a rank-one reflexive sheaf as +the intersection of its lattices at codimension-one points. For a finite +morphism between normal varieties, a prime divisor upstairs lies over a +prime divisor downstairs. The last assertion follows from +\((\pi_*\cO_{V'})^{\operatorname{Gal}(V'/V)}=\cO_V\) and the projection +formula. +\end{proof} + +\begin{lemma}[Finite pullback]\label{p1logcbfalt:finite} +If \(\pi:V'\to V\) is a dominant generically finite morphism of normal +projective varieties and \(D\) is a rational Cartier divisor on \(V\), +then +\[ +\kappa(V',\pi^*D)=\kappa(V,D). +\] +If \(\kappa(V,D)=0\), every nonzero space \(H^0(V,mD)\) has dimension +one. +\end{lemma} + +\begin{proof} +Stein factorization reduces the first assertion to a finite morphism, +since a proper birational morphism onto a normal variety has direct image +\(\cO_V\). After replacing \(D\) by a multiple, use the natural inclusion +of graded section rings +\[ +R(V,D)\ \subseteq\ R(V',\pi^*D). +\] +Each homogeneous section upstairs is integral over the ring downstairs. +Indeed its monic characteristic polynomial in the finite function-field +extension has coefficients which are rational sections of the appropriate +multiples of \(D\); they are regular in the required lattices at every +prime divisor, and hence are global sections by +Lemma~\ref{p1logcbfalt:section-test}. Thus the two nonzero graded rings have the +same transcendence degree, which gives equality of Iitaka dimensions. +If the ring downstairs has no positive-degree section, the same norm +argument excludes one upstairs. Finally, two independent sections in one +degree have a nonconstant ratio and define a positive-dimensional +rational image. This proves the last assertion. +\end{proof} + +\begin{lemma}[Logarithmic pullback]\label{p1logcbfalt:log-pullback} +Let \(\rho:V\to Y\) be a dominant generically finite morphism of smooth projective +varieties. Let \(D_Y\) be a reduced snc divisor, and let \(B\) be a +boundary on \(V\) satisfying \(B\ge(\rho^*D_Y)_{\red}\). Assume that the +boundary supports on the chosen models are snc. Then pullback of rational +pluricanonical forms gives +\[ +H^0(Y,m(K_Y+D_Y))\ \longrightarrow\ +H^0(V,m(K_V+B)) +\] +injectively, for divisible \(m\). +\end{lemma} + +\begin{proof} +A logarithmic differential on a smooth snc pair pulls back to a +logarithmic differential along the reduced inverse image. In local +coordinates this follows from +\(d(\rho^*z)/\rho^*z\), whose divisorial poles are simple. Taking top +exterior powers and then tensor powers proves the order inequalities. +Equivalently, at a prime of ramification index \(e\) over a boundary +component, the Jacobian contributes \(e-1\), and the coefficient-one +boundary supplies the remaining one. At exceptional-image primes the +same logarithmic differential calculation applies. Injectivity follows +from dominance and separability. +\end{proof} + +\begin{lemma}[Relative Iitaka fibration]\label{p1logcbfalt:iitaka} +Let \(p:V\to B\) be a dominant projective morphism, with \(V\) smooth +projective, and let \(D\) be a rational divisor whose restriction to the +geometric generic fiber has Iitaka dimension \(k\ge0\). After birational +modifications there is a relative Iitaka fibration +\[ +V'\xrightarrow{q}U\longrightarrow B +\] +in which \(q\) has geometrically connected generic fiber, such that +\[ +\dim U-\dim B=k,\qquad +\kappa(G,D'|_G)=0 +\] +for a very general fiber \(G\) of \(q\). Here \(D'=\mu^*D\), where +\(\mu:V'\to V\) is the modification. When \(D=K_V+B_V\), one may +instead take the full log divisor from Lemma~\ref{setup:log-modification}. + +If \(D'=K_{V'}+B_{V'}\) and the models are log smooth, then +\(D'|_G\sim_{\Q}K_G+B_G\). +\end{lemma} + +\begin{proof} +Apply the Iitaka fibration theorem over the function field of \(B\) +\cite[Section~6, Remark~1]{IitakaOriginal}, +take the relative algebraic closure of its section field in \(\C(V)\), +and spread out. Resolve the resulting rational maps and take their +Stein factorizations. The dimension of the relative section field is +\(k\); the restriction of the full divisor to the Iitaka fiber has +Iitaka dimension zero by the Iitaka fibration theorem. For the logarithmic +replacement, the proof of Lemma~\ref{setup:log-modification} gives the sheaf identity +\[ + \mu_*\cO_{V'}\bigl(m(K_{V'}+B_{V'})\bigr) + =\cO_V\bigl(m(K_V+B_V)\bigr) +\] +in every sufficiently divisible degree. Pushing to the original base +identifies the relative section systems, even if there are no global +sections. Apply the same full-divisor Iitaka theorem to the replacement +divisor; its relative section field identifies its fibration with the +one already constructed and gives the asserted Iitaka dimension zero +on the fiber. +Generic smoothness and adjunction give the last assertion. +The dimensions at very general complex fibers agree with the geometric +generic values: for each divisible degree use cohomology and base +change, and remove the union of the exceptional closed sets over the +countably many degrees. +\end{proof} + +\begin{lemma}[Easy addition]\label{p1logcbfalt:easy-addition} +Let \(p:V\to B\) be dominant and projective, with \(V\) smooth projective +and integral very general fiber \(G\), and let \(D\) be a rational +divisor on \(V\). Then +\[ +\kappa(V,D)\le \dim B+\kappa(G,D|_G). +\] +If \(D\) has a nonzero global section, its restriction to a general +fiber is nonzero. +\end{lemma} + +\begin{proof} +A nonzero section cannot vanish identically on all fibers over a dense +open subset. For each divisible \(m\) with sections, let \(\phi_m\) +be its rational map. The image of \((p,\phi_m)\) has dimension at most +\(\dim B+\kappa(G,D|_G)\), and dominates the image of \(\phi_m\). +Taking the maximum over \(m\) proves the inequality. If the fiber term +is \(-\infty\), the last assertion implies that the total term is +\(-\infty\) as well. +\end{proof} + +\subsection{Restricted valuations and the fixed original source} + +Let $p:H\to I$ be a dominant morphism of smooth projective varieties, +and fix the model $H$. A divisorial valuation $v$ of $\C(H)$ whose +restriction to $\C(I)$ is nontrivial restricts to a positive integral +multiple of a divisorial valuation of $\C(I)$. To see this, put +$n=\dim H$, $m=\dim I$, and denote the restricted valuation by $w$. +The value groups have rational rank one with finite-index inclusion. +The relative residue-transcendence-degree inequality gives +\[ + \operatorname{trdeg}_{\kappa(w)}\kappa(v)\le n-m. +\] +As $\operatorname{trdeg}_{\C}\kappa(v)=n-1$, this implies +$\operatorname{trdeg}_{\C}\kappa(w)\ge m-1$. The Abhyankar +inequality gives the reverse inequality. Thus $w$ is divisorial. +Finitely many such restrictions can be extracted on one smooth +projective birational model of $I$. + +Only finitely many prime divisors of the fixed $H$ have image of +codimension at least two in $I$: they lie in the complement of a +locus where $p$ is smooth and are components of its divisorial part. +Extract their restrictions as above. Original horizontal primes +remain horizontal, while original vertical primes then have +divisorial centers on the prepared base. These centers remain +divisorial on higher base models. Hence on any subsequent commuting +birational diagram, a source prime whose base image still has +codimension at least two is exceptional over the fixed $H$. + +For the rank-one section comparison, fix the original smooth pair +$(X,D_X)$, and let $X'\to X$ carry strict plus reduced exceptional +boundary $D'$. Let $x:X'\to W$ be the relative Iitaka fibration +on smooth projective models; its geometric generic fiber $(J,D_J)$ +has logarithmic Kodaira dimension zero. Let $s$ be a nonzero +relative rational $p$-canonical form generating the degree-$p$ +generic section space. For a source prime $E$, put +$d_E=\operatorname{coeff}_E D'$, and for a base prime $P$ define +\[ + t_P=\min_{E\mapsto P} + \frac{p^{-1}\operatorname{ord}^{pK_{X'/W}}_E(s)+d_E} + {\operatorname{ord}_E(x^*P)},\qquad + T_W=\sum_Pt_PP. +\] +Here $T_W$ is the entire minimum-rule base term, before its division +into discriminant and moduli parts. In divisible degrees, division by +$s^{m/p}$ gives +\begin{equation}\label{p1logcbfalt:source-descent} + H^0\bigl(X',m(K_{X'}+D')\bigr) + \lhook\joinrel\longrightarrow + H^0\bigl(W,m(K_W+T_W)\bigr). +\end{equation} +Indeed, on the geometric generic fiber the section is a constant +multiple of $s^{m/p}$; connectedness identifies that coefficient +with a rational $m$-canonical form $\sigma$ on $W$. At a component +$E$ dominating $P$, write $\sigma$ in a nonvanishing local frame of +$mK_W$, and put $a_E=\operatorname{ord}_E(x^*P)$. Its logarithmic +regularity upstairs is equivalent to +\[ + a_E\operatorname{ord}_P(\sigma) + +(m/p)\operatorname{ord}^{pK_{X'/W}}_E(s)+md_E\ge0. +\] +The inequalities for all components above $P$ are exactly +$\operatorname{ord}_P(\sigma)+mt_P\ge0$. Horizontal primes impose +no further condition because $s$ is an allowed generic logarithmic +pluriform. The same argument works over the generic point of the +original base $Y$. If the relative Iitaka dimension is $k$, then +$\dim(W/Y)=k$, so preservation of section ratios makes +$K_W+T_W$ big over $Y$. + +For the converse, extract on $W$ the restrictions of all source +divisors with exceptional image that survive as divisors on the +original $X$. Resolve the induced diagram and recompute the minimum +rule. Every divisor of the original $X$ is now tested either by the +generic horizontal condition or by a base prime. A section on the +right of \eqref{p1logcbfalt:source-descent}, multiplied by $s^{m/p}$, +therefore satisfies the logarithmic order inequality at every +original source prime. Any remaining untested pole lies on a +divisor exceptional over $X$. The codimension-one criterion on the +normal original source gives a section of $m(K_X+D_X)$; later +models preserve this criterion. Conversely the logarithmic +modification identity recovers its section on $X'$. The +identifications respect powers and multiplication. Under these +identifications, corresponding section ratios agree after pullback by +$x:X'\to W$. Their section fields and rational-map image dimensions +therefore agree. + +\subsection{The fixed zero divisor and the discrepancy-sheaf test} + +In the notation of Lemma~\ref{lem:reduced-cyclic-top}, the rational +divisor $Z=E/p$ is independent of the chosen nonzero logarithmic +pluriform and its degree. Indeed, if $\omega_m$ is a nonzero section +in another degree $m$, the degree-$mp$ sections $\omega_m^p$ and +$\omega^m$ are proportional, since that section space is +one-dimensional. Consequently +\[ + p\bigl(\operatorname{div}(\omega_m)+mD\bigr)=mE. +\] +The least positive integer trivializing $K_J+D-Z$ is also $p$. +Writing $\omega=b\eta^{\otimes p}$ in a rational canonical frame +gives $p(K_J+D-Z)=-\operatorname{div}(b)$. Conversely, a +trivialization of $r(K_J+D-Z)$ gives a rational $r$-canonical form +whose logarithmic zero divisor is $rZ\ge0$, hence a nonzero +logarithmic pluriform of degree $r$. Minimality gives $r\ge p$. + +Here is a second way to check the generic discrepancy-sheaf rank +condition in Corollary~\ref{cor:reduced-integral-moduli}. Take a log +resolution $J'\to J$, let $D_{J'}$ be the strict boundary plus the +reduced exceptional divisor, and put +\[ + E_{J'}=\operatorname{div}(\omega|_{J'})+pD_{J'}\ge0. +\] +The crepant transform of the auxiliary generic subboundary is +$D_{J'}-E_{J'}/p$. Its coefficients are at most one. If $A^*$ is +the discrepancy b-divisor with coefficient $-1$ replaced by zero, +the rounded trace is effective, and a positive coefficient can +occur only on $\operatorname{Supp}E_{J'}$. Coefficientwise, +\[ + 0\le \lceil A^*\rceil_{J'} + \le\lceil E_{J'}/p\rceil\le E_{J'}. +\] +The first inequality uses the omission of discrepancy $-1$. +Logarithmic birational invariance gives $\kappa(E_{J'})=0$. +The associated space of rational functions therefore has dimension +at most one and contains the constants. This proves rank one on +every sufficiently high resolution, including when the auxiliary +subboundary has negative coefficients. + +\subsection{A toroidal proof of logarithmic extension} + +The extension part of Lemma~\ref{lem:reduced-cyclic-top} also has a +direct coordinate proof on a toroidal resolution of the SNC +cyclic-cover construction. First resolve the supports on $J$, +retaining strict plus reduced exceptional logarithmic boundary. +The minimal index and the fixed-zero assertions are unchanged. +In a local SNC chart write +\[ + \omega=\varepsilon\prod_i z_i^{a_i-pd_i} + (dz_1\wedge\cdots\wedge dz_r)^{\otimes p}, + \qquad a_i\ge0,\quad d_i\in\{0,1\}, +\] +where $\varepsilon$ is a unit. Include all coordinates; outside +$D+\operatorname{Supp}E$ put $d_i=a_i=0$. After a local unramified +change the unit has a root and contributes no divisorial order. +Let $v$ be a toroidal prime on the resolved root cover, and put +$n_i=v(z_i)\ge0$. The nonzero ray defining this prime has at least +one positive $n_i$. Rewriting with logarithmic differentials gives +\begin{equation}\label{p1logcbfalt:toroidal-root-order} + \operatorname{ord}_v(\Omega) + =\sum_i n_i\left(1-d_i+\frac{a_i}{p}\right)-1. +\end{equation} +The logarithmic volume form has order $-1$ along a toroidal prime, +and the change of torus lattice multiplies it by a nonzero constant. +Every coefficient in the sum is nonnegative. It vanishes exactly +when $d_i=1$ and $a_i=0$, namely along the coefficient-one pole +boundary of the auxiliary subpair. + +If $v$ is outside the inverse image of this pole boundary, then +$n_i=0$ for each such zero coefficient. Some remaining $n_i$ is +positive, so the sum in \eqref{p1logcbfalt:toroidal-root-order} is +strictly positive. The order of the rational top form $\Omega$ is +integral, so that positive sum is at least one, giving regularity. +On the inverse pole boundary the nonnegative sum gives order at +least $-1$. Primes in the smooth unramified locus have the +regularity already checked on the finite normalization. Thus +$\Omega$ extends as a top form with logarithmic poles allowed only +along the inverse pole boundary. This provides the coordinate +alternative to the discrepancy-scaling proof. + +\subsection{The normalized discriminant on higher models} + +Let $x:X'\to W$ and $D'$ be the projective reduced-boundary +data of Lemma~\ref{lem:rankone-normalization}. Write $s$ for its +relative rational $p$-canonical generator, and set +\[ + t_P=\min_{E\mapsto P} + \frac{p^{-1}\operatorname{ord}^{pK_{X'/W}}_E(s)+d_E} + {\operatorname{ord}_E(x^*P)},\qquad + T_W=\sum_Pt_PP, +\] +\[ + \Delta_0=-p^{-1}\operatorname{div}_{K_{X'/W}}(s), + \qquad \Delta=\Delta_0+x^*T_W. +\] +In this subsection $T_W$ denotes the entire minimum-rule base +term, before its decomposition into discriminant and moduli parts; +the boundary denoted $T$ in Section~\ref{rankone:section} is the +discriminant part. We have +$K_{X'}+\Delta\sim_{\Q}x^*(K_W+T_W)$. +The auxiliary divisor is required to be sub-lc over the generic +point of $W$. Its coefficients at source divisors with image of +codimension at least two are not asserted to be at most one. + +For a prime $P\subset W$, let +\[ + c_P=\operatorname{lct}_{\eta_P}(X',\Delta;x^*P), + \qquad b_P=1-c_P, + \qquad B_W=\sum_Pb_PP. +\] +Over the generic point of $P$ the minimum rule gives +$\Delta\le D'$, including the horizontal divisors. Log canonicity +therefore gives $c_P\ge0$. A component $E$ attaining the minimum +has coefficient exactly $d_E\in\{0,1\}$ in $\Delta$, and hence +\[ + c_P\le\frac{1-d_E}{\operatorname{ord}_E(x^*P)}\le1. +\] +It follows that $0\le B_W\le1$. If $P$ lies in a required inverse +boundary from a base $Y$, all components over $P$ have $d_E=1$, +so $c_P=0$ and $b_P=1$. + +For the exact model comparison, let $\beta:W_1\to W_0$ be a +higher smooth base model and let $\pi:X_1\to X_0$ resolve the +induced diagram, with $x_i:X_i\to W_i$. Choose compatible +rational top forms for the canonical divisors. First form the +crepant transform +\[ + K_{X_1}+\Delta_{\mathrm{crepant}} + =\pi^*(K_{X_0}+\Delta_{\mathrm{old}}). +\] +Using the new logarithmic boundary and the same relative +trivialization $s$, recompute the minimum-rule divisor $T_{W_1}$ +and the normalized auxiliary divisor $\Delta_{\mathrm{new}}$. +Comparison of their definitions gives +\begin{equation}\label{p1logcbfalt:model-twist} + \begin{split} + Q&=K_{W_1}+T_{W_1}-\beta^*(K_{W_0}+T_{W_0}),\\ + \Delta_{\mathrm{new}} + &=\Delta_{\mathrm{crepant}}+x_1^*Q. + \end{split} +\end{equation} +Thus the twist includes the canonical discrepancy of the base. +Adding $x^*Q$ lowers the threshold at $P$ by +$\operatorname{coeff}_P Q$, adding $Q$ both to the discriminant +and to the total base term. It leaves the moduli part unchanged. +Birational pullback of the lc-trivial fibration also preserves its +moduli b-divisor; these compatibilities agree with the Hodge +extension description in \cite[Remarks~6.22 and~6.25]{BFMT}. + +Resolve the finite supports on $W$ and resolve the induced source +diagram. At an unchanged codimension-one point, logarithmic forms +satisfying the old tests remain logarithmic on the source +resolution, while the strict old minimizing components persist. +Thus the minimum and discriminant trace there are unchanged. +New support is contained in the base-exceptional divisor. Its union +with the old strict support is SNC on the chosen resolution; +recomputing and applying the preceding coefficient argument yields +an effective SNC discriminant with the same coefficient-one inverse +boundary and the same moduli b-divisor. + +Finally, if $m=ap$ and $s_m=c s^a$ on the generic fiber, with +$c\in\C(W)^*$, the minimum construction gives +\[ + T_W(s_m)=T_W(s)+m^{-1}\operatorname{div}(c). +\] +The two occurrences of $\operatorname{div}(c)$ cancel in the +definition of the normalized auxiliary subpair. It is unchanged. +The minimal-index cover is therefore compatible with all common +divisible degrees used for section descent. diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/period-positivity.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/period-positivity.tex new file mode 100644 index 0000000000000000000000000000000000000000..e2531d2853e99898c09c0b68d28de701601dbc1e --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/period-positivity.tex @@ -0,0 +1,923 @@ +\section{Adjoint positivity of an extreme Hodge line} +\label{period:section} + +The relative canonical bundle calculation gives a line in the highest +nonzero Hodge filtration step of one complex summand. Its curvature +is semipositive, but this alone does not compare it with a canonical +bundle. We construct a projective quotient on which the line descends +and prove that adding a positive multiple of it makes the canonical +class big. + +Here an \emph{extreme Hodge line} means the rank-one highest nonzero +filtration step of a complex Hodge direct summand. An integral +structure is a lattice in the ambient real local system. The chosen +summand need not be rational, and the ambient real polarization need +not be rational either. + +The \emph{parabolic extension} is obtained by local power substitutions +that make boundary monodromy unipotent, followed by Schmid extension +and descent with rational weights. The monodromy theorem gives +rational weights in the integral case. When the local monodromy has +a finite part, these weights are essential. We first resolve any +compactification whose boundary is not simple normal crossing. +Unipotent extension commutes with pullback, so the same is true of +the parabolic rational line after the power substitutions; see +\cite{Schmid,CKS} and \cite[Lemma~5.6]{BFMT}. We use version~2 +of \cite{BFMT} throughout. + + +We prove the adjoint-positivity theorem stated as +\Cref{period:adjoint}. The line in that statement is allowed to be a +complex character line inside an integral ambient variation. + +There are two maps in the proof. In a flat trivialization the +\emph{line map} records only the chosen highest line; the \emph{period +map} records all Hodge filtration steps. Their ranks can differ. +Curvature identifies the numerical dimension of the line with the +rank of the first map. We use the full periods of suitable rational +adjoint factors to construct the projective quotient. On that +quotient, logarithmic general type gives a big canonical class with +boundary. The remaining argument proves that the line loses numerical +rank on each marked boundary component, so a section count removes +the boundary after adding a multiple of the line. + +\subsection{Curvature, numerical dimension, and the boundary} + +We first relate the curvature on the open set to intersection numbers +on the compactification. For a nef rational line bundle $A$ on a compact +Kähler $n$-fold, we use +\[ + \nu(A)=\max\{k:c_1(A)^k\cdot[\omega]^{n-k}>0\}, +\] +where $\omega$ is any Kähler form. On a projective manifold this is +the usual numerical dimension. + +\begin{lemma}[Curvature and numerical dimension] +\label{period:numerics} +Let $T$ be a smooth compact Kähler manifold, let $E\subset T$ be a +simple normal crossing divisor, and let $A$ be the parabolic extension +of the rank-one highest step of a polarizable pure complex variation on +$T\setminus E$ with quasi-unipotent local monodromy. Then $A$ is nef. +Its numerical dimension equals the generic rank of the map that +remembers this line in a flat trivialization. + +The corresponding assertion for the Griffiths line, the tensor product +of the determinants of the Hodge filtration steps, uses the full period +map. In particular, its rational extension is big if the full period +map is generically immersive. +\end{lemma} + +The curvature-and-volume argument is closely related to the proofs of +\cite[Lemmas~6.15 and~6.17]{BakkerBrunebarbeTsimerman}; here we need +all mixed intersection numbers, since the chosen line need not be big. + +\begin{proof} +We may check the assertions after a finite cover and resolution on +which local monodromy is unipotent. This does not require an integral +lattice. Indeed, $\pi_1(T\setminus E)$ is finitely generated. Put +the entries of finitely many monodromy generators and their inverses +in a finitely generated $\Z$-algebra $R\subset\C$. Adjoin the +finite group of roots of unity generated by the eigenvalues of the +finitely many boundary monodromies, and invert every difference +$\zeta-1$ for a nonidentity element of this group. Reduction at a +maximal ideal with finite residue field gives a finite-index normal +congruence subgroup. Every element of a local boundary monodromy +group that lies in this subgroup is unipotent: its eigenvalues +belong to that finite group and reduce to one, so the inverted +differences force them to equal one. This also applies to products +of commuting boundary monodromies at crossings. The corresponding finite cover extends +over the compactification, and a Kähler resolution gives the desired +unipotent model; see \cite[Lemma~2.21]{VilladsenHodgeKahler}. +These covers preserve nefness, numerical dimension, and bigness of +the rational line. Let $h$ be its Hodge metric and $\Theta$ its curvature +on $T\setminus E$, normalized to represent $c_1(A)$. Griffiths' +curvature formula gives $\Theta\geq0$; its rank is the rank of the +line map, since its value on a tangent vector is the squared norm of +the second fundamental form of the highest step. The same formula +for the Griffiths line detects the differential of the full period +map. + +The Hodge norm estimates in extending frames give, in boundary +coordinates $t_1,\ldots,t_k$, weights whose absolute values are bounded +by +\begin{equation} +\label{period:loglog} + C\left(1+\sum_{j=1}^k + \log(-\log|t_j|)\right). +\end{equation} +The semipositive metric therefore extends as a positive current in +$c_1(A)$ with zero Lelong numbers. Regularization of positive +currents then gives nefness; see +\cite[Corollary~6.4]{DemaillyRegularization}. +This extends the nefness argument of \cite[Lemma~2.16(1)]{BFMT} +from a proper log smooth algebraic space to the present compact +Kähler setting. + +To identify numerical dimension, we must also show that no +intersection mass is lost at $E$. Write $\alpha=c_1(A)$ and fix +$\varepsilon>0$. Since $\alpha$ is nef, $\alpha+\varepsilon[\omega]$ +has a smooth Kähler representative +$\theta_\varepsilon$. Write +\[ + \Theta+\varepsilon\omega=\theta_\varepsilon+\ddc u, + \qquad \ddc=\frac{\sqrt{-1}}{2\pi}\partial\bar\partial. +\] +The global potential $u$ satisfies the local bounds +\eqref{period:loglog}. Put $n=\dim T$. If $E$ is empty, the potential +is smooth and there is no lost mass. Otherwise write +$E=E_1+\cdots+E_N$ and choose smooth divisor metrics, with defining +sections $s_j$, such that +\[ + u_j=-\log|s_j|^2,\qquad + \ddc(-u_j)=[E_j]-\eta_j, +\] +where the $\eta_j$ are smooth curvature forms. Choose $c>0$ with +$c\eta_j\leq\theta_\varepsilon$ for all $j$, and rescale the metrics +so that $cu_j\geq1$. Then $\phi_j=-cu_j$ is +$\theta_\varepsilon$-plurisubharmonic and $\phi_j\leq-1$. +For any fixed $00$ gives +\begin{equation} +\label{period:masses} + \int_{T\setminus E}\Theta^k\wedge\omega^{n-k} + =\alpha^k\cdot[\omega]^{n-k} + \qquad(0\leq k\leq n). +\end{equation} +If the generic curvature rank is $r$, the left side is positive for +$k\leq r$ and zero for $k>r$. This proves the numerical assertion. +For an immersive Griffiths line the top integral is positive, and +the volume criterion for a positive current on a compact Kähler +manifold gives bigness; see \cite{BoucksomVolume}. +\end{proof} + +\begin{lemma}[Restriction to a boundary component] +\label{period:boundary} +In the unipotent case of \Cref{period:numerics}, let $E_i$ be a smooth +boundary component and $N_i$ its monodromy logarithm. On the open +stratum of $E_i$, the limiting highest line occurs in a unique weight +grade of the limiting mixed variation. The restriction $A|_{E_i}$ +is the Schmid extension, across the remaining boundary of $E_i$, of +that graded Hodge line. Its numerical dimension is therefore the +generic rank of the graded line map. +\end{lemma} + +\begin{proof} +The extended filtration and $W(N_i)$ give the limiting mixed +variation along the open stratum. Since the highest step has rank +one, exactly one weight grade contains it. Changes of the normal +parameter act by $\exp(cN_i)$ and act trivially on the associated +graded. At a further crossing the relative monodromy filtration +identifies the extension of this graded variation with the graded +of the original extending filtration. Thus the identification holds +on the entire component. +For log smooth algebraic spaces it is stated in +\cite[\S2.5.3 and Lemma~2.16(3)]{BFMT}; its local proof uses the +multivariable nilpotent orbit theorem \cite{CKS} and applies here +as well. The filtrations and strictness also restrict to a complex +summand. Apply \Cref{period:numerics} on $E_i$. +\end{proof} + +\subsection{Rational adjoint periods and descent of the line} + +The period-quotient construction requires discrete monodromy. A +projection to one complex factor need not retain it. We therefore +construct a rationally polarized adjoint variation from the ambient +integral local system and keep whole rational factors. A tensor +construction then recovers a positive power of the original line, +including its scalar monodromy. + +\begin{lemma}[Rational adjoint reduction] +\label{period:adjoint-reduction} +In the setting of \Cref{period:adjoint}, let $G$ be the identity +component of the rational Zariski closure of monodromy. Then $G$ is +semisimple. Its adjoint local system is a rationally polarized +integral variation of weight zero. + +There is a monodromy-invariant collection of rational simple adjoint +factors, each containing a complex factor acting nontrivially on the +highest-line orbit, with the following properties. The resulting +adjoint variation $\mathbb A$ has discrete integral monodromy. After a +positive power and up to a constant one-dimensional complex Hodge +shift, the original line is the highest line of a complex +subvariation of a tensor construction on $\mathbb A$. On a finite +cover preserving the factors, that line splits into the tensor +product of the highest lines belonging to the individual complex +factors. All these highest steps have rank one. +\end{lemma} + +\begin{proof} +Semisimplicity and the theorem of the fixed part hold for polarizable +real and complex variations on a Zariski open subset of a compact +Kähler manifold. We may use a Kähler modification of $B$ to apply +them; see \cite[Lemma~2.3 and its proof]{PopaSchnellKahler} and +\cite[Theorem~3 and Remark~4]{BakkerMatsushita}. Thus the algebraic +monodromy group is reductive. Pass temporarily to a finite cover +on which it is connected. The irreducible complex constituents of +the local system carry polarizable complex variations. They can be +defined over a number field: the representation is rational and its +algebraic monodromy is reductive. The determinant of each constituent +is unitary, as are all its algebraic conjugates, which are again +constituents of the rational representation. Its monodromy values +are algebraic units by the integral lattice. Kronecker's theorem +makes these values roots of unity in a fixed number field, hence +makes the determinant character finite. It is consequently trivial +on $G$. This is Deligne's finite-determinant argument; its classical +algebraic formulation is \cite[Corollary~4.2.8(iii)(b)]{DeligneHodgeII}. +The connected center of $G$ acts by scalar characters on +the irreducible constituents; their determinants detect these +characters. Since the original representation is faithful, this +connected center is trivial. Therefore $G$ is semisimple. + +We next equip the monodromy Lie algebra with a rational polarization. +In every tensor construction, +the subspace of $G$-invariant tensors is the fixed part and has a +constant Hodge structure. In a flat trivialization the Hodge circles +$h_x$ and $h_{x_0}$ therefore induce exactly the same linear operator +on each such subspace at each parameter $z\in S^1$. Reductivity +identifies $G$ with the pointwise stabilizer of all its invariant +tensors. The circles preserve these subspaces, so normalize $G$, +and the equality of their actions gives +\[ + h_x(z)h_{x_0}(z)^{-1}\in G(\R). +\] +Here is a direct local conjugacy argument. Along a smooth path in +the base, write $a_t=\operatorname{Ad}\circ h_t:S^1\to G^{\rm ad}(\R)$ +and $u_t(z)=(\partial_t a_t(z))a_t(z)^{-1}$. Differentiating the +homomorphism identity gives +\[ + u_t(zz')=u_t(z)+\operatorname{Ad}(a_t(z))u_t(z'). +\] +Normalized Haar averaging, with $X_t=\int_{S^1}u_t(z')\,dz'$, yields +$u_t(z)=X_t-\operatorname{Ad}(a_t(z))X_t$. The solution of +$g_t'g_t^{-1}=X_t$, $g_0=1$, therefore gives +$a_t=g_ta_0g_t^{-1}$. Lift this path through the central isogeny +$G(\R)^+\to G^{\rm ad}(\R)^+$. The ratio of $h_t$ and +$g_th_0g_t^{-1}$ lies in $G(\R)$ by the invariant-tensor equality, +and centralizes $G$ by the equality of their adjoint actions. +It lies in the finite center of $G$; continuity in the circle +parameter makes it the identity. Thus $h_t=g_th_0g_t^{-1}$, +proving that the lifted period map lies in a single +$G(\R)^+$-orbit. + +It follows that $\mathfrak g_{\Q}=\operatorname{Lie}(G)$ is a Hodge +sub-local system of $\End(\mathbb V_{\Q})$, with its weight-zero +grading. The Weil involution $\theta$ preserves $\mathfrak g_{\R}$ +and is the restriction of the Cartan involution of the real +polarization isometry group. To check the restriction, use the +positive Hodge metric: on its Lie algebra $\theta(X)=-X^*$. +If $\mathfrak g_{\R}=\mathfrak k\oplus\mathfrak p$ is its +eigenspace decomposition, then $\mathfrak k\oplus i\mathfrak p$ +is a compact real form of $\mathfrak g_{\C}$, represented by +skew-Hermitian endomorphisms. Thus +$-B_{\mathfrak g}(X,\theta X)$ is positive definite, proving that +$\theta$ is Cartan on $\mathfrak g_{\R}$. The appropriately +signed Killing form therefore polarizes this weight-zero variation. +The form is rational. Moreover +\[ + \mathfrak g_{\Q}\cap\End(\mathbb V_{\Z}) +\] +is a monodromy-invariant lattice. This gives the rationally polarized +integral adjoint variation. + +Let $\ell$ be the highest line at a reference point of the chosen +complex summand. The non-lowering parabolic of the Hodge grading +preserves $\ell$, so a Borel subgroup preserves it. Its closed orbit +\[ + J=G_{\C}\ell +\] +is a flag variety, a product of flag varieties for the complex simple +factors. The span $W$ of this orbit is irreducible. Indeed, in a +semisimple decomposition of the representation, the nonzero +components of a Borel-eigenline all have the same highest weight; +their span is a single diagonal copy of that irreducible +representation. The line is its unique highest Hodge step. Indeed, at every point +the Hodge circle normalizes $G$ and preserves the highest line there; +it consequently preserves the span of its $G_{\C}$-orbit. Thus $W$ +is a Hodge subvariation. The lifted line map takes values in $J$, +and the original line is the pullback of the tautological line there, +equivariantly for monodromy. + +Call a complex simple factor \emph{visible} if it acts nontrivially +on $J$. Retain every rational simple adjoint factor containing a +visible factor. +This collection is invariant under the possibly disconnected +monodromy group. On its Lie algebra we obtain the variation +$\mathbb A$; its monodromy is contained in the automorphisms of a +lattice and is discrete. Notice that an entire rational factor is +kept. Projection to only one real factor would not ensure +discreteness. + +We now recover the line from this adjoint system. Choose a positive +integer $N$ for which the highest weight $N\lambda$ of $W$ belongs +to the root lattice. The Cartan component +$W_N\subset W^{\otimes N}$ is generated by the orbit of +$\ell^{\otimes N}$. It factors through the connected adjoint group, +and its highest Hodge line is $\ell^{\otimes N}$. The same +orbit-span argument just used shows that $W_N$ is a Hodge +subvariation. + +We must also account for disconnected monodromy before applying +tensor generation. Deck transformations take the lifted line to +another point of the same connected orbit, so they preserve $W$ and +its Cartan components. The kernel of the action on the retained +adjoint factors centralizes the $G$-action on $W$, and hence acts +scalarly by irreducibility. Its connected part is semisimple and +has no nontrivial scalar character. Its component group is finite. +Increasing $N$ therefore kills the entire kernel action on $W_N$. +Consequently $W_N$ is a representation of the full projected +algebraic monodromy group, not just of its identity component. +This step rules out a discarded nontorsion scalar local system. + +The adjoint representation of that projected group is faithful and +self-dual. By tensor generation and complete reducibility, +$W_N$ embeds as a flat direct summand in a tensor construction +$\mathbb T(\mathbb A)_{\C}$; see +\cite[Theorem~4.14]{MilneAlgebraicGroups}. To make the embedding +compatible with Hodge structures, apply the theorem of the fixed part +to +\[ + H^0\bigl(B^\circ, + \mathcal Hom(W_N,\mathbb T(\mathbb A)_{\C})\bigr). +\] +This nonzero space of flat maps has a constant complex Hodge +structure. A nonzero Hodge-homogeneous component of any nonzero +flat map is still flat. Its kernel is $G$-invariant, so the map is +injective because $W_N$ is irreducible. A constant one-dimensional +complex Hodge shift makes it a morphism of variations; that shift +changes neither the underlying holomorphic line nor its boundary +extension. This is the constituent and fixed-part argument of +\cite[Theorem~3]{BakkerMatsushita}. + +The complex summand carries a flat Hermitian polarization. In a +real tensor variation of weight $w$ polarized by $Q$, it is induced +by $i^wQ_{\C}(v,\bar u)$, up to the conventional overall sign. +Its restriction to a Hodge subvariation is nondegenerate because +it is definite with the prescribed sign on each Hodge component. +There is no need for the real bilinear form itself to restrict +nondegenerately to the chosen complex summand: it may pair that +summand with its conjugate. + +Finally, on a factor-preserving finite cover an irreducible +representation is a tensor product over the simple factors. The +dimension of its highest step is the product of the corresponding +dimensions. Since that dimension is one, each factor has a unique +rank-one highest step. This gives the asserted factor lines. Their +extensions are nef and functorial by +\Cref{period:numerics,period:boundary}. +\end{proof} + +\begin{lemma}[Generic stabilizers of adjoint periods] +\label{period:stabilizer} +Let $G$ be a connected semisimple adjoint group over $\Q$, and let +$\mathbb A_{\Q}$ be a rationally polarized integral variation whose +fiber is $\mathfrak g_{\Q}=\operatorname{Lie}(G)$ and whose connected +algebraic monodromy is $G$, acting by its adjoint representation. +No nonidentity rational Lie algebra automorphism of +$\mathfrak g_{\Q}$ fixes all generic lifted periods of $\mathbb A$. +The automorphism need not belong to the monodromy group. +\end{lemma} + +\begin{proof} +Suppose $\gamma$ fixes those periods. By equivariance, every +monodromy conjugate of $\gamma$ is everywhere of Hodge type $(0,0)$ +in the endomorphism variation. Their rational span is therefore a +rational subvariation of type $(0,0)$. Its induced polarization is +positive definite. Intersection with the endomorphism lattice is +a full monodromy-invariant lattice: clear one denominator of +$\gamma$; integral monodromy conjugation preserves that denominator. +A positive definite integral monodromy group is finite. Consequently +the connected group $G$ fixes $\gamma$, so $\gamma$ centralizes every +inner automorphism of $\mathfrak g$. The centralizer of the inner +group in $\operatorname{Aut}(\mathfrak g)$ is trivial: on each simple +ideal commutation with all adjoint operators makes an endomorphism +scalar, and compatibility with the Lie bracket makes this scalar +one; permutations of distinct ideals do not commute with their +separate inner actions. Thus $\gamma=1$. +\end{proof} + +\subsection{A projective quotient for the retained periods} + +We now quotient by connected fibers of the retained full period map. +The preceding reduction supplies both discrete integral monodromy +and a tensor description of a power of the line. The first property +allows us to compactify the quotient; the second will descend the +line with its exact parabolic extension. No bigness assertion about +$K_S+aP$ is used in this construction. + +\begin{proposition}[The projective period quotient] +\label{period:quotient} +In \Cref{period:adjoint}, after modifications there is a map +$p:B'\to S$ with connected fibers to a smooth projective variety +and a nef rational line $P$ with $\tau^*L=p^*P$. On a dense open +subset $S^\circ$, the retained adjoint variation of +\Cref{period:adjoint-reduction} descends, has the same connected +algebraic monodromy, and has a generically immersive period map. +The line $P$ is its parabolically extended highest-weight line, +up to the positive scaling already described. If the retained +collection is empty, $L$ is rationally trivial and $S$ may be a +point. +\end{proposition} + +The construction uses the compact-period-quotient approach of +\cite[Theorems~7.6--7.7]{BakkerBrunebarbeTsimerman}, whose +compactification input goes back to +\cite[Proposition~III and Remark~III-C]{SommesePeriodMapping}. +We use the precise class-$\mathcal C$ formulation of +\cite[Theorem~2.19]{VilladsenHodgeKahler} below. + +\begin{proof} +Choose a compact K\"ahler modification of the class-$\mathcal C$ +source. Then pass to a finite level at which the projected monodromy is +neat and preserves the factors. Finite covers of the open locus +extend to finite ramified covers of compactifications and then to +Kähler resolutions; see \cite[Lemma~2.21]{VilladsenHodgeKahler}. +Resolve the boundary to simple normal crossings. Extend the period +map over all finite-local-monodromy strata. At neat level their +monodromy is trivial and the nilpotent orbit theorem gives this +extension. + +The extended period map on this locus is proper; see +\cite[Proposition~9.11 and the proof of Theorem~9.6]{GriffithsPeriodsIII}. +Indeed, if a sequence tending to the boundary has period images in +a compact subset of the quotient, translate period lifts into a +compact set. +The punctured-disk Schwarz estimate makes the displacement of each +vanishing-coordinate monodromy tend to zero. The discrete group +acts properly, since its isotropy in the real period domain is +compact. The local monodromies must therefore be trivial at neat +level, so such a sequence cannot escape the extended locus. + +Take the Stein factorization of this proper map. Its first map has +compact connected fibers and its base is finite over the period +image. It extends after modifications to a map of compact +class-$\mathcal C$ spaces with Kähler resolutions by +\cite[Theorem~2.19]{VilladsenHodgeKahler} (version~1). +That theorem applies to proper connected-fiber maps from Zariski +open subsets of compact class-$\mathcal C$ spaces. In terms of +cycles, one first flattens the proper map, sends its irreducible general fibers +to a component of the Douady space of the compact source, and takes +the closure of that family. The component is compact and in class +$\mathcal C$. Near a general fiber the map to the cycle space is an +open embedding, since an irreducible compact cycle of the same +dimension in a sufficiently small neighborhood maps to a compact +analytic subset of a small Stein neighborhood in the base, hence +to a point, and must be the whole irreducible fiber. +The incidence map is consequently a modification of the source. +This produces the required compact quotient and also shows its +uniqueness as the quotient by the connected general period fibers. + +On a dense open subset of this quotient the period and its integral +variation descend. At neat level a constant period has trivial +stabilizer; local slices to the connected-fiber map give the descent +data. After shrinking to a proper smooth fibration, the map on +fundamental groups is surjective, so the descended variation has +the same connected algebraic monodromy. Its period differential is +generically injective by construction. On a Kähler compactification +its Griffiths line is big by \Cref{period:numerics}; equivalently, +one can apply \cite[Corollary~1.3]{BrunebarbeCadorel}. The quotient +therefore has a smooth projective model. This is also the argument +in \cite[proof of Corollary~2.22]{VilladsenHodgeKahler}. + +The finite deck group acts on the connected-period-fiber quotient. +Take its finite quotient and resolve. This again has a smooth +projective model. A general fiber before this finite quotient is +the union of connected level fibers indexed by one deck-group orbit. +The group acts transitively on these fibers, so the quotient is the +continuous image of any one of them and is connected. Thus the +induced map from the original source has connected general fibers, +hence connected fibers after its Stein factorization. The latter +does not change its function field. +All maps can be resolved using modifications of the original +source and of this projective model. + +There is an actual adjoint variation on a dense Zariski open subset +of the quotient before changing level. Write $\Gamma'$ for the +normal level subgroup of $\Gamma$. The induced action of +$\Gamma/\Gamma'$ on the level quotient is faithful. Indeed, if +$\delta\in\Gamma$ acts trivially there, discreteness shows locally +that its action on lifted periods agrees with a fixed +$\eta\in\Gamma'$. Then $\eta^{-1}\delta$ fixes a period germ, +hence the whole generic lifted period image by analytic +continuation. By \Cref{period:stabilizer} it is the identity, +so $\delta\in\Gamma'$. Resolve the finite group action +equivariantly and remove its proper algebraic fixed loci and the +branch locus. On the resulting dense Zariski open subset the level +quotient is free and unramified. Its equivariant integral adjoint +local system and Hodge filtration descend there. In particular, +this descent uses an open subset, not merely very general points. + +The highest-weight construction in +\Cref{period:adjoint-reduction} gives a line on that open subset +whose pullback is a fixed positive power of $L$. Take its +parabolic extension on a log smooth projective compactification +$S$ and divide by that power to obtain $P$. Nefness follows from +\Cref{period:numerics}. Pullback of the unipotent extensions, +followed by rational descent, gives +$\tau^*L=p^*P$ on the compact models, including the boundary. +Equivalently, this can be checked at level and then descended by +the norm: a line whose finite pullback is trivial is torsion in +the rational Picard group. The identity therefore includes the +boundary and persists under further modifications. + +If no factor is retained, the orbit is a point and the only possible +scalar monodromy is the finite scalar group treated in +\Cref{period:adjoint-reduction}. Its parabolic line is rationally +trivial, as asserted. +\end{proof} + +\subsection{Loss of numerical rank on the marked boundary} + +Fix the projective quotient of \Cref{period:quotient} and a log smooth +compactification $(S,E)$ of its descended variation. Let $D\leq E$ +be the reduced sum of components with nonidentity local monodromy +on the retained adjoint system, including finite monodromy. +Logarithmic general type will give bigness of $K_S+D$. The next +lemma is what permits us to remove $D$. + +\begin{lemma}[Strict loss of rank at the marked boundary] +\label{period:rankloss} +For every component $D_i$ of $D$, +\begin{equation} +\label{period:strictdrop} + \nu(P|_{D_i})<\nu(P). +\end{equation} +\end{lemma} + +\begin{proof} +Write $n=\dim S$ and $r=\nu(P)$, the generic rank of the line map +$j$ by \Cref{period:numerics}. The full retained period map has rank +$n$; the argument does not require $r=n$. +Work first on a finite Galois cover of the +open locus with connected algebraic monodromy and no factor +permutations. Its connected adjoint group is denoted by $G$. + +\smallskip\noindent +\emph{The Ax--Schanuel decomposition.} +Choose a Hodge-generic point, also general for the ranks of the +factor period maps and their joint differentials. There are only +finitely many rational normal-factor decompositions of $G$, so +these requirements can be imposed simultaneously. Let $T$ be a +local fiber germ of $j$ there, and $Z$ its irreducible Zariski +closure in the source of the variation. Thus $\dim T=n-r$. +Move within the germ to a smooth point of $Z$ and use a smooth +open subset of a resolution when applying Hodge theory. + +The restricted variation on $Z$ has the same generic Mumford--Tate +group as the ambient variation, since $Z$ contains the chosen +Hodge-generic point. The monodromy normality theorem therefore +makes its connected monodromy $H$ normal in $G$; see +\cite[\S5, Theorem~1]{AndreMonodromy}. In adjoint notation write +\[ + G=H\times Q. +\] +The $Q$-period is constant on the lifted $Z$, by the fixed-part +theorem after killing finite monodromy. The varying monodromy +domain on $Z$ is therefore the domain for $H$. Write $J_H$ for +its line-orbit projection and $d_H=\dim J_H$. +In local flat coordinates write $j=(j_H,j_Q)$, with targets $J_H$ +and $J_Q$, the products of line orbits for the two groups. We will +show that the $H$-line map has full rank $d_H$ on $Z$, whereas the +$Q$-line map and the full $Q$-period map have equal generic rank +on the ambient base. + +For the first assertion, put +$m=\dim Z$ and $d=\dim\check{\mathcal D}_H$, where +$\check{\mathcal D}_H$ is the compact dual of the connected-monodromy +period domain. The projection +$\check{\mathcal D}_H\to J_H$ is a surjective morphism of +homogeneous projective flag varieties. All its fibers have +dimension $d-d_H$. Let +$V\subset Z\times\check{\mathcal D}_H$ be the algebraic locus +where the projected line equals its value on $T$. It follows that +$\dim V=m+d-d_H$. The period graph has dimension $m$ in an +ambient variety of dimension $m+d$, so its expected intersection +dimension with $V$ is $m-d_H$. + +By \cite[Theorem~1.1]{BakkerTsimermanAxSchanuel}, an intersection +component of larger dimension projects into a proper algebraic weak +Mumford--Tate subvariety of $Z$. This is impossible for the component +containing the lifted period graph over $T$, since $T$ is Zariski +dense in $Z$. Conversely, a map +to $J_H$ has fibers of dimension at least $m-d_H$. At the chosen +smooth point the restricted fiber is exactly $T$, because the +ambient local fiber is $T\subset Z$. Equality follows, proving the +claim. The cited theorem applies to the rationally polarized +integral adjoint variation on a smooth algebraic base and uses +precisely its connected-monodromy domain. + +Let $q$ denote the full $Q$-period map and set +$s_Q=\rank(dq)$. Choose the smooth point $x\in T\cap Z_{\mathrm{reg}}$ +inside the ambient constant-rank neighborhood fixed above. Since +$T$ is Zariski dense in $Z$, such a point exists. The constant-rank +theorem gives $\ker(dj_x)=T_xT\subset T_xZ$, and hence +\[ + \rank(dj_x|_{T_xZ})=\dim Z-\dim T=d_H. +\] +The $Q$-period is constant on $Z$, so $T_xZ\subseteq\ker(dq_x)$. +Thus, at this same point, +\[ + \ker(dj)\subseteq\ker(dq),\qquad + dj_H(\ker(dq_x))=T_{j_H(x)}J_H. +\] +The first inclusion uses $T_xT\subset T_xZ$; the second is the +just-established rank calculation on $T_xZ$. +The differential $dj_Q$ factors through $dq$. If $dj_Q(v)=0$, +choose $w\in\ker(dq)$ with $dj_H(w)=dj_H(v)$. Then +$v-w\in\ker(dj)$, and therefore $dq(v)=0$. We conclude +\begin{equation} +\label{period:rankdecomposition} + \rank(dj_Q)=s_Q,\qquad r=d_H+s_Q. +\end{equation} +These are generic identities by our choice of the point. + +The factor-line construction, after a common positive rescaling +and a further finite cover, now gives nef lines with +\begin{equation} +\label{period:linedecomposition} + \pi^*P=P_H+P_Q,\qquad + \nu(P_H)=d_H, + \qquad P_Q=q^*B_Q. +\end{equation} +Here $q:\widetilde S\to S_Q$ is a resolved projective quotient +for the full $Q$-period, constructed by +\Cref{period:quotient}, and $\dim S_Q=s_Q$. The highest-line +rank on $S_Q$ is $s_Q$ by +\eqref{period:rankdecomposition}. Only the quotient construction, +not adjoint positivity, has been used to construct $S_Q$. + +\smallskip\noindent +\emph{Reduction for a boundary component.} +For a fixed marked $D_i$, use a common finite Galois cover that +preserves the factors and makes boundary monodromy unipotent. +Resolve it and the map $q$ equivariantly, and let $E_i$ be the +strict transform of a component above $D_i$ that is generically +finite over $D_i$. The inertia group at its generic point fixes +$E_i$ pointwise. Further rescaling does not change any numerical +dimension. Nefness gives +\[ + \nu(P_H|_{E_i})\leq d_H. +\] +For example, every intersection involving $E_i$ is bounded by +the corresponding intersection with a sufficiently ample +effective divisor; the vanishing of higher powers of $P_H$ +then gives this inequality. If $E_i$ does not dominate $S_Q$, +\[ + \nu(P_Q|_{E_i})\leq s_Q-1, +\] +and the binomial expansion for the two nef lines in +\eqref{period:linedecomposition} proves the strict drop. +We may therefore suppose $E_i$ dominates $S_Q$. At its generic +point the $Q$-variation is pulled back from the interior of +$S_Q$, so its local unipotent logarithm vanishes. + +\smallskip\noindent +\emph{Infinite local monodromy.} +If the original local monodromy is infinite, its logarithm at +unipotent level is a nonzero rational element +$N\in\mathfrak h$. Every nonzero rational element of a rational +simple factor has nonzero projection to every conjugate complex +factor. Each retained rational factor has a visible complex +factor. Hence $N$ acts nontrivially on $J_H$. + +Let $w$ be the pure weight of the representation supplying $P_H$, +and let $F^a=\C v$ be its limiting highest line at a general +point of $E_i$. This line has a unique weight contribution, +say of weight $w+k$. It is primitive and $k\geq0$: a +nonprimitive contribution, or a contribution below the middle +weight, would come by monodromy Lefschetz from a nonzero +filtration piece above $F^a$. +Moreover +\begin{equation} +\label{period:nilpotenttop} + N^{k+1}v=0,\qquad N^kv\ne0. +\end{equation} +To see that this statement holds for the actual vector rather +than only its weight-graded image, use the Deligne splitting +of the limiting mixed Hodge structure. The one-dimensional +highest step occupies a single summand, and $N$ has degree +$(-1,-1)$ for this functorial splitting. Primitive vanishing +on the graded piece therefore gives the first equality in +\eqref{period:nilpotenttop}; the Lefschetz isomorphism gives +the second. For a complex summand the same argument is made +in its conjugate-paired real variation and restricted to the +summand. + +In nilpotent-orbit coordinates $[v]\in J_H$. Since this orbit +is closed in projective space and invariant under $\exp(tN)$, +\begin{equation} +\label{period:fixedlocus} + [N^kv]=\lim_{t\to\infty}\exp(tN)[v] + \in J_H\cap\mathbb P(\ker N). +\end{equation} +This fixed locus is a proper closed subset of the irreducible +variety $J_H$, because $N$ acts nontrivially on it. Its dimension +is less than $d_H$. Restrict to a simply connected open part of the boundary stratum +on which $k$ is constant. In a flat trivialization, both $N$ and +$W(N)$ are constant. Project $N^kv$ to +$\Gr^{W(N)}_{w-k}$ and apply the inverse of the fixed Lefschetz +isomorphism +\[ + N^k:\Gr^{W(N)}_{w+k}\longrightarrow\Gr^{W(N)}_{w-k}. +\] +This recovers the weight-graded limiting highest line. Every actual +vector $N^kv$ lies in $W(N)_{w-k}$, and its image in that graded +piece is nonzero. Thus the graded line map factors through the +rational map on +\[ + J_H\cap\mathbb P(\ker N)\cap\mathbb P(W(N)_{w-k}) +\] +induced by this fixed linear quotient and inverse. This locus is +contained in the proper fixed locus in \eqref{period:fixedlocus}, +and the rational map is defined at every point under consideration. +Its rank is less than $d_H$. +By \Cref{period:boundary}, +\[ + \nu(P_H|_{E_i})0\}, +\] +where $H$ is ample. This is independent of $H$ and equals zero +on a point. The proof below fixes the nef coefficient before +taking the asymptotic section limit. + +\begin{lemma}\label{period:subtract} +Let $Z$ be a smooth projective variety, $J$ a nef rational divisor, +$A$ a rational divisor, and +$D=\sum_{i=1}^s d_iE_i$ an effective rational divisor with smooth +prime components. Suppose $A+D$ is big and +\[ + \nu(J|_{E_i})0$ and put $n=\dim Z$. +Choose an ample rational divisor $H_1$ with +$A+D-H_1$ rationally effective. For fixed rational $t\geq0$ +and sufficiently divisible $m$, +\[ + h^0(Z,m(A+D+tJ))\geq h^0(Z,m(H_1+tJ)). +\] +Since $H_1+tJ$ is ample, asymptotic Riemann--Roch makes the +leading coefficient on the right +$(H_1+tJ)^n/n!$. As a polynomial in $t$, this intersection has +degree $r$ and positive leading coefficient. + +Remove $mD$ one component at a time by divisor exact sequences. +Every lost quotient is a line bundle on an $E_i$ of the form +\[ + \mathcal O_{E_i}\left(m(A+D+tJ)-\sum_h k_hE_h\right), + \qquad 0\leq k_h\leq md_h. +\] +Choose an ample integral divisor $B_0$ so large that a fixed +denominator-clearing multiple of $B_0-A-D$ is globally generated. +For each $h$, choose an ample integral divisor $B_h$ such that +both $B_h$ and $B_h+E_h$ are globally generated. Put +$H_2=B_0+\sum_h d_hB_h$. The difference between +$m(H_2+tJ)$ and the displayed quotient divisor is +\[ + m(B_0-A-D)+ + \sum_h\bigl((md_h-k_h)B_h+k_h(B_h+E_h)\bigr). +\] +It is globally generated in these divisible degrees. A section +not vanishing identically on $E_i$ gives an injection into +$\mathcal O_{E_i}(m(H_2+tJ)|_{E_i})$. There are exactly +$md_i$ quotient terms on $E_i$. Hence +\[ + h^0(Z,m(A+tJ)) + \geq h^0(Z,m(H_1+tJ)) + -m\sum_i d_i h^0(E_i,m(H_2+tJ)|_{E_i}). +\] +For each fixed $t$, divide by $m^n/n!$ and let $m$ tend to +infinity through the specified multiples. All the comparison +divisors are ample, so the resulting lower bound is +\[ + (H_1+tJ)^n + -n\sum_i d_i(H_2+tJ)^{n-1}\cdot E_i. +\] +Each restriction polynomial has degree at most +$\nu(J|_{E_i})\leq r-1$, because the restricted divisor is nef. +The positive degree-$r$ term of the first polynomial therefore +dominates the loss for every sufficiently large $t$. The section +growth of $A+tJ$ is then positive in dimension $n$, which is +exactly bigness. +\end{proof} + +Taking $A=K_Z$ and $D$ reduced gives the required removal of the +period boundary once its restriction ranks have been established. +The weights also allow simultaneous removal of other effective +divisors satisfying the same rank inequality, for example a +divisorial ramification term on a generically finite cover. + + +\begin{proof}[Proof of \Cref{period:adjoint}] +Apply \Cref{period:quotient}. If its target is a point there +is nothing further to prove. Otherwise mark the divisor $D$ +as above. The adjoint variation extends across every unmarked +boundary component, including their intersections away from +$D$, because its local monodromies there are trivial. Its +period differential is generically injective. Consequently +\cite[Theorem~1.1]{BrunebarbeCadorel} gives +\[ + K_S+D\text{ big}. +\] +If an initial choice omitted codimension-two loci, apply the +theorem on a log resolution and push the sections down; the +resulting logarithmic canonical divisor pushes forward to +$K_S+D$. By \Cref{period:rankloss}, each marked component +has strictly smaller numerical rank. Apply \Cref{period:subtract} with $A=K_S$ and the marked reduced +divisor $D$. It gives $K_S+aP$ big. Together with the +rational line identity in \Cref{period:quotient}, this proves +\eqref{period:conclusion}. Since $P$ is nef, the same bigness +holds for every larger rational coefficient $a$. +\end{proof} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/reduced-root.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/reduced-root.tex new file mode 100644 index 0000000000000000000000000000000000000000..d4c201114a466ca052032519f133cd3057eba76d --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/reduced-root.tex @@ -0,0 +1,391 @@ +\section{Minimal root covers for reduced boundaries}\label{cyc:section} + +For a smooth projective pair of logarithmic Kodaira dimension zero +with reduced SNC boundary, the minimal cyclic cover has a +one-dimensional space of logarithmic pluriforms in every positive +degree. This strengthens +the character-line statement of Proposition~\ref{rankone:comparison}. +We prove the refinement together with its exact Hodge and divisorial +normalization. It gives a pure integral realization of the entire +highest line and fixes the normalization used in further +relative-Iitaka constructions. +The normalization lemmas below compare the minimum over actual fiber components +with the threshold over all divisorial valuations. They show that a +zero coefficient of the normalized base boundary yields a +multiplicity-one component, and that later permitted modifications +preserve the complete section spaces. These are the additional +normalization statements used by the variation companion. + +\begin{lemma}[The cyclic cover of a logarithmic Kodaira-zero pair] +\label{lem:reduced-cyclic-top} +Let $(J,D)$ be a smooth geometrically integral projective pair over a +field $K$ of characteristic zero. Assume that $D$ is reduced and has +simple normal crossings and that +$\kappa(J_{\overline K},K_{J_{\overline K}}+D_{\overline K})=0$. +Choose the least positive integer $p$ for which +$H^0(J,p(K_J+D))\ne0$, a nonzero section $\omega$ in that space, +and put +\[ + E=\operatorname{div}(\omega)+pD, + \qquad \Delta=D-\frac1pE. +\] +Write $\omega=b\eta^{\otimes p}$ in a rational canonical frame, +and let $\tau:N\to J$ be the normalization in +$K(J)[t]/(t^p-b)$. Let $\rho:\widetilde N\to N$ be a log +resolution, and set +\[ + P=\operatorname{Supp}(\Delta^{=1}),\qquad + D_{\widetilde N}=\bigl((\tau\circ\rho)^{-1}P\bigr)_{\mathrm{red}}. +\] +Then $N$ is geometrically integral, the rational top form +$\Omega=t\tau^*\eta$ extends logarithmically to +$(\widetilde N,D_{\widetilde N})$, and +\[ + H^0\bigl(\widetilde N,m(K_{\widetilde N}+D_{\widetilde N})\bigr) + =K\,\Omega^m\qquad(m\ge1). +\] +\end{lemma} + +\begin{proof} +Every nonzero plurilogarithmic section space on $J$ is +one-dimensional: two independent sections of one degree would +define a nonconstant rational map. Consequently $\kappa(E)=0$, +and every effective integral divisor supported on $E$ also has +exactly one section, since it is bounded above by a multiple of $E$. +These assertions and the minimal index $p$ are unchanged by field +extension, by flat base change for global sections. + +Over $\overline K(J)$, reducibility of $T^p-b$ would make $b$ +an $\ell$th power for some prime $\ell\mid p$. The corresponding +root of $\omega$ would be a nonzero logarithmic pluriform of +degree $p/\ell$, since its logarithmic zero divisor is $E/\ell$. +This contradicts minimality. Thus $N$ is geometrically integral. + +At a prime $Q$ of $N$ above a prime $R$ of $J$, write +\[ + a=\operatorname{coeff}_R E,\qquad + d=\operatorname{coeff}_R D\in\{0,1\},\qquad + e=\operatorname{ord}_Q(\tau^*R). +\] +The ramification formula for top differentials gives +\begin{equation}\label{eq:cyclic-root-order} + \operatorname{ord}_Q\Omega=\frac{ea}{p}-ed+e-1. +\end{equation} +If $a=0$, the cover is unramified at $R$, and this order is +$-d$. If $a>0$, then $q=ea/p$ is a positive integer: the +integrality follows from $t^p=b$ and $d\in\mathbf Z$. +The order is $q-1\ge0$ when $d=1$, and $q+e-1\ge0$ when +$d=0$. Thus the divisorial poles are exactly the reduced inverse +image $D_N=(\tau^{-1}P)_{\mathrm{red}}$. + +We also check extension across the exceptional divisors of $\rho$. +The subpair $(J,\Delta)$ is sub-log-canonical and is sub-klt +outside $P$. Indeed, its positive part is an SNC boundary bounded +by $D$, with all coefficients strictly below one outside $P$; +subtracting an effective divisor cannot worsen discrepancies. +Define the crepant finite pullback by +$K_N+\Delta_N=\tau^*(K_J+\Delta)$. With the compatible +canonical trivialization, $\Delta_N=-\operatorname{div}(\Omega)$. +For a prime divisor $V$ over $N$, let its restricted valuation +on $K(J)$ be $e_Vv$. The same ramification formula gives +\[ + 1+\operatorname{ord}_V\Omega + =e_V A_{J,\Delta}(v), +\] +where $A$ denotes log discrepancy. The right-hand side is +nonnegative, and is positive if the center lies outside $P$. +Since the left-hand side is integral, $\Omega$ has at most a +simple pole above $P$ and is regular elsewhere. This proves the +asserted logarithmic extension. + +Finally let $H_N=\operatorname{div}_N(\Omega)+D_N$. Equation +\eqref{eq:cyclic-root-order} gives the sharper bound +\[ + 0\le H_N\le\tau^*E. +\] +For completeness, when $a>0$ and $d=1$ its coefficient is +$q-1\le ea$; when $d=0$, it is +$q+e-1\le q+p-1\le pq=ea$, because $e\le p$ and $q\ge1$. +At $a=0$ its coefficient is zero. +Writing a logarithmic $m$-canonical form as $u\Omega^m$ and +testing its orders at the primes of the finite normalization gives +\[ + \operatorname{div}_N(u)+mH_N\ge0, + \qquad + u\in H^0(N,\mathcal O_N(m\tau^*E)). +\] +Finite pullback preserves Iitaka dimension, so +$\kappa(N,\tau^*E)=\kappa(J,E)=0$. Since $\tau^*E$ is +effective and $N$ is geometrically integral, the last space is +$K$. Conversely $\Omega^m$ is a logarithmic pluriform for every +$m\ge1$, proving the equality. +\end{proof} + +\begin{corollary}[An integral pure realization of the moduli line] +\label{cor:reduced-integral-moduli} +Let $x:X\to W$ be a projective fibration between smooth complex +projective varieties, with geometrically integral generic fiber +$J$. Let $D_X$ be a reduced SNC boundary whose restriction $D_J$ +satisfies the hypotheses of Lemma~\ref{lem:reduced-cyclic-top}. +Let $M_W$ be the rational Hodge divisor of the rank-one reduction, +with its rational frame and parabolic orders as in +Equation~\eqref{rankone:order}. On sufficiently high smooth models, +there is a polarizable integral pure variation +of Hodge structures on a dense smooth open $W^\circ\subset W$ +whose top nonzero Hodge filtration piece is a line, and +\[ + M_W\sim_{\mathbf Q} + \text{the rational canonical extension of that line}. +\] +In the canonical-bundle-formula normalization, this is the moduli +divisor of the auxiliary generic subpair +$\Delta_J=D_J-E_J/p$, which may have negative coefficients. +\end{corollary} + +\begin{proof} +Let $s$ be the minimal-degree logarithmic generator on the generic +fiber, and choose a rational relative canonical frame $\eta$ on +$X$. The rational function $s/\eta^{\otimes p}$ defines one +cyclic cover over a dense open of $W$, not merely a collection +of local character covers. Spread its resolution and pole boundary +over a smaller dense open $W^\circ$ so that they form a smooth +projective family of SNC pairs. Write $r=\dim J$, and let +$u:\widetilde X^\circ\setminus D_{\widetilde X}^\circ\to W^\circ$ +be the complementary open family. Its geometric variation +\[ + \mathbb V_{\mathbf Z}=R^ru_*\mathbf Z/\mathrm{torsion} +\] +is admissible and graded-polarizable. Logarithmic Hodge theory +identifies +\[ + F^{r+1}\mathbb V=0,\qquad + F^r\mathbb V_{\mathcal O} + =\widetilde x_*\omega_{\widetilde X^\circ/W^\circ} + (D_{\widetilde X}^\circ). +\] +Lemma~\ref{lem:reduced-cyclic-top} makes this entire top piece a +line. Strictness for the weight filtration therefore singles out +one weight $w$ with +\[ + F^rW_{w-1}\mathbb V=0, + \qquad F^rW_w\mathbb V=F^r\mathbb V, + \qquad + F^r\operatorname{Gr}^W_w\mathbb V\simeq F^r\mathbb V. +\] +The rational variation $\operatorname{Gr}^W_w\mathbb V$ is +polarizable. Intersecting its rational weight filtration with +$\mathbb V_{\mathbf Z}$ and removing torsion supplies an integral +lattice; after rescaling, its rational polarization is integral. +Thus this pure variation has precisely the required top line. + +After a finite cover making local monodromies unipotent, the +canonical extensions have locally free double gradeds for the +Hodge and weight filtrations. The displayed identification of top +lines consequently extends across the boundary; see +\cite[Proposition~3.12]{FujinoFujisawa}. Descent gives the same +identification for their rational canonical extensions. + +The global rational frame of this entire top line is +$s^{1/p}$. Equation~\eqref{rankone:order} computes its parabolic +order at every prime divisor $P$ as $\beta_P$, the order of the +same rational frame in $M_W$. Equality of these divisorial +orders identifies the rational extensions. In particular, it +identifies $M_W$ itself, with no additional factor of $p$. +We check the rank condition for the canonical-bundle-formula +comparison. On a log resolution $\mu:J'\to J$, let +$R'=\lceil A^*(J,\Delta_J)_{J'}\rceil$, where $A^*$ omits the +log-discrepancy-zero components of the discrepancy divisor. +The subpair is sub-log-canonical, so $R'\geq0$. At the strict +transform of each original prime its coefficient is zero if +$0\leq\Delta_J\leq1$, and is $\lceil-\Delta_J\rceil$ otherwise. +Thus $0\leq\mu_*R'\leq E_J$. Testing original primes gives +$H^0(J',\cO_{J'}(R'))\subseteq H^0(J,\cO_J(E_J))=\C(W)$, +and constants give equality. This is the required rank-one condition. +Extending the generic subpair by +$-p^{-1}\operatorname{div}_{K_{X/W}}(s)$ makes its log canonical +divisor rationally equivalent to $x^*K_W$. The generic sub-lc, +relative triviality, and rank conditions of +\cite[Definition~6.18]{BFMT} therefore hold. Adding a base pullback +to normalize vertical orders changes the total base term and the +discriminant equally, leaving the moduli divisor unchanged. + +The least positive index trivializing $K_J+\Delta_J$ is $p$. +Indeed, a smaller such index $q$ would produce a rational +$q$-canonical form with logarithmic zero divisor $qE_J/p\geq0$, +contradicting the minimality of $p$. Thus the cyclic field extension +in Construction-Definition~6.23 of \cite{BFMT} is the one used above. + +For its boundary comparison, put +$P_J=\operatorname{Supp}(\Delta_J^{=1})$ and first make a common +crepant log resolution of the auxiliary subpair and its root cover. The +coefficient-one support on the crepant model is contained in the +total inverse image of $P_J$: outside $P_J$ the auxiliary subpair is +sub-klt. Every additional component allowed by our pole boundary +has positive log discrepancy, so the ramification discrepancy +formula in Lemma~\ref{lem:reduced-cyclic-top} makes $\Omega$ +regular there. The top-form space for the BFMT pole boundary +therefore injects into the top-form space for our boundary and +contains $\Omega$; both are the line generated by $\Omega$. +After spreading over a common smooth base open, inclusion of the +open complements induces a morphism of their admissible geometric +mixed variations. This morphism is strict for the Hodge and weight +filtrations, including their limiting filtrations after a finite +cover making boundary monodromy unipotent. It therefore identifies +the top lines in the same pure weight grade. Functoriality +of canonical extension gives the same identification across the +boundary and hence for the parabolic rational lines. Theorem~6.28 of +\cite{BFMT} now identifies this Hodge divisor with the moduli +divisor using one common positive multiple on both sides. This +comparison permits negative coefficients in the generic subpair; +it does not use the separate semiampleness theorem requiring +generic effectiveness. +\end{proof} + +\begin{remark} +The hypothesis that $D$ is reduced is essential to +Lemma~\ref{lem:reduced-cyclic-top}. For five distinct points on +$\mathbf P^1$, the rational boundary +$D=\frac25(P_1+\cdots+P_5)$ satisfies $K_{\mathbf P^1}+D\sim_{\mathbf Q}0$. +Its index-five root cover has genus six and empty pole boundary. +Thus its entire top-form space has dimension six. The refinement therefore applies to reduced boundaries; the +character-line construction of Section~\ref{rankone:section} is used +for general rational orbifold boundaries. +\end{remark} + +\subsection{The original relative orders} + + +\begin{lemma}[Normalization of the rank-one boundary] +\label{lem:rankone-normalization} +Let $x:X'\to W$ be a projective fibration on smooth models, let $D'$ be +an effective reduced SNC divisor, and suppose that its geometric generic +fiber $(J,D_J)$ has logarithmic Kodaira dimension zero. Fix a nonzero +relative rational $p$-canonical form $s$ generating +$H^0(J,p(K_J+D_J))$. At a prime $P$ of $W$ set +\[ + a_E=\operatorname{ord}_E(x^*P),\qquad + r_E=p^{-1}\operatorname{ord}^{pK_{X'/W}}_E(s),\qquad + d_E=\operatorname{coeff}_E(D'),\qquad + t_P=\min_{E\mapsto P}\frac{r_E+d_E}{a_E}. +\] +Here the minimum is over actual components on the chosen source model. +Choose that model so that the generator divisor, boundary and inverse +image of $P$ have SNC support over the generic point of $P$; later +source modifications carry strict boundary plus reduced exceptional +boundary. The coefficients in the rank-one comparison satisfy +\[ + \alpha_P=t_P,\qquad + \beta_P=\inf_{E\mapsto P}\frac{1+r_E}{a_E}-1,\qquad + T_P=1-c_P, + \quad + c_P:=\inf_{E\mapsto P}\frac{1+r_E}{a_E}-t_P. +\] +The infima allow divisorial valuations on higher smooth source models. +Equivalently, $c_P$ is the log canonical threshold of $x^*P$ for the +normalized auxiliary subpair +\[ + \Delta^{\mathrm{aux}} + =-p^{-1}\operatorname{div}_{K_{X'/W}}(s)+x^*\sum_Qt_QQ. +\] +If $T_P=0$, an actual component attaining $t_P$ has +$a_E=1$, $d_E=0$, and attains the threshold infimum as well. +\end{lemma} + +\begin{proof} +Choose a local frame $\eta$ of $K_W$ that does not vanish at the generic +point of $P$. The canonical line-bundle identification +\[ + \mathcal O_{X'}(pK_{X'/W})\otimes x^*\mathcal O_W(pK_W) + \simeq\mathcal O_{X'}(pK_{X'}) +\] +identifies the order of $s$ with the order of +$s\wedge x^*\eta^{\otimes p}$. Thus the quantity $l_E$ in the +rank-one order formula is exactly $r_E$. This uses the actual relative +canonical line bundle: the Jacobian contribution is already present in +$r_E$, so no ramification term is added or subtracted here. +Consequently its two order formulas give +\[ + \alpha_P=t_P, + \qquad + \beta_P=\min_E\frac{r_E+1-a_E}{a_E}. +\] +Put +\[ + \Delta_0=-p^{-1}\operatorname{div}_{K_{X'/W}}(s), + \qquad \lambda_P=\min_{E\mapsto P}\frac{1+r_E}{a_E}, +\] +where the minimum is over actual vertical components. The indicated +SNC preparation and the generic logarithmic bound give $\Delta_0$ SNC +support and horizontal coefficients at most one. In +$\Delta_0+\lambda_Px^*P$, each vertical coefficient is +$-r_E+\lambda_Pa_E\leq1$. Thus this subpair is sub-log-canonical +over the generic point of $P$, and every higher divisorial valuation +satisfies +\[ + 1+r_E-\lambda_Pa_E\geq0. +\] +An actual component attaining $\lambda_P$ gives equality. Hence the +infimum over all such valuations equals $\lambda_P$, and +$\beta_P=\lambda_P-1$. + +For completeness, the first minimum also remains unchanged on higher +source models. A coefficient realizing $t_P$ makes +$t_Pa_E-r_E$ no greater than the original boundary coefficient at every +actual component over $P$. Equivalently, after clearing denominators, +the normalized generating form satisfies every logarithmic order test +over the generic point of $P$. Its pullback remains logarithmic with +strict plus reduced exceptional boundary. No new valuation can lower +the first minimum, while an old minimizing valuation persists. + +At any divisorial valuation above the generic point of $P$, the +crepant coefficient of $\Delta^{\mathrm{aux}}$ is $-r_E+a_Et_P$. +The threshold formula therefore gives +\[ + \operatorname{lct}_{\eta_P}(X',\Delta^{\mathrm{aux}};x^*P) + =\inf_E\frac{1+r_E-a_Et_P}{a_E}=c_P. +\] +Since $T_P=\alpha_P-\beta_P$, we obtain $T_P=1-c_P$. +Finally, if $T_P=0$ and $E$ attains the first minimum, then +\[ + 1=c_P\le\frac{1+r_E}{a_E}-t_P + =\frac{1-d_E}{a_E}\le1. +\] +All terms are equal. Since $a_E$ is a positive integer and +$d_E\in\{0,1\}$, this forces $a_E=1$ and $d_E=0$, as claimed. +\end{proof} + +\begin{lemma}[Permitted models after normalization] +\label{lem:normalized-models} +Suppose that $(W,T)$ is smooth with effective SNC rational boundary and +that the moduli divisor $M$ is a rational Cartier divisor on $W$. +For a smooth modification $\mu:W'\to W$, put +\[ + T'=\mu_*^{-1}T+\operatorname{Exc}(\mu)_{\mathrm{red}}, + \qquad M'=\mu^*M, +\] +and assume the resulting boundary is SNC. Then pullback identifies the +section spaces of every sufficiently divisible multiple of +$K_W+T+M$ and $K_{W'}+T'+M'$. At unchanged codimension-one points the +normalization in Lemma~\ref{lem:rankone-normalization} is preserved. +Every new exceptional prime has coefficient one in $T'$. +\end{lemma} + +\begin{proof} +Log canonicity of $(W,T)$ gives +\[ + K_{W'}+T'+M'=\mu^*(K_W+T+M)+E, +\] +where $E$ is effective and $\mu$-exceptional. For divisible $m$, +normality gives +\[ + \mu_*\mathcal O_{W'}(mE)=\mathcal O_W. +\] +The projection formula therefore identifies the section spaces. +The modification is an isomorphism at each old generic divisorial +point, so the old local order calculations are unchanged. The final +assertion is the definition of $T'$. +\end{proof} + +Appendix~\ref{p1logcbfalt:section} gives complementary projective +proofs of the section comparisons and further calculations with the +auxiliary subpair on higher models. diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/root-comparison.tex b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/root-comparison.tex new file mode 100644 index 0000000000000000000000000000000000000000..20c1f82eafec08d4aff2ec72d41ecfc6d3612497 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/root-comparison.tex @@ -0,0 +1,675 @@ +\subsection{The exact comparison for Kodaira-zero fibers} +\label{rankone:section} + +For fibers of logarithmic Kodaira dimension zero, the relative +pluricanonical forms determine a single Hodge line on the base. We +construct that line, compute its boundary orders, and use them to +identify the adjoint section spaces. Coefficient-one boundary +components introduce mixed Hodge structures, but the line itself +lies in one pure weight grade. + +A pure variation of Hodge structures carries a local system and a +varying Hodge filtration. The \emph{highest step} of a summand is its +highest nonzero filtration piece. The \emph{parabolic extension} of +such a line is obtained by local power substitutions making boundary +monodromy unipotent, extension of the unipotent Hodge filtration, and +descent with the resulting rational weights. We will compute those +weights using the actual relative pluricanonical form. + +\begin{proposition}[The rank-one comparison]\label{rankone:comparison} +Let $g:(X,\Delta)\to W$ be a fibration with connected fibers from a +smooth compact K\"ahler manifold to a smooth compact manifold in +class $\mathcal C$. Suppose that $\Delta$ is an SNC boundary with +rational coefficients in $[0,1]$ and that +$\kappa(G,K_G+\Delta_G)=0$ on a very general smooth fiber $G$. +After the modifications of +\Cref{setup:log-modification,setup:flattening}, fix a smooth compact +reference pair $(X_0,\Delta_0)$ and a modification +\[ + \pi:X\longrightarrow X_0,\qquad + \Delta=\pi_*^{-1}\Delta_0+\operatorname{Exc}(\pi)_{\mathrm{red}}, +\] +such that every prime divisor mapped by $g$ into codimension at least two +is $\pi$-exceptional. Then there are a rational line bundle $M$ +and a rational divisor $T$ on $W$ with the following properties. +\begin{enumerate}[label=\textup{(\roman*)}] +\item A positive integral multiple of $M$ is the parabolic highest Hodge +line of a complex direct summand of a pure, real-polarizable variation +with an integral lattice on a dense open subset of $W$. Its parabolic extension is the line to which the adjoint-positivity +theorem, \Cref{period:adjoint}, will apply. +\item The divisor $T$ is supported on an SNC divisor and satisfies +\begin{equation}\label{rankone:boundary} + B(g,\Delta)\leq T\leq 1. +\end{equation} +In particular, $T$ is effective. If every prime over a prime $D$ of $W$ +has $\Delta$-coefficient one, then $T_D=1$. +\item In sufficiently divisible degrees $q$, multiplication by the +relative generating forms gives linear isomorphisms +\begin{equation}\label{rankone:sections} + H^0\bigl(W,q(K_W+T+M)\bigr) + \simeq H^0\bigl(X,q(K_X+\Delta)\bigr) + \simeq H^0\bigl(X_0,q(K_{X_0}+\Delta_0)\bigr). +\end{equation} +In a common divisible grading they preserve products and ratios of +sections, and hence the dimensions +of the images of their linear systems. +\item The same comparison holds relatively. If $h:W\to Y$ and +$f_0:X_0\to Y$ are holomorphic maps with $f_0\pi=hg$, then +\begin{equation}\label{rankone:relative-sections} + h_*\mathcal O_W\bigl(q(K_W+T+M)\bigr) + \simeq (f_0)_*\mathcal O_{X_0}\bigl(q(K_{X_0}+\Delta_0)\bigr) +\end{equation} +for sufficiently divisible $q$. The comparison of ratios also applies +to systems over $Y$ and to their restrictions wherever generic base +change holds. The relative identity remains valid after tensoring by a +line bundle on $Y$, or a rational line bundle in degrees clearing its +denominator. +\end{enumerate} +\end{proposition} + +The Hodge construction and its divisorial calculation are independent +of the reference pair. The condition on $\pi$ is used in the final +section comparison, to descend forms across exceptional centers. + +\subsubsection{The root form and its eigenspace} + +The cyclic-root construction and its distinguished eigensheaf occur in +\cite[Remark~2.6]{FujinoMori}, +\cite[Lemma~5.2]{AmbroBoundary}, and +\cite[Remark~3.9]{FujinoGongyo}. Here the order calculations also +include coefficient-one boundary and keep the character line distinct +from the entire top Hodge space. + +Choose a positive integer $m$ clearing the boundary denominators such +that the generic rank of +\[ + g_*\mathcal O_X\bigl(m(K_{X/W}+\Delta)\bigr) +\] +is one. The same holds in every positive multiple of this degree: +powers of a nonzero section exist, whereas two independent sections +would contradict Kodaira dimension zero. Its reflexive hull +$\mathcal E_m$ is a line bundle since $W$ is smooth. A local frame $s$ of +$\mathcal E_m$ determines a meromorphic relative $m$-canonical form on +$X$ over the corresponding base open set. Two such frames differ by +an invertible base function. +The direct image records $m$-pluriforms. Positivity will come from the +Hodge extension of their rational root; comparing the two extension +orders will supply $T$. + +Resolve the horizontal zeros, poles, and boundary, then choose a dense +open set $W^\circ$ on which $g$ is smooth, the resulting divisors are +relatively SNC, and all their strata are smooth over the base. By +\Cref{setup:log-modification}, new exceptional components have +coefficient one, and the logarithmic Kodaira dimension of the general +fiber stays zero. On each base chart take the normalized cyclic cover +of $m$-th roots of $s$ as a relative pluriform and resolve it +functorially. Write $\tau$ for its tautological relative top form and +$d=\dim X-\dim W$. + +At a horizontal prime of boundary coefficient $\delta$, let $l$ be +the order of $s$ divided by $m$. +The inequality $l\geq-\delta\geq-1$ shows that $\tau$ has at most +logarithmic poles. Indeed, at a covering prime of ramification index +$j$, its order is +\begin{equation}\label{rankone:root-order} + j(1+l)-1. +\end{equation} +This number is integral. If $l>-1$ it is nonnegative; if $l=-1$ it +equals $-1$. On the resolved cover let $H$ be the reduced inverse +image of the horizontal primes with $l=-1$. It includes the +exceptional components above those primes that are needed in the log +resolution. Logarithmic pullback preserves the bound, and no +additional horizontal poles occur. We use the cohomology of the +complement of $H$. + +\begin{lemma}\label{rankone:eigenspace} +The tautological form spans the root-character part of +$F^dH^d(\widetilde G\setminus H,\C)$ on a general fiber of the cyclic +cover. In particular, that part has dimension one. +\end{lemma} + +\begin{proof} +For a compact K\"ahler SNC pair, logarithmic Hodge--de Rham degeneration +identifies this top Hodge space with its global logarithmic top +forms; it is also the smooth-fiber case of +\cite[Theorem~1.1]{FujinoFujisawaVMHS}, by duality. Let $\eta$ be +another form of the tautological character. Then $\eta/\tau$ is +deck-invariant and descends to a meromorphic function $v$ on $G$. + +Consider a prime where the logarithmic section $s$ has no zero. +There $l=-\delta$. For $l>-1$, the integer in +\eqref{rankone:root-order} lies between $0$ and $j-1$. A pole of +$v$ downstairs would lower the covering order by at least $j$, which +is impossible. For $l=-1$, the logarithmic pole allowance is already +exhausted, so the same conclusion holds. Thus the poles of $v$ can +occur only on the positive zero divisor of $s$ as a section of +$m(K_G+\Delta_G)$. A sufficiently high power $s^N$ absorbs those +poles. Both $s^N$ and $v s^N$ are then sections of +$Nm(K_G+\Delta_G)$. The assumption $\kappa=0$ forces them to be +dependent, and hence forces $v$ to be constant. + +The argument also applies to a disconnected cover: we retain the whole +cover with its $\mu_m$-action, whose invariant meromorphic functions +descend to $G$. +\end{proof} + +The ranks of the Hodge and weight graded bundles are locally constant +on $W^\circ$. The conclusion of \Cref{rankone:eigenspace}, proved on +a very general fiber, therefore holds throughout this smooth log +locus. Notice that neither the argument nor +\eqref{rankone:root-order} requires a boundary coefficient to have +the form $1-1/a$ with $a$ an integer. + +\subsubsection{Pure weights and descent of the local systems} + +We next place a power of the local Hodge line in a global pure +variation. This requires compatible polarizations and descent, +because the cyclic covers need not form a global family. + +First make the local normalized-cover constructions on charts of the +compact base $W$, using the meromorphic forms supplied by the frames +of $\mathcal E_m$. They extend across the missing base divisor as +finite covers; subsequent resolutions can be projective and +functorial in the analytic category +\cite[Theorem~1.1]{BierstoneMilmanFunctorial}. +The resulting maps to the K\"ahler source are projective, since the +finite covers are projective. Thus their total spaces are K\"ahler +over relatively compact source neighborhoods +\cite[Definition~4.1 and Lemma~4.4]{FujikiDouady}. Properness allows +such a neighborhood to contain the inverse image of a sufficiently +small disk transverse to a base divisor. +The divisor of the tensor is unchanged under multiplication +by a base unit. Thus the constructions on overlaps are identified +after taking a local root of that unit, and the resolutions and their +exceptional divisor classes are identified as well. + +Fix a K\"ahler class $\xi$ on $X$. On each resolved-cover chart +$q_i:Z_i\to X|_{U_i}$, choose the relatively ample line bundle $A_i$ +as follows. Functoriality matches the blowup center ideals and their +tautological line bundles under the root-change identifications. +Insert identity stages where a center is empty; the corresponding +tautological bundle is trivial there. A sufficiently weighted tensor +product gives compatible $A_i$, equivariant under deck transformations, +by descent of the centers, blowups, and tautological bundles. The classes +\[ + \theta_i=Nq_i^*\xi+c_1(A_i) +\] +are K\"ahler on the smooth compact fibers for $N$ sufficiently large; +their restrictions to the compact boundary strata are K\"ahler too. +At the intermediate finite covering stage one uses the pullback +K\"ahler \emph{class}; the literal pulled-back form can degenerate +at ramification. The relatively ample class supplies positivity +along the projective resolution. One may choose a single $N$: +carry out the construction over charts $U_i^+$ containing the closures +of finitely many relatively compact charts $U_i$ covering $W$. +Properness makes the inverse images of these closures compact, so +each admits a bound $N_i$ in this class construction. Taking +$N\geq\max_iN_i$ works on all charts, since increasing $N$ adds +the semipositive class $q_i^*\xi$. This is why the construction was +extended over the compact base before shrinking to $W^\circ$. +For the invariant K\"ahler-class construction on an equivariant +resolved cyclic cover, see also \cite[Lemma~8.1]{ChenLimits}. + +The classes $\theta_i$ come from total-space cohomology, so their +restrictions to the smooth fibers and strata are flat for the +Gauss--Manin connection. They agree under the overlap +identifications. Lefschetz decomposition and cup product consequently +give compatible flat real polarizations on the compact-stratum +cohomology systems. The weight spectral sequence, with its Tate +twists, gives the pure weight grades of the open-fiber cohomology as +subquotients of these systems. They are real-polarizable: a real Hodge +subvariation of a polarized pure variation is nondegenerate for the +polarization, and a quotient can be identified with an orthogonal +complement. These operations are compatible with the overlap maps. +The integral structures come from cohomology modulo torsion and the +saturated intersections with the rational weight filtration. We do +not assert that the real polarizations are rational. + +There is a unique weight $w$ for which +\[ + F^d\Gr^W_w H^d(\widetilde G\setminus H,\C)_\chi\ne0. +\] +Here $\chi$ is the tautological character. Uniqueness follows from +the rank-one assertion and strictness of the weight filtration. Let +$V_i$ be the pure integral variation given by this weight grade on +the $i$th chart; the chosen line is $F^d(V_i)_\chi$. + +\begin{lemma}\label{rankone:descent} +A positive tensor power of these local Hodge lines is the highest +Hodge line of a complex direct summand of a global real-polarizable +integral pure variation on $W^\circ$. +\end{lemma} + +\begin{proof} +Refine an overlap so that a root of its transition unit can be +chosen. It gives an isomorphism between the cyclic-cover families, +and therefore an isomorphism of their integral cohomology systems. +Another choice differs by a deck transformation. On a triple +overlap the cocycle defect is also a deck transformation. All these +maps commute with the distinguished $\mu_m$-action. + +Put +\[ + \mathbb U_i= + \bigl((V_{i,\Z}/\text{torsion})^{\otimes m}\bigr)^{\mu_m}, +\] +using the diagonal deck action. The ambiguity and the cocycle defect +act trivially on $\mathbb U_i$. Hence the $\mathbb U_i$ descend to +an integral local system $\mathbb U$; saturation only changes the +lattice by an isogeny. The compatible real polarizations, tensorized +and restricted to invariants, polarize this pure variation. + +Over $\C$ the character projector $e_\chi$ is a flat Hodge +endomorphism on each chart. The projector +$e_\chi^{\otimes m}$ restricts to the invariant tensor space, +because $\chi^m=1$. It commutes with all transition maps and +therefore defines a global complex direct summand with local fibers +$((V_i)_\chi)^{\otimes m}$. Its highest step is +$(F^d(V_i)_\chi)^{\otimes m}$, a line. In local rooted notation +this line is generated by $\tau^{\otimes m}$, which is identified +meromorphically with $s$. Thus no global root of $\mathcal E_m$ and +no global family of cyclic covers is needed. +\end{proof} + +\subsubsection{The two extension orders} + +Let $M$ denote one $m$th of the parabolic extension of the line just +constructed. We use $s^{1/m}$ for its local rational frame; the +notation becomes literal after the powers and finite boundary covers +above. We now compare this parabolic extension with the pole bounds +of the original pluriform. + +Fix a prime divisor $D$ on $W$ and use a local ordinary base volume +$\omega$ that does not vanish at its generic point. Work on a source +model on which the divisor of the generator, the boundary, and the +inverse image of $D$ have SNC support over that generic point, +using strict transforms plus reduced exceptional boundary as before. +All primes of this prepared source dominating $D$, including the +exceptional primes introduced by its resolution, enter the following +order tests. Put +\[ + l_E=\frac1m\ord_E(s\wedge g^*\omega^m),\qquad + a_E=\ord_E(g^*D),\qquad \delta_E=\Delta_E, +\] +and +\[ + \alpha_D=\min_{g(E)=D}\frac{l_E+\delta_E}{a_E}, + \qquad \beta_D=\ord_D^M(s^{1/m}). +\] +The notation $s\wedge g^*\omega^m$ means the relative-times-base +identification in the $m$th tensor power of the canonical bundle. +For $q$ divisible by $m$ and clearing the rational data, and a +meromorphic base function $\varphi$, the total pluriform +$\varphi\,\omega^q s^{q/m}$ has boundary-adjusted order +\[ + a_E\ord_D(\varphi)+q(l_E+\delta_E) +\] +at $E$. It therefore satisfies the source boundary bounds at all +these primes if and only if +\begin{equation}\label{rankone:base-bound} + \ord_D(\varphi)+q\alpha_D\geq0. +\end{equation} +The Hodge frame contributes $q\beta_D$ to the order on the base, +so the candidate boundary correction is $T_D=\alpha_D-\beta_D$. +Computing $\beta_D$ will reduce the required bounds on $T_D$ to +a comparison of two minima. We first justify computing that Hodge +order by logarithmic orders on a semistable cover, without changing +the chosen pure Hodge line. Global consistency of $T$ and descent +across the untested source primes will follow afterward. + +\subsubsection{The extension across a transverse disk} + +We use the analytic canonical-extension theorem +\cite[Theorem~1.1]{FujinoFujisawaVMHS}. It applies to a proper +surjective morphism of an analytic SNC pair $(Z,H)$ to a smooth base, +with $H$ reduced, every stratum K\"ahler and dominant, and all strata +smooth off a normal crossing discriminant. It supplies a +graded-polarizable real variation on compact-support cohomology, +locally free double Hodge/weight grades on the extension, and the +dual direct-image description by $\omega_{Z/B}(H)$. Admissibility +is established in \cite[Remark~4.8]{FujinoFujisawaVMHS}. +These conclusions apply to proper K\"ahler families; projectivity +is not required. + +Take a transverse disk at a general point of a base divisor. Resolve +the cyclic-cover pair together with the central fiber, leaving its +smooth logarithmic locus over the punctured disk unchanged. Near a +central point the resulting map has the form +\[ + t=\varepsilon x_1^{a_1}\cdots x_r^{a_r}, + \qquad H\subset\{x_{r+1}\cdots x_l=0\}, +\] +where $\varepsilon$ is a unit and $H$ is the horizontal boundary. Absorbing it +into a vertical coordinate gives a strict toroidal morphism. Compactness +of the central fiber and properness allow us to shrink the disk until +finitely many such charts cover its inverse image. The associated cone +complex is therefore finite. Apply the combinatorial semistable +subdivision theorem +\cite[Theorem~2.7]{AdiprasitoLiuTemkinSemistable}. Its base is one ray, +so the base lattice alteration is realized by $t=u^N$. Normalize this +base change, keeping all components, and realize the projective source +subdivision analytically +\cite[Lemmas~2.14--2.15, 2.18, and~2.31(4)] +{EnokizonoHashizumeSemistable}. Zero-image faces, which record the +horizontal boundary, are allowed in the combinatorial theorem. + +The semistable local form is +\[ + u=y_1\cdots y_q, + \qquad H\subset\{y_{q+1}\cdots y_k=0\}; +\] +see \cite[Lemma~2.21(4)]{EnokizonoHashizumeSemistable}. +Here $H$ includes every horizontal exceptional component in the +transformed boundary. The total space is smooth, the central fiber is +reduced, and its union with $H$ is SNC. On every horizontal +intersection stratum the displayed monomial is nonconstant; +properness makes its image the whole disk. The composite to the +initial K\"ahler total space is projective, so the new total space and +its smooth strata are K\"ahler after shrinking the disk +\cite[Lemma~4.4]{FujikiDouady}. + +These toroidal modifications preserve the open pair on the punctured +fibers. Its cohomology is therefore the pullback of the original +open-fiber cohomology, including the full direct sum if the cover +splits. Transport the original $\mu_m$-action through this +identification. The deck projector then acts on the variation and, +by functoriality, on its canonical extension; no action on a selected +component or on the chosen semistable model is needed. We apply the +extension theorem componentwise and take the direct sum. We use this +semistable pair itself: an arbitrary further resolution need not +preserve reducedness. + +For application of the cited theorem, $H$ consists of the horizontal +logarithmic boundary. The central fiber is not included in $H$: +it appears in the relative logarithmic complex, with $dt/t$ in the +base quotient. Write $f:Z\to B$ for this degeneration over the disk, +and $Z_0=f^{-1}(0)$ for its central fiber. The top relative +logarithmic sheaf is +\begin{equation}\label{rankone:log-volume} + \omega_{Z/B}\bigl(H+(Z_0)_{\mathrm{red}}-f^*[0]\bigr). +\end{equation} +On the semistable model the central fiber is reduced, so this is +$\omega_{Z/B}(H)$. For a smooth open fiber of dimension $d$, +Poincar\'e duality identifies +\[ + H^d_c(Z_t\setminus H_t)^*\simeq + H^d(Z_t\setminus H_t)(d). +\] +Accordingly the dual degree-zero Hodge quotient in +\cite[Theorem~1.1(iv)]{FujinoFujisawaVMHS} is precisely the top +$F^d$ step of ordinary $H^d$. The upper and lower canonical +extensions coincide after unipotentization. This proves that the +canonical extension of the top step is the direct image of +\eqref{rankone:log-volume}. The K\"ahler form of the +one-parameter limiting theorem is also explained in +\cite[Remarks~2.16, 4.4, and~4.8]{FujinoFujisawaVMHS}. + +\begin{lemma}\label{rankone:extension} +The projection of the rank-one top eigenspace to its nonzero pure +weight grade is an isomorphism on canonical extensions after +unipotentization. Consequently it is an isomorphism of parabolic +rational lines before that base change. +\end{lemma} + +\begin{proof} +Let $\overline{\mathcal V}_\chi$ be the extended mixed eigensystem +over the disk. The finite deck projector acts on the canonical +extension and its filtrations. By the cited extension theorem, +each +\[ + \Gr_F^p\Gr^W_j\overline{\mathcal V}_\chi +\] +is locally free and has its rank on the punctured disk. The +filtrations on the weight quotients are induced filtrations. +Thus the exact sequences +\[ + 0\longrightarrow F^dW_{j-1}\overline{\mathcal V}_\chi + \longrightarrow F^dW_j\overline{\mathcal V}_\chi + \longrightarrow F^d\Gr^W_j\overline{\mathcal V}_\chi + \longrightarrow0 +\] +have locally free quotients. There is no step above $F^d$, also +on the extension, by these constant-rank assertions. Exactly one +of the displayed quotients has rank one, namely that for $j=w$; +all the others have rank zero. Induction on the weight filtration +therefore gives +\[ + F^dW_{w-1}\overline{\mathcal V}_\chi=0,\qquad + F^dW_w\overline{\mathcal V}_\chi + =F^d\overline{\mathcal V}_\chi, +\] +and identifies the latter line with +$F^d\Gr^W_w\overline{\mathcal V}_\chi$ across the origin, without +an additional zero or pole. Descent gives the parabolic assertion. +\end{proof} + +\subsubsection{The divisorial order of the Hodge line} + +With the notation and prepared source above, we claim that the +parabolic order of the frame $s^{1/m}$ in $M$ is +\begin{equation}\label{rankone:order} + \beta_D=\ord_D^M(s^{1/m}) + =\min_{g(E)=D}\frac{l_E+1-a_E}{a_E}. +\end{equation} +Equivalently, $\beta_D$ is the largest rational number $b$ such that +$t^{-b}s^{1/m}$ extends at $D=(t=0)$ in the parabolic sense. + +It suffices to work over a transverse disk, suppressing the other +base coordinates. Write $\tau=s^{1/m}$ and +$\Omega=\tau\wedge g^*dt$. The multivalued total form +$\Omega/t$ has order $l_E-a_E$ at $E$. A logarithmic extension +allows order $-1$, so twisting by $t^{-b}$ imposes +\begin{equation}\label{rankone:inequality} + l_E+1-a_E-a_Eb\geq0. +\end{equation} +To verify this directly against the canonical extension, take a +finite base change $t=u^N$ and a semistable model of the root +cover. At a central prime over the strict transform of $E$, let +$e$ be the total ramification index of the composite map to $X$, +including the cyclic-cover ramification. Reducedness gives +$ea_E=\ord_{E'}(t)=N\ord_{E'}(u)=N$, where $E'$ is this new prime. +Orders of a total volume form shifted by one multiply by $e$. +Dividing its pullback by +$dt/du=Nu^{N-1}$ therefore gives +\[ + \ord(\tau\wedge du) + =e(l_E+1)-N + =e(l_E+1-a_E). +\] +The base twist subtracts $Nb=ea_Eb$. Regularity in +\eqref{rankone:log-volume} is exactly +\eqref{rankone:inequality}. By \Cref{rankone:extension}, testing +the logarithmic top form tests the chosen pure Hodge line as well. +The form being tested is the pullback of the original tautological +form on the full cover. The identification of open-fiber cohomology +above and the direct-image description of its canonical extension +identify it with that same Hodge section. Keeping all normalized +components ensures that a prime above every original $E$ is present; +the proper toroidal modifications retain its strict transform. + +To prove sufficiency, continue to write $X$ for the smooth source +over the transverse disk and put $n=d+1$. In SNC coordinates write +\[ + \Omega=h\prod_i x_i^{l_i}\,dx_1\wedge\cdots\wedge dx_n, + \qquad t=h'\prod_i x_i^{a_i}, +\] +where $h,h'$ are units and $a_i=0$ for horizontal components. +Put $c_i=l_i+1-(1+b)a_i$. The vertical inequalities +\eqref{rankone:inequality} and the horizontal logarithmic bounds +give $c_i\geq0$. For any additional divisorial valuation $v$, +write $A_X(v)$ for its log discrepancy over this smooth model, +that is, one plus the order of the Jacobian. Logarithmic pullback +of the SNC coordinate divisor gives +$A_X(v)\geq\sum_i v(x_i)$, exactly as in +\Cref{setup:log-modification}. Thus the shifted order of +$t^{-(1+b)}\Omega$ is +\[ + A_X(v)+\sum_i\bigl(l_i-(1+b)a_i\bigr)v(x_i) + \ \geq\ \sum_i c_i v(x_i)\ \geq\ 0. +\] +These inequalities persist under finite covers, since shifted orders +multiply by ramification indices. Further exceptional primes therefore +impose no stronger condition. +Horizontally, shifted orders are strictly positive outside the +designated log boundary; on the root cover these are positive +integers, so there are no poles there. Conversely, covering primes +above strict transforms detect every +inequality in \eqref{rankone:inequality}. Taking their minimum +proves \eqref{rankone:order}. + +\begin{remark}\label{rankone:collision} +The use of all primes on the resolved model in +\eqref{rankone:order} matters even for a simple logarithmic +family. On $\mathbf P^1_x$ over a disk, remove the horizontal +divisor $x^2=t$ and use $\tau=dx/(x^2-t)$. After $t=u^2$ and +$x=uz$, one has $\tau=u^{-1}dz/(z^2-1)$, so the parabolic order +is $-1/2$. Resolving the tangency of the horizontal divisor with +$t=0$ introduces a prime with $a_E=2$ and $l_E=0$, which gives +exactly $-1/2$ in \eqref{rankone:order}. The original central +prime alone would not detect it. +\end{remark} + +\subsubsection{The boundary and the comparison of sections} + +Define +\begin{equation}\label{rankone:alpha-T} + T_D=\alpha_D-\beta_D. +\end{equation} +Changing $s$ to $v s$, for a meromorphic base function $v$, adds +$a_E\ord_D(v)/m$ to $l_E$. Hence it adds +$\ord_D(v)/m$ to both $\alpha_D$ and $\beta_D$ and leaves +$T_D$ unchanged. This also checks the transformation rule for +the rational Hodge frame in \eqref{rankone:order}. Passing to +a higher divisible generating degree has the same effect: on the +generic fiber its generator is a base multiple of a power of $s$. +Thus the construction is independent of these choices. + +For the boundary inequality, write +\[ + x_E=\frac{l_E+\delta_E}{a_E},\qquad + c_E=1-\frac{1-\delta_E}{a_E}. +\] +Then $0\leq c_E\leq1$, and +\[ + T_D=\min_E x_E-\min_E(x_E-c_E). +\] +For any finite collections of real numbers $x_E,c_E$, this +difference lies between $\min_Ec_E$ and $\max_Ec_E$. +Moreover the inf-multiplicity convention gives +\[ + \min_Ec_E + =1-\max_{g(E)=D}\frac{1-\delta_E}{a_E} + =B(g,\Delta)_D. +\] +This proves \eqref{rankone:boundary}. If all $\delta_E=1$, +all $c_E=1$, so $T_D=1$. + +Choose the discriminant to include the vertical boundary and all +loci where the smooth log construction fails. Off it the family +and the generator line are smooth log-relative data, and there is +no vertical boundary. There $\alpha_D=\beta_D$; in a +nonvanishing generator frame both are zero. Thus $T$ has finite +support in the chosen discriminant, which may be resolved to be +SNC. On subsequent modifications one pulls back the variation and +its parabolic line and recomputes $\alpha_D$ and +\eqref{rankone:alpha-T}; the order computation is unchanged on the +resulting models. + +\begin{proof}[Completion of the proof of \Cref{rankone:comparison}] +Take $q$ divisible by $m$ and by the denominators of all rational +data. In a local base frame, a meromorphic section of +$q(K_W+T+M)$ is represented +by +\[ + \varphi\,\omega^q s^{q/m}. +\] +The order of the frame in $T+M$ is +$q(T_D+\beta_D)=q\alpha_D$. Consequently this section is +regular at $D$ exactly when \eqref{rankone:base-bound} holds. +The source-side order test already proved identifies this with +the boundary bounds at every source prime dominating $D$. +Along horizontal primes the required bound is automatic because +$s^{q/m}$ is a logarithmic +pluricanonical section on the general fiber. We have proved the +exact section comparison in codimension one on the base, and at +all source primes except those contracted by $g$ into codimension +at least two. + +Conversely, a section of $q(K_X+\Delta)$ restricts on a general +$g$-fiber to a multiple of $s^{q/m}$. Their ratio is a +meromorphic function constant on the connected general fibers, +and therefore descends meromorphically to $W$. The same +divisorial inequalities make it a section of $q(K_W+T+M)$ +outside codimension two; smoothness of $W$ extends it uniquely +across that set. This proves the converse comparison in all +divisible degrees under consideration. + +For the forward direction, a section on $W$ gives a meromorphic +adjoint form on $X$ whose only possible remaining poles lie on +$g$-contracted prime divisors. By hypothesis these divisors are +$\pi$-exceptional. Push the form to $X_0$. At every prime of +$X_0$ its transform was among the tested primes of $X$, so the +pushed form satisfies the $\Delta_0$ pole bound in codimension +one. Since $X_0$ is smooth and the adjoint multiple is a line +bundle, it extends across codimension two. Finally +\Cref{setup:log-modification} pulls it back to a section of +$q(K_X+\Delta)$, proving \eqref{rankone:sections}. + +If the diagram is over $Y$, apply this argument over an arbitrary +open subset $U\subset Y$. Its inverse image in $X$ is +$\pi^{-1}(f_0^{-1}U)=g^{-1}(h^{-1}U)$, and $\pi$ is still +proper over the smooth open space $f_0^{-1}U$. The exceptional +images still have codimension at least two. The construction and +the extension of sections commute with restriction to $U$, +giving \eqref{rankone:relative-sections}. Relative canonical +versions, when the auxiliary base is smooth, follow by the +projection formula. All comparisons multiply by the same relative +generator, with its powers in the common divisible grading, so they +preserve products and ratios of sections. The projection formula also +gives the identity after any pulled-back base twist, in degrees clearing +the denominator for a rational twist. This completes +\Cref{rankone:comparison}. +\end{proof} + +\subsection{Retaining the original base multiplicities} + +\begin{lemma}[Composition of inf-multiplicity bounds] +\label{application:composition} +Suppose that \(X\xrightarrow{g}W\xrightarrow{h}Y\) is a diagram of +surjective maps between smooth spaces, and \(\Delta,T,C\) are rational +boundaries on these spaces. If +\[ + T\geq B(g,\Delta),\qquad C\leq B(h\circ g,\Delta), +\] +then \(B(h,T)\geq C\). If, in addition, every prime divisor of \(X\) +mapped into codimension at least two in \(Y\) has coefficient one in +\(\Delta\), and \(T\leq1\), then every prime divisor of \(W\) mapped +into codimension at least two in \(Y\) has coefficient one in \(T\). +\end{lemma} + +This is the inf-multiplicity composition comparison of +\cite[Proposition~3.13(1)]{CampanaOrbifolds}; we include its order +calculation to track coefficient-one exceptional divisors. + +\begin{proof} +Let \(D\subset Y\) and \(V\subset W\) be prime divisors with +\(h(V)=D\). Write \(b_V=\ord_V(h^*D)\). If \(E\subset X\) is a +prime divisor dominating \(V\), put \(a_E=\ord_E(g^*V)\). At their +generic points orders multiply, so +\[ + \ord_E((h\circ g)^*D)=b_Va_E. +\] +The hypothesis on \(C\) implies +\(b_Va_E m_\Delta(E)\geq m_C(D)\) for every such \(E\). Taking the +minimum over them and then using the hypothesis on \(T\) gives +\[ + b_Vm_T(V)\ \geq\ b_Vm(g,\Delta;V)\ \geq\ m_C(D). +\] +Taking the minimum over \(V\) proves the first assertion. All these +inequalities are valid with the prescribed convention for infinity. + +For the second assertion, let \(V\) be mapped into codimension at least +two in \(Y\). Every prime \(E\) dominating it has +\(\Delta_E=1\), so \(m(g,\Delta;V)=\infty\). It follows that +\(B(g,\Delta)_V=1\), and hence \(T_V=1\). +\end{proof} diff --git a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9e786b26b31799737e111b24e1c9b7019fed5538 --- /dev/null +++ b/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b143366cc99615e19783fa075caafd8c9565e7bf5a554a612e086657c2aeeb79 +size 677817 diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/README.md b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..965a26b87458fb541e5a0a36d40804ab86cadad7 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [Ordinary two-point correlations of multiplicative functions](final.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026, + author = {{OpenAI}}, + title = {{Ordinary two-point correlations of multiplicative functions}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf}{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/introduction.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..da38786f2b27248f49985c1e7f3d8a95fd152f81 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/introduction.tex @@ -0,0 +1,278 @@ +\section{Introduction}\label{sec:introduction} + +For \(n\ge1\), let \(\Omega(n)\) be the number of prime factors of \(n\), +counted with multiplicity, and let \(\lambda(n)=(-1)^{\Omega(n)}\). +We study ordinary two-point averages: each integer up to the given +cutoff has weight one. Thus the correlation measures whether the two prime-factor counts +have the same parity more often than different parities on the full +initial interval. +The distinction from a logarithmic average is the distinction between +\[ + \frac1X\sum_{n\le X}a(n) + \qquad\text{and}\qquad + \frac1{\log X}\sum_{n\le X}\frac{a(n)}n. +\] +Logarithmic weighting combines information from many multiplicative +scales; an ordinary estimate must control the terminal scale itself. +Our quantitative conclusion concerns every fixed pair of +nonproportional affine forms. + +\begin{theorem}[A logarithmic saving for affine Liouville correlations] +\label{thm:q-affine} +There is an absolute constant \(c>0\) such that, for every fixed choice +of integers \(a_1,a_2\ge1\) and \(b_1,b_2\ge0\) with +\(a_1b_2-a_2b_1\ne0\), there is a constant +\(C_{a_1,a_2,b_1,b_2}\) for which +\[ + \left|\sum_{1\le n\le X} + \lambda(a_1n+b_1)\lambda(a_2n+b_2)\right| + \le C_{a_1,a_2,b_1,b_2}\frac{X}{(\log X)^c} + \qquad(X\ge3). +\] +The cutoff \(X\) may be any real number. No coprimality condition is +imposed on the coefficients. +\end{theorem} + +For $a_1=a_2=1$ and $b_1=0$, $b_2=h>0$, this resolves the +ordinary two-point Chowla conjecture, with a logarithmic saving. +The exponent is independent of the forms. The implied constants need +not be effective, and no uniformity for coefficients growing with \(X\) +is asserted. The proof first obtains the same saving for +\(\lambda(n)\lambda(n+h)\) in each fixed residue class modulo each fixed +positive integer, including nonunit and zero classes. Complete +multiplicativity then gives the affine conclusion. + +We also prove a qualitative statement for general multiplicative +functions. Let \(\mathbb D=\{z\in\C:|z|\le1\}\). A function +\(f:\N\to\mathbb D\) is multiplicative when +\(f(mn)=f(m)f(n)\) for coprime positive integers \(m,n\). +For a Dirichlet character \(\chi\), a real number \(t\), and \(X\ge2\), +define the nonnegative distance \(D\) by +\begin{equation}\label{eq:distance} + D(f,\chi n^{it};X)^2 + =\sum_{p\le X}\frac{1-\operatorname{Re} + (f(p)\overline{\chi(p)p^{it}})}p. +\end{equation} +All sums indexed by \(p\) are over primes. We call \(f\) +\emph{uniformly nonpretentious} if, for every fixed Dirichlet character, +\begin{equation}\label{eq:unp} + \inf_{|t|\le N}D(f,\chi n^{it};N)\longrightarrow\infty + \qquad(N\longrightarrow\infty). +\end{equation} +The prime cutoff and the height bound are the same \(N\). + +\begin{theorem}[Binary corrected Elliott]\label{thm:elliott} +Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, and suppose that +at least one satisfies \eqref{eq:unp}. For every fixed pair of distinct +nonnegative integers \(h_1,h_2\), +\[ + \frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0 + \qquad(N\longrightarrow\infty,\ N\in\N). +\] +\end{theorem} + +This proves the binary corrected Elliott conjecture for ordinary +averages. The functions may be complex, need not have modulus one, +and need not be completely multiplicative. There is no conjugation +in the displayed product; the conjugated version follows because +\eqref{eq:unp} is preserved by conjugation. We make no quantitative +rate claim for this class of functions. + +\begin{corollary}[Affine correlations]\label{cor:affine} +Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, with at least one +satisfying \eqref{eq:unp}. For all fixed integers \(a_1,a_2\ge1\) and +\(b_1,b_2\ge0\) with \(a_1b_2-a_2b_1\ne0\), +\[ + \frac1N\sum_{n=1}^N f_1(a_1n+b_1)f_2(a_2n+b_2) + \longrightarrow0. +\] +\end{corollary} + +Every fixed progression restriction is also permitted in +\Cref{thm:elliott}; \Cref{prop:affine-progression} gives the precise +statement through all real cutoffs and for every residue class. +These conclusions use the hypothesis on an original factor \(f_j\). +The finite local changes needed for dilations and residue restrictions +preserve that hypothesis. In particular, Liouville satisfies the exact +condition \eqref{eq:unp}; a quantitative distance bound is proved in +\Cref{lem:affine-liouville-unp}. + +\subsection{Historical context and prior work} + +Chowla's conjecture predicts vanishing Liouville correlations at every +fixed collection of distinct shifts \cite{Chowla65,MRT15}. Its binary +case asks whether the parities of the numbers of prime factors of +\(n\) and \(n+h\) become uncorrelated for each fixed \(h\ge1\). +Partial summation transfers ordinary cancellation to logarithmic +cancellation. An ordinary estimate at every cutoff requires more +than this implication. + +Elliott extended the correlation problem to bounded multiplicative +functions \cite{Elliott92}. The obstruction presented by the twists +\(n^{it}\) is already central to Hal\'asz's theory of one-point mean +values \cite{Halasz71}; see also the treatment of Granville, Harper, +and Soundararajan \cite{GHS19}. The distance in \eqref{eq:distance} +belongs to the pretentious approach developed by Granville and +Soundararajan \cite[Section~3]{GS07}. Matom\"aki, Radziwi\l\l, and Tao +showed why Elliott's original condition needs correction for complex +functions: a function can imitate different twists at successive +scales without imitating any one fixed twist globally. Their +corrected condition is equivalent to \eqref{eq:unp}; see +\cite[Section~1.1 and Appendix~B]{MRT15}. + +Matom\"aki and Radziwi\l\l\ proved that most short means of a real +bounded multiplicative function approximate its long mean +\cite[Theorem~1]{MR16}. They also obtained a nontrivial bound for each +fixed-shift Liouville correlation: the normalized absolute value is +at most \(1-\delta(h)\), for some \(\delta(h)>0\), at all sufficiently +large cutoffs \cite[Corollary~2]{MR16}. Matom\"aki, Radziwi\l\l, and +Tao then established cancellation averaged over arbitrarily slowly +growing windows of shifts. Their exponential-sum extension to +complex multiplicative functions supplies the short-interval +estimates used here \cite[Theorems~1.1, 1.3, and~1.7]{MRT15}. + +Tao proved the logarithmically averaged binary corrected Elliott +theorem for nonproportional affine forms +\cite[Theorem~1.3 and Corollary~1.5]{Tao16}. +For the logarithmically weighted Liouville sum, Helfgott and +Radziwi\l\l\ obtained +\[ + \left|\sum_{n\le x}\frac{\lambda(n)\lambda(n+1)}n\right| + \ll \frac{\log x}{\sqrt{\log\log x}} +\] +by divisibility-graph expansion \cite[Corollary~1.5]{HR21}. +Pilatte improved this bound to \((\log x)^{1-c_0}\) for an absolute +\(c_0>0\) \cite[Theorem~1.1]{Pilatte26}. +These estimates retain the logarithmic weight. + +The graph method of Helfgott and Radziwi\l\l\ uses high traces, +recurrence constraints, and cancellation at primes occurring only +once \cite[Sections~2--7]{HR21}. They also proposed composite steps +and nonbacktracking operators \cite[Sections~9.1--9.2]{HR21}. +Pilatte develops these directions through products of centered prime +factors, prohibited sequences, triangular arithmetic constraints, +and a rank-truncated intersection sieve +\cite[Sections~2--13 and Appendix~B]{Pilatte26}. +Our two graph arguments adapt these constructions to different +weights and orders of limits. Their trace estimates, local +expansions, and transfers to ordinary averages are proved below. + +For ordinary averages, +Tao and Ter\"av\"ainen obtained binary cancellation outside +a set of scales of logarithmic density zero, under the same +condition \eqref{eq:unp} on one factor +\cite[Corollary~1.13]{TT19}. Klurman, Mangerel, and Ter\"av\"ainen +relaxed the hypothesis to nonpretentiousness against each fixed +\(\chi(n)n^{it}\), +obtaining cancellation along a set of scales of full upper logarithmic +density \cite[Theorem~1.2]{KMT23}. The later quantitative progression +estimates of Tao and Ter\"av\"ainen +give logarithmic savings for affine Liouville correlations, +with coefficients allowed to grow slowly, outside a quantitatively +controlled exceptional set of scales +\cite[Theorem~3.1 and Remarks~3.2]{TT26}. The results here concern +every cutoff, with the affine forms fixed. + +The quantitative route also uses bounded independence for Boolean +circuits. This subject developed from the work of Linial and Nisan +\cite{LinialNisan90}, through Bazzi's depth-two theorem and +Razborov's simpler proof \cite{Bazzi09,Razborov08}, to Braverman's +theorem for every fixed depth \cite{Braverman10}. In applying that theorem, the +encoded residue law is first corrected to exact bounded independence. +The local Fourier calculation is a form of the almost-to-exact +independence construction of Alon, Goldreich, and Mansour +\cite[Theorem~2.1 and Section~3.1]{AGM03}. + +\subsection{Main ideas} + +Both arguments begin with multiplicative dilations. For a fixed +shift $h\ge1$, comparison of the correlations at $n,n+h$ and at +$dn,d(n+h)$ leads to a graph on the integers with edge displacement +$hd$. Short-interval Fourier estimates control the change in the +correlation when selected divisibility indicators are replaced by +signed expressions. Estimates for closed walks then bound the resulting +graph operator. The two proofs use different vertex weights, and their +cancellation mechanisms must be matched to those weights. + +In the quantitative argument, a step contains one prime from each of +$J$ disjoint supplies, each with reciprocal mass comparable to a large +fixed constant $W$. These primes contribute the factors +$\1_{p\mid n}-1/p$, which have mean zero under uniform residues. +An additional squarefree padding divisor uses a disjoint set of primes. +The centered supplies provide reciprocal mass at least $W^J$, while +their contribution to the operator bound is only $(C\sqrt W)^J$, with +an absolute $C$. Padding controls the concentration +of divisor weight in short multiplicative intervals. Writing $L$ for +a small fixed power of $\log X$, the edge normalization gives a factor +$1/L$ in the estimate for each interval, compensating for the factor +$L$ in the number of intervals. The remaining losses are exponential +in $J$ with absolute bases. Choosing $W$ large enough therefore gives +exponential decay in $J$; since $J$ grows proportionally to +$\log\log X$, this is a power-of-logarithm saving. + +Two issues underlie this comparison. First, the prime supplies grow +with $X$, so averaging over their full joint period would not give +a useful finite-scale estimate. We instead compare the residue tests +used in the proof with independent residues, correcting their encoded +bit law to exact bounded independence before applying Braverman's +theorem \cite{Braverman10}. Second, repeated primes constrain the +closed walks. Many independent arithmetic relations give a direct +saving. When few remain, a forest encodes the possible patterns of +repetition. Its pattern count is uniform over all padding values, +allowing the padding weights to be summed afterward in one common +residue environment. These steps preserve the absolute constants +needed for the choice of $W$. + +For general multiplicative functions, the short-interval input +contains the nonpretentiousness distance, whose divergence has no +prescribed rate. We therefore fix the graph before taking the long +average. At an auxiliary scale $B$, squarefree divisors are formed from +two bands of primes at most $e^B$. Primes in the higher band receive a +fixed weight greater than one, and those in the lower band receive +weight one; the weight of a divisor is the product of its prime weights. +After dividing the ordinary divisor average by the total reciprocal +weight of these divisors, a biased correlation subsequence produces a +raw average of size at least a constant times $B^{-1}$ in one short +multiplicative interval. The analytic centering error is $o(B^{-1})$, +and the graph estimate gives $O(B^{-1-c_* /2})$ for a fixed absolute +$c_*>0$. Taking the long limit with $B$ fixed, and then increasing $B$, +contradicts the bias. + +The higher prime band, called the core band, supplies decay through +restrictions on divisor size and on the number of core primes at a +vertex. The lower, center band provides the additional power saving +needed to beat the $B^{-1}$ scale. Its signed factor is +$\1_{p\mid n}-\theta/p$, with +$\theta$ chosen to match the vertex normalization; it is not itself +mean zero under uniform residues. For suitable primes appearing on +only one edge, the normalization factors at the two endpoints in the +closed-walk product make the leading divisibility contribution match +the subtracted constant term. +To retain this cancellation, +we organize a walk by the tree of its first visits and expand the +vertex-deletion conditions while leaving these center-prime residues +unconditioned. Their signed averages are taken before absolute values. +This yields the qualitative theorem without a Liouville-specific rate. + +\subsection{Organization} + +Part~I proves the quantitative progression estimate and then +\Cref{thm:q-affine}. +\Cref{q:setup,q:finite-law-section} construct its prime supplies and +establish the finite residue comparison; \Cref{q:analytic} bounds +centering and deletion errors. \Cref{q:paths-trace} introduces the +graph moment, and \Cref{q:rank-witness-section,q:forest-section} prove +it through arithmetic constraints and pattern counting. +\Cref{q:sec-spectral} returns to ordinary correlations, while +\Cref{q:affine-transfer} gives the affine deduction. + +Part~II proves \Cref{thm:elliott} with new parameters and prime sets. +\Cref{sec:graph-setup,sec:analytic-transfer,sec:analytic-centering} +state the graph estimate and prove the analytic reduction to it. +\Cref{sec:paths,sec:thinning} establish the geometry of retained walks; +\Cref{sec:cylinder-sieve,sec:trace-count} control their conditional +averages and sum the trace. \Cref{sec:localization} transfers that +estimate to long integer intervals and completes the theorem. +Part~III uses finite local expansions to prove the progression and +affine consequences for general functions, and verifies their +Liouville and M\"obius specializations. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/main.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..37b55a20b0b9e0eea5c44cc88dd256c7cd4d9a53 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/main.tex @@ -0,0 +1,90 @@ +\newcommand{\DoNotLoadEpstopdf}{} +\documentclass[11pt]{article} +\pdftrailerid{} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.08in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} +\usepackage{microtype,enumitem} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[hidelinks]{hyperref} +\usepackage[capitalise,noabbrev]{cleveref} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\AddToHook{env/theorem/begin}{\crefalias{theorem}{theorem}} +\AddToHook{env/proposition/begin}{\crefalias{theorem}{proposition}} +\AddToHook{env/lemma/begin}{\crefalias{theorem}{lemma}} +\AddToHook{env/corollary/begin}{\crefalias{theorem}{corollary}} +\AddToHook{env/definition/begin}{\crefalias{theorem}{definition}} +\AddToHook{env/remark/begin}{\crefalias{theorem}{remark}} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\T}{\mathbb R/\mathbb Z} +\newcommand{\E}{\mathbb E} +\newcommand{\Pbb}{\mathbb P} +\newcommand{\1}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\RePart}{Re} +\setlist{itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{1.5em} +\title{Ordinary two-point correlations of multiplicative functions} +\author{OpenAI} +\hypersetup{ + pdftitle={Ordinary two-point correlations of multiplicative functions}, + pdfauthor={OpenAI} +} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the ordinary two-point Chowla conjecture. For every fixed pair +of nonproportional affine forms, the Liouville correlation has a +power-of-logarithm saving at every cutoff, with an absolute exponent. +We also prove the binary corrected Elliott conjecture for ordinary +averages of complex multiplicative functions of modulus at most one, +under uniform nonpretentiousness of at least one original factor. +This qualitative conclusion holds in fixed residue classes and for +fixed nonproportional affine forms. +\end{abstract} +\tableofcontents +\input{introduction} +\part{A logarithmic saving for Liouville correlations} +\input{quantitative/01-setup} +\input{quantitative/02-finite-law} +\input{quantitative/03-analytic} +\input{quantitative/04-paths-trace} +\input{quantitative/05-rank-witness} +\input{quantitative/06-forest} +\input{quantitative/07-spectral} +\input{quantitative/08-affine} +\part{Qualitative correlations of general multiplicative functions} +\noindent +We now prove \Cref{thm:elliott}. The graph in this part uses a different +normalization and a different order of limits. Its parameters and prime +sets are defined afresh; none of the choices of \(A,W,L,J,\eta\) in +Part~I is in force here. The functions and their nonpretentiousness +condition remain those of the introduction. +\input{qualitative/graph-setup} +\input{qualitative/analytic-transfer} +\input{qualitative/paths} +\input{qualitative/thinning} +\input{qualitative/cylinder-sieve} +\input{qualitative/trace-count} +\input{qualitative/localization} +\part{Progressions and affine forms} +\input{qualitative/affine-corollaries} +\begingroup\small +\bibliographystyle{alpha} +\bibliography{refs} +\endgroup +\end{document} diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/affine-corollaries.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/affine-corollaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..8a3fc22dd9b42f902980305b736ee19ef127dee4 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/affine-corollaries.tex @@ -0,0 +1,363 @@ +\section{Local factors and affine correlations} +\label{sec:affine-corollaries} + +We now deduce correlations on arbitrary fixed progressions and affine +forms from \Cref{thm:elliott}. The main point is to handle the primes +dividing a progression modulus or a dilation without assuming complete +multiplicativity. A finite expansion at these primes will suffice. + +\subsection{Stability and finite local expansions} + +We first extend \Cref{lem:finite-prime-stability} to character twists +and complex conjugation. + +\begin{lemma}[Character twists and conjugation] +\label{lem:affine-stability} +Let $f,G\colon\N\to\mathbb D$ be multiplicative, let $\psi$ be a +fixed Dirichlet character, and let $S$ be a finite set of primes. +If $G(p)=f(p)\psi(p)$ for every $p\notin S$, then for every Dirichlet +character $\chi$, every real $t$, and every $X\ge2$, +\begin{equation} + \left|D(G,\chi n^{it};X)^2 + -D(f,(\chi\overline\psi)n^{it};X)^2\right| + \le 2\sum_{p\in S}\frac1p. + \label{eq:affine-stability} +\end{equation} +Thus uniform nonpretentiousness of $f$ implies that of $G$. +Complex conjugation also preserves uniform nonpretentiousness. +\end{lemma} + +\begin{proof} +The product $\chi\overline\psi$ is a Dirichlet character modulo the +least common multiple of the two moduli. At every prime outside $S$, +\[ + G(p)\overline{\chi(p)p^{it}} + =f(p)\overline{(\chi\overline\psi)(p)p^{it}}. +\] +This identity holds also at primes where a character vanishes. Each +prime in $S$ contributes at most $2/p$ to the difference of squared +distances, proving \eqref{eq:affine-stability}. Take the infimum over +exactly $|t|\le X$ on both sides of the resulting lower bound. For each +fixed $\chi$, the character $\chi\overline\psi$ is fixed, and the +bounded error cannot prevent divergence to infinity. +For conjugation the exact identity is +\[ + D(\overline f,\chi n^{it};X)^2 + =D(f,\overline\chi n^{-it};X)^2. +\] +The interval $|t|\le X$ is unchanged by $t\mapsto-t$. +\end{proof} + +A multiplicative function is either identically zero or has value one +at $1$: apply multiplicativity to $(1,n)$. Zero factors give zero +correlations, so in the following local construction we may work with +normalized functions. Write $v_p(m)$ for the exponent of $p$ in $m$. + +\begin{lemma}[A finite expansion for dilation] +\label{lem:affine-dilation} +Let $a\ge1$ be an integer and let $f\colon\N\to\mathbb D$ be +multiplicative with $f(1)=1$. Define +\begin{equation} + F_{a,f}(m)= + \begin{cases} + f(m/a),&a\mid m,\\ + 0,&a\nmid m. + \end{cases} + \label{eq:affine-dilation-function} +\end{equation} +Then $F_{a,f}$ is a signed sum of $2^{\omega(a)}$ normalized, +$1$-bounded multiplicative functions, each agreeing with $f$ at every +prime power whose prime does not divide $a$. If $f$ is uniformly +nonpretentious, every function in this sum is uniformly nonpretentious. +\end{lemma} + +\begin{proof} +Let $P(a)$ be the set of prime divisors of $a$. For $p\in P(a)$ set +$\alpha_p=v_p(a)$ and define +\begin{equation} + A_p(k)= + \begin{cases} + 0,&0\le k<\alpha_p,\\ + f(p^{k-\alpha_p}),&k\ge\alpha_p. + \end{cases} + \label{eq:affine-local-sequence} +\end{equation} +Factorization into pairwise coprime prime powers gives the exact identity +\begin{equation} + F_{a,f}(m)= + \prod_{p\in P(a)}A_p(v_p(m)) + \prod_{\substack{p\mid m\\p\notin P(a)}}f(p^{v_p(m)}). + \label{eq:affine-local-product} +\end{equation} +No identity between $f(p^k)$ and $f(p)^k$ is used. + +The obstruction to multiplicativity in this product is that +$A_p(0)=0$. Resolve it by setting +\begin{equation} + B_p(0)=C_p(0)=1,\qquad + B_p(k)=A_p(k),\quad C_p(k)=0\quad(k\ge1). + \label{eq:affine-b-minus-c} +\end{equation} +Then $A_p=B_p-C_p$ at every nonnegative exponent. For +$E\subseteq P(a)$ put +\[ + f_{a,E}(m)= + \prod_{p\in E}C_p(v_p(m)) + \prod_{p\in P(a)\setminus E}B_p(v_p(m)) + \prod_{\substack{p\mid m\\p\notin P(a)}}f(p^{v_p(m)}). +\] +All local factors have modulus at most one and value one at exponent +zero. If $u$ and $v$ are coprime, at most one of $v_p(u),v_p(v)$ is +positive at each prime; hence $f_{a,E}(uv)=f_{a,E}(u)f_{a,E}(v)$. +Expanding the finite product in \eqref{eq:affine-local-product} yields +\begin{equation} + F_{a,f}(m)= + \sum_{E\subseteq P(a)}(-1)^{|E|}f_{a,E}(m). + \label{eq:affine-dilation-expansion} +\end{equation} +Each summand agrees with $f$ at prime powers outside $P(a)$, so +\Cref{lem:finite-prime-stability} proves the last assertion. +\end{proof} + +For $a=1$ the expansion has just one term, namely $f$. For $a>1$ +and $m=1$, its signed sum is +$\sum_{E\subseteq P(a)}(-1)^{|E|}=0$, as required. Thus the expansion +also handles the value at $1$, despite normalizing every summand there. + +\begin{lemma}[Every residue class] +\label{lem:affine-residue} +Let $l\ge1$ and $0\le b1$, the +residue $c$ is a unit modulo $q$, so character orthogonality on +$(\Z/q\Z)^\times$ gives +\[ + \1_{m/d\equiv c\pmod q} + =\frac1{\varphi(q)}\sum_{\chi\bmod q} + \overline{\chi(c)}\chi(m/d). +\] +For a nonunit $m/d$ every character on the right vanishes. The case +$q=1$ is the trivial identity with the stated convention. The finite +expansion now follows from \Cref{lem:affine-dilation}. +\end{proof} + +In particular, if $p\mid d$ and $\alpha=v_p(d)$, the local sequence +in this application is +\[ + A_p(k)=0\quad(k<\alpha),\qquad + A_p(k)=\chi(p)^{k-\alpha}\quad(k\ge\alpha), + \qquad 0^0=1. +\] +If $p$ also divides $q$, then $\chi(p)=0$ and this sequence is one +at $k=\alpha$ and zero elsewhere. No character value is divided out. +When $b=0$ we have $d=l$ and $q=1$, so the same formula expands +$\1_{l\mid m}$; it includes $l=1$ and $m=1$. + +\subsection{Progressions and the affine deduction} + +The two expansions just proved reduce the desired averages to finitely +many applications of \Cref{thm:elliott}. We first incorporate a +progression restriction, placing its weight at the argument of a +nonpretentious factor. + +\begin{proposition}[Correlations on fixed progressions] +\label{prop:affine-progression} +Let $f_1,f_2\colon\N\to\mathbb D$ be multiplicative, with at least +one uniformly nonpretentious. For fixed distinct nonnegative integers +$h_1,h_2$, an integer $l\ge1$, and any integer $b$, +\begin{equation} + \frac1X\sum_{1\le n\le X} + f_1(n+h_1)f_2(n+h_2)\1_{n\equiv b\pmod l} + \longrightarrow0 + \qquad(X\longrightarrow\infty). + \label{eq:affine-progression-limit} +\end{equation} +The limit holds through all real $X$, including nonunit residue classes. +\end{proposition} + +\begin{proof} +We may assume both functions are nonzero. Choose $j\in\{1,2\}$ for +which $f_j$ is uniformly nonpretentious. Since +\[ + \1_{n\equiv b\pmod l} + =\1_{n+h_j\equiv b+h_j\pmod l}, +\] +\Cref{lem:affine-residue}, using the least nonnegative representative +of $b+h_j$ modulo $l$, expands the indicator as a function of +$m=n+h_j$ into finitely many bounded multiplicative functions, each +agreeing with a fixed Dirichlet character outside a fixed finite prime +set. Their products with $f_j$ are uniformly nonpretentious by +\Cref{lem:affine-stability}. Apply \Cref{thm:elliott} to each product +at shift $h_j$ and the unchanged other factor at its distinct shift, +and sum the finitely many conclusions. This proves +\eqref{eq:affine-progression-limit} at integer $X$. +For real $X$, replacing the normalization $X$ by $\lfloor X\rfloor$ +changes the average by at most $1/X$, proving the assertion. +\end{proof} + +\begin{proof}[Proof of \Cref{cor:affine}] +Zero factors again give an immediate conclusion. Otherwise set +\begin{equation} + \Delta=a_1b_2-a_2b_1,\qquad + l=a_1a_2,\qquad + c_0=\min(a_2b_1,a_1b_2),\qquad h=|\Delta|\ge1. + \label{eq:affine-parameters} +\end{equation} +For $\Delta>0$ put +$U=F_{a_2,f_1}$ and $V=F_{a_1,f_2}$; for $\Delta<0$ put +$U=F_{a_1,f_2}$ and $V=F_{a_2,f_1}$. In either case, +\begin{equation} + U(ln+c_0)V(ln+c_0+h) + =f_1(a_1n+b_1)f_2(a_2n+b_2). + \label{eq:affine-exact-identity} +\end{equation} +Indeed, when $\Delta>0$ the two arguments on the left are +$a_2(a_1n+b_1)$ and $a_1(a_2n+b_2)$; when $\Delta<0$ their order +is reversed. + +Expand $U$ and $V$ by \Cref{lem:affine-dilation}. Each pair of +multiplicative components has a uniformly nonpretentious factor, +inherited from the corresponding original function. Applying +\Cref{prop:affine-progression} termwise with shifts $0,h$ gives +\begin{equation} + \frac1X\sum_{1\le m\le X} + U(m)V(m+h)\1_{m\equiv c_0\pmod l}\longrightarrow0. + \label{eq:affine-expanded-limit} +\end{equation} +Put $T(m)=U(m)V(m+h)\1_{m\equiv c_0\pmod l}$ and $X=lN+c_0$. +The integers in this class satisfying $c_00$, +\[ + A_\chi(u):=\sum_{p\le u}\chi(p)\log p + =O_\chi\bigl(u\exp(-c_\chi\sqrt{\log u})\bigr). +\] +Partial summation gives +\[ + \sum_{p\le X}\frac{\chi(p)}p + =\frac{A_\chi(X)}{X\log X} + +\int_2^X A_\chi(u) + \frac{\log u+1}{u^2(\log u)^2}\,du + =O_\chi(1), +\] +since the integral converges absolutely as $X\to\infty$. +As $\lambda(p)=-1$, Mertens' estimate now gives +\[ + D(\lambda,\chi;X)^2 + =\sum_{p\le X}\frac{1+\chi(p)}p + =\log\log X+O_\chi(1). +\] +If instead $\chi$ is principal modulo $q$, then +\[ + D(\lambda,\chi;X)^2 + =2\sum_{p\le X}\frac1p + -\sum_{\substack{p\mid q\\p\le X}}\frac1p + =2\log\log X+O_q(1). +\] +In both cases \eqref{eq:affine-real-small-height} is at least +$\frac13\sqrt{\log\log X}-O_\chi(1)$. Combining this with +\eqref{eq:affine-real-large-height} proves +\eqref{eq:affine-liouville-unp} on the entire interval $|t|\le X$. +\end{proof} + +Thus $\lambda$ satisfies the exact hypothesis of the qualitative theorem. +Applying \Cref{prop:affine-progression,cor:affine} with +$f_1=f_2=\lambda$ independently recovers ordinary cancellation in +every fixed residue class and along every fixed nonproportional affine +pair. \Cref{q:progression,thm:q-affine} establish these conclusions +with an absolute power-of-logarithm saving. + +The qualitative theorem also applies to the M\"obius function, defined by +\[ + \mu(n)= + \begin{cases} + \lambda(n),&n\text{ is squarefree},\\ + 0,&n\text{ is not squarefree}. + \end{cases} +\] +This function is multiplicative and has $\mu(p)=\lambda(p)=-1$ at +every prime. Since the distance uses only prime values, +\[ + D(\mu,\chi n^{it};X)=D(\lambda,\chi n^{it};X) +\] +for every Dirichlet character $\chi$, real $t$, and $X\ge2$. +The preceding lemma therefore proves uniform nonpretentiousness of +$\mu$ with the same growing twist range. The conclusions of +\Cref{prop:affine-progression,cor:affine} hold for any choice +$f_1,f_2\in\{\lambda,\mu\}$. This proves ordinary cancellation for two-point M\"obius +and mixed M\"obius--Liouville correlations in every fixed +residue class and along every fixed nonproportional affine pair. +For products containing a M\"obius factor, this deduction gives +convergence without a quantitative rate. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/analytic-transfer.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/analytic-transfer.tex new file mode 100644 index 0000000000000000000000000000000000000000..e8bdbcdad26a7279ce958ec344102c6553a5244b --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/analytic-transfer.tex @@ -0,0 +1,671 @@ +\section{Reduction to the graph estimate}\label{sec:analytic-transfer} + +We now explain how \Cref{prop:graph-estimate} rules out a nonzero +multiplicative correlation. The first step produces a large sum with +ordinary divisibility indicators. We then state the estimate that permits +centering those indicators, and display the resulting contradiction. +The proof of the centering estimate occupies +\Cref{sec:analytic-centering}; the graph estimate itself will be proved +in \Cref{sec:localization}. + +\subsection{Absolute-value defects and a biased sequence} + +A multiplicative function satisfies $f(1)=f(1)^2$. If $f(1)=0$, then +$f(n)=f(n)f(1)=0$ for every $n$. We may therefore suppose that both +functions in \Cref{thm:elliott} take the value $1$ at $1$. +There is another case in which the conclusion follows without a graph. + +\begin{lemma}\label{lem:analytic-absolute-defect} +Let $f\colon\N\to\C$ be multiplicative and $|f|\le1$. If +\[ + \sum_p\frac{1-|f(p)|}{p}=\infty, +\] +then $N^{-1}\sum_{n\le N}|f(n)|\to0$. +\end{lemma} +\begin{proof} +Write $v_p(n)$ for the exponent of $p$ in $n$. For a finite set +$\mathcal P$ of primes, multiplicativity gives +\[ + |f(n)|\le + \prod_{\substack{p\in\mathcal P\\v_p(n)=1}}|f(p)|. +\] +The majorant is periodic modulo $\prod_{p\in\mathcal P}p^2$. +Its ordinary mean is +\[ + \prod_{p\in\mathcal P} + \left(1-(1-|f(p)|)\left(\frac1p-\frac1{p^2}\right)\right). +\] +These products tend to zero as $\mathcal P$ increases through the primes: +the sum of the subtracted quantities diverges, whereas +$\sum_p p^{-2}<\infty$. First take the long average with $\mathcal P$ +fixed, and then increase $\mathcal P$. +\end{proof} + +A fixed translation affects only finitely many terms of a bounded +average. Thus, after translating by the smaller shift and ordering the +functions accordingly, it suffices to prove cancellation of +$f_1(m)f_2(m+h)$ for a fixed $h\ge1$. +By \Cref{lem:analytic-absolute-defect}, we may assume +\begin{equation}\label{eq:analytic-finite-defects} + \sum_p\frac{1-|f_i(p)|}{p}<\infty\qquad(i=1,2). +\end{equation} +If the desired cancellation fails, there are $0<\gamma\le1$ and positive +integers $N_j\to\infty$ such that +\begin{equation}\label{eq:analytic-bias} + \left|S(N_j)\right|\ge\gamma N_j, + \qquad S(N)=\sum_{m\le N}f_1(m)f_2(m+h). +\end{equation} +We retain this same sequence throughout the argument. +The numbers $S(N_j)$ may have varying complex arguments. + +Fix $1<\tau<2$ sufficiently close to $1$ in terms of $\gamma$, and then +fix $T$ sufficiently large in terms of $\gamma$. Choose +$\phi\in C_c^\infty(\R)$ with $0\le\phi\le1$, supported in $[-T,T]$ +and equal to $1$ on $[-T/2,T/2]$. For every admissible divisor $d$, use +\begin{equation}\label{eq:analytic-cutoffs} + w_d(x)=\phi\left(\frac{k_d(x)-v_C}{\sqrt{v_C}}\right). +\end{equation} +These cutoffs satisfy the support and residue-dependence requirements +of \Cref{prop:graph-estimate}. + +\subsection{Capturing the bias in one divisor interval} + +Recall that $\mathcal S=\mathcal C\cup\mathcal Z$ is the finite prime set +at scale $B$, and that a squarefree product $d$ of these primes has +weight $A^{\omega_C(d)}$. The normalizing factor $L_0$ has the exact +expansion +\begin{equation}\label{eq:raw-divisor-law} + L_0=\sum_{\substack{d\ \mathrm{squarefree}\\p\mid d\Rightarrow p\in\mathcal S}} + \frac{A^{\omega_C(d)}}d. +\end{equation} +The term $d=1$ is included here. + +\begin{lemma}[A divisor interval carrying positive mass] +\label{lem:raw-bin} +Let $f_1,f_2\colon\N\to\C$ be $1$-bounded multiplicative functions +satisfying \eqref{eq:analytic-finite-defects} and $f_1(1)=f_2(1)=1$. +Fix $1<\tau<2$. There are constants $C_0\ge1$ and +$c>0$ such that, for every sufficiently large $B$, one can choose +$1\le H\le\exp(C_0B)$ and a family $\mathcal D$ of squarefree +$\mathcal S$-products satisfying +\begin{gather} + H1,\qquad + \omega(d)\le J=\lceil C_0\log B\rceil, + \qquad |f_1(d)f_2(d)|\ge\tfrac12, + \label{eq:raw-selected-divisors}\\ + \sum_{d\in\mathcal D}\frac{A^{\omega_C(d)}}d + \ge c\frac{L_0}{B}. + \label{eq:raw-bin-mass} +\end{gather} +The constants are independent of $B$ and of the long averaging variable. +\end{lemma} +\begin{proof} +Normalize the summands of \eqref{eq:raw-divisor-law} to a probability +law on divisors, and denote its expectation and probability by +$\E_{\mathrm{div}}$ and $\Pbb_{\mathrm{div}}$. Each prime $p$ in band $a$ +is included independently with +probability $A_a/(p+A_a)$. Prime harmonic estimates give +\[ + \E_{\mathrm{div}}\omega(d)\ll_A\log B, + \qquad + \E_{\mathrm{div}}\log d\le A\sum_{p\le e^B}\frac{\log p}{p}\ll_A B. +\] +Choose $C_0$ large enough that the probabilities of +$\omega(d)>\lceil C_0\log B\rceil$ and $\log d>C_0B$ are each at most +$1/8$, by Markov's inequality. + +For $r_p=|f_1(p)f_2(p)|$, the inequality +$1-\prod_j r_j\le\sum_j(1-r_j)$, valid when $0\le r_j\le1$, gives +\begin{align*} + \E_{\mathrm{div}}\bigl(1-|f_1(d)f_2(d)|\bigr) + &\le A\sum_{p>P_0}\frac{1-|f_1(p)f_2(p)|}{p}\\ + &\le A\sum_{p>P_0}\frac{(1-|f_1(p)|)+(1-|f_2(p)|)}p=o(1). +\end{align*} +Here squarefreeness permits the use of multiplicativity. +Another application of Markov's inequality shows that the probability +of $|f_1(d)f_2(d)|<1/2$ tends to zero. Also +$\Pbb_{\mathrm{div}}(d=1)=L_0^{-1}\to0$. +Thus the remaining divisors carry at least $L_0/2$ of the unnormalized +mass for large $B$. + +There are at most $1+C_0B/\log\tau$ intervals +$(\tau^r,\tau^{r+1}]$, $r\ge0$, meeting $(1,e^{C_0B}]$. +One of them carries at least $cL_0/B$ of that mass, for a fixed $c>0$. +Take its lower endpoint as $H$ and retain the divisors already satisfying +the preceding conditions. +\end{proof} + +For any such divisor family, any coefficients $|a_d|\le1$, and the +cutoffs \eqref{eq:analytic-cutoffs}, define the raw and centered sums by +\begin{align} + R_B(X)&=\sum_{x\le X}\sum_{d\in\mathcal D} + a_d A^{\omega_C(d)}f_1(x)f_2(x+hd) + w_d(x)w_d(x+hd)\1_{d\mid x}, + \label{eq:raw-sum}\\ + C_B(X)&=\sum_{x\le X}\sum_{d\in\mathcal D} + a_df_1(x)f_2(x+hd)g_d(x,x+hd). + \label{eq:analytic-centered-sum} +\end{align} +The first sum retains ordinary divisibility by all primes of $d$; +the second is the sum bounded by \Cref{prop:graph-estimate}. + +\begin{lemma}[Raw lower bound]\label{lem:raw-lower-bound} +Let $f_1,f_2\colon\N\to\C$ be $1$-bounded multiplicative functions, +with $f_1(1)=f_2(1)=1$, satisfying \eqref{eq:analytic-bias} and +\eqref{eq:analytic-finite-defects}. Choose $1<\tau<2$ and $T>0$ with +$\tau-1\le\gamma/64$ and $32/T^2\le\gamma/64$, and use a cutoff +$\phi$ as in \eqref{eq:analytic-cutoffs}, equal to $1$ on +$[-T/2,T/2]$. With $\mathcal D,H$ supplied by \Cref{lem:raw-bin}, set +\[ + a_d=\overline{f_1(d)f_2(d)},\qquad X_j=HN_j. +\] +There is $c_\gamma>0$, independent of $B,j$, such that, for every +sufficiently large fixed $B$, +\begin{equation}\label{eq:raw-lower-bound} + \liminf_{j\to\infty}\frac{|R_B(X_j)|}{X_j} + \ge c_\gamma\frac{L_0}{B}. +\end{equation} +\end{lemma} +\begin{proof} +Put +\[ + V_{\mathcal D}=\sum_{d\in\mathcal D}A^{\omega_C(d)},\qquad + K_{\mathcal D}=\sum_{d\in\mathcal D} + |f_1(d)f_2(d)|^2A^{\omega_C(d)}\ge\tfrac14 V_{\mathcal D}. +\] +For a fixed $d$, write $x=dm$. Unless +$\gcd(d,m)>1$ or $\gcd(d,m+h)>1$, ordinary multiplicativity gives +\[ + a_df_1(dm)f_2(d(m+h)) + =|f_1(d)f_2(d)|^2f_1(m)f_2(m+h). +\] +The exceptional set has ordinary density at most +$2\sum_{p\mid d}p^{-1}\le2J/P_0$. At fixed $B$ there are only finitely +many $d$, so all associated residue-counting errors vanish as +$j\to\infty$. + +For $p\in\mathcal C$ not dividing $d$, divisibility of $dm$ by $p$ is +equivalent to divisibility of $m$. Hence the mean and variance of +$k_d(dm)$ under uniform residues are +\[ + \mu_d=v_C-\sum_{\substack{p\mid d\\p\in\mathcal C}}\frac1p, + \qquad + \operatorname{Var}(k_d(dm)) + =\sum_{\substack{p\in\mathcal C\\p\nmid d}} + \frac1p\left(1-\frac1p\right)\le v_C. +\] +The same formulas hold with $m+h$ in place of $m$. For large $B$, +$|\mu_d-v_C|\le J/P_0\le(T/4)\sqrt{v_C}$. Chebyshev's inequality and +a union bound show that the proportion on which either cutoff is not +$1$ is at most $32/T^2$. Independence of the two endpoints is not needed. + +The length $X_j/d$ lies between $N_j/\tau$ and $N_j$. Comparing each +shorter sum with the sum up to $N_j$ therefore gives +\begin{equation}\label{eq:raw-comparison} + \begin{split} + |R_B(X_j)-K_{\mathcal D}S(N_j)| + \le N_jV_{\mathcal D}\left( + \tau-1+\frac{32}{T^2}+\frac{4J}{P_0}+o_{j\to\infty}(1)\right). + \end{split} +\end{equation} +The factor $4J/P_0$ allows a difference of size at most $2$ on each +coprimality failure. The endpoint rounding errors are included in $o(1)$. + +Choose $\tau-1\le\gamma/64$ and $32/T^2\le\gamma/64$, and then take +$B$ and $j$ large enough for the other two errors to be at most +$\gamma/64$ each. The reverse triangle inequality, valid for the +complex number $S(N_j)$, yields +\[ + |R_B(X_j)|\ge\frac\gamma8N_jV_{\mathcal D}. +\] +Finally, +$V_{\mathcal D}\ge H\sum_{d\in\mathcal D}A^{\omega_C(d)}/d +\ge cHL_0/B$. This proves the result with $c_\gamma=\gamma c/8$. +\end{proof} + +\subsection{The centering estimate and the contradiction} + +The remaining analytic task is to show that centering changes the raw +sum by less than its lower bound. Its statement does not require +\eqref{eq:analytic-finite-defects}. + +\begin{proposition}[Analytic centering estimate] +\label{prop:analytic-centering} +Let $f_1,f_2\colon\N\to\C$ be multiplicative, with $|f_i|\le1$, and +suppose at least one is uniformly nonpretentious. Fix +$h\ge1$, $1<\tau<2$, $C_0\ge1$, $T>0$, and a smooth function +$0\le\phi\le1$ supported in $[-T,T]$. +For each sufficiently large $B$, let $\mathcal D,H,J$ satisfy +\eqref{eq:divisor-family}, let $|a_d|\le1$, and use +\eqref{eq:analytic-cutoffs}. Define $R_B,C_B$ by +\eqref{eq:raw-sum} and \eqref{eq:analytic-centered-sum}. Then +\begin{equation}\label{eq:analytic-centering-bound} + \limsup_{X\to\infty}\frac{|R_B(X)-C_B(X)|}{X} + \ll L_0\left(B^{-1-\eps}+\frac{J}{P_0}\right). +\end{equation} +The implied constant may depend on the fixed parameters and $\phi$, +but is independent of the divisor family and coefficients. +The long-variable limit is taken with $B$ fixed. +\end{proposition} + +\begin{proof}[Reduction of \Cref{thm:elliott}] +We deduce \Cref{thm:elliott} from +\Cref{prop:graph-estimate,prop:analytic-centering}. +The zero-function and divergent-defect cases were settled above. +Otherwise suppose \eqref{eq:analytic-bias} holds. Choose +$\tau,T,\phi,C_0$ and, for each sufficiently large fixed $B$, the divisor +family of \Cref{lem:raw-bin}. Combining \Cref{lem:raw-lower-bound} with +the two propositions gives +\[ + c_\gamma\frac{L_0}{B} + \le C L_0\left(B^{-1-c_*/2}+B^{-1-\eps}+\frac{J}{P_0}\right). +\] +After dividing by $L_0/B$, this reads +\begin{equation}\label{eq:analytic-contradiction} + c_\gamma\le C\left(B^{-c_*/2}+B^{-\eps} + +BJ\exp(-B^{1-\eps})\right), +\end{equation} +which is impossible for sufficiently large $B$. + +In this comparison the functions, shift, and bias $\gamma$ are fixed +first; then $\tau,T,\phi,C_0$ are fixed. For each fixed $B$ the limit +is taken along the original sequence $N_j$, with $X_j=HN_j$. +Only after these inequalities hold does $B$ increase. +Thus the contradiction excludes every alleged biased sequence. +\end{proof} + +\section{The analytic centering estimate}\label{sec:analytic-centering} + +We prove \Cref{prop:analytic-centering}. Expanding the centered factors +leaves sums over integers with large prime factors. We shall bound the +Fourier multiplier of those integers and use short-interval cancellation +for whichever multiplicative function is nonpretentious. Combining short +exponential sums with a fourth-moment bound follows the centering +strategy in \cite[Appendix~C]{Pilatte26}; we prove the rough-number +estimates needed for the present weights. Throughout this section, +$\mathrm e(t)=\exp(2\pi it)$. + +\subsection{Expansion and finite-prime twists} + +A term other than the raw term in the expansion of the center-band +factors has $d=uw$, where $w>1$ is the product of the center primes +whose constant terms were selected. Thus $u,w$ are coprime squarefree +products, every core factor of $d$ belongs to $u$, and the coefficient +of the divisibility indicator $\1_{u\mid x}$ is +\[ + A^{\omega_C(u)}a_{uw}\frac{(-\theta)^{\omega(w)}}w. +\] +For each fixed $u$, put $c_w^{(u)}=a_{uw}(-\theta)^{\omega(w)}$ on +these admissible factorizations and set $c_w^{(u)}=0$ otherwise. +Then $|c_w^{(u)}|\le1$, and its support is independent of $x$. +Set $M=H/u$ and write $x=uz$. Whenever this support is nonempty, +its values of $w$ satisfy +\begin{equation}\label{eq:analytic-rough-range} + MP_0$. +Every such $w$ has no prime factor below $P_0$; we call integers with +this property \emph{$P_0$-rough}. + +Since $u$ and $d$ have exactly the same core factors, the first +cutoff becomes +\[ + w_{uw}(uz)=\phi\left( + \frac{\displaystyle\sum_{\substack{p\in\mathcal C\\p\nmid u}} + \1_{p\mid z}-v_C}{\sqrt{v_C}}\right). +\] +The second cutoff has the same expression with $z+hw$ in place of $z$. +Apart from +$\gcd(u,z(z+hw))>1$, multiplicativity gives +\[ + f_1(uz)f_2(u(z+hw))=f_1(u)f_2(u)f_1(z)f_2(z+hw). +\] +For a fixed $u,w$, the exceptional set has ordinary density at most +$2J/P_0$. Since +\begin{equation}\label{eq:analytic-sum-u} + \sum_{M0$ the sum on the right is $\binom{r-1}s\ge0$, and for +$r=0$ it is $1$. +Write $E_j((a_p))=\sum_{|\mathcal I|=j}\prod_{p\in\mathcal I}a_p$ +for the elementary symmetric sum over subsets of the primes $ps}\frac{\Lambda_0^j}{j!} + \le\frac{(e\Lambda_0/(s+1))^{s+1}}{1-\Lambda_0/(s+2)} + \le B^{-10}. +\] +Consequently the count in \eqref{eq:rough-brun} is at most +\begin{equation}\label{eq:rough-sieve-bound} + V\left(\prod_{pM$. +To estimate the fourth moment, extend $a_w=c_w/w$ by zero outside its +support. Fourier orthogonality gives +\[ + \int_{\T}|Q_M(\alpha)|^4\,d\alpha + =\sum_k\left|\sum_w a_{w+k}\overline{a_w}\right|^2. +\] +There is no factor depending on $h$: multiplication by the nonzero +integer $h$ preserves Haar measure on $\T$. +Let $R_k$ count pairs $w,w+k\in(M,\tau M]$ that are both +$P_0$-rough. Each inner absolute value is at most $R_k/M^2$. +We next bound $R_k$ for $k\ne0$; only $|k|\le2M$ can contribute. +The permitted interval for $w$ is an intersection of two intervals of +length less than $M$. Sieve out the residues $0,-k$ modulo $pt_B\}$. +By \Cref{lem:rough-multiplier}, +\begin{equation}\label{eq:rough-large-frequencies} + |E_B|\ll M^{-1}B^{8\eps}(\log B)^6, + \qquad \|Q_M\|_\infty\ll B^{-1+\eps}\log B. +\end{equation} +On $\T\setminus E_B$, Parseval and Cauchy--Schwarz give, for each $y$, +\[ + \int_{\T\setminus E_B}|Q_M A_y B_y|\,d\alpha + \le t_B\left(\int_{\T}|A_y|^2\right)^{1/2} + \left(\int_{\T}|B_y|^2\right)^{1/2} + \le t_B\sqrt{D_1D_2}. +\] +After integration in $y$ and division by $D_1Y$, the contribution is +$O_h(B^{-1-\eps})$. + +On $E_B$, suppose first that $f_1$ is uniformly nonpretentious. +Use $|B_y|\le D_2$, integrate first in $y$, and apply +\eqref{eq:mrt-fixed-length} to $A_y$ at each fixed frequency. The +normalized limsup is at most +\begin{equation}\label{eq:analytic-large-bound} + O\left(\|Q_M\|_\infty\,|E_B|\,D_2 + \frac{\log\log D_1}{\log D_1}\right). +\end{equation} +If instead $f_2$ is the nonpretentious factor, use $|A_y|\le D_1$ +and apply the same estimate to $B_y$; this replaces $D_1$ by $D_2$ +inside the logarithms in \eqref{eq:analytic-large-bound}. +In both cases Fubini's theorem is followed by the bound for the +already integrated exponential sum, with a supremum over its fixed +frequency. The frequency supremum has not been moved inside a +short-interval integral. + +The range \eqref{eq:analytic-rough-range} implies, for $i=1,2$, +\[ + \frac{D_i}{M}\ll_h1, + \qquad + \frac{\log\log D_i}{\log D_i} + \ll_{C_0,h,\tau}B^{-1+\eps}\log B. +\] +Using \eqref{eq:rough-large-frequencies} in +\eqref{eq:analytic-large-bound}, the large-frequency contribution is +\[ + O\left(B^{-2+10\eps}(\log B)^8\right) + =o(B^{-1-\eps}), +\] +since $11\eps<1$. Together with the small-frequency bound and +\eqref{eq:analytic-overlap}, this proves +\eqref{eq:analytic-rough-target}. The argument is uniform in the +Fourier twists, since \eqref{eq:mrt-fixed-length} is uniform in them. + +\begin{proof}[Completion of \Cref{prop:analytic-centering}] +Apply \eqref{eq:analytic-rough-target} with $c_w=c_w^{(u)}$ to each fixed +$u$ in the expansion at the start of this section. Dividing its contribution by +$X$ contributes the factor $A^{\omega_C(u)}/u$; the extracted factors +$f_1(u)f_2(u)$ have modulus at most $1$. +The two Fourier inversions cost at most $\|\widehat\phi\|_1^2$. +At fixed $B$ all sums over $u,w$ are finite, and the uniform estimate +therefore gives, by \eqref{eq:analytic-sum-u}, a total limsup +$O(L_0B^{-1-\eps})$. Adding the earlier coprimality error +$O(L_0J/P_0)$ proves \eqref{eq:analytic-centering-bound}. +\end{proof} + +The analytic comparison required for +\eqref{eq:analytic-contradiction} is now proved. It remains to establish +the graph estimate; the following sections develop its closed-line +bound and then pass from that bound to the long-average estimate. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/cylinder-sieve.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/cylinder-sieve.tex new file mode 100644 index 0000000000000000000000000000000000000000..c6ee84413e0d4358d36dd91537a627dc5b71cdba --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/cylinder-sieve.tex @@ -0,0 +1,402 @@ +\section{A conditional cylinder sieve}\label{sec:cylinder-sieve} + +The restrictions $n+b_j\in Y$ still involve the residue coordinates +outside the fixed set $\mathcal F$. We replace their product by a sum +of conditions from short lists of primitive specifications. In each +term the remaining coordinates are averaged before taking an absolute +value; this preserves the cancellation needed in the trace estimate. +The error will contain many specifications with distinct private prime +coordinates, which supply enough arithmetic constraints to apply +\Cref{lem:triangular-saving}. + +The intersection rank and the extraction of constraints from its +witnesses adapt Pilatte's construction +\cite[Definition~13.3 and Lemmas~13.4 and~13.6]{Pilatte26}. +The associated truncation belongs to the composite-modulus sieve +developed in \cite[Section~3]{HR21} and +\cite[Appendix~B]{Pilatte26}. We prove the finite-cylinder statement +and its application here, including the effect of conditioning. + +\subsection{Truncating an intersection poset} + +Let $\Omega=\prod_{p\in P}\Omega_p$ be a finite product, with +$|\Omega_p|\ge2$ for every $p$. An \emph{exact cylinder} specifies one +value at each coordinate in a subset of $P$ and imposes no condition at +the other coordinates. Its \emph{support} is the set of specified +coordinates, and its \emph{width} is the size of that support. + +Let $\mathscr E$ be a finite family of proper, nonempty exact cylinders. +Identify duplicate cylinders. Let $\mathscr P$ consist of the nonempty +intersections of subfamilies of $\mathscr E$, including the empty-family +intersection $\widehat0=\Omega$. Order $\mathscr P$ by reverse inclusion: +$I\le J$ means $I\supseteq J$. For $I\in\mathscr P$, define +\begin{equation}\label{eq:cylinder-rank} + \operatorname{rk}(I)= + \max\left\{|\mathscr A|:\begin{array}{l} + \mathscr A\subseteq\{E\in\mathscr E:I\subseteq E\},\\ + \text{each }E\in\mathscr A\text{ has a support coordinate}\\ + \text{absent from every other member of }\mathscr A + \end{array}\right\}. +\end{equation} +The empty family is allowed in this maximum. A coordinate as in +\eqref{eq:cylinder-rank} is called \emph{private} to its member of +$\mathscr A$. All events in such a family have compatible prescribed +values because they contain the nonempty intersection $I$. + +Write $\mu(\widehat0,I)$ for the M\"obius coefficients of this finite +poset, characterized by +\[ + \sum_{I\le J}\mu(\widehat0,I)=\1_{J=\widehat0}. +\] +Their role is inclusion--exclusion with equal intersections collected +into one term. This is M\"obius inversion in the incidence algebra of a +finite poset; see \cite[Section~3]{Rota1964}. The chain expansion used +below is the one in \cite[Section~3, Proposition~6]{Rota1964}. + +\begin{lemma}[Finite-cylinder truncation]\label{lem:cylinder-truncation} +Suppose that the cylinders in $\mathscr E$ have width at most $w\ge1$. +For an integer $t\ge1$, put $M=tw$, and let +\[ + \mathscr H_t=\{J\in\mathscr P:\operatorname{rk}(J)\ge t, + \ J\text{ is generated by at most }t\text{ members of }\mathscr E\}. +\] +Then the rank is nondecreasing on $\mathscr P$, every $I\in\mathscr P$ +is generated by at most $\operatorname{rk}(I)$ events, and +\begin{equation}\label{eq:cylinder-width} + |\operatorname{supp}(I)|\le w\operatorname{rk}(I). +\end{equation} +For every $x\in\Omega$, +\begin{equation}\label{eq:cylinder-truncation} + \left|\1_{x\notin\bigcup_{E\in\mathscr E}E} + -\sum_{\operatorname{rk}(I)0$. An admissible divisor +family $\mathcal D$ is any collection of squarefree products of primes +in $\mathcal S$ such that +\begin{equation}\label{eq:divisor-family} + \begin{gathered} + d>1,\qquad H0$, and use +\eqref{eq:graph-constants}--\eqref{eq:graph-weights}. For every sufficiently +large $B$, every admissible family $\mathcal D$, all permitted cutoffs +$w_d$, all coefficients $|a_d|\le1$, and all functions +$F,G:\N\to\mathbb D$, +\begin{equation}\label{eq:graph-estimate} + \limsup_{X\to\infty}\frac1X + \left|\sum_{1\le x\le X}\sum_{d\in\mathcal D} + a_dF(x)G(x+hd)g_d(x,x+hd)\right| + \ll L_0 B^{-1-c_*/2}. +\end{equation} +The implied constant and the threshold for $B$ may depend on +$h,\tau,C_0,T$, and the fixed constants in \eqref{eq:graph-constants}, +but not on $H,\mathcal D,w_d,a_d,F,G$. All graph parameters are fixed +when $X$ tends to infinity. +\end{proposition} + +The proof occupies \Cref{sec:paths,sec:thinning,sec:cylinder-sieve,sec:trace-count,sec:localization}. +The two bands have different roles in that proof. The core weights and +cutoffs give savings from the short divisor interval and from the +lower bound on $k_d(x)$ in \eqref{eq:core-cutoff}. +The center factors provide further cancellation in closed-walk products, +where each edge weight is divided by $W$ at its starting vertex. +The relation $\theta=\beta_Z^2$ matches the +centering constant to that normalization; its precise use is established +in \Cref{sec:trace-count}, once the relevant walk configurations have +been defined. + +The comparison scale is $L_0/B$. In the next section, one of $O(B)$ +short multiplicative intervals captures enough divisor weight to turn +any persistent correlation bias into a sum of this size with ordinary +divisibility indicators. The extra factor $B^{-c_*/2}$ in +\eqref{eq:graph-estimate} will contradict that +bias once the analytic cost of centering has been bounded. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/localization.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/localization.tex new file mode 100644 index 0000000000000000000000000000000000000000..9920de6a1aa2e4593b3992becf1ea37198e055b8 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/localization.tex @@ -0,0 +1,167 @@ +\section{From high traces to the graph estimate}\label{sec:localization} + +We now prove \Cref{prop:graph-estimate} from \Cref{thm:high-trace}. +The latter controls averages over a finite residue space. The present +step converts those averages into a bound for arbitrary test sequences +on long intervals, including complex coefficients $a_d$. + +Choose a power of two $D_0$ with +\[ + B^{10}H\le D_0<2B^{10}H, +\] +and partition the positive integers into blocks +$I_k=\{kD_0+1,\ldots,(k+1)D_0\}$, $k\ge0$. +For large $B$, all primes in $\mathcal S$ are odd, so +$\gcd(D_0,Q_B)=1$. Let $Y$ be the periodic vertex set from +\Cref{thm:high-trace}. On the coordinates of $I_k$, define the matrix +\[ + M_k(x,y)= + \begin{cases} + \displaystyle\1_Y(x)\1_Y(y) + \frac{a_dg_d(x,y)}{\sqrt{W(x)W(y)}}, + &y=x+hd,\quad d\in\mathcal D,\\[5pt] + 0,&\text{otherwise}. + \end{cases} +\] +This is a directed matrix and need not be self-adjoint. We therefore +use a moment of its singular values. + +\subsection{Averaging the singular-value moment} + +Set $m=\ell/2$. With $y_m=y_0$, matrix multiplication gives exactly +\begin{equation}\label{eq:localization-trace} + \operatorname{Tr}\bigl((M_k^*M_k)^m\bigr) + =\sum_{y_0,\ldots,y_{m-1}\in I_k}\ + \sum_{x_1,\ldots,x_m\in I_k} + \prod_{j=1}^m\overline{M_k(x_j,y_{j-1})}M_k(x_j,y_j). +\end{equation} +A nonzero term describes the closed line +\[ + y_0,x_1,y_1,x_2,y_2,\ldots,x_m,y_0, +\] +with alternating negative and positive steps of lengths $hd_i$. +Its scalar coefficient is +$\prod_{j=1}^m\overline{a_{d_{2j-1}}}a_{d_{2j}}$, of modulus at most one. +Symmetry and reality of $g_d$ identify its edge numerator with that of +$K_{\mathbf d}$ in \eqref{eq:trace-kernel}. For the successive vertices +$v_0,\ldots,v_\ell=v_0$, its denominator satisfies +\[ + \prod_{i=1}^{\ell}\sqrt{W(v_{i-1})W(v_i)} + =\prod_{i=0}^{\ell-1}W(v_i). +\] +Repeated vertices are counted with multiplicity in this identity. +The indicator factors reduce to $\prod_{i=0}^\ell\1_Y(v_i)$. + +Fix the initial position $r\in\{1,\ldots,D_0\}$ and write +$v_i=kD_0+r+b_i$. The block restrictions are precisely +$1\le r+b_i\le D_0$ for all $i$; they depend only on $r$ and the +ordered line. As $k$ runs modulo $Q_B$, the starting point $kD_0+r$ +is uniform modulo $Q_B$. Averaging \eqref{eq:localization-trace} over +this period, and taking absolute values after each line's residue +average, gives +\begin{align} + 0\le\frac1{Q_B}\sum_{k\bmod Q_B} + \operatorname{Tr}\bigl((M_k^*M_k)^m\bigr) + &\le D_0\sum_{\mathbf d} + \left|\E K_{\mathbf d}(n) + \prod_{i=0}^{\ell}\1_Y(n+b_i)\right|\notag\\ + &\le D_0 B^{-(1+c_*)\ell}.\label{eq:localization-moment} +\end{align} +A starting position and its ordered signed line determine all indices +in \eqref{eq:localization-trace}, so no further multiplicity occurs. + +Call a block exceptional if +$\|M_k\|>B^{-1-c_*/2}$, where the norm is the Euclidean operator norm. +Since $\|M_k\|^\ell\le\operatorname{Tr}((M_k^*M_k)^m)$, its fraction +$\delta$ among one period of blocks satisfies +\begin{equation}\label{eq:exceptional-blocks} + \delta\le D_0B^{-c_*\ell/2} + =\exp\bigl(-\Omega(B\log B)\bigr). +\end{equation} +Here $\log D_0\le C_0B+10\log B+O(1)$ and $\ell\sim2B$. +The threshold for this estimate may depend on $C_0$. + +\subsection{Discarded edges} + +We must control the weight on deleted vertices, exceptional blocks and +block boundaries before applying the operator norm. A common absolute +degree bound handles all three. Dropping cutoffs and extending the +sum to every squarefree product of primes in $\mathcal S$ gives, +separately for either sign, +\begin{align} + \sum_{d\in\mathcal D}|g_d(x,x\pm hd)| + &\le\prod_{p\in\mathcal C}(1+A\1_{p\mid x}) + \prod_{p\in\mathcal Z} + \left(1+\left|\1_{p\mid x}-\frac\theta p\right|\right) + \le W(x)e^{\theta v_Z}.\label{eq:absolute-degree} +\end{align} +For $p\in\mathcal Z$ dividing $x$, its factor is $2-\theta/p\le2$; +otherwise it is at most $e^{\theta/p}$. The same bound applies to +incoming edges at their terminal vertex, since the residues of both +endpoints agree at every prime in their divisor label. + +All parameters are now fixed. Periodic averages over the integers equal +the corresponding residue averages as $X\to\infty$. By +\Cref{thm:high-trace} and \eqref{eq:weight-moments}, the total absolute +edge weight incident to $Y^c$, divided by $X$, has limsup at most +\[ + 2e^{\theta v_Z}\E\bigl(W\1_{Y^c}\bigr) + \le 2e^{\theta v_Z}(\E W^2)^{1/2}\Pbb(Y^c)^{1/2} + \le B^{O_A(1)}P_0^{-1/2+o(1)}. +\] +The fixed extension of the interval to $X+\tau hH$ does not affect this +limsup. The last expression is smaller than every fixed negative power +of $B$. + +Let $E$ be the union of exceptional blocks. It has period $D_0Q_B$ and +density $\delta$. Cauchy--Schwarz over that full period gives +\[ + \lim_{X\to\infty}\frac1X\sum_{x\le X}W(x)\1_E(x) + \le (\E W^2)^{1/2}\delta^{1/2}. +\] +There is no independence assertion between the exceptional-block event +and the prime residues. By \eqref{eq:absolute-degree}, the within-block +terms in $E$ cost at most +\[ + e^{\theta v_Z}(\E W^2)^{1/2}\delta^{1/2} + \le B^{O_A(1)}D_0^{1/2}B^{-c_*\ell/4} + =\exp\bigl(-\Omega(B\log B)\bigr) +\] +after normalization by $X$ and passage to the limsup. + +A forward edge crossing a block boundary begins in the last +$\lceil\tau hH\rceil$ positions of its block. Position modulo $D_0$ +and residue modulo $Q_B$ are independent over a full period $D_0Q_B$. +Consequently the crossing cost is at most +\[ + \frac{\lceil\tau hH\rceil}{D_0}L_0e^{\theta v_Z} + \ll_h B^{-10}L_0e^{\theta v_Z} + =o\bigl(L_0B^{-1-c_*/2}\bigr). +\] +In the first two estimates we may also compare with this target, since +$L_0\ge1$. Thus all three discarded contributions are negligible. + +\subsection{The bilinear estimate} + +Use the standard complex inner product, conjugate-linear in its first +argument, and define +\[ + u(x)=\overline{F(x)}\sqrt{W(x)}\1_{1\le x\le X},\qquad + v(y)=G(y)\sqrt{W(y)}\1_{1\le y\le X+\tau hH}. +\] +Then $\langle u|_{I_k},M_kv|_{I_k}\rangle$ is exactly the desired +unnormalized sum over edges in $I_k$ with endpoints in $Y$. +On nonexceptional blocks, Cauchy--Schwarz in the block index yields +\begin{align*} + \left|\sum_{k\ \mathrm{nonexceptional}} + \langle u|_{I_k},M_kv|_{I_k}\rangle\right| + &\le B^{-1-c_*/2} + \left(\sum_{x\le X}W(x)\right)^{1/2} + \left(\sum_{y\le X+\tau hH}W(y)\right)^{1/2}\\ + &=(L_0+o_{X\to\infty}(1))X B^{-1-c_*/2}. +\end{align*} +The source cutoff in $u$ also handles the last partially used block. +Adding back the three negligible edge classes proves +\Cref{prop:graph-estimate}. This closes the deferred graph input in +\Cref{sec:analytic-transfer}, and hence completes the proof of +\Cref{thm:elliott}. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/paths.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/paths.tex new file mode 100644 index 0000000000000000000000000000000000000000..1c2a4223b5da91db47deb3e9491a0d11442ed320 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/paths.tex @@ -0,0 +1,381 @@ +\section{Forbidden paths and arithmetic savings}\label{sec:paths} + +We now prove the graph estimate stated in +\Cref{prop:graph-estimate}. Its main input is a bound for closed +walks after deleting a sparse periodic set of vertices. In this +section we define that set and establish the arithmetic estimates +used to discard exceptional walks. The parameters and kernels are +those of \Cref{sec:graph-setup}. In the graph proof, $B$ tends to +infinity with $h,\tau,C_0,T$ fixed. + +Deleting vertices that support short prohibited paths is part of the +divisibility-graph method of \cite[Section~4]{HR21}, developed for +composite labels in \cite[Sections~7 and~13.1]{Pilatte26}. +We use primitive specifications with an extra prime coordinate and prove +the resulting density and arithmetic bounds for the present graph. + +Put +\begin{equation}\label{eq:trace-parameters} + \ell=2\lfloor B\rfloor,\qquad + L=\lfloor B^{1-\rho}\rfloor,\qquad + t_0=\lceil B^{4\eps}\rceil. +\end{equation} +A \emph{closed line} is a list of signs and labels +$\boldsymbol d=((\varepsilon_i,d_i))_{i=1}^{\ell}$, where +$\varepsilon_i\in\{-1,1\}$, $d_i\in\mathcal D$, and the offsets +\[ + b_0=0,\qquad b_j=\sum_{i=1}^j\varepsilon_i h d_i + \quad(1\le j\le\ell) +\] +satisfy $b_\ell=0$. Repeated offsets are allowed. Define its +normalized product by +\begin{equation}\label{eq:trace-kernel} + K_{\boldsymbol d}(n) + =\prod_{i=1}^{\ell} + \frac{g_{d_i}(n+b_{i-1},n+b_i)}{W(n+b_{i-1})}. +\end{equation} +At each $B$, all residue expectations below are on the finite product +probability space of \Cref{sec:graph-setup}. + +\subsection{The deleted vertices} + +\begin{definition}[Path specifications]\label{def:primitive-specification} +A specification consists of a path with distinct integer offsets +$z_0=0,z_1,\ldots,z_m$, where $1\le m\le L$ and +\[ + z_i-z_{i-1}=\varepsilon_i h e_i, + \qquad e_i\in\mathcal D,\quad \varepsilon_i\in\{-1,1\}, +\] +an extra prime $p\in\mathcal S$ dividing none of the $e_i$, and an +index $i_0\in\{1,\ldots,m\}$ such that +$p\mid z_m-z_{i_0-1}$. It \emph{qualifies at $x$} when +\[ + p\mid x,\qquad e_i\mid x+z_{i-1}\quad(1\le i\le m). +\] +Its prime support $\Pi$ is the union of $\{p\}$ and the prime factors +of all its edge labels. Its \emph{cylinder} is the set of starting +integers $x$ satisfying these qualification conditions. The cylinder +is either empty or fixes one residue at each prime in $\Pi$; in the +latter case its measure is $\prod_{q\in\Pi}q^{-1}$. + +When testing a contiguous subpath as a specification, translate its +offsets to start at zero. If it starts at $z_a$, test qualification +at $x+z_a$. For the +reversed subpath, use the same convention with its new initial +vertex. The extra prime may be any member of $\Pi$ absent from the +subpath's edge labels. All these tests are constant as $x$ ranges +over a nonempty original cylinder, since they use only its fixed +prime coordinates. + +A specification is \emph{primitive} if its cylinder is nonempty and +no shorter nonempty contiguous subpath, in either orientation, can +be made into a specification, with an extra prime from $\Pi$, that +qualifies at its own initial vertex for $x$ in that cylinder. Set +\[ + Y=\{x\in\Z:\text{no primitive specification qualifies at }x\}. +\] +\end{definition} + +The set $Y$ is periodic modulo $\prod_{p\in\mathcal S}p$. +For a path translated to start at $x$, a prime $q$ is \emph{active} +at its $i$th vertex when $q\mid x+z_i$. + +\begin{lemma}[Descent to a primitive path]\label{lem:primitive-descent} +If a specification qualifies, a primitive specification using only +its prime support qualifies at one of its path vertices, along a +contiguous subpath in one of the two orientations. +\end{lemma} + +\begin{proof} +Full edge divisibility survives reversal: $e\mid x$ and +$x'=x\pm he$ imply $e\mid x'$. For a candidate subpath all residue +tests are constant on the original cylinder, since they involve only +its fixed prime coordinates. If the current specification is not +primitive, take a shorter qualifying contiguous subpath in an allowed +orientation. Its support is contained in the preceding support. +The positive length strictly decreases, so the procedure terminates. +\end{proof} + +We shall prove the following estimate. Its density assertion is +proved immediately; the closed-line bound is completed in +\Cref{sec:trace-count}. + +\begin{theorem}[High trace]\label{thm:high-trace} +For the divisor family, cutoffs, and kernels of +\Cref{sec:graph-setup}, the periodic set $Y$ in +\Cref{def:primitive-specification} satisfies, as $B\to\infty$, +\begin{align} + \Pbb(n\notin Y)&\le P_0^{-1+o(1)},\label{eq:deleted-density}\\ + \sum_{\boldsymbol d} + \left|\E K_{\boldsymbol d}(n) + \prod_{j=0}^{\ell}\1_Y(n+b_j)\right| + &\le B^{-(1+c_*)\ell}.\label{eq:high-trace} +\end{align} +The sum is over all closed lines of length $\ell$. The asymptotic +bounds are uniform over the admissible divisor families and cutoffs, +with $h,\tau,C_0,T$ fixed. +\end{theorem} + +\begin{lemma}[Density of the deleted set]\label{lem:bad-density} +The set $Y$ satisfies \eqref{eq:deleted-density}. +\end{lemma} + +\begin{proof} +It suffices to count all qualifying specifications. For fixed edge +labels, signs, and suffix, its displacement $D=z_m-z_{i_0-1}$ is +nonzero because the path vertices are distinct, and +$|D|\le Lh\tau H$. Sum the extra prime first, keeping these numerical +edge labels fixed. Since $p>P_0$, +\[ + \sum_{\substack{p\in\mathcal S\,,\ p\mid D}} + \frac1p + \le \frac{\log(|D|+2)}{P_0\log 2} + \ll \frac{\log(Lh\tau H+2)}{P_0}. +\] +The support weight for the remaining distinct edge primes is the +product of their reciprocals. There are at most $LJ$ edge-prime +slots. Choosing the length, signs, suffix, factor counts, and a +partition of these slots into equality classes costs +$\exp(O(LJ\log B))$. Indeed a partition of at most $LJ$ slots has +at most $(LJ)^{LJ}$ descriptions. Summing each remaining class over +its prime values costs at most $v_C+v_Z$, or $1+v_C+v_Z$ as a uniform +upper bound. At this point we may drop all bin and distinctness +restrictions in those positive sums. The resulting bound is +\[ + \Pbb(n\notin Y) + \le P_0^{-1} + \exp\bigl(O(LJ\log B)+O(LJ\log\log B)+O(\log B)\bigr). +\] +Since $LJ\log B=O(B^{1-\rho}\log^2B)=o(\log P_0)$, this proves +the assertion. +\end{proof} + +The deletion also provides useful constraints on the labels of a +primitive path. The next lemma identifies coefficients that remain +invertible even when a prime divides several edge labels. + +\begin{lemma}[Prime intervals and the last constant block] +\label{lem:primitive-properties} +For sufficiently large $B$, let a primitive specification have edge +labels $e_1,\ldots,e_m$, extra prime $p$, and suffix ending at $m$. +Then the following statements hold. +\begin{enumerate} +\item For every edge prime $q$, the indices $i$ with $q\mid e_i$ +form an interval of consecutive indices. +\item Let $b$ be the first index of the final maximal constant block +$e_b=\cdots=e_m$. Then $b>1$, the signs on this block are constant, +and there exists $r\mid e_b$ with $r\nmid e_{b-1}$. Every such $r$ +occurs only in the constant tail. In the specified suffix +displacement, its coefficient as a prime variable is nonzero modulo +$p$. This coefficient assertion holds for any edge prime whose +occurrences are confined to that tail. +\item If $q$ occurs before $b$ and $r$ is as in the preceding part, +the coefficient of $q$ in $z_{b-1}-z_0$ is nonzero modulo $r$. +\end{enumerate} +Coefficients here are obtained by fixing all other distinct prime +values; squarefreeness makes each displacement linear in the prime +being varied. +\end{lemma} + +\begin{proof} +Suppose successive occurrence blocks of $q$ are separated by edges +not containing $q$. The intervening path begins and ends at active +vertices for $q$, since adjacent qualifying $q$-edges have divisible +endpoints. It is a shorter simple $q$-free path with nonzero total +displacement divisible by $q$. With extra prime $q$ and its whole +path as suffix, it contradicts primitivity. This proves the first +part. + +Within a constant-label block, an adjacent change of sign would +repeat a vertex, so all signs agree. If $b=1$, the prescribed suffix +displacement is $\pm khe_m$ for some $1\le k\le L$. The prime $p$ +divides neither $e_m$ nor $h$, and $p>L$, so this is impossible. +Thus $b>1$. The unequal squarefree labels $e_b,e_{b-1}$ lie in one +bin of ratio less than $2$. If every prime of $e_b$ divided +$e_{b-1}$, their quotient would be an integer at least $2$, a +contradiction. Choose $r\mid e_b$ absent from $e_{b-1}$. By the +first part it is absent from the entire prefix. Its coefficient in +the suffix is +\[ + \pm kh\frac{e_b}{r},\qquad 1\le k\le L. +\] +This is nonzero modulo $p$, because $p>h,L$ and $p$ divides none of +the edge labels. The same reasoning applies to every prime confined +to the tail. + +Finally, intersect the occurrence interval of $q$ with the prefix +$1,\ldots,b-1$. The resulting nonempty interval is $i,\ldots,j$; +in particular, $j=b-1$ if $q$ also occurs in the tail. The sum of +these prefix terms is $S=z_j-z_{i-1}\ne0$. If $r\mid S$, reverse the path from +$z_{b-1}$ to $z_{i-1}$. Its edges avoid $r$, it starts at a vertex +active for $r$, and its suffix from $z_j$ to $z_{i-1}$ has +displacement divisible by $r$. Full divisibility survives reversal. +This shorter qualifying specification contradicts primitivity. +Since $q\ne r$ and $S$ is $q$ times the asserted coefficient, that +coefficient is invertible modulo $r$. +\end{proof} + +\subsection{Absolute counts and triangular constraints} + +We next record a deliberately coarse counting estimate. It is used +only when an additional arithmetic saving makes an entire class +negligible. The main contribution in \Cref{sec:trace-count} will +require a more economical enumeration. + +For a line occurrence $(i,p)$, $p\mid d_i$, call it \emph{lit} if +$p\mid n+b_{i-1}$ and \emph{unlit} otherwise. The condition is the +same at the arrival vertex. We may attach a specification at a +line vertex $n+b_j$; qualification then refers to that starting +point. A \emph{slot pattern} records the factor counts of all edge +labels, their signs, their bands, equality classes of prime slots, +and the lengths, attachments, extra-prime slots, and suffix indices +of attached specifications. Distinct classes represent distinct +prime variables. The pattern may also specify line occurrence +statuses and a tree on the visited vertices whose edges are recorded +line steps. + +\begin{lemma}[Absolute reciprocal counting]\label{lem:crude-count} +Consider a line of length $\ell$ together with at most $2t_0$ +attached qualifying specifications, each of length at most $L$. +Refine its slot pattern by recording the lit or unlit status of every +line occurrence. +There are $\exp(O(B\log^2B))$ slot patterns, including signs, +attachments, occurrence statuses, and recorded vertex identifications. +For each assigned pattern and its numerical prime values, let +$\mathfrak A$ be its attached specifications and let $E$ be any +residue event imposing the recorded occurrence statuses, possibly +with further restrictions. Then +\begin{equation}\label{eq:absolute-pattern-majorant} + \E\left[|K_{\boldsymbol d}(n)|\1_E(n) + \prod_{\sigma\in\mathfrak A} + \1_{\{\sigma\text{ qualifies at its assigned vertex}\}}\right] + \le A^{\ell J}\prod_{p\text{ used}}\frac1p. +\end{equation} +If the recorded statuses include at least $t_0$ unlit occurrences +among center primes occurring at least twice on the line, this majorant has the additional +factor $P_0^{-t_0/2}$. The total contribution of that class, summed +over all patterns and numerical assignments, is +$O(\exp(-B^{1+\eps/2}))$. +\end{lemma} + +\begin{proof} +The number of slots is at most +\[ + \ell J+2t_0(LJ+1)=O(B\log B), +\] +since $4\eps<\rho$. A partition of $R$ slots has at most $R^R$ +descriptions. All further indices have polynomially many choices +in $B$ per slot or per recorded vertex, giving the stated entropy. +Recording the destination of each step among at most $\ell+1$ +vertices includes possible vertex identifications and any chosen +tree of recorded steps within the same bound. + +Drop the cutoffs and $W^{-1}$, which lie in $[0,1]$. The powers of +$A$ from core occurrences are at most $A^{\ell J}$. A lit occurrence +of $p$, or qualification of any attached specification using $p$, fixes its +residue and supplies at most $1/p$. In the absence of either, $p$ +appears only on unlit line occurrences. An unlit core occurrence +vanishes; each unlit center occurrence supplies $\theta/p$, so at +least one factor $1/p$ is still available. The remaining conditions +may be discarded after taking these positive primewise bounds. +Independence of the prime residue coordinates proves +\eqref{eq:absolute-pattern-majorant}. + +If a repeated center prime has a lit occurrence, every unlit +occurrence supplies an additional reciprocal beyond the residue +probability. If all its $m\ge2$ line occurrences are unlit, they +supply $p^{-m}$, saving at least $p^{-(m-1)}$ beyond the displayed +majorant; here $m-1\ge m/2$. Additional qualification conditions can +only improve the estimate. At least $t_0$ repeated-unlit occurrences +therefore save $P_0^{-t_0/2}$. + +For each unrestricted prime variable its reciprocal sum is at most +$1+v_C+v_Z=O(1+\log B)$. Thus the total before the extra saving is +$\exp(O(B\log^2B))$, including $A^{\ell J}$. The saving has +logarithm at most $-\frac12 B^{1+3\eps}$, which proves the last +assertion for sufficiently large $B$. +\end{proof} + +The next lemma is the arithmetic alternative to having many unlit +occurrences. Its ordering hypothesis is essential: all selected +prime values cannot be summed independently without examining their +dependencies. + +\begin{lemma}[Triangular arithmetic saving]\label{lem:triangular-saving} +In the positive count of \Cref{lem:crude-count}, suppose every +configuration under consideration supplies at least $B^{2\eps}$ +distinct prime variables $q_1,\ldots,q_k$ and tests with the following +properties. Test $j$ uses only $q_j$, the earlier variables +$q_1,\ldots,q_{j-1}$, and nonselected variables, and is either +\begin{enumerate} +\item $q_j\mid D_j$, where $D_j\ne0$ is independent of $q_j$ and +$\log(|D_j|+2)=O(B\log B)$; or +\item $a_jq_j+c_j\equiv0\pmod{r_j}$, where $a_j,c_j,r_j$ are +independent of $q_j$, $r_j$ is an earlier or nonselected prime +variable, and $a_j\not\equiv0\pmod{r_j}$. +\end{enumerate} +Assume the choices of tests and their order have +$\exp(O(B\log^2B))$ descriptions. Then their total absolute +contribution, also allowing the attached specifications in +\Cref{lem:crude-count}, is +\begin{equation}\label{eq:negligible-constraints} + O\bigl(\exp(-B^{1+\eps/2})\bigr). +\end{equation} +The description bound applies in particular when each test uses a +bounded number of displacements along contiguous line or attached +paths, or along paths in a recorded tree of line steps, together with +one common specified integer offset of size $O(\ell H)$. +\end{lemma} + +\begin{proof} +Fix the nonselected variables. For the first type of test, +\[ + \sum_{\substack{q\in\mathcal S\,,\ q\mid D_j}}\frac1q + \le \frac{\log(|D_j|+2)}{P_0\log 2} + \ll \frac{B\log B}{P_0}. +\] +For the second type, invertibility leaves one residue class modulo +$r_j$. Comparison with the integral of $1/x$ along that progression +gives, even when the sum is enlarged from primes to integers, +\[ + \sum_{\substack{P_0=Stealth,thick, + vertex/.style={circle,draw,inner sep=2pt}, + every edge/.style={font=\small}] + \node[vertex] (o) at (0,0) {$0$}; + \node[vertex] (u) at (2.1,0) {$u$}; + \node[vertex] (v) at (4.3,1) {$v$}; + \node[vertex] (w) at (4.3,-1) {$w$}; + \draw[->] (o) to[bend left=12] node[above] {$1$} (u); + \draw[->,dashed] (u) to[bend left=12] node[below] {$6$} (o); + \draw[->] (u) to[bend left=12] node[above] {$2$} (v); + \draw[->,dashed] (v) to[bend left=12] node[pos=.25,below] {$3$} (u); + \draw[->] (u) to[bend right=12] node[below] {$4$} (w); + \draw[->,dashed] (w) to[bend right=12] node[pos=.25,above] {$5$} (u); +\end{tikzpicture} +\caption{First visits (solid) and returns (dashed) in the walk +$0\to u\to v\to u\to w\to u\to0$. Numbers give the chronological +step indices. The three solid edges form the first-visit tree. +This diagram records only the walk topology.} +\label{fig:first-visit-tree} +\end{figure} + +Recall that an occurrence $(i,p)$ is lit when +$p\mid n+b_{i-1}$. Set +\begin{equation}\label{eq:fixed-line-labels} + \mathcal F=\{p:\ p\text{ occurs on the line and either } + p\in\mathcal C\text{ or }p\text{ occurs at least twice}\}. +\end{equation} +Write $n_{\mathcal F}$ for these residue coordinates. We allow our +restrictions to depend on $n_{\mathcal F}$ and call the primes in +$\mathcal F$ the \emph{fixed labels}. Here ``fixed'' refers to the +residues: the numerical line already specifies every prime value. +The set includes core primes occurring once on the line, but excludes +background core primes absent from it. For a fixed tree prime $p$, +define its active vertices by +\[ + V_p=\{v\in V(\mathscr T):p\mid n+b(v)\}. +\] +The connected components of $V_p$ are those of the induced subgraph +on these vertices. Adjacent active vertices force $p$ to divide +their edge label, since $p\nmid h$ for sufficiently large $B$. + +A tree prime $p\notin\mathcal F$ is a center prime occurring on +exactly one step of the entire line. Let $e_p$ denote its tree edge. +Call it \emph{corrupted} if a tree vertex other than the endpoints +of $e_p$ has an offset congruent to those endpoints modulo $p$. +This is a condition on the numerical line alone. + +\begin{proposition}[Retained configurations]\label{prop:retained-lines} +For each closed line there is a predicate +$R_{\boldsymbol d}(n_{\mathcal F})\in\{0,1\}$ with the following +properties. Whenever it equals one: +\begin{enumerate} +\item all core line occurrences are lit; no primitive specification +with support contained in $\mathcal F$ qualifies at a trace vertex; +and fewer than $t_0$ repeated center occurrences are unlit; +\item every fixed tree prime has $O(B^\rho)$ active components; +fewer than $B^{1-2\rho}$ fixed tree primes have more than one +active component; +\item fewer than $B^{1-2\rho}$ nonfixed tree primes are corrupted. +\end{enumerate} +The implicit constant is uniform in the line. Moreover, +\begin{equation}\label{eq:retained-line-error} + \sum_{\boldsymbol d} + \E\left[|K_{\boldsymbol d}(n)| + \prod_{j=0}^{\ell}\1_Y(n+b_j) + (1-R_{\boldsymbol d}(n_{\mathcal F}))\right] + \ll \exp(-B^{1+\eps/2}). +\end{equation} +The same exceptional estimates used in this proof remain valid +after adding the qualification indicators of at most $2t_0$ +specifications and summing their recorded data, for the classes +discarded by an arithmetic saving. Configurations excluded because +of a fixed forbidden specification instead vanish with the original +$Y$ factors. +\end{proposition} + +We prove the proposition in three stages. First we make active +sets connected on a small number of blocks. Then a two-block +argument rules out many disconnected fixed labels. Finally we +handle the accidental offset coincidences of singleton labels. + +\subsection{Short gaps and connected blocks} + +\begin{lemma}[Hitting paths in a forest]\label{lem:forest-path-hitting} +Let a finite forest carry a finite family of nonempty edge paths. +If the family has no $t$ pairwise edge-disjoint paths, some set of +fewer than $2t$ edges meets every path in the family. +\end{lemma} + +\begin{proof} +Root each component. The top of a path is its vertex nearest that +root. Choose a path whose top has maximum depth, and mark its one +or two edges incident with the top. Every path sharing an edge +with the chosen path must contain a marked edge. Indeed, a path +sharing an edge below a mark and avoiding that mark remains wholly +in the component below it; its top would be strictly deeper. +Remove the paths hit by the marks and repeat. The paths chosen at +successive stages are pairwise edge-disjoint. There are fewer than +$t$ stages, so fewer than $2t$ marked edges suffice. +\end{proof} + +\begin{lemma}[Connected activity on blocks]\label{lem:activity-blocks} +Except for configurations of zero contribution in +\eqref{eq:retained-line-error} or total absolute contribution +$O(\exp(-B^{1+\eps/2}))$, one can cut $O(t_0)$ edges of the +first-visit tree and cover the resulting forest by $O(B^\rho)$ +connected blocks of diameter at most $L$. The blocks cover every +remaining edge and every vertex of the original tree. In each block, +the active set of every fixed tree prime is connected or empty. +Consequently each fixed tree prime has $O(B^\rho)$ whole-tree active +components. +\end{lemma} + +\begin{proof} +An unlit core occurrence makes $K_{\boldsymbol d}=0$. A primitive +specification supported on $\mathcal F$ and qualifying at a trace +vertex makes the corresponding $Y$ factor zero for every remaining +coordinate assignment. Discard these configurations. By +\Cref{lem:crude-count}, those with at least $t_0$ unlit repeated +center occurrences have the asserted negligible total. Each of +these decisions uses only the line and $n_{\mathcal F}$. + +Cut every tree edge having an unlit fixed occurrence. There are +fewer than $t_0$ such edges. In the remaining forest a \emph{gap +for $p$} is a path of length at most $L$ with active endpoints and +at least one interior vertex, all interior vertices being inactive, +for a fixed tree prime $p$. Such a gap has no $p$-edge: an active +endpoint of a $p$-edge forces its +other endpoint to be active, while a $p$-edge with inactive +endpoints was cut. + +Every gap contains a nonfixed prime label. Otherwise all its +edge labels are fixed and lit, so the path is fully divisible. +Its nonzero total displacement is divisible by $p$, and its edges +avoid $p$. Taking extra prime $p$ and the whole path as suffix +gives a qualifying specification. By +\Cref{lem:primitive-descent}, a primitive one with fixed support +qualifies at a vertex of that tree path. Every tree vertex is a +trace vertex, contrary to the preceding exclusion. + +If there are $t_0$ edge-disjoint gaps, choose a nonfixed prime $q$ +on each. Each chosen prime occurs once on the full line, hence +nowhere on the other selected gaps. The displacement of its gap +is linear in $q$ modulo the corresponding fixed prime $p$, with +coefficient $\pm h$ times the other prime factors of that edge. +This coefficient is invertible modulo $p$, since the gap is +$p$-free and $p\nmid h$. The selected constraints do not involve +one another's variables. Their paths and variables have the +description cost of \Cref{lem:triangular-saving}, which discards +this class. The inequality $t_0\ge B^{2\eps}$ holds for large $B$. + +Otherwise \Cref{lem:forest-path-hitting} supplies fewer than +$2t_0$ further cuts meeting every gap. For each remaining piece +traverse every edge twice and split that traversal into segments +of at most $L$ steps. The edges and vertices in a segment form +a connected subtree of diameter at most $L$; include a singleton +block for a piece with no edges. There are +\[ + O(\ell/L+t_0+1)=O(B^\rho) +\] +blocks. They may overlap. Any two active vertices in a block +have their joining path inside that block. An inactive portion +on it would give a surviving short gap, impossible after the cuts. +Thus activity is connected or empty on each block. Each block +meets at most one whole-tree active component, and every such +component has a vertex in a block. This proves the component bound. +\end{proof} + +\subsection{Comparing two blocks} + +The active sets are now connected within each block. To compare +different components for a fixed prime $p$, suppose blocks $1$ and +$2$ meet those components. Choose a root $c_j$ in block $j$ and an +active representative $x_p^j$ there. Since the representatives have +distinct integer offsets, +\begin{equation}\label{eq:two-block-constraint} + p\mid K+\bigl(b(x_p^2)-b(c_2)\bigr) + -\bigl(b(x_p^1)-b(c_1)\bigr)\ne0, + \qquad K=b(c_2)-b(c_1). +\end{equation} +To apply the divisibility case of \Cref{lem:triangular-saving}, we +seek roots and representatives for which both local paths contain +no edge label divisible by $p$. For many selected primes at once, +we also need a common order in which every selected prime appearing +on either path comes earlier than $p$. Once $K$ is specified separately, these +conditions give the required triangular dependence. The next lemma +supplies the choices. + +\begin{lemma}[Rerooting two blocks]\label{lem:reroot-constraints} +Let two connected blocks of a tree and $m$ distinct prime indices +$p$ be given. For each index let $V_p$ be a vertex subset, and let +$A_p^j$ be its intersection with block $j$. Suppose these +intersections are nonempty and connected. Edges have labels +containing at most $J$ prime factors. In either block, an edge has +both endpoints in $A_p^j$ if and only if its label contains $p$. +Assume also that $A_p^1$ and $A_p^2$ lie in distinct components of +the subgraph induced by $V_p$ in the whole tree. + +There are roots $c_j$ in block $j$, a set of +$\gg m^{1/8}/(2J+1)$ retained indices, and an order on those indices +with the following property. If $x_p^j$ is the nearest vertex of +$A_p^j$ to $c_j$, both paths from $c_j$ to $x_p^j$ avoid $p$, and +any retained prime label on either path occurs earlier in the order. +The two representatives $x_p^1,x_p^2$ are distinct. +\end{lemma} + +\begin{proof} +Start with arbitrary roots. A connected vertex set in a tree +contains the entire path between any two of its vertices. It +therefore has a unique nearest vertex to a chosen root. The +root-to-$x_p^j$ path avoids $p$-edges. If a retained label $q$ +occurs on this path, both ends of its edge lie in $A_q^j$; +the nearest vertex $x_q^j$ is then a strict ancestor of $x_p^j$. +Thus dependencies are contained in the two strict ancestor orders. +Indices with equal nearest points are incomparable. + +Every finite partially ordered set with $u$ elements has a chain +or an antichain of size at least $\sqrt u$: partition its elements +by the length of a longest chain ending there. Apply this to the +first ancestor order. If an antichain is obtained, there are no +dependencies from block 1 among its indices, so depth order in +block 2 suffices. Otherwise apply the same argument in block 2 +on the selected chain. An antichain there similarly suffices. +In the remaining case we have chains in both blocks with at least +$m^{1/4}$ indices. The Erd\H{o}s--Szekeres monotone-subsequence theorem +\cite[p.~467]{ErdosSzekeres1935} shows that two total orders have a +common agreeing or reversed subsequence of size at least the square +root of their length. To recall the elementary argument, assign to each position the lengths +of the longest increasing and decreasing subsequences ending +there. Two positions have different pairs, so if both lengths +are less than $\sqrt u$ there cannot be $u$ positions. We thus +retain at least a constant times $m^{1/8}$ indices. Agreeing +orders already give the conclusion. + +Suppose the orders are reversed. In block 2 take the segment +from its root to the farthest selected $x_p^2$. The intersection +of each $A_p^2$ with that segment is a closed vertex interval +beginning at $x_p^2$; these left endpoints are distinct. At a +vertex of the segment, each positive-length interval containing +it contains an incident segment edge. Each of the at most two +such edges has at most $J$ prime factors. At most one interval +can consist of that vertex alone. Hence at most $2J+1$ intervals +overlap at a vertex. Greedily coloring closed intervals in order +of their left endpoints uses at most $2J+1$ colors: an existing +color is available only if its preceding right endpoint is +strictly earlier than the new left endpoint. One color retains +at least a $1/(2J+1)$ fraction of pairwise vertex-disjoint intervals. + +Reroot block 2 at the far end of the segment. The nearest point +of $A_p^2$ is now the right endpoint of its interval. Indeed, for +any vertex of $A_p^2$ off the segment, its projection onto the +segment lies in that interval, because the path from $x_p^2$ to +the vertex lies in $A_p^2$. The path from the new root first +reaches $A_p^2$ at the right endpoint. This also proves that +off-segment branches cannot change the nearest point. + +The right endpoints of vertex-disjoint intervals have the same +order as their left endpoints along the original segment. Seen +from its other end, that order reverses. The two block orders +therefore agree after rerooting. The first paragraph applies to +the new root as well and proves the dependency assertion. Each +representative stayed in the same whole-tree component for its +prime, so the two representatives for that prime are still distinct. +\end{proof} + +\begin{lemma}[Few disconnected fixed labels]\label{lem:few-disconnected} +Among the configurations retained by \Cref{lem:activity-blocks}, +those with at least $B^{1-2\rho}$ fixed tree primes having multiple +active components have total absolute contribution +$O(\exp(-B^{1+\eps/2}))$. +\end{lemma} + +\begin{proof} +For every such prime choose two blocks meeting distinct whole-tree +active components. There are $O(B^{2\rho})$ ordered block pairs, +so one pair serves $m\gg B^{1-4\rho}$ primes. Inside each block, +the active set is nonempty and connected. Every edge carrying a +fixed prime has active endpoints after the earlier cuts, and the +converse follows from $p\nmid h$. Apply +\Cref{lem:reroot-constraints} and then specify the common integer +\[ + K=b(c_2)-b(c_1),\qquad |K|=O(\ell H). +\] +Each retained prime gives the test \eqref{eq:two-block-constraint}. +The expression is independent of $p$, since both local paths +avoid it. It involves only earlier retained variables by the +lemma. Its nonvanishing follows from distinctness of the two +representatives and the distinct integer offsets of tree vertices. +The number of tests is +\[ + \gg \frac{B^{(1-4\rho)/8}}{J} + =\frac{B^{1/10}}{O(\log B)}>B^{2\eps} +\] +for large $B$. + +Roots, endpoints, and prime choices can be recorded with the +entropy allowed in \Cref{lem:triangular-saving}. Choose $K$ only +after the rerooting; its $O(\ell H+1)$ possibilities cost +$\exp(O(B))$. Although its original value depends on the labels, +fixing it and dropping that defining equation enlarges a positive +sum. The local path expressions then have the required triangular +dependence. Retain their nonzero tests under this enlargement and +apply \Cref{lem:triangular-saving}. +\end{proof} + +\subsection{Accidental coincidences of singleton labels} + +The preceding argument concerned fixed residue coordinates. A +nonfixed tree prime requires a different test: its residue must +remain free, so we use only numerical coincidences between offsets. + +\begin{lemma}[Few corrupted singleton labels]\label{lem:few-corrupted} +Lines having at least $B^{1-2\rho}$ corrupted nonfixed tree primes +have total absolute contribution +$O(\exp(-B^{1+\eps/2}))$. +\end{lemma} + +\begin{proof} +For every corrupted prime choose a witnessing vertex outside its +edge endpoints, and put $D=\lfloor B^{1-3\rho}\rfloor$. Distance +from a vertex to an edge means its minimum tree distance to the two +endpoints. Suppose first that at least half the selected witnesses have distance at +most $D$ from their prime edge. Use the path from the nearer edge +endpoint to its witness. This path avoids the prime's sole edge, +so its nonzero displacement is independent of that prime and is +divisible by it. The expression contains at most $JD$ other +prime labels. + +On the selected primes draw a directed dependency edge $p\to q$ +if the test for $p$ contains $q$. Its outdegree is at most $JD$. +Every induced subgraph of the underlying undirected graph has +average degree at most $2JD$. Successively choose a vertex of +degree at most $2JD$ and delete its neighbors. The resulting +independent set has size at least a constant times +\[ + \frac{B^{1-2\rho}}{JD+1}\gg\frac{B^\rho}{J}>B^{2\eps}. +\] +Its tests have no selected-variable dependencies. They are +covered by \Cref{lem:triangular-saving}. + +Otherwise at least half the witnesses have distance greater than +$D$. Cover the full tree by $O(\ell/D+1)=O(B^{3\rho})$ connected +traversal blocks of diameter at most $D$, as in the earlier +construction. Assign each prime edge to a block containing that +edge and its witness to a block containing that vertex. One +ordered pair serves $\gg B^{1-8\rho}$ primes. No selected prime +edge has an endpoint in the second block: its corresponding +witness lies there, so that would put it at distance at most $D$. +Consequently paths inside the second block contain no selected +prime label. + +Root the two blocks at $c_1,c_2$. In block 1 let $x_p^1$ be the +nearer endpoint of the prime edge; in block 2 let $x_p^2$ be its +witness. The first local path avoids $p$. If it contains a +selected prime $q$, the nearer endpoint of the $q$-edge is a +strict ancestor of $x_p^1$. Thus the tests +\eqref{eq:two-block-constraint}, after specifying the common root +offset, are triangular in the depths of the first-block +endpoints. The second local paths introduce no selected +variables. Each tested displacement is nonzero because its +witness differs from the chosen edge endpoint. Since +$1-8\rho=3/5>2\eps$, there are more than enough tests for +\Cref{lem:triangular-saving}. All choices of witnesses, blocks, +roots, and paths fit its description bound. +\end{proof} + +\begin{proof}[Proof of \Cref{prop:retained-lines}] +Define $R_{\boldsymbol d}$ by retaining the conditions obtained +in \Cref{lem:activity-blocks,lem:few-disconnected,lem:few-corrupted}. +Every activity and occurrence status of a fixed label is determined +by $n_{\mathcal F}$; fixed forbidden qualifications use only those +coordinates; corruption uses only numerical offsets. Covers and +witnesses can be chosen by fixed finite ordering whenever they +exist. Thus $R_{\boldsymbol d}$ depends on no free center residue. + +The zero exclusions vanish with $K_{\boldsymbol d}\prod_j\1_Y$. +The other excluded classes have the stated absolute contribution +by the three lemmas, yielding \eqref{eq:retained-line-error}. +Their proofs used precisely the majorants of +\Cref{lem:crude-count,lem:triangular-saving}, which allow up to +$2t_0$ attached specifications. Their retained tests depend only +on the numerical line and fixed residues, and are unchanged when +qualification indicators are added. This proves the additional assertion as well. +\end{proof} + +We may therefore impose the retained predicate before estimating +the remaining conditional averages. This preserves the residue +coordinates of singleton center labels. The next section deals +with the remaining dependence of the factors $\1_Y$ on those +coordinates by expanding them into short lists of qualification +conditions. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/trace-count.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/trace-count.tex new file mode 100644 index 0000000000000000000000000000000000000000..7f673dfbb151cc6be3adb5eea1e4db46fdd0c827 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/trace-count.tex @@ -0,0 +1,626 @@ +\section{Summing the trace}\label{sec:trace-count} + +We now prove the high-trace estimate. The input from +\Cref{prop:retained-lines} is a description of the activities of the fixed +primes on the first-visit tree. The input from +\Cref{prop:cylinder-reduction} replaces vertex deletion by short lists of +qualifying primitive specifications. The remaining task is to sum the +resulting conditional averages without losing a constant for every prime +occurrence. An exact identity for weighted subtrees makes this possible. + +We retain the notation $\mathscr T$, $b(v)$ and $r_*$ for the first-visit +tree, its integer offsets and the number of returns. In this section a +subtree is a connected subgraph of $\mathscr T$; its \emph{top} is its +vertex nearest the root. Put +\[ + s(v)=\#\{\text{children of }v\},\qquad + M_0=t_0(LJ+1),\qquad N_{\mathrm{exc}}=B^{1-\rho/2}. +\] +We call a tree edge $u\longrightarrow v$ \emph{good} if +$s(u)=s(v)=1$, $u$ is not the root, and neither endpoint belongs to any +return step. Write $\mathcal G$ for the set of good edges. These +conditions ensure that an uncorrupted prime occurring just on such an +edge has exactly two normalization factors on its lit residue. + +For all but a small exceptional set of good edges, we will obtain the +factor $B^{-1+\eta}$ from a core-prime restriction to the divisor interval, +or a stronger penalty from a core cutoff. Returns receive a cutoff +penalty as well. Signed center-prime averages provide the further saving +needed in the high-trace estimate. The weighted-subtree identity below +will let us sum the prime labels without losing these gains. + +\subsection{Tree geometry and weighted subtrees} + +\begin{lemma}\label{lem:trace-topology} +For a closed line of length $\ell$ with $r_*$ returns, the topology of first +visits and returns, together with all step signs, has at most +$4^\ell(\ell+1)^{r_*}$ possibilities. Its tree satisfies +\begin{equation}\label{eq:trace-bad-edges} + |E(\mathscr T)\setminus\mathcal G|\le 10(r_*+1),\qquad + \ell=|E(\mathscr T)|+r_*. +\end{equation} +Once the labels of the tree edges and this topology are specified, every +return label is determined. +\end{lemma} + +\begin{proof} +At each step record whether its destination is new and record its sign. +At each return record one of at most $\ell+1$ previously visited vertices. +These data give the stated bound. A tree-edge label and its sign specify +the difference of its endpoint offsets; all offsets are consequently +determined, and a return label must be the absolute endpoint difference +divided by $h$. We retain only assignments for which the abstract vertices +have distinct offsets and all these return labels belong to $\mathcal D$. +There is no independent choice of a return label. + +Let $q$ be the number of leaves and $b$ the number of vertices with at +least two children. The adding steps occur in at most $r_*+1$ runs, +each a downward path, so $q\le r_*+1$. Also $b\le q-1$ and +\[ + \sum_{s(v)\ge2}s(v)=q+b-1. +\] +The number of edges incident with a leaf or such a branch vertex is +therefore at most $4q$. There are at most $2r_*$ endpoints of return +steps. An endpoint that has not already been counted is unary and is +incident with at most two tree edges, giving at most $4r_*$ further +edges. The root adds at most one edge unless it was already counted as a +branch vertex. These bounds imply \eqref{eq:trace-bad-edges}. +\end{proof} + +For $a\in\{C,Z\}$ and a subtree $Q$ with at least one edge define +\begin{equation}\label{eq:subtree-weights} + u_a(Q)=A_a^{|E(Q)|}\prod_{v\in V(Q)}\beta_a^{s(v)},\qquad + S_a(\mathscr T)=\sum_{v\in V(\mathscr T)}(1-\beta_a^{s(v)}). +\end{equation} +The degrees in this definition are those of the full tree. + +\begin{lemma}\label{lem:subtree-weight} +For every finite rooted tree and every $A_a>0$ with +$\beta_a=(1+A_a)^{-1}$, +\begin{equation}\label{eq:subtree-identity} + \sum_{Q:\,|E(Q)|\ge1}u_a(Q)=S_a(\mathscr T)\le |V(\mathscr T)|. +\end{equation} +\end{lemma} + +\begin{proof} +Start at a prescribed vertex $v$. At every reached vertex include each +child edge independently with probability $A_a\beta_a$ and exclude it +with probability $\beta_a$. The resulting connected subtree has top +$v$. A particular outcome $Q$ has probability +\[ + (A_a\beta_a)^{|E(Q)|} + \beta_a^{\sum_{w\in V(Q)}s(w)-|E(Q)|}=u_a(Q). +\] +The outcome with no edges has probability $\beta_a^{s(v)}$. +Sum the other probabilities, then sum over the possible tops $v$. +\end{proof} + +The exact sum in \eqref{eq:subtree-identity} will cancel the background +exponential in the primewise estimate below. To retain the additional +center-prime saving, we also use modified weights. These anticipate the +primewise bounds proved in \Cref{lem:trace-integration}: the reduction on +a single good edge comes from a signed average, while the increase on +other single edges permits an absolute bound. +Set $\widetilde u_C=u_C$. In the center +band keep $\widetilde u_Z(Q)=u_Z(Q)$ unless $Q$ consists of one edge; +for a single good edge set $\widetilde u_Z(Q)=\theta/2$, and for a single +nongood edge set $\widetilde u_Z(Q)=u_Z(Q)+\theta$. Since both endpoint +degrees of a good edge are one, its original weight is +$\beta_Z^2=\theta$. Thus +\begin{equation}\label{eq:modified-subtree-sum} + \sum_Q\widetilde u_Z(Q)-S_Z(\mathscr T) + =-\frac\theta2|\mathcal G| + +\theta|E(\mathscr T)\setminus\mathcal G|. +\end{equation} + +\subsection{Recording prime labels} + +Fix a retained line and a list $\mathfrak L$ of fewer than $t_0$ +primitive specifications attached at its vertices, as supplied by +\Cref{prop:cylinder-reduction}. Write $I_{\mathfrak L}$ for their joint +qualification indicator. Recall that a \emph{tree prime} occurs in +a tree-edge label. As before, the fixed label set $\mathcal F$ consists +of the core primes occurring on the line and all primes occurring at +least twice on the line. A fixed prime is active at $v$ when +$p\mid n+b(v)$. Active components include isolated vertices. + +Tag a tree prime if it satisfies at least one of the following conditions: +\begin{enumerate}[label=(\roman*),nosep] + \item it occurs among the primes of $\mathfrak L$; + \item it is fixed and has several active components, or has an unlit + occurrence anywhere on the line; + \item it is nonfixed and corrupted in the sense of + \Cref{prop:retained-lines}. +\end{enumerate} +These decisions depend only on the line, the list and the fixed +coordinates $n_{\mathcal F}$. + +We record tree labels by the following finite data, called \emph{tokens}. +For a tagged fixed prime, record one subtree token for each active +component having an edge, and one single-edge token for each unlit tree +occurrence. The latter is called a ghost token and is assigned weight +$2$. A subtree token of band $a$ and shape $Q$ has weight $u_a(Q)$. +For a tagged nonfixed prime, record its sole tree occurrence by a ghost +token. Tokens belonging to the same tagged prime are grouped together, +and the group is assigned that prime value. + +Every untagged tree prime receives a single token. If it is fixed, its +active set is connected and all its occurrences are lit; the token is +the subtree formed by its tree occurrences. If it is nonfixed, its token +is its single tree edge. An untagged token of band $a$ and shape $Q$ +has weight $\widetilde u_a(Q)$. List prime slots, including the extra +modulus prime of each specification, are recorded along with their +equalities to each other and to tagged groups. An untagged prime occurs +in no such group or slot. + +\begin{lemma}\label{lem:trace-token-record} +For retained data, these records have the following properties. +\begin{enumerate}[label=(\roman*)] + \item They recover every tree-edge label. Together with the topology + and the list data they consequently recover the full line and list. + \item The total number of tagged tokens and list prime slots is at + most $N_{\mathrm{exc}}$, for sufficiently large $B$. + \item An untagged fixed center prime cannot have a single good edge + as its token. +\end{enumerate} +\end{lemma} + +\begin{proof} +If adjacent vertices are active for $p$, their difference $\pm hd$ is +divisible by $p$. Since $p\nmid h$, the intervening label contains $p$. +Conversely, a lit occurrence has both endpoints active. Thus the +nontrivial active components record exactly the lit tree occurrences. +Ghosts record the other occurrences; isolated active vertices create no +label. Multiplying the distinct prime values of the tokens covering an +edge recovers its label, proving (i). + +By \Cref{prop:retained-lines}, the fixed primes with several active +components contribute $O(B^{1-2\rho}B^\rho)=O(B^{1-\rho})$ subtree +tokens. Unlit fixed occurrences contribute $O(t_0)$ ghosts and at most +that many newly tagged connected labels. Corrupted nonfixed primes +contribute $O(B^{1-2\rho})$ ghosts. Lists contribute $O(M_0)$ slots +and at most one new token per newly tagged connected tree prime: +disconnected primes and primes with unlit occurrences have already been +counted. Since +\[ + M_0=O(B^{1-\rho+4\eps}\log B),\qquad + 1-\rho+4\eps<1-\rho/2, +\] +the sum of these bounds is at most $N_{\mathrm{exc}}$ eventually. + +For (iii), such a prime is repeated on the line and all its occurrences +are lit. If its tree token consists of a single edge, its connected +active set contains only the two endpoints of that edge. A further +occurrence cannot be another tree edge, and hence must be a return +touching these endpoints. This contradicts goodness. An active return +endpoint elsewhere would give another active component and would have +tagged the prime. +\end{proof} + +We will use the divisor interval at a good edge by restricting the prime +value of an untagged core token whose top is its origin. First set aside +the tagged core tops: skip every good edge whose origin is the top of a +tagged core subtree token. At most $N_{\mathrm{exc}}$ edges are skipped. +Among the remaining good edges, let $\mathcal U$ be the set of origins +at which no untagged core token has its top. Thus no core subtree token +has top $u\in\mathcal U$, and $s(u)=1$. At these origins we will use +a cutoff penalty instead. The next estimate retains this penalty; +the subsequent sum over prime assignments will use the divisor interval +at the other non-skipped good origins. + +For a fixed numerical line and list, group the retained fixed-coordinate +configurations according to their complete assigned token records. +Within a group the token shapes, prime values and groupings are fixed; +unrecorded isolated activities may still vary. Order components and +tokens by any deterministic rule, so that these groups form a +partition, rather than a choice of representations for each residue +configuration. We next bound the integral of each such group. This is +the point at which the normalization of the graph produces cancellation. + +\subsection{Primewise integration} + +For an assigned record $R$, let $\1_R(n_{\mathcal F})$ denote its group +indicator. Its contribution to the conditional expression is +\[ + \mathcal I(R)= + \E_{\mathcal F}\left[ + \1_R\left| + \E_{\mathcal S\setminus\mathcal F} + K_{\boldsymbol d}(n)I_{\mathfrak L}(n) + \right|\right]. +\] +Empty or incompatible groups have contribution zero. Products over +tagged tokens below use the weights just defined, including weight $2$ +for ghosts. + +\begin{lemma}\label{lem:trace-integration} +Uniformly over retained lines, lists and assigned records, +\begin{align} + \mathcal I(R) + &\le + \exp\left\{ + o(1)-\kappa(v_C-T\sqrt{v_C})(r_*+|\mathcal U|)\right\} + \notag\\ + &\quad\times\exp\left\{ + -v_CS_C(\mathscr T)-v_ZS_Z(\mathscr T) + +v_C\beta_C(e^\kappa-1)|\mathcal U|\right\} + \notag\\ + &\quad\times\prod_{\substack{p\text{ distinct}\\ + p\text{ in tree or list}}}\frac1p + \prod_{\text{tagged tokens}}\mathrm{weight} + \prod_{\substack{\text{untagged tokens}\\(a,Q)}}\widetilde u_a(Q). + \label{eq:trace-integration} +\end{align} +Primes occurring only on returns need not appear in the product over +tree or list primes. The $o(1)$ tends to zero as $B$ tends to infinity, +uniformly in the line, list and record, with $h,\tau,C_0,T$ and the +constants in \eqref{eq:graph-constants} fixed. +\end{lemma} + +\begin{proof} +\emph{Order of integration.} +The set $\mathcal F$ does not contain every core prime. Put +$\mathcal C'=\mathcal C\setminus\mathcal F$ and +$\mathcal Z'=\mathcal Z\setminus\mathcal F$. At fixed +$n_{\mathcal F}$ use exactly the inequality +\begin{equation}\label{eq:trace-fubini} + \left|\E_{\mathcal C'}\E_{\mathcal Z'} + K_{\boldsymbol d}I_{\mathfrak L}\right| + \le \E_{\mathcal C'} + \left|\E_{\mathcal Z'} + K_{\boldsymbol d}I_{\mathfrak L}\right|. +\end{equation} +In the inner average hold \emph{all} core coordinates as parameters. +The cutoffs can depend on the background core coordinates in +$\mathcal C'$, but never on $\mathcal Z'$. Apart from these cutoffs, +the kernel and each compatible list cylinder separate by prime. +The inner center-prime averages therefore factor, with their signs +intact. We will subsequently use positive majorants in the other +coordinates. No enlargement of the fixed set in the cylinder +construction is involved. + +\emph{Charging the cutoffs.} +Use the origin cutoff at each return step. For each $u\in\mathcal U$, +use the endpoint cutoff at $u$ of its incoming tree edge, which exists +because a good origin is not the root. These are distinct cutoff +occurrences: vertices of $\mathcal U$ touch no return, and an incoming +tree edge has a unique child. A charged cutoff with label $d$ is at +most +\[ + \exp\{-\kappa(v_C-T\sqrt{v_C})\} + \exp\left\{\kappa + \sum_{\substack{p\in\mathcal C\\p\nmid d}} + \1_{p\mid n+b(u)}\right\}. +\] +This inequality holds also off its support. The other cutoff factors +are at most one. The constant parts give the first negative term in +the exponent of \eqref{eq:trace-integration}. + +For a return, allocate the normalization at its origin to that return. +For a core prime its multiplier is $A\beta_C$ when the prime is in the +label and lit, zero when it is in the label and unlit, and at most +$\max(1,\beta_Ce^\kappa)=1$ otherwise. Both +$A\beta_C<1$ and $\beta_Ce^\kappa<1$ follow from $A=e^{2\kappa}$. +Center-prime return multipliers also have absolute value at most one. +These bounds will be used except when integrating a good singleton, +where we retain the return factors until its signed average has been +taken. + +\emph{Fixed tree primes.} +After removing the bounded return factors, the core tree factors for +an active component $Q$ containing an edge are +\[ + A^{|E(Q)|}\prod_{v\in V(Q)}\beta_C^{s(v)}=u_C(Q). +\] +At a charged vertex $u\in\mathcal U$, activity in such a component +either continues through the incoming label, in which case that prime +is excluded from this charge, or begins a subtree token at $u$, which +is excluded by the definition of $\mathcal U$. Thus an additional +$e^\kappa$ can occur only at an isolated active vertex there. Its +normalization is $\beta_C^{s(u)}e^\kappa=\beta_Ce^\kappa\le1$. +Other isolated activities also contribute at most one. The remaining +bound is the product of the core subtree weights. + +A core tree prime has a lit occurrence. Choose one deterministically +from its tokens and retain its activity indicator. This is one +specified residue modulo $p$ and supplies the factor $1/p$ upon +integration. For a fixed center tree prime, the same argument gives +the product of its active-subtree weights; lit factors $1-\theta/p$ +are at most one, and ghost occurrences are bounded absolutely. If +there is a lit tree occurrence, retain one such residue indicator. +If there is none, at least one unlit tree occurrence supplies +$\theta/p\le1/p$ directly, without a residue restriction. Ghost +weight $2$ bounds all the remaining ghost factors. These constructions +are positive primewise majorants on the assigned group. + +\emph{Nonfixed tree primes.} +A tagged nonfixed tree prime has one occurrence. Its absolute +integral is at most $2/p$: the lit residue contributes at most $1/p$ +and the other residues at most $\theta/p$. If the list fixes its +residue, the bound only improves. An untagged nonfixed prime on a +nongood edge $Q$ contributes at most +\[ + \frac{u_Z(Q)}p+\frac\theta p + =\frac{\widetilde u_Z(Q)}p. +\] +Indeed its lit residue includes the normalizations of both endpoints, +and any additional activities can only reduce their product. + +Now let $p$ be an untagged nonfixed prime on a good edge $Q=(u,v)$. +It occurs nowhere else on the line and in no list. On its lit residue, +noncorruption says that no other tree vertex is active. Each endpoint +has exactly one tree departure and no return incidence. Its entire +prime multiplier, including return factors, is therefore exactly +\[ + (1-\theta/p)\beta_Z^2=\theta(1-\theta/p). +\] +Let $a_0$ be its lit residue, and for $a\in\Z/p\Z$ define +\[ + \nu_p(a)=\#\{0\le i<\ell:a+b_i\equiv0\pmod p\}. +\] +This counts departures with multiplicity, including return departures. +We have $\nu_p(a_0)=2$. On every other residue the entire prime +multiplier is exactly $-\theta\beta_Z^{\nu_p(a)}/p$. Consequently its +signed average is exactly +\begin{align*} + &\frac\theta p(1-\theta/p) + -\frac\theta{p^2}\sum_{a\ne a_0}\beta_Z^{\nu_p(a)}\\ + &\qquad=\frac\theta{p^2}\left( + 1-\theta+ + \sum_{\substack{a\ne a_0\\\nu_p(a)>0}} + (1-\beta_Z^{\nu_p(a)})\right). +\end{align*} +There are at most $r_p-1$ terms in the last sum, where +$r_p\le\ell+1$ is the number of residue classes represented by tree +vertices. The displayed quantity is nonnegative and at most +\[ + \frac{\theta r_p}{p^2} + \le\frac{\theta(\ell+1)}{p^2} + \le\frac{\theta}{2p} + =\frac{\widetilde u_Z(Q)}p +\] +for sufficiently large $B$. Thus $\theta=\beta_Z^2$ cancels the +terms of order $1/p$ before any absolute value is taken. +Removing return factors by an absolute estimate before taking this +average would not justify the calculation on the no-activity residues; +here they have been retained throughout. + +\emph{List primes and background primes.} +For a list prime absent from the tree, qualification supplies one +specified residue. All other factors, including charged factors, +are bounded by one: any activity in the tree is isolated, and at a +vertex of $\mathcal U$ its factor is again $\beta_Ce^\kappa\le1$. +This yields $1/p$ for that prime. + +For any remaining prime, discard return factors by the absolute bounds +already proved. The positive tree multiplier is +\[ + \prod_{v\text{ active}}\beta_a^{s(v)} + \exp\{\kappa\1_{a=C}\#(\mathcal U\cap\{v\text{ active}\})\} + \le1. +\] +If the tree offsets are pairwise distinct modulo $p$, its integral is +exactly +\begin{equation}\label{eq:trace-background} + 1-\frac{S_a(\mathscr T) + -\1_{a=C}\beta_C(e^\kappa-1)|\mathcal U|}{p}. +\end{equation} +For offset collisions we use the bound one. A prime occurring only on +a return divides the nonzero difference between that return's two +distinct vertices, so is automatically one of these collision primes. +It entails no independently chosen prime slot. + +The omitted background primes are tree or list primes, or divide one +of the $O(\ell^2)$ nonzero differences of size $O(\ell H)$. +There are $O(B\log B)$ tree and list slots. Since every prime is +at least $P_0$ and $\log(\ell H+2)=O(B+\log B)$, their reciprocal +mass, multiplied by $\ell+1$, is $o(1)$, uniformly in the valid +assignment. For example each nonzero difference has at most +$\log(O(\ell H)+2)/\log P_0$ prime divisors from $\mathcal S$. +Using $1-x\le e^{-x}$ in \eqref{eq:trace-background} and restoring +the omitted harmonic masses gives the remaining exponent in +\eqref{eq:trace-integration}, with an $o(1)$ error. The numerator in +\eqref{eq:trace-background} is nonnegative, since at every charged +vertex $\beta_C e^\kappa\le1$. + +Finally, the grouping does not assert independence of conditioned +coordinates. After \eqref{eq:trace-fubini} and the signed free-center +integrations, use the positive majorants above on the group. Whenever +a used-prime probability $1/p$ is claimed, its chosen activity or list +residue indicator is retained. Drop all other group restrictions and +integrate the resulting product over the remaining coordinates. This +is an enlargement of a nonnegative integral and hence is valid despite +any dependencies within the original group. It yields one bound per +assigned record, not one bound for each individual fixed-coordinate +configuration. Combining the primewise bounds proves the lemma. +\end{proof} + +\subsection{Prime assignments, factorials and scale constraints} + +We have bounded a record by a product of token weights and reciprocal +prime values. We now sum these products. The exact subtree identity +will cancel their background exponential. We must retain the factorials +for equal token types in order for this cancellation to remain exact. + +For a fixed topology, record the tagged tokens in an ordered list and +record all list metadata: lengths, signs, attachments, suffix choices, +band choices, prime-slot sizes and equalities. Apart from the token +shapes and prime values themselves, this costs +\begin{equation}\label{eq:trace-exceptional-entropy} + \exp(O(N_{\mathrm{exc}}\log B)). +\end{equation} +Indeed there are at most $N_{\mathrm{exc}}$ objects and slots; each +positional index has at most a fixed power of $B$ choices, and their +equality pattern has at most +$N_{\mathrm{exc}}^{N_{\mathrm{exc}}}$ possibilities. Multiple tagged +tokens may share a prime, and each resulting group has one prime +variable. None of these variables is shared with an untagged token. + +For the untagged tokens specify multiplicities $j_{a,Q}\ge0$ for their +types $(a,Q)$. The actual prime values of one type form an unordered +set of $j_{a,Q}$ distinct primes. Replace it temporarily by named +slots $1,\ldots,j_{a,Q}$ and divide the sum by +\begin{equation}\label{eq:trace-factorial} + \prod_{a,Q}j_{a,Q}!. +\end{equation} +Every valid prime assignment has exactly this number of named +realizations. Permuting values within one type preserves every tree +label, hence all return labels, the numerical line and the list. +In particular every bin condition is preserved. After obtaining this +identity, we may enlarge the nonnegative sum by dropping distinctness +and other compatibility conditions. Such an enlargement is a formal +sum of the established numerical majorants; it is not an application +of \Cref{lem:trace-integration} to new lines with coincident prime slots. + +\begin{lemma}\label{lem:trace-pivots} +Fix the topology, token shapes, tagged/list groupings and untagged +multiplicities. For every non-skipped good origin $u\notin\mathcal U$ +choose deterministically one named core slot whose token has top $u$. +When summing prime assignments, the bin conditions on these edges give +an extra factor $O(B^{-1+\eta})$ for each chosen slot, while retaining +the unrestricted harmonic factor $v_a$ for every prime variable. +Together with the charge exponent and its correction in +\eqref{eq:trace-integration}, these savings are bounded by +\begin{equation}\label{eq:trace-pivot-saving} + B^{-100r_*}\exp(O(\ell)) + B^{-(1-\eta)(|\mathcal G|-N_{\mathrm{exc}})}. +\end{equation} +\end{lemma} + +\begin{proof} +Choose one eligible type at each origin using any fixed order, and +then choose its first named slot. Distinct origins choose different +slots. Because $s(u)=1$, a nontrivial subtree with top $u$ contains +the edge leaving $u$. + +Fix all other prime values and order the selected slots ancestor-first +by their tops. If the token of a selected slot occurs on the edge +leaving $u$, its top is $u$ or an ancestor of $u$. Thus that edge's +product contains its selected prime and only earlier selected primes. +The bin requirement has the form +\[ + H\eps/10$ and +$\theta(\eps-\eta)<\eps$. + +It remains to check the numerical margin including the topology count. +Write $g=|\mathcal G|$, $b=|E(\mathscr T)\setminus\mathcal G|$ and +$r=r_*$. Then $\ell=g+b+r$ and $b\le10(r+1)$. +Set $\delta=\eps/10-\eta=9\eta$. Combining the preceding bounds, +\eqref{eq:trace-pivot-saving} and the count in +\Cref{lem:trace-topology}, the negative base-$B$ exponent is at least +\[ + (1+\delta)g+(99-o(1))r-\eps b-o(\ell). +\] +Here $4^\ell$ and the fixed constants per pivot contribute +$O(\ell/\log B)=o(\ell)$, and +$(\ell+1)^r=B^{(1+o(1))r}$. Substitution of $g=\ell-b-r$ and +$b\le10(r+1)$ gives the lower bound +\[ + (1+\delta)\ell + +(88-11\delta-10\eps-o(1))r + -10(1+\delta+\eps)-o(\ell). +\] +The coefficient of $r$ is positive, and +$\delta-2c_*=7\eta>0$. The slack absorbs the fixed last term, the +displayed $o(\ell)$ terms, and the sum over at most $\ell+1$ possible +values of $r$. This proves the claimed bound. +\end{proof} + +\begin{proof}[Completion of the proof of \Cref{thm:high-trace}] +The density assertion is \Cref{lem:bad-density}. Apply +\Cref{prop:retained-lines} to split the trace sum into retained and +discarded configurations. For the retained part, the triangle +inequality bounds the absolute full expectation by +$\E_{\mathcal F}R_{\boldsymbol d} + |\E_{\mathcal S\setminus\mathcal F} + K_{\boldsymbol d}\prod_j\1_Y(n+b_j)|$. +Now apply \Cref{prop:cylinder-reduction}. Their discarded +contributions are $O(\exp(-B^{1+\eps/2}))$. The retained conditional +main term is bounded by \Cref{lem:trace-summation}. Since +$\ell=2\lfloor B\rfloor$, +\[ + B^{-(1+2c_*)\ell}+O(\exp(-B^{1+\eps/2})) + \le B^{-(1+c_*)\ell} +\] +for sufficiently large $B$. This proves the trace assertion and +completes the theorem. +\end{proof} diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/01-setup.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/01-setup.tex new file mode 100644 index 0000000000000000000000000000000000000000..0a4b3fab1f06706cc4a10db00d0317bdef84d03e --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/01-setup.tex @@ -0,0 +1,199 @@ +\section{Prime supplies and the quantitative target}\label{q:setup} + +Fix integers \(h,l\ge1\) and \(b\in\Z\). Throughout Part~I, constants +in estimates may depend on these fixed data unless declared absolute. +We will prove the following intermediate statement. + +\begin{proposition}[Quantitative correlations in every progression] +\label{q:progression} +There is one absolute \(c>0\) such that, for every fixed \(h,l\ge1\) +and \(b\in\Z\), +\[ + F(X):=\frac1X\sum_{1\le n\le X} + \1_{n\equiv b\pmod l}\lambda(n)\lambda(n+h) + \ll_{h,l}(\log X)^{-c}\qquad(X\ge3). +\] +The estimate holds for every real \(X\), with no condition on +\(\gcd(b,l)\). +\end{proposition} + +There are only \(l\) residue classes, so the implied constant can be +chosen independently of the representative \(b\). +It suffices to work above a threshold depending on the fixed data: +bounded \(X\) can be absorbed into that constant. +Our first task is to find many divisors in short multiplicative intervals, +retaining a fixed positive fraction of their total reciprocal weight. +The construction separates primes used for centering from primes used +to supply those divisors. + +The proof uses two absolute constants. First choose \(A\ge1000\) +large enough for the finite-law comparison in +\Cref{q:finite-law}. Later choose \(W\ge10\) after the absolute +graph constants have been established. Set +\begin{equation}\label{q:parameters} + \begin{gathered} + L=(\log X)^{1/A},\qquad \delta=\frac1{200},\qquad + J=\left\lfloor\frac{\delta\log L}{6W}\right\rfloor,\\ + Y_i=L^{1-\delta}e^{6Wi}\quad(0\le i\le J),\qquad + H_0=e^{L^{1-\delta}},\qquad + \eta=e^{-J},\qquad K=e^{4J}. + \end{gathered} +\end{equation} +We may assume \(J\ge1\). Since \(6WJ\le\delta\log L\), one has +\(Y_J\le L\). + +\subsection{Centered primes and padding primes} + +For \(1\le i\le J\), let +\[ + P_i=\{p:p\equiv1\pmod5,\quad Y_{i-1}\le\log p0\) is absolute; the threshold and the implied constant +may depend on \(h,l\). In the second selection the possible prime +\(5\) contributes only a bounded term. + +Partial summation on the logarithmic interval +\([Y_{i-1},Y_i)\) gives +\[ + V_i=\frac14\log\frac{Y_i}{Y_{i-1}}+o(1) + =\frac32W+o(1), +\] +uniformly for \(1\le i\le J\). Indeed all these intervals begin at +least at \(L^{1-\delta}\), so their accumulated endpoint and +integrated error in \eqref{q:pnt} tends to zero uniformly. +Consequently, for sufficiently large \(X\), +\begin{equation}\label{q:prime-masses} + W\le V_i\le2W,\qquad + \log d\le\sum_{i=1}^JY_i + \le \frac{L}{1-e^{-6W}}<2L\quad(d\in\mathcal D). +\end{equation} + +Let \(\mathcal Q\) be the squarefree products of primes in \(Q\), +including \(1\), and put +\begin{equation}\label{q:padding-weights} + u(q)=4^{\omega(q)},\qquad + S=\sum_{q\in\mathcal Q}\frac{u(q)}q + =\prod_{p\in Q}(1+4/p),\qquad + w(n)=\sum_{\substack{q\in\mathcal Q\\q\mid n}}u(q) + =5^{\omega_Q(n)}. +\end{equation} +Here \(\omega\) counts distinct prime factors. For any finite prime +set \(\mathcal R\), write +\(\omega_{\mathcal R}(n)=\#\{p\in\mathcal R:p\mid n\}\). The factor \(q\) will be +called the padding factor. All sums over these finite prime sets +are finite, however large their cardinalities. + +\subsection{Short multiplicative intervals} + +For every integer \(j\ge0\) with \(j\eta\le100L\), define +\begin{equation}\label{q:bins} + \mathcal S_j= + \{(d,q)\in\mathcal D\times\mathcal Q: + \omega(q)\le100\log L,\quad + j\eta\le\log(dq)<(j+1)\eta\}, + \qquad T_j=Xe^{j\eta}. +\end{equation} +There are \(O(L/\eta)\) such intervals. Define their total harmonic +mass by +\[ + S_0=\sum_j\sum_{(d,q)\in\mathcal S_j}\frac{u(q)}{dq}. +\] +\begin{lemma}[Mass retained in the intervals]\label{q:bin-mass} +For sufficiently large \(X\), +\[ + \frac12SV_*\le S_0\le SV_*. +\] +\end{lemma} +\begin{proof} +Under the law \(u(q)/(qS)\), primes in \(Q\) are independently +included with probabilities \(4/(p+4)\). +Equation \eqref{q:pnt} and partial summation give +\[ + \mathbb E_q\log q\le4L,\qquad + \mathbb E_q\omega(q)\le4\log L +\] +for large \(L\). Markov's inequality therefore excludes +\(\log q>98L\) with probability at most \(4/98\), and excludes +\(\omega(q)>100\log L\) with probability at most \(4/100\). +Every remaining \(q\), together with every \(d\in\mathcal D\), +satisfies \(\log(dq)\le100L\) by \eqref{q:prime-masses}, and so lies +in one of the intervals \eqref{q:bins}. Their probability exceeds +\(1/2\). Summing the independent harmonic weights of \(d\), whose +total is \(V_*\), proves the lower bound. Dropping both restrictions +proves the upper bound. +\end{proof} + +\subsection{Centered divisor sums} + +The bins let us compare dilated correlations at essentially the same +cutoff. For each bin define +\begin{equation}\label{q:centered-average} +\begin{split} + C_j={}&\frac1{T_j}\sum_{1\le n\le T_j} + \sum_{(d,q)\in\mathcal S_j} + u(q)\1_{q\mid n}\1_{n\equiv bqd\pmod l}\\ + &\hspace{13mm}\cdot + \prod_{p\mid d}\left(\1_{p\mid n}-\frac1p\right) + \lambda(n)\lambda(n+hqd). +\end{split} +\end{equation} +For a fixed pair $(d,q)$, the term retaining every divisibility indicator +in this expansion forces $qd\mid n$. The substitution $n=qdm$ turns it into +\[ + \frac{u(q)}{qd}F\left(\frac{T_j}{qd}\right), + \qquad Xe^{-\eta}<\frac{T_j}{qd}\le X. +\] +Here $qd$ is coprime to $l$, and complete multiplicativity cancels +$\lambda(qd)^2=1$. Since the summands defining $F$ are bounded by one, +$F(T_j/(qd))=F(X)+O(\eta+X^{-1})$. Summing the weights and using +$S_0\le SV_*$, the full-divisibility contribution is therefore +\begin{equation}\label{q:full-divisibility} + S_0F(X)+O(SV_*(\eta+X^{-1})). +\end{equation} +\Cref{q:center} will bound the remaining terms. We will then estimate +the centered sums themselves by the graph argument, after removing +summands of small total absolute weight. + +\subsection{The independent residue space} + +The factor $\1_{p\mid n}-1/p$ has mean zero for a uniform residue +modulo $p$. To use this cancellation simultaneously at many primes, +we introduce the auxiliary product law with one independent uniform +residue modulo each $p\in P\cup Q$, and one independent uniform residue +modulo $l$. The latter is a single coordinate, even if $l$ is not +squarefree; for $l=1$ it is deterministic. The selected primes are all +coprime to $l$. Write $\E$ and $\Pbb$ for expectation and probability +under this law. A translated site $n+a$ uses these same coordinates +shifted by $a$, so divisibility, progression tests, and $w(n+a)$ are +defined without a common integer $n$. + +At each site $\E w=S$. The prime sets grow with $X$, so this model +must be compared with finite integer averages before it can be used to +estimate deletion costs and graph moments. The next section supplies +that comparison for the residue tests and truncated weights used below. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/02-finite-law.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/02-finite-law.tex new file mode 100644 index 0000000000000000000000000000000000000000..d85bcb7d6451c6220474513dedd156c24985a7c0 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/02-finite-law.tex @@ -0,0 +1,239 @@ +\section{Comparing finite integer averages with independent residues} +\label{q:finite-law-section} + +The auxiliary law makes the prime residue coordinates independent, +but the prime sets grow with $X$. Averaging over their full joint +period would therefore give no useful error at the required scale. +We instead compare the particular residue tests used in the proof. +The comparison rests on bounded independence for Boolean circuits; +we give the reduction from integer residues, including the small +correction needed to make all sufficiently small bit marginals exactly uniform. + +For a consecutive interval $I$ of integers, write $\E_I$ for the +average obtained by choosing the origin uniformly in $I$. +The symbol $\E$ continues to denote the independent residue law, +and $R_s$ denotes its coordinate modulo $s$. +A translated residue test at offset $a$ means +$R_s\equiv-a\pmod s$, using the same coordinate $R_s$ at every +translate. Different sites do not receive independent new coordinates. +A Boolean circuit here uses unbounded-fan-in AND and OR gates and +negations. Its depth counts AND and OR levels; for its size we count +both gates and input occurrences. An event is represented by a circuit +whose input bits are residue equalities. + +\begin{lemma}[Finite residue comparison]\label{q:finite-law} +There is an absolute constant $C_1$ with the following property. +Choose $A>C_1+20$. Let $l\ge1$ be fixed, and let $\mathcal R$ be any +set of primes at most $e^L$, none dividing $l$. On the coordinates +modulo $p\in\mathcal R$ and modulo $l$, let $\mu$ be the product of +the uniform laws; the modulus-one coordinate can be omitted. +Suppose that an event $\mathcal E$ is represented by a circuit of +depth at most $20$ and size at most $\exp(L^6)$ in these residue +equalities. For every consecutive integer interval $I$ with +$|I|\ge\exp(L^A/2)$, +\begin{equation}\label{q:finite-law-bound} + \bigl|\E_I\1_{\mathcal E}-\mu(\mathcal E)\bigr| + \le e^{-L^{10}} +\end{equation} +for sufficiently large $L$ depending on $l$. +The bound is uniform in the position of $I$ and in all offsets used +by the tests. The modulus $l$ need not be squarefree. +\end{lemma} + +\begin{proof} +We use Braverman's bounded-independence theorem +\cite[Corollary~2]{Braverman10}: for a fixed depth $d_0$, every +$t$-wise uniform distribution on Boolean bits fools circuits of +size $m$ to error $\varepsilon$ when +$t\ge(\log(m/\varepsilon))^{C(d_0)}$, with a suitable constant +depending only on the depth. Being $t$-wise uniform means that every +set of at most $t$ bits has exactly the uniform joint distribution. +We first encode the residue tests by bits, then correct the small +failure of exact independence in the integer average. + +Under the integer-residue law, $R_s$ denotes the residue of the +uniformly chosen origin modulo $s$; under $\mu$ it is the independent +uniform coordinate. In both cases represent $R_s$ by an integer in +$\{0,\ldots,s-1\}$. Put $B=\lceil L^{15}\rceil$. Independently at each modulus $s$, +adjoin a uniform random variable $U_s\in[0,1)$, independent also of +the origin, and take the first $B$ binary digits of +\[ + Z_s=\frac{R_s+U_s}{s}. +\] +Under $\mu$, all these bits are independent and unbiased. Decode +$R_s$ from its bits whenever their dyadic interval lies inside one +interval $[r/s,(r+1)/s)$; on ambiguous intervals make any fixed choice. +Conditional on any value of $R_s$, the probability of a decoding +error is at most $2s2^{-B}$, since only the two boundary cells can +cause an error. This bound holds under either origin law. Therefore +the probability of any error is at most +\begin{equation}\label{q:decoding-error} + 2^{1-B}\left(l+\sum_{p\in\mathcal R}p\right) + \le e^{3L}2^{-B} +\end{equation} +for large $L$. Correct decoding of one coordinate makes all its +translated equality tests correct simultaneously. + +A decoded equality is a Boolean function of $B$ bits. Its +truth-table disjunctive normal form has at most $2^B$ conjunctions, +each containing at most $B$ literals. Substituting these forms into +the original circuit gives depth at most $22$ and size at most +$\exp(L^{17})$ for large $L$. + +Fix any set of at most $t$ encoded bits. It involves at most $t$ +residue coordinates, whose moduli are pairwise coprime. Their +product $D$ satisfies $D\le l e^{tL}$. If $N=|I|$, each residue +modulo $D$ occurs either $\lfloor N/D\rfloor$ or $\lceil N/D\rceil$ +times in $I$. Thus its distribution has total variation distance +at most $D/N$ from uniform. Here total variation is one half of +the $\ell^1$ distance between probability masses. Adding independent +jitters and projecting to the selected bits cannot increase this +distance. Their marginal consequently differs from uniform by at +most +\begin{equation}\label{q:crt-bits} + \Delta\le \exp(2tL-L^A/2) +\end{equation} +for sufficiently large $L$. The only use of $l$ is as one modulus +coprime to the selected primes, so repeated prime factors in $l$ +cause no change. + +There are at most $n_{\mathrm{bit}}\le e^{2L}$ bits in all. +Let $f$ be the density of their integer-origin law relative to +uniform measure on this finite Boolean cube. For a set $S$ of bit +indices, let $\chi_S$ be the product of the signs $(-1)^{x_i}$ over +$i\in S$, and write $\widehat f(S)=\E_{\mathrm{unif}}f\chi_S$. +By \eqref{q:crt-bits}, +\[ + |\widehat f(S)|\le2\Delta + \qquad(1\le |S|\le t). +\] +Set $t=\lceil L^{C_1}\rceil$. The total absolute value of these +coefficients is bounded by +\begin{equation}\label{q:small-fourier-mass} + q:=\sum_{1\le|S|\le t}|\widehat f(S)| + \le 2\Delta(t+1)n_{\mathrm{bit}}^t + \le e^{-L^{11}}, +\end{equation} +provided $A>C_1+20$ and $L$ is large. + +The estimate \eqref{q:small-fourier-mass} allows a correction that +preserves nonnegativity, even where $f$ vanishes. This is a direct +Fourier form of the almost-to-exact independence argument of Alon, +Goldreich, and Mansour \cite[Theorem~2.1 and Section~3.1]{AGM03}. +We give the calculation with the error budget needed here. +Put $a=e^{-L^{11}}$ +and define +\[ + H=\sum_{1\le|S|\le t}\widehat f(S)\chi_S, + \qquad g=\frac{f-H+a}{1+a}. +\] +Since $|H|\le q\le a$, the function $g$ is nonnegative. Its integral +is one, and every nonconstant Fourier coefficient of order at most +$t$ vanishes. Fourier inversion on each marginal shows that the +law with density $g$ is exactly $t$-wise uniform. Moreover, +\begin{equation}\label{q:fourier-correction-tv} + d_{\mathrm{TV}}(f,g) + \le\frac{q+a\E_{\mathrm{unif}}|1-f|}{2(1+a)} + \le\frac{3a}{2}. +\end{equation} + +Choose the absolute constant $C_1$ large enough for Braverman's +theorem at depth $22$, size $e^{L^{17}}$, and error $e^{-L^{11}}$. +Then choose $A>C_1+20$. Applying Braverman's theorem to $g$ gives +error at most $a=e^{-L^{11}}$. Under either the integer-residue or +product-residue law, the original and encoded tests can disagree only +on the decoding event. The total comparison error is therefore at most +\[ + 2e^{3L}2^{-B}+d_{\mathrm{TV}}(f,g)+a + \le 2e^{3L}2^{-B}+\tfrac52e^{-L^{11}} + \le e^{-L^{10}} +\] +for sufficiently large $L$. This proves \eqref{q:finite-law-bound}. +All choices are absolute; the threshold may depend on the fixed modulus +$l$. +\end{proof} + +\begin{corollary}[Scalar expansions]\label{q:finite-expansion} +Under the hypotheses of \Cref{q:finite-law}, write $\E$ for +expectation under $\mu$. Suppose +$\Phi=\sum_\alpha c_\alpha\1_{\mathcal E_\alpha}$, where every event +has the stated circuit bounds and +$\sum_\alpha|c_\alpha|\le\exp(C L^5)$ for a fixed constant $C$. +Then, for sufficiently large $L$, +\begin{equation}\label{q:scalar-comparison} + |\E_I\Phi-\E\Phi|\le e^{-L^9}. +\end{equation} +This also applies to the sum of errors for a family of such expansions +when the same bound holds for their total absolute coefficient sum. +\end{corollary} + +\begin{proof} +Apply \eqref{q:finite-law-bound} term by term and sum +$|c_\alpha|e^{-L^{10}}$. This argument preserves the signs of the +coefficients until after taking each expectation. +\end{proof} + +\subsection{Low-degree residue states and truncated weights} + +To apply \Cref{q:finite-expansion} to weights, we first specify the +set of $Q$ primes dividing a site. When this set has bounded size, +its possible values give a manageable expansion into residue events. + +For an offset $a$, its \emph{active $Q$-set} is +\[ + Q_a=\{p\in Q:R_p+a\equiv0\pmod p\}. +\] +Write $m_Q=\lfloor400\log L\rfloor$. There are at most +\begin{equation}\label{q:q-state-count} + \sum_{u=0}^{m_Q}\binom{|Q|}{u} + \le(m_Q+1)(|Q|+1)^{m_Q} + =\exp(O(L\log L)) +\end{equation} +possible active sets with $|Q_a|\le m_Q$. Specifying one exactly +is a conjunction testing every prime in $Q$ as present or absent. +On this event $w(n+a)=5^{|Q_a|}$, with $n$ denoting the random +origin as in the auxiliary model. Every prescribed function or predicate +of this active set is then a scalar or a fixed decision. In +particular, any sum over squarefree divisors of the product of the +active primes can be evaluated at this stage; there is no restriction +on how many such divisors are used in defining that scalar. + +The event $|Q_a|>m_Q$ is also a small circuit: take the disjunction +over $(m_Q+1)$-element subsets of $Q$ of the conjunction asserting +that all their primes divide the site. Define $\omega_P(n+a)=\#\{p\in P:R_p+a\equiv0\pmod p\}$. +The same construction applies +to the condition $\omega_P(n+a)>6WJ$, since +$6WJ\le\delta\log L$. These circuits have size +$\exp(O(L\log L))$ and depth two. + +The restriction $|Q_a|\le m_Q$ must precede an arbitrary predicate +of the active set. It reduces that predicate to a decision on each of +the states counted in \eqref{q:q-state-count}. We will apply this +representation to the padding cutoffs and weight denominators when +they arise. Its simplest consequence is the following weight bound. + +\begin{corollary}[Truncated weight average]\label{q:truncated-weight} +For every offset $a$ and every interval $I$ allowed in +\Cref{q:finite-law}, +\begin{equation}\label{q:truncated-weight-average} + \E_I\bigl[w(n+a)\1_{\omega_Q(n+a)\le400\log L}\bigr] + \le S+e^{-L^9}\le2S +\end{equation} +for sufficiently large $L$. The analogous sum over $M$ fixed offsets +is at most $2MS$. +\end{corollary} + +\begin{proof} +Expand by exact active sets of size at most $m_Q$, with scalar +coefficient $5^{|Q_a|}$. The total coefficient sum is +$\exp(O(L\log L))$, so \Cref{q:finite-expansion} applies. +The product-law expectation is at most $\E w=S$, by independence +of the prime coordinates. Finally $S\ge1$. The bound is uniform +in the offset, and summing it proves the last assertion. +\end{proof} + +The comparison has now supplied the required passage to long integer +intervals for bounded residue tests and their explicit scalar +expansions. Moments of the unrestricted weight will be used only +inside the auxiliary product law; the integer averages use the +truncated weight in \Cref{q:truncated-weight}. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/03-analytic.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/03-analytic.tex new file mode 100644 index 0000000000000000000000000000000000000000..9699d3c3b43f56c3a52ba87d44986e04476d3a28 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/03-analytic.tex @@ -0,0 +1,567 @@ +\section{Centering and deleting a small edge mass}\label{q:analytic} + +We first compare the progression correlation with a sum whose prime +factors have mean zero in the independent residue model. The short +Fourier estimate controls the terms introduced by this centering. +We then show that restrictions on padding mass and prime degrees, together +with any suitably rare residue event, remove only a small total mass. +These two estimates prepare the correlation for the graph argument. + +\subsection{Centering the prime divisibility conditions} + +We first bound the terms other than full divisibility in the expansion +of $C_j$ from \eqref{q:centered-average}. Each such term leaves a +nonempty product of primes to be averaged as a shift. Our target is +an error that remains small after summing over all $O(L/\eta)$ bins. + +\begin{lemma}[Centering estimate]\label{q:center} +With the parameters and prime supplies fixed above, for every sufficiently +large real $X$ one has +\begin{equation}\label{q:center-estimate} + \left|\sum_j C_j-S_0F(X)\right| + \ll_{h,l}SV_*\left( + \eta+X^{-1}+\eta^{-1}2^JL^{-1/20}\right). +\end{equation} +\end{lemma} + +We prove this estimate after two Fourier bounds. The first concerns +rough integers; the second combines it with the short exponential-sum +theorem of Matom\"aki, Radziwi\l\l, and Tao. The resulting shift +average saves more than $L^{-1}$, which pays for the number of bins. +The fourth-moment treatment of the divisor exponential sum follows +\cite[Appendix~C]{Pilatte26}; we include the rough-number estimates and +their finite counting errors. Write $\mathrm e(t)=\exp(2\pi i t)$. + +\begin{lemma}[Fourier bounds for rough integers]\label{q:rough-fourier} +Let $L$ be sufficiently large, let $h\ge1$ be an integer, and suppose +\[ + \tfrac12\exp(L^{0.995})\le D\le\exp(2L). +\] +Let $\mathcal Z\subset[D,2D)$ consist of integers with no prime factor +at most $\exp(L^{0.99})$. For +\[ + B(\theta)=\sum_{z\in\mathcal Z}\frac{\mathrm e(hz\theta)}z + \qquad(\theta\in\mathbb R/\mathbb Z) +\] +one has, with absolute implied constants, +\begin{equation}\label{q:rough-bounds} + \|B\|_\infty\ll L^{-0.99},\qquad + \int_0^1|B(\theta)|^4\,d\theta\ll D^{-1}L^{-3.96}. +\end{equation} +\end{lemma} + +\begin{proof} +Put $y=\exp(L^{0.99})$. We need upper bounds for the number of integers +in $[D,2D)$ avoiding every prime at most $y$, and for the number of triples +in $[D,2D)^3$ for which all four forms +\[ + x_1,\quad x_2,\quad x_3,\quad x_1+x_2-x_3 +\] +avoid those primes. The last form is positive throughout this integer box. +In the first case put $m=1$, and in the second put $m=4$. For each prime $p$, +the proportion $\nu_m(p)$ of residue vectors where at least one form +vanishes satisfies +\begin{equation}\label{q:sieve-local-density} + \nu_m(p)=\frac mp+O(p^{-2}). +\end{equation} +For the four forms, every pair of defining hyperplanes is independent over +every prime field, including the field with two elements. Inclusion and +exclusion of pairs proves \eqref{q:sieve-local-density}. Moreover +$\nu_m(p)<1$ for every $p$: the residue $1$ in the first case and the vector +$(1,1,1)$ in the second avoid all the forms. + +For completeness, a finite inclusion--exclusion argument suffices here. +Let $r_0=2\lceil500\log L\rceil$ and truncate inclusion--exclusion of the +bad-prime events at this even order. This gives an upper bound for the +proportion avoiding all the events. For any selected prime set of product +$r$, the Chinese remainder theorem gives residue density +$\prod_{p\mid r}\nu_m(p)$. Its normalized count in the interval or box +differs from this density by $O(r/D)$ when $r\le D$. Indeed, each residue +coordinate occurs $D/r+O(1)$ times; summing the product of these counts over +at most $r$ or $r^3$ residue vectors proves the assertion. +There are at most +$(r_0+1)(1+\pi(y))^{r_0}$ terms, each with $r\le y^{r_0}$. +The total boundary error is therefore at most +\begin{equation}\label{q:sieve-boundary-error} + \frac1D\exp\bigl(O(L^{0.99}\log L)\bigr), +\end{equation} +which is smaller than every fixed negative power of $L$, by the lower bound +on $D$. + +In the independent residue model the difference between the even truncated +sum and the full product is at most the next elementary symmetric sum. +Partial summation of the prime number theorem and +\eqref{q:sieve-local-density} give +\[ + \sum_{p\le y}\nu_m(p)\le(4+o(1))\log L. +\] +Thus that difference is at most +\[ + \frac{\bigl(\sum_{p\le y}\nu_m(p)\bigr)^{r_0+1}}{(r_0+1)!} + \le + \left(\frac{e\sum_{p\le y}\nu_m(p)}{r_0+1}\right)^{r_0+1} + \ll L^{-100}. +\] +The full product satisfies +\[ + \prod_{p\le y}(1-\nu_m(p))\ll(\log y)^{-m}=L^{-0.99m}; +\] +the finitely many small primes contribute a positive constant, and the +remaining factors follow by taking logarithms in +\eqref{q:sieve-local-density}. Together with +\eqref{q:sieve-boundary-error}, this proves the respective counts +$O(DL^{-0.99})$ and $O(D^3L^{-3.96})$. + +The first count bounds $\sum_{z\in\mathcal Z}1/z$ and hence +$\|B\|_\infty$. Orthogonality expresses the fourth moment as the sum of +$(z_1z_2z_3z_4)^{-1}$ over +$z_1+z_2=z_3+z_4$ with all four variables in $\mathcal Z$. +The factor $h$ does not change this equality. Each triple determines $z_4$, +and each weight is at most $D^{-4}$. The second count proves the fourth +moment bound. +\end{proof} + +The analytic input keeps the frequency fixed while averaging the starting +point of a short interval. + +\begin{theorem}[Matom\"aki--Radziwi\l\l--Tao, +\cite{MRT15}, Theorem~1.3]\label{q:MRT-short} +For every pair of real numbers $10\le H\le Z$, +\[ + \sup_{\alpha\in\mathbb R} + \int_0^Z + \left|\sum_{x\le n\le x+H}\lambda(n)\mathrm e(\alpha n)\right|\,dx + \ll HZ\left(\frac{\log\log H}{\log H} + +(\log Z)^{-1/700}\right), +\] +with an absolute implied constant. The supremum is outside the integral. +\end{theorem} + +We next combine this theorem with Lemma~\ref{q:rough-fourier}. The conclusion +is uniform over arbitrary subsets of the rough integers, so it permits all +the bin and prime-supply restrictions imposed in +\eqref{q:centered-average}. + +\begin{lemma}[Correlation averaged over rough shifts]\label{q:rough-shifts} +Fix integers $h,l\ge1$ and $b_0\in\mathbb Z$. Let $A\ge1000$, let $L$ be +sufficiently large, and suppose $Y\ge\exp(L^A/2)$. Let $D$ be a positive +integer and let $\mathcal Z\subset[D,2D)$ satisfy the assumptions of +Lemma~\ref{q:rough-fourier}. Then +\begin{equation}\label{q:rough-correlation} + \left|\frac1Y\sum_{1\le v\le Y} + \sum_{z\in\mathcal Z}\frac1z + \1_{v\equiv b_0\pmod l}\lambda(v)\lambda(v+hz)\right| + \ll_{h,l}L^{-21/20}. +\end{equation} +\end{lemma} + +\begin{proof} +For each integer $1\le v\le Y$ define +\begin{align*} + F_v(\theta)&=\sum_{m=1}^D + \lambda(v+m)\1_{v+m\equiv b_0\pmod l}\mathrm e(\theta m),\\ + G_v(\theta)&=\sum_{a=1}^{(2h+1)D} + \lambda(v+a)\mathrm e(-\theta a). +\end{align*} +We first claim that, uniformly in $\theta$, +\begin{equation}\label{q:progression-short-sum} + \sum_{1\le v\le Y}|F_v(\theta)|\ll YD L^{-0.8}. +\end{equation} +For integer $D$, every $x\in(v,v+1)$ has precisely +$v+1,\ldots,v+D$ as the integers in $[x,x+D]$. +Apply Theorem~\ref{q:MRT-short} with $H=D$ and +$Z=\lfloor Y\rfloor+1$, discarding the unused initial unit interval. +The restriction to the residue class follows from +\[ + \1_{n\equiv b_0\pmod l} + =\frac1l\sum_{a=0}^{l-1}\mathrm e\left(\frac{a(n-b_0)}l\right). +\] +Its coefficients have total absolute value one. After the unimodular +factor arising from $n=v+m$ is removed, this applies the same theorem to +the frequencies $\theta+a/l$. In particular, no assumption that $b_0$ is +coprime to $l$ is needed. The scale hypotheses imply $10\le D\le Z$ and +\[ + \frac{\log\log D}{\log D}+(\log Z)^{-1/700} + \ll L^{-0.8}, +\] +which proves \eqref{q:progression-short-sum}. + +Average the sum on the left of \eqref{q:rough-correlation}, before taking +its absolute value, over translations by $m=1,\ldots,D$. Each translation +changes at most $2m$ prefix terms for each $z$. By +Lemma~\ref{q:rough-fourier}, the normalized endpoint error is at most +\[ + O\left(\frac D Y\sum_{z\in\mathcal Z}\frac1z\right)=O(D/Y). +\] +Let $B$ be the polynomial in Lemma~\ref{q:rough-fourier}. Orthogonality now +expresses the translated average as +\begin{equation}\label{q:short-convolution} + \frac1{YD}\sum_{1\le v\le Y} + \int_0^1B(\theta)F_v(\theta)G_v(\theta)\,d\theta. +\end{equation} +Indeed, expanding the integral imposes $a=m+hz$. Since +$1\le m\le D$ and $D\le z<2D$, this value of $a$ lies in the full range +used to define $G_v$. + +On the set where $|B|\le L^{-1.05}$, Parseval and Cauchy--Schwarz give +\[ + \int_0^1|F_vG_v|\le\|F_v\|_2\|G_v\|_2 + \le\sqrt{D}\sqrt{(2h+1)D}\ll_h D. +\] +Its contribution to \eqref{q:short-convolution} is therefore +$O_h(L^{-1.05})$. The complementary set, denoted by $\mathcal E$, satisfies +\[ + |\mathcal E|\le L^{4.2}\int_0^1|B|^4 + \ll D^{-1}L^{0.24}. +\] +On this set use $\|B\|_\infty\ll L^{-0.99}$, +$|G_v|\le(2h+1)D$, and \eqref{q:progression-short-sum}. Its contribution is +at most +\[ + \frac1{YD}|\mathcal E|\,L^{-0.99}(2h+1)D\,(YDL^{-0.8}) + \ll_h L^{0.24-0.99-0.8}=L^{-1.55}. +\] +Finally $D/Y\le\exp(2L-L^A/2)$ is negligible. This proves +\eqref{q:rough-correlation}. +\end{proof} + +\begin{proof}[Proof of \Cref{q:center}] +The full-divisibility contribution is \eqref{q:full-divisibility}. +For every other term in the expansion of \eqref{q:centered-average}, +let $I\subset\{1,\ldots,J\}$ be the nonempty set of +prime supplies whose factors have been replaced by $-1/p$. Put +\[ + z=\prod_{i\in I}p_i,\qquad d'=d/z. +\] +For fixed $q,d',j$, substitute $n=qd'v$ and put $Y=T_j/(qd')$. +After extracting the sign $(-1)^{|I|}$ and the factor $u(q)/(qd')$, the +remaining expression is +\begin{equation}\label{q:partial-centering} + \frac1Y\sum_{1\le v\le Y}\sum_z\frac1z + \1_{v\equiv bz\pmod l}\lambda(v)\lambda(v+hz), +\end{equation} +where $z$ ranges over the products from the supplies indexed by $I$ that +also satisfy the bin condition. This substitution is valid in every +residue class, since $qd'$ is invertible modulo $l$. + +The allowed $z$ lie in an interval of logarithmic width $\eta<\log2$. +Split them by residue modulo $l$ and by the at most two dyadic intervals +$[D,2D)$ that meet this interval. In each resulting piece the residue +$bz\pmod l$ is constant. Unique prime factorization and the disjointness +of the supplies show that each integer $z$ occurs at most once. Since +$I$ is nonempty, all its prime factors are at least +$H_0=\exp(L^{0.995})$, and +\[ + D\ge H_0/2,\qquad D\le\exp(2L). +\] +Furthermore, if the piece is nonempty, its bin condition implies +$Y>Xe^{-\eta}z$, so $\log Y\ge L^A/2$ for large $L$. +Lemma~\ref{q:rough-shifts} therefore bounds +\eqref{q:partial-centering} by $O_{h,l}(L^{-21/20})$. + +For each fixed $I$ and bin, the sum of the extraction weights is at most +\[ + \sum_q\frac{u(q)}q\sum_{d'}\frac1{d'} + \le S\prod_{i\notin I}V_i\le SV_*, +\] +where the last inequality uses $V_i>1$. There are +$O(L/\eta)$ bins and fewer than $2^J$ nonempty sets $I$. Hence the total +contribution of the partial centering terms is +\[ + O_{h,l}\bigl(SV_*\,2^J\eta^{-1}L^{-1/20}\bigr). +\] +Together with the full divisibility term this proves +\eqref{q:center-estimate}. +\end{proof} + +\subsection{The distribution of padding divisors} + +The centered averages still have occasional sites at which many padding +choices contribute to one bin. We next prove that deleting such sites +has small total cost. For a site $n$, a label $d\in\mathcal D$, and one of +the bin indices $j$, define +\begin{equation}\label{q:rho} + \rho_j(n,d)=\frac1{w(n)} + \sum_{\substack{q\in\mathcal Q,\ q\mid n\\ + j\eta\le\log(qd)<(j+1)\eta}}u(q). +\end{equation} +Here the sum includes all squarefree padding divisors in the bin, without +the restriction on $\omega(q)$ imposed in $\mathcal S_j$. +The denominator is positive because the divisor $1$ is always present. +Thus $0\le\rho_j\le1$ and $\sum_j\rho_j\le1$. +All these definitions also make sense in the auxiliary residue model. +At a translated site $n+a$, divisibility by $p$ means +$R_p+a\equiv0\pmod p$, using the same residue coordinate at every site. + +Define the tilted expectation at a single site by +\[ + \E^* H=\frac{\E(w(n)H)}S. +\] +This is a probability expectation since +\[ + \E w(n)=\prod_{r\in Q}(1+4/r)=S. +\] +The tilt preserves independence of the $Q$ coordinates and changes the +probability of $r\mid n$ from $1/r$ to $5/(r+4)$. + +\begin{lemma}[Padding anti-concentration]\label{q:padding} +For every $d\in\mathcal D$ and the bin width $\eta=e^{-J}$, +\begin{equation}\label{q:padding-square} + \E^*\sum_j\rho_j(n,d)^2\ll_{h,l}L^{-1}. +\end{equation} +The constant is independent of $d$, $W$, and the location of the bins. +\end{lemma} + +\begin{proof} +Conditionally on the primes of $Q$ dividing $n$, choose $q_1,q_2$ +independently with probabilities +\[ + \Pbb(q_a=q\mid n)=\frac{u(q)\1_{q\mid n}}{w(n)} + \qquad(a=1,2). +\] +Each available prime is selected with probability $4/5$. Under the +combined tilted law let $Z=\log(q_1/q_2)$. A prime $r$ contributes +$+\log r$ or $-\log r$ with probability $4/[5(r+4)]$ each, and contributes +zero otherwise. The contributions are independent, so its characteristic +function, with angular frequency $t$, is +\begin{equation}\label{q:padding-characteristic} + \phi(t)=\E^*e^{itZ} + =\prod_{r\in Q}\left(1-\frac8{5(r+4)} + (1-\cos(t\log r))\right). +\end{equation} +Every factor is nonnegative: even at $r=2$ it is at least $7/15$. +The inequality $1-x\le e^{-x}$ therefore gives +\[ + 0\le\phi(t)\le + \exp\left(-\frac85\sum_{r\in Q} + \frac{1-\cos(t\log r)}{r+4}\right). +\] +Replacing $r+4$ by $r$ changes the exponent by $O(1)$, uniformly in $t$. +For $|t|\le2$, partial summation of the fixed-modulus prime number theorem +in \eqref{q:pnt} gives +\begin{align} + \sum_{r\in Q}\frac{1-\cos(t\log r)}r + &=\frac34\int_{\log2}^L\frac{1-\cos(ty)}y\,dy+O_{h,l}(1) + \notag\\ + &=\frac34\log(1+L|t|)+O_{h,l}(1).\label{q:padding-pnt} +\end{align} +For completeness, in the partial summation the derivative of +$(1-\cos(t\log x))/(x\log x)$ is $O(1/(x^2\log x))$ uniformly for +$|t|\le2$. Its product with the prime-number-theorem error is integrable +on $[2,\infty)$. Deleting the finitely many prime divisors of $lh$ also +costs $O_{h,l}(1)$. To verify the second equality, set $u=|t|y$. +The integral over $u\le1$ is bounded, and integration by parts bounds +$\int_1^U\cos u\,du/u$ uniformly in $U\ge1$. The lower endpoint +$|t|\log2$ stays bounded. These observations also cover $t=0$. +It follows that +\begin{equation}\label{q:padding-decay} + 0\le\phi(t)\ll_{h,l}(1+L|t|)^{-6/5}\qquad(|t|\le2). +\end{equation} + +Conditionally on $n$, the probability that both divisors lie in the +$j$th bin is $\rho_j(n,d)^2$. Divisors in the same bin have +$|Z|\le\eta\le1$. Consequently the left side of +\eqref{q:padding-square} is at most +$\Pbb^*(|Z|\le1)$. The function +$\kappa(x)=(\sin x/x)^2$, with $\kappa(0)=1$, satisfies +\[ + \1_{[-1,1]}(x)\le\frac{\kappa(x)}{\sin^2 1},\qquad + \kappa(x)=\frac14\int_{-2}^2(2-|t|)e^{itx}\,dt. +\] +Taking expectations, using \eqref{q:padding-decay}, and integrating gives +\[ + \Pbb^*(|Z|\le1) + \ll_{h,l}\int_{-2}^2(1+L|t|)^{-6/5}\,dt + \ll_{h,l}L^{-1}. +\] +The common translation $\log d$ of the two padding logarithms disappeared +from $Z$, which proves the asserted uniformity. +\end{proof} + +\subsection{Deleting large degrees and prescribed rare sites} + +We formulate the deletion step with an explicit input for a family of rare +sites. The following section will construct that family from short +paths. For each bin $j$, choose a Boolean predicate of the residue coordinates +and write $n\in\mathcal B_j$ when it holds at site $n$. Translation +covariance means that $n+a\in\mathcal B_j$ is obtained from the same +predicate by adding $a$ to every residue coordinate. We assume +\begin{equation}\label{q:rare-input} + \Pbb(n\in\mathcal B_j)\le + \exp\bigl(-\tfrac12L^{1-\delta}\bigr), +\end{equation} +and that the event is represented by an AND--OR--NOT circuit of depth at +most $6$ and size at most $\exp(L^3)$, whose inputs are equalities in the +prime residue coordinates and the coordinate modulo $l$. +The event uses no value of the Liouville function. Define +$\sigma_j(n)=\1_{n\notin\mathcal B_j}$. For any prime set $R$, +write $\omega_R(n)=\#\{p\in R:p\mid n\}$. Define +\begin{align} + \tau_j(n,d)&= + \1_{\omega_Q(n)\le400\log L}\, + \1_{\rho_j(n,d)\le K/L}\,\sigma_j(n),\notag\\ + \chi_j(n,d)&= + \tau_j(n,d)\,\1_{\omega_P(n)\le6WJ}. + \label{q:tau} +\end{align} +The function $\tau_j$ omits the $P$-degree cut: the graph argument will +integrate the centered $P$ coordinates before applying that restriction +as a projection. At present we impose all four cuts at +both ends of each summand, defining +\begin{align} + C_j^\circ={}&\frac1{T_j}\sum_{1\le n\le T_j} + \sum_{(d,q)\in\mathcal S_j} + u(q)\1_{q\mid n}\1_{n\equiv bqd\pmod l} + \prod_{p\mid d}\left(\1_{p\mid n}-\frac1p\right) + \lambda(n)\lambda(n+hqd)\notag\\[-2pt] + &\hspace{35mm}\cdot\chi_j(n,d)\chi_j(n+hqd,d). + \label{q:cut-averages} +\end{align} + +\begin{lemma}[Deletion cost]\label{q:cuts} +Suppose the family $\mathcal B_j$ satisfies the covariance, probability, +and circuit hypotheses just stated. Then +\begin{equation}\label{q:cuts-bound} + \frac1{SV_*}\sum_j|C_j-C_j^\circ| + \ll_{h,l} + 2^J\bigl(K^{-1}+L^{-100}+e^{-2WJ}\bigr) + +\exp(-L^{0.9}). +\end{equation} +This estimate holds for the integer averages defining $C_j$ and +$C_j^\circ$, for every sufficiently large real $X$. +\end{lemma} + +\begin{proof} +We first bound the deletion costs in the auxiliary product model, after +dropping the Liouville factors and the progression indicator, and replacing +the absolute centered product by +\[ + a_d^+(n)=\prod_{p\mid d}\left(\1_{p\mid n}+\frac1p\right). +\] +At the end of the proof we transfer precisely these nonnegative costs +to integer averages by Lemma~\ref{q:finite-law}. No assertion about +untruncated integer moments of $w$ is needed. + +\paragraph{Either endpoint.} +For a fixed numerical pair $(d,q)$, translation by $hqd$ preserves the +auxiliary product law. Moreover divisibility by $q$ and by each prime +of $d$ is unchanged by this translation. Thus an upper-endpoint failure +has the same bound as a lower-endpoint failure. This argument does not +assert that the full weight $w$ is constant across an edge. It suffices +to analyze one endpoint and multiply the resulting bound by $2$. + +\paragraph{The two padding cuts.} +The $P$ coordinates are independent of the $Q$ coordinates and +$\E a_d^+=2^J/d$. Enlarge the eligible padding divisors to all divisors +in their respective bins. The total costs of the first two cuts, at +one endpoint and summed over bins, are at most +\begin{equation}\label{q:padding-deletion} + 2^JS\sum_{d\in\mathcal D}\frac1d + \left\{\Pbb^*(\omega_Q>400\log L) + +\E^*\sum_j\rho_j(n,d) + \1_{\rho_j(n,d)>K/L}\right\}. +\end{equation} +Indeed the sum of padding weights in a bin is $w(n)\rho_j(n,d)$, +and $\sum_j\rho_j\le1$. Under the tilted law the mean of $\omega_Q$ +is at most $4\log L$ for large $L$. Independence and exponential Markov +therefore give +\[ + \Pbb^*(\omega_Q>400\log L) + \le\exp\bigl((-400+4(e-1))\log L\bigr)\ll L^{-100}. +\] +Also $x\1_{x>K/L}\le(L/K)x^2$ for $x\ge0$. Applying +Lemma~\ref{q:padding} bounds the second term in braces by +$O_{h,l}(K^{-1})$. Since $\sum_d1/d=V_*$, this gives the first two +terms of \eqref{q:cuts-bound}. + +\paragraph{The $P$-degree cut.} +For fixed $d$, tilt the $P$ coordinates by $a_d^+$ divided by its mean +$2^J/d$. Primes outside $d$ retain their independent Bernoulli laws, +whereas the selected $J$ primes contribute at most $J$ to the count. +As $\sum_{p\in P}1/p=\sum_iV_i\le2WJ$, exponential Markov gives +\[ + \Pbb_{d,+}(\omega_P>6WJ) + \le\exp\bigl(-6WJ+J+2(e-1)WJ\bigr) + \le e^{-2WJ}\qquad(W\ge10). +\] +The padding coordinates remain independent. Summing their weights +$u(q)/q$ over all eligible pairs and bins is at most $S$ for each $d$. +Thus this deletion costs at most $2^JSV_*e^{-2WJ}$ per endpoint. + +\paragraph{The rare sites.} +For one bin, the entire absolute row weight is bounded by +\[ + B(n)=w(n)\prod_{i=1}^J(\omega_{P_i}(n)+V_i). +\] +To see this, sum all padding divisors for each $d$, and then sum each +prime slot separately. Independence gives +\[ + \E B(n)^2 + \le\prod_{r\in Q}(1+24/r) + \prod_{i=1}^J(4V_i^2+V_i) + \le C_{h,l}L^{C}(18W^2)^J + \le C_{h,l}L^{C'}. +\] +Here $C,C'$ are absolute: the first product is polynomial in $L$ by +partial summation, and +$J\log(18W^2)\le\delta\log L\,\log(18W^2)/(6W)$ is bounded by an +absolute multiple of $\log L$ for $W\ge10$. +Cauchy--Schwarz and \eqref{q:rare-input} therefore bound a single-bin +rare-site cost by +\[ + C_{h,l}L^{C'/2}\exp\bigl(-\tfrac14L^{1-\delta}\bigr). +\] +There are $O(L/\eta)$ bins, and $SV_*\ge1$. +Their total contribution is at most $\exp(-L^{0.9})$ for sufficiently +large $L$. + +\paragraph{Transfer to integer averages.} +We now justify applying Lemma~\ref{q:finite-law} to the preceding costs. +This also ensures that no independence claim is being made for arbitrary +integer functions. Use a union bound for the two endpoints, and for +the following four failures at either endpoint: +\[ + \omega_Q>400\log L;\qquad + \omega_Q\le400\log L\text{ and }\rho_j>K/L;\qquad + n\in\mathcal B_j;\qquad \omega_P>6WJ. +\] +This is the same deletion event, with the padding-mass failure tested +only after the low-$Q$ cut passes. In the product model its cost is +bounded by the estimates already proved, since dropping the extra +low-$Q$ condition can only increase a nonnegative cost. + +Fix a numerical $j,d,q$ and expand $a_d^+$ as a sum of divisibility +indicators. There are $2^J$ terms, with coefficients at most $1$. +The factor $\1_{q\mid n}$ is a conjunction of its prime divisibilities. +Each degree failure is an OR over subsets of primes of the required +cardinality, testing that all their residues vanish. Since there are +at most $e^L$ available primes, the low-$Q$ degree test has size +$\exp(O(L\log L))$. The $P$ threshold has the same bound, using +$6WJ\le\delta\log L$. + +For the padding-mass failure with low $Q$ degree, list all possible exact +sets of at most $400\log L$ primes of $Q$ dividing the site. For each +set, a conjunction asserts both the indicated presences and all other +absences. On that state, both $w$ and $\rho_j$ are fixed numbers; retain +the state precisely when $\rho_j>K/L$. There are +$\exp(O(L\log L))$ such states, each specified by at most $e^L$ tests. +The remaining event $\mathcal B_j$ has the circuit representation assumed +in the statement. Intersecting any of these events with the indicated +prime divisibilities has depth at most $10$ and size at most +$\exp(L^4)$ for large $L$. Translation to an upper endpoint changes +only the tested residues, not these bounds. + +Finally, the numerical pairs satisfy $dq\le\exp(100L+\eta)$. +The disjoint prime supports of $d$ and $q$ determine both from their +product, so there are $\exp(O(L))$ eligible pairs in all bins. Also +$u(q)\le4^{100\log L}$, and the expansion of $a_d^+$ has total +coefficient mass at most $2^J$. The total absolute coefficients in all +these comparisons are therefore at most $\exp(O(L^4))$; even the looser +bound permits all listed state expansions. The averaging interval has +length $\lfloor T_j\rfloor\ge\exp(\tfrac12L^A)$. +Lemma~\ref{q:finite-law} makes the aggregate comparison error at most +$\exp(-L^9)$. Replacing the integer-length normalization by $T_j$ costs +at most $T_j^{-1}\exp(O(L^4))$, which is smaller still. +Together with the product estimates, these errors prove +\eqref{q:cuts-bound}. +\end{proof} diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/04-paths-trace.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/04-paths-trace.tex new file mode 100644 index 0000000000000000000000000000000000000000..1ef7bf68dc0afb6c6d3c83bdaa77def99ef11c12 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/04-paths-trace.tex @@ -0,0 +1,421 @@ +\section{Prohibited paths and centered traces}\label{q:paths-trace} + +We now construct the sparse sets used in \Cref{q:cuts}. Their +removal will impose enough structure on closed walks to control a +matrix moment. Throughout this section the bin index $j$ is fixed, +and +\[ + s=\lfloor L^{1/10}\rfloor,\qquad k=\lfloor L\rfloor. +\] +All assertions are for sufficiently large $L$, with the fixed integers +$h,l$ and the absolute parameters $A,W$ held fixed. + +The centered graph and high-trace strategy was developed by Helfgott +and Radziwi\l\l\ \cite[Sections~2--7]{HR21}. +Pilatte developed its composite-label and nonbacktracking form +\cite[Sections~2--9]{Pilatte26}. Here each step carries a prime tuple +and a padding divisor; the trace expansion keeps the padding weights +and the common residue coordinates explicit. + +\subsection{Positive paths and prohibited words} + +A signed step is a displacement +$D_i=\varepsilon_i hq_i d_i$, where $\varepsilon_i\in\{-1,1\}$ +and $(d_i,q_i)\in\mathcal S_j$. +A word of $m$ steps, starting at $x$, visits the sites +$x+\sum_{a0}}. +\] + +If lit consistency fails, the designated term vanishes; set +$\Delta G=0$ in this case. Otherwise, starting from one common draw +of the prime residues, overwrite every +nonsingleton with a lit occurrence by its forced residue. For each +$H\subseteq\mathcal U$, also overwrite the singleton coordinates in +$H$ by the residues forced at their sole occurrences, leaving the +other singleton coordinates at their original values. Call the +resulting configuration a \emph{hybrid}, and let $G_H$ be the value of +$G$ there. The mixed difference, forced minus original in each +singleton coordinate, is +\[ + \Delta G=\sum_{H\subseteq\mathcal U}(-1)^{S_1-|H|}G_H. +\] + +\begin{lemma}[Exact centering before absolute values]\label{q:mixed-difference} +The absolute expectation of a designated term of +\eqref{q:word-product} is at most +\begin{equation}\label{q:mixed-majorant} + \1_{\mathrm{lit\ consistency}}\,b_{\mathcal L} + \E\bigl[R|\Delta G|\bigr]. +\end{equation} +In particular $|\Delta G|\le2^{S_1}$. +\end{lemma} + +\begin{proof} +For a uniform residue $Z$ modulo $p$, a fixed residue $a$, and any +function $F$ of that coordinate, +\[ + \E\left[\left(\1_{Z=a}-\frac1p\right)F(Z)\right] + =\frac1p\E\bigl[F(a)-F(Z)\bigr]. +\] +Apply this identity successively to the independent singleton +coordinates. At a nonsingleton, incompatible lit tests give zero; +otherwise they force one residue and supply $1/p$, while every unlit +choice supplies the scalar $-1/p$. These operations produce exactly +the factor $b_{\mathcal L}$ and the mixed difference, up to the signs +of the unlit scalars. The factor $R$ is unchanged in every operation. +Taking the absolute value after these integrations proves the bound. +The final assertion follows because $G$ takes values in $\{0,1\}$. +\end{proof} + +The square-root dependence on $W$ in \eqref{q:trace-bound} comes +from repeated prime labels. Among $2kJ$ tuple-prime occurrences there +are at most $Jk+S_1/2$ distinct labels, since every nonsingleton occurs +at least twice. For a fixed equality pattern, summing one reciprocal +per label therefore costs at most $(2W)^{Jk+S_1/2}$. +To use this count, we must still control the number of patterns and +sum the padding weights uniformly. We first discard terms where +extra unlit reciprocals, independent lit-consistency congruences, or +relations forced by many singletons give a stronger saving directly. +The forest count and uniform padding sum in \Cref{q:forest-section} +will handle the remaining terms. + +For negligible classes of words we will discard denominators and +cutoffs in $R$, but retain all padding divisibility tests. +Independence then gives one reciprocal per distinct $Q$ prime. +The factor $b_{\mathcal L}$ gives at least one reciprocal per distinct +$P$ prime. Together with \Cref{q:crude-count}, the total cost before +any additional saving is $\exp(O(L\log^2L))$, including +\eqref{q:external-dimension}, lit designations and $2^{S_1}$. +Padding values may be fixed first during a constrained sum over tuple +primes; their original bound $q_i\le e^{100L+1}$ is retained when +needed for coefficient-size estimates. + +\begin{lemma}[Many unlit occurrences]\label{q:many-unlit} +The total contribution to the moment majorant from designated terms +with $U>L^{1/50}$ is at most $\exp(-L^{1+\delta})$. +\end{lemma} + +\begin{proof} +If a nonsingleton has a lit occurrence, each of its unlit occurrences +supplies an extra reciprocal beyond its basic $1/p$. +If all its $m\ge2$ occurrences are unlit, it supplies $p^{-m}$ and +hence at least $m/2$ extra reciprocal powers beyond the basic one. +Thus at least $U/2$ extra powers remain, all at primes at least $H_0$. +The total is at most +\[ + \exp(O(L\log^2L))H_0^{-U/2} + \le\exp\bigl(O(L\log^2L)-\tfrac12L^{1.015}\bigr) + \le\exp(-L^{1+\delta}). +\] +\end{proof} + +We now turn to the two remaining sources of arithmetic savings: +independent congruences and singleton primes. diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/05-rank-witness.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/05-rank-witness.tex new file mode 100644 index 0000000000000000000000000000000000000000..ffdc4eb513e44b77b6c08bb87dd2817fb62d9962 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/05-rank-witness.tex @@ -0,0 +1,492 @@ +\section{Arithmetic savings for exceptional words}\label{q:rank-witness-section} + +The centered trace has been reduced to the majorants in +\Cref{q:mixed-difference}. We now discard two further classes. +Independent lit-consistency equations provide a rank saving, while many +singleton coordinates force positive witnesses whose relations can be +summed in a triangular order. Each saving makes its entire class +negligible before we count the remaining words. + +Recurrence congruences constraining prime labels play a central role in +\cite[Sections~6--7]{HR21}. We first prove a random-prime rank estimate +suited to the present coefficients. The later witness argument adapts +the active-prime and triangular-constraint constructions of +\cite[Lemmas~11.6, 13.1, and~13.6]{Pilatte26}. + +\subsection{Saving from independent congruences} + +We first discard the words whose lit-consistency conditions contain +many independent pairs of vectors. The point is to use rank over +$\mathbb R$, without assuming that a matrix remains invertible modulo +any of the primes being summed. The following probability estimate +isolates the argument. + +\begin{lemma}[A rank estimate for random prime divisors] +\label{q:rank-probability} +Let $\mathscr P$ be a finite nonempty set of primes, and let $\mu$ be a +probability measure on $\mathscr P$ with +$\max_{p\in\mathscr P}\mu(p)\le\alpha$. +Let $r\ge1$, let $B\in M_r(\mathbb Z)$ be nonsingular over +$\mathbb R$, and let $f:\mathscr P^r\to\mathbb Z^r$ be any function. +Suppose that $M\ge2$ and +\[ + \left|(B(y-y'))_i\right|\le M + \quad\text{for all }y,y'\in\mathscr P^r\text{ and }1\le i\le r. +\] +If the $2r$ coordinates of $c,y\in\mathscr P^r$ are independent with +law $\mu$, then +\begin{equation}\label{q:rank-probability-bound} + \Pbb\bigl(c_i\mid(By+f(c))_i\text{ for every }i\bigr) + \le \left[\alpha(1+d_M)\right]^{r/2}, + \qquad d_M=\left\lfloor\frac{\log M}{\log2}\right\rfloor. +\end{equation} +\end{lemma} + +\begin{proof} +Let $E(c,y)$ denote the divisibility event in the statement, and let +$y'$ be an independent copy of $y$, independent also of $c$. +Cauchy--Schwarz, applied to the conditional probability with $c$ fixed, +gives +\begin{align*} + \Pbb(E(c,y))^2 + &=\left(\E_c\Pbb_y(E(c,y)\mid c)\right)^2\\ + &\le \E_c\Pbb_y(E(c,y)\mid c)^2\\ + &=\Pbb(E(c,y)\cap E(c,y'))\\ + &\le\Pbb\bigl(c_i\mid (B(y-y'))_i\text{ for every }i\bigr). +\end{align*} +The two copies have the same controlling primes $c_i$, so subtraction +removes the entire vector $f(c)$. + +Write $v=B(y-y')$. For a subset $Z\subseteq\{1,\ldots,r\}$ of size +$a$, the rows of $B$ indexed by $Z$ have rank $a$. Choose an +invertible $a$-column minor in those rows. Conditional on $y'$ and on +the coordinates of $y$ outside that minor, the equations +$v_i=0$ for $i\in Z$ determine at most one vector of values for its +$a$ remaining coordinates. Independence and the atom bound imply +\begin{equation}\label{q:rank-zero-rows} + \Pbb(v_i=0\text{ for all }i\in Z)\le\alpha^a. +\end{equation} +This includes $a=0$, with the empty condition having probability one. + +Now fix $y,y'$. A nonzero integer of absolute value at most $M$ has +at most $d_M$ distinct prime divisors. Each nonzero coordinate $v_i$ +therefore satisfies $c_i\mid v_i$ with conditional probability at most +$d_M\alpha$. The controls are independent of each other and of $v$. +If the exact zero set of $v$ is $Z$, their joint conditional +probability is thus at most $(d_M\alpha)^{r-|Z|}$. +Using \eqref{q:rank-zero-rows} and summing over the possible exact zero +sets gives +\[ + \Pbb(c_i\mid v_i\text{ for every }i) + \le\sum_{Z\subseteq\{1,\ldots,r\}} + \alpha^{|Z|}(d_M\alpha)^{r-|Z|} + =\left[\alpha(1+d_M)\right]^r. +\] +Taking the square root proves \eqref{q:rank-probability-bound}. +\end{proof} + +We apply the lemma to one column of the prime tuples in a word of +length $2k$. Put +\[ + r=\lfloor L^{1/50}\rfloor. +\] +Fix a column $\nu\in\{1,\ldots,J\}$ and an equality pattern for its +prime slots. Let $\mathcal Z_\nu$ be the set of its abstract labels, +let $z_a\in\mathcal Z_\nu$ be the label at step $a$, and write $p_z$ +for the numerical prime assigned to $z$. Fix the padding values, +orientations, and the primes in every other column. In the real +vector space with basis $(e_z)_{z\in\mathcal Z_\nu}$ define +\begin{equation}\label{q:formal-offsets} + t_a=\frac{D_a}{p_{z_a}},\qquad + v(i)=\sum_{a0$ and sufficiently large $L$. +The bound includes the external dimension factors in the trace +expansion and is uniform in the bin. The threshold may depend on the +fixed parameters $W,h,l$. +\end{lemma} + +\begin{proof} +Use the positive reciprocal majorant and enumeration of +\Cref{q:crude-count}. Thus we may discard the denominators and +cutoffs, retain one reciprocal for every distinct prime label, and +include the factors $Lu(q_a)$ and $2^{S_1}$ in the crude count. We +retain the lit-consistency conditions. The total unrestricted count, +including external dimension factors, is +$\exp(O(L\log^2L))$. + +Fix a column, its equality pattern, and the other-column and padding +data. Choose $r$ jointly independent pairs, and denote their +controlling labels by $z_1,\ldots,z_r$ and their difference vectors +by $w_1,\ldots,w_r$. The controlling labels are distinct. Modulo +$C=\operatorname{span}(e_{z_1},\ldots,e_{z_r})$, the images of the +$w_i$ remain independent: a dependence would express a nontrivial +linear combination of the $w_i$ as a combination of the $e_{z_i}$, +contrary to joint independence. Therefore the row matrix of the +$w_i$, after deleting all controlling columns, has rank $r$. +Choose an invertible $r$-column minor $B_0$ of that matrix. + +The entries of the entire row matrix, and hence the choice of a +minor by any fixed deterministic rule, are independent of all the +numerical primes in column $\nu$. Fix the prime variables in that +column outside the controlling coordinates and the minor. Write +$c_i=p_{z_i}$ for the controls and $y$ for the vector of primes in +the minor. The selected lit-consistency conditions become +\begin{equation}\label{q:rank-congruences} + c_i\mid (B_0y+A_0c+d_0)_i\qquad(1\le i\le r), +\end{equation} +with fixed integer matrices $B_0,A_0$ and a fixed integer vector +$d_0$. No invertibility assertion modulo $c_i$ is being made. + +Enlarge the positive sum by dropping numerical distinctness among +abstract labels, as well as bin and closure restrictions. Continue +to attach one factor $1/p$ to each abstract label: if two coordinates +now take the same prime, their product weight is $1/p^2$. +The enlarged reciprocal sum is consequently a product sum. Its +normalized coordinates have independent law +\[ + \mu_\nu(p)=\frac{1}{V_\nu p}\quad(p\in P_\nu), + \qquad \max_p\mu_\nu(p)\le\frac{1}{V_\nu H_0} + \le\frac1{H_0}. +\] +This independence allows numerical coincidences. In particular, after +subtraction in \Cref{q:rank-probability}, the controls remain +independent of $B_0(y-y')$, even on a sample space that permits such +coincidences. + +We check the size hypothesis after these enlargements. The fixed +padding values came from eligible steps, so $q_a\le e^{100L+1}$. +The product of the primes in the other columns is at most $e^{2L}$ +by the geometric spacing of their bands. Hence +\[ + |t_a|\le h e^{102L+1}. +\] +Each row of the difference matrix has sum of absolute coefficients +at most $2kh e^{102L+1}$. Since every prime in the resampled column +is at most $e^L$, we obtain +\[ + |(B_0(y-y'))_i|\le 2kh e^{103L+1}=e^{O(L)}. +\] +For fixed $h$ and sufficiently large $L$, the constant in this last +exponent is absolute. In \Cref{q:rank-probability} we may therefore +take $d_M=O(L)$, $\alpha=(V_\nu H_0)^{-1}$, and +$f(c)=A_0c+d_0$. The probability of \eqref{q:rank-congruences} is at +most +\[ + \left(\frac{O(L)}{V_\nu H_0}\right)^{r/2} + \le \exp(O(r\log L))H_0^{-r/2}. +\] +Multiplying by the unrestricted harmonic masses restores the +unnormalized reciprocal sum with this same relative saving. +The estimate is uniform in every fixed outside variable, so these +variables may now be summed. + +The column, selected pairs and selected minor have at most +$J(2k)^{O(r)}$ descriptions. Their cost is absorbed in +$\exp(O(r\log L))$. Together with the crude enumeration, the total +is bounded by +\[ + \exp\left(O(L\log^2L)+O(r\log L)-\frac r2\log H_0\right). +\] +Since $\log H_0=L^{199/200}$ and +$r=L^{1/50}+O(1)$, its negative term is +$(\tfrac12+o(1))L^{203/200}$ and dominates both positive terms. +This proves the lemma. +\end{proof} + +We may therefore restrict the remaining trace sum to words for which, +in every column, a maximal jointly independent collection of pairs +\eqref{q:rank-pair} has fewer than $r$ members. This quantitative +restriction will constrain the equality patterns once the words with +many singleton labels have also been discarded. + +\subsection{Singleton primes and positive witnesses} + +We next dispose of words having many singleton primes. Without the +vertex deletion, integration of any singleton centered factor would give +zero. The mixed difference in \Cref{q:mixed-difference} measures the +failure of this cancellation caused by deletion. We show that a nonzero +mixed difference forces many positive witnesses, and that these witnesses +supply successively usable congruences on distinct prime labels. + +Put +\[ + T=\lfloor L^{1/12}\rfloor. +\] +\begin{lemma}[A common configuration of witnesses]\label{q:witness-cover} +Fix a numerical main word with singleton set $\mathcal U$, where +$S_1=|\mathcal U|>L^{1/4}$, and fix a draw of the residue coordinates. +If $\Delta G\ne0$, then some hybrid contains $T$ positive witnesses +attached at main departures, each of length at most $s$, with distinct +marked primes $y_1,\ldots,y_T\in\mathcal U$. The prime $y_j$ occurs +in witness $j$ and in none of the other $T-1$ witnesses. The combined +length of the main word and these witnesses is at most $4L$. +\end{lemma} + +\begin{proof} +For the fixed numerical word, let $\mathscr W$ be the finite set of all +possible witness tests at its departure sites. Write $I_w$ for the +indicator that the preselected numerical word $w$ is positive at its +assigned departure. Then +\[ + G=\prod_{w\in\mathscr W}(1-I_w). +\] +A repeated test in this product is harmless because its factors take +values in $\{0,1\}$. Expand this finite product algebraically and apply +the mixed difference in $\mathcal U$. If the union of the prime supports +of a subfamily $\mathscr A\subseteq\mathscr W$ omits a prime +$p\in\mathcal U$, then $\prod_{w\in\mathscr A}I_w$ is independent of +the $p$ coordinate. Its full mixed difference is therefore zero. +Consequently $\Delta G\ne0$ implies that some subfamily whose supports +cover $\mathcal U$ has a nonzero mixed difference. Its intersection +indicator is then one in at least one hybrid. All witnesses in that +subfamily are positive in this single configuration. + +Choose an inclusion-minimal subfamily covering $\mathcal U$ there. +Each member contains a prime of $\mathcal U$ occurring in no other +member, since otherwise that member could be removed. A witness has at +most $sJ$ distinct $P$ labels, so this subfamily has at least +$S_1/(sJ)$ members. For sufficiently large $L$, +\[ + \frac{S_1}{sJ}>\frac{L^{3/20}}{J}>T. +\] +Keep any $T$ members and one private prime in each. Their private +primes remain private in the selected subfamily. Finally, +\[ + 2k+sT\le 2L+L^{11/60}\le4L. +\] +The algebraic expansion above has been used only to deduce existence. +No bound is obtained by summing the absolute values of its terms, and +no enumeration of all its subfamilies is charged. +\end{proof} + +The cover gives each selected witness a private singleton prime, but +privacy alone does not give a congruence with an invertible coefficient +in that prime. We first identify labels having such a coefficient in +an internal relation. For a minimal prohibited word with steps +$E_1,\ldots,E_m$, call a $P$ label $x$ \emph{active} +if there are a $P$ label $c$ of the same word and an interval +$I\subseteq\{1,\ldots,m\}$ such that +\begin{equation}\label{q:active-definition} + c\mid\sum_{a\in I}E_a, + \qquad + \sum_{\substack{a\in I\\x\mid E_a}}E_a\not\equiv0\pmod c. +\end{equation} +In particular $x\ne c$. With every other numerical label fixed, the +first sum is linear in $x$ and its coefficient is invertible modulo +$c$. These are exactly the conditions needed for a reciprocal-prime +saving. + +\begin{lemma}[Active labels in a minimal witness]\label{q:active} +Let $E_1,\ldots,E_m$ be a minimal prohibited word. For every $P$ label +$y$ in this word, either $y$ is active or there are an active label $z$ +and a step $i$ containing $z$ such that +\begin{equation}\label{q:active-prefix} + \sum_{\substack{aa$. +Reverse the contiguous subword $E_a,\ldots,E_m$, negating all its steps. +Its first step contains $z$, its last does not, and its suffix consisting +of the reversed steps $E_b,\ldots,E_a$ has sum divisible by $z$. +This suffix starts strictly after the first step because $ba$. The interval appearances +and unequal consecutive tuples persist under reversal. The reversed +subword is therefore forward prohibited. Minimality forces $a=1$. + +Choose a label $z'$ newly entering on step $2$, which is possible because +the first two tuples differ. Let $b'$ be the smaller of $b$ and the last +index carrying $z'$. Its contribution to \eqref{q:active-zero-prefix} +is $E_2+\cdots+E_{b'}$. If that contribution were zero modulo $z$, +then $b'>2$, since $z\nmid E_2$. The reversed subword +$E_m,\ldots,E_2$ would then be forward prohibited, with controlling +prime $z$ and suffix $E_{b'},\ldots,E_2$. It is shorter than the +original word, contradicting minimality. Hence $z'$ is active through +\eqref{q:active-zero-prefix}. Since $y$ occurs on step $1$ and $z'$ does +not, its contribution before step $2$ is $E_1\not\equiv0\pmod{z'}$. +This proves \eqref{q:active-prefix} with $i=2$ and $z'$ in place of $z$. +\end{proof} + +We now have two ways to obtain a usable congruence. An active label +already appears with an invertible coefficient in an internal relation. +For a private label that is not active, Lemma~\ref{q:active} supplies a +nonzero prefix contribution modulo another active label. If that active +label also occurs in another witness, positivity lets us compare the +two departures to obtain a congruence. The next proof either selects +enough internal relations or arranges enough of these comparisons in +an order suitable for elimination. + +\begin{lemma}[Elimination of the singleton class]\label{q:singletons} +For a fixed bin, the total contribution to the absolute majorants in +\Cref{q:mixed-difference} from main words with $S_1>L^{1/4}$, including +the external factor $|\mathcal D|^2M$ in the trace expansion, is at most +\[ + \exp(-L^{21/20}) +\] +for sufficiently large $L$, with the fixed parameters held fixed. +\end{lemma} + +\begin{proof} +Whenever the integrand is nonzero, Lemma~\ref{q:witness-cover} supplies +$T$ witnesses positive in one common hybrid, with marked private +singletons. Order them as $w_1,\ldots,w_T$ by their attachment indices +$a_1\le\cdots\le a_T$ on the main word; order ties arbitrarily. +Write $t(y)$ for the unique main step carrying a main singleton $y$. +Thus the main position of $y$ has been passed at attachment $a$ exactly +when $t(y)0$ supplies every padding +divisibility. Thus every perfect step is positive in this hybrid, +and every main vertex satisfies $\sigma_j=1$. + +Within each half, split every maximal interval of consecutive perfect +positions into blocks of $s=\lfloor L^{1/10}\rfloor$ positions, followed +by one shorter block if necessary. Their number $B$ satisfies +\begin{equation}\label{q:block-count} + B\ll 1+k/s+S_1+U, +\end{equation} +with an absolute constant. Each block is a positive path whose +vertices all survive the prohibited-word deletion. + +\begin{lemma}[Geometry of perfect blocks]\label{q:block-intervals} +In each such block, the occurrences of every column prime form an +interval of consecutive positions. No nonempty subinterval of the +block has total displacement zero. +\end{lemma} + +\begin{proof} +If the interval assertion fails, choose across all columns two +consecutive occurrence groups having the shortest gap. Let $i$ be +the last position in the first group and $i'$ the first in the next, +and let $p$ be their common prime. The prime $p$ is absent from +$i+1,\ldots,i'-1$. Every label in the substring +$i,\ldots,i'-1$ has interval-shaped uses, since otherwise it would +give a shorter gap. Positivity at the departures of steps $i$ and +$i'$ gives +\[ + p\mid \sum_{a=i}^{i'-1}D_a, + \qquad\text{hence}\qquad + p\mid \sum_{a=i+1}^{i'-1}D_a. +\] +The length $i'-i$ cannot be two: the latter sum would be one +displacement whose tuple and padding both omit $p$, while $p\nmid h$. +The substring therefore has length at least three. It is a forward +prohibited word, with its suffix starting at its second step. +Pass to a minimal prohibited contiguous subword, allowing reversal. +Positivity survives these operations, and its starting vertex is +one of the main vertices. This contradicts $\sigma_j=1$ there. + +For the second assertion a single displacement is nonzero. In any +subinterval of length at least two, the first and last tuples differ. +Indeed, if they agreed in every column, the interval assertion would +make every tuple in that subinterval equal, contrary to the +nonbacktracking condition. Choose a first-step prime $p$ absent +from the last tuple. If the total displacement were zero, deleting +the first step would leave a suffix sum divisible by $p$. In a +two-step subinterval this is impossible, because $p$ does not divide +the last displacement. In length at least three it makes the +subinterval forward prohibited. Minimal descent gives the same +contradiction. A reversed subword may begin at the end of the block; +this endpoint is also a main vertex. At the end of the closed word +it is its initial vertex, so the survival condition still applies. +\end{proof} + +\subsection{A forest code independent of the coefficients} + +An equality pattern in a column is the partition of its $2k$ positions +according to equality of their prime labels; the numerical prime values +are not part of the pattern. In this subsection we count the union of +these patterns over all numerical label values, padding choices, signs, +and surviving hybrids that satisfy the indicated rank and occurrence +bounds. + +\begin{lemma}[Forest coding]\label{q:forest-code} +Suppose that +\[ + S_1\le L^{1/4},\qquad U\le L^{1/50},\qquad + r=\lfloor L^{1/50}\rfloor, +\] +and that, in every column, there are no $r$ pairs +\eqref{q:rank-pair} whose $2r$ vectors are jointly independent. Consider +lit-consistent terms of \Cref{q:mixed-difference} with a nonzero integrand. The +equality patterns that can occur in any one column belong to a set +of cardinality at most $\exp(Ck)$, for an absolute constant $C$. +This set can be chosen independently of all numerical coefficients +and padding choices. Consequently there are at most $\exp(CJk)$ +combined column patterns. +\end{lemma} + +\begin{proof} +Fix one realization and one column. Let $\mathcal Z$ be its set of +distinct labels and let $\mathcal V$ be the real vector space with +basis $(e_z)_{z\in\mathcal Z}$. If the label at position $a$ is +$z_a$, define its formal departure offsets by +\[ + v(i)=\sum_{aL^{1/50}$, with a high-rank column, or with $S_1>L^{1/4}$ are +$o(1)$ in total by \Cref{q:many-unlit,q:rank,q:singletons}, including +the external dimension factors of the trace expansion. We sum the +remaining contributions using \Cref{q:mixed-difference}. + +By \Cref{q:forest-code}, the combined column patterns lie in a single +family of at most $\exp(CJk)$ possibilities, independently of all +padding choices. Fix one such pattern, its numerical column labels, +the signs, and the lit or unlit designations. Replace +$|\Delta G|$ by $2^{S_1}$ and retain just one reciprocal per distinct +column prime in $b_{\mathcal L}$. Every nonsingleton provides at +least one such reciprocal: either it has a lit occurrence, or all +of its at least two occurrences are unlit. Dropping the other +reciprocals and lit consistency enlarges the nonnegative majorant. + +In $R$, drop closure, progression and block restrictions, arrival +cuts, and the low-$Q$ cuts. Retain the padding divisibilities, bin +conditions, departure weights, and departure $\rho_j$ cuts. The +closed-word identity for its denominators was established before +this enlargement. Conditional on the non-$P$ environment, +\Cref{q:padding-sum} with $m=2k$ bounds the full padding sum by +$K^{2k}$. All its factors are unchanged by the $P$-coordinate +overwrites, and the bound is uniform in the environment. Taking +its expectation preserves that bound. + +Let $m_i$ be the number of distinct labels in column $i$, and put +$m_P=\sum_i m_i$. Among the $2kJ$ column occurrences, exactly $S_1$ +belong to singleton labels and every other label occurs at least +twice. Hence +\[ + 2kJ\ge S_1+2(m_P-S_1),\qquad + m_P\le Jk+S_1/2. +\] +After the uniform padding bound, the numerical label sum is at most +\begin{equation}\label{q:column-mass} + \prod_{i=1}^J V_i^{m_i} + \le (2W)^{Jk+S_1/2} + \le (2W)^{Jk+L^{1/4}/2}. +\end{equation} +Here dropping distinctness among prime classes is simply an +enlargement of a positive reciprocal sum. The classes are already +identified by their first occurrences, so no further permutation +factor is introduced. + +There are at most $2^{2k}$ sign choices and $2^{2kJ}$ lit or unlit +designations. The mixed difference contributes $2^{S_1}$. The +external indices cost at most $|\mathcal D|^2M\le\exp(108L)$ for +large $L$, since $|\mathcal D|\le e^{2L}$ and +$M=\lceil e^{103L}\rceil$. Combining these bounds with the forest +count and \eqref{q:column-mass} gives +\begin{equation}\label{q:trace-preconstant} + \E\|H^k\|_{\mathrm{HS}}^2 + \le K^{2k}\exp(C_4Jk)(2W)^{Jk+L^{1/4}/2}+o(1), +\end{equation} +where $C_4$ is absolute. The factor $\exp(108L)$ is absorbed here +using $J\ge1$ and $k\ge L/2$. + +For every fixed $W$, once $L$ is sufficiently large, +$(2W)^{L^{1/4}/2}\le e^k$. This changes the required lower threshold +on $L$, but not the absolute constant in the exponential. An +absolute choice of $C_2$ therefore bounds the right-hand side of +\eqref{q:trace-preconstant} by +\[ + \bigl[K(C_2\sqrt W)^J\bigr]^{2k}. +\] +Finally, the finite-law comparison for the trace expansion changes +the expectation by at most $\exp(-L^9)$. Increasing the same +absolute constant $C_2$ absorbs this error as well and proves the +integer-interval assertion of \Cref{q:trace}. In particular, +$C_2$ is fixed before $W$ is selected; dependence on the fixed +parameters is confined to how large $X$ must be. +\end{proof} diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/07-spectral.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/07-spectral.tex new file mode 100644 index 0000000000000000000000000000000000000000..426cee9a7728ccde66b5e67864fa7458eddcf565 --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/07-spectral.tex @@ -0,0 +1,333 @@ +\section{Spectral transfer and the bound at every scale} +\label{q:sec-spectral} + +The moment bound for $H$ has only square-root dependence on each +prime supply's reciprocal mass. We now use it to control the retained +centered sums $C_j^\circ$. First we transfer control of $H$ to the +projected edge operator $E(\sum_d A_d)E$, then test against the +Liouville function with vertex weight $\sqrt w$. The factor $L$ in +the edge matrices yields a factor $1/L$ in the estimate for each bin, +compensating for the factor $L$ in the number $O(L/\eta)$ of bins. +After division by the retained mass $S_0\ge SV_*/2$, with $V_*\ge W^J$, +the spectral contribution will have the form +\[ + \frac K\eta\left(\frac C{\sqrt W}\right)^J +\] +for an absolute constant $C$. We prove the transfer and the required +finite-interval estimates before choosing $W$. + +The use of a nonbacktracking operator to control an adjacency operator +is exemplified by the weighted Ihara--Bass formula in +\cite[Section~4.2]{Pilatte26}. We prove the transfer needed here +directly for self-adjoint edge operators; distinct edge operators need +not commute. + +\subsection{A transfer lemma for self-adjoint edge matrices} + +\begin{lemma}\label{q:noncommuting} +Let $N\ge1$ and let $B_1,\ldots,B_N$ be self-adjoint operators on a finite-dimensional +complex Hilbert space $\mathcal H$, and let $P$ be an orthogonal +projection on $\mathcal H$. Define an operator $\mathscr B$ on +$\mathcal H^N$ by +\[ + \mathscr B_{i,j}=\1_{i\ne j}B_j. +\] +Suppose $a,b\ge0$, $\|B_i\|\le a$ for every $i$, and +\[ + \sum_{i=1}^N\|B_i v\|^2\le b^2\|v\|^2 + \qquad(v=Pv). +\] +Writing $\rho(\mathscr B)$ for its spectral radius, one has +\begin{equation}\label{q:noncommuting-bound} + \left\|P\left(\sum_{i=1}^N B_i\right)P\right\| + \le 3\max\{a,b,\rho(\mathscr B)\}. +\end{equation} +\end{lemma} + +\begin{proof} +Put $m=\max\{a,b,\rho(\mathscr B)\}$. If $m=0$, all $B_i$ vanish. +Otherwise set $t=(2m)^{-1}$. For real $|u|\le t$, the operators +$I-u\mathscr B$ and $I+uB_i$ are invertible. Define +\[ + F(u)=I-\sum_{i=1}^N uB_i(I+uB_i)^{-1}. +\] +Each summand is self-adjoint, since $B_i$ commutes with its own +resolvent. If $F(u)v=0$, set $z_i=(I+uB_i)^{-1}v$. Then +\[ + (u\mathscr Bz)_i + =\sum_{j\ne i}uB_jz_j + =v-uB_i z_i=z_i. +\] +A nonzero $v$ would give a nonzero $z$, contradicting the invertibility +of $I-u\mathscr B$. Hence $F(u)$ is invertible throughout $[-t,t]$. +It is positive definite there by continuity, because $F(0)=I$. + +The resolvent identity gives +\[ + F(u)=I-u\sum_i B_i+Q(u),\qquad + Q(u)=u^2\sum_i B_i^2(I+uB_i)^{-1}. +\] +Since $|u|\|B_i\|\le1/2$, the inverses in the last expression are +positive and bounded above by $2I$. For $v=Pv$, +\[ + 0\le\langle v,Q(u)v\rangle + \le2u^2\sum_i\|B_i v\|^2 + \le2u^2b^2\|v\|^2. +\] +Positivity of $F(t)$ and $F(-t)$ therefore yields +\[ + \left|\left\langle v,\sum_i B_i v\right\rangle\right| + \le \frac{1+2t^2b^2}{t}\|v\|^2 + \le3m\|v\|^2. +\] +Taking the supremum over unit vectors in the range of $P$ proves +\eqref{q:noncommuting-bound}. No step commutes two distinct $B_i$. +\end{proof} + +\subsection{Weighted row bounds and the degree projection} + +Fix a bin $j$ and one block of $M$ sites, with its matrices $A_d$, $H$ +and projection $E$ from the trace construction. Define +\[ + a_d(n)=\prod_{p\mid d}\left|\1_{p\mid n}-\frac1p\right|. +\] +If $A_d(n,n')\ne0$, the difference $n'-n$ is a multiple of $d$. +Thus $a_d(n)=a_d(n')$ along every edge of $A_d$. + +\begin{lemma}\label{q:weighted-rows} +On every block, in either the product model or the integer model, +\begin{align} + \|A_d\|&\le2K,\label{q:single-operator}\\ + \sum_{d\in\mathcal D}\|A_d v\|^2 + &\le4K^2(8W)^J\|v\|^2\qquad(v=Ev). + \label{q:square-operators} +\end{align} +\end{lemma} + +\begin{proof} +For each of the two edge orientations, the weighted absolute row sum +at $n$ is at most +\[ + a_d(n)\sum_q\frac{Lu(q)}{w(n)}\1_{q\mid n} + \1_{(d,q)\in\mathcal S_j}\1_{\rho_j(n,d)\le K/L} + \le K a_d(n). +\] +Indeed the sum without the last indicator is at most +$L\rho_j(n,d)$. For an incoming edge, $q$ divides one endpoint if +and only if it divides the other. All the other edge restrictions can +be discarded in this nonnegative bound. Consequently +\begin{equation}\label{q:weighted-row} + \sum_{n'}|A_d(n,n')| + \frac{\sqrt{w(n')}}{\sqrt{w(n)}}\le2K a_d(n). +\end{equation} + +The weighted Schur inequality for a real symmetric matrix $B$ follows +by applying +\[ + 2|v_n v_{n'}|\le + |v_n|^2\frac{\sqrt{w(n')}}{\sqrt{w(n)}}+ + |v_{n'}|^2\frac{\sqrt{w(n)}}{\sqrt{w(n')}} +\] +to its quadratic form. Apply it to $A_d$ on each level set of $a_d$; +these level sets are invariant under $A_d$. The resulting bound is +\begin{equation}\label{q:localized-square} + \|A_d v\|^2\le4K^2\sum_n a_d(n)^2|v_n|^2. +\end{equation} +In particular, $a_d(n)\le1$ proves \eqref{q:single-operator}. + +At a site retained by $E$, the total P-degree is at most $6WJ$. +Since $V_i\le2W$, the arithmetic-geometric mean inequality gives +\begin{align*} + \sum_d a_d(n)^2 + &\le\sum_d a_d(n) + \le\prod_{i=1}^J\bigl(\omega_{P_i}(n)+V_i\bigr)\\ + &\le\left(\frac{\omega_P(n)+\sum_iV_i}{J}\right)^J + \le(8W)^J. +\end{align*} +Sum \eqref{q:localized-square} over $d$. This proves +\eqref{q:square-operators} without requiring $E$ to commute with $A_d$. +\end{proof} + +\begin{proposition}\label{q:spectral} +There is an absolute constant $C_3$ with the following property. +For each bin $j$, average block origins over any integer interval to +which \Cref{q:trace} applies. Outside a fraction at most $e^{-2k}$ of +these origins, +\begin{equation}\label{q:compressed-norm} + \left\|E\left(\sum_d A_d\right)E\right\| + \le K(C_3\sqrt W)^J. +\end{equation} +\end{proposition} + +\begin{proof} +For every finite matrix, $\rho(H)^{2k}\le\|H^k\|_{\mathrm{HS}}^2$. +Thus \Cref{q:trace} and Markov's inequality show that +\[ + \rho(H)\le eK(C_2\sqrt W)^J +\] +except on a fraction at most $e^{-2k}$ of the origins. On each remaining +block apply \Cref{q:noncommuting} with +\[ + a=2K,\qquad b=2K(8W)^{J/2},\qquad P=E. +\] +For $J\ge1$ and $W\ge10$, the choice +\begin{equation}\label{q:C3-choice} + C_3=3\max\{2,2\sqrt8,eC_2\} +\end{equation} +gives \eqref{q:compressed-norm}. In particular, $C_3$ is independent of +$W,h,l$. +\end{proof} + +\subsection{Testing and averaging overlapping blocks} + +We now turn \eqref{q:compressed-norm} into an estimate for $C_j^\circ$, +the retained correlation in \Cref{q:cuts}. Its endpoint restrictions +are exactly those imposed by the $A_d$ and the projection $E$. + +\begin{proposition}\label{q:testing} +For every bin $j$ and all sufficiently large $X$, +\begin{equation}\label{q:bin-estimate} + |C_j^\circ|\ll \frac{S K(C_3\sqrt W)^J}{L}+e^{-L}. +\end{equation} +The threshold may depend on the fixed $h,l,W$, while $C_3$ is the +absolute constant in \eqref{q:C3-choice}. +\end{proposition} + +\begin{proof} +Write $T=T_j$ and $N=\lfloor T\rfloor$, and average over blocks +$I_t=\{t+1,\ldots,t+M\}$, $0\le tR_L$. If $M\le n\le N$, then +$m_N(n,r)=M-r$. The lower boundary and the extra sites $NC_1+20$, where $C_1$ is the absolute exponent in that comparison. +The trace proof gives an absolute $C_2$, and +\eqref{q:C3-choice} then gives an absolute $C_3$. Now choose $W\ge10$ +so large that +\begin{equation}\label{q:W-choice} + e^5 C_3/\sqrt W\le e^{-1}. +\end{equation} +Only after fixing these constants do we increase the threshold for +$X$, allowing it to depend on $h,l,W$. + +Recall $\eta=e^{-J}$, $K=e^{4J}$ and +$J\le\delta\log L/(6W)$ with $\delta=1/200$. +The spectral term in \eqref{q:finish} is at most $e^{-J}$ by +\eqref{q:W-choice}. The other terms satisfy +\begin{align*} + 2^JK^{-1}&=e^{-(4-\log2)J}\le e^{-J},\\ + 2^Je^{-2WJ}&=e^{-(2W-\log2)J}\le e^{-J},\\ + e^J2^J L^{-1/20}&\le e^{-J},\qquad + 2^J L^{-100}\le e^{-J}. +\end{align*} +For the last line, it is enough to note that +\[ + \frac{\delta(2+\log2)}{6W}<\frac1{20},\qquad + \frac{\delta(1+\log2)}{6W}<100. +\] +Also $e^{-L^{0.9}}+X^{-1}\ll e^{-J}$ for large $X$. Hence +$|F(X)|\ll_{h,l}e^{-J}$. Finally, +\[ + e^{-J}\le e L^{-\delta/(6W)} + =e(\log X)^{-c},\qquad c=\frac{\delta}{6WA}>0. +\] +All three constants $A,W,c$ are absolute. The argument applies to +every sufficiently large real $X$, with no excluded scales. On the +remaining bounded range $3\le X\le X_0(h,l)$, the estimate follows +from $|F(X)|\le1$ after enlarging the implied constant. This proves +\Cref{q:progression}. +\end{proof} diff --git a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/08-affine.tex b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/08-affine.tex new file mode 100644 index 0000000000000000000000000000000000000000..7f01444846c63eb67381c1f4b2b575bc78eff5de --- /dev/null +++ b/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/quantitative/08-affine.tex @@ -0,0 +1,51 @@ +\section{The quantitative affine bound}\label{q:affine-transfer} + +The progression estimate for Liouville gives the same logarithmic +exponent for every fixed affine pair. The finite initial interval +affects only the constant. + +\begin{proof}[Proof of \Cref{thm:q-affine}] +Fix $a_1,a_2\ge1$ and $b_1,b_2\ge0$ with nonzero determinant, and +put +\[ + l=a_1a_2,\qquad c_0=\min(a_2b_1,a_1b_2),\qquad + h=|a_1b_2-a_2b_1|>0. +\] +Multiplying the two affine arguments by $a_2$ and $a_1$, respectively, +and using complete multiplicativity gives +\begin{equation}\label{q:quant-affine-identity} + \lambda(a_1n+b_1)\lambda(a_2n+b_2) + =\lambda(l)\lambda(ln+c_0)\lambda(ln+c_0+h). +\end{equation} +Here $\lambda(l)^2=1$. For real $Y\ge0$ set +\[ + P(Y)=\sum_{1\le m\le Y} + \lambda(m)\lambda(m+h)\1_{m\equiv c_0\pmod l}. +\] +The sum is empty for $Y<1$. Let $c>0$ be the absolute exponent in +\Cref{q:progression}. That estimate applies to the residue $c_0$ +without any coprimality restriction and gives +\[ + |P(Y)|\ll_{l,h,c_0}\frac{Y}{(\log Y)^c}\qquad(Y\ge3). +\] +For every real $X\ge3$, the integers in the class $c_0\pmod l$ +with $c_00$, this Lefschetz operator +$\xi_X$ is nonzero. Lemma~\ref{norm:low-cohomology} gives +$H^2(X^{\mathrm B},\mathbb Q)=\mathbb Q e$ as a vector space, so the +image of $e$ in $\operatorname{Gr}^W_4$ is nonzero and is a nonzero +scalar multiple of $\xi_X$. Here $e\in W_4H^2$ because +$X^{\mathrm B}$ is smooth. Rescaling a Lefschetz operator preserves +all its isomorphisms. They hold over $\mathbb Q$, since the rational +maps in question become isomorphisms over $\mathbb C$. This proves +Theorem~\ref{betti:lefschetz}. No step took invariants under the +finite group $\Gamma$. diff --git a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/04-diagram.tex b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/04-diagram.tex new file mode 100644 index 0000000000000000000000000000000000000000..8cb0e267fde0b1e59a93271cb266e63ac1e78c18 --- /dev/null +++ b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/04-diagram.tex @@ -0,0 +1,299 @@ +\section{Weights for flat bundles of algebraic families} +\label{sec:diagram} + +Consider first a product $S\times X^{\mathrm B}$, with $S$ a closed +oriented smooth surface. The moving-point construction in the proof overview +requires integration over $S$ to lower cohomological degree by two while +preserving weights. On this product, give the K\"unneth factor +$H^a(S,\mathbb Q)$ pure weight zero and Hodge type $(0,0)$ in every degree, +and retain Deligne's mixed Hodge structure on $H^*(X^{\mathrm B},\mathbb Q)$. +Integration over $S$ then acts on the first factor without a Tate twist. +Even when $S$ is an algebraic curve, this convention differs from its +ordinary Deligne weights. + +The moving-puncture flag spaces need not be products. We therefore +extend this convention to bundles whose transition maps are locally +constant algebraic automorphisms of a fiber family. A finite diagram of +algebraic fibers will give the mixed Hodge structures together with their +compatibility with parallel transport and proper Gysin maps. Section~\ref{sec:logarithmic} +will construct the required bundles and identify the scalar unflagged +bundle as an actual flat product. The simplicial mixed-Hodge formalism +goes back to Deligne \cite[Section~8.3]{Deligne1974}; Shende uses weight-zero +simplicial cochains to compute weights of tautological classes on +character varieties \cite[proof of the main Theorem]{Shende2017}. +Here we supply the relative construction and the Gysin compatibility +needed for the moving-point correspondence. + +\subsection{The finite diagram} + +Let $V$ be a smooth connected complex algebraic variety and let +$p:H\to V$ be an algebraic morphism of finite type. We assume that +$Rp_*\mathbb Z_H$ has bounded, locally constant, finitely generated +cohomology sheaves and that its natural ordinary base-change maps +\begin{equation} + \label{diag:basechange} + i_t^*Rp_*\mathbb Z_H\longrightarrow + R\Gamma(H_t,\mathbb Z) +\end{equation} +are isomorphisms for all $t\in V$, where $i_t:\{t\}\hookrightarrow V$. +These are hypotheses about ordinary direct image; smoothness alone would +not imply them for a nonproper morphism. + +Let $S$ be a compact smooth manifold. A \emph{flat bundle of the family +$H\to V$ over $S$} means a space $\mathscr H$ with a map to $S\times V$ +and local identifications +\[ + \mathscr H|_{U_i\times V}\simeq U_i\times H +\] +whose transition maps have the form +$(s,h)\mapsto(s,g_{ij}(h))$, with $g_{ij}$ an algebraic automorphism +over $V$, locally constant in $s$. The maps $g_{ij}$ satisfy the cocycle +identity. Write $\mathscr H_t$ for the total space over $S$ at $t\in V$. +Thus $\mathscr H_t$ includes the parameter space $S$; it need not be an +algebraic variety. + +A map of flat bundles will mean one which, in simultaneous flat +trivializations, is the identity on $S$ times an algebraic map of the +fiber families. + +Choose a finite good cover $\mathcal U=\{U_i\}$ refining a flat +trivializing cover. Every nonempty intersection $U_I=\bigcap_{i\in I}U_i$ +is contractible. Choose a trivialization on each $U_I$. Inclusion of +intersections then gives a finite diagram of copies +$p_I:H_I\to V$ and algebraic isomorphisms over $V$. +The cocycle identity makes this a diagram of maps of varieties, before +passing to cohomology. Its index category is the poset of nonempty +intersections, ordered in the direction of cohomological restriction. + +We use the enhanced bounded derived category of mixed Hodge modules, +whose six operations and rational realization are compatible with +diagrams \cite[Section~2.1, Theorem~2.1.1, Remark~2.1.2, and +Proposition~4.2.6]{TubachStacks2025}. +For an algebraic variety $T$, let $\mathbb Q_T^H$ denote the constant +mixed-Hodge-module complex with underlying complex $\mathbb Q_T$, +without a dimension shift. Define +\begin{equation} + \label{diag:complex} + K_{\mathscr H} + =\mathop{\mathrm{holim}}_I Rp_{I*}\mathbb Q_{H_I}^H + \quad\text{in }D^b\mathrm{MHM}(V). +\end{equation} +Here the homotopy limit means the derived limit, retaining the +transition maps and their compatibilities. The index poset has a +finite-dimensional nerve, so this is a finite limit and remains +bounded. In particular, \eqref{diag:complex} uses the full derived +diagram, not only the cohomology groups of its terms. + +\begin{proposition}[Diagram weights] + \label{diag:variation} + Under the preceding hypotheses, $H^j(\mathscr H_t,\mathbb Q)$ carries + a graded-polarizable mixed Hodge structure, functorial for maps of flat + bundles that are algebraic in flat trivializations. As $t$ varies, + these structures form an admissible graded-polarizable variation of + mixed Hodge structures, with an integral underlying local system up + to torsion. They are computed by $H^j(i_t^*K_{\mathscr H})$. + + If $H^j(H_t,\mathbb Q)$ is Hodge--Tate for every $j$ and $t$, then + $H^j(\mathscr H_t,\mathbb Q)$ is Hodge--Tate as well. We call its + weight filtration the \emph{diagram weight filtration}. +\end{proposition} + +\begin{proof} + First we identify the underlying ordinary complex. Open-cover + descent computes the ordinary direct image of $\mathscr H\to V$ + from the spaces $U_I\times H_I$. Projection to $H_I$ induces a + quasi-isomorphism on direct images over $V$: over any open subset + of $V$, its fiber in this product is the contractible space $U_I$. + These projections commute with the transition maps. Their derived + limit therefore identifies the rational realization of + $K_{\mathscr H}$ with the ordinary direct image of $\mathscr H\to V$. + The same argument works with integral coefficients. + + A finite limit of bounded complexes with locally constant cohomology + again has locally constant cohomology. Its integral cohomology + sheaves have finite type. Point pullback commutes with finite limits, + and the termwise base-change maps are isomorphisms by + \eqref{diag:basechange}. The corresponding maps in mixed Hodge + modules are isomorphisms because rational realization is conservative. + It follows that $i_t^*K_{\mathscr H}$ computes the derived diagram of + the algebraic fibers $H_{I,t}$ and that its underlying cohomology is + $H^*(\mathscr H_t,\mathbb Q)$. + + Put $v=\dim V$. Since the ordinary cohomology sheaves of + $K_{\mathscr H}$ are locally constant, its perverse cohomology + ${}^pH^{j+v}(K_{\mathscr H})$ is a smooth mixed Hodge module. + Smooth mixed Hodge modules, after the dimension shift, are admissible + graded-polarizable variations + \cite[Section~3.2, Equation~(3.2.1)]{SaitoExtension1990}. + Applying point pullback to the perverse truncations shows that the + fiber of this variation is $H^j(i_t^*K_{\mathscr H})$: a smooth + perverse object has only one nonzero fiber degree, namely $-v$. + + Finally, mixed Hodge structures whose weight gradeds have only Hodge + types $(r,r)$ are closed under subobjects, quotients, and extensions. + The long exact sequence of a cone thus shows that complexes with + Hodge--Tate cohomology are closed under finite limits. Applying this + observation to the fiber diagram proves the last assertion. + Algebraic maps of the diagrams give the stated functoriality. + A common good refinement compares different choices of cover and + trivializations: its comparison maps are isomorphisms after ordinary + descent, hence also in mixed Hodge modules by conservativity. +\end{proof} + +The construction also preserves the cup product. Each algebraic +direct image of the constant complex has its functorial cup product, +and the transition pullbacks preserve it. The derived limit inherits +this product. Under the open-cover comparison in the proof of +Proposition~\ref{diag:variation}, it is the ordinary cup product on +$H^*(\mathscr H_t,\mathbb Q)$. Thus parallel transport in $V$ preserves +both the cohomology ring and its weight filtration. + +\subsection{Products and proper Gysin maps} + +We next identify the maps that preserve these weights. + +\begin{proposition}[Weight shifts] + \label{diag:gysin} + The diagram weight filtrations have the following properties. + \begin{enumerate} + \item For the product $S\times M$, with $M$ a complex algebraic + variety, the ordinary K\"unneth isomorphism identifies its mixed Hodge + structure with that on + \[ + H^*(S,\mathbb Q)\otimes H^*(M,\mathbb Q), + \] + where every $H^a(S,\mathbb Q)$ is pure of weight zero and type $(0,0)$, + and $H^*(M,\mathbb Q)$ has its Deligne mixed Hodge structure. + If $S$ is oriented and closed of real dimension $d_S$, projection + pullback and integration over $S$ preserve weight; the latter has + cohomological degree $-d_S$. + \item Pullbacks by maps of flat bundles preserve weight. + \item Suppose $u:\mathscr H_t\to\mathscr H'_t$ is a proper map of + flat bundles with smooth equidimensional algebraic fibers. Put + $c=\dim_{\mathbb C}H'_t-\dim_{\mathbb C}H_t$. With the complex + orientations on the fibers, its ordinary Gysin map satisfies + \[ + u_*:W_aH^j(\mathscr H_t)\longrightarrow + W_{a+2c}H^{j+2c}(\mathscr H'_t). + \] + If $S$ is oriented, this is the Gysin map for the induced orientations + on the total spaces. + \end{enumerate} +\end{proposition} + +\begin{proof} + For a product, the diagram is constant. Its derived limit is + \[ + C^*(N\mathcal U,\mathbb Q(0))\otimes R\Gamma(M,\mathbb Q_M^H), + \] + where $N\mathcal U$ is the finite nerve of the good cover. The + first complex has a copy of $\mathbb Q(0)$ for each simplex, with + its ordinary simplicial differential. All its cohomology therefore + has weight zero. The external-product comparison on the cover + gives exactly the ordinary K\"unneth map. Evaluation on a simplicial + fundamental cycle defines + \[ + C^*(N\mathcal U,\mathbb Q(0))\longrightarrow + \mathbb Q(0)[-d_S]. + \] + This is a morphism of complexes of mixed Hodge structures and + realizes ordinary integration over $S$. It has no Tate twist. + Projection pullback is induced by the unit of the same cochain + complex. This proves the first assertion; functoriality of algebraic + pullbacks proves the second. + + For the third assertion, fix $t$ and choose simultaneous + trivializations of the map. On each algebraic fiber map + $u_{I,t}:H_{I,t}\to H'_{I,t}$, smooth + dualizing complexes and the proper counit give the Gysin morphism + \begin{equation} + \label{diag:trace} + Ru_{I,t*}\mathbb Q^H_{H_{I,t}}\longrightarrow + \mathbb Q^H_{H'_{I,t}}[2c](c). + \end{equation} + Its degree and Tate twist give the displayed weight shift. + Complex orientations are preserved by the algebraic transition + isomorphisms, and the proper counit is natural. Consequently + \eqref{diag:trace} gives a coherent map of the finite diagrams and + hence of their derived limits. + + It remains to identify its ordinary realization. On + $U_I\times H_{I,t}$ the map is the identity on $U_I$ times $u_{I,t}$. + The orientation trace + for this product is the fiber trace, tensored with the identity on + $U_I$. It is compatible with restriction to intersections and with + the transition isomorphisms. Open-cover descent therefore identifies + the realized map with the ordinary topological Gysin map of the + total spaces. In particular, the construction gives the actual + Gysin map, not merely a map with the same source and target. +\end{proof} + +\subsection{Finite monodromy on weight gradeds} + +The diagram construction has now supplied a variation on the +cohomology of the total spaces. Hodge--Tate fibers and the integral +structure put a strong restriction on its monodromy. + +\begin{lemma} + \label{diag:finite} + Assume the Hodge--Tate hypothesis of + Proposition~\ref{diag:variation}. The combined monodromy action + on all $\operatorname{Gr}^W_aH^j(\mathscr H_t,\mathbb Q)$ has finite + image. For any loop with cohomology monodromy $T$, there is an integer + $M>0$ such that $T^M$ is unipotent and + \begin{equation} + \label{diag:log} + N=\frac{1}{M}\log(T^M) + \quad\text{satisfies}\quad N(W_a)\subset W_{a-2}. + \end{equation} + The logarithm is a finite polynomial. On the total cohomology ring, + $N$ is a derivation of cohomological degree zero. +\end{lemma} + +\begin{proof} + Each nonzero weight graded is a polarizable variation of pure type + $(r,r)$ and weight $2r$. After the conventional sign, its flat + polarization is positive definite on the underlying real vector + space. Let $\mathcal L$ be the torsion-free quotient of the integral + cohomology local system. Intersecting $\mathcal L$ with the rational + weight sub-local systems and taking successive quotients gives an + invariant full lattice in each rational graded piece. These + intersections are saturated because the subspaces are rational. + + The automorphisms of a lattice preserving a positive definite form + constitute a finite group: in a fixed lattice basis, every column of + such an automorphism has a prescribed bounded norm and hence lies + in a finite set. Thus each graded monodromy image is finite. Only + finitely many cohomological degrees and weights occur, which proves + the combined assertion. + + Choose $M$ killing the action of the specified loop on all weight + gradeds. Then + \[ + (T^M-1)W_a\subset W_{a-1}=W_{a-2} + \] + whenever $a$ is even; the identical conclusion for odd $a$ follows + because successive odd and even steps coincide. The operator + $T^M-1$ is nilpotent, since the weight filtration is finite. + Its finite logarithm has the same lowering bound. + + Parallel transport preserves cup products. For every integer + $k\geq0$, the polynomial operator $\exp(kMN)=T^{Mk}$ is therefore + a ring automorphism. For cohomology classes $\alpha,\beta$, the + identity + \[ + \exp(kMN)(\alpha\beta) + =\exp(kMN)(\alpha)\exp(kMN)(\beta) + \] + is an equality of polynomials in $k$. Comparing linear coefficients + proves $N(\alpha\beta)=N(\alpha)\beta+\alpha N(\beta)$. + Monodromy preserves cohomological degree, so $N$ does as well. +\end{proof} + +To apply these results, we will verify locally constant algebraic +transition maps for the family with a moving marked point, together +with simultaneous such transitions for its flag correspondence. +We will also identify the scalar unflagged bundle as a flat product. +Once those inputs are established, Proposition~\ref{diag:gysin} +controls its push-pull maps, and Lemma~\ref{diag:finite} supplies the +weight-lowering logarithm on the same ordinary cohomology groups. diff --git a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/05-logarithmic.tex b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/05-logarithmic.tex new file mode 100644 index 0000000000000000000000000000000000000000..3cbe79f69d01784df20feb5a8b30572d463ed428 --- /dev/null +++ b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/05-logarithmic.tex @@ -0,0 +1,559 @@ +\section{Moving logarithmic poles and cohomology comparisons} +\label{sec:logarithmic} + +We now construct a single family through which the Higgs and Betti +correspondences can be compared. The point carrying the full flag must move +over a compact curve. Its motion changes the determinant, so the first step +is to compensate for that change by a line bundle. The resulting comparison +will apply to the whole fixed-determinant cohomology. + +\subsection{The determinant correction and the scalar correspondence} + +Choose a point $x_0\in C$. For the construction we first take +$L=\mathcal O_C(d x_0)$; we return to an arbitrary determinant at the end of +this section. The line bundle $\mathcal O_C(d x)$ has a canonical +logarithmic connection $d_{d x}$: its canonical meromorphic section is +horizontal. Its residue at $x$ is $-d$. Multiplying a connection by +$\lambda$ produces a $\lambda$-connection, whose Leibniz rule is +$D(fs)=\lambda\,df\otimes s+fD(s)$. + +Pull back the finite \emph{\'etale} map +$[n]:\operatorname{Pic}^0(C)\to\operatorname{Pic}^0(C)$ by +\[ + C\longrightarrow\operatorname{Pic}^0(C),\qquad + x\longmapsto\mathcal O_C(d x-d x_0), +\] +and choose a connected component $S$ of this pullback. Then $S$ is a smooth +projective connected curve, and its map $x:S\to C$ is finite \emph{\'etale} +and surjective. Write $\Delta\subset C\times S$ for the graph of $x$. +A Poincar\'e bundle gives a line bundle $J$ on $C\times S$ such that +\begin{equation} + J^{\otimes n}\simeq + \mathcal O_{C\times S}\bigl(d\Delta-d(x_0\times S)\bigr) + \otimes\operatorname{pr}_S^*B + \label{log:root} +\end{equation} +for some line bundle $B$ on $S$. The possible base factor is the only +ambiguity in this equality of families. + +A line pulled back from $S$ has a canonical relative connection along $C$. +Equip the right side of \eqref{log:root} with the relative logarithmic +connection given by its canonical meromorphic section and this connection +on the base factor. Dividing its local connection forms by $n$ defines +a unique compatible connection on $J$. This division glues: the +transition equation for $J^{\otimes n}$ is $n$ times the equation for $J$. +Let $D_{J,\lambda}$ be the resulting relative $\lambda$-connection. Its +residues are $-\lambda d/n$ along $\Delta$ and $+\lambda d/n$ along +$x_0\times S$. + +Put +\[ + \mathfrak t_0=\{(u_1,\ldots,u_n)\in\mathbb A^n: + \textstyle\sum_i u_i=0\},\qquad + \Lambda=\mathbb A^1_\lambda\times\mathfrak t_0. +\] +Let $\mathcal X\to\mathbb A^1_\lambda$ parametrize stable logarithmic +$\lambda$-connections on rank-$n$ bundles, with determinant +$(\mathcal O_C(d x_0),\lambda d_{d x_0})$ and residue +$-\lambda(d/n)I$ at $x_0$. Stability here and below is slope stability for +subbundles preserved by the operator. At $\lambda=0$ the scalar residue +is zero, so the fiber $X_0$ is the ordinary trace-free Higgs moduli space $X$. + +Let $\mathcal Y\to S\times\Lambda$ parametrize the following data: +\begin{itemize} + \item a stable logarithmic $\lambda$-connection $(E,D)$ with its only + possible pole at $x(s)$ and determinant + $(\mathcal O_C(d x(s)),\lambda d_{d x(s)})$; + \item a full flag of $E_{x(s)}$, preserved by the residue, on whose + ordered graded lines $Q_i$ the residue acts by $-\lambda d/n+u_i$. +\end{itemize} +The flag does not enter the stability condition. We write $Y_{\lambda,u}$ +for the fiber over $(\lambda,u)\in\Lambda$, \emph{retaining all of $S$}. +Thus $Y_{\lambda,u}$ is itself a family over the moving-point curve. + +At $u=0$, let $\mathcal Z$ be the closed locus where the residue is scalar, +and let $\mathcal O$ be the same scalar-residue family with the flag +forgotten. The correspondence over $\mathbb A^1_\lambda$ is shown +fiberwise in Figure~\ref{log:scalar-correspondence}. + +\begin{figure}[htbp] +\centering +\begin{tikzcd}[column sep=large] + X_\lambda & O_\lambda\simeq S\times X_\lambda + \arrow[l,"\operatorname{pr}"'] & Z_\lambda + \arrow[l,"\pi"'] \arrow[r,hook,"\iota"] & Y_{\lambda,0} +\end{tikzcd} +\caption{The scalar correspondence. All moving-pole spaces retain the +curve $S$; $\operatorname{pr}$ has fiber $S$, $\pi$ is a +flag projection of relative dimension $b$, and $\iota$ has codimension +$b$. The residue on $Y_{\lambda,0}$ may have a strictly flag-lowering +nilpotent part in addition to its prescribed scalar part; $Z_\lambda$ +requires that nilpotent part to vanish. The derived morphisms and their cohomological and diagram-weight shifts +are recorded in Table~\ref{spec:shift-table}.} +\label{log:scalar-correspondence} +\end{figure} + +Here $\pi$ forgets the flag and $\iota$ is the scalar-residue inclusion. +The first map comes from the following product identification. + +\begin{lemma} +\label{log:scalar-product} +Tensoring by $(J,D_{J,\lambda})$ gives an algebraic isomorphism +$\mathcal O\simeq S\times\mathcal X$ over $S\times\mathbb A^1$. +At $\lambda=0$ it preserves the ordinary Hitchin coefficients. +\end{lemma} + +\begin{proof} +By \eqref{log:root}, tensoring fixed-pole data by $J$ changes its +determinant to $\mathcal O_C(d x(s))$, up to a base line, and changes its +determinant operator to the prescribed one. At $x_0$, the residue +$+\lambda d/n$ of $J$ cancels the residue $-\lambda d/n$ of the original +rank-$n$ connection. + +This cancellation also holds where the moving and fixed sections meet. +In a coordinate $z$ with moving section $z=a(s)$, the scalar polar part +contributed by $J$ is +\[ + -\frac{\lambda d}{n}\frac{dz}{z-a(s)} + +\frac{\lambda d}{n}\frac{dz}{z}. +\] +Adding the fixed-pole form leaves only the first term. The identity is an +identity of relative meromorphic forms over the whole local base, including +$a(s)=0$. Thus the tensor product has a single logarithmic pole at the +moving point. The inverse construction tensors by $J^{-1}$. + +Base-line twists do not change the induced map to coarse moduli, and +$\deg J_s=0$ preserves stability. When $\lambda=0$, the Higgs field of +$J$ is zero, so the characteristic polynomial is unchanged. +\end{proof} + +The relative quasiprojective moduli of logarithmic $\lambda$-connections +exist by the standard construction; we use the formulation in +\cite[Section~2.2]{dCFHLogHodge}. Fixed determinant is obtained by taking +the inverse image of its rank-one section. Adding a residue-invariant +full flag is projective over the unflagged moduli; its ordered residue +eigenvalues are then regular functions on the flag parameter space. +The prescribed-residue equations are closed. + +We will use universal bundles only \emph{\'etale} locally. Coprimality +makes semistable pairs stable, with scalar automorphisms. Their stack is +a scalar gerbe over its coarse space, or a $\mu_n$-gerbe after choosing a +determinant isomorphism; compare +\cite[Lemma~5.3 and Remark~5.4]{dCFHLogHodge}. +The universal projective bundle, $\operatorname{End}E$, and the line +ratios $Q_i\otimes Q_j^{-1}$ have trivial scalar action and descend. +Individual $Q_i$ need not descend, and no construction below requires them +to do so. + +\subsection{Smooth families and actual restriction isomorphisms} + +Our next objective is to identify the cohomology of all fibers by +restriction from the total space. Smoothness alone would not suffice, +because these moduli spaces are not proper. The scaling action supplies +the additional compactness property that is needed. + +\begin{proposition} +\label{log:families} +The morphism $\mathcal Y\to S\times\Lambda$ is smooth of relative +dimension $2R+2b$, and $\mathcal X\to\mathbb A^1$ is smooth of relative +dimension $2R$. Their total spaces are semiprojective for scaling the +operators, which scales $\lambda$ and every $u_i$ with weight one. +The same holds for the inverse image in $\mathcal Y$ of every linear +subspace of $\Lambda$ through zero. In +Figure~\ref{log:scalar-correspondence}, $\pi$ is a smooth proper flag +projection of relative dimension $b$, and $\iota$ is a regular embedding +of codimension $b$ with normal bundle $T_\pi^*$. +\end{proposition} + +\begin{proof} +We first establish zero limits for scaling. Given an unflagged object, +keep its bundle fixed and scale its operator. This extends the orbit to +the stack of all logarithmic $\lambda$-connections, without imposing +semistability on the special fiber. Semistable reduction +\cite[Proposition~4.48]{FHZ2025} replaces this DVR family by a semistable +one after a finite extension, without changing its generic fiber. +Its hypotheses hold for $\mathrm{GL}_n$ over $\mathbb C$. + +The determinant of the replacement and the prescribed determinant agree +generically, hence agree over the DVR by separatedness of the rank-one +moduli space. A generic flag extends because the flag variety of the +extended bundle at the pole is proper. Residue preservation and the +\emph{ordered} diagonal residue equations are closed, and therefore +persist at the special fiber. Coprimality makes the limiting semistable +pair stable. The same reasoning applies to the scalar-residue condition. + +A finite extension of the DVR is enough for a limit in the original +coarse space. Indeed, embed this quasiprojective space in projective +space as a locally closed subvariety. The original orbit has a unique +projective limit. The limit after finite base change is that point and +belongs to the coarse moduli space, so the original orbit extends there. +This proves the required zero-limit property. + +We next check smoothness over $\lambda=u=0$. At such a point put +\[ + \mathcal F=\operatorname{ParEnd}_0(E),\qquad + \mathcal G=\mathcal F^\vee\otimes K_C, +\] +where $\mathcal F$ is the sheaf of trace-free endomorphisms preserving +the flag at $x(s)$. Trace pairing identifies $\mathcal G$ with the +trace-free logarithmic Higgs fields having strictly flag-lowering +residue. The relative deformation complex is +\begin{equation} + \mathcal F\xrightarrow{[\theta,-]}\mathcal G. + \label{log:deformation} +\end{equation} + +Here is the lifting argument, including the parameter directions. +For a small Artin extension with prescribed lifts of $s,\lambda,u$, the +bundle, determinant isomorphism and flag lift because the obstruction +lies in $H^2(C,\mathcal F)=0$. In local frames respecting the lifted +flag, the operator lifts with its prescribed determinant and diagonal +residues: these are affine-linear conditions, and trace splits in +characteristic zero. Its gluing obstruction lies in +$H^1(C,\mathcal G)$ tensored with the square-zero ideal. Changing the +bundle and flag gluing changes this obstruction by the image of +$H^1(C,\mathcal F)$ under \eqref{log:deformation}. + +This map on $H^1$ is surjective. Its Serre dual is, up to sign, the map +on $H^0$ in \eqref{log:deformation}; its kernel consists of trace-free +automorphisms of the stable Higgs pair preserving the flag, hence is +zero. Thus every such lifting problem has a solution. Since +\[ + \chi(\mathcal F)=-(n^2-1)(g-1)-b=-R-b, +\] +the relative tangent dimension is $-2\chi(\mathcal F)=2R+2b$. +For the scalar unflagged family, replace $\mathcal F$ by +$\operatorname{End}_0(E)$ and obtain dimension $2R$ in the same way. +Finite scalar stabilizers act trivially on these deformations, so the +calculation also proves smoothness of the coarse spaces. + +The nonsmooth locus of the morphism is closed and scaling invariant. +If it contained a point, it would contain that point's zero limit, +contradicting the calculation over $\lambda=u=0$. This proves smoothness +everywhere. + +A scaling-fixed point has $\lambda=u=0$. Its logarithmic Hitchin +coefficients vanish, because their scaling weights are positive. +The logarithmic Hitchin map is proper; its base has coefficients in +$H^0(C,K_C(x(s))^{\otimes i})$ as in +\cite[Section~2.4, Equation~(4)]{dCFHLogHodge}. +The fixed locus is closed in its nilpotent fiber after adding flags +and imposing determinant and residue conditions. Since flags and $S$ +are proper, that fixed locus is proper. Together with the zero limits, +this proves semiprojectivity. The inverse image of a linear subspace +through zero is smooth by base change and inherits both properties. + +Finally $\pi$ is the full flag bundle of the universal projective bundle. +On the $u=0$ family, subtracting the prescribed scalar from the residue +gives a section of the bundle of strictly flag-lowering matrices. +Its zero scheme is $\mathcal Z$. The smoothness and dimensions just +proved show that it is a regular zero scheme of codimension $b$. +Trace pairing identifies this residue bundle along $\mathcal Z$ with +the cotangent bundle of the flag fibers, giving $N_\iota\simeq T_\pi^*$. +\end{proof} + +\begin{corollary} +\label{log:restriction} +For every $(\lambda,u)\in\Lambda$, restriction induces an isomorphism +\begin{equation} + \rho_{\lambda,u}:H^*(\mathcal Y,\mathbb Q) + \xrightarrow{\ \sim\ }H^*(Y_{\lambda,u},\mathbb Q). + \label{log:rho} +\end{equation} +Restriction to the inverse image of any line through zero in $\Lambda$ +is also an isomorphism, as is restriction from that inverse image to any +of its fibers. The analogous restriction isomorphisms hold for all +four families in Figure~\ref{log:scalar-correspondence} over the +$\lambda$-line. They commute with pullbacks and with the proper Gysin +maps in that correspondence, including integration in the $S$ factor. +\end{corollary} + +\begin{proof} +Recall that the core of a smooth semiprojective variety is the union +of the sets whose scaling orbits have limits as the parameter tends +to infinity. Its inclusion induces a cohomology isomorphism +\cite[Theorem~1.3.1]{HRV2015}. Because every coordinate on $\Lambda$ +has positive weight, a point of $\mathcal Y$ with such a limit must +lie over zero. Hence $\mathcal Y$ and its inverse image +$\mathcal Y_\ell$ of any line $\ell$ through zero have the same core. +Their restrictions to the core commute, so +$H^*(\mathcal Y)\to H^*(\mathcal Y_\ell)$ is an isomorphism. + +The map $\mathcal Y_\ell\to\ell$ is smooth and surjective. For +surjectivity, its image is a scaling-invariant open subset containing +zero, and therefore contains the whole affine line. Apply +\cite[Corollary~1.3.3]{HRV2015}: for a smooth surjection from a smooth +semiprojective variety to the affine line, equivariant for a positive +base weight, restriction to each fiber is an isomorphism. This proves +\eqref{log:rho} and its line version. The same proof applies to +$\mathcal X$, $\mathcal O$ and $\mathcal Z$. + +Pullback compatibility is functorial. For Gysin maps, the families +are smooth over the line, the scalar embedding is transverse to fiber +restriction, and the flag and product projections are smooth proper. +The transverse base-change identities for Gysin maps give the claimed +compatibilities. +\end{proof} + +\subsection{Riemann--Hilbert with a moving puncture} + +Let $V$, $q$ and $Y^{\rm B}\to V$ be the ordered eigenvalue family from +Section~\ref{sec:betti}. The exponent domain we need is +\begin{equation} + U=\left\{u\in\mathfrak t_0^{\rm an}: + t(u)=\bigl(\zeta e^{-2\pi\sqrt{-1}u_i}\bigr)_i\in V,\quad + u_i-u_j\notin\mathbb Z\setminus\{0\}\right\}. + \label{log:nonresonance} +\end{equation} +Notice that zero differences are allowed. In particular $0\in U$, +which is essential for the scalar correspondence. + +There is a flat bundle over $S$ with algebraic fiber family +$Y^{\rm B}\to V$. To define it, use local trivializations of the +punctured surfaces $C\setminus\{x(s)\}$ and record a full flag of local +subsystems near the puncture. We choose the peripheral-loop convention +in which its monodromy is the product of commutators used to define +$Y^{\rm B}$. Changes of surface markings act on the handle matrices +by group words and on the peripheral flag by the corresponding +conjugating word. They are algebraic maps over $V$. + +These transition maps are locally constant and satisfy the cocycle +condition as algebraic maps on the coarse spaces. To check this +precisely, let $\delta$ be the product of commutators in the free +fundamental group of the punctured surface. A change of marking +gives an outer automorphism represented by $\varphi$, with +$\varphi(\delta)=w\delta w^{-1}$. Word evaluation transforms a +representation to $\rho\varphi$ and its flag by $\rho(w)$. +Changing $\varphi$ by an inner automorphism gives simultaneous +conjugation. Changing $w$ multiplies it on the right by an element +centralizing $\delta$. The cyclically reduced word $\delta$ is not +a proper power: each positive handle generator occurs just once. +Its centralizer in the free group is therefore $\langle\delta\rangle$. +Thus this last ambiguity is a power of peripheral monodromy and +preserves every invariant flag, including those with repeated +eigenvalues or nonsemisimple monodromy. The resulting maps compose +strictly after passage to the geometric quotient. Consequently no +choice of eigenlines or higher homotopy of transitions is needed. +The scalar flag correspondence from +Section~\ref{sec:betti} gives a corresponding diagram of flat bundles +over $S$. + +\begin{proposition} +\label{log:rh} +Over $U$, Riemann--Hilbert identifies $Y_{1,u}$ with the total space +over $S$ of this flat bundle at parameter $t(u)$. These identifications +are homeomorphisms of families. At $u=0$ they identify the scalar +correspondence of Figure~\ref{log:scalar-correspondence} at $\lambda=1$ with +the flat Betti scalar correspondence, compatibly with its pullbacks +and Gysin maps. +\end{proposition} + +\begin{proof} +The local exponents of a connection are $r_i=-d/n+u_i$. +For the regular-singular correspondence and logarithmic lattices, including +Manin's extension theorem and Katz's nonresonance corollary, see +Deligne \cite[II, Proposition~5.4, Remark~5.5(ii), +Corollary~5.6, and Theorem~5.9]{Deligne1970}. We spell out the +nonresonant local construction to track the ordered flags and dependence +on the moving parameters. +In a disc coordinate $z$, write a logarithmic connection as +$d+A(z)\,dz/z$, with $A(0)=A_0$. The regular-singular reduction +to $d+A_0\,dz/z$ by a gauge equal to the identity at zero is obtained +recursively using the operators $k+\operatorname{ad}(A_0)$, $k\geq1$. +Their eigenvalues are $k+r_i-r_j$, so they are invertible under +\eqref{log:nonresonance}, even when $A_0$ has repeated eigenvalues or +a nonzero nilpotent part. On a small parameter neighborhood their +inverse norms are $O(1/k)$ for large $k$. The usual power-series +majorant for the analytic coefficients therefore gives convergence +on a common smaller disc and continuous dependence on parameters. + +In this constant-residue model a residue-invariant flag gives a +monodromy-invariant flag. Conversely, the prescribed exponents select +one logarithm for each monodromy eigenvalue: equal exponential values +have equal selected exponents, by \eqref{log:nonresonance}. Functional +calculus on each generalized eigenspace then gives a logarithm of +monodromy preserving its invariant flag. This constructs the inverse +logarithmic extension with the specified ordered residues. Coordinate +and logarithm changes in the constant-residue model act by functions +of the residue, and hence preserve the flag; the construction is +intrinsic. In rank one the same construction gives exactly +$(\mathcal O_C(d x(s)),d_{d x(s)})$, so the determinant is the required +one. The subproduct condition defining $V$ makes the representation +irreducible, and thus ensures stability. + +Holonomy along local moving generators, together with this local +normal form, makes the forward maps continuous in the bundle and +in $s,u$. Regard the Betti family as pulled back to $U$ via $t$; +there is no assertion that $t$ is globally injective. The forward +map is a continuous bijection between manifolds of equal real +dimension, by Proposition~\ref{log:families} and the Betti +smoothness. Invariance of domain gives its inverse continuity. +For a fixed point of $S$ it is holomorphic on the moduli fibers, +so it preserves their complex orientations. + +At $u=0$, scalar residues correspond to scalar monodromy, and +forgetting or retaining the flag commutes with this construction. +The proper maps in the correspondence have their usual complex +orientations in the algebraic fibers and the same orientation on +$S$. Consequently their topological Gysin maps agree under these +homeomorphisms. +\end{proof} + +\begin{lemma} +\label{log:scalar-flat-product} +The product identification in Lemma~\ref{log:scalar-product}, at +$\lambda=1$, identifies the scalar unflagged Betti family with +$S\times X^{\rm B}$ as a flat bundle with algebraic fiber. Its +projection to $X^{\rm B}$ corresponds to $\operatorname{pr}$. +\end{lemma} + +\begin{proof} +Away from the fixed and moving poles, the nth power of the connection +on $J$ is the canonically trivial connection. Its holonomies +therefore lie in $\mu_n$. Locally in $S$, take a disc containing +$x_0$ and the moving point, and choose the handle generators outside +that disc. Their line holonomies depend continuously on $s$ and +take values in the finite group $\mu_n$, so are locally constant. +Tensoring multiplies the handle matrices by these constant scalars, +an algebraic map on the representation variety. The argument also +applies in collision neighborhoods. Thus the algebraic product +identification of the scalar connection family is compatible with +the locally constant algebraic Betti transitions. +\end{proof} + +We have now obtained the actual flat diagrams to which +Section~\ref{sec:diagram} will be applied. In particular, when that +construction gives weight zero to the cochains of $S$, its product +and integration statements concern the same projection +$\operatorname{pr}$ as the logarithmic correspondence. + +\subsection{The nonabelian-Hodge comparison on full cohomology} + +The restriction isomorphisms for $\mathcal X$ and Riemann--Hilbert give +\begin{equation} + H^*(X^{\rm B})\xrightarrow{\ \sim\ }H^*(X_1) + \xleftarrow{\ \sim\ }H^*(\mathcal X) + \xrightarrow{\ \sim\ }H^*(X). + \label{log:comparison} +\end{equation} +We must check that this comparison transports the weight filtration +specified by nonabelian Hodge theory, including its nontrivial +$\Gamma$-character summands. Agreement only after taking +$\Gamma$-invariants would not suffice. + +\begin{lemma} +\label{log:finite-ambiguity} +Let a finite group $G$ act algebraically on a smooth connected complex +variety $B$, and let $q:B\to B/G$ be the quotient. Suppose +$g:A\to B$ is a homeomorphism and $f:A\to B$ is continuous with +$qf=qg$. Then $f=\gamma g$ for a single $\gamma\in G$. +\end{lemma} + +\begin{proof} +Divide out the kernel of the action. Smoothness and connectedness +make $B$ irreducible. Removing the fixed loci of the nonidentity +elements gives a dense connected Zariski open $B^\circ$ on which +the action is free. For $a\in g^{-1}(B^\circ)$ there is a unique +$\gamma(a)$ with $f(a)=\gamma(a)g(a)$. The free finite quotient +is a covering there, so $\gamma(a)$ is locally constant. It is +constant because this open set is connected. Continuity extends +the equality over its closure, which is all of $A$. +\end{proof} + +\begin{proposition} +\label{log:nah} +The comparison \eqref{log:comparison} transports the Deligne weight +filtration on the whole $H^*(X^{\rm B},\mathbb Q)$ in the same way as +the nonabelian-Hodge identification. The conclusion is unchanged +when $\mathcal O_C(d x_0)$ is replaced by an arbitrary +$L\in\operatorname{Pic}^d(C)$ and the Higgs spaces are identified by +tensoring with a degree-zero nth root of their determinant ratio. +\end{proposition} + +\begin{proof} +The full fixed-determinant nonabelian-Hodge diffeomorphism, including +its $\Gamma$-equivariance, is part of the twisted construction in +\cite[Sections~2 and~5]{HT2003}. We compare its cohomology map with +the particular logarithmic family above. + +For $(E,\theta)\in X$, choose a harmonic metric $H$ normalized to a +fixed metric on its determinant. Let $\partial_H$ denote the +$(1,0)$ part of its Chern connection. For real $0\leq\tau\leq1$ +consider the holomorphic structure and logarithmic $\tau$-operator +\begin{equation} + \bar\partial_E+\tau\theta^{\dagger_H},\qquad + \tau\partial_H+\theta+ + \frac{\tau}{n} + \bigl(d^{1,0}_{d x_0}-\partial_{\det H}\bigr)I. + \label{log:harmonic-path} +\end{equation} +Its projectivization is the usual harmonic preferred section +\cite[Section~4]{Simpson1997}. We verify the scalar adjustment. +The trace of $\theta$ and of $\theta^{\dagger_H}$ is zero, so the +determinant holomorphic structure stays fixed and the determinant +operator is exactly $\tau d_{d x_0}$. The trace-free +holomorphicity equation follows from +$\bar\partial_E\theta=0$, $\partial_H\theta^{\dagger_H}=0$ and +the projective Hitchin equation. The scalar equation is that of +the prescribed logarithmic determinant connection. The only +nonregular term is the scalar logarithmic term at $x_0$. + +For $\tau\ne0$, divide the operator by $\tau$. +Its puncture monodromy is $\zeta I$. A proper invariant subspace +of rank $r$ would have $\zeta^r=1$ by taking determinants of the +commutator relation, contrary to $\gcd(n,d)=1$. Thus these +connections are irreducible and stable. At $\tau=0$ we recover +the original Higgs pair. Hence \eqref{log:harmonic-path} lies in +$\mathcal X$ throughout. + +This construction is a continuous homotopy of maps $X\to\mathcal X$. +One may work in local families of stable Higgs pairs. The normalized +harmonic metrics vary smoothly: their trace-free metric linearization +is self-adjoint elliptic, and its kernel consists of infinitesimal +symmetries, which stability kills. This is also the parameter +dependence used for preferred sections in +\cite[Section~4]{Simpson1997}. To pass to the analytic topology of +coarse moduli, choose a sufficiently positive twist near any one +parameter so the underlying bundles are generated and have vanishing +$H^1$. The kernels of the corresponding Dolbeault operators then +give continuously varying bases of sections. Their evaluations +give continuous quotient presentations; the logarithmic operators, +viewed as morphisms from the first jet bundle with fixed symbol, +also have continuous coefficients in such presentations. Passing +to the stable quotient gives the asserted continuity, independently +of the local choices. + +At $\tau=0$ the homotopy is the inclusion $X\hookrightarrow\mathcal X$. +At $\tau=1$ it factors through $X_1$, and then through +Riemann--Hilbert to a map $F:X\to X^{\rm B}$. +The pullback $F^*$ is therefore precisely +\eqref{log:comparison}, because both restrictions from +$\mathcal X$ are isomorphisms. Projectivizing $F$ gives the +usual projective nonabelian-Hodge map. + +Let $\Phi:X\to X^{\rm B}$ be the fixed-determinant +nonabelian-Hodge diffeomorphism. The maps $F$ and $\Phi$ agree +after projectivization. Two determinant-one lifts of the same +projective representation differ on each handle generator by an +nth root of unity; thus their classes differ by the finite group +of determinant-preserving characters, identified with $\Gamma$. +The low-degree calculation of Section~\ref{sec:lefschetz} gives +$H^0(X^{\rm B})=\mathbb Q$, so $X^{\rm B}$ is connected; it is +smooth in the coprime setting. Lemma~\ref{log:finite-ambiguity} +therefore gives $F=\gamma\Phi$ for one $\gamma\in\Gamma$. +Its action on $X^{\rm B}$ is algebraic and preserves Deligne +weights. Consequently $F^*$ and $\Phi^*$ transport the weight +filtration identically on every cohomology group. No averaging +or projection to invariants occurs. + +Finally choose $M\in\operatorname{Pic}^0(C)$ with +$M^{\otimes n}\simeq L\otimes\mathcal O_C(-d x_0)$. +Tensoring by $M$ identifies the two Higgs moduli spaces +algebraically over the same Hitchin base. Their projective +Higgs bundles, and therefore their projective harmonic +correspondences, are unchanged. Applying the same finite-character +argument to the two fixed-determinant nonabelian-Hodge maps shows +that this identification also preserves the transported weight +filtration. It preserves the perverse filtration because it is +an isomorphism over the Hitchin base. +\end{proof} diff --git a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/06-translation.tex b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/06-translation.tex new file mode 100644 index 0000000000000000000000000000000000000000..2f4fa42666c6808f0bf2d4e22b1d340f775671c9 --- /dev/null +++ b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/06-translation.tex @@ -0,0 +1,178 @@ +\section{A common cohomological operator}\label{sec:translation} + +We now use the logarithmic families to turn an elementary modification of Higgs bundles into monodromy of Betti spaces. All cohomology identifications in this section are the restriction maps of Corollary~\ref{log:restriction}. This specifies a single operator before we estimate its effect on either filtration. + +\subsection{Modification of two eigenlines} + +Choose $v\in\sqrt{-1}\mathbb R^n$ with sum zero, distinct entries, and no vanishing proper nonempty partial sum. Such vectors exist because the excluded conditions form finitely many proper real hyperplanes. Set +\[ +\ell=(1,-1,0,\ldots,0). +\] +At a logarithmic operator with distinct residue eigenvalues, a positive +elementary modification allows a simple pole in a chosen residue eigenline; +a negative one takes the kernel of the projection to that eigenline at $x$. +With the convention that a positive shift replaces a local generator $s$ +by $z^{-1}s$, the $\lambda$-Leibniz rule changes its residue eigenvalue by +$-\lambda$. A negative shift changes it by $+\lambda$. +We perform the shifts specified by $\ell$ successively: first the positive +modification on the first labeled eigenline, then, after reselecting the +labeled eigenspaces of the new residue, the negative modification on the +second labeled eigenline. The ordered eigenspaces of the final residue +define the final flag. + +\begin{lemma}\label{trans:translation} +The two modifications define a continuous family of isomorphisms +\begin{equation}\label{trans:path} +F_a:Y_{a,v}\longrightarrow Y_{a,v-a\ell},\qquad 0\le a\le1. +\end{equation} +At $a=0$ they define an algebraic automorphism of $Y_{0,v}$. More generally, the same modification defines an automorphism over the punctured Higgs line $(\lambda,u)=(0,zv)$, $z\ne0$, preserving the logarithmic Hitchin map. At $a=1$, the Riemann--Hilbert identifications make $F_1$ the identity on the underlying Betti space. +\end{lemma} + +\begin{proof} +For each single modification, choose a local frame adapted to its current +residue eigenspaces. The off-diagonal coefficients involving the chosen +eigenline vanish at the pole, so the singly modified lattice remains +preserved by the logarithmic operator. The preceding calculation gives its +new residue eigenvalues. Their imaginary parts remain distinct throughout +\eqref{trans:path}, including after the first step. We can therefore +reselect the residue eigenspaces before performing the second modification. +The two-step composite preserves the determinant and its operator because +the lattice shifts have sum zero. Its inverse performs the opposite +single modifications in reverse order, again reselecting eigenspaces +between steps. These elementary transformations and their inverses vary +algebraically where the successive residue eigenvalues are distinct, +giving the asserted continuous family. + +We check stability, which is not supplied by lattice modification alone. On the Higgs line with $z\ne0$, a proper invariant saturated subbundle would have a trace Higgs differential with a single possible simple pole. Its residue would be $z\sum_{i\in I}v_i$ for a proper nonempty subset $I$. This is nonzero, contrary to the residue theorem. The modified Higgs bundles satisfy the same argument. Distinct residue eigenvalues split the local Higgs operator into formal eigenline summands; shifting these summands and shifting them back proves invertibility. + +For $a>0$, divide the operator by $a$. A proper invariant monodromy subspace would have determinant monodromy one around the puncture. The imaginary part of the sum of its exponents is $a^{-1}\sum_{i\in I}\operatorname{Im}v_i$, which is nonzero. Thus no such subspace exists. Differences of exponents have nonzero imaginary part, so no nonzero integral resonance occurs. The inverse modifications are valid and remain stable by the same tests. + +The characteristic polynomial on the Higgs line is unchanged away from the pole and hence unchanged as a meromorphic polynomial on $C$. Finally, modifications of a connection change only its logarithmic lattice at the puncture. At $a=1$ the exponent changes are integers, so the meromorphic flat bundle and the ordered monodromy eigenlines are unchanged. This is precisely the endpoint identification asserted in the lemma. +\end{proof} + +\subsection{Translation as eigenvalue monodromy} + +Write $T=F_0$ and transport its pullback to the total family: +\begin{equation}\label{trans:global-T} +\mathsf T=\rho_{0,v}^{-1}T^*\rho_{0,v} +\quad\text{on }H^*(\mathcal Y,\Q). +\end{equation} +To compare this operator with the endpoint $F_1$, we need constancy along +the two families in \eqref{trans:path}. Their fibers are nonproper; the +restriction isomorphisms, rather than properness, supply the following +criterion. + +\begin{lemma} +\label{log:compact-cycles} +Let $I$ be a connected real interval and let $p:M\to I$ and +$q:N\to I$ be smooth submersions, with finite-dimensional rational +cohomology on their fibers. Suppose there are spaces $A,B$ and +continuous maps $a:M\to A$, $b:N\to B$ whose restrictions induce +isomorphisms +\[ + \alpha_t:H^*(A,\mathbb Q)\xrightarrow{\sim}H^*(M_t,\mathbb Q), + \qquad + \beta_t:H^*(B,\mathbb Q)\xrightarrow{\sim}H^*(N_t,\mathbb Q) +\] +for every $t\in I$. If $F:M\to N$ is continuous over $I$, then +$\alpha_t^{-1}F_t^*\beta_t:H^*(B,\mathbb Q)\to H^*(A,\mathbb Q)$ +is independent of $t$. +\end{lemma} + +\begin{proof} +Fix a degree and a parameter $t_0$. Choose finitely many compact +singular cycles forming a rational homology basis of $M_{t_0}$. +A lift of the interval vector field near the compact union of their +carriers has a flow for a common short time. It transports these +cycles to nearby fibers; the resulting cycles sweep homologies in +$M$. Their images in $A$ are therefore homologous, so their +pairings with the restrictions of a fixed basis of $H^*(A)$ are +constant. The restriction isomorphisms show that the transported +cycles remain a basis of the fiber homology. + +Applying the continuous map $bF$ to these swept cycles gives +homologies in $B$. Hence their pairings with every fixed class +of $H^*(B)$ are also constant. These pairings are the matrix +coefficients of $\alpha_t^{-1}F_t^*\beta_t$, relative to the +fixed bases. The operator is thus locally constant, including +one-sided neighborhoods of interval endpoints, and therefore +constant on $I$. +\end{proof} + +Apply the lemma with $I=[0,1]$ and with $M,N$ the pullbacks of +$\mathcal Y\to\Lambda$ along +\[ + a\longmapsto(a,v),\qquad a\longmapsto(a,v-a\ell), +\] +respectively. These are smooth submersions by +Proposition~\ref{log:families}, and \eqref{trans:path} gives the continuous +map $F:M\to N$. Take both ambient spaces in the lemma to be +$\mathcal Y$, with the natural maps from the two pullbacks. Their fiber +restrictions induce the isomorphisms of +Corollary~\ref{log:restriction}. The constancy conclusion therefore gives +\begin{equation}\label{trans:endpoint} +\mathsf T=\rho_{1,v}^{-1}F_1^*\rho_{1,v-\ell}. +\end{equation} +In particular, this identification does not require the nonproper smooth families to be globally locally trivial. + +Recall the eigenvalue map +\[ +t(u)=(\zeta e^{-2\pi\sqrt{-1}u_i})_{i=1}^n. +\] +The path $u=v-a\ell$, $0\le a\le1$, maps to a loop in $V$. Every forbidden proper subproduct has modulus different from one because the corresponding partial sum of $v$ has nonzero imaginary part. The same condition excludes resonance along the entire path. By Proposition~\ref{log:rh}, this is a path in the actual moving-puncture Betti family. Restrictions of classes from $\mathcal Y$ are flat sections along this path: the Betti direct images satisfy ordinary base change and are locally constant. Since $F_1$ is the endpoint identity, \eqref{trans:endpoint} identifies $\mathsf T$ with the pullback convention for its eigenvalue monodromy. + +\begin{lemma}\label{trans:logarithm} +There is an integer $M>0$ such that +\begin{equation}\label{trans:N} +N=\frac1M\log(\mathsf T^M) +\end{equation} +is a well-defined nilpotent derivation of $H^*(\mathcal Y,\Q)$. Let +$N_{\lambda,u}=\rho_{\lambda,u}N\rho_{\lambda,u}^{-1}$. +On $H^*(Y_{1,0},\Q)$, equipped with the diagram weights of Section~\ref{sec:diagram}, one has +\[ +N_{1,0}W_j\subset W_{j-2}. +\] +\end{lemma} + +\begin{proof} +The full-flag Betti family has integral lisse direct images, ordinary base change, and Hodge--Tate fibers by Section~\ref{sec:betti}. The moving-puncture transition maps are algebraic and locally constant by Proposition~\ref{log:rh}. The finite-monodromy statement of Section~\ref{sec:diagram} therefore applies to this very family. Choose $M$ that kills the monodromy action on every weight-graded piece of its total cohomology. There are finitely many such pieces. Then $\mathsf T^M-1$ is nilpotent, and \eqref{trans:N} is a finite logarithmic sum. The weights are even, so $N$ lowers them by at least two at $(1,v)$. + +The path $(1,av)$, $0\le a\le1$, stays in the nonresonant inverse image of $V$. At $a=0$, the scalar point belongs to $V$ because $\zeta$ has order $n$. For $a>0$, the imaginary-part argument applies again. Parallel transport preserves the diagram weights and coincides with the restriction identifications. This gives the stated estimate at $(1,0)$. + +Lemma~\ref{diag:finite} makes the logarithm a derivation on the moving-puncture Betti cohomology. The restriction identifications are ring isomorphisms, so this property transports to $H^*(\mathcal Y,\Q)$. +\end{proof} + +\subsection{Flag push-pull and the weight bound} + +For the scalar correspondence at $\lambda$, put +\[ +\mathsf A_\lambda=\pi_{\lambda,*}\iota_\lambda^*, +\qquad +\mathsf B_\lambda=\iota_{\lambda,*}\pi_\lambda^*. +\] +Its unflagged moving-pole space is $S\times X_\lambda$, with projection $\pr_\lambda$. We define +\begin{equation}\label{trans:fprime} +f'=\pr_{0,*}\,\mathsf A_0 N_{0,0}^2\mathsf B_0\,\pr_0^* +:H^m(X,\Q)\longrightarrow H^{m-2}(X,\Q). +\end{equation} +The flag embedding and projection have opposite degree shifts, $2b$ and +$-2b$. Integration over the compact curve $S$ supplies the remaining +degree $-2$. Table~\ref{spec:shift-table} records the individual derived morphisms +and their degree and weight shifts. Section~\ref{sec:specialization} +derives the perverse bound from their composite over the Hitchin base. +We first prove the weight bound. + +\begin{proposition}\label{trans:weights} +Under nonabelian Hodge theory, the operator \eqref{trans:fprime} satisfies +\[ +f'W_jH^m(X^B,\Q)\subset W_{j-4}H^{m-2}(X^B,\Q). +\] +\end{proposition} + +\begin{proof} +The scalar correspondence and its Gysin maps commute with the restriction identifications of Corollary~\ref{log:restriction}. Thus \eqref{trans:fprime} transports to the same expression at $\lambda=1$. By Proposition~\ref{log:nah}, these identifications transport the ordinary Betti weight filtration on the full cohomology of $X^B$. The possible difference from a chosen nonabelian-Hodge identification is one algebraic $\Gamma$-action, which preserves every weight subspace, including those in variant cohomology. + +At $\lambda=1$, the scalar Betti space is the flat algebraic product $S\times X^B$ of Lemma~\ref{log:scalar-flat-product}. Its diagram mixed Hodge structure assigns weight zero to the cochains of $S$ and the ordinary Deligne weights to $H^*(X^B)$. Hence both $\pr_1^*$ and $\pr_{1,*}$ preserve weights. The flag maps are maps of the same flat transition diagrams. Their ordinary topological Gysin maps are the diagram Gysin maps, with weight shifts $2b$ for $\mathsf B_1$ and $-2b$ for $\mathsf A_1$. These shifts cancel, while $N_{1,0}^2$ lowers weight by four by Lemma~\ref{trans:logarithm}. +\end{proof} + +We have constructed the operator and proved its weight bound. The remaining tasks are to realize it over the Hitchin base, which gives the perverse bound, and to compute its double commutator with $e$. diff --git a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/07-specialization.tex b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/07-specialization.tex new file mode 100644 index 0000000000000000000000000000000000000000..736ad3ea0bf32973258b27fc5c4d7bdf9599d7a0 --- /dev/null +++ b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/07-specialization.tex @@ -0,0 +1,305 @@ +\section{Specialization over a Higgs line}\label{sec:specialization} + +We now prove the perverse-filtration bound for the operator $f'$ in +\eqref{trans:fprime}. The translation is defined over a logarithmic Hitchin +base away from zero on a Higgs line. Nearby cycles extend its action to the +zero fiber as a sheaf endomorphism. A final integration over the moving-point +curve supplies the shift by $-2$. The essential comparison with ordinary +cohomology uses the restriction isomorphisms of +Corollary~\ref{log:restriction}; it does not require the parameter morphism +to be proper. + +\subsection{The proper morphism and its relative translation} + +Keep the vector $v$ and the translation from Section~\ref{sec:translation}. +Let $p_C:C\times S\to S$ be the projection and let $\Delta\subset C\times S$ +be the graph of $x:S\to C$. The vector bundle of trace-free logarithmic +Hitchin coefficients is +\[ + \mathcal E= + \bigoplus_{i=2}^n + p_{C,*}\bigl(K_{C\times S/S}(\Delta)^{\otimes i}\bigr). +\] +We use the same notation for its total space. These direct images are locally +free and commute with base change: each twisting line on a fiber has degree +$i(2g-1)>2g-2$, so its first cohomology vanishes. The inclusion of ordinary +coefficient spaces defines a closed vector-subbundle embedding +\[ + j:S\times A\hookrightarrow\mathcal E. +\] +Indeed both the ordinary spaces and the logarithmic spaces commute with base +change, and their quotient has constant rank. + +Let $D=\mathbb A^1_z$, let $D^*=D\setminus\{0\}$, and define +\[ + \mathcal Y_v=\mathcal Y\times_{\Lambda}D, + \qquad z\longmapsto(0,zv). +\] +Thus its fiber at $z$ is $Y_{0,zv}$, including the moving-point coordinate in +$S$. Characteristic coefficients and the parameter define +\begin{equation}\label{spec:hitchin-line} + g_v:\mathcal Y_v\longrightarrow\mathcal E\times D. +\end{equation} +Write $g_0:Y_{0,0}\to\mathcal E$ for its zero fiber. + +\begin{lemma}\label{spec:geometry} +The morphism $g_v$ is proper, and $\mathcal Y_v\to D$ is smooth. The +translation defines an automorphism $T_v$ of $\mathcal Y_v|_{D^*}$ over +$\mathcal E\times D^*$. Scaling gives an isomorphism +\[ + \mathcal Y_v|_{D^*}\simeq Y_{0,v}\times D^* +\] +over $D^*$, and restriction from $H^*(\mathcal Y_v)$ to either $Y_{0,0}$ or +$Y_{0,v}$ is an isomorphism. +\end{lemma} + +\begin{proof} +Smoothness and the restriction assertions follow from +Proposition~\ref{log:families} and Corollary~\ref{log:restriction}. For properness, +start with the relative logarithmic Higgs Hitchin morphism over $S$. +Imposing determinant $\mathcal O_C(dx(s))$ and trace zero gives a closed +subspace. Adding a residue-invariant full flag is projective over that +subspace. Finally the requirement that the ordered diagonal residues equal +$zv_i$ is closed after adjoining $z$. The properness of the logarithmic +Hitchin morphism from Section~\ref{sec:logarithmic} therefore implies +properness of \eqref{spec:hitchin-line}. Coprimality ensures that the +semistable limits in this argument remain stable. + +For $z\ne0$ the residual eigenvalues $zv_i$ are distinct, so the elementary +modifications defining $T_v$ and their inverses are algebraic in $z$. +They remain in the stable moduli by the trace-residue argument of +Section~\ref{sec:translation}. A modified Higgs field agrees with the +original meromorphic Higgs field away from the pole. Their characteristic +coefficients are consequently equal as sections of +$K_C(x(s))^{\otimes i}$, which proves that $T_v$ is over $\mathcal E$. +Multiplying the Higgs field by $z$ identifies the fiber at $1$ with that at +$z$; division by $z$ is its inverse. This gives the claimed product over +$D^*$. The product need not fix the Hitchin coefficient in $\mathcal E$; +only the translation must do so. +\end{proof} + +\subsection{Specialization and ordinary cohomology} + +Here is the comparison needed for the noncompact spaces in +Lemma~\ref{spec:geometry}. Throughout, nearby cycles are unshifted; +we use the nearby-cycle and proper-base-change formalism recalled in +\cite[Sections~5.5 and~5.8]{dCM2009}. + +\begin{lemma}\label{spec:specialization} +Let $Y\to D=\mathbb A^1$ be a smooth morphism of complex algebraic varieties, +and let $g:Y\to B\times D$ be a proper morphism commuting with the maps to +$D$. Suppose that $Y|_{D^*}$ is +isomorphic to $Y_1\times D^*$ over $D^*$ and that the actual restriction maps +\[ + r_i:H^*(Y,\mathbb Q)\longrightarrow H^*(Y_i,\mathbb Q), + \qquad i=0,1, +\] +are isomorphisms. An automorphism $T$ of $Y|_{D^*}$ over $B\times D^*$ +then induces an endomorphism $\tau_0$ of $Rg_{0,*}\mathbb Q_{Y_0}$ whose +cohomological action is +\[ + H^*(\tau_0)=r_0r_1^{-1}T_1^*r_1r_0^{-1}. +\] +Neither $B$ nor $Y_i$ is required to be proper. +\end{lemma} + +\begin{proof} +Smoothness gives the canonical isomorphism +$\mathbb Q_{Y_0}\simeq\psi\mathbb Q_Y$. +Compatibility of nearby cycles with proper direct image gives +\[ + Rg_{0,*}\mathbb Q_{Y_0} + \simeq Rg_{0,*}\psi\mathbb Q_Y + \simeq\psi(Rg_*\mathbb Q_Y). +\] +Apply nearby cycles to the pullback endomorphism induced by $T$ on the +punctured direct image, and use these isomorphisms to define $\tau_0$. + +We verify its action on global cohomology without interchanging nearby +cycles with a nonproper direct image to a point. Put $K=Rg_*\mathbb Q_Y$ +on $B\times D$. Let $\widetilde D^*\to D^*$ be the universal cover and write +\[ + \widetilde j_B:B\times\widetilde D^*\longrightarrow B\times D, + \qquad i_B:B\times\{0\}\hookrightarrow B\times D, + \qquad \widetilde K=\widetilde j_B^*K. +\] +Set $\widetilde Y^*=Y|_{D^*}\times_{D^*}\widetilde D^*$. +Proper base change identifies $\widetilde K$ with the direct image of the +constant sheaf from $\widetilde Y^*$. The nearby-cycle formula downstairs +is $\psi K=i_B^*R\widetilde j_{B,*}\widetilde K$. Restriction of this +complex gives a natural morphism +\begin{equation}\label{spec:comparison-map} + \begin{split} + R\Gamma(\widetilde Y^*,\mathbb Q) + &\simeq R\Gamma(B\times D,R\widetilde j_{B,*}\widetilde K)\\ + &\longrightarrow R\Gamma(B,\psi K) + \simeq R\Gamma(Y_0,\mathbb Q). + \end{split} +\end{equation} +The last isomorphism uses proper nearby-cycle compatibility and smoothness +of $Y\to D$. The composite of \eqref{spec:comparison-map} with restriction +from $R\Gamma(Y,\mathbb Q)$ is ordinary restriction to $Y_0$. +Indeed the specialization unit for $K$ corresponds, under proper direct +image, to the unit $\mathbb Q_{Y_0}\to\psi\mathbb Q_Y$, which is the +canonical smooth isomorphism used above. + +Choose a lift of $1\in D^*$. The assumed product and the contractibility of +$\widetilde D^*$ identify $H^*(\widetilde Y^*)$ with $H^*(Y_1)$ by actual +restriction to that lift. Pullback from $H^*(Y)$ therefore becomes $r_1$ +and is an isomorphism. Since its composite with +\eqref{spec:comparison-map} is $r_0$, the latter map is also an isomorphism +on cohomology. + +The comparison was constructed on $B\times D$ so that it is equivariant +for the required operator: because $T$ is over $B\times D^*$, it induces +an endomorphism of $K|_{B\times D^*}$ and hence of +$\widetilde K$, $R\widetilde j_{B,*}\widetilde K$, and $\psi K$. +Every arrow in \eqref{spec:comparison-map} respects these actions. +Restricting at the chosen lift identifies the action on its domain with +$T_1^*$, and its action on the codomain is $H^*(\tau_0)$. +Thus the displayed formula for $H^*(\tau_0)$ follows. No extension of $T$ +to an automorphism of $Y$ is asserted or needed. +\end{proof} + +Apply Lemma~\ref{spec:specialization} to \eqref{spec:hitchin-line}. The +resulting endomorphism $\tau_0$ of $Rg_{0,*}\mathbb Q_{Y_{0,0}}$ acts on +cohomology as the translation transported to $Y_{0,0}$ by +Corollary~\ref{log:restriction}. In particular its action is the operator +whose logarithm is $N_{0,0}$ in Section~\ref{sec:translation}. + +A finite polynomial suffices to realize this logarithm over the Hitchin +base. Choose $M$ as in Lemma~\ref{trans:logarithm}, and choose an integer +$r\ge1$ for which $(\mathsf T^M-1)^r=0$ on total cohomology. Put +\[ + p(t)=\frac1M\sum_{a=1}^{r-1} + \frac{(-1)^{a+1}}{a}(t^M-1)^a. +\] +The polynomial $p(\tau_0)$ is a legitimate endomorphism of +$Rg_{0,*}\mathbb Q_{Y_{0,0}}$, and its action on cohomology is $N_{0,0}$. +No unipotence assertion about $\tau_0$ in the derived category is needed. + +\subsection{The derived shift} + +We have now realized the middle operator in $f'$ over the Hitchin base. +We next include the flag correspondence and the moving-point integration, +keeping every shift explicit. + +\begin{proposition}\label{spec:perverse-lowering} +The operator $f'$ of \eqref{trans:fprime} is induced by a morphism +\begin{equation}\label{spec:derived-lowering} + Rh_*\mathbb Q_X\longrightarrow Rh_*\mathbb Q_X[-2] +\end{equation} +in the constructible derived category on $A$. Consequently +\[ + f'(P_kH^m(X,\mathbb Q))\subseteq P_{k-2}H^{m-2}(X,\mathbb Q) +\] +for all integers $m,k$. +\end{proposition} + +\begin{proof} +Let $O_0=S\times X$ using the scalar product identification of +Lemma~\ref{log:scalar-product}, and write $q=\operatorname{id}_S\times h$. +Both maps in the scalar flag correspondence of +Figure~\ref{log:scalar-correspondence} at $\lambda=0$ are over $\mathcal E$. +The flag projection has relative complex dimension $b$, and the scalar +inclusion has complex codimension $b$. Thus their pullbacks and proper +Gysin maps give, with +\[ + \begin{gathered} + F_O=j_*Rq_*\mathbb Q_{O_0},\qquad + F_Y=Rg_{0,*}\mathbb Q_{Y_{0,0}},\\ + F_Z=R(g_0\circ\iota)_*\mathbb Q_{Z_0}, + \end{gathered} +\] +morphisms +\[ + \mathsf B_0:F_O\longrightarrow F_Y[2b], + \qquad + \mathsf A_0:F_Y\longrightarrow F_O[-2b]. +\] +Composing these with the appropriate shifts of $p(\tau_0)^2$ gives an +endomorphism of $F_O$ with zero net shift. Its cohomological action is +$\mathsf A_0N_{0,0}^2\mathsf B_0$. +Since direct image by the closed embedding $j$ is fully faithful, this is +an endomorphism of $Rq_*\mathbb Q_{O_0}$ over $S\times A$. + +Let $p_S:S\times A\to A$ be the projection, and write +$K=Rh_*\mathbb Q_X$. The product identification gives +\[ + Rp_{S,*}Rq_*\mathbb Q_{O_0} + \simeq K\otimes R\Gamma(S,\mathbb Q). +\] +Push the preceding endomorphism to $A$, precompose with the projection unit +$K\to K\otimes R\Gamma(S,\mathbb Q)$, and postcompose with the orientation +trace +\[ + K\otimes R\Gamma(S,\mathbb Q)\longrightarrow K[-2]. +\] +This constructs \eqref{spec:derived-lowering}. On ordinary cohomology the +unit is $\operatorname{pr}_0^*$, the trace is +$\operatorname{pr}_{0,*}$, and the middle map is the flag composite already +identified. Hence the resulting operator is exactly $f'$. + +For completeness, set $K_{\mathrm{norm}}=K[R]$, the normalization used in +the statement of the theorem. Functoriality of perverse truncation applied +to \eqref{spec:derived-lowering} gives +\[ + {}^p\tau_{\le k}K_{\mathrm{norm}} + \longrightarrow{}^p\tau_{\le k}(K_{\mathrm{norm}}[-2]) + =({}^p\tau_{\le k-2}K_{\mathrm{norm}})[-2]. +\] +Taking hypercohomology in degree $m-R$ and then its image in ordinary +cohomology proves the stated inclusion. This uses neither a splitting nor +multiplicativity of the perverse filtration. +\end{proof} + +We finish by recording the derived morphisms and their shifts. Retain the +complexes $F_O,F_Z,F_Y$ and $K$ from the proof, and put +$K_S=K\otimes R\Gamma(S,\mathbb Q)$. + +The first panel of Table~\ref{spec:shift-table} consists of morphisms on +$\mathcal E$. Compose the flag maps and $p(\tau_0)^2$ there to obtain an +endomorphism of $F_O$, descend it through $j$, and then apply $Rp_{S,*}$. +This produces the degree-zero endomorphism of $K_S$ in the second panel, +whose base is $A$. The unit and trace then give $K\to K[-2]$; it is this +complete morphism that yields the normalized perverse bound above. +No perverse exactness of $Rp_{S,*}$ is used. +The weight columns refer to the cohomological actions after transport +to $\lambda=1$ and use the diagram weights of Section~\ref{sec:diagram}. +In particular $p(\tau_0)$ acts as $N_{1,0}$ after this comparison, while +the unit and trace preserve weight because the cochains of $S$ have +weight zero. + +\begin{table}[htbp] +\centering +\small +\begin{tabular}{llrr} +\toprule +\multicolumn{4}{c}{Morphisms on $\mathcal E$}\\ +\midrule +Operation & Morphism & Degree & $\Delta W$\\ +\midrule +$\pi_0^*$ & $F_O\to F_Z$ & $0$ & $0$\\ +$\iota_{0,*}$ & $F_Z\to F_Y[2b]$ & $2b$ & $2b$\\ +$p(\tau_0)$ & $F_Y\to F_Y$ & $0$ & $-2$\\ +$p(\tau_0)^2$ & $F_Y\to F_Y$ & $0$ & $-4$\\ +$\iota_0^*$ & $F_Y\to F_Z$ & $0$ & $0$\\ +$\pi_{0,*}$ & $F_Z\to F_O[-2b]$ & $-2b$ & $-2b$\\ +Flag composite & $F_O\to F_O$ & $0$ & $-4$\\ +\midrule +\multicolumn{4}{c}{Morphisms on $A$}\\ +\midrule +Operation & Morphism & Degree & $\Delta W$\\ +\midrule +Projection unit & $K\to K_S$ & $0$ & $0$\\ +Pushed composite & $K_S\to K_S$ & $0$ & $-4$\\ +Orientation trace & $K_S\to K[-2]$ & $-2$ & $0$\\ +Complete $f'$ & $K\to K[-2]$ & $-2$ & $-4$\\ +\bottomrule +\end{tabular} +\caption{Derived degrees and diagram-weight bounds for the correspondence +maps. A weight bound $s$ means that $W_k$ maps into $W_{k+s}$ after +transport to the Betti side. The row for +$p(\tau_0)$ records $N$ separately; the flag composite uses its square.} +\label{spec:shift-table} +\end{table} diff --git a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/08-commutator.tex b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/08-commutator.tex new file mode 100644 index 0000000000000000000000000000000000000000..fc8176c2b6f9d397d6e8b8dff802b6fd2d61da4d --- /dev/null +++ b/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/build/sections/08-commutator.tex @@ -0,0 +1,298 @@ +\section{The translation class and the double commutator}\label{sec:commutator} + +The preceding sections give the two filtration bounds for the same operator +$f'$. We now compute its double commutator with $e$. The calculation has two +parts: an elementary modification differentiates the universal degree-two +class into a difference of flag classes, and flag integration converts the +square of that difference back into $e$. + +\subsection{The characteristic class under translation} + +The universal projective bundle on $C\times\mathcal Y$ defines a universal +endomorphism bundle. Let +\[ + e_{\mathcal Y} + =\frac1{2n}\int_C\operatorname{ch}_2(\operatorname{End}E) + \in H^2(\mathcal Y,\mathbb Q). +\] +Individual ordered flag lines $Q_i$ can be taken on local universal-bundle +charts; their ratios $Q_i\otimes Q_j^{-1}$ descend to the coarse moduli +space. We use the global classes +\[ + y=c_1(Q_1\otimes Q_2^{-1}),\qquad + t_C=x^*c_1(T_C), +\] +where the second is pulled back from $S$. We retain the same notation for +the restrictions of these classes to a fiber $Y_{\lambda,u}$. + +\begin{lemma}\label{calc:translation-formula} +For the translation $T:Y_{0,v}\to Y_{0,v}$ associated with +$\ell=(1,-1,0,\ldots,0)$ and every integer $k\ge0$, +\begin{equation}\label{calc:translation-polynomial} + T^{k*}e_{\mathcal Y}=e_{\mathcal Y}+ky+k^2t_C. +\end{equation} +Consequently the logarithm from Section~\ref{sec:translation} satisfies +\begin{equation}\label{calc:logarithm} + N(e_{\mathcal Y})=y +\end{equation} +on $H^*(\mathcal Y,\mathbb Q)$ and, by restriction, on every fiber. +\end{lemma} + +\begin{proof} +Work first on $Y_{0,v}$. Denote by $i:\Delta\hookrightarrow C\times Y_{0,v}$ +the graph of the moving point. Its normal line is the pullback of $x^*T_C$, +so +\[ + \mathcal O(\Delta)|_\Delta\simeq x^*T_C, + \qquad c_1(N_\Delta)=t_C. +\] +The residue eigenvalues $v_i$ are distinct. Consequently the Higgs operator +has uniquely labeled eigenline summands on the formal neighborhood of +$\Delta$. This follows by lifting its simple residual eigenvalues and +their idempotent projectors to every finite order. Multiplying a local +frame of the logarithmic canonical line by a unit does not change those +projectors. Thus the formal summands are intrinsic, and any finite number +of their jets suffices for an elementary modification. + +The $k$-fold translation replaces the $i$th formal line lattice by its twist +by $k\ell_i\Delta$. On the $(i,j)$ block of $\operatorname{End}E$, the +lattice shift is therefore +\[ + m_{ij}=k(\ell_i-\ell_j). +\] +If a line block has restriction $F$ to $\Delta$, its K-theory change under +an integer shift $m$ is +\begin{equation}\label{calc:signed-jets} + [\text{new block}]-[\text{old block}] + =\sum_{a=1}^{m}[i_*(F\otimes N_\Delta^{\otimes a})]. +\end{equation} +For $m<0$, the right side means the negative of the sum over $m0$ and set +\[ +f=\frac{(-1)^{b+1}}{4n(n-2)!\,\delta}\,f'. +\] +Propositions~\ref{trans:weights} and~\ref{spec:perverse-lowering} show that $f$ lowers the half-weight filtration and the perverse filtration by two. The calculation in Section~\ref{sec:commutator} gives $[[f,e],e]=-2e$. Apply Proposition~\ref{alg:comparison} to $P_k$ and $W_{2k}$. It follows that +$P_kH^m(X)=W_{2k}H^m(X^B)$ in every cohomological degree. The vanishing of odd weight-graded pieces gives $W_{2k}=W_{2k+1}$. + +For arbitrary $L\in\Pic^d(C)$, choose $M\in\Pic^0(C)$ with +$M^n\simeq L\otimes\mathcal O_C(-dx_0)$. Tensoring by $M$ identifies the Higgs spaces over the Hitchin base. Proposition~\ref{log:nah} proves that this identification transports the full Betti weight filtration as well. This proves the theorem for every determinant. +\end{proof} + +\begin{corollary} +Let $C$ be a smooth projective complex curve of genus $g\ge2$, let +$n\ge2$ and $d$ be coprime integers, and let $L\in\Pic^d(C)$. +For the moduli space $X$ of stable rank-$n$, determinant-$L$, trace-free +Higgs bundles and its corresponding twisted $\SL_n$ character variety +$X^B$, nonabelian Hodge theory identifies +\[ + P_k H^m(X,\Q)=W_{2k}H^m(X^B,\Q)=W_{2k+1}H^m(X^B,\Q) + \qquad\text{for all integers }m,k. +\] +The equality holds on full rational cohomology, including its +$\Pic^0(C)[n]$-variant part. +\end{corollary} +\begin{proof} +The composite ranks are Theorem~\ref{thm:main}; the prime ranks are \cite[Theorem~0.3]{MS2024}. +\end{proof} + +\begin{corollary}[Endoscopic weight compatibility]\label{cor:endoscopic-weights} +Keep $C,n,d,L$ as in the preceding corollary. Put $\Gamma=\Pic^0(C)[n]$, +let $\gamma\in\Gamma$ have order $m$, and let $\kappa\in\widehat{\Gamma}$ +be the character $\kappa(\gamma')=\langle\gamma,\gamma'\rangle_\Gamma$ +defined by the Weil pairing in \cite[Section~1.3]{MSEndoscopy2021}. Let +$\pi:C'\to C$ be the associated cyclic \'etale cover, set $r=n/m$, and put +\[ +d_\gamma=\operatorname{codim}_{\mathcal A^{K_C}}\mathcal A_\gamma^{K_C}, +\] +where $\mathcal A_\gamma^{K_C}$ is their endoscopic Hitchin base. +For $D=K_C$ and $D'=\pi^*K_C=K_{C'}$, let $\mathfrak p_\kappa^B$ denote +their nonabelian-Hodge-transported operator in +\cite[Section~5.4]{MSEndoscopy2021}. Write +$\widetilde{\mathcal M}_{r,d}^B(C')$ for the full degree-$d$ twisted +$\GL_r$ character variety and $\mathcal M_{n,L}^B(C)=X^B$ for the +fixed-determinant twisted $\SL_n$ character variety. On complexified +cohomology, for all integers $i,k\ge0$, +\begin{multline*} +\mathfrak p_\kappa^B\!\left(W_{2k}H^i(\widetilde{\mathcal M}_{r,d}^B(C'),\C)\right)\\ +=W_{2k+2d_\gamma}H^{i+2d_\gamma}(\mathcal M_{n,L}^B(C),\C)_\kappa, +\end{multline*} +where the subscript $\kappa$ denotes the pullback-action summand on which +$\gamma'^*\alpha=\kappa(\gamma')\alpha$ for every $\gamma'\in\Gamma$. +\end{corollary} +\begin{proof} +Since $r\mid n$, one has $\gcd(r,d)=1$, and \'etale Riemann--Hurwitz gives +$g(C')=m(g-1)+1\ge2$. The all-rank $\GL_r$ theorem +\cite[Theorem~0.2]{MS2024} therefore identifies $W_{2k}$ with $P_k$ on the +source, including when $r=1$. The preceding corollary identifies +$P_{k+d_\gamma}$ with $W_{2k+2d_\gamma}$ on the target after +complexification and passage to the $\Gamma$-$\kappa$ summand: the +$\Gamma$-action preserves both filtrations, and the target nonabelian-Hodge +identification is $\Gamma$-equivariant. With the common +defect normalization, \cite[Theorem~5.4]{MSEndoscopy2021} gives the exact +image of $P_kH^i$ as $P_{k+d_\gamma}H^{i+2d_\gamma}_\kappa$. +Writing $\eta_s,\eta_t$ for the source and target nonabelian-Hodge +cohomology identifications from Betti to Dolbeault, the definition +$\mathfrak p_\kappa^B=\eta_t^{-1}\mathfrak p_\kappa\eta_s$ then gives the +claim. Its nonzero scalar ambiguity does not affect images. +\end{proof} + +The standard implication from $P=W$ to perverse multiplicativity +\cite[Section~0.4.3, equation~(10)]{dCMSJAMS2022} now applies to the full +fixed-determinant cohomology. + +\begin{corollary}[Multiplicativity of the perverse filtration]\label{cor:perverse-multiplicativity} +In the setting of the all-rank corollary above, use the normalization +\eqref{intro:P}. For all $i,j\ge0$ and $a,b\in\Z$, +\[ +P_aH^i(X,\Q)\smile P_bH^j(X,\Q) +\subseteq P_{a+b}H^{i+j}(X,\Q). +\] +Put $\Gamma=\Pic^0(C)[n]$ and +$P_aH^i(X,\C)=P_aH^i(X,\Q)\otimes_{\Q}\C$. For the pullback-action +isotypic parts indexed by $\chi,\psi\in\widehat\Gamma +=\operatorname{Hom}(\Gamma,\C^\times)$, one also has +\[ +\bigl(P_aH^i(X,\C)\bigr)_\chi\smile +\bigl(P_bH^j(X,\C)\bigr)_\psi +\subseteq\bigl(P_{a+b}H^{i+j}(X,\C)\bigr)_{\chi\psi}. +\] +This includes trivial and distinct characters, with the same normalized +$P$ on every character summand. +\end{corollary} +\begin{proof} +The nonabelian-Hodge diffeomorphism induces a graded ring isomorphism +$\eta:H^*(X^B,\Q)\to H^*(X,\Q)$ \cite[Sections~2 and~5]{HT2003}. +Cup product is a morphism of mixed Hodge structures +\cite[Corollaire~(8.2.11)]{Deligne1974}, so it sends +$W_{2a}H^i(X^B,\Q)\smile W_{2b}H^j(X^B,\Q)$ into +$W_{2(a+b)}H^{i+j}(X^B,\Q)$. Transport by $\eta$ and the all-rank +$P=W$ equality give the first assertion. The filtration is +$\Gamma$-stable, and +$\gamma^*(\alpha\smile\beta)=\gamma^*\alpha\smile\gamma^*\beta$ +multiplies the two eigencharacters, giving the second assertion. +\end{proof} diff --git a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/README.md b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..4bbbb05075acaefde91d0ff8ff1e3f9c41e0b3ec --- /dev/null +++ b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [Paired states and Hamiltonian cycles in cubic bipartite planar graphs](paper.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026, + author = {{OpenAI}}, + title = {{Paired states and Hamiltonian cycles in cubic bipartite planar graphs}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf}{OAI:Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/paper.tex b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..01dd9e0c56b84f77c2051379e1202821bc2164cd --- /dev/null +++ b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/paper.tex @@ -0,0 +1,837 @@ +\documentclass[11pt]{article} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern,amsmath,amssymb,amsthm} +\pdfglyphtounicode{parenleftbig}{0028} +\pdfglyphtounicode{parenrightbig}{0029} +\pdfgentounicode=1 +\usepackage[margin=1in]{geometry} +\usepackage{microtype,enumitem,needspace} +\usepackage{tikz} +\usetikzlibrary{arrows.meta} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} +\newcommand{\doi}[1]{doi: \href{https://doi.org/#1}{\nolinkurl{#1}}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\ii}{\mathrm i} +\newcommand{\abs}[1]{\left|#1\right|} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\e}{\mathrm e} +\numberwithin{equation}{section} +\setlist{itemsep=3pt,topsep=5pt} +\hypersetup{ + pdftitle={Paired states and Hamiltonian cycles in cubic bipartite planar graphs}, + pdfauthor={OpenAI}, + pdfsubject={A proof of Barnette's conjecture by induced tree partitions} +} +\title{Paired states and Hamiltonian cycles\\ +in cubic bipartite planar graphs} +\author{OpenAI} +\date{September 24, 2026} + +\begin{document} +\maketitle +\begin{abstract} +We prove Barnette's conjecture: every finite simple cubic bipartite planar +\(3\)-vertex-connected graph has a Hamiltonian cycle. +\end{abstract} + +\section{Introduction} + +A graph is \emph{cubic} if every vertex has degree three, and +\emph{bipartite} if its vertices can be divided into two classes with +no edge inside either class. It is \emph{\(3\)-vertex-connected} if it +has at least four vertices and deleting any set of at most two vertices +leaves it connected. A \emph{Hamiltonian cycle} visits every vertex +exactly once. We consider finite simple undirected graphs throughout +the main statement. + +Barnette's conjecture, recorded by Gr\"unbaum in the 1969 Waterloo +proceedings \cite{Barnette1969}, asserts that every cubic bipartite polyhedral +graph is Hamiltonian, where polyhedral means planar and +\(3\)-vertex-connected. It belongs to the line of questions initiated +by Tait's conjecture for cubic polyhedral graphs, disproved by +Tutte \cite{Tutte1946}; see \citet{APSW2016,GSW2023} for the history +and equivalent formulations. + +Several partial results explain the role of face structure. +Goodey proved the case in which all faces are quadrilaterals or +hexagons \cite{Goodey1975}. Feder and Subi allowed arbitrarily +large faces in one class of a proper three-coloring of the faces, +requiring only quadrilaterals and hexagons in the other two classes +\cite{FederSubi2006}. Their argument uses proper quasi spanning trees of faces +after contracting a distinguished face class. Kardo\v{s} proved +the related Barnette--Goodey conjecture for cubic polyhedral graphs +with faces of size at most six, without assuming bipartiteness +\cite{Kardos2020}. More recently, Schnieders proved the +faces-at-most-eight case of Barnette's conjecture +\cite[Corollary~1.13]{Schnieders2025}. We remove the face-size +restriction within the full \(3\)-connected bipartite class. + +\begin{theorem}\label{thm:main} +Every finite simple undirected graph that is cubic, bipartite, +planar, and \(3\)-vertex-connected has a Hamiltonian cycle. +\end{theorem} + +The difficulty is to obtain one spanning cycle. A spanning subgraph +of degree two can have several cycle components, and local choices +alone need not join them. We instead construct an acyclic edge set +in a dual triangulation whose complement becomes one connected +spanning cycle. The construction also retains a prescribed local +pattern needed when the triangulation is cut into smaller pieces. + +\subsection{The dual problem and the proof strategy} + +For a graph satisfying Theorem~\ref{thm:main}, the faces of its sphere +dual are triangles. +A bipartition of the original graph colors these triangles black +and white, with opposite colors across every edge. The dual viewpoint goes back to Stein's region-partition formulation +\cite[Theorem~2.2]{Stein1971}. In the form proved by Alt, Payne, +Schmidt, and Wood \cite[Lemma~5]{APSW2016}, Hamiltonicity corresponds +to a partition of the triangulation's vertices into two sets, each +inducing a tree. Here an induced tree includes \emph{every} +edge between its vertices, so connectedness alone is insufficient. +Theorem~\ref{thm:partition} proves this partition with any chosen +facial edge's endpoints in one class and its third vertex in the +other. The corresponding Hamiltonian cycle avoids any specified +single edge, as shown in Corollary~\ref{cor:edge-avoidance}. This +single-edge avoidance property is a known equivalent global +strengthening of Barnette's conjecture, through the work of Kelmans +as explained by Hertel \cite[Corollary~3]{Hertel2005} and by +Gorsky, Steiner, and Wiederrecht \cite[Theorem~4.3]{GSW2023}. +Corollary~\ref{cor:pfaffian-paths} records a further consequence +through their published equivalence \cite[Theorem~4.16]{GSW2023}: +every three-edge path in a finite simple cubic \(3\)-vertex-connected Pfaffian +bipartite graph is contained in a Hamiltonian cycle. + +To explain the construction, choose one black triangle as the outer +face and call its three vertices the roots. The remaining black +triangles and the nonroot vertices have equal cardinality. A +\emph{state} assigns each remaining black triangle one of its vertices, +using every nonroot exactly once. These face--vertex incidence +matchings are Tutte states, arising in Tutte's theory of trinity +\cite{Tutte1975}. Hine and K\'alm\'an give a modern treatment and +relate the state moves to Kauffman's clock theorem \cite{HK2018}. +A \emph{pair} consists of two states making distinct +choices at every triangle. We prove all state properties needed here. + +For a second state \(s\), let \(Q_s\) contain the edge opposite its +chosen vertex in each remaining black triangle. We seek a pair for which +\(Q_s\) is a forest. When no triangle has vertices strictly on both sides, +a planar density inequality and an integral flow first provide a pair. +A disk count supplies the signed cycle identity used in the next step. + +We analyze one finite exponential sum over pairs in two different ways. +First fix \(s\). If \(Q_s\) has a cycle, reversing that cycle changes +only the first state \(r\). The disk identity shows that the two terms +have opposite phases, while their real weights agree because those +weights depend only on \(s\). Thus all pairs with cyclic \(Q_s\) cancel. +To prove that the whole sum is nonzero, instead fix the undirected edge +set \(D\) joining the two chosen vertices in each triangle. Its components +are cycles, and its pairs are exactly their independent orientations. +Here reversal exchanges the two state choices along a cycle. Join each +remaining black-face center to its three vertices by spokes. Positive circulations +on these spokes give real weights for which the counterclockwise +orientation of each cycle has the larger sum of weights. +Each cycle then contributes a simple zero at parameter zero. The disk +identity makes the leading phases agree for groups with the fewest +cycles, and their leading magnitudes are positive. These contributions +cannot cancel. Some pair with \(Q_s\) a forest must remain. + +A forest state has one root in each of three components. Complementary +plane-tree duality turns it into a tree on the white faces, which selects +one edge at every black and white face. The disk identity proves these +selected edges acyclic. Their complementary dual is connected and +has degree two everywhere; its two sides identify the required induced +trees. Finally, a separating triangle is shared by two smaller +triangulations. The prescribed facial pattern makes their partitions +agree on that triangle, so the trees glue along a vertex or an edge. + +Two established matching constructions help place these steps in +context. The passage between incidence matchings and plane trees +is related to Temperley's correspondence and its weighted extension +by Kenyon, Propp, and Wilson \cite{KPW2000}. Regrouping two matchings +by their union cycles is familiar from the double-dimer model +\cite[Lemma~1]{Kenyon2014}. We use this regrouping together with the +separate fixed-second-state cancellation and positive-circulation +argument proved here; neither cited correspondence is an input to +those conclusions. + +The weighted sum is finite and is used for existence. The argument +places no bound on the order or face lengths of the input graph; +it does not assert a polynomial-time construction of the cycle. + +\section{Plane trees and the triangulation statement}\label{sec:setup} + +All embeddings are on the sphere. After choosing an outer face +we work in the plane, with the point at infinity in that face. +Face lengths count edge-side incidences: a bridge contributes +twice to the length of its incident face. A \emph{sphere +triangulation} here is connected and simple, has at least four +vertices, and has a \(3\)-cycle as the boundary of every face. +A \emph{separating triangle} is a \(3\)-cycle with vertices strictly +on both sides. +A \emph{spanning forest} is an acyclic spanning subgraph, with +isolated vertices allowed. For a vertex set \(U\), the induced subgraph +\(T[U]\) contains every edge of \(T\) with both endpoints in \(U\). +A one-vertex graph is a tree. The plane graphs used for intermediate +dual constructions may have loops or parallel edges, with their usual +edge-side incidences; the triangulations themselves are simple. + +\begin{lemma}\label{lem:duality} +Let \(M\) be a connected graph embedded on the sphere. +The duals of the edges complementary to a spanning tree of \(M\) +form a spanning tree of \(M^*\). More generally, the duals of the +edges complementary to a spanning forest of \(M\) form a +connected spanning subgraph of \(M^*\). +\end{lemma} + +\begin{proof} +Contract a primal spanning tree to one vertex. Contraction +preserves faces and deletes the corresponding dual edges; the +dual of the remaining connected plane graph is connected. +The complementary dual edges number +\(\abs{E(M)}-\abs{V(M)}+1=\abs{F(M)}-1\), by Euler's formula, +so they form a tree. For a spanning forest, extend it to a +spanning tree; its complementary dual contains the tree just +described. +\end{proof} + +\begin{theorem}\label{thm:partition} +Let \(T\) be a sphere triangulation whose faces are colored black +and white so that each edge has one incident face of each color. +For any facial triangle \(t_0\) and any edge \(e_0\) of \(t_0\), +the vertex set admits a partition into two sets, each inducing +a tree, such that the endpoints of \(e_0\) are in one set and +the third vertex of \(t_0\) is in the other. +\end{theorem} + +We first prove Theorem~\ref{thm:partition} without separating +triangles. Put \(t_0\) on the outer face and exchange the face +colors if necessary to make it black. Its three vertices are +called the \emph{roots}. Let \(k\) be the number of black faces. +Counting sides of either color and applying Euler's formula gives +\begin{equation}\label{eq:counts} + \#\{\text{white faces}\}=k,\qquad + \abs{E(T)}=3k,\qquad \abs{V(T)}=k+2. +\end{equation} +Write \(A\) for the black faces other than \(t_0\), and \(X\) +for the nonroot vertices. Thus \(\abs{A}=\abs{X}=k-1\). + +A \emph{state} is a bijection \(r:A\longrightarrow X\) with +\(r(t)\in V(t)\), where \(V(t)\) is the vertex set of the +triangle \(t\). A \emph{pair} is an ordered pair of states +\((r,s)\) such that \(r(t)\ne s(t)\) for every \(t\in A\). +Equivalently, a pair consists of two edge-disjoint perfect +matchings in the bipartite incidence graph on \(A\) and \(X\). +This is the planar state convention of \citet[Section~1]{HK2018}, +with black and white exchanged; no vertex three-coloring or clock +theorem is needed below. + +\section{Two disjoint states}\label{sec:pairs} + +Our first task is to supply two distinct choices at every black face +while using each nonroot vertex exactly once in each choice. The capacity-one +incidence edges below enforce that the two states are disjoint. + +\begin{proposition}\label{prop:pairs} +If \(T\) has no separating triangle, then a pair of states exists. +\end{proposition} + +\begin{proof} +A degree-two spanning subgraph of the incidence graph splits +into two perfect matchings by alternating its bipartite cycles. +It is therefore enough to prove +\begin{equation}\label{eq:flow} + \sum_{t\in A}\min\bigl(2,\abs{V(t)\cap S}\bigr)\ge2\abs S + \qquad(S\subseteq X). +\end{equation} +Indeed, give capacity two to source-to-\(X\) and +\(A\)-to-sink edges, and capacity one to incidences from \(X\) +to \(A\). For a cut placing \(S\subseteq X\) on the source side, +minimizing over the placement of \(A\) gives capacity +\[ + 2(\abs X-\abs S)+ + \sum_{t\in A}\min\bigl(2,\abs{V(t)\cap S}\bigr). +\] +Thus \eqref{eq:flow} and integral max-flow give a flow of +\(2\abs X=2\abs A\), and hence the required degree-two subgraph. +The underlying max-flow/min-cut principle is due to +\citet{FF1956}. For completeness, the integral conclusion follows +by starting with zero +flow and augmenting along residual source--sink paths. Every augmentation +has a positive integral bottleneck, so at most \(2\abs X\) augmentations +are needed; if no path remains, the residual vertices reachable from the +source define a cut whose capacity equals the flow value. The displayed +cut bound therefore forces the full value. Every capacity-two outer edge +is saturated, and each capacity-one incidence carries zero or one, +giving degree two at every vertex of the incidence graph. + +Let \(U=V(T)\setminus S\); it contains the three roots. +Denote by \(e(U)\) the number of edges in \(T[U]\), and by +\(b(U)\) the number of actual black faces of \(T\) all of whose +vertices belong to \(U\). Include \(t_0\) in the sum in +\eqref{eq:flow}, with contribution zero. If a black triangle +has \(j\) vertices in \(U\), its shortfall from two is +\(\max(0,j-1)\). This is its number of edges in \(T[U]\), minus +one if \(j=3\). Since each edge has exactly one black incident +face, \eqref{eq:counts} shows that \eqref{eq:flow} is equivalent to +\begin{equation}\label{eq:density} + e(U)-b(U)\le2\abs U-4. +\end{equation} + +Consider a connected component \(K\) of \(T[U]\) with \(p\ge4\) +vertices, \(m\) edges, and \(f\) faces, with the inherited +sphere embedding. Every face has boundary length at least three. +Its triangular faces are exactly the actual triangular faces +of \(T\) whose vertices lie in \(K\). To justify the only +delicate direction, a triangular component face is bounded +by a \(3\)-cycle. As the cycle is not separating, one side is +an actual face of \(T\). The component has a vertex off the +cycle, on the other side, so its triangular facial side must +be the actual face. The converse follows because an actual +face contains no vertices or edges in its interior. + +Let \(b,w\) count these black and white triangular faces of +\(K\). Each other face region has length \(d\ge4\). Let +\(b',w'\) count all original faces of \(T\) in that region. +Signed edge-side counting gives +\[ + 3(b'-w')=\sum_{j=1}^{d}\sigma_j,\qquad + \sigma_j\in\{-1,1\}. +\] +Internal sides cancel in black/white pairs. This remains true +for repeated boundary edges and for regions containing other +components of \(T[U]\). Consequently +\begin{equation}\label{eq:face-defect} + \lvert b'-w'\rvert\le d-4. +\end{equation} +For \(d=4\), parity and divisibility by three force the signed +sum to be zero. For \(d=5\), its absolute value is at most +three. For \(d\ge6\), use \(d/3\le d-4\). + +The regions of \(K\) partition the original faces of \(T\). +Their total color balance therefore gives +\[ + w-b=\sum_{\text{nontriangular regions}}(b'-w') + \le\sum_{\text{nontriangular faces}}(d-4). +\] +It follows that +\[ + 2m=4f-b-w+\sum_{\text{nontriangular faces}}(d-4) + \ge4f-2b. +\] +Using \(p-m+f=2\) yields +\begin{equation}\label{eq:component} + m-b\le2p-4. +\end{equation} + +The component containing the roots satisfies +\eqref{eq:component} also when it has three vertices: +it is \(t_0\), with three edges and one actual black face. +Every other component satisfies the weaker bound +\(m-b_{\rm act}\le2p\), where \(b_{\rm act}\) counts its +actual black faces. For \(p\ge4\) this follows from +\eqref{eq:component}, and for smaller \(p\) from simplicity. +Summing over components proves \eqref{eq:density}. +\end{proof} + +\section{The disk identity}\label{sec:disk} + +The next identity controls both cancellation and the final forest +construction. It records more than a face-color imbalance: the state +bijection fixes exactly how many black faces can lie inside a directed +cycle. + +For adjacent vertices \(v,w\), define +\(\delta(v,w)=1\) if the black face of their edge is on the +left when the edge is traversed from \(v\) to \(w\), and +\(\delta(v,w)=-1\) otherwise. Thus \(\delta\) is antisymmetric, +and the counterclockwise sides of a bounded black face have +weight one. + +\Needspace{24\baselineskip} +\begin{lemma}[Disk identity]\label{lem:disk} +Let \(r\) be a state. In each \(t\in A\), choose one edge +directed from \(r(t)\) to another vertex of \(t\). +Every simple directed cycle in the chosen edges avoids the +roots. Its \(\delta\)-sum is \(-3\) if it is counterclockwise, +and \(3\) if it is clockwise. +\end{lemma} + +\begin{proof} +Only nonroots have outgoing edges, so the cycle avoids the +roots. Its bounded disk contains no root, because the roots +are on the outer-face boundary. Let \(h\) be the cycle length, +\(I\) the number of strictly interior vertices, \(B,W\) the +numbers of black and white faces in the disk, and \(l\) the +number of boundary edges with their black face in the disk. +Euler's formula for the triangulated disk and signed side +counting give, respectively, +\begin{equation}\label{eq:disk-counts} + B+W=2I+h-2,\qquad 3(B-W)=2l-h. +\end{equation} + +The state supplies the additional identity +\begin{equation}\label{eq:matching-disk} + B=I+l. +\end{equation} +Every strictly interior vertex is matched to an incident +black face in the disk. A boundary vertex is matched to the +black face of its outgoing cycle edge, which lies inside in +exactly the \(l\) counted cases. Conversely every black face +in the disk is matched to an interior or boundary vertex. +There is no double counting: the state is a bijection and +at most one chosen edge comes from each black face. + +Equations \eqref{eq:disk-counts}--\eqref{eq:matching-disk} +imply \(2l-h=-3\). This is the weight sum for a +counterclockwise traversal. A clockwise traversal reverses +the signs and has sum \(3\). +\end{proof} + +The matching condition in \eqref{eq:matching-disk} is essential. +The conclusion need not hold for arbitrary oriented cycles +of the triangulation. + +\section{A weighted cancellation argument}\label{sec:cancellation} + +For a state \(s\), let \(Q_s\) be the spanning subgraph of +\(T\) consisting of the edge opposite \(s(t)\) in every +\(t\in A\). These edges are distinct, since every edge of +\(T\) has a unique black incident face. +Figure~\ref{fig:local-triangle}(a) shows the two local edge choices. + +\begin{proposition}\label{prop:forest-state} +Whenever at least one pair exists, some pair \((r,s)\) has +\(Q_s\) a forest. +\end{proposition} + +We prove this by cancelling all cyclic \(Q_s\)'s from a +weighted sum and then proving that the sum is nonzero. +The cancellation works for arbitrary weights; we will choose them +afterwards to establish nonvanishing. + +For now, assign a real number \(a(t,v)\) to each incidence +\(t\in A\), \(v\in V(t)\), and write +\[ + \omega(s)=\sum_{t\in A}a(t,s(t)). +\] +For a pair, direct the edge in \(t\) from \(r(t)\) to +\(s(t)\). Every vertex of \(X\) has one edge in and one +out, so these edges form a union \(D\) of vertex-disjoint +cycles on \(X\). They are distinct even as undirected +edges, since each edge belongs to just one black face. +Thus the cycles are simple. By Lemma~\ref{lem:disk}, +\begin{equation}\label{eq:J} + J(r,s)=\frac13\sum_{t\in A}\delta(r(t),s(t)) + \quad\hbox{is an integer}. +\end{equation} +For real \(x\), define the finite exponential sum +\begin{equation}\label{eq:Z} + Z(x)=\sum_{(r,s)\ {\rm pair}} + \ii^{J(r,s)}\exp\bigl(x\omega(s)\bigr). +\end{equation} + +\subsection{Cancellation} + +\begin{lemma}\label{lem:cancel} +For fixed \(s\) with cyclic \(Q_s\), the terms of +\eqref{eq:Z} having second state \(s\) sum to zero. +\end{lemma} + +\begin{proof} +If \(s\) has no partner there is nothing to prove. +Otherwise choose a simple undirected cycle in \(Q_s\), +using \(s\) alone to make this choice. +For a partner \(r\), write \(u(t)\) for the third vertex +of \(t\), other than \(r(t),s(t)\), and orient the +opposite edge from \(r(t)\) to \(u(t)\). Every vertex +has outdegree at most one. The chosen cycle must be +directed: its edges need distinct tails at all its +vertices. In particular it contains no root. + +On the triangles supplying this cycle, replace \(r(t)\) +by \(u(t)\), and leave all other assignments unchanged. +Call the new state \(\widetilde r\). This cyclically +permutes the vertices assigned along the cycle, so +\(\widetilde r\) remains a bijection and a partner of \(s\). +It reverses the chosen cycle; applying it again restores +\(r\). Thus it is an involution without fixed points. + +On a changed triangle, the three cyclic directions give +\[ + \delta(u(t),s(t))-\delta(r(t),s(t)) + =2\delta(r(t),u(t)). +\] +Lemma~\ref{lem:disk}, applied to the directed opposite-edge +cycle, now yields +\[ + J(\widetilde r,s)-J(r,s)=\pm2. +\] +Hence \(\ii^{J(\widetilde r,s)}=-\ii^{J(r,s)}\). +The exponential weights coincide because \(s\) is fixed. +The involution cancels all terms. +\end{proof} + +\subsection{Positive spoke circulations} + +We now choose the incidence weights so that the two orientations of each +cycle of \(D\) have strictly ordered weights: the sum of +\(a(t,\mathrm{head})\) is larger in the counterclockwise orientation. +We establish this comparison below by replacing cycle edges with spokes. +Inside each bounded black triangle place a center and join +it to the three vertices by disjoint incidence spokes. +Let \(H_0\) be the resulting plane graph, allowing isolated +vertices. + +\Needspace{14\baselineskip} +\begin{lemma}\label{lem:circulation} +There are real numbers \(a(t,v)\) on the spokes of \(H_0\) +such that every counterclockwise simple spoke cycle has +strictly positive weight, where center-to-vertex traversal +has weight \(a(t,v)\) and reverse traversal its negative. +\end{lemma} + +\begin{proof} +Work separately in each connected component. Prescribe +signed boundary circulation one on every bounded face and +the negative of their number on the unbounded face, always +traversing with the face on the left. For each bounded face +take a simple path in the connected plane dual to the +unbounded face. Give each crossed edge an antisymmetric +weight contributing one to the face being left and minus +one to the face being entered. Along the path these +contributions cancel at intermediate faces. Summing over +bounded faces realizes the prescribed circulations. +Summing face boundaries inside any counterclockwise simple +cycle cancels internal sides and gives the positive number +of component faces inside it. A component with no bounded +face has no cycle and needs no nonzero weights. +\end{proof} + +Fix these weights for the remainder of the proof. + +\subsection{Nonvanishing} + +Group the pairs in \eqref{eq:Z} by their undirected edge +set \(D\). This uses the two-orientations-per-cycle regrouping +familiar from double-dimer expansions +\cite[Section~3.2.1, Lemma~1]{Kenyon2014}. Such a set determines the edge chosen in every +triangle of \(A\), because every edge belongs to a unique +black face. The pairs in its group are exactly the +independent orientation choices on its cycles: tails +give \(r(t)\), heads give \(s(t)\), and each choice +still gives two bijections on \(X\). + +For a cycle \(C\) of \(D\), let \(L_+(C)\) be the sum +of \(a(t,\mathrm{head})\) along its counterclockwise +orientation, and \(L_-(C)\) the corresponding clockwise +sum. We claim +\begin{equation}\label{eq:positive-difference} + L_+(C)-L_-(C)>0. +\end{equation} +For the counterclockwise traversal, replace each edge +by its two-spoke route from tail through the center of +its black triangle to head, as in Figure~\ref{fig:local-triangle}(b). +Its spoke weight is exactly +\(a(t,\mathrm{head})-a(t,\mathrm{tail})\), so the +closed spoke curve has weight \(L_+(C)-L_-(C)\). + +This spoke curve is simple and counterclockwise. Indeed, +there is at most one edge from each black triangle in +\(D\). Distinct detours have disjoint interiors, and +their centers are distinct; the only shared original +vertices are successive vertices of the original cycle. +Moreover the original edge and its detour enclose a +subdisk of the black triangle excluding its third +corner. Sliding the edge across that subdisk is an +isotopy through simple curves. This remains true if +the third corner occurs elsewhere on \(C\). Performing +the slides successively preserves the counterclockwise +orientation. Lemma~\ref{lem:circulation} therefore proves +\eqref{eq:positive-difference}. + +\input{triangle-figure} + +By Lemma~\ref{lem:disk}, a counterclockwise cycle +contributes \(-1\) to \(J\), and a clockwise cycle +contributes \(1\). The whole group contribution is +therefore +\begin{equation}\label{eq:group-product} + \prod_{C\subset D} + \left(-\ii\exp(xL_+(C))+\ii\exp(xL_-(C))\right). +\end{equation} +Each factor vanishes to exactly first order at zero. +If \(c(D)\) is the number of cycles of \(D\), the first +nonzero Taylor coefficient of \eqref{eq:group-product} +is in degree \(c(D)\) and equals +\begin{equation}\label{eq:leading-coefficient} + (-\ii)^{c(D)} + \prod_{C\subset D}\bigl(L_+(C)-L_-(C)\bigr). +\end{equation} +There is at least one group. In the smallest occurring +degree \(c_{\min}\), all contributing groups have phase +\((-\ii)^{c_{\min}}\) and strictly positive real +magnitudes by \eqref{eq:positive-difference}. +Groups with more cycles have no term of that degree. +Thus this coefficient of \(Z\) is nonzero. +Lemma~\ref{lem:cancel} cancels every second state with +cyclic \(Q_s\), so some pair has \(Q_s\) a forest. +This proves Proposition~\ref{prop:forest-state}. + +\section{Recovering the two induced trees}\label{sec:trees} + +We now convert the forest state into one selected edge at every face. +The prescription at \(t_0\) enters through the root of a tree on the +white faces. A second use of the disk identity will make the selected +edges acyclic; their complementary dual cycle will establish inducedness. + +Assume that \(T\) has no separating triangle. +By Propositions~\ref{prop:pairs} and \ref{prop:forest-state}, +choose a pair \((r,s)\) with \(Q_s\) a forest. +Orient its edges from \(r(t)\) to the third vertex. +The outdegrees are one on \(X\) and zero at the roots. +\begin{lemma}\label{lem:root-components} +For a pair \((r,s)\) with \(Q_s\) a forest, every component of +\(Q_s\) contains exactly one of the three roots. +\end{lemma} + +\begin{proof} +If a component has \(q\) vertices and \(a\) roots, its total +outdegree is \(q-a\), while its tree edge count is \(q-1\). +Each edge contributes one to the total outdegree, so \(a=1\). +This includes isolated vertices. The partner \(r\) supplies the +orientation and the required outdegrees. +\end{proof} + +\subsection{A tree on the white faces} + +The construction below uses a relation between matchings and plane +trees, a theme of the Temperley correspondence and its weighted +extension \cite[Theorem~1]{KPW2000}. We give the triangular-state +construction explicitly, including the prescribed outer edge. + +Let \(H\) be the sphere graph of incidence spokes from +\emph{all} black centers, including \(t_0\), to their +vertices. This graph is connected, since every edge +of \(T\) can be routed through its black center. +The faces of \(H\) correspond to the white triangles. +To see this, the spokes split each black triangle +into three sectors adjoining its sides. Removing +the original edges of \(T\) joins each white triangle +to the three sectors across its sides, and original +vertices prevent further merging. + +Delete the spokes \((t,s(t))\), \(t\in A\). +What remains is the forest \(Q_s\) with its edges +subdivided by their black centers, together with +the three spokes from the center of \(t_0\) to the +three roots. Since those roots lie in distinct +components, the remainder is a spanning tree of \(H\). +By Lemma~\ref{lem:duality}, the duals of the deleted +spokes form a spanning tree on the white triangles. + +Root this white-face tree at the white triangle \(y_0\) +across \(e_0\) from \(t_0\). Assign every tree edge to +its endpoint farther from \(y_0\). These endpoints +use each white triangle except \(y_0\) exactly once. +A spoke \((t,v)\) borders the white triangles across +the two sides of \(t\) incident with \(v\). +Thus for the deleted spoke \((t,s(t))\), select the +edge of \(t\) incident with \(s(t)\) bordering its +assigned white triangle. Finally select \(e_0\). + +Denote the selected set by \(P\), and put +\(P_0=P\setminus\{e_0\}\). +There is exactly one selected edge at each black +and at each white triangle. Selections from distinct +black faces cannot coincide, so \(\abs P=k\). +Orient each edge of \(P_0\) out from \(s(t)\) in the +black triangle that supplied it. + +\subsection{Acyclicity and inducedness} + +\begin{lemma}\label{lem:P-forest} +The selected set \(P\) is a forest. +\end{lemma} + +\begin{proof} +An undirected simple cycle in \(P_0\) must be +directed, since each vertex has outdegree at most one. +Use \(B,W,h,l\) for its disk data as in +Lemma~\ref{lem:disk}, now with state \(s\). +There is an additional selected-edge count: +\begin{equation}\label{eq:selection-count} + B-W=2l-h. +\end{equation} +Count the unique selected edge of every interior +face with sign \(+1\) for black and \(-1\) for white. +A selected edge whose relative interior lies inside the disk +contributes one of each sign and cancels. This includes a chord +with both endpoints on the boundary: both of its incident faces +lie in the disk. Every boundary edge +is selected, contributing \(+1\) when its black +face is inside and \(-1\) otherwise. This proves +\eqref{eq:selection-count}. + +But signed triangle-side counting gives +\(3(B-W)=2l-h\), so \eqref{eq:selection-count} +would imply \(2l-h=0\). The cycle would therefore have +\(\delta\)-sum zero in either orientation, contrary to +Lemma~\ref{lem:disk}. Thus \(P_0\) is a forest. +Its outdegrees are one at every nonroot and zero at +the roots, so the same component count used for +\(Q_s\) gives one root per component. +The edge \(e_0\) joins two different roots, hence +different components. Adding it preserves acyclicity. +\end{proof} + +Take the spanning dual subgraph formed by the edges +dual to \(E(T)\setminus P\). It is connected by +Lemma~\ref{lem:duality}. Every face of \(T\) has two +unselected sides, so this dual subgraph is 2-regular. +It has \(2k>2\) vertices and is one simple cycle. +Draw its edges crossing just their corresponding +primal edges, once transversely, and avoiding all +primal vertices. + +The resulting Jordan curve labels each vertex of +\(T\) by the side on which it lies. An edge has +same-label ends exactly when it is in \(P\): +an unselected edge crosses the curve once, while +a selected edge has no crossing. Both labels occur. +By \eqref{eq:counts}, the forest \(P\) has +\((k+2)-k=2\) components. Each has a constant label, +so there is exactly one component for each label. +The classes therefore induce trees. The unique +same-label edge at \(t_0\) is \(e_0\). +This proves Theorem~\ref{thm:partition} in the absence +of separating triangles. + +\section{Separating triangles and the Hamiltonian cycle}\label{sec:assembly} + +The facial prescription lets us glue partitions across separating +triangles. We then deduce Hamiltonicity and the edge-avoidance consequence. + +\begin{proof}[Proof of Theorem~\ref{thm:partition}] +Induct on the number of vertices, using the construction +above when there is no separating triangle. +Otherwise split along a separating \(3\)-cycle into +two sphere triangulations, capping each side by a +triangle on the separating cycle. Both pieces are +simple, have at least four vertices, and are smaller +than \(T\). + +The inherited face coloring extends to each cap. +Indeed, on one side let \(B,W\) count its original +black and white faces. Signed side counting makes +\(3(B-W)\) the sum of the three boundary signs, +each \(+1\) or \(-1\). Divisibility by three +forces all three signs to be equal. Color the +cap oppositely. + +Apply induction first to the piece containing +the prescribed original face \(t_0\), with its +specified edge \(e_0\). On that piece's cap both +tree labels occur, since a monochromatic triangle +would be a cycle in an induced tree. +Prescribe the same-label edge of the cap in the +other piece and apply induction there. Rename +the two labels to agree on the shared triangle. + +For each label the two induced trees intersect +in a connected subtree: one vertex, or one edge +with its endpoints. Their union is a tree. +It is induced in \(T\), since there are no edges +between the interiors of the two sides. +The original prescription is preserved, completing +the induction. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Let \(G\) satisfy the four hypotheses and fix a sphere +embedding. It has no edge cut of size one or two. +For otherwise each side of such a cut would have +at least three vertices: cubicity and simplicity +give at least three crossing edges for a singleton +and at least four for two vertices. +Deleting the at most two crossing endpoints on +one side would leave a nonempty remnant disconnected +from the other side, contrary to \(3\)-vertex-connectivity. + +The sphere dual \(T=G^*\) is simple. A dual loop +or a pair of parallel dual edges would be a Jordan +curve crossing one or two primal edges once, +giving an edge cut of that size. Its two primal +sides are nonempty, since every crossed edge +has its endpoints on opposite sides. +Cubicity gives each dual face a boundary walk +of length three, which in a simple graph is +a triangle. Euler's formula gives +\(\abs{V(T)}=2+\abs{V(G)}/2\ge4\). +The bipartition of \(G\) colors the faces of \(T\) +black and white with opposite colors at every edge. + +Apply Theorem~\ref{thm:partition}. The edges induced +by the two classes form a spanning forest. +Each triangle has exactly one such edge, since +neither tree class can contain all three vertices. +The complementary dual edges in \(G\) are connected +by Lemma~\ref{lem:duality} and give degree two at +every vertex. They form a single spanning cycle, +which is the required Hamiltonian cycle. +\end{proof} + +\begin{corollary}\label{cor:edge-avoidance} +For every graph \(G\) satisfying Theorem~\ref{thm:main} and every +edge \(e\in E(G)\), there is a Hamiltonian cycle of \(G\) avoiding \(e\). +\end{corollary} + +\begin{proof} +In the dual triangulation choose a facial triangle containing the dual +edge \(e^*\), and prescribe \(e_0=e^*\) in +Theorem~\ref{thm:partition}. That edge belongs to one of the induced +trees. The complementary dual Hamiltonian cycle constructed above +therefore omits \(e\). +\end{proof} + +Recall that a graph \(G\) is \emph{Pfaffian} if its edges admit an +orientation for which every even cycle \(C\) such that \(G-V(C)\) +has a perfect matching has an odd number of edges directed along either +cyclic traversal \cite[Section~4]{GSW2023}. + +\begin{corollary}[Prescribed paths in Pfaffian graphs]\label{cor:pfaffian-paths} +Let \(G\) be a finite simple undirected graph that is cubic, +\(3\)-vertex-connected, Pfaffian, and bipartite. Every path on four +vertices (three edges) in \(G\) is contained in a Hamiltonian cycle. +In other words, \(G\) is \(P_4\)-Hamiltonian; in particular, it is +Hamiltonian. +\end{corollary} + +\begin{proof} +Gorsky, Steiner, and Wiederrecht \cite[Theorem~4.16]{GSW2023} proved +the equivalence of Barnette's conjecture, their condition~(i), with +this \(P_4\)-Hamiltonicity statement, their condition~(v). +Theorem~\ref{thm:main} establishes condition~(i), so their implication +from~(i) to~(v) applies. Every planar graph is Pfaffian +\cite[Theorem~4.6]{GSW2023}, so the corollary includes the graphs of +Theorem~\ref{thm:main}, while also allowing nonplanar graphs. +\end{proof} + +\Needspace{12\baselineskip} +\begingroup +\small +\setlength{\bibsep}{3pt} +\bibliographystyle{plainnat} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/references.bib b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..4400fb02399e32a19f5cf4af43c2e9f474ce85ac --- /dev/null +++ b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/references.bib @@ -0,0 +1,150 @@ +@incollection{Barnette1969, + author = {Gr{\"u}nbaum, Branko}, + title = {Unsolved Problem 5}, + booktitle = {Recent Progress in Combinatorics: Proceedings of the Third + Waterloo Conference on Combinatorics, May 1968}, + editor = {Tutte, William T.}, + publisher = {Academic Press}, + address = {New York}, + year = {1969}, + pages = {343}, + note = {Conjecture attributed to David W. Barnette} +} + +@article{APSW2016, + author = {Alt, Helmut and Payne, Michael S. and Schmidt, Jens M. + and Wood, David R.}, + title = {Thoughts on {Barnette}'s Conjecture}, + journal = {Australasian Journal of Combinatorics}, + volume = {64}, + number = {2}, + year = {2016}, + pages = {354--365}, + url = {https://arxiv.org/abs/1312.3783v1}, + eprint = {1312.3783}, + archivePrefix = {arXiv}, + primaryClass = {math.CO} +} + +@misc{HK2018, + author = {Hine, Camden and K{\'a}lm{\'a}n, Tam{\'a}s}, + title = {Clock theorems for triangulated surfaces}, + year = {2018}, + howpublished = {arXiv:1808.06091v1}, + doi = {10.48550/arXiv.1808.06091}, + url = {https://arxiv.org/abs/1808.06091v1}, + eprint = {1808.06091}, + archivePrefix = {arXiv}, + primaryClass = {math.CO} +} + +@article{GSW2023, + author = {Gorsky, Maximilian and Steiner, Raphael and Wiederrecht, + Sebastian}, + title = {Matching theory and {Barnette}'s conjecture}, + journal = {Discrete Mathematics}, + volume = {346}, + number = {2}, + year = {2023}, + pages = {113249}, + doi = {10.1016/j.disc.2022.113249}, + url = {https://arxiv.org/abs/2202.11641v3}, + eprint = {2202.11641}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + note = {Theorem numbering follows arXiv:2202.11641v3, + 15 August 2022} +} + +@article{Tutte1946, + author = {Tutte, William T.}, + title = {On {Hamiltonian} Circuits}, + journal = {Journal of the London Mathematical Society}, + volume = {s1-21}, number = {2}, pages = {98--101}, year = {1946}, + doi = {10.1112/jlms/s1-21.2.98} +} +@article{Stein1971, + author = {Stein, Sherman K.}, + title = {{$B$}-sets and planar maps}, + journal = {Pacific Journal of Mathematics}, + volume = {37}, number = {1}, pages = {217--224}, year = {1971}, + url = {https://msp.org/pjm/1971/37-1/pjm-v37-n1-p20-s.pdf} +} +@article{Goodey1975, + author = {Goodey, Paul R.}, + title = {Hamiltonian circuits in polytopes with even sided faces}, + journal = {Israel Journal of Mathematics}, + volume = {22}, pages = {52--56}, year = {1975}, + doi = {10.1007/BF02757273} +} +@techreport{FederSubi2006, + author = {Feder, Tom{\'a}s and Subi, Carlos}, + title = {On {Barnette}'s conjecture}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR06-015}, year = {2006}, + url = {https://eccc.weizmann.ac.il/report/2006/015/} +} +@article{Kardos2020, + author = {Kardo{\v{s}}, Franti{\v{s}}ek}, + title = {A Computer-Assisted Proof of the {Barnette--Goodey} Conjecture: + Not Only Fullerene Graphs Are {Hamiltonian}}, + journal = {SIAM Journal on Discrete Mathematics}, + volume = {34}, number = {1}, pages = {62--100}, year = {2020}, + doi = {10.1137/140984737}, + eprint = {1409.2440}, archivePrefix = {arXiv}, primaryClass = {math.CO}, + url = {https://arxiv.org/abs/1409.2440v2} +} +@techreport{Hertel2005, + author = {Hertel, Alexander}, + title = {A Survey and Strengthening of {Barnette}'s Conjecture}, + institution = {Department of Computer Science, University of Toronto}, + year = {2005}, + url = {https://www.cs.toronto.edu/~ahertel/WebPageFiles/Papers/StrengtheningBarnette'sConjecture10.pdf} +} +@incollection{Tutte1975, + author = {Tutte, William T.}, + title = {Duality and trinity}, + booktitle = {Infinite and Finite Sets}, + editor = {Hajnal, Andr{\'a}s and Rado, Richard and S{\'o}s, Vera T.}, + series = {Colloquia Mathematica Societatis J{\'a}nos Bolyai}, + volume = {10}, pages = {1459--1472}, + publisher = {North-Holland}, address = {Amsterdam}, year = {1975}, + note = {Volume III; proceedings of the 1973 Keszthely colloquium} +} +@article{FF1956, + author = {Ford, Jr., Lester R. and Fulkerson, Delbert R.}, + title = {Maximal Flow Through a Network}, + journal = {Canadian Journal of Mathematics}, + volume = {8}, pages = {399--404}, year = {1956}, + doi = {10.4153/CJM-1956-045-5} +} +@article{KPW2000, + author = {Kenyon, Richard W. and Propp, James G. and Wilson, David B.}, + title = {Trees and matchings}, + journal = {Electronic Journal of Combinatorics}, + volume = {7}, number = {1}, pages = {R25}, year = {2000}, + doi = {10.37236/1503}, + eprint = {math/9903025}, archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/9903025v2} +} +@article{Kenyon2014, + author = {Kenyon, Richard}, + title = {Conformal Invariance of Loops in the Double-Dimer Model}, + journal = {Communications in Mathematical Physics}, + volume = {326}, number = {2}, pages = {477--497}, year = {2014}, + doi = {10.1007/s00220-013-1881-0}, + eprint = {1105.4158}, archivePrefix = {arXiv}, primaryClass = {math.PR}, + url = {https://arxiv.org/abs/1105.4158v2} +} + +@misc{Schnieders2025, + author = {Tobias Schnieders}, + title = {{Barnette} Graphs with Faces up to Size 8 are Hamiltonian}, + year = {2025}, + eprint = {2508.03531}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + doi = {10.48550/arXiv.2508.03531}, + url = {https://arxiv.org/abs/2508.03531v1}, + note = {Version 1, 5 August 2025} +} diff --git a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/triangle-figure.tex b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/triangle-figure.tex new file mode 100644 index 0000000000000000000000000000000000000000..7d55efd803c13f8a823915579ddf927bcf8e8a53 --- /dev/null +++ b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/build/triangle-figure.tex @@ -0,0 +1,59 @@ +\begin{figure}[tb] + \centering + \begin{tikzpicture}[ + font=\small, + line cap=round, + line join=round, + boundary/.style={draw=black!45,line width=.6pt}, + pair edge/.style={draw=blue!65!black,line width=1.1pt, + -{Stealth[length=2.1mm,width=1.5mm]}}, + opposite edge/.style={draw=black!75,line width=1pt,dashed, + -{Stealth[length=2.1mm,width=1.5mm]}}, + vertex/.style={circle,fill=black,inner sep=1.4pt} + ] + \begin{scope} + \coordinate (r) at (0,0); + \coordinate (s) at (3,0); + \coordinate (u) at (1.5,2.2); + \fill[black!5] (r)--(s)--(u)--cycle; + \draw[boundary] (r)--(s)--(u)--cycle; + \draw[pair edge,shorten >=2pt,shorten <=2pt] (r)--(s); + \draw[opposite edge,shorten >=2pt,shorten <=2pt] (r)--(u); + \node[vertex,label=below left:{$r(t)$}] at (r) {}; + \node[vertex,label=below right:{$s(t)$}] at (s) {}; + \node[vertex,label=above:{$u(t)$}] at (u) {}; + \node at (1.7,.9) {$t$}; + \node[align=center] at (1.5,-.85) + {\textbf{(a)} The two local edge choices}; + \end{scope} + \begin{scope}[xshift=6cm] + \coordinate (v) at (0,0); + \coordinate (w) at (3,0); + \coordinate (z) at (1.5,2.2); + \coordinate (c) at (1.5,.85); + \fill[black!5] (v)--(w)--(z)--cycle; + \fill[blue!12] (v)--(w)--(c)--cycle; + \draw[boundary] (v)--(w)--(z)--cycle; + \draw[draw=blue!65!black,line width=.85pt,dashed, + -{Stealth[length=2mm,width=1.4mm]},shorten >=2pt,shorten <=2pt] + (v)--(w); + \draw[pair edge,shorten >=2pt,shorten <=2pt] (v)--(c); + \draw[pair edge,shorten >=2pt,shorten <=2pt] (c)--(w); + \node[vertex,label=below left:{$v$}] at (v) {}; + \node[vertex,label=below right:{$w$}] at (w) {}; + \node[vertex,label=above:{$z$}] at (z) {}; + \node[circle,fill=blue!65!black,inner sep=1.7pt, + label=above:{$t$}] at (c) {}; + \node[align=center] at (1.5,-.85) + {\textbf{(b)} The local spoke detour}; + \end{scope} + \end{tikzpicture} + \caption{Two local constructions in a bounded black triangle. + (a) A paired state gives the edge $r(t)\to s(t)$ of $D$ + (solid) and the edge $r(t)\to u(t)$ of $Q_s$ (dashed). + (b) The directed edge $v\to w$ (dashed) is slid to the + two-spoke route $v\to t\to w$ (solid), with $t$ denoting + the face center. The shaded subdisk lies between the two routes + and excludes the third corner $z$.} + \label{fig:local-triangle} +\end{figure} diff --git a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e300b40a169850ecf2c018d593ed07b15f003195 --- /dev/null +++ b/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b519ff0c2b1f1554f34f01e644cd67651749826d43919862db6c42303701db18 +size 333997 diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/README.md b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..99a3dd25a05607047aac00b7af367430272fae12 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [Parabolic Intersections in Artin Groups](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Parabolic-intersections-in-Artin-groups-September-23-2026, + author = {{OpenAI}}, + title = {{Parabolic Intersections in Artin Groups}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf}{OAI:Parabolic-intersections-in-Artin-groups-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/figures/tied-path.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/figures/tied-path.tex new file mode 100644 index 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+\usepackage{microtype} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} +\usepackage{enumitem} +\usepackage{booktabs,array} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,calc,patterns} +\usepackage{xcolor} +\definecolor{linkcolor}{RGB}{34,75,87} +\usepackage[colorlinks=true,allcolors=linkcolor,breaklinks=true]{hyperref} +\usepackage{url} +\urlstyle{same} +\hypersetup{pdftitle={Parabolic Intersections in Artin Groups},pdfauthor={OpenAI}} +\setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*,itemsep=3pt} +\setlist[itemize]{leftmargin=*,itemsep=3pt} +\setlength{\emergencystretch}{2em} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\kk}{\mathbb C} +\newcommand{\RR}{\mathbb R} +\newcommand{\cC}{\mathscr C} +\newcommand{\wt}{\mathrm{wt}} +\newcommand{\transpose}{\mathsf T} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Cone}{Cone} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\soc}{soc} +\DeclareMathOperator{\Top}{top} +\DeclareMathOperator{\rank}{rank} + +\title{Parabolic Intersections in Artin Groups} +\author{OpenAI} +\date{September 23, 2026} + +\begin{document} +\maketitle +\begin{abstract} +We prove that every intersection of parabolic subgroups of any finite-rank +Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. +This resolves the Parabolic Intersection Conjecture affirmatively. +\end{abstract} +\tableofcontents + +\input{sections/01_intro} +\input{sections/02_category} +\input{sections/03_layers} +\input{sections/04_caps_twopoint} +\input{sections/05_null} +\input{sections/06_heights} +\input{sections/07_spherical} +\input{sections/08_stabilizers} + +\begingroup +\small +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/references.bib b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..dc92b96bea017683d772eaa81c8533a19f23dda0 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/references.bib @@ -0,0 +1,414 @@ +@book{Bourbaki2002, + author = {Bourbaki, Nicolas}, + title = {Lie Groups and Lie Algebras: Chapters 4--6}, + series = {Elements of Mathematics}, + publisher = {Springer-Verlag}, + address = {Berlin}, + year = {2002}, + note = {English translation of the 1968 French original}, + url = {https://link.springer.com/book/9783540691716} +} + +@phdthesis{VanDerLek1983, + author = {van der Lek, Harm}, + 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first posted in 2025}, + eprint = {2509.08382v2}, + archivePrefix = {arXiv}, + primaryClass = {math.GR}, + url = {https://arxiv.org/html/2509.08382v2} +} + +@article{HuerfanoKhovanov2001, + author = {Huerfano, Ruth Stella and Khovanov, Mikhail}, + title = {A category for the adjoint representation}, + journal = {Journal of Algebra}, + volume = {246}, + number = {2}, + pages = {514--542}, + year = {2001}, + doi = {10.1006/jabr.2001.8962}, + eprint = {math/0002060}, + archivePrefix = {arXiv}, + primaryClass = {math.QA}, + note = {Preprint version consulted: arXiv:math/0002060v2}, + url = {https://doi.org/10.1006/jabr.2001.8962} +} + +@article{KhovanovSeidel2002, + author = {Khovanov, Mikhail and Seidel, Paul}, + title = {Quivers, {Floer} cohomology, and braid group actions}, + journal = {Journal of the American Mathematical Society}, + volume = {15}, + number = {1}, + pages = {203--271}, + year = {2002}, + doi = {10.1090/S0894-0347-01-00374-5}, + eprint = {math/0006056}, + archivePrefix = {arXiv}, + primaryClass = {math.QA}, + note = {Preprint version consulted: arXiv:math/0006056v2}, + url = {https://doi.org/10.1090/S0894-0347-01-00374-5} +} + +@article{SeidelThomas2001, + author = {Seidel, Paul and Thomas, Richard}, + title = {Braid group actions on derived categories of coherent sheaves}, + journal = {Duke Mathematical Journal}, + volume = {108}, + number = {1}, + pages = {37--108}, + year = {2001}, + doi = {10.1215/S0012-7094-01-10812-0}, + eprint = {math/0001043}, + archivePrefix = {arXiv}, + primaryClass = {math.AG}, + note = {Preprint version consulted: arXiv:math/0001043v2}, + url = {https://doi.org/10.1215/S0012-7094-01-10812-0} +} + +@article{DuncanKazachkovRemeslennikov2007, + author = {Duncan, Andrew J. and Kazachkov, Ilya V. and Remeslennikov, Vladimir N.}, + title = {Parabolic and quasiparabolic subgroups of free partially commutative groups}, + journal = {Journal of Algebra}, + volume = {318}, + number = {2}, + pages = {918--932}, + year = {2007}, + doi = {10.1016/j.jalgebra.2007.08.032}, + eprint = {math/0702431}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0702431} +} + +@article{Blufstein2022, + author = {Blufstein, Mart{\'i}n Axel}, + title = {Parabolic subgroups of two-dimensional {Artin} groups and systolic-by-function complexes}, + journal = {Bulletin of the London Mathematical Society}, + volume = {54}, + number = {6}, + pages = {2338--2350}, + year = {2022}, + eprint = {2108.04929v2}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2108.04929v2} +} + +@article{Haettel2024, + author = {Haettel, Thomas}, + title = {Lattices, injective metrics and the {$K(\pi,1)$} conjecture}, + journal = {Algebraic \& Geometric Topology}, + volume = {24}, + number = {7}, + pages = {4007--4060}, + year = {2024}, + doi = {10.2140/agt.2024.24.4007}, + eprint = {2109.07891}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2109.07891} +} + +@article{CumplidoGavazziParis2024, + author = {Cumplido, Mar{\'i}a and Gavazzi, Federica and Paris, Luis}, + title = {Intersection of parabolic subgroups in {Euclidean} braid groups: a short proof}, + journal = {Comptes Rendus. Math{\'e}matique}, + volume = {362}, + pages = {1445--1448}, + year = {2024}, + doi = {10.5802/crmath.656}, + eprint = {2402.10919}, + archivePrefix = {arXiv}, + url = {https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.656/} +} + +@article{Godelle2023, + author = {Godelle, Eddy}, + title = {On parabolic subgroups of {Artin--Tits} groups}, + journal = {Journal of Algebra}, + volume = {632}, + pages = {520--534}, + year = {2023}, + eprint = {2207.06528v1}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2207.06528v1} +} + +@article{BlufsteinParis2023, + author = {Blufstein, Mart{\'i}n Axel and Paris, Luis}, + title = {Parabolic subgroups inside parabolic subgroups of {Artin} groups}, + journal = {Proceedings of the American Mathematical Society}, + volume = {151}, + number = {4}, + pages = {1519--1526}, + year = {2023}, + doi = {10.1090/proc/16289}, + eprint = {2204.05142v2}, + archivePrefix = {arXiv}, + note = {Preprint version consulted: version 2}, + url = {https://arxiv.org/abs/2204.05142v2} +} + +@article{HoltRees2012, + author = {Holt, Derek F. and Rees, Sarah}, + title = {{Artin} groups of large type are shortlex automatic with regular geodesics}, + journal = {Proceedings of the London Mathematical Society}, + series = {3}, + volume = {104}, + number = {3}, + pages = {486--512}, + year = {2012}, + doi = {10.1112/plms/pdr035}, + eprint = {1003.6007v1}, + archivePrefix = {arXiv}, + note = {Preprint version consulted: version 1}, + url = {https://arxiv.org/abs/1003.6007v1} +} + +@article{BlascoGarciaEtAl2026, + author = {Blasco-Garc{\'i}a, Rub{\'e}n and Cumplido, Mar{\'i}a and + Holt, Derek F. and Morris-Wright, Rose and Rees, Sarah}, + title = {Rewriting in {Artin} groups without {$A_3$} or {$B_3$} subdiagrams}, + journal = {Journal of Algebra}, + volume = {703}, + pages = {374--421}, + year = {2026}, + doi = {10.1016/j.jalgebra.2026.02.017}, + eprint = {2412.12195v2}, + archivePrefix = {arXiv}, + note = {Preprint version 2 and publisher proof consulted}, + url = {https://doi.org/10.1016/j.jalgebra.2026.02.017} +} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/01_intro.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/01_intro.tex new file mode 100644 index 0000000000000000000000000000000000000000..f5aa73222dfe3c6cfe723151a8e71a8d54c0925d --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/01_intro.tex @@ -0,0 +1,277 @@ +\section{Introduction} +\label{sec:intro} + +Let $S$ be finite, and let $M=(m_{st})_{s,t\in S}$ be a symmetric Coxeter +matrix. Thus $m_{ss}=1$ and, for $s\ne t$, +$m_{st}\in\{2,3,\ldots\}\cup\{\infty\}$. The associated Artin group is +\[ + A=A_M=\left\langle \sigma_s\ (s\in S)\ \middle|\ + \underbrace{\sigma_s\sigma_t\sigma_s\cdots}_{m_{st}} + =\underbrace{\sigma_t\sigma_s\sigma_t\cdots}_{m_{st}} + \quad(m_{st}<\infty)\right\rangle. +\] +For $X\subseteq S$, write $A_X=\langle\sigma_s:s\in X\rangle$, with +$A_\varnothing=\{1\}$. A \emph{parabolic subgroup} is a conjugate of +some $A_X$. The rank here refers to the cardinality of the defining set; +no finiteness assumption on the associated Coxeter group is understood. +The Coxeter group is obtained by imposing $\sigma_s^2=1$ for all $s$; +a type, or a parabolic of that type, is called \emph{spherical} when +its Coxeter group is finite. + +\begin{theorem}[Parabolic intersections] +\label{thm:main} +For every Coxeter matrix $M$ on a finite set $S$, subsets $X,Y\subseteq S$, and +$g,h\in A_M$, there are $Z\subseteq S$ and $k\in A_M$ such that +\[ + gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}. +\] +\end{theorem} + +In particular, the result applies to arbitrary conjugates of nonspherical +parabolics and to diagrams with infinite labels. The passage from pairwise intersections to arbitrary intersections and +parabolic closure is established in +\cite[Proposition~1.5 and Corollary~1.6]{MoellerParisVarghese2024}. +Our inside-parabolic description gives the following form, whose proof +in Section~\ref{sec:consequences} also records a finite witness bound. + +\begin{corollary}[Arbitrary intersections and parabolic closure] +\label{cor:arbitrary-intersections} +For every family $\mathcal F$ of parabolic subgroups of $A$, there is a +subfamily $\mathcal F_0\subseteq\mathcal F$ with $|\mathcal F_0|\le |S|$ such +that +\[ + \bigcap_{P\in\mathcal F}P=\bigcap_{P\in\mathcal F_0}P. +\] +Here the intersection of an empty family is $A$. In particular, arbitrary +intersections of parabolic subgroups are parabolic, and every subset +$E\subseteq A$ is contained in a unique smallest parabolic subgroup. +\end{corollary} + + +Writing $\operatorname{PC}(E)$ for this parabolic closure, the parabolic +subgroups form a complete lattice: their meet is intersection and the +join of a family is the parabolic closure of its union. + +The same stabilizer description gives an effective membership test. +Here the conjugating element and the tested element are supplied as +words in the standard generators and their inverses. + +\begin{corollary}[Word and parabolic membership problems] +\label{cor:membership} +There is an algorithm which, given a finite Coxeter matrix $M$, a subset +$X\subseteq S$, and words $u,v$ in the standard generators of $A_M$ and their inverses, decides +whether the element represented by $v$ belongs to $uA_Xu^{-1}$. +In particular, the word problem is decidable for every finite-rank +Artin group. +\end{corollary} + +The proof in Section~\ref{sec:algorithms} constructs the finite algebra +with exact algebraic coefficients and tests a single transported +projective by Gaussian elimination. + +The intersection theorem also supplies the hypothesis of a published +criterion for Artin groups with a visual splitting. We state the +result with the resulting trivial-intersection witness; its proof is +in Section~\ref{sec:consequences}. + +\begin{corollary}[Non-clique Artin-group dynamics] +\label{cor:nonclique-dynamics} +Suppose $M$ is irreducible, meaning that there is no nontrivial partition +$S=S_1\sqcup S_2$ with $m_{st}=2$ for every $s\in S_1$, $t\in S_2$, and +suppose $m_{st}=\infty$ for some pair. Then $A$ is acylindrically hyperbolic. +Every proper parabolic subgroup $P$ has a conjugate meeting it trivially: +there is $g\in A$ such that $P\cap gPg^{-1}=\{1\}$. +\end{corollary} + + +The acylindrical-hyperbolicity conclusion in this entire non-clique scope +already appears in \cite[Theorem~1.2]{KatoOguni2025}. Corollary~\ref{cor:nonclique-dynamics} +records the deduction through the parabolic-intersection criterion, not a +new acylindrical-hyperbolicity class. + +Theorem~\ref{thm:main} also removes the intersection hypothesis from +the fixed-point and locally compact group results of +\cite{MoellerParisVarghese2024}; their precise consequences are stated +in Section~\ref{sec:fixed-point-consequences}. + +\subsection{Context and method} + +Standard parabolic subgroups retain the Artin presentation of their +submatrix: this unrestricted embedding theorem is due to van der Lek +\cite{VanDerLek1983}, with subsequent treatments in +\cite{Paris1997,CharneyParis2014}. Charney and Paris also prove convexity of standard parabolics in the +standard word metric \cite[Theorem~1.2]{CharneyParis2014}. +Blufstein and Paris prove the general nesting theorem: an ambient +parabolic subgroup contained in a parabolic subgroup $Q$ is parabolic +as a subgroup of $Q$ \cite[Theorem~1.1]{BlufsteinParis2023}. +The intersection problem asks whether the intersection of two arbitrary +ambient parabolics is itself parabolic. It is the Parabolic Intersection Conjecture; precise +formulations appear in \cite[Conjecture~1]{Godelle2023} and +\cite[Section~3, Conjecture~4]{Cumplido2026Survey}. +Theorem~\ref{thm:main} resolves it affirmatively. + +Several approaches established this closure in substantial classes. +For right-angled Artin groups, Duncan, Kazachkov and Remeslennikov proved +even the arbitrary-family intersection statement +\cite[Proposition~2.8]{DuncanKazachkovRemeslennikov2007}. +In spherical type, Cumplido, Gebhardt, Gonz\'alez-Meneses and Wiest +proved pairwise closure using Garside theory and parabolic closures +\cite[Theorem~9.5]{CumplidoEtAl2019}. Their argument builds on the +finite-type positive-monoid structure developed by Brieskorn and Saito +and by Deligne \cite{BrieskornSaito1972,Deligne1972}. +Such positive fractions are available inside spherical subgroups; +they are not available as a general premise for the ambient groups +considered here. + +Geometric methods provide a different route. Cumplido, Martin and +Vaskou obtained arbitrary intersections in large type through fixed +paths in the Artin complex and stabilizer induction +\cite[Theorem~A and Theorem~11]{CumplidoMartinVaskou2023}. +Blufstein extended this approach to $(2,2)$-free two-dimensional Artin +groups \cite[Theorem~1.3]{Blufstein2022}. +Haettel proved the result for affine types $\widetilde A$ and +$\widetilde C$ using geodesic bicombings +\cite[Corollary~8.3]{Haettel2024}; Cumplido, Gavazzi and Paris later +gave a short alternative proof in type $\widetilde A$ +\cite[Theorem~1]{CumplidoGavazziParis2024}. + +The distinction between the ambient group and the two intersected +subgroups is especially important in FC type. Morris-Wright proved +the result for two spherical parabolics in an FC-type group, using its +CAT$(0)$ Deligne complex \cite[Theorem~3.1]{MorrisWright2021}. +M\"oller, Paris and Varghese showed that one spherical parabolic suffices +\cite[Corollary~1.2]{MoellerParisVarghese2024}. +Antol\'in and Foniqi proved full pairwise closure for even FC-type groups +by retractions and Bass--Serre theory +\cite[Theorem~1.1]{AntolinFoniqi2022}. +The full FC-type question also appears explicitly in +\cite[Question~3.6]{CharneyMartinMorrisWright2026}. +These results distinguish the closure problem for arbitrary conjugates +from the embedding and nesting properties of standard subgroups. + +The word problem has a parallel history. Brieskorn and Saito and +Deligne solved it in spherical type through positive normal forms +and the finite-type monoid structure +\cite[Section~6, especially Theorem~6.6]{BrieskornSaito1972}, +\cite[Section~4.20]{Deligne1972}. +For nonspherical examples, Holt and Rees construct shortlex normal +forms in large type \cite[Theorem~3.2]{HoltRees2012}. +Blasco-Garc\'ia, Cumplido, Holt, Morris-Wright and Rees give a +quadratic-time reduction to geodesic words when the Coxeter diagram +has no $A_3$ or $B_3$ subdiagram +\cite[Theorem~1.1]{BlascoGarciaEtAl2026}. +Their introduction explicitly poses the general decidability question; +Corollary~\ref{cor:membership} gives a uniform decision procedure with +the finite Coxeter matrix as part of the input. No complexity bound +is asserted here. + +Our proof also reduces intersections to stabilizers and then to smaller +parabolic types. The space on which we measure a stabilizer is supplied +by a categorical action and a relative harmonic height. The algebraic +construction belongs to the zigzag and braid-action framework of +Huerfano--Khovanov and Khovanov--Seidel +\cite{HuerfanoKhovanov2001,KhovanovSeidel2002}, with fusion-labelled +Artin actions as in Heng--Licata \cite{HengLicata2024}. +The positive-factor tests are related to Brav--Thomas +\cite{BravThomas2011}. We construct the required bimodule action and prove the layer and local +rigidity statements, using finite spherical tensor categories and the +standard long exact cohomology sequence as background inputs. In particular the proof does not use +asphericity of a Salvetti complex. + +Heng and Licata give a Gaussian-elimination algorithm for the word +problem in the image of their spherical-twist action +\cite[Proposition~7.11]{HengLicata2024}, and formulate faithfulness +of their original Artin-group action as Conjecture~7.10. +Our relative construction instead detects standard-parabolic +membership by a single framing projective. At the level of decidability, +this membership problem also follows from the word problem and +Charney--Paris convexity: for an input word of length $n$, test all +words of length at most $n$ in the specified parabolic generators and +their inverses. A supplied conjugate is handled by first conjugating +the input word. + +\subsection{From relative families to stabilizers} + +Fix $X\subseteq S$. Adjoin a vertex $f$ which commutes with the colors in +$X$ and has braid label $3$ with the colors in $S\setminus X$. +Section~\ref{sec:category} constructs a finite-dimensional graded +algebra $Z$, an action $x\mapsto B_x$ of $A$ on the homotopy category +of bounded complexes of graded projective $Z$-modules, and a distinguished finite family of +projectives $(P_{f,L})_L$. The index $L$ ranges over the simple labels +of a finite semisimple tensor category. Write +$E_L(x)=B_xP_{f,L}$ for the transported family. +For $Y\subseteq S$ put +\[ + H(Y,x)=\{a\in A_Y:E_L(ax)\simeq E_L(x)\text{ for every }L\}, +\] +where equivalence means graded chain homotopy equivalence. The two +claims that drive the proof are +\[ + H(S,1)=A_X, + \qquad + H(Y,x)\text{ is conjugate inside }A_Y\text{ to a standard parabolic}. +\] +The first claim translates membership in $A_X$ into a condition on the +initial family. Transport by $x$ then turns the intersection into +\[ + A_X\cap x^{-1}A_Yx=x^{-1}H(Y,x)x. +\] +The second claim therefore supplies the required parabolic type and +conjugator. The difficulty is to prove that claim when $Y$ is +nonspherical, without using positive fractions in $A_Y$. + +We measure this family by counting its projective generators, with +positive weights, separately at each color and each integer level. +The level is the sum of the generator's cochain and internal degrees. +On a spherical color set these counts have a harmonic interpolation: +the prescribed boundary values extend uniquely to a vector annihilated +there by the Coxeter Cartan matrix. Sections~\ref{sec:layers}--\ref{sec:null} +relate these numerical extensions to actual complexes and cosets. +The key obstruction rules out a nonzero isolated layer whose count +vector is a positive null vector. The relative frame requires this +obstruction at \emph{every integer level}; connected support supplies +the necessary boundary data. + +The resulting height is defined on cosets $A_Tz$, where $T\subseteq Y$ +is spherical and $z\in A_Yx$. Two vertices are joined when the types +and cosets are nested. Section~\ref{sec:heights} proves that a finite +path between minimum-height vertices can be lowered until it stays +entirely among minima. Comparable vertices can have equal height, +so this argument must also lower whole tied segments. Its conclusion +is connectedness of the minimum set. In spherical type, +Section~\ref{sec:spherical} supplies a different reduction when some +acted test has nonzero cohomology: a sequence of positive operations, +each using every color, cannot have bounded height. +A finite path bound for the algebra turns prolonged vanishing of +tested cohomology into an empty cochain placement, contradicting the +indecomposability of a transported projective. + +These two reductions have complementary roles. Harmonic minima either +place a stabilizer in a conjugate of smaller type or isolate a trivially +acting direct factor. If the operated set itself is spherical, however, +a minimum can have that whole set as its type, so harmonic containment +alone need not decrease rank. The positive-height argument gives a +proper-type reduction unless every test is acyclic and the entire group +already fixes the family. Rank induction then closes the argument +without assuming an intersection theorem at any intermediate step. + +\subsection{Organization and conventions} + +Sections~\ref{sec:category} and~\ref{sec:layers} construct the categorical +family and its layer calculus. Section~\ref{sec:caps} proves cap geometry +and the spherical two-point lemma. Section~\ref{sec:null} proves the +relative null-layer obstruction. Section~\ref{sec:heights} builds the +harmonic filtration and its stabilizer containment; +Section~\ref{sec:spherical} supplies the spherical positive-height +containment. Section~\ref{sec:stabilizers} proves the stabilizer theorem +and Theorem~\ref{thm:main}, then derives the further consequences. + +All complexes are bounded and have cochain differential of degree $1$. +Internal grading is separate from cochain grading; the shift $[1]$ lowers +cochain placements by one. A \emph{generator level} is the sum of its +cochain and internal placements. Unless explicitly qualified as ordinary +cochain degree, test heights use the sum of the two degrees. A coset +$A_Tz$ is a right coset. Every order on cosets below requires both +inclusion of types and actual inclusion of cosets. Connectivity of such +a poset always means finite-path connectivity of its undirected +comparability graph. diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/02_category.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/02_category.tex new file mode 100644 index 0000000000000000000000000000000000000000..bc6b4d5f0f0f1b5712c2a974df1406f6d5fbd0fe --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/02_category.tex @@ -0,0 +1,636 @@ +\section{The relative categorical family} +\label{sec:category} + +We construct an action on bounded complexes of graded projectives and a +family whose stabilizer will eventually be identified with $A_X$. Fusion +labels realize the possibly nonintegral entries of the Coxeter form by +an ordinary finite-dimensional algebra. The construction belongs to the +fusion-labelled zigzag framework of Heng and Licata +\cite[Sections~4--5]{HengLicata2024}. We supply the algebraic details +needed for our relative family, including braid relations as bimodule +chain homotopies. This strength will allow us to act on right simple +modules as well as on left projective complexes in Section~\ref{sec:caps}. + +\subsection{Classical inputs and the relative frame} + +Write $A=A_M$. For $T\subseteq S$, let $W_T$ denote the Coxeter group +defined by the restricted matrix, and call $T$ \emph{spherical} if +$W_T$ is finite. A positive word means a word in the generators +$\sigma_i$, with no inverses. Its support is the set of letters that +occur; positive braid relations preserve this set. + +\begin{theorem}[Classical Artin and Coxeter inputs] +\label{cat:classical} +The group defined by the Artin presentation on $T\subseteq S$ embeds +in $A$, with image $A_T$. No sphericity hypothesis is needed here. +For a spherical $T$, the positive Artin monoid $A_T^+$ embeds in +$A_T$, is cancellative, and has greatest common divisors on both +sides. Every element of $A_T$ is both a left fraction and a right +fraction of positive elements. Any two elements $a,b\in A_T$ thus +have a common positive extension $pa=qb$, with $p,q\in A_T^+$. + +Reduced expressions in any Coxeter system are related by braid moves. +They therefore define a positive lift $\iota(w)$ of a Coxeter element +$w$. If $T$ is spherical and $w_T$ is its longest element, the left +and right divisors of $\Delta_T=\iota(w_T)$ are precisely +$\{\iota(w):w\in W_T\}$. In the finite geometric representation, +root signs detect descents, the group acts simply transitively on +chambers, and the translates of a closed fundamental chamber cover +the entire dual space. +\end{theorem} + +The embedding statement is +\cite[Chapter~II, Section~4, Theorem~4.13(i)]{VanDerLek1983}; see also +\cite[Section~1, p.~2]{CharneyParis2014}. For the positive monoid, cancellation, +fractions, and greatest common divisors are +\cite[Propositions~2.3, 4.2, and~5.5; Theorem~5.6]{BrieskornSaito1972} +and \cite[Theorem~4.14]{Deligne1972}; the description of the divisors +of $\Delta_T$ is +\cite[Lemma~5.4 and Propositions~5.5 and~5.7]{BrieskornSaito1972}. +The common-extension assertion follows by writing +$ba^{-1}=q^{-1}p$. The Coxeter assertions are +\cite[Chapter~IV, \S1; Chapter~V, \S3, no.~3, and \S4, +nos.~4 and~6]{Bourbaki2002}; see also +\cite[Theorems~3.4.2(ii), 4.1.6(i), and Appendix~D]{Davis2008}. +Throughout the proof, positive fractions, +greatest common divisors, and the finite chamber covering are used +only after the relevant subsystem has been shown to be spherical. + +Fix $X\subseteq S$. Add a color $f$ with label $2$ to every color +of $X$ and label $3$ to every color of $S\setminus X$. Add one +further color $o$, with label $3$ to every color of $S\sqcup\{f\}$, +and put +\[ + \widehat S=S\sqcup\{f,o\}. +\] +Only colors of $S$ will act. The color $f$ specifies the initial +family, while $o$ is an auxiliary frame in the algebraic construction. +Set $C=2I-b$, where $b$ is symmetric, has zero diagonal, and +\[ + b_{ij}=2\cos(\pi/m_{ij})\quad(m_{ij}<\infty), + \qquad b_{ij}=2\quad(m_{ij}=\infty). +\] +Thus $b_{if}=0$ for $i\in X$, $b_{if}=1$ for +$i\in S\setminus X$, and $b_{io}=1$ for $i\ne o$. +Two colors are adjacent when $b_{ij}>0$. The boundary $\partial T$ +of a color set consists of its exterior neighbors in $\widehat S$. +All vector inequalities below are coordinatewise. + +On root-coefficient vectors, the geometric reflection at $i$ is +\begin{equation} + s_i v=v-(Cv)_i\varepsilon_i. + \label{cat:reflection} +\end{equation} +The restriction to a color subsystem is its geometric representation. +The finite-type criterion says that $T$ is spherical exactly when +$C_T$ is positive definite +\cite[Chapter~V, \S4, no.~8, Theorem~2]{Bourbaki2002}, +\cite[Theorem~6.12.9]{Davis2008}. + +\begin{lemma}[Positive and null Cartan data] +\label{cat:cartan} +For spherical $T$, the inverse $C_T^{-1}$ is nonnegative, and is +strictly positive in each connected block. Prescribed exterior +coordinates consequently have a unique extension $p$ satisfying +$(Cp)_T=0$. This extension is nonnegative if the prescribed +coordinates are nonnegative. + +If $G$ is nonempty and connected, $v_G>0$, and $C_Gv_G\geq0$, then +either $C_G$ is positive definite or $C_Gv_G=0$. +\end{lemma} + +\begin{proof} +For $C_T=2I-b_T>0$, the nonnegative symmetric matrix $b_T$ has +spectral radius less than $2$. Hence +\[ + C_T^{-1}=\frac12\sum_{n\geq0}(b_T/2)^n, + \qquad + p_T=C_T^{-1}b_{T,\widehat S\setminus T}p_{\widehat S\setminus T}. +\] +The first formula gives nonnegativity; a path between two vertices +in one connected block gives a strictly positive entry in some +power, proving strict positivity there. The second formula gives +the assertions about harmonic extension. + +For the final assertion, the following identity holds for every +real vector $u$ on $G$: +\begin{equation} + u^{\mathsf T}C_Gu + =\sum_{\substack{i0, + \qquad + F_1\otimes F_a\cong F_{a-1}\oplus F_{a+1}, +\] +where nonexistent endpoint summands are omitted. These are the +spherical semisimplifications of the planar pairing categories. +For the negligible quotient, finite semisimplicity and spherical +structure, see \cite[Sections~5.1--5.4, especially +Theorems~5.4.4--5.4.5 and the discussion following +Definition~5.4.7]{Chen2014}; the displayed simple list and fusion rule +also appear in \cite[Section~4.1, Equations~(10)--(11)]{HengLicata2024}. More +explicitly, the Jones--Wenzl idempotents with $a\leq m-2$ have the +positive dimensions displayed above, while the next idempotent has +trace zero and is killed in the semisimplification. Thus the +resulting category has finitely many simples, not merely finitely +many specified edge objects. + +Take the tensor product of these finitely many categories, or finite +dimensional vector spaces if there are no such edges. For an +oriented edge $(i,j)$ of label $m\geq4$, let $D^1_{ij}$ be $F_1$ in +its factor and the unit in the other factors. This per-edge $F_1$ +choice is a variation of the fusion-label choices in +\cite[Section~4.3, Equation~(15), and Remark~4.4]{HengLicata2024}. +On a label-$3$ edge +take $D^1_{ij}=\mathbf1$, and on an infinite edge take +$D^1_{ij}=\mathbf1\oplus\mathbf1$. Choose opposite edge objects +as pivotal duals. Then +\begin{equation} + \dim_{\cC}D^1_{ij}=b_{ij},\qquad + \dim_\kk\Hom_{\cC}(L,D^1_{ij}\otimes H)\leq1 + \quad\text{on every finite edge} + \label{cat:edge-labels} +\end{equation} +for simple $L,H$. The dimension bound follows from the displayed +fusion rule in a single factor. No such bound is required on +infinite edges. + +Let $\operatorname{Irr}(\cC)$ be a set of simple representatives, and +write +\[ + w_L=\dim_{\cC}L>0, + \qquad D_0=\sum_{L\in\operatorname{Irr}(\cC)}w_L^2. +\] +We always tensor labels on the right. No symmetry of the tensor +category is used. Duality and semisimplicity give the useful identity +\begin{equation} + \sum_L w_L[D\otimes L:H]=w_H\dim_{\cC}D + \qquad(H\text{ simple}), + \label{cat:weighted-labels} +\end{equation} +because the multiplicities on the left are those of $L$ in +$D^*\otimes H$. + +For each $i\in\widehat S$ and $D\in\cC$, introduce an additive +projective symbol $P_{i,D}$. Put +$D^0_{ii}=D^2_{ii}=\mathbf1$, use the edge objects in degree one, +and set all remaining $D^a_{ij}$ to zero. Define +\begin{equation} + \Hom^a(P_{i,D},P_{j,E}) + =\Hom_{\cC}(D,D^a_{ij}\otimes E). + \label{cat:projective-hom} +\end{equation} +Degree-zero maps compose as label maps. The product of two +degree-one maps is zero unless their colors backtrack; on a +backtrack, compose their label maps and apply the pivotal evaluation +$D^1_{ij}\otimes D^1_{ji}\to\mathbf1$. Opposite orientations use +the two pivotal evaluations. All other products of positive lengths +are zero. This multiplication is associative: compatibility with +degree zero is functoriality, while both bracketings of three +positive lengths vanish. + +An \emph{unfolded vertex} is a pair $\alpha=(i,L)$ with $L$ simple; +write $\widetilde T=T\times\operatorname{Irr}(\cC)$. The finite +basic graded algebra $Z$ is defined by +\[ + e_\alpha Z e_\beta=\Hom(P_\alpha,P_\beta) +\] +with multiplication in path order. Thus $P_\alpha=Ze_\alpha$ is +a left projective and +$\Hom_Z(P_\alpha,P_\beta)=e_\alpha Z e_\beta$. Let +$Q_\alpha=e_\alpha Z$ be the right projective and $U_\alpha$ its +simple top. For $\alpha=(i,L)$, write $t_\alpha$ for the +degree-two element corresponding to $\mathrm{id}_L$. We call it +the \emph{token}. The only path lengths are $0,1,2$, and lengths +zero and two each have a one-dimensional space on every diagonal +and vanish off the diagonal. + +\begin{lemma}[Trace and adjoint duality] +\label{cat:frobenius} +The functional +\[ + \tau(t_{i,L})=w_L, + \qquad \tau(Z^0\oplus Z^1)=0 +\] +is a symmetric Frobenius trace of degree $-2$. Opposite arrow +spaces pair perfectly by multiplication followed by $\tau$. The +radical of $Z$ is $Z^{>0}$, and the socle of every indecomposable +left or right projective is its token line. + +For a finite graded bimodule $N$, its adjoint dual satisfies +\begin{equation} + N^\vee:=\Hom_{Z\text{-left}}(N,Z) + \cong D_\kk(N)\langle2\rangle, + \qquad + (P_\alpha\otimes_\kk Q_\beta)^\vee + \cong(P_\beta\otimes_\kk Q_\alpha)\langle-2\rangle. + \label{cat:adjoint-dual} +\end{equation} +This is an involutive contravariant equivalence, fixes the regular +bimodule, and preserves additive radicals. +\end{lemma} + +\begin{proof} +Units and tokens pair perfectly. For arrows, pivotal adjunction +identifies a map $H\to D^1_{ji}\otimes L$ with a map +$D^1_{ij}\otimes H\to L$. Pairing it against +$L\to D^1_{ij}\otimes H$ is a semisimple trace pairing, nondegenerate +because the trace on each simple matrix block is multiplied by a +nonzero dimension. The two orders of a backtrack have the same +trace by the pivotal partial-trace identity. Hence +$\tau(ab)=\tau(ba)$, with no sign from internal degree. + +The positive-length ideal is nilpotent and its quotient is the +semisimple diagonal, proving the radical assertion. A token is +annihilated by the radical. A nonzero arrow pairs with an opposite +arrow, and a nonzero degree-zero element pairs with a token; neither +can lie in the projective socle. This proves the socle assertion. + +The map $f\mapsto\tau\circ f$ gives the first identification in +\eqref{cat:adjoint-dual}. A map of degree $k$ gives a functional +of degree $k-2$, which accounts for $\langle2\rangle$. With the +bimodule convention $(a f b)(n)=f(na)b$, symmetry gives +\[ + \tau(f(na)b)=\tau(bf(na))=\tau(f(bna)), +\] +exactly the vector-dual bimodule action. The trace identifies +$D_\kk(P_\alpha)=Q_\alpha\langle-2\rangle$ and +$D_\kk(Q_\beta)=P_\beta\langle-2\rangle$. Reversing the tensor +factors and then applying the shift $2$ proves the second formula. +Shifted vector duality squares to the identity. Any contravariant +equivalence preserves the additive radical, characterized by +invertibility of $1-gf$ for all opposite maps $g$. +\end{proof} + +\subsection{Finite unfoldings} + +Let $\widetilde C$ have diagonal entries $2$ and off-diagonal entries +$-\dim_\kk(e_\alpha Z^1e_\beta)$, and set +\begin{equation} + (\phi v)_{i,L}=w_Lv_i. + \label{cat:phi} +\end{equation} +The arrow pairing makes $\widetilde C$ symmetric, and +\eqref{cat:weighted-labels} gives +\begin{equation} + \widetilde C\phi=\phi C. + \label{cat:intertwining} +\end{equation} +Both statements remain true on every color submatrix. + +\begin{lemma}[Finite unfolding] +\label{cat:unfolding} +For spherical $T\subseteq S$, the matrix +$\widetilde C_{\widetilde T}$ is positive definite and simply laced. +Replacing an original reflection $s_i$ by the product of the +commuting reflections in its color gives a homomorphism from $W_T$ +to the resulting finite Coxeter group. Under this homomorphism, +\[ + D_L(\widetilde w)=\widetilde{D_L(w)}, + \qquad + D_R(\widetilde w)=\widetilde{D_R(w)}, +\] +where $D_L$ and $D_R$ are the left and right descent sets. Every +reduced expression lifts to a reduced expression. In particular, +the longest element lifts to the longest element. +\end{lemma} + +\begin{proof} +The strictly positive vector $v=\phi(C_T^{-1}\mathbf1)$ satisfies +$\widetilde C_{\widetilde T}v=\phi\mathbf1>0$. Applying +\eqref{cat:scaled-form} to each connected unfolded block proves +positive definiteness on the whole unfolded space, including +directions outside $\phi(\RR^T)$. An infinite edge cannot occur +inside $T$, since its two-by-two Cartan matrix is singular. +Equation~\eqref{cat:edge-labels} therefore makes every off-diagonal +entry of $\widetilde C_{\widetilde T}$ either $0$ or $-1$. The +finite Coxeter criterion applies to this simply laced matrix. + +There are no arrows between distinct vertices of one color. The +corresponding reflections commute, and their product acts on the +folded space as $s_i$, by \eqref{cat:intertwining}. Choose $p$ with +$C_Tp>0$. Then $\phi p$ is strictly inside a full unfolded +fundamental chamber. For an arbitrary word representing $w$, let $g$ +be its lifted product; then $g\phi p=\phi(wp)$. A root has coefficients either +all nonnegative or all nonpositive in the simple-root basis, so +pairing it with a chamber-interior point detects its sign. The +row at $(i,L)$ of $\widetilde C\phi(wp)$ is $w_L(Cwp)_i$. +These row signs are the signs of the pairings with +$g^{-1}\epsilon_{(i,L)}$ and $w^{-1}\epsilon_i$, respectively. +Consequently the unfolded left descents are exactly all vertices +over the original left descents. Applying the same reasoning to +inverse words proves the right-descent assertion. + +A word representing the identity fixes $\phi p$. The stabilizer +of a chamber-interior point in a finite Coxeter group is trivial, +so its lift is the identity. Thus the lifted product depends only +on $w$. If an original reduced expression is extended by an ascent, +every vertex of the added color is an ascent of the lifted prefix. +Acting at one of those vertices does not change the signs at the +others because they are mutually orthogonal. The whole block is +therefore a sequence of ascents. Induction proves reducedness of +the lift. The longest original element has every simple descent; +its lift has every unfolded simple descent, so is the longest +unfolded element. +\end{proof} + +\subsection{Twists as bimodule homotopies} + +The evaluation and coevaluation complexes below adapt the zigzag +bimodule twists of \cite[Proposition~2.4 and Theorem~2.5]{KhovanovSeidel2002} +and \cite[Section~4.6, Proposition~7]{HuerfanoKhovanov2001}; compare also +\cite[Definition~5.2 and Proposition~5.3]{HengLicata2024}. +We keep the full bimodule homotopies, because later tensor products +will involve nonprojective right simples. + +All complexes are bounded cochain complexes of finite internally +graded modules. The differential has cochain degree $1$ and +internal degree $0$. The shift $\langle a\rangle$ raises internal +placements by $a$, while $[1]$ lowers cochain placements by $1$. +Only cochain degrees contribute signs to tensor products and Hom +complexes. + +Set $M_\alpha=P_\alpha\otimes_\kk Q_\alpha$, with multiplication +$\mu_\alpha:M_\alpha\to Z$. Choose homogeneous dual bases +$p_a\in P_\alpha$, $q_a\in Q_\alpha$ for the trace pairing, so +$\tau(q_ap_b)=\delta_{ab}$. The element +$\sum_a p_a\otimes q_a$ is central and has internal degree $2$: +centrality is the invariance of the perfect pairing under the +two $Z$ actions. It defines a bimodule map +$\delta_\alpha:Z\to M_\alpha\langle-2\rangle$. Put +\begin{equation} + B_\alpha=[M_\alpha\xrightarrow{\mu_\alpha}Z], + \qquad + D_\alpha=[Z\xrightarrow{\delta_\alpha} + M_\alpha\langle-2\rangle], + \label{cat:elementary-twists} +\end{equation} +with $Z$ in cochain degree zero. Thus the projective terms lie +in cochain degrees $-1$ and $1$, respectively. + +\begin{proposition}[The categorical action] +\label{cat:action} +There are bimodule chain homotopy equivalences +$B_\alpha D_\alpha\simeq Z\simeq D_\alpha B_\alpha$. +For a color $i\in S$, define +\[ + B_i=\prod_{\alpha\in\widetilde{\{i\}}}B_\alpha, + \qquad + D_i=\prod_{\alpha\in\widetilde{\{i\}}}D_\alpha. +\] +These products are independent of their order up to bimodule +homotopy, are mutually inverse, and satisfy all the defining Artin +relations. They define an action of $A$ by invertible graded +bimodule homotopy classes, denoted $B_x$ for $x\in A$. +In particular, +\begin{equation} + F_i(E):=B_i\otimes_ZE + =\Cone\left( + \bigoplus_{\alpha\in\widetilde{\{i\}}} + P_\alpha\otimes_\kk e_\alpha E\longrightarrow E\right). + \label{cat:color-evaluation} +\end{equation} +Every $B_x$ preserves bounded projective complexes and preserves +the cohomology of their bigraded Hom complexes. Its adjoint dual +$B_x^\vee$ represents its inverse. All the action identities are +actual bimodule chain homotopy equivalences and remain valid after +tensoring with any finite module complex, whether or not projective. + +For $H\in\cC$, right tensoring of labels defines an additive functor +\[ + T_H(P_{i,D})=P_{i,D\otimes H} +\] +on graded projectives. It preserves path lengths, extends to +bounded homotopy categories, and commutes naturally up to homotopy +with $B_x\otimes_Z-$ for every $x\in A$. +\end{proposition} + +\begin{proof} +We first verify the inverse explicitly. The product +$B_\alpha D_\alpha$ has $M_\alpha$ in degree $-1$, +\[ + Z\oplus P_\alpha\otimes_\kk(e_\alpha Ze_\alpha) + \otimes_\kk Q_\alpha\langle-2\rangle +\] +in degree zero, and $M_\alpha\langle-2\rangle$ in degree $1$. +Write $\theta_\alpha=t_\alpha/w_L$ when $\alpha=(i,L)$, so that +$\tau(\theta_\alpha)=1$. Split the middle factor into its unit +and $\theta_\alpha$ lines. The incoming differential onto the +token summand is an isomorphism, up to the tensor sign: its +coefficient is the dual-basis identity +\[ + \sum_a\tau(qp_a)q_a=q. +\] +The token has degree $2$, so this summand is unshifted +$M_\alpha$. Cancel that isomorphism. The remaining unit summand +maps identically onto $M_\alpha\langle-2\rangle$ by multiplication +with $e_\alpha$; cancel it as well. Only the regular bimodule $Z$ +remains. In the opposite product $D_\alpha B_\alpha$, the +dual-basis identity gives the same incoming cancellation, followed +by the unit cancellation. These cancellations take place in +bimodule complexes, not just after evaluating on projectives. + +We next prove the elementary braid relation using the twist-conjugation +argument of \cite[Lemma~2.11 and Propositions~2.12--2.13]{SeidelThomas2001}. +For a bounded perfect +left complex $P$, let its evaluation twist be the bimodule cone +\[ + T_P=\Cone\bigl(P\otimes_\kk\Hom_Z(P,Z)\longrightarrow Z\bigr). +\] +It is unchanged by an internal or cochain shift of $P$. A homotopy +equivalence of $P$ identifies its evaluation map, and hence its +cone, up to homotopy. If $B$ is a product of the elementary +invertible complexes, each of its terms is projective on both +sides. Tensor--Hom adjunction identifies its tensor adjoint with +$B^\vee=\Hom_Z(B,Z)$; the evaluation and coevaluation identities +hold at chain level. Since $B$ is invertible, this adjoint is its +inverse in the bimodule homotopy category. Lemma~\ref{cat:frobenius} +keeps its terms in the same class. Tensor--Hom and evaluation now +give +\begin{equation} + B T_P B^{-1}\simeq T_{BP}. + \label{cat:twist-conjugation} +\end{equation} + +If distinct vertices $\alpha,\beta$ have no arrows between them, +$M_\alpha M_\beta=M_\beta M_\alpha=0$, so both +$B_\alpha B_\beta$ and $B_\beta B_\alpha$ are the simultaneous +evaluation cone with projective term $M_\alpha\oplus M_\beta$. +If there is exactly one arrow in each direction, the complexes +\[ + B_\alpha P_\beta + =[P_\alpha\langle1\rangle\longrightarrow P_\beta], + \qquad + D_\beta P_\alpha + =[P_\alpha\longrightarrow P_\beta\langle-1\rangle] +\] +have degrees $(-1,0)$ and $(0,1)$. Both maps are nonzero by the +perfect backtrack pairing, and their common arrow space is +one-dimensional. Rescaling one term gives +\[ + B_\alpha P_\beta\simeq + (D_\beta P_\alpha)\langle1\rangle[1]. +\] +Using $T_{P_\gamma}=B_\gamma$ in +\eqref{cat:twist-conjugation} therefore gives +$B_\alpha B_\beta D_\alpha\simeq D_\beta B_\alpha B_\beta$. +Multiplication on the left by $B_\beta$ and on the right by +$B_\alpha$ proves +\[ + B_\alpha B_\beta B_\alpha + \simeq B_\beta B_\alpha B_\beta. +\] + +Vertices within one color are orthogonal. Their twists commute, +their inverse products are the $D_i$, and all mixed terms in the +expansion vanish. This proves the simultaneous cone +\eqref{cat:color-evaluation}. For $m_{ij}<\infty$, the two +alternating color words of length $m_{ij}$ are reduced expressions +for the longest element of the spherical rank-two subsystem. +Lemma~\ref{cat:unfolding} lifts both to reduced expressions for +the same longest unfolded element. Matsumoto's Theorem relates +them by commutations and three-term braid moves, which we have +just proved as bimodule homotopies. For $m_{ij}=\infty$ there is +no relation to impose. This proves the action for arbitrary labels. + +It remains to verify the natural commutation with labels. On +\eqref{cat:projective-hom}, $T_H$ tensors the label map on the +right by $\mathrm{id}_H$. Evaluation of an edge and its dual +always occurs to the left of the label, so composition and path +length are preserved. On $P_{j,D}$, the evaluation term for +color $i$ has the canonical semisimple description +\[ + \bigoplus_{a,L}P_{i,L}\langle a\rangle\otimes_\kk + \Hom_{\cC}(L,D^a_{ij}\otimes D) + \cong\bigoplus_a P_{i,D^a_{ij}\otimes D}\langle a\rangle. +\] +Its evaluation is the tautological label map. Tensoring by $H$ +replaces $D$ by $D\otimes H$ in this description, naturally in +$D$ and in every projective morphism. Thus $T_H$ commutes with +the positive evaluation cone. Composing this natural +identification with the inverse equivalence on both sides proves +the inverse commutation without requiring $T_H$ to be invertible. + +Invertible functors preserve all morphism spaces, including both +shifts, and hence preserve bigraded Hom cohomology between +projective complexes. Finally, tensoring an explicit chain +homotopy with any module complex preserves its defining equations +with the usual cochain signs. No flatness assertion about the +additional module complex is needed. +\end{proof} + +\subsection{The transported family and folded counts} + +For $x\in A$ and a simple label $L$, define +\begin{equation} + E_L(x)=B_xP_{f,L}. + \label{cat:family-definition} +\end{equation} +The forthcoming layer construction is summarized here to specify +the data used throughout the proof. A minimal projective complex +has no path-length-zero entries in its differential. A generator in +cochain placement $p$ and internal placement $a$ has \emph{level} +$p+a$. A length-$j$ differential entry changes level by $1-j$. +The pure layer $Y^d(E)$ takes the generators of level $d$ and the +length-one differential. Lemma~\ref{lay:structure} proves the +existence and uniqueness of minimal models and the functorial layer +and truncation properties used below. Dimensions of pure layers +always count their projective generators, not the dimensions of +the underlying $Z$-modules. + +\begin{proposition}[Relative-family properties] +\label{cat:family} +There are nonnegative folded vectors $m^d(x)$ characterized by +\begin{equation} + \phi m^d(x)=\sum_Lw_L\dim Y^d(E_L(x)). + \label{cat:folded-counts} +\end{equation} +They have finite level support. Each individual coordinate ranges +over a locally finite subset of $\RR_{\geq0}$. Weighted counts of +tops and socles of the layers are likewise proportional within a +color: a nonzero top or socle in that color occurs at every +unfolded vertex of the color in some member of the family. +The same proportionality holds for families of subobjects obtained +functorially by twists and layer truncations. + +If only colors of $T\subseteq S$ act, all generator spaces with +their cochain and internal placements outside $\widetilde T$ stay +unchanged up to isomorphism. In particular, each $E_L(x)$ has +exactly its original $f$-generator in placements $(0,0)$, and +\[ + m_f^0(x)=1,\qquad m_f^d(x)=0\quad(d\ne0). +\] +Every $E_L(x)$ is indecomposable in the graded homotopy category. +The color support of its minimal complex is connected and contains +$f$. +\end{proposition} + +\begin{proof} +Proposition~\ref{cat:action} gives +$E_L(x)\simeq T_L(E_{\mathbf1}(x))$. Tensoring labels preserves +the positive path length of a minimal differential, and therefore +preserves minimality. If a multiplicity object $D$ occurs at a +fixed color and pair of placements in the unit-label complex, its +contribution in the $L$-member is $D\otimes L$. For each target +simple $H$, Equation~\eqref{cat:weighted-labels} says that the +weighted multiplicity is $w_H\dim_{\cC}D$. Summing over placements +of one level proves \eqref{cat:folded-counts}. + +Lemma~\ref{lay:labels} proves that label tensoring commutes with +layers, truncations, and the top and socle tests. Its proof uses +the exactness of tensoring in the semisimple category, together +with \eqref{cat:projective-hom} and +\eqref{cat:weighted-labels}. It gives the stated proportionality +for functorially constructed subobjects and for tops and socles. +Positivity of all label weights then proves the assertion about +occurrence at every vertex of a color. + +There are finitely many labels and finitely many terms in each +complex. To see local finiteness, fix one target label $H$. +For a bound $R$, Equation~\eqref{cat:folded-counts} expresses +$w_Hm_i^d\leq w_HR$ as a nonnegative integer combination of the +finitely many fixed positive numbers $w_L$. Each coefficient is +bounded separately. There are thus only finitely many possible +values below $R$. + +For an unfolded vertex $\beta$ outside color $i$, +$U_\beta e_\alpha=0$ for every $\alpha$ in that color. The +explicit complexes \eqref{cat:elementary-twists} consequently give +$U_\beta B_i=U_\beta=U_\beta D_i$. Tensoring a minimal left +complex with $U_\beta$ kills its radical differential, leaving +exactly its generator spaces at $\beta$ in their two placements. +This proves exterior invariance, and in particular the statement +about the $f$-generator and its folded counts. + +The initial projective $P_{f,L}$ is indecomposable, and an +equivalence preserves indecomposability, proving the first final +assertion. A disconnected color support would split its minimal +complex into the summands on different components: a differential +uses only arrows, which join adjacent colors, and tokens, which +stay at one color. Every nonzero minimal summand is +noncontractible, since reducing a putative contracting homotopy +equation modulo the radical would give an identity equal to zero. +Such a splitting contradicts indecomposability. Exterior +invariance ensures that $f$ belongs to the support. +\end{proof} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/03_layers.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/03_layers.tex new file mode 100644 index 0000000000000000000000000000000000000000..b17b1259ccfccfe2af4b04a7347e760fe0bb1b04 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/03_layers.tex @@ -0,0 +1,586 @@ +\section{Layers and positive divisors} +\label{sec:layers} + +We develop the layer calculus needed to compare representatives of a +parabolic coset. The underlying linear heart and twist calculations +are related to \cite[Sections~5 and~6.1]{HengLicata2024}. +We prove the statements used here, with particular attention to the degree conventions +and to the arbitrary inputs allowed in the cohomological divisor test. +None of the results in this section requires the initial projective to +be attached to every operated color. + +\subsection{The linear heart} + +Work in the homotopy category of bounded complexes of finite graded left +projective $Z$-modules. A differential has cochain degree $1$ and +internal degree $0$. The shift $\langle u\rangle$ raises internal +degrees by $u$, and $[1]$ lowers cochain placements by one. A generator +of $P_\alpha\langle u\rangle$ in cochain placement $c$ has +\emph{level} +\begin{equation} + d=c+u.\label{lay:level} +\end{equation} +Throughout, generator dimensions mean multiplicities of projective +summands, not dimensions of those projectives as vector spaces. + +A complex is \emph{minimal} if its differential entries belong to the +graded radical. Gaussian elimination removes an invertible entry +between shifted indecomposable projectives by splitting off a +contractible two-term complex. Since the number of summands is finite, +this produces a minimal model. A homotopy equivalence between minimal +models is an isomorphism modulo the radical in every cochain degree: +the homotopy terms in its composites with a homotopy inverse contain a +radical differential. Consequently its term maps are invertible. +Thus a minimal model is unique up to chain isomorphism, in particular +with uniquely determined generator multiplicities and placements. + +In a minimal model the differential has only path lengths $1$ and $2$. +An entry of length $a$ changes generator placements by $(1,-a)$, hence +changes the level by $1-a$. The linear differential preserves level, +and the token differential lowers it by one. Define $Y^d(E)$ to be the +generators of level $d$ with their linear differential. The path-length +$2$ part of the equation $\partial^2=0$ says that this is a complex. + +For clarity, its coefficient description has spaces +\[ + V^d_{\alpha,c}(E), + \qquad + P_\alpha\langle d-c\rangle\otimes V^d_{\alpha,c}(E) + \text{ in cochain placement }c. +\] +Each arrow entry gives a coefficient map from +$V^d_{\alpha,c}(E)$ to $V^d_{\beta,c+1}(E)$. At each vertex these +maps satisfy the backtrack relation obtained from the opposite-arrow +pairing in $Z^2$; all other products of two arrows in $Z$ vanish. +We call such a finite graded array a \emph{coefficient representation}. +Its arrow maps raise cochain placement and are therefore jointly +nilpotent. Its top and socle at $\alpha$ are, respectively, +\[ + \Top_\alpha M= + \frac{\bigoplus_c V_{\alpha,c}(M)} + {\text{images of incoming coefficient maps}}, + \qquad + \soc_\alpha M= + \bigcap\ker(\text{outgoing coefficient maps}), +\] +with the inherited placements. Write $\dim M$ for the vector of +generator multiplicities summed over placements. + +\begin{lemma}[Layer structure]\label{lay:structure} +The full subcategories defined by minimal generator levels at most $0$ +and at least $0$ form a bounded $t$-structure. The truncation +$\tau_{\le k}E$ is the subcomplex of levels at most $k$, and the +complementary quotient is $\tau_{\ge k+1}E$. + +The heart is the abelian category of finite graded coefficient +representations. In particular, subobjects of a pure layer are exactly +its graded subrepresentations. Its unshifted cohomology layer at $d$ +is $Y^d(E)$. +\end{lemma} + +\begin{proof} +Differentials do not raise level, so the indicated low terms form a +subcomplex. A degree-zero map from a generator of level $d$ to one of +level $e$ requires a path of length $d-e$. There are no such maps from +levels at most $0$ to levels at least $1$. This proves the required +orthogonality. The cochain shift $[1]$ lowers levels by one, giving the +shift-closure axioms, and the short exact sequence of low subcomplex, +full complex and high quotient gives the truncation triangle. Finitely +many occupied levels give boundedness. + +A degree-zero map between pure complexes at the same level has only +length-zero entries. It is precisely a map of graded coefficient +representations. A homotopy between two such maps would require path +length $-1$, so there are no homotopies to quotient out. Conversely a +finite coefficient representation gives a pure complex, since its +backtrack relations say exactly that the differential squares to zero. +Kernels and cokernels, formed in each coefficient space, retain the +relations. This identifies the heart, its subobjects and its +cohomology layers. +\end{proof} + +We use the long exact cohomology sequence associated with a distinguished +triangle in a $t$-structure \cite[Section~1.3, Theorem~1.3.6]{BBD1982}. +Denote its heart-valued cohomology by +$H^d_{\mathrm{lay}}$; with our conventions, +\[ + H^d_{\mathrm{lay}}(E)=Y^d(E)[d], + \qquad + H^d_{\mathrm{lay}}(E[1])=H^{d+1}_{\mathrm{lay}}(E). +\] +An exact functor has \emph{layer amplitude} $[a,b]$ if it sends a pure +layer at $d$ into levels $[d+a,d+b]$. Bounded truncation then gives the +corresponding bounds on any interval of input levels. + +\begin{lemma}[Adjacent layers]\label{lay:adjacent} +For an exact functor $F$ of layer amplitude $[0,1]$ there is a short +exact sequence of pure level-$d$ objects +\begin{equation} +\begin{split} +0\longrightarrow Y^d\bigl(F(Y^{d-1}(E))\bigr) +\longrightarrow Y^d(FE) +\longrightarrow Y^d\bigl(F(Y^d(E))\bigr) +\longrightarrow0. +\end{split}\label{lay:adjacent-positive} +\end{equation} +For amplitude $[-1,0]$ the corresponding sequence is +\begin{equation} +\begin{split} +0\longrightarrow Y^d\bigl(F(Y^d(E))\bigr) +\longrightarrow Y^d(FE) +\longrightarrow Y^d\bigl(F(Y^{d+1}(E))\bigr) +\longrightarrow0. +\end{split}\label{lay:adjacent-negative} +\end{equation} +All terms are shifted by $[d]$ to regard these as sequences in the +heart. In particular, generator dimensions are additive. +\end{lemma} + +\begin{proof} +In the positive case apply $F$ to the triangle +\[ + \tau_{\le d-1}E\longrightarrow\tau_{\le d}E + \longrightarrow Y^d(E). +\] +The first term after applying $F$ has levels at most $d$; the last has +levels $d,d+1$. Hence the flanking groups +$H^{d-1}_{\mathrm{lay}}F(Y^d(E))$ and +$H^{d+1}_{\mathrm{lay}}F(\tau_{\le d-1}E)$ vanish, giving a short +exact sequence at $d$. Input levels at most $d-2$ contribute nothing +there, identifying the left term in +\eqref{lay:adjacent-positive}; input levels at least $d+1$ contribute +nothing there, identifying its middle term. For the negative case +apply the same argument to +$\tau_{\le d}E\to\tau_{\le d+1}E\to Y^{d+1}(E)$. +After $F$, its first term has levels at most $d$ and its last has levels +$d,d+1$. Discarding input levels below $d$ and above $d+1$ gives +\eqref{lay:adjacent-negative}. +\end{proof} + +\begin{lemma}[Composition at the bottom]\label{lay:bottom} +Let $F_1,\ldots,F_r$ be exact functors with no negative change of +level. On input pure at $d$, the level-$d$ part of +$F_r\cdots F_1$ is obtained by retaining only level $d$ after every +operation. Relations between these composite functors induce the same +relations between their bottom-layer operations. +\end{lemma} + +\begin{proof} +After the first operation the quotient by the level-$d$ truncation +has levels at least $d+1$. All subsequent operations preserve that +bound. Its long exact layer sequence shows that discarding this +quotient does not change the final level $d$. Repeat after each +operation. +\end{proof} + +\subsection{Extremal tests and single twists} + +For a bigraded vector complex $V$, write $H^t(V)$ for the direct sum +of its cohomology spaces with cochain degree plus internal degree +equal to $t$. This \emph{total test degree} is distinct from the +cochain degree alone. Differentials raise total degree by one; only +cochain degrees determine their signs. + +\begin{lemma}[Extremal tests]\label{lay:extremal} +If $M$ is pure at level $d$, then +\begin{equation} + H^d(e_\alpha M)\cong\soc_\alpha M, + \qquad + H^{d+2}(e_\alpha M)\cong(\Top_\alpha M)\langle2\rangle. + \label{lay:extremal-formula} +\end{equation} +The second shift records the token's internal placement. More +generally, if a minimal complex $E$ has no generators above $d$ at +$\alpha$ or its neighbors, its degree-$(d+2)$ test is the top of +$Y^d(E)$ at $\alpha$, shifted internally by $2$, and all higher test +cohomology there vanishes. If there are no generators below $d$ at +those vertices, its degree-$d$ test is the socle of $Y^d(E)$ there. + +A nonzero pure layer supported on a vertex set has a nonzero top and +a nonzero socle somewhere in that set. The same assertions hold for +right projective complexes, tested by $Me_\alpha$. +\end{lemma} + +\begin{proof} +The test of a pure layer has path lengths $0,1,2$, in total degrees +$d,d+1,d+2$. Its length-zero cycles are the common kernels of the +outgoing coefficient maps. At the token end, the images of the +incoming differential are exactly the incoming coefficient images, +by the perfect opposite-arrow pairings. This proves +\eqref{lay:extremal-formula}. + +Every term of $e_\alpha E$ comes from a generator at $\alpha$ or a +neighbor, and its total degree is the generator level plus the path +length. Under the upper bound, only level-$d$ tokens can occur at +$d+2$. A token differential into them from length zero would require +a generator at level $d+1$, which is absent. The remaining boundaries +are the same incoming linear images. The lower statement follows +from the length-zero end in the same way. Finally, coefficient arrows +strictly raise cochain placement in a finite array. A nonzero array +therefore cannot have zero socle everywhere, or be generated by its +incoming images everywhere. Reversing module sides proves the right +statements. +\end{proof} + +Let $F_i=B_{\sigma_i}(-)$ be the color twist of +Proposition~\ref{cat:action}. Denote by $r_i$ the product of the commuting +coordinate reflections in its unfolded fiber. Thus, on folded +vectors, +\[ + r_i\phi(v)=\phi\bigl(v-(Cv)_i\varepsilon_i\bigr). +\] + +\begin{lemma}[Positive layers]\label{lay:positive} +Each $F_i$ has layer amplitude $[0,1]$. For a pure input $M$ at $d$, +its upper layer is supported on color $i$, has zero linear +differential, and has multiplicity $\dim\Top_\alpha M$ at each +$\alpha$ of that color. Its bottom layer has unchanged multiplicities +at all other vertices, and +\begin{equation} + \dim Y^d(F_iM)_\alpha + =\sum_\beta(2I-\widetilde C)_{\alpha\beta}\dim M_\beta + -\dim M_\alpha+\dim\Top_\alpha M + \quad(\alpha\text{ of color }i). + \label{lay:positive-count} +\end{equation} +The bottom layer has zero socle in color $i$. If $i\in J$ and $M$ +has zero socle on every color of $J$, its bottom layer after $F_i$ has +zero socle on all of $J$. + +For any bounded complex $E$ of graded left modules, with no +projectivity requirement on its terms, +\begin{equation} + e_\alpha F_iE\simeq(e_\alpha E)\langle2\rangle[1] + \quad(\alpha\text{ of color }i). + \label{lay:positive-shift} +\end{equation} +There is an identical homotopy equivalence for right actions and right +tests. Thus an acted test shifts up one in total degree and down one +in cochain degree. +\end{lemma} + +\begin{proof} +For pure input at $d$, the evaluation cone inserts generators in the +acted color at levels $d+a-1$, where $a=0,1,2$ is the path length. +The length-zero insertions at $d-1$ map by identities, modulo the +radical, to the old generators in that color. Cancelling those pairs +removes every generator below $d$. No generator was inserted above +$d+1$, and only the acted color can occur above $d$. + +For one vertex $\alpha$ of color $i$, put $A=e_\alpha E$. +Distinct vertices of that color are orthogonal, so the acted test is +\begin{equation} + \Cone\bigl(A\oplus A\langle2\rangle + \xrightarrow{(\mathrm{id},\theta_\alpha)} A\bigr). + \label{lay:token-cone} +\end{equation} +Projection to the token source $A\langle2\rangle[1]$ is a chain map +with contractible kernel $\Cone(\mathrm{id}_A)$, proving +\eqref{lay:positive-shift}. This is a homotopy calculation for any +$A$, so it neither assumes projectivity nor merely asserts a +quasi-isomorphism. The right calculation uses the same cone. + +For pure input, the degree-$(d+3)$ output test equals the old +degree-$(d+2)$ test. Lemma~\ref{lay:extremal} identifies the latter +with the old top. Since the upper layer is supported on a color with +no internal arrows, the former counts its generators. This proves +the upper-layer assertion. + +For the multiplicity formula, count generators with sign $(-1)^d$. +The signed contributions of a cancelled scalar pair sum to zero, +because its levels are consecutive. In the evaluation cone the +inserted signed contribution from a generator at $\beta$ to an acted +vertex $\alpha$ is minus the alternating path count +\[ + 2\delta_{\alpha\beta}-(2I-\widetilde C)_{\alpha\beta}. +\] +Consequently the total signed vector transforms by $r_i$. On pure +input the bottom vector minus the upper vector is therefore +$r_i\dim M$, proving \eqref{lay:positive-count}. The same calculation +also shows, for general inputs, that +\begin{equation} + \sum_d(-1)^d\dim Y^d(F_iE) + =r_i\sum_d(-1)^d\dim Y^d(E). + \label{lay:signed-reflection} +\end{equation} + +By \eqref{lay:positive-shift}, the degree-$d$ test at an acted vertex +is zero. Lemma~\ref{lay:extremal} identifies it with the socle of +the bottom layer. At another vertex $\beta$, the first term in the +tested evaluation triangle is a sum of +$(e_\alpha M)\langle1\rangle$, over adjacent acted vertices. +It has no degree-$d$ cohomology, and its degree-$(d+1)$ cohomology +is the sum of the old socles at those vertices. If these socles and +the old socle at $\beta$ vanish, the long exact sequence gives zero +degree-$d$ cohomology after the move. This proves preservation of +socle vanishing on $J$. +\end{proof} + +In particular, Lemma~\ref{lay:adjacent} says that the new layer at $d$ +under $F_i$ is just its bottom operation on $Y^d(E)$ whenever the +old top at color $i$ in level $d-1$ vanishes. By +Lemma~\ref{lay:bottom} and the braid relations, the bottom operations +of positive twists themselves satisfy the positive braid relations. + +\begin{lemma}[Inverse amplitude and exterior invariance] +\label{lay:inverse} +An exact equivalence of layer amplitude $[0,1]$ has inverse amplitude +$[-1,0]$. In particular $F_i^{-1}$ has that amplitude, and +\begin{equation} + e_\alpha F_i^{-1}E\simeq(e_\alpha E)\langle-2\rangle[-1] + \quad(\alpha\text{ of color }i). + \label{lay:negative-shift} +\end{equation} +This test identity is valid for arbitrary bounded module complexes +and also on the right. For either sign, acting in color $i$ leaves +the minimal generator spaces at other colors unchanged up to +isomorphism, with both placements retained. Formula +\eqref{lay:signed-reflection} also holds for $F_i^{-1}$. +\end{lemma} + +\begin{proof} +Let $F$ be the equivalence and let $N$ lie in the heart. For any $A$ +in levels at most $-2$, its image $FA$ has levels at most $-1$, so +\[ + \Hom(A,F^{-1}N)=\Hom(FA,N)=0. +\] +The orthogonal characterization of the $t$-structure implies that +$F^{-1}N$ has levels at least $-1$. If $B$ has levels at least $1$, +then so does $FB$, and +\[ + \Hom(F^{-1}N,B)=\Hom(N,FB)=0. +\] +Hence $F^{-1}N$ has levels at most $0$. Shifting proves the amplitude +bound on every pure input. Applying \eqref{lay:positive-shift} to +$F_i^{-1}E$ and using the bimodule homotopy inverse proves +\eqref{lay:negative-shift}, including for nonprojective inputs and +for right complexes. Applying \eqref{lay:signed-reflection} and +$r_i^2=1$ proves the inverse reflection formula. + +For a vertex $\beta$ outside color $i$, the right simple satisfies +\[ + U_\beta B_{\sigma_i}^{\pm1}\simeq U_\beta. +\] +Indeed the inserted projective bimodule terms have left vertex in +color $i$ and are annihilated by $U_\beta$. Tensoring a minimal left +projective complex with $U_\beta$ kills its radical differential and +records precisely the generators at $\beta$ with both placements. +The displayed equivalence proves the exterior assertion. +\end{proof} + +All assertions about projective layers have their right-handed +versions, obtained by using the opposite algebra. In particular the +inverse layer-amplitude statement does not assert a projective-layer +description for a nonprojective input; for such an input the test +identities \eqref{lay:positive-shift} and +\eqref{lay:negative-shift} still hold. The normalized layers containing +a residual right simple will be treated separately in +Section~\ref{sec:caps}. + +\begin{lemma}[Compatibility with labels]\label{lay:labels} +Layer truncations commute with the exact label functors $T_H$. +For the family $E_L(x)$ of Proposition~\ref{cat:family}, the weighted generator, +top and socle dimensions in each layer are proportional on every +color fiber. In particular, a top or socle occurring in one member +at a vertex of color $i$ occurs at every vertex of color $i$ in some +member of the family. The same proportionality holds for families +constructed from this one by twists and layer truncations. +\end{lemma} + +\begin{proof} +Tensoring a projective label by $H$ preserves its cochain and internal +placements. It preserves radical path lengths, hence commutes with +the low-level subcomplex and the high-level quotient. On coefficient +representations it is exact, since the label category is semisimple. +At a fixed color and placements, the coefficient spaces, tests and +their cohomology are obtained by applying +$\Hom_{\cC}(K,-)$ to the corresponding label objects. Exact label +tensoring also tensors these objects and their cohomology. The +weighted multiplicity identity +\[ + \sum_L w_L[D\otimes L:K]=w_K\dim_{\cC}D +\] +therefore gives proportionality for generators and for the extremal +tests, hence for tops and socles by Lemma~\ref{lay:extremal}. +All weights are positive, proving the occurrence assertion. +Twists and truncations commute with label tensoring, so the argument +applies to the constructed families as well. +\end{proof} + +\subsection{The highest-test divisor} + +We next recover a positive divisor from a highest cohomological test. +This is the mechanism used by Brav and Thomas to detect finite-type +Garside factors \cite[Proposition~3.1 and Lemma~3.3]{BravThomas2011}. +Here we prove the version for arbitrary bounded test inputs; right simple +modules in Section~\ref{sec:caps} will require precisely that generality. + +Fix a spherical color set $J$. By Lemma~\ref{cat:unfolding}, its +unfolded graph is finite simply laced. In the next two lemmas the +letters are individual unfolded vertices; $B_\alpha$ denotes their +single-vertex evaluation twist. For an input $E$ define +\begin{equation} + h_J(E)=\max\{t:H^t(e_\alpha E)\ne0 + \text{ for some }\alpha\in\widetilde J\}, + \label{lay:test-height} +\end{equation} +with value $-\infty$ when all these tests vanish. For right inputs +use $Ee_\alpha$. The following arguments require only bounded test +cohomology and the indicated action, not projective terms. They apply +in particular to bounded complexes containing right simples. + +\begin{lemma}[Change of test height]\label{lay:height-change} +If $h_J(E)$ is finite, then +\[ + h_J(E)\le h_J(B_\alpha E)\le h_J(E)+1. +\] +Height increases exactly when $\alpha$ was a highest-test vertex; +in that case it is the only new highest-test vertex. Vanishing of all +tests on $\widetilde J$ persists. The same statements hold for right +actions. Consequently height is nondecreasing under positive color +words in $J$. +\end{lemma} + +\begin{proof} +The acted test shifts up by one in total degree, and nonneighboring +tests are unchanged. At a neighboring vertex $\beta$ the evaluation +triangle is +\begin{equation} + (e_\alpha E)\langle1\rangle\longrightarrow e_\beta E + \longrightarrow e_\beta B_\alpha E. + \label{lay:neighbor-triangle} +\end{equation} +If the old maximum is $t$, the last term has no test cohomology above +$t$. Thus precisely the asserted condition produces height $t+1$, +and only at $\alpha$. If the old test at $\alpha$ has maximum $t-1$, +its shifted test reaches $t$. If it has maximum below $t-1$, an old +degree-$t$ test elsewhere survives: in +\eqref{lay:neighbor-triangle} the map into that degree has zero source. +This proves monotonicity. The same triangles preserve total +vanishing. Reversing sides proves the right case; a color twist is +a product of the commuting unfolded twists in its fiber. +\end{proof} + +\begin{lemma}[Top-divisor test]\label{lay:top-divisor} +Let $b$ be a positive word in the unfolded finite-type generators. +If $h_J(B_bE)>h_J(E)$, every vertex attaining the highest test +cohomology of $B_bE$ is a left divisor of $b$ in the positive Artin +monoid. For a right action, such a vertex is a right divisor of $b$. +These conclusions hold for arbitrary inputs with bounded test +cohomology, including right-simple complexes. +\end{lemma} + +\begin{proof} +We prove the left statement by induction on word length. Write +$b=\alpha b'$ and $N=B_{b'}E$. If the final action raises height, +its unique new highest vertex is $\alpha$ by +Lemma~\ref{lay:height-change}, and there is nothing to prove. +Otherwise +\[ + h_J(N)=h_J(B_\alpha N)=t>h_J(E). +\] +The vertex $\alpha$, when it is a highest vertex, already divides +$b$. If a highest vertex $\beta\ne\alpha$ is not adjacent to +$\alpha$, it was a highest vertex of $N$. By induction it divides +$b'$, and it commutes past $\alpha$. + +Suppose instead that $\beta$ is adjacent to $\alpha$. Since height +did not increase, $H^t(e_\alpha N)=0$. The long exact sequence of +\eqref{lay:neighbor-triangle} gives +\begin{equation} + H^t(e_\beta B_\alpha N) + \cong\operatorname{coker}\bigl( + H^{t-1}(e_\alpha N)\longrightarrow H^t(e_\beta N)\bigr). + \label{lay:divisor-cokernel} +\end{equation} +This cokernel is nonzero, so $\beta$ is a highest vertex of $N$. +Induction gives a positive factorization $b'=\beta v$. Put +$N_0=B_vE$, so $N\simeq B_\beta N_0$. + +We now identify the arrow in \eqref{lay:divisor-cokernel}; this is the +step that permits nonprojective inputs. Set +$A_0=e_\beta N_0$ and $C_0=e_\alpha N_0$, and choose opposite arrows +\[ + r\in e_\alpha Ze_\beta,\qquad + s\in e_\beta Ze_\alpha,\qquad + sr=\kappa\theta_\beta,\quad\kappa\ne0. +\] +The evaluated cones are +\[ +\begin{split} + e_\beta N&=\Cone\bigl(A_0\oplus A_0\langle2\rangle + \xrightarrow{(\mathrm{id},\theta_\beta)}A_0\bigr),\\ + e_\alpha N&=\Cone\bigl(A_0\langle1\rangle + \xrightarrow{r}C_0\bigr). +\end{split} +\] +Let $\pi:e_\beta N\to A_0\langle2\rangle[1]$ be the token +projection in \eqref{lay:token-cone}. Multiplication by $s$ gives +$(e_\alpha N)\langle1\rangle\to e_\beta N$. On its cone source, +the product $sr$ is $\kappa$ times the token, whereas its cone target +lands in the target $A_0$, killed by $\pi$. Therefore $\pi s$ is, +up to the cochain sign and the nonzero scalar $\kappa$, exactly the +connecting projection of the second cone. + +Since $\pi$ is a homotopy equivalence, the image of the map in +\eqref{lay:divisor-cokernel}, identified in +$H^{t-1}(e_\beta N_0)$, is +\[ + \ker\bigl(H^{t-1}(e_\beta N_0) + \longrightarrow H^t(e_\alpha N_0)\bigr). +\] +Its cokernel is nonzero, so this last arrow has nonzero image. +In particular $H^t(e_\alpha N_0)\ne0$. By height monotonicity, +$h_J(N_0)\le h_J(N)=t$, hence +$h_J(N_0)=t>h_J(E)$. Applying induction to $v$ gives +$v=\alpha v'$ in the positive monoid. The simply-laced braid relation +now extracts the required vertex: +\[ + b=\alpha\beta\alpha v'=\beta\alpha\beta v'. +\] +This completes the induction. On the right, use the right evaluation +cones and opposite arrows, with the final action on the right. The +same proof gives right divisibility. +\end{proof} + +\begin{remark} +Divisibility here is on the \emph{last-action side}: the left side for +left complexes and the right side for right complexes. Passing from +an unfolded vertex divisor to an original color divisor requires the +finite-type greedy argument of Lemma~\ref{two:greedy}; preservation of +descents for reduced words alone does not suffice for an arbitrary +positive word. +\end{remark} + +\begin{corollary}[Reduced-word amplitude]\label{lay:reduced} +For a spherical set $J$, the positive lift of a reduced Coxeter word +has layer amplitude $[0,1]$. The negative lift, obtained by replacing +each letter by its inverse without reversing the word, has amplitude +$[-1,0]$. These bounds hold on every pure projective input, and hence +on arbitrary bounded intervals of layers, on either module side. +\end{corollary} + +\begin{proof} +First consider a reduced unfolded word. Its test height can increase +at most once. Otherwise at a second increase the new letter was a +highest-test vertex of the preceding result, whose height already +exceeded the starting value. By Lemma~\ref{lay:top-divisor} it can be +extracted on the last-action side of that preceding positive word. +Adding it there would give two equal consecutive Coxeter generators, +contradicting reducedness. + +A pure input at $d$ has test height at most $d+2$. Positive twists +never produce levels below $d$. If the output had highest layer +$a>d+1$, exterior invariance would force that layer to be supported on +$\widetilde J$. It has a nonzero top there, so +Lemma~\ref{lay:extremal} gives a nonzero test at $a+2>d+3$. +This contradicts the single-increase bound. If the initial tests +vanish, they stay zero and give the same contradiction. This proves +amplitude $[0,1]$ for reduced unfolded words. + +By Lemma~\ref{cat:unfolding}, reduced color words unfold to reduced +words, proving the positive assertion. The inverse of a negative +reduced lift is the positive lift of the reversed reduced expression. +Lemma~\ref{lay:inverse} therefore proves the negative assertion. +The right argument has the same degree bounds. +\end{proof} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/04_caps_twopoint.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/04_caps_twopoint.tex new file mode 100644 index 0000000000000000000000000000000000000000..71e1c2f7d79201870330aa707d15ad0476b41317 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/04_caps_twopoint.tex @@ -0,0 +1,792 @@ +\section{Caps and two-point uniqueness} +\label{sec:caps} + +The next two sections supply the local input for the harmonic-height +argument. A cap controls the interaction with generators outside the +operated colors. On a spherical coset, this control gives both a convex +description of a layer and a uniqueness theorem for its possible minima. +We retain the family $E_L(x)=B_xP_{f,L}$ throughout. In particular, no +adjacency assumption between $f$ and the operated colors is made. +The two-point conclusion must identify the actual coset, rather than +only its numerical layer data. We first build convex subobjects of a +capped layer and then use right simple modules to recover that coset. + +\subsection{Accessibility of a cap} + +\begin{definition}[Cap] +\label{cap:definition} +For $q\in\mathbb Z$ and a color set $D$, the family at $x$ is +\emph{capped at $q$ on $D$} if +\[ + m_i^d(x)=0\qquad(i\in D,\ d>q). +\] +Equivalently, every family member has no generator in those colors +above level $q$. We write $\partial J$ for the exterior neighbors of +$J$ in $\widehat S$. +\end{definition} + +Positivity of the family weights proves the equivalence in the +definition. By Proposition~\ref{cat:family}, data outside $J$ are +fixed on $A_Jx$, so a cap on $\partial J$ is independent of its +representative. + +\begin{lemma}[A capped negative operation] +\label{cap:capped-negative} +Let $E$ be a bounded projective complex capped at $q$ on +$J\cup\partial J$, and let $G$ be any word of negative twists in $J$. +Then $GE$ has the same cap, and +\begin{equation} + Y^q(GE)\simeq Y^q\bigl(G(Y^q(E))\bigr). + \label{cap:negative-layer} +\end{equation} +\end{lemma} +\begin{proof} +The high quotient $R=\tau_{\ge q+1}E$ is supported outside +$J\cup\partial J$. Its tests at $J$ vanish termwise, so every +evaluation or coevaluation term in $J$ vanishes and $GR\simeq R$. +Apply $G$ to the truncation triangle +$\tau_{\le q}E\to E\to R$. Negative twists do not increase levels. +The low term therefore remains at levels at most $q$, while the +unchanged quotient is at levels at least $q+1$. The long exact +layer sequence identifies all higher layers with those of $R$ and +identifies level $q$ with that of $G\tau_{\le q}E$. Apply $G$ next +to $\tau_{\le q-1}E\to\tau_{\le q}E\to Y^q(E)$. +Its first term stays below $q$, proving \eqref{cap:negative-layer}. +\end{proof} + +\begin{lemma}[Cap accessibility] +\label{cap:access} +Let $J\subseteq S$ be spherical and suppose $A_Jx$ has a cap at $q$ +on $\partial J$. It has a representative capped at $q$ also on $J$. +Every negative letter in $J$ preserves this joint cap. A positive +letter in $J$ preserves it whenever the top at that color in the +level-$q$ layer vanishes in every family member. +\end{lemma} +\begin{proof} +Negative preservation is Lemma~\ref{cap:capped-negative}. For the +positive assertion use the same high quotient $R$. It is unchanged. +On the low truncation, the only possible level-$q+1$ output is the +upper part of the old level-$q$ layer. By Lemma~\ref{lay:positive} +this is precisely its top in the operated color; it vanishes by +hypothesis. + +For accessibility, let $d>q$ be the greatest occupied level on $J$. +The joint cap at $d$ holds. At that level the portion on $J$ is a +nonzero direct summand of a pure layer, since the boundary is absent. +It has a nonzero top somewhere on $J$. Apply a negative letter at +such a color and continue while level $d$ remains. + +The selected colors form a reduced Coxeter expression. Indeed, +suppose the letters already selected form a reduced word $w$, with +the last operation on the left. If the next color $j$ were a left +descent, Matsumoto moves could put $j$ at the last-action end of a +reduced expression for $w$. The corresponding negative words give +the same action. Every intermediate negative operation preserves +the joint $d$-cap. Immediately before the last inverse twist, the +tests in color $j$ vanish above $d+2$; after it they vanish at $d+2$ +as well, by the negative test shift. Lemma~\ref{lay:extremal} would +then make the current level-$d$ top at $j$ zero, contrary to its +selection. Thus each choice is a Coxeter ascent. The longest length +in $W_J$ bounds the number of choices before level $d$ disappears. +No larger level is recreated. Repeating at successive excess +levels reaches $q$ in finitely many stages. The empty set $J$ +requires no operations. +\end{proof} + +\subsection{Candidate cosets and the uniqueness problem} + +Fix a nonempty spherical set $T\subseteq S$, an element $x_0\in A$, +and an integer $q$. Suppose the family $(E_L(x_0))_L$ is capped at +$q$ on $\partial T$. Exterior invariance preserves this cap +throughout $A_Tx_0$. In the comparison below, all candidates refer to this +family, this coset and this level. + +\begin{definition}[Candidate] +\label{two:definition} +A candidate is $(J,A_Jz)$ with $J\subsetneq T$ and $z\in A_Tx_0$. +Put $V=T\setminus J$, let $p$ harmonically interpolate $m^q(z)$ +on $J$, and put $a=-(Cp)_T$. The requirements are: +\begin{enumerate} +\item the open colors $V$ are capped at $q$; +\item $a$ is strictly positive on $V$ and zero on $J$; +\item every representative $y\in A_Jz$ capped at $q$ on $T$ +satisfies $m_J^q(y)\ge p_J$. +\end{enumerate} +\end{definition} + +These are properties of the typed coset: its exterior counts, hence +$p$ and $a$, are fixed. The representatives in the third condition +exist by Lemma~\ref{cap:access}, since +$\partial J\subseteq V\cup\partial T$. + +\begin{proposition}[Two-point uniqueness] +\label{two:uniqueness} +Let $T\subseteq S$ be nonempty and spherical, let $x_0\in A$, and let +$q\in\mathbb Z$ cap $(E_L(x_0))_L$ on $\partial T$. +There is at most one typed coset $(J,A_Jz)$ satisfying +Definition~\ref{two:definition} in $A_Tx_0$ at level $q$. +\end{proposition} + +The proof first identifies the harmonic vectors of two candidates; +the positivity condition then identifies their open colors and types. +It must also show that their representatives differ by an element of +the common parabolic subgroup. Capped-layer subobjects will supply +the numerical comparison, while transported right simple modules will +detect this relation between representatives. + +\subsection{Convex subobjects of a capped layer} + +We first express the excess of a capped layer over its harmonic +interpolation through dimensions of subobjects. The weak form below +will apply to candidates; the strict form will also be needed in +Section~\ref{sec:null}. + +For a spherical $J$, harmonic interpolation of exterior data is +unique: $p$ agrees with those data outside $J$ and satisfies +$(Cp)_J=0$. It is nonnegative for nonnegative exterior data and is +fixed by $W_J$, by Lemma~\ref{cat:cartan}. + +\begin{proposition}[The capped-layer hull] +\label{cap:hull} +Suppose $x$ is capped at $q$ on $J\cup\partial J$, with $J$ spherical. +Let $p$ be the harmonic interpolation of $m^q(x)$ on $J$. Assume +\begin{equation} + m_j^q(y)\ge p_j\quad(j\in J) + \quad\text{for every jointly capped }y\in A_Jx. + \label{cap:universal} +\end{equation} +Put $e=m^q(x)-p$, which is supported on $J$. Then $e_J$ belongs to +the convex hull of finitely many +vectors $s_J$ for which $s$ is supported on $J$ and there are graded +subrepresentations +\begin{equation} + \mathcal S_L\subseteq Y^q(E_L(x)),\qquad + \supp\mathcal S_L\subseteq\widetilde J,\qquad + \sum_Lw_L\dim\mathcal S_L=\phi s. + \label{cap:subobjects} +\end{equation} +If every inequality in \eqref{cap:universal} is strict, $e_J$ is in +the ordinary interior of such a hull in $\RR^J$. For $J=\varnothing$ +this uses the convention that $\{0\}$ is open in $\RR^0$. +\end{proposition} +\begin{proof} +For each $w\in W_J$ choose a negative reduced lift $G_w$. Put +$M_L=Y^q(E_L(x))$. The reduced-word amplitude in +Corollary~\ref{lay:reduced} places $G_wM_L$ at levels $q-1,q$. +Apply $G_w^{-1}$ to its low/high truncation triangle, obtaining +\[ + A_L\longrightarrow M_L\longrightarrow C_L. +\] +The endpoint bounds are $[q-1,q]$ and $[q,q+1]$. The long exact +layer sequence forces the possible level $q-1$ of $A_L$ and level +$q+1$ of $C_L$ to vanish. Thus both are pure at $q$ and +\begin{equation} + 0\longrightarrow A_L\longrightarrow M_L\longrightarrow C_L + \longrightarrow0 + \label{cap:short-exact} +\end{equation} +is exact in the shifted heart. Its subobject is an actual graded +subrepresentation by Lemma~\ref{lay:structure}. The low part of +$G_wM_L$ is supported on $\widetilde J$, since outside-color counts +are unchanged, and applying $G_w^{-1}$ keeps that support. + +The inclusion in \eqref{cap:short-exact} is obtained by twisting, +canonical truncation, and inverse twisting. These operations commute +with label tensoring, including their morphisms into $M_L$. The +weighted fusion identity therefore gives \eqref{cap:subobjects} for +a vector $s_w$. This does not require compatible choices of Gaussian +eliminations among the family members. + +The quotient has folded dimension $m^q(x)-s_w$. Applying $G_w$ +recovers the high part at level $q$, whose signed generator count is +therefore $w(m^q(x)-s_w)$. Lemma~\ref{cap:capped-negative} identifies +it with the actual level-$q$ layer at the operated representative. +That representative is jointly capped, so \eqref{cap:universal} +and $wp=p$ give +\begin{equation} + \bigl(w(e-s_w)\bigr)_J\ge0. + \label{cap:chamber-inequality} +\end{equation} +In the strict case all these coordinates are strictly positive. + +On the root-coordinate space supported on $J$, finite Coxeter +chambers cover the dual space. Thus every real functional has the +form $\lambda(v)=\sum_{j\in J}c_j(wv)_j$ for some $w$ and $c_j\ge0$. +Equation \eqref{cap:chamber-inequality} gives +\[ + \lambda(e)\ge\lambda(s_w) + \ge\min_{s\in\mathcal V}\lambda(s), + \qquad \mathcal V=\{s_w:w\in W_J\}\cup\{0\}. +\] +Separation proves membership in this finite convex hull. If the +inequalities are strict and $\lambda\ne0$, some $c_j>0$, so the first +inequality is strict. A boundary point, including a point of a hull +in a proper affine subspace, would admit a nonzero supporting +functional with equality. This proves ordinary interior membership. +For $J=\varnothing$ all the vectors are zero. +\end{proof} + +The convex description permits a linear estimate proved for those +subobjects to be applied to the excess vector, even when that vector is +not itself the dimension vector of a subobject. + +\subsection{Right simples, duality, and parabolic detection} + +Fix a nonempty spherical $T\subseteq S$. For $\ell\in\widetilde T$ +and $h\in A_T$ let +\[ + R_\ell(h)=U_\ell B_h. +\] +Expanding a word for $B_h$ gives one residual simple $U_\ell$ in +placements $(0,0)$ and shifted right projectives $Q_\beta$ with +$\beta\in\widetilde T$. Minimize in the additive category generated +by these graded terms. Its indecomposable endomorphism rings are local, +so the usual radical cancellation and uniqueness argument applies. +The simple cannot cancel with a projective, which has both top and +token socle. Hence exactly one residual simple remains. Denote by +$K^r_{\ell\beta}(h)$ the number of projective generators of type +$Q_\beta$ at total generator degree $r$, summed over both placements. + +For representatives $z,z'$ of two candidates, set $h=z'z^{-1}$. +The identity +\[ + R_\ell(h)\otimes_ZE_H(z)\simeq U_\ell E_H(z') +\] +relates the transported right simple to the two left families. Later +we will use their caps to bound the projective terms of $R_\ell(h)$. +The detection lemma below states exactly which vanishing will suffice: +once the two open sets agree and are denoted by $V$, vanishing of +$K^r_{\ell\beta}(h)$ for all $r$ and +$\ell,\beta\in\widetilde V$ forces $h\in A_{T\setminus V}$. +We establish this criterion, together with the invariance of the block +under changing representatives, before estimating it. +\begin{lemma}[Right duality and exterior invariance] +\label{two:duality} +For all $\ell,\beta\in\widetilde T$ and $r\in\mathbb Z$, +\begin{equation} + K^r_{\ell\beta}(h)=K^{-2-r}_{\beta\ell}(h^{-1}). + \label{two:duality-equation} +\end{equation} +For $J,J'\subseteq T$, left multiplication of $h$ by $A_{J'}$ leaves all rows outside +$\widetilde J'$ unchanged. Right multiplication by $A_J$ leaves +all columns outside $\widetilde J$ unchanged. +\end{lemma} +\begin{proof} +First minimize the bimodule complex $B_h$. Before minimization it +has one regular bimodule $Z$ at $(0,0)$, and all other terms are +shifts of $P_\alpha\otimes_{\kk}Q_\beta$, with indices in +$\widetilde T$; tensoring two such terms has the same form because +their middle tensor factor is a graded path space. + +No indecomposable regular summand is isomorphic to a projective +bimodule of this form. A connected-component regular bimodule is +indecomposable: a central idempotent would separate its vertices, +contrary to the nonzero arrows connecting them. Modulo the left +and right radicals, its top contains the diagonal simple at every +vertex of that component. The top of $P_\alpha\otimes Q_\beta$ +contains only the pair $(\alpha,\beta)$. The only remaining +possibility is a singleton component with $\alpha=\beta$; there the +regular bimodule has dimension two and the projective bimodule has +dimension four. Shifts do not change this obstruction. Thus all +regular summands survive, only at $(0,0)$. + +A radical differential between projective bimodules is a sum of path +tensors with positive length on at least one side: a nonzero +scalar component with both lengths zero would be an invertible +entry and would have been canceled. Tensoring by $U_\ell$ kills +positive left lengths; every surviving projective differential has +positive right length. Maps involving the regular term specialize +to radical maps between the simple and a projective. There is no +differential between residual simples, since they occur only in +cochain degree zero. The specialized right complex is therefore +already minimal, and its counts are the bimodule counts with first +index $\ell$. + +The Frobenius trace gives adjoint duality +\[ + \Hom_{Z\text{-left}}(M,Z)\cong\Hom_{\kk}(M,\kk)\langle2\rangle. +\] +It preserves additive radicals and sends +$P_\ell\otimes Q_\beta$ to +$(P_\beta\otimes Q_\ell)\langle-2\rangle$. On complexes it reverses +cochain placements as well. Thus $(c,u)$ becomes $(-c,-u-2)$. +The adjoint of the invertible complex $B_h$ is its inverse +$B_{h^{-1}}$, by Proposition~\ref{cat:action}. Minimality and the +preceding specialization give \eqref{two:duality-equation}. + +If $\ell\notin\widetilde J'$, every letter of $A_{J'}$ fixes +$U_\ell$: its inserted bimodule terms tensor to zero. This proves +row invariance. Apply \eqref{two:duality-equation} to +$(ha)^{-1}=a^{-1}h^{-1}$ to obtain column invariance for $a\in A_J$. +\end{proof} + +To descend a cohomologically detected unfolded letter to a color, +we record the required finite-type greedy argument. A simple positive +element is a reduced-word lift of an element of $W_T$. By the +finite-type monoid facts in Theorem~\ref{cat:classical}, a +positive element $b$ has a greatest simple right divisor $t(b)$, +its right gcd with the Garside element. + +\begin{lemma}[Greedy divisibility and unfolding] +\label{two:greedy} +For nonidentity simple factors $d_1,\ldots,d_k$, repeated extraction +of the greatest simple suffix gives $d_1\cdots d_k$ precisely when +\begin{equation} + D_R(d_i)\subseteq D_L(d_{i+1})\qquad(1\le ir+2). + \label{two:right-top} +\end{equation} +Indeed, only level-$r$ projective tokens can occur in degree $r+2$. +Their incoming images are the arrow-coefficient images, by the perfect +backtrack pairing, together with the simple's token inclusions when +$r=-1$. An additional incoming token differential would require a +level-$(r+1)$ source, which is absent. The residual simple itself has +degree zero, below $r+2$. The same argument applies to a common bound +for a finite family of rows. + +\begin{lemma}[Negative right normalization] +\label{two:negative-bound} +A right inverse twist, followed by minimization, preserves an upper +normalized bound $r\ge-1$ for a complex with one residual simple +at $(0,0)$ and shifted right projectives. At its operated vertex +$\alpha$ the tests satisfy +\begin{equation} + (R D_\alpha)e_\alpha + \simeq (Re_\alpha)\langle-2\rangle[-1]. + \label{two:negative-test} +\end{equation} +Both assertions hold for a full color twist as well. +\end{lemma} +\begin{proof} +For an individual unfolded vertex the inverse bimodule is +\[ + D_\alpha=[Z\longrightarrow + (P_\alpha\otimes Q_\alpha)\langle-2\rangle], +\] +with its second term in cochain degree one. A projective at +$(c,u)$ and a middle path of length $a$ insert $Q_\alpha$ at +$(c+1,u+a-2)$, of level $c+u+a-1$. Only the insertions with +$a=2$ can exceed the old level, and they come from old +$Q_\alpha$ terms. + +These offending terms can be canceled simultaneously, even in the +presence of the old differential. In $\mathcal E=\operatorname{Tot} +(R D_\alpha)$ let $A$ be the sum of old $Q_\alpha$ terms in the +$Z$ column and $B$ their inserted length-two copies. The latter +are at $(c+1,u)$, and the token--unit part of coevaluation gives +an invertible block +$\Phi=\pi_Bd|_A:A\to B$, consisting of signed scalar identities. +The old differential stays within a tensor column and does not +enter this block. The subcomplex +\[ + \mathcal K=A\oplus d(A) +\] +is contractible: projection onto $B$ identifies $d(A)$ with $B$, +and its differential maps $A$ isomorphically to $d(A)$ and kills +the latter. It is a direct summand as a bigraded module. +Writing $\mathcal E=A\oplus B\oplus W$, the quotient is homotopy +equivalent to $\mathcal E$ and has underlying terms $W$ with +\begin{equation} + d'_W=d_{WW}-d_{WA}\Phi^{-1}d_{BW}. + \label{two:pivot} +\end{equation} +Thus scalar entries induced among the inserted terms cause no +problem; their absence was not assumed. + +The remaining projectives are the old terms of other vertices and +the length-zero and length-one insertions. Their levels are, +respectively, the old level, the old level minus one, and the old +level. The simple remains at normalized level $-1$; if its vertex +is $\alpha$, it also inserts $Q_\alpha$ at $(1,-2)$, again level +$-1$. All remaining terms have level at most $r$. Subsequent +minimization only removes summands and so preserves the bound. + +The right-handed positive test identity follows from the same +token-cone cancellation as its left-handed version in +Lemma~\ref{lay:positive}. Inverting it gives +\eqref{two:negative-test}, of total shift $-1$. This is a bimodule +homotopy before tensoring, so it holds for simple inputs too. +Finally the vertices in one color are orthogonal, and their twists +commute; successive application proves the color statements. +\end{proof} + +\subsection{Comparing two candidates} + +We now compare two candidates with data +$(J,V,p,a,z)$ and $(J',V',p',a',z')$. Initially choose $z$ capped +on $T$ by accessibility, while $z'$ need only have its prescribed +cap on $V'$. Put $h=z'z^{-1}\in A_T$. Use right rows +$R_\ell(h)$ for $\ell=(i,L)\in\widetilde V'$, with weights +$a'_iw_L$, and independently give the left member $E_{L'}(z)$ +weight $w_{L'}$. Since +\[ + R_\ell(h)\otimes_ZE_{L'}(z)\simeq U_\ell E_{L'}(z'), +\] +their weighted actual cohomology in total degree $d$ has dimension +\begin{equation} + D_0\sum_{i\in V'}a'_i m_i^d(z'). + \label{two:actual-cohomology} +\end{equation} +Indeed tensoring a minimal left complex with $U_\ell$ kills its +radical differential and counts its generators at $\ell$. Summing +over $L'$ first gives $w_Lm_i^d(z')$, and then the row sum gives +\eqref{two:actual-cohomology}. It vanishes for $d>q$. + +Let $r\ge-1$ be the highest normalized level of these right rows, +and let $N$ be their coefficient data at level $r$, retaining the +row index. Put $M_{L'}=Y^q(E_{L'}(z))$. The linear tensor complex +at the pair $(r,q)$ has columns of middle path lengths $0,1,2$, +with simple tensors assigned to column one. Denote its cohomology +by $H_j(N,M)$ and its weighted dimensions by $\dim_{\wt}$. + +\begin{lemma}[The highest tensor pair] +\label{two:tensor} +One has $H_2(N,M)=0$, and $H_1(N,M)$ survives in actual total +degree $r+q+1$. At any projective vertex the token cokernel +surjects onto the tensor product of the right and left top quotients. +In particular a highest right top in a color $j\in J$ forces all +level-$q$ left-family tops in that color to vanish. +\end{lemma} +\begin{proof} +Filter the entire tensor by middle path length. Radical projective +entries increase length by one or two. A top quotient into the +simple takes a length-zero tensor to its assigned length one; +a token inclusion out of it takes assigned length one to length +two. A left radical map tensored with a right simple vanishes. +Every differential therefore increases this index. + +For a pair of normalized levels $(u,v)$ and middle length $j$, +actual total degree is $u+v+j$. This also holds for simple tensors, +since their assigned right level and middle length are $-1$ and +$1$. The length-one differential preserves the two levels. +After its cohomology the only remaining possible differential goes +from column zero to column two. In particular every column-one +class survives. This assertion can also be seen directly: writing +the differential as $d_1+d_2$, the cycle conditions are +$d_1u_0=0$ and $d_2u_0+d_1u_1=0$; the only secondary obstruction +is the class of $d_2u_0$ in column two. + +Every nonzero tensor uses a left vertex in $T\cup\partial T$, +because right projectives are supported at vertices of $T$ and +only equal-vertex or adjacent paths exist. Hence all relevant +left levels are at most $q$. A secondary differential hitting +column two at $(r,q)$ would originate at a level pair of sum +$r+q+1$, since the differential raises total degree by one and +length by two. Right levels are at most $r$ and relevant left +levels at most $q$, so no such source exists. Therefore $H_2(N,M)$ +also survives. It lies in degree $r+q+2>q$, where +\eqref{two:actual-cohomology} is zero. + +Project the token tensors at a vertex to the tensor of its two +coefficient spaces modulo incoming images. Each length-one +boundary has an incoming coefficient on one side, so the projection +annihilates boundaries and is surjective onto the tensor of the top +quotients. A nonzero right top thus forces every left top at that +vertex to vanish. Family proportionality in +Proposition~\ref{cat:family} makes a left top occurring anywhere in +one color occur at every label in some member. Thus the entire +left-family top in that color vanishes. +\end{proof} + +\begin{lemma}[Removal of highest right tops] +\label{two:normalization} +Changing $z$ within $A_Jz$, while keeping its cap on $T$, one can +arrange that no highest right top occurs on $\widetilde J$. +\end{lemma} +\begin{proof} +For a highest right top in $j\in J$, Lemma~\ref{two:tensor} +licenses the cap-preserving positive move $z\mapsto\sigma_jz$. +It changes $h$ to $h\sigma_j^{-1}$ and applies the corresponding +inverse twist to the right rows. Lemma~\ref{two:negative-bound} +shows that $r$ cannot increase. Candidate data $p,a$ are unchanged, +and the tensor products still compute the same target at $z'$, so +the next move is licensed anew. + +At a fixed $r$, the chosen negative right moves form a reduced word +in $W_J$. Otherwise a right descent could be placed last by braid +moves. All intermediate negative moves in that expression preserve +the upper bound $r$. Its final acted-color test shifts down by one, +so its degree-$r+2$ test vanishes, contradicting the currently chosen +right top by \eqref{two:right-top}. Sphericity bounds the length of +such a sequence. Thus after finitely many moves either the relevant +tops disappear or $r$ decreases. The residual simple bounds $r$ +below by $-1$, so the procedure terminates. If $J$ is empty no move +is required. +\end{proof} + +\subsection{The weighted Euler estimate} + +Use the representative in Lemma~\ref{two:normalization} and retain +$r,N,M$. Put $m=m^q(z)$ and $e=m-p$. The universal candidate +inequality verifies Proposition~\ref{cap:hull} for $A_Jz$: the cap +on $T\setminus J$ is fixed, so its joint $J$-cap is equivalent to +the additional cap on $T$. Hence $e$ is a convex combination of +folded dimensions $s$ of subfamilies $\mathcal S\subseteq M$ +supported on $\widetilde J$. + +\begin{lemma}[Tensoring a subobject] +\label{two:subobject-tensor} +For each such subfamily, +\[ + H_0(N,\mathcal S)\lhook\joinrel\longrightarrow H_0(N,M), + \qquad H_2(N,\mathcal S)=0. +\] +\end{lemma} +\begin{proof} +Column-zero cohomology is a kernel with no incoming boundaries, +so inclusion of its vector spaces gives the first assertion. + +For the second work row by row and member by member, retaining +cochain placements. At a projective vertex of $\widetilde J$, +each right coefficient is a sum of incoming images because its +top vanishes. We use this fact inside the token cokernel. +Suppose an image at $\beta$ comes from a coefficient at $\gamma$ +through a right arrow entry. Choose a middle arrow dual to that +entry under the backtrack pairing and tensor it with the predecessor +coefficient and a left coefficient in $\mathcal S_\beta$. +Its differential has two possible token contributions: the chosen +right image paired with the unchanged left coefficient at $\beta$, +and the predecessor paired with the corresponding left-arrow image +at $\gamma$, with its cochain sign. Non-backtrack products vanish; +dual arrow bases isolate the selected coefficient even for arrow +spaces of dimension greater than one. Modulo boundaries, therefore, +the former token tensor equals the latter, up to a nonzero scalar +and sign. + +The new left coefficient remains in $\mathcal S$ by the +subrepresentation property. Each substitution decreases the right +cochain placement by one, so boundedness forces termination. At a +projective vertex outside $\widetilde J$ the left coefficient is +zero. An incoming image from the simple is a boundary directly: +the assigned length-one simple tensor maps to that token inclusion, +and its left differential is zero. Thus every token tensor dies +in the cokernel, proving $H_2(N,\mathcal S)=0$. +\end{proof} + +Define a nonnegative vector supported on $T$ by +\begin{equation} + n_j=\frac1{D_0}\sum_H w_H + \sum_{i\in V'}\sum_L a'_i w_L + K^r_{(i,L),(j,H)}(h). + \label{two:right-weight} +\end{equation} +For a left family of weighted dimension $\phi y$, its weighted +length Euler characteristic with $N$, divided by $D_0$, is +\begin{equation} + \chi(n,y)=n^{\mathsf T}Cy + -\mathbf1_{\{r=-1\}}(a')^{\mathsf T}y_T. + \label{two:euler-formula} +\end{equation} +For completeness, let +$N_\beta^{\wt}=\sum_{i\in V',L}a'_iw_LK^r_{(i,L),\beta}(h)$. +The projective terms contribute +\[ + \sum_{\beta,\gamma}N_\beta^{\wt}\widetilde C_{\beta\gamma} + (\phi y)_\gamma + =\sum_{j,H}N_{(j,H)}^{\wt}w_H(Cy)_j + =D_0n^{\mathsf T}Cy. +\] +The unit and token contribute plus signs, and the middle arrows +minus signs. We used $\widetilde C\phi=\phi C$. At $r=-1$ the +residual simples contribute in column one +$-\sum_{i,L}a'_iw_L(w_Ly_i)=-D_0(a')^{\mathsf T}y_T$. +At higher $r$ there is no residual simple. This proves +\eqref{two:euler-formula}, including its normalization. + +\begin{lemma}[The two-point Euler estimate] +\label{two:euler} +At the normalized representative, +\begin{equation} + \frac{\dim_{\wt}H_1(N,M)}{D_0} + \ge n_T^{\mathsf T}a+ + \mathbf1_{\{r=-1\}}(a')^{\mathsf T}p_T. + \label{two:estimate} +\end{equation} +\end{lemma} +\begin{proof} +For each hull vector $s$, Lemma~\ref{two:subobject-tensor} gives +\[ + \chi(n,s)=\frac{\dim_{\wt}H_0(N,\mathcal S) + -\dim_{\wt}H_1(N,\mathcal S)}{D_0} + \le\frac{\dim_{\wt}H_0(N,M)}{D_0}. +\] +Linearity gives the same upper bound for $\chi(n,e)$, since $e$ +is a convex combination of these vectors. Lemma~\ref{two:tensor} +gives +$\chi(n,m)=(\dim_{\wt}H_0(N,M)-\dim_{\wt}H_1(N,M))/D_0$. +Subtracting yields +\[ + \frac{\dim_{\wt}H_1(N,M)}{D_0} + \ge\chi(n,e)-\chi(n,m)=-\chi(n,p). +\] +Now $n$ is supported on $T$ and $(Cp)_T=-a$, so +\eqref{two:euler-formula} is exactly \eqref{two:estimate}. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{two:uniqueness}] +Normalize the first representative as above. If $r>-1$, its highest +right layer has nonzero projective coefficients and no residual +simple. Some coefficient lies in $\widetilde V$: otherwise all +would lie in $\widetilde J$, where their tops vanish, and iterating +incoming images toward smaller cochain placements would force the +entire layer to vanish. All row weights are positive, so +$n_T^{\mathsf T}a>0$. Equation \eqref{two:estimate} gives nonzero +middle cohomology surviving in degree $r+q+1>q$, contradicting +\eqref{two:actual-cohomology}. Therefore $r=-1$. + +The surviving middle part now lies in degree $q$ and has dimension +at most the actual degree-$q$ cohomology. Since $a'$ is supported +on $V'$, where $p'_i=m_i^q(z')$, we obtain +\begin{equation} + (a')^{\mathsf T}(p'_T-p_T)\ge n_T^{\mathsf T}a\ge0. + \label{two:forward} +\end{equation} +Interchanging the candidates gives +\begin{equation} + a^{\mathsf T}(p_T-p'_T)\ge0. + \label{two:reverse} +\end{equation} +The two normalizations may use different representatives, but their +harmonic vectors and open-color vectors belong to the candidate +cosets and are unchanged. The exterior values of $p,p'$ outside +$T$ agree. With $\delta=p'_T-p_T$ this gives $a'-a=-C_T\delta$. +Adding \eqref{two:forward} and \eqref{two:reverse} yields +$-\delta^{\mathsf T}C_T\delta\ge0$. Positive definiteness forces +$p=p'$, hence $a=a'$, $V=V'$ and $J=J'$, with equality throughout +and $n_T^{\mathsf T}a=0$. + +It remains to identify the cosets. Fix any original comparison +$h_0=z'z^{-1}$. Replacing the two representatives by $bz$ and +$cz'$, with $b\in A_J$, $c\in A_{J'}$, replaces $h_0$ by +$ch_0b^{-1}$. Lemma~\ref{two:duality} preserves its block with +rows in $\widetilde V'$ and columns in $\widetilde V$. These are +now the same set $\widetilde V$. + +The forward comparison has no right generators above level $-1$. +At level $-1$, equality $n_T^{\mathsf T}a=0$ and +\eqref{two:right-weight} imply +$K^{-1}_{\ell\beta}=0$ for $\ell,\beta\in\widetilde V$, since +every coefficient weighting such a count is strictly positive. +Block invariance therefore gives the same zero block for $h_0$ +in all degrees $r\ge-1$. The reverse comparison gives its zero +block for $h_0^{-1}$ in all degrees $r\ge-1$, regardless of its +chosen representatives. Duality \eqref{two:duality-equation} +transposes this into vanishing for $h_0$ in all degrees $r\le-1$. +Together the two ranges cover every integer. By +Lemma~\ref{two:detection}, $h_0\in A_{T\setminus V}=A_J$. +Thus $A_Jz'=A_Jz$, proving uniqueness of the typed candidate. +\end{proof} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/05_null.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/05_null.tex new file mode 100644 index 0000000000000000000000000000000000000000..164b064a3016ca88b3129b7fb04043ee41281f37 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/05_null.tex @@ -0,0 +1,452 @@ +\section{Null layers with a relative frame} +\label{sec:null} + +The next obstruction will resolve the null-vector ties in the harmonic +filtration. The frame $f$ need not neighbor every operated color, so its +fixed generator alone does not provide a boundary coordinate for a given +color set. We instead use the connected support of an individual family +member. This gives the required obstruction at every integer level. + +\begin{proposition}[Relative null-layer obstruction] +\label{null:obstruction} +Let $I\subseteq S$ be nonempty and connected, let $c_I>0$ satisfy +$C_Ic_I=0$, and extend $c$ by zero outside $I$. Let $J\subseteq I$ be +spherical, possibly empty, let $q\in\mathbb Z$, and fix $A_Jx$. +Suppose its fixed data on $(I\setminus J)\cup\partial I$ are capped at +$q$ and satisfy +\begin{equation} + m^q\big|_{(I\setminus J)\cup\partial I} + =c\big|_{(I\setminus J)\cup\partial I}. + \label{null:exterior} +\end{equation} +It is impossible that every representative $z\in A_Jx$ capped at $q$ +on $I$ satisfies +\begin{equation} + m_j^q(z)>c_j\qquad\text{for every }j\in J. + \label{null:strict} +\end{equation} +Thus, if $J\ne\varnothing$, some such representative has +$m_j^q(z)\le c_j$ for at least one $j\in J$. If $J=\varnothing$, +the exterior hypotheses themselves are impossible. +\end{proposition} + +We first develop the linear Hom calculation used in the proof. All +complexes, maps, and pairings retain both gradings, and all differential +signs use cochain degree. + +\subsection{Linear Hom and its permanent middle column} + +For pure projective layers $V,W$ of levels $l,k$, respectively, let +$\mathcal L_j^{n,v}(V,W)$ be the space of maps of path length $j$, +cochain degree $n$, and internal degree $v$. It is zero unless +$j\in\{0,1,2\}$ and +\begin{equation} + n+v=k-l+j. + \label{null:hom-degree} +\end{equation} +Indeed, between generator placements $(a,u)$ and $(a',u')$, these +degrees are $n=a'-a$ and $v=u'+j-u$. Write $\delta_V,\delta_W$ for +the linear differentials. The differential +\[ + D_1g=\delta_Wg-(-1)^n g\delta_V +\] +raises path length and cochain degree by one and preserves internal +degree. Denote its cohomology in the displayed space by +$\mathsf H_j^{n,v}(V,W)$, and put +\[ + h_j(V,W)=\sum_{n,v}\dim_{\kk}\mathsf H_j^{n,v}(V,W). +\] +The sums are finite. In length zero there are no incoming boundaries: +a cocycle is a homogeneous map of coefficient arrays commuting with +their arrows up to its cochain sign. Its image is a graded +subrepresentation, with the corresponding shift in placements. + +\begin{lemma}[Pairing and middle vanishing] +\label{null:hom} +For pure layers, including layers obtained from graded +subrepresentations, there are perfect pairings +\begin{equation} + \mathsf H_j^{n,v}(V,W)\ \times\ + \mathsf H_{2-j}^{-n,\,2-v}(W,V)\longrightarrow\kk. + \label{null:pairing} +\end{equation} +In particular, +\begin{equation} + \begin{split} + h_2(V,W)&=h_0(W,V),\\ + (\dim V)^{\mathsf T}\widetilde C\,\dim W + &=h_0(V,W)+h_0(W,V)-h_1(V,W), + \end{split} + \label{null:euler} +\end{equation} +where dimensions count projective generators, summed over placements. +For every $z\in A$, every pair of labels $L,H$, and all levels $l,k$, +\begin{equation} + h_1\bigl(Y^l(E_L(z)),Y^k(E_H(z))\bigr)=0. + \label{null:middle} +\end{equation} +The vanishing persists after replacing either layer by a direct +summand as a pure layer. +\end{lemma} + +\begin{proof} +The symmetric Frobenius trace pairs opposite paths of complementary +lengths, with total internal degree two. On a complex use the +cochain supertrace +\[ + \operatorname{Str}_W(a) + =\sum_c(-1)^c\tau\bigl(\operatorname{tr}_{\mathrm{mat}}(a|_{W^c})\bigr), +\] +where the matrix trace sums diagonal path entries and $\tau$ is the +Frobenius trace, not the ordinary trace of the underlying linear map. Composition +and this trace give perfect pairings between the corresponding length +spaces in Equation~\eqref{null:pairing}. Cyclicity gives +\[ + \langle D_1g,h\rangle+(-1)^n\langle g,D_1h\rangle=0 +\] +in complementary degrees. Finite-dimensional duality yields the +pairings on cohomology. Units and tokens each contribute the +equal-vertex pairing to the length Euler characteristic; arrows +contribute minus the adjacency multiplicities. This proves +Equation~\eqref{null:euler}. The argument only uses the linear +differentials and therefore also applies to subrepresentations. + +For Equation~\eqref{null:middle}, take minimal models of $E_L(z)$ and +$E_H(z)$ and filter their full bigraded Hom complex by paths of length +at least $j$. Its differential is $D_1+D_2$, with $D_a$ raising path +length by $a$. The first nonzero differential is $D_1$, whose +cohomology splits by the ordered source and target layer levels. +There are only three columns. After this linear page, the only +possible differential goes from length zero to length two; every +middle-column class is therefore permanent. + +The action in Proposition~\ref{cat:action} preserves actual bigraded +Hom cohomology. Between the initial projectives $P_{f,L},P_{f,H}$ +it is zero for $L\ne H$; for $L=H$ it is one-dimensional in each of +bidegrees $(0,0)$ and $(0,2)$ and zero elsewhere. Both diagonal +dimensions already have permanent representatives. The identity of +the full complex is a nonzero length-zero class: a boundary has no +length-zero component. For the other class, put a normalized token +on one diagonal projective summand of a nonzero minimal model and put +zero on the other entries. This is a cycle of bidegree $(0,2)$, +because a token multiplied by a radical entry is zero. Its +supertrace is $(-1)^c\ne0$ if that summand lies in cochain placement +$c$, whereas every Hom boundary has zero supertrace. It is thus a +nonzero actual class in filtration two; filtration three is zero. + +The identity and token exhaust the actual diagonal dimensions, and +the off-diagonal cohomology vanishes. Hence every permanent middle +class is zero, proving Equation~\eqref{null:middle}. A direct +summand of a pure layer gives a direct summand of its linear Hom +complex, so the same vanishing holds for it. This last assertion +does not require the summand to split off the full transported complex. +\end{proof} + +For finite label-indexed families $\mathcal V=(V_L)_L$ and +$\mathcal W=(W_H)_H$, define the weighted dimensions +\begin{equation} + h_j^{\mathrm{wt}}(\mathcal V,\mathcal W) + =\sum_{L,H}w_Lw_H h_j(V_L,W_H). + \label{null:weighted-dimension} +\end{equation} +They need not be integers. If the two weighted generator arrays are +$\phi v$ and $\phi y$, then +\[ + \sum_{L,H}w_Lw_H(\dim V_L)^{\mathsf T}\widetilde C\dim W_H + =D_0v^{\mathsf T}Cy. +\] +Equation~\eqref{null:euler} therefore becomes +\begin{equation} + D_0v^{\mathsf T}Cy + =h_0^{\mathrm{wt}}(\mathcal V,\mathcal W) + +h_0^{\mathrm{wt}}(\mathcal W,\mathcal V) + -h_1^{\mathrm{wt}}(\mathcal V,\mathcal W). + \label{null:weighted-euler} +\end{equation} + +\subsection{The capped secondary injection} + +Fix a representative $z$, and abbreviate +$\mathcal Y^d=(Y^d(E_L(z)))_L$. The following lemma uses a cap to +remove one of the two possible blocks of the secondary Hom differential. + +\begin{lemma}[Secondary injection] +\label{null:injection} +Suppose the family at $z$ is capped at $q$ on $I\subseteq S$. +For each $L$, let $M_L$ be a direct summand of $Y^q(E_L(z))$ as a +pure layer, supported on $\widetilde I$, and put +$\mathcal M=(M_L)_L$. For $d0$ and +$C_Ic_I=0$, and extend $c$ by zero. Let $J\subseteq I$. +Suppose $\mathcal M=(M_L)_L$ consists of direct summands of a single +layer family $\mathcal Y^q$, supported on $\widetilde I$, with +weighted generator array $\phi m$. Assume $m-c$ is supported on +$J$, and $(m-c)_J$ is an interior point in $\mathbb R^J$ of the +convex hull of finitely many vectors $s_J$. Each such $s$ is zero +off $J$, and $\phi s$ is the weighted generator array of a family +of graded subrepresentations $\mathcal S\subseteq\mathcal M$ +supported on $\widetilde J$. + +If the socle of $\mathcal Y^d$ vanishes on $J$, and its folded +weighted vector is $y=m^d(z)$, then +\begin{equation} + \begin{gathered} + y|_{\partial I}=0,\qquad (Cy)_J=0,\\ + h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y^d) + =h_0^{\mathrm{wt}}(\mathcal Y^d,\mathcal M)=0. + \end{gathered} + \label{null:separation-conclusions} +\end{equation} +The usual zero-dimensional interior convention is used when +$J=\varnothing$. +\end{lemma} + +\begin{proof} +Write $\mathcal Y=\mathcal Y^d$, and fix one of the subfamilies +$\mathcal S$. A nonzero homogeneous length-zero cocycle from a +member of $\mathcal S$ to one of $\mathcal Y$ has a nonzero graded +image supported on $\widetilde J$. That image has a socle: in its +highest occupied cochain placement every outgoing coefficient arrow +is zero, since arrows raise that placement. This gives a socle of +the target on $J$, contrary to the hypothesis. Hence +\[ + h_0^{\mathrm{wt}}(\mathcal S,\mathcal Y)=0. +\] +Inclusion of $\mathcal S$ in $\mathcal M$ also gives +\[ + h_0^{\mathrm{wt}}(\mathcal Y,\mathcal S) + \le h_0^{\mathrm{wt}}(\mathcal Y,\mathcal M), +\] +because length-zero cocycles have no boundaries to quotient out. +Apply Equation~\eqref{null:weighted-euler} to these two families. +Only nonnegativity of $h_1^{\mathrm{wt}}(\mathcal S,\mathcal Y)$ +is needed; for $\mathcal M$, middle vanishing does hold by +Lemma~\ref{null:hom}. Subtraction gives +\begin{equation} + D_0(m-s)^{\mathsf T}Cy + \ge h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y)\ge0. + \label{null:convex-inequality} +\end{equation} + +This inequality is affine-linear in $s$, so it holds on the entire +hull. At its interior point $e=m-c$, its left side is +$D_0c^{\mathsf T}Cy$. Symmetry of $C$ and the null equation give +\begin{equation} + c^{\mathsf T}Cy + =-\sum_{a\notin I}\sum_{i\in I}b_{ai}c_i y_a\le0. + \label{null:boundary-sign} +\end{equation} +It is strictly negative if $y$ meets $\partial I$. Comparing with +Equation~\eqref{null:convex-inequality} proves boundary vanishing, +$c^{\mathsf T}Cy=0$, and +$h_0^{\mathrm{wt}}(\mathcal M,\mathcal Y)=0$. + +For each $j\in J$, ordinary interior membership supplies the two +nearby hull points $e\pm\varepsilon\mathbf e_j$. Their inequalities, +whose value at $e$ is zero, force $(Cy)_j=0$. Therefore +\[ + m^{\mathsf T}Cy + =c^{\mathsf T}Cy+(m-c)_J^{\mathsf T}(Cy)_J=0. +\] +The Euler identity for $(\mathcal M,\mathcal Y)$ now gives the +remaining Hom vanishing. When $J$ is empty, $m=c$ and no coordinate +variation is required. +\end{proof} + +\subsection{Proof of the obstruction} + +\begin{proof}[Proof of Proposition~\ref{null:obstruction}] +Assume Equation~\eqref{null:strict} holds at every representative +capped on $I$. Since +$\partial J\subseteq(I\setminus J)\cup\partial I$, cap accessibility +in Lemma~\ref{cap:access} supplies a representative capped on $J$ +and hence on all of $I$. We will use only cap-preserving moves in +$J$, so the exterior data in Equation~\eqref{null:exterior} remain +fixed throughout. + +\emph{The convex data at each capped representative.} +Let $M_L$ be the part of $Y^q(E_L(z))$ supported on $\widetilde I$. +It is a direct summand of the pure layer, because the level-$q$ +coefficients on $\partial I$ vanish. This does not assert a splitting +of $E_L(z)$. Its weighted generator array is $\phi m$, where $m$ +agrees with $m^q(z)$ on $I$ and is zero elsewhere. In particular, +$m-c$ is supported on $J$. + +The harmonic prescription on $J$ is $c_J$: its boundary data are +those of $c$, and $C_Ic_I=0$. Remote coordinates outside +$I\cup\partial I$ do not enter those equations. The strict part of +Proposition~\ref{cap:hull} and the universal hypothesis +Equation~\eqref{null:strict} therefore give +\begin{equation} + (m-c)_J\in\operatorname{int}_{\mathbb R^J} + \operatorname{conv}\{s_J\}, + \label{null:interior} +\end{equation} +with the subrepresentation families required in Lemma~\ref{null:separation}. +They lie in $\mathcal M$ because their support is in $\widetilde J$. +The assertion is renewed after every capped move: both $\mathcal M$ +and its hull may change, but the strict hypothesis is quantified over +every capped representative. + +\emph{An occupied boundary below $q$.} +The set $I\setminus J$ is nonempty, since $C_J$ is positive definite +whereas $C_I$ has a positive null vector. Choose a color there. +Its prescribed positive level-$q$ multiplicity occurs in some member +$E_L(z)$. By the family properties in Proposition~\ref{cat:family}, +the minimal color support of that member is connected and contains +$f$, which lies outside $I$. A support path from the chosen color +to $f$ meets $\partial I$. There is consequently an occupied +boundary level. Let $d_b$ be the lowest such level over the family. +The boundary cap and zero boundary data at level $q$ give +\begin{equation} + d_b0,\quad C_Ic_I=0,\\ + c=0\ \text{outside }I,\qquad + m^q=c\ \text{on }(I\setminus T)\cup\partial I, + \quad\text{with a cap at }q\text{ there}. + \end{gathered} + \label{ht:closed-data} +\end{equation} +Then \(b_T^*=c_T\). The \emph{special closed sublevel} +\(\mathcal R_{\le}(T,A_Tx;q,I,c)\) has the same definition except that its +open inequalities are \(m_i^q(z)\le c_i\). In both definitions the type is +proper, and a cap bounds every higher level, not merely level \(q\). + +\begin{proposition}[Spherical sublevel connectivity] +\label{ht:sublevels} +The strict sublevel is empty or connected. Under +Equation~\eqref{ht:closed-data}, the special closed sublevel is nonempty +and connected. For \(T=\varnothing\), the strict sublevel is empty and +the special closed exterior data cannot occur. These statements hold +for every integer \(q\). +\end{proposition} + +\begin{proof} +For \(T=\varnothing\), there is no proper type. The closed exterior +data would prescribe \(m^q=c\) and a cap on \(I\cup\partial I\), contrary +to the empty-set case of Proposition~\ref{null:obstruction}. + +In the nonempty closed case, Lemma~\ref{cap:access} supplies a +representative capped on \(T\), hence on \(I\). If the closed sublevel +were empty, every representative capped on \(I\) would satisfy +\(m_j^q>c_j\) for every \(j\in T\): a coordinate with \(m_j^q\le c_j\) +could be opened alone, giving type \(T\setminus\{j\}\). +This contradicts Proposition~\ref{null:obstruction}. Thus this sublevel +is nonempty. + +We prove both connectivity assertions by induction on \(|T|\). +For a vertex \(v=(J,A_Jz)\) in either sublevel, let \(p_v\) be the +harmonic interpolation of \(m^q(z)\) on \(J\), and set +\[ + u_v=(p_v)_T-b_T^*. +\] +The open inequalities and nonnegative interpolation give +\begin{equation} + u_v\le0,\qquad + (u_v)_J=-C_J^{-1}C_{J,T\setminus J}(u_v)_{T\setminus J}, + \qquad (Cp_v)_T=C_Tu_v. + \label{ht:local-difference} +\end{equation} +Order vertices first by increasing \(\sum_{i\in T}(u_v)_i\). +At an equal negative sum put larger types first; at sum zero put +smaller types first. Refine the remaining ties by any well-order. +Only finitely many sums occur: open counts lie in bounded intervals +with finitely many possible values, the exterior is fixed, and there +are finitely many types. The resulting order is well founded. +Remaining ties are between incomparable vertices, since distinct +comparable cosets of the same type cannot occur. + +Consider first \(u=u_v\ne0\), and write \(V=T\setminus J\). Put +\[ + N=\{i\in V:(C_Tu)_i\ge0\}. +\] +An upper extension has the form \((K,A_Kz)\), with +\(J\subsetneq K\subsetneq T\), and is still allowed. The harmonic +change is zero outside \(K\) and on \(K\) equals +\begin{equation} + (p_{(K,A_Kz)}-p_v)_K=-C_K^{-1}(C_Tu)_K. + \label{ht:local-upper-change} +\end{equation} +The old rows on \(J\) vanish. If \(K\setminus J\) avoids \(N\), +the nonnegative inverse with positive diagonal makes the sum strictly +increase. If \(\varnothing\ne K\setminus J\subseteq N\), the change +is nonpositive; a strict decrease makes the extension earlier, and +equality favors its larger type at the negative sum. +Moreover \(N\ne V\), because otherwise \(C_Tu\ge0\) would imply +\(u=C_T^{-1}C_Tu\ge0\). + +Every earlier upper extension therefore contains a new color in \(N\). +Removing its other new colors gives a comparable earlier extension +whose type is contained in \(J\cup N\). These extensions all connect +through the maximum \((J\cup N,A_{J\cup N}z)\). Thus the earlier upper +part is empty or connected. + +An earlier lower neighbor of \(v\) must strictly decrease the sum. +Write it as \((R,A_Rw)\), where \(R\subsetneq J\), \(w\in A_Jz\). +The difference of its harmonic vector and \(p_v\) is zero outside \(J\), +has prescribed values +\[ + m_i^q(w)-(p_v)_i\qquad(i\in J\setminus R), +\] +and is harmonically interpolated on \(R\). +If all these values were nonnegative, the sum could not decrease. +Hence the set of newly opened colors satisfying +\(m_i^q(w)<(p_v)_i\) is nonempty. Close the other newly opened colors. +This produces a comparable earlier lower vertex in the strict sublevel +\(\mathcal R_{<}(J,A_Jz;q)\), with harmonic bound \(p_v\). + +The boundary of \(J\) is capped: its colors lie in \(V\) or +\(\partial T\). Every vertex of this inner strict sublevel is conversely +an allowed earlier lower vertex in the original sublevel, because +\(p_v\le b^*\) and its harmonic difference from \(p_v\) has strictly +negative sum. Induction makes the inner sublevel empty or connected. +Since each earlier lower vertex is comparable with a vertex in it, +the whole earlier lower part is empty or connected as well. + +If \(u_v=0\), we are in the special closed case. An upper extension +keeps the same harmonic vector and has larger type, so is later. +Every allowed lower neighbor is earlier, and these neighbors form +\(\mathcal R_{\le}(J,A_Jz;q,I,c)\). Its exterior data are supplied by +Equation~\eqref{ht:closed-data} together with the equalities and caps on +\(V\). The empty \(J\) case is impossible by +Proposition~\ref{null:obstruction}; otherwise induction makes this lower +part nonempty and connected. + +If both earlier parts are nonempty, every earlier lower vertex is below +every earlier upper vertex, so their union is connected. +Lemma~\ref{ht:vertex-filtration} now applies. +A vertex with no earlier neighbor must have \(u_v\ne0\), \(N=\varnothing\), +and empty inner strict sublevel. Therefore +\[ + -(Cp_v)_i>0\quad(i\in V),\qquad (Cp_v)_J=0. +\] +At every representative \(w\in A_Jz\) capped on \(T\) one also has +\(m_J^q(w)\ge(p_v)_J\); otherwise opening a failing coordinate alone +would give a vertex in the inner strict sublevel. +Together with the caps on \(V\), these are exactly the candidate +conditions of Proposition~\ref{two:uniqueness}. There is at most one +such vertex in the fixed spherical residue. +Every component has one by Lemma~\ref{ht:vertex-filtration}, so a +nonempty sublevel is connected. +\end{proof} + +\subsection{The global relative height} + +Fix \(Y\subseteq S\) and \(x\in A\), without a sphericity assumption on +\(Y\). Let \(\mathcal P(Y,x)\) be the poset of +\[ + v=(T,A_Tz),\qquad T\subseteq Y\ \text{spherical},\quad z\in A_Yx. +\] +Its graph is connected. Indeed singleton cosets \((\varnothing,\{z\})\) +are joined to their left-generator translates through the corresponding +rank-one cosets, and every vertex contains a singleton vertex. + +Choose \(d_*\le0\) at or below every occupied level of the exterior data +\(m^k(x)|_{\widehat S\setminus Y}\). +These data are fixed throughout \(A_Yx\). +For \(v=(T,A_Tz)\), let \(p^k(v)\) be the harmonic interpolation of +\(m^k(z)\) on \(T\). It is independent of \(z\) in its coset. + +For each \(k\ge d_*\), assign a sign \(\eta_i^k(v)\) as follows. +It is zero if \(p_i^k(v)=0\), and is \(-1\) at positive entries, except +that it is \(+1\) on a whole positive-support component \(I\) when +\begin{equation} + I\subseteq S,\qquad (Cp^k(v))_I=0,\qquad + p^d(v)|_{I\cup\partial I}=0\quad\text{for every }d>k. + \label{ht:null-sign} +\end{equation} +Here support components use adjacency in the full diagram +\(\widehat S\). Fix an ordering of those coordinates and form the block +\begin{equation} + B_k(v)= + \left( + \sum_{i\in\widehat S}p_i^k(v),\ + (p_i^k(v))_{i\in\widehat S},\ + (\mathbf1_{\{i\in T\}}\eta_i^k(v))_{i\in\widehat S} + \right). + \label{ht:block} +\end{equation} +Blocks are ordered lexicographically, with smaller earlier. +The \emph{height} of \(v\) compares these finite-support blocks from the +highest level downward, stopping at \(d_*\). +Distinct vertices may have equal height. + +The final coordinates of each block break numerical ties. When the +recorded interpolated vectors are unchanged, adding a positive color to +the type changes its coordinate from zero to \(\eta_i^k\): sign \(-1\) +favors closing that color, while sign \(+1\) favors opening it. +The exceptional \(+1\) case in Equation~\eqref{ht:null-sign} is an +isolated highest positive null component. An entire such component is +nonspherical, so it cannot all be added to a spherical type. Openings +whose counts are at most the prescribed null vector are instead +controlled by the special closed sublevel of +Proposition~\ref{ht:sublevels}. + +\begin{lemma}[Well-founded heights] +\label{ht:well-order} +The possible heights are well ordered. Their least block is the zero +block. +\end{lemma} + +\begin{proof} +All entries of \(p^k\) are nonnegative by Lemma~\ref{cat:cartan}. +Bounding a block's first +coordinate bounds every exterior count for its type. Those counts +have finitely many possible values in a bounded interval by +Proposition~\ref{cat:family}. There are +finitely many types, their harmonic interpolations are fixed linear +maps, and there are finitely many sign masks. +Thus the block alphabet has finite initial segments. Its least +element is zero. + +An infinite strictly descending sequence of heights would have no +nonzero block above the largest occupied level of its first term. +It would therefore involve only finitely many levels \(k\ge d_*\). +At the highest level the block can decrease only finitely often, so +eventually stays fixed. The same argument then applies to the next +level, and successively to all remaining levels, contradicting strict +descent. Equivalently, if \(r(B)\) is the finite rank of a block in its +alphabet, the heights are ordered by the finite Cantor sums +\(\sum_{k\ge d_*}\omega^{k-d_*}r(B_k)\), with decreasing exponents. +\end{proof} + +A neighbor is called \emph{descending} when it has strictly smaller +height. We analyze upper and lower neighbors separately. + +\subsection{Active upper colors} + +Fix \(v=(T,A_Tz)\). For \(i\in Y\setminus T\), let \(\ell_i\) be the +highest recorded level with a positive entry on \(i\) or one of its +neighbors, if such a level exists. Call \(i\) \emph{active} if +\begin{equation} + (Cp^{\ell_i}(v))_i>0 + \quad\text{or}\quad + (Cp^{\ell_i}(v))_i=0,\quad \eta_i^{\ell_i}(v)=-1. + \label{ht:active} +\end{equation} +An active color has a positive entry at that level: otherwise incidence +would make its row strictly negative. + +\begin{lemma}[Upper neighbors] +\label{ht:upper} +The union of \(T\) with all active colors is spherical. +The descending upper neighbors of \(v\) are empty or connected using +only descending upper neighbors. +\end{lemma} + +\begin{proof} +An upper neighbor has the form \(v'=(U,A_Uz)\), where \(T\subsetneq U\) +is spherical. Compare at the highest incidence level of its new colors. +Above that level, the new colors and their neighbors have zero entries, +so interpolation changes neither vectors nor signs. +At the comparison level \(k\), the change is zero outside \(U\) and is +\begin{equation} + (p^k(v')-p^k(v))_U=-C_U^{-1}(Cp^k(v))_U. + \label{ht:global-upper-change} +\end{equation} +Rows on \(T\) vanish. +If no new color is active, all the new rows at this level are +nonpositive. A negative row strictly increases the sum. If all +vanish, the vector and its intrinsic signs stay fixed, and the incident +positive new coordinates have sign \(+1\). Closing them increases +the sign mask. If the new colors have no recorded incidence at all, +the heights tie. +Consequently every descending upper neighbor adds an active color. + +Conversely, suppose only active colors are added. +At their highest incidence level all new rows are nonnegative. +A positive row strictly decreases the sum; if all vanish, the vector +is fixed and the positive new coordinates have sign \(-1\), so the +sign mask decreases. Thus such an extension is descending whenever +its type is spherical. + +Let \(G\) be a connected component of \(T\) together with all active +colors, and suppose \(G\) contains an active color. +Choose the highest recorded level \(k\) incident to \(G\). +All full rows on \(G\) are nonnegative: closed rows vanish, while an +active color either has its highest incidence at \(k\), or has zero +data on itself and its neighbors there. Hence +\begin{equation} + C_Gp_G^k=(Cp^k)_G+b_{G,\widehat S\setminus G}p_{\widehat S\setminus G}^k + \ge0. + \label{ht:active-cartan} +\end{equation} +The vector \(p_G^k\) is nonzero, since otherwise incidence would give a +negative full row. It is strictly positive by connectedness: a zero +coordinate adjacent to a positive one would again have a negative row. + +If \(G\) were nonspherical, Lemma~\ref{cat:cartan} would give +\(C_Gp_G^k=0\). Both nonnegative terms on the right of +Equation~\eqref{ht:active-cartan} would vanish. +Thus \(G\) is the entire positive-support component, and higher levels +vanish on \(G\cup\partial G\) by the choice of \(k\). +Equation~\eqref{ht:null-sign} assigns sign \(+1\) to \(G\). +Every active color in it has highest incidence \(k\) and zero full row, +contradicting Equation~\eqref{ht:active}. Therefore \(G\) is spherical. +Components with no active color lie in the old spherical type \(T\). +The whole union is consequently spherical. + +Every descending upper neighbor can now be joined to the one obtained +by removing all its nonactive new colors. The latter is descending +and still contains a new color. All these reduced extensions connect +through the extension by every active color, which is spherical by the +preceding argument. If there are no active colors, there is no +descending upper neighbor. +\end{proof} + +\subsection{Eligible lower colors and null blocks} + +A connected diagram component of \(T\) is \emph{visible} if its harmonic +entries are positive at some recorded level. +Whenever they are nonzero at a level, all its entries are positive, +by positivity of the inverse Cartan block. +It therefore has a common highest such level \(q\). +Its boundary data are capped there: at every higher level its zero +harmonic entries force all neighboring nonnegative entries to vanish. +Its entries also have a common sign at level \(q\). + +For a lower coset, call a newly opened color \(i\) \emph{eligible} when +it belongs to a visible component with highest level \(q\), its actual +data are capped at \(q\), and +\begin{equation} + m_i^q0\) for \(1\leq n0\). +The established range identity gives +\[ + (T_n^*T_n)^2=\lambda T_n^*T_n,\qquad + \mathsf Q_nT_n^*T_n=T_n^*T_n + =T_n^*T_n\mathsf Q_n. +\] +Consequently +\(\mathsf Q_n-\lambda^{-1}T_n^*T_n\) is self-adjoint +and idempotent. It has image in +\(\operatorname{im}\mathsf Q_n\cap\ker T_n\) and is +the identity on that subspace. This is the projection +\eqref{sph:eq:projection}, completing the induction. + +It remains to prove that $\mathcal E_n$ vanishes for some $n$. +The endpoint-dimension recurrence below will give that finite path bound. +Let \(\mathsf d_n(i,j)\) be the dimension of +\(\mathcal E_n\) with endpoints \(i,j\). +Extending a path of length \(n-1\) by one edge makes the +endpoint-dimension matrix of +\(\operatorname{im}\mathsf Q_n\) equal to +\(\mathsf d_{n-1}\mathsf B\). +The restricted surjection +\(T_n:\operatorname{im}\mathsf Q_n\to\mathcal E_{n-2}\) +preserves endpoints and has kernel \(\mathcal E_n\). +Hence, for \(2\leq nl+jN\), a possible token source from the next +higher layer at \(p-1\) is absent on that interval. +The same argument expresses the coefficient space +by incoming arrows, and then by \(N\)-step paths. +It removes this layer on \([l+(j+1)N,r]\). +The successive conclusions are +\[ + \begin{array}{c|c} + \text{layer}&\text{interval proved empty}\\ \hline + d_2 &[l+N,r]\\ + d_2-1 &[l+2N,r]\\ + \vdots&\vdots\\ + d_1 &[l+N(d_2-d_1+1),r]. + \end{array} +\] +On the last interval every possible generator layer +has been removed, proving the assertion. +\end{proof} + +\begin{proposition}\label{sph:no-chain} +Assume \(h(1)>-\infty\). +There is no sequence +\[ + a_0=1,\qquad a_{k+1}=w_k a_k,\qquad + w_k\in A_Y^+,\quad\operatorname{supp}(w_k)=Y, +\] +for which the integers \(h(a_k)\) have a common upper +bound. +\end{proposition} + +\begin{proof} +Suppose that \(h(a_k)\leq h_*\) for all \(k\). +Choose a common lower bound \(d_1\) on the generator +levels of the initial family \(E_L(x)\). +Positive amplitude, Lemma~\ref{lay:positive} and the +bounded-layer formalism of Lemma~\ref{lay:structure}, +preserves this bound at every stage. + +All generator placements outside \(Y\) are fixed by +the family properties in Proposition~\ref{cat:family}. +Let \(d_{\mathrm{ext}}\) be their largest occupied +level. The exterior includes the unchanged f +generator at placement \((0,0)\), so this is defined. +If a current global highest layer has degree +\(d>d_{\mathrm{ext}}\), it is supported entirely in +\(\widetilde Y\). A nonzero such layer has a nonzero +top: incoming coefficient maps increase cochain +placement in a finite array and cannot generate a +nonzero array indefinitely. Lemma~\ref{lay:extremal} +then gives a nonzero test in total degree \(d+2\). +It follows that every current generator has level +in the fixed interval +\[ + [d_1,d_2],\qquad + d_2=\max(d_{\mathrm{ext}},h_*-2). +\] + +By Lemma~\ref{sph:cochain-drop}, the common upper bound +on ordinary cochain test cohomology drops by at least +one at every step \(a_k\to a_{k+1}\). +It therefore tends to \(-\infty\). +Let \(p_{\mathrm{ext}}\) be the least exterior +generator cochain placement, over the initial finite +family. It is fixed, and \(p_{\mathrm{ext}}\leq0\) +because of the f generator. +Set +\[ + N=N(Y),\quad + K_0=N(d_2-d_1+1),\quad + r=p_{\mathrm{ext}}-1,\quad l=r-K_0. +\] +At a sufficiently late stage all tested cohomology +on \(\widetilde Y\) lies in cochain degrees strictly +below \(l\). Every generator in \([l,r]\) lies in +\(\widetilde Y\), since \(r-\infty\). +For that member there is a term in a cochain degree +below \(l\), while its unchanged f generator is in +degree \(0>r\). +The absent term in degree \(r\) splits its minimal +complex into two nonempty bounded minimal complexes, +one on either side of that degree. +Neither summand is contractible. Indeed a contracting +homotopy would express its identity as \(dh+hd\); +every component of the latter lies in the additive +radical because all entries of \(d\) do. Reduction +modulo the radical would give a nonzero identity +equal to zero. +This contradicts the indecomposability of +\(E_L(a_kx)\) from Proposition~\ref{cat:family}. +\end{proof} + +\subsection{Positive extensions and a proper support} + +\begin{lemma}\label{sph:common-extension} +For \(a,b\in A_Y\), there are \(p,q\in A_Y^+\) and +\(u\in A_Y\) such that +\[ + u=pa=qb,\qquad h(u)\leq\max(h(a),h(b)). +\] +The same upper bound holds for a common positive +left extension of any finite family. +\end{lemma} + +\begin{proof} +The finite-type fraction property in +Theorem~\ref{cat:classical} first gives \(u=pa=qb\) +with \(p,q\) positive. Whenever \(p\) and \(q\) +have a common left atom, write +\(p=\sigma_i p'\), \(q=\sigma_i q'\) and cancel it. +This replaces the common extension by +\(p'a=q'b\). +The sum of positive lengths decreases by two, so +the process terminates with no common left atom. + +Suppose the resulting extension satisfies +\(h(u)>\max(h(a),h(b))\). +Choose one label \(L\) and one unfolded vertex +\(\alpha\in\widetilde Y\) attaining \(h(u)\) for +\(E_L(ux)\). +For this same family member, its starting test +height on each side is smaller than \(h(u)\). +If one starting set of tests vanished, persistence +of vanishing under positive words would contradict +the final nonzero test. Otherwise +Lemma~\ref{lay:top-divisor} applies to each side. +It makes \(\alpha\) a left divisor of both unfolded +positive words. + +Lemma~\ref{two:greedy} reflects this divisibility +to a common original color. In its right-divisor +form this lemma is transferred to left divisors +by word reversal: every Artin braid relation is +preserved by reversal, and a color unfolds to a +product of commuting letters whose reversal +represents the same element. +Thus the original positive elements \(p,q\) have +a common left atom, a contradiction. +This proves the stated inequality. +Repeated pairwise application constructs a common +extension of any finite family, with height no +larger than the maximum of its starting heights. +\end{proof} + +Positive support is well defined on \(A_Y^+\): +the positive monoid embeds, and every positive +Artin relation preserves the set of colors +occurring in a word. In particular, the support +of a positive divisor is contained in the support +of the element it divides. + +\begin{lemma}\label{sph:proper-support} +Assume \(h(1)>-\infty\), and put \(h_0=h(1)\). +There exist \(p\in A_Y^+\) with \(h(p)\leq h_0\) +and one proper subset \(J\subsetneq Y\) such that +\[ + b\in A_Y^+,\quad h(bp)\leq h_0 + \quad\Longrightarrow\quad + \operatorname{supp}(b)\subseteq J. +\] +\end{lemma} + +\begin{proof} +If no such \(p\) existed, then for every positive +\(p\) with \(h(p)\leq h_0\) the union of supports +of all its permitted continuations would equal +\(Y\). +For each color \(i\in Y\), choose a continuation +\(b_i\) whose support contains \(i\) and for which +\(h(b_i p)\leq h_0\). +Lemma~\ref{sph:common-extension} gives a common +positive left extension of these endpoints: +\[ + u=c_i b_i p\quad(i\in Y),\qquad + c_i\in A_Y^+,\qquad h(u)\leq h_0. +\] +Cancelling \(p\) shows that \(w=up^{-1}=c_i b_i\) +is a positive element independent of \(i\). +Every \(b_i\) is a right divisor of \(w\), so +\(\operatorname{supp}(w)=Y\). +This constructs a new positive endpoint \(wp\) +with height at most \(h_0\). +At every intermediate letter of this operation +the height also stays at most \(h_0\), by positive +monotonicity. + +Starting from \(p=1\) and repeating gives an +infinite sequence forbidden by +Proposition~\ref{sph:no-chain}. +Therefore some \(p\) has a proper union of the +supports of all its permitted continuations. +Take \(J\) to be precisely that union. +This establishes one common proper subset, +not merely a separately missing color for +each continuation. +\end{proof} + +\begin{proposition}\label{sph:containment} +Let \(Y\subseteq S\) be spherical and \(x\in A\). +For the family stabilizer +\[ + H(Y,x)=\{s\in A_Y:E_L(sx)\simeq E_L(x) + \text{ for every }L\}, +\] +one of the following holds: +\begin{enumerate} + \item all tests on \(\widetilde Y\) at \(x\) + are acyclic and \(H(Y,x)=A_Y\); + \item there exist \(p\in A_Y^+\) and + \(J\subsetneq Y\) such that + \begin{equation}\label{sph:eq:containment} + H(Y,x)\subseteq p^{-1}A_Jp . + \end{equation} +\end{enumerate} +\end{proposition} + +\begin{proof} +Suppose first that some test is nonzero. +Use \(p,J\) from Lemma~\ref{sph:proper-support}. +For \(s\in H(Y,x)\), the equivalences of family +members are preserved by every prefix of the +action. In particular +\[ + h(ps)=h(p),\qquad h(bps)=h(bp) + \quad(b\in A_Y^+). +\] +Apply Lemma~\ref{sph:common-extension} to +\(p,ps\), obtaining +\[ + ap=bps,\qquad a,b\in A_Y^+,\qquad + h(ap)=h(bps)\leq h(p)\leq h_0. +\] +Thus \(a\) is a permitted continuation of \(p\). +The equality \(h(bp)=h(bps)\) shows that \(b\) +is also permitted. Both have support in \(J\), +so +\[ + s=p^{-1}b^{-1}ap\in p^{-1}A_Jp. +\] +This proves \eqref{sph:eq:containment}. + +If all the tests are acyclic, the evaluation +term for each positive color twist is a bounded +acyclic complex of graded vector spaces tensored +with projectives. It is contractible, since a +bounded acyclic vector-space complex splits +degreewise, in each internal degree. +The evaluation cone is consequently equivalent +to the original family member. +Every positive generator fixes the family, and +its inverse fixes it as well by applying the +inverse equivalence. +The generators therefore give +\(H(Y,x)=A_Y\). +This argument includes \(Y=\varnothing\), when +there are no tests and the group is trivial. +\end{proof} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/08_stabilizers.tex b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/08_stabilizers.tex new file mode 100644 index 0000000000000000000000000000000000000000..8520f2dba3fc46d5d0711d0dd2bdc0a430225683 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/build/sections/08_stabilizers.tex @@ -0,0 +1,311 @@ +\section{Stabilizers and parabolic intersections} +\label{sec:stabilizers} + +We now combine the two containment arguments. Throughout this section, +$X\subseteq S$ and its relative family are fixed. The element $x\in A$ +is arbitrary; in particular, it need not belong to the subgroup that +is currently acting. + +\begin{lemma}[Conjugating a stabilizer] +\label{stab:conjugation} +Let $D\subseteq Y\subseteq S$ and $b\in A_Y$. If +$H(Y,x)\subseteq b^{-1}A_Db$, then +\[ + bH(Y,x)b^{-1}=H(D,bx). +\] +\end{lemma} +\begin{proof} +For $c\in A_D$, invertibility of $B_b$ gives +\[ + E_L(cbx)\simeq E_L(bx)\quad\text{for all }L + \quad\Longleftrightarrow\quad + E_L(b^{-1}cbx)\simeq E_L(x)\quad\text{for all }L. +\] +Since $b^{-1}cb\in A_Y$, the right side says that +$b^{-1}cb\in H(Y,x)$. Thus +$H(D,bx)=A_D\cap bH(Y,x)b^{-1}$. The assumed containment removes the +intersection. +\end{proof} + +\begin{theorem}[Relative stabilizers are parabolic] +\label{stab:parabolic} +For every $Y\subseteq S$ and $x\in A$, there exist $R\subseteq Y$ and +$c\in A_Y$ such that +\[ + H(Y,x)=c^{-1}A_Rc. +\] +\end{theorem} +\begin{proof} +We induct on $|Y|$, allowing every $x\in A$ at every rank. The assertion +is immediate for $Y=\varnothing$. By Lemma~\ref{stab:conjugation}, it +suffices to place $H(Y,x)$ inside $b^{-1}A_Db$ for some $b\in A_Y$ and +proper $D\subsetneq Y$: the conjugated stabilizer is then $H(D,bx)$, +to which induction applies. + +Suppose first that $Y$ is spherical. If all its tests on the family at +$x$ are acyclic, every generator in $Y$ fixes every member, so +$H(Y,x)=A_Y$. Otherwise Proposition~\ref{sph:containment} gives the +required proper-type containment. + +Now suppose that $Y$ is nonspherical. Choose a minimum vertex +$P=(T,A_Tz)$ of the relative harmonic filtration of $A_Yx$. Let $K$ be +its invisible set and put $T_+=T\setminus K$, as in +Section~\ref{sec:heights}. Proposition~\ref{ht:containment} gives +\[ + x^{-1}H(Y,x)x\subseteq z^{-1}A_Dz, + \qquad D=T_+\cup K\subseteq Y. +\] +Since $z\in A_Yx$, the element $b=zx^{-1}$ belongs to $A_Y$, and this +is exactly $H(Y,x)\subseteq b^{-1}A_Db$. If $D\ne Y$, the induction +step is complete. + +It remains to treat $D=Y$. The sets $T_+$ and $K$ are disjoint and +nonadjacent, and $T_+$ is spherical because it is a subset of $T$. +We claim that the entire $K$-factor fixes the family at $x$. + +Let $j\notin Y$ be an exterior neighbor of $K$. At every recorded level, +the definition of $K$ makes its harmonic coordinate zero. Outside $Y$ +these coordinates are actual generator counts, fixed throughout $A_Yx$. +The cutoff was chosen at or below \emph{every} occupied exterior level, +so there are no generators of color $j$ at any level, for any member +of the family at any $w\in A_Yx$. There is also no +edge from $K$ to $T_+=Y\setminus K$. If a member $E_L(w)$ had a generator +in $K$, its connected color support, which contains $f\notin Y$, would +have to cross an exterior neighbor of $K$. We have just excluded every +such generator. This applies in particular at $w=x$: the family there +has no generators in $K$ or any of its neighbors, even at levels below +the cutoff. + +Every test at a color of $K$ is therefore zero termwise. The evaluation +cone shows that its positive generator fixes each member, and its inverse +does so as well. The Artin presentation and standard embedding give +\[ + A_Y=A_K\times A_{T_+}, + \qquad H(Y,x)=A_K\times H(T_+,x). +\] +Since $Y$ is nonspherical, the spherical subset $T_+$ is proper. +Induction supplies $R\subseteq T_+$ and $c\in A_{T_+}$ with +$H(T_+,x)=c^{-1}A_Rc$. Commutation with $A_K$ now gives +\[ + H(Y,x)=c^{-1}(A_K\times A_R)c=c^{-1}A_{K\cup R}c. +\] +This also covers $T_+=\varnothing$. +\end{proof} + +\begin{proposition}[The initial stabilizer] +\label{stab:initial} +The relative family associated with $X$ satisfies $H(S,1)=A_X$. +\end{proposition} +\begin{proof} +Every color of $X$ is nonadjacent to $f$. Its evaluation on every +$P_{f,L}$ is zero, so its twist fixes that projective. This proves +$A_X\subseteq H(S,1)$. + +Consider the filtration for $Y=S$ and $x=1$. The empty-type vertex +$P_0=(\varnothing,\{1\})$ has only one nonzero layer vector, namely +the unit vector at $f$ in level zero. This vertex has minimum height. +Indeed a competitor with any nonzero higher block is later. At level +zero, every competitor retains the exterior coordinate $p_f^0=1$, and +all coordinates are nonnegative. Its coordinate sum is therefore at +least one. Equality forces its entire vector to be the unit vector at +$f$, and then every closed-type sign coordinate is zero, since the +type lies in $S$. Every lower block of $P_0$ is zero and is least. + +At this minimum $T_+=\varnothing$. A color of $S$ is invisible precisely +when it is nonadjacent to $f$, so $K=X$. Applying +Proposition~\ref{ht:containment} with $z=x=1$ gives +$H(S,1)\subseteq A_X$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Simultaneous conjugation reduces the assertion to the intersection +$A_X\cap x^{-1}A_Yx$, with $x\in A$. For $a\in A_Y$, invertibility +of $B_x$ and Proposition~\ref{stab:initial} give +\begin{align*} + x^{-1}ax\in A_X + &\ \Longleftrightarrow\ + E_L(x^{-1}ax)\simeq E_L(1)\quad\text{for every }L\\ + &\ \Longleftrightarrow\ + E_L(ax)\simeq E_L(x)\quad\text{for every }L\\ + &\ \Longleftrightarrow\ a\in H(Y,x). +\end{align*} +Consequently +\[ + A_X\cap x^{-1}A_Yx=x^{-1}H(Y,x)x. +\] +By Theorem~\ref{stab:parabolic}, write $H(Y,x)=c^{-1}A_Zc$ with +$Z\subseteq Y$ and $c\in A_Y$. The displayed intersection is then +$(cx)^{-1}A_Z(cx)$, hence is parabolic. Conjugating back proves the +assertion for the originally specified $g$ and $h$. +\end{proof} + + +\subsection{Parabolic closure and dynamics} +\label{sec:consequences} + +The inside-parabolic description obtained above makes the rank decrease +in an intersection explicit. We use it to prove the introductory +consequences, then record the fixed-point and continuity implications. + +\begin{proof}[Proof of Corollary~\ref{cor:arbitrary-intersections}] +If $Q=gA_Yg^{-1}$ and $P=hA_Xh^{-1}$, put $x=g^{-1}h$. The proof of +Theorem~\ref{thm:main} and Theorem~\ref{stab:parabolic} give +\[ + Q\cap P=g c^{-1}A_Rc g^{-1} + \quad\text{for some }R\subseteq Y\text{ and }c\in A_Y. +\] +If this intersection is strictly smaller than $Q$, then $R\ne Y$, since +conjugation by $c\in A_Y$ preserves $A_Y$. Thus a strict intersection step +decreases the size of the displayed defining set. + +Start with $Q=A=A_S$ and no selected members of $\mathcal F$. Whenever $Q$ +is not contained in some $P\in\mathcal F$, replace $Q$ by $Q\cap P$ and +select that member. The preceding strict decrease permits at most $|S|$ +steps. The resulting $Q$ is contained in every member of $\mathcal F$ and +is the intersection of the selected subfamily, proving the displayed +equality and parabolicity. Finally, the family of parabolics containing +$E$ is nonempty because it contains $A$; its intersection is the unique +smallest such subgroup. +\end{proof} + +\begin{proof}[Proof of Corollary~\ref{cor:nonclique-dynamics}] +In the presentation graph in which every finite label is an edge, the +infinite-label pair gives the visual splitting over +$A_{S\setminus\{s,t\}}$ of +\cite[Definition~2.4]{CharneyMartinMorrisWright2026}. +Theorem~\ref{thm:main} makes intersections of two conjugates of this edge +group parabolic in $A$. Hence +\cite[Theorem~3.2]{CharneyMartinMorrisWright2026} gives acylindrical +hyperbolicity and weak malnormality of the edge group, and +\cite[Proposition~4.2]{CharneyMartinMorrisWright2026} gives weak malnormality +of every proper parabolic. + +The finite intersection supplied by weak malnormality is parabolic by +Theorem~\ref{thm:main}. A nontrivial parabolic contains a conjugate of +some $\sigma_s$, which has infinite order: the exponent-sum homomorphism +$A\to\mathbb Z$ sends every standard generator to $1$. Thus a finite +parabolic is trivial, giving the stated witness. +\end{proof} + +\subsection{Fixed-point subgroups and locally compact groups} +\label{sec:fixed-point-consequences} + +A parabolic subgroup is \emph{complete} if it is conjugate to $A_T$ +for a set $T$ with no infinite label; it need not be spherical. +A group has property $\mathrm{FA}'$ if, in every simplicial action on a +tree without inversions, each element fixes a vertex. A locally compact +Hausdorff group $L$ is \emph{almost connected} if $L/L^\circ$ is compact. +A discrete group is \emph{lcH-slender} if every abstract homomorphism +into it from a locally compact Hausdorff group is continuous. + +\begin{corollary}[Reduction to complete parabolics] +\label{cor:complete-parabolics} +Let $A=A_M$ with $S$ finite. +\begin{enumerate} +\item Every subgroup of $A$ with property $\mathrm{FA}'$ lies in a +complete parabolic subgroup. +\item Every abstract homomorphism $L\to A$ from an almost-connected +locally compact Hausdorff group has image in a complete parabolic. +\item If every complete standard parabolic of $A$ is lcH-slender, +then $A$ is lcH-slender. +\end{enumerate} +\end{corollary} +\begin{proof} +Theorem~\ref{thm:main} places $A$ in the class of groups whose parabolics +are closed under pairwise intersections. Apply +\cite[Proposition~1.3 and Theorem~1.7]{MoellerParisVarghese2024}. +\end{proof} + + +\subsection{Faithfulness and effective membership} +\label{sec:algorithms} + +The unit-label projective already detects membership in $A_X$. Indeed, +label compatibility in Proposition~\ref{cat:family} and the initial +stabilizer calculation give +\begin{equation} + B_wP_{f,\mathbf1}\simeq P_{f,\mathbf1} + \quad\Longleftrightarrow\quad w\in A_X. + \label{stab:single-detector} +\end{equation} +For the forward implication, tensoring labels by each simple $L$ gives +$B_wP_{f,L}\simeq P_{f,L}$, so Proposition~\ref{stab:initial} applies. +The reverse implication follows from the same proposition. +In particular, for $X=\varnothing$ the constructed action is faithful: +an element acting by the identity functor fixes $P_{f,\mathbf1}$ and +is therefore the identity of $A$. This is the action for our framed, +per-edge-labelled algebra; the faithfulness question for the original +unframed construction of \cite[Conjecture~7.10]{HengLicata2024} concerns +a different category. + +To make \eqref{stab:single-detector} an algorithm, we must compute in the +chosen algebra exactly. The related twist-group algorithm of +\cite[Proposition~7.11]{HengLicata2024} also uses Gaussian elimination. +We give the coefficient-field construction here, so finite-dimensionality +over $\mathbb C$ is not being used as an effectiveness assumption. + +\begin{proof}[Proof of Corollary~\ref{cor:membership}] +Fix the supplied $M$ and $X$, and use their relative construction. +Let $N$ be the least common multiple of the integers $2m$ for finite +edge labels $m\ge4$, and take $K=\mathbb Q(\exp(2\pi i/N))$; take +$K=\mathbb Q$ if there is no such label. Then $K$ contains every +parameter $q_m=\exp(\pi i/m)$. This field and its exact arithmetic are +computable from the finite matrix $M$. + +We realize each Temperley--Lieb--Jones factor in its planar model. +Loop-free pairing diagrams give finite bases for each required underlying +planar morphism space; stacking diagrams evaluates every closed loop as $q_m+q_m^{-1}$. +The Jones--Wenzl recursion computes the simple idempotents +$p_0,\ldots,p_{m-2}$ over $K$, with nonzero denominators in this range +\cite[Definition~2.1.1, Remark~2.1.3, and Theorem~2.3.2]{Chen2014}. +Negligible maps are kernels of finite trace-pairing matrices, so their +quotient spaces and basis reductions are obtained by exact linear +algebra. For idempotents $p,q$, their corner morphism space is the image +of $f\mapsto qfp$ in that quotient +\cite[Definition~2.4.1, Lemma~2.4.3, and Definition~5.2.1]{Chen2014}. +Composition, tensoring and pivotal evaluation are consequently +computable in these finite diagram spaces. These finite kernels, +quotients and corner images commute with scalar extension to +$\mathbb C$, recovering the model used in Section~\ref{sec:category}. + +The known simple list also makes decomposition effective. For any +object $U$ needed in a product, compute bases of +$\Hom(p_a,U)$ and $\Hom(U,p_a)$. Their composition pairing takes values +in $\End(p_a)=K$ and is nondegenerate; invert its matrix to obtain dual +inclusions and projections. The sum of the resulting simple-block +projections is $\mathrm{id}_U$, by the complete semisimple decomposition +of \cite[Theorems~5.4.4--5.4.5]{Chen2014}. These identities hold over +$K$: they hold after extension to $\mathbb C$, and scalar extension is +faithful. Thus no search for an unspecified categorical decomposition +is required. Finite products of the factors and direct sums are handled +by tensor products and matrices of these maps. + +It follows that every space +$\Hom(L,D^a_{ij}\otimes H)$ and every multiplication map defining $Z$ +has computable structure constants over $K$. Let $Z_K$ denote this +finite algebra. Its trace weights belong to $K$, and matrix inversion +computes trace-dual bases, hence the coevaluation maps as well as the +evaluation maps. All terms and differentials of $B_i,D_i$ are now +explicit over $K$. Unnormalized planar bases suffice; the square-root +normalization used only in the proof of Lemma~\ref{sph:path-bound} +is not needed for this construction. + +Given the input words $u,v$, form the word $w=u^{-1}vu$ and compute the +bounded projective complex $B_wP_{f,\mathbf1}$ by finite tensor products. +Cancel invertible scalar differential entries between equal shifted +projectives until none remain, as in Section~\ref{sec:layers}. Each +cancellation uses exact arithmetic in $K$ and removes two summands, +so the process terminates. + +The positive-path-length ideal of $Z_K$ is nilpotent and its quotient is +a product of copies of $K$. It is therefore the radical, and it remains +the radical after extension to $\mathbb C$. The computed minimal complex +stays minimal under that extension. By uniqueness of minimal models, +it is equivalent over $\mathbb C$ to $P_{f,\mathbf1}$ precisely when its +only generator is that projective in placements $(0,0)$. This is a finite +check of its summands; when it succeeds there is no differential left. +Equation~\eqref{stab:single-detector} identifies success with $w\in A_X$, +equivalently $v\in uA_Xu^{-1}$. Taking $X=\varnothing$ decides whether a +word is the identity, and applying this test to the quotient of two words +decides their equality. +\end{proof} diff --git a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..39aaf7393d89aee5a8dd8902a4d184bd85cc7a32 --- /dev/null +++ b/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4cd7fa316371447c2cb8a99d36bf9f48934c2366550609917210574e3f4f7e2e +size 655595 diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/README.md b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..47bc7131a5a5173f273d11c3e7994d65b0aae4a4 --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/README.md @@ -0,0 +1,18 @@ +# [Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence](paper.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026, + author = {{OpenAI}}, + title = {{Parity lifts and bounded-treewidth witnesses for Weisfeiler--Leman equivalence}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf}{OAI:Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/figures/helper-incidence.tex b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/figures/helper-incidence.tex new file mode 100644 index 0000000000000000000000000000000000000000..2aba0a5fc8d37abf0a8557be85b379b39e7d04ba --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/figures/helper-incidence.tex @@ -0,0 +1,14 @@ +\begin{tikzpicture}[ + vertex/.style={draw,rounded corners,fill=blue!4,inner sep=5pt}, + every node/.style={font=\small}, + test/.style={font=\scriptsize,fill=white,inner sep=2pt} +] +\node[vertex] (left) at (-4.2,0) {\((i,d)\)}; +\node[vertex] (middle) at (0,0) {\((j,e)\)}; +\node[vertex] (right) at (4.2,0) {\((\ell,f)\)}; +\node[vertex] (helper) at (0,2.8) {\((h_{ij;\ell},q)\)}; +\draw (helper) -- node[test,sloped,above] {\(q=\lambda_{ij}^i(d)\)} (left); +\draw (helper) -- node[test,pos=.67,right=3pt] {\(q=\lambda_{ij}^j(e)\)} (middle); +\draw (helper) -- node[test,sloped,above] {unrestricted} (right); +\draw (left) -- node[test,below=4pt] {\(\lambda_{ij}^i(d)=\lambda_{ij}^j(e)\)} (middle); +\end{tikzpicture} diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/main.tex b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..5f78a2cc327437581c8f257506b3e46ec0da3d14 --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/main.tex @@ -0,0 +1,54 @@ +\PassOptionsToPackage{hyphens}{url} +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{booktabs} +\usepackage{enumitem} +\usepackage{float} +\usepackage[numbers,square,sort&compress]{natbib} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,fit,calc,backgrounds} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\hypersetup{pdftitle={Parity lifts and bounded-treewidth witnesses for Weisfeiler--Leman equivalence},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\newcommand{\FF}{\mathbb F_2} +\newcommand{\NN}{\mathbb N} +\newcommand{\RR}{\mathbb R} +\newcommand{\ind}[1]{\mathbf 1_{\{#1\}}} +\title{Parity lifts and bounded-treewidth witnesses for Weisfeiler--Leman equivalence} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +For \(k\ge4\), we construct two uncolored graphs that are +\(k\)-dimensional Weisfeiler--Leman equivalent exactly when a prescribed +finite-domain choice system has no compatible choice. The system has +\(k+1\) domains for joint refinement and \(k\) for separate-coordinate +refinement. A successful choice is detected after two joint rounds or +one separate round. +Applied to sparse satisfiability, the reduction gives fixed-dimension +\(n^{\Omega(k)}\) time exclusions under positive-rate ETH, even for +deciding equality of these early histograms. +\end{abstract} + +\input{sections/01-introduction} +\input{sections/02-wl-tests} +\input{sections/03-parity-lifts} +\input{sections/04-separator} +\input{sections/05-complexity} + +\bibliographystyle{plainnat} +\bibliography{references} +\end{document} diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/references.bib b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..7912fb8499feed40aed6dcbd1fc8e468b80c7a74 --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/references.bib @@ -0,0 +1,184 @@ +@article{WL1968, + author = {Weisfeiler, Boris and Leman, Andrei A.}, + title = {The Reduction of a Graph to Canonical Form and the Algebra Which Appears Therein}, + journal = {Nauchno-Technicheskaya Informatsiya, Series 2, no. 9}, + pages = {12--16}, + year = {1968}, + note = {In Russian. English translation by Grigory Ryabov}, + url = {https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf} +} + +@article{CFI1992, + author = {Cai, Jin-yi and F{\"u}rer, Martin and Immerman, Neil}, + title = {An Optimal Lower Bound on the Number of Variables for Graph Identification}, + journal = {Combinatorica}, + volume = {12}, + number = {4}, + pages = {389--410}, + year = {1992}, + doi = {10.1007/BF01305232}, + url = {https://people.cs.umass.edu/~immerman/pub/opt.pdf} +} + +@article{AMRSSV2019, + author = {Atserias, Albert and Man{\v c}inska, Laura and Roberson, David E. and {\v S}{\'a}mal, Robert and Severini, Simone and Varvitsiotis, Antonios}, + title = {Quantum and Non-Signalling Graph Isomorphisms}, + journal = {Journal of Combinatorial Theory, Series B}, + volume = {136}, + pages = {289--328}, + year = {2019}, + doi = {10.1016/j.jctb.2018.11.002}, + url = {https://www.cs.upc.edu/~atserias/papers/quantumisomorphism/quantumisomorphisms.pdf}, + note = {Author-hosted manuscript dated October 25, 2018} +} + +@article{Grohe1999, + author = {Grohe, Martin}, + title = {Equivalence in Finite-Variable Logics Is Complete for Polynomial Time}, + journal = {Combinatorica}, + volume = {19}, + number = {4}, + pages = {507--532}, + year = {1999}, + doi = {10.1007/s004939970004}, + url = {https://link.springer.com/article/10.1007/s004939970004} +} + +@article{IP2001, + author = {Impagliazzo, Russell and Paturi, Ramamohan}, + title = {On the Complexity of {$k$}-{SAT}}, + journal = {Journal of Computer and System Sciences}, + volume = {62}, + number = {2}, + pages = {367--375}, + year = {2001}, + doi = {10.1006/jcss.2000.1727}, + url = {https://cseweb.ucsd.edu/~paturi/myPapers/pubs/ImpagliazzoPaturi_2001_jcss.pdf} +} + +@article{IPZ2001, + author = {Impagliazzo, Russell and Paturi, Ramamohan and Zane, Francis}, + title = {Which Problems Have Strongly Exponential Complexity?}, + journal = {Journal of Computer and System Sciences}, + volume = {63}, + number = {4}, + pages = {512--530}, + year = {2001}, + doi = {10.1006/jcss.2001.1774}, + url = {https://cseweb.ucsd.edu/~paturi/myPapers/pubs/ImpagliazzoPaturiZane_2001_jcss.pdf} +} + +@article{Dvorak2010, + author = {Dvo{\v r}{\'a}k, Zden{\v e}k}, + title = {On Recognizing Graphs by Numbers of Homomorphisms}, + journal = {Journal of Graph Theory}, + volume = {64}, + number = {4}, + pages = {330--342}, + year = {2010}, + doi = {10.1002/jgt.20461}, + url = {https://onlinelibrary.wiley.com/doi/10.1002/jgt.20461} +} + +@phdthesis{Berkholz2014, + author = {Berkholz, Christoph}, + title = {Lower Bounds for Heuristic Algorithms}, + school = {RWTH Aachen University}, + year = {2014}, + note = {Electronic publication 2015}, + url = {https://publications.rwth-aachen.de/record/462250/files/462250.pdf} +} + +@inproceedings{DGR2018, + author = {Dell, Holger and Grohe, Martin and Rattan, Gaurav}, + title = {{Lov{\'a}sz Meets Weisfeiler and Leman}}, + booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {107}, + pages = {40:1--40:14}, + publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, + year = {2018}, + doi = {10.4230/LIPIcs.ICALP.2018.40}, + url = {https://arxiv.org/abs/1802.08876v2}, + note = {Full version with Appendix B: arXiv:1802.08876v2} +} + +@article{BN2023, + author = {Berkholz, Christoph and Nordstr{\"o}m, Jakob}, + title = {Near-Optimal Lower Bounds on Quantifier Depth and {Weisfeiler--Leman} Refinement Steps}, + journal = {Journal of the ACM}, + volume = {70}, + number = {5}, + pages = {32:1--32:32}, + year = {2023}, + doi = {10.1145/3195257}, + url = {https://jakobnordstrom.se/docs/publications/QuantifierDepthWeisfeilerLeman_JACM.pdf} +} + +@article{GLNS2025, + author = {Grohe, Martin and Lichter, Moritz and Neuen, Daniel and Schweitzer, Pascal}, + title = {Compressing {CFI} Graphs and Lower Bounds for the {Weisfeiler--Leman} Refinements}, + journal = {Journal of the ACM}, + volume = {72}, + number = {3}, + pages = {21:1--21:27}, + year = {2025}, + doi = {10.1145/3727978}, + url = {https://arxiv.org/abs/2308.11970v2}, + note = {Conference version appeared at FOCS 2023} +} + +@inproceedings{LRS2025, + author = {Lichter, Moritz and Ra{\ss}mann, Simon and Schweitzer, Pascal}, + title = {Computational Complexity of the {Weisfeiler--Leman} Dimension}, + booktitle = {33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {326}, + pages = {13:1--13:22}, + publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, + year = {2025}, + doi = {10.4230/LIPIcs.CSL.2025.13}, + url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.13} +} + +@misc{DN2026, + author = {D{\"o}ring, Simon and Neuen, Daniel}, + title = {The Classical {Weisfeiler--Leman} Algorithm Stabilizes in {$O(n)$} Rounds}, + year = {2026}, + eprint = {2609.17364}, + archivePrefix = {arXiv}, + primaryClass = {cs.DS}, + note = {Preprint, version 1, 15 September 2026}, + url = {https://arxiv.org/abs/2609.17364v1} +} + +@article{Roberson2026, + author = {Roberson, David E.}, + title = {Oddomorphisms and Homomorphism Indistinguishability over Graphs of Bounded Degree}, + journal = {Journal of Combinatorial Theory, Series B}, + volume = {181}, + pages = {1--61}, + year = {2026}, + doi = {10.1016/j.jctb.2026.07.003}, + note = {Preprint: arXiv:2206.10321v1 (2022)}, + url = {https://arxiv.org/abs/2206.10321v1} +} + +@misc{CompanionUnconditional, + author = {{OpenAI}}, + title = {{Unconditional time lower bounds for Weisfeiler--Leman equivalence}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf}{OAI:Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026}}, + year = {2026} +} + +@misc{NS2026, + author = {Neuen, Daniel and Seppelt, Tim}, + title = {Distinguishing Graphs by Counting Homomorphisms from Sparse Graphs}, + year = {2026}, + eprint = {2601.18602}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + note = {Version 2, 10 September 2026}, + url = {https://arxiv.org/abs/2601.18602v2} +} diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/01-introduction.tex b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..b528ab7a91dad33d6ff302c15b24626056829c5e --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/01-introduction.tex @@ -0,0 +1,255 @@ +\section{Introduction}\label{sec:introduction} + +The Weisfeiler--Leman method compares graphs by repeatedly refining +colors of vertex tuples. Its dimension controls how much of a graph +each tuple can describe. A useful way to study this refinement is to +count graph homomorphisms from graphs of bounded treewidth: those +counts connect local information in a tree decomposition to the global +color distribution of the target graph. + +We give an explicit reduction from compatible choices in finite domains +to Weisfeiler--Leman equivalence. Its main ingredients are two parity +lifts of one base graph and a procedure that extracts compatible choices +from any distinguishing homomorphism test with sufficiently small bags. +All output graphs are finite, simple, undirected and uncolored. +The construction is independent of the computational applications in +the last section. + +\subsection{Refinement and simultaneous choices} + +Fix an integer \(k\ge4\). Initially, an ordered tuple +\(\mathbf x\in V(X)^k\) has its \emph{atomic type} as color: the +equalities and adjacencies between all positions. Write +\(\mathbf x[i\leftarrow y]\) for replacement of coordinate \(i\) by \(y\). +In \emph{joint} refinement, also called the correlated replacement-vector +convention, a round records the old color \(C_r(\mathbf x)\) and +\[ + \{\!\{\,\bigl(C_r(\mathbf x[1\leftarrow y]),\ldots, + C_r(\mathbf x[k\leftarrow y])\bigr): + y\in V(X)\,\}\!\}. +\] +In \emph{separate} refinement it records the old color and the \(k\) +multisets +\(\{\!\{C_r(\mathbf x[i\leftarrow y]):y\in V(X)\}\!\}\), one for +each coordinate \(i\). Colors are these data, with common names across +graphs. Two graphs are \(k\)-WL equivalent when their tuple-color +histograms agree at every round. Because each round retains the old +color, it suffices to compare until the common color partition on the +two tuple sets stabilizes. +We also consider refinement of all tuples in a disjoint union +\(X_0\sqcup X_1\), comparing the histograms on its two pure tuple sets +\(V(X_0)^k\) and \(V(X_1)^k\). + +The parameter used throughout the construction is +\begin{equation}\label{eq:bag-parameter} + t=\begin{cases} + k+1,&\text{joint refinement},\\ + k,&\text{separate refinement}. + \end{cases} +\end{equation} + +\begin{definition}[Simultaneous choices]\label{def:choice} +A \emph{choice system} consists of finite domains \(D_1,\ldots,D_t\) +and, for every \(i0\) rules out a +deterministic \(3\)-SAT algorithm with running time +\(2^{\delta N}\operatorname{poly}(L)\), where \(N\) is the number of +variables and \(L\) the formula encoding length. This is the hypothesis +for the following fixed-dimension application. + +\begin{theorem}[Fixed-dimension ETH consequence]\label{thm:eth} +Assume positive exponential-rate ETH. There are constants \(c>0\) +and \(K\ge4\) such that, for every fixed integer \(k\ge K\), no +deterministic algorithm decides \(k\)-WL equivalence of two +\(n\)-vertex graphs in time \(O(n^{ck})\). +This holds for both joint and separate refinement. The inputs are +explicit adjacency matrices of uncolored simple undirected graphs, +and time is measured on a multitape Turing machine. +\end{theorem} + +The hypothetical algorithm and its multiplicative constant may depend +arbitrarily on the fixed \(k\); the constants \(c,K\) do not. +Theorem~\ref{thm:eth} therefore excludes even an ineffectively selected +family of solvers whose running-time exponents are \(o(k)\). +Under the weaker premise excluding one +uniform \(2^{o(N)}\operatorname{poly}(L)\)-time \(3\)-SAT algorithm, we +instead prove the uniform \(f(k)n^{o(k)}\) exclusion in +Proposition~\ref{prop:uniform}. Its algorithm receives \(k\) as input. +These hypotheses are kept separate throughout the parameter analysis. + +Bounds on the number of refinement rounds concern a different question. +For each fixed \(k\ge2\), Grohe, Lichter, Neuen and Schweitzer +\citep[preprint version, Theorem~1 and Corollary~34]{GLNS2025} +construct colored graph pairs requiring \(\Omega(n^{k/2})\) rounds of +joint refinement. Such a bound constrains the refinement process itself; +it does not rule out a faster algorithm deciding its stable equivalence +relation. +The time exclusions proved here already apply to comparing tuple-color +histograms after two joint rounds or one separate round, under the same +respective hypotheses; Section~\ref{sec:complexity} gives the deduction. + +Lichter, Ra{\ss}mann and Schweitzer~\citep[Section~6]{LRS2025} +expressed the expectation that neither equivalence nor identification +admits \(n^{o(k)}\) time and suggested lower bounds based on ETH or +its strong form. +The application above concerns the equivalence question. +The companion article \emph{Unconditional time lower bounds for +Weisfeiler--Leman equivalence}~\citep[Theorem~1.1]{CompanionUnconditional} +proves a stronger time bound: for every sufficiently large fixed \(k\) +and every correct deterministic decider, an absolute exponent linear +in \(k\) is required at every sufficiently large graph order. +It also obtains connected graphs of diameter at most two. +Its unconditional theorem supersedes the conditional lower bound for +stable equivalence in Theorem~\ref{thm:eth}. +The contribution of this article is the explicit choice reduction +and its helper-based bounded-bag extraction, using the parity-lift and +noncancellation framework described above. No part of its proof uses +that companion theorem. + +For the ETH application, the sparsification lemma of Impagliazzo, +Paturi and Zane~\citep{IPZ2001} reduces a formula to sparse disjuncts. +Partitioning their clauses into \(t\) groups gives domains of size +\(2^{O(N/t)}\): each domain consists of the assignments satisfying +one group, and pair labels are restrictions to common variables. +We retain the sparsification rate, its dependent density constant, +the explicit matrix-writing cost and the simulator's input cost in +the full calculation. Extensions to other machine models require an +absolute-power Turing simulation, as specified in +Section~\ref{sec:complexity}. diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/02-wl-tests.tex b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/02-wl-tests.tex new file mode 100644 index 0000000000000000000000000000000000000000..fec59c6a3ca5a1169bbec532bcd4c6aac2c12697 --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/02-wl-tests.tex @@ -0,0 +1,199 @@ +\section{WL refinement and homomorphism tests}\label{sec:wl-tests} + +We use the refinement conventions and parameter \(t\) from +Section~\ref{sec:introduction}; throughout \(k\ge4\). +We will use homomorphism counts in two ways to connect the construction to WL. +Equality of all counts from sources with small bags will prove +equivalence. A difference for one explicit template will prove +inequivalence, already after two joint rounds or one separate round. + +A graph has \emph{bag size at most \(t\)} if it has a tree decomposition +on a finite tree with bags of size at most \(t\): the bags containing any vertex form a +nonempty connected subtree, and some bag contains both ends of each +edge. Equivalently, its treewidth is at most \(t-1\). +Write \(\hom(F,X)\) for the number of homomorphisms from \(F\) to \(X\), +that is, vertex maps preserving adjacency. + +\begin{definition}\label{def:template} +The template \(T=T_t\) has main vertices \([t]\) forming a clique. +For each \(i\hom(T,X_\star)\), and hence \(X_0,X_\star\) +are not \(k\)-WL equivalent. +\end{lemma} + +\begin{proof} +Given a successful choice \((d_i)_{i\in[t]}\), map main vertex \(i\) +to \((i,d_i)\) and each \(h_{ij;\ell}\) to +\((h_{ij;\ell},\lambda_{ij}^i(d_i))\). +The common pair label and the unrestricted edge toward \(\ell\) +make this a homomorphism \(\phi:T\to G\) with +\(\tau\circ\phi\) the identity. Set every vertex and edge weight to +one. Each vertex of \(T\) has exactly one neighbor of each adjacent +type, and exactly one source vertex has type \(1\), so +\eqref{eq:dual} holds. Apply Lemma~\ref{lem:lift-duality} and then +Lemma~\ref{lem:test-detected}. +\end{proof} + + +A successful choice has now supplied a noncancelling, explicitly detected +witness. It remains to prove that a small-bag witness cannot arise without +such a choice. The next section extracts one directly from the dual +weights of Lemma~\ref{lem:lift-duality}. diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/04-separator.tex b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/04-separator.tex new file mode 100644 index 0000000000000000000000000000000000000000..ce5385fe7410a2f136e68fa838a7c4ee7eb52fd1 --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/build/sections/04-separator.tex @@ -0,0 +1,208 @@ +\section{Extracting a choice from a bounded-bag witness} +\label{sec:separator} + +The dual equations force odd total weight at every type. +We now use small separators to locate one bag whose main vertices +give compatible choices. +The odd-side orientation below is closely related to the separator +argument for oddomorphisms of Neuen and Seppelt +\citep[Lemma~4.4 and proof of Theorem~4.1]{NS2026}. +Here the helper types additionally force compatibility of the +representatives; we retain the local orientation to control the +weighted contribution of every outer side. + +\begin{lemma}[Choice extraction]\label{lem:extract-choice} +Let \(F\) have a tree decomposition with bags of size at most \(t\). +Suppose that a homomorphism \(\phi:F\to G\) and weights +\(s:V(F)\to\FF\) and \(r:E(F)\to\FF\) satisfy +\eqref{eq:dual}, where \(a=\tau\circ\phi\). +Then the choice system has a successful choice. +\end{lemma} + +\begin{proof} +All sums in this proof are in \(\FF\). +We use the type masses \(m_u(Y)\) from \eqref{eq:type-mass}; +in particular, \(m_u(V(F))=1\) for every type \(u\). + +\smallskip +\noindent\emph{Orienting the decomposition.} +Fix a decomposition tree \(\mathcal D\), and contract edges joining +identical bags. Every remaining edge has an intersection \(S\) of +size at most \(t-1\): an intersection of size \(t\) would force both +incident bags to be that same set. +Deleting the tree edge partitions \(V(F)\) into \(S\) and two sides +\(A^0,A^1\). Indeed, the occurrence subtree of a vertex outside \(S\) +lies in just one of the two tree components. No graph edge joins the +two sides, since its endpoints occur together in a bag. + +At least one main type is absent from \(S\). +If two main types \(u,v\) are absent, all \(uv\)-edges incident with +either type on a side stay on that side. Summing \eqref{eq:dual} +over their two endpoint classes gives +\(m_u(A^\nu)=m_v(A^\nu)\) for \(\nu\in\{0,1\}\). +For any absent main type \(u\), +\[ +m_u(A^0)+m_u(A^1)=m_u(V(F))=1. +\] +Orient the tree edge toward the side of mass \(1\). +This is well defined whether there is one absent main type or several. + +Choose a sink of the oriented tree and denote its bag by \(B\). +For each incident edge let \(A\) be its outer side and \(S\subseteq B\) +its intersection. These outer sides partition \(V(F)\setminus B\). +To see this precisely, the occurrence subtree of a vertex outside +\(B\) avoids the sink node and lies in exactly one component of +\(\mathcal D\) minus that node. A vertex in \(B\) occurring in a branch +must belong to that branch's intersection \(S\), so is excluded from +its outer side. Thus vertices are counted once, regardless of how +many bags contain them. Moreover, every neighbor of \(A\) outside +\(A\) lies in \(S\), and the sink orientation gives +\[ +m_\ell(A)=0 +\qquad\text{for every main type \(\ell\) absent from \(S\).} +\] +If a main type were absent from \(B\), it would be absent from every +such \(S\) and have mass zero in the bag and in all outer sides, +contrary to its global mass \(1\). +Since \(|B|\le t\), the bag therefore consists of exactly one vertex +\(x_i\) of each main type \(i\), and no helper vertices. +No condition on the individual weights \(s(x_i)\) is needed. + +\smallskip +\noindent\emph{A global mismatch test.} +Write \(\phi(x_i)=(i,d_i)\), with \(d_i\in D_i\). +The bag \(B\) need not induce a clique, so compatibility of these +representatives does not follow from \(\phi\) alone. +Suppose, for a contradiction, that some pair \(i0\) there are a constant \(C_\epsilon\ge1\) and a +deterministic algorithm that expresses any \(3\)-CNF \(F\) on \(N\ge1\) +variables as a disjunction of at most \(2^{\epsilon N}\) \(3\)-CNFs on +the same variables, each with at most \(C_\epsilon N\) clauses. +The running time is \(2^{\epsilon N}\operatorname{poly}(|F|)\). +For positive rational \(\epsilon\), the procedure and a valid density +bound can be chosen effectively. +\end{lemma} + +The lemma is the imported sparsification theorem, not a consequence of +the parity construction. In its constructive set-system proof, the +universe consists of the \(2N\) literals: choosing rate \(\epsilon/2\) +there gives the displayed variable-based rate \(\epsilon\). The integer +threshold choice and fixed branching procedure provide the effective +rational-rate selection used below~\citep[pp.~520--523]{IPZ2001}. + +The stated input-length bound allows an initial polynomial-time +normalization: rename variables and remove tautologies and repeated +clauses. There are only \(O(N^3)\) distinct clauses of width at most three, +so the normalized encoding has length polynomial in \(N\). + +\begin{lemma}[Grouping sparse clauses] +\label{lem:sat-choice} +A \(3\)-CNF with \(m\) clauses yields a simultaneous-choice instance with +\(t\) domains, each domain and pair-label set having size at most +\[ +A=2^{3\lceil m/t\rceil}. +\] +A simultaneous choice exists if and only if the formula is satisfiable. +\end{lemma} +\begin{proof} +Partition the clauses into \(t\) groups of size at most \(\lceil m/t\rceil\). +Let \(V_i\) be the variables in group \(i\), and let \(D_i\) contain exactly +the assignments on \(V_i\) satisfying that group. Set +\[ +L_{ij}=\{0,1\}^{V_i\cap V_j}, +\qquad +\lambda_{ij}^i(d)=d|_{V_i\cap V_j}, +\qquad +\lambda_{ij}^j(d)=d|_{V_i\cap V_j}. +\] +Each scope has size at most \(3\lceil m/t\rceil\), proving the bounds. +Pairwise agreement assigns every shared variable one value, so the +choices join to an assignment satisfying all clauses; unused variables +are free. Conversely, restrict a satisfying assignment to each scope. +An empty group has its unique empty assignment, an unsatisfiable group +has empty domain, and an empty overlap has one label, the empty +assignment. Thus these cases require no exception. +\end{proof} + +Apply Theorem~\ref{thm:choice-reduction} with \(t=k+1\) for the +joint convention, or \(t=k\) for separate coordinate multisets. +For a sparse formula with \(m\le CN\), recall that +\[ +M_t=\sum_{a\in V(T)}2^{\deg_T(a)-1}=2^{O(k^2)}. +\] +There are at most \(A\) vertices in each base fiber, and precisely +\(2^{\deg_T(a)-1}\) tags per vertex of type \(a\). The two output graphs +therefore have a common order satisfying +\begin{equation} +n\le M_t\,2^{3\lceil m/t\rceil} + \le b_k\,2^{3CN/t},\qquad b_k=8M_t. +\label{eq:output-size} +\end{equation} +These are explicit graphs. Enumerate the scope and overlap assignments, +retain the satisfying domain assignments, and enumerate the parity +tags. A scan of all vertex pairs then writes both adjacency matrices: +each entry is determined by restrictions of assignments and a comparison +of tag bits. The vertex records have length \(O(N+k^2)\), so a direct +multitape implementation takes at most +\begin{equation} +b'_k\,(N+|F|)^{d}\,2^{B_0 CN/t}, +\qquad B_0=6, +\label{eq:construction-time} +\end{equation} +for an absolute \(d\) and a constant \(b'_k\) depending only on \(k\). +Indeed, squaring the assignment bound gives +\(2^{6\lceil m/t\rceil}\le64\,2^{6CN/t}\). +A fixed-power tape simulation only changes the absolute constant \(B_0\). + +\begin{proof}[Proof of Theorem~\ref{thm:eth}] +Use positive-rate ETH \citep{IP2001}: for some \(\delta>0\), no +deterministic \(3\)-SAT algorithm runs in +\(2^{\delta N}\operatorname{poly}(|F|)\) time. +Choose \(\epsilon<\delta/4\), and fix \(C=C_\epsilon\) from +Lemma~\ref{lem:sparsification}. +For the Turing-machine model set \(q=1\). More generally, suppose the +chosen machine model has a simulation taking at most +\(A_k(T+\ell+1)^q\) steps, where \(q\ge1\) is fixed by the model, +independently of the simulated program and \(k\). +Here \(T\) is the program's running time and \(\ell\) its input length; +\(A_k\) may depend on the chosen fixed program and \(k\). Choose +\[ +c=\frac{\delta}{24Cq}, +\qquad +K\ge4,\qquad +K>\frac{4C\max\{B_0,6q\}}{\delta}. +\] + +Fix any \(k\ge K\) and suppose a solver takes \(O(n^{ck})\) time. +Hardcode this solver and \(k\) into a \(3\)-SAT algorithm. +Sparsify the input, form the graph pair for every sparse disjunct, and +accept exactly when at least one pair is inequivalent. +Lemmas~\ref{lem:sparsification} and~\ref{lem:sat-choice}, together with +Theorem~\ref{thm:choice-reduction}, prove correctness. + +The matrix input has length \(\ell=O(n^2)\), so the simulated solver +takes \(O_k(n^{qck}+n^{2q}+1)\) time. Both terms matter, even when +\(ck<2\). By \eqref{eq:output-size} and \eqref{eq:construction-time}, +the total time is bounded, up to polynomial input factors and +fixed-\(k\) constants, by +\[ +2^{\epsilon N} +\left( + 2^{B_0 CN/t} + +2^{6qCN/t} + +2^{3CqckN/t} +\right). +\] +Since \(t\ge k\), the first two inner rates are below \(\delta/4\), +and the third is at most \(\delta/8\). +The resulting rate is below \(\delta/2\), contradicting ETH. +The solver's description, its running-time constant, \(A_k\), \(b_k\), +and the eventual threshold in \(N\) may all depend on this fixed \(k\); +\(c\) and \(K\) do not. Thus the contradiction applies to every fixed +\(k\ge K\), without a uniform way of selecting the alleged solvers. +\end{proof} + +\begin{remark}[Small dimensions on matrix inputs] +On a comparison of two edgeless \(n\)-vertex graphs, a deterministic algorithm must +inspect every potential edge: changing an unseen edge in one graph +destroys equivalence, by degrees for ordinary \(1\)-WL and by atomic +tuple types for \(k\ge2\). This gives an \(\Omega(n^2)\) bit-query bound. +Decreasing \(c\) so that \(ck<2\) for the finitely many +\(2\le k0\), first fix \(j\) with \(1/j<\eta/20\). +Then fix \(k\) so that both terms in the maximum defining +\(\rho_{j,k}\) are below \(\eta/20\). +This is possible since \(t\ge k\) and \(g(k)/k\to0\). +Thus \(\rho_{j,k}<\eta/10\); the fifth-power bound is +\(2^{\eta N}\) for all sufficiently large \(N\), after absorbing its +fixed constants and polynomial factors. The same pair works at all +these lengths. The machine need not recognize a good pair or know a +convergence modulus for \(g(k)/k\). +For precision, normalize first to a canonical formula on its occurring +variables. If none occur, decide the resulting constant formula +directly, without invoking sparsification or the racing machine. +Otherwise, if their number is \(1\le N'\le N\), its length is polynomial +in \(N'\). For every desired rate the same fixed pair bounds all +sufficiently large \(N'\); the finitely many canonical formulas with +smaller \(N'\) have a common finite maximum running time. Thus the +normalized solver has a uniform \(2^{o(N)}\) bound even when the original +encoding names many unused variables. Parsing and normalization cost +one fixed polynomial in the original input length, so the complete +algorithm contradicts the assumed uniform subexponential lower bound. +\end{proof} + +The same model qualification applies: simulation replaces \(g(k)\) by +at most \(\max\{qg(k),2q\}+1=o(k)\) when \(q\) is fixed. +For this uniform transfer, the simulation must itself be uniform in \(k\). +An ineffective family of separately supplied fixed-\(k\) solvers would +not define \(U\); Proposition~\ref{prop:uniform} makes no assertion +about such a family under its weaker premise. + +Given two \(n\)-vertex graphs, consider deciding whether their +\(k\)-tuple color histograms agree after two joint rounds, or after +one separate round, starting from atomic types. This includes separate +runs with common color names and pure-tuple comparison in the +disjoint-union run. On the graph pairs produced by the reduction, +Theorem~\ref{thm:choice-reduction} gives equality at every round when +there is no successful choice, and inequality by the specified round +when a choice exists. The inequality persists because each update +retains the old color. Thus the reductions proving +Theorem~\ref{thm:eth} and Proposition~\ref{prop:uniform} give the same +deterministic time exclusions for these fixed-round decisions, with +their respective hypotheses and parameter ranges. The difficulty is +already present in comparing these early histograms; it does not require +many rounds of stabilization. diff --git a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f4688fc84fd3a17c1c2458d9b0b1daf17ddce7b9 --- /dev/null +++ b/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:742b4b63acfcfeec728427c388b0704c5e152f79cd9a8b5e9d16573fcb8177e7 +size 423196 diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/README.md b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..513a8ea523029c70f6d9158af7ce30fdc0e731bf --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [Path selection and an elliptic inequality on degenerating Ricci-flat trees](paper.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026, + author = {{OpenAI}}, + title = {{Path selection and an elliptic inequality on degenerating Ricci-flat trees}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf}{OAI:Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/figures/tree-scales.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/figures/tree-scales.tex new file mode 100644 index 0000000000000000000000000000000000000000..3ebdf70daaafe26b622b056c0a82e52cd1cdb750 --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/figures/tree-scales.tex @@ -0,0 +1,21 @@ +\begin{tikzpicture}[>=Stealth,scale=.93,every node/.style={font=\small}] + \draw[thick] (0,0)--(10,0); + \draw[thick] (0,1.5)--(10,1.5); + \fill[blue!8] (0,0)rectangle(5,1.5); + \fill[green!8] (5,0)rectangle(10,1.5); + \draw[thick] (0,0)rectangle(10,1.5); + \draw[dashed,thick] (5,-.3)--(5,1.8); + \node at (2.5,.78){Parent punctured half}; + \node at (7.5,.78){Child exterior half}; + \draw[->] (.8,-.5)--(4.5,-.5)node[midway,below]{$\tau_p=-\log r_p$}; + \draw[->] (9.2,-.5)--(5.5,-.5)node[midway,below]{$\tau_c=\log r_c$}; + \node[above] at (5,1.8){Seam: $\tau_p=\tau_c=L_e$}; + \node[below] at (5,-1.18){$\rho=a_pr_p=a_cr_c,\qquad a_c=a_pe^{-2L_e}$}; + \node[circle,draw,fill=blue!8,inner sep=3pt] (r) at (2,-3){$a_{\rm root}=1$}; + \node[circle,draw,fill=green!8,inner sep=3pt] (c) at (5,-2.8){$a_c$}; + \node[circle,draw,inner sep=3pt] (d) at (8,-2.3){$a_d$}; + \node[circle,draw,inner sep=3pt] (f) at (8,-3.65){$a_f$}; + \draw[->] (r)--node[above]{$L_1$}(c); + \draw[->] (c)--node[above]{$L_2$}(d); + \draw[->] (c)--node[below]{$L_3$}(f); +\end{tikzpicture} diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/paper.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..d397ad7e1f40cfd951f193b1b2a8d1acc225f37f --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/paper.tex @@ -0,0 +1,34 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{Path selection and an elliptic inequality\\on degenerating Ricci-flat trees} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove a sequential elliptic inequality for the renormalized +Einstein--Hilbert functional on a fixed finite tree of four-dimensional +Ricci-flat spaces. As the joining lengths diverge and the scale-neutral +weighted Ricci error $M$ tends to zero, the functional satisfies $E=o(M)$. +The root end has decay exponent greater than one, and every joined quotient +group is nontrivial. We also prove the multivariable path-selection theorem +for asymptotic expansions used to obtain this estimate. +\end{abstract} +\begingroup +\small +\tableofcontents +\endgroup +\input{sections/01-introduction} +\input{sections/02-tree} +\input{sections/03-path} +\input{sections/04-family} +\input{sections/05-tails} +\input{sections/06-length} +\input{sections/07-observables} +\AddToHook{cmd/thebibliography/after}{\addcontentsline{toc}{section}{\refname}} +\begingroup +\small +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/preamble.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..52d8c3dc798ee59df59148a040fe20883f708a3c --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/preamble.tex @@ -0,0 +1,80 @@ +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{enumitem,booktabs,array} +\usepackage{microtype} +\usepackage{xcolor,graphicx,tikz} +\usepackage{placeins} +\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds} +\usepackage{xurl} +\usepackage[colorlinks=true,linkcolor=blue!45!black, + citecolor=green!35!black,urlcolor=blue!55!black, + bookmarksnumbered=true]{hyperref} + +\ifdefined\pdfglyphtounicode + \pdfgentounicode=1 + \pdfglyphtounicode{tfm:lmex10/parenleftbig}{0028} + \pdfglyphtounicode{tfm:lmex10/parenrightbig}{0029} + \pdfglyphtounicode{tfm:lmex10/braceleftbigg}{007B} + \pdfglyphtounicode{tfm:lmex10/bracerightbigg}{007D} + \pdfglyphtounicode{tfm:lmex10/parenleftBigg}{0028} + 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\pdfglyphtounicode{tfm:lmex10/hatwide}{02C6} + \pdfglyphtounicode{tfm:lmex10/radicalbig}{221A} + \pdfglyphtounicode{tfm:lmsy10/mapsto}{007C} +\fi + +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{notation}[theorem]{Notation} +\newtheorem{convention}[theorem]{Convention} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Z}{\mathbb{Z}} +\DeclareMathOperator{\Ric}{Ric} +\DeclareMathOperator{\Rm}{Rm} +\DeclareMathOperator{\Scal}{Scal} +\DeclareMathOperator{\Vol}{Vol} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Id}{Id} +\DeclareMathOperator{\rank}{rank} + +\setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt} +\setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{1.5em} +\setcounter{tocdepth}{2} +\hypersetup{pdfauthor={OpenAI}, + pdftitle={Path selection and an elliptic inequality on degenerating Ricci-flat trees}, + pdfsubject={Quasianalytic path selection, Ricci-flat bubble trees, and a renormalized Einstein-Hilbert functional}} diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/references.bib b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..b55eeca97e23928512180c34fbf97179d63f98e9 --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/references.bib @@ -0,0 +1,146 @@ +@article{BamlerZhangI, + author = {Bamler, Richard H. and Zhang, Qi S.}, + title = {Heat kernel and curvature bounds in {Ricci} flows with bounded scalar curvature}, + journal = {Advances in Mathematics}, + volume = {319}, + year = {2017}, + pages = {396--450}, + doi = {10.1016/j.aim.2017.08.025}, + url = {https://arxiv.org/abs/1501.01291}, + eprint = {1501.01291}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + note = {\href{https://arxiv.org/abs/1501.01291v3}{arXiv:1501.01291v3}} +} + +@article{BamlerZhangII, + author = {Bamler, Richard H. and Zhang, Qi S.}, + title = {Heat kernel and curvature bounds in {Ricci} flows with bounded scalar curvature---{Part II}}, + journal = {Calculus of Variations and Partial Differential Equations}, + volume = {58}, + number = {2}, + year = {2019}, + note = {Article 49, 14 pp.; \href{https://arxiv.org/abs/1506.03154v1}{arXiv:1506.03154v1}}, + doi = {10.1007/s00526-019-1484-5}, + url = {https://arxiv.org/abs/1506.03154}, + eprint = {1506.03154}, + archivePrefix = {arXiv}, + primaryClass = {math.DG} +} + +@article{BamlerConvergence, + author = {Bamler, Richard H.}, + title = {Convergence of {Ricci} flows with bounded scalar curvature}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {188}, + number = {3}, + year = {2018}, + pages = {753--831}, + doi = {10.4007/annals.2018.188.3.2}, + url = {https://annals.math.princeton.edu/2018/188-3/p02}, + note = {\href{https://doi.org/10.4007/annals.2018.188.3.2}{DOI}} +} + +@article{SpeisseggerQA, + author = {Speissegger, Patrick}, + title = {Quasianalytic {Ilyashenko} algebras}, + journal = {Canadian Journal of Mathematics}, + volume = {70}, + number = {1}, + year = {2018}, + pages = {218--240}, + doi = {10.4153/CJM-2016-048-x}, + url = {https://arxiv.org/abs/1606.02751}, + eprint = {1606.02751}, + archivePrefix = {arXiv}, + primaryClass = {math.CV}, + note = {\href{https://arxiv.org/abs/1606.02751v8}{arXiv:1606.02751v8}} +} + +@article{GalalKaiserSpeissegger, + author = {Galal, Zeinab and Kaiser, Tobias and Speissegger, Patrick}, + title = {{Ilyashenko} algebras based on transserial asymptotic expansions}, + journal = {Advances in Mathematics}, + volume = {367}, + year = {2020}, + note = {Article 107095; \href{https://arxiv.org/abs/1806.01732v2}{arXiv:1806.01732v2}}, + doi = {10.1016/j.aim.2020.107095}, + url = {https://arxiv.org/abs/1806.01732}, + eprint = {1806.01732}, + archivePrefix = {arXiv}, + primaryClass = {math.LO} +} + +@article{DeTurckKazdan1981, + author = {DeTurck, Dennis M. and Kazdan, Jerry L.}, + title = {Some regularity theorems in {Riemannian} geometry}, + journal = {Annales scientifiques de l'\'{E}cole Normale Sup\'{e}rieure}, + series = {4}, + volume = {14}, + number = {3}, + year = {1981}, + pages = {249--260}, + doi = {10.24033/asens.1405}, + url = {https://www.numdam.org/item/10.24033/asens.1405/}, + note = {\href{https://doi.org/10.24033/asens.1405}{DOI}} +} + + +@article{DeruelleOzuch2025, + author={Deruelle, Alix and Ozuch, Tristan}, + title={A {{\L}ojasiewicz} inequality for {ALE} metrics}, + journal={Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, + series={5},volume={26},number={4},year={2025},pages={1965--2042}, + doi={10.2422/2036-2145.202309_029}, + url={https://journals.sns.it/index.php/annaliscienze/article/view/6142} +} + +@article{Haslhofer2011, + author={Haslhofer, Robert}, + title={A renormalized {Perelman}-functional and a lower bound for the {ADM}-mass}, + journal={Journal of Geometry and Physics}, + volume={61},year={2011},pages={2162--2167}, + doi={10.1016/j.geomphys.2011.06.016}, + url={https://arxiv.org/abs/1101.1226v2} +} + +@article{LockhartMcOwen1985, + author={Lockhart, Robert B. and McOwen, Robert C.}, + title={Elliptic differential operators on noncompact manifolds}, + journal={Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, + series={4},volume={12},number={3},year={1985},pages={409--447}, + url={https://www.numdam.org/item/ASNSP_1985_4_12_3_409_0/} +} +@article{Biquard2011, + author={Biquard, Olivier}, + title={D\'esingularisation de m\'etriques d'{Einstein}. {I}}, + journal={Inventiones Mathematicae}, + volume={192},year={2013},pages={197--252}, + doi={10.1007/s00222-012-0410-7}, + url={https://arxiv.org/abs/1105.4904v1}, + note={The cited proposition locator refers to \href{https://arxiv.org/abs/1105.4904v1}{arXiv:1105.4904v1}} +} +@article{Ozuch2022II, + author={Ozuch, Tristan}, + title={Noncollapsed degeneration of {Einstein} 4-manifolds {II}}, + journal={Geometry \& Topology},volume={26},year={2022},pages={1529--1634}, + doi={10.2140/gt.2022.26.1529}, + url={https://arxiv.org/abs/1909.12960v2}, + note={The cited theorem locators refer to \href{https://arxiv.org/abs/1909.12960v2}{arXiv:1909.12960v2}} +} +@article{PerrucciRoy2017, + author={Perrucci, Daniel and Roy, Marie-Fran\c{c}oise}, + title={Elementary recursive quantifier elimination based on {Thom} encoding and sign determination}, + journal={Annals of Pure and Applied Logic},volume={168},number={8},year={2017},pages={1588--1604}, + url={https://arxiv.org/abs/1609.02879v2}, + note={The cited theorem locators refer to \href{https://arxiv.org/abs/1609.02879v2}{arXiv:1609.02879v2}} +} + +@misc{Ozuch2021Integrability, + author={Ozuch, Tristan}, + title={Integrability of {Einstein} deformations and desingularizations}, + year={2021}, + howpublished={\href{https://arxiv.org/abs/2105.13193v1}{arXiv:2105.13193v1}}, + url={https://arxiv.org/abs/2105.13193v1} +} diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/01-introduction.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..f30e9214e39607401c1a56f9ee52108c54c2472d --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/01-introduction.tex @@ -0,0 +1,193 @@ +\section{Introduction}\label{sec:introduction} + +Ricci-flat metrics can degenerate by developing regions at several widely +separated length scales. In four dimensions, an asymptotically locally +Euclidean end of one region can join a neighborhood of an orbifold point +in another. Repeating this configuration produces a finite rooted tree. +The geometry on every fixed vertex region may converge smoothly while +ratios of scales tend to zero. An estimate useful in this situation must +control the whole joined metric, including the regions between its +limiting pieces. + +We study a renormalized Einstein--Hilbert functional on such metrics. +The leading Euclidean divergence in the scalar-curvature density is +subtracted at the exterior end of the root. The resulting quantity, +denoted by $E$, has length degree two. Its differential is the integral +pairing with the Einstein tensor $\tfrac12 Rg-\Ric$. +A fixed root with decay $g-\delta=O(r^{-q})$, $q>1$, makes the +quadratic remainder and the first-variation pairing integrable. The +individual scalar integral and boundary flux need not converge. + +Renormalization of curvature energies on noncompact spaces has several +established forms. Haslhofer's renormalized Perelman functional treats +relative energies near steady solitons~\cite{Haslhofer2011}. +Deruelle and Ozuch subtract the mass contribution in their analytic +ALE functional and discuss explicitly the corresponding renormalized +Hilbert--Einstein expression~\cite[Definition~4.2, Remark~4.3, and +Proposition~4.4]{DeruelleOzuch2025}. Their +weighted \L{}ojasiewicz inequality near a fixed Ricci-flat ALE background +holds without an integrability hypothesis; integrability gives an +improved exponent~\cite[Theorem~1.3]{DeruelleOzuch2025}. The functional +$E$ here is the unminimized scalar-curvature-minus-flux expression, +whereas their $\lambda_{\mathrm{ALE}}$ also minimizes over a potential. +Our estimate concerns simultaneous degeneration of the joining scales +in a fixed tree, outside one fixed weighted neighborhood of a smooth +ALE metric. + +The main theorem, stated with the geometric hypotheses in +Section~\ref{sec:elliptic}, gives a strict improvement over the direct +first-variation estimate. Let $M$ control ten scaled derivatives of +the dimensionless Ricci tensor, with positive exponential decay on +joining half-ends and decay $r^{-q}$ at the root. For a fixed limiting +tree and sequences with $M\to0$ and all joining lengths tending to +infinity, Theorem~\ref{thm:tree-inequality} proves $E=o(M)$. +The metrics converge to the specified vertex metrics on compact sets, +and their end bounds are uniform through order twenty. Kernels and +cokernels of the linearized equations are retained in the argument. + +This is a sequential statement for one fixed tree. Joining lengths may +diverge at unrelated rates, and the proof does not require the +infinitesimal Ricci-flat deformations to integrate to a smooth moduli +space. These features are relevant to four-dimensional Ricci-flow +concentration, where the tree is obtained only after choosing a sequence +of times. For a compact four-dimensional Ricci flow on a finite time interval +with bounded scalar curvature, a curvature blow-up that converges +smoothly to an ancient flow has a Ricci-flat ALE limit +\cite[arXiv version, Corollary~1.9]{BamlerZhangI}. +Related distance-control and compactness results appear +in~\cite{BamlerZhangII,BamlerConvergence}. The present article proves +the static estimate; its hypotheses are stated independently of a flow. + +\subsection{Why an analytic path enters the proof} + +On a fixed finite-dimensional analytic family, a gradient inequality +relates a function to its differential near a critical set. The +parameters here include neck lengths $L_e\to\infty$. Their boundary +is outside an ordinary analytic coordinate neighborhood, and a change +such as $t_e=e^{-L_e}$ need not make the family analytic at $t_e=0$. +Asymptotic power-logarithm expansions can exist to every finite order +without defining a convergent power series. + +Our substitute is a path-selection statement for a precise class of +holomorphic functions. In each length these functions have expansions +in $e^{-dL}$ with polynomial coefficients in $L$, where the +nonnegative exponent sets are locally finite. The coefficients admit +the same expansions in the other lengths. Remainders are strictly +better than the requested exponential order, and all recursively +extracted coefficients are holomorphic on a common ordinary-parameter +neighborhood. These hypotheses are defined in +Definition~\ref{path:def-admissible}. + +Theorem~\ref{thm:path-selection} realizes finite sign conditions and +finite prescribed limits along a real analytic path. Every actual +length coordinate and every bounded ordinary coordinate is eventually +monotone or constant. Therefore the path has finite variation in the +weighted differential form +\[ + \sum_\alpha |dz_\alpha|+\sum_e e^{-cL_e}|dL_e|, + \qquad c>0. +\] +This finite variation is the exact property used in the elliptic +argument. The construction does not replace an all-order expansion +by a convergent series. + +Quasianalytic asymptotic algebras provide the ancestry of this approach. +Speissegger~\cite{SpeisseggerQA} develops one-variable Ilyashenko +algebras, and Galal, Kaiser, and Speissegger~\cite{GalalKaiserSpeissegger} +construct algebras based on transserial asymptotic expansions. Their +one-variable field and differentiation results provide precedents for +the clock calculus used here. Our simultaneous length specialization, ordinary +parameter normalization, and path selection are proved in +Section~\ref{sec:path}; no multivariable selection theorem is imported +from those works. + +\subsection{The elliptic reduction and its quantitative output} + +The analytic family is built in a gauge whose principal part is a +componentwise scalar elliptic operator. The cancellation underlying +this use of coordinates is part of the Ricci-coordinate method; +see DeTurck and Kazdan~\cite{DeTurckKazdan1981} for its classical +geometric setting. On each fixed vertex, finite core tests separate +the kernel, while finitely many compactly supported sources represent +the cokernel. On a joined edge, the two Cauchy matching conditions are +supplemented by the value in the critical spherical channel. Explicit +radial mode matrices then give an inverse whose bound is uniform as +the lengths diverge. + +This reduction has close predecessors in Einstein desingularization. +Biquard constructs metrics modulo a finite-dimensional obstruction +space in the Eguchi--Hanson gluing problem +\cite[arXiv version, Proposition~8.1]{Biquard2011}. +Ozuch develops uniform inverse estimates and nearby +Einstein-modulo-obstructions families for desingularization trees, +including ALE orbifolds +\cite[arXiv version, Proposition~11 and Theorem~4.6]{Ozuch2022II}. +The original development was also informed by Ozuch's treatment of +potentially nonintegrable Einstein deformations +\cite{Ozuch2021Integrability}. This is methodological motivation, not +an invocation of that paper's results for the tree inequality proved here. +Our proof retains that obstruction-space viewpoint. The additional +interfaces needed here are continuation in complex length regions, +recursive finite-order expansions with a common ordinary-parameter +domain, and a differential estimate for the renormalized observable. + +The ordinary parameters $z$ record these finite data. The length +parameters are continued to quadratic complex regions. A uniform +contraction there gives a small family $h_*(L,z)$ and, for a metric +$g$ in the theorem, the comparison +\[ + \|g-h_*\|\le CM,\qquad + \|\Ric(h_*)\|\le CM,\qquad + |E(g)-E(h_*)|\le CM^2. +\] +The norms and the gauge are specified in Section~\ref{sec:family}. +This comparison is why the finite-dimensional family is sufficient +even though the original metrics depend on infinitely many variables. + +Set $F=E(h_*)$. Let $J$ be the nonnegative squared Ricci norm defined +in Equation~\eqref{ell:observables}, with an additional weight on the +root tail. The first variation and the shrinking physical volumes of +descendant regions imply +\[ + |dF|\le C\sqrt J + \left(\sum_\alpha|dz_\alpha|+\sum_e e^{-cL_e}|dL_e|\right). +\] +The remaining analytic obligation is to place $F$ and $J$ in the +path-selection class. We establish this by opening one edge at a +time. Solutions on an opened end have integer exponential tails. +Finite incoming modes cancel the seam mismatch, and triangular linear +auxiliary equations produce each additional asymptotic coefficient. +Finite-part integration transfers the expansions to $F$ and $J$. +Every order uses the same small nonlinear branch and the same basic +inverse; coefficients with greater prescribed growth enter only +linear solves. This preserves one ordinary-parameter domain through +all orders. + +If the tree inequality failed, the comparison would produce a sequence +with $F\ne0$, $F\to0$, and $F^2\ge c_0J$ for some $c_0>0$. +Path selection would preserve those conditions on a path of finite +weighted variation. Dividing the differential estimate by $|F|$ +would then give finite variation of $\log|F|$, contradicting +$F\to0$. Corollary~\ref{path:variation} isolates this short final +argument from the substantial analytic work needed to apply it. + +\subsection{Organization and conventions} + +Section~\ref{sec:elliptic} defines the tree geometry, the Ricci error, +and the functional and states the main theorem. +Section~\ref{sec:path} proves path selection. +Section~\ref{sec:family} constructs the inverses, gauge, analytic family, +and comparison with the original sequence. +Sections~\ref{ell:tail-calculus} and~\ref{ell:length-expansion} +prove the end and length expansions. +Section~\ref{ell:scalar-calculus} establishes the scalar expansion +class and completes the theorem. + +We use $\Delta=\tr\nabla^2$. Metric and tensor components on a quotient +end are equivariant Cartesian components on its Euclidean cover; +all integrals are quotient integrals. Scaled derivative norms insert +one local length factor per derivative. Tensor magnitudes are physical +unless vertex units are explicitly specified. Constants may depend on +the fixed tree, its charts, and a fixed finite differentiation order. +Uniformity in diverging lengths or in an approximation index is +asserted at the point where it is needed. diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/02-tree.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/02-tree.tex new file mode 100644 index 0000000000000000000000000000000000000000..fadb54a125c0a23211313ca134cda70babf9f0e0 --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/02-tree.tex @@ -0,0 +1,156 @@ +\section{Tree geometry and the renormalized functional} +\label{sec:elliptic} + +The result of this section is sequential. Its constants, its auxiliary +analytic family, and its small exponents belong to one fixed limiting tree. +No uniform exponent over different trees is asserted or needed. + +\subsection{The geometric data and the functional} +\label{ell:setup} + +Let $\mathcal T$ be a finite rooted tree of four-dimensional Ricci-flat +spaces $(V_v,h_v)$. A vertex has one exterior end and one punctured end +for each child. Every end has Cartesian coordinates on a Euclidean +annulus modulo a finite subgroup of $O(4)$ acting freely on $S^3$. +The group on a joined end is nontrivial. The two groups on an edge +are identified; a fixed orthogonal identification is included in the +charts. Put $\tau=\log r$ on an exterior end and $\tau=-\log r$ at a +puncture. In these charts, in vertex units, +\begin{equation} + h_v=\delta+O(e^{-\mu\tau}) + \label{ell:end-decay} +\end{equation} +on joined ends, with derivatives at scale, for some $\mu>0$. +On the unjoined exterior end of the root we require instead +$h_v-\delta=O(r^{-q})$, where $q>1$. +All remaining parts are smooth compact manifolds with the indicated +boundary collars. Bounds through order $20$ suffice in the data below. + +For an edge $e$ from $p$ to $c$, truncate both end coordinates at +$\tau=L_e$ and identify the boundary collars by +\begin{equation} + x_p=e^{-2L_e}x_c,\qquad a_c=a_p e^{-2L_e},\qquad a_{\rm root}=1. + \label{ell:scales} +\end{equation} +Thus the metric tensor in vertex units is multiplied by $a_v^2$ in +physical units. The physical radius $\rho=a_vr$ agrees in both charts, +and $\tau_p+\tau_c=2L_e$. Gluing just the boundary faces, with their +compatible smooth collars, gives the same manifold. +Figure~\ref{fig:tree-scales} records how the two logarithmic coordinates +meet at one physical radius and how edge lengths determine descendant +scales. + +Consider a sequence of metrics $g$ for which every $L_e\to\infty$, +which converges to $h_v$ on each compact subset of every open vertex. +Assume the same end estimates as in Equation~\eqref{ell:end-decay} +on each joining half, and the stated root estimate, with uniform +finite derivative bounds. On a compact core put $\rho=a_v$. +Suppose $M\to0$ and +\begin{equation} + |\rho^2\Ric(g)|_j\le CM + \begin{cases} + e^{-\mu\tau},&\text{on joined half-ends},\\ + r^{-q},&\text{on the root exterior end},\\ + 1,&\text{on vertex cores}, + \end{cases} + \qquad j=10. + \label{eq:tree-ricci-error} +\end{equation} +Here the tensor magnitude is physical, and the subscript includes +covariant derivatives multiplied by the corresponding powers of +$\rho$. The metric bounds convert these to coordinate estimates. +Decreasing $\mu$ or $q$, while retaining $\mu>0$ and $q>1$, is allowed. + +The subtraction below is the linear Euclidean boundary flux in the +scalar-curvature density. Compare the mass-subtracted variational +construction in~\cite[Remark~4.3 and Proposition~4.4]{DeruelleOzuch2025}. +We give its normalization and first variation directly for the present +finite-tree geometry. Define, in physical root units, +\begin{equation} + E(g)=\lim_{D\to\infty}\left\{ + \int_{r1$. A global length scaling +by $s$, accompanied by a coordinate dilation at infinity restoring +the leading tensor $\delta$, multiplies $E$ by $s^2$. +Under Equation~\eqref{eq:tree-ricci-error}, $|E(g)|\le CM$. +\end{lemma} + +\begin{proof} +Write $g=\delta+p$ on the root end. The scalar-curvature density is +\[ + R(g)\sqrt{\det g}= + \partial_i(\partial_jp_{ij}-\partial_ip_{jj})+ + O(|p||\partial^2p|+|\partial p|^2). +\] +The last expression is $O(r^{-2q-2})$ and its radial integral is +bounded by $C\int_{r_0}^\infty r^{1-2q}\,dr$. This proves convergence. +The linear boundary term in the variation of scalar curvature +cancels the variation of the subtracted integral. Every remaining +boundary product is $O(D^{2-2q})$; it tends to zero. Integration by +parts therefore gives Equation~\eqref{ell:first-variation}. +For a Lie derivative, the contracted Bianchi identity makes the +interior divergence vanish. The remaining boundary term vanishes +with the stated displacement decay. Integration along the isotopy +gives invariance. Changing variables under a dilation in +Equation~\eqref{eq:energy-functional} gives the scaling law. + +Differentiate this law at $s=1$. Choose the compensating dilation +to be the identity inside a fixed outer root region. Its metric +variation equals $2g$ on the interior and decays like $r^{-q}$ on +the root tail. Its pairing with Ricci on the root is integrable +by $q>1$. On a punctured half the interior estimate costs +\[ + CM a_p^2\int_{\tau_0}^{L_e}e^{-(2+\mu)\tau}\,d\tau\le CM a_p^2. +\] +On a child exterior half the corresponding quantity is +\[ + CM a_c^2\int_{\tau_0}^{L_e}e^{(2-\mu)\tau}\,d\tau + \le CM a_p^2(1+L_e)e^{-\min(2+\mu,4)L_e}, +\] +with a harmless change of constant for the fixed lower endpoint. +The finite sum of core contributions is bounded as well, since +$a_v\le1$. Equation~\eqref{ell:first-variation} now gives +$2|E(g)|\le CM$. In particular $E=0$ when $\Ric=0$. +\end{proof} + +\begin{theorem}[Tree inequality] +\label{thm:tree-inequality} +Fix a finite rooted tree with the end identifications and order-$20$ metric bounds of Section~\ref{ell:setup}. Let $(g_j,L^{(j)},M_j)$ satisfy the compact vertex convergence, the same uniform half-end and root-end bounds, and Equation~\eqref{eq:tree-ricci-error} through order $10$, with every $L_e^{(j)}\to\infty$ and $M_j\to0$. Then, in physical root units, +\begin{equation} + E(g_j)=o(M_j). + \label{ell:little-oh} +\end{equation} +For an index with $M_j=0$, the assertion means $E(g_j)=0$. The little-oh is taken for this fixed tree; it does not assert a power exponent uniform over different trees. +\end{theorem} + +The proof constructs a finite-dimensional analytic family with +complex length parameters. It does not assume that Einstein +deformations are unobstructed, or that the given metrics or the +coefficients of a fixed core metric are spatially analytic. + + +\begin{figure}[htbp] +\centering +\input{figures/tree-scales} +\caption{A joined parent puncture and child exterior use opposite logarithmic coordinates. Their seam has physical radius $a_p e^{-L_e}=a_c e^{L_e}$ and $a_c/a_p=e^{-2L_e}$. The inset records a finite tree with independently diverging edge lengths; neither relative length ratios nor rates of divergence are fixed. The diagram is schematic.} +\label{fig:tree-scales} +\end{figure} +\FloatBarrier diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/03-path.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/03-path.tex new file mode 100644 index 0000000000000000000000000000000000000000..68abb8a085ed7b76d30b3d19b4c2d5e15deb4ac9 --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/03-path.tex @@ -0,0 +1,1320 @@ +\section{Selection of paths at several exponential ends}\label{sec:path} + +The reduction used to prove Theorem~\ref{thm:tree-inequality} needs a path on +which every original parameter has finite weighted variation. The parameters +include several independently diverging neck lengths. Ordinary analytic +curve selection does not directly apply at those ends. We give the precise +asymptotic class and the selection argument used below. In particular, the +argument includes the specialization of several lengths; one-variable +quasianalyticity alone would not supply that step. + +\subsection{The class of functions and the selection statement} + +Put +\[ + Q(A,c)=\{L\in\C:\Re L>A,\ |\Im L|0\), on +possibly smaller quadratic regions. Bounds are uniform on each fixed +smaller ordinary-parameter polydisc. +\item For every subset of expanded lengths, all its coefficient formulas, +at all orders, are holomorphic on one common tail band +\[ + \prod_{j\text{ unexpanded}} + \{\Re L_j>A',\ |\Im L_j|0, +\] +times one common open ordinary-parameter neighborhood. These domains may +be shrunk once when a finite number of families are used together. The +constants and thresholds in asymptotic \emph{estimates} may depend on the +order; the common domains of \emph{analyticity} do not. +\item If a subset of real parts has the form +\(\Re L_i=c_iU+o(U)\), where \(c_i>0\), the corresponding iterated +expansion can be grouped by weights \(\sum_i c_id_i\). The grouped +weights are locally finite, and each finite cutoff has a remainder of +strictly better exponential order in \(U\). The assertion holds on the +complex neighborhoods in \textup{(i)} and locally uniformly on the common +ordinary-parameter neighborhood. It is enough to establish it by +successive single-length expansions. +\end{enumerate} +When \(m=0\), admissible families are ordinary holomorphic germs. +\end{definition} + +The distinction in \textup{(iii)} will be used when an expansion has +infinitely many zero coefficients. It is not an assertion that all +remainder estimates hold beyond a single order-independent threshold. + +\begin{theorem}[Path selection]\label{thm:path-selection} +Let \(P_1,\ldots,P_s\) be an admissible finite family, and let a finite +Boolean combination of conditions \(P_j>0\), \(P_j=0\), and \(P_j<0\) +hold along a sequence +\[ + L_i^{(n)}\longrightarrow+\infty,\qquad z^{(n)}\longrightarrow z_0. +\] +There is a real analytic map \(u\mapsto(L(u),z(u))\), defined for all +sufficiently large real \(u\), with the same limits and satisfying those +conditions on its tail. Each actual coordinate \(L_i(u)\) and +\(z_a(u)\) is eventually monotone or constant. Given any additional +finite admissible list, its values on the path can also be required to be +eventually monotone or constant. Finite prescribed limits of functions +can be imposed at the same time. +\end{theorem} + +A path germ here means analyticity at every sufficiently large finite +\(u\); analyticity at the limiting endpoint is not asserted. To impose +\(P\to\ell\), introduce an ordinary variable \(v\to\ell\) and impose +\(P-v=0\). If there are no length variables, add an unused length. This +also provides positive infinitesimals for strict sign conditions at an +ordinary limiting point. + +\begin{corollary}[The variation consequence]\label{path:variation} +Suppose admissible functions \(F,J\), with \(J\ge0\), satisfy +\[ + |dF|\le C\sqrt J\left(\sum_a|dz_a| + +\sum_i e^{-cL_i}|dL_i|\right),\qquad c>0, +\] +on real parameter domains. There cannot be a sequence with +\(L_i\to\infty\), an interior ordinary-parameter limit, and +\(F\ne0\), \(F^2\ge c_0J\), \(F\to0\), where \(c_0>0\). +\end{corollary} +\begin{proof} +Theorem~\ref{thm:path-selection} realizes all these conditions on one +path with monotone actual coordinates. Along it, +\[ + |d\log|F||\le\frac C{\sqrt{c_0}} + \left(\sum_a|dz_a|+\sum_i e^{-cL_i}|dL_i|\right). +\] +Each \(z_a\) has finite total variation because it is monotone with +a finite limit. Each length tends monotonically to infinity on its +tail, and +\[ + \int_{u_0}^\infty e^{-cL_i(u)}|L_i'(u)|\,du + =c^{-1}e^{-cL_i(u_0)}<\infty. +\] +Thus \(\log|F|\) has finite total variation and a finite limit, whereas +\(F\to0\) forces \(\log|F|\to-\infty\), a contradiction. +\end{proof} + +The remaining subsections prove Theorem~\ref{thm:path-selection}. +The geometric construction resumes in Section~\ref{sec:family} and +uses Definition~\ref{path:def-admissible} and +Corollary~\ref{path:variation} as its analytic interface. + +\subsection{Sectors, uniqueness, and clocks} + +\begin{definition}\label{path:def-sector} +An admissible angular loss is a positive continuous function \(\omega(r)\), for large +\(r\), such that +\[ + \omega(r)\ge(\log r)^{-2},\qquad + \sum_{n\ge n_0}\sup_{2^nR,\ |\arg z|<\pi/2-\omega(|z|)\}\). +Every positive loss satisfying the displayed bounds has a continuous +admissible enlargement: at dyadic radii choose a summable majorant of +the neighboring shell suprema and interpolate linearly in \(\log r\). +Thus enlarging a loss to make its sector open only shrinks that sector. +Increasing a loss by a fixed factor or taking a continuous majorant of +the suprema over finitely many neighboring dyadic shells is permitted. Given finitely many losses we +can dominate all of them by another admissible loss. We can also arrange +that the domination ratio tends to infinity: if \(a_n\ge0\) is summable, +choose increasing constants \(c_n\to\infty\), sufficiently slowly that +\(\sum c_na_n<\infty\). +\end{definition} + +The principle that a bounded holomorphic function is determined by its +exponentially accurate expansion is central to Ilyashenko +quasianalyticity; see~\cite[Fact~1.5]{SpeisseggerQA} in the published +version. We prove the following variant with a summable angular loss, +which is the form preserved by the clock changes below. + +\begin{lemma}[Sector uniqueness]\label{path:sector-uniqueness} +Suppose \(f\) is bounded and holomorphic on an admissible sector and, on +the central subsector \(|\arg z|\le\pi/4\), satisfies +\(f(z)=O_B(e^{-B|z|})\) for every \(B>0\). Then \(f=0\). +\end{lemma} +\begin{proof} +Let \(w_n\) be the supremum of the prescribed loss on a fixed enlargement +of the \(n\)-th dyadic shell. Choose summable \(a_n\ge0\) with +\(a_n/w_n\to\infty\); enlarge them by a fixed factor if necessary. +On the right half-plane define +\[ + A(z)=\sum_n\frac{a_n}{1+z/2^n},\qquad \Phi(z)=z\exp A(z). +\] +The series is normally convergent there, \(A(z)\to0\) uniformly as +\(|z|\to\infty\), and \(|\Phi(z)|\asymp|z|\). For \(\Im z>0\), +every summand has nonpositive imaginary part. If +\(\pi/4\le\arg z\le\pi/2\), the terms with \(2^n\asymp|z|\) +have negative imaginary part of magnitude at least a fixed multiple of +\(a_n\). Thus the change in argument pushes the outer part of the +half-plane inside the prescribed sector. Reflection proves the assertion +below the real axis; on the central part it follows from \(A=o(1)\). +Translating the half-plane sufficiently far to the right puts its whole +image in the domain of \(f\). + +The bounded pullback \(F=f\circ\Phi\) is superexponentially small on a +fixed central sector, for example \(|\arg z|\le\pi/8\). In a +half-disc of radius \(r\), the harmonic measure at a fixed positive +interior point of the central part of its semicircular boundary is at +least \(c/r\). One can see this directly by scaling to the unit +half-disc: the Poisson kernel on a compact subarc away from its endpoints +is comparable to the distance of the rescaled axial point from the +straight boundary, namely a constant times \(1/r\). Write +\(M\) for a bound for \(|F|\). On that arc +\(\log|F|\le-Bc_1r+C_B\), and on the rest of the boundary +\(\log|F|\le\log M\). The subharmonic maximum principle gives +\[ + \log|F(z_*)|\le\log M-c_2B+O_B(r^{-1}). +\] +Zeros cause no difficulty: use the subharmonic function +\(\max\{\log|F|,-N\}\) and then let \(N\to\infty\). +First let \(r\to\infty\), then \(B\to\infty\). Hence \(F\) vanishes +at every fixed axial point. The Identity Theorem gives \(F=0\), and +then \(f=0\), since \(\Phi\) is nonconstant and its image is open. +\end{proof} + +We now describe the one-variable fields into which the selected paths +will be placed. Begin with a positive variable \(x_1\to\infty\) and +the field \(B_0\) of convergent meromorphic Puiseux germs in \(1/x_1\). +At stage \(j\), the clock \(x_j\) belongs to \(B_{j-1}\), and for +\(j>1\) it satisfies \(x_{j-1}=o(x_j)\). The first clock is \(x_1\), +or a positive constant multiple of it. A germ in \(B_j\) has an +expansion +\begin{equation}\label{path:eq-clock-series} + f\sim\sum_{\beta\in E}e^{\beta x_j}b_\beta, + \qquad b_\beta\in B_{j-1}, +\end{equation} +where the real set \(E\) is bounded above and finite above each cutoff. +The germ and the continued coefficients are holomorphic on admissible +sectors in the variable \(x_j\). After a finite cutoff the remainder +has a strictly better exponential rate in \(\Re x_j\), on a possibly +smaller admissible sector. This includes sector estimates for an empty +expansion. The same definition is used for holomorphic families on +fixed ordinary-parameter polydiscs, with estimates on smaller polydiscs. +At the Puiseux level a parameter family has a common finite +ramification denominator and pole order, holomorphic coefficient +functions on the ordinary polydisc, and a convergent series locally +uniform there; bounds are taken on smaller polydiscs. + +\begin{lemma}[Clock calculus and change of clock]\label{path:clock} +The preceding construction has the following properties, established +simultaneously by induction. +\begin{enumerate}[label=\textup{(\alph*)}] +\item Each \(B_j\) is a field, closed under differentiation in \(x_1\) +and conjugation. A nonzero scalar has a leading equivalent +\begin{equation}\label{path:eq-leading-monomial} + Cx_1^p\exp\!\left(\sum_{k=1}^j\beta_kx_k\right), + \qquad C\ne0, +\end{equation} +uniformly on a suitable admissible sector. Its real subfield is a Hardy +field: every nonzero real germ has a fixed eventual sign, and its +derivative either vanishes or has a fixed eventual sign. +\item If \(H\in B_j\) is positive and \(x_j=o(H)\), it is a valid +new sector variable. Finitely many prescribed earlier-clock sectors +can be reached by taking a sufficiently narrow admissible \(H\)-sector. +If \(f\in B_j\) is positive, unbounded, and \(f=O(H)\), then on such +a sector its modulus is comparable to its value at the corresponding +positive real \(H=|H|\); it maps into any finitely prescribed +admissible \(f\)-sectors. If \(f=o(H)\) on the real ray, then +\(\Re f=o(\Re H)\) throughout the narrowed sector. +\end{enumerate} +\end{lemma} +\begin{proof} +For \(B_0\), inversion and all assertions follow by convergent Puiseux +inversion. A positive unbounded Puiseux germ is a positive constant +times a positive power of the real variable, with a relative error +tending to zero. If \(f=O(H)\), the corresponding power is at most the +power for \(H\). A strictly smaller power leaves a fixed angular +margin. Equal powers require only an enlarged admissible loss. +Pullback of a loss under comparable positive power growth preserves +dyadic summability. + +Assume both assertions at lower levels, and write \(x=x_j\). Their +change-of-clock assertion permits evaluation of lower germs on +\(x\)-sectors and gives +\(\Re x_k=o(\Re x)\) for \(k0, + \quad E=O(e^{-\epsilon\Re x}). +\end{equation} +Since \(H\) is unbounded, \(b\ge0\). First suppose \(b>0\). +On bounded strips, +\((\log H)'=b+o(1)\). Analytic inversion along the arcs +\(H=\rho e^{i\theta}\), \(|\theta|\le\pi/2\), gives +\begin{equation}\label{path:eq-clock-inverse} + x(\rho e^{i\theta})=x(\rho)+i\theta/b+o(1),\qquad + \partial_\theta x=i/b+o(1), +\end{equation} +uniformly in \(\theta\). For completeness, solve the inverse equation +first on a fixed small \(\theta\)-interval by the Analytic Inverse +Theorem, using the uniform lower bound \(|(\log H)'|\ge b/2\). +Continue over finitely many such intervals. Integrating the inverse +derivative gives Equation~\eqref{path:eq-clock-inverse}, keeps the +solution in a bounded strip, and guarantees uniqueness on overlaps. +The resulting branches agree with the real inverse and hence agree +where their sector domains overlap. + +If \(f=e^{ax}g(1+O(e^{-\epsilon'\Re x}))\), where \(g>0\) is lower +level, then unboundedness and \(f=O(H)\) give \(0\le a\le b\). +When \(a0\), there +is a positive scalar \(\rho\) with \(\rho^n/a\) tending to a finite +positive constant. Scalars and reciprocals have magnitude at most +\(\exp(C\max_i L_i^*)\), where \(L_i^*\) are the current centers. +For an empty tuple the bound is constant. +\item The system is closed under finite sums and products, reflection, +ordinary holomorphic composition at finite limiting arguments, inversion +after a nonvanishing normalized limit has been specified, and Cauchy +integration on fixed compact contours. Integrals over fixed compact +segments are allowed by a finite cover of their holomorphic parameter +neighborhoods. Domain restrictions for these operations are part of +the finite data of the expression. +\item Given a domain and finitely many normalization, comparison, and +limit requirements, there are families of real analytic paths eventually +in that domain preserving them. The expressions used belong on each +path to a clock hierarchy of Lemma~\ref{path:clock}, with parameter +versions for any finite specified uses. The path families have real +constant parameters in open sets and are analytic in those constants +where defined. At every sufficiently large fixed path time, their +evaluation into the independent-variable domain is locally submersive. +The starting time may depend on the constants. +\end{enumerate} +Limits of families in \textup{(R1)} mean uniform limits on each fixed +smaller polydisc. A nonzero holomorphic limit need not be nonzero at +its central point. An inverse requires nonvanishing on the particular +polydisc or contour where that inverse is used. +\end{definition} + +The expression ``finite analytic operation'' always includes its inputs +and the domain data needed for them. This prevents an implicit request +to preserve infinitely many normalizing comparisons on a path. Two +elementary consequences will be useful. + +\begin{lemma}[Preparation on fixed contours]\label{path:prepared-contours} +Let \(F(q,\zeta,y)\) belong to a regular system. Suppose that after +scalar normalization its limiting germ is regular of degree \(d\) in +the \(y\)-direction at \((\zeta,y)=(0,0)\). Then its Weierstrass +polynomial, its unit, and division of any other family by that polynomial +are obtained by the operations in \textup{(R3)} on fixed contours and +fixed smaller parameter polydiscs. Arbitrarily high Taylor coefficients +of these operations use the same denominator zero sets and the same +contours. +\end{lemma} +\begin{proof} +Choose a circle \(|y|=r\) on which the limiting germ is nonzero and +inside which it has precisely \(d\) zeros at \(\zeta=0\), counted +with multiplicity. After shrinking the \(\zeta\)-polydisc and the +domain in the base, normalized \(F\) is uniformly nonzero on that +circle. The power sums of its enclosed roots are +\[ + s_\ell(q,\zeta)=\frac1{2\pi i}\int_{|y|=r} + y^\ell\frac{\partial_yF(q,\zeta,y)}{F(q,\zeta,y)}\,dy, + \qquad 1\le\ell\le d. +\] +Newton's identities give the monic polynomial \(W\) from these power +sums, by finite polynomial operations. The quotient \(F/W\) is a +holomorphic unit on a smaller disc by cancellation of the enclosed +zeros; its values there can also be expressed by a fixed-circle Cauchy +formula. A useful explicit division formula for a holomorphic family +\(G\) is +\[ + G(y)=W(y)\frac1{2\pi i}\int_{|w|=r} + \frac{G(w)}{W(w)(w-y)}\,dw + +\frac1{2\pi i}\int_{|w|=r} + \frac{G(w)}{W(w)}\frac{W(w)-W(y)}{w-y}\,dw. +\] +The second term is a polynomial in \(y\) of degree less than \(d\). +These formulas prove the asserted closure. Differentiating or expanding +them introduces powers of the existing nonzero contour denominators, +not new denominators with new zero sets. All Taylor orders are +holomorphic on the same smaller polydisc; their Cauchy constants may +depend on the order. +\end{proof} + +\begin{lemma}[Roots in a regular scalar system]\label{path:roots} +Suppose \textup{(R1)--(R3)} hold, including divisibility of positive +sizes. Then the complex scalar field is algebraically closed and its +real part is real closed. +\end{lemma} +\begin{proof} +We factor monic polynomials by induction on degree. Translate the +variable to remove the coefficient of degree \(d-1\). If all the +remaining coefficients vanish, the polynomial is a pure power and the +claim follows. Otherwise write it as +\[ + y^d+\sum_{j=2}^d a_jy^{d-j}. +\] +Choose, from finitely many positive root-size representatives of the +nonzero \(a_j\), one of largest size, denoted \(\rho\). Replacing +\(y\) by \(\rho v\) makes all coefficients bounded and makes at +least one non-leading coefficient have a nonzero limit. The limiting +depressed polynomial has at least two distinct roots: a polynomial +with a single root is \((v-c)^d\), and its vanishing degree-\(d-1\) +coefficient forces \(c=0\). + +Choose disjoint small fixed circles around the distinct limiting roots. +The contour power sums used in Lemma~\ref{path:prepared-contours} +give monic factors for the corresponding clusters. Their coefficients +are in the system, and all their degrees are strictly smaller than +\(d\). Induction factors each cluster completely. Undo the scaling +and translation. For a real polynomial, conjugation pairs the +nonreal factors and fixes every real-valued root. The real scalar +field is ordered by eventual sign. Every positive element has a real +square root, and every odd-degree real polynomial has a real root: +at the real points of a sufficiently small base domain these assertions +hold for the factorizations just constructed, and the finite sign and +conjugacy alternatives can be fixed on the domain base. These two +properties characterize a real closed ordered field. +\end{proof} + +\begin{lemma}[Detection of analytic identities]\label{path:detection} +Path families with the submersivity property in \textup{(R4)} detect +analytic germs. Namely, if an analytic function vanishes on the tail +of every path in such a family, then it vanishes as a domain germ. +The same holds for a holomorphic family in extra ordinary variables. +\end{lemma} +\begin{proof} +Let \(V\) be an open parameter set for one family and use integer path +times \(N\). At a fixed \(N\), evaluation is analytic on an open +subset \(V_N\subset V\). On the subset where its differential is +surjective, the composition with a nonzero domain germ is a nonzero +real analytic function on every connected parameter neighborhood: +otherwise submersivity would give an open set of zeros in the domain. +The zero set of a nonzero real analytic function has Lebesgue measure +zero. This last fact follows by induction on the number of variables: +in one variable zeros are discrete; in higher dimension separate the +points where a nonzero Taylor coefficient in the last variable is +nonzero and use the one-dimensional result and Fubini, with induction +on the exceptional coefficient-zero set. + +Every parameter lies, for all sufficiently large integers, in a +submersive evaluation domain and in the zero set of the composition. +If the original germ were nonzero, countably many local parameter +charts for these integer evaluations would thus cover \(V\) by sets +of measure zero, a contradiction. An open set of domain zeros makes +the germ zero by connectedness and the Analytic Identity Theorem. +For a family in extra variables, apply this argument to each of its +Taylor coefficients, or fix the extra variables in a countable dense +set and then use continuity. +\end{proof} + +\subsection{Adjoining a reciprocal fastest length} + +The next Lemma is the analytic specialization step. It is stated with +its quantitative separation hypothesis, since that hypothesis rules out +uncontrolled cancellation against the lower field. + +\begin{lemma}[One independent reciprocal]\label{path:adjoin-p} +Let \(\mathcal A\) be a regular system for a nonempty lower tuple +\(L'=(L'_1,\ldots,L'_r)\), with real scalar field \(K_{\R}\). +Assume that each \(L'_i\) is a positive unbounded element of +\(K_{\R}\), and put \(M'=1+\sum_{i=1}^rL'_i\). Assume in addition +that, for every finite use of its path property, the lower paths can +be chosen with last clock \(x_{\rm prev}=O(M')\). This is an +additional property of the length systems constructed below; it is +not part of the abstract conditions \textup{(R1)--(R4)}. +Let a new independent positive variable \(p\) have the distinguished limits +\[ + p\longrightarrow0,\qquad p\max L'\longrightarrow0, + \qquad |p-c|\gtrsim p^2\quad(c\in K_{\R}). +\] +The constant in the last comparison may depend on \(c\). Use domain +fibers in \(p\) which are open intervals cut out by finitely many +strict comparisons with lower scalars. There is a regular system +\(\mathcal A[p]\) containing the lower system and \(p,p^{-1}\), +closed under the finite analytic operations, with the following size +bound for each nonzero scalar and its reciprocal: +\begin{equation}\label{path:eq-p-size} + |a|+|a|^{-1}\le e^{C(1+\max L')}p^{-C}. +\end{equation} +Its finite path requirements can be realized by substituting for \(p\) +an expression in lower scalars, with an additional real constant varying +in an open interval, and preserving \(pM'\to0\). On the resulting +lower paths one therefore has \(px_{\rm prev}\to0\). + +For the empty lower tuple the same conclusion holds with ordinary +holomorphic germs and the choice \(p=\theta/x_1\), \(\theta>0\). +\end{lemma} +\begin{proof} +We give the chart construction, the normalization argument, and the +path argument separately. Real closedness of the lower field is +available from Lemma~\ref{path:roots}. + +\emph{Charts and their common refinements.} +For a real center \(c\in K_{\R}\), let \(s=|p-c|\); fix its sign on +the domain base. If \(s\) is comparable to a positive lower scalar, +use a disc coordinate +\[ + u=(p-c)/R +\] +with finite limiting value, where \(R>0\) is in \(K_{\R}\). If no +positive lower scalar has size comparable to \(s\), call the size +unmatched and use a ramified annular chart +\begin{equation}\label{path:eq-annulus} + u=(s/R)^{1/n},\qquad v=(r/s)^{1/n},\qquad u,v\longrightarrow0, +\end{equation} +where \(R,r\) are lower positive scalars, \(r=0\) is allowed, and the +positive real branches are used. In the latter chart permit scalar +prefactors \(ks^q\), with \(k\in K\), \(q\in\mathbb Q\). Functions +are lower analytic families in these bounded coordinates, at their +specified standard values, times such prefactors. Disc charts require +only lower scalar prefactors. + +Here is the finite refinement rule. Include zero and every center in +the finite data, and choose a center whose distance from \(p\) is +smallest in order. If its distance is matched, a disc at that scale +expresses all other required coordinates by bounded analytic +substitutions, after normalizing their nonzero distances. In the +unmatched case each nonzero difference of two centers is either +\(o(s)\) or much larger than \(s\): it cannot be comparable to \(s\), +since that would match \(s\) with a lower scalar. Write +\[ + p-c'=(p-c)+(c-c') +\] +as the larger term times one plus the smaller ratio. Choose annular +bounds \(r\ll s\ll R\) recording every one of these finitely many +ratios. Rational powers are analytic in the factor tending to one, +and a common root denominator handles all ramification. A required +bounded ratio of prefactors is treated in exactly this way: a ratio +\(ks^q\) has, for \(q\ne0\), a threshold at a lower root-size +representative of \(|k|^{-1/q}\). Add these finitely many thresholds +to the bounds \(r,R\). This proves the common-refinement rule, +including bounded analytic representation of every needed ratio. + +\emph{Normalization in an annulus.} +On the relation \(uv=\lambda=(r/R)^{1/n}\), decompose a bounded +lower analytic family \(H(u,v,\zeta)\) as +\[ + H_+(u,\zeta)+H_-(v,\zeta), +\] +where the second series has only positive powers of \(v\). This is +an actual convergent decomposition on a smaller bidisc. For example, +substitute \((w,\lambda/w)\) on a fixed circle in the original +\(u\)-disc and take its nonnegative Laurent part by Cauchy projection; +the negative part is the positive series in \(v\). Both belong to +the lower system, since \(\lambda\) is a bounded lower scalar and +the circle is fixed. If \(r=0\), set \(v=0\) and use just the first +series. + +Normalize a nonzero one of these families in the lower system, with +its series variable still ordinary. Let \(N\) be the first degree +whose coefficient has nonzero standard holomorphic limit in +\(\zeta\). The terms of degrees less than \(N\) have finitely many +lower scalar normalizers. The tail beginning at degree \(N\), +factored by that power, is bounded with nonzero standard limit at the +origin. Thus only finitely many candidate sizes occur. Distinct +powers of \(s\) cannot have comparable sizes after lower scalar +multiplication: such a comparison, together with divisibility of lower +sizes, would match \(s\). Choose the largest candidate size over +both Laurent parts. It is unique up to terms with the identical +power, which are combined before this comparison. All smaller +candidate ratios have zero limit, and the factored tails remain +bounded by the refinement rule. Division by the largest size +therefore gives a bounded analytic representation with nonzero +holomorphic standard limit. If both projected series vanish, the +original family vanishes. + +\emph{Normalization in a disc.} +Translate the disc coordinate to have limit zero and normalize the +family \(H(u,\zeta)\) in the lower system before substituting \(p\). +If it is nonzero, choose a Taylor coefficient \(b(u)\) in \(\zeta\) +whose standard limit is a nonzero germ in \(u\). Preparation and +division in \(u\), by Lemma~\ref{path:prepared-contours}, give +\begin{equation}\label{path:eq-disc-division} + H(u,\zeta)=b(u)G(u,\zeta)+S(u,\zeta), +\end{equation} +where \(G\) is bounded and \(S\) is a polynomial in \(u\). Also +\(b\) is a unit times a monic polynomial in \(u\). + +The finite-dimensional \(K\)-span of the coefficient germs in +\(\zeta\) of \(S\) has a basis \(\psi_1,\ldots,\psi_h\) whose +standard limits are linearly independent over \(\C\), with each +\(\psi_j\) bounded. To construct it, normalize a nonzero element, +choose one of its Taylor coefficients with nonzero limit, and normalize +that coefficient to be one. Pass to the kernel of the corresponding +coefficient functional in the original span and repeat. The dimension +drops by one at each step. The resulting triangular Taylor-coefficient +matrix has nonzero limiting diagonal, proving the assertion. Hence +\[ + S(u,\zeta)=\sum_{j=1}^h A_j(u)\psi_j(\zeta), +\] +where the \(A_j\) are polynomials over \(K\); its size after +specialization is the largest of their nonzero sizes. + +Factor these finitely many polynomials, together with the polynomial +for \(b\), over \(K\). Include the real parts of their roots, expressed +in the \(p\)-coordinate, among the centers of the refinement. The +imaginary parts are lower scalars; their sizes are additional comparison +thresholds. Every factor then has a bounded normalized analytic +representation, using either a disc or an annulus. The largest size +among \(b,A_1,\ldots,A_h\) normalizes \(H\). If an \(A_j\)-size +strictly dominates that of \(b\), independence of the standard +\(\psi_j\) prevents cancellation. Otherwise \(b\) itself is a +Taylor coefficient of \(H\), so a normalized limit of \(H\) cannot +vanish identically. The boundedness of the representation and Cauchy's +formula make this coefficient argument valid for holomorphic limits. + +\emph{Bounds and analytic closure.} +All resulting sizes are products of lower scalar sizes and rational +powers of distances and scales. Every nonzero distance from \(p\) +to a real lower center is at least a fixed multiple of \(p^2\). +Upper bounds for centers and scales, and bounds for their reciprocals, +are lower exponential bounds. The preceding finite factorizations +therefore give Equation~\eqref{path:eq-p-size}. Positive root-size +representatives follow from ramification and the lower representatives. +For a bounded scalar the chart formula gives its finite standard +limit; prefactor ratios are controlled by the recorded comparisons. +Nonzero real scalars have a fixed sign. A complex normalizer for a real +family can be replaced by a real or imaginary part of comparable size, +and then by a positive representative if desired. + +After common refinement, sums, products, ordinary composition, and +normalized unit inversion are the corresponding lower analytic +operations on bounded families. Contour integrals are treated by +finitely many chart neighborhoods covering the contour. The same +argument works for a compact segment. This proves +\textup{(R1)--(R3)} for the extension. Comparisons such as +\(|p-c|\gtrsim p^2\) can be expressed by interval endpoints in the +real lower field: factor the relevant real polynomial inequalities +using Lemma~\ref{path:roots}, order their roots, and choose the interval +containing the distinguished points. + +\emph{Finite specialization and path families.} +Refine using all chart centers, all endpoints of the required interval, +and zero. In a matched chart, with positive scale \(d\) and limiting +coordinate \(a\), choose +\begin{equation}\label{path:eq-matched-choice} + p=c+ad+\theta d\eta, +\end{equation} +where \(\eta>0\) is a lower infinitesimal and \(\theta\) varies in a +small open interval of constants. Its sign can be fixed to meet the +fiber constraints. Every included endpoint has nonzero normalized +distance at its own recorded scale. Choose \(\eta\) smaller than +the finitely many required relative margins. Then all comparisons +persist and every required coordinate has its exact limiting standard +part. If an endpoint were nearer than the chosen center-scale data, +it would already have been selected by the closest-center refinement; +thus this choice omits no cancellation relevant to the finite data. + +In an unmatched chart choose a lower scale strictly between all +required lower and upper bounds, for instance their geometric mean +after taking the largest lower and smallest upper bound. If the lower +bound is zero, multiply the upper bound by a positive lower +infinitesimal. Set \(s\) equal to this scale times \(\theta\) in a +bounded positive open interval, and set \(p=c\pm s\) with its recorded +sign. Thresholds required by bounded prefactors were already included, +so these substitutions preserve all finite normalization requirements. +They use only lower scalar and ordinary analytic operations. + +Include among the finite requirements \(p\to0\) and \(pM'\to0\). +The latter uses only the already defined positive lower scalar +\(M'\): it is the requirement \(p/(1/M')\to0\), handled by the +same disc or annular refinement. The lower system contains positive +infinitesimals, for example \(1/M'\). Apply the lower path property, +with its assumed clock bound, to these and all other expressions and +comparisons used in the chosen substitutions. Then +\(px_{\rm prev}=O(pM')\to0\); no comparison involving a clock +chosen only later is imposed on the domain base. +Formula~\eqref{path:eq-matched-choice}, or +its unmatched counterpart, has nonzero derivative in the new constant +\(\theta\). Together with submersivity of the lower evaluation, +the triangular differential gives submersivity into the extended +domain. Interval fibers over connected lower domains are connected +after ordering their endpoints; they remain the domain base under +these restrictions. This proves \textup{(R4)}. In the empty case, + \(p=\theta/x_1\) gives the stated meromorphic Puiseux families directly. +\end{proof} + +\subsection{Construction for several lengths} + +We now construct the regular system of +Definition~\ref{path:def-regular} for the original tuple of length +variables. The induction separates lengths of comparable size from +strictly slower differences, applies the reciprocal-length construction +when necessary, and preserves the common coefficient domains. Its +output will leave only the ordinary-variable sign conditions to solve +in the final subsection. + +For example, the admissible equation +\[ + e^{-L_1}-L_2e^{-L_2}=0 +\] +has the path \(L_2=e^u\), \(L_1=e^u-u\). +The lengths have ratio tending to one, but their difference is the +unbounded slower quantity \(-u=-\log L_2\). Thus grouping lengths by +their limiting ratios must retain the magnitudes of unbounded slower +differences as new centers. + +\begin{proposition}[Centers and regular systems]\label{path:centers} +For a distinguished sequence of \(m\) lengths tending to infinity, +there are centers \(L_1^*,\ldots,L_m^*\), differing from the sequence +values by bounded additive amounts, and a regular system on a domain +base with the following additional properties. +\begin{enumerate}[label=\textup{(\alph*)}] +\item The system contains the centers, \(e^{bL_i^*}\) for every real +\(b\), and every admissible family evaluated at these centers. +\item Every finite expression built with the allowed analytic +operations has coefficient formulas on a common connected domain of +the base and a common ordinary-parameter neighborhood. The same +formulas remain valid to all orders on paths preserving its finite +preparation data. Remainder sectors and thresholds may depend on +order. +\item On the path families, the grouped centers have the form +\(c_iX+D_i^*\), \(c_i>0\), where the positive magnitudes of the +unbounded \(D_i^*\) are slower centers. All required quadratic +holomorphy domains are reached after narrowing the clock sectors. +\item For a nonempty tuple, its path families can be chosen so that +the last clock is the selected fastest center \(X\). Every center +is \(O(X)\), and one of the centers equals \(X\). In particular +the last clock is comparable to \(\max_iL_i^*\), independently +of which finite expression requirements are imposed. +\end{enumerate} +\end{proposition} +\begin{proof} +We induct on \(m\), establishing all regularity and path assertions +together, including the last-clock invariant in \textup{(d)}. +The empty tuple has ordinary holomorphic germs, constant +scalars, and a point as its independent-variable domain. Its +normalization and contour assertions are immediate; the path assertion +has zero-dimensional target and is vacuous. + +\emph{Splitting the fastest block.} +Choose a fastest length \(X\), so that every ratio \(L_i/X\) has +a finite ultrafilter limit. This is possible by taking a maximal +one among the finitely many lengths. For the indices with positive +limit write +\[ + L_i=c_iX+D_i,\qquad c_i>0,\quad D_i=o(X). +\] +Ignore bounded \(D_i\) for now, replacing them by zero in the centers. +For each unbounded difference take its absolute value and record its +eventual sign. These magnitudes and the lengths with ratio zero to +\(X\) form a lower tuple of at most \(m-1\) entries. Indeed a +fastest block of size \(k\) contributes at most \(k-1\) differences, +and the other block has \(m-k\) lengths. Apply the induction +hypothesis to that tuple and denote the resulting positive centers by +\(L'\). Reinsert the recorded signs in the differences \(D_i^*\). + +If some real lower scalar approximates the original \(X\) to bounded +absolute error, replace \(X\) by it. Its positivity, its divergence, +and \(\max L'=o(X)\) are finite lower-system requirements. Otherwise +adjoin \(p=1/X\) as an independent variable. For every real lower +scalar \(c\), +\begin{equation}\label{path:eq-separation} + |p-c|\gtrsim p^2. +\end{equation} +In fact failure, or even a bounded ultrafilter value of +\(|p-c|/p^2\), would give \(c/p\to1\) and +\[ + X-1/c=\frac{c-p}{pc}=O(1), +\] +contrary to the case distinction. Thus Lemma~\ref{path:adjoin-p} +applies. In both cases call the resulting coefficient system +\(\mathcal A'\). Every nonzero scalar of \(\mathcal A'\), and its +reciprocal, is \(e^{o(X)}\), by +Equation~\eqref{path:eq-p-size}, \(\max L'=o(X)\), and +\(\log X=o(X)\). The final centers in the fastest block are +\(c_iX+D_i^*\), including \(X\) itself. All errors relative to the +distinguished original tuple are bounded. + +\emph{Initial grouped expansions and their domains.} +Adjoin \(e^{\alpha X}\), \(\alpha\in\R\), and the initial +admissible functions evaluated at the centers. Their fastest-block +expansions have the form +\begin{equation}\label{path:eq-grouped} + \sum_d e^{-dX}B_d,\qquad B_d\in\mathcal A', +\end{equation} +with locally finite weights and strict exponential remainder gains. +To obtain the coefficient, expand a term with fastest-block indices +\(d_i\) as +\[ + \prod_i e^{-d_i(c_iX+D_i^*)}(c_iX+D_i^*)^{k_i}. +\] +Its top exponential is \(e^{-(\sum_i c_id_i)X}\). Its remaining +factors are polynomials in \(X,D_i^*\), exponentials of the signed +lower centers, and recursively extracted initial coefficients in the +slower lengths. All are in \(\mathcal A'\). Definition~\ref{path:def-admissible} +supplies both the grouped remainder and the common analytic domains +of every coefficient formula. + +The inductive lower construction supplies +\(x_{\rm prev}=O(1+\sum_iL'_i)\), and contains the positive lower +centers as real scalars. Thus it meets the additional hypotheses of +Lemma~\ref{path:adjoin-p}. In the independent case that specialization +preserves \(p(1+\sum_iL'_i)\to0\); hence +\(x_{\rm prev}=o(X)\), where \(X=1/p\). In the dependent case +\(X\) is a lower scalar, and the already required comparison +\((1+\sum_iL'_i)/X\to0\) can be imposed using the lower path +property; it again gives \(x_{\rm prev}=o(X)\). Thus in either +case \(X\) belongs to the lower path field and is an admissible +next clock by Lemma~\ref{path:clock}. Declare it to be the last +clock of the new hierarchy. All fastest-block centers are +\(c_iX+o(X)\), the other centers are \(o(X)\), and one center +is \(X\), proving \textup{(d)}. When the lower tuple is empty, +the independent choice \(p=\theta/x_1\) gives the first clock +\(X=x_1/\theta\). Subsequent finite analytic operations and domain +requirements introduce no further clocks. +The slower centers and the positive magnitudes of the differences +satisfy +\[ + \Re L'=o(\Re X),\qquad |L'|=o(|X|) +\] +on a narrowed sector. Consequently +\(\Re(c_iX+D_i^*)\asymp\Re X\) and +\(|c_iX+D_i^*|=O(|X|)\). The admissible loss gives +\(\Re X\gtrsim |X|/(\log|X|)^2\), so for every fixed \(c>0\) +and sufficiently large \(|X|\), +\[ + |\Im(c_iX+D_i^*)| + \le C|X|0$ the inverse of Lemma~\ref{ell:seam-inverse} +continues to +\begin{equation} + \Re L_e>A,\qquad |\Im L_e|0$; this pays any polynomial contour loss. +On the unbent portion there is no such loss. The second +summand of the strengthened norm pays the polynomial losses +for the remaining source. + +To use a disconnected core inverse, solve forced data in the flat +part by these kernels, cut that solution off on a real collar, +and apply the real disconnected inverse to the remaining compact +error. Its decaying homogeneous far field is continued by the +same exponentials. This also constructs the inverse on an opened +end with a bend followed by a return to the real ray farther out. +If a bend starts increasingly far out, a cutoff for the forced +solve can be moved with it inside the flat region. The real +core estimate remains the fixed estimate, and the flat estimates +are unchanged. This proves the needed uniformity for such bends. + +Extend finite-half data through the seam and apply this +disconnected construction. Correct traces by the incoming modes +from Lemma~\ref{ell:seam-inverse}. They have polynomially bounded +norms relative to their weighted seam amplitudes. Their leading +trace matrices are unchanged because $\tau'=1$ at the endpoint. +The core correction is exponentially small. Its outgoing +homogeneous traces at every receiving seam have a strict extra +decay rate, absorbing the polynomial at that receiving length. +For the initially forced solution, data before $\ell_e/5$ reach +the seam with exponential gain; data after that point have the +strengthened norm on that very end. Increase $P$ once to pay +the subsequent trace correction as well. This proves a uniform +bound and a vanishing perturbation of the block trace map. +The same representation proves injectivity. + +Finally, a coefficient perturbation with size $O(e^{-\sigma u})$ +is small in the strengthened operator norm on a bent region, +since $(1+\ell_e)^Pe^{-\sigma\ell_e/5}\to0$. +Small perturbations on the remaining real portions are handled +by the real inverse. A Neumann series proves the perturbative +version that will be used below. This argument is unchanged if +any subset of the ends is left open. +\end{proof} + +Here and below ``uniform'' permits increasing the fixed lower +length threshold. The choices have the following order, which +avoids a circular truncation requirement. Fix weights, derivative +orders, the augmented operators at $h$, and their inverse bound +$C_0$. Fix a contraction radius $\delta>0$ so that the product +and inverse constants give a nonlinear Lipschitz constant at most +$1/4$ on the $2\delta$ ball. Choose $H$ so that the operator +perturbation is below $(2C_0)^{-1}$ and the reference defect is +below $\delta/(8C_0)$. Choose the finite parameter radius so +that its additional defect has the same bound. Finally choose +$A>10(H+1)$ large enough that all trace perturbations have norm +at most $1/4$ and that the required polynomial times exponential +bounds hold. The finite number of partial disconnections permits +the same choices for all of them. The inverse constants used to +choose $\delta$ are uniform in $H$ by +Lemma~\ref{ell:disconnected-inverse}. The scale-normalized tensors +$b_H$ have a uniform positive-definite lower bound, and $b_H$, +$b_H^{-1}$ and their derivatives through the fixed orders used +here have bounds independent of $H$: the transition uses a +fixed-width cutoff translated into regions where $h$ approaches +$\delta$. Consequently the local product and inverse-matrix +constants used to choose $\delta$ are independent of $H$ as well. +No new smallness radius is +chosen after increasing the expansion order. + +\begin{lemma}[Gauge for the real sequence] +\label{ell:gauge} +After a diffeomorphism of the invariance class in +Lemma~\ref{ell:functional}, each sufficiently late metric in the +sequence satisfies +\begin{equation} + W_b(g)=m\cdot\psi, + \label{ell:gauge-equation} +\end{equation} +where $\psi$ is the finite list of vector augmentation sources, +rescaled in physical units, and $m$ is small. The gauged metric +retains Equation~\eqref{eq:tree-ricci-error} in weaker admissible +weights and is close to $b$ at the joining weight $\gamma_g$. +\end{lemma} + +\begin{proof} +Solve for a map $f:(N,g)\to(N,b)$ near the identity with +$\tau(f;g,b)=-m\cdot\psi(f)$. Parametrize it by the background +exponential map of a displacement $V$, and impose zero core and +critical-seam tests on $V$. The derivative in $(V,m)$ is the +augmented vector isomorphism. On every coordinate patch of scale +$r$, this is the ordinary tension equation with bounded smooth +target coefficients. Sobolev multiplication at the chosen +orders, with locally comparable weights, bounds its quadratic +remainder and its Lipschitz difference. The uniform inverse and +the smallness of $g-b$ give a contraction. The latter smallness +follows from compact convergence and the integrable dominating +tail $e^{-(\mu-\gamma_g)\tau}$; the root argument uses $q-q_0>0$. + +Smallness of $V/r$ in scaled $C^1$ makes the entire displacement +isotopy locally invertible. It is proper, since the displacement +is small relative to radius and decays on the root, so it is a +degree-one covering and hence a diffeomorphism. The equation in +the image metric reads $-W_b(f_*g)=-m\cdot\psi$. +Seam matching of $V/r$ and its first derivatives gives a genuine +solution across the seam; the smooth equation then supplies the +higher compatibility. At infinity $V=O(r^{1-q_0})$, which is +admissible for Lemma~\ref{ell:functional}. Small scaled coordinate +changes preserve the end estimates after decreasing exponents. +\end{proof} + +We now write $g$ for the gauged metric. Choose the metric +augmentation sources $\varphi$ from +Lemma~\ref{ell:disconnected-inverse} and solve +\begin{equation} + P_b(h_*)+\tfrac12\mathcal L_{m\cdot\psi}h_*=d\cdot\varphi, + \qquad h_*=b+U. + \label{ell:family-equation} +\end{equation} +We make the scaling of the augmentation sources explicit. On a +vertex of physical length $a_v$, let $D_{a_v}(x)=a_vx$ denote +passage from vertex to physical coordinates. A vector source in the +tension equation has physical components $a_v^{-1}\psi_v$: +its pullback as a vector field is $a_v^{-2}\psi_v$. +A covariant two-tensor source has physical components +$a_v^{-2}\varphi_v$, so $D_{a_v}^*\varphi_{\mathrm{phys}}=\varphi_v$. +Consequently, writing $k_v$ for a metric in vertex coordinates, +\[ + D_{a_v}^*\bigl(\mathcal L_{m\psi_{\mathrm{phys}}}k_{\mathrm{phys}}\bigr) + =\mathcal L_{m\psi_v}k_v + \quad\text{when }D_{a_v}^*k_{\mathrm{phys}}=a_v^2k_v. +\] +These are equation-source scalings; a displacement itself scales as +$a_vV_v$. With these conventions the normalized bulk equations and +core tests have no length dependence. + +Here $d$ is an unknown finite vector. The core test values for +$U$, the vector $m$, and the coefficients of the critical seam +values +\begin{equation} + \pi U|_e=e^{-\lambda L_e}z_e + \label{ell:seam-prescription} +\end{equation} +make up a finite parameter vector $z$. Values and first +derivatives genuinely match on every remaining seam. On an +opened edge omit its matching and its critical prescription. + +\begin{lemma}[Analytic small branch] +\label{ell:analytic-family} +On a fixed complex polydisc in $z$ and the quadratic length domain +of Equation~\eqref{ell:quadratic-domain}, +Equation~\eqref{ell:family-equation} has a unique small solution +$(U,d)$. It obeys uniform bounds at both joining weights $\nu$ +and $\nu'$. Its augmented linearization has the same uniform +inverse bounds. These assertions hold for every partial +disconnection on common smaller domains. The family, its compact +observables, and its endpoint values are single-valued holomorphic +functions of the length parameters. +\end{lemma} + +\begin{proof} +On cores, inverse metric matrices and the tensor operations in +Equation~\eqref{ell:family-equation} are analytic in $U$ near zero; +the fixed spatial coefficients need only be smooth. On an +exactly flat end, in neutralized Cartesian components, +\begin{equation} + \mathcal P U=\mathcal N(U), + \label{ell:flat-equation} +\end{equation} +where $\mathcal P$ is the flat cylindrical component Laplacian +and $\mathcal N$ vanishes to second order. Its Taylor operations +contain at most two derivatives and have polynomial angular +coefficients. Indeed $r\partial_{x_i}$ is a combination of +$\partial_\tau$ and tangential derivatives with polynomial +coefficients in the unit Cartesian coordinates. Consequently +finite angular harmonic sums remain finite at each Taylor order. + +Apply Lemma~\ref{ell:complex-inverse} to the equation and all +tests. A quadratic term has one additional factor of decay +relative to the source weight. Its basic source norm is +bounded by $C\delta^2$, and the additional strengthened +contribution is bounded by +\[ + C\delta^2(1+\ell_e)^P e^{-\nu\ell_e/6}, +\] +because the cutoff support lies in $u\ge\ell_e/6$. +For differences the corresponding bound replaces $\delta^2$ +by $\delta\|U-V\|_{X^k_\nu}$. Taking the supremum over +$\ell_e$ gives constants independent of the truncation and +of all other lengths. +The seam data have size at most a polynomial times +$e^{-(\lambda-\nu)\ell_e}|z_e|$. The order of choices stated +above makes the equation a uniform contraction, at both $\nu$ +and $\nu'$. Uniqueness identifies the solutions. A Neumann +series around the augmented linear operator proves invertibility +of the branch linearization. Analytic convergence of the +contraction proves analyticity in ordinary parameters and in +local holomorphic variations of endpoints on a fixed contour. + +We specify why this is analyticity in $L$, despite a smooth rather +than holomorphic global choice of contours. In a local +parametrization of a flat contour, a holomorphic endpoint change +changes the coefficients analytically through +$\partial_\tau=(d\tau/du)^{-1}\partial_u$. +For an infinitesimal parametrization change $v(u)=\delta\tau(u)$ +fixing the endpoints, the candidate variation $v\partial_\tau U$ +satisfies the differentiated equations. The normal operator varies +by $\delta\partial_\tau=-(\partial_\tau v)\partial_\tau$, so +\[ + \delta(\partial_\tau U) + =\partial_\tau(v\partial_\tau U) + -(\partial_\tau v)\partial_\tau U + =v\partial_\tau^2U. +\] +Thus $v=0$ at each endpoint preserves both matching conditions and +the critical-value tests; on the fixed core portions it also preserves +the core tests. This variation belongs to $X^{k-1}_\nu=X^7_\nu$. +The lower-order inverse already supplied by +Lemmas~\ref{ell:disconnected-inverse}--\ref{ell:complex-inverse} +uses the same sources and tests, with value and critical traces in +$H^{13/2}$ and normal traces in $H^{11/2}$. The same small-perturbation +argument gives this inverse for the branch linearization. Its +uniqueness therefore identifies $v\partial_\tau U$ with the actual +solution variation, with zero variation of the compact unknowns. +Integrating it through a homotopy shows contour independence +where the coordinate is fixed. Monotone-real-part contours with +fixed endpoints are deformable through the same class; this +identifies the local holomorphic continuations and excludes +monodromy. For an open end take a bend returning to the real ray; +deformations are compactly supported in its flat region. Tail +limits on the unchanged ray are consequently unchanged. An +autonomous analytic density integrated with $d\tau$ changes by +the endpoint term $v$ times that density, and therefore its +integral is also invariant under fixed-endpoint deformations. +The same shape calculation applies to linearized solutions. +\end{proof} + +\begin{lemma}[Comparison with the actual metric] +\label{ell:comparison} +Choose $z$ from the core tests and gauge coefficients of $g$, and +from Equation~\eqref{ell:seam-prescription} at its seams. Then +\begin{equation} + \|g-h_*\|_{X^k_\nu}\le CM,\qquad + \|\Ric(h_*)\|_{Y^{k-2}_\nu}\le CM,\qquad + |E(g)-E(h_*)|\le CM^2. + \label{ell:comparison-estimates} +\end{equation} +\end{lemma} + +\begin{proof} +The seam parameters are small because $g-b$ has weight +$\gamma_g>\lambda$. All parameters lie strictly inside the +chosen small polydisc by taking the truncation and sequence +thresholds sufficiently large. In the gauge +Equation~\eqref{ell:gauge-equation}, the actual metric solves +Equation~\eqref{ell:family-equation} with $d=0$ and the extra source +$\Ric(g)$. Its source norm is $O(M)$ by +Equation~\eqref{eq:tree-ricci-error}. Subtraction and the uniform +inverse, absorbing the small nonlinear Lipschitz term, prove the +first estimate, including the difference in compact unknowns. +The local Ricci map from $X^k_\nu$ to $Y^{k-2}_\nu$ is Lipschitz +on the small ball, proving the second estimate. + +Along the segment from $g$ to $h_*$ the Ricci source and the +metric variation are both $O(M)$. In the interior the pairing +in Equation~\eqref{ell:first-variation} costs integrals of +$\rho^2$ against products of the joining weights; these are +bounded by the scale calculation in Lemma~\ref{ell:functional}. +On the root the radial integral is +$\int r^{1-2q_0}\,dr<\infty$. Integrating the first variation +proves the third estimate. +\end{proof} + +Define real functions on the real family by +\begin{equation} + F=E(h_*),\qquad + J=\int|\Ric(h_*)|_{h_*}^2\,d\mu_{h_*} + +\int_{\substack{\text{root end}\\r>r_0}} + r^{2q_1}|\Ric(h_*)|_\delta^2\,dx. + \label{ell:observables} +\end{equation} +For complex metrics the squares are analytic bilinear +contractions, without complex conjugation. These give bounded +holomorphic continuations of $F,J$ on the quadratic domains. +For $F$, integrate the quadratic subtracted scalar density on a +fixed root tail and retain its linear boundary contribution at +$r_0$. For $J$, the root weighted radial bound is +$\int r^{2q_1-2q_0-1}\,dr<\infty$. Interior Ricci-square +integrals are scale invariant in dimension four and integrable +in cylindrical measure. Scalar-volume integrals carry the factor +$a_v^2r^2$: on a child half this is +$a_p^2e^{-4L_e}e^{2\tau}$, whose modulus stays bounded. +Every contour polynomial is absorbed by the available exponential +decay. On the approximants in Lemma~\ref{ell:comparison}, +\begin{equation} + 0\le J\le CM^2. + \label{ell:J-small} +\end{equation} diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/05-tails.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/05-tails.tex new file mode 100644 index 0000000000000000000000000000000000000000..a30f8d06aa607b2765037b73543066abc731469d --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/05-tails.tex @@ -0,0 +1,213 @@ +\section{Tail expansions and prescribed growth} +\label{ell:tail-calculus} + +Bounded holomorphy alone does not allow the application of +Theorem~\ref{thm:path-selection}. We prove the all-order expansion +and common-domain assertions needed there. All expansions below +are finite to each requested precision. An infinite formal +power-logarithm sum is never used as an inverse or a solution. + +\begin{definition}[Tail expansion to finite precision] +\label{ell:tail-definition} +On an opened joined-type end, a field has an integer tail if, for +each nonintegral rate $B$, it can be written +\begin{equation} + Y(\tau,\theta)= + \sum_{lB}. + \label{ell:tail-series} +\end{equation} +Here $l\in\Z$. There is a lower bound for $l$, each $Y_{lj}$ is a finite +equivariant spherical harmonic sum, and the sum is finite. +The remainder belongs to the cylindrical Sobolev space with any +fixed weight strictly below $B$. A bound to finite precision +controls the finitely many displayed coefficients and that +remainder norm. Polynomial contour costs can be paid by a +strictly smaller remainder rate. For a growing field this is +a bound after subtraction of the finite nondecaying tail; it is +not a bound on the unsubtracted field at infinity. +\end{definition} + +The extraction of homogeneous exponential terms when a weight crosses +an indicial root is part of the classical change-of-weight method +\cite[Section~5]{LockhartMcOwen1985}. Here the componentwise flat +operators allow an explicit polynomial calculation, including the +resonant logarithmic terms. + +\begin{lemma}[Constant-operator tail improvement] +\label{ell:flat-tail} +Let $\mathcal P$ be either flat operator above. Suppose a field +has a finite growth bound and its source has an expansion to +rate $B$, where the intermediate remainder weights avoid the +indicial roots. Then the field has an expansion to that rate, +with free homogeneous coefficients recorded at the crossed +roots. Estimates involve the source coefficients and remainder, +a fixed inner trace, and the previous growth bound. They are +uniform on the contours of Lemma~\ref{ell:complex-inverse}, with +an arbitrarily small loss in a strictly available exponential +rate. +\end{lemma} + +\begin{proof} +In one angular channel, apply +$(\partial_\tau-\kappa_1)(\partial_\tau-\kappa_2)$ to +$e^{-l\tau}p(\tau)$. It gives $e^{-l\tau}$ times a constant +coefficient differential operator on the polynomial $p$. +If $-l$ is not a root, this is invertible on polynomials of a +given maximum degree by a triangular coefficient calculation. +If $-l$ is a root, integrate one polynomial factor, increasing +the degree by one; prescribe zero constant term for that +primitive and record the omitted constant as a free homogeneous +coefficient. The roots are distinct, so this exhausts resonance. +Only finitely many angular channels occur in the forced finite sum. + +Subtract these primitives. Extend the remaining forcing by a +fixed cutoff to a full-cylinder source and solve it by the root +kernels at a new noncritical weight. The difference from the +original field is homogeneous on the tail. The previous growth +bound excludes roots growing faster than it permits; the roots +between the old and new weights are recorded explicitly. The +remaining faster-decaying homogeneous modes are bounded by +the inner trace, using the exponential root gap and summing +their squared coefficients. This proves both the improvement +and its estimate. The kernel calculation on contours is the +one in Lemma~\ref{ell:complex-inverse}; polynomial losses are +absorbed by decreasing the remainder weight within its strict +gap. No parameter-dependent global inverse at the new weight +has been invoked. +\end{proof} + +\begin{lemma}[Nonlinear tails and finite prescribed growth] +\label{ell:prescribed-tails} +The small solution of Lemma~\ref{ell:analytic-family} on every +partial disconnection has integer tails with only $l>0$. +Its augmented linearization, denoted $\mathcal A_U$, can be +solved for a source with a finite-growth integer tail, with +arbitrary finitely supported prescriptions of free homogeneous +coefficients in the nondecaying indicial channels. Unspecified +coefficients are zero, so the prescribed growth has a finite maximum. +The result has an integer tail; its growth does not exceed the +maximum of zero, the forced growth, and the prescribed growth, +apart from polynomial factors. The solution is unique with +these prescriptions and the other augmented data. Estimates +to any fixed finite precision are linear in the data, with +constants allowed to depend on that precision and growth. +Every finite tail coefficient depends analytically on the +parameters on one common domain. +\end{lemma} + +\begin{proof} +First consider the small nonlinear solution. Its initial +positive-weight estimate and +Equation~\eqref{ell:flat-equation} give a source at twice a +slightly weaker positive decay rate. Apply +Lemma~\ref{ell:flat-tail}. Suppose finitely many powers have +already been extracted and the remainder has positive rate $B$. +In the difference between the nonlinear expression and its +finite Taylor expansion, every occurrence of that remainder +is multiplied by a field or derivative with a fixed positive +decay rate. The analytic Taylor remainder is controlled by +Sobolev multiplication, since $H^k$ is an algebra at the chosen +orders. Consequently the next source remainder has rate at +least $B+\epsilon$ for one fixed sufficiently small +$\epsilon>0$, using intervening noncritical weights when needed. +This improves indefinitely. The initial roots are integers; +Taylor products add their degrees. The angular property in +Lemma~\ref{ell:analytic-family} makes each resulting coefficient +a finite harmonic sum. The original positive decay excludes +all nonpositive degrees. This proves the first assertion, +including estimates on the complex contours. + +On an opened end the linearization is +$\mathcal A_U=\mathcal P+\mathcal B_U$, where the coefficients +of the differential operator $\mathcal B_U$ have positive +integer tails and decay. Start at the most growing required +power of the desired solution. Use the polynomial primitives +in Lemma~\ref{ell:flat-tail} to cancel it. Multiplication by +$\mathcal B_U$ improves the degree by a positive integer, so +the recursion is triangular in degree. At each indicial +channel fix the log-free homogeneous coefficient to be its +prescribed value; use the zero-constant polynomial primitive +for the forced part. Continue through finitely many degrees +until the residual decays strictly faster than the basic +weight $\nu$. This constructs a finite sum $T$, with no +growth greater than the allowed maximum. + +Choose a fixed cutoff $\zeta$ supported on the opened end and +equal to one sufficiently far out. With all finite seam data, +core tests, and compact source unknowns included in +$\mathcal A_U$, set +\begin{equation} + Y=\zeta T+ + \mathcal A_U^{-1}\bigl(f-\mathcal A_U(\zeta T)\bigr). + \label{ell:finite-growth-inverse} +\end{equation} +The term in parentheses is in the basic decaying source space. +The inverse is exactly the inverse already constructed in +Lemma~\ref{ell:analytic-family}. No smallness of $T$ or of this +source is required: this is a linear solve. The correction +decays and cannot change any prescribed nondecaying coefficient. +Applying Lemma~\ref{ell:flat-tail} to its equation, and using +the coefficient tails of $\mathcal B_U$, obtains all further +orders. If two solutions have the same data, their difference +has no nondecaying tail, hence belongs to the basic decaying +space and vanishes by its inverse. This proves uniqueness and +the growth assertion. + +All steps in Equation~\eqref{ell:finite-growth-inverse} are +finite polynomial operations, fixed cutoffs, and the same +analytic inverse family. A leading coefficient can alternatively +be extracted by multiplying the tail by its leading exponential +and removing its known polynomial terms, then taking its limit +at infinity. The remainder estimates are locally uniform in +the parameters, so these limits are holomorphic. Repeat only +finitely many times for any requested coefficient. The +parameter domain never shrinks as the precision or prescribed +growth increases; only estimates and a convenient tail-start +threshold may change. In particular, a cutoff moved farther +out on an opened infinite end does not change the admissible +range of another edge's finite length. +\end{proof} + +\begin{lemma}[Triangular auxiliary equations] +\label{ell:triangular} +Lemma~\ref{ell:prescribed-tails} applies successively to every +finite system appended to the small solution in which each +equation has diagonal operator $\mathcal A_U$ and source given +by Taylor differential operations on earlier fields. Allow +finite growth at already opened ends, and free asymptotic +conditions depending polynomially on finitely many earlier +tail coefficients. On still-finite seams impose homogeneous +matching and zero added critical tests. Require that sources +on those halves have at least two decaying factors, counting +the base field and its derivatives. On opening a new pair +of ends, start with decay there. The resulting fields and +the inverse of the triangular linearization have uniform +finite-precision bounds and the same analytic parameter domain. +\end{lemma} + +\begin{proof} +Solve the equations in their given order by +Equation~\eqref{ell:finite-growth-inverse}. Products of finitely +many fields with finite growth still have finite growth. +Forced nondecaying coefficients are determined by the equation; +only the free indicial constants are prescribed. Thus a forced +power is never incorrectly set to zero by an asymptotic boundary +condition. Conditions involving earlier coefficients are known +data at the relevant step. On a finite bent end the two +decaying factors supply the exponential surplus needed to pay +the polynomial strengthening on that same end. Linearizing +the finite triangular system leaves $\mathcal A_U$ on every +diagonal, so induction gives its inverse as well. + +At a newly opened end every earlier field initially decays. +The off-diagonal coefficients of this linearization therefore +decay there. This fact will preserve the leading incoming +mode calculation when this end is closed again. On older +opened ends the off-diagonal coefficients can grow, but their +growth is finite and is handled at each successive step. +Arbitrarily large bounds for a finite auxiliary field do not +affect the smallness of the base metric or the domain of its +inverse. This proves all the assertions. +\end{proof} + diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/06-length.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/06-length.tex new file mode 100644 index 0000000000000000000000000000000000000000..c735276cc24f3ab36e494fca883ceace42687358 --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/06-length.tex @@ -0,0 +1,166 @@ +\section{Expansion in one length} +\label{ell:length-expansion} + +Fix an edge length $L$ that is still finite. Open that edge, +retaining all other lengths and ordinary parameters. Let $Y_0$ +denote the resulting solution, including compact unknowns and +any finite triangular system already appended. At the two +new ends it is initially decaying. We will close those ends +by adding finitely many terms of the form +\begin{equation} + e^{-pL}L^jY_{pj}. + \label{ell:length-terms} +\end{equation} +The coefficient fields are solutions of the disconnected +triangular linearization. They may grow at the two new ends. +If their largest growth exponent there is $b$, assign the term +the usable order +\begin{equation} + s=p-\max(0,b). + \label{ell:usable-order} +\end{equation} +Take the maximum over both ends. Polynomial powers in $\tau$ +or $L$ do not change this order. All $p,s$ are on the grid +$\lambda\Z$, because the tail powers and indicial roots are +integers and the seam prescription uses $\lambda=1/n$. +The terms constructed below have $s>0$. + +Coefficient fields decay on every other still-finite half-end and +may have prescribed finite growth on older opened ends. At the +pair now being closed, the scalar factor in each length term +compensates its incoming growth through the usable order $s$. +The remainder estimate will keep these three bounds separate. + +\begin{lemma}[Finite length expansion] +\label{ell:one-length} +For every desired exponential precision, the actual small +solution and every appended field have an expansion by finitely +many terms in Equation~\eqref{ell:length-terms}, with an +exponentially better remainder. It is uniform in the remaining +finite lengths on their quadratic regions, in a fixed smaller +ordinary-parameter polydisc. For older opened ends the conclusion +holds in every specified finite tail norm. Coefficient fields +are obtained by the finite constructions in +Lemmas~\ref{ell:prescribed-tails} and~\ref{ell:triangular}. +\end{lemma} + +\begin{proof} +We give the induction and then its remainder estimate. Process +usable orders $s=\lambda,2\lambda,\ldots$. Products add lower +bounds for usable order, since +\[ + \max(0,b_1+b_2)\le\max(0,b_1)+\max(0,b_2). +\] +No initial bulk residual is present: $Y_0$ solves the opened +equations. Expand the remaining bulk equations by a finite +Taylor formula about $Y_0$, placing their full linearization +on the left. Core equations, compact coefficients, and +previously imposed asymptotic conditions are part of these +equations. At order $s$ only finitely many products of the +earlier positive-order terms occur. For each coefficient of +$e^{-pL}$ times a polynomial in $L$, cancel its complete forced +field by the disconnected linearized inverse, with zero free +incoming constants on the two new ends. A source assigned +order $s$ has growth at most $p-s$ there; by +Lemmas~\ref{ell:prescribed-tails} and~\ref{ell:triangular} its +correction has the same upper growth bound. When this bound +is negative the inverse may also contribute decaying +homogeneous terms; these still have positive usable order. +More uniformly, record each individual product with its +product lower bound and allow the bound $\max(0,p-s)$. +For the terms at issue $p\ge s$, by +Equation~\eqref{ell:usable-order}. The newly solved linear +terms vanish exactly. Their later bulk interactions are +higher-order products. + +Next evaluate the value, normal trace, and critical-test +residuals at $\tau=L$, using integer tails to finite precision. +At order $s$ they are finite harmonic sums times +$e^{-sL}$ and polynomials in $L$; include the prescribed +$e^{-\lambda L}z_e$ term. For every incoming root $b\ge0$ +there is a homogeneous disconnected variation with prescribed +leading mode $e^{b\tau}$ on its selected side. Prescribe zero +free nondecaying constants elsewhere. Its difference from +that leading mode has strictly smaller growth at the two +new ends, because the coefficients coupling channels decay +there. For $b=0$ all other new-end components decay. +Multiply this variation by $e^{-(s+b)L}$ times a polynomial +in $L$. Its leading seam data have order $s$. The block +matrices in Lemma~\ref{ell:seam-inverse} solve every leading +seam residual. Their lower tails contribute only later +orders. Matching at all other finite seams is preserved by +the disconnected inverse. This completes the induction step. +Finiteness follows from the positive minimum tag $\lambda$, +the finite harmonic support at each previously reached order, +and the integer root lists. In particular, no angular +analyticity estimate uniform in all expansion orders is used. + +We now justify that the construction approximates the actual +fields. Restrict each coefficient to the new halves +$\Re\tau\le\Re L$. Before applying the basic weight, a term +with usable order $s$ has field norm, or source norm with two +fewer derivatives, bounded by +\begin{equation} + C(1+\Re L)^C e^{-s\Re L}. + \label{ell:tag-bound} +\end{equation} +For a decaying coefficient this uses its whole half-cylinder +norm; for a growing coefficient it uses the finite tail and +$e^{b\Re\tau}\le e^{b\Re L}$. Products consequently cost +only polynomials in this unweighted estimate. Analytic Taylor +remainders beyond a fixed tag cutoff contain arbitrarily many +positive-order factors if the Taylor degree is chosen large +enough. The base nonlinear field and its corrections on these +halves are small, so those Taylor estimates are valid. +Apply the basic weight only after the product estimate; it +costs $e^{\nu\Re L}$ once, not once per factor. The fixed +polynomial source strengthening costs an arbitrarily small +additional exponential rate. The same reasoning applies +to seam tails and their first normal traces, which have been +expanded to a strictly higher rate before evaluation. +It follows that a sufficiently high finite tag cutoff gives +any requested residual order in all basic norms. + +On another still-finite half, the coefficient fields decay +in the fixed basic weight. Residual bulk products contain +the two factors required by Lemma~\ref{ell:triangular}, +which absorb the polynomial loss in that other length. +There is no seam residual there. On an older opened end, +every term retains its full scalar factor $e^{-pL}$. +Products there are estimated to a higher finite tail precision +to offset the finite growth of the other factors. For +unexpanded analytic operations on the base metric, its own +corrections at old opened ends remain decaying: their equations +and old-end conditions depend only on the base metric and +its own corrections. Thus Taylor remainders there are +controlled by extracting additional finite Taylor powers; +growing auxiliary fields appear only in finite differential +products and triangular equations. This proves residual +estimates in all required old-end tail norms as well. + +Compare the nonlinear approximate metric first in the basic +decaying spaces, allowing its seam jumps as data. The +small-branch inverse and the small nonlinear Lipschitz bound +give the same arbitrary exponential accuracy for its difference +from the true metric. On an old opened end the difference +equation has flat principal operator, decaying coefficients +with integer tails, and the residual just bounded to higher +precision. Apply Lemma~\ref{ell:flat-tail} repeatedly. It +upgrades the initial small error to the same exponential +accuracy for any specified tail coefficients and remainder. +For an appended field, work in triangular order: its difference +equation has the same diagonal inverse and errors from previous +members multiplied by at most finite growth. Apply +Lemma~\ref{ell:prescribed-tails}, also to differences of its +free prescriptions. This proves the asserted comparison in +finite tail norms, without ever trying to put a growing +field itself in a decaying space. + +All estimates use a fixed high Sobolev order. A Taylor source +uses at most two derivatives and its inverse regains those +two derivatives. Thus iteration creates no accumulating +differential loss. Traces and contour shape differentiation +may use one lower order, and parameter derivatives use Cauchy +estimates in the independent analytic variables. +\end{proof} + diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/07-observables.tex b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/07-observables.tex new file mode 100644 index 0000000000000000000000000000000000000000..40c8ea2464f0da603bd8300c55f9d48755ee2f4b --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/build/sections/07-observables.tex @@ -0,0 +1,307 @@ +\section{Scalar expansions and the tree inequality} +\label{ell:scalar-calculus} + +\begin{lemma}[Finite-part integration] +\label{ell:finite-parts} +Integration of the coefficient fields constructed in +Lemma~\ref{ell:one-length} preserves exponential-polynomial +length expansions for $F$ and $J$. Each resulting coefficient +is a finite sum of core or root-tail integrals, finite-part +integrals on opened ends, and evaluations of finite tail +coefficients, on the corresponding partial disconnection. +These operations are analytic and continuous in sufficiently +high finite tail precision. +\end{lemma} + +\begin{proof} +For clarity fix a normalization of finite parts. Let an +integrand density on a complete opened end, already integrated +over its cross-section, be $f(\tau)d\tau$. Subtract a fixed +cutoff times every nondecaying term of its finite tail, +and denote this finite sum by +$T(\tau)=\sum_{\alpha\ge0,j}c_{\alpha j}e^{\alpha\tau}\tau^j$. +Let $Q_{\alpha j}$ be a chosen polynomial-exponential primitive +of $e^{\alpha\tau}\tau^j$, with zero additive constant. +The remainder $f-\zeta T$ is integrable once its tail rate is +positive. There is then a uniquely normalized constant $C_f$ +such that +\begin{equation} + \int_{\tau_0}^{L}f(\tau)\,d\tau + =C_f+\sum_{\alpha\ge0,j}c_{\alpha j}Q_{\alpha j}(L) + -\int_L^\infty(f-T)(\tau)\,d\tau + \label{ell:finite-part-formula} +\end{equation} +when $L$ is beyond the cutoff. Define $C_f$ by integrating +$f-\zeta T$ and the fixed compact correction. The final +integral uses the exact tail after subtraction and can also +be expanded by subtracting its finitely many decaying terms. +If the remaining rate is $\eta>0$, its integral is bounded +by $C_\eta$ times that finite tail norm, by Cauchy--Schwarz +against $e^{-\eta u}$. This proves continuity. It also proves +analyticity by locally uniform dominated integration; the +coefficients themselves are analytic. On complex contours, +use $d\tau$ and the contour-independence argument from +Lemma~\ref{ell:analytic-family}. A strict exponential margin +pays the polynomial length of the contour. + +Apply this formula to each Taylor coefficient of the densities +in Equation~\eqref{ell:observables}. They are finite analytic +differential operations on the base field, or finite Taylor +operations on appended fields. The interior Ricci-square +density is scale neutral. A scalar-volume density has the +additional factor $a_v^2e^{\pm2\tau}$. On a newly opened +child half its endpoint factor is accompanied by $e^{-4L}$. +Thus all primitives still have integer exponential powers +and polynomial logarithmic factors, with the stated grid +after multiplication by the factors in +Equation~\eqref{ell:length-terms}. + +Core integration is continuous in the basic Sobolev norm. +Root-tail integration is continuous in the fixed root weights, +using $q_0>1$ for the subtracted density and $q_10$, then choose +the tail precision higher still if required by the finitely +many auxiliary growth exponents. The allowance $\nu$ pays +the basic weight, $2$ pays the worst radial factor, and the +strict surplus pays every polynomial factor. The actual +child factor $e^{-4L}$ only improves this bound. Thus the +formal integration and endpoint truncation have genuine +exponentially better remainders. +\end{proof} + +\begin{proposition}[All-order scalar expansion] +\label{ell:all-orders} +The functions $F,J$ satisfy the analytic expansion hypotheses +of Theorem~\ref{thm:path-selection}. Their length exponents +belong to the discrete nonnegative grid $\lambda\N\cup\{0\}$. +In any length, and recursively after taking any finite list +of coefficients, expansions have exponentially better +remainders at every cutoff. All recursively obtained +coefficients are analytic on a common fixed-width complex +tail band of the remaining lengths and one common open +complex neighborhood of the ordinary parameters. +The neighborhood does not decrease with the expansion order. +\end{proposition} + +\begin{proof} +Lemmas~\ref{ell:one-length} and~\ref{ell:finite-parts} give an +expansion in one length. Its coefficients are finite constructions +of the same kind: triangular field equations with finite free +asymptotic data, followed by core and root integrals, finite parts, +and tail-coefficient evaluations. On every still-finite seam the +added fields have homogeneous matching and zero critical tests. +Opening the next edge therefore uses the same lemmas; variations +of conditions at older opened ends expand their linear free-coefficient +equations in triangular order. + +We check that each such coefficient is bounded uniformly in the +lengths still finite. Fix its entire finite construction and a +sufficient finite tail precision. A prescribed growing tail is cut +off on an already opened end in +Equation~\eqref{ell:finite-growth-inverse}. The correction is obtained +from the fixed basic inverse, so it still decays on every finite +half-end. Repeating the finitely many triangular solves preserves +uniform field and tail bounds. Polynomial prescriptions in earlier +tail coefficients are bounded by the preceding steps, and the two +decaying factors on still-finite halves pay their strengthened source +norms. Constants may depend on this coefficient, its growth and its +chosen precision, but not on the remaining lengths or parameters +in a fixed smaller polydisc. + +The scalar operations preserve these bounds. Core and root integrals +are continuous in the fixed norms. Tail-coefficient evaluation and +finite-part integration on an older opened end are bounded maps at +the chosen sufficient precision, by Lemma~\ref{ell:finite-parts}. +On a still-finite edge keep the physical scale factors belonging to +unexpanded lengths. Scale-neutral Ricci-square terms have two +decaying factors and hence a uniformly bounded cylindrical integral. +For scalar-volume terms, the parent punctured half has the integrable +factor $a_p^2e^{-2\tau}$, while the child half satisfies +\[ + a_c^2\int_{\tau_0}^{L_e}e^{2\tau}\,d\tau + \le C a_p^2e^{-2L_e},\qquad a_c=a_pe^{-2L_e}. +\] +The bounded normalized fields multiply these factors. On complex +contours the corresponding bounds use real parts; polynomial contour +costs are absorbed by decay on the bent portion or by the displayed +child factor. Thus each recursively extracted scalar coefficient +has the required uniform bound on quadratic regions, with thresholds +and constants allowed to depend on its finite construction. + +Every scalar exponent lies on $\lambda\Z$. Group equal exponents. +For fixed real remaining parameters, boundedness as the expanded +length tends to infinity excludes a nonzero negative exponent and +a positive polynomial degree at exponent zero: the first such term +would dominate all later terms. The recursive bounds just proved +permit the same argument after any previous coefficient extraction. +Finite expansions are unique, since a nonzero polynomial cannot +decay exponentially after multiplication by the exponential of the +smallest differing exponent. Applying this at real remaining +parameters and then using analytic continuation makes the formulas +from different cutoffs agree. This gives consistent expansions on +the nonnegative grid. + +We give the common-domain argument separately, since merely +having an expansion to every finite order would not imply it. +For each of the finitely many partial disconnections, fix its +small solution and its augmented basic inverse from +Lemma~\ref{ell:analytic-family}. Take once the minimum of +their parameter radii and the maximum of their length thresholds, +and restrict to +a smaller polydisc compactly contained in all of them. +Their common quadratic regions include a band +\begin{equation} + \Re L_e>A_*,\quad |\Im L_e|0$. There are only finitely many +grid multi-indices with $\sum_i d_ic_i\le B$ for any fixed +$B$. Expand the first length sufficiently far that its +remainder has order greater than $(B+\epsilon)/c_1$. +For each of its finitely many retained coefficients, expand +the next length sufficiently far to put the resulting +weighted remainder above $B+\epsilon$, and continue. +Uniform coefficient bounds in the still unexpanded lengths +make this a legitimate finite procedure. Choose the +strict surplus before making the finitely many choices. +The $o(U)$ changes of the length ratios and all logarithmic +polynomials consume less than that surplus for large $U$. +This gives the required grouped weighted expansions with +strictly better remainders, including after previous +coefficients have been taken. These are precisely the +remaining expansion conditions in +Theorem~\ref{thm:path-selection}. +\end{proof} + +\subsection{The differential estimate and the contradiction} +\label{ell:conclusion} + +\begin{lemma}[Differential of the scalar observable] +\label{ell:differential} +On the real small family, for some $c>0$, +\begin{equation} + |dF|\le C\sqrt J\left(\sum_\alpha|dz_\alpha| + +\sum_e e^{-cL_e}|dL_e|\right). + \label{ell:dF} +\end{equation} +\end{lemma} + +\begin{proof} +For ordinary-parameter derivatives the uniform linearized +inverse gives the fixed weighted field bounds. Pair the +first variation with the first Ricci-square term of $J$ on +the physical interior. Its dual metric-variation $L^2$ +norm is uniformly bounded: core volumes scale by $a_v^4$, +and the half-end volume calculation uses +$a_v^4e^{\pm4\tau}d\tau$, with the factor $e^{-8L_e}$ on +a child side. The scalar part of the Einstein tensor costs +only a dimension-dependent multiple of $|\Ric|$. +On the root use instead the weighted part of $J$ and +Cauchy--Schwarz. The dual integral is bounded by +\[ + C\int_{r_0}^{\infty}r^{-2q_1}r^{-2q_0}r^3\,dr<\infty, +\] +since $q_0+q_1>2$. This proves the $dz$ part. + +Now differentiate one length $L_e$, initially at fixed vertex +and end coordinates. The normalized bulk equations and +the compact tests have no dependence on this length. Only +the seam data acquire an inhomogeneity: derivatives of the +prescribed $e^{-\lambda L_e}z_e$, and endpoint normal +derivatives of the solution in its matching conditions. +The solution has the stronger joining weight $\nu'>\nu$. +The trace theorem, using one fewer derivative if necessary, +therefore bounds these differentiated data in the basic +weighted norms by +\[ + C\bigl(e^{-(\nu'-\nu)L_e} + +e^{-(\lambda-\nu)L_e}\bigr)|dL_e|. +\] +The linearized inverse gives this same bound throughout +the normalized field. Its first-variation pairing is +bounded by $C\sqrt J e^{-c_1L_e}|dL_e|$ for any fixed +$00$. Then +$|E(g)|\ge\epsilon M$ for some fixed $\epsilon>0$. +By Lemma~\ref{ell:comparison} and +Equation~\eqref{ell:J-small}, late terms satisfy +\[ + F\ne0,\qquad F^2\ge c_2J,\qquad F\longrightarrow0, +\] +for some $c_2>0$. The finite parameter vectors have an +interior subsequential limit: all the earlier smallness +choices can be strict inside the parameter polydisc. +On real parameters, $J\ge0$ by Equation~\eqref{ell:observables}. +Proposition~\ref{ell:all-orders} supplies admissibility, and +Lemma~\ref{ell:differential} supplies Equation~\eqref{ell:dF}. +The resulting sequence therefore satisfies every hypothesis +excluded by Corollary~\ref{path:variation}, a contradiction. +This proves Equation~\eqref{ell:little-oh}. +\end{proof} diff --git a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5e22ec024ebec258df55fbb8c2283d3d09f54a27 --- /dev/null +++ b/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a49f3dc92b22b888e356c7d8e02620683b9e1c00ca1b28212cf800dd4d692327 +size 542345 diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/README.md b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..b937cf00f61a436e1a5fab19e6e3a2575c4f7fdf --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [Perfect completeness for 2-to-1 games](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Perfect-completeness-for-2-to-1-games-September-23-2026, + author = {{OpenAI}}, + title = {{Perfect completeness for 2-to-1 games}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf}{OAI:Perfect-completeness-for-2-to-1-games-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/00-introduction.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..0f3d5a761f25f731cc6dc288eea319846100f686 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/00-introduction.tex @@ -0,0 +1,254 @@ +\section{Introduction}\label{sec:introduction} + +A \emph{projection game} consists of a finite bipartite multigraph with disjoint +vertex sets $U,V$, a nonempty multiset $E$ of edge occurrences, finite answer +sets at its vertices, and a map $\pi_e$ from the left answer set to the right +answer set for each occurrence $e=(u,v)$. +A labeling chooses one answer at each vertex and satisfies $e$ when the right +answer equals the image of the left answer under $\pi_e$. +The \emph{value} of the game is the maximum fraction of satisfied occurrences. +Parallel occurrences may have different projection maps. + +A \emph{2-to-1 game} has a common left alphabet $[2q]$ and right alphabet $[q]$, +where $[m]=\{1,\ldots,m\}$, and +\[ + |\pi_e^{-1}(b)|=2 + \qquad(e\in E,\ b\in[q]). +\] +Khot introduced the games conjectures in the study of hardness of +approximation~\cite{Khot2002}. His original formulation allowed fibers +of size at most two; we use the exact-two convention. The 2-to-1 Games +Conjecture asks for +NP-hardness of distinguishing satisfiable games from games of arbitrarily +small fixed value. The satisfiable case is the \emph{perfect-completeness} +requirement: one labeling must satisfy every constraint. Completeness +$1-\eps$ for each fixed $\eps>0$ does not itself give this endpoint, since +those reductions need not produce any finite satisfiable game. +There is also an obstruction to amplifying a constant gap by ordinary +repetition: repeating an exact-two projection $k$ times gives $2^k$ +preimages of every right answer. We prove the conjecture while retaining +two-element fibers, in the following explicit form. + +\begin{theorem}\label{thm:main} +For every fixed rational $\delta\in(0,1)$ there are an integer $q=q(\delta)\ge2$ +and a deterministic polynomial-time reduction from $3$-SAT to 2-to-1 games +with alphabets $[2q]$ and $[q]$ such that: +\begin{enumerate}[label=\textup{(\roman*)}] +\item a satisfiable formula produces a game of value $1$; +\item an unsatisfiable formula produces a game of value at most $\delta$. +\end{enumerate} +The output contains a nonempty explicit list of unweighted edge occurrences +and an explicit table for every projection map. +Every table has exactly two preimages for each right answer. +The polynomial running-time bound is in the ordinary binary input length; +its degree and constants may depend on the fixed $\delta$. +\end{theorem} + +The theorem supplies the hypothesis of several perfect-completeness +reductions. We derive consequences for maximum $k$-colorable subgraph +and satisfiable Not-Two constraints below, identifying those already +known unconditionally. + +\subsection{Historical context and ingredients} + +Austrin, O'Donnell, Tan and Wright established perfect-completeness 2-to-1 +Label Cover hardness with alphabets of sizes $6$ and $3$ and soundness +$23/24+\eps$~\cite[Theorem~1.3, author manuscript]{AustrinODonnellTanWright2014}. +The remaining issue was to make soundness arbitrarily small while retaining +both perfect completeness and two-element fibers. + +The near-perfect-completeness theorem arose from the Grassmann and shortcode +program. Khot, Minzer and Safra introduced a Grassmann-based candidate +reduction~\cite{KhotMinzerSafra2017}; Dinur, Khot, Kindler, Minzer and Safra +developed the reduction to a Grassmann agreement hypothesis and the +associated expansion problem +\cite{DinurKhotKindlerMinzerSafra2018,DinurKhotKindlerMinzerSafra2021}. +Barak, Kothari and Steurer related the agreement problem to expansion through +the shortcode viewpoint~\cite{BarakKothariSteurer2018}. +The near-perfect expansion theorem of Khot, Minzer and Safra completed these +ingredients, giving completeness arbitrarily close to one and arbitrarily +small soundness~\cite{KhotMinzerSafra2018}. +More recently, Fei, Minzer and Wang proved perfect-completeness hardness for +4-to-1 games at arbitrarily small fixed soundness +\cite[Theorem~1.6]{FeiMinzerWang2026}. + +Our conclusion concerns ordinary 2-to-1 games. In the Rich 2-to-1 +framework of Braverman, Khot and Minzer, a uniformly random edge incident +to each fixed left vertex must induce a partition that is uniform among +all partitions of the left alphabet into pairs +\cite[Definition~5]{BravermanKhotMinzer2021}. No such condition is part +of Theorem~\ref{thm:main}. For any positive real soundness target, one may apply +the theorem with a smaller fixed rational target. + +Our external theorem inputs are a perfect-completeness PCP gap, +projection-game parallel repetition, and Grassmann expansion +\cite{Dinur2007,DinurSteurer2014,KhotMinzerSafra2018}. +From the last input we derive the inverse shortcode estimates using the +chart and averaging arguments of Barak, Kothari and Steurer +\cite{BarakKothariSteurer2018}; Section~\ref{sec:preliminaries} +gives the forms needed here. +The sampling comparisons have antecedents in the sub-code covering +framework of Khot and Safra~\cite{KhotSafra2013}. The passage from +agreement of restricted bilinear forms to a fixed form of bounded rank +has a close counterpart in Fei, Minzer and Wang +\cite[Lemmas~5.15--5.16]{FeiMinzerWang2026}. +We prove the comparisons needed for the function spaces and sampling +laws below. + +\subsection{Proof overview} + +We start with a clause-versus-variable game: the left player answers a +clause with a satisfying assignment to its variables, the right player +answers a variable with a bit, and the constraint checks their agreement +on that variable. A perfect-completeness PCP and parallel repetition +make the source value a fixed small constant on NO inputs. We place +independent copies of its questions at the leaves of a finite rooted tree. +Each leaf carries all binary functions on its local answer domain. +At an internal node we take the sum of the spaces spanned by products of +pairs of functions from each child. In particular, the product of two functions from one descendant space +belongs to every space above that descendant. + +At every nonleaf node we display an array, meaning a list of functions +from its space. A question is the unordered partition of the product of +all leaf-answer domains according to the joint evaluation of these arrays. +A legal answer is a nonempty part, so one assignment realizes all its +restored outputs simultaneously. We discard input coordinates that these +evaluations cannot distinguish; a clause coordinate seen only through one +variable is recorded by that variable. This identifies a question on +projected source domains with its pullback to the original domains. +Deleting one binary output direction merges at most two parts and gives +the preliminary at-most-two constraints. A satisfying source assignment +answers every constraint. The same joint partitions also remain unchanged +when an upper row is shifted by products of displayed lower rows, forcing +exact identities on all restored responses. Section~\ref{sec:construction} +defines the questions, their sampling law, and these identities. + +To analyze soundness, we sample a random root-to-leaf path. Arrays off +that path are uniform; arrays on it use a recursive sampler with fresh +randomness for every call. Stopping the calls at a node expresses every +ancestor array as a deterministic function of a bounded number of draws +there. Increasing the number of children hides the special path child +and makes the observed draws close to uniform. A separate comparison +replaces the source questions below a small random set of children by +their right questions. After lifting the resulting functions to the +original domains, this change also has small total variation. +Section~\ref{sec:sampling} proves these comparisons and bounds the chance +that a child selected from the observed functions was projected. + +Suppose a labeling satisfies a fixed positive fraction of the preliminary +constraints. With enough levels, some pair of levels has positive +simultaneous success. At the upper level an inverse shortcode theorem +finds one response occurring with positive density on an affine slice. +The table records responses at every lower-level descendant of the upper +node, so its range may be large; +Theorem~\ref{thm:inverse} has constants independent of that range and also +applies to small quotient spaces. Joint partition identities convert the +slice into a coset of linear functionals $z$ on the upper function space +with $z(1)=1$. Each functional +defines a product form $F(g,h)=z(gh)$ on the lower function space. +For the displayed lower rows $s_1,\ldots,s_\ell$ and their restored +response $y$, joint legality gives +\[ + \bigl(F(s_t,s_u)\bigr)_{t,u}=yy^{\mathsf t}, +\] +so this tested matrix has rank at most one. A Fourier decoder on the +projected side selects from a bounded list computed under a small-bias +law determined by its own local input (Lemma~\ref{lem:heavy-list}). +The two local forms agree after projection with fixed positive probability. +Section~\ref{sec:upper} constructs these decoders, with an agreement bound +independent of the lower row count. + +The remaining difficulty is that the left form depends on the very rows +on which its rank is tested. Passing that test therefore does not +immediately bound its full rank. Resampling within the lower omitted-direction +fiber gives equal restrictions with positive probability. The sparse-projection +posterior bound promotes this to equality of full forms, apart from a small +error. A second inverse application then finds one output form occurring densely +on an affine slice of row lists. A fixed high-rank form almost never has a tested +matrix of rank at most one on that slice. Thus a positive part of the +agreement must come from bounded-rank forms (Section~\ref{sec:lower}). + +A low-rank normalized product form selects a short odd list of answers at +each source leaf. On a projected leaf, agreement gives the parity +pushforward of that list: retain exactly those right answers having an +odd number of preimages in the left list. Some projected children have +all their sampled contributions equal to zero. After exposing the other +data, both decoders can reconstruct their original inputs from their own +questions at designated leaves in these children. The designated source +pairs remain independent because the probability of the zero event is +exactly independent of them (Lemmas~\ref{lem:clean-law} +and~\ref{lem:reconstruction}). Projection-game repetition then contradicts +positive agreement (Section~\ref{sec:clean}). + +Finally, Section~\ref{sec:assembly} chooses the constants in their dependency +order, completes at-most-two maps to exact 2-to-1 maps, and replaces rational +edge weights by an explicit unweighted multiset. + +\subsection{Applications} + +\begin{corollary}[Maximum $k$-colorable subgraph]\label{cor:max-k-colorable} +There are absolute constants $C>0$ and an integer $k_0\ge3$ such that, for +every fixed integer $k\ge k_0$, it is NP-hard to distinguish the following +cases for a finite unweighted graph $G=(W,F)$ with $F\ne\varnothing$: +\begin{enumerate}[label=\textup{(\roman*)}] +\item $G$ is $k$-colorable; +\item every coloring $\chi:W\to[k]$ satisfies +\[ + \frac{|\{\{u,v\}\in F:\chi(u)\ne\chi(v)\}|}{|F|} + \le 1-\frac1k+C\frac{\log k}{k^2}. +\] +\end{enumerate} +The fraction on the left is the proportion of edges properly colored by +$\chi$, or equivalently cut by its partition into $k$ color classes. +\end{corollary} +\begin{proof} +Theorem~\ref{thm:main} supplies the exact-two, perfect-completeness premise +of Guruswami and Sinop's reduction +\cite[Conjecture~3.4 and Theorem~3.26]{GuruswamiSinop2013}. +Their regularization and reduction preserve perfect completeness, giving +$k$-colorable YES graphs, and their Remark~2.6 transfers the hardness to +unweighted graphs. Their soundness bound +$1-1/k+O(\log k/k^2)$ is therefore unconditional. Choose an absolute +upper bound $C$ for the lower-order term and enlarge $k_0$ so the two +cases are disjoint. The reduction parameters, including the soundness +used in Theorem~\ref{thm:main}, are fixed once $k$ is chosen. +\end{proof} + +The next corollary recovers H{\aa}stad's unconditional hardness theorem +for satisfiable Not-Two and the corresponding three-query PCP consequence +\cite[Theorem~4.3]{Hastad2014NotTwo}, using the earlier reduction of +O'Donnell and Wu. + +\begin{corollary}[Satisfiable Not-Two and three-query PCPs]\label{cor:not-two-pcp} +Let $\operatorname{NTW}(a,b,c)$ be the Boolean predicate accepting exactly +zero, one, or three true inputs. For every fixed $0<\eps<3/8$, it is NP-hard +to distinguish a satisfiable instance with constraints +$\operatorname{NTW}(\ell_1,\ell_2,\ell_3)$ from one in which every assignment +satisfies at most a $5/8+\eps$ fraction of the constraints; each literal +$\ell_i$ may be a variable or its negation. +Consequently, every language in NP has a polynomial-time verifier using +$O(\log n)$ random bits and three nonadaptive proof-bit queries, with perfect +completeness and soundness at most $5/8+\eps$, where $n$ is the input length. +\end{corollary} +\begin{proof} +Apply O'Donnell and Wu's reduction +\cite[Theorem~2.1 and Corollary~2.2, author manuscript]{ODonnellWu2008}. +Their premise requires only at-most-$d$-to-1 projections for some fixed $d$. +Take $d=2$, use the uniform distribution on the explicit edge occurrences, +and choose a fixed rational $\delta\in(0,1)$ smaller than their required +soundness threshold. Theorem~\ref{thm:main} supplies this premise with +perfect completeness. Their folded verifier allows all eight literal-sign +variants of Not-Two and gives both conclusions. +\end{proof} + +Corollary~\ref{cor:not-two-pcp} also follows by applying the +O'Donnell--Wu reduction to Fei, Minzer and Wang's perfect 4-to-1 +theorem~\cite{FeiMinzerWang2026}. + +Through Guruswami and Sandeep's reduction at $d=2$ +\cite[Theorem~1]{GuruswamiSandeep2020}, Theorem~\ref{thm:main} also +recovers the known NP-hardness of distinguishing three-colorable graphs +from graphs that are not $c$-colorable, for every fixed integer $c\ge3$ +\cite[Corollary~1.7]{FeiMinzerWang2026}; the companion +\cite[Theorem~1.1]{OpenAIIndependentSets2026} separately proves stronger +independent-set soundness with three-colorable completeness. diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/01-preliminaries.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/01-preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..f87334d080d6090da5418242418801709feff378 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/01-preliminaries.tex @@ -0,0 +1,529 @@ +\section{Source games and inverse matrix structure} +\label{sec:preliminaries} + +This section supplies the three external ingredients of the reduction and +the precise matrix consequence that we will use. A perfect-completeness +PCP and parallel repetition give a source game of arbitrarily small +constant value. Grassmann expansion gives a dense constant-value slice +of a matrix table that often agrees across a rank-one step. We prove +the extensions needed for tables with arbitrarily many possible values, +for small column spaces, and for finding a slice by random row advice. + +Throughout, $\F$ denotes the field with two elements. All vector spaces are finite-dimensional +over $\F$, and all unspecified uniform distributions are on finite sets. +For probability measures $\mu,\nu$ on a common finite set, write +$\TV(\mu,\nu)=\frac12\sum_x|\mu(x)-\nu(x)|$; in particular, every event +has probabilities differing by at most $\TV(\mu,\nu)$. + +\subsection{Finite projection games and the source gap} + +A \emph{finite projection game} $G$ consists of disjoint finite question +sets $U,V$, nonempty finite answer sets $L_u,R_v$, a nonempty finite set +of edge occurrences $\mathcal E$, and a probability distribution $\mu$ +on $\mathcal E$. An occurrence $e$ has endpoints $u(e),v(e)$ and a +total map $\pi_e:L_{u(e)}\to R_{v(e)}$. Distinct occurrences may have +the same endpoints and different maps. A deterministic strategy is a +pair of functions selecting $a(u)\in L_u$ and $b(v)\in R_v$, and +\[ + \val(G)=\max_{a,b}\Pr_{e\sim\mu} + [\pi_e(a(u(e)))=b(v(e))]. +\] +Thus each player receives only its own question, and neither receives +the edge occurrence unless that information is already determined by +its question. Question-dependent legal answer sets are part of the +definition. Zero-weight occurrences may be discarded. Isolated +questions, when retained to define auxiliary tables, do not affect value. +Independent private randomness, or shared randomness independent of the +sampled edge, cannot increase value: fix complete answer functions for +all questions using the random tapes, and average their deterministic +success probabilities. + +For $k\geq1$, the game $G^{\otimes k}$ samples $k$ independent +occurrences. Each player receives its ordered tuple of questions and +returns a tuple of legal answers; acceptance requires all $k$ +constraints. The answer coordinates remain separate even when a +question is repeated. We also set $\val(G^{\otimes0})=1$. + +\begin{theorem}[Perfect-completeness PCP gap] +\label{thm:pcp} +There is a constant $\gamma_0>0$ and a deterministic polynomial-time +map taking any Boolean $3$-CNF formula $\varphi$ to a nonempty $3$-CNF +formula $\Psi$ such that every clause of $\Psi$ uses three distinct +variables. If $\varphi$ is satisfiable, then $\Psi$ is satisfiable. +If $\varphi$ is unsatisfiable, every assignment violates at least a +$\gamma_0$ fraction of the clauses of $\Psi$. +Clause occurrences are explicitly listed, and the binary encoding +length of $\Psi$ is polynomial in that of $\varphi$. +\end{theorem} + +\begin{proof} +The PCP theorem originates in the work of Arora--Safra and +Arora--Lund--Motwani--Sudan--Szegedy +\cite{AroraSafra1998,AroraLundMotwaniSudanSzegedy1998}. +We use Dinur's perfect-completeness formulation +\cite[Definition~8.1 and Theorem~8.1, February 13, 2007 author manuscript]{Dinur2007}. +It gives, in deterministic polynomial time, an explicitly listed family of +$N$ predicates of bounded arity over a fixed finite alphabet, with a +satisfying assignment in the YES case and a fixed rejected fraction +$\theta>0$ under every assignment in the NO case. +We spell out the conversion to clauses, since perfect completeness must +survive every local encoding. + +Encode each alphabet symbol injectively by a fixed number of bits. For +each predicate, require both that its queried bit blocks encode valid +symbols and that the decoded symbols satisfy the predicate. A satisfying +symbol assignment satisfies all these Boolean predicates. Conversely, +decode every invalid block to a fixed default symbol. Whenever an original +predicate rejects this decoded assignment, the corresponding Boolean +predicate rejects the bit assignment: either a queried block is invalid, +or all are valid and the original rejection is unchanged. Thus every +Boolean assignment still violates at least $\theta N$ predicates on a +NO input. After identifying repeated queried variables, each predicate +involves at most a fixed number $k_0$ of distinct bits. + +For each rejecting assignment of a Boolean predicate, form the clause that +excludes precisely that assignment. The conjunction of these at most +$2^{k_0}$ clauses is exactly the predicate. Each clause has width at most +$k_0$; a rejecting zero-bit predicate contributes the empty clause. +For a clause $\ell_1\vee\cdots\vee\ell_k$ with $k\ge4$, introduce fresh +variables $y_1,\ldots,y_{k-3}$ and replace it by +\[ + (\ell_1\vee\ell_2\vee y_1),\qquad + (\neg y_j\vee\ell_{j+2}\vee y_{j+1})\ (1\le j\le k-4),\qquad + (\neg y_{k-3}\vee\ell_{k-1}\vee\ell_k). +\] +If all $\ell_h$ are false, the first clause forces $y_1=1$, each middle +clause propagates this value, and the last clause fails. If the original +clause is true, setting $y_j=1$ exactly when +$\ell_1,\ldots,\ell_{j+1}$ are all false satisfies every displayed clause. +Use disjoint auxiliary variables for every such replacement. Clauses of +width at most three are retained. Consequently an accepting predicate +admits a satisfying extension, whereas a rejecting predicate forces a +violated clause for every extension. There is a fixed bound $B\ge1$ on +the number of clauses per predicate. The resulting formula has at most +$BN$ clauses and at least $\theta N$ violated clauses under every assignment +on a NO input, so its gap is at least $\widehat\gamma=\theta/B>0$. + +Here is the further normalization, including its gap loss. Remove +tautological clauses and simplify repeated literals in each remaining +clause. A resulting clause $C$ on $k\leq3$ distinct variables is +replaced by all $2^{3-k}$ clauses obtained by adjoining every sign +pattern on $3-k$ fresh variables, used only for this occurrence. If +$C$ is true, every extension is true. If $C$ is false, exactly one +extension is false for every assignment to the fresh variables. +This also handles an empty clause, using all eight sign patterns on +three fresh variables. Consequently every violated original clause +still contributes a violated clause, while the number of clauses +increases by at most eight. The gap is therefore at least +$\widehat\gamma/8$, and completeness remains one. If normalization +leaves no clauses, use a single satisfiable clause on three fresh +variables; this case cannot occur for a NO output of the gap reduction. +An input with no clauses can likewise be sent directly to this fixed +satisfiable formula. All steps are deterministic and polynomial. +\end{proof} + +\begin{theorem}[Weighted projection-game repetition] +\label{thm:repetition} +Let $G$ be any finite projection game with the conventions above. +For $00$, choose a fixed integer $t$ with +$(1-\gamma_0^2/144)^t\leq\sigma$ and take +\begin{equation} + G_{\mathrm{src}}=G_\Psi^{\otimes t}. + \label{eq:prelim-source-game} +\end{equation} +Theorem~\ref{thm:repetition} gives source soundness at most $\sigma$, +with perfect completeness. Its left question is an ordered tuple of +clause occurrences, its answer domain is the product of their +$\mathcal A_C$, its right question is the corresponding tuple of +variable IDs, and its right answer domain is always $\F^t$. +In particular, a repeated variable ID in a right question imposes no +equality condition on the corresponding answer coordinates. +Both answer-domain sizes, $7^t$ and $2^t$, are constants, and the +explicit game has polynomial size because $t$ is fixed. + +Each $\mathcal A_C$ projects onto both bits at any single position and +onto all four bit pairs at any two distinct positions. Indeed, a +clause excludes just one of the eight triples, and any prescribed pair +has two extensions. Taking products proves that every source +projection is surjective. The pair-surjectivity observation will also +make the support reduction of function queries unambiguous. + +\subsection{Matrix slices and the Grassmann input} + +Let $H$ be a finite vector space, put $K=\F^\ell$, and identify +$K\otimes H$ with $\Hom(H^*,K)$. Thus a matrix $M$ is a list of +$\ell$ rows in $H$, and $Mq\in K$ evaluates those rows at +$q\in H^*$. For a linear map $A:K\to\F^a$, the notation $AM$ +denotes its $a$ specified combinations of rows. A \emph{row and +column slice} is a nonempty affine set +\begin{equation} + \mathcal S= + \{M\in K\otimes H: AM=U,\ Mq_b=t_b\ (1\leq b\leq c)\}, + \label{eq:prelim-slice} +\end{equation} +where $U\in\F^a\otimes H$, $q_b\in H^*$, and $t_b\in K$. +There is no independence requirement on the rows of $A$ or on the +$q_b$. We count specified row combinations and column values, rather +than their ranks; redundant equations can always be removed. A slice +with no column constraints is a \emph{row fiber}. + +For $a\in K$ and $h\in H$, write $ah=a\otimes h$. +The matrix step used below is +\[ + M\longmapsto M+ah, + \qquad M\text{ uniform},\quad + a\text{ uniform in }K\setminus\{0\},\quad h\text{ uniform in }H, +\] +with the three choices independent. This step is stationary: both +endpoints are uniform. The possibility $h=0$ is retained. + +We recall the precise structural theorem behind the matrix argument. +The Grassmann graph on an $N$-dimensional binary vector space has +the $\ell$-dimensional subspaces as its vertices; distinct vertices +are adjacent when their intersection has dimension $\ell-1$. +For a nonempty vertex set $S$, its \emph{retention} is the probability +that a uniform vertex of $S$ has a uniformly chosen neighbor in $S$. + +\begin{theorem}[Grassmann expansion consequence] +\label{thm:prelim-grassmann} +For each $0<\zeta<1$, there are $\alpha>0$, an integer $r\geq1$, +and $\ell_0$ such that, for every $\ell\geq\ell_0$ and then every +sufficiently large $N$, the following holds. Every nonempty set of vertices +with retention at least $\zeta$ has density at least $\alpha$ in an +interval +\[ + \{L:A_0\subseteq L\subseteq B_0,\ \dim L=\ell\}, + \qquad \dim A_0+\operatorname{codim}B_0\leq r. +\] +The constants $\alpha,r,\ell_0$ depend only on $\zeta$; the +threshold for $N$ may also depend on $\ell$. +\end{theorem} + +For sets of density at most $1/2$, this is Theorem~1.12 of +Khot--Minzer--Safra in ECCC TR18-006, revision~2 +\cite{KhotMinzerSafra2018}, with expansion written as one minus retention; +Definitions~1.8 and~1.10 give the expansion and interval conventions. +If the set has density greater than $1/2$, take $A_0=\{0\}$ and +$B_0=\F^N$. This whole interval has co-order zero and density greater +than $1/2$, so replacing the cited density constant by its minimum with +$1/2$ proves the statement in that case too. +The order of the dimension quantifiers is important. We +next give the matrix-chart argument, as in +\cite[Section~3]{BarakKothariSteurer2018}, and then remove the +column-dimension requirement explicitly. + +\begin{lemma}[Matrix set extraction in large dimension] +\label{lem:prelim-matrix-set} +For every $0<\eta<1$ there are $\alpha>0$, $r\geq1$, and +$\ell_0$, depending only on $\eta$, such that for each +$\ell\geq\ell_0$ and all sufficiently large $m$ the following holds. +If a nonempty set $S\subseteq\Mat_{\ell,m}(\F)$ satisfies +\[ + \Pr_{M\text{ uniform in }S,\ a\ne0,\ h} + [M+ah^{\mathsf t}\in S]\geq\eta, +\] +where $a\in\F^\ell\setminus\{0\}$ and $h\in\F^m$ are +independent uniform vectors, then $S$ has density at least $\alpha$ +in a nonempty row and column slice with at most $r$ equations of each +kind. +\end{lemma} + +\begin{proof} +Increase the lower threshold on $m$ so that $2^{-m}\leq\eta/2$. +The step has $h=0$ with probability $2^{-m}$. Its retention +conditional on $h\ne0$ is therefore at least +$(\eta-2^{-m})/(1-2^{-m})\geq\eta/2$. + +Associate to $M$ its graph subspace +\[ + L_M=\{(x,M^{\mathsf t}x):x\in\F^\ell\} + \subseteq\F^\ell\oplus\F^m. +\] +These subspaces form the chart $\mathcal C$ on which projection to the +first summand is invertible, and $M\mapsto L_M$ is a bijection onto +that chart. The identity +$\dim(L_M\cap L_{M'})=\ell-\rank(M-M')$ shows that chart neighbors +are precisely the nonzero rank-one changes. Over $\F$, each nonzero +rank-one matrix has exactly one factorization $ah^{\mathsf t}$ with +both factors nonzero. The degree within the chart is thus +$(2^\ell-1)(2^m-1)$. + +For comparison, a vertex of the full Grassmann graph has +$(2^\ell-1)(2^{m+1}-2)$ neighbors: choose a hyperplane in the vertex, +then a one-dimensional extension in its quotient, excluding the +original vertex. Exactly half the neighbors of every chart vertex +therefore remain in the chart. The set +$\widetilde S=\{L_M:M\in S\}$ has full Grassmann retention at least +$\eta/4$. Apply Theorem~\ref{thm:prelim-grassmann} with +$\zeta=\eta/4$. It supplies an interval $[A_0,B_0]$ of bounded +co-order on which $\widetilde S$ has density at least $\alpha$. +Its intersection with $\mathcal C$ is nonempty, and restricting the +denominator to this intersection cannot decrease that density, +since $\widetilde S\subseteq\mathcal C$. + +It remains to identify the intersection as a slice. A basis of +$A_0$ consists of vectors $(d_b,s_b)$, and its containment in $L_M$ +is exactly $d_b^{\mathsf t}M=s_b^{\mathsf t}$ for every basis vector. +Using the standard dot product, a basis $(t_b,q_b)$ of $B_0^\perp$ +gives the equations $Mq_b=t_b$ for containment $L_M\subseteq B_0$. +Thus there are at most $r$ row equations and at most $r$ column +equations. The Grassmann theorem fixes $\alpha,r,\ell_0$ using +only $\eta$; after fixing $\ell$, its ambient-dimension threshold +is met by taking $m$ sufficiently large. +\end{proof} + +Our eventual matrices may have very small column spaces, including +quotient spaces of dimension zero. Adding unused columns preserves +the matrix test, but a dense slice in the enlarged space can mix real +and dummy columns. The next lemma describes its projection exactly. + +\begin{lemma}[Eliminating dummy columns] +\label{lem:padding} +Let $H,D,K$ be finite binary vector spaces, and write a matrix on +$H\oplus D$ as $(X,Y)$, with $X\in\Hom(H^*,K)$ and +$Y\in\Hom(D^*,K)$. Suppose the nonempty affine slice +\begin{equation} + \widetilde{\mathcal S}= + \{(X,Y):AX=U_H,\ AY=U_D,\ + Xq_b+Yd_b=t_b\ (1\leq b\leq c)\} + \label{eq:prelim-padded-slice} +\end{equation} +has $A:K\to\F^a$, $q_b\in H^*$, and $d_b\in D^*$. +Its image $\mathcal S$ under $(X,Y)\mapsto X$ is a nonempty row +and column slice with the same $a$ row combinations and at most $c$ +column values. Every $X\in\mathcal S$ has equally many lifts in +$\widetilde{\mathcal S}$. In particular, a uniform point of the +padded slice projects to a uniform point of $\mathcal S$. +\end{lemma} + +\begin{proof} +Put $W=\ker A$. Encode all dependencies among the column equations +by the linear maps +\[ + q:\F^c\to H^*,\quad q(\lambda)=\sum_b\lambda_bq_b, + \qquad + d:\F^c\to D^*,\quad d(\lambda)=\sum_b\lambda_bd_b, + \qquad + t(\lambda)=\sum_b\lambda_bt_b. +\] +We claim that the image consists exactly of the matrices $X$ obeying +\begin{equation} + AX=U_H,\qquad Xq(\lambda)=t(\lambda) + \quad(\lambda\in\ker d). + \label{eq:prelim-projected-slice} +\end{equation} +A basis of $\ker d$ gives at most $c$ column equations, whether or +not their real arguments are independent. Necessity follows by +taking linear combinations of the original column equations whose +dummy arguments vanish. + +For sufficiency, fix one point $(X_*,Y_*)$ of the original nonempty +slice, and let $X$ satisfy Equation~\eqref{eq:prelim-projected-slice}. +The row equations give $A(X_*-X)=0$. Define on $\im d$ the candidate +linear map +\begin{equation} + L_0(d(\lambda))=(X_*-X)q(\lambda). + \label{eq:prelim-dummy-extension} +\end{equation} +It is well-defined: if $d(\lambda)=0$, the projected equations for +$X$ and $X_*$ give $(X_*-X)q(\lambda)=0$. Its values lie in $W$ +because $A(X_*-X)=0$. Extend $L_0$ arbitrarily to a linear map +$L:D^*\to W$, and put $Y=Y_*+L$. Then $AY=U_D$, and +\[ + Xq(\lambda)+Yd(\lambda) + =Xq(\lambda)+Y_*d(\lambda)+(X_*-X)q(\lambda) + =t(\lambda) +\] +for every $\lambda$. This gives a lift. It also shows explicitly +why the row compatibility and all dependent column equations suffice. + +Every lift is obtained this way, since its difference from $Y_*$ +takes values in $W$ and must restrict to $L_0$ on $\im d$. The +number of such extensions is +\[ + 2^{\dim W\, (\dim D-\rank d)}, +\] +independent of $X$. Thus all projection fibers have the same size, +which proves the uniform-measure assertion. The argument permits +zero dimensions and rank-deficient $A$ throughout. +\end{proof} + +\subsection{Partial tables and many row fibers} + +A partial table with finite range $\mathcal Y$ is a function +$f:K\otimes H\to\mathcal Y\cup\{\bot\}$, where $\bot\notin\mathcal Y$ +means undefined. A \emph{defined equality} requires both endpoints to +have the same value in $\mathcal Y$; two undefined values never count. + +\begin{theorem}[Inverse theorem for a finite-range partial table] +\label{thm:inverse} +For every $0<\eta<1$, there are $\alpha>0$, integers $r\geq1$ and +$\ell_0$, depending only on $\eta$, with the following property. +For every $\ell\geq\ell_0$, every finite binary vector space $H$, +every finite set $\mathcal Y$, and every partial table +$f:K\otimes H\to\mathcal Y\cup\{\bot\}$, where $K=\F^\ell$, +if +\begin{equation} + \Pr_{M,a,h}[f(M)=f(M+ah)\ne\bot]\geq\eta, + \label{eq:prelim-defined-equality} +\end{equation} +for independent uniform $M$, $a\in K\setminus\{0\}$, and $h\in H$, +then some $y\in\mathcal Y$ has density at least $\alpha$ on a +nonempty slice with at most $r$ row combinations and at most $r$ +column values. The constants are independent of both $\dim H$ and +$|\mathcal Y|$. +\end{theorem} + +\begin{proof} +Use the constants of Lemma~\ref{lem:prelim-matrix-set} for $\eta$. +Add a dummy space $D$ so that $\dim(H\oplus D)$ meets its column +threshold, and define $\widetilde f(X,Y)=f(X)$. A uniform padded +matrix projects to a uniform real matrix; likewise, a uniform step +factor in $H\oplus D$ projects to a uniform factor in $H$. +Thus the defined-equality probability is unchanged, even when $H=0$. + +For a defined value $y$, let $S_y=\widetilde f^{-1}(y)$, let +$\mu_y$ be its uniform matrix measure, and let $\theta_y$ be its +retention under the padded matrix step. Empty fibers can be omitted. +Partitioning according to the first value gives exactly +\begin{equation} + \Pr[\widetilde f(\widetilde M) + =\widetilde f(\widetilde M+a\widetilde h)\ne\bot] + =\sum_{y\in\mathcal Y}\mu_y\theta_y. + \label{eq:prelim-fiber-average} +\end{equation} +Since $\sum_y\mu_y\leq1$, a value $y$ has $\theta_y\geq\eta$. +This averaging, also used in +\cite[Lemma~2.3]{BarakKothariSteurer2018}, incurs no factor for +the number of fibers. Lemma~\ref{lem:prelim-matrix-set} supplies a +padded slice on which that fixed value has density at least $\alpha$. +Project the slice using Lemma~\ref{lem:padding}. Its uniform measure +projects uniformly, and $\widetilde f$ depends only on the projected +matrix. The same value therefore has the same density on the +resulting slice in $K\otimes H$, with no increase in either count of +constraints. +\end{proof} + +We will need random row advice to encounter such a slice, rather than +merely the existence of one slice somewhere in the table. The next +erasure argument supplies exactly this averaged guarantee. + +\begin{lemma}[Many good row fibers] +\label{lem:many-fibers} +Fix $0<\eta<1$, and let $\alpha,r,\ell_0$ be constants from +Theorem~\ref{thm:inverse} with threshold $\eta/2$. +Suppose $\ell\geq\ell_0$ and a partial table on $K\otimes H$ has +defined-equality probability at least $\eta$. For each linear map +$A:K\to\F^r$, call a nonempty row fiber $\{M:AM=U\}$ +\emph{good} if there are at most $r$ column specifications and a +defined value $y$ such that the resulting slice is nonempty and has +$f=y$ on at least an $\alpha$ fraction of its uniform points. +Then +\begin{equation} + \Pr_M[\text{$M$ lies in a good row fiber for some $A$}] + \geq\eta/4. + \label{eq:prelim-good-union} +\end{equation} +Consequently, for independent uniform $A\in\Hom(K,\F^r)$ and +$M\in K\otimes H$, +\begin{equation} + \Pr_{A,M}[\text{the fiber of $AM$ for $A$ is good}] + \geq (\eta/4)2^{-r\ell}. + \label{eq:prelim-many-fibers} +\end{equation} +The witnesses can be chosen deterministically from $f,A,U$ alone. +For every good row fiber, any chosen nonempty slice defined by at most +$r$ column values has conditional uniform probability at least +$2^{-r\ell}$ inside that row fiber. +\end{lemma} + +\begin{proof} +Let $T$ be the union of all good row fibers, over all the +$2^{r\ell}$ possible row maps. Suppose its uniform measure were +less than $\eta/4$. Erase the table on $T$, obtaining $f_0$. +Stationarity of the matrix step gives +\[ + \Pr[f_0(M)=f_0(M+ah)\ne\bot] + \geq \Pr[f(M)=f(M+ah)\ne\bot] + -\Pr[M\in T]-\Pr[M+ah\in T] + >\eta/2. +\] +Theorem~\ref{thm:inverse} gives a defined value of $f_0$ of density +at least $\alpha$ on a nonempty slice with at most $r$ row and $r$ +column specifications. Pad its row map by zero output coordinates +to make its codomain $\F^r$. Because $f_0$ is a restriction of +$f$, this slice certifies that its row fiber is good for $f$. +That entire fiber belongs to $T$, where $f_0$ is undefined, a +contradiction. This proves Equation~\eqref{eq:prelim-good-union}. + +For each $M\in T$, at least one of the $2^{r\ell}$ maps has a +good fiber through $M$. Averaging the number of such maps gives +Equation~\eqref{eq:prelim-many-fibers}. No statement for every +particular row map is needed or asserted. + +All vector spaces and value sets involved are finite. After fixing +orders on them, select the first witnessing list of columns, values, +and $y$ for each $(f,A,U)$. This choice uses the row advice and +the table, without inspecting any further part of $M$. +Finally, the column equations restricted to a nonempty row fiber form +an affine linear system with at most $r\ell$ scalar equations. +Whenever it is consistent its solution set has relative size +$2^{-b}$, where $b\leq r\ell$ is the rank of that system. This +proves the stated conditional probability, including all dependent +and zero-rank cases. +\end{proof} + +All three matrix conclusions apply to quotient spaces without any +change of constants. In particular, when $H$ is replaced by $H/B$, +the column functionals produced above belong to $(H/B)^*$; their +pullbacks to $H$ annihilate $B$. The proof adds dummy columns to the +quotient itself and removes them by Lemma~\ref{lem:padding}, so no +functional outside this annihilator is introduced. diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/02-construction.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/02-construction.tex new file mode 100644 index 0000000000000000000000000000000000000000..8e15756ebaef07b34f6e7e054232775ed4974f10 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/02-construction.tex @@ -0,0 +1,521 @@ +\section{The tree of function spaces and the one-bit test} +\label{sec:construction} + +We first construct questions whose labels are nonempty parts of a joint-evaluation +partition. Deleting one output direction gives maps with fibers of size at +most two, and a satisfying source assignment supplies labels that satisfy +every such map. We then specify the distribution of tests, obtaining a +finite weighted game with perfect completeness. Its soundness will be proved +in the next four sections; the +conversion to the exact alphabets and unweighted format of +Theorem~\ref{thm:main} is deferred to Section~\ref{sec:assembly}. +The shared questions must forget both the names of their output values and +every input coordinate that their evaluations cannot distinguish. We give +this identification explicitly, since it is responsible for both exact +folding and the later comparison of clause and variable questions. + +\subsection{Formal slots and pointwise spaces} + +Let the source game be the fixed power of the clause--variable game from +Section~\ref{sec:preliminaries}, and let $t\geq1$ denote its power. +A source left question is an ordered tuple of $t$ clause occurrences. +For a clause occurrence $c$, write $v(c,1),v(c,2),v(c,3)$ for its three +distinct variable IDs in their specified order, and let +\[ + \Omega_c=\mathcal A_c=\{x\in\F^3:x\text{ satisfies }c\}. +\] +These are actual variable values, with the literal signs accounted for in +the satisfaction condition. Thus $|\Omega_c|=7$, and every coordinate map +$x\mapsto x_j$, and every pair of distinct coordinate maps, is surjective. +A source edge chooses one position in each clause. Its right question is +the resulting ordered tuple of variable IDs and its answer domain is +$\F^t$. Coordinates of this domain remain separate even when two IDs agree. + +Fix positive integers $d,n_1,\ldots,n_d$, row counts +$\ell_1,\ldots,\ell_d$, and product counts $R_1,\ldots,R_d$. +Let $\mathcal T$ be the rooted ordered tree with a single root at level $d$, +leaves at level $0$, and exactly $n_p$ children at every level-$p$ node. +Write $\mathcal T_p$ for its level-$p$ nodes, and set $K_p=\F^{\ell_p}$. +Every location in this fixed tree is part of its indexing, independently of +the question ID placed there. + +Let $E$ be a tuple of source left questions, one at each leaf. Its elementary slots +are the formal pairs $(C,k)$ with $C\in\mathcal T_0$ and $k\in[t]$. +If the clause occurrence at this slot is $c$, its domain is $\Omega_c$. +All slots are distinct factors of a Cartesian product: repeated clause or +variable IDs impose no equality constraint between their answer coordinates. +For any node $D$, let $\Omega_D(E)$ be the product of the elementary-slot +domains below $D$, and write $\Omega(E)$ for the root domain. + +We will also use a mixed tuple $O$ obtained by replacing some leaf left +questions by source right questions along specified source edges. +At a projected elementary slot the domain is $\F$, tagged with the selected +variable ID. Write $\Omega_D(O)$ and $\Omega(O)$ for the resulting products. +The selected coordinate maps give a surjection +\[ + \pi_D:\Omega_D(E)\longrightarrow\Omega_D(O). +\] +Neither the formal location nor the independence of the answer coordinates +is changed by projection. For example, the right question $(v,v)$ has the +legal answer $(0,1)$, although a completeness labeling will not use that +answer. + +\begin{definition}[Pointwise function spaces]\label{def:spaces} +For any such tuple $Q$ of clause and variable questions, define finite +vector spaces over $\F$ recursively. At a leaf $C$, set +$H_C(Q)=\F^{\Omega_C(Q)}$, the space of all functions from its answer domain +to $\F$. For every node $C$, set +\[ + H_C(Q)^2=\Span\{gh:g,h\in H_C(Q)\}, + \qquad (gh)(x)=g(x)h(x). +\] +At a nonleaf node $D$, put +\begin{equation}\label{eq:construction-spaces} + H_D(Q)=\sum_{C\text{ child of }D} H_C(Q)^2 + \ \subseteq\ \F^{\Omega_D(Q)}. +\end{equation} +Here a function on a descendant domain is extended by ignoring every +other argument. All sums, products and equalities are equalities of actual +functions on these product domains, rather than formal polynomials. +\end{definition} + +\begin{lemma}\label{lem:construction-space-properties} +The spaces in Definition~\ref{def:spaces} contain $1$ and satisfy +$H_C(Q)\subseteq H_C(Q)^2$. For every proper descendant $C$ of $D$, +\begin{equation}\label{eq:descendant-square-inclusion} + H_C(Q)^2\subseteq H_D(Q). +\end{equation} +Moreover, pullback by $\pi_D$ injectively identifies $H_D(O)$ with a +subspace of $H_D(E)$ and preserves pointwise multiplication. +\end{lemma} +\begin{proof} +Every elementary domain is nonempty, so extension by ignoring arguments +is injective. At a leaf, the constant $1$ belongs to the full function +space. Inductively, any child's constant $1=1\cdot1$ supplies the constant +at its parent. Multiplication by $1$ gives $H_C(Q)\subseteq H_C(Q)^2$. +For a child, \eqref{eq:descendant-square-inclusion} is the definition. +For a more distant descendant, repeatedly use +$H_C(Q)^2\subseteq H_{D'}(Q)\subseteq H_{D'}(Q)^2$ along the route to $D$. + +Each elementary projection onto a selected bit is surjective; their +product $\pi_D$ is therefore surjective. Consequently pullback is +injective on the entire function space and respects sums and products. +At a leaf its image lies in the full original function space. Applying +the recursive sum-of-products definition proves the asserted inclusion +at every other node. +\end{proof} + +We henceforth use these injective pullbacks without writing their symbols. +No assertion that the summands in \eqref{eq:construction-spaces} form a +direct sum is intended. In particular their copies of the constant +function coincide. + +\subsection{Joint evaluations and canonical questions} + +An array at a level-$p$ nonleaf node $D$ is an element +$M_D\in K_p\otimes H_D(E)$, equivalently an ordered list of $\ell_p$ rows +in $H_D(E)$. At $x\in\Omega(E)$, its evaluation is the vector +$M_D(x)\in K_p$. The full question function of an array family +$\mathbf M=(M_D)_{D\in\mathcal T,\,D\text{ nonleaf}}$ is +\begin{equation}\label{eq:joint-question-function} + \Psi_{E,\mathbf M}:\Omega(E)\longrightarrow + \prod_{p=1}^d\prod_{D\in\mathcal T_p}K_p, + \qquad x\longmapsto (M_D(x))_D. +\end{equation} +Every nonleaf output occurs in this tuple. + +For a nonleaf node $D$ at level $p$ and a nonzero direction $a\in K_p$, +let $Q_{D,a}$ act on the output product in +\eqref{eq:joint-question-function} by the quotient map +$K_p\to K_p/\Span\{a\}$ at the $D$ block and by the identity at every +other block. Define +\begin{equation}\label{eq:quotient-question-function} + \Psi^{D,a}_{E,\mathbf M}=Q_{D,a}\Psi_{E,\mathbf M}. +\end{equation} +The quotient merges at most two attained output values. To turn this +observation into a game constraint, we must first specify which representations +of a joint function give the same question and which answers are legal. + +We now turn an array-valued function into a question without retaining +unobservable metadata. The construction applies to any finite-valued +function $f:\Omega(Q)\to Y$ on a mixed product domain, and in particular +to both functions \eqref{eq:joint-question-function} and +\eqref{eq:quotient-question-function}. The codomain $Y$ may be a product +of vector spaces and quotient spaces. + +An elementary slot $s$ is \emph{unused by $f$} if changing only its +argument never changes $f$. A clause slot with domain $\Omega_c$ is +\emph{reducible to position $j$} if, for every fixed choice $x_{-s}$ of +all other arguments and every $u,u'\in\Omega_c$ with $u_j=u'_j$, +\begin{equation}\label{eq:global-singleton-factorization} + f(x_{-s},u)=f(x_{-s},u'). +\end{equation} +Thus one fixed position $j$ must work for the whole joint function, for +all choices of the other arguments. + +\begin{lemma}\label{lem:canonical-support} +If a used clause slot is reducible to a position, that position is unique. +The following reductions can be performed simultaneously: discard unused +slots; replace each used reducible clause slot by its selected bit; retain +every other used clause slot in full; and retain each used variable slot +as its bit. They give a surjection +$r_f:\Omega(Q)\to\Omega_f$ and a unique induced function +$\widetilde f:\Omega_f\to Y$ such that $f=\widetilde f\circ r_f$. +The reduced domain and its partition into fibers depend only on the +equality relation +\[ + x\sim_f x'\quad\Longleftrightarrow\quad f(x)=f(x') +\] +on the original product domain, not on names of the output values. +\end{lemma} +\begin{proof} +Suppose the clause slot factors through two distinct positions $j,k$. +Fix all other arguments. For this section, write the value both as +$g(u_j)$ and as $h(u_k)$. Pair-surjectivity gives +$g(b)=h(b')$ for every $(b,b')\in\F^2$. Both functions are constant, +so this section is constant. This holds for every choice of the other +arguments, making the slot unused. + +For each slot let its coordinate reduction be the constant map to a +one-point set, the selected coordinate map, or the identity, according +to the stated rule. Each is surjective, hence their product $r_f$ is +surjective. If $r_f(x)=r_f(x')$, replace the slots of $x$ by those of $x'$ +one at a time. At every replacement the value of $f$ is unchanged by +the definition of the reduction at that slot. This proves that $f$ is +constant on fibers of $r_f$, giving the unique $\widetilde f$. +One-point factors are removed in the notation $\Omega_f$; if all slots +disappear, $\Omega_f$ is a one-point domain. + +Both unusedness and \eqref{eq:global-singleton-factorization} are predicates +about pairs of arguments having equal $f$-values. They are therefore +determined by $\sim_f$. So are the coordinate reductions, by uniqueness +in the used singleton case. Surjectivity then shows that the fiber +partition of $\widetilde f$ is determined by the same relation. +\end{proof} + +The global quantifier in \eqref{eq:global-singleton-factorization} cannot +be replaced by a sectionwise choice of position. For example, on +$\Omega_c\times\F$ the function +$f(x,b)=(1+b)x_1+bx_2$ uses one clause position in each fixed-$b$ section, +but has no single position through which it factors globally. The +canonical reduction retains its clause slot. Pair-surjectivity proves +uniqueness when a global factorization exists; it does not turn these +different sectionwise choices into one factorization. + +The reduced domain has explicit typed coordinates. At a retained full +clause slot we record its formal location, clause occurrence ID and +triple domain. At a retained bit slot we record its formal location +and variable ID, with domain $\F$. When a clause slot is reduced to a +bit, its clause occurrence and selected position are discarded. +Unused locations are discarded. Retained locations are kept in the fixed +tree-and-slot order, so equal IDs in distinct slots are never merged. +Let +\begin{equation}\label{eq:canonical-partition} + \mathcal P_f= + \{\widetilde f^{-1}(y):y\in\im f\} +\end{equation} +be the \emph{unordered} partition of this reduced domain into nonempty +fibers. The canonical key $\mathfrak k_L(f)$ consists of a left-side +tag, the retained typed coordinates, and $\mathcal P_f$. +Define $\mathfrak k_R(f)$ with a distinct right-side tag. +The codomain $Y$, names $y$ of its values, array coordinates, source +projection positions and any sampling choices are not additional fields +of a key. The typed coordinates and finite subsets in +\eqref{eq:canonical-partition} have ordinary deterministic finite encodings. + +A legal label at this key is a part $P\in\mathcal P_f$. +At a particular representation $f$ of the key, define its restored output +by +\begin{equation}\label{eq:restored-output} + \operatorname{out}_f(P)=y + \quad\Longleftrightarrow\quad + P=\widetilde f^{-1}(y). +\end{equation} +This is a bijection from legal labels to $\im f$. +In particular every restored joint response is realized by one +$x\in\Omega(Q)$: choose a point of the part and lift it through $r_f$. +We never allow separately legal block outputs that do not belong to +the image of the whole joint function. + +\begin{lemma}[Sharing across projected domains]\label{lem:canonical-sharing} +Let $O$ be obtained from $E$ by the specified source projections, let +$\pi:\Omega(E)\to\Omega(O)$ be their product, and let +$f:\Omega(O)\to Y$ be any function. For either side tag, +the keys of $f$ and $f\pi$ coincide literally, with the same parts and +the same restored outputs. +\end{lemma} +\begin{proof} +At an unprojected slot, unusedness and every candidate singleton +factorization are the same for $f$ and $f\pi$, because the projection +of all other arguments is surjective. At a projected slot, unusedness +is likewise equivalent by surjectivity. If the slot is used by $f$, +then $f\pi$ factors through its selected position and is used. +Lemma~\ref{lem:canonical-support} makes that position its unique singleton +reduction. Both representations consequently retain the same formal +location and selected variable ID, with the same bit domain; neither +retains the original clause occurrence or projection position there. +These observations identify their reduced domains. On that common +domain the two induced functions are identical, since the relevant +product maps are surjective and both lift to $f\pi$. +Their fiber partitions and restored values therefore agree. +\end{proof} + +Take as vertices all keys of the indicated side obtainable from mixed +tuples of source questions, arbitrary arrays in their prescribed spaces, +and, on the right, one quotient at a nonleaf node in a nonzero direction. +Every key used by the sampling law below belongs to this set. Keys never used by a +test have zero incident weight and may be assigned any legal part. +Equivalently, a labeling of the positive-weight vertices can always be +extended to these auxiliary keys. Because the definition identifies +keys before choosing labels, all such extensions retain the exact sharing +identity of Lemma~\ref{lem:canonical-sharing}. + +\subsection{One-bit constraints and perfect completeness} + +Fix an array family on $E$, a nonleaf node $D$ at level $p$, and +$a\in K_p\setminus\{0\}$. Put $f=\Psi_{E,\mathbf M}$ and +$g=Q_{D,a}f$. The edge joins $\mathfrak k_L(f)$ to +$\mathfrak k_R(g)$ and has the following map on legal parts: +\begin{equation}\label{eq:legal-edge-map} + \pi_{f,D,a}(P)= + \operatorname{out}_g^{-1} + \bigl(Q_{D,a}\operatorname{out}_f(P)\bigr). +\end{equation} +The argument of the inverse belongs to $\im g$, since +$\operatorname{out}_f(P)\in\im f$. +These edge records define the constraints that our game may use. We specify +their sampling distribution at the end of this section. Distinct records +may have identical endpoints and may be combined only when their actual +maps agree. + +\begin{lemma}\label{lem:construction-two-fibers} +Every map \eqref{eq:legal-edge-map} is a well-defined surjection between +nonempty legal label sets and has at most two preimages per right label. +For a fixed labeling, acceptance of the edge is exactly equality of its +restored outputs through $Q_{D,a}$. +\end{lemma} +\begin{proof} +Restoration is a bijection between parts and attained values on each +side. Since $g=Q_{D,a}f$, every attained $g$-value has a preimage +among attained $f$-values. This proves surjectivity and the acceptance +description. The linear map $Q_{D,a}$ has kernel consisting of the +zero vector and the vector supported at $D$ with value $a$. +Hence every fiber on the entire output space has exactly two elements, +and its intersection with $\im f$ has at most two. +Applying the two restoration bijections proves the claimed fiber bound. +This argument is unchanged if the two question functions retain +different input supports. +\end{proof} + +The local label sets have a uniform finite bound once the parameters are +fixed. Indeed, if $N=|\mathcal T_0|$, every mixed answer domain has size at +most $7^{tN}$, so every legal partition has at most $7^{tN}$ parts. +The canonicalization and each edge map can be computed by enumerating +these finite domains. Quantitative polynomial-time and weight-removal +details will be given when the parameters have been chosen. + +\begin{proposition}[Perfect completeness]\label{prop:completeness} +If the source clause instance has a satisfying assignment, every constraint +just defined is satisfied by one common legal labeling, for every choice +of the positive integer parameters. Consequently every game supported on +these constraints has value one. +\end{proposition} +\begin{proof} +Let $\xi$ be a satisfying assignment to all source variables. +At a canonical key, assign each retained full clause coordinate the +corresponding satisfying triple from $\xi$, and each retained bit +coordinate its variable value under $\xi$. These choices form a point +of the reduced domain, even if IDs recur at distinct locations. +Label the vertex by the unique part containing this point. +This prescription depends only on the typed coordinates and partition +in the key, so it gives one consistent label at every shared vertex. + +For any original tuple $E$, the assignment $\xi$ gives a legal +point $x_\xi\in\Omega(E)$ whose support reduction is exactly the point +just prescribed at either endpoint. The restored outputs are therefore +$f(x_\xi)$ and $g(x_\xi)$, respectively. Their equality through +$Q_{D,a}$ follows from $g=Q_{D,a}f$, so +Lemma~\ref{lem:construction-two-fibers} shows that the edge accepts. +Every constraint therefore accepts, proving the assertion. +\end{proof} + +\subsection{Algebraic identities forced by joint answers} + +Evaluation tables and folding are standard mechanisms in PCP constructions; +see Bellare, Goldreich and Sudan~\cite[Section~3.3]{BellareGoldreichSudan1998}. +Here the canonical joint partition forces an exact algebraic identity on +every labeling. It allows an upper array to absorb products of already +displayed descendant rows, including all relations among those products. + +\begin{lemma}[Joint folding]\label{lem:folding} +Fix a full array question on $\Omega(Q)$ and a nonleaf node $D$. +For each row index $h$ of $M_D$, choose +\begin{equation}\label{eq:folding-shift} + b_h=c_h1+ + \sum_{C,u,v}\lambda_{h,C,u,v} + (M_C)_u(M_C)_v, + \qquad c_h,\lambda_{h,C,u,v}\in\F, +\end{equation} +where $C$ ranges over any collection of proper nonleaf descendants of $D$ +and $u,v$ over their displayed rows. Let $\widetilde{\mathbf M}$ replace +$(M_D)_h$ by $(M_D)_h+b_h$ and keep every other array fixed. +Then $\widetilde{\mathbf M}$ is a valid array family and its full question +has exactly the same canonical key as that of $\mathbf M$. +For any fixed label at this key, write $y_C$ and $\widetilde y_C$ for +its restored block outputs. They satisfy +\begin{equation}\label{eq:folding-output} + \widetilde y_C=y_C\quad(C\neq D),\qquad + (\widetilde y_D)_h=(y_D)_h+c_h+ + \sum_{C,u,v}\lambda_{h,C,u,v}(y_C)_u(y_C)_v. +\end{equation} +The same statement holds for a question with quotient blocks whenever +the shift uses only unchanged scalar rows actually displayed in that +question; a shift at a quotient block is interpreted through its quotient +map. +\end{lemma} +\begin{proof} +By Lemma~\ref{lem:construction-space-properties}, every term in +\eqref{eq:folding-shift} lies in $H_D(Q)$, proving validity. +On the full output product, define a transformation that fixes every +block other than $D$ and adds the displayed polynomial in those blocks +to block $D$. Applying this transformation twice is the identity: +its input blocks remain fixed, and addition is in characteristic two. +It is therefore a bijection, and the new joint function is this bijection +composed with the old joint function. The two functions have identical +equality relations on $\Omega(Q)$. Lemma~\ref{lem:canonical-support} +gives the same coordinate reduction and the same unordered partition. +Thus a fixed part represents precisely the same set of reduced arguments +in both questions, and evaluating either function on any such argument +gives \eqref{eq:folding-output}. + +For quotient blocks use the identical triangular transformation on the +product of displayed quotient spaces. Its inverse is again itself +because all arguments of the added expression are unchanged displayed +outputs. The previous proof applies verbatim. No action on labels is +assumed to be free, and no choice of a representative assignment is +needed to define the identity. +\end{proof} + +\begin{lemma}[Joint evaluation of product generators] +\label{lem:joint-evaluation} +Fix a legal response to a full array question and a node $D$. +For any collection $\mathcal C$ of proper nonleaf descendants of $D$, let +\[ + B=\Span\bigl(\{1\}\cup + \{(M_C)_u(M_C)_v:C\in\mathcal C,\ 1\leq u,v\leq + \ell_{\operatorname{level}(C)}\}\bigr) + \subseteq H_D(Q). +\] +If $y_C$ are the restored outputs, there is a unique linear functional +$b_y:B\to\F$ such that $b_y(1)=1$ and +$b_y((M_C)_u(M_C)_v)=(y_C)_u(y_C)_v$. +The functional $b_y$ extends to $H_D(Q)^*$. Every such extension $z$ has +value $1$ on the constant function, and its matrix on these products at each $C$ is +$y_Cy_C^{\mathsf t}$, of rank at most one. +\end{lemma} +\begin{proof} +The restored response has one realizing assignment $x\in\Omega(Q)$. +Point evaluation at $x$ restricts to a linear functional on $B$ and has +the stated generator values. This proves that every pointwise linear +relation between the generators is respected, including relations +involving different descendants. The generators span $B$, so this +functional is unique and depends only on the displayed $y_C$. +Extend a basis of $B$ to a basis of $H_D(Q)$ and assign arbitrary values +on the added basis vectors to obtain an extension. Any extension has +the same generator values, giving the asserted outer product at each descendant. +\end{proof} + +The use of all relevant descendant outputs in +Lemma~\ref{lem:joint-evaluation} will allow one upper quotient table to +handle every possible lower node. A label always supplies those +outputs together; selecting a single descendant is not part of the +vertex key. + +\subsection{The distribution of tests} + +The low-rank, Hadamard-bucket viewpoint is inspired by Golowich's +construction \cite{Golowich2023}, as mediated by Fei--Minzer--Wang +\cite{FeiMinzerWang2026}. The sampler here is a recursive-product +redesign with separate independent calls. We prove its sampling +estimates below; no expansion theorem or identical sampler from those +works is assumed. + +We now choose an original tuple $E$, an array family, a node, and an omitted +direction to sample one of the preceding constraints. Sample a source left +question independently at each leaf, using the source left marginal, and +denote the resulting tuple by $E$. + +Independently of $E$, sample a path +$\mathsf P=(D_d,D_{d-1},\ldots,D_0)$ by choosing $D_{p-1}$ uniformly +among the children of $D_p$, successively for $p=d,\ldots,1$. +We next define scalar laws conditional on $E$ and this whole path. +The procedure $\operatorname{Sample}(D_p)$ returns a function in $H_{D_p}(E)$. +At $p=0$ it returns an independent uniform element of $H_{D_0}(E)$. +At $p\geq1$, write $C_*=D_{p-1}$ for the path child and make the following +independent draws: +\[ + U_C\sim\operatorname{Unif}(H_C(E)^2)\quad(C\neq C_*), + \qquad + G_h,H_h\sim\operatorname{Sample}(C_*)\quad(1\leq h\leq R_p). +\] +Return +\begin{equation}\label{eq:scalar-sampler} + \operatorname{Sample}(D_p) + =\sum_{C\neq C_*}U_C+\sum_{h=1}^{R_p}G_hH_h. +\end{equation} +Its conditional law is denoted $\nu_{D_p}$; the dependence on $E$ and +the path below $D_p$ is suppressed in this notation. +An empty sum over ordinary children is zero. By +\eqref{eq:construction-spaces}, every returned function lies in the +claimed space. + +Every invocation of $\operatorname{Sample}$ uses a new random tape for +the entire recursive call tree. In particular the two factors in one +product, the factors in different products, and calls made on behalf of +different displayed arrays are independent conditional on $E,\mathsf P$. +Visiting the same geometric node in two calls does not reuse a value. +Uniform functions in ordinary children are drawn directly from the stated +spaces; they do not refer to the displayed arrays in those children. + +For every nonleaf path node $D_p$, $1\le p\le d$, independently take a fresh sample +$h_{p,v}\sim\nu_{D_p}$ for each $v\in K_p\setminus\{0\}$ and set +\begin{equation}\label{eq:array-bucket-sampler} + M_{D_p}=\sum_{v\in K_p\setminus\{0\}}v\otimes h_{p,v}. +\end{equation} +We refer to the summands indexed by $v$ as buckets; $v$ is a bucket +direction in $K_p$, not a row map. +For every nonleaf node off the path, take $M_D$ uniform in +$K_{\operatorname{level}(D)}\otimes H_D(E)$. +All these displayed arrays use separate random tapes. Thus, conditional +on $E,\mathsf P$, the displayed arrays are independent, including a path +array and an ancestor array whose scalar calls pass through that path +node. They need not be independent after the common path is averaged out. +The same definitions apply to any mixed tuple $O$ in its own function +spaces. Later modifications will explicitly replace calls at a specified +cut by uniform draws. + +Figure~\ref{fig:tree-cut}, beside the decoder input definitions, depicts the +two cuts and the distinction between a node and separate calls at that node. + +Only the canonical key of the joint function +\eqref{eq:joint-question-function} is supplied as the left question. The +path, buckets, and internal tapes determine its distribution and are not +appended to that key. + +To specify all candidate tests on the sampled array family, draw independently +\[ + a_p\sim\operatorname{Unif}(K_p\setminus\{0\}),\qquad 1\leq p\leq d, +\] +independently of the preceding data. At level $p$ use the quotient +$Q_{D_p,a_p}$; abbreviate it to $Q_{p,a_p}$. The actual test chooses $p$ +uniformly from $[d]$, independently, and uses the edge with left question +$\mathfrak k_L(\Psi_{E,\mathbf M})$, right question +$\mathfrak k_R(\Psi^{D_p,a_p}_{E,\mathbf M})$, and projection +\eqref{eq:legal-edge-map}. This finite distribution of edge records defines +the preliminary weighted game. Equivalently, one may draw only the direction +belonging to the chosen level. Sampling all directions specifies a joint law +of the candidate tests whose individual marginals are precisely this test. + +By Proposition~\ref{prop:completeness}, the game has perfect completeness. +The remaining sections prove its soundness by analyzing what a successful +labeling reveals about the sampled arrays. diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/03-sampling.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/03-sampling.tex new file mode 100644 index 0000000000000000000000000000000000000000..e8b8c9d7cc839a53349fa1388bbe174a08112055 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/03-sampling.tex @@ -0,0 +1,553 @@ +\section{Changing the sampling law without revealing the path} +\label{sec:sampling} + +The decoding argument will use uniform matrices at two different levels. +Here we prove the distributional comparisons that permit those changes. +There are two distinct conclusions: a total-variation comparison of the +\emph{displayed functions}, and a posterior bound for a child selected after +inspecting those functions. Neither conclusion permits disclosure of the +hidden path or of the projection mask. All statements below hold conditional +on the complete original question tuple $E$; consequently they also hold after +averaging over $E$, uniformly in the size of the source instance. +We first replace the recursive draws at a cut by uniform draws, then compare +that law with one in which a sparse set of children is projected. Each +comparison retains only the observation specified in its lemma. + +For probability measures on a finite set, use +\[ + \TV(P,Q)=\frac12\sum_x|P(x)-Q(x)|, + \qquad + \chi^2(Q\Vert P)=\E_P[(dQ/dP-1)^2] +\] +when $Q\ll P$. Cauchy--Schwarz gives +$\TV(P,Q)\le\tfrac12\sqrt{\chi^2(Q\Vert P)}$. +An observation formed by a deterministic map cannot increase total variation: +its discrepancy on an event is the discrepancy on that event's inverse image. +The same assertion holds on adjoining independent randomness and applying a +randomized map, by conditioning on that randomness. These elementary facts +will let us retain complete assembled queries without treating their +components as independent. + +\subsection{Stopping the ancestor generators} + +A scalar call means one invocation of the law $\nu_D$ defined in +Section~\ref{sec:construction}, with its own fresh random choices. Write +$b_p=2^{\ell_p}-1$ and $c_p=2R_p$. For a fixed path prefix ending at a node +$D_p$ of level $p$, a call at a higher path node branches into exactly $c_h$ +independent calls at its path child when it crosses level $h$. Its ordinary +child contributions are uniform functions supported outside that child and +require no calls farther down the distinguished path. + +\begin{lemma}[Stopped-call representation]\label{lem:stopped-calls} +Fix $E$ and the path from the root through $D_p$, including $D_p$ but no +choice of its child. The original sampling can be generated as follows. +First generate an exterior record $\Xi_p$ consisting of the random choices +outside $D_p$'s subtree, with every recursive call reaching $D_p$ left +unevaluated. Then generate the pending scalar calls at $D_p$, its own +buckets, and the arrays strictly below $D_p$. Every displayed array is a +deterministic function of this record and these remaining values. + +The pending ancestor calls together with the own buckets number at most +\begin{equation}\label{eq:stopped-count} + T_p=\sum_{k=p}^d b_k\prod_{h=p+1}^k c_h. +\end{equation} +An empty product is one. If $e_k$ additional fresh scalar draws are requested +at each level $k\ge p$, a sufficient bound is +\begin{equation}\label{eq:stopped-extra-count} + T_p(e)=\sum_{k=p}^d(b_k+e_k)\prod_{h=p+1}^k c_h. +\end{equation} +In particular, replacing one level-$k$ bucket uses one extra draw and adds +$\prod_{h=p+1}^k c_h$ calls at the cut. The count is independent of $n_p$ and +of all higher branching numbers. + +Conditional on the full path, distinct stopped calls have independent +randomness. Pending ancestor calls are independent of the tapes of the +actual arrays at or below the cut; own buckets are independent of the tapes +of actual descendant arrays. Calls for the own buckets and calls pending +from ancestors are distinct even +when they sample the same function space. The exterior record can be exposed +before sampling the continuation of the path. Already assembled ancestor +arrays are deterministic outputs of the construction and are not independent +exterior information. +\end{lemma} + +\begin{proof} +Expand each of the $b_k$ scalar buckets used for the actual matrix at $D_k$, +for every $k\ge p$. A call at level $h>p$ is a sum of ordinary-child uniform +functions and $R_h$ products of two fresh calls at the path child. Stop each +of those factors the first time it reaches $D_p$. Thus a bucket originating +at level $k$ produces exactly $\prod_{h=p+1}^k c_h$ stopped calls. Adding over +$k$ proves \eqref{eq:stopped-count}; expanding additional calls proves +\eqref{eq:stopped-extra-count}. Ordinary children at every step are disjoint +from $D_p$'s subtree, so their contributions can be generated before any path +choice below $D_p$. + +Give each recursive invocation a separate random tape, indexed by its +originating matrix, bucket and sequence of factor indices. These indices +are distinct even if two invocations visit the same node. The sampler's +fresh-call convention says precisely that these tapes and the tapes of +actual descendant matrices are independent, conditional on the shared path. +All unevaluated expressions are finite expressions involving sums and +pointwise products. Filling their stopped leaves therefore reconstructs +every ancestor array deterministically. The expression may involve many +exterior functions when higher branching is large, but its number of stopped +leaves is still \eqref{eq:stopped-count}. This proves all the assertions. +\end{proof} + +We will also use an equivalent representation after uniformizing the cut. +Instead of generating $M_{D_p}$ from $b_p$ uniform buckets, generate its +$\ell_p$ rows as separate uniform draws. The relevant number of calls is then +\begin{equation}\label{eq:direct-cut-count} + \widehat T_p=\ell_p+ + \sum_{k=p+1}^d b_k\prod_{h=p+1}^k c_h. +\end{equation} +We reserve $\widehat T_p$ for the displayed original call family. If a +comparison adds $e$ independent uniform rows, where $e$ is fixed, its +enlarged count is $\widehat T_p+e$. For instance, forming a uniform matrix +together with one rank-one step from it requires its rows and one additional +independent scalar draw, giving the count $\widehat T_p+1$. +These rows have separate tapes from the ancestor calls. If visibility of +upper buckets will later be chosen by a row map $A$, predetermine calls for +\emph{all} buckets before choosing which are visible. Selecting or discarding +calls afterward adds no dependence to the count. + +\subsection{Finite child blocks and a hidden special child} + +Fix a level-$p$ cut $D=D_p$ and an integer $t_{\mathrm{cut}}$ bounding all scalar draws +retained at that cut. For a child $C$ of $D$, put $V_C=H_C(E)^2$ and define +the following finite product space: +\begin{equation}\label{eq:raw-child-block} + \mathcal X_C=V_C^{t_{\mathrm{cut}}}\times + \prod_{\substack{T\subseteq C\\T\text{ nonleaf}}} + (K_{\operatorname{level}(T)}\otimes H_T(E)). +\end{equation} +Here $T\subseteq C$ means that $T$ is $C$ or a descendant of $C$. +A block $X_C$ records a separate child contribution to each of the $t_{\mathrm{cut}}$ cut +draws and the actual arrays throughout this child's subtree. The factors +used internally to generate a special contribution are not included in +$X_C$. Each recorded contribution is just a function in $V_C$. + +Let $P_C$ be uniform on $\mathcal X_C$. To construct independent uniform cut +draws, sample these blocks independently and, for each draw, add its child +contributions. This uses a latent product of spaces even though the actual +spaces $V_C$ share the constant functions. Indeed the map +\[ + \prod_{C\text{ child of }D}V_C\longrightarrow H_D(E), + \qquad (u_C)_C\longmapsto\sum_Cu_C +\] +is a surjective linear map. Every fiber is a coset of its kernel, so the +image of uniform measure is uniform. Separate copies give independent +uniform cut draws, independently of all descendant arrays. No direct-sum +decomposition, or recovery of the summands from their sum, is asserted. + +There is a uniform finite density bound for any law on $\mathcal X_C$. +For clarity, if the source's legal assignment domains have size at most $a$, +put $N_0=1$ and $N_k=\prod_{h=1}^k n_h$. Every space on a level-$k$ subtree +has dimension at most $a^{N_k}$. Thus, for children at level $p-1$, +\begin{equation}\label{eq:block-dimension-bound} + \dim\mathcal X_C\le + D_p(t_{\mathrm{cut}}):=a^{N_{p-1}}\left(t_{\mathrm{cut}}+ + \sum_{k=1}^{p-1}\ell_k\prod_{h=k+1}^{p-1}n_h\right). +\end{equation} +In particular we may use $L_p(t_{\mathrm{cut}})=2^{D_p(t_{\mathrm{cut}})}$. Every probability law $Q_C$ on +$\mathcal X_C$ has density $q_C=dQ_C/dP_C\le L_p(t_{\mathrm{cut}})$, since each $P_C$ atom has +mass $2^{-\dim\mathcal X_C}$. This bound uses the source alphabet, the fixed row +and repetition counts through $t_{\mathrm{cut}}$, and branching below $p$. It uses neither +$n_p$, higher branching, source question identities, nor the source instance +size. Any larger fixed common bound will also be denoted by $L_p$. + +\begin{lemma}[One hidden child]\label{lem:cut-tv} +In the original experiment, expose $E$, the path through $D_p$, and the +exterior record of Lemma~\ref{lem:stopped-calls}. Forget the entire +continuation of the path below $D_p$. Retain any fixed family of stopped +scalar values, the own buckets at $D_p$, and all actual arrays below $D_p$. +Let $\mathsf U_p$ be the \emph{unrestricted experiment}: replace the retained +cut scalar values by independent uniform draws in $H_{D_p}(E)$, and replace +all descendant arrays by independent uniform arrays in their respective +spaces, retaining the same exterior record and reconstruction expressions. +Then +\begin{equation}\label{eq:cut-tv-bound} + \TV(\mathsf A_p,\mathsf U_p)\le + \varepsilon_p^{\mathrm{cut}}:=\frac{L_p}{2\sqrt{n_p}} +\end{equation} +for the retained values and for every deterministic observation of them. +Here $\mathsf A_p$ denotes the corresponding retained original law, and +$L_p$ can be chosen using \eqref{eq:block-dimension-bound} with the count in +\eqref{eq:stopped-extra-count}. + +One may also append a fixed number of independent uniform cut rows, or a +fixed number of fresh bucket replacements, with the appropriate increased +count. The bound includes the reconstructed ancestor arrays and candidate +edge tests at levels at least $p$, with independent quotient directions. +It does not include a tag identifying the next path child, any later path +indices, or the internal factor tapes generating a special child block. +\end{lemma} + +\begin{proof} +Condition on $E$, the prefix and the exterior record. Let $J$ be the next +path child, uniform among $n=n_p$ children. Given $J=C$, every block other +than $X_C$ has law $P_{C'}$: all its scalar contributions are ordinary-child +uniforms, and all its subtree matrices are off the path and uniform. They +are independent of one another and of $X_C$. Within $X_C$, the scalar +contributions are sums of $R_p$ products from fresh factor calls, and its +subtree arrays use the continued path. Average over that continuation to +obtain a law $Q_C$ on \eqref{eq:raw-child-block}. Correlations among the +coordinates of this one block cause no difficulty. Its density $q_C$ is +bounded by $L_p$. + +Relative to $P=\prod_CP_C$, the original raw-block law has density +\[ + Z=\frac1n\sum_Cq_C(X_C). +\] +Under $P$ these summands are independent and have expectation one. Therefore +\[ + \E_P[(Z-1)^2] + =\frac1{n^2}\sum_C\E_{P_C}[(q_C-1)^2] + \le\frac{L_p^2}{n}. +\] +The total-variation bound follows by Cauchy--Schwarz. Summing the child +contributions and reconstructing all pending expressions is the same +deterministic map in both laws, so it preserves the bound. It remains valid +conditional on every exterior record and hence on averaging that record. +Fresh bucket replacements merely enlarge the finite family of scalar calls; +independent uniform cut rows add uniform coordinates to every block. Both +are covered by the same argument. A revealed value of $J$ would replace the +average density by $q_J$ and would lose the factor $n^{-1/2}$, which explains +the required information restriction. +\end{proof} + +In $\mathsf U_p$, the own buckets are separate from the pending ancestor +calls and the descendant matrices. The matrix +$M=\sum_{v\in K_p\setminus\{0\}}v h_v$ is uniform: the linear map from its +uniform buckets onto $K_p\otimes H_{D_p}(E)$ is surjective, since the standard +basis vectors occur among the indices. Replacing $h_a$ by an independent +uniform $h_a^*$, for an independent nonzero $a\in K_p$, gives +\begin{equation}\label{eq:unrestricted-step} + M^*=M+a h,\qquad h=h_a+h_a^*. +\end{equation} +Conditional even on all old buckets, $h$ is uniform and independent of the +old data. Thus this is exactly the uniform rank-one step, and $M$ is +independent of the other displayed arrays and the pending ancestor calls. +This conclusion would not follow by reusing a bucket as an ancestor call. + +\subsection{Sparse projection and its posterior} + +The next comparison is a form of sub-code covering: a sparse restriction +leaves the lifted query law nearly unchanged. This method appears in +Khot and Safra~\cite[Section~3.4]{KhotSafra2013}, with advice-conditioned +thinning estimates in Dinur, Khot, Kindler, Minzer and Safra +\cite[Lemma~5.5, full version]{DinurKhotKindlerMinzerSafra2018} and +Fei, Minzer and Wang~\cite[Propositions~4.7--4.8 and~5.5]{FeiMinzerWang2026}. +Here the independent units are entire child subtrees, whose records include +all stopped-call contributions and displayed descendant arrays. We prove +the comparison and posterior estimate for these finite blocks. + +At a level-$i$ cut $D_i$, independently mark each child with probability +$\beta_i=n_i^{-2/3}$; take $n_i>1$ to be a cube. At every leaf in a marked +child, sample a source projection conditional on its left question, and +replace that question by its right question. Leave all other questions +unchanged, and call the resulting tuple $O$. Pullback along these +projections identifies $H_T(O)$ with a subspace of $H_T(E)$ and preserves +pointwise multiplication. Define $\mathsf M_i$, the \emph{modified +experiment}, by stopping all ancestor calls at $D_i$ and making their values +independent uniform draws in $H_{D_i}(O)$. Also sample $M_{D_i}$ and all its +descendant matrices independently uniformly in the corresponding projected +spaces. Reconstruct all ancestors by the original stopped expressions. +Path choices below $i$ are unused. Independent extra cut rows and bucket +replacements are treated in the same way. The original unmodified law will +be denoted by $\mathsf A$. + +\begin{lemma}[Sparse projection comparison]\label{lem:sparse-tv} +Fix $E$, the path through $D_i$ and an exterior record generated without the +projections. Compare $\mathsf M_i$ and $\mathsf U_i$ as laws of lifted +functions on $E$, forgetting the marks, the projection choices, and the +projected-question tuple $O$. All raw child summands and descendant arrays +may be retained. For a bound $L_i$ from +\eqref{eq:block-dimension-bound}, +\begin{equation}\label{eq:sparse-tv-bound} + \TV(\mathsf M_i,\mathsf U_i) + \le \varepsilon_i^{\mathrm{spr}} + :=\frac12\sqrt{(1+\beta_i^2L_i^2)^{n_i}-1}. +\end{equation} +The bound holds for any fixed number of independent extra cut rows and fresh +higher-level bucket draws, with their enlarged call count. In particular it +holds for observations containing the assembled ancestor arrays and both +candidate tests at levels $ii$, require the observation to be a common deterministic function of the +exterior record $\Xi_j$, the retained scalar values at $D_j$ (including its +own buckets and any extra cut draws), all displayed arrays strictly below +$D_j$, and independent auxiliary choices. Here $E$ and the path through +$D_j$ are fixed. Such an observation may include reconstructed ancestors +and descendant arrays with their tree locations, but it cannot use the +hidden continuation of the path or the internal decompositions of the +retained scalar values. In particular, a selected lower-level test is +permitted only if it is determined by these data without using the hidden +path. The distance from the corresponding unrestricted-$j$ observation is +then at most +\begin{equation}\label{eq:two-cut-triangle} + \varepsilon_i^{\mathrm{spr}}+ + \varepsilon_i^{\mathrm{cut}}+\varepsilon_j^{\mathrm{cut}}. +\end{equation} +\end{lemma} + +\begin{proof} +For a child $C$ of $D_i$, let $P_C$ be uniform on the unprojected block +\eqref{eq:raw-child-block}. Conditional on marking $C$, average over all its +projection choices and lift the independent projected uniform contributions +and arrays to $E$; call the resulting law $Q_C$. It is a law on the same +finite set, so $q_C=dQ_C/dP_C\le L_i$. Different children use independent +marks, projections and tapes. Thus their lifted blocks have product law +\[ + \prod_C\bigl((1-\beta_i)P_C+\beta_iQ_C\bigr), +\] +whose density with respect to $\prod_CP_C$ is +$\prod_C(1+\beta_i(q_C-1))$. Squaring and using independence yields the +exact identity +\[ + 1+\chi^2\!\left( + \prod_C((1-\beta_i)P_C+\beta_iQ_C) + \,\middle\Vert\,\prod_CP_C\right) + =\prod_C\bigl(1+\beta_i^2\chi^2(Q_C\Vert P_C)\bigr) + \le(1+\beta_i^2L_i^2)^{n_i}. +\] +This proves \eqref{eq:sparse-tv-bound}. Uniform cut rows are obtained by the +same addition map in both laws. A uniform matrix can be represented either +by independent rows or by independent uniform buckets, as proved above; +choose the latter representation when retaining a bucket replacement. +Additional calls simply add coordinates to each finite block. + +All higher arrays, and their quotients in a candidate test, are functions of +the blocks, exterior record and independent test directions. Data processing +therefore proves their bounds, without conditioning on their realized values. +The comparison with $\mathsf A$ is the triangle through $\mathsf U_i$ and +Lemma~\ref{lem:cut-tv}. For \eqref{eq:two-cut-triangle}, take the common +observation just specified. First average the preceding +$i$-cut comparison over the prefix between levels $j$ and $i$, and then apply +Lemma~\ref{lem:cut-tv} at $j$ to the original law. A pending scalar value at +$j$ is an output of the $i$-cut reconstruction and is a retained cut draw in +the $j$-cut comparison, so it can be included throughout. The internal +factor choices within its subtree cannot be included in the latter +comparison. This establishes precisely the stated triangle. +\end{proof} + +\paragraph{Common observations in the three comparisons.}\label{sec:law-guide} +The laws just defined serve different purposes, and each comparison forgets +different information: +\begin{description}[style=nextline,leftmargin=1.5em,font=\normalfont\bfseries] +\item[Original to unrestricted at one cut.] +Lemma~\ref{lem:cut-tv} replaces the cut draws and descendant arrays by +uniforms after forgetting the path below that cut. The observation may +include the ancestors reconstructed from those draws. +\item[Modified to unrestricted at the lower cut.] +Lemma~\ref{lem:sparse-tv} compares lifted functions on $E$ after forgetting +$O$, the projection choices, and the marks. Raw child contributions and +all displayed descendant arrays may be retained. +\item[Modified lower cut to unrestricted upper cut.] +For $j>i$, Equation~\eqref{eq:two-cut-triangle} applies to a common function +of the upper-cut record specified in Lemma~\ref{lem:sparse-tv}. That record +retains pending scalar values there, but not their internal decompositions, +and all displayed descendant arrays. The observation cannot select among +those arrays using the hidden path. +\end{description} + +The sparse product calculation also gives a posterior bound for a child +selected after seeing the lifted matrices. This is stronger than a bound +for a child fixed in advance, and will allow the lower decoder to choose a +witness from its observed data. + +\begin{lemma}[Posterior bound for adaptive witnesses]\label{lem:posterior} +In $\mathsf M_i$, condition on $E$, the path through $D_i$ and a permissible +exterior record. Let $X=(X_C)_C$ consist of all lifted raw child blocks, +including any fixed number of independent uniform cut rows. Let +$\mathcal G$ be the sigma-algebra generated by these data and additional +independent random choices, such as a test direction or fixed full strategy +tapes. It contains neither projected-question metadata nor projection marks. +For every child $C$, almost surely, +\begin{equation}\label{eq:posterior-bound} + \Prob(C\text{ marked}\mid\mathcal G) + \le q_i:=\frac{\beta_i L_i}{1-\beta_i}. +\end{equation} +If a set $\mathcal C(X)$ of at most $m$ children is selected measurably from +these data, then +\begin{equation}\label{eq:adaptive-child-bound} + \Prob(\mathcal C(X)\text{ contains a marked child})\le m q_i. +\end{equation} +Both assertions remain valid for a coarsening of this information. + +In particular, let $F_0,F_1$ be bilinear forms on $H_{D_i}(E)$ determined by +these lifted data. Then +\begin{equation}\label{eq:adaptive-bilinear-error} + \Prob\bigl(F_0\ne F_1,\ + F_0|_{H_{D_i}(O)\times H_{D_i}(O)} + =F_1|_{H_{D_i}(O)\times H_{D_i}(O)}\bigr) + \le 2q_i, +\end{equation} +where the restrictions in this display mean restriction of both arguments +to $H_{D_i}(O)$. +No part of the statement conditions on strategy success, a right output, +a slice event, or a witness-matching event. +\end{lemma} + +\begin{proof} +Conditional on the fixed exterior data, child independence and Bayes's +formula give, simultaneously at every lifted block tuple with positive +probability, +\[ + \Prob(C\text{ marked}\mid X) + =\frac{\beta_iq_C(X_C)}{1-\beta_i+\beta_iq_C(X_C)} + \le\frac{\beta_iL_i}{1-\beta_i}. +\] +Independent extra choices do not change this conditional formula. For an +adaptively selected set, condition on $\mathcal G$ first and use +\[ + \E\left[\sum_{C\in\mathcal C(X)} + \one_{\{C\text{ marked}\}}\ \middle|\ \mathcal G\right] + \le |\mathcal C(X)|q_i\le m q_i. +\] +Taking expectations and a union bound proves +\eqref{eq:adaptive-child-bound}; the tower property proves the coarsened +versions. There is no factor for the total number of available children. + +For the bilinear consequence, write $\Delta=F_0-F_1$. Since +$H_{D_i}(E)=\sum_C H_C(E)^2$, if $\Delta\ne0$, then there are children +$C_1,C_2$ and functions $x\in H_{C_1}(E)^2$, $y\in H_{C_2}(E)^2$ with +$\Delta(x,y)=1$. Indeed decompose a pair on which $\Delta$ is nonzero into +such sums and expand bilinearly; at least one summand is nonzero. Fix orders +on the finite spaces and choose the first such witness from the lifted data. +The children may coincide. If neither is marked, their spaces are unchanged +and lie in $H_{D_i}(O)$, so the two restricted forms cannot be equal. The +bad event in \eqref{eq:adaptive-bilinear-error} is therefore contained in the +event that one of at most two adaptively selected children is marked. +Apply \eqref{eq:adaptive-child-bound} with $m=2$. +\end{proof} + +The distinction from success conditioning is substantive: after the +unconditional bad-event probability is bounded, it may be subtracted from +any success probability. The posterior estimate itself need not hold within +that success event. Exposing a direction $a_i$ to the analyst does not +provide it to the left strategy. Likewise, adding assembled ancestor values to the +lifted blocks changes no information, because they are functions of those +blocks and $\Xi_i$; treating those values as independently sampled exterior +data would not justify the product law used above. + +\subsection{Bias that is independent of dimension} + +Total variation above is obtained by increasing branching. The upper +Fourier argument instead requires a small-bias bound before any slice +conditioning, obtained by increasing the product counts. We use the +character-wise notion of small bias of Naor and Naor~\cite{NaorNaor1993}; +the required sampler estimate is proved below. For a law $\mu$ +on a finite binary vector space $V$, call it \emph{quarter-balanced} if every +nonzero $f\in V^*$ satisfies +$\Prob_{x\sim\mu}(f(x)=b)\ge1/4$ for both $b\in\F$. + +\begin{lemma}[Scalar bias and independent hidden buckets]\label{lem:bias} +There is an absolute integer $R_{\min}$ such that the following holds if +$R_p\ge R_{\min}$ at every nonleaf level. Fix the domains and the full path +in the original sampler, or fix $E,O$, all projection choices, and the path +through a modified cut in $\mathsf M_i$. Every scalar law is quarter-balanced. +For each nonzero $f\in H_{D_p}^*$ at an original path node, or +$f\in H_{D_p}(O)^*$ at a node strictly above the modified cut, +\begin{equation}\label{eq:scalar-bias} + \left|\E_{h\sim\nu_{D_p}}(-1)^{f(h)}\right| + \le (7/8)^{R_p}. +\end{equation} +At leaves, and at or below the uniformized cut, uniform scalar draws have +zero bias on nonzero functionals. + +More generally, fix $j>i$, a linear map $A:K_j\to\F^r$, and $W=\ker A$. +In the modified experiment let $h_v$, $v\ne0$, be the separate level-$j$ +buckets and set +\[ + X=\sum_{v\in W\setminus\{0\}}v h_v + \in W\otimes H_{D_j}(O). +\] +Conditional on the domain and path data just specified, all other actual +matrices, and all buckets with $Av\ne0$, every nonzero character of $X$ has +bias at most $(7/8)^{R_j}$. This remains so on revealing original questions +and any other independent sampling tapes. Its law is determined by $O$, +the path through the cut, $A$, and the fixed sampling parameters; it does +not require the original questions or the missing part of a lower matrix. +Revealing a full lower matrix here is an analytical exposure; the right +decoder still receives only its prescribed quotient. These conclusions +are not asserted after conditioning on a column slice, on successful +decoding, or on cleanliness. +\end{lemma} + +\begin{proof} +First suppose $g,h$ are independent draws from quarter-balanced laws on a +binary space $V$, and $B:V\times V\to\F$ is a nonzero bilinear form. The +linear map $g\mapsto B(g,\cdot)$ has proper kernel. Some nonzero linear +functional on $V$ vanishes on that kernel; quarter balance therefore implies +\[ + \Prob(B(g,\cdot)\ne0)\ge1/4. +\] +For each such $g$, quarter balance of the independent $h$ implies +$\Prob(B(g,h)=b\mid g)\ge1/4$ for each bit $b$. Hence each bit of $B(g,h)$ +has probability at least $1/16$, and +$|\E(-1)^{B(g,h)}|\le7/8$. + +Now apply induction from the leaves, or upward from the modified uniform +cut. Let $f$ be nonzero on the space at a path node $D_p$. If its +restriction to the square space of some ordinary child is nonzero, the +corresponding uniform summand makes $f(h)$ an exactly uniform bit. Otherwise +$f$ vanishes on every ordinary child's square space. Since these spaces +together with the special child's square space span $H_{D_p}$, its +restriction to the latter is nonzero. The bilinear form +\[ + B(g,h)=f(gh) +\] +on that child's row space is then nonzero, since pointwise products span its +square space. The induction hypothesis makes the two independent factor +laws quarter-balanced. Each product thus has sign expectation of absolute +value at most $7/8$, and the $R_p$ independent products give +\eqref{eq:scalar-bias}. Choose $R_{\min}$ so that +$(7/8)^{R_{\min}}\le1/2$. Then the resulting scalar law is again +quarter-balanced, closing the induction. Uniform cut and leaf laws provide +the initial step. The argument uses only whether linear or bilinear maps +are nonzero, and no dimension enters its bound. + +For the hidden sum, identify a character with a functional +$\Phi\in(W\otimes H_{D_j}(O))^*$. If $\Phi\ne0$, then for some nonzero +$v\in W$ the functional $h\mapsto\Phi(v\otimes h)$ is nonzero; simple +tensors span the tensor product. Conditional on the stated information, +every hidden bucket retains its own fresh tape, independently of the other +hidden buckets. This follows directly from the call indexing in +Lemma~\ref{lem:stopped-calls}; higher matrices use distinct calls even where +they visit the same spaces. Consequently +\[ + \E(-1)^{\Phi(X)}= + \prod_{v\in W\setminus\{0\}} + \E(-1)^{\Phi(v\otimes h_v)}. +\] +One factor has absolute value at most $(7/8)^{R_j}$ and the others at most +one. If $W=0$ there are no nonzero characters and the assertion is vacuous. +Every hidden bucket is generated on the projected domain $O$ by fresh +recursion ending with uniform draws at the cut. That recipe uses neither +$E$ nor the values of other actual matrices, proving the asserted law and +its invariance under the permissible additional exposure. +\end{proof} + +The order of accuracy choices follows from this proof. At descendants one +needs only the absolute quarter-balance guarantee, obtained by the common +$R_{\min}$. Once an upper level $j$ requires a tolerance $\lambda_j>0$, it +suffices to choose $R_j$ with $(7/8)^{R_j}\le\lambda_j$. No requested upper +accuracy imposes a finer accuracy at a lower level. After all row counts, +product counts and the source alphabet are fixed, choose branching upward. +At cut $p$, $L_p$ is already fixed by lower branching, and +\[ + \varepsilon_p^{\mathrm{cut}}\longrightarrow0, + \qquad + \varepsilon_p^{\mathrm{spr}}\longrightarrow0, + \qquad + \frac{2n_p^{-2/3}L_p}{1-n_p^{-2/3}}\longrightarrow0. +\] +For the middle limit, use +$(1+n_p^{-4/3}L_p^2)^{n_p}\le\exp(L_p^2n_p^{-1/3})$. +Enlarging higher branching changes only the exterior record and the +reconstruction map, so all three bounds already secured at $p$ persist. diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/04-upper-decoding.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/04-upper-decoding.tex new file mode 100644 index 0000000000000000000000000000000000000000..aaf7463914003e45cfc017a61d02cf569796f137 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/04-upper-decoding.tex @@ -0,0 +1,642 @@ +\section{Two successful levels and upper decoding}\label{sec:upper} + +We show that appreciable test acceptance produces two local bilinear-form +decoders. Their outputs agree on the projected lower space with a probability +depending only on the upper row count. The left output is always a normalized +product form whose restriction to its displayed lower rows has rank at most +one. Keeping this probability independent of the lower row count will allow +us to choose that count afterward. + +Fix a legal labeling of the partition vertices, extended to unused legal keys +as in Section~\ref{sec:construction}. All responses below are evaluations +restored from this fixed labeling. For a candidate test at level $p$, write +$\mathrm{Acc}_p$ for its acceptance event. Sample the directions for all candidate +tests independently, using the same left query and sampled path. + +\subsection{Selecting two levels and a quotient table} + +Suppose that the labeling accepts with probability at least $\eps$, where +$0<\eps<1$, and choose $d\eps\ge4$. If +$N=\sum_{p=1}^d\one_{\mathrm{Acc}_p}$, then +\[ + \E[N(N-1)]\ge (\E N)^2-\E N + \ge d^2\eps^2-d\eps. +\] +Consequently some $i0. +\end{equation} +All quantities $g,h_0,q_0,\rho,b$ depend only on $c,\alpha,r,\ell$. +\end{lemma} + +\begin{proof} +On $I$ the mark $U$ is one. Independence of $A$ and the measurability +just proved imply +\[ + \Prob_i(I\cap \mathrm{Acc}_i) + =\E_i\bigl[\one_I\Prob_i(\mathrm{Acc}_i\mid P,\bar M,A)\bigr] + \ge\kappa\Prob_i(I). +\] +Acceptance at level $i$ preserves the entire upper evaluation. Thus on +$I\cap \mathrm{Acc}_i$ the right response is $Mz_0+u$. Exact sharing of canonical +keys permits evaluating this right response directly on $O$. +Writing $h=\Prob_j^0(J)\ge q_0$, unconditional total variation and +\eqref{eq:upper-IJ} give +\[ + \Prob_i(I)\ge\alpha h-v,\qquad \Prob_i(J)\le h+v. +\] +The left side of \eqref{eq:upper-difference} is consequently at least +\[ + (\kappa\alpha-\rho)h-(\kappa+\rho)v + \ge\frac{3\kappa\alpha}{4}q_0 + -\kappa(1+\alpha/4)\frac{\alpha q_0}{16} + \ge b. +\] +This proves an unconditional inequality. We will disintegrate it below, +without reusing the usefulness estimate after a finer conditioning. +\end{proof} + +The affine witness now predicts the upper response with positive probability, +and every functional in its coset is normalized and passes the lower rank +test. To obtain two local decoders, we need a bounded collection of candidate +functionals, computed from the right input, that contains the restriction of +some functional in this coset to the projected space. The same collection +must work for every compatible left witness. + +\subsection{The local inputs and the original hidden law}\label{sec:right-hidden-law} + +Here and below the experiment is $\Prob_i$. Write +$H'=H_{D_j}(O)$, $H_i=H_{D_i}(E)$, $H_i'=H_{D_i}(O)$, and +$S=M_{D_i}$. Pullback identifies the primed spaces with subspaces of the +unprimed ones, preserving products. Put $W=\ker A$ and split the upper +matrix into +\begin{equation}\label{eq:upper-hidden} + M=M_{\rm known}+X,\qquad + M_{\rm known}=\sum_{v\in K\setminus W}v\otimes h_v,\qquad + X=\sum_{v\in W\setminus\{0\}}v\otimes h_v\in W\otimes H'. +\end{equation} +The known sum is determined without any bucket whose index lies in $W$. +In particular $AM=AM_{\rm known}$. + +We specify the two inputs exactly. Tree locations are retained in these +analytical inputs, even though the original partition keys omit unnecessary +sampling metadata. The left information is +\begin{equation}\label{eq:upper-left-input} + \mathcal L=\sigma\bigl(E,\text{path through }D_i,A, + (M_D)_{D\ne D_j},(h_v)_{v\in K\setminus W}\bigr), +\end{equation} +where every function is represented on its original $E$-domain. These +data supply $S$ in full, without supplying $a_i$, $O$, or the +projection choices as additional fields. The right information is +\begin{equation}\label{eq:upper-right-input} + \mathcal R=\sigma\bigl(O,\text{path through }D_i,A,a_i, + (M_D)_{D\ne D_j,D_i},S\bmod\langle a_i\rangle, + (h_v)_{v\in K\setminus W}\bigr), +\end{equation} +with functions represented on their $O$-domains. The right is supplied +only the displayed quotient of $S$, with no additional missing-component +ingredients. Neither side is supplied $X$. The left can compute $P$ and $A\bar M$ from +\eqref{eq:upper-left-input}, so it can compute the good-advice predicate +and its chosen witness without knowing $M$. +Figure~\ref{fig:tree-cut} summarizes these information restrictions. + +\input{figures/tree-cut} + +For clarity, the independence underlying these inputs is exact, not an +approximation. Conditional on $O$, the projection choices, and the path +through $D_i$, give each actual node matrix its own random tape. Give each +scalar bucket in a matrix a fresh recursive call tree, with independent +ordinary-child uniforms and independent factor calls at every product. +All calls that reach $D_i$ terminate in separate uniform draws in +$H_i'$. In particular, even a call used for an ancestor of $D_j$ and a +bucket of $M$ that both visit $D_j$ use disjoint random tapes. +The product factorization of these tapes implies +\begin{equation}\label{eq:upper-factorization} + (h_v)_{v\in W\setminus\{0\}}\mathrel{\perp\!\!\!\perp} + \bigl((h_v)_{v\notin W},(M_D)_{D\ne D_j}\bigr) + \quad\text{given the domains, path, and }A. +\end{equation} +The same statement remains true if the original questions $E$ and the +projection maps are additionally given: they determine the embeddings of +the spaces but do not change the intrinsic $O$-space sampler. This also +explains why analytical disclosure of the full $S$ is harmless for this +factorization, although $S$ is not part of the right input. + +Let $\mu_{\mathcal R}$ be the distribution on $W\otimes H'$ obtained by +sampling the hidden buckets in \eqref{eq:upper-hidden} with this original +modified-$i$ sampler. It is a function of the right input alone. Indeed, +the right knows $O$, the path down to the terminating cut, all spaces and +all sampler parameters. Independent recursive calls above that cut can +be enumerated on those spaces without knowing $E$, any projection map, +or the missing part of $S$. Formula~\eqref{eq:upper-factorization} shows +that $\mu_{\mathcal R}$ is also the actual conditional hidden law after +revealing all of the data in both inputs, including full $S$. + +For each fixed right input, define +\begin{equation}\label{eq:upper-G} + G_{\mathcal R}:W\otimes H'\longrightarrow K +\end{equation} +by assembling the level-$i$ right query with upper matrix +$M_{\rm known}+X$, consulting its fixed partition label, and restoring +the upper component. This is a well-defined local function: the lower +matrix in this query is precisely $S\bmod\langle a_i\rangle$. +The original labeling has already been extended to unused legal keys, so +this definition is total for every $X$ in the ambient space. + +Assume that every product count meets the absolute balance minimum of +Lemma~\ref{lem:bias}, and that +\begin{equation}\label{eq:upper-R} + (7/8)^{R_j}\le\lambda. +\end{equation} +Every nonzero character of $W\otimes H'$ then has bias at most $\lambda$ +under $\mu_{\mathcal R}$. To check this directly, identify a character +with a nonzero linear map $\Phi:W\longrightarrow(H')^*$. Some nonzero +$v\in W$ has $\Phi(v)\ne0$, so its expectation on the corresponding +bucket has absolute value at most $\lambda$ by Lemma~\ref{lem:bias}. +Independence factors the expectation over all hidden buckets, and all +remaining factors have absolute value at most one. This argument makes +no claim about small bias after a slice, a successful match, or any later +conditioning. + +\subsection{A bounded list for an arbitrary small-bias law} + +We record the Fourier calculation separately. It uses characters under +their actual sampling law rather than treating small bias as total-variation +closeness to uniform measure. The short-list principle is related to +Goldreich--Levin heavy-character decoding~\cite{GoldreichLevin1989}. +We prove the needed statement for the present nonuniform, own-question law; +no efficient decoding algorithm is required. If $V=W\otimes H'$, write +\[ + \chi_\Phi(X)=(-1)^{\langle\Phi,X\rangle},\qquad + \Phi\in V^*=\Hom(W,(H')^*). +\] +For $\sigma\in K^*$ and a fixed function $G:V\longrightarrow K$, put +$g_\sigma(X)=(-1)^{\sigma(G(X))}$. + +\begin{lemma}[One heavy list for all affine-slice witnesses] +\label{lem:heavy-list} +Let $K=\F^\ell$, let $W\subseteq K$ have codimension at most $r$, and +let $H'$ be a finite vector space. Let $\mu$ be a probability law on +$V=W\otimes H'$ with $|\E_\mu\chi_\Phi|\le\lambda$ for every nonzero +$\Phi\in V^*$. Fix $G:V\longrightarrow K$, $0<\rho\le1$, and set +\[ + h_0=2^{-r\ell},\qquad \tau=\rho h_0/4. +\] +Suppose +\begin{equation}\label{eq:upper-Fourier-conditions} + 2^{-(\ell-r)}<\rho/8, + \qquad 0\le\lambda\le\min\{h_0/2,\tau^2/2\}. +\end{equation} +For each $\sigma\in K^*$ define the list +\begin{equation}\label{eq:upper-heavy-definition} + \mathcal H_\sigma + =\{\Phi\in V^*:|\E_\mu[g_\sigma\chi_\Phi]|\ge\tau\}. +\end{equation} +Each list has size at most $2/\tau^2$. Moreover, let +$Q\subseteq(H')^*$ have dimension at most $r$, let +$t_Q\in\Hom(Q,W)$ specify the nonempty affine slice +$\mathcal C=\{X:X|_Q=t_Q\}$, and let +$z'\in(H')^*$ and $k_0\in K$. If +\begin{equation}\label{eq:upper-slice-equality} + \Prob_\mu(G(X)=Xz'+k_0\mid X\in\mathcal C)\ge\rho, +\end{equation} +then some $\sigma$ with $\sigma|_W\ne0$ has a list element in +\begin{equation}\label{eq:upper-heavy-coset} + (\sigma|_W)\otimes z'+W^*\otimes Q. +\end{equation} +The lists in \eqref{eq:upper-heavy-definition} depend on $\mu,G,W,H'$ and +$\tau$, and not on the witness $Q,t_Q,z',k_0$. +\end{lemma} + +\begin{proof} +Put $w=\dim W$ and $q=\dim Q$. The map +$V\longrightarrow\Hom(Q,W)$ given by $X\mapsto X|_Q$ is surjective: +choose a basis of $Q$, extend it to a basis of $(H')^*$, and extend any +specified linear map to all of $(H')^*$. Thus the linear equations for +$\mathcal C$ have rank $wq\le\ell r$. Their dual space is +$L=W^*\otimes Q\subseteq V^*$. Choose any $X_0\in\mathcal C$. The +exact character identity for the affine slice is +\begin{equation}\label{eq:upper-slice-expansion} + \one_{\mathcal C}(X) + =\frac1{|L|}\sum_{\Psi\in L}\chi_\Psi(X_0)\chi_\Psi(X). +\end{equation} +Indeed, the sum is one when $X-X_0$ annihilates $L$, and zero otherwise. +Consequently +\begin{equation}\label{eq:upper-slice-mass} + \mu(\mathcal C)\ge2^{-wq}-\lambda\ge h_0-\lambda\ge h_0/2>0. +\end{equation} +This proves the needed slice mass using unconditional bias. + +Expand the equality in \eqref{eq:upper-slice-equality} over $K^*$: +\[ + \Prob_\mu(G(X)=Xz'+k_0\mid\mathcal C) + =2^{-\ell}\sum_{\sigma\in K^*}(-1)^{\sigma(k_0)} + \E_\mu[g_\sigma(X)\chi_{(\sigma|_W)\otimes z'}(X) + \mid\mathcal C]. +\] +The fraction of $\sigma$ annihilating $W$ is $2^{-w}$, which is at most +$2^{-(\ell-r)}<\rho/8$. Each term has absolute value at most one, so +some remaining $\sigma$ has +\[ + \left|\E_\mu[g_\sigma\chi_{(\sigma|_W)\otimes z'}\mid\mathcal C]\right| + \ge\rho/2. +\] +Multiply by \eqref{eq:upper-slice-mass} and then insert +\eqref{eq:upper-slice-expansion}. The coefficients in that expansion +have total absolute value one. It follows that for some $\Psi\in L$, +\[ + \left|\E_\mu[g_\sigma + \chi_{(\sigma|_W)\otimes z'+\Psi}]\right| + \ge(\rho/2)(h_0/2)=\tau. +\] +This gives \eqref{eq:upper-heavy-coset}. Notice that the unknown target +offset contributes only a sign in the equality expansion. + +It remains to bound the lists under the possibly nonuniform law $\mu$. +Take $m$ distinct elements $\Phi_1,\ldots,\Phi_m$ of one list and choose +signs $\epsilon_a\in\{-1,1\}$ aligning their correlations with +$g_\sigma$. Cauchy--Schwarz gives +\begin{align*} + m^2\tau^2 + &\le\left(\E_\mu\left[g_\sigma + \sum_{a=1}^m\epsilon_a\chi_{\Phi_a}\right]\right)^2\\ + &\le\E_\mu\left[\left(\sum_{a=1}^m + \epsilon_a\chi_{\Phi_a}\right)^2\right] + \le m+m(m-1)\lambda\le m+m^2\lambda. +\end{align*} +For distinct indices the product character is nonzero, which justifies +the off-diagonal bound even after sign alignment. If $m>0$, the hypothesis +$\lambda\le\tau^2/2$ yields $m\le2/\tau^2$; the empty case is immediate. +Only the selected characters were summed, so no ambient-dimension factor +appears. All correlations defining the lists are unconditional under the +same law $\mu$. +\end{proof} + +\subsection{Constructing the two decoders} + +\begin{proposition}[Upper decoding with a lower rank test] +\label{prop:upper} +Suppose a legal labeling has acceptance at least $\eps$, $d\eps\ge4$, +and the cut comparisons have the errors required in +\eqref{eq:upper-c}, Lemma~\ref{lem:useful-equality}, and +Lemma~\ref{lem:positive-difference}. Choose upper inverse constants +$\alpha,r$ and $\ell=\ell_j$ as above, and assume +\eqref{eq:upper-Fourier-conditions} and \eqref{eq:upper-R}, with the +absolute balance minimum at every level. For the pair $i0. +\end{equation} +Every left output has the form +\[ + F_L(x,y)=z(xy),\quad z\in(H_i^2)^*,\quad z(1)=1, + \qquad \rank\bigl(F_L(S_t,S_u)\bigr)_{t,u}\le1. +\] +The constant $\gamma_j$ depends only on $c,\alpha,r,\ell_j$; it is +independent of the lower row count $\ell_i$, the branching numbers, the +source alphabet and instance size, and the bias accuracies chosen for +lower levels. The left strategy never receives $a_i$, and the right +strategy uses $S$ only through its quotient by $\langle a_i\rangle$. +\end{proposition} + +\begin{proof} +On good advice the left samples $z$ uniformly from $z_0+Z$, using fresh +private randomness, and returns $F_L(x,y)=z(xy)$ for $x,y\in H_i$. +The products belong to $H$ because $D_i$ is a proper descendant of $D_j$. +Equation~\eqref{eq:upper-valid} proves normalization and the rank bound +on $S$. On bad advice it evaluates at a fixed legal assignment on the +original subtree, obtaining the same two validity properties. These rules +also define valid defaults on any other well-typed left input: use the +stated construction whenever its witness is valid, and point evaluation +otherwise. The rule involves no omitted direction. + +For the right decoder, use $\mu=\mu_{\mathcal R}$ and +$G=G_{\mathcal R}$ from \eqref{eq:upper-G}, and compute all lists +\eqref{eq:upper-heavy-definition}. Choose $\sigma\in K^*$ uniformly. +If $\sigma|_W\ne0$ and its list is nonempty, choose an element $\Phi$ +uniformly from that list. Choose $h_\sigma\in W$ deterministically from +$(A,\sigma)$ with $\sigma(h_\sigma)=1$, for example the first such vector +in a fixed order, and put $z_R=\Phi(h_\sigma)\in(H')^*$. Return +\begin{equation}\label{eq:upper-right-form} + F_R(x,y)=z_R(xy),\qquad x,y\in H_i'. +\end{equation} +The inclusion $(H_i')^2\subseteq H'$ makes this a bilinear form. In the +remaining cases return the zero bilinear form. These defaults make the +algorithm total; normalization is required only of the left output. +No chosen witness from the left enters the right rule. + +We now prove agreement for these particular algorithms. For this analysis +only, expose $E,O$, all projection maps, the path through $D_i$, $A,a_i$, +all nonhidden upper buckets, and all other matrices including full $S$. +Call the resulting data $\mathcal D$. Do not expose $X$, any hidden +bucket ingredients, the slice event $J$, the representative-match event +$I$, a successful test, or either decoder's final random choices. +This exposure determines $P$, $M_{\rm known}$ and $A\bar M$, since +$AX=0$. Hence it determines goodness, $Z,z_0,u$ and the column witness. +By \eqref{eq:upper-factorization}, the conditional distribution of $X$ +given $\mathcal D$ remains exactly $\mu_{\mathcal R}$, with the bias +bound already proved. + +On good data let $Q=Z|_{H'}\subseteq(H')^*$ and $z'=z_0|_{H'}$. +The event $J$ either is impossible, or is an affine slice +\[ + \mathcal C_{\mathcal D}= + \{X\in W\otimes H':X(q_a|_{H'})=v_a-M_{\rm known}q_a + \text{ for every }a\}. +\] +In the latter case these equations define a well-defined linear +specification on $Q$, of dimension at most $r$. This also handles a +collapse of some columns on restriction to $H'$. The target in +\eqref{eq:upper-difference} is $Xz'+k_0$, where +$k_0=M_{\rm known}z_0+u$ is fixed by $\mathcal D$. + +Set $p_{\mathcal D}=\mu_{\mathcal R}(\mathcal C_{\mathcal D})$ on good +compatible data, and zero otherwise. When $p_{\mathcal D}>0$, let +\[ + q_{\mathcal D}= + \Prob_i\bigl(G_{\mathcal R}(X)=Xz'+k_0 + \mid\mathcal D,\ X\in\mathcal C_{\mathcal D}\bigr), +\] +and set $q_{\mathcal D}=0$ otherwise. Disintegrating the already proved +unconditional inequality \eqref{eq:upper-difference} gives +\begin{equation}\label{eq:upper-favorable-mass} + \E_i[p_{\mathcal D}(q_{\mathcal D}-\rho)]\ge b. +\end{equation} +Its integrand is nonpositive unless the data are good and compatible, +$p_{\mathcal D}>0$, and $q_{\mathcal D}\ge\rho$. The integrand is at +most one everywhere. The set of data with those properties consequently +has probability at least $b$. This is the sole use of the finer exposure: +we have not asserted that usefulness persists upon conditioning on it. + +Fix any such favorable $\mathcal D$. Lemma~\ref{lem:heavy-list} supplies +at least one nonannihilating $\sigma$ and one +\[ + \Phi\in\mathcal H_\sigma\cap + \bigl((\sigma|_W)\otimes z'+W^*\otimes Q\bigr). +\] +The right chooses such a pair with probability at least +$2^{-\ell}\tau^2/2$. Every such choice, with the already fixed +$h_\sigma$, has +\[ + z_R=\Phi(h_\sigma)\in z'+Q. +\] +The restriction map $Z\longrightarrow Q$ is surjective. Uniform measure +on $z_0+Z$ therefore pushes to uniform measure on $z'+Q$, so the fresh +left choice has restriction equal to this particular $z_R$ with probability +$2^{-\dim Q}\ge2^{-r}$. It is independent of the right's final choices +conditional on $\mathcal D$. Equality of these functionals on $H'$ +implies agreement of their product forms on $H_i'\times H_i'$. +Averaging over the favorable data proves \eqref{eq:upper-gamma}. + +For a fixed right input, the law $\mu_{\mathcal R}$, function +$G_{\mathcal R}$, lists $\mathcal H_\sigma$, and contraction vectors +$h_\sigma$ are identical for all compatible left inputs and witnesses. +Different favorable exposures may certify different elements of these same +lists. The preceding hit probability holds separately for each exposure, +so averaging introduces no factor for the number of witnesses. + +Finally, $\eta,\alpha,r$ were fixed from $c$, while $g,q_0,b,\rho,\tau$ +were fixed from these constants and $\ell_j$. Conditions +\eqref{eq:upper-Fourier-conditions} require only an upper row-count +lower bound and an upper bias tolerance. Condition~\eqref{eq:upper-R} +is met by choosing $R_j$ large enough; below $j$ it uses only the common +absolute balance minimum. The inverse statement applies to $H/B$ even +in small dimension by Lemma~\ref{lem:padding}. Thus none of these choices +depends on the later lower row count or source and branching dimensions. +\end{proof} + +The algorithms in Proposition~\ref{prop:upper} are now fixed local rules. +In particular, the right rule always computes its heavy lists from the +original law $\mu_{\mathcal R}$ just defined. A later conditioning may +change the physical law of the hidden buckets; it does not change that +algorithm. The subsequent rank and repetition arguments use these same +rules with their prescribed inputs. diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/05-lower-decoding.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/05-lower-decoding.tex new file mode 100644 index 0000000000000000000000000000000000000000..d2c4fa162816d15ba887dbf3ed9588fa0b49a40b --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/05-lower-decoding.tex @@ -0,0 +1,426 @@ +\section{From tested rank to full rank}\label{sec:lower} + +Proposition~\ref{prop:upper} provides two local decoders whose bilinear +forms agree after projection with probability at least $\gamma_j>0$. +The left form has rank at most one on the displayed lower rows. We now +show that a positive part of this agreement comes from left forms of +bounded rank on the entire lower function space. The distinction matters: +the form selected by the decoder can depend on the rows on which it is +tested. + +The passage from agreement of restricted forms to a fixed form of bounded +rank has a close predecessor in Fei, Minzer and Wang +\cite[Lemmas~5.15--5.16]{FeiMinzerWang2026}. Here we carry it out for the +tree function spaces, using the sparse-projection posterior bound and +an estimate on affine row and column slices. + +Fix the levels $is$, the first term can be replaced by +$2^{m-(s-2r')}$. +\end{lemma} + +\begin{proof} +The form $F$ is fixed throughout this proof. First remove dependent +row equations, leaving $a=\rank A_0\leq r'$ independent ones. +Nonemptiness of the slice guarantees the corresponding relations among +their prescribed values. Extend these $a$ row combinations to an +invertible matrix $T\in\Mat_{\ell\times\ell}(\F)$ and put $Y=TS$. +The first $a$ rows of $Y$ are fixed. Bilinearity gives the congruence +\begin{equation}\label{eq:lower-gram-congruence} + \operatorname{Gram}_F(Y) + =T\operatorname{Gram}_F(S)T^{\mathsf t}. +\end{equation} +It follows that the two Gram matrices have the same rank. We are +transforming the rows of a fixed form, and make no equivariance +assumption about any decoder that might have selected $F$. + +Remove dependent column functionals as well, leaving a basis +$q_1,\ldots,q_c$ of their span with $c\leq r'$. Again, the nonempty +slice guarantees every required dependence among the right-hand sides. +The column equations transform to $Yq_v=Tw_v$. Define +\[ + N=\bigcap_{v=1}^c\ker q_v,\qquad \operatorname{codim}_H N=c. +\] +After the first $a$ rows have been fixed, the equations on each remaining +row are independent of those on the other rows and specify a nonempty +translate of $N$. Thus the uniform slice measure makes its +$\ell-a\geq2m$ free rows independent, each uniform on a possibly +different affine space $u_k+N$. This conclusion also covers redundant +original equations, zero column span, and zero row rank. + +Let $F_N$ be the restriction of $F$ to $N\times N$, and put +$t=\rank F_N$. Regard a bilinear form as a map from its first argument +space to the dual of its second argument space. Restricting the +domain from $H$ to $N$ loses at most $c$ in rank. Restricting the +codomain from $H^*$ to $N^*$ also loses at most $c$, since the restriction +map has a kernel of dimension $c$. Consequently +\begin{equation}\label{eq:lower-rank-loss} + t\geq\rank F-2c\geq\rank F-2r'=t_0. +\end{equation} + +Choose two disjoint groups of $m$ free rows, denoted +$x_1,\ldots,x_m$ and $y_1,\ldots,y_m$. Write +$x_u=u_u+n_u$ with independent uniform $n_u\in N$. The full +functionals supplied by the first group are +\[ + \lambda_u=F(x_u,\cdot)|_N + =F(u_u,\cdot)|_N+F(n_u,\cdot)|_N\in N^*. +\] +Let $V\subseteq N^*$ be the image of $F_N$, of dimension $t$. +Each $\lambda_u$ is uniform on its specified translate of $V$, and the +$\lambda_u$ are independent. For any nonzero +$e=(e_1,\ldots,e_m)\in\F^m$, the sum +$\sum_u e_u\lambda_u$ is uniform on a translate of $V$: select one +index with $e_u=1$ and condition on all other summands. Its probability +of being zero is therefore either zero or $2^{-t}$. Taking the union +over the $2^m-1$ nonzero coefficient vectors gives +\begin{equation}\label{eq:lower-affine-dependence} + \Pr[\lambda_1,\ldots,\lambda_m\text{ are linearly dependent}] + \leq(2^m-1)2^{-t}. +\end{equation} +The offsets $F(u_u,\cdot)|_N$ are included in these functionals; no +claim about independence of only their variable parts is being made. + +Conditional on a realization of the first group for which the +$\lambda_u$ are linearly independent, the linear map +\[ + N\longrightarrow\F^m,\qquad + n\longmapsto(\lambda_1(n),\ldots,\lambda_m(n)) +\] +is surjective. A second-group row $y_v=v_v+n_v'$ has $n_v'$ uniform +in $N$, independently of the first group and of the other such rows. +Its cross column $(F(x_u,y_v))_{u=1}^m$ is a fixed offset plus the image +of $n_v'$ under this surjection. It is therefore uniform in $\F^m$. +The cross matrix +\[ + C=\bigl(F(x_u,y_v)\bigr)_{1\leq u,v\leq m} +\] +is consequently uniform in $\Mat_{m\times m}(\F)$, conditional on that +first group. + +Exactly one such matrix has rank zero. Every rank-one matrix has a +factorization $uv^{\mathsf t}$ with nonzero $u,v\in\F^m$, unique over +$\F$ because the only nonzero scalar is one. Hence exactly +$1+(2^m-1)^2$ matrices have rank at most one. If the full Gram matrix +of $Y$ has rank at most one, its cross submatrix $C$ does as well. +Combining this count with +Equations~\ref{eq:lower-gram-congruence}, +\ref{eq:lower-rank-loss} and~\ref{eq:lower-affine-dependence} proves +\eqref{eq:lower-rank-slice-bound}. Finally, $\rank F>s$ implies +$(2^m-1)2^{-t}\leq2^{m-(s-2r')}$, giving the stated simpler bound. +Neither symmetry nor nonalternation of $F$ was required. +\end{proof} + +\subsection{Positive agreement of bounded-rank forms} + +\begin{proposition}[Low-rank agreement]\label{prop:low-rank} +Fix a pair $i0$. Apply +Theorem~\ref{thm:inverse}, including Lemma~\ref{lem:padding}, at the +equality threshold +\[ + \eta_{\mathrm{low}}=\gamma^2/8, +\] +and let $\alpha'>0$, $r'$ and $\ell_0'$ be its density, constraint and +row-count constants. Choose integers $m\geq1$, $s\geq1$ and $\ell_i$ +such that +\begin{equation}\label{eq:lower-parameter-conditions} + \bigl[1+(2^m-1)^2\bigr]2^{-m^2}<\alpha'/2, + \qquad 2^{m-(s-2r')}<\alpha'/2, + \qquad \ell_i\geq\max\{\ell_0',2m+r'\}. +\end{equation} +Take $n_i$ sufficiently large that the data in +Lemma~\ref{lem:resampling} satisfy +\begin{equation}\label{eq:lower-error-conditions} + \frac{2\beta_iL_i}{1-\beta_i}\leq\gamma^2/64, + \qquad \upsilon_i\leq\gamma^2/64, + \qquad \beta_i=n_i^{-2/3}. +\end{equation} +Then the original randomized decoders satisfy, in the modified +experiment, +\begin{equation}\label{eq:lower-low-rank-mass} + \Pr\bigl[F_L|_{H_i'\times H_i'}=F_R, + \ F_L\text{ is admissible at }S, + \ \rank F_L\leq s\bigr]\geq\gamma/2. +\end{equation} +The rank cutoff and row-count requirements depend only on the upper +agreement constant. The likelihood and total-variation requirements +can be met after the source alphabet and lower branching numbers have +been fixed, without any dependence on higher branching numbers. +\end{proposition} + +\begin{proof} +Suppose instead that agreement from left outputs of rank greater than +$s$ has probability at least $\gamma/2$. Fix complete local tapes +preserving this high-rank agreement mass, and erase the left outputs +as above. Since the original left decoder is admissible, this leaves +nonerased agreement probability $h\geq\gamma/2$. By +Lemma~\ref{lem:resampling} and +Equation~\ref{eq:lower-error-conditions}, the unrestricted experiment +satisfies +\[ + \Pr_{\mathrm{unif}}[f(S)=f(S+a_i l)\ne\bot] + \geq\frac{\gamma^2}{4}-\frac{\gamma^2}{64} + -\frac{\gamma^2}{64} + =\frac{7\gamma^2}{32}>\eta_{\mathrm{low}}. +\] +Fix the domains and all remaining independent background data at a +value for which this equality probability is at least +$\eta_{\mathrm{low}}$. Conditional on this fixing, $S,l,a_i$ still +have the independent uniform laws required by the inverse theorem. +The output alphabet is the finite set of bilinear forms on this fixed +$H_i$; its size has no effect on the inverse constants. + +Theorem~\ref{thm:inverse} now selects one \emph{fixed} defined output +$F$ and a nonempty affine row/column slice, with at most $r'$ constraints +of either type, on which $f(S)=F$ has density at least $\alpha'$. +Every such occurrence satisfies $\rank F>s$ and +$\rank\operatorname{Gram}_F(S)\leq1$, by the pointwise definition of +the partial table. Only after fixing this output do we apply the row +operations in Lemma~\ref{lem:rank-slice}. That lemma and +Equation~\ref{eq:lower-parameter-conditions} show that the latter +rank event has density strictly less than $\alpha'$ on the same slice, +a contradiction. Dependence of $B(S)$ and the upper witness on $S$ +does not enter this step: at each occurrence of the selected form its +Gram rank condition holds on that occurrence's own list. + +Thus high-rank agreement has probability less than $\gamma/2$ for the +original randomized decoders. Subtracting it from the agreement mass +at least $\gamma$ proves Equation~\ref{eq:lower-low-rank-mass}. The +temporary fixing of tapes was used only to refute the hypothesized +high-rank mass; it has not replaced the original decoder algorithms +in the conclusion. + +For the parameter claims, the second summand in +Equation~\ref{eq:lower-rank-slice-bound} tends to zero as $m$ grows. +Choose $m$ first, then $s$, and then $\ell_i$ as in +\eqref{eq:lower-parameter-conditions}. All these choices depend only +on $\gamma$. The finite likelihood bound $L_i$ is fixed once the +source alphabet, upper row and product counts, and lower branching +numbers have been fixed; the extra resampling row changes only this +finite bound. Lemmas~\ref{lem:sparse-tv} and~\ref{lem:posterior} then +make both losses in \eqref{eq:lower-error-conditions} arbitrarily small +by increasing $n_i$. For the finitely many upper levels $j>i$, one +may meet all row requirements and take the maximum rank cutoff. +Increasing a cutoff only strengthens every high-rank exclusion already +proved. +\end{proof} + +The resulting agreement event supplies a normalized product form of +bounded full rank. Its restrictions to the atomic indicators at a +leaf will produce an odd list of at most $s$ legal answers. The next +section turns those lists into local strategies on many independent +source questions. diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/06-clean-repetition.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/06-clean-repetition.tex new file mode 100644 index 0000000000000000000000000000000000000000..44af07f2e3645f8e25322535380961c4dca5c20f --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/06-clean-repetition.tex @@ -0,0 +1,526 @@ +\section{Clean children and independent repetition} +\label{sec:clean} + +A normalized product form of bounded rank determines a short odd list +of answers at each source leaf. When the decoded forms agree after +projection, their lists satisfy a projection constraint. To apply +parallel repetition to these lists, we must recover independent source +edges and local answer rules despite their sampled advice. + +We do this by finding projected children whose contributions to every +sampled function are zero. The probability of this event is exactly +independent of their source questions. Revealing the remaining data +therefore leaves independent source edges at designated leaves, and +each player can reconstruct its original decoder input from its own +questions there. Repetition then bounds the low-rank agreement from +Proposition~\ref{prop:low-rank}. + +Fix levels $ii$ makes $2R_p$ calls in its path child. +All other contributions lie outside $D_i$, and distinct calls use fresh +randomness. If a recipe uses fewer calls, pad it with independent unused +calls to reach $T_i^{\mathrm{cl}}$. Thus the call count is fixed before any +visibility decision. The notation $T_i^{\mathrm{cl}}$ distinguishes this +family from those enlarged by extra rows in comparison experiments. + +Let $\Xi$ record the questions outside $D_i$, the path through $D_i$, +the exterior ingredients of the stopped expressions, and all other arrays +supported outside $D_i$. Include the independent choices $A$ and $a_i$ +after fixing the call family. Record each exterior function on the domain +of its supporting subtree, together with its location in the expression. +This representation requires no endpoint questions inside $D_i$; each +player will later extend it to its own product domain by ignoring the +other arguments. Assembled ancestor values are still functions of the +pending cut calls and are not exposed in $\Xi$. +Consequently $\Xi$ depends only on exterior source records and independent +exterior randomness, so the source samples inside $D_i$ retain their +independent laws conditional on $\Xi$. + +It is convenient to sample a complete source edge at every leaf, +even when its child will remain unprojected. This merely augments +the modified experiment: sample the left question and then a source +projection conditional on it, using that projection only if the +child is selected. For each child $C$ of $D_i$, independently let +$P_C$ be its projection flag, with +$\Pr(P_C=1)=\beta_i=n_i^{-2/3}$. Write $\boldsymbol e_C$ for the +tuple of source edge records at its leaves and $m_i$ for the number +of those leaves. Thus $\boldsymbol e_C$ has distribution +$\mu^{\otimes m_i}$. + +Generate cut call $q\in[T_i^{\mathrm{cl}}]$ by taking, independently for all +$q,C$, a uniform latent summand +\[ + U_{q,C}\in H_C(O)^2, + \qquad U_q=\sum_{C\text{ child of }D_i}U_{q,C}. +\] +The sum is uniform in $H_i'$ because the addition map from the +product of the child square spaces onto $H_i'$ is a surjective linear +map. Its fibers all have the same size. The resulting calls are +independent, and are independent of the uniform arrays at every +nonleaf proper descendant of $D_i$. The first $\ell_i$ calls can +therefore serve as the rows of $S$, with the rest substituted in the +stopped ancestor expressions. This is exactly the modified sampling +law on the visible arrays. No direct-sum assumption on the child +spaces is involved. + +Call $C$ \emph{clean} if $P_C=1$, every $U_{q,C}$ is zero, and every +sampled array at a nonleaf node in the subtree rooted at $C$ is zero. +Write $\mathcal C$ for the set of clean children. + +\begin{lemma}[Exact clean probability and conditional source law] +\label{lem:clean-law} +Conditional on any exterior datum $\Xi=\xi$ of positive probability, +the clean indicators are independent. In the padded construction +above each has probability $\beta_i p_{0,i}$, where +\begin{equation} + p_{0,i}= + 2^{-T_i^{\mathrm{cl}}\dim(H_C'^2) + -\displaystyle\sum_{\substack{D\preceq C\\D\text{ nonleaf}}} + \ell_{\operatorname{level}(D)}\dim H_D'}>0. + \label{eq:clean-zero-probability} +\end{equation} +Here $D\preceq C$ includes $D=C$, and $H_D'$ is the function space +on a fully projected subtree of the indicated shape. The exponent +has the same value for every child and every choice of source +questions and projection positions. It depends only on the source +power, the fixed row and product counts, and branching below level +$i$; it is independent of the instance, $n_i$, and all higher +branching numbers. + +After fixing $\mathcal C$, expose all source questions, projection +choices and latent data at nonclean children, and all source edge +records at nondesignated leaves of clean children. Denote this +additional exposure together with $\Xi$ and $\mathcal C$ by +$\mathcal D_{\mathrm{pub}}$. Conditional on every value of $\mathcal D_{\mathrm{pub}}$ of positive +probability, the source edge records at the designated clean leaves +are independent and each has distribution $\mu$. +\end{lemma} + +\begin{proof} +Fix $\Xi=\xi$ and a child edge tuple $\boldsymbol e_C$. Conditional +on $P_C=1$, the $T_i^{\mathrm{cl}}$ latent summands in that child are independent +uniform points of $H_C'^2$. Their probability of all being zero is +$2^{-T_i^{\mathrm{cl}}\dim(H_C'^2)}$. An independent uniform +$\ell_p$-row array in $H_D'$ is zero with probability +$2^{-\ell_p\dim H_D'}$. Multiplication over all such descendant +arrays proves Equation~\eqref{eq:clean-zero-probability}. +This calculation sets each latent summand to zero separately. +The possibility that nonzero summands could cancel, including shared +constants, has no effect on this event or on its probability. + +We next prove that the displayed probability is exactly independent +of $\boldsymbol e_C$. Each fully projected leaf has answer domain +$\F^t$. Given two projected subtrees of the same ordered shape, +identify these domains coordinatewise, retaining a separate factor +at every leaf and every source-power slot. Pullback by this product +bijection identifies their full leaf function spaces. It preserves +sums and pointwise products, so induction through the tree identifies +every corresponding $H_D'$ and $H_D'^2$. In particular their +dimensions agree. Neither the question IDs nor the selected +projection positions enter these identifications. Even a right +question $(v,v)$ has the full answer domain $\F^2$ in its two slots; +it does not identify their answer coordinates. + +Consequently, for every possible edge tuple $\boldsymbol e$, +\begin{equation} + \Pr(\boldsymbol e_C=\boldsymbol e, C\in\mathcal C\mid\Xi=\xi) + =\mu^{\otimes m_i}(\boldsymbol e)\,\beta_i p_{0,i}. + \label{eq:clean-factorization} +\end{equation} +The latent child blocks, edge tuples and flags are independent +across children conditional on $\Xi$. Thus the clean indicators are +independent, and the constant factor in +Equation~\eqref{eq:clean-factorization} cancels on conditioning that +a child is clean. The entire edge tuple in each clean child retains +its original product law. Positivity of a question-dependent zero +probability would not suffice for this cancellation; its constancy +has just been proved. + +Fixing the whole clean mask still leaves a product across children. +Exposing full data of nonclean children therefore imposes no +additional condition on any clean edge tuple. Within each clean +child, fixing the nondesignated edge records in its product tuple +leaves the designated edge with law $\mu$. This proves the asserted +law conditional on $\mathcal D_{\mathrm{pub}}$. Natural repetitions of an ID in +different samples are permitted throughout; this exposure neither +requires nor forbids a collision with an exposed ID. + +Finally, the child shapes involve only $n_1,\ldots,n_{i-1}$, and +Equation~\eqref{eq:clean-call-count} involves row and product counts +but no branching numbers. This proves the stated parameter +dependence. If $i=1$, the children are leaves and the descendant-array +sum is empty, with value zero, so the same argument applies. +\end{proof} + +\subsection{Reconstructing the original local strategies} + +The preceding lemma leaves independent source questions, but repetition +also requires local answer rules on those questions. Here the input +restriction is substantive: the left decoder in +Proposition~\ref{prop:upper} never receives $a_i$, while the right +decoder receives $S$ only modulo $a_i$. We now reconstruct precisely +those inputs after the final exposure. + +\begin{lemma}[Local reconstruction on clean children] +\label{lem:reconstruction} +Fix $\mathcal D_{\mathrm{pub}}=d_0$ of positive probability as in +Lemma~\ref{lem:clean-law}. From $d_0$ and its own designated source +questions in $\mathcal C$, each player can reconstruct its input to +the original decoder of Proposition~\ref{prop:upper}. Composing +these decoders with the total diagonal extraction rule defines local +strategies for $G_{\mathrm{odd},s}^{\otimes |\mathcal C|}$. +Whenever their original forms agree on $H_i'$ and the left form is +normalized, of product form, and of full rank at most $s$, these +strategies accept every designated clean edge. +\end{lemma} + +\begin{proof} +First reconstruct the two question tuples. The left player combines +its designated clause-tuple questions with all the exposed questions +to obtain $E$. It does not need the selected positions or variable +IDs at its designated clean leaves. The right player combines its +designated variable-tuple questions with the exposure to obtain $O$: +every clean child is projected, and the entire question and projection +data at every nonclean child are exposed. The right player does not +need the missing clause-tuple questions. Repeated IDs specify +separate slots in both reconstructions. + +We next reconstruct the functions appearing in the advice. At a +nonclean child the exposure supplies its actual latent summands and +descendant arrays, including the projections needed to pull them +back on the left. At a clean child substitute a zero symbol for +each latent summand and descendant row. Such a symbol specifies +only its call, row and tree location. The left interprets it as the +zero function on its own original domain; the right interprets it +as the zero function on its own projected domain. It carries no +missing endpoint ID, selected position or partner-domain metadata. +In either interpretation it equals the corresponding actual +function, because zero pulls back to zero under every projection. + +Sum these child contributions to obtain every cut call, including +all rows of $S$. Then evaluate the stopped expressions using the +exposed exterior ingredients. This yields all ancestor arrays and +all their individual buckets, hence every nonhidden bucket required +by the decoders. This is a reconstruction from the chosen sampled +decomposition, not an attempt to recover a unique decomposition from +the resulting function. Overlaps of constants between child +spaces, or other nonuniqueness of a sum representation, do not +alter the resulting sums. + +To check that no missing domain information enters these expressions, +consider the full algebra of functions on the product of all leaf +domains that ignore the arguments in clean children. It contains +constants and is closed under sums and pointwise products. Every +reconstructed cut call belongs to this algebra: its clean summands +are zero and its other summands are supported on nonclean children. +Every exterior ingredient also belongs to it, since its support is +outside $D_i$. Induction over each stopped expression therefore +shows that every reconstructed ancestor function ignores all clean +arguments. Descendant arrays in clean children are identically +zero. Thus products at higher levels cannot introduce dependence +on the unavailable endpoint questions. + +Each player now has its own domains and all the functions supplied +to its decoder. It can form their reduced-support partition keys, +consult the fixed labeling, and restore the evaluations of the +selected parts in its own domain, using the conventions of +Section~\ref{sec:construction}. For each reconstructed sampled joint function, every clean slot is +unused, so canonicalization discards that slot and its domain metadata. On nonclean projected +slots, the exposed projection supplies the necessary pullback; on +designated clean slots, the functions just proved independent of +those arguments require none. A decoder can also evaluate any +hypothetical matrices called for by its algorithm: it constructs +them on its own now-known domain. Such additional matrices need +not ignore the designated arguments, since the player already knows +its own questions there. + +Although this final public record may determine full $S$, the +simulated right decoder is called with its original input +$S\bmod a_i$. Similarly the left decoder is called without $a_i$, +even though the final public record may contain that direction. +Neither decoder receives the reconstructed hidden $M$ or additional +buckets that are absent from its specified input. Extra public +information in the simulation is simply omitted from the relevant +arguments. All information used to form the original arguments is +a function of $d_0$ and the player's own designated questions. + +The simulation runs the same fixed algorithms on these reconstructed +inputs. The left uses its fixed usefulness marks and deterministic witness +rule. The right computes the heavy lists in +Equation~\eqref{eq:upper-heavy-definition} using the original law and response +function determined by its own input in Section~\ref{sec:right-hidden-law}, +then applies the same selection and contraction rules. +Conditioning on $\mathcal D_{\mathrm{pub}}$ may change the physical distribution of the +sampled hidden buckets, but it does not change the reference law prescribed +by this algorithm. The local simulation therefore requires no small-bias +assertion under the clean conditioning. + +Finally, run the total diagonal extraction rule at the designated +clean leaves. Each player has exactly its own source question there, +so this is a legal local answer rule on the repeated game. The rules +are defined even on inputs on which the original decoder chooses a +default, produces a high-rank form, or fails the diagonal test. On +normalized low-rank agreement, Lemma~\ref{lem:parity-game} says that +the left supports are valid odd lists and their right supports are +their parity pushforwards. Both diagonal tests then pass, so every +designated clean edge accepts. This is an event inclusion, with no +conditioning on the forms' validity, their agreement, or their rank. +\end{proof} + +\subsection{The vanishing upper bound} + +We can now bound low-rank agreement in the original modified law. +All public exposures used for this purpose have already been +specified, and none includes a useful-advice event, a slice event, +a decoder output or a successful character match. + +\begin{proposition}[Clean repetition bound] +\label{prop:clean-bound} +Assume +$\val(G_{\mathrm{src}})<1/(2s^2)$. Let +$\mathcal L_{i,j,s}$ be the event in the modified experiment that the +two original decoder outputs agree on $H_i'$, while the left output +is a normalized product form on $H_i$ of full rank at most $s$. +Then, for the clean probability in +Equation~\eqref{eq:clean-zero-probability}, +\begin{equation} + \Pr(\mathcal L_{i,j,s}) + \le \left(1-\frac{\beta_i p_{0,i}}{64}\right)^{n_i} + \le \exp\!\left(-\frac{p_{0,i}}{64}n_i^{1/3}\right). + \label{eq:clean-final-bound} +\end{equation} +The bound holds for randomized decoders and is uniform over the +source instance and over the fixed labeling. +\end{proposition} + +\begin{proof} +Lemma~\ref{lem:parity-game} gives +$\val(G_{\mathrm{odd},s})<1/2$. Fix a public record +$\mathcal D_{\mathrm{pub}}=d_0$, and put $k=|\mathcal C|$. By +Lemma~\ref{lem:clean-law}, its designated edge records have exactly +the independent product distribution of $k$ source samples. By +Lemma~\ref{lem:reconstruction}, the original decoders and the total +extraction rules give local strategies for the corresponding odd-list +game. Theorem~\ref{thm:repetition}, with gap $1/2$, bounds their +probability of accepting all $k$ edges by $(63/64)^k$. +For $k=0$ the bound is interpreted as one. Private or shared random +tapes independent of the questions cause no difficulty: fixing +complete tapes gives deterministic local functions, to each of which +the same bound applies, and then one averages over those tapes. + +The event inclusion in Lemma~\ref{lem:reconstruction} therefore gives +\[ + \Pr(\mathcal L_{i,j,s}\mid\mathcal D_{\mathrm{pub}}=d_0) + \le (63/64)^{|\mathcal C|}. +\] +This inequality bounds the favorable agreement event; that event +has never been used to define the conditional source distribution +or the local strategies. Average over $\mathcal D_{\mathrm{pub}}$, and then use +independence of the clean indicators conditional on $\Xi$. In the +padded construction this yields the exact generating-function identity +\[ + \E\bigl[(63/64)^{|\mathcal C|}\mid\Xi\bigr] + =\prod_{C\text{ child of }D_i} + \left(1-\beta_i p_{0,i} + +\frac{63}{64}\beta_i p_{0,i}\right) + =\left(1-\frac{\beta_i p_{0,i}}{64}\right)^{n_i}. +\] +The second bound in Equation~\eqref{eq:clean-final-bound} follows from +$1-x\le e^{-x}$ and $n_i\beta_i=n_i^{1/3}$. +\end{proof} + +\begin{remark}[Exterior-dependent call families] +The padded family above gives one constant clean probability. The argument +also permits independent exterior recipe choices with +$T_i^{\mathrm{cl}}(\xi)\le T_i^{\mathrm{cl}}$ calls. Substituting this +count in Equation~\eqref{eq:clean-zero-probability} gives an exact +question-independent probability $p_{0,i}(\xi)\ge p_{0,i}$. +If the recipe differs between children, denote these probabilities by +$p_{0,C}(\xi)$. Conditional on $\Xi=\xi$, the clean indicators remain +independent, and every $p_{0,C}(\xi)$ is at least $p_{0,i}$. +The exact factors, rather than this lower bound, cancel in +Equation~\eqref{eq:clean-factorization}; hence the conditional source law +and the local reconstruction are unchanged. + +The same generating-function calculation then gives +\begin{equation} + \Pr(\mathcal L_{i,j,s}) + \le \E_\Xi\prod_{C\text{ child of }D_i} + \left(1-\frac{\beta_i p_{0,C}(\Xi)}{64}\right) + \le \exp\!\left(-\frac{p_{0,i}}{64}n_i^{1/3}\right). + \label{eq:clean-exterior-average} +\end{equation} +Only the second inequality uses the uniform positive lower bound. +\end{remark} + +For the decoders supplied by Proposition~\ref{prop:upper}, +Proposition~\ref{prop:low-rank} gives +$\Pr(\mathcal L_{i,j,s})\ge\gamma_j/2$. Once the source power and +the lower branching numbers have been fixed, +$p_{0,i}>0$ is already fixed independently of $n_i$ and of higher +branching. Taking, for example, +\[ + n_i^{1/3}>\frac{64}{p_{0,i}}\log\frac{4}{\gamma_j} +\] +makes Equation~\eqref{eq:clean-final-bound} smaller than +$\gamma_j/4$, a contradiction. There are only finitely many choices +of $j>i$, so one branching number at level $i$ accommodates all of +them. Enlarging a higher branching number adds only exterior data +and changes the deterministic stopped expressions; it changes +neither the number of calls in Equation~\eqref{eq:clean-call-count} +nor the projected child dimensions in +Equation~\eqref{eq:clean-zero-probability}. This is the uniform +contradiction needed for the parameter selection in +Section~\ref{sec:assembly}. + diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/07-parameters-output.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/07-parameters-output.tex new file mode 100644 index 0000000000000000000000000000000000000000..c301702186eeed390cd8ea35ad29a8cd5809175a --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/07-parameters-output.tex @@ -0,0 +1,277 @@ +\section{Choosing the parameters and completing the reduction} +\label{sec:assembly} + +We now combine the local statements into a reduction. +The parameters are constants depending on the requested soundness, but their +order of choice matters: the upper agreement guarantee and its constants +must be fixed before the lower row counts used to test rank. We choose row +counts from the root downwards, fix the source repetition +power, and then choose branching numbers from the leaves upwards. The +bounds used during these choices are uniform in every dimension and table +range not yet fixed; the decoders themselves are instantiated after all +choices. This establishes soundness for the preliminary weighted game. +Completing its at-most-two maps and rounding its weights will give the fixed alphabets and explicit +unweighted output of Theorem~\ref{thm:main}. + +\subsection{Row counts, rank cutoffs, and source repetition} + +Fix $\delta\in\mathbb Q\cap(0,1)$ and put $\eps=\delta/3$. +Choose a depth $d$ with $d\eps\ge4$. +The pair-counting argument of Section~\ref{sec:upper} allows a common positive +simultaneous-success threshold; for definiteness take +\[ + c=\eps^2/2,\qquad \kappa=c/2,\qquad + \eta=c^2/32. +\] +The losses below will be small enough for these choices. +Apply Theorem~\ref{thm:inverse} at the upper inverse threshold $\eta/2$, +and denote its density, number of constraints and minimum row count by +$\alpha,r,\ell_0$. +Set $\rho=\kappa\alpha/4$. + +Choose the row counts from the root downwards. +When choosing $\ell_j$, all agreement constants $\gamma_k$ with $k>j$ have +already been fixed. For every such $k$, apply the lower analysis of +Section~\ref{sec:lower} with agreement threshold $\gamma_k$: +fix its inverse constants, then a cross-matrix size, then a rank cutoff. +Choose $\ell_j$ large enough for each of these finitely many lower row +requirements, and also so that +\begin{equation}\label{eq:upper-row-choice} + \ell_j\ge\ell_0,\qquad + 2^{-(\ell_j-r)}<\rho/8. +\end{equation} +At the top level there are no preceding lower-test requirements. + +With $\ell_j$ fixed, the following conservative constants from +Section~\ref{sec:upper} are available: +\begin{align} + h_j&=2^{-r\ell_j}, + &g_j&=\frac{\eta^2}{4}\,2^{-r\ell_j}, + &q_j&=g_jh_j, \label{eq:upper-masses}\\ + b_j&=\kappa\alpha q_j/4, + &\tau_j&=\rho h_j/4, + &\gamma_j&=b_j\,2^{-\ell_j-r}\tau_j^2/2. + \label{eq:upper-agreement-choice} +\end{align} +Choose +\begin{equation}\label{eq:bias-choice} + \lambda_j\le\min\{h_j/2,\tau_j^2/2\},\qquad + (7/8)^{R_j}\le\lambda_j, +\end{equation} +and require every $R_j$ to exceed the absolute minimum giving the +quarter-balance premise in Lemma~\ref{lem:bias}. +The scalar induction uses only this lower balance minimum, not a lower +level's eventual accuracy $\lambda_i$. +Thus choosing $R_j$ creates no requirement to determine a lower source +dimension or a lower row count first. + +Carry this procedure down to level $1$. +Take $s\ge1$ to be the maximum of all the resulting rank cutoffs. +Increasing a cutoff only strengthens the high-rank exclusion estimate, so +the estimates for every pair $i=Latex, font=\small, + cut/.style={dashed,draw=black!50}, + treebox/.style={draw=blue!45!black,rounded corners=2pt,fill=blue!3, + align=center,inner sep=5pt}, + inputbox/.style={draw=black!65,rounded corners=2pt,fill=yellow!4, + align=left,text width=6.7cm,inner sep=7pt}] +\node[font=\small\bfseries] at (3.2,0) {Latent sampling structure}; +\node[font=\small\bfseries,align=center] at (11.1,0.1) {Original sampler:\\separate calls at the same node}; +\node at (3.2,-0.55) {ancestors}; +\node[treebox] (upper) at (3.2,-1.25) {$D_j$: upper matrix $M$}; +\draw[->] (3.2,-0.78)--(upper); +\draw[cut] (0,-1.25)--(1.0,-1.25); +\draw[cut] (5.4,-1.25)--(6.6,-1.25); +\node[align=left,font=\footnotesize] at (0.6,-1.78) {upper\\cut}; +\node[align=center] (middle) at (3.2,-2.05) {$\vdots$}; +\node[align=left,font=\footnotesize] at (5.45,-2.0) {$j>i$; not necessarily\\adjacent levels}; +\node[treebox] (lower) at (3.2,-2.85) {$D_i$: lower matrix $S$}; +\draw (upper)--(middle); +\draw (middle)--(lower); +\draw[cut] (0,-2.85)--(1.0,-2.85); +\draw[cut] (5.4,-2.85)--(6.6,-2.85); +\node[align=left,font=\footnotesize] at (0.6,-3.38) {lower\\cut}; +\node (ordinary) at (1.65,-3.95) {ordinary $C$}; +\node (special) at (4.7,-3.95) {$C_* = D_{i-1}$}; +\draw (lower)--(ordinary); +\draw[thick,->] (lower)--(special); +\node[align=center,text width=6.5cm,font=\footnotesize] at (3.2,-4.65) + {$C_*$ is the original sampler's latent child.\\ + The modified-$i$ law uses no path below $D_i$.}; +\node[treebox,text width=5.2cm] (own) at (11.1,-1.45) + {Own bucket at $D_p$\\$h_{p,v}\sim\nu_{D_p}$}; +\node[treebox,text width=5.2cm] (call) at (11.1,-3.35) + {Stopped ancestor call at $D_p$\\$h'\sim\nu_{D_p}$}; +\draw[->] (11.1,-0.5)--node[right,font=\footnotesize] + {fresh tape}(own.north); +\draw[->] (11.1,-2.4)--node[right,font=\footnotesize] + {fresh tape}(call.north); +\node[align=center,text width=5.8cm,font=\footnotesize] at (11.1,-4.65) + {Independent conditional on the domains and shared path;\\ + no call is reused in the other array.}; +\node[inputbox,anchor=north west] (left) at (-0.2,-5.35) { + \textbf{Left decoder: functions on $E$}\\[4pt] + $E$, path through $D_i$, and $A$\\[3pt] + Full $S$\\[3pt] + $(M_D)_{D\ne D_j,D_i}$ on $E$\\[3pt] + $(h_v)_{Av\ne0}$ on $E$\\[4pt] + \footnotesize No $a_i$ or projection metadata is supplied. +}; +\node[inputbox,anchor=north east] (right) at (14.5,-5.35) { + \textbf{Right decoder: functions on $O$}\\[4pt] + $O$, path through $D_i$, $A$, and $a_i$\\[3pt] + Only $S\bmod\langle a_i\rangle$\\[3pt] + $(M_D)_{D\ne D_j,D_i}$ on $O$\\[3pt] + $(h_v)_{Av\ne0}$ on $O$\\[4pt] + \footnotesize The missing component of $S$ is not supplied. +}; +\node[fit=(left)(right),inner sep=0pt] (cards) {}; +\node[align=center,text width=14cm,below=5mm of cards.south] + {Neither decoder is supplied the hidden upper sum + $X=\sum_{0\ne v\in\ker A}v\otimes h_v$.}; +\end{tikzpicture} +\caption{Cuts and the information supplied to the decoders. +The selected levels satisfy $j>i$. The tree depicts latent sampling +choices; the upper-cut comparison forgets the path below $D_j$. +The call panel shows the original sampler; at a modified cut these calls +are replaced by independent uniform draws in the cut space. +An own bucket and a stopped ancestor call at the same node use separate +tapes, independently conditional on the domains and shared path. +The cards summarize the analytical inputs +\eqref{eq:upper-left-input}--\eqref{eq:upper-right-input}, with $W=\ker A$. +The left receives full $S$ without $a_i$; the right receives $a_i$ and +only the quotient of $S$. Both receive upper buckets with $Av\ne0$ on +their own domains. These analytical records are distinct from the +original game's canonical partition keys.} +\label{fig:tree-cut} +\end{figure} diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/main.tex b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..281661b221e6e51e19658cad1b6df225cf7077fd --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/main.tex @@ -0,0 +1,64 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype,enumitem} +\usepackage{xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,fit,backgrounds,calc} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black]{hyperref} +\hypersetup{pdftitle={Perfect completeness for 2-to-1 games},pdfauthor={OpenAI}, + pdfsubject={Perfect-completeness hardness for exact 2-to-1 projection games}, + bookmarksdepth=2} +\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi +\newcommand{\F}{\mathbb F_2} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\newcommand{\one}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\TV}{TV} +\DeclareMathOperator{\val}{val} +\DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\Mat}{Mat} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\allowdisplaybreaks[1] +\setlength{\emergencystretch}{2em} +\title{Perfect completeness for 2-to-1 games} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the 2-to-1 Games Conjecture with perfect completeness. +For every fixed rational $\delta\in(0,1)$, it is NP-hard to distinguish +satisfiable 2-to-1 games from games of value at most $\delta$, with a fixed +alphabet and an explicitly listed unweighted multiset of constraints. +\end{abstract} +\input{00-introduction} +\input{01-preliminaries} +\input{02-construction} +\input{03-sampling} +\input{04-upper-decoding} +\input{05-lower-decoding} +\input{06-clean-repetition} +\input{07-parameters-output} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/references.bib b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..ef7fb73c2786c329d0aec0b98da31bb86e2e4682 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/build/references.bib @@ -0,0 +1,263 @@ + +@inproceedings{Khot2002, + author = {Khot, Subhash}, + title = {On the Power of Unique 2-Prover 1-Round Games}, + booktitle = {Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing}, + series = {STOC '02}, + pages = {767--775}, + publisher = {Association for Computing Machinery}, + year = {2002}, + doi = {10.1145/509907.510017}, + note = {\url{https://doi.org/10.1145/509907.510017}} +} + +@article{Dinur2007, + author = {Dinur, Irit}, + title = {The {PCP} Theorem by Gap Amplification}, + journal = {Journal of the ACM}, + volume = {54}, + number = {3}, + articleno = {12}, + numpages = {44}, + year = {2007}, + month = jun, + doi = {10.1145/1236457.1236459}, + note = {Article 12, 44 pages. 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Lemma 2.3 and Section 3 are cited in arXiv version 1: \url{https://arxiv.org/abs/1804.08662v1}} +} + +@article{KhotMinzerSafra2018, + author = {Khot, Subhash and Minzer, Dor and Safra, Muli}, + title = {Pseudorandom Sets in {Grassmann} Graph Have Near-Perfect Expansion}, + journal = {Annals of Mathematics}, + volume = {198}, + number = {1}, + pages = {1--92}, + year = {2023}, + doi = {10.4007/annals.2023.198.1.1}, + note = {Theorem 1.12 is cited in ECCC TR18-006, revision 2, 2018: \url{https://eccc.weizmann.ac.il/report/2018/006/revision/2/download/}} +} + +@techreport{FeiMinzerWang2026, + author = {Fei, Yumou and Minzer, Dor and Wang, Shuo}, + title = {On the Hardness of 4-to-1 Games with Perfect Completeness}, + institution = {Electronic Colloquium on Computational Complexity}, + type = {Report}, + number = {TR26-179}, + year = {2026}, + month = sep, + note = {September 14, 2026. \url{https://eccc.weizmann.ac.il/report/2026/179/}} +} + + +@article{AroraSafra1998, + author = {Arora, Sanjeev and Safra, Shmuel}, + title = {Probabilistic Checking of Proofs: A New Characterization of {NP}}, + journal = {Journal of the ACM}, volume = {45}, number = {1}, + pages = {70--122}, year = {1998}, doi = {10.1145/273865.273901}, + note = {Preliminary version in FOCS 1992. \url{https://doi.org/10.1145/273865.273901}} +} +@article{AroraLundMotwaniSudanSzegedy1998, + author = {Arora, Sanjeev and Lund, Carsten and Motwani, Rajeev and Sudan, Madhu and Szegedy, Mario}, + title = {Proof Verification and the Hardness of Approximation Problems}, + journal = {Journal of the ACM}, volume = {45}, number = {3}, + pages = {501--555}, year = {1998}, doi = {10.1145/278298.278306}, + note = {Preliminary version in FOCS 1992. \url{https://doi.org/10.1145/278298.278306}} +} +@article{Raz1998, + author = {Raz, Ran}, title = {A Parallel Repetition Theorem}, + journal = {SIAM Journal on Computing}, volume = {27}, number = {3}, + pages = {763--803}, year = {1998}, doi = {10.1137/S0097539795280895}, + note = {Preliminary version in STOC 1995. \url{https://doi.org/10.1137/S0097539795280895}} +} +@article{Holenstein2009, + author = {Holenstein, Thomas}, + title = {Parallel Repetition: Simplification and the No-Signaling Case}, + journal = {Theory of Computing}, volume = {5}, number = {8}, + pages = {141--172}, year = {2009}, doi = {10.4086/toc.2009.v005a008}, + note = {\url{https://doi.org/10.4086/toc.2009.v005a008}} +} +@article{Rao2011, + author = {Rao, Anup}, + title = {Parallel Repetition in Projection Games and a Concentration Bound}, + journal = {SIAM Journal on Computing}, volume = {40}, number = {6}, + pages = {1871--1891}, year = {2011}, doi = {10.1137/080734042}, + note = {Preliminary version in STOC 2008. \url{https://doi.org/10.1137/080734042}} +} +@article{BellareGoldreichSudan1998, + author = {Bellare, Mihir and Goldreich, Oded and Sudan, Madhu}, + title = {Free Bits, {PCPs}, and Nonapproximability---Towards Tight Results}, + journal = {SIAM Journal on Computing}, volume = {27}, number = {3}, + pages = {804--915}, year = {1998}, doi = {10.1137/S0097539796302531}, + note = {Preliminary version in FOCS 1995. \url{https://doi.org/10.1137/S0097539796302531}} +} +@article{AustrinODonnellTanWright2014, + author = {Austrin, Per and O'Donnell, Ryan and Tan, Li-Yang and Wright, John}, + title = {New {NP}-Hardness Results for 3-Coloring and 2-to-1 Label Cover}, + journal = {ACM Transactions on Computation Theory}, + volume = {6}, number = {1}, articleno = {2}, pages = {2:1--2:20}, + year = {2014}, + note = {Theorem 1.3 is cited in the author manuscript dated July 16, 2013: \url{https://www.cs.cmu.edu/afs/cs/Web/People/jswright/papers/coloring.pdf}} +} +@article{KhotMinzerSafra2017, + author = {Khot, Subhash and Minzer, Dor and Safra, Muli}, + title = {On Independent Sets, 2-to-2 Games and {Grassmann} Graphs}, + journal = {Theory of Computing}, volume = {21}, number = {10}, + pages = {1--55}, year = {2025}, doi = {10.4086/toc.2025.v021a010}, + note = {First circulated as ECCC TR16-124 in 2016; preliminary version in STOC 2017. \url{https://doi.org/10.4086/toc.2025.v021a010}} +} +@article{DinurKhotKindlerMinzerSafra2021, + author = {Dinur, Irit and Khot, Subhash and Kindler, Guy and Minzer, Dor and Safra, Muli}, + title = {On Non-Optimally Expanding Sets in {Grassmann} Graphs}, + journal = {Israel Journal of Mathematics}, volume = {243}, number = {1}, + pages = {377--420}, year = {2021}, doi = {10.1007/s11856-021-2164-7}, + note = {First circulated as ECCC TR17-094 in 2017; preliminary version in STOC 2018. \url{https://doi.org/10.1007/s11856-021-2164-7}} +} + +@inproceedings{GoldreichLevin1989, + author = {Goldreich, Oded and Levin, Leonid A.}, + title = {A Hard-Core Predicate for All One-Way Functions}, + booktitle = {Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing}, + pages = {25--32}, publisher = {Association for Computing Machinery}, year = {1989}, + note = {Author manuscript: \url{https://www.cs.bu.edu/fac/lnd/pdf/hard.pdf}} +} +@article{NaorNaor1993, + author = {Naor, Joseph and Naor, Moni}, + title = {Small-Bias Probability Spaces: Efficient Constructions and Applications}, + journal = {SIAM Journal on Computing}, volume = {22}, number = {4}, + pages = {838--856}, year = {1993}, doi = {10.1137/0222053}, + note = {\url{https://doi.org/10.1137/0222053}} +} +@inproceedings{BravermanKhotMinzer2021, + author = {Braverman, Mark and Khot, Subhash and Minzer, Dor}, + title = {On Rich 2-to-1 Games}, + booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {185}, pages = {27:1--27:20}, year = {2021}, + publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, + doi = {10.4230/LIPIcs.ITCS.2021.27}, + note = {\url{https://doi.org/10.4230/LIPIcs.ITCS.2021.27}} +} + +@article{GuruswamiSinop2013, + author = {Guruswami, Venkatesan and Sinop, Ali Kemal}, + title = {Improved Inapproximability Results for Maximum {$k$}-Colorable Subgraph}, + journal = {Theory of Computing}, + volume = {9}, number = {11}, pages = {413--435}, year = {2013}, + doi = {10.4086/toc.2013.v009a011}, + note = {\url{https://theoryofcomputing.org/articles/v009a011/v009a011.pdf}} +} + +@inproceedings{ODonnellWu2008, + author = {O'Donnell, Ryan and Wu, Yi}, + title = {Conditional Hardness for Satisfiable {3-CSPs}}, + booktitle = {Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing}, + series = {STOC '09}, + pages = {493--502}, + publisher = {Association for Computing Machinery}, + year = {2009}, + doi = {10.1145/1536414.1536482}, + note = {Theorem 2.1 and Corollary 2.2 are cited in the author manuscript dated November 17, 2008: \url{https://www.cs.cmu.edu/~odonnell/papers/3bit-hardness.pdf}} +} + +@inproceedings{GuruswamiSandeep2020, + author = {Guruswami, Venkatesan and Sandeep, Sai}, + title = {{$d$}-to-{$1$} Hardness of Coloring {$3$}-Colorable Graphs with {$O(1)$} Colors}, + booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {168}, pages = {62:1--62:12}, year = {2020}, + publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, + doi = {10.4230/LIPIcs.ICALP.2020.62}, + note = {Theorem 1. \href{https://doi.org/10.4230/LIPIcs.ICALP.2020.62}{Published paper}} +} + +@misc{OpenAIIndependentSets2026, + author = {{OpenAI}}, + title = {{Hardness of finding large independent sets in three-colorable graphs}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026.pdf}{OAI:Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026}}, + year = {2026}, + note = {Theorem 1.1.} +} + +@article{KhotSafra2013, + author = {Khot, Subhash and Safra, Muli}, + title = {A Two-Prover One-Round Game with Strong Soundness}, + journal = {Theory of Computing}, + volume = {9}, number = {28}, pages = {863--887}, year = {2013}, + doi = {10.4086/toc.2013.v009a028}, + note = {Preliminary version in FOCS 2011. \url{https://doi.org/10.4086/toc.2013.v009a028}} +} + +@article{DinurGuruswami2015, + author = {Dinur, Irit and Guruswami, Venkatesan}, + title = {{PCPs} via the low-degree long code and hardness for constrained hypergraph coloring}, + journal = {Israel Journal of Mathematics}, + volume = {209}, number = {2}, pages = {611--649}, year = {2015}, + doi = {10.1007/s11856-015-1231-3}, + note = {Preliminary version in FOCS 2013. Equation (1) and Fact 9 are cited in ECCC Report TR13-122: \url{https://eccc.weizmann.ac.il/report/2013/122/download/}} +} + +@article{KhotSaket2017, + author = {Khot, Subhash and Saket, Rishi}, + title = {Hardness of Coloring 2-Colorable 12-Uniform Hypergraphs with {$2^{(\log n)^{\Omega(1)}}$} Colors}, + journal = {SIAM Journal on Computing}, + volume = {46}, number = {1}, pages = {235--271}, year = {2017}, + doi = {10.1137/15100240X}, + note = {Preliminary version in FOCS 2014. Sections 2.1 and 7.2 are cited in ECCC Report TR14-051: \url{https://eccc.weizmann.ac.il/report/2014/051/download/}} +} + +@article{Hastad2014NotTwo, + author = {H{\aa}stad, Johan}, + title = {On the {NP}-Hardness of {Max-Not-2}}, + journal = {SIAM Journal on Computing}, + volume = {43}, number = {1}, pages = {179--193}, year = {2014}, + doi = {10.1137/120882718}, + note = {Preliminary version in APPROX 2012. \url{https://www.csc.kth.se/~johanh/sicompnottwo.pdf}} +} + +@techreport{Golowich2023, + author = {Golowich, Louis}, + title = {From {Grassmannian} to Simplicial High-Dimensional Expanders}, + institution = {Electronic Colloquium on Computational Complexity}, + type = {Report}, + number = {TR23-065}, + year = {2023}, + note = {\url{https://eccc.weizmann.ac.il/report/2023/065/}} +} diff --git a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8daf4a34a05723e45c8bc1a00b277a062b947063 --- /dev/null +++ b/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2691ba6bd2ac7159a2b0900faa180a222fa078e3df8520607b23adf8a3a38aa9 +size 746539 diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/README.md b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..2e447b490fdcacb70aa05801fc162c1a7d19c5d4 --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [Petty’s projection-volume conjecture in dimensions at least four](paper.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026, + author = {{OpenAI}}, + title = {{Petty's projection-volume conjecture in dimensions at least four}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf}{OAI:Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/bibliography.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/bibliography.tex new file mode 100644 index 0000000000000000000000000000000000000000..a3ec2dd4848adf3eb5f9a9f5ec0c8c4e03c22eee --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/bibliography.tex @@ -0,0 +1,132 @@ +\begingroup +\small +\begin{thebibliography}{99} + +\bibitem{Brouwer1911} +L.~E.~J. Brouwer, \"Uber Abbildung von Mannigfaltigkeiten, +\emph{Math. Ann.} \textbf{71} (1911), 97--115. +\url{https://doi.org/10.1007/BF01456931}. + +\bibitem{ChenEtAl2026} +S.~Chen, Y.~Feng, Y.~Li, D.~Xi, and L.~Xu, +\emph{Petty's conjectured projection inequality in dimension three}, +preprint, 7 August 2026, MathSciDoc:2608.23001, 20~pp. +\url{https://archive.ymsc.tsinghua.edu.cn/pacm_download/741/12780-pettyinequality.pdf}. + +\bibitem{Gardner2002} +R.~J. Gardner, The Brunn--Minkowski inequality, +\emph{Bull. Amer. Math. Soc. (N.S.)} \textbf{39} (2002), no.~3, 355--405. +\url{https://doi.org/10.1090/S0273-0979-02-00941-2}. + +\bibitem{Haddad2020} +J.~E. Haddad, +A convex body associated to the Busemann random simplex inequality +and the Petty conjecture, +\emph{J. Funct. Anal.} \textbf{281} (2021), no.~7, 109118. +\url{https://doi.org/10.1016/j.jfa.2021.109118}. +Preprint: arXiv:1904.10427v6, 10 January 2020. +\url{https://arxiv.org/abs/1904.10427v6}. + +\bibitem{HuangLutwakYangZhang2016} +Y.~Huang, E.~Lutwak, D.~Yang, and G.~Zhang, +Geometric measures in the dual Brunn--Minkowski theory and their associated +Minkowski problems, \emph{Acta Math.} \textbf{216} (2016), 325--388. +\url{https://doi.org/10.1007/s11511-016-0140-6}. + +\bibitem{HuangLutwakYangZhang2018} +Y.~Huang, E.~Lutwak, D.~Yang, and G.~Zhang, +The $L_p$-Aleksandrov problem for $L_p$-integral curvature, +\emph{J. Differential Geom.} \textbf{110} (2018), no.~1, 1--29. +\url{https://doi.org/10.4310/jdg/1536285625}. + +\bibitem{HuangZhao2018} +Y.~Huang and Y.~Zhao, On the $L_p$ dual Minkowski problem, +\emph{Adv. Math.} \textbf{332} (2018), 57--84. +\url{https://doi.org/10.1016/j.aim.2018.05.002}. + +\bibitem{Ivaki2018} +M.~N. Ivaki, A local uniqueness theorem for minimizers of Petty's conjectured +projection inequality, \emph{Mathematika} \textbf{64} (2018), no.~1, 1--19. +\url{https://doi.org/10.1112/S0025579317000444}. +Preprint: \url{https://arxiv.org/abs/1610.03796v2}. + +\bibitem{Ludwig2002} +M.~Ludwig, Projection bodies and valuations, +\emph{Adv. Math.} \textbf{172} (2002), no.~2, 158--168. +\url{https://dmg.tuwien.ac.at/ludwig/projection.pdf}. + +\bibitem{Lutwak1990} +E.~Lutwak, On quermassintegrals of mixed projection bodies, +\emph{Geom. Dedicata} \textbf{33} (1990), no.~1, 51--58. +\url{https://doi.org/10.1007/BF00147600}. + +\bibitem{LutwakPetty1990} +E.~Lutwak, On a conjectured projection inequality of Petty, +in \emph{Integral Geometry and Tomography} (Arcata, CA, 1989), +Contemporary Mathematics \textbf{113}, American Mathematical Society, +Providence, RI, 1990, pp.~171--182. + +\bibitem{MartiniMustafaev2008} +H.~Martini and Z.~Mustafaev, Extensions of a Bonnesen-style inequality +to Minkowski spaces, \emph{Math. Inequal. Appl.} \textbf{11} (2008), +no.~4, 739--748. +\url{https://doi.org/10.7153/mia-11-63}. + +\bibitem{MielkeSulz2026} +F.~Mielke-Sulz, \emph{The Petty conjecture for convex bodies of revolution}, +preprint, arXiv:2609.13517v1, 11 September 2026. +\url{https://arxiv.org/abs/2609.13517v1}. + +\bibitem{OlafssonPasqualeRubin2013} +G.~\'Olafsson, A.~Pasquale, and B.~Rubin, +Analytic and group-theoretic aspects of the cosine transform, +in \emph{Geometric Analysis and Integral Geometry}, +Contemporary Mathematics \textbf{598}, American Mathematical Society, +Providence, RI, 2013, pp.~167--188. +\url{https://doi.org/10.1090/conm/598/12009}. +Preprint: \url{https://arxiv.org/abs/1209.1822v1}. + +\bibitem{OrtegaMorenoSchuster2021} +O.~Ortega-Moreno and F.~E. Schuster, Fixed points of Minkowski valuations, +\emph{Adv. Math.} \textbf{392} (2021), 108017. +\url{https://doi.org/10.1016/j.aim.2021.108017}. +Author version: \url{https://www.dmg.tuwien.ac.at/schuster/FixedPointMinkVal.pdf}. + +\bibitem{Petty1971} +C.~M. Petty, Isoperimetric problems, +in \emph{Proceedings of the Conference on Convexity and Combinatorial Geometry} +(University of Oklahoma, Norman, 1971), +Department of Mathematics, University of Oklahoma, pp.~26--41. + +\bibitem{Saroglou2015} +C.~Saroglou, On the shape of a convex body with respect to its second projection body, +\emph{Adv. Appl. Math.} \textbf{67} (2015), 55--74. +\url{https://doi.org/10.1016/j.aam.2015.03.004}. +Preprint: \url{https://arxiv.org/abs/1409.4347v2}. + +\bibitem{SaroglouZvavitch2017} +C.~Saroglou and A.~Zvavitch, Iterations of the projection body operator and a remark +on Petty's conjectured projection inequality, +\emph{J. Funct. Anal.} \textbf{272} (2017), no.~2, 613--630. +\url{https://doi.org/10.1016/j.jfa.2016.08.015}. + +\bibitem{Spector2011} +D.~Spector, Simple proofs of some results of Reshetnyak, +\emph{Proc. Amer. Math. Soc.} \textbf{139} (2011), no.~5, 1681--1690. +\url{https://doi.org/10.1090/S0002-9939-2010-10593-2}. +Author version: +\url{https://cvgmt.sns.it/media/doc/paper/505/reshetnyakcontinuityrevised.pdf}. + +\bibitem{WangZhou2026} +H.~Wang and J.~Zhou, Uniqueness and continuity of the solution to +$L_p$ dual Minkowski problem, +\emph{Commun. Math. Stat.} \textbf{14} (2026), no.~1, 105--140. +\url{https://doi.org/10.1007/s40304-023-00374-2}. +Preprint: \url{https://arxiv.org/abs/2103.13075v1}. + +\bibitem{Zhang1999} +G.~Zhang, The affine Sobolev inequality, +\emph{J. Differential Geom.} \textbf{53} (1999), no.~1, 183--202. + +\end{thebibliography} +\endgroup diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/main.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..b9da292065462726707d8fef49c91231a1ce22a5 --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/main.tex @@ -0,0 +1,60 @@ +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{mathpazo} +\renewcommand{\ttdefault}{lmtt} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype,booktabs,array} +\usepackage{xcolor} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={Petty's projection-volume conjecture in dimensions at least four}, + pdfauthor={OpenAI}, + pdfsubject={The projection-volume inequality and its equality cases in every dimension at least four}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E_{\sigma}} +\newcommand{\Sph}{S^{n-1}} +\newcommand{\Id}{I_n} +\newcommand{\gradS}{\nabla_S} +\newcommand{\lapS}{\Delta_S} +\newcommand{\abs}[1]{\left|#1\right|} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\newcommand{\ip}[2]{\left\langle#1,#2\right\rangle} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\range}{range} +\DeclareMathOperator{\Ker}{ker} +\allowdisplaybreaks[1] +\title{Petty's projection-volume conjecture\\ +in dimensions at least four} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that ellipsoids uniquely minimize the volume of the projection body +among convex bodies of fixed volume in every dimension at least four. +This proves Petty's projection-volume conjecture in these dimensions. +\end{abstract} +\input{sections/01_introduction} +\input{sections/02_gauges} +\input{sections/03_spherical} +\input{sections/04_norm} +\input{sections/05_position} +\input{sections/06_conclusion} +\input{sections/07_consequences} +\input{bibliography} +\end{document} diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/01_introduction.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/01_introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..9a16c910eb53f85dfa1caf968e40cd2e3c13bda4 --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/01_introduction.tex @@ -0,0 +1,183 @@ +\section{Introduction} +\label{sec:introduction} + +For a convex body $K\subset\R^n$, its projection body $\Pi K$ is the +origin-symmetric convex body whose support function is +\[ + h_{\Pi K}(u)=\operatorname{vol}_{n-1} + \bigl(\operatorname{proj}_{u^\perp}K\bigr), + \qquad u\in S^{n-1}. +\] +Here a convex body is compact, convex, and has nonempty interior, and +$h_L(x)=\max_{y\in L}\ip{x}{y}$ denotes its support function. +Thus $\Pi K$ records the volumes of the orthogonal shadows of $K$ in +all directions. Write $|L|=\operatorname{vol}_n(L)$ and let $B_2^m$ +be the Euclidean unit ball in $\R^m$, with volume $\kappa_m$. +The normalized projection volume is +\[ + R_n(K)=\frac{|\Pi K|}{|K|^{n-1}}. +\] +It is invariant under translations and invertible linear maps: the +classical contravariance identity is +$\Pi(TK)=|\det T|T^{-t}\Pi K$; see +\cite[Equation~(1)]{Ludwig2002}. We verify this identity directly in +Section~\ref{sec:conclusion}. + +Petty's projection-volume conjecture asserts, for $n\ge3$, that +$R_n$ is minimized precisely by ellipsoids \cite{Petty1971}. +The ball satisfies $\Pi B_2^n=\kappa_{n-1}B_2^n$, so the proposed +minimum is fixed by the Euclidean case. This is an affine isoperimetric +problem in which directional projection data are assembled into a +second body before its volume is measured. The classical Petty +inequality for the \emph{polar} of the projection body concerns a +different functional; see \cite[Section~18.6]{Gardner2002}. +We prove the unpolarized conjecture in every dimension at least four. + +\begin{theorem}\label{thm:main} +For every integer $n\ge4$ and every convex body $K\subset\R^n$, +\begin{equation}\label{eq:main} + \frac{|\Pi K|}{|K|^{n-1}} + \ge \kappa_{n-1}^{\,n}\kappa_n^{\,2-n}. +\end{equation} +Equality holds if and only if $K=a+TB_2^n$ for some $a\in\R^n$ and some +invertible linear map $T$. +\end{theorem} + +The theorem includes bodies without symmetry or boundary regularity. +Its equality assertion is global and imposes no proximity to an +ellipsoid. + +\subsection*{Previous results and methods} + +One approach studies a body together with its second projection body. +The class-reduction inequality $R_n(\Pi K)\le R_n(K)$, together with +its equality condition, reduces the study of minimizers to bodies +homothetic to $\Pi^2K$; see \cite[Sections~1--2]{Saroglou2015}. +Saroglou and Zvavitch proved a local minimum theorem under the +hypothesis that the density of the surface-area measure, after an +invertible linear change of variables, is sufficiently close to that +of the ball in $L^\infty$ \cite[Theorem~1.3]{SaroglouZvavitch2017}. +Ivaki established local rigidity for $C^2$ solutions of +$\Pi^2K=cK$ near the ball after an invertible linear change of +variables \cite{Ivaki2018}. These results use the harmonic structure +of the projection operator and the special role of the directions +arising from linear changes of coordinates. + +Chen, Feng, Li, Xi, and Xu proved the unrestricted three-dimensional +inequality, including ellipsoid equality +\cite[Theorem~1.1]{ChenEtAl2026}. Mielke-Sulz proved the conjecture +for bodies of revolution in every dimension $n\ge3$ +\cite{MielkeSulz2026}. Theorem~\ref{thm:main} treats arbitrary convex +bodies in the remaining dimensions. Combining it with the separately +proved three-dimensional theorem of Chen et al. establishes the +conjecture for every $n\ge3$; that external theorem is not used in our +proof for $n\ge4$. + +Lutwak reformulated the conjecture as a lower bound for a two-body +integral with kernel $|u\cdot v|$ against surface-area measures +\cite{LutwakPetty1990}; see the explicit formulation in +\cite[Introduction, item~(i)]{Saroglou2015}. +Our reduction uses the same pairing for a broader class of positive +measures satisfying a lower bound on integrals of support functions. + +Our proof uses the classical first-variation and spherical-transform +frameworks, but requires estimates and an affine normalization that +apply to arbitrary bodies. The Wulff variations are closely related +to the Aleksandrov variations for dual curvature measures developed +by Huang, Lutwak, Yang, and Zhang +\cite{HuangLutwakYangZhang2016}. The variational representation of a +support functional belongs to the $L_p$ dual Minkowski framework; +compare Huang and Zhao \cite{HuangZhao2018}. We prove the needed +representation directly, including its logarithmic endpoint. +The cosine and Funk spectra are classical +\cite{OlafssonPasqualeRubin2013}; we derive their multipliers in our +normalization. The further tasks are a strict estimate for all norms +and a continuous choice of variational minimizers through changes of +matrix rank. The latter permits a global choice of affine coordinates, +where proximity to the ball is unavailable. + +\subsection*{The argument} + +The proof passes from a body to a positive measure on vectors. +After translating and normalizing $|K|=\kappa_n$, the first variation +of volume gives a bounded, spanning probability measure $\eta_K$ on +$\R^n$ such that +\[ + \int h_M\,d\eta_K\ge (|M|/\kappa_n)^{1/n} +\] +for every origin-symmetric convex body $M$. +The same measure represents $h_{\Pi K}(y)$ as a fixed multiple of +$\int|x\cdot y|\,d\eta_K(x)$. +Theorem~\ref{thm:bilinear} bounds the pairing +\[ + \iint|x\cdot y|\,d\eta(x)d\zeta(y) +\] +for any two bounded, spanning positive measures satisfying this +support-functional inequality. Applying the result to the measures +of $K$ and its volume-normalized projection body yields the desired +lower bound. Sharpness in the pairing returns equality to the +homothety case of the first variation of volume. + +Two ingredients have statements independent of projection bodies. +Theorem~\ref{thm:norm-estimate} bounds the higher spherical harmonics +of a power of an arbitrary norm by a difference of its spherical +means. The estimate is strict except for the Euclidean norm, and no +smoothness is needed. Proposition~\ref{prop:representation} represents +a positive support functional using the gradient of a variationally +selected norm. A spherical sign test bounds the pairing below by a +bilinear form involving the cosine and Funk transforms. The strict norm estimate +controls the contribution of harmonics of degree at least four. + +The degree-two component must be removed by choosing affine +coordinates; it is not a zero eigenspace of the sign-test operator. +Proposition~\ref{prop:position} makes this choice by extending the +family of variational minimizers continuously to singular matrices. +At that boundary the gauges become seminorms. An explicit calculation +in their missing directions gives an inward-pointing matrix field, +and a fixed-point argument yields an invertible matrix at which the +degree-two component vanishes. This position is needed for only one +of the two measures. The other is transformed contragrediently, which +preserves their pairing. These constructions and the strict norm +estimate also supply the equality case. + +Combining Theorem~\ref{thm:main} with the separate three-dimensional +result of Chen et al. gives several sharp inequalities in dimensions +$n\ge3$. Lutwak's implication yields Euclidean-ball lower bounds for +lower-degree projection-body ratios, in which intrinsic volumes replace +the volumes of shadows. Haddad's $p=1$ implication bounds the integral of +the absolute determinant of gradients below by the product of critical +Lebesgue norms. Its +diagonal case strengthens +the Sobolev--Zhang affine $L^1$ inequality. The theorem also gives the +sharp Holmes--Thompson isoperimetric inequality in normed spaces, where +equality requires an ellipsoidal norm ball and a body homothetic to it. +The full statements below distinguish the hypotheses and equality +conclusions of these consequences. + +Section~\ref{sec:gauges} establishes the volume and gauge facts used in +the reduction. Section~\ref{sec:spherical} derives the sign test and +its harmonic multipliers. Section~\ref{sec:norm} proves the strict norm +estimate, including the scalar inequalities in every dimension. +Section~\ref{sec:position} constructs the variational representation +and affine position. Section~\ref{sec:conclusion} proves the bilinear +inequality and deduces Theorem~\ref{thm:main}. +Section~\ref{sec:consequences} develops its consequences. We use the classical +Brunn--Minkowski inequality with its homothety equality case and +Brouwer's fixed-point theorem in their stated forms; the first-variation +reduction, harmonic estimates, variational construction, and equality +argument are proved below. + +\subsection*{Notation} + +Throughout Sections~\ref{sec:gauges}--\ref{sec:conclusion}, $n\ge4$, +$p=n-1$, and $\sigma$ is uniform probability on +$S^{n-1}$. The symbol $\E$ denotes integration against $\sigma$, and +\[ + b_n=\E|u_1|. +\] +Repeated spherical integrals use independent directions. +A symmetric body always means an origin-symmetric body. +When a function on the sphere is used as a degree-one homogeneous test +on vectors, it is extended by positive homogeneity and given value zero +at the origin. Euclidean gradients are denoted by $\nabla$; the +tangential gradient and Laplacian are $\gradS$ and $\lapS$. diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/02_gauges.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/02_gauges.tex new file mode 100644 index 0000000000000000000000000000000000000000..91203c53aa23bb39b193d9bfa219f8f8c2f263c2 --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/02_gauges.tex @@ -0,0 +1,281 @@ +\section{Gauges and projection measures}\label{sec:gauges} + +We first associate a measure to an arbitrary convex body. A volume +variation will give both the lower bound satisfied by this measure and +its relation to the projection body. No symmetry of the original body +is needed. + +\subsection{Differentiating a gauge} + +For a convex body $L$ containing the origin in its interior, its +\emph{gauge} is +\[ + g_L(x)=\inf\{a>0:x\in aL\}. +\] +It is convex and positively homogeneous, and $L=\{x:g_L(x)\le1\}$. +When $L$ is origin-symmetric, its gauge is a norm. A continuous function +$\phi$ on $\Sph$, when evaluated at a vector, will mean its positively +homogeneous extension +\[ + \phi(x)=|x|\phi(x/|x|)\quad(x\ne0),\qquad \phi(0)=0. +\] +This convention does not impose evenness on $\phi$. + +Varying the supporting constraints in this way is a Wulff variation. +The almost-everywhere radial differentiation below is the gauge form +of the Aleksandrov variation used in dual Brunn--Minkowski theory; +compare \cite[Lemmas~4.1--4.3 and Corollary~4.9]{HuangLutwakYangZhang2016}. We give the +argument directly because the later minimization uses arbitrary +continuous constraints of both signs, without boundary regularity. + +\begin{lemma}[Gauge variation]\label{lem:gauge-variation} +Let $s:\Sph\to(0,\infty)$ and $\phi:\Sph\to\R$ be continuous. For +sufficiently small $|t|$, define +\[ + W_t=\bigcap_{v\in\Sph}\{x\in\R^n:x\cdot v\le s(v)+t\phi(v)\}, + \qquad g_t=g_{W_t}. +\] +Then $W_t$ is a convex body containing the origin in its interior, and +\begin{equation}\label{eq:gauge-formulas} + g_t(x)=\max_{v\in\Sph}\frac{x\cdot v}{s(v)+t\phi(v)}, + \qquad + \frac{|W_t|}{\kappa_n}=\E g_t(u)^{-n}. +\end{equation} +For almost every $u\in\Sph$, the maximizing vector $v/s(v)$ at $t=0$ +is unique and equals the Euclidean gradient $\nabla g_0(u)$. At these +directions, +\begin{equation}\label{eq:gauge-derivative} + \left.\frac{d}{dt}\right|_{t=0}g_t(u) + =-g_0(u)\phi\bigl(\nabla g_0(u)\bigr). +\end{equation} +On the sphere the gauges are uniformly bounded above and away from +zero for small $|t|$, and the difference quotients +$(g_t-g_0)/t$ are uniformly bounded. In particular, +\begin{equation}\label{eq:constraint-volume-derivative} + \left.\frac{d}{dt}\right|_{t=0}|W_t| + =n\kappa_n\E\!\left[ + g_0(u)^{-n}\phi\bigl(\nabla g_0(u)\bigr)\right]. +\end{equation} +For $s=h_L$, one has $W_0=L$. +\end{lemma} + +\begin{proof} +Choose $00$. +\end{theorem} + + + +\begin{corollary}[First variation of volume]\label{cor:volume-first-variation} +Let $K,M\subset\R^n$ be convex bodies, with the origin interior to +$K$. Then +\begin{align} + \left.\frac{d}{dt}\right|_{0+}|K+tM| + &=n\kappa_n\E\!\left[g_K(u)^{-n} + h_M\bigl(\nabla g_K(u)\bigr)\right] + \label{eq:minkowski-volume-derivative}\\ + &\ge n|K|^{(n-1)/n}|M|^{1/n}.\nonumber +\end{align} +Equality in the inequality holds if and only if $K$ and $M$ are +homothetic up to translation. The derivative identity remains valid +when $M$ is an arbitrary nonempty compact convex set. +\end{corollary} + +\begin{proof} +Support functions add under Minkowski addition, so +Lemma~\ref{lem:gauge-variation}, with $s=h_K$ and $\phi=h_M$, +gives the derivative identity. This applies equally to a compact +convex set $M$ of lower dimension. For full-dimensional $M$, +Theorem~\ref{thm:brunn-minkowski} and homogeneity give +\[ + F(t):=|K+tM|^{1/n}\ge |K|^{1/n}+t|M|^{1/n} + \qquad(t\ge0). +\] +In addition, $F$ is concave, by applying that theorem to $K+sM$ +and $K+tM$. Its right derivative at zero therefore satisfies +$F'(0+)\ge |M|^{1/n}$, which is the asserted lower bound. + +If equality holds in this derivative bound, concavity gives +\[ + F(t)\le F(0)+tF'(0+)=|K|^{1/n}+t|M|^{1/n}. +\] +Thus the preceding volume inequality is an equality for every $t>0$. +The equality case of Theorem~\ref{thm:brunn-minkowski}, applied for +example to $K+M$, forces homothety. Conversely, if $M=a+cK$ +with $c>0$, then $K+tM=ta+(1+tc)K$, which verifies equality in +the derivative bound. +\end{proof} + +\subsection{The measure associated with a convex body} + +The construction below is a gauge-coordinate form of the classical +surface-area and mixed-volume description of projection bodies; +see \cite[Section~6]{Gardner2002} and \cite[Section~1]{Saroglou2015}. +The local segment calculation will fix its normalization and retain +the equality information from the first variation. + +For a finite positive Borel measure $\eta$ on $\R^n$ with bounded +support, write +\[ + \tau_\eta(M)=\int h_M(x)\,d\eta(x). +\] +\begin{definition}\label{def:admissible-measure} +We call $\eta$ \emph{admissible} if its support spans $\R^n$ and +\begin{equation}\label{eq:functional-condition} + \tau_\eta(M)\ge\left(\frac{|M|}{\kappa_n}\right)^{1/n} + \quad\text{for every origin-symmetric convex body }M\subset\R^n. +\end{equation} +The measure itself is not required to be symmetric. +\end{definition} + +\begin{proposition}[Projection measure]\label{prop:projection-measure} +Translate and dilate a convex body $K$ so that $0\in\operatorname{int}K$ +and $|K|=\kappa_n$. Define $\eta_K$ by +\begin{equation}\label{eq:eta-definition} + \int \psi(x)\,d\eta_K(x) + =\E\!\left[g_K(u)^{-n}\psi\bigl(\nabla g_K(u)\bigr)\right] +\end{equation} +for nonnegative Borel functions $\psi$; the choice of the gradient on +its null exceptional set is immaterial. Then $\eta_K$ is an +admissible probability measure of bounded support. Equality in +\eqref{eq:functional-condition} for $\eta_K$ and a symmetric body +$M$ holds if and only if $K$ is a translate of a positive dilate of +$M$. Moreover, +\begin{equation}\label{eq:projection-measure} + h_{\Pi K}(y)=\frac{n\kappa_n}{2} + \int |x\cdot y|\,d\eta_K(x) + \qquad(y\in\R^n), +\end{equation} +and +\begin{equation}\label{eq:eta-normalization} + \int h_K(x)\,d\eta_K(x)=1. +\end{equation} +For the unit ball, $\eta_{B_2^n}=\sigma$, and consequently +\begin{equation}\label{eq:ball-constant} + \frac{n\kappa_n b_n}{2}=\kappa_{n-1}. +\end{equation} +\end{proposition} + +\begin{proof} +By \eqref{eq:gauge-formulas}, $\E g_K^{-n}=1$, so +\eqref{eq:eta-definition} defines a probability measure. The active +vectors $v/h_K(v)$ lie in a bounded set because $h_K$ is positive +on the sphere. Hence $\eta_K$ has bounded support. Applying +Corollary~\ref{cor:volume-first-variation} gives, for every symmetric +convex body $M$, +\[ + \tau_{\eta_K}(M) + =\frac1{n\kappa_n} + \left.\frac{d}{dt}\right|_{0+}|K+tM| + \ge\left(\frac{|M|}{\kappa_n}\right)^{1/n}, +\] +with precisely the stated homothety equality condition. The same +derivative identity with $M=K$, using +$|K+tK|=(1+t)^n\kappa_n$, proves +\eqref{eq:eta-normalization}; this identity does not require $K$ +to be symmetric. + +Now let $y\ne0$ and take $M=[-y,y]$. Every nonempty fiber of +$K$ parallel to $y$ is a compact interval. Adding $t[-y,y]$ +increases its length by $2t|y|$, without changing its projection +onto $y^\perp$. Fubini's theorem therefore gives +\[ + |K+t[-y,y]|=|K|+ + 2t|y|\operatorname{vol}_{n-1} + (\operatorname{proj}_{y^\perp}K) + =|K|+2t h_{\Pi K}(y). +\] +Since the support function of the segment is $x\mapsto|x\cdot y|$, +the derivative identity yields \eqref{eq:projection-measure}. +The formula is also true at $y=0$. The projection of a +full-dimensional convex body has positive $(n-1)$-volume in every +direction. If $\supp\eta_K$ failed to span $\R^n$, a nonzero +vector perpendicular to its span would make the integral in +\eqref{eq:projection-measure} vanish. This contradiction proves +the spanning property and hence admissibility. + +Finally, for $K=B_2^n$ one has $g_K(u)=1$ and +$\nabla g_K(u)=u$ on the sphere, so $\eta_K=\sigma$. +Its projection volume in every unit direction is $\kappa_{n-1}$. +Substituting this into \eqref{eq:projection-measure} proves +\eqref{eq:ball-constant}. +\end{proof} diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/03_spherical.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/03_spherical.tex new file mode 100644 index 0000000000000000000000000000000000000000..10d14e6822238f224d52bcc6f5985be32f94bf17 --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/03_spherical.tex @@ -0,0 +1,359 @@ +\section{A spherical sign estimate}\label{sec:spherical} + +The estimate in this section separates the constant part of a spherical +function from its harmonics of degree at least four. The degree-two part +will later be removed by a linear change of coordinates. Throughout, +$p=n-1$, surface measure $\sigma$ has total mass one, and inner products +and norms of functions are those of $L^2(\sigma)$. + +\subsection{Harmonics and the spectral gap} + +Let $\mathcal H_l$ be the restrictions to $\Sph$ of homogeneous harmonic +polynomials of degree $l$, and put +\[ + \lambda_l=l(l+n-2)=l(l+p-1). +\] +For an even function, let $Q$ denote orthogonal projection onto the +closed sum of $\mathcal H_4,\mathcal H_6,\ldots$. + +\begin{lemma}\label{lem:spherical-harmonics} +The spaces $\mathcal H_l$ form an orthogonal decomposition of +$L^2(\sigma)$, and $\lapS Y=-\lambda_lY$ for $Y\in\mathcal H_l$. +Every even $H\in C^1(\Sph)$ satisfies +\begin{equation}\label{eq:spectral-gap} + \lambda_2\Var(H)+(\lambda_4-\lambda_2)\norm{QH}_2^2 + \le \E\abs{\gradS H}^2, + \qquad \lambda_2=2n,\quad \lambda_4=4(n+2). +\end{equation} +\end{lemma} + +\begin{proof} +The polar-coordinate formula for the Euclidean Laplacian gives +\[ + \Delta_{\R^n}\big(r^lY(u)\big) + =r^{l-2}\big(l(l+n-2)Y(u)+\lapS Y(u)\big). +\] +This proves the eigenvalue formula. Integration by parts on the sphere +then proves orthogonality of spaces with distinct degrees. + +We give the completeness argument explicitly. If $Y_l$ is homogeneous +and harmonic, direct differentiation yields +\[ + \Delta_{\R^n}\big(\abs{x}^{2j}Y_l(x)\big) + =2j(2l+2j+n-2)\abs{x}^{2j-2}Y_l(x). +\] +Induction on the degree now decomposes every homogeneous polynomial +into terms $\abs{x}^{2j}Y_l(x)$. Indeed, decompose its Laplacian by the +induction hypothesis and use the displayed identity to find a sum of +such terms with the same Laplacian. The difference is harmonic. +Restricting to the unit sphere shows that every polynomial restriction +belongs to the span of the $\mathcal H_l$. + +Polynomial restrictions uniformly approximate continuous functions on +$\Sph$. To see this without an approximation theorem, set +\[ + K_j(u,v)= + \frac{((1+u\cdot v)/2)^j} + {\displaystyle\int_{\Sph}((1+u\cdot w)/2)^j\,d\sigma(w)}. +\] +The denominator is positive and independent of $u$, so +$T_jH(u)=\int K_j(u,v)H(v)\,d\sigma(v)$ is a polynomial in $u$. +For $0<\delta<2$, let $m_\delta>0$ be the measure of the cap +$\{v:\abs{u-v}<\delta/2\}$. Since +$(1+u\cdot v)/2=1-\abs{u-v}^2/4$, +\[ + \int_{\abs{u-v}\ge\delta}K_j(u,v)\,d\sigma(v) + \le \frac1{m_\delta} + \left(\frac{1-\delta^2/4}{1-\delta^2/16}\right)^j. +\] +Uniform continuity of $H$ therefore gives $T_jH\to H$ uniformly. +Continuous functions are dense in $L^2(\sigma)$: approximate the sets +in a simple function from inside by compact sets and from outside by +open sets, then interpolate their indicators continuously using +distance to these sets. The squared error is bounded by the measure +of the intervening sets. This proves completeness. + +Choose real orthonormal bases $Y_{l,a}$ of the spaces $\mathcal H_l$, +with $Y_{0,1}=1$. Integration by parts shows that the tangent fields +$\gradS Y_{l,a}/\sqrt{\lambda_l}$, for $l\ge1$, are orthonormal. +Bessel's inequality, which follows by expanding the squared distance +from any finite orthogonal sum, consequently gives +\[ + \E\abs{\gradS H}^2 + \ge \sum_{l\ge1,a} + \left|\E\,\gradS H\cdot + \frac{\gradS Y_{l,a}}{\sqrt{\lambda_l}}\right|^2 + =\sum_{l\ge1,a}\lambda_l\left|\E(HY_{l,a})\right|^2. +\] +No differentiation of an infinite series is needed. If $H$ is even, +its odd-degree coefficients vanish. Completeness identifies its +variance with the sum of the squared nonconstant coefficients and +$\norm{QH}_2^2$ with the corresponding sum over even $l\ge4$. +Using $\lambda_l\ge\lambda_2$ for even $l\ge2$ and +$\lambda_l\ge\lambda_4$ for $l\ge4$ proves +\eqref{eq:spectral-gap}. +\end{proof} + +\subsection{The sign-test operator} + +For $k>0$ and an even differentiable function $f$ on $\Sph$, define +\begin{equation}\label{eq:vector-field} + X_f(u)=f(u)u+k^{-1}\gradS f(u). +\end{equation} +For even $C^1$ functions $f_1,f_2$, set +\begin{equation}\label{eq:sign-form} + \mathcal B(f_1,f_2) + =\frac1{b_n}\iint_{\Sph\times\Sph} + \sgn(u\cdot v)\,X_{f_1}(u)\cdot X_{f_2}(v) + \,d\sigma(u)d\sigma(v). +\end{equation} +The value assigned to $\sgn(0)$ is immaterial. Since +$|a|\ge\sgn(u\cdot v)a$ for every real $a$, the form gives the lower bound +\[ + \iint |X_{f_1}(u)\cdot X_{f_2}(v)|\,d\sigma(u)d\sigma(v) + \ge b_n\mathcal B(f_1,f_2). +\] +We estimate this lower bound using two spherical transforms. +Write $\sigma_{u^\perp}$ +for uniform probability on the equator $u^\perp\cap\Sph$, and define +\[ + (Cf)(u)=\frac1{b_n}\int_{\Sph}\abs{u\cdot v}f(v)\,d\sigma(v), + \qquad + (Pf)(u)=\int_{u^\perp\cap\Sph}f(v)\,d\sigma_{u^\perp}(v). +\] +These are the cosine and Funk transforms, here normalized to preserve +constants; compare \cite[Section~2, especially (2.3), (2.5)--(2.6)] +{OlafssonPasqualeRubin2013}. The local multiplier calculation below +uses this probability-measure convention throughout. +Both are self-adjoint contractions on $L^2(\sigma)$ and preserve +constants. For $C$ this follows from symmetry of its nonnegative +kernel and Jensen's inequality. For $P$, average first over the band +$\abs{u\cdot v}<\epsilon$, dividing by its measure. The resulting +kernel is symmetric and preserves constants. Spherical coordinates +show that, for continuous $f$, these band averages tend to $Pf$; +Jensen's inequality and passage to the limit give the same conclusions +for $P$, followed by extension to $L^2(\sigma)$. + +\begin{lemma}\label{lem:sign-operator} +For even $C^1$ functions, +\begin{equation}\label{eq:sign-operator} + \mathcal B(f_1,f_2) + =\left\langle f_1, + \left[C+\left(\frac{2p}{k}+\frac{p^2}{k^2}\right)(C-P)\right] + f_2\right\rangle. +\end{equation} +\end{lemma} + +\begin{proof} +Let $d_0$ be the density at zero of $u\cdot v$ for uniform $u$ and fixed +$v\in\Sph$. Equivalently, $d_0$ is the ratio of the surface areas of +$S^{n-2}$ and $S^{n-1}$. For a fixed ambient vector $w$, its tangential +projection has divergence $-p\,u\cdot w$. Integrating separately on +the two hemispheres gives +\begin{align} + &\int_{\Sph}\sgn(u\cdot v)\,\gradS f(u)\cdot w\,d\sigma(u) + \notag\\ + &\quad=p\int_{\Sph}\sgn(u\cdot v)f(u)\,u\cdot w\,d\sigma(u) + -2d_0\int_{v^\perp\cap\Sph}f(u)\,v\cdot w + \,d\sigma_{v^\perp}(u). + \label{eq:hemisphere-ibp} +\end{align} +The outward conormals on the equator are $-v$ and $v$, respectively; +the sign change between the hemispheres makes their contributions +add with the displayed negative sign. Taking $f=1$ and $w=v$ yields +\[ + 2d_0=p b_n. +\] + +The radial--radial term in \eqref{eq:sign-form}, before division by +$b_n$, is $b_n\langle f_1,Cf_2\rangle$. Apply +\eqref{eq:hemisphere-ibp} with $w=v$ and then integrate against +$f_2(v)$. Each of the two mixed radial--gradient terms, before its +factor $k^{-1}$, equals +\[ + p b_n\langle f_1,(C-P)f_2\rangle. +\] +For the gradient--gradient term, first integrate in $u$ with +$w=\gradS f_2(v)$. Its equator term vanishes because +$v\cdot\gradS f_2(v)=0$. What remains is +\[ + p\iint f_1(u)\sgn(u\cdot v)\, + u\cdot\gradS f_2(v)\,d\sigma(u)d\sigma(v). +\] +A second application of \eqref{eq:hemisphere-ibp}, now in $v$ with +$w=u$, makes this $p^2b_n\langle f_1,(C-P)f_2\rangle$. +Adding the four terms proves \eqref{eq:sign-operator}. +\end{proof} + +\subsection{Multipliers and passage to norms} + +\begin{lemma}\label{lem:spherical-multipliers} +On $\mathcal H_{2j}$ the operators $P$ and $C$ are scalar multipliers, +with +\begin{equation}\label{eq:transform-multipliers} + P_{2j}=(-1)^j + \frac{1\cdot3\cdots(2j-1)}{p(p+2)\cdots(p+2j-2)}, + \qquad + C_{2j}=\frac{pP_{2j}}{p-\lambda_{2j}}, +\end{equation} +where empty products equal one. In particular, +\begin{equation}\label{eq:cosine-recurrence} + \begin{gathered} + C_0=1,\qquad C_2=\frac1{p+2},\qquad + C_4=-\frac1{(p+2)(p+4)},\\[3pt] + \frac{C_{l+2}}{C_l}=-\frac{l-1}{l+p+2}\quad(l\ge2\text{ even}). + \end{gathered} +\end{equation} +\end{lemma} + +\begin{proof} +Fix $u\in\Sph$ and a homogeneous harmonic polynomial $Y_{2j}$. +Let $Z$ be a standard Gaussian in the $p$-dimensional space $u^\perp$. +Its direction is uniform on the equator and independent of its radius, +so polar integration gives +\[ + \mathbb E Y_{2j}(Z) + =p(p+2)\cdots(p+2j-2)\,(PY_{2j})(u). +\] +The radial moment here follows by successive integration by parts in +$\int_0^\infty r^{p+2j-1}e^{-r^2/2}\,dr$. +The one-dimensional Gaussian moment recurrence +$\mathbb E Z_1^{2a}=(2a-1)\mathbb E Z_1^{2a-2}$, together with +independence of coordinates, also gives +\[ + \mathbb E Y_{2j}(Z) + =\frac{(\Delta_{u^\perp})^jY_{2j}(0)}{2^j j!}. +\] +This identity follows term by term on monomials; monomials with an odd +exponent have both sides zero, and the other coefficients are exactly +the multinomial coefficients in the power of the Laplacian. +Harmonicity and commutation of constant-coefficient derivatives imply +\[ + (\Delta_{u^\perp})^jY_{2j} + =(-1)^j\partial_u^{2j}Y_{2j}, + \qquad + \partial_u^{2j}Y_{2j}(0)=(2j)!Y_{2j}(u). +\] +Since $(2j)!/(2^j j!)=1\cdot3\cdots(2j-1)$, the formula for $P_{2j}$ +follows. + +To obtain $C$, integrate by parts on the two hemispheres in the +variable $v$. Off the equator, +$\lapS\abs{u\cdot v}=-p\abs{u\cdot v}$; the jump of its normal +derivative contributes $2d_0$ times equator averaging. Thus, for any +smooth $Y$, +\[ + \int_{\Sph}\abs{u\cdot v}\lapS Y(v)\,d\sigma(v) + =-p\int_{\Sph}\abs{u\cdot v}Y(v)\,d\sigma(v) + +2d_0(PY)(u). +\] +For $Y\in\mathcal H_l$, division by $b_n$ gives +$-\lambda_l CY=p(PY-CY)$. The denominator +$p-\lambda_l=-(l-1)(l+p)$ is nonzero for every even $l$, proving the +second formula in \eqref{eq:transform-multipliers}. Its first three +values give those in \eqref{eq:cosine-recurrence}; division of successive +values, using $P_{l+2}/P_l=-(l+1)/(l+p)$, gives the recurrence. +\end{proof} + +Set +\begin{equation}\label{eq:tail-constant} + c_{n,k}= + \frac{1+\left(2/k+(n-1)/k^2\right)4(n+2)}{(n+1)(n+3)}. +\end{equation} + +\begin{proposition}\label{prop:sign-estimate} +Let $k>0$ and let $f_1,f_2$ be even $C^1$ functions on $\Sph$. +If $f_1$ has zero projection onto $\mathcal H_2$, then +\begin{equation}\label{eq:sign-bound} + \mathcal B(f_1,f_2) + \ge (\E f_1)(\E f_2) + -c_{n,k}\norm{Qf_1}_2\norm{Qf_2}_2. +\end{equation} +The operator identity \eqref{eq:sign-operator} and this estimate also +hold when $f_i=g_i^k$ for arbitrary norms $g_i$ on $\R^n$, with the +vector fields defined almost everywhere. In that case +\begin{equation}\label{eq:norm-vector-field} + X_{g_i^k}(u)=g_i(u)^{k-1}\nabla g_i(u). +\end{equation} +\end{proposition} + +\begin{proof} +By Lemma~\ref{lem:spherical-multipliers}, the operator in +\eqref{eq:sign-operator} has multiplier one on constants and, on degree +$l$, multiplier +\[ + \mu_l=C_l\left[1+\left(\frac2k+\frac p{k^2}\right)\lambda_l\right]. +\] +Indeed, $C_l-P_l=(\lambda_l/p)C_l$. +Put $a=2/k+p/k^2>0$. For even $l\ge4$, +\[ + (l+p+2)\lambda_l-(l-1)\lambda_{l+2} + =(p-1)l^2+(p^2-1)l+2(p+1)>0. +\] +Since also $l-10$ define +\begin{align} + e(z)&=\frac{z^{-n}-1}{n},& + M(z)&=z^k-1+k e(z),& + W(z)&=z^k-z^{k-1}+e(z).\label{eq:scalar-basic} +\end{align} +Since $M'(z)=k(z^{k-1}-z^{-n-1})$, the function $M$ decreases up to +$1$, increases thereafter, and satisfies $M(1)=M'(1)=0$. In particular, +$M\ge0$. For a norm with $\E g^{-n}=1$, the correction $e(g)$ +has mean zero. We will split $g^k-1$ into a term controlled by +convexity and a remainder controlled by its variance. Choose the constants +\begin{equation}\label{eq:scalar-constants} +\begin{array}{c|cc} + n&\beta&w\\ \hline + 4&1/2&3/14\\ + 5,6&2/3&1/3\\ + n\ge7&2/3&1/2 +\end{array} +\end{equation} +and split +\begin{equation}\label{eq:scalar-split} + G(z)=\beta M(z)\mathbf 1_{\{z>1\}},\qquad + D(z)=z^k-1-G(z). +\end{equation} +Thus $G$ vanishes for $z\le1$, while $D$ retains the remaining part +of $z^k-1$. To estimate $G(g)$ by convexity, +let $I(z)=0$ for $01$, let +\begin{align} + I(z)&=\int_1^z M'(a)^2\,da\notag\\ + &=k^2\left(\frac{z^{2k-1}-1}{2k-1} + -\frac{2(1-z^{k-1-n})}{n+1-k} + +\frac{1-z^{-2n-1}}{2n+1}\right). + \label{eq:scalar-I} +\end{align} +Both $G$ and $I$ are continuously differentiable across $1$; +$I\ge0$ and $G'^2=\beta^2 I'$. + +\begin{lemma}\label{lem:curvature} +For every norm $g$ on $\R^n$, +\begin{equation}\label{eq:norm-curvature} + 2n\Var(G(g))+d_*\|QG(g)\|_2^2 + \le p\beta^2\E[gI(g)],\qquad p=n-1. +\end{equation} +\end{lemma} + +\begin{proof} +Suppose first that $g$ is smooth away from the origin. The trace of its +Euclidean Hessian on the tangent space at $u\in\Sph$ is +$\lapS g(u)+p g(u)$, and it is nonnegative by convexity. Multiplying by +$I(g)\ge0$ and integrating by parts gives +\begin{align*} + \E|\gradS G(g)|^2 + &=\beta^2\E[I'(g)|\gradS g|^2] + =-\beta^2\E[I(g)\lapS g] + \le p\beta^2\E[gI(g)]. +\end{align*} +The spectral gap estimate \eqref{eq:spectral-gap} proves +\eqref{eq:norm-curvature} in this case. + +Here is a passage to arbitrary norms that does not assume the +approximants are homogeneous. Convolve the ambient norm with smooth, +even, nonnegative mollifiers of mass one and shrinking compact support, +and write the resulting smooth convex functions as $g_\varepsilon$. +On the unit sphere their tangent Hessian trace is +\[ + \lapS g_\varepsilon+p\partial_r g_\varepsilon\ge0. +\] +The preceding argument and \eqref{eq:spectral-gap} therefore give +\[ + 2n\Var(G(g_\varepsilon))+d_*\|QG(g_\varepsilon)\|_2^2 + \le p\beta^2\E[I(g_\varepsilon)\partial_r g_\varepsilon]. +\] +The functions $g_\varepsilon$ converge uniformly to $g$ on compact +sets, are uniformly Lipschitz there, and their gradients converge to +$\nabla g$ at every differentiability point of $g$. To see the last +assertion, every limit of their bounded gradients satisfies the +subgradient inequality for $g$ by convexity and uniform convergence; +differentiability makes that subgradient unique. The exceptional +directions have spherical measure zero, as in +Lemma~\ref{lem:gauge-variation}. Homogeneity of $g$ then gives +$\partial_r g_\varepsilon(u)\to\nabla g(u)\cdot u=g(u)$ almost +everywhere. On the sphere the approximants stay in a fixed positive +compact interval, while their radial derivatives are bounded. Uniform +convergence on the left and dominated convergence on the right prove +\eqref{eq:norm-curvature}. +\end{proof} + +\subsection{From scalar bounds to the norm estimate}\label{subsec:norm-deduction} + +Define, for $z>0$, +\begin{equation}\label{eq:scalar-L} + \mathcal L(z)=c\left( + \frac{p\beta^2 zI(z)-2nG(z)^2}{wd_*} + +\frac{D(z)^2}{1-w}\right). +\end{equation} + +\begin{lemma}\label{lem:scalar-estimates} +The constants and functions above satisfy +\begin{equation}\label{eq:scalar-mean-bound} + c\left(\frac{2n\beta^2}{wd_*} + -\frac{(1-\beta)^2}{1-w}\right) + \le\frac{2k-1}{k^2} +\end{equation} +and +\begin{equation}\label{eq:scalar-pointwise} + \mathcal L(z)\le2W(z)\qquad(z>0). +\end{equation} +Equality in \eqref{eq:scalar-pointwise} holds only at $z=1$. +\end{lemma} + + +We first deduce the norm estimate from the two scalar bounds. Their +verification, including the strict pointwise inequality in every +dimension, occupies the remainder of the section. + +\begin{proof}[Proof of Theorem~\ref{thm:norm-estimate}] +In the following expectations, $G,D,I$ denote their compositions with +$g$. Set +\[ + m=\E g^k-1,\qquad \ell=\E g^{k-1}-1,\qquad t=\E G. +\] +The normalization gives $\E e(g)=0$, so $m=\E M(g)\ge0$, +$\E D=m-t$, and $\E W(g)=m-\ell$. Since +$0\le G\le\beta M$, we have $0\le t\le\beta m$. +For every $a>0$, Jensen's inequality applied to the convex function +$y\mapsto y^{-a/n}$ yields +$\E g^a\ge(\E g^{-n})^{-a/n}=1$. +Together with +\[ + z^{k-1}\le\frac{(k-1)z^k+1}{k}, +\] +this gives +\begin{equation}\label{eq:norm-mean-range} + 0\le\ell\le\frac{k-1}{k}\,m. +\end{equation} +For $k=1$ this simply says $\ell=0$. + +Since $Q$ annihilates constants, +$Q(g^k)=QG+QD$. The Hilbert-space inequality +\[ + \|a+b\|^2\le\frac{\|a\|^2}{w} + +\frac{\|b\|^2}{1-w} + \qquad(00. +\] +This proves \eqref{eq:scalar-mean-bound} for every dimension. + +\medskip +\noindent\emph{The interval $01$ in dimensions $4,5,6$.} +Set $x=z-1>0$. Multiplication by a positive factor clears the negative +powers of $z$: if +\[ + P_n(z)=s_nz^{2n}\bigl(2W(z)-\mathcal L(z)\bigr), + \qquad (s_4,s_5,s_6)=(2880,712800,221130), +\] +then substitution of \eqref{eq:scalar-basic}--\eqref{eq:scalar-L} +and collection of equal powers give +\begin{align} + P_4(z)={}&462z^{10}-983z^9+93z^8 + +3399z^5-3378z^4+407,\notag\\ + P_5(z)={}&151525z^{14}-273570z^{12}-680800z^{11} + +938157z^{10}\notag\\ + &\quad+225720z^7-462264z^5+101232,\notag\\ + P_6(z)={}&48399z^{16}-75894z^{14}-218004z^{13} + +285961z^{12}\notag\\ + &\quad+56940z^8-127478z^6+30076. + \label{eq:small-polynomials} +\end{align} +For clarity, the coefficients needed from $P_n(1+x)$ are listed below; +the constant and linear coefficients are zero. +\[ +\begin{array}{c|rrr} + [x^j]&n=4&n=5&n=6\\ \hline + j=2&1728&623700&452790\\ + j=3&-1446&-1505900&679380\\ + j=4&-6711&-5802775&-132249\\ + j=5&1173&12849430&7800780\\ + j=6&17052&86314305&51269452\\ + j=7&20796&191506920&146058120 +\end{array} +\] +All remaining coefficients are nonnegative, as can be seen without +listing them. Let $a_nz^{d_n}$ be the leading term in +\eqref{eq:small-polynomials}. At $j=8$, its contribution +$a_n\binom{d_n}{8}$ minus the contributions of all negative terms is +\[ + 11943,\qquad 207280425,\qquad 114414300 + \quad\text{for } n=4,5,6, +\] +respectively. For every $ii$ it is zero. Thus the same leading term covers all negative +contributions for every $8\le j\le d_n$. + +It follows that $P_n(1+x)/x^2$ is bounded below by, respectively, +\begin{align*} + U_4(x)&=1728-1446x-6711x^2+17052x^4,\\ + U_5(x)&=623700-1505900x-5802775x^2+86314305x^4,\\ + U_6(x)&=452790-132249x^2+51269452x^4. +\end{align*} +These lower bounds are positive. Indeed +\begin{align*} + 1446x&\le400+1400x^2,& + 8111x^2&\le1000+17052x^4,\\ + 1505900x&\le200000+2900000x^2,& + 8702775x^2&\le230000+86314305x^4. +\end{align*} +Each follows from $dt\le a+bt^2$ when $d^2\le4ab$; the corresponding +values of $4ab-d^2$ are +\[ + 149084,\quad 2419679,\quad 52265190000,\quad 3670867899375. +\] +The first row gives $U_4(x)\ge328$ and the second gives +$U_5(x)\ge193700$. Finally $U_6$, regarded as a quadratic in $x^2$, +has positive leading coefficient and negative discriminant, since +\[ + 4\cdot452790\cdot51269452-132249^2=92839690886319>0. +\] +Therefore $P_n(z)>0$ for $z>1$ in all three dimensions. + +\medskip +\noindent\emph{The interval $z>1$ in all dimensions $n\ge7$.} +Put $H=18(n+4)/c$. Direct collection of powers in +\eqref{eq:scalar-L} now gives +\begin{equation}\label{eq:large-polynomial} + \frac{9(n+4)}{c}z^{2n}(2W-\mathcal L) + =z^{2n}(A_4z^4+A_2z^2+A_1z+A_0) + +z^n(B_2z^2+B_0)+T_0, +\end{equation} +where +\begin{align} + A_4&=\frac{2n-8}{3},& + A_2&=H-12n-32-\frac{64}{n},\notag\\ + A_1&=-H+\frac{16(n-1)}3+32-\frac{16(n-1)}{2n+1},&&\notag\\ + A_0&=-\frac Hn+6n+40+\frac{64}{n}-\frac{128}{n^2},&&\notag\\ + B_2&=16+\frac{64}{n},& + B_0&=\frac Hn-48-\frac{64}{n}+\frac{256}{n^2},\notag\\ + T_0&=\frac{16(n-1)}{2n+1}-\frac{128}{n^2}.&& + \label{eq:large-coefficients} +\end{align} +We prove that the polynomial on the right of +\eqref{eq:large-polynomial}, expressed in $x=z-1$, has nonnegative +coefficients and a positive quadratic coefficient. + +Write its two parentheses as $\sum_{i=0}^4q_ix^i$ and +$\sum_{i=0}^2r_ix^i$. Thus +\begin{align*} + q_0&=A_4+A_2+A_1+A_0,&q_1&=4A_4+2A_2+A_1,\\ + q_2&=6A_4+A_2,&q_3&=4A_4,\qquad q_4=A_4,\\ + r_0&=B_2+B_0,&r_1&=2B_2,\qquad r_2=B_2. +\end{align*} +In particular $q_3,q_4,r_1,r_2>0$. The constant and linear +coefficients of the full polynomial vanish. Its quadratic coefficient +is +\begin{equation}\label{eq:large-quadratic} + \frac{9(n+4)}c\,(n+3-8c)>0. +\end{equation} +One can obtain these three coefficients either from +\eqref{eq:large-coefficients} or directly from the order of vanishing at +$z=1$: $G=O(x^2)$, $I=O(x^3)$, and $D=2x+O(x^2)$, so that +$2W-\mathcal L=(n+3-8c)x^2+O(x^3)$. Positivity is +\eqref{eq:scalar-local-margin}. + +The remaining signs admit explicit polynomial certificates. Set +\begin{align*} + V_0(n)&=34n^5-95n^4-1666n^3-5701n^2-8052n-2688,\\ + V_1(n)&=2n^5+9n^4-162n^3-845n^2-1220n-448,\\ + V_2(n)&=5n^3+8n^2-41n-56,\\ + R(n)&=7n^4+8n^3-187n^2-748n-896. +\end{align*} +Substituting $H=18(n+4)(n+1)(n+3)/(n^2+5n+7)$ gives +\begin{align} + q_0+\frac{r_0}{8} + &=\frac{V_0(n)}{4n^2(2n+1)(n^2+5n+7)},\notag\\ + q_1&=\frac{2V_1(n)}{n(2n+1)(n^2+5n+7)},\notag\\ + q_2&=\frac{2(n+4)V_2(n)}{n(n^2+5n+7)},& + r_0&=-\frac{2R(n)}{n^2(n^2+5n+7)}. + \label{eq:large-signs} +\end{align} +The required signs hold throughout their stated ranges because, for +$t\ge0$, +\begin{align} + V_0(10+t)={}&34t^5+1605t^4+28534t^3 + +227319t^2\notag\\ + &\quad+698128t+130692,\notag\\ + V_1(10+t)={}&2t^5+109t^4+2198t^3+19695t^2 + +69280t+30852,\notag\\ + V_2(7+t)={}&5t^3+113t^2+806t+1764,\notag\\ + R(7+t)={}&7t^4+204t^3+2039t^2+7414t+4256. + \label{eq:shift-certificates} +\end{align} +Consequently $q_2>0$ and $r_0<0$ for every $n\ge7$, while +$q_0+r_0/8>0$ and $q_1>0$ for every $n\ge10$. + +Use the convention that $\binom mj=0$ when $j<0$ or $j>m$. For +$3\le j\le2n$, comparison of the factors in the binomial products gives +\begin{equation}\label{eq:binomial-comparison} + \binom nj\le 2^{-j}\binom{2n}{j}. +\end{equation} +Indeed $(n-a)/(2n-a)\le1/2$ for $0\le an$ the left +side is zero. For $n\ge10$, the sum of the $q_0$ and $r_0$ +contributions to the coefficient of $x^j$ is therefore at least +\[ + \left(q_0+\frac{r_0}{2^j}\right)\binom{2n}{j} + \ge\left(q_0+\frac{r_0}{8}\right)\binom{2n}{j}>0. +\] +All other contributions are nonnegative. For $j>2n$, the $q_0$ and +$r_0$ contributions vanish and every remaining term is nonnegative. +Together with \eqref{eq:large-quadratic}, this proves coefficient +positivity for every $n\ge10$. + +It remains to treat $n=7,8,9$, where $q_1$ can be negative. +For $3\le j\le2n$, divide the coefficient of $x^j$ by +$\binom{2n}{j}$, discard the nonnegative $q_3,q_4,r_2$ contributions, +and use \eqref{eq:binomial-comparison} for the negative $r_0$ term. +The resulting lower bound is +\begin{equation}\label{eq:small-large-dimension-bound} + \begin{split} + q_0+\frac{r_0}{2^j} + +\frac{j}{2n-j+1} + \left(q_1+q_2\frac{j-1}{2n-j+2}\right) + +r_1\frac{\binom n{j-1}}{\binom{2n}{j}}. + \end{split} +\end{equation} +Substitution in \eqref{eq:large-coefficients} gives the componentwise +lower bounds +\[ +\begin{array}{c|rrrrr} + n&q_0&q_1&q_2&r_0&r_1\\ \hline + 7&-2&-11&60&-2&50\\ + 8&-1&-6&72&-6&46\\ + 9&-1&-6&72&-6&46 +\end{array} +\] +All weights in \eqref{eq:small-large-dimension-bound} are nonnegative, +so these bounds may be substituted directly. They give the following +positive numbers at $j=3,4$; at $j=5$ we have already omitted the final +$r_1$ term: +\[ +\begin{array}{c|ccc} + n&j=3&j=4&j=5\ \text{without the final term}\\ \hline + 7&5/26&1233/1144&589/176\\ + 8&37/28&10707/3640&1153/208\\ + 9&101/136&3851/2040&395/112 +\end{array} +\] +These finite substitutions can also be checked from the exact tuples +$(q_0,q_1,q_2,r_0,r_1)$: +\begin{align*} + n=7:&\quad + \left(-\frac{5984}{3185},-\frac{4832}{455}, + \frac{792}{13},-\frac{1216}{637},\frac{352}{7}\right),\\ + n=8:&\quad + \left(-\frac{421}{629},-\frac{3736}{629}, + \frac{2688}{37},-\frac{145}{37},48\right),\\ + n=9:&\quad + \left(\frac{2416}{10773},-\frac{2192}{1197}, + \frac{100568}{1197},-\frac{57968}{10773},\frac{416}{9}\right). +\end{align*} +For $5\le j\le2n$, the parenthesis +$q_1+q_2(j-1)/(2n-j+2)$ is positive and increasing in $j$. +In fact the displayed componentwise bounds give at $j=5$ the positive +lower bounds $119/11$, $210/13$, and $66/5$, respectively. +The prefactor $j/(2n-j+1)$ is positive and increasing too, while +$r_0/2^j$ increases because $r_0<0$. The expression +\eqref{eq:small-large-dimension-bound} with its last term omitted is +therefore increasing throughout this range. Positivity at $j=5$ +proves positivity at every subsequent index through $2n$. + +For $j>2n$ all $r_i$ contributions vanish, since $n+2<2n+1$. +The only possibly negative term is the $q_1$ contribution at +$j=2n+1$. At this index the $q_1$ and $q_2$ contributions sum to +$q_1+2nq_2$, which the same componentwise bounds make at least +$829,1146,1290$ for $n=7,8,9$, respectively. For $j\ge2n+2$ only +the nonnegative $q_2,q_3,q_4$ terms remain. This handles every +coefficient, including the endpoints. + +We have proved that the polynomial in \eqref{eq:large-polynomial} +has nonnegative coefficients and a strictly positive coefficient of +$x^2$ for every $n\ge7$. It is consequently positive for every +$x>0$. Its multiplier $9(n+4)z^{2n}/c$ is positive, so +$2W(z)-\mathcal L(z)>0$ for $z>1$. This finishes the proof of +\eqref{eq:scalar-pointwise} and its strictness in every dimension. +\end{proof} + diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/05_position.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/05_position.tex new file mode 100644 index 0000000000000000000000000000000000000000..31ef447d1f09e412f7bc89009280fb5abdef03ca --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/05_position.tex @@ -0,0 +1,409 @@ +\section{Variational representation and affine position}\label{sec:position} + +We next represent a support functional by a vector field of the form +\eqref{eq:vector-field}, and choose coordinates in which the associated +norm power has no degree-two component. Throughout this section, $k$ has the value in +\eqref{eq:k-choice}, and $\eta$ is a finite positive Borel measure with +bounded support spanning $\R^n$. Neither symmetry of $\eta$ nor the +functional condition \eqref{eq:functional-condition} is needed here. + +Let $A$ be a nonzero symmetric positive semidefinite matrix, and write +$V=\range(A)$. For an origin-symmetric convex body $L$ in $V$, with +interior relative to $V$, put +\[ + \tau_A(L)=\int h_L(Ax)\,d\eta(x). +\] +We extend its gauge to $\R^n$ by $g=g_L\circ P_V$, where $P_V$ denotes +orthogonal projection onto $V$. This extension is a seminorm, with +kernel $V^\perp$. Define +\begin{equation}\label{eq:variational-objective} + J(g)= + \begin{cases} + \E\log g,&k=1,\\ + \E g,&k=2. + \end{cases} +\end{equation} +The logarithm is integrated outside the kernel, which is a spherical +null set. +When $A$ is positive definite, these objectives are chosen so that their +Wulff first variations involve $g^{k-1}\nabla g=X_{g^k}$, the vector +field in \eqref{eq:norm-vector-field}. + +This variational construction belongs to the framework for dual +curvature measures. The optimizer-to-measure argument corresponds to +the $L_1$ dual Minkowski problem with dual parameter $q=1-k$: +for $k=2$, compare \cite[Section~3.1, Lemmas~3.1--3.3]{HuangZhao2018}; +for $k=1$ and full-dimensional bodies, minimization under $\tau_A(L)=1$ +is equivalent, by polarity, to the $p=1$ entropy maximization in +\cite[Section~5, Lemmas~5.1--5.3]{HuangLutwakYangZhang2018}. +Wang and Zhou established uniqueness and continuity for the relevant +full-dimensional dual Minkowski problems +\cite[arXiv version, Theorems~1.1--1.3]{WangZhou2026}. +Here we prove the representation in our normalization, together with +uniqueness and continuity through changes of rank, where the gauges +become seminorms. This continuity is required to select affine +coordinates. The measures may have atoms, and all +variations remain valid for nonsmooth minimizing bodies. + +\begin{lemma}\label{lem:minimizer} +For every nonzero symmetric positive semidefinite $A$, there is a unique +origin-symmetric convex body $L_A$ in $V=\range(A)$ that minimizes +\eqref{eq:variational-objective} subject to $\tau_A(L_A)=1$. +Denote its extended gauge by $g_A$. +\end{lemma} + +\begin{proof} +The spanning hypothesis gives a norm on $V$: +\[ + q_A(y)=\int |y\cdot Ax|\,d\eta(x). +\] +Indeed, $q_A(y)=0$ implies that $Ay$ is orthogonal to $\supp\eta$, +and hence $Ay=0$; because $y\in V$, this forces $y=0$. +For any normalized competitor $L$ and $y\in L$, symmetry gives +$q_A(y)\le\tau_A(L)=1$. Thus all normalized competitors have a common +outer radius in $V$. + +We also need a bound on their gauges along a minimizing sequence. +For any nonzero seminorm $g$, let $R=\max_{\Sph}g$. By separation, +$g$ is the support function of the compact symmetric set +$\{z:z\cdot y\le g(y)\text{ for every }y\in\R^n\}$. +This set has maximal radius $R$: if $Re$ is a point of maximal norm +in it, then +\begin{equation}\label{eq:seminorm-coordinate-bound} + g(u)\ge R|u\cdot e|\qquad(u\in\Sph). +\end{equation} +Consequently, +\[ + J(g)\ge + \begin{cases} + \log R+\E\log|u_1|,&k=1,\\ + Rb_n,&k=2. + \end{cases} +\] +Here $\E|\log|u_1||^2<\infty$: the first coordinate has a density +proportional to $(1-t^2)^{(n-3)/2}$ on $(-1,1)$, and +$\int_0^{1/2}|\log t|^2\,dt<\infty$. In particular every nonzero +seminorm has a finite logarithmic objective. + +Set $b_A=\int|Ax|\,d\eta(x)>0$. The ball of radius $1/b_A$ in $V$ +is a normalized competitor. Comparison with this ball bounds $J$ +above along a minimizing sequence, so the preceding inequalities bound +$R$ above. The bodies therefore contain a common inner ball in $V$, +as well as lying in a common outer ball. Their gauges are uniformly +bounded and equicontinuous on the unit sphere of $V$. A subsequence +converges uniformly to the gauge of a body with the same two ball +bounds. The support functions converge uniformly as well, so the +normalization passes to the limit. The objective also passes to the +limit: in the logarithmic case the extended gauges are uniformly +comparable to $|P_Vu|$, whose logarithm is integrable by +\eqref{eq:seminorm-coordinate-bound}. This proves existence. + +For uniqueness, let $L_0,L_1$ be two minimizers and +$L_t=(1-t)L_0+tL_1$, where $00$. Thus the objective of $L_t$ is at most the +interpolated objectives, and the inequality is strict wherever +$\rho_0\ne\rho_1$. Distinct bodies have different radial functions +on an open set of directions in $V$, which gives a set of positive +spherical measure after projection. They would therefore give a +strictly smaller objective, a contradiction. +\end{proof} + +\begin{proposition}[Variational representation]\label{prop:representation} +Suppose $A$ is positive definite, and let $L$ be any positive dilate +of $L_A$, with gauge $g=g_L$. For every continuous even function +$\phi:\R^n\to\R$ that is positively homogeneous of degree one, +\begin{equation}\label{eq:representation} + \int\phi(Ax)\,d\eta(x) + =s\,\E\phi\bigl(X_{g^k}(u)\bigr), + \qquad + s=\frac{\tau_A(L)}{\E g^{k-1}}. +\end{equation} +Here, almost everywhere on the sphere, +\begin{equation}\label{eq:norm-power-field} + X_{g^k}=g^ku+g^{k-1}\gradS g=g^{k-1}\nabla g. +\end{equation} +\end{proposition} + +\begin{proof} +Normalization turns the minimization problem into minimizing +$J(\tau_A(L)g_L)$ over all symmetric bodies. Put +$\tau=\tau_A(L)$ and $I=\int\phi(Ax)\,d\eta(x)$. For small +positive or negative $t$, define the body +\[ + W_t=\bigl\{y\in\R^n: + y\cdot v\le h_L(v)+t\phi(v)\text{ for all }v\in\Sph\bigr\}, + \qquad g_t=g_{W_t}. +\] +The constraint is positive for sufficiently small $|t|$, so $W_t$ is +a symmetric convex body. Although that constraint need not be a +support function, it bounds the actual support function above. +Therefore +\[ + \tau_A(W_t)\le\tau+tI. +\] +Both quantities are positive for small $|t|$. Since $J$ increases +under pointwise increase of a positive gauge, minimality gives +\begin{equation}\label{eq:wulff-minimum} + J\bigl((\tau+tI)g_t\bigr) + \ge J\bigl(\tau_A(W_t)g_t\bigr) + \ge J(\tau g), +\end{equation} +with equality at $t=0$. + +Lemma~\ref{lem:gauge-variation} gives +\[ + \left.\frac{d}{dt}\right|_{t=0}g_t(u) + =-g(u)\phi(\nabla g(u)) +\] +almost everywhere, with uniform bounds that permit differentiation +of the objective. The left side of \eqref{eq:wulff-minimum} has a +two-sided minimum at zero. Its derivative is consequently zero: +\[ + \frac{I}{\tau}=\E\phi(\nabla g)\quad(k=1), + \qquad + I\E g=\tau\E\bigl[g\phi(\nabla g)\bigr]\quad(k=2). +\] +Equivalently, +\[ + \frac{I}{\tau} + =\frac{\E\bigl[g^{k-1}\phi(\nabla g)\bigr]}{\E g^{k-1}}. +\] +Euler's identity for the homogeneous gauge gives +\eqref{eq:norm-power-field}, and positive homogeneity of $\phi$ +then gives \eqref{eq:representation}. The argument applies to both +signs of $t$ without imposing convexity on the varied constraint. +\end{proof} + +To select a position we allow the matrix to approach the boundary of +the compact convex set +\begin{equation}\label{eq:matrix-domain} + \mathcal A=\{A=A^t\ge0:\tr A=1\}. +\end{equation} +The minimizers extend continuously to this boundary, even though +their ambient unit sets may become cylinders. + +\begin{lemma}\label{lem:matrix-continuity} +The map $A\mapsto g_A|_{\Sph}$ is continuous on $\mathcal A$ in +the uniform norm. +\end{lemma} + +\begin{proof} +Let $A_j\to A$ in $\mathcal A$, and write +$V_j=\range(A_j)$, $g_j=g_{A_j}$, and +$b_j=\int|A_jx|\,d\eta(x)$. Then +$b_j\to b=\int|Ax|\,d\eta(x)>0$. The normalized ball in $V_j$ +has extended gauge $b_j|P_{V_j}u|$, and hence objective at most +$\log b_j$ if $k=1$, and at most $b_j$ if $k=2$. +By \eqref{eq:seminorm-coordinate-bound}, this bounds +$R_j=\max_{\Sph}g_j$ uniformly above. The minimizing body contains +the ball of radius $1/R_j$ in $V_j$, so its normalization gives +$1\ge b_j/R_j$. Thus $R_j\ge b_j$, bounded away from zero. + +The seminorms $g_j$ have uniformly bounded Lipschitz constants. +Every subsequence therefore has a further subsequence converging +uniformly on $\Sph$ to a nonzero seminorm $g$. Along such a +subsequence, +\begin{equation}\label{eq:objective-limit} + J(g_j)\longrightarrow J(g). +\end{equation} +For $k=2$ this is immediate. For $k=1$, choose unit vectors $e_j$ +with $g_j(u)\ge R_j|u\cdot e_j|$. The positive parts of $\log g_j$ +are uniformly bounded, and their negative parts have uniformly +bounded squared integrals, by rotation invariance and +$\E|\log|u_1||^2<\infty$. These logarithms are therefore uniformly +integrable. They converge almost everywhere to $\log g$, since +the kernel of a nonzero seminorm is a proper subspace. Truncating +the logarithms below at $-M$ and then sending $M$ to infinity proves +\eqref{eq:objective-limit}; the discarded integrals are uniformly +$O(M^{-1})$ by the squared-integral bound. + +We identify this subsequential limit. Set +$Z=\{g\le1\}$ and $Z_j=\{g_j\le1\}$, allowing infinite values +for their support functions. If $g(y)<1$, then $y\in Z_j$ for all +sufficiently large $j$. Testing the support functions against $y$ +and then taking the supremum gives, for every $x$, +\begin{equation}\label{eq:cylinder-support-limit} + h_Z(Ax)\le\liminf_j h_{Z_j}(A_jx). +\end{equation} +The supremum over $g(y)<1$ equals $h_Z$, by scaling towards zero. +Since $Z_j=L_{A_j}+V_j^\perp$ and $A_jx\in V_j$, its support +function at $A_jx$ equals that of $L_{A_j}$. All these support +functions are nonnegative, so Fatou's lemma yields +\begin{equation}\label{eq:cylinder-normalization} + \int h_Z(Ax)\,d\eta(x)\le1. +\end{equation} + +Let $V=\range(A)$ and $L_0=\overline{P_VZ}$. This projection is +bounded despite the possible unboundedness of $Z$. Indeed, for +every $z\in Z$, symmetry and \eqref{eq:cylinder-normalization} give +\[ + q_A(P_Vz)=\int|z\cdot Ax|\,d\eta(x) + \le\int h_Z(Ax)\,d\eta(x)\le1. +\] +The norm $q_A$ bounds the radius in $V$. Moreover $Z$ contains a +Euclidean ball, so $L_0$ has interior in $V$. Its extended gauge +$\bar g=g_{L_0}\circ P_V$ satisfies $\bar g\le g$, and +\begin{equation}\label{eq:projected-normalization} + 00$. On the kernel this +follows from \eqref{eq:kernel-field}; on the range it follows from +the positive definiteness of the restriction of $A$. Its trace is +one. The same perturbation is allowed at a positive definite $A$ +for sufficiently small $\epsilon$. + +Use the Euclidean inner product $\langle U,W\rangle=\tr(UW)$ on +symmetric matrices, and consider the continuous map +\[ + A\longmapsto + \operatorname{proj}_{\mathcal A}\bigl(A-\mathcal F(A)\bigr). +\] +Lemma~\ref{lem:fixed-point} provides a fixed point $A$. The +nearest-point condition there is +\[ + \langle\mathcal F(A),B-A\rangle\ge0 + \qquad(B\in\mathcal A). +\] +We may take $B=A-\epsilon\mathcal F(A)$, as just shown. This +gives $-\epsilon\tr(\mathcal F(A)^2)\ge0$, hence +$\mathcal F(A)=0$. Formula~\eqref{eq:kernel-field} rules out +singularity, so $A$ is positive definite. + +For every symmetric traceless matrix $H$, the vanishing field gives +\[ + \E\bigl[g_A(u)^k u^tHu\bigr]=0. +\] +These quadratics are exactly $\mathcal H_2$, since the Euclidean +Laplacian of $x^tHx$ is $2\tr H$. Thus the required degree-two +component vanishes. + +Finally, for $a>0$ the correspondence $L\mapsto L/a$ between +normalized competitors for $A$ and $aA$ gives +\begin{equation}\label{eq:minimizer-scaling} + g_{aA}=a g_A. +\end{equation} +Indeed, $\tau_{aA}(L/a)=\tau_A(L)$, while multiplying a gauge by +$a$ adds $\log a$ to the logarithmic objective and multiplies +the linear objective by $a$. The minimizers therefore correspond. +Taking $a=(\det A)^{-1/n}$ makes the determinant one and preserves +the vanishing degree-two component. +\end{proof} + +Only one measure needs the position of Proposition~\ref{prop:position}. +For a second measure, Proposition~\ref{prop:representation} provides +the required vector field in the resulting coordinates without a +simultaneous position condition. diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/06_conclusion.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/06_conclusion.tex new file mode 100644 index 0000000000000000000000000000000000000000..0f71c3a9c453ea83f20d95f47fe84b3244f48e5b --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/06_conclusion.tex @@ -0,0 +1,160 @@ +\section{The bilinear inequality and equality cases} +\label{sec:conclusion} + +We now combine the spherical and variational estimates. Recall that an +admissible measure satisfies the support-functional inequality +\eqref{eq:functional-condition}, and need not itself be symmetric. + +\begin{theorem}\label{thm:bilinear} +Let $\eta,\zeta$ be finite positive measures on $\R^n$ with bounded, +spanning support. Suppose that both satisfy +\eqref{eq:functional-condition}. Then +\begin{equation}\label{eq:bilinear} + \iint |x\cdot y|\,d\eta(x)d\zeta(y)\ge b_n. +\end{equation} +If equality holds, there is an origin-centered ellipsoid $E_0$ such that +\[ + \int h_{E_0}\,d\eta=(|E_0|/\kappa_n)^{1/n}. +\] +\end{theorem} + +\begin{proof} +Choose $k$ as in \eqref{eq:k-choice} and write $c=c_{n,k}$. +Apply Proposition~\ref{prop:position} to $\eta$, obtaining a symmetric +positive definite matrix $A$ with determinant one. +Replace $\eta$ by its image under $A$ and $\zeta$ by its image under +$A^{-1}$. Their pairing is unchanged, since +$(Ax)\cdot(A^{-1}y)=x\cdot y$. +Their support-functional inequalities also persist: for any linear map +$T$, +\[ + h_M(Tx)=h_{T^tM}(x), +\] +and $|T^tM|=|M|$ when $|\det T|=1$. + +For the first transformed measure, the support functional at the +identity is precisely the original $\tau_A$. Its unique minimizing +body is therefore $L_A$, with the degree-two cancellation supplied by +Proposition~\ref{prop:position}. +For each transformed measure choose its minimizing body as in +Proposition~\ref{prop:representation}, and rescale that body to volume +$\kappa_n$. Let the resulting gauges be $g_1,g_2$, and put +\[ + f_i=g_i^k,\qquad t_i=\E g_i^{k-1},\qquad + s_i=\frac{\tau_i(L_i)}{t_i},\qquad + a_i=s_i\E f_i\quad(i=1,2), +\] +where $\tau_i$ is the functional of the corresponding transformed +measure. Thus +\begin{equation}\label{eq:normalized-parameters} + \E g_i^{-n}=1,\qquad s_it_i\ge1,\qquad a_i\ge1. +\end{equation} +For the last inequality, if $k=1$ then $\E g_i\ge1=t_i$ by the +negative-moment normalization. If $k=2$, the same normalization gives +$\E g_i\ge1$, and $\E g_i^2\ge(\E g_i)^2\ge\E g_i=t_i$. +The first function $f_1$ has no degree-two harmonic component; scalar +rescaling does not alter this property. + +Apply \eqref{eq:representation} to $|x\cdot y|$ successively in both +variables. These are even continuous homogeneous tests, so the formula +applies despite any asymmetry of the measures. It gives +\begin{align} + \iint |x\cdot y|\,d\eta(x)d\zeta(y) + &=s_1s_2\iint + |X_{f_1}(u)\cdot X_{f_2}(v)|\,d\sigma(u)d\sigma(v) + \notag\\ + &\ge b_ns_1s_2\mathcal B(f_1,f_2). + \label{eq:pairing-sign} +\end{align} +The pointwise inequality in the last line uses the test +$\sgn(u\cdot v)$. Proposition~\ref{prop:sign-estimate} now gives +\[ + b_n^{-1}\iint |x\cdot y|\,d\eta(x)d\zeta(y) + \ge a_1a_2-s_1s_2c\norm{Qf_1}_2\norm{Qf_2}_2. +\] +By Theorem~\ref{thm:norm-estimate} and +\eqref{eq:normalized-parameters}, +\begin{equation}\label{eq:error-absorption} + s_i^2c\norm{Qf_i}_2^2 + \le a_i^2-s_i^2t_i^2\le a_i^2-1. +\end{equation} +Consequently the last lower bound is at least +\[ + a_1a_2-\sqrt{(a_1^2-1)(a_2^2-1)}\ge1. +\] +Indeed $a_1a_2-1\ge0$ and +\[ + (a_1a_2-1)^2-(a_1^2-1)(a_2^2-1)=(a_1-a_2)^2\ge0. +\] +This proves \eqref{eq:bilinear}. + +Suppose equality holds. If $a_i>1$, the inequality +$s_i^2c\norm{Qf_i}_2^2\le a_i^2-1$ is strict: this follows from +Theorem~\ref{thm:norm-estimate} when $g_i$ is nonconstant, and from its +zero left side when $g_i$ is constant. If both $a_i$ exceed one, the +product error in \eqref{eq:error-absorption} is therefore strictly +smaller than the displayed square root. If exactly one exceeds one, +the error product vanishes and $a_1a_2>1$. +Both alternatives contradict equality, so $a_1=a_2=1$. +In particular, $\E g_1^k\le\E g_1^{k-1}$. +For $k=1$, this forces $\E g_1=1$ and equality in the negative-moment +power-mean bound; hence $g_1=1$. +For $k=2$, the chain +\[ + 1\le\E g_1\le(\E g_1)^2\le\E g_1^2\le\E g_1 +\] +again forces constancy and then $g_1=1$. +Thus $s_1=1$, and the support-functional condition of the transformed +first measure is sharp on $B_2^n$. Undoing the transformation makes the +original condition sharp on $A^tB_2^n$, which is an origin-centered +ellipsoid of volume $\kappa_n$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Translate $K$ so that the origin is in its interior, and first rescale +it to satisfy $|K|=\kappa_n$. +Proposition~\ref{prop:projection-measure} supplies the admissible +measure $\eta_K$. Set +\[ + r=\left(\frac{|\Pi K|}{\kappa_n}\right)^{1/n}, + \qquad D=\frac1r\Pi K. +\] +The symmetric body $D$ has volume $\kappa_n$, so it too has an +admissible measure $\eta_D$. Its own support-functional integral is +one: the volume derivative under $D+tD=(1+t)D$ gives +\[ + \int h_D\,d\eta_D + =\frac1{n\kappa_n}\left.\frac{d}{dt}\right|_{0+}|(1+t)D|=1. +\] +Using \eqref{eq:projection-measure} and Theorem~\ref{thm:bilinear}, +\[ + 1=\frac{n\kappa_n}{2r} + \iint|x\cdot y|\,d\eta_K(x)d\eta_D(y) + \ge\frac{n\kappa_nb_n}{2r}. +\] +Equation~\eqref{eq:ball-constant} yields $r\ge\kappa_{n-1}$, or +$|\Pi K|\ge\kappa_n\kappa_{n-1}^n$ under the current normalization. +The projection body scales as $\Pi(aK)=a^{n-1}\Pi K$ for $a>0$. +Undoing the volume normalization proves \eqref{eq:main}. + +If equality holds, the pairing of $\eta_K$ and $\eta_D$ equals $b_n$. +Theorem~\ref{thm:bilinear} gives an ellipsoid on which the +support-functional bound for $\eta_K$ is sharp. +The equality assertion of Proposition~\ref{prop:projection-measure} +then makes $K$ a translate of a positive dilate of that ellipsoid. + +For the converse, the quotient is invariant under translations and +invertible linear maps. To verify the latter directly, the segment +volume derivative gives, for every vector $y$ and every invertible $T$, +\begin{align*} + 2h_{\Pi(TK)}(y) + &=\left.\frac{d}{dt}\right|_{0+}|TK+t[-y,y]|\\ + &=|\det T|\left.\frac{d}{dt}\right|_{0+} + |K+t[-T^{-1}y,T^{-1}y]|\\ + &=2|\det T|h_{\Pi K}(T^{-1}y). +\end{align*} +Hence $\Pi(TK)=|\det T|T^{-t}\Pi K$ and +$|\Pi(TK)|=|\det T|^{n-1}|\Pi K|$, exactly matching the factor in +$|TK|^{n-1}$. Finally, $\Pi B_2^n=\kappa_{n-1}B_2^n$ gives equality +for the ball, and therefore for every ellipsoid. +\end{proof} diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/07_consequences.tex b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/07_consequences.tex new file mode 100644 index 0000000000000000000000000000000000000000..ce889945ba1aa95cf555468caa4e8a442b080bdd --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/build/latex/sections/07_consequences.tex @@ -0,0 +1,240 @@ +\section{Consequences}\label{sec:consequences} + +Theorem~\ref{thm:main}, together with the separate three-dimensional +theorem of Chen, Feng, Li, Xi, and Xu \cite[Theorem~1.1]{ChenEtAl2026}, +gives Petty's projection-volume inequality and its ellipsoid equality +case in every dimension $n\ge3$. We now apply this combined conclusion +to lower-degree projection bodies, affine functional inequalities, and +isoperimetry in normed spaces. The implications and equality assertions +are specified separately in each setting. + +\subsection{Lower-degree projection bodies} + +For a convex body $L\subset\R^m$, use the standard intrinsic volumes +$V_j(L)$, normalized by the Steiner formula +\[ + \operatorname{vol}_m(L+tB_2^m) + =\sum_{j=0}^m\kappa_{m-j}V_j(L)t^{m-j}, + \qquad t\ge0,\qquad \kappa_0=1. +\] +This is the normalization in +\cite[author version, Section~2, p.~6]{OrtegaMorenoSchuster2021}; +on a hyperplane we use its induced Euclidean structure. For +$1\le i\le n-1$, the degree-$i$ projection body $\Pi_iK$ is defined by +\[ + h_{\Pi_iK}(u)=V_i(K|u^\perp),\qquad u\in S^{n-1}, +\] +where $K|u^\perp=\operatorname{proj}_{u^\perp}K$. In particular, +$\Pi_{n-1}K=\Pi K$ because $V_{n-1}$ on $u^\perp$ is its +$(n-1)$-dimensional volume. + +\begin{corollary}[Lutwak--Petty lower-degree inequalities] +\label{cor:lutwak-petty} +Let $n\ge3$ and $1\le i\le n-1$ be integers, and let +$K\subset\R^n$ be a convex body with nonempty interior. Then +\begin{equation}\label{eq:lutwak-petty} +\begin{aligned} + \frac{V_{i+1}(\Pi_iK)}{V_{i+1}(K)^i} + &\ge + \frac{V_{i+1}(\Pi_iB_2^n)}{V_{i+1}(B_2^n)^i}\\ + &= + \frac{\left(\displaystyle\binom{n-1}{i} + \frac{\kappa_{n-1}}{\kappa_{n-i-1}}\right)^{i+1}} + {\left(\displaystyle\binom{n}{i+1} + \frac{\kappa_n}{\kappa_{n-i-1}}\right)^{i-1}}. +\end{aligned} +\end{equation} +The lower bound is sharp, since $K=B_2^n$ attains it. +\end{corollary} + +\begin{proof} +Theorem~\ref{thm:main} and the separately proved three-dimensional +result \cite[Theorem~1.1]{ChenEtAl2026} establish Petty's conjecture for +every $n\ge3$. Lutwak's conditional result \cite{Lutwak1990}, in the +explicit formulation of Ortega-Moreno and Schuster +\cite[author version, Introduction, p.~2]{OrtegaMorenoSchuster2021}, +states that Petty's conjecture implies the Euclidean-ball lower bound +for these ratios for every $1\le i\le n-1$. Applying that implication +gives the inequality. For its explicit value, the Steiner normalization +gives +\[ + V_j(B_2^m)=\binom{m}{j}\frac{\kappa_m}{\kappa_{m-j}}, + \qquad + \Pi_iB_2^n=V_i(B_2^{n-1})B_2^n. +\] +The $(i+1)$-homogeneity of $V_{i+1}$ now gives the displayed ball +ratio. Only the inequality conclusion of the conditional result is used. +\end{proof} + +At $i=n-1$, the displayed constant is +$\kappa_{n-1}^{\,n}\kappa_n^{\,2-n}$, exactly the constant in +Equation~\eqref{eq:main}. The case $i=1$ was already established by +Lutwak, as recalled in +\cite[author version, Introduction, p.~2]{OrtegaMorenoSchuster2021}. +Thus the additional range furnished here is $2\le i\le n-2$ for +$n\ge4$. No classification of equality cases is asserted for +Corollary~\ref{cor:lutwak-petty}. + +\subsection{A sharp affine \texorpdfstring{$L^1$}{L1} Sobolev consequence} + +For functions $f_1,\ldots,f_n\in C_c^1(\R^n)$, define the mixed +determinant-gradient integral +\[ + \mathcal D_n(f_1,\ldots,f_n) + =\int_{(\R^n)^n} + \left|\det\bigl(\nabla f_1(x_1),\ldots,\nabla f_n(x_n)\bigr)\right| + \,dx_1\cdots dx_n. +\] +The determinant is the volume of the spanned parallelepiped, with no +factor $1/n!$. Under $f_i(x)\mapsto f_i(Ax+b)$ with $A$ invertible, +this integral and the product of the $L^{n/(n-1)}$ norms below both +scale by $|\det A|^{1-n}$. + +\begin{corollary}[Mixed determinant-gradient and affine Sobolev inequalities] +\label{cor:affine-sobolev} +Let $n\ge3$ be an integer, put $q=n/(n-1)$, and let +$f_1,\ldots,f_n\in C_c^1(\R^n)$ be nonnegative. Then +\begin{equation}\label{eq:mixed-affine-sobolev} + \mathcal D_n(f_1,\ldots,f_n) + \ge n!\,\kappa_{n-1}^{\,n}\kappa_n^{\,2-n} + \prod_{i=1}^n\norm{f_i}_q. +\end{equation} +In particular, every nonnegative $f\in C_c^1(\R^n)$ satisfies the +affine $L^1$ Sobolev inequality +\begin{equation}\label{eq:affine-sobolev} + \mathcal D_n(f,\ldots,f)^{1/n} + \ge (n!)^{1/n}\kappa_{n-1}\kappa_n^{(2-n)/n}\norm{f}_q. +\end{equation} +Both constants are sharp in this class, through smooth approximation +of ellipsoid indicators in $BV(\R^n)$; no equality assertion for nonzero +$C_c^1$ functions is made. +\end{corollary} + +\begin{proof} +Write $c_n=\kappa_{n-1}^{\,n}\kappa_n^{\,2-n}$. +Haddad's geometric functional +$\widetilde I_1(K_1,\ldots,K_n)$ integrates the absolute determinant +against the surface-area measures of the convex bodies $K_i$. +He uses +\[ + h_{\Pi K}(u)=\frac12\int_{S^{n-1}}|\ip{u}{v}|\,dS_K(v), +\] +which is the projection-volume normalization above by Cauchy's +projection formula. His mixed-volume identity and the +Aleksandrov--Fenchel inequality give +\[ + \frac1{n!}\widetilde I_1(K_1,\ldots,K_n) + =V(\Pi K_1,\ldots,\Pi K_n) + \ge\prod_{i=1}^n|\Pi K_i|^{1/n}. +\] +Here $V$ denotes mixed volume, and Haddad's $\omega_m$ is our +unit-ball volume $\kappa_m$; see +\cite[arXiv version, Introduction, p.~5]{Haddad2020}. +Theorem~\ref{thm:main} for $n\ge4$ and the separately proved +three-dimensional case \cite[Theorem~1.1]{ChenEtAl2026} therefore give +\[ + \widetilde I_1(K_1,\ldots,K_n) + \ge n!c_n\prod_{i=1}^n|K_i|^{(n-1)/n}. +\] + +Haddad's Lemma~5.1 and the equivalence of Equations~(18)--(21) +\cite[arXiv version, Section~5, pp.~16--17]{Haddad2020} +transfer this $p=1$ geometric inequality to the functional inequality +for $\mathcal D_n$, with critical exponent $p^*=n/(n-1)=q$. +At this endpoint his normalized weak profile for $K$ is $\mathbf1_K$, +with the same surface-area measure as +$K$ and $\norm{\mathbf1_K}_q=|K|^{(n-1)/n}$; see +\cite[arXiv version, Equation~(9) and Definition~2.2, pp.~8--9]{Haddad2020}. +Consequently the functional and geometric sharp constants agree, as +also follows from the constant relations after his Equation~(21). +This proves Equation~\eqref{eq:mixed-affine-sobolev}; taking all +$f_i=f$ gives Equation~\eqref{eq:affine-sobolev}, the $p=1$ case of +his Equation~(22). + +For sharpness, take nonnegative compactly supported smooth +mollifications $f_j$ of the indicator of an ellipsoid $E$. They converge to +$\mathbf1_E$ strictly in $BV(\R^n)$ and in $L^q$. Define finite measures +$\nu_j$ on the sphere by +\[ + \int\psi\,d\nu_j + =\int_{\{\nabla f_j\ne0\}} + \psi\!\left(-\frac{\nabla f_j}{|\nabla f_j|}\right) + |\nabla f_j|\,dx, + \qquad \psi\in C(S^{n-1}). +\] +Reshetnyak's continuity theorem +\cite[author version, Theorem~1.3, p.~3]{Spector2011} gives +$\nu_j\rightharpoonup S_E$; the minus sign selects the outward normal +in $D\mathbf1_E$. The product measures also converge weakly on the +compact product sphere. Since the absolute determinant is continuous +and homogeneous in each variable, this gives +$\mathcal D_n(f_j,\ldots,f_j)\to +\widetilde I_1(E,\ldots,E)=n!|\Pi E|$. Since +$\norm{\mathbf1_E}_q^n=|E|^{n-1}$, ellipsoid equality in Petty's +inequality gives the stated constants in the limit. This is the weak +$p=1$ interpretation of sharpness. +\end{proof} + +To compare the diagonal statement with Zhang's affine Sobolev +inequality, let $f\ne0$ be as in Corollary~\ref{cor:affine-sobolev} +and define the origin-symmetric convex body $Z_f$ by +\[ + h_{Z_f}(u)=\frac12\int_{\R^n}|u\cdot\nabla f(x)|\,dx. +\] +This integral is positive for every unit vector $u$: otherwise $f$ +would be constant on every line parallel to $u$, contradicting compact +support and $f\ne0$. Thus $Z_f$ has nonempty interior. Haddad's +zonoid identity gives $|Z_f|=\mathcal D_n(f,\ldots,f)/n!$, and his +Blaschke--Santal\'o comparison +\cite[arXiv version, Equations~(24)--(25), p.~17]{Haddad2020} yields +\[ + |Z_f^\circ|^{-1/n} + \ge \left(\frac{\mathcal D_n(f,\ldots,f)}{n!\kappa_n^2}\right)^{1/n} + \ge \frac{\kappa_{n-1}}{\kappa_n}\norm f_q. +\] +The resulting bound on $|Z_f^\circ|$ is Zhang's sharp affine Sobolev +inequality \cite{Zhang1999}. Thus Equation~\eqref{eq:affine-sobolev} +is a stronger affine $L^1$ inequality; the case $f=0$ is immediate. +This application uses only $p=1$ and gives no conclusion for +Haddad's $p>1$ problems. + +\subsection{Holmes--Thompson isoperimetry} + +\begin{corollary}[Holmes--Thompson isoperimetry] +\label{cor:holmes-thompson} +Let $n\ge3$, let $B=-B\subset\R^n$ be the unit ball of a real normed +space, and let $C\subset\R^n$ be a convex body. Write $B^\circ$ for the +polar unit ball and put $I_B=\Pi(B^\circ)/\kappa_{n-1}$. Normalize the +Holmes--Thompson volume and boundary area by +\[ + \mathcal V_B(C)=\frac{|B^\circ|}{\kappa_n}|C|, + \qquad + \mathcal A_B(\partial C)=nV(C[n-1],I_B), +\] +where $V(C[n-1],I_B)$ is mixed volume with $n-1$ copies of $C$, +normalized by $V(C[n])=|C|$. Then +\begin{equation}\label{eq:holmes-thompson} + \mathcal A_B(\partial C)^n + \ge n^n\kappa_n\,\mathcal V_B(C)^{n-1}. +\end{equation} +Equality holds if and only if $B$ is an ellipsoid and $C=x+tB$ for some +$x\in\R^n$ and $t>0$. +\end{corollary} + +\begin{proof} +The fixed-norm isoperimetrix formula and the equivalence with symmetric +unpolarized Petty are classical; see Martini--Mustafaev +\cite[Definition~1(i), Equation~(2), Section~5, and Theorem~12]{MartiniMustafaev2008}. +Minkowski's mixed-volume inequality gives +\[ + \frac{\mathcal A_B(\partial C)^n}{\mathcal V_B(C)^{n-1}} + \ge n^n\frac{\kappa_n^{n-1}}{\kappa_{n-1}^{n}}R_n(B^\circ) + \ge n^n\kappa_n. +\] +The last step applies Theorem~\ref{thm:main} to $B^\circ$ for $n\ge4$, +and the separate three-dimensional result +\cite[Theorem~1.1]{ChenEtAl2026} for $n=3$. Equality in the first step +means that $C$ is homothetic to $I_B$, while equality in the last makes +$B^\circ$, hence $B$, an ellipsoid. Then $I_B$ is homothetic to $B$, +giving exactly the stated equality cases. +\end{proof} diff --git a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1201f9e327d2b018de249f78e710ca66df86e55c --- /dev/null +++ b/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ce25657d3f42aebbc5c6623367c257e46ff79cde62069bebc3324daa282d98e1 +size 298355 diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/README.md b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..a97aacbd9a8a806343a00d4ceeb9e1a695eaf6d8 --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/README.md @@ -0,0 +1,20 @@ +# [Planar Graph Metrics Embed into $L_1$ with Constant Distortion](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +To cite this manuscript: + +```bibtex +@misc{OAI:Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026, + author = {{OpenAI}}, + title = {{Planar Graph Metrics Embed into $L_1$ with Constant Distortion}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf}{OAI:Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/column-realization.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/column-realization.tex new file mode 100644 index 0000000000000000000000000000000000000000..901ae7168b335621d8a90b20ddb20b1612e4499c --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/column-realization.tex @@ -0,0 +1,49 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,font=\small, + vertex/.style={circle,fill=black,inner sep=1.7pt}, + twig/.style={blue!65!black,line width=1.1pt}, + identify/.style={blue!65!black,densely dashed,line width=.8pt}] + \node[vertex,label=below:$o$] (o) at (0,0) {}; + \node[vertex,label=left:$u$] (u) at (-1.35,1.55) {}; + \node[vertex,label=right:$v$] (v) at (1.35,2.3) {}; + \draw[thick] (o)--node[left] {$2$} (u); + \draw[thick] (o)--node[right] {$3$} (v); + \draw[twig] (u)--node[above] {$\ell_e=3$} (v); + \node[align=center] at (0,-1.05) {(a) A weighted graph\\$h(u)=2,\quad h(v)=3$}; + + \draw[->,thick] (2.05,1.6)--(3,1.6); + + \begin{scope}[shift={(4.2,0)},y=.76cm] + \draw[gray,densely dotted] (0,-.45)--(0,0); + \draw[gray,densely dotted] (2.7,-.45)--(2.7,0); + \draw[thick] (0,0)--(0,2); + \draw[thick] (2.7,0)--(2.7,3); + \draw[twig] (0,2)--node[left] {$2$} (0,4); + \draw[twig] (2.7,3)--node[right] {$1$} (2.7,4); + \draw[gray,densely dotted] (0,4)--(0,4.5); + \draw[gray,densely dotted] (2.7,4)--(2.7,4.5); + \draw[identify] (0,0)--node[below] {identify at $h=0$} (2.7,0); + \draw[identify] (0,4)--node[above] {identify at $H_e=4$} (2.7,4); + \node[vertex,label=left:$o$] at (0,0) {}; + \node[vertex,label=right:$o$] at (2.7,0) {}; + \node[vertex,label=left:$u$] at (0,2) {}; + \node[vertex,label=right:$v$] at (2.7,3) {}; + \node[vertex] at (0,4) {}; + \node[vertex] at (2.7,4) {}; + \draw[->,gray] (3.8,-.05)--(3.8,4.5) node[above,black] {$h$}; + \foreach \h in {0,2,3,4} { + \draw[gray] (3.73,\h)--(3.87,\h) node[right,black] {$\h$}; + } + \end{scope} + \node[align=center] at (5.55,-1.05) {(b) Two columns with zero-cost switches\\Only vertical motion contributes length}; +\end{tikzpicture} +\caption{An exact example of Proposition~\ref{geo:realization}. +The black edges form a shortest-path tree rooted at $o$. +The non-tree edge is replaced by two ascending twigs of lengths +$(\ell_e+h(v)-h(u))/2=2$ and $(\ell_e+h(u)-h(v))/2=1$. +Their tips have the same height $H_e=4$ and are identified. +Dashed horizontal segments denote identifications, not extra edges; +the dotted vertical tails do not shorten original distances.} +\label{fig:column-realization} +\end{figure} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/surviving-seeds.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/surviving-seeds.tex new file mode 100644 index 0000000000000000000000000000000000000000..90b1ffb1715d9f7cc0b6c26768d184b116b3e3fe --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/surviving-seeds.tex @@ -0,0 +1,67 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,font=\small, + vertex/.style={circle,fill=black,inner sep=1.6pt}, + seed/.style={circle,draw=black,fill=white,inner sep=2pt}, + low/.style={circle,fill=gray!70,inner sep=2.2pt}] + \draw[gray!45] (0,0) circle (2.2); + \draw[gray!45,densely dashed,line width=2pt] (310:2.36) arc (310:590:2.36); + \node[align=center,fill=white,inner sep=2pt] at (0,2.72) {one circular gap of $S$}; + \coordinate (z) at (90:2.2); + \coordinate (wm) at (170:2.2); + \coordinate (q) at (190:2.2); + \coordinate (wp) at (10:2.2); + \coordinate (a) at (65:2.2); + \coordinate (b) at (25:2.2); + \coordinate (w) at (45:2.2); + \coordinate (partner) at (210:2.2); + \coordinate (s1) at (230:2.2); + \coordinate (s2) at (270:2.2); + \coordinate (s3) at (310:2.2); + \draw[gray!70,line width=1.4pt] (s1)--(s2)--(s3); + \draw[blue!65!black,line width=1.05pt] (z)--node[left,pos=.62] {$P_-$} (q)--(wm); + \draw[blue!65!black,line width=1.05pt] (z)--(a)--(b)--(wp); + \node[blue!65!black] at (.45,1.85) {$P_+$}; + \draw[orange!75!black,dashed,line width=1.05pt] (w)--(a); + \node[vertex,label=above:$z$] at (z) {}; + \node[seed,label=left:$w_-$] at (wm) {}; + \node[vertex] at (q) {}; + \node[seed,label=right:$w_+$] at (wp) {}; + \node[vertex,label=above right:$a$] at (a) {}; + \node[vertex,label=right:$b$] at (b) {}; + \node[seed,label=above right:$w$] at (w) {}; + \node[seed,label=below left:$w'$] at (partner) {}; + \foreach \s in {s1,s2,s3} {\node[low] at (\s) {};} + \node[gray!80!black,below] at (0,-2.48) {connected low set $S$ in $G_I$}; + + \begin{scope}[shift={(5.25,.05)}] + \node[align=center] at (.9,2.15) {At the skipped link $ab$}; + \draw[thick] (0,-.7)--(0,1.45); + \draw[thick] (2.1,-.7)--(2.1,1.45); + \node[above] at (0,1.5) {column $a$}; + \node[above] at (2.1,1.5) {column $b$}; + \draw[blue!65!black,densely dashed,line width=.9pt] (0,1.05)--(2.1,1.05); + \node[above,fill=white,inner sep=1pt] at (1.05,1.05) {tip $e$ at $H_e$}; + \node[vertex] at (0,1.05) {}; + \node[vertex] at (2.1,1.05) {}; + \node[vertex,label=left:$v$] at (0,-.1) {}; + \node[seed,label=right:$w$] (wi) at (1.4,-.65) {}; + \draw[orange!75!black,dashed,line width=1.05pt] (wi) to[bend left=18] node[below left,pos=.6] {$p$} (0,-.1); + \draw[<->,gray] (-.52,-.1)--node[left,black] {$\le a_1$} (-.52,1.05); + \node[align=center] at (.8,-1.35) {$p\le j+\epsilon+a_1$\\[3pt]$d_{J_1}(w,e)\le j+\epsilon+2a_1$}; + \node[align=center,font=\footnotesize] at (.8,-2.25) {Horizontal switch: zero cost.\\Inset not to scale.}; + \end{scope} +\end{tikzpicture} +\caption{The key step in Proposition~\ref{pack:bound}, shown for a +surviving seed $w$. In a gap of the connected low set $S$, an extreme +path skips the column of $w$ by the terminal link $ab$. The witness +partner $w'$ is outside the small ball and lies strictly outside the +closed interval $[a,b]$. +The descending path from $w$ must first reach $a$ or $b$; call that point +$v$. Only this first prefix is shown in orange dashes. +Survival of the low-tip test gives $s(e)\ge s(w)-j$, forcing the +short prefix estimated in the inset. Blue paths use $G_{J_2}$; connections +of $S$ may use the larger graph $G_I$. The circular drawing records column +order, not metric lengths; extreme paths need not be monotone in that order.} +\label{fig:surviving-seeds} +\end{figure} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/two-portals.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/two-portals.tex new file mode 100644 index 0000000000000000000000000000000000000000..0973f672aa7e8a0e785503b2f92c448e21e3a2d8 --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/figures/two-portals.tex @@ -0,0 +1,39 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,font=\small, + vertex/.style={circle,fill=black,inner sep=1.6pt}, + interior/.style={circle,fill=blue!65!black,inner sep=1.8pt}] + \draw[fill=gray!10,rounded corners=8pt] (-.32,-1.45) rectangle (.32,1.45); + \draw[fill=gray!10,rounded corners=8pt] (3.28,-1.45) rectangle (3.92,1.45); + \draw[thick] (0,-1.1)--(0,1.1); + \draw[thick] (3.6,-1.1)--(3.6,1.1); + \foreach \y in {-1.1,0,1.1} { + \draw[blue!65!black,thick] (0,\y)--(3.6,\y); + \node[vertex] at (0,\y) {}; + \node[vertex] at (3.6,\y) {}; + \node[interior] at (1.2,\y) {}; + \node[interior] at (2.4,\y) {}; + } + \node at (0,1.8) {$C$}; + \node at (3.6,1.8) {$L$}; + \node[align=center] at (1.8,-2) {Three disjoint connectors\\with nonempty interiors}; + \draw[->,thick] (4.45,0)--(5.45,0) node[midway,above] {contract}; + + \node[vertex,label=left:$C$] (C) at (6.25,0) {}; + \node[vertex,label=right:$L$] (L) at (9.15,0) {}; + \foreach \y/\k in {1.2/1,0/2,-1.2/3} { + \node[interior,label=above:$q_\k$] (q\k) at (7.7,\y) {}; + \draw[blue!65!black,thick] (C)--(q\k)--(L); + } + \node at (7.7,-2) {$K_{2,3}$: a forbidden minor}; +\end{tikzpicture} +\caption{The hypothetical obstruction in Lemma~\ref{geo:portals}. +Three sufficiently separated entrance points would give vertex-disjoint +paths from the connected high set $C$ to the connected low set $L$. +After trimming, each connector has a nonempty interior avoiding both +sets. Contracting $C$, $L$, and the three interiors gives $K_{2,3}$, +which cannot be a minor of the outerplanar column graph. +The drawing records this contraction; it is not an outerplanar drawing +of a configuration that the column model permits.} +\label{fig:two-portals} +\end{figure} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/main.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..fbd7835bb4730a01d7e448eb045a2da883795c8a --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/main.tex @@ -0,0 +1,51 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern,microtype} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{enumitem,booktabs} +\usepackage{xcolor,tikz} +\usetikzlibrary{arrows.meta,calc,positioning,fit,decorations.pathreplacing} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[nameinlink,capitalise,noabbrev]{cleveref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\1}{\mathbf 1} +\DeclareMathOperator{\Lip}{Lip} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\setlist{itemsep=3pt,topsep=5pt} +\hypersetup{pdftitle={Planar Graph Metrics Embed into L1 with Constant Distortion},pdfauthor={OpenAI}} +\title{Planar Graph Metrics Embed into $L_1$\\with Constant Distortion} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that every finite connected planar graph with arbitrary positive +real edge lengths embeds into real $L_1$ with a universal distortion bound. +This resolves the planar embedding conjecture positively. +\end{abstract} +\tableofcontents + +\input{sections/introduction} +\input{sections/geometry} +\input{sections/sets} +\input{sections/interpolation} +\input{sections/scale-setup} +\input{sections/main-packing} +\input{sections/ordered} +\input{sections/secondary} +\input{sections/assembly} +\input{sections/consequences} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/references.bib b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..1600bea88b35341f23cc7fe7d907843a99981ee8 --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/references.bib @@ -0,0 +1,223 @@ +@article{LLR95, + author = {Linial, Nathan and London, Eran and Rabinovich, Yuri}, + title = {The geometry of graphs and some of its algorithmic applications}, + journal = {Combinatorica}, + volume = {15}, + number = {2}, + pages = {215--245}, + year = {1995}, + doi = {10.1007/BF01200757}, + url = {https://link.springer.com/article/10.1007/BF01200757} +} + +@article{GNRS04, + author = {Gupta, Anupam and Newman, Ilan and Rabinovich, Yuri and Sinclair, Alistair}, + title = {Cuts, Trees and {$\ell_1$}-Embeddings of Graphs}, + journal = {Combinatorica}, + volume = {24}, + number = {2}, + pages = {233--269}, + year = {2004}, + doi = {10.1007/s00493-004-0015-x}, + url = {https://link.springer.com/article/10.1007/s00493-004-0015-x}, + note = {Preliminary version in FOCS 1999, pp. 399--408} +} + +@inproceedings{Rao99, + author = {Rao, Satish}, + title 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Two estimates let us use that chance in a complete +schedule: coarser scales rarely destroy the common charts required by the +test, and finer scales move too little label mass to erase its certificate. +We first collect these estimates, then choose the scale spacing and write +the resulting finite cut embedding. + +\subsection{The one-step estimates} +All constants in this section depend only on the fixed choices in +Section~\ref{sets:parameters}. In particular they are independent of the +number of columns, band components, chart identifiers, and scales. + +\begin{proposition}[Movement and success]\label{pad:success} +There are constants $M<\infty$ and $b,p>0$ such that the following hold +at every step of scale $r$. +\begin{enumerate}[label=(\roman*)] +\item At every point of $X$, in either system, the symmetric-difference +movement of the unrefined set is at most $Mr$. +\item Given any past for which a vertex pair at distance in $[r,2r)$ is +eligible, the step has probability at least $p$ of producing one of these +certificates: +\begin{enumerate}[label=(\alph*)] +\item unrefined symmetric-difference separation at least $br$ in one +system; +\item a non-neighbor tip $e$ and a set of labels of measure at least $br$ +in system $A$ on which both pair vertices are active and $e$ is inactive, +with the vertices strictly on opposite sides of that link. +\end{enumerate} +\end{enumerate} +\end{proposition} +\begin{proof} +Lemma~\ref{main:validity} bounds movement in the main clipping. +Secondary movement is bounded by Lemma~\ref{sec:validity}; increments are bounded by +Lemma~\ref{sec:increment-bound}. These estimates apply even at points +outside the finite vertex list. There are no simultaneous main updates +at a point except at cell overlaps, where every update is unchanged. +Secondary updates take a maximum, whose movement is bounded without +summing over groups. Taking the largest of the bounds gives $M$. + +Work in scale units. If one of the cutoffs in +\eqref{sets:cutoffs} fails, no change preserves separation at least the +minimum of $10^{-4}a_2,\kappa_AE_A,\kappa_B10^{-4}R$. +Otherwise the interior requirements hold with probability at least $1/4$. +Conditional on the bands and cells, a missing descending path is handled +by Corollary~\ref{main:pair-separation}: keep that endpoint's cell and suppress the other, +giving positive separation with probability at least $1/4$ conditional +on the main mode. If both paths exist, an excluded pair has the component +certificate on a label set of measure at least $\kappa_Aj/4$, which the +no-change mode preserves. + +For a surviving pair, admission at an endpoint is handled by +Lemma~\ref{sec:admitted-success}, including its no-change alternative. +If neither orientation admits either endpoint, apply +Lemma~\ref{sec:remaining}. Each case uses one of four modes, at most one +fair orientation, and at most $2M_{\mathrm{pack}}+1$ prescribed group coins. It follows, +for example, that $p=2^{-(2M_{\mathrm{pack}}+10)}$ is a permissible lower bound after +all interior and mode probabilities. Choose $b>0$ below all the finitely +many displayed gains and certificate masses. These choices depend only +on the fixed parameters. +\end{proof} + +\subsection{The probability of splitting a small ball} +The following estimate concerns chart identifiers, not cut components. +System $A$ is refined only after all the updates. + +\begin{proposition}[Chart persistence]\label{pad:persistence} +There exist constants $C_0<\infty$ and $\rho_0>0$ such that the following +holds. Condition on the past at a step of scale $r$. Suppose a fixed +closed ball $B_d(v,\rho r)$ lies in a single old region in each system. +For $0<\rho\le\rho_0$, the probability that this ball is split between +different new regions in either system is at most +\begin{equation}\label{pad:loss} + C_0(\rho+\eta_0). +\end{equation} +Here $\eta_0$ is the relative mesh allowance from +Section~\ref{sets:setup}. +\end{proposition} +\begin{proof} +Normalize $r=1$ and write $B=B_d(v,\rho)$. Every point of $B$ has +a shortest path to $v$ contained in $B$. Joining two such paths gives +a path of length at most $2\rho$ between any two points of $B$. +We will use these paths for all band-distance comparisons. + +There are two ways an update could give different chart identifiers on +$B$: its domain could end inside the ball, or different formulas could +win there. Strict margins at domain boundaries prevent the first kind +of change. The comparison estimate of Corollary~\ref{int:comparisons} +controls the second. + +\paragraph{Height bands.} +The probability that an endpoint of either random height tiling lies +within $\rho$ of $h(v)$ is $O(\rho+\eta_0)$. Exclude this event and +condition on the tilings. The ball lies wholly in one component of a +broad band and one component of a narrow band: all its center paths +are paths in those bands. +For clipping, each such band is either processed in its entirety or left unchanged. +A narrow band not contained in a broad band has no secondary seeds, +so secondary clipping also leaves it unchanged. + +\paragraph{Main clipping.} +Condition further on the radial cells and their coins. Suppose first +that $B$ meets more than one width-$D$ cell, including through different +representatives of a quotient point. For any touched cell $C$ and any +$w\in B\cap C$, a path through the center to a point in another cell +reaches a frontier point $z\in B\cap\partial C$ within length $2\rho$. +If $C$ is available, its old comparison field is the common old field +on $B$. Thus +\[ + F_C(w)\le s(z)+(1-\zeta_A)d_{J_1}(w,z)-3E_A + \le s(w)+4\rho-3E_A. +\] +For $\rho_00$. +\end{corollary} + +\begin{proof} +The singleton case is immediate. Otherwise, replace each input edge length +by the distance between its endpoints; this preserves the metric and gives +reduced positive lengths. Lee--Sidiropoulos Theorem~1.7 stochastically embeds +this metric with distortion at most $D$ into finite $1$-sums of planar graphs +and graphs of treewidth at most $h_0$. Since the companion indexes $C_t$ by +bag cardinality, enlarging $h_0$ to $\widehat h$ gives the applicable constant +$C_{\widehat h+1}$. Put $M=\max\{C_{\mathrm{pl}},C_{\widehat h+1}\}$. + +For a connected target piece with nonnegative edge lengths, add $\varepsilon>0$ +to every edge and apply Theorem~\ref{thm:main} or the companion theorem. +As $\varepsilon\downarrow0$, the shortest-path matrices converge, so the +corresponding $L_1$ distance matrices have a subsequential limit in the closed +finite cut cone, retaining exactly the bound $M$ and identifying zero-distance +pairs. The finite expected expansion +ensures that all images of the finite input lie in a single finite-distance +target component almost surely. Lemma~1.5 of~\cite{LS09} retains the bound +$M$ on finite $1$-sums, and its Lemma~1.2 composes this bound with the +stochastic distortion $D$, giving $K=DM$. Applying the same cut-cone limit +to input lengths, followed by the fixed-graph equality +of~\cite[Section~3, Theorem~3.2]{GNRS04}, gives the flow--cut conclusion. +\end{proof} + +This corollary concerns only the cited fixed-parameter almost-embeddable +families. It does not establish closure under general clique sums of order +greater than one, and therefore does not prove the full GNRS conjecture. diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/geometry.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..c60f60fef464a3409308460d0503a5c8b9491336 --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/geometry.tex @@ -0,0 +1,452 @@ +\section{Columns and thin bands}\label{geo:section} + +We first replace a plane graph by a family of vertical lines. This separates +the length of a path from the planar restriction on its possible switches. +The replacement preserves the original metric exactly. We then work in +a thin height band and prove two estimates for the later local updates: +entrances to a connected outer region lie near at most two fixed points, +and within each radial layer, connected groups of columns admit short +joining paths in the band. +All these estimates use the width of the band and the circular order of +the columns, with constants independent of their number. + +\begin{definition}\label{geo:column-model} +A \emph{column model} consists of a finite cyclically ordered set $V_0$ of +indices, a copy $\{i\}\times\R$ of the height line for each $i\in V_0$, and +a finite collection of links. A link $ij$ has a prescribed closed bounded +set $S_{ij}\subset\R$ of switch heights. At $H\in S_{ij}$ we identify +$(i,H)$ with $(j,H)$. The resulting quotient is denoted by $X$, and its +height function by $h$. Paths consist of finitely many vertical segments +and permitted switches; their length is the sum of the lengths of the +vertical segments. The distance $d$ is the infimum of these lengths. + +The links are \emph{noncrossing} if no two have four distinct endpoints +alternating in the cyclic order. We require all column models below to have +noncrossing links. We additionally require that every link +between nonconsecutive indices is a switch at a single height $H_e$, and +that neither endpoint column has any switch above $H_e$. Such a switch +will be called a \emph{terminal switch}. +Its quotient point is called the \emph{tip} of the link. + +For a closed interval $I\subset\R$, possibly unbounded, write $X_I=h^{-1}(I)$. +The intrinsic distance $d_I$ only permits paths with heights in $I$; +different components have distance $+\infty$. The active column graph +$G_I$ has vertex set $V_0$ and the links whose switch sets meet $I$ as edges. +\end{definition} + +We write $B_{d_I}(x,r)$ for the closed ball and +$d_I(x,S)=\inf_{z\in S}d_I(x,z)$ for distance to a set, with +$d_I(x,\varnothing)=+\infty$. The same conventions apply to $d$. + +\begin{lemma}[Realization and shortest paths]\label{geo:geodesics} +The function $d_I$ is a metric on each component of $X_I$. Its finite +distances are attained by paths using each column at most once. Components +of $X_I$ correspond exactly to components of $G_I$. If $I$ is bounded, each +component of $X_I$ is compact. For arbitrary $I$, bounded closed subsets of +each component are compact. In particular distance from a point to a +nonempty compact set is attained. +\end{lemma} + +\begin{proof} +If a path leaves a column and later returns to it, replace the intervening +part by the vertical segment joining its two visits. This does not increase +length: total vertical variation is at least the difference of the two +heights. The replacement stays in $I$. Repeating the operation leaves +a path with no repeated column and at most $|V_0|-1$ switches. + +There are finitely many possible such column words. For a fixed word, its +switch heights belong to a product of closed bounded sets $S_{ij}\cap I$. +The length, including the initial and final vertical segments, is a +continuous function of these heights. Thus every feasible word attains +its minimum, and minimizing over the finitely many words proves attainment. +A path of length zero has all its switches at the common endpoint height, +so its endpoints are already identified in the quotient. This proves +positive definiteness. The other metric axioms follow by reversing and +concatenating paths. + +A permitted path projects to a walk in $G_I$. Conversely any graph walk +can be realized by choosing an available height for each edge and moving +vertically between consecutive switches. This proves the component claim. +The quotient map from any bounded closed piece of a column to $X_I$ is +$1$-Lipschitz and hence continuous. Finitely many compact column pieces +therefore have compact image. Finally $|h(x)-h(y)|\le d_I(x,y)$, so a bounded +metric set has bounded heights and is contained in such a compact image. +A closed subset of that image is compact. +\end{proof} + +\begin{proposition}[Planar graphs admit column models]\label{geo:realization} +Every finite connected planar graph with positive edge lengths has an +isometric realization of its geometric edge metric in a noncrossing column +model satisfying the terminal-switch condition. The added parts of the +model consist of separate infinite tails and do not shorten distances in +the original graph. +\end{proposition} + +\begin{proof} +Fix a plane embedding and a root $o$. Let $T$ be a shortest-path spanning +tree and put $h(v)=d_G(o,v)$. Every tree edge is monotone in height and has length +equal to the difference of its endpoint heights. + +For each non-tree edge $e=uv$ of length $\ell_e$, attach a new branch to +$u$ of length +\[ + \frac{\ell_e+h(v)-h(u)}2 +\] +and a separate new branch to $v$ of length +\[ + \frac{\ell_e+h(u)-h(v)}2. +\] +Both lengths are nonnegative because $|h(u)-h(v)|\le\ell_e$. Their tips +have the common height +\[ + H_e=\frac{\ell_e+h(u)+h(v)}2. +\] +The enlarged abstract tree is denoted by $\widetilde T$. Its branches may +have zero metric length; retain their incidences and their plane order +until taking the metric quotient. Identify the two tips belonging to each +non-tree edge. This replaces that edge by two intervals of total length +$\ell_e$, with no additional identifications, and hence recovers the +geometric graph metric. + +Take one column for each root-to-leaf path of $\widetilde T$. Place these +columns in the cyclic contour order of its plane leaves. For neighboring +leaves include a link on the heights of their common root prefix. These +links make exactly the identifications prescribed by the tree. Indeed, +at each tree point the descendant leaves form a consecutive block; adjacent +leaves in that block identify all of its copies. Conversely neighboring +leaves cease to be linked when their tree paths diverge. + +For completeness, the relevant cyclic order can be obtained by taking a +small closed disk neighborhood of a topological drawing of +$\widetilde T$. Draw each new branch as an initial portion of its +corresponding non-tree edge. Zero metric lengths do not prevent drawing +these portions as distinct topological arcs. The remaining portions of +the non-tree edges are mutually disjoint arcs in the complementary disk +on the sphere. Their endpoints on the boundary are the paired leaf tips. +Two disjoint arcs in a disk cannot have alternating endpoints: one arc, +together with either boundary arc joining its endpoints, separates the +disk, and a second arc with alternating endpoints would have to cross it. +Thus the tip links are noncrossing. The common-prefix links join cyclic +neighbors and cannot cross any link. + +Identify each pair of matching tips by its switch at $H_e$. A leaf column +has no common-prefix switch above its own tip and belongs to no other tip +pair. Consequently every non-neighbor tip link satisfies the stated +terminal condition. Extend every column below the root and above its +leaf by its own separate tail. These tails have no additional switches. +An excursion into a tail must return through its point of attachment and +can be deleted. It therefore cannot shorten any distance in the original +geometric graph. The degenerate singleton case can be treated directly; +a one-column tree is simply a path. +\end{proof} + +If the resulting model has one column, height embeds the original graph +metric isometrically into $\R$; the singleton case is immediate as well. +We henceforth assume that the column model has at least two columns. + +\input{figures/column-realization} + +\subsection{Consequences of the circular order} + +The circular order gives separation statements used later for ordered +fields and the seed packing argument. These statements concern the column +graph, so they hold simultaneously for all height bands. Its outerplanarity +will also give the two-portal estimate in the next subsection. + +\begin{lemma}[Chord separation]\label{geo:chord-separation} +Let $uv$ be an edge of a noncrossing circular graph. Any graph path from +the strict interior of one circular $u$--$v$ arc to the strict interior of +the other visits $u$ or $v$. +\end{lemma} + +\begin{proof} +A path avoiding $u,v$ and changing arcs has an edge whose endpoints lie +in the two strict interiors. Its endpoints alternate with $u,v$, which +contradicts noncrossing. +\end{proof} + +\begin{lemma}[Gaps of a connected set]\label{geo:gap} +Let $S$ be a nonempty connected vertex set in a noncrossing circular graph. +Every graph path avoiding $S$ lies in one of the open circular gaps between +consecutive vertices of $S$. Here the complement of a singleton is regarded +as one gap. Connectivity of $S$ and the path may be witnessed using +different subgraphs of the same noncrossing graph. +\end{lemma} + +\begin{proof} +Suppose an edge $uv$ of the path joined distinct gaps. Then each strict +circular $u$--$v$ arc contains a vertex of $S$. A path inside $S$ joining +these two vertices would have to visit $u$ or $v$ by +Lemma~\ref{geo:chord-separation}. Both endpoints lie outside $S$, a +contradiction. Thus each edge stays in one gap, and so does the whole path. +\end{proof} + +\begin{lemma}[Skipping a column]\label{geo:skipping} +Give an open circular arc its linear order, and assume its complementary +closed arc contains a column index. Let a vertex walk stay in the open +arc, and let an index $i$ lie strictly between its initial and final indices. +Either the walk visits $i$, or one of its edges has endpoints strictly on +opposite sides of $i$. In the latter case the edge is a non-neighbor link. +If a second path starts strictly between the endpoints of this edge and +leaves that strict interval, it visits an endpoint of the edge. +\end{lemma} + +\begin{proof} +If the walk avoids $i$, take its first edge changing from one side of $i$ +to the other. The index $i$ lies on one strict circular arc between its +endpoints, and an index in the complementary closed arc lies on the other. +They are therefore not cyclic neighbors. The final assertion is +Lemma~\ref{geo:chord-separation}. +\end{proof} + +\begin{lemma}[Outerplanarity]\label{geo:outerplanar} +Every active graph $G_I$ is outerplanar and has no $K_{2,3}$ minor. +\end{lemma} + +\begin{proof} +Place the indices on a circle and draw each edge as a straight chord. +The noncrossing condition makes this an outerplanar drawing. Add a new +vertex in the complementary disk on the sphere and join it to every +column vertex by a disjoint radial star. The resulting cone graph is +planar. If $G_I$ had a $K_{2,3}$ minor, perform its deletions and contractions +while retaining the new vertex and one edge from it to every contracted +branch set. The resulting planar minor contains the cone over $K_{2,3}$. +That cone contains $K_{3,3}$: add the cone vertex to the part of size two. +This is impossible, since a simple bipartite planar graph on six vertices +has at most eight edges, whereas $K_{3,3}$ has nine. +\end{proof} + +\begin{lemma}[Connected distance sublevels]\label{geo:sublevel} +Fix a component of $X_I$, a source $o$ in it, and a height $H\in I$. +Put $q_i=d_I(o,(i,H))$, taking $q_i=+\infty$ outside that component. +For every real $t$, the nonempty set $\{i:q_i\le t\}$ is connected in +$G_I$. +\end{lemma} + +\begin{proof} +For an index $i$ in the set, take a shortest path from $o$ to $(i,H)$. +If it visits a column $j$ at height $H'$, stop there and move vertically +to $(j,H)$. The cost of this new ending is $|H-H'|$, at most the vertical +variation of the discarded suffix. Thus $q_j\le q_i\le t$. The projected +path therefore lies in the sublevel set and joins $i$ to the source column. +Taking these paths for all $i$ proves connectivity. This argument also +shows that every source-column representative belongs to the sublevel +whenever the sublevel is nonempty. +\end{proof} + +\subsection{Portals and radial cells} + +We will cover the entrances to a connected radial superlevel set by at +most two controlled balls. This estimate will then bound distances +within the connected pieces of each radial layer. + +In this subsection $J$ is a closed band of width $a>0$, $H_*$ is its +midheight, and we work in a fixed component of $X_J$. Choose a basepoint +$o$ and define +\[ + b_i=d_J(o,(i,H_*)). +\] +For a point $v=(i,H)$ in the band, +\begin{equation}\label{geo:radial-estimates} + \bigl|d_J(o,v)-b_i\bigr|\le a/2, + \qquad |b_i-b_j|\le a\quad\text{if }ij\in G_J. +\end{equation} +The second estimate follows by traveling from the two midpoints to an +available switch, at cost at most $a$. We occasionally use the weaker +bound $2a$ without further comment. + +\begin{lemma}[Two portals]\label{geo:portals} +Let $C$ be a component of the subgraph of $G_J$ induced by $\{i:b_i\ge t\}$. +An \emph{entrance column} of $C$ is an index in $C$ adjacent to an index +with radial value below $t$. If entrance columns exist, there are at most +two of their midpoints, called portals, such that every point on every +entrance column lies within $200a$ in $d_J$ of one of the portals. +Every portal has radial value in $[t,t+a)$. If $t>a/2$ and $C$ is nonempty, +entrance columns exist. +\end{lemma} + +\begin{proof} +An entrance column has $t\le b_ia$, while adjacent columns differ by at most $a$. +Contract the connected sets $C$ and $L$, and contract each of the three +nonempty connector interiors to a separate vertex. This gives a +$K_{2,3}$ minor, contrary to Lemma~\ref{geo:outerplanar}. + +A maximal set of entrance midpoints at mutual distances greater than +$100a$ thus has cardinality at most two. Maximality covers every entrance +midpoint within $100a$, and the extra vertical distance is at most $a/2$. +This proves the stated $200a$ bound. Finally a representative of $o$ has +radial value at most $a/2$. If $t>a/2$, any nonempty high component is a +proper subset of the connected graph and has an entrance. +\end{proof} + +\input{figures/two-portals} + +\begin{definition}[Radial cells]\label{geo:cell-definition} +Fix $D>0$ and an offset $\theta\in[0,D)$. Partition the indices into +layers by assigning $i$ to the unique half-open interval +\[ + [\theta+nD,\theta+(n+1)D),\qquad n\in\mathbb Z, +\] +containing $b_i$. Split each layer into the connected components of its +induced subgraph of $G_J$. A \emph{cell} is the image in $X_J$ of all +intervals $\{i\}\times J$ belonging to one such component. Its +\emph{frontier} $\partial C$ is the union of their images for indices +adjacent in $G_J$ to an index outside the cell. +\end{definition} + +This is a partition of the column indices, not necessarily a disjoint +partition of their quotient images. If two cells share a quotient point, +that point belongs to the frontier of each: a switch chain identifying +representatives in the two cells has a first edge leaving either cell. +Consequently formulas that agree with the unchanged data on all frontiers +give unambiguous values at such shared points. + +Here a \emph{weak diameter} bound measures distances in the whole band; +the paths witnessing those distances may leave the cell. + +\begin{lemma}[Diameter of cells]\label{geo:cells} +Assume $a\le D/1000$. Every radial cell is compact and path-connected, +and has weak diameter at most $10D$ in $d_J$. Its frontier is compact, +possibly empty. +\end{lemma} + +\begin{proof} +Each cell is the image of finitely many closed column intervals. Its +column graph is connected, which makes that image path-connected. The +same finite-union argument proves compactness of the cell and frontier. + +Let $q$ be the lower threshold of its layer. If $q\le30a$, every point +of the cell has distance less than $D+31a$ from $o$, so the diameter bound +follows. Suppose $q>30a$ and let $C^+$ be the component of $\{b_i\ge q\}$ +containing the cell. It has at most two entrance portals by +Lemma~\ref{geo:portals}. + +Take $v$ in the cell and follow a shortest path to $o$ until the first +switch below threshold $q$. Before that switch the projected path stays +in $C^+$: two adjacent high columns belong to the same high component. +The last high column is an entrance, and the switch point has radial +distance at least $q-a/2$. Since $d_J(o,v)\rho+2a, +\end{equation} +then every point reachable from $v$ by a band path of length at most +$\rho$ has all its representatives in the same cell as $i$. Moreover +$d_J(v,\partial C)>\rho$ for that cell. + +For uniform $\theta\in[0,D)$, the probability that +\eqref{geo:padding-condition} fails is at most +\[ + \min\{1,\,2(\rho+2a)/D\}. +\] +There is no factor for the number of representatives of $v$: their +radial values differ pairwise by at most $a$. + +In particular, if $a\le D/1000$ and the radial values of two points are +at distance greater than $0.02D$ from the thresholds, then their distances +to their respective frontiers are greater than $0.01D$. If they belong +to different cells, their mutual $d_J$-distance is greater than $0.01D$. +\end{lemma} + +\begin{proof} +If a band path from $v$ of length at most $\rho$ visits column $k$, join +its visit point to $(k,H_*)$ and join $v$ to $(i,H_*)$. The triangle +inequality gives +\[ + |b_k-b_i|\le\rho+a. +\] +Thus every visited column stays in $i$'s layer under +\eqref{geo:padding-condition}; its projected path puts it in the same +connected component of that layer. The same argument applies after any +zero-cost switch chain at an endpoint, proving the assertion about all +representatives. + +Every frontier column has its radial value within $a$ of one of the +cell's layer thresholds: its neighbor outside the cell cannot belong +to the same layer. If $z$ lies on a frontier column $k$, then +\[ + d_J(v,z)\ge |b_i-b_k|-a + \ge \dist(b_i,T)-2a>\rho. +\] +The formula remains true for an empty frontier with distance $+\infty$. +The offset estimate is the length, at most $2(\rho+2a)$, excluded in one +period. Two representatives at height $H$ have midpoint distance at +most $2|H-H_*|\le a$, since their points at height $H$ are identical. +This proves the representative assertion without a union bound. + +For the numerical conclusion use +$0.02D-2a\ge0.018D>0.01D$. A path to a different cell must first visit +a frontier column of the initial cell, so the same lower bound applies +to its length. +\end{proof} + +The cell estimates will control the regions where the local updates act. +For interpolation we also need the entrance estimate at successive radial +thresholds. The resulting components form a rooted tree, with controlled +distances between the portals at successive levels. + +\begin{lemma}[Successive portal levels]\label{geo:portal-levels} +Fix a step $\delta>1000a$. At depth $n\ge1$ take the components of +$\{b_i\ge n\delta\}$ and give each its entrance portals from +Lemma~\ref{geo:portals}. The depth-zero node is the whole component, +with its sole portal $o$. Each positive-depth node has a unique parent. +For a node at depth $n$, every point on one of its columns with +$b_i<(n+1)\delta$, and every portal of one of its children, is within +$\delta+220a$ of one of its own portals. Each child portal is at distance +at least $\delta-10a$ from every own portal. +\end{lemma} + +\begin{proof} +A connected component at a higher threshold is contained in a unique +component at the preceding threshold, proving the parent assertion. +For $n\ge1$, repeat the root-geodesic argument in +Lemma~\ref{geo:cells} at lower threshold $n\delta$. A point of the +node's own layer has radial distance below $(n+1)\delta+a/2$. A child +portal has radial distance in +$[(n+1)\delta,(n+1)\delta+a)$. In either case its geodesic reaches an +entrance of the parent node after length at most $\delta+2a$, and an +own portal is within a further $200a$. The asserted upper bound follows. + +An own portal has radial distance in $[n\delta,n\delta+a)$, whereas a +child portal has radial distance at least $(n+1)\delta$. Their distance +is therefore at least $\delta-a$. At depth zero the upper bounds follow +directly by taking $o$, and a depth-one portal is at distance at least +$\delta$ from it. These estimates imply the stated bounds in every case. +\end{proof} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/interpolation.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/interpolation.tex new file mode 100644 index 0000000000000000000000000000000000000000..979088a7200fac7dad121a244a1a1d55f4b3ba88 --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/interpolation.tex @@ -0,0 +1,454 @@ +\section{Interpolation by single-source distance functions} +\label{int:section} + +All distances in this section are intrinsic to one connected component +$Y$ of a closed band $X_J$. We write $d_J$ for this distance. The width +of $J$ is denoted by $a$, and its midpoint by $H_*$. The active column +graph and its radial coordinate $b_i=d_J(o,(i,H_*))$ are as in +Section~\ref{geo:section}; we choose $o$ at height $H_*$. The choices of +$o$ and of all portals below are made before any random parameters in +this section are sampled. + +The construction will repeatedly use signed point-distance formulas as +local data. We therefore approximate a function of Lipschitz constant +strictly below one by formulas $c\pm d_J(k,\cdot)$, while controlling where the +selected formula changes. Each formula has a fixed source and sign, a +random additive constant, and a name. When the construction reuses a +formula, it retains that name. Stability on a small ball means that one +name is selected throughout the ball, not merely that the values agree +at the interfaces. + +\begin{theorem}[Thin-band interpolation]\label{int:interpolation} +Let $0<\zeta<1$, let $E>0$, and set +\begin{equation}\label{int:constants} + \begin{gathered} + \delta=E/8,\qquad a_0=10^{-6}\zeta\delta,\qquad + q=\zeta\delta/20,\\ + \rho_0=\zeta\delta/1000,\qquad + M=\lceil15/\zeta\rceil+3,\qquad N=6(M+1). + \end{gathered} +\end{equation} +Suppose $013\delta\quad\text{for every }k\in K_\mu. +\end{equation} +Equations \eqref{int:tentative-bounds}--\eqref{int:lower-atom-bound} +give +\[ + T_\mu(v)>p(v)+13\delta,\qquad + A_\mu(v)>p(v)+7\delta,\qquad + B_\mu(v)p(v)+7\delta. +\] +In particular, a value within $6\delta$ of $p(v)$ cannot be obtained +at this node by selecting its parent atom. Notice that the lower +bound on the tentative formula is required here as well as the two +clamping bounds. + +Let $n$ be the depth of $\nu$. If $n\le M$, every ancestor is already +in the asserted list. Otherwise consider its ancestor $\mu$ of +depth $n-M\ge1$. For a point $v$ of the ball, +Lemma~\ref{int:common-formula}, radial comparison, and +\eqref{int:portal-radial} give +\[ + d_J(k,v)\ge M\delta-3a\qquad(k\in K_\mu). +\] +The definition of $M$ implies \eqref{int:far-portals}. An atom +originating strictly earlier than $\mu$ could be selected by $F_\nu(v)$ +only by being inherited through every intervening node. All these +inherited values would equal $F_\nu(v)=Q(v)$, whose error is at most +$6\delta$. In particular $\mu$ would select its parent atom with +such a value, contradicting the preceding paragraph. This excludes +all earlier origins. Each of the remaining $M+1$ nodes contributes +at most six atoms, proving the count $N$. +\end{proof} + +\subsection{Conditional comparisons and probability bounds} + +We have a list of at most $N$ possible names on the ball, fixed before the +noises are sampled. Their intercepts can depend on earlier noises. +The next lemma identifies an independent noise in each comparison, so +this dependence does not require a union bound over the whole node tree. + +\begin{lemma}[Fresh-noise comparison]\label{int:near-tie} +For any two distinct named atoms in the deterministic list of +Lemma~\ref{int:depth}, any fixed $x\in Y$, and $t\ge0$, +\begin{equation}\label{int:near-tie-bound} + \Pr\bigl(|A(x)-A'(x)|\le t\bigr)\le(2t+\eta)/q. +\end{equation} +The same bound holds when one atom is replaced by any real quantity +independent of all current interpolation noises. +\end{lemma} + +\begin{proof} +For two atoms originating at different depths, use the later atom's +own noise. Its intercept, whether $p(k)$ or $F_\pi(k)$, does not +depend on that noise, because parent formulas use only earlier nodes. +The earlier atom does not depend on it either. Thus, after conditioning +on every other noise, the difference has the form $c+\varepsilon\alpha$ +with $\varepsilon\in\{-1,1\}$ and $\alpha$ uniform on $[q,2q]$. +For two atoms at the same node their own noises are different and +their parent intercepts depend on neither, so the same argument +applies. The candidate list lies on one ancestor chain, so these +are all cases. An independent comparison quantity can likewise be +included in the conditioning. + +For a continuous uniform noise an interval of length $2t$ has +probability at most $2t/q$. For a grid with spacing $\eta'\le\eta$, +it contains at most $2t/\eta'+1$ grid points; division by the total +number $q/\eta'+1$ gives at most $(2t+\eta')/q$. +\end{proof} + +\begin{proof}[Completion of Theorem~\ref{int:interpolation}] +Take the deterministic list furnished by Lemma~\ref{int:depth}. +The union bound and Lemma~\ref{int:near-tie}, with $t=4\rho$, show +that the probability of a pair satisfying +$|A(x)-A'(x)|\le4\rho$ is at most +$N^2(8\rho+\eta)/q$. On the complementary event, two atoms in the +list cannot be equal anywhere on the ball, since both are +$1$-Lipschitz. The continuous function $Q=F_\nu$ selects a value +from this finite list at every point. Where the list values are +distinct its selected name is locally constant: continuity and the +positive minimum gap to the other finitely many values give a +neighborhood with the same selection. A ball is connected by paths +through its center, so the name is constant throughout the ball. +This proves \eqref{int:probability}. The path version follows by +applying the ball assertion to the paths, all of whose points lie +in the ball. +\end{proof} + +\begin{corollary}[Comparison with other branches]\label{int:comparisons} +Condition on arbitrary prior data. Let $W$ be the union of a nonempty +family of band paths issuing from a fixed point $x$, each of length at +most $\rho$. The set $W$ and these paths are fixed by the prior data. +Suppose that $Q_1,\ldots,Q_s$ are interpolation constructions, each +on a band component containing all these paths. Their geometries, +inputs, and portals are fixed by the prior data, and their fresh noise +families are mutually independent. The bands and their intrinsic metrics +may differ; each of the specified paths must be permitted in every band. +Denote the constants of $Q_i$ in \eqref{int:constants} by +$q_i,N_i,\rho_{0,i}$. + +Allow inputs $c_i+\varepsilon_iQ_i$, with +$\varepsilon_i\in\{-1,1\}$ and $c_i$ fixed by the prior data. +Also allow one fixed branch $f$ and $t$ branches $g_j+\beta_j$, +where $f$ and the $g_j$ are fixed functions whose restrictions to each +specified path, parametrized by arclength, are $L$-Lipschitz. The +$\beta_j$ are fresh independent uniform noises on intervals of length +$w_j>0$, independent also of the interpolation noises. All functions respect the switches used by the paths. +Set +\[ + L_* =\max(1,L),\qquad + S=1+t+\sum_{i=1}^sN_i,\qquad + q_* =\min\bigl(\{q_i:1\le i\le s\}\cup + \{w_j:1\le j\le t\}\bigr). +\] +Assume that there is at least one randomized input and that +$\rho\le\min_i\rho_{0,i}$ when $s>0$. Any fixed expression formed +from these inputs by finitely many minima and maxima selects one +branch and, for a selected interpolation, one atom throughout $W$, +except with conditional probability at most +\begin{equation}\label{int:comparison-probability} + \frac{S^2}{q_*}(8L_*\rho+\eta). +\end{equation} +Continuous noises have $\eta=0$; for equally spaced endpoint grids +$\eta$ can be the largest spacing of any noise used here. The +expression uses each interpolation with a single chosen sign and +shift; repeated appearances of that same input are given the same +name. +\end{corollary} + +\begin{proof} +Every specified path lies in $B_{d_{J_i}}(x,\rho)$ for each interpolation +band $J_i$. Use the deterministic bounded atom list for $Q_i$ on that +ball, with its chosen sign and shift. Adjoin the shifted $g_j$ branches +and, if present, $f$. +For pairs from one interpolation use Lemma~\ref{int:near-tie}. +For pairs from different interpolations condition on all noises +except the own noise of an atom in one interpolation; independence +of the constructions makes the other atom independent of that noise. +The same argument applies to a shifted $g_j$ or the fixed branch +$f$. There is at most one branch without a fresh noise. Therefore +every comparison of distinct names at $x$ has the interval bound +$(2u+\eta)/q_*$ at tolerance $u$. + +Take $u=4L_*\rho$ and sum over at most $S^2$ pairs. Off the event +in \eqref{int:comparison-probability}, no two listed values agree +anywhere on $W$: their changes along each path from $x$ are bounded +by $L_*\rho$. The expression is a continuous selection from the +resulting finite list along each path. Its selected name is therefore +constant on each path and, because all paths begin at $x$, throughout +$W$. +\end{proof} + +\begin{remark}\label{int:conditional-scope} +The conditional independence requirements in +Corollary~\ref{int:comparisons} are substantive. In particular, a +branch defined using current interpolation noises cannot be treated +as a fixed comparison branch, and two arbitrary fixed branches need +not have a small probability of meeting. The later applications use +one old branch, plus independently shifted height or ordered branches, +after conditioning on the bands, old charts, groups, and admissions. +All choices of geometric portals and interpolation inputs precede +the noises being tested. With this conditioning, the displayed +bounds are uniform in all prior choices and in the size of the +column graph. +\end{remark} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/introduction.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..b4d28e92e4fdce849528f8fe18d71adf00137b2f --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,239 @@ +\section{Introduction} +\label{intro} + +Let $G=(V,E)$ be a finite connected undirected graph with positive real +edge lengths. Its graph metric $d_G$ is the shortest-path distance on $V$. +An embedding of a finite metric space $(V,d)$ into real $L_1$ has +\emph{distortion at most $C$} if, after multiplication by a positive +scalar, its distances lie between $d$ and $Cd$. +The planar embedding conjecture asks whether one constant $C$ works for +every planar graph metric, independently of the number of vertices and +the edge lengths. We prove that it does. + +\begin{theorem}\label{thm:main} +There is a finite universal constant $C\ge1$ with the following property. +For every finite connected planar graph $G=(V,E)$ with strictly positive real +edge lengths, there exist a measure space $(\Omega,\mathcal F,\mu)$ and a map +$f:V\to L_1(\Omega,\mathcal F,\mu;\R)$ such that +\[ + d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y) + \qquad(x,y\in V). +\] +\end{theorem} + +This proves the planar case of the conjecture of Gupta, Newman, +Rabinovich, and Sinclair, which predicts uniformly bounded $L_1$ +distortion in every proper minor-closed family~\cite{GNRS04}. +In particular, the bound is independent of the number of distance +scales in the graph metric. + +\subsection{Cuts, multicommodity flow, and the embedding problem} +\label{intro:flow} + +The relationship with cuts explains why $L_1$ is a natural target. +For $S\subseteq V$, put +$\delta_S(x,y)=|\1_S(x)-\1_S(y)|$. +A nonnegative combination +\begin{equation}\label{intro:cut-metric} + D(x,y)=\sum_{S\subseteq V}w_S\delta_S(x,y),\qquad w_S\ge0, +\end{equation} +is the $\ell_1$ distance between the vectors +$(w_S\1_S(x))_{S\subseteq V}$ and $(w_S\1_S(y))_{S\subseteq V}$. +Conversely, threshold sweeps through real coordinates give such a +representation for every $L_1$ metric on a finite set. Thus we seek +weighted cuts that separate each pair in proportion to its graph +distance while charging each edge at most a constant times its length. +The edge bound then extends to every pair along a shortest path. + +Linial, London, and Rabinovich established the connection between +metric embeddings and multicommodity flow~\cite{LLR95}; Aumann and +Rabani independently obtained a general approximate max-flow/min-cut +theorem~\cite{AR98}. To state the corresponding planar consequence, +give the edges nonnegative capacities and the unordered pairs of +distinct vertices nonnegative demands, not all zero. Let $\lambda_*$ +be the largest common multiplier of the demands that can be routed +fractionally. Let $\phi_*$ be the minimum ratio of crossing capacity +to separated demand over cuts with positive separated demand. For +$\lambda_*>0$, the ratio $\phi_*/\lambda_*$ is the \emph{flow--cut gap}. +Gupta et al. showed that its worst value on a fixed graph equals the +worst $L_1$ distortion over all weighted shortest-path metrics on that +graph~\cite[Section~3, Theorem~3.2]{GNRS04}. Hence +Theorem~\ref{thm:main} also gives a universal bound for the undirected +fractional flow--cut gap on planar graphs, with arbitrary demands. + +\subsection{Earlier results and the remaining obstacle} + +The general preceding bound for $n$-vertex planar metrics is +$O(\sqrt{\log n})$, due to Rao~\cite{Rao99}. His construction embeds +first into Euclidean space and then into $L_1$; it builds on the +decomposition methods of Klein, Plotkin, and Rao for graphs excluding +a fixed minor~\cite{KPR93}. Newman and Rabinovich showed that even +series-parallel metrics can require $\Omega(\sqrt{\log n})$ distortion +into Euclidean space~\cite{NR03}. Thus a constant planar $L_1$ bound +requires a different route. Abraham, Filtser, Gupta, and Neiman later +recovered Rao's planar bound through recursive shortest-path +decompositions~\cite{AFGN22}. + +Constant distortion was known under several topological restrictions. +Gupta et al. proved it for series-parallel graphs~\cite{GNRS04}. +Chakrabarti, Jaffe, Lee, and Vincent obtained the sharp upper bound +$2$~\cite{CJLV08}, with a matching lower bound due to Lee and +Raghavendra~\cite{LR10}. Chekuri, Gupta, Newman, Rabinovich, and +Sinclair gave a bound exponential in the outerplanarity +parameter~\cite{CGNRS06}: the number of rounds needed to delete all +vertices by successive outer-face removals in a planar drawing. +This gives a uniform bound when that parameter is fixed. + +Another line of work restricts the pairs whose distances must be +preserved. The theorem of Okamura and Seymour, and its metric +formulation by Hurkens, Schrijver, and Tardos, give an isometric +$L_1$ representation of the metric on vertices of one +face~\cite{OS81,HST88}. If a terminal set is covered by $\gamma$ +faces in a fixed drawing, Krauthgamer, Lee, and Rika obtained a +logarithmic bound in $\gamma$~\cite{KLR19}. Filtser improved this to +$O(\sqrt{\log(\gamma+1)})$~\cite{Filtser25}. Kumar constructed a +single nonexpansive map on all vertices that preserves every +cofacial pair within a universal factor, even when different pairs +lie on different faces~\cite{Kumar25}. + +Geometric restrictions lead to a different family of results. +Sidiropoulos proved constant distortion for planar metrics realized +in simply connected surfaces of nonpositive curvature in the sense +of Busemann~\cite{Sidiropoulos13}. Chalopin, Chepoi, and Naves proved +that Busemann surfaces themselves embed isometrically into +$L_1$~\cite{CCN15}. These are geodesic spaces homeomorphic to the +plane whose distance is convex along pairs of geodesics. + +The difficulty is to combine separation at different scales while +keeping a uniform upper bound. Sidiropoulos addressed this difficulty +by evolving a distribution of monotone cuts from coarse scales to +fine scales~\cite[Sections~4--5]{Sidiropoulos13}. Our construction +also preserves monotonicity through successive local changes. +For unrestricted planar metrics, two additional mechanisms provide +the needed control: component separation limits which pairs need +further updates, and interpolation makes local formulas agree +exactly at their interfaces. The first yields a packing bound only +for the surviving pairs; the second lets those local updates be +combined without a loss proportional to the number of scales. + +The structural reductions of Lee and Sidiropoulos relate the planar +problem to the larger GNRS conjecture and a separate problem of +closure under graph sums~\cite{LS09}. Together with the +bounded-treewidth companion~\cite[Theorem~1.1]{Treewidth108}, our +theorem also gives constant distortion on their fixed-parameter +almost-embeddable families; see Corollary~\ref{cor:almost-embeddable}. +The general clique-sum closure required for the full GNRS conjecture +is not established here. + +\subsection{The construction and its main ideas} + +We construct two families of measurable label sets, denoted by +$\mathcal A_v$ and $\mathcal B_v$ at a point $v$. +A label determines the cut of points whose sets contain it, so +$\mu(\mathcal A_x\mathbin\triangle\mathcal A_y)$ is the resulting +cut distance. The random construction chooses these entire families; +the labels index their cuts. Every outcome has the required upper +bound. Random local changes supply the lower bound. + +\paragraph{Columns and the upper bound.} +Section~\ref{geo:section} represents the geometric graph isometrically +by vertical columns with zero-cost switches at specified heights. +Choose a shortest-path tree rooted at a vertex and use distance from +the root as height. Replace each non-tree edge by two ascending +branches of the same total length and identify their tips. The +plane embedding gives a circular order of columns in which the +switching links do not cross. Figure~\ref{fig:column-realization} +illustrates this realization. + +For a fixed label, the active points on each column form an upper +interval. Switches preserve membership, and the measure of labels +whose membership changes along a vertical segment equals its length. +Consequently every path bounds the cut distance between its endpoints, +regardless of the number of updates. Section~\ref{sets:section} +formulates this invariant for both systems. The problem is to obtain +lower separation while retaining it. + +\paragraph{Common local formulas.} +Local updates use \emph{charts}: regions on which one scalar field +$P$ and common label variables describe the thresholds. Their form is +$h(v)+\sigma\kappa P(v)$, where $h$ is height, $\sigma$ is a fair +sign, and $\kappa>0$ is fixed. Averaging over the two signs detects +both height differences and field differences. A sufficiently small +vertical Lipschitz constant for $\kappa P$ preserves the upper-interval +property. + +Within a thin height band, use distances from a basepoint to the +column midpoints to form successive radial layers. Noncrossing links +restrict the entrances to a connected region beyond a layer boundary +to neighborhoods of at most two points. This two-portal property, proved in +Section~\ref{geo:section}, underlies the interpolation in +Section~\ref{int:section}. A field with Lipschitz constant strictly +below one is approximated by pieces of the form $c\pm d_I(k,\cdot)$, +where $d_I$ is distance within the band and $k$ is one point in it. +At each entrance, the neighboring regions select exactly the same +point-distance formula, including its source and its name. +Strictness of the input slope bounds +the number of candidate formulas that can govern a small ball. +Fresh independent offsets then make a change of selected +formula across that ball unlikely. + +\paragraph{The first system selects the remaining pairs.} +At scale $r$, system $A$ tries to raise the field near one endpoint +of a pair at distance comparable to $r$, leaving the other endpoint +unchanged. Section~\ref{main:section} shows that failure to produce +a definite rise supplies a short path on which the old field +decreases at almost the greatest rate its Lipschitz bound permits. + +System $A$ has a second separation mechanism. For each label, refine +its active set into connected components of active columns. This +refinement retains the upper bound. A sufficiently low inactive tip +of a switching link separates active points on opposite sides and +therefore certifies a lower bound. Only pairs with descending paths +at both ends and without this certificate contribute \emph{seeds}, +the vertices near which the second system may update. + +Proposition~\ref{pack:bound} places the seeds in a small narrow-band +ball close to two short paths. Their distance from cell boundaries +then separates their groups along those paths, bounding the number +of groups that can affect one local update. +Figure~\ref{fig:surviving-seeds} shows the decisive case: when a path +skips a seed's column, failure of the low-tip certificate forces that +seed close to the skipping link. The ambient metric need not have +a uniform packing bound. + +\paragraph{The second system separates the surviving pairs.} +The circular order supplies fields that increase along both arcs +between two column indices. Section~\ref{ord:section} constructs +such a field below a point-distance function and equal to it at a +chosen target. Equal values +can occur on opposite arcs; a signed transverse field distinguishes +the arcs and vanishes where a switch can join them. + +Section~\ref{sec:secondary} uses these fields in local changes near +the seeds. Either a change can be admitted while agreeing with the +old field at the boundary, or its failure in both orientations +supplies short paths along which the ordered field rises and falls. +If this happens at both endpoints, Lemma~\ref{ord:two-side} puts them on opposite +arcs, where a transverse increment separates them. The packing +bound limits the number of competing randomized proposals, both +for the separation probability and for chart persistence. + +\paragraph{Sparse schedules preserve separation.} +Section~\ref{pad:assembly} combines three estimates uniform in the +graph and the update history. At scale $r$, a point's label set moves +by at most $Mr$; an eligible pair at distance in $[r,2r)$ gains a +separation certificate with fixed positive probability; and a ball of +radius $\rho$ already in one chart in +each system is split between new charts with probability +$O(\rho/r)$. Here eligibility requires neighborhoods of both +endpoints, of radius a fixed multiple of $r$, to share the same +chart data in each system. + +Within a schedule, successive scales differ by a sufficiently large +constant factor $Q$. The coarser chart-loss probabilities have a +small geometric sum, so a pair reaches its own scale with positive +probability. Later movement has another small geometric sum and +cannot erase all the gained separation. For a component certificate, +it suffices to preserve activity at the two endpoints and inactivity +at one fixed tip. Averaging finitely many interleaved schedules +covers every pair distance and completes the embedding. diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/main-packing.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/main-packing.tex new file mode 100644 index 0000000000000000000000000000000000000000..5982ac56f0b20acf69a7b2950aa5ad5f32903249 --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/main-packing.tex @@ -0,0 +1,510 @@ +\section{The main update and a packing bound for secondary seeds} +\label{main:section} + +This section gives system $A$ its two separation mechanisms and extracts +the geometric information available when both fail. Main clipping either +separates a pair or supplies short descending paths at its endpoints. A +low separating tip supplies a component certificate. The remaining +endpoints satisfy a packing bound that lets one secondary proposal act +alone with positive probability and bounds the number of formulas +competing near a point. + +All lengths in this section are measured in the units of the current +scale. Thus the pairs under consideration satisfy +$1\le d(x,y)\le2$. We use the parameters and the random bands of +Sections~\ref{sets:parameters} and~\ref{sets:setup}. In particular, +$J_2\subset J_1$ denotes a narrow band contained in its broad band. +The cells used below are the connected column cells of +Lemma~\ref{geo:cells}. The frontier of a cell $C$ is the union, within +$J_1$, of all its column intervals whose columns have an active link +to a column outside $C$. When two cells have a common quotient point, +that point is in the frontier of both cells. Frontier values will +therefore always be left unchanged. + +An ordinary chart of system A has field $P=C_0+\tau s$, where +$\tau\in\{-1,1\}$ and $s=d_I(o,\cdot)$ on a component of an old band +$I$, or $s=0$ in the constant case. Within that chart we perform all +comparisons in the $s$ orientation. Replacing $s$ by $s'$ means +replacing $P$ by $C_0+\tau s'$ in the original label variables. + +\subsection{The main update} +\label{main:update-section} + +Let $C$ be a cell of width $D$ in $J_1$. We make it available for an +update only when its entire closed $d$-neighborhood of radius $1$ +is contained in one old ordinary chart of system A. This condition +allows every path of length at most $10D$ starting in $C$ to be +evaluated in that chart: all its points are in the indicated +neighborhood, and hence in the band component on which the old field +is defined. In particular, +\begin{equation} + |s(v)-s(w)|\le d_{J_1}(v,w)\le10D + \qquad(v,w\in C). + \label{main:cell-field-variation} +\end{equation} +Here and below an intrinsic shortest path of length at most $10D$ +is allowed to leave the cell. Its length bound, rather than intrinsic +connectedness of the cell, ensures its availability in the old chart. + +The frontier bounds how far a strictly Lipschitz proposal can rise at an +interior point. If a frontier constraint prevents a definite rise, a +shortest path to that constraint will be almost maximally descending +for the old field. A downward shift makes the proposal strictly inactive +at the frontier, which will also protect small balls crossing cell boundaries. + +Fix $v_C\in C$ and put $s_C=s(v_C)$. On the entire $J_1$-component +containing $C$, define +\begin{equation} + p_C(v)=\min\left\{s_C+1, + \inf_{z\in\partial C} + \bigl(s(z)+(1-\zeta_A)d_{J_1}(v,z)\bigr)\right\}. + \label{main:frontier-extension} +\end{equation} +The infimum over an empty frontier is $+\infty$. The frontier is a +finite union of compact column intervals, so that a finite infimum +is attained. The function $p_C$ is $(1-\zeta_A)$-Lipschitz for +$d_{J_1}$. Apply Theorem~\ref{int:interpolation}, with error $E_A$, +to obtain $\widetilde p_C$, and set +\begin{equation} + F_C=\widetilde p_C-4E_A. + \label{main:F} +\end{equation} +In particular, +\begin{equation} + p_C-5E_A\le F_C\le p_C-3E_A. + \label{main:F-error} +\end{equation} + +For a fresh offset $\alpha_C\in[0,E_A]$, define a function of height +$p_{h,C}$ as follows. It is $s_C-1+\alpha_C$ at either endpoint of +$J_1$, is $s_C+2+\alpha_C$ on the interval obtained by deleting +the outer portions of width $0.09a_1$, and is linear on the two +remaining portions. Extend it constantly beyond the two endpoints +of $J_1$. Its Lipschitz constant is $3/(0.09a_1)$. +If the independent fair coin of $C$ is kept, make the replacement +\begin{equation} + s'(v)=\max\{s(v),\min\{F_C(v),p_{h,C}(h(v))\}\} + \qquad(v\in C). + \label{main:replacement} +\end{equation} +Otherwise leave the cell unchanged. Cells that are unavailable +are also left unchanged. + +\begin{lemma}[Validity and size of the main replacement] +\label{main:validity} +The simultaneous replacements~\eqref{main:replacement} respect +switches, preserve vertical nesting and the vertical upper bound +for system A, and admit ordinary charts on every new branch. +At any point the change in the system-A label set has measure at most +$\kappa_A(1+10D)$. +\end{lemma} + +\begin{proof} +At a frontier point $z$, the term with source $z$ in +\eqref{main:frontier-extension} gives $p_C(z)\le s(z)$. Thus +$F_C(z)\le s(z)-3E_A$. At either endpoint of the height band, +\eqref{main:cell-field-variation} gives +\[ + p_{h,C}\le s-1+10D+E_As(w)+2E_A. +\] +It follows that $F_C(w)\le s(w)+2E_A$, and +\eqref{main:F-error} then gives +\begin{equation} + p_C(w)\le s(w)+7E_A. + \label{main:small-extension} +\end{equation} +The constant branch $s_C+1$ is strictly greater than the right-hand +side of~\eqref{main:small-extension}. The frontier is therefore +nonempty, and there is a minimizing $z\in\partial C$ with +\[ + s(z)+(1-\zeta_A)d_{J_1}(w,z)\le s(w)+7E_A. +\] +Put $d_0=d_{J_1}(w,z)$. The radial padding conclusion of +Lemma~\ref{geo:padding} gives $d_0\ge0.01D=10^{-5}$. +The weak diameter estimate gives $d_0\le10D<0.02$. +Choose an intrinsic shortest path from $w$ to $z$. It remains in +the old chart band by availability. Since +\[ + \zeta_A d_0+7E_A + \le (\epsilon/100)(10D)+7\epsilon/1000<\epsilon, +\] +this path satisfies~\eqref{main:ray-endpoint}. + +For the last assertion, apply the old field's Lipschitz bound to +the remaining subpath, of length $d_0-t$: +\[ + s(\gamma(t))\le s(z)+(d_0-t) + \le s(w)-t+\epsilon. +\] +\end{proof} + +\begin{corollary}[Separation supplied by the main test] +\label{main:pair-separation} +Let $(x,y)$ be an eligible pair satisfying the interior requirements, +$1\le d(x,y)\le2$, and $|s(x)-s(y)|\le E_A$. If either endpoint +has no descending path, then, conditional on main-update mode, +the independent cell coins give probability at least $1/4$ of +system-A symmetric-difference separation at least $\kappa_A E_A$ +at this step. +\end{corollary} + +\begin{proof} +The two endpoints lie in different width-$D$ cells, since a cell +has $d$-diameter at most $10D<1$. Each of these cells is available: +its radius-$1$ neighborhood lies in the radius-$5$ neighborhood +of its endpoint, by the cell diameter bound and eligibility. +Keep the coin at an endpoint with no descending path and suppress +the coin of the other endpoint. This has probability $1/4$. +Lemma~\ref{main:ray} raises the first comparison value by more +than $2E_A$ and leaves the second unchanged, so their new field +difference has magnitude at least $E_A$. Evaluate both new sets +in their common old label variables. The common-chart +symmetric-difference formula gives the asserted bound. +\end{proof} + +\subsection{The component barrier and the seed set} +\label{pack:seed-definition-section} + +The main test has reduced the problem to pairs with two descending paths. +Before using those paths to locate nearby groups, we remove pairs already +separated by a low terminal tip. The resulting certificate records only +three memberships, so later changes elsewhere cannot invalidate it. + +For the rest of the section, retain only eligible pairs $(x,y)$ in +the distance range $[1,2]$ that satisfy all the interior requirements +and both field-difference cutoffs, lie in the same bands $J_2\subset +J_1$, and possess descending paths at both endpoints. In particular, +their common ordinary system-A field $s$ is nonconstant: a constant +field cannot satisfy~\eqref{main:ray-endpoint} with +$d_0\ge10^{-5}>\epsilon$. +Choose once and for all a representative column for every vertex +under consideration. A chord separates two representatives +strictly if they lie in the strict interiors of the two complementary +circular arcs determined by its endpoint columns. + +\begin{lemma}[Label mass at a separating tip] +\label{pack:barrier} +Suppose a retained pair has a non-neighbor link separating its +representatives strictly, with tip $e$ at a height in $J_2$, such +that $d(e,x)\le1$ or $d(e,y)\le1$. If +\begin{equation} + s(e)t. +\] +All comparisons take place in $I$, as verified above. The +noncrossing gap property, Lemma~\ref{geo:gap}, now places all +these path words and their endpoint representatives in one +linear circular gap of $S$. Fix a representative of $z$ to +order this gap. Equivalent representatives also lie in the +same gap, since a zero-length switch path satisfies the same +avoidance argument. + +Choose one representative seed from each group meeting $B$. +In the linear order of the gap, choose a seed of minimum index +among those to the left of the center index, if there is one, +and similarly a seed of maximum index on the right. Let +$P_-$ and $P_+$ be shortest $J_2$-paths from $z$ to these extreme +seeds. When one side has no seed, use the constant path at $z$ +on that side. Each path has length at most $4R$. +For any chosen seed $w$, at least one of these two path words +either visits its representative column or makes a switch +strictly over its index. This is the elementary skipping +property of Lemma~\ref{geo:skipping}, applied between the +center and the extreme index on that side. + +If the path visits the column of $w$, a vertical interval of +length at most $a_1$ connects $w$ to a point of that path in +$J_1$. Consider instead a switch strictly over its index. +Let $a,b$ be the switching columns, ordered within the gap so +that the representative of $w$ is in $(a,b)$. The switch is +not a circular-neighbor link, because its strict interval +within the gap contains a column and the complementary arc +contains the nonempty set $S$. It is consequently a single +switch at a tip $e$, by the column-model hypothesis. This tip +is on $P_-$ or $P_+$, has height in $J_2$, and satisfies +\begin{equation} + d(w,e)\le8R<1. + \label{pack:tip-near} +\end{equation} +The interval $(a,b)$ is disjoint from $S$. + +We claim that the witness partner $w'$ has representative +strictly outside the closed interval $[a,b]$ in the circle. +If its representative were in $(a,b)$, its descending path, +whose terminal column belongs to $S$, would have to visit +column $a$ or $b$. This follows from noncrossing, or directly +from Lemma~\ref{geo:chord-separation}. Join that visit +vertically to $e$, then follow the appropriate extreme path +back to $z$ and a shortest center-to-$w$ path. The resulting +path has length at most +\[ + 0.02+a_1+8R<1, +\] +contradicting $d(w,w')\ge1$. If $w'$ itself has representative +$a$ or $b$, the vertical connection to $e$ gives the same +contradiction without the descending segment. This proves +the claim. + +The link at $e$ thus separates the two witness representatives +strictly. Because the witness pair was not excluded and +\eqref{pack:tip-near} holds, its failure to satisfy the +barrier-drop condition implies +\begin{equation} + s(e)\ge s(w)-j. + \label{pack:tip-not-low} +\end{equation} +The descending path from $w$ must likewise hit column $a$ or +$b$, since it begins in $(a,b)$ and ends in $S$. Let $v$ be +its first such hit and let $p$ be the arclength to $v$. +Lemma~\ref{main:ray} and the vertical comparison with $e$ give +\[ + s(v)\le s(w)-p+\epsilon, + \qquad + s(v)\ge s(e)-a_1\ge s(w)-j-a_1. +\] +It follows that $p\le j+\epsilon+a_1$. The descending prefix +and the vertical interval from $v$ to $e$ both lie in $J_1$, +so +\begin{equation} + d_{J_1}(w,e)\le j+\epsilon+2a_1. + \label{pack:near-path} +\end{equation} + +We have proved that every chosen seed is within +$\rho_{\mathrm{pack}}=j+\epsilon+2a_1$ in $d_{J_1}$ of +$P_-\cup P_+$. Distinct chosen groups have distinct width-$g$ +cells: their bands, band component, and two old identifiers +have already been shown to coincide. Their seed points are +radially padded, so Lemma~\ref{geo:padding} gives pairwise +$d_{J_1}$-separation at least $b_0=0.01g$. + +For each chosen seed, take one of the path points just +constructed within $\rho_{\mathrm{pack}}$, and assign it to one path +containing it. Path points assigned to different seeds have +$d_{J_1}$-distance at least $b_0-2\rho_{\mathrm{pack}}\ge b_0/2$. +Their arclength parameters on either path are therefore +separated by at least $b_0/2$. A path of length $4R$ has at +most $1+8R/b_0$ such points. The two paths together have at +most +\[ + 2+16R/b_0=2+1600R/g +\] +assigned points. This proves~\eqref{pack:number}. +\end{proof} + +Figure~\ref{fig:surviving-seeds} shows the skipping-link argument and the +short descending prefix that places a surviving seed near an extreme path. + +\input{figures/surviving-seeds} + +\begin{remark} +\label{pack:purpose} +The bounded number of nearby seed groups has two uses. Independent +group coins can isolate one group's proposal at a pair, and only +boundedly many families of random formulas can compete on a small ball. +The first use gives secondary separation; the second preserves the +common charts needed at later scales. The conclusion applies to seeds +of the surviving witness pairs. It does not assert bounded overlap of +all broad-band cells or their full neighborhoods: padded cells separate +the chosen seeds, and exclusion of the component barriers brings those +seeds close to the two short paths. +\end{remark} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/ordered.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/ordered.tex new file mode 100644 index 0000000000000000000000000000000000000000..bc6ebdbeb7d39e08cdbc70fce2da6acc3de15b9f --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/ordered.tex @@ -0,0 +1,406 @@ +\section{Ordered fields and separation on the two arcs} +\label{ord:section} + +The packing bound limits the number of secondary proposals that can +affect any one small ball. We now supply the scalar fields used by those +proposals and by +their final separation test. We use the circular order and the +noncrossing links of the column model. +All fields in this section are defined on a whole band $X_I$, including +its separate components. A bound for vertical Lipschitz constants, +together with compatibility at switches, implies the same bound for +the intrinsic distance $d_I$ wherever that distance is finite. + +This section provides two tools. First, a point-distance function has a +supporting field that is monotone toward a specified target along both +circular arcs and retains the exact distance at that target. Second, +short paths on which the field rises and falls in a sufficiently thin band +force a distant pair with nearly equal field values onto opposite arcs. A +signed transverse field will then distinguish the pair. + +\subsection{Ordered supports of point distances} + +\begin{definition}\label{ord:definition} +Let $a,b$ be two distinct column indices. A real field $s$ on $X_I$ is +\emph{ordered from $a$ to $b$}, with slope bound $M$, if it respects +every switch, is $M$-Lipschitz on every vertical column, and, at each +height $H\in I$, is nondecreasing on each of the two circular arcs +oriented from $a$ to $b$. In particular, the value on column $a$ is a +global minimum at that height, and the value on column $b$ is a global +maximum. We call the two open arcs the positive and negative arcs, +with either assignment of these names. +\end{definition} + +Adding a constant to an ordered field preserves its order and slope +bound. Multiplication by a positive constant multiplies the slope +bound by that constant. Its negative is ordered after interchanging +the two boundary columns. A field depending only on height is +ordered for any choice of distinct boundary columns. + +The supporting field below is obtained by taking running minima toward +the target column on each arc. Connected distance sublevels are what make +this operation compatible with the switches. + +\begin{lemma}[Ordered support of a point distance]\label{ord:support} +Suppose that $k,w$ belong to one component of $X_I$. There is an +ordered field $L$ of slope bound $1$ such that +\[ + L(k)=0,\qquad L(w)=d_I(k,w),\qquad + L(v)\le d_I(k,v)\quad(v\in X_I), +\] +where the last inequality uses the convention that distance to a +different component is infinite. +\end{lemma} + +\begin{proof} +Put $D=d_I(k,w)$. If $D=0$, the constant zero field suffices. If $k,w$ +have representatives on the same column, then +$D=|h(w)-h(k)|$ and the field +\[ + L(v)=\min\{|h(v)-h(k)|,D\} +\] +suffices, with any two distinct indices as boundary columns: vertical travel attains this distance on that column, and +every path has length at least its absolute height change. + +Otherwise choose a representative column $a$ of $k$ and a different +representative column $b$ of $w$. Write +\[ + f_i(H)=\min\{d_I(k,(i,H)),D\},\qquad H\in I, +\] +where $f_i=D$ on any component not containing $k$. Every $f_i$ is +$1$-Lipschitz, and the values respect switches. Moreover, +\[ + f_a(H)=\min\{|H-h(k)|,D\}=\min_i f_i(H). +\tag{\ref{ord:support}.1}\label{ord:root-minimum} +\] + +For fixed $H$ and $t\le D$, every nonempty strict sublevel +\[ + S_t(H)=\{i:f_i(H)32\eta, +\] +then +\[ + d(x,y)\le A_x+A_y+2\eta. +\] +\end{lemma} + +\begin{proof} +Restrict and orient each path so that its endpoint values are +$u$ and $v$. Fix $H_*\in J$ and write $q_i=s(i,H_*)$. +The values $q_i$ are nondecreasing in the linear order on this +open arc, and +\[ + |s(i,H)-q_i|\le4\eta\qquad(H\in J). +\tag{\ref{ord:overlap}.1}\label{ord:profile-error} +\] +Choose a point on $\Gamma_x$ of value $c=(u+v)/2$, with +representative column $i$. Then $|q_i-c|\le4\eta$. +The initial column of $\Gamma_y$ has reference value at most +$u+4\etaq_i$. Its column word therefore starts +strictly before $i$ and ends strictly after $i$. + +If $\Gamma_y$ visits column $i$, connect the two visits on +that column vertically at cost at most $\eta$, giving the +claim. Otherwise Lemma~\ref{geo:skipping} supplies a switch +on $\Gamma_y$ with endpoint columns $p16\eta$, these lie +strictly below $q_p$ and strictly above $q_q$, respectively. +Thus the middle point on column $i$ is strictly inside the +interval bounded by the link $pq$, whereas the endpoints +of $\Gamma_x$ are strictly outside it. +Lemma~\ref{geo:chord-separation} implies that $\Gamma_x$ +visits $p$ or $q$. Connect this visit vertically to height +$H_0$ and, if needed, use the zero-cost switch $pq$. +This joins a point of $\Gamma_x$ to a point of $\Gamma_y$ +at cost at most $2\eta$ (in fact $\eta$ suffices when +using the visited endpoint of that switch). The assumed +connections to $x,y$ complete the proof. +\end{proof} + +\begin{lemma}[Quantitative separation on the two arcs]\label{ord:two-side} +Set $R=10^{-8}$. Let $s$ be an ordered field of slope bound +$4$ on $X_I$. Suppose that $x,y\in X_I$ satisfy +\[ + 1\le d(x,y)\le2,\qquad d_I(x,y)=d(x,y), + \qquad |s(x)-s(y)|\le10^{-3}R. +\] +Let $J\subset I$ be a common height interval of width +\[ + \eta\le10^{-6}R +\] +containing $x,y$. For each $z\in\{x,y\}$, suppose that +there are two paths $\gamma_z^-,\gamma_z^+$ in $X_J$, +starting at $z$, of length at most $2R$, whose terminal +points $z^-,z^+$ satisfy +\[ + s(z^-)s(z)+0.4R. +\] +Then $t_s(x),t_s(y)$ have opposite signs and +\[ + 0.01R\le |t_s(x)|,|t_s(y)|\le8. +\] +\end{lemma} + +\begin{proof} +Write $s_0=s(x)$ and $H_x=h(x)$. We first prove that $Z_s$ +does not meet the rectangle +\[ + \mathcal Q= + [s_0-0.2R,s_0+0.2R]\times + [H_x-0.02R,H_x+0.02R]. +\tag{\ref{ord:two-side}.1}\label{ord:empty-rectangle} +\] +Suppose otherwise that $(c,H_0)\in Z_s\cap\mathcal Q$. +At the reference height $H_0$, the endpoint columns of +each low path have field value strictly below $c$, and +the endpoint columns of each high path have value strictly +above $c$. Indeed their terminal values are respectively +below $s_0-0.399R$ and above $s_0+0.399R$, and moving +their values to height $H_0$ changes them by at most +\[ + 4(0.02R+\eta)\le0.080004R. +\] +These inequalities leave a strict margin from the interval +$[s_0-0.2R,s_0+0.2R]$. + +If $(c,H_0)$ is the image of a boundary column, these +strict inequalities contradict that column's extremality. +Otherwise it is the image of a crosslink with one endpoint +on each open arc. At height $H_0$, monotonicity puts every +column of value less than $c$ on the root side of this +crosslink and every column of value greater than $c$ on +its opposite side. The projected concatenation +$(\gamma_z^-)^{-1}\gamma_z^+$ must therefore visit an +endpoint of the crosslink by +Lemma~\ref{geo:chord-separation}. Each such visit is +reachable from $z$ in length at most $2R$, since it lies +on one of the two original paths. Move vertically from +the two visits to height $H_0$ and use the crosslink if +needed. This produces an $x$--$y$ path of length at most +\[ + 4R+2(0.02R+\eta)<1, +\] +contradicting $d(x,y)\ge1$. This proves +\eqref{ord:empty-rectangle}. + +The maximum-norm ball of radius $0.01R$ around either +$(s(x),h(x))$ or $(s(y),h(y))$ is contained in +$\mathcal Q$. Hence +\[ + |t_s(x)|,|t_s(y)|\ge0.01R. +\tag{\ref{ord:two-side}.2}\label{ord:positive-magnitude} +\] +In particular both centers have representatives on open +arcs, with a well-defined choice of arc. + +Consider the value strip +\[ + \mathcal S=\{v\in X_J: s_0-0.18R\le s(v) + \le s_0+0.18R\}. +\] +No representative of a point of $\mathcal S$ is on a +boundary column, and no switch in $\mathcal S$ joins +opposite open arcs. Either event would give an image in +$Z_s\cap\mathcal Q$, because its height belongs to $J$ +and its value is within $0.18R$ of $s_0$. Thus any path +contained in $\mathcal S$ remains on a single open arc. + +For $z=x,y$, the concatenation +$(\gamma_z^-)^{-1}\gamma_z^+$ starts below the strip +and ends above it. The scalar field is continuous along +this path, including at switches. Take its first visit +to the upper strip boundary, and its last visit to the +lower strip boundary preceding that visit. The intervening +subpath $\Gamma_z$ stays in $\mathcal S$, traverses its +full value interval, and lies on one open arc. Every +point on it is reachable from $z$ in length at most $2R$. +If $\Gamma_x,\Gamma_y$ were on the same arc, +Lemma~\ref{ord:overlap}, with common interval of length +$0.36R>32\eta$, would give +\[ + d(x,y)\le4R+2\eta<1. +\] +Consequently their two arcs are opposite. + +It remains to relate these full traversals to the centers. +From either center $z$, follow either one of its two +original paths until its first exit from $\mathcal S$. +The resulting initial subpath $\Pi_z$ lies on the +center's arc and its value range contains the interval +between $s(z)$ and the strip boundary it first reaches. +This interval has length at least $0.179R$. All its +points are reachable from $z$ in length at most $2R$. +If the two centers had the same arc, exactly one of the opposite +full traversals $\Gamma_x,\Gamma_y$ would lie on that +arc. If it were $\Gamma_x$, compare it with $\Pi_y$; if +it were $\Gamma_y$, compare it with $\Pi_x$. In either case +the paths come from different centers and their value ranges overlap in +an interval +of length at least $0.179R>32\eta$. Another application +of Lemma~\ref{ord:overlap} gives the same contradiction. +Thus the centers have opposite arcs and, by +\eqref{ord:positive-magnitude}, $t_s(x),t_s(y)$ have +opposite signs. + +Finally Lemma~\ref{ord:transverse-lipschitz} and the +assumed equality of band and unrestricted distances give +\[ + |t_s(x)|+|t_s(y)| + =|t_s(x)-t_s(y)| + \le4d_I(x,y)=4d(x,y)\le8. +\] +This proves both upper bounds. +\end{proof} + + +\subsection{Ordered charts in system \texorpdfstring{$B$}{B}} +\label{ord:charts} + +Besides ordinary charts, system $B$ uses \emph{ordered charts}. Such a +chart has field +\[ + P=s+\lambda u, +\] +where $s$ is ordered with vertical Lipschitz constant at most $4$ and +$u$ has Lipschitz constant at most $K$ for $d_I$. At the creation of the +chart, $u=0$. The update in Section~\ref{sec:increment} changes $u$ while +preserving this bound. Our choice of $\lambda$ gives +$\Lip(P)\le4+\lambda K<5$ for finite $d_I$, as required by the chart +invariant of Section~\ref{sets:section}. + +The band $I$, the ordered part $s$, its two boundary columns and choice +of positive arc, and the label functions $Z,\sigma$ remain fixed in original units for the +lifetime of an ordered identifier. In particular its transverse field +$t_s$ remains fixed. Its region can shrink as other charts replace parts +of it, and its auxiliary field $u$ changes through the prescribed +increments. Thus the current full field $P$ may change without changing +the identifier or the ordered data it records. diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/scale-setup.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/scale-setup.tex new file mode 100644 index 0000000000000000000000000000000000000000..32fd9571dc3bfc77c42d1f14e70bd2af4853d21e --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/scale-setup.tex @@ -0,0 +1,138 @@ +\section{Scales and the local experiment} +\label{scale:section} + +We now specify the scales, geometric randomizations, and common inputs +for the local updates. The next sections construct those updates and +prove the separation and persistence estimates stated above. + +\subsection{Fixed constants and scales}\label{sets:parameters} +The geometric constants satisfy +$\epsilon\ll j\ll g\ll R\ll D\ll1$. The table records their roles; +the explicit choices below supply every numerical inequality used later. + +\begin{center} +\renewcommand{\arraystretch}{1.12} +\begin{tabular}{@{}p{0.19\textwidth}p{0.77\textwidth}@{}} +\toprule +Parameters & Role \\ +\midrule +$D$, $R$, $g$ & Main radial cell width, secondary update radius, and radial width for grouping seeds. \\ +$j$, $\epsilon$ & Detectable drop at a separating tip, and error in a descending path. \\ +$a_1$, $a_2$ & Broad and narrow band widths; control vertical transfers and interpolation. \\ +$E_A$, $E_B$ & Interpolation errors in the two systems. \\ +$\kappa_A$, $\kappa_B$ & Convert field variation into label mass while preserving vertical nesting. \\ +$\lambda$ & Size of the auxiliary correction in system $B$ (Section~\ref{sec:increment}). \\ +$Q$ & Scale ratio; controls earlier chart splitting and later movement. \\ +\bottomrule +\end{tabular} +\end{center} + +Fix the following numerical constants: +\[ + R=10^{-8},\quad D=10^{-3},\quad g=10^{-14},\quad + j=10^{-20},\quad \epsilon=10^{-24},\quad + E_A=\epsilon/1000,\quad E_B=10^{-3}R. +\] +Set $\zeta_A=\epsilon/100$ and $\zeta_B=1/10$. Choose $a_1>0$ within the +width bound of Theorem~\ref{int:interpolation} for $(\zeta_A,E_A)$, within the bounds of Lemmas~\ref{geo:cells} and~\ref{geo:padding} for $D$ and $g$, and with $a_1\le10^{-6}E_A$. +Set $\kappa_A=a_1/10^6$. Choose $a_2>0$ within the bound of Theorem~\ref{int:interpolation} for +$(\zeta_B,E_B)$ and with +\[ + a_2\le10^{-6}\min\{a_1,\kappa_Aj\}, + \qquad \kappa_B=a_2/10^6. +\] +Reserve $K=100$ as a Lipschitz bound for the auxiliary field in system +$B$; Lemma~\ref{sec:increment-bound} will supply this bound. Set +$\lambda=R/(10^{12}K)$. If necessary decrease $a_1,a_2$ further to meet the +explicit thin-band hypotheses of the geometric lemmas, and then choose +their dependent constants in the order just specified. All inequalities +in this choice are upper bounds on a newly chosen positive number. +The scale spacing $Q=2^m$ is chosen last. + +Choose finitely many dyadic scales so that every nonzero vertex distance +belongs to $[r,2r)$. For each residue class of exponents modulo $m$, run one +schedule in decreasing order of its scales. For the analysis of a step, +divide distances, heights, fields, and their additive constants by its +scale $r$. Multiply the resulting threshold and field changes by $r$ +before evaluating them in the original label space. In particular a field +change $\Delta P$ in scale units moves label mass +$r\kappa|\Delta P|$. The label coordinates are not resampled or +renormalized. + +\subsection{The data of a step}\label{sets:setup} +At a step tile the height line by intervals $J_1$ of width $a_1$ with a +common random shift, and independently by intervals $J_2$ of width $a_2$. +For the two clipping constructions, process every broad or narrow band +meeting the height interval +\[ + [\min_{v\in V}h(v)-6,\ \max_{v\in V}h(v)+6] +\] +in the current scale units, and make no clipping update in other bands. This is a +finite collection and contains every band needed for a radius-$5$ +neighborhood of an original vertex. The decision whether to process a +band is common to that entire band. This convention does not truncate the +ordered increments of Section~\ref{sec:increment}, which act on their +current chart regions. Given the bands, fix the radial basepoints in +their components before sampling any radial layer origins. In every +broad-band component form both the width-$D$ radial cells +and the width-$g$ radial cells, using independently shifted origins. +The narrow bands will instead determine the domains of the secondary +proposals and the balls in the packing estimate. Interpolation +basepoints and portals are likewise fixed before +any interpolation noises. + +A vertex is \emph{interior} if its narrow band is contained in its broad +band, its height avoids the outer tenths of both bands, and all its column +representatives have radial values farther than $0.02D$ and $0.02g$ from +the respective layer interfaces. A pair with height difference less than +$10^{-4}a_2$ has probability at least $1/4$ of satisfying these requirements +in common bands. Indeed each of the two height tests has failure probability +at most $0.21$, and each radial test, for two vertices together, has failure +probability at most $0.081$, with the remaining allowance covering the +column-width and grid errors. A chosen representative controls all the +others within an error at most $a_1$, without a union bound over columns. +Interior in the broad band implies containment of the narrow band because +$a_2\le10^{-6}a_1$. + +Recall that eligibility at the current scale requires the two closed +$d$-balls of radius $5$ to lie in one common old region in each system. +At this scale use the complete finite list +of eligible original vertex pairs with $1\le d(x,y)<2$. For the detailed +tests impose +\begin{equation}\label{sets:cutoffs} + |h(x)-h(y)|<10^{-4}a_2,\quad + |s_A(x)-s_A(y)|0.03\kappa_BR$, and in +particular at least $0.02\kappa_BR$. +Each mode has probability $1/4$, so both alternatives have uniform +positive probability. +\end{proof} + +\subsection{The increment in ordered charts}\label{sec:increment} +For an ordered identifier, the ordered part $s$, its band $I$, and its +arc assignment stay fixed while the auxiliary field $u$ changes and the +current region $\mathcal R$ may shrink. Its transverse field $t_s$ is +therefore fixed as well. We define the increments in original length +units, so these stored fields do not change when a step is analyzed in +units of its scale. + +Define a continuous trapezoidal function $H:[0,\infty)\to[0,1]$ which vanishes +off $[0.005R,10]$, equals one on $[0.01R,8]$, and is linear on the two +remaining intervals. At scale $r$, in original units, add to $u$ the +function +\begin{equation}\label{sec:increment-formula} + \pm t_s(v) H(|t_s(v)|/r) + \min\{1,d(v,X\setminus\mathcal R)/r\}. +\end{equation} +The sign is fair and common to the old identifier. When the complement is +empty its distance factor is one. The formula is defined on the chart's +whole band, with changes to the sets effective only in its current region. +Old terms of $u$ are retained without modification. + +\begin{lemma}[Increment bound]\label{sec:increment-bound} +If $Q>2000/R$, all increments of one ordered identifier together have +$d_I$-Lipschitz constant at most $K=100$. They preserve the nested sets, +introduce no chart interfaces, and move each point by at most a universal +constant times the scale. +\end{lemma} +\begin{proof} +The function $t_s$ is $4$-Lipschitz in $d_I$. The scalar function +$t\mapsto tH(|t|/r)$ has Lipschitz constant at most $6$ and magnitude +at most $10r$. The distance factor has Lipschitz constant at most $1/r$ +for $d$, hence for $d_I$. Each increment has Lipschitz constant at most +$24+10=34$. + +Its support is contained in +$0.005Rr<|t_s|<10r$. Since $t_s$ is fixed for this identifier, the +support intervals at distinct scheduled scales are disjoint when +$Q>2000/R$. In a comparison of two points at most two summands are +nonzero at either endpoint. Subtracting their zero values at the other +endpoint if necessary bounds the difference by $68d_I(v,w)$. Thus +$\Lip(u)\le680.4R +\] +along these paths from each endpoint. + +A shortest $x$--$y$ path has length at most $2$ and stays in the eligible +radius-$5$ ball, hence in the old band $I$. This proves +$d_I(x,y)=d(x,y)$ and bounds the change of $u$ by $2K$. The $B$ cutoff +therefore gives +\[ + |s(x)-s(y)|\le10^{-4}R+2\lambda K<0.001R. +\] +The four excursion paths lie in the common band $J_2$, whose width +satisfies Lemma~\ref{ord:two-side}. Applying that lemma gives opposite +signs of $t_s$ and $0.01R\le |t_s(x)|,|t_s(y)|\le8$. + +These bounds put both endpoints on the plateau of the trapezoidal +factor. Eligibility also puts their radius-$5$ balls in the current +region, so the distance factor in \eqref{sec:increment-formula} equals +one at both endpoints. For one of the two increment signs, +\[ + |P_{\mathrm{new}}(x)-P_{\mathrm{new}}(y)| + \ge \lambda|t_s(x)-t_s(y)|\ge0.02\lambda R. +\] +Formula~\eqref{sets:distance} proves separation at least +$0.02\kappa_B\lambda R$. The favorable sign has probability at least +$1/2$. +\end{proof} diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/sets.tex b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/sets.tex new file mode 100644 index 0000000000000000000000000000000000000000..299e7e3625a27f4ce5c95ea9e9d42796be3265bd --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/build/sections/sets.tex @@ -0,0 +1,240 @@ +\section{Threshold sets and local coordinates} +\label{sets:section} + +We construct cuts by assigning a measurable set of labels to every point +of the column model. Along a column these sets will grow with height, +with symmetric-difference measure equal to the vertical length. At a +switch the sets will agree exactly. Every resulting cut distance is +therefore bounded by the length of every column path. The local updates +must produce lower separation while preserving these two properties. + +There are two systems, called $A$ and $B$. System $A$ also uses the +connected components of a label's active columns to separate points. +System $B$ supplies a second local separation test. Evolving distributions +of monotone cuts through smaller scales is a central mechanism in +Sidiropoulos's nonpositively curved planar construction~\cite[Sections~4--5]{Sidiropoulos13}. +Here the updates must also preserve the exact sets where local formulas meet. + +\subsection{Charts on a fixed label space} + +For each system fix a label space +\[ + (\Omega,\mu)=\bigl(\R\times\{-1,1\}^{\N}, + \mathcal L\otimes\nu\bigr), +\] +where $\mathcal L$ is Lebesgue measure and $\nu$ is the product of fair sign +measures. Every set in that system is a subset of this same $\Omega$. +The random choices of updates are independent of the label coordinates; +we condition on those choices when describing the sets. + +A \emph{chart} consists of a region $\mathcal R\subset X$, a height band +$I$ containing its heights, a field $P$ on the corresponding component of +$X_I$ (or on all of $X_I$), and measurable functions +$Z:\Omega\to\R$ and $\sigma:\Omega\to\{-1,1\}$ whose joint pushforward +measure is Lebesgue measure times a fair sign. For $v\in\mathcal R$ set +\begin{equation}\label{sets:formula} + \mathcal A_v=\{\omega\in\Omega: + Z(\omega)0$ will try to separate original vertex pairs +whose distances lie in $[r,2r)$. At scale $r$, a pair $(x,y)$ is +\emph{eligible} if the two closed balls $B_d(x,5r)$ and $B_d(y,5r)$ +lie in one common old chart region in system $A$ and in one common +old chart region in system $B$. The common region may differ between +the two systems. This condition supplies common field and label data +for all short paths used in the pair's local tests. + +The construction below will give constants $M,C_0<\infty$ and +$b,p,\rho_0>0$, independent of the graph and the update history, +with three properties. +\begin{enumerate} +\item At every point of the full column space $X$, an update at scale +$r$ changes each unrefined label set by symmetric-difference measure +at most $Mr$. +\item For a vertex pair with $d(x,y)\in[r,2r)$, conditional on any past +in which $(x,y)$ is eligible, the update +has probability at least $p$ of producing either unrefined symmetric-difference separation +at least $br$ in one system, or a non-neighbor link separating +representatives of $x,y$ strictly and a system-$A$ label set of measure +at least $br$ on which $x,y$ are active and the link's tip is inactive. +\item Suppose a fixed ball $B_d(v,\rho r)$ lies in one old chart region in each +system, where $0<\rho\le\rho_0$. Conditional on the past, the probability +that this ball meets different new chart regions in either system is +at most $C_0(\rho+\eta_0)$. +\end{enumerate} +Here $\eta_0$ is a dimensionless upper bound for the relative mesh of +the finite grids used to sample update parameters. It can be chosen +arbitrarily small after the finite set of scales is fixed; the +sampling convention is given in Section~\ref{sets:setup}. + +Propositions~\ref{pad:success} and~\ref{pad:persistence} establish +these estimates after the local construction is complete. The third +estimate keeps a pair eligible until its own scale with positive +probability. The first then prevents sufficiently smaller scales from +erasing the second estimate's gain. For a component certificate, this +requires movement control at the tip as well as at the two vertices; +that is why the first estimate concerns all of $X$. Section~\ref{pad:assembly} +combines the estimates by using schedules whose successive scales +differ by a sufficiently large fixed factor. diff --git a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0316f1a2107eef10b8c9510654e7de6497c808f --- /dev/null +++ b/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:11b3a102e7333bf836a1317d774647394e580a5078027d1b9153a0b7fbc9e0e5 +size 578840 diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/README.md b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..86adebca15a5a0f7f642addb01c2d162948ebae5 --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/README.md @@ -0,0 +1,18 @@ +# [Pointwise Multiple Ergodic Averages for Mixing Transformations](multiple-ergodic-averages.pdf) + +**Author:** OpenAI + +**Date:** October 4, 2026 + +## Citation + +```bibtex +@misc{OAI:Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026, + author = {{OpenAI}}, + title = {{Pointwise Multiple Ergodic Averages for Mixing Transformations}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf}{OAI:Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/figures/square.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/figures/square.tex new file mode 100644 index 0000000000000000000000000000000000000000..8046d0ccaf841d859bd1666820acef46412e35a9 --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/figures/square.tex @@ -0,0 +1,36 @@ +\begin{figure}[tbp] +\centering +\begingroup +\setlength{\arraycolsep}{10pt} +\renewcommand{\arraystretch}{1.45} +\[ +\begin{array}{c|ccccc|l} + &1&\cdots&b&\cdots&n&\text{row parameter}\\ \hline + 1&y_{11}&\cdots&\colorbox{blue!10}{$y_{1b}$}&\cdots&y_{1n}&E_1\\ + \vdots&\vdots&&\colorbox{blue!10}{$\vdots$}&&\vdots&\vdots\\ + b&\colorbox{green!12}{$y_{b1}$}& + \colorbox{green!12}{$\cdots$}& + \fcolorbox{black}{yellow!25}{$y_{bb}$}& + \colorbox{green!12}{$\cdots$}& + \colorbox{green!12}{$y_{bn}$}&E_b\\ + \vdots&\vdots&&\colorbox{blue!10}{$\vdots$}&&\vdots&\vdots\\ + n&y_{n1}&\cdots&\colorbox{blue!10}{$y_{nb}$}&\cdots&y_{nn}&E_n +\end{array} +\] +\[ + \underbrace{\mathbf z_b=(y_{bj})_{j\in I}}_{\text{row }b: + \text{ old parameters }R(E_b)} + \qquad + \underbrace{(y_{ib})_{i\in I}}_{\text{column }b: + \text{ old parameters }p} + \qquad + w=p_b=R(E_b)_b. +\] +\endgroup +\caption{The ordered square under $\lambda_Z$. The highlighted row +and column meet at $y_{bb}$, whose old distal parameter is $w$. +Both have law $\lambda_Y$: the row by the single-slot marginal, +the column by the factor map $Z\to Y$. No invariance under +transposition is asserted.} +\label{fig:square} +\end{figure} diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/main.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..21f161885a25426839573c811cbef40aa9c113aa --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/main.tex @@ -0,0 +1,74 @@ +\newcommand{\DoNotLoadEpstopdf}{} +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs,array} +\usepackage{xcolor} +\usepackage{tikz} +\usetikzlibrary{matrix,fit,positioning,backgrounds} +\usepackage{needspace} +\usepackage[hidelinks,unicode]{hyperref} +\input{glyphtounicode} +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\pdftrailerid{} +\hypersetup{pdftitle={Pointwise Multiple Ergodic Averages for Mixing Transformations},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\C}{\mathbb C} +\newcommand{\T}{\mathbb T} +\newcommand{\E}{\mathbb E} +\newcommand{\dd}{\,d} +\newcommand{\law}{\mathcal L} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\ip}[2]{\langle#1,#2\rangle} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Ran}{Ran} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\spanop}{span} +\newcommand{\articleid}[1]{\begingroup\hyphenchar\font=-1\relax\textnormal{#1}\endgroup} +\setlist[enumerate]{itemsep=3pt,topsep=5pt} +\setlist[itemize]{itemsep=3pt,topsep=5pt} +\numberwithin{equation}{section} +\title{Pointwise Multiple Ergodic Averages\\for Mixing Transformations} +\author{OpenAI} +\date{October 4, 2026} +\begin{document} +\maketitle +\begin{abstract} +Let $T$ be an invertible mixing probability-preserving transformation. +For every integer $n\ge2$ and every fixed tuple of bounded measurable +functions, we prove that the consecutive multiple ergodic averages of +length $n$ converge almost everywhere to the product of the integrals, +as the averaging length tends to infinity through all positive integers. +The probability space need not be standard, and no rate of mixing is +required. +\end{abstract} +\input{sections/introduction} +\input{sections/inputs} +\input{sections/selected} +\input{sections/channels} +\input{sections/correlations} +\input{sections/square} +\appendix +\input{sections/packing} +\bibliographystyle{amsplain} +\begingroup\raggedright +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..9ab6929fedc3622dccab79a70a589e23d15591d4 --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/references.bib @@ -0,0 +1,183 @@ +@misc{MixingCompanion, + author = {{OpenAI}}, + title = {{Rokhlin's multiple-mixing problem for one transformation}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf}{OAI:Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026}}, + year = {2026} +} + +@misc{FourfoldCompanion, + author = {{OpenAI}}, + title = {{Pointwise convergence of fourfold ergodic averages for mixing transformations}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf}{OAI:Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026}}, + year = {2026} +} + +@misc{SlopesCompanion, + author = {{OpenAI}}, + title = {{Triple ergodic averages with distinct integer slopes}}, + howpublished = {OpenAI Math Release preprint + 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Math.}, + volume = {20}, + number = {3}, + year = {1976}, + pages = {373--409}, + doi = {10.1215/ijm/1256049780} +} + +@article{Jamneshan2023, + author = {Jamneshan, A.}, + title = {An uncountable {Furstenberg--Zimmer} structure theory}, + journal = {Ergodic Theory Dynam. Systems}, + volume = {43}, + number = {7}, + year = {2023}, + pages = {2404--2436}, + doi = {10.1017/etds.2022.43}, + note = {Corrected version: arXiv:2103.17167v4} +} + +@article{JamneshanCorrigendum2026, + author = {Jamneshan, A.}, + title = {An uncountable {Furstenberg--Zimmer} structure theory---{Corrigendum}}, + journal = {Ergodic Theory Dynam. Systems}, + volume = {46}, + number = {1}, + year = {2026}, + pages = {211--217}, + doi = {10.1017/etds.2025.10214} +} + +@article{Fernique1974, + author = {Fernique, X.}, + title = {Minorations des fonctions al{\'e}atoires gaussiennes}, + journal = {Ann. Inst. Fourier (Grenoble)}, + volume = {24}, + number = {2}, + year = {1974}, + pages = {61--66}, + doi = {10.5802/aif.506} +} + +@article{Sudakov1971, + author = {Sudakov, V. N.}, + title = {{Gaussian} random processes and measures of solid angles in {Hilbert} space}, + journal = {Dokl. Akad. Nauk SSSR}, + volume = {197}, + number = {1}, + year = {1971}, + pages = {43--45}, + note = {Russian} +} + +@article{Tropp2018, + author = {Tropp, J. A.}, + title = {Second-order matrix concentration inequalities}, + journal = {Appl. Comput. Harmon. Anal.}, + volume = {44}, + number = {3}, + year = {2018}, + pages = {700--736}, + doi = {10.1016/j.acha.2016.07.005} +} diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/channels.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/channels.tex new file mode 100644 index 0000000000000000000000000000000000000000..d098cf0e18d0949eb286d6c94dd93840c484edba --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/channels.tex @@ -0,0 +1,288 @@ +\section{Channels and closure under selected laws}\label{sec:channels} + +The selected length is determined by the original point in the suspension. +It can therefore change the joint law of other coordinates, even when +their unconditioned law is a product. We organize those coordinates in +three groups. Independence of the original point from any two groups +will control the effect of selecting a length on all three groups. +This channel construction and its closure argument adapt +\cite[Section~5]{FourfoldCompanion}; the proof below permits every fixed +finite set of slots $I=\{1,\ldots,n\}$. + +\begin{definition}[Channel]\label{def:channel} +A \emph{channel} is a flow $Y$ over the suspension $C$, with factor map +$\pi_Y:Y\to C$, together with a finite list of probability-preserving +maps $D_\gamma:Y\to C$, called \emph{outputs}, and positive rational +numbers $q_\gamma$, such that +\[ + D_\gamma(S_Y^t y)=S_C^{q_\gamma t}D_\gamma(y) +\] +for each fixed $t$, modulo null sets. Partition the output indices into +three groups, numbered $1,2,3$; empty groups are allowed. Set +\[ + x_0=\xi\circ\pi_Y,\qquad x_\gamma=\xi\circ D_\gamma, + \qquad d_Y=(x_\gamma)_\gamma, +\] +and let $d_{Y,H}$ be the sublist from groups in +$H\subseteq\{1,2,3\}$. Let $V$ collect the phase coordinates of +$\pi_Y$ and all the $D_\gamma$. Under the measure $m_Y$ require +\begin{equation}\label{eq:channel-laws} + \begin{split} + \law(d_Y\mid V)&=\bigotimes_\gamma\mu,\\ + \law(x_0,d_{Y,H}\mid V)&= + \mu\otimes\bigotimes_{\gamma\text{ in groups }H}\mu, + \qquad H\subsetneq\{1,2,3\}. + \end{split} +\end{equation} +An empty spatial list has the one-point law. +\end{definition} + +Thus the spatial outputs have a product law independent of the phases. +The original spatial coordinate has a product law with the outputs in +any proper collection of groups, but no independence from the complete +output list is required. The suspension $C$, with no outputs, is a +channel. + +\begin{lemma}[Conditioning and extensions]\label{lem:channel-conditioning} +For a channel $Y$, both identities in \eqref{eq:channel-laws} hold with +conditioning on its maximal measure-distal factor $W_Y$ in place of +$V$. Every flow extension of $Y$ over $C$ is a channel with the pulled +back output data, and the same strengthening holds there. +\end{lemma} + +\begin{proof} +Choose a positive integer $h$ such that $hq_\gamma$ is an integer for +each output. The phase vector $V$ is a factor fixed by time $h$, and +its time-one action is distal, so $V$ belongs to $W_Y$. For any spatial +tuple appearing in \eqref{eq:channel-laws}, its joint factor with $V$ +has product law over $V$. At time $h$ the spatial coordinates evolve +by a product of positive integer powers of $T$. This product is mixing: +the assertion follows first for product tests from mixing of each +power, and then for all bounded tests by approximation. Consequently +this factor is relatively weakly mixing over $V$ for time $h$. + +On the other hand, $W_Y$ is distal for time $h$ and relatively distal +over $V$. Relative disjointness, in the form of +\cite[Proposition~2.2]{FourfoldCompanion}, makes the spatial factor and +$W_Y$ relatively independent over $V$. Its conditional spatial law +given $W_Y$ is therefore the product law in +\eqref{eq:channel-laws}. Pullback to an extension preserves the joint +laws of the original data, proving the channel identities there; the +same argument gives their stronger conditioning. +\end{proof} + +We now prove the closure property that will permit repeated extensions. +Recall from Lemma~\ref{lem:normalized-joining} that +$\mathsf M_b(Y)$ has state space $Y^I$, measure $\lambda_Y$, and flow +$(y_i)_i\mapsto(S_Y^{it/b}y_i)_i$. + +\begin{proposition}[Channel closure]\label{prop:channel-closure} +Let $Y$ be a channel and $b\in I$. On $\mathsf M_b(Y)$ take +$\pi_Y(y_b)$ as original factor and take every $D_\gamma(y_i)$, +$i\in I$, as an output, in the old group of $\gamma$. +These data make $\mathsf M_b(Y)$ a channel. +\end{proposition} + +\begin{proof} +Each new output is probability-preserving and has positive rational +speed $iq_\gamma/b$. It remains to prove the two conditional laws. +Test them with products of bounded spatial functions and products of +bounded phase functions. Such tests determine the laws. Expanding +each spatial function into its mean and centered part, and scaling, +reduces the proof to real spatial functions that are either $1$ or +centered and bounded in absolute value by $1$. We may also bound every +phase function by $1$. Whenever at least one spatial factor is +centered, the tested integral must be zero. A test using the new +original coordinate uses outputs from only a proper set of groups. + +\paragraph{Time intervals and offsets.} +Use the selected-law formula \eqref{eq:selected-law} and condition on +the old phase vector $V$ under $m_Y$. Fix an integer $h>0$ clearing +all output-speed denominators and write +$t=hl+s$, where $l\ge0$ is an integer and $0\le s0$, after omitting finitely many initial indices, the +families $(u_l)$ and $(v_l)$ each have absolute pair inner products at +most $\eta$ outside a graph of bounded maximum degree $L_0$. +All these bounds are uniform in the offset patterns. To verify this, +in each of the two groups choose a centered occurrence at some output +index $\gamma$. Its coordinate integral in a pair inner product is +small by mixing of all orders whenever the times in the two copies of +\eqref{eq:channel-times} are sufficiently separated. The times within +each copy separate once $l,l'$ exceed a fixed threshold. Failure of +separation between copies then requires one of finitely many inequalities +\[ + \abs{e_\gamma i l-e_\gamma i'l'+a'}\le R. +\] +For fixed $l$, each inequality permits only a bounded number of +integers $l'$, because $e_\gamma,i'>0$; the same holds with $l,l'$ +interchanged. Their union is the required bounded-degree graph. +The coordinate integrals for other output indices are bounded by $1$. +The separation threshold and $L_0$ may depend on $\eta$, but no rate +of mixing is needed. + +\paragraph{Conditioning on a selected length.} +For fixed selector index $a$, partition into the events +$N_a=c2^k$, $c\in F$, $k\ge a$, choosing one representation when a +length has more than one. Given $V$, write their probabilities as +$p_{k,c}(V)$. On one such event of probability $p>0$, the conditional +spatial law of any two groups, and of each single group, is unchanged. +Indeed the event is determined by $V,x_0$, which is independent of any +proper set of groups by \eqref{eq:channel-laws}. The full three-group +law may change. Its density $\rho$ relative to the product law satisfies +\begin{equation}\label{eq:channel-density} + 0\le p\rho\le1,\qquad + \int p\rho=p,\qquad \norm{p\rho}_2\le\sqrt p. +\end{equation} +The reason is that the joint measure of the outputs and the event is +dominated by the unconditioned product measure. These statements hold +for almost every $V$; the length events are countable. + +We choose the rank bound to meet two requirements: the packing +bound must be $o(m)$, and the reciprocal bounds must be summable over +dyadic length scales. For $m=\lfloor c2^k/h\rfloor$ set +\begin{equation}\label{eq:channel-dimension} + d_m=\left\lceil(1+\log m)^{3/2}\right\rceil, + \qquad z_l=u_l\otimes v_l\quad(0\le l0$ there is such a profile with +\[ + \norm{\varphi-c^{-1}\mathbf1_{(0,c]}}_{L^1(\R)}<\epsilon. +\] +\end{proposition} +The approximation is \cite[Lemma~3.9]{SlopesCompanion}. Neither part +requires mixing or ergodicity. We will not assume pointwise convergence +of longer averages on arbitrary systems. + +\subsection{Relative weak mixing and distal factors} + +We recall the structure facts in the form needed for the auxiliary +systems. An extension of an invertible system $R$ over a factor $B$ +is \emph{relatively weakly mixing} if, for bounded $f,g$ with +$\E[f\mid B]=0$, +\begin{equation}\label{eq:rwm} + \frac1H\sum_{h=1}^H + \norm{\E[\overline f(g\circ R^h)\mid B]}_2^2\longrightarrow0. +\end{equation} +An extension is \emph{relatively compact} if the invariant finitely +generated $L^\infty(B)$-modules span a dense subspace of its $L^2$ +space. A module consists of finite sums $\sum_j a_jF_j$, with +$a_j\in L^\infty(B)$ and fixed $F_j\in L^2$; invariance is under +composition with both $R$ and $R^{-1}$. Individual modules need not be +closed; see the corrected module characterization in +\cite[Theorem~2.5(ii)$'$]{JamneshanCorrigendum2026}. +A \emph{relatively distal} extension is a tower of relatively +compact extensions and inverse limits. A system is \emph{distal} if +it is distal over the trivial factor. A joining is an invariant +probability law on a product with the prescribed marginals. + +\begin{proposition}[Relative structure]\label{prop:structure} +The following statements hold for standard probability-preserving +systems, without an ergodicity assumption. +\begin{enumerate} +\item Every system $Y$ has a maximal measure-distal factor $W_Y$, +and $Y\to W_Y$ is relatively weakly mixing. Every distal factor +pulls into the maximal distal factor of an extension. Automorphisms +commuting with the transformation preserve $W_Y$. +\item Factors and joinings of distal systems are distal. A distal +system is relatively distal over each of its factors. Distality is +equivalent for a transformation and any of its nonzero integer powers. +\item Relative weak mixing passes to nonzero integer powers. The +conditional product of relatively weakly mixing fibre laws over an +invariant joining of their base laws is relatively weakly mixing +over that joining. +\item A relatively weakly mixing extension and a relatively distal +extension of the same base are relatively disjoint: every invariant +joining identifying their bases is the conditional product over +that base. +\item For a distal transformation $R$, every $d\ge1$, and bounded +$g_1,\ldots,g_d$, the averages +\[ + \frac1N\sum_{l=1}^N\prod_{i=1}^d g_i\circ R^{il} +\] +converge almost everywhere. +\end{enumerate} +For a flow, the maximal distal factor of time one is preserved by +the flow, and its action at every nonzero rational time is distal. +\end{proposition} + +These are the conclusions of +\cite[Proposition~2.2]{FourfoldCompanion}. They include the +nonergodic form of the Furstenberg--Zimmer structure theorem +\cite{Zimmer1976,Jamneshan2023} and the pointwise theorem for +distal systems \cite[Theorem~C]{HuangShaoYe2019}. The cited proposition +gives the component-disintegration argument needed for the latter +in the nonergodic setting. In the flow assertion, invariance of +$W_Y$ follows because each time commutes with time one. If +$r=p/q\ne0$ is rational, the $q$th power of its action is the +distal action at integer time $p$, so the power assertion applies. + +The following short argument, from +\cite[Lemma~2.3]{FourfoldCompanion}, will also fix the norm-convergence +input explicitly. +\begin{lemma}[Relative norm convergence]\label{lem:relative-norm} +Suppose $Y\to B$ is relatively weakly mixing for $R$. For distinct +nonzero integers $a_1,\ldots,a_d$ and bounded $G_1,\ldots,G_d$, +\begin{equation}\label{eq:relative-norm} + \left\|\frac1N\sum_{l=1}^N + \left(\prod_{i=1}^dG_i\circ R^{a_il} + -\prod_{i=1}^d\E[G_i\mid B]\circ R^{a_il}\right)\right\|_2 + \longrightarrow0. +\end{equation} +\end{lemma} +\begin{proof} +By telescoping and scaling, it suffices to show that the average of +$v_l=\prod_i G_i\circ R^{a_il}$ tends to zero when all functions +are bounded by one and one of them, $G_s$, is centered over $B$. +Induct on $d$. For fixed $h\ge1$ put +$F_{i,h}=\overline{G_i}(G_i\circ R^{a_ih})$. Invariance gives +\[ + \ip{v_l}{v_{l+h}} + =\int_Y F_{s,h}\prod_{i\ne s} + F_{i,h}\circ R^{(a_i-a_s)l}\dd m_Y. +\] +The induction hypothesis replaces the factors indexed by $i\ne s$ +in their average by their conditional expectations over $B$. The +remaining product is $B$-measurable and bounded by one; hence +\[ + \limsup_{N\to\infty} + \left|\frac1N\sum_{l=1}^N\ip{v_l}{v_{l+h}}\right| + \le\norm{\E[F_{s,h}\mid B]}_1. +\] +For $d=1$ this bound holds directly. The Hilbert-space van der +Corput inequality now bounds the limiting squared norm by a constant +times +\[ + \frac1H+\frac1H\sum_{h=1}^H\norm{\E[F_{s,h}\mid B]}_1. +\] +Relative weak mixing for $R^{a_s}$ and Cauchy--Schwarz make this +expression tend to zero as $H\to\infty$. +\end{proof} + +For positive slopes, inserting functions $1$ in omitted positions +and using Proposition~\ref{prop:structure}(5) shows that the +distal-factor averages have an $L^2$ limit. Thus +Lemma~\ref{lem:relative-norm} gives an $L^2$ limit for arbitrary +bounded tuples with distinct positive slopes on every standard +system. On the original mixing system, the trivial factor already +satisfies \eqref{eq:rwm}; the limit is the product of the means. + +\subsection{Triple convergence along all lengths} + +\begin{lemma}[Triple pointwise convergence]\label{lem:triple-convergence} +On any standard invertible probability-preserving system, averages +of at most three bounded functions at distinct positive integer +slopes converge almost everywhere along all positive integer lengths. +\end{lemma} +\begin{proof} +Add constant-one functions if necessary to have three distinct +positive slopes, write them as $gb_1,gb_2,gb_3$ with +$\gcd(b_1,b_2,b_3)=1$, and apply +Proposition~\ref{prop:smooth-input} to $S=R^g$. By scaling, suppose +the three functions are bounded by one, and put +$F_l=\prod_{j=1}^3g_j\circ S^{b_jl}$. + +For each rational $c\in[1,2]$, choose a sequence of profiles tending +in $L^1$ to $h_c=c^{-1}\mathbf1_{(0,c]}$. On a common conull set +all the corresponding smooth dyadic averages converge. For +$N_k=\lfloor c2^k\rfloor$, the difference between the ordinary +average at $N_k$ and $V_k^\varphi$ is at most +\[ + \frac1{c2^k} + +2^{-k}\sum_{l\in\Z}|h_c(l/2^k)-\varphi(l/2^k)|. +\] +The last sum tends to $\norm{h_c-\varphi}_1$ by Riemann sums. +Arbitrarily close profiles therefore give the Cauchy property on +each of these scales. The $L^2$ limit established above identifies +all their almost-everywhere limits with the same function. + +For any bounded sequence $|F_l|\le1$, changing the averaging length +from $N$ to $M$ changes the average by at most +\begin{equation}\label{eq:length-change} + \frac{2|M-N|}{\max(M,N)}. +\end{equation} +For $2^k\le N<2^{k+1}$, compare $N$ with +$\lfloor c2^k\rfloor$ on a finite rational grid in $[1,2]$. +Equation~\eqref{eq:length-change} makes the limiting error +arbitrarily small as the grid mesh tends to zero. This proves +convergence along all integer lengths. +\end{proof} diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a6df9e8a343b9ec676ebac165d69a36af7fca396 --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/introduction.tex @@ -0,0 +1,117 @@ +\section{Introduction} +\label{sec:introduction} + +Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$ +be invertible, bimeasurable, and measure preserving. We say that $T$ +is \emph{mixing} if +\begin{equation}\label{eq:mixing} + \mu(A\cap T^{-r}B)\longrightarrow\mu(A)\mu(B) + \qquad (|r|\to\infty) +\end{equation} +for every $A,B\in\mathcal F$. For a fixed tuple of bounded measurable +functions, the consecutive multiple ergodic averages are +\[ + A_N(f_1,\ldots,f_n)(x) + =\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x). +\] +The problem is to determine their almost-everywhere behavior, not merely +their limit in norm. We prove the following result. + +\begin{theorem}\label{thm:main} +Suppose that $T$ satisfies \eqref{eq:mixing}. For every integer +$n\ge2$ and every fixed $f_1,\ldots,f_n\in L^\infty(\mu)$, +\[ + A_N(f_1,\ldots,f_n)(x) + \longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu + \qquad\text{for $\mu$-almost every }x, +\] +as $N$ tends to infinity through all positive integers. The probability +space need not be standard, and no rate of mixing is required. +\end{theorem} + +The exceptional null set may depend on the system and the fixed tuple. +The theorem concerns the times $k,2k,\ldots,nk$ and does not require +a common exceptional set for all bounded functions. + +\subsection{Context and prior work} + +Arithmetic-progression averages became central to ergodic Ramsey theory +through Furstenberg's proof of Szemer\'edi's theorem +\cite{Furstenberg1977}. The distinction between norm and pointwise +convergence is substantial. Host and Kra proved $L^2$ convergence of +these averages for every finite length on arbitrary +probability-preserving systems \cite[Theorem~1.1]{HostKra2005}. +Ziegler obtained another proof through universal characteristic +factors \cite[Corollary~1.8]{Ziegler2007}. +For pointwise convergence, Bourgain's double recurrence theorem treats +two factors \cite{Bourgain1990}. Higher-length results have been +obtained under additional structural assumptions: K-systems +\cite[Theorem~4.2]{DerrienLesigne1996}, weakly mixing systems with +singular spectrum on their Pinsker factor \cite{Assani1998}, and +weakly mixing systems whose pairwise independent self-joinings are +independent \cite[Theorem~3.4]{GutmanHuangShaoYe2018}. Huang, Shao, +and Ye proved pointwise convergence for every finite length on +ergodic measure-distal systems \cite[Theorem~C]{HuangShaoYe2019}. + +The present proof extends the fourfold mixing theorem of +\cite[Theorem~1.1]{FourfoldCompanion}. It uses two substantial +companion results. The first proves that ordinary mixing implies +mixing of every finite order, the assertion of Rokhlin's +multiple-mixing problem \cite[Theorem~1.1]{MixingCompanion}. +The second gives an oscillation estimate for triple averages at +distinct integer slopes on arbitrary invertible probability-preserving +systems \cite[Proposition~3.5]{SlopesCompanion}; together with its +profile approximation, this yields pointwise triple convergence +at distinct positive integer slopes without a mixing assumption. +Section~\ref{sec:inputs} states their exact forms and derives the +required triple pointwise theorem along all lengths. These companion +proofs are not reproduced here. The selected-law construction, +arbitrary-length extension argument, and tensor estimate used below +are proved in full. + +\subsection{The arbitrary-length step} + +We briefly describe what must change when the number of factors +increases. A failure of pointwise convergence allows an averaging +length to be chosen measurably at each starting point so that a +centered average stays positive. After passing to a suspension flow, +these choices define a probability law on $n$ output coordinates. +The law has the correct individual marginals and is invariant under +different speeds in the different coordinates. The same construction +works on extensions of the flow. + +The method of \cite[Sections~3--6]{FourfoldCompanion} organizes +additional outputs into three groups. Their spatial coordinates are +independent, and the original spatial coordinate is independent of +any two groups; independence from all three groups is not imposed. +We retain this three-group structure even when there are $n$ slots. +Higher-order mixing and a tensor-packing estimate show that these +independence properties survive the selected joining construction. + +Two changes make this approach work for arbitrary $n$. First, we +minimize the number of slots supporting a nonzero centered correlation. +This gives independence for every smaller collection of the specific +spatial lists being tested. A slot may read both the original spatial +coordinate and the outputs in any proper subset of the three groups. +That flexibility permits the outputs to be regrouped without imposing +independence on arbitrary collections of whole states. + +Second, an extension is chosen so that conditioning on all distal +parameters together with a single state cannot improve under any +further extension. This is a conditional-expectation energy +construction, related to the sated-extension method +\cite{Austin2015} and adapted from +\cite[Section~6]{FourfoldCompanion}. Applying the joining construction +again gives a square array. The single-state conditioning identity +controls its distinguished row and column. Their parameter laws and +a Hilbert-space argument then produce a new nonzero correlation with +the same number of slots but one more output-only slot. Maximizing +that number gives the contradiction. Only the pointwise theorem for +at most three slots on arbitrary systems is used. + +Section~\ref{sec:inputs} records the dynamical inputs. +Section~\ref{sec:selected} constructs the selected laws and their +conditional marginals. Section~\ref{sec:channels} proves preservation +of the three-group structure. Sections~\ref{sec:correlations} +and~\ref{sec:square} carry out the minimal-correlation and square +arguments. Appendix~\ref{sec:packing} proves the tensor estimate. diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/packing.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/packing.tex new file mode 100644 index 0000000000000000000000000000000000000000..8482892e28ff63658f4c2e07fe5c77d187c8b28a --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/packing.tex @@ -0,0 +1,222 @@ +\section{Tensor packing}\label{sec:packing} + +The closure proof uses a deterministic fact: a subspace of moderately +growing dimension captures a substantial part of only a small fraction +of an almost orthogonal tensor family. We reproduce the argument of +\cite[Theorem~4.1]{FourfoldCompanion}, with the same quantitative +estimate. Throughout this appendix, Hilbert spaces are real, tensor +products are Hilbert tensor products, and logarithms are natural. + +\begin{theorem}[Tensor packing]\label{thm:packing} +There is a universal constant $C$ with the following property. Let +$0<\varepsilon\le1$, and let $d\ge1$ and $L\ge0$ be integers. +Let $x_1,\ldots,x_m$ and $y_1,\ldots,y_m$ lie in the unit balls of +real Hilbert spaces $H_1,H_2$. For each family suppose that a graph +on $\{1,\ldots,m\}$ of maximum degree at most $L$ contains every +off-diagonal pair whose absolute inner product exceeds $\eta$, where +\[ + 0\le\eta\le\varepsilon^2/10^4. +\] +The two exceptional graphs may differ. For every orthogonal projection +$P$ on $H_1\otimes H_2$ of rank at most $d$, +\begin{equation}\label{eq:packing-quantitative} + \#\{j:\norm{P(x_j\otimes y_j)}\ge\varepsilon\} + \le C(1+L)\varepsilon^{-2} + \exp\!\left(C\varepsilon^{-2}\sqrt d\log(2+d)\right). +\end{equation} +For $d=d_m$ as defined in \eqref{eq:channel-dimension}, this is +$o_{\varepsilon,L}(m)$, uniformly over the Hilbert spaces, vectors, +projections, and allowed values of $\eta$. +\end{theorem} + +The proof separates directions on which the testing space has large +variance and applies Gaussian comparison to the remaining directions. +The comparison is the finite Sudakov--Fernique inequality; see +\cite[Section~1, Equation~(1.2)]{Fernique1974} and +\cite{Sudakov1971}. The following matrix estimate uses the Gaussian +integration-by-parts moment argument behind matrix Khintchine +inequalities; compare \cite[Lemma~6.1 and Proposition~7.3]{Tropp2018}. +We include its proof to retain dependence on the covariance trace, +rather than on the ambient dimensions. + +\begin{lemma}[Gaussian matrix estimate]\label{lem:packing-gaussian} +Let $B_1,\ldots,B_r$ be matrices between two finite-dimensional real +Hilbert spaces, and put +\[ + H_j=\begin{pmatrix}0&B_j\\B_j^*&0\end{pmatrix}, + \qquad S=\sum_{j=1}^rH_j^2. +\] +If $\norm{S}_{\rm op}\le\sqrt d$ and $\tr S\le2d$ for some +$d\ge1$, then for independent standard normal variables $g_j$, +\begin{equation}\label{eq:packing-gaussian-bound} + \E\left\|\sum_{j=1}^rg_jB_j\right\|_{\rm op} + \le C d^{1/4}\sqrt{\log(2+d)}. +\end{equation} +The constant is independent of the dimensions of the two spaces. +\end{lemma} + +\begin{proof} +Write $H=\sum_jg_jH_j$. Gaussian integration by parts, followed by +differentiation of the matrix power, gives for every integer $q\ge2$ +\[ + \E\tr H^{2q} + =\sum_{j=1}^r\sum_{a=0}^{2q-2} + \E\tr(H_jH^aH_jH^{2q-2-a}). +\] +For a fixed self-adjoint $H$ with eigenvalues $\lambda_u$ and a +self-adjoint matrix $K$, the absolute value of such a trace is at most +\[ + \sum_{u,v}|K_{uv}|^2 + |\lambda_v|^a|\lambda_u|^{2q-2-a} + \le\tr(K^2|H|^{2q-2}). +\] +The inequality follows from weighted arithmetic--geometric mean and +the symmetry $|K_{uv}|=|K_{vu}|$. Summing over $j$ gives +\begin{equation}\label{eq:packing-moment-recursion} + \E\tr H^{2q} + \le(2q-1)\norm{S}_{\rm op}\E\tr|H|^{2q-2}. +\end{equation} +Since $\E\tr H^2=\tr S$, iteration yields +\[ + \E\tr H^{2q} + \le(2q-1)!!\,\norm{S}_{\rm op}^{q-1}\tr S. +\] +The operator norm of $H$ equals that of $\sum_jg_jB_j$. +Taking a $2q$-th root and using the hypotheses bounds its expectation by +\[ + C\sqrt q\,d^{1/4}(2\sqrt d)^{1/(2q)}. +\] +Choose an integer $q\ge2$ comparable to $\log(2+d)$ to obtain +\eqref{eq:packing-gaussian-bound}. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:packing}] +We first record an elementary consequence of the exceptional-graph +hypothesis. If unit-ball vectors $v_j$ have absolute inner products +at most $\eta$ outside a graph of degree $L$, then for every nonempty +set $A$ of indices, +\begin{equation}\label{eq:packing-cluster} + \left\|\frac1{|A|}\sum_{j\in A}v_j\right\|^2 + \le\frac{1+L}{|A|}+\eta. +\end{equation} +Indeed the expansion contains at most $(1+L)|A|$ ordered diagonal or +exceptional pairs. A lower bound on the norm of an average therefore +bounds the number of vectors in that average. + +It suffices to work in the finite spans of the $x_j$ and $y_j$. +To see this, let $Q$ project onto their tensor product and replace the +testing subspace $\operatorname{Ran}P$ by $Q(\operatorname{Ran}P)$. +For any vector $z$ in that tensor product, the norm of its projection +onto the new subspace is at least $\norm{Pz}$: in the supremum defining +$\norm{Pz}$, replace each unit testing vector by its $Q$-image, whose +norm is at most one and whose pairing with $z$ is unchanged. The +dimension does not increase. + +Identify tensors with matrices from $H_2$ to $H_1$. Choose a +Hilbert--Schmidt orthonormal basis $A_1,\ldots,A_r$ of the testing +space, where $r\le d$. The matrices +\[ + R_1=\sum_jA_jA_j^*,\qquad R_2=\sum_jA_j^*A_j +\] +have trace $r$. Let $E_1,E_2$ be their spectral subspaces for +eigenvalues strictly greater than $\sqrt d$; each has dimension at most +$\sqrt d$. + +Discard indices for which the projection of $x_j$ onto $E_1$ has norm +greater than $\varepsilon/10$, and do the same for $y_j,E_2$. +Cover either exceptional subspace's unit ball by at most +$(1+80/\varepsilon)^{\sqrt d}$ balls of radius $\varepsilon/40$. +If one ball contains $s$ of the discarded projections, their average +has norm at least $\varepsilon/20$. Applying +\eqref{eq:packing-cluster} to the original vectors, with +$\eta\le\varepsilon^2/10^4$, gives +$s\le C(1+L)\varepsilon^{-2}$. The number of discarded indices is +therefore at most +\begin{equation}\label{eq:packing-discarded} + C(1+L)\varepsilon^{-2} + \exp\!\bigl(\sqrt d\log(1+80/\varepsilon)\bigr). +\end{equation} + +Let $Q_1,Q_2$ project onto $E_1^\perp,E_2^\perp$, and put +$B_j=Q_1A_jQ_2$. The coefficient map +$z\mapsto(\ip{A_j}{z}_{\rm HS})_{j=1}^r$ is a contraction. +For a remaining index, +\[ + \norm{x_j\otimes y_j-Q_1x_j\otimes Q_2y_j} + \le\norm{(1-Q_1)x_j}+\norm{(1-Q_2)y_j} + \le\varepsilon/5. +\] +Thus each remaining index counted in +\eqref{eq:packing-quantitative} has coefficient vector +\[ + v_j=(\ip{x_j}{B_ay_j})_{a=1}^r, + \qquad\norm{v_j}\ge\varepsilon/2. +\] +The map from $x_j\otimes y_j$ to $v_j$ remains contractive. +Outside the union of the two exceptional graphs, the tensor inner +products have absolute value at most $\eta^2$, and that union has +degree at most $2L$. A collection of these coefficient vectors in a +ball of radius $\varepsilon/8$ centered at one of them has average +norm at least $3\varepsilon/8$. Applying +\eqref{eq:packing-cluster} to its preimage tensors bounds its size by +$C(1+L)\varepsilon^{-2}$. A maximal $\varepsilon/8$-separated subset +therefore has cardinality $l$ such that the number of indices still +to be counted is at most $C(1+L)\varepsilon^{-2}l$. + +We now bound this separated set. For $G=\sum_ag_aB_a$, the centered +Gaussian variables $Z_j=\ip{x_j}{Gy_j}$ satisfy +\[ + \E|Z_j-Z_{j'}|^2=\norm{v_j-v_{j'}}^2. +\] +Gaussian comparison gives, when $l\ge2$, +\begin{equation}\label{eq:packing-comparison} + c\varepsilon\sqrt{\log l} + \le\E\max_j Z_j\le\E\norm{G}_{\rm op}. +\end{equation} +For completeness, the comparison of finite centered Gaussian families +follows by applying Gaussian integration by parts to +$\beta^{-1}\log\sum_j e^{\beta z_j}$ along their independent Gaussian +interpolation. Its derivative is +\[ + \frac\beta4\E\sum_{j,j'}p_jp_{j'} + \left(\E|Z_j-Z_{j'}|^2-\E|Z'_j-Z'_{j'}|^2\right), + \qquad p_j=\frac{e^{\beta z_j}}{\sum_a e^{\beta z_a}}. +\] +It is nonnegative when the first increment variances dominate. +Letting $\beta\to\infty$ compares the expected maxima. For +\eqref{eq:packing-comparison}, take the $Z'_j$ to be independent +normals of variance $\varepsilon^2/128$ and use +$\E\max_{j\le l}g_j\ge c\sqrt{\log l}$. + +Compression gives +\[ + \sum_jB_jB_j^*\le Q_1R_1Q_1,\qquad + \sum_jB_j^*B_j\le Q_2R_2Q_2. +\] +These operators have norms at most $\sqrt d$ and traces at most $d$. +Lemma~\ref{lem:packing-gaussian} therefore applies. Together with +\eqref{eq:packing-comparison} it yields +\[ + \log l\le C\varepsilon^{-2}\sqrt d\log(2+d). +\] +The resulting bound holds for $l\le1$ as well. Combine it with +\eqref{eq:packing-discarded} and absorb +$\log(1+80/\varepsilon)\le C\varepsilon^{-2}$ to prove +\eqref{eq:packing-quantitative}. + +For $d=d_m$, the exponent is +$O_\varepsilon((\log m)^{3/4}\log\log m)=o(\log m)$, +which proves the asserted $o(m)$ bound. +\end{proof} + +Since projection norms are at most one, the estimate also gives +\begin{equation}\label{eq:packing-average} + \frac1m\sum_{j=1}^m\norm{P(x_j\otimes y_j)} + \le\varepsilon+ + \frac{C(1+L)}{m\varepsilon^2} + \exp\!\left(C\varepsilon^{-2}\sqrt{d_m}\log(2+d_m)\right) +\end{equation} +when $\operatorname{rank}P\le d_m$. Here $\varepsilon$ and then +$\eta$ are fixed before $m$ increases; $L$ may depend on that chosen +$\eta$. This is exactly the order of quantifiers supplied by mixing +in Proposition~\ref{prop:channel-closure}. diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/selected.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/selected.tex new file mode 100644 index 0000000000000000000000000000000000000000..ab15d2f60ab5359169e78e2a0cfc4537847a547f --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/selected.tex @@ -0,0 +1,349 @@ +\section{Selected lengths and their joining laws}\label{sec:selected} + +We assume that the asserted pointwise convergence fails for the fixed +set of slopes $I=\{1,\ldots,n\}$. This section records that failure in +a probability law and constructs compatible laws on all flow extensions. +The construction retains a positive correlation while forcing every +one-, two-, or three-slot marginal to have its ordinary conditional law. +We follow the selector and joining construction in +\cite[Sections~3--6]{FourfoldCompanion}, keeping the extension and +conditioning properties explicit for the arbitrary number of slots. + +\subsection{A positive witness at selected lengths} + +Expanding each function into its mean and centered part, and then into +real and imaginary parts, gives a nonempty $J_0\subseteq I$ and real +functions $f_i$, $i\in J_0$, such that +\[ + \int_X f_i\,\dd\mu=0,\qquad \norm{f_i}_\infty\le1, + \qquad + B_L(x)=\frac1L\sum_{l=1}^L\prod_{i\in J_0}f_i(T^{il}x) +\] +does not tend to zero on a set of positive measure. Here and below +$B_L$ has an integer subscript. Lemma~\ref{lem:relative-norm}, applied +over the trivial factor, gives $B_L\to0$ in $L^2(\mu)$. After changing +the sign of one $f_i$, there are $\delta>0$ and a measurable set +$E_*\subseteq X$ of positive measure such that +\begin{equation}\label{eq:failure-witness} + B_L(x)>2\delta\quad\text{for arbitrarily large integers }L, + \qquad x\in E_*. +\end{equation} + +We use the roof-one suspension +\begin{equation}\label{eq:suspension} + C=X\times[0,1),\qquad m_C=\mu\otimes\dd u, + \qquad + S_C^t(x,u)=\bigl(T^{\lfloor u+t\rfloor}x,\{u+t\}\bigr). +\end{equation} +Write $\xi(x,u)=x$ and $\upsilon(x,u)=u$, regarding $\upsilon$ also as +a coordinate in the circle $\R/\Z$. + +\begin{lemma}[Selected positive witness]\label{lem:selected-witness} +There are a finite set $F\subseteq[1,2]$ containing $1$ and measurable +functions $N_a:C\to[2^a,\infty)$, $a\in\N$, independent of the phase, +with +\begin{equation}\label{eq:selector-grid} + N_a(x,u)\in\{c2^k:c\in F,\ k\ge a\},\qquad + N_a(x,u)=2^a\quad(x\notin E_*), +\end{equation} +such that $B_{\lfloor N_a(x,u)\rfloor}(x)>\delta$ on $E_*$. +Moreover, there is a continuous nonnegative function $\Psi$ on +$(\R/\Z)^I$ for which +\begin{equation}\label{eq:selected-positive-limit} + \liminf_{a\to\infty}\int_C\frac1{N_a(z)} + \int_0^{N_a(z)} + \Psi\bigl((\upsilon(S_C^{it}z))_{i\in I}\bigr) + \prod_{i\in J_0}f_i\bigl(\xi(S_C^{it}z)\bigr) + \,\dd t\,\dd m_C(z)>0. +\end{equation} +\end{lemma} + +\begin{proof} +By~\eqref{eq:length-change}, changing the length from $L$ to $M$ +changes an average of a sequence bounded by one by at most +$2\abs{M-L}/\max(M,L)$. Choose $F$ to be a sufficiently fine finite +grid in $[1,2]$. For every sufficiently large integer $L$, some +$c\in F$ and $k=\lfloor\log_2L\rfloor$ satisfy +\[ + \abs{B_L(x)-B_{\lfloor c2^k\rfloor}(x)}<\delta + \qquad\text{for every }x. +\] +Equation~\eqref{eq:failure-witness} therefore supplies, for each +$x\in E_*$ and $a$, a pair $(c,k)$ with $k\ge a$ at which the latter +average exceeds $\delta$. Choose the first such pair in a fixed +enumeration. This is a measurable choice from a countable set and +defines $N_a$ on $E_*$; use $2^a$ elsewhere. + +Choose continuous nonnegative circle functions $\alpha,\beta$, neither +identically zero, supported respectively inside $(1/4,1/2)$ and +$(0,1/(4n))$. Set +\begin{equation}\label{eq:phase-witness} + \Psi(v)=\alpha(2v_1-v_2)\beta(v_2-v_1), + \qquad v=(v_i)_{i\in I}\in(\R/\Z)^I. +\end{equation} +For $z=(x,u)$ and $t=l+s$, where $l\in\Z$ and $0\le s<1$, the +phase factor is $\alpha(u)\beta(s)$. If it is nonzero, then +$00. +\] +On $X\setminus E_*$ its integral tends to zero because $N_a=2^a$ +there and $B_{2^a}\to0$ in $L^2(\mu)$. The uniform error is +$O(2^{-a})$, proving~\eqref{eq:selected-positive-limit}. +\end{proof} + +\subsection{Selected laws on all extensions} + +Fix these selectors and a free ultrafilter $\mathcal U$ on $\N$. +A \emph{flow over $C$} is a standard probability space $(Y,m_Y)$ +with a jointly Borel probability-preserving action $S_Y^t$, $t\in\R$, +and a probability-preserving factor map $\pi_Y:Y\to C$ satisfying +$\pi_Y S_Y^t=S_C^t\pi_Y$ modulo null sets for each fixed $t$. +An extension $\rho:Y'\to Y$ of such flows is a probability-preserving +factor map, equivariant in the same sense, with +$\pi_{Y'}=\pi_Y\rho$ modulo null sets. We do not require a common +conull set on which every real-time factor identity holds. Joint +measurability and Fubini justify their uses inside time integrals. + +\begin{proposition}[Selected joining laws]\label{prop:selected-law} +Every flow $Y$ over $C$ has a unique probability law $\lambda_Y$ on +$Y^I$ satisfying, for all bounded measurable functions $G_i$ on $Y$, +\begin{equation}\label{eq:selected-law} + \int_{Y^I}\prod_{i\in I}G_i(y_i)\,\dd\lambda_Y + =\lim_{a\to\mathcal U}\int_Y\frac1{N_a(\pi_Yy)} + \int_0^{N_a(\pi_Yy)}\prod_{i\in I}G_i(S_Y^{it}y) + \,\dd t\,\dd m_Y(y). +\end{equation} +Its marginals are $m_Y$, and it is invariant under +$(y_i)_i\mapsto(S_Y^{it}y_i)_i$, $t\in\R$. +If $\rho:Y'\to Y$ is an extension over $C$, then +$(\rho^I)_*\lambda_{Y'}=\lambda_Y$. + +Countable inverse limits of extensions over $C$ exist in this +category. The selected law at such a limit projects to the selected +law at every stage. Pullbacks of the stage selected-law spaces have +dense union in $L^2$ of the limit selected law. +\end{proposition} + +\begin{proof} +For each $a$ the expression before the ultralimit defines a genuine +probability measure $\lambda_{Y,a}$ on $Y^I$, by integrating the +measurable orbit map over the indicated finite time interval. We +first show that each of its marginals converges against every bounded +measurable test $G$ to $m_Y$. The one-slot continuous-time averages +\[ + \frac1L\int_0^L G(S_Y^{it}y)\,\dd t +\] +converge almost surely as $L\to\infty$. For example, apply Birkhoff's +theorem for $S_Y^i$ to the function +$y\mapsto\int_0^1G(S_Y^{is}y)\,\dd s$ and discard the incomplete +unit interval. Their almost-sure limit has integral $\int G\,\dd m_Y$, +since every deterministic average has that integral. As +$N_a(\pi_Yy)\ge2^a$, bounded convergence gives the same integral +limit at the selected lengths. + +Embed $Y$ as a Borel subset of a compact metrizable space $K$. +Ultralimits of the $\lambda_{Y,a}$ against continuous functions on +$K^I$ give a positive normalized functional, hence a probability +measure on $K^I$. Its marginals are $m_Y$, by the preceding paragraph, +so it is concentrated on $Y^I$. Call the resulting law $\lambda_Y$. +To obtain~\eqref{eq:selected-law} for arbitrary bounded slot functions, +normalize their bounds to one and approximate each in $L^1(m_Y)$ by +continuous functions on $K$ with the same bound. The difference +between the two product integrals is bounded by the sum of the +single-slot errors. Under $\lambda_Y$ these errors are the +$L^1(m_Y)$ errors, while under $\lambda_{Y,a}$ they converge to those +errors by the marginal convergence already proved. Letting the +errors tend to zero proves the formula. Product tests determine a +probability measure on $Y^I$, giving uniqueness. In particular, all +these identities depend only on the almost-everywhere classes of +their slot functions. + +Applying the product action at a fixed time $t_0$ shifts the time +interval in~\eqref{eq:selected-law} by $t_0$ without changing its +length or its initial-point selector. For slot functions bounded by +one the error is at most $2\abs{t_0}2^{-a}$, proving invariance. +For an extension $\rho$, substitute $G_i\circ\rho$ in the formula. +Equivariance, the equality of the selectors, and Fubini identify its +right-hand side with that for $Y$, proving functoriality. + +For a tower $Y_0\leftarrow Y_1\leftarrow\cdots$, take the full +countable product of the underlying standard spaces, with the +consistent inverse-limit probability measure and coordinatewise +flow. This is a jointly Borel action. A cylinder through stage $r$ +has probability computed from $m_{Y_r}$ by the finitely many bonding +maps. At any fixed time their equivariance and preservation of +$m_{Y_r}$ leave that probability unchanged. Cylinders determine the +measure, so every fixed time preserves it. The measure is +concentrated on compatible sequences; using +the full product avoids any need for one compatibility set invariant +at all real times. The stage projections are factor maps modulo +null sets, and projection through $Y_0$ gives the factor onto $C$. +Functoriality now gives the asserted selected-law projections. +Finally, the stage coordinates generate the inverse-limit +sigma-algebra. The compatibility relations hold under the selected +law slot by slot because its marginals are the inverse-limit +probability measure. Thus the pulled-back stage sigma-algebras on +the selected-law space are increasing and generate it modulo null +sets. The usual $L^2$ approximation by an increasing sequence of +sigma-algebras proves density. +\end{proof} + +\subsection{Distal parameters and small marginals} + +For a flow $Y$ over $C$, let +$\omega_Y:Y\to W_Y$ be its maximal measure-distal factor for time one, +and let $\nu_Y=(\omega_Y)_*m_Y$. Write $m_{Y,w}$ for its conditional +probability measures, so that +\[ + m_Y=\int_{W_Y}m_{Y,w}\,\dd\nu_Y(w). +\] +Under $\lambda_Y$ put +\begin{equation}\label{eq:distal-parameters} + P_i=\omega_Y(y_i),\qquad P=(P_i)_{i\in I}. +\end{equation} +Proposition~\ref{prop:structure} ensures that $W_Y$ is preserved by +every flow time. Such statements about factor sigma-algebras are +understood modulo null sets; all time integrals below are evaluated +on the original jointly Borel flow. If $\rho:Y'\to Y$ is an extension, +the pullback of $W_Y$ is distal for time one and hence is contained +in $W_{Y'}$. Consequently the old parameter array is a measurable +function of the new parameter array on selected-law spaces. + +\begin{proposition}[Parameter and small-marginal laws] +\label{prop:selected-parameters} +For every flow $Y$ over $C$ and bounded measurable $H_i$ on $W_Y$, +\begin{equation}\label{eq:deterministic-parameters} + \E_{\lambda_Y}\prod_{i\in I}H_i(P_i) + =\lim_{L\to\infty}\int_Y\frac1L\int_0^L + \prod_{i\in I}H_i\bigl(\omega_Y(S_Y^{it}y)\bigr) + \,\dd t\,\dd m_Y(y). +\end{equation} +For every nonempty $J\subseteq I$ with $\abs{J}\le3$, +\begin{equation}\label{eq:small-conditional-law} + \law_{\lambda_Y}\bigl((y_i)_{i\in J}\mid P\bigr) + =\bigotimes_{i\in J}m_{Y,P_i}. +\end{equation} +\end{proposition} + +\begin{proof} +We first explain the passage from discrete to continuous time used +in both assertions. Write $t=l+s$, with $l\ge0$ an integer and +$0\le s<1$. For each fixed $s$, a continuous-time product becomes +the discrete product at slopes $i$ for the shifted slot functions +$G_i\circ S_Y^{is}$. Whenever the corresponding discrete averages +converge almost surely, Fubini and bounded convergence show that +their integrals in $s$ converge almost surely. Starting the discrete +average at $l=0$ rather than $l=1$, and dropping an incomplete final +unit interval, each has an error tending to zero. The resulting +continuous-time convergence holds along all real lengths $L$. + +Apply this observation to functions from $W_Y$. Their shifts still +belong to $W_Y$, and pointwise convergence on the distal factor is +part of Proposition~\ref{prop:structure}. Thus the time averages +in~\eqref{eq:deterministic-parameters} converge almost surely along +all lengths, and therefore at the selected lengths as well. Bounded +convergence and~\eqref{eq:selected-law} prove that equation. + +For $\abs{J}\le3$, Lemma~\ref{lem:triple-convergence} and the same +continuous-time argument show that the selected $J$-marginal equals +its deterministic-length integrated law. The extension +$Y\to W_Y$ is relatively weakly mixing. Apply +Lemma~\ref{lem:relative-norm} to the distinct integer slopes +$i\in J$ and the shifted functions $G_i\circ S_Y^{is}$, for each +fixed $s$. Conditional expectation commutes with these shifts +because $W_Y$ is preserved. Integration in $s$, using the uniform +norm bound and bounded convergence, gives +\begin{equation}\label{eq:small-replacement} + \E_{\lambda_Y}\prod_{i\in J}G_i(y_i) + =\E_{\lambda_Y}\prod_{i\in J} + \E_{m_Y}[G_i\mid W_Y](y_i). +\end{equation} +Indeed the difference of the deterministic averages tends to zero +in $L^2(m_Y)$, so also in their integrated values. Applying this +identity to $G_i$ multiplied by arbitrary bounded functions of +$P_i$ yields +\begin{equation}\label{eq:small-own-parameters} + \law_{\lambda_Y}\bigl((y_i)_{i\in J}\mid(P_i)_{i\in J}\bigr) + =\bigotimes_{i\in J}m_{Y,P_i}. +\end{equation} + +Conditioning on the full parameter array requires one more step; +it does not follow from~\eqref{eq:small-own-parameters} alone. +Use the invariant transformation with speed $i$ in slot $i$. +By the stability under nonzero powers and conditional products in +Proposition~\ref{prop:structure}, the $J$-marginal system +in~\eqref{eq:small-own-parameters} is relatively weakly mixing over +its parameter law. The full $P$-system is a joining of the distal +systems $W_Y$ at these integer powers. It is distal, and is +therefore relatively distal over the subarray $(P_i)_{i\in J}$. +Their joint law under $\lambda_Y$ is an invariant joining +identifying that common subarray. Relative disjointness in +Proposition~\ref{prop:structure} makes this joining conditionally +independent over the subarray, which is precisely +\eqref{eq:small-conditional-law}. +\end{proof} + +The full parameter law is thus unaffected by the selectors, although +the full state law need not be. We now turn that state law into a new +flow while preserving one chosen slot as the original system. + +\begin{lemma}[Normalized joining flow]\label{lem:normalized-joining} +For $b\in I$, define +\begin{equation}\label{eq:normalized-joining} + \mathsf M_b(Y)= + \left(Y^I,\lambda_Y, + S_{\mathsf M_b(Y)}^t(y_i)_i=(S_Y^{it/b}y_i)_i\right). +\end{equation} +This is a standard probability-preserving flow over $C$ via +$(y_i)_i\mapsto\pi_Y(y_b)$, and projection onto slot $b$ is an +extension onto $Y$. The array $P$ belongs to the maximal +measure-distal factor of $\mathsf M_b(Y)$ for time one. +\end{lemma} + +\begin{proof} +Invariance follows from Proposition~\ref{prop:selected-law}, after +rescaling the time parameter by $1/b$. Slot $b$ has speed one and +marginal $m_Y$, giving the stated factor maps. In slot $i$ the +$W_Y$-factor is distal for time $i/b$, by +Proposition~\ref{prop:structure}. The law of $P$ is an invariant +joining of these rational-time distal systems, hence is distal. +Maximality of the distal factor gives the last assertion. +\end{proof} + +\begin{lemma}[The retained positive correlation]\label{lem:positive-witness} +With the functions and phase test of Lemma~\ref{lem:selected-witness}, +\begin{equation}\label{eq:positive-witness} + \int_{C^I}\Psi\bigl((\upsilon(z_i))_i\bigr) + \prod_{i\in J_0}f_i\bigl(\xi(z_i)\bigr)\,\dd\lambda_C>0. +\end{equation} +The phase array in this integral is measurable in the distal +parameter array of $C^I$. +\end{lemma} + +\begin{proof} +Continuous functions on the compact product of circles are uniformly +approximable by finite sums of products of continuous one-coordinate +functions. Apply this to $\Psi$, use~\eqref{eq:selected-law} for +each resulting product, and then use the strictly positive lower +limit in~\eqref{eq:selected-positive-limit}. Finally, time one fixes +the phase on $C$, so the phase factor is distal and lies in $W_C$. +\end{proof} diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/square.tex b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/square.tex new file mode 100644 index 0000000000000000000000000000000000000000..7c2ccfbf8bcbcd6df4cb1839dd914b1595019310 --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/build/sections/square.tex @@ -0,0 +1,292 @@ +\section{The square argument} +\label{sec:square} + +We now use the saturated channel $Y$, its nonzero correlation +\eqref{eq:role-correlation}, and the regrouped channel +$Z=\mathsf M_b(Y)$ from Section~\ref{sec:correlations}. The selected +law $\lambda_Z$ is a law on an ordered square of $Y$-states. Its +$b$th row and $b$th column both have law $\lambda_Y$, but for different +reasons. Saturation will relate their correlations without requiring +any symmetry of the square or any pointwise theorem for many slots +on an arbitrary extension. + +\subsection{A common corner and two conditional expectations} + +On $(Y^I,\lambda_Y)$, put +\begin{equation} + q((y_i)_{i\in I})= + \prod_{j\in J\setminus\{b\}}h_j(L_j(y_j)), + \qquad + g(P,y_b)=\E_{\lambda_Y}[q\mid P,y_b]. + \label{eq:q-and-g} +\end{equation} +The function $g$ is not zero in $L^2(\lambda_Y)$: otherwise the +nonzero correlation would vanish after conditioning on $(P,y_b)$. +On the same state space regarded as $Z$, the function $q$ reads only +the newly added outputs. Write $e=\omega_Z(z)$ for the new distal +parameter and define +\begin{equation} + u(e,y_b)=\E_{m_Z}[q\mid e,y_b]. + \label{eq:u} +\end{equation} +All these functions have bounded real Borel versions. + +By Lemma~\ref{lem:normalized-joining}, the old array $P$ is a factor +of the new distal parameter. Write +\[ + R(e)=P, + \qquad \kappa(e)=R(e)_b. +\] +Thus $\kappa\circ\omega_Z$ is the old distal parameter of the +slot-$b$ factor $Z\to Y$. + +A point of the selected law space $(Z^I,\lambda_Z)$ has the form +\[ + \mathbf z_i=(y_{ij})_{j\in I},\qquad + E_i=\omega_Z(\mathbf z_i),\qquad E=(E_i)_{i\in I}. +\] +The first index specifies the row. Define the column states and their +parameters by +\begin{equation} + y_i=y_{ib},\qquad + p=(\kappa(E_i))_{i\in I},\qquad + w=\kappa(E_b). + \label{eq:square-parameters} +\end{equation} +In particular $y_b=y_{bb}$ is the common corner. The column +$(y_i)_{i\in I}$ has law $\lambda_Y$, by functoriality for $Z\to Y$. +The row $\mathbf z_b$ also has law $\lambda_Y$, because the one-slot +marginal of $\lambda_Z$ is $m_Z=\lambda_Y$. +Figure~\ref{fig:square} records these distinct identifications. + +\input{figures/square} + +\begin{lemma}[The row--column identity] +\label{lem:row-column} +Under $\lambda_Z$, +\begin{equation} + \E\big[q((y_i)_i)q(\mathbf z_b)\mid E\big] + =\int_Y g(p,y)u(E_b,y)\,\dd m_{Y,w}(y). + \label{eq:row-column} +\end{equation} +Moreover, in a single $Z$-state with law $m_Z$, the conditional law +of its slot-$b$ state given $e$ is $m_{Y,\kappa(e)}$. +\end{lemma} + +\begin{proof} +Apply saturation \eqref{eq:saturation} with datum index $0$ to the +extension $Z\to Y$. It says that the conditional law of the entire +column given $E$ is the law of $(Y^I,\lambda_Y)$ given $p$. +This assertion follows first for $L^2$ tests, and then as an identity +of conditional kernels by a countable determining class. By the +singleton case of \eqref{eq:small-conditional-law}, the conditional +law of $y_b$ given $E$ is consequently $m_{Y,w}$. Conditioning down +to $E_b$ gives the same law for the slot-$b$ state of row $b$ given +$E_b$. Since row $b$ has marginal $m_Z$, this proves the last +assertion of the lemma. + +Saturation with datum index $b$ gives the stronger identity +\begin{equation} + \E_{\lambda_Z}[q((y_i)_i)\mid E,\mathbf z_b]=g(p,y_b). + \label{eq:column-single-row} +\end{equation} +Indeed the old datum is $(P,y_b)$ and the new datum is +$(E,\mathbf z_b)$. Multiply \eqref{eq:column-single-row} by +$q(\mathbf z_b)$ and condition on $E$. The singleton case of +\eqref{eq:small-conditional-law}, now for $Z$, states that the law +of $\mathbf z_b$ given $E$ is the $m_Z$-fibre law over $E_b$. +Within that fibre, \eqref{eq:u} conditions $q(\mathbf z_b)$ on its +slot-$b$ state, whose law is $m_{Y,w}$ by the first part of the proof. +This gives \eqref{eq:row-column}. +\end{proof} + +\subsection{Independence of the two parameter descriptions} + +The integral in \eqref{eq:row-column} pairs a column function with +a row function. Its nonvanishing will follow from the fact that, +apart from the parameter at their common corner, the two parameter +descriptions are independent. + +\begin{lemma}[Parameter independence] +\label{lem:parameter-independence} +Under $\lambda_Z$, the random variables $p$ and $E_b$ are +conditionally independent given $w$. +\end{lemma} + +\begin{proof} +Let $F_0$ be a bounded real test on $W_Z$. In the single-state +parameter law $(\omega_Z)_*m_Z$, let $\overline F_0$ satisfy +\[ + \overline F_0(\kappa(e))=\E[F_0(e)\mid\kappa(e)]. +\] +We prove that $F_0(E_b)$ can be replaced by +$\overline F_0(w)$ when tested against any product +$\prod_i H_i(p_i)$ of bounded real coordinate tests. All the +variables are functions of $E$, so the deterministic parameter-law +identity \eqref{eq:deterministic-parameters} on $Z$ computes their +joint integral. At each fixed averaging time $t$, its state integral +is +\[ + \int_Z F_0(\omega_Z(S_Z^{bt}z)) + \prod_{i\in I}H_i\big(\kappa(\omega_Z(S_Z^{it}z))\big) + \,\dd m_Z(z). +\] +Make the measure-preserving change of variable $z'=S_Z^{bt}z$. +The product of the $H_i$ becomes +\[ + \prod_{i\in I}H_i\big( + \kappa(\omega_Z(S_Z^{(i-b)t}z'))\big). +\] +For each fixed $t$, this is measurable with respect to +$\kappa(\omega_Z(z'))$. To see this, the slot-$b$ map $Z\to Y$ +is a flow factor, and the old factor $W_Y$ is preserved by every +flow time. Hence the old parameter at time $(i-b)t$ is a measurable +function of the old parameter at time zero. + +Conditional expectation in this single-state integral therefore +replaces the first factor by +\[ + \overline F_0\bigl(\kappa(\omega_Z(z'))\bigr). +\] +Reversing the change of +variable and taking the deterministic averaging limit gives +\[ + \E_{\lambda_Z}\left[F_0(E_b)\prod_i H_i(p_i)\right] + =\E_{\lambda_Z}\left[\overline F_0(w)\prod_i H_i(p_i)\right]. +\] +The fixed-time identities suffice under the time integral by +measurability and Fubini; no simultaneous pointwise model for the +factor action is needed. The $E_b$-marginal is the single-state +parameter law, so $\overline F_0(w)=\E[F_0(E_b)\mid w]$. +Products of the $H_i$ determine the $p$-law, and $w=p_b$ is itself +a coordinate of $p$. The displayed identity is exactly conditional +independence given $w$. +\end{proof} + +\subsection{Why the pairing cannot vanish} + +Let $\alpha_w$ be the conditional law of the column parameter +array $p$ over its $b$th parameter $w$. Let $\beta_w$ be the +conditional law of $E_b$ over $w$. These are regular conditional +laws on standard spaces. Lemma~\ref{lem:parameter-independence} +gives +\begin{equation} + \law(p,E_b\mid w)=\alpha_w\otimes\beta_w, + \qquad R_*\beta_w=\alpha_w. + \label{eq:parameter-disintegrations} +\end{equation} +The second identity uses the row marginal $m_Z=\lambda_Y$: +its old full parameter array $R(e)$ has the same law as the column +parameter array, with the same distinguished $b$th coordinate. + +For almost every $w$, work in the separable real Hilbert space +\[ + \mathcal H_w=L^2(m_{Y,w};\R). +\] +The bounded functions $g(p,\cdot)$ and $u(e,\cdot)$ define +square-integrable random vectors in this space, with laws of their +parameters respectively $\alpha_w$ and $\beta_w$. There is no +measurability ambiguity in this fibrewise assertion. Bounded Borel +versions of $g$ and $u$ are fixed; disintegration and Fubini yield +their sections for almost every $w$. A countable generating algebra +of the standard Borel space $Y$ has rational simple span dense in +every $L^2(m_{Y,w})$. Tests against that span give weak measurability +of the sections, and separability gives strong measurability. + +The conditional mean of the second random vector recovers the first: +\begin{equation} + \E_{\beta_w}[u(e,\cdot)\mid R(e)] + =g(R(e),\cdot) + \quad\text{in }\mathcal H_w. + \label{eq:fibre-tower} +\end{equation} +Here is the conditional-law justification. In a single $Z$-state, +write $y$ for its slot-$b$ state. Lemma~\ref{lem:row-column} gives, +conditional on $w$, +\[ + \law(e,y\mid w)=\beta_w(\dd e)m_{Y,w}(\dd y). +\] +Conditioning \eqref{eq:u} down on $(R(e),y)$ gives +$g(R(e),y)$ by \eqref{eq:q-and-g}, because $m_Z=\lambda_Y$. +Under the preceding product law, conditioning $e$ on $(R(e),y)$ +is the same as conditioning it on $R(e)$. This proves +\eqref{eq:fibre-tower}. One can impose the scalar conditional +identities first for countably many generating tests and then use +Fubini and density to obtain the stated Hilbert-space identity for +almost every $w$. + +Suppose that the right side of \eqref{eq:row-column} vanished +almost surely. By \eqref{eq:parameter-disintegrations}, for almost +every $w$ this would give +\[ + \langle g(p,\cdot),u(e,\cdot)\rangle_{\mathcal H_w}=0 + \quad(\alpha_w\otimes\beta_w)\text{-almost surely}. +\] +Condition on $R(e)=p'$ and use \eqref{eq:fibre-tower}. We obtain +\begin{equation} + \langle g(p,\cdot),g(p',\cdot)\rangle_{\mathcal H_w}=0 + \quad(\alpha_w\otimes\alpha_w)\text{-almost surely}. + \label{eq:independent-orthogonality} +\end{equation} +A square-integrable random vector in a separable real Hilbert +space cannot be orthogonal almost surely to an independent copy +unless it vanishes almost surely. Indeed, otherwise its norm is at +least some $\epsilon>0$ with positive probability. By a countable +cover with balls of radius $\epsilon/4$, this event has positive +probability in one such ball. Two independent copies then have +positive probability of both lying in that ball and having norm +at least $\epsilon$. Their distance is less than $\epsilon/2$, so +\[ + \langle v,v'\rangle + =\tfrac12(\|v\|^2+\|v'\|^2-\|v-v'\|^2)>0, +\] +a contradiction. Apply this observation to +\eqref{eq:independent-orthogonality}. It forces $g(p,\cdot)=0$ +for $\alpha_w$-almost every $p$, for almost every $w$. +The singleton conditional law for the column then implies +$g(P,y_b)=0$ in $L^2(\lambda_Y)$, contrary to +\eqref{eq:q-and-g}. Thus \eqref{eq:row-column} does not vanish +almost surely. + +\subsection{Increasing the number of output roles} + +Choose a bounded real function $\Theta(E)$ that detects the +nonzero conditional expectation in \eqref{eq:row-column}; its sign +is one such choice. Then +\begin{equation} + \E_{\lambda_Z}\left[ + \Theta(E)q((y_i)_i)q(\mathbf z_b)\right]\ne0. + \label{eq:improved-correlation} +\end{equation} +We interpret this as a correlation on the regrouped channel $Z$. +For $i\in J\setminus\{b\}$, the factor +$h_i(L_i(y_{ib}))$ retains its old role. An old output role uses +only the retained outputs from slot $b$. An old input role uses +the new original coordinate and retained outputs from its old +proper subset $H_i$ of group numbers. It may ignore the additional +outputs in those groups. Each such test is centered for its new +product spatial law, since its marginal on the coordinates it +actually uses is the old product law. + +At row $b$, the factor $q(\mathbf z_b)$ reads only the added outputs +with spatial lists $Q_j$, $j\in J\setminus\{b\}$, and thus has an +output role. It is centered: under the product spatial law of all +new outputs these lists are independent product lists, so the mean +of this product of centered tests is zero. Hence +\eqref{eq:improved-correlation} is a nonzero correlation of support +size $k$ with $r+1$ output roles. This contradicts the maximal +choice of $r$. + +\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}] +If the required pointwise limit failed for a fixed tuple and a +fixed $n\geq2$, the reduction and selectors of +Section~\ref{sec:selected} would give the positive correlation +\eqref{eq:positive-witness}. The minimum-support argument rules this +out immediately when $n<4$, and the saturated, regrouped square +argument rules it out when $n\geq4$. Therefore the averages converge +almost everywhere to the product of the means, along all positive +integer lengths. Finally, the countable-coordinate coding reduction +pulls this conclusion back to the original, possibly nonstandard, +probability space. The tuple and $n$ were fixed throughout, as allowed +in the statement of the theorem. +\end{proof} diff --git a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d2c086f61d4b91e4f906c12e75441016fb860190 --- /dev/null +++ b/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b9d8fcffac455475a2d846d1e28904f920e10b73dff498b718783023524f7d77 +size 435847 diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/README.md b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..da2866b3b7d5b4532e8a6c4c5190969d70bd2ec4 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/README.md @@ -0,0 +1,20 @@ +# [Pointwise convergence of fourfold ergodic averages for mixing transformations](fourfold-ergodic-averages.pdf) + +**Author:** OpenAI + +**Date:** October 4, 2026 + +## Citation + +To cite this manuscript: + +```bibtex +@misc{OAI:Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026, + author = {{OpenAI}}, + title = {{Pointwise convergence of fourfold ergodic averages for mixing transformations}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf}{OAI:Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/main.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..b9b6b3bb2b0931e256d16e55113e213457356fdc --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/main.tex @@ -0,0 +1,63 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{booktabs,array} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds} +\usepackage{xcolor} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={Pointwise convergence of fourfold ergodic averages for mixing transformations},pdfauthor={OpenAI}} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\newcommand{\1}{\mathbf 1} +\newcommand{\dd}{\,d} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\ip}[2]{\langle#1,#2\rangle} +\newcommand{\CE}[2]{\E[#1\mid #2]} +\DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\dist}{dist} +\setlength{\emergencystretch}{2em} +\title{Pointwise convergence of fourfold ergodic averages\newline for mixing transformations} +\author{OpenAI} +\date{October 4, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that fourfold ergodic averages along the times $n,2n,3n,4n$ +converge almost everywhere to the product of the integrals for every +invertible, bimeasurable, mixing probability-preserving transformation +and every fixed choice of bounded measurable inputs. Convergence holds +along all positive integer averaging lengths. No quantitative mixing +rate is assumed, and the probability space need not be standard. +\end{abstract} +\input{sections/01-introduction} +\input{sections/02-structure} +\input{sections/03-selected} +\input{sections/04-packing} +\input{sections/05-channels} +\input{sections/06-activation} +\appendix +\input{sections/07-lower-length} +\input{references} +\end{document} diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/references.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..56c27dda11b8dfe4115a838dcb636cd1c2cefc51 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/references.tex @@ -0,0 +1,159 @@ +\begin{thebibliography}{99} +\raggedright + +\bibitem{Assani} +I. 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N. Sudakov, +\emph{Gaussian random processes and measures of solid angles in Hilbert space}, +Dokl. Akad. Nauk SSSR \textbf{197} (1971), no.~1, 43--45 (Russian). + +\bibitem{Tropp} +J. A. Tropp, +\emph{Second-order matrix concentration inequalities}, +Appl. Comput. Harmon. Anal. \textbf{44} (2018), no.~3, 700--736. +\href{https://doi.org/10.1016/j.acha.2016.07.005}{doi:10.1016/j.acha.2016.07.005}. + +\bibitem{Ziegler} +T. Ziegler, +\emph{Universal characteristic factors and Furstenberg averages}, +J. Amer. Math. Soc. \textbf{20} (2007), no.~1, 53--97. + +\bibitem{Zimmer} +R. J. Zimmer, +\emph{Extensions of ergodic group actions}, +Illinois J. Math. \textbf{20} (1976), no.~3, 373--409. +\href{https://doi.org/10.1215/ijm/1256049780}{doi:10.1215/ijm/1256049780}. + +\end{thebibliography} diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/01-introduction.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a32cf31677863ab226bac0b14e991bff6380b216 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/01-introduction.tex @@ -0,0 +1,161 @@ +\section{Introduction}\label{sec:introduction} + +Let $(X,\mathcal F,\mu,T)$ be an invertible probability-preserving system. +For bounded measurable functions $f_1,\ldots,f_d$, the multiple ergodic +averages are +\[ +A_N(f_1,\ldots,f_d)(x) +=\frac1N\sum_{n=1}^N\prod_{j=1}^d f_j(T^{jn}x). +\] +The general pointwise question asks whether these averages converge +almost everywhere for every finite $d$ on every invertible +probability-preserving system. Here we treat four functions under +the mixing hypothesis. We say that $T$ is mixing if +\begin{equation}\label{eq:mixing} + \mu(A\cap T^{-n}B)\longrightarrow\mu(A)\mu(B) + \qquad (|n|\longrightarrow\infty) +\end{equation} +for every $A,B\in\mathcal F$. The functions are fixed throughout each +convergence assertion. No rate in \eqref{eq:mixing} is assumed. + +\begin{theorem}\label{thm:main} +Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$ +be invertible, bimeasurable, measure preserving, and mixing in the sense +of \eqref{eq:mixing}. For every $f_1,f_2,f_3,f_4\in L^\infty(\mu)$, +\[ + \frac1N\sum_{n=1}^N + f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x)f_4(T^{4n}x) + \longrightarrow \prod_{j=1}^4\int_X f_j\dd\mu +\] +for $\mu$-almost every $x$, along all positive integers $N$. +The original probability space need not be standard. +\end{theorem} + +\subsection*{Context and inputs} + +Birkhoff's pointwise ergodic theorem treats a single observation along +an orbit \cite{Birkhoff}. Furstenberg's ergodic proof of Szemer\'edi's +theorem made products at arithmetic-progression times a central object +of study \cite{Furstenberg}. For weakly mixing transformations, these +averages converge in $L^2$ to the product of the integrals. For arbitrary +transformations, Host and Kra established norm convergence of every +finite consecutive length using nilpotent characteristic factors +\cite[Theorem~1.1]{HostKra}; Ziegler independently developed universal +characteristic factors and another proof \cite{Ziegler}. Norm +convergence alone does not control the exceptional lengths at an +individual point. + +Bourgain proved pointwise convergence for two powers of one +transformation \cite{Bourgain}. Higher-order pointwise results include +$K$ systems \cite[Theorem~4.2]{DerrienLesigne}, weakly mixing systems +with suitable spectral restrictions \cite{Assani}, and weakly mixing +systems whose pairwise-independent finite self-joinings are all +product measures \cite{GutmanHuangShaoYe}. Huang, Shao and Ye proved +pointwise convergence of every finite consecutive length on ergodic +measure-distal systems \cite[Theorem~C]{HuangShaoYe}, extending the +distal investigation of Lesigne \cite{Lesigne}. Recent pointwise +theorems for polynomial iterates include the bilinear result of +Krause, Mirek and Tao \cite{KrauseMirekTao} and the distinct-degree +multilinear result of Kosz, Mirek, Peluse, Wan and Wright +\cite{KoszMirekPeluseWanWright}. Their degree hypotheses do not cover +the four linear iterates considered here. Kuca's survey \cite{Kuca} +gives a broader account of the norm and pointwise questions. + +Theorem~\ref{thm:main} uses three companion results. The multiple-mixing +theorem of \cite[Theorem~1.1]{MixingCompanion} supplies mixing of every finite order +from \eqref{eq:mixing}. The analytic construction of +\cite{TripleCompanion}, based on the local estimates of +\cite{HilbertCompanion}, supplies an oscillation estimate for threefold +averages. Its oscillation assertion holds on arbitrary invertible +systems; mixing is used separately to identify limits. We need the +threefold estimate at each three-element subset of $\{1,2,3,4\}$. +Appendix~\ref{sec:lower-length} proves the required coefficient +adaptation and the resulting pointwise convergence lemma, including its +nonergodic formulation. The local analytic theorems themselves are cited +with their hypotheses and locations. Thus the contribution here is the +fourfold deduction from these specified inputs. + +The structural part uses the Furstenberg--Zimmer theory of relatively +compact and relatively weakly mixing extensions. We record the +nonergodic and power-stability statements at their point of use in +Section~\ref{sec:structure}. The construction also uses extensions on +which certain conditional expectations cannot improve. Such energy +maximization belongs to the same general method as sated extensions +\cite{Austin}; the data being saturated here come from selected +four-coordinate laws. + +\subsection*{The difficulty caused by selected lengths} + +Suppose that the conclusion fails. After subtracting means, there are +four centred real functions and a positive-measure set of points at +which their averages remain positively separated from zero along +arbitrarily late lengths. We choose one such length $N_a(x)$ after +scale $2^a$. Averaging the configurations +\[ + (T^nx,T^{2n}x,T^{3n}x,T^{4n}x),\qquad 1\le n\le N_a(x), +\] +and passing to a limit preserves a nonzero correlation. A constant-roof +suspension makes these limit laws invariant under a flow and permits +the changes of speed needed later. + +Selection is the main analytic obstruction. Conditioning on a length +chosen from the initial state can change the joint law of independent +auxiliary variables. Ordinary norm convergence, with no +rate, does not by itself bound all the resulting errors. We introduce +three groups of auxiliary spatial coordinates. Their joint law is a +product, and any two groups together are independent of the original +coordinate, after conditioning on the suspension phases. A length +event may change the full three-group law, but preserves the +two-group marginals. This is precisely the amount of independence used +in Section~\ref{sec:channels}. For a finite model of the distinction, +let $U,V,W$ be independent symmetric signs. Their product is independent +of each pair, but conditioning on $UVW=1$ constrains the triple. + +The estimate making this possible is a finite-dimensional packing +statement proved in Section~\ref{sec:packing}. If two bounded families +of Hilbert-space vectors have small pairwise inner products outside a +graph of bounded degree, few of their tensor products can have large +projection onto a subspace of slowly growing dimension. A Gaussian +comparison and a trace-moment estimate give a quantitative form of +this assertion. Applied to the leading covariance subspace, it controls +the projected part of each selected average. The complementary part +has a square-summable error over the possible lengths. The estimate is +independent of the ambient Hilbert-space dimensions and requires no +quantitative mixing rate. + +\subsection*{From a correlation to a contradiction} + +Each auxiliary system carries both an original spatial coordinate and +the three groups of additional coordinates. We call such a system a +channel after specifying its exact independence conditions in +Definition~\ref{def:channel}. In its selected four-coordinate law, +consider nonzero correlations in which each slot uses either a +centred function of the original coordinate or a centred function of +the additional coordinates. Bounded functions of the distal parameters +are allowed as weights. The initial failure gives a correlation with +no additional-coordinate slots. + +Choose a channel and a correlation maximizing the number of slots of +the second kind. Independence excludes four such slots. An iterated +array construction excludes three: products along different axes +through a fixed corner are orthogonal, but each has the same nonzero +correlation with the function at that corner. Bessel's inequality +rules this out. At most two slots can therefore be of the second kind. + +The final construction increases that number. First enlarge the +channel until the conditional expectations needed by the argument +are stable under further extensions. Then regard one selected +four-coordinate law as a new system and select once more. The result +is a square of old states. Its distinguished row and column meet at +one corner; saturation supplies their exact conditional laws. A +reanchoring identity on the distal parameters shows that the row and +column correlation cannot vanish. Regrouping the other coordinates +therefore replaces one original-coordinate slot by an +additional-coordinate slot, contradicting maximality. + +Sections~\ref{sec:structure} and~\ref{sec:selected} establish the +structural inputs and selected laws. The packing estimate and channel +preservation are proved in Sections~\ref{sec:packing} +and~\ref{sec:channels}. Section~\ref{sec:activation} carries out the +maximality argument and completes the proof. The coefficient-dependent +analytic work is collected in Appendix~\ref{sec:lower-length}. diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/02-structure.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/02-structure.tex new file mode 100644 index 0000000000000000000000000000000000000000..8fd33eac3731b58658350176690907f1d684a9c0 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/02-structure.tex @@ -0,0 +1,278 @@ +\section{Convergence inputs and relative structure} +\label{sec:structure} + +The auxiliary systems in the proof will be extensions and joinings of a +suspension of the original transformation. They need not be ergodic. +We therefore formulate the structural facts in the nonergodic setting, +and distinguish the pointwise input for three functions from the norm +estimate available for any finite number of functions. + +\subsection{Standard systems and the pointwise inputs} + +Put $I=\{1,2,3,4\}$. It suffices to prove +Theorem~\ref{thm:main} on a standard probability space. Indeed, for a +given tuple of bounded functions, the map recording all their integer +translates takes values in a countable product of compact scalar discs. +The pushforward measure and the shift give a standard factor, the +functions descend to its coordinate functions, and mixing passes to +this factor. Almost-sure convergence there pulls back to the original +space. We henceforth write $(X,\mu,T)$ for this standard mixing system. +All factors and conditional expectations are understood modulo null +sets; probability spaces may be completed. + +We use \cite[Theorem~1.1]{MixingCompanion} in the following +form. For every fixed finite family $g_1,\ldots,g_s\in L^\infty(\mu)$, +\begin{equation}\label{eq:all-orders-input} + \int_X\prod_{j=1}^s g_j(T^{n_j}x)\,\dd\mu(x) + \longrightarrow \prod_{j=1}^s\int_Xg_j\,\dd\mu + \quad\text{if}\quad \min_{j\ne l}|n_j-n_l|\longrightarrow\infty. +\end{equation} +The bounded-function formulation follows from the set formulation by +simple-function approximation. All uses of this input involve fixed +functions; no rate of mixing is assumed. + +The other companion input will be used on systems without mixing. +Its coefficient dependence is important, so we isolate its exact scope. + +\begin{lemma}[Lower-length convergence]\label{lem:lower-length} +Let $(Y,m,R)$ be an invertible probability-preserving transformation on +a standard probability space. If $\varnothing\ne J\subsetneq I$ and +$g_i\in L^\infty(m)$ for $i\in J$, then +\[ + \frac1N\sum_{n=1}^N\prod_{i\in J}g_i(R^{in}y) +\] +converges for $m$-almost every $y$. +\end{lemma} + +The proof, including the changes of coefficients in the analytic +estimates of \cite{TripleCompanion,HilbertCompanion}, is given in +Appendix~\ref{sec:lower-length}. Neither ergodicity nor mixing is an +assumption of Lemma~\ref{lem:lower-length}. + +\subsection{Distal factors and relatively weakly mixing extensions} + +Let $(Y,m,R)$ be a standard probability-preserving system, and let +$B$ be a factor. We use the conditional-correlation characterization +of \emph{relative weak mixing}: for bounded $f,g$, with +$\E[f\mid B]=0$, +\begin{equation}\label{eq:rwm-correlations} + \frac1H\sum_{h=1}^H + \bigl\|\E[\overline f\,(g\circ R^h)\mid B]\bigr\|_2^2 + \longrightarrow0. +\end{equation} +For $F_1,\ldots,F_r\in L^2(m)$, the associated finitely generated +$L^\infty(B)$-module is +\[ + \left\{\sum_{j=1}^r a_j F_j:a_j\in L^\infty(B)\right\}. +\] +It is invariant if composition with $R$ and with $R^{-1}$ preserves +this space. An extension is \emph{relatively compact} if its $L^2$ +space is generated by these invariant modules. +The individual modules need not be closed in $L^2$; closure is taken +after their union is spanned. This is the compact-module +characterization in its corrected form +\cite[Theorem~4.1]{Jamneshan}\cite[Theorem~2.5]{JamneshanCorrigendum}. +A +\emph{relatively distal} extension is a tower of relatively compact +extensions and inverse limits. Distality without a specified base +means distality over the trivial factor. These definitions include +nonergodic systems; in particular an identity action is compact. + +\begin{proposition}[Relative structure]\label{prop:structure} +The following statements hold for standard probability-preserving +systems. +\begin{enumerate} +\item There is a maximal measure-distal factor $W_Y$. It contains the +invariant factor, and $Y\to W_Y$ is relatively weakly mixing. A distal +factor pulls back into the maximal distal factor of any extension. +Every measure-preserving automorphism commuting with $R$ preserves +$W_Y$. +\item Factors and joinings of distal systems are distal. A distal +system is relatively distal over each of its factors. For every +nonzero integer $q$, $R$ is distal if and only if $R^q$ is distal. +\item Relative weak mixing passes to nonzero integer powers and to +conditionally independent base change. More generally, the product +of relatively weakly mixing fibre laws over any invariant joining of +their base laws is relatively weakly mixing over that joining. +\item A relatively weakly mixing extension and a relatively distal +extension of the same base are relatively disjoint: the only invariant +joining which identifies their base coordinates is the relatively +independent joining. +\item If $R$ is distal, then, for every $d\ge1$ and bounded +$g_1,\ldots,g_d$, the averages +\[ + \frac1N\sum_{n=1}^N\prod_{i=1}^d g_i(R^{in}y) +\] +converge almost surely. +\end{enumerate} +If $(S_Y^t)_{t\in\R}$ is a probability-preserving flow, the maximal +distal factor $W_Y$ of $S_Y^1$ is a flow factor. Its action at every +nonzero rational time is distal. +\end{proposition} + +\begin{proof} +The existence assertion and the relatively weakly mixing extension +are the Furstenberg--Zimmer structure theorem +\cite{Furstenberg,Zimmer}; its nonergodic probability-algebra formulation +is given in \cite[Theorems~1.1 and~1.2]{Jamneshan}. +We recall the consequences needed here, with particular attention to +changes of base and of time. + +At a compact stage, an invariant finitely generated module remains such a +module after the base is enlarged. One may use bounded generators: +for a vector of generators transforming by unitary matrices over the +base, truncate by its invariant pointwise length. Products of bounded +generators lie in finitely generated tensor modules. Consequently the join +of compact stages is compact over the joined base, and the same +assertion for distal towers follows by passing to closures at limit +stages. This proves closure under joinings and the assertion about +enlarging the base to any factor of a distal system. It also shows +that distal factors of a system lie in its maximal distal factor. +Applying this observation to an automorphism commuting with $R$, and +then to its inverse, shows that it preserves $W_Y$. The invariant +factor is compact and therefore belongs to $W_Y$. + +Compact modules for $R$ are compact modules for $R^q$, so distality +passes to powers. Conversely, suppose $q>0$ and $R^q$ is distal. +Its canonical compact tower is preserved by $R$, which commutes with +$R^q$. At each successor stage replace an $R^q$-invariant finitely +generated module by the sum of its first $q$ translates under $R$. +This is a finitely generated $R$-invariant module over the same base. +The resulting tower +proves distality for $R$. Negative powers give the same conclusion +by inversion. + +For relative weak mixing, the nonnegative summands in +\eqref{eq:rwm-correlations} show that restriction to multiples of a +fixed positive integer still has Ces\`aro limit zero. Invariance and +complex conjugation handle negative powers. To verify the assertion +about joined bases, let $Y_j\to B_j$ be the extensions and let $\sigma$ +be an invariant joining of the base measures. On the resulting space +give the fibres the law +\[ + \int\bigotimes_j m_j(\dd y_j\mid b_j)\,\dd\sigma((b_j)_j). +\] +For bounded tensor tests, their conditional correlation over the full +base has the form +\begin{equation}\label{eq:product-rwm-correlation} + a(b)\,c(S_B^h b) + \prod_j\E[\overline{f_j}\,(g_j\circ R_j^h)\mid B_j](b_j). +\end{equation} +If one $f_j$ is conditionally centred, the squared $L^2(\sigma)$ norm +is bounded by a constant times the squared norm of that factor in +$L^2(B_j)$. Here only the $j$th marginal of $\sigma$ is used. +Its Ces\`aro mean tends to zero. Subtracting the conditional +expectation of a tensor and telescoping shows that such centred +tensors span a dense subspace of the orthocomplement of the full +base. Bounded approximation now proves the claim. The same +calculation with an arbitrary extra base coordinate proves stability +under conditionally independent base change. + +For relative disjointness we argue with conditional modules. The +absolute weak-mixing disjointness principle goes back to +Furstenberg~\cite[Part~I, Section~7]{FurstenbergDisjointness}. +First take a compact extension of the base. A joining defines a +base-linear, time-intertwining conditional +expectation operator from its $L^2$ space to the relatively weakly +mixing side. It takes each finitely generated invariant module into +another such module. Relative weak mixing says that the latter lies +in the base, so the joining is relatively independent. Induct along +a distal tower: once independence has been proved at a stage, the +extension on the other side is the conditionally independent base +change of the original relatively weakly mixing extension. The +preceding calculation allows the compact-stage argument to be +repeated. Conditional expectations pass to the limit at limit +stages. This proves relative disjointness without an ergodicity +assumption. It also gives closure of distal systems under factors: +if $D$ is distal and $B$ is its factor, then $D$ is distal over the +pullback of $W_B$; relative disjointness of $D\to W_B$ and +$B\to W_B$ forces the latter extension to be trivial. + +For the pointwise assertion, the ergodic case is +\cite[Theorem~C]{HuangShaoYe}, whose distality is measure-theoretic. +For a nonergodic distal system, start its compact tower with the +invariant factor and disintegrate over that factor. The tower is +countable on a standard space: its genuinely increasing successor +stages give nonzero orthogonal differences in a separable $L^2$ +space. At each stage choose a countable generating algebra. Approximate +each of its indicators by functions in the compact modules at a +successor stage, or in the preceding factors at a limit stage, with +summable squared $L^2$ errors. Fubini makes these errors summable in +the conditional $L^2$ spaces on almost every ergodic component, +simultaneously for all the chosen approximations. The restricted +generating algebras are dense for every component measure. The +countably many identities expressing invariance of the chosen modules +also disintegrate. Thus the successor stages remain compact and the +limit stages remain generated by their predecessors on a common +conull set of components. Every intermediate factor has the same invariant +algebra, so each component tower consists of ergodic systems; its +compact extensions are isometric by the compact-extension structure +theorem. Each component is thus an ergodic distal system. +The cited theorem applies to the given tuple on almost every +component, proving the assertion. + +Finally, every flow time commutes with time one and therefore +preserves $W_Y$. If $r=p/q\ne0$ is rational, the $q$th power of +its action on $W_Y$ is the distal action at integer time $p$. +The power assertion gives distality at time $r$. +\end{proof} + +The following norm estimate explains why proper subaverages have a +particularly simple law over a distal factor. Its proof uses relative +weak mixing alone. + +\begin{lemma}[A relatively weakly mixing characteristic factor] +\label{lem:relative-norm} +Suppose $Y\to B$ is relatively weakly mixing for an invertible +transformation $R$. If $a_1,\ldots,a_d$ are distinct nonzero integers +and $g_1,\ldots,g_d$ are bounded, then +\begin{equation}\label{eq:relative-norm} + \left\|\frac1N\sum_{n=1}^N + \left(\prod_{i=1}^d g_i\circ R^{a_i n} + -\prod_{i=1}^d\E[g_i\mid B]\circ R^{a_i n}\right)\right\|_2 + \longrightarrow0. +\end{equation} +\end{lemma} + +\begin{proof} +By telescoping and scaling, it suffices to consider functions bounded +by one, one of which, say $g_s$, has conditional mean zero. We induct +on $d$, with the empty product as the initial case. Set +$v_n=\prod_i g_i\circ R^{a_i n}$ and, for a fixed $h\ge1$, put +$F_{i,h}=\overline{g_i}(g_i\circ R^{a_i h})$. +Invariance gives +\[ + \langle v_n,v_{n+h}\rangle + =\int F_{s,h}\prod_{i\ne s} + F_{i,h}\circ R^{(a_i-a_s)n}\,\dd m. +\] +The other coefficients are distinct and nonzero. The induction +hypothesis replaces their factors, in the average over $n$, by +their conditional expectations over $B$, with an error tending to +zero. Those remaining factors are $B$-measurable and bounded by +one, so +\[ + \limsup_{N\to\infty} + \left|\frac1N\sum_{n=1}^N\langle v_n,v_{n+h}\rangle\right| + \le \|\E[F_{s,h}\mid B]\|_1. +\] +For $d=1$ the same inequality holds directly. The Hilbert-space +van der Corput inequality bounds the squared norm of the averages +of $v_n$ by an absolute constant times +\[ + \frac1H+\frac1H\sum_{h=1}^H\|\E[F_{s,h}\mid B]\|_1 +\] +in the limit superior as $N\to\infty$. The second term tends to +zero by \eqref{eq:rwm-correlations} for $R^{a_s}$ and +Cauchy--Schwarz. Letting $H\to\infty$ proves the induction. +\end{proof} + +In particular, weak mixing of $T$ gives $L^2$ convergence to the +product of the means for all finite distinct nonzero linear powers +on $X$. In a flow, with $B$ preserved by every time, the analogue of +\eqref{eq:relative-norm} for deterministic continuous-time averages +follows by writing $t=n+s$, $0\le s<1$. Apply the lemma for each +fixed $s$ to $g_i\circ S_Y^{a_i s}$, and integrate the resulting +$L^2$ estimate in $s$. Conditional expectation commutes with these +time shifts. Bounded convergence and the final interval of length +less than one complete the passage to continuous time. diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/03-selected.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/03-selected.tex new file mode 100644 index 0000000000000000000000000000000000000000..00ba97d717b78593cc27e5aa71814745ad8dbab3 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/03-selected.tex @@ -0,0 +1,265 @@ +\section{Selected laws and their distal parameters} +\label{sec:selected} + +A failure of pointwise convergence allows the averaging length to +depend on the initial state. We encode this failure in a probability +law on four outputs. The selection can change their joint law, but +the convergence results of Section~\ref{sec:structure} will determine +every proper marginal, even after all distal parameters are given. + +\subsection{From failure to a sequence of selectors} + +Let $C=X\times[0,1)$ have measure $m_C=\mu\otimes\dd u$ and the +roof-one suspension flow +\begin{equation}\label{eq:suspension} + S_C^t(x,u)=\bigl(T^{\lfloor u+t\rfloor}x,\{u+t\}\bigr). +\end{equation} +Write $\xi:C\to X$ for its spatial coordinate and +$\upsilon:C\to\R/\Z$ for its phase. The use of a flow will let +us normalize the action at any one of the four outputs. + +\begin{proposition}[Selectors witnessing failure]\label{prop:selectors} +If Theorem~\ref{thm:main} fails, there are real functions +$f_i\in L^\infty(\mu)$ with $\|f_i\|_\infty\le1$ and +$\int f_i\,\dd\mu=0$, a finite set $F\subset[1,2]$, measurable +functions +\[ + N_a:C\longrightarrow\{c2^k:c\in F,\ k\ge a\},\qquad a\ge1, +\] +a continuous real function $\Psi$ on $(\R/\Z)^I$, and +$\epsilon_0>0$ such that, for all sufficiently large $a$, +\begin{equation}\label{eq:selected-positive} + \int_C\frac1{N_a(z)}\int_0^{N_a(z)} + \Psi\bigl((\upsilon(S_C^{it}z))_{i\in I}\bigr) + \prod_{i\in I}f_i\bigl(\xi(S_C^{it}z)\bigr) + \,\dd t\,\dd m_C(z)\ge\epsilon_0. +\end{equation} +The selectors may be chosen independent of the initial phase. +\end{proposition} + +\begin{proof} +Expand each function in the failed assertion into its mean and its +centred part. Lemma~\ref{lem:lower-length} and the $L^2$ conclusion +of Lemma~\ref{lem:relative-norm} show that every proper subaverage +converges almost surely to the product of its means. Thus a +fourfold average of centred functions fails to converge to zero. +Expanding real and imaginary parts and scaling gives a real tuple +bounded by one with the same property. Its averages +\[ + A_N(x)=\frac1N\sum_{n=1}^N\prod_{i\in I}f_i(T^{in}x) +\] +converge to zero in $L^2$. After changing the sign of one function +if necessary, there are $\delta>0$ and a measurable set $E$ of +positive measure such that $A_N(x)>2\delta$ for arbitrarily large +$N$, for every $x\in E$. + +For bounded summands, changing the averaging length from $N$ to $M$ +changes the average by at most $2|N-M|/\max(N,M)$. Choose a finite +grid $F\subset[1,2]$ fine enough that arbitrarily late inequalities +$A_N(x)>2\delta$ yield arbitrarily late lengths $L=c2^k$ satisfying +$A_{\lfloor L\rfloor}(x)>\delta$. The rounding error tends to zero +uniformly as $k\to\infty$. For $x\in E$, choose measurably, from a +fixed enumeration, one such length with $k\ge a$. Set $N_a(x,u)$ +equal to that choice. For $x\notin E$, set $N_a(x,u)=2^a$, including +$1$ in $F$. Countability of the available lengths gives measurable +selectors. + +Choose nonnegative continuous functions $\alpha,\beta$ on $\R/\Z$, +not identically zero, with supports respectively in $(1/4,1/2)$ and +$(0,1/16)$. Define +\[ + \Psi(v_1,v_2,v_3,v_4) + =\alpha(2v_1-v_2)\,\beta(v_2-v_1). +\] +For output phases $v_i=u+it\pmod1$, this equals +$\alpha(u)\beta(\{t\})$. On its support, writing $t=n+s$ with +$0\le s<1$ gives $00$ means a measure-preserving map $D:Y\to C$ satisfying +$D(S_Y^t y)=S_C^{qt}D(y)$ for every $t$, modulo the usual null sets. + +\begin{definition}\label{def:channel} +A \emph{channel} is a flow extension +$\pi_Y:(Y,m_Y,S_Y^t)\to C$ together with a finite list of output factors +$D_\alpha:Y\to C$, of positive rational speeds $q_\alpha$, partitioned +into three groups $\mathcal D_1,\mathcal D_2,\mathcal D_3$. +Empty groups are allowed. Let $x_0$ be the spatial coordinate of +$\pi_Y$, let $x_\alpha$ be that of $D_\alpha$, and let $V$ collect +the phases of $\pi_Y$ and of all the $D_\alpha$. The following +conditional laws are required: +\begin{align} + \mathcal L((x_\alpha)_\alpha\mid V) + &=\bigotimes_\alpha\mu, + \label{eq:channel-all}\\ + \mathcal L\bigl(x_0,(x_\alpha)_{\alpha\in\cup_{j\in J}\mathcal D_j} + \mid V\bigr) + &=\mu\otimes + \bigotimes_{\alpha\in\cup_{j\in J}\mathcal D_j}\mu + \qquad(J\subsetneq\{1,2,3\}). + \label{eq:channel-proper} +\end{align} +The product laws on the right do not depend on $V$. +\end{definition} + +In particular, the original spatial variable is independent of every +proper collection of groups, even after the phases are given. The +definition does not require its independence from all three groups +jointly. The flow $C$ with no outputs is a channel. + +\begin{lemma}\label{lem:channel-distal} +For a channel $Y$, both conditional product identities in +\eqref{eq:channel-all}--\eqref{eq:channel-proper} also hold when +conditioning on its maximal measure-distal factor $W_Y$. +Every flow extension of $Y$, equipped with the pulled-back original +factor and outputs, is a channel and satisfies the same assertion for +its own maximal distal factor. +\end{lemma} + +\begin{proof} +The phase factor $V$ is a factor of a finite product of circle rotations, +and hence belongs to $W_Y$. Choose a positive integer $h$ clearing all +the output speed denominators. For the time-$h$ map the phases are +fixed. Each spatial tuple appearing on the right of +\eqref{eq:channel-all} or \eqref{eq:channel-proper}, conditionally on +these phases, has its product measure and evolves under a product of +positive integer powers of $T$. Such a product is mixing. Consequently +the tuple together with $V$ is a relatively weakly mixing extension of +the fixed phase factor. + +The factor $W_Y$ remains distal for time $h$, and is relatively distal +over $V$. Relative disjointness in Proposition~\ref{prop:structure} +therefore says that the joining of this spatial tuple with $W_Y$ over +$V$ is the conditionally independent joining. This is exactly the +claimed conditional product law over $W_Y$. + +Passing to an extension does not change the joint distribution in the +definition of a channel. Apply the same argument to its possibly +larger maximal distal factor to obtain the last assertion. +\end{proof} + +Recall that $\lambda_Y$ is the selected law on $Y^I$, and that +$\mathsf M_b(Y)$ equips this law with the flow whose speed in slot +$i$ is $i/b$. Projection to slot $b$ is then an extension of $Y$. + +\begin{proposition}[Closure under selected laws]\label{prop:channel-closure} +Let $Y$ be a channel and $b\in I$. On $\mathsf M_b(Y)$ use +$\pi_Y(y_b)$ as original factor. For each $j\in\{1,2,3\}$, let +the new group $j$ contain the outputs $D_\alpha(y_i)$ with +$\alpha\in\mathcal D_j$ and $i\in I$. With these factors, +$\mathsf M_b(Y)$ is a channel. +\end{proposition} + +The new output speeds are $iq_\alpha/b$, so they remain positive and +rational. The content of the proposition is independence. Its proof +first treats a proper collection of groups, for which conditioning on +the selected length preserves independence from the input. For all +three groups, the selected law can change, and tensor packing provides +the needed estimate. + +\begin{proof} +We test the desired laws by products of bounded spatial functions and +bounded continuous functions of the new phases. Products of separate +phase tests determine the phase distribution, and fit the slot-test +definition of $\lambda_Y$. Linear combinations and bounded +approximation then suffice for all tests in +\eqref{eq:channel-all}--\eqref{eq:channel-proper}. +By taking real and imaginary parts, expanding each spatial test into +its mean and centered part, and scaling, we may take every spatial +factor to be either $1$ or a real centered function bounded by one. +It remains to prove that the tested integral tends to zero whenever +at least one such centered factor occurs. + +We work first under the input measure $m_Y$, conditionally on its phase +vector $V$. Choose a positive integer $h$ such that +$hq_\alpha$ is integral for every output, and split the time in the +averaging formula as +\begin{equation}\label{eq:channel-time-split} + t=hn+s,\qquad 0\le s0$, the families $x_n,y_n$ satisfy the small-pair +hypothesis of Theorem~\ref{thm:packing}, after finitely many initial +indices are removed. To see this, expand a pair inner product into +expectations over the independent outputs in that group. One output +has a centered occurrence. Mixing of all orders makes its expectation +small when the distinct times from the two indices are all sufficiently +far apart. Within either index, this holds for large $n$ and $n'$. +A remaining obstruction requires +\[ + |e_\alpha i n-e_\alpha i'n'+a|\le R +\] +for one of finitely many choices of $i,i',a$ and a fixed $R$. +For each $n$ there are boundedly many $n'$ satisfying any such +inequality, since $e_\alpha i'>0$. The obstructing pairs therefore +form a graph of bounded degree $L$. The constants and the initial +cutoff depend on $\eta$ and the fixed tests; the finite offset +patterns make them uniform in $V,s$. + +Condition further on an event +$E_{k,c}=\{N_a=c2^k\}$ with conditional probability +$p_{k,c}=\Prob(E_{k,c}\mid V)>0$. Repeated values of $c2^k$ are assigned +one index so that these events partition the space. Write $p=p_{k,c}$. +The event is determined by $x_0$ once $V$ is fixed. By +\eqref{eq:channel-proper}, conditioning on it preserves the product +marginal on any two groups and the marginal on each single group. +The three-group law need not remain a product. Its density $\rho$ +with respect to the original three-group product measure satisfies +\begin{equation}\label{eq:channel-density} + 0\le p\rho\le1,\qquad \int p\rho=p, + \norm{p\rho}_2\le\sqrt p. +\end{equation} +Indeed, for any measurable set of group coordinates its probability +jointly with $E_{k,c}$ is at most its original product probability. + +For the $m$ terms corresponding to this length, put +$z_n=x_n\otimes y_n$. We choose a dimension small enough that the +packing bound is $o(m)$ and large enough that its reciprocal is +summable over the length classes. The choice +$d_m=\lceil(1+\log m)^{3/2}\rceil$ has both properties: for +$m\asymp2^k$, its reciprocal is $O((1+k)^{-3/2})$. +Let $P_m$ project onto $d_m$ orthonormal +eigenvectors of $\sum_n z_nz_n^*$ corresponding to its largest +eigenvalues, counted with multiplicity. If the range has smaller dimension, +take the whole range; inside a tied eigenspace make any fixed choice. +Fix $0<\varepsilon\le1$, and choose +$0<\eta\le\varepsilon^2/10^4$ first. +Equation~\eqref{eq:packing-average} gives +\begin{equation}\label{eq:channel-leading-average} + \frac1m\sum_n\norm{P_mz_n} + \le\varepsilon+o_{\varepsilon,L}(1). +\end{equation} +If an initial cutoff of $n_0$ indices was needed, apply the general +bound \eqref{eq:packing-quantitative} to the $m-n_0$ remaining vectors +with the original rank bound $d_m$. The numerator in that bound is +still $m^{o(1)}$, and the discarded indices cost at most $n_0/m$. +This proves \eqref{eq:channel-leading-average} with the cutoff as +well. All norms here are for the original product measures. + +Under the event law the first two groups still have their product +marginal, and the third retains its original marginal. Cauchy--Schwarz +under this law therefore bounds the contribution from a term +$(P_mz_n)w_n$, including the probability of the event, by +$p\norm{P_mz_n}$. Thus its averaged contribution is bounded by +$p$ times the right side of \eqref{eq:channel-leading-average}. + +Write $z'_n=(1-P_m)z_n$. The covariance has trace at most $m$, so its +$(d_m+1)$-st eigenvalue is at most $m/(d_m+1)$ whenever a residual is +present. The residual Gram matrix $A=(\ip{z'_n}{z'_{n'}})_{n,n'}$ +has norm at most $m/d_m$. The Gram matrix +$B=(\ip{w_n}{w_{n'}})_{n,n'}$ is positive with trace at most $m$. +Consequently +\begin{equation}\label{eq:channel-residual} + \left\|\frac1m\sum_n z'_n\otimes w_n\right\|_2^2 + =\frac{\tr(AB)}{m^2}\le\frac1{d_m}. +\end{equation} +Combining \eqref{eq:channel-density} and +\eqref{eq:channel-residual} bounds the residual contribution from +this event by $\sqrt{p/d_m}$. + +It remains to sum over every possible length. For the fixed integer +$h$, we have $m=\lfloor c2^k/h\rfloor\asymp2^k$ at large $k$, +uniformly for $c\in F$. Thus $d_m\asymp(1+k)^{3/2}$, and +\begin{equation}\label{eq:channel-tail} + \sum_{k\ge a,\ c\in F}\sqrt{p_{k,c}/d_{m(k,c)}} + \le\left(\sum_{k,c}p_{k,c}\right)^{1/2} + \left(\sum_{k\ge a,\ c\in F}d_{m(k,c)}^{-1}\right)^{1/2} + \longrightarrow0. +\end{equation} +The first factor is one, and the second is a tail of a convergent +series. The leading-space contributions sum to at most +$\varepsilon+o_{\varepsilon,L}(1)$ because the event probabilities +sum to one. First let $a\to\infty$, then let $\varepsilon\downarrow0$. +This proves the all-group product rule. + +All estimates were uniform over the finitely many offset patterns. +For each pattern and each integer $m$, the finite-dimensional +projections can be chosen once, so their use causes no measurability +problem. We may now integrate in $V,s$ and restore the phase test. +The uniformly vanishing endpieces do not affect the selected law. +Both defining channel identities have been proved. +\end{proof} + +\begin{corollary}\label{cor:channel-array-distal} +In Proposition~\ref{prop:channel-closure}, let +$P=(\omega_Y(y_i))_{i\in I}$ be the array of old distal parameters. +The spatial coordinates of all copied outputs are independent with +their product law conditionally on $P$. The corresponding assertion +holds after any finite iteration of that proposition, conditionally +on the complete array of old $W_Y$-parameters. +\end{corollary} + +\begin{proof} +Lemma~\ref{lem:parameter-distal} places the old parameter array in +the maximal distal factor of the new flow, including after any finite +iteration. Repeated application of +Proposition~\ref{prop:channel-closure} supplies its channel structure. +Lemma~\ref{lem:channel-distal} gives the conditional product law over +that maximal distal factor, hence over the old parameter array. +\end{proof} + +The channel and its tests are fixed before the selected limit is +taken. The packing estimate and the summation over length classes +require no uniform mixing estimates over different channels or over +all iteration depths. diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/06-activation.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/06-activation.tex new file mode 100644 index 0000000000000000000000000000000000000000..35bce89d14c84f528851c4a5d42ce747a9c49ad3 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/06-activation.tex @@ -0,0 +1,558 @@ +\section{Activating an additional output role} +\label{sec:activation} + +The channel construction supplies independent spatial outputs even after +the averaging length has been selected. We now use that independence to +eliminate the nonzero correlation supplied by Proposition~\ref{prop:selectors}. +The argument counts how many of its four factors can be read from channel +outputs. After bounding this number and choosing a correlation with the +maximal number of output roles, we take an extension that stabilizes +certain conditional expectations. This will permit a correlation with +one additional output role, contradicting maximality. + +Throughout this section the selectors, and hence the assignment +$Y\mapsto\lambda_Y$, are fixed. For a channel $Y$, write $d_Y(y)$ for +its full list of spatial outputs and $x_Y(y)$ for the spatial coordinate +of its factor $\pi_Y(y)\in C$. If the output list has length $m$, its +spatial product law is $\mu^{\otimes m}$; when $m=0$ this is the +one-point probability space. All functions used below are real and +bounded. + +\subsection{Correlations and an upper bound on output roles} + +For $A\subseteq I$, consider a correlation +\begin{equation}\label{eq:role-correlation} + \int \Phi(P)\prod_{i\in A}h_i(d_Y(y_i)) + \prod_{i\notin A}h_i(x_Y(y_i))\,\dd\lambda_Y, + \qquad P=(\omega_Y(y_i))_{i\in I}. +\end{equation} +Here $\Phi$ is an arbitrary bounded measurable function of the complete +distal parameter array. Each $h_i$ is centred for its spatial law: +$\mu^{\otimes m}$ when $i\in A$ and $\mu$ otherwise. We call the +indices in $A$ the \emph{output roles}. The functions in an output role +may depend jointly on the whole output list. Lemma~\ref{lem:channel-distal} +implies that every factor in \eqref{eq:role-correlation} is also centred +conditionally on its own parameter $P_i$. + +The channel with no outputs is permitted by Definition~\ref{def:channel}. +The phase test in Proposition~\ref{prop:selectors} is measurable in the +distal parameters of the suspension. Thus that proposition gives a +nonzero correlation \eqref{eq:role-correlation} with $A=\varnothing$. +Consequently, under the supposition of pointwise failure, there is a +largest number $r\in\{0,1,2,3,4\}$ of output roles in any nonzero +correlation of this form, over all channels and all such tests. + +\begin{proposition}\label{prop:role-bound} +Every nonzero correlation \eqref{eq:role-correlation} has at most two +output roles. In particular, the maximal number $r$ just defined +satisfies $r\le2$. +\end{proposition} + +\begin{proof} +Apply Proposition~\ref{prop:channel-closure} to $Y$, with any +normalizing slot $b$. In the channel $\mathsf M_b(Y)$, all the copied +outputs from its four slots have their spatial product law. This +continues to hold conditionally on the old parameter array $P$, by +Corollary~\ref{cor:channel-array-distal}. This makes +\eqref{eq:role-correlation} zero +when $|A|=4$. + +Suppose now that $|A|=3$, and let $b$ be the remaining slot. Starting +with $Y_0=Y$, define +\[ + Y_{n+1}=\mathsf M_b(Y_n),\qquad n\ge0, +\] +using at each step the outputs and group assignment in +Proposition~\ref{prop:channel-closure}. The state of $Y_n$ is an array +of $Y$-states indexed by $I^n$. The projection to slot $b$ at each new +axis respects the flows, since that slot has speed $b/b=1$. + +In this array, consider the $n$ coordinate lines through the corner +$(b,\ldots,b)$. Each line has law $\lambda_Y$. To check this without +any symmetry assertion, first fix all but one coordinate at $b$. +Projection along a coordinate fixed at $b$ is an equivariant factor +map. If the remaining coordinate is the newly added one, functoriality +of the selected law projects it to $\lambda_Y$; if it is an earlier +coordinate, the new slot-$b$ marginal is the preceding array law. +Induction proves the assertion for every line. + +All spatial output lists throughout the array are independent +conditionally on its full array of old $W_Y$-parameters, again by +Corollary~\ref{cor:channel-array-distal}. Write +$(y_i^{(a)})_{i\in I}$ for coordinate line $a$ and +$P^{(a)}=(\omega_Y(y_i^{(a)}))_i$ for its parameter array. Put +\[ + H_a=\Phi(P^{(a)})\prod_{i\ne b}h_i(d_Y(y_i^{(a)})). +\] +The products for two different lines use disjoint output lists, since +neither uses the common corner. Conditional independence and centring +give $\E[H_aH_{a'}]=0$ for $a\ne a'$. Also $\norm{H_a}_2\le K$ for a +constant $K$ depending only on the fixed tests. If +$H=h_b(x_Y(\text{corner}))$ and $c\ne0$ is the original correlation, +the line-marginal identity gives $\ip{H}{H_a}=c$ for every $a$. +Bessel's inequality therefore yields +\[ + n|c|^2\le K^2\norm H_2^2. +\] +As $n$ is arbitrary, this is impossible. +\end{proof} + +\subsection{Stabilizing conditional expectations under extension} + +The next extension construction prevents extra information in a later +system from changing the conditional laws that we shall use. It is an +energy-increment construction of the kind used for sated extensions +\cite[Theorem~3.11 and Lemma~3.12]{Austin}; we give the argument for the selected-law assignment +needed here. + +On $(Y^I,\lambda_Y)$ introduce the five sigma-algebras +\begin{equation}\label{eq:saturation-data} + \mathcal D_0(Y)=\sigma(P),\qquad + \mathcal D_l(Y)=\sigma\bigl(P,(y_i)_{i\ne l}\bigr),\quad l\in I. +\end{equation} +If $\rho:Y'\to Y$ is an extension over $C$, functoriality gives the +factor map $\rho^I:(Y'^I,\lambda_{Y'})\to(Y^I,\lambda_Y)$. +The pullback of each $\mathcal D_l(Y)$ is contained in +$\mathcal D_l(Y')$, since the old distal factor is contained in the +new one. + +\begin{proposition}\label{prop:saturation} +Every standard flow over $C$ has a standard extension $Y$ with the +following property. For every further standard extension +$\rho:Y'\to Y$ over $C$, every $f\in L^2(\lambda_Y)$, and every +$l\in\{0\}\cup I$, +\begin{equation}\label{eq:saturation-identity} + \E_{\lambda_{Y'}}[f\circ\rho^I\mid\mathcal D_l(Y')] + =\E_{\lambda_Y}[f\mid\mathcal D_l(Y)]\circ\rho^I. +\end{equation} +If the initial flow is a channel, the extension is a channel with the +same outputs pulled back to it. +\end{proposition} + +\begin{proof} +For a test $f$ on a selected joining and a datum type $l$, call +\[ + \bigl\|\E[f\mid\mathcal D_l]\bigr\|_2^2 +\] +its energy. Under extensions this number increases and remains bounded +by $\norm f_2^2$. Starting from the given system $Y_0$, build a sequence +of extensions $Y_0\leftarrow Y_1\leftarrow\cdots$. At each stage +introduce a countable dense subset of the +new selected joining's $L^2$ space and retain all previously introduced +tests by pullback. Schedule every retained test and every datum type +infinitely often, with positive tolerances tending to zero. At a +scheduled step, take an extension whose energy is within the prescribed +tolerance of the supremum over all standard extensions of the present +system. Such an extension exists by the definition of a supremum. + +Let $Y$ be the countable inverse limit. It is a standard flow over $C$. +By Proposition~\ref{prop:selected}, its selected law projects to all +the stage laws. The union of the stage $L^2$ spaces is dense in +$L^2(\lambda_Y)$: the sigma-algebra on the fourfold inverse limit is +generated by its stage coordinate factors. + +Fix a retained test $f$, a datum type $l$, and an extension $Y'$ of +$Y$. If stage $n+1$ optimized this pair with error $\epsilon_n$, then +$Y'$ is also an extension available over stage $n$. Hence +\begin{align*} + \operatorname{energy}_{Y'}(f) + &\le\sup_{R\to Y_n}\operatorname{energy}_{R}(f)\\ + &\le\operatorname{energy}_{Y_{n+1}}(f)+\epsilon_n + \le\operatorname{energy}_{Y}(f)+\epsilon_n. +\end{align*} +Along the scheduled subsequence $\epsilon_n\to0$. Monotonicity gives +the opposite inequality, so the two energies are equal. The old +conditional-expectation subspace is contained in the new one; equality +of the two projection norms therefore implies equality of the +projections. This proves \eqref{eq:saturation-identity} for the retained +tests. Conditional expectation is an $L^2$ contraction, so density +proves it for every $f\in L^2(\lambda_Y)$. + +This argument uses only inclusion of the datum spaces under extension; +it does not require an inverse-limit identity for maximal distal +factors. Finally, Lemma~\ref{lem:channel-distal} shows that retaining +the outputs makes every stage and the limit a channel. +\end{proof} + +We call a system satisfying Proposition~\ref{prop:saturation} +\emph{saturated}. A nonzero correlation persists under the passage to +a saturated extension: its functions pull back, and its old parameter +test remains measurable in the enlarged distal parameters. + +\subsection{A channel carrying the three other factors} + +Choose a nonzero correlation with the maximal number $r$ of output +roles, and replace its channel by a saturated extension $Y$. Retain +the notation $A,\Phi,h_i$ for its tests. Proposition~\ref{prop:role-bound} +gives $r\le2$, so choose $b\in I\setminus A$. Set +\[ + Z=\mathsf M_b(Y),\qquad Z\longrightarrow Y,\quad + (y_j)_{j\in I}\longmapsto y_b. +\] +Its state law is $\lambda_Y$. We give $Z$ a new channel structure +adapted to this particular correlation. Keep the full old output list +from slot $b$, and for every $j\ne b$ add the list +\begin{equation}\label{eq:added-output-lists} + Q_j= + \begin{cases} + d_Y(y_j),&j\in A,\\ + x_Y(y_j),&j\notin A, + \end{cases} +\end{equation} +where in the second case the actual flow factor added is the copy of +$C$ in that slot. Index the new three groups by $j\in I\setminus\{b\}$. +Put $Q_j$ in group $j$ and assign the three old groups from slot $b$ +bijectively to these new groups. All speeds are positive rationals. + +\begin{lemma}\label{lem:regrouped-channel} +With these outputs and groups, $Z$ is a channel over its slot-$b$ +copy of $C$. +\end{lemma} + +\begin{proof} +Condition first on the entire old parameter array $P$ in $\lambda_Y$. +The full output collection consists of one list at each slot: $Q_j$ +at $j\ne b$, and the old list $d_Y(y_b)$ at $b$. Every proper collection +of these slot lists has its spatial product law by +Proposition~\ref{prop:selected} and Lemma~\ref{lem:channel-distal}. +To verify the full product law, test by a product of one bounded spatial +function of each list and expand each function into its mean and +centred part. Every term except the all-centred term is already +determined by a proper marginal. If the all-centred conditional term +were nonzero, some bounded test of $P$ would have nonzero integral +against it. This would be a correlation \eqref{eq:role-correlation} +on $Y$ with output roles $A\cup\{b\}$, contradicting maximality of $r$. +Thus the full collection has product law conditionally on $P$. + +Now take a proper subset of the three new groups together with the +original spatial coordinate $x_Y(y_b)$. At least one slot $j\ne b$ is +absent. At slot $b$, only a proper subset of the old three groups has +been retained. The conditional independence of the corresponding +proper set of slots, followed by the old channel rule at slot $b$, +gives the required spatial product law conditionally on $P$. +All phases of the new channel are functions of $P$. Integrating the +two product laws down to those phases proves the two conditions in +Definition~\ref{def:channel}. +\end{proof} + +Write +\begin{equation}\label{eq:outside-product} + q((y_j)_{j\in I})= + \prod_{j\in A}h_j(d_Y(y_j)) + \prod_{j\notin A\cup\{b\}}h_j(x_Y(y_j)). +\end{equation} +It is a function of the three added lists $Q_j$ in a single $Z$-state. +With $y=y_b$, define +\begin{equation}\label{eq:g-and-u} + g(P,y)=\E_{\lambda_Y}[q\mid P,y_b],\qquad + u(e,y)=\E_{m_Z}[q\mid\omega_Z,y_b],\quad e=\omega_Z. +\end{equation} +The original nonzero correlation equals +$\E[\Phi(P)h_b(x_Y(y_b))g(P,y_b)]$, so $g$ is nonzero in $L^2$. +The two conditional expectations in \eqref{eq:g-and-u} use the same +state law, but different parameter information: $e$ contains the old +array $P$ and may contain more. + +\subsection{The square and its conditional laws} + +The selected law $\lambda_Z$ is a law on a square of $Y$-states. +Write $\mathbf z_i=(y_{ij})_{j\in I}$ for row $i$, a $Z$-state, and set +\[ + P_{ij}=\omega_Y(y_{ij}),\qquad + E_i=\omega_Z(\mathbf z_i),\qquad E=(E_i)_{i\in I}. +\] +Each $E_i$ contains the complete old parameter row $(P_{ij})_j$, by +Lemma~\ref{lem:parameter-distal}. In column $b$ put +\[ + y_i=y_{ib},\qquad p=(P_{ib})_{i\in I},\qquad w=P_{bb}. +\] +Functoriality for $Z\to Y$ says that this column has law $\lambda_Y$. +Row $b$ has the same law because a single $Z$-state has law $\lambda_Y$. +Our aim is to show that +\[ + \E\bigl[q((y_i)_i)q(\mathbf z_b)\mid E\bigr]\not\equiv0. +\] +The row factor $q(\mathbf z_b)$ is read from the added outputs, so it +is the candidate for the additional output role at $b$. To establish +nonvanishing, saturation will first identify the column law given $E$ +and the conditional expectation of the row factor given that column. +Figure~\ref{fig:activation-square} distinguishes these two uses of the +old selected law. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[x=1.22cm,y=0.88cm, + every node/.style={font=\small}, + cell/.style={draw=black!35,minimum width=1.13cm,minimum height=0.74cm}] + \foreach \i in {1,2,3,4}{ + \foreach \j in {1,2,3,4}{ + \node[cell] at (\j,-\i) {}; + } + } + \fill[blue!10] (1.54,-4.41) rectangle (2.46,-0.59); + \fill[orange!17] (0.54,-2.41) rectangle (4.46,-1.59); + \fill[violet!22] (1.54,-2.41) rectangle (2.46,-1.59); + \foreach \i in {1,2,3,4}{ + \foreach \j in {1,2,3,4}{ + \draw[black!35] (\j-0.46,-\i-0.41) rectangle + (\j+0.46,-\i+0.41); + } + } + \node at (1,-0.15) {$j_1$}; + \node at (2,-0.15) {$b$}; + \node at (3,-0.15) {$j_2$}; + \node at (4,-0.15) {$j_3$}; + \node at (0.05,-1) {$i_1$}; + \node at (0.05,-2) {$b$}; + \node at (0.05,-3) {$i_2$}; + \node at (0.05,-4) {$i_3$}; + \node at (2,-1) {$y_{i_1}$}; + \node at (2,-2) {$y_b$}; + \node at (2,-3) {$y_{i_2}$}; + \node at (2,-4) {$y_{i_3}$}; + \node at (1,-2) {$y_{b j_1}$}; + \node at (3,-2) {$y_{b j_2}$}; + \node at (4,-2) {$y_{b j_3}$}; + \node[font=\scriptsize] at (1,-1) {$P_{i_1j_1}$}; + \node[font=\scriptsize] at (3,-1) {$P_{i_1j_2}$}; + \node[font=\scriptsize] at (4,-1) {$P_{i_1j_3}$}; + \node[font=\scriptsize] at (1,-3) {$P_{i_2j_1}$}; + \node[font=\scriptsize] at (3,-3) {$P_{i_2j_2}$}; + \node[font=\scriptsize] at (4,-3) {$P_{i_2j_3}$}; + \node[font=\scriptsize] at (1,-4) {$P_{i_3j_1}$}; + \node[font=\scriptsize] at (3,-4) {$P_{i_3j_2}$}; + \node[font=\scriptsize] at (4,-4) {$P_{i_3j_3}$}; + \node[anchor=west] at (4.7,-1) {$E_{i_1}\supseteq(P_{i_1j})_j$}; + \node[anchor=west] at (4.7,-2) {$E_b\supseteq(P_{bj})_j$}; + \node[anchor=west] at (4.7,-3) {$E_{i_2}\supseteq(P_{i_2j})_j$}; + \node[anchor=west] at (4.7,-4) {$E_{i_3}\supseteq(P_{i_3j})_j$}; + \node[blue!65!black,anchor=north] at (2,-4.7) + {$p=(P_{ib})_i$}; + \node[orange!65!black,anchor=north] at (5.4,-4.7) + {$q(\mathbf z_b)\text{ uses row }b\text{ off the corner}$}; +\end{tikzpicture} +\caption{The square under $\lambda_Z$. Blue marks column $b$, whose +old parameter array is $p$; orange marks the row containing the added +outputs used by $q$. Their common corner has parameter $w=P_{bb}$. +The new parameters $E_i$ contain the respective old parameter rows. +The distinguished slot is drawn in second position for legibility. +The figure records marginal and factor relations, not exchangeability +of the two axes.} +\label{fig:activation-square} +\end{figure} + +\begin{lemma}\label{lem:square-conditional} +In the square just defined, the conditional law of column $b$ given +$E$ is its $\lambda_Y$ law given $p$. Moreover, +\begin{equation}\label{eq:square-row-locality} + \E_{\lambda_Z}[q(\mathbf z_b)\mid E,(y_i)_{i\in I}] + =u(E_b,y_b). +\end{equation} +In particular, the conditional law of $y_b$ given $E$ is +$m_Y(\dd y\mid w)$; in a single $Z$-state, the conditional law of its +slot-$b$ state given $e=\omega_Z$ is the same fibre law over its old +slot-$b$ parameter. +\end{lemma} + +\begin{proof} +Apply saturation of $Y$ to the extension $Z\to Y$. Under the induced +map $Z^I\to Y^I$, the old joining is precisely column $b$, and the new +datum $\mathcal D_0(Z)$ is $\sigma(E)$. Equation~\eqref{eq:saturation-identity} +for $l=0$, first on a countable determining family, gives the first +conditional-law assertion. The singleton case of +Proposition~\ref{prop:selected} then gives +\begin{equation}\label{eq:corner-fibre-law} + \mathcal L(y_b\mid E)=m_Y(\cdot\mid w). +\end{equation} +Taking the row-$b$ marginal and applying the tower property from $E$ +to its coordinate $E_b$ gives +the stated single-state assertion. + +For \eqref{eq:square-row-locality}, fix $l\ne b$ and denote by +$\mathbf z_{-l}$ all rows except row $l$. Given $E$, these three rows +are independent with their $Z$ fibre laws, by +Proposition~\ref{prop:selected} applied to $Z$. In particular, +\begin{equation}\label{eq:omit-one-column-state} + \E[q(\mathbf z_b)\mid E,(y_i)_{i\ne l}]=u(E_b,y_b). +\end{equation} +It remains to justify adding $y_l$ to the conditioning. For every +bounded measurable test $F$ of a $Y$-state, saturation for the datum +type $l$ gives +\begin{equation}\label{eq:missing-state-markov} + \E[F(y_l)\mid E,\mathbf z_{-l}] + =\E_{\lambda_Y}[F(y_l)\mid p,(y_i)_{i\ne l}]. +\end{equation} +The right side is measurable in $(E,(y_i)_{i\ne l})$. Thus, given this +smaller datum, $y_l$ is conditionally independent of the complete rows +$\mathbf z_{-l}$. For completeness, multiplying +\eqref{eq:missing-state-markov} by any bounded function of +$\mathbf z_{-l}$ and conditioning on the smaller datum factors their +conditional product expectations. A monotone-class argument then gives +the asserted conditional independence. Since $q(\mathbf z_b)$ is +measurable in $\mathbf z_{-l}$, adjoining $y_l$ does not alter +\eqref{eq:omit-one-column-state}. This proves +\eqref{eq:square-row-locality}. +\end{proof} + +We can now compute the correlation between the two highlighted lines. +First condition on the column in Lemma~\ref{lem:square-conditional}, +then condition within its old law on its corner state. The result is +\begin{equation}\label{eq:square-product} + \E\bigl[q((y_i)_i)q(\mathbf z_b)\mid E\bigr] + =\int g(p,y)u(E_b,y)\,m_Y(\dd y\mid w). +\end{equation} +The right side is not manifestly positive. The next lemma supplies the +parameter independence that will show it cannot vanish identically. + +\begin{lemma}\label{lem:parameter-reanchoring} +Under $\lambda_Z$, the variables $p$ and $E_b$ are conditionally independent +given $w=P_{bb}$. +\end{lemma} + +\begin{proof} +Let $\kappa:W_Z\to W_Y$ be the factor obtained from the slot-$b$ +map $Z\to Y$. This is a flow factor, and +$P_{ib}=\kappa(E_i)$. Proposition~\ref{prop:selected} identifies the +$E$ law with the ordinary deterministic-length Furstenberg law on +$W_Z$. For a bounded test $F$ of $E_b$ and a rectangle test +$\prod_i H_i(P_{ib})$, its integrated average is therefore computed +from +\[ + \int_{W_Z} F(S_{W_Z}^{bt}e) + \prod_i H_i\bigl(\kappa(S_{W_Z}^{it}e)\bigr)\, + \dd m_{W_Z}(e) +\] +at each fixed $t$. Substitute $e'=S_{W_Z}^{bt}e$. The expression becomes +\[ + \int_{W_Z} F(e') + \prod_i H_i\bigl(S_{W_Y}^{(i-b)t}\kappa(e')\bigr)\, + \dd m_{W_Z}(e'). +\] +All factors other than $F$ depend only on $\kappa(e')$. Thus $F$ may +be replaced, at every $t$, by its conditional expectation given +$\kappa$. Averaging and taking the deterministic limit gives +\[ + \E\left[F(E_b)\prod_i H_i(P_{ib})\right] + =\E\left[\E[F(E_b)\mid w]\prod_i H_i(P_{ib})\right]. +\] +Rectangle tests generate the $p$ sigma-algebra, and $w$ is one of +its coordinates. The monotone-class theorem yields the required +conditional independence. +\end{proof} + +\subsection{Nonvanishing and the additional role} + +We record the elementary Hilbert-space fact used to finish the +argument. Its separability hypothesis is automatic for the probability +fibres occurring here. + +\begin{lemma}\label{lem:independent-hilbert-vectors} +Let $V$ be a square-integrable random vector in a separable real +Hilbert space, and let $V'$ be an independent copy. If +$\ip V{V'}=0$ almost surely, then $V=0$ almost surely. +\end{lemma} + +\begin{proof} +If $\Prob(\norm V\ge\epsilon)>0$ for some $\epsilon>0$, a countable +cover by balls of radius $\epsilon/4$ contains one ball meeting that +event with positive probability. With positive probability both $V$ +and $V'$ lie in this intersection. On that event their norms are at +least $\epsilon$ and their distance at most $\epsilon/2$, so +\[ + \ip V{V'} + =\tfrac12\bigl(\norm V^2+\norm{V'}^2-\norm{V-V'}^2\bigr)>0. +\] +This is a contradiction. +\end{proof} + +\begin{proposition}\label{prop:activation} +For the saturated channel $Y$ and maximal nonzero correlation chosen +above, the regrouped channel $Z$ has a nonzero correlation +\eqref{eq:role-correlation} with output roles $A\cup\{b\}$. +\end{proposition} + +\begin{proof} +We first prove that the conditional quantity in +\eqref{eq:square-product} is nonzero. In a single $Z$-state let $R(e)$ +denote its old $W_Y$-parameter array; it is a function of $e=\omega_Z$. +Let $w$ denote its slot-$b$ parameter. Given $w$, write $\alpha_w$ for +the law of the old parameter array in $\lambda_Y$, and $\beta_w$ for +the law of $e$ in $m_Z$. Then $R_*\beta_w=\alpha_w$. Both old arrays, +the column array $p$ and the row array $R(E_b)$, have this same +conditional marginal. Lemma~\ref{lem:parameter-reanchoring} says that +the pair $(p,E_b)$ has conditional law $\alpha_w\otimes\beta_w$. + +Work for almost every fixed $w$ in the separable real Hilbert space +$H_w=L^2(m_Y(\cdot\mid w))$. The bounded kernels +$g(p,\cdot)$ and $u(e,\cdot)$ are measurable $H_w$-valued random +vectors. To justify these assertions on a common conull set of $w$, +choose a countable generating algebra on the standard Borel space $Y$. +Its rational simple span is dense in every $H_w$; the scalar tests +against this span and Fubini give the required strong measurability. +The single-state assertion of +Lemma~\ref{lem:square-conditional} says that, conditionally on $e$, +the state $y$ has precisely the fixed law $m_Y(\cdot\mid w)$. Hence +conditioning on $R(e)$ does not introduce a $y$-dependent weight into +the distribution of $e$. The tower property for +\eqref{eq:g-and-u} gives the Hilbert-valued identity +\begin{equation}\label{eq:average-u-is-g} + \E_{\beta_w}\bigl[u(e,\cdot)\mid R(e)\bigr] + =g(R(e),\cdot). +\end{equation} +This can also be checked against a countable dense set in $H_w$: +the joint law of $(e,y)$ over $w$ is +$\beta_w(\dd e)m_Y(\dd y\mid w)$, and conditioning the defining +expectation of $u$ further on $(R(e),y)$ gives the defining expectation +of $g$. + +Suppose now that \eqref{eq:square-product} were zero almost surely. +Conditional parameter independence would imply +\[ + \ip{g(p,\cdot)}{u(e,\cdot)}_{H_w}=0 + \quad\text{for }\alpha_w\otimes\beta_w\text{-almost every }(p,e). +\] +Average in $e$ conditionally on $R(e)$ and use +\eqref{eq:average-u-is-g}. We obtain +\[ + \ip{g(p,\cdot)}{g(p',\cdot)}_{H_w}=0 + \quad\text{for }\alpha_w\otimes\alpha_w\text{-almost every }(p,p'). +\] +Lemma~\ref{lem:independent-hilbert-vectors} forces +$g(p,\cdot)=0$ almost surely for almost every $w$. This contradicts +the previously established nonvanishing of $g$. + +It follows that some bounded measurable real test $\Psi(E)$ has +\begin{equation}\label{eq:activated-correlation} + \E_{\lambda_Z}\bigl[ + \Psi(E)q((y_i)_i)q(\mathbf z_b)\bigr]\ne0. +\end{equation} +For example, take the sign of the nonzero conditional expectation +in \eqref{eq:square-product}. We identify its four roles. In every +row $i\ne b$, the factor from $q((y_i)_i)$ reads the old slot-$b$ +state: when $i\in A$ it is a function of the old outputs retained in +$Z$, and when $i\notin A$ it is a function of $Z$'s original spatial +coordinate. Their spatial means remain zero. In row $b$, the factor +$q(\mathbf z_b)$ is a function of the three added lists $Q_j$. +Under the full spatial product law of the outputs of $Z$, these lists +are independent and each of its three factors is centred. Therefore +$q(\mathbf z_b)$ is itself centred for that spatial product law. +Finally $\Psi$ is a test of the full new distal parameter array $E$. +Thus \eqref{eq:activated-correlation} has exactly the form +\eqref{eq:role-correlation}, with output roles $A\cup\{b\}$, as +asserted. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +If the required pointwise conclusion failed, the standard coding +reduction and Proposition~\ref{prop:selectors} would supply the fixed +selectors and a nonzero correlation with no output roles. A maximal +role number $r$ would therefore exist. Proposition~\ref{prop:role-bound} +gives $r\le2$. Proposition~\ref{prop:saturation} preserves a maximal +witness in a saturated channel, and Proposition~\ref{prop:activation} +then produces a nonzero correlation with $r+1$ output roles. This +contradicts the definition of $r$. Pointwise failure is impossible. +The limiting value is the product of the four integrals, as in the +norm convergence of Lemma~\ref{lem:relative-norm}. Pulling back from +the standard coded factor proves the assertion on the original +probability space. +\end{proof} diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/07-lower-length.tex b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/07-lower-length.tex new file mode 100644 index 0000000000000000000000000000000000000000..d6decf219694df94f816182f342757e3ad435992 --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/build/sections/07-lower-length.tex @@ -0,0 +1,756 @@ +\section{Pointwise convergence for the proper subsets of four times} +\label{sec:lower-length} + +We prove Lemma~\ref{lem:lower-length}. The systems to which that lemma +is applied in Section~\ref{sec:selected} are extensions of a suspension +and need not be mixing or ergodic. Accordingly, the input needed here +is a pointwise convergence theorem on arbitrary invertible probability +preserving systems. Its analytic basis is the oscillation estimate in +\cite[Proposition~2.2]{TripleCompanion}, denoted by P in this appendix. We also +write H for \cite{HilbertCompanion}. + +P states this oscillation estimate for the times $n,2n,3n$ before using +mixing to identify its limit. We need the three additional triples +contained in $\{1,2,3,4\}$. We obtain them by modifying the local +four-linear construction in H and its extension to inputs varying with +depth in P. This is a use of those analytic proofs, including their +intermediate estimates; the boundedness theorem for the trilinear +Hilbert transform alone would not supply the required oscillation +estimate. We specify the imported statements first, then verify the +coefficient changes, and finally give the passage to arbitrary systems +and all integer averaging lengths. + +\subsection{The local estimates used from H and P} + +For this appendix, a \emph{role set} is a set $S$ of four distinct +integer coefficients. Initially $S=\{0\}\cup J$, where +$J\subset\{1,2,3,4\}$ has three elements. If $K$ is a dyadic interval +of length $r$, put +\begin{equation} + \mathcal H_{K,S}(z) + =r^{-2}\int_{\R^2}w_K(x,t)\prod_{j\in S}z_j(x+jt)\,\dd x\,\dd t. + \label{eq:ll-local-form} +\end{equation} +Fix $c_0>0$ and $C_0\ge1$. The admissible kernels are the smooth, +compactly supported functions $w_K$ for which every $x+jt$ on the +support lies in $K$ at distance at least $c_0r$ from its boundary, +\begin{equation} + \int_{\R}w_K(v-jt,t)\,\dd t=0 + \quad(v\in\R, j\in S), + \qquad + \|\partial_x^a\partial_t^b w_K\|_\infty + \le r^{-a-b}\bigl(C_0(a+b+2)\bigr)^{C_0(a+b+2)}. + \label{eq:ll-kernels} +\end{equation} +Kernels may be chosen independently at the different intervals. + +H's \emph{Uniform local estimate} +\cite[Theorem~2.2]{HilbertCompanion} +asserts, for $S=\{0,1,2,3\}$ and a suitable $q\in(2,3)$, that +\begin{equation} + \sum_{K\in\mathcal K}|K|\,|\mathcal H_{K,S}(z)| + \le C\int_{\R}\prod_{j\in S}M_qz_j(y)\,\dd y, + \qquad M_qz=(M(|z|^q))^{1/q}, + \label{eq:ll-fixed-local} +\end{equation} +for every finite family of dyadic intervals and compactly supported +scalar $L^q$ inputs. Here $M$ is the Hardy--Littlewood maximal +operator. Its proof, in H, Sections~3--10, supplies the quantitative +chart, projection, separation, counting, and side estimates used in P. + +The extension needed for oscillation is P's \emph{Local estimate with +variation in depth}, Theorem~3.2. To state it, let $K_*$ be a root +interval and let $s(K)=\log_2(|K_*|/|K|)$ be the depth of a descendant. +For a finite integer block $B=\{a,\ldots,b\}$ and a scalar sequence +$(z_s)$, write +\[ + \mathcal V_B(z)(y) + =\max_{s\in B}|z_s(y)|+ + \sum_{s=a}^{b-1}|z_{s+1}(y)-z_s(y)|. +\] +For the original role set $\{0,1,2,3\}$, P proves the assertion below. +We establish it for the four required role sets. + +\begin{proposition}[The local analytic input with changed coefficients] +\label{prop:ll-depth} +Let $S=\{0\}\cup J$, where $J$ is a three-element subset of +$\{1,2,3,4\}$. Let $\mathcal T$ be a finite rooted dyadic tree with +root $K_*$ and largest depth $d$, and let $\mathcal K\subset\mathcal T$ +have kernels satisfying \eqref{eq:ll-kernels}. For each $j\in S$, let +$(z_{j,s})_{0\le s\le d}$ be measurable scalar functions on $K_*$. +Extend these functions by zero outside $K_*$. +Suppose that $\{0,\ldots,d\}$ has a deterministic partition into +$W_j\ge1$ nonempty consecutive blocks, on each of which +\[ + \mathcal V_B(z_j)(y)\le W_j^{-1/2} + \quad\text{for almost every }y\in K_*. +\] +The partitions may differ between the four roles. Then +\begin{equation} + \sum_{K\in\mathcal K}|K| + \bigl|\mathcal H_{K,S}((z_{j,s(K)})_{j\in S})\bigr| + \le C(c_0,C_0)|K_*|. + \label{eq:ll-depth} +\end{equation} +The constant is independent of the tree, kernels, depth range, and +block partitions. It can be chosen uniformly over the four sets $S$. +\end{proposition} + +The proof of Proposition~\ref{prop:ll-depth} is the proof-by-modification +below. The analytic inputs are H's local estimate and its proofs of the +intermediate estimates, and P's depth-local construction in Sections~4--7. +Proposition~\ref{prop:ll-depth} is the sole analytic input to the +local-to-orbit argument in Subsection~\ref{subsec:ll-transfer}; +Subsection~\ref{subsec:ll-convergence} then proves +Lemma~\ref{lem:lower-length}. The intervening subsections establish +the proposition by checking, in order, the coordinate and quadratic +identities, the integer-lift and counting arithmetic, and their +use in P's depth scheduling. +We retain their definitions of quadratic chart atoms, inherited +projection families, curvature tags, and finite progression tests. The +following computations specify their dependence on the four role +coefficients. In particular, the proposition is not an assertion of +uniform continuity of a Hilbert-transform bound in its slopes. + +\subsection{Coordinate changes and the quadratic cancellation vector} +\label{subsec:ll-algebra} + +Every set $S$ above contains an adjacent pair. Translate all role +labels by the same integer so that one such pair is $0,1$. This is the +change of variables $x'=x+mt$ in \eqref{eq:ll-local-form}, with +Jacobian one. The four spatial coordinates themselves are unchanged. +Consequently the support and cancellation conditions remain the same, +and the derivative bounds retain their form after increasing the fixed +constant $C_0$. During the local argument we use these translated +labels. Their pairwise differences have absolute value at most four, +and their diameter is either three or four. Thus +\begin{equation} + |t|\le |K|/3 + \quad\text{whenever every }x+jt\text{ lies in }K. + \label{eq:ll-support-t} +\end{equation} + +The vector $(-1,3,-3,1)$ in H is replaced by a primitive integer vector +$(c_j)_{j\in S}$ proportional to +\[ + \left(\prod_{\ell\in S\setminus\{j\}}(j-\ell)\right)^{-1}. +\] +The overall sign is immaterial. Before translation, convenient choices +are +\begin{equation} +\begin{array}{c|rrrr} + S&\multicolumn{4}{c}{(c_j)_{j\in S}\text{ in increasing role order}}\\ \hline + \{0,1,2,3\}&-1&3&-3&1\\ + \{0,1,2,4\}&-3&8&-6&1\\ + \{0,1,3,4\}&-1&2&-2&1\\ + \{0,2,3,4\}&-1&6&-8&3 +\end{array} +\label{eq:ll-coefficient-table} +\end{equation} +Lagrange interpolation of the polynomials $1,u,u^2$ gives +\begin{equation} + \sum_{j\in S}c_j=\sum_{j\in S}jc_j= + \sum_{j\in S}j^2c_j=0. + \label{eq:ll-moments} +\end{equation} +These identities are preserved by translation of the role labels. +Every $c_j$ is nonzero. All coefficients and denominators arising +from this finite list of coordinate systems are bounded absolute +constants. + +We distinguish numerical role labels from enumerations of four +objects. A coefficient such as $j$, $j-i$, or $c_j$ is changed according +to $S$. An ordering of four input sizes, four leaves, or four positions +in a multilinear expansion still enumerates four objects. This +convention leaves those parts of H's proofs unchanged. + +\subsection{Chart separation, detection, and the scale estimates} + +The local absolute estimate, H, Lemma~2.1, uses the map +$(x,t)\mapsto(x+it,x+jt)$, whose determinant is $j-i$. Its endpoint +bounds and interpolation therefore hold for the new roles. In +particular the initial bound for the finite-depth constant in +\eqref{eq:ll-fixed-local}, and H's stopping reduction to normalized +local sizes, remain valid. The coefficient and projection bounds, +compression and height-bin estimates, H, Lemmas~3.2--3.5, are +statements about finite families of vectors or real frequencies and do +not specify a progression coefficient. + +We next check the part of H, Section~4, that converts two chart atoms +in distinct roles into separated factors. A chart phase has the form +\[ + \lambda y^2+(\theta y)^TCd+\beta y, +\] +where $\theta$ is a vector of real frequencies, $C$ is the rational +chart matrix, and $d$ is a bounded lift of $\theta y$ modulo the +integer lattice. The source definition also includes a smooth +amplitude on the lift. Its one-variable Weyl estimates, Fourier +expansion, and uniform tail estimates are unchanged. + +For H, Lemma~4.5 (\emph{Progression separation}), fix source roles $i,k$ +and spectator roles $a,b$. Set $\alpha_i=1$, $\alpha_k=0$, and solve +\[ + \sum_{j\in S}\alpha_j=\sum_{j\in S}j\alpha_j=0 +\] +for $\alpha_a,\alpha_b$. This is possible because $a\ne b$. +Writing $y_j=x+jt$, insert the same lift partitions as in H at +$\theta y_a$, $\theta y_b$, and $\theta t$. If their lifts are +$d_a,d_b,d$, then +\[ + k_j=d_j-d_a-(j-a)d +\] +is an integer vector. For instance, if $d_j=\theta y_j-m_j$ and +$d=\theta t-m$, the identity $y_j=y_a+(j-a)t$ gives +$k_j=-m_j+m_a+(j-a)m$. Thus the equality cutoffs in H still test +integer carries exactly. Their support bounds change only by a +fixed factor. With $A_2=\sum_jj^2\alpha_j$, direct expansion gives +\begin{equation} + \begin{split} + \sum_{j\in S}\alpha_j + [\lambda y_j^2+(\theta y_j)^TCd_j] + &=A_2[\lambda t^2+(\theta t)^TCd]\\ + &\quad+\sum_{j\in S}\alpha_j(\theta y_j)^TCk_j. + \end{split} + \label{eq:ll-chart-identity} +\end{equation} +The last sum is linear in $x,t$ and hence separates in $y_a,y_b$. +The smooth amplitude on the bounded lift box has the same Gevrey +bounds up to fixed constants. Fourier expansion therefore gives +H's absolutely convergent separation into two spectator atoms and one +atom in $t$, with the same coefficient-tail exponents and +quasipolynomial complexity bounds. H, Lemma~4.6 (\emph{Single-scale separation}), then uses $t=(y_b-y_a)/(b-a)$; every added rational height is +bounded by an absolute constant. + +For the generalized von Neumann bound, H, Lemma~4.11, let $i$ be +the role whose input is controlled by its $U^3$ norm. +Apply Cauchy--Schwarz in the directions $(-j,1)$ for the other +three roles, beginning with the role containing the $L^2$ input. +Each direction holds $x+jt$ fixed and changes $x+it$ by +$(i-j)h$. Multiplication by the nonzero integer $i-j$ preserves Haar +measure on a circle. The three independent circle parameters +therefore produce the same cube integral as in the source proof. +Character weights split using the marked pair $x=y_0$, +$t=y_1-y_0$. Enlarging the fixed periodic boxes accommodates all +translated role sets. This verifies the coefficient-dependent part +of the generalized von Neumann bound. In H, Lemma~4.12 +(\emph{Local detection}), the potentially unbounded $L^2$ residual +occupies a different role. After clipping the opposing inputs, the +source proof decomposes two of them using its continuous inverse +theorem and bounds the uniform errors by Lemma~4.11. The separated +chart moments then detect the residual. Those separations were +verified above; the clipping, inverse theorem, and mixed-norm +exponents are unchanged. + +H, Lemma~5.1 (\emph{A scale sum for one separated term}), +uses the spectator coordinates $(y_a,y_b)$. Holding one fixed, the +integral of the kernel over the other is zero by +\eqref{eq:ll-kernels}, with a constant Jacobian factor $|b-a|$. +The double primitive of that kernel consequently has the same +compact support and smooth bounds. Its tensor expansion into +mean-zero bumps, the Bessel estimates for those bumps, and the +frequency-height bounds apply with the new pair. The only additional +demodulations have frequencies $\pm\gamma/(b-a)$; their rational +multipliers have bounded height, although $\gamma$ can be arbitrary +real. This also verifies the coefficient changes +in the finite predictor list and the delayed predictor replacement. +The scale bands, accuracy gaps, and projection-energy bounds in the +remaining base construction depend on these estimates and the +four-slot multilinearity, rather than the values of the labels. +There is also a multivariable use of the chart estimates in H, +Lemma~5.7 (\emph{Few active scales for a fixed triple}). Squaring +the transpose at the remaining role $\ell$ gives six atoms with +arguments $h+(j-\ell)t$ and $h+(j-\ell)t'$. These are still fixed +integer linear forms in three variables, so the three-variable Weyl +alternative and chart expansion of H, Lemmas~4.3--4.4, apply with +the same complexity bounds. Its final kernel cancellation is +the marginal at anchor $\ell$ in \eqref{eq:ll-kernels}. + +\subsection{Curvature tags, short blocks, and integer heights} + +H's localization in Section~6 groups chart atoms by their normalized +quadratic phase and horizontal chart data, called their curvature +tags. Normalize the parameters $(\lambda,C)$ in role $j$ by +dividing both by $c_j$, and retain the horizontal vector $\theta$. +The coefficient calculation in H, Lemma~6.1 (\emph{Separation of different tags}), can +then be checked directly. For source roles $i,j$ and spectator roles +$a,b$, put +\[ + \xi_\ell=\frac{b-\ell}{b-a},\qquad + \zeta_\ell=\frac{\ell-a}{b-a}; + \qquad y_\ell=\xi_\ell y_a+\zeta_\ell y_b. +\] +The barycentric formula gives +\begin{equation} + c_i\xi_i\zeta_i=-c_j\xi_j\zeta_j\ne0. + \label{eq:ll-tag-identity} +\end{equation} +Indeed the factors $(i-a)(i-b)$ cancel from the product defining +$c_i$, leaving a nonzero multiple of $(i-j)^{-1}$; interchanging +$i,j$ changes its sign. The return calculation in that lemma +therefore has the same nonzero prefactor and the same separation +conclusion. Its period is enlarged to clear these fixed denominators. + +The residue-template calculation in H, Lemma~6.6 +(\emph{A uniform same-tag test}), uses the +marked adjacent pair. Its lifts satisfy +\begin{equation} + d_\ell=(1-\ell)d_0+\ell d_1+k_\ell, + \qquad k_\ell\in\Z^d. + \label{eq:ll-template-lift} +\end{equation} +Substitution of \eqref{eq:ll-template-lift} and +\eqref{eq:ll-moments} cancels the common quadratic terms and leaves +only linear phases from the carries. Their Fourier expansion has +the same residue-template bound after a fixed enlargement of the +lift box. It introduces no new curvature tag. Thus the source +localization proof receives the same tag-separation and transfer +estimates. Its graph distances, color histories, masks, and +projection bounds are then used with those estimates and their +original parameter order. + +H, Lemma~7.2 (\emph{Short-block comparison}), uses two sampling +windows to compare a finite lattice form with a form on a connected +torus. Place those windows on roles $0,1$. The lattice and lifted +torus coordinates in the other roles are respectively +\[ + n_j=(1-j)n_0+jn_1, + \qquad w_j=(1-j)u+jv. +\] +Both formulas have integer coefficients. A fixed torus of length +$64$ and lift boxes in $[-12,12]$ accommodate the present role sets; +the source support parameter is decreased by a fixed factor if +necessary. Maps from a pair of variables to two distinct role +coordinates are injective on their lattice images and surjective on +connected tori. These are the pair-coordinate properties needed in +the comparison. + +The quadratic randomization in role $j$ becomes +$e(\gamma m^2/c_j)$, where $e(v)=\exp(2\pi i v)$ and $\gamma$ is +uniform modulo a common multiple of all $|c_j|$; the value $24$ works +for \eqref{eq:ll-coefficient-table}. On the Fourier support of the +four-linear form, +\[ + \sum_jm_j=\sum_jjm_j=0. +\] +The two vectors $c=(c_j)$ and $(jc_j)$ are independent and span this +two-dimensional nullspace. Hence $m_j=c_j(v+jv')$ for some real +$v,v'$, and +\[ + \sum_j\frac{m_j^2}{c_j} + =\sum_jc_j(v+jv')^2=0 +\] +by \eqref{eq:ll-moments}. The randomization preserves the form. +For each single role, averaging its fourth moment in the spatial +variable and in $\gamma$ imposes equality of a pair's sum and sum of +squares; the same two pairings as in H remain. Thus its fourth-moment +and clipping estimates, including their powers of $q-2$, are +unchanged. + +The return from lattice points to small intervals also holds with +these coordinates. If four real role coordinates are within $e$ of +integers, the marked adjacent pair reconstructs an integer progression. +The discrepancy at any other role is an integer of magnitude less +than $8e$. Choosing $e<1/100$ forces it to vanish. The admissible +real offsets have positive area, computed explicitly below in +\eqref{eq:ll-polygon}. All support and overlap losses in the +short-block comparison are therefore fixed constants. + +The remaining arithmetic change is the height base. In H, +Sections~8--9, replace $L_h=6^h$ by $L_h=12^h$, and replace the +factor $36$ in the finite progression tests by $144$. Anchor changes +still have integer coefficients. The common progression multiplier +in H, Section~8.3, uses exactly \eqref{eq:ll-template-lift} and +\eqref{eq:ll-moments}, so its smoothness and Fourier truncation are +unchanged in type. For the divisibility step, a coordinate in the +free lattice quotient in H, Lemma~8.3 (\emph{Height stabilization}), +is an integral rational multiple of $12^{R'}$. Here the quotient +is the integer coordinate lattice modulo its intersection with the +source's small rational subspace, $R'$ is the height gap, and $C_*$ +bounds the denominators before multiplication by $12^{R'}$. +The intersection is saturated, so this quotient is free abelian. +Writing $\Delta$ for the required drop in height, its $2$- and $3$-adic +valuations are at least +\[ + 2R'-\log_2 C_*,\qquad R'-\log_3 C_*. +\] +Thus $R'>\log_2C_*+2\Delta$ ensures divisibility by $12^\Delta$. +No assumption that the initial denominator has only primes $2,3$ is +needed: the resulting coordinate is already integral. The reserves +and the later growth constant $A_0$, used in the source bounds +$E_h=A_0^{h+1}$, are chosen using the new base. + +\subsection{Progression counting and the depth-varying extension} + +We finish the coefficient check at the finite counting step because +it requires more than the identities for quadratic phases. In H, +Section~9, eliminating a role $\ell'$ gives the three coefficients +\[ + \alpha_i=i-\ell',\qquad i\in S\setminus\{\ell'\}. +\] +They are nonzero and have absolute value at most four. The +horizontal frequency tests use sums of role frequencies, while their +anchored tests use the weights $i-\ell'$. Their kernel cancellations +are precisely \eqref{eq:ll-kernels}. Accordingly the source's +Fourier expansions, coefficient tails, and annular tests keep their +forms after these replacements. + +In H, Lemma~9.1 (\emph{Sparse mass of separated progression tests}), +the shift data consist of a vector $\zeta\in\R^d$ and integer +vectors $n$ with $\|n\|_\infty\le B$. Replace the shifts +$36n\cdot\zeta$ in its annular relations by $144n\cdot\zeta$, +and the neighborhood shifts $6n\cdot\zeta$ by $12n\cdot\zeta$. +The parameter $N_0$ is the coefficient reserve for the latter +neighborhoods. The proof uses a fine common grid; choose its step +on a multiple clearing the integers $1,2,3,4$. + +For a four-step path, alternately add its four annular relations. +The intermediate label in role $i$ and the two labels in a second +role cancel, leaving $\alpha_i(a-a')$ and an expression in the +four color labels. Dividing by $\alpha_i$ constructs the source's +difference cover. The required lattice compatibility is +\begin{equation} + \frac{144}{\alpha_i}\in12\Z, + \qquad \frac{12\alpha_j}{\alpha_i}\in\Z. + \label{eq:ll-counting-divisibility} +\end{equation} +Both assertions hold because $|\alpha_i|$ divides $12$. +The integer vector obtained by adding four original shifts has +norm at most $4B$. In units of the new neighborhood shift $12$, +division by $\alpha_i$ therefore gives a vector of norm at most +$4B\cdot12/|\alpha_i|\le48B$. +The reserve $N_0\ge64B$ accommodates it. The scalar error is at most +four times one relation's error after division by a nonzero integer. + +For the slab cover, write the first two coordinates as their box +centers plus shifts $12n_i\cdot\zeta$ and scalar interval errors. +Substitute them in the three-role relation and solve for the third +coordinate. The resulting shift coefficients are +$12\alpha_1/\alpha_3$, $12\alpha_2/\alpha_3$, and +$144/\alpha_3$. By \eqref{eq:ll-counting-divisibility} they are +integers, and the resulting shift vector has norm at most +$C(d+1)^2B$ as in H. Choose the separation reserve and +the excluded pure-shift interval constants larger by the fixed +factors arising from these substitutions. The covering iterations +use the same orders of sumsets, the graph count uses the same four +roles, and the block and scale counts use the same covers. Their +exponents are consequently unchanged. In particular the source +factor $L^{29}\delta^{1/2080}$ remains valid, where $L$ bounds the +number of scale occurrences of a fixed pair of labels and $\delta$ +is the sparse tuple-density threshold. In the application, let +$\xi$ be the vector of frequencies obtained from the height +construction and take $\zeta=\xi/144$, so the available heights start at +$12^{\Delta-2}$ instead of $6^{\Delta-2}$. The preceding height +reserves supply the required shift sizes. Pair determination in +the application divides only by role differences, and the +short-block windows were checked above. This verifies the +coefficient-dependent hypotheses of the counting estimate and its +application, not merely its formal conclusion. + +We can now apply the remaining operator and summation arguments of +H. The main projection families, Fourier modes, and counting tests +have the same bounds in powers of logarithms of their budgets. The +coordinate changes have inserted only fixed rational heights, +fixed derivative losses, and the replacement of one fixed +exponential height base by another. None changes a budget exponent +as a function of the final $q$. H's Section~10 chooses those +exponents before taking $q\in(2,3)$ sufficiently close to two; +its absorption of the finite-depth constant therefore applies to +the modified estimates. This proves \eqref{eq:ll-fixed-local} for +the four role sets together with the intermediate estimates used +in P. + +It remains to carry these inputs through P's depth-local proof. +The \emph{Maximal estimate for local catalog rows}, +\emph{The depth-varying row estimate}, and +\emph{Smooth mean-zero rows on depth inputs} in P, Section~4, +concern Hilbert-space operators supported on nested dyadic cells. +Here a catalog consists of finitely many vector-valued column +functions, each with a dyadic birth cell; descendants use restrictions +of the same functions. A local projection is the orthogonal +projection in the cell's $L^2$ space onto constant-coefficient +combinations of a specified set of available columns. +Their hypotheses specify cellwise constant coefficients in a +persistent catalog, an ordinary fixed-input Bessel bound, and +nondecreasing input-read indices. They contain no role coefficients. +These are P, Lemmas~4.3, 4.4, and 4.6. The replacement just proved +supplies their same scalar and Bessel inputs. + +More concretely, P, Lemma~5.3, Corollary~5.4, and Lemma~5.5, +use +$(y_a,y_b)$ and the demodulations $\pm\gamma/(b-a)$ already checked. +Their cutoff equal to one on $[-1/3,1/3]$ still covers the support +by \eqref{eq:ll-support-t}. Separating the kernel gives exactly +mean-zero bump rows, with fixed complete carriers. Clump replacement +and predictor rows have the same Bessel bounds because the +cancellation and height estimates have just been verified. The +support and observation partitions of the rows, and their +nondecreasing reads of the depth inputs, are unchanged. In P, +Section~6, use the normalized tags from \eqref{eq:ll-moments}; the +mask, template-transfer, short-block, and counting estimates are +then the versions established above. Scheduling the column +updates and summing their row bounds introduces no new arithmetic +operation on a role label. + +For completeness, the final parameter dependence can be seen in +the retained estimates of H, Section~10.3, and P, Section~7. +Write $k$ for the accuracy logarithm, $\mathfrak p$ for their +growth parameter, $K_0$ for the resulting depth-block length, and +$G_0$ for the strong energy and projection overhead. The term +requiring $q$ close to two has the form +\[ + \mathfrak p^{-1/2}G_0(K_0+2)^{C_{19}(q-2)},\qquad + K_0\le C_q(1+k+\mathfrak p)^{C_5},\qquad + G_0=\exp\!\bigl(C_q(1+\log(2k))^C\bigr). +\] +Here we have renamed the source parameter $P$ to avoid confusion +with the manuscript P. The degrees $C,C_5,C_{19}$ have absolute +bounds fixed before $q$. The short-block estimate feeding this +term has the explicit exponent $11(q-2)/(q-1)$ in H, Lemma~7.2; +the coefficient changes above affect its fixed constants, not this +exponent. Taking $q$ so close to two that +\[ + \max\{C_{19}^{\rm H}C_5^{\rm H}, + C_{19}^{\rm P}C_5^{\rm P}\}(q-2)<1/8 +\] +is therefore permitted in both proofs. Here the accuracy scheme +fixes $0<\theta<1$ before $q$, and the positive constant $\alpha$ +is chosen sufficiently small after $q$, as in the source proofs. +Their later choice $\mathfrak p=\lceil\exp(\alpha k^\theta)\rceil$ gives +$G_0=\mathfrak p^{o(1)}$ and makes the displayed term at most +$C_q\mathfrak p^{-1/4}$. The remaining accuracy exponents and +summations retain their source order of choice. P's depth-local +proof consequently gives \eqref{eq:ll-depth}. + +Translating the role labels back restores $S=\{0\}\cup J$. +P's depth-local theorem permits variation in any role, and the +local constructions allow any role as testing slot. Thus this +translation does not require the testing role to remain the marked +role $0$ during the local proof. +This completes the proof of Proposition~\ref{prop:ll-depth}. + +\subsection{From local forms to orbit oscillation} +\label{subsec:ll-transfer} + +An \emph{averaging profile} is a real compactly supported smooth +function $\varphi$ satisfying +\begin{equation} + \int\varphi=1,\qquad \int t^a\varphi(t)\,\dd t=0\quad(1\le a\le3), + \qquad + \|\varphi^{(n)}\|_\infty\le + (G(n+2))^{G(n+2)}\quad(n\ge0) + \label{eq:ll-profile} +\end{equation} +for some $G\ge1$. Profiles may have either sign. Write +$\varphi_A(t)=A^{-1}\varphi(t/A)$. These are the profiles of P, +Section~2. + +\begin{proposition}[Oscillation for the four triples] +\label{prop:ll-oscillation} +Let $(Y,\nu,R)$ be an invertible probability preserving system, +let $J\subset\{1,2,3,4\}$ have three elements, and let the functions +$g_j:Y\to\C$, $j\in J$, be measurable with $|g_j|\le1$. +For an averaging profile set +\[ + V_k^\varphi(y)=\sum_{n\in\Z}\varphi_{2^k}(n) + \prod_{j\in J}g_j(R^{jn}y),\qquad k\ge0. +\] +There is $C_\varphi<\infty$ such that, for all deterministic integers +$0\le k_1<\cdots32B$. For $S=\{0\}\cup J$ put +\[ + F(z,\sigma)=u(z)D^{-3}g(D\sigma),\qquad + \mathcal W(z,\sigma)= + \prod_{j\in S}(\partial_\sigma-j\partial_z)F(z,\sigma). +\] +The four differential operators commute. Each is differentiation +along a line on which one role coordinate is fixed, giving all +cancellations in \eqref{eq:ll-kernels}. On the support, +$|z+j\sigma|\le1/8+4B/D<3/8$. Thus the rescaled kernel on an +interval of length $r=DA$ has the required interior margin. +The derivative bounds have the form \eqref{eq:ll-kernels}, with +constants depending only on the profile. Integration in $z$ kills +every term containing a $z$ derivative, leaving +\begin{equation} + \frac1r\int_\R\mathcal W(z,t/r)\,\dd z + =A^{-1}b(t/A)=\varphi_{2A}(t)-\varphi_A(t). + \label{eq:ll-kernel-normalization} +\end{equation} + +Let $\Delta_k=\varphi_{2^{k+1}}-\varphi_{2^k}$ and let $f_j$, +$j\in J$, be bounded by one and supported in an interval of length +$L$. Suppose the testing functions $f_{0,k}$ have the same support +and satisfy the block variation bound $W^{-1/2}$ over a fixed +partition of $k_-\le k\le k_+$ into $W$ blocks. Apply +Proposition~\ref{prop:ll-depth} to dyadic cells of length $D2^k$, +then average the translated lattice over one period of its largest +scale. All nonzero cells lie below roots of total length at most +$L+2D2^{k_+}$. Scale order is the reverse of depth order, which +preserves block variation. The normalized translation average +of a scale's kernels is \eqref{eq:ll-kernel-normalization}. +We obtain +\begin{equation} + \left|\sum_{k=k_-}^{k_+}\int_{\R^2} + f_{0,k}(x)\prod_{j\in J}f_j(x+jt)\Delta_k(t) + \,\dd x\,\dd t\right| + \le C_\varphi(L+D2^{k_+}). + \label{eq:ll-continuous-test} +\end{equation} + +For clarity, integer transfer does not require consecutive roles. +Plant each integer sequence on intervals of radius $e<1/200$ about +the integers. If $x+jt$ lies in such intervals in all four roles, +use an adjacent pair to reconstruct integers $m,n$. Every other +integer center must equal $m+jn$, since its discrepancy has +magnitude less than $8e$. Write $x=m+\xi$, $t=n+\eta$. +The offset domain is +\[ + \Omega_{S,e}=\{(\xi,\eta):|\xi+j\eta|0$ there is +an averaging profile with +\begin{equation} + \|\varphi-h_c\|_1<\epsilon, + \qquad h_c(t)=c^{-1}\mathbf1_{(0,c]}(t). + \label{eq:ll-profile-approximation} +\end{equation} +The use of signed profiles is essential to this assertion. One +first smooths $h_c$, preserving integral one. The other three +moments can be canceled, without changing the integral, by four +broad translated bumps at distance and width comparable to a large +parameter $A$. Their moment matrix for orders $0,1,2,3$, after +division of row $j$ by $A^j$, is a fixed invertible Vandermonde-type +matrix. The prescribed zeroth correction is zero and the remaining +normalized corrections are $O(A^{-1})$. The correcting coefficients +are therefore $O(A^{-1})$, and their total $L^1$ cost tends to zero. The bumps +retain the derivative bounds in \eqref{eq:ll-profile} with a +profile-dependent constant. + +Fix a triple $J$ and write +$a_n(y)=\prod_{j\in J}g_j(R^{jn}y)$, so $|a_n(y)|\le1$. +For $N_k=\lfloor c2^k\rfloor$, the Riemann-sum comparison gives, +uniformly in $y$, +\begin{equation} + \limsup_{k\to\infty} + \left|\frac1{N_k}\sum_{n=1}^{N_k}a_n(y)-V_k^\varphi(y)\right| + \le\|h_c-\varphi\|_1. + \label{eq:ll-profile-error} +\end{equation} +To see this, first replace $N_k^{-1}$ by $(c2^k)^{-1}$, at cost +$O(2^{-k})$, then bound the difference by +$2^{-k}\sum_{n\in\Z}|h_c(n2^{-k})-\varphi(n2^{-k})|$. +This converges to the displayed integral because the integrand is +compactly supported and piecewise continuous. + +Choose countably many profiles with errors tending to zero for each +rational $c\in[1,2]$. On their common conull set, the existence of +all the smooth-profile limits and \eqref{eq:ll-profile-error} show +that the ordinary averages along $N_k$ are Cauchy. The +Host--Kra norm convergence theorem \cite[Theorem~1.1]{HostKra}, +stated for arbitrary invertible probability preserving systems, +gives an $L^2$ limit for the averages at all integer lengths: apply +it to the four consecutive powers, inserting the constant function +one at the omitted role. The bounded pointwise limit along each +$N_k$ must equal this $L^2$ limit almost everywhere, by bounded +convergence and uniqueness of norm limits. Thus all rational-$c$ +subsequences have the same limit on one conull set. No identification +of that limit and no mixing assumption is needed. + +Finally, for $M\le N$ and $|a_n|\le1$, +\[ + \left|\frac1N\sum_{n=1}^Na_n-\frac1M\sum_{n=1}^Ma_n\right| + \le 2\frac{N-M}{N}. +\] +For any rational mesh of $[1,2]$, every sufficiently large integer +$N$ lies within its relative mesh size, up to $O(N^{-1})$, of some +$\lfloor c2^k\rfloor$ from that finite mesh. Its finitely many +subsequences have the common pointwise limit. The preceding +inequality, followed by meshes of size tending to zero, proves +convergence along all positive integers. Homogeneity removes the +normalization $|g_j|\le1$. A proper subset with fewer than three +roles is contained in one of the triples already treated; assign +the constant function one to its additional roles. This proves +Lemma~\ref{lem:lower-length} in its full stated scope. diff --git a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5a62ba6bc8ca8ee51651035823f033c8f139921c --- /dev/null +++ b/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dbb86e1a3da44a8bcccb1b8ab6bd0b8083ebfbebd64c67f880db59f8ddbada7e +size 541428 diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/README.md b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..f8377cce68146c3efade829f5197c0efeab5a6c0 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/README.md @@ -0,0 +1,18 @@ +# [Pointwise convergence of triple ergodic averages for mixing transformations](pointwise-triple-ergodic-averages-mixing-transformations.pdf) + +**Author:** OpenAI + +**Date:** October 4, 2026 + +## Citation + +```bibtex +@misc{OAI:Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026, + author = {{OpenAI}}, + title = {{Pointwise convergence of triple ergodic averages for mixing transformations}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf}{OAI:Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/figures/frozen-depths.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/figures/frozen-depths.tex new file mode 100644 index 0000000000000000000000000000000000000000..7954599ce834592f38df2db4ccf1bd334f598c97 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/figures/frozen-depths.tex @@ -0,0 +1,19 @@ +\begin{tikzpicture}[x=0.95cm,y=1cm,every node/.style={font=\small}] + \draw[-{Latex[length=2mm]}] (0,0.8)--(11.7,0.8) node[right]{depth}; + \foreach \x in {0.5,4.5,8.5,10.5} {\draw (\x,0.66)--(\x,0.94);} + \draw[fill=black!8] (8.5,1.1) rectangle (10.5,1.6); + \draw[densely dashed] (9.5,0.8)--(9.5,1.6); + \node[fill=black!8,inner sep=1.5pt] at (9.5,1.35) {$s$}; + \node at (9.5,1.86) {actual uses}; + \draw[{Latex[length=1.7mm]}-{Latex[length=1.7mm]}] (0.65,2.25)--(8.35,2.25); + \node[fill=white,inner sep=2pt] at (4.5,2.25) {$4K$ depths}; + + \node[anchor=north,align=center] at (0.5,0.45) + {$u_b=b-4K$\\freeze $Z_{l,G,u_b}$}; + \node[anchor=north,align=center] at (4.5,0.45) + {$v_b=b-2K$\\common test cell}; + \node[anchor=north,align=center] at (8.5,0.45) + {$b$\\block begins}; + \node[anchor=north,align=center] at (10.5,0.45) + {$t_b=b+K$\\block ends}; +\end{tikzpicture} diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/figures/row-chronology.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/figures/row-chronology.tex new file mode 100644 index 0000000000000000000000000000000000000000..7cb6a62d260e8c9dccbf17c4d2029f84e7d594e8 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/figures/row-chronology.tex @@ -0,0 +1,27 @@ +\begin{tikzpicture}[x=0.92cm,y=1cm,box/.style={draw,minimum height=6mm},every node/.style={font=\small}] + \node[anchor=east] at (-0.25,2.7) {support cells $\pi_n$}; + \node[anchor=east] at (-0.25,1.5) {observation cells $\sigma_n$}; + \node[anchor=east] at (-0.25,0.3) {support cells $\pi_{n+1}$}; + + \draw[fill=black!3] (0,2.4) rectangle (8,3.0); + \node at (4,2.7) {one row on $Q$, assigned to bucket $b$}; + + \draw[fill=black!3] (0,1.2) rectangle (4,1.8); + \draw[fill=black!13] (4,1.2) rectangle (8,1.8); + \node at (2,1.5) {no new hit}; + \node at (6,1.5) {a new hit}; + + \draw[fill=black!3] (0,0) rectangle (2,0.6); + \draw[fill=black!3] (2,0) rectangle (4,0.6); + \draw[fill=black!13] (4,0) rectangle (6,0.6); + \draw[fill=black!13] (6,0) rectangle (8,0.6); + \node at (1,0.3) {$b$}; + \node at (3,0.3) {$b$}; + \node at (5,0.3) {$b+1$}; + \node at (7,0.3) {$b+1$}; + + \draw[densely dashed,black!35] (0,0.6)--(0,1.2) (4,0.6)--(4,1.2) (8,0.6)--(8,1.2); + \draw[densely dashed,black!35] (0,1.8)--(0,2.4) (8,1.8)--(8,2.4); + \draw[-{Latex[length=2mm]},black!65] (8.45,2.7)--(8.45,0.3); + \node[anchor=west,align=left] at (8.6,1.5) {chronological\\order}; +\end{tikzpicture} diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/main.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..6679a9d0da8518a942c2bda9f6a9e5355dfdf3c8 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/main.tex @@ -0,0 +1,33 @@ +\documentclass[11pt,reqno]{amsart} +\input{preamble} +\title[Pointwise convergence of triple averages]{Pointwise convergence of triple ergodic averages for mixing transformations} +\author{OpenAI} +\date{October 4, 2026} +\subjclass[2020]{37A44, 42B20, 42B25} +\keywords{Multiple ergodic averages, pointwise convergence, mixing, oscillation inequalities, trilinear Hilbert transform} +\begin{document} +\begin{abstract} +We prove pointwise convergence of triple ergodic averages for every +invertible mixing probability preserving transformation \(T\) of an +arbitrary probability space \((X,\mathcal F,\mu)\), with measurable inverse. +For every triple of bounded measurable functions +\(f_1,f_2,f_3:X\to\mathbb C\), +\[ + \frac1N\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x) + \longrightarrow \prod_{j=1}^3\int_X f_j\,d\mu +\] +for \(\mu\)-almost every \(x\) as \(N\to\infty\) through all positive +integers. +\end{abstract} +\maketitle +\tableofcontents +\input{sections/introduction} +\input{sections/ergodic} +\input{sections/transfer} +\input{sections/rows} +\input{sections/base} +\input{sections/localization} +\input{sections/completion} +\bibliographystyle{amsplain} +\bibliography{references} +\end{document} diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/preamble.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..1040a19922523e029e3f6748cfc23ae3e49000b7 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/preamble.tex @@ -0,0 +1,46 @@ +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage[margin=1.04in]{geometry} +\usepackage{graphicx,booktabs,array} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,calc,positioning} +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{28,68,110} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue,bookmarksnumbered=true]{hyperref} +\hypersetup{pdftitle={Pointwise convergence of triple ergodic averages for mixing transformations},pdfauthor={OpenAI}} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\Z}{\mathbb Z} +\newcommand{\Q}{\mathbb Q} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\1}{\mathbf 1} +\newcommand{\ip}[2]{\langle #1,#2\rangle} +\newcommand{\norm}[1]{\lVert #1\rVert} +\newcommand{\abs}[1]{\lvert #1\rvert} +\newcommand{\phys}{\mathrm{phys}} +\newcommand{\Heq}[1]{\textup{(H.#1)}} +\newcommand{\fint}{\mathop{\mathchoice{\mkern2mu\int\mkern-15mu-\mkern7mu}{\mkern2mu\int\mkern-13mu-\mkern6mu}{\mkern2mu\int\mkern-11mu-\mkern5mu}{\mkern2mu\int\mkern-9mu-\mkern4mu}}\nolimits} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\spanop}{span} +\DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\poly}{poly} +\setlength{\emergencystretch}{1.5em} +\allowdisplaybreaks[2] +\pdfgentounicode=1 +\input{glyphtounicode} diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/references.bib b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..8e6dc2b90ec285aae56e43df18e2bdfff5fb2c6c --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/references.bib @@ -0,0 +1,449 @@ + +@article{Calderon1968, + author = {A. P. Calder{\'o}n}, + title = {Ergodic theory and translation-invariant operators}, + journal = {Proceedings of the National Academy of Sciences of the United States of America}, + volume = {59}, + number = {2}, + pages = {349--353}, + year = {1968}, + doi = {10.1073/pnas.59.2.349}, + url = {https://pmc.ncbi.nlm.nih.gov/articles/PMC224676/} +} + +@article{Lepingle1976, + author = {D. L{\'e}pingle}, + title = {La variation d'ordre $p$ des semi-martingales}, + journal = {Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte Gebiete}, + volume = {36}, + pages = {295--316}, + year = {1976}, + doi = {10.1007/BF00532696}, + url = {https://link.springer.com/article/10.1007/BF00532696} +} + +@article{Bourgain1989, + author = {Jean Bourgain}, + title = {Pointwise ergodic theorems for arithmetic sets}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {69}, + pages = {5--41}, + year = {1989}, + doi = {10.1007/BF02698838}, + url = {https://www.numdam.org/articles/10.1007/BF02698838/} +} + +@article{Bourgain1990, + author = {Jean Bourgain}, + title = {Double recurrence and almost sure convergence}, + journal = {Journal f{\"u}r die reine und angewandte Mathematik}, + volume = {404}, + pages = {140--161}, + year = {1990}, + doi = {10.1515/crll.1990.404.140}, + url = {https://www.degruyterbrill.com/document/doi/10.1515/crll.1990.404.140/html} +} + +@article{JKRW1998, + author = {Roger L. Jones and Robert Kaufman and Joseph M. Rosenblatt and M{\'a}t{\'e} Wierdl}, + title = {Oscillation in ergodic theory}, + journal = {Ergodic Theory and Dynamical Systems}, + volume = {18}, + number = {4}, + pages = {889--935}, + year = {1998}, + doi = {10.1017/S0143385798108349}, + url = {https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/abs/oscillation-in-ergodic-theory/55839C7F1B35F3C4900BD5A04F69233F} +} + +@article{Demeter2007, + author = {Ciprian Demeter}, + title = {Pointwise convergence of the ergodic bilinear {Hilbert} transform}, + journal = {Illinois Journal of Mathematics}, + volume = {51}, + number = {4}, + pages = {1123--1158}, + year = {2007}, + doi = {10.1215/ijm/1258138536}, + eprint = {math/0601277}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + url = {https://arxiv.org/abs/math/0601277} +} + +@article{KMT2022, + author = {Ben Krause and Mariusz Mirek and Terence Tao}, + title = {Pointwise ergodic theorems for non-conventional bilinear polynomial averages}, + journal = {Annals of Mathematics}, + volume = {195}, + number = {3}, + pages = {997--1109}, + year = {2022}, + doi = {10.4007/annals.2022.195.3.4}, + eprint = {2008.00857}, + archivePrefix = {arXiv}, + primaryClass = {math.DS}, + url = {https://annals.math.princeton.edu/2022/195-3/p04} +} + +@article{LT1997, + author = {Michael Lacey and Christoph Thiele}, + title = {{$L^p$} estimates on the bilinear {Hilbert} transform for $2\delta\), there is a bounded chart atom +\(\varphi\) such that +\[ + \left|\fint_I r\overline\varphi\right|\ge\rho,\qquad + \log(1/\rho),\ \log(\operatorname{complexity}(\varphi)) + \le C_q\bigl(1+\log C_1+\log(1/\delta)\bigr)^C. +\] +We may decrease \(\rho\) to lie in \((0,1]\). The form norm of two +specified factors below is the supremum against two normalized +\(L^2(I)\) inputs in their complementary slots. + +\begin{lemma}[One capped insertion trial] +\label{lem:hbase-trial} +Let \(R\) be an attempt root with a restricted normalized input +schedule \(F_s\). Let the candidate request cells belong to a fixed finite +original request tree below \(R\). Fix +\(0<\rho\le1\), \(D_0\ge2\), and an integer cap \(T\ge1\). +Process candidates in nondecreasing request-cell depth, allowing +repetitions at one cell. Stop before the first request that would +exceed \(T\) successful requests on its spatial path, and remove +that cut cell and its candidate descendants from the trial. + +For each successful request \(v\) at \(I_v\), suppose there is a +set \(\mathcal R_v\) of charged row indices, with these sets pairwise +disjoint, such that +\[ + \sum_{\nu\in\mathcal R_v} + \|\Delta_\nu F_{s(\nu)}\|_2^2 + \ge c\rho^2|I_v|. +\] +Assume all charged rows are a Bessel-norm-one subfamily of one +realized finite nested sequence of full-partition projections and +satisfy every hypothesis of Lemma~\ref{lem:row-depth}. Suppose its +completed catalog has budget +\[ + D_T\le C D_0(T+1)^C, +\] +where \(D_0\) includes all predetermined columns, including their +future admissions on this attempt. Then the disjoint first-excess +cells \(\mathcal C_T\), which are original request-tree cells, obey +\begin{equation} + \sum_{I\in\mathcal C_T}|I| + \le C T^{-1}\rho^{-2}\log^2(2D_T)|R|. \label{eq:hbase-trial-cut} +\end{equation} +If +\[ + a=1+\log(2D_0)+\log(1/\rho),\qquad + T=\left\lceil C_*\rho^{-4}(1+a)^8\right\rceil, +\] +then a sufficiently large fixed \(C_*\) makes the right side at +most \(|R|/4\), and \(\log(2T)\le C(a+\log C_*)\). +\end{lemma} + +\begin{proof} +Freeze the completed rows and catalog. By their disjoint ownership +and Equation~\eqref{eq:h-row-use}, +\[ + c\rho^2\sum_v|I_v| + \le\sum_{\nu\ {\rm charged}}\|\Delta_\nu F_{s(\nu)}\|_2^2 + \le C\log^2(2D_T)|R|. +\] +Set \(n(y)=\#\{v:y\in I_v\}\). Earlier request cells meeting a +candidate contain it or equal it, because request depths are +nondecreasing. Thus their count is constant on that candidate. +Each first-excess cell lies in \(\{n\ge T\}\), and removing its +subtree makes these cells disjoint. Tonelli's identity +\(\int_R n=\sum_v|I_v|\) proves +Equation~\eqref{eq:hbase-trial-cut}. + +For the stated cap, the budget bound gives +\(\log(2D_T)\le C'(a+\log C_*)\). Consequently the coefficient +in Equation~\eqref{eq:hbase-trial-cut} is at most +\[ + C C_*^{-1}\rho^2(1+a)^{-8}(a+\log C_*)^2, +\] +which is at most \(1/4\) for fixed sufficiently large \(C_*\). +The logarithmic cap bound follows from the same formula. +\end{proof} + +We realize the Lemma's nested sequence in both applications by one +rule. At each projection depth, refine and locally refit the entire +root partition, admit the predetermined columns prescribed there, +and then perform simultaneous insertion rounds. Cells with no +request retain their projections. The next depth refines the entire +partition, including branches with no later tests for this attempt. +Refinements, admissions, and identity transitions remain in the +nested sequence. Splitting an insertion increment on all cells of +its partition preserves its joint Bessel norm, by the splitting +identity in Corollary~\ref{cor:row-projections}. The following two +applications specify the charged supports, observations, and reads. + +Apply this construction first to the +\emph{Inherited adaptive approximation} Lemma of +\cite[Section~5]{H}, at accuracy \(e^{-k}\), for the fixed provisional +\(q\) and sufficiently large \(k\). Every attempt starts with an +empty fit catalog. On an assigned original candidate cell \(I\) of +depth \(s\), test +\[ + r=f_{j,s}-(PF_{j,s})_{\rm phys}, +\] +where \(P\) is the current local projection. Contraction gives +\(\|r\|_{2,I}\le2\). On failure of the \(\mathfrak d_{j,I}\) +test, Local detection supplies a bounded atom with moment \(\rho\), +where \(\log(1/\rho)\le k^{O(1)}\). Adjoin +\((\varphi,\mathbf e_v)\) with a fresh private direction and refit. +For the new projection \(P^+\), +\begin{equation} + \|(P^+-P)F_{j,s}\|_{L^2(I)}^2 + \ge + \frac{\left|\int_I + (f_{j,s}-(PF_{j,s})_{\rm phys})\overline\varphi\right|^2} + {\|(\varphi,\mathbf e_v)\|_{L^2(I)}^2} + \ge c\rho^2|I|. \label{eq:hbase-fresh-gain} +\end{equation} +Indeed, projection of the fresh column off the old span preserves +its pairing with the old residual and can only reduce its norm. + +Here each request owns its insertion row on \(I\). The support and +observation partitions are the full depth-\(s\) partition, and every +round at this depth reads \(F_{j,s}\). Their refinement and read +order therefore satisfy Lemma~\ref{lem:row-depth}. The initial +budget \(D_0\) is fixed, and a path cap \(T\) adds at most \(T\) +bounded columns of private weight one, giving the required polynomial +budget \(D_T\). Lemma~\ref{lem:hbase-trial} makes the cut cells have +total length at most one quarter of the attempt root. + +On every child root the restricted schedule is still normalized; +the fit again starts with no columns and has the same detection, +gain, full-partition chronology, and budget. The one-trial Lemma +therefore applies anew. Its length bound prevents a child from +cutting itself, so all new roots are proper original request-tree +descendants. The finite recursion terminates, and the attempt-root +lengths sum to at most +\(\sum_{n\ge0}4^{-n}|R|=4|R|/3\), as in H's packing argument. +Each original summand belongs to one attempt. The old projection +family continues by inheritance and local refitting only for its +old summands and their operator estimates. + +On a retained cell the fit has passed: +\[ + g_I=P_IF_{j,s},\qquad + \mathfrak d_{j,I}\bigl(f_{j,s}-(g_I)_{\rm phys}\bigr)\le e^{-k}, + \qquad \|g_I\|_{2,I}\le1. +\] +Within a segment, masks are inherited. The cap and Local detection +bound path counts and atom complexities by \(\exp(k^{O(1)})\). +Private weights are one, so the coefficient \(\ell^2\) norm is at +most the normalized projection norm; Cauchy--Schwarz over the path +labels gives the same \(\exp(k^{O(1)})\) physical supremum bound. +All these statements hold anew on a normalized subroot. + +We now use these fits cyclically in all four slots. Let +\(\iota(t)=(t-1)\bmod4\) be the slot updated at accuracy level \(t\). +Choose an absolute \(A>1\), a sufficiently large initial \(k_1\), +and set \(k_{t+1}=k_t^A\). Write \(g_{t,I}=P^t_IF_{\iota(t),s}\) +for the fit at level \(t\), and set \(g_{t-4,I}=0\) before that +slot's first update. For any fixed bounded number of levels, +intersect their segment families. Every common root is a root of +one constituent family or the original root, so their total length +remains bounded. + +\begin{lemma}[The two earliest increments on depth inputs] +\label{lem:hbase-X-output} +On each original summation cell \(I\), put \(s=s(I)\), its original +spatial depth in \(J\). At a finite terminal level \(n\), write \(g_{e,n,I}\) for the last +fit in slot \(e\) at or before \(n\), or zero if there is none, and +put \(r_{e,n,I}=F_{e,s}-g_{e,n,I}\). For \(uw\) in distinct slots, replacing their pair of +projections by their actual inputs has form-norm error at most +\begin{equation} + C_q\bigl(e^{-k_v}\exp(C_qk_w^{d_{\rm det}})+e^{-k_w}\bigr), \label{eq:hbase-pair-comparison} +\end{equation} +where \(d_{\rm det}\) is absolute. First test the residual at \(v\) +against the physical supremum bound of \(g_w\) as the \(L^q\) +opponent, then test the residual at \(w\) against the normalized +actual input. Choose \(A>d_{\rm det}\) with fixed reserve. The +same calculation with an actual input in one position makes the +form norm of two terminal residuals tend to zero. The two remaining +factors of each remainder term have bounded \(L^2\) size. Since +\(\mathcal U\) is finite, its absolute remainder sum tends to zero. + +For the pair bound assume \(t\ge12\). Each of \(X_j,X_a,X_b\) +has coefficient sum zero after its existing fitted operands are +replaced by the corresponding actual input: their preceding fits +occur at \(t-4\) and, for \(a,b\), among \(t-3,t-2,t-1\), so all exist. +Thus every pair contains a cancelling later factor; the early \(X_i\) +need not cancel when \(u\le4\). If \(uw\). +Pairs excluding \(i\) have no projection index below \(t-4\). + +If \(u\ge t-4\), the distinct cyclic slots imply \(u\ge t-3\). +Every existing fitted operand in the tuple then has index at least \(t-7\). +For such a pair, and for the pairs excluding \(i\) in the preceding +case, expand into projection/actual pairs and apply +Equation~\eqref{eq:hbase-pair-comparison}. The two-actual contribution +vanishes because at least one factor has coefficient sum zero. +The bounded number of errors is at most +\(C_qe^{-c_qk_{t-7}}\). Since \(k_{t-7}=k_t^{A^{-7}}\), the stated +choice of \(\theta_0\) gives the required bound for large \(k_1\). +Contraction and the triangle inequality give the local sizes. +\end{proof} + +\subsection{The high replacement target} + +Fix one tuple from Equation~\eqref{eq:hbase-X}, put \(k=k_t\), and +write \(J_0\) for the normalized root on which its construction is +run (initially \(J_0=J\)). Let \(\mathcal I_0\subseteq\mathcal I\) +be the original summands under consideration on \(J_0\). The original +request-tree candidates are those already fixed for this construction. +Fix a common segment of its older fit families. The positions \(i,j\) are now called low because their +two signed fitted differences will be expanded into original scalar +labels. In the residual positions \(e=a,b\), we seek an integer +\(K_0\) and further inherited projections for which +\[ + X_e=F_{e,s}-g_{e,\mathrm{prev},I} + \quad\hbox{can be replaced by}\quad + P^{\rm new}_{e,s+K_0}F_{e,s}-g_{e,\mathrm{prev},I} +\] +with summable error. A fixed label from each low position has a +common birth root: the deeper birth cell, or the segment root when +inherited there. Their fitted coefficients remain outside the +scalar approximation. The following tools are stated for a +generic fixed scalar pair, independently of the tuple notation. + +\subsection{Two fixed scalar labels and their predictor list} + +Fix distinct slots \(i,j\), let \(a,b\) be the complementary slots, +and fix bounded chart atoms \(\varphi_i,\varphi_j\) on their common +birth root \(J'\). Their two-input form is +\[ + \mathcal L_I^{\varphi_i,\varphi_j}(z_a,z_b) + :=|I|^{-2}\int_{\mathbb R^2}w_I(x,t) + \varphi_i(y_i)\varphi_j(y_j)z_a(y_a)z_b(y_b)\,dx\,dt. +\] +For Hilbert-valued inputs, the displayed \(z_e\) denotes its physical +component. The \emph{Progression separation} Lemma of +\cite[Section~4]{H} expands the scalar low product into terms +\(\psi_a(y_a)\psi_b(y_b)Q(t)\). With the scalar coefficient removed, +\(\psi_a,\psi_b,Q\) are fixed bounded chart atoms of complexity at +most \(H\); unrestricted fixed linear modulations do not enter \(H\). +For one such term define +\[ + \begin{split} + H_I(\psi_a z_a,\psi_b z_b;Q) + :=|I|^{-2}\int_{\mathbb R^2}&w_I(x,t)Q(t) + \psi_a(y_a)z_a(y_a)\\ + &\qquad\cdot\psi_b(y_b)z_b(y_b)\,dx\,dt . + \end{split} +\] +Choose the cutoff allowed by the \emph{Single-scale separation} Lemma +of \cite[Section~4]{H} with +\[ + \zeta\in C_c^\infty(\mathbb R),\qquad + 0\le\zeta\le1,\qquad \zeta=1\ \hbox{on }[-1/3,1/3], + \qquad \operatorname{supp}\zeta\subset(-1/2,1/2). +\] +This is permitted because \(x,x+3t\in I\) on the kernel support +imply \(|t|/|I|\le1/3\). Set +\[ + A_r=\sup_{\xi\in\mathbb R} + \left|\int_{\mathbb R}Q(ru)\zeta(u)e(-\xi u)\,du\right|. +\] +The normalization gives \(0\le A_r\le\|\zeta\|_1\le1\). +The plateau also gives \(\int\zeta\ge2/3\), so the source scalar +test retains a uniform constant. + +For a dyadic \(K\subseteq J'\) and a selected finite use collection +\(\mathcal U\subseteq\mathcal I\) below \(K\), call an opposing +sequence \(z_{e,I}\) admissible on \(K\) if +\[ + z_{e,I}=\sum_{\ell=1}^{m_e}c_{e,\ell}V_{e,I}^{\ell},\qquad + m_e\in\{0,1,2,3\},\qquad |c_{e,\ell}|\le1, +\] +where the scalar coefficients \(c_{e,\ell}\) are fixed independently +of \(I\) and \(s(I)\), and each \(V_{e,I}^{\ell}\) is one of these types: +\[ + F_{e,s(I)}|_I,\qquad P_{e,I}F_{e,s(I)},\qquad Z_{e,I}. +\] +All raw and projected constituents in slot \(e\) use the same full +normalized schedule \(F_{e,s}\); the two slots may have different +schedules. Each projected constituent uses one inherited current +family common to all its cells, with the finitely many families under +a joint budget \(D_{\rm h}\). Missing cells are completed by +inheritance and local refitting. An energy constituent is supported +on its cell and satisfies the total bound +\[ + \sum_{I\in\mathcal U}|I|\|Z_{e,I}\|_{2,I}^2\le L^2|K|. +\] +Bilinearity and the triangle inequality reduce the estimates with two +opponents to at most nine constituent pairs, and the delayed estimate +to at most three opposing terms. These absolute factors are absorbed +into constants. All families used in an operator estimate are fixed +as realized families. + +\begin{lemma}[A separated scale sum on depth inputs] +\label{lem:hbase-scale} +For the fixed separated term above, let both opponents be admissible +on \(K\) with parameters \(D_{\rm h},L\). For +\(m\in\mathbb Z_{\ge0}\), let \(\mathcal U_m\subseteq\mathcal U\) +consist of any selected cells on which +\(e^{-m-1}h_*\), +\begin{align} + \int_I r_{h_*}^{[s]}\omega_I + ={}&\int_I\mu_{h_*}^{[s]}\omega_I + +\sum_{h=h_*}^{n-1}\int_I + (r_h^{[s]}-r_{h+1}^{[s]})(\omega_I-E_h\omega_I) + \nonumber\\ + &+\sum_{h=h_*}^{n-1}\int_I + (\mu_{h+1}^{[s]}-\mu_h^{[s]}) + (E_{h+1}\omega_I-E_h\omega_I) + +\int_I r_n^{[s]}(\omega_I-E_n\omega_I). \label{eq:hbase-telescope} +\end{align} +Indeed, telescope +\(\int_I r_h^{[s]}(\omega_I-E_h\omega_I)\). In the pairing with +\(E_{h+1}\omega_I-E_h\omega_I\), conditional expectation replaces +\(r_{h+1}^{[s]}\) by \(\mu_{h+1}^{[s]}\), while the pairing with +\(\mu_h^{[s]}\) vanishes. The terminal term tends to zero because +\[ + \|r_n^{[s]}\|_{L^2(I)}\le2\|F_{a,s}\|_{L^2(I)},\qquad + \|\omega_I-E_n\omega_I\|_{L^2(I)} + \le C|I|^{1/2}2^{s-n}. +\] +For each finite \(n\), continue the family by retaining its labels +and locally refitting on the needed descendants. These bounds then +dispose of the terminal term as \(n\) increases. + +To sum the telescope across use depths, recall the following +projection energy bounds. Let \(R\subseteq J\) be a normalized dyadic +subroot with its restricted normalized schedule \(F_{j,s}\). Take one +full inherited continuation \((P_h)\) on its complete depth partitions +over the needed finite range, with catalog budget \(D\). All depths +are the original depths in \(J\), with spatial origin \(s(R)\). Put +\(\Delta_h=P_{h+1}-P_h\). For a fixed integer offset \(\ell\), an +integer \(L_{\rm proj}\ge0\), and selected disjoint depth-\(s\) use cells +\(\mathcal U_s\) inside \(R\), Corollary~\ref{cor:row-projections} gives +\begin{equation} + \begin{gathered} + \sum_s\sum_{I\in\mathcal U_s} + \|\mathbf1_I\Delta_{s+\ell}F_{j,s}\|_2^2 + \le C\log^2(2D)|R|,\\ + \sum_s\sum_{I\in\mathcal U_s} + \|\mathbf1_I(P_{s+L_{\rm proj}}-P_s)F_{j,s}\|_2^2 + \le C(1+L_{\rm proj})^2\log^2(2D)|R|. + \end{gathered} \label{eq:h-projection-energy} +\end{equation} +Only valid projection depths are summed. These are +Equations~\eqref{eq:row-increment} and \eqref{eq:row-delay}, and they +apply anew on every subroot. The present Lemma uses \(R=K\) and +\(D=D_{\rm h}\). In every finer evaluation belonging to a use at depth +\(s\), the operand remains \(F_{j,s}\). + +We first sum the mean-zero tensor part. Both conditional-expectation +errors in the two sums have supremum \(O(2^{s-h})\). Normalize each +bump test by \(|I|^{-1/2}\); a length-weighted tensor form is a +product of two such tests, up to the fixed coordinate Jacobian. +At \(h=s+\ell\), the first normalized test is bounded by +\[ + C2^{-\ell}\|\mathbf1_I\Delta_{s+\ell}F_{a,s}\|_2, +\] +and the second has the additional factor \(\sqrt\epsilon\) from +Equation~\eqref{eq:hbase-predictor-moment}. Apply the fixed-offset +part of Equation~\eqref{eq:h-projection-energy} at each \(\ell\) +and then Minkowski's inequality over \(\ell\ge K_0\). The Hilbert +sum of these remainder tests is at most +\begin{equation} + C2^{-K_0}\log(2D_{\rm h})|K|^{1/2}. \label{eq:hbase-predictor-remainder} +\end{equation} +This bound includes every fine cell of an increment and remains +valid after selecting the stated coarse uses. The allowed bump +bounds multiply it. + +When \(\int_I\omega_I=0\), the initial term may use +\(\mu_{h_*}^{[s]}-\mu_s^{[s]}\), since \(\mu_s^{[s]}\) is constant +on \(I\). By Equation~\eqref{eq:hbase-predictor-moment}, it is a +sum of the \(K_0\) private increments at offsets \(0,\ldots,K_0-1\). +Equation~\eqref{eq:h-projection-energy} bounds the Hilbert sum of +these normalized tests by +\[ + CK_0\sqrt\epsilon\log(2D_{\rm h})|K|^{1/2}. +\] +Expand \(w_I\) into the mean-zero tensors used in the proof of +Lemma~\ref{lem:hbase-scale} and pair the error tests with the +opposing rows. At one fixed tensor index, a raw opponent has the +singleton fixed carrier \(\psi_{b,\alpha}\) and one row per selected +use cell. The singleton case of the ordinary mean-zero Bessel +calculation invoked in H's predictor proof, followed by +Lemma~\ref{lem:smooth-depth}, bounds it. For a projected constituent, +apply the calculation in Equation~\eqref{eq:hbase-projected-rows} +to this fixed singleton array and its one inherited current family; +its full support and observation partitions and its use-depth reads +are the same. An energy constituent uses the per-interval Bessel +bound and its total energy. Rapid tensor coefficient decay absorbs +the polynomial bump costs. This proves the asserted estimate for +the part of \(T_I^\alpha\) in which \(e(\lambda_\alpha t^2)\) is +replaced by one. + +For the quadratic-difference part, hold \(s\) fixed. Put +\(x_s=|\lambda_\alpha||I|^2\) at this depth. A smooth tensor +expansion of \(w_I(e(\lambda_\alpha t^2)-1)\) has coefficient and +fixed-derivative cost at most +\[ + x_s\operatorname{poly}(D_{\rm pred}),\qquad x_s\le D_{\rm pred}. +\] +These bumps may have nonzero mean. Ordinary orthogonality of the +full nested sequence on the one operand \(F_{a,s}\) gives +\[ + \sum_{h\ge h_*}\|\Delta_hF_{a,s}\|_2^2 + \le\|\mathbf1_KF_{a,s}\|_2^2. +\] +The local test bounds above and Cauchy--Schwarz in \(h\) consequently +give Hilbert-sum norm +\(C2^{-K_0}\|\mathbf1_KF_{a,s}\|_2\) for the two remainder sums +over the disjoint depth-\(s\) use cells. For the initial mean, +\(\mu_{h_*}^{[s]}=\sqrt\epsilon(P_{a,h_*}F_{a,s})_v\); +contraction instead gives +\(C\sqrt\epsilon\|\mathbf1_KF_{a,s}\|_2\). +The opposing normalized tests at this depth have Hilbert-sum norm +at most \(C(1+L)|K|^{1/2}\), by disjointness, contraction, or the +energy bound. Their total contribution is therefore at most +\[ + x_s\operatorname{poly}(D_{\rm pred}) + \bigl(2^{-K_0}+\sqrt\epsilon\bigr)(1+L)|K|. +\] +For this one fixed \(\lambda_\alpha\), +Equation~\eqref{eq:hbase-quadratic-weight} gives +\(\sum_{s:x_s\le D_{\rm pred}}x_s\le CD_{\rm pred}\). +This proves the quadratic-difference part. + +Finally, on each use cell the delayed projection decomposes exactly as +\[ + P_{b,s+K_0}F_{b,s} + =P_{b,s}F_{b,s} + +(P_{b,s+K_0}-P_{b,s})F_{b,s}. +\] +The second term, restricted to each use cell as one whole constituent, +has total energy parameter at most \(C(1+K_0)\log(2D_{\rm h})\) by +Equation~\eqref{eq:h-projection-energy}. Together with the current +first term it gives two admissible constituents; subtracting one older +current constituent gives the permitted third. +\end{proof} + +\subsection{High fits and the replacement tuple} + +Return to the fixed tuple and one common older segment +\(R_{\rm old}\subseteq J_0\). We follow the +\emph{Truncating the two high positions} Proposition of +\cite[Section~5]{H}. The low columns, their path counts, their +coefficient sums, and their atom complexities are bounded by +\(D_{\rm old}=\exp(k^{O(1)})\). Expand the two low differences +only for this replacement. Index the available scalar pairs by +\(p\), and let \(J_p\) be the deeper birth root, rebased to +\(R_{\rm old}\) when inherited there. Pairs with disjoint births +have no common use. Their coefficient products on a use cell +are bounded by a fixed power of \(D_{\rm old}\), and +\begin{equation} + n_{\rm pair}(y):=\#\{p:y\in J_p\}\le D_{\rm old}^C,\qquad + \sum_p|J_p|=\int_{R_{\rm old}}n_{\rm pair}(y)\,dy + \le D_{\rm old}^C|R_{\rm old}|. \label{eq:hbase-low-packing} +\end{equation} + +For every \(p\), choose its scalar list by +Corollary~\ref{cor:hbase-predictor-list}, with \(B\) a sufficiently +large fixed power of \(k\). Choose one common numerical bound +\(D_{\rm pred}\) from that Corollary at this \(B\) and the common upper +bound on low atom complexity. It bounds the count, coefficient sizes +at every scale, atom complexities, and quadratic cutoff of each +pair-specific list; its logarithm is a fixed polynomial in \(k\). +The lists remain distinct on their roots \(J_p\), and the predictor +count on a path uses only the pair roots on that path. In high slot +\(e\), admit the conjugate predictors for each pair's list at that pair's +first joint availability, with distinct private directions of weight +\(\sqrt\epsilon\). Pairs inherited at an attempt root have their +predictors present there; later pairs retain their predetermined +first-availability admissions. The predictor path count is at most +\(D_{\rm old}^C D_{\rm pred}\). After these lists and their common bound +are fixed, choose the integer \(K_0\) and \(\log(1/\epsilon)\) as +sufficiently large fixed powers of \(k\). + +Here is the error rate that these choices must supply. For a single +completed owner family with numerical budget \(D_{\rm h}\), put +\[ + \Lambda=C(1+K_0)\log(2D_{\rm h}), +\] +and +\[ + \begin{aligned} + \mathfrak e_k(D_{\rm h})={}& + e^{-B}\operatorname{poly}\bigl(\log(2D_{\rm h}),1+\Lambda\bigr)\\ + &+\bigl(2^{-K_0}+(1+K_0)\sqrt\epsilon\bigr) + \operatorname{poly} + \bigl(D_{\rm pred},\log(2D_{\rm h}),1+\Lambda\bigr). + \end{aligned} +\] +The first term is the finite-list error on both sides of a change +of high input. The second is the delayed error, summed over the +bounded list coefficients and entries. In a sequential change, +the undelayed opponent is raw, optionally minus one previous current +constituent. +The delayed opponent is current plus one whole energy constituent with +parameter \(\Lambda\), optionally minus that one previous current +constituent. Thus these are exactly the opposing classes of the two +scalar results. + +Choose \(B\) to absorb the fixed low coefficient and pair-count +powers. This fixes \(D_{\rm pred}\). Choose \(K_0\) and +\(\log(1/\epsilon)\) afterward to absorb the fixed powers of +\(D_{\rm pred}\) and the same low costs, retaining an \(e^{-2k}\) +reserve. Once the completed \(\log D_{\rm h}\) is a fixed power +of \(k\), its remaining factors in \(\mathfrak e_k\) are fixed +powers of \(k\). Increasing the initial \(k_1\) then gives +\(D_{\rm old}^C\mathfrak e_k(D_{\rm h})\le Ce^{-k}\), for the fixed +power \(C\) needed below. We now verify that the high catalogs +have this logarithmic size. + +On one high attempt root \(R_{\rm fit}\), build +\(P^{\rm new}_{e,h}\) in projection-depth order. The restricted +older data fix the previous approximants and the predictor admissions; +the high family's extra-fit catalog begins empty. At each \(h\), +refine and locally refit the full preceding partition and admit all +predetermined predictors becoming available there. This is done +also on branches with no further tests. A candidate coarse use +cell at this step has depth \(s=h-K_0\); it is an assigned original +request-tree cell. Depths with no such candidate still carry the +full refinements and admissions. + +For a chosen precision \(\chi_e\), test on each candidate \(I\) +\[ + r_{e,I}=f_{e,s}-(P^{\rm new}_{e,h}F_{e,s})_{\rm phys} +\] +in \(\mathfrak d_{e,I}\) at threshold \(e^{-\chi_e}\). +The fine direct sum is an \(L^2(I)\) contraction, so this residual +has bounded normalized size. Local detection gives on failure a +bounded atom \(\varphi\) with coarse moment at least \(\rho'\), +where \(\log(1/\rho')\) and its complexity log are bounded by a +fixed polynomial in \(k,\chi_e\). + +Insert one common label \((\varphi,\mathbf e_v)\), born on \(I\), +into the mask of every depth-\(h\) subcell \(Q\subseteq I\), and +retain it on their descendants. Its scalar function and private +direction are common, while its fitted coefficients may differ +between the \(Q\)'s. With +\(m_Q=\fint_Q r_{e,I}\overline\varphi\), the fresh-column estimate +on each fine cell and Cauchy--Schwarz give +\begin{equation} + c\sum_{\substack{Q\subseteq I\\s(Q)=h}}|Q||m_Q|^2 + \ge c|I|\left|\fint_I r_{e,I}\overline\varphi\right|^2 + \ge c(\rho')^2|I|. \label{eq:hbase-high-gain} +\end{equation} +One request owns the distinct insertion rows on all these \(Q\)'s. +The label counts once at every point of \(I\). + +After the admissions, perform simultaneous insertion rounds at \(h\). +Each still-failing coarse \(I\) may request one such label; every +other fine cell retains its projection. The charged fine rows +have support and observation cells \(Q/Q\) in the full depth-\(h\) +partition and read \(F_{e,h-K_0}\). Repeated rounds read the same +input. The uncharged refinements and admissions remain in the +global nested sequence, and splitting on all fine cells preserves +Bessel norm one. These data meet Lemma~\ref{lem:row-depth} and +the disjoint charged-row hypothesis of Lemma~\ref{lem:hbase-trial}. + +The predetermined budget \(D_0\) includes all predictor admissions, +including future ones, and the bounded older data needed for the +joint budget. Its logarithm is polynomial in \(k\). +A coarse path cap \(T\) adds at most \(T\) common bounded labels +of private weight one. Thus \(D_T\le C D_0(T+1)^C\), and +\(\log(2D_T)\le k^{O(1)}+C\log(2T)\). +The one-trial Lemma chooses a cap with logarithm polynomial in +\(k,\chi_e\) and first-excess cells of total length at most +\(|R_{\rm fit}|/4\). Only original coarse request cells are cut. + +The two families at a cut have separate roles. + +\emph{Old family.} If a coarse cell at depth \(s\) is cut while +\(h=s+K_0\), every strict ancestor was already evaluated at a +smaller delayed depth. On the cut branch, keep the old family's +current depth-\(h\) fine-cell masks unchanged through the remaining +rounds at \(h\). On finer descendants inherit every label in those +masks, including extra-fit labels, and locally refit. Its predictors +and their private directions persist, supplying every finer +projection in Equation~\eqref{eq:hbase-telescope} for the old +ancestor summands. No further residual test on this branch is +needed for those summands. + +\emph{New attempt.} Send the cut coarse summands and their original +candidate descendants to a fresh attempt. It retains the restricted +predetermined older data and the prescribed predictors: inherited +pairs have their predictors at the new root, and later pairs keep +their first-availability admissions. Its extra-fit catalog has no +labels from the old extra fit. This is the fresh-extra-catalog +choice permitted in the proof of the source high-truncation +Proposition. The new \(D_0\) covers only its predetermined data +and admissions, and its cap counts only its own extra requests. + +On each new root, restriction preserves the input normalization; +rebasing and local refitting preserve the predetermined path and +private budgets. The same common-label gain, distinct fine rows, +and full-partition read chronology apply there, with the new empty +extra catalog. Thus Lemma~\ref{lem:hbase-trial} applies anew and +again gives the quarter fraction. The cut roots are proper original +request-tree descendants, and the finite recursion has total root +length at most \(4|R_{\rm fit}|/3\). This realizes the continuation +and restart scheme of \cite[Section~5]{H} with the scheduled rows. + +Apply this algorithm in the two high slots successively. Let +\(S_{\rm old}=\exp(k^{O(1)})\) dominate the older supremum and +predictor costs. Choose \(\chi_1\) with +\(e^{-\chi_1}S_{\rm old}\le e^{-2k}\), and construct the first high +fit and its capped catalog. For a depth-\((s+K_0)\) cell \(Q\subset I\), +\[ + \|F_{e,s}\|_{2,Q}\le2^{K_0/2}\|F_{e,s}\|_{2,I}. +\] +The private-coordinate coefficient bound therefore bounds the first +fit's physical supremum on \(I\) by \(2^{K_0}\) times a fixed power +of its count and inverse private weights. Its logarithm is +polynomial in \(k,\chi_1\); denote the resulting bound by \(S_1\). +Now choose \(\chi_2\ge\chi_1\) and construct the second fit so that +\[ + e^{-\chi_1}S_{\rm old}\le e^{-2k},\qquad + e^{-\chi_2}S_1\le e^{-2k}. +\] +This also gives \(e^{-\chi_2}S_{\rm old}\le +e^{-\chi_1}S_{\rm old}\le e^{-2k}\). Since \(\chi_1\) and \(\log S_1\) +are fixed powers of \(k\), both precision logs and both completed +catalog logs remain fixed powers of \(k\). In particular their individual budgets have a common +numerical upper bound \(D_{\rm h}=\exp(k^{O(1)})\). For any one +common owner piece below, a fixed enlargement of this number also +bounds the joint catalog of its two high families and the bounded +number of older families. These +tuple-specific precision logs do not enter \(k_{t+1}=k_t^A\). + +Intersect the two high ownership trees with the bounded number of +older segment trees. Their maximal common refinement changes +owner only at a constituent segment root. Its piece roots +\(\mathcal S(R_{\rm old})\) have total length at most +\(C|R_{\rm old}|\). Each original summand in this older segment +is assigned to one piece \(S\), even when piece roots are nested. +On its assigned cells define +\begin{equation} + Y_i=X_i,\qquad Y_j=X_j,\qquad + Y_e=P^{\rm new}_{e,s+K_0}F_{e,s} + -g_{e,\mathrm{prev},I}\quad(e=a,b). \label{eq:hbase-Y} +\end{equation} +Each projection here is the family of that cell's owner, with its +own full continuation. + +We verify the aggregate replacement across these owners. For a +low pair \(p\) and piece \(S\), put \(K=J_p\cap S\) when this is +nonempty; dyadic nesting makes \(K\) one cell. Select only the +original coarse summands owned by \(S\) on which that pair is +available. Use the original scalar list and coefficients chosen +on \(J_p\), and the one owner family in each high slot on \(S\), +with its complete descendant continuation. On \(K\), rebase an +inherited pair and its predictors when needed and locally refit +the owner family. This rebasing uses the same masks and retained spans +on each used cell, hence the same local owner projection there. +The scalar results apply anew on \(K\): +their source-selected raw rows are the original array restricted +to these cells, with the same Bessel bound by zero extension of +tests from \(K\), and their projected rows use this owner's +individual catalog. The normalized schedules and row energies +have root budget \(|K|^{1/2}\). Apply the finite-list identity +before and after each high change and the delayed estimate to the +retained entries. With the low coefficient products outside, +the resulting cost for this pair and owner is at most a fixed +power of \(D_{\rm old}\) times +\(\mathfrak e_k(D_{\rm h})|K|\). + +Writing \(n_{\rm seg}(y)=\#\{S\in\mathcal S(R_{\rm old}):y\in S\}\), +the intersection lengths satisfy +\[ + \sum_{p,S}|J_p\cap S| + =\int_{R_{\rm old}}n_{\rm pair}(y)n_{\rm seg}(y)\,dy + \le D_{\rm old}^C\sum_{S\in\mathcal S(R_{\rm old})}|S|. +\] +Here empty intersections contribute zero. Combining the coefficient +and pair-count powers into \(D_{\rm old}^C\) proves +\begin{align} + &\sum_{S\in\mathcal S(R_{\rm old})} + \sum_{\substack{I\ {\rm owned}\\{\rm by}\ S}} + |I|\,|H_I(X_I)-H_I(Y_I)| + \nonumber\\ + &\qquad\le + D_{\rm old}^C\mathfrak e_k(D_{\rm h}) + \sum_{S\in\mathcal S(R_{\rm old})}|S| + \le Ce^{-k}|R_{\rm old}|. \label{eq:hbase-high-errors} +\end{align} +This sums fresh estimates using each owner's continuation and the +original scalar list. + +The high residuals pass their \(\mathfrak d\) tests on the coarse +cells. The pair comparison used for \(X\) applies with the two high +precision logs ordered after the old logs. If the earliest low +index is separated from \(t\), test the later residual difference +against each existing early fitted operand; otherwise every existing +old fitted operand has index at least \(t-7\). For \(t\ge12\), each of +\(Y_j,Y_a,Y_b\) cancels when its existing fitted operands are replaced +by the same \(F_{e,s}\), so every pair has a cancelling later factor, +as in the proof of Lemma~\ref{lem:hbase-X-output}. The high-truncation comparison in +\cite[Section~5]{H} therefore supplies an absolute exponent +\(\theta_{\rm av}>0\) for which each pair of \(Y\) factors has form +norm at most \(e^{-k^{\theta_{\rm av}}}\), when \(t\ge12\). +Contraction gives a fixed bound \(C_{\rm size}\) for every local +\(L^2(I)\) size. Choose once +\begin{equation} + 0<\theta<\min\{1,\theta_{\rm av},\theta_0\},\qquad + \eta=e^{-k^\theta}. \label{eq:hbase-quality} +\end{equation} +After increasing \(k_1\), both the pair norm and +\(C_{\rm size}^2e^{-k^{\theta_{\rm av}}}\) are at most \(\eta\). +Thus every full \(Y\) pair has norm at most \(\eta\), and +\(|H_I(Y_I)|\le\eta\) for \(t\ge12\). + +Across all common older segments, let \(\mathcal R_{\rm base}\) be the +final base segments and let \(\mathcal I_R\) be the preimage of \(R\) +under their owner map. The sets \(\mathcal I_R\) partition this fixed +\(\mathcal I_0\), each +\(I\in\mathcal I_R\) lies in \(R\), and no auxiliary projection +cell is a summand. Summing Equation~\eqref{eq:hbase-high-errors} +over the older segments gives the separate global statement +\begin{equation} + \sum_{R\in\mathcal R_{\rm base}}|R|\le C|J_0|,\qquad + \sum_{R\in\mathcal R_{\rm base}}\sum_{I\in\mathcal I_R} + |I|\,|H_I(X_I)-H_I(Y_I)|\le Ce^{-k}|J_0|. \label{eq:hbase-high-replacement} +\end{equation} + +\subsection{The constructed tuple and its active patterns} + +\begin{proposition}[Base data and active reduction] +\label{prop:hbase-data} +Fix the constructed tuple with later accuracy \(t\), put \(k=k_t\), +and use its final segments \(R\in\mathcal R_{\rm base}\) and original owner +sets \(\mathcal I_R\) from Equation~\eqref{eq:hbase-high-replacement}. +On an owned cell of depth \(s\), define +\begin{equation} + \begin{gathered} + Y_l=B_l+U_l,\qquad B_i=X_i,\quad B_j=X_j,\quad U_i=U_j=0,\\ + B_e=P^{\rm new}_{e,s}F_{e,s}-g_{e,\mathrm{prev},I},\qquad + U_e=(P^{\rm new}_{e,s+K_0}-P^{\rm new}_{e,s})F_{e,s} + \quad(e=a,b),\\ + \|B_{l,I}\|_{2,I},\ \|U_{l,I}\|_{2,I},\ \|Y_{l,I}\|_{2,I}\le C,\\ + \Lambda_k=C(1+K_0)\log(2D_{\rm base})\le k^C,\qquad + \sum_{I\in\mathcal I_R}|I|\|U_{e,I}\|_{2,I}^2 + \le\Lambda_k^2|R|\quad(e=a,b). + \end{gathered} \label{eq:h-base-data} +\end{equation} +Each \(B_l\) is a signed sum of at most two current nested-catalog +projections applied to \(F_{l,s}\). Their basic catalogs have +distinct private directions and inherited masks. Their path counts, +column sizes, inverse private weights, bounded scalar atom +complexities, and current coefficient sums are bounded by +\(D_{\rm base}\), with +\[ + e^k\le D_{\rm base}\le\exp(k^C). +\] +For \(t\ge12\), the associated full \(Y\) tuple has the pair bound +and pointwise form bound \(\eta\) of Equation~\eqref{eq:hbase-quality}. + +For three available basic scalar columns \(\boldsymbol\varphi\) in +distinct slots, let \(\operatorname{act}_I(\boldsymbol\varphi)\) +be their form norm against a normalized \(L^2(I)\) input in the +fourth slot. For a sufficiently large fixed \(C_{\rm act}\), put +\[ + \mathcal A_R= + \left\{I\in\mathcal I_R: + \operatorname{act}_I(\boldsymbol\varphi)>D_{\rm base}^{-C_{\rm act}} + \text{ for some available triple }\boldsymbol\varphi\right\}. +\] +This common active set has path multiplicity +\[ + \sum_{I\in\mathcal A_R}\mathbf1_I(y)\le M\le D_{\rm base}^C. +\] +Define three patterns in their original slots by +\[ + \begin{array}{lll} + z^\varnothing_l=B_l&\text{for every }l,\\ + z^a_a=U_a,&z^a_l=B_l& (l\ne a),\\ + z^b_b=U_b,&z^b_l=B_l& (l\ne b), + \end{array} + \qquad \mathcal P=\{z^\varnothing,z^a,z^b\}. +\] +For \(z\in\mathcal P\), let +\begin{equation} + \mathfrak M_R(z) + :=\sum_{I\in\mathcal A_R}|I| + \min\{\eta,|H_I(z_I)|\}. \label{eq:h-active-pattern-target} +\end{equation} +Then for \(t\ge12\), +\begin{equation} + \sum_{I\in\mathcal I_R}|I|\,|H_I(Y_I)| + \le C\eta\Lambda_k^2|R|+e^{-k}|R| + +\sum_{z\in\mathcal P}\mathfrak M_R(z). \label{eq:h-active-minimum} +\end{equation} +The \(e^{-k}|R|\) term is the total inactive error for the three +patterns. The finitely many leading accuracy levels have a bound +independent of the number of spatial depths. These base data and +their estimates hold anew on normalized subroots with the inherited +basic lists and their local continuations. +\end{proposition} + +\begin{proof} +The definitions give \(Y=B+U\). Contraction bounds the local sizes. +Choose \(D_{\rm base}\) to dominate the bounded number of basic +catalogs used by this tuple and enlarge it to be at least \(e^k\). +The preceding logarithmic bounds give \(D_{\rm base}\le\exp(k^C)\). +On \(R\), use its owner families with their full inherited +continuations. Equation~\eqref{eq:h-projection-energy}, applied +anew on this root, gives the stated \(U\) energies. + +The scalar input for the active set is the +\emph{Few active scales for a fixed triple} Lemma of +\cite[Section~5]{H}. For three fixed bounded basic scalar columns +on their common birth subtree and \(0<\delta\le1\), it gives on +every spatial path +\[ + \#\{I:\operatorname{act}_I(\boldsymbol\varphi)>\delta\} + \le\operatorname{poly}\bigl(\log D_{\rm base}+\log(1/\delta)\bigr), +\] +and +\[ + \sum_{\substack{I\ {\rm on\ the\ path}\\ + \operatorname{act}_I(\boldsymbol\varphi)\le\delta}} + \operatorname{act}_I(\boldsymbol\varphi) + \le\delta\operatorname{poly} + \bigl(\log D_{\rm base},\log(1/\delta)\bigr). +\] +The bounds are uniform in the path location and unrestricted real +oscillation parameters. Every detected atom and predictor used +here is a fixed scalar label at birth. At most \(D_{\rm base}^3\) +available triples lie on one path. With +\(\delta=D_{\rm base}^{-C_{\rm act}}\), the first source bound gives +\(M\le D_{\rm base}^C\) after enlarging the fixed power. + +On an inactive owned cell, expand three \(B\) factors of a pattern +into their basic scalar columns. Their coefficients are constant +on that cell and their sums are bounded by a fixed power of +\(D_{\rm base}\); the fourth factor has bounded normalized \(L^2\) +size. For each fixed triple, the second source bound controls the +inactive path sum by +\(D_{\rm base}^{-C_{\rm act}} + \operatorname{poly}(\log D_{\rm base},C_{\rm act})\). +Bound the varying coefficients by their uniform sizes, sum the +pathwise triples and the three patterns, and integrate over path +points in \(R\). The total absolute length-weighted inactive sum +is at most +\[ + D_{\rm base}^{C-C_{\rm act}} + \operatorname{poly}(\log D_{\rm base},C_{\rm act})|R| + \le e^{-k}|R| +\] +when \(C_{\rm act}\) dominates the fixed power costs and \(k_1\) +is sufficiently large. + +It remains to derive the displayed minimum. Let \(z^{UU}\) have +\(Y_i=B_i\) and \(Y_j=B_j\) in the low slots and \(U_a,U_b\) in +the other two. For \(t\ge12\), the complementary \(Y_i,Y_j\) +pair bound and Cauchy--Schwarz give +\[ + \sum_{I\in\mathcal I_R}|I|\,|H_I(z_I^{UU})| + \le C\eta + \left(\sum_{I\in\mathcal I_R}|I|\|U_{a,I}\|_{2,I}^2\right)^{1/2} + \left(\sum_{I\in\mathcal I_R}|I|\|U_{b,I}\|_{2,I}^2\right)^{1/2} + \le C\eta\Lambda_k^2|R|. +\] +At each owned cell, multilinearity gives +\(H_I(Y)=\sum_{z\in\mathcal P}H_I(z)+H_I(z^{UU})\). +The full-tuple bound \(|H_I(Y)|\le\eta\), established before the +pattern split, therefore gives +\[ + |H_I(Y)| + \le\min\left\{\eta,\sum_{z\in\mathcal P}|H_I(z)| + +|H_I(z^{UU})|\right\} + \le |H_I(z^{UU})| + +\min\left\{\eta,\sum_{z\in\mathcal P}|H_I(z)|\right\}. +\] +Sum over \(\mathcal I_R\), use the absolute inactive bound outside +\(\mathcal A_R\), and on the active cells apply +\(\min(\eta,\sum_z a_z)\le\sum_z\min(\eta,a_z)\). +This proves Equation~\eqref{eq:h-active-minimum}. It assigns no +pair bound or pointwise \(\eta\) bound to an individual pattern. + +For a leading level, the Local absolute estimate bounds the +two-\(U\) term by \(C\Lambda_k^2|R|\) without \(\eta\). +At each of the finitely many such levels, \(D_{\rm base}\), +\(\Lambda_k\), and \(M\) are constants independent of spatial +depth. The local sizes and +\(\sum_{I\in\mathcal A_R}|I|\le M|R|\) bound the active contribution +independently of the number of depths. + +Finally, on a normalized subroot the original schedule restrictions +remain normalized, basic masks remain inherited with the same +budgets, and local refitting gives the required full continuations. +The row estimate applied there supplies the \(U\) energies anew. +The arguments handed to localization are the whole \(B,U\) factors +above, together with the associated \(Y\) pair bound; the scalar +pair expansion was used only in the high replacement and the triple +expansion only on inactive cells. +\end{proof} diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/completion.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/completion.tex new file mode 100644 index 0000000000000000000000000000000000000000..fee63a2cf15fcb01c5fb921924addffd707d09d3 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/completion.tex @@ -0,0 +1,225 @@ +\section{Completion of the local estimate} +\label{sec:completion} + +\subsection{Restarts on the original summands} + +We complete the target for a base segment \(R_0\) by the +construction in \cite[Section~10, One attempt and its restarts]{H}. +On an attempt \(R\), the main realization and actual uses precede +the prospective-offset calculation; the chosen whole sides then +precede helper closure and its dependent size and counting cuts. +The parent functions and operators remain fixed when original +summands are transferred at a cut. + +The cuts accumulate nonnegative partial path counts, component +square sizes, or partial ancestor suprema. Their integrated +bounds permit the thresholds to be increased by fixed factors. +Together with the initial marking-failure cuts, the union of first +stopped cells can then be chosen to have total length at most \(|R|/4\). +For the strong stack thresholds, only auxiliary choices contribute +a cardinality overhead, absorbed in \(G_0\); all full indices +\(\nu=(G,b,\chi)\) are controlled jointly by the aggregate energy +and full-hull bounds. The polynomially many configurations in +\(P,k\) occur in the power budgets. The hull construction +includes every potentially needed ancestor within a block before +the common cuts, and its maximal functions remain defined on +whole retained hull cells, including portions below a cut. +Large-frozen-complement flags are charged to the powered error +sum; only a first crossing of their accumulated budget creates +a restart root. + +At a stopped cell \(R'\), send the original base summands there +and at its descendants to a fresh attempt. Their original +depths, input schedules, and basic expressions remain assigned +to them. The restrictions \(F_{j,s}|_{R'}\) satisfy the same +block normalization in each slot. On \(R'\), the input size bound +\eqref{eq:h-input-size} gives the ordinary \(L^2\) budget +\(|R'|^{1/2}\), and \eqref{eq:h-input-variation} has right side +\(5a|R'|\). Rebase the inherited basic columns with their +private constants unchanged and use genuine local refits and +inherited continuations on the new root. The old continuations +needed by the parent's summands remain fixed. + +All root bounds are obtained anew on \(R'\). +Equation~\eqref{eq:h-projection-energy} gives the basic rolling +energy with factor \(|R'|\). The proofs of +\eqref{eq:h-certified-variation} and +\eqref{eq:h-frozen-composition} apply to the new main and mode +constructions, and their newly constructed linear sides satisfy +\eqref{eq:h-linear-energy} on \(R'\). In particular, this +linear energy is proved by the new-root construction, with no +inference of proportional child energy from a parent's arbitrary +family. The local sizes, column budgets, and inherited label +properties used in those proofs hold on \(R'\) as well. The displayed +cross-depth bounds and local sizes thus have the same constants on +every new root, independently of the remaining number of depths. + +The packing-of-attempts Lemma of \cite[Section~3]{H} now gives +\[ + \sum_{\substack{R\ {\rm attempt\ root}\\ + {\rm for\ the\ segment}\ R_0}}|R| + \le |R_0|\sum_{n\ge0}4^{-n}=\tfrac43|R_0|. +\] +The recursion is finite on the original finite depth tree; +equivalently, induction on the remaining depths gives this +geometric bound. + +\subsection{The estimate at one accuracy} + +At each \(I\in\mathcal A_R^{\rm ret}\), apply the minimum to the +absolute value of its whole contribution. Use +\eqref{eq:h-localization} and subadditivity to separate the fixed +search leaves while keeping the group sum inside the cell. +In each leaf expand \eqref{eq:h-core-side}. Terms with at least +three cores use \eqref{eq:h-core-estimate}; terms with at least +three sides use the \(p\)th-power estimate of the side calculation, +where \(p=1-\varepsilon>2/3\) is supplied by the core calculation. +Exactly two sides use the transferred main term and its freeze, +large-complement, and restoration estimates. These cases exhaust +the four-position expansion. + +For each powered within-cell contribution use +\[ + \min(\eta,x)\le\eta^{1-p}x^p\qquad(x\ge0). +\] +The near terms use \eqref{eq:h-near-band} at power one, with +their coupled band sums kept inside the cell. Physical and +numerical errors are also estimated at power one. Increase +\(C_4\) to include the powered costs and the smaller main cost +\(\eta P^C|R|\). The overhead from source leaves and auxiliary +choices fits \(G_0\) for the strong terms; the polynomially many +power-budget choices fit the enlarged \(C_4\). The strong energy +and hull bounds already sum all groups and blocks. Thus on the +retained cells of one attempt, +\begin{equation} + \begin{aligned} + \sum_{I\in\mathcal A_R^{\rm ret}}|I| + \min(\eta,|H_I(z_I)|) + &\le C_q\eta^\varepsilon P^{C_4}|R|\\ + &\quad +(1+C_N(q))C_qP^{-1/2}G_0 + (K+2)^{C_{19}(q-2)}|R|\\ + &\quad +O(e^{-k}|R|). + \end{aligned} + \label{eq:h-retained-estimate} +\end{equation} +This is the conditional retained-cell calculation of +\cite[Section~10]{H}, with its basic square, variation, and +frozen-row hypotheses supplied above. Summing it over the +attempt roots by the preceding packing bounds the full segment +target on \(\mathcal A_{R_0}\) by the same right-hand expression +with \(|R_0|\), up to a fixed factor. + +\subsection{\texorpdfstring{Choice of \(q\), \(\alpha\), and starting accuracy}{Choice of q, alpha, and starting accuracy}} + +The degrees in \eqref{eq:h-retained-estimate} and +\eqref{eq:h-budget-hierarchy} can be fixed independently of +\(q\). The main-row and stack calculation following +\eqref{eq:h-stacks} records the norm and squared losses: +row logarithms, Compression of \(O(\log k)\) main factors, +and the strong stack factors fit \(G_0\); the larger \(D_*\) +is confined to the mode construction, Bessel, and counting estimates. The base +row, smooth, and delay estimates add fixed polynomial degrees +in \(k\). Powers of \(K\) enter the allowed component and top +budgets and the displayed short-block factor. These facts +preserve absolute upper bounds on \(C_4,C_5,C_{19}\). + +We need the fixed-input constant at the same \(q\). +The \emph{Uniform local estimate} of \cite[Section~2]{H} is +stated for some \(q\in(2,3)\), and its proof permits the choice +required here. The \emph{Local detection} Lemma of +\cite[Section~4]{H} holds for every \(q>2\), with an absolute +logarithmic degree and \(q\)-dependence only in its coefficient. +The \emph{Short-block comparison} Lemma of +\cite[Section~7]{H} holds for every \(20\) with +\(\alpha C_4<\varepsilon/2\); \(C_4\) already dominates the +powered side costs. For this fixed \(\alpha\), +\(P=\lceil e^{\alpha k^\theta}\rceil\) dominates every fixed +power of \(k\) and \(G_0=P^{o(1)}\). By +\eqref{eq:h-common-q}, after increasing the starting \(k\), +\[ + \begin{gathered} + P^{-1/2}G_0(K+2)^{C_{19}(q-2)}\le C_qP^{-1/4},\\ + \eta^\varepsilon P^{C_4}\le C_q\eta^{\varepsilon/2}, + \qquad P^{-1/4}\le\eta^{\alpha/4}. + \end{gathered} +\] +Indeed the \(K\) power costs less than \(P^{1/8}\), and +\(G_0\le P^{1/8}\) for large \(k\). Since \(0<\theta<1\), +the numerical error \(e^{-k}\) is smaller than the resulting +quality power. Substitute \eqref{eq:h-uniform-fixed} into +\eqref{eq:h-retained-estimate} and sum the attempt roots. +For a fixed \(c_*>0\), this gives the full segment bound +\begin{equation} + \sum_{I\in\mathcal A_{R_0}}|I| + \min(\eta,|H_I(z_I)|) + \le C_q\eta^{c_*}|R_0|. + \label{eq:h-quality-gain} +\end{equation} +The fixed constant \(C_{\rm H}(q)\) is included in \(C_q\). + +\subsection{Summing the accuracy expansion} + +Use the final accuracy summation of +\cite[Section~10, Summing the base expansion]{H}. For a tuple +with later level \(t\), the earlier level has \(O(t)\) choices +and there are only a bounded number of slot patterns. The +intersections of the tuple's bounded number of basic segment +families have total root length at most a fixed multiple of +\(|J|\); the localization attempt packing preserves that bound. +The noninitial costs in \eqref{eq:h-quality-gain} therefore +sum because +\[ + \sum_{t\ge12}t\,e^{-c_*k_t^\theta}<\infty,\qquad + k_{t+1}=k_t^A,\quad A>1. +\] +The separate global replacement errors in +\eqref{eq:hbase-high-replacement} and the two-\(U\) errors have the +same summable multiplicity. The +finitely many initial levels have the depth-independent bound +from the base active-scale reduction. + +Finally take the two-earliest-increments expansion at a finite +accuracy cutoff. At each fixed cell all its accuracy updates +were evaluated on the same actual \(F_{j,s(I)}\); the per-cell +pair comparisons in the base proof therefore make the terminal +residual tend to zero. There are finitely many original cells, +so this limit commutes with their finite sum. The triangle +inequality, the uniform summable bounds, and passage to that +cutoff limit give \(C|J|\) for the original sum. + +The cross-depth estimates +\eqref{eq:h-input-variation}, \eqref{eq:h-row-use}, +\eqref{eq:h-projection-energy}, and \eqref{eq:h-smooth-use}, +together with the single-depth size bound +\eqref{eq:h-input-size}, have constants independent of the four +input block partitions and their numbers \(W_j\). The resulting +constant is therefore uniform in the input schedules, as +required by Theorem~\ref{thm:depth-local}. diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/ergodic.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/ergodic.tex new file mode 100644 index 0000000000000000000000000000000000000000..04ab6daa97471b9715951766067d74b976ef7442 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/ergodic.tex @@ -0,0 +1,557 @@ +\section{Smooth orbit averages and the pointwise theorem} +\label{sec:ergodic} + +This section deduces Theorem~\ref{thm:main} from a finite oscillation +estimate for smooth orbit averages. The oscillation estimate forces +almost everywhere convergence on every invertible probability preserving +system. The mixing hypothesis \eqref{eq:mixing} identifies the limit, and +signed approximations then recover all positive integer averaging lengths. + +We use the notion of an invertible probability preserving system fixed in +the introduction. Every integer power of \(T\) is measurable and measure +preserving: positive powers follow by iteration, and applying invariance to +the measurable set \(TA\) gives \(\mu(TA)=\mu(A)\). + +\begin{definition}[Averaging profiles] +\label{def:averaging-profile} +An averaging profile is a real function +\(\varphi\in C_c^\infty(\mathbb R;\mathbb R)\) such that, for some +\(G\geq1\), +\begin{equation} + \|\varphi^{(n)}\|_\infty + \leq\bigl(G(n+2)\bigr)^{G(n+2)} + \qquad(n=0,1,2,\ldots), + \label{eq:profile-derivatives} +\end{equation} +and +\begin{equation} + \int_{\mathbb R}\varphi(t)\,dt=1,\qquad + \int_{\mathbb R}t^j\varphi(t)\,dt=0\quad(1\leq j\leq3). + \label{eq:profile-moments} +\end{equation} +The function may have either sign. For \(a>0\), write +\(\varphi_a(t)=a^{-1}\varphi(t/a)\). +\end{definition} + +Section~\ref{sec:transfer} uses the moment and derivative conditions to +construct local kernels for adjacent smooth scale differences. +Lemma~\ref{lem:ergodic-moments} shows that these restrictions still permit +signed profiles arbitrarily close in \(L^1\) to +\(h_c(t)=c^{-1}\mathbf1_{(0,c]}(t)\) for every \(c\in[1,2]\). + +\begin{proposition}[Finite oscillation of smooth orbit averages] +\label{prop:orbit-oscillation} +For every averaging profile \(\varphi\), there is a constant +\(C_\varphi<\infty\) with the following property. On every invertible +probability preserving system, let \(f_1,f_2,f_3:X\to\mathbb C\) be +measurable with \(|f_j(x)|\leq1\) for all \(x\), and define +\[ + V_k^\varphi(x)=\sum_{n\in\mathbb Z}\varphi_{2^k}(n) + \prod_{j=1}^3f_j(T^{jn}x), + \qquad k=0,1,2,\ldots . +\] +For every integer \(w\geq1\) and every deterministic list of integers +\(0\leq k_1<\cdots0\}=\bigcup_{h\geq1}\{\omega>2/h\}\), there are +\(\delta>0\) and a measurable set +\(E=\{\omega>2\delta\}\) of measure \(d>0\). For every \(K\geq0\) and +\(x\in E\), +\[ + \sup_{\ell\geq K}|Z_\ell(x)-Z_K(x)| + \geq\frac12\sup_{p,q\geq K}|Z_p(x)-Z_q(x)| + \geq\frac12\omega(x)>\delta. +\] +The integral of the first supremum is at least \(\delta d\). +For \(L\geq K\), the measurable finite maxima +\[ + M_{K,L}(x)=\max_{K\leq\ell\leq L}|Z_\ell(x)-Z_K(x)| +\] +increase to this supremum. By monotone convergence there is a +deterministic \(L>K\) with \(\int_XM_{K,L}\,d\mu\geq\delta d/2\). +Starting with \(k_1=0\), choose \(k_{i+1}>k_i\) successively with this +property. The assumed estimate gives +\(w\delta d/2\leq C\sqrt w\) for every \(w\), a contradiction for large +\(w\). Hence \(\omega=0\) almost everywhere. +\end{proof} + +Proposition~\ref{prop:orbit-oscillation} and +Lemma~\ref{lem:ergodic-cauchy} already give an almost everywhere limit +for each fixed averaging profile. We now identify it for mixing systems. + +\subsection{The product limit in norm} + +Suppose for this subsection that \(T\) satisfies the mixing condition +\eqref{eq:mixing}. +For every fixed bounded measurable \(g,h:X\to\mathbb C\), this implies +\begin{equation} + \int_X g(x)h(T^m x)\,d\mu(x) + \longrightarrow + \left(\int_Xg\,d\mu\right)\left(\int_Xh\,d\mu\right) + \qquad(|m|\to\infty). + \label{eq:ergodic-function-mixing} +\end{equation} +For finite complex simple functions this is the finite expansion into +the set correlations in \eqref{eq:mixing}. Every bounded +complex measurable function has uniform simple-function approximations, +obtained by partitioning its bounded range into finitely many small sets. +If \(g_0,h_0\) are such approximations, the correlation error is at most +\[ + \|g-g_0\|_\infty\|h\|_\infty+ + \|g_0\|_\infty\|h-h_0\|_\infty +\] +uniformly in \(m\), and the product of integrals has the same continuity +property. Taking the mixing limit for fixed approximations and then +letting their errors tend to zero proves +\eqref{eq:ergodic-function-mixing}. + +We include the classical van der Corput induction for multiple averages, +specialized to mixing; see \cite[Section~3]{FKO1982} and +\cite[Section~3]{Bergelson1987}. The finite Hilbert space van der Corput +inequality used below is the one recorded in \cite[Lemma~D.1]{HostKra2005}; +the criterion follows by taking limits. +\begin{lemma}[Hilbert space van der Corput criterion] +\label{lem:ergodic-vdc} +Let \(\mathcal H\) be a complex Hilbert space, with inner product linear +in the first variable. Let \(u_n\in\mathcal H\), \(n\geq1\), satisfy +\(\|u_n\|\leq M<\infty\). Suppose that, for each fixed integer \(h\geq1\), +\[ + \gamma_h=\lim_{N\to\infty} + \frac1N\sum_{n=1}^{N-h}\langle u_{n+h},u_n\rangle +\] +exists, with a sum over an empty range interpreted as zero. If +\(\gamma_h\to0\) as \(h\to\infty\), then +\(\|N^{-1}\sum_{n=1}^Nu_n\|\to0\). +\end{lemma} + +\begin{proof} +For fixed integers \(N,H\geq1\), extend \(u_n\) by zero outside +\(\{1,\ldots,N\}\). The identity +\[ + H\sum_{n=1}^Nu_n + =\sum_{m=1}^{N+H-1}\sum_{j=0}^{H-1}u_{m-j} +\] +and Cauchy's inequality give +\begin{equation} +\begin{split} + \left\|\frac1N\sum_{n=1}^Nu_n\right\|^2 + \leq\frac{N+H-1}{N^2H^2}\bigg[ + &H\sum_{n=1}^N\|u_n\|^2\\ + &+2\sum_{h=1}^{H-1}(H-h) + \operatorname{Re}\sum_{n=1}^{N-h} + \langle u_{n+h},u_n\rangle\bigg]. +\end{split} +\label{eq:ergodic-finite-vdc} +\end{equation} +Indeed, after Cauchy's inequality the expression to expand is +\((N+H-1)\sum_m\|\sum_{j=0}^{H-1}u_{m-j}\|^2\). Each diagonal term +occurs \(H\) times, and each pair of positions at difference \(h\) +occurs \(H-h\) times, giving the displayed bracket. + +Fix \(H\) and let \(N\to\infty\) in +\eqref{eq:ergodic-finite-vdc}. Then +\begin{align*} + \limsup_{N\to\infty}\left\|\frac1N\sum_{n=1}^Nu_n\right\|^2 + &\leq\frac{M^2}{H} + +\frac2{H^2}\sum_{h=1}^{H-1}(H-h)\operatorname{Re}\gamma_h\\ + &\leq\frac{M^2}{H}+\frac2H\sum_{h=1}^{H-1}|\gamma_h|. +\end{align*} +The right side tends to zero as \(H\to\infty\), because +\(|\gamma_h|\to0\) implies convergence of its Ces\`aro means to zero. +\end{proof} + +\begin{lemma}[Ces\`aro norm convergence at distinct slopes] +\label{lem:ergodic-cesaro} +On every invertible probability preserving system satisfying +\eqref{eq:mixing}, for every integer \(r\geq1\), every +list of pairwise distinct nonzero integers +\(\lambda_1,\ldots,\lambda_r\), and every list of bounded measurable +\(f_1,\ldots,f_r:X\to\mathbb C\), +\[ + \left\| + \frac1N\sum_{n=1}^N\prod_{j=1}^r f_j(T^{\lambda_jn}\,\cdot) + -\prod_{j=1}^r\int_Xf_j\,d\mu + \right\|_{L^2(\mu)} + \longrightarrow0\qquad(N\to\infty). +\] +\end{lemma} + +\begin{proof} +Write \(U^m f=f\circ T^m\), a unitary operator on \(L^2(\mu;\mathbb C)\). +Use the inner product \(\langle f,g\rangle=\int_Xf\overline g\,d\mu\). +We induct on \(r\), taking the empty product to be \(1\), so the +assertion for \(r=0\) is immediate. Suppose it is known for \(r-1\). +First assume that at least one \(f_j\) has integral zero. Put +\[ + u_n=\prod_{j=1}^rU^{\lambda_jn}f_j,\qquad + g_{j,h}=(U^{\lambda_jh}f_j)\overline{f_j},\qquad n,h\geq1. +\] +The vectors \(u_n\) are uniformly bounded in \(L^2\). Invariance gives +\begin{align*} + \langle u_{n+h},u_n\rangle + &=\int_X\prod_{j=1}^rU^{\lambda_jn}g_{j,h}\,d\mu\\ + &=\int_Xg_{1,h}\prod_{j=2}^r + U^{(\lambda_j-\lambda_1)n}g_{j,h}\,d\mu . +\end{align*} +The second equality shifts all times by \(-\lambda_1n\). +For fixed \(h\), the remaining slopes +\(\lambda_j-\lambda_1\), \(2\leq j\leq r\), are pairwise distinct and +nonzero, and the functions \(g_{j,h}\) are fixed and bounded. Induction +therefore gives +\[ + \frac1{N-h}\sum_{n=1}^{N-h} + \prod_{j=2}^rU^{(\lambda_j-\lambda_1)n}g_{j,h} + \longrightarrow\prod_{j=2}^r\int_Xg_{j,h}\,d\mu + \quad\text{in }L^2(\mu),\qquad N\to\infty,\ N>h. +\] +For \(r=1\) both products are \(1\), so this assertion is immediate. +The functional \(A\mapsto\int_Xg_{1,h}A\,d\mu\) is continuous on \(L^2\) +by Cauchy's inequality. Hence, using \((N-h)/N\to1\), +\[ + \lim_{N\to\infty}\frac1N\sum_{n=1}^{N-h} + \langle u_{n+h},u_n\rangle + =\prod_{j=1}^r c_j(h),\qquad + c_j(h)=\int_X(U^{\lambda_jh}f_j)\overline{f_j}\,d\mu . +\] +By \eqref{eq:ergodic-function-mixing} and \(\lambda_j\neq0\), +\[ + c_j(h)\longrightarrow\left|\int_Xf_j\,d\mu\right|^2 + \qquad(h\to\infty). +\] +The factors are bounded by \(\|f_j\|_\infty^2\), and at least one limit +is zero. Their product tends to zero, so +Lemma~\ref{lem:ergodic-vdc} gives +\(\|N^{-1}\sum_{n=1}^Nu_n\|_2\to0\). + +For a general list, let \(m_j=\int_Xf_j\,d\mu\). The identity +\[ + \prod_{j=1}^rU^{\lambda_jn}f_j-\prod_{j=1}^rm_j + =\sum_{i=1}^r + \left(\prod_{ji}m_j\right) +\] +expresses the difference as a finite sum of products with a centered +factor. Regard the later constants as constant functions at their +corresponding slopes. The centered case just proved applies to every +summand and completes the induction. + +For each fixed \(h\), induction was used only for the fixed functions +\(g_{j,h}\). The van der Corput argument fixes its finite lag range +\(H\) before sending \(N\) to infinity, and sends \(H\) to infinity +afterwards. No convergence rate uniform in \(h\) is needed. +\end{proof} + +\begin{lemma}[Norm convergence with smooth weights] +\label{lem:ergodic-smooth-norm} +On every invertible probability preserving system satisfying +\eqref{eq:mixing}, fix an integer +\(r\geq1\), pairwise distinct nonzero integers +\(\lambda_1,\ldots,\lambda_r\), and bounded measurable +\(f_1,\ldots,f_r:X\to\mathbb C\). Set +\[ + F_n=\prod_{j=1}^r f_j(T^{\lambda_jn}\,\cdot),\qquad + p=\prod_{j=1}^r\int_Xf_j\,d\mu,\qquad n\in\mathbb Z. +\] +For every \(\varphi\in C_c^1(\mathbb R;\mathbb C)\) with +\(\int_{\mathbb R}\varphi=1\), +\[ + \left\|\sum_{n\in\mathbb Z}a^{-1}\varphi(n/a)F_n-p\right\|_2 + \longrightarrow0\qquad(a\to\infty,\ a\geq1). +\] +In particular the conclusion holds for every averaging profile. +\end{lemma} + +\begin{proof} +Apply Lemma~\ref{lem:ergodic-cesaro} to the slopes \(\lambda_j\) and +then to the slopes \(-\lambda_j\). For the \(L^2\) vectors +\[ + R_N^\pm=\sum_{n=1}^N(F_{\pm n}-p) +\] +this gives \(\|R_N^\pm\|_2/N\to0\) for both signs. +Choose \(B\geq1\) with +\(\operatorname{supp}\varphi\subseteq[-B,B]\). For \(a\geq1\), put +\(M_a=\lfloor Ba\rfloor+1\) and +\(w_n^\pm=a^{-1}\varphi(\pm n/a)\), so \(w_{M_a}^\pm=0\). +Vector-valued summation by parts gives +\[ + \sum_{n=1}^{M_a}w_n^\pm(F_{\pm n}-p) + =\sum_{n=1}^{M_a-1}(w_n^\pm-w_{n+1}^\pm)R_n^\pm. +\] +Since \(|w_n^\pm-w_{n+1}^\pm|\leq a^{-2}\|\varphi'\|_\infty\), the norm +of this sum is at most +\[ + \frac{\|\varphi'\|_\infty}{a^2} + \sum_{n=1}^{M_a-1}\|R_n^\pm\|_2. +\] +For any \(\epsilon>0\), choose \(N_0\) with +\(\|R_n^\pm\|_2\leq\epsilon n\) for both signs and every \(n\geq N_0\). +As \(M_a/a\leq B+1\), +\[ + \frac1{a^2}\sum_{n=1}^{M_a-1}\|R_n^\pm\|_2 + \leq\frac1{a^2}\sum_{n=1}^{N_0-1}\|R_n^\pm\|_2 + +\frac{\epsilon}{a^2}\sum_{n=1}^{M_a-1}n + \leq o(1)+\frac{\epsilon}{2}(B+1)^2. +\] +Letting \(a\to\infty\) and then \(\epsilon\downarrow0\) proves convergence +to zero for the positive-time and negative-time parts of the weighted sum +of \(F_n-p\). +The term at \(n=0\) has norm at most +\(a^{-1}|\varphi(0)|\|F_0-p\|_2\), which also tends to zero. +Finally the Riemann sums +\(S_a=a^{-1}\sum_n\varphi(n/a)\) tend to \(\int\varphi=1\), and +\[ + \sum_n a^{-1}\varphi(n/a)F_n-p + =\sum_n a^{-1}\varphi(n/a)(F_n-p)+p(S_a-1). +\] +This proves the assertion for signed and complex weights alike. +\end{proof} + +\subsection{Moment approximation and all lengths} + +We need averaging profiles arbitrarily close in \(L^1\) to normalized +interval indicators. We first fix the smooth bump used in their +construction and later in the local kernels. + +\begin{lemma}[A compact Gevrey bump] +\label{lem:ergodic-bump} +There is a real nonnegative \(\rho\in C_c^\infty(\mathbb R)\), supported +in \([-1,1]\), with integral \(1\), whose derivatives obey a bound of +the form \eqref{eq:profile-derivatives} for some constant \(G\geq1\). +\end{lemma} + +\begin{proof} +Start with +\[ + \rho_0(x)= + \begin{cases}e^{-1/(1-x^2)},&|x|<1,\\0,&|x|\geq1.\end{cases} +\] +For \(|x|<1\), put \(d=1-x^2\). If \(|z-x|\leq c d\) in the complex +plane, with a sufficiently small fixed \(c>0\), then +\(1-z^2=d(1-\theta)\) with \(|\theta|\leq1/4\). Consequently +\(\operatorname{Re}(1/(1-z^2))\geq c'/d\) for a fixed \(c'>0\). +Cauchy's estimate gives +\[ + |\rho_0^{(n)}(x)|\leq n!(cd)^{-n}e^{-c'/d}. +\] +For every fixed \(n\), the right side tends to zero as \(x\) approaches +either endpoint, so the derivatives extend there by zero and the +extension is smooth. Maximizing \(d^{-n}e^{-c'/d}\) for \(00\), there is an averaging +profile \(\varphi\) with +\[ + \|\varphi-h_c\|_{L^1(\mathbb R)}<\epsilon,\qquad + h_c(t)=c^{-1}\mathbf 1_{(0,c]}(t). +\] +The support and derivative constant may depend on \(c,\epsilon\). +\end{lemma} + +\begin{proof} +Use the bump \(\rho\) from Lemma~\ref{lem:ergodic-bump}. For +\(\delta>0\), set \(\rho_\delta(t)=\delta^{-1}\rho(t/\delta)\) and +\(\psi=h_c*\rho_\delta\). It is real, compactly supported, smooth, +and has mass \(1\). Its derivatives satisfy +\[ + \|\psi^{(n)}\|_\infty + \leq\|h_c\|_1\|\rho_\delta^{(n)}\|_\infty + =\delta^{-n-1}\|\rho^{(n)}\|_\infty, +\] +which has the form \eqref{eq:profile-derivatives} for fixed \(\delta\). +Moreover, +\[ + \|\psi-h_c\|_1 + \leq\int\rho_\delta(s)\|h_c(\,\cdot-s)-h_c\|_1\,ds + \leq\frac{2\delta}{c}\int|v|\rho(v)\,dv, +\] +because \(\|h_c(\,\cdot-s)-h_c\|_1=2\min(|s|,c)/c\). +Choose \(\delta\) to make this error less than \(\epsilon/2\), fix +\(\psi\), and let \(m_j=\int t^j\psi(t)\,dt\), \(1\leq j\leq3\). + +We correct these moments with broad translated bumps: their \(j\)th moments +will be \(D^j\) times fixed numbers, allowing a linear system with a fixed +coefficient matrix to cancel the three moments with total coefficient size +tending to zero as \(D\to\infty\). +Fix distinct real numbers \(z_0,z_1,z_2,z_3\), and for \(D\geq1\) put +\[ + \rho_{i,D}(t)=D^{-1}\rho(t/D-z_i),\qquad + P_j(z)=\int(z+v)^j\rho(v)\,dv,\qquad 0\leq i,j\leq3. +\] +Then \(\int t^j\rho_{i,D}(t)\,dt=D^jP_j(z_i)\). +Each \(P_j\) is monic of degree \(j\). The matrix +\(\mathsf M=(P_j(z_i))_{j,i=0}^3\) is therefore the Vandermonde matrix +\((z_i^j)_{j,i}\) multiplied on the left by a lower triangular matrix +with diagonal entries \(1\). Its determinant is +\(\prod_{i0\) +chosen in Section~\ref{sec:completion}. For fixed \(\alpha\), +\(\log(2P)=O(k)\). The main catalog consists of +the basic columns and the additional columns used by H's local +filters; its budget \(D_{\rm m}\) also anticipates the later helper +columns. It bounds path counts, column sizes, coefficient bounds, +inverse private weights, and main-use multiplicities. We use +\[ + \log D_{\rm base},\ \log D_{\rm m}\le C_qk^C,\qquad + \mathcal L=k^{O(1)},\qquad + G_0=\exp\!\bigl(C_q(1+\log(2k))^C\bigr). +\] +There are \(\mathcal L\) colors, and the relevant main histories have +\(O(\log\mathcal L)\) factors. The value of \(G_0\) may increase +within this displayed class. + +We describe the main data of \cite[Section~6]{H} before checking +their realization for the present inputs. A present vertex is an +occurrence of an original basic column, with H's full normalized +curvature tag, including its horizontal chart data. The present +lists grow by inheritance along downward paths. In slot \(l\), +the quadratic chart phase is \(\kappa_l\) times the phase represented +by that tag, where \(\kappa=(-1,3,-3,1)\); linear modulation is not +part of the tag. Distances are measured in H's ambient graph on all +hypothetical tags satisfying its bounds. They depend on the tags and +absolute depth, and do not increase as depth increases. Every added main +or helper column is attached to a present vertex and has its tag; +these insertions add no graph vertices. + +A group \(G\) is a source index with a fixed color and a set of +actual central uses. Its pad mask \(\mathcal M_{G,h}(I)\) is a set of +present vertices on the cell \(I\), retaining historical central +uses. Such masks may continue by inheritance after the group has +ceased to be current on a branch. At a given rank, the associated +pad projection is the orthogonal projection in \(L^2(I;\mathcal H)\) onto +constant-coefficient combinations of the available columns of at +most that rank whose tags belong to the mask. Pads are nested on a +fixed cell, and each individual pad family is inherited on the +complete depth partitions and unused-cell continuations needed +below. A positive smeared filter is the normalized average of its +\(P\) pad projections. A named history is a specified ordered +history of these source filters and their complements. After the +catalogs are fixed it is a fixed bounded operator; expanding its +complements and choosing individual pads gives ordered products of +inherited orthogonal projections. + +Usability is relative to the pad mask supplying a range: that mask +is nonempty, and the cell has an actual use of \(G\) at or below it. +The \emph{Mask inclusion, closure, and separation} Lemma of +\cite[Section~6]{H} gives three properties. At an actual use, the +pad-\(h\) mask contains every present vertex at graph distance at +most \(h\) from the center; its pad-\(h+1\) mask contains every +neighbor of a pad-\(h\) vertex. For distinct groups of the same +color, vertices in their respective pad masks on overlapping usable +endpoints are separated by graph distance greater than one at the +finer endpoint. + +We now verify the closure input. At the cell \(I\) of depth \(s\) +being tested, hold fixed its one current basic argument +\(b=z_{l,s}\), of type \(B\) or \(U\). For two specified positions, +let \(\mathfrak d_I(v,w)\) be the supremum of the absolute form with +those entries and arbitrary normalized \(L^2(I)\) functions in the +other positions. At rank \(r\), H requires, for every named target +history and each of its prescribed individual-pad choices, +\begin{equation} + \mathfrak d_I\bigl(\hbox{one lower-rank source}, + (1-P_{r,\mathcal M})\mathcal H b\bigr) + \le \frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}}\|b\|_{2,I}, + \qquad \tau_3=D_{\rm base}^{-C_\tau}. + \label{eq:h-closure-test} +\end{equation} +The source is one pure available lower-rank column whose tag lies +in the target mask, and \(n_{r-1}\) bounds the path count through +the preceding rank. This is Equation~\Heq{13} of +\cite[Section~6]{H}; the individual tests also control the +normalized averages. + +In the \emph{Catalog closure and its budget} Lemma of H, the tested +state \(X_i=\mathcal H_i b\) changes with the insertion index \(i\) +because its projections acquire columns, while \(b\) stays fixed +on \(I\). The \emph{Uniform same-tag test} Lemma supplies on failure +a fresh target-column increment satisfying +\[ + \|(P_{i,\rm new}-P_{i,\rm old})X_i\|_{2,I} + \ge c\frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}H_0}\|b\|_{2,I}, + \qquad H_0\le\exp(C_qk^C). +\] +For one named history and mask in a trial of at most \(T\) +insertions, the target increments are orthogonal rows and every +underlying positive projection is nested in \(i\). Expanding +complements and applying the products-of-prefixes calculation of +\cite[Section~3]{H} gives +\begin{equation} + \sum_i\|(P_{i,\rm new}-P_{i,\rm old})\mathcal H_i b\|_{2,I}^2 + \le (C\log(2T))^{O(\log\mathcal L)}\|b\|_{2,I}^2. + \label{eq:h-closure-prefix} +\end{equation} +This is a fixed-vector estimate on \(b\), so it applies when \(b\) +contains \(F_{l,s}\): the depth \(s\) is fixed during the cell test. +The lower bound and \eqref{eq:h-closure-prefix} cancel the norm of +\(b\). Counting named masks and histories and taking +\(n_r/n_{r-1}=\exp(k^{C_2})\) through \(\mathcal L+5\) ranks gives +\(\log n_{\max}\le C_qk^C\), using the active path count from the +base reduction. + +This is an a priori count from names of contexts, including the +anticipated number of helper bases; it is independent of the +numeric edge height \(H_{\rm ed}\) and later mode precision. As +in H, choose the mismatch accuracy after this count and then choose +\(H_{\rm ed}\) and realize compatible graph, mask, and main-catalog +data. We fix these main projections, color histories, ball masks, +and actual central uses, including the small marking-failure cuts +and the inherited continuations with local refits on unused branches +needed for operator estimates. The actual helper selections will +be made after their bases have been constructed. + +The \emph{Bessel separation of usable endpoint spaces} Lemma of +\cite[Section~6]{H} supplies the following operator interface. For +one fixed slot and color, let +\[ + h_G=\sum_{v,U}a_{v,G,U}\mathbf1_U\Phi_v +\] +be finite sums of available pure main columns. Each coefficient is +constant on an entire usable endpoint cell \(U\), and the particular +pad mask certifying that term contains the indicated vertex bearing its tag. +If the group, label, and inverse-private-weight counts on a path are +bounded by \(\exp(C_qk^C)\), H's chosen mismatch accuracy gives +\begin{equation} + \left\|\sum_Gh_G\right\|_2^2\le C\sum_G\|h_G\|_2^2. + \label{eq:h-usable-synthesis} +\end{equation} +Labels and cells may be repeated at arbitrary endpoint times. The +constant does not count these repetitions or the depth of a future +certificate. If an output is restricted to a union \(E\) of finer +cells, then \((\mathbf1_E P_U)^*=P_U\mathbf1_E\); its adjoint range +is still the column span on the whole \(U\). In particular, for a +common depth \(t\), certified unions \(E_G\) of \(t\)-cells, and +individual ball projections \(L_{G,t}\), duality gives +\begin{equation} + \sum_G\|\mathbf1_{E_G}L_{G,t}h\|_2^2\le C\|h\|_2^2 + \quad\text{for one shared input }h. + \label{eq:h-usable-rows} +\end{equation} + +The binary localization proof in H now applies to the one current +tuple \((z_{0,s},\ldots,z_{3,s})\) on each original cell in +\(\mathcal A_R^{\rm main}\). Each pad in a smeared filter acts on +that same operand. The closure tests are \eqref{eq:h-closure-test}, +the mismatch estimates concern fixed column tags, and the other +factors use only local \(L^2\) sizes. The +\emph{Localization into current groups} Proposition of +\cite[Section~6]{H}, Equation~\Heq{14}, therefore gives at most +\(\operatorname{poly}(k)\) fixed search leaves and, for every +\(I\in\mathcal A_R^{\rm main}\), +\begin{equation} + \begin{gathered} + H_I(z_{\cdot,s})= + \sum_\lambda\ \sum_{G\ {\rm current}} + H_I(Z^\lambda_{0,G,s},\ldots,Z^\lambda_{3,G,s})+e_I,\\ + Z^\lambda_{l,G,s}=\overline L^\lambda_{l,G,s} + \mathcal B^\lambda_{l,s}z_{l,s},\qquad + \sum_{I\in\mathcal A_R^{\rm main}}|I||e_I|\le Ce^{-k}|R|. + \end{gathered} + \label{eq:h-localization} +\end{equation} +Here \(\overline L\) is one positive average of ball projections, +and \(\mathcal B\) is a source-ordered history of cumulative filters +and complements of length \(O(\log\mathcal L)\), independent of +\(G\). The history and slot choices are fixed by the search leaf +throughout the tree. Only groups with an actual current central +use on \(I\) are summed in each leaf, and that sum may be empty. +The sum over groups stays inside +the contribution of the original cell; subadditivity distributes +the minimum with \(\eta\) only over the fixed leaves. + +\subsection{\texorpdfstring{Variation of the localized \(B\) output}{Variation of the localized B output}} + +Fix one slot, one search leaf, and its color, and suppress their +subscripts. Write the full basic expression as +\[ + B_s=\sum_{\tau=1}^{h_0}\varepsilon_\tau Q_s^\tau F_s, + \qquad h_0\le2, +\] +with its signs and constituent basic projection families fixed. +Define the full endpoint output +\[ + Z_{G,s}=\overline L_{G,s}\mathcal B_sB_s +\] +using the source positive ball average and the fixed +group-independent history of the leaf. We evaluate this expression +on every needed endpoint of the complete inherited families within +the attempt, including endpoints where \(G\) has no current central +use. All individual pad families and unused-cell continuations are +fixed for the calculation. The same pad choice is coupled across +compared depths in each average. + +We also use a shared output \(Z_s^{\rm sh}\) from H's +group-independent current-depth coarse histories on \(F_s\). +These are the histories of the fixed source leaf and the remaining +input-side histories obtained when a source transfer step removes +its prescribed positive output-side filters in decreasing rank. +They retain the source order, \(O(\log\mathcal L)\) main factors, +normalized positive pad averages or controlled complements, and at +most two fixed signed basic constituents. Their complement expansions +have the source bound \(2^{O(\log\mathcal L)}\). The identity is included. +The output has no outer ball and is counted once across its uses. + +\begin{lemma}[Certified variation with depth inputs] +\label{lem:h-certified-variation} +For the full endpoint outputs +\(Z_{G,s}=\overline L_{G,s}\mathcal B_sB_s\) just defined, let +\(\mathcal J\) be a common finite collection of depth intervals +\([r,t]\) with edge-overlap multiplicity at most \(a\). +For each \(G,[r,t]\), let \(E_{G,r,t}\) be a union of \(t\)-cells +having a downstream actual-use certificate for \(G\). Then +\begin{equation} + \sum_{G,[r,t]\in\mathcal J} + \|\mathbf1_{E_{G,r,t}}(Z_{G,t}-Z_{G,r})\|_2^2 + \le aG_0|R|. + \label{eq:h-certified-variation} +\end{equation} +For one shared output \(Z_s^{\rm sh}\) in the stated source class, +let \(E_{r,t}\) be the union of the required \(t\)-cells across all +its uses. Then +\[ + \sum_{[r,t]\in\mathcal J} + \|\mathbf1_{E_{r,t}}(Z_t^{\rm sh}-Z_r^{\rm sh})\|_2^2 + \le aG_0|R|. +\] +This shared assertion has no group sum or group certificate. +All outputs before the attempt root are zero. +\end{lemma} + +\begin{proof} +Expand the complements, choose one basic constituent, and choose +one individual pad in each average, coupled at all depths. One +term of the full ball output is +\begin{equation} + \widehat Z_{G,s}=L_{G,s}B'_sF_s,\qquad + B'_s=P_s^m\cdots P_s^1. + \label{eq:h-current-product} +\end{equation} +Here \(L_{G,s}\) is an individual nested ball projection and the +\(P^j_s\) are nested main projection families shared across groups. +Every nonzero \(B\) constituent has its specified basic projection +as the input-side factor \(P_s^1\). An empty product is allowed for +a shared history. + +Color the intervals into at most \(a\) collections with disjoint +interiors. Within one color, omit zero intervals and order the +others as \(r_1\frac{\Xi}{2}\max(|I|^{-1},|I'|^{-1}) + \quad(n\in\mathbb Z^{\dim\theta_z},\ |n|_\infty\le E_h). + \label{eq:h-representative-separation} +\end{equation} +The identification across depths is the persistent mode label +and its value \(\omega_a(x_*)\). + +For an actual cell \(I\) and a successful component +\(\mathcal F\), H uses +\[ + \sigma_{\mathcal F}^2 + =\sum_{\substack{\mathcal D\subseteq\mathcal F\\ + \mathcal D\ {\rm minimal\ CURRENT}}} + \left(\|A_{\mathcal D}\|_{2,I} + +D_*^{-30}\!\!\sum_{\substack{m\ {\rm core}\\m\in\mathcal D}} + |\alpha_m|\right)^2. +\] +Here \(A_{\mathcal D}\) is the complete core subtotal within +\(\mathcal D\); side labels in that component are kept separately. +The \emph{Component-size comparison} Lemma of +\cite[Section~8]{H} gives the same-cell bounds +\[ + \sigma_{\mathcal F}^2 + \le C\bigl(\|P_{\mathcal F,I}\widetilde Z_{l,G,b}\|_{2,I}^2 + +\|V_{\mathcal F}\|_{2,I}^2+D_*^{-50}\bigr), + \qquad + \sum_{\mathcal F}\|V_{\mathcal F}\|_{2,I}^2 + \le C\|V_{l,G,b}\|_{2,I}^2. +\] +The projection \(P_{\mathcal F,I}\) uses all mode labels of +\(\mathcal F\) on \(I\), and \(V_{\mathcal F}\) is their exact +side subtotal. This is an almost-orthogonality estimate for +arbitrary coefficient subtotals on one cell. + +For a fixed source role, upper height, type, and dyadic threshold +\(D_*^{-1}\le\sigma\le D_*^2\), H takes, for each group and +representative, the first actual-use cells on each spatial branch +for which \(\sigma_{\mathcal F}\ge\sigma\). Several incomparable +tops for the same group and representative are allowed. The +resulting top rows +\(P_{\mathcal F,I}\) are separated mode rows. They are supported +on \(I\subseteq U_e\), the \(u_b\)-cell of the frozen list, and +the top itself is a downstream certificate. Applying +Lemma~\ref{lem:h-frozen-composition} gives squared energy +\(G_0|R|\) for their frozen projections. Summing the side +term of the size comparison first at one depth and then through +a block gives at most +\(K\sum_{G,b}\|V_{l,G,b}\|_2^2\le KG_0|R|\). The +\(D_*^{-50}\) floor is absorbed because +\(\sigma^2\ge D_*^{-2}\). Hence +\[ + \sigma^2\sum_{\rm tops}|I|\le C(1+K)G_0|R|. +\] +There are polynomially many choices in \(P,k\) of height, type, +and threshold. The first-crossing construction of the +\emph{Top count} Proposition in \cite[Section~8]{H} gives, on +retained paths, +\begin{equation} + \sum_G|\mathcal A_j^G| + \le\min(D_*,P^C\sigma_j^{-2}). + \label{eq:h-top-count} +\end{equation} +Here \(\mathcal A_j^G\) is the set of distinct representatives +actually used in the bin \([\sigma_j,2\sigma_j)\) along the retained +path in question, for the fixed role, height, and type. The full +label budget supplies the +minimum for \(\sigma_j2\) give +\[ + \sum_s(a_sb_sc_s)^p + \le + \left(\sum_sa_s^2\right)^{p/2} + \left(\sum_sb_s^2\right)^{p/2} + \left(\sum_sc_s^2\right)^{p/2}. +\] +Integrating the path caps therefore gives total length-weighted +\(p\)th-power cost \(P^C|R|\). The power is taken after the +group sum at each original cell. + +For exactly two sides, replace their two core complements in +\eqref{eq:h-core-side} by the whole current \(Z\) arguments. +The corrections have a third side, apart from the physical errors +already estimated. We recall the source filter algebra for these +whole complements. Let the side positions be \(a,b\) and the +complementary positions be \(c,d\). Expand each complementary +factor \(1-\overline Q\) as identity minus a positive average. +The \emph{Outgoing filters and near-band defects} Lemma of +\cite[Section~6]{H} removes the positive filters from \(c,d\) +in decreasing rank and transfers helper averages to the fixed +whole state \(S_a\). Its final main term is +\[ + H_I(\mathcal D_{a,s}S_a,S_b,B_c,B_d), +\] +where \(\mathcal D_{a,s}\) has \(O(\log\mathcal L)\) helper +factors. + +For one step from \(c\) to \(a\), write \(Q_{l,p}\), +\(1\le p\le P\), for the outgoing and corresponding helper +projections, and put +\[ + Q_{l,0}=0,\quad Q_{l,P+1}=1,\quad + A_l^p=Q_{l,p}-Q_{l,p-1},\quad + \overline Q_l=P^{-1}\sum_{p=1}^P Q_{l,p}. +\] +Here \(x_c\) is the exact remaining input-side source history +on \(B_c\), \(x_d\) is the whole remaining history on \(B_d\), +\(x_a\) is the already transferred helper history on \(S_a\), +and \(x_b=S_b\). The exact difference is +\begin{equation} + H_I(x_a,x_b,\overline Q_cx_c,x_d) + -H_I(\overline Q_ax_a,x_b,x_c,x_d) + =\sum_{p,q'=1}^{P+1}\frac{p-q'}P + H_I(A_a^p x_a,x_b,A_c^{q'}x_c,x_d). + \label{eq:h-transfer} +\end{equation} +The pad projections are nested on this cell, so the \(A_l^p\) +are orthogonal bands summing to the identity. The average has +weight \((P-p+1)/P\) on band \(p\), which proves the identity. +H's closure and tag-separation estimates bound the sum with +\(|p-q'|>2\) by +\(D_{\rm base}^{-C_3}\|S_a\|_{2,I}\|S_b\|_{2,I}\), for any +prescribed fixed \(C_3\) after choosing its numerical accuracies. +The helper closure here uses the already fixed whole \(S_a\) +as its base in \eqref{eq:h-closure-test}. For the retained terms, +keep the entire band sum at each fixed offset \(q'-p\). + +In the final main term, write \(B_c=Y_c-U_c\) and +\(B_d=Y_d-U_d\). Each correction has a third basic side. +The remaining complementary pair \(Y_c,Y_d\) has form norm at +most \(\eta\) against the two side inputs, by the full-pair +statement of Proposition~\ref{prop:hbase-data}. Contraction of +the helper factors and Cauchy--Schwarz in the side sizes give +absolute main cost \(\eta P^C|R|\). The numerical transfer +errors are summable with the chosen inverse-base-budget accuracy. + +For a retained offset, write +\(C_{G,s}^{q'}=A_c^{q'}x_{c,s}\) for the outgoing complementary +band stack and \(D_{G,s}=x_{d,s}\) for the other complement. +Freeze both at the block start \(b\), using their full endpoint +evaluations there. These endpoints differ from the mode +freezing depth \(u_b\). Corollary~\ref{cor:h-transfer-endpoint-variation} +applies to exactly these source histories. For a ball transfer, +a nonterminal band is the difference of two eligible positive +ball endpoints; its terminal band is the shared inner history +minus the positive \(P\)-pad endpoint. For a cumulative transfer, +the outer ball has already been removed and all the complementary +endpoints are shared. The other complement has its one source +positive outer ball or is shared. Thus each band costs a fixed +multiple of its endpoint estimates, and the shared histories +are counted once across their uses. Summing the \(P+1\) bands +and the \(O(K)\) overlap of \([b,s]\) retains the allowed +polynomial cost \(P^C|R|\). +Their nonnegative partial path sums enter the power-cost first-crossing +cuts. + +At a required endpoint the pad bands are orthogonal, and the +remaining source histories are contractions of bounded basic +inputs. Choose \(C_{\rm fr}\ge1\) to dominate the retained +current local norm of the band stack and of the other complement. +A freeze error has a third path-square size by the preceding +endpoint bounds. If the other frozen complement is large, +write it as its bounded current value plus its difference. +Likewise, if a frozen local size exceeds \(2C_{\rm fr}\) while +the current one is at most \(C_{\rm fr}\), then the difference +exceeds \(C_{\rm fr}\), and the flag is at most +\(C_{\rm fr}^{-2}\) times the squared difference size. Its square root is therefore +another path-square size. Certified variation applies on the whole ancestor cells with downstream +certificates; propagation through a block costs at most \(K\) +in this power budget. H's freeze, large-complement, and +restoration errors therefore have total \(p\)th-power cost +\(P^C|R|\). A large-complement flag is charged to that error +sum; only a first crossing of its accumulated budget makes a +restart cut. On the remaining uses the frozen complements have +bounded local sizes on their entire ancestor hulls in the block. + +\subsubsection{Fixed stacks for the two side inputs} + +Fix a source transfer step, a near offset, and a search leaf. +Under these fixed data, let \(\chi\) record the source auxiliary +side-component and binary-length names for the two side stacks. +Pairing their choices raises the auxiliary count to a fixed power, +so a common auxiliary list has at most \(G_0\) entries after enlarging \(G_0\) within +its stated class; unused combinations are set to zero. Write +\(\nu=(G,b,\chi)\) for the full stack index. Concrete binary +interval positions and band labels \(p\) are stack coordinates, +and individual pads remain in coupled averages. For one full +index, a stored side stack is a function into a finite Hilbert +direct sum: +\[ + \begin{gathered} + \mathrm{stack}_{\mathrm{untouched},\nu}(y) + =(W_{\nu,i}(y))_{i\in\mathcal I_\nu} + \in\bigoplus_{i\in\mathcal I_\nu}\mathcal H,\\ + \|\mathrm{stack}_{\mathrm{untouched},\nu}\|_2^2 + =\sum_i\|W_{\nu,i}\|_2^2. + \end{gathered} +\] +The banded stack \(\mathrm{stack}_{a,\nu}\) has in addition the +band index \(p\in\{1,\ldots,P+1\}\). Here \(\mathcal H\) is the +augmented ambient Hilbert space of the stored coordinates. Each +whole coordinate is restricted, when formed, to the union of the +use cells selecting its position and valid band. Later cuts +remove summands without changing that stored function. At a use, +each side selects one position; the banded side retains all valid +matched band indices at the fixed offset. + +The source stack construction gives the squared energies +\begin{equation} + \sum_\nu\|\mathrm{stack}_{a,\nu}\|_2^2\le PG_0|R|, + \qquad + \sum_\nu\|\mathrm{stack}_{\mathrm{untouched},\nu}\|_2^2\le G_0|R|. + \label{eq:h-stacks} +\end{equation} +Both sums range over all full indices \(\nu=(G,b,\chi)\). +The \(G_0\) cardinality overhead counts only the auxiliary labels +\(\chi\); both energies sum all groups and blocks. +Their quantitative purpose is the following. The whole-hull +maximal and sparse calculation below turns these energies into +a root-size sum with cost \(P^{1/2}G_0|R|\). The coefficient +\(O(P^{-1})\) in the matched transfer bands then leaves the +gain \(P^{-1/2}\). We first check the fixed-vector arguments +behind \eqref{eq:h-stacks}. + +For a sequence \(W_s\) in one block, write +\[ + W_s=W_b+\sum_{[r,t]\in\mathcal P(b,s)}(W_t-W_r), +\] +where \(\mathcal P(b,s)\) is the binary decomposition of the +prefix. At a fixed length its intervals are disjoint and at +most one is selected at a use. Store each resulting function +on its selecting-use union as above. For a basic-\(U\) side, +store each already formed depth function directly; +\eqref{eq:h-side-square} bounds its energy, and subsequent +helper products are local contractions. + +For a linear side \(V_{G,b}\), let \(T_s\) be the product of +the subsequent helper projections, including one endpoint of +a band difference when needed. The products-of-prefixes +estimate of \cite[Section~3]{H}, on at most \(K+1\) times, gives +for each fixed binary length +\[ + \|T_bV_{G,b}\|_2^2+ + \sum_{[r,t]\ {\rm at\ that\ length}} + \|(T_t-T_r)V_{G,b}\|_2^2 + \le (C\log(K+2))^{O(m)}\|V_{G,b}\|_2^2. +\] +The input \(V_{G,b}\) is the one fixed function on the block. +Use \eqref{eq:h-linear-energy}, sum the logarithmic length +choices, and pay at most \(P+1\) band indices for the first +stack. + +For a rolling side put \(W_s=T_s(Z_{G,s}-Z_{G,u_b})\). +Its base value uses \eqref{eq:h-certified-variation} on +\([u_b,b]\), whose overlap across blocks is bounded. Its +increments split exactly as +\[ + W_t-W_r=T_t(Z_{G,t}-Z_{G,r}) + +(T_t-T_r)(Z_{G,r}-Z_{G,u_b}). +\] +The first term uses certified variation on the whole endpoint +cells containing its selected use cells. For the second, also +decompose \([u_b,r]\) into binary intervals. Fix an outer +length, an inner length, and an inner interval \([a,v]\). +Every outer position selecting this interval acts on the one +fixed function +\[ + X=\mathbf1_\Omega(Z_{G,v}-Z_{G,a}), +\] +where \(\Omega\) is the union of its required \(r\)-cells. +Since \(v\le r\), locality of \(T_t-T_r\) gives the required +outputs from \(X\). Enlarge \(\Omega\) to the containing whole +\(v\)-cells to bound its norm; those cells have downstream +certificates. The fixed-vector product-jump estimate sums the +outer positions by +\((C\log(K+2))^{O(m)}\|X\|_2^2\). Then +\eqref{eq:h-certified-variation} sums these squared inputs over +groups and the disjoint inner intervals at that length. The +\(O(K)\)-depth windows for different blocks have bounded +overlap. The two endpoint outputs in \(X\) use \(F_v\) and +\(F_a\), but \(X\) itself is fixed once \(a,v\) are fixed, as +required for the outer product-jump estimate. + +This proves \eqref{eq:h-stacks}. Coupled pad averages are restored +by Jensen's inequality. When the fixed search-leaf and transfer-step +choices are restored later, their auxiliary overhead fits an enlarged +\(G_0\). + +We collect the main-row and stack costs here. For +\(m=O(\log\mathcal L)=O(\log k)\), the Compression factor in +\eqref{eq:h-compression} satisfies +\[ + \log\mathfrak C(m,D_{\rm m}) + \le C(m+1)\bigl(\log(m+1)+\log\log(2D_{\rm m})\bigr) + \le C_q(1+\log(2k))^C. +\] +A row application loses \(\log(2D_{\rm m})\) in norm, hence +\(\log^2(2D_{\rm m})\) in squared energy. This loss, the +controlled signed expansions, and the telescoping counts fit +\(G_0\). The factors \((C\log(K+2))^{O(m)}\) and the binary +length choices also fit \(G_0\) by +\eqref{eq:h-budget-hierarchy}. Thus no fixed power of \(K\) +enters \eqref{eq:h-stacks}. Its powers in the component and +top bounds stay in the allowed \(P^C\) budgets. The larger +\(D_*\) is used in the mode construction, Bessel, and counting arguments; +the final row adjoints in the two new lemmas use \(D_{\rm m}\). + +\subsubsection{The fixed-input comparison on a short block} + +For \(20\), and +\(\sum_\nu|\Omega_\nu|\le M_0|R|\), where +\(M_0\le\exp(C_qk^C)\) follows from the active cell/group path +counts and the auxiliary-choice overhead, with all groups and +blocks included in this support sum. The dyadic maximal calculation +in +\cite[Section~7]{H} gives +\[ + \begin{aligned} + \sum_\nu|\{m_\nu^{\rm full}>T\}| + &\le\min(M_0|R|,CE|R|/T),\\ + \sum_\nu\int_R\min(m_\nu^{\rm full},C_1E) + &\le CE(1+\log(2+C_1M_0))|R|. + \end{aligned} +\] +For a fixed sufficiently large \(C_1\), first-crossing cuts on the +individual and then joint partial ancestor suprema yield the retained caps +\[ + \sum_\nu m_{a,\nu}(x)\le PG_0,\qquad + \sum_\nu m_{\mathrm{untouched},\nu}(x)\le G_0. +\] +Here \(m_{a,\nu},m_{\mathrm{untouched},\nu}\) use the retained hull cells, still +taken whole. Include every potentially needed hull ancestor +within the block before these common cuts. At a crossing the +cell is removed before its value enters a retained cap; below a +cut only earlier retained ancestors contribute. The caps +therefore hold also on portions of retained hull cells below a +cut, with the small relative length cost supplied by the full +maximal bounds. + +On each retained hull, make the source sparse decomposition: +from a root \(Q\), stop where either squared side average first +exceeds sixteen times its average on \(Q\). A zero root average gives +zero contribution. The stopped cells +have total length at most \(|Q|/8\), so the disjoint remainders +\(E_Q\) have measure at least \(7|Q|/8\). Complete each piece +by children immediately below its included cells. A child's +squared average is at most twice its parent's, which gives the +leaf bounds in \eqref{eq:h-short-block}; the frozen complementary +sizes are bounded on the whole hull and its boundary leaves. +If \(a_Q,b_Q\) are the side sizes at the piece roots, the +whole-hull convention puts \(Q\) in both suprema at every +\(x\in E_Q\), even below a cut. Hence +\[ + \begin{split} + \sum_{\nu,Q}|Q|a_Qb_Q + &\le C\int_R\sum_\nu\sqrt{m_{a,\nu}m_{\mathrm{untouched},\nu}}\\ + &\le C\int_R + \left(\sum_\nu m_{a,\nu}\right)^{1/2} + \left(\sum_\nu m_{\mathrm{untouched},\nu}\right)^{1/2} + \le P^{1/2}G_0|R|. + \end{split} +\] +This is the promised root-size cost from the stack energies and +the whole-hull cuts. Combining it with +\eqref{eq:h-short-block} and the matched coefficient \(O(P^{-1})\) +gives the near-band form of Equation~\Heq{19} of +\cite[Section~7]{H}: +\begin{equation} + (1+C_N(q))C_qP^{-1/2}G_0 + (K+2)^{C_{19}(q-2)}|R|. + \label{eq:h-near-band} +\end{equation} +It bounds the absolute sum of original cell contributions with +the prescribed band sum retained inside each contribution at a +fixed offset. The logarithmic block factors fit \(G_0\); the +displayed power of \(K+2\) comes from the fixed-input +short-block comparison. + +\subsection{The core calculation} + +The remaining terms have at least three core positions. We +specialize the conditional calculation in +\emph{Counts for the tested core tuples} and +\emph{Powered estimate for the core terms} of +\cite[Section~9]{H}. Its good-height predicates, shifted +annular tests, pure-shift gap, and numerical separation and +height reserves remain exactly those of the source construction. +We check the inputs that can be affected by the changed endpoint +operands. + +Fix a retained path point \(x_*\), a group, an upper height, +role types, frequency bins \((\mu,\nu,\Lambda)\), and amplitude +bins \(\sigma_j\). Let \(N_s^G\) be H's exact count of tested +core representative-label tuples at depth \(s\) for these data. +In a core role put \(M_j^G=|\mathcal A_j^G|\). In the optional +side role put +\[ + M_l^G=\sigma_l^{-2}\sum_{s:\,G\ {\rm used}} + \|S_{l,G,s}\|_{2,I_s}^2,\qquad + M'=\max_jM_j^G. +\] +Omit groups with no nonzero count, so \(M'\ge1\). H's cell +calculation gives, apart from total absolute error +\(O(e^{-k}|R|)\), terms of the form +\[ + C_L\mu\nu^{-L}\Lambda^{-L} + \left(\prod_{j=0}^3\sigma_j\right)N_s^G. +\] +Here \(0<\mu\le1\le\nu\) and \(\Lambda\ge1\) are dyadic, with +\(\Lambda=1\) for four core positions. The good-height masks +select the full core subtotals within the minimal CURRENT +components, with the coefficient floor already included in +\(\sigma_{\mathcal F}\). + +The preceding arguments provide the three inputs to the source count +that depend on endpoint data. First, a +representative label has the fixed value \(\omega_a(x_*)\). +Distinct labels in one role obey +\eqref{eq:h-representative-separation}. Together with H's exact +tested-tuple constraints, this ensures that two labels in distinct +core roles determine at most one tested tuple, and +the source height-bin estimate bounds its scale multiplicity. +Second, the frozen-row, component-size, and top argument gives +\eqref{eq:h-top-count}, including the \(D_*\) cap in each core +role. Third, a side role uses the actual same-cell function +\(S_{l,G,s}\) in its Fourier windows. Parseval gives its +incidence bound +\[ + w_s=\sigma_l^{-2}\|S_{l,G,s}\|_{2,I_s}^2, +\] +whose sum is \(M_l^G\) and whose path budget is supplied by +\eqref{eq:h-side-cap}. + +The remaining hypotheses of the +\emph{Sparse mass of separated progression tests} Lemma of +\cite[Section~9]{H} are exactly H's shifted frequency tests +and pure-shift gap, retained from its construction. As in its proof of +\emph{Counts for the tested core tuples}, every successful +component contains a nonexceptional primary, whose +pure-horizontal alternatives exclude the required intermediate +range for every permitted shift. This is a property of the +persistent mode graph and its parameters. The finite Lemma +therefore applies to the persistent labels and the actual +side incidences and gives, for an absolute \(p_0\in(0,1)\), +\begin{equation} + \begin{gathered} + \sum_sN_s^G\le L_0\min_{i\ne j}M_i^GM_j^G,\qquad + L_0\le P^C(1+\log(\nu/\mu))^C,\\ + \sum_s(N_s^G)^{p_0} + \le P^C(1+\nu+\mu^{-1}+\Lambda)^C(M')^{2p_0}. + \end{gathered} + \label{eq:h-core-counts} +\end{equation} +This source calculation identifies representative labels +across depths and uses same-cell side data; coefficients frozen +in different blocks may be unrelated. + +Finally \eqref{eq:h-top-count} and \eqref{eq:h-side-cap} give +\(\sum_GM_j^G\le P^C\sigma_j^{-2}\) in every role, with the +additional \(D_*\) cap in a core role. H interpolates the two +inequalities in \eqref{eq:h-core-counts}, keeps \(N_s^G\) +inside its power, and sums the amplitude and frequency bins. +Its \emph{Powered estimate for the core terms}, +Equation~\Heq{37} of \cite[Section~9]{H}, gives absolute +constants \(\varepsilon>0\) and \(C_4\), with +\(p=1-\varepsilon>2/3\), and total contribution +\begin{equation} + C_q\eta^\varepsilon P^{C_4}|R|+O(e^{-k}|R|) + \label{eq:h-core-estimate} +\end{equation} +for the terms with at least three core positions on this +attempt's retained cells. The original cell minimum is applied +before subadditivity and the group count remains inside the +power. Numerical errors are estimated absolutely. The +conditional source calculation thus uses the supplied persistent +labels, full core-subtotal masks, role counts, and actual side +Fourier data, with no further cross-depth estimate on an +original input. diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/rows.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/rows.tex new file mode 100644 index 0000000000000000000000000000000000000000..f0920ac31dd03020198e6bc625e5bb0848825635 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/rows.tex @@ -0,0 +1,881 @@ +\section{Bessel rows with inputs varying in depth} +\label{sec:rows} + +We shall apply fixed-input operator estimates to rows that read different +members of one input sequence. The input is allowed bounded total variation +on each of finitely many consecutive depth blocks. The additional structure +of the rows is spatial: their adjoints have constant coefficients in a +persistent catalog of columns with private coordinates. We first prove the +result for such rows, and then treat smooth mean-zero rows with fixed +carriers. + +Throughout this section $R$ is a bounded dyadic interval of positive length. +Unqualified $L^2$ norms use Lebesgue measure, without normalization. +The inner product of a complex Hilbert space is linear in its first +variable. A dyadic interval is taken half open. A finite dyadic partition +of an interval $R$ consists of pairwise disjoint dyadic subintervals whose +union is $R$; a partition refines another if each of its cells is contained +in one cell of the other. Thus cells of a partition may have different +depths. Choose an integer $s_R$ and define the spatial depth of a dyadic +cell $I\subset R$ by +\[ + s(I)=s_R+\log_2\frac{|R|}{|I|}. +\] +For an integer $h\ge s_R$, the full depth-$h$ partition consists of all +cells in $R$ of depth $h$. On a dyadic subroot $R'\subset R$, retain +$s_{R'}=s(R')$, so the spatial depths of its cells are unchanged. + +\subsection{The input size and variation bounds} + +Let +\[ + \mathsf S=\{s_-,s_-+1,\ldots,s_+\} +\] +be a nonempty finite interval of integers. Partition $\mathsf S$ into +$W\ge1$ nonempty consecutive blocks $\mathsf S_1,\ldots,\mathsf S_W$, +in that order. This partition is fixed independently of the spatial +point. For strongly measurable functions $F_s:R\to\mathcal H$ into a +Hilbert space, the block normalization will mean +\begin{equation} + \max_{s\in\mathsf S_b}\|F_s(y)\|_{\mathcal H} + +\sum_{\substack{s,s+1\in\mathsf S_b}} + \|F_{s+1}(y)-F_s(y)\|_{\mathcal H} + \le W^{-1/2} + \quad (1\le b\le W) + \label{eq:row-input-normalization} +\end{equation} +for almost every $y\in R$. Since the data are finite, one may take the +same exceptional null set for all blocks. + +\begin{lemma}[Size and variation of the depth inputs] +\label{lem:depth-input-size} +Assume Equation~\eqref{eq:row-input-normalization}. For every measurable +$E\subset R$ of positive measure, every $s\in\mathsf S$, and +$1\le q\le\infty$, one has +\[ + \|F_s\|_{q,E}\le W^{-1/2}\le1, + \qquad + \|\mathbf1_EF_s\|_2\le W^{-1/2}|E|^{1/2}\le |E|^{1/2}. +\] +Here the first norm uses normalized measure on $E$, with the usual +essential-supremum interpretation when $q=\infty$. + +Let $\mathcal J$ be a finite collection of pairs $[r,t]$ with +$r,t\in\mathsf S$ and $r\le t$. Its edge overlap is +\[ + \mathfrak m= + \max_{s_-\le j0$, +\begin{equation} + |\{y\in R:M(y)>\lambda\}| + \le \min\left\{|R|, + \frac{\mathfrak C_D^2 B^2 E_c}{4\lambda^2}\right\}, + \qquad + \|M\|_{L^1(R)} + \le \mathfrak C_D B\,|R|^{1/2}E_c^{1/2}. + \label{eq:row-maximal} +\end{equation} +\end{lemma} + +\begin{proof} +Restrict to the rounds $p,\ldots,q$ and relabel them starting at one. +Fix the test vector $c$ and $\lambda>0$. Choose representatives of +the finitely many tested adjoints in their asserted cellwise forms, +and remove their exceptional null sets together with the catalog null +set. On a cell $K\in\sigma_n$, the resulting function $g_n$ has a +unique coefficient vector $a_n(K)\in\ell^2(\mathcal V)$, supported +on $\mathcal V(K)$, by Lemma~\ref{lem:row-catalog}. The locality +identity also gives +$A_\nu^*=\mathbf1_{Q_\nu}A_\nu^*$. Thus on $K$ only rows supported +on the unique cell of $\pi_n$ containing $K$ contribute to $a_n(K)$. + +Set +\[ + T=2^{\lceil5\log_2D\rceil},\qquad + \tau=\frac{\lambda}{4D^{3/2}}. +\] +We will select $T$ disjoint subfamilies of whole rows so that, away from +paths with $T$ hits, every partial sum is a prefix of completed bucket +sums plus a remainder of norm at most $\lambda/4$. The upper catalog +bound controls the remainder; the lower bound and the original Bessel +inequality control the measure of paths with $T$ hits. A binary-prefix +estimate then costs only $\log T$. + +We assign whole rows to buckets $1,\ldots,T$ as follows. At the +start of the first round, every cell has current bucket $1$ and +current coefficient vector zero. Inductively these two states at +the start of round $n$ are constant on each cell $Q\in\pi_n$, +and the vector is supported on $\mathcal V(Q)$. +Assign every row with support $Q$ to its current bucket, unless +that bucket is $T+1$, in which case omit the row. On each +$K\in\sigma_n$ contained in $Q$ with current bucket at most $T$, +add $a_n(K)$ to the current coefficient vector. If the norm of the resulting vector is +greater than $\tau$, declare a hit, move to the next bucket, and +set the current vector to zero. Otherwise retain the resulting +vector and the current bucket. Once bucket $T$ has a hit, keep +state $(T+1,0)$ and make no further additions. + +This induction is well defined on whole cells. Indeed, the new +states are constant on cells of $\sigma_n$. All their coefficients +refer to the same global label space: older columns and their +coefficients restrict unchanged to $K$, and their labels remain +available there. Since $\pi_{n+1}$ refines $\sigma_n$, the states +are constant on each support cell at the start of the next round. +In particular a row is assigned one bucket for its entire support, +and is never split by the set on which a hit has occurred. All +hit sets are finite unions of cells and are measurable. + +Figure~\ref{fig:row-chronology} illustrates the refinement that keeps this +assignment on whole rows. +\begin{figure}[ht] +\centering +\input{figures/row-chronology} +\caption{Schematic view of one branch of the spatial partitions in consecutive +rounds. The row on $Q$ has already entered bucket $b$ before a new hit is tested. +Hits may differ on the observation cells. Each support cell in the next +round lies inside one observation cell, so every row in that round still +receives one bucket; the bottom labels give those bucket numbers.} +\label{fig:row-chronology} +\end{figure} + +Let $\mathcal B_i$ be the rows assigned to bucket $i$, and set +\[ + v_i=\sum_{\nu\in\mathcal B_i} A_\nu^*c_\nu,\qquad 1\le i\le T. +\] +For every subset $J\subset\{1,\ldots,T\}$, the adjoint of the +joint operator in the hypothesis gives +\begin{equation} + \left\|\sum_{i\in J}v_i\right\|_2^2 + \le B^2\sum_{\nu\in\bigcup_{i\in J}\mathcal B_i} + \|c_\nu\|_{\mathcal Y_\nu}^2. + \label{eq:row-bucket-bessel} +\end{equation} +Although the sets $\mathcal B_i$ depend on the fixed test vector, +they are actual subsets of the original row indices. Thus +Equation~\eqref{eq:row-bucket-bessel} is the original operator +inequality applied to a test vector with some coordinates set to +zero. In particular +\[ + \sum_{i=1}^T\|v_i\|_2^2\le B^2E_c. +\] + +Let $\Omega_T$ be the set of points whose path has $T$ hits during +these rounds. At any point $y$, a tested adjoint from $\mathcal B_i$ +whose support does not contain $y$ vanishes there. If its support +contains $y$, it was assigned $i$ exactly when the path's current +bucket was $i$; conversely, before stopping, every tested adjoint +whose support contains $y$ enters that current bucket. +Thus, on $\Omega_T$, $v_i(y)$ is exactly the sum accumulated in +bucket $i$ up to and including the round that causes its hit. +The coefficient vector of this sum has norm greater than $\tau$. All its labels remain +available on the final observation cell containing $y$, so +Equation~\eqref{eq:row-catalog-comparison} yields +$\|v_i(y)\|_{\mathcal H}>D^{-1}\tau$ for $1\le i\le T$. +Consequently +\begin{equation} + |\Omega_T| + \le \frac{B^2E_c}{TD^{-2}\tau^2} + =\frac{16D^5B^2E_c}{T\lambda^2}. + \label{eq:row-many-hits} +\end{equation} + +Outside $\Omega_T$, at any intermediate round the partial sum +equals a prefix of the completed bucket sums $v_i(y)$ plus the +current unfinished sum. The latter has coefficient norm at most +$\tau$, and hence has Hilbert norm at most +$D^{3/2}\tau=\lambda/4$. This statement also includes the instant +of a hit, by putting the newly completed bucket in the prefix and +taking unfinished sum zero. Thus, with +\[ + M_T(y)=\max_{0\le j\le T}\left\|\sum_{i=1}^jv_i(y)\right\|_{\mathcal H}, +\] +one has +\begin{equation} + M(y)\le M_T(y)+\lambda/4\qquad(y\notin\Omega_T) + \label{eq:row-prefix-domination} +\end{equation} +outside the fixed null set. + +It remains to estimate $M_T$ using +Equation~\eqref{eq:row-bucket-bessel}. We use the binary-prefix argument +of Rademacher--Menshov type; compare \cite[Lemma~2.7]{KrauseLacey2020}. +For each +$0\le k\le\log_2T$, let $\mathcal D_k$ be the partition of +$\{1,\ldots,T\}$ into consecutive dyadic index intervals of length +$2^k$, and put $v_J=\sum_{i\in J}v_i$. Every prefix is the disjoint +union of at most one interval from each $\mathcal D_k$. The triangle +inequality therefore gives pointwise +\[ + M_T\le + \sum_{k=0}^{\log_2T} + \left(\sum_{J\in\mathcal D_k}\|v_J\|_{\mathcal H}^2\right)^{1/2}. +\] +For a fixed $k$, Equation~\eqref{eq:row-bucket-bessel} and the +disjointness of the index intervals imply +\[ + \int_R\sum_{J\in\mathcal D_k}\|v_J(y)\|_{\mathcal H}^2\,dy + =\sum_{J\in\mathcal D_k}\|v_J\|_2^2 + \le B^2E_c. +\] +Minkowski's inequality in $L^2(R)$ now gives +\begin{equation} + \|M_T\|_2\le (1+\log_2T)B E_c^{1/2}=L_D B E_c^{1/2}. + \label{eq:row-binary-prefix} +\end{equation} + +Equations~\eqref{eq:row-many-hits}--\eqref{eq:row-binary-prefix} +and Chebyshev's inequality show that +\[ + \begin{split} + |\{M>\lambda\}| + &\le |\Omega_T|+|\{M_T>3\lambda/4\}|\\ + &\le \frac{16B^2E_c}{\lambda^2} + \left(\frac{D^5}{T}+\frac{L_D^2}{9}\right) + \le \frac{16(9+L_D^2)B^2E_c}{9\lambda^2}, + \end{split} +\] +because $T\ge D^5$. This is the stated weak estimate, as +$\mathfrak C_D^2/4=16(9+L_D^2)/9$. Integrating the minimum of +this estimate and $|R|$ over $\lambda>0$ proves the $L^1$ estimate: +for $H=(\mathfrak C_D/2)BE_c^{1/2}>0$, +\[ + \int_0^\infty\min\{|R|,H^2/\lambda^2\}\,d\lambda + =2H|R|^{1/2}. +\] +If $H=0$, the weak estimate gives $M=0$ almost everywhere instead. +Finally, put $x=\log_2D\ge1$. Since $L_D\le5x+2$ and +$9+(5x+2)^2\le25(x+1)^2$, the asserted upper bound on +$\mathfrak C_D$ follows. +\end{proof} + +\begin{lemma}[The depth-varying row estimate] +\label{lem:row-depth} +Let the root, catalog, partitions, and rows satisfy every hypothesis +of Lemma~\ref{lem:row-maximal}. Let $(F_s)_{s\in\mathsf S}$ satisfy +Equation~\eqref{eq:row-input-normalization} on $R$. Choose +$s_n\in\mathsf S$ with +$s_1\le s_2\le\cdots\le s_N$, and let every row of round $n$ +read the same input $F_{s_n}$. These read indices label the input +schedule independently of the spatial depths of the support cells. Then +\begin{equation} + \left(\sum_{n=1}^N\sum_{\nu\in\mathcal R_n} + \|A_\nu F_{s_n}\|_{\mathcal Y_\nu}^2\right)^{1/2} + \le \mathfrak C_D B\,|R|^{1/2} + \le \frac{40}{3} B\log_2(2D)\,|R|^{1/2}. + \label{eq:row-depth} +\end{equation} +The constant is independent of the number of rows, depths, and +input blocks. +\end{lemma} + +\begin{proof} +Test the output vector against arbitrary +$c=(c_\nu)_{\nu\in\mathcal R}$ in the finite Hilbert direct sum of +the row spaces. The rounds whose read index lies in a particular +block $\mathsf S_b$ are consecutive, or absent, because the +$s_n$ are nondecreasing. On a nonempty such range $p\le n\le q$, +form $g_n,S_n,M$ as in Lemma~\ref{lem:row-maximal}. Pointwise Abel +summation, with the stipulated inner-product convention, is +\[ + \sum_{n=p}^q\langle F_{s_n},g_n\rangle_{\mathcal H} + =\langle F_{s_q},S_q\rangle_{\mathcal H} + +\sum_{n=p}^{q-1} + \langle F_{s_n}-F_{s_{n+1}},S_n\rangle_{\mathcal H}. +\] +The absolute value of this expression is at most +\[ + \left(\|F_{s_q}\|_{\mathcal H} + +\sum_{n=p}^{q-1}\|F_{s_{n+1}}-F_{s_n}\|_{\mathcal H}\right)M + \le W^{-1/2}M. +\] +For the last inequality, the nondecreasing reads traverse each +edge in the block at most once; repeated reads contribute zero. +Integrating and using Equation~\eqref{eq:row-maximal} bounds this +block's pairing by +\[ + \mathfrak C_D B\,W^{-1/2}|R|^{1/2} + \left(\sum_{n=p}^q\sum_{\nu\in\mathcal R_n} + \|c_\nu\|_{\mathcal Y_\nu}^2\right)^{1/2}. +\] +All sums are finite and each pairing is integrable by the $L^2$ +inequality, so the pointwise identity may be integrated. Write +$E_b$ for the squared coefficient sum in the last display, with +$E_b=0$ for an absent block. The adjoint identity and +Cauchy--Schwarz over the $W$ blocks give +\[ + \begin{split} + \left|\sum_{n=1}^N\sum_{\nu\in\mathcal R_n} + \langle A_\nu F_{s_n},c_\nu\rangle_{\mathcal Y_\nu}\right| + &\le \mathfrak C_D B\,|R|^{1/2} + W^{-1/2}\sum_{b=1}^W E_b^{1/2}\\ + &\le \mathfrak C_D B\,|R|^{1/2} + \left(\sum_{\nu\in\mathcal R}\|c_\nu\|_{\mathcal Y_\nu}^2\right)^{1/2}. + \end{split} +\] +Duality in the finite Hilbert direct sum proves +Equation~\eqref{eq:row-depth}. +\end{proof} + +The realized finite rows, partitions, and catalog may depend on the given +depth inputs, provided they satisfy all preceding hypotheses and are held +fixed while their ordinary operator inequality is tested on arbitrary inputs. + +\subsection{Projection increments} + +Here is a useful consequence for the inherited local projections +of H. It also records why their splitting into support cells +preserves the original Bessel bound. + +\begin{corollary}[Increments of inherited projections] +\label{cor:row-projections} +Use the catalog of Lemma~\ref{lem:row-catalog}. On a finite +interval of integers $h\ge s_R$, let $\mathcal P_h$ be the full +partition of $R$ into cells of depth $h$. On each +$Q\in\mathcal P_h$, let $P_{h,Q}$ be the orthogonal projection in +$L^2(Q;\mathcal H)$ onto constant-coefficient combinations of +a subset $\mathcal M_h(Q)\subset\mathcal V(Q)$. Assume inheritance: +if $Q'\in\mathcal P_{h+1}$ is contained in $Q$, then +$\mathcal M_h(Q)\subset\mathcal M_{h+1}(Q')$. Let $P_h$ be the +direct sum of the $P_{h,Q}$ on $L^2(R;\mathcal H)$. + +Suppose $(F_s)_{s\in\mathsf S}$ satisfies +Equation~\eqref{eq:row-input-normalization}. For any integer $J\ge1$ and +\[ + r_1\le t_1\le r_2\le t_2\le\cdots\le r_J\le t_J +\] +in the projection depth range and nondecreasing +$s_1,\ldots,s_J\in\mathsf S$, one has +\begin{equation} + \sum_{j=1}^J\|(P_{t_j}-P_{r_j})F_{s_j}\|_2^2 + \le \mathfrak C_D^2|R|. + \label{eq:row-disjoint-increments} +\end{equation} +If the intervals can be divided into $\mathfrak m$ such ordered +families whose reads are nondecreasing, the right side is multiplied +by $\mathfrak m$. + +For each $s\in\mathsf S$ with $s\ge s_R$, let $\mathcal U_s$ be any +subcollection of the depth-$s$ cells of $R$. In particular fix an integer +$\ell$ and retain only those $s$ for which $s+\ell$ and $s+\ell+1$ are in +the projection range. Then +\begin{equation} + \sum_s\sum_{I\in\mathcal U_s}|I|\, + \|(P_{s+\ell+1}-P_{s+\ell})F_s\|_{2,I}^2 + \le \mathfrak C_D^2|R|. + \label{eq:row-increment} +\end{equation} +For every integer $L\ge0$, retaining only $s\in\mathsf S$ with $s\ge s_R$ +for which $s$ and $s+L$ are in the projection range, +\begin{equation} + \left(\sum_s\sum_{I\in\mathcal U_s}|I|\, + \|(P_{s+L}-P_s)F_s\|_{2,I}^2\right)^{1/2} + \le L\mathfrak C_D|R|^{1/2} + \le (1+L)\mathfrak C_D|R|^{1/2}. + \label{eq:row-delay} +\end{equation} +\end{corollary} + +\begin{proof} +The range of $P_h$ is contained in the range of $P_{h+1}$: +restrict a constant-coefficient combination on $Q$ to its +children and use inheritance. Thus the $P_h$ are nested +orthogonal projections. For $r\le t$, $P_t-P_r$ is an +orthogonal projection, and the projections corresponding to +intervals with disjoint interiors have pairwise orthogonal ranges. + +For a cell $Q\in\mathcal P_r$, both $P_r$ and $P_t$ commute with +multiplication by $\mathbf1_Q$, since both of their defining +partitions refine $\mathcal P_r$. The row +\[ + A_{j,Q}=(P_{t_j}-P_{r_j})\mathbf1_Q: + L^2(R;\mathcal H)\longrightarrow L^2(R;\mathcal H), + \qquad Q\in\mathcal P_{r_j}, +\] +has both input and output supported in $Q$. Consequently +\[ + \sum_{Q\in\mathcal P_{r_j}}\|A_{j,Q}z\|_2^2 + =\|(P_{t_j}-P_{r_j})z\|_2^2, + \qquad + \sum_{j,Q}\|A_{j,Q}z\|_2^2\le\|z\|_2^2. +\] +This establishes the joint Bessel bound after the splitting. +The adjoint of $A_{j,Q}$ is +$\mathbf1_Q(P_{t_j}-P_{r_j})$. On a cell of +$\mathcal P_{t_j}$ its first term is a constant-coefficient +combination there, while its second term is the restriction of +such a combination on an $r_j$-cell. Both use labels available +on the $t_j$-cell. Therefore the support and observation +partitions in Lemma~\ref{lem:row-depth} are +$\mathcal P_{r_j}$ and $\mathcal P_{t_j}$, respectively. +The stated order of intervals gives the required refinement, +and Lemma~\ref{lem:row-depth} with $B=1$ proves +Equation~\eqref{eq:row-disjoint-increments}. Summing the +estimates for $\mathfrak m$ subfamilies proves the overlap version. +After discarding the zero increments $r=t$, intervals with edge overlap +$\mathfrak m$ can be so divided by the usual greedy coloring in increasing left +endpoint, whenever the reads in that order are nondecreasing: +at a left endpoint fewer than $\mathfrak m$ colors are occupied +by earlier intervals whose right endpoints are still larger. + +For fixed $\ell$, the intervals $[s+\ell,s+\ell+1]$ are +consecutive and their reads $s$ increase. Apply +Equation~\eqref{eq:row-disjoint-increments}. At each $s$ the +cells in $\mathcal U_s$ are disjoint, and +$|I|\|z\|_{2,I}^2=\|\mathbf1_Iz\|_2^2$. Restricting the +outputs to those cells can only decrease their squared sum, +which proves Equation~\eqref{eq:row-increment}. Finally, +\[ + (P_{s+L}-P_s)F_s + =\sum_{\ell=0}^{L-1}(P_{s+\ell+1}-P_{s+\ell})F_s. +\] +Minkowski's inequality in the Hilbert direct sum over $(s,I)$ +and Equation~\eqref{eq:row-increment} prove +Equation~\eqref{eq:row-delay}. For $L=0$ both sides of its +first inequality are zero. +\end{proof} + +\subsection{Smooth rows with fixed carriers} + +The remaining rows have smooth mean-zero test functions on their +support cells, multiplied by a fixed function that may oscillate +arbitrarily. Dyadic differences make their coefficients constant +on finer cells. Private coordinates can then be adjoined at a +controlled cost. + +\begin{lemma}[Smooth mean-zero rows on depth inputs] +\label{lem:smooth-depth} +Let $R$ be a finite dyadic root and let $\mathcal N$ be a finite +set of carrier labels. For each $\nu\in\mathcal N$ fix one +measurable function $m_\nu:R\to\mathbb C$ with +$\|m_\nu\|_\infty\le1$. This function is the same for every +interval using the label $\nu$. Let $\mathcal E$ be a finite +set of pairs $(I,\nu)$, where $I\subset R$ is a dyadic interval +of depth $s(I)$ and $\nu\in\mathcal N$. Thus at most one row +is selected for a given pair $(I,\nu)$. + +For each selected pair, let $\omega_{I,\nu}\in C^1(\overline I)$ +satisfy +\[ + \int_I\omega_{I,\nu}(y)\,dy=0,\qquad + \|\omega_{I,\nu}\|_\infty+ + |I|\|\omega_{I,\nu}'\|_\infty\le A +\] +for one $A\ge0$. Suppose the selected physical rows +\[ + \mathcal W_{I,\nu}f + =|I|^{-1/2}\int_I m_\nu(y)\omega_{I,\nu}(y)f(y)\,dy +\] +satisfy, for one $B_H\ge0$, +\[ + \sum_{(I,\nu)\in\mathcal E}|\mathcal W_{I,\nu}f|^2 + \le B_H^2\|f\|_{L^2(R)}^2\qquad(f\in L^2(R)). +\] +Let the scalar functions $(f_s)_{s\in\mathsf S}$ satisfy +Equation~\eqref{eq:row-input-normalization}, with absolute +values in place of Hilbert norms, and assume that every +$s(I)$ belongs to $\mathsf S$. Put +\[ + d=2+\#\mathcal N,\qquad h_0=\lceil10\log_2(2d)\rceil. +\] +Then +\begin{equation} + \begin{split} + \left(\sum_{(I,\nu)\in\mathcal E} + |\mathcal W_{I,\nu}f_{s(I)}|^2\right)^{1/2} + &\le \mathfrak C_d\left[ + (h_0+1)(B_H+3A) + +2A\sqrt d\,(h_0+3)2^{-h_0}\right]|R|^{1/2}\\ + &\le 760(B_H+A)\bigl(\log_2(2d)\bigr)^2|R|^{1/2}. + \end{split} + \label{eq:smooth-depth} +\end{equation} +The estimate uses only the displayed Bessel inequality for the +selected set $\mathcal E$, not for any larger array of intervals +and carriers. +\end{lemma} + +\begin{proof} +The assertion is immediate if $\mathcal E$ is empty. Extend each +$\omega_{I,\nu}$ by zero off $I$. Let $\mathbb E_h$ be dyadic +conditional expectation on the full depth-$h$ partition of $R$; +we use it for every integer $h\ge s_R$. These are nested orthogonal +projections in $L^2(R)$. For $s=s(I)$, exact mean zero and the zero +extension give $\mathbb E_s\overline{\omega_{I,\nu}}=0$. Define the +difference pieces for $h\ge1$ and the initial sum by +\[ + \begin{aligned} + q_{I,\nu,h} + &=|I|^{-1/2}(\mathbb E_{s+h}-\mathbb E_{s+h-1}) + \overline{\omega_{I,\nu}}\quad(h\ge1),\\ + q_{I,\nu,0} + &=|I|^{-1/2}\mathbb E_{s+h_0}\overline{\omega_{I,\nu}} + =\sum_{h=1}^{h_0}q_{I,\nu,h}. + \end{aligned} +\] +Every $q_{I,\nu,h}$ is supported in $I$, and for $h\ge1$ it +is constant on depth-$(s+h)$ cells and lies in the martingale +difference space +$\operatorname{range}(\mathbb E_{s+h}-\mathbb E_{s+h-1})$. +The initial sum is constant on depth-$(s+h_0)$ cells. +The zero extension causes no difficulty: cells at these depths +do not cross the boundary of $I$. + +On a child of a depth-$(s+h-1)$ cell, the difference between +its average of $\omega_{I,\nu}$ and the parent's average is +at most the oscillation on the parent. The derivative bound +therefore gives +\begin{equation} + \|q_{I,\nu,h}\|_2\le 2A\,2^{-h}\qquad(h\ge1). + \label{eq:smooth-difference-size} +\end{equation} +For a fixed $\nu$ and $h$, these functions are pairwise +orthogonal as $I$ varies through the selected pairs. Indeed, +different support depths give different martingale difference +spaces. At one depth the distinct intervals are disjoint, +and there is at most one selected row for $(I,\nu)$. +Thus their analysis operator has norm at most $2A2^{-h}$. +Since multiplication by the one fixed $m_\nu$ is a contraction, +the physical lag-$h$ rows, whose adjoints are +$q_{I,\nu,h}\overline{m_\nu}$, have joint operator norm at most +\begin{equation} + B_{\mathrm{phys},h}\le 2A\sqrt{\#\mathcal N}\,2^{-h} + \le 2A\sqrt d\,2^{-h}. + \label{eq:smooth-physical-lag} +\end{equation} +This argument uses the fixedness and boundedness of $m_\nu$, +and requires no regularity or frequency separation for it. + +The derivative bound implies uniform convergence on $I$ +of $\mathbb E_{s+H}\omega_{I,\nu}$ to $\omega_{I,\nu}$ as +$H\to\infty$. Consequently +\[ + |I|^{-1/2}\overline{\omega_{I,\nu}} + =q_{I,\nu,0}+\sum_{h>h_0}q_{I,\nu,h}. +\] +The associated physical row series also converges in operator norm, +by Equation~\eqref{eq:smooth-physical-lag}. +Subtracting its tail from the original selected row operator +therefore proves +\[ + B_{\mathrm{phys},0} + \le B_H+2A\sqrt{\#\mathcal N}\,2^{-h_0} + \le B_H+A. +\] +For the last inequality use +$2^{-h_0}\le(2d)^{-10}$ and $d\ge2$. + +Adjoin one private direction for each complete carrier. +Specifically take +$\mathcal H^{\mathrm{aug}}=\mathbb C\oplus\ell^2(\mathcal N)$ +and the root-born columns +\[ + \Phi_\nu(y)=(\overline{m_\nu(y)},\mathbf e_\nu),\qquad y\in R. +\] +Their norms are at most $\sqrt2$, their private constants have +size one, and the number of labels on any path is +$\#\mathcal N\le d$. They satisfy +Equation~\eqref{eq:row-catalog-budget} with $D=d$. +For $h=0$ and for $h>h_0$, define the augmented scalar row by +\[ +\widetilde{\mathcal W}_{I,\nu,h}Z + =\langle Z,q_{I,\nu,h}\Phi_\nu\rangle_{L^2(R;\mathcal H^{\mathrm{aug}})}. +\] +Write $\widetilde{\mathcal W}_h$ for the joint operator consisting +of these rows over $(I,\nu)\in\mathcal E$. +The adjoint of the individual row sends $c$ to +$c\,q_{I,\nu,h}\Phi_\nu$. On a physical input $Z=(f,0)$, +\[ + \widetilde{\mathcal W}_{I,\nu,h}(f,0) + =\langle m_\nu f,q_{I,\nu,h}\rangle_{L^2(R)}, +\] +which is exactly the physical row for the corresponding bump piece. + +For a private input $z=(z_\nu)_{\nu\in\mathcal N}$, the lag-$h$ +analysis map has entries $\langle z_\nu,q_{I,\nu,h}\rangle$. +The fixed-$\nu$ orthogonality already proved, followed by +orthogonality of the private coordinates, gives its norm at +most $2A2^{-h}$. The physical and private input summands are +orthogonal. Applying Cauchy--Schwarz to their two norm bounds +and using Equation~\eqref{eq:smooth-physical-lag} gives +\begin{equation} + \|\widetilde{\mathcal W}_h\| + \le 2A\sqrt{\#\mathcal N+1}\,2^{-h} + \le 2A\sqrt d\,2^{-h}\qquad(h>h_0). + \label{eq:smooth-augmented-lag} +\end{equation} +For the initial sum, the identity +$q_{I,\nu,0}=\sum_{h=1}^{h_0}q_{I,\nu,h}$ and the operator +triangle inequality bound its private part by +$2A\sum_{h=1}^{h_0}2^{-h}\le2A$. Its physical part has the +bound $B_H+A$ above. Thus +\begin{equation} + \|\widetilde{\mathcal W}_0\|\le B_H+3A. + \label{eq:smooth-augmented-main} +\end{equation} + +The two pieces, each indexed by the same selected set $\mathcal E$, have +the following data: +\begin{center} +\small +\begin{tabular}{@{}lccc@{}} +\hline +Piece & Observation depth & \shortstack{Joint augmented\\operator norm} +& \shortstack{Residue modulus\\for support depths}\\ +\hline +Initial sum $q_0$ & $s+h_0$ & $B_H+3A$ & $h_0+1$\\ +Lag $q_h$, $h>h_0$ & $s+h$ & $2A\sqrt d\,2^{-h}$ & $h+1$\\ +\hline +\end{tabular} +\end{center} +For either piece, let $\ell$ denote its observation offset, namely $h_0$ +or $h$. Each row is supported on its whole cell $I$ of depth $s=s(I)$, +and its adjoint coefficient is constant on depth-$(s+\ell)$ cells. +Divide the selected support depths into residue classes modulo $\ell+1$. +Within one class, group all selected rows of one depth $s$ in one round, +using the full depth-$s$ support partition and the full depth-$(s+\ell)$ +observation partition of $R$; unused cells carry no rows. Successive +support depths differ by at least $\ell+1$, so the next support partition +refines the preceding observation partition. All rows in a round read +the same $F_s=(f_s,0)$, and these indices increase in $\mathsf S$. +The catalog has $D=d$, and restricting either displayed array to a +residue class preserves its operator bound. Lemma~\ref{lem:row-depth} +therefore applies in each of the at most $\ell+1$ classes. + +Write $V_0\in\ell^2(\mathcal E)$ for the outputs of the initial sum on +their respective $F_{s(I)}$, and $V_h\in\ell^2(\mathcal E)$ for the +analogous lag-$h$ vector. Extending each residue-class vector by zero +to the full selected row space and using its triangle inequality yields +\[ + \|V_0\|_{\ell^2(\mathcal E)} + \le (h_0+1)\mathfrak C_d(B_H+3A)|R|^{1/2}, + \qquad + \|V_h\|_{\ell^2(\mathcal E)} + \le (h+1)\mathfrak C_d\,2A\sqrt d\,2^{-h}|R|^{1/2}. +\] +These bounds are summable over $h>h_0$. The uniform bump +expansion gives rowwise +$\mathcal W_{I,\nu}f_{s(I)} += (V_0)_{I,\nu}+\sum_{h>h_0}(V_h)_{I,\nu}$. +There are finitely many selected rows; alternatively the last +summable bounds give convergence directly in their Hilbert +sum. Minkowski's inequality and +\[ + \sum_{h>h_0}(h+1)2^{-h}=(h_0+3)2^{-h_0} +\] +prove the first inequality in Equation~\eqref{eq:smooth-depth}. +For its last inequality put $l=\log_2(2d)\ge2$. Then +$h_0+1\le11l$, $h_0+3\le12l$, and +$\sqrt d\,2^{-h_0}\le1$. The bracket there is at most +$57(B_H+A)l$, while +$\mathfrak C_d\le(40/3)l$. Their product is at most +$760(B_H+A)l^2$, as asserted. +\end{proof} diff --git a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/transfer.tex b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/transfer.tex new file mode 100644 index 0000000000000000000000000000000000000000..bc66d8615a535bd78b3e09f44af08729128b0704 --- /dev/null +++ b/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/build/paper/sections/transfer.tex @@ -0,0 +1,506 @@ +\section{Local forms and the orbit oscillation estimate} +\label{sec:transfer} + +We prove Proposition~\ref{prop:orbit-oscillation} from the local estimate +for inputs depending on spatial depth. The proof constructs admissible +kernels for the difference of two smooth scales, averages over translations +of one dyadic lattice, and then transfers the resulting estimate first to +integer sequences and then to finite orbit segments. + +\subsection{Admissible local forms} + +Dyadic lattice intervals are taken half-open. If \(J\) is a lattice +interval and \(I\subseteq J\) is a descendant, its depth in \(J\) is +\(s(I)=\log_2(|J|/|I|)\). A finite rooted tree contains every ancestor +between each of its intervals and its root. + +The kernel class is the one used in \cite[Section~2]{H}. +\begin{definition}[Admissible local kernels] +\label{def:transfer-local-kernels} +Fix \(c_0>0\) and \(C_0\geq1\). For an interval \(I\) of length +\(r=|I|\), a kernel \(w_I\in C_c^\infty(\mathbb R^2;\mathbb C)\) is +admissible if the following hold. +\begin{enumerate} +\item For every \((x,t)\in\operatorname{supp}w_I\) and + \(j\in\{0,1,2,3\}\), + \(x+jt\in I\) and + \(\operatorname{dist}(x+jt,\partial I)\geq c_0r\). +\item For every \(v\in\mathbb R\) and \(j=0,1,2,3\), + \[ + \int_{\mathbb R}w_I(v-jt,t)\,dt=0. + \] +\item For all nonnegative integers \(p,q\), with \(n=p+q\), + \[ + \|\partial_x^p\partial_t^q w_I\|_\infty + \leq r^{-n}\bigl(C_0(n+2)\bigr)^{C_0(n+2)}. + \] +\end{enumerate} +For measurable scalar functions \(z_j:\mathbb R\to\mathbb C\), define +the associated local form, whenever its integral is finite, by +\[ +H_I(z_0,z_1,z_2,z_3) + =r^{-2}\int_{\mathbb R^2}w_I(x,t) + \prod_{j=0}^3z_j(x+jt)\,dx\,dt. +\] +For a Hilbert-valued argument with a distinguished physical scalar +coordinate, the form uses that coordinate. We write +\(\|z\|_{p,I}=(|I|^{-1}\int_I\|z\|^p)^{1/p}\) for the normalized interval +norm, with the usual essential supremum when \(p=\infty\). Norms without +an interval subscript use ordinary Lebesgue measure. +\end{definition} + +For a nonempty consecutive integer block \(B=\{a,\ldots,b\}\) and a +sequence of complex functions \(h_s\) on any common domain, write +\[ + \mathcal V_B(h)(y) + =\max_{s\in B}|h_s(y)| + +\sum_{s=a}^{b-1}|h_{s+1}(y)-h_s(y)|. +\] +The sum is zero for a singleton block. + +\begin{theorem}[Local estimate with variation in depth] +\label{thm:depth-local} +Fix the admissibility parameters \(c_0,C_0\). Let \(J\) be an interval +in any dyadic lattice, let \(\mathcal T\) be a finite rooted tree below +\(J\), and let \(d\) be its largest depth in \(J\). Let +\(\mathcal I\subseteq\mathcal T\), with an admissible kernel on each of +its intervals. For each \(j\in\{0,1,2,3\}\), let +\(f_{j,s}:J\to\C\), \(0\le s\le d\), be measurable, and write +\(f_j=(f_{j,s})_{s=0}^d\). Suppose that +\(\{0,\ldots,d\}\) is partitioned into \(W_j\ge1\) deterministic nonempty +consecutive blocks and that, on every block \(B\), +\[ + \mathcal V_B(f_j)(y)\le W_j^{-1/2} + \quad\text{for almost every }y\in J. +\] +The four partitions may differ. Extend the inputs by zero outside \(J\). +Then +\begin{equation} + \sum_{I\in\mathcal I}|I| + \bigl|H_I(f_{0,s(I)},f_{1,s(I)},f_{2,s(I)},f_{3,s(I)})\bigr| + \le C_{\mathrm{loc}}(c_0,C_0)|J|. + \label{eq:depth-local} +\end{equation} +The constant is independent of the lattice, root, tree, kernels, depth +range, blocks, and their numbers. +\end{theorem} + +Sections~\ref{sec:rows}--\ref{sec:completion} prove +Theorem~\ref{thm:depth-local} from the specified analytic results of H. +Values are prescribed also on depths +without summation intervals; that full schedule is used in the proof. + +The application below varies only slot zero. The other three inputs use +the same functions at every depth and one block each, so their block condition +is simply the bound \(1\). Theorem~\ref{thm:depth-local} applies uniformly +to translations of the lattice. + +\subsection{Kernels and the continuous test} + +For an averaging profile \(\varphi\) and \(a>0\), +Lemma~\ref{lem:transfer-kernel} constructs an admissible kernel whose +normalized spatial integral is \(\varphi_{2a}-\varphi_a\). +The unit-scale difference \(\varphi_2-\varphi\) has zero moments through +order three, hence a compactly supported fourth antiderivative. The +construction multiplies a rescaling of this primitive by a smooth spatial +bump, then differentiates in the four cancellation directions. + +\begin{lemma}[Exact realization of a smooth scale difference] +\label{lem:transfer-kernel} +For every averaging profile \(\varphi\) in +Definition~\ref{def:averaging-profile}, there are a dyadic integer +\(D_\varphi=2^d\), \(d\in\mathbb Z_{\geq0}\), a constant +\(C_{0,\varphi}\geq1\), and +\(\mathcal W_\varphi\in C_c^\infty(\mathbb R^2;\mathbb R)\) such that +the following holds. For every \(a>0\) and every interval \(I\) of +length \(r=D_\varphi a\), with midpoint \(m_I\), the kernel +\[ + w_I(x,t)=\mathcal W_\varphi\left(\frac{x-m_I}{r},\frac{t}{r}\right) +\] +is admissible with \(c_0=1/8\) and \(C_0=C_{0,\varphi}\). Moreover, for +every \(t\in\mathbb R\), +\begin{equation} + \frac1r\int_{\mathbb R}\mathcal W_\varphi(z,t/r)\,dz + =\varphi_{2a}(t)-\varphi_a(t). + \label{eq:transfer-kernel-normalization} +\end{equation} +The parameters are fixed once \(\varphi\) is fixed. +\end{lemma} + +\begin{proof} +Set \(b=\varphi_2-\varphi\). For \(0\leq j\leq3\), +\[ + \int t^jb(t)\,dt=(2^j-1)\int t^j\varphi(t)\,dt=0, +\] +using the zero factor for \(j=0\) and \eqref{eq:profile-moments} for +the other moments. Choose \(B\geq1\) with +\(\operatorname{supp}b\subseteq[-B,B]\), and define +\[ + g(t)=\frac16\int_{-\infty}^t(t-v)^3b(v)\,dv. +\] +It vanishes for \(t<-B\). For \(t>B\), expansion of the cubic and the +four zero moments show that it again vanishes. Differentiating gives +\(g^{(4)}=b\), so \(g\in C_c^\infty(\mathbb R)\) with support in +\([-B,B]\). For \(0\leq j\leq3\), +\[ + g^{(j)}(t)=\frac1{(3-j)!}\int_{-\infty}^t(t-v)^{3-j}b(v)\,dv, + \qquad + \|g^{(j)}\|_\infty + \leq\frac{(2B)^{4-j}}{(4-j)!}\|b\|_\infty. +\] +For \(n\geq4\), \(g^{(n)}=b^{(n-4)}\). The derivative bounds for +\(\varphi\) therefore give bounds of the form +\eqref{eq:profile-derivatives} for all derivatives of \(g\), with a +constant depending on \(\varphi\). + +Let \(\rho\) be the bump from Lemma~\ref{lem:ergodic-bump} and put +\(u(z)=8\rho(8z)\). It has integral \(1\), support in \([-1/8,1/8]\), +and the same type of derivative bounds. Choose a dyadic integer +\(D=D_\varphi>12B\). Define +\[ + F(z,\sigma)=u(z)D^{-3}g(D\sigma),\qquad + L_j=\partial_\sigma-j\partial_z,\qquad + \mathcal W_\varphi=L_0L_1L_2L_3F. +\] +The operators commute, and differentiation does not enlarge support. +Thus, on the support of \(\mathcal W_\varphi\), +\[ + |z+j\sigma|\leq\frac18+\frac{3B}{D}<\frac38 + \qquad(0\leq j\leq3). +\] +For the scaled kernel this places every \(x+jt\) at distance strictly +greater than \(r/8\) from the endpoints of \(I\). + +For \(j=0,1,2,3\), put \(G_j=\prod_{\ell\neq j}L_\ell F\). The chain +rule gives, for every \(v,t\in\mathbb R\), +\[ + w_I(v-jt,t) + =r\,\frac{d}{dt} + G_j\left(\frac{v-jt-m_I}{r},\frac{t}{r}\right). +\] +The differentiated function is compactly supported in \(t\), so the +integral of its derivative is zero. This proves each of the four +cancellation conditions for every \(v\). + +Expansion of the operators gives +\[ + \mathcal W_\varphi(z,\sigma) + =D u(z)g^{(4)}(D\sigma)-6u'(z)g^{(3)}(D\sigma) + +11D^{-1}u''(z)g''(D\sigma)-6D^{-2}u'''(z)g'(D\sigma). +\] +A derivative \(\partial_z^p\partial_\sigma^q\) of total order \(n\) +is a sum of four products of derivatives of \(u,g\) of orders at most +\(n+4\), with factors bounded by a constant times \(D^{n+1}\). +For fixed \(D\), their derivative bounds yield +\[ + \|\partial_z^p\partial_\sigma^q\mathcal W_\varphi\|_\infty + \leq\bigl(C_{0,\varphi}(n+2)\bigr)^{C_{0,\varphi}(n+2)} +\] +for one sufficiently large \(C_{0,\varphi}\) and all \(p,q\). +Rescaling gives the required factor \(r^{-n}\) for derivatives in +\(x,t\). Finally, integration in \(z\) kills all three terms containing +derivatives of \(u\), and hence +\[ + \int_{\mathbb R}\mathcal W_\varphi(z,\sigma)\,dz + =Dg^{(4)}(D\sigma)=Db(D\sigma). +\] +As \(r=Da\), division by \(r\) gives +\(a^{-1}b(t/a)=\varphi_{2a}(t)-\varphi_a(t)\). +\end{proof} + +For the rest of this section fix an averaging profile \(\varphi\), and +write \(D=D_\varphi\), \(b=\varphi_2-\varphi\), and +\[ + \Delta_k(t)=\varphi_{2^{k+1}}(t)-\varphi_{2^k}(t) + =2^{-k}b(2^{-k}t),\qquad k\in\mathbb Z. +\] +Constants with subscript \(\varphi\) may depend on this profile, the +fixed choices in Lemma~\ref{lem:transfer-kernel}, and +\(C_{\mathrm{loc}}(1/8,C_{0,\varphi})\). They are independent of +the scales, blocks, inputs, and probability system. + +\begin{lemma}[A continuous test obtained by translation averaging] +\label{lem:transfer-continuous} +For the fixed profile \(\varphi\) and \(D=D_\varphi\), there is +\(C_\varphi^{\mathrm c}<\infty\) with the following property. +Let \(k_-\leq k_+\) be integers, and partition +\(\{k_-,\ldots,k_+\}\) into \(w\geq1\) deterministic nonempty consecutive +blocks. Let \(K\subset\mathbb R\) be an interval of length \(L>0\). +Let \(f_1,f_2,f_3:\mathbb R\to\mathbb C\) and +\(f_{0,k}:\mathbb R\to\mathbb C\), \(k_-\leq k\leq k_+\), be measurable +and vanish almost everywhere outside \(K\). Suppose +\(|f_j|\leq1\) almost everywhere for \(j=1,2,3\), and +\(\mathcal V_B(f_0)(x)\leq w^{-1/2}\) for almost every \(x\in\mathbb R\) +and every block \(B\). +Then +\begin{equation} + \left|\sum_{k=k_-}^{k_+}\int_{\mathbb R^2} + f_{0,k}(x)\prod_{j=1}^3f_j(x+jt)\Delta_k(t)\,dx\,dt\right| + \leq C_\varphi^{\mathrm c}(L+D2^{k_+}). + \label{eq:transfer-continuous} +\end{equation} +\end{lemma} + +\begin{proof} +Set \(r_k=D2^k\) and \(R=r_{k_+}\). For \(0\leq h0. +\] +Distinct integer pairs give disjoint regions up to null boundaries, +because their zeroth and first centers are unique. Therefore the +continuous planted integral at scale \(k\) is exactly +\[ + I_k=\sum_{m,n\in\mathbb Z} + f_{0,k}(m)\prod_{j=1}^3f_j(m+jn) + \int_{Q_e}\Delta_k(n+\eta)\,d\xi\,d\eta. +\] + +Choose \(B_b\geq1\) with \(\operatorname{supp}b\subseteq[-B_b,B_b]\). +At scale \(a=2^k\geq1\), the mean value inequality gives +\[ + |\Delta_k(n+\eta)-\Delta_k(n)| + \leq2e\|b'\|_\infty a^{-2}\qquad((\xi,\eta)\in Q_e). +\] +If the difference is nonzero, then \(|n|\leq B_ba+2e\), so there are at +most \((2B_b+4e+1)a\) possible integers \(n\). There are at most \(L\) +contributing integers \(m\), and every coefficient product has absolute +value at most \(1\), since +\(\mathcal V_B(f_0)(m)\leq w^{-1/2}\leq1\). Consequently, with +\[ + D_k=\sum_{m,n\in\mathbb Z} + f_{0,k}(m)\prod_{j=1}^3f_j(m+jn)\Delta_k(n), +\] +we have +\[ + |I_k-A_eD_k| + \leq A_e\,2e\|b'\|_\infty(2B_b+4e+1)L\,2^{-k}. +\] +The restriction \(k\geq0\) makes the errors summable, since +\(\sum_{k=k_-}^{k_+}2^{-k}\leq2\). Apply +Lemma~\ref{lem:transfer-continuous} to the planted inputs, sum this error +bound, and divide by \(A_e\). The resulting error is at most a constant +depending only on \(\varphi\) times \(L\), proving +\eqref{eq:transfer-discrete}. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:orbit-oscillation}] +We first obtain the corresponding finite estimate on integer sequences. +Let \(\mathcal K\) be an integer interval of cardinality \(L\geq1\), +let \(\mathcal E\subseteq\mathcal K\) be a nonempty integer interval, and +let \(f_1,f_2,f_3:\mathbb Z\to\mathbb C\) be supported in \(\mathcal K\) +and bounded in absolute value by \(1\). Put +\[ + \mathsf V_k(m)=\sum_{n\in\mathbb Z}\varphi_{2^k}(n) + \prod_{j=1}^3f_j(m+jn), + \qquad k\geq0,\quad m\in\mathbb Z. +\] +Fix deterministic integers \(0\leq k_1<\cdots=Latex,font=\small,x=1cm,y=1cm] +\fill[blue!8] (0,-.23) rectangle (3.3,.23); +\fill[blue!8] (3.0,-2.23) rectangle (6.3,-1.77); +\draw[->] (-.2,0)--(6.6,0) node[right] {$a$}; +\draw[->] (-.2,-2)--(6.6,-2) node[right] {order in $z-i$}; +\foreach \x/\lab in {0/0,3.3/A,6.3/{N-1}} { + \draw (\x,-.08)--(\x,.08); + \node[above=4pt] at (\x,0) {$\lab$}; +} +\draw (0,-2.08)--(0,-1.92); +\node[below=4pt] at (0,-2) {$\Delta$}; +\draw (3,-2.08)--(3,-1.92); +\node[below=4pt] at (3,-2) {$M-1-A$}; +\draw (6.3,-2.08)--(6.3,-1.92); +\node[below=4pt] at (6.3,-2) {$M-1$}; +\draw[->,blue!60!black,thick] (.5,-.3)--(5.8,-1.68); +\draw[->,blue!60!black,thick] (2.8,-.3)--(3.5,-1.68); +\node[anchor=south,font=\footnotesize] at (1.65,.45) {possible surviving indices}; +\node[align=center,anchor=south] at (3.15,.95) {A unit coefficient with $a\leq A$ forces minimum $t$-valuation $\leq A$.}; +\node[anchor=south] at (3.2,1.5) {$c_a(z-i)^a\ \longmapsto\ c_at^a(z-i)^{M-1-a}$}; +\end{tikzpicture} +\caption{Taylor reversal in the expansion argument (schematic, not to scale). For a normalized input with integral Taylor coefficients and a unit among $a\leq A$, every term surviving in the leading reduction of its image has $a\leq A$. The reduction therefore vanishes to order at least $M-1-A$ at $i$. The proof applies this to every vector in a $j$-dimensional subspace, with $A=a_{ij}$.} +\label{fig:taylor-reversal} +\end{figure} diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/main.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..9441fb0b42400dab762cd1858533d58af2868282 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/main.tex @@ -0,0 +1,55 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{xcolor} +\usepackage{booktabs,enumitem} +\usepackage{needspace,etoolbox} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[capitalise,noabbrev]{cleveref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\BeforeBeginEnvironment{theorem}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{lemma}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{proposition}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{corollary}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{remark}{\Needspace{6\baselineskip}} +\renewcommand{\bibsection}{\section*{References}\addcontentsline{toc}{section}{References}} +\newcommand{\Q}{\mathbb Q} +\newcommand{\C}{\mathbb C} +\DeclareMathOperator{\Mat}{Mat} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Span}{span} +\setlist{itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{2em} +\numberwithin{equation}{section} +\title{Polynomial Hitting Lists for Noncommutative Rational Formulas} +\author{OpenAI} +\date{September 24, 2026} +\hypersetup{pdftitle={Polynomial Hitting Lists for Noncommutative Rational Formulas},pdfauthor={OpenAI},pdfsubject={A uniform construction over the rational numbers}} +\begin{document} +\maketitle +\begin{abstract} +We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over $\Q$ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded. +\end{abstract} +\input{sections/introduction} +\input{sections/expansion} +\input{sections/division} +\input{sections/encoding} +\input{sections/generator} +\input{sections/rank} +\bibliographystyle{plainnat} +\bibliography{references} +\end{document} diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/references.bib b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..bc95ad1c221f1172f7248bf37fc0bc2453ba9b98 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/references.bib @@ -0,0 +1,234 @@ +@article{HW15, + author = {Hrube{\v{s}}, Pavel and Wigderson, Avi}, + title = {Non-Commutative Arithmetic Circuits with Division}, + journal = {Theory of Computing}, + volume = {11}, + number = {14}, + pages = {357--393}, + year = {2015}, + doi = {10.4086/toc.2015.v011a014}, + url = {https://theoryofcomputing.org/articles/v011a014/} +} + +@inproceedings{GGOW16, + author = {Garg, Ankit and Gurvits, Leonid and Oliveira, Rafael and Wigderson, Avi}, + title = {A Deterministic Polynomial Time Algorithm for Non-commutative Rational Identity Testing}, + booktitle = {Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS)}, + pages = {109--117}, + publisher = {IEEE Computer Society}, + year = {2016}, + doi = {10.1109/FOCS.2016.95}, + url = {https://www.math.ias.edu/~avi/PUBLICATIONS/GargGuOlWi_FOCS_2016.pdf} +} + +@article{GGOW20, + author = {Garg, Ankit and Gurvits, Leonid and Oliveira, Rafael and Wigderson, Avi}, + title = {Operator Scaling: Theory and Applications}, + journal = {Foundations of Computational Mathematics}, + volume = {20}, + pages = {223--290}, + year = {2020}, + doi = {10.1007/s10208-019-09417-z}, + url = {https://www.math.ias.edu/~avi/PUBLICATIONS/GargGOW20.pdf} +} + +@article{IQS18, + author = {Ivanyos, G{\'a}bor and Qiao, Youming and Subrahmanyam, K. V.}, + title = {Constructive Non-commutative Rank Computation Is in Deterministic Polynomial Time}, + journal = {Computational Complexity}, + volume = {27}, + number = {4}, + pages = {561--593}, + year = {2018}, + doi = {10.1007/s00037-018-0165-7}, + url = {https://arxiv.org/abs/1512.03531v6}, + note = {Cited definition and section numbers refer to arXiv:1512.03531v6} +} + +@misc{ACM25, + author = {Arvind, V. and Chatterjee, Abhranil and Mukhopadhyay, Partha}, + title = {Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time}, + year = {2025}, + eprint = {2309.15647}, + archivePrefix = {arXiv}, + primaryClass = {cs.CC}, + url = {https://arxiv.org/abs/2309.15647v4}, + note = {Version 4, 11 July 2025. Preliminary version in STOC 2024} +} + +@article{ACM26, + author = {Arvind, V. and Chatterjee, Abhranil and Mukhopadhyay, Partha}, + title = {Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time}, + journal = {SIAM Journal on Computing}, + year = {2026}, + doi = {10.1137/24M1689508}, + url = {https://epubs.siam.org/doi/10.1137/24M1689508}, + note = {Published online 1 June 2026} +} + +@article{DM17, + author = {Derksen, Harm and Makam, Visu}, + title = {Polynomial Degree Bounds for Matrix Semi-Invariants}, + journal = {Advances in Mathematics}, + volume = {310}, + pages = {44--63}, + year = {2017}, + doi = {10.1016/j.aim.2017.01.018}, + url = {https://sites.lsa.umich.edu/hderksen/wp-content/uploads/sites/614/2018/05/A.I.a.54.pdf} +} + +@inproceedings{FS13, + author = {Forbes, Michael A. and Shpilka, Amir}, + title = {Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs}, + booktitle = {Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS)}, + pages = {243--252}, + publisher = {IEEE Computer Society}, + year = {2013}, + doi = {10.1109/FOCS.2013.34}, + url = {https://arxiv.org/abs/1209.2408v2} +} + +@article{GK16, + author = {Guruswami, Venkatesan and Kopparty, Swastik}, + title = {Explicit Subspace Designs}, + journal = {Combinatorica}, + volume = {36}, + pages = {161--185}, + year = {2016}, + doi = {10.1007/s00493-014-3169-1}, + url = {https://eccc.weizmann.ac.il/report/2013/060/}, + note = {Cited Section 5.1 refers to ECCC TR13-060, 10 April 2013} +} + +@article{Amitsur66, + author = {Amitsur, S. A.}, + title = {Rational Identities and Applications to Algebra and Geometry}, + journal = {Journal of Algebra}, + volume = {3}, + pages = {304--359}, + year = {1966}, + doi = {10.1016/0021-8693(66)90004-4} +} + +@book{Cohn06, + author = {Cohn, P. M.}, + title = {Free Ideal Rings and Localization in General Rings}, + publisher = {Cambridge University Press}, + year = {2006} +} + +@article{CR94, + author = {Cohn, P. M. and Reutenauer, C.}, + title = {A Normal Form in Free Fields}, + journal = {Canadian Journal of Mathematics}, + volume = {46}, + pages = {517--531}, + year = {1994} +} + +@misc{KVV10, + author = {Kaliuzhnyi-Verbovetskyi, Dmitry S. and Vinnikov, Victor}, + title = {Noncommutative Rational Functions, Their Difference-Differential Calculus and Realizations}, + year = {2010}, + eprint = {1003.0695}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1003.0695} +} + +@inproceedings{Nisan91, + author = {Nisan, Noam}, + title = {Lower Bounds for Non-Commutative Computation (Extended Abstract)}, + booktitle = {Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC)}, + pages = {410--418}, + year = {1991}, + doi = {10.1145/103418.103462} +} + +@article{RS05, + author = {Raz, Ran and Shpilka, Amir}, + title = {Deterministic Polynomial Identity Testing in Non-Commutative Models}, + journal = {Computational Complexity}, + volume = {14}, + number = {1}, + pages = {1--19}, + year = {2005}, + doi = {10.1007/s00037-005-0188-8} +} + +@inproceedings{BW05, + author = {Bogdanov, Andrej and Wee, Hoeteck}, + title = {More on Noncommutative Polynomial Identity Testing}, + booktitle = {Proceedings of the 20th Annual IEEE Conference on Computational Complexity (CCC)}, + pages = {92--99}, + year = {2005}, + doi = {10.1109/CCC.2005.13}, + url = {https://www.di.ens.fr/~wee/pubs/ncpit-ccc05.pdf} +} + +@inproceedings{ACM22, + author = {Arvind, V. and Chatterjee, Abhranil and Mukhopadhyay, Partha}, + title = {Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time}, + booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {245}, + pages = {23:1--23:22}, + year = {2022}, + doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.23} +} + +@article{BD10, + author = {Bostan, Alin and Dumas, Philippe}, + title = {Wronskians and Linear Independence}, + journal = {The American Mathematical Monthly}, + volume = {117}, + number = {8}, + pages = {722--727}, + year = {2010}, + doi = {10.4169/000298910X515785}, + url = {https://mathexp.eu/dumas/publications/Bostan_Dumas_2010_WLI.pdf}, + note = {Cited Lemmas 1--3 refer to the linked author version} +} + +@inproceedings{FG15, + author = {Forbes, Michael A. and Guruswami, Venkatesan}, + title = {Dimension Expanders via Rank Condensers}, + booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, + series = {Leibniz International Proceedings in Informatics}, + volume = {40}, + pages = {800--814}, + year = {2015}, + doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.800} +} + +@article{FR04, + author = {Fortin, Marc and Reutenauer, Christophe}, + title = {Commutative/Noncommutative Rank of Linear Matrices and Subspaces of Matrices of Low Rank}, + journal = {S\'eminaire Lotharingien de Combinatoire}, + volume = {52}, + pages = {Article B52f, 12 pp.}, + year = {2004}, + url = {https://www.mat.univie.ac.at/~slc/s/s52reut.pdf} +} + +@article{IQS17, + author = {Ivanyos, G{\'a}bor and Qiao, Youming and Subrahmanyam, K. V.}, + title = {Non-commutative {Edmonds'} Problem and Matrix Semi-Invariants}, + journal = {Computational Complexity}, + volume = {26}, + number = {3}, + pages = {717--763}, + year = {2017}, + doi = {10.1007/s00037-016-0143-x}, + url = {https://arxiv.org/abs/1508.00690v2}, + note = {Cited section numbers refer to arXiv:1508.00690v2} +} + +@inproceedings{GRX18, + author = {Guruswami, Venkatesan and Resch, Nicolas and Xing, Chaoping}, + title = {Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs}, + booktitle = {33rd Computational Complexity Conference (CCC 2018)}, + year = {2018}, + note = {Article 4}, + doi = {10.4230/LIPIcs.CCC.2018.4}, + url = {https://doi.org/10.4230/LIPIcs.CCC.2018.4} +} diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/division.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/division.tex new file mode 100644 index 0000000000000000000000000000000000000000..16974b03d7f35e6fec8b7445f5334cc2780dc7f1 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/division.tex @@ -0,0 +1,288 @@ +\section{Deformation inside a division algebra}\label{sec:division} + +The rank bound from Proposition~\ref{prop:rectangular} is close to full +rank. We now deform the test matrices so that every possible rank is a +multiple of their block size. The same deformation will also turn a +defined nonzero value of a rational formula into an invertible value over +an extension field. +The construction is a form of the generic cyclic, or symbol, +division algebra. Explicit split representations and block-rank +divisibility appear in the constructive regularity methods of +Ivanyos, Qiao, and Subrahmanyam~\citep[Sections~4.3 and~5.2 of the +preprint]{IQS17}; see also their later regularity +algorithm~\citep[Section~4]{IQS18}. +Division-algebra witnesses and invertible hits are also used in +rational-formula black-box testing by Arvind, Chatterjee, and +Mukhopadhyay~\citep{ACM22,ACM25}. +Here the needed deformation and rank-divisibility statement have +a direct proof using two explicit matrices. This matrix embedding +is distinct from the rational multiplication representation in +Section~\ref{sec:generator}. + +\begin{lemma}[A matrix lift into a division algebra]\label{lem:division-lift} +Let $d\geq 1$, let $K$ be a field of characteristic zero, and let +$\omega\in K$ be a primitive $d$th root of unity. On the standard basis +$e_0,\ldots,e_{d-1}$ of $K^d$, with subscripts taken modulo $d$, set +\[ + De_j=\omega^j e_j, + \qquad Se_j=e_{j+1}. +\] +For algebraically independent indeterminates $p,r$ over $K$, there is a +division algebra +\[ + \mathcal D\subseteq\Mat_d\bigl(K(p,r)\bigr) +\] +containing, for every $R\in\Mat_d(K)$, the matrix +\begin{equation}\label{eq:division-lift} + R^*=\sum_{a,b=0}^{d-1}c_{ab}p^a r^bD^aS^b, + \qquad + c_{ab}=\frac1d\sum_{j=0}^{d-1}\omega^{-aj}R_{j,j-b}. +\end{equation} +The matrix $R^*$ specializes to $R$ at $p=r=1$. +Moreover, $\mathcal D$ has dimension $d^2$ over the central subfield +$K(p^d,r^d)$, and every matrix with blocks in $\mathcal D$ has rank over +$K(p,r)$ divisible by $d$. +\end{lemma} + +\begin{proof} +For each $b$, the only potentially nonzero entries of $D^aS^b$ have the +form +\[ + (D^aS^b)_{j,j-b}=\omega^{aj}. +\] +The identity +\[ + \sum_{j=0}^{d-1}\omega^{(a-a')j} + =\begin{cases}d,&a=a',\\0,&a\ne a',\end{cases} + \qquad 0\leq a,a'(q-1)M. +\] +Choose a nonzero minor of order greater than $(q-1)M$. +It remains nonzero over $\Q(\omega)(t)$. The corresponding minor +of the lifted matrix specializes to it at $p=r=1$, so the lifted +minor is not identically zero. The lifted matrix consequently has +rank greater than $(q-1)M$ over $\Q(\omega)(t,p,r)$. + +Its $M\times M$ blocks belong to the division algebra of +Lemma~\ref{lem:division-lift}: each is a rational linear combination +of the $R_i^*$. Its rank is thus a multiple of $M$. As the whole +matrix has order $qM$, its rank must be $qM$, proving +\eqref{eq:deformed-pencil-determinant}. + +The entries of $R_i(t)$ lie in $\mathbb Z[t]$ and have degree at +most $N-1$. The Fourier coefficients in +\eqref{eq:division-lift} introduce only the constant denominator +$M$, while the factors $p^ar^b$ have total degree at most $2(M-1)$. +Every lifted entry is therefore a polynomial of total degree at +most $N-1+2(M-1)<3M$. A determinant of order $qM$ has total degree +at most $3qM^2\leq3BM^2$. + +For the last assertion, apply the result to the order-one constant +pencil $L=1$, whose coefficient $A_0$ is $1$ and whose other +coefficients vanish. Its determinant in +\eqref{eq:deformed-pencil-determinant} is exactly $\det R_0^*$. +\end{proof} + +At any specialization at which $R_0^*$ is invertible, set +$X_i=(R_0^*)^{-1}R_i^*$ for $1\leq i\leq n$. The identity +\begin{equation}\label{eq:division-left-normalization} + A_0\otimes I_M+\sum_{i=1}^n A_i\otimes X_i + = + \bigl(I_q\otimes(R_0^*)^{-1}\bigr) + \left(\sum_{i=0}^n A_i\otimes R_i^*\right) +\end{equation} +shows that simultaneous nonvanishing of the determinant constraints +in Proposition~\ref{prop:pencil-deformation}, including the +constant-pencil constraint, makes the corresponding affine pencils +invertible on this normalized tuple. The order of multiplication +in \eqref{eq:division-left-normalization} is part of the identity. + +\begin{lemma}[An invertible witness for a nonzero formula] +\label{lem:invertible-witness} +Every admissible nonzero rational formula over $\Q$ has a defined +evaluation at matrices over an extension field of $\Q$ for which +its value is invertible. +\end{lemma} + +\begin{proof} +By the definition of nonzero, there is a tuple +$W=(W_1,\ldots,W_n)$ of rational matrices of some common dimension +$d\geq1$ at which the given formula is defined and has nonzero +value. Choose a primitive $d$th root $\omega$, put $K=\Q(\omega)$, +and lift each $W_i$ to $W_i^*$ by +Lemma~\ref{lem:division-lift}. + +We first verify the domain of the actual formula tree independently +of the division-algebra argument. Inside $K(p,r)$ consider the +commutative ring +\[ + \mathcal O= + \left\{\frac{f}{g}: f,g\in K[p,r],\ g(1,1)\ne0\right\}. +\] +Evaluation at $(1,1)$ is a ring homomorphism +$\varepsilon:\mathcal O\to K$. An element of $\mathcal O$ whose +image under $\varepsilon$ is nonzero is a unit in $\mathcal O$: +if it is written $f/g$, then $f(1,1)\ne0$ and its inverse is $g/f$. + +We claim, by induction up the original tree, that each subtree +evaluated at the lifted inputs gives a matrix over $\mathcal O$ +whose image under $\varepsilon$ is its value at $W$. +This holds for leaves by the specialization property of the lift, +and is preserved by addition and ordered multiplication. At an +inverse gate, let $G\in\Mat_d(\mathcal O)$ be the already +constructed value of the child. Since the original evaluation is +defined, $\varepsilon(G)$ is invertible. Consequently +\[ + \varepsilon(\det G)=\det\varepsilon(G)\ne0, +\] +so $\det G$ is a unit of $\mathcal O$. The formula +\[ + G^{-1}=\frac{\operatorname{adj}(G)}{\det G} +\] +then constructs its inverse in $\Mat_d(\mathcal O)$ and gives the +correct specialized inverse. This proves the claim at every +inverse gate, regardless of its nesting depth. + +The lifted formula is therefore defined over $K(p,r)$. Its final +matrix is nonzero, because one of its entries specializes to a +nonzero entry of the original final value. + +Now use the division algebra $\mathcal D$ from +Lemma~\ref{lem:division-lift}. Every lifted input and every +rational scalar constant belongs to $\mathcal D$. A second +induction shows that every intermediate value belongs to +$\mathcal D$: sums and products stay in it, and an operand at an +inverse gate is already known to be invertible as an ambient +matrix, hence is nonzero in $\mathcal D$. Its inverse in +$\mathcal D$ equals its ordinary matrix inverse by uniqueness. +The final nonzero value is thus a nonzero element of a division +algebra, and is invertible as a matrix over $K(p,r)$. +\end{proof} + +Only those entries proved to lie in $\mathcal O$ are specialized in +this argument. No specialization map on the whole division algebra +is needed. In particular, the inverse of the final value may have +a pole at $(1,1)$ when the original nonzero value is singular; its +invertibility is asserted only at the lifted tuple over the +extension field. If $d=1$, the lift leaves every input scalar +unchanged, and a nonzero scalar final value is already invertible. +The witness dimension $d$ in Lemma~\ref{lem:invertible-witness} is +purely existential and need not be bounded. The matrices used in +the construction instead have the predetermined block size $M$. diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/encoding.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/encoding.tex new file mode 100644 index 0000000000000000000000000000000000000000..0e6e35794985c3e96ae1ce27e6ec322653fa8df1 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/encoding.tex @@ -0,0 +1,240 @@ +\section{Encoding a formula and its evaluation domain} +\label{sec:encoding} + +We now turn the conditions imposed by a rational formula into finitely many +affine-pencil invertibility conditions. We keep the original formula tree: +an inverse gate must have an invertible operand even when its contribution +later cancels or is multiplied by zero. +Affine-matrix representations of free-field elements appear in +Cohn and Reutenauer~\citep[Section~3.1]{CR94}. +Linear realizations by block inverses and tests for the original domain +appear in Hrube\v{s} and Wigderson~\citep[Theorem~2.6, Section~6, +and Proposition~7.1]{HW15}. Their domain criterion retains pencils +for all gate subformulas. We give the recursions and the domain +argument in full for the precise tree-size convention used here, +and show why the inverse-node pencils together with one root +augmentation suffice. The argument concerns the given tree and +requires no domain-preserving balancing transformation. + +For a subtree $h$, let $|h|$ denote its number of vertices, let $\lambda(h)$ +denote its number of leaves, and let $\iota(h)$ denote its number of inverse +gates. Each occurrence of a subtree is treated separately. A pencil of +order $q$ has the form +\[ + L(x)=A_0+\sum_{i=1}^n A_i x_i, + \qquad A_0,\ldots,A_n\in\Mat_q(\Q). +\] +At a tuple $X=(X_1,\ldots,X_n)$ of matrices of order $d$ over a field +extension of $\Q$, its evaluation is +\[ + L(X)=A_0\otimes I_d+\sum_{i=1}^n A_i\otimes X_i. +\] +When a constant row $u$ or column $v$ is used with such an evaluation, we +write $u$ and $v$ for $u\otimes I_d$ and $v\otimes I_d$, respectively. +Thus the expression $uL(X)^{-1}v$ is a matrix of order $d$. + +\begin{lemma}[Affine realization of a subtree] +\label{lem:formula-realization} +For each subtree $h$ there are an affine pencil $L_h$ of order $q_h$, a +constant rational row $u_h$ of length $q_h$, and a constant rational column +$v_h$ of length $q_h$, such that +\begin{equation} + q_h=2\lambda(h)+\iota(h)=|h|+1. + \label{eq:realization-order} +\end{equation} +At every matrix tuple over every field extension of $\Q$ where the +original subtree $h$ is defined, $L_h$ is invertible and +\begin{equation} + h=u_hL_h^{-1}v_h. + \label{eq:realization-invariant} +\end{equation} +\end{lemma} + +\begin{proof} +Here are the constructions, followed by verification of the invariant. +All zero blocks have the sizes prescribed by the adjacent blocks. + +\emph{Leaf.} If $h$ is a variable or a rational constant, set +\[ + L_h=\begin{pmatrix}1&-h\\0&1\end{pmatrix}, + \qquad u_h=\begin{pmatrix}1&0\end{pmatrix}, + \qquad v_h=\begin{pmatrix}0\\1\end{pmatrix}. +\] + +\emph{Addition.} For $h=f+g$, set +\[ + L_h=\begin{pmatrix}L_f&0\\0&L_g\end{pmatrix}, + \qquad u_h=\begin{pmatrix}u_f&u_g\end{pmatrix}, + \qquad v_h=\begin{pmatrix}v_f\\v_g\end{pmatrix}. +\] + +\emph{Ordered multiplication.} For $h=fg$, set +\[ + L_h=\begin{pmatrix}L_f&-v_fu_g\\0&L_g\end{pmatrix}, + \qquad u_h=\begin{pmatrix}u_f&0\end{pmatrix}, + \qquad v_h=\begin{pmatrix}0\\v_g\end{pmatrix}. +\] + +\emph{Inversion.} For $h=g^{-1}$, set +\[ + L_h=\begin{pmatrix}L_g&v_g\\u_g&0\end{pmatrix}, + \qquad u_h=\begin{pmatrix}0&-1\end{pmatrix}, + \qquad v_h=\begin{pmatrix}0\\1\end{pmatrix}, +\] +where the zero part of each selector has length $q_g$. Equivalently, +$u_h=-e_{q_g+1}^{\mathsf T}$ and $v_h=e_{q_g+1}$. + +The rows and columns just specified are constant. In particular, +$v_fu_g$ is a constant matrix, so every displayed pencil is affine in +the original variables. No multiplication of variable entries is +introduced into a pencil. + +Fix an evaluation tuple and prove the invariant by induction on the +subtree. A leaf pencil has inverse +$\left(\begin{smallmatrix}I_d&h\\0&I_d\end{smallmatrix}\right)$, and its selectors +give $h$. At an addition gate, definedness of the gate implies +definedness of both children. Their pencils are invertible by induction, +and block-diagonal inversion gives the sum of the two represented values. +At a multiplication gate, the same induction gives +\[ + L_h^{-1}= + \begin{pmatrix} + L_f^{-1}&L_f^{-1}v_fu_gL_g^{-1}\\ + 0&L_g^{-1} + \end{pmatrix}. +\] +The selected block is consequently +\[ + u_fL_f^{-1}v_fu_gL_g^{-1}v_g + =(u_fL_f^{-1}v_f)(u_gL_g^{-1}v_g)=fg, +\] +with the required order of multiplication. + +At an inverse gate $h=g^{-1}$, definedness means that $g$ is defined and +its value is invertible. By induction $L_g$ is invertible and +$u_gL_g^{-1}v_g=g$. The block factorization +\begin{equation} + \begin{pmatrix}L_g&v_g\\u_g&0\end{pmatrix} + = + \begin{pmatrix}I&0\\u_gL_g^{-1}&I\end{pmatrix} + \begin{pmatrix}L_g&0\\0&-g\end{pmatrix} + \begin{pmatrix}I&L_g^{-1}v_g\\0&I\end{pmatrix} + \label{eq:inverse-schur-factorization} +\end{equation} +shows that $L_h$ is invertible. The lower-right block of its inverse is +$(-g)^{-1}=-g^{-1}$, so the negative row selector gives precisely +$g^{-1}$. This proves the invariant in all four cases. + +The orders are $2$ at leaves, add at binary gates, and increase by one +at inverse gates. Hence $q_h=2\lambda(h)+\iota(h)$. A rooted tree whose +internal vertices have one or two children has $\lambda(h)-1$ binary +vertices: counting edges both by vertices and by children gives this +identity. Thus +\[ + |h|=\lambda(h)+(\lambda(h)-1)+\iota(h) + =2\lambda(h)+\iota(h)-1, +\] +which proves~\eqref{eq:realization-order}. +\end{proof} + +For the root formula $\Phi$, define its \emph{root augmentation} by +\begin{equation} + J_\Phi= + \begin{pmatrix}L_\Phi&v_\Phi\\u_\Phi&0\end{pmatrix}. + \label{eq:root-augmentation} +\end{equation} +This has the same block form as an inverse-node pencil, but no inverse +gate is being added to the formula or counted in its size. + +\begin{proposition}[Pencil certificates for the original formula tree] +\label{prop:formula-certificates} +Let $\Phi$ be a formula of size at most $s$, and put $B=2s+1$. Form the +indexed collection $\mathcal C_\Phi$ consisting of $L_h$ for every +inverse-node occurrence $h$ of $\Phi$, together with $J_\Phi$. +Then: +\begin{enumerate} + \item The collection contains at most $s+1$ affine pencils over $\Q$, + each of positive order at most $B$. + \item At every tuple of matrices of a common positive order over a + field extension of $\Q$, all pencils in $\mathcal C_\Phi$ are + invertible if and only if the original formula tree $\Phi$ is + defined there and its value is invertible. + \item If $\Phi$ is admissible and nonzero, there is a single tuple over + some field extension of $\Q$ at which every pencil in + $\mathcal C_\Phi$ is invertible. +\end{enumerate} +Adding the order-one constant pencil $1$ for normalization gives at +most $s+2$ pencils, still of order at most $B$, with the same simultaneous +feasibility property in the third assertion. +\end{proposition} + +\begin{proof} +There are at most $s$ inverse-node occurrences. By +Lemma~\ref{lem:formula-realization}, each corresponding pencil has order +$|h|+1\le s+1\le B$. The augmentation has order +$|\Phi|+2\le s+2\le2s+1=B$, where the last inequality uses $s\ge1$. +This proves the first assertion. The extra constant pencil has order +one and evaluates to an identity matrix at every tuple. + +For one implication in the second assertion, assume that +$\Phi$ is defined and its value is invertible. Each subtree is then +defined, so Lemma~\ref{lem:formula-realization} makes every inverse-node +pencil invertible. The same lemma makes $L_\Phi$ invertible and gives +$u_\Phi L_\Phi^{-1}v_\Phi=\Phi$. Applying +factorization~\eqref{eq:inverse-schur-factorization} at the root shows +that $J_\Phi$ is invertible, because its Schur complement is $-\Phi$. + +Conversely, suppose all pencils in $\mathcal C_\Phi$ are invertible. +First establish that the original tree is defined, proceeding from +the leaves toward the root. Leaves are always defined. At an addition +or multiplication gate, both children have already been shown defined, +so the gate is defined. At an inverse gate $h=g^{-1}$, all gates in +the child subtree have already been handled; in particular $g$ is +defined. Lemma~\ref{lem:formula-realization} therefore makes $L_g$ +invertible and represents $g$ by $u_gL_g^{-1}v_g$. The pencil $L_h$ is +one of the assumed invertible pencils. The two outer factors in +\eqref{eq:inverse-schur-factorization} are invertible, so the middle +factor is invertible as well. It follows that $g$ is invertible, which +is exactly the condition needed for this inverse gate to be defined. + +This induction proves definedness of the entire original tree. It +also explains why no separate constraint on the pencil of an ordinary +subtree is needed: whenever that pencil is used, its invertibility +already follows from the established definedness of its subtree. +Now $L_\Phi$ is invertible by the realization lemma. The assumed +invertibility of $J_\Phi$, whose Schur complement is $-\Phi$, forces +the final value $\Phi$ to be invertible. This proves the reverse +implication without simplifying the formula or enlarging its domain. + +Finally, if $\Phi$ is admissible and nonzero, +Lemma~\ref{lem:invertible-witness} supplies an evaluation over a field +extension of $\Q$ at which the original tree is defined and its value +is invertible. The second assertion makes every pencil in +$\mathcal C_\Phi$ invertible at this one evaluation. The constant +pencil $1$ is invertible there as well. This proves the third assertion +and the normalization statement. +\end{proof} + +\begin{remark}[Why every inverse occurrence is retained] +Consider the scalar expression +\[ + a=x^{-1},\qquad b=a^{-1},\qquad \Phi=b+1. +\] +It is undefined at $x=0$, although its simplified rational function is +$x+1$. For the pencils above, direct computation gives +\[ + \det L_a=-x,\qquad \det L_b=1,\qquad \det J_\Phi=-(x+1). +\] +Thus the outer inverse pencil and the root augmentation alone would +accept $x=0$. The retained inner inverse pencil $L_a$ rejects it. +Likewise, the formula $1+0\cdot x^{-1}$ retains the requirement that +$x$ be invertible. The certificate collection is indexed by gates of +the given tree, including gates in branches whose later contributions +cancel. +\end{remark} + +The witnesses and pencils in this section are used to prove the hitting +property; the generator will receive only $n,s$ and will not construct +them from an unknown formula. The order and count bounds depend only +on the number of vertices. They impose no restriction on inverse +nesting or on the bit lengths of the rational leaf constants. diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/expansion.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/expansion.tex new file mode 100644 index 0000000000000000000000000000000000000000..0d6f348eb6faa9b51ce823bb901e821d5f015669 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/expansion.tex @@ -0,0 +1,371 @@ +\section{Rectangular tests from subspace expansion} +\label{sec:expansion} + +Fix the variable and formula-size bounds $n,s\geq1$. We construct +rectangular matrices $T_0,\ldots,T_n$ such that +$\sum_{i=0}^n A_i\otimes T_i$ is injective whenever +$A_0+\sum_{i=1}^n A_i x_i$ is a feasible affine pencil of order +at most $2s+1$. The construction uses an indeterminate $t$; no +specialization of a formal Laurent series will be needed. Set +\begin{equation}\label{eq:parameters} +\begin{gathered} +w=n+1,\qquad B=2s+1,\qquad \Delta=2w^3B,\\ +M=\min\{2^a:a\in\mathbb Z_{\geq0},\ 2^a>B\Delta\}, +\qquad N=M-\Delta. +\end{gathered} +\end{equation} +The bound $B$ will accommodate all certificate pencils in +Proposition~\ref{prop:formula-certificates}. The padding width $\Delta$ +provides the expansion margin, while $M>B\Delta$ makes the eventual +rank deficit smaller than one block. These choices give +$1\leq N(q-1)M. + \label{eq:padded-rank} +\end{equation} +\end{proposition} + +We prove the proposition by showing that the maps $T_i$ jointly expand +every nonzero subspace of dimension at most $B$. Two elementary facts will +control the vanishing orders of their leading coefficients. +The ordinary-Wronskian multiplicity method used in the subspace designs +of Guruswami and Kopparty~\citep[Section~5.1 of the preprint]{GK16} +is a predecessor of this argument. Their construction bounds +intersections with multiplicity subspaces; here Taylor reversal +relates the vanishing sequences of the input and image spaces. +Forbes and Guruswami~\citep{FG15} explain broader connections among +subspace designs, rank condensers, and dimension expansion. +The lossless and unbalanced dimension expanders of Guruswami, Resch, +and Xing~\citep{GRX18} provided a comparison for the required +small-subspace expansion parameters during development. Their +linearized-polynomial construction is not used here. +We prove the order estimate and the rectangular expansion statement +directly for the maps above. + +\subsection{Vanishing orders and integral reduction} + +Let $V\subseteq\mathcal P_d(\Q)$ have dimension $h\geq1$. Row reduction +of Taylor coefficient vectors about $z=i$ gives a basis with distinct +vanishing orders +\[ + 0\leq d_{i1}<\cdots(w-1)k. + \label{eq:small-expansion} +\end{equation} +\end{lemma} + +\begin{proof} +Put $W=\sum_{i=0}^nT_iU$ and $\ell=\dim_FW$. Since $T_0$ is +injective, $\ell\geq k$. By Lemma~\ref{lem:integral-reduction}, +$\overline U\subseteq\mathcal P_N(\Q)$ has dimension $k$ and +$\overline W\subseteq\mathcal P_M(\Q)$ has dimension $\ell$. +Write their increasing vanishing orders at $i$ as +\[ + a_{i1}<\cdotsA$ one has +$\nu_t(c_{ia}(f))+a\geq a>A\geq m$. Thus the leading reduction of +$T_i f$ uses only output powers $(z-i)^{M-1-a}$ with $a\leq A$. +These powers are independent, so their leading coefficients cannot +all cancel. Since Taylor coordinate changes preserve minimum +valuation, $m$ is also the minimum valuation of the output in monomial +coordinates. Its leading reduction consequently vanishes at $i$ +to order at least $M-1-A$. Figure~\ref{fig:taylor-reversal} illustrates this reversal. + +\input{figures/taylor-reversal} + +This conclusion controls the whole reduction $\overline{T_iU'}$. +Indeed, every nonzero $v\in T_iU'$ has a preimage $f\in U'$. Normalizing +$f$ multiplies $v$ by the same power of $t$, so the leading reduction +of $v$ is exactly the leading reduction already considered. An integral +$v$ reduces either to this leading reduction or to zero. We have not +assumed that $T_i(U'\cap\mathcal O^N)$ is the full integral lattice of +$T_iU'$. By injectivity of $T_i$ and +Lemma~\ref{lem:integral-reduction}, $\overline{T_iU'}$ has dimension $j$. +It lies in $\overline W$ and in its vanishing filtration of order at +least $M-1-A$. Formula~\eqref{eq:vanishing-filtration} now gives +\eqref{eq:order-reversal}. + +For each fixed $i$, summing \eqref{eq:order-reversal} over $j=1,\ldots,k$ +uses the largest $k$ distinct orders of the same space $\overline W$. +The remaining orders are nonnegative. Although the auxiliary spaces +$U'$ may depend on $j$, these are bounds on the fixed filtration of +$\overline W$ and can therefore be summed. Applying +Lemma~\ref{lem:wronskian-orders} to both reduced spaces gives +\begin{align*} + \ell(M-1)+w\binom\ell2 + &\geq\sum_{i=0}^n\sum_{h=1}^{\ell}b_{ih}\\ + &\geq wk(M-1)-\sum_{i=0}^n\sum_{j=1}^k a_{ij}\\ + &\geq wk(M-1)-k(N-1)-w\binom k2\\ + &=(w-1)k(M-1)+k\Delta-w\binom k2. +\end{align*} +If $\ell\leq(w-1)k$, this implies +\[ + k\Delta\leq w\left(\binom k2+\binom\ell2\right) + \leq\frac w2\bigl(1+(w-1)^2\bigr)k^2 + \leq\frac{w^3}{2}Bk + <2w^3Bk=k\Delta, +\] +a contradiction. This proves the strict expansion bound. +\end{proof} + +\subsection{From expansion to the rank bound} + +The proof uses the standard shrunk-subspace obstruction: if every +coefficient matrix sends a common subspace into one of smaller +dimension, no matrix substitution can be invertible. +This is the elementary obstruction direction of the +noncommutative-rank characterization of Fortin and +Reutenauer~\citep[Section~2, Theorem~1]{FR04}; see also +\citet[Definition~1.1 and Section~5.1]{IQS18}. +We derive the obstruction from expansion and give the required +extension-field comparison explicitly. + +\begin{proof}[Proof of Proposition~\ref{prop:rectangular}] +Suppose that $\sum_iA_i\otimes T_i$ has a nonzero kernel vector, +written as $q$ components $f_1,\ldots,f_q\in\mathcal P_N(F)$. Let +$U$ be their span, of dimension $1\leq k\leq q\leq B$, and choose +a basis $u_1,\ldots,u_k$ of $U$. Write +\[ + f_\alpha=\sum_{j=1}^k P_{\alpha j}u_j, + \qquad P\in\Mat_{q\times k}(F). +\] +Since the $f_\alpha$ span $U$, the matrix $P$ has rank $k$. Each +row of the concatenated matrix +\[ + C=[\,A_0P\ \ A_1P\ \ \cdots\ \ A_nP\,] +\] +is a relation among the $wk$ vectors $T_i u_j$: this is exactly the +corresponding component of the kernel equation. Those vectors span +$\sum_iT_iU$, whose dimension is greater than $(w-1)k$ by +Lemma~\ref{lem:small-expansion}. Their relation space therefore has +dimension less than $k$, and hence $\rank C0, +\] +which proves \eqref{eq:padded-rank}. +\end{proof} diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/generator.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/generator.tex new file mode 100644 index 0000000000000000000000000000000000000000..61c87b1abf080f127b97cda06ed3131c5abfad33 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/generator.tex @@ -0,0 +1,413 @@ +\section{A rational hitting list and its bit complexity} +\label{sec:generator} + +Proposition~\ref{prop:pencil-deformation} supplies polynomial matrices +$R_i^*(t,p,r)$ for which $\sum_i A_i\otimes R_i^*$ is invertible +over $\Q(\omega)(t,p,r)$ whenever the pencil +$A_0+\sum_{i=1}^n A_i x_i$ is feasible and has order at most~$B$. +We now replace the root-of-unity coefficients and specialize the +three parameters to obtain a single list of rational matrix tuples. +The parameters $w,B,\Delta,M,N$ retain their earlier definitions. + +To remove the root of unity, we represent multiplication in a +finite-dimensional rational algebra. Over $\C$ this representation +is a direct sum of root evaluations, so our invertibility conditions +must hold simultaneously at all roots. Set +\begin{equation} + \ell=\frac M2,\qquad D=M\ell=\frac{M^2}{2},\qquad + H=1+3(s+2)BM^3, + \qquad + \mathcal R=\Q[\xi]/(\xi^\ell+1). + \label{eq:generator-parameters} +\end{equation} +Here and below $\xi$ also denotes its residue class. In particular, +$\xi^\ell=-1$ and $\xi^M=1$, so every integral power of~$\xi$, including +a negative power, reduces to a signed member of the basis +$1,\xi,\ldots,\xi^{\ell-1}$. + +\subsection{The rational representation and the generator} + +Let $\Omega$ be the set of complex roots of $X^\ell+1$. Since $M$ is a +power of two, these are the $\ell$ distinct numbers +\[ + \exp\!\left(\frac{2\pi\mathrm{i}(2j+1)}{M}\right), + \qquad 0\le j<\ell. +\] +Every one is a primitive $M$th root of unity: its exponent $2j+1$ is +coprime to~$M$. Evaluation at these roots is an algebra isomorphism +\begin{equation} + \mathcal R\otimes_{\Q}\C + \longrightarrow \prod_{\omega\in\Omega}\C, + \qquad f\longmapsto \bigl(f(\omega)\bigr)_{\omega\in\Omega}. + \label{eq:cyclotomic-splitting} +\end{equation} +Indeed, a polynomial of degree less than $\ell$ is uniquely determined +by its values at $\ell$ distinct points, and interpolation gives a +polynomial with any prescribed values there. + +For $a\in\mathcal R$, let $\rho(a)\in\Mat_\ell(\Q)$ be the matrix of +multiplication by~$a$ in the power basis. This is a unital algebra +homomorphism. It is computed using only rational arithmetic and the +relation $\xi^\ell=-1$. To make its splitting explicit, let $V$ be the +evaluation matrix whose $(\omega,j)$ entry is $\omega^j$, for +$\omega\in\Omega$ and $0\le j<\ell$. Interpolation says that $V$ is +invertible, and +\begin{equation} + V\rho(a)V^{-1} + =\operatorname{diag}\bigl(a(\omega):\omega\in\Omega\bigr). + \label{eq:regular-representation-splitting} +\end{equation} +If $Q\in\Mat_d(\mathcal R)$, write $\rho_d(Q)$ for the rational +$d\ell$ by $d\ell$ matrix obtained by applying $\rho$ to every entry. +First changing basis by $I_d\otimes V$, and then reordering the basis +to group together coordinates belonging to the same root, gives +\begin{equation} + \rho_d(Q)\sim_{\C} + \bigoplus_{\omega\in\Omega}Q(\omega). + \label{eq:matrix-regular-splitting} +\end{equation} +In particular, $\rho_d(Q)$ is invertible if and only if every +$Q(\omega)$ is invertible. These complex basis changes are used to +prove this assertion; the generator will not compute them. + +We use the lift of Lemma~\ref{lem:division-lift}, now with $\xi$ in +place of a primitive root. For clarity, it can be computed directly +over~$\mathcal R$. Number the coordinates from $0$ to $M-1$ and set +\[ + D_\xi e_j=\xi^j e_j,\qquad Se_j=e_{j+1}, +\] +with indices taken modulo~$M$. For $0\le i\le n$ define +\begin{align} + \widetilde c_{i,a,b}(t) + &=\frac1M\sum_{j=0}^{M-1} + \xi^{-aj}(R_i(t))_{j,j-b}, + &&0\le a,bk$, the remaining Gaussian entry is +\begin{equation} + A_{uv}-A[u,\{1,\ldots,k\}]\, + B_k^{-1}A[I_k,v] + =\frac{ + \det\begin{pmatrix} + B_k&A[I_k,v]\\ + A[u,\{1,\ldots,k\}]&A_{uv} + \end{pmatrix}} + {\det B_k}. + \label{eq:generator-pivot-minors} +\end{equation} +For $k=0$ the denominator is~$1$ and the entry is $A_{uv}$. +The determinant identity follows by subtracting the appropriate +linear combination of the first $k$ block rows from the last row. +It also shows inductively that ordinary Gaussian updates compute +the displayed entries. + +If the next pivot column has a nonzero remaining entry, swap its +row into the pivot position and append that original row index to +$I_k$. These swaps involve only unselected rows, so they do not +change any previously selected submatrix. Writing the next pivot +as~$p_{k+1}$, equation~\eqref{eq:generator-pivot-minors} gives +\[ + p_{k+1}=\frac{\det B_{k+1}}{\det B_k}, + \qquad p_1\cdots p_k=\det B_k. +\] +If the entire next remaining column is zero, the Schur complement +is singular and $\det A=0$. Otherwise all $D$ pivots can be taken, +and their product, multiplied by the sign of the accumulated row +permutation, equals~$\det A$. + +Both the stored Gaussian entries and the accumulated pivot +products, when reduced to fractions, therefore have +$O(\Lambda)$-bit numerators and denominators: they are ratios of +minors or minors themselves. Each individual update combines only +a constant number of these fractions, so its intermediate products +and sums also have $O(\Lambda)$ bits before reduction. Integer +addition, multiplication, and division on such operands have +polynomial bit cost by the elementary digit algorithms. Gcd +reduction does as well: in the Euclidean algorithm the nonzero +remainders decrease by a factor of at least two within every two +steps, so there are $O(\Lambda)$ polynomial-cost divisions. +There are $O(D^3)$ Gaussian arithmetic updates. Pivot searches, +row swaps, and sign bookkeeping also have polynomial cost. +This proves polynomial bit time for each determinant, including +singular matrices and all the column-replaced matrices used by +Cramer's rule. + +At each grid triple, at most $nD^2+1$ determinants suffice for the +singularity test and all output entries. Equations +\eqref{eq:generator-dimension-bounds}--\eqref{eq:generator-minor-bit-bound} +therefore give a fixed polynomial bound on the total computation. +The complete list contains at most $H^3 nD^2$ rational entries, so +its encoding length is +\begin{equation} + O(H^3 nD^2\Lambda)=O(m^{87}\log m). + \label{eq:generator-output-bit-bound} +\end{equation} +Dimensions, signs, and fixed delimiters fit within the same bound. +A fixed delimited encoding allows the list to be written as its +entries are produced; if its length is placed first, one may instead +buffer the polynomial-length list or make a preliminary counting +pass. + +Finally, these arithmetic and loop bounds give a Turing-machine +bound as well. All arrays have polynomially many entries of +polynomial bit length. They may be stored as indexed binary records +on work tapes; even scanning all records for each requested access +adds only a fixed polynomial factor. The loops, elementary digit +algorithms, and delimiters are fixed in advance. Thus one +deterministic Turing machine performs the construction and writes +the whole list within $C(n+s+1)^k$ bit operations for some fixed +positive integers $C,k$. None of its arithmetic uses the constants +or coefficients of an unknown formula. +\end{proof} + +\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}] +Use the generator just defined. Proposition~\ref{prop:generator-hitting} +gives the universal hitting property, with invertible and therefore +nonzero output values. Proposition~\ref{prop:generator-complexity} +gives both required polynomial bounds, after increasing the same +fixed constants $C,k$ if necessary. The generator is deterministic +and depends only on its two unary inputs. +\end{proof} diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/introduction.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..62a23dfad01beab83efb0013c922226b3ffc01ad --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/introduction.tex @@ -0,0 +1,181 @@ +\section{Introduction} +\label{sec:introduction} + +A noncommutative rational formula is a rooted tree whose leaves are variables or rational constants, whose binary gates are addition or ordered multiplication, and whose unary gates are inversion. Its size is its number of vertices, including leaves and inverse gates. At a tuple of square rational matrices, a scalar is interpreted as a scalar multiple of the identity. Evaluation is defined only when every inverse gate encounters an invertible matrix. + +Call a formula \emph{admissible} if it has a defined rational matrix evaluation of some positive dimension, and \emph{nonzero} if one of its defined rational matrix evaluations is nonzero. These operational definitions suffice for the hitting-list theorem; its proof does not need a construction of the free skew field. In particular, algebraic cancellations do not enlarge a formula's domain of evaluation. + +Identity testing asks whether an admissible formula is zero at every +defined evaluation. Scalar substitutions cannot answer this question: +for example, $x_1x_2-x_2x_1$ vanishes on all scalar tuples but is +nonzero on matrices. A matrix hitting list is a fixed finite list of +tuples that supplies a defined nonzero evaluation for each nonzero +formula in the prescribed class. Such a list permits testing through +evaluations without access to the formula tree. With inverse gates, +the same tuple must satisfy every domain condition imposed by that +tree, including conditions in branches that later cancel. +In oracle terminology, an undefined evaluation is reported as such; +only defined outputs are tested for zero. + +\begin{theorem}\label{thm:main} +There are a deterministic Turing machine $G$ and fixed positive integers $C,k$ with the following property. Given $1^n,1^s$, where $n,s\geq1$, the machine outputs a finite list $\mathcal H_{n,s}$ of $n$-tuples of rational square matrices, in at most $C(n+s+1)^k$ bit operations. The entire output, with signed binary numerators and positive binary denominators, has at most $C(n+s+1)^k$ bits. For every nonzero admissible rational formula $\Phi$ in $n$ variables of size at most $s$, some tuple in $\mathcal H_{n,s}$ is a defined evaluation of $\Phi$ with invertible value. + +The same list works for all such formulas, regardless of the bit lengths of their rational constants or their inverse depth. The generator receives no formula, oracle, randomness, or advice. +\end{theorem} + +The invertible-value conclusion is stronger than the nonzero-value requirement. All tuples produced below even have one common dimension depending only on $n,s$. + +\subsection{Context} +The algebraic setting comes from the theory of rational identities +and universal fields of fractions~\citep{Amitsur66,Cohn06}. +Affine-matrix representations of free-field elements appear in +Cohn and Reutenauer~\citep[Section~3.1]{CR94}; +Kaliuzhnyi-Verbovetskyi and Vinnikov give a systematic account of +matrix evaluation domains and rational functions~\citep[Section~2]{KVV10}. +The distinction between a rational function and an expression +representing it matters here: equivalent expressions can impose +different conditions on their intermediate inverses. Our operational +formulation retains the original expression throughout. + +For division-free computation, Nisan's coefficient-matrix +characterization made rank a central tool in noncommutative +branching-program complexity~\citep{Nisan91}. +Raz and Shpilka gave deterministic polynomial-time identity testing +with access to a formula or branching program~\citep{RS05}. +Bogdanov and Wee developed randomized black-box matrix evaluation +for bounded-degree noncommutative polynomials~\citep{BW05}, and +Forbes and Shpilka constructed deterministic quasipolynomial-size +matrix hitting sets for noncommutative algebraic branching programs, +a model including division-free formulas~\citep{FS13}. + +For computation with division, Hrube\v{s} and Wigderson developed +the rational-formula model and its reduction to linear-matrix +singularity testing~\citep{HW15}. Their inverse-matrix realizations +and tests for the domains of gate subformulas are direct predecessors +of our certificate construction. Deterministic polynomial-time +algorithms with access to the input formula followed from operator +scaling and constructive noncommutative-rank +computation~\citep{GGOW16,GGOW20,IQS18}. These algorithms use the +given coefficients, with their encoding lengths included in the +bit model. Derksen and Makam established polynomial bounds on the +dimensions needed for an admissible rational formula to have a +defined evaluation, and derived randomized polynomial-time rational +identity testing~\citep[Section~6]{DM17}. Small witnesses and +algorithms that depend on the input coefficients leave open the +construction of a universal deterministic list from size bounds alone. + +Arvind, Chatterjee, and Mukhopadhyay obtained deterministic +quasipolynomial-size hitting sets for rational formulas of inversion +height two~\citep[Theorem~1.1]{ACM22}. +Their subsequent general result handles unrestricted inverse +nesting with quasipolynomial-size rational matrix hitting +sets~\citep{ACM26}; the quantitative bounds used in this comparison +are Theorem~2, Corollary~3, and the paragraph following Remark~52 +of the specified preprint version~\citep{ACM25}. Thus rational matrix outputs and arbitrary +inverse nesting already occur in the preceding literature. +Invertible evaluations are also used in the division-algebra methods +of these works~\citep{ACM22,ACM25}. Their general treatment states a +depth reduction that preserves the input formula's evaluation +domain~\citep[Fact~1]{ACM25}. Theorem~\ref{thm:main} gives polynomial +bounds on construction time, total binary output length, and matrix +dimension. +Its complexity bound concerns generation of the list; the cost of +evaluating an unknown formula with arbitrarily large rational +constants is not included. + +The result resolves, over $\Q$, the explicit polynomial-size hitting-set +problem for this formula model, and gives the polynomial matrix +dimension asked for in the conclusion of~\citet{ACM25}. +The proof of Theorem~\ref{thm:main} below establishes all the algebraic and algorithmic +ingredients it uses. + +The proof also draws on established methods. The multiplicity +argument for explicit subspace designs of Guruswami and +Kopparty~\citep{GK16}, and the connection between subspace designs, +rank condensers, and dimension expansion studied by Forbes and +Guruswami~\citep{FG15}, provide context for our +construction. The underlying Wronskian calculation is +classical~\citep{BD10}. The common-subspace obstruction to +invertibility belongs to noncommutative-rank +theory~\citep{FR04}, while cyclic division algebras and rank +divisibility underlie the regularity methods of Ivanyos, Qiao, +and Subrahmanyam~\citep{IQS17,IQS18}. We prove the precise expansion, +deformation, and domain statements needed here, without importing +the stronger general algorithms from those works. + +\subsection{The construction and its logic} + +Write $\Mat_q(F)$ for the $q$ by $q$ matrices over a field $F$. +The bridge from formula evaluation to linear algebra is an +\emph{affine linear pencil}, a matrix +\[ + L(x)=A_0+\sum_{i=1}^n A_ix_i,\qquad A_i\in\Mat_q(\Q), +\] +of order $q$. Its evaluation at matrices $X_i\in\Mat_e(E)$ over an +extension field $E/\Q$ is $A_0\otimes I_e+\sum_iA_i\otimes X_i$. +Call the pencil \emph{feasible} if one such evaluation is invertible. +Tensor products and ranks are taken over the indicated field. + +A size-$s$ formula gives at most $s+1$ pencils of order at most +$B=2s+1$. Their simultaneous invertibility certifies that every +inverse occurrence in the original tree is defined and that the +final value is invertible. A nonzero admissible formula has a +matrix evaluation over an extension field satisfying all these +conditions. Section~\ref{sec:encoding} proves this reduction, including +the exact conditions at the inverse gates. The construction therefore +seeks a universal rational list on which each such finite family of +feasible pencils is simultaneously invertible. + +Two obstacles remain: obtaining full rank at matrices independent of +the pencil coefficients, and keeping all conditions valid when the +parameters become rational numbers. The proof handles them in the +following order. +\begin{enumerate} +\item Section~\ref{sec:expansion} constructs $n+1$ rectangular +matrices over $\Q[t]$. They act on coefficient vectors of polynomials, +reversing the Taylor expansion about the distinct points $0,\ldots,n$ +and weighting the Taylor coefficients by powers of $t$. +A Wronskian bound shows that, for every nonzero subspace of dimension +at most $B$, the sum of its images has dimension greater than $n$ +times its dimension, even when the subspace has Laurent-series +coefficients. The resulting test for a feasible +pencil is injective. Padding the test matrices to a common square +order $M$ gives matrices $R_i(t)$ for which every feasible pencil of +order $1\leq q\leq B$ satisfies +\[ + \rank\Bigl(\sum_{i=0}^n A_i\otimes R_i(t)\Bigr)>(q-1)M. +\] +\item Section~\ref{sec:division} uses two further parameters $p,r$ +to lift these matrices into a division algebra represented by +$M\times M$ matrices. Ranks of block matrices over this algebra are +multiples of $M$, so the strict inequality forces full rank. +Dividing on the left by the lifted $R_0$ restores the constant term +of an affine pencil. A separate application of the lift turns any +defined nonzero formula evaluation into a defined invertible one +over an extension field. This supplies the feasibility assertion +needed for the certificates in Section~\ref{sec:encoding}. +\item Section~\ref{sec:generator} represents multiplication in a +finite-dimensional rational algebra by rational matrices, thereby +replacing the root of unity used in the lift. All root components +must be invertible simultaneously. For any fixed formula, the +product of their determinant conditions is a nonzero polynomial +of bounded degree in $t,p,r$. A fixed three-dimensional integer +grid detects it. Exact rational arithmetic bounds both the +generation time and the total binary output length. +\end{enumerate} + +The witnesses, subspaces, and formula-specific determinant products in this proof certify the construction. None is computed by $G$. In particular, the dimension of an initial witness for a formula has no effect on the generator's resource bounds. + +The construction also hits every feasible rational affine pencil +of bounded order; see Corollary~\ref{cor:pencil-hitting}. +In particular, it answers the rational-coefficient version of the +black-box \textsc{Singular} construction question for homogeneous +pencils posed by Garg, Gurvits, Oliveira, and +Wigderson~\citep[Section~6, p.~280]{GGOW20}. +Section~\ref{sec:rank} then treats rectangular affine pencils. +Their \emph{noncommutative rank} is their rank over the free skew +field. Corollary~\ref{cor:rank-detecting-list} constructs a universal +nonadaptive list whose normalized evaluation ranks recover this rank +for every rectangular rational affine pencil with prescribed row, +column, and variable counts. It permits exact matrix-value or +ordinary-rank responses and accounts for their processing costs. diff --git a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/rank.tex b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/rank.tex new file mode 100644 index 0000000000000000000000000000000000000000..78c166d5c3ffa3efb68ddf09347c8526cf9f4cc0 --- /dev/null +++ b/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/build/sections/rank.tex @@ -0,0 +1,172 @@ +\section{A universal rank-detecting list} +\label{sec:rank} + +The hitting list and its complexity bounds are now established. +We use the pencil-hitting consequence to recover the exact +noncommutative rank of a rectangular affine pencil from one list +that is independent of all coefficient values. The additional input +is the characterization of free-field rank by normalized +matrix-evaluation ranks, recalled in the proof. + +\begin{corollary} +\label{cor:rank-detecting-list} +Let $a,b\geq1$ and $k\geq0$, and put $P=a+b+k+1$. From these +parameters in unary, a deterministic generator constructs a nonempty +finite list $\mathcal R_{a,b,k}$ of pairs +\[ + (e,X),\qquad e\geq1,\quad + X=(X_1,\ldots,X_k)\in\Mat_e(\Q)^k, +\] +in a fixed polynomial in $P$ bit operations. Its number of entries, +total binary encoding length, and every matrix dimension $e$ are +bounded by fixed polynomials in $P$. The list depends only on $a,b,k$. + +For every rectangular rational affine pencil +\[ + A(x)=A_0+\sum_{i=1}^k A_i x_i,\qquad A_i\in\Q^{a\times b}, +\] +write $\operatorname{ncrk}A$ for its rank over the free skew field on +$x_1,\ldots,x_k$ over $\Q$, with ordinary rational rank when $k=0$. +Then the exact identity +\begin{equation} + \operatorname{ncrk}A + =\max_{(e,X)\in\mathcal R_{a,b,k}} + \frac{\rank_{\Q}\!\left(A_0\otimes I_e+ + \sum_{i=1}^k A_i\otimes X_i\right)}{e} + \label{eq:rank-detecting-list} +\end{equation} +holds for all choices and bit lengths of the coefficients. + +Suppose that, at each requested pair $(e,X)$, an oracle returns +either the complete exact rational $ae$ by $be$ matrix in +\eqref{eq:rank-detecting-list}, in binary rational encoding, or its +exact ordinary rank. Querying the whole list nonadaptively and taking +the maximum of the exact quotients returns $\operatorname{ncrk}A$. +If $L$ is the total binary length of the responses, list generation, +query writing, and response processing use $\operatorname{poly}(P+L)$ +bit operations, in addition to the sum of the oracle's costs. +The response lengths and oracle costs may depend on the coefficient +heights. If $a=0$ or $b=0$, the rank is zero and no queries are needed. +\end{corollary} + +\begin{proof} +\emph{Affine evaluations and free-field rank.} +Put +$H(z)=\sum_{i=0}^k A_i z_i$, with a new variable $z_0$. +The coefficient-span invariance of Fortin and +Reutenauer~\citep[Section~2, Corollary~1, and Section~3, Lemma~1]{FR04} +includes the constant coefficient: the free-field rank of a +rectangular affine linear matrix depends only on the span of all its +coefficient matrices. The coefficient spans of $A$ and $H$ are both +$\Span_{\Q}\{A_0,\ldots,A_k\}$, so their free-field ranks agree. +This uses coefficient-span invariance, not a specialization $z_0=1$ +in a free skew field. + +For $e\geq1$, let $h_e$ be the maximum rank of +$\sum_{i=0}^k A_i\otimes Z_i$ over $Z_i\in\Mat_e(\Q)$, and let +$\alpha_e$ be the maximum rank of +$A_0\otimes I_e+\sum_{i=1}^k A_i\otimes X_i$ over rational $X_i$. +Zero-padding the $A_i$ to square matrices of order $\max(a,b)$ +preserves both free-field rank and every ordinary blow-up rank. +The homogeneous blow-up characterization and its attainment over +$\Q$~\citep[arXiv version, Definition~1.2 and its discussion, +Theorem~1.5]{IQS18} therefore give +\[ + r:=\operatorname{ncrk}A=\max_{e\geq1}\frac{h_e}{e}, +\] +with the maximum attained. The tensor order used in that reference +differs from $A_i\otimes Z_i$ only by row and column permutations. +Here padding is used to preserve ranks, not to test invertibility. + +For each fixed $e$, taking $Z_0=I_e$ gives $\alpha_e\leq h_e$. +If $h_e>0$, some $h_e$ by $h_e$ minor is a nonzero polynomial $p$ +in the entries of the $Z_i$. The polynomial $p\det Z_0$ is nonzero; +since $\Q$ is infinite, it is nonzero at a rational tuple. There +$Z_0$ is invertible and the rank is $h_e$. Right multiplication gives +\[ + \left(\sum_{i=0}^k A_i\otimes Z_i\right) + (I_b\otimes Z_0^{-1}) + =A_0\otimes I_e+\sum_{i=1}^k A_i\otimes(Z_iZ_0^{-1}), +\] +without changing rank. The case $h_e=0$ is immediate, so +$\alpha_e=h_e$ for every $e$. Consequently +\begin{equation} + r=\max_{e\geq1}\frac{\alpha_e}{e} + \label{eq:affine-normalized-rank} +\end{equation} +is attained: every individual affine evaluation has rank at most +$re$, and some rational evaluation has rank exactly $re$. No bound +on the dimension or entry heights of this existence witness +will be used. + +\emph{Square completions for rank thresholds.} +For $0\leq t\leq\min(a,b)$, form the square affine pencil +\[ + C_t=\begin{pmatrix}A&U_t\\ V_t&0\end{pmatrix}, + \qquad U_t:a\times(a-t),\quad V_t:(b-t)\times b. +\] +The entries of $U_t,V_t$ are independent variables, so their +dimension-$e$ evaluations may be arbitrary matrices of the displayed +block sizes multiplied by $e$. For a fixed rational tuple $X$, +there exist rational choices of these completion blocks making +$C_t(X,U_t,V_t)$ invertible if and only if +$\rank A(X)\geq te$. Necessity follows by splitting the +completed matrix into its $A(X)$ and two off-diagonal blocks: +\[ + (a+b-t)e=\rank C_t(X,U_t,V_t) + \leq\rank A(X)+(a-t)e+(b-t)e. +\] +For sufficiency, choose a $te$-dimensional subspace +$K\subseteq\Q^{be}$ on which $A(X)$ is injective. Take $V_t$ to be +surjective with kernel $K$, and take $U_t$ to map isomorphically onto +a complement of $A(X)K$ in $\Q^{ae}$. A vector in the kernel of the +completed matrix then has its first component in $K$, and the direct +sum $A(X)K\oplus\operatorname{im}U_t$ makes both components zero. +This proves invertibility. It includes $t=0$ and all empty-block +cases; in particular $C_0$ can always be made invertible. + +\emph{A coefficient-independent list.} +Reserve $a^2$ variables $u_{ij}$ and $b^2$ variables $v_{ij}$, +disjoint from the first $k$ variables. For $C_t$, use the first +$a-t$ columns of the $u$-pool for $U_t$ and the first $b-t$ rows of +the $v$-pool for $V_t$. The pools can be reused for different $t$. +Thus every $C_t$ is a pencil in the same +$N=k+a^2+b^2$ variables, with unused variables allowed, and has +order $a+b-t\leq a+b$. Set $s=a+b$, so this order is at most +$B=2s+1$. Both $N$ and $s$ are positive. Apply the single list +$\mathcal H_{N,s}$ from Corollary~\ref{cor:pencil-hitting}, and +project each output onto its first $k$ matrices, retaining its +dimension $e$ and any repetitions. This defines $\mathcal R_{a,b,k}$; +it is nonempty because $C_0$ is feasible. + +For $t=r$, equation~\eqref{eq:affine-normalized-rank} and the +completion criterion make $C_r$ feasible. A hit for $C_r$ therefore +has projected rank at least $re$, while every projected evaluation +has rank at most $re$. This proves~\eqref{eq:rank-detecting-list} +exactly, without rounding; individual ranks need not be divisible by +$e$. The argument includes $r=0$. When $k=0$, the projected tuple is +empty but still carries $e$, and the identity also follows directly +from $\rank(A_0\otimes I_e)=e\rank A_0$. + +Since $N+s+1$ is polynomial in $P$, Proposition~\ref{prop:generator-complexity} +and the dimension bound~\eqref{eq:generator-dimension-bounds} give +the stated generation, query-encoding, and dimension bounds. The +generator never constructs $C_t$, learns $r$, or reads the +coefficients: $C_r$ is used only in the existence proof. For complete +matrix responses, exact rational elimination is polynomial in the +encoded response lengths and dimensions. For ordinary-rank responses, +only exact comparison of the rank/dimension quotients is needed. +These observations give the stated cost accounting. +\end{proof} + +The assertion concerns exact matrix values or ordinary ranks, not +singularity-bit access to the original pencil. It makes no claim for +arbitrary rational circuits, finite fields, or total oracle-computation +bit time independent of coefficient heights. If complete scalar value +queries at zero and the standard basis points are also available, then +$k+1$ such queries reveal all coefficients, after which known +coefficient-aware rank algorithms apply with those bit lengths +counted~\citep[arXiv version, Theorem~1.5]{IQS18}. The distinct +conclusion here is the universal nonadaptive coefficient-independent +list, which also works with ordinary-rank responses. + diff --git a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/README.md b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..56d52f6c788c2b0535f385be604b5df836a74576 --- /dev/null +++ b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/README.md @@ -0,0 +1,18 @@ +# [Polynomial mixing of the switch chain for every graphical degree sequence](main.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026, + author = {{OpenAI}}, + title = {{Polynomial mixing of the switch chain for every graphical degree sequence}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf}{OAI:Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/main.tex b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..b3fb78b08b178a75ad3e6f6b29011ea3d83766d6 --- /dev/null +++ b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/main.tex @@ -0,0 +1,44 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{booktabs,array} +\usepackage{microtype} +\usepackage{xcolor} +\usepackage[colorlinks=true,linkcolor=blue!55!black,citecolor=blue!55!black,urlcolor=blue!55!black]{hyperref} +\hypersetup{pdftitle={Polynomial mixing of the switch chain for every graphical degree sequence},pdfauthor={OpenAI}} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\title{Polynomial mixing of the switch chain\\for every graphical degree sequence} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the simple-undirected form of the Kannan--Tetali--Vempala +conjecture: the switch chain on simple undirected graphs mixes in polynomial +time for every graphical degree sequence. For a lazy chain that proposes +switches uniformly on four vertices, the total-variation mixing time at +distance $1/4$ is at most $2n^8$. +We also give an exactly uniform sampler for every graphical labeled degree +vector. It uses unbiased random bits, terminates almost surely, and has +expected polynomial bit running time. +\end{abstract} +\input{sections/introduction} +\input{sections/setup} +\input{sections/rows} +\input{sections/triples} +\input{sections/disjoint} +\input{sections/slice} +\input{sections/mixing} +\input{sections/exact-sampling} +\input{sources/references} +\end{document} diff --git a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/disjoint.tex b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/disjoint.tex new file mode 100644 index 0000000000000000000000000000000000000000..d7b438aaf9eed51b8e9c96d034e284566835dcb6 --- /dev/null +++ b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/disjoint.tex @@ -0,0 +1,271 @@ +\section{Disjoint pair resamplings} +\label{dis:section} + +The resamplings indexed by disjoint vertex pairs need not commute: +they both alter the four edges between the two pairs. Carstens, Berger, +and Strona already noted this dependence of undirected +trades~\cite[Section~2.3 of the arXiv version]{CBS18}. We show that +one interaction is bounded below by a cross-edge switch term minus two +errors. Each error is a projection onto an equal-assignment indicator +in one pair fiber. Section~\ref{sl:section} will bound the sum of these +errors by the available switch energy. + +For disjoint vertex pairs $a,b$, let $T_{a,b}$ toggle the four edges between +$a$ and $b$ when their $2$-by-$2$ cross matrix has every row and column sum +one, and leave the graph unchanged otherwise. This map is a degree-preserving +involution, and hence acts as a self-adjoint isometry on functions. Define +\begin{equation} + g_{a,b}=\frac{I-T_{a,b}}2=g_{b,a}. + \label{sl:def-cross} +\end{equation} +Thus $g_{a,b}$ is an orthogonal projection. + +Fix an $a$-fiber $\mathcal F$, and write $b=\{k,l\}$. If both $k$ and $l$ +are singleton vertices in this fiber, let $x_k,x_l$ indicate assignment to +the first endpoint of $a$, and set +\begin{equation} + z_{kl}^{\mathcal F} + =\mathbf 1_{\{x_k=x_l\}} + -\mathbb P(x_k=x_l\mid\mathcal F). + \label{sl:def-equality} +\end{equation} +On $\mathcal F$, define $R_{a,b}$ to be orthogonal projection onto the span +of $z_{kl}^{\mathcal F}$, or zero if this function is zero. If either vertex +is not a singleton, also use the zero operator. Taking the direct sum over +all $a$-fibers defines an orthogonal projection $R_{a,b}$ on the graph +space. Unlike $g_{a,b}$, this projection is directed: $R_{a,b}$ and +$R_{b,a}$ use different fibers. Its definition is independent of the chosen +ordering of the two endpoints of $a$. + +\begin{proposition}[Disjoint-pair interaction]\label{dis:interaction} +For disjoint vertex pairs $a,b$ and every real function $f$ on +$\Omega_d$, +\begin{equation}\label{dis:bound} + 2\langle h_a f,h_b f\rangle + \ge 2\|g_{a,b}f\|^2 + -\|R_{a,b}f\|^2-\|R_{b,a}f\|^2. +\end{equation} +\end{proposition} + +\begin{proof} +Write $a=\{i,j\}$ and $b=\{k,l\}$, with $i$ and $k$ the first +endpoints in the orders used to define the slice projections, and put +$O=V\setminus(a\cup b)$. Partition $\Omega_d$ by recording the graph +on $O$, the edges $ij,kl$, and, separately for each $w\in O$, its +numbers of neighbors in $a$ and in $b$. Both pair resamplings preserve +these data. In particular, each full $a$-fiber or $b$-fiber is contained +in a cell of this partition. We prove the inequality on each nonempty +cell, using its conditional uniform measure throughout. + +\paragraph{The cell and its two conditional expectations.} +Let $A_0,B_0\subseteq O$ be the outside vertices with exactly one +neighbor in $a,b$, respectively. A graph $G$ in the cell is encoded by +\[ + M=\begin{pmatrix} + \mathbf 1_{\{ik\in E(G)\}}&\mathbf 1_{\{il\in E(G)\}}\\ + \mathbf 1_{\{jk\in E(G)\}}&\mathbf 1_{\{jl\in E(G)\}} + \end{pmatrix}, + \qquad U\subseteq A_0,\qquad W\subseteq B_0, +\] +where $U$ contains those outside singletons assigned to $i$, and $W$ +those assigned to $k$. The total $s$ of the four entries of $M$ is fixed +by the sum of the degrees in $a$ and the recorded data. Set +\[ + r=M_{ik}+M_{il},\qquad c=M_{ik}+M_{jk}. +\] +For fixed integers $q_0,q_1$, the remaining degree constraints are +exactly +\begin{equation}\label{dis:sizes} + |U|=q_0-r,\qquad |W|=q_1-c. +\end{equation} +Indeed, these equations enforce the degrees of $i,k$; the degrees of +$j,l$ then follow from the fixed pair totals. Each outside degree is +already enforced by the recorded graph and counts. This also proves +that every binary matrix of total $s$ and every pair of subsets +satisfying \eqref{dis:sizes} give a valid graph in the cell. The sets +$A_0,B_0$ may overlap: the two assignments use different edges and +preserve the two recorded neighbor counts separately. Thus overlap +adds no constraint. All these encodings have the same probability. + +Within the cell, the full $a$-fiber data are equivalent to +$X=(c,W)$. Namely, $c$ and $s-c$ specify the counts from $k,l$ to $a$, +and $W$ specifies all remaining edges not incident to $a$. Together +with the cell data this is exactly the defining information of an +$a$-fiber. Similarly, the full $b$-fiber data are $Y=(r,U)$. Therefore +\begin{equation}\label{dis:conditional} + E_a=\mathbb E[\,\cdot\mid X],\qquad + E_b=\mathbb E[\,\cdot\mid Y]. +\end{equation} +Let $\mathcal A,\mathcal B$ be the subspaces of functions of $X,Y$, +respectively, with constants included, and let +$\mathcal J=(\mathcal A+\mathcal B)^\perp$. +The involution defining $g_{a,b}$ preserves $X$ and $Y$. Hence its +antisymmetric projection kills both subspaces, and, by +self-adjointness, +\begin{equation}\label{dis:g-range} + \operatorname{ran}g_{a,b}\subseteq\mathcal J. +\end{equation} + +\paragraph{Independent cases.} +If $s=0$ or $s=4$, the cross matrix is fixed and $X,Y$ are independent. +If $s=1$ or $s=3$, every feasible combination of its row total $r$ and +column total $c$ determines a unique matrix: it specifies the position +of its unique one or its unique zero. Since the subset restrictions +in \eqref{dis:sizes} are separate, $X,Y$ again have a product law. + +For $s=2$, all six cross matrices appear in the following display, +placed according to their first row sum $r$ and first column sum $c$: +\begin{equation}\label{dis:table} +\begin{array}{c|ccc} + &c=0&c=1&c=2\\ \hline + r=0&\varnothing& + \begin{psmallmatrix}0&0\\1&1\end{psmallmatrix}&\varnothing\\[8pt] + r=1&\begin{psmallmatrix}0&1\\0&1\end{psmallmatrix}& + \begin{psmallmatrix}1&0\\0&1\end{psmallmatrix},\quad + \begin{psmallmatrix}0&1\\1&0\end{psmallmatrix}& + \begin{psmallmatrix}1&0\\1&0\end{psmallmatrix}\\[8pt] + r=2&\varnothing& + \begin{psmallmatrix}1&1\\0&0\end{psmallmatrix}&\varnothing +\end{array}. +\end{equation} +Let $m_{rc}$ count the matrices in cell $(r,c)$ of this display: +$m_{11}=2$, the four adjacent cells have count one, and the corners +have count zero. Write $D_r=\{r\ne1\}$ and $D_c=\{c\ne1\}$. +The top and bottom matrices realize $D_r$, and the side matrices +realize $D_c$, so the two events are disjoint. Call $r=c=1$ the center; +its two matrices are exchanged by $T_{a,b}$. If both events have positive +probability, the center is feasible: occurrence of $D_r$ implies +feasibility of a subset $W$ of size $q_1-1$, and occurrence of $D_c$ +implies feasibility of a subset $U$ of size $q_0-1$. Together these +subsets realize either central matrix. Thus, if the center is absent, +at most one of the events occurs, and the support lies in a single +row or column of \eqref{dis:table}; the law of $X,Y$ is a product. +If the center is present but one of the events has probability zero, +the same conclusion holds. For example, if $r=1$ always, then $U$ is +uniform of fixed size, independently of $c,W$; the multiplicity in +\eqref{dis:table} depends only on $c$. + +Whenever $X,Y$ are independent, both $E_aE_b$ and $E_bE_a$ equal the +projection onto constants in the cell. Consequently $h_a h_b$ is the +orthogonal projection onto $\mathcal J$, and +\[ + 2\langle h_a f,h_b f\rangle + =2\|\operatorname{proj}_{\mathcal J}f\|^2 + \ge 2\|g_{a,b}f\|^2 +\] +by \eqref{dis:g-range}. This is stronger than \eqref{dis:bound}. + +\paragraph{The remaining correlation has rank one.} +It remains to consider $s=2$ when both events and the center have +positive probability. Put $p_r=\mathbb P(D_r)$ and +$p_c=\mathbb P(D_c)$, and define centered unit vectors +\[ + u=\frac{\mathbf 1_{D_c}-p_c}{\sqrt{p_c(1-p_c)}}\in\mathcal A, + \qquad + v=\frac{\mathbf 1_{D_r}-p_r}{\sqrt{p_r(1-p_r)}}\in\mathcal B. +\] +Their correlation is +\begin{equation}\label{dis:rho} + \langle u,v\rangle=-\rho,\qquad + \rho=\sqrt{\frac{p_cp_r}{(1-p_c)(1-p_r)}}\in(0,1), +\end{equation} +since $D_r,D_c$ are disjoint and $1-p_c-p_r$ is the positive center +probability. + +With the counts $m_{rc}$ associated with \eqref{dis:table}, for every +occurring $X=(c,W)$ and each feasible subset $U$ of size $q_0-r$, +\[ + \mathbb P\bigl(Y=(r,U)\mid X=(c,W)\bigr) + =\frac{m_{rc}}{\displaystyle\sum_{t=0}^2 + \binom{|A_0|}{q_0-t}m_{tc}}, +\] +where a binomial coefficient is zero when its lower index is outside +$\{0,\ldots,|A_0|\}$. Thus the conditional law of $Y$ given $X$ +depends only on whether $c=1$. When $c\ne1$, necessarily $r=1$ and +$U$ is uniform of size $q_0-1$; when $c=1$, the displayed formula has +no dependence on $W$. +Symmetrically, the conditional law of $X$ given $Y$ depends only on +whether $r=1$. It follows that +\[ + E_a\mathcal B\subseteq\operatorname{span}(1,u),\qquad + E_b\mathcal A\subseteq\operatorname{span}(1,v),\qquad + E_a v=-\rho u,\quad E_b u=-\rho v. +\] +Let $\mathcal A'$ be the centered functions in $\mathcal A$ +orthogonal to $u$, and define $\mathcal B'$ symmetrically. The +conditional-expectation inclusions show that +$\mathcal A'\perp\mathcal B$ and $\mathcal B'\perp\mathcal A$. +We therefore have the orthogonal decomposition +\begin{equation}\label{dis:decomposition} + \operatorname{span}(1)\ \oplus\ \mathcal A'\ \oplus\ + \mathcal B'\ \oplus\ \operatorname{span}(u,v)\ \oplus\ \mathcal J. +\end{equation} +All these summands are invariant under $E_a,E_b$. The first three +contribute zero to $\langle h_a f,h_b f\rangle$, and $\mathcal J$ +contributes $\|\operatorname{proj}_{\mathcal J}f\|^2$. + +On the plane in \eqref{dis:decomposition}, write the component of +$f$ as $xu+yv$. Its images under $h_a$ and $h_b$, respectively, are +\[ + y(v+\rho u),\qquad x(u+\rho v). +\] +The parenthesized vectors have squared norm $1-\rho^2$ and mutual +inner product $\rho(1-\rho^2)$. Hence the plane contribution obeys +\begin{equation}\label{dis:plane} + 2xy\rho(1-\rho^2) + \ge -(x^2+y^2)(1-\rho^2) + =-\|y(v+\rho u)\|^2-\|x(u+\rho v)\|^2. +\end{equation} +Here $2|xy|\le x^2+y^2$ and $0<\rho<1$ justify the inequality +for either sign of $xy$. + +\paragraph{The error lines belong to the slice projections.} +The vector $y(v+\rho u)$ is the orthogonal projection of the full +function $f$ onto the line spanned by $v+\rho u$: the spanning vector is +orthogonal to $u$, has inner product $1-\rho^2$ with $v$, and +all other summands of \eqref{dis:decomposition} are orthogonal to +the plane. The symmetric assertion holds for $x(u+\rho v)$. + +We claim that the first line belongs to +$\operatorname{ran}R_{a,b}$. Examine +$v+\rho u=v-E_a v$ on each $X$-slice, which, by +\eqref{dis:conditional}, is a full global $a$-fiber. If $c\ne1$, +then $D_r$ is deterministically false and this difference is zero. +If $c=1$, both $k,l$ are singleton vertices for the pair $a$. +Moreover, $D_r$ occurs exactly when their assignments to the two +endpoints of $a$ are equal. Writing these assignments as $x_k,x_l$, +on this fiber we have +\[ + v-E_a v= + \frac{\mathbf 1_{\{x_k=x_l\}}- + \mathbb P(x_k=x_l\mid X)}{\sqrt{p_r(1-p_r)}}. +\] +This is a scalar multiple of the centered function defining +$R_{a,b}$, including when that function vanishes. Thus the line +belongs to its fiberwise direct-sum range. Likewise the other line +belongs to $\operatorname{ran}R_{b,a}$. Orthogonal projection onto +a subspace has no larger norm than projection onto a containing +subspace: if $\Pi$ projects onto a line contained in +$\operatorname{ran}R$, then $\Pi R=\Pi$ and +$\|Rf\|^2\ge\|\Pi Rf\|^2=\|\Pi f\|^2$. Consequently +\[ + \|y(v+\rho u)\|^2\le\|R_{a,b}f\|^2,\qquad + \|x(u+\rho v)\|^2\le\|R_{b,a}f\|^2. +\] +Combining these inequalities with \eqref{dis:plane} and +\eqref{dis:g-range} proves \eqref{dis:bound} on the cell. + +Finally, $E_a,E_b,g_{a,b}$ preserve the cells. So do $R_{a,b}$ and +$R_{b,a}$, since their defining full pair fibers are contained in +the cells. Averaging the conditional inequalities proves the +asserted global inequality. +\end{proof} + +We retain the positive term in Proposition~\ref{dis:interaction} to +control the two directed error terms after summation. The estimate needed for that step is, for each fixed $a$, +\[ + \sum_{b:\,b\cap a=\varnothing}\|R_{a,b}f\|^2 + \le \sum_{b:\,b\cap a=\varnothing}\|g_{a,b}f\|^2. +\] +We prove this estimate next, using the uniform subset structure of +each $a$-fiber. diff --git a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/exact-sampling.tex b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/exact-sampling.tex new file mode 100644 index 0000000000000000000000000000000000000000..8e5820e0c48a99008b0cb1c952b477e299068070 --- /dev/null +++ b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/exact-sampling.tex @@ -0,0 +1,171 @@ +\section{Exact uniform sampling} +\label{sec:exact-sampling} + +We use the rare residual-mixture construction of G\"obel, Liu, +Manurangsi, and Pappik~\cite[Section~3.1, Algorithm~1 and +Theorem~6]{GLMP2024}. Their Corollary~15 computes the approximate law +by powers of the transition matrix. Exact tabulation is also used in +the contingency-table sampler~\cite[Sections~6.3--6.4]{OpenAIContingency2026}. Here we +establish the tabulation and bit-cost bounds directly for the rational +switch chain. + +The idea is to sample from a very accurate finite-time law $p$ most of +the time, and to correct its remaining bias on a rare branch. For a +suitably small $\delta>0$, we will have $(1-\delta)p\le\pi_d$ pointwise. +Then +\[ + r=\frac{\pi_d-(1-\delta)p}{\delta} +\] +is a probability law, and the mixture $(1-\delta)p+\delta r$ is exactly +$\pi_d$. Only the branch of probability $\delta$ enumerates the state +space and tabulates $p$ to sample from $r$. We choose $\delta$ +exponentially small enough to pay for that enumeration in expectation; +the chain still reaches the required accuracy in polynomially many +steps. + +\begin{proof}[Proof of Corollary~\ref{cor:exact-sampling}] +We use the usual explicit binary encoding of the degree vector, with +delimiters between its entries. Let $n$ be the number of entries of +$d$. For $n<4$, +enumerate the at most $2^3$ simple graphs and retain those with the +specified labeled degrees. The unbiased integer draw described below +samples uniformly from this nonempty constant-size set. We may +therefore assume $n\ge4$ and use the notation of +Section~\ref{sec:introduction}. Havel's greedy degree-realization +procedure~\cite{Havel55}, with deterministic tie-breaking, constructs +one graph $G_0\in\Omega_d$ in polynomial bit time. + +\paragraph{An accurate rational law.} +Set +\begin{equation}\label{exact:parameters} +\begin{gathered} + B=\binom n2,\qquad M=|\Omega_d|\le2^B,\qquad + q=12\binom n4,\\ + D=4B+10,\qquad \delta=2^{-D},\qquad + t=\left\lceil\gamma_0^{-1}(B+2D+2)\right\rceil, +\end{gathered} +\end{equation} +where $\gamma_0=[24n^2\binom n4]^{-1}$ is from +Equation~\eqref{mix:gamma}. Thus $D=O(n^2)$ and $t=O(n^8)$. +Write $p_G=P_d^t(G_0,G)$. If $M>1$, Equation~\eqref{mix:TV} gives +\[ + \|p-\pi_d\|_{\mathrm{TV}} + \le\frac12\sqrt M\,e^{-\gamma_0t} + \le\frac12\,2^{B/2}e^{-(B+2D+2)} + \le2^{-2D}=\delta^2. +\] +The last inequality follows from $21$, the spectral gap of $P_d$ is at least +$[24n^2\binom n4]^{-1}$. +\end{theorem} + +\begin{samepage} +\begin{corollary}[Exact uniform sampling]\label{cor:exact-sampling} +There is a uniform randomized algorithm that, given any finite graphical +labeled degree vector $d=(d_1,\ldots,d_n)$, uses unbiased random bits, +terminates almost surely, and outputs a graph chosen exactly uniformly +from the simple undirected graphs on $\{1,\ldots,n\}$ with labeled +degree vector $d$. Its expected bit running time is bounded by a fixed +polynomial in the binary encoding length of $d$. +\end{corollary} + +The corollary includes vectors of length less than four, which are +handled directly. Section~\ref{sec:exact-sampling} proves it by +correcting the finite-time law of the chain. The time bound is in +expectation; no polynomial bound on every random execution is asserted. +\end{samepage} + +\paragraph{History and significance.} +Uniform sampling with prescribed degrees provides a basic null model +for networks and a natural problem in approximate counting. + +Kannan, Tetali, and Vempala initiated the switch-chain program, treated +regular bipartite sequences, and raised the possibility of broad +applicability~\cite{KTV99}. For simple undirected graphs, Cooper, Dyer, +and Greenhill established rapid mixing for regular sequences; their +corrigendum preserves the polynomial bound with a corrected +exponent~\cite{CDG07,CDG12}. Feder, Guetz, Mihail, and Saberi studied +early irregular families~\cite{FGMS06}, and Greenhill and Sfragara +proved rapid mixing under quantitative restrictions on the degrees, +correcting and extending Greenhill's earlier irregular-degree +argument~\cite{Greenhill15,GS18}. Amanatidis and Kleer developed a route through +strong stability~\cite{AK20}. + +Stability conditions for degree-constrained sampling were developed by +Jerrum and Sinclair~\cite{JS90} and by Jerrum, McKay, and +Sinclair~\cite{JMS92}. Erd\H{o}s, Greenhill, Mezei, Mikl\'os, +Solt\'esz, and Soukup subsequently +proved rapid mixing for every P-stable family~\cite{Unified22}. +P-stability requires a polynomial bound, uniform over the family, on +the ratio between the number of realizations of nearby perturbed +sequences and that of the original sequence. Recent work of +Erd\H{o}s, Mikl\'os, and Soukup~\cite{EMS25} and of Erd\H{o}s, Lippner, +Nevo, and Soukup~\cite{ELNS25} gives further P-stable families from +regions in which every admissible degree vector is graphical. +These results explain substantial classes of degrees; the theorem +here requires only that the individual degree vector be graphical. +The modern Kannan--Tetali--Vempala conjecture predicts rapid mixing +for every realizable degree specification in each of the bipartite, directed, and +simple-undirected models; see \cite[Conjecture~1.1]{Unified22}. +Theorem~\ref{thm:main} resolves its simple-undirected formulation +affirmatively. Our off-diagonal transition probabilities agree with +the simple-undirected kernel displayed in +\cite[Equation~(3.1)]{Unified22}. + +Fu, Qin, and Wang proved the bipartite formulation, equivalently the +fixed-margin binary-matrix problem~\cite{FQW26}. They compare the +swap chain with a two-row heat bath and reduce its spectral analysis +to a three-row conditional-projection inequality. We reprove the +equivalent variance inequality of their Theorem~5.1, retaining the +separation of label assignments and block counts and the adjacent-count +coupling developed in their Lemmas~5.2--5.4. A simple undirected graph +has a symmetric adjacency matrix, so an arbitrary binary-matrix +resampling does not preserve its state space. Our pair resamplings +preserve that symmetry by redistributing neighbors between two vertices. + +These pair updates belong to the Curveball framework. For binary +matrices with fixed margins, Verhelst developed a multiple-swap +procedure~\cite{Verhelst08}. Strona, Nappo, Boccacci, Fattorini, and +San-Miguel-Ayanz later independently formulated a version called the +Curveball algorithm~\cite{Strona14,Strona16,CK18}; Carstens, Berger, +and Strona developed its undirected version~\cite{CBS18}. +They also observed that undirected trades on disjoint vertex pairs can +be dependent. The additional estimate here gives quantitative control +of those interactions: although an individual disjoint-pair term may +be negative, their full sum is positive semidefinite. + +For prescribed degrees on arbitrary explicitly listed simple undirected +unweighted hosts, OpenAI~\cite[Corollary~10.1]{OpenAIPerfectMatchings2026} +gives a fully polynomial randomized approximation scheme for counting +labeled subgraphs with those degrees and, when feasible, a sampler within +total-variation distance $\tau$ of their uniform law. For rational +$0<\tau<1$, the sampling bit cost on every execution is polynomial in the +full input length and $\tau^{-1}$. These broader-host guarantees do not +assert mixing of a host-restricted switch chain: Theorem~\ref{thm:main} +concerns the complete host, while Corollary~\ref{cor:exact-sampling} +gives exact uniform sampling with expected polynomial bit cost. + +\paragraph{Proof strategy.} +For an unordered vertex pair $a$, we resample which of its two endpoints +receives each neighbor adjacent to exactly one of them. All other +adjacencies and the number assigned to each endpoint remain fixed. +Each conditional space is therefore a uniform subset slice: all subsets +of one fixed size from the available neighbors. Let $E_a$ be its +averaging projection, $h_a=I-E_a$, and $H=\sum_a h_a$. +The central estimate is $H^2\succeq H$, where $\succeq$ denotes the +order of quadratic forms. Once the kernel of $H$ is shown to consist +of constants, comparison with individual switches gives the stated +gap and mixing time. + +The reduction from $H$ to its restrictions on triples follows the +squared-generator method of Caputo~\cite[Section~1.1]{Caputo08}, also +used for binary matrices by Fu, Qin, and Wang~\cite[Lemma~4.1]{FQW26}. +The expansion groups terms according to whether the two indexing pairs +intersect. Caputo's disjoint terms are controlled by commutation +\cite[Equation~(1.11)]{Caputo08}. +Here the undirected trades need a separate estimate; this is where +the slice comparison and the disjoint-pair calculation enter. + +First, all three pair resamplings on a vertex triple can be represented +by updates of a three-row binary matrix with fixed margins. When needed, +one extra column records the internal edges of the triple. We prove that the +variance of a sum of three row functions is at most twice the sum of +their variances. To see the structure behind this estimate, discard +columns of sum zero or three. Each remaining column chooses either +the row containing its unique one or the row containing its unique zero. +We first control these label assignments with their counts fixed, +then control the random counts themselves. The resulting variance +bound controls all overlapping pair resamplings. + +Second, after fixing the data preserved by two disjoint pair +resamplings, the dependence between their conditional data has rank +at most one. Its negative contribution is bounded by projections onto +centered indicators that two neighbors receive the same endpoint. +A slice inequality bounds the sum of these projections by the switch +energy available on the same slice. Summing cancels the negative +contributions and completes the estimate for $H$. The three-row +variance inequality and the slice projection comparison are stated +separately from the graph argument, making their scope explicit. + +\paragraph{Organization.} +Section~\ref{sec:setup} defines the pair resamplings and their conditional +projections. Section~\ref{row:section} proves the three-row inequality, +and Section~\ref{tri:section} applies it to overlapping updates. +Section~\ref{dis:section} identifies the errors in one disjoint-pair +interaction. Section~\ref{sl:section} controls their sum by a slice +comparison and proves positivity of the full disjoint contribution. +Section~\ref{mix:section} combines these estimates, proves connectivity, +and tracks the comparison and proposal constants through the mixing +bound. Section~\ref{sec:exact-sampling} then gives an exact uniform +sampler with expected polynomial bit cost. All proof ingredients are +supplied in the paper. diff --git a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/mixing.tex b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/mixing.tex new file mode 100644 index 0000000000000000000000000000000000000000..6efd9de2f26245732513346bb0954052a5309ce5 --- /dev/null +++ b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/mixing.tex @@ -0,0 +1,259 @@ +\section{From pair resampling to the switch chain} +\label{mix:section} + +We now combine the local estimates, identify the kernel of the resulting +operator, and compare it with the chain in Theorem~\ref{thm:main}. + +\subsection{The global operator inequality} + +We use the triangle regrouping of Caputo's squared-generator method +\cite[Section~1.1, Equations~(1.13)--(1.15)]{Caputo08}. +The exact correction counts how often each pair belongs to a triple. +Here Corollary~\ref{dis:nonnegative} supplies the disjoint-term +positivity that commutation supplies in that framework. + +\begin{proposition}\label{mix:global} +The pair-resampling operator satisfies +\begin{equation}\label{mix:H-square} +H^2\succeq H. +\end{equation} +\end{proposition} + +\begin{proof} +Let $D$ be the ordered disjoint-pair sum from +Equation~\eqref{dis:D}. Every ordered pair of distinct intersecting vertex +pairs belongs to the unique triple formed by their union. Each single +vertex pair belongs to $n-2$ triples, and $h_a^2=h_a$. Expanding the +squares therefore gives the exact identity +\begin{equation}\label{mix:expansion} +H^2=\sum_{T\in\binom V3} H_T^2-(n-3)H+D. +\end{equation} +Proposition~\ref{tri:gap} and Corollary~\ref{dis:nonnegative} imply +\[ +H^2\succeq\sum_{T\in\binom V3}H_T-(n-3)H +=(n-2)H-(n-3)H=H. +\] +\end{proof} + +\subsection{Connectivity and the kernel} + +We give a direct connectivity argument to ensure that the operator bound +controls every centered function. It uses Havel's classical +degree-realization switching argument~\cite[Theorem~3]{Havel55}, +with deterministic tie-breaking to select the same realization from +every starting graph. + +\begin{lemma}\label{mix:connectivity} +Any two graphs in $\Omega_d$ can be joined by a sequence of valid switches. +\end{lemma} + +\begin{proof} +We transform every realization into the same recursively prescribed +graph. Suppose a set of vertices has already been frozen, and let $R$ be +the remaining vertex set. The edges involving frozen vertices will no +longer change. Write $d_i^R$ for the degrees in the current induced graph +on $R$. + +Choose the first vertex $v\in R$ in label order. Prescribe its desired +neighbors to be the $d_v^R$ vertices of $R\setminus\{v\}$ of highest +residual degree, breaking ties by label. If the current neighbors differ +from this prescribed set, there are a missing desired neighbor $j$ and +an undesired neighbor $i$. Their residual degrees satisfy +$d_j^R\ge d_i^R$. With neighborhoods taken in the induced graph on $R$, +put +\[ +A=N_R(j)\setminus\{i\},\qquad +B=N_R(i)\setminus\{j\}. +\] +Removing a possible edge $ij$ subtracts the same amount from both +neighborhood sizes, so $|A|\ge |B|$. Moreover $v\in B\setminus A$. +Consequently $A\setminus B$ is nonempty. A vertex $k$ in this set is +distinct from $v,i,j$, and the membership conditions say precisely that +$vi,jk$ are edges while $vj,ik$ are absent. The valid switch +\[ +\{vi,jk\}\longmapsto\{vj,ik\} +\] +changes no other edge at $v$, so it retains every previously attained +desired neighbor and increases their number by one. Repetition therefore +realizes the prescribed neighborhood. If fewer than four residual +vertices remain, a discrepancy would still force the four distinct +vertices just constructed; hence no discrepancy can occur. + +Freeze $v$ and continue on $R\setminus\{v\}$. Its residual degree vector +is now determined by the previous vector and the prescribed neighborhood, +independently of the starting realization. It is graphical because the +current graph realizes it. Inductively, the prescription fixes the same +edges from every starting graph. All switches occur entirely among +unfrozen vertices, and the procedure terminates after all vertices are +frozen. Thus every realization is connected to the same graph. Reversing +one of these paths proves the claim. +\end{proof} + +\begin{corollary}\label{mix:poincare} +The kernel of $H$ consists exactly of the constant functions. For every +real function $f$ on $\Omega_d$, +\begin{equation}\label{mix:H-poincare} +\operatorname{Var}_{\pi_d}(f)\le \langle f,Hf\rangle. +\end{equation} +\end{corollary} + +\begin{proof} +Since the $h_a$ are orthogonal projections, +\[ +\langle f,Hf\rangle=\sum_{a\in\binom V2}\|h_af\|^2. +\] +Its vanishing forces $f$ to be constant on every pair fiber. Every valid +switch exchanges two singleton assignments in a pair fiber: take either +opposite vertex pair of the toggled four-cycle. Thus $f$ is constant along +switches, and Lemma~\ref{mix:connectivity} makes it constant on $\Omega_d$. +The converse is immediate. + +The operator $H$ is positive semidefinite. By +Proposition~\ref{mix:global}, each eigenvalue $\lambda$ satisfies +$\lambda^2\ge\lambda$, and hence is either zero or at least one. +Its kernel has just been identified, so its restriction to the centered +subspace is bounded below by the identity. Subtracting the mean of $f$ +gives Equation~\eqref{mix:H-poincare}. +\end{proof} + +\subsection{Comparison with single switches} + +We compare a whole-fiber resampling with exchanges of two singleton +assignments inside that fiber. This is the fiberwise heat-bath +comparison principle used by Carstens and Kleer for bipartite switch +and Curveball chains~\cite[Theorems~2 and~7 and Appendix~A]{CK18}. +The argument below proves the required undirected comparison with +its own constant and counts the multiplicity of each switch. + +Let $L$ be the unnormalized Laplacian of the switch graph: +\begin{equation}\label{mix:L} +(Lf)(G)=\sum_{G'\text{ a switch neighbor of }G} +\bigl(f(G)-f(G')\bigr). +\end{equation} +Each distinct neighbor appears once in this sum. + +For a vertex pair $a$, define $J_a$ on each of its fibers by +\[ +J_a=\sum_{1\le k<\ell\le N}(I-\tau_{k\ell}), +\] +where the $N$ singleton vertices have been indexed and $\tau_{k\ell}$ +exchanges their assignments. Combining these operators over all fibers +defines $J_a$ on the full graph space. + +\begin{lemma}\label{mix:comparison} +The operators above satisfy +\begin{equation}\label{mix:comparison-bounds} +h_a\preceq n^2J_a,\qquad +\sum_{a\in\binom V2}J_a=2L,\qquad +H\preceq2n^2L. +\end{equation} +\end{lemma} + +\begin{proof} +First work in one nonconstant uniform $q$-subset fiber with $N$ singleton +labels. Let $S_0$ be a uniform $q$-subset and let $\sigma$ be an independent +uniform permutation of the $N$ labels. Then $\sigma S_0$ is uniform and +independent of $S_0$, so +\[ +\operatorname{Var}(f)=\frac12\mathbb E +\bigl(f(S_0)-f(\sigma S_0)\bigr)^2. +\] +Fix, for each permutation, a decomposition into at most $N$ +transpositions, obtained for example by decomposing its cycles. Pad with +identities to length $N$. Conditional on $\sigma$, every intermediate +subset is uniform. Cauchy--Schwarz along the resulting path gives +\begin{align*} +\operatorname{Var}(f) +&\le \frac{N^2}{2}\sum_{k<\ell} +\mathbb E_{\mathrm{slice}} +\bigl(f-\tau_{k\ell}f\bigr)^2\\ +&=N^2\left\langle f, +\sum_{k<\ell}(I-\tau_{k\ell})f\right\rangle. +\end{align*} +Indeed, Cauchy--Schwarz costs a factor $N$, and each of the $N$ steps has +conditional squared difference bounded by the sum of all transposition +squared differences displayed above. Identity steps contribute zero. +Constant fibers require no estimate. Since $N\le n$, averaging over +fibers proves $h_a\preceq n^2J_a$. + +Each switch has two pair-trade realizations, as observed in +Lemma~2 of~\cite{CBS18}. To check this count in the present notation, +consider a particular transition +\[ +\{uv,xy\}\longmapsto\{ux,vy\}. +\] +The four distinct vertices form the toggled cycle $u,v,y,x,u$. Taking +$a=\{u,y\}$ exchanges the assignments of singleton vertices $v,x$; +taking $a=\{v,x\}$ exchanges those of singleton vertices $u,y$. These +are two occurrences of this transition. They are also the only ones: +a singleton transposition toggles precisely the four edges between its +indexing pair and the two exchanged singleton vertices. Thus its +indexing pair must be one of the two parts of the unique bipartition of +the toggled four-cycle. Each choice determines its transposition uniquely. +It follows pointwise that $\sum_aJ_a=2L$. Summing the first comparison +now proves the last one. +\end{proof} + +Combining Corollary~\ref{mix:poincare} and Lemma~\ref{mix:comparison}, we +obtain +\begin{equation}\label{mix:L-poincare} +\operatorname{Var}_{\pi_d}(f)\le 2n^2\langle f,Lf\rangle. +\end{equation} +In particular, if $|\Omega_d|>1$, the spectral gap of $L$ is at least +$1/(2n^2)$. + +\subsection{The prescribed chain and its mixing time} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +If $|\Omega_d|=1$, the mixing time is zero. Suppose henceforth that +$|\Omega_d|>1$. For any distinct switch neighbors $G,G'$, their transition +uniquely specifies its four-element vertex set and the ordered pair of +removed and added matchings. Under the proposal rule of +Section~\ref{sec:introduction}, +\[ +P_d(G,G')=\frac12\,\frac1{\binom n4}\,\frac16 +=\frac1{12\binom n4}. +\] +The reverse transition has the same probability, so $\pi_d$ is +reversible, and the diagonal probabilities give +\begin{equation}\label{mix:normalization} +I-P_d=\frac{L}{12\binom n4}. +\end{equation} +By Equation~\eqref{mix:L-poincare}, the spectral gap of $P_d$ is at least +\begin{equation}\label{mix:gamma} +\gamma_0=\frac1{24n^2\binom n4}. +\end{equation} + +The holding probability also ensures nonnegative eigenvalues: write +$P_d=(I+Q)/2$, where $Q$ is the reversible Markov kernel consisting of +the proposal step, including rejected proposals. Jensen's inequality and +stationarity show that $Q$ is a contraction on $L^2(\pi_d)$, so its +eigenvalues lie in $[-1,1]$. Thus every eigenvalue of $P_d$ on the centered +subspace lies in $[0,1-\gamma_0]$, and for a centered function $f$, +\[ +\|P_d^t f\|\le(1-\gamma_0)^t\|f\| +\le e^{-\gamma_0t}\|f\|. +\] +The density of a point mass at $G$, minus one, has squared norm +$|\Omega_d|-1$. Reversibility evolves this density by $P_d$. Therefore +Cauchy--Schwarz yields, uniformly over starting states, +\begin{equation}\label{mix:TV} +\|P_d^t(G,\cdot)-\pi_d\|_{\mathrm{TV}} +\le\frac12\sqrt{|\Omega_d|}\,e^{-\gamma_0t}. +\end{equation} +In particular, total variation is at most $1/4$ when +\[ +t=\left\lceil +24n^2\binom n4\left(\log2+\frac12\log|\Omega_d|\right) +\right\rceil. +\] +Here and throughout, logarithms are natural. Since there are at most +$2^{\binom n2}$ labeled simple graphs, +\[ +\log2+\frac12\log|\Omega_d| +\le \left(1+\frac{n(n-1)}4\right)\log2\le n^2 +\qquad(n\ge4). +\] +Also $24n^2\binom n4\le n^6$. The displayed time is consequently at most +$n^8+1\le2n^8$, as claimed. +\end{proof} diff --git a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/rows.tex b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/rows.tex new file mode 100644 index 0000000000000000000000000000000000000000..0d5a2fbb3792aaa9e24cd6be861b1d009f2e9995 --- /dev/null +++ b/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/build/sections/rows.tex @@ -0,0 +1,211 @@ +\section{A variance inequality for three rows}\label{row:section} + +The estimate in this section concerns arbitrary binary matrices with three +rows and prescribed margins. It will control resamplings on a vertex +triple, but does not require a graph interpretation. The equivalent +conditional-projection inequality was proved by Fu, Qin and Wang +\cite[Theorem~5.1]{FQW26}. We give a proof using label transpositions +and conditional variance, separating the randomness of the labels from +that of the block sizes. The count distribution and adjacent-total +coupling below correspond to their Lemmas~5.2--5.4; all steps are +included here. + +The transposition and invariant-coordinate strategy for the fixed-block +argument was inspired by the symmetric-group heat-flow method of +Carlen, Lieb and Loss \cite[Section~2]{CLL06}. The variance proof below +is self-contained; the three-row inequality and count coupling retain +the Fu--Qin--Wang attribution above. + +\begin{lemma}[Fixed block sizes]\label{row:fixed-blocks} +Let $(U_1,U_2,U_3)$ and $(W_1,W_2,W_3)$ be independent uniform ordered +partitions of finite labeled sets $U$ and $W$, with all six block sizes +prescribed. For arbitrary real functions $g_i=g_i(U_i,W_i)$, +\begin{equation}\label{row:fixed-variance} + \operatorname{Var}\Bigl(\sum_{i=1}^3 g_i\Bigr) + \le 2\sum_{i=1}^3\operatorname{Var}(g_i). +\end{equation} +\end{lemma} +\begin{proof} +Subtract the mean from each $g_i$. On the finite conditional space, let +\[ + J=\sum_{\tau}(I-T_\tau), +\] +where $\tau$ ranges over transpositions of two labels within $U$ or +within $W$, and $T_\tau$ acts on functions by permuting those labels. +Every $T_\tau$ is a self-adjoint involution, so +\begin{equation}\label{row:transposition-energy} + \langle f,Jf\rangle + =\frac12\sum_\tau\mathbb E(f-T_\tau f)^2. +\end{equation} +The permutations of $U$ and $W$ act transitively on the ordered +partitions with their prescribed sizes. Since transpositions generate +these permutations, the kernel of $J$ consists precisely of constants. +This also covers empty blocks and label sets; a one-state space has no +nonzero centered functions. + +The space of functions of $(U_i,W_i)$ is invariant under $J$. +Indeed, a transposition within $U$ changes such a function only when it +exchanges a label in $U_i$ with one outside $U_i$. The resulting block +is determined by $U_i$ and the two labels, regardless of which other +block contains the outside label. The same observation applies to $W$. +Every spectral projection of the finite-dimensional operator $J$ is a +polynomial in $J$, and therefore preserves each row-function space. +Write $g_i=\sum_{\lambda>0}g_{i,\lambda}$ for its orthogonal spectral +decomposition. For a fixed transposition, at most two of the three +block tuples change. Hence, pointwise, +\[ + \left(\sum_i(g_{i,\lambda}-T_\tau g_{i,\lambda})\right)^2 + \le 2\sum_i(g_{i,\lambda}-T_\tau g_{i,\lambda})^2. +\] +Using \eqref{row:transposition-energy} gives +\[ + \lambda\left\|\sum_i g_{i,\lambda}\right\|^2 + \le 2\lambda\sum_i\|g_{i,\lambda}\|^2. +\] +Divide by $\lambda>0$ and sum over the orthogonal eigenspaces to obtain +\eqref{row:fixed-variance}. +\end{proof} + +\begin{lemma}[Random block sizes]\label{row:counts} +Let $m,\alpha_1,\alpha_2,\alpha_3$ be nonnegative integers. Give the +triples $x=(x_1,x_2,x_3)$ of nonnegative integers with +$x_1+x_2+x_3=m$ probabilities proportional to +\[ + \prod_{i=1}^3 w_i(x_i),\qquad + w_i(t)=\frac1{t!(t+\alpha_i)!}. +\] +For arbitrary real functions $b_i$ of one coordinate, +\begin{equation}\label{row:count-variance} + \operatorname{Var}\Bigl(\sum_{i=1}^3b_i(x_i)\Bigr) + \le 2\sum_{i=1}^3\operatorname{Var}(b_i(x_i)). +\end{equation} +\end{lemma} +\begin{proof} +The case $m=0$ is immediate. Suppose $m>0$. +We first compare the distributions of two coordinates with adjacent +totals, using the coupling of Fu, Qin and Wang \cite[Lemma~5.2]{FQW26}. +We include the argument. Let $(Y_s,Z_s)$ have probabilities proportional to +$w_j(y)w_k(z)$ on $y+z=s$. For $s\ge1$ and $0\le y\le s-1$, the +ratio of the probability of $Y_s=y$ to that of $Y_{s-1}=y$ is a positive +constant times +\[ + \frac{w_k(s-y)}{w_k(s-1-y)} + =\frac1{(s-y)(s-y+\alpha_k)}, +\] +which is nondecreasing in $y$. Reweighting by a nondecreasing function +increases expectations of nondecreasing functions: for two such +functions $a,b$ and independent identically distributed $Y,Y'$, this +follows from +\[ + \operatorname{Cov}(a(Y),b(Y)) + =\frac12\mathbb E\bigl[(a(Y)-a(Y'))(b(Y)-b(Y'))\bigr]\ge0. +\] +Applying this observation on the common support, and then adding the +possible mass at $y=s$, proves $Y_{s-1}\le_{\mathrm{st}}Y_s$. +Interchanging the coordinates also gives +$Z_{s-1}\le_{\mathrm{st}}Z_s$, or equivalently +$Y_s\le_{\mathrm{st}}Y_{s-1}+1$. +Use the same uniform random variable in the inverse distribution +functions of $Y_{s-1}$ and $Y_s$. Writing $Q$ for these quantile +functions, the identity $Q_{Y_{s-1}+1}(u)=Q_{Y_{s-1}}(u)+1$ shows that +both stochastic inequalities hold simultaneously: +\begin{equation}\label{row:covering-coupling} + Y_{s-1}\le Y_s\le Y_{s-1}+1. +\end{equation} +Since these variables are integer-valued, setting $Z$ from its +respective total yields a coupling in which exactly one of the two +coordinates increases by one. + +Now let $A$ map the direct sum of the three centered marginal function +spaces to the centered joint space by +$A(b_1,b_2,b_3)=\sum_i b_i(x_i)$. +The direct sum has squared norm $\sum_i\|b_i\|^2$. +To prove \eqref{row:count-variance}, it suffices to bound the largest +eigenvalue $\lambda$ of $A^*A$ by $2$. A corresponding nonzero +centered eigenvector satisfies +\begin{equation}\label{row:count-eigenvector} + \lambda b_i(t) + =\mathbb E\left[\sum_{j=1}^3b_j(x_j)\,\middle|\,x_i=t\right], + \qquad 0\le t\le m. +\end{equation} +Every value $0,\ldots,m$ has positive marginal probability. Therefore +\[ + M=\max_{i,\,0\le t0. + \end{aligned} + \label{sl:p} +\end{equation} +All norms of functions in this proof use the uniform measure on the slice. + +Let $M=\binom N2$ and give the pair coefficient space $\mathbb R^M$ its +usual Euclidean inner product. Define +\[ + Bc=\sum_{i,black!70] (-8.13,.60) -- (2.25,.60) + node[right,font=\small] {$x$}; + \node[font=\scriptsize] at (-4,1.32) {exterior: $d<0$}; + \node[font=\scriptsize] at (1,1.32) {interior: $d>0$}; + + \geometryrow{0}{\textbf{Main: initial}\\before the exchange} + \geometryband{0}{-8}{0}{C} + \geometryband{0}{0}{1}{A} + \geometryband{0}{1}{2}{B} + + \geometryrow{-.82}{\textbf{Auxiliary}\\three nested births} + \geometrybandat{-.82}{-8}{-6}{C}{-7.22} + \geometryband{-.82}{-6}{-5}{B} + \geometryband{-.82}{-5}{-4}{A} + \geometrybandat{-.82}{-4}{-3}{C}{-3.72} + \geometryband{-.82}{-3}{-2}{A} + \geometryband{-.82}{-2}{-1}{B} + \geometryband{-.82}{-1}{2}{C} + \draw[lens] (-7,-1.07) rectangle (-3.5,-.57); + \node[below,inner sep=0pt,font=\scriptsize,text=red!65!black] at (-5.25,-1.09) + {exchange lens $Y=(-7,-7/2)$}; + + \geometryrow{-1.64}{\textbf{Main: exchange}\\auxiliary retired} + \geometryband{-1.64}{-8}{-6}{C} + \geometryband{-1.64}{-6}{-5}{B} + \geometryband{-1.64}{-5}{-4}{A} + \geometryband{-1.64}{-4}{0}{C} + \geometryband{-1.64}{0}{1}{A} + \geometryband{-1.64}{1}{2}{B} + + \geometryrow{-2.46}{\textbf{Main: first death}\\$C\to A$ on $(-4,0)$} + \geometryband{-2.46}{-8}{-6}{C} + \geometryband{-2.46}{-6}{-5}{B} + \geometryband{-2.46}{-5}{1}{A} + \geometryband{-2.46}{1}{2}{B} + + \geometryrow{-3.28}{\textbf{Main: second death}\\$A\to B$ on $(-5,1)$} + \geometryband{-3.28}{-8}{-6}{C} + \geometryband{-3.28}{-6}{2}{B} + + \geometryrow{-4.10}{\textbf{Main: birth}\\$B\to C$ on $(-3,0)$} + \geometryband{-4.10}{-8}{-6}{C} + \geometryband{-4.10}{-6}{-3}{B} + \geometryband{-4.10}{-3}{0}{C} + \geometryband{-4.10}{0}{2}{B} + + \geometryrow{-4.92}{\textbf{Main: final death}\\$B\to C$ on $(-6,-3)$} + \geometryband{-4.92}{-8}{0}{C} + \geometryband{-4.92}{0}{2}{B} +\end{tikzpicture} +\endgroup +\caption{Normal-band words at one fixed $0,thick] (-1.1,0)--(1.1,0); + \node[above,align=center,font=\footnotesize] at (0,0.12) + {fill the square\\with $P$}; + \begin{scope}[shift={(3.5,0)}] + \fill[gray!12] (-2,-2) rectangle (2,2); + \fill[blue!18] (0,0) rectangle (2,2); + \fill[orange!22] (0,-2) rectangle (2,0); + \fill[gray!12] (-1,-1) rectangle (1,1); + \draw[dashed,gray] (-1,-1) rectangle (1,1); + \draw[thick] (0,2)--(0,1)--(1,1)--(1,-1)--(0,-1)--(0,-2); + \draw[thick] (1,0)--(2,0); + \draw[red!75!black,thick] (0.84,-0.16) rectangle (1.16,0.16); + \draw[red!75!black,dashed] (0.68,-0.32) rectangle (1.32,0.32); + \fill (1,0) circle (1.5pt); + \draw[gray] (0,0) circle (1.5pt); + \node at (-1.45,0.55) {$P$}; + \node at (0,0.45) {$P$}; + \node at (1.5,1.5) {$A$}; + \node at (1.5,-1.5) {$B$}; + \node[below] at (0,-2.18) {After clearing}; + \node[above,align=center,font=\footnotesize] at (0,2.15) + {New true vertex\\on the rim}; + \end{scope} +\end{tikzpicture} +\caption{A point treatment for three distinct labels, shown in units of +$t$. The dashed gray square is the radius-one treated square; solid +black lines are actual label interfaces. After filling with $P$, the +only true vertex in this example is the meeting of $P,A,B$ on the right +rim. The other ray/rim contacts involve only two labels. Red solid +and dashed squares illustrate the inner and outer holes at the actual +true vertices, enlarged for visibility. In the protocol their radii are +$h_n$ and $2h_n$, with one uniformly small $\epsilon_0$; the same +construction handles the finitely many ray patterns in the proof.} +\label{fig:geometry-points} +\end{figure} diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/main.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..c1af9c25f5539e734c0a323a48b3cc7e352e50f0 --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/main.tex @@ -0,0 +1,73 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\input{glyphtounicode} +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{booktabs,array,enumitem} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,calc,patterns,positioning,decorations.pathreplacing} +\usepackage{xurl} +\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black,hypertexnames=false]{hyperref} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\setlength{\emergencystretch}{2em} +\setcounter{tocdepth}{2} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\id}{id} +\newcommand{\norm}[1]{\lVert #1\rVert} +\newcommand{\abs}[1]{\lvert #1\rvert} +\title{Polynomial PEPS approximation of gapped\texorpdfstring{\\}{ }square-grid ground states} +\author{OpenAI} +\date{September 24, 2026} +\hypersetup{pdftitle={Polynomial PEPS approximation of gapped square-grid ground states},pdfauthor={OpenAI}} +\begin{document} +\maketitle +\begin{abstract} +We prove that the unique ground state of a uniformly gapped nearest-neighbor +Hamiltonian on an $L\times L$ square lattice admits a projected entangled-pair +state approximation with bond dimension polynomial in $L$ and global vector +error at most $L^{-1}$ after normalization. Only the gap of the full Hamiltonian is assumed. +The result is an existence theorem, with constants uniform over +Hamiltonians of fixed local dimension, interaction strength, and gap. +\end{abstract} +\tableofcontents +\clearpage +\input{sections/00-introduction} +\input{sections/01-preliminaries} +\input{sections/02-information} +\input{sections/03-patches} +\input{sections/04-compression} +\input{sections/05-frames} +\input{sections/06-geometry} +\input{sections/07-assembly} +\appendix +\input{sections/08-companion-interface} +\begingroup +\small +\let\originalthebibliography\thebibliography +\renewcommand{\thebibliography}[1]{% + \originalthebibliography{#1}% + \addcontentsline{toc}{section}{\refname}% +} +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/references.bib b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..9228dbece20bd895b71a2c89c9ad2c626d6481a9 --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/references.bib @@ -0,0 +1,338 @@ +@article{LiebRuskai1973, + author = {Lieb, Elliott H. and Ruskai, Mary Beth}, + title = {Proof of the strong subadditivity of quantum-mechanical entropy}, + journal = {Journal of Mathematical Physics}, + volume = {14}, + number = {12}, + pages = {1938--1941}, + year = {1973}, + doi = {10.1063/1.1666274} +} + +@book{EtingofEtAl2011, + author = {Etingof, Pavel and Golberg, Oleg and Hensel, Sebastian and Liu, Tiankai and Schwendner, Alex and Vaintrob, Dmitry and Yudovina, Elena}, + title = {Introduction to Representation Theory}, + series = {Student Mathematical Library}, + volume = {59}, + publisher = {American Mathematical Society}, + year = {2011}, + url = {https://math.mit.edu/~etingof/reprbook.pdf} +} + +@article{NachtergaeleOgataSims2006, + author = {Nachtergaele, Bruno and Ogata, Yoshiko and Sims, Robert}, + title = {Propagation of Correlations in Quantum Lattice Systems}, + journal = {Journal of Statistical Physics}, + volume = {124}, + pages = {1--13}, + year = {2006}, + doi = {10.1007/s10955-006-9143-6}, + eprint = {math-ph/0603064}, + archivePrefix = {arXiv} +} + +@article{NachtergaeleSims2006, + author = {Nachtergaele, Bruno and Sims, Robert}, + title = {{Lieb--Robinson} Bounds and the Exponential Clustering Theorem}, + journal = {Communications in Mathematical Physics}, + volume = {265}, + pages = {119--130}, + year = {2006}, + doi = {10.1007/s00220-006-1556-1}, + eprint = {math-ph/0506030}, + archivePrefix = {arXiv} +} + +@article{Winter2016, + author = {Winter, Andreas}, + title = {Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints}, + journal = {Communications in Mathematical Physics}, + volume = {347}, + number = {1}, + pages = {291--313}, + year = {2016}, + doi = {10.1007/s00220-016-2609-8}, + eprint = {1507.07775}, + archivePrefix = {arXiv} +} + +@article{ChristandlEtAl2007, + author = {Christandl, Matthias and K{\"o}nig, Robert and Mitchison, Graeme and Renner, Renato}, + title = {One-and-a-half quantum {de Finetti} theorems}, + journal = {Communications in Mathematical Physics}, + volume = {273}, + number = {2}, + pages = {473--498}, + year = {2007}, + doi = {10.1007/s00220-007-0189-3}, + eprint = {quant-ph/0602130}, + archivePrefix = {arXiv} +} + +@misc{CarlenVershynina2019, + author = {Carlen, Eric A. and Vershynina, Anna}, + title = {Recovery map stability for the Data Processing Inequality}, + year = {2019}, + eprint = {1710.02409}, + archivePrefix = {arXiv}, + note = {Version 3}, + url = {https://arxiv.org/pdf/1710.02409v3} +} + +@article{Hastings2007, + author = {Hastings, M. 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ETH No. 16242}, + eprint = {quant-ph/0512258}, + archivePrefix = {arXiv} +} + +@article{DupuisBertaWullschlegerRenner2014, + author = {Dupuis, Fr{\'e}d{\'e}ric and Berta, Mario and + Wullschleger, J{\"u}rg and Renner, Renato}, + title = {One-Shot Decoupling}, + journal = {Communications in Mathematical Physics}, + volume = {328}, + pages = {251--284}, + year = {2014}, + doi = {10.1007/s00220-014-1990-4}, + eprint = {1012.6044}, + archivePrefix = {arXiv} +} + +@article{Isserlis1918, + author = {Isserlis, L.}, + title = {On a Formula for the Product-Moment Coefficient of Any Order + of a Normal Frequency Distribution in Any Number of Variables}, + journal = {Biometrika}, + volume = {12}, + number = {1--2}, + pages = {134--139}, + year = {1918}, + doi = {10.1093/biomet/12.1-2.134} +} + +@article{FassinoPistoneRogantin2019, + author = {Fassino, Claudia and Pistone, Giovanni and Rogantin, Maria Piera}, + title = {Computing the Moments of the Complex Gaussian: + Full and Sparse Covariance Matrix}, + journal = {Mathematics}, + volume = {7}, + number = {3}, + pages = {263}, + year = {2019}, + doi = {10.3390/math7030263} +} + +@article{BarthelKlieschEisert2010, + author = {Barthel, Thomas and Kliesch, Martin and Eisert, Jens}, + title = {Real-space renormalization yields finite correlations}, + journal = {Physical Review Letters}, + volume = {105}, + pages = {010502}, + year = {2010}, + doi = {10.1103/PhysRevLett.105.010502}, + eprint = {1003.2319v2}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1003.2319v2}, + note = {Section and equation numbers refer to arXiv:1003.2319v2} +} diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/00-history.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/00-history.tex new file mode 100644 index 0000000000000000000000000000000000000000..1e7d61c3688e6fd65ccdd1a031a0bde69919122f --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/00-history.tex @@ -0,0 +1,83 @@ +Projected entangled-pair states extend matrix product states to +higher-dimensional lattices by contracting virtual pairs through local +maps~\cite{VerstraeteCirac2004PEPS}. Their local tensor structure makes them +natural candidates for describing ground states whose entanglement is +concentrated near region boundaries. In one dimension, quantitative +Schmidt-tail truncation estimates connect entanglement spectra with global +MPS approximation~\cite{VerstraeteCirac2006MPS}. Hastings established the +entropy area law for unique ground states of uniformly gapped spin chains +with bounded finite-range interactions and obtained polynomial-dimension +MPS approximations~\cite{Hastings2007}. The higher-dimensional question +requires a representation compatible across many intersecting cuts. + +There are several distinct PEPS approximation questions. The global +formulation in \cite[Section~2.2, Questions~7--8]{CiracGarreRubioPerezGarcia2019} +asks whether a spectral gap ensures polynomial bond dimension at +inverse-polynomial Hilbert-space norm error. That formulation uses +translation-invariant finite-range Hamiltonians on a torus. +Theorem~\ref{thm:main} addresses the corresponding assertion for arbitrary +bounded on-site and nearest-neighbor interactions on an open square. +Local-observable formulations instead control expectations on fixed-size +regions; the stronger structural formulation in +\cite[Conjecture~2]{SchwarzBuerschaperEisert2017} also requires an injective +approximant with a gapped parent Hamiltonian. The theorem here concerns the +full vector and imposes no such structural requirements on the approximant. + +A positive local approximation result is due to Huang +\cite[Theorem~3]{Huang2026Local}. For states satisfying a R\'enyi entropy +area law of fixed order $0<\alpha<1$ across every rectangle, he obtains +PEPS of bond dimension $e^{O(1/\delta)}$, independent of lattice size, +that approximate norm-one local-observable expectations to error $\delta$. +The observable support scale is held fixed in this bound. The R\'enyi +hypothesis is stronger than a von Neumann entropy area law, and the +conclusion controls local expectations rather than global vector error. + +An entropy bound and a representation theorem are different statements. +For arbitrary chain-state families, bounded von Neumann block entropy +does not ensure efficient global MPS approximation +\cite{SchuchWolfVerstraeteCirac2008}. In higher dimensions, Ge and Eisert +constructed families satisfying strong area laws that have no efficient +classical tensor-network descriptions +\cite[Definition~7 and Corollary~9]{GeEisert2016}. Their description model +includes a bound on the classical complexity of tensor entries; it should +be distinguished from a bound on virtual dimension with arbitrary complex +entries. These examples concern general states, rather than the uniformly +gapped ground-state class in Theorem~\ref{thm:main}. They explain why one +must control compatible operations and their accumulated error, in addition +to the spectra of separate marginals. + +Thermal tensor networks provide another approach. Hastings constructed +local spectral filters and tensor-network descriptions of Gibbs operators +\cite{Hastings2006Local}. Moln\'ar, Schuch, Verstraete, and Cirac refined the +thermal approximation and derived quasipolynomial-dimension ground-state +PEPS under a gap together with a low-energy density-of-states bound +\cite[Section~III.C]{MolnarSchuchVerstraeteCirac2015}. At inverse-polynomial +accuracy their bond bound is exponential in a squared logarithm of system +size. The low-energy counting condition is an additional hypothesis. +A different cooling and postselection construction approximates the unique +ground state of a gapped $d$-dimensional local Hamiltonian at the boundary +of a $(d+1)$-dimensional PEPS +\cite[Section~V]{SchuchWolfVerstraeteCirac2007Complexity}; transferring that +representation to the original lattice is a further problem. + +Recent tensor-network results also distinguish the dimension and the +approximation task. Arad, Firanko, and Jain obtained two-dimensional +mutual-information bounds for locally gapped frustration-free systems and +polynomial-bond matrix product operator approximations to maximally mixed +ground states in one dimension +\cite[Corollaries~2.7 and~2.9]{AradFirankoJain2026}. Such entropy or +one-dimensional operator bounds do not provide the simultaneous +original-grid PEPS conclusion needed here. + +Our analytic inputs are the entropy area law and the stronger collar +estimates of the companion \cite{RepoAreaLaw}. The present proof adapts its +nested-filter and two-family ideas, and then develops compatible changes +of ownership, compression of a distributed density network, and routing on +the prescribed grid. The representation theorem uses the full companion +interface recorded in Appendix~\ref{app:companion}, rather than the +cutwise entropy inequality alone. Its polynomial bond bound permits +arbitrary tensor entries and is an existence statement. The distinction +from efficient contraction is substantive: general PEPS contraction has +worst-case complexity obstructions +\cite{SchuchWolfVerstraeteCirac2007Complexity}, although that fact alone does +not classify the particular approximants constructed here. diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/00-introduction.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..0f6f4dcdd46d9e764b4f1e52718bf330cdc4313a --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/00-introduction.tex @@ -0,0 +1,164 @@ +\section{Introduction}\label{sec:introduction} + +A projected entangled-pair state (PEPS) represents a many-body vector by local +tensors joined along the edges of its interaction lattice. The central +approximation question is whether ground states of gapped local Hamiltonians +admit such representations with modest virtual dimensions. An entropy area +law controls every bipartite cut, but a global vector approximation requires +more: the descriptions at different cuts must be compatible, their errors +must remain small through the entire lattice, and the final tensor network +must use the prescribed edges. + +We prove a uniform polynomial bound for square lattices with arbitrary +bounded on-site and nearest-neighbor interactions. The only spectral +assumption is a unique ground vector and a positive gap for the full +Hamiltonian. We use the entropy area law and the collar estimates of +\cite{RepoAreaLaw}, always with their full-system hypotheses. The new argument +establishes the stronger representation conclusion. + +\subsection{Statement of the result} + +For an integer $L\ge2$, let $\Lambda_L=\{1,\ldots,L\}^2$, with the +nearest-neighbor square-lattice edge set $E(\Lambda_L)$ and open boundary +conditions. Put +\[ + \mathcal H_{\Lambda_L}=\bigotimes_{v\in\Lambda_L}\mathbb C^q. +\] +A PEPS on this grid is the contraction obtained by placing +$\sum_{a=1}^{D_e}|a\rangle\otimes|a\rangle$ on each edge $e$ and applying +at each vertex an arbitrary linear map +\[ + A_v:\bigotimes_{e\ni v}\mathbb C^{D_e}\longrightarrow\mathbb C^q. +\] +Its maximum bond dimension is $\max_eD_e$. The maps need not be isometries, +and the resulting vector need not be normalized. + +\begin{theorem}\label{thm:main} +Fix an integer $q\ge2$ and constants $J,\Delta>0$. There are constants +$C,c>0$, depending only on $q,J,\Delta$, with the following property. +For every $L\ge2$, consider any Hermitian Hamiltonian +\begin{equation}\label{eq:model} + H=\sum_{v\in\Lambda_L}h_v+\sum_{e\in E(\Lambda_L)}h_e, + \qquad \norm{h_v},\norm{h_e}\le J, +\end{equation} +where each summand is supported on its designated site or edge. +Suppose that $H$ has a unique normalized ground vector $\Omega$, up to +phase, with ground energy $E_0$, and +\begin{equation}\label{eq:global-gap} + H-E_0I\ge\Delta\bigl(I-|\Omega\rangle\langle\Omega|\bigr). +\end{equation} +Then there is a nonzero PEPS $\Phi$ on the same grid with maximum bond +dimension at most $CL^c$ such that +\begin{equation}\label{eq:target-error} + \min_{\theta\in\mathbb R} + \left\|\frac{\Phi}{\norm{\Phi}}-e^{i\theta}\Omega\right\| + \le L^{-1}. +\end{equation} +\end{theorem} + +Theorem~\ref{thm:main} gives an affirmative resolution of the polynomial +PEPS approximation assertion for the stated open-square Hamiltonian class. +The relation to other formulations of the PEPS conjecture is explained below. No gap for restrictions to subregions, +frustration freeness, commutativity, translation invariance, or path from a +product state is assumed. The error in \eqref{eq:target-error} concerns the +entire vector. The theorem makes no claim that its tensors can be found or +contracted efficiently; their entries may depend arbitrarily on $H$ and +$\Omega$. + +\subsection{History and significance} +\input{sections/00-history} + +\subsection{Proof strategy} + +The main intermediate object is a distribution of the physical registers +among auxiliary parties. A party owns a collection of registers and may +perform arbitrary linear contractions on them privately. At the start, +one party holds the entire vector $\Omega$; at the end, each physical +site has its own party. Preparing the initial vector is permitted because +we seek a representation, rather than an algorithm for finding it. +The constraint is on communication: each party must participate in only +boundedly many operations with other parties over its whole lifetime. +Each such operation must also have a polynomial-size expansion built from +private contractions and normalized bipartite vectors or covectors. The +bipartite states themselves may have arbitrarily large Schmidt rank. + +\paragraph{Distributing the state while preserving a reference vector.} +The parties are indexed by dyadic squares. We pass from the ownership +assignment on one scale to that on the next by giving each child square +its physical registers. Intermediate assignments need not be dyadic, and +we use additional independent copies of $\Omega$. Each copy has its own +registers and ownership assignment; its reference vector remains $\Omega$ +in the original site coordinates, up to the small-patch encodings described +below. The operations approximate these reference vectors with an error +small enough to sum over the whole construction. + +There are two analytic inputs. First, information between regions becomes +negligible when their separation tapers linearly toward finitely many +points but stays above a sufficiently large multiple of $\log L$ +(Proposition~\ref{prop:angular}). Small mutual information lets the owner +of a separating buffer split its purification approximately into two +factors. This implements a change of owner for the interior region, or +an exchange of assignments between two copies, using private maps and +bipartite resources. To prove the information estimate, an ordered +partition into two tile families bounds the conditional-information cost +of adding a cell. Gaussian filtering with the full-system gap then resets +that cost to an inverse power of $L$ after each annular step. + +Second, near a logarithmic-size square we can project with high accuracy +onto a sum of mutually orthogonal subspaces. Writing +$\mathcal H_A=\bigotimes_{v\in A}\mathbb C^q$ for the registers in +$A\subseteq\Lambda_L$, these subspaces have the form +\[ + \operatorname{span}\{v_{j\ell}\}_{\ell=1}^{d_j} + \otimes\mathcal H_{\Lambda_L\setminus X_j}, + \qquad \sum_jd_j\le CL^c, +\] +where the $X_j$ are nested square samples within the specified patch and +$v_{j\ell}\in\mathcal H_{X_j}$ are orthonormal inside vectors +(Proposition~\ref{prop:patch}). The radius is allowed to depend on $j$. +Recording $(j,\ell)$ in a small register and replacing the corresponding +inside vector by a fixed product vector gives the encoding used near +junctions. The proof combines optimized nested filters, the gap, and the +entropy area law; it does not require an accurate truncation at a single +prescribed boundary. Section~\ref{sec:frames} uses these encodings to +express suitable changes on small patches in terms of pair states and +effects, even when their underlying Hilbert spaces are large. + +The remaining geometric problem is to arrange that these operations +suffice. Along a long edge, our separation estimate may not justify +changing an old owner directly to a new one: a third owner can touch the +opposite side of the region being changed. Exchanging part of an auxiliary copy first supplies the +surrounding regions needed for successive changes of owner. Small patches +handle the finitely many endpoints and junctions, where the separation +would otherwise vanish. Section~\ref{sec:geometry} assembles these moves +so that a block party participates only in boundedly many nearby +repaintings at each of two adjacent scales. Its total participation in +operations with other parties is therefore bounded. + +\paragraph{Compressing the density and returning to a vector.} +The distribution initially uses pair resources of unrestricted dimension. +Theorem~\ref{thm:compression} replaces them in the density calculation by +polynomial sums of product operators. To bound the error from +replacing a chosen set of resources, we expand only operations involving +the parties that hold their endpoints. Bounded lifetime participation +limits the resulting coefficient cost. A Gaussian +second-moment bound then controls the trace-norm error independently of +all private dimensions and resource Schmidt ranks. This produces an +operator tensor network with polynomial virtual dimensions on the party +graph. The compression theorem is independent of the planar construction. + +The protocol retains norm close to one, so its density and the compressed +operator both approximate $|\Omega\rangle\langle\Omega|$ in absolute trace +norm. One product-basis column of the compressed operator gives a nonzero +ket approximation after normalization. Finally, the links join parties +at comparable nearby dyadic scales. Routing them along rows and columns +through their dyadic anchors has constant edge congestion, and therefore +preserves polynomial bond dimension on the original square grid. +Section~\ref{sec:assembly} proves these last steps and the main theorem. + +Section~\ref{sec:preliminaries} fixes the conventions and states the area +law. Sections~\ref{sec:information} and~\ref{sec:patches} prove the two +analytic inputs above; Appendix~\ref{app:companion} records their exact +companion interfaces. All error powers, geometric constants, and +polynomial exponents are fixed before taking $L$ large, uniformly in the +individual Hamiltonian. diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/01-preliminaries.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/01-preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..462bc866b038792fcb706e1641b627430fac225f --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/01-preliminaries.tex @@ -0,0 +1,161 @@ +\section{Conventions and entropy inputs}\label{sec:preliminaries} + +Fix the parameters of Theorem~\ref{thm:main}, and abbreviate +$\Lambda=\Lambda_L$. Unsubscripted norms denote the operator norm for maps +and the Euclidean norm for vectors; $\norm{\cdot}_1$ and +$\norm{\cdot}_2$ denote trace and Hilbert--Schmidt norms. All Hilbert spaces +are finite dimensional. Empty tensor products have dimension one. +Constants denoted by $C,c$ may change between occurrences. They depend only +on $q,J,\Delta$ and explicitly fixed numerical powers, geometric templates, +or tolerances. A polynomial bound means $CL^c$ with this uniform dependence. + +Plane distances are sup-norm distances. For a plane set $T$, write +$T^{+r}=\{x:\dist_\infty(x,T)\le r\}$; sampling always means intersection +with the lattice, and physical sampling includes intersection with $\Lambda$. +Graph neighborhoods, when used for Hamiltonian locality, are neighborhoods +in the induced physical lattice and lie in the corresponding ambient +sup-norm enlargement. Distances to the empty set are infinite. Region +complements in entropy formulas refer to the full physical system, unless +another system is specified. Adjacent region letters denote their disjoint +union. + +For a normalized state $\omega$ and a region $A$, let $\rho_{\omega,A}$ +be its marginal, and set +\begin{align*} + S_\omega(A)&=-\Tr\rho_{\omega,A}\log\rho_{\omega,A},\\ + I_\omega(A:C)&=S_\omega(A)+S_\omega(C)-S_\omega(AC),\\ + I_\omega(A:C\mid B)&=S_\omega(AB)+S_\omega(BC) + -S_\omega(B)-S_\omega(ABC). +\end{align*} +Logarithms are natural and $0\log0=0$. We omit the state subscript when it +is $\Omega$. + +\subsection{The imported area law} + +The following is the range-one square-domain specialization of the companion +theorem. Stating its uniformity here is useful because later auxiliary +regions are not themselves assigned gapped Hamiltonians. + +\begin{proposition}[Entropy area law, {\cite[Theorem~1.1 and Corollary~1.2]{RepoAreaLaw}}] +\label{prop:area-law} +Under \eqref{eq:model}--\eqref{eq:global-gap}, there is +$C_A=C_A(q,J,\Delta)$ such that every $A\subseteq\Lambda$ satisfies +\[ + S_\Omega(A)\le C_A|\partial_\Lambda A|, + \qquad + \partial_\Lambda A=\{\{x,y\}\in E(\Lambda):x\in A, + \ y\notin A\}. +\] +The same constant works for every $L$ and every Hamiltonian in the class. +\end{proposition} + +The companion theorem allows any finite induced square-lattice domain, with one +bounded Hermitian summand per designated nonempty support of bounded graph +diameter. In \eqref{eq:model} the diameter bound is one. Its only spectral +assumption is the unique full-system ground vector and +\eqref{eq:global-gap}. These are precisely the hypotheses used here. +For a physical sample $X$ of a square of radius $r$, this implies +$S_\Omega(X)\le C(1+r)$, including squares clipped by the outer boundary. + +We also use the companion's sublinear collar and amplification propositions. +Their geometric and exponent requirements are checked explicitly in +Proposition~\ref{prop:coarse}; neither is a PEPS approximation statement. +Appendix~\ref{app:companion} records their exact interfaces, including the +positive constraints and two-family construction underlying the companion +argument. The complete companion is \cite{RepoAreaLaw}. + +\subsection{Finite-dimensional information facts} + +We use strong subadditivity, the chain rule, monotonicity of mutual +information under discarding systems, and, for a pure state on $ABCD$, +the duality +\[ + I(A:C\mid B)=I(A:C\mid D). +\] +These are standard entropy identities and inequalities; see +\cite{LiebRuskai1973} and the preliminary section of \cite{RepoAreaLaw}. +For example $|S(A\mid B)|\le\log\dim\mathcal H_A$, so the entropy cost +of a bounded set of individual sites is bounded by its cardinality times +$\log q$. + +For normalized density operators define +$F(\rho,\sigma)=\norm{\sqrt\rho\sqrt\sigma}_1$ and +$D(\rho\Vert\sigma)=\Tr\rho(\log\rho-\log\sigma)$, with the usual +infinite value when the support condition fails. + +We recall the relative-entropy estimate through quantum affinity +\cite[Equation~(2.9)]{CarlenLieb2014} and Uhlmann's purification-overlap +theorem \cite[Sections~2 and~5]{Uhlmann1976}. The finite-dimensional proof +below fixes our root-fidelity convention and includes nonminimal purifying +spaces. + +\begin{lemma}\label{lem:fidelity} +For normalized finite-dimensional states, +\begin{equation}\label{eq:fidelity-information} + F(\rho,\sigma)\ge e^{-D(\rho\Vert\sigma)/2}, + \qquad + D(\rho_{AC}\Vert\rho_A\otimes\rho_C)=I(A:C), +\end{equation} +Moreover, after enlarging the +purifying space if necessary, an isometry on that space realizes overlap +$F(\rho,\sigma)$ between purifications. In particular a +mutual-information bound $\delta$ gives purifications of the joint state +and product state at vector distance at most $\sqrt\delta$, after phase +choice, since $2(1-e^{-\delta/2})\le\delta$. +\end{lemma} +\begin{proof} +Write the eigenvalues and orthonormal eigenvectors of $\rho,\sigma$ as +$(p_i,|i\rangle)$ and $(q_j,|j'\rangle)$. When $D(\rho\Vert\sigma)$ is +finite, $w_{ij}=p_i|\langle i|j'\rangle|^2$ is a probability distribution +on its positive entries, and +\[ + D(\rho\Vert\sigma)=\sum_{i,j}w_{ij}\log\frac{p_i}{q_j}. +\] +Convexity of the exponential gives +\[ + \Tr\sqrt\rho\sqrt\sigma + =\sum_{i,j}w_{ij}\exp\!\left(-\frac12\log\frac{p_i}{q_j}\right) + \ge e^{-D(\rho\Vert\sigma)/2}. +\] +Terms with $p_i=0$ contribute zero. The trace norm is at least the +absolute trace, proving the first inequality; the infinite case is +immediate. The relative-entropy identity follows by taking the trace +against $\log(\rho_A\otimes\rho_C) +=\log\rho_A\otimes I+I\otimes\log\rho_C$ on its support. + +Canonical purifications have coefficient matrices $\sqrt\rho$ and +$\sqrt\sigma$. Varying a unitary on a common sufficiently large +purifying space makes their overlap range over +$\Tr(\sqrt\rho\sqrt\sigma\,U)$, up to a harmless transpose convention. +Polar decomposition gives the maximum absolute value +$\norm{\sqrt\rho\sqrt\sigma}_1$. Every other purification is obtained +from the canonical one by an isometry on the purifying factor. Enlarge +the common factor to accommodate the entire original purifying space, +extend the isometries to unitaries there, and restrict the maximizing +unitary to the original factor. This proves the stated isometry form, +including unused directions in a nonminimal purification. Finally, +the squared vector distance of unit purifications with positive overlap +$F$ is $2(1-F)$. +\end{proof} + +If $\frac12\norm{\rho-\sigma}_1\le\epsilon\le1$, conditional entropy +obeys +\begin{equation}\label{eq:entropy-continuity} + |S(A\mid B)_\rho-S(A\mid B)_\sigma| + \le2\epsilon\log\dim\mathcal H_A + +(1+\epsilon)h\!\left(\frac{\epsilon}{1+\epsilon}\right), +\end{equation} +where $h$ is binary entropy \cite[Lemma~2]{Winter2016}. The dimension is +that of $A$, regardless of the purifying or conditioning space. We may +also use the weaker consequence $C\sqrt\epsilon\log(e\dim\mathcal H_A)$. +All retained physical logarithmic dimensions are $O(L^2)$; hence increasing +a fixed error power absorbs the continuity loss. + +Finally, the gap directly implies +\begin{equation}\label{eq:energy-vector} + \min_\theta\norm{\psi-e^{i\theta}\Omega}^2 + \le\frac{2}{\Delta}\langle\psi,(H-E_0I)\psi\rangle +\end{equation} +for every normalized $\psi$. Indeed the excitation energy bounds +$\Delta(1-|\langle\Omega,\psi\rangle|^2)$, while phase minimization gives +$2(1-|\langle\Omega,\psi\rangle|)$. diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/02-information.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/02-information.tex new file mode 100644 index 0000000000000000000000000000000000000000..4f9a5431ac655e68a80cdfd5fdeedfce046f83ae --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/02-information.tex @@ -0,0 +1,786 @@ +\section{Information bounds for separated regions} +\label{sec:information} + +Throughout this section, distances in the plane are measured in the sup +norm. The distance used in a Hamiltonian propagation estimate is the +nearest-neighbor graph distance on the physical square; every graph ball +is contained in the corresponding ambient sup-norm ball. As in Section~\ref{sec:preliminaries}, $F^{+r}$ denotes the ambient +radius-$r$ enlargement of a plane set $F$. Its physical sample is +$\Lambda\cap F^{+r}$. Ambient rectangles and squares include their +unoccupied locations as well. Constants +may depend on $q,J,\Delta$ and on explicitly fixed geometric parameters, +but not on $L$ or on the Hamiltonian. A lower threshold on $L$, or on a +geometric scale, always has the same uniformity. + +The purpose of this section is to prove that information is negligible +between regions whose separation decreases linearly toward a fixed +number of points, provided the separation is never smaller than a +sufficiently large multiple of $\log L$. We first extract a coarse +collar estimate from the companion results, then prove a conditional +information bound for one cell. A separate consequence of the global +gap, using Gaussian spectral filtering as in \cite{Hastings2006Local}, reduces an information cost to an arbitrarily small inverse +power of $L$ after shrinking the regions by a width proportional to that +cost plus $\log L$. Applying that reduction after each annular step gives the +required logarithmic minimum separation. + +\subsection{The coarse collar interface} + +We specify the additional companion statements used here, beyond the +entropy area law in Proposition~\ref{prop:area-law}. The constants +$C_{\rm tpl}$ and $D_0$ below are fixed template constants from the companion. +In the terminology +of \cite[Sections~9 and~10]{RepoAreaLaw}, a template is a nonempty ambient +lattice set +\[ + T=\bigcup_{a=1}^{N_0}(P_a\cap\mathbb Z^2), + \qquad + n\ge C_{\rm tpl}N_0(s_0+1), + \qquad \diam_\infty T\le n, +\] +where the pieces are closed rectangles or triangles, have sup diameter +at most $s_0$, and have sides of slopes $0,\infty,1,-1$. The companion also +fixes a physical cut $A\subseteq\Lambda$ and its set $Z_A$ of +crossing-edge endpoints. Its template-clearance assumption is +$\dist_\infty(T,Z_A)>4D_0s_0$. Put +\[ + \varepsilon_{\rm col}=10^{-5},\qquad + \ell=\lfloor n^{1-\varepsilon_{\rm col}}\rfloor, + \qquad T_\ell=T+([-\ell,\ell]^2\cap\mathbb Z^2). +\] +The Proposition ``A sublinear collar bound'' in +\cite[Section~9]{RepoAreaLaw} states that, when $\ell\le s_0$ and the +scale is sufficiently large, +\begin{equation} + \begin{split} + X&=A\cap T,\qquad S_0=A\cap T_\ell,\\ + I_\Omega(X:\Lambda\setminus S_0)&\le Cn^{1-\varepsilon_{\rm col}}. + \end{split} + \label{eq:info-repository-collar} +\end{equation} +It also supplies $|S_0|\le Cn^2$ and $Cn$ upper bounds for the numbers +of physical crossing edges of all three cuts $X,S_0,S_0\setminus X$. +All constants and thresholds are uniform in the domain, cut, and +admissible template. + +The Proposition ``Amplification'' in \cite[Section~10]{RepoAreaLaw} +uses precisely these size, three-cut, and information hypotheses, with +any fixed common upper-bound constant. Its fixed exponents are +\[ + \beta_0=1-2\cdot10^{-6},\qquad + \alpha=\frac{k}{k+1}>1-10^{-6}, +\] +where $k$ is a sufficiently large fixed integer. For every fixed integer +$M\ge1$, it gives a graph radius $r$ such that +\begin{equation} + \ell+2r=o(n^{\beta_0}),\qquad + I_\Omega(X:E)\le C_Mn^{-M} + \quad\text{whenever }E\subseteq\Lambda\setminus N_{2r}(S_0). + \label{eq:info-repository-amplification} +\end{equation} +Here $N_{2r}$ is a physical graph neighborhood. The positive system +required by that Proposition is furnished by the Proposition ``Positive +replacement of the Hamiltonian'' in \cite[Section~4]{RepoAreaLaw}, for +each prescribed integer $k$. That replacement assumes the original +full-system eigenvector and gap inequality. Its constants may depend +on $k$, which is fixed here. It does not assume a gap for any restricted +Hamiltonian. Our on-site and nearest-neighbor Hamiltonian has +induced-graph range one and one bounded term per designated support, so +it satisfies the bounded-interaction hypotheses of all these results. + +\begin{proposition}[Coarse information outside a sublinear collar] +\label{prop:coarse} +Fix an integer $N_0\ge1$ and a constant $A_0\ge1$. There are +$\beta<1$, $C<\infty$, and $u_0<\infty$ such that the following holds. +Let $u\ge u_0$ and let $T$ be the full ambient lattice sample of a union +of at most $N_0$ axis-parallel rectangles, with +$\diam_\infty T\le A_0u$. For every physical set $E\subseteq\Lambda$ +satisfying +\[ + \dist_\infty(E,T)>u^\beta, +\] +one has +\[ + I_\Omega(\Lambda\cap T:E)\le C. +\] +The same statement holds for any subset of $\Lambda\cap T$ in its first +argument, for rectangles with consistently assigned open or closed +sides, and after a common translation of the lattice coordinates. +\end{proposition} + +\begin{proof} +An empty ambient sample gives the assertion immediately. Otherwise +omit the empty pieces, choose an integer $s_0$ comparable to $u$ and +large enough to bound every piece diameter, and then choose an integer +$n$ comparable to a sufficiently large fixed multiple of $u$. These +choices ensure both template inequalities. The prescribed cut is +$A=\Lambda$. Consequently $Z_A=\varnothing$, so all template-clearance +conditions are vacuous. Since the ratio $n/s_0$ is fixed, +\[ + \lfloor n^{1-\varepsilon_{\rm col}}\rfloor\le s_0 +\] +for all sufficiently large $u$. Equation~\eqref{eq:info-repository-collar} +therefore supplies every hypothesis of +\eqref{eq:info-repository-amplification}; fix its common constant once +and apply it, for example, with $M=1$. + +Choose $\beta$ strictly between $\beta_0$ and $1$. Because $n$ is a +fixed multiple of $u$, +\[ + \ell+2r0$. Tile the plane by an axis-parallel grid of squares of side +$s\ge1$. Let $G$ be the physical sample of one cell, let $B$ be the +physical sample of any union of other cells, and let +$C\subseteq\Lambda\setminus(B\cup G)$ satisfy +\[ + \dist_\infty(C,\overline{G}_{\rm amb})\ge cs, +\] +where $\overline{G}_{\rm amb}$ denotes the full closed ambient cell. +There are fixed constants $C_1,K<\infty$ such that +\begin{equation} + I_\Omega(G:C\mid B)\le C_1(1+\log s)^K. + \label{eq:info-cell-cmi} +\end{equation} +The constants are independent of which exterior cells comprise $B$ +and of the shape of $C$ away from the cell. +\end{lemma} + +\begin{proof} +Outside the chosen cell, color all cells of $B$ by forbidden color $0$ +and all other cells by forbidden color $1$. In particular this colors +also the physical sites outside $B,G,C$. The interior of the chosen +cell is initially allowed. We construct two ordered tile families +$\mathcal F_0,\mathcal F_1$, together with a set of exceptional sites. +The order in a family is its order of creation. Whenever an allowed +region is assigned to family $i$, give it forbidden color $i$ for all +subsequent stages. A tile $U$ in family $i$ will satisfy +\begin{equation} + I_\Omega\bigl(U:C\cup F_i\cup + \{\text{earlier tiles of family }i\}\bigr)\le C_2, + \label{eq:info-tile-cost} +\end{equation} +where $F_i$ is the physical exterior of the initial cell of forbidden +color $i$. Tile notation in an entropy denotes its physical sample. + +Assume for the moment that such a partition has been constructed. Its +information cost bounds the desired conditional information as follows. +Apply the chain rule +for $I(G:C\mid B)$ by listing first $\mathcal F_0$ in creation order, +then all exceptional sites, then $\mathcal F_1$ in reverse creation +order. For a tile $U$ the chain-rule term is $I(U:C\mid Z)$, where +$Z$ is $B$ together with previously listed sites. In family $0$, +\[ + I(U:C\mid Z)=I(U:CZ)-I(U:Z)\le I(U:CZ), +\] +and $Z$ consists of forbidden-$0$ exterior sites and earlier +family-$0$ tiles. Equation~\eqref{eq:info-tile-cost} applies. +For a family-$1$ tile, purity on the whole physical system gives +\[ + I(U:C\mid Z)=I\bigl(U:C\mid(UCZ)^c\bigr). +\] +The complementary conditioning set contains only forbidden-$1$ +exterior sites and as-yet unprocessed family-$1$ tiles. Because this +family is processed in reverse order, those tiles were created earlier +than $U$. Bounding the last conditional mutual information by the +ordinary mutual information with $C\cup(UCZ)^c$ again invokes +\eqref{eq:info-tile-cost}. Each exceptional one-site term is at most +$2\log q$, by the conditional entropy dimension bound. Thus the desired +bound follows once the number of tiles and exceptional sites is at most +$C(1+\log s)^K$. This explains both the two exterior +colors and the opposite orders in which the two families will be read. + +We now construct that partition by an ambient recursion. Fix +\[ + \beta<\gamma<\gamma'<1, +\] +with $\beta$ from Proposition~\ref{prop:coarse} for bounded rectangular +unions. At a given generation, all allowed residual material lies in +active axis-parallel squares of a common side $u$. Distinct active +squares have mutual distance greater than $10u$. In the ambient +radius-$u$ padding of each active square, the three labels (allowed, +forbidden $0$, forbidden $1$) are constant on chambers of boundedly +many horizontal and vertical lines. The square sides are included, +and distinct parallel lines are at least $u/8$ apart. Initially there +is one active square, the original grid cell, with $u=s$. Its padded +window meets only boundedly many grid lines, which verifies the +invariant regardless of the exterior color choices. + +Choose a sufficiently large fixed stopping threshold $u_*$. At a +nonterminal square put $t_0=u^\gamma$. Increase $u_*$ so that all +inequalities below hold for $u\ge u_*$. Around each represented line +intersection within distance $10t_0$ of the square, reserve a closed +square of radius $4t_0$ for the next generation. Reserving a hole does +not change a previously fixed color; only still-allowed material is +reserved. The hole squares are disjoint. Every allowed point of the +active square outside these holes can see at most one forbidden color +within distance $t_0$. Indeed it cannot see two parallel represented +lines, because $2t_0\beta$, both distances exceed the +collar required by Proposition~\ref{prop:coarse} after increasing $u_*$. +This proves \eqref{eq:info-tile-cost} for every new tile. In particular, +it bounds the information with the union of all earlier tiles of that +family, not just with any one earlier tile. + +The new active squares are the reserved holes that contain allowed +material. Their side is +\[ + u'=8t_0. +\] +Distinct represented intersections are separated by at least $u/8$. +Thus, for sufficiently large $u_*$, the new squares are at mutual +distance greater than $10u'$. Squares descending from different +parents inherit even larger separation. To verify the chamber +invariant, consider a child's radius-$u'$ padded window, of radius +$12t_0$ about its center. The old line spacing implies that the only +old horizontal and vertical lines in this window are the lines through +its center. New boundaries in the window occur only at offsets +\[ + 0,\quad \pm t_0,\quad \pm4t_0 +\] +from those two lines: these account for the distance test, the reserved +hole, and any clipping by the old active-square sides. Other holes +and old parallel lines are outside the window. Consequently the new +parallel-line spacing is at least $t_0=u'/8$, and the number of lines +is uniformly bounded. This proves the invariant at the next generation. +The same verification covers a reserved hole centered on or slightly +outside a parent side; old fixed colors are retained and only the +allowed intersection is processed. + +At scale below $u_*$, stop and make every remaining physical site +exceptional. There are boundedly many children per active square and +boundedly many sites in each terminal square. Choose $u_*$ also so that +\[ + 8u^\gamma\le u^{\gamma'}\qquad(u\ge u_*). +\] +The number of generations is $O(1+\log\log(s+e))$. Bounded branching +therefore bounds the total number of tiles and exceptional sites by +$C(1+\log s)^K$ for some fixed $C,K$. If $s0$ there is $C_p<\infty$ with the following +property for all sufficiently large $L$. Let $R,C\subseteq\Lambda$ +be disjoint, let $R_0\subseteq R$ and $C_0\subseteq C$, and suppose +\[ + I_\Omega(R:C)\le b,\qquad b\ge0. +\] +If +\begin{equation} + \dist_\infty(R_0,\Lambda\setminus R)>w,\qquad + \dist_\infty(C_0,\Lambda\setminus C)>w, + \qquad w=C_p(1+b+\log L), + \label{eq:info-reset-width} +\end{equation} +then +\[ + I_\Omega(R_0:C_0)\le L^{-p}. +\] +Only the original full-system gap is assumed. +\end{lemma} + +\begin{proof} +If either core is empty there is nothing to prove. Use two replicas +and put +\[ + \Psi=\Omega^{\otimes2},\qquad F=F_R,\qquad \Phi=F\Psi, +\] +where $F_R$ exchanges the two replicas on $R$. Our aim is to construct +a unitary $U$, supported outside $R_0\cup C_0$ in both replicas, for +which $U\Psi$ is close to $\Phi$. Such a unitary leaves the joint core marginal in one replica +unchanged. In $\Phi$, however, $R_0$ and $C_0$ come from different +original replicas, so this marginal is $\rho_{R_0}\otimes\rho_{C_0}$. +Closeness of the two vectors will therefore force small information +between the cores. + +We first obtain a controlled overlap using an operator on the physical +buffer $Q=(R\cup C)^c$ of the two replicas. Spectral filtering and +localization will turn it into a two-sided approximation to the desired +map; its polar factor will supply $U$ on the same physical support. + +The relative entropy of $\rho_{RC}$ from +$\rho_R\otimes\rho_C$ is $I_\Omega(R:C)$. The required support +inclusion holds: positivity implies that a joint state is supported +on the tensor product of the supports of its marginals. For example, +the expectation of the positive joint state on the kernel projector +of either marginal is zero, so that projector annihilates the joint +state. Equation~\eqref{eq:info-relative-entropy-overlap} and the +purification formula therefore give an isometry +\[ + V:\mathcal H_Q\longrightarrow\mathcal H_{b_R}\otimes\mathcal H_{b_C} +\] +and normalized vectors $a_{Rb_R}$, $a'_{Cb_C}$ such that +\begin{equation} + \bigl|\langle a\otimes a',(\id_{RC}\otimes V)\Omega\rangle\bigr| + \ge e^{-b/2}. + \label{eq:info-split-overlap} +\end{equation} +The factors $b_R,b_C$ may first be enlarged by blank ancillary factors +so that the product purification and an isometric embedding of the +whole original buffer both fit. Let +$\widetilde\Omega=(\id_{RC}\otimes V)\Omega$. +Its largest Schmidt probability across $Rb_R\mid Cb_C$ is at least +$e^{-b}$, hence +\[ + \Tr\rho_{\widetilde\Omega,Rb_R}^{2}\ge e^{-2b}. +\] +Pull the swap on $b_R$ back to the original doubled buffer: +\[ + W=(V^\dagger\otimes V^\dagger)F_{b_R}(V\otimes V). +\] +It is a Hermitian contraction on the two physical copies of $Q$. +The auxiliary spaces have now disappeared. Since $V$ acts outside +$R$, the swap identity for purity gives the real positive number +\begin{equation} + z:=\langle\Phi,W\Psi\rangle + =\langle\widetilde\Omega^{\otimes2}, + F_RF_{b_R}\widetilde\Omega^{\otimes2}\rangle + =\Tr\rho_{\widetilde\Omega,Rb_R}^{2} + \ge e^{-2b}. + \label{eq:info-reset-overlap} +\end{equation} + +Write $\mathbb H$ for the sum of the original Hamiltonian on the two +replicas and $\mathbb H'=F\mathbb H F^\dagger$. Treat the pair of +replica registers at a site as one site. Both Hamiltonians have range +one and uniformly bounded interaction degree and strength. Their +unique ground vectors are $\Psi,\Phi$, their common ground energy is +$\mathcal E=2E_0$, and their gaps are at least $\Delta$. Their +difference is supported on endpoints of edges crossing $R$: terms +entirely inside $R$ merely exchange the two identical replica terms, +and all other noncrossing terms are unchanged. + +Put $B_*=1+b+\log L$ and choose a fixed error exponent +$r_*=2p+10$. For $h>0$, let +\[ + g_h(t)=(2\pi h)^{-1/2}e^{-t^2/(2h)},\qquad + M=\int_{\mathbb R}g_h(t)e^{it\mathbb H'}W e^{-it\mathbb H}\,dt. +\] +This is a contraction. The spectral theorem gives +\[ + M\Psi=e^{-h(\mathbb H'-\mathcal E)^2/2}W\Psi, + \qquad + M^\dagger\Phi=e^{-h(\mathbb H-\mathcal E)^2/2}W^\dagger\Phi. +\] +Consequently +\begin{equation} + \norm{M\Psi-z\Phi},\quad\norm{M^\dagger\Phi-z\Psi} + \le e^{-h\Delta^2/2}. + \label{eq:info-gaussian-two-sided} +\end{equation} +Choose $h=A B_*$ with $A\Delta^2/2\ge10+r_*+20$. The errors in +\eqref{eq:info-gaussian-two-sided} are then bounded by +$e^{-10b}L^{-r_*}$ up to a harmless fixed factor. Truncate the integral +to $|t|\le T=T_0B_*$. The resulting operator-norm error is at most +\[ + 2e^{-T^2/(2h)}=2e^{-T_0^2 B_*/(2A)}. +\] +Choosing $T_0$ after $A$ with $T_0^2/(2A)\ge10+r_*+20$ gives the +same error bound. + +For completeness we prove the finite-time locality estimate needed +to localize this truncated integral. It is the usual commutator +argument for bounded finite-range interactions; see also +\cite{NachtergaeleOgataSims2006}. For a norm-one single-site operator +$B_y$, define +\[ + c_y(S,t)=\sup_{\substack{\supp A\subseteq S\\\norm A\le1}} + \norm{[\tau_t(A),B_y]}, + \qquad \tau_t(A)=e^{itK}Ae^{-itK}, +\] +where $K$ is either doubled Hamiltonian. If its interaction terms are +$k_Z$, then, for $t\ge0$, +\begin{equation} + c_y(S,t)\le c_y(S,0)+ + 2\sum_{Z:Z\cap S\ne\varnothing}\norm{k_Z} + \int_0^t c_y(Z,v)\,dv. + \label{eq:info-commutator-recursion} +\end{equation} +Indeed $[K,A]$ contains only interactions meeting the initial support +$S$. Differentiate $[\tau_t(A),B_y]$ and use Jacobi's identity. +The homogeneous part is commutation by the Hermitian operator +$\sum_{Z:Z\cap S\ne\varnothing}\tau_t(k_Z)$ and is removed by a +unitary change of frame. The remaining term for $Z$ has norm at most +$2\norm A\norm{k_Z}c_y(Z,t)$, proving +\eqref{eq:info-commutator-recursion}. Negative times have the same +bound. + +Iteration produces chains of overlapping interaction supports with +factorial time denominators. Only chains reaching $y$ contribute an +initial commutator. The first support has at most $C|S|$ choices, +each subsequent support has at most a fixed number of choices, and a +chain reaching distance $d_K(S,y)$ must have length at least a fixed +multiple of that distance, up to an additive constant. Here $d_K$ +is the physical site-graph distance; for either doubled Hamiltonian +it is the original nearest-neighbor distance. Bounding the resulting +exponential series with a weight on chain length yields constants +$v,\mu,C>0$ such that +\begin{equation} + \norm{[\tau_t(A),B_y]} + \le C|S|\norm A\exp\bigl(v|t|-\mu d_K(S,y)\bigr). + \label{eq:info-lr} +\end{equation} +The remainder of the iteration vanishes by its factorial denominator. +The constants depend only on the fixed local interaction bounds. + +For a site set $D$, let $\mathcal P_D$ be conditional expectation +onto operators supported on $D$, obtained by averaging over independent +unitaries at all sites outside $D$. It contracts operator norm and +preserves Hermitian operators. Successively averaging single sites +and using \eqref{eq:info-lr} gives +\begin{equation} + \norm{\tau_t(A)-\mathcal P_{N_\ell(S)}\tau_t(A)} + \le C|S|L^2\norm A\,e^{v|t|-\mu\ell}. + \label{eq:info-operator-localization} +\end{equation} +No Hilbert-space dimension appears in this averaging bound. + +Let $Z_R$ be the endpoints of edges crossing $R$, and put +\[ + V(t)=e^{it\mathbb H'}e^{-it\mathbb H}. +\] +Its left generator is +\[ + \frac{d}{dt}V(t)=iG(t)V(t),\qquad + G(t)=e^{it\mathbb H'}(\mathbb H'-\mathbb H)e^{-it\mathbb H'}. +\] +The difference $\mathbb H'-\mathbb H$ is a sum of $O(L^2)$ uniformly +bounded terms, each on at most two sites of $Z_R$. Apply +\eqref{eq:info-operator-localization} to those terms separately and +replace $G(t)$ by +$G_\ell(t)=\mathcal P_{N_\ell(Z_R)}G(t)$. This replacement is +Hermitian, and +\[ + \norm{G(t)-G_\ell(t)}\le CL^4e^{v|t|-\mu\ell}. +\] +The evolution $V_\ell(t)$ generated by $G_\ell(t)$, with +$V_\ell(0)=\id$, is unitary and supported on $N_\ell(Z_R)$. Duhamel's formula for two unitary +evolutions therefore gives +\[ + \norm{V(t)-V_\ell(t)}\le CL^4|t|e^{v|t|-\mu\ell}. +\] +Localize $e^{it\mathbb H'}We^{-it\mathbb H'}$ by conditional +expectation onto $N_\ell(Q)$. It stays a contraction, and its error +is at most $CL^4e^{v|t|-\mu\ell}$. The factorization +\[ + e^{it\mathbb H'}We^{-it\mathbb H} + =\bigl(e^{it\mathbb H'}We^{-it\mathbb H'}\bigr)V(t) +\] +thus has a contraction approximation supported on +\[ + S=N_\ell(Q\cup Z_R) +\] +with operator-norm error at most +$CL^4(1+T)e^{vT-\mu\ell}$ uniformly for $|t|\le T$. +If $Q$ or $Z_R$ is empty, its factor is already a scalar or the +identity and needs no localization. + +Take $\ell=\lceil K_{\rm loc}B_*\rceil$. Because +$\log L\le B_*$ and $\log(1+T)\le C+B_*$, a sufficiently large fixed +$K_{\rm loc}$, chosen after $T_0$, makes the last error at most +$Ce^{-10b}L^{-r_*}$. Averaging the localized integrands against the +truncated Gaussian gives a square operator $M_{\rm loc}$ supported on +$S$, with $\norm{M_{\rm loc}}\le1$. Combining the three errors, we +have +\begin{equation} + \norm{M_{\rm loc}\Psi-z\Phi},\quad + \norm{M_{\rm loc}^\dagger\Phi-z\Psi} + \le C e^{-10b}L^{-r_*}. + \label{eq:info-local-two-sided} +\end{equation} + +Set $D=M_{\rm loc}/z$. We arranged both estimates in +\eqref{eq:info-local-two-sided} so that, besides mapping $\Psi$ close +to $\Phi$, $D$ is almost isometric on $\Psi$: applying $D^\dagger$ +to the first error and adding the second controls +$(D^\dagger D-\id)\Psi$. By +\eqref{eq:info-reset-overlap} and \eqref{eq:info-local-two-sided}, +\begin{align} + \norm D&\le e^{2b},\\ + \norm{D\Psi-\Phi},\quad\norm{D^\dagger\Phi-\Psi} + &\le C e^{-8b}L^{-r_*},\\ + \norm{(D^\dagger D-\id)\Psi} + &\le C e^{-6b}L^{-r_*}. + \label{eq:info-reset-polar-errors} +\end{align} +For the last estimate, write the defect as +$D^\dagger(D\Psi-\Phi)+(D^\dagger\Phi-\Psi)$. +The polar partial isometry of $D$, as a square operator on the finite +space $\mathcal H_S$, extends to a unitary $U$ on that same space. +Indeed its two support subspaces have the same dimension, as do their +orthogonal complements. Tensor this unitary with the identity outside +$S$. If $|D|=(D^\dagger D)^{1/2}$, then the scalar inequality +$(\sqrt{x}-1)^2\le(x-1)^2$ for $x\ge0$ gives +\[ + \norm{(U-D)\Psi}=\norm{(\id-|D|)\Psi} + \le\norm{(\id-D^\dagger D)\Psi}. +\] +Consequently +\begin{equation} + \norm{U\Psi-\Phi}\le C e^{-6b}L^{-r_*}\le CL^{-r_*}. + \label{eq:info-reset-unitary} +\end{equation} +In particular, the large norm of $D$ has been explicitly paid for in +\eqref{eq:info-reset-polar-errors}; no unitary dilation on an additional +system is used. + +We check the support against both cores. Every site of $Q$ is more +than $w$ from $R_0$ and from $C_0$. For an edge crossing $R$, write its +endpoints as $x\in R$ and $y\notin R$. The point $y$ is more than $w$ +from $R_0$, so $x$ is more than $w-1$ from $R_0$. Since $R\cap C$ is +empty, $x\in C^c$ is more than $w$ from $C_0$, so $y$ is more than +$w-1$ from $C_0$. Thus every site of $Z_R$ is more than $w-1$ from +both cores. Choose $C_p$ in \eqref{eq:info-reset-width} larger than +$K_{\rm loc}+3$. Then $S=N_\ell(Q\cup Z_R)$ is disjoint from both cores in +both replicas, including the rounding of $\ell$. + +The unitary $U$ therefore leaves their joint two-copy marginal unchanged. +In $\Phi=F_R\Psi$, retaining $R_0$ and $C_0$ from one replica gives +exactly $\rho_{R_0}\otimes\rho_{C_0}$, because $R_0$ came from the +other original replica and $C_0$ did not. Equation~\eqref{eq:info-reset-unitary} +and trace-norm contractivity under partial trace show that the original +$R_0C_0$ marginal is within trace distance $CL^{-r_*}$ of this product. +The conditional entropy continuity bound +\[ + |S(A\mid B)_\rho-S(A\mid B)_\sigma| + \le 2\delta\log\dim\mathcal H_A + +(1+\delta)h\left(\frac{\delta}{1+\delta}\right) + \le C\sqrt\delta\log(e\dim\mathcal H_A), +\] +where $\delta=\tfrac12\norm{\rho-\sigma}_1$ and $h$ is binary entropy, +is valid for normalized finite-dimensional states +\cite{Winter2016}. Applied to $I(R_0:C_0)=S(R_0)-S(R_0\mid C_0)$, +it costs at most $C\sqrt\delta\,L^2$, since $q$ is fixed. We conclude +\[ + I_\Omega(R_0:C_0)\le CL^{2-r_*/2}=CL^{-p-3}\le L^{-p} +\] +for sufficiently large $L$. This proves the Lemma. +\end{proof} + +\subsection{Regions tapering toward finitely many points} + +\begin{proposition}[Information under angular separation] +\label{prop:angular} +Fix $a>0$, $A_0\ge1$, an integer $m_0\ge0$, and $p>0$. +There is $D<\infty$ such that the following holds for all sufficiently +large $L$. Let $\mathcal V$ be a set of at most $m_0$ marked points +in the plane, let $X,E\subseteq\Lambda$ be disjoint, and let +\[ + D\log L\le n\le A_0L, + \qquad \diam_\infty X\le A_0n. +\] +Put $s_{\min}=D\log L$ and +$d_{\mathcal V}(x)=\min_{v\in\mathcal V}|x-v|_\infty$, with +$d_{\varnothing}=+\infty$. Suppose that for every $x\in X$, +\begin{equation} + \dist_\infty(x,E) + \ge a\max\{s_{\min},\min(n,d_{\mathcal V}(x))\}. + \label{eq:info-angular-separation} +\end{equation} +Then +\[ + I_\Omega(X:E)\le L^{-p}. +\] +The constant $D$ and the scale threshold are uniform in the marked +points, physical sets, and Hamiltonian. +\end{proposition} + +\begin{proof} +An empty target or exterior is immediate. Fix a small constant +$a'>0$, for example $a'<\min(a/100,1/100)$, and then fix +$00. + \label{eq:info-cell-clearance} +\end{equation} +This is a fixed positive multiple of its side $ds$. In particular +$R$ and $C$ are disjoint. Add the remaining cells one at a time. +The previously included physical sites are a union of whole cells, +so the chain rule and Lemma~\ref{lem:cell-cmi} apply at each addition. +For all sufficiently large $L$, $ds\ge dD\log L$ is above that +Lemma's fixed scale threshold. Thus, after absorbing the bounded +number of additions and the initial $L^{-p}$, +\begin{equation} + I_\Omega(R:C)\le C_b(1+\log s)^K. + \label{eq:info-before-reset} +\end{equation} +Here $C_b,K$ are fixed independently of $D,L$ and of the current layer. + +Apply Lemma~\ref{lem:reset} with +\[ + R_0=\Lambda\cap A_j^{+a's},\qquad C_0=\Lambda\cap E^{+a's}. +\] +The set $R$ contains the physical radius-$1.5a's$ enlargement of +$A_j$. Both cores therefore have distance at least $a's/2$ from +the respective physical complements of $R,C$. We can safely use +any strictly smaller available width $c_*s$, for instance $c_*=a'/3$. +In view of \eqref{eq:info-before-reset}, the width required by the +reset is bounded by +\begin{equation} + w(s)\le C_3(1+\log s)^K+C_4\log L. + \label{eq:info-reset-scale-cost} +\end{equation} +We verify uniformly over all layers that this is less than $c_*s$. +First fix all constants in the cell bound and reset, including $K$, +$C_3,C_4,c_*$. Choose the fixed number +\[ + D\ge\frac{4C_4}{c_*}. +\] +The function $f(s)=(1+\log s)^K/s$ has logarithmic derivative +\[ + \frac{f'(s)}{f(s)} + =\frac1s\left(\frac K{1+\log s}-1\right), +\] +so it is decreasing for $s>e^{K-1}$. Increase the fixed threshold +$L_0$ so that, for all $L\ge L_0$, +\[ + D\log L\ge e^{\max(K,1)},\qquad + C_3\frac{(1+\log(D\log L))^K}{D\log L}\le\frac{c_*}{2}. +\] +Such a threshold exists because +$(\log\log L)^K/\log L\to0$ for every fixed $K$. +For every scale $s\ge D\log L$ it follows that +\[ + \frac{w(s)}s + \le C_3 f(D\log L)+\frac{C_4}{D} + \le\frac{3c_*}{4}0\) and \(p>0\). There are constants \(C,c>0\), depending +only on \(q,J,\Delta,B_0,p\), with the following property. For every +\(L\geq2\), center \(z\in\mathbb R^2\), and +\(0j\), and let \(B\) be any anti-Hermitian operator on \(X_j\). +Vary \(x_j\) by \(x_j(s)=e^{sB}x_j e^{-sB}\), keeping all other +variables fixed. Functional calculus gives +\[ + L_j(s)=e^{sB}L_j e^{-sB},\qquad + \left.\frac{dL_j(s)}{ds}\right|_{s=0}L_j^{-1} + =B-L_j B L_j^{-1}. +\] +For a differentiable invertible \(M(s)\), its logarithmic norm +derivative on the fixed input \(\Omega\) is +\[ + \left.\frac{d}{ds}\log\norm{M(s)\Omega}\right|_{s=0} + =\operatorname{Re} + \langle\phi,M'(0)M(0)^{-1}\phi\rangle . +\] +Here \(M'(0)M(0)^{-1}\) is \(B-L_j B L_j^{-1}\) conjugated +by \(L_m\cdots L_{j+1}\). Strip those outer conjugations using +\eqref{eq:patch-trace-stripping}. Stationarity at the minimum yields +\[ + 0=\operatorname{Re}\Tr\bigl(\rho_j(B-L_j B L_j^{-1})\bigr) + =-\operatorname{Re}\Tr\bigl(L_j^{-1}\rho_jL_j B\bigr). +\] +The term \(\Tr(\rho_jB)\) is purely imaginary. An operator whose +real trace pairing with every anti-Hermitian operator vanishes is +Hermitian. Consequently +\[ + L_j^{-1}\rho_jL_j=L_j\rho_jL_j^{-1}, + \qquad + \rho_jL_j^2=L_j^2\rho_j . +\] +Positivity of \(L_j\) gives commutation with \(L_j\); functional +calculus then gives commutation with \(x_j\). This starts at \(j=m\) +with no outer filters and proves the induction. Notice that no +commutation between two different filters has been claimed. + +In a common eigenbasis of \(x_j\) and \(\rho_j\), write \(x_i\) +and \(p_i\) for their eigenvalues. A diagonal variation of \(x_j\) +within the probability simplex has logarithmic norm derivative +\begin{equation} + -\frac{a_j}{2}\sum_i \frac{p_i}{x_i+b}\,dx_i. + \label{eq:patch-simplex-derivative} +\end{equation} +Again, this follows by stripping the outer conjugations; the +diagonal variation commutes with \(x_j+b\id\). +For two indices with \(x_i,x_k>0\), mass can be moved in either +direction, so the ratios \(p_i/(x_i+b)\) agree. Moving mass from +an index with \(x_i>0\) to any zero coordinate gives the one-sided +inequality. Thus there is \(\lambda>0\) such that +\begin{equation} + \frac{p_i}{x_i+b}=\lambda\quad(x_i>0), + \qquad + \frac{p_i}{b}\leq\lambda\quad(x_i=0). + \label{eq:patch-kkt} +\end{equation} +Indeed, if the common ratio on the positive coordinates were zero, +the second inequality would force every \(p_i\) to be zero. +Writing \(r\) for the number of positive \(x_i\), we also have +\[ + \lambda(1+br)=\sum_{x_i>0}p_i\leq1, + \qquad\hbox{so}\qquad 0<\lambda\leq1. +\] +Equivalently, +\begin{equation} + x_i+b=\max\{p_i/\lambda,b\}. + \label{eq:patch-clipped-eigenvalues} +\end{equation} +If \(\ell_i=(x_i+b)^{-a_j/2}\) is the corresponding eigenvalue +of \(L_j\), then for \(p_i,p_k>0\), +\begin{equation} + \abs{\log(\ell_i/\ell_k)} + \leq \frac{a_j}{2}\abs{\log(p_i/p_k)} . + \label{eq:patch-log-ratio} +\end{equation} +This follows because \(t\mapsto\max\{t,\log b\}\) is +\(1\)-Lipschitz, applied to \(\log(p_i/\lambda)\). +The regularization keeps all \(\ell_i\) finite, including on zero +marginal eigenspaces; no ratio of zero probabilities is used. + +\emph{A dimension-independent energy estimate.} +We next show that the minimizing vector has uniformly small +excitation energy. The two distances +\(\abs{x-z}_\infty\) and \(\abs{y-z}_\infty\) at the endpoints +of a nearest-neighbor edge differ by at most one. A closed square +of radius \(r\) cuts that edge only when +\[ + \min\{\abs{x-z}_\infty,\abs{y-z}_\infty\} + \leq r + <\max\{\abs{x-z}_\infty,\abs{y-z}_\infty\}. +\] +Since selected radii differ by at least two, an interaction crosses +at most one selected square. If it crosses \(X_j\), its support +is disjoint from every \(X_k\) with \(kj\). This argument includes exact boundary +equalities and repeated clipped samples. + +The ground equation gives +\[ + M H M^{-1}\phi=E_0\phi. +\] +For an interaction \(h\) crossing \(X_j\), remove the conjugations +by the containing outer filters using +\eqref{eq:patch-trace-stripping}. The smaller filters commute +with \(h\) by disjointness. Therefore +\begin{equation} + \langle\phi,MhM^{-1}\phi\rangle + =\langle\phi,L_j h L_j^{-1}\phi\rangle. + \label{eq:patch-single-crossing} +\end{equation} +For a term crossing none of the selected squares the same argument +leaves its original expectation unchanged: strip the containing +filters, then commute past all disjoint ones. On-site terms never +cross a square. + +Consider now a Hermitian nearest-neighbor term \(h\) crossing +\(X_j\). Use the Schmidt decomposition +\[ + \phi=\sum_{i:p_i>0}\sqrt{p_i}\, + e_i\otimes f_i +\] +across \(X_j\) and its complement, where the \(e_i\) are chosen +in a simultaneous eigenbasis of \(L_j\) and \(\rho_j\). Put +\[ + h_{ik}=\langle e_i\otimes f_i,\, + h(e_k\otimes f_k)\rangle. +\] +Hermiticity gives \(h_{ki}=\overline{h_{ik}}\). Pairing the +\((i,k)\) and \((k,i)\) terms consequently gives +\begin{align} + &\operatorname{Re}\langle\phi,L_j h L_j^{-1}\phi\rangle + -\langle\phi,h\phi\rangle \notag\\ + &\qquad = + \sum_{i,k:p_i p_k>0} + \sqrt{p_i p_k} + \bigl(\cosh(\log(\ell_i/\ell_k))-1\bigr) + \operatorname{Re}h_{ik}. + \label{eq:patch-hermitian-pairing} +\end{align} +For \(0\leq a\leq1\), the power series of the hyperbolic cosine +shows that +\[ + \cosh(ay)-1\leq a^2(\cosh y-1)\leq a^2\cosh y. +\] +Using \eqref{eq:patch-log-ratio} and +\(\sqrt{p_i p_k}\cosh(\tfrac12\log(p_i/p_k)) +=(p_i+p_k)/2\), the nonnegative coefficient in +\eqref{eq:patch-hermitian-pairing} is bounded by +\begin{equation} + \sqrt{p_i p_k} + \bigl(\cosh(\log(\ell_i/\ell_k))-1\bigr) + \leq \frac{a_j^2}{2}(p_i+p_k). + \label{eq:patch-quadratic-coefficient} +\end{equation} +Zero Schmidt probabilities contribute zero to the expectation +and need no separate limiting argument. + +Here is the step that prevents a Schmidt-rank factor from entering. +For operators \(A\) on \(X_j\) and \(B\) on its complement, +let \(A_{ik}=\langle e_i,Ae_k\rangle\) and +\(B_{ik}=\langle f_i,Bf_k\rangle\), restricted to the displayed +Schmidt vectors. Cauchy--Schwarz in a row and in a column gives +\[ + \sum_k\abs{A_{ik}B_{ik}}\leq\norm{A}\norm{B}, + \qquad + \sum_i\abs{A_{ik}B_{ik}}\leq\norm{A}\norm{B}. +\] +It follows that +\begin{equation} + \sum_{i,k}(p_i+p_k)\abs{A_{ik}B_{ik}} + \leq2\norm{A}\norm{B}. + \label{eq:patch-weighted-row-column} +\end{equation} +Expand \(h\) in matrix units on its endpoint inside \(X_j\): +\[ + h=\sum_{a,b=1}^q + \lvert a\rangle\langle b\rvert\otimes h_{ab}, + \qquad \norm{h_{ab}}\leq\norm{h}\leq J. +\] +The other factors are extended by identities within their respective +sides of the cut. Combining +\eqref{eq:patch-hermitian-pairing}--\eqref{eq:patch-weighted-row-column} +term by term yields the explicit bound +\begin{equation} + \abs{\operatorname{Re}\langle\phi,L_j h L_j^{-1}\phi\rangle + -\langle\phi,h\phi\rangle} + \leq q^2J\,a_j^2. + \label{eq:patch-one-term-energy} +\end{equation} + +The ambient sample of a square of radius at most \(2u\) has at most +\(16u+4\) crossing lattice edges. Intersecting with the physical +square cannot add a physical crossing edge. Hence, using the +ground equation, \eqref{eq:patch-single-crossing}, and +\eqref{eq:patch-one-term-energy}, +\begin{align} + 0\leq\langle\phi,(H-E_0\id)\phi\rangle + &\leq q^2J(16u+4)\sum_{j=1}^m a_j^2 \notag\\ + &\leq 36q^2J\,\kappa^2. + \label{eq:patch-energy-bound} +\end{align} +The last inequality uses \(m\geq u/2\) and \(u\geq4\). +This bound is independent of \(R\), the private dimensions of the +filters, and the choice of a minimizer. + +Set \(C_E=36q^2J\), and now fix +\begin{equation} + 0<\kappa\leq + \min\left\{\frac12,\sqrt{\frac{\Delta}{1024C_E}}\right\}. + \label{eq:patch-kappa-choice} +\end{equation} +The full-system gap implies +\[ + 1-\abs{\langle\Omega,\phi\rangle}^2 + \leq \frac{C_E\kappa^2}{\Delta}. +\] +In particular, up to a phase, +\(\norm{\phi-\Omega}\leq +\sqrt{2C_E/\Delta}\,\kappa\). +The trace norm distance of the two pure-state density operators is +\(2\sqrt{1-\abs{\langle\Omega,\phi\rangle}^2}\). +After partial trace, \eqref{eq:patch-kappa-choice} therefore gives +\begin{equation} + \norm{\rho_j-\rho_{\Omega,X_j}}_1\leq\frac1{16} + \quad\hbox{for every }j. + \label{eq:patch-marginal-closeness} +\end{equation} +Only the original full-system gap has been used. + +\emph{Constant-tail ranks from the area law.} +Proposition~\ref{prop:area-law}, together with the crossing-edge +count, gives a constant \(C_S\geq1\), depending only on the physical +parameters, such that +\[ + S_\Omega(X_j)\leq C_S(1+u). +\] +Let \(Q_j\) be the spectral projection of \(\rho_{\Omega,X_j}\) +onto eigenvalues at least +\(\exp[-16C_S(1+u)]\). Summing the eigenvalues shows +\[ + \rank Q_j\leq e^{C_Q(1+u)},\qquad C_Q=16C_S. +\] +Markov's inequality for the marginal surprisal gives +\[ + \Tr\bigl(\rho_{\Omega,X_j}(\id-Q_j)\bigr) + \leq\frac1{16}. +\] +By \eqref{eq:patch-marginal-closeness}, +\begin{equation} + \Tr\bigl(\rho_j(\id-Q_j)\bigr)\leq\frac18. + \label{eq:patch-constant-tail} +\end{equation} +These \(Q_j\) come from the original ground state. We have not +applied entropy continuity to the filtered state. + +At a minimizer define the commuting positive product +\[ + T_j=\rho_j\,b(x_j+b\id)^{-1}. +\] +Equation~\eqref{eq:patch-kkt} shows that its eigenvalues obey +\[ + 0\leq \frac{p_i b}{x_i+b}\leq b, + \qquad + \frac{p_i b}{x_i+b}\leq p_i. +\] +Thus \(0\leq T_j\leq b\id\) and \(T_j\leq\rho_j\). +The projector \(Q_j\) need not commute with \(T_j\); the positive +operator inequalities suffice to give +\begin{align} + \Tr T_j + &=\Tr(Q_jT_j)+\Tr((\id-Q_j)T_j) \notag\\ + &\leq b\,\rank Q_j+ + \Tr((\id-Q_j)\rho_j) + \leq e^{-R+C_Q(1+u)}+\frac18. + \label{eq:patch-regulator-trace} +\end{align} +Choose, for example, +\[ + C_0=C_Q+\log8,\qquad R_{\mathrm{start}}=C_0(1+u). +\] +Then \(\Tr T_j\leq1/4\) whenever \(R\geq R_{\mathrm{start}}\), +uniformly over all minimizers. + +\emph{The nonsmooth minimum and its growth.} +The regulator estimate will bound the growth of $\log N(R)$. Its rate +must stay below $\kappa/2$, the exponential decay rate of the +inverse-filter remainder described above. Write +\[ + f(R,x)=\log\norm{M(R,x)\Omega}, + \qquad g(R)=\log N(R)=\min_x f(R,x). +\] +We justify the envelope calculation without choosing a continuous +family of minimizers. For fixed \(x\), +\[ + \frac{\partial L_j}{\partial R} + =\frac{a_j}{2}\,b(x_j+b\id)^{-1}L_j, + \qquad + \norm{\frac{\partial L_j}{\partial R}} + \leq\frac{a_j}{2}\norm{L_j}. +\] +On \(0\leq R\leq T\), the product rule and +\eqref{eq:patch-elementary-norm-bounds} therefore give +\[ + \norm{\frac{\partial M}{\partial R}} + \leq\frac{\kappa}{2}e^{\kappa T/2}, + \qquad + \norm{M\Omega}\geq2^{-\kappa/2}. +\] +Consequently +\begin{equation} + \abs{\partial_R f(R,x)} + \leq \frac{\kappa}{2}\,2^{\kappa/2}e^{\kappa T/2} + \quad(0\leq R\leq T). + \label{eq:patch-uniform-local-lipschitz} +\end{equation} +The bound holds for every feasible \(x\) and is independent of all +Hilbert-space dimensions and of \(m\). The minimum \(g\) inherits +the same Lipschitz bound on each bounded interval, and hence is +locally absolutely continuous. + +Fix \(R\) and any minimizing tuple \(x_R\). For \(h>0\), +\[ + g(R+h)-g(R) + \leq f(R+h,x_R)-f(R,x_R). +\] +The upper right Dini derivative of \(g\) is therefore at most +\(\partial_R f(R,x_R)\). At the minimizer the descending +trace rule removes the outer filters in each term of the product +derivative, giving +\[ + \partial_R f(R,x_R) + =\sum_{j=1}^m\frac{a_j}{2} + \Tr\bigl(\rho_j b(x_j+b\id)^{-1}\bigr). +\] +By \eqref{eq:patch-regulator-trace}, this is at most +\(\kappa/8\) for \(R\geq R_{\mathrm{start}}\). +At almost every point where \(g\) is differentiable its derivative +has this upper bound, so absolute continuity permits integration. +Using \(g(R_{\mathrm{start}})\leq +\kappa R_{\mathrm{start}}/2\) from +\eqref{eq:patch-elementary-norm-bounds}, we obtain the convenient +estimate +\begin{equation} + g(R)\leq\frac{\kappa}{2} + \left(R_{\mathrm{start}}+\frac R4\right) + \quad(R\geq R_{\mathrm{start}}). + \label{eq:patch-minimum-growth} +\end{equation} +Neither differentiability of a minimizing tuple nor uniqueness of +that tuple is required. + +\emph{An ordered expansion and a small all-tail remainder.} +Choose \(R\geq R_{\mathrm{start}}\) and a minimizing tuple at +that value. Let \(\Pi_j\) be the spectral projection of \(x_j\) +onto eigenvalues at least \(b=e^{-R}\). Split the inverse filter, +on its inside space, as +\[ + F_j=L_j^{-1},\qquad + F_j^{\mathrm h}=F_j\Pi_j,\qquad + F_j^{\mathrm t}=F_j(\id-\Pi_j). +\] +Since \(\Tr x_j=1\), +\begin{equation} + \rank F_j^{\mathrm h}\leq b^{-1}, + \qquad + \norm{F_j^{\mathrm t}}\leq(2b)^{a_j/2}. + \label{eq:patch-head-tail-bounds} +\end{equation} +The rank in this display is on \(\mathcal H_{X_j}\), before +extension by the outside identity. + +Inverting the ordered filter product gives +\(\Omega=N(R)F_1\cdots F_m\phi\). +Expand by the first head in this left-to-right order: +\begin{equation} + F_1\cdots F_m + =\sum_{j=1}^m + F_1^{\mathrm t}\cdots F_{j-1}^{\mathrm t} + F_j^{\mathrm h}F_{j+1}\cdots F_m + +F_1^{\mathrm t}\cdots F_m^{\mathrm t}. + \label{eq:patch-first-head-expansion} +\end{equation} +This identity uses distributivity only and never changes the order +of noncommuting factors. Every factor preceding the \(j\)-th head +acts inside \(X_j\). Hence the \(j\)-th summand applied to \(\phi\) +lies in +\[ + \mathcal C_j=S_j\otimes\mathcal H_{\Lambda\setminus X_j}, + \qquad + \dim S_j\leq b^{-1}, +\] +where \(S_j\) is the range on \(\mathcal H_{X_j}\) of +\(F_1^{\mathrm t}\cdots F_{j-1}^{\mathrm t}F_j^{\mathrm h}\). +The preceding factors may alter the head subspace, but cannot +increase its dimension. + +The remaining vector has norm at most +\[ + N(R)\prod_j\norm{F_j^{\mathrm t}} + \leq N(R)(2b)^{\kappa/2}. +\] +By \eqref{eq:patch-minimum-growth}, its logarithm is at most +\begin{equation} + \frac{\kappa}{2} + \left[C_0(1+u)+\log2-\frac{3R}{4}\right]. + \label{eq:patch-all-tail-log} +\end{equation} +To make this choice explicit and uniform, set +\begin{equation} + R_0=\frac43\left[ + C_0\left(B_0+\frac1{\log2}\right) + +1+\frac{2p}{\kappa}\right], + \qquad R=R_0\log L. + \label{eq:patch-explicit-radius} +\end{equation} +For \(L\geq2\) and \(u\leq B_0\log L\), +\[ + C_0(1+u)\leq + C_0\left(B_0+\frac1{\log2}\right)\log L, + \qquad \log2\leq\log L. +\] +Thus \(R\geq R_{\mathrm{start}}\), and +\eqref{eq:patch-all-tail-log} is at most \(-p\log L\). +Writing \(\mathcal E=\sum_j\mathcal C_j\), we have consequently +proved +\begin{equation} + \dist(\Omega,\mathcal E)\leq L^{-p}, + \qquad + \dim S_j\leq L^{R_0}. + \label{eq:patch-distance-to-cylinders} +\end{equation} + +\emph{Orthogonalization at variable radii.} +Let \(P\) be the orthogonal projection onto \(\mathcal E\). +We finish by putting it in the required form, keeping the radius +associated with each new summand. Process the cylinders in +increasing \(j\). By nesting, the earlier span can be written +\[ + \mathcal E_{j-1}:=\sum_{i0$, there +is an operator tensor network $\sigma$ satisfying +\[ + \norm{\sigma-\rho}_1\le L^{-p}. +\] +It has one node per party, a bounded number of incident virtual links at +each node, and uniformly polynomial link dimensions. Every link joins +two parties which participate in a common nonprivate gate of the +original construction. The physical legs at a node are the ket and bra +legs of that party's intended output. +The constants in these bounds depend only on $p$ and the fixed uniform +data in the assumptions, including $b,r$ and the polynomial bounds, +and not on private dimensions or source Schmidt ranks. + +The same conclusion holds if the gate expansions in the third +assumption are available to every fixed inverse-polynomial +operator-norm accuracy, with uniform polynomial bounds at each fixed +accuracy. +\end{theorem} + +\begin{proof} +We give the estimates for a target error $0<\epsilon\le1$ whose inverse +is uniformly polynomial; take $\epsilon=L^{-p}$ at the end. Expand the +small messages as above. This preserves all constant incidence and +participation bounds and gives a uniformly polynomial absolute +coefficient sum for every gate. + +After eliminating effects, we will replace each source operator +$|\eta\rangle\langle\widetilde\eta|$ in the density calculation by an +unbiased sum of $k$ product operators. We estimate the resulting physical +output, allowing the sampled factors themselves to have large norms. +Expanding the error by the positions where a centered correction is used +makes the lifetime participation bound decisive: $s$ corrected positions +have at most $2s$ endpoint parties, which together participate in at most +$2bs$ nonprivate gates. We expand only those gates into monomials; all +other gates remain aggregate contractions. Thus the coefficient and +physical-output choices cost a fixed polynomial per corrected position, +rather than a product over every gate in the construction. + +For each fixed choice, the remaining task is a mean trace-norm estimate +of $k^{-s/2}$. Normalized probability weights from the corrected endpoints +and the exact sources crossing the boundary of their party set make this +estimate +independent of all private dimensions. Summing over correction positions +will then give a small error with polynomial $k$. + +\paragraph{Reduction to source-only gates.} +Let $M$ bound the number of nonprivate gates. If $M=0$, the output is a +product of private densities and the assertion is immediate. Otherwise +choose $\delta=\epsilon/(8M)$. Apply Lemma~\ref{lem:effects} at every +nonprivate gate, choosing one sufficiently large polynomial integer $m$ +so that its error is at most $\delta$. This choice is made using the +\emph{original} coefficient sum and number of effects, before expanding +the permutation averages. Afterward there are polynomially many +monomials, and their absolute coefficient sum has not increased. Thus +the choice of $m$ is not circular. + +The gate $\widetilde G_m$ has norm at most $1+\delta$. Replace it by +$\widetilde G_m/(1+\delta)$, a contraction within $2\delta$ of the +original gate with its fixed garbage appended. Keep these garbage +registers idle. The augmented original circuit has the same physical +output density as the original circuit, since all the appended garbage +vectors are fixed and normalized. Telescoping products of contractions +shows that the two final pure vectors differ by at most $2M\delta$. +For vectors of norm at most one, +\[ + \norm{|v\rangle\langle v|-|w\rangle\langle w|}_1 + \le(\norm{v}+\norm{w})\norm{v-w} + \le2\norm{v-w}. +\] +Partial trace does not increase this norm. Hence the source-only +circuit's physical density, denoted $\rho'$, satisfies +\begin{equation} + \norm{\rho'-\rho}_1\le4M\delta\le\epsilon/2. + \label{eq:compression-effect-circuit-error} +\end{equation} +The number of gates and the list of participating parties have not +changed. The new polynomial exponents are now fixed. The sampling +parameter introduced below is chosen only after this reduction. + +Each nonprivate gate of this new circuit can be written +\begin{equation} + G_g=\sum_{\xi\in\Xi_g}c_{g,\xi} + \Bigl(\bigotimes_{P}B_{g,P,\xi}\Bigr) + \left[\Bigl(\bigotimes_{e\in E_g} + |\eta_{g,e,\xi}\rangle\Bigr) + \otimes\id_{\mathrm{in}}\right], + \qquad \norm{G_g}\le1. + \label{eq:compression-source-gate} +\end{equation} +Here $E_g$ contains one slot for each participating pair of parties, +each $\eta_{g,e,\xi}$ is normalized, and every $B_{g,P,\xi}$ is a local +contraction. Unused pairs have a one-dimensional source. To obtain +this form, prepare a monomial's sources in fresh registers before their +first use, and tensor the preceding local maps with identities on those +registers. This preserves their norms and their chronological action. +Then combine sources on each pair, and compose the local maps at each +party. This procedure stores fresh registers; it does not commute an +existing source half through an earlier map. + +\paragraph{Common private slot spaces.} +The dimensions of a source slot can depend on $\xi$. For each endpoint +$P$ of $e$, embed its half-space for branch $\xi$ isometrically into +\[ + \mathcal H_{g,P,e} + =\bigoplus_{\xi\in\Xi_g}\mathcal H_{g,P,e,\xi}. +\] +Embed the branch source accordingly, and extend $B_{g,P,\xi}$ by +orthogonal projection onto its branch's source sectors. Its norm +remains at most one. All external gate input and output spaces are +already fixed by the memory layout and, in the preceding reduction, by +the common inventory of stacks. Formula +\eqref{eq:compression-source-gate} therefore has one fixed ambient +space for each slot. These direct sums are private spaces; no bound on +their dimensions will be used. + +Fix uniformly polynomial bounds $B(L)$ and $d(L)$ such that +\begin{equation} + \sum_{\xi\in\Xi_g}|c_{g,\xi}|\le B(L), + \qquad + \dim\mathcal H_{P,\mathrm{phys}}\le d(L) + \label{eq:compression-polynomial-bounds} +\end{equation} +for all gates and parties. The cardinalities $|\Xi_g|$ are also +uniformly polynomial. We may increase $B,d$ to be at least one. + +\paragraph{A random replacement for one density source.} +For a ket branch $\xi$ and bra branch $\zeta$ at a slot, write Schmidt +decompositions, in their respective bases, as +\[ + |\eta\rangle=\sum_a\sqrt{\lambda_a} + |u_a\rangle|v_a\rangle, + \qquad + |\widetilde\eta\rangle=\sum_c\sqrt{\mu_c} + |\widetilde u_c\rangle + |\widetilde v_c\rangle. +\] +Both probability lists sum to one; their lengths need not agree. Let +$g_{ac}^{(j)}$, for $1\le j\le k$, be independent standard circular +complex Gaussians, normalized by $\mathbb E|g_{ac}^{(j)}|^2=1$. Put +\begin{align} + U_j&=\sum_{a,c}(\lambda_a\mu_c)^{1/4}g_{ac}^{(j)} + |u_a\rangle\langle\widetilde u_c|,\nonumber\\ + V_j&=\sum_{b,d}(\lambda_b\mu_d)^{1/4} + \overline{g_{bd}^{(j)}} + |v_b\rangle\langle\widetilde v_d|,\nonumber\\ + \widehat X&=\frac1k\sum_{j=1}^k U_j\otimes V_j, + \qquad X=|\eta\rangle\langle\widetilde\eta|, + \qquad \Delta=\widehat X-X. + \label{eq:compression-random-source} +\end{align} +These matrices act between the corresponding bra and ket source +supports in the common ambient half-spaces. No bound on their sampled +operator norms is claimed or needed. Since +$\mathbb E[g_{ac}\overline{g_{bd}}]=\delta_{ab}\delta_{cd}$, +$\mathbb E\widehat X=X$. + +The centered coefficient of $\Delta$ with ket endpoint indices $(a,b)$ +and bra endpoint indices $(c,d)$ is +\[ + (\lambda_a\lambda_b\mu_c\mu_d)^{1/4} + \left(\frac1k\sum_{j=1}^k + g_{ac}^{(j)}\overline{g_{bd}^{(j)}} + -\delta_{ab}\delta_{cd}\right). +\] +For composite Gaussian indices $\alpha,\beta,\gamma,\delta$, the +complex Gaussian pairing formula \cite{Isserlis1918,FassinoPistoneRogantin2019} +gives the fourth-moment identity +\begin{equation} + \mathbb E\left[ + (g_\alpha\overline{g_\beta}-\delta_{\alpha\beta}) + \overline{(g_\gamma\overline{g_\delta} + -\delta_{\gamma\delta})}\right] + =\delta_{\alpha\gamma}\delta_{\beta\delta}. + \label{eq:compression-gaussian-covariance} +\end{equation} +Indeed independence, circular symmetry, and $\mathbb E|g_\alpha|^4=2$ +give the two Gaussian pairings +$\delta_{\alpha\beta}\delta_{\gamma\delta} + +\delta_{\alpha\gamma}\delta_{\beta\delta}$, and centering removes +the first. Thus the coefficient covariance is diagonal and its +diagonal entry is +\begin{equation} + \frac1k\sqrt{\lambda_a\lambda_b\mu_c\mu_d}. + \label{eq:compression-slot-variance} +\end{equation} +Take fresh independent samples for each slot location and each ordered +pair of monomial labels at that gate. + +\paragraph{Expansion by corrected positions.} +Let $\mathcal E=\{(g,e):e\in E_g\}$ be the set of slot locations and +$N=|\mathcal E|$. It is uniformly polynomial. In the density +calculation for each gate, substitute $X+\Delta$ for each source +operator inside its sum over ket and bra labels. Expanding these +products throughout the circuit gives +\begin{equation} + \widehat\rho-\rho' + =\sum_{\varnothing\ne S\subseteq\mathcal E}F_S. + \label{eq:compression-subset-expansion} +\end{equation} +A set $S$ specifies which \emph{positions} use centered corrections; +it does not specify their monomial labels. At a gate with no selected +position, summing its labels gives exactly $G_g(.)G_g^\dagger$. + +Fix $S$, write $s=|S|$, and let $\mathcal A$ be the parties which are +endpoints of its slots. Then $|\mathcal A|\le2s$. Expand the ket and +bra labels at every nonprivate gate touching $\mathcal A$, whether or +not that gate has a selected slot. There are at most $2bs$ such gates. +All other gates lie entirely in the exterior and remain their exact +aggregate contractions. Also fix ket and bra basis entries $x,y$ of +the physical outputs at parties in $\mathcal A$. The total weighted +cost of these choices is at most +\begin{equation} + B(L)^{4bs}d(L)^{4s}=Q(L)^s, + \qquad Q(L):=B(L)^{4b}d(L)^4. + \label{eq:compression-choice-cost} +\end{equation} +Here each expanded gate contributes at most +$(\sum_\xi|c_{g,\xi}|)^2$; the number of physical matrix entries is at +most $d(L)^{2|\mathcal A|}$. Scalar coefficients are removed from the +maps and charged in this bound. + +From now on fix one such deterministic choice of labels and physical +entries. The samples used at its different corrected positions are +independent. There is no sample-dependent selection of a branch. We +will apply the triangle inequality over these choices, so no +cross-branch covariance assumption is necessary. It remains to bound the +mean trace norm for this fixed choice by $k^{-s/2}$, with no private-dimension +factor; the normalized source weights will provide that bound. + +\paragraph{The two separated contractions.} +Both ends of every corrected source lie in $\mathcal A$. All sources +crossing from $\mathcal A$ to its exterior are therefore exact. Their +tensor product is a normalized pure vector on the ket side and a +possibly different normalized pure vector on the bra side. In their +Schmidt bases let the probability matrices be $\tau$ and +$\widetilde\tau$. Their diagonal entries sum to one. Empty lists of +sources are represented by a one-dimensional space and the scalar +probability one. + +For every corrected slot retain both ket endpoint indices as free +indices; collect them in $i$. Similarly collect the bra endpoint +indices in $l$. Define normalized product probability lists $p_i$ and +$\widetilde p_l$ by giving a corrected slot the respective factors +$\lambda_a\lambda_b$ and $\mu_c\mu_d$. In particular the two endpoint +indices are independent summation indices, even though an exact pure +source would identify them. If $z_{il}$ is the product of the centered +coefficients at these slots, independence and +\eqref{eq:compression-slot-variance} give +\begin{equation} + \mathbb E[z_{il}\overline{z_{i'l'}}] + =k^{-s}\sqrt{p_i\widetilde p_l}\, + \delta_{ii'}\delta_{ll'}. + \label{eq:compression-product-covariance} +\end{equation} + +Regard all corrected endpoint registers and all affected-side halves of +exact crossing sources as free inputs supplied at the beginning of the +separated calculation. This again means keeping fresh registers idle +until first use. Prepare all exact sources internal to either side as +normalized vectors on that side. Every previously expanded crossing +gate now consists of local contractions and these exact sources. Thus +there is no remaining communication between the two sides. Gates +wholly in the exterior which were not expanded remain their aggregate +contractions; their internal branch sums are not opened. + +Let $\mathcal I$ be the ket space with basis $i$ and $\mathcal T$ the +affected-side ket Schmidt-support space with basis $t$ for $\tau$. +Use tildes for their bra analogues. Including final physical basis +projection, the affected calculations give contractions +\[ + A_x:\mathcal I\otimes\mathcal T\longrightarrow\mathcal F, + \qquad + \widetilde A_y:\widetilde{\mathcal I}\otimes + \widetilde{\mathcal T}\longrightarrow\mathcal F. +\] +The codomain $\mathcal F$ contains all affected discard registers. It +is the same on ket and bra because the external memory spaces are fixed; +the inventory construction ensured this also for the added garbage. +These are contractions on \emph{all} free inputs: each operation in +their construction is a contraction, an isometric preparation, an +identity on an idle register, or a physical basis projection. No +normalization of an output vector has occurred. Therefore +\begin{equation} + O:=\widetilde A_y^\dagger A_x: + \mathcal I\otimes\mathcal T\longrightarrow + \widetilde{\mathcal I}\otimes\widetilde{\mathcal T}, + \qquad \norm{O}\le1. + \label{eq:compression-block-contraction} +\end{equation} +Write its blocks as $O_{li}:\mathcal T\to\widetilde{\mathcal T}$. + +The exterior calculations similarly give contractions from their +crossing-source half-spaces to the exterior physical output and +garbage. In the matching Schmidt coordinate spaces denote them by +$K$ and $\widetilde K$. They can differ on ket and bra but do not +depend on $i,l$. Those indices are free inputs only on the affected +side, and fixing all crossing-gate labels removed every other means +of communication across the division. For any rectangular operator +$Z$ between the two exterior input spaces, +\begin{equation} + \norm{\Tr_{\mathrm{garbage}}(KZ\widetilde K^\dagger)}_1 + \le\norm{Z}_1. + \label{eq:compression-exterior-contraction} +\end{equation} +To see this, test the output against an arbitrary matrix $M$ of operator +norm at most one. The pulled-back test is +$\widetilde K^\dagger(M\otimes\id)K$, of norm at most one. +Trace-norm duality proves the inequality even when $Z$ is neither +Hermitian nor positive. + +\paragraph{The second-moment bound.} +Density-weighted comparisons between trace and Hilbert--Schmidt norms +also occur in privacy amplification and decoupling +\cite[Lemma~5.1.3]{Renner2005Security} \cite[Lemma~3.7]{DupuisBertaWullschlegerRenner2014}. +Here we use a two-weight rectangular form of Schatten H\"older, proved below. +Tracing the affected garbage gives the following exterior input +operator for the fixed labels and $x,y$: +\begin{equation} + Z=\sum_{i,l}z_{il}\, + \tau^{1/2}O_{li}^{T}\widetilde\tau^{1/2}. + \label{eq:compression-exterior-input} +\end{equation} +Indeed its $(t,t')$ coefficient is +$\sum_{i,l}z_{il}\sqrt{\tau_t\widetilde\tau_{t'}} + \langle\widetilde A_y(l,t'),A_x(i,t)\rangle$. +The transpose in \eqref{eq:compression-exterior-input} is the ordinary +full matrix transpose in these bases, including when the matrix is +rectangular; it preserves all Schatten norms. + +For an operator $W$ between these coordinate spaces, Schatten +H\"older with exponents $4,2,4$ gives +\begin{equation} + \norm{\tau^{1/2}W\widetilde\tau^{1/2}}_1 + \le\norm{\tau^{1/4}}_4\, + \norm{\tau^{1/4}W\widetilde\tau^{1/4}}_2\, + \norm{\widetilde\tau^{1/4}}_4 + =\norm{\tau^{1/4}W\widetilde\tau^{1/4}}_2. + \label{eq:compression-quarter-powers} +\end{equation} +The subscript $2$ here denotes the Hilbert--Schmidt norm, and the +quarter-power Schatten-$4$ norms equal one because the probability +matrices have trace one. Applying this and transposing the middle +matrix in \eqref{eq:compression-exterior-input} yields +\[ + \norm{Z}_1\le + \norm{\sum_{i,l}z_{il} + \widetilde\tau^{1/4}O_{li}\tau^{1/4}}_2. +\] +By \eqref{eq:compression-product-covariance}, the expected square of +the right side is +\begin{equation} + k^{-s}\sum_{i,l}\sqrt{p_i\widetilde p_l}\, + \norm{\widetilde\tau^{1/4}O_{li}\tau^{1/4}}_2^2. + \label{eq:compression-block-second-moment} +\end{equation} +Define trace-one positive diagonal matrices on the full free input +spaces by +\[ + R=\operatorname{diag}(p_i)\otimes\tau, + \qquad + \widetilde R=\operatorname{diag}(\widetilde p_l) + \otimes\widetilde\tau. +\] +The sum in \eqref{eq:compression-block-second-moment} is exactly +$\norm{\widetilde R^{1/4}OR^{1/4}}_2^2$: its $(l,i)$ block is +$\widetilde p_l^{1/4}p_i^{1/4} + \widetilde\tau^{1/4}O_{li}\tau^{1/4}$, and the +Hilbert--Schmidt squares of disjoint blocks add. Another application +of Schatten H\"older and \eqref{eq:compression-block-contraction} gives +\begin{equation} + \norm{\widetilde R^{1/4}OR^{1/4}}_2 + \le\norm{\widetilde R^{1/4}}_4\norm{O}\norm{R^{1/4}}_4 + \le1. + \label{eq:compression-dimension-free} +\end{equation} +This bound contains no rank or dimension of a source, an input, or a +discard space. Cauchy--Schwarz for the expectation now proves +\begin{equation} + \mathbb E\norm{Z}_1\le k^{-s/2}. + \label{eq:compression-one-choice} +\end{equation} +Applying the exterior map preserves this bound by +\eqref{eq:compression-exterior-contraction}. Reinserting the affected +physical matrix unit $|x\rangle\langle y|$ preserves the trace norm, +since that matrix unit has trace norm one. + +\paragraph{Summation and realization as a network.} +Summing over the fixed choices with +\eqref{eq:compression-choice-cost} gives +\[ + \mathbb E\norm{F_S}_1\le Q(L)^{|S|}k^{-|S|/2}. +\] +Consequently +\begin{equation} + \mathbb E\norm{\widehat\rho-\rho'}_1 + \le\left(1+\frac{Q(L)}{\sqrt k}\right)^N-1. + \label{eq:compression-total-error} +\end{equation} +If $N=0$, there is no sampling error. Otherwise choose an integer +$k$ with +\[ + \frac{Q(L)}{\sqrt k}\le\frac{\epsilon}{8N}. +\] +It is uniformly polynomial. The right side of +\eqref{eq:compression-total-error} is at most +$e^{\epsilon/8}-1\le\epsilon/4$. Hence there is a realization with +$\norm{\widehat\rho-\rho'}_1\le\epsilon/2$, for example by Markov's +inequality. Fix one such realization and set $\sigma=\widehat\rho$. +Together with \eqref{eq:compression-effect-circuit-error}, this proves +the asserted error. + +It remains to check the network dimensions and link locality. At a +gate, the common label $(\xi,\zeta)$ ranges over a polynomial set. +Choose one participating party and share this label along a star to +the other participating parties, using one virtual index of that +polynomial dimension on each star edge. Each slot's sample index +$j\in\{1,\ldots,k\}$ is shared directly between its two endpoint +parties. The local source matrices in +\eqref{eq:compression-random-source} are determined by these labels +and the fixed samples. Put the gate's scalar coefficient and averaging +factors into any one local tensor. All local maps, private source +preparations, intermediate memories, and final garbage traces can then +be contracted within each party to give a single tensor with its +physical ket and bra legs. In particular, coordinates in the common +private direct sums are contracted locally and are not virtual link +indices. + +There are at most $b-1$ star links and $\binom b2$ sample links per +gate. Every party belongs to at most $b$ nonprivate gates, so the +resulting network has bounded incidence. Each link joins two +participants of one original gate. Parallel links can be left +separate or combined; only boundedly many meet a party, so combining +them preserves a polynomial dimension bound. Sampled tensor entries +need not be bounded. Their existence and link dimensions, rather than +their numerical size or efficient computation, are what is required. + +Finally, suppose only approximate gate expansions are provided. At +any prescribed fixed accuracy exponent their operator-norm errors can +be made smaller than $\epsilon$ divided by a sufficiently large +polynomial bound on the number of gates. A near-contraction can be +rescaled by its known bound $1+\delta$ to be a contraction at an +additional error at most $\delta$. The same telescoping estimate as +above absorbs these errors. Apply the proved result at a fixed +smaller error budget. All subsequent choices remain uniform +polynomials at that precision, which proves the last assertion. +\end{proof} + +\begin{remark}[Absolute error and retained norm] +\label{rem:compression-retained-norm} +Theorem~\ref{thm:compression} approximates the subnormalized density +produced by the contraction circuit. It does not normalize that +density. In its later application, the construction separately ensures +that its physical output is close in trace norm to a normalized pure +state; in particular, the retained norm is close to one. The theorem's +absolute error can then be added directly to that approximation error. +No bound on the private dimensions or on the Schmidt ranks of the +supplied pair vectors is needed for this use. +\end{remark} diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/05-frames.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/05-frames.tex new file mode 100644 index 0000000000000000000000000000000000000000..e5aace5b02cf32814272e3e7405be5c7b1ad3dc9 --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/05-frames.tex @@ -0,0 +1,570 @@ +\section{Encoded frames and changes of ownership}\label{sec:frames} + +We next construct the elementary operations that will distribute the ground +vector among the parties of Theorem~\ref{thm:compression}. These are +contractions between specified register layouts. A change of layout is part +of the map to be constructed: a raw register cannot be transferred to a +different party merely by changing its name. Private operations, in contrast, +may act on all the registers of one party, irrespective of their spatial +positions. + +\subsection{Sheets and adaptive encodings} + +For a physical set $A\subseteq\Lambda$, write +$\mathcal H_A=\bigotimes_{x\in A}\mathbb C^q$. A \emph{sheet} has one raw +register $\mathbb C^q$ at each $x\in\Lambda$ and a specified owner for each +register. Different sheets have distinct registers. Their canonical raw +reference vectors are independent copies of $\Omega$, with sites indexed by +position. Thus ownership does not permute the positions in the definition +of the reference vector. + +Fix a constant $B_0>0$. In this section all applications of +Proposition~\ref{prop:patch} use accuracy +\begin{equation}\label{eq:frame-epsilon} + \varepsilon=L^{-60}. +\end{equation} +For a center $c\in\mathbb R^2$ and $00$, an operator-norm +approximation of error $O(L^{-a})$ is allowed, with a polynomial expansion +whose exponent can depend on $a$. Scaling such an approximation by a +factor $1+O(L^{-a})$ restores contractivity. + +\subsection{A tensor estimate for bounded patch networks} + +The following elementary tensor estimate permits us to replace a complicated +bounded patch network by pair resources without a dependence on the +dimensions of its open wires. A normalized vector shared by three or more +parties is not itself one of the permitted pair resources. We therefore +need products of vectors whose factors each belong to at most two parties. +The estimate below first produces a controlled number of product terms, +keeping all open legs of each original tensor together as one group. +In the patch rewrite that follows, we will verify that each such group +belongs to at most two owners. These groups are determined by the +individual input bras and output kets, rather than by the parties. + +For a matrix $M$, write $\norm{M}_1$ for its +sum of singular values and $\norm{M}_2$ for its Hilbert--Schmidt norm. The +Hilbert-space norm of a tensor equals the Hilbert--Schmidt norm of every +matrix reshaping of that tensor. + +\begin{lemma}[Whole-group tensor bound]\label{lem:group-tensor} +Let a finite graph carry at each vertex a Hilbert-space tensor of norm at +most one. Contract its internal edges by the usual coordinate pairing, +and suppose there are no edges from a vertex to itself. Group the open +legs according to their original vertices. The resulting tensor $Z$ has +norm at most one and has matrix nuclear norm at most one across every +bipartition into whole open-leg groups. + +If there are $g$ nonempty open-leg groups, then, for every integer $k\ge1$, +there are subspaces of dimension at most $k$ in the respective group spaces +whose tensor-product projection sends $Z$ to a tensor $Z_k$ with +\begin{equation}\label{eq:group-truncation} + \norm{Z-Z_k}\le\frac{g}{\sqrt{k}}. +\end{equation} +In orthonormal bases of those subspaces, $Z_k$ has at most $k^g$ product +terms, with coefficients of absolute value at most one. +\end{lemma} + +\begin{proof} +Merge two vertices, contracting all edges between them at the same time. +After grouping their common indices, the new tensor is a matrix product +up to a transpose, and +\[ + \norm{AB}_2\le\norm{A}_2\norm{B}_2. +\] +If there are no common edges, use a tensor product instead. Continue by +merging components and always contracting all edges between the two +components being merged. No internal self-contraction is left behind. +This proves the norm bound, and proves the same bound for every partially +contracted subgraph with its remaining legs left open. It applies equally +to graphs with cycles. + +For a bipartition of the whole open-leg groups, assign vertices without +open legs to either side. Contract each side internally. The two tensors +so obtained have norms at most one. If all cross edges are grouped into +one index, the desired flattening is $AB^{\mathsf T}$. Hence +\begin{equation}\label{eq:group-nuclear} + \norm{AB^{\mathsf T}}_1\le\norm{A}_2\norm{B}_2\le1. +\end{equation} +No dimension enters this estimate. + +Flatten $Z$ across one open-leg group and its complement. Let +$s_1\ge s_2\ge\cdots\ge0$ be its singular values. By +\eqref{eq:group-nuclear}, $\sum_js_j\le1$, and consequently +\[ + \sum_{j>k}s_j^2 + \le s_{k+1}\sum_{j>k}s_j\le\frac{1}{k+1}. +\] +Projection onto its first $k$ left singular directions thus has error at +most $k^{-1/2}$. Do this for every original open-leg group. The projections +act on disjoint tensor factors, so telescoping their product gives +\eqref{eq:group-truncation}. The resulting tensor has norm at most one; +every coefficient in any product orthonormal basis therefore has absolute +value at most one. Empty or one-group cases have the same conclusion, +with the natural scalar or one-factor interpretation. +\end{proof} + +\subsection{Changing a frame on small patches} + +For a frame $\mathcal F$, let $H^-(\mathcal F)$ and $H^+(\mathcal F)$ be the +unions of the physical samples of its inner and outer hole squares. A tag +is unchanged only if its hole, its encoding, and its owner are unchanged. + +\begin{lemma}[Small-patch rewrite]\label{lem:small-rewrite} +Consider old and new one-sheet frames. Only a bounded number of their holes +are affected; all other holes, encodings, and tag owners agree. Suppose +there are $m=O(1)$ additional square patches $(c_i,u_i)$, with +$00$, $M$ admits a contractive approximation $M_a$ with +$\norm{M_a-M}\le L^{-a}$ and a polynomial sum expansion of the kind required +by Theorem~\ref{thm:compression}, using only the specified parties. In +particular the rewrite is a bounded change with reference-vector error +$O(L^{-20})$. +\end{lemma} + +\begin{proof} +Write $K_{\rm old}$ and $K_{\rm new}$ for the products of just the affected +hole encodings, and $K_0$ for the common untouched encodings. For the +additional patches, let $P_1,\ldots,P_m$ be the projectors of +Proposition~\ref{prop:patch}, again with error $\varepsilon$. Fix any order +of the patches, and set +\begin{equation}\label{eq:rewrite-map} + M=K_{\rm new}P_m\cdots P_1K_{\rm old}^{\dagger}. +\end{equation} +This formula uses canonical raw indices between its factors and interprets +the final raw and tag indices with their new owners. It is the identity on +all other registers. Every factor is contractive. + +The untouched encodings commute through the displayed factors, by the +first condition. Moreover, +$K_{\rm old}^{\dagger}K_{\rm old}$ is the product of the affected old-hole +projectors. For arbitrary contractions $A_1,\ldots,A_s$, the elementary +identity +\[ + A_s\cdots A_1-\id + =\sum_{j=1}^s A_s\cdots A_{j+1}(A_j-\id) +\] +holds; its first summand is $A_s\cdots A_2(A_1-\id)$ and its last is +$A_s-\id$. Thus +\[ + \norm{(A_s\cdots A_1-\id)\Omega} + \le\sum_{j=1}^s\norm{(A_j-\id)\Omega}. +\] +Apply this to all the patch and old-hole projectors in the order in which +they occur. Multiplication by $K_0$ and $K_{\rm new}$ can only decrease +the norm, giving \eqref{eq:rewrite-reference-error}. This estimate does not +assume that projectors of overlapping patches commute. + +We now construct the required sum expansion of \eqref{eq:rewrite-map}. +Expand each tag and each patch projector using +\eqref{eq:hole-projector}--\eqref{eq:hole-encoder}. There are polynomially +many complete branch choices, because the number of factors is bounded. +Fix one such branch. On each selected old-hole square the branch of +$K_{\rm old}^{\dagger}$ consumes the input tag by its basis bra, consumes +the raw inputs by product zero-state bras, and supplies a decoded normalized +ket $v_{j\ell}$. On a new-hole square the branch of $K_{\rm new}$ contracts +against a normalized bra $\langle v_{j\ell}|$ and supplies product zeros +and a fixed tag. A selected patch term has a normalized bra and a +normalized ket on its selected square; keep these as \emph{two distinct +tensor vertices}. + +Factor out the fixed tag bras and kets, the one-site zero bras and kets, and +every raw wire that continues directly from an input to an output without +encountering any selected action. The first two sorts of factors are +private norm-one maps. Every direct wire is private as well: a changed +site belongs to some patch inner square, so it belongs to that patch's +selected square in every radius branch and cannot be such a wire. The +old and new owner of a remaining direct wire are therefore equal. In +particular, no factor $\id_{\mathcal H_A}$ with a large Hilbert--Schmidt +norm needs to enter a tensor norm estimate. + +Follow the other site wires in their chronological order through the +selected actions. Each one connects a normalized ket vertex to a later +normalized bra vertex, or gives a raw open input or output. Actions that +do not contain the site simply let its wire continue. This leaves a graph +of boundedly many normalized tensors and no self-contractions. We claim +that each nonempty group of open legs at an original vertex is either +entirely input or entirely output, and belongs to at most two parties. + +First, every leg of an old-hole decoded ket encounters a later patch bra. +Indeed its selected square lies in that hole's outer footprint, the whole +of which is covered by patch inner squares. Thus this decoded ket has no +open output legs. Dually, every leg of a new-hole encoding bra encounters +an earlier patch ket, so that bra has no open input legs. These assertions +hold for every choice of the old and new radii, including unequal radii +and overlaps between old and new holes. + +The only remaining vertices with open legs are patch bras and patch kets. +A patch bra can have only raw input legs, and a patch ket can have only raw +output legs. An input site inside an old inner hole is always consumed by +the old decoder's zero-state bra, since every selected hole square contains +its inner square. It is therefore not an open input of a patch bra. +Consequently all open inputs of a patch bra lie in that patch's outer +square outside the old inner holes, and the third condition limits their +old owners to two. The corresponding output argument uses the new +encoder's zero-state kets and gives at most two new owners for the open +legs of each patch ket. Intersections of selected patches only add +internal ket-to-later-bra wires; they do not change these statements about +open legs. This establishes the claim branch by branch, without any +disjointness assumption on the patches. + +Let $Z$ be the remaining open tensor, with groups given by these original +vertices. Lemma~\ref{lem:group-tensor} supplies an approximation $Z_k$ with +error $g/\sqrt{k}$, where $g=O(1)$, and a sum of at most $k^g$ product terms. +An output group vector in such a term is a normalized state on one party +or a normalized bipartite state on its two owners. An input group vector, +reshaped with the conjugate coordinates as appropriate, is a normalized +private or bipartite covector. Their order can be chosen by first +contracting all input effects and then appending all output states, since +the groups have disjoint open legs. After restoring the factored private +maps and direct identities, each product term is precisely a monomial +allowed by Theorem~\ref{thm:compression}. It uses only boundedly many pair +states and effects, and its scalar coefficient has absolute value at +most one. A scalar residual tensor is already a coefficient of absolute +value at most one. + +For completeness, the Hilbert-space error on $Z$ bounds the operator-norm +error of this branch. Consume its factored input bras, apply the residual +tensor reshaped as a map, tensor that map with the direct identities, and +append the factored output kets. All the outer maps are contractions, +and the operator norm of a matrix is at most its Hilbert--Schmidt norm. +Tensoring with a direct identity preserves its operator norm. Hence the +branch error is at most $g/\sqrt{k}$, with no factor involving the dimensions +of the bypassing wires. + +Let $N\le L^{C_1}$ bound the number of branches, absorbing fixed prefactors +by enlarging $C_1$ for $L\ge2$, and let $g\le g_0$. Summing their errors +gives an operator-norm error at most $Ng_0/\sqrt{k}$. Choose a polynomial +$k$ so this quantity is at most $\tfrac12L^{-a}$, and call the resulting +sum $\widetilde M$. Since $M$ is contractive, +$\norm{\widetilde M}\le1+\tfrac12L^{-a}$. Then +\[ + M_a=\frac{\widetilde M}{1+\tfrac12L^{-a}} + \quad\hbox{satisfies}\quad + \norm{M_a}\le1,\qquad \norm{M_a-M}\le L^{-a}. +\] +The total number of monomials is at most $Nk^{g_0}$, their coefficients are +bounded, and their participating parties are the prescribed ones. Taking, +for example, $a=30$ proves the final assertion. +\end{proof} + +\subsection{Splitting a raw buffer and changing a homogeneous owner} + +The next two operations use small mutual information. We state the +purification consequence with its dimension-independent error explicitly. + +\begin{lemma}[Splitting consequence]\label{lem:splitting} +Let $T,U,E$ partition $\Lambda$, and let +$b=I_\Omega(T:E)$. There are finite-dimensional spaces $\mathcal B_T$ and +$\mathcal B_E$, an isometry +\[ + V:\mathcal H_U\longrightarrow\mathcal B_T\otimes\mathcal B_E, +\] +and normalized vectors +$s\in\mathcal H_T\otimes\mathcal B_T$ and +$s'\in\mathcal H_E\otimes\mathcal B_E$ such that +\begin{equation}\label{eq:frame-splitting} + \norm{(\id_{TE}\otimes V)\Omega-s\otimes s'} + \le\sqrt{2(1-e^{-b/2})}\le\sqrt b. +\end{equation} +There is no bound on the dimensions of the two new spaces. In particular, +if $b\le L^{-60}$, the error is at most $L^{-30}$. +\end{lemma} + +\begin{proof} +Set $\rho=\rho_{TE}$ and $\sigma=\rho_T\otimes\rho_E$. Positivity shows +that $\supp\rho\subseteq\supp\sigma$: the expectation of the projector +onto the kernel of either marginal, tensored with identity, is zero, so +that projector annihilates the support of $\rho$. In particular +$D(\rho\Vert\sigma)=b$ is well defined on this support. + +Choose separate purifications $s$ of $\rho_T$ and $s'$ of $\rho_E$. Enlarge +their purifying factors, if necessary by appending pure ancillas, so that +$\mathcal B_T\otimes\mathcal B_E$ also has room for an isometric image of +the entire $\mathcal H_U$. Lemma~\ref{lem:fidelity} and +\eqref{eq:fidelity-information} give a purification overlap +\[ + F(\rho,\sigma)\ge e^{-D(\rho\Vert\sigma)/2}=e^{-b/2}. +\] +Its purification characterization supplies an isometry $V$ on the support +used by $\Omega$, attaining this overlap, with real positive phase, with +$s\otimes s'$. Extra unused dimensions allow extension to an isometry on +all of $\mathcal H_U$; this extension does not change the overlap. Both +vectors have norm one, so their squared distance is at most +$2(1-e^{-b/2})\le b$. +\end{proof} + +The isometry in Lemma~\ref{lem:splitting} will always act privately: all of +$U$, and both of its enlarged output factors, will be at one party. + +\begin{lemma}[Homogeneous birth and death]\label{lem:birth} +In an encoded frame, let $T$ be a set of raw sites owned by $P_\circ$ and +disjoint from every outer hole footprint. The proposed birth changes only +the owner of $T$, from $P_\circ$ to $Q_\circ$. Keep every hole, its encoding, +and its tag owner fixed. Define +\begin{equation}\label{eq:birth-partition} + E=\{x:\text{the old raw owner of }x\ne P_\circ\} + \cup H^+(\mathcal F),\qquad + U=\Lambda\setminus(T\cup E). +\end{equation} +If $I_\Omega(T:E)\le L^{-60}$, the birth is a bounded change involving +only $P_\circ,Q_\circ$, with reference-vector error at most $L^{-30}$. +The reversed death obeys the same bound whenever these ownership and +information conditions hold in its surrounding frame, namely the frame +\emph{after} the death. +\end{lemma} + +\begin{proof} +Put $\mathcal V=\id_{TE}\otimes V$, with $V$ from +Lemma~\ref{lem:splitting}, and use its normalized vectors $s,s'$. All of +$T,U$ initially belongs to $P_\circ$. At that party apply $V$ on $U$, +contract $T\mathcal B_T$ with $\langle s|$, and supply the normalized pair +state $s$ on $T\mathcal B_T$ with $T$ now held by $Q_\circ$ and +$\mathcal B_T$ still held by $P_\circ$. Finally apply $V^{\dagger}$ to +$\mathcal B_T\mathcal B_E$ at $P_\circ$. The associated canonical map is +\begin{equation}\label{eq:birth-map} + B=\mathcal V^{\dagger} + \left(|s\rangle\langle s|_{T\mathcal B_T} + \otimes\id_{E\mathcal B_E}\right) + \mathcal V. +\end{equation} +It is a Hermitian contraction. Writing $\Pi$ for the middle projector, +\[ + \norm{(B-\id)\Omega} + \le\norm{(\Pi-\id)\mathcal V\Omega} + =\norm{(\Pi-\id)(\mathcal V\Omega-s\otimes s')} + \le L^{-30}. +\] +Every hole encoder is supported in $E$, on whose raw registers this map +acts as the identity. It also acts as the identity on tags. Commuting the +encoders through the map therefore gives the same error bound on the +encoded reference vectors, by their contractivity. + +For the reversed operation, first apply $V$ privately on $U$ at +$P_\circ$, use the normalized bipartite effect $\langle s|$ on +$T\mathcal B_T$ held by $Q_\circ,P_\circ$, and reinsert $s$ wholly at +$P_\circ$. Applying $V^{\dagger}$ gives the same canonical operator +$B=B^{\dagger}$, with the reverse input and output ownership. This has +one pair effect rather than one pair source. Both procedures have the +required bounded expansion. If the two owner labels agree, all these +maps are private; the identity also realizes the ownership change. +\end{proof} + +In the geometric construction the information hypothesis is supplied by +Proposition~\ref{prop:angular}, with its exponent set to $60$. In +particular, for finitely many marked points and a fixed clearance constant, +the logarithmic floor $t=D\log L$ can be chosen so that any eligible +tapering pair $T,E$ satisfies the hypothesis. The set $E$ in +\eqref{eq:birth-partition} includes outer hole footprints even when all +their raw owners happen to be $P_\circ$; this keeps the splitting isometry +off every encoding support. + +\subsection{Exchanging assignments between two sheets} + +\begin{lemma}[Two-sheet exchange]\label{lem:exchange} +Consider two encoded frames, and a plane region $Y$. Suppose every hole's +entire outer square lies on one side of $\partial Y$. Inside $Y$, exchange +the two raw ownership assignments and the two lists of encoded holes, +including their tag owners; outside $Y$, keep both frames unchanged. +Let $P_\circ$ be a party, and on the physical square put +\begin{align} + Z&=\{x:\text{the two old raw owners of }x + \text{ are not both }P_\circ\} + \cup H^+(\mathcal F_1)\cup H^+(\mathcal F_2), + \label{eq:exchange-Z}\\ + T&=Z\cap Y,\qquad E=Z\setminus Y,\qquad + U=\Lambda\setminus Z.\label{eq:exchange-partition} +\end{align} +If $I_\Omega(T:E)\le L^{-60}$, the exchange can be implemented by private +contractions and register renaming, with reference-vector error at most +$4L^{-30}$. It uses $P_\circ$ and the owners of the registers and tags +being renamed. Thus, when these parties form a bounded list, it is a +bounded change in the sense of Theorem~\ref{thm:compression}. +\end{lemma} + +\begin{proof} +Let $K_{\rm in}$ and $K_{\rm out}$ denote the products of the two frames' +encoders before and after the exchange. For $A\subseteq\Lambda$, write +$F_A$ for the unitary swapping the two raw sheets at the positions in $A$. +First rename the two sheets inside $Y$, together with each tag of a hole +there, leaving every physical register at its present party. Call this +map $\mathcal R$. This is a private identification of tensor factors: the +output owner of sheet 1 at $x\in Y$ is precisely the old owner of sheet 2 +there, and conversely. The same assertion holds for a tag moved with its +whole hole. In canonical raw coordinates the exact intertwining identity +is +\begin{equation}\label{eq:exchange-intertwining} + \mathcal R K_{\rm in}=K_{\rm out}F_{Y\cap\Lambda}. +\end{equation} +It follows by checking each site outside holes and each whole hole +encoding separately. For a hole within $Y$, all selected raw squares and +its tag change sheet together; for a hole outside $Y$, nothing changes. +The assumption on the outer square ensures this for every selected radius. + +All registers of $U$ on both sheets are raw, outside every hole, and held +by $P_\circ$ both before and after this renaming. Apply +Lemma~\ref{lem:splitting} to the one-copy partition $T,U,E$, and again write +$\mathcal V=\id_{TE}\otimes V$. Let +\begin{equation}\label{eq:exchange-buffer-correction} + D_U=(V^{\otimes2})^{\dagger}F_{\mathcal B_T}V^{\otimes2} +\end{equation} +on the two copies of $U$, where $F_{\mathcal B_T}$ swaps the two enlarged +$\mathcal B_T$ factors. The whole map is a private contraction at +$P_\circ$: apply the two copies of $V$, swap the indicated factors, and +apply their adjoints. + +Set $A=Y\cap U$. After $\mathcal R$, first apply the private swap $F_A$, +and then $D_U$. These two corrections need not commute, so we retain this +order. Both corrections commute through $K_{\rm out}$, since they act +only on raw sites of $U$, away from every hole, and act trivially on tags. +As $Y\cap\Lambda=T\sqcup A$, their net action on raw two-copy coordinates +is +\begin{align} + D_UF_AF_{Y\cap\Lambda} + &=D_UF_T\notag\\ + &=(\mathcal V^{\otimes2})^{\dagger} + F_{T\mathcal B_T}\mathcal V^{\otimes2}. + \label{eq:exchange-net-map} +\end{align} +The last equality uses that $F_T$ acts outside $U$ and hence commutes +through its isometries. Consequently the implemented encoded map +$C=D_UF_A\mathcal R$ satisfies +\begin{equation}\label{eq:exchange-exact-map} + CK_{\rm in} + =K_{\rm out}(\mathcal V^{\otimes2})^{\dagger} + F_{T\mathcal B_T}\mathcal V^{\otimes2}. +\end{equation} + +Put $\zeta=s\otimes s'$ and +$\delta=\norm{\mathcal V\Omega-\zeta}\le L^{-30}$. Both vectors are +normalized, so +\[ + \norm{(\mathcal V\Omega)^{\otimes2}-\zeta^{\otimes2}} + \le2\delta. +\] +The vector $\zeta^{\otimes2}$ is fixed by $F_{T\mathcal B_T}$, which swaps +the two identical $s$ factors. Since +$(\mathcal V^{\otimes2})^{\dagger}\mathcal V^{\otimes2}=\id$, we obtain +\begin{align*} + &\left\| + (\mathcal V^{\otimes2})^{\dagger}F_{T\mathcal B_T} + \mathcal V^{\otimes2}\Omega^{\otimes2}-\Omega^{\otimes2} + \right\|\\ + &\hspace{2em}\le + \norm{(F_{T\mathcal B_T}-\id) + ((\mathcal V\Omega)^{\otimes2}-\zeta^{\otimes2})} + \le4\delta. +\end{align*} +Multiplication by the contraction $K_{\rm out}$ and +\eqref{eq:exchange-exact-map} give the claimed error between the product +of the old encoded reference vectors and the product of the new ones. +All actual operations are private at the indicated parties; the apparent +partial raw swap was induced by the change of sheet names and corrected +using the common owner's buffer. No raw register is transferred by this +renaming. +\end{proof} + +Proposition~\ref{prop:angular} will again verify the one-copy information +bound in Lemma~\ref{lem:exchange}. Including holes from \emph{both} sheets +in \eqref{eq:exchange-Z} ensures that the private correction commutes with +every encoding. Exchanging whole outer squares ensures that all tag and +radius branches use the same register identification. The second sheet +may be discarded after the exchange and any subsequent point treatments; +its encoded reference vector is retained in the error comparison until +that discard is taken. diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/06-geometry.tex b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/06-geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..966ae4a61b91bb9ceb2157b0ec642b063d67c7ea --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/build/sections/06-geometry.tex @@ -0,0 +1,681 @@ +\section{A finite planar distribution protocol} +\label{sec:geometry} + +We now construct a schedule to which the compression rule applies. +The parties will be attached to dyadic squares. Their memories need not +be spatially localized: a party may initially hold an entire auxiliary +copy of $\Omega$, and may act on all of its own registers. What we must +control is the number of parties involved in a gate, each party's total +participation in gates with other parties, and the distances between the +squares to which those parties are attached. + +\begin{proposition}[Distribution through a dyadic hierarchy] +\label{prop:protocol} +For all sufficiently large $L$, there is a protocol with the following +properties. All constants in this statement are uniform in the +Hamiltonian and $L$. +\begin{enumerate} +\item +There are $O(L^2)$ parties, one for each square in a padded dyadic +hierarchy and at most a fixed number of additional parties for each +repainting of a square. Initialization is a pure product across +parties. One party initially holds the main copy of $\Omega$; all +auxiliary copies are initialized privately at their designated parties. +\item +The schedule has at most $C L^3$ changes. Each change is implemented by +a fixed number of contraction gates involving at most a fixed number of +parties and having the bipartite-resource expansions established in +Lemmas~\ref{lem:small-rewrite}, \ref{lem:birth}, and +\ref{lem:exchange}. The expansions can be made accurate to any fixed +inverse power of $L$ in operator norm where an approximation is needed. +Each party participates in at most a fixed number of gates involving +other parties over its entire lifetime. +\item +Every linked gate involves parties at the same or adjacent dyadic +scales. Parties at a gate have addresses at distance $O(n)$ from one +another, where $n$ is either endpoint scale; additional parties have the +scale and address of the square to whose repainting they belong. +\item +There is a designated reference vector at every stage, formed from the +main encoded frame and the auxiliary encoded frames. All its factors +have norm at most one. Each change takes its preceding reference vector +within $C L^{-20}$ in vector norm of its following reference vector. +At the end, the reference is +\[ + \Omega\otimes\Xi, + \qquad 1-C L^{3-60}\le \norm{\Xi}\le1, +\] +where $\Xi$ contains only registers to be discarded. Every original +physical site is a raw register at its own unit-square party. No other +party has a physical output, so every party's physical output dimension +is at most $q$. +\end{enumerate} +\end{proposition} + +The conservative exponent $3$ in the schedule and reference-norm bounds +will be convenient for error accounting. The construction below in fact +has $O(L^2)$ changes. We prove the proposition by specifying all its +guides and verifying the hypotheses of the three frame operations. + +\subsection{Ambient guides, holes, and the hierarchy} + +Translate the lattice so that its sites are the cell centers +$(i+\tfrac12,j+\tfrac12)$, $0\le i,j0$ and +$\epsilon_0>0$, and write +\begin{equation} + t=D\log L, + \qquad h_n=\epsilon_0\min(n,t). + \label{eq:geometry-scales} +\end{equation} +A \emph{true vertex} of a guide is a point lying in the closures of at +least three distinct actual labels. The guides we construct have only +finitely many such points. At every true vertex use the encoded hole +of Definition~\ref{def:frame}, with inner radius $h_n$ and outer radius +$2h_n$, and with test exponent $60$. We will choose $\epsilon_0$ so +that the outer squares are disjoint within each guide. For a given +center and radius, make the same choice of the adaptive constraint on +every sheet. Thus each branch of a hole uses a square with radius +between $h_n$ and $2h_n$. + +Choose the tag holder canonically among the actual incident labels +other than the placeholder, for example by a fixed order on genuine +block parties. The placeholder is omitted from this choice, regardless +of its current identity. A true vertex has at least two other incident +labels, so the rule is always possible. An empty physical outer square +uses the trivial one-branch encoding. When a guide is exchanged between +sheets, use the same tag rule; its tag then accompanies the whole hole. + +Before entering a new hierarchy level, resize existing main-sheet holes +to its value of $h_n$, one hole at a time, without changing the guide. +These resizings will also be small-patch rewrites. A fresh auxiliary +sheet has a uniform guide on the whole plane and therefore no holes. +An auxiliary sheet that has been retired is kept idle, with its +registers and tags at their current parties, until the final discard. +Its holes are not resized at subsequent levels. + +\subsection{The edge construction before point treatment} + +Choose a large fixed constant $K_0$. For $n\le K_0t$ we will repaint +the square directly by a small-patch rewrite. We first describe the +large-scale construction, for $n>K_0t$, before treating its finitely +many contact points. + +Let $S$ be the $n$-square being repainted, with old label $A$ and final +label $B$. For any of its edges let $s\in(0,n)$ be the coordinate +parallel to that edge, measured from one endpoint, and let $d$ be the +normal coordinate, positive into $S$. Set +\begin{equation} + w(s)=10^{-3}\min(s,n-s), + \qquad x=\frac{d}{w(s)}. + \label{eq:geometry-band} +\end{equation} +An interval in $x$ always refers only to this parallel-coordinate span. +These intervals define polygonal bands, with their only bends at +$s=n/2$. + +First birth $B$ in the central part of $S$, leaving the four inward +edge bands $01, +\] +in the neighborhood of the edge. Here $C$ is the label on the unique +exterior neighboring $n$-square, or the placeholder at an exterior +edge. Nominal names in all the words below are permitted to coincide; +adjacent equal labels are simply one actual region. The marked points +for an edge operation are its two endpoints. + +The reason for using an auxiliary sheet is already visible when $A,B,C$ +are distinct. A direct death of the $A$ strip into $B$ would leave the +surrounding word $C\mid B$. The closure of the changed strip meets $C$ +along the entire original edge, so it has zero separation from the +non-$B$ owners. The angular-information estimate therefore does not +supply the hypothesis needed for Lemma~\ref{lem:birth}. We instead prepare +additional $A$ and $B$ strips on an auxiliary sheet and exchange them into +the exterior. The resulting $C$ strip is surrounded by $A$ and can die +into it; the combined $A$ strip is then surrounded by $B$ and can die in +turn. A final birth and death restore the original exterior assignment. +The construction below supplies these buffers away from the endpoints; +the subsequent point treatment supplies the logarithmic clearance there. + +Perform the following finite sequence. +\begin{enumerate} +\item +Initialize a fresh auxiliary $\Omega$ sheet uniformly at $C$. On this +sheet birth $B$ in $-60$ such that every unmodified elementary birth or +death above has changed region $R$ of diameter $O(n)$ and surrounding +guide $f$ satisfying +\begin{equation} + \dist\bigl(y,\overline{\{f\ne P_\circ\}}\bigr) + \ge a_0\min\{n,d_{\mathcal V}(y)\}, + \qquad y\in\overline R. + \label{eq:geometry-birth-clearance} +\end{equation} +For an unmodified lens exchange, every point in the closure of the +positions that are not common $C$ on its two before guides satisfies +\begin{equation} + \dist(y,\partial Y) + \ge a_0\min\{n,d_{\mathcal V}(y)\}. + \label{eq:geometry-exchange-clearance} +\end{equation} +Here and throughout this section distances are ambient sup distances. +\end{lemma} + +\begin{proof} +For the central birth the closed changed region is separated from the +exterior of $S$ by the four edge bands, except at the four corners. +Near a corner its two distances from the adjacent edges are bounded +below by a fixed positive fraction of its sup distance from that +corner. This gives the asserted conical separation there. Away from +the corners the margin is a fixed positive fraction of $n$. + +The first auxiliary insertion has a uniform surrounding guide. The +second and third have closed normal intervals strictly inside the +surrounding intervals $(-6,-1)$ and $(-5,-2)$, respectively. For the +first main death the surrounding $A$ interval is $(-5,1)$ and the +changed interval is $(-4,0)$. For the second main death the surrounding +$B$ sector includes $(-6,2)$ and the changed interval is $(-5,1)$. +For the last birth the surrounding $B$ sector again includes +$(-6,2)$, while the inserted interval is $(-3,0)$. For the last death +the surrounding $C$ sector includes $(-8,0)$ and the changed interval +is $(-6,-3)$. All these margins are strict. The enlarged edge bands +avoid unrelated edges and blocks by the preceding disjointness check. + +For the exchange, the main guide is $C$ throughout the open exterior +band. Both lens thresholds $-7$ and $-7/2$ lie strictly inside $C$ +sectors of the auxiliary word in \eqref{eq:geometry-aux-word}. +Hence the lens boundary can meet the closed noncommon-$C$ region only +at its two marked endpoints. + +In a fixed neighborhood of every marked point, in units of $n$, all +these interfaces are straight rays. The relevant closed conical sets +have no common direction: away from their center they are disjoint by +the strict interval margins just checked. Their intersections with a +unit sup sphere therefore have positive separation. Homogeneity gives +\eqref{eq:geometry-birth-clearance} and +\eqref{eq:geometry-exchange-clearance} near the marked points. On +the remaining bounded region, compactness after scaling by $n$ gives +a positive minimum separation. Points far from the bounded lens are +harmless for \eqref{eq:geometry-exchange-clearance}. + +Only finitely many scaled configurations occur: the baseline nearby +$n$-squares have a finite number of label-equality patterns, and the +band prescriptions are fixed. Thus one $a_0$ works for all of them. +Identifying two nominal labels only removes parts of the obstructing +sets or makes an operation an identity, and never worsens a separation +already proved. This includes equality with the single placeholder +label. +\end{proof} + +All true vertices of an unmodified guide are separated on scale $n$. +Indeed the baseline vertices are grid corners. The new band interfaces +are disjoint except at their indicated endpoints. Their bends at +$s=n/2$ have only two incident labels and so are not true vertices. +In a fixed-radius neighborhood of a marked point, in units of $n$, +the only possible true vertex is the marked point itself. + +\subsection{Point treatment and the actual bulk operations} + +Before implementing one elementary birth or death, homogenize each +involved sheet inside the radius-$t$ square around every mark to its +surrounding label $P_\circ$. Make the same modification in the target +guide of that elementary operation. Thus its actual bulk recoloring +is truncated by those squares. For an exchange, homogenize both +sheets there to $C$; the lens $Y$ itself need not be altered because +the assignments coincide throughout the homogenized portions. After +the bulk operation, remove the point treatment by changing to the +specified target guide. We prove below that every homogenization and +every removal is a small-patch rewrite. + +Choose $K_0$ large enough that, when $n>K_0t$, the tenfold enlargements +of the treated squares are disjoint, contain no other unmodified true +vertices, and lie inside the straight-ray charts used above. In a +treated neighborhood, any newly created true vertex is at the +intersection of a ray with the square rim. The set of possible rim +intersection points is finite after scaling by $t$. Coincident +points count as one point. There is therefore a positive minimum +separation between distinct possible centers, uniformly over the +finite pattern family. Take $\epsilon_0$ small compared with these +separations, with the ordinary grid separations, and with $a_0$. +This makes the outer holes disjoint, including in the temporary guides. + +Figure~\ref{fig:geometry-points} illustrates how a true vertex can +move to the rim during a point treatment. +\input{figures/geometry-point-treatment} + +For a birth or death, a point of the remaining closed changed region +lies outside the interiors of the treated squares. Consequently +$d_{\mathcal V}(y)\ge t$, and homogenization to the surrounding label +can only remove its other-label obstructions. By +\eqref{eq:geometry-birth-clearance}, after decreasing a fixed constant +if necessary, its separation from the other labels is bounded below by +\begin{equation} + a_1\max\{t,\min(n,d_{\mathcal V}(y))\}, + \qquad a_1>0. + \label{eq:geometry-floor} +\end{equation} +Every true vertex of the surrounding temporary guide belongs to the +closure of positions of labels different from $P_\circ$. Enlarging +such a center to its outer hole square subtracts at most $2h_n$ from +this separation. Since $h_n=\epsilon_0t$ in the present large-scale +case, a further fixed decrease of $a_1$ preserves +\eqref{eq:geometry-floor} for all outer hole sites too. + +There are no changed true vertices in the bulk recoloring. To see +this also at the treated rims, the closure of the truncated changed +region has a positive neighborhood containing only $P_\circ$ in the +surrounding guide. A birth or reversed death introduces only one +additional label in that neighborhood. Thus any newly drawn +interface, including where it meets a treated-square rim, has at most +two incident labels. Existing true vertices are separated from this +closed change and retain their guides, holes, and tag holders. The +raw target $T$ is therefore outside all holes, compactly located in +diameter $O(n)$, and satisfies the tapering hypothesis against the set +$E$ consisting of other owners and all outer hole sites. The remaining +raw buffer has a single owner. Lemma~\ref{lem:birth}, in its forward +or reverse form, implements the bulk change. + +For the exchange, homogenization removes all noncommon-$C$ positions +inside the treated squares. By +\eqref{eq:geometry-exchange-clearance}, every remaining such position +has the same floor at $t$ in its distance from $\partial Y$. This +also holds for new rim vertices: a rim true vertex belongs to the +closure of an outside sector that is not $C$, and the original angular +estimate applies at that point. Thus $\partial Y$, including each of +its intersections with a treated-square rim, lies in an open +common-$C$ corridor. No new true vertex is created by exchanging +the guides along that boundary. + +Every true vertex on either sheet belongs to the closure of a +noncommon-$C$ region. Its distance from $\partial Y$ is at least a +fixed multiple of $t$ in a treated neighborhood, and obeys the larger +angular bound elsewhere. The full outer hole square, of radius +$2\epsilon_0t$, therefore lies strictly on one side of the lens +boundary. The same tapered bound holds at every point of an outer hole, not only +at its center: both $y\mapsto\dist(y,\partial Y)$ and +$y\mapsto d_{\mathcal V}(y)$ are $1$-Lipschitz. If $c$ is the center +and $|x-c|_\infty\le2\epsilon_0t$, put +$Q_x=\max\{t,\min(n,d_{\mathcal V}(x))\}$. A center clearance +$a_*\max\{t,\min(n,d_{\mathcal V}(c))\}$ gives +\[ + \dist(x,\partial Y)\ge a_*Q_x-2\epsilon_0(a_*+1)t + \ge \tfrac12a_*Q_x +\] +after fixing $\epsilon_0$ small enough. In particular, \emph{every} +selected variable-radius square in that hole lies on the same side, +since its radius is at most $2h_n$. Exchanging a neighborhood of such a vertex substitutes one +complete sheet guide and carries its entire hole and tag. It does +not splice sectors of two guides at a vertex. + +For Lemma~\ref{lem:exchange}, let $T$ be the positions inside $Y$ +that are not common $C$, together with all inside outer hole sites, +and let $E$ be the analogous set outside $Y$. The set $T$ has +diameter $O(n)$. Any segment from a point of $T$ to a point of $E$ +crosses $\partial Y$, so its length is at least the distance of that +point of $T$ from the boundary. The preceding clearance bounds, +including the outer-hole enlargement, therefore give the required +tapering separation of $T$ from $E$. On the remaining positions both +raw copies are held by $C$. This proves all hypotheses of the exchange +lemma, including its condition on whole holes and their tags. + +All statements use actual labels. Coalescing nominal labels cannot +create a meeting of three actual owners: the closure of an identified +region is the union of its finitely many former closures. If a nominal +vertex disappears under such an identification, at most two actual +owners meet there and no vertex encoding is required. The same +argument covers the exterior placeholder. Empty physical targets +need no information estimate. + +Choose one fixed $a>0$ smaller than all the surviving constants in +these tapering bounds. Proposition~\ref{prop:angular}, at accuracy +$L^{-60}$, then supplies a sufficiently large fixed $D$ in +\eqref{eq:geometry-scales}. The birth and exchange errors are +$O(L^{-30})$, as required by their lemmas. The constants used to +choose $a$ depend only on the fixed geometric templates, not on $D$. + +\subsection{The small-patch covers} + +We verify every invocation of Lemma~\ref{lem:small-rewrite}. The +following compactness observation makes the two-owner condition +uniform, including along the boundary of an inner hole. + +Fix a compact working neighborhood and one scaled guide. For each +actual label $c$, let $F_c$ be the closure of its region in that +neighborhood, with all open inner hole squares removed. Each $F_c$ +is compact. No three $F_c$ with distinct actual labels have a common +point, because every true vertex lies strictly inside its own inner +hole. It follows that there is a positive footprint diameter such +that no smaller footprint meets three of these sets. Otherwise take +a sequence of shrinking footprints, pass to constant label choices +and convergent points in the compact working neighborhood, and obtain +a point common to three distinct $F_c$. + +This remains uniform for our small-scale and resizing radius +parameters. After scaling, these parameters range over compact +intervals bounded away from zero. The joint sets of a parameter and +a point belonging to $F_c$ are closed: being outside an open square +is a non-strict distance inequality. The same convergent-subsequence +argument applies to the joint sets. There are only finitely many +label-equality types, so take the minimum positive footprint bound +over those types. Apply the test separately to the old and new +guides, then take the smaller bound. Different old and new hole +centers do not mix the two tests. + +Notice that the \emph{center} of a footprint may lie inside a hole. +The condition counts labels only at footprint points outside the +relevant inner holes. Thus this observation permits a net of small +patches to cover the \emph{whole outer} hole footprints, as required +by Lemma~\ref{lem:small-rewrite}. + +\paragraph{Point homogenization and removal.} +For one mark in the large-scale case, work in its radius-$2t$ +neighborhood. Include as affected every old and new hole at the +marked center or on the treated-square rim. These are all possible +changed true vertices, and their outer squares lie strictly inside +that neighborhood for small $\epsilon_0$. There are no other holes +nearby. In units of $t$, the old and new guides are from a finite +family of ray patterns, with or without the square clearing, and the +inner hole radius is $\epsilon_0$. + +Choose a fixed sufficiently small $\nu>0$. Patches with inner radius +$u=\nu t$, centered on a square net of mesh at most $u$, cover the +treated square and all affected outer holes. Take the centers within +$u$ of the set to be covered. With $\nu$ small, their outer squares +are still inside the working neighborhood and avoid every untouched +hole. The compactness observation gives at most two old owners +outside the old inner holes and at most two new owners outside the +new inner holes in each outer patch. The number of patches is fixed: +the working diameter is $O(t)$ and the patch radius is $\nu t$. +Their raw owners and affected tag holders are among the bounded list +of parties in this local diagram. Apply +Lemma~\ref{lem:small-rewrite}, one mark and one sheet at a time. +The separated marked neighborhoods do not change one another's +diagrams. The same verification applies when removing the point +treatment after the bulk operation, using its actual target guide. + +\paragraph{Direct repainting when $n\le K_0t$.} +Before and after a direct repainting, true vertices are grid corners. +Only the four corners of the square being repainted can change their +true-vertex status or their tag holder. Treat every old or new hole +at those corners as affected. The hole radius obeys +\begin{equation} + \frac{\epsilon_0}{\max(K_0,1)} + \le \frac{h_n}{n}\le\epsilon_0. + \label{eq:geometry-small-radius} +\end{equation} +In units of $n$, the old/new label patterns form a finite collection, +and the possible radius ratios lie in the displayed compact positive +interval. The compactness observation gives a uniform sufficiently +small patch radius $u=\nu n$. A fine net covers $S$ and the entire +affected outer holes with a fixed number of such patches. Every +untouched grid-corner hole is a fixed scaled distance from the set to +be covered: corners of $S$ have all been included, and any other grid +corner is at distance at least $n$ from $S$. Choosing $\epsilon_0$ +and then $\nu$ small makes all outer patches avoid those holes. All +hypotheses of Lemma~\ref{lem:small-rewrite} follow. This +directly repaints $S$ without using an auxiliary sheet. + +\paragraph{Hole resizing.} +Entering level $n$, the old and new radii are +$\epsilon_0\min(2n,t)$ and $\epsilon_0\min(n,t)$. If they differ, +then $n0$, and then choose $D$ from +Proposition~\ref{prop:angular} at information power $60$. +The largest patch scale constant $B_0$ is selected afterwards to cover +all patches of scale at most a fixed multiple of $D\log L$. + +Proposition~\ref{prop:patch} then determines a fixed rank exponent. +Only boundedly many such cylinder expansions occur in a change, so their +total branch count is polynomial. Choose the singular-subspace dimensions +in Lemma~\ref{lem:small-rewrite} next, to obtain operator error +$L^{-25}$. Choose the effect-inventory stack lengths from the +\emph{original} weighted expansion error in Lemma~\ref{lem:effects}; +only afterwards expand the permutation averages. Finally choose the +Gaussian sample count in Theorem~\ref{thm:compression}. None of these +algebraic choices changes an earlier geometric support or clearance. +The resulting $C_1,r$ and the congestion $\chi$ in +\eqref{eq:final-bond} depend only on the fixed parameters and templates. + +Increase a fixed threshold $L_0$ so that every preceding asymptotic +inequality holds and the error $CL^{-5}$ is at most $L^{-1}$ for +$L\ge L_0$. For the finitely many smaller sizes, every vector has an exact +PEPS on a spanning tree of the square grid. One explicit construction +uses a common index for the $q^{L^2}$ product-basis configurations on all +tree edges, equality tensors at tree vertices that also output the local basis state +of that common configuration, and puts each expansion coefficient at one +chosen root. Non-tree edges have dimension one. +This bond bound is finite and independent of the Hamiltonian at each +fixed $L$. Enlarging the single prefactor $C$ handles all $2\le L4D_0s_0. +\] +The $P_a$ are closed convex rectangles or triangles of sup diameter at most +$s_0$, with side slopes $0,\infty,1,-1$; $n,s_0$ are positive integers and +$s_0\ge1$. With $\epsilon=10^{-5}$, set +$\ell=\lfloor n^{1-\epsilon}\rfloor\le s_0$, +$T_\ell=T+[-\ell,\ell]^2_{\mathbb Z}$, $X=A\cap T$, +$S_0=A\cap T_\ell$, and $Y=S_0\setminus X$. +At all sufficiently large scales the input guarantees +\[ + I_\Omega(X:\Gamma\setminus S_0)\le Cn^{1-\epsilon},\qquad + |S_0|\le Cn^2,\qquad + |\partial_\Gamma X|,|\partial_\Gamma Y|,|\partial_\Gamma S_0|\le Cn. +\] + +The Proposition ``Amplification'' in \cite[Section~10]{RepoAreaLaw} +requires these three bounds with any fixed common constant $C_0$ and +$\alpha=k/(k+1)>1-10^{-6}$. Put $\beta_0=1-2\cdot10^{-6}$. +For every fixed integer $M\ge1$ it supplies a radius +\[ + r\le C_M n^{(1-\epsilon)/\alpha},\qquad + \ell+2r=o(n^{\beta_0}),\qquad + I_\Omega(X':E)\le C_M n^{-M} +\] +simultaneously for every $X'\subseteq X$ and +$E\subseteq\Gamma\setminus N_{2r}(S_0)$. The radius and constants are +independent of the remote set and total volume. If also +$\dist_\infty(T,Z_A)>\ell+n^{\beta_0}$, then +$N_{2r}(S_0)\subseteq A$. In Proposition~\ref{prop:coarse} we take +$A=\Lambda$, so $Z_A=\varnothing$, choose $s_0,n$ comparable to the target +scale with fixed ratios, and use $M=1$. A graph neighborhood lies in the +corresponding ambient enlargement; this is the precise conversion used to +obtain the ambient-separation statement. + +\paragraph{The two-family interface and its adaptation.} +The Lemma ``Two-family entropy cancellation'' in +\cite[Section~11]{RepoAreaLaw} states that, for a pure state and a partition +$A=D\mathbin{\dot\cup}\bigcup_iX_i$ into two ordered families, bounds +\[ + I\left(X_i:A^c\cup\bigcup_{j0$ and any fixed lower +scale $n_*$. It gives $|D|\le Cb$, $X_i\subseteq A\cap T_i$, admissible +separated templates with $n_i\ge n_*$ and +$\lfloor n_i^{1-\epsilon}\rfloor\le s_i$, and +\[ + \dist_\infty(T_i,Z_A)>2n_i^{\beta_0},\qquad + \dist_\infty\left(T_i,\bigcup_{j2n_i^{\beta_0}, + \qquad \sum_i n_i^{-100}\le Cb. +\] +The constant may depend on $n_*$ and the template constants but not on the +cut or domain. These statements are used within the companion's area-law +proof. Lemma~\ref{lem:cell-cmi} adapts their ordering principle to a cell +with arbitrary exterior conditioning and proves that local variant in +full. In particular, we do not infer a conditional-information bound merely +by conditioning an ordinary mutual-information inequality. + +The companion's replica limit is taken at each fixed finite physical +instance and schedule, before the geometric scale tends to infinity. +Its exported bounds concern the original one-copy state with the uniform +constants above. The later tapering construction, adaptive patch +constraints, density compression, encoded protocol, and routing are proved +in this article. diff --git a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f1d200a00015273acead7ec2e779d45e308467dd --- /dev/null +++ b/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0b55ba0b87ad04680d7ec5be5873deb20b20060dfd84193edc10dfa939a7e756 +size 671703 diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/README.md b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..2024c03484037ddf27a2f1d2e99aea62455c39cd --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [Polynomial-Time Unitary Synthesis from a Boolean Oracle](paper.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026, + author = {{OpenAI}}, + title = {{Polynomial-Time Unitary Synthesis from a Boolean Oracle}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf}{OAI:Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/figures/recursion.tex b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/figures/recursion.tex new file mode 100644 index 0000000000000000000000000000000000000000..5d1f0cfeb2d4470394a282f5a6e8fbec3bb787ca --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/figures/recursion.tex @@ -0,0 +1,35 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[>=Latex, font=\small, + box/.style={draw,rounded corners=2pt,minimum height=10mm,align=center}, + state/.style={align=center,font=\footnotesize}] + \node[box,minimum width=19mm] (e) at (0,0) {$\mathsf U_E$}; + \node[box,minimum width=39mm] (f) at (4.8,0) + {$\mathsf C$\,: one recursive call}; + \node[box,minimum width=19mm] (r) at (9.6,0) {$\mathsf U_R^*$}; + \draw[->] (-1.8,0) -- (e.west); + \draw[->] (e.east) -- (f.west); + \draw[->] (f.east) -- (r.west); + \draw[->] (r.east) -- (11.4,0); + \node[state] at (-1.45,.58) {$\iota x$}; + \node[state] at (2.05,.58) {$\jmath\mathcal E_Lx$}; + \node[state] at (7.55,.58) {$\jmath\mathcal R_LVx$}; + \node[state] at (11.02,.58) {$\iota Vx$}; + \node[state] at (0,-.92) {local encoding}; + \node[state] at (4.8,-.92) {order $k0. +\] +Choose any rotation when $P(p)=0$. After $u$ such slots the amplitude at $x$ +is $|a_x|$. One diagonal slot then supplies the phases of the $a_x$, choosing +arbitrary phases at zero entries. Including $\lambda$ among the controls +makes all these choices simultaneously. This is the usual conditional +rotation construction for state preparation; here an entire uniformly +controlled rotation is counted as one slot; see~\cite{MVBS2005,BVMS2005}. +\end{proof} + +\begin{lemma}[Basis permutations]\label{slot:permutation} +Any family of permutations of the $2^u$ computational basis states can be +implemented with $5u$ slots and $u$ temporary qubits that start and finish +at zero. The skeleton is independent of the permutations. +\end{lemma} + +\begin{proof} +For the permutation $\pi_\lambda$, use $u$ slots to write its image bitwise +into a temporary register: +\[ + \ket{\lambda}\ket{x}\ket{0} + \longmapsto \ket{\lambda}\ket{x}\ket{\pi_\lambda(x)}. +\] +Each slot chooses either $I$ or the bit flip, controlled on $\lambda$ and +$x$. Next use $u$ slots to xor $\pi_\lambda^{-1}(y)$ into the first register, +controlled on the second register $y$. The two registers become +$\ket{0}\ket{\pi_\lambda(x)}$. Swap them using three CNOTs per pair of bits. +\end{proof} + +The same convention makes a Fourier transform inexpensive. We use the +positive-phase transform, so that a frequency basis vector becomes its +character in the computational basis. + +\begin{lemma}[Fourier transforms]\label{slot:fourier} +The unitary +\[ + \ket{x}\longmapsto 2^{-b/2}\sum_{z=0}^{2^b-1} + e^{2\pi i xz/2^b}\ket{z} +\] +on $b\ge1$ qubits has a uniformly generated skeleton with $O(b)$ slots. +Products of such transforms have a number of slots linear in their total +register width. +\end{lemma} + +\begin{proof} +Write the input integer as $x$. Process its bits in decreasing order of +place value. At the bit of place value $2^{b-1-j}$, apply $H$, then a +diagonal slot controlled on the still unprocessed lower bits. Give the +state $1$ the additional phase that, together with its sign from $H$, +makes the ratio of its two amplitudes equal to +$e^{2\pi i x/2^{b-j}}$. Bits above the current bit contribute integral +multiples of $2\pi$ to this expression, and are no longer needed as +controls. The resulting bit is the Fourier output bit of place value +$2^j$. A final reversal of bit order gives the stated transform. There are +two slots per processed bit and $O(b)$ CNOTs for the reversal. +\end{proof} + +These constructions retain the scalar phase of every label branch. This is +essential: phases that would be irrelevant for one isolated unitary become +relative phases when the same operation is controlled by a quantum register. diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/03-contraction.tex b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/03-contraction.tex new file mode 100644 index 0000000000000000000000000000000000000000..3574a24583aa503c66fce43425a1ab134b778adc --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/03-contraction.tex @@ -0,0 +1,217 @@ +\section{Producing a contracting block}\label{sec:contraction} + +The reduction will remove a small group of indices from the synthesis problem. +For its geometric series to converge rapidly, the corresponding diagonal block +of the unitary must have norm bounded away from one. We now obtain such a block +using only permutations, diagonal signs, and Walsh--Hadamard transforms. All +these transformations have short programmable-slot implementations. + +We first record the matrix estimate needed to choose the signs. A +\emph{Rademacher variable} takes the values $1$ and $-1$ with equal probability. +The following elementary moment argument is a finite-dimensional form of the +noncommutative Khintchine estimate; see Tropp~\cite[Sections~6--7]{Tropp2018} +for the Gaussian integration-by-parts and trace-product method. We include +the proof with the constants used below. + +\begin{lemma}[A matrix sign estimate]\label{contract:moment} +Let $L_1,\ldots,L_q$ be Hermitian $h\times h$ matrices, and let +$\varepsilon_1,\ldots,\varepsilon_q$ be independent Rademacher variables. +Set $v=\norm{\sum_i L_i^2}$. For every integer $a\geq1$, +\begin{equation}\label{contract:sign-bound} + \E\norm{\sum_{i=1}^q\varepsilon_iL_i} + \leq 2h^{1/(2a)}\sqrt{(2a-1)v}. +\end{equation} +\end{lemma} + +\begin{proof} +We first replace the signs by independent standard real Gaussians $g_i$ and +write $X=\sum_i g_iL_i$. Gaussian integration by parts, applied to this matrix +polynomial, gives +\begin{equation}\label{contract:ibp} + \E\Tr X^{2a} + =\sum_{i=1}^q\sum_{l=0}^{2a-2} + \E\Tr\bigl(L_iX^lL_iX^{2a-2-l}\bigr). +\end{equation} +Indeed, write the first factor of $X^{2a}$ as $\sum_i g_iL_i$, use +$\E[g_iP(g)]=\E[\partial_iP(g)]$, and differentiate each of the remaining +$2a-1$ factors in turn. + +Put $m=2a-2$. For a fixed Hermitian $X$, diagonalize it with real eigenvalues +$\lambda_j$. In this basis, for $m>0$, +\begin{align*} + \Tr(L_iX^lL_iX^{m-l}) + &=\sum_{j,k}|(L_i)_{jk}|^2\lambda_k^l\lambda_j^{m-l}\\ + &\leq\sum_{j,k}|(L_i)_{jk}|^2 + \left(\frac lm|\lambda_k|^m+ + \frac{m-l}{m}|\lambda_j|^m\right) + =\Tr(L_i^2X^m). +\end{align*} +The inequality is the weighted arithmetic--geometric mean inequality after +taking absolute values. The last equality uses the symmetry +$|(L_i)_{jk}|^2=|(L_i)_{kj}|^2$ and the evenness of $m$. +For $m=0$ the same assertion is equality. Since $X^m$ is positive semidefinite +and $\sum_iL_i^2\preceq vI$, equation~\eqref{contract:ibp} implies +\[ + \E\Tr X^{2a}\leq(2a-1)v\,\E\Tr X^{2a-2}. +\] +Iterating from $\Tr I=h$ yields +\[ + \E\Tr X^{2a} + \leq h(2a-1)!!\,v^a + \leq h(2a-1)^a v^a. +\] +Here $(2a-1)!!=1\cdot3\cdots(2a-1)$. Therefore +\begin{equation}\label{contract:gaussian-bound} + \E\norm X + \leq\bigl(\E\norm X^{2a}\bigr)^{1/(2a)} + \leq h^{1/(2a)}\sqrt{(2a-1)v}. +\end{equation} + +To return to signs, write $g_i=\varepsilon_i t_i$, where the $t_i=|g_i|$ are +independent of the signs and have common mean $\mu=\sqrt{2/\pi}$. Conditional +Jensen gives +\[ + \mu\,\norm{\sum_i\varepsilon_iL_i} + \leq\E_t\norm{\sum_i\varepsilon_i t_iL_i}. +\] +Average over the signs, apply~\eqref{contract:gaussian-bound}, and use +$\mu^{-1}<2$. +\end{proof} + +Fix throughout the construction +\begin{equation}\label{contract:parameters} + M=4096(n+1)^2,\qquad r=\frac34. +\end{equation} +For an integer $2\leq d\leq2^n$, let $q$ be the largest power of two not +exceeding $d$, and define +\begin{equation}\label{contract:orders} + s=\max\{1,\lfloor q/M\rfloor\},\qquad k=d-s. +\end{equation} +We call the first $k$ indices \emph{ports} and the last $s$ indices +\emph{internal indices}. Both groups are nonempty. + +\begin{proposition}[A block of uniformly small norm]\label{contract:block} +For every unitary $U$ on $\C^d$, there are unitaries $P_{\mathrm L}$ and +$P_{\mathrm R}$ on $\C^d$ such that +\begin{equation}\label{contract:block-form} + V=P_{\mathrm L}UP_{\mathrm R} + =\begin{pmatrix}A&B\\C&D\end{pmatrix}, + \qquad \norm D\leq r, +\end{equation} +where the block sizes are $k,s$ from~\eqref{contract:orders}. +The transformations $P_{\mathrm L},P_{\mathrm R}$ and their inverses have +implementations using $O(n)$ programmable slots and $O(n)$ clean working +qubits, with a structure determined by $n,d$. The same structure works +simultaneously for any family $U_\lambda$, using the label $\lambda$ as an +additional control. +\end{proposition} + +\begin{proof} +If $s=1$, the last column of $U$ has an entry of modulus at most $1/\sqrt d$. +A row permutation moves this entry to the last position. The resulting +one-dimensional block satisfies $\norm D\leq1/\sqrt d<3/4$, and the permutation +has the claimed slot implementation by Lemma~\ref{slot:permutation}. + +Suppose $s>1$. Let $M_0$ be the last-by-last $q\times q$ submatrix of $U$; +it is a contraction. Set +$H_q=H^{\otimes\log_2q}$, and let $W$ consist of its last $s$ columns. We +will choose diagonal sign matrices $E_1,E_2$ on $\C^q$ and use +$H_qE_1$ on the left and $E_2H_q$ on the right, acting as the identity on the +remaining $d-q$ indices. The resulting internal block is +\begin{equation}\label{contract:D-signs} + D=W^*E_1C',\qquad C'=M_0E_2W. +\end{equation} +We choose $E_2$ first to bound the entries of $C'$, then choose $E_1$ to +bound the operator norm of $D$. + +For independent signs in $E_2$, each entry of $C'$ is a sign sum +$\sum_j\varepsilon_j\alpha_j$ with $\sum_j|\alpha_j|^2\leq1/q$. +For completeness, a real sign sum with squared coefficient sum at most $1/q$ +satisfies +\[ + \E\exp\left(t\sum_j\varepsilon_j a_j\right) + =\prod_j\cosh(ta_j)\leq\exp(t^2/(2q)). +\] +Markov's inequality, optimized over $t$, bounds its two-sided tail at $u$ by +$2e^{-qu^2/2}$. Applying this to the real and imaginary parts shows +\[ + \Pr\left(\left|\sum_j\varepsilon_j\alpha_j\right|\geq u\right) + \leq4e^{-qu^2/4}. +\] +There are $qs$ entries. With $u=4\sqrt{(n+1)/q}$, the union-bound failure +probability is at most +\[ + 4qs e^{-4(n+1)}\leq4\cdot2^{2n}e^{-4(n+1)}<1. +\] +We may therefore fix $E_2$ such that every entry of $C'$ has modulus at most +$4\sqrt{(n+1)/q}$. Also $\norm{C'}\leq1$. + +Let $h_i,c_i$ be row $i$ of $W,C'$, respectively, regarded as row vectors. +Then +\[ + \norm{h_i}^2=s/q,\qquad + \norm{c_i}^2\leq v:=16s(n+1)/q. +\] +Choosing independent signs $\varepsilon_i$ for $E_1$, write +\[ + D=\sum_i\varepsilon_i h_i^*c_i, + \qquad + L_i=\begin{pmatrix}0&h_i^*c_i\\c_i^*h_i&0\end{pmatrix}. +\] +The two diagonal blocks of $\sum_iL_i^2$ satisfy +\[ + \sum_i\norm{c_i}^2h_i^*h_i\preceq vW^*W=vI_s, + \qquad + \sum_i\norm{h_i}^2c_i^*c_i + =\frac sq C'^*C'\preceq\frac sq I_s\preceq vI_s. +\] +The norm of $\sum_i\varepsilon_iL_i$ equals $\norm D$. Apply +Lemma~\ref{contract:moment} with $h=2s$ and $a=n+1$. Since +$(2s)^{1/(2n+2)}\leq\sqrt2$, we obtain +\[ + \E\norm D + \leq 2\sqrt2\sqrt{2(n+1)\,16s(n+1)/q} + =16(n+1)\sqrt{s/q} + \leq\frac14. +\] +The last inequality uses $s\leq q/M$. Some choice of $E_1$ consequently +satisfies $\norm D\leq1/4$, which is more than required. + +It remains to check the circuit structure. A permutation on the $n$-bit +index register moves the last $q$ valid indices to the first $q$ valid +indices, an aligned $q$-element block. On that block, $H_q$ uses at most +$n$ Hadamard slots, controlled on +the remaining bits; each diagonal sign matrix uses one diagonal slot. Undo +the permutation afterward. Lemma~\ref{slot:permutation} supplies clean +$O(n)$-slot implementations of the permutations. All operations preserve +the valid $d$-dimensional subspace and are extended unitarily to unused index +states. Labels change only the row permutation or diagonal-sign tables, +so the same slot structure works for every $U_\lambda$. +\end{proof} + +The random signs are used only to prove that suitable tables exist. +Each label may use its own choices; the resulting slot circuit is deterministic. +The size of the removed block is large enough to make recursion short. +\begin{lemma}[Dimension decrease]\label{contract:depth} +For the parameters in~\eqref{contract:orders}, $s\geq d/(4M)$. +Starting at any $d\leq2^n$ and repeatedly replacing $d\geq2$ by $d-s$, one +reaches $d=1$ in fewer than +\begin{equation}\label{contract:J} + J=8M(n+1) +\end{equation} +steps. +\end{lemma} + +\begin{proof} +If $q/M\geq2$, then +$s=\lfloor q/M\rfloor\geq q/(2M)>d/(4M)$, because $d<2q$. +If $q/M<2$, then $s=1$ and $d<2q<4M$, with the same conclusion. +Thus every nontrivial step multiplies the order by at most $1-1/(4M)$. +If there were $J$ such steps, their final order would be at most +\[ + 2^n(1-1/(4M))^J + \leq2^n e^{-J/(4M)} + =2^n e^{-2(n+1)}<1, +\] +contrary to its being a positive integer. +\end{proof} diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/04-fields.tex b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/04-fields.tex new file mode 100644 index 0000000000000000000000000000000000000000..a1c44363da10207f4bd3bd82788b0ae1a521dcdc --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/04-fields.tex @@ -0,0 +1,248 @@ +\section{Fourier fields for a smaller unitary}\label{sec:fields} + +Let $d=k+s\le 2^n$, with $k,s\ge1$, and suppose that +\begin{equation}\label{fields:block} + V=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in U(d), + \qquad \norm{D}\le r<1,\quad r\ge0. +\end{equation} +The first $k$ coordinates are the ports and the last $s$ coordinates are +internal. We will associate to $V$ a family of $k$-dimensional unitaries +$F(z)$ and two nearly isometric maps from $\C^d$ into the space of +port-valued functions of $z$. These maps intertwine $V$ with the controlled +application of $F(z)$. Their Fourier coefficients will have few nonzero +entries in each row, which makes the maps implementable by slots. + +\subsection{Closing the internal coordinates} + +For the moment let $Z$ be any diagonal unitary of order $s$. Given a port +input $u\in\C^k$, feed the internal output $v\in\C^s$ back into the +internal input through $Z$. If $y\in\C^k$ is the port output, the equation +$V(u,Zv)=(y,v)$ becomes +\[ + y=Au+BZv,\qquad v=Cu+DZv. +\] +Since $\norm{DZ}<1$, the second equation has the unique solution $v=Xu$, +and the first then gives $y=Fu$, where +\begin{equation}\label{fields:feedback} + X=(I-DZ)^{-1}C,\qquad F=A+BZX, + \qquad + E=\begin{pmatrix}I_k\\ ZX\end{pmatrix},\qquad + O=\begin{pmatrix}F\\ X\end{pmatrix}. +\end{equation} +Thus $Eu$ and $Ou$ collect the full input and output of $V$ after the +internal coordinates have been closed. +For a scalar phase $Z=zI$, this is the standard transfer function of a +unitary block matrix; see \cite[Section~3]{FKK2009}. The same conservation +of norm proves unitarity with independently chosen diagonal phases. + +\begin{lemma}\label{fields:unitary} +For the unitary block matrix in \eqref{fields:block} and any diagonal +unitary $Z$, the matrix $F$ in \eqref{fields:feedback} is unitary. +If +\begin{equation}\label{fields:R} + R=OF^*=\begin{pmatrix}I_k\\ Y\end{pmatrix}, + \qquad Y=XF^*, +\end{equation} +then +\begin{equation}\label{fields:exact} + FE^*=R^*V, + \qquad + Y=(I-Z^*D^*)^{-1}Z^*B^*. +\end{equation} +\end{lemma} + +\begin{proof} +The definitions give $VE=O$. Consequently, +\[ + I_k+X^*X=E^*E=O^*O=F^*F+X^*X, +\] +so $F^*F=I_k$. Because $F$ is square, it is unitary. Thus $OF^*$ has +the stated top block, and $V^*R=EF^*$. Taking adjoints proves the first +identity in \eqref{fields:exact}. Comparing the bottom blocks of +$V^*R=EF^*$ gives +\[ + B^*+D^*Y=ZY. +\] +Multiplication by $Z^*$ and $\norm{Z^*D^*}\le r<1$ give the second identity. +\end{proof} + +\subsection{Truncation and averaging} + +Fix an integer truncation length $L\ge1$, and set +\begin{equation}\label{fields:parameters} + b=nL+\lceil\log_2(L+1)\rceil,\qquad + \mathcal G=(\Z/(2^b))^{L+1}. +\end{equation} +For $w,z\in\mathcal G$, write +\[ + \chi_w(z)=\exp(2\pi i\,w\cdot z/2^b). +\] +Number the internal coordinates by $i=1,\ldots,s$ and put +\begin{equation}\label{fields:phases} + v_i=(1,i,i^2,\ldots,i^L)\in\mathcal G, + \qquad Z(z)=\diag\bigl(\chi_{v_1}(z),\ldots,\chi_{v_s}(z)\bigr). +\end{equation} +The constant first coordinate of $v_i$ gives all diagonal entries of $Z(z)$ +a common phase when the first coordinate of $z$ varies. Averaging that phase +will separate terms of different lengths in the truncated series. The +remaining coordinates record power sums: the Fourier frequency of a product +of at most $L$ phases will determine its multiset of internal indices, as proved in +Proposition~\ref{fields:sparsity}. These two properties will give the norm +bound and the row bound, respectively. + +Apply Lemma~\ref{fields:unitary} at each $z$ to obtain $F(z),E(z),R(z)$. +Truncate the two Neumann expansions by defining +\begin{equation}\label{fields:truncations} + E_L(z)=\begin{pmatrix} + I_k\\[2pt] \displaystyle\sum_{\ell=0}^{L-1}Z(DZ)^\ell C + \end{pmatrix},\qquad + R_L(z)=\begin{pmatrix} + I_k\\[2pt] \displaystyle\sum_{\ell=0}^{L-1}(Z^*D^*)^\ell Z^*B^* + \end{pmatrix}. +\end{equation} +Here and below, products containing $Z$ are evaluated at the same $z$. +Since every block of a unitary has norm at most one, +\begin{equation}\label{fields:tails} + \norm{E_L(z)-E(z)},\ \norm{R_L(z)-R(z)} + \le \tau,\qquad \tau=\frac{r^L}{1-r}. +\end{equation} + +Define the field maps $\mathcal E_L,\mathcal R_L: +\C^d\longrightarrow\C^{\mathcal G}\otimes\C^k$ by +\begin{equation}\label{fields:maps} + \mathcal E_L x=\frac1{\sqrt{|\mathcal G|}} + \sum_{z\in\mathcal G}\ket z\otimes E_L(z)^*x, + \qquad + \mathcal R_L x=\frac1{\sqrt{|\mathcal G|}} + \sum_{z\in\mathcal G}\ket z\otimes R_L(z)^*x. +\end{equation} +Although a single matrix $E_L(z)^*$ need not be a contraction, +averaging its squared norm over $z$ nearly preserves the norm of every input. + +\begin{proposition}\label{fields:gram} +For either $\mathcal T=\mathcal E_L$ or $\mathcal T=\mathcal R_L$, +\begin{equation}\label{fields:gram-bound} + (1-r^{2L})I_d\ \le\ \mathcal T^*\mathcal T\ \le\ I_d. +\end{equation} +In addition, let $\widetilde F$ apply $F(z)$ to the port register, +controlled on $z$. Then +\begin{equation}\label{fields:intertwine} + \norm{\widetilde F\mathcal E_L-\mathcal R_L V}\le2\tau. +\end{equation} +\end{proposition} + +\begin{proof} +The Gram matrix of $\mathcal E_L$ is the average of $E_LE_L^*$. +Fix the last $L$ coordinates of $z$ and vary its first coordinate. +Writing $Z=\omega Z'$, the scalar $\omega$ ranges uniformly over the +$2^b$-th roots of unity, while $Z'$ stays fixed. The summand of index +$\ell$ in the lower block of $E_L$ is +\[ + \omega^{\ell+1}Z'W^\ell C,\qquad W=DZ'. +\] +The degrees $0,1,\ldots,L$ are distinct modulo $2^b$. Their orthogonality +under averaging shows that the port block of $\E_\omega E_LE_L^*$ is +$I_k$, that its off-diagonal blocks vanish, and that its internal block is +\[ + Z'\left(\sum_{\ell=0}^{L-1}W^\ell CC^*(W^*)^\ell\right)(Z')^*. +\] +Unitarity of $V$ gives $CC^*=I-DD^*=I-WW^*$. Therefore the sum telescopes: +\begin{equation}\label{fields:telescope} + \sum_{\ell=0}^{L-1}W^\ell(I-WW^*)(W^*)^\ell + =I-W^L(W^*)^L. +\end{equation} +As $\norm{W}\le r$, the internal block lies between +$(1-r^{2L})I_s$ and $I_s$. Averaging the remaining coordinates of $z$ +preserves these inequalities. + +For $R_L$, set $W=(Z')^*D^*$ and $C_0=(Z')^*B^*$. Its lower summand is +$\omega^{-(\ell+1)}W^\ell C_0$. The negative degrees are again distinct +from each other and from zero. Now +\[ + C_0C_0^*=(Z')^*(I-D^*D)Z'=I-WW^*, +\] +so the internal Gram block is exactly the sum in +\eqref{fields:telescope}, with these choices of $W,C_0$. +This proves \eqref{fields:gram-bound} for both maps. + +Finally, Lemma~\ref{fields:unitary} and \eqref{fields:tails} give, pointwise, +\[ + \norm{F(z)E_L(z)^*-R_L(z)^*V} + \le \norm{E_L(z)-E(z)}+\norm{R_L(z)-R(z)}\le2\tau. +\] +Apply this bound to a fixed vector and average its squared norm over $z$ +to obtain \eqref{fields:intertwine}. +\end{proof} + +\subsection{Few possible input indices at each Fourier frequency} + +The norm bounds supply nearly isometric maps. To implement them, we also +need a restriction on their matrix entries. Use the Fourier basis +\[ + \ket{\widehat w}=\frac1{\sqrt{|\mathcal G|}} + \sum_{z\in\mathcal G}\chi_w(z)\ket z, + \qquad w\in\mathcal G, +\] +for the first output register of either field map. A row of the resulting +matrix is indexed by a frequency $w$ and a port $a\in\{1,\ldots,k\}$; +its columns are the $d$ input coordinates. + +\begin{proposition}\label{fields:sparsity} +In these Fourier coordinates, every row of $\mathcal E_L$ and of +$\mathcal R_L$ has at most $L$ nonzero entries. +\end{proposition} + +\begin{proof} +First, the sum of the vectors $v_i$ over any multiset of at most $L$ +internal indices determines that multiset uniquely. Indeed, every coordinate +of such a sum, regarded as an ordinary nonnegative integer, is at most +$L2^{nL}<2^b$, so reduction modulo $2^b$ loses no information. Its first +coordinate records the length $m$, and the next $m$ coordinates record the +power sums $p_j=\sum_{t=1}^m i_t^j$. The elementary symmetric polynomials +are recovered successively from Newton's identities +\[ + e_0=1,\qquad + j e_j=\sum_{t=1}^j(-1)^{t-1}e_{j-t}p_t\quad(1\le j\le m). +\] +These identities follow by taking the logarithmic derivative of +$\prod_{t=1}^m(1+i_t x)=\sum_{j=0}^m e_jx^j$ and comparing coefficients. +Consequently the monic polynomial +$T^m-e_1T^{m-1}+\cdots+(-1)^me_m$, and hence its multiset of roots, +is determined. + +An input port contributes only to frequency zero and the same output port. +For an internal input $i$, the lower columns of $E_L^*$ are sums of +\[ + C^*Z^*,\quad C^*Z^*D^*Z^*,\quad\ldots, +\] +whereas those of $R_L^*$ are sums of +\[ + BZ,\quad BZDZ,\quad BZDZDZ,\quad\ldots. +\] +In both cases the rightmost diagonal factor acts on input $i$. Thus every +monomial contains the phase belonging to $i$, and its frequency is the +signed sum of the $v_j$ over a multiset of between $1$ and $L$ internal +indices containing $i$. The sign is negative for $\mathcal E_L$ and +positive for $\mathcal R_L$. For example, the second displayed term on +the $R$ side has entry +\[ + (BZDZ)_{ai}=\sum_j B_{aj}D_{ji}\, + \chi_{v_j+v_i}(z). +\] + +Fix a nonzero frequency and an output port. For $\mathcal E_L$, negate +the frequency first to recover the corresponding positive tuple sum. +By the uniqueness just proved, +all its contributing monomials have the same multiset of internal indices. +Only indices belonging to that multiset can therefore occur as nonzero +input columns. There are at most $L$ of them. The ordering of indices in +a monomial can change its coefficient but cannot enlarge this column set. +No internal monomial has frequency zero, since its first coordinate has +absolute value between $1$ and $L$ modulo $2^b$. At frequency zero the +single port column gives the required bound as well. +\end{proof} + +The role of the power-sum phases is now explicit: a frequency remembers +an unordered list of internal visits, so it permits only a short list of +possible input indices. The next section uses exactly this list to erase +the input register coherently. diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/05-encoders.tex b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/05-encoders.tex new file mode 100644 index 0000000000000000000000000000000000000000..7f6545f5f304ea7fc220592c6511b7abce3b9d87 --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/05-encoders.tex @@ -0,0 +1,186 @@ +\section{Implementing the field encodings}\label{sec:encoders} + +A matrix with few nonzero entries in each row admits a useful slot +construction. First prepare each column while retaining its input index. +Then, conditional on the output row, erase that index with an amplitude +determined only by the maximum row size. Two reflections amplify the +result. When the matrix is nearly isometric, all its singular directions +require almost the same amplification angle. +The reflection argument is amplitude amplification +\cite[Section~2]{BHMT2002}. Preparing states to realize a matrix as a block +of a larger unitary is also a standard block-encoding method +\cite[Lemmas~47--48]{GSLW2019}. Here arbitrary table-controlled column +preparation permits us to use sparsity only in the rows. + +\subsection{A construction for a nearly isometric matrix} + +Let $\mathcal U$ be a set of rows encoded in a $w$-qubit register, and +let $T:\C^d\to\C^{\mathcal U}$ have entries $a_{ui}$, where $d\le2^n$. +Fix an integer $L\ge1$. Suppose that every row has at most $L$ nonzero +entries and +\begin{equation}\label{encode:assumptions} + (1-\varepsilon)I_d\le T^*T\le I_d, + \qquad L\ge1,\quad 0\le\varepsilon<1. +\end{equation} +All these data may depend on a label register, with the same dimensions +and bounds for every label. The construction below treats labels as controls. + +Extend the $n$-qubit input index by +$h=\lceil\log_2(L+1)\rceil$ high bits, and add the row register and two +one-bit flags. Let $\iota$ embed $\C^d$ with all these new qubits zero. +Let $\jmath$ embed $\C^{\mathcal U}$ with the extended index zero and +both flags equal to one. Write $P_0=\iota\iota^*$ and +$Q=\jmath\jmath^*$ for the corresponding projections. Thus $Q$ allows +any valid row, while $P_0$ allows any of the $d$ input indices. +We call the range of $Q$ the \emph{good subspace}. + +\begin{lemma}\label{encode:row-sparse} +Under \eqref{encode:assumptions}, an exact-slot unitary $\mathsf U_T$ +satisfies +\begin{equation}\label{encode:generic-error} + \norm{\mathsf U_T\iota-\jmath T}\le5\varepsilon. +\end{equation} +It uses $O(L(n+w+h))$ slots and $w+h+2$ fresh qubits. Its slot structure +depends only on the register sizes, $d$, and $L$, and it works coherently +for any family of matrices indexed by labels. +\end{lemma} + +\begin{proof} +Set +\begin{equation}\label{encode:angle} + \theta=\frac\pi{4L+2},\qquad t_0=\sin\theta. +\end{equation} +We first construct a unitary $\mathsf B$ with the prescribed block +\begin{equation}\label{encode:block} + \jmath^*\mathsf B\iota=t_0T. +\end{equation} +For each valid input $i$, the column norm is at most one. Conditional on +$i$, use Lemma~\ref{slot:preparation} to prepare the row register and +first flag in a unit vector whose flag-one part is +\[ + \sum_{u\in\mathcal U}a_{ui}\ket u\ket1. +\] +The remaining squared norm is placed in the flag-zero subspace. The +input index is unchanged during this preparation. + +For each row $u$, choose a list $I_u$ of exactly $L$ distinct extended +indices containing every $i$ with $a_{ui}\ne0$. Such lists exist because +the extended index space has dimension +$2^{n+h}\ge2^n(L+1)>L$. Padding indices need not be valid inputs. +Conditional on the row, apply the inverse of a state preparation for +\[ + \frac1{\sqrt L}\sum_{i\in I_u}\ket i +\] +on the extended index register. Its matrix element from any $i\in I_u$ +to zero is $1/\sqrt L$. Finally rotate the second flag so that its +amplitude at one is $t_0\sqrt L$. This is possible because +$t_0\sqrt L\le\theta\sqrt L\le1$. Prescribe arbitrary unitary extensions +for invalid indices and rows. Projecting onto index zero and flags one +now multiplies each relevant $a_{ui}$ by +$(1/\sqrt L)(t_0\sqrt L)=t_0$, proving \eqref{encode:block}. +All three operations together form $\mathsf B$. + +Starting with $\mathsf B$, perform $L$ rounds of +\begin{equation}\label{encode:round} + (2\mathsf B P_0\mathsf B^*-I)(I-2Q). +\end{equation} +These are implementable with $\mathsf B$, its inverse, and diagonal +reflections. We verify their action on every singular direction of $T$. +Let $v_j$ be an orthonormal basis of right singular vectors and write +\[ + Tv_j=\sigma_j e_j,\qquad + \sqrt{1-\varepsilon}\le\sigma_j\le1, +\] +where the $e_j$ are orthonormal. In this calculation identify the input +and output spaces with their embeddings $\iota$ and $\jmath$. +By \eqref{encode:block}, +\begin{equation}\label{encode:plane} + \mathsf Bv_j=\sin\beta_j\,e_j+\cos\beta_j\,e'_j, + \qquad \beta_j=\arcsin(t_0\sigma_j), +\end{equation} +for a unit vector $e'_j$ in the kernel of $Q$. The $e'_j$ are mutually +orthogonal: this follows by taking pairwise inner products in +\eqref{encode:plane}, using unitarity of $\mathsf B$ and orthogonality of +the $e_j$. Each $e'_j$ is also orthogonal to every $e_i$. + +For completeness, both reflections preserve the plane spanned by +$e_j,e'_j$. The first reflection is about the range of $\mathsf B P_0$, +which is spanned by the mutually orthogonal vectors $\mathsf Bv_i$. +Within the $j$th plane it is therefore reflection about +$\sin\beta_j\,e_j+\cos\beta_j\,e'_j$; the second reflection changes the +sign of $e_j$ and fixes $e'_j$. Their product in the order +\eqref{encode:round} sends +\[ + \sin\alpha\,e_j+\cos\alpha\,e'_j + \quad\hbox{to}\quad + \sin(\alpha+2\beta_j)e_j+\cos(\alpha+2\beta_j)e'_j. +\] +After $L$ rounds, the angle is $(2L+1)\beta_j$. As +$(2L+1)\theta=\pi/2$, the distance of this vector from $\sigma_j e_j$ +is at most +\begin{align*} + (2L+1)(\theta-\beta_j)+(1-\sigma_j) + &\le (2L+1)\,2t_0(1-\sigma_j)+(1-\sigma_j)\\ + &\le(\pi+1)(1-\sigma_j) + \le5\varepsilon. +\end{align*} +Here $t_0\le1/2$, so the derivative of $\arcsin$ is at most $2$ on +$[0,t_0]$, and +$1-\sigma_j\le1-\sqrt{1-\varepsilon}\le\varepsilon$. +The error vectors lie in mutually orthogonal planes. The same bound +therefore holds in operator norm on the whole input space, proving +\eqref{encode:generic-error}. + +The first preparation uses $O(w+1)$ slots and the second $O(n+h)$, so +$\mathsf B$ uses $O(n+w+h)$ slots. The complete circuit contains +$2L+1$ occurrences of $\mathsf B$ or $\mathsf B^*$ and $2L$ diagonal +reflections. A diagonal reflection is one slot, as explained in +Section~\ref{sec:slots}. No additional workspace is needed. All prescriptions +can be made separately for each label using the same slot positions and +controls. Since the resulting maps are block diagonal in the labels, +the bounds hold for coherent superpositions of labels as well. +\end{proof} + +\subsection{Applying the construction to the Fourier fields} + +In the preceding section, a row is a frequency and a port. Let +\begin{equation}\label{encode:register-size} + K=b(L+1),\qquad h=\lceil\log_2(L+1)\rceil. +\end{equation} +Encode the frequency in $K$ qubits and the port in $n$ qubits, using +the first $k$ port states. The two field maps have the same row space +and therefore use the same output embedding $\jmath$: extended input +index zero, both flags one, and a valid port. This shared layout will +allow the second encoding to be reversed after the recursive call. + +\begin{proposition}\label{encode:implementation} +Under the hypotheses and definitions of Section~\ref{sec:fields}, there +are exact-slot unitaries $\mathsf U_E,\mathsf U_R$ such that +\begin{equation}\label{encode:field-error} + \norm{\mathsf U_E\iota-\jmath\mathcal E_L}\le5r^{2L}, + \qquad + \norm{\mathsf U_R\iota-\jmath\mathcal R_L}\le5r^{2L}. +\end{equation} +Each uses $O\bigl(L(K+n+h)+K\bigr)$ slots and $K+n+h+2$ fresh qubits, +with slot structure independent of $V$. The construction works coherently +with any additional label register. +\end{proposition} + +\begin{proof} +Write either field map in Fourier coordinates. Proposition~\ref{fields:sparsity} +gives at most $L$ nonzero entries in each row, and +Proposition~\ref{fields:gram} supplies \eqref{encode:assumptions} with +$\varepsilon=r^{2L}$. Apply Lemma~\ref{encode:row-sparse}, taking +$w=K+n$, to obtain the corresponding coefficient map with error at most +$5r^{2L}$. Then apply the positive Fourier transform to each of the +$L+1$ frequency factors, so that $\ket w$ becomes +$|\mathcal G|^{-1/2}\sum_z\chi_w(z)\ket z$. +The same physical register now holds the phase label $z$, rather than the +Fourier frequency $w$. Lemma~\ref{slot:fourier} implements these transforms +using $O(K)$ slots. +They preserve the range of $Q$, because every frequency is allowed there +and the index, flags, and port are unchanged. Thus they convert the +coefficient map into the field map in the same embedding $\jmath$, +while preserving the error norm on the full space. Their addition +gives \eqref{encode:field-error} and the stated resource bounds. +\end{proof} diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/06-recursion.tex b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/06-recursion.tex new file mode 100644 index 0000000000000000000000000000000000000000..65bf0450f460311b13a11a136db6ada4520cb6f0 --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/06-recursion.tex @@ -0,0 +1,163 @@ +\section{Assembling the recursion}\label{sec:recursion} + +The two field encodings now reduce synthesis of $V$ to synthesis of the +smaller controlled unitary $F(z)$. The reduction uses the smaller circuit +exactly once. This fact keeps the total circuit size polynomial even though +the number of recursion levels grows with $n$. + +\begin{lemma}[One synthesis step]\label{rec:step} +Let $V$ be a unitary on $\C^{k+s}$ in the block form of +Proposition~\ref{contract:block}, and let $\norm D\leq r<1$. +Use the fields and encoders of Sections~\ref{sec:fields} and~\ref{sec:encoders} +with truncation length $L$. Suppose a circuit synthesizes the family of +$k$-dimensional unitaries $F(z)$, controlled by $z$, with clean-workspace +operator norm error $\delta'$. Then one use of that circuit, preceded by +$\mathsf U_E$ and followed by $\mathsf U_R^*$, synthesizes $V$ with +clean-workspace error at most +\begin{equation}\label{rec:step-error} + \delta'+\frac{2r^L}{1-r}+10r^{2L}. +\end{equation} +The assertion also holds coherently for a family $V_\lambda$, with the +smaller circuit controlled by both $\lambda$ and $z$. +\end{lemma} + +\begin{proof} +Let $\iota$ embed the current index space into the full register layout by +initializing all working qubits to zero. Include the fresh working qubits +of the smaller circuit in this definition. Let $\jmath$ embed the field space +$\C^{\mathcal G}\otimes\C^k$ into the encoder's good subspace, again with +the smaller circuit's fresh workspace zero. Write $\mathsf C$ for the +smaller synthesis circuit acting on its target, control, and fresh working +registers, and as the identity on the remaining registers. Its approximation +guarantee implies +\begin{equation}\label{rec:restricted-error} + \norm{\mathsf C\jmath-\jmath\widetilde F}\leq\delta'. +\end{equation} +Indeed, the good subspace has a valid port index, and $\widetilde F$ applies +$F(z)$ with the phase register as control. + +Put $\tau=r^L/(1-r)$. Proposition~\ref{encode:implementation} gives +\[ + \norm{\mathsf U_E\iota-\jmath\mathcal E_L}\leq5r^{2L}, + \qquad + \norm{\mathsf U_R\iota-\jmath\mathcal R_L}\leq5r^{2L}. +\] +The field maps are contractions by Proposition~\ref{fields:gram}, and their +intertwining bound is~\eqref{fields:intertwine}. Consequently, +\begin{align*} + &\norm{\mathsf C\mathsf U_E\iota-\mathsf U_R\iota V}\\ + &\quad\leq + \norm{\mathsf C(\mathsf U_E\iota-\jmath\mathcal E_L)} + +\norm{(\mathsf C\jmath-\jmath\widetilde F)\mathcal E_L}\\ + &\qquad\quad + +\norm{\jmath(\widetilde F\mathcal E_L-\mathcal R_LV)} + +\norm{(\jmath\mathcal R_L-\mathsf U_R\iota)V}\\ + &\quad\leq5r^{2L}+\delta'+2\tau+5r^{2L}. +\end{align*} +Multiplication by $\mathsf U_R^*$ proves~\eqref{rec:step-error}. +This calculation invokes the smaller circuit's approximation only on the +ideal field-map output, whose norm is at most one. Any component of the +actual encoded state outside the good subspace is covered by the first +error term and is not enlarged by $\mathsf C$. + +For additional labels $\lambda$, all maps are block diagonal in the label +basis and all bounds are uniform over the labels. Their operator norms on +the full controlled space are the maxima of the block norms. The same +calculation therefore applies to arbitrary coherent superpositions of labels. +\end{proof} + +Figure~\ref{fig:recursion} shows the comparison made in this proof. The +amplification inside $\mathsf U_E$ and $\mathsf U_R$ repeats only their own +state-preparation circuits. It never repeats $\mathsf C$. +\input{figures/recursion} + +We can now choose one set of parameters for all recursion levels. Keep +$M,r,J$ from~\eqref{contract:parameters} and~\eqref{contract:J}, and set +\begin{equation}\label{rec:parameters} + L=10000J,\qquad + b=nL+\lceil\log_2(L+1)\rceil,\qquad + K=b(L+1). +\end{equation} +Thus $K$ is the number of qubits in each field's phase register. + +\begin{proposition}[Uniform programmable synthesis]\label{rec:skeleton} +Let $n\geq1$, $1\leq d\leq2^n$, and $t\geq0$. There is a programmable-slot +circuit structure, determined by $n,d,t$, with the following property. +For every family $(U_\lambda)_{\lambda\in\{0,1\}^t}$ of unitaries on +$\C^d$, its slot tables can be chosen so that the circuit implements the +controlled unitary $U_\lambda$, preserves the labels, and has clean-workspace +operator norm error less than $1/100$ in the sense of~\eqref{slot:clean-error}. +The circuit uses +\begin{equation}\label{rec:resources} + t+O((n+1)^{10})\quad\text{qubits}, + \qquad O((n+1)^{13})\quad\text{slots}. +\end{equation} +Its structure, including each slot's ordered control list, is generated +deterministically in time polynomial in $n+t$ from $n,d,t$. +\end{proposition} + +\begin{proof} +We recurse on the valid order $d$, retaining $n$-bit index and port registers +at every level. If $d=1$, each $U_\lambda$ is a scalar phase, implemented +exactly by one slot on an index bit that is zero throughout the valid input +subspace. + +For $d\geq2$, Proposition~\ref{contract:block} gives +$V_\lambda=P_{\mathrm L,\lambda}U_\lambda P_{\mathrm R,\lambda}$ with +internal order $s$ and port order $k=d-s$. Apply the one-step construction +of Lemma~\ref{rec:step}, prescribing the recursive family to be +$F_\lambda(z)$ and adding $z$ to the existing control labels. Supply fresh +working qubits for this recursive circuit. To implement $U_\lambda$, apply +$P_{\mathrm R,\lambda}^*$ before this construction and +$P_{\mathrm L,\lambda}^*$ afterward. These transformations have exact slot +implementations, so they introduce no further error. Their order is correct +because $P_{\mathrm L,\lambda}^*V_\lambda P_{\mathrm R,\lambda}^*=U_\lambda$. + +At each step, Lemma~\ref{rec:step} adds at most +\[ + \frac{2r^L}{1-r}+10r^{2L} + =8r^L+10r^{2L}\leq18r^L +\] +to the error of the smaller circuit. There are fewer than $J$ steps by +Lemma~\ref{contract:depth}. Bernoulli's inequality gives +$(4/3)^L\geq1+L/3$, and hence $r^L\leq3/L$. The total error is therefore +\begin{equation}\label{rec:total-error} + 18Jr^L\leq\frac{54J}{L}=\frac{54}{10000}<\frac1{100}. +\end{equation} +In particular, all working qubits, including those introduced at inner +recursion levels, return to zero in the ideal comparison map. + +We next count the circuit resources. Write $N=n+1$ and +$h=\lceil\log_2(L+1)\rceil$. Our parameters satisfy +\[ + J,L=O(N^3),\qquad b=O(N^4),\qquad K=O(N^7),\qquad h=O(N). +\] +Proposition~\ref{encode:implementation} uses $O(K+n+h)$ fresh qubits for a +pair of encoders and $O(L(K+n+h)+K)$ slots for each encoder. The two encoders +at one level share the same register layout. The pre- and post-conditioning +operations require only $O(n)$ more clean working qubits and slots. +All registers from an outer level remain allocated during an inner call, +so summing over the at most $J$ levels gives +\[ + t+O(J(K+n+h))=t+O(N^{10}) +\] +total qubits. Summing the slots gives +\[ + O\bigl(J[L(K+n+h)+K+n]\bigr)=O(N^{13}). +\] +These sums suffice because there is only one recursive call at a level. +The local amplification repetitions have already been included in the +factor $L$ in the encoder cost. + +Finally, the structure can be generated without computing any of its table +contents. The successive orders $d,q,s,k$ have $O(n)$-bit descriptions and +are computed in polynomial time. Register allocations, state-preparation +bit order, Fourier transforms, reflections, and the fixed number $L$ of +amplification rounds depend only on these sizes. Every operation depending +on $U_\lambda$---the signs, permutations, state-preparation amplitudes, +support lists, and recursively prescribed matrices---changes only a slot +table. At most $t+O(N^{10})$ register positions occur in any ordered control +list, so writing all such lists for $O(N^{13})$ slots still takes polynomial +time in $n+t$. No list of possible label values is generated. This proves +both the uniformity assertion and~\eqref{rec:resources}. +\end{proof} diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/07-compilation.tex b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/07-compilation.tex new file mode 100644 index 0000000000000000000000000000000000000000..869be8c325d6d149bf8825c1191b33732d89c2d2 --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/build/sections/07-compilation.tex @@ -0,0 +1,268 @@ +\section{Compilation into one Boolean oracle}\label{sec:compilation} + +The skeleton of Proposition~\ref{rec:skeleton} specifies only polynomially +many slots, although each slot may have a large table. We now encode finite +gate words for those tables in one Boolean function. Two details require +care: the words must approximate matrices with their actual phases, and the +resulting circuit must be generated without finding those words. The need to +retain scalar phases under coherent control is already present in the general +controlled-gate decompositions of~\cite{Barenco1995}. + +\subsection{Phase-sensitive one-qubit approximation} + +We first prove the approximation fact needed for the compilation. The +argument combines the irrational-rotation proof of universality for this +gate set~\cite{Boykin2000} with the commutator refinement underlying the +Solovay--Kitaev theorem~\cite{DN2006}. The weaker polynomial bound below is +sufficient for our purpose. + +\begin{lemma}\label{compile:words} +There is an absolute positive integer $K_{\mathrm w}$ such that every $S\in SU(2)$ and every +$0<\xi<1$ admit a word in $H,T,T^\dagger$ of length at most $K_{\mathrm w}\xi^{-3}$ +whose matrix has determinant one and differs from $S$ by at most $\xi$ in +operator norm. +\end{lemma} + +\begin{proof} +Write $X_p,Y_p,Z_p$ for the Pauli matrices, with $Y_p=iX_pZ_p$, and put +$\boldsymbol\sigma=(X_p,Y_p,Z_p)$. We first show that the determinant-one +words are dense in $SU(2)$. Up to their scalar factors, +\[ + T=e^{i\pi/8}e^{-i(\pi/8)Z_p},\qquad + HTH=e^{i\pi/8}e^{-i(\pi/8)X_p}. +\] +Thus $W=(HTH)T=e^{i\pi/4}V$, where $V\in SU(2)$ has trace +\[ + \Tr V=2\cos^2(\pi/8)=1+\frac{\sqrt2}{2}. +\] +Recall that an algebraic integer is a root of a monic polynomial with +integer coefficients; equivalently, its monic minimal polynomial over +$\mathbb Q$ has integer coefficients. The displayed trace is not an algebraic +integer, because its monic irreducible polynomial is $x^2-2x+1/2$. +Consequently the eigenvalues of $V$ cannot be roots of unity: roots of unity +are algebraic integers, and sums of algebraic integers are algebraic integers. +Writing its eigenvalues as $e^{\pm i\theta}$, we have +$\theta/\pi\notin\mathbb Q$. The word $W^4=-V^4$ has determinant one, +and its powers are dense in the one-parameter circle about the rotation +axis of $V$. + +For completeness, every matrix in $SU(2)$ has the form +$\cos t\,I+i\sin t\,\mathbf v\cdot\boldsymbol\sigma$, with +$\mathbf v\in\R^3$ a unit vector, except that the axis can be chosen +arbitrarily when $\sin t=0$. Multiplying the two rotations defining $V$ +shows that its axis is not parallel to the $Z_p$ axis. Conjugation by $T$ +rotates this axis through $\pi/4$ about $Z_p$, giving a nonparallel axis. +The closure of the determinant-one words contains both corresponding +circles: conjugation of a determinant-one word by $T$ is again such a word. +Rotating the second axis slightly about the first moves it out of the +plane they span. Conjugation by an element of the first circle therefore +gives a third circle with a linearly independent axis. +Let $\mathbf v_1,\mathbf v_2,\mathbf v_3$ be the three axes. The derivative +at zero of +\[ + (t_1,t_2,t_3)\longmapsto + \prod_{j=1}^3\exp(i t_j\mathbf v_j\cdot\boldsymbol\sigma) +\] +sends $(t_1,t_2,t_3)$ to +$i(\sum_jt_j\mathbf v_j)\cdot\boldsymbol\sigma$, an isomorphism onto the +three-dimensional tangent space of $SU(2)$ at the identity. The inverse +function theorem shows that these products contain a neighborhood of the +identity. The closure of our words is thus an open subgroup. Every other +coset is open as well, so connectedness of $SU(2)$ forces this subgroup to +be all of $SU(2)$. Here connectedness also follows from the preceding +representation of $SU(2)$ as the unit sphere in $\R^4$. + +We next obtain a uniform length bound. Density and compactness provide a +finite $\varepsilon_0$-net of determinant-one words of some fixed maximum +length $\ell_0$, where $\varepsilon_0>0$ will be chosen sufficiently small. +Suppose that words of length at most $\ell$ form an $\varepsilon$-net, and +choose such a word $W_0$ within $\varepsilon$ of a target $S$. The residual +$S W_0^\dagger$ has the form +\[ + S W_0^\dagger=\exp(i\mathbf a\cdot\boldsymbol\sigma), + \qquad |\mathbf a|\le 2\varepsilon, +\] +for sufficiently small $\varepsilon$. The Pauli commutator identity gives +skew-Hermitian matrices +\[ + P=i\mathbf u\cdot\boldsymbol\sigma,\qquad + Q=i\mathbf v\cdot\boldsymbol\sigma, + \qquad [P,Q]=i\mathbf a\cdot\boldsymbol\sigma, +\] +with $\norm{P},\norm{Q}=O(\sqrt\varepsilon)$: take perpendicular vectors +with $\mathbf u\times\mathbf v=-\mathbf a/2$ and equal lengths. +For $A=e^P$ and $B=e^Q$, expansion of the four exponential series through +degree two gives +\[ + ABA^\dagger B^\dagger=I+[P,Q]+O(\varepsilon^{3/2}) + =S W_0^\dagger+O(\varepsilon^{3/2}). +\] +All errors here are in operator norm with absolute constants; the remaining +terms of the exponential series are bounded by a constant times +$(\norm P+\norm Q)^3$. + +Choose net words $\widetilde A,\widetilde B$ within $\varepsilon$ of +$A,B$. The gain in precision survives this substitution. Indeed, +\begin{align*} + \norm{ABA^\dagger B^\dagger- + \widetilde A B\widetilde A^\dagger B^\dagger} + &\le 2\varepsilon\norm{B-I},\\ + \norm{\widetilde A B\widetilde A^\dagger B^\dagger- + \widetilde A\widetilde B\widetilde A^\dagger\widetilde B^\dagger} + &\le 2\varepsilon\norm{\widetilde A-I}. +\end{align*} +The first inequality follows by writing $B=I+(B-I)$ and cancelling the +identity terms; the second is the same calculation with the roles reversed. +Both right sides are $O(\varepsilon^{3/2})$. Therefore the word +\[ + \widetilde A\widetilde B\widetilde A^\dagger + \widetilde B^\dagger W_0 +\] +approximates $S$ to $C\varepsilon^{3/2}$ and has length at most $5\ell$. +Inverse words are available because $H^\dagger=H$ and both $T$ and +$T^\dagger$ belong to the alphabet. + +Choose the fixed initial net so small that +$C\sqrt{\varepsilon_0}\le1/2$. Repeating the refinement $j$ times gives +error at most $2^{-j}\varepsilon_0$ and length at most $5^j\ell_0$. +Taking $j$ just large enough for the former quantity to be at most $\xi$ +gives a length bounded by $K_{\mathrm w}\xi^{-3}$, since $\log_2 5<3$. +\end{proof} + +Only the existence of the fixed integer $K_{\mathrm w}$ will be used. A generator can +hardcode this integer; it does not need the net or a procedure +that selects approximating words. Those selections will be stored in the +oracle. + +\subsection{Lookup, control, and restoration of scratch space} + +\begin{proposition}[Compilation of slots]\label{compile:slots} +Let a skeleton have $S$ slots on $Q$ qubits, and put $\eta=1/h$ for an integer +$h\ge2$. +There is a deterministic procedure, polynomial in $Q,S,1/\eta$, that produces +a circuit over $H,T,T^\dagger,\mathrm{CNOT}$ and one Boolean oracle, with +polynomially many qubits, gates, calls, and address bits. For every assignment +of the slot tables, some contents of that oracle make the circuit differ +from the prescribed skeleton by at most $S\eta$ on inputs with its added +scratch qubits zero, with the comparison circuit keeping those scratch +qubits zero. +\end{proposition} + +\begin{proof} +First consider one table entry $W_y\in U(2)$. Choose a real number $\gamma_y$ +and a matrix $S_y\in SU(2)$ such that +\[ + W_y=e^{i\gamma_y}S_y. +\] +By Lemma~\ref{compile:words}, choose words within $\eta/2$ of $S_y$ and of +\[ + P_y=\diag(e^{i\gamma_y},e^{-i\gamma_y}). +\] +Pad every word with identities to the common length +$\ell=\lceil K_{\mathrm w}(2/\eta)^3\rceil$. Apply the first word to the slot target +and the second to a fresh scratch qubit initially in $\ket0$. Their ideal +actions on this input multiply to $W_y$ on the target and return scratch +to zero. The two approximation errors add to at most $\eta$. Since +different $y$ occupy orthogonal control blocks, this is an operator-norm +bound for the entire controlled slot. In particular, the scalar factor +$e^{i\gamma_y}$ has been retained as a relative phase between control +values. + +Use two bits to encode each symbol in $\{I,H,T,T^\dagger\}$. A single +Boolean function stores all symbol bits at addresses containing +\[ + (\text{slot number},\ y,\ \text{which word},\ + \text{word position},\ \text{symbol bit}). +\] +Pad the field for $y$ to $Q$ bits. This makes every address have a common +length $O(Q+\log(S+1)+\log(\ell+1))$. For each word position, copy the +slot's controls into the address by CNOTs, set the fixed address bits, and +query the two symbol bits into zero qubits. Apply the specified symbol +conditionally, then erase the symbol bits by querying again and reverse +the address preparation. The original slot controls do not change under +any of these target operations, so the erasure is exact even in a +superposition of control values. + +We verify that symbol selection uses only the allowed elementary gates. +Bit flips are available as $HT^4H=X_p$. On bits $a,b,c$, the integer identity +\begin{equation}\label{compile:parity} + 4abc=a+b+c-(a\oplus b)-(a\oplus c)-(b\oplus c) + +(a\oplus b\oplus c) +\end{equation} +implements a doubly controlled $Z_p$ using $T,T^\dagger$: compute each +parity into clean scratch, apply its indicated phase, and uncompute it. +Conjugating the target by $H$ gives Toffoli. Thus equality tests on symbol +bits and all required reversible address operations are exact. +To apply $T$ conditionally, compute the AND of its control and target into +a clean scratch bit, apply $T$ there, and uncompute the AND; the same +construction applies to $T^\dagger$. + +For a controlled $H$, define +\[ + D_0=(T^2H)T(T^2H)^\dagger. +\] +Because $(T^2H)Z_p(T^2H)^\dagger=Y_p$, we have +$D_0=e^{i\pi/8}e^{-i(\pi/8)Y_p}$ and hence +\[ + D_0Z_pD_0^\dagger=(X_p+Z_p)/\sqrt2=H. +\] +Apply $D_0^\dagger$ to the target, then a controlled $Z_p$, then $D_0$. +When the control is zero the two unconditional gates cancel; when it is +one their product is exactly $H$. A controlled $Z_p$ itself is a CNOT +conjugated on its target by $H$. This proves that every selected symbol, +including its phase, can be applied exactly. + +It remains to compose the approximate slots. Let $A_j$ denote the actual +implementation of slot $j$, and let $V_j$ denote the ideal slot acting +trivially on all added scratch. The phase scratch may be reused. Although +its actual state need not be exactly zero after a slot, the identity +\[ + A_S\cdots A_1-V_S\cdots V_1 + =\sum_{j=1}^S A_S\cdots A_{j+1}(A_j-V_j)V_{j-1}\cdots V_1 +\] +uses the error $A_j-V_j$ only after an ideal prefix. Such a prefix keeps +all added scratch at zero, precisely where the one-slot bound applies. +The unitary suffix preserves its norm. The total error is therefore at +most $S\eta$ on the claimed input subspace. + +There are two words per slot, $O(\ell)$ symbol positions per word, and a +polynomial number of elementary operations per lookup. The addresses, +scratch, and total circuit size are thus polynomial in $Q,S,1/\eta$. +Generating the circuit requires only the skeleton, the fixed length bound, +and ordinary integer arithmetic. In particular, it never selects a word +or computes a table entry. Each target merely determines one admissible +Boolean function storing all choices. Adjoint slots can have their own +addresses and approximating words, or can reverse and invert stored +words; either convention uses the same single oracle. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Apply Proposition~\ref{rec:skeleton} with $d=2^n$ and no initial labels. +Its skeleton has polynomially many qubits and slots, and approximates the +target unitary, including restoration of its working qubits, to error +less than $1/100$. If it has $S$ slots, compile it using +\[ + \eta=\frac{1}{100(S+1)}. +\] +Proposition~\ref{compile:slots} adds error less than $1/100$. The entire +circuit is therefore within $1/10$ in operator norm, on zero-workspace +inputs, of applying $U$ and returning every other qubit to zero. The same +polynomial bounds its width, number of gates and calls, oracle address +length, and the running time of its deterministic generator, after +increasing fixed coefficients if necessary. + +Take the original index qubits as the designated outputs. For a pure input +state $\ket\psi$ together with a reference register, let $a$ and $b$ be the +actual and ideal final unit vectors before discarding workspace. The +operator-norm estimate gives $\norm{a-b}\le1/10$. Since the trace norm of +a rank-one matrix $uv^\dagger$ is $\norm u\norm v$, +\[ + \norm{aa^\dagger-bb^\dagger}_1 + \le\norm{(a-b)a^\dagger}_1+\norm{b(a-b)^\dagger}_1 + \le2\norm{a-b}\le\frac15. +\] +Partial trace cannot increase the trace norm of a Hermitian matrix, and +convexity gives the same bound for mixed input states. In particular the +supremum in Theorem~\ref{thm:main}, with its $n$-qubit reference, is at most +$1/5<1/2$. +\end{proof} diff --git a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6e9b4e6a0cdc9d42cd405c55a15430e994ddf9a9 --- /dev/null +++ b/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:198849e472386b7a14e8cd7ff52e8ec305c1b6ebc596ec846bcd947bb04b101c +size 450775 diff --git a/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/README.md b/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..0c8a143e1d5d3ff8ea74d6949c4442afe4d41098 --- /dev/null +++ b/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/README.md @@ -0,0 +1,20 @@ +# [Polynomial removal fails for ordered binary matrices](paper.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +To cite this manuscript: + +```bibtex +@misc{OAI:Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026, + author = {{OpenAI}}, + title = {{Polynomial removal fails for ordered binary matrices}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf}{OAI:Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/figures/tree-orders.tex b/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/figures/tree-orders.tex new file mode 100644 index 0000000000000000000000000000000000000000..55335f3cb309f4ffda02938387063f35ed7fd2f1 --- /dev/null +++ b/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/figures/tree-orders.tex @@ -0,0 +1,30 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[x=1cm,y=1cm, + parent/.style={draw,rounded corners=2pt,minimum width=1.65cm,minimum height=.65cm,fill=black!4}, + child/.style={draw=blue!45!black,rounded corners=2pt,minimum width=3.25cm,minimum height=.65cm,fill=blue!4,font=\small}, + >={Stealth[length=4pt]}] +\node[anchor=east,font=\small] at (-.1,1.2) {Rows}; +\node[parent] (rm) at (1,1.2) {$R_i^-(p)$}; +\node[child] (r0) at (4.1,1.2) {subtree of $2p$}; +\node[child] (r1) at (8.0,1.2) {subtree of $2p+1$}; +\node[parent] (rp) at (11.1,1.2) {$R_i^+(p)$}; +\draw[->] (rm.east)--(r0.west); +\draw[->] (r0.east)--(r1.west); +\draw[->] (r1.east)--(rp.west); +\node[anchor=east,font=\small] at (-.1,0) {Columns}; +\node[parent] (cp) at (1,0) {$C_i^+(p)$}; +\node[child] (c0) at (4.1,0) {subtree of $2p$}; +\node[child] (c1) at (8.0,0) {subtree of $2p+1$}; +\node[parent] (cm) at (11.1,0) {$C_i^-(p)$}; +\draw[->] (cp.east)--(c0.west); +\draw[->] (c0.east)--(c1.west); +\draw[->] (c1.east)--(cm.west); +\end{tikzpicture} +\caption{The two orders at an internal node $(i,p)$, where $i0$ such that, for every +integer $n\ge1$, every $0<\epsilon<1$, and every binary $n\times n$ matrix $A$, +\[ + N_H(A)\ge c_H\epsilon^{C_H}n^{2k} + \quad\text{whenever}\quad \operatorname{dist}_H(A)\ge\epsilon. +\] +We disprove this conjecture with an explicit pattern of size $66$. + +\begin{theorem}\label{thm:main} +There is a fixed binary $66\times66$ matrix $H$ with the following +property. For every integer $h\ge1$, put +\[ + n_h=(386h+2)2^h,\qquad \epsilon_h=(386h+2)^{-2}. +\] +There is a binary $n_h\times n_h$ matrix $A_h$ such that +\[ + \operatorname{dist}_H(A_h)\ge\epsilon_h, + \qquad + \frac{N_H(A_h)}{n_h^{132}}\le\epsilon_h2^{-h}. +\] +Consequently, no positive constants $c,C$ give a polynomial removal +bound for this fixed $H$. +\end{theorem} + +The examples also give a lower bound for canonical sampling +(Corollary~\ref{cor:canonical-testing}). A tester that independently +chooses uniform $q$-element row and column sets and rejects only when +the sampled ordered submatrix contains $H$ needs +$q\ge\exp(\Omega(\epsilon_h^{-1/2}))$ for any fixed positive rejection +probability along this sequence. Here $q$ is the number of sampled +positions on each axis; the submatrix has $q^2$ entries. + +\paragraph{History and significance.} +Alon, Fischer and Newman raised the ordered-matrix question after proving +polynomial-query testing for fixed finite forbidden families when row +and column order is ignored~\cite[Section~7]{AlonFischerNewman2007}. +Alon and Ben-Eliezer formulated the polynomial removal question for +finite families of binary matrices~\cite[Problem~1.4]{AlonBenEliezer2020}. +The singleton version appears explicitly as +Conjecture~4.5 in the first arXiv version of the survey by Gishboliner and +Shapira~\cite{GishbolinerShapira2025}. +Theorem~\ref{thm:main} resolves that conjecture negatively and answers +the finite-family question negatively already for one square pattern. + +Qualitative ordered matrix removal was established by Alon, +Ben-Eliezer and Fischer~\cite[Theorem~1.8 in the full +version]{AlonBenEliezerFischer2017}: a positive copy-density bound exists +for each fixed positive distance. Our examples concern its quantitative +dependence. Polynomial bounds also hold for finite families closed under +row permutations, or under column permutations by +transposition~\cite[Theorem~1.5]{AlonBenEliezer2020}. +The polynomial induced removal classification for ordered graphs of Gishboliner +and Tomon~\cite[Theorem~1.1]{GishbolinerTomon2022} concerns a +different model, with a single vertex order and symmetric adjacency. + +Nonpolynomial lower bounds were already known for ternary hosts: +Fischer and Rozenberg obtained them for a fixed two-symbol $2\times2$ +pattern, with row and column order ignored~\cite{FischerRozenberg2007}. +Alon and Ben-Eliezer give a related +ternary-matrix construction based on progression-free +sets~\cite[Section~5.1]{AlonBenEliezer2020}. These examples use a third +symbol in the host matrix; our construction uses only $0$ and $1$. + +A particularly relevant positive result is the polynomial lower bound +for matrices containing many entry-disjoint copies of a fixed +pattern~\cite[Theorem~1.2]{AlonBenEliezer2020}. It yields a small set of +cells meeting all existing copies when the copy density is small. +Changing those cells can create new copies. Our construction makes +this distinction explicit: all original copies meet a set of $2^h$ +cells, while any binary repair changes at least $4^h$ cells. + +\paragraph{Proof strategy.} +The last two rows and columns of $H$ meet in the pattern +\[ + P=\begin{pmatrix}1&0\\1&1\end{pmatrix}. +\] +Its first $64$ rows and columns specify which pairs of rows and columns +can play these two roles. The host matrix contains many copies of this +$64\times64$ anchor, each imposing a different instance of the same +$2\times2$ restriction. Section~\ref{sec:construction} gives the pattern +and the host entries explicitly. + +The host has rows and columns indexed by a binary tree, with a plus +block and a minus block at each internal node on each axis. At each +leaf, the two blocks on each axis coincide. One root entry is $1$ and +the other is $0$. Avoiding $P$ in the designated bodies propagates +these values down a common path, first to the next row level and then +to the next column level. At the leaves the two signs share their +positions, so they cannot retain opposite values. + +We first prove the distance bound in Section~\ref{sec:distance}. +Fix an arbitrary edited matrix and sample a tree path together with +anchor positions and auxiliary ``dummy'' positions, whose entries +against the tree positions are all $1$ in the original host. Each axis is +partitioned into +classes of equal size $m=2^h$. For every cell needed to keep the sampled +anchors, dummy incidences and root values unchanged, its row and column +are selected with probability $m^{-2}$. Summing over the actual edited +cells shows that fewer than $m^2$ changes leave a suitable selection. +All other entries may have changed; avoidance of $H$ itself forces the +contradictory propagation in the edited matrix. This cellwise sampling +argument avoids a loss depending on the number of classes and gives +the inverse-square distance in Theorem~\ref{thm:main}. + +The copy bound concerns the unedited host and requires locating every +copy, beyond the designated anchors used in the distance proof. +In Section~\ref{sec:copies}, dense rows and columns, together with a +five-column block whose rows realize all binary words, force every +anchor copy to occupy one of the prescribed anchor blocks. The remaining rows and columns must then have +that block's designated roles. The order and threshold rules then force +every copy to use a diagonal leaf cell, giving the small copy count. +Section~\ref{sec:conclusion} combines the two estimates and derives the +sampling consequence. diff --git a/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/02-construction.tex b/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/02-construction.tex new file mode 100644 index 0000000000000000000000000000000000000000..71519a32e5ff4d7b368dfd93ac5cac3152a52f28 --- /dev/null +++ b/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/02-construction.tex @@ -0,0 +1,202 @@ +\section{A fixed pattern and its tree matrices} +\label{sec:construction} + +We specify a binary pattern of size $66$ and an ordered binary matrix +$A_h$ for each integer $h\ge1$. The pattern consists of a $64\times64$ +block and two additional rows and columns. The matrices $A_h$ place +copies of that block before positions indexed by a binary tree. + +\subsection{The pattern} + +Set $s=64$. For $0\le a<32$ and $1\le b\le5$, let +\[ + \eta_b(a)=\left\lfloor\frac{a}{2^{5-b}}\right\rfloor\bmod2, +\] +so $(\eta_1(a),\ldots,\eta_5(a))$ lists the five-bit words in +lexicographic order. Define $S\in\{0,1\}^{[s]\times[s]}$ by +\[ + S(u,v)= + \begin{cases} + \eta_{v-59}(u-33),&33\le u\le64,\quad60\le v\le64,\\ + \mathbf1[u\ne v],&\text{otherwise}. + \end{cases} +\] +Let $e_1,e_2$ be the first two standard basis vectors of $\mathbb R^s$. +The fixed pattern and its lower-right block are +\begin{equation} + H= + \begin{pmatrix} + S&e_1&e_2\\ + e_1^{\mathsf T}&1&0\\ + e_2^{\mathsf T}&1&1 + \end{pmatrix}, + \qquad + P=\begin{pmatrix}1&0\\1&1\end{pmatrix}, + \qquad k=s+2=66. + \label{eq:pattern-definition} +\end{equation} +We call $S$ the \emph{anchor} and $P$ the \emph{body} of $H$. + +\subsection{The ordered positions} + +Fix $h\ge1$ and put $m=2^h$. At depth $i$ the full binary tree has +nodes indexed by $p\in\{0,\ldots,2^i-1\}$, with children $2p,2p+1$. +For $0\le i0$. As $h$ tends to infinity, +\[ + \frac{N_H(A_h)}{\epsilon_h^C n_h^{132}} + \leq (386h+2)^{2C-2}2^{-h} + \longrightarrow 0. +\] +Choose an integer $h$ for which the right-hand side is less than $c$. +Then $0<\epsilon_h<1$, the matrix $A_h$ is $\epsilon_h$-far from being +$H$-free, and +$N_H(A_h)0. +\] +\end{proof} + +\paragraph{Reference weights and uniform comparisons.} +For weights on \(s\ge0\), zeros of the degree-\(d\) orthogonal polynomial are eigenvalues of compression of multiplication by \(s\) to degrees below \(d\): the eigen-equation states orthogonality of \((s-\rho)P\). An increasing likelihood ratio shifts each zero up by min-max, since every Rayleigh quotient increases by monotone tilting. We also compare diagonal kernels by their extremal property, +\[ + K_d^w(u,u)=\sup_{\substack{\deg Q0\). Orthonormal polynomials of degree \(j\) in \(s\) have values, up to sign, given by coefficient \(2j\) in +\[ + (1-v^2)^{-1/2}\big((1+v)/(1-v)\big)^{iy}. +\] +Indeed integrating a product of two generating functions gives \(1/(1+uv)\) by \(\int e^{iay}dy/\cosh(\pi y)=1/\cosh(a/2)\) (shift contour by \(i\)); use just even indices. At \(s=0\) the magnitudes are of order \((j+1)^{-1/2}\); at \(s=-1\) and \(r=\alpha/\pi\in(1/2,1)\), of order \((j+1)^{r-1/2}\). Indeed the coefficients needed there are those of \((1-v)^{-1/2-r}(1+v)^{r-1/2}\). The first factor has increasing coefficients of order \((k+1)^{r-1/2}\) with ratio tending to 1 (binomial series); the second has summable coefficients with nonzero sum, giving the claim by convolution. + +The all-poles lower bound for \(w\) comes from density \(w_\infty(p)=x(2\cosh2x-1)/\sinh(8x)\) before pushing forward. Its ratio to the reference density at \(\alpha=6\lambda\) is increasing on \(p\ge0\): up to a constant this ratio is +\(x(2-1/\cosh2x)/(\tanh2x+\tanh6x)\). Removing poles only increases the upward tilt. Conversely \(w\) relative to reference \(\alpha=2\lambda\) has decreasing ratio since \(\mu\) times \(\cosh2x\) has density proportional to \((2\cosh2x-1)/\cosh4x\), decreasing. It follows that +\[ + K_N^w(0,-1)\le C\sqrt{\log(2N)}\ N^{3/4}. +\] +(Indeed \(w\) dominates a constant times the \(6\lambda\) reference uniformly, so the two diagonal kernel values by the supremum characterization are at most constant times the reference sums of squared coefficients; now use Cauchy--Schwarz.) +Moreover, for every orthogonal polynomial \(P_j\) of degree \(j\) for any of the conditional weights \(w\), +\[ + 1\le P_j(-1)/P_j(0)\le C(j+1)^{3/4} +\] +by positivity and comparison of zeros with \(\alpha=6\lambda\). + +The upper kernel bound holds for every pole set. The lower bound will use pole sets that contain all low poles; the next lemma shows that these account for asymptotically all of the probability in \eqref{eq:pv:pole-representation}. + +\begin{lemma}[Low-pole saturation]\label{lem:pv:pole-saturation} +Fix \(0<\delta<1/2\) and put \(J=N^{1-\delta}\). Under the law of Lemma~\ref{lem:pv:pole-representation}, let \(\mathcal E_J\) be the event that \(A\) and \(B\) each contain every pole in their respective sets with index \(l\le J\). Then +\[ + \mathbb P(\mathcal E_J)=1-o(1). +\] +For either subset, the probability of missing such a pole is \(o(1)\) uniformly conditional on the other subset. +\end{lemma} + +\begin{proof} +Fix the other subset. If an index \(j\le J\) is missing, replace the smallest occupied \(k\ge2J\) in the chosen subset by \(j\). Such a \(k\) exists for all large \(N\), since the subset has \(N/2\) elements and only \(O(J)=o(N)\) indices lie below \(2J\). For fixed \(k\) the replacement is injective. The ratio of old to new weight is +\[ + (k/j)^2\prod_{l\ne k}\left(\frac{k^2-l^2}{j^2-l^2}\right)^2 + \ \mathbb E\prod_{i=1}^N\frac{s_i+(8j)^2}{s_i+(8k)^2} +\] +(the product is over other indices of that subset, and the expectation is under the new polynomial ensemble proportional to \(\Delta^2\prod_i dw_{\rm new}(s_i)\)). The first factors cost at most \(\exp(CJ)(k/J)^{CJ}\). Larger indices than \(k\) cost nothing; by minimality of \(k\), smaller occupied indices lie below \(2J\), where +\[ + \sum_l\big[\log^+(J/l)+\log^+(J/|j-l|)\big]\le CJ +\] +by unit spacing, excluding \(j\). + +The expectation is the ratio of degree-\(N\) monic orthogonal polynomial values at \(-(8j)^2,-(8k)^2\) for weight \(w_{\rm new}/(s+(8k)^2)\). Indeed, averaging \(\prod_{i=1}^N(u-s_i)\) against a normalized \(\Delta^2\)-ensemble gives its monic orthogonal polynomial. Its integral against \(u^r\), \(r0\) small makes \(w\) bounded above by a constant times \(\nu_{\alpha'}\). This pointwise comparison need not hold on the whole line. Put \(d=\lfloor N^{1-2\delta}\rfloor\). What the kernel estimate requires, and what does hold uniformly for all saturated pole sets, is the bound +\begin{equation}\label{eq:pv:saturated-quadratic-form} + \int |Q(s)|^2\,dw(s) + \le C\int |Q(s)|^2\,d\nu_{\alpha'}(s) + \qquad(\deg Q\epsilon J\) is then at most +\[ + C\exp(C_r d-c\epsilon J)\,\|Q\|_{L^2(\nu_{\alpha'})}^2. +\] +Since \(d=o(J)\), this proves \eqref{eq:pv:saturated-quadratic-form} together with the comparison on the central interval. + +The zeros of every \(P_j\) lie in \((0,\infty)\), so \(P_j(0)P_j(-1)>0\). The bound \(P_j(-1)/P_j(0)\le C(j+1)^{3/4}\) and the extremal kernel comparison from \eqref{eq:pv:saturated-quadratic-form} therefore give, on \(\mathcal E_J\), +\[ + K_N^w(0,-1)\ge C^{-1}d^{-3/4} K_d^w(-1,-1) + \ge c\,d^{2\alpha'/\pi-3/4}. +\] +Here the second inequality uses the reference sum of squared values at \(-1\). The event has probability \(1-o(1)\) by Lemma~\ref{lem:pv:pole-saturation}, and the denominator in \eqref{eq:pv:pole-representation} is bounded above and below. Together with the uniform upper bound, letting \(\delta\downarrow0\) and \(\alpha'\uparrow6\lambda\) proves \eqref{eq:pv:boundary-kernel-power}. + +\subsection{Comparing the strip tails}\label{c01-c1-comparing-the-strip-tails} + +The total half-plane arch weight to the right is now exactly \(1/(2\cos(3\lambda))\): this was the upper bound by flux, and \(G_N\) tends to it. Equations~\eqref{eq:pv:excluded-arch-mass} and~\eqref{eq:pv:boundary-kernel-power} give +\[ + d_N:=\frac1{2\cos(3\lambda)}-G_N=N^{-1/4+o(1)} +\] +for even \(N\). This is the mass excluded by displacement or failure of simple cylinder projection. We first transfer its exponent to a spatial maximum, and then use a second tilted direction to force the same exponent for height. + +\paragraph{Two tilted flux identities.} + +Use these positive-endpoint arches above a horizontal bottom from coordinate 0, and write their endpoint as \(D>0\), height maximum as \(Y\), and +\[ + M_+=\max(A x+B y),\quad M_-=\max(A x-B y),\qquad A=\sqrt3/2,\ B=1/2 +\] +in planar coordinates. Write \(P\) for their unnormalized weighted law. At large positive levels \(r\) on the respective tiling grids, +\[ + P(M_\pm-A D\ge r)=q_\pm P(M_\pm\ge r),\quad + q_\pm=\frac{\sin(\sigma(\pi-\phi_\pm))}{\sin(\sigma(\pi+\phi_\pm))},\qquad + (\phi_+,\phi_-)=(\pi/3,2\pi/3). +\] +Indeed, in a large convex exhaustion add the cut \(Ax\pm By\le r\) to the domain. Subtracting the two identities of Lemma~\ref{lem:pv:flux} puts the phased sum over old exits crossing that cut on the ray \(e^{-i\sigma\phi_\pm}\) by convexity. Tiling cuts of differing directions never share ports. Old non-bottom exits vanish in weight in the exhaustion by the now saturated arch mass: bottom real parts increase to 1 in total, and all real parts control absolute weights. Bottom exits crossing the cut have phases \(e^{-i\sigma\pi}\) to the right and \(e^{i\sigma\pi}\) to the left, with the latter events corresponding by path reversal and translation to the shifted maxima above. Taking the imaginary part after rotating the ray to angle zero gives the identity. Note \(0T$. + +We bound this remainder absolutely, without an estimate on $m$. +A fixed-source shape $\beta_1$ has at most $|\beta_1|$ translations +placing a visited vertex at $v$, and +$|\beta_1|\le C(h+1)(X_1+1)$, where $X_i=X(\beta_i)$. +For each such translate, a second shape has at most +$C(X_1+X_2+1)$ translations hitting it. Hence, uniformly in $v$, +\begin{align*} + R_h&\le C(h+1) + \iint_{\max(X_1,X_2)>T}(1+X_1+X_2)^2 + \,d\mu_h(\beta_1)\,d\mu_h(\beta_2)\\ + &\le C(h+1)(T+h+1)^2e^{-cT/(h+1)}=O(h^{-100}). +\end{align*} +The second line follows by integrating the exponential tail in +Lemma~\ref{lem:pv:lateral-tail} for the first two moments of $X$, +using $b_h\le1$ and splitting according to which $X_i>T$. +Since $b_h=h^{-1/4+o(1)}$, we obtain +\begin{equation}\label{eq:pv:bridge-pair-upper} + J_{CC}\le h^{3/4+o(1)}m+O(h^{-100}). +\end{equation} + +\subsection{Translation hits on a deep arch} + +The upper pair bound pays one bridge crossing mass for each allowed +horizontal translation. We need a matching lower bound for bridges +hitting each arch through the marked vertex. Here the error must be +uniform in the arch, whose length and horizontal span are unrestricted. + +\begin{lemma}[Translated bridge hits]\label{lem:pv:translation-hits} +Let $\alpha$ be any same-side arch meeting the central third of the +strip. Let $\mathcal H_h(\alpha)$ be the total weight of strip bridges, +summed over all bottom sources, that intersect $\alpha$. For every +fixed $\epsilon\in(0,1)$, +\[ + \mathcal H_h(\alpha)\ge h^{3/4-\epsilon-o(1)}, +\] +uniformly over all such arches. In particular +$\mathcal H_h(\alpha)\ge h^{3/4-o(1)}$ uniformly. +\end{lemma} + +\begin{proof} +By reflection in the strip midline, it suffices to treat a bottom +arch. Translate its start to the fixed bottom source, put +$W=h^{1-\epsilon}$, and consider its prefix up to first crossing +layer $r=\lfloor h/4\rfloor$. Such a crossing exists because the arch +meets the central third. + +\emph{Equal-height contact intervals.} +We first compare two prefixes $\alpha_0,\beta_0$ from the bottom to +their first crossings of the same cut. Parametrize them on $[0,1]$; +their heights lie strictly between the two cut heights internally. +There is a compact connected set of equal-height parameter pairs +containing $(0,0)$ and $(1,1)$, whose projections onto both parameter +intervals are all of $[0,1]$. Consequently its horizontal-difference +image is an interval of translations giving contact. It contains the +start and final offsets, and also an offset matching any prescribed +point on either prefix with a point on the other. + +This is the piecewise-linear mountain-climbing construction +\cite[Theorem~5.5]{Pak2010}; we give the contact argument needed here. +On the parameter square, triangulate so that the height difference is +piecewise linear. For sufficiently small positive regular values its +level set has exactly two boundary endpoints, near the diagonal +corners, joined by an arc. Indeed the two prefix heights are strictly +monotone near their extremal endpoints and stay away from these +extremal heights on every compact interior parameter interval. +A subsequential Hausdorff limit of the compact arcs is connected, +lies in the zero set, and contains both corners. Such a subsequence +can also be obtained by diagonal selection of the dyadic boxes met by +the arcs. Each coordinate projection is connected and contains $0$ +and $1$, hence equals $[0,1]$. Taking horizontal differences proves +the interval assertion. At integer translations, geometric contact +of the lattice walks gives a shared vertex. A contact at the +intermediate cut is also a shared vertex of the full walks, since +that port lies inside their lattice edges. + +\emph{Two direct alternatives.} +Suppose first that at some intermediate cut the arch prefix has two +crossings at horizontal distance at least $W$. Every bridge shape +then has at least $W-O(1)$ hitting translates. To see this, use the +bridge crosscut's left/right side indicator at the two points, with +any fixed convention for points on it. Their separation is an integer. +Summing the signed indicator difference over integer translates +telescopes to that integer in absolute value. Any discrepancy forces +the intervening arch segment to meet the bridge. Summing bridge +shapes therefore gives hitting mass at least $(W-O(1))b_h$. + +Suppose next that the arch's displacement $a$ at first crossing of +some layer $1,\ldots,r$ satisfies $|a|\ge W$. A bridge and its +reflection about its start have first-crossing displacements $d$ and +$-d$ there, and +\[ + \max\{|a-d|,|a+d|\}\ge |a|\ge W. +\] +For at least one member of this reflection pair, the contact interval +between the start and final offsets yields $W-O(1)$ hitting +translates. Reflection preserves weights, so again the hitting mass +is at least a constant times $Wb_h$. + +In the remaining case every first-crossing displacement is less than +$W$ and every same-cut crossing spread is less than $W$. Thus all +cut crossings of the arch prefix lie within $2W$ of its start. +The full prefix stays within $2W+O(1)$: inside a cut-to-cut layer a +walk between ports follows the internal zigzag monotonically. + +\emph{Wide bridge prefixes against a confined arch.} +We will show that fixed-source bridges of weight at least +$h^{-1/4-o(1)}$ make lateral distance greater than $4W$ before their +first crossing of layer $r$. We first bound the exceptional confined +pieces that enter the marking argument. + +For $h/10\le j\le h/6$, let $e_j$ be the weight of fixed-source +$j$-bridges confined to lateral distance $4W$ from their start. Then +\begin{equation}\label{eq:pv:tube-exception} + e_j\le C\exp(-ch/W). +\end{equation} +Surround the width-$8W$ tube by a tube of width $O(W)$ whose side +walls alternate the two slanted upward tiling steps between successive +horizontal layers, outside the desired confinement with a safety +margin. Join these walls horizontally on top and bottom. Starting +east along the positive bottom boundary, its unwrapped tangent +angles stay in $[0,2\pi]$: on the right wall they are +$\pi/3,2\pi/3$, and on the left $4\pi/3,5\pi/3$. For a bottom +source, closing an exit by the boundary gives turn $B-\pi$, where +$B\in[0,2\pi]$ is the cumulative tangent turn from source to exit. +Thus all original exits have turns in $[-\pi,\pi]$ and positive +comparable real weights. This supplies the needed positivity even +though the tube is not convex. + +Now cut by lines with the two slanted upward normals, of angles +$\pm\pi/3$ from vertical. Boundary-source leaf cancellation still +holds after either cut, since a polygon cannot surround the bottom +source. The genuinely new exits lie on the cut and stay in the convex +wedge between its half-plane and the half-plane above the bottom, +so their turns are fixed. The lost exit sums consequently lie on two +separated rays. Place the intercepts at the source's lateral +coordinate near $s+C'(W+1)$, stagger them by $O(W+1)$, and round to +tiling lines. Since all lateral coordinates are restricted to +$O(W+1)$, the two lost events are nested between reaching heights +$s+C(W+1)$ and $s$. The separated-ray argument of +Lemma~\ref{lem:pv:lateral-tail}, with the positive original exit +weights just established, gives fixed-factor decay per such height +increment, starting at $O(W+1)$. The confined $j$-bridges are among +the top exits of this tube. This proves +\eqref{eq:pv:tube-exception}. + +Under the fixed-source bridge measure $\mu_h$ defined above, let +$N(\gamma)$ count layers +$j\in[\lceil h/10\rceil,\lfloor h/6\rfloor]$ that it crosses +exactly once and whose initial $j$-bridge reaches lateral distance +greater than $4W$. At a single-crossing cut, the two bridge pieces +occupy disjoint open slabs. Their weights multiply exactly, since +ports carry no weight. Proposition~\ref{prop:pv:boundary} and +\eqref{eq:pv:tube-exception} give +\begin{align*} + \int N\,d\mu_h + &=\sum_j b_{h-j}(b_j-e_j)=h^{1/2+o(1)},\\ + \int N^2\,d\mu_h + &\le2\sum_{j_1\le j_2} + b_{j_1}b_{j_2-j_1}b_{h-j_2} + \le h^{5/4+o(1)}. +\end{align*} +For the second line, dropping the lateral excursion restrictions +leaves three bridge pieces. Use $b_0=1$ on the diagonal and +$\sum_{d\le h}b_d=h^{3/4+o(1)}$; the two outside factors each have +order $h^{-1/4+o(1)}$, uniformly over the chosen layers. Hence +unnormalized Cauchy--Schwarz yields +\[ + \mu_h(N>0)\ge + \frac{\big(\int N\,d\mu_h\big)^2}{\int N^2\,d\mu_h} + \ge h^{-1/4-o(1)}. +\] +Every bridge counted here has the required lateral excursion before +first crossing $r$. Apply the contact-interval construction to the +two prefixes ending at $r$, matching that prescribed excursion point +of the bridge to the confined arch prefix. The resulting offset +differs from the initial offset by at least $2W-O(1)$, and hence gives +at least order $W$ integer hitting translates. Their total weight is +at least $W h^{-1/4-o(1)}=h^{3/4-\epsilon-o(1)}$. + +All three alternatives give this bound with constants and errors +independent of the arch. For each fixed $\epsilon$, first let +$h\to\infty$; only then let $\epsilon\downarrow0$. This proves the +uniform exponent assertion. +\end{proof} + +\subsection{The bridge first moment} + +Lemma~\ref{lem:pv:translation-hits} gives +$J\ge h^{3/4-o(1)}u$. Combining this with +\eqref{eq:pv:pair-target} and \eqref{eq:pv:bridge-pair-upper} yields +\[ + u\le h^{o(1)}m+O\big(h^{-100-3/4+o(1)}\big). +\] +On the other hand, $2(u+m)=A_{S_h}(v)$: the ordered excursion +convention counts each path twice. Lemma~\ref{lem:pv:strip-visits} +therefore gives $u+m=h^{1/12+o(1)}$ in the central third, uniformly +in $v$. The negligible additive error above cannot carry this mass, +so +\begin{align*} + m(v)&=h^{1/12+o(1)} + &&\text{uniformly in the central third},\\ + m(v)&\le h^{1/12+o(1)} + &&\text{throughout the strip}. +\end{align*} + +To pass from a marked vertex to a fixed source, let $\mathcal V_h$ +contain one representative of each horizontal-translation orbit of +strip vertices. Translating a bridge's bottom source to the fixed +source identifies $m(v)$ with the weighted number of visits to the +orbit of $v$. Thus the unnormalized fixed-source length mass is +exactly +\[ + \ell_h=\sum_{v\in\mathcal V_h}m(v). +\] +There are $O(h)$ such orbits, with order $h$ represented in the +central third. The pointwise upper and lower estimates give +$\ell_h=h^{13/12+o(1)}$. Finally, dividing by +$b_h=h^{-1/4+o(1)}$ gives +\[ + \mathbb E|\gamma|=\frac{\ell_h}{b_h}=h^{4/3+o(1)}. +\] +Lemma~\ref{lem:pv:lateral-tail} makes both the discarded mass and +discarded first moment at horizontal range $h\log^2h$ negligible, +so the same conclusion holds with that confinement. This completes +Theorem~\ref{thm:pv:bridge}. diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/complex.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/complex.tex new file mode 100644 index 0000000000000000000000000000000000000000..69f17e9bb464006ca229cc36773751facd3ccac9 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/complex.tex @@ -0,0 +1,693 @@ +\section{Asymptotics of the balanced observable}\label{c01-c1-asymptotics-of-the-balanced-observable} + +The boundary exponent alone does not determine marked bulk mass. We estimate the balanced polygon observable by comparing its color-sum numerator with the squared homogeneous empty component. Their leading exponential factors cancel; control of the remaining powers must be uniform in the complex twist. + +\subsection{Color formula and small-cycle gas}\label{c01-c1-color-formula-and-small-cycle-gas} + +For the balanced homogeneous case (circumference \(2n\)) the target is the following estimate, with \(C\) uniform on each compact subset of \(|\Im s|<\pi\): +\[ + |X_{n,n}(\kappa=e^s)|\le C\ n^{\,1/6+2\Re(s^2)/(3\pi^2)},\qquad |\Im s|<\pi , +\] +The normalization is \(X_{n,n}=1\) at \(s=i\pi/2\). Together with positivity the estimate will give the matching exponent at \(s=0\) and analytic control near zero polygon fugacity. The calculation uses (i) a finite color interpolation formula converted to residues; (ii) a one-sided homogeneous normalization estimate for the Pfaffian; (iii) analytic radial periodization of the residues; upper-circle resummation creates hole variables, after which only a small defect gas around an explicit pressure remains. We bound that gas after a Gaussian count summation and recover the nonperiodized value by Cauchy at zero radial nome. The radial period \(L\) is an auxiliary \textbf{spectral} period, not the circumference of the physical SAW cylinder. Matrix and scalar abbreviations will be local to the computations. + +Here is a rational product sum for +\[ + \mathcal P=S_N^2 X_{n,m}/\prod_{i0\) sufficiently small. Indeed subtract from the generating function the two singular terms \(2^{-1+iy}(1-v)^{-1-iy},2^{-1-iy}(1+v)^{-1+iy}\). The derivative of the remainder in the disk is bounded by \(C(1+y^2)e^{C|y|}/|1-v^2|\) by a first-order estimate of the regular factor at each end. On radius \(1-1/m\) the angular integral of \(1/|1-v^2|\) is \(O(\log m)\), proving the additive error via coefficient extraction of the derivative. The multiplicative error follows by the convergent logarithmic products (tails after subtracting first-order terms are \(O(y^2/m)\)). Also \(|H(y)|^2=\sinh(\pi y)/(\pi y)\). + +Drop the range \(|y|>c_1\log m\) by positivity in the trace lower bound. The remaining integral per even index is at least +\[ + \frac{1}{\pi(m+1)}\int_{\mathbb R} f(y/\delta) + \left[1+\Re\big((2m)^{2iy}H(y)/\overline{H(y)}\big)\right]dy +\] +up to summable errors: integrate the square of the coefficient against \(2y f(y/\delta)\,dy/[(m+1)\sinh(\pi y)]\); \(|H|^2\) cancels its weight, and \(f\) decays exponentially (also to restore the omitted range on the right). The oscillatory integrals are \(O((\log m)^{-2})\) before the prefactor, by parts twice; the first two logarithmic derivatives of the phase factor \(H/\bar H\) have polynomial growth by its product. Finally \(\int f(p)dp=5\pi/9\), by expanding \(\log(1-e^{-3u})-\log(1-e^{-u})-\log(1+e^{-2u})\), \(u=\pi|p|/4\) (using \(\sum k^{-2}=\pi^2/6\), e.g. by Parseval on the linear function). The even-index sum now gives the claimed lower bound. + +\end{proof} + +The denominator is now fixed. We return to the numerator integral and deform it without changing its physical value at zero radial nome. + +\subsection{Radial periodization of the small-cycle integral}\label{c01-c1-radial-periodization-of-the-small-cycle-integral} + +Write \(\alpha=\pi/4,\ \kappa=e^s\), and use coordinates \(x=\log(z/t)\), periods \(L,2\pi i\), radial nome \(\mathfrak q=e^{-L/2}\), and \(P=\mathfrak q^2\). Define +\[ + S(x)=2\sinh(x/2)\prod_{a\ge1}(1-P^a e^x)(1-P^a e^{-x}). +\] +Use kernels \(\prod_j S(x-i\alpha j)^{d_j}\) specified by these divisors (actual integer shifts, so \(4,-4\) denote two factors): +\[ + h:\ d_{\pm1}=-1,\ d_{\pm4}=1;\quad + K_s:\ d_0=2,\ d_{\pm1}=-1,\ d_{\pm2}=1;\quad + K_x:\ d_{\pm3}=-2,\ d_{\pm4}=1. +\] +They are for the external field in \(x\) and for pair differences respectively. At \(P=0\) these agree with the small-cycle kernels with multipliers \(t^{-1},t\) stripped from the pair kernels; these phases would contribute \(t^{M+\#D}\) in total. Set \(C_L^2=\lim_{x\to0}K_s(x)/x^2\) (square root positive at \(P=0\)), \(k_L={\rm Res}_{x=i\alpha}K_s(x)\). Use E activity \(a_L=-i C_L e^{-s/2}\) in each type and D activity \(-k_L a_L^2\) (incorporating the phases at \(P=0\); indeed \(C_\infty^2=D_0\) and \(t A_A A_B=-D_0/\kappa\) there). Multiply the integrand by +\[ + \Theta(C)^2,\quad C=\exp(\sum_\nu q_\nu x_\nu),\quad + \Theta(C)=\sum_{b\in\mathbb Z} e^{-Lb^2/2} C^b +\] +where \(\nu\) ranges over virtual variables including both constituents of D, and \(q_\nu\) is their color sign. D constituents are now \(x,x-i\alpha\). Integrands are elliptic in each object's top coordinate, by neutrality and the shift formula \(S(x+L)=-e^{x+L/2}S(x)\) (the multiplier of the pair kernels moving one object of charge \(p_i\) is \(C^{-2p_i}\exp(-p_i^2L)\), canceled by \(\Theta(C)^2\)). + +Denote by \(\mathcal N_n(\mathfrak q,s)\) the resulting small-cycle gas around tops \(x=-i\alpha\), summed over \(0\le M\le2n\) with its factorial measures and deformed activities, and multiplied by \(B_n\). This includes the factor \(\kappa^n\) in \(B_n\). The numerator \(\mathcal N_n\) is analytic for small \(\mathfrak q\); at zero nome the homogeneous monomial denominator in \(\mathcal P\) is one, so +\[ + \begin{aligned} + \mathcal N_n(0,s)&=\mathcal P(1,\ldots,1;\kappa=e^s)\\ + &=S_{2n}(1,\ldots,1)^2 X_{n,n}(\kappa=e^s). + \end{aligned} +\] +Even if the small-cycle sum is extended, any fixed \(M>2n\) contributes zero: resolve \(h(x)^n\) into a product with slightly displaced fields and use distinct top poles as before, then pass to the limit. + +Here a single sufficiently small contour size and disk of analyticity in \(\mathfrak q\) work independently of \(n\). For top differences near zero the same-color string products have orders \(2,1,2\) for EE, ED, DD, and the opposite-color ones have no poles; other real-period images stay away. So after splitting the external poles each color selects distinct top poles (at most \(n\), length at most 2 each). Products defining \(S,\Theta\) converge normally where needed, and \(C_L,k_L\) are analytic deformations of their values at zero radial nome. + +We describe term transformations first at fixed \(M\), large real \(L\). Write a radial cycle of shift \(\theta\) as \(x=r-i\alpha\theta,\ -L/20\), with reversed shifts at negative frequency. Winding powers here can be grouped to integer powers for whole objects (off the walls this follows from even total divisor order per kernel; at the final shifts the same-contour EE walls have even order, and for DD the \(+1,-1\) differences of constituents enter together along with the two zero differences). One direct derivation uses, for each shifted sine at \(r-i\alpha u\), +\[ + S=\mathrm{sgn}(r)e^{(|r|-i\alpha u\,\mathrm{sgn}(r))/2} + \prod_{l\in\mathbb Z}(1-e^{-|r-lL|} e^{i\alpha u\,\mathrm{sgn}(r-lL)}) + \qquad (|r|\le L/2). +\] +Signs cancel; the log jump at 0 after summing and Gaussian compensation is \(-2\pi i k\), giving the indicated phase. The mean of the product log per shifted factor is \(-2\pi^2 B_2(\{u/8\})/L\), \(B_2(w)=w^2-w+1/6\). At nonzero frequency expand the logs; integrating on the two half-lines gives sums of \(1/(a^2+\omega^2)\) against \(\cos(a\alpha u)\) and \((\omega/a)\sin(a\alpha u)\), \(\omega=2\pi m/L\). Combining with the exponential and winding terms yields the formula for \(W\) (equivalently use the Fourier series of \(e^{\omega(x-\pi)}/(2\sinh\pi\omega)\) on \(00,\ v=m\varepsilon\), with \(T_\pm\) the columns of sums of upper \(y^{\pm m}\), the quadratic term +\[ + m^{-1}\big[T_-^t(1-A)T_+ + -T_-^t(nH\mathbf1+\sum_i J_i^+y_i^m) + -(nH\mathbf1+\sum_i J_i^- y_i^{-m})^t T_+\big]. +\] +The remaining nonzero-mode log is +\[ + -m^{-1}\big[\sum_{ij}W_{ij} y_j^m y_i^{-m} + -\sum_i W_{ii}(+\infty)(1-e^{-v}) + +n\sum_i(h_i^+y_i^m+h_i^- y_i^{-m})\big] +\] +per mode. + +Indeed from U to a lower object the winding \(k\) is 1 in the same color and 0 in the other, and for its external field is \(-|p_i|\). Fourier sums with coincident cycles are understood away from coincidences or with the pure log singularities factored. Including self terms requires the factor \(\exp(\Gamma_i)\) per lower object (note \(W_{ii}^{[0]}=0\) by reversal symmetry), where +\[ + \Gamma_i=-p_i^2 L/12+\varepsilon W_{ii}^{[1]}/2+ + \sum_{m>0}[W_{ii}(m\varepsilon)-W_{ii}(+\infty)(1-e^{-m\varepsilon})]/m=O(1) +\] +uniformly at large \(\Re L\), bounded \(\Im L\) (subtract \(p_i^2/(4m\varepsilon)\) in the series to cancel the linear growth). At U instead subtract 1 from the same-color coefficient at every frequency for the Haar separation: the resulting self log correction is exactly \(-\log(LC_L/(2\pi))\) by the limiting same-color kernel divided by \(|1-e^{i\phi}|^2\). Thus it cancels the signed measure conversion with \(a_L\), modulo the length fugacity already counted. Lower objects still have factorial measures and activities with \(e^{-sM}\) removed. These fixed-\(M\) identities continue to complex \(L\) (radial cycles along the period) near large positive real parts with bounded imaginary parts. + +In this bookkeeping \(\sum_{i0}(A_{ss}(m\varepsilon)-1)/m . +\] +Negative half of this is the self correction for writing the smooth residual U interactions with both indices independently summed. The kernel at this alignment is positive on the real cycle away from its zeros, so there is no extra phase in the conversion. + +\subsection{Upper-edge resummation}\label{c01-c1-upper-edge-resummation} + +The upper contours still contain an arbitrarily large number of points. The following circle determinant expansion sums them first and replaces them by holes on two separated contours. The dependence of its sign on the winding and circle size must be retained: it is what later permits the Gaussian sum over the original count \(M\). + +The zero-winding case is closely related to the Toeplitz--Fredholm identity of Borodin and Okounkov~\cite[Theorem~1]{BorodinOkounkov2000}. The calculation below retains the winding and its size-dependent sign; its analytic interchange with the Hubbard fields is justified separately. + +Here is the winding version of the expansion. Take potentials \(V_+(y),V_-(y)\) with strictly positive and negative powers respectively, analytic on a neighborhood of a closed annulus spanning the unit circle. For a circular unitary expectation of size \(u\) insert winding \(y^{-d}\) and \(\exp(V_++V_-)\) on each variable. Expand in additional points called holes, with signs \(\tau=+1\) (outer circle \(>1\), O) or \(-1\) (inner \(<1\), I), counts \(r,l\) with \(r-l=d\). Use normalized positive circle measures divided by factorials. The scalar prefactor is +\[ + (-1)^{ud}\exp\sum_{m>0}m [y^m]V_+ [y^{-m}]V_- . +\] +Hole factors are +\[ + \prod Y_h^{-u\tau_h}\exp\sum_h\tau_h(V_-(Y_h)-V_+(Y_h))\ + \prod_{h0}t_k z^k)\) for the respective coefficient list. Transposing partitions gives \((-1)^{ud}\sum_{\nu_1\le u}s_{(u^d,\nu)}(-V_+)s_\nu(-V_-)\). Indeed the partition determinant is a Toeplitz minor on columns \(u-i\), rows \(u-i+\lambda_i\); the complementary inverse minor on a sufficiently large finite index interval starting at 0 uses the complement \(u+j-1-\lambda^t_j\) and \(h^{-1}\), giving the transpose and sign. + +Explicitly \(\det[y_j^{u-i}]\prod_j h(y_j)\) expands by matrix multiplication into determinants with rows \(y_j^{u-i+\mu_i}\) for partitions \(\mu\) (padded to size \(u\)), with precisely these minor coefficients. The reciprocal-variable determinant expands likewise. Their product integrates with \(\prod_j y_j^{-d}\) and the circle factorial measures by orthogonality of monomials, requiring positive-list indices to be the negative-list indices shifted up by \(d\); this gives the asserted first sum. In the transposed sum \((u^d,\nu)\) means prepend \(d\) parts of size \(u\). + +The charged-wedge calculation is the classical boson--fermion correspondence~\cite{Frenkel1981}; see the Clifford-operator and Schur-function treatment in \cite[Theorems~8.1 and~8.3]{FrenkelJingWang2000}. We give its coefficient calculation to fix the charge convention and the projection onto permitted occupied levels, before taking any analytic specialization. + +Use wedges with decreasing occupied positions \(c+\lambda_i-i\), charge \(c\), vacancy of the partition denoted \(|c\rangle\). Insert/remove level \(j\) by \(\psi_j,\psi^*_j\) (sign by exterior order). The currents \(A_k=\sum_j\psi_{j-k}\psi_j^*\ (k\ne0)\) satisfy +\[ + [A_k,\psi_j]=\psi_{j-k},\quad [A_k,\psi^*_j]=-\psi^*_{j+k},\quad + [A_k,A_h]=k\delta_{h,-k}. +\] +Indeed terms act finitely on each state, and the last bracket commutes with all insertions/removals so is scalar (commute first with occupation projectors); evaluate on the vacancy, using the \(k\) possible moves up by \(k>0\). The partition amplitudes of \(\Gamma_-(t)|c\rangle\) with \(\Gamma_\pm(t)=\exp(\sum_{k>0}t_k A_{\pm k})\) are the minors above by exterior multiplication with the positive shift generating function \(h\) (can truncate the sea deep below all possible movements at any fixed weighted degree). Thus after removing the transpose sign our sum is +\[ + \langle d|\Gamma_+(-V_+)\psi_{u+d-1}\cdots\psi_u + \prod_{j\ge u+d}(1-\psi_j\psi_j^*)\Gamma_-(-V_-)|0\rangle . +\] +Expand the projectors (finitely effective in each weighted degree) and use generating currents \(\psi(w)=\sum\psi_k w^k,\ \psi^*(v)=\sum\psi_k^* v^{-k}\), with creations outside, annihilations inside. Commuting the \(\Gamma\)'s past the currents gives precisely the exponential factors stated. The vacancy expectation with \(r=d+l\) creations and \(l\) annihilations (ordered separately, signs from reordering depending only on counts) is up to sign the determinant with columns \(w_i^a\ (0\le a0\) by the explicit \(A\) (eigenvalues asymptotic to \((9/8)v,1/(2v)\) at zero and \(1+O(e^{-v})\) at infinity). At fixed \(\varepsilon\), subtract \((1-e^{-cv})1\) for very small \(c>0\), leaving positive smooth Fourier forms whose missing diagonal terms cost at most constants per hole. The subtracted kernels have pairwise gain at most \(O(c\varepsilon)\) per pair (sum the logarithms), which is absorbed by the favorable Cauchy mean \(-2\varepsilon h_0\sum_c r_c l_c\) up to constants since count differences are fixed. The factorial measures now ensure absolute convergence. + +Here the small subtraction is the identity on both color and shift indices. Its distinct-point terms only concern identical shifts in one color (hence equal charge signs), with pair gain bounded by \(2\log\sup_{|z|=1}|(1-z)/(1-e^{-c\varepsilon}z)|\le c\varepsilon\). It preserves positivity if \(c\) is small: at high modes the Hessian tends exponentially to identity, and on the rest \(v\ge\varepsilon\) use strict positivity. Completing the residual (positive) ordered sums with their diagonals then only pays \(\sum_{m>0}e^{-cm\varepsilon}/m\) per hole: the Gaussian gains already have their diagonals, and the original missing Cauchy diagonal coefficient 1 was split into \(1-e^{-cv}\) (kept on distinct points only) and \(e^{-cv}\). Also \(\binom{r_c}{2}+\binom{l_c}{2}\le r_c l_c+O(1)\) at fixed count difference for the two same-shift classes in each color (the subtraction constant can be chosen below \(2h_0\)). Gaussian integrations can also be truncated in modes first (for each fixed number of points the high coefficients are exponentially small, giving integrable domination of that tail). + +The absolute bounds work uniformly over the lower radial positions here at fixed preceding counts and indices. With Gaussian cutoffs the high-mode gain terms in the real Hessian simply drop out, preserving the bound, and the Gaussian cross terms with the deterministic sources are bounded by a constant per hole by high-mode decay on the tiny annulus. For each fixed number of holes the Gaussian formula passes to the infinite product as well (summable coefficient tails; one can take absolute moments of order slightly above 1 using strict Gaussian stiffness). + +\subsection{The residual kernels}\label{c01-c1-the-residual-kernels} + +The circle determinant has eliminated the \(\mathrm U\) variables. We first carry out the remaining transformations for a fixed nonnegative integer \(M\) and a large real period \(L\). Here \(M\) is the common total number of virtual integration variables per color in the original small-cycle gas, including variables assigned to lower objects. It is unrelated to the visited-vertex length of a walk. If \(g_c\) is the number of lower virtual variables of color \(c\), the eliminated upper ensemble had size \(u_c=M-g_c\). + +We now give the remaining interactions as a matrix on a fixed list of species. The exact one-object measures and factorials are retained, together with the implicit constant phase signs. The one-object prefactors obey the stated uniform bounds and are independent of \(M\) at fixed defect data; the implicit phases have modulus one and depend only on the discrete configuration. The position-dependent phases remain explicit below. Section~\ref{c01-c1-estimates-and-summation-of-the-defect-gas} will justify absolute summability of the assembled expression, complex-period continuation, and the subsequent sum over \(M\). + +The hole sign \(\tau=1,-1\) refers to the outer or inner contour of an eliminated upper ensemble. Both signs occur in each gas color, whose sign is \(q\). In the moves below we continue the Fourier coefficients and their winding powers as functions of the offsets, re-expanding a pure singular factor when its Fourier branch changes. The upper circles have already been integrated out. + +After integrating the Hubbard fields move hole contours O to \(h=3/8\); then move I to \(h=-1/4\). In the second move the pole with a D of the same color creates an object \(\mathrm P=(\mathrm D,\mathrm I)\) at equality of their radial variables, with \(d=h\) (here \(2+d\) is the lower top shift of D). Put free D at \(d=1/8\) and P at \(d=h=0\). Their unshifted radial variables within the period are denoted \(x_i\) in what follows (so \(y_i=\exp(2\pi i x_i/L)=e^{i\phi_i}\), and e.g. a hole has \(Y=e^{\varepsilon h}y_i\)). + +Use \({\bf 1}=(1,1)^t, Q=(1,-1)^t\), and unit columns by color. Let \(t_i=\tau_i\) times the corresponding unit column for holes, otherwise zero. For a lower gas object write \(p_i\) for its signed length, \(z_i=\sum q_\nu\theta_\nu\) for its virtual angular moment, \(\lambda_i\) for its color unit column; all three are zero on holes. Use addition of data for P. The following table collects the resulting five species in either color of sign \(q\). Here \(a_i\) means respectively \(\ell,d,h,h,d=h\), the jump charge is \(j_i={\bf1}^t(\lambda_i-t_i)\), and the external-drive coefficient is \(\xi_i=|p_i|-j_i\). +\[ +\begin{array}{c|l|rrrr} + \text{species}&\text{constituents}&a_i&p_i/q&j_i&\xi_i\\\hline + \mathrm L&\text{one lower variable}&1/2&1&1&0\\ + \mathrm D&\text{two lower variables}&1/8&2&1&1\\ + \mathrm O&\text{outer hole}&3/8&0&-1&1\\ + \mathrm I&\text{inner hole}&-1/4&0&1&-1\\ + \mathrm P&\text{bound D and I}&0&2&2&0 +\end{array} +\] +For fixed species, counts and positions, we will isolate all \(M\)-dependence in a Gaussian count factor. We first define its center and the residual interactions, state the resulting factorization, and then verify the contour and scalar bookkeeping. + +Continue \(J_{\rm D}^\pm\) by retaining the initial branches (coefficient dependence \(e^{\pm d v}\)). Define columns +\[ +\begin{gathered} + F_i^+=J_i^++t_i e^{h_i v}=T_i/v+f_i+r_i v+\cdots,\\ + F_i^-=J_i^- -t_i e^{-h_i v}=-F_i^+(-v),\qquad + T_i=Qp_i/4,\quad f_i=Qz_i/4-\lambda_i+t_i. +\end{gathered} +\] +(terms only on applicable constituents). The dressed oriented Fourier kernel \(G(v)\) is +\[ + G_{ij}=W^{g}_{ij}-(F_i^-)^t A^{-1}F_j^+ + +t_i^t t_j\,\operatorname{sgn}(h_i-h_j)e^{(h_j-h_i)v}+C_{ij}. +\] +Here \(W^g\) is just the preceding bare \(W\) between lower gas parts, with actual lower-lower shift branches. The crossing correction for each I,D pair with relative offsets \(h,d\) satisfying \(d>h\), writing \(R=1\) on equal colors and \(R=-2\) otherwise, is +\[ + C_{\rm ID}=R e^{(d-h)v},\qquad C_{\rm DI}=-R e^{(h-d)v}. +\] +Use half at equality, zero for \(d0}(\det A(m\varepsilon))^{-1}\, + \exp\!\left[n^2\varepsilon\Big(\tfrac12 H_*(0)+\sum_{m>0} H_*(m\varepsilon)\Big)\right], + \qquad H_*(v)=\frac{2\cosh3v-2}{v(2\cosh v-1)\sinh4v}. +\] +\emph{Count Gaussian and twist.} The count factor is +\[ + \exp[-2\varepsilon(b_1-b_2)^2-a\varepsilon(M-M_0)^2 + -s(M-n)+i(M-M_0)\mathcal S],\qquad + \mathcal S=\sum_i j_i\phi_i. +\] +\emph{External drive.} Its logarithm is +\[ + -n\sum_i\xi_i\left[i\phi_i+a_i\varepsilon+ + \sum_{m>0}\frac{\rho(m\varepsilon)}{m} + \big(e^{a_i m\varepsilon}y_i^m-e^{-a_i m\varepsilon}y_i^{-m}\big)\right], + \qquad \rho(v)=\frac1{2\cosh v-1}. +\] +\emph{Interaction.} Its logarithm, in addition to a linear phase, is +\[ + -\frac\varepsilon2\sum_{ij} [v]G_{ij} + -\sum_{m>0}\frac{\sum_{ij}G_{ij}(m\varepsilon)y_j^m/y_i^m-\sum_i(1-e^{-m\varepsilon})}{m}. +\] +Its additional linear phase is precisely \(-i\sum_{ij}\phi_j(K_0)_{ij}\). + +The displayed count twist and linear phase retain their position dependence; only the implicit constant phases are suppressed. The verification below is termwise at fixed counts. Section~\ref{c01-c1-a-kernel-positivity-decomposition} bounds the interaction matrix frequency by frequency. Section~\ref{c01-c1-estimates-and-summation-of-the-defect-gas} then proves absolute summability, justifies the assembled fixed-sector identity, continues it to complex periods, and permits the sum over the original count. + +\paragraph{Contour crossings and one-object factors.} +The growing term in the I-to-D \(- (F_i^-)^t A^{-1} F_j^+\) is \(-R e^{(d-h)v}\). Transferring the corresponding singular factor to the opposite Fourier direction adds \(C\) as above. Between two P objects at equal shifts both directions are transferred or averaged together (the powers are integer powers of the combined two-sided factor, up to a sign); on the internal I,D pair the singular factor is omitted by the residue. + +Indeed at high positive \(v\), the only growing or nondecaying singular part of \(J_{\rm D}^+\) when crossing is the column with entries \(R e^{d v}\), whereas \(J_{\rm D}^-\) decays at least as fast as \(O(e^{-(1+d)v})\), and \(J_{\rm L}^\pm=O(e^{-v/2})\). This follows by summing the given divisors. Since \(A^{-1}=1+O(e^{-v})\), and D always stays at offset strictly inside O, the troublesome product among these factors is precisely \(-R e^{(d-h)v}\) from I to D, giving \((1-e^{(d-h)\varepsilon}y_D/y_I)^{-R}\); other than the direct hole pair and gas factors all the remaining mode sums are analytic across the moves. Re-expansion gives the correction, mean and phase above (log mean changes by \(-\varepsilon R(d-h)\)). The pole is simple and taking its residue removes this factor and the normalized I circle measure. Bound pairs have no further poles in these moves: equal-color DD and II each have a double zero at coincidence, compensating the possible simple pole with an unbound object or the two poles between bound pairs. Opposite colors had a zero rather than a pole; remaining bare factors between lower contours have no poles for these offset differences. Thus only disjoint matchings are taken, with factorial denominators now those for the combined species and signs depending on the defect data. This reasoning can move contours successively with separated radii. The natural expansions of lower-lower and hole-hole factors after each move can always be taken directly. Constants and phases for the continued upper-lower modes track the retained branches: under an offset increment \(\delta\), a coefficient continued by \(e^{\delta v}\) changes its mean \(p\alpha^2\theta^2/L-\varepsilon[v]W\) by exactly \(\varepsilon k\delta\). In paired-paired terms at equal offsets the two transferred factors have opposite orientations and can take the symmetric boundary values together up to signs. + +More explicitly the leading terms of \(J_{\rm D}^+\) come from index \(-2\) at the top (same color) or \(-3\) at the bottom (opposite color), retained with wrapped value \(8+d\); the rest is \(O(e^{-(1-d)v})\). One can cross by moving the I contours while all D have \(d=-1/8\), and then move free D's and bound pairs (\(h=d\)) to final positions. Gas top relative offsets between two D gas parts then have size \(<1\) (the possible same-color constituent poles at zero top difference already cancel); differences from L to the two constituents have size \(<1\). Two P's on one contour combine their I-to-D pure factors to \(((1-y'/y)(1-y/y'))^{-R}\), agreeing with the averaged expansions. The gas windings group to integers (at equal D shifts the \(+1,-1\) constituent differences occur together), as do the crossing powers (the equal-offset halves occur in opposite directions together). + +Self subtractions in the final series are \(G_{ii}(+\infty)(1-e^{-v})\) (always \(G_{ii}(+\infty)=1\)); self and measure prefactors besides powers already isolated are bounded by \(C\,|dx_i|\) per object, uniformly for large \(\Re L\) and bounded \(\Im L\). Indeed the lower self corrections were given earlier. An unbound hole has a missing 1 on the diagonal from the circle-pair log. Using \(1-e^{-v}\) instead leaves an activity \(\exp(\sum e^{-m\varepsilon}/m)\) with its measure \(dx_i/L\). For a bound pair, removing the pure pole adds 1 on its effective diagonal, so the two old subtractions (\(1-e^{-v}\) for D and 1 for I) now give exactly \(1-e^{-v}\) without extra activity. The residue has only the D radial measure. At infinity the joint kernel has same-color limit the identity in species, opposite-color limit 2 at P,P only, by the divisor rules. + +\paragraph{Zero modes and count dependence.} +The direct constant of \(G\) at zero is \((p_i z_j-z_i p_j)/4-k_{ij}\) by symmetry of divisor shifts. Dressing contributes \(-2T_i^tf_j+2 f_i^tT_j\) and \((-r_i^tP_+f_j+f_i^tP_+r_j)/a\) there. For means during the contour moves, retaining a branch with offset increase \(\delta\) multiplies its coefficient by \(e^{\delta v}\); the winding-power increment of the constant log is then as above, using \(W^{[0]}=p\theta/4-k\) on a virtual pair. In total the \(z\)-square term precisely accounts for changes in all Gaussian offsets and the theta winding offsets by neutrality. Natural re-wrappings just use the actual direct factors (hole Cauchy, bare gas, or transferred pole). + +Indeed the retained \([v]\) changes by \((p\theta/4-k)\delta+p\delta^2/8\) per virtual pair; the sum of \(p\) times squared shift differences changes by minus the change of \((\sum z_i)^2\). The winding monomials from the old formulas change by the same required scalar when an old radial argument is moved (\(y_{\rm old}=e^{\varepsilon\delta}y_{\rm new}\) for a single increment). On the internal bound pair \([v]C\) costs zero; in the nonzero modes deleting the pure ID pole subtracts \(1/m\) per frequency after identification, giving the self subtraction specified above. + +For external insertions the dressed columns (rows similarly) couple with \(h_i^+-H{\bf1}^tA^{-1}F_i^+=\xi_i \rho e^{a_i v}\), reversed \(-\xi_i\rho e^{-a_i v}\); the exterior-exterior kernel is \(2H\rho=-vH_*\). Indeed \(-A^{-1}H{\bf1}=\rho {\bf1}\). On gas divisors, cyclic neighbor sum minus the divisor itself for \(h\) equals minus the sum of the two pair divisors. In multiplying \(h_i^+\) by \(2\cosh v-1\) the unwrapped neighbor exponents therefore leave only terms passing the seam. At the initial shift branches with \(d<0\) none do so for L or D bottom; at D top the \(-1\) index passes down through zero, leaving \(+e^{dv}\). This continues identically. For zero modes before dressing the bracket in the mean, now including the factors \(Y^{-\tau u_c}\), is +\[ + 2|f_{\rm tot}|^2+4(b_1-b_2)^2 + +[v]\big\{u^tAu+2u^t(nH{\bf1}+\textstyle\sum F_i^+)+B(v)\big\}. +\] +Here \(f_{\rm tot}=(\sum z_i/4+b)Q\), the norm square is real, and \(B(v)\) denotes the sum of the direct kernels \(W^g,K^{\rm hole}=t_i^t t_j\operatorname{sgn}(h_i-h_j)e^{(h_j-h_i)v},C\) and direct external \(n(h_i^++h_i^-)\). Hole means and crossings indeed match these coefficients of \(v\). Replacing \(B(v)\) by +\[ + B(v)-(nH{\bf1}+\sum F_i^-)^t A^{-1}(nH{\bf1}+\sum F_i^+) +\] +incorporates all the other terms except \(4(b_1-b_2)^2+2a(M-M_0)^2\). This is completion of the square coefficientwise, using \(P_-u=-2\sum T_i, H=-av+O(v^3),\ \sum f_i=f_{\rm tot}\). Finally the winding phases on \(y_j\) have powers \(\sum_i k_{ij}+u^t(\lambda_j-t_j)-n|p_j|-bp_j\), with count signs independent of \(M\) as seen above. Splitting \(u\) by \(M_0\), the identity +\(\sum_i((K_0)_{ij}+k_{ij})=b p_j-\sum_i\nu_i^t(\lambda_j-t_j)\) +gives exactly the stated phases. + +Explicitly in the square completion the minus-channel pieces with \(P_-u=-2T_{\rm tot}\) leave \(-2|f_{\rm tot}|^2\) at order \(v\), canceling the explicit norm square. The plus-channel part is \(a|P_+u+P_+ r_{\rm tot}/a-n{\bf1}|^2\) since \(P_+f_{\rm tot}=0\). + +To reiterate the count-sign point, the upper cycle measure was \(-a_L\,dx/(2\pi i)=C_L e^{-s/2}dx/(2\pi)\) with exactly the positive-sign self conversion on real periods. The signs from \((-y_i)^{u_{c(i)}}\) over gas parts and \((-1)^{\sum u_c(e_c+bq_c)}\) from the winding determinant combine to a sign independent of \(M\) (\(e_c\) here counts those gas parts). Transferring pure factors and taking their normalized circle residues uses only signs fixed by defect data. These constant phase signs can simply be kept unchanged in continuation. + +\subsection{A kernel positivity decomposition}\label{c01-c1-a-kernel-positivity-decomposition} + +Our next task is to bound the residual particle sum uniformly in the spectral period. A positive Fourier form will control high local populations; a separate pole term will control the cumulative integer charge. The following decomposition supplies both controls. + +In this calculation only the five species (per color) index the matrices; the symmetric part of \(G(v)\) for \(v>0\) is meant. One can subtract the following pieces: +\par\noindent\textbullet\quad \(j j^t\operatorname{sech}^2(Kv)/(2av)\), with \(K\) a sufficiently large constant, all colors; +\par\noindent\textbullet\quad on the opposite-color blocks only, the entrywise repulsion +\[ + R(v)=\tanh(v/2)\big(R_1/\cosh v+R_2/\cosh(v/2)\big)+2\tanh v\ e_5 e_5^t , +\] +where +\[ + 8 R_1=\begin{pmatrix}6&15&0&3&16\\15&22&0&9&26\\0&0&2&0&0\\3&9&0&0&5\\16&26&0&5&5\end{pmatrix},\qquad + 8 R_2=2(e_2e_5^t+e_5e_2^t)+5(e_4e_5^t+e_5e_4^t); +\] +\par\noindent\textbullet\quad on the diagonal, \(\tanh(c v)\), for sufficiently small \(c>0\). + +The result divided by \(v\) is an even, smooth, exponentially decreasing positive Fourier form; it even dominates \(c_1\operatorname{sech}(c_2 v)\) times the identity, for some positive constants. The repulsions (second and third pieces divided by \(v\)) have entrywise nonnegative Fourier transforms. Indeed \(\tanh(c v)/v=\int_0^c\operatorname{sech}^2(u v)\,du\), and \(\operatorname{sech}\) has positive Fourier transform (by shifting through one pole), so squares and products here give positive convolutions. + +Here are verification details of the matrix inequality. Work in either color channel \(h=\pm1\), i.e. the eigenblock same plus \(h\) times opposite, and first subtract only \(hR(v)\). Call the resulting symmetric matrix \(S_h(v)\). With \(x=e^{v/8}, w=x^8\), put \(D=w^4-w^{-4}\) and +\[ + A_n=w^{-4}(w^2-w+1)(w^3-h)^2. +\] +Thus the single upper eigenvalue is \(A_n/D\). After removal of the offsets (factor \(x^{8\beta},\ \beta=a_j-a_i\)), the oriented entry of \(G\) in this channel is +\[ + m_{ij}(w)/D - p_i(w^{-1})p_j(w)/(D A_n)+b_{ij}. +\] +Here \(m_{ij}\) is the divisor exponential sum for bare gas parts, \(p_i\) that from an upper variable plus \(\tau_i D\), and \(b_{ij}\) the hole-direct and crossing entries; explicit arrays are given below. At every \(w^8=1\), \(m_{ij}A_n=p_i(w^{-1})p_j(w)\), since wrapping and the branch choice do not change any powers. At \(w=1, h=1\) the difference vanishes to order at least two because \(p_i(1)=0\) and \(A_n\) has a double zero. Thus, putting \(z=w^{-1}\), +\[ + \begin{split} + N&=44\ (h=1),\quad40\ (h=-1),\\ + \Delta_+&=(1-z)(1+z)^2(1+z^2)(1+z+z^2)^2(1-z+z^2),\\ + \Delta_-&=(1+z)^2(1+z^2)(1-z+z^2)^3 , + \end{split} +\] +the matrix \(4 x^N\Delta_h S_h\) is a Laurent polynomial of parity \(h\) under \(x\mapsto1/x\), with powers at most \(N\) (indeed \(S_h\to1\) at infinity). Denote its coefficients at \(x^k,\ k\ge0\) by \(P_k\), taking only half the constant coefficient. In the Taylor expansion in \(\log x\) the relevant matrices are +\[ + B_p=\sum_{k=0}^N (k/N)^p P_k +\] +for even \(p\) in the plus and odd \(p\) in the minus channel. \(B_0=128 j j^t\) for the plus channel (here \(j\) is the five-component vector). All the other relevant matrices are positive definite. + +We include small algebra details for checking this last sign test. Take \(K_i=(2),(2,3),\varnothing,\varnothing,(2,3)\), respectively, and \(\tau_i=0,0,1,-1,-1\). Let \(d_l=(2,-1,1,-2h,2h,-2h,1,-1)_l\) for residues starting at 0. Put +\[ + L_b(m)= + \begin{cases} + z^{8-(m\bmod8)}&m\not\equiv0,\\ + z^8, 1, (1+z^8)/2&m\equiv0,\quad b>0, b<0, b=0\ \text{respectively}. + \end{cases} +\] +Then +\[ + m'_{ij}=z^4 m_{ij}(w)=\sum_{r\in K_i,s\in K_j,l}d_l L_\beta(l+s-r),\quad + u_j=z^4 p_j(w)=\tau_j(1-z^8)+\sum_{s\in K_j,l}d_l L_{\sigma_j}(l+s), +\] +where \(\sigma_j>0\) on L, negative on D or P. Finally +\[ + b_{ij}=-\tau_i\tau_j\operatorname{sgn}\beta+(1-2h) + [\mathbf1_{\tau_i=-1,\,|K_j|=2}\chi(\beta)- + \mathbf1_{|K_i|=2,\,\tau_j=-1}\chi(-\beta)] +\] +with \(\chi\) the positive step (half at zero). + +The following table gives integer parts of the five successive leading minors; the last row in each block instead gives integer parts of \(100\sum_{k0\) instead and multiply by \(100/N^p\). These coefficient formulas use only polynomials of small degree. + +For some further detail on the common denominators: at an eighth root \(w\) put \(U=\sum_l d_l w^{l-4}=\sum_l d_l w^{4-l}=A_n\). Then \(m_{ij}=U(\sum_{r\in K_i}w^{-r})(\sum_{s\in K_j}w^s)\), and \(p_i(w)=U\sum_{r\in K_i}w^r\) there. Once the \(D\) poles cancel this leaves the factor \((1-z+z^2)(1-hz^3)^2\), with at most a simple pole at \(z=1\) in the plus channel; the repulsion denominators use \((1+z)^2(1+z^2)\). Their combination thus divides \(\Delta_h\). The offsets only give integral powers of \(x\). Symmetrization is odd under \(v\mapsto-v\) before multiplication by \(4x^N\Delta_h\); this gives the stated parity. The plus constant Taylor value is \(2B_0\) and also follows immediately from \(\Delta_+\sim72v\) and the pole. For moments at positive index \(p\), only coefficients strictly above power zero in \(x\) matter, so \((N-1)/8+\max|a_i-a_j|\) controls the truncation in \(z\). + +To pass from these coefficient tests to every \(v>0\), expand the Laurent polynomial in \(\log x=v/8>0\). Its parity and the half-constant convention give +\[ + \begin{split} + 4x^{44}\Delta_+S_+&=2\sum_{r\ge0} + \frac{(44\log x)^{2r}}{(2r)!}B_{2r},\\ + 4x^{40}\Delta_-S_-&=2\sum_{r\ge0} + \frac{(40\log x)^{2r+1}}{(2r+1)!}B_{2r+1}, + \end{split} +\] +where each line uses its own channel's \(B_p\). All scalar weights are nonnegative, and \(\Delta_h>0\) for \(x>1\), proving that \(S_h\) is positive definite. Moreover \(B_1\) gives positivity of \(S_-/v\) at zero, while \(B_2\) gives the positive order-\(v\) coefficient of \(S_+\) on the orthogonal complement of its pole; the denominator corrections to \(B_0=128jj^t\) remain in the pole direction. Subtracting the indicated smoothed pole leaves a positive definite order-\(v\) coefficient for sufficiently large \(K\); away from zero this subtraction can be made arbitrarily small by further increasing \(K\). At infinity the result is still \(I+O(e^{-c_3 v})\), \(c_3>0\), so subtracting \(\tanh(c v)\) times identity for sufficiently small \(c\) is also allowed and leaves the asserted lower bound at all frequencies. + +Indeed in the plus block the pole removed is \(j j^t\operatorname{sech}^2(Kv)/(a v)\), improving the regular linear coefficient in its pole direction arbitrarily as \(K\) increases. Once this works in a neighborhood of zero, increasing \(K\) only improves the pointwise form there. Similarly one can decrease \(c\) further on compact intervals after ensuring the inequalities near zero and infinity (at infinity first take \(2c0\) deform the line into \(v=u+i(\delta+2\sigma_0|u|)\), small fixed \(\delta,\sigma_0>0\), \(\sigma\le\sigma_0\); similarly reflect for the other end. The integrals for real nonzero \(t\) allow the deformation also for decay of order 1 by oscillatory limits. Poisson summation in the form +\(\varepsilon\sum_{d\in\mathbb Z} f(d\varepsilon)e^{i d\varepsilon x/\alpha}=\sum_k \widehat f((x+kL)/\alpha)\) +then gives the kernels above, first for real periods and positions (for a logarithmic singularity by Fourier series off the diagonal, or smoothing by a narrow Gaussian first), then by continuation. Here the degree-1 mean of the original regular numerator supplies the half-weighted value of \(f(0)\). + +On the real comparison line of period \(R\), the smooth positive form costs at least \(c_4\sum n_d^2\). Here is the passage from frequency positivity to this population bound. Subtract its diagonal minorant \(c_1\operatorname{sech}(c_2v)I\); the ordered point-measure quadratic form of the remainder, including its diagonal and zero mode, is nonnegative by the periodized Fourier series. The transform of the subtracted scalar kernel is nonnegative at every real distance and bounded below for two points in a box of width at most 2. If \(n_{d,t}\) counts type \(t\) in box \(d\), the retained energy is therefore at least +\[ + c\sum_d\sum_{t=1}^{10}n_{d,t}^2 + \ge \frac{c}{10}\sum_d n_d^2. +\] +This uses matrix positivity for the smooth remainder, not entrywise positivity of its transform. The separate repulsions between distinct objects are nonnegative in space, and the diagonal self correction costs only \(O(m)\). Changing to complex \(\gamma\) costs only \(C\sigma\sum n_d^2\): every lifted real difference changes by at most \(\sigma\) times its size, so the derivative bounds above give losses \(C\sigma\) times exponentially decreasing pair kernels. Their sums are bounded by the box square sum (Cauchy--Schwarz). Coincident positions can be disregarded. Thus these parts of the real log are bounded by \(-c_5\sum n_d^2+C m\). + +For the antisymmetric part \(K=(G-G^t)/2\), even, smooth and exponentially decreasing, include the additional linear phase involving \(K_0\) in the interaction. This gives on each ordered pair a primitive, vanishing at difference zero, of +\[ + -\frac{i}{2\alpha}\sum_k \widehat K((x+kL)/\alpha). +\] +Indeed the linear phase inserts half the zero mode of the derivative of the sine sum. It is imaginary on the real comparison. On the complex curve its extra real part again costs only \(C\sigma\sum n_d^2\), applying the same endpoint comparisons to the primitives of each image term (subtract at \(kL\)), whose derivatives have the exponential bound. Thus small enough slopes allow absorption. + +Here the lower endpoint comparison at \(k=0\) has exactly zero change, and those at nonzero \(k\) cost \(O(\sigma e^{-c'R})\) summed per pair, harmless also for the box bound. For the transform arguments with \(1/|v|\) tails, the nonnegative real transforms used above are locally integrable functions (use the displayed positive mixtures of scaled \(\operatorname{sech}^2\) transforms, with convolutions as applicable), with the pointwise bounds off zero; Poisson's formula there also follows by periodic Fourier coefficients in \(L^1\). + +For example on a real period these series off coincidences converge in the ordinary paired-frequency sense too, since at positive frequencies one only has a constant over frequency plus a rapidly decreasing remainder at infinity. On straight complex periods, with \(x/L\) fixed real in the Poisson formula, both sides continue (for the image transforms keep the positive or negative real-end branch according to the lifted real difference). For the antisymmetric primitive, real parts can be compared image by image using a line primitive real on the real axis, evaluated at a lifted difference and at \(kL\), before multiplying by the imaginary prefactor. The joining paths can be chosen on the appropriate wedge side, since for differences inside one period the endpoint real parts do not straddle zero strictly. + +\paragraph{Integer-charge variance and contour deformation.} +The population bound does not yet localize the objects along the period. For this we retain the pole term, whose charges are the integer jumps \(j_i\). Let \(w(r)\) on the real parametrizing period have initial value zero at \(-R/2\), with jumps \(j_i\) at the positions \(r_i\), periodically extended. Smooth this step function on the curve by convolution in contour length coordinate (complex differential \(dz\)) with +\[ + H(z)=\frac1{2\pi\alpha}\sum_k \widehat{\operatorname{sech}(K\,\cdot)}((z+kL)/\alpha) +\] +(here \(K\) in \(\operatorname{sech}\) is the pole smoothing constant). Denote the result by \(w_*\), and its average by \(\bar w_*\); this average is that of the steps using complex differential, \(-\mathcal S/(2\pi)\). The nonzero log of the smoothed pole is exactly +\[ + -c_0\int_\gamma(w_*-\bar w_*)^2 dz,\qquad c_0=\pi^2/(2\pi\alpha a). +\] +On straight contours these facts follow from Fourier series, differentiating the steps to the jump masses. On bent contours the identities continue: convolutions and integrals of steps use arcs with moving endpoints, the line kernels analytic with exponential bounds in neighborhoods of all the contour differences as contours and endpoints move together. Thus the resulting integrals continue holomorphically in endpoint data along the deformation (locally deform arcs by Cauchy's theorem). Equivalently the nonzero pole pair potential has second derivative given by a periodized smooth line kernel (numerator \(\operatorname{sech}^2\)) minus its constant mode. + +In fact the unsmoothed step has nonzero Fourier coefficient \(\sum_i j_i e^{-i b\phi_i}/(2\pi i b)\) at integer \(b\ne0\), by integration by parts, and the smoothing multiplier is \(\operatorname{sech}(K b\varepsilon)\) (unit at zero). + +We deform the product torus of spatial contours jointly. Three checks make this passage precise. + +\emph{Coincidences.} Every lifted pair difference remains in the preceding wedge. Near a contact modulo $L$, separate the high-frequency constant of the \emph{combined} pair coefficient from its exponentially decreasing remainder. The former exponentiates to the nonnegative integer power $G_{ij}(+\infty)$ of $(1-y_j/y_i)(1-y_i/y_j)$; the remainder converges normally in a joint neighborhood. Thus the full exponential, rather than each separated logarithm, extends across contact loci. The external kernel used below has no singularity along this homotopy. + +\emph{Period seams.} The original winding factors show periodicity: for each integer value of the original count $M$, the powers from winding and crossings are integers after constituents are combined. Continue the same straight-torus germ while following both representatives at a period seam. Their values remain equal. Small joint Cauchy deformations therefore create no vertical-end term. The common graph has a global Lipschitz bound on its imaginary part, with endpoint difference $\Im L$. + +\emph{Moving endpoints.} For the step integrals, fix the order of distinct jump positions by real part and split every convolution arc at those positions. Expanding the convolutions gives iterated analytic-kernel integrals; all integration-point differences remain in the same analytic neighborhoods. Short tails added to the arcs show local holomorphy in their endpoints, including the smoothing variables. The arcs and their endpoints can consequently move together in small product neighborhoods. This continues the straight-contour identities needed at distinct jumps. These three checks justify the joint deformation without separating non-single-valued logarithmic terms. + +Denote the smoothed step on the real comparison period by \(w_R\), and use \(b(r)\) here for a local density bound of the form \(\sum_{i,k} e^{-c_6|r-r_i+kR|}\), with \(c_6>0\) sufficiently small. The elementary smoothing estimates are +\[ + |w_R-w|\le C b(r),\qquad |w_*(\gamma(r))-w_R(r)|\le C\sigma b(r). +\] +Indeed both kernels integrate to one, so one convolves differences from the value at \(r\), bounded along a lift by the jumps passed. The kernels including differentials differ by at most a periodized exponential times \(C\sigma\). Thus \(\int b\le C m,\ \int b^2\le C\sum n_d^2\). Also, writing \(w_R^0\) for the real average, one has +\[ + |\bar w_* - w_R^0|\le C\sigma R^{-1/2}(\|w_R-w_R^0\|_2+\|b\|_2) +\] +by averaging the unsmoothed steps. + +\paragraph{Sharp contour and absolute assembly.} +The external expression in brackets in the drive is a primitive (zero at 0) with derivative \((i/\alpha)\sum_k\widehat\rho((x+kL)/\alpha)\), evaluated at \(x=x_i-i\alpha a_i\). Here +\(\widehat\rho(t)\) is a positive constant times \(\sinh(2\pi t/3)/\sinh(\pi t)\), by shifting the integration line by \(2\pi i\) and taking the two poles. It decays as \(e^{-\pi |\Re t|/3}\), uniformly at large real ends, and for \(|\Im t|\le3/8\) has real part bounded below by a positive constant times the same decay (multiply by the conjugate denominator). Only D,O,I have nonzero drive. In general its real log on the contours is bounded above by \(C_n m\). For the sharp bound take +\[ + R=(3/2)\log n. +\] +This choice makes the weakest external drive, at the period edges, have order one: \(n e^{-2R/3}=1\). It also matches the pressure calculation in Section~\ref{c01-c1-bulk-calibration-and-puncture-exponent}: after removing its analytic correction \(c_*n^2e^{-L}\), the error is \(O(n^2e^{-4R/3})=O(1)\). The removed term itself has order \(\sqrt n\), so it cannot be included in the bounded error. + +Bend so that \(\chi=0\) except within fixed width of the edge \(\{-R/2,R/2\}\), using sufficiently large constant width to achieve the small slope. At \(x_i=r_i\) the main line term integrated to \(r_i\) is imaginary, and on going vertically to the shifted position it gives signed real part at least \(c_7 e^{-4|r_i|/3}\), since \(\xi_i a_i>0\) for active objects. Other images contribute \(O(e^{-4(R-|r_i|)/3})\); the bend changes the result also by at most \(O(1/n)\) per primitive, by the fixed width and the decay. Hence the real log drive bound here is +\[ + C m-c_8\sum_{i:\xi_i\ne0}\exp(4\,\mathrm{dist}(r_i,\mathrm{edge})/3). +\] + +Before resumming \(M\), these estimates, omitting common pressure and determinant factors, give (besides the \(-2\Re\varepsilon(b_1-b_2)^2\) log and factorial measures) bounds of the form +\[ + \exp[C_n m+C R-c_9\sum n_d^2-c_{10}(M-M_0)^2/R], +\] +where \(C_n\) can be taken constant in the sharp contour setting. Indeed the real part of the complex pole variance integral is bounded below (before its negative multiplier) by a constant times \(\|w_R-w_R^0\|_2^2-O(\sigma^2\|b\|_2^2)\), by the smoothing bounds. Also \(|\Im\mathcal S|\) is controlled by the complex average bound. Its cost multiplied by \(|M-M_0|\) is absorbed by a fraction of the count Gaussian and variance and box costs. The real tilt is bounded by completing a square, since \(M_0=n+O(m)\). + +These bounds prove absolute summability after the stated transformations, even over all values of the original count \(M\) (the sum \(b_1+b_2\) is already determined by hole and lower counts). They upgrade the termwise reduction to an assembled fixed-sector identity. More explicitly, on the real periods one can first perform the circle expansion and shifts as above separately per fixed \(M,b_1,b_2\) and initial lower data, moving contours term by term at fixed hole counts; disjoint bindings just convert factorials by the matching choices. The bounds apply absolutely including these terms. On straight contours in a complex strip of large real periods all the expressions continue analytically with locally uniform convergence by the non-sharp bounds. Thus the identities for each original count \(M\) continue there, including vanishing after summing a sector \(M>2n\). We can bend terms as stated; their sum over \(M\) with only the lower constraint \(M\ge\max_c g_c\) is exactly \(\mathcal N_n(e^{-L/2},s)\). Extending this count sum to all integers is a subsequent approximation, with the same count-independent prefactors and negligible error in the sharp setting (using the unresummed bounds). If \(m\) is at most a small fraction of \(n\), the omitted Gaussian tail costs order \(n^2/R\); otherwise keep part of the box cost, at least a constant times \(m^2/R\). Unlocalized factorial measures even with \(\exp(C m)\) sum to \(\exp(O(R))\), so either error is negligible. + +Indeed binding \(p\) indistinguishable pairs converts the factorials for the two counts to those for the residual counts and \(p!\); given the final counts they and the winding differences impose no new \(M\) constraint. Both error bounds here (relative to the common unsummed bulk factors) are negligible also against any \(\exp(-C R)\) on compact tilt ranges. Straight-contour holomorphy can first be used at fixed angular parametrizations, factoring the pure contact zeros and summing exponentially decreasing Fourier remainders locally uniformly for each fixed term, before continuing the torus as above. + +\paragraph{Count summation and localization.} +Poisson summation in \(M\) now gives the prefactor \((\pi/(a\varepsilon))^{1/2}e^{-s(M_0-n)}\) and a sum over integers \(l\) with phases of modulus one (\(M_0\) real). Its exponential combined with the pole log is +\[ + \exp\left[-c_0\int_\gamma(w_*-l-is/(2\pi))^2 dz\right]. +\] +Extract \(\exp(s^2/(4a\varepsilon))\). What is left costs in real log at least +\[ + c_{11}\|w_R-l\|_2^2-C m-C\sigma^2\|b\|_2^2 +\] +(as suppression). Indeed the real linear coefficient relative to the square for real arguments, including the differential \(1+i\chi'\), has absolute value bounded strictly below 1, since \(|\Im s|<\pi\). For any such coefficient \(t\), \(x^2+t x\ge c_{12}x^2-C\,\mathrm{dist}(x,\mathbb Z)\) on the real line. The smoothed real step is within \(Cb\) of integer values, and replacing it by the complex smoothing costs only the controlled small errors above and a fraction of the square. Box costs again absorb quadratic errors. + +Indeed (choosing the sign of \(l\)) the count transform's square exponent is \((s-i\mathcal S-2\pi i l)^2/(4a\varepsilon)=-c_0L(\bar w_*-l-is/(2\pi))^2\), completing the variance term. After the \(s^2\) extraction the real linear coefficient on replacing \(w_*\) by \(w_R\) is \(t=(\Im s+\chi'\Re s)/\pi\); slopes can be taken small depending on the prescribed compact tilt set. The additional replacement errors in the square and linear terms integrate against \(1+i\chi'\), and are bounded by a small fraction of \(\|w_R-l\|_2^2\) plus \(O(\sigma^2\|b\|_2^2+m)\). + +Finally use \(g_0(r)=2\log(2+\mathrm{dist}(r,\mathrm{edge}))\). Integration by parts against the jumps gives +\[ + \sum j_i g_0(r_i)=-\int(w-l)g_0'\,dr + \le\tfrac12 c_{11}\|w_R-l\|_2^2+C+C m . +\] +Only O has negative \(j\), and the external cost for O absorbs the wrong sign and leaves localization for it as well. Thus one keeps factors \(\exp(-g_0(r_i))\) on all objects, of bounded integral each. The unused variance penalty sums boundedly over \(l\). All defect sums with their bounded activities and factorials are now \(O(1)\). The Gaussian sum over \(b_1-b_2\) costs \(O(\sqrt R)\), offset together with the Poisson prefactor by the original \(2\pi/L\). This completes the bound, leaving only the bulk factors and the extracted harmonic twist. + +\subsection{Bulk calibration and puncture exponent}\label{c01-c1-bulk-calibration-and-puncture-exponent} + +The defect sum is bounded after extracting its explicit pressure, determinant and harmonic twist. We now compute these factors and compare them with Proposition~\ref{prop:pv:denominator}; this completes Theorem~\ref{thm:pv:puncture}. + +For the overall determinant, Euler summation gives +\[ + -\sum_{m>0}\log\det A(m\varepsilon)=L/4+O(1). +\] +Indeed the eigenvalues are \((2\cosh v-1)(\cosh3v\mp1)/\sinh4v\); the negative log determinant extends smoothly at zero, decays exponentially, and its integral on the positive line is \(\pi^2/8\), by integrating +\(2[\log(1-e^{-8v})-\log(1-e^{-6v})-\log(1-e^{-v}+e^{-2v})]\). +Bounded Euler error follows also on the slightly rotated rays by the derivative bound there. + +Put \(I=\int_0^\infty H_*(v)dv\). Poisson summation for the even extension shows +\[ + \varepsilon\left(\tfrac12 H_*(0)+\sum_{m>0}H_*(m\varepsilon)\right) + =I+c_* e^{-L}+O(e^{-4\Re L/3}) +\] +for an absolute constant \(c_*\). The transform at positive frequencies \(2\pi k/\varepsilon\) can use the same tilted line shift as above, here passing the first two simple poles at \(i\pi/4,i\pi/3\) (removable poles, if any, simply give zero), which gives this estimate including bounded imaginary parts of the period. For the bulk integral use \(y=e^{-v}\): +\[ + v H_*(v)=\frac{2y^2(1-y^3)^2}{(1-y+y^2)(1-y^8)} + =\frac{P(y)}{1-y^{24}} +\] +with symmetric coefficients \(P_j=P_{24-j}\), sum zero; at \(j<12\) the nonzero ones are \(P_2=P_3=2,\ P_5=P_6=-6,\ P_8=P_9=8,\ P_{10}=2,P_{11}=-6\). Integrate groupwise against \(dv/v\) (absolute convergence by the double zero of \(P\)). Pairing the resulting log products symmetrically about 12 gives +\[ + I=-\sum_{j<12} P_j\log\sin(\pi j/24),\qquad + \exp I=8D_0^3/81. +\] +Here one uses \(\prod_{l\ge0}(l+x)(l+1-x)/(l+1/2)^2=\sin(\pi x)\); e.g. log differentiation gives \(\pi\cot(\pi x)\) by its elementary partial fractions (residue integration on expanding half-integer squares), and the normalization is at \(1/2\). To simplify, the sines at 2 and 10 have product \(1/4\), at 5 and 11 product \((1+\sqrt2)/4\), at 3 and 9 product \(\sqrt2/4\). + +Multiply the analytic numerator \(\mathcal N_n(\mathfrak q,s)\) by \(\exp(-c_* n^2\mathfrak q^2)\). On \(|\mathfrak q|=n^{-3/4}\) the bounds proved above apply around the entire circle, with the pressure error after multiplication bounded in log. The correcting factor equals one at zero, so Cauchy's formula gives +\[ + \mathcal N_n(0,s)=\frac{1}{2\pi i} + \int_{|\mathfrak q|=n^{-3/4}} + e^{-c_*n^2\mathfrak q^2}\mathcal N_n(\mathfrak q,s) + \frac{d\mathfrak q}{\mathfrak q}. +\] +This bounds the original numerator \(S_{2n}(1)^2X_{n,n}(\kappa=e^s)\). The \(\kappa\)-free part of \(B_n\) times the bulk growth is exactly \(D_0^{-2n}(4D_0^2/9)^{2n^2}\), the squared exponential factor in the denominator bound. The remaining factors besides constants are bounded by \(\left|\exp[L/4+s^2/(4a\varepsilon)]\right|\). Since \(a=9/8\), \(\varepsilon=2\pi\alpha/L\), and \(\alpha=\pi/4\), their powers on this circle satisfy, uniformly on each compact tilt set, +\[ + \left|\exp[L/4+s^2/(4a\varepsilon)]\right| + \le C n^{3/8+2\Re(s^2)/(3\pi^2)},\qquad + \frac38-2\cdot\frac5{48}=\frac16. +\] +Thus division by the squared homogeneous Pfaffian gives, locally uniformly in \(|\Im s|<\pi\), +\[ + |E_n(s)|\le C,\quad + E_n(s)=X_{n,n}(\kappa=e^s)\,\exp[-(1/6+2s^2/(3\pi^2))\log n]. +\] +At \(s=i\pi/2\) this analytic function is exactly 1. Hence all these puncture polynomials as functions of \(\eta=2\cosh s\) are zero-free in a fixed disk about \(\eta=0\), with Taylor coefficients of each fixed order in \(\log X_{n,n}\) bounded by \(O_k(1+\log n)\) (Cauchy estimates for \(\log E_n\) near the normalization point). Also +\[ +X_{n,n}(1)=n^{1/6+o(1)} +\] +where 1 denotes \(\kappa=1\), thus fugacity 2. For the lower bound take any subsequence and a further locally uniform limit of \(E_n\) in the strip. It is nonzero at the normalization point and hence nonzero also at points \(s=i t\) with \(t>0\) arbitrarily small. There \(2\cos t<2\), so coefficient positivity gives the lower bound with exponent \(1/6-2t^2/(3\pi^2)\). The upper bound uses \(s=0\) directly. + +Here compact subsequential convergence follows e.g. from derivative bounds by Cauchy's estimates followed by a compact exhaustion and diagonal selection, with holomorphy of the limit. The same bounds at \(i\pi/2\) give a neighborhood where \(|E_n-1|<1/2\) uniformly, and \(\eta\) has a local analytic inverse parameter there since its derivative is nonzero. diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/introduction.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..0ff97954a466ea89d6dd2cb74d586504953b86a8 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/introduction.tex @@ -0,0 +1,236 @@ +\section{Critical bridge length and a polynomial polygon observable} +\label{sec:pv:introduction} + +A walk can cross a wide strip while making many returns to a much smaller +region. Its total crossing weight therefore does not determine its mean +length. We study this distinction for critical self-avoiding walks on the +honeycomb lattice. Our initial boundary point is fixed and the terminal +point is summed over the opposite wall. The main result is a mean-length +exponent of $4/3$ under this all-length measure. + +We use the honeycomb lattice dual to the equilateral triangular tiling of +side one. Its edges have length $1/\sqrt3$. A \emph{port} is the midpoint +of an edge cut by a tiling line. A path between ports visits distinct +honeycomb vertices, and its length $|\gamma|$ is their number. Its critical +weight is $x_*^{|\gamma|}$, where +\[ + x_*=(2\cos(\pi/8))^{-1}=(2+\sqrt2)^{-1/2}. +\] +The terminal half-edges have no additional weight. A strip of $h$ layers +has physical height $h\sqrt3/2$. With a bottom source at horizontal +coordinate zero, its attainable top ports lie in $\mathbb Z+h/2$. +Write $\mathcal B_h(e)$ for the paths confined to that strip from the +source to top port $e$, and put +\[ + b_h(e)=\sum_{\gamma\in\mathcal B_h(e)}x_*^{|\gamma|},\qquad + b_h=\sum_e b_h(e),\qquad + \ell_h=\sum_e\sum_{\gamma\in\mathcal B_h(e)} + |\gamma|x_*^{|\gamma|}. +\] +Thus $\ell_h$ is a first-length insertion, rather than a partition +function conditioned on one length. The bridge law assigns each such +path probability $x_*^{|\gamma|}/b_h$. + +\begin{theorem}[Strip mass and bridge length]\label{thm:pv:bridge} +As the integer layer height $h$ tends to infinity, +\[ + b_h=h^{-1/4+o(1)},\qquad + \ell_h=h^{13/12+o(1)},\qquad + \mathbb E_h|\gamma|=h^{4/3+o(1)}. +\] +All three sums are finite. These exponents are unchanged if the paths +are also confined to horizontal distance $h\log^2h$ from their initial +port. +\end{theorem} + +Throughout, $f(h)=h^{a+o(1)}$ means that $f(h)>0$ for all sufficiently +large $h$ and $\log f(h)/\log h\to a$. In particular, the theorem gives +subpower precision; it does not assert bounds by fixed multiples of the +displayed powers. Its last conclusion uses a logarithmically widening +strip, and is distinct from confinement to a fixed multiple of $h$. + +The same proof determines a finite free-boundary chord law. In a regular +lattice hexagon of integer side $R$, sum critical weights over paths +joining ordered boundary ports, retaining only paths of diameter at +least $R/10$. Reversal gives a second ordered path and leaves the +normalized law unchanged. Denote the resulting probability by +$\mathbb P_R$. + +\begin{theorem}[Hexagon visits]\label{thm:pv:hexagon} +The mass normalizing $\mathbb P_R$ is $R^{3/4+o(1)}$. Uniformly at +vertices in the concentric hexagon of side $R/3$, +\[ + \mathbb P_R(v\in\gamma)=R^{-2/3+o(1)}. +\] +The corresponding upper bound holds at every vertex, and +$\mathbb E_R|\gamma|=R^{4/3+o(1)}$. +\end{theorem} + +There is an intermediate strip law in which the distinction between +arches and bridges matters. An \emph{arch} returns to the wall where it +started. From one fixed bottom port, allow both arches and bridges and +condition the path to reach at least $\vartheta h$ layers above the +bottom, where $0<\vartheta<1/2$ is fixed. + +\begin{proposition}[Mixed exits]\label{prop:pv:mixed} +The conditioned event has unnormalized mass $h^{-1/4+o(1)}$ and +first-length mass $h^{13/12+o(1)}$. Hence its normalized mean length is +$h^{4/3+o(1)}$. Restricting horizontal distance to $h\log^2h$ preserves +these exponents. The total first-length mass of all exits, without the +height condition, is at most $h^{13/12+o(1)}$. +\end{proposition} + +The mixed mean follows directly from a finite vertex-visit identity. +The bridge mean requires an additional argument that transfers mass +from returning arches to crossing paths. It is this argument, rather +than normalization alone, that completes Theorem~\ref{thm:pv:bridge}. +Both theorems concern means in positive critical measures; fixing a +length or prescribing both ports asks a different question. + +\subsection{Critical crossing mass and related laws} + +Nienhuis's analysis of the dilute $O(n)$ model led to the honeycomb +critical point and the critical-exponent predictions associated with planar +self-avoiding walk~\cite{Nienhuis1982}. Duminil-Copin and Smirnov proved +the connective constant $\sqrt{2+\sqrt2}$ by a parafermionic observable +and its boundary cancellation~\cite{DuminilCopinSmirnov2012}. We use that +local cancellation, prove the particular finite flux identities needed +here, and keep the visited-vertex convention explicit. + +The sharper crossing-mass power $h^{-1/4}$ is recorded as a prediction +in \cite[Section~4]{DuminilCopinSmirnov2012}. Beaton, Bousquet-M\'elou, +de Gier, Duminil-Copin and Guttmann proved that the critical bridge +mass tends to zero~\cite{BeatonEtAl2014}. Glazman and +Manolescu~\cite{GlazmanManolescu2017} obtained a quantitative subsequence +bound, as well as invariance of boundary two-point functions in +columnwise rhombic half-planes with Yang--Baxter weights. Krachun and +Panagiotis proved a polynomial upper bound on the critical bridge +mass~\cite{KrachunPanagiotis2026}. These crossing-mass estimates do not +by themselves control a first-length insertion. In particular, the +all-length means considered here are distinct from displacement laws +for uniform walks of a fixed length. + +The companion paper \cite[Theorem~1.1]{DiskTransfer2026} proves the same +all-length bridge-mean exponent and retains it inside a fixed corridor +of sufficiently large width $Ch$. It also obtains bridge mass +comparable to $h^{-1/4}$. Our proof of the strip and hexagon results is +given independently below; we do not use the companion's mean theorem. + +\subsection{A polygon observable and its construction} + +The first-length insertion will be obtained by summing the mass of paths +through a marked vertex. The corresponding algebraic observable counts +surrounding polygons instead of paths. Identify horizontal translates by +$2n$, forming an infinite honeycomb cylinder, and mark two face centers +on one horizontal cut, separated by $n$ sites. Let $F_n(\eta)$ sum over +systems of pairwise vertex-disjoint unoriented simple polygons, each +separating the two marks. A polygon $P$ has weight $\eta x_*^{|P|}$, and +the empty system has weight one. Each polygon crosses the intervening +marked interval, so this is a polynomial in $\eta$; its coefficients are +convergent weighted sums over polygon shapes on the infinite cylinder. + +\begin{theorem}[Normalized separator pairing]\label{thm:pv:puncture} +For every compact set $K\subset\{s\in\mathbb C:|\Im s|<\pi\}$ there +is $C_K<\infty$ such that, for all $n\ge1$ and $s\in K$, +\[ + |F_n(2\cosh s)|\le + C_K n^{1/6+2\Re(s^2)/(3\pi^2)}. +\] +There is a disk about $\eta=0$, independent of $n$, in which $F_n$ has +no zeros. For each fixed $k$, the order-$k$ Taylor coefficient of +$\log F_n$ at zero is $O_k(1+\log n)$. At the positive fugacity two, +\[ + F_n(2)=n^{1/6+o(1)}. +\] +\end{theorem} + +To construct $F_n$, record at each of $N=2n$ edges on a cylinder cut +whether the edge is empty or carries either orientation of a path. +The oriented pieces carry local turn phases, and adding a row acts +linearly on the resulting three-state tensors. We construct a common +eigenvector of eigenvalue one whose entries are polynomials in +parameters attached to those edges. Dividing by its empty component +gives the normalized \emph{vacuum}: at the physical honeycomb parameters +its entries are the amplitudes of the half-cylinder path systems. +Pairing two such states with a twist between the marked faces cancels +nonseparating loops and assigns each separating loop weight +$\eta=2\cosh s$. + +The normalization $F_n(0)=1$ is essential. Both the finite color formula +for the numerator and the Pfaffian denominator have exponential growth +in $n^2$. The polynomial power survives only after those factors cancel +in the same normalization. Uniformity for complex $s$ also has a +positive-measure consequence: the zero-free disk controls polygon +incompatibilities when the cylinder is opened into the plane. + +This construction uses polynomial exchange equations and reductions at +special values of neighboring parameters. Di Francesco and +Zinn-Justin~\cite{DiFrancescoZinnJustin2005} used these ideas for a +multiparameter sum rule in the dense $O(1)$ model. Garbali and +Nienhuis~\cite{GarbaliNienhuis2017Ground,GarbaliNienhuis2017Sum} +developed $q$KZ and fusion recurrences and determinant expressions for +the dilute $O(1)$ model with open boundaries; they also discussed the +polynomial construction at loop weight zero. Here the periodic +twist-$i$ state, its empty-component normalization and its physical +separator pairing are constructed explicitly. The companion's disk +states represent the same homogeneous separator sum; +Section~\ref{c01-c1-the-balanced-puncture-observable} compares the common +scalar polynomial and physical observable. The inhomogeneous vectors +and normalized complex estimates here are proved separately. + +The local transfer weights belong to the dilute-loop and three-state +vertex-model setting of Zhou and Batchelor~\cite{ZhouBatchelor1997}. +Related finite recurrences and corner-transfer calculations for +restricted-height dilute models were developed by Warnaar, Pearce, +Seaton and Nienhuis~\cite{WarnaarPearceSeatonNienhuis1994}; Batchelor +and Yung~\cite{BatchelorYung1995} obtained integrable boundary weights +for the ordinary and special transitions of honeycomb polymers. +Bleher and Fokin's six-vertex analysis with domain-wall boundaries +likewise connects determinant partition functions to +orthogonal-polynomial asymptotics~\cite{BleherFokin2006}. These are +methodological comparisons, not inputs for the vacuum or complex-twist +estimate above. + +\subsection{Route through the proof} + +The proof first obtains two different mass estimates. After the flux +and convergence bounds in Section~\ref{c01-c1-planar-and-cylindrical-setup}, +Sections~\ref{c01-c1-the-row-matrices-and-two-fusions} +and~\ref{c01-c1-the-balanced-puncture-observable} construct the polynomial +vacuum and identify its pairing with $F_n$. Differentiating a row at +zero loop fugacity then gives the boundary mass $b_h=h^{-1/4+o(1)}$ +in Section~\ref{c01-c1-the-boundary-exponent}. This calculation is +independent of the bulk pairing asymptotics. + +Section~\ref{c01-c1-asymptotics-of-the-balanced-observable} proves +Theorem~\ref{thm:pv:puncture}. A finite color sum becomes contour +integrals, with a separate Pfaffian lower bound having bounded +logarithmic error. Periodizing the integrals in an +auxiliary spectral direction permits a circle-determinant expansion. +After that expansion, a positive Fourier decomposition controls the +remaining interacting variables. Their sum stays bounded after its +explicit pressure and twist factors are removed. This spectral period +is distinct from the physical cylinder circumference $2n$. + +Section~\ref{c01-c1-localization-of-surrounding-polygons} uses the +zero-free disk and six lattice rotations to localize polygons around +one face: their planar partition function at radius $R$ has exponent +$1/12$. The finite exploration identity in +Section~\ref{c01-c1-the-two-leg-mass} converts this nesting mass into +vertex-visit mass. Summing over $h$ layers and using horizontal +translation gives mixed-exit first-length mass $h^{1+1/12+o(1)}$. +In the hexagon, dividing visit mass by the free-boundary normalizer +$R^{3/4+o(1)}$ gives the visit probability $R^{-2/3+o(1)}$ and hence +the stated mean. + +For the strip, the mixed mass could still be carried by returning +arches. Section~\ref{c01-c1-transfer-to-bridges} compares intersecting +bridge--arch pairs with bridge--bridge pairs through a marked tripod. +A translation argument supplies sufficiently many bridge hits on each +deep arch, so bridges retain the same first-length exponent. Only then +do we divide by $b_h$, obtaining +\[ + \mathbb E_h|\gamma| + =\ell_h/b_h + =h^{13/12+1/4+o(1)}=h^{4/3+o(1)}. +\] diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/lateral-tail.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/lateral-tail.tex new file mode 100644 index 0000000000000000000000000000000000000000..7774f7634824c128d28a8e766c1cba74ed0d1247 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/lateral-tail.tex @@ -0,0 +1,62 @@ +\subsection{Exponential lateral tails in a strip} + +Before normalizing the finite chord laws, we control excursions that +travel sideways while remaining in a thin strip. This estimate also +justifies the subsequent lateral exhaustions with a length insertion. + +\begin{lemma}[Lateral tail]\label{lem:pv:lateral-tail} +Consider critical exits from a fixed bottom port in a strip of $h$ +layers, or in a convex tiling-line truncation of that strip. Let +$X(\gamma)$ be the largest horizontal distance of a visited vertex +from the source. There are absolute constants $C,c>0$ such that +\[ + \sum_{X(\gamma)\ge t}x_*^{|\gamma|} + \le C\exp\{-ct/(h+1)\}\qquad(t\ge0). +\] +The same bound, with a polynomial prefactor, holds with a first-length +insertion. In particular the contribution with +$X\ge h\log^2h$ to any first-length mass is smaller than every +fixed inverse power of $h$ as $h\to\infty$. +\end{lemma} + +\begin{proof} +For the right tail, take two further tiling cuts with outward normals +$(\sqrt3/2,1/2)$ and $(\sqrt3/2,-1/2)$, in coordinates along and +above the bottom side. Let $E_+,E_-$ be the sets of old exits lost when +the corresponding half-plane is imposed. Flux subtraction puts their +phased sums on the distinct rays +$e^{-i\sigma\pi/3}$ and $e^{-2i\sigma\pi/3}$: the turn from the +source to a new cut is fixed. Differently directed cuts do not share +ports. Since vertical coordinates range over $O(h+1)$, their +intercepts can be staggered by $O(h+1)$ and rounded to the tiling grid +so that +\[ + \{X_+>r+C(h+1)\}\subset E_-\subset E_+ + \subset\{X_+>r\}, +\] +where $X_+$ is maximum rightward displacement. All old exit weights +have phased real parts comparable to their positive weights by +convexity. Separation of the two rays implies that the absolute +weight of $E_+\setminus E_-$ is at least a fixed positive fraction +of the weight of $E_-$. This gives fixed-factor decay in each +increment $C(h+1)$, starting from the uniformly bounded total exit +mass. Reflection gives the left tail. + +In this subtraction, an exit is lost exactly when it visits a removed +vertex. A newly clipped exit attempts such a vertex along the new +cut; an old exit still valid contributes identically to the two flux +sums, even if its port belongs to both boundaries. Thus no boundary +term was omitted by the nesting of the cuts. The constants are +uniform in a convex lateral truncation, and monotone exhaustion +gives the full-strip statement. + +Finally a simple path in $h$ layers and horizontal range $X$ visits +at most $C(h+1)(X+1)$ vertices. The tail integral of the preceding +exponential bound consequently gives, after adjusting $C,c$, +\[ + \sum_{X(\gamma)\ge t}|\gamma|x_*^{|\gamma|} + \le C(h+1)(h+t+1)e^{-ct/(h+1)}. +\] +At $t=h\log^2h$ this is smaller than any fixed power of $h^{-1}$. +\end{proof} + diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/localization-figure.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/localization-figure.tex new file mode 100644 index 0000000000000000000000000000000000000000..2a88c82858e62e119471edd1f1d8715ee4bcda6d --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/localization-figure.tex @@ -0,0 +1,25 @@ +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[>=stealth, every node/.style={align=center}] +\node[text width=5.4cm] (physical) at (-3.35,2.3) + {Physical plane \(\mathbb C\)\\terminal radius \(|z|\)}; +\node[text width=5.4cm] (cone) at (-3.35,0) + {Cone sphere \(O\)\\\(A,B,C\): orders \(6,3,2\)\\\(q\): regular mark}; +\node[text width=5.4cm] (torus) at (3.35,2.3) + {Auxiliary torus \(\mathcal T\)\\doubled arc has homology \(u\)}; +\node[text width=5.4cm] (pillow) at (3.35,0) + {Pillowcase \(\mathcal P\)\\four branch corners\\labelled \(A,B,C,q\)}; +\draw[->] (physical) -- node[right, font=\small] + {quotient by\\\(p\Lambda\rtimes\langle\omega\rangle\)} (cone); +\draw[->] (torus) -- node[right, font=\small] + {double cover\\branched at four corners} (pillow); +\draw[<->] (cone.east) -- (pillow.west); +\node[font=\small] at (0,-1.1) + {Topological identification of marked spheres only}; +\end{tikzpicture} +\caption{The two covering constructions in the radius bound. The auxiliary +torus classifies the endpoint arc by a primitive vector \(u\); its physical +terminal radius is measured in the plane on the left. The identification +of the marked spheres does not identify their metrics or their covers.} +\label{fig:pv:localization-covers} +\end{figure} diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/localization.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/localization.tex new file mode 100644 index 0000000000000000000000000000000000000000..2f848d2e921d4b4769e52f773b9eba5a25569e69 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/localization.tex @@ -0,0 +1,440 @@ +\section{Localization of surrounding polygons}\label{c01-c1-localization-of-surrounding-polygons} + +We pass from the cylinder observable to polygons surrounding one face in the +plane. Write \(Z(R)\) for the planar gas at fugacity 2 of mutually disjoint +honeycomb polygons enclosing the face center 0 and contained within distance +\(R\) of it. Changing the disk cutoff by a fixed factor will not affect the +exponent. + +\begin{proposition}\label{prop:pv:planar-localization} +The disk partition function satisfies +\[ + Z(R)=R^{1/12+o(1)}. +\] +\end{proposition} + +There are two comparisons with the balanced cylinder of circumference +\(p=2n\). Two small disk systems, one near each marked face, give the upper +bound directly from \(F_n(2)=n^{1/6+o(1)}\). For the lower bound we must remove +the cylinder polygons that cannot be put into two disk systems of radius +\(p^{1+\delta}\), for fixed \(\delta>0\). These are the essential polygons +and the contractible polygons with oversized planar lifts. We will show that +their total single-polygon weight is bounded, so removing them costs only a +bounded multiplicative factor. The difficulty is the second family: a +contractible polygon may travel far before meeting a horizontal translate. + +\subsection{Weighted incompatibility from the zero-free disk} + +Let \(r_0>0\) be a common radius of a zero-free disk for \(F_n(\eta)\) about +\(\eta=0\). Available polygons on a fixed balanced cylinder have weights +\(w(P)=x_*^{|P|}\). Write \(I(P,Q)\) for incompatibility, including +\(I(P,P)=1\). + +\begin{lemma}\label{lem:pv:weighted-degree} +For every nonempty finite family \(\mathcal H\) of available polygons, +\[ + M_{\mathcal H}=\sum_{P\in\mathcal H}w(P),\qquad + d_{\mathcal H} + =\frac{\sum_{P,Q\in\mathcal H}w(P)I(P,Q)w(Q)}{M_{\mathcal H}} + \le r_0^{-1}. +\] +In particular, the bound does not depend on the circumference or the family. +\end{lemma} + +\begin{proof} +The formal logarithm of a hard-core gas is the connected-graph expansion; +see Fern\'andez and Procacci~\cite[Eqs.~(2.2)--(2.4)]{FernandezProcacci2007}. +For a tuple \((P_1,\ldots,P_k)\), its incompatibility graph has the distinct +tuple positions as vertices, even when polygons repeat. Its contribution to +the coefficient of \(\eta^k\) in \(\log F_n(\eta)\) is +\(\prod_iw(P_i)/k!\) times the alternating sum over its connected spanning +edge sets. This follows by expanding the products of \(1-I\) and grouping +components. Finite total polygon weight makes the sums absolutely finite +coefficientwise. + +The alternating sum, multiplied by \((-1)^{k-1}\), counts acyclic +orientations with unique source at vertex 1, by the fixed-root +chromatic-polynomial identity of Greene and +Zaslavsky~\cite[Theorem~7.3]{GreeneZaslavsky1983}. In particular, all tuples +contribute with the same sign. One can check the identity by +deletion--contraction at edges incident to the root: the alternating sum is +the linear chromatic coefficient, and the orientation count has the same +addition rule. Delete a root edge when its other endpoint has another +incoming edge; contract it otherwise. These give the corresponding rooted +orientations, with parallel edges simplified, and the recursion ends at the +isolated-root base cases. + +Every Hamilton path starting at vertex 1 supplies one such orientation, by +orienting all edges in its path order. Different paths give different +orientations, since the directed Hamilton path forces the total order. +Summing the \((k-1)!\) orders of the remaining tuple positions therefore gives +\begin{equation}\label{eq:pv:degree-coefficient} + \bigl|[\eta^k]\log F_n(\eta)\bigr| + \ge \frac1k + \sum_{P_1,\ldots,P_k\in\mathcal H} + \prod_{i=1}^k w(P_i) + \prod_{i=1}^{k-1}I(P_i,P_{i+1}). +\end{equation} +Thus a log coefficient bounds a weighted walk count in the incompatibility +graph, not merely the mass of individual incompatible pairs. + +For completeness, use the weighted inner product +\(\langle f,g\rangle_w=\sum_{P\in\mathcal H}w(P)f(P)g(P)\) and the symmetric +operator +\[ + (Af)(P)=\sum_{Q\in\mathcal H}I(P,Q)w(Q)f(Q). +\] +The sum in \eqref{eq:pv:degree-coefficient} is +\(\langle\mathbf1,A^{k-1}\mathbf1\rangle_w\). For odd \(k\), the spectral +moment inequality for the even power \(k-1\) gives +\[ + \langle\mathbf1,A^{k-1}\mathbf1\rangle_w + \ge M_{\mathcal H}d_{\mathcal H}^{\,k-1}. +\] +Now freeze both \(\mathcal H\) and its containing cylinder, and let odd +\(k\) tend to infinity. Since \(\log F_n\) is analytic in \(|\eta|0\). Suppose \(\mathcal U\) is a family of planar honeycomb +polygons enclosing 0 whose interiors are disjoint from their nontrivial +translates by a common pure lattice period of length \(p\). For all +sufficiently large \(p\), +\[ + \sum_{\substack{P\in\mathcal U\\ + \max_{z\in P}|z|>p^{1+\delta}}}x_*^{|P|}=O(1). +\] +The constant is independent of the family and the period direction. +\end{corollary} + +\begin{proof} +For the geometric argument, rotate coordinates to make the given period +horizontal; the six-rotation lemma is unchanged by this rotation. For the +weighted-degree argument, use an ordinary balanced cylinder in the original +lattice coordinates. Take a finite subset of oversized polygons, of mass +\(B\), and put it and all six lattice rotations into this cylinder with a +sufficiently large circumference, so that every polygon encloses one marked +face and not the other and has its original weight. This containing cylinder +need not have the given period, and its circumference may depend on the +finite subset. Lemma~\ref{lem:pv:weighted-degree} is uniform in that choice. + +For two chosen polygons \(P,Q\) with area of \(P\) at least area of \(Q\), +suppose all rotations of \(Q\) are compatible with \(P\). For each rotation, +its Jordan boundary and that of \(P\) are disjoint and enclose 0, so its +closed disk lies strictly inside the disk of \(P\), by the area ordering. +The interior of \(P\) can +therefore serve as \(U\) in Lemma~\ref{lem:pv:six-rotations}, giving +\(D_Q\le Cp\log(D_Q+2)\). For fixed \(\delta>0\) and large \(p\), this +contradicts \(D_Q>p^{1+\delta}\). + +The area-ordered pairs have total weight at least \(B^2/2\). Assign to each +one an incompatible rotation of \(Q\). A pair in the union with rotations +has at most six preimages, so its incompatibility numerator is at least +\(B^2/12\), while the union has mass at most \(6B\). The weighted-degree +bound gives \(B\le72r_0^{-1}\). Exhaustion by finite subsets proves the +claim. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:pv:planar-localization}] +On the cylinder of circumference \(p=2n\), essential polygons separating +the marked faces fall into two families, according to which cylinder end +is on the same side as each mark. Opposite families cannot coexist, and +translation by \(n\), swapping the marks, exchanges their weights. If a +finite subset of one family has mass \(B\), its union with its translated +image has mass \(2B\) and incompatibility numerator at least \(2B^2\). +Lemma~\ref{lem:pv:weighted-degree} gives \(B\le r_0^{-1}\). Thus the total +single-polygon weight of essential polygons is \(O(1)\). + +Every remaining cylinder polygon is contractible. Lift its disk and +translate the enclosed marked face to 0. Its interior is disjoint from all +nonzero translates by the horizontal period \(p\). For each of the two +marked-face families, Corollary~\ref{cor:pv:oversized-polygons} shows that +the polygons whose lifted radius exceeds \(p^{1+\delta}\) have total weight +\(O(1)\). + +For the upper comparison, put two independent planar disk gases of radius +\(cp\), for a fixed sufficiently small \(c>0\), around the two marked +faces. Their disks are disjoint on the cylinder, so all their polygons +qualify and +\begin{equation}\label{eq:pv:disk-cylinder-upper} + Z(cp)^2\le F_n(2)=n^{1/6+o(1)}. +\end{equation} +For the lower comparison, remove essential and oversized polygons from a +cylinder configuration. Ignoring every avoidance restriction involving the +removed polygons bounds their possible contribution at fugacity 2 by +\[ + \prod_{P\text{ removed}}(1+2w(P)) + \le \exp\left(2\sum_{P\text{ removed}}w(P)\right)=O(1). +\] +The retained polygons lift into the two planar disk systems of radius +\(p^{1+\delta}\). Dropping restrictions between these systems yields +\begin{equation}\label{eq:pv:disk-cylinder-lower} + F_n(2)\le C_\delta Z(p^{1+\delta})^2. +\end{equation} +The upper comparison gives exponent at most \(1/12\), and the lower one +gives exponent at least \(1/(12(1+\delta))\). Monotonicity fills the radii +between the cylinder scales. First let the radius tend to infinity for +fixed \(\delta\), then let \(\delta\downarrow0\), proving the proposition. +\end{proof} + +In the strip application below, a transverse lattice period of length +comparable to the strip height makes each polygon interior period-free. +Corollary~\ref{cor:pv:oversized-polygons} therefore supplies the same +oversized-polygon removal before Proposition~\ref{prop:pv:planar-localization} +is applied to the retained disk system. diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/model.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/model.tex new file mode 100644 index 0000000000000000000000000000000000000000..9b17b6f12821844df32b43a1931624443e6a4402 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/model.tex @@ -0,0 +1,83 @@ +\section{Planar and cylindrical setup}\label{c01-c1-planar-and-cylindrical-setup} + +We begin with the positive path bounds that justify all later limits of finite cylinders. The same flux identity also verifies the critical normalization of the vertex weight. + +Use the honeycomb lattice dual to the equilateral triangle tiling of side 1. Paths can begin/end at mid-edges ("ports"). A triangle visited has weight +\[ + x_*=(2\cos\lambda)^{-1},\qquad \lambda=\pi/8. +\] +Otherwise its weight is 1. Edges to/from terminal midpoints are traversed halfway; the paths counted are simple (in particular they cannot reverse immediately along an edge, nor repeat a lattice vertex (triangle)). Absolute SAW weights in what follows are products, without dividing out any partition function unless specified. A strip of height \(h\) (in layers, physical height \(h\sqrt3/2\)) has bottom and top on parallel tiling lines. Write \(A_h,b_h\) for the total weights from a given bottom port to bottom and to top respectively, inside the strip; the trivial path is excluded. All translated and reflected versions have the same totals. + +On cuts formed by tiling lines we stop at the port just before leaving the allowed triangles. Centers of triangles are never on these cuts. The half-edge from a center up to (but not including) a port is inside that triangle, and the port is on its side but not at a tiling vertex. Thus a tiling line cutting a walk meets it through a perpendicular edge, not at a turn, and different nonparallel cuts never share a port. + +The finite-domain identity below controls absolute path weights because convexity keeps all its phases in a fixed open half-plane. + +\begin{lemma}[Critical flux]\label{lem:pv:flux} +For an outer boundary port of a finite convex domain bounded by tiling lines, start a path inwards and let it exit anywhere. Then +\[ + \sum_\gamma x_*^{|\gamma|} e^{i\sigma W(\gamma)}=1,\qquad \sigma=3/8 +\] +where the sum is over the nontrivial walks to exit, \(|\gamma|\) counts vertices, and \(W\) is total turning angle along honeycomb edges. Every such walk has \(|W|\le\pi\), so its real phased weight is at least \(\cos(\sigma\pi)x_*^{|\gamma|}\). +\end{lemma} + +\begin{proof} +At each interior lattice vertex take the sum of the phased weights of paths to its adjacent ports signed + when pointing away from that vertex and - when pointing towards it there (the initial empty path at the source points towards the interior). Sum also over vertices; internal terms cancel. Locally, paths pointing in towards an unused vertex each have two extensions out, exactly balancing them (\(2x_*\cos(\sigma\pi/3)=1\)). + +The remaining negative terms occur when the vertex was used earlier; the tail describes one of the two ways round a polygon coming back towards that vertex on its unused edge. Pair them by reversing the traversal after the stem, thus changing the terminal port but keeping the same number of vertices. The stem lies outside the polygon, since it connects to the source without crossing it. Relative to the end of the stem just before the vertex, the two extra turning angles are \(\pm(2\pi-2\pi/3)\): on a positively oriented traversal the turn at the vertex would be \(+\pi/3\) (edge of the stem outside), and is replaced by \(-\pi/3\). Their phases therefore cancel. This uses only trivalence and the polygon turning-angle theorem. + +Paths to the outer boundary are simple arcs in a convex polygon, so \(|W|\le\pi\): close them by the counterclockwise boundary arc and subtract from \(2\pi\) its intervening corner turns and the two right angles. In particular all real parts compare to absolute weights by the constant \(\cos(\sigma\pi)>0\). +\end{proof} + +Arches to the right from a horizontal bottom side have \(W=-\pi\), those to the left \(W=\pi\), and opposite-side exits have \(W=0\). Passing from finite convex domains to an infinite strip gives the following normalization and preliminary bounds. + +\begin{corollary}[Strip flux and crude bridge bounds]\label{cor:pv:strip-flux} +The strip masses \(A_h,b_h\) are finite and satisfy +\begin{equation}\label{eq:pv:strip-flux} + b_h+\cos(\sigma\pi)A_h=1. +\end{equation} +Moreover, +\[ + A_{h+1}-A_h\le x_*^{-1} b_{h+1}^2, + \qquad \frac{c}{h+1}\le b_h\le1. +\] +\end{corollary} + +\begin{proof} +Cut the strip off at great lateral distances by tiling lines whose outward normals are rotated by \(\pm\pi/3\) from vertical upward. Lemma~\ref{lem:pv:flux} first gives finiteness of \(A_h,b_h\). To obtain equality in \eqref{eq:pv:strip-flux}, continue each side exit from its port into the part cut away, strictly increasing both height and outward projection, by a zigzag of vertical up and upward-outward edges to the top in the untruncated strip. The boundary edge crossing the cut is itself upward-outward. This costs a bounded factor at fixed \(h\), without repetition or unbounded multiplicity, since the first hit of the cut recovers the original exit. The weight of bridges so obtained tends to zero by finiteness, so the side-exit mass vanishes. + +An arch visiting the last layer uses a vertex immediately below a top port: it enters the layer by the unique downward edge of a lower triangle there, and next goes to an upper triangle. Split there into two bridges sharing that vertex and ending at that port, and drop their avoidance. The first bridge starts at the fixed source; for each endpoint of the first, the reverse of the second can go to any bottom port, again with total weight bounded by \(b_{h+1}\). The shared vertex accounts for the factor \(x_*^{-1}\), proving the difference bound. By \eqref{eq:pv:strip-flux}, +\[ + \frac1{b_{h+1}}-\frac1{b_h} + \le x_*^{-1}\cos(\sigma\pi)\frac{b_{h+1}}{b_h} + \le x_*^{-1}\cos(\sigma\pi). +\] +Summation gives the lower bound; the upper bound follows directly from the strip identity. +\end{proof} + +\paragraph{Criticality of the weight.} +The unfolding argument follows the bridge-decomposition method of Hammersley and Welsh~\cite{HammersleyWelsh1962}, as used on the honeycomb lattice in~\cite[Section~3]{DuminilCopinSmirnov2012}. We include the geometric normalization needed for our ports. A vertex-to-vertex weak bridge of positive span (intermediate heights between those of its endpoints inclusively) can be extended at start and end to bracketing layer cuts using only strictly more extreme vertices, at most one extra vertex per end. Hence by the strip upper bound the weighted sum for each span and fixed start is uniformly bounded. + +Count vertex SAWs of length \(L\) from a fixed start by splitting at a lowest vertex and considering the two weak half-space walks from there. On each, cut at the successive last maximum, last minimum of the remainder, and so on. This gives weak bridges of alternating sign with decreasing positive spans (the first two can be equal), in units of one third of a layer, with sum at most \(2L\) per half; there are no zero-height steps. There are \(O(\sqrt L)\) pieces and \(\exp(O(\sqrt L\log(L+1)))\) span lists. One can enumerate the two walks relative to the splitting vertex (two translation types suffice) and then the translate back to the given start is fixed. Thus multiply the uniform bounds allowing also a bounded weight factor per joint. SAW counts have exponential rate at most \(1/x_*\). + +A strictly smaller rate would make the vertex SAW susceptibility (weighted sum over all lengths at \(x_*\)) finite, hence by trimming port half-edges would make \(\sum b_h\) finite, contradicting the bridge lower bound. The rate exists, e.g. by submultiplicativity and lattice transitivity. Thus the weight \(x_*\) is critical. + +A flat cylinder identifies horizontal translates by an integer \(N>0\). The following fixed-width estimate will justify the row-transfer limits. Its constants may depend on \(N\). + +\begin{lemma}[Fixed-width cylinder convergence]\label{lem:pv:cylinder-decay} +Fix \(N\). Let \(b_h^{(N)}\) and \(A_h^{(N)}\) denote the cylindrical versions of the strip masses from a fixed bottom port, with terminal port summed on the top and bottom cut respectively. Then \(\sum_h b_h^{(N)}<\infty\), the limit \(A_\infty^{(N)}\) is finite, and there are \(C_N,c_N>0\) such that +\[ + b_h^{(N)}+A_\infty^{(N)}-A_h^{(N)} + \le C_N e^{-c_N h}. +\] +The same bounds apply when a terminal port is prescribed instead of summed. +\end{lemma} + +\begin{proof} +A cylindrical bridge lifts to a plane strip bridge. Cut strip bridges at each interior horizontal layer line crossed just once. The weights of positive-height indecomposable pieces summed over all heights in the plane, from a fixed bottom start, total at most 1 by concatenation and \(b_h\le1\). More explicitly, all sequences concatenate freely by translations. If the total restricted to some finite height bound \(H\) exceeded 1, their weight after \(k\) concatenations would grow exponentially in \(k\), but would be bounded by \(\sum_{h\le kH}b_h\le kH\). + +The total mass of pieces whose projections are simple on the cylinder is strictly less than 1: already in a single layer one can take a bottom-to-top bridge by following the internal zigzag arbitrarily far horizontally, and some such pieces repeat a vertex after projection. Summing the resulting geometric series bounds the total cylindrical bridge mass over all heights, after dropping avoidance between distinct pieces. Same-boundary arc weights are bounded by lifting and the uniform bound on planar arches. + +It remains to strengthen summability to exponential decay. The sequences \(b_h^{(N)}\) and \(A_{h+1}^{(N)}-A_h^{(N)}\) have rational generating functions. Reveal layer by layer, retaining the occupied ports and the connectivity of dangling ends and prescribed terminals; keep only diagrams of paths, rejecting a closed cycle as soon as it forms. The update has finitely many states and constant coefficients, including periodic identification in each layer. A complete arc below the current line can be retained in a completed state. This counts single arcs with fixed or free terminal along the final cut, and also confined arcs starting and ending on the first cut. Translation by a lattice period in the layer direction identifies successive updates. + +These sequences are nonnegative and summable. Rationality and absolute summability exclude poles on or inside the unit circle, so both sequences decay exponentially. Summing the second sequence over heights at least \(h\) gives the asserted same-boundary tail. Prescribing a terminal only reduces these positive sums. +\end{proof} diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/pairing.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/pairing.tex new file mode 100644 index 0000000000000000000000000000000000000000..f7703016de2a9639672cdf79adda514ed8fc705a --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/pairing.tex @@ -0,0 +1,215 @@ +\section{The balanced puncture observable}\label{c01-c1-the-balanced-puncture-observable} + +A contraction of the two vacua will count polygons separating two specified cylinder faces. We first establish that positive interpretation and then the interpolation data that characterize the contraction. + +Divide the ordered sites into a first block A of size \(n\) and remaining block B of size \(m=N-n\). Define +\[ + X_{n,m}(X;\kappa)=\Omega(X^{-1})^t D_{\kappa/\zeta}^{\rm A}\,\Omega(X) +\] +where the diagonal multiplies by \((\kappa/\zeta)^{\sum_{j\in{\rm A}}h_j}\), for \(\kappa\ne0\). +\begin{samepage} +\begin{proposition}[Physical separator pairing]\label{prop:pv:physical-pairing} +For two homogeneous blocks of equal size $n$, +\[ + X_{n,n}(1,\ldots,1;\kappa)=F_n(\kappa+\kappa^{-1}). +\] +Here the left side is the bilinear vacuum contraction just defined and the right side is the positive polygon polynomial from Section~\ref{sec:pv:introduction}. +\end{proposition} +\end{samepage} + +\begin{proof} +Work first at finite height with homogeneous sites. If \(|0\rangle\) +is the empty tensor, the lower and upper boundary states are +\[ + |u_H\rangle=\mathcal T_\zeta(t)^H|0\rangle, + \qquad + \langle v_H|=\langle0|\mathcal T_{\zeta^{-1}}(t)^H. +\] +Every empty-to-empty entry is 1 by loop cancellation. +Lemma~\ref{lem:pv:physical-eigenline} and the normalized left and right +eigenvectors therefore give +\[ + |u_H\rangle\longrightarrow\Omega(1,\ldots,1),\qquad + \langle v_H|\longrightarrow\Omega(1,\ldots,1)^t. +\] +The upper state is obtained by rows between the gluing cut and an empty +upper boundary; its twist is \(\zeta^{-1}\). This identifies the +reciprocal left eigenvector geometrically by the rank-one limit. +At nonsingular physical data we always use these analytic states, +including when their rational formulas are evaluated by a limit. + +The finite contraction \(\langle v_H|D_{\kappa/\zeta}^{\rm A}|u_H\rangle\) +glues matching occupied ports to make disjoint polygons. Internal gauge +factors telescope under this gluing. It has three +directed seams: from the lower boundary to \(f\), with weight +\(\zeta\); from \(f\) to the upper boundary, with weight +\(\zeta^{-1}\); and from the block-division face \(f'\) to \(f\) +along block A, with weight \(\kappa/\zeta\). See +Figure~\ref{fig:pv:seams}. Positive crossing is toward the left of a +directed seam. For an oriented simple polygon, let \(L(a)\) indicate +whether \(a\) lies on its left. The signed crossing number of a seam +\(a\to b\) is \(L(a)-L(b)\), so its three seam factors multiply to +\[ + \zeta^{L(-\infty)-L(f)} + \zeta^{-L(f)+L(+\infty)} + (\kappa/\zeta)^{L(f')-L(f)}. +\] +Here the two end indicators may be taken at the empty finite boundaries. +For a contractible polygon oriented with interior left, the turn factor +is \(i=\zeta\) and both end indicators are zero. For an essential +polygon oriented with upper end left, the turn factor is 1 and the +end indicators are zero and one. In both cases the total factor is +\[ + \zeta^{1-L(f)-L(f')}\kappa^{L(f')-L(f)}. +\] +The reverse orientation contributes its inverse. Equal indicators +therefore give \(\zeta+\zeta^{-1}=0\), whereas unequal indicators +give \(\eta=\kappa+\kappa^{-1}\). Exactly the separating polygons +survive, with their critical vertex weights. + +At each finite height the contraction is consequently a polynomial +in \(\eta\) with nonnegative coefficients. Its degree is at most +\(n\), since every separating polygon uses a distinct port of block A. +The coefficients increase as the height grows. At any fixed positive +\(\eta\), convergence of the two boundary states bounds the +contractions, and hence bounds every coefficient. Their finite limits +are exactly the infinite-cylinder polygon sums. Passing to the limit +coefficientwise proves the claimed identity for every \(\kappa\ne0\). + +\end{proof} + +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,every node/.style={font=\small}] + \draw[dashed] (-3,-1.8)--(-3,1.8); + \draw[dashed] (3,-1.8)--(3,1.8); + \draw (-3,-1.8)--(3,-1.8); + \draw (-3,1.8)--(3,1.8); + \draw[gray] (-3,0)--(3,0); + \draw[thick,->] (1.5,-1.65)--(1.5,-.12); + \draw[thick,->] (1.5,.12)--(1.5,1.65); + \draw[thick,->] (-1.5,0)--(1.35,0); + \fill (1.5,0) circle (1.8pt); + \fill (-1.5,0) circle (1.8pt); + \node[below right] at (1.5,0) {$f$}; + \node[below left] at (-1.5,0) {$f'$}; + \node[right] at (1.5,-.85) {$\zeta$}; + \node[right] at (1.5,.85) {$\zeta^{-1}$}; + \node[above] at (0,0) {block A: $\kappa/\zeta$}; + \node[above] at (0,1.8) {empty upper cut}; + \node[below] at (0,-1.8) {empty lower cut}; + \node[rotate=90] at (-3.3,0) {periodic identification}; + \node[rotate=90] at (3.3,0) {periodic identification}; +\end{tikzpicture} +\caption{The finite cylinder, cut open along the two identified dashed +sides. The middle line is the gluing cut; the arrows are directed twist +seams, not polygon segments. Block A follows the site order from $f$ +toward $f'$, while its seam is directed from $f'$ to $f$. The empty +upper and lower cuts recede to the cylinder ends in the limit.} +\label{fig:pv:seams} +\end{figure} + +\begin{remark}[Comparison of the scalar and physical pairings]\label{rem:pv:scalar-comparison} +For $N=2n$, reversing both the numerator and Vandermonde conventions +leaves the empty polynomial unchanged: +\[ + S_{2n}(X)=\prod_{i0\), +\[ + X_{n,n}(a_1,\ldots,a_n;Ta_n,\ldots,Ta_1)=(1+\eta)^n. +\] +\end{proposition} + +\begin{proof} +Separate block symmetry follows from exchange and its inverse, +transpose symmetry of \(F\), and charge conservation. Reciprocity +of \(S_N\) shows that \(S_N^2X_{n,m}\) is the contraction of two +polynomial vectors, each of degree at most \(N-1\) per site; this +gives the degree bound. + +For the extreme reductions, the complementary vanishing of nonzero +spins on the two sides leaves only the spin-zero limit in +Corollary~\ref{cor:pv:normalized-reductions}. At an adjacent cup the +right vector inserts \(C\). In the reciprocal vector, the projection +onto the same channel inserts \(C/r_0(T)\): exchanging its reversed +pair into cup order acts by \(F(T)=r_0(T)P_0\). Contraction cancels +the scalar because \(C^tC=r_0(T)=-n_0\), and the block twist +preserves the pair. At the other fusion, use +\(F(t^2)=n_0P_1\) and \(I^tI=n_0\,1\) in the same way. +The block twist on the image of \(I\) is the single-site twist. +These are rational identities on the generic domain. + +At the balanced specialization, successive nested cups in the right +vector force spins \(h\) on A with opposite mates in B and amplitudes +\(\prod q^{5h}\). Rotate B to the beginning in the reciprocal vector. +Its cups now face the other way, so in the original order its amplitudes +are \(\prod\zeta^h q^{-5h}\). Together with the block insertion, +each pair contributes \(\sum_{h=0,\pm1}\kappa^h=1+\eta\). +This gives the last specialization. + +For uniqueness, suppose two families agree at smaller sizes, and let +\(A(X)\) be their difference at \((n,m)\), multiplied by \(S_N^2\). +The zero and infinity reductions give respectively +\(x_i\mid A\) and \(\deg_{x_i}A\le2N-3\) at each site. The +cup and second fusion, in both orders, give four factors for each +intra-block pair. Thus +\[ + Q(X)=\left(\prod_{i=1}^Nx_i\right) + \prod_{P\in\{\mathrm A,\mathrm B\}} + \prod_{\substack{i] (4.2,.9)--(3.4,.9); + \draw[->] (.8,.9)--(-.1,.9); + \draw[->] (1.5,-.55)--(1.5,.25); + \draw[->] (2.7,1.55)--(2.7,2.35); + \node[right] at (4.2,.9) {$i$}; + \node[left] at (-.1,.9) {$k$}; + \node[below] at (1.5,-.55) {$j$}; + \node[above] at (2.7,2.35) {$l$}; + \draw (.48,0) arc[start angle=0,end angle=56.31,radius=.48]; + \node at (.76,.26) {$\theta$}; + \node at (2.1,.9) {$F(z)_{lk,ij}$}; +\end{tikzpicture} +\caption{Port and tensor order for the checked rhombus matrix. +The horizontal stream enters at the right and leaves at the left; +the vertical stream enters below and leaves above. The arrows specify +the positive spin directions, not a particular allowed path pairing. +The local phase convention is imposed on each pairing separately.} +\label{fig:pv:row} +\end{figure} diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/rows.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/rows.tex new file mode 100644 index 0000000000000000000000000000000000000000..c64938174be649cf77dfb8dee00a3bf9908bea5b --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/rows.tex @@ -0,0 +1,253 @@ +\section{Row matrices and polynomial vacua}\label{c01-c1-the-row-matrices-and-two-fusions} + +The algebra has two outputs. Commuting transfer rows and their simple +physical eigenline will supply the boundary susceptibility in +Section~\ref{c01-c1-the-boundary-exponent}. A polynomial representative +of that eigenline will supply the polygon pairing. We first establish +the local relations that both constructions use. + +The rhombus weights are the dilute-loop weights of +\cite[Eq.~(1.2)]{ZhouBatchelor1997}, divided by the empty weight. +Related dilute Yang--Baxter operators admit the two-colour braid--monoid +formulation of Grimm and Pearce~\cite[Sections~4.2, 4.5--4.6]{GrimmPearce1993}. +We specify the gauge here and verify the local and transfer identities +needed below. + +\subsection*{The checked tile and its rank reductions} + +Write \(q=e^{i\lambda},\ t=q^2,\ T=t^3,\ V=\mathbb C^{\{0,+1,-1\}}\). +We denote standard tensors by lists of spins in kets. The charge of a +tensor is the sum of its spins. Transpose throughout this construction +involves no conjugation. Put \(n_0=\sqrt2-1\) and define a vector \(C\) +and a map \(I:V\to V^{\otimes2}\), whose columns are \(I_h\): +\[ + C=|00\rangle+\sum_{h=\pm1}q^{5h}|h,-h\rangle,\quad + I_0=|00\rangle+x_*\sum_{h=\pm1}q^{4h}|h,-h\rangle,\quad + I_h=x_*(q^h|h,0\rangle+q^{-h}|0,h\rangle). +\] +Direct multiplication gives +\[ + I^tI=n_0\,1_V,\qquad C^tC=-n_0,\qquad C^tI=0. +\] +Thus \(P_1=II^t/n_0,\ P_0=-CC^t/n_0,\ P_2=1-P_1-P_0\) +are mutually orthogonal idempotents for this bilinear form. The +\emph{checked} matrix is +\[ + F(z)=\sum_{j=0}^2 r_j(z)P_j,\qquad + (r_0,r_1,r_2)=(a,b,ab),\quad + a=\frac{t^2-z}{t^2z-1},\quad b=\frac{T-z}{Tz-1}. +\] +It acts between ordered tensor slots of parameters \(u,w\) with \(z=u/w\), +and outputs in order \(w,u\). It preserves charge, is symmetric, and satisfies +\[ + F(1)=1,\qquad F(z)F(1/z)=1,\qquad + F(T)=CC^t=-n_0P_0,\qquad F(t^2)=II^t=n_0P_1. +\] +The last two specializations reduce two spin slots to a cup, or to one +spin slot embedded by \(I\). + +Here is its geometric realization. On a rhombus of angle \(\theta\) (bottom edge to side at the left, counterclockwise), set \(v=3\theta/8,\ z=e^{2iv}\), \(D=\sin(2\lambda+v)\sin(3\lambda+v)\). Allowed unoriented pieces have these weights: +\par\noindent\textbullet\quad empty 1; a single arc joining right to top or left to bottom has weight \(u_1=\sin(2\lambda)\sin(3\lambda-v)/D\); +\par\noindent\textbullet\quad a single arc round either other corner has weight \(u_2=\sin(2\lambda)\sin v/D\); opposite-edge connection \(o=\sin(3\lambda-v)\sin v/D\); +\par\noindent\textbullet\quad a pair of disjoint arcs at the first two corners has weight \(w_1=\sin(3\lambda-v)\sin(2\lambda-v)/D\); at the other two, \(w_2=\sin(v-\lambda)\sin v/D\). + +Orient all arcs, giving them the phase \(e^{\mathrm i W/4}\) for their turns (normal incidence, turns in \((-\pi,\pi)\)). Horizontal-stream labels are \(i,k\) at right, left respectively (+ pointing leftwards across), vertical-stream \(j,l\) at bottom, top (+ towards top); vacant means 0. Multiply by the further (gauge) factor \(e^{5\mathrm i v(i-k)/3}=e^{5\mathrm i v(l-j)/3}\). The resulting sum is \(F(z)_{l k,i j}\). Indeed the blocks on \((|h0\rangle,|0h\rangle)\) have diagonals \(u_1 e^{\mathrm i hv},u_1 e^{-\mathrm i hv}\) and off-diagonal \(o\); on \(|00\rangle,|+,-\rangle,|-,+\rangle\) the matrix is symmetric, first diagonal 1, the couplings to \(|h,-h\rangle\) are \(u_2 q^{2h}e^{\mathrm i hv}\), the latter diagonals \((w_1+t^{2h}w_2)e^{2\mathrm i hv}\), and their mutual coupling \(w_2\); on charge \(\pm2\) it is \(w_1\). These entries also follow by substituting \(r_j\) above (\(a,b=\sin(k\lambda-v)/\sin(k\lambda+v)\), \(k=2,3\), giving +\(F=w_1\,1+x_*^{-2} o II^t-x_*^{-1}\sin(2\lambda-v)\sin v\, CC^t/D\)). At \(v=\lambda\) the rhombus splits into two triangles and \(u_1=x_*,\ u_2=o=w_1=x_*^2,\ w_2=0\), exactly giving the disjoint honeycomb paths. Reflection gives the corresponding realization at \(v=2\lambda\). From the same entries the matrix at \(z=0\) has \(k\ge i\), at \(z=\infty\) has \(k\le i\). Its diagonal in stream order (\(k=i,l=j\)) at 0 has value 1 if either spin vanishes, \(t^5\) if they are equal nonzero, \(t^3\) if opposite nonzero; at infinity the inverse values. + +\input{sections/row-figure} + +For any of the nondegenerate real rhombi used geometrically the allowed pairings can indeed be drawn disjointly with these turns: perturb straight chords between ports by giving them short inward-normal segments at their endpoints (the chord points strictly into the first supporting line's half-plane and out towards the last line, so the two direction changes each have magnitude \(<\pi/2\)). In the two-triangle case the actual honeycomb connections have the same turns. + +\begin{lemma}[Local slide and interchange relations]\label{lem:pv:local-relations} +As rational identities, the maps from \(V\) and \(V^{\otimes2}\), respectively, +to \(V^{\otimes3}\) satisfy +\[ + F_{12}(z)F_{23}(Tz)(C\otimes1)=1\otimes C,\qquad + F_{12}(z)F_{23}(t^2 z)(I\otimes1)=(1\otimes I)F(tz). +\] +The checked matrices also satisfy the Yang--Baxter equation +\[ + F_{12}(y)F_{23}(zy)F_{12}(z) + =F_{23}(z)F_{12}(zy)F_{23}(y). +\] +\end{lemma} + +\begin{proof} +We first prove the slides by projecting onto the three channels. Write +\(m=(1,1+1/\sqrt2,1/\sqrt2)\), \(c=(1/n_0,\sqrt2,2+\sqrt2)\). +The needed small contractions are +\[ + (1\otimes C^t)(C\otimes1)=1,\quad + (1\otimes C^t)(I\otimes1)=(C^t\otimes1)(1\otimes I)=I^t,\quad + (1\otimes I)C=(I\otimes1)C, +\] +\[ + (1\otimes I^t)(I\otimes1)/n_0=\sum m_j P_j,\qquad + 1\otimes C-C\otimes1=(2+\sqrt2)(1\otimes I-I\otimes1)I. +\] +All are direct products of the tensors above; e.g. one can use \(C_{ab}=\delta_{a,-b}q^{5a}\) and \(I_{ab,h}=\delta_{h,a+b}J(a,b)\), with \(J(0,0)=1, J(h,-h)=x_*q^{4h},J(h,0)=x_*q^h,J(0,h)=x_*q^{-h}\) for \(h\ne0\) (invalid labels zero); the last identity reads +\[ + \delta_{ah} C_{bc}-C_{ab}\delta_{ch} + =x_*^{-2}\delta_{a+b+c,h} + [J(b,c)J(a,b+c)-J(a,b)J(a+b,c)] . +\] +Similarly \((1\otimes I^t)(I\otimes1)=x_*^2(1-CC^t)+II^t\) reduces entrywise at \(a+k=h+c\) to +\[ + J(a,h-a)J(h-a,c)=x_*^2(\delta_{ah}-\delta_{h,-c}q^{5(a+h)})+J(a,k)J(h,c), +\] +using \(q^8=-1,\ q+q^{-1}=x_*^{-1}\). +Consequently \(P_{j,12}(1\otimes C)=c_j P_{j,12}(1\otimes I)I\) (first project onto 0 and 1, then subtract). Project the first slide onto \(P_{j,12}\); the scalar required, relative to its right side, is +\[ + r_j(z)\big[-\delta_{j0}n_0 r_2(Tz) + -(r_0-r_2)(Tz)/n_0+(r_1-r_2)(Tz)/(n_0 c_j)\big]=1 . +\] +For the second slide sandwich between \(P_{j,12}\) and \(P_k\) on the source. Relative to \(P_{j,12}(1\otimes I)P_k\) the multipliers on the two sides are +\[ + r_j(z)\big[\delta_{j1}r_2(t^2 z)/m_k-\delta_{k1}c_j (r_0-r_2)(t^2 z)+m_k(r_1-r_2)(t^2 z)\big]=r_k(tz) +\] +whenever \(j=1\) or \(k=1\) or \(j=k=2\). The scalar equalities are substitutions of \(a,b\) above: with \(p=t^2\) use the common denominator \((pu-1)(Tu-1)\) for \((r_2,r_0-r_2,r_1-r_2)(u)\), with numerators \((p-u)(T-u),(p-u)(T+1)(u-1),(T-u)(p+1)(u-1)\). At \(u=Tz,pz\) the denominators are \(T(p-z)(T-z),(1+z)(1+tz)\) respectively; \(a(tz)=(p-tz)/(Tz-1)\), \(b(tz)=-(T-tz)/(z+1)\). In the remaining cases the tensor factor vanishes by the contractions. This proves the slides. Inverse and transpose slides follow as well, using unitarity as rational identities. + +Explicitly, denoting the three numerators just given by \(D_2,D_0',D_1'\), expansion using \(p^2=-1,\ t=(1+p)/\sqrt2,\ T=(p-1)/\sqrt2\) reads (indices ordered \(0,1,2\)) +\[ + \left[-\delta_{j0}n_0 D_2-D'_0/n_0+D'_1/(n_0 c_j)\right](Tz) + =T\begin{pmatrix}(T-z)(p z-1)\\(p-z)(Tz-1)\\(p z-1)(Tz-1)\end{pmatrix}_j , +\] +{\small +\[ + \left[\delta_{j1}D_2/m_k-\delta_{k1}c_j D'_0+m_k D'_1\right](pz) + = + \begin{pmatrix} + * & -t(1+tz)(p z-1)&*\\ + -p(1+z)(t-z)&p(p-z)(Tz-1)&T(p-z)(t-z)\\ + * &p(p z-1)(Tz-1)&T(t-z)(p z-1) + \end{pmatrix}_{j k} +\] +} +where \(*\) entries are not needed. + +For Yang--Baxter, clear the denominators and interpolate in \(z\). +For this use of degeneration values in dilute-loop Yang--Baxter +relations, compare~\cite[Section~4]{MorinDuchesnePearce2019}. +Both sides have degree at most four, and for generic \(y\) the five +points +\[ + 1,\quad T,\quad t^2,\quad T/y,\quad t^2/y +\] +are distinct. Equality at \(z=1\) is immediate. At \(z=T,t^2\), the +\(z\) factor is \(EE^t\), with \(E=C,I\), respectively; sliding \(E\) +on the left and \(E^t\) on the right gives equality. Here the +reversed-order slide before transposing is the inverse slide at +\(1/(zy)\). The cases \(zy=T,t^2\) follow from the preceding cases +at \((zy,1/y)\) by unitarity. This proves the rational identity. +\end{proof} + +\subsection*{Twisted rows and the physical eigenline} + +For \(X=(x_1,\ldots,x_N)\) and \(\kappa\ne0\), bring an auxiliary +slot of parameter \(z\) past the sites in their listed order and trace +its output against its input, weighting the identified spin \(h\) by +\(\kappa^h\). This defines the row transfer +\(\mathcal T_\kappa(z;X)\) on \(V^{\otimes N}\). Explicitly, for +input spins \(j_1,\ldots,j_N\) and output spins \(l_1,\ldots,l_N\), +\[ + [\mathcal T_\kappa(z;X)]_{\boldsymbol l,\boldsymbol j} + =\sum_{i_0,\ldots,i_{N-1}\in\{0,\pm1\}} + \kappa^{i_0}\prod_{a=1}^N + F(z/x_a)_{l_a i_a,i_{a-1}j_a},\qquad i_N=i_0. +\] +Thus site order follows the horizontal auxiliary stream, from right to +left in the tile convention of Figure~\ref{fig:pv:row}. + +\begin{lemma}[Row relations]\label{lem:pv:row-relations} +At fixed twist, the row transfers commute in their row arguments. +Exchanging adjacent sites by \(F\) intertwines the corresponding rows. +At generic sites, and hence meromorphically, +\[ + \mathcal T_{\kappa}(x_j)\mathcal T_{\kappa}(Tx_j)=1,\qquad + \mathcal T_{\kappa}(x_j)\mathcal T_{\kappa}(t^2x_j)=\mathcal T_{\kappa}(tx_j). +\] +For adjacent sites \((x,Tx)\), insertion of \(C\) intertwines the row +with the row omitting those sites. For \((x,t^2x)\), insertion of \(I\) +intertwines it with the row having the single site \(tx\). On charge zero, +\(\mathcal T_\kappa(0)=(1+\kappa+\kappa^{-1})1\). +\end{lemma} + +\begin{proof} +For commutation, insert a generic invertible crossing of two auxiliary +lines, move it through the sites by Yang--Baxter, and remove it in the +trace; charge conservation preserves the product twist. The same +identity intertwines a site exchange. + +For the two row fusion identities, take the indicated auxiliary pair +in that order. The second auxiliary crosses each site first. The slide +preserves its embedded \(C\) or \(I\) subspace throughout, and at the +site \(x_j\) the rank reduction followed by a checked identity sends +the whole auxiliary space into that subspace. The twisted trace is +therefore the fused trace: the product twist on \(C\) is 1, and on +the \(I\) image it is the single twist under the identification. +The inverse slides give the corresponding statements for a pair of +sites. Finally, at row argument zero, triangularity forces the auxiliary +spin to remain constant around the trace. The stated diagonal values +of \(F(0)\), together with charge zero, give +\(1+\kappa+\kappa^{-1}\). +\end{proof} + +\begin{lemma}[Physical eigenline]\label{lem:pv:physical-eigenline} +At \(X=(1,\ldots,1)\), \(z=t\), and \(\kappa=\pm i\), powers of +the row transfer converge exponentially to a rank-one projection whose +empty-to-empty entry is 1. In particular, eigenvalue 1 is simple and +all other eigenvalues have modulus less than 1. The same conclusions +hold at the reflected honeycomb point \(z=t^2\). + +Near either set of physical sites and twist, the rows have a common +analytic eigenline of charge zero. Its eigenvalue is rational in the +row argument and analytic in the other data near regular arguments. +At \(\kappa=\pm i\) this row eigenvalue is identically 1. +\end{lemma} + +\begin{proof} +The empty basis tensor has all spins zero. At the homogeneous point, +closed loops cancel after orientation summation: contractible loops +have turns \(\pm2\pi\), with trace twist net 1, while simple essential +loops have winding \(\pm1\), trace twist \(\kappa^{\pm1}\), and total +turn zero. The latter total turn follows from the exterior-angle sum +on the annular region between the curve and a distant parallel geodesic, +which is zero by triangulation and Euler's formula. The winding assertion +also follows by identifying the cylinder with a punctured plane. Gauge +factors telescope, so only prescribed boundary-to-boundary arcs remain. + +At least one crossing arc gives an exponentially small contribution +by the finite-width cylinder bounds in Lemma~\ref{lem:pv:cylinder-decay}. +With no crossing arc, +the systems along the two cuts factor in the limit into half-cylinder +amplitudes. Their error is exponentially small, since a system preventing +factorization must reach halfway across the cylinder height. There are +only finitely many boundary pairings and orientations, and dropping +mutual avoidance bounds each of them by products of arc weights. This +gives the rank-one limit. Its empty-to-empty entry is 1 because every +nonempty closed system cancels already at finite height. Reflection +gives the same argument at the other honeycomb realization. + +Finite-dimensional perturbation of \(\mathcal T_\kappa(t)\) now gives +a simple analytic eigenline near the physical data. Commutation makes +it a common eigenline, and charge conservation keeps it in charge zero. +For fixed sites and twist its row eigenvalue is rational in \(z\). + +To identify this eigenvalue at \(\kappa=\pm i\), first take real nearby +spectral angles \(z=e^{2iu},x_j=e^{2i\alpha_j}\). Lay the rows using +unit rhombus side vectors with directions \(8u/3\) and bottom vectors +with directions \(8\alpha_j/3\). The slightly zigzag spatial boundary +repeats by translation and projects monotonically onto the direction +perpendicular to the row side vector, so the rows do not overlap. +Loop turns and twists still cancel. The gauge factors telescope because +\(v=u-\alpha_j\) and \(i-k=l-j_{\rm spin}\) at every crossing: the +\(u\) term sums along auxiliary strands and the site-angle term along +site strands. Hence every empty-to-empty power remains 1. + +Analyticity extends this equality to complex nearby data. With \(z\) +also near \(t\), all other eigenvalues remain strictly inside the unit +disk and the empty entry of the near-1 projection is nonzero. The +constant power entry therefore forces the distinguished eigenvalue +and that projection entry to equal 1. Rational continuation in \(z\) +finishes the proof. +\end{proof} diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/tripod-figure.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/tripod-figure.tex new file mode 100644 index 0000000000000000000000000000000000000000..84cea49bbbcd8d99d10645447397cbbc9bbe22d9 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/tripod-figure.tex @@ -0,0 +1,31 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[scale=.9, every node/.style={font=\small}] + \draw[thick] (-.4,0)--(8.4,0); + \draw[thick] (-.4,3.4)--(8.4,3.4); + \node[below] at (1,0) {$b$}; + \node[below] at (7,0) {$c'$}; + \node[above] at (4,3.4) {$a$}; + \draw[very thick,blue!60!black] (1,0) + .. controls (1.2,1.3) and (2.6,.8) .. (4,1.5); + \draw[very thick,blue!60!black] (4,1.5) + .. controls (5.5,2.1) and (6.6,.6) .. (7,0); + \draw[very thick,red!65!black] (4,3.4) + .. controls (4.8,2.8) and (3.5,2.1) .. (4,1.5); + \fill (4,1.5) circle (2.5pt); + \node[left] at (3.9,1.65) {center}; + \fill (2.48,.99) circle (2pt); + \node[below] at (2.5,.91) {$v$}; + \node[blue!60!black] at (6.7,1.7) {spectator}; + \node[red!65!black] at (6.4,2.8) {first-contact prefix}; + \draw[->] (1.5,-.5)--(2.5,-.5); + \node[below] at (2,-.5) {positive boundary order}; +\end{tikzpicture} +\caption{A contact tripod, shown schematically. The arch from $b$ to +$c'$ is the spectator and a prefix from the opposite port $a$ reaches +it for the first time at the center. The center is weighted once. +Reassigning the active arm gives the two bridge-spectator terms. +The vertex $v$ may lie on any arm or at the center; the restrictions +in the proof keep these cases separate.} +\label{fig:pv:tripod} +\end{figure} diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/vacuum.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/vacuum.tex new file mode 100644 index 0000000000000000000000000000000000000000..11c0e62fd36d90ac7095f91cc3a07605dc1ddf8a --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/vacuum.tex @@ -0,0 +1,388 @@ +\subsection{Polynomial vacuum}\label{c01-c1-polynomial-vacuum} + +Polynomial exchange and fusion constructions occur in the dense and dilute $O(1)$ models~\cite{DiFrancescoZinnJustin2005,GarbaliNienhuis2017Ground,GarbaliNienhuis2017Sum}. We construct the present normalized vacuum directly by polynomial interpolation and determine its empty component. Its Pfaffian formula supplies the denominator of the polygon observable. + +\begin{proposition}[Polynomial vacuum]\label{prop:pv:vacuum} +Use twist \(\zeta=t^2=i\). Let \(\rho X=(x_N,x_1,\ldots,x_{N-1})\), and \(\Pi_\kappa\) the corresponding rotation of spin slots with factor \(\kappa^{h_N}\) from the last (now first) spin. We construct a charge-zero vector polynomial \(\Psi_N(X)\) with +\[ + F_{i,i+1}(x_i/x_{i+1})\Psi(X)=\Psi(s_i X),\qquad + \Pi_\zeta\Psi(X)=\Psi(\rho X) +\] +(\(s_i\) adjacent transposition), homogeneous of degree \(\binom N2\), +with degree at most \(N-1\) in each variable. For a spin component +\(\Psi_{N,\boldsymbol h}\), the refined bounds are +\[ + h_j=+1\ \Longrightarrow\ \deg_{x_j}\Psi_{N,\boldsymbol h}\le N-2, + \qquad + h_j=-1\ \Longrightarrow\ x_j\mid\Psi_{N,\boldsymbol h}. +\] +It has the adjacent-pair reduction +\[ + \Psi_N(\ldots,a,Ta,\ldots)=s_C(a;Y)\ \iota_C \Psi_{N-2}(Y), + \qquad + s_C(a;Y)=(1+T)a\prod_{x\in Y}(x-t^5a)(x-t^6a), +\] +where \(Y\) is the remaining list and \(\iota_C\) inserts a \(C\) tensor in the indicated positions; and at last parameter zero its projection onto last spin zero is \((\prod_{j\alpha_{i+1}\) in the earlier embedding. From left to +right the upper boundary has site steps of directions +\(8\alpha_{i+1}/3,8\alpha_i/3\). Adding the rhombus above those +steps exchanges their order, entering from right and bottom with +parameters \(x_i,x_{i+1}\), without crossing the twist seam. +With empty final output, all gauge factors telescope and all loops +cancel even at finite height. The added crossing therefore takes one +normalized half-cylinder state to the other with no scalar factor. +Comparing with polynomial exchange gives \(S_N(s_iX)=S_N(X)\). +This extends polynomially from the open set of real angles. + +\begin{proposition}[Empty component]\label{prop:pv:empty-component} +Put \(D(a,b)=a^2+\sqrt2 ab+b^2\) and +\(K(a,b)=(a^2-b^2)/D(a,b)\). The symmetric empty component is +\[ + S_N(X)=\prod_{i0$ such that +\[ + c\sum_{f\sim v}Z_D(f)\le T_D(v)\le C\sum_{f\sim v}Z_D(f). +\] +If $D$ is cut out by admissible tiling lines in a convex polygon and +$A_D(v)$ is the positive ordered excursion weight through $v$, then +\[ + \cos(3\pi/8)A_D(v)\le T_D(v)\le A_D(v). +\] +\end{proposition} + +\begin{proof} +At a fresh vertex reached along one edge, take either nonbacktracking +continuation with weight $x_*e^{i\sigma\pi/3}$ or +$x_*e^{-i\sigma\pi/3}$. Their sum is one. Stop at an exit port or just +before choosing a continuation at the first repeated vertex. The +exploration tree is finite and replacing a node by its children +preserves total weight. A prefix stopped on first arrival at a vertex +has not yet taken its weight or its turn. + +Start at a boundary port. A first repeat consists of a simple stem +followed by a polygon attached at its end. The stem is outside the +polygon: otherwise the outside neighbor of its starting port would +be inside that polygon, contradicting filling. The two orientations +of the polygon add turns $\pm4\pi/3$ relative to the stem. Their +phase sum is $2\cos(4\pi\sigma/3)=0$. This pairing also cancels after +restricting to histories that have visited $v$, because reversal +preserves their vertices. Continuing a first-arrival prefix at $v$ +has total terminal weight one. Hence the sum of first-arrival prefixes +equals the excursion weight through $v$. Summing starting ports gives +$T_D(v)$. + +Now start at $v$ along each of its three edges, with initial weight one +and with neither weight nor turn at $v$. Stop at an exit, a return to +$v$, or the first repeat of another vertex. The three explorations +have total weight $3$. Their exit terms reverse the prefixes just +counted, so their sum is the conjugate of $T_D(v)$ and hence $T_D(v)$. +A return to $v$ traces a polygon $P$ through it. Its turn omits exactly +$\tau_v(P)$ from the total $2\pi$, and its weight omits exactly one +factor $x_*$. The two orientations therefore contribute +\[ + 2x_*^{|P|-1}\cos\big(\sigma(2\pi-\tau_v(P))\big). +\] + +For a first repeat away from $v$, let $P$ be the attached polygon. If +$v$ is outside $P$, exterior-stem cancellation applies. If $v$ is +inside, the unused edge at the attachment points into the polygon. +Arriving from the stem replaces the counterclockwise turn $-\pi/3$ +there by $+\pi/3$. The two traversals add turns $\pm8\pi/3$, +both with phase $-1$, and contribute $-2x_*^{|P|}$ times the stem +weight. For fixed $P$, these stems are exactly paths from $v$, with +no weight or turn there, to boundary ports of $P^\circ$. Filling gives +$P^\circ\subset D$; planarity shows that an edge leaving its interior +ends on $P$. Completing the last half-edge therefore introduces no +extra weighted vertex. Prefix reversal in this smaller filled set +gives stem sum $T_{P^\circ}(v)$. The terminal balance is +\[ + 3=T_D(v)+\sum_{P\ni v}2x_*^{|P|-1} + \cos\big(\sigma(2\pi-\tau_v(P))\big) + -\sum_{v\in P^\circ}2x_*^{|P|}T_{P^\circ}(v), +\] +which proves the recursion. + +If $P$ encloses one of the three faces incident to $v$, its turn there +is $+\pi/3$; if it encloses two, the turn is $-\pi/3$. The omitted-turn +angles are $5\pi/3$ and $7\pi/3$. Set +\[ + c_1=-\cos(5\pi/8)>0,\qquad c_2=-\cos(7\pi/8)>0. +\] +Iterate the recursion. It terminates since a strict interior has fewer +vertices than its container. The result sums outer-to-inner chains +of disjoint polygons strictly enclosing $v$, with factor $2x_*^{|P|}$ +for each. In the final interior $D'$ the remaining multiplier is +\[ + 3+\frac2{x_*}\sum_{\substack{Q\subset D'\\v\in Q}} + c_{m_Q}x_*^{|Q|},\qquad m_Q\in\{1,2\}. +\] +This positive expansion proves both positivity and filled-set +monotonicity: inclusion only adds eligible chains and through-polygons. + +The same strict chains enumerate $\sum_{f\sim v}Z_D(f)$. A polygon +strictly enclosing $v$ encloses all three incident face centers. A +polygon through $v$ encloses exactly $m_Q$ of them. Conversely, a +polygon enclosing an incident face has $v$ in its interior or on its +boundary, because the open face does not meet the polygon and +approaches $v$. A vertex-disjoint system has at most one polygon +through $v$, and it must be innermost. Thus its final multiplier is +\[ + 3+2\sum_{\substack{Q\subset D'\\v\in Q}}m_Qx_*^{|Q|}. +\] +The two ratios $c_m/(x_*m)$ are fixed positive numbers. Termwise +comparison proves the incident-face bounds. + +Finally close a convex-domain excursion by the complementary +counterclockwise boundary arc. Its turn is the lifted change of +outward boundary normal from the initial port to the terminal one, +minus $\pi$. That change lies in $[0,2\pi]$, so $|W|\le\pi$. +Each phased real part is therefore between $\cos(3\pi/8)$ and one +times its positive weight. Summation proves the last assertion. +\end{proof} + +The proposition estimates visits without choosing the wall on which +the excursion ends. We first apply it to hexagon chords and mixed +strip exits; only afterward do we isolate bridges. + diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/visits.tex b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/visits.tex new file mode 100644 index 0000000000000000000000000000000000000000..e7f45a5c9225ae45745dc09c6d2d049187bd5f85 --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/build/sections/visits.tex @@ -0,0 +1,122 @@ +\subsection{Macroscopic chords in a hexagon}\label{sec:pv:hexagon-proof} + +For specificity take regular lattice hexagons of integer side \(R\to\infty\). Say an arc is macroscopic when its diameter is at least \(R/10\). Their absolute critical mass over all boundary sources is +\[ + R^{3/4+o(1)}. +\] +For the lower bound use sources in the central third of one side. Exits not to the initial bottom side all have macroscopic reach. Flux and comparison of bottom returns with arches in a containing flat strip of \(O(R)\) layers give non-bottom mass at least \(b_{O(R)}\). + +For the upper bound per source, impose a cut parallel to its starting side at height $h=\lfloor R/\log^2R\rfloor$. Flux subtraction bounds the mass of exits removed by this cut by a constant times $b_h$, since its new exit prefixes are strip bridges. A retained macroscopic path must reach horizontal distance comparable to $R$, so Lemma~\ref{lem:pv:lateral-tail} bounds its weight by $Ce^{-c\log^2R}$. Summing over the $O(R)$ boundary sources proves the upper normalizing exponent. + +Under the normalized macroscopic arc weights, bulk vertices (say in the central third) now have visit probability +\[ + R^{-2/3+o(1)}. +\] +Indeed all visiting paths there qualify, and the surrounding-polygon sums in Proposition~\ref{prop:pv:visits} lie between disk sums at comparable radii. Proposition~\ref{prop:pv:planar-localization} gives their exponent. The same upper bound holds throughout the hexagon. Summing the probabilities yields the mean visited mass, or length, +\[ + \mathbb E |\gamma|=R^{4/3+o(1)} . +\] + +\subsection{A fixed-source strip version}\label{c01-c1-a-fixed-source-strip-version} + +There is also an annealed dimension statement with one actual boundary point fixed. In the full planar strip of \(h\) layers take all the critical exits from a fixed bottom port (thus bridges to the top \textbf{or} arches back to bottom), conditioned to get at least one quarter of the way up. Then +\[ + \mathbb E |\gamma|=h^{4/3+o(1)} . +\] +Changing this to any fixed positive fraction strictly below one half gives the same result. The exit mass of this ensemble before normalization is \(h^{-1/4+o(1)}\): lower by the bridges, upper for the arches by the height-tail bound. This conditioning could also confine horizontal range to \(h\log^2 h\) about the start without changing the exponent. For bridges alone the argument gives the mean upper bound \(h^{4/3+o(1)}\) (and hence the usual in-probability upper-exponent bound), not by itself the matching bridge lower bound. + +The pointwise estimate behind this statement will also be needed for the +bridge lower bound. It sums over all boundary positions, rather than +fixing a source. + +\begin{lemma}[Strip visit weights]\label{lem:pv:strip-visits} +Let $A_{S_h}(v)$ be the positive weight of ordered strip excursions +through $v$, with both endpoints free on the two horizontal boundaries. +Uniformly over strip vertices, +\[ + A_{S_h}(v)\le h^{1/12+o(1)}. +\] +For every fixed $\delta>0$, uniformly over vertices at least +$\delta h$ layers from both horizontal boundaries, +\[ + A_{S_h}(v)=h^{1/12+o(1)}. +\] +The same upper bound holds for the ordered excursion weight $A_D(v)$ +in every convex lateral truncation $D$, including excursions with +lateral endpoints. Its lower bound holds uniformly when, in addition +to the horizontal clearance above, $v$ is at distance at least +$\delta h$ from every lateral side. +\end{lemma} + +\begin{proof} +Convexly truncate the strip at lateral distance $W$ on both sides using +tiling lines of the two slanted side orientations. For a face within +bounded distance of the strip, fugacity-$2$ systems using strip vertices +and surrounding that face have sum at most $h^{1/12+o(1)}$, uniformly in +the truncation. Indeed every enclosing polygon interior lies in the +slab and is disjoint from its translates by nonzero multiples of +$p\omega$, for a sufficiently large integral $p\asymp h$. +Corollary~\ref{cor:pv:oversized-polygons} applies to this transverse +period: its geometric argument rotates the period direction to +horizontal, while its degree bound uses a sufficiently large balanced +cylinder containing any finite test subset and all its rotations. Loops of +radius greater than $p^{1+\alpha}$ therefore cost at most a bounded +exponential factor. The remaining loops are controlled by +Proposition~\ref{prop:pv:planar-localization}. First let $h\to\infty$ +for fixed $\alpha>0$, then let +$\alpha\downarrow0$. For faces at least a fixed multiple of $h$ from +the walls and the lateral ends, an inscribed disk gives the matching lower +exponent. + +Proposition~\ref{prop:pv:visits} now gives both estimates in every +finite convex truncation. We also need a tail bound for its side +sources. Use the two slanted tiling cuts facing toward the far part of +the strip, with both outward normals pointing laterally that way. +Their distinct angles from the initial inward direction lie strictly +between $-\pi$ and $\pi$, so the separated-ray argument of +Lemma~\ref{lem:pv:lateral-tail} bounds the weight reaching lateral +distance $t$ by $Ce^{-ct/(h+1)}$. Displacement toward the near end is +limited by $O(h+1)$, and length is bounded by +$C(h+1)(1+\text{lateral span})$. Thus each side source has bounded +length mass at fixed $h$, uniformly in the truncation width. + +Let $W\to\infty$ at fixed $h$. Excursions between horizontal boundary +ports exhaust the strip excursions. For a fixed marked vertex, paths +from the $O(h)$ distant side ports have vanishing weight by the same +tail bound. This proves the stated full-strip estimates. +\end{proof} + +We finish Proposition~\ref{prop:pv:mixed} by averaging over horizontal +sources. In a finite truncation $D$, counting every visited vertex gives +the exact identity +\[ + \sum_{v\in D} A_D(v) + =\sum_{\gamma\text{ an ordered excursion in }D} + |\gamma|x_*^{|\gamma|}. +\] +Divide by horizontal width and let that width tend to infinity at fixed +$h$. Lemma~\ref{lem:pv:strip-visits} bounds the result above by +$h^{13/12+o(1)}$. A central band of $\asymp h$ layers supplies the same +lower exponent. For the ensemble conditioned to reach $\vartheta h$ +layers above its starting wall, where $0<\vartheta<1/2$ is fixed, +choose this band symmetric about the midline and more than +$\vartheta h$ layers from either wall. Every +path contributing a visit to that band then meets the conditioning +event, from either horizontal starting side. + +The slanted side sources disappear in this average: there are only +$O(h)$ of them, and the tail estimate in the preceding proof bounds +each one's length mass at fixed $h$. Their contribution divided by +width therefore vanishes. + +Each horizontal wall has one source port per unit length. Translation +invariance and top--bottom symmetry therefore identify the limiting +average with twice the fixed-source length weight. One can first +restrict to bounded path range, pass to +the large-width limit, and then remove that restriction using the +exponential tails. Thus the conditioned ensemble has unnormalized +length weight $h^{13/12+o(1)}$. Its mass is $h^{-1/4+o(1)}$ by +Proposition~\ref{prop:pv:boundary}, giving the asserted mean. The same +tails make the discarded first moment at horizontal range +$h\log^2 h$ smaller than every fixed power of $h$, proving the optional +confinement statement. diff --git a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c172cd5e481963841f305c6c9790d0b41cb8d6df --- /dev/null +++ b/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:69885ae7308d1f03e44116c1c26114005825913669e398ee388a5d06c2e579b4 +size 767701 diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/README.md b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..5d41f6d1a5bb79ba867a44de6c402bfebc8386eb --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [Positive Metric Entropy for the Standard Map at Large Parameters](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026, + author = {{OpenAI}}, + title = {{Positive Metric Entropy for the Standard Map at Large Parameters}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf}{OAI:Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/figures/bridge-sweep.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/figures/bridge-sweep.tex new file mode 100644 index 0000000000000000000000000000000000000000..923638fad6eceb9b7ee05636670a466879e06d56 --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/figures/bridge-sweep.tex @@ -0,0 +1,37 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=0.82cm,y=0.82cm,>=Latex,font=\small] + \definecolor{leafblue}{RGB}{28,75,118} + \definecolor{sweeporange}{RGB}{167,82,36} + \draw[gray!65] (0,0) rectangle (7,4.5); + \draw[gray!65,->] (0,0) -- (7.35,0) node[right] {$x$}; + \draw[gray!65,->] (0,0) -- (0,4.85) node[above] {$y$}; + \draw[gray!55,densely dotted] (0,2.25) -- (7,2.25); + \node[left] at (0,2.25) {$y_c$}; + \draw[gray!60,densely dashed] (5.4,0) -- (5.4,4.5); + \node[below] at (5.4,0) {$u$}; + \draw[sweeporange,thick] + (0.2,1.17) .. controls (2.5,1.26) and (4.6,1.43) .. (6.8,1.51); + \draw[sweeporange,thick] + (0.2,2.02) .. controls (2.5,2.13) and (4.6,2.40) .. (6.8,2.46); + \draw[sweeporange,thick] + (0.2,2.91) .. controls (2.5,3.02) and (4.6,3.16) .. (6.8,3.24); + \draw[leafblue,very thick] + (3.0,0.18) .. controls (2.86,1.35) and (3.18,2.69) .. (3.06,4.33); + \fill[leafblue] (3.07,2.25) circle (1.7pt); + \node[below right,leafblue] at (3.07,2.25) {$s$}; + \draw[leafblue,thick,{Bar[width=4pt]}-{Bar[width=4pt]}] + (2.95,1.34) -- (3.13,3.11); + \node[left,leafblue,fill=white,inner sep=1pt] at (2.75,3.13) {$D_s$}; + \fill[sweeporange] (5.4,2.40) circle (1.7pt); + \node[right,sweeporange,fill=white,inner sep=1pt] + at (5.4,2.40) {$\Theta_s(u,y)$}; + \node[above,leafblue] at (3.18,4.32) {forward $Q$-level}; + \node[above,sweeporange] at (5.74,3.25) {backward $F$-levels}; +\end{tikzpicture} +\caption{Coordinate schematic for the bridge change of variables, not +an orbit plot. A forward level is labelled at $y=y_c$ by $s$; the +eligible slice $D_s$ is swept across the square along transverse +backward levels to the coordinate $(u,\Theta_s(u,y))$.} +\label{bridge:sweep-figure} +\end{figure} diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/paper.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..d685d6f3d32384a77d2de437b5660d2846015574 --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/paper.tex @@ -0,0 +1,75 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{booktabs,array,graphicx} +\usepackage{enumitem} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,patterns,positioning} +\usepackage[section]{placeins} +\usepackage{needspace} +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{28,75,118} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref} +\hypersetup{ + pdftitle={Positive Metric Entropy for the Standard Map at Large Parameters}, + pdfauthor={OpenAI}, + pdfsubject={Lebesgue-area metric entropy of the standard sine map} +} +\usepackage[nameinlink,capitalise,noabbrev]{cleveref} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\AddToHook{env/theorem/before}{\Needspace{4\baselineskip}} +\AddToHook{env/lemma/before}{\Needspace{4\baselineskip}} +\AddToHook{env/proposition/before}{\Needspace{4\baselineskip}} +\AddToHook{env/corollary/before}{\Needspace{4\baselineskip}} +\AddToHook{env/definition/before}{\Needspace{4\baselineskip}} +\AddToHook{env/remark/before}{\Needspace{4\baselineskip}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\T}{\mathbb{T}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\PP}{\mathbb{P}} +\newcommand{\dd}{\,\mathrm{d}} +\newcommand{\one}{\mathbf{1}} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\newcommand{\abs}[1]{\left\lvert#1\right\rvert} +\DeclareMathOperator{\Law}{Law} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\diag}{diag} +\setlength{\emergencystretch}{1.5em} +\title{Positive Metric Entropy for the Standard Map\\at Large Parameters} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that the standard sine map of the two-dimensional torus has +positive metric entropy with respect to normalized area for every +sufficiently large positive parameter. This gives a full parameter tail, +and hence answers Sinai's positive-parameter-measure conjecture +affirmatively. +\end{abstract} +\tableofcontents +\input{sections/introduction} +\input{sections/setup} +\input{sections/cancellation} +\input{sections/local-scarcity} +\input{sections/bridge} +\input{sections/scales} +\input{sections/geometry} +\input{references} +\end{document} diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/references.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..09d9bed18a92c7ed81435672104452698d8bc339 --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/references.tex @@ -0,0 +1,122 @@ +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{References} + +\bibitem{BergerCarrasco2014} +P.~Berger and P.~D.~Carrasco, +\emph{Non-uniformly hyperbolic diffeomorphisms derived from the standard map}, +Communications in Mathematical Physics \textbf{329} (2014), no.~1, 239--262. +\href{https://doi.org/10.1007/s00220-014-1932-1} +{doi:10.1007/s00220-014-1932-1}. + +\bibitem{BergerTuraev2019} +P.~Berger and D.~Turaev, +\emph{On Herman's positive entropy conjecture}, +Advances in Mathematics \textbf{349} (2019), 1234--1288. +\href{https://doi.org/10.1016/j.aim.2019.04.002} +{doi:10.1016/j.aim.2019.04.002}. + +\bibitem{Bestvina1997} +M.~Bestvina, +\emph{Real trees in topology, geometry, and group theory}, +arXiv:math/9712210v1 (1997). +\href{https://arxiv.org/abs/math/9712210v1}{arXiv:math/9712210v1}. + +\bibitem{BloorLuzzatto2009} +K.~Bloor and S.~Luzzatto, +\emph{Some remarks on the geometry of the standard map}, +International Journal of Bifurcation and Chaos \textbf{19} (2009), +no.~7, 2213--2232. +\href{https://doi.org/10.1142/S0218127409024025} +{doi:10.1142/S0218127409024025}. + +\bibitem{Blumenthal2021} +A.~Blumenthal, +\emph{Statistical properties for compositions of standard maps with increasing coefficient}, +Ergodic Theory and Dynamical Systems \textbf{41} (2021), no.~4, 981--1024. +\href{https://doi.org/10.1017/etds.2019.115}{doi:10.1017/etds.2019.115}. + +\bibitem{BlumenthalXueYoung2017} +A.~Blumenthal, J.~Xue, and L.-S.~Young, +\emph{Lyapunov exponents for random perturbations of some area-preserving +maps including the standard map}, +Annals of Mathematics \textbf{185} (2017), no.~1, 285--310. +\href{https://doi.org/10.4007/annals.2017.185.1.5} +{doi:10.4007/annals.2017.185.1.5}. + +\bibitem{Chirikov1979} +B.~V.~Chirikov, +\emph{A universal instability of many-dimensional oscillator systems}, +Physics Reports \textbf{52} (1979), no.~5, 263--379. +\href{https://doi.org/10.1016/0370-1573(79)90023-1} +{doi:10.1016/0370-1573(79)90023-1}. + +\bibitem{Duarte1994} +P.~Duarte, +\emph{Plenty of elliptic islands for the standard family of area preserving maps}, +Annales de l'Institut Henri Poincar\'e. Analyse non lin\'eaire +\textbf{11} (1994), no.~4, 359--409. +\href{https://www.numdam.org/item/AIHPC_1994__11_4_359_0/} +{Official article record}. + +\bibitem{Gorodetski2012} +A.~Gorodetski, +\emph{On stochastic sea of the standard map}, +Communications in Mathematical Physics \textbf{309} (2012), no.~1, 155--192. +\href{https://doi.org/10.1007/s00220-011-1365-z} +{doi:10.1007/s00220-011-1365-z}. + +\bibitem{HollandLuzzatto2006} +M.~Holland and S.~Luzzatto, +\emph{Stable manifolds under very weak hyperbolicity conditions}, +Journal of Differential Equations \textbf{221} (2006), no.~2, 444--469. +\href{https://doi.org/10.1016/j.jde.2005.07.013}{doi:10.1016/j.jde.2005.07.013}. + +\bibitem{KatokBurns1994} +A.~Katok, in collaboration with K.~Burns, +\emph{Infinitesimal Lyapunov functions, invariant cone families and +stochastic properties of smooth dynamical systems}, +Ergodic Theory and Dynamical Systems \textbf{14} (1994), 757--785. +\href{https://doi.org/10.1017/S0143385700008142} +{doi:10.1017/S0143385700008142}. + +\bibitem{Mane1981} +R.~Ma\~n\'e, +\emph{A proof of Pesin's formula}, +Ergodic Theory and Dynamical Systems \textbf{1} (1981), no.~1, 95--102. +\href{https://doi.org/10.1017/S0143385700001188} +{doi:10.1017/S0143385700001188}. + +\bibitem{Mane1983Errata} +R.~Ma\~n\'e, +\emph{Errata to A proof of Pesin's formula}, +Ergodic Theory and Dynamical Systems \textbf{3} (1983), no.~1, 159--160. +\href{https://doi.org/10.1017/S0143385700001863} +{doi:10.1017/S0143385700001863}. + +\bibitem{Obata2021} +D.~Obata, +\emph{Uniqueness of the measure of maximal entropy for the standard map}, +Commentarii Mathematici Helvetici \textbf{96} (2021), no.~1, 79--111. +\href{https://doi.org/10.4171/CMH/508}{doi:10.4171/CMH/508}. + +\bibitem{Oliveira2024} +F.~Oliveira, +\emph{Positive topological entropy for the Standard Map}, +arXiv:2404.02209v2 (2024). +\href{https://arxiv.org/abs/2404.02209}{arXiv:2404.02209}. + +\bibitem{Oseledets1968} +V.~I.~Oseledets, +\emph{A multiplicative ergodic theorem. +Lyapunov characteristic numbers for dynamical systems}, +Transactions of the Moscow Mathematical Society \textbf{19} (1968), 197--231. +\href{https://www.mathnet.ru/eng/mmo214}{Official bibliographic record}. + +\bibitem{Pesin1977} +Ya.~B.~Pesin, +\emph{Characteristic Lyapunov exponents and smooth ergodic theory}, +Russian Mathematical Surveys \textbf{32} (1977), no.~4, 55--114. +\href{https://doi.org/10.1070/RM1977v032n04ABEH001639} +{doi:10.1070/RM1977v032n04ABEH001639}. + +\end{thebibliography} diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/bridge.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/bridge.tex new file mode 100644 index 0000000000000000000000000000000000000000..6107748bba7993070c1634f1ed6fe2206fca6ca2 --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/bridge.tex @@ -0,0 +1,531 @@ +\section{A product estimate across a fast bridge} +\label{bridge:section} + +The second estimate compares observations near opposite ends of a long +interval on which the transfer matrix grows almost at its maximum rate. +The comparison comes from two transverse families of contracting graphs +in the middle of the interval. All measures in this section are the +normalized area measure, expressed in the coordinates +$(q_i,q_{i-1})$ when convenient, whenever they define probabilities +or expectations. Label lengths and slice integrals use ordinary +one-dimensional Lebesgue measure in their displayed coordinates. + +For distinct integers $d,e$, write +\begin{equation} + \mathfrak h_{d,e} + =1-\frac{\log_M\norm{T_{d,e}}}{\abs{d-e}}. + \label{bridge:shortfalls} +\end{equation} +These quantities belong to $[0,1]$. An observation with bound $(R,L,B)$ +means a function of at most $R$ such quantities whose defining function +on $[0,1]^R$ has absolute value at most $B$ and Lipschitz constant at most +$L$ for the maximum norm. Allowing fewer than $R$ arguments is harmless. +The defining functions and the integer indices may vary from one +application to another, provided these bounds are preserved. +Equivalently, these bounds may be imposed only on the realized range +$\mathcal R\subseteq[0,1]^R$ of the arguments. Indeed, a function $f$ +on that range extends by +$u\mapsto\inf_{v\in\mathcal R}(f(v)+L\norm{u-v}_\infty)$; +clipping this extension to $[-B,B]$ preserves its values on +$\mathcal R$ and its Lipschitz bound. + +\begin{theorem}[Product estimate across a fast bridge] +\label{bridge:theorem} +There are absolute constants $c_b\in(0,1)$, $\kappa>0$, and $\sigma>0$ +with the following property. Fix observation bounds $(R,L,B)$ and a +threshold $h>0$. For sufficiently large $M$ and sufficiently large +$n=b-a$, let $W,H$ be observations with these bounds, with $W\geq0$. +All indices occurring in one observation are within $\kappa n$ of $a$, +and all indices occurring in the other are within $\kappa n$ of $b$. +Either assignment of $W,H$ to the two ends is allowed. Then +\begin{equation} + \E\!\left[ + \one_{\{\log_M\norm{T_{a,b}}\geq c_b n\}} + W\one_{\{H>h\}} + \right] + \leq + C\bigl(\E W+M^{-\sigma n}\bigr)\PP(H>h/2). + \label{bridge:product} +\end{equation} +The constants $c_b,\kappa,\sigma$ are independent of the observation +bounds and threshold. The constant $C$ and the lower size requirements +for $M,n$ may depend on $(R,L,B)$ and $h$, but not on the indices, +the defining functions, or the bridge location. +\end{theorem} + +We first construct the graphs needed in the proof. Finite-time +contracting directions and curves have been developed in stable-manifold +constructions under weak hyperbolicity hypotheses +\cite{HollandLuzzatto2006}. Here the graphs are +level sets of a distant orbit coordinate, obtained from finite +Dirichlet problems; we prove their extent and transverse distortion +directly. We use $c=0.98$ throughout the construction. A direction is +understood modulo sign. + +\begin{lemma}[Contracting graphs and their distortion] +\label{bridge:graphs} +For sufficiently large $M$, suppose $B_+\geq2$ and, at a fixed state with central coordinates +$(x,y)=(q_0,q_{-1})$, +\begin{equation} + D_l:=\norm{T_{0,l}}\geq M^{cl} + \qquad(1\leq l\leq B_+). + \label{bridge:prefix-growth} +\end{equation} +For at least one choice $j\in\{B_+-1,B_+\}$ there is a smooth graph +through this state, parametrized by $q_{-1}=y+r$, $\abs r\leq1/4$, +on which the lifted coordinate $q_j$ is fixed. On this graph, +\begin{equation} + \abs{q_p(r)-q_p(0)}+ + \abs{\frac{\dd q_p(r)}{\dd r}} + \leq C M^{-0.8(p+1)} + \qquad(0\leq pl} + M^{l-0.9(p+1)+1-1.6(p+1)}. +\] +Here $\abs{t_p}\leq M^p$ and \cref{bridge:linear-dirichlet} were +used. After multiplying by $M^{0.8(l+1)}$, the exponents are, +respectively, +\[ + -0.1(l+1)-0.6(p+1), + \qquad + 1.8(l+1)-2.5(p+1). +\] +Their sums tend uniformly to zero as $M$ tends to infinity, +independently of $j$. The linear term $Ur$ has bounded weighted norm +for $\abs r\leq1/4$. Choose $R_0$ larger than twice this bound and +then choose $M$ large. The right side of \cref{bridge:fixed-point} +maps this ball to itself and is a contraction there. Its unique fixed +point gives the desired graph on the whole parameter interval. + +The same estimates show that +$I-G\diag(F'_p(\xi_p))$ has a uniformly bounded inverse in this +weighted norm. Multiplication by the original Dirichlet matrix +identifies it with the varied linearized Dirichlet problem. +Consequently that problem is invertible, and the finite-dimensional +implicit function theorem gives smooth dependence on $r$. +Differentiating \cref{bridge:fixed-point} solves with right side $U$, +which has bounded weighted norm. This proves +\cref{bridge:graph-contraction}. The contraction and its fixed point +also depend continuously on the eligible initial data: all the +coefficients are continuous there, and the contraction constant and +ball are uniform. This proves the asserted continuous dependence. + +If $j=B_+$, the endpoint pair estimate follows directly from the +fixed boundary value and \cref{bridge:graph-contraction}. If +$j=B_+-1$, the recurrence and the fixed value of $q_{B_+-1}$ give +$\xi_{B_+}=-\xi_{B_+-2}$, proving the same estimate without an extra +factor of $M$. + +It remains to control transverse distortion. At the varied orbit let +$t_p(r)$ again have initial values $t_{-1}(r)=0$, $t_0(r)=1$. +Invertibility of the varied Dirichlet problem gives $t_j(r)\ne0$. +The graph derivative $U_p(r)=\dd q_p(r)/\dd r$ has boundary values +1 and 0 at indices $-1,j$ and solves that varied problem. Its Green +diagonal is therefore $t_p(r)U_p(r)$. Differentiating the determinant +of the varied Dirichlet matrix gives the exact identity +\begin{equation} + \frac{\dd}{\dd r}\ln\abs{t_j(r)} + =\sum_{p=0}^{j-1}t_p(r)U_p(r)^2\phi''(q_p(r)). + \label{bridge:determinant-distortion} +\end{equation} +The derivative of a diagonal entry is +$\phi''(q_p(r))U_p(r)$, explaining the second factor of $U_p$. +Since $\abs{t_p(r)}\leq M^p$ for every varied orbit, +\cref{bridge:graph-contraction} bounds the absolute value of +\cref{bridge:determinant-distortion} by +\[ + C\sum_{p\geq0}M^p M^{-1.6(p+1)}M + =C\sum_{p\geq0}M^{-0.6(p+1)}\leq C. +\] +Integration over $\abs r\leq1/4$ proves the bounded-ratio claim. + +For backward evolution set $\widetilde q_p=q_{-1-p}$. The recurrence +has the same form in these variables. At step $l$ its ordered pair +is the swapped original pair at time $-l$, and the initial pair is +$(q_{-1},q_0)$. Swapping coordinates is an isometry, so the backward +growth hypotheses and all norm estimates are identical. The resulting +graphs vary $q_0$ and have small slope in the $q_{-1}$ coordinate. +\end{proof} + +\begin{proof}[Proof of \cref{bridge:theorem}] +We first find many usable middle times. On the bridge event choose a +unit vector realizing $\norm{T_{a,b}}$ and follow its norms along the +bridge. The increments of their logarithms to base $M$ are at most 1, +and their total shortfall from 1 is at most $(1-c_b)n$. By +\cref{setup:shortfall-count}, if $c_b$ is sufficiently close to 1, +at least $\eta n$ indices $i$ in the middle third, for an absolute +$\eta>0$, satisfy +\begin{equation} + \begin{aligned} + \norm{T_{i,i+l}}&\geq M^{cl} + &&(1\leq l\leq b-i),\\ + \norm{T_{i,i-l}}&\geq M^{cl} + &&(1\leq l\leq i-a). + \end{aligned} + \label{bridge:middle-eligibility} +\end{equation} +In detail, remove indices admitting a forward interval starting there +or a forward interval ending there whose average log increment is +less than $c$. The counting bound removes at most +$C(1-c_b)n/(1-c)$ indices. At a retained index, the chosen vector +bounds the forward operator norm on either kind of interval from +below. On an interval ending at $i$, use also equality of the forward +and inverse operator norms to obtain the second inequality in +\cref{bridge:middle-eligibility}. For large $n$, the integer rounding +of the middle third does not affect a fixed positive proportion. + +It suffices to establish \cref{bridge:product}, with a uniform +constant, when the bridge indicator is replaced by the indicator of +\cref{bridge:middle-eligibility} for one fixed middle index $i$. +Indeed, the bridge indicator is at most $(\eta n)^{-1}$ times the +sum of these middle-index indicators; multiplying by the nonnegative +weight and summing then costs only an absolute constant. + +Shift $i$ to time 0. Put $B_+=b-i$ and $B_-=i-a$; both are at least +$n/3-O(1)$. Apply \cref{bridge:graphs} forward and backward. There +are two possible endpoint-coordinate choices in each direction. +We work with one fixed pair of choices and eventually sum over the +four possibilities. Write $Q(x,y)$ for the lifted forward coordinate +held fixed, and $F(x,y)$ for the lifted backward coordinate held +fixed. On forward graphs $Q_x\ne0$; on backward graphs $F_y\ne0$. +For large $M$, all forward slopes $\abs{\dd x/\dd y}$ and backward +slopes $\abs{\dd y/\dd x}$ are less than a common absolute number +$a_0<1/2$. The relevant transverse partial derivatives have uniformly +bounded absolute ratios along each graph. + +\paragraph{Transport of endpoint observations.} +Along a forward graph the endpoint pair at time $b$ changes by at +most $CM^{-0.8(B_+-1)}$. A single forward or inverse map is +$M$-Lipschitz on the lifted coordinates. Thus pair differences +throughout the window of radius $\kappa n$ about $b$ are at most +\[ + C M^{\kappa n+O(1)}M^{-0.8(B_+-1)}. +\] +The one-step matrix difference is at most $CM$ times the position +difference. Since forward and inverse product norms agree, the +endpoints of each observation interval may first be put in increasing +order. For a product of length at most $2\kappa n$, the +telescoping product identity therefore bounds its matrix difference +by a length factor times +\[ + C M^{3\kappa n+O(1)}M^{-0.8(B_+-1)}. +\] +Every product norm compared here is at least 1. The elementary +inequalities $\abs{\norm A-\norm B}\leq\norm{A-B}$ and +$\abs{\log u-\log v}\leq\abs{u-v}$ for $u,v\geq1$ consequently +give the same exponential control for the normalized logarithms in +\cref{bridge:shortfalls}. The positive integer interval lengths in +their denominators cause no additional loss. + +Choose fixed $\kappa,\sigma>0$ with +\begin{equation} + 3\kappa+\sigma<0.8/3. + \label{bridge:error-exponents} +\end{equation} +The strict margin absorbs the length factor and the bounded powers +of $M$ when $n$ is large. After increasing the size requirements to +account for the observation bounds, each observation at the forward +end changes by at most +\begin{equation} + \delta_n=C M^{-\sigma n} + \label{bridge:observation-error} +\end{equation} +along a forward graph. The same statement holds for an observation +at the backward end along a backward graph. All these estimates are +uniform in the chosen middle index, endpoint-coordinate choices, +and observation indices. Increase $M,n$ further so that +$\delta_nh/2$ by +\cref{bridge:observation-error}. The identity +\[ + Q(X(s,y),y)=Q(s,y_c) +\] +and implicit differentiation give +\begin{equation} + X_s(s,y)=\frac{Q_x(s,y_c)}{Q_x(X(s,y),y)}. + \label{bridge:label-jacobian} +\end{equation} +The derivative in \cref{bridge:label-jacobian} is interpreted through +a local smooth extension when $E$ has empty interior; the existence +of that extension is justified below. The distortion assertion of +\cref{bridge:graphs} bounds its absolute value above and below by +positive constants. + +The map $(s,y)\mapsto(X(s,y),y)$ on $E\times I_y$ is injective and +has Jacobian comparable to 1. Its image stays in a fixed bounded +lift: the small slope bounds the displacement from the square by an +absolute constant. Projection of this bounded lift to the torus has +bounded multiplicity. Change of variables therefore yields +\begin{equation} + \abs E\leq C\PP(H>h/2). + \label{bridge:label-area} +\end{equation} +The factor $\abs{I_y}=s_0$ has been absorbed into the constant. + +\paragraph{A transverse sweep for the weight.} +Fix $s\in E$. Let $D_s\subseteq I_y$ consist of the parameters $y$ +for which $z(y)=(X(s,y),y)$ lies in the square and is backward-eligible +in the fixed backward endpoint choice. This is a compact set. Carry +each $z(y)$ along its backward graph to first coordinate $u\in I_x$, +and denote the resulting second coordinate by $\Theta_s(u,y)$. +The carry is defined throughout $I_x$, with a margin. + +For each fixed $u$, the map $y\mapsto\Theta_s(u,y)$ is injective on +$D_s$. If two carried points agreed, their backward graphs would +coincide throughout $I_x$ by regular-level uniqueness. A single such +graph cannot meet the chosen forward graph in two distinct points: +two intersections would satisfy both +$\abs{\Delta x}\leq a_0\abs{\Delta y}$ and +$\abs{\Delta y}\leq a_0\abs{\Delta x}$, forcing equality of the +points. This proves the asserted injectivity. + +The equation defining the carry is +\[ + F(u,\Theta_s(u,y))=F(X(s,y),y). +\] +Differentiation with respect to the starting parameter gives +\begin{equation} + \partial_y\Theta_s(u,y) + =\frac{F_y(z(y))+F_x(z(y))X_y(s,y)} + {F_y(u,\Theta_s(u,y))}. + \label{bridge:sweep-jacobian} +\end{equation} +The ratio of the two $F_y$ terms is uniformly bounded above and +below along the backward graph. Moreover $\abs{F_x/F_y}\leq a_0$ +at its starting point, since $-F_x/F_y$ is its slope, and +$\abs{X_y}\leq a_0$. Hence the absolute value of +\cref{bridge:sweep-jacobian} is uniformly bounded above and below +by positive constants. + +The endpoint observation estimate gives, for all $u\in I_x$ and +$y\in D_s$, +\[ + W(X(s,y),y)\leq W(u,\Theta_s(u,y))+\delta_n. +\] +Apply change of variables to the injective map +$(u,y)\mapsto(u,\Theta_s(u,y))$ on $I_x\times D_s$. +Its image lies in a bounded lift. Since $W\geq0$, its integral over +that image is at most a fixed multiple of $\E W$. Using +\cref{bridge:sweep-jacobian} and then dividing by $\abs{I_x}$ gives +\begin{equation} + \int_{D_s}W(X(s,y),y)\,\dd y + \leq C\bigl(\E W+M^{-\sigma n}\bigr). + \label{bridge:conditional-weight} +\end{equation} + +\paragraph{Changes of variables on the eligible subsets.} +Here are details ensuring that the preceding changes of variables +do not require the eligible sets to have interior. Consider first +one forward graph with label $s$. It is a compact regular level arc +over $I_y$ and extends with a margin beyond this interval. The +partial derivative $Q_x$ is nonzero on the arc. The implicit +function theorem in finitely many overlapping charts along the arc +therefore continues it smoothly for intercepts in an open +neighborhood of $s$. The continued level is prescribed by +$Q(s',y_c)$ for each nearby intercept $s'$. Uniqueness in the +overlapping charts makes these local continuations consistent. +Every nearby eligible label follows the same continuation, because +its regular graph passes through that intercept. Thus $X$ on +$E\times I_y$ is locally the restriction of a smooth function of +$(s,y)$, and \cref{bridge:label-jacobian} holds at all required +points. + +For the transverse sweep, fix $s$ and a point of $D_s$. Its backward +graph is likewise a compact regular arc, with $F_y\ne0$, extending +with margin over $I_x$. The same continuation argument applies to +the smoothly varying starting points $(X(s,y),y)$ on the fixed +forward graph. It gives a local smooth extension of +$\Theta_s(u,y)$ in both variables, agreeing with all nearby +eligible carries by uniqueness. The maps used in both area +comparisons are consequently restrictions of local smooth +diffeomorphisms, with the displayed nonvanishing Jacobians. + +A countable cover by such charts can be split into disjoint Borel +pieces. On each piece ordinary change of variables applies to its +measurable subset; the images of the pieces are disjoint because +the full restricted maps are injective. Summing proves the formulas +on the compact eligible domains. This also establishes the +measurability of all integrands and swept regions used above. +No extension of eligibility to a neighborhood is assumed. + +\paragraph{Completion of the area comparison.} +Every point of the fixed square satisfying both eligibility +conditions and $H>h$ belongs to a forward graph with label in $E$. +Change variables from $x$ to this label at each $y$, using +\cref{bridge:label-jacobian}. Drop conditions other than square +membership and backward eligibility, which is allowed because the +weight is nonnegative. The desired weighted integral over this +square and this pair of choices is bounded by +\[ + C\int_E\int_{D_s}W(X(s,y),y)\,\dd y\,\dd s. +\] +Combining \cref{bridge:conditional-weight,bridge:label-area} bounds +this by the right side of \cref{bridge:product}. + +If $H$ is at the backward end, begin instead with the labels of the +almost horizontal backward graphs and sweep them in the $x$ +direction to bound their label measure by $\PP(H>h/2)$. For each +such graph, transport its eligible points along the almost vertical +forward graphs to estimate $W$. The proof just given applies with +$x,y$ and $Q,F$ interchanged: the graph extents, distortion bounds, +small-slope injectivity argument, and local chart construction are +the same. This proves the estimate for the other assignment of the +two observations as well. + +Finally sum over the finitely many squares and the at most four +endpoint-coordinate choices. The constants remain uniform. Average +over the usable middle indices as explained after +\cref{bridge:middle-eligibility}. This establishes +\cref{bridge:product} and completes the proof. +\end{proof} diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/cancellation.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/cancellation.tex new file mode 100644 index 0000000000000000000000000000000000000000..cd750b8cf578c7ad0f6f943ee8f5a39b8955c41e --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/cancellation.tex @@ -0,0 +1,472 @@ +\section{Nearly full-rate cancellations} +\label{can:section} + +The first estimate concerns two long products which expand rapidly when +viewed from their common endpoint, although their concatenation has little +growth. Such a configuration forces a linearized solution to be small at +both distant endpoints. We use this solution to locate many pairs of +comparable expansion scales on the two sides of the common endpoint. + +\begin{theorem}[Cancellation estimate]\label{can:theorem} +There are constants $\delta>0$, $C<\infty$, $\gamma>0$, and $M_0<\infty$ +such that, for every $M\ge M_0$, every $i\in\Z$, and every $n\in\N$, +\begin{equation}\label{can:event-bound} +\PP\left\{ +\begin{array}{l} +\min\bigl(\log_M\norm{T_{i,i-n}}, + \log_M\norm{T_{i,i+n}}\bigr)\ge(1-\delta)n,\\[2pt] +\log_M\norm{T_{i-n,i+n}}\le\delta n +\end{array} +\right\} +\le C\exp(-\gamma n). +\end{equation} +All four constants are independent of the center, the length, and the +parameter $M$ in the indicated range. +\end{theorem} + +Throughout this section and the next, fix +\begin{equation}\label{can:c} +c=\frac{99}{100}. +\end{equation} +Constants used before the final choice of $\delta$ are independent of +$\delta$, $n$, and $M$, unless a dependence is stated explicitly. + +\subsection{A solution small at both endpoints} + +By stationarity it is enough to consider $i=0$. Suppose the event in +\cref{can:event-bound} occurs. Write +\[ +D_- = \norm{T_{0,-n}},\qquad D_+ = \norm{T_{0,n}}, +\] +and let $a_-$ and $a_+$ be unit most-expanded input directions for these +two products. If $b$ is a unit vector perpendicular to $a_-$, then +\[ +\norm{T_{0,-n}b}=D_-^{-1},\qquad +\norm{T_{0,n}b}\ge D_+\abs{a_-\wedge a_+}. +\] +The first equality uses determinant one: the singular values of either +product are its norm and the reciprocal of its norm. Applying the full +product to the vector $T_{0,-n}b$ gives +\[ +\norm{T_{-n,n}} +\ge D_-D_+\abs{a_-\wedge a_+}, +\qquad +\abs{a_-\wedge a_+}\le M^{-(2-3\delta)n}. +\] +Resolving $b$ in the singular input directions of $T_{0,n}$ also gives the +upper bound +\[ +\norm{T_{0,n}b} +\le D_+^{-1}+D_+\abs{a_-\wedge a_+} +\le 2M^{-(1-3\delta)n}. +\] +The left endpoint has the smaller bound $D_-^{-1}$. Thus the solution +of the linearized recurrence with central pair $b$ is small at both +endpoints. One of its entries at times $0$ and $-1$ has absolute value +at least $1/\sqrt2$. Choose that time $e\in\{0,-1\}$ and divide the +whole solution by its value there. The normalized solution satisfies +\begin{equation}\label{can:small-solution} +u_e=1,\qquad \norm{(u_0,u_{-1})}\le\sqrt2,\qquad +\norm{(u_n,u_{n-1})},\ \norm{(u_{-n},u_{-n-1})} +\le 3M^{-(1-3\delta)n}. +\end{equation} +This construction will be used only as an existence statement; no +measurable selection of a singular direction is required. + +\subsection{Outward paths and good indices} + +Fix $e$ and $q_e$. Use $\theta=q_{e-1}$ as the other circle coordinate, +so that $q_{e+1}=\phi(q_e)-\theta$ modulo one. The two outward paths are +\[ +q_j^-=q_{e-j},\qquad q_j^+=q_{e+j}\qquad(j\ge0). +\] +On either path the positions and every linearized solution obey the same +second-order recurrence. In calculations involving just one path, omit +its sign and write $q_j,u_j,v_j=v(q_j)$, with $u_0=1$. Define +\begin{equation}\label{can:t-definition} +t_0=0,\qquad t_1=1,\qquad t_{j+1}=v_jt_j-t_{j-1}. +\end{equation} +For local real lifts of the positions, +\[ +\partial_\theta q_j^-=t_j^-,\qquad +\partial_\theta q_j^+=-t_j^+. +\] +The sign difference comes solely from the initial positions at index one. +It does not change the recurrence or its Wronskian identity: +\begin{equation}\label{can:wronskian} +u_{j-1}t_j-u_jt_{j-1}=1\qquad(j\ge1). +\end{equation} +Indeed the expression on the left is one at $j=1$ and the recurrence +preserves it. A common normalized solution on the two paths has +\begin{equation}\label{can:center-equation} +u_1^-+u_1^+=v(q_e). +\end{equation} +Conversely, two outward solutions with value one at index zero and +satisfying \cref{can:center-equation} assemble into a solution of the +original recurrence across the center. + +On each path the two small endpoint entries in +\cref{can:small-solution} are indexed by $N-1,N$. More explicitly, the +forward and backward lengths are $N_+=n-e$ and $N_-=n+e+1$, +respectively: one is $n$ and the other is $n+1$. Define pair-growth +logarithms +\[ +\tau_j=\log_M\max(\abs{t_j},\abs{t_{j-1}}),\qquad +\upsilon_j=\log_M\max(\abs{u_j},\abs{u_{j-1}}) +\quad(1\le j\le N). +\] +Both pairs are nonzero, so these logarithms are defined. The bounds for +one step and its inverse in the maximum norm give +\begin{equation}\label{can:log-increments} +\abs{\tau_{j+1}-\tau_j}\le1,\qquad +\abs{\upsilon_{j+1}-\upsilon_j}\le1. +\end{equation} + +\begin{definition}\label{can:good-definition} +An index $2\le j\le N-2$ is \emph{good} for the selected normalized +solution if +\begin{equation}\label{can:good} +\begin{aligned} +\tau_j-\tau_l&\ge c(j-l)&& (1\le l0$, $c_1>0$, $C_1<\infty$, and +$n_*<\infty$ with the following property. If $0<\delta\le\delta_0$ +is fixed, increasing $n_*$ if necessary, every configuration in +\cref{can:small-solution} with $n\ge n_*$ has at least $c_1n$ pairs +$(j_m^+,j_m^-)$ of good indices such that both index sequences increase +strictly and +\begin{equation}\label{can:matching-condition} +\abs{\tau_{j_m^+}^+-\tau_{j_m^-}^-}\le\frac35. +\end{equation} +The collection of all possible increasing index-pair patterns for a +given $n$ has cardinality at most $\exp(C_1n)$. +\end{lemma} + +\begin{proof} +Choose a small fixed $\beta>0$, for example $\beta=10^{-3}$. By +\cref{can:bad-count}, choosing $\delta_0$ sufficiently small and then +$n_*$ sufficiently large makes the bad-index count on each path at most +$\beta n$. The total-shortfall estimate also gives, at every index, +\[ +j-1-3\delta n-\log6\le\tau_j\le j-1. +\] +Shrink $\delta_0$ once more and increase $n_*$ so that, for +$n/3\le j\le n/2$, the corresponding $\tau_j$ lies in +$[n/4,3n/4]$, while on the minus path +\[ +\tau_{N-2}^-\ge\tau_N^--2>3n/4+1. +\] +There are at least $n/6-2-\beta n$ good plus indices in this range. + +For each such index put $a=\tau_j^+$. Starting from $\tau_1^-=0$, +let $k$ be the first minus index with $\tau_k^-\ge a$. We have +$\tau_{k-1}^-1$. A fixed bad minus +index can therefore be responsible for at most two discarded plus +indices. At least $n/6-2-3\beta n$ candidates remain. + +Thin these candidates in increasing plus order by retaining the first +one, then retaining the next whose plus logarithm is at least two larger +than the last retained logarithm, and continuing. Since successive +candidate logarithms are separated by at least $c$, each retention +removes at most three candidates. A fixed positive fraction of $n$ +remains; for example, decreasing the fraction if necessary, one may fix +$c_1=10^{-2}$. The matched minus logarithms increase strictly, because +each is within $1/2$ of the corresponding plus logarithm and the retained +plus logarithms have gaps at least two. For two good minus indices, +increasing index implies strictly increasing logarithm by +\cref{can:good}. Strict increase of their logarithms therefore forces +strict increase of their indices as well. The matching error is at most +$1/2$, which implies \cref{can:matching-condition}. + +Each path contains at most $n+1$ available indices. An increasing paired +pattern is determined by a subset of the indices on each path, with the +two subsets of equal cardinality paired in order. There are at most +$2^{2n+2}$ such choices. Enlarging an absolute $C_1$ gives the stated +exponential bound. +\end{proof} + +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth] + \draw[->] (0,0)--(-5,0); + \draw[->] (0,0)--(5,0); + \foreach \x in {-4.6,-2,0,2.6,4.6} + \fill (\x,0) circle (1.5pt); + \node[above] at (0,0) {$q_e$}; + \node[above] at (-2,0) {$q_{e-j_m^-}$}; + \node[above] at (2.6,0) {$q_{e+j_m^+}$}; + \node[above] at (-4.6,0) {left endpoint}; + \node[above] at (4.6,0) {right endpoint}; + \draw[densely dashed] (-2,-.15)--(-2,-.65)--(2.6,-.65)--(2.6,-.15); + \node[below,align=center] at (0,-.65) + {a matched pair: $\abs{\log_M L_{j_m^-}^- -\log_M L_{j_m^+}^+}\le3/5$}; +\end{tikzpicture} +\caption{The two paths are followed outward from a fixed central position. +Matched indices need not be equally far from the center; their derivatives +with respect to the other central coordinate have comparable logarithmic +size.} +\label{can:wing-figure} +\end{figure} + +\subsection{The local input} + +The analytic fact used at each matched pair is the following uniform +scarcity statement. Its proof occupies \cref{loc:section}. + +\begin{lemma}[Local scarcity]\label{can:local-scarcity} +For every $\eta>0$ there are $\eta'>0$ and $M_\eta<\infty$ such that +the following holds for $M\ge M_\eta$. Fix $q_e$, a base coordinate +$\theta_*$, and indices $j_+,j_-\ge2$ on the two outward paths. +Suppose both indices are good at $\theta_*$ for a common normalized +solution, and their logarithms satisfy +\cref{can:matching-condition}. Let +\[ +L_+=\abs{t_{j_+}^+(\theta_*)},\qquad +L_-=\abs{t_{j_-}^-(\theta_*)},\qquad L=\max(L_+,L_-). +\] +Let $\mathcal S\subset[-1,1]$ consist of the $h$ for which, at +$\theta=\theta_*+h/L$, the same two indices are good for some common +solution with $u_e=1$. The solution is allowed to depend on $h$, and +neither endpoint smallness nor a new matching condition is required at +these other points. Then +\begin{equation}\label{can:scarcity-bound} +\abs{\{x\in\R:\dist(x,\mathcal S)<\eta'\}}\le\eta. +\end{equation} +Here and below the size of a subset of the line or circle denotes its +Lebesgue measure. The constants are uniform over the central position, +the base coordinate, and the selected indices. +\end{lemma} + +\subsection{From local scarcity to exponential decay} + +\begin{proof}[Proof of \cref{can:theorem}] +Fix once and for all a sufficiently small $\delta>0$ allowed by +\cref{can:matching}. Its constants $c_1,C_1,n_*$ are now fixed. +Choose $\eta>0$ so small that +\begin{equation}\label{can:eta-choice} +5\eta<1,\qquad +c_1\log\frac1{5\eta}>C_1+1. +\end{equation} +Obtain $\eta'$ and $M_\eta$ from \cref{can:local-scarcity}. Increase +a fixed lower bound $M_0$ so that all estimates above hold, that +$M_0\ge M_\eta$, and that +\begin{equation}\label{can:radius-choice} +M_0^{-c}<\frac14,\qquad M_0^{-c}/5<\eta'. +\end{equation} +These choices are independent of $n$, of the center, and of $M\ge M_0$. + +Fix $e\in\{0,-1\}$ and $q_e$. It is enough to consider patterns +containing exactly $r=\lceil c_1n\rceil$ pairs, since a longer pattern +can be truncated. For a fixed such increasing pattern, let +$\mathcal F$ be the set of $\theta$ admitting one common normalized +solution with all of the following properties: the original central +pair has norm at most $\sqrt2$; both endpoint pairs satisfy +\cref{can:small-solution}; and every pair of selected indices is good +and satisfies \cref{can:matching-condition}. We emphasize that one +solution must satisfy all conditions of a pattern at once. + +The set $\mathcal F$ is measurable, in fact compact. To see this without +selecting solutions, parameterize the original central pair by its entry +other than the fixed entry $u_e=1$. The norm bound puts that parameter in +$[-1,1]$. All later solution entries and all recurrence coefficients +depend continuously on this parameter and on the circle coordinate +$\theta$. Each pair entering a logarithm is nonzero, by invertibility +of the recurrence. The endpoint, goodness, and matching conditions are +closed inequalities on the compact product of the circle with +$[-1,1]$. Their projection is therefore compact. + +For $\theta\in\mathcal F$ and $1\le m\le r$, define +\[ +L_m(\theta)=\max\left( +\abs{t_{j_m^+}^+(\theta)},\abs{t_{j_m^-}^-(\theta)}\right), +\qquad s_m(\theta)=L_m(\theta)^{-1}, +\] +and let +\[ +V_m=\bigcup_{\theta\in\mathcal F} + B\bigl(\theta,s_m(\theta)/10\bigr) +\] +be an open subset of the circle. Goodness at the later of two selected +indices gives, separately on both paths, +\[ +\abs{t_{j_{m+1}^\pm}^\pm(\theta)} +\ge M^{c(j_{m+1}^\pm-j_m^\pm)} + \abs{t_{j_m^\pm}^\pm(\theta)}. +\] +Since both index sequences increase strictly, +\begin{equation}\label{can:radius-decay} +s_{m+1}(\theta)\le M^{-c}s_m(\theta), +\qquad s_m(\theta)\le M^{-c}. +\end{equation} +For fixed $M,n$, these positive radii also have a positive lower bound: +the one-step norm bound gives $\abs{t_j}\le M^{j-1}\le M^n$ at all +indices in question, so $s_m\ge M^{-n}$. + +We claim +\begin{equation}\label{can:cover-contraction} +\abs{V_{m+1}}\le5\eta\abs{V_m}\qquad(1\le m0$, uniformly in the fixed +coordinate $q_e$. The coordinate pair $(q_e,q_{e-1})$ has normalized +product Lebesgue law. Integrating the conditional bound over $q_e$ and +summing the two choices of $e$ therefore gives the same form of bound +for the original event, for every $n\ge n_*$. Every point of that event +is covered, by \cref{can:small-solution,can:matching}. Finally enlarge +$C$ to cover the finitely many positive integers $n0$ such that +\begin{equation}\label{shape:forbidden-triple} + d(r-t,r)>(1-\delta)t,\qquad + d(r,r+t)>(1-\delta)t,\qquad + d(r-t,r+t)<\delta t. +\end{equation} +\item If $s\in P_d$ and $t\in Q_d$, then +\begin{equation}\label{shape:slow-pair} + d(s,t)\le c_b|t-s|. +\end{equation} +\end{enumerate} +The same statements hold almost everywhere for every positive measure +appearing in the decomposition of \cref{scale:decomposition}. +\end{proposition} + +\begin{proof} +First fix $r,t\in\mathcal D$, $t>0$. For all sufficiently large $K$, +$nr$ and $nt$ are integers at every scale $n\ge n_0$ in our sums. +The event \cref{shape:forbidden-triple} for $g_n$ is contained in the +cancellation event of \cref{can:theorem}, at center $nr$ and spacing +$nt$. Consequently its $\mu_K$-mass is at most +\[ + \frac{C}{\epsilon}\sum_{n_0\le n0$. Fix a power-of-two interval $I$ in the +allowed window about one endpoint and another such interval $I'$ in the +allowed window about the other endpoint. The two endpoints can be +interchanged. Put +\[ + W=J(I)+V(I),\qquad H=h(I'), +\] +and fix a threshold $\eta>0$. These are fixed bounded Lipschitz +functions of finitely many shortfalls, and $W\ge0$. All their endpoints +align with the integer times at every scale in the sum once $K$ is large. +By \cref{scale:individual-test-bound} and Markov's inequality, +\[ + \sup_{n_0\le n\eta/2) + \le C_{I',\eta}\epsilon. +\] +Applying \cref{bridge:theorem} at the integer bridge length $rn$, summing, +and using \cref{scale:local-mass}, we obtain +\begin{align} + &\int \one_{\{d(u,v)>c_b r,\ H>\eta\}}W\,\dd\mu_K + \nonumber\\ + &\qquad\le C\epsilon\left[ + \int W\,\dd\mu_K+ + \frac1\epsilon\sum_{n_0\le n0$. In fact exact unit distance between the endpoints of any +interval forces exact unit distance on every subinterval, by the triangle +inequality and $d(x,y)\le|x-y|$; hence an interval containing $t$ in its +interior cannot have zero shortfall. + +Choose a positive rational $\eta0$ +therefore has zero $\mu$-mass. There are only countably many choices of +the bridge, the two tests, their orientation, and the rational threshold. +Every violation is covered by one of these choices, proving +\cref{shape:slow-pair} simultaneously for all real $s,t$. + +Both properties are invariant under every translation and positive +dilation of time, including the corresponding division of distances by +the dilation factor. The measure inequality +$(S^{-1})_*\nu\le\mu$ from \cref{scale:decomposition} consequently transfers +them to $\nu$. All the other component measures are dominated by $\mu$, +so they inherit them as well. This also provides a measurable full-measure +set: the exceptional sets were covered by countably many measurable +strict tests before extension to real arguments. +\end{proof} + +The local-to-global propagation below uses this zero-hyperbolic +Gromov-product framework \cite[Section~2.2]{Bestvina1997}, with an +explicit argument on the semimetric arrays rather than an invocation +of a theorem about geodesic real trees. + +\begin{proposition}[Shape classification]\label{shape:classification} +Every nonaffine array in $X_0$ satisfying \cref{shape:exclusions} has one +of the following forms: +\begin{enumerate} +\item every secant has speed at most $c_b$; +\item there is one exterior ray of exact unit speed, and the complementary +closed half-line has all secant speeds at most $c_b$; +\item there are two exterior rays of exact unit speed, separated by a +closed interval $[a,b]$ of positive length, and every secant contained in +$[a,b]$ has speed at most $c_b$. +\end{enumerate} +The second possibility includes either orientation. These descriptions +do not assert additivity of distances across the slow part. +The three temporal configurations are illustrated in \cref{shape:figure}. +\end{proposition} + +\begin{proof} +We first show that a connected component $U$ of $\R\setminus P_d$ is +affine with a single speed throughout. Its local speeds agree on +overlapping neighborhoods, and hence are a common $w\in[0,1]$ by +connectedness. A compact subinterval of $U$ has a sufficiently fine +partition $x_0<\cdots0$, propagate additivity +inductively. Suppose it holds from $x_0$ through $x_j$, and use the +four-point inequality of \cref{scale:tree-inequality} at base $v=x_j$ in its +limiting form +\begin{equation}\label{shape:tree-inequality} + B_v(x,z)\ge\min\{B_v(x,y),B_v(y,z)\},\qquad + B_v(x,z)=d(v,x)+d(v,z)-d(x,z). +\end{equation} +The inductive additivity gives +$B_v(x_0,x_{j-1})=2d(x_{j-1},x_j)>0$, whereas local additivity gives +$B_v(x_{j-1},x_{j+1})=0$. Apply +\cref{shape:tree-inequality} to the latter pair with intermediate point +$x_0$. Nonnegativity of $B_v$ forces $B_v(x_0,x_{j+1})=0$, exactly the +next additivity statement. The first two steps are already locally +affine. Thus $d(x_0,x_m)=w(x_m-x_0)$, and varying the compact subinterval +proves the claim. In particular $P_d\ne\varnothing$ for a nonaffine +array. + +Suppose a component $U$ has speed $w>c_b$. It has at least one finite +endpoint $s\in P_d$, since $U=\R$ would imply affinity. If $w<1$, every +interior point $t\in U$ belongs to $Q_d$. Continuity at the endpoint gives +$d(s,t)=w|t-s|$, contradicting \cref{shape:slow-pair}. Therefore $w=1$. +Such a component cannot have two finite endpoints: they would both belong +to $P_d\subset Q_d$, and their unit mutual speed would again contradict +\cref{shape:slow-pair}. Hence every component of speed greater than +$c_b$ is an exterior unit ray; there are at most two. + +Let $I$ be the closed complement of these rays. It is the whole line, a +closed half-line, or a closed bounded interval. For $x0$, and, with the fixed constant +\begin{equation}\label{shape:cross-constant} + C_*=2+\frac6\delta, +\end{equation} +every $z\in\R$ and $t>0$ satisfy +\begin{equation}\label{shape:cross-bound} + 0\le 1-\frac{d(z,z+t)}t\le\frac{C_*L}{t}. +\end{equation} +The shortfall is zero whenever the interval is wholly in either closed +unit ray. +\end{lemma} + +\begin{proof} +For the moment permit $L=0$. Write +$x_u=a-u$, $y_v=b+v$, where $u,v\ge0$, and put $D=d(a,b)\le L$. +Set +\[ + s=\frac{u+v-D-d(x_u,y_v)}2. +\] +If $s\le0$, the absolute shortfall of $[x_u,y_v]$ is at most +$L+D\le2L$. Suppose $s>0$. The reverse triangle inequalities imply +$d(x_u,y_v)\ge u-D-v$ and $d(x_u,y_v)\ge v-D-u$, so that +$s\le\min\{u,v\}$. At base $a$ the unit rays and triangle inequalities +give +\begin{align*} + B_a(x_u,x_s)&=2s,\\ + B_a(x_u,y_v)&\ge u+(v-D)-d(x_u,y_v)=2s,\\ + B_a(y_v,y_s)&\ge(v-D)+(s-D)-(v-s)=2s-2D. +\end{align*} +Twice applying \cref{shape:tree-inequality} yields +$B_a(x_s,y_s)\ge2s-2D$. As $d(a,x_s)=s$ and +$d(a,y_s)\le D+s$, it follows that +\begin{equation}\label{shape:matched-ray-distance} + d(x_s,y_s)\le3D\le3L. +\end{equation} + +Use the temporal midpoint $m=(a+b)/2$ and spacing $t_*=s+L/2$. +Its two wing distances obey +\[ + d(x_s,m),\ d(m,y_s)\ge s-L/2. +\] +If $L>0$ and $s>3L/\delta$, these bounds are strictly greater than +$(1-\delta)t_*$ and \cref{shape:matched-ray-distance} is strictly less +than $\delta t_*$. This is the forbidden triple of +\cref{shape:forbidden-triple}. If $L=0$, every $s>0$ produces that same +strict contradiction. Thus always $s\le3L/\delta$. A full crossing +interval has absolute shortfall +\[ + (u+L+v)-d(x_u,y_v)=L+D+2s\le C_*L. +\] +An interval meeting only part of the slow interval has absolute +shortfall at most twice the length of that part. For example, if +$x0$. +\end{proof} + +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=1.05cm,y=1cm,>=Latex,font=\small] + \node[anchor=east] at (-4.4,2.7) {No unit ray}; + \draw[gray!65,line width=3pt] (-4,2.7)--(4,2.7); + \node[fill=white,inner sep=2pt] at (0,2.7) {all secants have speed $\le c_b$}; + \node[anchor=east] at (-4.4,1.5) {One unit ray}; + \draw[blue!65!black,line width=1.1pt,->] (-1,1.5)--(-4,1.5); + \draw[gray!65,line width=3pt] (-1,1.5)--(4,1.5); + \fill (-1,1.5) circle (1.6pt); + \node[above] at (-2.6,1.5) {unit speed}; + \node[above] at (1.5,1.5) {slow half-line}; + \node[below] at (-1,1.5) {$a$}; + \node[anchor=east] at (-4.4,0) {Two unit rays}; + \draw[blue!65!black,line width=1.1pt,->] (-1,0)--(-4,0); + \draw[gray!65,line width=3pt] (-1,0)--(1,0); + \draw[blue!65!black,line width=1.1pt,->] (1,0)--(4,0); + \draw[<->] (-1,.55)--(1,.55) node[midway,above] {$L=b-a>0$}; + \foreach \x in {-3.3,-1,0,1,3.3}{\fill (\x,0) circle (1.6pt);} + \node[below] at (-3.3,0) {$x_s=a-s$}; + \node[below] at (-1,0) {$a$}; + \node[below] at (0,0) {$m$}; + \node[below] at (1,0) {$b$}; + \node[below] at (3.3,0) {$y_s=b+s$}; +\end{tikzpicture} +\caption{The possible temporal regions in \cref{shape:classification}; +the one-ray picture may also be reflected. A slow region need not be +affine. In the bottom picture, excessive loss between the exterior rays +would produce the symmetric forbidden triple $(x_s,m,y_s)$ used in +\cref{shape:crossloss}. The diagram describes time regions, not an +embedding of the semimetric.} +\label{shape:figure} +\end{figure} + +\FloatBarrier +\section{A capped deficit and the final balance} +\label{end:section} + +The cap will compare the slow interval of a limiting shape at fine and +coarse scales. It must equal one on uniformly slow secants. For a +two-ray shape, \cref{shape:cross-bound} makes the shortfall of a long +crossing interval small; a sufficiently small slope of the cap near +zero will make its averaged coarse contribution smaller than the +fine-scale occupancy of the slow interval. This is an integrated +comparison, not a pointwise sign assertion for the observable below. + +Choose a fixed $\alpha>0$ so small that $10C_*\alpha<1$. Choose a smooth +nondecreasing function $\Phi:[0,1]\to[0,1]$ such that +\begin{equation}\label{end:cap} + \Phi(u)=\alpha u\ \text{near }0,\qquad + \Phi(u)=1\quad(1-c_b\le u\le1). +\end{equation} +For completeness, one can choose a nonnegative smooth derivative equal +to $\alpha$ near zero, zero near $1-c_b$, and with integral one on +$[0,1-c_b]$, by adding a nonnegative bump supported strictly between the +two endpoint neighborhoods. Integrating and continuing constantly gives +\cref{end:cap}. Positivity near zero and compactness give constants +$00. +\end{equation} +\end{proposition} + +\begin{proof} +Stationarity makes the two half-interval terms in +\cref{end:G-definition} have equal expectation. Since +$h_2(0;g_n)=h_1(0;g_{2n})$, summing over the dyadic scales gives exactly +\[ + \int G\,\dd\mu_K + =\int\Phi(h_1(0))\,\dd\nu_K + -\frac1\epsilon\E\Phi(h_1(0;g_{n_0})). +\] +The final term tends to zero by +\cref{scale:initial-test-bound,end:linear-bounds}. Thus these integrals +converge to $H_\Phi$. We must show that this mass does not escape to +$\mathcal A$ under vague convergence. + +The measures $J\mu_K$ and $V\mu_K$ have uniformly bounded total mass on +the compact space $X$, and the signed measures $G\mu_K$ have uniformly +bounded total variation by \cref{end:G-bound}. Take simultaneous weak +limits +\[ + J\mu_K\longrightarrow\lambda_J,\qquad + V\mu_K\longrightarrow\lambda_V,\qquad + G\mu_K\longrightarrow\eta. +\] +On $\mathcal U$ they agree with $J\mu$, $V\mu$, and $G\mu$ respectively: +multiply any compactly supported continuous test there by the indicated +continuous weight. We show that $\eta|_{\mathcal A}=0$ by treating all +three ranges of affine speeds. Write $d_w(s,t)=w|s-t|$. + +\paragraph{Speeds $0\le w\le c_b$.} +Both $\Phi'$ and $\Phi''$ vanish at every shortfall $1-w$ in this compact +speed range, including its boundary $w=c_b$. For each $\rho>0$ there is +an open neighborhood $U_\rho$ of +$\{d_w:0\le w\le c_b\}$ on which the Taylor argument for +\cref{end:G-bound} gives +\[ + |G|\le\rho(J+V). +\] +Indeed all arguments between the three relevant shortfalls are then +uniformly close to $[1-c_b,1]$, where both derivatives vanish. +For a continuous test supported in $U_\rho$, pass this domination to +the weak limits. It implies +$|\eta||_{U_\rho}\le\rho(\lambda_J+\lambda_V)|_{U_\rho}$. +The latter positive measure is finite on $X$. Letting $\rho$ decrease +to zero shows that $\eta$ has zero restriction to this compact set of +affine arrays. + +\paragraph{Speeds $c_b2$. Put $W=J+V$ and +$H=h([R,R+1])$. For each positive rational $\eta_0$, the open set +\[ + U_{\eta_0}=\{d:d(0,R)>c_bR,\ h([R,R+1])>\eta_0\} +\] +has $W\mu_K$-mass tending to zero by +\cref{shape:weighted-bridge}. The open-set inequality for weak +convergence of positive measures gives +$(\lambda_J+\lambda_V)(U_{\eta_0})=0$. Each $d_w$ in the current +speed range belongs to such a set by taking $0<\eta_0<1-w$. +This countable cover and \cref{end:G-bound} show that $\eta$ vanishes +on all intermediate affine speeds. Notice that the threshold is fixed +separately for each test; no threshold uniform as $w\uparrow1$ is needed. + +\paragraph{The speed $w=1$.} +Near $d_1$, all three arguments in \cref{end:G-definition} are in the +linear part of $\Phi$, so $G=\alpha J$. It remains to prove +$\lambda_J(\{d_1\})=0$. Fix an inverse power of two $q\in(0,1)$. +For the product over $[0,2n]$, with center $n$, put +\[ + D=n-\min\{g(0,n),g(n,2n)\},\qquad + \ell=g(0,n)+g(n,2n)-g(0,2n)=2nJ(g_n). +\] +The quantities $D/n$ and $\ell/n$ are continuous tests of $g_n$, +both zero at $d_1$. Thus the condition +$2q-2D/n>\ell/n$ defines an open neighborhood $U_q$ of $d_1$. +Eventually $s=qn$ is an integer at every scale in the sum. +On $U_q$, the truncation estimate of \cref{scale:truncation} gives, +with its fixed additive constant $A$, +\[ + \ell\le\ell_{\rm cut}+A, + \qquad + J\le qJ([1-q,1+q])+\frac{A}{2n}. +\] +The shorter-interval test telescopes exactly, by stationarity: +\[ + \int J([1-q,1+q])\,\dd\mu_K + =\frac{e_{qN}-e_{qn_0}}\epsilon\le1, +\] +where dyadic monotonicity gives $e_{qN}\le e_N=\epsilon$. +Also $\epsilon^{-1}\sum_n n^{-1}=O((\epsilon n_0)^{-1})=o(1)$. +Since $J\ge0$, +\[ + \int_{U_q}J\,\dd\mu_K\le q+o(1). +\] +The open-set inequality now gives $\lambda_J(U_q)\le q$, so +$\lambda_J(\{d_1\})\le q$. First take the fixed weak limit and then +let $q$ decrease through inverse powers of two to obtain zero. The +local identity $G=\alpha J$ transfers this conclusion to $\eta$. + +These cases cover $\mathcal A$. Weak convergence on compact $X$ +preserves the total mass of the signed measures, so +$\eta(X)=H_\Phi$. As $\eta|_{\mathcal A}=0$ and +$\eta|_{\mathcal U}=G\mu$, this is \cref{end:balance-equation}. +\end{proof} + +\subsection{The terminal measure and its backward dilates} + +The decomposition in \cref{scale:decomposition}, together with +\cref{end:G-bound}, gives an absolutely convergent identity +\begin{equation}\label{end:integrated-decomposition} + \int G\,\dd\mu + =\int G\,\dd\mu_\infty + +\sum_{j=1}^{\infty}\int G(S^{-j}d)\,\dd\nu(d). +\end{equation} +More precisely, the sum of the integrals of $|G(S^{-j}d)|$ is at most +$\int|G|\,\dd\mu$, so all the subsequent rearrangements of these +component integrals are justified. + +\begin{lemma}[Strict bound for the terminal contribution] +\label{end:terminal} +The series in \cref{end:integrated-decomposition} satisfies +\begin{equation}\label{end:terminal-strict} + \sum_{j=1}^{\infty}\int G(S^{-j}d)\,\dd\nu(d)0$. +Consequently the subtracted integral in \cref{end:terminal-fatou} is +positive. It is finite by Fatou's lemma and the bound $A_m\le C_\Phi$. +This proves \cref{end:terminal-strict} in all cases. +\end{proof} + +\subsection{The translation- and dilation-invariant remainder} + +We next give the measurable and local-finiteness details needed to +integrate over shapes. A dyadic interval whose endpoint distance equals +its length has exact unit speed throughout, by the triangle inequality +and the Lipschitz bound. Thus the local unit-speed set is the union of +the interiors of those countably many intervals. More directly, define +extended-real endpoint functions on $X_0$ by +\begin{align} + a(d)&=\sup\{r\in\mathcal D: + d(p,q)=q-p\text{ for all }p,q\in\mathcal D, pk_0$, the top exponent is positive on a set of positive +normalized area. Increasing the threshold gives the same statement +for every $k\ge k_0$. The entropy formula \cref{main:pesin} gives +$h_m(f_k)>0$ for every such $k$. In particular the measurable parameter +interval $[k_0+1,k_0+2]$ has positive one-dimensional Lebesgue measure +and satisfies the required assertion. +\end{proof} diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/introduction.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..b0f39abb16bb3c2e239143090446910ed5186bbf --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,191 @@ +\section{Introduction} +\label{sec:introduction} + +Let $m$ be normalized Lebesgue area on $\T^2=\R^2/\Z^2$. +The standard sine map is the area-preserving diffeomorphism +\begin{equation}\label{main:map} + f_k(x,y)=\bigl(x+y+k\sin(2\pi x),\;y+k\sin(2\pi x)\bigr) + \pmod{\Z^2},\qquad k\in\R. +\end{equation} +We consider its Kolmogorov--Sinai entropy $h_m(f_k)$ for this fixed +invariant measure. The distinction between topological complexity and +chaos visible to area is essential: a hyperbolic set may have zero area, +and even full Hausdorff dimension does not imply positive area. + +\begin{theorem}\label{main:entropy} +There exists $k_0>0$ such that +\[ + h_m(f_k)>0\qquad\text{for every }k\ge k_0. +\] +Equivalently, for every such $k$ the largest Lyapunov exponent of $f_k$ +is positive on a set of positive $m$-measure. +\end{theorem} + +In particular the parameter set with positive metric entropy contains +an interval of positive length. Thus \cref{main:entropy} resolves +affirmatively Sinai's positive metric entropy conjecture in the +formulation of Berger and Turaev +\cite[Conjecture~0.4]{BergerTuraev2019}, and gives the stronger +all-sufficiently-large-parameter conclusion. This large-parameter +formulation is also recorded by Obata \cite{Obata2021}. +Berger and Turaev's recurrence +$(x,y)\mapsto(2x-y+a\sin(2\pi x),x)$ is conjugate to +\cref{main:map} by $(x,y)\mapsto(x,x-y)$, with $a=k$. +The assertion concerns both parameter measure and, at each parameter, +phase-space area. It makes no claim of ergodicity, almost-everywhere +positive exponents, or a uniform positive lower bound on entropy. + +\subsection*{Context} + +The standard map arose as a model of conservative instability +\cite{Chirikov1979}. Its large-parameter regime combines strong local +expansion with recurrent losses of derivative growth. Duarte's +large-parameter analysis constructs hyperbolic basic sets and exhibits +elliptic-island and tangency phenomena in the standard family +\cite{Duarte1994}. Gorodetski \cite{Gorodetski2012} obtained transitive +invariant sets of full Hausdorff dimension for a residual set of large +parameters. These results reveal extensive deterministic structure, +but neither hyperbolic sets nor full dimension imply positive area; +residual parameter sets also need not have positive Lebesgue measure. + +Other entropy results use different measures or notions. Obata +\cite{Obata2021} established uniqueness of the measure of maximal +entropy for sufficiently large standard-map parameters, without +identifying that measure with area. Oliveira +\cite[Theorem~5.1]{Oliveira2024} proved positive topological entropy +for every $k>0$ with $k\ne2/\pi$ in the normalization of +\cref{main:map}. Positive topological entropy does not by itself give positive +entropy for the specific invariant measure $m$. Berger and Turaev +\cite{BergerTuraev2019} proved Herman's positive-entropy conjecture by +conservative perturbations, including perturbations near standard +maps; these are not restricted to amplitude changes in \cref{main:map}. +Related hyperbolicity is available for Anosov-driven skew products +with standard-map fibers +\cite{BergerCarrasco2014} and randomly perturbed standard maps +\cite{BlumenthalXueYoung2017}. Coupling, noise, and smooth +perturbation change the dynamical system or its probability law. The +question here concerns Lebesgue area for the exact, fixed sine map at +each sufficiently large amplitude. + +We use the established entropy formula, not a new entropy theory. +For a smooth area-preserving diffeomorphism of the compact torus, +the multiplicative ergodic theorem \cite{Oseledets1968} gives +almost-everywhere Lyapunov exponents. The determinant-one condition +makes them $\lambda_+(z)$ and $-\lambda_+(z)$, with +$\lambda_+(z)\ge0$. The entropy formula of Pesin, in the absolutely +continuous invariant-measure form +\cite{Pesin1977,Mane1981,Mane1983Errata}, gives +\begin{equation}\label{main:pesin} + h_m(f_k)=\int_{\T^2}\lambda_+(z)\dd m(z). +\end{equation} +The map is $C^\infty$, the manifold is compact without boundary, +and $m$ is an absolutely continuous invariant probability. The +derivative and inverse-derivative logarithms are bounded at each fixed +parameter, so all integrability hypotheses hold. Neither ergodicity +nor a prior assumption of nonzero exponents is required. Consequently +the task is to prove positivity of $\lambda_+$ on positive area. + +\begin{corollary}[Nonuniformly hyperbolic ergodic component] +\label{main:components} +For each $k\ge k_0$ there is a measurable $f_k$-invariant set $E=E(k)$ +with $m(E)>0$ such that $(E,m(E)^{-1}m|_E,f_k)$ is ergodic and $f_k$ has +one positive and one negative Lyapunov exponent almost everywhere on $E$. +For some integer $N=N(k,E)\ge1$, there is a measurable partition +$E=E_0\sqcup\cdots\sqcup E_{N-1}$ such that +$f_k(E_j)=E_{j+1\bmod N}$ and +$(E_j,m(E_j)^{-1}m|_{E_j},f_k^N)$ is Bernoulli for every $j$. +All invariance and partition statements are modulo $m$-null sets; +each $E_j$ has area $m(E)/N>0$. +\end{corollary} + +\begin{proof} +Fix $k\ge k_0$ and let $H_k$ be the invariant Oseledets set where +$\lambda_+>0$. It has positive area by \cref{main:entropy}, while +$\det Df_k=1$ makes the other exponent $-\lambda_+<0$ there. +The normalized restriction $\nu_k=m(H_k)^{-1}m|_{H_k}$ is an absolutely +continuous invariant probability on the compact torus, with no zero +Lyapunov exponents. Pesin's ergodic and Bernoulli decomposition theorems, +as stated in \cite[Theorems~3.3--3.4]{KatokBurns1994}, apply to $\nu_k$ +and give a positive-$\nu_k$ ergodic component $E$ and its finite cyclic +partition. Since $\nu_k(E)=m(E)/m(H_k)>0$, the component has positive +area; normalizing $\nu_k$ on $E$ or on a partition piece agrees with +normalizing $m$. Area preservation gives $m(E_j)=m(E)/N$. +\end{proof} + +\subsection*{Proof mechanism} + +The proof has two quantitative inputs and a multiscale conclusion. +In transfer coordinates the derivative is a product of determinant-one +matrices $T_{i,j}$. With the one-step bound $M=2\pi k+4$, put +$g(i,j)=\log_M\norm{T_{i,j}}$. The bound $0\le g(i,j)\le|i-j|$ +and the triangle inequality make $g$ a stationary semimetric on orbit +time. We track the expected shortfall $1-\E g(0,n)/n$ from maximal +growth at each length $n$. + +First, \cref{can:theorem} bounds the probability that two long segments +grow nearly at the maximal rate but their concatenation almost +completely cancels. A cancellation supplies a linearized solution that +is small at both endpoints. Matched expansion scales on its two sides +give many local tests. A uniform analytic scarcity estimate, including +the degenerate cases of nearly cancelling reciprocal cosines, converts +those tests into exponential probability decay. Finite-time direction +and tangency geometry in the standard family has precedents in +\cite{BloorLuzzatto2009}; the matched-scale scarcity estimate used +here is proved below, rather than imported from that work. + +Second, \cref{bridge:theorem} gives a weighted product estimate for +observations near opposite ends of a rapidly growing segment. Finite +Dirichlet problems construct two transverse contracting graph families. +Their uniformly controlled slice Jacobians yield the estimate by +changes of variables. This step does not presume mixing or independence +of orbit segments. For comparison, Blumenthal +\cite{Blumenthal2021} obtains fixed-parameter correlation +bounds that are useful over a finite parameter-dependent time range. +Those bounds do not supply the weighted estimate across fast bridges in +\cref{bridge:product}, which applies at arbitrary sufficiently large +bridge lengths; its restriction to fast bridges is essential. + +Suppose now that arbitrarily large parameters had zero exponent almost +everywhere. The shortfall at every fixed length tends to zero as the +parameter grows, whereas at each fixed zero-exponent parameter it tends +to one as the length grows. Cancellation scarcity bounds how much this +shortfall can jump when the length doubles. We can therefore select a +range of dyadic lengths over which it first increases and normalize by +the small terminal shortfall. Rescaling the distance arrays produces +measures that are locally finite away from the affine arrays +$d(s,t)=w|s-t|$, where every interval grows at one common linear rate. +The bridge estimate restricts which slow intervals can coexist with a +fast connecting segment. Together with cancellation scarcity and a +four-point inequality, it leaves three kinds of nonaffine limit arrays: +an entirely slow array, or one with one or two exterior unit-speed rays. + +The final comparison uses a capped shortfall: it is linear with a small +slope near zero, and equals one on uniformly slow intervals. Sum its +whole-interval value minus the average of its two half-interval values +over the selected dyadic lengths. This telescopes to a positive +normalized terminal value, since the initial contribution vanishes. +The limiting sum decomposes into the backward dilates of the terminal +measure and a translation- and dilation-invariant remainder. The first +contribution is strictly less than that same terminal value: a nonzero +terminal measure retains a slow interval visible at arbitrarily fine +scales. The invariant remainder's contribution is nonpositive. For its two-ray shapes, +translation and dilation averaging compare fine-scale occupancy of the +slow interval with the smaller capped loss over long crossing intervals. +These two bounds contradict the positive telescoping balance. + +Passing to the limit in that balance is itself a proof obligation. +Vague convergence away from affine arrays does not control weighted mass +approaching them. The cap is flat at slow affine speeds; the bridge +estimate removes intermediate affine speeds; and cancellation truncation +controls the remaining unit-speed array. Thus the same quantitative +inputs govern both the nonaffine shapes and the mass that might otherwise +escape them. + +\Cref{sec:setup} fixes the transfer coordinates and growth conventions. +\Cref{can:section,loc:section} prove cancellation scarcity and its local +analytic estimate. \Cref{bridge:section} supplies the finite-graph +product estimate. \Cref{sec:scales} constructs the dyadic limit measures; +\cref{shape:section} classifies their support, and \cref{end:section} +completes the capped-deficit balance. These quantitative ingredients +are proved in the form needed here; no external standard-map entropy +assertion or independence hypothesis is imported. diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/local-scarcity.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/local-scarcity.tex new file mode 100644 index 0000000000000000000000000000000000000000..749580b33ae67443173c5ec99b5a6c7d85f2c7de --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/local-scarcity.tex @@ -0,0 +1,553 @@ +\section{Local scarcity at matched scales} +\label{loc:section} + +This section proves \cref{can:local-scarcity}. We use its two outward +paths and the notation established in the cancellation argument. In +particular, $c=99/100$, the central coordinate is fixed, and the variable +is $\theta=q_{e-1}$. The selected indices are denoted by $j_+$ and +$j_-$. Every assertion concerning an interval in $\theta$ uses real +lifts of the recurrence. The constants below are independent of the +selected indices and of the base configuration. + +The central relation in \cref{can:center-equation} will become, after rescaling, an approximate +identity between a cubic polynomial and two shifted reciprocal-cosine +terms. A positive-measure Hausdorff limit of the closures of successful +sets would force an analytic identity. We rule out that identity even when the two +reciprocal terms nearly cancel or one phase slope tends to zero. The +derivative estimates in the first two subsections provide the uniform +remainders needed for these two degeneracies. + +\subsection{Control on a single rescaled path} + +Fix either outward path and write $j$ for its selected index. Put +\[ + L_j=\abs{t_j(\theta_*)},\qquad + \xi=L_j(\theta-\theta_*),\qquad D=\frac{\dd}{\dd\xi}. +\] +The initial position $q_0$ is constant and $q_1$ is affine in $\xi$, +with derivative $\pm L_j^{-1}$. By \cref{can:good-t}, the base +linearized solution satisfies +\begin{equation} + \frac{\abs{t_l(\theta_*)}}{L_j} + \le M^{-c(j-l)}\quad(1\le l\le j). + \label{loc:base-growth} +\end{equation} +We will prove, throughout $\abs{\xi}\le1$, that +\begin{align} + \abs{q_l(\xi)-q_l(0)}+\abs{Dq_l(\xi)} + &\le C M^{-c(j-l)}, \label{loc:first-derivatives}\\ + \abs{D^p q_l(\xi)} + &\le C_p M^{-pc(j-l)}M^{1-pc} + \quad(2\le p\le5), \label{loc:higher-derivatives} +\end{align} +for $0\le l\le j$. The terms with $l=0$ vanish. + +Write $B_l=M^{-c(j-l)}$ and $\Delta_l=q_l(\xi)-q_l(0)$. +Taylor's formula for the recurrence, followed by propagation with the +base linearized matrices, gives +\begin{equation} + \abs{\Delta_l} + \le \abs{\xi}\frac{\abs{t_l(\theta_*)}}{L_j} + +C\sum_{d=1}^{l-1}M^{l-d}\abs{\Delta_d}^2. + \label{loc:nonlinear-comparison} +\end{equation} +Here the quadratic error at site $d$ is at most $CM\abs{\Delta_d}^2$, +and the remaining $l-d-1$ steps have norm at most $M^{l-d-1}$. +For $l=1$ the estimate $\abs{\Delta_l}\le2B_l$ follows directly +from \cref{loc:base-growth}. If it holds for all earlier indices, +the sum in \cref{loc:nonlinear-comparison} is bounded by +\[ + 4C B_l^2\sum_{a\ge1}M^{(1-2c)a}. +\] +Since $B_l\le1$ and $1-2c<0$, this is at most $B_l$ for all +sufficiently large $M$. Induction proves the required displacement +bound, uniformly in $j$. + +Compare $Dq_l(\xi)$ with the base variation +$\pm t_l(\theta_*)/L_j$. Their initial data agree. Their difference +is propagated by the base matrices with forcing +\[ + \bigl(v(q_d(\xi))-v(q_d(0))\bigr)Dq_d(\xi), +\] +whose absolute value is at most $CM\abs{\Delta_d}\abs{Dq_d}$. +Induction with $\abs{Dq_d}\le2B_d$ therefore uses the same geometric +sum and proves \cref{loc:first-derivatives}. + +For completeness, the higher derivative induction uses the recurrence +at the varied orbit. For $p\ge2$ it has the form +\[ + D^p q_{l+1}=v(q_l)D^p q_l-D^p q_{l-1}+F_{p,l}, +\] +where $F_{p,l}$ is a finite sum, with coefficients depending only on +$p$, of terms +\[ + \phi^{(s)}(q_l)\prod_{i=1}^{s}D^{m_i}q_l, + \qquad 2\le s\le p,\quad m_i\ge1,\quad + \sum_{i=1}^{s}m_i=p. +\] +Each $m_i$ is strictly smaller than $p$. The first derivative bound +and the induction hypothesis give +$\abs{D^{m_i}q_l}\le C_{m_i}B_l^{m_i}$, since +$M^{1-m_ic}\le1$ whenever $m_i\ge2$. Consequently +$\abs{F_{p,l}}\le C_p M B_l^p$. The initial higher derivatives +at sites $0,1$ vanish. Propagation by the varied matrices gives +\[ + \abs{D^p q_l} + \le C_p\sum_{d=1}^{l-1}M^{l-d}B_d^p + \le C_p B_l^p\sum_{a\ge1}M^{(1-pc)a} + \le C'_p B_l^p M^{1-pc}. +\] +This proves \cref{loc:higher-derivatives} successively for +$p=2,3,4,5$. + +At the selected site $Dq_j(0)=\pm1$. The case $p=2$ shows that +\[ + Dq_j(\xi)=Dq_j(0)+O(M^{1-2c}). +\] +It follows, for all sufficiently large $M$, that +\begin{equation} + \abs{t_j(\xi)}\ge L_j/2,\qquad + \abs{t_l(\xi)}\le C L_j M^{-c(j-l)}\quad(l2+4/5$, this proves \cref{loc:boundary-fourth}. + +\subsection{The cubic relation at successful points} + +Return to the two paths and put +$L=\max(L_{j_+},L_{j_-})$, $h=L(\theta-\theta_*)$. +Number the paths so that the first has the larger base scale. Write +$Q_i(h)$ for their selected positions and $E_i(h)$ for the quotients +in \cref{loc:quotient}. With $b_i=Q_i'(0)$ and $d_i=Q_i''(0)$, +the preceding estimates imply +\begin{align} + Q_i(h)&=z_i+b_ih+\tfrac12d_ih^2+\mathcal E_i(h), + &\norm{\mathcal E_i}_{C^1([-1,1])} + &\le C\abs{b_i}^3M^{1-3c}, \label{loc:phase-taylor}\\ + \abs{d_i}&\le C\abs{b_i}^2M^{1-2c}, + &\abs{b_1}&=1,\quad + \abs{b_2}=r\in[M^{-3/5},1], \label{loc:phase-slopes}\\ + \abs{\partial_h^pE_i}&\le C_pM^{-c}\abs{b_i}^p + &&(0\le p\le4). \label{loc:scaled-quotient} +\end{align} +The scale change is $\xi=(L_{j_i}/L)h=\abs{b_i}h$. +It also gives, for $2\le p\le5$, +\begin{equation} + \abs{Q_i^{(p)}}\le C_p\abs{b_i}^pM^{1-pc},\qquad + \abs{Q_i'}\asymp\abs{b_i}. + \label{loc:scaled-phase-derivatives} +\end{equation} + +At a successful point there is a common solution with central value +$1$, and its two outward values at site $1$ add to $v(q_e)$. +Apply \cref{can:boundary-expansion} to both paths at that point. +Since $\abs{t_{j_i}(\theta)}=L\abs{Q_i'(h)}$ and +$v(Q_i)-E_i=K(\cos(2\pi Q_i)+(2-E_i)/K)$, multiplication by +$KL^2$ yields +\[ + KL^2\left(v(q_e)-\sum_{i=1}^{2}R_{j_i}(\theta)\right) + =\sum_{i=1}^{2} + \frac{1}{(Q_i')^2(\cos(2\pi Q_i)+(2-E_i)/K)} + +O(r^{-2}M^{1-3c}). +\] +The error bound uses \cref{loc:scaled-phase-derivatives} to compare +the varied $t_{j_i}$ with their base sizes. Replace the left side by +its cubic Taylor polynomial $P(h)$ at $0$. The contribution of each +$R_{j_i}$ to the Taylor remainder is at most +\[ + C K L^2\abs{b_i}^4 L_{j_i}^{-2}M^{-2-\rho} + =C K\abs{b_i}^2M^{-2-\rho}. +\] +Consequently every successful $h\in[-1,1]$ satisfies +\begin{equation} + P(h)=\sum_{i=1}^{2} + \frac{1}{(Q_i'(h))^2 + (\cos(2\pi Q_i(h))+(2-E_i(h))/K)} + +O(M^{-1-\rho}+r^{-2}M^{1-3c}). + \label{loc:cubic-relation} +\end{equation} +The polynomial may depend on the configuration. Its degree is at +most three. The displayed error is uniform on the successful set, +and tends to zero uniformly over all configurations, since +$r^{-2}M^{1-3c}\le M^{-77/100}$. + +\subsection{Polynomial extraction from a Hausdorff limit} + +Consider any sequence of configurations with $M\to\infty$ and +nonempty successful sets $\mathcal S_M$. Pass to a subsequence on +which their closures converge in Hausdorff distance to a nonempty +compact set $F\subset[-1,1]$. We prove that $F$ has measure zero. +Suppose to the contrary that it has positive measure. + +We record precisely the compactness argument for the polynomials +that will be used below. Suppose, after an indicated subtraction +or multiplication of \cref{loc:cubic-relation}, a polynomial $P_M^*$ +of degree at most three agrees at successful points, up to an error +tending to zero, with functions converging uniformly on every compact +subinterval of $[-1,1]$ away from a fixed finite set of limiting poles +in that interval. Choose four +distinct points of $F$ outside those poles. Hausdorff convergence +and approximation of points in the closure supply successful points +converging to these four points. The values of $P_M^*$ there are +bounded. The associated Vandermonde matrices converge to an invertible +matrix, so all four coefficients of $P_M^*$ are bounded. Pass to a +further subsequence on which the polynomials converge coefficientwise. +For every point of $F$ away from the poles the same approximation +argument shows equality with the limiting function. In particular, +equality holds on a set of positive measure. Clearing the finitely +many denominator factors used in that case gives an identity between +real analytic functions on $\R$: the difference has an accumulation +point of zeros and hence vanishes identically. Passing to further +subsequences throughout does not change the fixed Hausdorff limit $F$. + +\subsection{Nonzero limiting slopes and opposite cosines} + +First suppose that $r$ has a positive limit $r_0$. Take convergent +subsequences of the phases modulo integers and of the slopes. The +first slope has a fixed sign after a further subsequence, so write +its value as $b\in\{-1,1\}$. By +\cref{loc:phase-taylor,loc:scaled-quotient}, the two terms on the +right of \cref{loc:cubic-relation} converge, away from their limiting +poles, to +\[ + \sec(2\pi(z+bh))+ + r_0^{-2}\sec(2\pi(z'+b'h)),\qquad \abs{b'}=r_0. +\] +Polynomial extraction and multiplication by both cosines show that +the zero sets of these cosines on $\R$ coincide: at a zero of either +cosine, the other must vanish because its coefficient is positive. +Their zero sets are arithmetic progressions with respective spacings +$1/(2\abs{b})$ and $1/(2\abs{b'})$, so $r_0=1$. +The two cosines are therefore equal or opposite. Equality would leave +twice a secant, which cannot equal a polynomial, as multiplication by +the cosine and evaluation at one of its zeros shows. Only the opposite +case remains. + +Reflect the second phase if necessary and translate it by an odd +half-integer. The resulting phase $\widetilde Q_2$ has the same +limiting affine phase as $Q_1$, while +\[ + \cos(2\pi Q_2)=-\cos(2\pi\widetilde Q_2),\qquad + (Q_2')^2=(\widetilde Q_2')^2. +\] +By \cref{loc:phase-taylor}, +\begin{equation} + \widetilde Q_2-Q_1=p(h)+O_{C^1}(M^{1-3c}),\qquad + \deg p\le2,\qquad \norm{p}_{\rm coeff}\longrightarrow0. + \label{loc:phase-difference} +\end{equation} +Here $\norm{p}_{\rm coeff}$ is the maximum absolute value of its +three coefficients. Define +\[ + \ell=\max(\norm{p}_{\rm coeff},K^{-1}). +\] +Then $\ell\to0$, but every error of order $M^{1-3c}=M^{-197/100}$ +is $o(\ell)$. The error in \cref{loc:cubic-relation} is also +$o(\ell)$ in the present case, since $r\to1$ and $\rho>0$. + +We expand on a compact subinterval away from the zeros of the common +limiting cosine. Put +\[ + f(x,y)=y^{-2}\sec(2\pi x),\qquad + A=2\pi(z_1+b_1h). +\] +The denominators in \cref{loc:cubic-relation} stay uniformly away +from zero there for large $M$. Their shifts are +$2/K+O(M^{-c}/K)$. Expanding the reciprocals, with their respective +cosine signs, gives the sum of the two terms as +\begin{align*} + &f(Q_1,Q_1')-f(\widetilde Q_2,\widetilde Q_2')\\ + &\quad-\frac2K\left((Q_1')^{-2}\sec^2(2\pi Q_1) + +(\widetilde Q_2')^{-2}\sec^2(2\pi\widetilde Q_2)\right) + +o(\ell). +\end{align*} +The reciprocal expansion has error $O(K^{-2}+M^{-c}/K)=o(\ell)$. +It is essential to subtract the two $f$ terms at the actual common +phase before replacing that phase by an affine one. Taylor expansion +of $f$, using \cref{loc:phase-difference}, gives +\[ + f(Q_1,Q_1')-f(\widetilde Q_2,\widetilde Q_2') + =-f_x(Q_1,Q_1')p-f_y(Q_1,Q_1')p'+o(\ell). +\] +Indeed the quadratic error is $O(\ell^2)$ and the phase remainder +is $O(M^{1-3c})$. Moreover +$(Q_1,Q_1')=(z_1+b_1h,b_1)+O(M^{1-2c})$; replacing the derivatives +of $f$ by their values at this affine pair produces only +$O(\ell M^{1-2c})=o(\ell)$. Similarly, the preceding $K^{-1}$ term +can be evaluated at the common affine pair with error $o(K^{-1})$. +We conclude that \cref{loc:cubic-relation} becomes +\begin{equation} + P(h)=\frac{2p'}{b_1^3}\sec A + -\frac{2\pi p}{b_1^2}\sin A\,\sec^2 A + -\frac4{K b_1^2}\sec^2 A+o(\ell). + \label{loc:opposite-expansion} +\end{equation} + +Divide by $\ell$ and take a subsequence such that +$p/\ell\to\bar p$ coefficientwise and $K^{-1}/\ell\to\beta$. +At least one is nonzero, because +$\max(\norm{p/\ell}_{\rm coeff},K^{-1}/\ell)=1$. +Polynomial extraction now applies to $P/\ell$. Multiply its limiting +identity by $b^2\cos^2(2\pi(z+bh))$ and evaluate at every zero of +that cosine. One obtains +\begin{equation} + 2\pi\bar p(h)\sin(2\pi(z+bh))+4\beta=0 + \quad\hbox{at every cosine zero.} + \label{loc:alternating-obstruction} +\end{equation} +The zeros form an unbounded arithmetic progression and their sine +values alternate between $1$ and $-1$. Thus $\bar p$ is bounded on +that progression and must be constant. The alternating signs then +force both that constant and $\beta$ to be zero, contradicting their +normalization. This excludes every positive limiting slope. + +\subsection{A vanishing second slope} + +It remains to treat $r\to0$. After a subsequence, the first term of +\cref{loc:cubic-relation} converges away from its poles to +$\sec(2\pi(z+bh))$, with $\abs{b}=1$. Write the second term as +\begin{equation} + g(h)=\frac1{r^2S(h)},\qquad + S(h)=\left(\frac{Q_2'(h)}r\right)^2 + \left(\cos(2\pi Q_2(h))+\frac{2-E_2(h)}K\right), + \qquad w=\max(\abs{S(0)},r). + \label{loc:slow-denominator} +\end{equation} +Uniformly on $[-1,1]$, +\begin{equation} + \abs{S^{(m)}}\le C_m r^m\quad(1\le m\le4). + \label{loc:slow-derivatives} +\end{equation} +To see every derivative involved, set $a=Q_2'/r$. Then $a$ is +bounded and bounded away from zero, and +\[ + \abs{a^{(m)}}\le C_m r^m M^{1-(m+1)c}\le C_m r^m + \quad(1\le m\le4) +\] +by \cref{loc:scaled-phase-derivatives}. The derivatives through +order four of the cosine factor are $O(r^m)$ by the chain rule; +those of the shift are bounded by $C_mr^mM^{-c}/K$ by +\cref{loc:scaled-quotient}. The product rule proves +\cref{loc:slow-derivatives}. This is the point requiring the fifth +derivative of $Q_2$. + +If $w\to0$, there is also the sharper fact +\begin{equation} + \frac{\abs{S'(0)}}r\longrightarrow2\pi. + \label{loc:slow-nonzero-slope} +\end{equation} +Indeed $a(0)^2=1$ and $\abs{Q_2'(0)}=r$. The cosine with its shift +equals $S(0)\to0$, so $\abs{\sin(2\pi Q_2(0))}\to1$. +Differentiating the factor $a^2$ contributes +$O(rwM^{1-2c})=o(r)$; differentiating the shift contributes +$O(rM^{-c}/K)=o(r)$. The remaining contribution is +$-2\pi\sin(2\pi Q_2(0))Q_2'(0)$, which proves the assertion. + +Take a subsequence on which $r/w$ has a limit. If this limit is +positive, then $w\to0$, and Taylor's formula gives +\[ + \frac{S(h)}w=\frac{S(0)}w+\frac{S'(0)}w h+O(r^2/w). +\] +Here $r^2/w\le r\to0$. The coefficients are bounded; by +\cref{loc:slow-nonzero-slope}, a further subsequence gives a limiting +affine polynomial $A_0(h)$ with nonzero slope. Multiply +\cref{loc:cubic-relation} by $r^2w$. Its first term and its error +tend to zero locally away from the fast poles, while its second term +converges to $1/A_0(h)$ away from the one possible zero of $A_0$. +Polynomial extraction would therefore give a polynomial of degree +at most three equal to $1/A_0$. Multiplication by $A_0$ and evaluation +at its real zero is a contradiction. + +We may consequently assume $r/w\to0$. Then $w=\abs{S(0)}$ +eventually, and +\begin{equation} + \frac{S(h)}{S(0)}=1+O(r/w)\longrightarrow1 + \quad\hbox{uniformly on }[-1,1]. + \label{loc:almost-constant-denominator} +\end{equation} +In particular $S$ has no zeros there for large $M$. +Let $T_3g$ be the cubic Taylor polynomial of $g$ at $0$. +If $w$ stays bounded away from zero, the reciprocal derivative formula +and \cref{loc:slow-derivatives} give $g^{(4)}=O(r^2)$ uniformly. +Thus $g-T_3g=O(r^2)$. Subtracting $T_3g$ from +\cref{loc:cubic-relation} leaves only the fast secant in the limit. +It cannot agree with a cubic polynomial, by multiplication by its +cosine and evaluation at a zero. + +The last case is $w\to0$ and $r/w\to0$. To track also the possibly +large remainder of $g$, differentiate explicitly: +\begin{equation} + g^{(4)}=\frac1{r^2}\left( + \frac{24(S')^4}{S^5} + -\frac{36(S')^2S''}{S^4} + +\frac{6(S'')^2+8S'S'''}{S^3} + -\frac{S''''}{S^2}\right). + \label{loc:reciprocal-fourth} +\end{equation} +By \cref{loc:slow-derivatives,loc:slow-nonzero-slope}, +$S'(h)/S'(0)=1+O(r)$ uniformly. Together with +\cref{loc:almost-constant-denominator}, this makes the first term +in \cref{loc:reciprocal-fourth} equal to +\[ + (24+o(1))\frac{S'(0)^4}{r^2S(0)^5}. +\] +Relative to that term, the remaining three displayed contributions +are respectively $O(w)$, $O(w^2)$, and $O(w^3)$: for example the +first ratio is bounded by $C\abs{S}\abs{S''}/\abs{S'}^2\le Cw$. +All these comparisons are uniform on $[-1,1]$. Put +\[ + a_M=\frac{S'(0)^4}{r^2S(0)^5}. +\] +Taylor's formula with integral remainder, valid on both sides of +zero, now yields +\begin{equation} + g(h)-T_3g(h)=a_Mh^4+o(\abs{a_M}) + \quad\hbox{uniformly on }[-1,1]. + \label{loc:quartic-remainder} +\end{equation} +If $\abs{a_M}\to\infty$, subtract $T_3g$ from +\cref{loc:cubic-relation} and divide by $a_M$. The fast term tends +to zero away from its poles and the original error also tends to +zero after this division. Polynomial extraction would identify a +polynomial of degree at most three with $h^4$, which is impossible. +Otherwise pass to a subsequence with $a_M\to a\in\R$. +Without division, the same subtraction gives a limiting identity +between a cubic polynomial and +\[ + \sec(2\pi(z+bh))+ah^4. +\] +Move the quartic term to the polynomial side and multiply by the +cosine. Evaluation at any of its zeros again gives a contradiction. +These alternatives exhaust the vanishing-slope case. + +\subsection{Uniform neighborhoods of successful sets} + +We have proved that every Hausdorff limit of nonempty closures of +successful sets along $M\to\infty$ has measure zero. This implies +the uniform assertion of \cref{can:local-scarcity}, including the +neighborhood width in its statement. Indeed, if the assertion failed +for some $\eta>0$, then for every integer $n$ one could choose +$M_n\ge n$ and a successful set $\mathcal S_n\subset[-1,1]$ such +that +\[ + \operatorname{Leb}_{\R}\{x:\dist(x,\mathcal S_n)<1/n\}>\eta. +\] +These sets are nonempty. Compactness of the nonempty compact subsets +of $[-1,1]$ for Hausdorff distance supplies a subsequence of their +closures with limit $F$. For every $\varepsilon>0$, eventually +\[ + \{x:\dist(x,\mathcal S_n)<1/n\} + \subset\{x:\dist(x,F)<\varepsilon\}. +\] +Thus every $\varepsilon$-neighborhood of $F$ has measure at least +$\eta$. Continuity of Lebesgue measure from above for these bounded +neighborhoods implies $\operatorname{Leb}_{\R}(F)\ge\eta$, +contrary to the result just proved. The empty successful set satisfies +the claimed estimate automatically. This completes the proof of +\cref{can:local-scarcity}. diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/scales.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/scales.tex new file mode 100644 index 0000000000000000000000000000000000000000..4dbef3451536114ccd280b415d1d607847be31e9 --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/scales.tex @@ -0,0 +1,571 @@ +\section{Product distances and measures across scales} +\label{sec:scales} + +Suppose, toward a contradiction, that there is a sequence of parameters +$k\to+\infty$ for which the top Lyapunov exponent vanishes almost +everywhere. All limits in this section are taken along this sequence, +passing to subsequences when indicated, except where a fixed-parameter +limit in orbit length is explicitly specified. The structural constants +in \cref{can:theorem,bridge:theorem} are fixed throughout; the bridge +estimate's auxiliary constants may depend on its observation bounds. +Use the growth semimetric $g$ and expected shortfall $e_n$ from +\cref{scale:distance-deficit}; their parameter dependence remains +suppressed. At each parameter in the sequence, the zero-exponent +assumption and bounded convergence give +\begin{equation}\label{scale:long-deficit} +e_n\longrightarrow1\qquad(n\to\infty). +\end{equation} +By contrast, the shortfall at every fixed length tends to zero as the +parameter grows. We will select a dyadic range in which the shortfall +first increases, and sum the loss between each whole interval and its +two halves over that range. This sum telescopes, but its limiting mass +must still be controlled near arrays with one constant growth rate. +The local tests and measures below are designed to retain that mass. + +\Needspace{8\baselineskip} +\subsection{Four-point comparison and truncation} + +\begin{lemma}[Four-point comparison]\label{scale:fourpoint} +Fix a base index $i$ and write +\[ + D_j=\norm{T_{i,j}},\qquad + R_{jp}=M^{g(j,p)-g(i,j)-g(i,p)}. +\] +There is an absolute constant $C_0\ge1$ such that, for every orbit, +every sufficiently large parameter, and every choice of integer indices, +\begin{equation}\label{scale:R-fourpoint} + R_{jp}\le C_0\max\{R_{jl},R_{lp}\}. +\end{equation} +\end{lemma} + +\begin{proof} +Choose a most expanded unit input direction $a_j$ for $T_{i,j}$; +if $D_j=1$, any singular direction may be used. Set +\[ + Q_{jp}=\max\{|a_j\wedge a_p|,D_j^{-2},D_p^{-2}\}. +\] +We claim +\begin{equation}\label{scale:R-comparison} + Q_{jp}\le R_{jp}\le3Q_{jp}. +\end{equation} +To obtain the sine lower bound, use a unit input at the base +perpendicular to $a_p$. Its norm at $p$ is $D_p^{-1}$, whereas its +norm at $j$ is at least $D_j|a_j\wedge a_p|$. The transfer from $p$ +to $j$ therefore has norm at least $D_jD_p|a_j\wedge a_p|$. +The inequalities +\[ + \norm{T_{p,j}}\ge D_j/D_p,\qquad + \norm{T_{p,j}}\ge D_p/D_j +\] +give the other two lower bounds; the second uses equality of the norms +of a determinant-one matrix and its inverse. + +For the upper bound, transport a unit vector from $p$ back to the +base. Its components along $a_p$ and $a_p^\perp$ have absolute values +at most $D_p^{-1}$ and $D_p$, respectively. Also +\[ + \norm{T_{i,j}a_p^\perp} + \le D_j|a_j\wedge a_p|+D_j^{-1}. +\] +It follows that +\[ + R_{jp}\le D_p^{-2}+|a_j\wedge a_p|+D_j^{-2}, +\] +proving \cref{scale:R-comparison}. By +\cref{setup:sine-triangle}, +$Q_{jp}\le2\max\{Q_{jl},Q_{lp}\}$, so $C_0=6$ is permissible. +\end{proof} + +\begin{lemma}[Truncation of a cancellation]\label{scale:truncation} +Fix integers $i$, $n\ge1$, and $1\le s\le n$, and put +\[ + L=i-n,\quad R=i+n,\quad L'=i-s,\quad R'=i+s, +\] +\[ + D=n-\min\{g(i,L),g(i,R)\},\qquad + \ell=g(i,L)+g(i,R)-g(L,R). +\] +Use the quantities $R_{uv}$ of \cref{scale:fourpoint} with base $i$. +Then +\begin{align} + g(i,L'),g(i,R')&\ge s-D, + \label{scale:cut-wings}\\ + R_{L'R'}&\le C_0^2\max\{R_{LR},M^{-2s+2D}\}. + \label{scale:cut-R} +\end{align} +There is a constant $A$ independent of the parameter and indices such +that the cut cancellation loss satisfies +\begin{equation}\label{scale:cut-loss} + g(i,L')+g(i,R')-g(L',R') + \ge\min\{\ell,2s-2D\}-A, +\end{equation} +and consequently +\begin{equation}\label{scale:cut-full} + g(L',R')\le2s-\min\{\ell,2s-2D\}+A. +\end{equation} +\end{lemma} + +\begin{proof} +The triangle inequality and \cref{scale:distance-bound} give +\[ + g(i,L')\ge g(i,L)-g(L,L')\ge(n-D)-(n-s)=s-D, +\] +and similarly on the right. In particular, +\[ + R_{LL'}\le M^{(n-s)-(n-D)-(s-D)}=M^{-2s+2D}, + \qquad R_{RR'}\le M^{-2s+2D}. +\] +Apply \cref{scale:R-fourpoint} twice along the chain $L',L,R,R'$ to +obtain \cref{scale:cut-R}. Taking negative base-$M$ logarithms gives +\cref{scale:cut-loss}, with any fixed upper bound for +$2\log_M C_0$ as $A$. Such a bound is uniform after imposing, for +example, $M\ge2$. Finally, each cut wing has length at most $s$, +which yields \cref{scale:cut-full}. +\end{proof} + +\begin{lemma}[Doubling bound]\label{scale:doubling} +For all sufficiently large parameters and every integer $n\ge1$, +\begin{equation}\label{scale:doubling-equation} + e_n\le e_{2n}\le C e_n+\frac{C}{n}, +\end{equation} +with a constant independent of $n$ and the parameter. +\end{lemma} + +\begin{proof} +Use the notation of \cref{scale:truncation}. Both $D$ and $\ell$ are +nonnegative, and $\ell\le2n$. Let $\delta>0$ be the fixed constant +in \cref{can:theorem}. Choose a fixed $H$ sufficiently large that +\[ + \ell>H(1+D),\qquad s=\lfloor\ell/2\rfloor + \quad\Longrightarrow\quad + s\ge1,\quad 2D+A\le\delta s. +\] +This is possible since $s\ge\ell/2-1$. Also $s\le n$ and +$2s\le\ell$. Thus \cref{scale:cut-wings,scale:cut-full} imply +\[ + \min\{g(i,i-s),g(i,i+s)\}\ge(1-\delta)s, + \qquad g(i-s,i+s)\le2D+A\le\delta s. +\] +This is the event bounded by \cref{can:theorem}, with spacing $s$. +On the part where this integer $s$ is selected, $\ell<2s+2$. +Writing the constants of that theorem as $C_{\rm can},\gamma>0$, +we obtain +\[ + \E\bigl[\ell\,\one_{\{\ell>H(1+D)\}}\bigr] + \le C_{\rm can}\sum_{s=1}^{\infty}(2s+2)e^{-\gamma s} + \le C. +\] +The complementary contribution is at most $H(1+\E D)$, hence +\begin{equation}\label{scale:mean-cancellation} + \E\ell\le C(1+\E D). +\end{equation} +Stationarity gives +\[ + \E D\le\E[n-g(i,L)]+\E[n-g(i,R)]=2ne_n, + \qquad e_{2n}=e_n+\frac{\E\ell}{2n}. +\] +These identities and \cref{scale:mean-cancellation} prove the result. +In particular the lower inequality uses only $\ell\ge0$. +\end{proof} + +\Needspace{8\baselineskip} +\subsection{Selecting the first growing deficit} + +\begin{lemma}[Critical scales]\label{scale:selection} +Along the assumed zero-exponent parameter sequence, there are powers +of two $n_0=n_0(K)$ and $N=N(K)>n_0$, and numbers $\epsilon=e_N$, +such that +\begin{equation}\label{scale:selected-limits} + n_0\to\infty,\qquad \epsilon\to0,\qquad + \epsilon n_0\to\infty, +\end{equation} +and, for every fixed integer $m\in\Z$ (once the lengths below are +positive integers), +\begin{equation}\label{scale:selected-boundaries} + e_{2^m n_0}=o(\epsilon),\qquad + e_{2^m N}=O_m(\epsilon). +\end{equation} +All scale sums below run over powers of two. +\end{lemma} + +\begin{proof} +First, for every fixed integer $a\ge1$, we have $e_a\to0$ as +$K\to\infty$. To see this, fix $0<\zeta<1$. The uniform marginal +law of each $q_i$ implies +\[ + \PP\{|v(q_i)|\theta_K$. It satisfies $N>n_0$ and +$e_{N/2}\le\theta_K$. Applying \cref{scale:doubling} once gives +\[ + \theta_K<\epsilon=e_N\le C\theta_K+C/n_0\longrightarrow0. +\] +Also $\epsilon n_0\ge\theta_K n_0\ge\sqrt{n_0}\to\infty$. +For fixed $m\ge0$, iteration of \cref{scale:doubling} yields +\[ + e_{2^m n_0}\le C_m(e_{n_0}+n_0^{-1}) + =C_m\theta_K^2=o(\epsilon), +\] +and +\[ + e_{2^m N}\le C_m(\epsilon+N^{-1})=O_m(\epsilon). +\] +For a fixed negative $m$, the lengths $2^m n_0$ and $2^mN$ are +eventually positive integers. Dyadic monotonicity bounds their +deficits by $e_{n_0}=o(\epsilon)$ and $e_N=\epsilon$, respectively. +This proves every assertion. +\end{proof} + +\subsection{A compact space of distance arrays} + +The next construction records shortfall increments as measures on +rescaled distance arrays. Its output will be locally finite limits +away from affine arrays, together with a dilation balance; the +discarded affine part requires separate control in the final argument. + +Let $\mathcal D=\Z[1/2]$ denote the dyadic rational numbers. Define +$X$ to be the set of symmetric semimetric arrays +$d:\mathcal D\times\mathcal D\to[0,\infty)$ satisfying +\begin{equation}\label{scale:ambient-bound} + d(s,t)\le|s-t|+2. +\end{equation} +Give $X$ the topology of pointwise convergence. It is a closed subset +of a countable product of compact intervals, and hence is compact +and metrizable. Its compact subset of affine arrays is +\[ + \mathcal A=\{d_w:0\le w\le1\},\qquad d_w(s,t)=w|t-s|. +\] +These are precisely the arrays of this form that belong to $X$: +the bound at arbitrarily large separations forces $w\le1$. +Write $\mathcal U=X\setminus\mathcal A$. + +For a positive integer $n$, define the random array +\begin{equation}\label{scale:rescaled-array} + g_n(s,t)=\frac{g(\lfloor ns\rfloor,\lfloor nt\rfloor)}{n} + \qquad(s,t\in\mathcal D). +\end{equation} +It belongs to $X$, since its distances are at most +$|s-t|+1/n$. Introduce the finite measures +\begin{equation}\label{scale:prelimit-measures} + \mu_K=\epsilon^{-1}\sum_{\substack{n_0\le n0$ +such that +\begin{equation}\label{scale:finite-test-cover} + \sum_{a=1}^{b}\bigl(J(I_a)+V(I_a)\bigr)\ge c_F + \quad\hbox{on }F. +\end{equation} +\end{lemma} + +\begin{proof} +Every affine array annihilates these tests. Conversely, suppose they +all vanish. For one of the stated intervals $I$, the equalities +$J(I)=V(I)=0$ say that its two halves have the same speed as $I$ +and that their distances add to the full distance. Repeating this +argument shows that every edge of every dyadic subdivision of $I$ +has distance $w_I$ times its length, with +$w_I=d(s,t)/(t-s)$. The triangle inequality gives the same upper +bound between any two subdivision nodes. Applying the triangle +inequality across the full interval, using those nodes as intermediate +points, gives the reverse bound, and therefore equality. Every dyadic +point of $I$ is a subdivision node at some level. Apply this argument +to $[-2^a,2^a]$ for arbitrarily large integers $a$. The common pair +$0,1$ fixes one speed on all these intervals, so the array is globally +affine. The ambient bound forces that speed to lie in $[0,1]$. + +For the last assertion, each point of $F$ has a positive test. The +corresponding open sets cover $F$, and a finite subcover suffices. +Their sum has a positive minimum on the compact set $F$. +\end{proof} + +\Needspace{8\baselineskip} +\subsection{Locally finite limits and their symmetries} + +The limiting inequality \cref{scale:tree-inequality} is the +zero-hyperbolic Gromov-product condition: after identifying points at +zero semimetric distance, $B_v(s,t)/2$ is the Gromov product based at $v$; +see \cite[Section~2.2, Definition~2.10]{Bestvina1997}. +The determinant-one estimate and the passage to the limiting arrays +are proved here; no real-tree representation theorem is used. + +\begin{proposition}[Limits away from affinity]\label{scale:limits} +After passing to a subsequence, the restrictions of $\mu_K$ and +$\nu_K$ to $\mathcal U$ converge vaguely to locally finite +nonnegative Borel measures $\mu$ and $\nu$, respectively. Moreover, +\begin{equation}\label{scale:initial-negligible} + \xi_K\longrightarrow0\quad\hbox{vaguely on }\mathcal U. +\end{equation} +The measures $\mu$ and $\nu$ are supported on the closed class +$X_0\subset X$ consisting of arrays satisfying $d(s,t)\le|s-t|$ +and the inequalities +\begin{equation}\label{scale:tree-inequality} + B_v(s,t)\ge\min\{B_v(s,p),B_v(p,t)\},\qquad + B_v(s,t)=d(v,s)+d(v,t)-d(s,t) +\end{equation} +for every $v,s,t,p\in\mathcal D$. Every member of $X_0$ extends +uniquely to a continuous semimetric on $\R$, obeying the same +inequalities at all real arguments. Both limiting measures are +invariant under +\[ + (\tau_r d)(s,t)=d(s+r,t+r)\qquad(r\in\mathcal D). +\] +For every interval $I$ as in \cref{scale:local-mass}, +\begin{equation}\label{scale:limiting-test-bound} + \int(J(I)+V(I))\,\dd\mu<\infty, + \qquad + \int(h(I)+J(I)+V(I))\,\dd\nu<\infty. +\end{equation} +In particular, +\begin{equation}\label{scale:terminal-shortfall} + \int h([0,1])\,\dd\nu\le1. +\end{equation} +\end{proposition} + +\begin{proof} +For each compact $F\subset\mathcal U$, combine +\cref{scale:finite-test-cover} with +\cref{scale:sum-test-bound,scale:individual-test-bound} to obtain +uniform bounds on $\mu_K(F)$ and $\nu_K(F)$. Using +\cref{scale:initial-test-bound} instead gives $\xi_K(F)=o(1)$. +The latter proves \cref{scale:initial-negligible}, since a continuous +compactly supported test is bounded and has compact support. + +Here is an explicit way to extract the local limits. Fix a compatible +metric on $X$ and let $a(d)=\dist(d,\mathcal A)$. Choose a continuous +nondecreasing function $\chi:[0,\infty)\to[0,1]$ equal to zero on +$[0,1/2]$ and to one on $[1,\infty)$. Set +$\chi_j(d)=\chi(2^j a(d))$. These functions increase to one on +$\mathcal U$, have compact support there, and satisfy +$\chi_{j+1}=1$ on $\supp\chi_j$. The finite measures +$\chi_j\mu_K$ and $\chi_j\nu_K$ have bounded total masses for +each $j$. Weak compactness for finite measures on compact metric +spaces, followed by a diagonal extraction, gives weak limits for +all of them. Their restrictions agree on the open regions where the +corresponding cutoffs are one: this follows by testing against +continuous functions supported in such a region, for which the +prelimit integrals agree. These compatible restrictions define +locally finite measures on $\mathcal U$. Every compactly supported +continuous function is supported where some $\chi_j$ is one, so +these are the asserted vague limits. + +For completeness, the support restrictions survive despite the +possibly unbounded total masses. The arrays $g_n$ satisfy +$g_n(s,t)\le|s-t|+1/n$. Taking logarithms in +\cref{scale:R-fourpoint}, then dividing by $n$, shows that their +version of \cref{scale:tree-inequality} has error at most +$(\log_M C_0)/n$. These errors are bounded uniformly by $C/n_0$ +at every sampled scale. Any fixed strict violation, by a fixed +positive margin, is thus absent from every sampled array once $K$ +is large. Testing on compact subsets of its open violation set +shows that it has zero limiting measure. There are countably many +dyadic arguments and positive rational margins, which proves the +stated support assertion. For an array with $d(s,t)\le|s-t|$, +the semimetric inequality gives +\[ + |d(s,t)-d(s',t')|\le|s-s'|+|t-t'|. +\] +It therefore has a unique jointly continuous extension to real +arguments; all its inequalities extend by continuity. + +For fixed $r\in\mathcal D$, each prelimit law is invariant under +$\tau_r$ once $n_0r$ is an integer. Indeed +$\lfloor n(s+r)\rfloor=\lfloor ns\rfloor+nr$, and stationarity +applies simultaneously to the array. The map $\tau_r$ is a +homeomorphism of $X$ preserving $\mathcal A$, so it preserves +compact support in $\mathcal U$. Invariance therefore passes to +the vague limits. + +Finally, multiply any nonnegative test $J(I)+V(I)$ by $\chi_j$, +pass to the vague limit, and use +\cref{scale:sum-test-bound}. Monotone convergence as $j\to\infty$ +gives the first bound in \cref{scale:limiting-test-bound}. For $\nu$ +use \cref{scale:individual-test-bound} in the same way. Although +$h(I)$ is not nonnegative everywhere on $X$, it is nonnegative on +the aligned samples and on the supported limiting class $X_0$; +these facts justify the cutoff bounds and monotone convergence for +that term as well. In the case $I=[0,1]$ the uncut terminal +integral is exactly $e_N/\epsilon=1$, giving +\cref{scale:terminal-shortfall}. +\end{proof} + +Define the dilation +\begin{equation}\label{scale:dilation} + (Sd)(s,t)=\frac{d(2s,2t)}2. +\end{equation} +It is a continuous map from $X$ into $X$ and fixes every affine +array. Its inverse need not map all of $X$ into $X$, but on $X_0$ +it is the homeomorphism +\[ + (S^{-1}d)(s,t)=2d(s/2,t/2). +\] +Both maps preserve $X_0\cap\mathcal U$. + +\begin{proposition}[Positive dilation decomposition]\label{scale:decomposition} +On $\mathcal U$, with the measures supported on $X_0$ as above, +\begin{equation}\label{scale:measure-balance} + S_*\mu=\mu+\nu. +\end{equation} +There is a locally finite nonnegative measure $\mu_\infty$, invariant +under $S$ and every dyadic translation, such that +\begin{equation}\label{scale:measure-decomposition} + \mu=\mu_\infty+\sum_{j=1}^{\infty}(S^{-j})_*\nu, + \qquad S_*\mu_\infty=\mu_\infty. +\end{equation} +The series is a series of positive measures dominated by $\mu$. +\end{proposition} + +\begin{proof} +The floor convention in \cref{scale:rescaled-array} gives the exact +identity $Sg_n=g_{2n}$. Hence, as finite measures on $X$, +\begin{equation}\label{scale:finite-measure-balance} + S_*\mu_K=\mu_K+\nu_K-\xi_K. +\end{equation} +To pass to the local limits, observe that $S^{-1}(\mathcal A) +=\mathcal A$: multiplication by two permutes the dyadic rationals, +so $Sd=d_w$ implies $d=d_w$. If $F\subset\mathcal U$ is compact, +its preimage under $S$ is closed in compact $X$ and disjoint from +$\mathcal A$, hence compact in $\mathcal U$. Thus $S$ is proper +on $\mathcal U$, and composition with $S$ preserves continuous +compactly supported tests there. Vague convergence and +\cref{scale:initial-negligible} now give +\cref{scale:measure-balance}. + +We may henceforth work on $X_0\cap\mathcal U$, where $S$ is +invertible. Applying its inverse to \cref{scale:measure-balance} +and iterating gives, for every $b\ge1$, +\[ + \mu=(S^{-b})_*\mu+\sum_{j=1}^{b}(S^{-j})_*\nu. +\] +The increasing positive partial sums are dominated by $\mu$. +Their limit +$\lambda=\sum_{j\ge1}(S^{-j})_*\nu$ is therefore a locally finite +measure. On relatively compact Borel sets define +$\mu_\infty=\mu-\lambda$. The definitions agree on overlaps and +extend by exhaustion to a locally finite nonnegative measure; +this construction avoids subtracting infinite total masses. +Monotone convergence for the positive series yields +\[ + S_*\lambda=\nu+\lambda. +\] +Subtract this identity from \cref{scale:measure-balance} on +relatively compact sets to obtain $S_*\mu_\infty=\mu_\infty$. + +For $r\in\mathcal D$ and $j\ge1$, the identity +\[ + \tau_r S^{-j}=S^{-j}\tau_{r/2^j} +\] +and translation invariance of $\nu$ show that every summand of +$\lambda$ is invariant under $\tau_r$. The same holds for their +sum and, by local subtraction from the translation-invariant +measure $\mu$, for $\mu_\infty$. This proves all assertions. +\end{proof} diff --git a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/setup.tex b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/setup.tex new file mode 100644 index 0000000000000000000000000000000000000000..91eade583e0d511f46616113811000188bcfb030 --- /dev/null +++ b/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/build/sections/setup.tex @@ -0,0 +1,127 @@ +\section{Transfer matrices and growth shortfalls} +\label{sec:setup} + +Throughout the proof the parameter $k$ is positive and sufficiently large. +Set +\begin{equation}\label{setup:parameters} + K=2\pi k,\qquad M=K+4,\qquad + \phi(x)=2x+k\sin(2\pi x),\qquad + v(x)=\phi'(x)=2+K\cos(2\pi x). +\end{equation} +Growth logarithms are to base $M$. Natural logarithms are retained in the +definitions of entropy and Lyapunov exponents. Constants denoted by $C$ +may increase from one occurrence to the next. Unless a dependence is +specified, constants in large-$M$ estimates do not depend on the orbit, +the integer times, or $M$. + +Write $(x_i,y_i)=f_k^i(z)$ for all $i\in\Z$, and put $q_i=x_i\in\T$. +The identity $x_i-y_i=x_{i-1}$ shows that the fixed torus automorphism +$(x,y)\mapsto(x,x-y)$ replaces the state by $(q_i,q_{i-1})$. +It preserves normalized area, and the recurrence becomes +\begin{equation}\label{setup:recurrence} + q_{i+1}=\phi(q_i)-q_{i-1}\pmod1. +\end{equation} +When varying an orbit locally we choose initial real lifts and iterate +this same recurrence over $\R$, without reducing modulo one. Its linear +recurrence is +\begin{equation}\label{setup:transfer} + \binom{u_{i+1}}{u_i} + =A_i\binom{u_i}{u_{i-1}},\qquad + A_i=\begin{pmatrix}v(q_i)&-1\\1&0\end{pmatrix}. +\end{equation} +For $a,b\in\Z$, let $T_{a,b}$ carry the linearized pair at time $a$ to +the pair at time $b$. Thus $T_{a,c}=T_{b,c}T_{a,b}$ and +$T_{b,a}=T_{a,b}^{-1}$. + +All matrix and vector norms are Euclidean unless otherwise stated. +Every transfer matrix has determinant one. A real two-dimensional +determinant-one matrix has singular values $D,D^{-1}$ with $D\ge1$; +in particular +\begin{equation}\label{setup:norm-bounds} + \norm{T_{a,b}}=\norm{T_{b,a}},\qquad + 1\le\norm{T_{a,b}}\le M^{|b-a|}. +\end{equation} +Indeed both $A_i$ and $A_i^{-1}$ have Euclidean norm at most $M$. +The same bound holds in the maximum norm: their largest absolute row sum +is at most $K+30$ tending to infinity such that +\[ + H\Psi_j=E_j\Psi_j. +\] +In particular, $\sigma_{\mathrm p}(H)\cap(0,\infty)$ is infinite. +\end{theorem} + +The mechanism is visible in the quartic bosonic potential +\[ + V(x)=\sum_{at_2>t_3>|t_4|$, so $t$ is regular dominant for type $D_4$, +including when $t_4=0$. The coefficient of $Z^t$ in this expression +is therefore $m_\nu$: only $\eta=\nu$ and the identity Weyl element +contribute. In the determinant in \eqref{eq:branch-character}, row +$i$ involves only $Z_i$, so coefficient extraction acts row by row. +If the coordinate lattices of $\lambda$ and $\nu$ differ, the +coefficient is zero. Otherwise it is +\[ + \det\bigl(\mathbf1_{\{|t_i|\leq L_j\}}\bigr)_{i,j=1}^4. +\] +The $L_j$ strictly decrease, and so do the $|t_i|$. Each row is an +initial string of ones followed by zeros, and the string lengths are +nondecreasing down the rows. A nonzero determinant has neither a zero +row nor two equal rows, so these four lengths must be $1,2,3,4$. +In particular, +$t_1>L_2$, or $\nu_1>\lambda_2-1$. The difference +$\nu_1-\lambda_2$ is integral because the lattices agree. Hence +$\nu_1\geq\lambda_2$. +\end{proof} + +For each $u$, choose a rotation carrying $e_9$ to $\widehat u$. +It conjugates $K$ to $K_u$; transport the torus and positive roots +by this conjugation. Lemma~\ref{lem:branching} then holds for $K_u$ +in the transported weight coordinates. + +\begin{lemma}[Types on a transverse slice] +\label{lem:slice} +Let $\Psi\in\Core\cap\Hphys_\lambda$. For the fixed parameters in +\eqref{eq:slice}, the function +\[ + \Phi(z)=\Psi(x(u,s,z)) +\] +belongs, as a vector in $L^2((u^\perp)^2;\F)$, only to $K_u$ +types whose highest weights satisfy $\nu_1\geq\lambda_2$. +The $K_u$ action here rotates $z$ and acts by $U$ on $\F$. +\end{lemma} + +\begin{proof} +To restrict at fixed parameters, we first use the finite-dimensional +span of the smooth translates of $\Psi$. +The isotypic representation has the form +$\Hphys_\lambda\simeq V_\lambda\widehat\otimes\mathcal M_\lambda$, +where $\mathcal M_\lambda$ is a multiplicity Hilbert space. Writing +$d=\dim V_\lambda$, any individual vector has an expansion +$\sum_{j=1}^d e_j\otimes m_j$. Its span under $G$ is contained in +$V_\lambda\otimes\operatorname{span}\{m_1,\ldots,m_d\}$ and is +therefore finite dimensional. Apply this to $\Psi$, and denote its +translate span by $W_\Psi$. + +Every vector in $W_\Psi$ has a smooth compactly supported +representative, because it is a finite linear combination of translates +of $\Psi$. An equality between these representatives in $L^2$ holds +pointwise by continuity. Thus restriction at the fixed $u,s_b$ is a +well-defined linear map from $W_\Psi$ to +$C_c^\infty((u^\perp)^2;\F)$. It intertwines $K_u$, which fixes +$u$ and the longitudinal coordinates $s_b$. Its image can have only +types occurring in $W_\Psi|_{K_u}$. Lemma~\ref{lem:branching} gives +the claimed bound. The $K_u$ action on the fiber $L^2$ space is +unitary, so distinct isotypic subspaces there are orthogonal. +\end{proof} + +Choose $M\geq0$ so that the first coordinate of every $K$ weight of +$\F$ is at most $M$. Such an $M$ exists because $\F$ is finite +dimensional. The same bound holds for $K_u$ in the transported +coordinates. + +The factor $1/4$ in the next estimate uses half of the kinetic term +in \eqref{eq:form}; the remaining quarter will provide the local +$H^1$ bound needed for compactness. + +\begin{lemma}[Transverse oscillator bound] +\label{lem:oscillator} +Let $m\geq0$ be an integer and suppose +\begin{equation} + \lambda_2>m+M. + \label{eq:large-weight} +\end{equation} +For every slice $\Phi$ in Lemma~\ref{lem:slice}, +\begin{equation} + \int_{(u^\perp)^2} + \left(\tfrac14|\nabla_z\Phi|^2+|u|^2|z|^2\norm{\Phi}^2\right)\,dz + \ \geq\ (m+9)|u|\int_{(u^\perp)^2}\norm{\Phi}^2\,dz. + \label{eq:fiber-oscillator} +\end{equation} +\end{lemma} + +\begin{proof} +For $r=|u|>0$, the oscillator +\[ + h_r=-\tfrac14\Delta_z+r^2|z|^2 +\] +has energies $r(n+8)$, $n=0,1,\ldots$. Indeed, the change of variables +$\xi=\sqrt{2r}\,z$ gives +$h_r=\frac r2(-\Delta_\xi+|\xi|^2)$ in sixteen dimensions. +Each vector at level $n$ is an invariant Gaussian times a polynomial +of total degree at most $n$. Tensor this level with $\F$. + +The transverse coordinates are two copies of the vector representation +of $K_u$, whose weights are $\pm\epsilon_1,\ldots,\pm\epsilon_4$. +Every weight in a polynomial of degree at most $n$ has first +coordinate at most $n$, and every weight in the level tensored with +$\F$ consequently has first coordinate at most $n+M$. In particular, +the first coordinate of the highest weight of any constituent is at +most $n+M$. + +Let $\Pi_n$ be the projection onto level $n$ tensored with $\F$. +It commutes with $K_u$ and hence with every $K_u$-isotypic projection +on the fiber $L^2$ space. By Lemma~\ref{lem:slice}, $\Pi_n\Phi$ can +therefore contain only types with $\nu_1\geq\lambda_2$. Since it +lies in level $n$ tensored with $\F$, its types also satisfy +$\nu_1\leq n+M$. For $n\leq m$ these bounds are incompatible by +\eqref{eq:large-weight}, so $\Pi_n\Phi=0$. Expanding in the complete +Hermite basis, the least remaining energy is $r(m+9)$, which proves +\eqref{eq:fiber-oscillator}. +\end{proof} + +\begin{proposition}[Confinement in a rotation sector] +\label{prop:confinement} +Let $C$ be as in \eqref{eq:fermion-bound}, and choose an integer +$m\geq0$ such that +$c_m=(m+9)/3-C>0$. If $\lambda_2>m+M$, every +$\Psi\in\Dom(q)\cap\Hphys_\lambda$ belongs to +$H^1(\R^{27};\F)$, satisfies $|x|^{1/2}\Psi\in L^2(\R^{27};\F)$, +and obeys +\begin{equation} + q(\Psi)\ \geq\ \tfrac14\norm{\nabla\Psi}_2^2 + +c_m\int_{\R^{27}}|x|\norm{\Psi(x)}^2\,dx. + \label{eq:confinement} +\end{equation} +The operator $H_\lambda$ has compact resolvent and +$\ker H_\lambda=\{0\}$. +\end{proposition} + +\begin{proof} +First take $\Psi\in\Core\cap\Hphys_\lambda$ and fix a color $a$. +For fixed $u$, the decomposition of the two other color variables +into $s_b,z_b$ is an orthogonal linear change of variables with unit +Jacobian. The transverse gradient uses only derivatives in those two +variables, so $|\nabla_z\Phi|^2\leq|\nabla\Psi|^2$. No derivative of +the decomposition with respect to $u$ is taken. Fubini's theorem, +using local orthogonal frames for $u^\perp$, therefore applies to +\eqref{eq:fiber-oscillator}; the set $u=0$ has measure zero. +Together with \eqref{eq:slice-potential}, it gives +\begin{equation} + \tfrac14\norm{\nabla\Psi}_2^2+ + \int V(x)\norm{\Psi(x)}^2\,dx + \ \geq\ (m+9)\int |x^a|\norm{\Psi(x)}^2\,dx. + \label{eq:one-color} +\end{equation} +Average \eqref{eq:one-color} over the three colors; its left side +is unchanged. Splitting the kinetic term in \eqref{eq:form} into +two quarters and using \eqref{eq:fermion-bound} then gives +\[ + q(\Psi)\geq\tfrac14\norm{\nabla\Psi}_2^2+ + \int_{\R^{27}}\left(\tfrac{m+9}{3}\sum_a|x^a|-C|x|\right) + \norm{\Psi(x)}^2\,dx. +\] +Since $\sum_a|x^a|\geq|x|$, this proves \eqref{eq:confinement} +on the core. + +We now pass to the restricted form domain. By +Lemma~\ref{lem:rotations}, any vector in that domain is a form-norm +limit of vectors in +$\Core\cap\Hphys_\lambda$. Applied to differences of these vectors, +\eqref{eq:confinement} shows convergence of their gradients in $L^2$ +and of their products with $|x|^{1/2}$ in $L^2$. The gradient and +multiplication by $|x|^{1/2}$ are closed operators. Their limits are +therefore the gradient and weighted product of the form limit. +Passing to the limit proves \eqref{eq:confinement} on the full +restricted form domain. + +Its form-norm unit ball is bounded in $H^1(\R^{27};\F)$. Moreover, +for $R>0$, \eqref{eq:confinement} gives the uniform tail estimate +\[ + \int_{|x|>R}\norm{\Psi(x)}^2\,dx + \leq\frac{q(\Psi)}{c_m R}. +\] +Rellich compactness on bounded balls applies componentwise in the +finite-dimensional space $\F$. A diagonal subsequence and the tail +estimate give compactness in $L^2(\R^{27};\F)$. Hence the inclusion +of the restricted form domain into $\Hphys_\lambda$ is compact. +The resolvent $(H_\lambda+1)^{-1}$ maps the Hilbert space boundedly +into that form domain, so it is compact. + +If $\Psi\in\ker H_\lambda$, then $q(\Psi)=0$. The positive weighted +term in \eqref{eq:confinement} vanishes, forcing $\Psi=0$ because +$|x|>0$ almost everywhere. This proves the kernel assertion. +\end{proof} + +\section{Physical rotation sectors and positive eigenvalues} +\label{sec:sectors} + +It remains to produce an infinite-dimensional physical sector meeting +\eqref{eq:large-weight}. We first obtain a gauge-invariant fermionic +vector, then multiply a rotation highest vector by gauge-invariant +spatial polynomials. + +\begin{lemma} +\label{lem:fermion-invariant} +The subspace $\F^{\SU(2)}$ is nonzero and invariant under $G$. +\end{lemma} + +\begin{proof} +For $1\leq j\leq8$ and $1\leq a\leq3$, put +\[ + c_j^a=\frac{\theta_{2j-1}^a+i\theta_{2j}^a}{\sqrt2}. +\] +These twenty-four operators and their adjoints satisfy the canonical +anticommutation relations +$\{c_j^a,c_k^b\}=0$ and +$\{c_j^a,(c_k^b)^*\}=\delta_{jk}\delta_{ab}$. +The occupation operators $N_j^a=(c_j^a)^*c_j^a$ are commuting +projections. Start with a nonzero joint eigenvector. Applying an +occupied mode's annihilator removes that occupation and preserves all +others. The result is nonzero, since +$\norm{c_j^a v}^2=\ip{v}{N_j^a v}=\norm{v}^2$. Repeating this gives +a nonzero vector +annihilated by every $c_j^a$. For each subset of modes, apply its +creation operators in a fixed order. The resulting vectors are +nonzero and have distinct occupation patterns, hence are orthogonal. +Their span is invariant under every creation and +annihilation operator, so irreducibility makes it all of $\F$. +The common kernel of the annihilators is therefore exactly the +original line. + +Gauge transformations mix the color indices and preserve the span +of the annihilators for each $j$. They therefore preserve the common +kernel line. Their action on that line is a continuous character of +$\SU(2)$ and is trivial. Thus $\F^{\SU(2)}\ne0$. Its $G$ +invariance follows from the commutation in Lemma~\ref{lem:rotations}. +\end{proof} + +\begin{lemma} +\label{lem:large-sectors} +There are $G$ types $\lambda$ with arbitrarily large $\lambda_2$ +for which $\Core\cap\Hphys_\lambda$ is infinite dimensional. +\end{lemma} + +\begin{proof} +Choose a nonzero highest vector $v$ of some irreducible $G$ type +$\mu$ in the finite-dimensional representation $\F^{\SU(2)}$. + +We construct a gauge-invariant polynomial that raises the first two +highest-weight coordinates. In the complexified vector representation, set +\[ + w_1=e_1+i e_2,\qquad w_2=e_3+i e_4, +\] +choosing the torus convention in which their weights are +$\epsilon_1,\epsilon_2$. Use the complex bilinear extension of the +Euclidean dot product, and put +\[ + \zeta_a=w_1\cdot x^a,\qquad \eta_a=w_2\cdot x^a,\qquad + P(x)=\sum_{a0$, one has +$P(x)=t^4$ and $|x|=\sqrt2\,t$. Thus $P$ is nonzero on every +nonempty radial shell. + +For the action on spatial polynomials by pullback, +\[ + (w\cdot R_g^{-1}x)=(R_gw)\cdot x. +\] +Thus $w\mapsto(w\cdot x^a)$ intertwines the complexified vector +representation with linear polynomials, with the same weight sign. +The analogous map on exterior squares sends $w_1\wedge w_2$ to +$\zeta_a\eta_b-\zeta_b\eta_a$. The bivector $w_1\wedge w_2$ is +highest of weight $\epsilon_1+\epsilon_2$: a positive-root operator +either kills both factors or sends $w_2$ to a multiple of $w_1$, +and in the latter case its action on the wedge also vanishes. +Hence each minor is highest of that weight. Root operators act as +derivations on polynomials, so $P$ is highest of weight +$2(\epsilon_1+\epsilon_2)$. Its nonzero power $P^k$ is highest of +weight $2k(\epsilon_1+\epsilon_2)$. + +Let $\chi\in C_c^\infty((0,\infty))$ be nonzero. The function +\[ + \Psi_\chi(x)=\chi(|x|)P(x)^k v +\] +is in $\Core$: it is smooth because its support avoids the origin, +and its factors are gauge invariant. The preceding ray shows that +$\Psi_\chi\ne0$. Multiplication by the radial function is $G$ +equivariant. The translates of $\Psi_\chi$ lie in +the finite-dimensional representation obtained from homogeneous +polynomials of degree $4k$ tensored with $\F$. The vector is +highest of the dominant weight +\[ + \lambda=2k(\epsilon_1+\epsilon_2)+\mu + =(2k+\mu_1,\,2k+\mu_2,\,\mu_3,\,\mu_4). +\] +In a decomposition of that finite representation into irreducibles, +each nonzero component of this vector is a weight-$\lambda$ vector +killed by all positive-root operators. It must be a highest vector, +so its summand has type $\lambda$. Therefore +$\Psi_\chi\in\Hphys_\lambda$. + +Choosing $\chi$ with pairwise disjoint radial supports gives infinitely +many linearly independent vectors in the same sector. Finally, +$\lambda_2=2k+\mu_2$ tends to infinity with $k$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Choose an integer $m\geq0$ with $c_m=(m+9)/3-C>0$. +Lemma~\ref{lem:large-sectors} supplies a type $\lambda$ such that +$\lambda_2>m+M$ and $\Hphys_\lambda$ is infinite dimensional. +By Proposition~\ref{prop:confinement}, $H_\lambda$ has compact +resolvent and no zero eigenvector. + +Since $H_\lambda$ is nonnegative with compact resolvent on an +infinite-dimensional Hilbert space, the compact-resolvent spectral +theorem gives an orthonormal eigenbasis $(\Psi_j)$ with eigenvalues +$E_j\to\infty$. Its trivial kernel makes every $E_j$ positive. +By Lemma~\ref{lem:rotations}, $H_\lambda$ is the part of $H$ on a +reducing subspace. These vectors therefore belong to $\Dom(H)$ and +satisfy $H\Psi_j=E_j\Psi_j$ in the full physical Hilbert space. +\end{proof} + +\begin{thebibliography}{99} + +\bibitem{ArefevaKoshelevMedvedev} +I. Ya. Aref'eva, A. S. Koshelev, and P. B. 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Smilga, +\emph{Comments on thermodynamics of supersymmetric matrix models}, +Nuclear Physics B \textbf{818} (2009), 101--114. +\url{https://doi.org/10.1016/j.nuclphysb.2009.03.023}. + +\end{thebibliography} + +\end{document} diff --git a/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf b/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c82314dd480f85040a54834e2e01f89749ab7262 --- /dev/null +++ b/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3501f8a562e65fc545fbc82070b0d5260ee0df6893cf232db3777015be1f7164 +size 364093 diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/README.md b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..ba9151952de9e217a005f40f15a987833c8a93b5 --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/README.md @@ -0,0 +1,18 @@ +# [Positive lower density of large prime gaps](main.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026, + author = {{OpenAI}}, + title = {{Positive lower density of large prime gaps}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf}{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/main.tex b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..ba08b3e235bd602c82c8f82ee1146ab5fdf703bf --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/main.tex @@ -0,0 +1,44 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{enumitem} +\usepackage[hidelinks]{hyperref} +\usepackage{microtype} +\hypersetup{ + pdftitle={Positive lower density of large prime gaps}, + pdfauthor={OpenAI}, + pdfsubject={Ordinary lower density of consecutive prime gaps}} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\DeclareMathOperator{\Var}{Var} +\title{Positive lower density of large prime gaps} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +For every fixed $C>0$, we prove that a positive proportion of consecutive +prime gaps exceed $C\log p$, where $p$ is the smaller prime. The proportion +is bounded below for every sufficiently large initial segment of the +prime sequence, with a constant depending on $C$. It follows that the +indices at which $p_n/n$ increases have positive lower asymptotic density, +answering a question of Erd\H{o}s and Prachar. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-counting} +\input{sections/04-weights} +\input{sections/05-cancellation} +\input{sections/03-moments} +\bibliographystyle{plain} +{\raggedright\bibliography{references}} +\end{document} diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/references.bib b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..56e58dbbdc71193c558c98976f6f4d255dd6701c --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/references.bib @@ -0,0 +1,215 @@ +@article{ErdosPrachar1962, + author = {Erd{\H o}s, Paul and Prachar, Karl}, + title = {S{\"a}tze und {Probleme} {\"u}ber {$p_k/k$}}, + journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg}, + volume = {25}, + year = {1962}, + pages = {251--256}, + doi = {10.1007/BF02992930}, + url = {https://www.renyi.hu/~p_erdos/1961-21.pdf} +} + +@article{BFI1986, + author = {Bombieri, Enrico and Friedlander, John B. and Iwaniec, Henryk}, + title = {Primes in arithmetic progressions to large moduli}, + journal = {Acta Mathematica}, + volume = {156}, + year = {1986}, + pages = {203--251}, + doi = {10.1007/BF02399204}, + url = {https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6385-11511_2006_Article_BF02399204.pdf} +} + +@misc{GY2005, + author = {Goldston, Daniel A. and Y{\i}ld{\i}r{\i}m, Cem Y.}, + title = {Small Gaps Between Primes {I}}, + year = {2005}, + eprint = {math/0504336}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/math/0504336v1}, + howpublished = {\href{https://arxiv.org/abs/math/0504336v1}{arXiv:math/0504336v1}}, + note = {Version 1, 16 April 2005. Published as {Higher correlations of + divisor sums related to primes III: small gaps between primes}, + Proc. London Math. Soc. (3) 95 (2007), no. 3, 653--686, + \href{https://doi.org/10.1112/plms/pdm021}{doi:10.1112/plms/pdm021}} +} + +@article{Maynard2015, + author = {Maynard, James}, + title = {Small gaps between primes}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {181}, + number = {1}, + year = {2015}, + pages = {383--413}, + doi = {10.4007/annals.2015.181.1.7}, + eprint = {1311.4600}, + archivePrefix = {arXiv}, + url = {https://annals.math.princeton.edu/wp-content/uploads/annals-v181-n1-p07-p.pdf}, + note = {\href{https://annals.math.princeton.edu/wp-content/uploads/annals-v181-n1-p07-p.pdf}{Published version}} +} + +@article{Gallagher1976, + author = {Gallagher, Patrick X.}, + title = {On the distribution of primes in short intervals}, + journal = {Mathematika}, + volume = {23}, + number = {1}, + year = {1976}, + pages = {4--9}, + doi = {10.1112/S0025579300016442}, + note = {Corrigendum concerning Theorem 2: Mathematika 28 (1981), no. 1, + 86, \href{https://doi.org/10.1112/S0025579300015382}{doi:10.1112/S0025579300015382}} +} + +@article{BLZ2010, + author = {Bazzanella, Danilo and Languasco, Alessandro and Zaccagnini, Alessandro}, + title = {Prime numbers in logarithmic intervals}, + journal = {Transactions of the American Mathematical Society}, + volume = {362}, + year = {2010}, + pages = {2667--2684}, + doi = {10.1090/S0002-9947-09-05009-0}, + eprint = {0809.2967}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/0809.2967} +} + +@misc{Tao2019, + author = {Tao, Terence}, + title = {Answer to: Positive proportion of logarithmic gaps between consecutive primes}, + year = {2019}, + month = may, + howpublished = {MathOverflow, \href{https://mathoverflow.net/a/332888}{answer 332888}}, + url = {https://mathoverflow.net/a/332888}, + note = {Accessed 25 September 2026} +} + +@article{GoldstonYildirim2003Triple, + author = {Goldston, D. A. and Y{\i}ld{\i}r{\i}m, C. Y.}, + title = {Higher correlations of divisor sums related to primes {I}: Triple correlations}, + journal = {Integers}, + volume = {3}, + year = {2003}, + pages = {A05}, + note = {66 pp.}, + url = {https://math.colgate.edu/~integers/d5/d5.pdf} +} + +@article{Bombieri1965, + author = {Bombieri, Enrico}, + title = {On the large sieve}, + journal = {Mathematika}, + volume = {12}, + number = {2}, + year = {1965}, + pages = {201--225}, + doi = {10.1112/S0025579300005313} +} + +@article{Vinogradov1965, + author = {Vinogradov, A. I.}, + title = {The density hypothesis for {Dirichlet} {$L$}-series}, + journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, + volume = {29}, + number = {4}, + year = {1965}, + pages = {903--934}, + url = {https://www.mathnet.ru/eng/im3080}, + note = {In Russian; correction, vol. 30 (1966), no. 3, pp. 719--720, + \href{https://www.mathnet.ru/eng/im2856}{Math-Net correction record}} +} + +@misc{Jha2026, + author = {Jha, Abhishek}, + title = {The {Poisson Tail Conjecture} for primes in short intervals}, + year = {2026}, + eprint = {2605.23014}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/2605.23014v2}, + howpublished = {\href{https://arxiv.org/abs/2605.23014v2}{arXiv:2605.23014v2}}, + note = {Version 2, 12 September 2026} +} + +@article{GKM2021, + author = {Granville, Andrew and Koukoulopoulos, Dimitris and Maynard, James}, + title = {Sieve weights and their smoothings}, + journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure}, + series = {4}, + volume = {54}, + number = {5}, + pages = {1089--1177}, + year = {2021}, + doi = {10.24033/asens.2478}, + url = {https://arxiv.org/abs/1606.06781v4} +} + +@article{Westzynthius1931, + author = {Westzynthius, E.}, + title = {{\"U}ber die Verteilung der Zahlen, die zu den $n$ ersten Primzahlen teilerfremd sind}, + journal = {Commentationes Physico-Mathematicae}, + volume = {5}, + number = {25}, + year = {1931}, + pages = {1--37} +} + +@article{Erdos1935, + author = {Erd{\H o}s, Paul}, + title = {On the difference of consecutive primes}, + journal = {The Quarterly Journal of Mathematics}, + series = {Oxford Series}, + volume = {6}, + year = {1935}, + pages = {124--128}, + doi = {10.1093/qmath/os-6.1.124} +} + +@article{Rankin1938, + author = {Rankin, R. A.}, + title = {The difference between consecutive prime numbers}, + journal = {Journal of the London Mathematical Society}, + volume = {13}, + number = {4}, + year = {1938}, + pages = {242--247}, + doi = {10.1112/jlms/s1-13.4.242} +} + +@article{FordGreenKonyaginTao2016, + author = {Ford, Kevin and Green, Ben and Konyagin, Sergei and Tao, Terence}, + title = {Large gaps between consecutive prime numbers}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {183}, + number = {3}, + year = {2016}, + pages = {935--974}, + doi = {10.4007/annals.2016.183.3.4} +} + +@article{Maynard2016Large, + author = {Maynard, James}, + title = {Large gaps between primes}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {183}, + number = {3}, + year = {2016}, + pages = {915--933}, + doi = {10.4007/annals.2016.183.3.3} +} + +@article{FordGreenKonyaginMaynardTao2018, + author = {Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence}, + title = {Long gaps between primes}, + journal = {Journal of the American Mathematical Society}, + volume = {31}, + number = {1}, + year = {2018}, + pages = {65--105}, + doi = {10.1090/jams/876}, + eprint = {1412.5029}, + archivePrefix = {arXiv} +} diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/01-introduction.tex b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..f7a4ee7102c3d70c2c437924419a0557044af10c --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/01-introduction.tex @@ -0,0 +1,137 @@ +\section{Introduction} + +Let $p_n$ denote the $n$th prime and write $d_n=p_{n+1}-p_n$. +The prime number theorem identifies $\log p$ as the average scale of +prime gaps near $p$. Individual gaps can be much larger. +Westzynthius proved that $d_n/\log p_n$ is unbounded +\cite{Westzynthius1931}; the constructions of Erd\H{o}s and Rankin +gave increasingly strong quantitative lower bounds for exceptionally +large gaps \cite{Erdos1935,Rankin1938}. +Modern refinements were obtained independently by Ford, Green, +Konyagin and Tao and by Maynard +\cite{FordGreenKonyaginTao2016,Maynard2016Large}, followed by their +joint work \cite{FordGreenKonyaginMaynardTao2018}. +These results concern the size of very large gaps. A lower bound for +the largest gap does not by itself give a positive proportion of +gaps exceeding a fixed multiple of $\log p$. + +We study this frequency question with equal weight on the prime +indices. For a set $A$ of positive integers, its lower asymptotic +density is $\liminf_{N\to\infty}|A\cap[1,N]|/N$. + +\begin{theorem}\label{thm:main} +For every fixed real $C>0$, there are constants $c(C)>0$ and $N_0(C)$ +such that, for every integer $N\ge N_0(C)$, +\[ + \#\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N. +\] +\end{theorem} + +The density is ordinary lower density: the bound holds for every +sufficiently large initial segment, and both constants may depend +on the fixed threshold $C$. + +One consequence concerns successive values of $p_n/n$. +Erd\H{o}s and Prachar asked whether the indices at which this ratio +increases have positive lower density \cite[p.~256]{ErdosPrachar1962}. + +\begin{corollary}\label{cor:ratios} +The set +\[ + \left\{n\ge1:\frac{p_n}{n}<\frac{p_{n+1}}{n+1}\right\} +\] +has positive lower asymptotic density. +\end{corollary} +\begin{proof} +The displayed inequality is equivalent to +$p_{n+1}-p_n>p_n/n$. +The prime number theorem gives $p_n/n\sim\log p_n$. +Apply Theorem~\ref{thm:main} with $C=2$ and discard the finitely many +indices for which $p_n/n\ge2\log p_n$. +\end{proof} + +Thus the corollary answers the Erd\H{o}s--Prachar question affirmatively. + +\subsection*{Gap frequency and empty intervals} + +The distinction between counting intervals and counting prime gaps is +essential. A long gap contains many integer starting points of +prime-free short intervals. Consequently a positive proportion of +empty intervals may be accounted for by a sparse set of very long +gaps. Bazzanella, Languasco and Zaccagnini distinguish the two counting +measures explicitly \cite{BLZ2010}. Their Theorem~5 gives positive +prime-start proportions for each fixed threshold below +$2/3.454=0.579038\ldots$, whereas their Theorem~3 reaches some +intervals longer than the average prime spacing by counting integer +starts. Tao's empty-interval argument \cite{Tao2019} also identifies +the additional control required to pass from interval measure to +gap counts. + +Conditional results describe much more of the distribution. +Gallagher derived Poisson statistics for prime counts in +integer-start intervals of length $\lambda\log X$, for fixed +$\lambda>0$, from a uniform +Hardy--Littlewood prime-tuple hypothesis +\cite[Theorem~1]{Gallagher1976}. +The Poisson-tail results of Jha +\cite[Theorems~1.4 and~1.6]{Jha2026} treat growing parameter ranges +under a stronger uniform tuple hypothesis. +The counting measures and uniformity assumptions matter in comparing +these conclusions with Theorem~\ref{thm:main}. + +\subsection*{The adjacent-interval construction} + +We control the number of distinct gaps by requiring a prime just +before an empty interval. Fix $\lambda>0$, put +$h=\lfloor\lambda\log X\rfloor$, and average over integers +$X0$, write +\[ + L=\log X,\qquad h=\lfloor\lambda L\rfloor,\qquad + J=\{1,\ldots,h\},\qquad I=\{h+1,\ldots,2h\}. +\] +We use the uniform average +$\mathbb E_X A=X^{-1}\sum_{X0$ says exactly that the corresponding interval +contains a prime. Moreover, $V_B\ge1$ whenever $V_B>0$, since all +its prime arguments exceed $X$. + +\begin{proposition}[Weights for adjacent intervals]\label{prop:weights} +For each fixed $\lambda>0$ there is a finite constant $K_\lambda>0$ +with the following property. For every $\varepsilon>0$, there are +weights $W_X(m)\ge0$, defined for integers $Xh$. +\end{lemma} +\begin{proof} +For each such $m$, let $p(m)$ be the last prime in $(m,m+h]$. +There is no prime in $(p(m),m+2h]$, so +$p(m)^+>m+2h\ge p(m)+h$. +For a fixed selected prime $p$, the condition $m0$ and choose $\lambda>\max\{C,1\}$. +Take $\varepsilon=\lambda^2/(2K_\lambda)$ in +Proposition~\ref{prop:weights}, and fix the resulting weight family. +There is a finite constant $M>0$ such that +$\mathbb E_XW_X^2\le M$ for every sufficiently large $X$. +Let $A_X=\{V_J>0\}$ and $B_X=\{V_I>0\}$. +Weighted Cauchy--Schwarz gives +\[ + \mathbb E_X(W_X\mathbf1_{A_X}) + \ge \frac{\bigl(\mathbb E_X(W_XV_J)\bigr)^2} + {\mathbb E_X(W_XV_J^2)} +\] +for all sufficiently large $X$; its denominator is positive because +the numerator tends to $\lambda^2>0$. +By \eqref{eq:weight-first}--\eqref{eq:weight-primes}, +\[ + \liminf_{X\to\infty}\mathbb E_X(W_X\mathbf1_{A_X}) + \ge\frac{\lambda^2}{K_\lambda}. +\] +Since $\mathbf1_{B_X}\le V_I$ and $W_X\ge0$, it follows that +\[ + \liminf_{X\to\infty}\mathbb E_X + (W_X\mathbf1_{A_X\setminus B_X}) + \ge \frac{\lambda^2}{K_\lambda}-\varepsilon + =:\Delta>0. +\] +Another use of Cauchy--Schwarz yields, for every sufficiently large $X$, +\[ + \frac{|A_X\setminus B_X|}{X} + \ge\frac{\bigl(\mathbb E_X + (W_X\mathbf1_{A_X\setminus B_X})\bigr)^2} + {\mathbb E_XW_X^2} + \ge\frac{\Delta^2}{4M}=:\delta>0. +\] +This is an ordinary proportion of integer starting points. +Lemma~\ref{lem:adjacent-gaps} therefore gives at least +\[ + \frac{\delta X}{h}\ge\frac{\delta X}{\lambda\log X} +\] +distinct prime starts $p\in(X,2X+h]$ with consecutive gap exceeding $h$. +Uniformly on this range, +$\log p=\log X+O(1)$ and $h=\lambda\log X+O(1)$. +The strict inequality $\lambda>C$ consequently gives +$h>C\log p$ for every sufficiently large $X$. + +For an arbitrary sufficiently large integer $N$, choose +$X=\lfloor p_N/3\rfloor$. Then $2X+h0$ is admissible. +All choices preceding $X$ depend only on the fixed threshold. +\end{proof} + +The rest of the paper proves Proposition~\ref{prop:weights}. +The task is to make the weighted prime mass in $I$ arbitrarily small +without losing either the mass of the square or the uniform detector +bound in the adjacent block $J$. diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/03-moments.tex b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/03-moments.tex new file mode 100644 index 0000000000000000000000000000000000000000..47895e529a42e8fac970d2d03ef1908764b9b84e --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/03-moments.tex @@ -0,0 +1,618 @@ +\section{Analytic estimates for the square}\label{sec:moments} + +It remains to prove the moment identities and detector bound in +Proposition~\ref{prop:weight-moments}. The products in $Z^2$ contain two +divisor-sum factors, those in $Z^4$ contain four, and the detector estimate +uses six. We first evaluate all these products in one formula that allows +repeated shifts and one prime mark. We then average the resulting singular +series over the shifts in $I$ and $J$, and apply the formulas to the square. +All dimensions, support budgets and functions remain fixed as $X$ tends +to infinity. + +The leading coefficient separates into a test-function constant, determined +by which coordinates share a shift, and a singular series carrying the +numerical shifts. This separation is what permits the subset expansions +of $Z$ to be summed uniformly. + +For a finite set $\mathcal H$ of distinct integers, put +\[ + \nu_p(\mathcal H)=|\mathcal H\bmod p|,\qquad + \mathfrak S(\mathcal H)=\prod_p + (1-\nu_p(\mathcal H)/p)(1-1/p)^{-|\mathcal H|}. +\] +We set $\mathfrak S(\varnothing)=1$. For a nonempty set the product converges: +beyond its diameter, the factors are $1+O_{|\mathcal H|}(p^{-2})$. It is nonnegative +and may be zero. The product is one when $|\mathcal H|=1$. + +\begin{proposition}[Uniform mixed moments]\label{prop:mixed-moments} +Put $L=\log X$, where $X$ tends to infinity through integers, and let +$\mathbb E_X$ denote the average over $X0$ and $B\ge1$ are fixed, +and set $b_{r,\ell}=a_{\kappa(r,\ell)}$. Choose $\delta\in\{0,1\}$. +In the marked case take one more +shift $a_0$, distinct from these and satisfying the same bound. Put +\[ + (\chi_\delta(m),\mathcal H)= + \begin{cases} + (1,\{a_1,\ldots,a_t\}),&\delta=0,\\ + (\vartheta(m+a_0),\{a_0,a_1,\ldots,a_t\}),&\delta=1, + \end{cases} + \qquad \vartheta(n)=(\log n)1_{\{n\text{ prime}\}}. +\] +Assume $\sum_r\rho_r<1$ if $\delta=0$, and $\sum_r\rho_r<1/2$ if +$\delta=1$. Then there is a real constant $\mathcal C$, depending only on +the functions and $\kappa$, such that +\begin{equation}\label{eq:mixed-moment} + \mathbb E_X\left[\chi_\delta(m)\prod_rD_{F_r}(m;\mathbf b^{(r)})\right] + =L^{-t}\left\{\mathfrak S(\mathcal H)\mathcal C + +O\!\left(L^{-1/2}(\log L)^A\right)\right\}. +\end{equation} +Here $A$ is a fixed finite number depending only on $\sum_rj_r$; the implied +constant may depend on all the fixed data, but is uniform in the actual +shifts. In particular the error is additive when $\mathfrak S(\mathcal H)=0$. + +If each $|\mathcal M_i|$ is at most two, the constant is +\begin{equation}\label{eq:mixed-derivative-constant} + \mathcal C= + \int\prod_r\left((-1)^{j_r}\partial_1\cdots\partial_{j_r}F_r\right) + \big((y_{\kappa(r,\ell)})_{\ell=1}^{j_r}\big) + \prod_{i:\,|\mathcal M_i|=2}dy_i, +\end{equation} +where $y_i=0$ for $|\mathcal M_i|=1$, and each integrated variable ranges over +$[0,\infty)$. The zero-dimensional derivative is the original scalar. +When $\Gamma$ is empty, take $t=0$ and $\mathcal C=\prod_rF_r$. +\end{proposition} + +The integer $t$ counts distinct numerical shifts, whereas +$\sum_r j_r$ counts divisor coordinates with repetitions. The power +$L^{-t}$ is determined by the former. Repetitions are recorded by the +fibers $\mathcal M_i$ and affect the constant $\mathcal C$. For two +divisor-sum factors whose individual shift tuples have distinct entries, +these fibers have size at most two. Products of four such factors +may have larger fibers; their estimates will use the general +finiteness assertion rather than the displayed derivative formula. +The marked formula retains the same power $L^{-t}$, consistent with +$\vartheta(m+a_0)$ having average tending to one. The additional factor +$L^{-1}$ in $V_B$ is supplied when the marked shifts are summed. + +The scalar divisor-sum correlations of Goldston--Y{\i}ld{\i}r{\i}m +\cite{GoldstonYildirim2003Triple,GY2005} and Tao's smooth Fourier formulation, +described by Maynard \cite[Section~6, p.~404]{Maynard2015} motivate this calculation. We include its +proof because coupled smooth functions, arbitrary equality patterns and +an additive error uniform in logarithmic shifts are needed here. +The choice of smooth cutoffs was also informed by the analysis of +smoothing and high moments of divisor sums by +Granville--Koukoulopoulos--Maynard \cite{GKM2021}. This is motivation +for the cutoff choice, not a source of the mixed-moment proposition +proved here. + +\subsection{Residue classes and the prime mark} +Our objective is to replace a product of divisor sums, possibly weighted by one shifted prime, by an explicit finite density sum. We use the ordinary endpoint Bombieri--Vinogradov theorem in the form recorded by Bombieri, Friedlander and Iwaniec \cite[p.~206, equation~(1.4)]{BFI1986}; their $\psi$ notation is defined on p.~204. The additional divisor weights and endpoint shifts required here are handled below. + +We retain the notation of Proposition~\ref{prop:mixed-moments} and put +$H_X=DL^B$. The two arithmetic lemmas below also allow $B\ge0$. +Negative shifts cause no difficulty, since all shifted arguments are +positive for sufficiently large $X$. +We write $\Lambda$ for the von Mangoldt function, $\varphi(q)$ for Euler's +totient, $\omega(q)$ for the number of distinct prime divisors of $q$, +$d(q)$ for the number of positive divisors, and $\zeta$ for the Riemann +zeta function. + +\begin{lemma}[Weighted distribution with uniform shifted endpoints]\label{lem:arithmetic-1} + +Assume the Bombieri--Vinogradov theorem: for every $A>0$ there is $b(A)>0$ such that +\begin{equation} +\sum_{q\le x^{1/2}(\log x)^{-b(A)}} + \max_{(c,q)=1}\left|\psi(x;q,c)-\frac{x}{\varphi(q)}\right| + \ll_A x(\log x)^{-A}. +\label{eq:arithmetic-BV} +\end{equation} +Here $\psi(x;q,c)=\sum_{n\le x,\ n\equiv c\pmod q}\Lambda(n)$, and $q=1$ is included. Fix $0\le\sigma<1/2$, $K\ge1$, and define +\[ +E_X(q)=\max_{\substack{a\in\mathbb Z,\ |a|\le H_X\\(c,q)=1}} +\left|\sum_{\substack{X+a0$, +\begin{equation} +\sum_{q\le X^\sigma}\mu^2(q)K^{\omega(q)}E_X(q) + \ll_{A,\sigma,K,D,B}X L^{-A}. +\label{eq:arithmetic-1} +\end{equation} +The conventions are $\omega(1)=0$, $K^0=1$, and $\varphi(1)=1$. + +\end{lemma} +\begin{proof} At either endpoint $x=X,2X$, the range $q\le X^\sigma$ is contained in the range of \eqref{eq:arithmetic-BV} for every fixed logarithmic saving once $X$ is sufficiently large. Removing proper prime powers from a $\psi$-sum costs at most +\[ +\sum_{p^j\le3X,\ j\ge2}\log p\ll X^{1/2}L^2 +\] +in each residue class. Indeed there are $O(X^{1/2}L)$ pairs $p,j\ge2$, and each summand is $O(L)$. Translating the two endpoints through any $|a|\le H_X$ costs $O((H_X+1)L)$, uniformly in the residue class. Thus, writing $Q=\lfloor X^\sigma\rfloor$, for every $A_0>0$, +\begin{equation} +\sum_{q\le Q}E_X(q) + \ll_{A_0}X L^{-A_0} + +Q\{X^{1/2}L^2+(H_X+1)L\} + \ll_{A_0,\sigma,D,B}X L^{-A_0}. +\label{eq:arithmetic-2} +\end{equation} +The last absorption uses the fixed strict inequality $\sigma<1/2$. The same bound controls the maximum over all shifts at once; no union bound over shifts is used. + +An interval of length $X$ contains at most $X/q+1$ integers in one residue class. Also +\[ +\frac q{\varphi(q)}=\prod_{p\mid q}\frac p{p-1} + \le2^{\omega(q)}\le d(q). +\] +Since $q\le Q0$, uniformly in the allowed shifts, +\begin{equation} + \mathbb E_X\left[\chi_\delta(m) + \prod_rD_{F_r}(m;\mathbf b^{(r)})\right] + =\mathcal A_\delta+O_A(L^{-A}). +\label{eq:arithmetic-6} +\end{equation} +The implied constant may depend on the fixed functions, dimensions, +support budgets, $D,B,A$. For $\delta=0$ and $M\ge1$, the error is +\begin{equation} + O\bigl(X^{\sigma-1}(1+\log X)^{M-1} + \prod_r\|F_r\|_\infty\bigr). +\label{eq:arithmetic-7} +\end{equation} + +\end{lemma} +\begin{proof} Expand the product of divisor sums. M\"obius factors kill all nonsquarefree divisors. Whenever the product of the $F_r$'s is nonzero, the support hypothesis gives +\begin{equation} +\prod_{\ell=1}^{j_r}d_{r,\ell}\le X^{\rho_r} +\quad\text{for every }r, +\qquad +q\le\prod_\gamma d_\gamma\le X^\sigma. +\label{eq:arithmetic-8} +\end{equation} +This uses the budget of the sum of the coordinates in each factor, and does not impose a budget on the dimension. + +For a squarefree tuple, the conditions $d_\gamma\mid m+b_\gamma$ prescribe at prime $p\mid q$ the residues $m\equiv-b_\gamma\pmod p$ for all selected coordinates. These congruences are inconsistent unless the selected shifts agree modulo every $p\mid q$. When they agree, the Chinese remainder theorem prescribes exactly one class $m\pmod q$. In the unmarked case this is precisely compatibility, and the count is $X/q+O(1)$, with an absolute constant in the $O(1)$. + +When $M\ge1$, the number of positive integer tuples whose product is at most $Y\ge1$ is at most +\begin{equation} +Y\left(\sum_{d\le Y}\frac1d\right)^{M-1} + \le Y(1+\log Y)^{M-1}. +\label{eq:arithmetic-9} +\end{equation} +Indeed, fix the first $M-1$ entries and bound the number of possible last entries by $Y$ divided by their product, then enlarge each preceding summation to $d\le Y$. Apply \eqref{eq:arithmetic-9} with $Y=X^\sigma$, multiply the $O(1)$ count error by $\prod_r\|F_r\|_\infty$, and divide by $X$. This proves \eqref{eq:arithmetic-7}, hence \eqref{eq:arithmetic-6} in the unmarked case. If $M=0$, the unmarked average and \eqref{eq:arithmetic-5} are both exactly $\prod_rF_r$. + +For the marked case, put $n=m+a_0$. On a residue class satisfying these divisor congruences, the residue of $n\pmod p$ is $a_0-b_\gamma$ whenever $p\mid d_\gamma$. Consequently this class is reduced modulo $q$ exactly when the additional marked compatibility condition holds. If it is not reduced, a prime $n$ in that class would have to be a prime divisor of $q$, hence satisfy $n\le q$. But \eqref{eq:arithmetic-8} and $\sigma<1/2$ give +\[ +n\ge X-H_X> X^\sigma\ge q +\] +for large $X$. The marked sum on a nonreduced class is therefore exactly zero, not an unestimated exceptional term. On a reduced class, the marked sum is +\[ +\frac X{\varphi(q)}+O(E_X(q)). +\] +Since $q$ is squarefree, $\varphi(q)=\prod_{p\mid q}(p-1)$, proving the main density factor in \eqref{eq:arithmetic-5}. + +For fixed squarefree $q$, each prime dividing $q$ must be assigned to a nonempty subset of the $M$ divisor coordinates. There are at most $(2^M-1)^{\omega(q)}\le (2^M)^{\omega(q)}$ choices if $M\ge1$; support and compatibility only reduce this count. The empty tuple, if $M=0$, has $q=1$ and is handled directly by the same bound with $K=1$. Thus the total normalized error is at most +\[ +\frac{\prod_r\|F_r\|_\infty}{X} + \sum_{q\le X^\sigma}\mu^2(q)(2^M)^{\omega(q)}E_X(q), +\] +which is $O_A(L^{-A})$ by Lemma~\ref{lem:arithmetic-1}. This proves the marked case, including all scalar factors and the wholly zero-dimensional case. \end{proof} + +\subsection{Fourier evaluation of the density sum} + +The residue-class calculation has controlled the prime-counting error. +It remains to evaluate its finite density sum, separating a constant +determined by the test functions from the singular series of the shifts. + +\begin{proof}[Proof of Proposition~\ref{prop:mixed-moments}] +Let $M=|\Gamma|$. By Lemma~\ref{lem:arithmetic-2}, it remains to +evaluate $\mathcal A_\delta$ from \eqref{eq:arithmetic-5}: the arithmetic +error is $O_N(L^{-N})$ for every fixed $N$. +If $M=0$, that sum is exactly $\prod_rF_r$, and the assertion follows. +We henceforth assume $M>0$. + +Choose a smooth compact extension $\widetilde F_r$ of each positive-dimensional +function and define its Fourier transform by +\[ + \Phi_r(\mathbf u)=(2\pi)^{-j_r}\int_{\mathbb R^{j_r}} + e^{\sum_\ell v_\ell}\widetilde F_r(\mathbf v) + e^{\mathrm i\mathbf u\cdot\mathbf v}\,d\mathbf v. +\] +The function $\Phi_r$ is Schwartz, and Fourier inversion gives exactly +\begin{equation}\label{eq:fourier-inversion} + F_r(\mathbf v)=\int_{\mathbb R^{j_r}}\Phi_r(\mathbf u) + e^{-\sum_\ell(1+\mathrm i u_\ell)v_\ell}\,d\mathbf u + \qquad(\mathbf v\in[0,\infty)^{j_r}). +\end{equation} +For scalar factors use the scalar itself. Put $\Phi=\prod_r\Phi_r$, +$z_\gamma=(1+\mathrm i u_\gamma)/L$, and +$z_S=\sum_{\gamma\in S}z_\gamma$. + +Insert \eqref{eq:fourier-inversion} into the density sum +\eqref{eq:arithmetic-5}, retaining the sum over all squarefree tuples. +The following bound justifies moving that sum inside the integrals: +\begin{equation}\label{eq:absolute-euler-sum} + \sum_{\mathbf d}\rho_\delta(\mathbf d)\prod_\gamma d_\gamma^{-1/L} + \le\prod_p(1+C_Mp^{-1-1/L})\ll_M L^{C_M}. +\end{equation} +Indeed $1/(p-\delta)\le2/p$, every nonempty selected subset contributes at +most $p^{-1/L}$, and there are at most $2^M-1$ subsets. Moreover +$\sum_pp^{-1-\varepsilon}\le\log\zeta(1+\varepsilon) +\le\log(1+1/\varepsilon)$. Multiplication by the finite integral +$\int|\Phi|$ proves absolute convergence of the sum-integral interchange. + +Call a nonempty subset $S\subset\Gamma$ allowed at $p$ if its shifts satisfy +the same compatibility and mark-exclusion conditions. The exact Euler factor +is then +\[ + P_p(\mathbf z)=1+\frac1{p-\delta} + \sum_{S\text{ allowed at }p}(-1)^{|S|}p^{-z_S}, + \qquad + \mathcal A_\delta=\int\Phi(\mathbf u)\prod_pP_p(\mathbf z)\,d\mathbf u. +\] +By \eqref{eq:absolute-euler-sum} and Schwartz decay, the part of this integral +where $\max_\gamma|u_\gamma|>\sqrt L$ is $O_N(L^{-N})$ for every fixed $N$. +All subsequent comparisons are made on the complementary region. + +The factors carrying the zeta poles are +\[ + B_p(\mathbf z)=\prod_{i=1}^t\prod_{\varnothing\ne S\subset\mathcal M_i} + (1-p^{-1-z_S})^{-(-1)^{|S|}},\qquad + H_p(\mathbf z)=P_p(\mathbf z)/B_p(\mathbf z). +\] +For $\Re z_\gamma\ge0$, all denominators in this expression have absolute +value at least $1/2$, and $|H_p(\mathbf z)|\le\exp(C_M/p)$. If distinct shifts +in $\mathcal H$ have distinct residues modulo $p$, the allowed subsets are +exactly the nonempty subsets of the individual fibers $\mathcal M_i$. +Matching the terms of order $p^{-1}$ gives +\begin{equation}\label{eq:generic-local-factor} + H_p(\mathbf z)=1+O_M(p^{-2}), +\end{equation} +uniformly for $\Re z_\gamma\ge0$. + +At $\mathbf z=0$, summing signs over the nonempty subsets of each occupied +allowed residue class gives +\[ + P_p(0)=\frac{p-\nu_p(\mathcal H)}{p-\delta},\qquad + B_p(0)=(1-1/p)^t. +\] +It follows that $H_p(0)$ is the local factor of $\mathfrak S(\mathcal H)$. +This identity holds also when that factor is zero. + +We now compare the product of the $H_p$'s additively with its value at zero. +Take $Y=\max(2,2DL^B)$. For $p\le Y$, differentiation along the segment from +zero to $\mathbf z$ gives +\[ + |H_p(\mathbf z)-H_p(0)|\ll_M L^{-1/2}\frac{\log p}{p}. +\] +For example, $|p^{-z_S}-1|\le|z_S|\log p$ on that segment, while the denominators +stay bounded away from zero. Each mixed partial product in the additive +telescoping identity is at most +\[ + \exp\left(C_M\sum_{p\le Y}1/p\right)\ll_M(\log Y)^{C'_M}. +\] +Using even the elementary bound +$\sum_{p\le Y}(\log p)/p\le\sum_{n\le Y}(\log n)/n=O((\log Y)^2)$, we obtain +an error $O(L^{-1/2}(\log L)^A)$ for the finite product. For $p>Y$ there are +no collisions of distinct shifts, so \eqref{eq:generic-local-factor} shows +that both remaining tail products are $1+O_M(1/Y)$. Consequently +\begin{equation}\label{eq:zero-safe-comparison} + \prod_pH_p(\mathbf z)=\mathfrak S(\mathcal H) + +O\!\left(L^{-1/2}(\log L)^A\right),\qquad + \mathfrak S(\mathcal H)\ll(\log L)^A. +\end{equation} +This argument never divides by a possibly zero local singular-series factor. + +It remains to evaluate the zeta factors. Absolute Euler products for +$\Re z_\gamma>0$ give +\[ + \prod_pB_p(\mathbf z)=\prod_{i=1}^t + \prod_{\varnothing\ne S\subset\mathcal M_i} + \zeta(1+z_S)^{(-1)^{|S|}}. +\] +Write $w_\gamma=1+\mathrm i u_\gamma$ and +\[ + K_i(\mathbf u)=\prod_{\varnothing\ne S\subset\mathcal M_i} + \left(\sum_{\gamma\in S}w_\gamma\right)^{(-1)^{|S|+1}}. +\] +The pole expansion $\zeta(1+z)=z^{-1}(1+O(|z|))$ and the identity +$\sum_{\varnothing\ne S\subset\mathcal M_i}(-1)^{|S|}=-1$ imply +\begin{equation}\label{eq:zeta-kernel} + \prod_pB_p(\mathbf z) + =L^{-t}\prod_iK_i(\mathbf u)\{1+O_M(L^{-1/2})\}. +\end{equation} +Every denominator in $K_i$ has positive integer real part, so the product of +the $K_i$ has at most polynomial growth. Thus +\[ + \mathcal C=\int_{\mathbb R^M}\Phi(\mathbf u)\prod_iK_i(\mathbf u)\,d\mathbf u +\] +is absolutely convergent. Integrating \eqref{eq:zero-safe-comparison} and +\eqref{eq:zeta-kernel} against the Schwartz transform, then restoring the +negligible frequency tails, proves \eqref{eq:mixed-moment}. The constant is +real by conjugation under $\mathbf u\mapsto-\mathbf u$. + +Finally, a singleton fiber contributes $w_\gamma$, and a double fiber +contributes +\[ + \frac{w_\gamma w_{\gamma'}}{w_\gamma+w_{\gamma'}} + =w_\gamma w_{\gamma'}\int_0^\infty + e^{-(w_\gamma+w_{\gamma'})y_i}\,dy_i. +\] +After taking absolute values, the new variables contribute +$e^{-2\sum_i y_i}$; the remaining factor is a Schwartz function times a fixed +polynomial. Fubini therefore applies. Recombining the Fourier integrals by +differentiating \eqref{eq:fourier-inversion} proves +\eqref{eq:mixed-derivative-constant}. +\end{proof} + +\subsection{Averaging the singular series}\label{sec:singular-average} + +We next average the numerical-shift factor in +Proposition~\ref{prop:mixed-moments}. The following form allows different +coordinates to range over either of the two blocks. It is a box version +of Gallagher's singular-series mean \cite[p.~5, equation~(3)]{Gallagher1976}. +We give a self-contained proof by truncating the Euler product; +no statement about the distribution of primes in individual short intervals +is assumed. + +\begin{lemma}[Singular series in boxes]\label{lem:singular-average} +Fix an integer $s\ge0$ and a constant $K\ge1$. For each positive integer $h$, +let $I_1,\ldots,I_s$ be intervals of $h$ consecutive integers, all contained +in an interval of diameter at most $Kh$. Then, as $h\to\infty$, +\[ + \sum_{\substack{a_i\in I_i\ (1\le i\le s)\\a_1,\ldots,a_s\text{ distinct}}} + \mathfrak S(\{a_1,\ldots,a_s\})=h^s(1+o(1)). +\] +The error is uniform in the intervals subject to this condition, with $s,K$ +fixed. Here $A_s$ denotes a fixed finite constant depending only on $s$. In fact the normalized sum differs from one by +\[ + O_{s,K}(1/\log\log h)+O_s\big(h^{-1/2}(\log\log h)^{A_s}\big). +\] +\end{lemma} + +\begin{proof} +The cases $s=0,1$ are immediate. Otherwise let $y=(1/4)\log h$ and +$Q=\prod_{p\le y}p$. The prime number theorem gives $Q\le h^{1/2}$ for large +$h$. For all tuples, including those with equal entries, define +\[ + \mathfrak S_y(\mathbf a)=\prod_{p\le y} + (1-\nu_p(\{a_1,\ldots,a_s\})/p)(1-1/p)^{-s}. +\] +The exponent remains $-s$ even on tuples with collisions. For independent +uniform residues $a_1,\ldots,a_s$ modulo a prime $p$, averaging over an +additional uniform residue $x$ gives +\[ + \mathbb E(1-\nu_p/p) + =\mathbb P(a_i\ne x\text{ for every }i)=(1-1/p)^s. +\] +The Chinese remainder theorem therefore shows that $\mathfrak S_y$ has +exactly mean one on complete residue tuples modulo $Q$. + +A uniform integer in a length-$h$ interval has residue distribution modulo +$Q$ within total variation $O(Q/h)$ of uniform. The elementary bound $\sum_{p\le y}1/p=O(\log\log(3y))$ used above gives +\[ + 0\le\mathfrak S_y(\mathbf a) + \le\prod_{p\le y}(1-1/p)^{-s}\ll_s(\log y)^{A_s}. +\] +It follows that +\[ + h^{-s}\sum_{a_i\in I_i}\mathfrak S_y(\mathbf a) + =1+O_s\big((Q/h)(\log y)^{A_s}\big). +\] +There are $O_s(h^{s-1})$ tuples with equal entries, so deleting them changes +this normalized mean by at most $O_s(h^{-1}(\log y)^{A_s})$. + +For a distinct tuple put $\Delta=\prod_{i2s$ every factor of the tail +\[ + T_y(\mathbf a)=\prod_{p>y}(1-\nu_p/p)(1-1/p)^{-s} +\] +is positive. A prime not dividing $\Delta$ has $\nu_p=s$ and a local factor +$1+O_s(p^{-2})$; the logarithm of a remaining factor is $O_s(1/p)$. Hence +\[ + |\log T_y|\ll_s \frac1y+\sum_{\substack{p>y\\p\mid\Delta}}\frac1p + \le C_s\left(\frac1y+\frac{\log\Delta}{y\log y}\right). +\] +Since $\log\Delta\le\binom s2\log(Kh)$, this gives, uniformly in the distinct +tuples, +\[ + T_y(\mathbf a)=1+O_{s,K}(1/\log\log h). +\] +Now use the identity $\mathfrak S(\{a_1,\ldots,a_s\})=\mathfrak S_y(\mathbf a)T_y(\mathbf a)$ +and the nonnegative truncated mean. This identity is valid also if a +small-prime factor vanishes; no quotient by $\mathfrak S_y$ is taken. +The stated error follows from $Q\le h^{1/2}$ and $\log y\asymp\log\log h$. +\end{proof} + +When the shifts are summed, an equality pattern with $t$ distinct unmarked shifts has +$O(h^t)$ assignments in fixed blocks of length $h\asymp L$. Thus the uniform +error in Proposition~\ref{prop:mixed-moments}, summed over that pattern, +is $O(L^{-1/2}(\log L)^A)=o(1)$. If one additional distinct prime shift is +summed and its weight divided by $L$, there are $O(h^{t+1})$ assignments and +the same conclusion holds. Only finitely many patterns occur for a fixed +family of functions. The family is chosen before $X$ tends to infinity. + +\subsection{Evaluation of the square}\label{sec:weight-moment-proof} + +We now apply the correlation formula and singular-series mean to the +functions constructed in Section~\ref{sec:weights}. The signed complete +mixed derivative of each cumulative function is $f_j$, as in +\eqref{eq:cumulative-derivative}; its vanishing on coordinate faces will +leave only matching subsets in the quadratic moments. + +The support restrictions in the moment formula apply to the sum of +the support budgets of the factors, independently of their dimensions. +For the present choice $\tau=1/8$, the required uses are as follows: +\begin{center} +\begin{tabular}{lll} +Quantity & Moment used & Total support budget\\ +\hline +$\mathbb E_XZ^2$ & unmarked & $2\tau<1$\\ +$\mathbb E_X(Z^2V_B)$, $B=I,J$ & one prime mark & $2\tau<1/2$\\ +$\mathbb E_XZ^4$ & unmarked & $4\tau<1$\\ +Detector off-diagonal terms & unmarked & $6\tau<1$ +\end{tabular} +\end{center} +The last line uses divisor-sum majorants for the two possible primes +in $J$. It therefore requires no asymptotic formula with two prime marks. + +\begin{proof}[Proof of Proposition~\ref{prop:weight-moments}] +First we check the smoothness and support needed by the moment +formula. Extend $f_j$ by zero to $\mathbb R^j$. Its cumulative integral +is smooth on $\mathbb R^j$, and multiplication by smooth coordinate +cutoffs that equal one near $[0,\infty)$ and vanish below $-1$ makes it +compactly supported without changing its orthant values or derivatives. +Indeed, the cumulative integral already vanishes when any coordinate +exceeds an upper support bound of $f_j$. On the nonnegative orthant, +$F_j$ vanishes if $\sum_i v_i\ge\tau$, and its signed complete mixed derivative is given by +\eqref{eq:cumulative-derivative}. +This derivative is zero on every coordinate face. For $j\ge1$, +$Tf_{j+1}$ has the same smoothness, interior-support, and face-vanishing +properties. + +Expand $Z^2$ over pairs of subsets $S,T\subset I$. Any shift in +$S\mathbin\triangle T$ occurs once and sets a coordinate of one of the +functions in \eqref{eq:cumulative-derivative} to zero. Its main +coefficient is therefore zero. If $S=T$ has size $j$, the coefficient +is $\|f_j\|_2^2$. Averaging the singular series and using +\[ + \frac{\binom hj}{L^j}\longrightarrow\alpha_j +\] +proves the first limit in \eqref{eq:weight-unmarked}. With a prime mark +in $J$, the same matching rule applies and +\[ + \frac{h\binom hj}{L^{j+1}}\longrightarrow\lambda\alpha_j. +\] +The two divisor factors have total support $2\tau<1/2$, so the marked +moment formula proves the first limit in \eqref{eq:weight-marked}. + +For an internal mark $a\in I$, the exact deletion identity +\eqref{eq:prime-deletion} replaces $F_j$ by +$\widetilde F_j=F_j+F_{j+1}(\cdot,0)$ on the prime event. +Deletion is a bijection, so no extra dimension factor occurs. The +support radius of $\widetilde F_j$ is still $\tau$, and its signed +complete mixed derivative is $f_j+Tf_{j+1}$. It vanishes on coordinate +faces when $j\ge1$. Apply the external marked formula to these +functions. Only equal subsets $U$ survive, and +\[ + \frac{h\binom{h-1}{j}}{L^{j+1}}\longrightarrow\lambda\alpha_j. +\] +This proves the formula for $v$, including the scalar term +$\lambda|f_0+\int f_1|^2$. + +The equality-pattern estimate preceding this subsection justifies +summing the errors in these limits. Applied to $Z^4$, the same argument +uses four divisor factors, total support +$4\tau<1$, and only finiteness of their pattern constants. Bounded +averages of the singular series then give the second assertion in +\eqref{eq:weight-unmarked}. + +It remains to prove a detector estimate whose constant is independent +of the chosen family. Write $q_X=\log(2X+2h)/L=1+o(1)$. At a +relevant prime $m+b$, we have $D_G(m;b)=1$. Consequently, pointwise, +\[ + \frac{\vartheta(m+b)}L\le q_XD_G(m;b)^2, + \qquad + \left(\frac{\vartheta(m+b)}L\right)^2 + \le q_X\frac{\vartheta(m+b)}L. +\] +The first inequality holds at composites as well, since its left side +is zero. Because $Z^2\ge0$, these inequalities give +\begin{align*} + \mathbb E_X(Z^2V_J^2) + &\le q_X\mathbb E_X(Z^2V_J)\\ + &\quad+q_X^2\sum_{\substack{b,c\in J\\b\ne c}} + \mathbb E_X\bigl[Z^2 D_G(m;b)^2D_G(m;c)^2\bigr]. +\end{align*} +An off-diagonal summand contains six divisor factors, with total +support $6\tau<1$. The shifts $b,c$ are paired and lie outside $I$. +The two subsets from $Z^2$ must again be identical. For their common +size $j$, the coefficient is $c_G^2\|f_j\|_2^2$. The pair $(b,c)$ is +ordered, and +\[ + \frac{h(h-1)\binom hj}{L^{j+2}} + \longrightarrow\lambda^2\alpha_j. +\] +Thus the off-diagonal sum tends to $\lambda^2c_G^2w$. +Together with \eqref{eq:weight-marked}, this proves +\eqref{eq:detector-bound}. Although the convergence rates may depend +on the fixed family, the resulting $C_*$ does not. +\end{proof} diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/04-weights.tex b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/04-weights.tex new file mode 100644 index 0000000000000000000000000000000000000000..92350e62d6a9d583aec295b9b9de7bd806b3c854 --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/04-weights.tex @@ -0,0 +1,131 @@ +\section{A square with a small marked moment} +\label{sec:weights} + +We construct the weight in Proposition~\ref{prop:weights} as the square +of a signed sum of smooth divisor sums. We first define that square +and state the exact analytic estimates it satisfies. The next section +uses their formulas to choose the coefficients and functions; the +proofs of the analytic estimates follow in Section~\ref{sec:moments}. + +Fix $\lambda>0$, and write +\[ + L=\log X,\qquad h=\lfloor\lambda L\rfloor,\qquad + J=\{1,\ldots,h\},\qquad I=\{h+1,\ldots,2h\}. +\] +Here $X$ tends to infinity through integers, and $\mathbb E_X$ denotes +the average over the integers $X1>\tau$, so its contribution vanishes by the support +condition on $F_j$. +Put $\widetilde F_j=F_j+F_{j+1}(\cdot,0)$. Deleting $a$ from each +subset that contains it therefore gives the exact identity +\begin{equation}\label{eq:prime-deletion} + Z(m)=\sum_{j=0}^k + \sum_{\substack{U\subset I\setminus\{a\}\\|U|=j}} + D_{\widetilde F_j}(m;U) + \qquad\text{when $m+a$ is prime}. +\end{equation} +Each subset occurs once, so deletion introduces no dimension factor. +The signed complete mixed derivative of $\widetilde F_j$ is +$f_j+Tf_{j+1}$. This explains why the marked form $v$ couples neighboring +dimensions. + +The analytic proof evaluates the products in these squares by pairing +equal shifts. Any unmatched shift sets a coordinate of a complete +mixed derivative to zero, so the face vanishing in +\eqref{eq:cumulative-derivative} removes its main term. Only matching +subsets remain in the limiting quadratic forms, producing the squared +norms in $w$ and $v$. The same proof controls the errors and establishes +the two different second-moment bounds. + +For a family with $w>0$, the normalization $W_X=Z^2/w$ therefore has +all the properties in Proposition~\ref{prop:weights} except the +prescribed smallness of the prime mass in $I$. That last requirement +is exactly $v/w\le\varepsilon$. The next section constructs families +with $w\to1$ and $v\to0$, while the detector constant $C_*$ remains +unchanged. diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/05-cancellation.tex b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/05-cancellation.tex new file mode 100644 index 0000000000000000000000000000000000000000..266761414ff27383037ca7a81e1fc29683eb168b --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/build/sections/05-cancellation.tex @@ -0,0 +1,181 @@ +\section{Cancellation between adjacent dimensions}\label{sec:cancellation} + +We now choose the functions so that the adjacent-dimension terms in +$v$ almost cancel. Product functions restricted to a simplex, with +the truncation controlled by moments of their squared density, also +occur in Maynard's sieve construction \cite[Section~7]{Maynard2015}. +Here we combine such functions at a short range of dimensions with +alternating signs. + +\begin{lemma}[A one-dimensional profile] +\label{lem:small-first-moment} +For every $\lambda>0$ and $\beta>0$, there exists a nonnegative +$g\in C_c^\infty((0,\infty))$ such that +\[ + \int_0^\infty g(u)^2\,du=1,\qquad + \int_0^\infty g(u)\,du=\sqrt\lambda,\qquad + \mu_g:=\int_0^\infty u g(u)^2\,du<\beta. +\] +\end{lemma} + +\begin{proof} +For $R\ge2$, choose a smooth $\phi_R$ on $(0,\infty)$, equal to $1/u$ +on $[1,R]$, with $0\le\phi_R(u)\le1/u$ and support in $[1/2,2R]$. +Put +\[ + A_R=\int\phi_R,\qquad B_R=\int u\phi_R(u)^2\,du, + \qquad N_R^2=\int\phi_R^2. +\] +Then $A_R\ge\log R$, $B_R\le\log(4R)$, and +$1/2\le N_R^2\le2$. Set $g_0=\phi_R/N_R$, +$c=\lambda/(\int g_0)^2$, and $g(u)=c^{-1/2}g_0(u/c)$. +Changes of variable give exactly +\[ + \int g^2=1,\qquad \int g=\sqrt\lambda, + \qquad + \mu_g=\lambda\frac{B_R}{A_R^2} + \le\lambda\frac{\log(4R)}{(\log R)^2}. +\] +Choose $R$ sufficiently large and then hold it fixed. +\end{proof} + +The profile just constructed may have a very large support endpoint +and variance. We need only their finiteness for this one fixed +profile: the dimension will be chosen afterwards. The auxiliary +limit in the next proposition concerns the constants $w$ and $v$ +attached to finite families, before any asymptotic in $X$ is taken. + +\begin{proposition}[Cancellation between high dimensions] +\label{prop:dimension-cancellation} +For every fixed $\lambda>0$, there are families of the type used in +Proposition~\ref{prop:weight-moments}, with increasing finite maximum +dimension $k$, for which $w\to1$ and $v\to0$. +\end{proposition} + +\begin{proof} +Fix $0<\beta<\gamma<\tau$ and choose $g$ from +Lemma~\ref{lem:small-first-moment}. Write $d=\beta-\mu_g>0$. +Let $B$ be an upper endpoint of the support of $g$, and let $s_g^2$ +be the variance of the probability density $g^2$. All these +quantities are fixed and finite; they need not be bounded uniformly +in $\lambda$. + +For a large integer $k$ and $0\le j\le k$, define +\[ + Q_j(\mathbf t)=k^{j/2}\prod_{i=1}^j g(kt_i),\qquad + S_j=\sum_{i=1}^j t_i, +\] +with $Q_0=1$ and $S_0=0$. Before imposing the simplex cutoff, these +products satisfy +\[ + \|Q_j\|_2=1,\qquad TQ_{j+1}=\sqrt{\lambda/k}\,Q_j\quad(0\le j\beta-B/k)\le\frac{4s_g^2}{kd^2}. +\] +On the complementary event every cutoff in +\eqref{eq:interior-cancellation} equals one. The bracket is then +$1-\sqrt{(j+1)/k}$, whose square is at most $(r/k)^2$. Everywhere, +both terms in the bracket lie in $[0,1]$, since $g\ge0$, +$\int g=\sqrt\lambda$, and $j+1\le k$; its absolute value is at most +one. After summing the $r-1$ interior terms, their total contribution +to $v/\lambda$ is at most +\[ + (r/k)^2+\frac{4s_g^2}{kd^2}. +\] + +Here the probability estimate is with respect to $Q_j^2$, while the +coordinate-deletion integral contains $g(u)\,du$. The passage between +them uses no comparison of these measures: on the complementary event, +$S_j+u/k\le\beta$ holds for every $u$ in the support of $g$. +Thus the whole integral has its cutoff equal to one pointwise. + +Exactly two further indices can contribute. At the top, $j=k$, the +term is $\alpha_k\|f_k\|_2^2\le1/r$. At the bottom, +$j=a-1=k-r$, only $Tf_a$ occurs, and +\[ + \sqrt{\alpha_{a-1}}Tf_a + =\frac{(-1)^aQ_{a-1}}{\sqrt r} + \sqrt{\frac a{\lambda k}} + \int_0^\infty g(u)\chi(S_{a-1}+u/k)\,du. +\] +Its squared norm is at most $a/(kr)\le1/r$. Consequently +\begin{equation} + 0\le\frac v\lambda + \le\frac2r+\left(\frac rk\right)^2+\frac{4s_g^2}{kd^2} + \longrightarrow0. + \label{eq:v-cancellation-bound} +\end{equation} +Together with \eqref{eq:w-concentration}, this proves the proposition. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:weights}] +Fix $\lambda>0$ and one function $G$ as in +Proposition~\ref{prop:weight-moments}. Set +\[ + K_\lambda=C_*=\lambda+\lambda^2c_G^2. +\] +This choice is independent of $\varepsilon$. Given $\varepsilon>0$, +choose $\beta$, $\gamma$, $g$, and $\chi$ as in the preceding construction. +By Proposition~\ref{prop:dimension-cancellation}, one can select a +\emph{single finite} $k$ and its family with $w>0$ and +$v/w\le\varepsilon$. Freeze this family and define +\[ + W_X(m)=\frac{Z(m)^2}{w}. +\] +All conclusions follow from Proposition~\ref{prop:weight-moments} by +division by $w$, or by $w^2$ for the second moment. Once the construction +is chosen in terms of $\lambda$ and $\varepsilon$, the fixed-family +second-moment bound may be written $O_{\lambda,\varepsilon}(1)$. + +The parameter order is essential: the auxiliary limit $k\to\infty$ +produces one family, and only then does $X\to\infty$. No divisor-sum +asymptotic uniform in $k$ is required. For an arbitrary fixed gap +threshold $C>0$, the application in Section~\ref{sec:counting} chooses a fixed +$\lambda>\max\{1,C\}$ before making these choices. +\end{proof} diff --git a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..eafa91fc746a6edf59b72ea9858f27625c53b445 --- /dev/null +++ b/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7a4a37fa92b30f4e9749ed16a6e8d715eb98ab0c0332cc5061ac6b2a067009ed +size 434255 diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/README.md b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..1bfdebe1664e3376303d8d7205c29c7284b7b171 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/README.md @@ -0,0 +1,16 @@ +# [Positive scalar curvature and uniform codimension-two width](paper.pdf) + +OpenAI · September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026, + author = {{OpenAI}}, + title = {{Positive scalar curvature and uniform codimension-two width}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/minimizers.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/minimizers.tex new file mode 100644 index 0000000000000000000000000000000000000000..8f2c443800eb13e1a44698a6d41661d834c85d3d --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/minimizers.tex @@ -0,0 +1,1187 @@ +\section{Weighted minimizing cuts and the forms on retained sides}\label[appendix]{sec:cut} + +Prescribed-curvature perimeter minimization and convergence of its +boundaries have classical precedents in Massari and Massari--Pepe +\citep{massari1975,massariPepe1975}. For scalar-curvature applications, +prescribed-mean-curvature minimization is the $\mu$-bubble method +described by \citet[\S\(5\frac56\)]{gromov1996}. +We prove here the weighted, +varying-metric and product-end versions required by the cutting +construction, including the forms on both retained sides. + +Fix a finite stage $r$ and the label $j_0$ to be processed. The +packet and profile hypotheses are those of \cref{packet:category,packet:invariant}. +For the finite-horizon formulation in this appendix, only finitely many +labels and finitely many nonunit old warps occur in each incident diagram +under consideration. The processed potential has a positive lower bound +\[ + \Pi_{j_0}\ge\pi_{\rm cut}>0. +\] +A common lower bound for all labels at all stages is not required. +The constants satisfy $a_r\le .63$, $b_r\ge1.07$, and +$\lambda_r>\lambda_{r+1}$. The extension to a countable collection of +labels is proved in the main text; every argument below uses only the +current pressure and the indicated finite lists. + +Throughout, $E$ denotes a parent packet of base dimension $m\ge2$. +After the radial split it has the compact truncations, finite incidence, +and exact product ends of \cref{packet:category}. At the initial split +we instead work on the complete original base, and use the locally finite +compact radial barriers of \cref{packet:initialization}. We assume no +volume growth or bounded geometry on that initial base. + +On a finite incident diagram, insert unit warps where necessary so that +the old warp lists agree. Their metric is +\[ + \widehat g=g+\sum_{i=1}^q f_i^2\,\dd\theta_i^2, + \qquad \omega=\prod_{i=1}^q f_i, + \qquad \dd\mu=\omega\,\dd\vol_g. +\] +Empty products are one. Fiber volume constants are suppressed throughout. +Adding a unit warp changes neither $\omega$ nor any formula below. +At the initial radial stage the list need only be finite on each compact +window; the compatible local expressions then agree on overlaps. No +infinite-dimensional torus is used. In particular all minimizing sets, +BV compactness, tangent cones, and singular-dimension bounds concern the +$m$-dimensional base, regardless of the length of the warp list. +Write $R=\Scal_{\widehat g}$ and +$\Div_\omega Y=\omega^{-1}\Div_g(\omega Y)$. +For a set $A$ of locally finite perimeter define its weighted perimeter by +\[ + P_\omega(A;U)=\int_{U\cap\partial^*A}\omega\,\dd\mathcal H_g^{m-1}. +\] +The reduced-boundary representation is the framework developed by +De Giorgi \citep{deGiorgi1954,deGiorgi1955}; the current formulation +and slicing originate in Federer and Fleming \citep{federerFleming1960}. +We use the precise representation and slicing statements in +\citet[Chapter~3, Section~4, and Chapter~6, Section~4]{simonGMT}. +All set equalities and inclusions involving measurable phases are understood +up to $\mu$-null sets. The phase representative used for geometry will be +specified below. Constants in local estimates may depend on a compact +window; after the radial split they can be chosen uniformly on each fixed +packet, without any uniformity between packets. No scalar-curvature lower bound +is used to construct the cuts or to prove their local regularity. The +parent scalar bound enters only at the end of this appendix, and only on +the required region $D_E$ of \cref{packet:required-set}. + +\subsection{Smooth finite pressures and absolute minimizers} + +\begin{lemma}[Realizing the processed profile]\label{cut:pressure} +There is a smooth finite compatible function $p_0$ and a number $c_*>0$ +such that every $p=p_0+c$, $|c|0$. +All collars used here are buffered inside a relatively compact ambient +chart. The functions under consideration are integrable and have +finite total variation on each smaller collar reaching $T$; for the +restrictions of our finite-perimeter phases this follows from local +finite perimeter in the ambient chart. Local BV regularity only on +sets bounded away from $T$ would not suffice. + +Let $w$ be such a BV function on the plus half-collar, and let $D_t w$ +denote its distributional derivative in the $t$ direction. +Choose a compact tangential box $K$ and an open box $K'$ with +$K\Subset K'\Subset Q$, and fix $a_00$, gives +\begin{equation}\label{cut:fixed-face-slices} + \int_K |w(y,t)-w(y,s)|\,\dd y + \le |D_t w|\bigl(K'\times(s,t)\bigr), + \qquad 00$, \eqref{cut:barrier-volume} would imply +$V(R)\ge V(R_0)e^{(R-R_0)/2}$, a contradiction. Thus $G$ is null. +For a relatively compact radial barrier, simply add all of $D$; no +slicing terms occur, and the same comparison gives $\mu(G)=0$ directly. +The complement of $A$ minimizes pressure $-p$, proving the negative +assertion. The domains contain the core closures in their interiors; +the packetwise positive separation follows from exact products and the +compact-part property. A phase equal to one or zero on an open set has +no perimeter support there. +\end{proof} + +\begin{proposition}[Generic uniqueness on all required products] +\label{cut:generic-uniqueness} +After the radial split one common shift $c$, with $|c|0$, use that competitor +over $(-L,L)^d$ and the phase $F$ outside, retaining the original phase +outside the compact base discrepancy. The energy gain is +$\eta(2L)^d$. The interfaces at the Euclidean box boundary cost at most +$O(L^{d-1})$ times the weighted volume of that compact discrepancy. +For large $L$ this contradicts minimality. Thus $F$ minimizes on $E$, +and base uniqueness identifies it with $A_E$. The case $d=0$ needs +no product argument. +\end{proof} + +At the radial split set $c=0$ and choose any minimizer supplied by +\cref{cut:bv-compactness}. Its compact barriers confine its boundary to +the locally finite compact transition shells. No uniqueness or +polynomial-growth assertion on the initial manifold is used. We call +the selected minimizing boundary a \emph{true cut}; it will later be +replaced by a compatible regular wall near the required region. + +\subsection{The precise local regularity input} + +\begin{theorem}[Local GMT properties used here]\label{cut:gmt} +Let $A$ satisfy \eqref{cut:local-minimality} on a smooth $m$-manifold, +$m\ge2$, with smooth positive $\omega$ and smooth $p$. Put +$S=\supp|D\chi_A|$. Then $S=S_{\rm reg}\cup\Sigma$, where $S_{\rm reg}$ +is a smooth embedded hypersurface, $\Sigma$ is relatively closed, and +\begin{equation}\label{cut:singular-dimension} + \dim_{\mathcal H}\Sigma\le m-8. +\end{equation} +A negative dimension bound means that $\Sigma$ is empty. The outward +measure-theoretic normal extends smoothly on $S_{\rm reg}$. With +$B(u,v)=\langle\nabla_u\nu,v\rangle$, its base mean curvature satisfies +\begin{equation}\label{cut:base-mean-curvature} + H_g=p-\partial_\nu\log\omega. +\end{equation} +The unweighted multiplicity-one perimeter varifold has locally bounded +generalized mean curvature. Its density exists at every support point; +monotonicity gives local uniform area upper bounds and positive area +lower bounds at support points. Under the convergences of +\cref{cut:bv-compactness}, densities are upper semicontinuous at +converging points, provided the local generalized-mean-curvature bounds +are uniform. The supports converge locally, and on every compact +subset of the regular limiting hypersurface convergence is smooth and +single-sheeted when the data converge smoothly. On compact subsets of +the interiors of the limiting phases there is eventually no boundary. + +Every tangent at a point of $S$ is a multiplicity-one locally +perimeter-minimizing boundary cone in the Euclidean tangent space. +A flat tangent implies regularity. A tangent at a singular point is +therefore nonflat, and its translation-invariance spine has dimension +at most $m-8$. +\end{theorem} + +\begin{proof}[Proof by first variation, strict blowups, and regularity] +Put $d=m-1$. The classical inputs are the bounded-mean-curvature +monotonicity theorem and Allard's graphical theorem +\citep[Chapter~4, Theorem~3.17; Chapter~5, hypotheses~5.1 and +Theorem~5.2]{simonGMT}, and the singular-dimension theorem for +\emph{exactly} minimizing boundary currents +\citep[Chapter~7, Theorem~5.8]{simonGMT}. The minimizing-boundary dimension bound follows from the work of +Simons and Federer \citep{simons1968,federer1970}. The last theorem will be applied +only to constant-metric blowup limits. We verify the transfer to the +present smooth weighted problem, including the singular-set compactness +needed for the Hausdorff-dimension conclusion. + +\emph{First variation and local area bounds.} +First variation is available before any regularity conclusion. The +flow of a compactly supported smooth field $Z$ is an admissible +comparison in \eqref{cut:local-minimality}, so +\[ + \int_{\partial^*A}\omega + \bigl(\Div_S Z+\langle\grad\log\omega,Z\rangle\bigr) + \,\dd\mathcal H_g^d + =\int_{\partial^*A}p\omega\langle Z,\nu\rangle + \,\dd\mathcal H_g^d. +\] +Here tangent planes and $\nu$ are the measure-theoretic ones. Substituting +$Z=W/\omega$ cancels the tangential weight derivatives and gives +\[ + \int_{\partial^*A}\Div_S W\,\dd\mathcal H_g^d + =\int_{\partial^*A} + (p-\partial_\nu\log\omega)\langle W,\nu\rangle + \,\dd\mathcal H_g^d. +\] +Thus the unweighted perimeter varifold has intrinsic generalized +mean-curvature vector +$-(p-\partial_\nu\log\omega)\nu$, locally bounded by +$|p|+|\grad\log\omega|$. Its multiplicity is one almost everywhere. + +On a relatively compact coordinate patch, first pass to a smaller +patch and extend its metric to a closed manifold. Explicitly, place +its coordinate closure inside a coordinate ball of a flat torus and +blend the given positive definite metric with the flat metric outside +the smaller patch; the convex combination stays positive definite. +The resulting metric is smooth and the ambient manifold is compact, +so the smooth isometric embedding theorem applies +\citep[Theorem~2]{nash1956}. The preceding intrinsic first variation then gives +Euclidean generalized mean curvature with the additional bounded term +$\tr_{TS}\mathrm{II}$, where $\mathrm{II}$ is the second fundamental +form of this fixed embedding. Indeed decompose a Euclidean test field +into its ambient-manifold tangent and normal parts and differentiate +that decomposition along $TS$. Its normal contribution is exactly the +trace of $\mathrm{II}$. Consequently Euclidean monotonicity applies +locally. Intrinsic and embedded density limits agree, since the +embedding is isometric to first order. Monotonicity gives density +existence and upper semicontinuity, including at nonreduced support +points; their densities are at least one, by approximation by reduced +points. Together with the clearing-ball upper bound from +\cref{cut:bv-compactness}, it gives uniform upper and positive lower +area bounds on compact patches. + +\emph{A common ambient model for varying metrics.} +We record carefully the uniform version when the metric changes. +Suppose $g_i\to g$ smoothly on a fixed patch, and all other data have +the local bounds in the theorem. Any claimed failure of regular-point +persistence or support convergence has a subsequence on which the +failure persists. Pass further so that, on a slightly larger compact +patch, +\[ + \|g_i-g\|_{C^i}\le 2^{-i^2}. +\] +Set $t_i=2^{-i}$ and take disjoint smooth bump functions $\beta_i(t)$, +equal to one on a neighborhood of $t_i$, supported in intervals of +width comparable to $t_i$, and satisfying +$|\beta_i^{(b)}|\le C_b t_i^{-b}$. After omitting finitely many terms, +\[ + g_t=g+\sum_i\beta_i(t)(g_i-g),\qquad G=\dd t^2+g_t +\] +is a smooth positive metric on a product neighborhood, with $g_0=g$ +and $g_{t_i}=g_i$. For each pair of fixed orders $a,b$, the summands +after $a$ spatial and $b$ temporal derivatives are bounded by +$C_b2^{ib-i^2}$ for all sufficiently large $i$. These bounds, divided +by any fixed power of $t_i$, still tend to zero. Thus all mixed +derivatives extend at $t=0$. The plateaus also give +$\partial_tg_t=0$ at $t_i$, so the slices carrying $g_i$ are totally +geodesic in $G$. + +Extend this \emph{one} smooth metric $G$ across the edge of a smaller +product patch to a closed manifold and isometrically embed it once. +The images of the slices $t=t_i$ converge smoothly to the slice +$t=0$, and the second fundamental form of this single embedding is +bounded on their common compact neighborhood. The preceding +first-variation calculation therefore gives one Euclidean +mean-curvature bound for all the embedded perimeter varifolds. +Strict perimeter convergence from \cref{cut:bv-compactness} becomes +varifold convergence in this same Euclidean space. This construction +proves the needed sequential uniformity; it does not assume that +independently chosen isometric embeddings have uniform bounds. + +\emph{Exact minimizing tangent cones.} +Next consider a blowup of one phase at $x\in S$. In smooth coordinates, +at scale $\rho$, the normalized data are +\[ + A_{x,\rho}=\rho^{-1}(A-x),\quad + g_{x,\rho}(y)=g(x+\rho y),\quad + \omega_{x,\rho}(y)=\omega(x+\rho y),\quad + p_{x,\rho}(y)=\rho p(x+\rho y). +\] +The metric here has been divided by the square of the physical scale: +perimeter scales by $\rho^d$, volume by $\rho^{d+1}$, and this accounts +for the factor $\rho$ in the pressure. On every fixed window the data +converge smoothly to $g(x),\omega(x),0$. The strict compactness lemma +therefore identifies every subsequential tangent varifold with the +unit-multiplicity perimeter varifold of a phase minimizing perimeter +in the constant metric $g(x)$. A constant weight can be divided out. +After one linear isometry this is an ordinary Euclidean +perimeter-minimizing phase. It is also a minimizing boundary current. +Indeed, orient this Euclidean chart. A compactly supported integral +current filling a boundary competitor has top dimension and is given +by an integer-valued BV function $q$. The competitor is the boundary +of $\chi_A+q$. Clipping that integer-valued function to $[0,1]$ leaves +it unchanged outside the compact discrepancy and gives a characteristic +function without increasing variation. Any current competitor with +smaller mass would therefore give a smaller-perimeter set competitor. +This verifies the precise minimizing-boundary hypothesis of +\citet[Chapter~7, Theorem~5.8]{simonGMT}. All orientations in this +argument are local. + +The support lower bound makes this limit nonzero at the origin. +Monotonicity based at $x$ makes its mass ratio constant on all radii; +the equality case, or the stationary radial identity in +\citet[Chapter~4, Equation~(3.6)]{simonGMT}, makes it conical. +Equivalently, after a fixed local isometric embedding, dilations of +the embedded ambient manifold converge smoothly to its tangent +space, its second fundamental form is multiplied by $\rho$, and +the same blowups give this cone. Its multiplicity is one as a +perimeter varifold, although its density at the vertex need not +be one. + +\emph{Regularity and persistence under convergence.} +We now apply Allard's graphical regularity theorem \citep{allard1972}. In its Euclidean form used here, +fix an exponent $q>d$. For a rectifiable $d$-varifold of multiplicity +at least one, on a ball centered at a support point and contained in +the first-variation domain, sufficiently small excess of the normalized +mass above one and sufficiently small +\[ + \rho^{\,1-d/q} + \left(\int_{B_\rho}|\mathbf H|^q\,\dd\mu_V\right)^{1/q} +\] +give a single $C^{1,1-d/q}$ graph on a smaller ball +\citep[Chapter~5, Theorem~5.2]{simonGMT}. A unit flat tangent supplies +the small mass excess. The bounded mean curvature supplies the other +small quantity, since it is $O(\rho)$ under the local area bound. +Thus a flat tangent makes the original point regular as a +$C^{1,\alpha}$ graph. The graph satisfies the weak prescribed-curvature +equation +$H_g=p-\partial_\nu\log\omega$; its coefficients and right-hand side +are smooth functions of position, graph height, and graph gradient. +The bounded slope gives uniform ellipticity. We spell out the passage +from a weak graph to a smooth one. In a relatively compact chart write +the equation as +\[ + \partial_i A^i(x,u,Du)=B(x,u,Du). +\] +Here $A$ is the gradient, in the slope variables, of the smooth +positive weighted graph-area integrand. Thus $A_p$ is symmetric and +uniformly positive on the bounded slope range; $B$ is smooth in its +arguments. For a compactly supported cutoff $\eta$, test translated +differences with $\eta^2\delta_hu$. The difference of $A$ has the form +$a_hD\delta_hu+c_h\delta_hu+b_h$, where $a_h$ is uniformly elliptic +and all three coefficients are uniformly bounded. Since $u$ is +Lipschitz, $\delta_hu$ is bounded. On the right move the difference +quotient from $B$ to the test function, using +$\|\delta_{-h}\phi\|_2\le\|D\phi\|_2$. +Ellipticity and Cauchy's inequality then give +\[ + \int\eta^2|D\delta_hu|^2\le C_\eta +\] +independently of small $h$. Hence $u\in H^2_{\mathrm{loc}}$. +For $v=\partial_ku$, weak differentiation now gives +\[ + \partial_i(a^{ij}\partial_jv)=\partial_iF^i,\qquad + a^{ij}=A^i_{p_j},\qquad + F^i=\delta_{ik}B-A^i_{x_k}-A^i_uv, +\] +with coefficients evaluated at $(x,u,Du)$. Both $a$ and $F$ are +$C^{0,\alpha}$ because $u$ is $C^{1,\alpha}$. + +To justify the divergence Schauder estimate for this weak $H^1$ +solution, mollify $a,F$ on a slightly larger ball and solve the +coercive linear Dirichlet problems on the ball with the same trace as +$v$. Their solutions $v_\epsilon$ satisfy +\[ + \lambda\|D(v_\epsilon-v)\|_2 + \le \|F_\epsilon-F\|_2 + +\|a_\epsilon-a\|_\infty\|Dv\|_2, +\] +where $\lambda>0$ is a common ellipticity constant. Thus they converge +in $H^1$. Smooth-coefficient Sobolev regularity makes each approximant +smooth in the interior; the local $L^2$--$L^\infty$ estimate and then +the a priori divergence Schauder estimate give uniform interior +$C^{1,\alpha}$ bounds +\citep[Lecture~6, Theorem~1 and Corollary~1; Lecture~18, Theorem~2; +Lecture~12, Theorem~1]{simonPDE}. +These bounds use only the common ellipticity and H\"older bounds, +not higher derivatives of the mollified coefficients. Passing to +the limit gives $v\in C^{1,\alpha}_{\mathrm{loc}}$, hence +$u\in C^{2,\alpha}_{\mathrm{loc}}$. Repeating the difference-quotient +and weak Schauder argument for differentiated equations makes the +graph smooth. The estimates are uniform for families with common +$C^{1,\alpha}$ graph bounds and uniformly smooth geometric data. + +Likewise, near a regular point of a limiting phase choose a sufficiently +small ball with mass ratio as close to one as needed, with strict slack +in the Allard threshold, and with small scaled mean-curvature bound. +Choose its radius also so that its sphere has zero limiting perimeter +measure; almost every sufficiently small radius has this property. +For smoothly convergent data, use the single ambient embedding +construction above. Weak perimeter-measure convergence now passes the +mass bound on this ball to the approximating phases. To center at a +nearby approximating support point, decrease the radius by a fixed +arbitrarily small factor: the recentered ball lies in the chosen ball +for all large indices, and the strict slack pays the change in the +normalizing power of the radius. Allard therefore gives one graph in +each smaller ball for all sufficiently large indices. The graph equation and smooth data convergence then give +smooth convergence on smaller compact patches. This also proves +\[ + x_i\in\Sigma_i,\quad x_i\longrightarrow x + \quad\Longrightarrow\quad x\in\Sigma +\] +for interior limits: a regular limiting point has just been excluded, +and a point outside the limiting support is excluded by the uniform +support lower bound and perimeter convergence. The same lower bound +and weak convergence prove local support convergence; they also exclude +stray support in compact subsets of phase interiors. The regular set +is open in the support, so its complement $\Sigma$ is relatively closed. + +\emph{Transfer of the singular-dimension bound.} +For the singular-dimension transfer we use the content argument in +\citet[Chapter~1, Theorem~3.26(2), and Appendix~A, proof of Theorem~A.4, +step~(1)]{simonGMT}. +Its hypotheses in the present application have all been established: +on every compact coordinate patch and at every center, each vanishing +scale sequence has a strict tangent subsequence; these tangents are +homogeneous minimizing boundaries; singular sets commute exactly with +translations and dilations; and, on compact windows of every such +convergent sequence, singular sets eventually lie in every open +neighborhood of the limiting singular set. For the last assertion, +a contrary sequence of points has a convergent subsequence, contradicting +regular-point persistence or support convergence. The data on the +rescaled windows have uniform local derivative bounds, with derivatives +of order $j$ acquiring factors $\rho^j$ and pressure derivatives acquiring +$\rho^{j+1}$. Thus the compactness and regularity argument applies to +all these sequences. We use only the cited content-transfer step, and +spell it out next; no identification with the scalar-function family +of Appendix~A is needed. + +Suppose $m\ge8$ and, for some +$s>m-8$, the singular set in an interior window had positive +$\mathcal H^s$ measure. Let $\mathcal H^s_\infty$ denote Hausdorff +content, defined by covering with balls and summing their radii to +the power $s$; normalization constants are immaterial. +The Hausdorff-content density selection in the proof of Theorem~A.4 +provides an interior $x\in\Sigma$, scales $\rho_i\downarrow0$, and +$c>0$ such that +\[ + \rho_i^{-s}\mathcal H^s_\infty + (\Sigma\cap\overline B_{\rho_i}(x))\ge c. +\] +Pass to a strict tangent limit $C$ along these scales. The exact +minimizing-boundary theorem gives +$\mathcal H^s(\Sigma_C)=0$, because its hypersurface dimension is +$d=m-1$ and its bound is $d-7=m-8$. +Cover $\Sigma_C\cap\overline B_{3/2}$ by finitely many open balls +whose sum of $s$th powers of radii is less than $c/2$; the cover can +first be chosen countable with this bound and then reduced to a finite +subcover by compactness. Singular-set persistence puts every +$\rho_i^{-1}(\Sigma-x)\cap\overline B_1$ in that same union for large +$i$: otherwise a convergent sequence in its compact complement would +give a singular limiting point outside the cover. Scaling of +Hausdorff content now contradicts the preceding lower bound. +This proves $\mathcal H^s(\Sigma)=0$ for every $s>m-8$. +It is the density selection and this finite-cover passage, in addition +to tangent information, that transfer the dimension bound. + +If $m\le7$, the exact minimizing-boundary theorem says that every tangent +cone is regular at its vertex and hence is a hyperplane. Flat-tangent +regularity then makes $\Sigma$ empty. Finally a nonflat minimizing +cone is singular at its vertex, and translation invariance carries +that vertex to each point of its spine. The spine is therefore +contained in its singular set, giving the asserted spine bound. +The preceding monotonicity argument, applied in the common embeddings +for convergent data, also gives upper semicontinuity of densities at +moving points by first passing to the limit at a fixed radius and then +letting the radius decrease to zero. The smooth graph's coorientation +comes from the phase, and its first-variation equation is +\eqref{cut:base-mean-curvature}. All constructions are local in the +base and require no ambient orientability or torus-dimensional +regularity theorem. +\end{proof} + + +\begin{corollary}[True cuts converge at packet ends] +\label{cut:end-convergence} +For the common generic shift, let $A_E$ be the true phase on each packet +and let $A_B$ be that on a face. On translated windows escaping in a +product collar over a compact part of $B$, the phases $A_E$ converge to +$A_B\times\R^d$ in $L^1_{\rm loc}$, their perimeter measures and +varifolds converge, and their regular boundaries converge smoothly on +compact sets off the limiting singular set. Their boundary supports +converge locally as well. The same conclusions hold after passage to +any deeper product furnished by \cref{packet:escape}. +\end{corollary} + +\begin{proof} +The metric, weights, and shifted pressure in the translated expanding +windows are exactly the data pulled from $B$. Local compactness +produces a subsequential limiting phase on $B\times\R^d$, and it is +an absolute minimizer by \cref{cut:bv-compactness}. The uniqueness and +factorization of \cref{cut:generic-uniqueness} identify it with +$A_B\times\R^d$. Every subsequence has this same possible limit, which +proves convergence of the full sequence. Apply \cref{cut:gmt} for the +support and smooth assertions. The deeper-face statement follows by +using the commuting product coordinates in \cref{packet:escape}. +\end{proof} + +\subsection{Stability on a wall and the retained-side estimate} + +We now choose the representative of each phase which is open away from +$S$ and agrees near $S_{\rm reg}$ with the corresponding smooth closed +side when a trace is required. Write $S=S_{\rm reg}$ in integrals below. +Every test has compact support away from $\Sigma$; joined-side tests +also avoid the cylinder over $\Sigma$. The old torus variables remain +invariant, and $\dd\mu_S=\omega\,\dd\mathcal H_g^{m-1}$ is the wall +measure. The intrinsic scalar curvature $R_S$ and the second +fundamental form $B$ refer to $S\times\T^q$ with its induced old +stabilized metric. Its torus principal curvature in direction $i$ is +$\partial_\nu\log f_i$, so \eqref{cut:base-mean-curvature} gives +\[ + H=\tr B=H_g+\partial_\nu\log\omega=p +\] +for the outward normal of the plus phase. The invariant wall Laplacian +is $\omega^{-1}\Div_S(\omega\grad_S\,\cdot\,)$, so its integration by +parts uses precisely $\dd\mu_S$. + +\begin{lemma}[The wall stability form]\label{cut:wall-stability} +For every compactly supported smooth invariant test $v$ on $S$, +\begin{equation}\label{cut:wall-form} + \int_S\bigl(2|\grad v|^2+R_Sv^2\bigr)\,\dd\mu_S + \ge\int_S\bigl(R+p^2+|B|^2+2\partial_\nu p\bigr)v^2\,\dd\mu_S. +\end{equation} +In particular its right side is at least +$\int_S(R-1.02\Pi_{j_0})v^2\,\dd\mu_S$. +\end{lemma} + +\begin{proof} +Weighted perimeter is the area of the old torus product for invariant +phases. The invariant normal variation with speed $v$ is compactly +supported in the smooth regular part and comes from an admissible base +variation. Thus its energy has a minimum at parameter zero. +For completeness the variation identities hold in every dimension: +differentiating the induced metric gives $\dot g_{ab}=2vB_{ab}$, +differentiating the unit normal gives $\dot\nu=-\grad_Sv$, and +commuting the two ambient covariant derivatives in the derivative of +$B_{ab}$ gives, after tracing and differentiating the inverse metric, +\[ + \dot H=-\lap_S v-(\Ric_{\widehat g}(\nu,\nu)+|B|^2)v. +\] +The area derivative is $Hv\,\dd\mu_S$, while the pressure-volume +variation is $pv\,\dd\mu_S$. The first energy derivative is therefore +$\int_S(H-p)v\,\dd\mu_S$. At $H=p$ the derivatives of the area +measure and of the chosen speed multiply zero, and +$\dot p=v\partial_\nu p$. Hence the nonnegative second derivative is +\[ + \int_S\left[-v\lap_Sv- + (\Ric_{\widehat g}(\nu,\nu)+|B|^2+\partial_\nu p)v^2\right] + \,\dd\mu_S. +\] +Weighted integration by parts gives +\[ + \int_S|\grad v|^2\,\dd\mu_S + \ge\int_S(\Ric_{\widehat g}(\nu,\nu)+|B|^2+ + \partial_\nu p)v^2\,\dd\mu_S. +\] +Compare the first variation in \citet[Lemma~13]{chodoshLi} and the +second-variation calculation in the proof of its Lemma~14. The displayed +calculation proves the identity in the present stabilized setting; the +stability and Gauss-equation mechanism is that of +\citet[Section~1]{schoenYau1979Structure}. +The derivation here uses neither Chodosh and Li's dimension restriction +for smooth existence nor a regularity assertion for the torus product. +Only the already smooth base patch and its invariant variations enter. +The Gauss identity +\[ + R_S=R-2\Ric_{\widehat g}(\nu,\nu)+H^2-|B|^2 +\] +gives \eqref{cut:wall-form}. By \eqref{cut:pressure-gradient}, +\[ + p^2+2\partial_\nu p + \ge p^2-2|\grad p| + \ge .33p^2-1.02\Pi_{j_0}, +\] +which proves the final statement. +\end{proof} + +For either phase side $\Omega$, put $p_s=p$ on the plus side and +$p_s=-p$ on the minus side. The side is itself an absolute minimizing +phase for the signed pressure $p_s$, and its outward stabilized mean +curvature is $H_{\rm out}=p_s$. + +\begin{lemma}[Deletion and the seam estimate]\label{cut:side-estimate} +Every nonnegative compactly supported smooth function $f$ on the +closed regular side, with support avoiding the singular set, satisfies +\begin{equation}\label{cut:deletion} + \int_S f\,\dd\mu_S + \le\int_\Omega\bigl(|\grad f|+p_sf\bigr)\,\dd\mu. +\end{equation} +For every smooth real test $v$ of the same type, +\begin{equation}\label{cut:side-form} + 3.05\int_\Omega|\grad v|^2\,\dd\mu + +2\int_S H_{\rm out}v^2\,\dd\mu_S + \ge-1.02\int_\Omega\Pi_{j_0}v^2\,\dd\mu. +\end{equation} +Both statements extend to compactly supported Lipschitz tests by +approximation. +\end{lemma} + +\begin{proof} +For almost every $t>0$, delete $\Omega\cap\{f>t\}$ from the minimizing +phase. The removed boundary area is the area on $S\cap\{f>t\}$; +the newly introduced boundary is the level surface of $f$ in the +interior of $\Omega$. The pressure-volume term increases by +$\int_{\Omega\cap\{f>t\}}p_s\,\dd\mu$. Minimality gives the +corresponding perimeter inequality. Integrating in $t$, the layer-cake +formula and weighted coarea give \eqref{cut:deletion}. All comparisons +are compact and occur on regular patches; the perimeter slicing formula +justifies the same argument at exceptional critical levels by discarding +a null set of levels. + +Apply \eqref{cut:deletion} to $f=2(-p_s)_+v^2$, initially by Lipschitz +approximation. Its right side is bounded above by +\[ + \int_{\Omega\cap\{p_s<0\}} + \bigl[(2|\grad p_s|-2p_s^2)v^2 + +4|p_s|\,|v|\,|\grad v|\bigr]\,\dd\mu. +\] +Young's inequality gives +\[ + 4|p_s|\,|v|\,|\grad v| + \le3.05|\grad v|^2+\frac4{3.05}p_s^2v^2. +\] +The pressure estimate holds unchanged for $p_s$, and +\begin{equation}\label{cut:young-margin} + 2-\frac4{3.05}-.67=\frac{113}{6100}>0. +\end{equation} +Discard the resulting nonpositive multiple of $p_s^2v^2$. Thus +\[ + 2\int_S(-p_s)_+v^2\,\dd\mu_S + \le3.05\int_\Omega|\grad v|^2\,\dd\mu + +1.02\int_\Omega\Pi_{j_0}v^2\,\dd\mu. +\] +Since $2p_s\ge-2(-p_s)_+$ on the boundary and +$H_{\rm out}=p_s$, this is \eqref{cut:side-form}. +\end{proof} + +\subsection{Raw joined forms and the numerical reserve} + +Attach $S\times[0,\infty)_t$ to the regular boundary of either side, +freezing all induced metric and old warp data on the cylinder. This +is a raw joined space: it is smooth on its two pieces, and the seam +retains its scalar corner contribution. An admissible test is a pair +$v=(v_b,v_c)$, smooth separately up to regular seam patches, compactly +supported away from singularities and their cylinders, satisfying +$v_b|_S=v_c|_{t=0}$. Compact Lipschitz versions are allowed by local +approximation in regular seam charts. For the wall alone the tests are +compact smooth functions on $S$ away from $\Sigma$. + +Set $C=3.99$. On a wall define +\[ + Q_S(v)=\int_S(C|\grad v|^2+R_Sv^2)\,\dd\mu_S. +\] +On a joined side define +\begin{align}\label{cut:raw-form} + Q_\Omega(v) + ={}&\int_\Omega(C|\grad v_b|^2+Rv_b^2)\,\dd\mu + +2\int_S H_{\rm out}v_b^2\,\dd\mu_S\notag\\ + &+\int_0^\infty\!\int_S + \bigl(C(|\grad_Sv_c|^2+|\partial_tv_c|^2) + +R_Sv_c^2\bigr)\,\dd\mu_S\,\dd t. +\end{align} +The sign in the seam term can also be checked directly: in outward +arclength $s$ the mean curvature changes from $H_{\rm out}$ on the +bulk side to zero on the frozen cylinder, and the scalar expression +contains $-2\partial_sH$. Its distributional contribution is +$+2H_{\rm out}$ at the seam. The smooth realization and its form +convergence will be established in \cref{sec:regularization}. + +\begin{proposition}[Raw scalar domination]\label{cut:raw-domination} +Let $V=R-1.02\Pi_{j_0}$, restricted to the wall and pulled back to the +cylinder when necessary. Then +\begin{align}\label{cut:wall-domination} + Q_S(v)&\ge\int_S Vv^2\,\dd\mu_S + +1.99\int_S|\grad v|^2\,\dd\mu_S,\\ + Q_\Omega(v)&\ge\int_\Omega Vv_b^2\,\dd\mu + +\int_0^\infty\!\int_S Vv_c^2\,\dd\mu_S\,\dd t\notag\\ + &\quad+.94\int_\Omega|\grad v_b|^2\,\dd\mu + +1.99\int_0^\infty\!\int_S|\grad v_c|^2 + \,\dd\mu_S\,\dd t. + \label{cut:side-domination} +\end{align} +Here $|\grad v_c|^2=|\grad_Sv_c|^2+|\partial_tv_c|^2$. +\end{proposition} + +\begin{proof} +The wall assertion follows by adding $(C-2)|\grad v|^2$ to +\cref{cut:wall-stability}. For the bulk use +\cref{cut:side-estimate}; the remaining gradient coefficient is +$C-3.05=.94$. Apply the wall assertion to every cylinder slice and +integrate in $t$. The horizontal coefficient is $C-2=1.99$, while the +vertical coefficient is still $C=3.99$, so lowering the latter to $1.99$ +gives \eqref{cut:side-domination}. The seam trace in the bulk estimate +is precisely the matching trace in \eqref{cut:raw-form}. +\end{proof} + +\begin{definition}[Reserved form and test completion]\label{cut:reserve} +For the wall or a joined side define, respectively, +\begin{align*} + \mathcal N_S(v)&=1.7\int_S|\grad v|^2\,\dd\mu_S,\\ + \mathcal N_\Omega(v) + &=.8\int_\Omega|\grad v_b|^2\,\dd\mu + +1.7\int_0^\infty\!\int_S|\grad v_c|^2 + \,\dd\mu_S\,\dd t. +\end{align*} +We write $\mathcal N$ when the tested space is understood. The squared norm +$\|v\|_2^2$ is the sum of the corresponding piecewise weighted integrals. +With +\[ + e_0=(\lambda_r-\lambda_{r+1})/100>0, + \qquad \mathfrak h(v)=\mathcal N(v)+e_0\|v\|_2^2, +\] +the test completion means completion of the admissible compact tests +just specified in the norm $\mathfrak h^{1/2}$. In particular its seam +condition is matching traces, with no zero trace imposed there. +\end{definition} + +The coefficients have a reserve even after the later shielding charge +$.9\mathfrak h$. Specifically, +\begin{equation}\label{cut:remaining-gradient} + .94-.9(.8)=.22, + \qquad 1.99-.9(1.7)=.46. +\end{equation} +The potential term in \cref{cut:raw-domination}, combined with the +parent inequality, satisfies +\begin{equation}\label{cut:localized-potential} + R-1.02\Pi_{j_0} + \ge\lambda_r+b_r\sum_{j\ne j_0}\Pi_j + +(b_r-1.02)\Pi_{j_0} + \qquad\hbox{on }D_E. +\end{equation} +Here $b_r\ge1.07>1.02$. The raw form inequalities themselves hold for +all admissible compact tests; the replacement of their potential by +the right side of \eqref{cut:localized-potential} is permitted only +where the test is supported over the old required region. +The count inclusions of \cref{packet:counts} supply exactly this +localization for the later required regions. These are the forms and +constants used in shielding and regularization. The minimizing cut, +its pressure, and the old weight remain fixed while the other profiles +and their cores are changed. diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/packets.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/packets.tex new file mode 100644 index 0000000000000000000000000000000000000000..04765f735825efd7cbf3960eeea422ee47484087 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/packets.tex @@ -0,0 +1,598 @@ +\section{Packets, finite horizons, and the induction invariant}\label[appendix]{sec:packets} + +This appendix gives the geometry and bookkeeping for a finite number of +partitions. The minimization, shielding, and regularization arguments +in the following appendices construct the regular cuts to which this +geometry applies. The passage to countably many labels is proved in +\cref{countable:construction}. + +The packet diagrams, finite stage constants, analytic invariant, and +radial initialization are stated in \Cref{sec:finite-cuts}. We prove here +their geometric properties and the compatibility of finite reconstructions. + +\subsection{Product ends and regular splitting} + +\begin{lemma}[Compatible splitting and finite truncations]\label{packet:split} +Suppose a packet diagram after the radial stage has the properties in +\cref{packet:category}. On every packet let $g_E$ be a smooth compatible +function, with $0$ a regular value, including on every face. +Keep $L_E=g_E^{-1}(0)$ as the new hit space. To each closed side +$\{\pm g_E\ge0\}$ attach $L_E\times[0,\infty)$ along its boundary. +Use smooth normal collars across the attachment, chosen as products +in all sufficiently far old tails. Connected components of these +spaces, with the appended statuses, form a new packet diagram. + +It has finite face and coface incidence, properly embedded closed tails, +and compact remainders. Each packet admits a compact truncation which +is a smooth manifold with corners and is a strong deformation retract +of the full packet. Truncations can be chosen simultaneously on every +finite downward diagram, with their faces equal to the corresponding +truncated packets. If thresholds are in the product range of $g_E$, +the old retractions preserve its zero set and both closed sides. +These conclusions also hold at the first radial stage provided that, +before attachment, its wall and closed-side components are compact and +form a locally finite family in $X$. +\end{lemma} + +\begin{proof} +First work in the finite downward diagram of one old packet. Raise +all entry thresholds until every overlap is in a commuting quadrant, +and until all the functions in this diagram are products there. +For an elementary normal coordinate $t_i$ choose a larger horizon $h_i$. +On its tail use +$t_i\mapsto\min(t_i,h_i)$ and the homotopy +$t_i\mapsto(1-s)t_i+s\min(t_i,h_i)$; use the identity elsewhere. +The formulas agree continuously at $t_i=h_i$. In an overlap the +other normal coordinates are unchanged and the formulas commute. +One may therefore perform these clamps in any order. Their joint image +is compact: remove the far elementary tails, and then inductively +truncate each of the finitely many face tails. This gives a finite +union of compact products. Its boundary strata have precisely the +claimed product descriptions. Each clamp fixes the cutting function, +since that function is a face pullback throughout the moving region. + +The regular zero set in this compact manifold with corners is transverse +to every corner stratum. Indeed on a far old face the derivative in +face directions is the derivative of the regular face function. +Consequently the compact zero set and the two compact closed sides are +manifolds with corners. Each has finitely many components: every point +has a connected neighborhood in the relevant manifold with corners, +and compactness supplies a finite such cover. Clamping shows that a +full zero or side component has a nonempty connected intersection with +the truncation. Conversely a connection in the full set clamps to a +connection in the truncation. Thus no additional components or +connections occur only at infinity. + +On an old tail the whole operation is literally the same operation on +its face times the old normal intervals. In particular the cylinder +attached to a face zero set is the same subset in every order of +passing to that face. For the new sign-to-hit relation the tail is +exactly the newly attached cylinder. These observations prove all +product and commutation rules, including the unique occurrence for +a prescribed coarser status: the outgoing product of a connected face +packet is connected and belongs to a unique coarser component. +A new packet has finitely many faces by the compact truncation just +constructed. A face has at most one incident packet for each coarser +status, of which there are finitely many. Applying these assertions +through the finite status depth proves finiteness in both directions. + +For clarity, properness of tails is part of this argument, not an +inference from a metric completeness assertion. Fresh closed cylinder +tails are proper by the attachment topology. The continuous collapse +of a new space to its old parent sends a compact set to a compact set. +On an inherited tail, old properness then bounds its old parameters +in a compact subset of the old tail; the fresh cylinder length is +also bounded on the given compact set. The remaining zero and side +conditions are closed. Thus the inverse image of that compact set +in the inherited closed tail is compact. Equivalently, it is contained +in the union of closed old pieces and bounded fresh cylinders over +compact zero pieces. This also shows that the complement of all new +far tails is compact. Clamp the fresh cylinder coordinate as well +to obtain the asserted new compact truncations and retractions. + +At the radial stage there are initially no old tails. The hypotheses +give compact wall and side pieces, with finite incidence on each +compact piece, and the fresh cylinders start the preceding induction. +All diagrams remain countable, because all manifolds involved are +second countable and the radial family is locally finite. + +For later metric uses, normal collars can be obtained from any +compatible smooth parent metric. In an old product tail, both the +regular defining function and the metric are pulled back from the +face, so the unit normal and its normal geodesics split off the old +normal coordinates. On a compact truncation of the zero set choose +a positive tubular radius by finitely many normal-coordinate charts. +Induction through its finite face diagram extends this radius to a +positive uniform radius on that parent packet: any failure would +be in a product over a face where the radius was already chosen. +Thus these collars are products on all old ends; at the radial step +the same choice is made separately on each incident compact block. +\end{proof} + +\subsection{Completeness, growth, and product limits} + +\begin{lemma}[Metric geometry of a packet]\label{packet:geometry} +Equip a packet diagram after the radial stage with compatible smooth +base metrics and finitely many compatible smooth positive warp +functions. Each connected packet $E$, separately, is complete and +has bounded geometry: positive injectivity radius and bounds on +curvature and every covariant derivative. Every warp and all its +covariant derivatives are bounded on $E$, and every warp has a positive +lower bound. For $\omega_E$ the product of the warps there are constants +$A_E<\infty$ and an integer $d_E\le\dim E$ such that +\begin{equation}\label{packet:growth} + \int_{B_E(x,R)}\omega_E\,\dd\vol_E + \le A_E(1+R)^{d_E} + \qquad(x\in E,\ R\ge0). +\end{equation} +The constant is uniform over centers in this packet. The same assertions +hold on $E\times\R^a$, with the product metric, pulled-back warps, +and growth exponent at most $d_E+a$, for every finite $a$. +\end{lemma} + +\begin{proof} +Induct upward from packets with no faces, which are compact. +A full elementary product tail over a complete face has no finite-distance +exit through its face coordinate. More precisely, a finite-length +curve in that product whose normal coordinate stays above the entry +level has Cauchy face and normal coordinates. Completeness of the face +and properness of the closed tail keep its limit in the product. +It follows that, after fixing a closed-tail entry level $T_i$, the +function +\[ + \rho_i=\max(t_i-T_i,0)\quad\hbox{on that tail},\qquad + \rho_i=0\quad\hbox{outside it} +\] +is continuous and globally $1$-Lipschitz. The only exit is through +$t_i=T_i$, and its differential has norm at most one wherever defined. +A Cauchy sequence has all the finitely many $\rho_i$ bounded. +The compact-truncation assertion of \cref{packet:split} puts such a +sequence in a compact set. Hence the packet is complete. + +Recursively subdividing each coordinate into a bounded interval and +its far range gives a finite cover of $E$ by compact sets and sectors +\begin{equation}\label{packet:sector} + K_B\times[T_1,\infty)\times\cdots\times[T_d,\infty), +\end{equation} +where $K_B$ is a compact subset of a further face and the metric and +warps are products. The recursion terminates because a face decreases +base dimension. On such a sector each displayed normal coordinate is +also the corresponding elementary coordinate on $E$, by commutation. +In an $R$-ball its range therefore has length at most $2R$, by the +Lipschitz functions $\rho_i$ (enlarge by a fixed bounded interval at +an entry if necessary). The sector's weighted volume in that ball +is bounded by its compact base volume times a constant multiple of +$(1+R)^d$. Summing the finitely many bounds proves +\cref{packet:growth}, for all centers, including centers outside the +sector. Notice that $d\le\dim E$. + +Bounds on smooth data follow from the same finite sector decomposition: +on compact pieces they are finite, and on a product they are precisely +face bounds. For bounded geometry one can equivalently use uniform +charts. Any failure of a uniform chart radius or of a fixed-order +chart bound has an escaping sequence; a subsequence lies in a far +product over a compact part of a further face, as detailed in +\cref{packet:escape}. Fixed-radius neighborhoods there are product +neighborhoods with uniform smooth chart bounds and a positive +injectivity radius. Compact remaining regions have the same +properties by a finite chart cover. This proves bounded geometry +and all warp bounds. The product assertions follow either by the +same proof or from +$B_{E\times\R^a}((x,z),R)\subset B_E(x,R)\times B_{\R^a}(z,R)$. +\end{proof} + +\begin{lemma}[Escaping sequences and product windows]\label{packet:escape} +If a sequence escapes all compact sets of a fixed packet $E$, a +subsequence has coordinates $(x_\nu,t_{1,\nu},\ldots,t_{d,\nu})$ +in one further-face product, where $d\ge1$, $x_\nu$ converges in a +compact subset of the face $B$, and every $t_{i,\nu}\to\infty$. +After translating these normal coordinates, the product charts contain +each fixed metric neighborhood of the sequence for all sufficiently +large $\nu$. They exhaust $B\times\R^d$ locally. All compatible +smooth data equal the corresponding product data on these windows. +\end{lemma} + +\begin{proof} +Choose a sector from the finite cover in +\cref{packet:sector} containing a subsequence. Pass to a subsequence +so each of its normal coordinates either converges finitely or tends +to infinity. The compact base coordinate also converges. Absorb the +bounded normal coordinates into the base of the intermediate face +specified by the remaining, unbounded coordinates. Commutation +identifies this with the corresponding full product occurrence. +At least one unbounded coordinate remains, since otherwise the +sequence lies in a compact truncation. A path of length at most $R$ +changes each normal coordinate by at most $R$. It cannot escape +through a face coordinate, by completeness of the face as in +\cref{packet:geometry}. Thus a fixed-radius neighborhood lies in the +product once all its normal coordinates exceed the entry levels by +more than $R$. Translating yields the stated exhaustion. This +argument uses only the completeness part of \cref{packet:geometry}, +which was established before its bounded-geometry conclusion. +\end{proof} + +The initial metric and warps need not have bounded geometry on $X$. +At the radial split each bulk piece and each wall piece is compact +before cylinder attachment. The smooth data have finite derivative +bounds there, the positive warps have positive minima, and the +cylinders freeze those data. This initializes \cref{packet:geometry}. +For a locally finite initial warp list, a compact radial block and its +compact buffer meet only finitely many nonunit factors; the same +argument applies after omitting the unit factors. Polynomial growth +is a conclusion for a single post-radial packet, never an assumption +on the original complete manifold. + +\subsection{Compatible choices} + +\begin{lemma}[Compatible choices and finite tolerance allocation]\label{packet:approximation} +On a packet diagram after the radial stage, choices of smooth cutoffs and approximations can +be made from deeper faces to cofaces, retaining exact products outside +finite thresholds. More precisely, fix a finite derivative order and, +on each packet, positive error tolerances for any finite collection of +smooth data or approximations. A datum on a face may additionally be +required to meet the tolerances of all its incident cofaces, including +through chains. These requirements can be replaced in advance by +strictly positive face tolerances, and then met by compatible blending, +provided the individual local approximations and their required +convergence to the face data are available. +In particular there exist compatible smooth positive functions +arbitrarily small in any prescribed finite derivative norm on each +packet. Also, if compatible closed sets $F_E$ and compatible open sets +$O_E$ satisfy $F_E\subset O_E$ on every packet, then +$\dist_E(F_E,E\setminus O_E)>0$ on each fixed packet whenever both +sets are nonempty. +\end{lemma} + +\begin{proof} +There are only finitely many requests on a particular face, because +there are only finitely many coface chains. Starting with the given +requests, propagate stricter tolerances down these chains, allowing +for the finitely many product, derivative, and cutoff constants. +Take finite minima; no minimum over an infinite adjacency component +is taken. This finishes all tolerance choices before construction +begins. A countable collection of packets causes no problem, since +each one has finitely many faces and cofaces and the depth is finite. + +For the construction on one packet, enumerate its elementary tails. +Choose smooth cutoffs $f_i(t_i)$ rising from zero to one sufficiently +far out, with bounded derivatives through the chosen order, and +extend them by zero off their tails. Write $u_0$ for an individual +approximation and $u_i$ for the already chosen face copy. The weights +\begin{equation}\label{packet:blend} + w_0=\prod_i(1-f_i),\qquad + w_i=f_i\prod_{a0$. Each compact wall component +has its strip chosen separately, with the same error bound +$\varepsilon_r$ on every component. Disjoint strips give a supremum +bound, rather than a sum over components. The resulting error after +finitely many stages is at most $\sum_{jT_0+3\}. +\] +Their closures lie strictly inside the respective pole interiors; +they contain $A_i$ and $B_i$. The requested finite separations are +available from \cref{rounds:wide} after $N_*$, and hence $T_0$, are +fixed. Their size does not change the original radius $D$. + +We give the radial construction without a bounded-geometry +hypothesis. Assume first that $X$ is noncompact, fix $o\in X$, and +put $d(x)=d_h(o,x)$. Choose a countable locally finite cover by +relatively compact coordinate neighborhoods on which the metric +oscillation is small, and a subordinate smooth partition of unity +$\{\chi_j\}_{j\ge1}$ whose supports are compact in those neighborhoods. +Convolution in a slightly larger coordinate neighborhood gives +smooth $d_j$ with +\[ + |d_j-d|<\epsilon_j,\qquad |\dd d_j|_h\le\tfrac32 + \quad\hbox{near }\supp\chi_j. +\] +The derivative estimate follows from the local Lipschitz bound for +$d$ and the small metric oscillation; the convolution radius can +then be reduced to make its value error arbitrarily small. +Let $C_j=\sup|\dd\chi_j|_h<\infty$, and choose +$0<\epsilon_j<\min\{1/2,2^{-j-2}/(1+C_j)\}$. +The locally finite sum $\rho=2+\sum_j\chi_jd_j$ is smooth and +$|\rho-(2+d)|<1/2$. Moreover, since $\sum_j\dd\chi_j=0$, +\[ + \dd\rho=\sum_j\chi_j\dd d_j + +\sum_j(d_j-d)\dd\chi_j, + \qquad |\dd\rho|_h\le\tfrac32+\sum_j2^{-j-2}<2. +\] +Completeness makes $d$ proper, so $\rho$ is proper and bounded below. +Thus a smooth proper radius with bounded gradient has been constructed +using only local smoothness and completeness. + +Choose successive plateau intervals of the values of $\rho$, with +alternating signs and positive-width cores strictly inside each +plateau. Separate successive plateaus by transition intervals of +width greater than $2T_0$. On such an interval, a copy of the crossing +$y$, parametrized by $(\rho-a)/L$ with $L>2$ and with sign reversed +on alternate intervals, obeys the same profile inequality because +$|\dd\rho|_h/L\le1$. Include an initial plateau extending below the +minimum of $\rho$. Core closures have positive value margins to +both ends of their plateaus, except at the harmless lower end of the +initial plateau. Sard's theorem supplies regular values in those +margins. The sublevel domain at the initial plateau, and the domains +between two such regular levels at later plateaus, are compact smooth +barriers enclosing the corresponding core closures and lying in the +same pole interiors. These barriers and the transition blocks form +locally finite families, since their value intervals tend to infinity. +Connectedness and properness imply that the image of $\rho$ contains +all sufficiently large values, so the separating subbands are nonempty. + +A zero wall respecting the signs misses every separating core +subband. On any connected wall component, the continuous function +$\rho$ therefore takes values in one bounded intervening block. +The component is closed in $X$, and properness of $\rho$ makes it +compact. For a fixed sign, successive core subbands of the opposite +sign bound the $\rho$-values of every connected component of the +closed side; the central component is bounded by the first opposite +separator. These closed-side components are compact as well. +In a compact block, regularity gives a finite cover by charts in +which the wall is a single sheet and each closed side is a half-space +or the full chart. Only finitely many components can meet that block. +The blocks are locally finite, which proves the required local +finiteness of wall and side components. + +If $X$ is compact, set the radial profile identically $+\infty$, with +$U_{\mathrm{rad}}^+=X$ and $U_{\mathrm{rad}}^-=\varnothing$. +The constant positive defining function realizes its empty wall; +the single nonempty closed side is $X$, which is compact. All radial +profile and confinement assertions then hold directly. + +Finally, the initial scalar requirement holds everywhere, since +\[ + \lambda_0+b_0\sum_{j\in J_0}\Pi_j + =\frac14+1.12N_*\frac{1}{10N_*} + =.362<.4\le R_X. +\] +The profile, core, and forced-preimage conditions have already been +verified. This proves the invariant at stage zero. +\end{proof} + +The geometric lemmas require compatible regular defining functions. +The minimizing boundaries used to obtain them may initially be +singular. The remaining appendices first construct these minimizing +partitions, enlarge the remaining cores until the required guards +avoid the singular sets, and then produce the regular compatible functions to which +\cref{packet:split} applies. The curvature restriction to the required +sets is respected precisely by \cref{packet:counts}. diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/regularization.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/regularization.tex new file mode 100644 index 0000000000000000000000000000000000000000..bb4cb3df4c3fe5b01c4a0c0eff7ed9b463aed7a8 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/regularization.tex @@ -0,0 +1,975 @@ +\section{Regularization and restoration of the curvature invariant}\label[appendix]{sec:regularization} + +The minimizing partition is smooth away from a singular set, whereas the +next cutting step needs smooth complete packets. Shielding has made the +closed guard disjoint from the singular set; the first lemma below obtains +the regular tubes needed for comparison. We construct globally smooth +partitions that accurately approximate the true partitions near these +guards. We then transfer the scalar form to the new walls and joined +sides, smooth the cylinder seams, and convert the form bound into a +pointwise bound by adding circle factors. Throughout this section a stage +$r$ and its current label $j_0$ are fixed. The profiles and +cores of the remaining labels have already been updated by +\Cref{shield:global}. Write +\[ +\begin{gathered} + \widetilde\Pi_j=\Pi_j+D_j,\qquad + G'_j=E\setminus(U_j^{+,\mathrm{updated}}\cup U_j^{-,\mathrm{updated}}),\\ + H_E^*=\left\{x\in E: + \#\{j\in J_r\setminus\{j_0\}:x\in G'_j\}\ge m-2\right\}. +\end{gathered} +\] +Here $m=\dim E$ is the parent base dimension. The set $H_E^*$ is closed +and misses the singular set of the true boundary. A nonpositive counting +threshold means the whole packet. Scalar curvatures, mean curvatures, +Laplacians, and measures include the old torus factors; functions and tests +are invariant in those factors. Constants in this section may depend on a +packet and its finite incidence diagram. At the first, radial step the +corresponding constants are assigned on enlarged compact radial blocks. +In particular no uniform geometry bound for the original manifold is used. + +Use adjacent parameters $0\le t_r0\}$, the function $g+c\chi$ has regular zeros +there. Where $\chi=0$, its differential is that of $g$, since +$d\chi=0$ there. Bounded derivatives of $\chi$ and small $c$ preserve +all strict error and sign requirements. On the initial space use a +positive smooth $\chi$ with locally prescribed small derivatives and +the same regular-value argument; the radial separators are preserved. + +If the perturbed wall is $t=\alpha(x)$ in a true normal tube, the map +\[ + (x,t)\longmapsto (x,t-\chi_1(x,t)\alpha(x)) +\] +with $\chi_1=1$ on the needed graph region and supported inside a larger +regular buffer is a diffeomorphism for sufficiently small graph norm. +It carries the graph to $t=0$, preserves its two local sides, and tends +to the identity in $C^{J-1}$. Choose the buffer cutoff on the regular +wall slightly beyond the projection of the smaller guard neighborhood +and extend by the identity. Uniform regular charts give bounded +cutoff derivatives. The same formula restricts to the wall used as a +cylinder base. No product property of this auxiliary comparison is +asserted or needed. +\end{proof} + +Perform the split of \Cref{packet:split} using these defining functions. +For the moment use the raw metric: the old metric on the bulk and the +frozen induced metric on each fresh infinite cylinder. Let $p_A$ denote +restriction/inclusion on a wall or bulk and cylinder projection on a +fresh cylinder. Pull back the remaining profiles, cores, and +$\widetilde\Pi_j$ by $p_A$. These data are continuous at a seam and +compatible in old ends. Define the new required set $D_A$ from the +remaining cores and the new base dimension. +The raw forms of \Cref{cut:raw-domination} were defined on the true +minimizing wall and its joined sides. We first estimate those forms, +then use the comparison map to transfer the estimate to the raw metrics +of the smooth split just constructed. + +\begin{lemma}[Counts and transfer of the raw form]\label{regular:transfer} +The defining-function accuracies can be chosen so that on a uniform +neighborhood of $D_A$ in each new packet the raw scalar form controls +\begin{equation}\label{regular:raw-target} + \lambda_{r+1}+(1+3\eta_r)b_{r+1}\sum_{j\ne j_0}\widetilde\Pi_j, +\end{equation} +with a strictly positive mass and gradient reserve. The form has +gradient coefficient $C=3.99$ and includes the seam term +$2\int H_{\rm out}u(0)^2$ on a joined side. Tests are compactly +supported, with matching seam traces. +\end{lemma} + +\begin{proof} +For a side packet $\dim A=m$, a point in $D_A$ projects to at least +$m-1$ remaining undecided labels, hence into $H_E^*$ and the parent's +old required set. For a hit packet $\dim A=m-1$, it projects to at +least $m-2$ remaining undecided labels. The wall lies in $G_{j_0}$, +which supplies the additional old undecided label. The projection is +again in both required sets. These inclusions are unchanged on the +fresh cylinder, where all relevant data are pulled back from its base. +All these maps decrease lengths in the raw metrics; therefore a small +neighborhood of $D_A$ projects into small neighborhoods of the two +parent sets. The closed-set count, rather than a continuity claim for +the counting function, is what is used here. + +On compared true pieces in a sufficiently small neighborhood of the +old required set, continuity and the bounded derivatives of the fixed +parent data give +\[ + R\ge\lambda_r+b_r\sum_j\Pi_j-e_0, + \qquad e_0=\frac{\lambda_r-\lambda_{r+1}}{100}. +\] +By \Cref{cut:raw-domination,cut:reserve}, the raw form on these pieces +dominates $R-1.02\Pi_{j_0}$, with spare gradient coefficients $.94$ +on bulk and $1.99$ on a wall or cylinder. The bound in +\Cref{shield:global} pays for the additional potential +$1.16\sum_{j\ne j_0}D_j$ from these spare gradient terms and a mass +loss of at most $.9e_0$. Since +$b_r>1.02$, dropping the remaining nonnegative current-label term +leaves +\begin{align} + Q_{\rm true}(u)&\ge + \int\left(\lambda_r-2e_0+ + b_r\sum_{j\ne j_0}\Pi_j+ + 1.16\sum_{j\ne j_0}D_j\right)u^2 + +.2\int|\nabla u|^2.\label{regular:true-budget} +\end{align} +Indeed the unrounded gradient remainders are +$.94-.9\cdot.8=.22$ and $1.99-.9\cdot1.7=.46$, and the mass loss is +at most $1.9e_0$. The exact numerical margins are +\begin{equation}\label{regular:eta} +\begin{aligned} + a_{r+1}-(1+\eta_r)a_r + &=\frac{3\Delta_r(180-t_r)}{20000} + \ge .02685\Delta_r>0,\\ + b_r-(1+3\eta_r)b_{r+1} + &=\frac{\Delta_r[664+15(t_r+\Delta_r)]}{20000} + \ge .0332\Delta_r>0,\\ + 1.16-(1+3\eta_r)b_{r+1} + &=\frac{800+1000t_r+664\Delta_r+ + 15\Delta_r(t_r+\Delta_r)}{20000}>0. +\end{aligned} +\end{equation} +These identities use $0\le t_r0$ the discrepancy of the compared forms +is bounded by +\[ + \delta\left[\int(u^2+|\nabla u|^2)+\int_{\rm seam}u(0)^2\right]. +\] +The last term is controlled using the first unit interval of the raw +product cylinder: +\begin{equation}\label{regular:trace} + \int_{\rm seam}u(0)^2 + \le 2\int_{S\times[0,1]}(u^2+|\partial_tu|^2). +\end{equation} +This follows by writing $u(0)=u(t)-\int_0^t\partial_tu$ and integrating +in $t$. It needs no zero boundary condition. Choosing all approximation +errors sufficiently small leaves, for example, a reserve of $90e_0$ +in mass and $.15$ in gradient. These choices can be prescribed on +parent buffers before the individual new components are known: the +projection inclusions place every relevant point there. At the radial +step the enlarged compact blocks supply the bounds; preserved +separators trap each new component in one such block before cylinder +attachment. Finally impose all coface requests through the finite +tolerance allocation already used in \Cref{regular:partition}. +\end{proof} + +\subsection{Seam smoothing and restriction of profiles} + +The proof below uses scalar-curvature concentration at a mean-curvature +jump, as in the corner smoothing of +\citet[Sections~2--3]{miao2002Corners}. Here the smoothing width varies +tangentially, and the form estimate must preserve compatible product ends. + +\begin{lemma}[Smoothing a seam with variable width]\label{regular:seam} +The raw joined metrics admit smooth compatible metrics with arbitrarily +small $C^0$ error, and their scalar forms converge to the raw forms in +the following quantitative sense. On each joined packet and for every +$\delta>0$, the smoothing can be chosen so that the absolute form +error is at most +\[ + \delta\int(u^2+|\nabla u|^2) +\] +for every compact matching test, using the raw metric to measure the +right side. The old warped-product structure and every sufficiently +far old product end are preserved. The choices on a wall can satisfy +the requests of both incident sides at once. +\end{lemma} + +\begin{proof} +In a normal collar of a smooth wall write the old stabilized metric +as $ds^2+\beta_s$ for $s\le0$, where $s$ increases outward. Here +$\beta_s$ includes the old torus factors. The raw joined metric is +$ds^2+\beta_{\min(s,0)}$. Choose a smooth concave function $\Phi$ +equal to $t$ for $t\le-1$ and to $0$ for $t\ge1$, with +$0\le\Phi'\le1$. Such a function is obtained by integrating a smooth +decreasing step function symmetric about $0$. It satisfies +$\Phi\le0$ and $\int\Phi''=-1= -\int|\Phi''|$. + +For a smooth positive function $\epsilon(x)$ on the wall put +\[ + \phi(x,s)=\epsilon(x)\Phi(s/\epsilon(x)),\qquad + G_\epsilon=ds^2+\beta_{\phi(x,s)}. +\] +It equals the raw metric outside $|s|<\epsilon(x)$ and differs from it +in $C^0$ by $O(\sup\epsilon)$. In a fixed tangential chart the extra +derivatives of $\phi$ satisfy +\begin{align*} + |\partial_x\phi|&\le c|d\epsilon|,\\ + |\partial_x^2\phi|&\le + c\bigl(|d^2\epsilon|+|d\epsilon|^2/\epsilon\bigr). +\end{align*} +Choose $\epsilon=\zeta^2$ with $\zeta>0$ as small in $C^2$ as +prescribed. Then $|d\epsilon|^2/\epsilon=4|d\zeta|^2$ and +$d^2\epsilon=2\zeta d^2\zeta+2d\zeta\otimes d\zeta$. +Thus these tangential errors tend to zero. On a packet, positive +compatible choices have a positive infimum and bounded derivatives by +the compact-part/product-end description. Their size, but not the +resulting higher metric bounds, can be made arbitrarily small. Normal +collar widths are also positive packetwise by the same description. + +Let $B^{\rm old}_\tau,H^{\rm old}_\tau$ be the slice second +fundamental form and mean curvature of $ds^2+\beta_s$. Exactly, +\[ + B_s=\phi_sB^{\rm old}_{\phi},\qquad + H_s=\phi_sH^{\rm old}_{\phi}. +\] +For a metric $ds^2+\beta(x,s)$ the Gauss and normal variation +identities give +\[ + R(G_\epsilon)=R(\beta(x,s))-|B_s|^2-H_s^2-2\partial_sH_s. +\] +The tangential derivative bounds just proved show that all terms on +the strip are uniformly bounded except +\begin{equation}\label{regular:spike} + -2\phi_{ss}H^{\rm old}_{\phi}. +\end{equation} +In particular the formula differentiates $H_s$ only in $s$; +potentially large mixed $xs$ derivatives do not give another scalar +spike. The remaining contribution from this derivative is +$-2\phi_s^2\partial_\tau H^{\rm old}_\phi$, which is bounded. + +Let $J_\epsilon(x,s)$ be the density of $\beta_\phi$ relative to +the zero slice. Uniformly on the strip, +$J_\epsilon=1+O(\epsilon)$ and +$H^{\rm old}_\phi=H^{\rm old}_0+O(\epsilon)$. +For each fixed $x$, +\[ + \int-\phi_{ss}\,ds=1,\qquad + \int|\phi_{ss}|\,ds=1. +\] +Writing $E_x=\int_{-\epsilon(x)}^{\epsilon(x)}|\partial_su|^2ds$, +the fundamental theorem of calculus gives +\[ + |u(s)-u(0)|^2\le\epsilon(x)E_x, + \quad + |u(s)^2-u(0)^2| + \le2\sqrt{\epsilon(x)}|u(0)|E_x^{1/2}+\epsilon(x)E_x. +\] +After integrating against $|\phi_{ss}|ds$, the error in replacing +$u(s)^2$ by $u(0)^2$ is at most +$c\sqrt{\sup\epsilon}\,(u(0)^2+E_x)$, for $\sup\epsilon\le1$. +The density and mean-curvature errors cost at most +$c\sup\epsilon\,(u(0)^2+E_x)$. Thus the form contribution of +\Cref{regular:spike} converges to +$2\int H^{\rm old}_0u(0)^2$, with the outward sign required in +\Cref{cut:side-estimate}. + +The bounded strip terms have vanishing form cost because +\[ + \int_{-\epsilon(x)}^{\epsilon(x)}u(s)^2ds + \le c\epsilon(x)u(0)^2+c\epsilon(x)^2E_x. +\] +The metric and density errors in the gradient and fixed bounded +potential terms tend to zero in $H^1$ form norm. Integrate these +estimates in $x$ and use \Cref{regular:trace}; the stated error bound +follows. One first proves it for smooth matching tests and then for +the matching $H^1$ completion by density. + +Choose $\zeta$ on new hit packets, from deeper faces upward, after +propagating the two incident-side requests through their finite +diagrams. The function is an exact pullback in old product ends, so +the displayed smoothing is also an exact product there. It is +unchanged far out in the fresh cylinder coordinate. Since the normal +coordinate and old metric already split off every old normal factor, +the operation commutes with all the inherited collar identifications. +\end{proof} + + +\begin{lemma}[Approximation that reproduces constants]\label{regular:constant-preservation} +A continuous scalar function on a packet can be approximated by smooth +functions, with any prescribed positive value error, by operators that +reproduce constants. For compatible input on a packet diagram, the +approximations can be chosen compatible in all sufficiently far product +ends of that diagram. If the input and all face copies +used at a packet equal the same constant $c$, the result equals $c$ +there exactly. The same assertion holds componentwise for a finite list +of functions and for a list having only finitely many nonconstant entries +on each prescribed finite directed face/coface diagram. +\end{lemma} + +\begin{proof} +Choose a locally finite coordinate cover, relatively compact smaller +charts, and a smooth partition of unity $\{\psi_\alpha\}$ subordinate +to the smaller charts. Extend each chart slightly and average the +function against a smooth nonnegative kernel of integral one, at a +sufficiently small chart-dependent scale. Call the resulting local +function $T_\alpha f$. Uniform continuity on the chart closure makes +$|T_\alpha f-f|$ smaller than the prescribed error on +$\supp\psi_\alpha$. Thus +\[ + Tf=\sum_\alpha\psi_\alpha T_\alpha f +\] +is a smooth approximation with that error. Every local average sends +$c$ to $c$, and $\sum\psi_\alpha=1$, so $Tc=c$ exactly. Locally +prescribed derivative accuracy is obtained in the same way wherever +$f$ is already smooth, using a slightly larger regular buffer. + +Now use the finite tolerance allocation of +\Cref{packet:approximation}, and blend from deeper faces upward with +the weights $w_i$ in the proof of \Cref{regular:partition}. Their sum +is one. Consequently blending preserves a common constant as well as +exact face identities. The finite exceptional union over the prescribed +directed dependencies is chosen before applying this construction. +For every other entry the entire local input and all participating face +copies coincide with the same constant, so all its averaging and +blending errors are zero. No minimum of tolerances over these constant +entries is needed. On the initial radial space the construction is +local on buffered compact blocks and uses a locally finite chart cover. +It has the same constant-reproduction property there. +\end{proof} + +On a new packet $A$, write $R_{\rm old}$ and $\lap_{\rm old}$ for the +scalar curvature and invariant Laplacian of the smoothed split metric +with its retained torus factors, before any new circle factors are added. + +\begin{lemma}[Restricted profiles and smooth potentials]\label{regular:profiles} +After the seam smoothing the pulled-back profiles and cores satisfy +the stage $r+1$ profile conditions with smooth compatible potentials +$\Pi_j^{\rm new}$ chosen so that +\[ + (1+\eta_r)\widetilde\Pi_j + \le\Pi_j^{\rm new}\le(1+2\eta_r)\widetilde\Pi_j. +\] +If +\begin{equation}\label{regular:Vstar} + V_A^*=\lambda_{r+1}+b_{r+1}\sum_{j\ne j_0}\Pi_j^{\rm new},\qquad + L_A^0=-C\lap_{\rm old}+R_{\rm old}-V_A^*, +\end{equation} +then on some positive uniform neighborhood of $D_A$ its form is bounded +below by a positive multiple of $\|u\|_2^2$. +\end{lemma} + +\begin{proof} +We justify restriction to a hypersurface, which does not follow merely +by restricting an ambient almost-everywhere inequality. If $f$ is +locally Lipschitz on an open Riemannian set and $|df|\le F$ almost +everywhere for continuous $F$, then at every point its local Lipschitz +slope is at most $F$. In a small coordinate ball, convolve $f$ on a +slightly larger ball. The derivative bound of the convolutions is at +most the supremum of $F$ there times a metric distortion factor tending +to $1$ as the ball shrinks. Integrating this bound along straight +segments, passing to the uniform limit, and shrinking the ball proves +the assertion. It follows also that every rectifiable curve satisfies +the corresponding integrated bound, by subdividing into such balls. + +Apply this on each open finite band of a profile, with +$F=(a_r\mu_j^2+\widetilde\Pi_j)/2$. Restriction to the wall, +inclusion of bulk, and projection on a cylinder therefore preserve the +bound everywhere in local slope, and hence almost everywhere for the +restricted locally Lipschitz function. Finite truncations remain +locally Lipschitz, including where a pole is present. Across a seam +the same conclusion follows by splitting a path into its bulk and +cylinder portions; traces agree. A sufficiently small $C^0$ metric +change increases all these slopes by at most $1+\eta_r$. + +For each label $j$, the raw potential has its own positive lower +bound $p_j^*>0$. Apply \Cref{regular:constant-preservation} to +$(1+3\eta_r/2)\widetilde\Pi_j$, with strict value error smaller than +$\eta_r p_j^*/2$. This proves the displayed relative bounds. Different +labels may require different accuracies; a common positive lower bound +for an infinite label list is unnecessary. When the raw potential is +a spatial constant $c_j$ on the whole relevant incident diagram, choose +its new value to be exactly $(1+3\eta_r/2)c_j$. By +\Cref{regular:eta}, +\[ + 2|\nabla\mu_j|_{\rm new} + \le (1+\eta_r)(a_r\mu_j^2+\widetilde\Pi_j) + \le a_{r+1}\mu_j^2+\Pi_j^{\rm new}. +\] +Continuous projections preserve the inclusion of every core closure +inside its pole interior and preserve all prescribed $q$ constraints. + +Choose the seam form error small relative to the reserves of +\Cref{regular:transfer}. The potential in \Cref{regular:Vstar} is at +most that in \Cref{regular:raw-target}; the extra $\eta_r$ supplies +additional margin for the potential smoothing. Metric equivalence +allows the required-set neighborhood to be slightly reduced after +smoothing. Thus one may retain, for example, an $e_0\|u\|_2^2$ +lower bound for the form of $L_A^0$ on tests in that neighborhood. +No compatibility of the neighborhoods themselves is asserted. +\end{proof} + +We have obtained a smooth packet metric and the next profile +inequalities. The scalar lower bound is still only a quadratic-form +bound near the new required set. We next extend its positive gap to +the whole packet by a correction that vanishes near that set, solve a +bounded elliptic equation, and approximate its solutions compatibly. +The final group of $399$ circle factors converts this form information +into the pointwise scalar inequality. + +\subsection{Compatible corrections and positive supersolutions} + +At this point the smooth split packets have compact truncations and +exact product ends by \Cref{packet:split}. At the radial step the +preserved separating bands give the compact initial pieces. Hence +\Cref{packet:geometry} already applies to these metrics and retained +warps, before the following correction is constructed. In particular +all bounds used below hold separately on each whole packet; they are +not assumptions on the original manifold. + +\begin{lemma}[A compatible correction with a global gap]\label{regular:correction} +There are smooth nonnegative bounded compatible potentials $K_A$, +with all derivatives bounded packetwise, such that $K_A=0$ in a +positive uniform neighborhood of $D_A$ and +\begin{equation}\label{regular:global-gap} + Q_A(u):=\int\bigl(C|\nabla u|^2+ + (R_{\rm old}-V_A^*+K_A)u^2\bigr) + \ge g_A\|u\|_2^2\qquad(g_A>0) +\end{equation} +for every compact smooth test. +Let $\omega_A^{\rm old}$ be the product of the retained circle length +functions for the smoothed metric on $A$, before the new circle +factors at this split are appended. With $K_A$ acting by +multiplication, define +\[ + L_A=L_A^0+K_A + =-C\lap_{\rm old}+R_{\rm old}-V_A^*+K_A. +\] +Its quadratic form in the measure +$\omega_A^{\rm old}\,\dd\vol_A$ is $Q_A$. +In an exact end over $B$, the base metric is a product and every +retained warp pulls back from $B$. Consequently +\[ + \lap_{{\rm old},A}=\lap_{{\rm old},B}+\lap_{\rm normal}, + \qquad L_A=L_B-C\lap_{\rm normal}. +\] +\end{lemma} + +\begin{proof} +First disregard compatibility on a fixed packet. Put +$W_A=R_{\rm old}-V_A^*$. Choose a smooth angle $\theta$ equal to +$0$ near $D_A$ and $\pi/2$ outside the neighborhood of local positivity, +with bounded derivatives and with the support of $\cos\theta$ inside +that neighborhood. Uniform charts and the positive neighborhood +width give such a function by smoothing a distance cutoff with +buffers. Set +\[ + K_{\rm ind}=J\sin^2\theta+C|\nabla\theta|^2, + \qquad J>\max\{0,-\inf W_A\}. +\] +The identity for the squared partition $\cos\theta,\sin\theta$ +shows that $Q_A^0+K_{\rm ind}$ has gap +$\gamma_A=\min\{e_0,J+\inf W_A\}>0$. Indeed the gradient +partition error is exactly $C|\nabla\theta|^2u^2$; local positivity +controls the cosine part and $W_A+J>0$ controls the sine part. +The correction is zero on a smaller uniform neighborhood of $D_A$. + +Proceed from deeper faces upward. In sufficiently far elementary +collars the already constructed $K_B$ copies agree on intersections: +on a common quadrant each is the pullback of the same further-face +potential. They consequently define a \emph{single} nonnegative +function $K_{\rm end}$ on the collar union. Copies are not added +together on an intersection. The metric, measure, and $V_A^*$ are +likewise exact products there. In particular $V^*$ has the same +formula on a face: $C$, $\lambda_{r+1}$, $b_{r+1}$, the label set, +and the old torus list do not depend on base dimension, after padding +unused flat factors on each finite compared diagram. + +For the finitely many elementary collars choose angles $\theta_i$ +which rise from $0$ to $\pi/2$ over a normal interval of length $L$, +inside the valid common product ranges. Use the squared partition +\[ + \eta_0=\prod_i\cos\theta_i,\qquad + \eta_i=\sin\theta_i\prod_{a0$, the finite IMS identity gives +\[ + Q_A(u)\ge(g_*-Cc_A L^{-2})\|u\|_2^2. +\] +Choose $L$ so that the error is at most $g_*/2$. All quantities used +for this choice are fixed before solving any elliptic equation. +Boundedness and the derivative assertions follow from compact parts +and finitely many product copies. Empty required sets or absent faces +cause no change: one may take $\theta=\pi/2$ or omit the corresponding +cutoff. This completes the compatible induction and proves the +operator identity. +\end{proof} + +\begin{lemma}[The bounded gapped resolvent]\label{regular:resolvent} +Let $A$ be one of the resulting complete packets, with its weighted +base measure. Suppose $W$ is smooth and bounded with bounded +derivatives, and $L=-C\lap_\omega+W$ has form gap $g>0$. Then +there is a unique bounded smooth solution $s$ of $Ls=1$. +It satisfies $0<\inf s\le\sup s<\infty$ and has bounded derivatives +of every fixed order. Bounded smooth supersolutions of $Lw\ge0$ are +nonnegative. The same conclusions hold on Euclidean products of $A$. +\end{lemma} + +\begin{proof} +Use the bounded geometry, positive bounded weight, and center-uniform +polynomial volume estimate of \Cref{packet:geometry}. The form gap +and boundedness of $W$ make the closed form coercive on $H^1$: +combine $Q\ge g\|u\|_2^2$ with +$Q\ge C\|\nabla u\|_2^2-\|W_-\|_\infty\|u\|_2^2$. +It therefore has a self-adjoint inverse on $L^2$, of norm at most +$g^{-1}$. If $f\ge0$, testing $Lu=f$ with the negative part gives +$Q(u_-)\le0$, so the inverse is positive. + +Here is the decay estimate needed for bounded sources. We use an +Agmon-type exponential-weight argument \citep{agmon1982}, proved here +for the weighted packet operator. If $f$ is +supported in a metric unit ball $I$ and $u=L^{-1}f$, put +$\chi=\exp(\delta\min\{d(\cdot,I),D\})$, where $C\delta^20$ and choose $a>0$ with +$Ca^2\|s_B\|_\infty\le1/2$. The function +\[ + b(x,t)=e^{-a(t-T)}s_B(x) +\] +satisfies $L_Ab=e^{-a(t-T)}(1-Ca^2s_B)\ge\tfrac12e^{-a(t-T)}$. +Comparison on the half collar, using its sliced Dirichlet gap and +the Dirichlet comparison argument in the proof of +\Cref{regular:resolvent}, yields +\begin{equation}\label{regular:end-decay} + |s_A-s_B|(x,t)\le + \frac{\|s_A-s_B\|_\infty}{m_B}\, + e^{-a(t-T)}s_B(x). +\end{equation} +The boundary inequalities hold at $t=T$ for both signs. Uniform +interior elliptic estimates for the homogeneous difference give the +same exponential decay in $C^2$ on $t\ge T+1$. All constants are +uniform in $x\in B$, including its deeper ends. Thus the estimate holds +on each elementary collar, also where collars overlap. Exact agreement +of the approximations on those overlaps will come from the compatible +face copies used in the following construction. + +All operator coefficients and every exact $s_A$ are now fixed. Let +$M_A$ be a finite bound for $L_A:C^2\to C^0$ in uniform charts, and +choose a target +\[ + \tau_A<\min\{\tfrac14\inf s_A,\ (4M_A)^{-1}\}. +\] +The finite family of collar blending cutoffs has a $C^2$ operator +bound $H_A\ge1$, independent of translation of their starts. Include in $H_A$ the fixed finite factor for the number of copies. +For each packet $B$, let $\varepsilon_B$ be the minimum of +\[ + \frac{\tau_A}{\prod_{i=0}^{\ell-1}(4H_{A_i})} + \qquad + (A=A_0>A_1>\cdots>A_\ell=B), +\] +over all directed coface-to-face chains ending at $B$, including the +length-zero chain. There are finitely many such chains, so this +minimum is positive. Appending the edge $A>B$ to a minimizing chain +for $A$ gives $\varepsilon_B\le\varepsilon_A/(4H_A)$. +All these numbers are fixed after the corrections and operator norms, +and before constructing any $v_A$. + +Construct $v_A$ from deeper faces upward. Keep $s_A$ in the central +part and use the ordinary weights in the proof of +\Cref{regular:partition} to blend to the already constructed copies +of $v_B$. Move the starts until +$\|s_A-s_B\|_{C^2}<\varepsilon_A/(4H_A)$ on all participating +collars, using \Cref{regular:end-decay}. The persistent difference +$v_B-s_B$ is paid by its previously allocated tolerance, not by moving +the start. Start also beyond the finitely many compatibility +thresholds for the $v_B$'s, so their copies agree on overlaps. +The derivative bounds for the weights then give +\[ + \|v_A-s_A\|_{C^2}\le\varepsilon_A\le\tau_A. +\] +The result is an exact face copy sufficiently far out, is smooth, +and has bounded higher derivatives by compact parts and finitely +many smooth products. Finally $v_A\ge\tfrac34\inf s_A>0$ and +$L_Av_A\ge1-M_A\tau_A\ge3/4$, which implies +\Cref{regular:strict-supersolution}. No geometry radius, correction, +or operator is changed during this approximation. +\end{proof} + +\subsection{Restoring the invariant} + +The passage from a positive solution to scalar-curvature control by a +circle warp appears in +\citet[Theorem~4\textup{(ii)}]{fischerColbrieSchoen1980}. +Here bounded positive supersolutions must also be exactly compatible +with every packet end; \Cref{regular:solutions} supplies these functions. + +\begin{proposition}[The full invariant step]\label{regular:step} +Assume the stage $r$ invariant and the conclusion of +\Cref{shield:global}. For every nonempty parent packet of base +dimension at least two, processing $j_0$ produces the compatible +smooth split packets of +\Cref{packet:split}, with the required forced signs and continuous +projection maps. There are smooth compatible old metrics and $399$ +new torus factors on every packet such that the profiles, cores, +potentials, and scalar curvature satisfy the stage $r+1$ invariant. +This includes a possible one-dimensional output, on which the scalar +lower bound is global. After the radial split the packets have the +complete compact-part/product-end geometry of +\Cref{packet:geometry}. +\end{proposition} + +\begin{proof} +Use \Cref{regular:partition,regular:seam,regular:profiles} for the +smooth base metrics and the profile data, and +\Cref{regular:correction,regular:solutions} for $v_A$. Append +$k=399$ circle variables, each with length function +$v_A^{2/(k+1)}$. The warped-product identity is +\[ + R_{\rm new}=R_{\rm old}-\frac{4k}{k+1} + \frac{\lap_{\rm old}v_A}{v_A}. +\] +Indeed a common scale $f$ contributes +$-2k f^{-1}\lap_{\rm old}f-k(k-1)f^{-2}|df|^2$; substitution of +$f=v_A^{2/(k+1)}$ cancels the separate $|dv_A|^2$ term. +The old Laplacian on invariant functions is the weighted base +Laplacian, including the product of all old warp lengths. Here $k$ counts only the newly appended circle factors. Any unit +factors used earlier to compare old lists on a finite diagram have +zero logarithmic derivative and leave both $\lap_{\rm old}$ and this +coefficient unchanged. Since $4k/(k+1)=C$, \Cref{regular:Vstar,regular:strict-supersolution} give +\[ + R_{\rm new}=V_A^*-K_A+\frac{L_Av_A}{v_A} + \ge V_A^*\quad\hbox{on }D_A. +\] +Here $K_A=0$ on a neighborhood of $D_A$. Adding these factors does +not change base distances or profile gradients. The new factors +are positive, bounded above and away from zero, and exactly compatible. +The same number is appended in every status at the stage, so all +end identities compare the same list of factors. This proves the +analytic invariant, including when the new required threshold is zero. +The componentwise metric operations preserve every unused old circle +length identically equal to $1$: restriction, product extension, and +$\beta_s\mapsto\beta_{\phi(x,s)}$ all fix that constant entry. The +$399$ appended lengths are the only common new entries. Likewise +restriction and cylinder pullback preserve a constant pole profile and +its full core, and \Cref{regular:constant-preservation} preserves the +specified multiplied constant potentials. + +The zero wall lies in $\operatorname{int}G_{j_0}$ and all current +forced signs hold by \Cref{regular:partition}; earlier forced cores +are carried by the continuous projection and retain their pole +neighborhoods. Thus this is exactly the regular compatible splitting +operation required by the packet construction. At the radial split, +the preserved alternating compact barrier bands separate all wall +components into bounded transition shells and trap every closed +sign-side component between two separating bands (with a compact +initial block). These components are compact before prolongation. +The normal collars and smooth metrics used above are products in all +old ends. \Cref{packet:split,packet:geometry} therefore supply the +claimed geometry after prolongation. The main argument applies the +one-dimensional obstruction and reconstructs the compatible partitions in the original manifold. +\end{proof} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/shield-global.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/shield-global.tex new file mode 100644 index 0000000000000000000000000000000000000000..7849a1d477d16a2270e144cc2d5903f468f92885 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/appendices/shield-global.tex @@ -0,0 +1,661 @@ +\section{Global shielding and extension to cofaces}\label[appendix]{sec:shield-global} + +The local annular cost does not add once for every singular piece: +a concentration argument bounds a new shrinking cost by the maximum +of the old and local form coefficients. We prove that assertion first, +then transport each chosen addition to cofaces, and finally show that +only finitely many additions are needed on a whole packet. +The finite-label hypotheses in this appendix are extended in +\cref{countable:finite-data,countable:step}. Each label may already +have its own positive potential floor $p_j^*$; none of the estimates +uses a positive lower bound common to all countably many labels. + +We keep the true cuts fixed throughout this section. Only profiles of +remaining labels, their cores, and their potentials are changed. In +particular, none of the convergence assertions below concerns minimizers +for a changing pressure. + +\begin{definition}[The matching form domain]\label{shield:domain} +Let $E$ be a parent packet of dimension $m$, with true wall $S$ and singular +set $\Sigma$. A raw piece $Z$ means either $S_{\rm reg}$ or one of the two +joined spaces +\[ + Z=\Omega\mathbin{\cup}_{S_{\rm reg}} + (S_{\rm reg}\times[0,\infty)). +\] +The tests on a joined space are compactly supported, piecewise smooth +functions whose two traces agree at the regular seam and whose support +avoids $\Sigma$ and the cylinders over $\Sigma$. The wall has the +corresponding compact tests away from $\Sigma$. All measures include the +old warp weight. Put +\begin{equation}\label{shield:hilbert} + \mathfrak h_Z(v)=\mathcal N_Z(v)+e_0\norm{v}_{L^2(Z)}^2, + \qquad e_0=\frac{\lambda_r-\lambda_{r+1}}{100}>0, +\end{equation} +where $\mathcal N$ is the reserved gradient form in \cref{cut:reserve}: +its gradient coefficients are $.8$ on the bulk and $1.7$ on the wall or +cylinder. Write $\mathcal H_Z$ for the completion of these tests in +$\mathfrak h_Z^{1/2}$. Pure cylinders $S_{\rm reg}\times\R$ use the +coefficient $1.7$ throughout. The definitions also apply after adjoining +finitely many Euclidean product coordinates, with the same coefficients +and $e_0$. Such products are used only to identify limits on translated +windows; they do not append further circle warps or change the weight. +A compact window means a fixed compact ambient base window together with +a bounded cylinder interval, when a cylinder is present. +\end{definition} + +The trace matching in this definition is essential. There is no +Dirichlet condition at the seam. A pure-cylinder test can be translated +into the cylindrical part of a joined space, with zero bulk part. +Consequently every form inequality on that joined space for potentials +pulled back from its base also holds on the pure cylinder. + +\begin{lemma}[Compactness on joined windows]\label{shield:compactness} +Assume $m\ge8$. On any fixed compact window, a bounded sequence in +$\mathcal H_Z$ has a strongly convergent $L^2$ subsequence. The following +moving version also holds. Work in fixed smooth ambient geometry on each expanding window, +as is the case after translation in an exact product end. Suppose +positive weights and pressures converge smoothly there, and the true +phases and walls converge as in \cref{cut:gmt,cut:end-convergence}. +Consider normalized tests on the corresponding raw pieces $Z_i$, all +supported in a fixed compact window strictly inside a larger window of +convergence. After passage to a subsequence there is a function $v$ on +the limiting raw piece $Z_\infty$ such that: +\begin{enumerate}[label=\textup{(\roman*)}] +\item On every regular compact subwindow, the tests converge weakly in +$H^1$ and strongly in $L^2$ in the smooth graphical identifications. At +a limiting regular seam the two traces agree. +\item No $L^2$ mass is lost in neighborhoods of the limiting singular +set. In particular, integrals of $v_i^2$ converge to those of $v^2$, +including integrals against any fixed bounded continuous ambient +potential and its pullback to the wall and cylinder. +\item The limit belongs to $\mathcal H_{Z_\infty}$ and +\begin{equation}\label{shield:liminf} + \mathfrak h_{Z_\infty}(v) + \le\liminf_i\mathfrak h_{Z_i}(v_i). +\end{equation} +\end{enumerate} +These assertions include pure cylinders, translated cylinder intervals, +and Euclidean product limits. Constants are local to the enlarged +window; no uniform bound among all packets is asserted. +If the limiting wall is smooth on the enlarged window, the same +conclusions follow from ordinary Rellich compactness and the bounded +trace maps in the smooth graphical charts, in every base dimension. +\end{lemma} + +\begin{proof} +We give the estimates for tests first; they extend to the completions by +density. On the enlarged window the weights are uniformly comparable +to a positive constant, the ambient geometry is uniformly smooth, and +the unweighted base mean curvature of the regular walls is bounded. +The latter bound follows from the prescribed weighted mean curvature +and the bounded first derivative of the logarithm of the weight. +The perimeter varifolds have density at least one and no additional +boundary term on this interior window, by \cref{cut:gmt}. They therefore +meet the hypotheses of the Sobolev inequality used below. +No bound on the individual principal curvatures is needed. + +We use the submanifold Sobolev inequality in the following precise form. +For a $d$-dimensional Euclidean rectifiable varifold of density at least +one with generalized mean curvature $H$, and a nonnegative compactly +supported test $f$, +\begin{equation}\label{shield:submanifold-sobolev} + \left(\int f^{d/(d-1)}\right)^{(d-1)/d} + \le C_d\int\bigl(\abs{\nabla f}+\abs H f\bigr). +\end{equation} +This holds in arbitrary codimension; see \citet[Chapter~4, Section~5, +Theorem~5.7]{simonGMT}. This is the submanifold Sobolev inequality +of Michael and Simon \citep{michaelSimon1973}. A smooth extension of the fixed ambient chart +metric to a compact manifold and a fixed smooth isometric embedding +\citep{nash1956} give the corresponding local inequality in our charts. +The extra mean curvature of the embedding is bounded on this fixed +compact geometry. For the moving-domain assertion the embedding is chosen once in the +fixed ambient geometry; its constants are therefore independent of the +wall in the sequence. Applying +\cref{shield:submanifold-sobolev} to a power of the test gives the usual +$H^1$ Sobolev exponent on the wall. Applied to +$S_{\rm reg}\times\R$, it gives, with +\[ + q=\frac{2m}{m-2},\qquad b=\frac{2(m-1)}{m-2}, + \qquad 2(b-1)=q, +\] +a local $L^q$ bound on the cylinder. A test on the half-cylinder is +reflected evenly at its base before this application. The wall alone +has the stronger exponent $2(m-1)/(m-3)>2$. + +Here and below cutoffs are taken on the enlarged window and are pulled +back to both seam sides. The cutoff errors are bounded by the local +$L^2$ norm. Normalize $\mathfrak h_Z(v)\le1$. Choose a cylinder cutoff +which is one at $s=0$ and vanishes at the outer end of a fixed bounded +interval. The fundamental theorem of calculus and H\"older give +\begin{equation}\label{shield:trace} + \int_{S_{\rm reg}}\abs{v(0)}^b + \le C\int_{S\times[0,L]} + \bigl(\abs v^b+\abs v^{b-1}\abs{\partial_s v}\bigr) + \le C. +\end{equation} +The last step uses the cylinder $L^q$ bound and $2(b-1)=q$. + +The bulk need not have a uniform smooth boundary, so we use a BV +calculation there. Extend $\abs{v_{\rm bulk}}^b$ by zero in the ambient +chart, including its compact cutoff. For an initial test its support +misses the singular set, and the BV jump formula gives precisely the +boundary contribution $\int_{S_{\rm reg}}\abs{v(0)}^b$ in addition to the +interior variation. This is also obtained directly by integration by +parts against unit vector fields on the regular side. The ambient +$L^1$ Sobolev inequality and Cauchy--Schwarz therefore yield, for +$J=\int_\Omega\abs v^q$, +\begin{equation}\label{shield:bulk-bv} + J^{(m-1)/m} + \le C\left( + \int_\Omega\abs v^b+ + \int_\Omega\abs v^{b-1}\abs{\nabla v} + +\int_{S_{\rm reg}}\abs{v(0)}^b\right) + \le C(1+J^{1/2}). +\end{equation} +Here $bm/(m-1)=q$; the first integral on the right is at most +$J^{1/2}\norm v_2$, because $b=(q+2)/2$. Since $(m-1)/m>1/2$, \cref{shield:bulk-bv} bounds $J$. +Rescaling removes the normalization. Thus all pieces have a local +$L^p$ bound for some $p>2$ controlled by their matching form norm. +This argument extends a power as a BV function; it constructs no +$H^1$ extension operator for the bulk. + +We next exclude concentration in the moving version. On a slightly +larger compact window let $F$ be the compact portion of the limiting +singular set. Its ambient volume and its limiting perimeter measure +are zero. For a closed $\delta$-neighborhood of $F$, perimeter measure +convergence and the closed-set upper bound for weakly convergent +measures imply +\[ + \limsup_i\mu_{S_i}(\overline{U_\delta(F)}) + \le \mu_{S_\infty}(\overline{U_\delta(F)}), +\] +with a fixed outer compact cutoff if necessary. The right side tends +to zero as $\delta\downarrow0$. The same conclusion for bulk measure +follows from ambient volume bounds, and for a cylinder follows by +multiplying the wall measure by the bounded interval length. H\"older +and the local $L^p$ estimates give +\[ + \int_{A_i}\abs{v_i}^2 + \le C\mu_i(A_i)^{1-2/p}. +\] +Thus the order is first $i\to\infty$ at a fixed singular neighborhood, +then $\delta\downarrow0$. Neither disappearing components nor a thin +boundary layer can retain $L^2$ mass. + +Off $F$, graphical convergence and the absence of stray wall support +supply smooth charts for both bulk phases and their seams. Rellich +compactness in those charts gives the asserted weak $H^1$ and strong +$L^2$ limits. The trace map on each fixed regular chart is bounded and +linear; weak convergence of both traces preserves their equality. +Combining these regular limits with the no-concentration conclusion +proves all the stated $L^2$ and bounded-potential convergence. Exhausting +the regular pieces gives lower semicontinuity of their gradient +integrals. + +It remains to justify membership in the limit completion, rather than +assume it. On this one limiting geometry, monotonicity and the local +area bound give +\[ + \mu_S(B_{2a}(x))\le C a^{m-1} +\] +for sufficiently small balls in the fixed window. Since +$\dim_{\rm H}\Sigma\le m-80$, a sufficiently small $\eps>0$ gives simultaneously +\begin{equation}\label{shield:maximum-bound} + \int_Z(V+A_\eps)v^2 + \le\bigl(\max\{\theta,.78\}+\eta\bigr)\mathfrak h_Z(v). +\end{equation} +It suffices that $V$ be bounded on the compact base windows used below. +In particular the assertion applies to a finite sum of compact +additions on the initial radial parent. +\end{lemma} + +\begin{proof} +First $\int A_\eps v^2\to0$ for every fixed $v\in\mathcal H_Z$. +Indeed \cref{shield:local-input} bounds these forms uniformly by a +constant times $\mathfrak h_Z$, whereas for a defining test the integral +is zero for small $\eps$, since its support misses $F$. Density and +Cauchy--Schwarz for the nonnegative form give the assertion. + +We detail the localization needed on noncompact pieces. If +$\sum_\alpha\chi_\alpha^2=1$ is a locally finite Lipschitz partition +which has the same seam traces, then +\begin{equation}\label{shield:ims} + \sum_\alpha\mathfrak h_Z(\chi_\alpha v) + =\mathfrak h_Z(v)+ + \int_Z c_Z\sum_\alpha\abs{\nabla\chi_\alpha}^2v^2, + \qquad + c_Z=\begin{cases}.8&\text{on bulk},\\1.7&\text{on wall or cylinder}. + \end{cases} +\end{equation} +The mass term $e_0\norm v_2^2$ allows the last integral to be at most +$\tau\mathfrak h_Z(v)$ for any prescribed $\tau>0$. For example use a +sine/cosine pair of a Lipschitz cutoff angle, equal to $\pi/2$ on a +fixed compact base set containing $\supp A_\eps$ for all small $\eps$, +and zero outside a sufficiently large compact neighborhood. Its transition width makes its derivative +arbitrarily small. The cosine part sees only $V$. Its bound is the assumed global +form inequality for $V$; no compact support assumption on $V$ is needed. +The sine part has compact base support, where the old potential is bounded. +On a joined side +also use a squared partition into cylinder intervals of a sufficiently +large fixed length. The first member is identically one near $s=0$ +and extends as one on the bulk; all the others vanish near the seam +and extend as zero there. Their squared gradients have an arbitrarily +small sum. All cutoffs in the base are pulled back to the cylinder. + +Choose the partition error so small that a violation of +\cref{shield:maximum-bound} forces one localized test to have quotient +larger than $\max\{\theta,.78\}+\eta/2$. This follows directly by +summing the potential integrals and \cref{shield:ims}; for instance it +suffices to have +$(\max\{\theta,.78\}+\eta/2)(1+\tau) + <\max\{\theta,.78\}+\eta$. +The selected test has compact base support and a bounded cylinder +interval. Passing to a subsequence, its interval either stays in a +fixed bounded range, or escapes to infinity. In the latter case it is +translated to a fixed interval on the pure cylinder. Both the old +bound and \cref{shield:local-input} hold there by translating its tests +into the original joined side. Thus in all cases the selected sequence +lies in one fixed Hilbert space, in a fixed compact window. + +Suppose now that $\eps_i\downarrow0$ gives such violating tests $v_i$, +normalized by $\mathfrak h(v_i)=1$. By \cref{shield:compactness} and +weak Hilbert compactness, $v_i\rightharpoonup v$ in this fixed space and +$v_i\to v$ strongly in $L^2$. Write $w_i=v_i-v$. Then +\[ + \mathfrak h(w_i)=1-\mathfrak h(v)+o(1),\qquad \norm{w_i}_2\to0. +\] +The bounded old potential satisfies +$\int Vv_i^2\to\int Vv^2\le\theta\mathfrak h(v)$. The new form +vanishes on $v$, and its mixed term vanishes by Cauchy--Schwarz and the +uniform bound on $w_i$. Consequently \cref{shield:local-input} gives +\begin{align*} + \limsup_i\int(V+A_{\eps_i})v_i^2 + &\le\theta\mathfrak h(v)+.78\limsup_i\mathcal N(w_i)\\ + &\le\theta\mathfrak h(v)+.78(1-\mathfrak h(v)) + \le\max\{\theta,.78\}. +\end{align*} +This contradicts the selected quotient. There are only three raw +pieces, so the same sufficiently small scale works for all of them. +\end{proof} + +We next extend a single chosen addition. The scale in +\cref{shield:maximum} is fixed before this next limit is taken. + +\begin{lemma}[Extension with late fronts]\label{shield:extension} +Let $B$ be a packet, let $D\ge0$ be a fixed smooth compactly supported +potential addition on $B$, and let $V$ denote the old cumulative +potential on every packet. Suppose these old potentials are exactly +compatible and obey the common form bound $\theta\mathfrak h$. +Suppose also that +\[ + \int(V_B+1.16D)v^2\le\beta\mathfrak h_B(v),\qquad \beta\ge\theta, +\] +on the wall and joined sides over $B$. In each of the finitely many +coface occurrences $B\times(T,\infty)^d\subset E$, introduce a smooth +potential $D_E$ with +\begin{equation}\label{shield:extension-support} + 0\le D_E\le D\circ\pi_B,\qquad + \supp D_E\subset K\times\prod_{i=1}^d[R_i,\infty), + \qquad K\Subset B. +\end{equation} +Use consistent functions on the normal coordinates identified by +commuting face chains, so that these extensions are compatible. +For every $\eta>0$ all starts can be chosen sufficiently large that +\begin{equation}\label{shield:extended-bound} + \int(V_E+1.16D_E)v^2\le(\beta+\eta)\mathfrak h_E(v) +\end{equation} +on every affected raw piece. The conclusion concerns the same fixed +true cuts; it uses their product convergence from +\cref{cut:end-convergence}. +\end{lemma} + +\begin{proof} +Only finitely many occurrences are involved by \cref{packet:category}. +Here and below a source is extended along the directed face-to-coface +relations, through their finite depth. Closing under arbitrary +alternating face and coface walks is neither required nor performed. +For a common normal coordinate take the largest of the finitely many +requested starts, so commuting chains give identical occurrences. +It is enough to prove the assertion for any sequence of their fronts +with $\min_iR_i\to\infty$, passing to a fixed occurrence and piece if +needed. Choose every localization width below using $\eta$ and $e_0$ +first, independently of the front positions. + +Separate the new support from its complement with a sine/cosine pair +as in \cref{shield:ims}. More explicitly, an angle can be $\pi/2$ +times a product of a compact base cutoff, equal to one on $K$, and +normal cutoffs which equal one on $[R_i,\infty)$ and vanish before +$R_i-L$. Enlarge the compact base window and $L$ until the sum of +squared derivatives is sufficiently small. Extend the angle by zero +outside the product sector. Taking the fronts beyond all collar +thresholds makes this extension well defined. The cosine test misses +the addition and has the old bound $\theta$. Partition the sine part +further into fixed large boxes in every normal coordinate, and into +fixed large intervals in cylinder height when applicable. The total +localization error is as small a fraction of $\mathfrak h$ as required. + +If \cref{shield:extended-bound} failed along fronts tending out, the +same quotient selection as in \cref{shield:maximum} would give +normalized tests on selected boxes with quotient at least +$\beta+\eta/2$. Such a box must meet the new potential support, +since otherwise the old bound $\theta\le\beta$ applies. Its center +therefore has +\[ + t_i\ge R_i-O(L)\longrightarrow\infty + \quad\text{for every }i=1,\ldots,d. +\] +Its base window is fixed and compact. Recenter every normal coordinate. +A bounded cylinder interval stays in a bounded-height joined window, +or, after another subsequence, is translated to a pure cylinder. The +ambient geometry on the recentered windows is exactly the product +geometry, and the true cuts converge to those over $B$ times $\R^d$. +The old potential is exactly $V_B\circ\pi_B$ there, because the fronts +also exceed its fixed product thresholds. The new potential is bounded +above by the fixed bounded function $D\circ\pi_B$. + +Apply \cref{shield:compactness}. For its limit $v$, +\begin{equation}\label{shield:extension-limit} + \limsup\int(V_E+1.16D_E)v_E^2 + \le\int(V_B+1.16D)\circ\pi_B\,v^2, + \qquad \mathfrak h_{\rm lim}(v)\le1. +\end{equation} +The face form bound applies to the right side by slicing the $\R^d$ +coordinates. For defining tests this is Fubini's theorem: the gradient +terms in the extra directions are nonnegative, and all coefficients +$.8,1.7,e_0$ are the same in every base dimension. Approximation in the +limit completion, proved in \cref{shield:compactness}, gives the result +for $v$. In a pure-cylinder limit use the face joined bound after +translation in cylinder height. Thus the right side of +\cref{shield:extension-limit} is at most $\beta$, a contradiction. +Finite incidence permits simultaneous choices for all occurrences. +This limit compares bounded potential integrals on moving domains; +it uses no subtraction of vectors belonging to different form spaces. +\end{proof} + +\begin{lemma}[Compatible caps and cores]\label{shield:caps} +A compact forcing update from \cref{local:shell} extends to all its +cofaces so as to satisfy \cref{shield:extension-support}, preserve the +profile gradient inequality with coefficient $a_r$, and enlarge only +the designated cores. The extension is an exact product sufficiently +far in every old face collar and is made once per occurrence, independent +of the order of commuting face relations. +\end{lemma} + +\begin{proof} +Consider a plus addition on $B$, with forcing profile $u$ and shell +potential $D$. Choose a compact base window $K\Subset\{h_j>t_j\}$ +containing both $\overline{\{u> -\infty\}}$ and $\supp D$ in its +interior. The first set includes the inner pole region; keeping only +$\supp D$, which is annular, would not suffice for this extension. +In a coface product over $K$ put +\begin{equation}\label{shield:cap-formula} + u_E(x,t)=\min\{u(x),w_1(t_1),\ldots,w_d(t_d)\},\qquad + D_E(x,t)=D(x)\prod_{i=1}^d\chi_i(t_i). +\end{equation} +Each cap $w_i$ is $-\infty$ up to a late front, then crosses to +$+\infty$, and remains there, with +\[ + 2\abs{w_i'}\le .58w_i^2+p_j^* +\] +on its finite band. For example integrate $2w_i'=.58w_i^2+p_j^*$ on the finite +band and extend by its poles. The crossing length is +$2\pi/\sqrt{.58p_j^*}<\infty$; finite truncations remain locally +Lipschitz at both ends. Only this label's positive floor is used. +Each smooth cutoff $\chi_i$ takes values in +$[0,1]$ and \emph{finishes} increasing to one before $w_i$ begins its +crossing. Choose these cutoff fronts to satisfy +\cref{shield:extension} for the already fixed $D$ and shell scale. +Outside this product occurrence use the inactive value $-\infty$. +The old profile is replaced by its maximum with $u_E$. + +For locally Lipschitz finite truncations, a finite minimum or maximum +has the derivative of one active branch almost everywhere (on a tie, +the branch derivatives agree almost everywhere on the tie set). +If the active finite branch is $u$, every cap has already left its +$-\infty$ plateau; hence every $\chi_i=1$ and the entire shell +potential $D$ is available. If it is a cap, its derivative cost is +paid by the preexisting baseline: +\[ + .58w_i^2+p_j^*\le a_rw_i^2+\Pi_j+D_E. +\] +The old branch retains its old inequality. Thus no sum of normal +cap derivatives, and no extra cap potential, enters the budget. +For a minus update negate $u_E$ and take a minimum with the old profile. +The strict anchor regions for opposite signs are disjoint and contain +no opposite old poles by \cref{local:anchors}. + +Write $T_i^+$ for the end of the $i$th cap crossing. Enlarge the +plus core by the open set +\[ + \{x:\delta_B(x)<\varepsilon/2\}\times + \prod_i(T_i^++b_i,\infty),\qquad b_i>0. +\] +The properly embedded closed tails ensure that its closure in the +coface has $\delta_B\le\varepsilon/2$ and $t_i\ge T_i^++b_i$. +It is therefore contained in the open set +$\{\delta_B<\varepsilon\}\times\prod_i(T_i^+,\infty)$, where every +branch of the minimum is $+\infty$. This proves closure confinement, +including at the cap fronts, with the corresponding statement for a +minus core. The front positions are also chosen beyond the exact anchor-product +thresholds, so its support retains the designated strict anchor sign. + +Use the same caps and cutoffs for the same normal coordinates in every +identified occurrence. On a face obtained by hitting one of these +coordinates, that coordinate's cap is eventually $+\infty$ and its +cutoff is one, and that factor disappears from +\cref{shield:cap-formula}. If an unrelated coordinate is hit, an +intersection sufficiently far in its collar would require $x$ to +escape into a further face of $B$, which the compact support forbids. +Finite incidence allows all relevant intersection thresholds to be +exceeded at once. These observations prove compatibility for all +chains and show why a commuting product occurrence is not counted +more than once. +\end{proof} + +\begin{proposition}[The global shielding output]\label{shield:global} +At a fixed stage $r$, let $j_0$ be the current label and keep its true +minimizers fixed. There are updates of every remaining profile and +core, and compatible smooth additions $D_j\ge0$, with the following +properties on every parent packet $E$ of dimension $m$: +\begin{enumerate}[label=\textup{(\roman*)}] +\item The updated profiles have locally Lipschitz finite truncations +and satisfy, almost everywhere on their finite bands, +\begin{equation}\label{shield:profile-output} + 2\abs{\nabla\mu'_j}\le a_r(\mu'_j)^2+\Pi_j+D_j. +\end{equation} +All these data are exact compatible products. Each core only enlarges, +and its closure remains in the interior of its designated updated pole +set. The original forced sets and the anchor sign restrictions are +preserved. +\item With $G'_j=E\setminus(U_j^{\prime+}\cup U_j^{\prime-})$, the set +\begin{equation}\label{shield:guard-output} + H_E^*=\left\{x:\#\{j\ne j_0:x\in G'_j\}\ge m-2\right\} +\end{equation} +is closed and misses $\Sigma_E$. In an exact collar over a face $B$, +its projection is contained in $H_B^*$. This holds for every further +face and every simultaneous product limit. +\item For one common $\theta_*<.9$, all walls and joined sides satisfy +\begin{equation}\label{shield:form-output} + \int 1.16\sum_{j\ne j_0}D_jv^2 + \le\theta_*\bigl(\mathcal N(v)+e_0\norm v_2^2\bigr) + \le .9\bigl(\mathcal N(v)+e_0\norm v_2^2\bigr). +\end{equation} +The additions are bounded with their derivatives on each post-radial +packet. There are only finitely many local additions on each such +packet and only finitely many extensions into it. +\end{enumerate} +For the initial radial parent the same conclusions hold with locally +finite additions over its compact transition shells; boundedness is +asserted on the corresponding compact blocks and +\cref{shield:form-output} holds on every compact test. No convergence +assertion at an end of the original manifold is needed. +\end{proposition} + +\begin{proof} +First consider the packet system after the radial cut. Process its +finite set of depths from the deepest faces upward. The anchors and +thresholds of \cref{local:anchors} are fixed once, before any updates. +Initially the cumulative addition is zero. Maintain a single common +upper form bound at least $.78$; it is harmless to start this bookkeeping +bound at $.78$ although the initial form vanishes. Enumerate the +countable pairs consisting of a packet and a possible successive local +addition number by distinct integers $\nu\ge1$. Reserve +$\gamma_\nu=.1\,2^{-\nu}$ for that pair, allocating at most half to +\cref{shield:maximum} and half to the simultaneous extensions of +\cref{shield:extension}. The common bound after any collection of +completed updates is at most +\[ + .78+\sum_{\nu\text{ used}}\gamma_\nu\le .88<.9. +\] +We may consequently take $\theta_*=.88$. There is no requirement +that a shell scale or its extension fronts be uniform in the packet +or in the addition number. + +Suppose all strict further faces of $E$ have been treated and their +updates extended. The remaining risky set +\[ + K_E=\Sigma_E\cap H_E^* +\] +is compact. To prove this, if points in it escaped, the packet escape +property \cref{packet:escape} would give a subsequence over a deeper +face $B$, with compact convergent base coordinates and every escaping +normal coordinate tending to infinity. The updated cores are exact +products there. There are finitely many relevant labels, so pass to a further +subsequence with a fixed subset of $m-2$ undecided labels. +Membership in each of their closed sets $G'_j$ passes to the base +limit. Thus the limit base point has at least $m-2$ undecided labels. +This implies membership in $H_B^*$, because +$\dim B-2\le m-2$. By \cref{cut:end-convergence} the translated true +cuts converge to the true cut over $B$ times the normal Euclidean +factor. Support convergence puts the limiting base point on the true +boundary over $B$. A regular limiting boundary point would have a neighborhood +of smooth graphical convergence, which contains no singular points of +the approximating walls. The limit must therefore be singular on the +face. This contradicts its completed shielding. Thus $K_E$ has no +escaping sequence; it is closed, and the compact-part property proves +compactness. If $m<8$ it is already empty. + +Denote this initial compact set by $K_E^{(0)}$. For each remaining +label and each strict anchor sign, cover $K_E^{(0)}$ in that open +region by the countable compact flat pieces of \cref{local:flat-pieces}. +These pieces need not themselves be an open cover. Enumerate the +signed pieces; their eventual inner forcing neighborhoods will provide +the open cover needed for termination. For each, +choose its shell sufficiently small by \cref{shield:maximum}, then +hold that scale fixed and extend its caps, potential, and core by +\cref{shield:extension,shield:caps}. Supports can be as small as needed +to stay within the prescribed strict anchor region. Opposite updates +for one label therefore never conflict. Profiles retain +\cref{shield:profile-output} by their active-branch inequalities, and +cores enlarge monotonically. + +To prove finite termination, after $N$ additions set +\[ + O_N=\{x\in E:\#\{j\ne j_0:x\in G_j^{(N)}\}t_j\}\cap\{\mu_j=-\infty\}=\varnothing, + \qquad + \{h_jt_j$, proving the last assertions. +\end{proof} + +The guard $H_E^*$ in \eqref{packet:guard} requires at least $m-2$ +remaining undecided labels. At $x\in\Sigma$, enlarging the positive +core for every label with $h_j(x)>t_j$ and the negative core for every +label with $h_j(x)0$. +The set $\Sigma\cap O$ is covered by countably many compact sets +$F\subset\Sigma\cap O$, each contained in a buffered coordinate chart, +with the following property. There is $r_F>0$ such that, for every +$x\in F$ and $00$. There are +$d_x>0$ and $r_x>0$ such that, at every radius $r\Theta(x)-d_x$. +Thus the group itself has the plane containment at every one of its +centers and every radius less than $1/a$. Work within one buffered chart at a time. Intersect these groups +with a countable family of compact subpatches $Q\Subset O$ exhausting +that chart's portion of $O$, and take +$F=\overline{Q\cap\text{group}}$. Relative closedness of $\Sigma$ +and the compact buffer ensure $F\subset\Sigma\cap O$. +These compact sets still cover $\Sigma\cap O$. + +To check the geometric condition at a closure center $x$, let +$x_i$ be group points tending to $x$ and fix $r<1/(3a)$. Apply the +group estimate in $B_{2r}(x_i)$. The planes pass through $x_i$, so a +subsequence converges to a plane through $x$. Every group point in +$B_r(x)$ is in these larger balls eventually. Passing first to the +limit of the planes and then to closure points gives containment with +width $2\eta'r$. Taking, for example, $\eta'<\eta/3$ proves +\eqref{local:flatness}; choose $r_F$ below $1/(3a)$ and the buffered +chart radius. This argument transfers the containment +itself; no density continuity at a closure point is required. +\end{proof} + +The next construction fixes the pressure and potential to be estimated. +Distances denoted by $\delta$ from now on are ambient base distances. + +\begin{lemma}[A shell forcing profile]\label{local:shell} +Let $F$ be a compact subset of $\Sigma\cap\{h_j>t_j\}$. +For every sufficiently small $\varepsilon>0$ there is an extended-real +profile $u_\varepsilon$, equal to $+\infty$ for $\delta\le\varepsilon$ +and $-\infty$ for $\delta\ge R_\varepsilon$, where +\[ + \delta(x)=\dist(x,F),\qquad R_\varepsilon=e^{5\pi}\varepsilon, +\] +whose finite truncations are locally Lipschitz and which satisfies +\begin{equation}\label{local:shell-gradient} + 2|\nabla u_\varepsilon| + \le .58u_\varepsilon^2+2\delta^{-2} + \quad (\varepsilon<\deltat_j\}. + \end{aligned} +\end{equation} +Replacing $\mu_j$ by $\max(\mu_j,u_\varepsilon)$, adding +$D_\varepsilon$ to $\Pi_j$, and adding +$\{\delta<\varepsilon/2\}$ to $U_j^+$ preserves the profile inequality +with coefficient $a_r$. There is an identical negative construction +in $\{h_j0$ and $c_0>0$ with the +following property. Suppose $F$ satisfies \eqref{local:flatness} +with $\eta\le\eta_0$ in a sufficiently accurate buffered chart. +There are $d_F>0$ and $C_F<\infty$ such that, on the wall and on +either joined side, +\begin{equation}\label{local:rough-estimate} + c_0\int_{\deltad$ separately, +gives +\begin{equation}\label{local:density-integral} + \int_{B_d(x)}\delta^{-5.5}\,dx\le C d^{m-5.5}. +\end{equation} +Indeed the sum of the bounds is a geometric series with ratio +$2^{-2}$. In particular this density is locally integrable. + +Choose a nonnegative smooth chart cutoff $\chi$, equal to one on a +neighborhood of $F$, and, using Euclidean coordinates, define +\begin{equation}\label{local:rough-potential} + U(x)=1+\int\chi(y)\delta(y)^{-5.5}|x-y|^{-(m-4)}\,dy. +\end{equation} +For small $\delta(x)>0$, the part $|x-y|0$ such that +\begin{equation}\label{local:rough-U-bounds} + c\delta^{-1.5}\le U\le C\delta^{-1.5}, + \qquad |\nabla U|\le C\delta^{-2.5}. +\end{equation} + +Set $a=m-4$ temporarily. For any Euclidean $d'$-plane $T$, direct +differentiation of $K(z)=|z|^{-a}$ gives +\[ + -\tr_T D^2K(z) + =a|z|^{-a-2}\left(d'-(a+2)|\operatorname{proj}_T(z/|z|)|^2\right) + \ge a(d'-a-2)|z|^{-a-2}. +\] +For $d'=m$ or $m-1$ the last coefficient is positive. A sufficiently +small, dimension-dependent metric distortion preserves a fixed +positive fraction of it for traces in metric orthonormal frames. +The connection and weight terms in the weighted Laplacian involve +only first derivatives of $U$. On $S_{\rm reg}$ the restriction +formula adds its mean-curvature vector paired with $\nabla U$. +This vector is locally bounded, since +$H_{\rm base}=p-\partial_\nu\log\omega$ by \cref{cut:gmt}. +There is no dependence here on individual principal curvatures. +Integrating the positive second-derivative kernel over +$|x-y|0$ sufficiently small, using the +universal constants in \eqref{local:U-superharmonic}, so that this +is absorbed by half the positive term of order $b\delta^{-2}$. +All remaining localization terms cost $C_F\|v\|_2^2$. +This proves \eqref{local:rough-estimate} on a joined side. The +wall proof is the same calculation without the seam or the vertical +field. Taking the smaller leading coefficient makes one $c_0$ +valid for the wall and both sides. +\end{proof} + +\begin{proposition}[The local shielding estimate]\label{local:sharp} +There is a sufficiently small flatness number $\eta>0$, depending +only on $m$ and the fixed numerical coefficients, with the following +property. Let $F$ be any piece from \cref{local:flat-pieces} for this +flatness number, in a sufficiently accurate chart. For the potential +of \cref{local:shell}, there are $\varepsilon_F>0$ and $C_F<\infty$ +such that, for every $0<\varepsilon<\varepsilon_F$, +\begin{equation}\label{local:sharp-estimate} + 1.16\int D_\varepsilon v^2 + \le .78\mathcal N(v)+C_F\|v\|_2^2. +\end{equation} +This holds on the true wall and either joined side, and hence on +translated compact tests in a pure wall cylinder. The constants +$\varepsilon_F,C_F$ may depend on the local smooth data and $F$, +but are independent of $\varepsilon$. No vanishing trace condition +is imposed at a joined seam. +\end{proposition} +\begin{proof} +We give the parameter order first. Fix the rough number $\eta_0$ +and coefficient $c_0$ from \cref{local:rough}. Below, constants in +partition gradient bounds depend only on dimension and the fixed +chart distortion. Choose $\xi>0$ so small that the rough estimate +pays their total squared-gradient error with at most $.04$ of +$\mathcal N$. Next choose a large fixed $K$, then a small fixed +$\alpha>0$, and finally a strong flatness $\eta\le\eta_0$, as +specified below. Only after the piece $F$ has been selected for +this $\eta$ do we choose its small distance threshold and then +$\varepsilon_F$. In particular $C_F$ is never used to choose +$\xi$, $K$, or $\eta$. + +Put $a_-=1/2$, $a_+=2e^{5\pi}$ and $R_1=K\varepsilon$. +The shell supporting $D_\varepsilon$ lies in +$a_-\varepsilon<\delta9.8\quad\text{in the cylinder}, +\] +and $1.7\cdot6=10.2$ on the wall alone. With the explicit error +bounds just imposed, the coefficients are at least $.8\cdot4.9=3.92$ +in bulk, $1.7\cdot5.9-(.8^2/1.7)1.005>9.6$ in the cylinder, +and $1.7\cdot5.9>10$ on the wall. In particular all exceed $3.8$. +Applying the vector inequality to $q\phi_iv$ gives +\begin{equation}\label{local:box-bound} + \mathcal N(q\phi_iv)\ge + 3.8\int\zeta^{-2}(q\phi_iv)^2. +\end{equation} +This is a bound throughout the localized support; its nonnegative +right side may therefore be restricted to the shell. + +We finally compare $\zeta$ with $\delta$ on that shell. A nearest +point $z\in F$ lies in the ball used for the plane condition, so +\[ + \zeta(x)\le 1.005\delta(x)+L_m\eta K\varepsilon + +\alpha\varepsilon. +\] +The first factor is ensured by the fixed chart accuracy and its +buffer, after shrinking the scale so that a minimizing short path +to $F$ stays in the chart. Since $\delta\ge a_-\varepsilon$ there, +choose $\alpha/a_-\le .005$ and then +$L_m\eta K/a_-\le .015$. These choices give +$\zeta\le1.03\delta$. Notice that $K$ was already fixed using the +rough coefficient, and that the strong flatness was chosen before +selecting $F$. Since $3.8/(1.03)^2>3.5$, \eqref{local:box-bound} +controls $3.5\delta^{-2}$ on the shell. Summing over boxes uses +$q=1$ there and $\sum_i\phi_i^2=1$, and then +\eqref{local:localization} gives +\[ + 3.5\int_{\supp D_\varepsilon}\delta^{-2}v^2 + \le1.04\mathcal N(v)+C_F\|v\|_2^2. +\] +Together with \eqref{local:shell-potential}, this proves the stronger +coefficient bound +\[ + 1.16\int D_\varepsilon v^2 + \le\frac{1.16\cdot2.12\cdot1.04}{3.5}\mathcal N(v) + +C_F\|v\|_2^2, + \qquad + \frac{1.16\cdot2.12\cdot1.04}{3.5}<.731<.78. +\] +This yields \eqref{local:sharp-estimate}. A compact test on a pure +cylinder can be translated to positive height in a joined cylinder +and extended by zero onto the bulk, so it obeys the same estimate. +\end{proof} + +\begin{remark}\label{local:choice-order} +The choices can be summarized without a hidden dependence on the +compact piece: dimension and chart distortion; rough flatness +$\eta_0$ and coefficient $c_0$; localization parameter $\xi$; +logarithmic taper and $K$; $\alpha$ and the stronger flatness $\eta$; +the compact pieces supplied for that $\eta$; finally their individual +distance thresholds and shell scales. Shrinking a threshold absorbs +large-scale kernel tails and smooth geometric errors. It never +changes the leading coefficient used to select the stronger +flatness. All estimates are weighted base estimates, so the number +of previously appended circle factors does not replace $m$ in any +dimension or rank count. +\end{remark} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/figures/packet-split.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/figures/packet-split.tex new file mode 100644 index 0000000000000000000000000000000000000000..281d3467713cea5c05b83ea132f2aa343af19093 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/figures/packet-split.tex @@ -0,0 +1,23 @@ +\begin{figure}[!b] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,font=\small] + \fill[blue!7] (0,0) rectangle (2.3,2.2); + \fill[orange!12] (2.3,0) rectangle (5,2.2); + \draw[thick] (0,0) -- (5,0); + \draw[thick] (0,2.2) -- (5,2.2); + \draw[densely dashed,thick] (2.3,0) -- (2.3,2.2); + \node[align=center] at (1.1,1.2) {retained\\closed side}; + \node[align=center] at (3.6,1.2) {$L\times[0,\infty)$\\attached\\cylinder}; + \draw[->,thick] (4.4,.5) -- (5.3,.5) node[right] {$s$}; + \node[align=center,below] at (2.3,-.1) {internal attachment seam}; + \draw[very thick,teal!70!black] (7,0) -- (7,2.2); + \node[align=center,right] at (7.2,1.1) {separate hit space $L$\\base dimension $m-1$}; + \node[above] at (2.5,2.7) {either sign side: base dimension $m$}; +\end{tikzpicture} +\caption{A regular cut retains each closed sign side with a full cylinder +attached, and also retains the wall as a separate hit space of one lower base +dimension. The attachment seam is internal to the extended side. In old +product ends this entire operation is a product. The raw metric and its +later smoothing are defined in Appendices~\ref{sec:cut} and~\ref{sec:regularization}.} +\label{fig:regular-packet-split} +\end{figure} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/main.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..a70db4d84fb7d602576006b4e1cdf48700b47325 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/main.tex @@ -0,0 +1,101 @@ +\pdfinfoomitdate=1 +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem,booktabs,longtable} +\usepackage{graphicx,tikz} +\usetikzlibrary{arrows.meta,positioning,calc,patterns,fit,backgrounds} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[hidelinks]{hyperref} +\usepackage[capitalise,nameinlink,noabbrev]{cleveref} +\makeatletter +\newcommand{\psc@thmrefstepcounter}[2][]{% + \let\psc@saved@refstepcounter\cref@old@refstepcounter + \let\cref@old@refstepcounter\Hy@theorem@refstepcounter + \refstepcounter[#1]{#2}% + \let\cref@old@refstepcounter\psc@saved@refstepcounter +} +\patchcmd{\cref@thmoptarg}{\refstepcounter}{\psc@thmrefstepcounter}{} + {\PackageError{psc-width}{Shared-counter theorem patch failed}% + {The installed cleveref/hyperref combination needs a compatibility review.}} +\patchcmd{\cref@thmnoarg}{\refstepcounter}{\Hy@theorem@refstepcounter}{} + {\PackageError{psc-width}{Base-counter theorem patch failed}% + {The installed cleveref/hyperref combination needs a compatibility review.}} +\makeatother +\setlength{\emergencystretch}{2em} +\setlist{itemsep=3pt,topsep=5pt} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{convention}[theorem]{Convention} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\T}{\mathbb T} +\newcommand{\Fcal}{\mathcal F} +\newcommand{\dd}{\mathrm d} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\Scal}{Scal} +\DeclareMathOperator{\Div}{div} +\DeclareMathOperator{\grad}{grad} +\DeclareMathOperator{\lap}{\Delta} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Ric}{Ric} +\DeclareMathOperator{\Hess}{Hess} +\DeclareMathOperator{\Id}{Id} +\DeclareMathOperator{\UW}{UW} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\title{Positive scalar curvature and uniform codimension-two width} +\author{OpenAI} +\date{September 23, 2026} +\hypersetup{pdftitle={Positive scalar curvature and uniform codimension-two width},pdfauthor={OpenAI}} +\begin{document} +\maketitle +\begin{abstract} +We prove the quantitative continuous form of Gromov's scalar-curvature +conjecture in every dimension $n\ge4$. Every complete connected smooth +boundaryless $n$-manifold with scalar curvature at least one admits a +continuous map to a simplicial complex of dimension at most $n-2$ whose +entire fibers have diameter bounded only in terms of $n$. We also obtain +the continuous macroscopic-dimension conclusion for +universal covers of closed manifolds with positive scalar curvature in +every dimension $n\ge2$. +\end{abstract} +\begingroup\small\tableofcontents\endgroup +\clearpage +\input{sections/introduction} +\input{sections/pressure} +\input{sections/local-step} +\input{sections/stretching-rounds} +\input{sections/finite-cuts} +\input{sections/countable-partitions} +\input{sections/reconstruction} +\input{sections/closed-covers} +\appendix +\input{appendices/packets} +\input{appendices/minimizers} +\input{appendices/shield-local} +\input{appendices/shield-global} +\input{appendices/regularization} +\clearpage +\phantomsection +\addcontentsline{toc}{section}{References} +\begingroup\raggedright +\bibliographystyle{plainnat} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/references.bib b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..15243b654058c26cc73ac147e658af73e818fead --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/references.bib @@ -0,0 +1,430 @@ +@incollection{gromov1996, + author = {Gromov, M.}, + title = {Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures}, + booktitle = {Functional Analysis on the Eve of the 21st Century, Volume II}, + editor = {Gindikin, Simon and Lepowsky, James and Wilson, Robert L.}, + series = {Progress in Mathematics}, + volume = {132}, + pages = {1--213}, + publisher = {Birkh{\"a}user Boston}, + year = {1996}, + doi = {10.1007/978-1-4612-4098-3_1}, + url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/positivecurvature.pdf} +} + +@article{dranishnikov2014, + author={Dranishnikov, Alexander}, + title={On macroscopic dimension of universal coverings of closed manifolds}, + journal={Transactions of the Moscow Mathematical Society}, + volume={74}, year={2013}, pages={229--244}, + doi={10.1090/S0077-1554-2014-00221-1}, + eprint={1307.1029}, archivePrefix={arXiv}, primaryClass={math.GT}, + url={https://arxiv.org/pdf/1307.1029v1}, + note={Definitions 1.1--1.2 cited from arXiv:1307.1029v1} +} + +@article{bolotovDranishnikov2010, + author={Bolotov, Dmitry and Dranishnikov, Alexander}, + title={On {Gromov}'s scalar curvature conjecture}, + journal={Proceedings of the American Mathematical Society}, + volume={138}, number={4}, year={2010}, pages={1517--1524}, + eprint={0901.4503}, archivePrefix={arXiv}, primaryClass={math.GT}, + url={https://arxiv.org/pdf/0901.4503v1}, + note={Theorem 4.3 cited from arXiv:0901.4503v1} +} + +@misc{simonGMT, + author = {Simon, Leon}, + title = {Introduction to Geometric Measure Theory}, + year = {2014}, + howpublished = {Author-hosted lecture notes}, + note = {Tsinghua lecture notes, draft updated 3 December 2014}, + url = {https://web.stanford.edu/class/math285/ts-gmt.pdf} +} + +@article{gromov2019Mean, + author = {Gromov, Misha}, + title = {Mean Curvature in the Light of Scalar Curvature}, + journal = {Annales de l'Institut Fourier}, + volume = {69}, + number = {7}, + pages = {3169--3194}, + year = {2019}, + doi = {10.5802/aif.3347}, + url = {https://www.numdam.org/articles/10.5802/aif.3347/} +} + +@misc{simonPDE, + author = {Simon, Leon}, + title = {Lectures on {PDE}}, + year = {2015}, + howpublished = {Author-hosted lecture notes}, + note = {Updated 5 March 2015}, + url = {https://math.stanford.edu/~lms/lecs-on-pde.pdf} +} + +@article{nash1956, + author = {Nash, John}, + title = {The Imbedding Problem for {Riemannian} Manifolds}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {63}, + number = {1}, + pages = {20--63}, + year = {1956}, + doi = {10.2307/1969989}, + url = {https://mccuan.math.gatech.edu/courses/6455/nash55.pdf} +} + +@article{chodoshLi, + author = {Chodosh, Otis and Li, Chao}, + title = {Generalized soap bubbles and the topology of manifolds with positive scalar curvature}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {199}, + number = {2}, + pages = {707--740}, + year = {2024}, + doi = {10.4007/annals.2024.199.2.3}, + url = {https://arxiv.org/abs/2008.11888v4}, + note = {Numbering follows arXiv:2008.11888v4, 20 September 2023} +} + +@article{schoenYau1979Structure, + author = {Schoen, Richard and Yau, Shing-Tung}, + title = {On the structure of manifolds with positive scalar curvature}, + journal = {Manuscripta Mathematica}, + volume = {28}, + pages = {159--183}, + year = {1979}, + doi = {10.1007/BF01647970} +} + +@article{gromovLawson1983, + author={Gromov, Mikhael and Lawson, Jr., H. 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Published in Journal of Topology and Analysis; doi:10.1142/S1793525321500084}, + url={https://arxiv.org/abs/1904.00846v1} +} + +@article{liokumovichWang2023, + author={Liokumovich, Yevgeny and Wang, Zhichao}, + title={On the waist and width inequality in complete 3-manifolds with positive scalar curvature}, + journal={Cambridge Journal of Mathematics}, + volume={14}, number={2}, year={2026}, pages={349--375}, + doi={10.4310/CJM.260514230515}, + eprint={2308.04044}, archivePrefix={arXiv}, primaryClass={math.DG}, + note={Theorem 1.1 cited from arXiv version 1 (2023)}, + url={https://arxiv.org/abs/2308.04044v1} +} + +@incollection{liokumovichMaximo2023, + author = {Liokumovich, Yevgeny and Maximo, Davi}, + title = {Waist inequality for 3-manifolds with positive scalar curvature}, + booktitle = {Perspectives in Scalar Curvature}, + volume = {2}, + publisher = {World Scientific}, + pages = {799--831}, + year = {2023}, + eprint = {2012.12478}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + url = {https://arxiv.org/abs/2012.12478v4}, + note = {Theorem 1.1 cited from arXiv:2012.12478v4} +} + +@article{deGiorgi1954, + author = {De Giorgi, Ennio}, + title = {Su una teoria generale della misura {$({r}-1)$}-dimensionale in uno spazio ad {$r$} dimensioni}, + journal = {Annali di Matematica Pura ed Applicata}, + series = {4}, + volume = {36}, + pages = {191--213}, + year = {1954}, + doi = {10.1007/BF02412838}, + url = {https://doi.org/10.1007/BF02412838} +} + +@article{deGiorgi1955, + author = {De Giorgi, Ennio}, + title = {Nuovi teoremi relativi alle misure {$({r}-1)$}-dimensionali in uno spazio ad {$r$} dimensioni}, + journal = {Ricerche di Matematica}, + volume = {4}, + pages = {95--113}, + year = {1955} +} + +@article{federerFleming1960, + author = {Federer, Herbert and Fleming, Wendell H.}, + title = {Normal and Integral Currents}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {72}, + number = {3}, + pages = {458--520}, + year = {1960}, + doi = {10.2307/1970227}, + url = {https://doi.org/10.2307/1970227} +} + +@article{allard1972, + author = {Allard, William K.}, + title = {On the First Variation of a Varifold}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {95}, + number = {3}, + pages = {417--491}, + year = {1972}, + doi = {10.2307/1970868}, + url = {https://doi.org/10.2307/1970868} +} + +@article{simons1968, + author = {Simons, James}, + title = {Minimal varieties in {Riemannian} manifolds}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {88}, + number = {1}, + pages = {62--105}, + year = {1968}, + doi = {10.2307/1970556}, + url = {https://doi.org/10.2307/1970556} +} + +@article{federer1970, + author = {Federer, Herbert}, + title = {The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension}, + journal = {Bulletin of the American Mathematical Society}, + volume = {76}, + number = {4}, + pages = {767--771}, + year = {1970}, + doi = {10.1090/S0002-9904-1970-12542-3}, + url = {https://doi.org/10.1090/S0002-9904-1970-12542-3} +} + +@article{michaelSimon1973, + author = {Michael, J. H. and Simon, L. M.}, + title = {Sobolev and mean-value inequalities on generalized submanifolds of {$R^n$}}, + journal = {Communications on Pure and Applied Mathematics}, + volume = {26}, + number = {3}, + pages = {361--379}, + year = {1973}, + doi = {10.1002/cpa.3160260305}, + url = {https://doi.org/10.1002/cpa.3160260305} +} + +@misc{antonelli2026Volume, + author = {Antonelli, Gioacchino}, + title = {Universal Volume Growth Bounds from Positive Intermediate Curvature}, + year = {2026}, + eprint = {2608.14507}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + version = {2}, + note = {arXiv:2608.14507v2, August 19, 2026}, + doi = {10.48550/arXiv.2608.14507}, + url = {https://arxiv.org/abs/2608.14507v2}, + urldate = {2026-09-23} +} + +@misc{kumarSen2026Width, + author = {Kumar, Aditya and Sen, Balarka}, + title = {{Urysohn} Width and Macroscopic Scalar Curvature}, + year = {2026}, + eprint = {2601.14669}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + version = {2}, + note = {arXiv:2601.14669v2, February 1, 2026}, + doi = {10.48550/arXiv.2601.14669}, + url = {https://arxiv.org/abs/2601.14669v2}, + urldate = {2026-09-23} +} + +@article{nabutovskyRotmanSabourau2021, + author = {Nabutovsky, Alexander and Rotman, Regina and Sabourau, St{\'e}phane}, + title = {Sweepouts of closed {Riemannian} manifolds}, + journal = {Geometric and Functional Analysis}, + volume = {31}, + pages = {721--766}, + year = {2021}, + doi = {10.1007/s00039-021-00575-3}, + eprint = {2007.14954}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + url = {https://arxiv.org/abs/2007.14954v1}, + note = {Definitions 3.1 and 6.1 and equation (6.1) cited from arXiv:2007.14954v1} +} + +@misc{gromov2017Dozen, + author = {Gromov, Misha}, + title = {A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature}, + year = {2017}, + howpublished = {Author-hosted preprint}, + note = {19 September 2017}, + url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/dozen_problems_Sept_2017.pdf} +} + +@misc{openai2026RationalInessentiality, + author = {{OpenAI}}, + title = {{Positive scalar curvature forces rational inessentiality}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026}}, + year = {2026} +} + +@book{agmon1982, + author = {Agmon, Shmuel}, + title = {Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of {N}-Body {Schr{\"o}dinger} Operators}, + series = {Mathematical Notes}, + volume = {29}, + publisher = {Princeton University Press}, + year = {1982}, + url = {https://api.pageplace.de/preview/DT0400.9781400853076_A23704097/preview-9781400853076_A23704097.pdf} +} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/closed-covers.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/closed-covers.tex new file mode 100644 index 0000000000000000000000000000000000000000..1373f337eea89d537db17905a2da29e41514db65 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/closed-covers.tex @@ -0,0 +1,177 @@ +\section{Universal covers and the uniform simplex metric} +\label{sec:closed-covers} + +We use the continuous macroscopic dimension of a metric space $X$: +$\dim_{\mathrm{mc}}X\le d$ means that a continuous map from $X$ to a +simplicial complex of dimension at most $d$, with its uniform simplex +metric, has uniformly bounded entire fibers. Here each simplex is a +standard Euclidean simplex with the same edge length. This is the +continuous convention in \citet[Definitions~1.1--1.2]{dranishnikov2014}. +We first convert the map already constructed to this target metric. +We then supply the two lower dimensions using the classical geodesic +index argument and the width theorem of Liokumovich and Wang. + +\begin{lemma}[Changing the simplex metric without changing fibers] +\label{cover:uniform} +Let $k\ge0$ be an integer. Suppose continuous functions $t_i:Y\to[0,1]$ have locally finite +supports, at least one equals $1$ at each point, and at most $k$ are +strictly between $0$ and $1$. Their coordinate map admits an injective +conversion to a simplicial complex of dimension at most $k$ with its +uniform simplex metric. The converted map is continuous, and two of +its values agree if and only if all the original coordinates agree. +\end{lemma} +\begin{proof} +Use the cubes $Q(S,A)$ and staircase simplices of +\eqref{topology:staircase}, where $S,A$ are disjoint finite sets, +$S\ne\varnothing$, and $|A|\le k$. Realize the abstract simplices +as standard simplices, and send each coordinate simplex to its new +realization with the same barycentric coordinates. The rules agree +on common faces. Equality of two images identifies their minimal +simplex and its positive barycentric coordinates; the affine +combination of the corresponding finite indicator vectors recovers +the same original coordinate vector. Thus the conversion is injective. + +For $k\ge1$, give each simplex edge length $1$ and define $d_K$ +as the infimum of the total Euclidean lengths of finite piecewise +straight paths in the simplices. This infimum is finite: vertices +$\mathbf1_S$ and $\mathbf1_T$ are joined by adding the coordinates +in $T\setminus S$ and then removing those in $S\setminus T$, always +retaining a nonempty set of coordinates equal to $1$. Every point +is joined to a vertex in its simplex. For a vertex $v$, its +barycentric coordinate $b_v$, set to zero on simplices not containing +$v$, agrees on faces and is $\sqrt2$-Lipschitz on each unit-edge +simplex. Indeed such a simplex is realized by the vectors +$e_v/\sqrt2$ in a Euclidean coordinate space. Hence +\[ + |b_v(x)-b_v(y)|\le\sqrt2\,d_K(x,y). +\] +The barycentric coordinates distinguish points, so $d_K$ is a genuine +finite metric. For $k=0$ use the discrete vertex metric. + +Near a point of $Y$, only a finite set $F$ of coordinates can be +nonzero. The coordinate map there lands continuously in the finite +cubical subcomplex with all other coordinates zero. The conversion +is continuous on each of its closed simplices and hence on the finite +complex. Inclusion into the full uniform realization is continuous +on a simplex because distance there is at most the length of its +straight segment. Finite pasting therefore proves continuity on +the entire finite subcomplex, and thus near every point of $Y$. +The same local finite argument applies for $k=0$. No equality between two topologies on an arbitrary +infinite complex is required. +\end{proof} + +\begin{lemma}[Level components of a proper Morse function] +\label{cover:reeb} +Let $u:N\to\R$ be a proper Morse function on a connected smooth +manifold without boundary. There is a locally finite simplicial graph $K$ and a +continuous map $q:N\to K$, with $K$ carrying its uniform simplex +metric, whose point fibers are precisely the connected components +of the levels of $u$. +\end{lemma} +\begin{proof} +Choose an auxiliary smooth Riemannian metric on $N$ to define gradients. +The critical set is closed and its points are isolated. Properness +therefore makes the number of critical points in any compact slab +$u^{-1}([a,b])$ finite. Thus the critical values form a locally finite +subset of $\R$, even if several critical points have the same value. +Every level is compact and locally path connected: regular charts +give a hypersurface, while the Morse normal form at a critical point +gives the zero set of a nondegenerate quadratic form, which is a cone +or an isolated point. A compact locally path connected space has finitely +many components, since its components are open and form an open cover. + +Let $S$ consist of the critical values and the integers. It is +locally finite. At each $a\in S$ put a vertex for every component +of $u^{-1}(a)$. Between consecutive values $a0$ and $X=\widetilde M$. The lifted metric +is complete. If $X$ is compact, use a constant map to a vertex, +whose only nonempty fiber has finite diameter. + +For $n\ge4$, the complete metric $s\widetilde h$ has scalar +curvature at least $1$. Apply \Cref{thm:main} using the coordinates +of \eqref{topology:coordinates}, then \Cref{cover:uniform}. Local +finiteness of their supports was proved in +\Cref{sec:reconstruction}, and the conversion preserves their +entire fibers. Scaling back gives the bound $C_n/\sqrt{s}$ in +the original metric. + +For $n=2$, the Gaussian curvature of the cover is at least $s/2$. +If a unit-speed minimizing geodesic had length $L>\pi\sqrt{2/s}$, +take a parallel unit normal $E$ along it and the endpoint-zero +variation field $V(t)=\sin(\pi t/L)E(t)$. Its index form obeys +\[ + I(V,V)=\int_0^L\bigl(|\nabla_tV|^2-K|V|^2\bigr)\,dt + \le \frac{\pi^2}{2L}-\frac{sL}{4}<0, +\] +contradicting minimality. Completeness supplies minimizing geodesics, +so the cover has diameter at most $\pi\sqrt{2/s}$ and is compact +by Hopf--Rinow. The constant map therefore proves this case. + +Finally let $n=3$ and $X$ be noncompact. Apply +\citet[Theorem~1.1, arXiv version~1]{liokumovichWang2023} to the +complete metric $(s/6)\widetilde h$, whose scalar curvature is at +least $6$. It gives a proper Morse function such that every +connected component of every level, including critical levels, has +ambient diameter less than $18\pi$ in that metric. Apply +\Cref{cover:reeb}. Each whole fiber of its graph map is one such +level component, so in the original lifted metric it has diameter +less than $18\pi\sqrt{6/s}$. This is a continuous map to a +one-dimensional uniform simplicial complex, as required. +\end{proof} + +The low-dimensional conclusions use their stated classical and public +inputs. The quantitative theorem in dimensions at least four has +the separate proof given above. All map statements here are continuous; +changing the simplex metric does not supply a Lipschitz bound. diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/countable-partitions.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/countable-partitions.tex new file mode 100644 index 0000000000000000000000000000000000000000..5d134b7e6ec00ca759c6d69446aee31764c5e8e2 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/countable-partitions.tex @@ -0,0 +1,707 @@ +\section{Countably many compatible partitions}\label{sec:countable} + +The separation widths in \Cref{rounds:wide} may depend on the label. +We now use this freedom to construct the partitions. The analytic +construction is performed at finite stages. Its essential finiteness +property concerns each \emph{whole packet}, including all its infinite +product ends; local finiteness merely on compact subsets would not +suffice. The finite-label procedure is stated in \Cref{finite:cut}, and its detailed +proof is in \Cref{sec:packets,sec:cut,sec:local,sec:shield-global,sec:regularization}. +We adapt that proof using individual positive floors and finite +exceptional data on each whole packet. +This section establishes their hypotheses for the countable construction +and proves that those hypotheses persist. + +\subsection{Initialization and the inductive data} + +Let $D$, the centers $x_i$, and the pairs $(A_i,B_i)$ be as in +\Cref{rounds:wide}. We assume every $B_i$ is nonempty, since otherwise +$\diam_g M\le4D$. There is one label for each center, and an additional +label $0$ used first for radial confinement. The set of labels is finite +or countable; write it as $I=\{0,1,2,\ldots\}$, stopping the enumeration +if necessary. Choose +\begin{equation}\label{countable:floors} + p_i^*>0,\qquad \sum_{i\in I}p_i^*=\frac1{20}. +\end{equation} +Use \Cref{rounds:wide} with +\begin{equation}\label{countable:widths} + L_i=10+2\pi\sqrt{2/p_i^*}\qquad(i\ne0). +\end{equation} +We henceforth use the resulting complete metric $h\ge g$, its locally +finite list of log warps $p_\alpha$, and its stabilized scalar lower +bound $2/5$. + +The initial weighted base measure is globally well defined: +\[ + \omega\,\dd\vol_h,\qquad + \omega=\exp\Bigl(\sum_\alpha p_\alpha\Bigr)>0. +\] +The sum is locally finite. On a buffered compact set, only finitely many +nonunit factors $e^{p_\alpha}$ occur, so ordinary finite warped-product +formulas compute the stabilized scalar, weighted variations, and +invariant test forms there. These formulas agree on overlaps: adjoining +a unit factor changes none of them. In particular, the initial problem +is a weighted problem on the $n$-dimensional base $M$. + +Fix numbers +\[ + 0=t_0T(p_i^*)+3\}. +\] +Their closures lie in the respective pole interiors. The stipulated +width puts $B_i$ strictly inside the negative core. In particular, on +the complement of the original closed ball $\overline B_g(x_i,2D)$, +the profile is the negative pole and the negative core is full. + +There exists a smooth proper function $\rho:M\to\R$, bounded below, +with $|\rho-\dist_h(o,\cdot)|<1$ and $|\grad\rho|_h\le2$. +Here no uniform geometric assumption is necessary. To construct it, +use relatively compact coordinate balls in a locally finite cover, +convolve the distance on slightly larger balls, and blend with a +smooth partition of unity. The metric distortion and local convolution +errors can be made arbitrarily small on each ball. In differentiating +the blend, write each local approximation as the distance plus its +error; the terms involving derivatives of the partition sum to zero +on the common distance. Choose the value errors so that the remaining +partition-derivative terms have sum less than a fixed small number at +each point. A locally finite refinement and successively smaller +positive errors achieve this, while keeping the total value error +below one. Properness follows from the bounded value error and +properness of the complete metric $h$. + +Choose alternating positive and negative plateaus in the $\rho$-axis, +with transition intervals wider than $2T(p_0^*)$. Rescale the preceding +crossing to these intervals, reversing its sign on alternate intervals. +The gradient bound for $\rho$ gives the profile inequality with +coefficient $.5$ and potential $p_0^*$. Choose the core closures +strictly inside the plateaus. Regular values in the remaining margins +give compact smooth barrier annuli, and a compact central barrier, +whose closures stay in the same pole plateaus. Their value bands tend +to infinity and are locally finite. + +A wall respecting the signs cannot cross a separating plateau band; +each component is closed in one compact radius block and hence compact. +A component of either closed side is trapped between two separator +bands of the opposite sign, or in the initial compact block. +Regularity of the defining function gives manifold-with-boundary +charts for these sides. A compact block therefore contains only +finitely many components. The same charts prove local finiteness of +the component families. When $M$ is compact, use a radial label with +constant positive pole and full positive core; its split has no wall +and its only side is compact. + +The scalar requirement initially holds everywhere because +\[ + \lambda_0+b_0\sum_i p_i^*=.25+1.12/20=.306<.4. +\] +Finally a compact block with a compact buffer meets only finitely many +original closed outer balls and finitely many initial warp supports. +Every other center label and warp has the asserted constant data there. +\end{proof} + +\subsection{Finite exceptional data on entire packets} + +A \emph{downward diagram} of a packet consists of it and all its further +faces. An \emph{upward diagram} consists of it and all its cofaces. +Each is finite at a fixed stage, by \Cref{packet:split}. A finite union +of either type is finite. The constructions below make finitely many +directed passes through such diagrams. They never ask for closure +under arbitrary alternating face and coface steps. + +Radial confinement gives the geometric reason to expect finite data on +a whole packet. At the first split, a compact radial side or wall +component is prolonged by cylinders; its entire collapse image stays +in that original compact component. At every later split a child $A$ +of $E$ has +\begin{equation}\label{countable:collapse-recursion} + q_A=q_E\circ p_A,\qquad + p_A=\begin{cases} + \text{inclusion},&\text{on the wall or bulk},\\ + \text{projection to the wall},&\text{on a fresh cylinder}. + \end{cases} +\end{equation} +Hence $q_A(A)\subset q_E(E)$, including in all simultaneous normal +directions. A finite incidence diagram therefore has image in a finite +union of compact original radial blocks. Only finitely many original +outer balls and initial nonunit warp supports meet that image. +Shielding changes no collapse map, and regularization chooses new +walls inside old parents, so both preserve this confinement. + +This argument controls the initial supports. Later smooth data need +not factor through $q_A$: they also use data copied from faces. We +must therefore retain finite exceptional lists through those directed +dependencies, as well as preserve every other entry exactly. + +\begin{definition}[Finite exceptional data]\label{countable:finite-data} +At stage $r\ge1$, on each whole packet $E$ there are finite sets +$I_E\subset J_r$ of label indices and $F_E$ of old warp indices such +that, for $j\notin I_E$, +\begin{equation}\label{countable:passive} + \mu_j\equiv-\infty,\qquad U_j^-=E,\qquad U_j^+=\varnothing, + \qquad \Pi_j\equiv c_rp_j^*. +\end{equation} +For every old warp index outside $F_E$, its length is identically one. +The number $c_r$ is the same on all packets and is prescribed by +\begin{equation}\label{countable:passive-multiplier} + c_0=1,\qquad c_{r+1}=(1+3\eta_r/2)c_r. +\end{equation} +For any finite diagram used in a comparison, take the union of its +finite exceptional sets, adding the current label explicitly. This +one finite union then works simultaneously on all its whole packets. +At stage zero the analogous property is required on every compact +radial block and buffer, with $c_0=1$. +\end{definition} + +The sets $I_E$ need not consist exactly of the nonconstant entries: an +entry may be retained in the exceptional list even if it happens to be +constant. This convention avoids any need to discard previous exceptions. +Only the existence of a finite list matters. + +\begin{lemma}[Consequences and directed dependencies]\label{countable:locality} +Suppose the packet geometry and \Cref{countable:finite-data} hold at a +finite stage. Then the following assertions hold. +\begin{enumerate}[label=(\roman*),leftmargin=*] +\item The sum of potentials on a whole packet is a smooth function +with bounded derivatives of every fixed order. The undecided counts, +required sets, and guards use only a finite list and are closed. +\item On each whole packet and each finite comparison diagram, the old +weight, stabilized scalar, stability forms, and weighted Laplacian are +computed with one finite list of warps, after padding with unit factors. +The bounded geometry and polynomial weighted growth conclusions of +\Cref{packet:geometry} apply, including on any finite Euclidean product. +\item Compatible choices with positive packetwise tolerances can be +made in the following order: fix individual requests, propagate numerical +requests down the finite coface chains, and then construct data upward +from deeper faces. Enlarging a coface's exceptional list by the union +over its downward source diagram suffices for the last pass. No data +are propagated back from a newly constructed coface to an already +constructed face. +\end{enumerate} +\end{lemma} + +\begin{proof} +On $E$, the sum of potentials is exactly +\begin{equation}\label{countable:potential-sum} + \sum_{j\in J_r}\Pi_j + =\sum_{j\in I_E}\Pi_j+c_r\sum_{j\in J_r\setminus I_E}p_j^*. +\end{equation} +The second term is a finite spatial constant. Indeed +\[ + 1\le c_r\le\exp\Bigl(\frac32\sum_{s0$, and $e_0=e_r=\Delta_r/800$. +Its stage margins \eqref{regular:eta} were proved for these general +adjacent parameters, so they do not require equally spaced stages. +The potential calculation also applies: on each finite comparison +diagram, \eqref{countable:potential-sum} expresses the old potential +sum as finitely many smooth functions plus one spatial constant. +Only finitely many shielding additions occur there. Thus, with +$\widetilde\Pi_j=\Pi_j+D_j$, the same raw form and comparison estimates +control +\begin{equation}\label{countable:raw-target} + \lambda_{r+1}+(1+3\eta_r)b_{r+1} + \sum_{j\ne r}\widetilde\Pi_j +\end{equation} +near the new required set, with the positive mass and gradient reserves +proved there. The constant tail has no derivative error; potential +approximation errors concern only the finite exceptional list. At the +radial step these estimates are made on buffered compact blocks, as +in that proof. +The seam and potential smoothing can consequently retain an $e_r$ +local mass gap. This positive number is fixed for the current stage; +all geometric tolerances may depend on the packet. No infimum over +future stages or over all packets is taken. + +\emph{Exact passive constants in regularization.} +Restrictions, bulk inclusions, and cylinder projections preserve +identically constant functions. Seam smoothing of an old identically +unit warp leaves it one. For the remaining potentials, smooth +$(1+3\eta_r/2)\widetilde\Pi_j$ as in +\Cref{regular:profiles,regular:constant-preservation}, so that +\begin{equation}\label{countable:smooth-potentials} + (1+\eta_r)\widetilde\Pi_j\le\Pi_j^{\rm new} + \le(1+2\eta_r)\widetilde\Pi_j. +\end{equation} +On a finite exceptional list, allocate the required relative errors +separately, using $p_j^*>0$ for its $j$th entry. If an aggregate error +is prescribed, subdivide it among that finite list. If an entry is +passive on the whole source diagram, set its output \emph{exactly} to +$c_{r+1}p_j^*$. This is compatible with the smoothing procedure: +local convolution of a constant is that constant, partition weights +sum to one, and every copied end value is the identical constant. +Equivalently this entry can simply be omitted from all smoothing and +restored as that constant afterward. We only impose this rule on +whole constant inputs and their agreeing copies, so it makes no claim +about preserving the edge of an arbitrary nonsmooth plateau. + +Before blending on a coface, enlarge its exceptional list by the union +over its entire finite downward source diagram. Thus a face-active +entry is recorded even if the coface's individual input was passive. +The remaining copies agree identically. Coface requests merely adjust +numeric tolerances; they do not introduce new source values on faces. +This proves exact passive reproduction on the completed diagram and +precludes a cascade through infinitely many same-depth neighbors. +At the radial step choose a compact buffer for each compact radial +block, enlarged to contain the finitely many chart closures and shell +supports used by its approximations. Only finitely many original +outer balls and initial nonunit warp supports meet this buffer. +Enlarge the exceptional list by precisely these indices. Every other +entry is the same constant on every averaging neighborhood and is +therefore reproduced exactly before cylinder attachment. + +The local-slope argument of \Cref{regular:profiles} justifies restriction +of profiles to the wall: an almost-everywhere gradient bound by a +continuous function gives the corresponding local Lipschitz slope +bound at every point of each finite band. Restriction and cylinder +projection then preserve it. The small metric change increases slopes +by at most $1+\eta_r$, and the first inequality in \eqref{regular:eta} +together with \eqref{countable:smooth-potentials} gives the next profile +inequality. +Continuous pullback and core enlargement preserve all pole-neighborhood +and forced-preimage conditions. + +\emph{Corrections, supersolutions, and the new warp group.} +The sum in \eqref{countable:potential-sum} is now smooth, bounded, and +exactly compatible, as are its finite-order derivatives. On every +required finite diagram, pad the old warps by units. In a product end +over $B$ the operator of \Cref{regular:correction} therefore satisfies +exactly +\[ + L_A=L_B-C\Delta_{\rm normal},\qquad + L_A=-C\Delta_{\rm old}+R_{\rm old}-V_A^*+K_A, + \qquad + V_A^*=\lambda_{r+1}+b_{r+1}\sum_{j\ne r}\Pi_j^{\rm new}. +\] +In particular the potential formula does not depend on the base +dimension or on how many unit factors were padded. The correction is +zero near the new required set and produces a positive form gap on +each whole packet. The proof of \Cref{regular:correction} uses finite +minima of the gaps on the packet's face diagram. Its bounded resolvent +and the compatible positive supersolutions in +\Cref{regular:resolvent,regular:solutions} apply with the packetwise +bounded geometry and polynomial weighted volume already proved. +Their tolerances are chosen after the operators and their individual +exact solutions are fixed, and before the final upward blending pass. +Thus their finite-dependency hypothesis is precisely (iii) of +\Cref{countable:locality}. + +Let $v_A>0$ be the resulting supersolution with $L_Av_A\ge1/2$. +Append the common group of $k=399$ circle lengths +$f=v_A^{2/(k+1)}=v_A^{1/200}$. The scalar identity in +\Cref{regular:step} gives +\[ + R_{\rm new}=R_{\rm old}-C\frac{\Delta_{\rm old}v_A}{v_A} + =V_A^*-K_A+\frac{L_Av_A}{v_A} + \ge V_A^*\quad\hbox{on }D_A. +\] +Only the number of \emph{new} factors enters the coefficient $C$. +The old finite list may vary from packet to packet; unit padding has +no effect. Each stage adds just this one finite group, with the same +indices in every status. All its lengths are positive and bounded +above and away from zero on a whole packet, and exactly compatible. +The initial lists and their finite propagated exceptions remain finite, +so the complete old list at stage $r+1$ is still finite on each whole +packet and each finite comparison diagram. + +Finally the regular split preserves compact truncations and finite +incidence by \Cref{packet:split}. Initially the radial separators make +all components compact before prolongation. Compatible smooth metrics +and finite relevant warps then give the complete packet geometry by +\Cref{packet:geometry}. The recursion +\eqref{countable:collapse-recursion} preserves confinement in original +radial blocks. All exceptional-list enlargements just described are +finite, and all passive constants have multiplier $c_{r+1}$. +This completes the induction step, including the first radial step. +\end{proof} + +\subsection{Termination in dimension and finite-stage walls} + +\begin{lemma}[The one-dimensional obstruction]\label{countable:dimension-one} +No nonempty one-dimensional packet can satisfy the invariant at a +finite stage of the construction. +\end{lemma} + +\begin{proof} +Such a packet is complete by \Cref{packet:geometry} and has only +finitely many relevant positive smooth warps $f_1,\ldots,f_q$ on +its whole base. Its arclength universal cover is $\R$, including +when the packet itself is a circle. On this line put +$u_\alpha=\log f_\alpha$ and $P=\sum_{\alpha=1}^q u_\alpha$. +The warped-product scalar formula reduces to +\[ + R=-2P''-(P')^2-\sum_{\alpha=1}^q(u_\alpha')^2. +\] +The required count in dimension one is zero, so the invariant holds +everywhere and gives $R\ge\lambda_r>0$. Consequently +\[ + -2P''\ge\lambda_r+(P')^2,\qquad + \frac{\dd}{\dd s}\arctan\frac{P'}{\sqrt{\lambda_r}} + \le-\frac{\sqrt{\lambda_r}}2. +\] +Integration over a line interval longer than +$2\pi/\sqrt{\lambda_r}$ contradicts the range of the arctangent. +If $q=0$, the scalar formula is simply $R=0$. The argument takes +place at one finite stage and uses an ordinary finite sum. +\end{proof} + +\begin{proposition}[The countable compatible construction]\label{countable:construction} +All labels can be processed successively. At every finite stage the +packet diagram has the invariant above and no nonempty packet of +dimension at most one. After the radial stage it has finite exceptional +data on whole packets; at stage zero this assertion holds on buffered +compact radial blocks. +Its finite reconstruction $X_r$ has a proper collapse $q_r:X_r\to M$ +and closed signed sides $P_i^{(r)},N_i^{(r)}$ for each processed center +label, with +\[ + P_i^{(r)}\cup N_i^{(r)}=X_r,\qquad + P_i^{(r)}\cap N_i^{(r)}=W_i^{(r)}, +\] +where $W_i^{(r)}$ is its wall. The forced preimages satisfy +\[ + q_r^{-1}(A_i)\subset P_i^{(r)}\setminus W_i^{(r)},\qquad + q_r^{-1}(B_i)\subset N_i^{(r)}\setminus W_i^{(r)}. +\] +These inclusions hold in open phases and persist at every subsequent +finite stage. No $n-1$ distinct walls, even including the radial wall, +have a common point. Each center wall lies over its original open +outer ball, and its components are compact and locally finite. + +Old horizons can be enlarged by identifications preserving collapse, +all walls, and all signed statuses. For any prescribed sequence +$\eps_r>0$ with $\sum_r\eps_r<1$, the next finite collar compression +can be chosen with collapse displacement less than $\eps_r$, preserving +the previously decided wall strata and signed sides. These are +finite-stage conclusions; the limiting fixed-copy construction is +made in the next section. +\end{proposition} + +\begin{proof} +Initialization is \Cref{countable:initial}. If the construction has +reached stage $r$, its parents have dimension at least two: a first +nonempty output of smaller dimension would have dimension exactly +one, since a hit decreases dimension by one and a sign preserves it. +\Cref{countable:step} produces the next invariant, and +\Cref{countable:dimension-one} excludes that first output. Thus the +induction continues through every finite stage. There is no passage +to an infinite-stage analytic metric, weight, or torus. + +Apply \Cref{packet:reconstruction} to each finite initial segment. +Its hypotheses are the regular compatible splits and the compact +radial components already established. Its status pieces are exactly +the corresponding compact packet truncations. If $n-1$ walls met, +assign those labels hit status and choose a closed sign status for +each other processed label containing the point. The corresponding +nonempty truncation would have dimension one, a contradiction. +Forced signs persist under the continuous collapse recursion and +enlargement of the cores. In particular +\[ + q_r(W_i^{(r)})\subset M\setminus(A_i\cup B_i) + \subset B_g(x_i,2D). +\] +Properness of $q_r$ and compactness of the original closed outer ball +make each wall component compact; regular local wall charts give +local finiteness, as in \Cref{packet:reconstruction}. + +The horizon identifications in \Cref{packet:reconstruction} preserve +collapse, walls, and signed statuses. Its collar compressions preserve +the old strata and their sides, and \Cref{packet:small-displacement} +allows their collapse oscillations to be bounded by the prescribed +$\eps_r$. These choices are made separately on the locally finite +compact component family; no uniform collar width or horizon at +infinity is required. The next section chooses these finite-stage +homeomorphisms coherently on one fixed copy of $M$. +\end{proof} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/finite-cuts.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/finite-cuts.tex new file mode 100644 index 0000000000000000000000000000000000000000..63b32e98e88f40aaddd031f08ab0360a7513143b --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/finite-cuts.tex @@ -0,0 +1,353 @@ +\section{Finite cuts with compatible ends}\label{sec:finite-cuts} + +The stretching construction supplies separated sets and a scalar-curvature +reserve. We now describe the second input: a cutting procedure that retains +both sides of every cut and controls how later cuts meet them. Its output is +a family of smooth walls and complete manifolds obtained by attaching +cylinders to the closed sides. The definitions below make its hypotheses +and output explicit; Appendices~\ref{sec:packets}--\ref{sec:regularization} +give the proofs. Section~\ref{sec:countable} will separately justify its use +with countably many labels. + +Let $X=M$ denote the original complete manifold, let $g$ be its original +metric, and write $d_0=d_g$ for its distance. The metric used in an analytic +packet can change; the map to $X$ always records its position in the +original manifold. Fix finitely many of the supplied ball pairs +\[ + A_i=\overline B_{d_0}(x_i,D),\qquad + B_i=X\setminus B_{d_0}(x_i,2D),\qquad 1\le i\le\ell, +\] +with $A_i,B_i$ nonempty. There is one label for each ball pair; these +labels are also called \emph{colors}. An additional first label is radial. +Thus $N_*=\ell+1$, and the order is +radial, then $1,\ldots,\ell$. Properness of $(X,g)$ makes each closed +outer ball compact. No covering-space or quotient hypothesis is used. + +The analytic input at stage zero is a complete smooth metric $h\ge g$ on $X$ +and finitely many smooth positive circle warps for which the stabilized +scalar curvature is at least $.4$. Each prescribed distance +$\dist_h(A_i,B_i)$ can be required to exceed any given finite number. +\cref{rounds:wide} supplies these data. For finitely many requested +pairs its construction uses only finitely many compactly supported +increments and hence finitely many circle factors. The countable +version has a locally finite list of nonunit warps and is used in +\cref{countable:construction}; the geometric statements below apply to +each finite collection of the relevant factors. No uniform geometry +bounds on $h$, or uniform packetwise constants, are assumed. + +For a single smooth cut, let $u$ be a smooth function on an +$m$-manifold $E$ without boundary, with $0$ a regular value, and put +$L=u^{-1}(0)$. +Attach $L\times[0,\infty)$ to each closed side $\{\pm u\ge0\}$ +along its boundary, and also retain $L$ as a separate space. +The two extended sides have dimension $m$, while the wall has dimension +$m-1$. The cylinders replace the new boundaries by ends. +The following definition records these spaces and their incidences +after repeated cuts. + +\begin{definition}[Packet diagram]\label{packet:category} +At stage $r$, a status is $\alpha\in\{+,-,*\}^{r}$. +Its entries record the first $r$ processed cuts. +A sign $+$ or $-$ selects the corresponding retained side of a cut, +and a star $*$ selects the wall, called a \emph{hit}. +The space $E_\alpha$ is a smooth second countable manifold without boundary, +possibly empty or disconnected; when nonempty its base dimension is +$m(\alpha)=n-\#\{i:\alpha_i=*\}$. Its connected components are called +\emph{packets}. Replacing signs by stars gives the face order. +If $B$ is a packet with $d$ additional stars relative to a coarser status, +then for that coarser status it has exactly one incident packet $E$ and +one corresponding occurrence of a full product tail +\begin{equation}\label{packet:tail} + \iota_{BE}:B\times(T_1,\infty)\times\cdots\times(T_d,\infty) + \longrightarrow E. +\end{equation} +The normal coordinates are labelled by the changed status entries. +Thresholds may depend on the incidence. For a single entry, tails +from distinct face components are disjoint. Intersections of sufficiently +far tails for different entries are exactly the tails over their common +further faces. The identifications commute: composing face maps in +any order gives the same occurrence and the same individually labelled +normal coordinates. After slightly raising the entry thresholds, +each closed tail is properly embedded. + +Each packet $E$ is equipped with a continuous map $q_E:E\to X$. +After the radial stage we require the following additional properties. +Every packet has only finitely many incident faces and cofaces, including +through chains; the complement of its sufficiently far elementary tails +is compact. On each sufficiently far face product these maps satisfy +$q_E(\iota_{BE}(b,\mathbf t))=q_B(b)$. +A metric, scalar function, warp, or set is \emph{compatible} if, in every +sufficiently far tail, it is exactly the product or pullback of the +corresponding face datum. For metrics the normal factors are +$\dd t_1^2+\cdots+\dd t_d^2$; all warps are independent of the normal +coordinates there. The thresholds for different data may differ. +\end{definition} + +There is no boundary in an analytic packet. Its face is represented by +an entire cylindrical end, not by a boundary subset. Finite boundaries +appear only when we truncate these ends. + +A \emph{regular split} applies the preceding construction using +compatible smooth defining functions whose zero sets are regular on +every status space. Each zero set is the new hit space. +In inherited ends the operation is the corresponding face operation +times the old normal factors. The proof that these operations preserve +the packet diagram is \cref{packet:split}. + +For the diagrams constructed here, these maps are defined recursively, +starting with $q_X=\Id_X$. In the split of \cref{packet:split}, let +$A$ be a new stage-$(r+1)$ packet with stage-$r$ parent $E$. +Define $p_A:A\to E$ by inclusion on a hit space or the bulk of a joined +side, and by $p_A(x,t)=x$ on each newly attached cylinder. +Set $q_A=q_E\circ p_A$. Thus $q_A$ is pulled back from the base on +each fresh cylinder. In a sufficiently far inherited tail of $A$, +let $C$ be the corresponding new face with old parent $B$, and let +$I$ denote the old normal intervals. Under the identifications of +these tails with $C\times I$ and $B\times I$, the map $p_A$ is +$p_C\times\Id_I$; explicitly, +\[ + p_A\circ\iota_{CA}=\iota_{BE}\circ(p_C\times\Id_I),\qquad + q_A\bigl(\iota_{CA}(c,\mathbf t)\bigr)=q_C(c). +\] +Thus the recursion preserves compatibility. The proper maps $q_r$ +on finite reconstructions are constructed in +\cref{packet:reconstruction}. + +Figure~\ref{fig:regular-packet-split} depicts one extended side and the +separate hit space. +\input{figures/packet-split} + +The label set and the status depth are finite in this section. The family +of connected packets may be countable; finiteness is required only for +the directed face and coface chains incident to each packet. The +downward diagram of a packet consists of that packet and all its further +faces. Numerical +requests are allocated downward through these chains before actual data +are constructed upward from faces. This does not require finite +components of the undirected incidence graph. + +We now specify the analytic data that must survive each split. +For the finite-label statements in this section and the appendices we set $\sigma=1$ +and use the following constants: +\begin{equation}\label{packet:constants} +\begin{gathered} + \lambda_r=\frac14-\frac{r}{8N_*},\qquad + b_r=1.12-\frac{.05r}{N_*},\qquad + a_r=.60+\frac{.03r}{N_*}\quad(0\le r\le N_*),\\ + \pi_0=\frac{1}{10N_*},\qquad k=399,\qquad + C=\frac{4k}{k+1}=3.99,\qquad + e_0=\frac{\lambda_r-\lambda_{r+1}}{100} + =\frac{1}{800N_*}\quad(rT_0+4,\qquad + T_0=2\pi\sqrt{2/\pi_0}. +\] +Then the data of \cref{packet:invariant} exist at stage zero with +$\Pi_j=\pi_0$. On a noncompact $X$ the radial cores can be chosen +as alternating separating subbands of a smooth proper function, +with compact smooth barrier domains around their closures lying in +the same pole plateaus. Every regular radial partition respecting +these signs has locally finite compact wall components and compact +closed-side components before collar attachment. On a compact $X$ +a decided radial label with empty wall has the same properties. +\end{lemma} + + +\begin{theorem}[Finite compatible cutting]\label{finite:cut} +Fix a finite number $N_*$ of labels and the stage constants in +\eqref{packet:constants}. Suppose a stage-$r$ packet diagram is obtained by +the regular splitting construction of \cref{packet:split}, with compatible +smooth base metrics and a finite common list of positive circle warps, +and satisfies \cref{packet:invariant}. After the radial stage, require the +complete compact-part/product-end geometry of \cref{packet:category,packet:geometry}. +Let $j_0$ be the next label, and suppose every nonempty parent packet has +base dimension at least two. Then $j_0$ can be processed as follows. + +On every parent there is a compatible smooth regular defining function +with the prescribed signs on the current forced core closures. Keep its +zero wall as the new hit space. To each closed sign side attach the full +product cylinder over its boundary. These pieces form a compatible smooth +stage-$(r+1)$ packet diagram with the continuous projections described +above. The wall on a parent $E$ lies in $\operatorname{int}G_{j_0}$. +The remaining profiles may first be modified, their cores enlarged, +and their potentials increased in the parents; these data are then carried to the new pieces by restriction and cylinder projection. +There are compatible smooth metrics and potentials, and exactly $399$ +new positive circle factors on each new packet, for which the entire +stage-$(r+1)$ invariant holds. Earlier forced signs are preserved. +The new packets have complete compact-part/product-end geometry. This +assertion includes a possible one-dimensional output, on which its scalar +lower bound holds everywhere. + +The operation fixes every old warp identically equal to $1$. For a +remaining label that is passive on the finite downward source diagrams +and the face copies used in their comparisons---constant pole profile, full corresponding core, and spatially +constant potential $c_j$ agreeing on compared faces---the construction +can preserve the pole and core and choose its new potential to be exactly +$(1+3\eta_r/2)c_j$, where $\eta_r=1/(200N_*)$. + +At stage zero the single parent may instead be an arbitrary complete +smooth manifold with no product-end or bounded-geometry assumption. Subject to the same scalar and separation assumptions, if its +data are the radial initialization of \cref{packet:initialization}, the +same conclusion holds for the first split. Every resulting wall component and every closed-side component is compact before the cylinders are attached, and these components form locally finite +families. +\end{theorem} + +\begin{proof} +Realize the processed profile by a smooth finite pressure and take a +local absolute minimizer of weighted perimeter minus its pressure-volume +term, as in \cref{cut:pressure,cut:bv-compactness}. The phase has the +forced signs of \cref{cut:barriers}; its boundary, called the true cut, +has the regular-part geometry of \cref{cut:gmt} but may still have +singularities. After the +radial cut, the generic pressure choice in +\cref{cut:generic-uniqueness} makes these true cuts converge to the +corresponding face products at packet ends +(\cref{cut:end-convergence}). On the initial radial parent use shift +zero and a phase from compactness; compact barriers replace this +post-radial growth and uniqueness argument. + +The wall, and each retained side with its attached cylinder, +carry the quadratic-form lower bounds of \cref{cut:raw-domination}. +These bounds include a positive gradient reserve and, on a joined +side, the scalar contribution of its internal attachment seam. +They are not yet pointwise scalar bounds on smooth new packets. +Keep these true cuts and their pressures fixed. The constructions of +\Cref{local:anchors,local:sharp,shield:global} change only the remaining +profiles, enlarge their cores, and add nonnegative potentials. Their +form cost is paid by the gradient reserve and the gap +$\lambda_r-\lambda_{r+1}$, and the updated guard $H_E^*$ of +\eqref{packet:guard} misses the singular set. Thus the future required +regions can be controlled through regular portions of the true cut. +When the parent dimension is at most seven the +singular set is empty, and no singular forcing is needed; compactness on +the resulting smooth windows is the smooth case of +\cref{shield:compactness}. More generally this smooth-window case applies +whenever the limiting wall is smooth, including products of smooth faces +with Euclidean factors even when their total dimension is at least eight. +At the first radial split the same arguments +are applied on the locally finite compact transition blocks, as specified +in \cref{shield:global}. + +The guarded approximation in \cref{regular:partition} now replaces +the true cut by a smooth regular wall. Its defining functions are +exact products in the old ends. The comparison with the true cut on +the guard, followed by the seam smoothing of \cref{regular:seam}, +transfers the form bound to the smooth split. The positive +supersolutions of \cref{regular:correction,regular:solutions} +then supply $399$ compatible new circle factors; the scalar identity +in \cref{regular:step} converts that form bound into the pointwise +stage-$(r+1)$ scalar inequality. Throughout this transfer the parent +scalar inequality is used only over $D_E$: \cref{packet:counts} +places every new required region inside that old region as well as +inside the guard. + +\Cref{packet:split,packet:geometry} give the +new geometry; \cref{packet:initialization} gives the radial confinement. +Unit entries are fixed by restriction, product extension, and seam +smoothing. For a remaining label passive in the sense stated above, +we omit its forcing updates, and +\cref{regular:constant-preservation,regular:profiles} reproduce its chosen +constant potential exactly. The choice of $\eta_r$ satisfies the strict +stage margins in \eqref{regular:eta}. +\end{proof} + +This theorem is a finite-label statement. Its common positive floor and finite +list of warp factors do not provide a common floor or a global finite +list for the construction that follows. The next section proves the +stronger locality property actually needed there: on each whole packet +and each directed comparison diagram, only finitely many labels and warp +factors are nonconstant, and the remaining potential sum is a spatial +constant reproduced exactly by the construction. diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/introduction.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..d5533c67aaa135bb41f06467ecde2dd1eabc7725 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,315 @@ +\section{Introduction}\label{sec:introduction} + +A lower bound for scalar curvature is a local condition. Gromov's +macroscopic dimension conjecture predicts a global consequence: at a +controlled scale, two dimensions can be collapsed. We prove the +quantitative form of this assertion for arbitrary complete manifolds +in dimensions at least four. + +Throughout, scalar curvature means the ordinary pointwise Riemannian +invariant. Fixed-scale macroscopic scalar curvature uses ball-volume +comparison at a prescribed positive radius +\citep[Definitions~1.2--1.3]{kumarSen2026Width}. + +For a metric space \(X\) and an integer \(d\ge0\), define its +\emph{Urysohn \(d\)-width} by +\[ + \UW_d(X)=\inf_f\ \sup_{y\in K}\diam_X f^{-1}(y), +\] +where the infimum runs over continuous maps \(f:X\to K\) to simplicial +complexes of dimension at most \(d\). The complexes have their usual +polyhedral topology. The diameter is that of the entire fiber, including +all its components; empty fibers may be omitted. + +\begin{theorem}\label{thm:main} +For every integer \(n\ge4\), there is a finite constant \(C_n\) such +that every connected complete smooth Riemannian \(n\)-manifold +\((M,g)\), without boundary and with \(\Scal_g\ge1\), admits a +continuous map \(f:M\to K\) to a simplicial complex of dimension at +most \(n-2\) satisfying +\[ + \diam_g f^{-1}(y)\le C_n\qquad(y\in K). +\] +In particular, \(\UW_{n-2}(M,g)\le C_n\). +\end{theorem} + +The constant is independent of \(M\), its topology, and its geometry. +The theorem applies to compact and noncompact manifolds, with no +orientability, spin, covering, or bounded geometry assumption. If +\(\Scal_g\ge\sigma^2>0\), applying the theorem to \(\sigma^2g\) gives +\(\UW_{n-2}(M,g)\le C_n/\sigma\). + +For a closed connected Riemannian manifold, let +\(\operatorname{FillRad}(M,g)\) be Gromov's filling radius +\citep{gromov1983Filling}, defined by the Kuratowski embedding and the +fundamental class, with coefficients in +\(\Z\) if \(M\) is orientable and in \(\Z_2\) otherwise +\citep[Definition~3.1]{nabutovskyRotmanSabourau2021}. +The next corollary gives the closed-manifold case in dimensions at least +four of Gromov's filling-radius Conjecture~\(A^{-}\) +\citep[\S3]{gromov2017Dozen}. + +\begin{corollary}[Closed-manifold filling-radius bound] +\label{cor:filling-radius} +Let \((M,g)\) be a closed connected smooth Riemannian \(n\)-manifold +with \(n\ge4\) and \(\Scal_g\ge\sigma^2\) for some \(\sigma>0\). Then +\[ + \operatorname{FillRad}(M,g)\le \frac{C_n}{2\sigma}, +\] +where \(C_n\) is the constant in Theorem~\ref{thm:main}. +\end{corollary} + +\begin{proof} +Gromov's comparison, in the form recorded by Nabutovsky, Rotman, and +Sabourau \citep[Definition~6.1 and equation~(6.1)]{nabutovskyRotmanSabourau2021}, +uses the same continuous-map and entire-fiber width convention as above. +A target of dimension at most \(n-2\) is also admissible for +\(\UW_{n-1}\). Since \(M\) is closed, the comparison and the scaled +form of Theorem~\ref{thm:main} give +\[ + \operatorname{FillRad}(M,g) + \le \tfrac12\UW_{n-1}(M,g) + \le \tfrac12\UW_{n-2}(M,g) + \le \frac{C_n}{2\sigma}. +\] +\end{proof} + +For the closed-cover consequence we also use the \emph{continuous +macroscopic dimension}: $\dim_{\mathrm{mc}}X\le d$ means that there is a +continuous map to a simplicial complex of dimension at most $d$, with its uniform simplex metric +and a finite uniform bound on the diameters of its entire fibers. +Each simplex has the same standard Euclidean shape and edge length. + +\begin{corollary}[Continuous macroscopic dimension of universal covers] +\label{cover:main} +Let $(M,h)$ be a closed connected smooth Riemannian manifold of +dimension $n\ge2$ with positive scalar curvature. Its universal cover, +with the lifted original metric $\widetilde h$, satisfies +\[ + \dim_{\mathrm{mc}}(\widetilde M,\widetilde h)\le n-2. +\] +\end{corollary} + +\begin{corollary}[Aspherical obstruction]\label{cor:aspherical} +No closed smooth aspherical manifold of dimension \(n\ge4\) admits +a Riemannian metric of strictly positive scalar curvature. +\end{corollary} + +\begin{proof} +It suffices to consider a connected manifold; passing to its orientation +double cover if necessary preserves the hypotheses. Suppose such a metric +exists. Its scalar curvature has a positive minimum, so the lifted metric +on the contractible universal cover \(X\) is complete with a uniform +positive scalar-curvature lower bound. The deck group acts isometrically +and cocompactly. The scaled form of Theorem~\ref{thm:main} gives a +continuous map from \(X\) to an \((n-2)\)-complex with uniformly bounded +entire fibers. The width and filling-radius implications in Gromov's +noncompact argument \(A^+\Rightarrow B\) +\citep[\S3, pp.~6--8]{gromov2017Dozen} exclude such a map on a +contractible universal cover with a cocompact deck action. This gives +the contradiction. +\end{proof} + +An independent proof in every dimension follows from the stronger +rational-inessentiality theorem of +\citet[Theorem~1.1 and Corollary~1.2]{openai2026RationalInessentiality}. + +\subsection{History and significance} + +Gromov introduced macroscopic dimension in his study of positive scalar +curvature \citep[\S2]{gromov1996}. His quantitative conjecture already asks +whether a complete manifold with a positive scalar-curvature lower bound +can be mapped continuously to a polyhedron of codimension two with all +point inverses bounded at the curvature scale +\citep[\S\(2\frac12\)]{gromov1996}; see also +\citep[p.~3180, Comments after Conjecture~3.2]{gromov2019Mean}. +The diameter condition concerns each entire fiber, including all its +components. In the universal cover of a closed manifold the corresponding +macroscopic-dimension assertion asks only for some finite bound. +The continuous invariant and the version requiring a Lipschitz map to a +uniformly metrized complex are distinct +\citep[Definitions~1.1--1.2]{dranishnikov2014}. + +The pointwise curvature hypothesis cannot be replaced by its fixed-scale +macroscopic counterpart. Kumar and Sen construct closed manifolds in +every dimension at least four with a prescribed positive fixed-scale +macroscopic scalar-curvature lower bound and arbitrarily large +Urysohn codimension-two width \citep[Theorem~A]{kumarSen2026Width}. + +The low-dimensional cases have a substantial earlier history. +Gromov and Lawson obtained geometric width bounds for closed three-manifolds +\citep[Corollary~10.11]{gromovLawson1983}, and Bolotov proved that the +universal cover of a closed three-manifold cannot have macroscopic +dimension two \citep{bolotov2003}. +For closed three-manifolds with positive scalar curvature, Liokumovich +and Maximo constructed Morse functions with curvature-scale bounds for +the areas and ambient diameters of connected level components +\citep[Theorem~1.1]{liokumovichMaximo2023}. +Liokumovich and Wang constructed proper Morse functions on +complete noncompact three-manifolds with uniformly positive scalar +curvature, with uniform diameter bounds for connected level components +\citep[Theorem~1.1]{liokumovichWang2023}. +We use that result and an explicit passage to the Reeb graph for the +three-dimensional closed-cover consequence; the surface case follows +from the Bonnet--Myers argument. + +The aspherical obstruction in dimensions four and five was established +by Chodosh and Li \citep[Theorem~2]{chodoshLi}, with an independent +proof in dimension five by Gromov +\citep[\S7, Main Corollary]{gromov2020Aspherical}. + +For the macroscopic-dimension conjecture, Bolotov proved the closed spin +case under the Strong Novikov Conjecture and a cohomological-dimension condition +\citep{bolotov2009}. Bolotov and Dranishnikov then obtained the +codimension-two conclusion for closed spin manifolds under the +Strong Novikov Conjecture and +injectivity of the connective-to-periodic real K-homology map +\citep[Theorem~4.3]{bolotovDranishnikov2010}. +For closed orientable totally non-spin manifolds of dimension at least +five, they also showed that a macroscopic dimension drop by one implies +a drop by two \citep[Theorem~3.4]{bolotovDranishnikov2016}. +Daher and Dranishnikov established the analogous implication for closed +orientable totally non-spin four-manifolds with residually finite +fundamental group +\citep[Theorem~3.3]{daherDranishnikov2019}. +These topological advances concern universal covers and finite bounds; +the present quantitative statement is formulated for arbitrary complete +manifolds with a bound depending only on dimension after scalar normalization. + + +The geometric method uses the minimal-hypersurface approach of Schoen +and Yau, in which stability and the Gauss equation pass scalar-curvature +information to hypersurfaces \citep[Section~1]{schoenYau1979Structure}, +and the torical symmetrization mechanism described by Gromov +\citep[Section~2]{gromov2018Metric}. Gromov's \(\mu\)-bubble method +uses prescribed mean curvature to localize the hypersurfaces entering +scalar-curvature descent \citep[\S\(5\frac56\)]{gromov1996}. +The weighted variational formulation and its relation to warped-circle +geometry and stability are developed by Chodosh and Li +\citep[Section~3 and Remark~11]{chodoshLi}. + +Removing singular regions while retaining scalar-curvature information +is also central to Lohkamp's splitting theory +\citep[\S1.2 and Theorem~3.15]{lohkamp2022SplittingsII}. +In the construction below, profile updates exclude singular points from +the regions where the next scalar bound is required; the wall and both +retained sides are then regularized together. + +Recent results also address quantitative width and singular descent. +Antonelli proves width bounds under a positive scalar-curvature lower +bound, an almost-nonnegative Ricci lower bound, and uniform noncollapse, +with constants depending on the volume lower bound +\citep[Theorem~1.4]{antonelli2026Volume}. +Bi and Zhu develop singular scalar-curvature descent for enlargeability +obstructions and cubical-width inequalities +\citep[Theorem~B and Section~4]{biZhu2026}. +The present construction retains both bulk sides and the wall through +countably many cuts, with compatible product ends. + +\subsection{The proof and its main constructions} + +Fix an original length \(D\), depending only on \(n\), and a separated +net \((x_i)\) in \((M,g)\). The inner balls of radius \(D\) cover \(M\), +and the closed outer balls of radius \(2D\) are locally finite. Each +center has its own label. We construct a continuous collapse map +$q:M\to M$ with $d_g(q(x),x)<1$, together with compatible walls +separating the inverse images of each inner ball and the exterior of +its outer ball. The map $q$ records original positions while cylinders +are inserted along the cuts. The final +walls have no common intersection of \(n-1\) distinct labels. Continuous +sign coordinates $t_i:M\to[0,1]$ are chosen to equal $1$ on the +$q$-preimage of the inner ball and $0$ on the $q$-preimage of the exterior +of its outer ball. Their transition regions lie in wall neighborhoods +chosen simultaneously to preserve the intersection bound. They give the +desired map: at most \(n-2\) coordinates +are fractional at a point, and some coordinate is one. Equality of all +coordinates confines both collapse images to the same original outer +ball, giving a bound for the entire fiber. + +Three points are essential to make this construction uniform and to +allow arbitrary complete metrics. + +First, the separations must become as wide as prescribed without +spending curvature in proportion to the number of labels. For a metric +\(h\) and a smooth function \(t\), put +\begin{equation}\label{intro:drift-scalar} + \mathcal D(h,t)=\Scal_h+2\Delta_h t-|\dd t|_h^2. +\end{equation} +Fix a background metric \(h_0\) throughout a round. The local stretching +estimate changes \(h\) monotonically and increases \(t\), while paying +for its loss with a decrease in \(\tr_h h_0\). +The pressure law in \Cref{pressure:law} provides a quadratic budget in +\(t\). This makes the initial scale depend only on \(n\), even when +there is no bound on the overlap multiplicity of the original balls. +Trace losses telescope over a countable locally finite round. Each +label stops after its own finite number of rounds, so the entire +construction stabilizes on every compact set +(\Cref{rounds:wide}). Diagonal circle warps convert the drift scalar +bound to an ordinary scalar-curvature inequality on every required +finite product. + +Second, a cut produces both extended bulk sides and a wall of one lower +base dimension. We keep all these pieces, called \emph{packets}, with +compatible product collars. Each label has two open \emph{forced cores}, +one of each sign, on which that sign must be retained. A label is +\emph{decided} at a point when the point lies in one of its cores. +On an \(m\)-dimensional packet, positive +scalar curvature is required only where at least \(m-1\) remaining +labels are outside their forced cores. Minimizing walls may be +singular in high dimensions. Near the dangerous singular region, +additional pressure profiles make the remaining labels decided. +Hardy estimates, compactness across the bulk--cylinder seam, and a +maximum estimate for concentrating potentials pay for this shielding. +A guarded regularization and positive supersolutions then restore +smooth scalar-curvature control. The full finite-diagram arguments are +proved in \Cref{sec:packets,sec:cut,sec:local,sec:shield-global,sec:regularization}. +They include the stability and variation computations used here; +classical elliptic and geometric-measure-theory results are cited with +their hypotheses at their points of use. + +Third, a countable list needs more than finiteness on compact subsets. +After a first radial cut, only finitely many labels and warp factors +are nonconstant on any \emph{whole} packet and on each finite incident +diagram needed by a step. All other labels are already forced; their +potentials form a summable list of spatial constants. Approximations +reproduce those constants exactly. This invariant permits the finite +packet arguments to be iterated with successively smaller positive +budgets, without a common positive floor for all labels. An intersection +of \(n-1\) distinct walls would produce a complete one-dimensional +packet. The positive scalar-curvature invariant on its finite warped +product yields a Riccati inequality that cannot hold on the complete +arclength line (\Cref{countable:dimension-one}), giving the required +wall-intersection bound. + +Finally, the analytic cylinders are used only through finite +truncations in the topological reconstruction. Successive collar +compressions are supported over the corresponding original outer +balls. On one fixed copy of \(M\), the collapse maps and the old wall +statuses are therefore eventually unchanged near every point. This +local stabilization, together with a locally finite thickening of the +walls, gives continuity in the ordinary polyhedral topology. The +resulting constant is +\[ + C_n=4D+2, +\] +where the original scale \(D\) is chosen in terms of \(n\) before any +metric-dependent analytic estimates. The construction does not aim to optimize this constant. + +The proof follows two separate inputs. Sections~\ref{pressure:section}--\ref{sec:rounds} +construct the geometric stretching and its scalar reserve. +Section~\ref{sec:finite-cuts} states the finite-label cutting procedure, +whose complete proof occupies Appendices~\ref{sec:packets}--\ref{sec:regularization}. +Section~\ref{sec:countable} proves its countable-label adaptation, and +Section~\ref{sec:reconstruction} constructs the width map. +Section~\ref{sec:closed-covers} verifies the uniform simplex metric and +proves the closed-cover consequence in dimensions two and three. + +\begin{convention}\label{conv:signs} +The Laplacian is \(\Delta=\Div\grad\), and round spheres have positive +scalar curvature. For a hypersurface with unit normal \(\nu\), its +second fundamental form is +\(B(v,w)=\langle\nabla_v\nu,w\rangle\), and its mean curvature is +\(\tr B\). Integration and divergence use Riemannian densities, so no +orientation is needed. Distances in the final fiber bound are always +measured in the original metric \(g\). +\end{convention} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/local-step.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/local-step.tex new file mode 100644 index 0000000000000000000000000000000000000000..3a11553876dfc0df0f487a445508858e91f98455 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/local-step.tex @@ -0,0 +1,766 @@ +\section{A compactly supported stretching step} +\label{step:section} + +For a metric $h$ and a smooth function $t$, put +\[ + \mathcal D(h,t)=\Scal_h+2\lap_h t-|\dd t|_h^2. +\] +The next theorem enlarges the distance from a compact set while preserving +a quadratic reserve in this quantity. Its error has two parts: a trace +decrement, which telescopes under successive metric enlargements, and an +arbitrarily small quadratic loss. + +\begin{theorem}[Local stretching]\label{step:main} +Fix $n\ge2$ and $0<\delta<1/10$, and put $L_*=1.004$. There are +constants $W,T\ge1$, depending only on $n,\delta$, with the following +property. Let $h_0$ be a complete smooth metric on a manifold $M^n$ +without boundary, let $h\ge h_0$ be smooth, and let $t\ge T$ be smooth. +Let $A$ be a nonempty compact set and $B$ a nonempty closed set such that +\[ + \dist_{h_0}(A,B)\ge W. +\] +For every $00$ also put +\begin{equation}\label{step:chi} + \chi=\left(1+\frac{a^2s_0^2}{\theta}\right)^{1/2},\qquad + m=\frac{\dd\log\chi}{\dd\log\sqrt\theta} + =-\frac{a^2s_0^2}{\theta+a^2s_0^2}. +\end{equation} +Here $\theta$ is strictly increasing. The following bounds will be used +repeatedly: +\begin{equation}\label{step:switch-bounds} +\begin{gathered} + 0\le -s\beta_s\le2,\qquad -1\le m\le0,\qquad + 1\le\chi\le2\quad(s\ge s_0),\\ + m\le-\tfrac12,\quad \chi\ge s_0/s,\quad + \theta\chi^2\le2a^2s_0^2\quad(00). +\end{equation} +To check this, write $b_*=-\beta$. If $\theta\le1/2$, then +$b_*\le\theta$, and the left side is at most $3t$. Otherwise use +$b_*\le\tfrac12\log(1+s^2)\le\log(1+s)\le\sqrt s$ to bound it by +$4t\sqrt s+2s\le4(1+s)t$. + +For the provisional height $u$, define the raw metric and drift by +\begin{equation}\label{step:raw-output} + h_u=h+\frac{\theta}{(1-\theta)s^2}\,\dd u^2 + =h+\frac{\mu}{1+(1-\mu)s^2}\,\dd u^2, + \qquad t_u=t-\beta. +\end{equation} +These expressions are smooth also at $s=0$, and $h_u\ge h$, $t_u\ge t$. +To see explicitly the smoothness needed at zero slope, note that near $s=0$ +the coefficient of $\dd u^2$ is +$a^2/[1+(1-a^2)s^2]$, while the drift increment is +$-\tfrac12\log([1+(1-a^2)s^2]/[1+s^2])$. +Both are smooth functions of the first jet of $u$; therefore +$C^3$ convergence of $u$ to zero gives $C^2$ convergence of these +metric and drift increments to zero. +For $s\ge5s_0$, $h_u=h+\dd u^2$. At points of nonzero slope in the original manifold, let +$e=\grad_hu/s$. With the given background metric $h_0$, the rank-one +inverse formula gives +\begin{equation}\label{step:trace} + \tr_hh_0-\tr_{h_u}h_0=\theta b^2, + \qquad b^2=h_0(e,e)\le1. +\end{equation} + +We now fix all constants that are required to be independent of the local +geometry and of $d$. Let +\begin{equation}\label{step:constants} + A_0=1000,\qquad P_0=20,\qquad C_1=s_0^{-1},\qquad + b_0=\frac{\delta s_0^2}{1000},\qquad + 0<\eps\le\min\left\{\frac{s_0}{100},\frac1{8n},\frac1{8C_1}\right\}. +\end{equation} +Apply \cref{pressure:law} with these constants, obtaining $F,S_0,T$. +In particular $S_0\le1$ and $T\ge1$. Put $C_2=2C_1b_0$. +Only after a loss $d$ has been specified do we choose $a$ so that +\begin{equation}\label{step:a-choice} + 0W/5,\qquad |\dd\psi|_{h_0}\le1, + \qquad\supp\psi\subset\{r_A0$ as above and every $00$ and $t(N)$ is compact, the uniform pole +limits of $F$ permit one choice $S_10$, retaining its +original values on a slightly larger interval than $[-S_1,S_1]$. For +example, extend the positive even function $F_x$ by a smooth cutoff to a +positive constant and integrate from zero. Solve this globally defined +equation. The same barriers force every solution into $[-S_1,S_1]$, so +that the extension is never used in the resulting solution. + +We give the gradient estimate, including the dependence of the pressure +on the graph angle. If $\max_N|\dd u|_h\le1$ there is nothing to prove. +Otherwise, near a point of maximum slope, $\mu=1$ and the graph in +$(N,h)\times\R$ is an ordinary graph. Write +\[ + w=(1+|\dd u|_h^2)^{-1/2},\qquad + \nu=(N_u,-w),\qquad N_u=w\grad_hu, +\] +for its downward unit normal. Denote its second fundamental form and +mean curvature by $\mathrm{II}$ and $H$, with $H=\Div\nu$. Extend +$t$ constantly in the vertical direction and put $X=-\grad t$ and +$K=-\partial_z$. The equation is +\[ + H+\langle X,\nu\rangle=p(x,z,w),\qquad + p(x,z,\omega)=\sqrt{1-\omega^2+a^2s_0^2}\, + F\left(\frac{z-\tau\psi(x)}{\eta(x)},t(x)\right). +\] +Let $K^\top=K-w\nu$. Under a normal variation with speed $v$, +\begin{align*} + \dot H&=-\lap v-(|\mathrm{II}|^2+\Ric(\nu,\nu))v,\\ + \frac{\dd}{\dd q}\langle X,\nu\rangle + &=-\Hess t(\nu,\nu)v-\langle X^\top,\grad v\rangle,\\ + \dot w&=-\langle K^\top,\grad v\rangle. +\end{align*} +Thus the linearization of the equation, with all terms moved to the +left, is $-\mathcal Lv$, where +\begin{equation}\label{step:angle-operator} + \mathcal L=\lap+ + \langle X^\top-p_\omega K^\top,\grad\,\cdot\,\rangle + +|\mathrm{II}|^2+\Ric(\nu,\nu)+\Hess t(\nu,\nu) + +\partial_\nu^{\mathrm{fixed}\ \omega}p. +\end{equation} +The last derivative differentiates the base and height variables while +holding the angle fixed. Translation in direction $K$ leaves the angle, +$H$, and $\langle X,\nu\rangle$ unchanged and differentiates the +residual by $p_z$. Its normal speed is $w$; its tangential component +differentiates an identically zero residual. We obtain the exact identity +\begin{equation}\label{step:angle-Jacobi} + \mathcal Lw=-p_z. +\end{equation} +In particular the term $-p_\omega K^\top$ in the drift has not been +discarded. + +Put $\xi=(u-\tau\psi)/\eta$. On the graph, +\[ + p_z=\frac{JF_x}{\eta},\qquad + \partial_\nu^{\mathrm{fixed}\ \omega}p + =J\left\{ + \frac{F_x}{\eta}\bigl[-w-\tau\dd\psi(N_u)-\xi\dd\eta(N_u)\bigr] + +F_t\dd t(N_u)\right\}. +\] +The height bound puts $F,F_x,F_t$ in a fixed compact range. Moreover +$|N_u|\le1$, $|\dd\psi|_h\le1$, and $|\dd\eta|_h\le s_0/4$. +The zero-order coefficient in \cref{step:angle-operator}, after omitting +the nonnegative term $|\mathrm{II}|^2$, is therefore at least $-C/\eta$, +with $C$ independent of $r,\tau$. At the minimum of $w$ under +consideration, the slope is at least one, so $J\ge1/\sqrt2$ and +$p_z\ge A_0T/(\sqrt2\eta)$. Since $\grad w=0$ and $\lap w\ge0$, +\cref{step:angle-Jacobi} implies +\[ + \frac{A_0T}{\sqrt2\eta}\le\frac C\eta w. +\] +This gives a positive lower bound for $w$, and hence the asserted $G$. + +We finish the fixed-$r$ continuation argument in enough detail to specify +its regularity inputs. A classical $C^{2,\alpha}$ solution is smooth before +we apply estimates requiring smoothness. Indeed, writing its equation as +$a^{ij}u_{ij}=f$, the coefficient functions $a^{ij}$ and $f$ are +$C^{1,\alpha}$. On nested coordinate balls, a difference quotient $u_h$ +satisfies +\[ + a^{ij}(x)\partial_{ij}u_h + =f_h-(a^{ij})_h\partial_{ij}u(x+h). +\] +For each nonzero $h$, $u_h$ is already $C^{2,\alpha}$; its right side has +a $C^{0,\alpha}$ bound independent of $h$. The classical interior Schauder +estimate and passage to $h=0$ give $u\in C^{3,\alpha}$, and iteration gives +smoothness. These qualitative estimates may depend on the existing +$C^{2,\alpha}$ norm; the following a priori estimates do not. At $\tau=0$, $u=0$ solves the equation. For fixed +$r>0$, the height and gradient bounds put $(x,u,\dd u)$ in a compact set +on which the principal coefficients are uniformly elliptic and all +explicit coefficient derivatives are bounded. In a coordinate chart +write the divergence-form equation as +\[ + \partial_i\mathfrak A^i(x,Du)=\mathfrak B(x,u,Du). +\] +Differentiating with $v=\partial_ku$ gives +\begin{equation}\label{step:differentiated} + \partial_i(\mathfrak A^i_{p_j}\partial_jv) + -\mathfrak B_{p_j}\partial_jv + =\mathfrak B_u v+\mathfrak B_{x_k} + -\partial_i\mathfrak A^i_{x_k}. +\end{equation} +The $x$ derivatives on the right are explicit derivatives of the +coefficient functions. All nondifferentiated terms and the vector field +inside the last divergence are bounded using only the compact jet range; +no second derivative estimate has been assumed. The local Hölder +estimate for uniformly elliptic divergence-form equations, with bounded +lower-order coefficients and bounded divergence data, now gives a +uniform $C^{1,\alpha}$ bound for $u$ on a finite cover. The equation in +nondivergence form then has Hölder coefficients and right side, so local +Schauder estimates give $C^{2,\alpha}$ bounds, followed by bounds of +every order. The H\"older and interior Schauder estimates used here +are respectively \citet[Lecture~18, Theorem~3, and Lecture~12, +Theorem~2]{simonPDE}. +All their uses are on compact coordinate balls with a positive +ellipticity constant; their constants may depend on fixed $r$. + +Fix the floor $r>0$, and choose $0<\alpha<1$ smaller, if necessary, +than the exponent in the preceding H\"older estimate. +The linearization in $u$ has strictly negative zero-order coefficient +$-JF_x/\eta$; angle and gradient derivatives affect only its principal +and first-order terms. We verify that this linearization is an +isomorphism from $C^{2,\alpha}(N)$ to $C^{0,\alpha}(N)$. + +First consider $\lap_h-1$. For $f\in L^2(N)$, minimize on $H^1(N)$ +the strictly convex coercive functional +\[ + \frac12\int_N\bigl(|\dd v|_h^2+v^2\bigr) + \,\dd\operatorname{vol}_h + +\int_N fv\,\dd\operatorname{vol}_h. +\] +Its Euler equation is $(\lap_h-1)v=f$ in the weak sense. +Testing the homogeneous difference with itself proves uniqueness +among $H^1$ weak solutions. +For $f\in C^{0,\alpha}(N)$ choose smooth $f_j$ converging uniformly +to $f$, with uniformly bounded $C^{0,\alpha}$ norms, using coordinate +mollification and a finite partition of unity. Weak interior +regularity makes the corresponding weak solutions $v_j$ smooth +\citep[Lecture~6, Theorem~1 and Corollary~1]{simonPDE}. +The maximum principle gives $\|v_j\|_{C^0}\le\|f_j\|_{C^0}$. +A finite interior cover and the Schauder estimate +\citep[Lecture~12, Theorem~2]{simonPDE} then bound $v_j$ uniformly +in $C^{2,\alpha}(N)$. Passing to a subsequence in +$C^{2,\alpha'}(N)$, where $0<\alpha'<\alpha$, gives a solution for $f$. +The common $\alpha$-H\"older bound for the second derivatives passes +to the limit, so this solution belongs to $C^{2,\alpha}(N)$. +Weak uniqueness identifies it with the energy solution, and the +maximum principle also gives uniqueness in the classical class. + +Let $L_q$, $0\le q\le1$, interpolate the coefficients of +$\lap_h-1$ and the linearization written in nondivergence form. +Uniform ellipticity and a uniform +strictly negative zero-order bound persist along this path. The +maximum principle and the same finite-cover Schauder estimate give +\[ + \|v\|_{C^{2,\alpha}(N)} + \le C\|L_qv\|_{C^{0,\alpha}(N)}, + \qquad 0\le q\le1. +\] +Invertibility is open by the Neumann series. To see closedness, +suppose $q_j\to q$ through invertible parameters and, for a fixed +$f\in C^{0,\alpha}(N)$, put $v_j=L_{q_j}^{-1}f$. +The estimate above bounds $v_j$ and gives +\[ + \|v_j-v_k\|_{C^{2,\alpha}} + \le C\|(L_{q_k}-L_{q_j})v_k\|_{C^{0,\alpha}} + \longrightarrow0. +\] +Thus $v_j$ converges in $C^{2,\alpha}$ to a solution of +$L_qv=f$; injectivity follows from the same estimate. +This proves the required linearized isomorphism. + +The implicit function theorem now gives openness of the nonlinear +continuation in $\tau$. The preceding difference-quotient bootstrap +applies to the $C^{2,\alpha}$ solutions obtained by implicit inversion. +The uniform a priori $C^{2,\alpha}$ bound makes its higher derivative +bounds uniform along continuation at this fixed positive floor. +In particular these bounds give compactness in +$C^{2,\alpha}(N)$, so a convergent sequence of soluble parameters +has a solution at its limit. The height bound keeps the pressure +argument in its fixed compact interval. The soluble set is therefore +closed as well as open and contains $0$, hence equals $[0,1]$. +The local bootstrap makes the resulting solution smooth. +\end{proof} + +\subsection{Decay on the region of constant floor} + +\begin{lemma}[Locality]\label{step:locality} +Let $u_r$ be the solutions from \cref{step:existence} at $\tau=1$. +On any compact subset of the fixed open region where $\psi=0$ and +$\eta_1=0$, the functions $u_r$ tend to zero in every $C^k$ norm as +$r\downarrow0$. +\end{lemma} + +\begin{proof} +Here $\eta=r$ and $|u_r/r|\le S_1$. Expanding the left side of +\cref{step:PDE} and using $F(0,t)=0$ writes its equation as +\begin{equation}\label{step:mass-equation} + a^{ij}(x)\partial_{ij}u_r+b^i(x)\partial_i u_r + -\frac{c_r(x)}r u_r=0, + \qquad + c_r=J\int_0^1F_x(qu_r/r,t)\,\dd q\ge as_0A_0T>0. +\end{equation} +The coefficients $a^{ij}$ are uniformly elliptic and bounded, and $b^i$ +are bounded, independently of $r$. Indeed they are smooth functions of +$(x,Du_r)$ on the fixed compact jet range from \cref{step:C01}; the +explicit $x$ part of the flux vanishes at zero covector and is therefore +a bounded first-order term. Also $c_r$ is bounded above independently +of $r$. No derivatives of these coefficients are needed for the next +comparison. + +Take a coordinate box compactly contained in the constant-floor region. +Let $\ell_f$ be its inward affine distance from face $f$ in these +coordinates and set +\[ + V_r=M_0\sum_f\exp(-\gamma\ell_f/\sqrt r), +\] +where $M_0$ dominates $|u_r|$ on the boundary for every $r$. +Let $\Lambda$ bound $a^{ij}(\partial_i\ell_f)(\partial_j\ell_f)$, +let $c_*>0$ be the lower bound for $c_r$, and let $B_*$ bound the +first-order contributions. The operator in \cref{step:mass-equation} +applied to each summand of $V_r$ is at most that summand times +\[ + \frac{\Lambda\gamma^2-c_*}{r}+\frac{B_*\gamma}{\sqrt r}. +\] +Choose $\gamma>0$ with $\Lambda\gamma^28b$, the same estimates give +\[ + E\ge(7s/4-20\eps)F_x\ge sF_x. +\] +It remains to consider $s_0\le s\le8b$. The possible deficit from +$sF_x$ is at most $10F_x$: indeed +$2\chi(s-b)-s\ge-4$ and $20\eps<1$. +On the favorable pressure branch, however, +\[ + E-sF_x\ge[20\chi^2+2\chi(s-b)-s]F_x\ge0, +\] +since $s\le8b\le8$. Thus a deficit is possible only when +$F_x\le b_0t^2$. In that case $b\ge s_0/8$ and our choice of $b_0$ +gives $10b_0\le\delta(s_0/8)^2$. We conclude throughout $s\ge s_0$ +that +\begin{equation}\label{step:large-slope} + E\ge sF_x-\delta b^2t^2 + \ge4(1+s)t-\delta b^2t^2. +\end{equation} +The last inequality follows from $F_x\ge1000t$ and $s\ge.01$. +Combining \cref{step:beta-budget,step:E,step:trace} proves the raw +inequality in this region, even without its $d$ term. + +Now suppose $00$. +At a zero of $\dd u$ it follows by continuity unless $\dd u$ vanishes +on a neighborhood; on such a neighborhood the raw output equals the +input and the inequality is immediate. +\end{proof} + +\subsection{Cutting off the deformation and obtaining distance} + +Define on the original manifold +\begin{equation}\label{step:final-output} + h'=h+\zeta(h_u-h),\qquad t'=t+\zeta(t_u-t), +\end{equation} +and extend by $(h,t)$ outside the auxiliary neighborhood. The result +is smooth, $h'\ge h$, $t'\ge t$, and its changes have compact support +in $\{r_A0$ there. Off the annulus +the output is either raw or unchanged, and the same desired inequality +holds directly. This proves \cref{step:conclusion}. In particular, +no rank-one trace identity for the uncut metric has been substituted +for the trace of the cutoff metric. + +It remains to prove the distance gain, including for curves that revisit +the cutoff annulus or leave the auxiliary neighborhood. Introduce the +globally defined effective height +\[ + v_{\mathrm{eff}}=\zeta u, +\] +extended by zero outside the neighborhood. On $A$ the height barrier +and \cref{step:cutoffs} give $v_{\mathrm{eff}}>W/5-1$, whereas on $B$ +it is zero. Decrease $r$ further, if necessary, so that +\begin{equation}\label{step:height-cutoff} + |\dd v_{\mathrm{eff}}|_h\le .01 +\end{equation} +throughout $\mathcal A_\zeta$; locality justifies this choice. +Consider a piecewise smooth curve from $A$ to $B$ of $h'$ length at most +$L_*W$. Its $h$ length is at most $L_*W$. Partition its parameter interval according to $0<\zeta<1$, +$\zeta=1$, and $\zeta=0$. At points where $\zeta$ equals either endpoint, +$\dd\zeta=0$ because $\zeta$ is smooth and valued in $[0,1]$. +The total variation of +$v_{\mathrm{eff}}$ on $\{0<\zeta<1\}$ is at most $.01L_*W$, +counting all repeated visits. On the region where $\zeta=1$ and +$s\le5s_0=.05$, its variation is at most $.05L_*W$. Where $\zeta=0$ +it has zero variation. Therefore the variation on +\[ + \Omega_{\mathrm{graph}}=\{\zeta=1,\ s\ge .05\} +\] +is greater than +\[ + (1/5-.06L_*)W-1>.11W. +\] +The strict last inequality holds for our fixed sufficiently large $W$. +On this region $h'=h+\dd u^2$, while everywhere $h'\ge h$. +If $\gamma$ is parametrized on an interval $I$, the integral triangle +inequality in $\R^2$ yields +\begin{align*} + \operatorname{Length}_{h'}(\gamma) + &\ge\int_I\sqrt{|\dot\gamma|_h^2+ + \mathbf1_{\Omega_{\mathrm{graph}}}|(u\circ\gamma)'|^2} + \,\dd q\\ + &\ge\left\{ + \operatorname{Length}_h(\gamma)^2+ + \left(\int_{\gamma^{-1}(\Omega_{\mathrm{graph}})} + |(u\circ\gamma)'|\,\dd q\right)^2 + \right\}^{1/2}\\ + &>W\sqrt{1+.11^2}>L_*W. +\end{align*} +Here $\operatorname{Length}_h(\gamma)\ge W$ follows from $h\ge h_0$ +and the original separation of $A,B$. This contradiction proves +\cref{step:distance}; curves with infinite length are irrelevant. +Finally, a smooth metric dominating a complete metric is complete: +an $h'$-Cauchy sequence is $h_0$-Cauchy, and local equivalence of the +two smooth metrics supplies convergence in $h'$. The proof of +\cref{step:main} is complete. diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/pressure.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/pressure.tex new file mode 100644 index 0000000000000000000000000000000000000000..ca55a522b179148553ecbcd519c89dcb8a55f921 --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/pressure.tex @@ -0,0 +1,239 @@ +\section{A pressure law with a quadratic budget} +\label{pressure:section} + +The local metric deformation will use a pressure depending both on a height +variable $x$ and on the drift parameter $t$. Its derivative in $x$ must be +large, while a large positive derivative in $x$ must be accompanied by a +favorable derivative in $t$. The following construction supplies these +properties with constants chosen before any local geometric data. + +\begin{theorem}[Pressure law]\label{pressure:law} +For every $A_0,P_0,\eps,b_0>0$ there exist $00$, and satisfies +\begin{equation}\label{pressure:estimates} +\begin{gathered} + F_x\ge A_0t,\qquad |F_t|\le\eps|F|,\qquad FF_t\le\eps F_x,\\ + F_x\le b_0t^2\quad\text{or}\quad -FF_t\ge P_0F_x,\\ + F_x\le b_0t^2+\eps F^2. +\end{gathered} +\end{equation} +For each fixed $t\ge T$, +\[ + \lim_{x\to-S_0}F(x,t)=-\infty, + \qquad \lim_{x\to S_0}F(x,t)=+\infty. +\] +These limits are uniform when $t$ ranges over a compact subinterval of +$[T,\infty)$. The choices depend only on $A_0,P_0,\eps,b_0$ and fixed +numerical constants. In particular, they are independent of any manifold, +metric, local cutoff, or subsequent deformation parameter. The lower bound +$T$ may be increased after the construction. +\end{theorem} + +\begin{proof} +We construct $F$ by prescribing its inverse in the first variable. For +the moment, let $k>0$ and let $g$ be a positive smooth function on +$[0,\infty)$, both to be chosen. Write $f>0$ for an independent pressure +value and, for $t>0$, set +\begin{equation}\label{pressure:inverse} + X(f,t)=\frac1k\arctan\frac{f^{1/3}}{g(y)},\qquad + q=f^{2/3},\qquad y=\frac tq. +\end{equation} +Put $A(y)=g(y)+2yg'(y)$. Differentiation at fixed $f$ or at fixed $t$, +respectively, gives +\begin{equation}\label{pressure:inverse-derivatives} + fX_t=-\frac{g'(y)}{k(1+g(y)^2/q)},\qquad + X_f=\frac{A(y)}{3kq^2(1+g(y)^2/q)}. +\end{equation} +Thus $A>0$ will make $X$ invertible in $f$. For the inverse, +$F_x=1/X_f$ and $-FF_t/F_x=fX_t$. We will arrange the favorable +pressure branch on a decreasing part of $g$, and obtain the quadratic +branch from a lower bound for $t^2X_f$. We choose $g$ to decrease for +small $y$ and to equal $y$ for large $y$, while keeping $A$ positive +through the transition. + +We construct $g$ as a smooth rounded maximum with a controlled negative +derivative. Fix a nonnegative smooth probability density $\rho$ supported +in $[1,2]$, positive in $(1,2)$, and vanishing to infinite order at its +endpoints. For example, normalize +\[ + \rho(r)= + \begin{cases} + c_\rho\exp\bigl(-1/((r-1)(2-r))\bigr),&10$ set +\begin{equation}\label{pressure:rounded-max} + h(y)=D_g(1+y)^{-1/4},\qquad + g(y)=\int_1^2\max\{y,rh(y)\}\rho(r)\,\dd r, + \qquad y\ge0. +\end{equation} +Here and below smoothness at $y=0$ means smooth extendibility to a +neighborhood of $[0,\infty)$. To verify smoothness, put +\[ + M(z)=\int_1^2\max\{z,r\}\rho(r)\,\dd r. +\] +Then $M''=\rho$, with $\rho$ extended by zero, and +$g(y)=h(y)M(y/h(y))$. Thus $g$ is smooth; moreover $M(z)=\mu$ for +$z\le1$ and $M(z)=z$ for $z\ge2$. Consequently $g>0$, $g\ge y$, and +$g(y)=y$ for all sufficiently large $y$. + +For a direct derivative estimate, let +\[ + L(z)=\int_{r\le z}\rho(r)\,\dd r, + \qquad R(z)=\int_{r\ge z}r\rho(r)\,\dd r. +\] +At $z=y/h(y)$ the identities +\[ + g=yL(z)+hR(z),\qquad + g'=L(z)-\frac{hR(z)}{4(1+y)} +\] +give +\begin{equation}\label{pressure:g-bounds} + g\ge y,\qquad + -\frac{g}{4(1+y)}\le g'\le1,\qquad + A(y)=g(y)+2yg'(y)\ge\frac{y+2}{2(y+1)}g(y)\ge\frac12g(y). +\end{equation} +In particular the lower bound for $A/g$ is numerical and does not +deteriorate as $D_g$ increases. + +We now specify the order of all choices. Choose +\begin{equation}\label{pressure:k-Y} + k\ge\max\{A_0,\eps^{-1},\pi/2\},\qquad + Y\ge1\quad\text{with}\quad Y^{-3}\le\frac{b_0}{12k}. +\end{equation} +Choose $D_g$ only after $k,Y$, large enough that +\begin{equation}\label{pressure:D-choice} + D_g\ge\max\left\{ + Y(1+Y)^{1/4},\, + \frac{8kP_0(1+Y)^{5/4}}{\mu} + \right\}. +\end{equation} +Since $y(1+y)^{1/4}$ is increasing, $y/h(y)\le1$ on $[0,Y]$. +Hence throughout this interval +\begin{equation}\label{pressure:decreasing-range} + g(y)=\mu D_g(1+y)^{-1/4},\qquad + g'(y)=-\frac{\mu D_g}{4(1+y)^{5/4}}\le-2kP_0. +\end{equation} +The equality cases at the endpoints are covered by the smooth matching +already proved. + +With this $g$ fixed, define the finite constants +\[ + g_0=\min_{0\le y\le Y}g(y)>0,\qquad + G_0=\max_{0\le y\le Y}g(y),\qquad + B=\sup_{y>Y}\frac{g(y)}y<\infty. +\] +For example, +\[ + 1\le B\le\max\left\{1,\frac{2D_g}{Y(1+Y)^{1/4}}\right\}, +\] +because $g(y)\le\max\{y,2D_g(1+y)^{-1/4}\}$. +Finally choose +\begin{equation}\label{pressure:T-choice} + T\ge\max\left\{ + 1,\frac6\eps,\frac{12kB^2}{b_0},\, + YG_0^2,\frac{12kY}{\eps g_0} + \right\}. +\end{equation} +Thus the order is $k$, then $Y$, then $D_g$ and $g$, and then $T$; +none of the preceding choices depends on a later one. + +We now invert the map $X$ in \cref{pressure:inverse}. +By \cref{pressure:g-bounds}, $X_f>0$. As $f\downarrow0$, the argument +$y$ tends to infinity, where $g(y)=y$, and $X(f,t)\to0$. As +$f\to\infty$, $y\to0$, $g(y)\to g(0)>0$, and +$X(f,t)\to S_0:=\pi/(2k)\le1$. Thus for every $t>0$, $X(\cdot,t)$ +maps $(0,\infty)$ bijectively onto $(0,S_0)$ with positive derivative. +Its inverse $f=F(x,t)$ is smooth by the implicit function theorem. + +On a neighborhood of $f=0$ the exact identity $g(y)=y$ gives +\[ + X(f,t)=\frac1k\arctan(f/t),\qquad F(x,t)=t\tan(kx). +\] +Define $F(0,t)=0$ and extend $F$ oddly to $-S_00$. The inverse construction also +proves the asserted pole limits. For uniformity, if $I\subset[T,\infty)$ +is compact and $H>0$, then $\max_{t\in I}X(H,t)H$ for every $t\in I$. +The negative pole follows by oddness. + +It remains to prove \cref{pressure:estimates}. For $x>0$ set +$f=F(x,t)$ and use the variables in \cref{pressure:inverse}. Implicit +differentiation of $X(F(x,t),t)=x$ yields +\begin{equation}\label{pressure:signed-identities} + F_x=\frac1{X_f},\qquad F_t=-\frac{X_t}{X_f},\qquad + \frac{FF_t}{F_x}=\frac{g'}{k(1+g^2/q)},\qquad + \frac{F_t}{F}=\frac{3yg'}{t(g+2yg')}. +\end{equation} +In particular the sign in the last identity is the sign of $g'$. +Since $A\le g+2y\le3g$ and $y\le g$, we have +\begin{equation}\label{pressure:lower-slope} + tX_f=\frac{yA}{3k(q+g^2)}\le\frac1k. +\end{equation} +This proves $F_x\ge kt\ge A_0t$ without a constant depending on $g$. +Also $|yg'|\le g$: for $g'\ge0$ this follows from $g'\le1$ and +$y\le g$, whereas for $g'<0$ the lower derivative bound in +\cref{pressure:g-bounds} even gives $|yg'|\le g/4$. Therefore +\[ + \left|\frac{F_t}{F}\right|\le\frac6t\le\eps, + \qquad + \frac{FF_t}{F_x}\le\frac1k\le\eps. +\] +For the second inequality a negative numerator is already favorable; +for a nonnegative numerator use $g'\le1$. + +Consider first $y>Y$. Using $A\ge g/2\ge y/2$ and $g/y\le B$ gives +\begin{align} + t^2X_f + &=\frac{y^2A}{3k(1+yg^2/t)}\notag\\ + &\ge\frac{y^3}{6k(1+B^2y^3/T)} + =\frac1{6k(y^{-3}+B^2/T)} + \ge\frac1{6k(Y^{-3}+B^2/T)} + \ge\frac1{b_0}. + \label{pressure:large-y} +\end{align} +The last inequality uses \cref{pressure:k-Y,pressure:T-choice}. +This bound remains uniform as $y\to\infty$, and it proves +$F_x\le b_0t^2$ throughout this region. + +If $0Y$ +this proves the final inequality of \cref{pressure:estimates}. + +Oddness preserves all five inequalities on $x<0$, since $F_x$, $F^2$, +and $FF_t$ are even and $|F_t|,|F|$ are even. At $x=0$ they follow +by continuity; more explicitly, $F=F_t=0$ and $F_x=kt$, while +$T\ge12kB^2/b_0\ge k/b_0$ gives the small-$F_x$ branch there. +All claims follow. +\end{proof} + +\begin{remark}\label{pressure:parameter-independence} +When $A_0,P_0,\eps,b_0$ are prescribed using only the dimension and fixed +numerical constants, the same is true of $S_0,T$ and the entire pressure +law. In particular, the small parameters denoted by $a$ and $d$ in the +local deformation may be chosen after this theorem has been applied, +without modifying the pressure or increasing $T$. +\end{remark} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/reconstruction.tex b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/reconstruction.tex new file mode 100644 index 0000000000000000000000000000000000000000..9f26d2803f16f720668db01717161b2f80578e8d --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/build/sections/reconstruction.tex @@ -0,0 +1,373 @@ +\section{Reconstruction and the map to a complex}\label{sec:reconstruction} + +All distances in this section are measured in the original metric $g$; +write $d_0$ for that distance. Put $D=WR_0$, as in +\cref{rounds:wide}. The centers $x_i$ form a maximal $D$-separated set, +and the color pairs are +\[ + A_i=\overline B_0(x_i,D),\qquad + B_i=M\setminus B_0(x_i,2D). +\] +The $A_i$ cover $M$. Every family of balls with these centers and a +fixed finite radius is locally finite: a compact set has a compact +neighborhood of that radius by completeness, and a compact metric +space contains only finitely many $D$-separated points. The family is +countable, since $M$ is second countable. If some $B_i$ is empty, +$\diam_{g}M\le 4D$ and a constant map proves the theorem. We therefore +assume that all $B_i$ are nonempty. + +\subsection{A limit on the original manifold} + +The analytic packets have infinite product ends. Their finite +reconstructions in \cref{packet:reconstruction} are manifolds +homeomorphic to $M$; only finite intervals of the product ends are used +in these reconstructions. We now choose those homeomorphisms +coherently. This is the point at which local finiteness of the color +supports is essential. + +\begin{proposition}[Reconstructed partitions]\label{topology:partitions} +The compatible cuts of \cref{countable:construction} give a continuous +proper map $q:M\to M$, and for every color $i$ closed sets $P_i,N_i,W_i$ +such that +\begin{align} + &d_0(q(x),x)<1,\label{topology:displacement}\\ + &P_i\cup N_i=M,\qquad P_i\cap N_i=W_i, + \qquad W_i\subset q^{-1}\bigl(B_0(x_i,2D)\bigr), + \label{topology:sides}\\ + &q^{-1}(A_i)\subset P_i\setminus W_i, + \qquad q^{-1}(B_i)\subset N_i\setminus W_i. + \label{topology:forced} +\end{align} +The wall family $(W_i)$ is locally finite, and no $n-1$ distinct walls +have a common point. +\end{proposition} + +\begin{proof} +Choose positive numbers $\eps_0,\eps_1,\ldots$ with +$\sum_{r\ge0}\eps_r<1$. Here $\eps_0$ is assigned to the single radial +stage; the other numbers are assigned to successive color stages. +In this proof only, index the post-radial reconstruction by $r=0$. +For a finite reconstruction write +\[ + q_r:X_r\longrightarrow M,\qquad + H_r:X_r\longrightarrow M,\qquad + Q_r=q_r\circ H_r^{-1}. +\] +The map $q_r$ is the proper collapse of +\cref{packet:reconstruction}, whereas $H_r$ is a homeomorphism to a +fixed copy of the original manifold. We maintain +\begin{equation}\label{topology:finite-error} + d_0(Q_r(x),x)\le E_r, + \qquad E_r\le\sum_{j=0}^{r}\eps_j<1. +\end{equation} +Walls and their two closed sides will also be viewed on this fixed copy +using $H_r$. + +For the radial stage, the cut has a locally finite family of compact +components. Choose pairwise disjoint, locally finite collars of these +components on which the collapse varies by less than $\eps_0$ in each +normal segment. Compactness of each component and continuity of the +collapse permit the collars to be this small, independently from one +component to another. Compress the finite inserted cylinders into +these collars. Since their supports are disjoint, the displacement +bound is $\eps_0$, rather than a sum over components. This constructs +$H_0$ and proves \eqref{topology:finite-error} initially. The finite +length of an inserted cylinder may be arbitrarily large; its length +does not impose a lower bound on the width of the collar into which it +is compressed. + +We describe a subsequent color stage $i$. First raise any old finite +horizons needed for the new compatible defining functions. Denote the +resulting reconstruction by $\widehat X_r$. The horizon +reparameterization +\[ + \Phi_r:\widehat X_r\longrightarrow X_r +\] +preserves collapse and every old wall and signed side. Indeed, on an +old product end the collapse is independent of the normal coordinate, +and an increasing change of that coordinate preserves its sign and its +zero section. Compatible products of these changes agree at corners. +Thus we may set +\begin{equation}\label{topology:horizon} + \widehat q_r=q_r\circ \Phi_r,\qquad + \widehat H_r=H_r\circ \Phi_r, + \qquad + \widehat q_r\circ\widehat H_r^{-1}=Q_r. +\end{equation} +In particular, increasing a horizon contributes no displacement on the +fixed copy. + +The new wall lies over $B_0(x_i,2D)$. Choose its collar strip $S_i$ in +$\widehat X_r$ so that +\begin{equation}\label{topology:strip} + S_i\subset\widehat q_r^{-1}\bigl(B_0(x_i,2D+1)\bigr), +\end{equation} +and so that collapse oscillation along every segment used by the +compression is less than the error assigned to this stage. These +choices are possible because the relevant wall components in a finite +reconstruction are compact and locally finite. More specifically, +their union lies in the inverse image of the compact ball +$\overline B_0(x_i,2D)$ under the proper finite collapse; hence only +finitely many components are involved in this color stage. + +The strips can be chosen to preserve every old stratum, including the +radial strata. To check this assertion, work near a point of a new wall +on an old stratum, and denote the new defining function by $u$. +The horizons were chosen beyond the product thresholds, so the terminal +collars glued at each old hit wall lie in the product range. In the +resulting product chart, $u$ is pulled back from the corresponding face +and is independent of the old signed normal coordinates. Its restriction +to that face is regular at zero. In the corresponding product chart choose a vector field +$V$ in the face directions with $V(u)=1$, and with zero component in +every old normal direction. The chart may avoid the old walls not +passing through the point. A partition of unity combines these fields: +the linear identity $V(u)=1$ and tangency to every old wall both survive. +Its short flow supplies a collar preserving the old walls and their +sides. On a fresh cylinder, each old stratum is the product of its +intersection with the new wall and the cylinder interval. The +compression uses the same increasing normal reparameterization on +all these products. It consequently preserves every old signed side +as well as all their intersections. + +Let +\[ + c_r:X_{r+1}\longrightarrow\widehat X_r + \quad\hbox{and}\quad + h_r:X_{r+1}\longrightarrow\widehat X_r +\] +be, respectively, the new cylinder collapse and the homeomorphism that +compresses those cylinders into the chosen strips. They agree off +the inverse image of $S_i$. Figure~\ref{fig:topology-compression} shows +the two maps schematically in a normal coordinate: $c_r$ collapses the +inserted interval, while $h_r$ compresses it into the chosen old collar. +Define +\begin{equation}\label{topology:recursion} + q_{r+1}=\widehat q_r\circ c_r, + \qquad H_{r+1}=\widehat H_r\circ h_r, + \qquad + Q_{r+1}=\widehat q_r\circ c_r\circ h_r^{-1} + \circ\widehat H_r^{-1}. +\end{equation} +The oscillation choice gives +$d_0(Q_{r+1}(x),Q_r(x))<\eps_{r+1}$ wherever they differ, proving +\eqref{topology:finite-error} at the next stage. Their difference is +supported in $\widehat H_r(S_i)$. If +$x=\widehat H_r(z)$ belongs to this set, then +\eqref{topology:horizon}, \eqref{topology:strip}, and +\eqref{topology:finite-error} imply +\begin{equation}\label{topology:fixed-support} + d_0(x,x_i) + \le d_0(x,Q_r(x))+d_0(\widehat q_r(z),x_i) + <1+(2D+1)=2D+2. +\end{equation} +This support radius is independent of the stage. + +\begin{figure}[!b] +\centering +\begin{tikzpicture}[x=1.25cm,y=1.05cm,>=Latex,font=\small] + \fill[blue!6] (-3,-1) rectangle (3,1); + \fill[red!8] (-1,-2.25) rectangle (1,2.35); + \draw[->] (-3.2,0)--(3.5,0) node[right] {$s$}; + \draw[->] (0,-2.3)--(0,2.65) node[above] {$t$}; + \draw[densely dashed,gray] (-3,1)--(3,1); + \draw[densely dashed,gray] (-3,-1)--(3,-1); + \draw[very thick,purple!75!black] + (-3,-2)--(-1,0)--(1,0)--(3,2); + \draw[thick,orange!85!black] + (-3,-2)--(-2,-1)--(-1,-.35)--(0,0)--(1,.35)--(2,1)--(3,2); + \draw (-1,.07)--(-1,-.07) node[below=4pt,fill=white,inner sep=1pt] {$-L$}; + \draw (1,.07)--(1,-.07) node[below=4pt] {$L$}; + \node[left,fill=white,inner sep=1pt] at (-.08,1) {$\delta$}; + \node[left,fill=white,inner sep=1pt] at (-.08,-1) {$-\delta$}; + \node[align=left,anchor=west,text=purple!75!black] at (3.55,1.25) + {collapse $c_r$}; + \node[align=left,anchor=west,text=orange!85!black] at (3.55,.55) + {compression $h_r$}; + \node[align=left,anchor=west] at (3.55,-.55) + {$c_r=h_r$ outside the strip}; + \node[align=left,anchor=west] at (3.55,-1.35) + {old strata follow\\the same reparameterization}; + \node[below,align=center] at (0,-2.4) + {finite inserted cylinder: $-L\le s\le L$}; +\end{tikzpicture} +\caption{The two maps used in a finite reconstruction, shown schematically +in local normal coordinates: $s$ on the reconstructed domain $X_{r+1}$ +and $t$ on the old target $\widehat X_r$. The target strip $|t|<\delta$ +represents the chosen collar $S_i$. +The collapse is constant on the inserted cylinder; +the homeomorphism squeezes it into an arbitrarily narrow old collar. +Where the maps differ, both images lie in the target strip. Infinite analytic ends +are truncated before this topological operation.} +\label{fig:topology-compression} +\label{packet:collar-figure} +\end{figure} + +The balls in \eqref{topology:fixed-support} form a locally finite +family. Consequently, on every relatively compact neighborhood in +the fixed copy, the maps $Q_r$ are exactly unchanged after finitely +many stages. Their locally stationary limit $q$ is continuous and +satisfies \eqref{topology:displacement}. It is proper: the inverse +image of a compact set is a closed subset of its compact closed +$1$-neighborhood in $(M,d_0)$. + +At each finite stage the signed sides are the closed reconstructed +sides, with their intersection equal to the central wall. The +stratum-preserving identifications just constructed preserve every +previously installed side and wall exactly. Newly installed color +walls are confined to the fixed balls in +\eqref{topology:fixed-support}. Thus all the partition data are locally +eventually unchanged. Their final closed sets satisfy +\eqref{topology:sides} and \eqref{topology:forced}, since these are the +finite-stage forced-sign properties of +\cref{countable:construction}, and both collapse and statuses are +stationary near each point. The wall family is locally finite by the +same support bound. Finally, a point on $n-1$ distinct final color +walls would, on a neighborhood where these data have stabilized, lie +on the same $n-1$ finite-stage walls. This contradicts +\cref{countable:construction}. +\end{proof} + +\subsection{Thickening the walls without increasing their multiplicity} + +Put $k=n-2$. We need open neighborhoods of the whole walls, rather +than independently chosen neighborhoods of their individual compact +pieces. + +\begin{lemma}[Simultaneous wall neighborhoods]\label{topology:thicken} +There are locally finite open sets $T_i\supset W_i$ such that +\begin{equation}\label{topology:thickening} + T_i\subset q^{-1}\bigl(B_0(x_i,2D)\bigr) + \setminus q^{-1}(A_i\cup B_i), + \qquad + \#\{i:x\in T_i\}\le k\quad(x\in M). +\end{equation} +\end{lemma} + +\begin{proof} +Set $O_i=q^{-1}(B_0(x_i,2D))$ and +$F_i=q^{-1}(A_i\cup B_i)$. By +\cref{topology:partitions}, $W_i$ is disjoint from the closed set $F_i$ +and lies in the open set $O_i$. At a point $x$, local finiteness and +closedness of the walls give an open neighborhood meeting only walls +that contain $x$. Shrink it to a neighborhood $U_x$ contained in +$O_i\setminus F_i$ for every such wall. Therefore $U_x$ meets at most +$k$ walls, and +\begin{equation}\label{topology:cover-property} + U_x\cap W_i\ne\varnothing\quad\Longrightarrow\quad + U_x\subset O_i\setminus F_i. +\end{equation} + +Choose a locally finite open star refinement $\mathcal V$ of this +cover: the union of the members of $\mathcal V$ meeting any fixed +member is contained in some $U_x$. For completeness, such a refinement +can be obtained directly with the metric $d_0$. The function +\[ + \rho(x)=\sup_U\min\{1,d_0(x,M\setminus U)\}, + \qquad U\in\{U_x:x\in M\}, +\] +is positive and $1$-Lipschitz, with distance to the empty set truncated +to $1$. The balls $V_x=B_0(x,\rho(x)/8)$ star-refine the original +cover. Indeed, if $V_y$ meets $V_x$, the Lipschitz inequality gives +$d_0(x,y)<2\rho(x)/7$, and hence +$V_y\subset B_0(x,25\rho(x)/56)$. The latter ball is contained in +$B_0(x,\rho(x)/2)$, which lies in a member of the original cover. +Fix $o\in M$. Obtain a locally finite open refinement of $(V_x)$ by covering each +compact annulus +$\overline B_0(o,m)\setminus B_0(o,m-1)$ with finitely many open subsets +of members of $(V_x)$ lying in +$\{m-20\), depending only on \(n\), so small that +\begin{equation}\label{rounds:delta} + 2(n+1)\delta<1-L_*^{-2}. +\end{equation} +Let \(W,T\) be the constants in \Cref{step:main}. Enlarge \(T\) if necessary +to make \(T\ge1\), and choose +\begin{equation}\label{rounds:original-scale} + R_0\ge\sqrt2\,T,\qquad D=WR_0. +\end{equation} +All of these constants depend only on \(n\). + +\begin{lemma}[A complete round]\label{rounds:round} +Suppose that \(h_0\) is a complete metric, \(t_0\ge T\) is smooth, and +\[ + \mathcal D(h_0,t_0)\ge c+t_0^2 +\] +for a constant \(c\). Let \((A_i,B_i)\) be a finite or countable family of +nonempty closed pairs, with each \(A_i\) compact and +\(\dist_{h_0}(A_i,B_i)\ge W\). Suppose there are locally finite compact +sets \(K_i\) such that the open \(h_0\)-distance-\(W\) neighborhood of +\(A_i\) is contained in \(K_i\setminus B_i\). +Then there are smooth \(h_1\ge h_0\) and \(t_1\ge t_0\), obtained by a +locally finite sequence of compactly supported changes inside the +\(K_i\), such that +\begin{align} + \dist_{h_1}(A_i,B_i)&\ge L_*W \quad\text{for every }i,\label{rounds:separation}\\ + \mathcal D(h_1,t_1)&\ge c+ + \bigl(1-(n+1)\delta\bigr)t_1^2.\label{rounds:curvature} +\end{align} +The metric \(h_1\) is complete. +\end{lemma} + +\begin{proof} +Enumerate the labels and choose \(d_i>0\) with +\(\sum_i d_i<\delta\). Apply \Cref{step:main} in this order, keeping +\(h_0\) as the background metric throughout. Earlier changes increase +distances, so every pair still meets the separation hypothesis when it +is processed. If \((h^{(j)},t^{(j)})\) denotes the data after \(j\) changes, +write +\[ + Q_j=\tr_{h^{(j)}}h_0,\qquad + A_j=1-\delta(n-Q_j)-\sum_{i\le j}d_i. +\] +Thus \(Q_0=n\), \(00\). +This proves \eqref{rounds:trace-induction}. Its coefficient is greater +than \(1-(n+1)\delta\), which proves \eqref{rounds:curvature}. +Each processed separation satisfies \eqref{rounds:separation}, and +later changes preserve it. + +The supports lie in the locally finite family \((K_i)\). Hence every +point has a neighborhood on which the sequence is eventually constant; +its limit is smooth and satisfies the same inequalities. Completeness +follows from \(h_1\ge h_0\): an \(h_1\)-Cauchy sequence converges in +\(h_0\), and near its limit the two smooth metrics are equivalent. +\end{proof} + +The passage from a positive scalar-curvature form to circle warps uses +torical symmetrization; see \citet[Section~2]{gromov2018Metric}. The +following identity fixes the finite-dimensional convention used here. + +\begin{lemma}[Scalar curvature of diagonal circle warps]\label{rounds:warp-identity} +For smooth functions \(p_1,\ldots,p_k\) on \((M,h)\), the metric +\(\widehat h=h+\sum_{\alpha=1}^k e^{2p_\alpha}\dd\theta_\alpha^2\) +on \(M\times\T^k\) has scalar curvature +\[ + \Scal_{\widehat h}=\Scal_h-2\Delta_h P-|\dd P|_h^2 + -\sum_{\alpha=1}^k|\dd p_\alpha|_h^2, + \qquad P=\sum_{\alpha=1}^k p_\alpha. +\] +\end{lemma} + +\begin{proof} +Let \(E_\alpha=e^{-p_\alpha}\partial_{\theta_\alpha}\) be the unit +vertical fields. For horizontal fields \(X,Y\), the Koszul formula gives +\[ + \widehat\nabla_XY=\nabla^h_XY,\quad + \widehat\nabla_XE_\alpha=0,\quad + \widehat\nabla_{E_\alpha}X=(Xp_\alpha)E_\alpha,\quad + \widehat\nabla_{E_\alpha}E_\alpha=-\grad_h p_\alpha, +\] +and \(\widehat\nabla_{E_\alpha}E_\beta=0\) for \(\alpha\ne\beta\). +Therefore, for a horizontal unit vector \(X\), the mixed sectional +curvatures are +\[ + K(X,E_\alpha)=-\Hess_h p_\alpha(X,X)-(Xp_\alpha)^2, + \qquad + K(E_\alpha,E_\beta)=-\langle\dd p_\alpha,\dd p_\beta\rangle_h. +\] +Summing twice over all orthonormal two-planes gives the identity. +\end{proof} + +\begin{theorem}[Prescribed widths for a locally finite family]\label{rounds:wide} +Let \((M,g)\) be complete, connected and smooth, with +\(\Scal_g\ge1\). There is a countable set of centers \((x_i)\) such that +\[ + A_i=\overline B_g(x_i,D),\qquad + B_i=M\setminus B_g(x_i,2D) +\] +have the following properties: the \(A_i\) cover \(M\), and the closed +balls \(\overline B_g(x_i,2D)\) are locally finite. Suppose that every +\(B_i\) is nonempty. Given arbitrary finite numbers \(L_i>0\), there +exist a complete smooth metric \(h\ge g\) and a countable family of +smooth functions \((p_\alpha)\), whose supports form a locally finite +family of compact sets, for which +\begin{align} + \dist_h(A_i,B_i)&>L_i \quad(i\ge1),\label{rounds:prescribed}\\ + \Scal_h-2\Delta_h\sum_\alpha p_\alpha + -\left|\dd\sum_\alpha p_\alpha\right|_h^2 + -\sum_\alpha|\dd p_\alpha|_h^2&\ge\frac25.\label{rounds:stabilized} +\end{align} +Every sum in \eqref{rounds:stabilized} is locally finite. If some +\(B_i\) is empty, then \(\diam_g M\le4D\). +\end{theorem} + +\begin{proof} +A complete connected Riemannian manifold is proper. Choose a maximal +\(D\)-separated set of centers. Maximality gives the covering by closed +\(D\)-balls. A compact set contains only finitely many centers; more +generally, centers whose closed \(2D\)-balls meet a fixed compact set +lie in a fixed compact neighborhood of that set. This proves local +finiteness of the outer balls and countability of the net. The triangle +inequality gives \(\dist_g(A_i,B_i)\ge D\). The last assertion follows +immediately if \(M=B_g(x_i,2D)\). + +We keep a scalar reserve of \(1/2\) in the original units. Put +\(R_j=L_*^jR_0\), start with \(h^{[0]}=g\), \(t^{[0]}=T\), and work in +round \(j\) with the metric \(h^{[j]}/R_j^2\). At the start of that +round we require +\begin{equation}\label{rounds:start} + \mathcal D(h^{[j]}/R_j^2,t^{[j]}) + \ge \frac{R_j^2}{2}+(t^{[j]})^2. +\end{equation} +For \(j=0\), this follows from \(\Scal_g\ge1\) and +\(R_0^2/2\ge T^2\). + +A label remains active until its separation exceeds its prescribed +\(L_i\). At the start of round \(j\), every active pair has separation +at least \(WR_j\) in \(h^{[j]}\); this is initially true and follows +inductively from \eqref{rounds:separation}. Apply \Cref{rounds:round} to +the active pairs in the working metric, with reserve \(c=R_j^2/2\). +The required support condition follows from +\(\dist_{h^{[j]}/R_j^2}(A_i,B_i)\ge W\): the open distance-\(W\) +neighborhood of \(A_i\) misses \(B_i\), and is therefore contained in +the original outer ball \(B_g(x_i,2D)\). We may take +\(K_i=\overline B_g(x_i,2D)\). + +Return the output metric to original units and denote it by +\(h^{[j+1]}\). It dominates \(h^{[j]}\). Scalar curvature, the +Laplacian and squared gradient norms all multiply by \(R^2\) when a +metric is divided by \(R^2\). Consequently, on passing from working +scale \(R_j\) to \(R_{j+1}\), the round estimate becomes +\[ + \mathcal D(h^{[j+1]}/R_{j+1}^2,t^{[j+1]}) + \ge \frac{R_{j+1}^2}{2} + +L_*^2\bigl(1-(n+1)\delta\bigr)(t^{[j+1]})^2. +\] +By \eqref{rounds:delta}, the coefficient of the final square is greater +than one. Thus \eqref{rounds:start} is restored. In original units the +reserve remains exactly \(1/2\); it has not been spent by the change +of scale. + +Each fixed label becomes inactive after finitely many rounds, since +its separation is at least \(WR_j\) as long as it remains active and +\(R_j\to\infty\). A compact set meets only finitely many of the +original outer balls. Each of their labels is processed only finitely +many times. Thus there are only finitely many changes on a neighborhood +of any compact set over the \emph{entire} construction, not merely in +each round. The limiting data \((h,t)\) are smooth, satisfy +\[ + h\ge g,\qquad t\ge T,\qquad \mathcal D(h,t)\ge\frac12, +\] +and satisfy \eqref{rounds:prescribed}. The same comparison argument as +in \Cref{rounds:round} proves completeness of \(h\). + +It remains to realize the drift term by ordinary finite-dimensional +warped products on each compact window. Enumerate the nonzero compactly +supported increments of \(t\) as \(\tau_\nu\ge0\), so that +\(t-T=\sum_\nu\tau_\nu\). The supports are locally finite. For each +\(\nu\), choose an integer \(m_\nu\ge1\) such that +\[ + \frac1{m_\nu}\sup_M|\dd\tau_\nu|_h^2\le\frac{2^{-\nu}}{10}. +\] +The supremum is finite because the support is compact. Use \(m_\nu\) +fresh log warps, all equal to \(-\tau_\nu/m_\nu\). Listing these +functions as \((p_\alpha)\), we obtain +\[ + \sum_\alpha p_\alpha=-(t-T),\qquad + \sum_\alpha|\dd p_\alpha|_h^2\le\frac1{10}. +\] +On a neighborhood meeting only finitely many supports, equip a finite +product of circles with metric +\(h+\sum_\alpha e^{2p_\alpha}\dd\theta_\alpha^2\), omitting all unused +unit factors. Its scalar curvature is the left side of +\eqref{rounds:stabilized}; this is the identity in \Cref{rounds:warp-identity}. Substitution gives +\(\mathcal D(h,t)-\sum_\alpha|\dd p_\alpha|_h^2\ge1/2-1/10=2/5\). +This proves \eqref{rounds:stabilized}. +\end{proof} + +\begin{remark}\label{rounds:no-infinite-torus} +The list of warps is bookkeeping for compatible finite-dimensional +calculations. We never construct an infinite-dimensional torus. +Adding unit factors changes neither the weight nor the scalar formula. +After the radial partition in \Cref{countable:construction}, only +finitely many initial warp supports occur on any whole packet; later +stages add finite groups of factors. This stronger finiteness is used +in the one-dimensional obstruction, not just local finiteness of a +series on the original manifold. +\end{remark} diff --git a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7f4b7518849d336ad8464a2942632116c0f42c8c --- /dev/null +++ b/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f67c30579306bd4dd256ece4d24cdad18ba149930fa97f9ab31a1e6a768cef89 +size 864828 diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/README.md b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..e80f4bee8d203d8c64ea34c4ad0faacbcd0bc5b3 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/README.md @@ -0,0 +1,17 @@ +# [Positive scalar curvature forces rational inessentiality](paper.pdf) + +OpenAI +September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026, + author = {{OpenAI}}, + title = {{Positive scalar curvature forces rational inessentiality}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/figures/degree-triad.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/figures/degree-triad.tex new file mode 100644 index 0000000000000000000000000000000000000000..4062e9e850d42ddbe5a04713ec710745467f7b43 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/figures/degree-triad.tex @@ -0,0 +1,28 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\small] + \fill[blue!3] (-3,-1.55) rectangle (3,1.55); + \draw[very thick,blue!65!black] (-3,-1.55)--(-3,1.55); + \draw[very thick,blue!65!black] (3,-1.55)--(3,1.55); + \draw[very thick,orange!75!black] (-3,-1.55)--(3,-1.55); + \draw[very thick,orange!75!black] (-3,1.55)--(3,1.55); + \draw[thick,->] (-3,-.35) .. controls (-1.8,.5) and (1.8,-.5) .. (3,.35); + \draw[thick,->,purple!75!black] (-.45,-1.55) .. controls (.8,-.5) and (-.8,.5) .. (.45,1.55); + \fill (0,0) circle (2pt); + \node[above right] at (.15,.1) {intersection}; + \node[below] at (-1.9,-.03) {$j(N)$}; + \node[left,purple!75!black] at (-.45,-1) {$A_+$}; + \node[left,blue!65!black] at (-3,0) {$E_R$}; + \node[right,blue!65!black] at (3,0) {$E_R$}; + \node[above,orange!75!black] at (1.9,1.55) {$E_B$}; + \node[below,orange!75!black] at (1.9,-1.55) {$E_B$}; + \node at (-2.1,1.05) {$P$}; +\end{tikzpicture} +\caption{A schematic transverse section of the compact triad. +The relative cycle $j(N)$ has boundary on the radial side $E_R$ and avoids +$E_B$. Its dual representative $A_+$ has boundary on $E_B$ and is pushed +off $E_R$. Their nonzero intersection becomes a local degree after the +parameter is pulled back and doubled. The drawing records incidence and +boundary conditions, not the dimensions of the cycles.} +\label{fig:degree-triad} +\end{figure} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/figures/direction-splitting.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/figures/direction-splitting.tex new file mode 100644 index 0000000000000000000000000000000000000000..a829efd172473f44d5568592cb34c6e852db442b --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/figures/direction-splitting.tex @@ -0,0 +1,25 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[>=Latex,font=\small, + block/.style={draw,rounded corners=2pt,minimum height=1.2cm,align=center}] + \node[block,fill=blue!5,minimum width=3.2cm] (m) at (0,0) + {Original fibre directions\\$\widetilde M$, dimension $n$}; + \node[block,fill=orange!7,minimum width=3.3cm] (a) at (3.65,0) + {Parameter directions\\$A$, dimension $q-n$}; + \node[block,fill=purple!5,minimum width=3.0cm] (t) at (7.2,0) + {Stabilizing circles\\$\T^{kq}$, dimension $kq$}; + \draw[thick] (-1.6,-.8)--(-1.6,-.95)--(5.3,-.95)--(5.3,-.8); + \node[align=center] (z) at (1.85,-1.45) + {Base $Z$, dimension $q$}; + \draw[->] (z.south)--node[right] {$F_k$, nonzero degree} (1.85,-2.65); + \node at (1.85,-2.9) {$S^q$}; + \node[align=center] at (7.2,-1.6) + {Scalar curvature is controlled\\on $Z\times\T^{kq}$}; +\end{tikzpicture} +\caption{The local direction splitting after $k$ finite passes. The base +$Z$ is locally a product of original and parameter directions; no global +product decomposition is asserted. The inverse-metric trace uses only the +$n$ original directions. The sphere map uses the full base of dimension +$q$, which is fixed before the number $kq$ of circles is chosen.} +\label{fig:direction-splitting} +\end{figure} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/main.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..10015487a5cad66ae3b5e8cc1b8491fa50e2e70f --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/main.tex @@ -0,0 +1,146 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\input{glyphtounicode} +\pdfglyphtounicode{angbracketleftBigg}{27E8} +\pdfglyphtounicode{angbracketleftbig}{27E8} +\pdfglyphtounicode{angbracketrightBigg}{27E9} +\pdfglyphtounicode{angbracketrightbig}{27E9} +\pdfglyphtounicode{arrowhookleft}{02D3} 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+\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\Q}{\mathbb Q} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\T}{\mathbb T} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\Scal}{Scal} +\DeclareMathOperator{\Ric}{Ric} +\DeclareMathOperator{\Hess}{Hess} +\DeclareMathOperator{\diver}{div} +\DeclareMathOperator{\Vol}{Vol} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\sgn}{sgn} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\DeclarePairedDelimiter{\ip}{\langle}{\rangle} +\setlist[itemize]{topsep=4pt,itemsep=2pt} +\setlist[enumerate]{topsep=4pt,itemsep=2pt} +\setlength{\emergencystretch}{1.5em} +\allowdisplaybreaks[2] +\title{Positive scalar curvature forces rational inessentiality} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that every closed connected oriented smooth manifold admitting +strictly positive scalar curvature is rationally inessential: its rational +fundamental class maps to zero under the classifying map. No spin or +fundamental-group hypothesis is needed, and there is no upper dimension bound. +This proves the Gromov--Lawson aspherical conjecture; moreover, every +nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. +We also show that every closed oriented smooth manifold of positive dimension +with nonnegative scalar curvature has zero real simplicial volume, proving the +qualitative vanishing consequence of Gromov's conjectural comparison. +\end{abstract} +\begingroup +\small +\tableofcontents +\endgroup +\input{sections/01-introduction} +\input{sections/02-reductions} +\input{sections/03-comparison} +\input{sections/04-graph-calculus} +\input{sections/05-graph-existence} +\input{sections/06-shortening} +\input{sections/07-obstruction} +\input{sections/08-proof} +\begingroup +\small +\phantomsection +\addcontentsline{toc}{section}{References} +\let\originalthebibliography\thebibliography +\renewcommand{\thebibliography}[1]{% + \originalthebibliography{#1}\setlength{\itemsep}{1pt}} +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/references.bib b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..281bd7aefe9d3841c4830a15153f4c2e89ea578e --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/references.bib @@ -0,0 +1,420 @@ + +@article{Gromov1982, + author = {Gromov, Mikhael}, + title = {Volume and bounded cohomology}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {56}, + year = {1982}, + pages = {5--99}, + doi = {10.1007/BF02698699}, + url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/435.pdf}, + note = {\href{https://doi.org/10.1007/BF02698699}{DOI}} +} + +@incollection{Gromov1986, + author = {Gromov, Mikhael}, + title = {Large {Riemannian} manifolds}, + booktitle = {Curvature and Topology of Riemannian Manifolds}, + editor = {Shiohama, Katsuhiro and Sakai, Takashi and Sunada, Toshikazu}, + series = {Lecture Notes in Mathematics}, + volume = {1201}, + publisher = {Springer}, + address = {Berlin}, + year = {1986}, + pages = {108--121}, + doi = {10.1007/BFb0075649}, + url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/452.pdf}, + note = {\href{https://doi.org/10.1007/BFb0075649}{DOI}} +} + +@incollection{Gromov2023, + author = {Gromov, Misha}, + title = {Four lectures on scalar curvature}, + booktitle = {Perspectives in Scalar Curvature. Vol. 1}, + publisher = {World Scientific}, + address = {Hackensack, NJ}, + year = {2023}, + pages = {1--514}, + doi = {10.1142/9789811273223_0001}, + eprint = {1908.10612}, + archivePrefix = {arXiv}, + note = {Section locators refer to the author's version: \href{https://arxiv.org/abs/1908.10612v6}{arXiv:1908.10612v6}. \href{https://doi.org/10.1142/9789811273223_0001}{DOI}}, + url = {https://arxiv.org/abs/1908.10612v6} +} + +@incollection{Hanke2011, + author = {Hanke, Bernhard}, + title = {Positive scalar curvature, {$K$}-area and essentialness}, + booktitle = {Global Differential Geometry}, + editor = {B{\"a}r, Christian and Lohkamp, Joachim and Schwarz, Matthias}, + series = {Springer Proceedings in Mathematics}, + volume = {17}, + publisher = {Springer}, + address = {Berlin}, + year = {2012}, + pages = {275--302}, + doi = {10.1007/978-3-642-22842-1_10}, + url = {https://arxiv.org/abs/1011.3987v2}, + note = {Definition and conjecture numbers refer to \href{https://arxiv.org/abs/1011.3987v2}{arXiv:1011.3987v2}. \href{https://doi.org/10.1007/978-3-642-22842-1_10}{DOI}} +} + +@article{CecchiniSchick2021, + author = {Cecchini, Simone and Schick, Thomas}, + title = {Enlargeable metrics on nonspin manifolds}, + journal = {Proceedings of the American Mathematical Society}, + volume = {149}, + number = {5}, + year = {2021}, + pages = {2199--2211}, + doi = {10.1090/proc/14706}, + url = {https://arxiv.org/abs/1810.02116v4}, + note = {\href{https://doi.org/10.1090/proc/14706}{DOI}; author's version: \href{https://arxiv.org/abs/1810.02116v4}{arXiv:1810.02116v4}} +} + +@misc{MaWangXieYuZhu2026, + author = {Ma, Qiaochu and Wang, Jinmin and Xie, Zhizhang and Yu, Guoliang and Zhu, Bo}, + title = {{Gromov}'s simplicial volume vanishing conjecture for positive scalar curvature}, + year = {2026}, + eprint = {2606.29135}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + note = {\href{https://arxiv.org/abs/2606.29135v4}{arXiv:2606.29135v4}, 28 August 2026}, + url = {https://arxiv.org/abs/2606.29135v4} +} + +@misc{BiZhu2026, + author = {Bi, Yuchen and Zhu, Jintian}, + title = {Positive scalar curvature obstructions via singular dimension descent}, + year = {2026}, + eprint = {2606.20528}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + note = {\href{https://arxiv.org/abs/2606.20528v3}{arXiv:2606.20528v3}, 11 September 2026}, + url = {https://arxiv.org/abs/2606.20528v3} +} + +@article{Thom1954, + author = {Thom, Ren{\'e}}, + title = {Quelques propri{\'e}t{\'e}s globales des vari{\'e}t{\'e}s diff{\'e}rentiables}, + journal = {Commentarii Mathematici Helvetici}, + volume = {28}, + year = {1954}, + pages = {17--86}, + doi = {10.1007/BF02566923}, + note = {\href{https://doi.org/10.1007/BF02566923}{DOI}} +} + +@book{Hatcher2002, + author = {Hatcher, Allen}, + title = {Algebraic Topology}, + publisher = {Cambridge University Press}, + address = {Cambridge}, + year = {2002}, + isbn = {0-521-79540-0}, + url = {https://pi.math.cornell.edu/~hatcher/AT/AT.pdf}, + note = {\href{https://pi.math.cornell.edu/~hatcher/AT/AT.pdf}{Author's text}} +} + +@article{Hamilton1982, + author = {Hamilton, Richard S.}, + title = {Three-manifolds with positive {Ricci} curvature}, + journal = {Journal of Differential Geometry}, + volume = {17}, + number = {2}, + year = {1982}, + pages = {255--306}, + doi = {10.4310/jdg/1214436922}, + note = {\href{https://doi.org/10.4310/jdg/1214436922}{DOI}} +} + +@book{Topping2006, + author = {Topping, Peter}, + title = {Lectures on the {Ricci} Flow}, + series = {London Mathematical Society Lecture Note Series}, + volume = {325}, + publisher = {Cambridge University Press}, + address = {Cambridge}, + year = {2006}, + doi = {10.1017/CBO9780511721465}, + url = {https://warwick.ac.uk/fac/sci/maths/people/staff/peter_topping/rfnotes/topping_rf_mar06.pdf}, + note = {Theorem locators refer to the author's notes of 9 March 2006. \href{https://doi.org/10.1017/CBO9780511721465}{DOI}} +} + +@book{GilbargTrudinger2001, + author = {Gilbarg, David and Trudinger, Neil S.}, + title = {Elliptic Partial Differential Equations of Second Order}, + edition = {second}, + series = {Classics in Mathematics}, + publisher = {Springer}, + address = {Berlin}, + year = {2001}, + isbn = {978-3-540-41160-4}, + doi = {10.1007/978-3-642-61798-0}, + note = {\href{https://doi.org/10.1007/978-3-642-61798-0}{DOI}} +} + +@misc{SchoenYau2017, + author = {Schoen, Richard and Yau, Shing-Tung}, + title = {Positive scalar curvature and minimal hypersurface singularities}, + year = {2017}, + eprint = {1704.05490}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + note = {\href{https://arxiv.org/abs/1704.05490v1}{arXiv:1704.05490v1}}, + url = {https://arxiv.org/abs/1704.05490v1} +} + +@article{GromovLawson1983, + author = {Gromov, Mikhael and Lawson, Jr., H. Blaine}, + title = {Positive scalar curvature and the {Dirac} operator on complete {Riemannian} manifolds}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {58}, + year = {1983}, + pages = {83--196}, + doi = {10.1007/BF02953774}, + url = {https://www.numdam.org/item/PMIHES_1983__58__83_0/}, + note = {\href{https://doi.org/10.1007/BF02953774}{DOI}} +} + +@article{SchoenYauStructure1979, + author = {Schoen, Richard and Yau, Shing-Tung}, + title = {On the structure of manifolds with positive scalar curvature}, + journal = {Manuscripta Mathematica}, + volume = {28}, + year = {1979}, + pages = {159--183}, + doi = {10.1007/BF01647970}, + note = {\href{https://doi.org/10.1007/BF01647970}{DOI}} +} + +@article{SchoenYauIncompressible1979, + author = {Schoen, Richard and Yau, Shing-Tung}, + title = {Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {110}, + number = {1}, + year = {1979}, + pages = {127--142}, + doi = {10.2307/1971247}, + url = {https://annals.math.princeton.edu/1979/110-1/p04}, + note = {\href{https://doi.org/10.2307/1971247}{DOI}} +} + +@article{ChodoshLi2024, + author = {Chodosh, Otis and Li, Chao}, + title = {Generalized soap bubbles and the topology of manifolds with positive scalar curvature}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {199}, + number = {2}, + year = {2024}, + pages = {707--740}, + doi = {10.4007/annals.2024.199.2.3}, + url = {https://annals.math.princeton.edu/2024/199-2/p03}, + eprint = {2008.11888}, + archivePrefix = {arXiv}, + note = {Author's version: arXiv:2008.11888v4} +} + +@incollection{ChodoshLiSurvey2023, + author = {Chodosh, Otis and Li, Chao}, + title = {Recent results concerning topological obstructions to positive scalar curvature}, + booktitle = {Perspectives in Scalar Curvature. Vol. 2}, + chapter = {6}, + publisher = {World Scientific}, + year = {2023}, + pages = {215--230}, + doi = {10.1142/9789811273230_0006}, + url = {https://web.stanford.edu/~ochodosh/PSC-aspherical-survey.pdf}, + note = {Numbered locators refer to the \href{https://web.stanford.edu/~ochodosh/PSC-aspherical-survey.pdf}{authors' survey}. \href{https://doi.org/10.1142/9789811273230_0006}{DOI}} +} + +@misc{GromovAsphericalFive2020, + author = {Gromov, Misha}, + title = {No metrics with positive scalar curvatures on aspherical 5-manifolds}, + year = {2020}, + eprint = {2009.05332}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + doi = {10.48550/arXiv.2009.05332}, + url = {https://arxiv.org/abs/2009.05332v1}, + note = {Version 1, 11 September 2020; \href{https://arxiv.org/abs/2009.05332v1}{arXiv:2009.05332v1}} +} + +@article{BrunnbauerHanke2010, + author = {Brunnbauer, Michael and Hanke, Bernhard}, + title = {Large and small group homology}, + journal = {Journal of Topology}, + volume = {3}, + number = {2}, + year = {2010}, + pages = {463--486}, + doi = {10.1112/jtopol/jtq014}, + eprint = {0902.0869}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/0902.0869v2}, + note = {Theorem numbers refer to arXiv:0902.0869v2} +} + +@article{BraunSauer2021, + author = {Braun, Sabine and Sauer, Roman}, + title = {Volume and macroscopic scalar curvature}, + journal = {Geometric and Functional Analysis}, + volume = {31}, + year = {2021}, + pages = {1321--1376}, + doi = {10.1007/s00039-021-00588-y}, + url = {https://doi.org/10.1007/s00039-021-00588-y}, + note = {\href{https://doi.org/10.1007/s00039-021-00588-y}{DOI}} +} + +@article{Guth2011, + author = {Guth, Larry}, + title = {Volumes of balls in large {Riemannian} manifolds}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {173}, + number = {1}, + year = {2011}, + pages = {51--76}, + doi = {10.4007/annals.2011.173.1.2}, + url = {https://annals.math.princeton.edu/2011/173-1/p02}, + note = {\href{https://doi.org/10.4007/annals.2011.173.1.2}{DOI}} +} + +@article{Cairns1934, + author = {Cairns, Stewart S.}, + title = {On the Triangulation of Regular Loci}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {35}, + number = {3}, + year = {1934}, + pages = {579--587}, + doi = {10.2307/1968752}, + note = {\href{https://doi.org/10.2307/1968752}{DOI}} +} + +@article{Whitehead1940, + author = {Whitehead, J. H. C.}, + title = {On {$C^1$}-complexes}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {41}, + number = {4}, + year = {1940}, + pages = {809--824}, + doi = {10.2307/1968861}, + note = {\href{https://doi.org/10.2307/1968861}{DOI}} +} + +@book{Munkres1966, + author = {Munkres, James R.}, + title = {Elementary Differential Topology}, + edition = {revised}, + series = {Annals of Mathematics Studies}, + volume = {54}, + publisher = {Princeton University Press}, + address = {Princeton, NJ}, + year = {1966} +} + +@article{Sard1942, + author = {Sard, Arthur}, + title = {The measure of the critical values of differentiable maps}, + journal = {Bulletin of the American Mathematical Society}, + volume = {48}, + number = {12}, + year = {1942}, + pages = {883--890}, + doi = {10.1090/S0002-9904-1942-07811-6}, + url = {https://webhomes.maths.ed.ac.uk/~v1ranick/papers/sard.pdf}, + note = {\href{https://doi.org/10.1090/S0002-9904-1942-07811-6}{DOI}} +} + +@article{Milnor1968, + author = {Milnor, John}, + title = {A note on curvature and fundamental group}, + journal = {Journal of Differential Geometry}, + volume = {2}, + year = {1968}, + pages = {1--7}, + doi = {10.4310/jdg/1214501132}, + note = {\href{https://doi.org/10.4310/jdg/1214501132}{DOI}} +} + +@article{CheegerGromoll1971, + author = {Cheeger, Jeff and Gromoll, Detlef}, + title = {The splitting theorem for manifolds of nonnegative {Ricci} curvature}, + journal = {Journal of Differential Geometry}, + volume = {6}, + number = {1}, + year = {1971}, + pages = {119--128}, + doi = {10.4310/jdg/1214430220}, + note = {\href{https://doi.org/10.4310/jdg/1214430220}{DOI}} +} + +@incollection{GromovDozen2018, + author = {Gromov, Misha}, + title = {A dozen problems, questions and conjectures about positive scalar curvature}, + booktitle = {Foundations of Mathematics and Physics One Century After Hilbert}, + editor = {Kouneiher, Joseph}, + publisher = {Springer}, + address = {Cham}, + year = {2018}, + pages = {135--158}, + doi = {10.1007/978-3-319-64813-2_6}, + note = {Section locators refer to the author's version of 19 September 2017. \href{https://doi.org/10.1007/978-3-319-64813-2_6}{DOI}; \href{https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/dozen_problems_Sept_2017.pdf}{author's PDF}}, + url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/dozen_problems_Sept_2017.pdf} +} + +@article{BishopONeill1969, + author = {Bishop, R. L. and O'Neill, B.}, + title = {Manifolds of negative curvature}, + journal = {Transactions of the American Mathematical Society}, + volume = {145}, + year = {1969}, + pages = {1--49}, + doi = {10.1090/S0002-9947-1969-0251664-4}, + note = {\href{https://doi.org/10.1090/S0002-9947-1969-0251664-4}{DOI}} +} + +@incollection{Wei2007, + author = {Wei, Guofang}, + title = {Manifolds with a lower {Ricci} curvature bound}, + booktitle = {Surveys in Differential Geometry. Vol. XI}, + series = {Surveys in Differential Geometry}, + volume = {11}, + publisher = {International Press}, + address = {Somerville, MA}, + year = {2007}, + pages = {203--227}, + doi = {10.4310/SDG.2006.v11.n1.a7}, + url = {https://web.math.ucsb.edu/~wei/paper/06survey.pdf}, + note = {\href{https://doi.org/10.4310/SDG.2006.v11.n1.a7}{DOI}} +} + +@article{Yau2000, + author = {Yau, Shing-Tung}, + title = {Review of geometry and analysis}, + journal = {Asian Journal of Mathematics}, + volume = {4}, + number = {1}, + year = {2000}, + pages = {235--278}, + doi = {10.4310/AJM.2000.v4.n1.a16}, + url = {https://archive.intlpress.com/site/pub/files/_fulltext/journals/ajm/2000/0004/0001/AJM-2000-0004-0001-a016.pdf}, + note = {\href{https://doi.org/10.4310/AJM.2000.v4.n1.a16}{DOI}} +} + +@article{SchoenYau1981, + author = {Schoen, Richard and Yau, Shing-Tung}, + title = {Proof of the positive mass theorem. {II}}, + journal = {Communications in Mathematical Physics}, + volume = {79}, + pages = {231--260}, + year = {1981}, + note = {\href{https://archive.ymsc.tsinghua.edu.cn/pacm_download/59/11069-Shing-Tung_Yau_105.pdf}{Journal article}} +} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/01-introduction.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..6048f1045211e4063da0824151592107e05c6979 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/01-introduction.tex @@ -0,0 +1,331 @@ +\section{Introduction} +\label{sec:introduction} + +Let $M^n$ be a closed connected oriented smooth manifold, let +$\Gamma=\pi_1(M)$, and let $c_M:M\to B\Gamma$ be a classifying map +for its universal cover. Here $B\Gamma$ is a CW $K(\Gamma,1)$: a connected +CW complex with fundamental group $\Gamma$ and contractible universal cover. +The manifold is +\emph{rationally essential} if +\[ + (c_M)_*[M]\ne0\quad\text{in }H_n(B\Gamma;\Q), +\] +and \emph{rationally inessential} otherwise. The fundamental class in this +definition has rational coefficients. The condition is independent of the +choice of classifying map. It records whether the fundamental class remains +visible after passing from the manifold to the topology of its fundamental +group. + +We prove the rational-essentiality obstruction to positive scalar curvature +formulated in \cite[Conjecture~2.9]{Hanke2011}. + +\begin{theorem}[Rational inessentiality] +\label{thm:main} +Let $M^n$ be a closed connected oriented smooth manifold of dimension +$n\ge2$. If $M$ admits a Riemannian metric of strictly positive scalar +curvature, then +\[ + (c_M)_*[M]=0\qquad\text{in }H_n(B\pi_1(M);\Q). +\] +\end{theorem} + +The theorem has no spin, universal-cover spin, or fundamental-group +hypothesis, and no upper dimension bound. In particular, the group may +have torsion and its classifying space need not have a finite-dimensional +model. + +A manifold is \emph{aspherical} if its universal cover is contractible. + +\subsection{From a detected class to contracting maps} + +In the later geometric convention, a closed oriented Riemannian +$n$-manifold is \emph{enlargeable} if, for every $\varepsilon>0$, some cover +with the lifted metric admits an $\varepsilon$-Lipschitz map of nonzero degree +to the unit sphere $S^n$ that is constant outside a compact set. This +definition imposes no spin condition. Rational essentiality supplies a +group-cohomology class whose pullback pairs nontrivially with $[M]$, but it +does not guarantee these contracting maps. Brunnbauer and Hanke construct +rationally essential manifolds which are not enlargeable, as well as +enlargeable manifolds whose universal covers alone do not admit such maps +at every scale, that is, are not hyperspherical +\cite[Theorems~1.4--1.5]{BrunnbauerHanke2010}. Their examples are not aspherical. + +Guth and Braun--Sauer give a different comparison between topology and +metric largeness: at every radius, they guarantee a ball in the universal +cover whose volume is at least a dimensional multiple of the radius to the +$n$th power, for closed aspherical and rationally essential manifolds, +respectively \cite{Guth2011,BraunSauer2021}. These bounds do not have the +sharp Euclidean leading constant, so they do not determine the scalar-curvature +term in the small-ball volume expansion. + +Our proof converts the cohomology pairing into a local degree by adding a +parameter manifold, then changes the metric to contract the resulting map +while retaining positive scalar curvature. The main estimate makes the scalar +cost grow only logarithmically with the number of comparison coordinates. +Suppose for contradiction that $\Scal_M\ge\kappa>0$ and that $M$ is +rationally essential. The construction has four stages. + +\paragraph{Finite comparison and local degree.} +At each sufficiently large finite word-metric comparison scale $R$, an +auxiliary closed oriented manifold $C$ with a principal left $\Gamma$-cover +$\widehat C\to C$ gives the cover +$X=(\widetilde M\times\widehat C)/\Gamma\to M\times C$. The detecting cocycle +yields a compactly supported cohomology class on $X$ that evaluates +nontrivially on its locally finite fundamental class. Distance profiles supply a preliminary +map $\mathcal V:X\to\R^q$, for a finite integer +$q>n$. Together with the parameter projection $\bar o:X\to C$, it defines +the map $x\mapsto(\bar o(x),\mathcal V(x))$ into $C\times\R^q$. + +The construction transfers the compact pairing to a relative cohomology class +on a compact region of this product, evaluated nontrivially on a relative +cycle carried by the preceding map. Relative duality gives a complementary +relative rational homology class, and rational realization represents a nonzero multiple +by a compact oriented $(q-n)$-manifold with boundary mapped to +$C\times\R^q$. Its transverse intersections with the preceding map have +nonzero signed count. Doubling the parameter manifold and reflecting its map +gives a closed oriented manifold $A$ with a map $(o_0,y_0):A\to C\times\R^q$. +Pulling back $\widehat C\to C$ along $o_0$ gives a principal cover +$\widehat A\to A$ and the fibre product +\[ + Z=X\times_C A=(\widetilde M\times\widehat A)/\Gamma + \longrightarrow M\times A. +\] +This cover has dimension $q=n+\dim A$, and both $\mathcal V$ and $y_0$ pull +back to $\R^q$-valued maps on it. +On a precompact open set $\Omega$ over one half of $A$, the zeros of their +scaled difference $(\mathcal V-y_0)/R$ are precisely the preceding +intersections. A cutoff equal to one on $\Omega$ makes this difference a +smooth compactly supported map $G:Z\to\R^q$ satisfying +\[ + \deg(G,\Omega,0)\ne0, + \qquad \|G\|_\infty>3\quad\text{on }\partial\Omega, +\] +where $\|\cdot\|_\infty$ is the maximum of the absolute target coordinates. +The scalar coordinates satisfy +\begin{equation}\label{eq:introduction-data} + \max_i|d_{\widetilde M}G_i|\le J/R, + \qquad \log q\le JR, +\end{equation} +with $J$ independent of $R$. Here $d_{\widetilde M}$ differentiates in the +fibre directions, holding the parameter in $A$ fixed, and its norm uses the +lifted metric of $M$. Parameter derivatives are bounded at each fixed $R$ +and are controlled later by metric enlargement. +\Cref{sec:comparison} constructs these data without assuming that +$\widetilde M$ is contractible or that the cocycle is bounded. + +\paragraph{A graph and one warped circle per coordinate.} +A graph change increases the fibre metric in the direction of one function's +differential, and a reciprocal warped circle supplies favorable scalar terms. +A nonlinear elliptic equation chooses a replacement coordinate close to the +old one while shortening its differential. Processing all $q$ coordinates +once is called a \emph{pass}. Paying an independent worst-case scalar cost +for every coordinate would introduce a factor $q$ and destroy the scalar +lower bound. Instead, the decreases of the inverse metric telescope on the +$n$-dimensional fibre cotangent space. Their sum has trace at most $n$ in the +metric at the start of the pass, independently of the numbers of coordinates +and circles. The scalar identity in \cref{sec:graph-calculus}, the graph +solution in \cref{sec:graph-existence}, and the summation in +\cref{sec:shortening} bound the fibre scalar-curvature loss by +$C(n+1)(2+\log q)L^2/w^2$, for an absolute constant $C$, when the input row +norms are at most $L$ and each replacement is allowed coordinate error +$\pi w/2$. + +\paragraph{Finite shortening with a summable cost.} +Successive passes divide $L$ by four and $w$ by two, so their coordinate +errors and fibre scalar losses form summable series. The initial error +allowance makes the uniform total error in each coordinate less than $\pi/10$, +preserving the maximum-norm boundary gap. Composing the shortened coordinate +map on $\Omega$ with a degree-one map $\R^q\to S^q$ that is constant outside +the unit ball, and extending constantly outside $\Omega$, preserves this +degree. By \eqref{eq:introduction-data}, +the total fibre scalar loss is $O((2+\log q)R^{-2})=O(R^{-1})$. +At each fixed finite stage, enlarging +the metric on $A$ makes the parameter part of the differential small and +makes the assembled scalar curvature uniformly close to the fibre scalar +curvature. For $R$ sufficiently large and every finite integer $k\ge1$, +there are a metric $g_k$ on $Z$, positive warps $m_{k,\nu}$, and a smooth map +$F_k:Z\to S^q$, constant outside a compact set and of nonzero degree, such +that +\[ + \Scal_{g_k+\sum_{\nu=1}^{kq}m_{k,\nu}^2d\theta_\nu^2}\ge\kappa/2, + \qquad + |dF_k|_{g_k,\mathrm{op}}\le C(q)4^{-k}J/R. +\] +The scalar bound concerns $Z\times\T^{kq}$, while the differential bound +uses all the base directions in $Z$. +\Cref{fig:direction-splitting} distinguishes these directions from the +stabilizing circles. + +\input{figures/direction-splitting} + +\paragraph{An obstruction independent of the circle count.} +Pulling back a torus tube in $S^q$ gives a compact band $W\subset Z$ carrying +the degree. Doubling $W$ and retaining the circle factors gives a closed +oriented manifold $\operatorname{dbl}(W)\times\T^{kq}$ mapped with nonzero +degree to $S^1\times\T^{q-1}\times\T^{kq}$. The local torical theorem of +Cecchini--Schick requires a +scalar lower bound and a bound on the full differential only on the full +preimage of one connected open target arc. The doubled metric therefore needs +to agree with the constructed metric only there. Constant enlargement of the +stabilizing circles controls their joint projection without scalar loss. +\Cref{sec:obstruction} derives +an obstruction threshold depending on $q$ and $\kappa$, but not on the +number $kq$ of circles. We first choose $R$ large enough to control the total +fibre scalar loss, thereby fixing $q$ and all comparison data, and then choose a +sufficiently large finite $k$ to contradict this obstruction. + +\subsection{Consequences} + +For a connected aspherical manifold, the classifying map is a homotopy +equivalence. Thus every closed connected oriented aspherical manifold is +rationally essential. The main theorem gives the following consequence, with the +nonorientable case obtained from the orientation double cover. + +\begin{corollary}[The Gromov--Lawson aspherical conjecture] +\label{cor:aspherical} +No closed smooth aspherical manifold, whether orientable or not, admits a +Riemannian metric of strictly positive scalar curvature. +\end{corollary} + +For nonnegative scalar curvature, the corresponding statement determines +the given metric. + +\begin{corollary}[Aspherical nonnegative-scalar-curvature rigidity] +\label{cor:aspherical-flat} +Let $g$ be a Riemannian metric of nonnegative scalar curvature on a closed +smooth aspherical manifold $M$, not necessarily oriented. Then $g$ is flat. +\end{corollary} + +Both aspherical statements apply componentwise and have no spin assumption. +In dimensions zero and one scalar curvature vanishes +identically, and every metric is flat. Gromov already noted the rigidity +implication conditionally on the aspherical positive-scalar-curvature +obstruction \cite[Section~3, following Conjecture~B]{GromovDozen2018}. +Here the proof combines \cref{lem:scalar-reduction} with the compact-manifold +consequence of the Cheeger--Gromoll splitting theorem. + +A nonzero-degree map to an aspherical target forces rational essentiality by +naturality of classifying maps; no injectivity or surjectivity condition on +the induced map of fundamental groups is needed. The following is the +oriented form of the target-map conjecture recorded by Chodosh and Li +\cite[Conjecture~18]{ChodoshLiSurvey2023}. Their survey also states the +connected-sum obstruction in dimensions four and five as a consequence of +Gromov's work \cite[Theorem~17]{ChodoshLiSurvey2023}. + +\begin{corollary}[Oriented target-map obstruction] +\label{cor:degree-aspherical} +Let $M^n$ and $N^n$ be closed connected oriented smooth manifolds of +dimension $n\ge2$, with $N$ aspherical. If a continuous map $f:M\to N$ +has nonzero ordinary oriented degree $d\in\Z\setminus\{0\}$, then $M$ +admits no Riemannian metric whose scalar curvature is strictly positive +everywhere. + +In particular, if $P^n$ is any closed connected oriented smooth manifold, +the oriented connected sum $N\#P$ admits no such metric. +\end{corollary} + +For a closed connected oriented $n$-manifold, its \emph{real simplicial +volume} is +\[ + \|M\|=\inf\left\{\sum_i|a_i|: + \sum_i a_i\sigma_i\text{ is a real singular cycle representing }[M]\right\}. +\] +Positive simplicial volume implies rational essentiality, by the +bounded-cohomology mapping theorem and duality for this seminorm +(\cref{lem:essentiality}). Therefore rational inessentiality implies zero +simplicial volume. The converse fails: a positive-dimensional torus is +rationally essential but has zero simplicial volume, as follows from its +self-covers of degree greater than one. Its flat metric also shows why +strict positivity in \cref{thm:main} cannot be replaced by nonnegative +scalar curvature. For simplicial volume the nonnegative hypothesis does +suffice. + +\begin{corollary}[Simplicial-volume vanishing] +\label{cor:simplicial} +Every closed oriented smooth manifold of positive dimension +admitting a Riemannian metric of nonnegative scalar curvature has zero +real simplicial volume. +\end{corollary} + +On each connected component, \cref{sec:reductions} reduces a +nonnegative-scalar-curvature metric to either a positive-scalar-curvature +metric or a Ricci-flat metric. The positive branch uses \cref{thm:main}; +the Ricci-flat branch uses volume comparison and amenability, without +requiring rational inessentiality in that case. + +\subsection{The obstruction problem and earlier work} + +Two approaches have shaped scalar-curvature obstructions. Schoen and Yau +used stable two-sided minimal hypersurfaces and conformal deformation to +transfer positive scalar curvature to lower dimensions +\cite{SchoenYauStructure1979,SchoenYauIncompressible1979}. Their original +obstructions requiring no spin assumption apply in dimensions at most seven, +including to manifolds mapping with nonzero degree to tori. Gromov and Lawson +developed Dirac-operator obstructions using enlargeability and relative +indices on complete manifolds \cite{GromovLawson1983}. Their 1983 +enlargeability convention requires oriented spin coverings, and their compact +obstruction has no upper dimension bound. The later geometric convention +used above omits this spin requirement. The spin condition in geometric +index applications and the regularity of minimizing hypersurfaces are thus +different constraints. + +More recently, Chodosh and Li established the aspherical obstruction in +dimensions four and five using generalized soap bubbles +\cite{ChodoshLi2024}. Gromov independently treated the closed +five-dimensional case, and also complete-manifold domination problems under +properness and uniform-positivity or decay hypotheses +\cite{GromovAsphericalFive2020}. +\Cref{cor:aspherical} establishes the closed-aspherical statement in every +dimension. + +The homological formulation in \cref{thm:main} concerns more manifolds than +the aspherical class. Hanke explicitly formulates the conjecture that +rationally essential manifolds do not admit positive scalar curvature +\cite[Definition~2.8 and Conjecture~2.9]{Hanke2011}. It is also the +classifying-map form, for closed oriented sources, of Gromov's conjecture +that a nontorsion homology class in the classifying space of a countable group +cannot be the image of the fundamental class of a positive-scalar-curvature manifold +\cite[Section~1.5, \(\lbrack\mathrm{Sc}\not>0\rbrack\), p.~19]{Gromov2023}. +The numbered sections cited from that work refer throughout to the author's +arXiv version~6. Yau also posed this homological obstruction in his survey +\cite[p.~265]{Yau2000}. + +Gromov introduced simplicial volume and its bounded-cohomology description +in \cite{Gromov1982}. His quantitative scalar-curvature/simplicial-volume +conjecture asks for a dimension-dependent upper bound on simplicial volume in +terms of Riemannian volume under a fixed scalar-curvature lower bound +\cite[Section~3.13]{Gromov2023}. Rescaling a nonnegative-scalar-curvature +metric in such a comparison would force zero simplicial volume. +\Cref{cor:simplicial} proves this qualitative vanishing consequence. +The quantitative comparison remains a separate question. + +Recent work of Ma, Wang, Xie, Yu and Zhu proves that nonnegative scalar +curvature forces zero real simplicial volume for closed oriented manifolds +with spin universal cover \cite[Corollary~1.2]{MaWangXieYuZhu2026}. Their +stated version has no further group-decay assumption. This conclusion +controls the simplicial norm, rather than the rational classifying image. +Bi and Zhu develop singular hypersurface descent and rule out positive scalar +curvature on closed oriented enlargeable and overtorical manifolds, where an +$n$-manifold is overtorical if it admits a nonzero-degree map to $\T^n$ +\cite[Theorems~A and~B]{BiZhu2026}. Their theorem for complete oriented +sources and complete oriented enlargeable targets retains properness, +nonzero-degree and Lipschitz hypotheses and rules out a uniform positive +scalar lower bound. +Under strict positive scalar curvature, +\cref{thm:main} gives rational classifying-image vanishing without these +geometric-largeness or spin-cover hypotheses. + +The terminal geometric input here is the local torical theorem of Cecchini +and Schick \cite[Theorem~1.14(1)]{CecchiniSchick2021}. It has no spin assumption +or upper dimension bound, and requires both the scalar lower bound and the +full differential bound only on the full preimage of one connected open target +arc. Its higher-dimensional proof uses the minimal-slicing theory of Schoen +and Yau \cite{SchoenYau2017}. + +\Cref{sec:reductions} records the degree conventions and the nonnegative +scalar-curvature reduction. \Cref{sec:comparison} constructs the comparison +data, and \cref{sec:graph-calculus,sec:graph-existence,sec:shortening} +establish the graph stabilization and shortening estimates. +\Cref{sec:obstruction} supplies the stabilized obstruction. +Finally, \cref{sec:main-proof} combines these results and proves the four +corollaries. diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/02-reductions.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/02-reductions.tex new file mode 100644 index 0000000000000000000000000000000000000000..c56197745ca9e7391198d716e5b650e40f5b8ee4 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/02-reductions.tex @@ -0,0 +1,138 @@ +\section{Conventions and the nonnegative scalar-curvature reduction} +\label{sec:reductions} + +We record the degree conventions for the comparison construction and reduce +nonnegative scalar curvature to the positive and Ricci-flat cases used in the +corollaries. All manifolds and metrics are smooth. A closed manifold is compact +and has no boundary. Unless stated otherwise, manifolds are oriented. +Fundamental classes are taken with the indicated coefficients; we use the same +symbol $[M]$ for +an integral fundamental class and its images under extension of coefficients. +The Laplacian is $\Delta=\diver\nabla$, and a round sphere has positive scalar +curvature. For a hypersurface with unit normal $\nu$, our second fundamental +form is $\mathrm{II}(X,Y)=\langle\nabla_X\nu,Y\rangle$ and its mean curvature +is its trace. These conventions fix the signs in the graph calculations below. +We give $S^d$ its unit round metric and write +$\T^b=(\R/2\pi\Z)^b$ for the standard flat torus. + +\subsection{Fundamental classes and degree} + +Throughout the proof, $\Gamma=\pi_1(M)$ and $c_M:M\to B\Gamma$ +have the meaning fixed in \cref{sec:introduction}. No finite-dimensional +model for $B\Gamma$ is assumed. Disconnected closed manifolds are treated +one component at a time; compactness leaves only finitely many components. + +For a possibly noncompact oriented $d$-manifold $Y$, a smooth map +$F:Y\to S^d$ that is equal to one value $*\in S^d$ outside a compact set has +a well-defined degree. One may compute it as the signed number of preimages +of any regular value different from $*$. Those preimages are contained in a +compact set. Equivalently, pull back a compactly supported top-degree class on +$S^d\setminus\{*\}$ and evaluate it on the locally finite fundamental class +of $Y$. This definition also permits disconnected $Y$: only finitely many +components can meet the compact region on which $F$ is nonconstant. We use +\emph{constant outside a compact set} in this sense, rather than assigning a +linear meaning to compact support for a sphere-valued map. + +We also use the ordinary oriented local degree $\deg(G,\Omega,0)$ for a +smooth map from an oriented $q$-manifold to $\R^q$, where $\Omega$ is a +precompact open set and $0\notin G(\partial\Omega)$. Smoothness of +$\partial\Omega$ is unnecessary. A homotopy whose boundary values avoid zero +preserves this degree. More generally, if they avoid a fixed ball, composition +with a degree-one map $\R^q\to S^q$ constant outside that ball gives a +sphere-valued map of the same degree, extending constantly across the boundary +of $\Omega$. + +\begin{lemma}[Simplicial volume detects rational essentiality] +\label{lem:essentiality} +If a closed connected oriented manifold $M^n$ has positive real simplicial +volume, then $(c_M)_*[M]\ne0$ in $H_n(B\Gamma;\Q)$. +\end{lemma} + +\begin{proof} +The bounded-cohomology mapping theorem identifies real bounded cohomology of +$M$ with that of $B\Gamma$ by pullback along $c_M$. The duality between +bounded cohomology and the $\ell^1$-seminorm implies that +$\|M\|>0$ supplies a bounded class whose ordinary cohomology image evaluates +nontrivially on $[M]$. These are the mapping and duality results of +\cite{Gromov1982}; neither requires a spin structure. + +If $(c_M)_*[M]$ vanished rationally, its real image would vanish. Every bounded +class pulled back from $B\Gamma$ would then evaluate trivially on $[M]$, a +contradiction. In dimension one, real bounded cohomology already vanishes and +the assertion is vacuous; the direct circle argument below also suffices. +\end{proof} + +\subsection{The scalar-flat alternative} + +\begin{lemma}[Positive scalar curvature or Ricci flatness] +\label{lem:scalar-reduction} +Let $M$ be closed and connected, not necessarily oriented, and suppose that +$g$ has nonnegative scalar curvature. Either $M$ admits a metric of strictly positive scalar curvature, +or $g$ is Ricci-flat. In the Ricci-flat case, if $M$ is oriented and has +positive dimension, then $\|M\|=0$. +\end{lemma} + +\begin{proof} +Run the short-time Ricci flow starting at $g$. Short-time existence on closed +manifolds and the scalar evolution equation give +\begin{equation}\label{eq:scalar-evolution} + \partial_t\Scal=\Delta\Scal+2|\Ric|^2. +\end{equation} +See \cite[Theorem~5.2.1 and Proposition~2.5.4]{Topping2006} for these +statements in arbitrary dimension, and \cite{Hamilton1982} for the +Ricci-flow framework. The maximum principle +preserves nonnegativity. If the initial scalar curvature is positive at one +point, the strong maximum principle makes it positive everywhere at any +sufficiently small later time. If it is initially identically zero but the +Ricci tensor is nonzero at a point, \eqref{eq:scalar-evolution} gives a +strictly positive initial time derivative there. At a sufficiently small positive time $t_1$, scalar curvature is +therefore positive at that point. Applying the strong maximum principle +from $t_1$ produces a metric of positive scalar curvature everywhere at +a later time. Thus failure of this alternative +forces $\Ric_g=0$. + +For the final assertion, suppose that $g$ is Ricci-flat, that $M$ is +oriented, and that $n=\dim M>0$. We use the orbit-packing argument +of Milnor \cite[Theorem~1]{Milnor1968}. Its complete universal Riemannian cover +$(\widetilde M,\widetilde g)$ has nonnegative Ricci curvature. Fix +$x\in\widetilde M$ and a finite generating set of $\Gamma$. There is a +constant $D$ such that +\[ + d(x,\gamma x)\le D|\gamma| + \qquad(\gamma\in\Gamma), +\] +where $|\gamma|$ is word length. Choose a positive radius $r$ for which the +balls $B(\gamma x,r)$, $\gamma\in\Gamma$, are disjoint. To choose $r$, take a sufficiently small metric ball about the image of +$x$ contained in an evenly covered neighborhood. The radius-$r$ balls +about its lifts lie in distinct sheets and are therefore disjoint. All these +balls have the same positive volume, say $v_r$. The orbit balls with +$|\gamma|\le k$ lie in $B(x,Dk+r)$. Bishop--Gromov comparison \cite[Theorem~4.1]{Wei2007} therefore gives +\[ + \#\{\gamma:|\gamma|\le k\}\,v_r + \le \Vol B(x,Dk+r)\le \omega_n(Dk+r)^n. +\] +Thus $\Gamma$ has polynomial word growth. Write +$b(k)=\#B_\Gamma(1,k)$. There are integers $k_j\to\infty$ such that +$b(k_j+1)/b(k_j)\to1$. Otherwise, some $\varepsilon>0$ would satisfy +$b(k+1)/b(k)\ge1+\varepsilon$ for all sufficiently large $k$, forcing +exponential growth. For each generator $s$, +\[ + \#\bigl(sB_\Gamma(1,k_j)\mathbin\triangle B_\Gamma(1,k_j)\bigr) + \le2\bigl(b(k_j+1)-b(k_j)\bigr). +\] +These balls form a F\o lner sequence, so $\Gamma$ is amenable. This +uses only the growth bound, without a structural classification of groups +of polynomial growth. Real bounded +cohomology of an amenable group vanishes in positive degrees +\cite[Sections~3.0--3.1]{Gromov1982}. The mapping +theorem and duality used in \cref{lem:essentiality} now imply +$\|M\|=0$. +\end{proof} + +Every closed connected oriented one-manifold is a circle. Its self-cover of +degree two and functoriality of the simplicial seminorm give +$2\|S^1\|\le\|S^1\|$, hence $\|S^1\|=0$. Consequently, to prove the +simplicial-volume theorem in every positive dimension, it remains to show +that a positive-scalar-curvature manifold of dimension at least two is +rationally inessential. Compactness lets us write its scalar lower bound as +$\Scal_M\ge\kappa>0$. diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/03-comparison.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/03-comparison.tex new file mode 100644 index 0000000000000000000000000000000000000000..acbf4a53aff51fa88e9b68068e0f6038d4234914 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/03-comparison.tex @@ -0,0 +1,870 @@ +\section{Compact pairings and comparison coordinates} +\label{sec:comparison} + +The construction in this section is topological. A Riemannian metric is +used only to measure derivatives in the direction of the original +manifold. In particular, no curvature hypothesis enters the following +statement. For a vector, $\norm{\cdot}_\infty$ is the maximum of the +absolute coordinate values. Bounds asserted on a set are pointwise; +uniform norms take the supremum over the domain. + +\begin{theorem}[Comparison data from rational essentiality] +\label{thm:comparison-data} +Let $(M^n,g_M)$ be a closed connected oriented Riemannian manifold, +$n\ge2$, let $\Gamma=\pi_1(M)$, and suppose that its classifying map +$c_M:M\to B\Gamma$ satisfies +\[ + (c_M)_*[M]\ne0\quad\text{in }H_n(B\Gamma;\Q). +\] +There are constants $J>0$ and $R_0\ge1$ such that, for every real +$R\ge R_0$, there exist a closed oriented smooth manifold $A$, a +principal left $\Gamma$-cover $\widehat A\to A$, and an integer +$q=n+\dim A>n$ with the following properties. The diagonal cover +\[ + Z=(\widetilde M\times\widehat A)/\Gamma\longrightarrow M\times A +\] +is oriented and has dimension $q$. There are a smooth compactly +supported map $G:Z\to\R^q$ and a relatively compact open set +$\Omega\subset Z$ such that +\begin{equation} +\label{eq:comparison-degree} + \deg(G,\Omega,0)\ne0, + \qquad \norm{G}_\infty>3\quad\text{on }\partial\Omega, +\end{equation} +and +\begin{equation} +\label{eq:comparison-bounds} + \max_{1\le i\le q}\sup_Z\abs{d_{\widetilde M}G_i}_{g_M} + \le\frac JR, + \qquad \log q\le JR. +\end{equation} +Write $\pi_Z:Z\to M\times A$ for the covering projection. Its +differential gives a global splitting +\[ + TZ=E_M\oplus E_A,\qquad + E_M=(d\pi_Z)^{-1}(TM\oplus0),\qquad + E_A=(d\pi_Z)^{-1}(0\oplus TA). +\] +The bundle $E_M$ is tangent to the $\widetilde M$ fibres of $Z\to A$, +and $E_A$ is the flat lift of the parameter directions. We write +$d_{\widetilde M}G_i=dG_i|_{E_M}$. For each fixed $R$, after choosing any +smooth metric on $A$, all derivatives of $G$ in the lifted product +geometry are bounded. No bound uniform in $R$ is asserted for +parameter derivatives or for derivatives of order greater than one. +The covers and the parameter manifolds need not be connected. +\end{theorem} + +We use rational coefficients for the chain constructions in this +section. A principal left $\Gamma$-cover is a covering on which $\Gamma$ +acts freely by deck transformations and transitively on each fibre. Every +diagonal quotient below uses left actions in both factors. Orientations +on covers are lifted from the base, so all deck transformations preserve +them. + +The construction proceeds as follows. An auxiliary closed manifold $C$ +provides a finite parameter space for a compactly supported group-cohomology +pairing. Distance profiles then supply $q$ scalar coordinates, with $q$ +initially only a row count. Relative duality produces a complementary +rational class of degree $q-n$. Oriented realization of a nonzero multiple, +followed by doubling, gives the parameter manifold $A$. Thus the final cover has +exactly the dimension needed for a local degree, $n+\dim A=q$; the topology +of the realized $A$ does not alter the previously chosen row count. + +\subsection{Probability maps and a finite parameter manifold} + +Fix a finite symmetric generating set of $\Gamma$, and write +$d_\Gamma$ for its left-invariant word metric. For $a\ge0$, let +$\mathcal K_a$ be the Rips complex realized as finite probability vectors +$t=(t_\gamma)_{\gamma\in\Gamma}$ whose nonempty support has diameter at +most $a$. This complex is locally finite, finite dimensional, and +cocompact under left translation. The action is proper, although it +need not be free when $\Gamma$ has torsion. + +\begin{lemma}[A proper probability map] +\label{lem:probability-map} +There are $s\ge1$, $H<\infty$, a smooth equivariant probability map +$h:\widetilde M\to\mathcal K_s$, and an equivariant simplicial map +$h_0:\widetilde M\to\mathcal K_s$ such that +\begin{enumerate} +\item the coordinate functions of $h$ are smooth and locally supported +on finitely many labels, and $\abs{dh}_{\ell^1}\le H$; +\item $h$ and $h_0$ are joined by an equivariant homotopy that is proper +as a map $\widetilde M\times[0,1]\to\mathcal K_s$; +\item there is an invariant alternating homogeneous rational group +$n$-cocycle $\omega$ whose evaluation on the labelled fundamental +simplices of $M$ is nonzero. +\end{enumerate} +\end{lemma} + +\begin{proof} +Choose a finite smooth triangulation of $M$ by the Cairns--Whitehead +theorem \cite{Cairns1934,Whitehead1940}; see also +\cite[Theorem~10.6]{Munkres1966}. Label the vertex lifts +equivariantly by elements of $\Gamma$, making one arbitrary choice for +each vertex downstairs. Barycentric coordinates define the labelled +simplicial map $h_0$. A smooth partition of unity subordinate to the +open vertex stars, lifted equivariantly, gives $h$; weights with the same +label are added. At each point the labels that occur belong to the +vertices of one lifted simplex. Cocompactness bounds their word +diameters by a fixed $s$, and it also bounds the $\ell^1$-operator norm +of $dh$ by a fixed $H$. Linear interpolation of the weights stays in +$\mathcal K_s$. + +For completeness, this entire homotopy is proper. If $Q\subset +\widetilde M$ is compact with $\Gamma Q=\widetilde M$, its image during +the homotopy is compact. Given a compact $B\subset\mathcal K_s$, +properness of the $\Gamma$-action allows only finitely many translates +of that image to meet $B$. The inverse image of $B$ is therefore a +closed subset of finitely many translates of $Q\times[0,1]$. + +Choose a class in $H^n(B\Gamma;\Q)$ pairing nontrivially with +$(c_M)_*[M]$, and represent it on the homogeneous bar resolution. On +tuple chains, signed permutation averaging is an equivariant chain map +lifting the identity of the augmentation. It is equivariantly chain +homotopic to the identity on this free resolution. Antisymmetrizing +the representative thus gives a cocycle $\omega$ in the same class. +It vanishes on repeated labels and restricts to a simplicial cocycle on +every $\mathcal K_a$. + +Give the vertices of $M$ a global order. Sending an ordered lifted +simplex to its tuple of labels gives an augmentation-preserving +equivariant chain map into the bar resolution. Such a map is unique +up to equivariant chain homotopy: inductively, the difference in each +degree is a cycle in the exact target resolution, and freeness of the +source chains permits equivariant fillings. Comparison with the chain +map induced by a classifying map shows that evaluation of $\omega$ on +these labelled fundamental simplices is precisely its pairing with +$(c_M)_*[M]$, and hence is nonzero. +\end{proof} + +The detecting cocycle is now fixed. To construct a finite smooth parameter +space carrying its compact pairing, we need a free auxiliary group action. +The ordinary Rips action may have torsion stabilizers; keeping the order +of every simplex removes them without requiring contractibility. + +Fix henceforth $R\ge s+1$ and set +\begin{equation} +\label{eq:comparison-scales} + L=100R. +\end{equation} +All auxiliary spaces constructed at this scale may depend on $L$. +Let $\mathcal O_L$ be the semisimplicial complex with one simplex for +each ordered tuple of distinct group elements of diameter at most $L$; +faces are obtained by deleting slots, and different orderings are not +identified. Projection of a tuple to its probability simplex gives +$\pi:\mathcal O_L\to\mathcal K_L$. On chains it has the equivariant +section +\begin{equation} +\label{eq:ordered-section} + \sigma[v_0,\ldots,v_j] + =\frac1{(j+1)!}\sum_{\varpi\in S_{j+1}} + \sgn(\varpi)(v_{\varpi(0)},\ldots,v_{\varpi(j)}). +\end{equation} +Counting the positions of a deleted entry verifies that $\sigma$ is a +chain map, and $\pi_\#\sigma=\id$. + +\begin{lemma}[A parameter thickening] +\label{lem:parameter-thickening} +Put $v_0=\dim\mathcal O_L$. There are a closed oriented smooth +$d$-manifold $C$, with $d\le2v_0+2$, a principal left $\Gamma$-cover +$\widehat C\to C$, and equivariant maps +\[ + \mathcal O_L\xrightarrow{i}\widehat C + \xrightarrow{r}\mathcal K_L, + \qquad r i=\pi. +\] +Moreover $v_0\le\abs{B_\Gamma(1,L)}$. +\end{lemma} + +\begin{proof} +A point in the realization of $\mathcal O_L$ belongs to the interior of +a unique minimal-dimensional ordered simplex. A group element fixing +the point must preserve that ordered simplex, hence must fix its labels +slot by slot. It is therefore the identity. This also proves freeness +at points represented on faces; face identifications do not identify +distinct orderings of the minimal simplex. Local finiteness and the +word-metric bound make the action proper and its quotient finite. +Thus $\mathcal O_L\to\mathcal O_L/\Gamma$ is a principal cover, including +when $\Gamma$ has torsion. The quotient has a finite polyhedral +subdivision. A simplex has at most $\abs{B_\Gamma(1,L)}$ vertices after +one of its vertices is translated to the identity, proving the +dimension bound. + +Embed the finite polyhedron $\mathcal O_L/\Gamma$ in +$\R^{2v_0+1}$ and choose an open neighborhood retracting onto it. +Choose a compact smooth domain inside that neighborhood containing the +polyhedron in its interior, and double the domain. Folding the double +to the domain followed by the neighborhood retraction gives a +retraction from a closed oriented manifold $C$ onto +$\mathcal O_L/\Gamma$. Its dimension satisfies the stated bound. +Pull back the principal cover along this retraction. The inclusion +of the polyhedron lifts to $i$, while the pullback projection followed +by $\pi$ gives $r$. Their composition has the required value. +\end{proof} + +Fix a smooth metric on $C$ and lift it to $\widehat C$. A useful +consequence of continuity and cocompactness is the following uniform +common-label property: sufficiently small subsets of $\widehat C$ have +images under $r$ with a label common to every positive support. Indeed, +at any point a positive coordinate remains positive on a neighborhood; +a finite cover downstairs gives a uniform neighborhood size upstairs. + +\begin{lemma}[A controlled map to dual chains] +\label{lem:dual-carrier} +One can triangulate $C$ so finely that the lifted triangulation has +the following properties: +\begin{enumerate} +\item the union of the supports of $r(o)$ over any closed primal +simplex has diameter at most $2L$; +\item there is an augmentation-preserving equivariant chain map +\[ + \mathcal F:C_*(\mathcal O_L)\longrightarrow + C_*^{\mathrm{dual}}(\widehat C) +\] +with finite output, such that if a dual cell to a primal simplex +$\tau$ occurs in $\mathcal F(e)$, then every label of every $r(o)$, +$o\in\tau$, is within $2L$ of every vertex of $e$. +\end{enumerate} +\end{lemma} + +\begin{proof} +Use a sufficiently fine smooth triangulation, so that both primal +simplices and closed dual blocks have arbitrarily small mesh. If all +images over a primal simplex share a label, every pair of labels in +their union is within $2L$. This gives the first assertion. + +We describe the chain construction explicitly. The regular dual-cell +decomposition has closed cells that are balls and cellular subcomplexes. +Subdivide it barycentrically. A simplex $s'$ of this subdivision is a +flag of dual cells; let $B(s')$ be the closure of its largest cell. +This is the unique minimal closed dual cell containing $s'$. For a +face $t'$ of $s'$ we have $B(t')\subset B(s')$, so these closed balls +form a face-compatible acyclic carrier. + +There is an augmentation-preserving chain map from the subdivision +chains to dual-cell chains carried by $B$. To construct it, send a +vertex to a dual zero-cell in its carrier with coefficient one. If +the map has been defined below dimension $j$, the image of the boundary +of a $j$-simplex is an augmented cellular cycle in its carrier, because +all face carriers are contained there. Fill the cycle within this +contractible cellular subcomplex. Make all choices on simplex orbits +and translate. The deck action on the subdivision is free, so this is +equivariant. Each carrier is finite, and hence each output is finite. + +Now subdivide $\mathcal O_L$ equivariantly and approximate the actual +map $i$ closely by a simplicial map into the subdivision of the dual +decomposition. Choose the target stars inside sufficiently small evenly +covered balls. Simplicial approximation on the finite quotients can then +be joined to the original map by a homotopy staying in those stars. +Lifting that homotopy from $i$ gives an equivariant simplicial endpoint +that remains uniformly close to $i$ upstairs. Equivalently, one may choose +the stars directly on free vertex orbits. Compose source subdivision, +this simplicial chain map, and the carrier map just constructed. This +defines $\mathcal F$. Contractibility of $\mathcal O_L$ is not needed: +the only fillings take place inside the target carrier balls. + +Consider a dual cell occurring on an original simplex $e$. Its +carrier lies within a small mesh error of $i(z)$ for some $z\in e$. +The primal simplex $\tau$ dual to that cell meets it at its barycenter; +small primal mesh therefore puts every $o\in\tau$ within another small +error of such an $i(z)$. Choose all meshes and the approximation close +enough to apply the common-label property to the pair $o,i(z)$. A +common label $v$ belongs both to $r(o)$ and to $r(i(z))$. Since +$ri=\pi$, the latter support consists of vertices of $e$. For a label +$\gamma$ of $r(o)$ and a vertex $w$ of $e$, we obtain +\[ + d_\Gamma(\gamma,w) + \le d_\Gamma(\gamma,v)+d_\Gamma(v,w)\le L+L. +\] +This proves the second assertion. +\end{proof} + +\subsection{A compactly supported pairing} + +For $0\le a\le L$ define +\[ + D_a=(\mathcal K_a\times\widehat C)/\Gamma, + \qquad + X=(\widetilde M\times\widehat C)/\Gamma. +\] +Both spaces project to $C$. The oriented manifold $X$ has dimension +$n+d$, and the probability map induces the graph map +\[ + \begin{array}{ccc} + X&\xrightarrow{\Phi_h}&D_L\\ + \downarrow&&\downarrow\\ + C&=&C + \end{array} + \qquad + \Phi_h([\widetilde x,o])=[h(\widetilde x),o]. +\] +This map is proper, as is the graph homotopy induced by the homotopy in +\cref{lem:probability-map}. To see joint properness, cover compact $C$ +by finitely many compact subsets contained in trivializing charts of +$\widehat C$. In each chart a compact subset of $D_L$ has compact +first-factor projection. Joint properness of the probability homotopy +controls the preimage in the $\widetilde M$ factor, and the parameter subset is compact. +A finite union gives the assertion. + +The chain construction has a specific target: cap each simplex with the +detecting cocycle, retain the correct augmentation in degree $n$, and keep +the complementary dual cells close to the original labels. The following +lemma packages these properties before cellular duality is applied. + +\begin{lemma}[A controlled cap operation] +\label{lem:controlled-cap} +There are equivariant linear maps with finite output +\begin{equation} +\label{eq:cap-chain-map} + \begin{gathered} + T_j:C_j(\mathcal K_L;\Q)\longrightarrow + C_{j-n}^{\mathrm{dual}}(\widehat C;\Q),\qquad j\ge0,\\ + \partial T_j=(-1)^nT_{j-1}\partial,\qquad j\ge1, + \end{gathered} +\end{equation} +where chain groups in negative degrees are zero. For every oriented +$n$-simplex $e$, the augmentation of $T_n(e)$ is $\omega(e)$. For every +oriented $j$-simplex $e$, if a dual cell to a primal simplex $\tau$ occurs +in $T_j(e)$, then every label of +$r(o)$, for every $o\in\tau$, is within $3L$ of every vertex of $e$. +\end{lemma} + +\begin{proof} +On an oriented $j$-simplex $e=[v_0,\ldots,v_j]$ define the averaged +Alexander--Whitney diagonal by +\[ + \Delta_{\mathrm{AW}}e + =\frac1{(j+1)!}\sum_{\varpi\in S_{j+1}}\sgn(\varpi) + \sum_{k=0}^j + [v_{\varpi(0)},\ldots,v_{\varpi(k)}] + \otimes[v_{\varpi(k)},\ldots,v_{\varpi(j)}]. +\] +Each ordering gives the usual diagonal on that simplex and its faces. +Restriction of a uniformly chosen ordering to a face is again uniformly +distributed, so the average commutes with the boundary. It is equivariant, +and both factors in every term are faces of $e$. + +Apply $\omega$ to the first factor of degree $n$ and $\mathcal F\sigma$ +to the second factor, and call the resulting graded map $T$. Its outputs +are finite. In the tensor boundary, the sum of terms from the first factor +vanishes after evaluation by $\omega$ because $\delta\omega=0$. The boundary in the +second factor carries the sign $(-1)^n$, proving +\eqref{eq:cap-chain-map}. When $j=n$, each contributing second factor is a +vertex. Its image under $\mathcal F\sigma$ has augmentation one, while +alternation makes every signed first-factor evaluation equal to $\omega(e)$. +Thus the augmentation of $T_n(e)$ is exactly $\omega(e)$. + +Finally, every output of $T_j(e)$ comes from a face of $e$. By +\cref{lem:dual-carrier}, labels over its corresponding primal simplex are +within $2L$ of every vertex of that face. Since $e$ has diameter at most +$L$, they are within $3L$ of every vertex of $e$. +\end{proof} + +Cellular duality will turn the finite output of $T$ in dual chains into primal +cochains. Its support bound will make the resulting product class compactly supported, +and its augmentation will recover the original cocycle pairing. + +\begin{proposition}[Compact diagonal pairing] +\label{prop:compact-pairing} +There are a compact set $K_*\subset D_L$ and a class +\[ + u\in H^{n+d}(D_L,D_L\setminus K_*;\Q) +\] +such that +\begin{enumerate} +\item if $[t,o]\in K_*$, every label of $t$ is within $7L$ of +every label of $r(o)$; +\item the compactly supported evaluation +\begin{equation} +\label{eq:compact-pairing-nonzero} + \ip{\Phi_h^*u,[X]_{\mathrm{lf}}}\ne0 +\end{equation} +is nonzero. +\end{enumerate} +Here $[X]_{\mathrm{lf}}$ is the locally finite fundamental class. +\end{proposition} + +\begin{proof} +Cellular duality on the oriented cover identifies finite dual chains +of degree $k$ with compactly supported primal cochains of degree +$d-k$. Write this identification as $I$, choosing the degree signs +so that $\delta I=I\partial$. At degree zero choose the sign for which +evaluation on the oriented locally finite fundamental cycle is the +augmentation; the remaining degree signs are then determined +successively by the boundary--coboundary identity. On product cells, +define a cochain $U$ of degree $n+d$ by +\begin{equation} +\label{eq:product-cocycle} + \begin{split} + U(e^j,e'^{\,n+d-j})&=a_j I(T(e))(e'),\\ + a_n&=1,\qquad + a_j=(-1)^{j+n+1}a_{j-1}\quad(j>n), + \end{split} +\end{equation} +and set it to zero outside the indicated chain degrees. The cochain +is invariant under simultaneous translation and thus descends to +$D_L$. The product-cell structure is valid even if the action on +$\mathcal K_L$ has inversions: the freely lifted second-factor cells +give trivial stabilizers of product cells. + +On a product cell of bidegree $(j,n+d+1-j)$, the two terms in +$\delta U$ are $a_{j-1}IT(\partial e)$ and +$(-1)^ja_jI\partial T(e)$. By \cref{eq:cap-chain-map}, their +coefficients relative to $IT(\partial e)$ are +$a_{j-1}$ and $(-1)^{j+n}a_j$, which cancel. Thus $U$ is a cocycle. + +Call a product cell \emph{far} if all distances between vertices of +its first simplex and labels of $r$ over its closed second simplex +are greater than $4L$. This condition is inherited by faces and is +translation invariant, so the far cells form a subcomplex +$D_{\mathrm{far}}\subset D_L$. The support bound in +\cref{lem:controlled-cap} shows that +$U$ vanishes on it. Let $K_*$ be the union of the closures of all +non-far cells in the quotient. There are only finitely many such +cells: after fixing a second-factor simplex from finitely many orbit +representatives, non-farness puts some first-factor vertex within +$4L$ of a label in a fixed finite set; the other vertices lie within +$L$ of that vertex. Local finiteness of the word metric gives only +finitely many choices. + +The same argument gives the required support bound. One pair of +labels in a non-far product is within $4L$, the first simplex has +diameter at most $L$, and the union of second-factor labels has +diameter at most $2L$. Every pair in its closure is consequently +within $4L+L+2L=7L$. Furthermore +$D_L\setminus K_*\subset D_{\mathrm{far}}$, so the relative cellular +class of $U$ induces the required class $u$. + +It remains to compute the pairing. By proper homotopy invariance of +compactly supported cohomology, replace $h$ by $h_0$ and write +$\Phi_0$ for the resulting graph map. On the product cell structure +of $X$, each open cell maps to an open product cell, possibly of +smaller dimension. Thus +$X_{\mathrm{far}}=\Phi_0^{-1}(D_{\mathrm{far}})$ is a subcomplex. +Every cell not in it lies in the compact set $\Phi_0^{-1}(K_*)$, and +there are finitely many such cells by local finiteness. The locally +finite fundamental class maps successively to +\[ + H_{n+d}(X,X\setminus\Phi_0^{-1}(K_*);\Q) + \longrightarrow H_{n+d}(X,X_{\mathrm{far}};\Q). +\] +In the latter group it is represented by the finite product +fundamental cellular chain modulo $X_{\mathrm{far}}$. Its coefficient +on each remaining top cell is the local orientation coefficient, +as follows by restricting further to local homology at an interior +point. Hence evaluation by the relative cellular cocycle computes +the desired compact pairing. + +Only bidegree $(n,d)$ contributes. For each oriented fundamental +top simplex of $M$, choose one lift and sum over all oriented +fundamental top simplices of $\widehat C$. The latter sum evaluates +$IT(e)$ on the locally finite parameter fundamental cycle and equals +the augmentation of $T(e)$, namely $\omega(e)$. Collapsed labelled +top simplices contribute zero. The total is exactly the nonzero +evaluation from \cref{lem:probability-map}, proving +\cref{eq:compact-pairing-nonzero}. +\end{proof} + +\subsection{Distance profiles} + +The compact pairing detects the original fundamental class. We next encode +its supporting region by finitely many scalar coordinates whose fibre +derivatives have a scale-independent bound. Their row count will determine +the dimension of the final parameter manifold only after these estimates +have been fixed. + +Choose a finite cover of $C$ by small trivializing charts $U_\lambda$, +with $d+1$ colors and with charts of each color pairwise disjoint. +One obtains this by taking open stars of the vertices of a sufficiently +fine barycentric subdivision and coloring them by the dimensions of +the original simplices. Shrink this finite cover and choose smooth +functions $0\le w_\lambda\le1$, supported in $U_\lambda$, so that at +least one equals one at each point of $C$. On every sheet over a chart +choose an anchor $g_\lambda(o)\in\Gamma$, constant on the sheet and +equivariant under left translation, which is a common label of $r$ +over that sheet. Thus it is within $L$ of every label of every $r(o)$ +there. + +For each color and each $b\in B_\Gamma(1,1000L)$ use one row of a +vector $V_o(t)$. Outside the charts of that color its value is zero; +on the sheet of a chart $U_\lambda$ it is +\begin{equation} +\label{eq:distance-profile-row} + w_\lambda(\bar o)\sum_{\gamma\in\Gamma}t_\gamma + \bigl(d_\Gamma(\gamma,g_\lambda(o)b) + -d_\Gamma(g_\lambda(o),g_\lambda(o)b)\bigr), + \qquad \bar o=[o]. +\end{equation} +The disjointness within each color makes this definition unambiguous. +Compact support of each cutoff inside its chart makes it smooth in +the parameter wherever the probability coordinates are fixed. Left +invariance of the word metric makes the vector invariant under +simultaneous translation, and hence it is defined on $D_L$. + +Pad the vector by zero rows if necessary so that its number $q$ of +coordinates exceeds $n$. Since the word balls have at most exponential +growth and $d\le2\abs{B_\Gamma(1,L)}+2$, there is a constant $J_1$, +independent of $R$, such that +\begin{equation} +\label{eq:profile-count} + \log q\le J_1R. +\end{equation} +Choose an even integer $p_R\ge2$ with $q^{1/p_R}\le3$. The norm +$\norm{\cdot}_{p_R}$ is smooth off zero and satisfies +\begin{equation} +\label{eq:profile-norms} + \norm{z}_\infty\le\norm{z}_{p_R}\le3\norm{z}_\infty. +\end{equation} + +\begin{lemma}[Profile estimates] +\label{lem:distance-profiles} +The map $V:D_L\to\R^q$ has the following properties. +\begin{enumerate} +\item It is continuous and proper, and +$\norm{V}_{p_R}\le24L$ on $K_*$. +\item If $t,t'\in\mathcal K_s$ satisfy +\[ + \max\{\norm{V_o(t)}_\infty,\norm{V_o(t')}_\infty\}\le110L, + \qquad + \norm{V_o(t)-V_o(t')}_\infty\le14R, +\] +then the union of their supports has diameter at most $L$. +\item The map +$\mathcal V=V\Phi_h:X\to\R^q$ is smooth. With $K_0=sH$, +each of its rows satisfies +\[ + \sup_X\abs{d_{\widetilde M}\mathcal V_i}\le K_0, +\] +where $K_0$ is independent of $R$. +\end{enumerate} +\end{lemma} + +\begin{proof} +Continuity follows from the coordinate definition and local finiteness. +At any point choose a chart whose cutoff equals one, and use its +$b=1$ row. This row is the nonnegative average distance to its +anchor $g$. For $t\in\mathcal K_L$ and any +$\gamma\in\supp t$, the triangle inequality gives +\[ + d_\Gamma(\gamma,g) + \le\sum_{\gamma'}t_{\gamma'}d_\Gamma(\gamma',g)+L + \le\norm{V_o(t)}_\infty+L. +\] +Over finitely many trivializing charts, a bound on $V$ therefore +restricts the first-factor labels to finitely many possibilities. +Closed inverse images of compact subsets are consequently compact. +On $K_*$, every input label is within $7L+L=8L$ of the active anchor. +Each distance difference in \cref{eq:distance-profile-row} is bounded +by that distance to the anchor, giving +$\norm{V}_\infty\le8L$, and hence the stated $24L$ bound. + +For the second assertion, use a chart of weight one common to both +probabilities and let $g$ be its anchor. Every label of either +probability is within $110L+s$ of $g$. In particular a label +$\gamma\in\supp t$ equals $gb$ for one of the rows being used. +In this row the average of $d_\Gamma(\,\cdot\,,\gamma)$ over $t$ +is at most $s$. The row comparison implies that the average over +$t'$ is at most $s+14R$. Since $\supp t'$ has diameter at most $s$, +every $\gamma'\in\supp t'$ satisfies +\[ + d_\Gamma(\gamma',\gamma)\le14R+2s4R\ \text{on }E_B, + \qquad \rho<7R\ \text{throughout }P. +\end{equation} + +There is a map $\psi:P\to D_L$ over $C$ with +\begin{equation} +\label{eq:witness-approximation} + \norm{V\psi(\bar o,y)-y}_\infty<7R. +\end{equation} +Indeed, at each point choose a witness in $\mathcal K_s$ satisfying +this inequality, and continue it as a constant probability vector in +a small sheet trivialization. It remains a witness on a neighborhood. +A finite partition of unity gives a convex mixture of these local +witnesses, interpreted in the common fiber. Each witness has profile +norm at most $T_0+7R<110L$, and two witnesses at the same point have +profiles differing by less than $14R$. By +\cref{lem:distance-profiles}, all labels in their union have pairwise +distance at most $L$. Thus every mixture belongs to +$\mathcal K_L$. Linearity of $V$ in probability coordinates proves +\cref{eq:witness-approximation}. + +On $E_R$ its image satisfies +\[ + \norm{V\psi}_{p_R}\ge T_0-21R>24L, +\] +so $\psi(E_R)$ avoids $K_*$. Consequently +\[ + u_P=\psi^*u\in H^{n+d}(P,E_R;\Q) +\] +is defined. The compact domain +\[ + N=\{x\in X:\norm{\mathcal V(x)}_{p_R}\le T_0\} +\] +has smooth boundary and maps by +\[ + j=(\bar o,\mathcal V):(N,\partial N)\longrightarrow(P,E_R). +\] +Its image is disjoint from $E_B$, since $\rho$ is zero there. At a +point of $N$, add $h(\widetilde x)$ to the family of witnesses used +by $\psi j$. The same support test shows that linear interpolation +between $\psi j$ and $\Phi_h|_N$ stays in $D_L$. On $\partial N$, +the profile norm throughout this homotopy is at least $T_0-21R>24L$. +Since $\Phi_h^{-1}(K_*)$ is in the interior of $N$, restriction and +excision, followed by this homotopy, give +\begin{equation} +\label{eq:relative-pairing} + \ip{u_P,j_*[N,\partial N]}\ne0. +\end{equation} + +\Cref{fig:degree-triad} records the complementary boundary conditions +used in the following duality argument. + +\begin{lemma}[A transverse parameter cycle] +\label{lem:relative-degree} +There is a compact oriented smooth $(q-n)$-manifold $A_+$ with +boundary, together with a smooth map +\[ + (o_0,y_0):A_+\longrightarrow C\times\R^q, +\] +such that $\norm{y_0}_{p_R}4R$ on $\partial A_+$, and its intersections with +$j:N\to C\times\R^q$ are transverse, interior, and have nonzero +total oriented count. The component maps may be taken constant in +the inward collar coordinate near $\partial A_+$. +\end{lemma} + +\begin{proof} +Poincar\'e--Lefschetz duality for the triad +$(P;E_R,E_B)$ \cite[Theorem~3.43]{Hatcher2002} sends $u_P$ to a class +\[ + \alpha\in H_{q-n}(P,E_B;\Q), +\] +because $\dim P=d+q$. Its relative intersection with +$j_*[N,\partial N]$ is, up to the fixed orientation convention, +\cref{eq:relative-pairing}. + +We spell out the relative realization using the absolute theorem of +Thom \cite[Theorem~III.4]{Thom1954}. Write $k=q-n$ and clear +denominators in $\alpha$. Choose an integral relative cycle $z$ in +$(P,E_B)$ representing the resulting class. Form the double +$D=P\cup_{E_B}P'$ along the angular face. Compatible product collars +at $E_R\cap E_B$ give $D$ the structure of a smooth compact manifold +with boundary; the seam $E_B$ is a separating hypersurface meeting the +remaining boundary neatly. If the seam is empty, this double is the +disjoint union of the two copies. Write $z'$ for the reflected chain. +Because the boundary chains on the seam agree, $z-z'$ is an absolute +cycle in $D$. The restriction and excision map +\[ + H_k(D;\Z)\longrightarrow H_k(D,P';\Z) + \cong H_k(P,E_B;\Z) +\] +sends $[z-z']$ to $[z]$. + +The compact manifold $D$ has finite polyhedral type. Transfer the doubled +class to a finite polyhedron homotopy equivalent to $D$, apply Thom's +theorem there, and compose back to $D$. This supplies a closed oriented +smooth $k$-manifold $B$ and a map +$f:B\to D$ whose fundamental class maps to a nonzero integer multiple +of $[z-z']$. Reverse the orientation if necessary to make this +multiple positive. Push $f$ away from the remaining boundary of $D$ +by an inward collar deformation, chosen tangent to the seam. Smooth +$f$ and perturb it transversely to the seam. These operations are +homotopies in $D$, and compactness keeps the image away from the +boundary of the seam. Thus $Q=f^{-1}(E_B)$ is a closed smooth +hypersurface, and $B_+=f^{-1}(P)$ and $B_-=f^{-1}(P')$ are compact +oriented smooth manifolds with common boundary $Q$. Give +$B_+$ the orientation restricted from $B$. The map +$H_k(B;\Z)\to H_k(B,B_-;\Z)\cong H_k(B_+,Q;\Z)$ sends $[B]$ to +$[B_+,Q]$: collar excision and the local orientation classes on the +interior of $B_+$ characterize this restriction. Components wholly +in $B_-$ vanish in the relative group. Naturality for the map of +pairs $(B,B_-)\to(D,P')$, followed by the displayed target +excision map, therefore gives +\[ + (f|_{B_+})_*[B_+,\partial B_+] + =\text{a positive integer multiple of }\alpha + \quad\text{in }H_k(P,E_B;\Q). +\] +This proves the required relative realization without prescribing +boundary data or requiring the representing map to be an embedding. + +Write $a_+:B_+\to P$ for this representative and push its image off +$E_R$ by a collar deformation respecting $E_B$. At a corner one uses a +field pointing inward from the radial side and tangent to the angular +side; transversality of the two defining functions supplies such a field. +Compactness now separates $a_+(B_+)$ from $E_R$, and already separates +$j(N)$ from $E_B$. Consequently all coincidences of $a_+$ and $j$ lie in +a compact subset of the source interiors, with target values in +$\operatorname{int}P$. + +Perturb $a_+$ only near this compact coincidence set. More precisely, +choose finitely many target coordinate neighborhoods with closures in +$\operatorname{int}P$ and use small flows of vector fields supported there +to make $a_+\times j$ transverse to the diagonal in $P\times P$. +Compactness permits the perturbation to be small enough that it creates +no coincidences outside these neighborhoods. The whole homotopy remains +a map of pairs $(B_+,\partial B_+)\to(P,E_B)$, is fixed near +$\partial B_+$, and keeps the image separated from $E_R$. Thus it +preserves the relative class and its pairing with $j_*[N,\partial N]$. +The dimension sum is $(q-n)+(n+d)=q+d$, so the transverse intersections +are isolated and have the asserted nonzero signed count. Set $A_+=B_+$ +and write $(o_0,y_0)$ for the perturbed map into $E=C\times\R^q$. + +Finally take a sufficiently small boundary collar, free of +intersections, and precompose the map there by a smooth collar +retraction that is constant in the inward collar coordinate near the +boundary and is the identity away from a slightly larger collar. +The image on that collar stays in a region where the strict +separation inequalities hold. The resulting map is smooth, has the +same interior intersections, and has the claimed collar form. +\end{proof} + +\input{figures/degree-triad} + +\begin{proof}[Proof of \cref{thm:comparison-data}] +Double $A_+$ along its boundary, reversing the orientation on the +second copy, to obtain a closed oriented manifold $A$. Reflect both +component maps $o_0,y_0$ across the seam. The collar conclusion of +\cref{lem:relative-degree} makes the resulting maps smooth. Pull back +$\widehat C\to C$ by $o_0:A\to C$, obtaining $\widehat A\to A$, and +form +\[ + Z=X\times_C A=(\widetilde M\times\widehat A)/\Gamma. +\] +The projection $X\to C$ is a submersion, so $Z$ is a smooth +$q$-manifold even though $o_0$ need not be a submersion. Local sheet +trivializations identify it with a product of the +$\widetilde M$ fibre directions and the $A$ directions, with their product +orientation. This also exhibits it as the stated cover of $M\times A$. +Continue to write $\mathcal V$ for the pulled-back profile map. + +Choose once and for all a smooth function $\eta:[0,\infty)\to[0,1]$ +equal to one on $[0,2]$ and zero on $[3,\infty)$, and define +\begin{equation} +\label{eq:comparison-map} + G=\eta\bigl(\norm{\mathcal V}_{p_R}/T_0\bigr) + \frac{\mathcal V-y_0}{R}. +\end{equation} +Properness of the profile map and compactness of $A$ imply that its +support is compact. It is smooth, including at $\mathcal V=0$, +because the cutoff is constant in a neighborhood of that locus. +Write $a:Z\to A$ for the parameter projection, and let +\[ + \Omega=\{z\in Z:\ a(z)\in\operatorname{int}A_+, + \norm{\mathcal V(z)}_{p_R}<2T_0\}. +\] +This is relatively compact. On its angular boundary the defining +boundary inequality for $A_+$ gives +$\norm{\mathcal V-y_0}_\infty>4R$. On its other boundary, +\[ + \norm{\mathcal V-y_0}_{p_R} + \ge2T_0-\norm{y_0}_{p_R}>T_0, +\] +and hence $\norm{\mathcal V-y_0}_\infty>T_0/3$. +The cutoff equals one on both parts of the boundary. In view of +$T_0>100L=10000R$, these bounds prove the boundary assertion in +\cref{eq:comparison-degree}. + +The zeros in $\Omega$ are exactly the intersections furnished by +\cref{lem:relative-degree}: if $\mathcal V=y_0$, their common +$p_R$-norm is less than $T_0$, so the point belongs to $N$. +We check orientations to identify their signed count. In oriented +local lifts, the difference of the two map differentials, in tangent +variables $(v_M,v_C,v_A)$ and target rows $(C,\R^q)$, is +\[ + \begin{pmatrix} + 0&I&-do_0\\ + d_{\widetilde M}\mathcal V&d_C\mathcal V&-dy_0 + \end{pmatrix}. +\] +Substitute $v_C=v'_C+do_0(v_A)$. This substitution has determinant +one. Moving the $C$ variables to the front introduces only the fixed +dimensional sign $(-1)^{nd}$, and eliminating their identity block +leaves the differential of the pulled-back $\mathcal V-y_0$ on $Z$. +Thus the intersection signs agree with its zero signs up to one +fixed sign, independent of the point, sheet, and component. Division +by the positive number $R$ does not change them. Their nonzero sum +is therefore $\deg(G,\Omega,0)$ up to that sign. Doubling the parameter +may cancel a global intersection count between its two copies; it +does not change the local degree over the first copy used here. + +It remains to check the derivative bound. By +\cref{lem:distance-profiles,eq:profile-norms}, +\[ + \abs{d_{\widetilde M}\mathcal V_i}\le K_0, + \qquad + \abs{d_{\widetilde M}\norm{\mathcal V}_{p_R}}\le3K_0 + \quad(\mathcal V\ne0). +\] +In the cutoff transition region, +$\norm{\mathcal V-y_0}_\infty<4T_0$. Writing +$E_\eta=\sup\abs{\eta'}$, differentiation of +\cref{eq:comparison-map} gives for every row +\[ + \abs{d_{\widetilde M}G_i} + \le\frac{K_0}{R} + +\frac{3K_0E_\eta}{T_0}\frac{4T_0}{R} + \le\frac{K_0(1+12E_\eta)}{R}. +\] +The estimate also holds where the cutoff is constant. Together with +\cref{eq:profile-count}, it proves \cref{eq:comparison-bounds} with +$J=\max\{J_1,K_0(1+12E_\eta),1\}$, increasing $R_0$ if necessary. +Finally $G$ is a smooth compactly supported map on the lifted product +geometry of the cover $Z\to M\times A$; each of its covariant +derivatives therefore has finite supremum for this fixed $R$. +\end{proof} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/04-graph-calculus.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/04-graph-calculus.tex new file mode 100644 index 0000000000000000000000000000000000000000..9e9ba5d9422a65c84e85cbd8e173bf2a36141e5a --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/04-graph-calculus.tex @@ -0,0 +1,191 @@ +\section{A scalar-curvature identity for graph stabilization} +\label{sec:graph-calculus} + +One shortening step increases a metric in the direction of a single +function's differential and adds one circle. The circle warp compensates +for the scalar terms introduced by the graph. Warped-circle formulas are +also central to the stabilization used in minimal slicing +\cite[Lemma~2.5]{SchoenYau2017}; here we compute their combination with a +rank-one graph change before choosing the graph by a nonlinear equation. + +We use the Laplacian $\Delta=\diver\nabla$ and the convention for which +round spheres have positive scalar curvature. The base $(B,g)$ may have +any finite dimension and may already contain warped circle factors. All +derivatives not explicitly assigned another metric are taken with respect +to $g$. + +Let $f\in C^\infty(B)$ and let $N:[0,\infty)\to(0,\infty)$ be smooth and constant near zero. Set +\begin{equation}\label{eq:graph-quantities} + \begin{gathered} + t=\abs{df}_g,\qquad W=(1+N(t)^2t^2)^{1/2},\qquad v=W^{-1},\\ + U=vN(t)\nabla f,\qquad r=\abs{U}_g=vN(t)t,\qquad + D=-\diver_g U. + \end{gathered} +\end{equation} +Although $t$ and $r$ need not be smooth where $df=0$, the functions $N(t)$ +and $v$ and the vector field $U$ are smooth there. Indeed, $N(t)$ is +constant near such points, and the other two are smooth functions of $df$. +We use identities involving $\log t$ or $\log r$ only where $t>0$; the +resulting scalar estimates extend by continuity. Consider +\begin{equation}\label{eq:graph-metric-update} + g^+=g+N(t)^2df^2+v^2d\theta^2 + \quad\text{on }B\times\T. +\end{equation} +The first two terms form the metric induced on the graph of $f$ in +$(B\times\R,g+N(t)^2dz^2)$, where $N(t)$ is regarded as a fixed function of the base point, independent of $z$. + +\begin{proposition}[Exact scalar-curvature identity]\label{prop:scalar-update} +At a point where $t>0$, choose a $g$-orthonormal frame +$e_1=\nabla f/t,e_2,\ldots,e_{\dim B}$, and put +\[ + \mathfrak p=\frac{tN'(t)}{N(t)},\qquad + A=d\log t,\qquad + C_{jk}=t^{-1}\Hess_g f(e_j,e_k)\quad(j,k>1). +\] +Then +\begin{equation}\label{eq:scalar-exact} + \begin{split} + \Scal_{g^+}-\Scal_g + ={}&D^2-2U(D)+r^2\abs{C}^2+(e_1r)^2\\ + &+2r^2(1+\mathfrak p)(v^2-\mathfrak p r^2) + \sum_{j>1}A(e_j)^2. + \end{split} +\end{equation} +In particular, the final term is nonnegative whenever +$\mathfrak p\ge0$ and $\mathfrak p N(t)^2t^2\le1$. +For $N\equiv1$, one also has, everywhere, +\begin{equation}\label{eq:ordinary-scalar-update} + \Scal_{g^+}-\Scal_g=\abs{II}^2+D^2-2U(D), +\end{equation} +where $II$ is the second fundamental form of the ordinary product graph. +The inverse graph metric on base covectors and the volume form of $g^+$ are +\begin{equation}\label{eq:graph-inverse-volume} + P=g^{-1}-U\otimes U, + \qquad dV_{g^+}=dV_g\,d\theta. +\end{equation} +\end{proposition} + +\begin{proof} +Write $E=N^{-1}\partial_z$ and give the graph the upward unit normal +$\nu=-U+vE$. Our convention for the second fundamental form is +$II(X,Y)=\ip{\nabla_X\nu,Y}$, so its trace is +\begin{equation}\label{eq:graph-mean-curvature} + H=D-U(\log N). +\end{equation} +The vertical Killing field $\partial_z$ has normal speed $Nv$ along the +graph, and its tangential part projects to $NvU$ on the base. Vertical +translation preserves the graph's mean curvature as a function of that +base point. The normal and tangential variation terms must therefore +cancel, giving +\begin{equation}\label{eq:graph-jacobi} + \bigl(\Delta_{\rm gr}+\abs{II}^2+\Ric_{\rm amb}(\nu,\nu)\bigr)(Nv) + =Nv\,U(H). +\end{equation} +Here a normal variation of speed $\psi$ changes $H$ by the negative of the +Jacobi operator applied to $\psi$, while a tangential variation +differentiates $H$ in its tangential direction. This fixes the sign on the +right of \eqref{eq:graph-jacobi}. + +The classical warped-product formulas \cite[Section~7]{BishopONeill1969}, +together with the Gauss equation, give, respectively, +\[ + \begin{aligned} + \Scal_{\rm amb}&=\Scal_g-2\Delta_gN/N,\\ + \Scal_{\rm gr}&=\Scal_{\rm amb}-2\Ric_{\rm amb}(\nu,\nu) + +H^2-\abs{II}^2,\\ + \Scal_{g^+}&=\Scal_{\rm gr}-2\Delta_{\rm gr}v/v. + \end{aligned} +\] +Expanding $\Delta_{\rm gr}(Nv)$ in \eqref{eq:graph-jacobi} therefore yields +\begin{equation}\label{eq:scalar-before-cancellation} + \begin{split} + \Scal_{g^+}-\Scal_g + ={}&\abs{II}^2+H^2-2U(H)\\ + &+2\frac{(\Delta_{\rm gr}-\Delta_g)N}{N} + +4\ip{d\log N,d\log v}_{P}. + \end{split} +\end{equation} +The formula for $P$ in \eqref{eq:graph-inverse-volume} follows directly by inverting the rank-one addition $g+N^2df^2$. + +Put $b_0=d\log N$. For a function on the base, +$\Delta_{\rm gr}N$ is its ambient Hessian traced over the graph, minus $H\nu(N)$. Since +$\Hess_{\rm amb}N(E,E)=N\abs{b_0}^2$, this gives +\begin{equation}\label{eq:graph-laplacian-correction} + \frac{(\Delta_{\rm gr}-\Delta_g)N}{N} + =-\Hess_g\log N(U,U)-b_0(U)^2 + +r^2\abs{b_0}^2+H b_0(U). +\end{equation} +Substitute $H=D-b_0(U)$ in \eqref{eq:scalar-before-cancellation}. The Hessian terms cancel because +\[ + U\bigl(b_0(U)\bigr) + =\Hess_g\log N(U,U)+b_0(\nabla^g_UU). +\] +The result is +\begin{equation}\label{eq:scalar-intermediate} + \begin{split} + \Scal_{g^+}-\Scal_g + ={}&\abs{II}^2+D^2-2U(D) + +2b_0(\nabla^g_UU)-3b_0(U)^2\\ + &+2r^2\abs{b_0}^2+4\ip{b_0,d\log v}_{P}. + \end{split} +\end{equation} +When $N\equiv1$, all terms involving $b_0$ vanish. This proves +\eqref{eq:ordinary-scalar-update} without any condition on $D$ other than its definition. + +For the remaining calculation use the graph-orthonormal frame +$T=ve_1+rE,e_2,\ldots,e_{\dim B}$. For base vector fields lifted independently of $z$, the ambient connection satisfies +\[ + \nabla_X^{\rm amb}E=0,\qquad + \nabla_E^{\rm amb}X=b_0(X)E,\qquad + \nabla_E^{\rm amb}E=-b_0^{\sharp}. +\] +Differentiating $\nu=-U+vE$ gives +\begin{equation}\label{eq:graph-second-form} + \begin{aligned} + II(T,T)&=-r\bigl[v^2(1+\mathfrak p)+\mathfrak p\bigr]A(e_1),\\ + II(T,e_j)&=-rv(1+\mathfrak p)A(e_j),\qquad j>1,\\ + II(e_j,e_k)&=-rC_{jk},\qquad j,k>1. + \end{aligned} +\end{equation} +Moreover, +\begin{equation}\label{eq:graph-log-derivatives} + \begin{gathered} + b_0=\mathfrak p A,\qquad + d\log r=v^2(1+\mathfrak p)A,\qquad + d\log v=-r^2(1+\mathfrak p)A,\\ + \nabla^g_UU=r^2\left(v^2(1+\mathfrak p)A(e_1)e_1 + +\sum_{j>1}A(e_j)e_j\right). + \end{gathered} +\end{equation} +For the last equality, symmetry of $\Hess f$ gives +$\nabla_{e_1}e_1=\sum_{j>1}A(e_j)e_j$. + +After substitution in \eqref{eq:scalar-intermediate}, the $C$ terms give +$r^2\abs{C}^2$. The remaining terms are squares of the component of $A$ +along $e_1$ and of its components orthogonal to $e_1$. After division by +$r^2$, the coefficient of $A(e_1)^2$ is +\[ + \begin{split} + &\bigl[v^2(1+\mathfrak p)+\mathfrak p\bigr]^2 + -2\mathfrak p v^2(1+\mathfrak p)-\mathfrak p^2\\ + &\hspace{35mm}=v^4(1+\mathfrak p)^2. + \end{split} +\] +For every $j>1$, the corresponding coefficient is +\[ + 2v^2(1+\mathfrak p)^2-2\mathfrak p(1+\mathfrak p) + =2(1+\mathfrak p)(v^2-\mathfrak p r^2). +\] +Since $e_1r=rv^2(1+\mathfrak p)A(e_1)$, these are precisely +\eqref{eq:scalar-exact}. Also +\[ + v^2-\mathfrak p r^2=v^2\bigl(1-\mathfrak p N(t)^2t^2\bigr), +\] +which proves the positivity assertion. Finally, +$\det(g+N^2df^2)=W^2\det g$ and $vW=1$ prove the volume identity. +\end{proof} + +The inverse-metric decrease will later let us sum the costs of many graph +changes in a fixed $n$-dimensional space. First we must choose each graph +so that the remaining term $D^2-2U(D)$ can be bounded from below. +The next section proves existence for the equation used for this choice. diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/05-graph-existence.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/05-graph-existence.tex new file mode 100644 index 0000000000000000000000000000000000000000..48cb2fbf575e933faa87e252a19f9d57abd89df1 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/05-graph-existence.tex @@ -0,0 +1,256 @@ +\section{Solving the graph equation on a cover} +\label{sec:graph-existence} + +The scalar identity alone does not construct a graph. We now solve a +nonlinear equation prescribing its divergence in terms of the error from +a given function. The strict tangent band controls that error, while +uniqueness will allow the solutions to descend through the flat-bundle +transition maps and remain invariant under previously added circles. + +The graph regularization and gradient-estimate strategy was inspired by +the Jang-equation capillarity method of Schoen--Yau +\cite[Section~4, pp.~251--253]{SchoenYau1981}. The variable flux, tangent +forcing, stabilization identity, and fixed-rank shortening estimates +needed here are developed in this paper; their existence theorem is not +being applied to the present equation. + +We specify the uniformity needed for the nonlinear equation. Let $B$ be a Riemannian cover of a closed manifold, with lifted reference metric $g_{\rm ref}$. Fix nested coordinate charts obtained by lifting a finite collection of relatively compact reference charts, with the smaller charts covering $B$. For $0<\alpha<1$, uniform $C^{j,\alpha}$ norms are the suprema of the usual chart norms on these smaller charts; the larger charts provide room for interior estimates. Equivalent choices give equivalent norms. A metric $g$ is \emph{uniformly controlled} if it is uniformly equivalent to $g_{\rm ref}$ and its coordinate coefficients have bounded derivatives of every order. The metric need not be invariant under deck transformations. We write $C_b^{j,\alpha}$ for these uniform H\"older spaces and $C_b^\infty$ for smooth functions with bounded derivatives of every order. + +Products of such covers with any fixed finite torus have the same properties. In all assertions in this Section, constants may depend on the particular finite-dimensional geometry and the fixed equation coefficients. + +\begin{lemma}[A uniform linear inverse]\label{lem:uniform-linear} +Suppose that, in the reference charts, a scalar operator has the form +\[ + L\xi=-a^{ij}\partial_i\partial_j\xi+b^i\partial_i\xi+c\xi, +\] +with uniformly $C^{0,\alpha}$ coefficients, uniformly bounded drift, uniformly positive definite principal matrix, and $c\ge c_0>0$. Then +\[ + L:C_b^{2,\alpha}(B)\longrightarrow C_b^{0,\alpha}(B) +\] +is an isomorphism. Its inverse is bounded in terms of the indicated uniform bounds. If the coefficients have bounded derivatives of every order, the inverse also maps $C_b^{j,\alpha}$ boundedly into $C_b^{j+2,\alpha}$ for every $j\ge0$. +\end{lemma} + +\begin{proof} +It suffices to prove the assertion when $\dim B\ge2$. If $d=\dim B<2$, +put $T_0=\T^{2-d}$ with its flat product metric and consider +$\overline L=L-\Delta_{T_0}$ on $B\times T_0$, with right-hand side +pulled back from $B$. Its principal matrix is uniformly positive and its +zeroth-order coefficient is still at least $c_0$. The result in dimension +two gives a unique solution; translation invariance of the data and +uniqueness make that solution independent of $T_0$. It therefore descends +to the required solution on $B$. Restriction to a fixed torus point gives +the same uniform estimates, and lifting two possible solutions proves +uniqueness on $B$. + +Take a smooth compact exhaustion and solve the zero-boundary Dirichlet problem on each member. Bounded-domain elliptic Dirichlet theory applies since the principal matrix is uniformly positive and the zeroth-order term has the indicated sign; see, for example, the linear Schauder theory in \cite[Theorems~6.2 and~6.14]{GilbargTrudinger2001}. The maximum principle gives, for right-hand side $F$, +\begin{equation}\label{eq:uniform-inverse-sup} + \norm{\xi}_{\infty}\le c_0^{-1}\norm{F}_{\infty}, +\end{equation} +independently of the exhaustion domain. On every smaller reference chart whose enlargement is contained in that domain, the interior estimate gives +\[ + \norm{\xi}_{C^{2,\alpha}(\text{smaller chart})} + \le C\bigl(\norm{F}_{C^{0,\alpha}(\text{larger chart})} + +\norm{\xi}_{\infty}\bigr), +\] +with one constant for all charts. Local compactness and a diagonal subsequence produce a solution on $B$. Each fixed enlarged chart eventually lies in the exhaustion, so the same estimate holds at every point of the limit. Taking the supremum over charts proves that the solution belongs to $C_b^{2,\alpha}$ with a bounded norm. No estimate at the moving exhaustion boundary is needed. If $B$ is compact, one instead uses the closed-manifold version of the same theory. + +For uniqueness, suppose $L\xi=0$ and $S=\sup_B\xi>0$. Center uniform charts at a sequence of points where $\xi\to S$. The pulled-back coefficients have a convergent local $C^{0,\beta}$ subsequence, and the functions a convergent $C^{2,\beta}$ subsequence, for $0<\beta<\alpha$. At the center of the limiting chart the limiting function attains the maximum $S$. There its gradient vanishes and its Hessian is negative semidefinite, so +\[ + L\xi\ge c_0S>0, +\] +a contradiction. Apply the same argument to $-\xi$. This also proves the global maximum principle used in \eqref{eq:uniform-inverse-sup} for any solution in the uniform class. Higher-order interior estimates give the final assertion. +\end{proof} + +\begin{theorem}[Uniform graph solvability]\label{thm:uniform-graph} +Let $(B,g)$ be uniformly controlled as above, and let $\ell\in C_b^\infty(B)$. Fix $\eps,w,B_0,s>0$ and $0\le\beta<1$. Suppose $N$ is smooth, positive, nondecreasing, constant near zero, and equal to $1$ for $t\ge1/2$. Define the quantities in \eqref{eq:graph-quantities} and +\[ + \lambda(r)=B_0(r^2+s^2)^{\beta/2}. +\] +There is a unique solution in the uniform $C^{2,\alpha}$ class with a uniform band margin to +\begin{equation}\label{eq:graph-equation} + -\diver_g U + =\lambda(r)\tan\left(\frac{\ell-\eps f}{w}\right), + \qquad \abs{\ell-\eps f}<\frac{\pi w}{2}. +\end{equation} +This solution lies in $C_b^\infty(B)$, and the band has a positive uniform margin. +The metric \eqref{eq:graph-metric-update} is uniformly controlled and complete. + +The solution is invariant under every isometry preserving the equation's +data. The following parameter statement will be used below. Fix the +reference cover, its metric and nested charts, and the equation parameters +$\eps,w,B_0,s,\beta,N$. Let $(g_a,\ell_a)$ be a jointly smooth family over +a finite-dimensional parameter manifold $A$. For every relatively compact +subset $V'\Subset V$ of a parameter chart, assume common constants +$c_{V'},C_{V'}>0$ such that +\[ + c_{V'}g_{\rm ref}\le g_a\le C_{V'}g_{\rm ref}\qquad(a\in V'), +\] +and common bounds in the fixed reference charts for every mixed derivative +of the coefficients of $g_a$ and of $\ell_a$, including order zero. Then +$a\mapsto f_a$ is smooth in every $C_b^{j,\alpha}$ norm, for $j\ge0$ and +$0<\alpha<1$. If $A$ is compact, +all mixed derivatives of $f_a$ are uniformly bounded and the tangent band +has a common positive margin. For a bundle of such covers, the local +solutions glue whenever fixed reference deck transformations carry one +local data set to another. +\end{theorem} + +\begin{proof} +We may assume $\dim B\ge2$. Once the theorem is proved in those dimensions, +the lower-dimensional cases follow by taking the product with +$T_0=\T^{2-\dim B}$, using the product metric and pulling back $\ell$. +The solution on $B\times T_0$ is invariant under $T_0$ translations by +uniqueness, and hence is pulled back from a function on $B$. For such a +function the flux is $(U,0)$ and its product divergence is $\diver_g U$, +so the descended function solves exactly \eqref{eq:graph-equation}. +Restriction to a fixed torus point gives the uniform and parameter bounds, +and lifting two possible solutions gives uniqueness. The graph metric +claims then follow from the same final bounds below. + +Replace $\ell$ by $\tau\ell$, $0\le\tau\le1$, and start at $f=0$ for $\tau=0$. We use $C_b^{2,\alpha}(B)$, for a fixed $0<\alpha<1$, as the solution space. + +\smallskip\noindent\textbf{Linearization and openness.} +As a function of the covector $df$, the flux $U$ has transverse eigenvalue $r(t)/t$ and radial eigenvalue +\begin{equation}\label{eq:graph-flux-ellipticity} + r'(t)=\frac{N(t)(1+\mathfrak p(t))} + {(1+N(t)^2t^2)^{3/2}}>0. +\end{equation} +Both extend positively and smoothly through $t=0$, since $N$ is constant there. Hence the equation is uniformly elliptic on every bounded gradient range. + +On the open set of $C_b^{2,\alpha}$ functions with a positive uniform margin in the tangent band, the equation is a smooth map into $C_b^{0,\alpha}$. The apparent nonsmoothness of $r=\abs{U}$ at zero causes no problem: $\lambda$ is a smooth function of $r^2$, and the flux is smooth in $df$. At an individual solution, moving the right-hand side to the left gives the linearization +\begin{equation}\label{eq:graph-linearization} + -\diver_g\bigl(a_f(d\xi)\bigr)+\mathbf b(d\xi) + +\frac{\eps\lambda(r)}w + \sec^2\left(\frac{\tau\ell-\eps f}{w}\right)\xi. +\end{equation} +Here $a_f$ is the derivative of the flux. After expansion, the coefficients are uniformly H\"older, and the zeroth-order coefficient is at least +$\eps B_0s^\beta/w>0$. The individual $C_b^{2,\alpha}$ bound supplies a bounded gradient range, so \cref{lem:uniform-linear} makes the linearization invertible. The Banach implicit function theorem proves openness in $\tau$. + +\smallskip\noindent\textbf{Comparison and individual regularity.} +Two solutions in this class agree. To see this, take a positive supremum of their difference, if one exists, and pass to a limiting uniform chart as in \cref{lem:uniform-linear}. At the limiting maximum their gradients agree and their Hessians are ordered. At that common gradient the right-hand side of the equation is strictly decreasing in height, with derivative at most $-\eps B_0s^\beta/w$. The two equations are therefore incompatible. The same argument compares a solution with a strict upper or lower barrier. + +Each individual solution has uniform bounds in every higher derivative before we seek bounds independent of $\tau$. Indeed its given $C_b^{2,\alpha}$ norm and band margin make the equation uniformly elliptic with smooth nonlinearities on a bounded range. In a chart, write its nondivergence form as +\[ + a^{ij}(x,df)f_{ij}=H(x,f,df). +\] +For $z=\partial_k f$, differentiation, initially by difference quotients, gives +\begin{equation}\label{eq:individual-graph-bootstrap} + \begin{split} + a^{ij}\partial_i\partial_j z + +\bigl(a^{ij}_{p_l}f_{ij}-H_{p_l}\bigr)\partial_lz-H_fz + =H_{x_k}-a^{ij}_{x_k}f_{ij}. + \end{split} +\end{equation} +The coefficients and source are uniformly $C^{0,\alpha}$ using only the individual $C^{2,\alpha}$ bound. The difference quotients have bounded sup norms, so interior Schauder estimates on smaller charts give $f\in C_b^{3,\alpha}$. Iterating gives $C_b^\infty$. These constants may at this stage depend on the individual solution. + +\smallskip\noindent\textbf{Uniform tangent-band barriers.} +For $\delta>0$ consider +\[ + f_\pm=\eps^{-1}\bigl(\tau\ell\pm(\pi/2-\delta)w\bigr). +\] +Their gradient and divergence terms are bounded uniformly in $\tau$ and in small $\delta$. At $f_+$ the tangent term is negative and diverges uniformly to $-\infty$ as $\delta\downarrow0$; at $f_-$ it diverges uniformly to $+\infty$. This uses $\lambda\ge B_0s^\beta>0$. Choose a sufficiently small fixed $\delta$ so that these are strict upper and lower barriers. Comparison gives, for every solution along the homotopy, +\begin{equation}\label{eq:uniform-graph-band} + f_-\le f\le f_+, + \qquad + \abs{\tau\ell-\eps f}\le(\pi/2-\delta)w. +\end{equation} +In particular the heights and the band margins are now controlled independently of $\tau$. + +\smallskip\noindent\textbf{A uniform gradient bound.} +Fix one solution along the continuation path. If $\sup_B t\le1$, there is +nothing to prove. Otherwise choose points where $t$ approaches its +supremum. The individual higher bounds just established allow the metric, +data, and solution to converge in uniform charts, in enough derivatives +for the graph equation and its Jacobi identity. The limiting slope attains +its maximum at the center. This passage uses finite bounds for that one +solution; the estimate obtained at the maximum will depend only on the +fixed data and the uniform band margin. + +At this maximum, $t>1$, so $N=1$ on a neighborhood. The graph is locally an ordinary product graph, $v=(1+t^2)^{-1/2}$ has a minimum, and $dr=0$. Put +\[ + h_e=\frac1w\left(\lambda+\frac{D^2}{\lambda}\right) + =\frac{\lambda}{w} + \sec^2\left(\frac{\tau\ell-\eps f}{w}\right). +\] +Differentiating the right-hand side of the equation along $U$, and using $dr=0$ and $vU(f)=r^2$, the vertical translation identity becomes +\begin{equation}\label{eq:gradient-jacobi} + \bigl(\Delta_{\rm gr}+\abs{II}^2+\Ric_{g+dz^2}(\nu,\nu)\bigr)v + =v h_e\tau U(\ell)-\eps h_e r^2. +\end{equation} +The band bound implies fixed constants $01/2$ there and $\abs{U(\ell)}\le\norm{d\ell}_{\infty}$, it follows that +\[ + \frac{\eps h_{\min}}2 + \le v\bigl(C_{\rm Ric}+h_{\max}\norm{d\ell}_{\infty}\bigr). +\] +This bounds $v$ below by a positive constant and hence bounds $t$ above, uniformly in $\tau$. One may add $1$ to the denominator when writing an explicit positive bound. All constants in this final estimate depend only on the fixed data and the already uniform band margin; the individual higher norms used to attain the limiting maximum have disappeared. + +\smallskip\noindent\textbf{Uniform higher estimates and closedness.} +After the gradient estimate, the flux is uniformly elliptic on a fixed bounded range. Absorbing the metric volume density, write +\[ + -\partial_i\mathcal A^i(x,\partial f) + =\mathcal H(x,f,\partial f). +\] +Every coordinate derivative $z=\partial_k f$ satisfies +\begin{equation}\label{eq:graph-divergence-bootstrap} + -\partial_i\bigl(\mathcal A^i_{p_j}\partial_jz\bigr) + =\mathcal H_{p_j}\partial_jz+\mathcal H_f z+\mathcal H_{x_k} + +\partial_i\mathcal A^i_{x_k}. +\end{equation} +The drift coefficient and the nondivergence source are uniformly bounded; +the last term is the divergence of a uniformly bounded vector field. +Interior H\"older regularity for scalar divergence-form equations gives a +uniform $C^{0,\alpha'}$ bound for $z$ on smaller charts, for some +$\alpha'>0$; we use \cite[Theorem~8.24]{GilbargTrudinger2001}. In that +theorem's notation take $b^i=d=0$, retain the bounded drift, and treat +$\mathcal H_fz+\mathcal H_{x_k}$ as scalar data and $\mathcal A^i_{x_k}$ as +divergence data, with the signs determined by +\eqref{eq:graph-divergence-bootstrap}. On a fixed chart these bounded data +satisfy its $L^{r/2}$ and $L^r$ hypotheses for any finite $r>\dim B$. +The estimate uses the local $L^2$ norm of $z$, already controlled by the +gradient bound; it requires no bound for the derivatives of $z$. Thus its +constants depend only on ellipticity, the fixed chart size, the displayed +data bounds, and the gradient bound. We obtain a uniform +$C^{1,\alpha'}$ bound for $f$. + +The original equation now has uniformly $C^{0,\alpha'}$ nondivergence coefficients and right-hand side. Interior Schauder estimates give $C^{2,\alpha'}$ bounds, and \eqref{eq:individual-graph-bootstrap} gives higher uniform bounds by iteration. In particular these imply uniform $C^{2,\alpha}$ bounds for the originally chosen $\alpha<1$, by using a bound on one further derivative. Given a sequence of solvable parameters tending to $\tau_*$, local compactness on an exhaustion produces a solution at $\tau_*$ with the same uniform bounds and band margin. The solvable set is therefore closed as well as open, and contains all of $[0,1]$. + +\smallskip\noindent\textbf{Symmetries and parameter dependence.} +An isometry preserving the data takes a solution to a solution; uniqueness +gives invariance. More generally, an isometry carrying one set of data to +another carries their solutions to one another by pullback. The uniform +solution class is preserved: covariant H\"older norms for the uniformly +controlled metric are equivalent to the reference-chart norms, and these +intrinsic norms are unchanged under an isometry. In particular all old torus translations preserve a solution when they preserve the input data. + +For parameter dependence it is essential to use the stated mixed bounds. If $h(x,a)$ is any input coefficient in a local parameter chart, bounds on its mixed derivatives imply smoothness of $a\mapsto h(\cdot,a)$ in every $C_b^{j,\alpha}$ norm. Explicitly, Taylor's formula in $a$, applied also to $x$ derivatives through order $j+1$, gives +\[ + \norm{h(\cdot,a+u)-h(\cdot,a)-D_a h(\cdot,a)u}_{C_b^{j,\alpha}} + \le C\abs{u}^2. +\] +The additional $x$ derivative controls the H\"older seminorm on the uniformly bounded charts. Applying the same argument to every parameter derivative proves the claimed Banach-space smoothness. + +The equation is therefore a smooth Banach map in $a$ as well as $f$. +The fixed reference charts supply the Banach spaces, while the common +metric equivalence on $V'$ supplies locally uniform ellipticity on bounded +gradient ranges. At each parameter, +the invertible linearization gives a locally smooth solution by the +implicit function theorem. Applying the higher-order part of +\cref{lem:uniform-linear} gives the same conclusion in every higher norm, +and uniqueness identifies all the local branches. Compactness of $A$ +supplies uniform mixed bounds and a common band margin using finitely many +such neighborhoods. + +For gluing, suppose a fixed reference deck transformation $\psi$ carries +one local data set to another, so that +$g_a^{(2)}=\psi^*g_a^{(1)}$ and $\ell_a^{(2)}=\psi^*\ell_a^{(1)}$. +It is then an isometry between those data sets. Naturality and uniqueness +give $f_a^{(2)}=\psi^*f_a^{(1)}$. Extending $\psi$ by the identity on old +circle factors gives the compatibility needed for the flat bundles below. + +Finally, $df$ is bounded, $N$ is positive and bounded on its attained range, and $v$ is bounded below. Thus $g^+$ is uniformly equivalent to the reference product metric and has bounded coefficient derivatives of every order. Its completeness follows from that equivalence to a complete reference metric. +\end{proof} + +We have obtained a smooth graph for each fixed finite set of data, together +with the symmetry and parameter control needed to repeat the construction. +The estimates in the next section will use its equation to bound scalar +loss independently of these potentially large analytic regularity constants. diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/06-shortening.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/06-shortening.tex new file mode 100644 index 0000000000000000000000000000000000000000..be7d776aa26bcdf64f38c2ff5b65aef1e3dbfa9e --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/06-shortening.tex @@ -0,0 +1,302 @@ +\section{Shortening all comparison coordinates} +\label{sec:shortening} + +Return to the flat bundle +\[ + Z=(\widetilde M\times\widehat A)/\Gamma\longrightarrow A, + \qquad q=n+\dim A, +\] +where $M^n$ and $A$ are closed and $\widehat A\to A$ is a principal left +$\Gamma$-cover. The covering projection to $M\times A$ induces the splitting +$TZ=E_M\oplus E_A$ described in \cref{thm:comparison-data}. Here $E_M$ is +tangent to the $\widetilde M$ fibres, while $E_A$ is the flat lift of the +parameter directions. Local choices of sheets over $A$ identify the fibres +with $\widetilde M$; the transition maps are fixed deck transformations. +We first change the metric in the fibre directions and add circles. The +metric on $E_A$ is introduced only in the final assembly step. + +A metric with $b$ stabilizing circles has, locally over $A$, the form +\begin{equation}\label{eq:fibre-stabilized-metric} + g_h(x,a)+\sum_{\nu=1}^b m_\nu(x,a)^2d\theta_\nu^2. +\end{equation} +Here $g_h$ acts on the $n$ fibre directions, the positive warps depend only on $x,a$, and the metric is invariant under all circle translations. At every fixed finite stage, the coefficients and all mixed derivatives will be bounded, and the metric will be uniformly equivalent to the lifted reference product. These bounds may depend on the stage. + +Until the parameter metric is assembled in +\cref{lem:parameter-scaling}, every differential, gradient and scalar +curvature is computed on the fibre $\widetilde M\times\T^b$ with +$a\in A$ fixed. The functions and metrics may vary smoothly with $a$; +this variation is measured separately by the mixed derivative bounds. +Thus a torus-invariant fibre differential lies in the original $n$ +fibre directions, even though its full differential on $Z$ may +have a nonzero parameter component. + +\subsection{The cost of one pass} + +A pass processes each of $q$ comparison functions once. In this section, +$L$ denotes a bound for their current fibre differentials. The scalar +estimate for a pass depends on $n$ and $\log q$, even though the full fibre +dimension includes all the circles already introduced. + +\begin{lemma}[One pass]\label{lem:one-pass} +Let $q\ge2$ and $L,w>0$. On a uniformly controlled metric of the form +\eqref{eq:fibre-stabilized-metric}, suppose smooth bounded torus-invariant +functions $\ell_1,\ldots,\ell_q$ have bounded derivatives of every order. +For a smooth family over a compact parameter manifold, assume in each of +finitely many parameter charts common bounds for every mixed fibre and +parameter derivative of the metric coefficients, warps, and functions, +and a common equivalence to the reference fibre product. Suppose +\[ + \abs{d\ell_i}_{g_*}\le L, +\] +where $g_*$ denotes the metric on $E_M$ at the start of the pass. Then one +can add $q$ circles and replace these functions by $\ell_i^+$ so that +\begin{equation}\label{eq:pass-function-bounds} + \abs{\ell_i^+-\ell_i}<\frac\pi2w, + \qquad \abs{d\ell_i^+}_{g_h^+}\le\frac L4, + \qquad g_h^+\ge g_*. +\end{equation} +For an absolute constant $C_0$, the full fibre scalar curvature satisfies +\begin{equation}\label{eq:pass-scalar-bound} + \Scal_{g^+}\ge\Scal_g + -C_0(n+1)(2+\log q)\frac{L^2}{w^2}. +\end{equation} +All data at the new finite stage remain uniformly controlled. For a compact +parameter family satisfying the stated hypotheses, the new mixed derivatives +are also uniformly bounded. +\end{lemma} + +\begin{proof} +Set +\begin{equation}\label{eq:pass-parameters} + \eps=L/4,\qquad a_0=(2+\log q)^{-1},\qquad + \beta=1-a_0,\qquad s=q^{-3},\qquad B_0=\frac{4L}{a_0w}. +\end{equation} +Choose a smooth nondecreasing cutoff $\phi$ which is zero for $t\le1/4$, one for $t\ge1/2$, and satisfies $0\le\phi'\le10$. Let +\begin{equation}\label{eq:pass-cutoff} + N(t)^2=s^2+(1-s^2)\phi(t). +\end{equation} +Writing $\mathfrak p=tN'/N$, we have +\begin{equation}\label{eq:pass-cutoff-positivity} + \mathfrak p\ge0,\qquad + \mathfrak p N(t)^2t^2 + =\tfrac12 t^3(1-s^2)\phi'(t)\le5/8<1. +\end{equation} +The expression vanishes outside $[1/4,1/2]$, which justifies the displayed bound. + +For one coordinate, denote the original function by $\ell$ and the current full fibre metric by $g$. Solve \eqref{eq:graph-equation} with these choices and replace +\[ + g\longmapsto g^+=g+N(t)^2df^2+v^2d\theta^2, + \qquad \ell\longmapsto\ell^+=\eps f. +\] +\Cref{thm:uniform-graph} applies on $\widetilde M\times\T^b$. Its +uniqueness gives torus invariance and compatibility with the parameter +trivializations. Consequently $df$ vanishes on the circle directions, so +the metric retains the form \eqref{eq:fibre-stabilized-metric}. Its block +on $E_M$ has only increased. Since +\[ + \abs{df}_{g^+}=vt\le1 +\] +(the inequality is immediate for $t<1/2$, and $N=1$ for $t\ge1/2$), the new coordinate has derivative norm at most $L/4$. Subsequent steps can only improve this bound. The tangent band gives the height estimate in \eqref{eq:pass-function-bounds}. + +It remains to estimate the scalar loss in a way that can be summed. At $t>0$, put $z=e_1r$. Differentiating $\lambda(r)=B_0(r^2+s^2)^{\beta/2}$ along $U=re_1$ gives +\[ + \frac{U(\lambda)}{\lambda}=\gamma z, + \qquad \gamma=\frac{\beta r^2}{r^2+s^2}\in[0,\beta]. +\] +The part of $-2U(D)$ contributed by this derivative is $-2\gamma zD$. Hence +\begin{equation}\label{eq:pass-square-absorption} + D^2+z^2-2\gamma zD + \ge(1-\gamma^2)D^2\ge a_0D^2. +\end{equation} +By \eqref{eq:pass-cutoff-positivity}, the final term of \eqref{eq:scalar-exact} is nonnegative. + +Measure the vector $U$ in the metric at the start of the pass, and set +\[ + b_U=\abs{U}_{g_*}. +\] +Since the current metric on $E_M$ is at least $g_*$, one has $b_U\le r$. +The input coordinate still satisfies $\abs{d\ell}_{g_*}\le L$, so +$U(\ell)\le Lb_U$, while $U(f)=rt$. Differentiating the tangent in the +equation therefore contributes an adverse term bounded above by +\begin{equation}\label{eq:pass-adverse-term} + \frac2w\left(\lambda+\frac{D^2}{\lambda}\right) + (Lb_U-\eps rt)_+. +\end{equation} +Since $0\le r<1$ and $\beta<1$, +\[ + \lambda=B_0(r^2+s^2)^{\beta/2}\ge B_0r. +\] +Thus the $D^2$ part of \eqref{eq:pass-adverse-term} is at most +\[ + \frac{2L}{wB_0}D^2=\frac{a_0}{2}D^2, +\] +which is absorbed by \eqref{eq:pass-square-absorption}. + +For the remaining part, first suppose $r0$. If instead $r\ge s$, then $t\ge1/4$: for $t<1/4$ one has $N=s$ and $r< s/4$. On the adverse region where $Lb_U-\eps rt>0$, it follows that +\[ + b_U>\frac{rt}{4}\ge\frac r{16}. +\] +Moreover, +\[ + r^{-a_0}\le s^{-a_0} + =\exp\left(\frac{3\log q}{2+\log q}\right)\le e^3. +\] +Consequently, on that region, +\begin{equation}\label{eq:pass-large-slope} + \lambda b_U\le\sqrt2 B_0r^{1-a_0}b_U + \le16\sqrt2 e^3B_0b_U^2. +\end{equation} +Combining these estimates and substituting $B_0$ from \eqref{eq:pass-parameters} gives the one-coordinate inequality +\begin{equation}\label{eq:coordinate-scalar-cost} + \Scal_{g^+}\ge\Scal_g + -C_0\frac{L^2}{a_0w^2}\left(\frac1q+b_U^2\right). +\end{equation} +For example $C_0=128\sqrt2 e^3$ is sufficient. The inequality extends by continuity to the boundary of $\{t>0\}$. On the interior of $\{t=0\}$ the graph is constant and the update adds a constant circle, so it holds there as well. + +Finally sum \eqref{eq:coordinate-scalar-cost} over the $q$ coordinates. +At a fixed point of $Z$, the inverse metrics of the $E_M$ blocks all act +on the same $n$-dimensional space $E_M^*$. If $U_i$ is the vector used at the $i$th +step, \eqref{eq:graph-inverse-volume} gives the telescoping identity +\begin{equation}\label{eq:pass-trace-telescope} + \sum_{i=1}^q U_i\otimes U_i + =g_*^{-1}-(g_h^+)^{-1}\le g_*^{-1}. +\end{equation} +Taking its $g_*$-trace gives $\sum_i b_{U_i}^2\le n$. The $q$ terms $1/q$ sum to $1$, proving \eqref{eq:pass-scalar-bound}. In particular the growing number of old circle directions does not appear in this trace. Uniform control and smooth parameter dependence at the new stage follow from \cref{thm:uniform-graph} applied successively a finite number of times. +\end{proof} + +\subsection{Finite iteration and the parameter directions} + +We shall use the following elementary assembly observation. + +\begin{lemma}[Magnifying the parameter metric]\label{lem:parameter-scaling} +Fix a finite-stage smooth family of metrics of the form +\eqref{eq:fibre-stabilized-metric}, equivariant under the flat transition maps over the closed manifold $A$, with all mixed derivatives bounded and uniform equivalence to a reference product. For any fixed metric $g_A$ on $A$, set +\[ + g_P=g_h+P^2g_A+\sum_{\nu=1}^b m_\nu^2d\theta_\nu^2 + \quad\text{on }Z\times\T^b, +\] +using the flat splitting and no mixed parameter terms. As $P\to\infty$, $\Scal_{g_P}$ converges uniformly to the scalar curvature of the fibre metrics. If $H:Z\to\R^q$ has bounded mixed derivatives, its differential in the parameter directions has norm tending uniformly to zero. Each $g_P$ is complete and uniformly controlled. +\end{lemma} + +\begin{proof} +In a parameter chart about $a_*$ write $a=a_*+P^{-1}y$. The parameter block becomes $g_A(a_*+P^{-1}y)$, and the other metric coefficients are the corresponding fibre coefficients evaluated at $a_*+P^{-1}y$. All parameter derivatives through order two tend uniformly to zero. Their limits, at $y=0$, are the two-jets of the fibre metric with $a=a_*$ frozen, times the constant Euclidean parameter block $g_A(a_*)$. Inverses remain uniformly bounded for this fixed stage. The scalar-curvature formula is a continuous expression in these metric two-jets and their inverses, so its limit is uniformly the fibre scalar curvature. The finitely many parameter charts and the assumed bounds make the convergence uniform over all of $Z$. + +The parameter differential of $H$ acquires the factor $P^{-1}$, proving the second assertion. The reference product on $Z$ is the pullback of a metric on the closed manifold $M\times A$ and is complete. The assembled metric, with its finite number of circle blocks, is uniformly equivalent to this complete reference product and has bounded coefficient derivatives. +\end{proof} + +\begin{theorem}[Logarithmic shortening]\label{thm:shortening} +Suppose $\Scal_M\ge\kappa>0$. On the above bundle $Z$, let +$G:Z\to\R^q$ be smooth and bounded with all mixed derivatives bounded, and suppose $L_0>0$ and +\[ + \sup_{1\le i\le q}\abs{d_{\widetilde M}G_i}\le L_0, + \qquad q\ge2. +\] +There is a constant $C_n$ depending only on $n$ such that, if +\begin{equation}\label{eq:shortening-smallness} + C_n(2+\log q)L_0^2\le\kappa/4, +\end{equation} +then for every finite integer $k\ge1$ there are a complete stabilized metric +\[ + \mathfrak g_k=h_k+\sum_{\nu=1}^{kq}m_{k,\nu}^2d\theta_\nu^2 + \quad\text{on }Z\times\T^{kq} +\] +and a smooth bounded map $G^{(k)}:Z\to\R^q$ satisfying +\begin{equation}\label{eq:shortening-output} + \Scal_{\mathfrak g_k}\ge\kappa/2,\qquad + \norm{G^{(k)}-G}_{\infty}<\pi/10,\qquad + \abs{dG^{(k)}}_{\rm op}\le\sqrt{q+1}\,4^{-k}L_0. +\end{equation} +The infinity norm in the middle inequality is the target coordinate maximum, followed by the supremum over $Z$. All metric coefficients, warps, and maps have uniform bounds of every order at each fixed finite stage; their equivalence and derivative constants may depend on that stage. +\end{theorem} + +\begin{proof} +Start with the lifted metric of $M$ and no circles, and perform passes indexed by $j=0,\ldots,k-1$ with +\begin{equation}\label{eq:pass-schedule} + L_j=4^{-j}L_0,\qquad w_j=2^{-j}/10. +\end{equation} +\Cref{lem:one-pass} applies inductively because every coordinate derivative is divided by at least four in each pass. Every stage is finite, so the uniform control required for its successor follows from the last assertion of that Lemma. All local constructions match under the flat parameter transitions by uniqueness of the graph solution. + +For any number of passes, the total scalar loss is at most +\[ + C_0(n+1)(2+\log q)\sum_{j\ge0}\frac{L_j^2}{w_j^2} + =\frac{400}{3}C_0(n+1)(2+\log q)L_0^2. +\] +Choose $C_n=(400/3)C_0(n+1)$. Under +\eqref{eq:shortening-smallness}, all the fibre scalar curvatures are at least $3\kappa/4$. Each coordinate is modified exactly once per pass, so +\[ + \norm{G^{(k)}-G}_{\infty} + <\frac\pi2\sum_{j=0}^{k-1}w_j<\frac\pi{10}. +\] +No coordinate count enters this estimate. The fibrewise Euclidean operator norm is at most $\sqrt q\,L_k$, since each of the $q$ rows has norm at most $L_k=4^{-k}L_0$. + +At this fixed stage apply \cref{lem:parameter-scaling}, making the parameter scale so large that the scalar-curvature loss from assembly is at most $\kappa/4$ and the parameter part of $dG^{(k)}$ has operator norm at most $L_k$. The domain's fibre and parameter blocks are orthogonal. Cauchy--Schwarz therefore bounds the full differential by +$\sqrt{qL_k^2+L_k^2}$, proving \eqref{eq:shortening-output}. The resulting base metric on $Z$ is $h_k$; all circle warps remain positive and independent of circle variables. The completeness and finite-stage bounds follow from the same assembly Lemma. +\end{proof} + +\subsection{A compact degree region} + +\begin{corollary}[Stabilized contraction at a fixed scale] +\label{cor:stabilized-contraction} +Assume $\Scal_M\ge\kappa>0$ and take the comparison data of +\cref{thm:comparison-data} at scale $R$, so that +\[ + L_0=J/R,\qquad \log q\le JR. +\] +For all sufficiently large $R$, fix these data once and for all. Then for every finite $k\ge1$ there is a stabilized metric $\mathfrak g_k$ on $Z\times\T^{kq}$ with scalar curvature at least $\kappa/2$, and a smooth map +\[ + F_k:Z\longrightarrow S^q +\] +equal to a single value outside a compact set, with nonzero degree, such that +\begin{equation}\label{eq:stabilized-contraction-bound} + \abs{dF_k}_{\rm op}\le C(q)4^{-k}J/R. +\end{equation} +The sphere has its standard unit metric. The constant $C(q)$ is independent of $k$. If desired, the circle lengths may be increased so that the projection to the standard $\T^{kq}$ also has operator norm at most $4^{-k}J/R$; this does not change scalar curvature or the bound on $dF_k$. +\end{corollary} + +\begin{proof} +The left-hand side of \eqref{eq:shortening-smallness} is at most +\[ + C_n(2+JR)J^2/R^2\longrightarrow0. +\] +Choose a sufficiently large finite $R$ once. In particular $q$, $A$, and the original comparison map are now fixed while $k$ remains arbitrary. Apply \cref{thm:shortening} at this fixed scale. + +Choose a smooth map $Q_q:\R^q\to S^q$ which is constant outside the Euclidean unit ball and has degree one on the one-point compactification. It can be chosen with the north pole as a regular value having the sole preimage $0$. For example, in polar coordinates use +\[ + y\longmapsto + \left(\sin\rho(\abs y)\frac{y}{\abs y},\cos\rho(\abs y)\right), +\] +where $\rho(r)$ is a positive linear function near zero, increases to $\pi$, and is smoothly constant at $\pi$ in a collar of $r=1$ and beyond; choose the orientation, or compose with a reflection, to give degree one. The expression is smooth at zero because $\rho$ is linear there. Let $D_q<\infty$ bound its differential. + +By the comparison theorem there is a precompact open $\Omega\subset Z$ with +$\deg(G,\Omega,0)\ne0$ and $\norm{G}_{\infty}>3$ on $\partial\Omega$. The homotopy +\[ + G+u(G^{(k)}-G),\qquad 0\le u\le1, +\] +stays outside the unit ball on $\partial\Omega$, by +\eqref{eq:shortening-output}. Define +\[ + F_k=Q_q\circ G^{(k)}\quad\text{on }\Omega, + \qquad F_k=Q_q(\infty)\quad\text{on }Z\setminus\Omega. +\] +The boundary is compact and its image is separated from the unit ball. Thus $Q_q\circ G^{(k)}$ is constant on a neighborhood of the boundary, and this definition is smooth even if $\partial\Omega$ is not smooth. The map is constant outside the compact set $\overline\Omega$. + +Homotopy invariance of local degree gives +\[ + \deg(G^{(k)},\Omega,0)=\deg(G,\Omega,0)\ne0. +\] +The degree of $F_k$ is this same integer: count preimages of a regular target sufficiently near the north pole, where $Q_q$ is an orientation-preserving local diffeomorphism, or equivalently compose the relative degree classes. Hence $F_k$ has nonzero degree. Its differential is bounded by +$D_q\sqrt{q+1}\,4^{-k}J/R$, establishing +\eqref{eq:stabilized-contraction-bound} with $C(q)=D_q\sqrt{q+1}$. + +For the last assertion, multiply each circle warp by a sufficiently large positive constant. Because all coefficients are independent of the circle variables and the circle blocks are mutually orthogonal, this is locally a constant change of each circle coordinate. It preserves scalar curvature. The base metric and the norm of $dF_k$ are unchanged. At the fixed stage all warps have positive uniform lower bounds, so their constant multipliers can make every inverse circle length at most $4^{-k}J/R$. The operator norm of the projection to the product flat torus is the maximum of these inverse lengths, with no factor depending on the number of circles. +\end{proof} + +We have now constructed maps of nonzero degree from $Z$ to $S^q$. +Their differentials tend to zero as the finite pass count grows, with +$q$ fixed. The scalar lower bound holds on $Z\times\T^{kq}$. The +remaining obstruction must therefore be independent of the number of +stabilizing circles. We establish it in \cref{sec:obstruction}. diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/07-obstruction.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/07-obstruction.tex new file mode 100644 index 0000000000000000000000000000000000000000..f56da3c2160d5c472b633923d02d46e3ee0023e8 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/07-obstruction.tex @@ -0,0 +1,166 @@ +\section{The stabilized obstruction from the torical theorem} +\label{sec:obstruction} + +The shortening construction varies the number of circle factors while +keeping the base dimension fixed. We prove the required obstruction with +a threshold independent of that circle count. Its only external geometric +input is the local torical theorem of Cecchini and Schick; the reduction +below checks its compactness, degree and local metric hypotheses. + +\begin{definition} +\label{def:torical-metric} +A \emph{torus-stabilized metric} on a smooth manifold $Y$ is a metric +\begin{equation} +\label{eq:stabilized-metric} + \mathcal G=g+\sum_{\nu=1}^{b}m_\nu^2\,d\theta_\nu^2 + \qquad\text{on }Y\times\T^b, +\end{equation} +where $b\geq0$ is finite, $g$ is a smooth metric on $Y$, and the functions +$m_\nu:Y\to(0,\infty)$ are smooth. Each circle is +$\R/(2\pi\Z)$. We call $g$ the base metric. +Scalar curvature always refers to the full metric $\mathcal G$. +Differentials of base maps have the norm determined by $g$. +\end{definition} + +For a map $F:Y^d\to S^d$ equal to a fixed value $y_\infty$ outside a +compact set, degree means evaluation against a smooth top-degree form +supported away from $y_\infty$ and of integral one. Equivalently, it is the +signed preimage count at a regular value different from $y_\infty$. +This convention also applies when $Y$ is disconnected. + +\begin{theorem}[Stabilized torical obstruction] +\label{thm:torical-obstruction} +For every integer $d\geq2$ and every $\chi>0$ there is a number +$\delta(d,\chi)>0$ for which the following data do not exist: +\begin{enumerate} + \item a smooth oriented $d$-manifold $Y$ without boundary, possibly + disconnected, with a torus-stabilized metric $\mathcal G$ satisfying + $\Scal_{\mathcal G}\geq\chi$; + \item a smooth map $F:Y\to S^d$, equal to one value outside a compact + set, such that $\deg F\ne0$ and + $\norm{dF}_{\mathrm{op}}\leq\delta(d,\chi)$ everywhere. +\end{enumerate} +The constant is independent of the finite circle count in +\cref{eq:stabilized-metric}. Neither the base metric nor the stabilized +metric is assumed complete. +\end{theorem} + +Cecchini--Schick prove that if a closed oriented Riemannian $D$-manifold $X$ +has a smooth map $f:X\to S^1\times\T^{D-1}$ of nonzero degree, and a +connected open arc $J\subset S^1$ satisfies +\[ + \Scal_X\ge\chi>0,\qquad |df|_{\mathrm{op}}\le\epsilon + \quad\text{on }f^{-1}(J\times\T^{D-1}), +\] +then +\begin{equation}\label{eq:cecchini-schick} + 2\pi\epsilon\ge\sqrt\chi\,\operatorname{Length}(J). +\end{equation} +This is \cite[Theorem~1.14(1)]{CecchiniSchick2021}. The statement has no +upper dimension bound and no spin hypothesis. Its notion of contraction is +a pointwise bound on the full differential. In high dimensions its proof uses the +minimal-slicing input identified in that paper; we invoke the +published theorem at this interface. We use its flat torus normalization +in the target; any fixed change from our angular convention is absorbed +in the tube-projection bound and the constant circle enlargements below. + + +\begin{proof}[Proof of \cref{thm:torical-obstruction}] +Fix $d\ge2$ and $\chi>0$. Suppose that the data in the theorem exist, +and write $b$ for the finite number of circles. Set $D=d+b$. +Thus $Y^d$ carries the stabilized metric +\[ + g+\sum_{\nu=1}^b m_\nu^2d\theta_\nu^2 +\] +of scalar curvature at least $\chi$, and the nonzero-degree map +$F:Y\to S^d$ has differential bounded by a number $\delta>0$ in the +base metric $g$. We derive a positive lower bound for $\delta$ which +is independent of $b$. + +Choose a fixed two-sided embedded $\T^{d-1}$ in $S^d$ away from the exterior +value of $F$, with a closed tubular neighborhood parametrized by +$\T^{d-1}\times[-2,2]$. Such embeddings can be built inductively in a +Euclidean chart: add one ambient dimension and take the boundary of a small +trivial two-dimensional normal disk bundle around the preceding torus. +After a rotation, the target tube and all its projection bounds depend only +on $d$. + +Choose regular interval levels +$s_-\in(-5/4,-3/4)$ and $s_+\in(3/4,5/4)$ and pull back the strip +between them. This gives a compact smooth band $W\subset Y$, with a +map to that strip of nonzero relative degree. Compactness follows because +the closed target tube avoids the exterior value. Regularity of the two +levels gives a smooth boundary; nonzero relative degree follows by +counting preimages of a regular interior target of $F$. + +Write the target strip as $\T^{d-1}\times[s_-,s_+]$. Double +$W$ with the opposite orientation on the second copy. On both copies +retain the $\T^{d-1}$ component of the target map, and send the interval +component monotonically along complementary semicircles of $S^1$. +To make this map smooth across the seams, first choose disjoint source +boundary collars and precompose the \emph{entire base map}, including its +torus component, with a smooth collar reparametrization that is constant +in the normal variable near the boundary. On the first copy, choose the circle angle $\alpha(t)$ to be +$\pi/2+\pi t/2$ for $|t|\le1/2$, extend it monotonically to $0$ and +$\pi$ at the two walls, and make it flat near the endpoints. On the +second copy use the complementary semicircle, with the same endpoint +values in $S^1$. The two torus components then +agree and are constant in the normal variable at each seam, while the +circle components agree there as well. Hence they define a smooth map +\[ + f:\operatorname{dbl}(W)\times\T^b + \longrightarrow S^1\times\T^{d-1}\times\T^b, +\] +with the identity on the final torus. + +All collar changes can be confined near the walls, away from $|t|\le1/2$. +Choose the open target arc +\[ + J=\{e^{i\alpha}:3\pi/8<\alpha<5\pi/8\}. +\] +Its full preimage in the first copy is precisely $|t|<1/4$, where the +map is unchanged and its interval slope is the fixed number $\pi/2$. +The second copy maps to the complementary semicircle and has no preimage +over $J$. Counting preimages of a regular target over $J$ therefore +recovers the nonzero relative degree of the original band map. The map +$f$ has nonzero degree, on at least one connected component if the +double is disconnected. + +Each $m_\nu$ has a positive minimum on the compact band. Enlarge its circle +length by a constant factor until the joint projection to the standard +$\T^b$ has operator norm at most $\delta$ on $W$. This changes no local +scalar curvature: a constant change of a circle coordinate is a local +isometry between the two warped metrics. The operator norm of the joint +projection is the maximum inverse circle length, rather than a sum over +$b$. + +Give the closed double any smooth metric agreeing with the enlarged +stabilized metric throughout the chosen middle substrip and a neighborhood +of its closure. On the preimage of $J\times\T^{d+b-1}$ it then has scalar +curvature at least $\chi$. The base part of $f$ has differential at most +$C_d\delta$, because only the fixed tube projection and a fixed middle-arc +parametrization are used there. Its torus part has differential at most +$\delta$. The orthogonal block structure therefore gives +\[ + |df|_{\mathrm{op}}\le\max(C_d,1)\delta + \quad\text{on }f^{-1}(J\times\T^{d+b-1}), +\] +independently of $b$. No metric or curvature control elsewhere on the +double is required by the cited theorem. + +Applying \eqref{eq:cecchini-schick} gives +\[ + 2\pi\max(C_d,1)\delta + \ge\sqrt\chi\,\operatorname{Length}(J). +\] +The fixed tube and middle-arc choices depend only on $d$. In particular +we may take +\[ + \delta(d,\chi)= + \frac{\sqrt\chi\,\operatorname{Length}(J)} + {4\pi\max(C_d,1)}. +\] +A map with differential at most this number contradicts the preceding +lower bound. The threshold depends only on $d$ and $\chi$, independently +of the finite circle count $b$. +\end{proof} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/08-proof.tex b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/08-proof.tex new file mode 100644 index 0000000000000000000000000000000000000000..9fb15fb860480292dab4a39a641b7d651301e5f5 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/build/sections/08-proof.tex @@ -0,0 +1,116 @@ +\section{Proof of the main results} +\label{sec:main-proof} + +We combine the comparison data, finite shortening, and the stabilized +obstruction at one fixed scale, then derive the four corollaries. + +\begin{proof}[Proof of \cref{thm:main}] +Suppose, to the contrary, that $M^n$, $n\ge2$, has a positive-scalar-curvature +metric and that $(c_M)_*[M]\ne0$ in $H_n(B\Gamma;\Q)$. Fix the metric and +choose $\kappa>0$ with $\Scal_M\ge\kappa$. + +\Cref{thm:comparison-data} supplies comparison data at arbitrarily large +finite scales $R$, with a constant $J$ independent of $R$. Choose one +sufficiently large finite $R$ for \cref{cor:stabilized-contraction}, and keep +all resulting comparison data $A,Z,q,G,\Omega$ fixed. +For every finite integer $k\ge1$, that corollary gives a smooth metric $g_k$ +on $Z$ and positive smooth warps $m_{k,\nu}$ such that +\[ + g_k+\sum_{\nu=1}^{kq}m_{k,\nu}^2d\theta_\nu^2 + \quad\text{has scalar curvature at least }\kappa/2. +\] +It also gives a smooth map $F_k:Z\to S^q$, constant outside a compact set +and of nonzero degree, satisfying +\[ + |dF_k|_{g_k,\mathrm{op}}\le C(q)4^{-k}\frac{J}{R}. +\] +The metric coefficients and the parameter enlargement may depend on the +finite pass count $k$. + +The sphere map uses the full base $Z$, with its $n$ fibre directions and +$\dim A$ parameter directions. The relevant instance of +\cref{thm:torical-obstruction} has $d=q=n+\dim A$, $b=kq$, and scalar lower +bound $\chi=\kappa/2$. +Its threshold $\delta(q,\kappa/2)$ is independent of $b$. Since $q$ is +fixed, choose a finite $k$ so large that +\[ + C(q)4^{-k}J/R\le\delta(q,\kappa/2). +\] +The resulting stabilized metric and nonzero-degree map contradict the +obstruction. This proves rational inessentiality. +\end{proof} + +\begin{proof}[Proof of \cref{cor:aspherical}] +Let $M$ be one connected component of a closed smooth aspherical manifold. +Scalar curvature is identically zero in dimensions zero and one, so suppose +$n\ge2$. If $M$ is oriented, its classifying map is a homotopy equivalence +because $M$ is aspherical. It therefore sends the nonzero rational fundamental +class to a nonzero class, contradicting \cref{thm:main} if $M$ admits positive +scalar curvature. + +If $M$ is not orientable, its orientation double cover is a closed connected +oriented smooth manifold with the same contractible universal cover. A +positive-scalar-curvature metric on $M$ lifts to such a metric on this +aspherical cover, contradicting the oriented case. +\end{proof} + +\begin{proof}[Proof of \cref{cor:aspherical-flat}] +Work on one connected component of $M$. In dimensions zero and one the +full Riemann curvature tensor vanishes, so suppose that $\dim M\ge2$. +Write $(\widehat M,\widehat g)=(M,g)$ if $M$ is orientable. Otherwise, +let $\widehat M\to M$ be its connected orientation double cover and let +$\widehat g$ be the pullback of $g$. In either case $\widehat M$ is closed, +oriented and aspherical, and $\widehat g$ has nonnegative scalar curvature. + +By \cref{lem:scalar-reduction}, either $\widehat M$ admits a metric of strictly +positive scalar curvature or the lifted given metric $\widehat g$ is Ricci-flat. +\Cref{cor:aspherical} excludes the first alternative, so +$\Ric_{\widehat g}=0$. The compact-manifold consequence of the Cheeger--Gromoll +splitting theorem \cite{CheegerGromoll1971} gives an isometric splitting +of the universal Riemannian cover +\[ + (\widetilde M,\widetilde g) + \cong (\R^k,g_{\mathrm{Eucl}})\times (N,h), +\] +where $N$ is a compact connected manifold without boundary and +$\widetilde g$ is the lift of the original metric $g$. Since $M$ is aspherical, +$\widetilde M$ is contractible. The factor $N$ is a deformation retract +of $\R^k\times N$, hence is contractible as well. If $m=\dim N>0$, +then $H_m(N;\Z/2\Z)\ne0$ by the mod-two fundamental class, a contradiction. +Thus $N$ is a point, $\widetilde g$ is Euclidean, and $g$ is flat because +the covering maps are local isometries. +\end{proof} + +\begin{proof}[Proof of \cref{cor:degree-aspherical}] +Choose compatible basepoints and put +$\phi=f_*:\pi_1(M)\to\pi_1(N)$. The $K(\pi,1)$ mapping property gives +\[ + B\phi\circ c_M\simeq c_N\circ f; +\] +see \cite[Theorem~1B.8 and Proposition~1B.9]{Hatcher2002}. Since $N$ is +aspherical, $c_N$ is a homotopy equivalence. Naturality and the degree +identity therefore give +\[ + (B\phi)_*(c_M)_*[M] + =(c_N)_*f_*[M] + =d(c_N)_*[N]\ne0 + \qquad\text{in }H_n(B\pi_1(N);\Q). +\] +Thus $M$ is rationally essential, and \cref{thm:main} rules out the +stated metric. No injectivity or surjectivity condition on $\phi$ is used. + +For the connected sum, the pinch $N\#P\to N\vee P$ followed by projection +to $N$ has degree $+1$: a point on the unchanged $N$-side has one +orientation-preserving local preimage. This construction is valid for every +$n\ge2$, including surfaces, so the first assertion applies. +\end{proof} + +\begin{proof}[Proof of \cref{cor:simplicial}] +Simplicial volume is the sum of the simplicial volumes of the finitely many +connected components, so it suffices to treat one component. The circle case +was proved in \cref{sec:reductions}. In dimensions at least two, +\cref{lem:scalar-reduction} either supplies a positive-scalar-curvature metric +or gives zero simplicial volume directly in the Ricci-flat case. In the former +alternative, \cref{thm:main} and \cref{lem:essentiality} give the same +conclusion. +\end{proof} diff --git a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0c199662f47cf4f5d49fa42b749b791430215df5 --- /dev/null +++ b/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dae319c1cc2582086b82ff65f2c210a972975ad321b340cff4c9d5211dcb14dc +size 498415 diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/README.md b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..625530f8c55824b5b8f804b407913ffe4d568305 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/README.md @@ -0,0 +1,27 @@ +# [Positively curved Einstein four-manifolds](paper.pdf) + +OpenAI · September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Positively-curved-Einstein-four-manifolds-September-23-2026, + author = {{OpenAI}}, + title = {{Positively curved Einstein four-manifolds}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf}{OAI:Positively-curved-Einstein-four-manifolds-September-23-2026}}, + year = {2026} +} +``` + +## Verification + +The exact-arithmetic checker requires Python 3.10 or later and only the standard library. Run from this paper directory: + +```sh +python3 -B verification/verify.py --check +``` + +Success means exit status zero and `all_passed: true`, with 53 identities, 15 reported univariate certificate rows plus 6 supplementary rows, 12 triangle certificate rows, and 24 additional exact-value assertions. The command recomputes the complete deterministic result and compares it with `verification/results.json`. + +These are finite arithmetic checks, not a proof-assistant formalization of the whole argument. The precise scope and its distinction from the mathematical reductions proved in the paper are recorded in [verification/COVERAGE.md](verification/COVERAGE.md). Additional package-level reproduction instructions are in [verification/README.md](verification/README.md). diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/figures/proof-map.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/figures/proof-map.tex new file mode 100644 index 0000000000000000000000000000000000000000..1bec4d850e7cfba8d574812171a8d5877c248353 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/figures/proof-map.tex @@ -0,0 +1,33 @@ +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[ + box/.style={draw=blue!40!black,rounded corners=2pt,fill=blue!3, + text width=4.15cm,align=center,inner sep=5pt,font=\footnotesize}, + arr/.style={-{Latex[length=2mm]},draw=blue!55!black,thick}, + note/.style={font=\scriptsize,fill=white,inner sep=2pt,align=center}] + \node[box] (scalar) at (0,0) {Scalar and upper volume estimates\\Sections~\ref{sec:analytic}--\ref{sec:volume}}; + \node[box] (hessian) at (6.2,0) {Weighted Hessian inequality\\Section~\ref{sec:hessian}}; + \node[box] (balance) at (0,-1.55) {Equal second moments\\Section~\ref{sec:volume}}; + \node[box] (integral) at (6.2,-1.55) {$\mathbf E\mathcal I\ge0$ for the two blocks\\Section~\ref{sec:coupling}}; + \node[box] (signs) at (0,-3.1) {Exact pointwise bounds\\Sections~\ref{sec:coupling}--\ref{sec:certificates}}; + \node[box] (vanish) at (6.2,-3.1) {Half-conformal flatness\\Strictness in Section~\ref{sec:classification}}; + \node[box] (lower) at (0,-4.65) {Conditional\\lower volume bound\\Section~\ref{sec:classification}}; + \node[box] (parallel) at (6.2,-4.65) {Parallel curvature and model tensor\\Section~\ref{sec:classification}}; + \node[box] (simply) at (0,-6.2) {Simple connectivity\\Appendix~\ref{app:topology}}; + \node[box] (global) at (6.2,-6.2) {Global model isometry\\Section~\ref{sec:classification}}; + \draw[arr] (scalar)--(balance); + \draw[arr] (hessian)--(integral); + \draw[arr] (balance)--node[note,above]{variance\\term}(integral); + \draw[arr] (integral)--(vanish); + \draw[arr] (signs)--node[note,above]{pointwise\\signs}(vanish); + \draw[arr] (vanish)--(parallel); + \draw[arr] (lower)--(parallel); + \draw[arr] (parallel)--(global); + \draw[arr] (simply)--(global); +\end{tikzpicture} +\caption{Assembly of the proof. Moment balance enters the nonnegative +integral; the exact pointwise bounds and active-gradient strictness force +one-block vanishing. The conditional lower bound identifies the remaining +parallel tensor, and simple connectivity permits global continuation.} +\label{fig:proof-map} +\end{figure} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/figures/spectral-domains.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/figures/spectral-domains.tex new file mode 100644 index 0000000000000000000000000000000000000000..cf0597c320a6f14c6d8565a8feaed6cea46a44a6 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/figures/spectral-domains.tex @@ -0,0 +1,33 @@ +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[scale=1.7, every node/.style={font=\small}] + \begin{scope} + \fill[blue!4] (0,0)--(2,0)--(0,2)--cycle; + \draw[thick] (0,0)--(2,0)--(0,2)--cycle; + \draw (0,0)--(1,1)--(1,0); + \node[below left] at (0,0) {$O$}; + \node[below] at (1,0) {$S$}; + \node[below right] at (2,0) {$U$}; + \node[above right] at (1,1) {$C$}; + \node[above left] at (0,2) {$W_0$}; + \node at (1,-0.48) {$\mathcal D$: certificate $\mathcal A$}; + \end{scope} + \begin{scope}[xshift=3.6cm] + \fill[blue!4] (0,0)--(2,0)--(1,1)--cycle; + \draw[thick] (0,0)--(2,0)--(1,1)--cycle; + \draw (0.5,0.5)--(1,0)--(1.5,0.5); + \draw (1,0)--(1,1); + \draw (0.5,0.5)--(1.5,0.5); + \node[below left] at (0,0) {$O$}; + \node[below] at (1,0) {$S$}; + \node[below right] at (2,0) {$U$}; + \node[above] at (1,1) {$C$}; + \node[above left] at (0.5,0.5) {$A$}; + \node[above right] at (1.5,0.5) {$B$}; + \node[above right,inner sep=2pt] at (1,0.5) {$E$}; + \node at (1,-0.48) {$\mathcal D_+$: certificate $\mathcal B$}; + \end{scope} +\end{tikzpicture} +\caption{The full and ordered curvature domains, with the subdivisions +used in the triangular coefficient table.} +\end{figure} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/paper.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..f3b4adc52704b8b0f90d9751a71a6a815a6737c6 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/paper.tex @@ -0,0 +1,56 @@ +\documentclass[11pt]{article} +\pdfglyphtounicode{summationtext}{2211} +\pdfglyphtounicode{summationdisplay}{2211} +\pdfglyphtounicode{integraltext}{222B} +\pdfglyphtounicode{integraldisplay}{222B} +\pdfglyphtounicode{productdisplay}{220F} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{geometry,microtype,booktabs,longtable,array,enumitem,xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\geometry{margin=1in} +\usepackage[hypertexnames=false,colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\hypersetup{pdftitle={Positively curved Einstein four-manifolds},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\allowdisplaybreaks[2] +\raggedbottom +\setlength{\emergencystretch}{1em} +\setlist{itemsep=3pt,topsep=5pt} +\title{Positively curved Einstein four-manifolds} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini--Study metric, or real projective four-space with its round metric. No orientability assumption is needed. +\end{abstract} +\input{sections/01-introduction.tex} +\clearpage +\begingroup +\small +\tableofcontents +\endgroup +\clearpage +\input{sections/02-preliminaries.tex} +\input{sections/03-analytic.tex} +\input{sections/04-volume.tex} +\input{sections/05-hessian.tex} +\input{sections/06-coupling.tex} +\input{sections/07-certificates.tex} +\input{sections/08-classification.tex} +\appendix +\input{sections/appendix-a-analytic.tex} +\input{sections/appendix-b-topology.tex} +\clearpage +\input{references.tex} +\end{document} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/references.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..e88e1c293b65ac52d21320765da8e840b3b6e81e --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/references.tex @@ -0,0 +1,142 @@ +\begin{thebibliography}{99} +\raggedright +\addcontentsline{toc}{section}{\refname} + +\bibitem{AbreschMeyer1997} +U.~Abresch and W.~T.~Meyer, \emph{Injectivity Radius Estimates and Sphere Theorems}, in \emph{Comparison Geometry}, MSRI Publications \textbf{30}, Cambridge University Press, 1997, 1--47. +\href{https://library.slmath.org/books/Book30/files/abresch.pdf}{Publisher-hosted text}. + +\bibitem{Ambrose1956} +W.~Ambrose, \emph{Parallel Translation of Riemannian Curvature}, Ann. of Math. 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J. \textbf{13} (1966), no.~4, 459--469. +\href{https://doi.org/10.1307/mmj/1028999604}{doi:10.1307/mmj/1028999604}. + +\bibitem{Roy2007} +M.-F.~Roy, \emph{Certificates of Positivity in the Bernstein basis}, Oberwolfach Reports, report 14/2007, 800--804. Report on joint work with F.~Boudaoud, F.~Caruso, and R.~Leroy. \href{https://ems.press/content/serial-article-files/46100}{Report}. + +\bibitem{Synge1936} +J.~L.~Synge, \emph{On the Connectivity of Spaces of Positive Curvature}, Quart. J. Math. Oxford Ser. \textbf{7} (1936), 316--320. +\href{https://doi.org/10.1093/qmath/os-7.1.316}{doi:10.1093/qmath/os-7.1.316}. + +\bibitem{Yang2000} +D.~Yang, \emph{Rigidity of Einstein 4-manifolds with positive curvature}, Invent. Math. \textbf{142} (2000), no.~2, 435--450. +\href{https://doi.org/10.1007/PL00005792}{doi:10.1007/PL00005792}. + +\end{thebibliography} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/01-introduction.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..f47ba33c207d730780fa3f3d5b449006ccdb1aad --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/01-introduction.tex @@ -0,0 +1,180 @@ +\section{Introduction}\label{sec:introduction} + +A Riemannian metric is Einstein if its Ricci tensor is a constant +multiple of the metric. This fixes the trace of the sectional curvatures +while allowing individual curvatures to vary. The classification +conjecture in dimension four asks whether the round $S^4$, the round +$\mathbb{RP}^4$, and Fubini--Study $\mathbb{CP}^2$ exhaust the connected +closed Einstein four-manifolds with strictly positive sectional curvature, +up to positive scaling and isometry. Yang explicitly lists these three +models in Conjecture~1 of~\cite{Yang2000}. We prove the classification +in the closed setting. + +\begin{theorem}\label{thm:classification} +Let $(M,g)$ be a connected smooth closed Einstein four-manifold with +strictly positive sectional curvature, without an orientability +assumption. There is a constant $a>0$ such that $(M,ag)$ is isometric to +the standard round $S^4$, to $\mathbb{CP}^2$ with its Fubini--Study +metric, or to $\mathbb{RP}^4$ with its standard round metric. +\end{theorem} + +Strict sectional positivity makes the Ricci tensor positive definite, +so the Einstein constant is positive. The oriented classification is the +principal task. Until the final subsection of +Section~\ref{sec:classification}, we therefore work on a connected +oriented smooth closed four-manifold, choose an orientation, and rescale +the metric so that $\operatorname{Ric}=3g$. The final orientation-cover +argument uses the oriented conclusion proved there and determines the +possible nonorientable quotient. + +\paragraph{Earlier results.} +The problem combines rigidity of the Einstein equation with the global +restrictions of positive sectional curvature. Berger's early rigidity +work under strict quarter pinching~\cite{Berger1961} was followed by +quantitative sectional-curvature bounds of Yang, Costa, and +Cao--Tran~\cite{Yang2000,Costa2004,CaoTran2018}. These bounds impose +additional quantitative hypotheses, some expressed as upper bounds +relative to the Einstein constant rather than as pinching ratios. +Gursky--LeBrun prove the Fubini--Study conclusion for compact oriented +Einstein manifolds with nonnegative sectional curvature and a nonzero +positive-definite intersection form~\cite[Theorem~A]{GurskyLeBrun1999}. +Koca proves the Fubini--Study conclusion for a complete positively sectionally +curved Einstein metric Hermitian for an integrable complex +structure~\cite[Theorem~2]{Koca2014}. Under the different condition of +nonnegative isotropic curvature, the four-dimensional local-symmetry +result is due to Micallef--Wang~\cite[Theorem~4.4]{MicallefWang1993}; +Brendle extends the Einstein rigidity result to higher dimensions +\cite[Theorem~1]{Brendle2010}. + +Recent results impose other geometric or topological conditions. +Dameno obtains half-conformal flatness from fibrewise constancy of the +scalar curvature of an associated twistor metric, and obtains the +standard models in the compact positive-scalar case +\cite[Theorem~1.1 and Corollary~1.2]{Dameno2026}. +Catino--Dameno use separation of the largest and middle eigenvalues +of one Weyl block to obtain conformal K\"ahler geometry after at most a double cover +\cite[Theorems~1.1 and~1.4]{CatinoDameno2026}. +Di Cerbo treats the closed oriented nonflat case with nonnegative +sectional curvature and equality everywhere in a Weyl-norm bound, +obtaining a locally symmetric $S^2\times S^2$ universal cover +\cite[Proposition~3.2]{DiCerbo2026}. + +Write $\chi$ for the Euler characteristic and $\tau$ for the signature. +Cheng proves the oriented positive-sectional classification under +$\chi\le3$, and consequently in the presence of a nontrivial Killing +field~\cite[Theorem~1.2 and Corollary~1.4]{Cheng2026}. His argument +combines radial Riccati comparison with Weyl-norm gaps and +characteristic-number identities. Gursky--Malchiodi prove the classification for closed oriented Einstein +four-manifolds with positive sectional curvature under +$2\chi-3|\tau|\le4$, using a Green-function comparison and ADM mass +\cite[Theorem~1.1]{GurskyMalchiodi2026}. These cited results retain +additional curvature, geometric, topological, or symmetry hypotheses. +The argument below addresses the interaction of the two Weyl blocks +without such an additional restriction. + +\paragraph{The two-block problem.} +The Hodge star on two-forms has eigenvalues $+1$ and $-1$; its two +rank-three eigenbundles are denoted by $\Lambda^+$ and $\Lambda^-$. +In the normalization $\operatorname{Ric}=3g$, the curvature operator +restricts to $\operatorname{Id}+T$ and $\operatorname{Id}+U$ on these +bundles, where $T,U$ are symmetric trace-free endomorphisms. They are +the two Weyl curvature blocks. The Einstein equation removes the +off-diagonal Ricci block but does not force either Weyl block to vanish. +Our central intermediate conclusion is \emph{half-conformal flatness}: +\[ + T\equiv0\qquad\text{or}\qquad U\equiv0. +\] + +Use Hilbert--Schmidt norms and put +\[ + v=|T|/\sqrt6,\qquad b=|U|/\sqrt6,\qquad + x=-\lambda_{\min}(T),\qquad y=-\lambda_{\min}(U). +\] +The exact sectional-positivity condition is $x+y<2$. The inequalities +$v\le x\le2v$ and $b\le y\le2b$ imply $v+b<2$, but this weaker norm +condition alone is insufficient for the tensor estimate below. +Gursky--LeBrun use the corresponding nonstrict cone and norm inequalities +and analyze the case of two nonzero Weyl blocks +\cite[Lemma~1 and Theorem~B]{GurskyLeBrun1999}. +For a function $f$, write $\mathbf E f$ for its integral divided by total +Riemannian volume. The gap and its parallel equality case are established +in Gursky--LeBrun~\cite[Theorem~1]{GurskyLeBrun1999}: a nonzero block +satisfies $\mathbf E v^2\ge1$, or the exchanged inequality for $b$. +We derive this statement in our normalization, including zeros of a block. + +Scalar coupling of the two blocks also has a clear antecedent. Yang +chooses a constant $t$ for which $|W^+|-t|W^-|$ has mean zero and combines +the Weyl equation, refined Kato inequality, and Poincar\'e inequality +\cite[\S6]{Yang2000}. Cao--Tran similarly balance regularized powers of +the norms~\cite[Proposition~2.2]{CaoTran2018}. Here a volume and moment +argument forces equality of the two second moments, and a weighted +Hessian square supplies the additional tensor estimate. + +\paragraph{From moment balance to half-conformal flatness.} +The characteristic identities relate the two second moments to volume +and to the dimensions of harmonic self-dual and anti-self-dual two-form +spaces. An upper polar-volume deficit, the two gaps, and a strengthened +scalar variance inequality force +$\mathbf E v^2=\mathbf E b^2$ when both blocks are nonzero +(Proposition~\ref{prop:balanced-moments}). This balance controls the mean +of $v-b$, enabling the variance inequality to contribute the sign needed +in the final integral argument. + +The main differential estimate uses the Bianchi identity twice. It first +places the gradient of a Weyl block in one irreducible tensor type. +The curvature commutator then determines all but one component of its +full, generally nonsymmetric Hessian. A curvature-weighted quadratic +form is positive on the remaining component because its lower bound is +$4-2x-2y>0$. Completing its square with different scalar weights for the +two blocks and integrating by parts produces inequalities involving only +first derivatives (Proposition~\ref{prop:weighted-hessian}). The proof +retains both the derivatives of the scalar weights and the derivative of +the curvature operator acting on the matrix slot. This local estimate +does not use moment balance. + +The two weighted inequalities, a scalar identity, and the variance bound +produce an integrand $\mathcal I$ with nonnegative mean. Explicit rational +comparison functions give $c\ge1/10$, nonnegative quantities $r,j$ built +from the norm gradients, and a scalar $\Theta$ such that +\[ + \mathbf E\mathcal I\ge0,\qquad + \mathcal I\le-\frac{(\Theta-cr)^2}{c}-c(j-r^2), + \qquad r^2\le j. +\] +The pointwise bounds follow from the weighted Hessian calculation and +exact finite polynomial signs. Moreover, $r^20$ on positively curved orthonormal +two-planes. All tensor connections are induced by the Levi--Civita +connection. We take covariant derivatives before choosing any +pointwise eigenbasis. + +\subsection{The two Weyl blocks} + +The Hodge-star eigenspaces $\Lambda^+$ and $\Lambda^-$ have eigenvalues +$+1$ and $-1$, respectively. On an oriented Euclidean four-space, choose +triples of skew orthogonal +complex structures $I_1,I_2,I_3$ and $J_1,J_2,J_3$ satisfying +\[ + I_1I_2=I_3=-I_2I_1,\qquad + J_1J_2=J_3=-J_2J_1, +\] +and their cyclic analogues, with every $I_s$ commuting with every $J_t$. +The two triples can be constructed from left and right quaternionic +multiplication. As two-forms, $I_s/\sqrt2$ and $J_s/\sqrt2$ are +orthonormal bases of $\Lambda^+$ and $\Lambda^-$, respectively. +Vector and covector indices are identified by the metric. + +\begin{lemma}[Curvature blocks and sectional positivity] +\label{lem:curvature-blocks} +There are symmetric trace-free endomorphisms +$T\in\operatorname{Sym}_0(\Lambda^+)$ and +$U\in\operatorname{Sym}_0(\Lambda^-)$ such that the curvature operator is +\[ + \mathcal R_{\Lambda^2} + =\begin{pmatrix}\operatorname{Id}+T&0\\ + 0&\operatorname{Id}+U\end{pmatrix}. +\] +At a point where these endomorphisms are diagonal, with eigenvalues +$t_s,u_s$, the curvature tensor is +\[ + R_{ikjl} + =\frac12\sum_s(1+t_s)I_{s,ik}I_{s,jl} + +\frac12\sum_s(1+u_s)J_{s,ik}J_{s,jl}. +\] +Put +\[ + x=-\min_s t_s,\qquad y=-\min_s u_s,\qquad + v=\frac{|T|}{\sqrt6},\qquad b=\frac{|U|}{\sqrt6}, +\] +where the matrix norms are Hilbert--Schmidt norms. Then +\[ + \sec_g>0\quad\Longleftrightarrow\quad x+y<2, + \qquad + v\le x\le2v,\qquad b\le y\le2b. +\] +Both Weyl blocks, viewed as curvature four-tensors, have zero +divergence. In particular, for $Z=\nabla T$, whose matrix entries are +one-forms, the differential Bianchi constraint is +\[ + \sum_s I_s Z_{sr}=0\qquad(r=1,2,3). +\] +\end{lemma} + +\begin{proof} +The first Bianchi identity says that the traces of the two diagonal +blocks of the curvature operator agree. Their sum is half the scalar +curvature, hence is $6$. Contraction identifies the off-diagonal block +with the trace-free Ricci tensor: its nine entries give the coefficients +in the independent symmetric trace-free matrices $I_sJ_t$. +Thus the Einstein condition makes this block zero, and each diagonal +block has trace $3$. Expanding in the orthonormal two-form bases gives +the displayed curvature tensor. + +A unit simple two-form has self-dual and anti-self-dual parts of norm +$1/\sqrt2$. Conversely, a unit two-form whose two parts have these +norms has exterior square zero, and is simple: in its orthogonal +normal form $\alpha=a\,e^1\wedge e^2+b\,e^3\wedge e^4$, the condition +$\alpha\wedge\alpha=0$ is $ab=0$. Therefore all pairs of plus and minus +parts of norm $1/\sqrt2$ occur. The smallest sectional curvature is +$1-(x+y)/2$, proving the equivalence. + +For the norm comparison, write the eigenvalues of $T$ as +$-x,a,x-a$. Minimality of $-x$ gives $-x\le a\le2x$, and hence +\[ + \frac32x^2 + \le x^2+a^2+(x-a)^2 + \le6x^2. +\] +Division by $6$ gives $v\le x\le2v$. The proof for $U$ is identical. + +Contracting the differential Bianchi identity shows that the curvature +tensor is divergence-free when Ricci is parallel. Project its last +pair of indices onto $\Lambda^+$ or $\Lambda^-$; these projections +are parallel. Subtracting the parallel scalar-curvature contribution +proves divergence freedom of each Weyl block. In a quaternionic basis, +the resulting constraint on $\nabla T$ is exactly the stated identity. +\end{proof} + +\subsection{Algebra of the derivative representations} + +Let $\Sigma_s$ denote the infinitesimal action of $I_s$ on the plus +indices of a tensor representation. On $\Lambda^+$ it is rotation +about axis $s$ with speed $2$. A complexified irreducible representation +has \emph{spin $j$} when +\[ + \sum_{s=1}^3\Sigma_s^2=-4j(j+1)\operatorname{Id}. +\] +Pairs $(j,\ell)$ specify the spins of the two commuting rotation +algebras. This notation describes associated tensor bundles of the +oriented orthonormal frame bundle; it requires no spin structure. + +\begin{lemma}[Representation rules] +\label{lem:spin-algebra} +A spin-$j$ representation has weights +$-j,-j+1,\ldots,j$ for $\Sigma_3/(2i)$, each once, and +\[ + H_j\otimes H_\ell + \cong H_{j+\ell}\oplus H_{j+\ell-1}\oplus\cdots\oplus H_{|j-\ell|}. +\] +In particular, the tangent, plus two-form, and plus trace-free +symmetric-matrix representations have types +\[ + \left(\frac12,\frac12\right),\qquad (1,0),\qquad (2,0). +\] +The Casimir operators and the orthogonal projections onto these +irreducible types are parallel bundle endomorphisms. +\end{lemma} + +\begin{proof} +The generators are skew-Hermitian and satisfy +$[\Sigma_1,\Sigma_2]=2\Sigma_3$, cyclically. Diagonalize +$\Sigma_3/(2i)$ and take the usual conjugate raising and lowering +operators, which shift its weight by one. If a unit highest vector +has weight $j$, commuting the raising and lowering operators gives +the squared lowering coefficient at weight $j-r$ as +\[ + (r+1)(2j-r). +\] +Nonnegativity and termination of the chain imply that $2j$ is a +nonnegative integer and give the listed weights and Casimir. +Taking invariant orthogonal complements decomposes an arbitrary +representation. Multiplying the weight lists, and successively +removing the chain of highest remaining weight, proves the tensor +product rule with multiplicity one. Commuting actions operate on the +corresponding multiplicity spaces. The same chain description shows +that an intertwiner between single irreducible copies is a scalar. + +The actions on the tangent space and on $\Lambda^+$ give the first +two types. The symmetric square of the spin-one representation is +spin two plus the scalar trace, giving the third. Finally, these +splittings are defined by invariant algebraic operators. The +connection induced by the orthonormal frame bundle preserves every +such operator. Thus the projections are parallel even though a +chosen local frame, or a curvature eigenbasis, need not be parallel. +\end{proof} + +Write $F=\operatorname{Sym}_0(\Lambda^+)$. For a symmetric trace-free +matrix $A$, define the self-adjoint operator on $F$ +\[ + B_F(A)S=\frac{AS+SA}{2} + -\frac{\operatorname{tr}(AS)}3\operatorname{Id}. +\] +It is characterized by +$\langle B_F(A)S,S\rangle=\operatorname{tr}(AS^2)$. +On any plus spin-$j$ representation with $j\ge1$, define +\[ + Q_j(A)=\frac{\sum_{r,s}A_{rs}\Sigma_r\Sigma_s}{4j(2j-1)}. +\] +For diagonal $A$ the numerator is $\sum_s a_s\Sigma_s^2$. + +\begin{lemma}[Curvature-weighted operators] +\label{lem:quadratic-curvature-action} +If $A$ has smallest eigenvalue $-x$, then +\[ + Q_j(A)\le \frac{x(j+1)}{2j-1}\operatorname{Id}. +\] +Moreover, $Q_2(A)=B_F(A)$. The compression of $B_F(A)$, acting on an +$F$ factor, to a highest-spin coupling of total plus spin $j$ is +$Q_j(A)$, provided the operator acts trivially on any additional +irreducible minus factor. +\end{lemma} + +\begin{proof} +In a diagonal basis, $a_s+x\ge0$ and every $\Sigma_s^2$ is +nonpositive. Thus +\[ + \sum_s a_s\Sigma_s^2 + \le -x\sum_s\Sigma_s^2 + =4xj(j+1)\operatorname{Id}, +\] +which proves the bound. + +To identify $Q_2$, diagonalize $S=\operatorname{diag}(d_1,d_2,d_3)$. +The speed-two matrix action gives +$|\Sigma_sS|^2=8(d_j-d_k)^2$, with $s,j,k$ cyclic; these three +derivatives are mutually orthogonal. Consequently, allowing $A$ +to be nondiagonal in this basis, +\[ + \begin{aligned} + \langle Q_2(A)S,S\rangle + &=-\frac13\sum_s A_{ss}(d_j-d_k)^2\\ + &=-\frac13\sum_s A_{ss} + \bigl(2\operatorname{tr}(S^2)-3d_s^2\bigr) + =\operatorname{tr}(AS^2). + \end{aligned} +\] +Here trace-freeness of both matrices gives the middle identity and +removes the scalar term. Polarization proves the operator equality. + +By Lemma~\ref{lem:spin-algebra}, spin two occurs once in +$\operatorname{End}(H_j)\cong H_j\otimes H_j$. Hence an equivariant +operator family linear in $A$ is unique up to scalar. The same is +true with an irreducible minus factor when the operators commute +with its action. Test the scalar at +$A=\operatorname{diag}(-1,-1,2)$ on a highest vector. At weight $m$, +\[ + Q_j(A)=\frac{j(j+1)-3m^2}{j(2j-1)}, +\] +so its highest-weight eigenvalue is $-1$. The $F$ highest vector +also has $B_F(A)$ eigenvalue $-1$. A product of highest vectors is +the highest vector of the stretched coupling, proving the claimed +compression. For the same reason, compression of a single generator +from a spin-$\ell$ factor to a stretched spin-$j$ product is +$\ell/j$ times the total generator. +\end{proof} + +\begin{proposition}[Differential equations for the Weyl blocks] +\label{prop:weyl-derivative-equations} +The tensor $Z=\nabla T$ belongs to the parallel subbundle +$G$ of type $(5/2,1/2)$. On this bundle, put +$B_{\mathrm{op}}=Q_{5/2}(T)$. Then +\[ + \Delta T=6(T-N),\qquad + N=(T^2)_0=T^2-2v^2\operatorname{Id}, +\] +and +\begin{equation} + \Delta Z=(13-20B_{\mathrm{op}})Z. + \tag{1}\label{eq:1} +\end{equation} +Every statement has an exchanged counterpart for $U$. +\end{proposition} + +\begin{proof} +The derivative space $T^*M\otimes F$ has types +$(5/2,1/2)$ and $(3/2,1/2)$. The divergence takes values in a space +whose plus spins are at most $3/2$. It is nonzero: a nonzero covector +tensored with $\operatorname{diag}(1,-1,0)$ has nonzero divergence. +It therefore vanishes on the first summand and is injective on the +second, by irreducibility. Lemma~\ref{lem:curvature-blocks} puts $Z$ +in the first summand. + +Let +\[ + C=\sum_s I_s\otimes\Sigma_s +\] +on $T^*M\otimes F$. Expansion of the combined Casimir gives +$C=-4$ on $G$ and $C=6$ on its complement. In particular +$P_G=(6-C)/10$ is a parallel orthogonal projector. The bundle +curvature on $F$ is +\[ + F_{ik}=\frac12\sum_s(1+t_s)I_{s,ik}\Sigma_s +\] +at a point of diagonalization. Taking the covariant divergence of +$CZ=-4Z$, using skewness of $I_s$ to take a commutator and +$\sum_{ik}I_{s,ik}I_{r,ik}=4\delta_{sr}$, gives +\[ + -4\Delta T=\sum_s(1+t_s)\Sigma_s^2T + =-24T+24B_F(T)T=-24(T-N). +\] + +To differentiate this equation, note first that +$\nabla N=2B_F(T)Z$. For each derivative index $j$, the covariant +commutation formula reads +\[ + \Delta Z_j + =\nabla_j\Delta T+\nabla^i(F_{ij}T) + +F_{ij}\nabla^iT + +\operatorname{Ric}_{ja}Z_a. +\] +The divergence of $F$ vanishes because it is the induced action of +the divergence-free Riemann curvature. Thus the differentiated +equation contributes $6Z-12B_F(T)Z$, the Ricci term contributes +$3Z$, and the remaining bundle term is +$-\sum_s(1+t_s)I_s\Sigma_sZ$. Its constant part is $-CZ=4Z$. + +The left side already lies in $G$, since $P_G$ is parallel and +$P_GZ=Z$. We may therefore compress the right side to $G$, +although its individual summands need not preserve $G$. +Lemma~\ref{lem:quadratic-curvature-action} gives +$P_GB_F(T)P_G=Q_{5/2}(T)$. On $G$, the total plus generators are +$I_s+\Sigma_s$, so +\[ + 40Q_{5/2}(T) + =P_G\sum_s t_s(I_s+\Sigma_s)^2P_G. +\] +The $I_s^2$ term vanishes by $\sum_s t_s=0$, and the $\Sigma_s^2$ +term compresses to $24Q_{5/2}(T)$. Hence +\[ + P_G\sum_s t_s I_s\Sigma_sP_G=8Q_{5/2}(T). +\] +The scalar contributions sum to $13$, while the two curvature +contributions sum to $-20Q_{5/2}(T)$, proving \eqref{eq:1}. +For a nondiagonal $T$ the same calculation uses the symmetric +contraction $\sum_{r,s}T_{rs}(I_r+\Sigma_r)(I_s+\Sigma_s)$: +its $I_rI_s$ part is zero by symmetry and trace-freeness. +Thus the computation is tensorial and entails no differentiation +of a moving eigenframe. +\end{proof} + +\subsection{The gradient parametrization and the norm equation} + +Write +\[ + p=\nabla v,\qquad q=\nabla b,\qquad + e=\frac{|\nabla T|^2}{6},\qquad + \bar e=\frac{|\nabla U|^2}{6}. +\] +The norms $v,b$ are Lipschitz. Their gradients are understood almost +everywhere and vanish almost everywhere on the respective zero sets. + +Where $v>0$, diagonalize $T$ and put, cyclically in $i,j,k$, +\[ + t_i=vs_i,\qquad + \widetilde s_i=\frac{s_j-s_k}{\sqrt3},\qquad + \sigma=\frac16\sum_i s_i^3,\qquad + \rho=\frac16\sum_i s_i^2\widetilde s_i. +\] +Here $\sum s_i=0$ and $\sum s_i^2=6$. On the plane of triples with +zero sum, the map $s\mapsto\widetilde s$ is a skew-adjoint orthogonal +quarter-turn. Thus +\[ + \sum_i\widetilde s_i=0,\qquad + \sum_i\widetilde s_i^2=6,\qquad + \sum_i s_i\widetilde s_i=0,\qquad + s_i^2+\widetilde s_i^2=4. +\] +For later use, these identities imply +\[ + \sigma^2+\rho^2=1,\qquad + \frac{N}{v^2} + =\operatorname{diag}(\sigma s_i+\rho\widetilde s_i), + \qquad + \frac16\sum_i\widetilde s_i^3=-\rho. +\] +Indeed, the trace-free triple $n_i=s_i^2-2$ has squared norm $6$: +use $\sum_i s_i^4=(\sum_i s_i^2)^2/2=18$. +Its coefficients in the orthogonal basis $s,\widetilde s$ are +$\sigma,\rho$, proving the first two assertions. The last follows +by factoring the alternating cubic +$\sum_i s_i^2(s_j-s_k)$. If $r=x/v$, the other two eigenvalues +besides $-r$ have sum $r$ and product $r^2-3$, so +\[ + 1\le r\le2,\qquad + \sigma=\frac{3r-r^3}{2}. +\] +Normalized quantities are used only where the corresponding block +is nonzero. Polynomial products such as +$\sigma v^3=\operatorname{tr}(T^3)/6$ have their natural extensions +across its zero set. Bars denote the analogous quantities for $U$. + +The next formula gives, in particular, the refined Kato inequality +$|\nabla T|^2\ge(5/3)|\nabla|T||^2$ for a half-Weyl block. +Gursky--LeBrun derive this estimate from the same irreducible Bianchi +gradient constraint~\cite[Lemma~4, equations~(10)--(12)]{GurskyLeBrun1999}; +Calderbank--Gauduchon--Herzlich give its general representation-theoretic +setting~\cite[equation~(7.1)]{CalderbankGauduchonHerzlich2000}. +We retain all three vector parameters: the excess over the norm-gradient +energy will be needed in the coupled estimate. + +\begin{lemma}[Parametrization of the Bianchi gradient space] +\label{lem:gradient-parametrization} +There are unique vectors $k,l$ such that +\begin{equation} + \begin{gathered} + Z_{ii}=s_i p-\sqrt3\,\widetilde s_i k,\qquad + Z_{jk}=-I_i\left(l+s_i k+\frac{\widetilde s_i}{\sqrt3}p\right), + \qquad (i,j,k)\ \text{cyclic},\\ + e=\frac53|p|^2+5|k|^2+|l|^2. + \end{gathered} + \tag{2}\label{eq:2} +\end{equation} +More generally, for any fixed normalized trace-free diagonal triple +$s$, the same formulas parametrize the entire algebraic space $G$ +by three arbitrary real four-vectors. +\end{lemma} + +\begin{proof} +Write $D_i=Z_{ii}$ and $a_i=I_iZ_{jk}$. Quaternion multiplication +turns the three divergence equations into +$D_i+a_j-a_k=0$. Trace-freeness and the orthogonal basis +$s,\widetilde s$ give uniquely +\[ + p=\frac16\sum_i s_iD_i,\qquad + k=-\frac1{6\sqrt3}\sum_i\widetilde s_iD_i,\qquad + D_i=s_ip-\sqrt3\,\widetilde s_i k. +\] +Since +$s_j-s_k=\sqrt3\,\widetilde s_i$ and +$\widetilde s_j-\widetilde s_k=-\sqrt3\,s_i$, +the vectors $s_i k+\widetilde s_i p/\sqrt3$ have exactly the +differences required of the $a_i$. Their difference from $a_i$ is +therefore a common vector $l$, proving the formulas and their +converse. This also establishes their full algebraic scope. + +Orthogonality of $s,\widetilde s$ gives +\[ + \sum_i|Z_{ii}|^2=6|p|^2+18|k|^2,\qquad + 2\sum_{j0\}$, pair the equation for $\Delta T$ with $T$ to obtain +\[ + \Delta(v^2)=12(v^2-\sigma v^3)+2e. +\] +This is globally an identity of smooth functions. Put +$v_\varepsilon=\sqrt{v^2+\varepsilon^2}$. Direct differentiation gives +\[ + \Delta v_\varepsilon + =\frac{6(v^2-\sigma v^3)}{v_\varepsilon} + +q_\varepsilon,\qquad + q_\varepsilon=\frac e{v_\varepsilon} + -\frac{v^2|p|^2}{v_\varepsilon^3}. +\] +The refined derivative estimate in \eqref{eq:2} yields +\[ + \frac25\frac e{v_\varepsilon} + \le q_\varepsilon\le\frac e{v_\varepsilon}. +\] +These assertions hold almost everywhere, including the zero set: +every smooth component of $T$ has derivative zero almost everywhere +on its zero set, hence $Z=0$ almost everywhere on $\{T=0\}$. +One can see this elementary fact at density points of a level set; +a nonzero derivative would instead make that scalar level set a +local hypersurface. + +The first term in $\Delta v_\varepsilon$ is uniformly bounded in +absolute value by $6(v+v^2)$. Its integral is the negative of the +integral of $q_\varepsilon$. Consequently +$\mathbf E(e/v_\varepsilon)$ is uniformly bounded. +Monotone convergence proves the asserted $L^1$ bound for $e/v$. +Off the zero set, $q_\varepsilon$ is dominated by $e/v$ and tends +to $(e-|p|^2)/v$; on the zero set both are zero almost everywhere. +Dominated convergence therefore gives an $L^1$ limit, rather than +only a weak limit of measures. The zeroth-order term converges in +$L^1$ as well. Since $v_\varepsilon\to v$ uniformly, their +Laplacians converge distributionally, proving \eqref{eq:3} without +a residual singular measure. + +Finally, approximate a bounded Lipschitz test function uniformly and +in $H^1$ by smooth functions, using local mollification and a +partition of unity. Uniform convergence controls its pairing with +the $L^1$ right side, and $H^1$ convergence controls its gradient +pairing with $\nabla v$. This proves the asserted testing rule. +\end{proof} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/03-analytic.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/03-analytic.tex new file mode 100644 index 0000000000000000000000000000000000000000..ac2e87de5b5ab19939880574a562d3377b5628f4 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/03-analytic.tex @@ -0,0 +1,245 @@ +\section{Scalar estimates and the characteristic identities} +\label{sec:analytic} + +This section supplies two kinds of global information: analytic bounds +for the sizes of the Weyl blocks, and integer identities relating their +second moments to volume. The former will control derivatives in the +coupled inequality; the latter will restrict the possible imbalance +between the two blocks. + +The standing setting is the closed, connected, oriented Einstein +four-manifold with $\operatorname{Ric}=3g$ fixed in the introduction. +Appendix~\ref{app:analytic} supplies the analytic foundations used here; +its proofs depend only on this setting and the curvature conventions of +Section~\ref{sec:preliminaries}. We retain the weighted variance argument +and the complete Weyl-gap proof in the main text because their precise +weights and behavior at zeros are used in the two-block argument. + +\subsection{A strengthened scalar variance estimate} + +\begin{proposition}[Weighted variance] +\label{prop:weighted-variance} +For every real $a\in H^1(M)$, +\begin{equation} + \operatorname{Var}(a) + :=\mathbf E\bigl[(a-\mathbf E a)^2\bigr] + \le\mathbf E\frac{|\nabla a|^2}{K}, + \qquad K=4+v^2+b^2. + \tag{4}\label{eq:4} +\end{equation} +\end{proposition} + +\begin{proof} +First let $\phi$ be smooth and write +$S=\operatorname{Hess}_0\phi$. The integrated Bochner identity +and $\operatorname{Ric}=3g$ give +\[ + \mathbf E\bigl[(\Delta\phi)^2\bigr] + =\mathbf E\left(4|\nabla\phi|^2+\frac43|S|^2\right). +\] +Define operators on covariant two-tensors by +\[ + (T_D)_{ikjl}=\frac12\sum_s t_s I_{s,ik}I_{s,jl}, + \qquad + (U_D)_{ikjl}=\frac12\sum_s u_s J_{s,ik}J_{s,jl}, +\] +where $ij$ is the row index and $kl$ is the column index. +These tensorial definitions may be evaluated after pointwise +diagonalization. They annihilate metric trace. On the symmetric +trace-free tensors, the nine matrices $I_iJ_j$ form an orthogonal +basis, with $T_D$ and $U_D$ eigenvalues $t_i$ and $u_j$. +For example, $T_D$ acts by +$-\frac12\sum_s t_s I_s(\,\cdot\,)I_s$; the quaternion relations +give the eigenvalue $t_i$ on $I_iJ_j$. +Thus +\[ + T_D+U_D\le 2(x+y)\operatorname{Id}<4\operatorname{Id} +\] +on this space. + +The integrated curvature contraction has a useful sign and a precise +coefficient. Since $T_D$ annihilates trace, replace $S$ by +$\operatorname{Hess}\phi$ in its quadratic form. Integrate the +derivative $k$, use $\nabla_k(T_D)_{ikjl}=0$, and use skewness in +$i,k$ to obtain +\[ + \begin{aligned} + \mathbf E\langle T_DS,S\rangle + &=-\mathbf E (T_D)_{ikjl}\phi_l\nabla_k\nabla_i\phi_j\\ + &=\frac12\mathbf E (T_D)_{ikjl}R_{ikja}\phi_a\phi_l. + \end{aligned} +\] +In this contraction the minus curvature block is orthogonal to the +plus block. Substituting their expansions gives +\[ + \frac12\sum_{ikj}(T_D)_{ikjl}R_{ikja} + =\frac12\sum_s t_s(1+t_s) + \sum_j I_{s,jl}I_{s,ja} + =3v^2\delta_{la}. +\] +The final equality uses $\sum t_s=0$ and $\sum t_s^2=6v^2$. +Consequently +\[ + \mathbf E\langle(T_D+U_D)S,S\rangle + =3\mathbf E\bigl[(v^2+b^2)|\nabla\phi|^2\bigr]. +\] +The pointwise upper operator bound, inserted into the positive +$|S|^2$ term of the Bochner identity, now yields +\[ + \mathbf E\bigl[(\Delta\phi)^2\bigr]\ge\mathbf E K|\nabla\phi|^2. +\] + +For smooth $a$, the closed-manifold Poisson statement in +Appendix~\ref{app:analytic-preliminaries} gives a unique smooth +zero-mean solution of $\Delta\phi=a-\mathbf E a$. Integration +by parts and Cauchy--Schwarz with the pointwise weight $K$ give +\[ + \begin{aligned} + \operatorname{Var}(a) + &=-\mathbf E\langle\nabla a,\nabla\phi\rangle\\ + &\le + \left(\mathbf E\frac{|\nabla a|^2}{K}\right)^{1/2} + \left(\mathbf E K|\nabla\phi|^2\right)^{1/2}\\ + &\le + \left(\mathbf E\frac{|\nabla a|^2}{K}\right)^{1/2} + \operatorname{Var}(a)^{1/2}. + \end{aligned} +\] +Division proves the assertion when the variance is nonzero; the +zero case is immediate. Smooth functions are dense in $H^1$ by the +local mollification argument recalled in +Appendix~\ref{app:analytic-preliminaries}. Since $K$ is bounded above +and below by positive constants, this approximation also passes the +weighted energy to the limit and proves the assertion on $H^1$. +\end{proof} + +\subsection{The sharp scalar inequality and the Weyl gap} + +The following is the four-dimensional Einstein case of Ilias's sharp +positive-Ricci Sobolev inequality~\cite[Theorem~3]{Ilias}. Its proof in +the present normalization is given in Appendix~\ref{app:sharp-sobolev}. +The nonlinear Hessian argument +belongs to the Bochner approach to elliptic rigidity; compare the +systematic treatment in~\cite{BidautVeronVeron}. + +\begin{proposition}[Normalized Sobolev inequality] +\label{prop:sharp-sobolev} +For every real $f\in H^1(M)$, +\[ + \bigl(\mathbf E|f|^4\bigr)^{1/2} + \le\mathbf E\left(f^2+\frac12|\nabla f|^2\right). +\] +\end{proposition} + +The next proposition is the Weyl-norm gap established in +Gursky--LeBrun~\cite[Theorem~1]{GurskyLeBrun1999}; we give the derivation in +our normalization, including the equality case and the possible +zeros of the Weyl block. + +\begin{proposition}[Weyl-norm gap] +\label{prop:weyl-gap} +For either Weyl block, +\begin{equation} + v\equiv0\qquad\text{or}\qquad \mathbf E v^2\ge1. + \tag{5}\label{eq:5} +\end{equation} +If $T\not\equiv0$ and $\mathbf E v^2=1$, then +$\nabla T=0$ and $v=\sigma=1$. The corresponding statements hold +for $U$. +\end{proposition} + +\begin{proof} +Put $A_\varepsilon=v^2+\varepsilon^2$ and +$f_\varepsilon=A_\varepsilon^{1/6}$. These are smooth functions. +The equation for $\Delta(v^2)$ and \eqref{eq:2} give +\[ + \begin{aligned} + \Delta f_\varepsilon + =A_\varepsilon^{-5/6} + \left[ + 2(v^2-\sigma v^3)+\frac13e + -\frac59\frac{v^2}{A_\varepsilon}|p|^2 + \right] + \ge 2A_\varepsilon^{-5/6}(v^2-v^3). + \end{aligned} +\] +Indeed, $\sigma\le1$, $v^2/A_\varepsilon\le1$, and +$e\ge5|p|^2/3$. Multiply by $f_\varepsilon$ and integrate: +\[ + \mathbf E|\nabla f_\varepsilon|^2 + \le 2\mathbf E\frac{v^3-v^2}{A_\varepsilon^{2/3}} + \le 2\mathbf E v^{5/3}. +\] +Their $L^2$ norms are uniformly bounded as well. +Thus $f_\varepsilon$ is bounded in $H^1$, and its uniform limit +$f=v^{1/3}$ belongs to $H^1$ by weak compactness. This bound is +obtained directly from the regularized equation; it requires no +positive lower bound on $v$. + +Apply Proposition~\ref{prop:sharp-sobolev} to $f_\varepsilon$ +and use the preceding energy bound: +\[ + \begin{aligned} + \bigl(\mathbf E f_\varepsilon^4\bigr)^{1/2} + &\le \mathbf E\left[ + A_\varepsilon^{1/3} + +\frac{v^3-v^2}{A_\varepsilon^{2/3}}\right]\\ + &=\mathbf E\frac{\varepsilon^2+v^3}{A_\varepsilon^{2/3}}. + \end{aligned} +\] +The term $\varepsilon^2/A_\varepsilon^{2/3}$ is at most +$\varepsilon^{2/3}$, while +$v^3/A_\varepsilon^{2/3}\le v^{5/3}$. +Dominated convergence therefore gives +\[ + \bigl(\mathbf E f^4\bigr)^{1/2} + \le\mathbf E(f^2v) + \le\bigl(\mathbf E f^4\bigr)^{1/2} + \bigl(\mathbf E v^2\bigr)^{1/2}. +\] +When $T\not\equiv0$, the common factor is positive, proving +\eqref{eq:5}. + +If $\mathbf E v^2=1$, equality holds in Cauchy--Schwarz. +Thus $v=\alpha f^2$ almost everywhere for a constant $\alpha>0$. +It follows that $v$ takes only the values $0$ and $\alpha^3$ +almost everywhere, and hence everywhere by continuity. +Connectedness and $T\not\equiv0$ force the positive constant value +on all of $M$. The moment condition gives $v=1$. +Equation~\eqref{eq:3} then becomes +$0=6(1-\sigma)+e$. Both terms are nonnegative, so +$\sigma=1$ and $e=0$. Smoothness gives $\nabla T=0$ everywhere. +\end{proof} + +\subsection{The Euler and signature formulas} + +\begin{proposition}[Characteristic identities] +\label{prop:characteristic-identities} +Let $n_+$ and $n_-$ be the finite nonnegative integer dimensions of +the spaces of harmonic self-dual and anti-self-dual two-forms. The Euler +characteristic and signature of $M$ are, respectively, +\[ + \chi=2+n_++n_-,\qquad \tau=n_+-n_-. +\] +Then +\begin{equation} + \begin{aligned} + 8\pi^2\chi + &=6\operatorname{Vol}(M) + \bigl(1+\mathbf E(v^2+b^2)\bigr),\\ + 12\pi^2\tau + &=6\operatorname{Vol}(M)\,\mathbf E(v^2-b^2). + \end{aligned} + \tag{6}\label{eq:6} +\end{equation} +If $U=0$, then $n_-=0$; likewise $T=0$ implies $n_+=0$. +\end{proposition} + +These are the Euler and signature formulas in the present curvature +normalization. Appendix~\ref{app:characteristic-identities} identifies +the topological integers through Hodge theory and proves the formulas +by the form-complex heat-supertrace method of +McKean--Singer~\cite{McKeanSinger1967}; compare the general formulations +and local coefficients in Gilkey~\cite[Theorems~1.7.6 and +4.8.16]{Gilkey1984}. That appendix also proves the zero-block kernel +vanishing statements, which will be needed after one-block vanishing. diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/04-volume.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/04-volume.tex new file mode 100644 index 0000000000000000000000000000000000000000..7294ecdcc62b3dd0b7849a0f2f8c554af713dec1 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/04-volume.tex @@ -0,0 +1,626 @@ +\section{Upper volume deficit and balanced Weyl moments} +\label{sec:volume} + +Write $V_M=\operatorname{Vol}(M,g)$ and $s_0=\mathbf E(v^2+b^2)$. +Our goal in this section is to prove that the two second moments agree +whenever neither Weyl block vanishes. The characteristic identities +turn an upper polar-volume deficit into an integer restriction; a +weighted scalar variance estimate then excludes the sole remaining +unequal-moment case. + +Radial comparison also plays a central role in Cheng's classification +under the additional assumption $\chi(M)\le3$ +\cite[Proposition~3.1]{Cheng2026}. His Riccati argument does not require +curvature matrices at different times to commute. We use a discrete +Jacobi determinant to retain their order and obtain the following +moment-dependent deficit. + +\begin{proposition}[Upper volume deficit]\label{prop:upper-volume-deficit} +One has +\[ + \frac{V_M}{8\pi^2/3}\le 1-\frac{3}{40}s_0. + \tag{7}\label{eq:7} +\] +\end{proposition} + +Here is how the estimate will be used. If both blocks are nonzero, their +gaps and $v+b<2$ give $2\le s_0<4$. Equations~\eqref{eq:6} and +\eqref{eq:7} then imply $\chi<7$ and $|\tau|/\chi<4/15$. +The parity and nonnegativity constraints from $n_\pm$ leave only +$(\chi,|\tau|)=(5,1)$ when $\tau\ne0$. The last part of this section +rules out that case and proves Proposition~\ref{prop:balanced-moments}. +We first establish the upper deficit and the common radial machinery; +the conditional lower comparison needed after one-block vanishing is +proved in Section~\ref{sec:classification}. + +Along a unit-speed geodesic, choose a parallel orthonormal normal frame +and let $\mathcal R(t)$ be the normal Jacobi curvature matrix. Set +\[ + A(t)=\mathcal R(t)-\mathrm{Id},\qquad + \mathcal J''+\mathcal R\mathcal J=0,\qquad + \mathcal J(0)=0,\quad \mathcal J'(0)=\mathrm{Id},\qquad + D(t)=\det\mathcal J(t). +\] +Positive sectional curvature and $\operatorname{Ric}=3g$ give +$0<\mathcal R<3\mathrm{Id}$ and $\operatorname{tr}A=0$. +Let $\mathcal E$ denote integration over the unit tangent bundle against +normalized Riemannian measure on the base and normalized spherical +measure in the fibers. + +\begin{lemma}[Geodesic averages]\label{lem:geodesic-averages} +At every time along the geodesic flow, +\[ + \mathcal E|A|^2=\frac32s_0,\qquad + \mathcal E|A'|^2\le\frac32\,\mathcal E|A|^2. + \tag{8}\label{eq:8} +\] +\end{lemma} +\begin{proof} +The indicated probability measure is invariant under geodesic flow. +Indeed, in cotangent coordinates the Hamiltonian flow of kinetic energy +preserves canonical volume: its divergence vanishes by equality of the +mixed second derivatives of the Hamiltonian. Applying this observation +to energy shells gives invariance of the measure on the unit-energy +level. In orthonormal fiber coordinates this is precisely Riemannian +base volume times spherical measure. + +The plus and minus contributions to $A$ are orthogonal conjugates of +$T/2$ and $U/2$, using the normal bases $I_s\dot\gamma$ and +$J_s\dot\gamma$. The same description applies to their derivatives in +parallel frames. The mixed contractions have zero spherical average: +they are invariant bilinear pairings between the inequivalent types +$(2,0)$ and $(0,2)$, or between $(5/2,1/2)$ and $(1/2,5/2)$. +Consequently, +\[ + \mathcal E|A|^2=\frac14\mathbf E(|T|^2+|U|^2),\qquad + \mathcal E|A'|^2 + =\frac1{16}\mathbf E(|\nabla T|^2+|\nabla U|^2). +\] +Here the second factor $1/4$ in the derivative identity is the spherical +average of a squared directional derivative in four dimensions. +Integrating the equation for $\Delta T$ yields +\[ + \mathbf E e=6\mathbf E(\sigma v^3-v^2)\le6\mathbf E v^2, +\] +since $\sigma\le1$ and $v<2$. The exchanged identity gives the same +bound for $\bar e$. The stated formulas follow. +\end{proof} + +\subsection{Minimizing geodesics and upper polar integration} + +The global input is the classical index-form and closed-geodesic argument +of Myers, Synge, and Klingenberg +\cite{Myers1941,Synge1936,Klingenberg1959}; see also +\cite[Theorem~1.5 and Remark~1.2(ii)]{AbreschMeyer1997}. +Its full proof is in Appendix~\ref{app:topology}, under all the standing +oriented, closed, normalized positivity hypotheses. + +\begin{lemma}[Diameter, injectivity, and topology]\label{lem:volume-topology} +The diameter of $M$ is at most $\pi$, its injectivity radius is at least +$\pi/\sqrt3$, and $M$ is simply connected. Every minimizing geodesic +segment has no point conjugate to its initial point in its interior. +\end{lemma} + +The last assertion allows a conjugate endpoint. It is needed in the +upper polar argument even for segment lengths between $\pi/\sqrt3$ and +$\pi$. + +Put $D_c(t)=D(t)$ before the first conjugate time and $D_c(t)=0$ at and +after it. + +\begin{lemma}[Upper polar bound]\label{lem:upper-polar} +One has +\[ + V_M\le2\pi^2\int_0^\pi\mathcal E D_c(t)\,dt. +\] +\end{lemma} +\begin{proof} +Fix a center $p$ and take all vectors of length +less than $\pi$ whose radial geodesics have no conjugate point at or +before their endpoints. This is an open set: the Jacobi determinant +divided by $t^3$, continued as $1$ at zero, has a positive minimum on +each such radial interval. Every point is the exponential image of a +minimizing vector of length at most $\pi$, by +Lemma~\ref{lem:volume-topology}. A minimizing vector is in this open +set unless it is critical for $\exp_p$, has length $\pi$, or is zero. + +The exponential map is smooth with bounded derivative on the closed +radius-$\pi$ ball. The area formula with multiplicity states that the +integral of its absolute Jacobian over a measurable set equals the +integral of the number of preimages over its image. In particular, +critical vectors and the boundary sphere have images of zero +four-dimensional volume: their respective Jacobian integral and domain +volume are zero. On the open preconjugate set the determinant is +positive. The area formula therefore bounds $V_M$ by the integral of +$D$ over this set, with multiplicities only increasing that integral. +Nonconjugate cut vectors are included in the set; no identification of +cut points with conjugate points is used. Polar coordinates give the +claimed bound at each center, since the area of the unit +three-sphere is $2\pi^2$. Average over centers and use Tonelli's theorem. + +\end{proof} + +\subsection{Discrete determinant comparisons} + +The upper polar bound reduces the volume problem to estimates for a +matrix Jacobi determinant. We approximate the Jacobi equation by a finite +Dirichlet matrix. Its Schur complements and sine-mode compressions +retain the order of matrix multiplication, without requiring simultaneous diagonalization along the geodesic. +The same discrete determinant will be used again for the conditional +lower comparison in Section~\ref{sec:classification}. + +\begin{lemma}[The discrete Jacobi determinant]\label{lem:discrete-jacobi} +Fix a smooth symmetric normal curvature matrix on $[0,t]$, put $a=t/N$, +and let $\mathcal L_N$ be the block tridiagonal matrix on nodes +$1,\ldots,N-1$ with diagonal blocks $2\mathrm{Id}-a^2\mathcal R(ia)$ +and off-diagonal blocks $-\mathrm{Id}$. Then +\[ + a^3\det\mathcal L_N\longrightarrow D(t). +\] +If $t$ is before the first conjugate time, $\mathcal L_N$ is positive +definite for all sufficiently large $N$. The Jacobi convergence is +uniform over families with uniformly bounded coefficients and first +derivatives. +\end{lemma} +\begin{proof} +Define matrices without using inverses by +\[ + Y_0=0,\quad Y_1=\mathrm{Id},\qquad + Y_{i+1}=(2\mathrm{Id}-a^2\mathcal R(ia))Y_i-Y_{i-1}. +\] +The position $aY_i$ and backward velocity $Y_i-Y_{i-1}$ evolve by +first changing velocity by $-a\mathcal R(ia)(aY_i)$ and then changing +position by $a$ times the new velocity. On the exact Jacobi solution +one step has error $O(a^2)$, while the update amplifies errors by at most +$1+Ca$. Iteration proves convergence of position and velocity, uniformly +on the fixed interval. The constants are uniform under the stated +coefficient bounds. + +Suppose first that $t$ is preconjugate. Away from zero the exact Jacobi +matrix is invertible with uniformly bounded inverse on compact time +intervals, so the converging $aY_i$ are invertible there. Near zero use +the identity, still without inverses, +\[ + Y_i=i\mathrm{Id}-a^2\sum_{r2/25$; the cubic power $S^3$ +is essential in each row. + +\begin{proof}[Proof of Proposition~\ref{prop:upper-volume-deficit}] +For the sine-squared probability weight defining $B$, its time variance +is $t^2(1/12-1/(2\pi^2))$. The variance identity and \eqref{eq:8} give +\[ + \mathcal E|B|^2\ge\mathcal E|A|^2 + \left(1-\frac32t^2\left(\frac1{12}-\frac1{2\pi^2}\right)\right). +\] +Indeed, parallel identification along a geodesic and Cauchy--Schwarz +show +$\mathcal E|A(s)-A(r)|^2\le(s-r)^2\mathcal E|A'|^2$. +With $u=t/\pi$ the factor in parentheses is at least $1-u^2/2$. +For $0\le u<1$, furthermore, +\[ + f(u)\le\min\{1,\beta^2\max(1,3-\beta)\},\qquad + \beta=\frac{u^2}{1-u^2}. +\] +At $u=1$ we give the capped comparison function its continuous value $1$. +For $u\le5/8$, comparison with $\beta^2(3-\beta)$ reduces to +$6-13u^2+6u^4\ge0$. On this interval $f$ is increasing and +$f(5/8)=7511875/8388608<1$, which also verifies the cap by $1$. The comparison function $\min\{1,\beta^2\max(1,3-\beta)\}$, as a function of $u$, is nondecreasing and remains $1$ after it first reaches $1$; at $u=5/8$ its value is +$57500/59319>15/16$, which handles the rest of the interval. +Lemmas~\ref{lem:upper-polar} and \ref{lem:upper-radial} therefore show +that the coefficient of $s_0$ saved from the round volume is at least +\[ + \eta_+=\frac{9\pi}{48}\int_0^1 + f(u)(1-u^2/2)\sin^3(\pi u)\,du. +\] +The first three rational bounds above give $\eta_+>2/25>3/40$, proving +\eqref{eq:7}. + +\end{proof} + +\subsection{A moment obstruction} + +The upper volume deficit is proved. Together with the characteristic +identities and the two norm gaps, it leaves a single +possible unequal-moment integer case when both blocks are nonzero, up to +exchanging the blocks. The next scalar lemma +will exclude that case. Its proof reduces a constrained probability +measure to two support points; all numerical signs are established by +explicit rational coefficients. + +For $0\le z\le2$, define +\[ + H(z)=\sqrt z\left(1+\frac{z^2}{40}-\frac{z^4}{1152}\right),\qquad + \Phi(z)=16H(z)^2+9z(1-z). +\] + +\begin{lemma}[Two constrained moments]\label{lem:moment-obstruction} +Every probability measure $\mu$ on $[0,2]$ satisfying +\[ + \int z^2\,d\mu\ge1,\qquad + \int(28z+6z^2)\,d\mu\le25 +\] +satisfies +\[ + 16\left(\int H\,d\mu\right)^2-\int\Phi\,d\mu<0. +\] +\end{lemma} +\begin{proof} +The feasible measures form a nonempty weakly compact convex set. +Suppose the continuous functional +$J(\mu)=16(\int H\,d\mu)^2-\int\Phi\,d\mu$ has a nonnegative maximum, +attained at $\mu$, and set $d=\int H\,d\mu$. First $d\ge11/20$. +Indeed, +\[ + 1+z^2/40-z^4/1152\ge1+(31/1440)z^2, +\] +and +\[ + 25+\frac{31}{45}z^2-\frac{346}{25}z + \ge\frac{17}{225}>0\qquad(0\le z\le2). +\] +The last quadratic is decreasing on this interval and has the stated +value at $2$. It follows that +$\Phi(z)\ge16(11/20)^2z^2$. The moment constraint and $J(\mu)\ge0$ +now give the claim about $d$. + +For another feasible measure $\nu$, direct expansion with +$\ell=32dH-\Phi$ gives +\[ + J((1-t)\mu+t\nu)-J(\mu) + =t\left(\int\ell\,d\nu-\int\ell\,d\mu\right) + +16t^2\left(\int H\,d\nu-d\right)^2. +\] +For $t\downarrow0$, maximality shows that $\mu$ maximizes the linear +functional $\int\ell$. This conclusion does not require concavity of +$J$. + +Put $g_1(z)=28z+6z^2-25$ and $g_2(z)=1-z^2$. +The measure with mass $9/32$ at $2$ and the rest at $0$ has respective +constraint expectations $-5/2$ and $-1/8$, so both constraints are +strictly feasible. The compact convex set of triples +\[ + \left\{\left(\int g_1\,d\nu,\int g_2\,d\nu,\int\ell\,d\nu\right): + \nu\text{ a probability measure on }[0,2]\right\} +\] +is disjoint from the open set of triples with first two coordinates +negative and third coordinate greater than $\int\ell\,d\mu$. +Separation gives coefficients $-\alpha,-\gamma,\beta$, with +$\alpha,\gamma,\beta\ge0$. The last coefficient cannot be zero: +otherwise the strictly feasible triple would violate separation. +Normalize $\beta=1$. Evaluating on $\mu$ and on point masses shows +that $\mu$ is supported on the global maxima of +\[ + F_d(z)=32dH(z)-\Phi(z) + -\alpha(28z+6z^2)-\gamma(1-z^2), + \qquad \alpha,\gamma\ge0. +\] +For completeness, separation bounds +$\int(\ell-\alpha g_1-\gamma g_2)\,d\nu$ by +$\int\ell\,d\mu$ for every $\nu$; for $\mu$ the reverse inequality +follows from its nonpositive constraints. Equality and then point +masses give precisely the asserted support condition. + +We next verify that $F_d'''(z)>0$ on $(0,2]$. Set +\[ + L(z)=\frac38+\frac{3z^2}{64}-\frac{35z^4}{1024},\qquad + V(z)=\frac{24}{5}-\frac{16z^2}{15}-\frac{7z^4}{48} + +\frac{7z^6}{1152}. +\] +Direct differentiation gives $H'''(z)=L(z)z^{-5/2}$ and +$\Phi'''(z)=V(z)$. For $x=\sqrt{z/2}$, the polynomial inequalities +\[ + L(2x^2)>0,\qquad + D_*(x):=\frac{88}{5}L(2x^2)-\frac{17}{3}x^5V(2x^2)>0 + \qquad(0\le x\le1) +\] +are certified below. Since $32d\ge88/5$ and $4\sqrt2<17/3$, they imply +$32dL(z)-z^{5/2}V(z)>0$ when $V(z)\ge0$; when $V(z)<0$ it is immediate. +The constraint polynomials have zero third derivative, proving the +claim. + +Also $F_d'(0+)=+\infty$. Since $F_d''$ is strictly increasing, $F_d'$ +can cross from positive to negative at most once. There is at most +one interior local maximum; an interval of maxima is impossible. +The endpoint $0$ is not a maximum, so the support contains at most +one interior point and the endpoint $2$. A point mass is infeasible: +its first moment condition would give $z\ge1$, and then +$28z+6z^2\ge34$. Thus the support is $\{w,2\}$, with weight $t$ at $2$. +Necessarily $09\bigl((1-t)(w^2-w)+2t\bigr) + \qquad(t=t_-(w),\ t_+(w)). +\] +To verify these inequalities exactly, put +\[ + x=\sqrt{w/2}\le\frac{37}{100},\qquad + R_*(x)=1+\frac{x^4}{10}-\frac{x^8}{72}. +\] +For each of +\[ + (A_-,B_-)=(1-w^2,4-w^2),\qquad + (A_+,B_+)=(25-28w-6w^2,80-28w-6w^2),\quad w=2x^2, +\] +the required inequality, after multiplication by its positive +$B_\pm^2$, is $P_\pm(x)>0$, where +\[ + P_\pm(x)=32A_\pm(B_\pm-A_\pm) + \bigl(R_*(1)-xR_*(x)\bigr)^2 + -9\bigl((B_\pm-A_\pm)(w^2-w)+2A_\pm\bigr)B_\pm. +\] +This uses $H(2)-H(w)=\sqrt2(R_*(1)-xR_*(x))$. +The endpoint $w=10/37$, at which the weight interval can collapse, is +included in the following strict certificates. + +Here are all the coefficient bounds used in this proof. If +$p(r+(s-r)y)=\sum_{j=0}^n a_jy^j$, define +\[ + h_i=\sum_{j=0}^i a_j\frac{\binom{i}{j}}{\binom nj} + \quad(0\le i\le n). +\] +Binomial expansion gives +$p(r+(s-r)y)=\sum_i h_i\binom ni y^i(1-y)^{n-i}$; thus positive $h_i$ +give positivity on $[r,s]$. If the original coefficients are +$p(x)=\sum p_kx^k$, they are obtained explicitly from +\[ + a_j=(s-r)^j\sum_{k=j}^n p_k\binom{k}{j}r^{k-j}. +\] +The degrees and strict lower bounds for the resulting coefficients are +\begin{center} +\begin{tabular}{@{}llll@{}} +\toprule +Polynomial & Degree & Interval & Lower bound\\ +\midrule +$L(2x^2)$ &8&$[0,1]$&$1/100$\\ +$D_*(x)$ &17&$[j/4,(j+1)/4]$, $j=0,1,2,3$&$2/5$\\ +$P_-(x)$ &22&$[0,37/100]$&$1$\\ +$P_+(x)$ &22&$[0,37/100]$&$1$\\ +\bottomrule +\end{tabular} +\end{center} +For direct reproduction, the first two polynomial inputs are +\[ + L(2x^2)=\frac38+\frac3{16}x^4-\frac{35}{64}x^8, + \qquad + V(2x^2)=\frac{24}{5}-\frac{64}{15}x^4-\frac73x^8+\frac7{18}x^{12}. +\] +For example the least coefficient of $L(2x^2)$ is $1/64$; the least +coefficient of $D_*$ over its four intervals is +$2459668689/5793382400>2/5$. For $P_-$ and $P_+$ the respective least +coefficients are +\[ + \begin{aligned} + \frac{58454914732902747377800095783827783501884053} + {50000000000000000000000000000000000000000000}&>1,\\ + \frac{132428032291164570287168821949356607977047} + {770000000000000000000000000000000000000}&>1. + \end{aligned} +\] +All inputs and endpoints in these coefficient calculations are rational. +This proves the derivative and endpoint signs. The strict endpoint +inequalities say exactly that $J(t_-(w)),J(t_+(w))<0$, contradicting its +assumed nonnegative maximum. +\end{proof} + +\begin{proposition}[Balanced Weyl moments]\label{prop:balanced-moments} +If neither Weyl block is identically zero, then +$\tau=0$ and $\mathbf E v^2=\mathbf E b^2$. +\end{proposition} +\begin{proof} +By \eqref{eq:5} and $v+b<2$, +$2\le s_0<4$. Equations \eqref{eq:6} and \eqref{eq:7} give +\[ + \chi\le2(1-3s_0/40)(1+s_0)<7, + \qquad + \frac{|\tau|}{\chi} + =\frac23\frac{|\mathbf E v^2-\mathbf E b^2|}{1+s_0} + \le\frac23\frac{s_0-2}{1+s_0}<\frac4{15}. +\] +Here the first polynomial is increasing on $[2,4]$ and equals $7$ at +$4$. The integers $\chi=2+n_++n_-$ and $\tau=n_+-n_-$ have the same +parity. If $\tau\ne0$, the only possibility is $\chi=5$, $|\tau|=1$. +Exchange the blocks so that $P=\mathbf E b^2$ is the smaller moment. +The ratio in \eqref{eq:6} then gives +\[ + \mathbf E v^2=\frac{13P+3}{7}. +\] +Combining this with $v^2<(2-b)^2$ yields +\[ + P\ge1,\qquad28\mathbf E b+6P\le25. +\] + +Integrating \eqref{eq:3} for $b$, using $\bar\sigma\le1$ and +$\bar e\ge5|\nabla b|^2/3$, gives +\[ + P-\mathbf E b\ge\frac49\mathbf E|\nabla\sqrt b|^2. +\] +The integrability of $\bar e/b$ proved with \eqref{eq:3} implies +$c=\sqrt b\in H^1$ by regularization. Put +\[ + G(c)=c+\frac{c^5}{40}-\frac{c^9}{1152},\qquad H(b)=G(c). +\] +The range of $c$ is bounded, and $G$ has bounded derivative there, so the +Sobolev chain rule applies, including at $c=0$. On $0\le b\le2$, +\[ + G'(c)=1+\frac{b^2}{8}-\frac{b^4}{128} + \in\left[0,\frac{\sqrt{4+b^2}}2\right] + \subseteq\left[0,\frac{\sqrt K}2\right]. +\] +For the first upper bound, put $z=b^2/4\in[0,1]$ and square the positive +quantity $1+z/2-z^2/8$; its square is +$1+z-z^3/8+z^4/64\le1+z$. +The variance inequality \eqref{eq:4}, extended by Sobolev approximation, +therefore gives +\[ + \operatorname{Var}(H(b)) + \le\frac14\mathbf E|\nabla\sqrt b|^2 + \le\frac9{16}(P-\mathbf E b). +\] +Equivalently, +\[ + J:=16(\mathbf E H(b))^2-\mathbf E\Phi(b)\ge0, + \qquad \Phi(b)=16H(b)^2+9b(1-b). + \tag{9}\label{eq:9} +\] +The distribution of $b$ is a probability measure on $[0,2]$ satisfying +the two moment constraints. Lemma~\ref{lem:moment-obstruction} +contradicts \eqref{eq:9}. Thus $\tau=0$, and \eqref{eq:6} gives equality +of the two second moments. +\end{proof} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/05-hessian.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/05-hessian.tex new file mode 100644 index 0000000000000000000000000000000000000000..391e9a7e58eaaf11b33863bc2575be1bc989a18c --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/05-hessian.tex @@ -0,0 +1,615 @@ +\section{A weighted Hessian inequality for the Weyl blocks} +\label{sec:hessian} + +We continue the case in which neither Weyl block vanishes identically. +The preceding argument gives +\(\mathbf E v^2=\mathbf E b^2\). This equality will enter the coupling +in Section~\ref{sec:coupling}; the weighted Hessian inequality proved here +does not itself require it. Our task is to extract a first-derivative +inequality from the Bianchi-constrained second derivatives. Unequal +weights on the two blocks will create a useful interaction, but their +derivatives have costs that must be retained exactly. +Set \(\delta=2-v-b\), and retain the notation +\(Z=\nabla T\), \(p=\nabla v\), \(q=\nabla b\), and +\(e=|Z|^2/6\). +The argument is stated for $T$; exchanging the two orientations gives +its companion for $U$. All expectations use normalized Riemannian volume. +The local square will first be proved for an arbitrary global positive +Lipschitz weight. We will choose the two fixed weights only when their +derivative costs enter. + +\subsection{The constrained Hessian and its curvature operator} + +Retain $F=\operatorname{Sym}_0(\Lambda^+)$ and the Bianchi gradient +bundle $G\subset T^*M\otimes F$ from Section~\ref{sec:preliminaries}. +Their types are $(2,0)$ and $(5/2,1/2)$. The irreducible gradient +constraint and the refined Kato coefficient used below are established +in Gursky--LeBrun~\cite[Lemma~4, equations~(10)--(12)]{GurskyLeBrun1999}; +Calderbank--Gauduchon--Herzlich~\cite{CalderbankGauduchonHerzlich2000} +place them in the general generalized-gradient framework. We use the +parallel projector $P_G=(6-C)/10$ and the compression identities already +proved in Proposition~\ref{prop:weyl-derivative-equations}. They are +pointwise tensor identities, including at repeated eigenvalues, and do +not require a smoothly chosen eigenbasis. + +\begin{lemma}[The two Hessian components] +\label{lem:hessian-components} +Let \(\mathcal H=T^*M\otimes G\), and write +\(X=\nabla Z\), so \(X_{ij}=\nabla_i\nabla_jT\). +Then +\[ + X=Y+D_0, + \qquad + Y\text{ has type }(3,1),\qquad + D_0\text{ has type }(2,0). +\] +If \(\operatorname{Tr}:\mathcal H\to F\) is contraction of the +two derivative slots and \(\operatorname{Tr}^*\) its adjoint, then +\[ + \operatorname{Tr}\operatorname{Tr}^*=\frac{12}{5}\mathrm{Id}, + \qquad + D_0=\frac5{12}\operatorname{Tr}^*\Delta T, + \qquad + |D_0|^2=\frac5{12}|\Delta T|^2. +\] +All the type projections in this statement are parallel. +\end{lemma} + +\begin{proof} +The product rule gives four inequivalent, multiplicity-one summands +\[ + \mathcal H=(3,1)\oplus(3,0)\oplus(2,1)\oplus(2,0). +\] +Skewing the derivative slots takes values in +\[ + \Lambda^2T^*M\otimes F=(3,0)\oplus(2,0)\oplus(1,0)\oplus(2,1). +\] +For the actual Hessian $X=\nabla\nabla T$, its skew difference is +$[\nabla_i,\nabla_j]T=F_{ij}T$, where $F_{ij}$ is the curvature of the +bundle $F$. It is essential that the tensor on which this curvature +acts is the same Weyl block $T$ that occurs in $F_{ij}$. The minus +rotation algebra acts trivially on $F$, so there is no minus component. +The dependence on $T$ is an equivariant linear term and an equivariant +term in its symmetric square. The latter has plus spins $4,2,0$; +its only common type with $\Lambda^+\otimes F=(3,0)\oplus(2,0)\oplus(1,0)$ +is $(2,0)$. The linear term has that type as well. Thus the skew +commutator has neither a $(3,0)$ nor a $(2,1)$ component. This conclusion +is specific to the Hessian of $T$, not to the derivative of an arbitrary +section of $G$. + +Skewing is injective on each of those two domain summands. The +\((3,0)\) summand is entirely skew: its stretched plus coupling makes the +two derivative plus spin halves symmetric, while their minus coupling +to spin zero is alternating. To treat \((2,1)\), suppose that a tensor +of this type is symmetric in its full derivative slots. Its derivative +minus pair has spin \(1\) and is symmetric, so its derivative plus pair +must also be symmetric and have spin \(1\). The inner Bianchi constraint +makes the coupling operator \(\sum I_s^{(j)}\Sigma_s\) equal to \(-4\). +Symmetry in the derivative slots gives the same value for +\(\sum I_s^{(i)}\Sigma_s\). The total plus Casimir would therefore be +\[ + -8-24+2(-4-4)=-48, +\] +whereas total spin \(2\) requires \(-24\). This contradiction proves +injectivity. Moreover, the two images lie in distinct target isotypic +components, so they cannot cancel one another. The corresponding +components of \(X\) must vanish separately. + +For the trace calculation, let +\(K:F\to T^*M\otimes T^*M\otimes F\) be +\(K(S)_{ij}=\delta_{ij}S\). The adjoint of the restricted trace is +\(\operatorname{Tr}^*=P_{G}^{(j)}K\), where the projector acts +on the inner derivative and matrix slots. Now +\[ + \operatorname{Tr}K=4\mathrm{Id}, + \qquad + \operatorname{Tr}\bigl(C^{(j)}K(S)\bigr) + =\sum_{i,s}(I_s)_{ii}\Sigma_sS=0. +\] +Thus \(\operatorname{Tr}\operatorname{Tr}^*=12\mathrm{Id}/5\). +By equivariance, trace vanishes on \((3,1)\) and is supported on the +unique \((2,0)\) summand. The displayed nonzero composition makes its +restriction to that five-dimensional summand an isomorphism onto +\(F\). It therefore determines the whole of \(D_0\), including +its skew part, and gives both the formula and the norm identity. +One can also see explicitly that the trace component retains the +curvature commutator. At a point, its adjoint is +\[ + (\operatorname{Tr}^*S)_{ij} + =\frac35\delta_{ij}S+\frac1{10}\sum_s I_{s,ij}\Sigma_sS. +\] +Hence +\[ + (D_0)_{ij}-(D_0)_{ji} + =\frac1{12}\sum_s I_{s,ij}\Sigma_s\Delta T + =\frac12\sum_s(1+t_s)I_{s,ij}\Sigma_sT. +\] +For the last equality use $\Delta T=6(T-N)$ and, in a pointwise +diagonal frame, $\Sigma_sN=-t_s\Sigma_sT$. The invariant version is +$\Sigma_rN=-\sum_sT_{rs}\Sigma_sT$; no eigenframe has been +differentiated. Thus the displayed skew difference is exactly the +bundle-curvature action. + +Finally, the type projections are invariant expressions in the combined +Casimirs, and trace is a metric contraction. They are parallel and do +not depend on any curvature eigenframe. The spin labels describe tensor +representations of the oriented frame group; no spin structure is used. +\end{proof} + +The derivative-slot operators $T_D,U_D$ were defined in the weighted +variance proof, and $B_F(T)$ acts on the matrix slot. Compress all three +orthogonally to $\mathcal H=T^*M\otimes G$. The useful combination is +visible before the quadratic form is chosen. Let +$\Sigma_s^{\mathrm{tot}}$ be the total plus generator on $\mathcal H$ and put +\[ + \widetilde Q=\frac1{60} + \sum_{r,s}T_{rs}\Sigma_r^{\mathrm{tot}}\Sigma_s^{\mathrm{tot}}. +\] +Compressing the inner derivative generator to $G$ gives +$I_s^{(j)}=\Sigma_s^G/5$: equivariance fixes the spin-one family up to a +scalar, and the highest weight gives $(1/2)/(5/2)=1/5$. At a point where +$T$ is diagonal, expansion of the total square therefore gives +\[ + T_D=\frac1{10}\sum_s t_s I_s^{(i)}\Sigma_s^G + =3\widetilde Q-2Q_{5/2}(T),\qquad + B_F(T)=Q_{5/2}(T) +\] +on the compressed inner factor. The inner square contributes +$40Q_{5/2}(T)$, while the trace-free outer half-spin square is zero. +Consequently +\[ + T_D+2B_F(T)=3\widetilde Q. +\] +This combined identity cancels the inner curvature action. On the free +$(3,1)$ type, $\widetilde Q=Q_3(T)\le4x/5$ and $U_D\le2y$. Thus the choice +\[ + \mathcal P=4-\frac56\bigl(T_D+2B_F(T)\bigr)-U_D + =4-\frac52\widetilde Q-U_D +\] +has its plus and minus losses bounded by $2x$ and $2y$, respectively. +The scalar $4$ then meets the sectional boundary $x+y=2$. This explains +the useful coefficients; no uniqueness of the choice is asserted. + +\begin{lemma}[Positivity on the highest Hessian type] +\label{lem:hessian-operator} +The combined compressed operator $\mathcal P$ preserves all four types +in $\mathcal H$. It is positive definite on $(3,1)$, and on the trace +type it satisfies +\[ + \mathcal P\operatorname{Tr}^* + =\operatorname{Tr}^*(4-B_F(T)). +\] +The individual derivative-slot and matrix-slot summands need not preserve +the four types. +\end{lemma} + +\begin{proof} +The total plus generators preserve each joint type. Similarly, +$\sum_su_s(J_s^{(i)}+J_s^{(j)})^2=4U_D$, so $U_D$ vanishes on minus +spin zero and is $Q_1(U)$ on minus spin one. Its latter spectrum is the +spectrum of $U$. The displayed combined formula for $\mathcal P$ therefore +preserves the joint types. On $(3,1)$, +\[ + \mathcal P\ge4-2x-2y>0. +\] +On $(2,0)$ the denominator in $Q_2$ is $24$, so under the trace +identification $\widetilde Q=(2/5)Q_2(T)=(2/5)B_F(T)$ and $U_D=0$. +This proves the stated trace action. +\end{proof} + +\subsection{The weighted square and its first-derivative form} + +The undetermined Hessian component $Y$ is now isolated, and its +curvature-weighted quadratic form is positive. We complete its weighted +square and integrate by parts to replace the Hessian by first derivatives. +The prescribed trace component must be subtracted in full. + +Set +\[ + B=\frac{\langle B_F(T)Z,Z\rangle}{6},\qquad + C_N=\frac{\langle B_F(N)Z,Z\rangle}{6},\qquad + E_B=\frac{|Q_{5/2}(T)Z|^2}{6}, + \qquad N=(T^2)_0. +\] +For a scalar weight \(m\), write \(L_m=dm\otimes Z\), with no +symmetrization, and let \((L_m)_Y\) denote its \((3,1)\) projection. + +\begin{proposition}[Weighted Hessian inequality] +\label{prop:weighted-hessian} +For every global Lipschitz scalar weight \mbox{$m\ge m_0>0$}, +\begin{equation} + 0\le\mathbf E\left[ + mG_v(p,k,l) + -\frac56\sum_i m_i\, + \frac{\operatorname{tr}\bigl(Z_i\sum_jZ_j^2\bigr)}6 + +\frac{\langle\mathcal P(L_m)_Y,(L_m)_Y\rangle}{24m} + \right], + \tag{10}\label{eq:10} +\end{equation} +where \(i,j\) in the middle term are derivative indices and +\[ + G_v=\left(-42-3b^2-\frac{10}{3}v^2\right)e + +75B+\frac{25}{3}C_N-\frac{100}{3}E_B + +5v^2|p|^2. +\] +\end{proposition} + +\begin{proof} +By Lemma~\ref{lem:hessian-operator}, +\[ + \mathbf E\left[ + \frac{m}{6}\left\langle + \mathcal P\left(Y+\frac{(L_m)_Y}{2m}\right), + Y+\frac{(L_m)_Y}{2m}\right\rangle\right]\ge0. +\] +The last term in its expansion is the last term of \eqref{eq:10}. +Since \(\mathcal P\) preserves the type splitting and +\(X=Y+D_0\), the other terms are +\[ + \frac16\mathbf E\bigl[ + m\langle\mathcal PX,X\rangle+\langle\mathcal PX,L_m\rangle + -m\langle\mathcal PD_0,D_0\rangle-\langle\mathcal PD_0,L_m\rangle + \bigr]. +\] +We compute these terms for smooth \(m\) first. Compression may be omitted +inside any of the displayed inner products, because both arguments +belong to \(\mathcal H\). + +\paragraph{The identity term.} +Integration by parts and \eqref{eq:1} give +\[ + \frac16\mathbf E\left[m|\nabla Z|^2 + +\sum_i m_i\langle\nabla_iZ,Z\rangle\right] + =-\frac16\mathbf E\bigl[m\langle\Delta Z,Z\rangle\bigr] + =\mathbf E\bigl[m(-13e+20B)\bigr]. +\] +This will be multiplied by the coefficient \(4\) in \(\mathcal P\). + +\paragraph{The derivative-slot curvature terms.} +Write \(A_{ikjl}=(T_D)_{ikjl}\). The two terms to integrate are +\[ + \mathbf E\left[ + mA_{ikjl}\langle\nabla_iZ_j,\nabla_kZ_l\rangle + +A_{ikjl}\langle\nabla_iZ_j,m_kZ_l\rangle\right]. +\] +Integrate the derivative \(k\) on the second paired tensor. +The derivative of \(m\) cancels the second term exactly. +Moreover \(\nabla_kA_{ikjl}=0\), by divergence freedom and skewness in +\(i,k\). The result is +\[ + -\mathbf E\bigl[mA_{ikjl}\langle\nabla_k\nabla_iZ_j,Z_l\rangle\bigr] + =\frac12\mathbf E\bigl[ + mA_{ikjl}\langle[\nabla_i,\nabla_k]Z_j,Z_l\rangle\bigr]. +\] +This calculation uses neither symmetry of \(X\) nor parallelness of +\(T_D\), and the factor \(1/2\) comes only from skewing the two +covariant derivatives. + +The curvature in the derivative slot has the contraction +\[ + \frac12(T_D)_{ikjl}R_{ikjh}=3v^2\delta_{lh}. +\] +To see its normalization, substitute the quaternion expansions: the +contraction over \(i,k\) removes all mixed plus/minus terms and gives +\(\frac12\sum_s t_s(1+t_s)\delta_{lh}=3v^2\delta_{lh}\). +The matrix-slot curvature gives the quadratic form of +\[ + -\frac12\sum_s t_s(1+t_s)I_s\Sigma_s. +\] +Since \(t_s^2=N_{ss}+2v^2\), its compression is +\[ + -\frac12\bigl(8Q_{5/2}(T)+8Q_{5/2}(N)+2v^2C\bigr) + =4v^2-4Q_{5/2}(T)-4Q_{5/2}(N). +\] +After division by \(6\), the full \(T_D\) contribution is therefore +\[ + \mathbf E\bigl[m(7v^2e-4B-4C_N)\bigr]. +\] +For \(U_D\), the same integration gives \(3\mathbf E[mb^2e]\). +Its matrix-slot curvature contraction is zero, because the minus forms +are orthogonal to the plus forms defining the curvature action on +\(F\). + +\paragraph{The nonparallel matrix-slot coefficient.} +Put \(A=B_F(T)\) temporarily. One integration by parts gives +\[ + \mathbf E\left[m\langle A\nabla_iZ,\nabla_iZ\rangle + +m_i\langle A\nabla_iZ,Z\rangle\right] + =-\mathbf E\bigl[m\langle A\Delta Z,Z\rangle\bigr] + -\mathbf E\bigl[m\langle(\nabla_iA)\nabla_iZ,Z\rangle\bigr]. +\] +The operator \(\nabla_iA\) is symmetric, so the last term is +\[ + \frac12\mathbf E\left[ + \left\langle\left(m\Delta A+\sum_i m_i\nabla_iA\right)Z,Z\right\rangle + \right]. +\] +Here \(\nabla_iA=B_F(Z_i)\) and +\(\Delta A=6B_F(T)-6B_F(N)\), since \(B_F\) is a linear parallel +construction. Also, pairing \(B_F(T)Q_{5/2}(T)Z\) with \(Z\) gives +\(|Q_{5/2}(T)Z|^2\), because the latter vector belongs to +\(G\). After division by \(6\), the contribution is +\[ + \mathbf E\left[ + m(-10B-3C_N+20E_B) + +\frac1{12}\sum_i m_i\langle B_F(Z_i)Z,Z\rangle\right]. +\] +Since +\(\langle B_F(Z_i)Z,Z\rangle + =\operatorname{tr}(Z_i\sum_jZ_j^2)\), +multiplication by \(-5/3\) produces exactly the middle term in +\eqref{eq:10}. In particular, the variation of \(B_F(T)\) has been +retained explicitly; \(\mathcal P\) is not assumed parallel. + +\paragraph{The prescribed trace component.} +By Lemmas~\ref{lem:hessian-components} and \ref{lem:hessian-operator}, +\[ + \mathcal PD_0=\frac5{12}\operatorname{Tr}^*(4-B_F(T))\Delta T, + \qquad \operatorname{Tr}L_m=\sum_i m_iZ_i. +\] +The characteristic polynomial of a trace-free \(3\times3\) matrix gives +\(B_F(T)T=N\) and \(B_F(T)N=v^2T\). Thus +\[ + B_0=(4-B_F(T))(T-N)=(4+v^2)T-5N. +\] +Using \(\Delta T=6(T-N)\), the two trace subtractions, before division +by \(6\), are +\[ + -\frac52\mathbf E\left[ + m\langle B_0,\Delta T\rangle+\sum_i m_i\langle B_0,Z_i\rangle\right] + =\frac52\mathbf E\bigl[m\langle\nabla B_0,Z\rangle\bigr]. +\] +Since \(\nabla N=2B_F(T)Z\) and +\(\langle T,Z_i\rangle=6vp_i\), their normalized contribution is +\[ + \mathbf E\left[ + m\left(\left(10+\frac52v^2\right)e-25B+5v^2|p|^2\right)\right]. +\] +Multiplying the preceding contributions by their coefficients in +\(\mathcal P\) and collecting gives \(G_v\) as stated. + +Finally, a global Lipschitz weight $m\ge m_0>0$ can be approximated +uniformly and in $H^1$ by smooth positive weights with a common positive +lower bound. The first derivatives converge in $L^2$, so their quadratic +products in the square cost converge in $L^1$; the uniformly bounded +reciprocals of the weights pass to the same limit. All other terms use +only the weight and its first derivative. Hence the identity and inequality +extend to every weight in the stated class. +\end{proof} + +\begin{lemma}[The quadratic-form matrices] +\label{lem:hessian-matrices} +At a point where \(v>0\), use the three vector parameters \(p,k,l\) in +\eqref{eq:2}. Then \(G_v\) is the quadratic form, with scalar entries +multiplying vector dot products, whose matrix is +\begin{equation} + \begin{pmatrix} + -70-5b^2+100\sigma v-25v^2 + &-100\sqrt3\rho v&-35\rho v^2/\sqrt3\\ + -100\sqrt3\rho v + &-210-15b^2-300\sigma v-190v^2&-75v-35\sigma v^2\\ + -35\rho v^2/\sqrt3 + &-75v-35\sigma v^2&-42-3b^2-10v^2 + \end{pmatrix}. + \tag{11}\label{eq:11} +\end{equation} +\end{lemma} + +\begin{proof} +Expanding the diagonal and off-diagonal matrix entries in \eqref{eq:2} +gives +\[ + \frac{B}{v}\ \longleftrightarrow\ + \begin{pmatrix} + 4\sigma/3&-4\rho/\sqrt3&0\\ + -4\rho/\sqrt3&-4\sigma&-1\\ + 0&-1&0 + \end{pmatrix}, + \qquad + \frac{C_N}{v^2}\ \longleftrightarrow\ + \begin{pmatrix} + 4/3&0&-\rho/\sqrt3\\ + 0&-4&-\sigma\\ + -\rho/\sqrt3&-\sigma&0 + \end{pmatrix}. +\] +Here one uses +\(\sum s_i\widetilde s_i=0\), +\(\sum\widetilde s_i^3=-6\rho\), and +\(N/v^2=\operatorname{diag}(\sigma s_i+\rho\widetilde s_i)\). +The energy metric in these coordinates is +\(H=\operatorname{diag}(5/3,5,1)\). +If \(\mathsf B\) denotes the matrix of \(B\), the matrix of \(E_B\) is +\(\mathsf B H^{-1}\mathsf B\): \(B_F(T)\) compressed to +\(G\) is the self-adjoint operator whose quadratic form is +\(B\). Using \(\sigma^2+\rho^2=1\), this gives +\[ + \frac{E_B}{v^2}\ \longleftrightarrow\ + \begin{pmatrix} + 16/15&0&4\rho/(5\sqrt3)\\ + 0&21/5&4\sigma/5\\ + 4\rho/(5\sqrt3)&4\sigma/5&1/5 + \end{pmatrix}. +\] +Substitution into the definition of \(G_v\) gives \eqref{eq:11} by +matrix addition. The invariant expression for \(G_v\) is defined +everywhere; on the zero set of \(T\), all its derivative parameters +vanish almost everywhere, as established for the norm identity. +\end{proof} + +\subsection{The two costs of a variable weight} + +For the remaining argument choose the fixed positive weights +\begin{equation} + M=1-\frac38(v-b),\qquad m=M^2, + \qquad + \overline M=1-\frac38(b-v),\qquad \overline m=\overline M^2. + \tag{12}\label{eq:12} +\end{equation} +They are one rational choice for the later coupling argument. Since +$v+b<2$, both $M$ and $\overline M$ lie between $1/4$ and $7/4$. +Almost everywhere, +\[ + dm=-\frac34M(p-q),\qquad d\overline m=-\frac34\overline M(q-p). +\] +Proposition~\ref{prop:weighted-hessian} applies to these global Lipschitz +weights. Their derivatives produce the two costs estimated next. + +The Hessian has been eliminated from the integrated inequality. Two +terms caused by the weight remain: a cubic contraction multiplied by +$dm$, and a projected quadratic form in $dm\otimes Z$. We now bound +both by first-derivative energies, while keeping the sign needed for +the later scalar relaxation. + +\begin{lemma}[The cubic contraction] +\label{lem:cubic-contraction} +For every real tangent vector \(a\), with +\(Z_a=\sum_i a_iZ_i\), +\begin{equation} + \left|\frac{\operatorname{tr}\bigl(Z_a\sum_jZ_j^2\bigr)}6\right| + \le \frac45\sqrt{\frac35}\,|a|e^{3/2}. + \tag{13}\label{eq:13} +\end{equation} +\end{lemma} + +\begin{proof} +By homogeneity, assume \(|a|=1\); the case \(Z_a=0\) is immediate. +Put \(\nu=|Z_a|/\sqrt6\), and diagonalize the normalized matrix +\(s=Z_a/\nu\). Apply the algebraic parametrization \eqref{eq:2} with +this \(s\), rather than with \(T/v\), and denote its vector parameters +by \(\widehat p,\widehat k,\widehat l\). +The diagonal entries of \(Z_a=\nu s\) imply +\(\widehat p_a=\nu\) and \(\widehat k_a=0\). +The part corresponding to the single component +\(\widehat p_a a\) has energy +\(e_1=5\nu^2/3\). It is orthogonal in the energy metric to the +remaining parameters. In the displayed matrix for \(B/v\), it has no +cross term with those parameters, because \(\widehat k_a=0\) and the +\(\widehat p,\widehat l\) entry is zero. Its contribution to +\(\frac16\sum_j\operatorname{tr}(sZ_j^2)\) has absolute value at most +\(4e_1/5\). + +For any symmetric trace-free \(3\times3\) matrix \(S\), its +characteristic polynomial gives +\[ + |(S^2)_0|^2=\frac{|S|^4}{6}. +\] +Since \(|s|=\sqrt6\), Cauchy--Schwarz yields +\(|\operatorname{tr}(sS^2)|\le |S|^2\). +Applied to the remaining tensor, this bounds its contribution by +\(e-e_1\). Therefore +\[ + \left|\frac{\operatorname{tr}(Z_a\sum_jZ_j^2)}6\right| + \le\nu\left(e-\frac{e_1}{5}\right) + =\sqrt{\frac35}\,e^{3/2} + \sqrt{\frac{e_1}{e}}\left(1-\frac{e_1}{5e}\right). +\] +For \(0\le r\le1\), the function \(\sqrt r(1-r/5)\) is increasing +and has maximum \(4/5\). This proves \eqref{eq:13}. +\end{proof} + +\begin{lemma}[Real-vector projection identities] +\label{lem:real-vector-projections} +Let \(P_+\) be the projection in \(\mathcal H\) onto total plus spin +\(3\), retaining both possible minus spins. For any real tangent vector +\(a\) and any \(Z\in G\), +\[ + |P_+(a\otimes Z)|^2=\frac7{12}|a|^2|Z|^2, +\] +\[ + \left\langle\widetilde QP_+(a\otimes Z),P_+(a\otimes Z)\right\rangle + =\frac12|a|^2\langle Q_{5/2}(T)Z,Z\rangle. +\] +\end{lemma} + +\begin{proof} +Complexify at the point, writing the outer tangent representation as +\(A\otimes B\), with \(A,B\) the plus and minus spin halves, and +\(G_{\mathbb C}=H_{5/2}\otimes B'\). +For a real vector \(a\), its reduced density on \(A\) is +\(\frac12|a|^2\mathrm{Id}_A\). Indeed, the three traceless Hermitian +operators on \(A\) are scalar multiples of \(-iI_s\), and each has +zero expectation because \(a^{\mathsf T}I_sa=0\). +Its trace is \(|a|^2\), which fixes the scalar part. + +The complete vector is \(a\otimes Z\), a product between the outer +factor and the entire inner factor. Consequently, for every operator +\(O\) acting only on \(A\otimes H_{5/2}\), +\[ + \langle a\otimes Z,O(a\otimes Z)\rangle + =\frac{|a|^2}{2} + \langle Z,((\operatorname{Tr}_A O)\otimes\mathrm{Id}_{B'})Z\rangle. +\] +This partial trace retains every correlation of the inner plus factor +with \(B'\); neither \(a\) nor \(Z\) is assumed to be a product within +its own plus and minus factors. + +The operator \(\operatorname{Tr}_A P_+\) is scalar on the irreducible +space \(H_{5/2}\). Its trace is the dimension \(7\) of spin \(3\), +whereas \(H_{5/2}\) has dimension \(6\). Thus it equals +\(7\mathrm{Id}/6\), proving the first identity. + +The family +\(\operatorname{Tr}_A(P_+\widetilde Q(T)P_+)\) is equivariant and +linear in the spin-\(2\) variable \(T\). Since spin \(2\) occurs once +in \(\operatorname{End}(H_{5/2})\), it is a scalar multiple of +\(Q_{5/2}(T)\). To determine the scalar, take +\(T=\operatorname{diag}(-1,-1,2)\) and the inner highest weight +\(5/2\). The outer weight \(1/2\) gives total weight \(3\) and +\(Q_3(T)\)-eigenvalue \(-1\). The outer weight \(-1/2\) projects to +total weight \(2\), where that eigenvalue is +\((12-3\cdot2^2)/15=0\). The normalized outer trace is therefore +\(-1/2\). Since \(Q_{5/2}(T)\) has eigenvalue \(-1\) on its highest +weight, the second identity has coefficient \(1/2\), as asserted. +\end{proof} + +\begin{proposition}[The projection cost] +\label{prop:hessian-projection-cost} +For the weight \eqref{eq:12}, +\begin{equation} + \frac{\langle\mathcal P(L_m)_Y,(L_m)_Y\rangle}{24m} + \le\frac9{64}|p-q|^2 + \left[\frac7{12}(4+y)e-\frac54B\right]. + \tag{14}\label{eq:14} +\end{equation} +The bracket on the right is nonnegative. Both assertions remain valid +under the closed inequalities \(v+b\le2\) and \(x+y\le2\). +\end{proposition} + +\begin{proof} +Let \(P_-\) be the projection onto total minus spin \(1\), so the +highest Hessian projection is \(P_+P_-\). On that type, +\(-U_D\le y\), and hence \(\mathcal P\le A\), where +\[ + A=4+y-\frac52\widetilde Q. +\] +This operator acts only on the plus factors and commutes with \(P_-\). +On the entire high-plus subspace, +\[ + A\ge4+y-2x\ge0; +\] +the last inequality follows from \(x+y\le2\). +It follows in operator order that \(P_-AP_-\le A\) there. +Thus, for every real \(a\), +\[ + \langle\mathcal P(a\otimes Z)_Y,(a\otimes Z)_Y\rangle + \le \langle AP_+(a\otimes Z),P_+(a\otimes Z)\rangle + =6|a|^2\left[\frac7{12}(4+y)e-\frac54B\right], +\] +where the last equality is Lemma~\ref{lem:real-vector-projections}. +This operator argument is valid for arbitrary correlations after the +plus projection and does not discard a mixed minus-factor term. + +Almost everywhere, +\(dm=-\frac34M(p-q)\). +Substituting \(a=dm\), dividing by \(24m\), and using \(m=M^2\) +gives \eqref{eq:14}. +Finally, for any unit real \(a\), the same equality and the lower bound +for \(A\) give +\[ + \frac7{12}(4+y)e-\frac54B + \ge \frac7{12}(4+y-2x)e\ge0. +\] +In particular, subsequent upper estimates may replace \(|p-q|^2\) by +\((|p|+|q|)^2\) in \eqref{eq:14} without reversing the inequality. +\end{proof} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/06-coupling.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/06-coupling.tex new file mode 100644 index 0000000000000000000000000000000000000000..399edfc168db0ff6c86614a8683545f292146bb8 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/06-coupling.tex @@ -0,0 +1,613 @@ +\section{Coupling the two Weyl blocks}\label{sec:coupling} + +Assume that neither Weyl block vanishes identically and that their second +moments agree: +\[ + \mathbf E v^2=\mathbf E b^2. +\] +We retain the notation +\[ + \delta=2-v-b,\qquad K=4+v^2+b^2,\qquad + M=1-\frac38(v-b),\qquad m=M^2, +\] +and place a bar over an expression to exchange the two blocks. In +particular, $\bar M=1-\frac38(b-v)$ and $\bar m=\bar M^2$. + +The weights in \eqref{eq:12} have opposite slopes along $v+b=2$. +Their derivatives introduce quartic terms in the first derivatives of +curvature. We combine the two Hessian inequalities with one identity that +controls those costs, the balanced variance defect, and tests of the weak +norm equations. This produces a single integrand with nonnegative mean. +The task is then to bound its quartic, quadratic, and potential parts by +three expressions that complete one square. + +\subsection{The combined integral inequality} + +Start with the sum of the two integrands in \eqref{eq:10}. The three +additions below are written explicitly in \eqref{eq:16}--\eqref{eq:18}. +Here $h\ge0$ is a scalar coefficient and $\kappa$ is a scalar multiplier, +whose rational values are specified below. Their roles and integral signs +are as follows; the degrees refer to first +derivatives of curvature in the resulting integrand. +\begin{center} +\small +\begin{tabular}{@{}p{0.24\textwidth}p{0.42\textwidth}p{0.25\textwidth}@{}} +\toprule +Added expression & Integral and derivative degrees & Purpose\\ +\midrule +$\Delta\delta$ identity & +$\dfrac h3\mathbf E[\delta(|Y|^2+|\bar Y|^2)]\ge0$; degrees $4,2,0$ & +Controls the quartic weight costs\\ +Balanced variance defect & Nonnegative integral; degrees $2,0$ & +Controls $v-b$ using moment balance\\ +$\kappa$ norm tests & Zero integral; degrees $2,0$ & +Adjusts the quadratic and potential terms\\ +\bottomrule +\end{tabular} +\end{center} + +We use the following verified rational choice for the constants and +functions in this combination and its final comparison. It is one +sufficient choice; no optimality or uniqueness is asserted. +Set +\[ + D=158,\qquad h=\frac{17}{10},\qquad + c=\frac25-\frac3{10}vb, + \qquad a=v-1,\quad s=b-1,\quad + n=\frac\delta2,\quad t=\frac{v-b}{2}. +\] +Thus $c\ge 1/10$ on $v,b\ge0$, $v+b\le2$. Define +\begin{equation} +\begin{aligned} + \kappa(v,b)={}&-\frac D6(a-s)+z_1a^2 + +\left(60-\frac{2D}{9}+2z_1\right)as+z_2s^2\\ + &+\sum_{i=0}^3z_{3+i}a^is^{3-i} + +\sum_{i=0}^4z_{7+i}a^is^{4-i},\\ + 4(z_1,\ldots,z_{11})={}& + (-43,-45,-63,21,183,208,61,46,101,79,-41),\\ + 4\Theta(v,b)={}&n^2(1-n)(105+5n) + +t^2(4+360n-586n^2)-4t^4,\\ + R_v={}&\frac{4(1+v)}{1+6v+v^2},\qquad + R_b=\frac{4(1+b)}{1+6b+b^2}. +\end{aligned} +\tag{15}\label{eq:15} +\end{equation} +Here and below an exchanged expression is obtained by exchanging all +block variables, and $\bar\kappa=\kappa(b,v)$. +The multiplier $\kappa$ changes the integrand through a zero-integral +norm test. The functions $\Theta$, $c$, $R_v$, and $R_b$ enter only the +comparison for the final square; they are not additional integrands. +Neither $\kappa$ nor $\Theta$ is required to have a sign. + +The first addition is +\begin{equation} +\begin{aligned} + h\bigg[&(e+\bar e) + \left(\lambda-\frac{e-|p|^2}{v} + -\frac{\bar e-|q|^2}{b}\right) + +40\delta(B+\bar B)\\ + &\hspace{24mm} + -30\delta\sum_{\pm}v^2(1-2\sigma v+v^2)\bigg], +\end{aligned} +\tag{16}\label{eq:16} +\end{equation} +where +\[ + \lambda=6(\sigma v^2+\bar\sigma b^2-v-b)-26\delta. +\] +The notation $\sum_\pm$ includes the displayed term and its exchange. +To verify its sign, use +\[ + \Delta e=\frac{2}{6}|\nabla Z|^2+26e-40B +\] +and \eqref{eq:3} in +\[ + \mathbf E\bigl[(e+\bar e)\Delta\delta\bigr] + =\mathbf E\bigl[\delta\Delta(e+\bar e)\bigr]. +\] +The decomposition $\nabla Z=Y+D_0$ gives +$|D_0|^2=(5/12)|\Delta T|^2$. Moreover, +\[ + \frac{2|D_0|^2}{6} + =30v^2(1-2\sigma v+v^2). +\] +Consequently the integral of \eqref{eq:16} equals +$\frac h3\mathbf E[\delta(|Y|^2+|\bar Y|^2)]$, which is nonnegative. + +The second addition is +\begin{equation} + D\left[\frac{|p-q|^2}{K} + -\left(1-\frac{\delta^2}{4}\right)(v-b)^2\right]. +\tag{17}\label{eq:17} +\end{equation} +Indeed, the equality of the second moments implies +\[ + \mathbf E(v-b)=\frac12\mathbf E[\delta(v-b)], +\] +and hence +\[ + (\mathbf E(v-b))^2\le\frac14\mathbf E[\delta^2(v-b)^2]. +\] +Applying \eqref{eq:4} to $v-b$ proves that \eqref{eq:17} has +nonnegative integral. Finally add the zero-integral expression +\begin{equation} + \sum_\pm\left[ + \kappa(v,b)\left\{6v(1-\sigma v)+\frac{e-|p|^2}{v}\right\} + +\nabla\kappa(v,b)\cdot p\right]. +\tag{18}\label{eq:18} +\end{equation} +This is the weak identity \eqref{eq:3} tested against $\kappa(v,b)$. + +Let $\mathcal I$ denote the resulting total integrand. Thus +$\mathbf E\mathcal I\ge0$. All these integrals are legitimate without +a positive lower bound for either norm: $e/v$ and $\bar e/b$ are +integrable by the regularization proving \eqref{eq:3}, while the energies +and all scalar coefficients are bounded. Bounded Lipschitz test functions +are admissible by smooth approximation, uniformly and in $H^1$. + +For the fixed weights in \eqref{eq:12}, $dm$ is linear in $p,q$. +Thus the cubic contraction multiplied by $dm$ and the quadratic form +in $L_m=dm\otimes Z$ have degree four in first derivatives of curvature. +Grouping the exact total integrand by this degree gives +$\mathcal I=\mathcal I_4+\mathcal I_2+\mathcal I_0$, where +\[ +\begin{aligned} +\mathcal I_4={}& + \sum_\pm\left[ + -\frac56\sum_i m_i\, + \frac{\operatorname{tr}\bigl(Z_i\sum_j Z_j^2\bigr)}6 + +\frac{\langle\mathcal P(L_m)_Y,(L_m)_Y\rangle}{24m} + \right]\\ + &-h(e+\bar e) + \left(\frac{e-|p|^2}{v}+\frac{\bar e-|q|^2}{b}\right),\\[1mm] +\mathcal I_2={}& + \sum_\pm\left[ + mG_v+h\lambda e+40h\delta B + +\kappa(v,b)\frac{e-|p|^2}{v} + +\nabla\kappa(v,b)\cdot p + \right] + +\frac DK|p-q|^2,\\[1mm] +\mathcal I_0={}& + -D\left(1-\frac{\delta^2}{4}\right)(v-b)^2\\ + &+\sum_\pm\left[ + 6\kappa(v,b)v(1-\sigma v) + -30h\delta v^2(1-2\sigma v+v^2) + \right]. +\end{aligned} +\] +Here $\lambda$ is the original symmetric expression in \eqref{eq:16}. +The exchange in $\sum_\pm$ includes the operator, the $Y$-projection, +the weight, the multiplier and its arguments, and all derivative labels. +On a block's zero set, only its corresponding derivative summands vanish +almost everywhere; the exchanged contributions remain. The quotients +use the zero value assigned in \eqref{eq:3}, and $G_v$ uses its invariant +expression there. Products such as $\sigma v^2$ use their continuous +zero value; no continuity is asserted for the singular coefficients. + +\begin{samepage} +\begin{proposition}[Three bounds for the combined integrand] +\label{prop:coupling-estimates} +For this decomposition, define +\[ + r=R_v|p|^2+R_b|q|^2,\qquad + j=(|p|^2+|q|^2) + \left(\frac{|p|^2}{v}+\frac{|q|^2}{b}\right). +\] +Then, almost everywhere, +\[ + \mathcal I_4\le-cj,\qquad + \mathcal I_2\le2\Theta r,\qquad + \mathcal I_0\le-\frac{\Theta^2}{c}. +\] +The corresponding derivative summands are assigned value zero on a +zero set, as in \eqref{eq:3}. The total integrand and the bounds are +integrable, and $\mathbf E\mathcal I\ge0$. +\end{proposition} +\end{samepage} + +Section~\ref{sec:classification} combines these estimates with $r^2\le j$ to obtain +\[ + \mathcal I\le-\frac{(\Theta-cr)^2}{c}-c(j-r^2). +\] +The strictness needed for vanishing is proved there. +We prove the three estimates in turn. The computations first take place +on $v,b>0$, $v+b\le2$, $x+y\le2$. The polynomial certificates used in +the proof are supplied in Section~\ref{sec:certificates}. + +\subsection{The quartic terms} + +Put +\[ + e_0=\frac53|p|^2,\qquad e_d=e-e_0\ge0,\qquad z=|p|+|q|. +\] +Since $dm=-\frac34M(p-q)$, \eqref{eq:13} bounds the cubic contraction +multiplied by $dm$ in \eqref{eq:10} by +$\frac12\sqrt{3/5}\,Mz e^{3/2}$. The nonnegative bracket in +\eqref{eq:14} is bounded above by $g e_0+H_d e_d$, where +\[ + g=\frac7{12}(4+y)-\sigma v+\frac{\rho^2v^2}{4},\qquad + H_d=\frac7{12}(4+y)+\frac54v+4. +\] +In fact, the pure defect contribution to $-5B/4$ is at most +$(5/4)v e_d$, and its mixed contribution is at most +$2v|\rho|\sqrt{e_0e_d}$. The latter is at most +$(\rho^2v^2/4)e_0+4e_d$. Nonnegativity of the original bracket permits +the replacement $|p-q|^2\le z^2$. We obtain +\begin{equation} +\begin{aligned} + \mathcal I_4\le{}& + \sum_\pm\left[ + \frac12\sqrt{\frac35}\,zMe^{3/2} + +\frac9{64}z^2(g e_0+H_de_d)\right]\\ + &-h(e+\bar e) + \left[\frac{(2/5)e_0+e_d}{v} + +\frac{(2/5)\bar e_0+\bar e_d}{b}\right]. +\end{aligned} +\tag{19}\label{eq:19} +\end{equation} + +We first show that this upper bound decreases when either defect +increases, with $p,q$ fixed. Denote the right-hand side by $\mathcal Q$. +For $e>0$, direct differentiation gives +\[ +\begin{aligned} + \partial_{e_d}\mathcal Q={}& + \frac34\sqrt{\frac35}\,zM\sqrt e + +\frac9{64}z^2H_d\\ + &-h\left[ + \frac{(2/5)e_0+e_d}{v} + +\frac{(2/5)\bar e_0+\bar e_d}{b} + +\frac{e+\bar e}{v}\right]. +\end{aligned} +\] +Set $u=\sqrt{\bar e/e}$. The bounds +\[ +\begin{gathered} + z\le\sqrt{\frac35}(\sqrt e+\sqrt{\bar e}),\qquad + M\le\frac74-\frac34v,\qquad H_d\le\frac{15}{2}+\frac23v,\\ + \frac25e_0+e_d\ge\frac25e,\qquad + \frac25\bar e_0+\bar e_d\ge\frac25\bar e, + \qquad \frac vb\ge\frac{v}{2-v} +\end{gathered} +\] +therefore imply +\[ + \frac ve\partial_{e_d}\mathcal Q + \le A_0(1+u)^2+L_0(1+u) + -h\left[\frac75+ + \left(1+\frac{2v}{5(2-v)}\right)u^2\right], +\] +where +\[ + A_0=\frac{27}{320}v\left(\frac{15}{2}+\frac23v\right), + \qquad L_0=\frac9{20}v\left(\frac74-\frac34v\right). +\] +All substitutions in the negative bracket use lower bounds; thus they +give the required upper bound after multiplication by $-h$. + +Write +\[ + C_0=\frac75h-A_0-L_0,\qquad + D_0^{\mathrm{sc}}=h-A_0+\frac{2hv}{5(2-v)}. +\] +Lemma~\ref{lem:scalar-certificates} gives $C_0>0$ and +\[ + 4C_0\left((h-A_0)(2-v)+\frac25hv\right) + -(2-v)(2A_0+L_0)^2>0. +\] +For $0(2A_0+L_0)^2$; hence the homogenized +quadratic +\[ + -C_0+(2A_0+L_0)u-D_0^{\mathrm{sc}}u^2 +\] +is negative for every real $u$. No ordering of the two energies is +needed. Before taking an endpoint limit, write the derivative bound as +\[ + \partial_{e_d}\mathcal Q\le\frac1v + \left[-C_0e+(2A_0+L_0)\sqrt{e\bar e} + -D_0^{\mathrm{sc}}\bar e\right]. +\] +It extends continuously to $e=0$, where necessarily $e_0=e_d=0$. +At $e=\bar e=0$ its value is zero; otherwise the derivative remains +negative. The exchanged argument applies to $\bar e_d$. Decreasing +the defects one at a time consequently gives +\[ + \mathcal Q(e_0+e_d,\bar e_0+\bar e_d) + \le\mathcal Q(e_0,\bar e_0+\bar e_d) + \le\mathcal Q(e_0,\bar e_0). +\] + +We may therefore put both defects equal to zero. Write +\[ + V=x-v,\qquad W=y-b,\qquad + 0\le V\le v,\quad0\le W\le b,\quad V+W\le\delta. +\] +We claim +\[ + g\le g_v:=\frac7{12}(4+b+W)-v+2V, + \qquad + g_b=\frac7{12}(4+v+V)-b+2W. +\] +For $r_0=x/v\in[1,2]$, use +\[ + \sigma=\frac{3r_0-r_0^3}{2},\quad + 1-\sigma=\frac{(r_0-1)^2(r_0+2)}2,\quad + 1+\sigma=\frac{(2-r_0)(r_0+1)^2}{2}. +\] +Since $r_0v\le2$, +\[ + \frac{(r_0-1)(r_0+2)}2 + \left(1+\frac{v(1+\sigma)}4\right)\le2. +\] +After substituting $v\le2/r_0$, the gap in this inequality is +\[ + \begin{aligned} + &2-\frac{(r_0-1)(r_0+2)}2 + \left(1+\frac{1+\sigma}{2r_0}\right)\\ + &\quad=\frac{2-r_0}{8r_0} + \left[4r_0(r_0+3)-(r_0-1)(r_0+2)(r_0+1)^2\right]\ge0. + \end{aligned} +\] +The bracket is concave on $[1,2]$ and has endpoint values $16$ and $4$. +This also covers $r_0=2$ without division by $2-r_0$. Multiplying the preceding bound +by $V=v(r_0-1)$ proves the claim for $g$, using +$\rho^2=1-\sigma^2$. + +Set $P=\sqrt{e_0}$ and $Q=\sqrt{\bar e_0}$. We will prove +\begin{equation} +\begin{aligned} + &\left(\frac{67}{50}+\frac{27}{100}vb\right) + (P^2+Q^2)(bP^2+vQ^2)\\ + &\qquad\ge + \frac{27}{128}vb(P+Q)^2(g_vP^2+g_bQ^2) + +\frac34vb(P+Q)(MP^3+\bar M Q^3). +\end{aligned} +\tag{20}\label{eq:20} +\end{equation} +At zero defects, $z=\sqrt{3/5}(P+Q)$. Thus the positive terms of +\eqref{eq:19} are at most $2/5$ times the right-hand side of +\eqref{eq:20}, divided by $vb$. Since +\[ + \frac{10}{9}\left(h-\frac{67}{50}-\frac{27}{100}vb\right)=c, +\] +\eqref{eq:20} gives the first estimate in +Proposition~\ref{prop:coupling-estimates}; the factor $9/25$ that remains +is exactly the conversion from $(e_0,\bar e_0)$ to $(|p|^2,|q|^2)$. + +To prove \eqref{eq:20}, exchange the blocks, if necessary, so that +$P>0$ and $0\le u=Q/P\le1$; the case $P=Q=0$ is immediate. +In $g_v+g_bu^2$ the coefficients of $V,W$ are nonnegative, and their +difference is $(17/12)(1-u^2)\ge0$. Replacing $(V,W)$ by +$(\delta,0)$ therefore enlarges this expression. This is a relaxation, +not an assertion that the new pair is geometrically attained. The +enlarged values are +\[ + g_v=\frac{19}{3}-3v-\frac{17}{12}b, + \qquad g_b=\frac72-\frac{19}{12}b. +\] +Put $\omega=27/128$, $\mu=9/32$, and define +\[ +\begin{aligned} + A&=\frac{67}{50}(1+u^2),\\ + G&=\frac{27}{100}(1+u^2)u^2 + +3\omega(1+u)^2+\mu(1+u)(1-u^3),\\ + H&=\frac{27}{100}(1+u^2) + +\frac\omega{12}(17+19u^2)(1+u)^2 + -\mu(1+u)(1-u^3),\\ + L&=\omega(1+u)^2\left(\frac{19}{3}+\frac72u^2\right) + +\frac34(1+u)(1+u^3). +\end{aligned} +\] +Dividing the difference between the left and enlarged right sides of +\eqref{eq:20} by $vbP^4$ gives exactly +\[ + \frac Av+\frac{Au^2}{b}+Gv+Hb-L. +\] +Here $A,L>0$, and Lemma~\ref{lem:scalar-certificates} gives $G,H>0$. +For +\[ + Z_*=L^2-4A(G+u^2H) +\] +the same lemma gives $Z_*\le0$ on $[0,1/16]$ and +$Z_*^2\le64A^2u^2GH$ on $[1/16,1]$. In the latter interval, +$Z_*\le|Z_*|\le8Au\sqrt{GH}$. In both intervals it follows that +\[ + L\le2\sqrt A(\sqrt G+u\sqrt H). +\] +Taking square roots is legitimate because both sides are nonnegative. +The arithmetic--geometric mean inequalities applied to $A/v+Gv$ and +$Au^2/b+Hb$ finish the proof of \eqref{eq:20}. + +\subsection{The quadratic terms} + +The quartic bound is established. We next bound the quadratic part by its \emph{axial value}: the algebraic +comparison with $\sigma=\bar\sigma=1$, $\rho=\bar\rho=0$, and +$k=l=\bar k=\bar l=0$, holding $v,b,p,q$ fixed. The corresponding spectral +shapes are $(-v,-v,2v)$ and $(-b,-b,2b)$. In the plus contribution, replacing +$\bar\sigma$ by $1$ raises the expression by +$6h(1-\bar\sigma)b^2e\ge0$. After this replacement, the shape-dependent +quadratic expression is +\[ + mG_v+h\lambda_v e+40h\delta B + +\frac{\kappa}{v}(e-|p|^2) + +(\partial_v\kappa)|p|^2+(\partial_b\kappa)p\cdot q, +\] +where $\lambda_v=6(\sigma v^2+b^2-v-b)-26\delta$. The last two terms and +the separate variance term are unchanged by the shape comparison. +Define +\[ + P_0=300m+30hv+160h\delta,\qquad + P_k=300m+160h\delta,\qquad P_l=35mv. +\] +On the closed spectral triangle, $m\ge1/16$ and $h,\delta,v\ge0$; +therefore $P_0=P_k+30hv>0$. This supplies the positive denominator in +the Schur comparison below. +By \eqref{eq:11} and \eqref{eq:16}, the loss in the $pp$ entry when +$\sigma<1$ is $(1-\sigma)vP_0/3$, and the mixed entries with $(k,l)$ +are $-v\rho(P_k,P_l)/\sqrt3$. The negative of the pure $(k,l)$ matrix +is exactly +\[ + \mathsf D_\sigma= + \begin{pmatrix}K_1&K_{12}\\K_{12}&K_2\end{pmatrix} + -\frac\kappa v\operatorname{diag}(5,1), +\] +where +\[ +\begin{aligned} + K_2&=m(42+3b^2+10v^2)-h\lambda_v,\\ + K_1&=5K_2+m(300v\sigma+140v^2)+160h\delta v\sigma,\\ + K_{12}&=m(75v+35v^2\sigma)+40h\delta v. +\end{aligned} +\] +Thus the required completion of squares follows from +\begin{equation} + P_0\left[ + v\begin{pmatrix}K_1&K_{12}\\K_{12}&K_2\end{pmatrix} + -\kappa\operatorname{diag}(5,1)\right] + -(1+\sigma)v^2 + \begin{pmatrix}P_k\\P_l\end{pmatrix} + \begin{pmatrix}P_k&P_l\end{pmatrix}\succeq0. +\tag{21}\label{eq:21} +\end{equation} + +Let $\mathsf A_\sigma$ be this matrix and $w=(P_k,P_l)^T$. +The identity $P_0=P_k+30hv$ gives +\[ + -\partial_\sigma\mathsf A_\sigma + =v^2\left[ + \begin{pmatrix}30hv\\-P_l\end{pmatrix} + \begin{pmatrix}30hv&-P_l\end{pmatrix} + +\operatorname{diag}(0,6hvP_0)\right]\succeq0. +\] +Lemma~\ref{lem:matrix-certificates} proves \eqref{eq:21} at +$\sigma=1$, and this matrix monotonicity proves it for every +$-1\le\sigma\le1$. When $\sigma<1$, put +$\ell=(1-\sigma)vP_0/3>0$ and $d=-v\rho w/\sqrt3$. +The Schur complement is +\[ + \mathsf D_\sigma-\frac{dd^T}{\ell} + =\mathsf D_\sigma-\frac{(1+\sigma)v}{P_0}ww^T + =\frac{\mathsf A_\sigma}{P_0v}\succeq0. +\] +At $\sigma=1$, one has $\rho=0$ and +\[ + \mathsf D_1=\frac{\mathsf A_1+2v^2ww^T}{P_0v}\succeq0, +\] +so no division by $1-\sigma$ is needed. No sign of $\kappa$ is assumed. +Applying the exchanged argument proves the announced axial reduction. + +At the axial comparison, \eqref{eq:2} and the matrix for $B/v$ give +$e_{\rm ax}=5|p|^2/3$ and $B_{\rm ax}=4v|p|^2/3$. Put +$\lambda_{\rm ax}=6(v^2+b^2-v-b)-26\delta$. Since $\delta=-a-s$, +\[ +\begin{aligned} +\bigl(h\lambda e+40h\delta B\bigr)_{\rm ax} + &=\frac53h\bigl(\lambda_{\rm ax}+32v\delta\bigr)|p|^2\\ + &=\frac53h\bigl(6(a^2+s^2)+32a\delta\bigr)|p|^2. +\end{aligned} +\] +Together with the axial $pp$ entry of \eqref{eq:11}, this gives +$X_v|p|^2$ for the axial part of $mG_v+h\lambda e+40h\delta B$, where +\[ + X_v=m(-70+100v-25v^2-5b^2) + +\frac53h\bigl(6(a^2+s^2)+32a\delta\bigr), + \qquad X_b=X_v(b,v). +\] +Collecting the remaining quadratic terms gives the desired estimate +precisely when +\begin{equation} +\begin{gathered} + \begin{pmatrix}\mathcal Q_v&J\\J&\mathcal Q_b\end{pmatrix} + \preceq2\Theta\operatorname{diag}(R_v,R_b),\\ + \mathcal Q_v=X_v+\frac DK+\partial_v\kappa+\frac{2\kappa}{3v}, + \qquad + \mathcal Q_b=X_b+\frac DK+\partial_b\bar\kappa + +\frac{2\bar\kappa}{3b},\\ + J=\frac{\partial_b\kappa+\partial_v\bar\kappa}{2}-\frac DK. +\end{gathered} +\tag{22}\label{eq:22} +\end{equation} +The polynomial matrix certificate in Lemma~\ref{lem:matrix-certificates} +will prove this inequality. + +The leading terms of $\kappa$ can now be read in their intended role. +Along the formal axial boundary $v=1+t$, $b=1-t$, direct substitution +in the displayed formulas gives +\[ + \mathcal Q_v,\mathcal Q_b,J=O(t^2)\quad(t\to0), + \qquad \Theta=t^2-t^4. +\] +Thus the constant and tangential linear terms cancel at the meeting +point of the two axial blocks. This explains the local shape of the +rational choice; the certificate proves its inequality on the full +stated domain. + +\subsection{The potential and the zero sets} + +The quadratic estimate has been reduced to its explicit matrix certificate. +It remains to bound the derivative-free potential and to extend the +three inequalities across the zero sets. The potential is symmetric under exchanging the two blocks. For its +pointwise estimate, suppose $v\ge b$ and define +\[ +\begin{aligned} + E_0={}&D\left(1-\frac{\delta^2}{4}\right)(v-b)^2 + +30h\delta(v^2a^2+b^2s^2) + +6(av\kappa+sb\bar\kappa),\\ + F={}&cE_0-\Theta^2,\qquad + L_v=10h\delta v-\kappa,\qquad + L_b=10h\delta b-\bar\kappa. +\end{aligned} +\] +Direct collection from \eqref{eq:16}--\eqref{eq:18} gives +\[ + \mathcal I_0=-E_0-6v^2(1-\sigma)L_v + -6b^2(1-\bar\sigma)L_b. +\] +Since $y-b\le\delta$ and $r_0=y/b\le2$, +\[ + b^2(1-\bar\sigma) + =\frac12b^2(r_0-1)^2(r_0+2)\le2\delta^2. +\] +It therefore suffices to establish +\begin{equation} + L_v\ge0,\qquad F\ge0,\qquad + F+12c\delta^2L_b\ge0\qquad(v\ge b). +\tag{23}\label{eq:23} +\end{equation} +Indeed, if $L_b\ge0$ then $\mathcal I_0\le-E_0\le-\Theta^2/c$. +If $L_b<0$, the displayed spectral bound instead gives +\[ + \mathcal I_0\le-E_0-12\delta^2L_b\le-\Theta^2/c. +\] +Lemmas~\ref{lem:matrix-certificates} and +\ref{lem:potential-certificate} prove all three inequalities in +\eqref{eq:23}. + +The use of an ordering in \eqref{eq:22} and \eqref{eq:23} is purely +algebraic. All weights and their derivatives have already been formed +from the fixed global labels of the two blocks. Their summed targets +are symmetric. At a point with $v0$ and all minus spectral and derivative +data fixed. Choose axial plus spectra $(-v,-v,2v)$, for which $x=v$, +with $00,\qquad + 4C_0\left((h-A_0)(2-v)+\frac25hv\right) + -(2-v)(2A_0+L_0)^2>0\quad(0\le v\le2),\\ + G,H>0\quad(0\le u\le1),\qquad + Z_*\le0\quad(0\le u\le1/16),\\ + Z_*^2\le64A^2u^2GH\quad(1/16\le u\le1). +\end{gathered} +\] +\end{lemma} + +\begin{proof} +Apply the interval formula of Lemma~\ref{lem:bernstein-conversion} in +the degrees and on the intervals indicated below. Every coefficient +has at least the displayed positive lower bound. In the first two rows +the bound applies on each of the two intervals separately. +\begin{center} +\small +\begin{tabular}{@{}lccc@{}} +\toprule +Polynomial & Degree & Interval & Lower bound\\ +\midrule +$C_0$ & $2$ & $[0,1],\ [1,2]$ & $1/2$\\ +$4C_0((h-A_0)(2-v)+\frac25hv) + -(2-v)(2A_0+L_0)^2$ + & $5$ & $[0,1],\ [1,2]$ & $1/2$\\ +$G$ & $4$ & $[0,1]$ & $9/10$\\ +$H$ & $4$ & $[0,1]$ & $1/4$\\ +$-Z_*$ & $8$ & $[0,1/16]$ & $13/100$\\ +$64A^2u^2GH-Z_*^2$ & $16$ & $[1/16,1]$ & $12/100$\\ +\bottomrule +\end{tabular} +\end{center} +For the first two rows the coefficients are in fact strictly greater +than $1/2$. For example, their respective minima on the two intervals are +\[ + \left(\frac{3971}{3200},\frac{1063}{1600}\right), + \qquad + \left(\frac{101027}{20000},\frac{283729}{400000}\right). +\] +All entries in this table are obtained by substituting the displayed +formulas for $A_0,L_0,C_0$ and for $A,G,H,L,Z_*$ into the coefficient sum in +Lemma~\ref{lem:bernstein-conversion}. Thus each bound is an inequality +between rational coefficients. The lemma proves the required signs. +\end{proof} + +\subsection{The two matrix certificates} + +For the quadratic inequality \eqref{eq:22}, introduce +\[ +\begin{aligned} + r_v&=1+6v+v^2,\qquad r_b=1+6b+b^2, + \qquad \alpha=\frac54,\\ + H_v&=8vK\Theta(1+v) + -r_v\left\{vD+K\left[v(X_v+\partial_v\kappa) + +\frac{2\kappa}{3}\right]\right\},\\ + H_b&=H_v(b,v),\\ + S_*&=4(\alpha+\alpha^{-1})vb + +\frac{\alpha b(1-v)^2+\alpha^{-1}v(1-b)^2}{2}. +\end{aligned} +\] +The diagonal differences in \eqref{eq:22} are +$H_v/(vKr_v)$ and $H_b/(bKr_b)$. Since +$r_v=8v+(1-v)^2$ and $r_b=8b+(1-b)^2$, weighted arithmetic--geometric +mean gives +\[ + \sqrt{vb r_vr_b} + \le\frac{\alpha b r_v+\alpha^{-1}v r_b}{2}=S_*. +\] +It is therefore sufficient to prove +\[ + \begin{pmatrix}H_v&S_*KJ\\S_*KJ&H_b\end{pmatrix}\succeq0. +\] +Indeed its diagonal entries are nonnegative and its determinant gives +$H_vH_b\ge S_*^2K^2J^2\ge vb r_vr_bK^2J^2$. Dividing by the positive +denominators proves that the matrix of differences in \eqref{eq:22} +is positive semidefinite. This argument uses $J^2$ and imposes no sign +condition on $J$. + +For a symmetric $2\times2$ matrix, record its entries in the order +$(11,22,12)$ as a triple $(r,h',s')$. In this notation the two triples +to be certified are +\[ +\begin{aligned} + \mathcal A={}&P_0\bigl(v(K_1,K_2,K_{12})-\kappa(5,1,0)\bigr) + -2v^2(P_k^2,P_l^2,P_kP_l),\qquad \sigma=1,\\ + \mathcal B={}&(H_v,H_b,S_*KJ). +\end{aligned} +\] +The first triple is precisely the matrix in \eqref{eq:21} at +$\sigma=1$. + +\begin{lemma}[Matrix and boundary certificates] +\label{lem:matrix-certificates} +The symmetric matrix represented by $\mathcal A$ is positive semidefinite on +\[ + \mathcal D=\{(v,b):v,b\ge0,\ v+b\le2\}. +\] +The symmetric matrix represented by $\mathcal B$ is positive semidefinite on +\[ + \mathcal D_+=\{(v,b)\in\mathcal D:v\ge b\}. +\] +On $\mathcal D_+$ one also has +\[ + L_v\ge0,\qquad F+12c\delta^2L_b\ge0. +\] +\end{lemma} + +\begin{proof} +Use the vertices +\[ +\begin{gathered} + O=(0,0),\quad S=(1,0),\quad C=(1,1),\quad + U=(2,0),\quad W_0=(0,2),\\ + A=(1/2,1/2),\quad B=(3/2,1/2),\quad E=(1,1/2). +\end{gathered} +\] +A three-letter word specifies the vertices of an ordered triangle. +The triangles $OSC$, $OCW_0$, and $SCU$ cover $\mathcal D$. +The triangles $OAS$, $ASE$, $ACE$, $SBE$, $CBE$, and $SUB$ cover +$\mathcal D_+$; the latter is also the single triangle $OCU$, or the +union $OSC\cup SCU$. All edges are included. The two subdivisions are +shown below. + +\input{figures/spectral-domains.tex} + +For a matrix row, the last two columns below bound, respectively, +\[ + \min([r]_{ij},[h']_{ij}),\qquad + [r]_{ij}[h']_{ij}-[s']_{ij}^{2}. +\] +Thus the last column is the determinant of each coefficient matrix. +For a scalar row, the first bound applies directly to its coefficient. +The bounds apply to every nonzero coefficient; a zero matrix exception +means that all three of its entries vanish. +\begin{center} +\small +\begin{tabular}{@{}lcccrr@{}} +\toprule +Object & Triangle & $N$ & Zeros & + \shortstack{Diagonal or\\scalar bound} & Determinant bound\\ +\midrule +$\mathcal A$ & $OSC$ & $8$ & $0$ & $16000$ & $53000000$\\ + & $OCW_0$ & $8$ & $0$ & $890$ & $3900000$\\ + & $SCU$ & $8$ & $0$ & $2400$ & $3500000$\\ +\midrule +$\mathcal B$ & $OAS$ & $9$ & $0$ & $43$ & $1050$\\ + & $ASE$ & $9$ & $0$ & $43$ & $490$\\ + & $ACE$ & $10$ & $1$ & $2$ & $21$\\ + & $SBE$ & $9$ & $0$ & $43$ & $1300$\\ + & $CBE$ & $10$ & $2$ & $13/100$ & $4$\\ + & $SUB$ & $9$ & $0$ & $5$ & $290$\\ +\midrule +$L_v$ & $OCU$ & $6$ & $1$ & $9/10$ & ---\\ +$F+12c\delta^2L_b$ & $OSC$ & $9$ & $6$ & $19/100$ & ---\\ + & $SCU$ & $9$ & $7$ & $9/1000$ & ---\\ +\bottomrule +\end{tabular} +\end{center} + +Here are explicit expansion instructions for all entries of the table. +They also fix the coefficient ordering. Write +$a=V-1$, $s=W-1$ in \eqref{eq:15} and form $\kappa(V,W)$ and +its two partial derivatives. A partial derivative is obtained by +reducing the appropriate exponent of each monomial and multiplying by +that exponent. Differentiate before substituting the affine expressions +$V,W$. Put +\[ + \kappa_* =\kappa(V,W),\qquad \bar\kappa_* =\kappa(W,V). +\] +For clarity, the derivative of the exchanged polynomial with respect to +its first variable is $(\partial_2\kappa)(W,V)$, and its derivative +with respect to its second variable is $(\partial_1\kappa)(W,V)$. + +Ordinary coefficient arrays are multiplied by convolution: +\[ + (PQ)_{de}=\sum_{f=0}^d\sum_{g=0}^e P_{fg}Q_{d-f,e-g}. +\] +Use +\[ +\begin{aligned} + m_*&=\left(1-\frac38(V-W)\right)^2, + &d_0&=2-V-W,\\ + P_{k,*}&=300m_*+160h d_0, + &P_{l,*}&=35m_*V,\\ + Y_*&=m_*(42+3W^2+10V^2) + -h\bigl[6(V^2+W^2-V-W)-26d_0\bigr]. +\end{aligned} +\] +For $\mathcal A$, apply $s_{ij}$ from +Lemma~\ref{lem:bernstein-conversion} to the three entries of +\[ +\begin{aligned} + &(P_{k,*}+30hV) + \bigl\{V(A_1,Y_*,A_{12})-\kappa_*(5,1,0)\bigr\}\\ + &\hspace{22mm}-2V^2(P_{k,*}^2,P_{l,*}^2,P_{k,*}P_{l,*}), +\end{aligned} +\] +where +\[ + A_1=5Y_*+m_*(300V+140V^2)+160h d_0V, + \qquad + A_{12}=m_*(75V+35V^2)+40h d_0V. +\] +These are the original entries of \eqref{eq:21} at $\sigma=1$, expressed +only in the affine polynomials $V,W$. + +For $\mathcal B$, put +\[ +\begin{aligned} + K_*&=4+V^2+W^2,\qquad \Theta_* =\Theta(V,W),\\ + X_*&=m_*(-70+100V-25V^2-5W^2) + +\frac53h\bigl(6(a^2+s^2)+32a d_0\bigr),\\ + H_*&=8VK_*\Theta_*(1+V)\\ + &\quad-(1+6V+V^2) + \left\{VD+K_*\left[V\bigl(X_*+(\partial_1\kappa)(V,W)\bigr) + +\frac{2\kappa_*}{3}\right]\right\}. +\end{aligned} +\] +Apply $s_{ij}$ to $H_*$, to its exchange in $V,W$, and to +\[ + S_*(V,W)\left\{ + \frac{K_*}{2}\bigl((\partial_2\kappa)(V,W) + +(\partial_2\kappa)(W,V)\bigr)-D\right\}. +\] +For example, on $CBE$ the triples at $(i,j)=(8,0),(8,1),(8,2)$ are +\[ + \left(\frac{11041}{810},\frac{4063}{810},\frac{60557}{16200}\right), + \quad + \left(\frac{4328}{405},\frac{839}{405},\frac{3977}{2700}\right), + \quad + \left(\frac{191}{81},\frac{191}{81},-\frac{1189}{1350}\right). +\] + +For the two scalar rows, apply $s_{ij}$ to +$10h d_0V-\kappa_*$ and, respectively, to +\[ +\begin{aligned} + c(V,W)\bigg\{&D\left(1-\frac{d_0^2}{4}\right)(V-W)^2 + +30h d_0(V^2a^2+W^2s^2)\\ + &+6(Va\kappa_*+Ws\bar\kappa_*) + +12d_0^2(10h d_0W-\bar\kappa_*)\bigg\}-\Theta_*^2. +\end{aligned} +\] +Together with the coefficient-sum formula, these prescriptions produce +the displayed rational bounds by addition and multiplication of finite +arrays. + +The zero exceptions are as follows, with the third homogeneous index +always $N-i-j$. For $\mathcal B$, they are the vertex $C$ in $ACE$ +and $CBE$, namely $(0,10,0)$ and $(10,0,0)$, and also $(9,1,0)$ in +$CBE$. For $L_v$ the only zero is $(0,6,0)$, the vertex $C$ in $OCU$. +For $F+12c\delta^2L_b$ on $OSC$, the zeros are exactly $i+j\le2$. +On $SCU$ they are exactly $9-j\le2$, together with $(i,j)=(0,6)$. +There are no other exceptions. + +Every nonzero matrix coefficient has positive diagonal entries and +positive determinant, and each exceptional matrix is zero. Thus all +coefficient matrices are positive semidefinite. The scalar coefficients +are all nonnegative. Lemma~\ref{lem:bernstein-conversion} and the stated +domain coverage prove the assertions. +\end{proof} + +\subsection{The remaining potential polynomial} + +\begin{lemma}[The polynomial $F$]\label{lem:potential-certificate} +The polynomial $F=cE_0-\Theta^2$ is nonnegative on $\mathcal D$. +\end{lemma} + +\begin{proof} +Use $v=1-n+t$, $b=1-n-t$. The full domain becomes +\[ + 0\le n\le1,\qquad |t|\le1-n. +\] +Write $F_*(n,t)=F(1-n+t,1-n-t)$ for the pullback to these coordinates. +It is even in $t$, and direct expansion gives +$F_*=\sum_{j=0}^4 f_j(n)t^{2j}$, where +\[ +\begin{aligned} + 240f_0={}&n^3(1-n)\bigl(53511n^4-151593n^3 + +222537n^2-83607n+10256\bigr),\\ + 60f_1={}&n\bigl(19389n^5-418926n^4+523719n^3 + -289776n^2+67782n+10564\bigr),\\ + 60f_2={}&-1260129n^4+1370598n^3-451164n^2+23442n+74,\\ + 10f_3={}&-12200n^2+2601n+75,\qquad + f_4=-\frac{23}{5}. +\end{aligned} +\] +For $n<1$, put $z=t^2/(1-n)^2\in[0,1]$. The degree-four Bernstein +coefficients in $z$ are +\[ + h_i(n)=\sum_{j=0}^i + \frac{\binom ij}{\binom4j}f_j(n)(1-n)^{2j}, + \qquad 0\le i\le4. +\] +Define +\[ + d_0(n)=n^3(1-n),\qquad d_1(n)=n(1-n),\qquad + d_2(n)=d_3(n)=d_4(n)=1-n. +\] +Each quotient $q_i=h_i/d_i$ is a polynomial. Applying the interval +conversion formula to it on each of the four intervals +\[ + [0,1/4],\quad[1/4,1/2],\quad[1/2,3/4],\quad[3/4,1] +\] +gives the following lower bounds for every coefficient: +\begin{center} +\begin{tabular}{@{}cccc@{}} +\toprule +$i$ & Divisor $d_i$ & Degree of $q_i$ & Lower bound\\ +\midrule +$0$ & $n^3(1-n)$ & $4$ & $3$\\ +$1$ & $n(1-n)$ & $6$ & $31$\\ +$2$ & $1-n$ & $7$ & $1/5$\\ +$3$ & $1-n$ & $7$ & $12/5$\\ +$4$ & $1-n$ & $7$ & $9/5$\\ +\bottomrule +\end{tabular} +\end{center} +The complete prescription for each entry is to insert the displayed +$f_j$ into $h_i$, cancel the indicated polynomial factor, and apply +the coefficient sum in Lemma~\ref{lem:bernstein-conversion} in the +listed degree. In particular, the identities $h_i=d_iq_i$ hold as +polynomial identities, including at the endpoints. + +Since every $d_i$ is nonnegative on $[0,1]$, every $h_i$ is +nonnegative. Therefore +\[ + F_*=\sum_{i=0}^4h_i(n)\binom4i z^i(1-z)^{4-i}\ge0 + \qquad(n<1). +\] +At $n=0$, the coordinate $z=t^2$ still covers the entire permitted +interval $|t|\le1$; the vanishing of $h_0,h_1$ causes no difficulty. +At $n=1$, the domain contains only $t=0$, and directly +$F_*(1,0)=F(0,0)=f_0(1)=0$. Thus no numerical division by a vanishing factor +is required, and all of $\mathcal D$ is covered. +\end{proof} + +Lemmas~\ref{lem:scalar-certificates}, \ref{lem:matrix-certificates}, +and \ref{lem:potential-certificate} establish every polynomial +inequality used in Section~\ref{sec:coupling}. Hence the three estimates +of Proposition~\ref{prop:coupling-estimates} hold on the stated +positive-variable spectral domain. The compatible limit in +Section~\ref{sec:coupling} extends them to the zero sets of the standing +strictly positive geometry. diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/08-classification.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/08-classification.tex new file mode 100644 index 0000000000000000000000000000000000000000..e41c7b11cbab0f87c9d8c508b2f606a64e5addf9 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/08-classification.tex @@ -0,0 +1,372 @@ +\section{Half-conformal flatness and the metric classification} +\label{sec:classification} + +The coupled estimates have been proved for the standing strictly +positive geometry, including its zero sets. Their strictness now rules +out two nonzero Weyl blocks. We then isolate the remaining harmonic +alternative, exclude it by a conditional lower volume estimate, and +identify the parallel curvature tensor. The orientation-cover argument +comes only after the oriented conclusion has been established. + +\begin{proposition}[Half-conformal flatness]\label{prop:one-block-vanishes} +At least one of $T,U$ vanishes identically; equivalently, the metric is +half-conformally flat. +\end{proposition} +\begin{proof} +Suppose otherwise. Proposition~\ref{prop:balanced-moments} gives $\mathbf E v^2=\mathbf E b^2$, so Proposition~\ref{prop:coupling-estimates} applies. Its integrable total expression $\mathcal I$ has nonnegative integral and satisfies +\[ + \mathcal I\le-cj+2\Theta r-\Theta^2/c + =-\frac{(\Theta-cr)^2}{c}-c(j-r^2),\qquad c\ge\frac1{10}, +\] +where +\[ + r=R_v|p|^2+R_b|q|^2,\qquad + j=(|p|^2+|q|^2)(|p|^2/v+|q|^2/b). +\] +The zero-set conventions established with \eqref{eq:3} apply to the quotients. For $v>0$, +\[ + 1+6v+v^2-4\sqrt v(1+v)=(\sqrt v-1)^4, +\] +so $R_v\le1/\sqrt v$, with equality only at $v=1$. Cauchy--Schwarz therefore gives $r^2\le j$. + +This inequality is strict almost everywhere on the set where $p$ or $q$ is nonzero. Indeed a Lipschitz function has zero gradient almost everywhere on each of its level sets: at a differentiability and density point of a level set, a nonzero differential would contradict density. Thus, outside a null set, an active gradient $p$ has $v>0$ and $v\ne1$, and similarly for $q$. At such a point +\[ + r< |p|^2/\sqrt v+|q|^2/\sqrt b\le\sqrt j. +\] +Terms with vanishing gradients are read as zero. It follows that $\mathcal I\le0$ almost everywhere, and that $\mathcal I<0$ almost everywhere where either gradient is nonzero. Since $\mathbf E\mathcal I\ge0$, both gradients vanish almost everywhere. Connectedness and continuity make $v,b$ constant. Both constants are positive, so the separate gaps \eqref{eq:5} imply $v+b\ge2$, contradicting $v+b\le x+y<2$. +\end{proof} + +Once one Weyl block vanishes, $(M,g)$ belongs to the classically +classified compact half-conformally flat Einstein class with positive +scalar curvature~\cite{Hitchin1981,FriedrichKurke1982,Obata1962}. +We retain the lower volume and parallel curvature argument below as an +independent proof of this final step under our standing hypothesis of +strictly positive sectional curvature. + +\subsection{The remaining harmonic alternatives} + +After reversing orientation if necessary, one-block vanishing leaves +$U=0$. If $T$ also vanishes, the sectional curvature is constantly one. +For a nonzero surviving block, the characteristic identities reduce the +possibilities to two integer values. + +\begin{lemma}[The one-block harmonic alternatives] +\label{lem:one-block-alternatives} +Suppose $U=0$ and $T\not\equiv0$. Then +\[ + \frac{V_M}{\pi^2}=\frac{8-2n_+}{3},\qquad + s_0=\mathbf E v^2=\frac{3n_+}{4-n_+},\qquad n_+=1\text{ or }2. +\] +In the second case, $s_0=3$ and $V_M=4\pi^2/3$. +\end{lemma} +\begin{proof} +Proposition~\ref{prop:characteristic-identities} gives $n_-=0$ and +$\chi=2+n_+$, $\tau=n_+$. Solving its two identities yields the displayed +volume and moment formulas. Positivity of volume gives $n_+<4$; the +nonzero-block gap gives $s_0\ge1$, while $v<2$ gives $s_0<4$. +The finite nonnegative integer $n_+$ is therefore $1$ or $2$. +Substitution gives the stated values in the second case. +\end{proof} + +\subsection{Excluding the exceptional harmonic count} + +The value $n_+=2$ would force exactly $s_0=3$ and $V_M=4\pi^2/3$. +We now prove a strict lower bound under that same global moment +hypothesis. The statement does not require one-block vanishing; that +conclusion is used only to produce the exceptional value to which the +bound will be applied. + +\begin{proposition}[Conditional lower volume bound] +\label{prop:conditional-lower-volume} +If $s_0=\mathbf E(v^2+b^2)=3$, then $V_M>4\pi^2/3$. +\end{proposition} + +Recall from Section~\ref{sec:volume} that $\mathcal E$ is expectation over +the unit tangent bundle, and that $\mathcal R$ is the normal Jacobi +curvature matrix along its geodesics. With +$\mathcal J''+\mathcal R\mathcal J=0$, $\mathcal J(0)=0$ and +$\mathcal J'(0)=\mathrm{Id}$, put $D(t)=\det\mathcal J(t)$. +We use the topology statement in Lemma~\ref{lem:volume-topology}, the +geodesic averages in Lemma~\ref{lem:geodesic-averages}, and the discrete +determinant in Lemma~\ref{lem:discrete-jacobi}. + +\begin{lemma}[Lower polar bound]\label{lem:lower-polar} +One has +\[ + V_M\ge2\pi^2\int_0^{\pi/\sqrt3}\mathcal E D(t)\,dt. +\] +\end{lemma} +\begin{proof} +The open ball of radius $\pi/\sqrt3$ lies in the injectivity domain at +every center by Lemma~\ref{lem:volume-topology}. Its volume is the +corresponding integral of $D$ and is at most $V_M$. Averaging over centers +gives the bound. Increasing open radii justify the endpoint; it does not +change the integral. +\end{proof} + +\begin{lemma}[Lower radial comparison]\label{lem:lower-radial} +If $s_0=3$ and $04/3$. + +\begin{proof}[Proof of Proposition~\ref{prop:conditional-lower-volume}] +Write the first bound in +Lemma~\ref{lem:lower-radial} relative to $t^3\sin\theta/\theta$. +Its exponential factor is +\[ + \exp\left[\frac16\left( + -\log\frac{\sin\theta}{\theta}-\frac{\theta^2}{6}\right)\right] + \ge1+\frac16\left[\frac12\left(\frac\theta\pi\right)^4 + +\frac13\left(\frac\theta\pi\right)^6\right]. +\] +Integrate Lemma~\ref{lem:lower-polar}, first up to any +$T<\pi/\sqrt3$ and then let $T$ increase to the endpoint. All integrands +are nonnegative, so monotone convergence suffices; no uniform endpoint +bound for a logarithm is required. The changes of variable +$\theta=\sqrt3t$, $u=\theta/\pi$ give +\[ + \frac{V_M}{\pi^2} + \ge\frac29(\pi^2-4) + +\frac{\pi^3}{27}\int_0^1 + u^2(u^4/2+u^6/3)\sin(\pi u)\,du. +\] +The last two rational integral bounds and their constant term make +this strictly greater than $1343/1000>4/3$, proving the assertion. +\end{proof} + +Lemma~\ref{lem:one-block-alternatives} makes the $n_+=2$ case equal to +$V_M=4\pi^2/3$ under $s_0=3$, contrary to +Proposition~\ref{prop:conditional-lower-volume}. Hence only $n_+=1$ +remains when exactly one block is nonzero. + +\subsection{Parallel curvature and global identification} + +\begin{proposition}[The remaining parallel tensors]\label{prop:parallel-models} +Up to reversing orientation, either $T=U=0$, or $U=0$ and $T$ is parallel +with eigenvalues $2,-1,-1$. +\end{proposition} +\begin{proof} +In the nonzero one-block case, the preceding exclusion gives $n_+=1$, +so Lemma~\ref{lem:one-block-alternatives} gives $\mathbf E v^2=1$. +The equality case of Proposition~\ref{prop:weyl-gap} yields +$\nabla T=0$ and $v=\sigma=1$, hence eigenvalues $2,-1,-1$. +The exchanged orientation is identical. If both blocks vanish, the +curvature-operator decomposition gives constant sectional curvature one. +\end{proof} + +The model calculation is the Hopf-submersion curvature computation, +following the framework of O'Neill~\cite{ONeill1966}. We include it to +fix the metric scaling precisely. + +\begin{lemma}[Curvature of the standard models]\label{lem:model-curvature} +The two tensors in Proposition~\ref{prop:parallel-models} are the curvature tensors of the unit round sphere and of twice the Fubini--Study metric induced by the Hopf quotient of the unit sphere $S^5\subset\mathbb C^3$. +\end{lemma} +\begin{proof} +The first assertion follows immediately from the curvature-operator decomposition. In the second case choose at one point the orthogonal complex structure $I$ corresponding to the eigenline of $T$ with eigenvalue $2$. On an orthonormal pair $X,Y$, the sectional curvature is +\[ + K(X,Y)=\tfrac12\bigl(1+3\langle IX,Y\rangle^2\bigr). +\] +Sectional curvatures determine an algebraic curvature tensor by polarization. Only this choice at a single tangent space is needed; no global complex structure is assumed. + +For comparison, let $g_H$ be the quotient metric on $S^5/S^1=\mathbb{CP}^2$ obtained by isometric horizontal lift. At $z\in S^5$, the unit vertical field is $iz$. For horizontal lifts of base vector fields, the horizontal part of the sphere connection is the lifted base connection by the metric formula, while +\[ + (\nabla_XY)^{\mathrm{vert}}=-\langle iX,Y\rangle iz. +\] +This follows by differentiating $\langle Y,iz\rangle=0$. Substitution in the curvature formula yields +\[ + K_{g_H}(X,Y)=1+3\langle iX,Y\rangle^2. +\] +To track the factor three, the vertical bracket is +$[X,Y]^{\mathrm{vert}}=-2\langle iX,Y\rangle iz$ and invariant horizontal lifts satisfy $\nabla_{iz}Y=iY$. The differentiated vertical term and the vertical bracket therefore contribute, respectively, one and two copies of the same squared inner product. Equivalently, the extra horizontal term in the sphere curvature expression is $3\langle iX,Y\rangle iY$, whose pairing with $X$ is $-3\langle iX,Y\rangle^2$ relative to the base expression. Scaling $g_H$ by two gives the required sectional curvatures. + +The quotient metric has parallel curvature: a unitary transformation that fixes a complex line and is minus the identity on its orthogonal complement induces an isometry fixing that point with tangent differential $-\operatorname{Id}$. It acts by $-1$ on the covariant five-tensor $\nabla R$, so $\nabla R=0$. Transitivity gives this conclusion at every point. The round sphere likewise has parallel curvature. +\end{proof} + +Parallel curvature turns matching tensors at one point into matching +Jacobi equations along paths. This is the continuation principle behind +curvature-based isometry results such as Ambrose's~\cite{Ambrose1956}. +We give the continuation and its inverse explicitly in the present +compact, simply connected setting. + +\begin{lemma}[Global continuation of the model isometry]\label{lem:global-isometry} +The local identifications furnished by Lemma~\ref{lem:model-curvature} extend to global isometries. +\end{lemma} +\begin{proof} +Lemma~\ref{lem:volume-topology} shows that $M$ is simply connected. Both model spaces are simply connected as well. For the Hopf quotient this can be seen directly: local sections lift any loop to a path in $S^5$. Join its endpoints in the circle fibre to close the path, contract the resulting loop in $S^5$, and project the contraction. The added fibre path projects to a constant, so the original loop contracts. + +Choose a tangent-space isometry matching the full curvature tensor at one point. On sufficiently small normal balls the exponential maps give an isometry: in parallel frames the Jacobi equations have identical coefficients because both curvature tensors are parallel. At a new point, the differential again matches curvature and defines the same construction on a new normal ball. Compactness supplies a common positive radius for these local constructions on the two manifolds. + +An isometry germ is determined by its value and differential through exponential maps. Hence it continues uniquely along each path. Subdivide a homotopy square into small pieces lying in normal balls; uniqueness on overlaps shows that continuation is invariant under homotopy. Simple connectivity therefore gives a global local isometry. Continue the inverse germ in the same way on the model. The two compositions agree with the identity near the original points and then everywhere by continuation. They are inverse global isometries. +\end{proof} + +\begin{corollary}[The oriented classification] +\label{cor:oriented-classification} +Every connected oriented smooth closed Einstein four-manifold with +strictly positive sectional curvature is, after positive scaling, +isometric to the round $S^4$ or to $\mathbb{CP}^2$ with its +Fubini--Study metric. +\end{corollary} +\begin{proof} +Strict sectional positivity gives a positive Einstein constant, so the +metric can be normalized to $\operatorname{Ric}=3g$. +Proposition~\ref{prop:parallel-models} and +Lemmas~\ref{lem:model-curvature}--\ref{lem:global-isometry} identify +that normalized metric with the unit round sphere or twice the Hopf +quotient metric. Undoing the positive normalization proves the stated +oriented conclusion. +\end{proof} + +\subsection{The orientation cover and the third model} + +\begin{proof}[Completion of Theorem~\ref{thm:classification}] +Strict sectional positivity makes the Ricci tensor positive definite, +so the Einstein constant is positive. +If $M$ is orientable, choose an orientation and apply +Corollary~\ref{cor:oriented-classification}. + +Otherwise let $p:\widetilde M\to M$ be the orientation double cover. +It is connected, closed and oriented; for the standard construction and +connectedness see~\cite[Proposition~3.25]{Hatcher2002}. +The pulled-back metric $\widetilde g=p^*g$ has the same Einstein and +strict sectional-positivity properties. The nontrivial deck map +$\tau$ switches the two local orientations over each point, so it is +a free orientation-reversing involution, and $p\circ\tau=p$ makes it +an isometry of $\widetilde g$. After positive scaling, +Corollary~\ref{cor:oriented-classification} identifies this cover with one of its +two standard models. Conjugating $\tau$ by the model isometry preserves +its freeness and its orientation sign. + +The $\mathbb{CP}^2$ case is impossible. Its integral cohomology ring is +$\mathbb Z[h]/(h^3)$ with $|h|=2$, so $h^2$ generates +$H^4(\mathbb{CP}^2;\mathbb Z)$ +\cite[Theorem~3.19]{Hatcher2002}. Any self-diffeomorphism sends $h$ to +$\pm h$ and therefore fixes $h^2$. It has degree $+1$, contradicting +the required orientation reversal. + +On the unit round $S^4\subset\mathbb R^5$, distance determines ambient +inner products by $\langle x,y\rangle=\cos d(x,y)$. An isometry therefore +preserves these inner products: its values on the five coordinate unit +vectors determine an $A\in O(5)$, and comparison with their images gives +$\tau(x)=Ax$ for every $x\in S^4$. Since $\tau^2=\operatorname{id}$, +we have $A^2=I$. An orthogonal involution has an orthogonal decomposition +into its $+1$ and $-1$ eigenspaces. A nonzero $+1$ eigenvector would give +a fixed point on the sphere, so freeness forces $A=-I$. The scaled base +is consequently the round quotient +$S^4/\{x\sim -x\}=\mathbb{RP}^4$. + +The antipodal map has degree $(-1)^5=-1$ on $S^4$, so it reverses +orientation~\cite[Section~2.2, property~(f)]{Hatcher2002}. +Its quotient is a closed connected smooth manifold +and inherits the Einstein and positive-sectional-curvature properties +from the round sphere through the local isometry. Thus this third model +does occur, and it is the only nonorientable case. +\end{proof} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/appendix-a-analytic.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/appendix-a-analytic.tex new file mode 100644 index 0000000000000000000000000000000000000000..251c53a06345698b0b55a2419bd15778c6adedfe --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/appendix-a-analytic.tex @@ -0,0 +1,457 @@ +\section{Analytic foundations and characteristic formulas} +\label{app:analytic} + +This appendix proves the analytic inputs stated in +Section~\ref{sec:analytic}. Throughout, $M$ is the standing closed, +connected, oriented four-manifold with $\operatorname{Ric}=3g$ and +strictly positive sectional curvature. Only that normalized geometry +and the conventions of Section~\ref{sec:preliminaries} are used; none +of the volume, Hessian, coupling, or classification conclusions enters. + +\subsection{Elliptic facts, density, and Poisson solvability} +\label{app:analytic-preliminaries} + +We use the following basic analytic facts on a closed smooth +four-manifold. The inclusion $H^1\subset L^4$ is continuous, and +$H^1\subset L^s$ is compact for $2\le s<4$. Smooth scalar and bundle +operators with Laplacian leading symbol satisfy the local +$W^{2,p}$ estimates for $10$. Strong zero-boundary solvability and coercive energy +uniqueness identify that component with its $W^{2,2}$ solution. +Once this initial regularity is established, componentwise interior +estimates and interpolation absorb the smooth first-order couplings; +differentiation gives the higher regularity used for Poisson solutions +and eigenfunctions. + +Smooth functions are dense in $H^1(M)$: use local mollification in a +finite coordinate cover and a smooth partition of unity. The same +construction gives uniform and $H^1$ approximation for a Lipschitz +function, as used in the weak norm testing rule. + +For completeness, every smooth mean-zero function $h$ has a unique smooth +mean-zero solution of $\Delta\phi=h$. On the mean-zero subspace of $H^1$, +the Dirichlet energy is coercive. Indeed, otherwise a sequence with +unit $L^2$ norm and gradients tending to zero has, by compactness in +$L^2$, a mean-zero limit with unit norm and zero weak gradient. On the +connected manifold that limit is constant, a contradiction. The +variational equation +\[ + \mathbf E\langle d\phi,d\psi\rangle=-\mathbf E(h\psi) +\] +therefore has a unique mean-zero weak solution by the Hilbert-space +representation theorem. The local regularity above makes it smooth. +The same energy identity proves uniqueness. This is the Poisson fact +used in Proposition~\ref{prop:weighted-variance}. + +\subsection{The sharp Sobolev inequality} +\label{app:sharp-sobolev} + +We give the normalized four-dimensional Einstein case of Ilias's +inequality~\cite[Theorem~3]{Ilias}. Its nonlinear Hessian proof belongs +to the Bochner approach to elliptic rigidity; compare +Bidaut-V\'eron--V\'eron~\cite{BidautVeronVeron}. + +\begin{proof}[Proof of Proposition~\ref{prop:sharp-sobolev}] +Fix $24/3. +\] +The strong Dirichlet existence theorem +\cite[Theorem~9.15]{GilbargTrudinger} gives a solution +$w_D\in W^{2,r_0}(B)\cap W^{1,r_0}_0(B)$ with this right side. +The domain and coefficients are smooth, the operator is uniformly +elliptic, and its zeroth-order coefficient is $-2$. +In dimension four, $r_0>4/3$ implies +$W^{2,r_0}\subset W^{1,q}$ for some $q>2$; compatibility of Sobolev +traces therefore gives $w_D\in H^1_0(B)$. +The difference $z-w_D\in H^1_0(B)$ solves +$(\Delta_g-2)(z-w_D)=0$ weakly. Testing by $z-w_D$ shows that it +vanishes. Thus $u$ is locally $W^{2,r_0}$, and the strong interior +estimates and integrability improvement now apply +\cite[Theorem~9.11 and Lemma~9.16]{GilbargTrudinger}. + +Initially $u\in L^4$, while $u^{s-2}\in L^r$ with +$r=4/(s-2)>2$. If $u\in L^q$, the equation has right side in +$L^p$ with $1/p=1/q+1/r$. The $W^{2,p}$ estimate and Sobolev +embedding improve $1/q$ by a fixed positive amount smaller than +$1/2-1/r$. Initially $p=4/(s-1)>1$. Iteration reaches $p>2$, +giving boundedness; at a borderline exponent $p=2$, use any +finite larger $q$ before the next iteration. + +After boundedness, the equation gives $u\in W^{2,p}$ for every finite +$p$. Since $s-1>1$, the derivative of $t^{s-1}$ is bounded on the +bounded nonnegative range of $u$. Differentiating the equation once, +and applying the same local regularity argument to each first +coordinate derivative, gives $u\in W^{3,p}$ for every finite $p$. +In particular, $u\in C^{2,\alpha}$ for some $\alpha>0$. +For $C\ge2$ we have +\[ + (-\Delta_g+C)u=(C-2)u+2u^{s-1}\ge0. +\] +The classical strong maximum principle gives $u>0$, since +$u\ge0$ and $u\not\equiv0$. A direct barrier proof uses a positive +exponential radial function on an annulus tangent to the zero set: +the weak maximum principle forces a nonzero inward derivative at the +tangency point, contradicting nonnegativity and differentiability. +Compactness now gives a positive minimum. The nonlinearity is smooth +on this positive range, so further differentiation and elliptic +regularity give smoothness of $u$. + +Put $F=u^{-1}$. Its equation is +\[ + F\Delta F-2|\nabla F|^2+2F^2=2F^{4-s}. +\] +Since +$\operatorname{div}(\operatorname{Hess}_0F) + =\frac34\,d(\Delta F+4F)$, differentiation gives +\[ + \begin{aligned} + \operatorname{div}(F^{-3}\operatorname{Hess}_0F) + &=\frac34F^{-4} + \bigl[F\,d\Delta F+(\Delta F)dF + -4\operatorname{Hess}F(\nabla F,\,\cdot\,)+4F\,dF\bigr]\\ + &=\frac32(4-s)F^{-1-s}\,dF . + \end{aligned} +\] +Pairing with $dF$ and integrating yields +\[ + -\mathbf E F^{-3}|\operatorname{Hess}_0F|^2 + =\frac32(4-s)\mathbf E F^{-1-s}|dF|^2. +\] +Both sides have opposite signs, so $dF=0$. The positive constant +solution of the equation for $u$ is $u=1$. The normalization of +$w$ then gives $w=1$ and $m_s=1$. + +Thus the claimed inequality holds with $4$ replaced by every +$s\in(2,4)$ and outer exponent $2/s$. For fixed $f\in H^1$, +the ordinary inclusion $H^1\subset L^4$ and dominated convergence +give $\|f\|_{L^s(d\nu)}\to\|f\|_{L^4(d\nu)}$ as $s\uparrow4$. +This proves the stated critical inequality. +\end{proof} + +\subsection{Harmonic forms and the characteristic identities} +\label{app:characteristic-identities} + +We derive the Euler and signature formulas using the complex of forms and +an explicit signed heat computation. This is the heat-supertrace method +of McKean--Singer~\cite{McKeanSinger1967}; general formulations and local +coefficient calculations appear in Gilkey~\cite[Theorems~1.7.6 and +4.8.16]{Gilkey1984}. + +We first record the spectral heat construction for a smooth symmetric +Laplace-type operator $L=-\Delta+E$ on a Euclidean bundle over $M$. Adding a sufficiently +large constant gives a coercive quadratic form on $H^1$. +Its weak inverse is compact and self-adjoint on $L^2$ by +compact embedding, and elliptic regularity makes its +eigenfunctions smooth. The spectrum is discrete with finite +multiplicities and tends to infinity. Spectral heat evolution +is smoothing for every positive time: polynomial powers of +the shifted eigenvalue times $e^{-t\lambda}$ are bounded, +and elliptic estimates for powers of $L$ identify the +corresponding Sobolev bounds. + +For completeness, these estimates also justify smooth kernel +and trace expansions. For a sufficiently large integer $k$, +evaluation of any prescribed finite derivative after +$(L+C)^{-k}$ is a bounded functional on $L^2$, uniformly in +its base point. Applied to an orthonormal eigenbasis, the +Riesz representation theorem gives a uniform bound for the +sum of the squared derivative values divided by +$(\lambda+C)^{2k}$. Cauchy--Schwarz and the exponential factor +then give uniform convergence of the spectral kernel series +and all its derivatives for $t$ in compact subintervals of +$(0,\infty)$. Integrating the diagonal gives the usual heat +trace. Uniqueness with prescribed initial point mass follows +by testing the kernel against the eigenfunctions. + +\begin{lemma}[The form-complex Laplacians] +\label{lem:form-complex-laplacians} +Consider the elliptic complex +\[ + \Lambda^0\xrightarrow{\ d\ }\Lambda^1 + \xrightarrow{\ \sqrt2\,d^+\ }\Lambda^+, + \qquad d^+=P_+d. +\] +Its three Laplacians are the Hodge Laplacians on the indicated +bundles. Written as $-\Delta+E$, their potentials are +\[ + E_0=0,\qquad E_1=3\operatorname{Id},\qquad + E_+=4\operatorname{Id}-2T. +\] +Its alternating heat trace is $1+n_+$. +\end{lemma} + +\begin{proof} +The middle complex Laplacian is +\[ + dd^*+2(d^+)^*d^+ + =dd^*+d^*d+d^*\!\star d. +\] +The last term is zero by $d^2=0$ and the formula for the +codifferential. On self-dual two-forms, Hodge star exchanges +$dd^*$ and $d^*d$, so $2P_+dd^*$ is their sum. This identifies +the last Laplacian; the scalar one is immediate. Their elliptic +principal symbols are consequently those of the metric Laplacian. + +Commuting covariant derivatives in the one-form Hodge Laplacian +gives the Ricci endomorphism, equal to $3\operatorname{Id}$. +For the plus potential, write a self-dual form as +$\eta=\sum_s\eta_s I_s/\sqrt2$. Its Hodge energy is +$2\mathbf E|d^*\eta|^2$, which equals +\[ + \mathbf E\left|\sum_s I_s\nabla\eta_s\right|^2 + =\mathbf E\left( + |\nabla\eta|^2+\frac12\langle C\nabla\eta,\nabla\eta\rangle + \right), +\] +where $C=\sum I_s\otimes\Sigma_s$ now uses the spin-one +generators on the plus coefficients. The equality follows by +expanding the quaternion products. Integrating the $C$ term +and taking its skew derivative commutator gives the potential +\[ + -\frac12\sum_s(1+t_s)\Sigma_s^2. +\] +In the spin-one representation, +$\Sigma_s^2=4(e_se_s^{\mathsf T}-\operatorname{Id})$. +The potential is therefore $4\operatorname{Id}-2T$, +as asserted. + +The scalar kernel consists of the constants. A harmonic one-form +satisfies +\[ + 0=\mathbf E\bigl(|\nabla\eta|^2+3|\eta|^2\bigr), +\] +so the middle kernel vanishes. The last kernel has dimension +$n_+$. On each positive eigenspace of the complex Laplacian, +the complex differential commutes with the Laplacian, and its +adjoint divided by the eigenvalue is a contracting homotopy. +That finite-dimensional complex has zero alternating dimension. +Summing the heat factors over the positive spectrum leaves only +the kernel contribution $1+n_+$. +\end{proof} + +\begin{lemma}[Local signed heat coefficient] +\label{lem:signed-heat-coefficient} +For the three operators in +Lemma~\ref{lem:form-complex-laplacians}, let $\operatorname{str}$ +denote the trace on $\Lambda^0$ minus that on $\Lambda^1$ plus +that on $\Lambda^+$. Write $\Omega$ for their connection +curvatures. The constant local coefficient of their alternating +heat trace, including the factor $(4\pi)^{-2}$, is +\[ + \frac1{16\pi^2} + \left(\frac12\operatorname{str}E^2 + +\frac1{12}\operatorname{str}\sum_{k,i}\Omega_{ki}^2\right). +\] +\end{lemma} + +\begin{proof} +Fix the source point of the kernel. In normal coordinates +$z$ around it, use radially parallel orthonormal frames on +each of the three bundles and set +$\mu=\sqrt{\det g}$, $h=\mu^{-1/2}$. The ansatz +\[ + (4\pi t)^{-2}e^{-|z|^2/(4t)} + \sum_{j\ge0}t^jU_j(z) +\] +gives $U_0=h\operatorname{Id}$ and the transport equations +\[ + \left(z\cdot\partial+j+\frac12z\cdot\partial\log\mu\right)U_j + =-LU_{j-1}. +\] +Here the radial connection coefficient is zero. +The scalar part follows also from +$\Delta|z|^2=8+2z\cdot\partial\log\mu$. +Solving the radial equation gives +\[ + U_j(z)=-h(z)\int_0^1 s^{j-1} + \bigl(h^{-1}LU_{j-1}\bigr)(sz)\,ds. +\] +These transport solutions compute the actual local asymptotics. +Indeed, truncate at any large $N$ and apply a cutoff inside a +normal ball. The resulting kernel has initial delta mass, +and the transport equations leave a residual uniformly +$O(t^{N-2})$; derivatives of the cutoff give exponentially +small terms. The heat evolution of a metric connection +with bounded potential has a bounded sup norm on bounded +time intervals, by the maximum principle for the squared +norm. Duhamel integration therefore makes the difference +from the true kernel $O(t^{N-1})$. Compactness permits a +uniform normal-ball radius and uniform constants in the +source point. Taking $N$ large justifies the diagonal +coefficient calculation below. + +In these frames write the connection as $\partial_i+A_i$. +Every $A_i$ is skew-symmetric and radial gauge gives +\[ + A_i(z)=\frac12z^k\Omega_{ki}(0)+O(|z|^2). +\] +For example, the exact radial-gauge identity +$A_i(z)=\int_0^1s z^k\Omega_{ki}(sz)\,ds$ follows by contracting +the curvature equation with $z^k$ and integrating along +the radius. +The signed ranks and potentials satisfy, at every point, +\[ + \operatorname{str}\operatorname{Id}=1-4+3=0, + \qquad + \operatorname{str}E=-12+\operatorname{tr}(4\operatorname{Id}-2T)=0. +\] +In coordinate form, +\[ + L=-g^{ij}\bigl[ + (\partial_i+A_i)(\partial_j+A_j) + -\Gamma^k_{ij}(\partial_k+A_k)\bigr]+E. +\] +Apply this to $U_0=h\operatorname{Id}$. All purely scalar terms +have zero supertrace. Every term linear in $A_i$ or a +coordinate derivative of $A_i$ has zero ordinary trace in +each bundle, by skewness. The potential has zero supertrace +pointwise. Thus the exact surviving expression is +\[ + \operatorname{str}(LU_0)(z) + =-h(z)g^{ij}(z)\operatorname{str}(A_i(z)A_j(z)). +\] +In particular, derivatives of the varying potential have +already disappeared before the radial integral is taken. +The transport formula now gives +\[ + \begin{aligned} + \operatorname{str}U_1(z) + &=h(z)\int_0^1 g^{ij}(sz) + \operatorname{str}(A_i(sz)A_j(sz))\,ds\\ + &=\frac1{12}z^kz^\ell + \operatorname{str}\sum_i\Omega_{ki}(0)\Omega_{\ell i}(0) + +O(|z|^3). + \end{aligned} +\] +The coefficient is +$\frac14\int_0^1s^2\,ds=\frac1{12}$. +Scalar and metric corrections do not change this quadratic term. + +At the center, $A_i(0)=0$, $\Gamma(0)=0$, and +$\sum_i\partial_iA_i(0)=\frac12\sum_i\Omega_{ii}(0)=0$ +as an endomorphism. Consequently +\[ + U_1(0)=c\operatorname{Id}-E(0),\qquad + (LU_1)(0)=-\sum_i\partial_i^2U_1(0)+E(0)U_1(0), +\] +where $c$ is the same scalar for all three bundles. +Thus there are no additional connection terms at the center. +Taking the Laplacian of the signed quadratic expression gives +\[ + \operatorname{str}\sum_i\partial_i^2U_1(0) + =\frac16\operatorname{str}\sum_{ki}\Omega_{ki}^2, + \qquad + \operatorname{str}(EU_1(0))=-\operatorname{str}E^2. +\] +Since the transport equation at the center gives +$U_2(0)=-\frac12(LU_1)(0)$, we obtain +\[ + \operatorname{str}U_2(0) + =\frac12\operatorname{str}E^2 + +\frac1{12}\operatorname{str}\sum_{ki}\Omega_{ki}^2. +\] +The lower coefficients have zero signed diagonal trace. +Multiplication by the heat prefactor $(4\pi t)^{-2}$ +now proves the assertion. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:characteristic-identities}] +By Hodge theory, harmonic forms represent de Rham cohomology. The +scalar kernel is one-dimensional, and the one-form kernel vanishes by +Lemma~\ref{lem:form-complex-laplacians}. Hodge star gives the corresponding +dimensions in degrees four and three. On harmonic two-forms, the +intersection form is positive definite on the self-dual part and negative +definite on the anti-self-dual part. Thus its signature is $n_+-n_-$, +and the alternating sum of the Betti numbers is $2+n_++n_-$. This +identifies the integers in Proposition~\ref{prop:characteristic-identities} +with the topological signature and Euler characteristic without using +simple connectivity. + +The potential-square supertrace is +\[ + \operatorname{str}E^2 + =-\operatorname{tr}(9\operatorname{Id}_{\Lambda^1}) + +\operatorname{tr}\bigl((4\operatorname{Id}_{\Lambda^+}-2T)^2\bigr) + =12+24v^2. +\] +On one-forms, the connection curvature has plus and minus +generators $I_s,J_s$. Orthogonality of the two-form bases, +together with $I_s^2=J_s^2=-\operatorname{Id}$, gives +\[ + \operatorname{tr}_{\Lambda^1}\sum_{ki}\Omega_{ki}^2 + =-4\sum_s\bigl((1+t_s)^2+(1+u_s)^2\bigr) + =-24(1+v^2+b^2). +\] +On plus two-forms only the plus generators act, and each +speed-two spin-one generator has square trace $-8$. +Therefore +\[ + \operatorname{tr}_{\Lambda^+}\sum_{ki}\Omega_{ki}^2 + =-8\sum_s(1+t_s)^2=-24(1+2v^2). +\] +The scalar connection is flat. Hence +$\operatorname{str}\sum\Omega_{ki}^2=24(b^2-v^2)$, and +Lemma~\ref{lem:signed-heat-coefficient} gives the signed +constant density +\[ + \frac1{16\pi^2}(6+10v^2+2b^2). +\] +Integrate this against $d\operatorname{vol}_g$ and compare +with the constant alternating heat trace from +Lemma~\ref{lem:form-complex-laplacians}: +\[ + 16\pi^2(1+n_+) + =\operatorname{Vol}(M)\, + \mathbf E(6+10v^2+2b^2). +\] +Reverse orientation to get the analogous formula with +$n_-,b,v$. Adding and subtracting the two formulas gives +exactly \eqref{eq:6}. + +On anti-self-dual forms the potential is +$4\operatorname{Id}-2U$. If $U=0$, the energy of a harmonic +anti-self-dual form is the integral of +$|\nabla\eta|^2+4|\eta|^2$, so its kernel is zero. +The assertion for $T=0$ follows in the same way. +\end{proof} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/appendix-b-topology.tex b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/appendix-b-topology.tex new file mode 100644 index 0000000000000000000000000000000000000000..44998ab166988371263b5cc81fd679cae36bc5ef --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/appendix-b-topology.tex @@ -0,0 +1,109 @@ +\section{Minimizing geodesics and topology} +\label{app:topology} + +We prove Lemma~\ref{lem:volume-topology} in the standing setting: +$M$ is a closed, connected, oriented smooth four-manifold with +$\operatorname{Ric}=3g$ and strictly positive sectional curvature. +Along a unit-speed geodesic, let $\mathcal R$ be the normal Jacobi +curvature matrix in a parallel orthonormal frame. The hypotheses give +\[ + 0<\mathcal R<3\operatorname{Id},\qquad + \operatorname{tr}\mathcal R=3. +\] +The trace supplies the diameter estimate, the upper bound supplies the +conjugate-radius estimate, and strict positivity supplies the shortening +variations. Orientation is used in the normal-holonomy argument. These +are the classical methods of Myers, Synge, and Klingenberg +\cite{Myers1941,Synge1936,Klingenberg1959}; the proof below retains the +endpoint variations and proves simple connectivity independently of +all volume and classification conclusions. + +\begin{proof}[Proof of Lemma~\ref{lem:volume-topology}] +For a unit-speed geodesic of length $L$, the index form on normal fields +with zero endpoint values is +\[ + I(Y,Y)=\int_0^L\bigl(|Y'|^2-\langle\mathcal RY,Y\rangle\bigr)\,dt. +\] +It is nonnegative on a minimizing segment, by the second variation of +energy. Summing it on three parallel normal unit fields multiplied by +$\sin(\pi t/L)$ gives +\[ + \frac{3L}{2}\left(\frac{\pi^2}{L^2}-1\right), +\] +so a minimizing segment has length at most $\pi$. +It cannot contain an interior conjugate point: a nonzero Jacobi field +vanishing at the initial point and that interior point, extended by +zero, has index zero but is not in the nullspace of the index form. +The latter assertion follows by integration by parts against a field +whose value pairs nontrivially with the jump in its derivative. +A nonnegative quadratic form, however, pairs a zero-energy vector to +zero against every vector, a contradiction. + +The ordinary Dirichlet Poincar\'e inequality and $\mathcal R<3\mathrm{Id}$ +show that a nonzero field with zero endpoints has positive index when +$L\le\pi/\sqrt3$. Thus no conjugate point occurs at such a distance. +We show that no cut point occurs at a smaller distance either. +Let $c$ be the infimum of all cut radii; compactness and normal balls give +$c>0$. Suppose $c<\pi/\sqrt3$ and choose cut segments with lengths tending +to $c$, all below the conjugate-radius bound. Each endpoint has two +distinct minimizing segments from its starting point. To see this, +extend a cut segment a little and take limits of minimizers to the +extended endpoints. If the limiting minimizer were the original +segment alone, local invertibility of the exponential map would force +the extended segment to remain minimizing, contrary to its cut time. + +After passing to a subsequence, compactness gives two length-$c$ +minimizing segments with common endpoints $p,q$. Their initial vectors +cannot coalesce: the endpoint map +\[ + (p,w)\longmapsto(p,\exp_p w) +\] +is a local diffeomorphism at every vector in question, so two converging +branches in the same inverse-function neighborhood would coincide. +This map also supplies two distinct smooth local geodesic branches for +nearby pairs of endpoints. + +Their terminal unit velocities must be opposite. If these velocities +are $u_1,u_2$ with $u_2\ne-u_1$, move $q$ initially in direction +$-(u_1+u_2)$. The first length variation of both branches is +$-(1+\langle u_1,u_2\rangle)<0$. Both branches would then have length +less than $c$. Every geodesic of length less than the global cut infimum +is minimizing and lies in the injectivity domain, so two distinct such +branches are impossible. Moving $p$ instead proves that their initial +velocities are opposite as well. The first segment followed by the +reverse of the second is therefore a smooth closed geodesic. + +Parallel transport around this closed geodesic fixes the tangent and +preserves orientation. Its action on the three-dimensional normal +space belongs to $\mathrm{SO}(3)$, and hence fixes a nonzero vector. +Let $V$ be the resulting periodic parallel normal field. Move both +endpoints along geodesics with initial velocities $V(p),V(q)$, so their +accelerations at the variation origin vanish. The local geodesic +branches vary smoothly. Their variation fields $J$ are normal: their +tangential components are affine and have zero boundary values. +For either branch, $W=V-J$ has zero boundary values and +$I(J,W)=0$, while $I(W,W)\ge0$ because the unvaried segment is minimizing. +Thus +\[ + I(J,J)\le I(V,V)=-\int\langle\mathcal RV,V\rangle<0. +\] +The first length variations vanish, and these negative index values +are the second length variations, since there are no endpoint +acceleration or tangential terms. Both branch lengths therefore become +less than $c$. They remain distinct in their disjoint inverse-function +neighborhoods, and lengths below $c$ force them to be minimizing. This +is the same contradiction as above, proving the injectivity bound. + +Finally, a nontrivial free homotopy class would have a shortest +representative. Indeed, constant-speed minimizing sequences have a +uniform Lipschitz bound; compactness gives a uniformly convergent +subsequence, and sufficiently close loops are freely homotopic by the +short normal geodesics between corresponding points. A uniform normal +ball radius also gives a positive lower bound on lengths of nontrivial +loops. Local geodesic replacement shows that the minimizing loop is +a nonconstant smooth closed geodesic. Its normal holonomy again fixes +a nonzero parallel field, whose periodic variation has strictly +negative second length variation. The variation preserves its free +homotopy class, a contradiction. Hence the fundamental group is +trivial. +\end{proof} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1f857668c99711feaeaafae8d3ac1d1cc241b5ae --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2eb05d3e177c39f3aaf6084352f658f5afcda414110db8417aec4d565d6ce407 +size 720539 diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/COVERAGE.md b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/COVERAGE.md new file mode 100644 index 0000000000000000000000000000000000000000..4a4c0d249f0f0d7401beb911024ad7bf480fee75 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/COVERAGE.md @@ -0,0 +1,8 @@ +# Verification coverage + +The complete finite results and asserted bounds are in `results.json`; their +definitions and checking recipe are in `verify.py` and `polynomial.py`. +The verifier checks finite polynomial, coefficient, and rational-integral +claims, not the manuscript's geometric or analytic reductions. + +See [README.md](README.md) for the command and success criteria. diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/README.md b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/README.md new file mode 100644 index 0000000000000000000000000000000000000000..bae2f570927eb28b0524eab947f0044ec1f1534f --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/README.md @@ -0,0 +1,23 @@ +# Exact arithmetic verification + +Requires Python 3.10 or later and only the standard library. No network +connection or external data is required. + +From this directory: + +```sh +python -B verify.py --check +``` + +The command uses `verify.py`, `polynomial.py`, and `results.json`. It +recomputes the finite results and compares the complete output with +`results.json` without modifying it. Checks remain active under Python `-O`. +Success requires exit status 0, a `PASS` line, and +`"all_passed": true` in the printed summary. + +## Other files + +- `COVERAGE.md`: location and limits of the checked data. +- `dependencies.json`: runtime requirements. +- `manifest.json`: current file sizes and hashes. +- `portability_check.py` and `portability-results.json`: optional diagnostics. diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/dependencies.json b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/dependencies.json new file mode 100644 index 0000000000000000000000000000000000000000..e2a89dcbbd768284c77b09cfbdaf1ea1db7a9f26 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/dependencies.json @@ -0,0 +1,21 @@ +{ + "author": "OpenAI", + "runtime": { + "implementation": "Python 3", + "minimum_version": "3.10", + "tested_version": "3.12.13" + }, + "third_party_dependencies": [], + "network_required": false, + "external_data_required": false, + "standard_library_modules": [ + "argparse", "fractions", "functools", "hashlib", "json", "math", + "pathlib", "platform", "shutil", "subprocess", "sys", "tempfile" + ], + "main_command": "python -B verify.py --check", + "regenerate_command": "python -B verify.py --write", + "portability_command": "python -B portability_check.py", + "runtime_files": ["verify.py", "polynomial.py", "results.json"], + "arithmetic": "Exact fractions.Fraction coefficients and integer combinatorics", + "floating_point_decisions": false +} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/manifest.json b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/manifest.json new file mode 100644 index 0000000000000000000000000000000000000000..be9b325877c0cbc7a6029a26afa948a882b5210b --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/manifest.json @@ -0,0 +1,42 @@ +{ + "author": "OpenAI", + "files": { + "COVERAGE.md": { + "bytes": 377, + "sha256": "228873d8e28b77a71db0205168313667417d4f5b099d66d94f5a320a8150f177" + }, + "README.md": { + "bytes": 768, + "sha256": "ab912852b62dbaf165c2ccc26f3aa895de127504048043cd79053ab13b588966" + }, + "dependencies.json": { + "bytes": 765, + "sha256": "651d44144b0b433aec6ea42b7ae4ab461e9f789cbf1761bdbaaf87dde1503082" + }, + "polynomial.py": { + "bytes": 10896, + "sha256": "848b98bf1f6cef533c1e2320b74afadfa9c4f00428a275c4663da4e7c53f19bb" + }, + "portability-results.json": { + "bytes": 1828, + "sha256": "5deacb4cf4155605166e0f82696dd57ca3cb5b0b41b4f37aeb89766acdf3f8f2" + }, + "portability_check.py": { + "bytes": 6050, + "sha256": "4bce1e140f6933d97d0b005747c38a28be4ae0dc8be1bf659430fc21fc4f4fea" + }, + "results.json": { + "bytes": 447073, + "sha256": "d84273376cedd109c7cbf46b77cdc9ae84eac99daced9e5cd3dcbb2f20b2e7dc" + }, + "verify.py": { + "bytes": 28511, + "sha256": "11060c858a42c96e5d4f5f70228e7a2264a38d2083dca687ce2b7fbc75af6dcc" + } + }, + "format_version": 1, + "main_command": "python -B verify.py --check", + "network_required": false, + "scope": "Supporting exact arithmetic package only; no theorem certification claim", + "third_party_dependencies": [] +} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/polynomial.py b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/polynomial.py new file mode 100644 index 0000000000000000000000000000000000000000..ca8b3faa8c56873fcd31b902dd9871711af59ce4 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/polynomial.py @@ -0,0 +1,272 @@ +"""Small exact rational polynomial arithmetic for the accompanying checks. + +Author: OpenAI. Standard library only; no symbolic or numerical dependencies. +""" +from fractions import Fraction +from functools import lru_cache +from math import comb, factorial + + +def rational(value): + if isinstance(value, float): + raise TypeError("Floating-point coefficients are not permitted") + return Fraction(value) + + +def require(condition, message): + """An assertion that remains active when Python is invoked with -O.""" + if not condition: + raise ArithmeticError(message) + + +class Poly: + __slots__ = ("variables", "terms") + + def __init__(self, variables, terms=None): + self.variables = tuple(variables) + self.terms = {} + for exponent, value in (terms or {}).items(): + exponent = tuple(exponent) + require(len(exponent) == len(self.variables), "Wrong monomial dimension") + require(all(isinstance(i, int) and i >= 0 for i in exponent), + "Polynomial exponents must be nonnegative integers") + value = rational(value) + if value: + self.terms[exponent] = value + + @classmethod + def constant(cls, variables, value): + return cls(variables, {(0,) * len(variables): rational(value)}) + + def _coerce(self, other): + if isinstance(other, Poly): + require(other.variables == self.variables, "Polynomial ring mismatch") + return other + return Poly.constant(self.variables, other) + + def __add__(self, other): + other = self._coerce(other) + terms = self.terms.copy() + for exponent, value in other.terms.items(): + terms[exponent] = terms.get(exponent, Fraction(0)) + value + return Poly(self.variables, terms) + + __radd__ = __add__ + + def __neg__(self): + return Poly(self.variables, {e: -c for e, c in self.terms.items()}) + + def __sub__(self, other): + return self + -self._coerce(other) + + def __rsub__(self, other): + return self._coerce(other) + -self + + def __mul__(self, other): + other = self._coerce(other) + terms = {} + for e, c in self.terms.items(): + for f, d in other.terms.items(): + exponent = tuple(a + b for a, b in zip(e, f)) + terms[exponent] = terms.get(exponent, Fraction(0)) + c * d + return Poly(self.variables, terms) + + __rmul__ = __mul__ + + def __truediv__(self, value): + if isinstance(value, Poly): + require(value.degree() <= 0, "Use divide_exact for polynomial division") + value = value.terms.get((0,) * len(self.variables), Fraction(0)) + value = rational(value) + require(value != 0, "Division by zero") + return self * (1 / value) + + def __pow__(self, exponent): + require(isinstance(exponent, int) and exponent >= 0, + "Only nonnegative integral powers are polynomial") + result = Poly.constant(self.variables, 1) + base = self + while exponent: + if exponent % 2: + result = result * base + exponent //= 2 + if exponent: + base = base * base + return result + + def __eq__(self, other): + try: + other = self._coerce(other) + except (ArithmeticError, TypeError, ValueError): + return False + return self.terms == other.terms + + def degree(self, variable=None): + if variable is None: + return max((sum(e) for e in self.terms), default=-1) + index = self.variables.index(variable) + return max((e[index] for e in self.terms), default=-1) + + def derivative(self, variable, order=1): + index = self.variables.index(variable) + result = self + for _ in range(order): + terms = {} + for exponent, coefficient in result.terms.items(): + if exponent[index]: + new_exponent = list(exponent) + new_exponent[index] -= 1 + terms[tuple(new_exponent)] = coefficient * exponent[index] + result = Poly(self.variables, terms) + return result + + def substitute(self, values): + require(len(values) == len(self.variables), "Wrong substitution length") + target = next((p.variables for p in values if isinstance(p, Poly)), None) + if target is None: + return self.evaluate(values) + values = [p if isinstance(p, Poly) else Poly.constant(target, p) for p in values] + require(all(p.variables == target for p in values), "Substitution ring mismatch") + powers = [] + for index, value in enumerate(values): + row = [Poly.constant(target, 1)] + for _ in range(max((e[index] for e in self.terms), default=0)): + row.append(row[-1] * value) + powers.append(row) + result = Poly.constant(target, 0) + for exponent, coefficient in self.terms.items(): + term = Poly.constant(target, coefficient) + for index, power in enumerate(exponent): + term = term * powers[index][power] + result = result + term + return result + + def evaluate(self, values): + require(len(values) == len(self.variables), "Wrong evaluation length") + values = list(map(rational, values)) + answer = Fraction(0) + for exponent, coefficient in self.terms.items(): + term = coefficient + for value, power in zip(values, exponent): + term *= value ** power + answer += term + return answer + + def coefficient(self, variable, power): + index = self.variables.index(variable) + variables = self.variables[:index] + self.variables[index + 1:] + terms = {e[:index] + e[index + 1:]: c for e, c in self.terms.items() + if e[index] == power} + return Poly(variables, terms) + + def divide_exact(self, divisor): + divisor = self._coerce(divisor) + require(bool(divisor.terms), "Polynomial division by zero") + leading = lambda terms: max(terms, key=lambda e: (sum(e), e)) + de = leading(divisor.terms) + dc = divisor.terms[de] + remainder = self + quotient = Poly.constant(self.variables, 0) + while remainder.terms: + re = leading(remainder.terms) + require(all(a >= b for a, b in zip(re, de)), "Nonzero polynomial remainder") + exponent = tuple(a - b for a, b in zip(re, de)) + term = Poly(self.variables, {exponent: remainder.terms[re] / dc}) + quotient = quotient + term + remainder = remainder - term * divisor + require(quotient * divisor == self, "Exact polynomial division failed") + return quotient + + def reduce_circle(self, cosine="sigma", sine="rho"): + """Reduce modulo sine^2+cosine^2-1 without numerical roots.""" + si = self.variables.index(sine) + ci = self.variables.index(cosine) + terms = {} + for exponent, coefficient in self.terms.items(): + pairs, residual = divmod(exponent[si], 2) + for j in range(pairs + 1): + new_exponent = list(exponent) + new_exponent[si] = residual + new_exponent[ci] += 2 * j + new_exponent = tuple(new_exponent) + value = coefficient * comb(pairs, j) * (-1) ** j + terms[new_exponent] = terms.get(new_exponent, Fraction(0)) + value + return Poly(self.variables, terms) + + def integral(self, left, right): + require(len(self.variables) == 1, "Interval integration requires one variable") + left, right = rational(left), rational(right) + return sum((c * (right ** (e[0] + 1) - left ** (e[0] + 1)) / (e[0] + 1) + for e, c in self.terms.items()), Fraction(0)) + + def data(self): + return {"variables": list(self.variables), "degree": self.degree(), + "terms": [{"powers": list(e), "coefficient": str(c)} + for e, c in sorted(self.terms.items())]} + + +def ring(*variables): + variables = tuple(variables) + answer = [] + for index in range(len(variables)): + exponent = [0] * len(variables) + exponent[index] = 1 + answer.append(Poly(variables, {tuple(exponent): Fraction(1)})) + return tuple(answer) + + +@lru_cache(maxsize=None) +def _interval_basis(degree): + (z,) = ring("z") + return tuple(comb(degree, i) * z ** i * (1 - z) ** (degree - i) + for i in range(degree + 1)) + + +def interval_coefficients(poly, left, right, degree=None): + require(len(poly.variables) == 1, "Expected a one-variable polynomial") + degree = poly.degree() if degree is None else degree + require(0 <= poly.degree() <= degree, "Univariate degree truncation prohibited") + (z,) = ring("z") + affine = poly.substitute([rational(left) + (rational(right) - rational(left)) * z]) + coeffs = [] + for i in range(degree + 1): + coeffs.append(sum((c * Fraction(comb(i, e[0]), comb(degree, e[0])) + for e, c in affine.terms.items() if e[0] <= i), Fraction(0))) + reconstructed = sum((c * p for c, p in zip(coeffs, _interval_basis(degree))), + Poly.constant(("z",), 0)) + require(reconstructed == affine, "Univariate Bernstein reconstruction failed") + return coeffs + + +@lru_cache(maxsize=None) +def _triangle_basis(degree): + z, w = ring("z", "w") + answer = {} + for i in range(degree + 1): + for j in range(degree + 1 - i): + k = degree - i - j + multinomial = factorial(degree) // (factorial(i) * factorial(j) * factorial(k)) + answer[i, j] = multinomial * z ** i * w ** j * (1 - z - w) ** k + return answer + + +def triangle_coefficients(poly, vertices, degree): + require(len(poly.variables) == 2, "Expected a two-variable polynomial") + require(poly.degree() <= degree, "Triangle degree truncation prohibited") + z, w = ring("z", "w") + p1, p2, p3 = [tuple(map(rational, p)) for p in vertices] + v = p3[0] + (p1[0] - p3[0]) * z + (p2[0] - p3[0]) * w + b = p3[1] + (p1[1] - p3[1]) * z + (p2[1] - p3[1]) * w + affine = poly.substitute([v, b]) + coeffs = {} + for i in range(degree + 1): + for j in range(degree + 1 - i): + coeffs[i, j] = sum((c * Fraction(comb(i, d) * comb(j, e), + comb(degree, d + e) * comb(d + e, d)) + for (d, e), c in affine.terms.items() + if d <= i and e <= j), Fraction(0)) + reconstructed = sum((coeffs[index] * basis + for index, basis in _triangle_basis(degree).items()), + Poly.constant(("z", "w"), 0)) + require(reconstructed == affine, "Triangle Bernstein reconstruction failed") + return coeffs diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/portability-results.json b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/portability-results.json new file mode 100644 index 0000000000000000000000000000000000000000..35bf558692a3b280920dd50b340b095b338a0370 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/portability-results.json @@ -0,0 +1,64 @@ +{ + "all_passed": true, + "author": "OpenAI", + "checks": [ + { + "command": "python -B ../relocated-package/verify.py --check", + "name": "relocated_run_from_different_cwd", + "passed": true, + "stdout": [ + "PASS: all exact identities, coefficient bounds, zero patterns, and rational volume margins.", + "{\"all_passed\": true, \"exact_value_assertions\": 24, \"identities\": 53, \"interval_certificate_rows\": 21, \"reported_interval_rows\": 15, \"triangle_certificate_rows\": 12}" + ] + }, + { + "name": "optimized_python_checks_remain_active", + "passed": true + }, + { + "name": "altered_saved_results_rejected", + "passed": true + }, + { + "name": "incorrect_printed_triple_rejected_under_O", + "passed": true + }, + { + "name": "incorrect_zero_locations_rejected", + "passed": true + }, + { + "name": "violated_coefficient_floor_rejected", + "passed": true + }, + { + "name": "univariate_degree_truncation_rejected", + "passed": true + }, + { + "name": "triangle_degree_truncation_rejected", + "passed": true + }, + { + "name": "inexact_polynomial_division_rejected", + "passed": true + }, + { + "name": "floating_point_coefficient_rejected", + "passed": true + }, + { + "name": "original_runtime_files_unchanged", + "passed": true + } + ], + "network_used": false, + "python_version": "3.12.13", + "runtime_sha256": { + "polynomial.py": "848b98bf1f6cef533c1e2320b74afadfa9c4f00428a275c4663da4e7c53f19bb", + "results.json": "d84273376cedd109c7cbf46b77cdc9ae84eac99daced9e5cd3dcbb2f20b2e7dc", + "verify.py": "11060c858a42c96e5d4f5f70228e7a2264a38d2083dca687ce2b7fbc75af6dcc" + }, + "temporary_files_outside_package": false, + "third_party_dependencies": [] +} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/portability_check.py b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/portability_check.py new file mode 100644 index 0000000000000000000000000000000000000000..75a9ac4818079b27a11488e56c920863fe438b9d --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/portability_check.py @@ -0,0 +1,108 @@ +#!/usr/bin/env python3 +"""Test relocation and selected rejection paths without third-party packages. + +Author: OpenAI. All temporary output stays inside this package directory. +""" +import sys +sys.dont_write_bytecode = True +import hashlib +import json +from pathlib import Path +import platform +import shutil +import subprocess +import tempfile + +from polynomial import Poly, interval_coefficients, require, ring, triangle_coefficients + +ROOT = Path(__file__).resolve().parent +RUNTIME_FILES = ("verify.py", "polynomial.py", "results.json") + + +def sha(path): + return hashlib.sha256(path.read_bytes()).hexdigest() + + +def main(): + checks = [] + originals = {name: sha(ROOT / name) for name in RUNTIME_FILES} + with tempfile.TemporaryDirectory(prefix=".portability-", dir=ROOT) as temporary: + base = Path(temporary) + package, other_cwd = base / "relocated-package", base / "different-cwd" + package.mkdir() + other_cwd.mkdir() + for name in RUNTIME_FILES: + shutil.copyfile(ROOT / name, package / name) + command = [sys.executable, "-B", str(package / "verify.py"), "--check"] + run = subprocess.run(command, cwd=other_cwd, capture_output=True, text=True, check=False) + require(run.returncode == 0, "Relocated verification failed: " + run.stderr) + require({name: sha(package / name) for name in RUNTIME_FILES} == originals, + "Read-only --check changed the runtime files") + checks.append({"name": "relocated_run_from_different_cwd", "passed": True, + "command": "python -B ../relocated-package/verify.py --check", + "stdout": run.stdout.splitlines()}) + # Ensure optimized Python does not disable assertions. + optimized = subprocess.run([sys.executable, "-O", "-B", str(package / "verify.py"), "--check"], + cwd=other_cwd, capture_output=True, text=True, check=False) + require(optimized.returncode == 0, "Optimized interpreter verification failed") + checks.append({"name": "optimized_python_checks_remain_active", "passed": True}) + results = package / "results.json" + pristine_results = results.read_bytes() + results.write_bytes(pristine_results + b" ") + modified = subprocess.run(command, cwd=other_cwd, capture_output=True, text=True, check=False) + require(modified.returncode != 0 and "differ from bundled" in modified.stderr, + "Altered bundled results were not rejected") + results.write_bytes(pristine_results) + checks.append({"name": "altered_saved_results_rejected", "passed": True}) + verifier = package / "verify.py" + pristine_verifier = verifier.read_text(encoding="utf-8") + require("Q(11041, 810)" in pristine_verifier, "Printed triple mutation target missing") + verifier.write_text(pristine_verifier.replace("Q(11041, 810)", "Q(11042, 810)", 1), encoding="utf-8") + wrong_triple = subprocess.run([sys.executable, "-O", "-B", str(verifier), "--check"], + cwd=other_cwd, capture_output=True, text=True, check=False) + require(wrong_triple.returncode != 0 and "eq22.printed_CBE.8.0" in wrong_triple.stderr, + "Incorrect printed CBE triple was not rejected") + checks.append({"name": "incorrect_printed_triple_rejected_under_O", "passed": True}) + verifier.write_text(pristine_verifier, encoding="utf-8") + mutations = [ + ("incorrect_zero_locations_rejected", "{(9, 1), (10, 0)}", "{(8, 1), (10, 0)}", "Exact zero pattern failed: eq22.B.CBE"), + ("violated_coefficient_floor_rejected", "8, 16000, 53000000", "8, 20000, 53000000", "Matrix diagonal bound failed: eq21.A.OSC"), + ] + for name, old, new, expected_error in mutations: + require(old in pristine_verifier, "Mutation target missing: " + name) + verifier.write_text(pristine_verifier.replace(old, new, 1), encoding="utf-8") + rejected = subprocess.run([sys.executable, "-O", "-B", str(verifier), "--check"], + cwd=other_cwd, capture_output=True, text=True, check=False) + require(rejected.returncode != 0 and expected_error in rejected.stderr, + "Invalid certificate was not rejected: " + name) + checks.append({"name": name, "passed": True}) + verifier.write_text(pristine_verifier, encoding="utf-8") + x, y = ring("x", "y") + (u,) = ring("u") + rejection_cases = [ + ("univariate_degree_truncation_rejected", lambda: interval_coefficients(u ** 3, 0, 1, 2)), + ("triangle_degree_truncation_rejected", lambda: triangle_coefficients(x ** 3, [(0, 0), (1, 0), (0, 1)], 2)), + ("inexact_polynomial_division_rejected", lambda: (u ** 2 + 1).divide_exact(u)), + ("floating_point_coefficient_rejected", lambda: Poly.constant(("u",), 0.1)), + ] + for name, operation in rejection_cases: + try: + operation() + except (ArithmeticError, TypeError): + checks.append({"name": name, "passed": True}) + else: + raise ArithmeticError("Rejection check unexpectedly succeeded: " + name) + require({name: sha(ROOT / name) for name in RUNTIME_FILES} == originals, + "Portability test changed the original runtime files") + checks.append({"name": "original_runtime_files_unchanged", "passed": True}) + output = {"author": "OpenAI", "python_version": platform.python_version(), + "third_party_dependencies": [], "network_used": False, + "temporary_files_outside_package": False, "runtime_sha256": originals, + "checks": checks, "all_passed": True} + (ROOT / "portability-results.json").write_text(json.dumps(output, indent=2, sort_keys=True) + "\n", encoding="utf-8") + print("PASS: relocation, read-only verification, and all rejection checks.") + print(json.dumps({"checks": len(checks), "all_passed": True}, sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/results.json b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/results.json new file mode 100644 index 0000000000000000000000000000000000000000..4070f5457afdbfa2372dafd7fb40481660c21558 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/results.json @@ -0,0 +1,20672 @@ +{ + "arithmetic": "Python standard library fractions.Fraction; no floating-point decisions", + "author": "OpenAI", + "exact_values": { + "engine.normalizations": { + "passed": true, + "passed_examples": 3 + }, + "eq19.printed_minima.C0": { + "exact_minima": [ + "3971/3200", + "1063/1600" + ], + "passed": true, + "printed_minima": [ + "3971/3200", + "1063/1600" + ] + }, + "eq19.printed_minima.determinant": { + "exact_minima": [ + "101027/20000", + "283729/400000" + ], + "passed": true, + "printed_minima": [ + "101027/20000", + "283729/400000" + ] + }, + "eq22.printed_CBE.8.0": { + "index": [ + 8, + 0 + ], + "passed": true, + "triple_11_22_12": [ + "11041/810", + "4063/810", + "60557/16200" + ] + }, + "eq22.printed_CBE.8.1": { + "index": [ + 8, + 1 + ], + "passed": true, + "triple_11_22_12": [ + "4328/405", + "839/405", + "3977/2700" + ] + }, + "eq22.printed_CBE.8.2": { + "index": [ + 8, + 2 + ], + "passed": true, + "triple_11_22_12": [ + "191/81", + "191/81", + "-1189/1350" + ] + }, + "eq9.endpoint_range_and_radical": { + "comparison_square": "289/9", + "endpoint_upper_square": "1369/10000", + "passed": true, + "required_square": "5/37" + }, + "eq9.moment_lower_bound": { + "passed": true, + "q_at_2": "17/225", + "q_prime_at_2": "-2494/225", + "q_second": "62/45" + }, + "eq9.printed_minimum.D_star": { + "exact_minimum": "2459668689/5793382400", + "passed": true, + "printed_minimum": "2459668689/5793382400" + }, + "eq9.printed_minimum.L": { + "exact_minimum": "1/64", + "passed": true, + "printed_minimum": "1/64" + }, + "eq9.printed_minimum.endpoint_lower": { + "exact_minimum": "58454914732902747377800095783827783501884053/50000000000000000000000000000000000000000000", + "passed": true, + "printed_minimum": "58454914732902747377800095783827783501884053/50000000000000000000000000000000000000000000" + }, + "eq9.printed_minimum.endpoint_upper": { + "exact_minimum": "132428032291164570287168821949356607977047/770000000000000000000000000000000000000", + "passed": true, + "printed_minimum": "132428032291164570287168821949356607977047/770000000000000000000000000000000000000" + }, + "eq9.printed_strictly_feasible_measure": { + "first_constraint": "-5/2", + "mass_at_2": "9/32", + "passed": true, + "second_constraint": "-1/8" + }, + "volume.cutoff_comparisons": { + "beta_squared_3_minus_beta": "57500/59319", + "f_at_cutoff": "7511875/8388608", + "passed": true + }, + "volume.lower": { + "asserted_bound": "1343/1000", + "exact_value": "325513803292178267556854357335457/242308366210310400000000000000000", + "passed": true, + "required_bound": "4/3", + "sine_power": 1 + }, + "volume.printed_lower_constant": { + "exact_constant": "4883/3750", + "passed": true + }, + "volume.printed_piece.lower_1": { + "degree": 15, + "exact_integral_with_prefactor": "131507100516520529090488603/188273788819200000000000000000", + "integrand": { + "degree": 15, + "terms": [ + { + "coefficient": "157/100", + "powers": [ + 7 + ] + }, + { + "coefficient": "-79249/51450", + "powers": [ + 9 + ] + }, + { + "coefficient": "-34944726658847/77175000000000", + "powers": [ + 11 + ] + }, + { + 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+ "passed": true, + "upper_floor_sum": "40781/500000", + "upper_printed_bound": "163/2000" + }, + "volume.printed_sine_positive_floor": { + "exact_floor": "19063149197/10376641800", + "passed": true, + "strict_lower_bound": "9/5" + }, + "volume.upper_saved": { + "asserted_bound": "163/2000", + "exact_value": "35568414570146708970215164653657279684840568049287583618357776852209994707811168145719343/436079249052527988124474363762562231471002196310491136000000000000000000000000000000000000", + "passed": true, + "required_bound": "3/40", + "sine_power": 3 + } + }, + "format_version": 1, + "identities": [ + { + "key": "engine.binomial_product", + "passed": true + }, + { + "key": "engine.derivative", + "passed": true + }, + { + "key": "engine.exact_division", + "passed": true + }, + { + "key": "eq9.H_third_derivative", + "passed": true + }, + { + "key": "eq9.Phi_third_derivative", + "passed": true + }, + { + "key": "eq9.H_derivative_ratio", + "passed": true + }, + { + "key": "eq9.R_lower_remainder", + "passed": true + }, + { + "key": "eq9.printed_moment_quadratic", + "passed": true + }, + { + "key": "eq9.weight_interval_nonempty", + "passed": true + }, + { + "key": "eq9.moment_ratio_clearing", + "passed": true + }, + { + "key": "eq9.printed_L_input", + "passed": true + }, + { + "key": "eq9.printed_V_input", + "passed": true + }, + { + "key": "eq9.endpoint_clearing.lower", + "passed": true + }, + { + "key": "eq9.endpoint_clearing.upper", + "passed": true + }, + { + "key": "eq20.one_minus_sigma", + "passed": true + }, + { + "key": "eq20.one_plus_sigma", + "passed": true + }, + { + "key": "eq20.enlarged_difference_cleared", + "passed": true + }, + { + "key": "eq20.quartic_coefficient", + "passed": true + }, + { + "key": "eq12.inverse_metric_square_scaled.00", + "passed": true + }, + { + "key": "eq12.full_matrix_scaled.00", + "passed": true + }, + { + "key": "eq12.inverse_metric_square_scaled.01", + "passed": true + }, + { + "key": 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"coefficient": "138635/256", + "powers": [ + 7, + 0 + ] + }, + { + "coefficient": "184801/2560", + "powers": [ + 7, + 1 + ] + }, + { + "coefficient": "-354025/4096", + "powers": [ + 8, + 0 + ] + } + ], + "variables": [ + "v", + "b" + ] + } + ], + "vertices": [ + "S", + "C", + "U" + ], + "zero_indices": [ + [ + 0, + 6 + ], + [ + 0, + 7 + ], + [ + 0, + 8 + ], + [ + 0, + 9 + ], + [ + 1, + 7 + ], + [ + 1, + 8 + ], + [ + 2, + 7 + ] + ] + } + ] +} diff --git a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/verify.py b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/verify.py new file mode 100644 index 0000000000000000000000000000000000000000..09c85987349b6526df18f27951c0d4259d568c74 --- /dev/null +++ b/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/verify.py @@ -0,0 +1,503 @@ +#!/usr/bin/env python3 +"""Reproduce the manuscript's finite exact arithmetic certificates offline. + +Author: OpenAI. Run: python -B verify.py --check +All arithmetic decisions use fractions.Fraction. This is not a proof assistant. +""" +import sys +sys.dont_write_bytecode = True +import argparse +from fractions import Fraction as Q +import json +from math import comb +from pathlib import Path + +from polynomial import Poly, interval_coefficients, rational, require, ring, triangle_coefficients + +ROOT = Path(__file__).resolve().parent +RESULTS_FILE = ROOT / "results.json" +RESULTS = { + "format_version": 1, + "author": "OpenAI", + "scope": "Finite polynomial identities, Bernstein coefficient certificates, and rational volume margins", + "arithmetic": "Python standard library fractions.Fraction; no floating-point decisions", + "identities": [], + "interval_certificates": [], + "triangle_certificates": [], + "exact_values": {}, +} + + +def identity(key, difference): + require(difference == 0, "Identity failed: " + key) + RESULTS["identities"].append({"key": key, "passed": True}) + + +def scalar_claim(key, condition, detail): + require(condition, "Exact assertion failed: " + key) + RESULTS["exact_values"][key] = {"passed": True, **detail} + + +def interval(key, poly, intervals, floor, degree=None, strict=False, supplementary=False): + degree = poly.degree() if degree is None else degree + floor = rational(floor) + record = {"key": key, "polynomial": poly.data(), "bernstein_degree": degree, + "coefficient_floor": str(floor), "strict": strict, + "supplementary": supplementary, "intervals": []} + for left, right in intervals: + coeffs = interval_coefficients(poly, left, right, degree) + require(all(c > floor if strict else c >= floor for c in coeffs), + "Coefficient bound failed: " + key) + minimum = min(coeffs) + record["intervals"].append({ + "left": str(rational(left)), "right": str(rational(right)), + "coefficients": [str(c) for c in coeffs], "minimum": str(minimum), + "minimizing_indices": [i for i, c in enumerate(coeffs) if c == minimum], + "zero_indices": [i for i, c in enumerate(coeffs) if c == 0], "passed": True}) + record["passed"] = True + RESULTS["interval_certificates"].append(record) + return record + + +VERTICES = { + "O": (0, 0), "S": (1, 0), "C": (1, 1), "U": (2, 0), "W0": (0, 2), + "A": (Q(1, 2), Q(1, 2)), "B": (Q(3, 2), Q(1, 2)), "E": (1, Q(1, 2)), +} + + +def triangle(key, entries, names, degree, floor, determinant_floor=None, zeros=()): + vertices = [VERTICES[name] for name in names] + coefficients = [triangle_coefficients(p, vertices, degree) for p in entries] + expected_zeros = set(zeros) + actual_zeros = {ij for ij in coefficients[0] + if all(row[ij] == 0 for row in coefficients)} + require(actual_zeros == expected_zeros, "Exact zero pattern failed: " + key) + floor = rational(floor) + determinant_floor = None if determinant_floor is None else rational(determinant_floor) + output = [] + diagonal_minima, determinants = [], [] + for ij in coefficients[0]: + values = [row[ij] for row in coefficients] + record = {"index": [ij[0], ij[1], degree - sum(ij)], + "entries": [str(value) for value in values], + "zero": ij in actual_zeros} + if ij not in actual_zeros: + if len(values) == 1: + require(values[0] >= floor, "Scalar triangle bound failed: " + key) + diagonal_minima.append(values[0]) + else: + r, h, off = values + determinant = r * h - off * off + require(min(r, h) >= floor, "Matrix diagonal bound failed: " + key) + require(determinant >= determinant_floor, "Matrix determinant bound failed: " + key) + diagonal_minima.extend([r, h]) + determinants.append(determinant) + record["determinant"] = str(determinant) + output.append(record) + RESULTS["triangle_certificates"].append({ + "key": key, "vertices": list(names), + "coordinates": [[str(rational(x)), str(rational(y))] for x, y in vertices], + "bernstein_degree": degree, "polynomials": [p.data() for p in entries], + "entry_order": ["scalar"] if len(entries) == 1 else ["11", "22", "12"], + "coefficient_floor": str(floor), + "determinant_floor": None if determinant_floor is None else str(determinant_floor), + "zero_indices": [list(ij) for ij in sorted(actual_zeros)], + "minimum_diagonal_or_scalar": str(min(diagonal_minima)), + "minimum_determinant": str(min(determinants)) if determinants else None, + "coefficients": output, "passed": True}) + return coefficients + + +def swap(poly): + values = list(ring(*poly.variables)) + vi, bi = poly.variables.index("v"), poly.variables.index("b") + values[vi], values[bi] = values[bi], values[vi] + return poly.substitute(values) + + +def definitions(v, b): + D, h = Q(158), Q(17, 10) + a, s, delta = v - 1, b - 1, 2 - v - b + n, t = delta / 2, (v - b) / 2 + z = [Q(k, 4) for k in (-43, -45, -63, 21, 183, 208, 61, 46, 101, 79, -41)] + kappa = (-D * (a - s) / 6 + z[0] * a ** 2 + (60 - 2 * D / 9 + 2 * z[0]) * a * s + + z[1] * s ** 2 + sum(z[2 + i] * a ** i * s ** (3 - i) for i in range(4)) + + sum(z[6 + i] * a ** i * s ** (4 - i) for i in range(5))) + kappab = swap(kappa) + M = 1 - Q(3, 8) * (v - b) + m, c, K = M ** 2, Q(2, 5) - Q(3, 10) * v * b, 4 + v ** 2 + b ** 2 + theta = (n ** 2 * (1 - n) * (105 + 5 * n) + + t ** 2 * (4 + 360 * n - 586 * n ** 2) - 4 * t ** 4) / 4 + Pk, Pl = 300 * m + 160 * h * delta, 35 * m * v + P0 = Pk + 30 * h * v + Xv = m * (-70 + 100 * v - 25 * v ** 2 - 5 * b ** 2) + Q(5, 3) * h * (6 * (a ** 2 + s ** 2) + 32 * a * delta) + rv, rb = 1 + 6 * v + v ** 2, 1 + 6 * b + b ** 2 + alpha = Q(5, 4) + Sstar = 4 * (alpha + 1 / alpha) * v * b + (alpha * b * (1 - v) ** 2 + v * (1 - b) ** 2 / alpha) / 2 + KJ = K * (kappa.derivative("b") + kappab.derivative("v")) / 2 - D + Hv = 8 * v * K * theta * (1 + v) - rv * (v * D + K * (v * (Xv + kappa.derivative("v")) + 2 * kappa / 3)) + E0 = D * (1 - delta ** 2 / 4) * (v - b) ** 2 + 30 * h * delta * (v ** 2 * a ** 2 + b ** 2 * s ** 2) + 6 * (a * v * kappa + s * b * kappab) + F, Lv, Lb = c * E0 - theta ** 2, 10 * h * delta * v - kappa, 10 * h * delta * b - kappab + return locals() + + +def matrix_A(v, b, sigma): + f = definitions(v, b) + m, h, d, kap = f["m"], f["h"], f["delta"], f["kappa"] + P0, Pk, Pl = f["P0"], f["Pk"], f["Pl"] + lam = 6 * (sigma * v ** 2 + b ** 2 - v - b) - 26 * d + K2 = m * (42 + 3 * b ** 2 + 10 * v ** 2) - h * lam + K1 = 5 * K2 + m * (300 * v * sigma + 140 * v ** 2) + 160 * h * d * v * sigma + K12 = m * (75 * v + 35 * v ** 2 * sigma) + 40 * h * d * v + return (P0 * (v * K1 - 5 * kap) - (1 + sigma) * v ** 2 * Pk ** 2, + P0 * (v * K2 - kap) - (1 + sigma) * v ** 2 * Pl ** 2, + P0 * v * K12 - (1 + sigma) * v ** 2 * Pk * Pl) + + +def check_engine(): + x, y = ring("x", "y") + identity("engine.binomial_product", (x + y) ** 2 - x ** 2 - 2 * x * y - y ** 2) + identity("engine.derivative", ((x + y) ** 3).derivative("x") - 3 * (x + y) ** 2) + identity("engine.exact_division", ((x ** 3 - y ** 3).divide_exact(x - y)) - (x ** 2 + x * y + y ** 2)) + (u,) = ring("u") + require(interval_coefficients(u ** 2, 0, 1, 2) == [Q(0), Q(0), Q(1)], "Engine univariate normalization") + tri = triangle_coefficients(x * y, [(1, 0), (0, 1), (0, 0)], 2) + require(tri[1, 1] == Q(1, 2) and sum(tri.values()) == Q(1, 2), "Engine triangle normalization") + require((u ** 2).integral(0, 1) == Q(1, 3), "Engine integration") + scalar_claim("engine.normalizations", True, {"passed_examples": 3}) + + +def check_scalar_certificates(): + (b,) = ring("b") + Rb = 1 + b ** 2 / 40 - b ** 4 / 1152 + Hterms = [(Q(1, 2), Q(1)), (Q(5, 2), Q(1, 40)), (Q(9, 2), -Q(1, 1152))] + Hthird_scaled = sum(co * ex * (ex - 1) * (ex - 2) * b ** int(ex - Q(1, 2)) for ex, co in Hterms) + Hratio = sum(2 * ex * co * b ** int(ex - Q(1, 2)) for ex, co in Hterms) + L9 = Q(3, 8) + Q(3, 64) * b ** 2 - Q(35, 1024) * b ** 4 + V9 = Q(24, 5) - Q(16, 15) * b ** 2 - Q(7, 48) * b ** 4 + Q(7, 1152) * b ** 6 + Phi = 16 * b * Rb ** 2 + 9 * b * (1 - b) + identity("eq9.H_third_derivative", Hthird_scaled - L9) + identity("eq9.Phi_third_derivative", Phi.derivative("b", 3) - V9) + identity("eq9.H_derivative_ratio", Hratio - (1 + b ** 2 / 8 - b ** 4 / 128)) + identity("eq9.R_lower_remainder", Rb - (1 + Q(31, 1440) * b ** 2) - b ** 2 * (4 - b ** 2) / 1152) + q = 16 * (1 + Q(62, 1440) * b ** 2) + 9 - (9 + 16 * Q(11, 20) ** 2) * b + identity("eq9.printed_moment_quadratic", q - (25 + Q(31, 45) * b ** 2 - Q(346, 25) * b)) + scalar_claim("eq9.moment_lower_bound", q.derivative("b", 2).evaluate([0]) > 0 and q.derivative("b").evaluate([2]) < 0 and q.evaluate([2]) == Q(17, 225), + {"q_at_2": str(q.evaluate([2])), "q_prime_at_2": str(q.derivative("b").evaluate([2])), "q_second": str(q.derivative("b", 2).evaluate([0]))}) + mass = Q(9, 32) + scalar_claim("eq9.printed_strictly_feasible_measure", 80 * mass - 25 == -Q(5, 2) + and 1 - 4 * mass == -Q(1, 8), + {"mass_at_2": str(mass), "first_constraint": "-5/2", "second_constraint": "-1/8"}) + identity("eq9.weight_interval_nonempty", (25 - 28 * b - 6 * b ** 2) * (4 - b ** 2) + - (1 - b ** 2) * (80 - 28 * b - 6 * b ** 2) - (2 - b) * (10 - 37 * b)) + X, P = ring("X", "P") + identity("eq9.moment_ratio_clearing", 15 * (Q(2, 3) * (X - P) - Q(1, 5) * (1 + X + P)) + - (7 * X - 13 * P - 3)) + (x,) = ring("x") + Lx, Vx = L9.substitute([2 * x ** 2]), V9.substitute([2 * x ** 2]) + identity("eq9.printed_L_input", Lx - (Q(3, 8) + Q(3, 16) * x ** 4 - Q(35, 64) * x ** 8)) + identity("eq9.printed_V_input", Vx - (Q(24, 5) - Q(64, 15) * x ** 4 - Q(7, 3) * x ** 8 + Q(7, 18) * x ** 12)) + quarters = [(Q(i, 4), Q(i + 1, 4)) for i in range(4)] + L_record = interval("eq9.L", Lx, [(0, 1)], Q(1, 100), 8) + D_record = interval("eq9.third_derivative_sign", Q(88, 5) * Lx - Q(17, 3) * x ** 5 * Vx, quarters, Q(2, 5), 17) + for name, record, expected in [("L", L_record, Q(1, 64)), + ("D_star", D_record, Q(2459668689, 5793382400))]: + least = min(Q(part["minimum"]) for part in record["intervals"]) + scalar_claim("eq9.printed_minimum." + name, least == expected, + {"exact_minimum": str(least), "printed_minimum": str(expected)}) + R = 1 + x ** 4 / 10 - x ** 8 / 72 + w = 2 * x ** 2 + for name, A, B in [("lower", 1 - w ** 2, 4 - w ** 2), + ("upper", 25 - 28 * w - 6 * w ** 2, 80 - 28 * w - 6 * w ** 2)]: + gap = R.evaluate([1]) - x * R + p = 32 * A * (B - A) * gap ** 2 - 9 * ((B - A) * (w ** 2 - w) + 2 * A) * B + # This second form is the numerator obtained by multiplying the rational expression by B^2. + numerator = 32 * (A * B - A ** 2) * gap ** 2 - 9 * ((B ** 2 - A * B) * (w ** 2 - w) + 2 * A * B) + identity("eq9.endpoint_clearing." + name, numerator - p) + endpoint_record = interval("eq9.endpoint." + name, p, [(0, Q(37, 100))], 1, 22) + expected = { + "lower": Q(58454914732902747377800095783827783501884053, + 50000000000000000000000000000000000000000000), + "upper": Q(132428032291164570287168821949356607977047, + 770000000000000000000000000000000000000), + }[name] + least = Q(endpoint_record["intervals"][0]["minimum"]) + scalar_claim("eq9.printed_minimum.endpoint_" + name, least == expected, + {"exact_minimum": str(least), "printed_minimum": str(expected)}) + interval("eq9.endpoint_denominator." + name, B, [(0, Q(37, 100))], 0, strict=True, supplementary=True) + scalar_claim("eq9.endpoint_range_and_radical", Q(5, 37) < Q(37, 100) ** 2 and Q(17, 3) ** 2 > 32, + {"endpoint_upper_square": str(Q(37, 100) ** 2), "required_square": str(Q(5, 37)), "comparison_square": str(Q(17, 3) ** 2)}) + (v,) = ring("v") + h = Q(17, 10) + A0 = Q(27, 320) * v * (Q(15, 2) + Q(2, 3) * v) + L0 = Q(9, 20) * v * (Q(7, 4) - Q(3, 4) * v) + C0 = Q(7, 5) * h - A0 - L0 + det = 4 * C0 * ((h - A0) * (2 - v) + Q(2, 5) * h * v) - (2 - v) * (2 * A0 + L0) ** 2 + for name, poly, degree, expected in [ + ("C0", C0, 2, [Q(3971, 3200), Q(1063, 1600)]), + ("determinant", det, 5, [Q(101027, 20000), Q(283729, 400000)])]: + record = interval("eq19.defect_" + name, poly, [(0, 1), (1, 2)], Q(1, 2), degree, strict=True) + minima = [Q(part["minimum"]) for part in record["intervals"]] + scalar_claim("eq19.printed_minima." + name, minima == expected, + {"exact_minima": [str(c) for c in minima], "printed_minima": [str(c) for c in expected]}) + (r,) = ring("r") + sigma = (3 * r - r ** 3) / 2 + identity("eq20.one_minus_sigma", 1 - sigma - (r - 1) ** 2 * (r + 2) / 2) + identity("eq20.one_plus_sigma", 1 + sigma - (2 - r) * (r + 1) ** 2 / 2) + auxiliary = 4 * r * (r + 3) - (r - 1) * (r + 2) * (r + 1) ** 2 + interval("eq20.g_majorant_auxiliary", auxiliary, [(1, 2)], 4, 4, supplementary=True) + (u,) = ring("u") + omega, mu = Q(27, 128), Q(9, 32) + A = Q(67, 50) * (1 + u ** 2) + G = Q(27, 100) * (1 + u ** 2) * u ** 2 + 3 * omega * (1 + u) ** 2 + mu * (1 + u) * (1 - u ** 3) + H = Q(27, 100) * (1 + u ** 2) + omega * (17 + 19 * u ** 2) * (1 + u) ** 2 / 12 - mu * (1 + u) * (1 - u ** 3) + L = omega * (1 + u) ** 2 * (Q(19, 3) + Q(7, 2) * u ** 2) + Q(3, 4) * (1 + u) * (1 + u ** 3) + Z = L ** 2 - 4 * A * (G + u ** 2 * H) + interval("eq20.G", G, [(0, 1)], Q(9, 10), 4) + interval("eq20.H", H, [(0, 1)], Q(1, 4), 4) + interval("eq20.minus_Z", -Z, [(0, Q(1, 16))], Q(13, 100), 8) + interval("eq20.radical", 64 * A ** 2 * u ** 2 * G * H - Z ** 2, [(Q(1, 16), 1)], Q(3, 25), 16) + v, b, ur = ring("v", "b", "u") + lift = lambda poly: poly.substitute([ur]) + gv, gb = Q(19, 3) - 3 * v - Q(17, 12) * b, Q(7, 2) - Q(19, 12) * b + M, Mb = 1 - Q(3, 8) * (v - b), 1 - Q(3, 8) * (b - v) + direct_num = ((Q(67, 50) + Q(27, 100) * v * b) * (1 + ur ** 2) * (b + v * ur ** 2) + - v * b * (omega * (1 + ur) ** 2 * (gv + gb * ur ** 2) + Q(3, 4) * (1 + ur) * (M + Mb * ur ** 3))) + claimed_num = lift(A) * b + lift(A) * ur ** 2 * v + v * b * (lift(G) * v + lift(H) * b - lift(L)) + identity("eq20.enlarged_difference_cleared", direct_num - claimed_num) + identity("eq20.quartic_coefficient", Q(10, 9) * (h - Q(67, 50) - Q(27, 100) * v * b) - (Q(2, 5) - Q(3, 10) * v * b)) + + +def check_matrix_identities(): + # Congruence by diag(sqrt(3),1,1) removes all irrational entries. + # This is the coordinate change p=sqrt(3) P; its norm matrix is diag(5,5,1). + v, b, sigma, rho = ring("v", "b", "sigma", "rho") + zero = 0 * v + B = [[4 * v * sigma, -4 * v * rho, zero], + [-4 * v * rho, -4 * v * sigma, -v], [zero, -v, zero]] + CN = [[4 * v ** 2, zero, -rho * v ** 2], + [zero, -4 * v ** 2, -sigma * v ** 2], [-rho * v ** 2, -sigma * v ** 2, zero]] + metric, inverse = [Q(5), Q(5), Q(1)], [Q(1, 5), Q(1, 5), Q(1)] + EB = [[sum(B[i][k] * inverse[k] * B[k][j] for k in range(3)) for j in range(3)] for i in range(3)] + EB_expected = [[Q(16, 5) * v ** 2, zero, Q(4, 5) * rho * v ** 2], + [zero, Q(21, 5) * v ** 2, Q(4, 5) * sigma * v ** 2], + [Q(4, 5) * rho * v ** 2, Q(4, 5) * sigma * v ** 2, v ** 2 / 5]] + G_expected = [ + [-210 - 15 * b ** 2 + 300 * sigma * v - 75 * v ** 2, -300 * rho * v, -35 * rho * v ** 2], + [-300 * rho * v, -210 - 15 * b ** 2 - 300 * sigma * v - 190 * v ** 2, -75 * v - 35 * sigma * v ** 2], + [-35 * rho * v ** 2, -75 * v - 35 * sigma * v ** 2, -42 - 3 * b ** 2 - 10 * v ** 2], + ] + for i in range(3): + for j in range(i, 3): + identity(f"eq12.inverse_metric_square_scaled.{i}{j}", (EB[i][j] - EB_expected[i][j]).reduce_circle()) + G = 75 * B[i][j] + Q(25, 3) * CN[i][j] - Q(100, 3) * EB[i][j] + if i == j: + G += (-42 - 3 * b ** 2 - Q(10, 3) * v ** 2) * metric[i] + if i == j == 0: + G += 15 * v ** 2 + identity(f"eq12.full_matrix_scaled.{i}{j}", (G - G_expected[i][j]).reduce_circle()) + f = definitions(v, b) + h, m, d, P0, Pk, Pl = [f[key] for key in ("h", "m", "delta", "P0", "Pk", "Pl")] + X = m * G_expected[0][0] / 3 + Q(5, 3) * h * (6 * (sigma * v ** 2 + b ** 2 - v - b) - 26 * d) + 40 * h * d * B[0][0] / 3 + Xaxial = X.substitute([v, b, 1, rho]) + identity("eq21.axial_Xv", Xaxial - f["Xv"]) + identity("eq21.pp_loss", Xaxial - X - (1 - sigma) * v * P0 / 3) + identity("eq21.pk_cross_scaled", m * G_expected[0][1] + 40 * h * d * B[0][1] + v * rho * Pk) + identity("eq21.pl_cross_scaled", m * G_expected[0][2] + 40 * h * d * B[0][2] + v * rho * Pl) + identity("eq21.P0_relation", P0 - Pk - 30 * h * v) + A_sigma = matrix_A(v, b, sigma) + target = (v ** 2 * (30 * h * v) ** 2, + v ** 2 * (Pl ** 2 + 6 * h * v * P0), + -v ** 2 * 30 * h * v * Pl) + for index, (entry, expected) in enumerate(zip(A_sigma, target)): + identity(f"eq21.sigma_monotonicity.{index}", -entry.derivative("sigma") - expected) + v, b = ring("v", "b") + f = definitions(v, b) + K, rv, rb, theta, kap, Xv, D = [f[key] for key in ("K", "rv", "rb", "theta", "kappa", "Xv", "D")] + cleared = 8 * v * K * theta * (1 + v) - v * K * rv * Xv - v * D * rv - v * K * rv * kap.derivative("v") - Q(2, 3) * K * rv * kap + identity("eq22.diagonal_clearing.v", cleared - f["Hv"]) + identity("eq22.diagonal_clearing.b", swap(cleared) - swap(f["Hv"])) + alpha, star = f["alpha"], f["Sstar"] + identity("eq22.Sstar_AMGM_pair", 2 * star - alpha * b * rv - v * rb / alpha) + identity("eq22.Sstar_square_majorant", star ** 2 - v * b * rv * rb - (alpha * b * rv - v * rb / alpha) ** 2 / 4) + v, b, sigma, sigmab = ring("v", "b", "sigma", "sigmab") + f = definitions(v, b) + d, h, kap, kappab = f["delta"], f["h"], f["kappa"], f["kappab"] + potential = (-30 * h * d * (v ** 2 * (1 - 2 * sigma * v + v ** 2) + b ** 2 * (1 - 2 * sigmab * b + b ** 2)) + - f["D"] * (1 - d ** 2 / 4) * (v - b) ** 2 + + 6 * kap * v * (1 - sigma * v) + 6 * kappab * b * (1 - sigmab * b)) + identity("eq23.potential_decomposition", potential + f["E0"] + 6 * v ** 2 * (1 - sigma) * f["Lv"] + 6 * b ** 2 * (1 - sigmab) * f["Lb"]) + (z,) = ring("z") + identity("final.R_strictness", 1 + 6 * z ** 2 + z ** 4 - 4 * z * (1 + z ** 2) - (z - 1) ** 4) + + +def check_triangle_certificates(): + v, b = ring("v", "b") + f = definitions(v, b) + A = matrix_A(v, b, 1) + B = (f["Hv"], swap(f["Hv"]), f["Sstar"] * f["KJ"]) + Fplus = f["F"] + 12 * f["c"] * f["delta"] ** 2 * f["Lb"] + rows = [ + ("eq21.A.OSC", A, ("O", "S", "C"), 8, 16000, 53000000, set()), + ("eq21.A.OCW0", A, ("O", "C", "W0"), 8, 890, 3900000, set()), + ("eq21.A.SCU", A, ("S", "C", "U"), 8, 2400, 3500000, set()), + ("eq22.B.OAS", B, ("O", "A", "S"), 9, 43, 1050, set()), + ("eq22.B.ASE", B, ("A", "S", "E"), 9, 43, 490, set()), + ("eq22.B.ACE", B, ("A", "C", "E"), 10, 2, 21, {(0, 10)}), + ("eq22.B.SBE", B, ("S", "B", "E"), 9, 43, 1300, set()), + ("eq22.B.CBE", B, ("C", "B", "E"), 10, Q(13, 100), 4, {(9, 1), (10, 0)}), + ("eq22.B.SUB", B, ("S", "U", "B"), 9, 5, 290, set()), + ("eq23.Lv.OCU", (f["Lv"],), ("O", "C", "U"), 6, Q(9, 10), None, {(0, 6)}), + ("eq23.Fplus.OSC", (Fplus,), ("O", "S", "C"), 9, Q(19, 100), None, + {(i, j) for i in range(3) for j in range(3 - i)}), + ("eq23.Fplus.SCU", (Fplus,), ("S", "C", "U"), 9, Q(9, 1000), None, + {(i, j) for i in range(10) for j in range(10 - i) if j >= 7} | {(0, 6)}), + ] + cbe = None + for key, entries, vertices, degree, floor, det_floor, zeros in rows: + coefficients = triangle(key, entries, vertices, degree, floor, det_floor, zeros) + if key == "eq22.B.CBE": + cbe = coefficients + triples = { + (8, 0): (Q(11041, 810), Q(4063, 810), Q(60557, 16200)), + (8, 1): (Q(4328, 405), Q(839, 405), Q(3977, 2700)), + (8, 2): (Q(191, 81), Q(191, 81), -Q(1189, 1350)), + } + for index, expected in triples.items(): + actual = tuple(row[index] for row in cbe) + scalar_claim("eq22.printed_CBE." + ".".join(map(str, index)), actual == expected, + {"index": list(index), "triple_11_22_12": list(map(str, actual))}) + return f + + +def check_F_expansion(f): + n, t = ring("n", "t") + Fnt = f["F"].substitute([1 - n + t, 1 - n - t]) + (nn,) = ring("n") + fj = [ + nn ** 3 * (1 - nn) * (53511 * nn ** 4 - 151593 * nn ** 3 + 222537 * nn ** 2 - 83607 * nn + 10256) / 240, + nn * (19389 * nn ** 5 - 418926 * nn ** 4 + 523719 * nn ** 3 - 289776 * nn ** 2 + 67782 * nn + 10564) / 60, + (-1260129 * nn ** 4 + 1370598 * nn ** 3 - 451164 * nn ** 2 + 23442 * nn + 74) / 60, + (-12200 * nn ** 2 + 2601 * nn + 75) / 10, + 0 * nn - Q(23, 5), + ] + identity("eq23.F_full_even_expansion", Fnt - sum(p.substitute([n]) * t ** (2 * j) for j, p in enumerate(fj))) + # Full equality rules out ALL unlisted powers, including odd powers and powers above eight. + require(Fnt.degree("t") == 8, "Unexpected F degree in t") + divisors = [nn ** 3 * (1 - nn), nn * (1 - nn), 1 - nn, 1 - nn, 1 - nn] + degrees, floors = [4, 6, 7, 7, 7], [3, 31, Q(1, 5), Q(12, 5), Q(9, 5)] + quarters = [(Q(i, 4), Q(i + 1, 4)) for i in range(4)] + for i in range(5): + numerator = sum(Q(comb(i, j), comb(4, j)) * fj[j] * (1 - nn) ** (2 * j) for j in range(i + 1)) + quotient = numerator.divide_exact(divisors[i]) + identity(f"eq23.F_factor_divisibility.{i}", quotient * divisors[i] - numerator) + interval(f"eq23.F_quarters.{i}", quotient, quarters, floors[i], degrees[i]) + + +def check_volume_and_supplementary_bounds(): + (z,) = ring("z") + a0, b0 = Q(157, 50), Q(22, 7) + sine = a0 * z - (b0 * z) ** 3 / 6 + (a0 * z) ** 5 / 120 - (b0 * z) ** 7 / 5040 + identity("volume.printed_sine_polynomial", sine - (Q(157, 50) * z - Q(5324, 1029) * z ** 3 + + Q(95388992557, 37500000000) * z ** 5 - Q(155897368, 259416045) * z ** 7)) + elementary_sine_floor = a0 - b0 ** 3 / 24 - b0 ** 7 / 322560 + scalar_claim("volume.printed_sine_positive_floor", elementary_sine_floor > Q(9, 5), + {"exact_floor": str(elementary_sine_floor), "strict_lower_bound": "9/5"}) + quotient = sine.divide_exact(z) + interval("volume.S_over_z_positive", quotient, [(0, Q(1, 2))], 0, 6, strict=True, supplementary=True) + (u,) = ring("u") + f = u ** 4 * (3 + 5 * u ** 2 + 6 * u ** 4) + factor = 6 - 13 * u ** 2 + 6 * u ** 4 + identity("volume.f_comparison_numerator", u ** 4 * (3 - 4 * u ** 2) - f * (1 - u ** 2) ** 3 - u ** 10 * factor) + interval("volume.f_comparison_factor", factor, [(0, Q(5, 8))], 0, 4, strict=True, supplementary=True) + beta = Q(25, 39) + scalar_claim("volume.cutoff_comparisons", f.evaluate([Q(5, 8)]) <= 1 + and beta ** 2 * (3 - beta) == Q(57500, 59319) > Q(15, 16), + {"f_at_cutoff": str(f.evaluate([Q(5, 8)])), "beta_squared_3_minus_beta": str(beta ** 2 * (3 - beta))}) + Sleft, Sright = sine.substitute([u]), sine.substitute([1 - u]) + lower_integrand = u ** 2 * (u ** 4 / 2 + u ** 6 / 3) + pieces = [ + ("upper_1", Q(9, 48) * a0, f * (1 - u ** 2 / 2) * Sleft ** 3, + 0, Q(1, 2), 31, Q(11869, 10 ** 6)), + ("upper_2", Q(9, 48) * a0, f * (1 - u ** 2 / 2) * Sright ** 3, + Q(1, 2), Q(5, 8), 31, Q(30175, 10 ** 6)), + ("upper_3", Q(9, 48) * a0, Q(15, 16) * (1 - u ** 2 / 2) * Sright ** 3, + Q(5, 8), 1, 23, Q(39518, 10 ** 6)), + ("lower_1", a0 ** 3 / 27, lower_integrand * Sleft, + 0, Q(1, 2), 15, Q(698, 10 ** 6)), + ("lower_2", a0 ** 3 / 27, lower_integrand * Sright, + Q(1, 2), 1, 15, Q(40554, 10 ** 6)), + ] + values = {} + for name, prefactor, poly, left, right, degree, floor in pieces: + value = prefactor * poly.integral(left, right) + scalar_claim("volume.printed_piece." + name, poly.degree() == degree and value > floor, + {"integrand": poly.data(), "prefactor": str(prefactor), + "left": str(rational(left)), "right": str(rational(right)), + "degree": degree, "exact_integral_with_prefactor": str(value), + "strict_lower_bound": str(floor)}) + values[name] = value + upper_saved = sum(values[name] for name in ("upper_1", "upper_2", "upper_3")) + lower_constant = Q(2, 9) * (a0 ** 2 - 4) + scalar_claim("volume.printed_lower_constant", lower_constant == Q(4883, 3750), + {"exact_constant": str(lower_constant)}) + lower = lower_constant + values["lower_1"] + values["lower_2"] + upper_floor_sum = sum(row[-1] for row in pieces[:3]) + lower_floor_sum = lower_constant + sum(row[-1] for row in pieces[3:]) + scalar_claim("volume.printed_piece_floor_sums", upper_floor_sum > Q(163, 2000) > Q(2, 25) + and lower_floor_sum > Q(1343, 1000) > Q(4, 3), + {"upper_floor_sum": str(upper_floor_sum), "upper_printed_bound": "163/2000", + "lower_floor_sum_with_constant": str(lower_floor_sum), "lower_printed_bound": "1343/1000"}) + expected_upper = Q(35568414570146708970215164653657279684840568049287583618357776852209994707811168145719343, + 436079249052527988124474363762562231471002196310491136000000000000000000000000000000000000) + expected_lower = Q(325513803292178267556854357335457, 242308366210310400000000000000000) + scalar_claim("volume.upper_saved", upper_saved == expected_upper and upper_saved > Q(163, 2000) > Q(2, 25) > Q(3, 40), + {"exact_value": str(upper_saved), "asserted_bound": "163/2000", "required_bound": "3/40", "sine_power": 3}) + scalar_claim("volume.lower", lower == expected_lower and lower > Q(1343, 1000) > Q(67, 50) > Q(4, 3), + {"exact_value": str(lower), "asserted_bound": "1343/1000", "required_bound": "4/3", "sine_power": 1}) + (b,) = ring("b") + ratio = 1 + b ** 2 / 8 - b ** 4 / 128 + identity("eq9.H_ratio_square_bound", 1 + b ** 2 / 4 - ratio ** 2 - b ** 6 * (32 - b ** 2) / 16384) + interval("eq9.H_ratio_positive", ratio, [(0, 2)], 0, 4, strict=True, supplementary=True) + + +def compute(): + check_engine() + check_scalar_certificates() + check_matrix_identities() + family = check_triangle_certificates() + check_F_expansion(family) + check_volume_and_supplementary_bounds() + RESULTS["summary"] = { + "identities": len(RESULTS["identities"]), + "interval_certificate_rows": len(RESULTS["interval_certificates"]), + "reported_interval_rows": sum(not r["supplementary"] for r in RESULTS["interval_certificates"]), + "triangle_certificate_rows": len(RESULTS["triangle_certificates"]), + "exact_value_assertions": len(RESULTS["exact_values"]), + "all_passed": True, + } + return RESULTS + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + modes = parser.add_mutually_exclusive_group() + modes.add_argument("--check", action="store_true", help="Recompute and compare to the bundled results (default)") + modes.add_argument("--write", action="store_true", help="Recompute and replace results.json beside this script") + args = parser.parse_args() + result = compute() + encoded = json.dumps(result, indent=2, sort_keys=True) + "\n" + if args.write: + temporary = ROOT / ".results.json.tmp" + temporary.write_text(encoded, encoding="utf-8") + temporary.replace(RESULTS_FILE) + else: + require(RESULTS_FILE.is_file(), "Bundled results.json is missing; run --write to create it") + require(RESULTS_FILE.read_text(encoding="utf-8") == encoded, + "Recomputed exact results differ from bundled results.json") + print("PASS: all exact identities, coefficient bounds, zero patterns, and rational volume margins.") + print(json.dumps(result["summary"], sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/README.md b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..56e794bf48de6eb127b29e908bdb4b56212b403b --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [Positivity of Serre's Intersection Multiplicity](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Positivity-of-Serres-Intersection-Multiplicity-September-23-2026, + author = {{OpenAI}}, + title = {{Positivity of Serre's Intersection Multiplicity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf}{OAI:Positivity-of-Serres-Intersection-Multiplicity-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/paper.tex b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..fd3959c9f33835c5f1aa1c235bdf5057ddc23d17 --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/paper.tex @@ -0,0 +1,53 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{enumitem} +\usepackage{microtype} +\usepackage{xcolor} +\usepackage{hyperref} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\hypersetup{colorlinks=true,linkcolor=black,citecolor=black,urlcolor=blue, + pdftitle={Positivity of Serre's Intersection Multiplicity},pdfauthor={OpenAI}} +\setlength{\emergencystretch}{2em} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\Perf}{\operatorname{Perf}} +\newcommand{\Spec}{\operatorname{Spec}} +\newcommand{\Tor}{\operatorname{Tor}} +\newcommand{\rank}{\operatorname{rank}} +\newcommand{\length}{\operatorname{length}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\m}{\mathfrak{m}} +\newcommand{\K}{K} +\newcommand{\cL}{\mathcal{L}} +\title{Positivity of Serre's Intersection Multiplicity} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove Serre's positivity conjecture, including ramified mixed +characteristic. If two nonzero finitely generated modules over a regular local +ring have tensor product of finite length and complementary dimensions, then +their intersection multiplicity is strictly positive. +\end{abstract} + +\input{sections/introduction} +\input{sections/lengths} +\input{sections/descent} +\input{sections/asymptotic} +\input{sections/positivity} +\input{references} +\end{document} diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/references.tex b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..b8f472b2e128cd7d5b3287a90c69d1b6032b0402 --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/references.tex @@ -0,0 +1,227 @@ +\begin{thebibliography}{99} +\raggedright + +\bibitem{BalmerSchlichting} +P.~Balmer and M.~Schlichting, +Idempotent completion of triangulated categories, +\emph{J. Algebra} \textbf{236} (2001), 819--834. +\href{https://doi.org/10.1006/jabr.2000.8529}{doi:10.1006/jabr.2000.8529}. + +\bibitem{Gabber} +P.~Berthelot, +Alt\'erations de vari\'et\'es alg\'ebriques [d'apr\`es A.~J.~de~Jong], +\emph{Ast\'erisque} \textbf{241} (1997), Exp.~815, 273--311; +Section~6.1, Theorem~6.1.1 (Gabber). +\url{https://www.numdam.org/item/SB_1995-1996__38__273_0.pdf}. + +\bibitem{BHM} +B.~Bhatt, M.~Hochster, and L.~Ma, +Lim Cohen--Macaulay sequences of modules, +\emph{J. Reine Angew. Math.} \textbf{835} (2026), 111--169. +Numbered references follow +\href{https://arxiv.org/abs/2410.18372v2}{arXiv:2410.18372v2}, +August~25, 2026. + +\bibitem{BhattScholze} +B.~Bhatt and P.~Scholze, +Prisms and prismatic cohomology, +\emph{Ann. of Math. 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Pure Appl. Algebra} \textbf{223} (2019), no.~4, 1801--1816. +\href{https://doi.org/10.1016/j.jpaa.2018.07.008}{doi:10.1016/j.jpaa.2018.07.008}. +Numbered references follow +\href{https://arxiv.org/abs/1510.05146v2}{arXiv:1510.05146v2}, +July~31, 2018. + +\bibitem{SkalitUnramified} +C.~Skalit, +Intersection multiplicity of Serre in the unramified case, +\emph{J. Algebra} \textbf{428} (2015), 91--121. +\href{https://doi.org/10.1016/j.jalgebra.2014.12.040}{doi:10.1016/j.jalgebra.2014.12.040}. +Numbered references follow +\href{https://arxiv.org/abs/1409.3616v2}{arXiv:1409.3616v2}, +December~10, 2014. + +\bibitem{Stacks} +The Stacks Project Authors, +\emph{The Stacks Project}, +\href{https://stacks.math.columbia.edu/tag/0815}{Tag 0815}, +\href{https://stacks.math.columbia.edu/tag/03QH}{Tag 03QH}, +\href{https://stacks.math.columbia.edu/tag/0BRI}{Tag 0BRI}, and +\href{https://stacks.math.columbia.edu/tag/0BRJ}{Tag 0BRJ}, +accessed September~23, 2026. + +\bibitem{ThomasonTrobaugh} +R.~W.~Thomason and T.~Trobaugh, +Higher algebraic $K$-theory of schemes and of derived categories, +in \emph{The Grothendieck Festschrift}, Vol.~III, +Progress in Mathematics \textbf{88}, Birkh\"auser Boston, 1990, 247--435. +\href{https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Thomason-Trobaugh.pdf}{Original-edition text}. + +\end{thebibliography} diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/asymptotic.tex b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/asymptotic.tex new file mode 100644 index 0000000000000000000000000000000000000000..c7bacee58273cee0a1038ac7cd5198bf855f9031 --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/asymptotic.tex @@ -0,0 +1,584 @@ +\section{Supported asymptotic complexes} +\label{sec:asymptotic} + +The goal is an analogue of Lemma~\ref{ds:ordinary-norm} for sequences of +bounded free complexes modulo free sequences of negligible normalized +rank, proved in Theorem~\ref{as:norm}. Two points require care: support +may arise only after passing to the quotient, and supported objects may +be idempotent summands whose projectors do not lift before quotienting. +Their homology lengths therefore cannot in general be computed from the +raw homologies of arbitrary lifts. We first construct nonnegative lengths +on uniformly supported lifts, extend them through the supported quotient +and its idempotent completion, and prove their compatibility with scalar +extension on every supported summand. + +Throughout this section, $(A,J)$ is an excellent regular local +$\Z_{(p)}$-algebra of dimension $e>0$ with residue characteristic $p$, +where $J$ is the maximal ideal, and $z_1,\ldots,z_e$ is a regular system +of parameters. Let $T$ be a finite +local domain over $A$ with the same residue field, and put +$r=\rank_A T$. Let $C$ be a $T$-algebra equipped with a nonnegative +additive length $\lambda$ on its modules annihilated by powers of $J$. +The assumptions on this length that are used in this section are +\begin{equation} + \lambda H_j(z_1^v,\ldots,z_e^v;C)=0\quad(j>0), + \qquad + \lambda\bigl(C/(z_1^v,\ldots,z_e^v)C\bigr)=v^e r + \quad(v\geq 1). + \label{as:length-hypotheses} +\end{equation} +The length is allowed to vanish on nonzero modules. In the applications, +Theorem~\ref{len:input} supplies these hypotheses, as well as the stronger +parameter properties used in the other sections. Write $K_v$ for the +Koszul complex $K(z_1^v,\ldots,z_e^v;A)$, in homological degrees +$0,\ldots,e$. + +\subsection{The rank quotient and its support condition} + +Fix positive integers $s_n$ and a nonprincipal ultrafilter $\mathcal U$ +on the positive integers. Rank estimates $O(s_n)$ and $o(s_n)$ always +refer to the ordinary limit, not to $\mathcal U$. If $(a_n/s_n)$ is +bounded, write +\[ + \overline a=\lim_{n\to\mathcal U}\frac{a_n}{s_n}. +\] +For $S=A$ or $T$, consider the additive category of sequences +$(S^{a_n})_n$ with $a_n=O(s_n)$, whose morphisms are arbitrary +sequences of matrices over $S$. Let $\mathcal D_S$ be its perfect +hull: take complexes in one common bounded degree range for the whole +sequence, and then idempotent completion. Let +$\mathcal N_S$ be the thick subcategory generated by the free +sequences with $a_n=o(s_n)$, and define +\[ + \mathcal P_S=(\mathcal D_S/\mathcal N_S)^{\natural}. +\] +Here $(-)^{\natural}$ denotes idempotent completion. The quotient +functor is denoted by $q_S$. + +A thick subcategory is closed under shifts, cones, and direct summands. +The Verdier quotient inverts maps whose cones lie in that subcategory; +idempotent completion then adjoins the images of idempotent endomorphisms. + +An object $X$ in $\mathcal D_S$ or $\mathcal P_S$ is called +\emph{supported on $J$} if there is an integer $v\geq1$ such that +$z_j^v\operatorname{id}_X=0$ for every $j$. The identity is taken +in the category containing $X$. Thus in $\mathcal D_S$ this is a +uniform homotopical support condition on the sequence, whereas in +$\mathcal P_S$ the actions need vanish only after the rank-small +quotient. Write $\mathcal S_S\subseteq\mathcal D_S$ and +$\mathcal P_S^J\subseteq\mathcal P_S$ for the resulting full +subcategories. They are thick: nilpotence of central actions is +preserved by shifts, cones, and retracts, allowing larger powers +for cones. We identify this condition with central-localization +support before applying descent at the end of the section. + +\begin{remark} +\label{as:nonuniform-warning} +Raw homology lengths need not survive the rank quotient. Over a +discrete valuation ring $A=T$ with parameter $z$, take $s_n=n$ and +the rank-one free resolution of $T/(z^n)$. This sequence belongs to +$\mathcal N_T$, whereas its raw normalized degree-zero homology +length is $1$. It has no uniform annihilating power before +quotienting. For each fixed $v$, however, its homology lengths after +tensoring with $K(z^v;T)$ are bounded independently of $n$ and hence +become zero after normalization. This is why we construct homology +lengths on uniformly supported lifts first, and then extend them to +supported quotient objects. +\end{remark} + +\subsection{The supported subquotient} + +The next lemma supplies that extension's categorical starting point: +every supported quotient object is a summand of a uniformly supported +lift, and morphisms between such lifts can be computed using supported +roofs. Neither assertion requires the summand projector to lift. + +\begin{lemma} +\label{as:supported-quotient} +The natural exact functor +\[ + \mathcal S_S/(\mathcal S_S\cap\mathcal N_S) + \longrightarrow \mathcal D_S/\mathcal N_S +\] +is fully faithful. Its idempotent completion is +$\mathcal P_S^J$. +\end{lemma} + +\begin{proof} +First let $X\in\mathcal S_S$, $Y\in\mathcal N_S$, and +$f:X\to Y$. Choose $v$ such that $z_j^v\operatorname{id}_X=0$ +for every $j$. Centrality gives $z_j^vf=0$. The exact Hom sequence +of the triangle +\[ + \operatorname{fib}(z_j^v:Y\to Y)\longrightarrow Y + \xrightarrow{z_j^v}Y +\] +therefore lifts $f$ through the first object. Repeat this operation +for all the parameters, always using the zero action on the original +source $X$. The result factors $f$ through +$K_v\otimes_A Y[-e]$. This object belongs to $\mathcal N_S$ by +thickness, and to $\mathcal S_S$ by the usual Koszul homotopies. +Thus every map from a supported object to a small object factors +through a supported small object. + +Here is the resulting roof argument. A morphism between $X,Y$ in +$\mathcal S_S$, computed in $\mathcal D_S/\mathcal N_S$, has a +representative +\[ + X\xleftarrow{s}Z\xrightarrow{f}Y, + \qquad \operatorname{cofib}(s)=N_0\in\mathcal N_S. +\] +In the triangle $Z\to X\xrightarrow{u}N_0$, factor $u$ through +$N_1\in\mathcal S_S\cap\mathcal N_S$ by the preceding paragraph. +Let $Z_1$ be the fiber of $X\to N_1$. The induced morphism of +triangles gives $c:Z_1\to Z$ and $s_1:Z_1\to X$ with $sc=s_1$. +Now $Z_1\in\mathcal S_S$ and $\operatorname{cofib}(s_1)=N_1$, +so $X\xleftarrow{s_1}Z_1\xrightarrow{fc}Y$ is a representative +entirely in the supported subquotient. This proves fullness. + +For faithfulness, suppose a map $f:Z\to Y$ between supported +objects becomes zero in $\mathcal D_S/\mathcal N_S$. The calculus +of Verdier roofs gives a map $t:W\to Z$ with small cone and $ft=0$. +The triangle of $t$ makes $f$ factor through that small cone. +Factoring its first leg through a supported small object as above +shows that $f$ is already zero in the supported subquotient. Apply +this to the numerator of any roof to obtain faithfulness. + +Finally take $X\in\mathcal P_S^J$. It is a retract of $q_S(Y)$ +for some $Y\in\mathcal D_S$. Choose $v$ with all $z_j^v$ acting +as zero on $X$. The cone of each such zero map splits, so $X$ is a +retract of $K_v\otimes_A X$. The latter is a retract of +$q_S(K_v\otimes_A Y)$, whose lift is uniformly supported. Hence +$X$ belongs to the idempotent completion of the fully faithful +supported subquotient. The reverse inclusion follows because +support is closed under retracts. +\end{proof} + +\subsection{Nonnegative homology lengths} + +Every supported quotient object is now accessible through a uniformly +supported lift and a summand. We next construct its nonnegative homology +lengths and show that they give additive Euler evaluations. + +For ordinary length, let $\mathcal A_T$ be the abelian category +of sequences of finite $T$-modules annihilated by some common power +of $J$ and with lengths $O(s_n)$. For the $C$ evaluation, let +$\mathcal A_C$ be the category of sequences of $C$-modules with +the same uniform torsion condition and with finite $\lambda$-lengths +$O(s_n)$. The power and the implied constant can depend on the +object. Kernels, cokernels, and extensions preserve these conditions, +so both categories are abelian. In each case the sequences of +normalized ultralimit length zero form a Serre subcategory. Denote +the Serre quotients by $\overline{\mathcal A}_T$ and +$\overline{\mathcal A}_C$, and their nonnegative additive lengths +by $\overline\length_T$ and $\overline\lambda$, respectively. +These lengths are finite on every object. +Here negligible length means zero ultralimit; the free sequences +generating $\mathcal N_T$ still satisfy the stronger ordinary estimate +$o(s_n)$. We will show that supported objects killed by the rank +quotient have zero homology in these abelian quotients. + +\begin{proposition} +\label{as:evaluations} +There are homological functors +\[ + \mathsf H_i^{\mathrm{ord}}:\mathcal P_T^J + \longrightarrow\overline{\mathcal A}_T, + \qquad + \mathsf H_i^C:\mathcal P_T^J + \longrightarrow\overline{\mathcal A}_C +\] +whose values on a uniformly supported lift are its ordinary homology +sequence and its homology sequence after tensoring with $C$, +respectively. For each object $X$, their lengths +\[ + b_i^{\mathrm{ord}}(X)=\overline\length_T + \mathsf H_i^{\mathrm{ord}}(X), + \qquad + b_i^C(X)=\overline\lambda\mathsf H_i^C(X) +\] +are finite nonnegative real numbers, zero outside a finite range. +Their Euler sums define homomorphisms +\[ + \chi_{\mathrm{ord}},\chi_C:K_0(\mathcal P_T^J)\longrightarrow\mathbb R. +\] +If $F=(F_n)$ is a bounded free sequence whose image is supported, +and $v$ makes all $z_j^v$ act as zero on that image, then for either +choice of evaluation, writing $b_i(F)=b_i(q_TF)$, +\begin{equation} + (1+t)^e\sum_i b_i(F)t^i + =\sum_i t^i\lim_{n\to\mathcal U} + \frac{\operatorname{len} + H_i(K_v\otimes_A F_n)}{s_n}. + \label{as:koszul-polynomial} +\end{equation} +Here $\operatorname{len}=\length_T$ in the ordinary case; in the +$C$ case the complex on the right is further tensored over $T$ +with $C$, and $\operatorname{len}=\lambda$. The identity is one +of finite Laurent polynomials. +\end{proposition} + +\begin{proof} +We first bound homology on $\mathcal S_T$, then show that it kills +$\mathcal S_T\cap\mathcal N_T$, and finally extend it to the +idempotent-completed quotient. + +Begin with a bounded free sequence $F$ in $\mathcal S_T$. Choose +uniform powers annihilating its identity. Splitting their cones +gives an isomorphism in the homotopy category +\begin{equation} + K_v\otimes_A F\simeq + \bigoplus_{j=0}^e F[j]^{\oplus\binom ej}. + \label{as:split-koszul} +\end{equation} +For ordinary length, every Koszul homology $H_j(K_v\otimes_A T)$ +has finite length. Taking Koszul homology first in the bounded +double complex bounds the homology lengths of $K_v\otimes_A F_n$ +by fixed constants times the ranks of the terms of $F_n$. They +are therefore $O(s_n)$. For $C$, the same spectral sequence uses +Equation~\eqref{as:length-hypotheses}: positive Koszul rows have +length zero and the zero row has finite length. Additivity and +nonnegativity give the same bound. The splitting in +Equation~\eqref{as:split-koszul} bounds the homologies of $F_n$ +themselves. They have a common primary annihilator, because a power +of every $z_j$ annihilates them. + +A general object $X\in\mathcal S_T$ may be a retract in +$\mathcal D_T$ of a bounded free sequence $F$ that is not supported. +Choose $v$ killing the parameter actions on $X$. Then $X$ is a retract +of $K_v\otimes_A X$, which is a retract of the uniformly supported +bounded free sequence $K_v\otimes_A F$. The preceding bounds apply +to this sequence and hence to the homology summands defining $X$. +This is a retract before taking the rank quotient. Thus homology on +$\mathcal S_T$ takes +values in the indicated abelian length quotients. + +Fix $v$. Koszul tensoring a rank-small free generator has homology +of normalized length zero by the same bounds. The class of objects +$Y\in\mathcal D_T$ with this property for $K_v\otimes_A Y$ +is thick: use homology long exact sequences, nonnegativity, and +retracts. It therefore contains $\mathcal N_T$. If +$X\in\mathcal S_T\cap\mathcal N_T$, choose $v$ killing its +parameter actions. Then $X$ is a retract of $K_v\otimes_A X$, +so its homology is zero in both length quotients. + +The homology functors consequently invert maps whose cones are +supported and small. They descend to +$\mathcal S_T/(\mathcal S_T\cap\mathcal N_T)$, preserving +homology long exact sequences. Now consider summands after quotienting. +By Lemma~\ref{as:supported-quotient}, +every object of $\mathcal P_T^J$ is a summand of an object of this +subcategory. Extend a homology functor to a summand $(X,a)$ by +taking $\operatorname{im}\mathsf H_i(a)$ in its abelian target. +This does not require a sequence of chain projectors lifting $a$. +Distinguished triangles in an idempotent completion are summands +of distinguished triangles in the original triangulated category +\cite[Definition~1.10 and Theorem~1.12]{BalmerSchlichting}. +Taking images of compatible idempotents preserves exact sequences +in an abelian category, so the extended functors are still +homological. + +Each object is a summand of a fixed bounded uniformly supported +Koszul lift. Its homology objects therefore vanish outside a finite +range, and their lengths are bounded by those of that lift. This +proves finiteness, nonnegativity, and boundedness of the $b_i$. +The finite long exact sequences prove additivity of the Euler sums. + +For the last assertion, the splitting in +Equation~\eqref{as:split-koszul} now takes place in $\mathcal P_T$, +since the chosen powers act as zero there. Applying the extended +homological functors gives the left side of +Equation~\eqref{as:koszul-polynomial}. Its Koszul tensor has a +uniformly supported lift, whose actual homology computes the right +side. This proves the identity without defining its coefficients +by formal division. +\end{proof} + +In particular, Equation~\eqref{as:koszul-polynomial} assigns zero +to the quotient object in Remark~\ref{as:nonuniform-warning}, as +required. Its nonnegative coefficients come from homological +functors, not from an assumed positivity of a polynomial quotient. + +\subsection{Scalar extension and nonlifting idempotents} + +Construct $\overline{\mathcal A}_A$ and its length in the same way +using ordinary finite $A$-modules. The preceding construction over +$A$ with ordinary length defines $\chi_A$ on $\mathcal P_A^J$. +Scalar extension gives an exact functor +$\mathcal P_A^J\to\mathcal P_T^J$. + +The supported Euler evaluations are defined on every object of +$\mathcal P_T^J$. To apply descent, it remains to prove that they agree +on all classes coming from $\mathcal P_A^J$, including idempotent +summands. + +The normalized evaluation uses the following consequence of the +length hypotheses. We state it for the present base $A$, which need +not be complete. + +\begin{lemma} +\label{as:finite-tor} +For every finite-length $A$-module $V$, +\[ + \lambda\Tor_j^A(V,C)=0\quad(j>0), + \qquad + \lambda(V\otimes_A C)=r\length_A V. +\] +All these modules are annihilated by a power of $J$. +\end{lemma} + +\begin{proof} +The Koszul and composition-series proof of +Lemma~\ref{ds:finite-length-basechange} applies here: it uses regularity +and the length hypothesis, but not completeness. A power of $J$ +annihilating $V$ also annihilates its derived tensor homology. +\end{proof} + +\begin{proposition} +\label{as:base-equality} +For every $X\in\mathcal P_A^J$, +\[ + \chi_{\mathrm{ord}}(X\otimes_A^{\mathbf L}T) + =r\chi_A(X) + =\chi_C(X\otimes_A^{\mathbf L}T). +\] +These equalities hold for summands defined by idempotents in the +Verdier quotient, without assuming that the idempotents lift to +chain maps before localization. +\end{proposition} + +\begin{proof} +The two evaluations recover the factor $r$ differently: the ordinary +one uses the alternating rank of a resolution of $T$, while the +normalized one uses the vanishing of positive Tor lengths. To make +both calculations on an arbitrary summand, we descend their finite +spectral sequences through supported Verdier roofs before taking +idempotent images. + +Let $Y$ be a uniformly supported bounded free sequence over $A$ and +let $a$ be an idempotent endomorphism of its image in +$\mathcal S_A/(\mathcal S_A\cap\mathcal N_A)$. Every supported +object of $\mathcal P_A$ has this form as a summand, by +Lemma~\ref{as:supported-quotient}: in its proof one may enlarge the +initial lift to a bounded free sequence before taking the Koszul +tensor. Put +\[ + V_b=\operatorname{im}\mathsf H_b^A(a) + \quad\text{in }\overline{\mathcal A}_A. +\] +These images exist in the abelian quotient even when $a$ has no +chain-level projector representing it. + +Choose a fixed bounded finite free $A$-resolution $P_\bullet$ of +$T$; such a resolution exists because $A$ is regular. The double +complex $Y\otimes_A P_\bullet$ computes +$Y\otimes_A^{\mathbf L}T$. Filtering by the degree of $P$ gives +a finite spectral sequence with first page +\begin{equation} + E^1_{u,b}=H_b(Y)\otimes_A P_u. + \label{as:ordinary-page} +\end{equation} +It is natural in homotopy-category maps from this page onward: +homotopies in $Y$ preserve the filtration and act trivially on +the first page. On retracts in the prequotient category the same +construction is defined by taking the corresponding idempotent +images on every page and on the abutment filtration. Its terms +and finite abutment filtration can be +regarded as $A$-modules. The abutment's ordinary $A$-length equals +its $T$-length, since the residue fields agree. + +A map with supported small cone is an isomorphism on $H_b(Y)$ +in $\overline{\mathcal A}_A$. It is thus an isomorphism on +Equation~\eqref{as:ordinary-page}, on every subsequent page, and +on the finite filtered abutment in the length quotient. The +spectral sequence therefore descends through all Verdier roofs +in the supported subquotient. The natural augmentation +$\operatorname{Tot}(Y\otimes_A P_\bullet)\to Y\otimes_A T$ +identifies its abutment with the homology of scalar extension, +and this identification also descends through those roofs. +Thus the induced action of $a$ on the abutment is precisely the +scalar-extended idempotent, not a separately chosen action. + +In particular $a$ acts as an actual +idempotent on every page and on that filtered abutment. The +differentials commute with this action. Taking its images commutes +with passage to homology, since an idempotent decomposes a complex +in an abelian category into its image and complementary image. +It also preserves the finite exact sequences defining the +abutment filtration: for a filtration preserved by an idempotent +$a$, one has $\operatorname{im}(a)\cap F_u=a(F_u)$ and hence +$\operatorname{gr}_u\operatorname{im}(a) +=\operatorname{im}(\operatorname{gr}_u a)$. +Consequently the Euler length on the +scalar-extended summand is the Euler length of the images on +the first page. They are $\rank_A(P_u)$ copies of $V_b$, and hence +this value is +\begin{equation} + \left(\sum_u(-1)^u\rank_A P_u\right) + \sum_b(-1)^b\overline\length_A V_b + =r\sum_b(-1)^b\overline\length_A V_b. + \label{as:ordinary-idempotent} +\end{equation} +The equality of alternating ranks with $r$ follows by tensoring +the resolution with the fraction field of $A$. + +For $C$, use the finite hyper-Tor spectral sequence +\begin{equation} + {}^CE^2_{u,b}=\Tor_u^A(H_b(Y),C) + \ \Longrightarrow\ H_{u+b}(Y\otimes_A^{\mathbf L}C). + \label{as:c-page} +\end{equation} +It can be constructed from the functorial finite Postnikov +filtration of $Y$. It has bounded $b$ and $0\leq u\leq e$, so +there is no unbounded convergence issue. Lemma~\ref{as:finite-tor} +makes every positive Tor term zero in +$\overline{\mathcal A}_C$. It also implies that +$V\mapsto V\otimes_A C$ induces an exact functor +\[ + \overline{\mathcal A}_A\longrightarrow + \overline{\mathcal A}_C +\] +that multiplies length by $r$. Indeed, on a short exact sequence +its possible failure of left exactness is the image of a +positive Tor module of zero length. Negligible-length sequences +map to negligible-length sequences by the same lemma. This +explains both exactness after the Serre quotient and descent of +the functor to that quotient. + +It follows that a supported-small-cone map induces isomorphisms +on the second page of Equation~\eqref{as:c-page} and its finite +filtered abutment in the length quotient. Thus this spectral +sequence too is functorial through the supported Verdier roofs. +Taking the images of the same idempotent $a$ gives only the zero +Tor row, namely the images of $V_b$ under this exact tensor +functor. Their Euler length is +$r\sum_b(-1)^b\overline\length_A V_b$, exactly the value in +Equation~\eqref{as:ordinary-idempotent}. Finally this last sum +is $\chi_A(Y,a)$ by construction. This proves both equalities +for every summand and hence for every $X\in\mathcal P_A^J$. +\end{proof} + +\subsection{Connective coefficients and the norm comparison} + +We have constructed the supported Euler evaluations and proved their +agreement after scalar extension from $A$. To extend that agreement +to conjugate sums, we now put the rank quotient in the form required +by Theorem~\ref{ds:descent}. The coefficient algebra must be +connective, and the support condition must be central-localization +support. + +\begin{lemma}[Connective coefficients] +\label{as:connective} +There is a connective associative derived $A$-algebra $H$ and +compatible equivalences +\[ + \mathcal P_A\simeq\Perf(H), + \qquad + \mathcal P_T\simeq\Perf(H\otimes_A^{\mathbf L}T). +\] +The equivalences respect central localization by elements of $A$. +\end{lemma} + +\begin{proof} +The sequence $E_S=(S^{s_n})_n$ is an additive generator. Indeed, if +$a_n=O(s_n)$, there is a single integer $c$ such that $a_n\leq cs_n$ +for every $n$, after increasing $c$ to handle the finitely many +exceptional indices. Coordinate inclusions and projections make +$(S^{a_n})_n$ a summand of $E_S^c$. Consequently +\[ + \mathcal D_S\simeq\Perf(B_S), + \qquad B_S=\prod_n M_{s_n}(S). +\] + +The $A$-module underlying $B_A$ is flat. One can check this by the +ideal criterion: every finitely generated ideal of the Noetherian +ring $A$ is finitely presented, so its tensor product with this +product of finite free modules is the corresponding product of its +tensor products; each required injection is then a product of +injections. Moreover, $T$ is finitely presented over $A$, so tensoring +with $T$ commutes with that product. Thus +\begin{equation} + B_A\otimes_A^{\mathbf L}T + \simeq B_A\otimes_A T + \cong B_T. + \label{as:product-base-change} +\end{equation} + +The same product argument makes every morphism module between the +original free sequences flat over $A$. Hence the explicit dg quotient +of this degree-zero category by its full rank-small subcategory computes +the derived quotient \cite[Section~3.1 and Theorem~3.4]{Drinfeld}. +Its morphisms are sums of paths formed from tensor products of the +original morphism modules and contracting homotopies in homological +degree $1$. They are concentrated in nonnegative degrees and degreewise +$A$-flat, hence are $K$-flat. Passing to the perfect hull and idempotent +completion kills exactly $\mathcal N_A$. The image of $E_A$ remains a +generator, and its endomorphism algebra +$H=\operatorname{End}_{\mathcal P_A}(q_AE_A)$ is connective, giving +$\mathcal P_A\simeq\Perf(H)$. This assertion concerns the chosen +generator, not arbitrary shifts of perfect objects. + +Since $T$ is finitely presented over $A$, tensoring each original +morphism module with $T$ gives the corresponding module of morphisms +between free $T$-sequences. Tensoring the path formula therefore gives +exactly the $T$-linear dg quotient by the rank-small free sequences. +The same flatness criterion applies over the Noetherian ring $T$. +The upstairs endomorphism algebra is consequently +$H\otimes_A T\simeq H\otimes_A^{\mathbf L}T$, and its perfect hull +and idempotent completion are $\mathcal P_T$. This also agrees with +the preservation of exact quotients under scalar extension +\cite[Lemma~3.12]{LT}. All constructions are centrally $A$-linear and +preserve central localization. +\end{proof} + +Under these equivalences, the support condition used throughout this +section is exactly the one in Theorem~\ref{ds:descent}. Indeed, for +a perfect module $X$, localization at a central element $z_j$ is +the telescope of multiplication by $z_j$ in the module category. +Compactness implies that $X[1/z_j]=0$ if and only if some power +$z_j^v$ kills its identity. A common power can be chosen for the +finitely many parameters. The same argument for $\Perf(B_S)$ +identifies $\mathcal S_S$ with the supported objects before the +rank quotient. In particular, this comparison does not require a +uniformly supported lift of an object in $\mathcal P_S^J$. + +Suppose now that a finite group $G$ acts on $T$ over $A$, that +$\Spec T\to\Spec A$ is surjective, and that $G$ is transitive +on every geometric fiber, as in Theorem~\ref{ds:descent}. It acts +on $\mathcal P_T^J$ by scalar twisting. We write $gX$ for the +functor induced by $g^*P=P\otimes_{T,g}T$, with the convention of +Section~\ref{sec:descent}. + +\begin{theorem}[Asymptotic norm comparison] +\label{as:norm} +Let $A,J,T,C,\lambda$ satisfy the setup of this section and +\eqref{as:length-hypotheses}. Suppose a finite group $G$ acts on $T$ +over $A$, the map $\Spec T\to\Spec A$ is surjective, and $G$ is +transitive on every geometric fiber. Then, for every +$X\in\mathcal P_T^J$, +\[ + \sum_{g\in G}\chi_{\mathrm{ord}}(gX) + =\sum_{g\in G}\chi_C(gX). +\] +Neither invariance of $\lambda$ under $G$ nor equality of the two +evaluations on individual classes is assumed. +\end{theorem} + +\begin{proof} +Lemma~\ref{as:connective} identifies the two supported categories +with the supported perfect categories of $H$ and +$H\otimes_A^{\mathbf L}T$, where $H$ is connective. +The identification over $T$ is compatible with the $G$-action: +scalar twisting acts on the $T$ factor in the coefficient algebra. +Theorem~\ref{ds:descent} therefore gives a surjection +\[ + K_0(\mathcal P_A^J)\otimes\Q + \longrightarrow + \bigl(K_0(\mathcal P_T^J)\otimes\Q\bigr)^G. +\] +The norm class $\sum_g[gX]$ is invariant. The homomorphism +$\chi_{\mathrm{ord}}-\chi_C$, extended rationally to the +Grothendieck group, vanishes on the image from $A$ by +Proposition~\ref{as:base-equality}. Applying it to a preimage +of this norm class proves the assertion. +\end{proof} diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/descent.tex b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/descent.tex new file mode 100644 index 0000000000000000000000000000000000000000..cc548ab93ea92c27833aaa5f12a6f1be6ed03c67 --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/descent.tex @@ -0,0 +1,532 @@ +\section{Rational descent with closed supports} +\label{sec:descent} + +We compare ordinary and normalized Euler lengths through rational +$K$-theory with support. In the regular-base applications, the strategy +is to compare the two evaluations on classes pulled back from the base, +then show that sums of conjugate classes come from rational classes over +that base. The latter step is the descent theorem below; the final +subsection combines it with the normalized-length input of +Section~\ref{sec:lengths}. We retain connective associative coefficients +because Section~\ref{sec:asymptotic} applies the same descent theorem to +the quotient by negligible-rank sequences. + +All derived algebras in this section are indexed homologically. Thus +connective means that their homotopy groups vanish in negative degrees. + +Let $A$ be a Noetherian commutative $\Z_{(p)}$-algebra of finite Krull +dimension, let $J\subset A$ be an ideal containing $p$, and let +$H$ be a connective associative $A$-algebra. The image of $A$ acts +centrally. For a commutative $A$-algebra $B$, write +\[ + \Perf_J(H\otimes_A^{\mathbf L}B) +\] +for the category of perfect modules whose central localizations vanish +off $V(JB)$. Its nonconnective $K$-theory spectrum and its zeroth +homotopy group are denoted by +$\K^J(H\otimes_A^{\mathbf L}B)$ and +$K_0^J(H\otimes_A^{\mathbf L}B)$, respectively. Rationalization of a +spectrum is indicated by a subscript $\Q$. +For a finite group action, $(-)^{hG}$ denotes homotopy fixed points +(derived invariants), whereas $(-)^G$ denotes ordinary invariants. + +\begin{theorem}[Supported rational descent] +\label{ds:descent} +Let $A$ be a commutative Noetherian $\Z_{(p)}$-algebra of finite Krull +dimension, let $J\subset A$ be an ideal containing $p$, and let $H$ be a connective +associative $A$-algebra with central $A$-action. +Let $T$ be a finite commutative $A$-algebra with an action of a finite +group $G$ over $A$. Suppose that $\Spec T\to\Spec A$ is surjective and +that $G$ acts transitively on the points of every geometric fiber. +Then pullback induces an equivalence +\begin{equation} + \K^J(H)_\Q + \simeq + \bigl(\K^J(H\otimes_A^{\mathbf L}T)_\Q\bigr)^{hG}. + \label{ds:spectral-descent} +\end{equation} +In particular, the map +\begin{equation} + K_0^J(H)\otimes\Q + \longrightarrow + \bigl(K_0^J(H\otimes_A^{\mathbf L}T)\otimes\Q\bigr)^G + \label{ds:kzero-descent} +\end{equation} +is surjective. +\end{theorem} + +Neither flatness nor tame ramification of $A\to T$ is required. +The coefficient algebra $H$ need not be commutative, finite over $A$, +or bounded in positive homological degrees. Its connectivity is +essential to the argument. + +\subsection{The truncating invariant and its supports} + +We use the following precise form of the results of Land--Tamme. +A localizing invariant takes exact sequences of stable categories to +fiber sequences. It is called \emph{truncating} if its value on a +connective associative algebra is unchanged by passage to $\pi_0$. +Write $\operatorname{fib}$ for homotopy fiber and +$\operatorname{HN}(-/\Q)$ for negative cyclic homology over $\Q$. +Rational infinitesimal $K$-theory +\[ + I(U)=\operatorname{fib}\bigl( + \K(U)_\Q\longrightarrow + \operatorname{HN}(U\otimes\Q/\Q)\bigr) +\] +is truncating by Goodwillie's relative comparison theorem +\cite[Main Theorem]{Goodwillie}, with the passage from simplicial rings +to connective associative ring spectra explained in +\cite[proof of Corollary~3.9 and Remark~3.10]{LT}. Truncating localizing +invariants satisfy cdh descent, also after tensoring with a fixed +connective associative algebra over the base +\cite[Theorem~A.2 and Appendix~B, especially Theorem~B.3]{LT}. +The cdh topology is generated by Nisnevich covers and abstract blowup +squares: a proper map and a closed center, with the proper map an +isomorphism off that center. + +For a derived $\Z_{(p)}$-algebra $U$, consider +\begin{equation} + E(U)=\operatorname{fib}\bigl( + \K(U)_\Q\longrightarrow\K(U[1/p])_\Q\bigr). + \label{ds:relative-invariant} +\end{equation} +The negative cyclic homology terms in the two character maps are +naturally identical, since +$U\otimes\Q\simeq U[1/p]\otimes\Q$. Taking fibers of the character +square therefore gives +\begin{equation} + E(U)\simeq\operatorname{fib}\bigl(I(U)\longrightarrow I(U[1/p])\bigr). + \label{ds:character-cancellation} +\end{equation} +It follows that $E$ is a truncating localizing invariant of +$\Z_{(p)}$-linear categories. Notice that this argument does not +identify $\K(U)_\Q$ with $\K(U\otimes\Q)$. + +Tensoring with $H$ preserves the localizing property. It also +preserves truncation: if $B$ is connective, then +$H\otimes_A^{\mathbf L}B$ is connective and +\[ + \pi_0(H\otimes_A^{\mathbf L}B) + =\pi_0(H)\otimes_A\pi_0(B). +\] +Thus replacing $B$ by $\pi_0(B)$ does not change the value of $E$. +For a finite-type $A$-scheme $Y$, let $H_Y$ be the derived pullback +of $H$ and put $Z_Y=Y\times_A V(J)$. Define +\begin{equation} + \cL(Y)=\operatorname{fib}\bigl( + E(Y;H)\longrightarrow E(Y\setminus Z_Y;H)\bigr). + \label{ds:supported-functor} +\end{equation} +Here the coefficient notation means evaluation on the category of +perfect $H_Y$-modules. The coefficient theorem just cited gives cdh +descent for $E(-;H)$. Restriction to the open complement of $Z_Y$ +preserves the squares defining this topology, and limits commute +with limits. Hence $\cL$ also satisfies cdh descent. + +Because $p\in J$, the support $Z_Y$ disappears after inverting $p$. +Expanding the two fibers in \eqref{ds:supported-functor} and using +perfect-complex localization consequently identifies +\begin{equation} + \cL(Y)\simeq\K\bigl(\Perf_{Z_Y}(H_Y)\bigr)_\Q. + \label{ds:support-identification} +\end{equation} +In particular, this is the desired rational supported $K$-theory, +not a cdh replacement of it. The functor is nilinvariant as well. + +We recall why the support condition is compatible with the +coefficient categories used here. If $P$ is perfect and $z\in A$, +compactness identifies localization of its endomorphism spectrum +with the endomorphism spectrum of $P[1/z]$. Therefore +$P[1/z]=0$ if and only if some power $z^v$ kills +$\operatorname{id}_P$ in the homotopy category. Applying this to +finitely many generators of $J$ gives the central support +criterion. The corresponding Koszul tensors generate the compact +kernel of restriction to the complement. Perfect localization, +including idempotent completion, thus computes the fiber in +\eqref{ds:support-identification}. + +The geometric localization input is Thomason--Trobaugh's perfect +localization theorem \cite[Propositions~5.2.2--5.2.4 and +Theorem~7.4]{ThomasonTrobaugh}. Tensoring its exact sequence with the +perfect coefficient category, using \cite[Lemmas~3.12 and B.2]{LT} +and the preceding supported-kernel identification, gives the +coefficient version. Nonconnective $K$-theory sends this exact sequence +to a fiber sequence by \cite[Definition~8.1, Theorems~8.7 and 9.8, +and Section~9.3]{BGT}. + +\subsection{Descent for finite surjections} + +The functor $\cL$ is now a nilinvariant cdh sheaf computing actual +rational supported $K$-theory. To apply it to the finite map in +Theorem~\ref{ds:descent}, we need descent for finite surjections, which +need not be flat. Rational transfers give the finite flat case; +flattening by blowup and induction on closed subsets will give the +general case. + +\begin{lemma} +\label{ds:finite-flat} +The functor $\cL$ satisfies ordinary \v{C}ech descent for finite +faithfully flat covers, universally under base change. +\end{lemma} + +\begin{proof} +Let $f:X\to Y$ be finite faithfully flat. Since the schemes are +Noetherian, $f_*\mathcal O_X$ is a vector bundle of positive rank. +Restriction of scalars preserves perfect coefficient modules and +their supports: locally the scalar-extended coefficient algebra is +a finite projective module over the original one. It therefore +defines a transfer. Finite-flat base change and the projection +formula give natural transformations of presheaves on $Y$-schemes +\[ + \cL\xrightarrow{f^*}\cL_X + \xrightarrow{\operatorname{tr}_f}\cL, + \qquad + \cL_X(W)=\cL(W\times_Y X), +\] +whose composite is tensoring with the pullback of +$f_*\mathcal O_X$. + +Work first on a Zariski open of $Y$ on which that vector bundle is +free of rank $a>0$, and fix a trivialization there. Additivity of +$K$-theory identifies this composite with multiplication by $a$. +The same trivialization pulls back on every $Y$-scheme, so this +identity is natural on the whole presheaf diagram. Rationally, +$a^{-1}\operatorname{tr}_f$ makes $\cL$ a retract of $\cL_X$. + +After base change to $X$, the augmented \v{C}ech nerve of $f$ has +a section and hence an extra degeneracy. Its image under any +presheaf has split descent. Thus $\cL_X$ satisfies descent for +$f$. The descent comparison for $\cL$ is a retract of this +equivalence, and is itself an equivalence. Finally, both its source +and its totalization are Zariski sheaves. The conclusion therefore +glues over the trivializing cover of $Y$. All constructions commute +with base change. +\end{proof} + +\begin{lemma} +\label{ds:finite-surjection} +The functor $\cL$ satisfies ordinary \v{C}ech descent for every +finite surjection of finite-type $A$-schemes. +\end{lemma} + +\begin{proof} +Let $\tau$ be the topology generated by cdh covers and finite +faithfully flat covers. These cover families are stable under base +change. By Lemma~\ref{ds:finite-flat} and cdh descent, $\cL$ is a +$\tau$-sheaf. We show that every finite surjection $X\to Y$ is a +$\tau$-cover. + +Reductions and finite closed decompositions are cdh-local +operations. For clarity, if a reduced scheme is a union +$Y=Y_1\cup Y_2$ of reduced closed subschemes, the proper map +$Y_1\to Y$ is an isomorphism off $Y_2$ and gives the abstract +blowup square with intersection $Y_1\cap Y_2$. This reduces closed +unions to their pieces. We may consequently reduce to integral +$Y$, with induction on dimension for proper closed subsets. + +Choose a reduced irreducible component $X_0$ of $X$ dominating +$Y$. The map $X_0\to Y$ is finite and surjective. On a dense open +$U\subset Y$ it is finite flat of positive rank. Flattening by +blowup, applied to $\mathcal O_{X_0}$, gives a blowup +$Y'\to Y$, an isomorphism over $U$, for which the strict transform +$X'_0\to Y'$ is flat and finitely presented +\cite[Tag~0815]{Stacks}. The strict transform is a closed subscheme +of $X_0\times_Y Y'$, so it is finite over $Y'$. The blowup $Y'$ is +integral, and the image of $X'_0$ contains its generic point. +Finiteness makes that image closed. Thus $X'_0\to Y'$ is +surjective, and is a finite faithfully flat cover factoring through +$X\times_Y Y'$. + +Let $Z\subsetneq Y$ be the blowup center. The family +$\{Y'\to Y,Z\to Y\}$ is a cdh cover. By induction on dimension, +the finite surjection $X\times_Y Z\to Z$ is a $\tau$-cover. +Together with $X'_0\to Y'$, these maps give a $\tau$-covering +family of $Y$ that factors through $X$. Hence $X\to Y$ itself +generates a covering sieve. The induction starts in dimension zero, +where a reduced integral base is a field and any nonzero finite +algebra is finite faithfully flat. This proves the assertion, and +therefore the required \v{C}ech descent. +\end{proof} + +The argument uses actual retracts and sheaf limits; it requires +neither hyperdescent nor convergence of an unbounded descent +spectral sequence. + +\subsection{The geometric orbit argument} + +We have established descent for finite surjections. It remains to use +geometric transitivity to identify the homotopy $G$-invariants with the +value on the base. After arranging a section and reducing the schemes, +we compare its translates away from their coincidence loci and use +induction along those loci. + +\begin{proof}[Proof of Theorem~\ref{ds:descent}] +Write $X=\Spec T$ and $S=\Spec A$. On finite-type $S$-schemes +consider the natural comparison +\begin{equation} + \cL(Y)\longrightarrow + \cL(X\times_S Y)^{hG}. + \label{ds:comparison} +\end{equation} +Both sides are sheaves for the topology used in +Lemma~\ref{ds:finite-surjection}: base change preserves the cover +families, and homotopy fixed points commute with limits. Applying +finite-surjection descent to $X\to S$, it suffices to prove +\eqref{ds:comparison} on the nonaugmented terms of its \v{C}ech +nerve. Over each such term, the pulled-back $X$-cover admits a +section, given by a projection to one of its $X$-coordinates. The +geometric transitivity hypothesis is stable under base change. +We are therefore reduced to covers admitting a section. + +We prove this case by induction on the dimension of $Y$. +Nilinvariance and the closed-union squares described in the proof +of Lemma~\ref{ds:finite-surjection} reduce the base to an integral +reduced scheme. Indeed both sides of \eqref{ds:comparison} have +these descent properties; the component reduction uses finitely +many closed pieces and lower-dimensional intersections. + +Let $s:Y\to X\times_S Y$ be a section. Its finitely many +$G$-translates cover the source on points. To see this, extend the +residue field at any point of $Y$ to an algebraic closure and use +transitivity on that geometric fiber. Each translate is a closed +immersion, because the source is separated over $Y$, and each +factors through the reduction +\[ + V=(X\times_S Y)_{\mathrm{red}}. +\] +Retain the distinct translated sections and index them by a +transitive finite $G$-set $I$. Two distinct sections have a proper +closed coincidence locus in $Y$: their equalizer is closed, and +agreement at the generic point of integral reduced $Y$ would imply +agreement everywhere. Let $Z\subsetneq Y$ be the reduced union +of these coincidence loci, with $Z=\varnothing$ if there is only +one section. + +The map +\[ + q:W=\coprod_{i\in I}Y\longrightarrow V +\] +is finite and surjective. Over $Y\setminus Z$ the section images +are pairwise disjoint and cover the reduced scheme $V$. Their +disjoint union is therefore isomorphic to $V$ there. Consequently +$q$ is a proper modification and the square +\[ + \begin{array}{ccc} + W\times_Y Z&\longrightarrow&W\\ + \big\downarrow&&\big\downarrow\\ + V\times_Y Z&\longrightarrow&V + \end{array} +\] +is an abstract blowup square. It is $G$-equivariant. + +The $G$-action on $W$ merely permutes its copies of $Y$. +The canonical restriction maps give a $G$-equivariant equivalence +$\cL(W)\simeq\prod_{i\in I}\cL(Y)$. +A stabilizer of one section acts identically on that copy, including +on the coefficient algebra pulled back from $Y$. This is a +canonical identity of coefficient-bearing $Y$-schemes, so the +action is the ordinary permutation action, with coherently trivial +stabilizer action. The same statement holds after restriction to $Z$. +Since $\cL$ is rational, averaging over a finite group is exact. +It follows that +\begin{equation} + \cL(W)^{hG}\simeq\cL(Y), + \qquad + \cL(W\times_Y Z)^{hG}\simeq\cL(Z). + \label{ds:permutation-invariants} +\end{equation} +For example, if $I=G/G_s$, the first fixed-point spectrum is +$\cL(Y)^{hG_s}$ for the trivial $G_s$-action, and is $\cL(Y)$. +This identity holds without boundedness: the trivial +$\Q[G_s]$-module $\Q$ is projective, so derived invariants are +ordinary invariants even on unbounded rational complexes. + +Apply $\cL$ to the abstract blowup square and take homotopy +$G$-invariants. The inductive hypothesis over $Z$ identifies +\[ + \cL(V\times_Y Z)^{hG}\simeq\cL(Z). +\] +Here nilinvariance allows one to pass between $V\times_Y Z$ and +the corresponding reduced or unreduced pullback of the original +cover. Combining this with \eqref{ds:permutation-invariants}, the +pullback square gives +\[ + \cL(V)^{hG} + \simeq\cL(Y)\mathop{\times}_{\cL(Z)}\cL(Z) + \simeq\cL(Y). +\] +Nilinvariance restores $X\times_S Y$ in place of $V$. In +dimension zero there are no proper nonempty coincidence loci, and +the same proof is simply the calculation for a disjoint union of +section copies. This starts the induction and proves +\eqref{ds:comparison}. + +Taking $Y=S$ and using \eqref{ds:support-identification} yields +\eqref{ds:spectral-descent}. Finally, exact rational invariants give +\[ + \pi_0\Bigl(\bigl(\K^J(H\otimes_A^{\mathbf L}T)_\Q\bigr)^{hG}\Bigr) + =\bigl(K_0^J(H\otimes_A^{\mathbf L}T)\otimes\Q\bigr)^G, +\] +which proves \eqref{ds:kzero-descent}, in fact as an isomorphism. +\end{proof} + +\subsection{Comparison of ordinary and normalized Euler lengths} + +We now isolate the consequence needed before introducing asymptotic +categories. Let $(A,\m_A,k)$ be a complete regular local +$\Z_{(p)}$-algebra of dimension $e>0$, with residue characteristic +$p$, and let $z_1,\ldots,z_e$ be regular parameters. Let $T$ be a +finite local domain extension of $A$ with the same residue field. +Put $r=\rank_A T$. Suppose that $C$ is a $T$-algebra equipped with +an additive nonnegative length $\lambda$ on $C$-modules annihilated +by powers of $\m_A$, with $\lambda(0)=0$. Infinite length is +allowed for general modules. Assume that, for every $v\geq1$, +\begin{equation} + \lambda H_i(z_1^v,\ldots,z_e^v;C)=0\quad(i>0), + \qquad + \lambda\bigl(C/(z_1^v,\ldots,z_e^v)C\bigr)=v^e r. + \label{ds:length-contract} +\end{equation} +Theorem~\ref{len:input} supplies exactly these hypotheses in our +applications, since the multiplicity of the base parameters on +$T$ is its generic rank over the regular base. +The next lemma is the formal version of +Corollary~\ref{len:finite-length-base-change}, using only +\eqref{ds:length-contract}. + +\begin{lemma} +\label{ds:finite-length-basechange} +For every finite-length $A$-module $V$, +\begin{equation} + \lambda\Tor_i^A(V,C)=0\quad(i>0), + \qquad + \lambda(V\otimes_A C)=r\length_A(V). + \label{ds:finite-length-formula} +\end{equation} +All these lengths are finite. +\end{lemma} + +\begin{proof} +The Koszul resolution on $z_1,\ldots,z_e$ resolves $k$ over +$A$, so \eqref{ds:length-contract} with $v=1$ proves the +assertion for $V=k$. Induct on the length of $V$ using a +composition series. In the long exact Tor sequence associated to +a short exact sequence with quotient $k$, nonnegativity and +additivity show that every positive Tor term has length zero. +The image of the connecting map into degree zero also has length +zero. Additivity in degree zero then gives the asserted factor +$r$. This simultaneously proves finiteness at every step. +\end{proof} + +If a finite group $G$ acts on $T$ over $A$, then for $g\in G$ scalar +twisting means $g^*P=P\otimes_{T,g}T$, +where the left $T$-action on the last factor is through $g$. +On a free complex this applies $g$ to every differential matrix entry. + +\begin{lemma}[Ordinary norm comparison] +\label{ds:ordinary-norm} +Assume, in addition, that a finite group $G$ acts on $T$ over $A$ +and is transitive on every geometric fiber of $\Spec T\to\Spec A$. +For a perfect $T$-complex $P$ supported on $V(\m_A T)$, the +quantities +\[ + \chi_T(P)=\sum_i(-1)^i\length_T H_i(P), + \qquad + \chi_C(P)=\sum_i(-1)^i + \lambda H_i(P\otimes_T^{\mathbf L}C) +\] +are finite and define homomorphisms on $K_0^{\m_A}(T)$. +They agree on every norm sum: +\begin{equation} + \sum_{g\in G}\chi_T(g^*P) + =\sum_{g\in G}\chi_C(g^*P). + \label{ds:ordinary-norm-identity} +\end{equation} +More generally, the two homomorphisms agree on the image of +$K_0^{\m_A}(A)$ and on every $G$-invariant rational class. +\end{lemma} + +\begin{proof} +We first establish finiteness and additivity, then compare the +evaluations on pullbacks before applying descent to the norm class. +Represent $P$ by a bounded finite free $T$-complex. Its ordinary +homology is finite and supported at the closed point, hence has +finite length. For the $C$-evaluation, choose a common $v$ such +that each $z_j^v$ acts nullhomotopically on $P$. This is possible +by the support criterion above. Set +$K_v=K(z_1^v,\ldots,z_e^v;T)$. Successively splitting the cones +of these nullhomotopic actions gives +\begin{equation} + K_v\otimes_T P\otimes_T C + \simeq + \bigoplus_{j=0}^e + (P\otimes_T C)[j]^{\oplus\binom ej}. + \label{ds:koszul-splitting} +\end{equation} +The homotopies may be tensored with $C$ and with the other Koszul +factors, so this splitting respects the module structure on which +$\lambda$ is defined. + +In the opposite order, taking Koszul homology first gives a finite +spectral sequence whose terms are finite direct sums of +$H_j(z_1^v,\ldots,z_e^v;C)$. Their lengths are finite by +\eqref{ds:length-contract}. Hence every homology module on the +left side of \eqref{ds:koszul-splitting} has finite length. +The splitting and nonnegativity then give finiteness for every +$H_i(P\otimes_T C)$. The homology modules and spectral-sequence terms +to which $\lambda$ is applied are annihilated by fixed powers of +$\m_A$. The Euler sums are therefore defined, and the long exact +homology sequence of a triangle proves their additivity. + +Let $Q$ now be a perfect $A$-complex supported at $\m_A$. +Its homology modules have finite length. The finite hyper-Tor +spectral sequence for $Q\otimes_A^{\mathbf L}C$, together with +Lemma~\ref{ds:finite-length-basechange}, gives +\begin{equation} + \chi_C(Q\otimes_A^{\mathbf L}T) + =r\sum_j(-1)^j\length_A H_j(Q). + \label{ds:normalized-base-euler} +\end{equation} +Indeed every positive Tor row has length zero and the zeroth row +has the displayed scaled lengths. + +For the ordinary evaluation, choose a bounded finite free +$A$-resolution $L_\bullet$ of $T$, possible because $A$ is +regular. Its alternating rank is $r$, as is seen after tensoring +with the fraction field of $A$. For every finite-length +$A$-module $V$, the finite complex $V\otimes_A L_\bullet$ +therefore gives +\[ + \sum_i(-1)^i\length_A\Tor_i^A(V,T) + =\sum_i(-1)^i\rank_A(L_i)\length_A(V) + =r\length_A(V). +\] +The hyper-Tor spectral sequence for +$Q\otimes_A^{\mathbf L}T$ consequently yields +\begin{equation} + \chi_T(Q\otimes_A^{\mathbf L}T) + =r\sum_j(-1)^j\length_A H_j(Q). + \label{ds:ordinary-base-euler} +\end{equation} +We have used that finite-length $T$-modules have the same +$A$-length and $T$-length, because both residue fields are $k$. +Equations~\eqref{ds:normalized-base-euler} and +\eqref{ds:ordinary-base-euler} prove agreement on classes pulled +back from $A$. + +Apply Theorem~\ref{ds:descent} with $H=A$ and $J=\m_A$. +Every invariant class in $K_0^{\m_A}(T)\otimes\Q$ is the +pullback of a rational class downstairs. Extending the two Euler +homomorphisms $\Q$-linearly, they therefore agree on that +invariant subspace. The class $\sum_{g\in G}[g^*P]$ is +invariant, which gives \eqref{ds:ordinary-norm-identity}. +\end{proof} + +\begin{remark} +\label{ds:no-length-invariance} +No invariance of $\lambda$ under $G$ is asserted or needed. +The equality is for the sum of the conjugate classes, not for each +individual class. This distinction allows the two length algebras +used later to be chosen independently. +\end{remark} diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/introduction.tex b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..06d6def95bfc89c9a5ce215f450981a01dbc0a6b --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,261 @@ +\section{Introduction}\label{sec:introduction} + +Let $(R,\m)$ be a commutative Noetherian regular local ring of dimension +$d$. For finite $R$-modules $M,N$ whose tensor product has finite length, +Serre's intersection multiplicity is the integer +\[ + \chi^R(M,N)=\sum_{i=0}^{d}(-1)^i + \length_R\Tor_i^R(M,N). +\] +The support condition makes every term finite, and regularity makes the +higher Tor groups vanish. This Euler characteristic measures the local +intersection of the supports while accounting for higher Tor. Its +positivity is not a formal consequence of the nonzero degree-zero term: +the higher terms enter with alternating signs. + +\begin{theorem}\label{thm:main} +Let $(R,\m)$ be a commutative Noetherian regular local ring of dimension +$d$, and let $M,N$ be nonzero finitely generated $R$-modules. If +$\length_R(M\otimes_R N)<\infty$ and +$\dim_R M+\dim_R N=d$, then +\[ + \chi^R(M,N)>0. +\] +\end{theorem} + +Here regularity means that $\m$ can be generated by $d$ elements, and +module dimension means the Krull dimension of its support. Theorem +\ref{thm:main} proves Serre's positivity conjecture without a restriction +on characteristic or ramification. In particular, it does not require +either module to lift to an unramified regular local ring.\footnote{A +public post by Editan, dated December~9, 2025 and credited +to Editan, Copilot, and Gemini, also asserts the prime-quotient form of +Serre's positivity conjecture. We record this as a same-scope claim, +not as an established theorem; see +\href{https://note.com/editan/n/n35dec2c1ddbe}{the author's post}.} + +\subsection{Background and significance} + +Serre introduced intersection multiplicities through the Euler +characteristic of Tor. His reduction to the diagonal proves positivity +in equicharacteristic and in the unramified setting; the same argument +also treats formal power series rings over an arbitrary complete +discrete valuation ring \cite[Chapter~V]{Serre}. Thus ramification alone +does not separate the known cases from the general problem. +Roberts and, independently, Gillet--Soul\'e proved vanishing when the +tensor product has finite length and the sum of dimensions is smaller +than the ring dimension \cite{Roberts,RobertsRennes,GilletSoule}. +Their arguments use local Chern characters and Adams operations, +respectively. Gabber proved nonnegativity in the remaining +mixed-characteristic case using alterations and an intersection +calculation on a normal bundle +\cite[Section~6.1, Theorem~6.1.1]{Gabber}. Vanishing removes +lower-dimensional contributions in the reductions below. For +complementary-dimensional pairs, strict positivity must additionally +rule out zero. + +Further progress in ramified mixed characteristic includes Skalit's +positivity theorem for regular local rings essentially smooth over a +two-dimensional regular local base \cite{Skalit}. KC and Soto Levins +prove positivity for modules satisfying a Serre-liftability condition +\cite[Theorem~A]{KCSotoLevins}. In their positivity statement, +$R=S/(f)$ for an unramified regular local ring $S$ and +$f\in\m_S\setminus\m_S^2$, and $M$ has a Serre lift $L$: +a finite $S$-module with $M=L\otimes_S R$ and +$\dim S-\dim L=\dim R-\dim M$; it need not satisfy the usual +Tor-vanishing condition for a lift. Theorem~\ref{thm:main} imposes +neither this liftability condition nor the smoothness hypothesis. + +The proof combines normalized-length methods with algebraic $K$-theory. +Faltings introduced the normalized-length formalism in almost +mathematics \cite[Appendix~2]{Faltings}; Gabber--Ramero developed its +almost-ring-theoretic foundations further \cite{GabberRamero}. +The perfectoid length results of Cai--Lee--Ma--Schwede--Tucker +\cite{CLMST}, together with their characteristic-$p$ counterparts +\cite{Ma}, give algebras on which parameter Koszul complexes are exact +in positive degrees after passing to normalized length. The mixed-characteristic +construction uses perfectoidization and almost purity, as developed by +Bhatt--Scholze \cite{BhattScholze}. Land--Tamme's +descent theorem for truncating invariants \cite{LT} supplies the +categorical descent input. Bhatt--Hochster--Ma show that the existence +of lim Cohen--Macaulay sequences for both quotient domains implies +positivity \cite[Theorem~6.1]{BHM}. Ishiro--Shimomoto obtain such +sequences of algebras from perfectoid towers over complete +mixed-characteristic Noetherian local domains with $F$-finite residue field +\cite[Theorem~3.17]{IshiroShimomoto}. These are conditional routes +through finite modules. Here no such tower on either quotient domain +is assumed: the proof uses bounded complexes modulo free sequences +of negligible normalized rank. + +\subsection{Proof strategy} + +After standard reductions, it is enough to consider two complete local +domains $D=R/P$ and $E=R/Q$ of positive dimensions $h,l$, where +$h+l=d$ and $P+Q$ is $\m$-primary. Each domain has a finite extension, +denoted $D'$ or $E'$, with a finite Galois action over a regular +normalization base $A_D$ or $A_E$, that is, a regular local subring +over which the extension is finite. Associated algebras $C_D,C_E$ carry additive +nonnegative lengths $\lambda_D,\lambda_E$. Their parameter quotients +have positive length, whereas positive parameter Koszul homology has +length zero. + +There are two distinct comparisons in the argument. First, rational +$K$-theory with support at the closed point shows that ordinary and +normalized Euler lengths agree after summing over all Galois +conjugates. Write $G_D,G_E$ for the two groups, and +$a_D=\rank_D D'$, $a_E=\rank_E E'$. For $\sigma\in G_D$, let +$c_\sigma$ be the normalized Tor Euler characteristic of $C_D$ and +$E'$ over $R$, where the $R$-action on $C_D$ is twisted by $\sigma$. +The first comparison identifies the sum of the $c_\sigma$ with +$|G_D|a_Da_E\chi^R(D,E)$. It therefore suffices to show that +each $c_\sigma$ is positive. Fix one such twist. + +To analyze that value, choose a regular sequence $x$ in $Q$ that is a +parameter system on $D$. The module $U=C_D/xC_D$ has positive finite +normalized length over $B=R/(x)$. It can be approximated by finite-length +$B$-modules $M_n$ killed by one fixed power of the maximal ideal of $B$, +with ordinary lengths +divided by integers $s_n$ converging to the relevant normalized lengths. +Their minimal resolutions have ranks $o(s_n)$ above degree $l$. +Truncating at degree $l$ therefore produces supported complexes over +$E'$ in the category that kills free sequences whose ranks divided by +$s_n$ tend to zero. + +The second comparison takes place in this category. Its supported +objects have nonnegative homology lengths, even when support appears +only after passing to the quotient. The two Euler evaluations again +agree on sums of conjugates. Evaluation using $C_E$ leaves only degree +zero, whose length is bounded below by a positive constant times the +normalized number of generators of $M_n$. The common annihilator and +the positive limiting length of $M_n$ make this lower bound strictly +positive. The second Galois average proves positivity for each value +in the first average, completing the argument. + +Figure~\ref{fig:two-comparisons} records the two interfaces. Here +$b_{\sigma,\gamma}$ denotes the $C_E$-evaluation of the truncated +sequence after the additional twist $\gamma\in G_E$. Choose $a$ with +$\m_B^aM_n=0$ for all $n$, and write $\beta_0(M_n)$ for the minimal +number of generators of $M_n$. Then +$\length_B M_n\leq\beta_0(M_n)\length_B(B/\m_B^a)$. +Thus the positive normalized length of the $M_n$ forces a positive +normalized number of generators; it is this uniform estimate that +makes every $b_{\sigma,\gamma}$ strictly positive. + +\begin{figure}[htbp] +\centering +\renewcommand{\arraystretch}{1.5} +\begin{tabular}{c} +\fbox{\strut Ordinary supported comparison over $A_D\subset D'$}\\ +$\displaystyle |G_D|a_Da_E\,\chi^R(D,E) + =\sum_{\sigma\in G_D}c_\sigma$\\ +$\Big\uparrow$\quad prove positivity separately for each $\sigma$\\ +\fbox{\strut Asymptotic supported comparison over $A_E\subset E'$}\\ +$\displaystyle |G_E|c_\sigma + =\sum_{\gamma\in G_E}b_{\sigma,\gamma}>0$ +\end{tabular} +\caption{Two norm comparisons, not invariance of individual evaluations. +The first is Lemma~\ref{pos:first-average}; the second is +Equation~\eqref{pos:second-average}. The uniform annihilator and +generator bound give the strict inequality in +Lemma~\ref{pos:positive-evaluation}.} +\label{fig:two-comparisons} +\end{figure} + +The reusable technical points are the finite-surjection descent +argument with arbitrary connective algebra coefficients and the +construction of supported Euler evaluations after quotienting by +negligible ranks. In particular, the comparison is proved on +idempotent summands in the quotient, not only on complexes that already +have uniform support before taking the quotient. No invariance of a +normalized-length function under either Galois group is assumed. + +Section~\ref{sec:lengths} establishes the precise normalized-length +inputs. Section~\ref{sec:descent} proves supported rational descent and +the first Euler comparison. Section~\ref{sec:asymptotic} constructs the +rank quotient and its supported evaluations. Section~\ref{sec:positivity} +performs the approximation and the two comparisons to prove +Theorem~\ref{thm:main}. + +\subsection{Consequences} + +Theorem~\ref{thm:main} also has consequences for symbolic powers. +For a ramified mixed-characteristic regular local ring $(R,\m)$ and +primes $P,Q$ with $\sqrt{P+Q}=\m$ and +$\operatorname{ht}P+\operatorname{ht}Q=\dim R$, the implication of +Kurano--Roberts \cite[Theorem~3.2]{KuranoRoberts} now gives +\[ + P\cap Q^{(r)}\subseteq\m^{r+1}\qquad(r\geq1), + \qquad Q^{(r)}:=Q^rR_Q\cap R. +\] +Here ramified means that the residue characteristic $p$ lies in +$\m^2$. This is a direct application of their positivity-to-symbolic-power +theorem, rather than an additional ingredient in our proof. + +The next consequence applies Skalit's finite-support blowup argument +\cite[Corollary~D]{Skalit} with Theorem~\ref{thm:main} as its positivity +input. Dutta studied the lower bound in this finite-support setting +\cite{DuttaFiniteSupport}. We include the argument to exhibit how +strict positivity at the points of the blowup controls equality. + +\begin{corollary}[Finite strict-transform intersection]\label{cor:tangent-cone} +Let $(A,\m)$ be a commutative Noetherian regular local ring with residue +field $k$, and let $P,Q$ be prime ideals such that $A/(P+Q)$ has finite +length and $\dim(A/P)+\dim(A/Q)=\dim A$. Write $e_\m$ for Hilbert--Samuel +multiplicity and put +\[ + T=\operatorname{gr}(A/P)\otimes_{\operatorname{gr}A}\operatorname{gr}(A/Q), +\] +where all associated graded rings use the maximal-ideal filtrations. +If $\dim T\leq1$, then +\[ + \chi^A(A/P,A/Q)\geq e_\m(A/P)e_\m(A/Q), +\] +with equality if and only if $\dim T=0$. +\end{corollary} + +\begin{proof} +If $\dim A=0$, then $A=k$, $P=Q=0$, and both sides are $1$ with $T=k$. +Otherwise, let $\widetilde X$ be the blowup of $X=\Spec A$ at its closed +point, and let $\widetilde Y,\widetilde Z$ be the strict transforms of +$Y=\Spec(A/P)$ and $Z=\Spec(A/Q)$. By \cite[Lemma~3.5]{SkalitUnramified}, +the underlying set $S$ of $\widetilde Y\cap\widetilde Z$ identifies with +that of $\operatorname{Proj}T$. Thus $S$ is a finite set of closed points +of the exceptional divisor, empty exactly when $\dim T=0$. +Fulton's blowup formula and Skalit's finite-support calculation +\cite[Theorem~3.4 and proof of Proposition~3.10]{SkalitUnramified} give +\[ + \chi^A(A/P,A/Q)-e_\m(A/P)e_\m(A/Q) + =\sum_{x\in S}[\kappa(x):k]\,\chi^{B_x}(M_x,N_x), +\] +where $B_x=\mathcal O_{\widetilde X,x}$, +$M_x=\mathcal O_{\widetilde Y,x}$, and $N_x=\mathcal O_{\widetilde Z,x}$. +Each residue degree is finite and positive since the exceptional divisor +is projective over $k$. The blowup $\widetilde X$ is regular +\cite[Proposition~A.4(1)]{Jannsen}. The modules $M_x,N_x$ are nonzero and +finite over $B_x$, with finite-length tensor product because their +intersection is supported on $S$. The local domains $A/P$ and $A/Q$, +as well as $A$, are formally equidimensional and universally catenary. +Thus \cite[Lemma~3.8 and proof of Proposition~3.10]{SkalitUnramified} +gives $\dim M_x=\dim(A/P)$, $\dim N_x=\dim(A/Q)$, and +$\dim B_x=\dim A$. Theorem~\ref{thm:main} therefore makes every local +intersection multiplicity in the sum strictly positive. The asserted +inequality follows, and equality holds exactly when $S$ is empty. +\end{proof} + +\subsection{Conventions} + +We fix universes large enough for the module and sequence categories +under consideration. +Complexes are indexed homologically. An associative derived algebra is +\emph{connective} when its homotopy groups in negative degrees vanish. +Perfect modules over associative algebras are right modules. +All scalar localizations of coefficient algebras are along the central +image of the commutative base. We use nonconnective algebraic +$K$-theory of perfect complexes; a subscript $\Q$ denotes rationalization. +For a sequence $a=(a_1,\ldots,a_r)$, write $K(a;V)$ for its homological +Koszul complex with coefficients in $V$, and put +$H_i(a;V)=H_i(K(a;V))$. The notation $a^v$ denotes +$(a_1^v,\ldots,a_r^v)$, not the $v$th power of the ideal it generates. +Normalized lengths are allowed to vanish on nonzero modules; all uses +of their additivity involve finite values or nonnegative extended +values for which the indicated operations are defined. diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/lengths.tex b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/lengths.tex new file mode 100644 index 0000000000000000000000000000000000000000..750bb4c3ec2d7aeb636ae6959663de1b0a34e4a6 --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/build/sections/lengths.tex @@ -0,0 +1,414 @@ +\section{Residue fields and normalized length}\label{sec:lengths} + +The comparison arguments require algebras on which positive parameter +Koszul homology has length zero and parameter quotients have prescribed +positive length. The finite approximations in Section~\ref{sec:positivity} +also require that this length be defined on underlying modules over a +flat tower, without an action of the full algebra. We construct these +objects after arranging a perfect residue field; neither unramifiedness +nor Cohen--Macaulayness of the domain is required. +Both characteristics are needed: a prime quotient of the +mixed-characteristic ambient ring may itself have characteristic $p$, +even though ambient equicharacteristic positivity is already known. + +\subsection{Extending the residue field} + +\begin{lemma}[Residue-field extension]\label{len:residue-extension} +Let $(B,\mathfrak b,k)$ be a complete Noetherian local ring, and let +$\overline{k}$ be an algebraic closure of $k$. There is a complete +Noetherian local ring $(B',\mathfrak b',\overline{k})$ and a flat local map +$B\to B'$ such that $\mathfrak bB'=\mathfrak b'$. +For every finite $B$-module $V$ and every $a\geq1$, +\[ + \length_{B'}\bigl((V\otimes_B B')/(\mathfrak b')^a + (V\otimes_B B')\bigr) + =\length_B(V/\mathfrak b^aV). +\] +Consequently module dimensions are preserved, as are finite lengths. +If $B$ is regular, then $B'$ is regular. For finite $B$-modules $M,N$ +whose tensor product has finite length, every Tor length, and hence +their intersection multiplicity when defined, is preserved. +\end{lemma} + +\begin{proof} +In equal characteristic, a coefficient field gives a presentation +$B=k[[X_1,\ldots,X_t]]/I$. Put +$B'=\overline{k}[[X_1,\ldots,X_t]]/I\overline{k}[[X_1,\ldots,X_t]]$. +The map of power-series rings is flat, by the local flatness criterion +for the variables and the flat coefficient-field extension. Its base +change to the quotient has the asserted residue field and maximal ideal. + +In the remaining case the residue characteristic is a prime $p$. +Cohen structure supplies a presentation $B=V[[X_1,\ldots,X_t]]/I$, +where $V$ is a complete discrete valuation ring with uniformizer $p$ +and residue field $k$. We construct a discrete valuation extension +with unchanged uniformizer and residue field $\overline{k}$, without +assuming that $k$ is perfect. Given a simple algebraic residue-field +extension, lift its monic irreducible polynomial to $f\in V[X]$. +The algebra $V[X]/(f)$ is finite free over $V$, its reduction modulo $p$ +is the desired field, and all its maximal ideals lie over $(p)$. +It is therefore local with maximal ideal generated by $p$. +Since $p$ is a nonzerodivisor, it is a one-dimensional regular local +ring, hence a discrete valuation ring. This argument also applies +to inseparable residue extensions. + +Perform this construction along a tower of simple algebraic extensions +exhausting $\overline{k}$, taking unions at limit stages. In every such +union, a nonzero element is a unit times a nonnegative integral power +of $p$. Every nonzero ideal is generated by an element of least +valuation, so the union is still a discrete valuation ring. Its +completion $V'$ has uniformizer $p$ and residue field $\overline{k}$. +The map $V\to V'$ is flat because $V'$ is torsion-free over $V$. +Now take +\[ + B'=V'[[X_1,\ldots,X_t]]/IV'[[X_1,\ldots,X_t]]. +\] +The same local flatness criterion proves the required flatness. +This is the usual residue-extension construction; compare +\cite[Lemma~2.2]{Ma}. + +For either construction, tensor a composition series of a finite-length +$B$-module with $B'$. Flatness preserves exactness, and every simple +factor becomes $B'/\mathfrak b'$. This proves the length assertions +and, applied to $V/\mathfrak b^aV$, the displayed identity. Equality of +Hilbert--Samuel functions gives equality of module dimensions. If $B$ +is regular of dimension $d$, its $d$ maximal-ideal generators generate +$\mathfrak b'$, and $\dim B'=d$, so $B'$ is regular. Finally flat base +change identifies the Tor modules after extension; their finite +lengths are preserved by the preceding argument. +\end{proof} + +\subsection{Regular bases and the tower length} + +For a perfect field $k$ of characteristic $p$, let $W(k)$ denote its +ring of Witt vectors. + +\begin{lemma}[Regular normalization bases]\label{len:normalization} +Let $(D,\mathfrak d,k)$ be a complete Noetherian local domain of +dimension $h>0$, where $k$ is perfect of characteristic $p>0$. +There is a finite, generically separable inclusion $A\subseteq D$, +inducing the identity on residue fields, with +\[ + A=k[[t_1,\ldots,t_h]]\quad\text{if }\operatorname{char}D=p, + \qquad + A=W(k)[[t_2,\ldots,t_h]]\quad\text{if }\operatorname{char}D=0. +\] +In the second case set $t_1=p$. In both cases $t_1,\ldots,t_h$ +is a parameter system of $D$ and a regular parameter system of $A$. +\end{lemma} + +\begin{proof} +The positive-characteristic assertion is the Cohen--Gabber +normalization theorem \cite{CohenGabber}. In mixed characteristic, +choose a coefficient ring $W(k)\subseteq D$ and extend $p$ to a +parameter system $p,t_2,\ldots,t_h$. Completeness and the parameter +condition make the resulting power-series map module finite. +Its kernel is zero: the source and target have dimension $h$, whereas +a nonzero ideal of the regular local domain +$W(k)[[t_2,\ldots,t_h]]$ lowers dimension. The fraction-field +extension is separable in characteristic zero. +\end{proof} + +Fix one of the bases in Lemma~\ref{len:normalization} and compatible +$p$-power roots of its parameters. Write +\begin{equation}\label{len:tower-definition} + A_n=A[t_1^{1/p^n},\ldots,t_h^{1/p^n}],\qquad + \Lambda=\varinjlim_n A_n. +\end{equation} +The tower is uncompleted. Only in mixed characteristic do we also +use its $p$-adic completion, denoted $A_\infty$. + +\begin{lemma}[The tower and its length]\label{len:tower} +Each $A_n$ is a complete regular local ring with residue field $k$. +For $b\geq a$, the map $A_a\to A_b$ is finite free of rank +$p^{(b-a)h}$; in particular $A\to A_n$ has rank $p^{nh}$. +On the category of $\Lambda$-modules killed by some power of +$\mathfrak m_A$ there is a nonnegative additive function +$\lambda_\infty$ with the following properties. +\begin{enumerate} +\item A finitely presented module descends to a finite-length + $A_n$-module $V_n$, and + \begin{equation}\label{len:finite-stage} + \lambda_\infty(V_n\otimes_{A_n}\Lambda) + =p^{-nh}\length_{A_n}(V_n). + \end{equation} +\item For a finitely generated module $V$, its length is the infimum + of the lengths of finitely presented modules in this category + admitting a surjection onto $V$. +\item For an arbitrary module $V$ in this category, + \begin{equation}\label{len:supremum} + \lambda_\infty(V)= + \sup\{\lambda_\infty(V_0):V_0\subseteq V + \text{ is finitely generated}\}. + \end{equation} +\end{enumerate} +The value on an arbitrary module may be infinite. In mixed +characteristic, restriction from $A_\infty$ to $\Lambda$ gives the same +length on modules killed by a fixed power of $\mathfrak m_A$. +\end{lemma} + +\begin{proof} +In characteristic $p$, the stages are power-series rings in the chosen +roots. In mixed characteristic they are power-series rings over the +complete discrete valuation ring $W(k)[p^{1/p^n}]$. +The monomials in the successive roots, with each exponent less than +the relevant root degree, form bases for all transition extensions. +This proves the assertions about regularity, ranks, and residue fields. + +Finite presentations over the union descend to a stage. The additional +relations expressing annihilation by a fixed power of +$\mathfrak m_A$ also descend after increasing that stage, because +there are finitely many generators and relations to consider. The +resulting finite-stage module has finite length. Formula +\eqref{len:finite-stage} is unchanged on enlarging the stage, by +flatness and the computed free ranks. The infimum and supremum +extensions, including their additivity and compatibility with this +formula, are Faltings' normalized-length formalism +\cite[Appendix~2]{Faltings}, in the formulation of +\cite[Definition~2.2.1 and Propositions~2.2.3, 2.2.6(a)]{CLMST}. +The same residue field implies that $A_n$-length and $A$-length +agree on finite-length $A_n$-modules. + +In mixed characteristic $\Lambda$ is $p$-torsion-free, and the natural +map induces $\Lambda/p^a\Lambda\simeq A_\infty/p^aA_\infty$ for every +$a\geq1$. A module killed by $\mathfrak m_A^a$ is killed by $p^a$. +Its submodules and finite generating sets are therefore identical +over the two bases. In computing the infimum over finitely presented +covers of such a module, one may replace a cover $V$ by +$V/\mathfrak m_A^aV$; this is still finitely presented. These covers +also correspond over the identical quotient rings, proving the last +assertion. +\end{proof} + +\begin{lemma}[Zero-length limits]\label{len:zero-length-limits} +Suppose all modules under consideration are killed by one fixed power +of $\mathfrak m_A$. Sums, direct sums, and directed colimits of +zero-length modules have length zero. For an increasing union, +\[ + \lambda_\infty\Bigl(\bigcup_a V_a\Bigr) + =\sup_a\lambda_\infty(V_a). +\] +If $(V_a)_{a\geq1}$ is a direct system with +$\lambda_\infty(V_a)\to0$, then +$\lambda_\infty(\varinjlim_a V_a)=0$, without any injectivity +assumption on the transition maps. +\end{lemma} + +\begin{proof} +A finite set of elements of a direct sum is contained in a finite +subsum. Additivity and \eqref{len:supremum} give the assertion for +direct sums; sums and directed colimits are quotients of direct sums. +For an increasing union, every finite generating set lies in one +stage. Finally a finitely generated submodule of $\varinjlim_a V_a$ +is contained in the image of every sufficiently late $V_a$, by lifting +its generators to a common stage. Monotonicity bounds its length by +$\lambda_\infty(V_a)$ at all such stages. Apply +\eqref{len:supremum}. The common annihilator ensures throughout +that these modules belong to the length category. +\end{proof} + +\subsection{Length algebras and parameter homology} + +The tower length will now measure modules over an algebra of the given +domain. Its parameter identities have two uses: the base parameters +control the Euler comparisons, while arbitrary parameter systems give +the positive quotients used in Section~\ref{sec:positivity}. +For an ideal $I$ primary to the maximal ideal of a local ring $D$, write +$e(I,D)$ for its Hilbert--Samuel multiplicity. + +Related normalized-length constructions, in which parameter-quotient +lengths recover multiplicities and positive parameter Koszul homology has +normalized length zero, appear in \cite[Theorem~3.1]{OpenAILech2026}. +Here we retain the chosen base $A\subseteq D$ and the finite-stage, +infimum, and supremum properties on underlying tower modules. + +\begin{theorem}[Normalized-length input]\label{len:input} +Let $(D,\mathfrak d,k)$ and a chosen finite generically separable base +$A\subseteq D$ be as in Lemma~\ref{len:normalization}, and let +$\Lambda$ be the tower \eqref{len:tower-definition}. There is a +$D$-algebra $C_D$, carrying a compatible $\Lambda$-action, such that +restriction of scalars defines an additive nonnegative length +$\lambda_D=\lambda_\infty$ on its modules annihilated by a power of +$\mathfrak d$. For every parameter system $z_1,\ldots,z_h$ of $D$ +and every integer $v\geq1$, +\begin{align} + \lambda_D H_i(z_1^v,\ldots,z_h^v;C_D)&=0\qquad(i>0), + \label{len:koszul-vanishing}\\ + \lambda_D\bigl(C_D/(z_1^v,\ldots,z_h^v)C_D\bigr) + &=v^h e((z_1,\ldots,z_h),D). + \label{len:parameter-length} +\end{align} +The right side is positive and finite. The algebra and length are +independent of the parameter system $z$. +For the regular parameters of $A$, in particular, +\begin{equation}\label{len:base-parameter-length} + \lambda_D\bigl(C_D/(t_1^v,\ldots,t_h^v)C_D\bigr) + =v^h\rank_A D. +\end{equation} +The finite-stage, infimum, and supremum properties of +Lemma~\ref{len:tower} remain available on underlying tower modules; +they do not require that the relevant submodules or covers be +$C_D$-modules. +\end{theorem} + +\begin{proof} +We give the positive-characteristic construction and then specify the +mixed-characteristic literature inputs. + +Assume first that $D$ has characteristic $p$, and put +$C_D=D_{\mathrm{perf}}=\bigcup_n D^{1/p^n}$ inside a fixed algebraic +closure of its fraction field. Here $\Lambda=A_{\mathrm{perf}}$. +Write $K=\operatorname{Frac}(A)$, $L=\operatorname{Frac}(D)$, and +$K'=K_{\mathrm{perf}}$. For $q=p^n$ put +\[ + B_q=\Lambda\otimes_{A^{1/q}}D^{1/q}. +\] +Flatness of $\Lambda$ over $A^{1/q}$ and torsion-freeness of +$D^{1/q}$ embed this tensor product into +$K'\otimes_{K^{1/q}}L^{1/q}$. Generic separability makes the latter +a field, by linear disjointness from a purely inseparable extension. +Thus $B_q=\Lambda D^{1/q}\subseteq C_D$, and these subrings increase +to $C_D$. + +Let $H$ be any $\mathfrak d$-primary ideal and set +$V_q=B_q/HB_q$ and $W_q=\operatorname{im}(V_q\to C_D/HC_D)$. +Write $H^{[q]}=(a^q:a\in H)$ for its Frobenius power. +Finite-stage normalization and the $q$-th-power isomorphism give +\[ + \lambda_\infty(V_q)=q^{-h}\length_D(D/H^{[q]}). +\] +Although $B_q\subseteq C_D$, the map on these quotients need not be +injective. To recover the length of $C_D/HC_D$ from the displayed +formula, we control the kernels of $V_q\to W_q$ by a single trace +denominator. There is a nonzero $g\in A$ with +\begin{equation}\label{len:conductor} + gC_D\subseteq B_1,\qquad g^{1/q}C_D\subseteq B_q. +\end{equation} +Indeed, choose a $K$-basis $b_1,\ldots,b_r$ of $L$ in $D$ and its +trace-dual basis $\beta_1,\ldots,\beta_r$. A common denominator +$g\in A\setminus\{0\}$ makes every $g\beta_j$ lie in $D$. +The field $LK'$ is perfect and equals $L_{\mathrm{perf}}$. +The ring $\Lambda$ is normal, since an integral equation and the +fractions occurring in it descend to a normal stage $A^{1/q}$. +For $c\in C_D$, each +$\operatorname{Tr}_{LK'/K'}(g\beta_jc)$ is integral over $\Lambda$ +and belongs to $K'$, hence belongs to $\Lambda$. The trace-dual +expansion of $gc$ proves the first inclusion in +\eqref{len:conductor}. Taking $q$-th roots proves the second. + +The kernel of $V_q\to W_q$ is killed by $g^{1/q}$, by +\eqref{len:conductor}. Kernel and cokernel of an endomorphism of a +finite-length object have equal length. Applied to multiplication +by $g^{1/q}$, this yields +\[ + \lambda_\infty\ker(V_q\to W_q) + \leq q^{-h}\length_D D/(H^{[q]},g)=O(q^{-1}). +\] +For the estimate, if $H$ has $s$ generators then +$H^{s(q-1)+1}\subseteq H^{[q]}$. Thus $H^{[q]}$ contains a power +of $\mathfrak d$ with exponent $O(q)$, whereas +$\dim D/(g)\leq h-1$ if $g$ is a nonunit. Hilbert--Samuel growth +gives the asserted bound; for a unit $g$ the quotient is zero. +The $W_q$ increase to $C_D/HC_D$ and are uniformly primary-torsion. +The Hilbert--Kunz limit and Lemma~\ref{len:zero-length-limits} show +\begin{equation}\label{len:perfection-quotient} + \lambda_D(C_D/HC_D)=e_{\mathrm{HK}}(H,D). +\end{equation} +Here we use Monsky's existence theorem for Hilbert--Kunz multiplicity +\cite{Monsky}; see also \cite[Section~2.2]{Ma}. + +We have identified primary-quotient lengths with Hilbert--Kunz +multiplicities. For a parameter ideal, it remains to obtain the +Hilbert--Samuel value and zero positive Koszul homology. If $z$ is +a parameter system of $D$, finite-stage flatness gives +\[ + \lambda_\infty H_i(z;B_q) + =q^{-h}\length_D H_i(z_1^q,\ldots,z_h^q;D). +\] +The Koszul complex $K(z;D)$ has finite free terms in degrees +$0,\ldots,h$ and finite-length homology. The Frobenius homology +theorem of Roberts, in the form \cite[Theorem~2.4]{Ma}, says that +the displayed quantity tends to zero for $i>0$; the original result +is \cite{RobertsIntersectionTheorems}. Its hypotheses +require precisely a complete characteristic-$p$ ring and a +finite-free complex of length its dimension with finite-length +homology. Filtered colimits commute with homology, and these +Koszul homology modules have the common annihilator $(z)$. +Lemma~\ref{len:zero-length-limits} proves +\eqref{len:koszul-vanishing}, also with $z$ replaced by $z^v$. +Finally the Koszul Euler characteristic formula gives +\[ + \sum_{i=0}^h(-1)^i\length_D H_i(z_1^q,\ldots,z_h^q;D) + =q^h e((z),D). +\] +After division by $q^h$, the positive-degree terms tend to zero. +Together with \eqref{len:perfection-quotient} and the multiplicity +power rule this proves \eqref{len:parameter-length}. + +Now suppose $D$ has mixed characteristic. Let $A_\infty$ be the +$p$-adic completion of $\Lambda$ and define +\[ + C_D=(D\otimes_A A_\infty)_{\mathrm{perfd}}, +\] +the perfectoidization of \cite[Notation~4.0.1]{CLMST}, built on +Bhatt--Scholze's construction \cite[Proposition~8.5]{BhattScholze}. +The stated hypotheses there are exactly that $D$ is a complete +mixed-characteristic local domain with perfect residue field and +the chosen finite Witt power-series base $A$. +Choose $g\in A\setminus\{0\}$ so that $A[1/g]\to D[1/g]$ is +finite etale. Corollary~4.0.4 of \cite{CLMST} annihilates positive +Koszul homology for every parameter system of $A_n\otimes_A D$ +by the almost ideal $(g)_{\mathrm{perfd}}$. +At $n=0$ this includes every parameter system of $D$. +Proposition~4.0.10 of that paper makes these uniformly +$\mathfrak m_A$-power-torsion homology modules have normalized +length zero. Proposition~4.0.14, again at $n=0$, identifies the +length of the parameter quotient with its Hilbert--Samuel +multiplicity. These two results, applied also to $z^v$, prove +\eqref{len:koszul-vanishing} and \eqref{len:parameter-length} +for this single algebra $C_D$. Lemma~\ref{len:tower} permits +restriction to the uncompleted tower throughout. + +In either characteristic, a module killed by a power of +$\mathfrak d$ is killed by a power of $\mathfrak m_A$, so all +claimed lengths are defined. Additivity and the tower properties +are those already established. Lastly the multiplicity rank formula, +and equality of the residue fields, give +$e((t),D)=\rank_A D\,e((t),A)=\rank_A D$. +This proves \eqref{len:base-parameter-length}. +\end{proof} + +The base-parameter case extends to every finite-length module over +$A$. The same parameter-quotient positivity also prevents the residue +quotient of $C_D$ from having length zero; this will supply the positive +residue-length factor in the final degree-zero estimate. + +\begin{corollary}[Finite-length base change]\label{len:finite-length-base-change} +In Theorem~\ref{len:input}, put $r=\rank_A D$. +For every finite-length $A$-module $V$, +\[ + \lambda_D\Tor_i^A(V,C_D)=0\quad(i>0),\qquad + \lambda_D(V\otimes_A C_D)=r\length_A(V). +\] +Moreover $\lambda_D(C_D/\mathfrak d C_D)>0$. +\end{corollary} + +\begin{proof} +For $V=k$, the first two assertions are +\eqref{len:koszul-vanishing} and \eqref{len:base-parameter-length}, +using the Koszul resolution over regular $A$. +Induct on a composition series of $V$. The Tor long exact +sequences and length additivity discard all positive-degree terms +and show that degree-zero length increases by $r$ at each step. +All modules here have bounded primary torsion. + +For the last assertion choose a parameter ideal $H$ of $D$. +Tensor a composition series of $D/H$ with $C_D$. +Right exactness gives a filtration of $C_D/HC_D$ whose factors +are quotients of $C_D/\mathfrak dC_D$. Therefore +\[ + 00,\qquad l=\dim E>0,\qquad h+l=d, +\] +where $R$ is complete with algebraically closed residue field $k$ of +characteristic $p>0$, $P,Q$ are prime, and $P+Q$ is primary to $\m_R$. +\end{lemma} + +\begin{proof} +The equicharacteristic case is Serre's positivity theorem +\cite{Serre}. We may therefore assume that the residue characteristic +is positive. Completion, followed by the extension in +Lemma~\ref{len:residue-extension}, reduces to a complete regular local +ring with algebraically closed residue field. For clarity, a flat +local extension $R\to R_1$ with $\m_RR_1=\m_{R_1}$ preserves the +length of every finite-length module: a composition series becomes a +filtration with factors $R_1/\m_{R_1}$. Flat base change identifies the +Tor modules after extension. The same argument applied to +$M/\m_R^nM$ preserves the Hilbert--Samuel function, and hence the +dimension of $M$; it applies equally to $N$. It also preserves +regularity, since it preserves the dimension and the minimal number +of generators of the maximal ideal. + +We use the vanishing theorem of Roberts and Gillet--Soul\'e in its +general regular local form \cite{Roberts,GilletSoule}: +\begin{equation}\label{pos:vanishing} + \chi^R(V,W)=0 + \quad\text{if}\quad + \length_R(V\otimes_R W)<\infty + \quad\text{and}\quad \dim V+\dim W0$. +\end{proof} + +Fix a pair as in Lemma~\ref{pos:reduction}. By +Lemma~\ref{len:normalization}, choose regular complete normalization +bases +\[ + A_D\subset D,\qquad A_E\subset E +\] +of dimensions $h,l$, with residue field $k$, such that the extensions +are finite and generically separable. + +Let $D'$ and $E'$ be the integral closures of these bases in finite +Galois closures of the respective fraction-field extensions. They +are finite over their bases by excellence, and are finite over $D,E$ +as well. They are local: a finite algebra over a complete local ring +is a product of complete local algebras, and a domain has only one +factor \cite[Tag~03QH]{Stacks}. Their residue fields are finite extensions of $k$, hence are +$k$. Write +\[ + G_D=\operatorname{Gal}(\operatorname{Frac}(D')/ + \operatorname{Frac}(A_D)),\qquad + G_E=\operatorname{Gal}(\operatorname{Frac}(E')/ + \operatorname{Frac}(A_E)). +\] +Normality of the bases gives $(D')^{G_D}=A_D$ and +$(E')^{G_E}=A_E$. For either extension write $T^G=A$. +Let $\Omega$ be an algebraically closed $A$-field and put +$S=T\otimes_A\Omega$. For $u=\sum_i t_i\otimes\omega_i\in S$, +the norm $\prod_{g\in G}g(u)$ is scalar: the coefficients of +$\prod_{g\in G}(\sum_i g(t_i)X_i)$ lie in $T^G=A$, and we +specialize $X_i=\omega_i$. If the finite Artinian $\Omega$-algebra +$S$ had more than one orbit of points, its product decomposition +would give a nontrivial $G$-invariant idempotent $e$, equal to one on +one orbit and zero on the others. But its norm is $e^{|G|}=e$ and is +scalar, so $e=0$ or $e=1$, a contradiction. Thus every geometric fiber +is transitive, without any assertion that invariants commute with +base change. The prime and residue-field formulation is also given +in \cite[Tags~0BRI and 0BRJ]{Stacks}. Thus the hypotheses of +Theorem~\ref{ds:descent} apply to both extensions. + +Use Theorem~\ref{len:input} to choose the algebras and normalized +lengths +\[ + (C_D,\lambda_D)=(C_{D'},\lambda_{D'}),\qquad + (C_E,\lambda_E)=(C_{E'},\lambda_{E'}). +\] +They remain fixed throughout the proof. Set +$a_D=\rank_D D'$ and $a_E=\rank_E E'$. For $\sigma\in G_D$, +define +\[ + \iota_\sigma:R\longrightarrow D' + \xrightarrow{\sigma}D'. +\] +Give $D'$ this $R$-action, and give $C_D$ the action obtained by +composing $\iota_\sigma$ with $D'\to C_D$. Define +\begin{equation}\label{pos:chi-sigma} + c_\sigma + =\sum_{i=0}^d(-1)^i + \lambda_D\bigl(\Tor_i^R(C_D,E')\bigr), +\end{equation} +where this $R$-action on $C_D$ is understood. + +\begin{lemma}\label{pos:first-average} +The lengths in Equation~\eqref{pos:chi-sigma} are finite, and +\begin{equation}\label{pos:first-average-equation} + |G_D|a_Da_E\,\chi^R(D,E)=\sum_{\sigma\in G_D}c_\sigma. +\end{equation} +\end{lemma} + +\begin{proof} +Take a finite $R$-free resolution of $E'$ and extend it to $D'$. +The resulting bounded free complex is supported at the closed point: +after localizing its homology away from $\m_R$, the supports of +$D'$ and $E'$ are disjoint. The same holds after every twist, whose +kernel on $R$ is still $P$. Lemma~\ref{ds:ordinary-norm} supplies +finite normalized homology lengths and equality of the ordinary and +normalized Euler evaluations after summing all conjugates. + +Applying $\sigma$ entrywise gives a semilinear isomorphism from the +untwisted complex to its conjugate, so all ordinary Euler +characteristics in that sum are equal to $\chi^R(D',E')$. Ordinary +lengths over $D'$ and over $R$ agree here because both residue fields +are $k$. + +Choose an injection $D^{a_D}\to D'$ with torsion cokernel. The +cokernel has dimension less than $h$, and therefore its intersection +characteristic with $E'$ vanishes by Equation~\eqref{pos:vanishing}. +Additivity gives +\[ + \chi^R(D',E')=a_D\chi^R(D,E'). +\] +The analogous argument for $E\subset E'$ gives +$\chi^R(D,E')=a_E\chi^R(D,E)$. The norm comparison now yields +Equation~\eqref{pos:first-average-equation}. +\end{proof} + +It remains to show $c_\sigma>0$ for each $\sigma$. Fix one +$\sigma$ until the final paragraph, and write $C=C_D$ and +$\lambda=\lambda_D$ with the chosen $R$-action. + +\subsection{A parameter quotient and its Tor groups} + +Choose +\begin{equation}\label{pos:parameters} + x_1,\ldots,x_h\in Q +\end{equation} +which form an $R$-regular sequence and whose images are a system of +parameters of $D$. Here is the simultaneous prime-avoidance +argument. After $i-1$ choices, the associated primes of +$R/(x_1,\ldots,x_{i-1})$ have height $i-1$, since $R$ is +Cohen--Macaulay. None contains $Q$, whose height is $h\ge i$. +Also avoid the primes in $D/(x_1,\ldots,x_{i-1})$ which obstruct the +next parameter dimension drop. Each has positive dimension; if its +inverse image contained $Q$, it would contain $P+Q$ and hence would +be the maximal ideal, a contradiction. Prime avoidance inside $Q$ +therefore makes both choices possible at every step. + +Put +\[ + B=R/(x_1,\ldots,x_h),\qquad + I_\sigma=(\iota_\sigma(x_1),\ldots, + \iota_\sigma(x_h))D',\qquad U=C/I_\sigma C. +\] +The ideal $I_\sigma$ is a parameter ideal of $D'$, since finiteness +and then the automorphism $\sigma$ preserve that property. Thus +Theorem~\ref{len:input} gives +\begin{equation}\label{pos:U-positive} + 0<\lambda(U)=e(I_\sigma,D')<\infty, + \qquad + \lambda\bigl(H_q(x;C)\bigr)=0\quad(q>0). +\end{equation} +Here the notation $x$ in a complex over $C$ uses $\iota_\sigma$. +Some integer $a\ge1$ satisfies +\begin{equation}\label{pos:uniform-annihilator} + \m_B^a(D'/I_\sigma)=0. +\end{equation} +In particular it annihilates $U$ and every module over $D'/I_\sigma$. + +\begin{lemma}\label{pos:tor-bound} +For every finite $B$-module $L$, all the following lengths are finite, +and +\begin{equation}\label{pos:tor-identification} + \tau_i(L):=\lambda\bigl(\Tor_i^B(U,L)\bigr) + =\lambda\bigl(\Tor_i^R(C,L)\bigr), + \qquad \tau_i(L)=0\quad(i>l). +\end{equation} +\end{lemma} + +\begin{proof} +The complexes in question have compatible $C$-actions. The Koszul +homology $H_q(x;C)$ is annihilated by $x$ and by $P$, and is therefore +bounded-primary-torsion. For $q>0$ its length is zero by +Equation~\eqref{pos:U-positive}. A free $B$-resolution of $L$, +with each term of finite rank, gives +\[ + \lambda\bigl(\Tor_p^B(H_q(x;C),L)\bigr)=0\quad(q>0). +\] +Indeed every such homology is a subquotient of a finite direct sum +of a length-zero module. For $q=0$ these lengths are finite because +$\lambda(U)<\infty$. The change-of-rings spectral sequence +\[ + \Tor_p^B\bigl(\Tor_q^R(C,B),L\bigr) + \ \Longrightarrow\ \Tor_{p+q}^R(C,L) +\] +now gives the asserted equality of lengths. Only finitely many +terms enter each total degree. + +For the sharper bound, choose an $R$-free resolution $P_L$ of $L$ +in degrees $0,\ldots,d$ and consider +\[ + K(x;R)\otimes_R C\otimes_R P_L. +\] +Taking Koszul homology first, all positive rows have length zero, +and the zero row is $U\otimes_R P_L$. Consequently the total +homology has length zero in degrees greater than $d$. On the other +hand, $xL=0$, so in the derived category of $R$-modules +\[ + K(x;R)\otimes_R^{\mathbf L}L + \simeq\bigoplus_{j=0}^h L[j]^{\binom hj}. +\] +Tensoring this identity derivedly with $C$ preserves the splitting +as a statement about $C$-complexes. The total homology in degree +$i+h$ therefore contains $\Tor_i^R(C,L)$ as a direct summand. If +$i>l=d-h$, that total degree exceeds $d$, proving the bound. This +argument also shows directly that all the displayed lengths are +finite. +\end{proof} + +\subsection{Finite approximations preserving the actions} + +We need finite-length $B$-modules whose normalized Tor lengths approach +those of $U$, while retaining the single annihilator +$\m_B^a$. No finite generation or finite presentation of $U$ is +assumed. The approximation will take place among modules carrying +both its tower action and its $B$-action. In this subsection, +$\lambda$ denotes $\lambda_\infty$ on underlying tower modules, +as permitted by Theorem~\ref{len:input}; the approximating modules +need not carry a $C$-action. + +Let $A=A_D$, let $A_j$ be its flat regular tower from +Lemma~\ref{len:tower}, and let $\Lambda=\bigcup_j A_j$. +Thus +\[ + s(j):=\rank_A A_j=p^{jh}. +\] +Set +\begin{equation}\label{pos:approximation-algebras} + S_j=A_j\otimes_A(D'/I_\sigma),\qquad + S=\Lambda\otimes_A(D'/I_\sigma). +\end{equation} +The commuting $\Lambda$- and $D'/I_\sigma$-actions on $U$ combine +into an $S$-action. Working with $S$-modules retains both of these +actions. Descent to $S_j$ retains the existing $B$-action by +restriction along $B\to D'/I_\sigma$ induced by $\iota_\sigma$; +no new $B$-action is chosen at a finite stage. + +\begin{lemma}\label{pos:approximation} +There are finite-length $B$-modules $M_n$ and positive integers +$s_n\to\infty$, with $\m_B^aM_n=0$, such that +\begin{align} + \frac{\length_B(M_n)}{s_n}&\longrightarrow\lambda(U), + \label{pos:approximation-length}\\ + \frac{\length_B\Tor_i^B(M_n,L)}{s_n} + &\longrightarrow\tau_i(L) + \quad(L=k,E',\ i\ge0). + \label{pos:approximation-tor} +\end{align} +Writing $\beta_0(M_n)=\dim_k(M_n/\m_BM_n)$, we also have +\begin{equation}\label{pos:generator-bound} + \frac{\beta_0(M_n)}{s_n}\longrightarrow\nu_\sigma, + \qquad + \nu_\sigma\ge + \frac{\lambda(U)}{\length_B(B/\m_B^a)}>0. +\end{equation} +\end{lemma} + +\begin{proof} +The algebra $S$ is finitely presented as a $\Lambda$-module and +has finite normalized length. Every $S_j$ is finite over the +twisted image of $B$: $D'$ is finite over the twisted image of +$R/P$, and $A_j$ is finite free over $A$. Equation +\eqref{pos:uniform-annihilator} annihilates $S_j$ and $S$. +Moreover, $S_j$ is Artinian local with residue field $k$. To see +locality, the tower parameter roots are nilpotent over the +Artinian quotient $D'/I_\sigma$, and reduction leaves the common +residue field $k$. Thus finite $S_j$-modules have finite lengths +over both $B$ and $A_j$, and these lengths agree: a composition +series over $S_j$ has factors $k$, of length one for either ring. +This argument is unchanged if a residue-field identification is +composed with an automorphism. + +Given $\epsilon>0$, choose a finitely generated $S$-submodule +$V\subset U$ with +$\lambda(U/V)<\epsilon$. This follows from the supremum property +of normalized length: the $S$-span of a finitely generated +$\Lambda$-submodule is still finitely generated over $\Lambda$. +To obtain a finite presentation as well, choose a surjection $S^r\to V$ +with kernel $K$. Its kernel has finite length, bounded by +$r\lambda(S)$. Choose a finitely generated $S$-submodule $K_0\subset K$ such +that $\lambda(K/K_0)<\epsilon$. Then +\[ + P_\epsilon=S^r/K_0\longrightarrow U +\] +has kernel and cokernel of length less than $\epsilon$. +It is finitely presented over $S$ and over $\Lambda$, and is +still annihilated by $\m_B^a$. + +Fix $B$-resolutions of $k$ and $E'$ with finite free terms, and write +$r_i(L)$ for the rank in degree $i$, with $r_{-1}(L)=0$. +If $K_\epsilon$ and $W_\epsilon$ are the kernel and cokernel of +$P_\epsilon\to U$, the two long exact homology sequences give +\begin{align} + &\left|\lambda\Tor_i^B(P_\epsilon,L) + -\lambda\Tor_i^B(U,L)\right| \notag\\ + &\quad\le + \bigl(r_i(L)+r_{i-1}(L)\bigr)\lambda(K_\epsilon) + +\bigl(r_{i+1}(L)+r_i(L)\bigr)\lambda(W_\epsilon). + \label{pos:tor-approximation-error} +\end{align} +For example, each homology of the complex obtained by tensoring +the resolution with $K_\epsilon$ has length at most the rank of +the corresponding term times $\lambda(K_\epsilon)$; the same +holds for $W_\epsilon$. All these quantities are finite. Also +$|\lambda(P_\epsilon)-\lambda(U)|<2\epsilon$. + +A finite $S$-presentation of $P_\epsilon$ descends to an +$S_j$-module $P_j$ for some $j$, with +$P_\epsilon=\Lambda\otimes_{A_j}P_j$. Flatness of +$\Lambda/A_j$ gives, for every $i$, +\[ + \Tor_i^B(P_\epsilon,L) + =\Lambda\otimes_{A_j}\Tor_i^B(P_j,L). +\] +The finite-stage homology on the right is finite over $S_j$. +The normalization formula in Theorem~\ref{len:input}, and the +length comparison above, therefore give exactly +\[ + \lambda\Tor_i^B(P_\epsilon,L) + =\frac{\length_{A_j}\Tor_i^B(P_j,L)}{s(j)} + =\frac{\length_B\Tor_i^B(P_j,L)}{s(j)}. +\] +The same formula holds for $P_\epsilon$ itself. + +At step $n$, choose $\epsilon$ so that all errors in +Equation~\eqref{pos:tor-approximation-error}, for $i\le n$ and +$L=k,E'$, and the length error are less than $1/n$. +Descend this presentation and enlarge its stage if necessary so +that $j\ge n$. Further flat tower extension leaves every +normalized value unchanged. Take $M_n=P_j$ and $s_n=s(j)$. +This proves Equations~\eqref{pos:approximation-length} and +\eqref{pos:approximation-tor}, including the common annihilator +and $s_n\to\infty$. + +The case $i=0,L=k$ gives the limit in +Equation~\eqref{pos:generator-bound}. A minimal generating set and the +annihilator give a surjection +$(B/\m_B^a)^{\beta_0(M_n)}\to M_n$. Consequently +\[ + \length_B(M_n) + \le\beta_0(M_n)\length_B(B/\m_B^a). +\] +Divide by $s_n$ and take limits to obtain the stated positive +lower bound. +\end{proof} + +\subsection{Bounded complexes and their ordinary evaluation} + +The approximations provide two kinds of control. In minimal +resolutions, the Tor bound $\tau_i(k)=0$ for $i>l$ makes the +corresponding ranks negligible. The common annihilator then makes +the truncated sequences +supported after quotienting by negligible ranks. That same annihilator +has already given the positive normalized generator bound +\eqref{pos:generator-bound}, which will make the second evaluation +strictly positive. + +Choose minimal free $B$-resolutions $F_n^\infty$ of $M_n$, indexed +homologically in nonnegative degrees. Let $F_n$ be their brutal +truncations to degrees $0,\ldots,l$. Minimality and +Equation~\eqref{pos:approximation-tor} give, in each fixed degree, +\begin{equation}\label{pos:rank-estimates} + \rank_B F_{n,i}^\infty=O(s_n),\qquad + \rank_B F_{n,i}^\infty=o(s_n)\quad(i>l), +\end{equation} +because these ranks equal $\length_B\Tor_i^B(M_n,k)$. + +Let $z_1,\ldots,z_l$ be regular parameters of $A_E$. For +$\gamma\in G_E$, extend $F_n$ through +\[ + B\longrightarrow E\longrightarrow E' + \xrightarrow{\gamma}E', +\] +and denote the resulting complex by $F_{n,\gamma}'$. +Use the normalization $s_n$ in the asymptotic category over $E'$. +These numbers come from the $D$-side approximation. Section~\ref{sec:asymptotic} +allows any positive integer sequence $s_n$, so no identification or +compatibility between the parameter-root towers of $A_D$ and $A_E$ +is needed. + +\begin{lemma}\label{pos:supported-truncation} +The sequence $(F_{n,\gamma}')_n$ is supported at the closed point +in that category. Its ordinary Euler evaluation is $c_\sigma$ +for every $\gamma\in G_E$. +\end{lemma} + +\begin{proof} +Every element $b\in\m_B^a$ acts nullhomotopically on +$F_n^\infty$, because it annihilates $M_n$. Restrict a chosen +homotopy to the brutal truncation. Its only remaining error is +in degree $l$, where it is +\begin{equation}\label{pos:truncation-error} + F_{n,l}\xrightarrow{h_l}F_{n,l+1}^\infty + \xrightarrow{d_{l+1}}F_{n,l}. +\end{equation} +This is a factorization by chain maps through +$F_{n,l+1}^\infty[l]$: the map induced by $d_{l+1}$ is a chain map +because $d_ld_{l+1}=0$, and the map induced by $h_l$ is a chain map +because the truncation has no degree $l+1$ term. The middle module has rank +$o(s_n)$ by Equation~\eqref{pos:rank-estimates}, and is therefore killed +in the asymptotic quotient. These identities persist after +extension to $E'$, without a flatness assumption over $B$. + +The ideal $\m_BE'$ for each indicated structural map is primary +to $\m_{E'}$. Choose $v$ so large that every $z_j^v$ belongs to +the extension of $\m_B^a$ for all $\gamma$. This is one fixed +choice, since the group is finite. Expressing $z_j^v$ as a +finite $E'$-linear combination of elements of $\m_B^a$ proves +that it acts by zero on the sequence in the quotient. The +same combinations give genuine nullhomotopies on the full +extended resolutions. This proves support. + +Apply the ordinary version of +Equation~\eqref{as:koszul-polynomial}. In its right-hand side, one may +replace the truncated extended resolutions by the full ones in +each fixed homological degree. Indeed, the quotient of the +full resolution by its brutal truncation has only terms of +degrees at least $l+1$, all of rank $o(s_n)$ in each fixed degree. +After tensoring that tail with $K(z^v;E')$, taking Koszul +homology first bounds each fixed-degree homology by finite sums +of +\[ + \rank_B F_{n,i}^\infty\, + \length_{E'} H_q(z^v;E')=o(s_n). +\] +The Koszul homology lengths are fixed and finite, and only +finitely many terms contribute to each total degree. The long +exact homology sequence thus proves the replacement assertion. + +On each full extended resolution the powers $z_j^v$ act +nullhomotopically, as established above. Koszul tensoring +therefore gives the direct sum of shifts with multiplicities +$\binom lj$. Its underlying homology has ordinary lengths +$\length_B\Tor_i^B(M_n,E')$, unchanged by conjugating the +$E'$-action: applying $\gamma$ entrywise is a semilinear +isomorphism. Equations~\eqref{pos:approximation-tor} and +\eqref{pos:tor-identification}, followed by coefficient +comparison in Equation~\eqref{as:koszul-polynomial}, now identify the +ordinary homology-length coefficients with +\[ + \tau_i(E')=\lambda\Tor_i^R(C,E'). +\] +Their alternating sum is exactly $c_\sigma$. +\end{proof} + +\subsection{A positive normalized evaluation} + +The ordinary evaluation has recovered $c_\sigma$ on every conjugate. +We now evaluate the same supported object using $C_E$. The parameter +Koszul identities will leave only its nonnegative degree-zero +coefficient, and minimality of the resolutions will bound that +coefficient below using \eqref{pos:generator-bound}. + +Corollary~\ref{len:finite-length-base-change} gives +\begin{equation}\label{pos:residue-positive} + 0<\eta_E:=\lambda_E(C_E/\m_{E'}C_E)<\infty. +\end{equation} +We record a quantitative bound for use below. Set +$q_E=(z_1,\ldots,z_l)E'$, +$r_E=\rank_{A_E}E'$, and +$t_E=\length_{E'}(E'/q_E)$. The length input gives +$\lambda_E(C_E/q_EC_E)=r_E>0$. +A composition series of $E'/q_E$, tensored right-exactly with +$C_E$, shows +\begin{equation}\label{pos:residue-bound} + r_E\le t_E\eta_E. +\end{equation} +At each step the image of the tensor of the submodule has +length at most that tensor's length; injectivity after tensoring +is not required. Finiteness of $\eta_E$ follows from the +surjection $C_E/q_EC_E\to C_E/\m_{E'}C_E$. + +\begin{lemma}\label{pos:positive-evaluation} +The $\lambda_E$ Euler evaluation of $(F_{n,\gamma}')_n$ is a +number $b_{\sigma,\gamma}$ satisfying +\begin{equation}\label{pos:strict-bound} + b_{\sigma,\gamma} + \ge\nu_\sigma\eta_E + \ge + \frac{\lambda_D(U)}{\length_B(B/\m_B^a)} + \frac{r_E}{t_E} + >0. +\end{equation} +\end{lemma} + +\begin{proof} +Let $b_i$ be the nonnegative homology-length coefficients of +this evaluation in Equation~\eqref{as:koszul-polynomial}. Consider the +finite double complex +\[ + K(z^v;E')\otimes_{E'}F_{n,\gamma}' + \otimes_{E'}C_E. +\] +Taking Koszul homology first, Theorem~\ref{len:input} makes all +positive Koszul rows have length zero. The remaining row has +complex degrees $0,\ldots,l$. Thus the normalized homology +lengths of this double complex vanish in total degrees greater +than $l$, as well as in negative degrees. Equation +\eqref{as:koszul-polynomial} says that their polynomial is +$(1+t)^l\sum_i b_it^i$. All $b_i$ are nonnegative. Boundedness +and comparison of the least degree rule out negative-degree +coefficients. Any $b_i>0$ with $i>0$ would then give a positive +coefficient in degree $i+l>l$, a contradiction. Consequently +only $b_0$ can be nonzero, and the Euler evaluation equals $b_0$. + +In degree zero, the displayed double complex has homology +\[ + C_E^{\beta_0(M_n)}\big/ + \left(\operatorname{im}(d_1)+(z_1^v,\ldots,z_l^v) + C_E^{\beta_0(M_n)}\right). +\] +Since $l>0$, $d_1$ is the degree-one presentation differential +of $M_n$. Minimality over $B$ puts all its entries in $\m_B$, +whose image under any twist is contained in $\m_{E'}$. +The base parameter powers are also in $\m_{E'}$. There is +therefore a surjection from this degree-zero homology onto +\[ + (C_E/\m_{E'}C_E)^{\beta_0(M_n)}. +\] +Its normalized length is at least +$\beta_0(M_n)\eta_E/s_n$. The degree-zero coefficient in +Equation~\eqref{as:koszul-polynomial} is $b_0$, so taking the ultralimit +and using the ordinary limit in Equation~\eqref{pos:generator-bound} +gives $b_0\ge\nu_\sigma\eta_E$. Equations +\eqref{pos:generator-bound} and \eqref{pos:residue-bound} give +the explicit remaining inequalities in +Equation~\eqref{pos:strict-bound}. Define $b_{\sigma,\gamma}=b_0$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +By Lemma~\ref{pos:reduction}, use the domain pair fixed above. +For each $\sigma\in G_D$, carry out the construction with its +specified $R$-action. Apply Theorem~\ref{as:norm} to the supported +sequence of +Lemma~\ref{pos:supported-truncation}, over $A_E\subset E'$. +It equates the sums of the ordinary and normalized evaluations +of all $G_E$-conjugates. The former evaluation is $c_\sigma$ +for every conjugate, whereas the latter is +$b_{\sigma,\gamma}$. Hence +\begin{equation}\label{pos:second-average} + |G_E|c_\sigma + =\sum_{\gamma\in G_E}b_{\sigma,\gamma}>0 +\end{equation} +by Lemma~\ref{pos:positive-evaluation}. +Finally, Lemma~\ref{pos:first-average} gives +\[ + |G_D|a_Da_E\,\chi^R(D,E) + =\frac{1}{|G_E|} + \sum_{\sigma\in G_D}\sum_{\gamma\in G_E} + b_{\sigma,\gamma}>0. +\] +Both ranks and both group orders are positive. Thus +$\chi^R(D,E)>0$, and the reduction proves the theorem. +Neither norm comparison requires a Galois action on $C_D$ or +$C_E$, nor invariance of either normalized length under a twist. +\end{proof} diff --git a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..49bbfd915ff4d2851f41a31665213852d7d44ce2 --- /dev/null +++ b/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7985cfbc11547cf659efdd6abc076a0c0371e9a38e8026aa09b23cbb042af917 +size 531450 diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/README.md b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..fc387532002be4d4b98e0d46e374d9cbbfc5462e --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/README.md @@ -0,0 +1,16 @@ +# [Posterior replicas and conditional information in Gaussian regression](paper.pdf) + +OpenAI · September 27, 2026 + +## Citation + +```bibtex +@misc{OAI:Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026, + author = {{OpenAI}}, + title = {{Posterior replicas and conditional information in Gaussian regression}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf}{OAI:Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/main.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..085ff68ded3d93b04cd2b54a0134e07dd0d6265d --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/main.tex @@ -0,0 +1,31 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{Posterior replicas and conditional information in Gaussian regression} +\author{OpenAI} +\date{September 27, 2026} +\begin{document} +\maketitle +\begin{abstract} +For a signal with density bounded by $L$ relative to uniform probability +on $S^{d-1}$, we bound the information in a finite message $W$ formed +from exact Gaussian measurements, conditional on an independent projection +revealed only to the analyst. For explicit row counts proportional to $d$, +the bound is $O(H(W)/d+d+\log(2+\log L))$. +Consequently, a finite-state learner with $o(d^2)$ persistent bits and a +deterministic sample horizon needs $\Omega(d\log(1/\epsilon))$ fresh +noiseless Gaussian measurements for constant-probability angular accuracy +$0<\epsilon\le1/10$ under the uniform spherical prior. +\end{abstract} +\tableofcontents +\input{sections/00-introduction} +\input{sections/01-model-endpoint} +\input{sections/02-replicas} +\input{sections/03-gaussian} +\input{sections/04-distance-bins} +\input{sections/08-streaming} +\input{sections/05-haar} +\input{sections/06-synthetic} +\input{sections/07-incidence} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/preamble.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..463f2b8587b111db443b2ead10286e4953a53209 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/preamble.tex @@ -0,0 +1,42 @@ +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem,booktabs} +\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{claim}[theorem]{Claim} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\newcommand{\sphere}{S^{d-1}} +\newcommand{\norm}[1]{\lVert #1\rVert} +\newcommand{\ip}[2]{\langle #1,#2\rangle} +\newcommand{\ind}{\mathbf 1} +\newcommand{\KL}{D_{\mathrm{KL}}} +\newcommand{\op}{\mathrm{op}} +\DeclareMathOperator{\aff}{aff} +\DeclareMathOperator{\col}{col} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\tc}{tc} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\diag}{diag} +\hypersetup{ + pdftitle={Posterior replicas and conditional information in Gaussian regression}, + pdfauthor={OpenAI}, + pdfsubject={Conditional information from exact Gaussian observations}, + bookmarksopen=true,bookmarksnumbered=true +} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/references.bib b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..2ca083cab7534d2870826d4cee10fc09c542f53a --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/references.bib @@ -0,0 +1,208 @@ +@inproceedings{SteinhardtDuchi2015, + author = {Steinhardt, Jacob and Duchi, John}, + title = {Minimax Rates for Memory-Bounded Sparse Linear Regression}, + booktitle = {Proceedings of the 28th Conference on Learning Theory}, + series = {Proceedings of Machine Learning Research}, + volume = {40}, + year = {2015}, + pages = {1564--1587}, + url = {https://proceedings.mlr.press/v40/Steinhardt15.html} +} + +@article{LelargeMiolane2019, + author = {Lelarge, Marc and Miolane, L{\'e}o}, + title = {Fundamental limits of symmetric low-rank matrix estimation}, + journal = {Probability Theory and Related Fields}, + volume = {173}, + year = {2019}, + pages = {859--929}, + doi = {10.1007/s00440-018-0845-x}, + note = {Proposition 16 refers to the author version arXiv:1611.03888v3, 30 March 2017. \url{https://arxiv.org/abs/1611.03888v3}} +} + +@inproceedings{BassilyEtAl2018, + author = {Bassily, Raef and Moran, Shay and Nachum, Ido and Shafer, Jonathan and Yehudayoff, Amir}, + title = {Learners that Use Little Information}, + booktitle = {Proceedings of Algorithmic Learning Theory}, + series = {Proceedings of Machine Learning Research}, + volume = {83}, + year = {2018}, + pages = {25--55}, + url = {https://proceedings.mlr.press/v83/bassily18a.html}, + note = {Appendix A.1, Lemmas 14--15, in the published PMLR version} +} + +@misc{OpenAIMemoryPrecision2026, + author = {{OpenAI}}, + title = {{Memory and precision in noiseless Gaussian regression}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf}{OAI:Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026}}, + year = {2026} +} + +@misc{OpenAIReplacing2026, + author = {{OpenAI}}, + title = {{Replacing Gaussian observations in memory-constrained inference}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf}{OAI:Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026}}, + year = {2026} +} + +@misc{Raz2016, + author = {Raz, Ran}, + title = {Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning}, + year = {2016}, + eprint = {1602.05161}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1602.05161v1}, + note = {arXiv:1602.05161v1, version 1, 16 February 2016. \url{https://arxiv.org/abs/1602.05161v1}} +} + +@inproceedings{Raz2017, + author = {Raz, Ran}, + title = {A Time-Space Lower Bound for a Large Class of Learning Problems}, + booktitle = {Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science}, + year = {2017}, + pages = {732--742}, + doi = {10.1109/FOCS.2017.73}, + url = {https://eccc.weizmann.ac.il/report/2017/020/}, + note = {ECCC Report TR17-020, 12 February 2017; theorem and section locators refer to that report} +} + +@inproceedings{SSV2019, + author = {Sharan, Vatsal and Sidford, Aaron and Valiant, Gregory}, + title = {Memory-Sample Tradeoffs for Linear Regression with Small Error}, + booktitle = {Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing}, + year = {2019}, + pages = {890--901}, + doi = {10.1145/3313276.3316403}, + url = {https://arxiv.org/abs/1904.08544v2}, + note = {Full version arXiv:1904.08544v2, 10 October 2020; theorem and section locators refer to that version} +} + +@inproceedings{DKS2019, + author = {Dagan, Yuval and Kur, Gil and Shamir, Ohad}, + title = {Space Lower Bounds for Linear Prediction in the Streaming Model}, + booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, + series = {Proceedings of Machine Learning Research}, + volume = {99}, + year = {2019}, + pages = {929--954}, + url = {https://proceedings.mlr.press/v99/dagan19b.html}, + note = {Theorem 3 and Theorem 10 locators refer to the published PMLR version} +} + +@article{Watanabe1960, + author = {Watanabe, Satosi}, + title = {Information Theoretical Analysis of Multivariate Correlation}, + journal = {IBM Journal of Research and Development}, + volume = {4}, + number = {1}, + year = {1960}, + pages = {66--82}, + doi = {10.1147/rd.41.0066}, + url = {https://doi.org/10.1147/rd.41.0066} +} + +@article{Austin2020, + author = {Austin, Tim}, + title = {Multi-variate correlation and mixtures of product measures}, + journal = {Kybernetika}, + volume = {56}, + number = {3}, + year = {2020}, + pages = {459--499}, + doi = {10.14736/kyb-2020-3-0459}, + url = {https://www.kybernetika.cz/content/2020/3/459} +} + +@article{Federer1959, + author = {Federer, Herbert}, + title = {Curvature measures}, + journal = {Transactions of the American Mathematical Society}, + volume = {93}, + number = {3}, + year = {1959}, + pages = {418--491}, + doi = {10.1090/S0002-9947-1959-0110078-1}, + url = {https://doi.org/10.1090/S0002-9947-1959-0110078-1} +} + +@article{Drury1984, + author = {Drury, S. W.}, + title = {Generalizations of {Riesz} potentials and {$L^p$} estimates for certain {$k$}-plane transforms}, + journal = {Illinois Journal of Mathematics}, + volume = {28}, + number = {3}, + year = {1984}, + pages = {495--512} +} + +@article{EdelmanRao2005, + author = {Edelman, Alan and Rao, N. Raj}, + title = {Random matrix theory}, + journal = {Acta Numerica}, + volume = {14}, + year = {2005}, + pages = {233--297}, + doi = {10.1017/S0962492904000236}, + url = {https://math.mit.edu/~edelman/publications/random_matrix_theory.pdf}, + note = {Page locators refer to the author manuscript at \url{https://math.mit.edu/~edelman/publications/random_matrix_theory.pdf}, accessed 27 September 2026} +} + +@article{EdelmanAriasSmith1998, + author = {Edelman, Alan and Arias, Tom{\'a}s A. and Smith, Steven T.}, + title = {The geometry of algorithms with orthogonality constraints}, + journal = {SIAM Journal on Matrix Analysis and Applications}, + volume = {20}, + number = {2}, + year = {1998}, + pages = {303--353}, + doi = {10.1137/S0895479895290954}, + url = {https://math.mit.edu/~edelman/publications/geometry_of_algorithms.pdf} +} + +@article{HardyLittlewood1930, + author = {Hardy, G. H. and Littlewood, J. E.}, + title = {A maximal theorem with function-theoretic applications}, + journal = {Acta Mathematica}, + volume = {54}, + year = {1930}, + pages = {81--116}, + doi = {10.1007/BF02547518}, + url = {https://doi.org/10.1007/BF02547518} +} + +@article{SteinStromberg1983, + author = {Stein, E. M. and Str{\"o}mberg, J.-O.}, + title = {Behavior of maximal functions in {$\mathbb{R}^n$} for large {$n$}}, + journal = {Arkiv f{\"o}r Matematik}, + volume = {21}, + year = {1983}, + pages = {259--269}, + doi = {10.1007/BF02384314}, + url = {https://doi.org/10.1007/BF02384314} +} + +@article{DupuisMao2022, + author = {Dupuis, Paul and Mao, Yixiang}, + title = {Formulation and properties of a divergence used to compare probability measures without absolute continuity}, + journal = {ESAIM: Control, Optimisation and Calculus of Variations}, + volume = {28}, + year = {2022}, + note = {Article 10}, + doi = {10.1051/cocv/2022002}, + url = {https://www.numdam.org/articles/10.1051/cocv/2022002/} +} + +@article{FranklMaehara1990, + author = {Frankl, Peter and Maehara, Hiroshi}, + title = {Some Geometric Applications of the Beta Distribution}, + journal = {Annals of the Institute of Statistical Mathematics}, + volume = {42}, + number = {3}, + year = {1990}, + pages = {463--474}, + doi = {10.1007/BF00049302}, + note = {\url{https://doi.org/10.1007/BF00049302}} +} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/00-introduction.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..45ed0fbb65dddb2e9ffc72ac31e629bb54532917 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/00-introduction.tex @@ -0,0 +1,241 @@ +\section{Introduction}\label{sec:introduction} + +Consider an unknown unit vector $s\in S^{d-1}$ and a stream of independent +Gaussian rows $X_j\sim N(0,I_d)$ with exact labels +$Y_j=\langle X_j,s\rangle$. If all raw observations can be retained, $d$ +rows determine $s$ almost surely, because their matrix is invertible. +A learner with finite persistent memory must instead replace each observed +pair by an update of its state. The question is how this restriction changes +the number of observations needed for a prescribed angular accuracy. +Exact labels make the question particularly delicate: an individual real +label has no prescribed bit precision. + +We study this problem through a conditional information bound. A finite +message is formed from one block of exact observations, and a separate, +independent projection of the signal is then revealed to the analyst. +We bound the information that remains in the message after this revelation. +The projection is an analysis variable throughout; it is not available to +the learner or to the rule forming the message. + +\subsection*{Related memory--sample bounds} + +Steinhardt and Duchi established memory and communication lower bounds for +noisy sparse regression, including a minimax squared-error lower bound +$\Omega(\sigma^2\min\{kd/(Bn),1\})$ in their model with sparsity $k$, +noise scale $\sigma$, and $n$ observations. Here $B$ bounds memory in bits +or average communication per observation +\cite[Theorem~2]{SteinhardtDuchi2015}. +For learning over finite fields, Raz proved that learning a parity requires +either quadratic memory or exponentially many samples, and extended the +branching-program method to classes of finite learning problems defined by +spectral conditions~\cite[Theorem~1]{Raz2016}\cite[Theorem~1]{Raz2017}. +These results distinguish persistent memory from unrestricted computation +on a newly observed sample. + +A closely related continuous regression result is due to Sharan, Sidford, +and Valiant~\cite[Theorem~1 and Section~2]{SSV2019}. With Gaussian rows, +independent uniform noise in $[-2^{-d/5},2^{-d/5}]$, and at most $d^2/4$ +bits, their bound is $\Omega(d\log r)$ observations for Euclidean accuracy +$\epsilon=d^{-r}$ in the stated range $r\le O(d/\log d)$. Their +$\Omega(d\log\log(1/\epsilon))$ consequence is stated for +$\epsilon\le d^{-\Omega(\log d)}$ within that range. +Their model already permits nonuniform finite-width +branching programs on real-valued data. The result here concerns exact +labels and $o(d^2)$ memory, and gives the single-logarithmic precision cost +$\Omega(d\log(1/\epsilon))$. These memory and noise hypotheses differ: +a lower bound with noisy labels alone does not imply a lower bound with +exact labels, since a learner receiving exact labels can add noise itself. + +Dagan, Kur, and Shamir proved quadratic-space lower bounds for finding +an approximate null vector of independent Gaussian rows and for achieving +small empirical residual on a random-order consistent linear system +\cite[Theorems~3 and~10]{DKS2019}. Their tasks explain another way that +linear algebra can demand quadratic storage, but use different recovery +criteria and observation laws from the fixed-signal stream considered here. + +Two companion papers give complementary treatments of exact observations. +\emph{Memory and precision in noiseless Gaussian regression} +\cite[Theorem~1.2]{OpenAIMemoryPrecision2026} obtains the same precision +scale for every fixed $A>0$ and memory $M\le Ad^2$, with a constant depending +on $A$, by propagating bounds on future success backward through blocks. +\emph{Replacing Gaussian observations in memory-constrained inference} +\cite[Proposition~7.1 and Lemma~7.2]{OpenAIReplacing2026} develops a +conditional-information comparison based on a finite two-point fiber +measure. The present paper instead develops the multi-replica block +comparisons stated below and proves its streaming consequence from them. + +\subsection*{Conditional information from one block} + +Let $\sigma_d$ denote uniform probability on $S^{d-1}$. In a block +experiment, the signal $S$ has law $p=f\sigma_d$, where $f$ is a bounded +nonnegative Borel density. Independently draw a standard Gaussian matrix +$A$, and observe $(A,AS)$. A finite-valued message $W$ is drawn from a +measurable kernel of these data, conditionally independently of $S$. +Finally draw a standard Gaussian matrix $B$ independently of the entire +experiment producing $(S,A,AS,W)$. Our principal quantity is +\[ + I(S;W\mid B,BS). +\] +All logarithms and entropies below are in nats. + +\begin{theorem}[Gaussian replica block bound]\label{thm:gaussian-block} +For all sufficiently large $d$, set +\[ + k=2\lfloor d/16\rfloor,\qquad m=k,\qquad t=m+1,\qquad \ell=4k. +\] +Suppose $p=f\sigma_d$ is a probability with $0\le f\le L<\infty$. +In the block experiment above, let $A$ have $k$ rows and let $B$ have +$\ell$ rows. There is an absolute constant $C$ such that +\begin{equation}\label{eq:gaussian-block} + I(S;W\mid B,BS) + \le \frac{H(W)}{t}+Cd+C\log(2+\log L). +\end{equation} +The bound is uniform over the prior, the density bound $L$, and the message +kernel. The labels used to produce $W$ are the exact values $AS$. +\end{theorem} + +The entropy cost is divided by a number $t$ proportional to $d$. +The weak dependence on $L$ matters when the theorem is applied to a +streaming state: conditioning a uniform signal on a state value $v$ of +probability $p_v>0$ gives a density at most $1/p_v$. Rare states may +therefore have large density bounds, even when their average information +cost is small. + +The proof couples several possible signals with the same observed data. +Given $(A,AS)$, draw $S_1,\ldots,S_t$ independently from the posterior, +independently also of $W$ given the data. Each pair $(S_i,W)$ has the +original signal--message law. The replicas are dependent after the data +are forgotten: they satisfy $AS_1=\cdots=AS_t$ for the common hidden +matrix. Write $K$ for the relative entropy of their joint law from +$p^{\otimes t}$, and $K'$ for the corresponding relative entropy of +$(BS_1,\ldots,BS_t)$ from the product of its marginals, conditional on +$B$. These are their total correlations before and after projection. +Lemma~\ref{lem:replica-information} gives +\[ + tI(S;W\mid B,BS)\le H(W)+K-K'. +\] +Thus the proof must bound the \emph{loss} of correlation. Bounding $K$ +alone would discard the geometric dependence still visible to the analyst. + +The exact-label geometry supplies the comparison. For the difference +matrix $D=[S_2-S_1,\ldots,S_t-S_1]$, the replica likelihood contains +$\det(D^{\mathsf T}D)^{-k/2}$. +Theorem~\ref{thm:equal-label-measure} identifies the underlying finite +measure, including its normalization and density values on the constraint. +An observable test of the projected replicas supplies a matching +inverse-volume term. +Section~\ref{sec:gaussian} then bounds the remaining projection-density +ratio by a dyadic decomposition, proving Theorem~\ref{thm:gaussian-block}. + +\subsection*{A streaming consequence} + +The learner receives the fresh pairs $(X_j,Y_j)$ in order, cannot choose +the rows, and cannot revisit them. At each index it has at most $2^M$ +persistent states. Its transition and stopping rules may be arbitrary +measurable randomized functions of the current state and current pair; +all information retained for later indices must be in the state. +It stops at some $\tau\in\{0,\ldots,T\}$ for a prescribed deterministic +horizon $T$. Its unit-vector output uses only the terminal state, the rule +at index $\tau$, and fresh randomness. Thus the final observation affects +the output only through that state. Rules may depend on the index, dimension, +and target accuracy. Shared randomness and initialization are jointly +independent of the signal and samples, and fresh random choices are +independent before use. The entire experiment is required to be jointly +measurable. Definition~\ref{def:streaming-model} records the full model and +the completed-measurability convention. + +\begin{theorem}[Uniform-prior sample lower bound]\label{thm:streaming} +Let $M(d)=o(d^2)$ be nonnegative integer-valued, and let +$0<\epsilon(d)\le1/10$. Suppose a family of learners in +Definition~\ref{def:streaming-model} satisfies +\[ + \Prob\{\arccos\langle\widehat S,S\rangle\le\epsilon(d)\}\ge\frac23 + \qquad\text{when }S\sim\sigma_d, +\] +where the probability includes the signal, samples, and learner randomness. +Then there is an absolute $c>0$ such that +\[ + T(d)\ge c\,d\log\frac1{\epsilon(d)} +\] +for all sufficiently large $d$. The eventual threshold may depend on the +memory sequence. +\end{theorem} + +The theorem assumes success averaged over the uniform sphere. A guarantee +of success at least $2/3$ for every fixed signal implies that hypothesis +by integration, so the same lower bound holds for such learners. +Section~\ref{sec:streaming-consequences} proves the result by conditioning +on the current state, applying a block estimate, and averaging its prior +cost. The independent projection leaves a residual sphere of dimension +proportional to $d$; accurate recovery still requires order +$d\log(1/\epsilon)$ conditional information on that sphere. + +\subsection*{Information and geometric comparisons} + +The total-correlation argument uses Watanabe's measure of multivariate +dependence~\cite{Watanabe1960} and its general probability-space chain +rules~\cite[Sections~3.3 and~6]{Austin2020}. Posterior resampling preserves +the joint law with the observed data, as in the general Nishimori identity +recorded by Lelarge and Miolane~\cite[Proposition~16]{LelargeMiolane2019}. +Here its role is to preserve the signal--message pair law while introducing +dependence among replicas. The projected tests use the +relative-entropy variational principle; a convenient statement is +\cite[Equations~(1.1)--(1.2)]{DupuisMao2022}. The equal-label measure is +related to coarea~\cite[Theorem~3.1]{Federer1959} and to affine +simplex-volume changes of variables~\cite[Lemma~1]{Drury1984}. +We prove the Gaussian probability normalization and the required +evaluations at constrained labels directly. Thus these classical tools +provide the background for the exact-fiber calculation, rather than an +unproved formula for its particular probability law. + +The subsequent comparisons retain different information from that geometry. +For a prior density at most $e^b$, Theorem~\ref{thm:distance-block} uses +$q=\lfloor d/8\rfloor$ replicas and gives the alternative cost +$H(W)/q+C(d+b/d)$. Its test records successive simplex heights in distance +bins and compares the observed bin with a mixture over fixed bins. +The direct linear dependence on $b$ makes averaging over state values +immediate. The other routes use different comparison laws: +Table~\ref{tab:comparisons} records their geometric inputs and the state +information fixed during iteration. Each route verifies its own +normalization and exceptional sets. They offer distinct tests and ways to +average the prior cost, rather than a succession of stronger asymptotic +sample bounds. + +\begin{table}[htbp] +\centering +\small +\setlength{\tabcolsep}{4pt} +\renewcommand{\arraystretch}{1.2} +\begin{tabular}{@{}p{.18\textwidth}p{.24\textwidth}p{.34\textwidth}p{.15\textwidth}@{}} +\toprule +Comparison & Geometric identity & Projected comparison & Conditioning \\ +\midrule +Gaussian selector & Spherical equal-label measure & +Normalized determinant likelihood & Current state \\ +Distance bins & Spherical equal-label measure & +Fixed mixture of height bins & Current state \\ +Haar frame & Successive Haar height insertion & +Distance distribution functions & Current state \\ +Synthetic Gaussian & Spherical and Gaussian equal-label measures & +Projected-prior likelihood with random covariance & Full history \\ +Stiefel incidence & Regular-incidence coarea & +Conditional product law and density ratio & Full history \\ +\bottomrule +\end{tabular} +\caption{The five block comparisons. The last column records the information +conditioned on before applying a block estimate. Every streaming proof also +uses a residual-sphere endpoint as in Section~\ref{sec:model-endpoint}.} +\label{tab:comparisons} +\end{table} + +One complete route occupies Sections~\ref{sec:model-endpoint}--\ref{sec:streaming-consequences}. +Section~\ref{sec:model-endpoint} supplies the learner reductions and the +residual-sphere information requirement. Section~\ref{sec:replicas} +establishes the replica inequality, exact equal-label measure, and +logarithmic integrability. Sections~\ref{sec:gaussian} +and~\ref{sec:distance-bins} prove the two block estimates, and +Section~\ref{sec:streaming-consequences} gives their current-state +iterations. The final three sections develop the other geometric +comparisons and their streaming applications; the synthetic and incidence +iterations condition on full state histories. All applications use the +same exact-row, finite-state, deterministic-horizon learner model. diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/01-model-endpoint.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/01-model-endpoint.tex new file mode 100644 index 0000000000000000000000000000000000000000..3bf5cd5d8ac7ff820f96bb1fea9b3a8334cb90b9 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/01-model-endpoint.tex @@ -0,0 +1,445 @@ +\section{Finite states and the information required at the endpoint} +\label{sec:model-endpoint} + +The block estimates will bound information in a state after an independent +projection is known. After specifying the learner model, we show why an +accurate state must still carry order $d\log(1/\epsilon)$ information in +that experiment. We also record +the reductions for randomized rules, rare states, and bounded stopping, +so that later applications need only specify their block dimensions and +their way of charging the clock. + +\subsection{Finite-state learners} + +\begin{definition}[Finite-state noiseless regression]\label{def:streaming-model} +An unknown unit vector $s\in S^{d-1}$ is fixed before sampling. At index +$j\ge1$ the learner receives +\[ + X_j\sim N(0,I_d),\qquad Y_j=\langle X_j,s\rangle, +\] +where the $X_j$ are independent. It cannot choose a row or revisit an earlier +one. At each index its persistent state has at most $2^M$ values. Its +transition and stopping rule may be any measurable randomized kernel of +the current state and the entire current pair $(X_j,Y_j)$. Computation +within a transition is unrestricted, and every piece of information retained +for later indices must be in the finite state. + +The learner stops at an index $\tau\in\{0,\ldots,T\}$, where the integer +$T$ is a prescribed deterministic horizon. Its unit-vector output is drawn +using only the terminal state, the rule at index $\tau$, and fresh randomness. +In particular, the last observation can affect the output only through the +terminal state. Rules may depend on $d$, the target accuracy, and the index, +and may be selected by a shared random seed. The initial state and that +seed are jointly independent of the signal and the entire sample sequence. +Fresh transition and output random variables may also be used; each is +independent of that pair, the signal, the sample sequence, and all earlier +fresh random variables before it is used. Memory is measured in bits, +not real registers. + +The construction from the signal, pre-generated sample rows, shared seed, +initialization, and fresh transition and output randomness through the +state path, stopping index, and output must form a jointly measurable +probability experiment. Per-seed measurability of the sample kernels alone +is not a definition of the mixture over seeds. +\end{definition} + +For each fixed seed, the transition and stopping coordinates may be +measurable in the completion of the Borel sigma field under +$\lambda(dx,dy)=\gamma_{1,d}(dx)\,dy$ on $\R^d\times\R$, where +$\gamma_{1,d}$ is standard Gaussian probability on $\R^d$. +Lemma~\ref{lem:borel-rules} supplies Borel versions for the fixed-seed +uniform-prior experiment. The original randomized experiment must still +be jointly measurable as stated in the definition. + +\subsection{Spherical caps and residual fibers} + +\begin{lemma}[Spherical balls]\label{lem:spherical-ball} +Let $\sigma_q$ be uniform probability on $S^{q-1}$, where $q\ge2$. +For every $z\in S^{q-1}$ and $r>0$, +\begin{equation}\label{eq:spherical-ball} + \sigma_q\{u:\norm{u-z}\le r\}\le r^{q-1}. +\end{equation} +In particular an angular cap of radius $0<\epsilon\le\pi$ has mass at most +$\epsilon^{q-1}$. +\end{lemma} + +\begin{proof} +For $r\le1$, rotate $z$ to the north pole. A point in the chordal ball has +last coordinate at least $1-r^2/2\ge1/2$. As a graph over its first $q-1$ +coordinates, the surface has area element at most twice Lebesgue measure, +and its projection lies in the $(q-1)$-ball of radius $r$. The whole sphere +has area at least twice the volume of the unit $(q-1)$-ball, by projecting +both hemispheres. This proves \eqref{eq:spherical-ball}. For $r>1$, total +mass one suffices. Angular distance at most $\epsilon$ implies chordal +distance $2\sin(\epsilon/2)\le\epsilon$, proving the last assertion. +\end{proof} + +\begin{lemma}[Residual sphere and radius bounds]\label{lem:residual-sphere} +Let $S\sim\sigma_d$ and let $G$ be independent of $S$, with $b$ rows, +where $0\le b\le d-2$. For $b\ge1$, let $G$ be either a standard Gaussian +matrix or a uniform Haar frame of $b$ orthonormal rows in $\R^d$. For $b=0$, +interpret $G$ and $GS$ as empty. Conditional on $(G,GS)$, the signal is uniform on an affine sphere +parallel to $\ker G$, with ambient kernel dimension $d-b$ and radius +\[ + R=\norm{P_{\ker G}S}. +\] +For $b=0$, this is the original unit sphere and $R=1$ deterministically. +For $1\le b\le d-2$, the squared radius has distribution +\begin{equation}\label{eq:residual-beta} + R^2\sim\operatorname{Beta}\bigl((d-b)/2,b/2\bigr). +\end{equation} +For $04d/3\}\le\frac{18}{d}. +\] +Outside these events $R^2\ge1/4$. Their union gives $90/d$. For $b=0$ +the probability is zero, as already established. +\end{proof} + +The next estimate tests success against a law in which the signal and +output are conditionally independent. This is the binary-relative-entropy +event method; see Bassily et al.~\cite[Appendix~A.1, Lemmas~14--15]{BassilyEtAl2018} +for finite-space information bounds. Here the reference signal lies on +a residual sphere. + +\begin{lemma}[Information on a residual sphere]\label{lem:residual-information} +Let $S\sim\sigma_d$, let $V$ be a random variable with values in a standard +Borel space, jointly distributed with $S$, and let $G$ be independent of +$(S,V)$, with $0\le b\le d-2$ rows. For $b\ge1$, $G$ may be either a standard +Gaussian matrix or a uniform Haar frame of $b$ orthonormal rows; at $b=0$ +it is empty. Let $\widehat S\in S^{d-1}$ +be generated from $V$ by a kernel conditionally independent of $(S,G)$. +Write +\[ + p=\Prob\{\arccos\langle\widehat S,S\rangle\le\epsilon\}, + \qquad R=\norm{P_{\ker G}S}. +\] +For $2\epsilon0$, then +\begin{equation}\label{eq:residual-information} + I(S;V\mid G,GS)\ge + \alpha(d-b-1)\log\frac{r_0}{2\epsilon}-\log2. +\end{equation} +\end{lemma} + +\begin{proof} +Put $Z=(G,GS)$. Compare the true law of $(S,V,Z)$ with +$P_ZP_{S\mid Z}P_{V\mid Z}$, and in both laws generate $\widehat S$ using +the same kernel from $V$. Adding that kernel preserves the divergence, +which is $I(S;V\mid Z)$. + +Fix side data with residual radius $R\ge r_0$ and fix an output. If the +set of successful points on the fiber is nonempty, choose one such point. +Angular success implies Euclidean distance at most $\epsilon$ from the +output, so any two successful fiber points are at Euclidean distance at +most $2\epsilon$. After translating and rescaling the fiber to its unit +sphere, the successful set therefore lies in a chordal ball of radius +$2\epsilon/R\le2\epsilon/r_0<1$ centered on that sphere. By +Lemma~\ref{lem:spherical-ball}, its conditional mass is at most +$(2\epsilon/r_0)^{d-b-1}$. Under the comparison law, the output and signal +are conditionally independent given $Z$, so the event of success and +$R\ge r_0$ has probability at most this quantity. Under the true law it has +probability at least $\alpha$. + +For Bernoulli probabilities $a,c$, relative entropy obeys +\[ + \KL(\operatorname{Bern}(a)\Vert\operatorname{Bern}(c)) + \ge a\log(1/c)-\log2; +\] +this follows by dropping the nonnegative term +$(1-a)\log(1/(1-c))$ and using binary entropy at most $\log2$. +Apply data processing to the event just described. If the comparison +probability is zero the conclusion is immediate; otherwise its upper bound +and the true lower bound give \eqref{eq:residual-information}. +\end{proof} + +We record three numerical forms used below. The first uses an angular +cap bound: a chordal radius $4\epsilon$ corresponds to angular radius +$2\arcsin(2\epsilon)\le5\epsilon$ for $\epsilon\le1/10$. + +\begin{corollary}[Three endpoint substitutions]\label{cor:replica-endpoints} +Assume $0<\epsilon\le1/10$ in Lemma~\ref{lem:residual-information}. +For sufficiently large $d$ the following bounds hold. +\begin{enumerate}[label=(\roman*)] +\item If $G$ is Gaussian, $p\ge3/5$, and $0\le b\le d/2$, the beta-variance bound (with +the separate deterministic case at $b=0$) and the angular cap just described +give +\begin{equation}\label{eq:endpoint-gaussian} + I(S;V\mid G,GS)\ge + \tfrac12(d-b-1)\log\frac1{5\epsilon}-\log2. +\end{equation} +\item If $G$ is a Haar frame, $p\ge2/3$, and $b=\lfloor d/4\rfloor$, Markov's bound +$\Prob\{R<1/2\}\le4b/(3d)\le1/3$ gives +\begin{equation}\label{eq:endpoint-haar} + I(S;V\mid G,GS)\ge + \tfrac13(d-b-1)\log\frac1{4\epsilon}-\log2. +\end{equation} +\item If $G$ is Gaussian, $p\ge3/5$, and $b=\lfloor d/2\rfloor$, the direct Gaussian bound +$\Prob\{R<1/2\}\le90/d$ gives +\begin{equation}\label{eq:endpoint-bins} + I(S;V\mid G,GS)\ge + \tfrac12(d-b-1)\log\frac1{4\epsilon}-\log2. +\end{equation} +\end{enumerate} +Each right side is at least $c\,d\log(1/\epsilon)$ for an absolute $c>0$ +and all sufficiently large $d$. +\end{corollary} + +\begin{proof} +Use $r_0=1/2$. In (i), the failure probability is at most $8/(d+2)$, so +the true event has mass at least $1/2$ for large $d$. The angular cap bound +is $(5\epsilon)^{d-b-1}$, which gives the displayed version of the binary +entropy calculation. In (ii), Markov's estimate leaves event mass at least +$1/3$, and the chordal cap is $(4\epsilon)^{d-b-1}$. In (iii), the stated +$90/d$ estimate leaves mass at least $1/2$. The last assertion follows from +$d-b-1\ge d/2-1$ and +\[ + \log\frac1{c_0\epsilon}\ge + \left(1-\frac{\log c_0}{\log10}\right)\log\frac1\epsilon + \qquad(c_0=4\text{ or }5). +\] +The positive multiple of $d\log(1/\epsilon)$ absorbs $\log2$ eventually. +\end{proof} + +\subsection{Borel rules, randomness, and bounded stopping} + +\begin{lemma}[Borel versions for the averaged experiment]\label{lem:borel-rules} +Fix $d\ge2$, a finite horizon $T$, and finite state sets. For fixed +data-independent rule parameters, suppose each transition and stopping +coordinate is measurable in the completion of the Borel sigma field on +$\R^d\times\R$ under +$\lambda(dx,dy)=\gamma_{1,d}(dx)\,dy$, where $\gamma_{1,d}$ is standard +Gaussian probability on $\R^d$. The output law depends only on the +terminal state and stopping index. There are Borel transition and stopping +kernels, with the same state sets and output laws, having the same joint law +of the signal, state path, stopping index, and output when $S\sim\sigma_d$. +\end{lemma} + +\begin{proof} +For each state and index combine the next-state and stopping alternatives +into a finite destination set. Choose a Borel version of every coordinate +of this finite probability vector. Such versions exist for completed- +measurable real functions: approximate by simple functions and replace +their countably many level sets by Borel sets modulo null sets. On the Borel +null set where the resulting vector is negative in a coordinate or fails +to sum to one, replace it by a fixed point mass. The finite union over all +states and indices is contained in one Borel $\lambda$-null set $N$. + +For a nonzero fixed $x$, the scalar $\langle x,S\rangle$ has a Lebesgue +density when $S\sim\sigma_d$ and $d\ge2$. Rotation and the one-coordinate +spherical law show this directly. The event $x=0$ is Gaussian-null. +Consequently each pre-generated sample pair +$(X_j,\langle X_j,S\rangle)$ has law absolutely continuous with respect +to $\lambda$, and it avoids $N$ almost surely. This remains true on the +event that any particular state is reached, because that event intersected +with the pair lying in $N$ is contained in the marginal null event. +The finite union over sample indices is still null. Coupling each original +finite kernel and its Borel version by the same fresh uniform random +variable now makes their state paths and stopping decisions agree by +induction. Their output laws agree at the identical terminal pair. +\end{proof} + +This lemma preserves the experiment averaged over the uniform signal; it +does not claim pointwise preservation at every fixed signal. We may therefore +apply it after passing to the uniform-prior hypothesis of +Theorem~\ref{thm:streaming}. +When a shared seed is present, apply the replacement separately in its +fixed-seed experiments. The original jointly measurable law supplies +well-defined conditional success probabilities and their average. No +jointly measurable selection of the replacement Borel versions is needed. + +\begin{lemma}[Fixing randomness and padding the clock]\label{lem:padding} +Consider a learner in Definition~\ref{def:streaming-model} under the uniform +prior. If its average success is at least $p$, some fixed shared-seed +conditional experiment has success at least $p$. Its initialization remains +independent of the signal and samples, and its fixed-seed Borel replacement +has the same success. In this Borel experiment the initialization and all +remaining randomized rules can be represented by data-independent variables +chosen before sampling, followed by deterministic state-based rules. For +every $p'0$, and set +$\beta=\log(1/\epsilon)$ for $0<\epsilon\le1/10$. +At each dimension, the terminal output capacity satisfies +\begin{equation}\label{eq:output-capacity} + p_0\le (T+1)2^M\epsilon^{d-1}. +\end{equation} +On the branch $T\le d\beta$, if $M=o(d^2)$, then +\begin{equation}\label{eq:early-clock} + \beta=o(d),\qquad T=o(d^2),\qquad h_T=o(d^2). +\end{equation} +\end{lemma} + +\begin{proof} +First fix a shared seed in the original conditional experiment. +There are at most $(T+1)2^M$ terminal state/index pairs. For a fixed pair, +write $\nu_{j,v}$ for its prescribed output law and +$r_{j,v}(s)$ for the conditional probability of reaching it given $S=s$. +Its contribution to success is +\[ + \int r_{j,v}(s)\int + \ind_{\{\arccos\langle u,s\rangle\le\epsilon\}} + \,\nu_{j,v}(du)\,\sigma_d(ds) + \le\epsilon^{d-1}, +\] +because $r_{j,v}\le1$ and each angular cap has mass at most +$\epsilon^{d-1}$ by Lemma~\ref{lem:spherical-ball}. +Summing gives \eqref{eq:output-capacity} for its +conditional success. The same numerical bound holds for almost every seed, +so averaging the original conditional success probabilities gives the +stated bound for the jointly measurable mixture. + +On $T\le d\beta$, the inequalities $\beta\ge\log10$ and +$T+1\le2d e^\beta$ imply +\[ + (d-2)\beta\le M\log2+\log(2d)+\log(1/p_0). +\] +Dividing by $d$ proves $\beta=o(d)$ for a subquadratic memory sequence. +The branch then gives $T=o(d^2)$, and \eqref{eq:padded-entropy} gives +$h_T=o(d^2)$. +\end{proof} + +\begin{lemma}[A raw-sample endpoint]\label{lem:raw-sample-bound} +Let $0<\epsilon\le1/10$, and let an estimator use all exact measurements +$(G,GS)$ from $b\le\lfloor d/2\rfloor$ independent Gaussian rows, together +with independent randomness, when $S\sim\sigma_d$. Its angular success +probability is at most +\begin{equation}\label{eq:raw-sample-bound} + \frac{8}{d+2}+(4\epsilon)^{d-b-1}. +\end{equation} +The separate Gaussian-norm estimate gives the also valid bound +$90/d+(4\epsilon)^{d-b-1}$. In particular, uniform-prior success at least +$3/5$ requires $T>\lfloor d/2\rfloor$ for all sufficiently large $d$, +even if all raw data up to the horizon are retained. +\end{lemma} + +\begin{proof} +For $d\le2$, both numerical bounds exceed one. We may therefore assume +$d\ge3$, when $b\le\lfloor d/2\rfloor\le d-2$. +Condition on all the data. The estimator and the signal are then +conditionally independent, and the conditional signal is the residual +sphere in Lemma~\ref{lem:residual-sphere}. On radius at least $1/2$, +the successful points have relative chordal cap mass at most +$(4\epsilon)^{d-b-1}$, by the argument in +Lemma~\ref{lem:residual-information}. Add the corresponding failure +probability from \eqref{eq:residual-chebyshev}. The zero-row case uses +the deterministic radius one. + +If a learner stops by $T\le\lfloor d/2\rfloor$, grant it all of the first +$\lfloor d/2\rfloor$ rows and labels, including unused ones. Its output is +a kernel of those data and independent randomness. The displayed bound +tends to zero uniformly for $\epsilon\le1/10$, contradicting success +$3/5$. This proves the last assertion. The angular-cap version used in +the first replica route follows as well by replacing $4\epsilon$ by +the weaker $5\epsilon$. +\end{proof} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/02-replicas.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/02-replicas.tex new file mode 100644 index 0000000000000000000000000000000000000000..bdf1d0622043655e31d47f1dcf28a7580b056871 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/02-replicas.tex @@ -0,0 +1,498 @@ +\section{Posterior replicas and exact equal-label measures} +\label{sec:replicas} + +The information estimate for a block has two parts. First, a general +identity converts information in one finite message into the loss of +dependence among several signals. Second, exact Gaussian observations give +an explicit law for signals sampled from the same posterior. This section +proves both statements. The subsequent sections estimate how much of that +dependence remains visible after another projection. + +All probability spaces used below are standard Borel. Conditional laws mean +measurable regular conditional probabilities, and conditional relative +entropies include integration over the conditioning variable. We write +$\KL(P\Vert Q)$ for relative entropy, with value $+\infty$ when $P$ is +not absolutely continuous with respect to $Q$. Logarithms and entropies are +in nats. + +\subsection{Information in a message and loss of correlation} + +For a joint law $Q$ of $(S_1,\ldots,S_t)$ with common marginal $p$, its +\emph{total correlation} is +\[ + K=\KL(Q\Vert p^{\otimes t}). +\] +This quantity was introduced by Watanabe~\cite{Watanabe1960}; see +Austin~\cite[Sections 3.3 and 6]{Austin2020} for its measure-theoretic +form and information identities. Let $B$ be independent of the tuple and +let $U_i=\phi_B(S_i)$, where $(B,s)\mapsto\phi_B(s)$ is measurable. +Writing $p_B=(\phi_B)_\#p$, define the correlation visible after projection by +\[ + K'=\E_B\KL\bigl(Q_{U_1,\ldots,U_t\mid B}\Vert p_B^{\otimes t}\bigr). +\] + +\begin{lemma}[Replica information inequality]\label{lem:replica-information} +Let $t\ge1$ and suppose $K<\infty$. Let $W$ be finite-valued and suppose +that every pair $(S_i,W)$ has the same law as $(S_1,W)$. In the conclusion +write $S=S_1$. If $B$ is independent of $(S_1,\ldots,S_t,W)$, then +\begin{equation}\label{eq:replica-information} + t I(S;W\mid B,\phi_B(S))\le H(W)+K-K',\qquad 0\le K'\le K. +\end{equation} +The same statement holds in a conditional experiment when all of these +hypotheses hold there, including the matching pair laws and the independence +of $B$. Equivalently, the left side may use a separate representative of +the common signal--message pair law if it is coupled independently with $B$. +\end{lemma} + +\begin{proof} +Let $K_W$ be the average total correlation conditional on $W$, computed +relative to the product of the conditional one-coordinate marginals. +Decomposing the relative entropy of $(S_1,\ldots,S_t,W)$ from +$p^{\otimes t}\otimes Q_W$ in the two orders gives +\begin{equation}\label{eq:replica-chain-before} + tI(S;W)=I((S_i)_{i=1}^t;W)+K-K_W. +\end{equation} +The relative entropy being decomposed is at most $K+H(W)$, so every +nonnegative chain-rule term in this display is finite. Define $K'_W$ in +the same way after projection, conditionally on $(B,W)$. The corresponding +identity after projection is +\begin{equation}\label{eq:replica-chain-after} + tI(\phi_B(S);W\mid B) + =I((U_i)_{i=1}^t;W\mid B)+K'-K'_W. +\end{equation} +Coordinatewise data processing gives $K'\le K$ and $K'_W\le K_W$. +Independence of $B$ and the fact that $\phi_B(S)$ is determined by $(B,S)$ +also give +\[ + I(S;W\mid B,\phi_B(S))=I(S;W)-I(\phi_B(S);W\mid B). +\] +Subtracting \eqref{eq:replica-chain-after} from +\eqref{eq:replica-chain-before} therefore gives the exact identity +\[ + tI(S;W\mid B,\phi_B(S)) + =I((S_i)_{i=1}^t;W\mid B,(U_i)_{i=1}^t) + +(K-K')-(K_W-K'_W). +\] +The conditional tuple-information is at most $H(W)$, and +$K_W-K'_W\ge0$. This proves the inequality. +The remaining assertion is nonnegativity of relative entropy. +\end{proof} + +The coupling used in this paper has precisely the pair-law property in the +lemma. Suppose that data $(A,Y)$ are generated from $S$, and that $W$ is +drawn from a kernel of the data, conditionally independently of $S$. +Given the same $(A,Y)$, draw $S_1,\ldots,S_t$ independently from the +posterior law of $S$, independently also of $W$. Conditioning on the data +shows that $(S_i,W)$ has the original signal--message law for every $i$. +We call these signals \emph{posterior replicas}. Equal one-signal marginals +alone would not be sufficient for Lemma~\ref{lem:replica-information}. +The invariance of the joint data--signal law under posterior resampling +is the Nishimori identity; see Lelarge and +Miolane~\cite[Proposition 16]{LelargeMiolane2019} for a general formulation. +When an independent $B$ is added to both experiments, equality of the pair +laws gives equality of the corresponding $(S_i,W,B)$ laws as well, so the +lemma's left side is the information in the original signal--message pair. + +We shall use the following entropy test. If $Q\ll R$ are probabilities and +$F\ge0$ is measurable with $\E_R F\le1$, then +\begin{equation}\label{eq:entropy-test} + \KL(Q\Vert R)\ge\E_Q\log F +\end{equation} +whenever the right side is defined. For a bounded positive test this follows +by tilting $R$ by $F/\E_R F$ and using nonnegativity of relative entropy; +bounded truncations give the stated form. This is the entropy variational +principle in the form used here; see +Dupuis and Mao~\cite[Equations (1.1)--(1.2)]{DupuisMao2022}. +In particular, if the relative entropy is finite, applying the bounded form +to truncations of $(1+F)/2$ and then monotone convergence yields +\begin{equation}\label{eq:entropy-positive-log} + \E_Q\log(1+F)\le\KL(Q\Vert R)+\log2. +\end{equation} +This supplies positive-log integrability for the likelihood tests below. + +\subsection{Projection fibers and inverse affine volumes} + +Let $\sigma_d$ be uniform probability on $S^{d-1}$ and let $\gamma_{k,d}$ +be the law of a $k\times d$ matrix with independent standard Gaussian +entries. For a subspace $F\subset\R^d$, let $\gamma_{k,d}^{F}$ denote the +law of a matrix whose rows are independent standard Gaussians in $F^\perp$. +For a tuple $\mathbf u=(u_1,\ldots,u_t)$ with $m=t-1$, set +\begin{equation}\label{eq:equal-label-geometry} + D_{\mathbf u}=[u_2-u_1,\ldots,u_t-u_1],\qquad + F_{\mathbf u}=\col(D_{\mathbf u}),\qquad + J(\mathbf u)=\det(D_{\mathbf u}^{\mathsf T}D_{\mathbf u})^{1/2}. +\end{equation} +At full column rank $J$ is the product of the successive distances to +predecessor affine spans. For two distinct points, the difference +$v=u_2-u_1$ has $Av\sim N(0,\norm v^2I_k)$, whose density at zero is +$(2\pi)^{-k/2}\norm v^{-k}$. This is the inverse-volume factor for two +points. With $m$ differences, each row of $AD_{\mathbf u}$ has covariance +$D_{\mathbf u}^{\mathsf T}D_{\mathbf u}$, giving the factor +$(2\pi)^{-km/2}J(\mathbf u)^{-k}$. The next lemma provides the integrability +needed to identify an exact measure on the constraint $Au_1=\cdots=Au_t$. + +\begin{lemma}[Projection fibers and inverse affine volumes] +\label{lem:projection-fibers} +Let $1\le k\le d-2$ and let $P^0$ be either $\sigma_d$ or standard +Gaussian probability on $\R^d$. Every full-row-rank $k\times d$ matrix +$A$ projects $P^0$ to a Lebesgue density $p_A^0$. With +$A^\dagger=A^{\mathsf T}(AA^{\mathsf T})^{-1}$, the spherical density is +\begin{equation}\label{eq:spherical-projection-density} + p_A^0(y)=\frac{\Gamma(d/2)}{\pi^{k/2}\Gamma((d-k)/2)} + \frac{(1-\norm{A^\dagger y}^2)^{(d-k-2)/2}} + {\sqrt{\det(AA^{\mathsf T})}} + \ind_{\{\norm{A^\dagger y}<1\}}. +\end{equation} +For $\norm{A^\dagger y}<1$, its conditional law $P^0_{A,y}$ is uniform +on the sphere centered at $A^\dagger y$, of radius +$\sqrt{1-\norm{A^\dagger y}^2}$, in the affine space parallel to $\ker A$. +For the Gaussian base, $p_A^0$ is the density of $N(0,AA^{\mathsf T})$, +and the conditional point is $A^\dagger y$ plus a standard Gaussian in +$\ker A$. These conditional laws have jointly measurable versions; +unused fibers may be assigned any fixed probability law. + +If $m\ge1$, $t=m+1$, and $k+m\le d-2$, then $J>0$ almost surely under +$(P^0)^{\otimes t}$ and +\begin{equation}\label{eq:inverse-affine-volume} + \int J(\mathbf u)^{-k}\,(P^0)^{\otimes t}(d\mathbf u)<\infty. +\end{equation} +If $k+m\le d-3$, the same assertion holds with exponent $k+1$. +\end{lemma} + +\begin{proof} +Split a standard Gaussian vector into the row space and kernel of $A$. +Its squared lengths in those spaces are independent chi-squares with $k$ +and $d-k$ degrees of freedom. After dividing by the total length, their +ratio has the beta density. Polar integration in orthonormal row coordinates +gives \eqref{eq:spherical-projection-density}; changing those coordinates +to $y$ supplies the determinant factor. The remaining direction is +independent and uniform in the kernel. This is the spherical beta +projection law, discussed geometrically by Frankl and +Maehara~\cite{FranklMaehara1990}. Without normalization, orthogonal Gaussian +independence gives the Gaussian assertion. For measurability, generate the +kernel component by projecting an extra independent Gaussian with the Borel +matrix $I-A^\dagger A$, and normalize its length in the spherical case. +Assign fixed values on rank failures and on the null event of a zero projected +Gaussian. + +For the inverse moment, first note that an affine $j$-plane $V$ satisfies +\begin{equation}\label{eq:affine-tube} + \sigma_d\{u:\dist(u,V)\le r\}\le C^d r^{d-1-j},\qquad 0k$. Integrating its tail bounds the inverse $k$th moment +of each distance uniformly in the preceding points: for a nonnegative +distance $X$, +\[ + \E X^{-k}\le1+k\int_0^1 r^{-k-1}\Prob\{X0. +\end{equation} +The conditional probability on an unused fiber may be arbitrary. In the +spherical case the displayed density is zero outside the open projection +ellipsoid. The constructions in Lemma~\ref{lem:projection-fibers} make these +versions jointly measurable. + +The equality below distinguishes two natural descriptions. The left first +selects a matrix and a common label, with the label weighted by $p_A^t$, +then selects points independently on that fiber. The right first selects +independent points and then Gaussian rows annihilating their differences, +with an inverse affine-volume weight. These are finite measures, not the +two unweighted probability experiments. When $\nu$ is a probability, its +actual common label has density $p_A$, so recovering posterior replicas +will require multiplying this identity by $p_A^{1-t}$ on positive fibers. +We first identify the measures and their null sets, before that division. +The general background is coarea +\cite[Theorem 3.1]{Federer1959}; related affine simplex-volume Jacobians +occur in Drury~\cite[Lemma 1]{Drury1984}. The following proof fixes the +normalization and the versions used at the exact constrained labels. + +\begin{theorem}[Finite measure for equal Gaussian labels] +\label{thm:equal-label-measure} +Let $k,m\ge1$, $t=m+1$, and $k+m\le d-2$. Let $P^0$ be either uniform +probability on $S^{d-1}$ or standard Gaussian probability on $\R^d$, +and let $\nu=fP^0$ for a bounded nonnegative Borel function $f$. +With the notation in \eqref{eq:equal-label-geometry} and +\eqref{eq:finite-tilt}, for every nonnegative Borel function +$H(A,y,\mathbf u)$, +\begin{align} + &\int\gamma_{k,d}(dA)\int_{\R^k}p_A(y)^t + \int H(A,y,\mathbf u)\,\nu_{A,y}^{\otimes t}(d\mathbf u)\,dy + \notag\\ + &\quad=(2\pi)^{-km/2}\int J(\mathbf u)^{-k}\,\nu^{\otimes t}(d\mathbf u) + \int H(A,Au_1,\mathbf u)\,\gamma_{k,d}^{F_{\mathbf u}}(dA). + \label{eq:equal-label-measure} +\end{align} +Both sides are finite measures before the test $H$ is inserted. They are +supported on $Au_i=y$ for every $i$, full-column-rank $D_{\mathbf u}$, +and full-row-rank $A$. In the spherical case the common fiber has positive +radius. Thus \eqref{eq:equal-label-measure} is an identity of finite measures +on $(A,y,\mathbf u)$; the right side places $y$ at $Au_1$. + +For a full-rank $A$, alternatively define $p_A(y)$ by the lower limit of +its radius-$1/n$ Lebesgue ball averages or by the lower limit of its Gaussian +convolutions with covariance $n^{-2}I_k$. Each choice is jointly measurable +and agrees with \eqref{eq:finite-tilt} for Lebesgue-almost every $y$. +All three versions agree at $y=Au_1$, with a finite positive value, almost +everywhere for the finite reference measure +\begin{equation}\label{eq:equal-label-reference} + R(d\mathbf u,dA)=\nu^{\otimes t}(d\mathbf u) + \gamma_{k,d}^{F_{\mathbf u}}(dA). +\end{equation} +When $\nu=0$ the last assertion is vacuous. The equality is obtained using +either ball or Gaussian approximate identities in both integration orders +for the base law $P^0$, then multiplying the resulting measure identity by +the bounded density factors. +\end{theorem} + +\begin{proof} +We first take $\nu=P^0$. Let $\varphi_\eta$ be either the uniform density +on the radius-$\eta$ $k$-ball or the density of $N(0,\eta^2 I_k)$. For a +continuous compactly supported test $G(\mathbf u,A)$ whose matrix support +lies in the open full-row-rank set, integrate +\begin{equation}\label{eq:equal-label-approximation} + G(\mathbf u,A)\prod_{i=2}^t\varphi_\eta(A(u_i-u_1)) +\end{equation} +against $(P^0)^{\otimes t}(d\mathbf u)\gamma_{k,d}(dA)$. + +Fix $A$ first and disintegrate each point by its projected label. Given the +first label $y$, the other labels approach $y$ under either probability +approximate identity. The base density is continuous in the interior of its +support, and the conditional fiber law varies weakly continuously there. +The spherical boundary has zero projected probability. Thus the integral +converges to +\[ + \int\gamma_{k,d}(dA)\int (p_A^0(y))^t + \int G(\mathbf u,A)(P^0_{A,y})^{\otimes t}(d\mathbf u)\,dy. +\] +To justify the passage, condition first also on $u_1$. The absolute value +of the remaining integral is at most +$\norm G_\infty(\sup p_A^0)^m$. The density suprema are uniformly bounded +on the compact matrix support: this follows from +\eqref{eq:spherical-projection-density}, whose exponent is nonnegative, +and from the Gaussian density formula. The bound is global in all the other +labels, including labels near or outside the spherical support, because +each $\varphi_\eta$ is a probability density. Dominated convergence applies +for both kernels. + +In the other order, fix a tuple of full affine rank. Each row of +$AD_{\mathbf u}$ is Gaussian with covariance +$D_{\mathbf u}^{\mathsf T}D_{\mathbf u}$. Its joint density has maximum +at zero, with value +\[ + w_{k,m}(\mathbf u)=(2\pi)^{-km/2}J(\mathbf u)^{-k}. +\] +Conditional on $AD_{\mathbf u}=Z$, the matrix $A$ is a draw from +$\gamma_{k,d}^{F_{\mathbf u}}$ plus +$Z(D_{\mathbf u}^{\mathsf T}D_{\mathbf u})^{-1}D_{\mathbf u}^{\mathsf T}$. +This translation tends to zero with $Z$. The integral in +\eqref{eq:equal-label-approximation}, conditioned on the tuple, therefore +converges to +\[ + w_{k,m}(\mathbf u)\int G(\mathbf u,A) + \gamma_{k,d}^{F_{\mathbf u}}(dA). +\] +For either kernel its absolute value is at most +$\norm G_\infty w_{k,m}(\mathbf u)$. Indeed, the joint density of the +whole matrix $Z=AD_{\mathbf u}$ is bounded by $w_{k,m}$, while +$\prod_{i=2}^t\varphi_\eta(Z_{:,i-1})$ has Lebesgue integral one on +$\R^{k\times m}$. This argument does not require the columns of $Z$ to be +independent. The bound is integrable by Lemma~\ref{lem:projection-fibers}. +Dominated convergence gives the second integration order and equates these +two limits. + +For clarity, the left measure is locally finite on the open full-row-rank +matrix set before we identify it. If a compact set in that space has matrix +projection $K_A$, its left mass is at most +\[ + \int_{K_A}\!\int (p_A^0(y))^t\,dy\,\gamma_{k,d}(dA) + \le \sup_{A\in K_A}(\sup_y p_A^0(y))^m<\infty, +\] +because $\int p_A^0(y)\,dy=1$. Thus both sides are locally finite Borel +measures on this locally compact second-countable space; the right one is +globally finite by \eqref{eq:inverse-affine-volume}. Continuous compactly +supported tests identify them on that open set. Exhausting it shows that +the left total mass there equals the finite right total mass. The right +measure gives no mass to remaining matrices: its rows live in a space of +dimension $d-m\ge k+2$. The left measure also has no such mass because +$\gamma_{k,d}$ is full rank almost surely. Equality therefore extends to +all nonnegative Borel tests. The right measure is +supported on full affine rank by Lemma~\ref{lem:projection-fibers}, so the +same holds on the left. In the spherical case its label is in the open +ellipsoid, and its fiber radius is positive. Equivalently, a boundary fiber +could contain only one unit vector and hence could not contain a full-affine- +rank tuple with $t\ge2$. + +Now multiply the base equality by $\prod_{i=1}^t f(u_i)$. The formula +\eqref{eq:finite-tilt} converts the fiber side to $p_A(y)^t\nu_{A,y}^{\otimes t}$; +the other side becomes $w_{k,m}\nu^{\otimes t}\gamma_{k,d}^{F_{\mathbf u}}$. +On a fiber with $h_A(y)=0$, the measure $fP^0_{A,y}$ is zero, so both +expressions on the fiber side are zero regardless of the unused conditional +probability. This conversion never divides on a zero fiber. +Boundedness of $f$ preserves finiteness. Inserting a test +$H(A,Au_1,\mathbf u)$ gives \eqref{eq:equal-label-measure}, since both sides +are supported on equal labels. No normalization by $\nu(\R^d)$ has been +made. In particular, the two equal-label limit calculations were made only +for the base law; no continuity of $f$ or of the reweighted fiber kernel +is used in them. + +It remains to justify density evaluation on that constraint. For either +choice of $\varphi$, set +\[ + \widetilde p_A(y)=\liminf_{n\to\infty} + \int\varphi_{1/n}(Au-y)\,\nu(du). +\] +The integral and lower limit are jointly measurable. Lebesgue +differentiation for the ball and Gaussian approximate identities makes each +version agree with \eqref{eq:finite-tilt} at almost every $y$ for each +full-rank $A$. On the left of \eqref{eq:equal-label-measure}, the common label +has a measure with Lebesgue density $p_A(y)^t$ and hence avoids the null sets +where the versions disagree or are not finite and positive. Equality of the +undivided finite measures transfers this fact to the right. There the weight +$w_{k,m}$ is finite and strictly positive at full affine rank. The reference +$R$ has finite mass $\nu(\R^d)^t$ and full affine rank almost everywhere, +because $\nu^{\otimes t}\ll(P^0)^{\otimes t}$. Restricting successively to +$\{1/j\le w_{k,m}\le j\}$ therefore transfers the same null statement to +the unweighted reference \eqref{eq:equal-label-reference}. The labels where +$p_A=0$ have zero left mass because their weight is $p_A^t$, so this +argument does not divide on them. If $\nu=0$, both finite measures and $R$ +are zero. The only additional limiting assertion for a possibly +discontinuous $f$ is the ordinary almost-everywhere differentiation of its +projected $L^1$ density used in this paragraph. This proves the claimed +versions before any division by a projection density. +\end{proof} + +\begin{corollary}[Posterior replica likelihood]\label{cor:replica-likelihood} +Under the hypotheses of Theorem~\ref{thm:equal-label-measure}, suppose +$p=fP^0$ is a probability. Draw $A\sim\gamma_{k,d}$ independently of +$S\sim p$, put $Y=AS$, and conditionally on $(A,Y)$ draw $t$ independent +replicas from $p_{A,Y}$. Let $Q$ be the law of their tuple together with +$A$, and let $R=p^{\otimes t}(d\mathbf u)\gamma_{k,d}^{F_{\mathbf u}}(dA)$. +Then $R$ is a probability and +\begin{equation}\label{eq:replica-likelihood} + \frac{dQ}{dR}(\mathbf u,A) + =(2\pi)^{-km/2}J(\mathbf u)^{-k}p_A(Au_1)^{-m}. +\end{equation} +In particular, the tuple marginal of $Q$ is absolutely continuous relative +to $p^{\otimes t}$. +\end{corollary} + +\begin{proof} +Multiply the equal finite measures by $p_A(Au_1)^{-m}$ on their common +finite positive set, first with bounded truncations and then by monotone +convergence. The exponent $t=m+1$ on the left becomes one, so that side is +exactly the probability experiment defining $Q$. The right side is the +displayed density relative to $R$. Marginalization gives the last assertion. +\end{proof} + +\begin{remark}[The uniform instance used with a further inverse moment] +\label{rem:extra-inverse-moment} +Suppose the replica count is $t\ge2$ and $k+t+1k$. Since $k$ is even, direct integration of +the chi-square density gives +\[ + \E(\chi_\nu^2)^{-k/2} + =\prod_{j=1}^{k/2}(\nu-2j)^{-1}\le(2k)^{-k/2}. +\] +In particular $Z_{k,m}<\infty$. The same densities give absolute +logarithmic integrability, and Jensen gives +$\E\log\det(G^{\mathsf T}G)\le m\log\ell$. Consequently +\begin{equation}\label{eq:gaussian-wishart-remainder} + \log Z_{k,m}+\frac{k}{2}\E\log\det(G^{\mathsf T}G)\le Ckm. +\end{equation} + +Before taking logarithms of the test, we control the positive logarithm +of its projection density. For a fixed tuple put +\[ + \delta(s')=\norm{P_{F^\perp}(s'-S_1)}. +\] +The ball-average density version in +Theorem~\ref{thm:equal-label-measure} holds at +$\widetilde A S_1$ under the independent-tuple reference and hence under +the replica law. The coordinates of $\widetilde A(s'-S_1)$ are +independent centered Gaussians with variance $\delta(s')^2$. +A ball has probability at most its volume times the maximum Gaussian +density. Apply this bound to the ball averages and then use Fatou's lemma: +\begin{equation}\label{eq:gaussian-negative-moment} + \E_{\widetilde A}p_{\widetilde A}(\widetilde A S_1) + \le (2\pi)^{-k/2}\int\delta(s')^{-k}\,p(ds'). +\end{equation} +Here and below such conditional estimates hold for almost every tuple. +The affine space $S_1+F$ has dimension $m$, so +\eqref{eq:affine-tube} gives +\[ + p\{\delta\le r\}\le LC^d r^{d-1-m},\qquad 0k$, tail integration bounds the inverse moment uniformly +over tuples for fixed $d,L$. Thus +$\log^+p_{\widetilde A}(\widetilde A S_1)$ is integrable. + +The projected law has divergence $K'\le K<\infty$ from the +independent-signal reference. Normalization of the test and +\eqref{eq:entropy-positive-log} imply that $(\log\Lambda)^+$ is +integrable. Its defining identity is +\begin{equation}\label{eq:gaussian-test-log} + \begin{aligned} + \log\Lambda + &=\log c-\log Z_{k,m}-k\log J + -\frac{k}{2}\log\det(G^{\mathsf T}G)\\ + &\qquad-m\log p_{\widetilde A}(\widetilde A S_1). + \end{aligned} +\end{equation} +The negative part of this expression is bounded by a constant and the +positive logarithms of $J$, the Gaussian determinant, and the last +density. Those are integrable by Lemma~\ref{lem:replica-logs}, the +chi-square calculation, and \eqref{eq:gaussian-negative-moment}. +Thus $\log\Lambda$ is absolutely integrable. Solving +\eqref{eq:gaussian-test-log} for the last logarithm proves its absolute +integrability as well. + +We may now apply the entropy test \eqref{eq:entropy-test} to obtain +$K'\ge\E\log\Lambda$. Subtract this bound from +\eqref{eq:gaussian-correlation-upper}. Equation +\eqref{eq:gaussian-test-log} displays the exact cancellation of +$-k\E\log J$, and \eqref{eq:gaussian-wishart-remainder} bounds the +remaining determinant terms by $Ckm$. This gives +\eqref{eq:gaussian-correlation-loss}. +\end{proof} + +\subsection{Comparing the remaining projection densities} + +Both the actual $A$ and the observable $\widetilde A$ annihilate $F$. +The preceding lemma has removed the affine volume from the information +bound. It remains to compare the two densities in +\eqref{eq:gaussian-correlation-loss}. We first bound the conditional mean +of the new density by a sum of averages of the actual density. The radii +in that sum will depend only on $A$, allowing us to average the actual +label with its original density $p_A$. + +\begin{lemma}[Dyadic density comparison]\label{lem:gaussian-density} +In the replica experiment with the independent side matrix and +$\widetilde A=C_0B$ defined above, +\begin{equation}\label{eq:gaussian-density-comparison} + \E\!\left[\log p_{\widetilde A}(\widetilde A S_1)-\log p_A(Y)\right] + \le Ck+C\log(2+\log L). +\end{equation} +\end{lemma} + +\begin{proof} +Fix the tuple and its actual $(A,Y)$, and again write +$\delta(s')=\norm{P_{F^\perp}(s'-S_1)}$. Conditional on these data, +$\widetilde A$ has law $\gamma_{k,d}^{F}$. We refine the inverse-moment +bound \eqref{eq:gaussian-negative-moment} by using $A$. +Choose the deterministic integer +\begin{equation}\label{eq:gaussian-cutoff} + J_0=1+\left\lceil + \frac{\log_2 L+C_1d}{d-1-m-k}\right\rceil +\end{equation} +with a sufficiently large absolute $C_1$. The denominator is bounded +below by a positive absolute multiple of $d$. The contribution of +$\delta\le2^{-J_0}$ to the inverse moment, including the Gaussian constant, +is at most +\[ + (2\pi)^{-k/2} + \sum_{j\ge J_0}2^{(j+1)k}LC^d2^{-j(d-1-m)}\le1. +\] +For the remaining distances use the bins +$\delta\in(r/2,r]$, where $r=2,1,\ldots,2^{-J_0+1}$. +These cover the remaining values because $\delta(s')\le\norm{s'-S_1}\le2$. +Since $A$ annihilates $F$, +\[ + \norm{As'-Y}\le\norm A_{\op}\,\delta(s'). +\] +Let $v_k$ be the unit $k$-ball volume, and let +$\overline p_A^{\,a}(y)$ be the average of $p_A$ over the radius-$a$ ball +centered at $y$. The contribution of one bin is at most +\[ + (2/r)^k p\{\delta\le r\} + \le 2^k v_k\norm A_{\op}^{k} + \overline p_A^{\,\norm A_{\op}r}(Y). +\] +Integrating $e^{-\norm z^2/2}$ on the radius-$\sqrt{k}$ ball gives +$v_k\le(2\pi e/k)^{k/2}$. Hence +\begin{equation}\label{eq:gaussian-density-sum} + \E_{\widetilde A}p_{\widetilde A}(\widetilde A S_1) + \le1+\left(\frac{C\norm A_{\op}}{\sqrt{k}}\right)^k + \sum_r\overline p_A^{\,\norm A_{\op}r}(Y). +\end{equation} +The right side depends on the tuple only through $(A,Y)$. + +Divide by the finite positive density $p_A(Y)$ and apply Jensen first to +$\widetilde A$; all logarithms are integrable by +Lemma~\ref{lem:gaussian-test}. Only after obtaining +\eqref{eq:gaussian-density-sum} do we average over $Y$ given $A$. +Its law is then the original $p_A(y)\,dy$, and for every radius fixed by $A$, +\[ + \E_{Y\mid A}\frac{\overline p_A^{\,a}(Y)}{p_A(Y)}\le1,\qquad + \E_{Y\mid A}\frac1{p_A(Y)} + \le v_k\norm A_{\op}^{k}. +\] +The first inequality integrates an averaged probability density over +$\{p_A>0\}$, and the second is the volume of that set, contained in the +radius-$\norm A_{\op}$ ball. A second application of Jensen thus bounds +the expected log ratio by +\begin{equation}\label{eq:gaussian-density-jensen} + k\E\log\!\left(C\left(1+\frac{\norm A_{\op}}{\sqrt{k}}\right)\right) + +\log(J_0+2). +\end{equation} +This order of averaging is essential: before the tuple has disappeared, +its conditional label need not have density $p_A$ under further conditioning. + +For completeness, choose $1/4$-nets of the unit spheres in $\R^d$ and +$\R^k$, with at most $9^d$ and $9^k$ points. The operator norm is at most +twice the maximum absolute Gaussian bilinear form on these nets. The +expected maximum of $N$ absolute standard normals is at most +$\sqrt{2\log(2N)}$, by the exponential-moment bound; their independence is +not required. Thus $\E\norm A_{\op}\le C\sqrt d$. Since $k$ is a fixed +positive fraction of $d$, Jensen bounds the first term of +\eqref{eq:gaussian-density-jensen} by $Ck$. The cutoff +\eqref{eq:gaussian-cutoff} gives +$\log(J_0+2)\le C\log(2+\log L)$, proving the lemma. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:gaussian-block}] +Generate $W$ conditionally on $(A,Y)$ independently of the replicas. +Each pair $(S_i,W)$ has the original pair law, and the side matrix $B$ is +independent of the whole construction. Lemma~\ref{lem:replica-information}, +followed by Lemmas~\ref{lem:gaussian-test} and +\ref{lem:gaussian-density}, gives +\[ + tI(S;W\mid B,BS) + \le H(W)+Ckm+m\bigl(Ck+C\log(2+\log L)\bigr). +\] +Here the representative signal is coupled independently with $B$, exactly +as in the original block experiment. Since $t=m+1$ and $k$ is proportional +to $d$, division by $t$ proves \eqref{eq:gaussian-block}. +\end{proof} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/04-distance-bins.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/04-distance-bins.tex new file mode 100644 index 0000000000000000000000000000000000000000..c488e61a6ccb11f15656efa3db6cc3b32a25e8c5 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/04-distance-bins.tex @@ -0,0 +1,355 @@ +\section{Shared data and distance-bin comparisons} +\label{sec:distance-bins} + +The determinant test in the preceding section pays a double logarithm of +the largest prior density. We instead encode the dyadic scale of each +projected simplex height. Under a prior bounded by $e^b$, these scales +have entropy $O(1+b/d)$. The projected correlation recovers the height +and label-density terms, leaving this scale cost. Since the chosen scale +depends on the side matrix, the central step compares it with a mixture +over fixed scales before averaging the matrix. + +Throughout this section, for sufficiently large $d$, set +\begin{equation}\label{eq:distance-dimensions} + k=\lfloor d/16\rfloor,\qquad q=\lfloor d/8\rfloor,\qquad + p=\lfloor d/2\rfloor. +\end{equation} +Here $k$ is the number of actual rows, $q$ is the number of replicas, +and $p$ is the number of side rows. In particular $k\ge d/32$, +$q\ge d/16$, $q\ge2$, and $p-i+2\ge2k$ for $2\le i\le q$. + +\begin{theorem}[Information from shared exact data]\label{thm:distance-block} +Let $P=f\sigma_d$ be a probability with a nonnegative Borel density +$f\le e^b$, where $b\ge0$. Let $S\sim P$, let $A$ be an independent +$k\times d$ standard Gaussian matrix, and put $Y=AS$. Let a finite message +$W$ be drawn from a kernel of $(A,Y)$, conditionally independently of $S$. +Let $B$ be a $p\times d$ standard Gaussian matrix independent of the +entire joint experiment producing $(S,A,Y,W)$. Then +\begin{equation}\label{eq:distance-block} + I(S;W\mid B,BS)\le\frac{H(W)}q+C\left(d+\frac bd\right) +\end{equation} +for an absolute $C$, uniformly over the prior and message kernel. +\end{theorem} + +The order of \eqref{eq:distance-block} also follows from +Theorem~\ref{thm:gaussian-block}: append $k$ ignored fresh rows to $A$ +and first use the initial $8k$ rows of $B$. The Gaussian theorem gives +the entropy denominator $2k+1\ge q$, and $8k\le p$. Adding the remaining +independent side rows can only decrease the conditional information, +because independence of the side matrix gives +\[ + I(S;W\mid B,BS)=I(S;W)-I(W;B,BS). +\] +Finally $\log(2+b)\le\log(2+d)+b/d\le Cd+b/d$. The direct proof below +identifies the entropy of the chosen distance scales as the prior cost. + +\subsection{The shared-data law and its affine heights} + +Write $g_A^0$ and $g_A$ for the projected densities of $\sigma_d$ and $P$, +using the versions in \eqref{eq:finite-tilt}. Put +$\overline f(A,y)=g_A(y)/g_A^0(y)$ on the open ellipsoid. At labels with +$g_A(y)>0$, the posterior density on the corresponding uniform fiber is bounded by +$\exp(b_{A,y})$, where +\begin{equation}\label{eq:distance-posterior-budget} + b_{A,y}=b-\log\overline f(A,y)\ge0,\qquad + \E b_{A,Y}=b-\E_A\KL(g_A\Vert g_A^0)\le b. +\end{equation} +These quantities are integrable: the positive density-ratio logarithm is +bounded by $b$, and its negative part is controlled by +$-z\log z\le1/e$ against the probability density $g_A^0$. + +Given $(A,Y)$, draw $S_1,\ldots,S_q$ independently from the posterior. +Under this shared-data law, $AS_i=Y$, and each pair $(S_i,A)$ has the +original independent signal--matrix law. Define +\[ + F_i=\aff(S_1,\ldots,S_{i-1}),\qquad + \rho_i=\dist(S_i,F_i),\qquad 2\le i\le q. +\] +The next estimate is stronger than merely proving logarithmic +integrability: its prior cost is divided by the ambient dimension. + +\begin{lemma}[Logarithmic cost of a height]\label{lem:distance-heights} +The heights are positive almost surely and obey +\begin{equation}\label{eq:distance-height} + \rho_i\le2,\qquad + \E\log^-\rho_i\le C\left(1+\frac bd\right), + \qquad \log^-x:=\max\{0,-\log x\}. +\end{equation} +\end{lemma} + +\begin{proof} +We first record a uniform-sphere distance estimate. Let $U$ be uniform on +$S^{n-1}$ with $n\le d$, and let $E$ be a linear subspace of dimension +$j$ whose codimension $h=n-j$ is at least $d/2$. Write a normalized +Gaussian representation with independent components $X\in\R^h$ and +$Z\in\R^j$. For $t\ge\log2$, +\[ + \dist(U,E)k$ and $\beta>0$, radial integration in $\R^k$ gives the exact +kernel mass +\begin{equation}\label{eq:distance-kernel-mass} + \int_{\R^k}\min\{1,(\beta/\norm u)^m\}\,du + =v_k\beta^k\frac{m}{m-k}. +\end{equation} +Indeed the inside ball contributes $v_k\beta^k$, and the exterior contributes +$kv_k\beta^k/(m-k)$. Let $a_{A,j}$ be the probability density obtained +by convolving $g_A$ with this normalized kernel, using +$m=m_i$ and $\beta=4\cdot2^j\norm A_{\op}$. Since $m_i\ge2k$ and +$v_k\le(2\pi e/k)^{k/2}$, \eqref{eq:distance-tube-kernel} becomes +\begin{equation}\label{eq:distance-fixed-bin} + 2^{-kj}\E_B P_B(E_j) + \le\left(\frac{C\norm A_{\op}}{\sqrt k}\right)^k a_{A,j}(Y). +\end{equation} +For each fixed $i$, this density depends on $A$ and $j$, but not on the +replica tuple and not on $Y$ as a parameter of the density. + +Define +\[ + a_A=\sum_{j\in\mathbb Z}\pi_j a_{A,j},\qquad + T_B=\sum_{j\in\mathbb Z}\pi_j2^{-kj}P_B(E_j). +\] +The first is a probability density depending only on $A$ and the fixed +index $i$. Conditional on the shared-data outcome, +\eqref{eq:distance-fixed-bin} gives +\[ + \E_B T_B\le + \left(\frac{C\norm A_{\op}}{\sqrt k}\right)^k a_A(Y). +\] +The $j=K_i$ summand gives the pointwise inequality +\begin{equation}\label{eq:distance-mixture} + \log P_B(E_{K_i}) + \le kK_i\log2-\log\pi_{K_i} + +\log T_B. +\end{equation} + +Before averaging logarithms, we verify their integrability. If +$Z_{A,j}=v_k(4\cdot2^j\norm A_{\op})^k m_i/(m_i-k)$, then for $y$ in +the support of $g_A$, +\[ + \frac{\pi_0}{Z_{A,0}}\le a_A(y)\le\sup g_A. +\] +The lower bound holds because the support has diameter at most +$2\norm A_{\op}$, so the unnormalized kernel at $j=0$ equals one throughout +that support. The upper bound follows by convolving a bounded density with +a probability density. The positive log of $\sup g_A$ is integrable by the +uniform projected-density formula and the Gaussian log-determinant bound. +The absolute log of $Z_{A,0}$ is integrable: $\log^+\norm A_{\op}$ is +controlled by its finite mean, and +$\log^-\norm A_{\op}\le\log^-\norm{Au_0}$ for any fixed unit $u_0$, whose +norm is chi-distributed. Thus $\log a_A(Y)$ is absolutely integrable. + +The pointwise inequality +\eqref{eq:distance-mixture} bounds $\log^-T_B$ by the sum of +$-\log P_B(E_{K_i})$, $k|K_i|\log2$, and $-\log\pi_{K_i}$, all integrable. +For the positive part, +$\E_B\log^+T_B\le\log(1+\E_BT_B)$, and +\eqref{eq:distance-fixed-bin} bounds its right side using $a_A(Y)$ and +$\norm A_{\op}$ as above. Thus $\log T_B$ is absolutely integrable as well. + +Conditional Jensen now gives +\[ + \E_B\log T_B\le + k\log\left(\frac{C\norm A_{\op}}{\sqrt k}\right)+\log a_A(Y). +\] +The right side depends only on $(A,Y)$. In the full experiment +$Y\mid A$ has density $g_A$, and nonnegativity of +$\KL(g_A\Vert a_A)$ therefore gives +\begin{equation}\label{eq:distance-reference-density} + \E\log a_A(Y)\le\E\log g_A(Y)=-h_k. +\end{equation} +This is where the fact that $a_A$ depends only on $A$ is needed. +Also $\E\log(C\norm A_{\op}/\sqrt k)\le C$ by the Gaussian norm +estimate in Section~\ref{sec:gaussian}, since $k\ge d/32$. Consequently +$\E\log T_B\le Ck-h_k$. Combining this with +\eqref{eq:distance-mixture} and \eqref{eq:distance-bin-budget} proves +\eqref{eq:distance-tube-goal}. + +Insert \eqref{eq:distance-tube-goal} in +\eqref{eq:distance-bin-information}, use the entropy bound +in \eqref{eq:distance-bin-budget}, and sum the conditional divergences +over $i=2,\ldots,q$. The projected correlation satisfies +\begin{equation}\label{eq:distance-correlation-lower} + \tc(C_1,\ldots,C_q\mid B) + \ge(q-1)h_k-k\sum_{i=2}^q\E\log\rho_i + -C(q-1)\left(k+1+\frac bd\right). +\end{equation} + +\begin{proof}[Proof of Theorem~\ref{thm:distance-block}] +Generate $W$ from the shared data independently of the replicas conditional +on $(A,Y)$. Each replica has the original signal--message pair law, and +$B$ is independent of this coupling. The finite correlation bound +\eqref{eq:distance-correlation-upper} therefore permits +Lemma~\ref{lem:replica-information} with $t=q$: +\[ + qI(S;W\mid B,BS) + \le H(W)+\tc(S_1,\ldots,S_q)-\tc(C_1,\ldots,C_q\mid B). +\] +Subtract \eqref{eq:distance-correlation-lower} from +\eqref{eq:distance-correlation-upper}. The measurement entropy and all +height logarithms cancel. Since $k\le d/16$, division by $q$ gives +\eqref{eq:distance-block}. Section~\ref{sec:streaming-consequences} applies +this bound while conditioning only on the current state and retaining its +separate late elimination of the clock cost. +\end{proof} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/05-haar.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/05-haar.tex new file mode 100644 index 0000000000000000000000000000000000000000..2f7afc136752c5e1ad388756f652cafd540c0ab0 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/05-haar.tex @@ -0,0 +1,795 @@ +\section{Haar projections and posterior replica information} +\label{sec:haar} + +This comparison uses Haar frames rather than the Gaussian side matrices +of Sections~\ref{sec:gaussian} and~\ref{sec:distance-bins}. We express the +replica measure by successive simplex heights. The normalization changes +at each insertion because the remaining ambient dimension shrinks. +A projected-distance distribution-function test, valid even with atoms, +recovers a fraction of the height cost. We choose its exponent below one +as a function of $d$ and keep the resulting entropy remainder explicit. + +Proposition~\ref{haar:block} gives that remainder in terms of the state +entropies. In particular, the block cost is $O(d)$ when all state +alphabets have logarithmic width $o(d^2)$. We first prove the fixed-prior +height identity and cancellation, then average over state values. A Gaussian observation +block is converted to its orthonormal row frame only in that final step; +the learner's observations are still the original exact Gaussian pairs. + +Write $\sigma=\sigma_d$ and $\log^+x=\max\{0,\log x\}$ for $x>0$. +A $j$-frame is a +$j\times d$ matrix with orthonormal rows; its Haar law is the probability +law invariant under orthogonal changes of ambient coordinates. Throughout +the section set +\begin{equation}\label{haar:dimensions} + d\ge64,\qquad k=r=\lfloor d/16\rfloor,\qquad m=r+1, + \qquad \ell=\lfloor d/4\rfloor. +\end{equation} +Here $k$ is the actual block row count, $r$ is the number of height +insertions, $m=r+1$ is the replica count, and $\ell$ is the side-frame +row count. +At the $i$th insertion, $1\le i\le r$, the two dimensions that +will matter are +\begin{equation}\label{haar:dimension-slack} + n_i=d-i+1,\qquad q_i=\ell-i+1. +\end{equation} +They satisfy +\begin{equation}\label{haar:dimension-bounds} + n_i>k+2,\qquad q_i\ge\frac{3d}{16}>k, + \qquad \frac{q_i}{q_i-k}\le\frac32, + \qquad \frac{n_i}{q_i}\le\frac{16}{3},\qquad \frac d k\le32. +\end{equation} +Indeed, $q_i\ge\ell-r+1\ge3d/16$ and $k\le d/16$; also +$k\ge d/32$ for $d\ge64$. The other inequalities follow at once. In +particular, none of the denominators in \eqref{haar:dimension-bounds} +vanishes. + +\subsection{An exact identity for the replica measure} + +We begin with the normalization of spherical projection. For integers +$1\le j0. + \end{split} +\end{equation} +The posterior on the unused set $\{p_A=0\}$ may be any fixed measurable +probability kernel. These are measurable versions. For example, a +fiber direction can be generated by projecting a standard Gaussian onto +$\ker A$ and normalizing it. If $u$ is that unit direction, the fiber +point is $A^{\mathsf T}y+\sqrt{1-\lVert y\rVert^2}\,u$. A fixed +coordinate convention handles the null normalization failures. This +construction +also makes $p_A(y)$ jointly measurable in $(A,y)$. We have +$p_A\le Kp^0$ and, for every $A$, $p_A$ is a probability density. + +For a tuple $\boldsymbol s=(s_0,\ldots,s_r)$, define +\[ + D_i=\operatorname{span}\{s_1-s_0,\ldots,s_i-s_0\},\qquad D_0=\{0\}, + \qquad + \delta_i=\operatorname{dist}(s_i-s_0,D_{i-1}). +\] +Thus $\delta_i$ is the height added by the $i$th point. Write +$\mathrm dA_D$ for Haar probability on the $k$-frames in $D^\perp$. +All subspaces arising here have $\dim D^\perp\ge d-r>k$. A measurable +version of this frame kernel is obtained by projecting independent +standard Gaussians onto $D^\perp$ and applying a fixed Gram--Schmidt +rule. We use $\mathrm dA$ when $D=\{0\}$. + +\begin{lemma}[Haar simplex-height identity]\label{haar:coarea} +For the dimensions in \eqref{haar:dimensions} and every probability +$P=g\sigma$ with $0\le g\le K<\infty$, the following finite measures on +the frame and the whole tuple are equal: +\begin{equation}\label{haar:coarea-identity} + \mathrm dA\int_{\mathbb R^k}p_A(y)^m\,\mathrm dy + \prod_{i=0}^r P_{A,y}(\mathrm ds_i) + =\left(\prod_{i=0}^rP(\mathrm ds_i)\right) + \left(\prod_{i=1}^r\alpha_{d-i+1,k}\delta_i^{-k}\right) + \mathrm dA_{D_r}. +\end{equation} +Equality means equality after integration against every bounded Borel +function of $(A,\boldsymbol s)$. Define the displayed product density +to be zero on degenerate tuples. The measure on the right is supported +on tuples with $\dim D_i=i$ for all $1\le i\le r$. +\end{lemma} + +\begin{proof} +We first take $P=\sigma$. Fix $s_0,\ldots,s_{i-1}$ with +$\dim D_{i-1}=i-1$, and put $H=D_{i-1}^\perp$, so $\dim H=n_i=d-i+1$. +For a bounded continuous function $\varphi(A,s)$, consider +\begin{equation}\label{haar:small-ball-insertion} + I_t=\int\mathrm dA_{D_{i-1}}\int\sigma(\mathrm ds)\, + \varphi(A,s) + \frac{\mathbf 1_{\{\lVert A(s-s_0)\rVert\le t\}}}{v_kt^k}, + \qquad t>0. +\end{equation} +We compute its limit as $t\downarrow0$ in both orders of integration. + +Fix $A$ first. Disintegration of $\sigma$ under $s\mapsto As$ rewrites +the inner integral as the average over the ball of radius $t$ about $As_0$ +of +\[ + y\longmapsto p^0(y)\int\varphi(A,s)\,U_{A,y}(\mathrm ds). +\] +This function is continuous in the open unit ball. Indeed, use the fiber +representation following \eqref{haar:posterior} with a common uniform +direction and apply bounded convergence. Because $d>k+2$, the factor +$p^0$ tends to zero at the boundary, so the product extends continuously +by zero to $\mathbb R^k$. The average therefore tends to +\begin{equation}\label{haar:first-order-limit} + p^0(As_0)\int\varphi(A,s)\,U_{A,As_0}(\mathrm ds). +\end{equation} +Its absolute value, and that of every preceding average, is at most +$\lVert\varphi\rVert_\infty\alpha_{d,k}$. Dominated convergence thus +permits the frame integration as well. + +For the other order, fix $s$ and put +$\delta=\lVert\operatorname{proj}_H(s-s_0)\rVert$. When $\delta>0$, +the vector $A(s-s_0)$ under $\mathrm dA_{D_{i-1}}$ has density +\begin{equation}\label{haar:insertion-density} + z\longmapsto\delta^{-k}\alpha_{n_i,k} + \left(1-\frac{\lVert z\rVert^2}{\delta^2}\right)^{(n_i-k-2)/2} + \mathbf 1_{\{\lVert z\rVert<\delta\}}. +\end{equation} +This is \eqref{haar:sphere-density} in the current ambient space $H$, +not in the original $d$-dimensional space. Since $n_i>k+2$, the density +is bounded by its value $\alpha_{n_i,k}\delta^{-k}$ at zero. + +The conditional frame law also has a concrete limit. Put +$e=\operatorname{proj}_H(s-s_0)/\delta$ and $w=z/\delta$. If $C$ is a +Haar $k$-frame in $H\cap e^\perp$, then the conditional law of $A$ given +$A(s-s_0)=z$, for $\lVert w\rVert<1$, is the law of +\begin{equation}\label{haar:conditional-frame} + A=w e^{\mathsf T}+(I_k-ww^{\mathsf T})^{1/2}C. +\end{equation} +Its rows are orthonormal and $Ae=w$. Invariance under rotations fixing +$e$ identifies the remaining factor as Haar; conversely the displayed +formula describes every such frame. The formula is continuous at $w=0$. +At zero the frame law is $\mathrm dA_{D_i}$, since +$H\cap e^\perp=D_i^\perp$. It follows that the frame integral in +\eqref{haar:small-ball-insertion} tends, for this $s$, to +\begin{equation}\label{haar:second-order-limit} + \alpha_{n_i,k}\delta^{-k} + \int\varphi(A,s)\,\mathrm dA_{D_i}. +\end{equation} + +The bound $\lVert\varphi\rVert_\infty\alpha_{n_i,k}\delta^{-k}$ +dominates the frame averages for every $t$. It is integrable in $s$. +Choose a $(k+1)$-dimensional subspace +$L\subset(D_{i-1}+\operatorname{span}\{s_0\})^\perp$; the available +dimension is at least $d-i\ge d-r>k+1$. Then +$\delta\ge\lVert\operatorname{proj}_L s\rVert$. The latter vector has +a bounded density by \eqref{haar:sphere-density}, because $d>k+3$. +In $k+1$ dimensions its inverse $k$th power is integrable at zero: +the radial integral is a constant times +$\int_0^1 u^k u^{-k}\,\mathrm du<\infty$. This also shows that +$\delta>0$ for $\sigma$-almost every $s$. Dominated convergence proves +the second limit after integration in $s$. + +Equating the two integrated limits in \eqref{haar:first-order-limit} and +\eqref{haar:second-order-limit} gives the insertion identity. Bounded +continuous functions determine finite Borel measures on the compact +frame--sphere space, so the identity also holds for bounded Borel tests. +The frame and fiber constructions above make the kernels measurable in +the preceding tuple, allowing successive integration. + +Start with $\sigma(\mathrm ds_0)\mathrm dA$ and insert, for +$i=1,\ldots,r$, the kernels +$p^0(As_0)U_{A,As_0}(\mathrm ds_i)$. Each insertion changes the frame +law from $\mathrm dA_{D_{i-1}}$ to $\mathrm dA_{D_i}$ and contributes +$\alpha_{d-i+1,k}\delta_i^{-k}$. On the original side, disintegration +of $s_0$ contributes one more factor $p^0(y)$, so its total power is +$r+1=m$. These measures are finite at every step because +$p^0\le\alpha_{d,k}$; in particular their final mass is +$\int p^0(y)^m\,\mathrm dy\le\alpha_{d,k}^{m-1}$. The preceding +almost-sure positivity of every inserted height gives the full-rank +support. This proves \eqref{haar:coarea-identity} for $\sigma$. + +Finally multiply this equality of measures by the bounded Borel function +$\prod_{i=0}^r g(s_i)$. Formula \eqref{haar:posterior} changes the +left side into the stated measure for $P$, while the right side becomes +the product of $P$ measures in \eqref{haar:coarea-identity}. The bounded +tilt preserves finiteness. No continuity of $g$ or of $p_A$ was used. +\end{proof} + +We now pass from the finite measure to the actual posterior experiment. +Draw $(A,Y)$ with law $\mathrm dA\,p_A(y)\mathrm dy$, and, conditionally +on $(A,Y)$, draw $S_0,\ldots,S_r$ independently from $P_{A,Y}$. Write +$Q$ for the law including the frame and tuple, and $Q_S$ for its tuple +marginal. Each $(A,S_i)$ has law $\mathrm dA\,P(\mathrm ds)$, and all +the replicas satisfy $AS_i=Y$ almost surely. + +Define the probability measure +$\Lambda(\mathrm dA,\mathrm d\boldsymbol s) + =P^{\otimes m}(\mathrm d\boldsymbol s)\mathrm dA_{D_r}$. +The degenerate tuples have zero +$P^{\otimes m}$-measure, by the positivity proved during insertion; fix +any measurable convention there. Multiply +\eqref{haar:coarea-identity} by $p_A(As_0)^{-r}$ on its positive set, +and by zero on its zero set. This operation is justified first for the +minimum with a finite constant and then by monotone convergence. On the +left it changes $p_A^m$ to $p_A$, since $m=r+1$ and the used posterior +is supported on $As_0=y$. Consequently +\begin{equation}\label{haar:replica-density} + \frac{\mathrm dQ}{\mathrm d\Lambda}(A,\boldsymbol s) + =\left(\prod_{i=1}^r\alpha_{d-i+1,k}\delta_i^{-k}\right) + p_A(As_0)^{-r} +\end{equation} +on the positive-density, full-rank set, with value zero elsewhere. + +For a fresh Haar $\ell$-frame $B$, independent of this whole experiment, +put $Z_i=BS_i$ and let $P_B$ be the image of $P$ under $B$. Define the +two total correlations +\begin{equation}\label{haar:correlations} + J_S=D_{\mathrm{KL}}(Q_S\Vert P^{\otimes m}),\qquad + J_Z=\mathbb E_B D_{\mathrm{KL}} + (Q_{Z_0,\ldots,Z_r\mid B}\Vert P_B^{\otimes m}). +\end{equation} + +\begin{lemma}[Finite upper correlation]\label{haar:upper-correlation} +The correlations satisfy $0\le J_Z\le J_S\le U<\infty$, where +\begin{align} + U&=\sum_{i=1}^r\left[ + \log\alpha_{d-i+1,k}+k\mathbb E_Q\log(1/\delta_i) + -\mathbb E_{A,S\sim P}\log p_A(AS)\right], + \label{haar:upper-U}\\ + 0\le U&\le Crd(\log K+\log d).\label{haar:crude-U} +\end{align} +Here $\mathbb E_{A,S\sim P}$ draws the Haar frame independently of a +vector with law $P$. Every logarithm in \eqref{haar:upper-U} is +integrable. +\end{lemma} + +\begin{proof} +The positive part of $\log p_A$ under $p_A$ is bounded uniformly in $A$ +by $\log^+(K\alpha_{d,k})$. Its negative part is also integrable +uniformly: the density is supported in a ball of finite volume, and +$-x\log x\le1/e$ for $00$. +The conditional residual direction of $S_i$ in $\ker A$ has density at +most +\[ + K'=\frac{Kp^0(Y)}{p_A(Y)}\ge1 +\] +relative to uniform probability on $S^{n-1}$. Choose a unit vector +$e\in\ker A$ perpendicular to the preceding residual directions, +including that of $S_0$. There are at most $i\le r0}\frac{1}{v_kt^k} + \int_{\lVert u'-u\rVert\le t} f(u')\,\mathrm du'. +\] +For fixed $f\in L^1$, the ball integral is continuous in its center and +positive radius; the supremum may therefore be restricted to positive +rational radii. In particular $\mathcal M f$ is measurable. For the +jointly measurable family $p_A$, the same rational supremum and Tonelli's +theorem give joint measurability in $(A,u)$. + +For historical background on one-dimensional maximal averages, see Hardy +and Littlewood~\cite[Section~III]{HardyLittlewood1930}. Stein and +Str\"omberg~\cite[p.~259]{SteinStromberg1983} discuss the exponential +dimension cost of customary covering proofs. We prove the centered weak +estimate needed here, including its constant. + +\begin{lemma}[Logarithmic maximal estimate]\label{haar:maximal} +Suppose $f$ is a probability density supported in the unit ball of +$\mathbb R^k$ and $f\le K\alpha_{d,k}$. Then +\begin{equation}\label{haar:maximal-log} + \int f(u)\log\frac{\mathcal M f(u)}{f(u)}\,\mathrm du + \le Cd+\log(1+Cd+\log K). +\end{equation} +Both logarithms are integrable under $f(u)\mathrm du$. +The integrand is taken as zero on $\{f=0\}$. +\end{lemma} + +\begin{proof} +For every $\lambda>0$, +\begin{equation}\label{haar:weak-maximal} + \bigl|\{u:\mathcal M f(u)>\lambda\}\bigr| + \le\frac{3^k}{\lambda}. +\end{equation} +To prove this, take a compact subset of the displayed superlevel set. +Open and closed balls have the same $f$-mass, since their boundaries have +Lebesgue measure zero. +At each point choose an open ball centered there whose $f$-mass exceeds +$\lambda$ times its volume. The strict inequality persists under a +sufficiently small enlargement, so these balls give an open cover. +Take a finite subcover and greedily keep disjoint balls in decreasing +order of radius. Every discarded ball meets a retained ball of at least +its radius, and is contained in that ball's threefold dilation. Hence +the compact set has volume at most $3^k$ times the sum of the retained +volumes. The retained balls are disjoint and have total $f$-mass at most +one, so their volumes sum to less than $1/\lambda$. Inner regularity +proves \eqref{haar:weak-maximal}. + +Set $b=3^k$ and $L_f=K\alpha_{d,k}$. Since $\mathcal M f\le L_f$, +layer integration over the unit ball gives +\begin{align*} + \int_{\lVert u\rVert\le1}\mathcal M f(u)\,\mathrm du + &\le\int_0^{L_f}\min(v_k,b/\lambda)\,\mathrm d\lambda\\ + &\le b\bigl(1+\log^+(L_fv_k/b)\bigr). +\end{align*} +On the support of $f$, a radius-two ball contains the whole unit ball, +so $\mathcal M f\ge(v_k2^k)^{-1}$ there. Its logarithm is bounded on +that support, while $f\log f$ is integrable by the bounded-density +argument in Lemma~\ref{haar:upper-correlation}. Jensen's inequality +under the probability density $f$ now yields +\[ + \int f\log(\mathcal M f/f) + \le\log\int_{\{f>0\}}\mathcal M f + \le k\log3+\log\bigl(1+\log^+(K\alpha_{d,k}v_k/3^k)\bigr). +\] +Use \eqref{haar:gamma-bounds} and $d/k\le32$ to obtain +\eqref{haar:maximal-log}. +\end{proof} + +The next test uses only the projected points, including when their +conditional distance distribution has atoms. + +\begin{lemma}[Lower projected correlation]\label{haar:lower-correlation} +For the replica law above and every $00$ almost surely under its own law. + +Apply the relative-entropy variational inequality to the test +$z\mapsto F_i(R_i(z))^{-a}$ under the conditional one-vector reference +law $P_B$. +For an unbounded test, use its minimum with a finite constant and pass +monotonically to the limit; \eqref{haar:cdf-moment} bounds its reference +expectation. The chain rule for $J_Z$ has one term for each conditional +law of $Z_i$ given $(B,Z_0,\ldots,Z_{i-1})$, and its $i=0$ term is zero. +It follows that +\begin{equation}\label{haar:cdf-test} + J_Z\ge a\sum_{i=1}^r\mathbb E_{Q,B}\log\frac1{F_i(\rho_i)} + +r\log(1-a). +\end{equation} +The logarithmic expectations are finite: the variational bound after +averaging controls their nonnegative parts by the finite correlation +$J_Z\le J_S$ from Lemma~\ref{haar:upper-correlation}. + +We next estimate $F_i(\delta_i)$ while holding the entire replica +outcome $(A,S_0,\ldots,S_r)$ fixed. Put $D=D_{i-1}$ and +$E=\operatorname{row}(B)$. The isometry $B^{\mathsf T}:\mathbb R^\ell +\to E$ gives the exact identity +\begin{equation}\label{haar:intersection} + \operatorname{dist}(Bu,BD) + =\lVert\operatorname{proj}_{E\cap D^\perp}u\rVert. +\end{equation} +Indeed, the orthogonal complement within $E$ of +$\operatorname{proj}_E D$ is precisely $E\cap D^\perp$. +The projection of the fixed $(i-1)$-space $D$ into a Haar $\ell$-space +$E$ has full rank almost surely: generate $E$ with independent standard +Gaussians and restrict them to an orthonormal basis of $D$, obtaining an +$\ell\times(i-1)$ Gaussian matrix of full column rank. Thus +$\dim(E\cap D^\perp)=q_i=\ell-i+1$. Rotations preserving $D$ show +that this intersection is a uniform $q_i$-space in the $n_i$-space +$D^\perp$. + +For fixed $u$, let $v=\operatorname{proj}_{D^\perp}u$. Spherical +projection in that $n_i$-space, followed by +\eqref{haar:gamma-bounds}, gives +\begin{equation}\label{haar:projected-small-ball} + \mathbb P_B\{\operatorname{dist}(Bu,BD)\le t\} + \le\min\left(1,\left(\frac{C_0t}{\lVert v\rVert}\right)^{q_i}\right), +\end{equation} +with value one if $v=0$. The constant $C_0$ is absolute because +$n_i/q_i\le16/3$; the density formula is applicable because +$n_i-q_i=d-\ell>2$. In the replica experiment $A$ annihilates $D$. +Therefore for $u=s-S_0$, +$\lVert v\rVert\ge\lVert A(s-S_0)\rVert$. Also +$\rho_i\le\delta_i$, since $B$ is a contraction and the distance +minimization permits all vectors in $D$. + +Set $u_0=AS_0$ and $q=q_i$. The identity +\[ + \min(1,(b/x)^q)=\int_1^\infty qt^{-q-1} + \mathbf 1_{\{x\le bt\}}\,\mathrm dt + \qquad(b>0,\ x\ge0) +\] +also holds at $x=0$. Integrating \eqref{haar:projected-small-ball} +against $s\sim P$ and using Tonelli's theorem yields +\begin{align} + \mathbb E_B F_i(\delta_i) + &\le\int_1^\infty qt^{-q-1} + P\{\lVert As-u_0\rVert\le C_0\delta_i t\}\,\mathrm dt\notag\\ + &\le v_k(C_0\delta_i)^k\mathcal M p_A(u_0) + \int_1^\infty qt^{k-q-1}\,\mathrm dt\notag\\ + &=\frac{q_i}{q_i-k}v_k(C_0\delta_i)^k\mathcal M p_A(u_0) + \le C_1^kv_k\delta_i^k\mathcal M p_A(u_0). + \label{haar:projected-maximal} +\end{align} +The integral converges because $q_i>k$, and the last step uses the exact +bound $q_i/(q_i-k)\le3/2$. One absolute choice of $C_1$ works for every +$k\ge1$. + +Now $F_i(\rho_i)\le F_i(\delta_i)$. Jensen's inequality for $-\log$ +and \eqref{haar:projected-maximal}, first for each fixed replica outcome, +give +\[ + \mathbb E_{Q,B}\log\frac1{F_i(\rho_i)} + \ge k\mathbb E_Q\log(1/\delta_i)-\log(C_1^kv_k) + -\mathbb E_{A,S\sim P}\log\mathcal M p_A(AS). +\] +Here the last expectation uses the individual marginal $(A,S_0)$. +The right side is finite by Lemmas~\ref{haar:upper-correlation} and +\ref{haar:maximal}; Jensen may also be read in the extended sense before +averaging. Substitution into \eqref{haar:cdf-test} proves +\eqref{haar:lower-Z}. +\end{proof} + +\subsection{Cancellation and the block information bound} + +The preceding estimates can now be subtracted. Retaining every term of +$U$ gives, for every $00$. Its conditional signal +law has the bounded density +\[ + g_v(s)=\frac{\mathbb P\{V=v\mid S=s\}}{p_v}\le K_v:=\frac1{p_v} +\] +relative to $\sigma$. The new Gaussian design matrix is independent of +$(S,V)$. Its row Gram--Schmidt factorization is $RA$ with an invertible +triangular $R$ and a Haar $k$-frame $A$, almost surely. The exact labels +are $R(AS)$, so, conditional on the design, observing them is equivalent +to observing $Y=AS$. Because the whole design is independent of $S$ +conditional on $V=v$, its coefficients $R$ add no information about $S$ +once $(A,Y)$ is specified. + +Generate the replicas conditionally independently from this posterior, +and independently of $W$ given $v$ and the complete block data. A single +replica then has the same joint law with the block data and $W$ as the +original signal: both use the posterior given the block, and $W$ has no +further dependence on the signal once those data and $v$ are fixed. +In particular each pair $(S_i,W)$ has the same conditional law as +$(S,W)$, the auxiliary $B$ is independent of the entire tuple and $W$, +and the original pair used on the left is also coupled independently +with $B$. The common map $(B,s)\mapsto Bs$ is continuous, and the +unprojected total correlation is finite by +Lemma~\ref{haar:upper-correlation}. These are precisely the application +hypotheses of Lemma~\ref{lem:replica-information}. Applied in the +conditional experiment $V=v$, it gives +\[ + mI(S;W\mid V=v,B,BS) + \le H(W\mid V=v)+J_S(v)-J_Z(v). +\] +No conditional independence of the replicas given $W$ is asserted or +needed. + +Insert \eqref{haar:fixed-prior-budget} with $K=K_v$ and average over $v$. +The identity +\[ + \sum_v p_v\log K_v=\sum_vp_v\log(1/p_v)=H(V) +\] +accounts exactly for rare state values. Jensen's inequality bounds the +average of $\log(1+Cd+\log K_v)$ by $\log(1+Cd+H(V))$. This proves +\eqref{haar:state-budget}. Under the width hypothesis, +$H(W\mid V)/m=o(d)$, the last term is $o(1)$, and the logarithmic term +is $O(\log d)$ for all sufficiently large $d$. The stated $Cd$ bound +follows. +\end{proof} + +\subsection{The terminal fiber and the streaming conclusion} + +The block comparison is complete. We apply the residual-sphere endpoint +of Section~\ref{sec:model-endpoint} with the present Haar side frame, +then compare the resulting information with the number of blocks. + +\begin{lemma}[The terminal Haar fiber]\label{haar:terminal} +Let $S\sim\sigma$, let $V_f$ be finite-valued, and let the unit-valued +output $\widehat S$ be generated from $V_f$ by a measurable kernel using +fresh randomness, conditionally independently of $S$. Let $B$ be a Haar +$\ell=\lfloor d/4\rfloor$ frame independent of $(S,V_f,\widehat S)$. +If +$\mathbb P\{\arccos\langle\widehat S,S\rangle\le\epsilon\}\ge2/3$ +for $0<\epsilon\le1/10$, then +\begin{equation}\label{haar:terminal-information} + I(S;V_f\mid B,BS) + \ge\frac{d-\ell-1}{3}\log\frac1{4\epsilon}-\log2. +\end{equation} +\end{lemma} + +\begin{proof} +Apply Lemma~\ref{lem:residual-information} with $G=B$, $V=V_f$, +$b=\ell$, and $r_0=1/2$. Its hypotheses hold because $B$ is independent of +$(S,V_f,\widehat S)$. Explicitly, the Markov +bound for the residual radius $h$ is +$\mathbb P\{h<1/2\}\le4\ell/(3d)\le1/3$. Success together with +$h\ge1/2$ therefore has probability at least $1/3$, while the conditional +prior cap probability is at most $(4\epsilon)^{d-\ell-1}$. These are exactly +the constants in \eqref{haar:terminal-information}. +\end{proof} + +\begin{proposition}[Streaming consequence]\label{haar:streaming} +For each dimension $d$, let $S\sim\sigma$ and consider a learner in +Definition~\ref{def:streaming-model}, with deterministic horizon $T=T(d)$ +and nonnegative integer memory $M=M(d)$. If $M(d)=o(d^2)$, +$0<\epsilon(d)\le1/10$, and its angular success probability averaged over +the signal, samples, and learner randomness is at least $2/3$, then +\[ + T(d)=\Omega\bigl(d\log(1/\epsilon(d))\bigr). +\] +The implicit constant is absolute; the dimension threshold may depend on +the memory sequence. +\end{proposition} + +\begin{proof} +Apply the fixed-shared-seed part of Lemma~\ref{lem:padding} to the learner +in Definition~\ref{def:streaming-model}. The resulting Borel experiment +has data-independent initialization and uniform-prior success at least +$2/3$, with fresh transition and output randomness retained. This is +the success threshold in Lemma~\ref{haar:terminal}. + +First, success forces $T=\Omega(d)$ without any memory restriction. +Suppose $T\le d/8$, and grant an estimator all $T$ raw pairs, filling +in ignored samples after any early stopping. Lemma~\ref{lem:residual-sphere} +with $b=T$ gives a uniform residual sphere in kernel dimension $d-T$; +\eqref{eq:residual-markov} bounds its probability of radius below $1/2$ +by $4T/(3d)$. At $T=0$ the radius is one deterministically. On radius +at least $1/2$, the cap argument of Lemma~\ref{lem:raw-sample-bound} +bounds conditional success by $(4\epsilon)^{d-T-1}$. Thus +\begin{equation}\label{haar:raw-sample-bound} + \mathbb P\{\text{angular error}\le\epsilon\} + \le\frac{4T}{3d}+(4\epsilon)^{d-T-1} + \qquad(T\le d/8). +\end{equation} +The output has no further dependence on $S$ conditional on all the raw +data, so the bound also covers randomized estimators. Its right side is +at most $1/6+(0.4)^{7d/8-1}<2/3$ for all sufficiently large $d$. +Hence $T>d/8$ on those dimensions. + +Put $L=\log(1/\epsilon)\ge\log10$. If $T\ge dL$, the conclusion already +holds. On the remaining branch $Td/8$ give +\[ + d\lceil T/k\rceil\le32T+d\le40T. +\] +Comparison with \eqref{haar:information-clock} proves the proposition. +\end{proof} + +A learner that succeeds with probability at least $2/3$ for every fixed +signal also has uniform-prior success at least $2/3$ by integration. +Thus Proposition~\ref{haar:streaming} separately gives the corresponding +worst-signal lower bound, with the same exact observations and finite-state +model. diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/06-synthetic.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/06-synthetic.tex new file mode 100644 index 0000000000000000000000000000000000000000..1541dd357859692ca019d361a298f26c427d4a47 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/06-synthetic.tex @@ -0,0 +1,610 @@ +\section{Synthetic Gaussian projections}\label{sec:synthetic} + +The Gaussian test in Section~\ref{sec:gaussian} uses an orthonormal +selector of the projected difference space's complement. Here we instead +draw additional Gaussian rows in that complement. Their composition with +the side matrix has a random covariance, which must be averaged in the +likelihood comparison. To normalize that comparison, we apply the +Gaussian-base equal-label identity to the projected prior, rather than +applying only its spherical instance to the original prior. + +The inverse-determinant scale of this random covariance cancels the scale +of the projected simplex volume. A critical inverse-distance kernel then +gives a double-logarithmic density cost. Although the cost has the same +order as in Section~\ref{sec:gaussian}, the comparison law and cancellation +are different. We use this bound while conditioning on an entire state +history. + +We use the spherical and Gaussian notation of Section~\ref{sec:replicas}. +Fix \(d\ge16\) and put +\begin{equation}\label{syn:parameters} + k=\ell=\left\lfloor\frac d8\right\rfloor,\qquad + u=\ell-1,\qquad m=\left\lfloor\frac d2\right\rfloor,\qquad q=m-u. +\end{equation} +In particular, +\begin{equation}\label{syn:dimension-margins} + k+u\le m-2,\qquad q-k-1=m-2k>0,\qquad + d-1-u-k=d-2k\ge\frac{3d}{4},\qquad + \frac{m}{q-k-1}\le2. +\end{equation} +The letter \(m\) in this section is the dimension of the side projection. +The difference count in Theorem~\ref{thm:equal-label-measure} will be \(u\). + +\begin{proposition}[Information beyond an independent Gaussian projection] +\label{syn:block} +Let \(P=f\sigma_d\) be a probability, where \(f\) is a nonnegative Borel +function satisfying \(f\le\Lambda<\infty\), with \(\Lambda\ge1\). +Draw \(X\sim P\) and \(A\sim\gamma_{k,d}\) independently. Let +\(z=z(A,AX)\) be a measurable function taking at most \(K\ge1\) values. +Independently of this entire experiment draw \(B\sim\gamma_{m,d}\), and put +\(Y=BX\). With the parameters in \eqref{syn:parameters}, +\begin{equation}\label{syn:block-bound} + I(X;z\mid B,Y) + \le \frac{\log K}{\ell} + +C\bigl[d+\log(1+\log\Lambda)\bigr], +\end{equation} +where \(C\) is absolute and independent of \(P,\Lambda,K\), and the message +rule. +\end{proposition} + +The proof couples \(\ell\) signals by giving them the same actual +measurements. Corollary~\ref{cor:replica-likelihood} describes their joint +law. We compare its total correlation before and after applying \(B\). +For the comparison after \(B\), a Gaussian matrix is constrained to +annihilate the projected differences. We first construct the resulting +probability law, then estimate its density while keeping its dependence +on \(B\) explicit. + +\subsection{The actual and projected replica laws} + +For a tuple \(\mathbf x=(x_1,\ldots,x_\ell)\) in \(\R^h\), where +\(h\) will be \(d\) or \(m\), write +\[ + D_{\mathbf x}=[x_2-x_1,\ldots,x_\ell-x_1],\qquad + L_{\mathbf x}=\col(D_{\mathbf x}),\qquad + \Delta(\mathbf x)=\det(D_{\mathbf x}^{\mathsf T}D_{\mathbf x})^{1/2}. +\] +Thus \(L_{\mathbf x}\) and \(\Delta(\mathbf x)\) are the objects +\(F_{\mathbf x}\) and \(J(\mathbf x)\) of +\eqref{eq:equal-label-geometry}, with difference count \(u\). At full +affine rank define +\begin{equation}\label{syn:tuple-weight} + w(\mathbf x)=(2\pi)^{-ku/2}\Delta(\mathbf x)^{-k}. +\end{equation} + +We need the same version of a projection density before and after a +composition of matrices. If \(\nu\) is a finite measure and \(C\) has +\(r\) rows, write +\begin{equation}\label{syn:density-version} + g_C^\nu(v)=\liminf_{j\to\infty} + \int\varphi_{1/j,r}(Cs-v)\,\nu(ds),\qquad + \varphi_{\eta,r}(v)=(2\pi\eta^2)^{-r/2} + \exp\!\left(-\frac{\norm v^2}{2\eta^2}\right). +\end{equation} +This formula defines a jointly measurable function even for matrices of +deficient rank. In every use below the matrix has full row rank almost +surely, and Theorem~\ref{thm:equal-label-measure} identifies this version +with the projection density and transfers its finite positive values to the +constrained evaluations. We abbreviate \(g_C^P\) to \(g_C\). + +Fix the data of Proposition~\ref{syn:block}. In +Corollary~\ref{cor:replica-likelihood}, take ambient dimension \(d\), +\(k\) rows, difference count \(u\), and prior \(P\). The condition +\(k+u\le d-2\) follows from \eqref{syn:dimension-margins}. +Draw \(A\sim\gamma_{k,d}\), then a common label \(b\) with density +\(g_A\), and finally draw +\(X_1,\ldots,X_\ell\) independently from \(P\) conditional on \(AX=b\), +and set \(z=z(A,b)\). Denote this law by \(Q\). Its tuple--matrix +likelihood is +\begin{equation}\label{syn:actual-replica} + Q(d\mathbf x,dA) + =P^{\otimes\ell}(d\mathbf x)\gamma_{k,d}^{L_{\mathbf x}}(dA) + w(\mathbf x)g_A(Ax_1)^{-u}. +\end{equation} +Every pair \((X_i,z)\) has the original signal--message law. +Independently draw \(B\) and put \(Y_i=BX_i\). For fixed \(B\), let +\(P^B=B_\#P\), and define +\[ + \mathcal C_X=\KL(Q_{\mathbf X}\Vert P^{\otimes\ell}),\qquad + \mathcal C_Y=\E_B\KL(Q_{\mathbf Y\mid B}\Vert(P^B)^{\otimes\ell}). +\] +Lemma~\ref{lem:replica-logs}, with the same substitution and density +bound \(L=\Lambda\), proves absolute integrability of +\(\log g_A(b)\) and \(\log w(\mathbf X)\). It gives +\begin{equation}\label{syn:correlation-upper} + \mathcal C_X + \le \E\log w(\mathbf X)-u\,\E\log g_A(b)<\infty. +\end{equation} +Data processing under the coordinatewise map \(x_i\mapsto Bx_i\) gives +\(0\le\mathcal C_Y\le\mathcal C_X\). + +We next construct a probability density against which to test +\(\mathcal C_Y\). Almost every \(B\) has full row rank. For such a fixed +\(B\), the spherical projection formula +\eqref{eq:spherical-projection-density}, with \(m\) rows, and +\(P\le\Lambda\sigma_d\) show that \(P^B\) has bounded Lebesgue density +on a compact ellipsoid. Here \(d-m-2\ge0\). The standard Gaussian density +has a positive minimum on that ellipsoid, so \(P^B\) has bounded density +relative to standard Gaussian probability on \(\R^m\). This bound is +finite for each such \(B\) and is allowed to depend on \(B\). + +Apply Corollary~\ref{cor:replica-likelihood} in ambient dimension \(m\), +again with \(k\) rows and difference count \(u\). Its dimension condition +is \(k+u\le m-2\), as in \eqref{syn:dimension-margins}. The resulting +same-label tuple law has density +\begin{equation}\label{syn:comparison-density} + R_B(\mathbf y) + =w(\mathbf y)\, + \E_{F\sim\gamma_{k,m}^{L_{\mathbf y}}}g_{FB}(Fy_1)^{-u} +\end{equation} +relative to \((P^B)^{\otimes\ell}\). Indeed, the density version agrees +pointwise under composition: +\begin{equation}\label{syn:composition-version} + g_F^{P^B}(v) + =\liminf_j\int\varphi_{1/j,k}(Fy-v)\,P^B(dy) + =g_{FB}^{P}(v). +\end{equation} +The finite-measure theorem transfers version, zero, and infinity +exceptions before the negative power is taken. Thus \(R_B\) is finite and +positive almost everywhere for the product law and integrates to one. +It can be chosen jointly measurable in \(B,\mathbf y\): on the +full-affine-rank set, realize the projected Gaussian rows with the Borel +orthogonal projection onto \(L_{\mathbf y}^{\perp}\), and assign fixed +values on the null complement. + +The actual law \(Q_{\mathbf Y\mid B}\) is absolutely continuous relative +to \((P^B)^{\otimes\ell}\), because \(Q_{\mathbf X}\ll P^{\otimes\ell}\) +and \(B\) is independent of the tuple. At the actual \((B,\mathbf Y)\), +draw \(F\) with the conditional law in \eqref{syn:comparison-density} +and put +\[ + \Xi=g_{FB}(FY_1). +\] +The same null-set transfer makes \(\Xi\) finite and positive almost surely +in this enlarged experiment. + +\subsection{The synthetic density and its critical kernel} + +The comparison density \(R_B\) contains the inverse \(u\)th moment of +\(\Xi\). To compare its logarithm with the actual likelihood in +\eqref{syn:correlation-upper}, we will control \(\Xi\) using the actual +matrix \(A\) and label \(b\). The next lemma first averages the synthetic +Gaussian covariance at a fixed tuple. It then bounds the resulting +inverse-distance integral by a function \(U_A(b)\) whose total mass can +be estimated independently of that tuple. + +\begin{lemma}[Synthetic likelihood and critical kernel]\label{syn:synthetic} +Let \(P=f\sigma_d\) be as in Proposition~\ref{syn:block}. Fix a tuple +\(\mathbf x\in(S^{d-1})^\ell\) of affine rank \(u\), and put +\[ + x=x_1,\qquad L=L_{\mathbf x},\qquad + \delta(s)=\dist(s-x,L). +\] +Draw \(B\sim\gamma_{m,d}\). Given \(B\), draw +\(F\sim\gamma_{k,m}^{BL}\), so each row of \(F\) is a standard Gaussian +in \((BL)^\perp\). Then +\begin{equation}\label{syn:synthetic-bound} + \E_{B,F}g_{FB}(FBx) + \le (2\pi)^{-k/2}a_q\int\delta(s)^{-k}\,P(ds),\qquad + a_q=\E\det(HH^{\mathsf T})^{-1/2}\le(q-k-1)^{-k/2}, +\end{equation} +where \(H\) is a standard Gaussian \(k\times q\) matrix. + +There is an absolute \(C_0>0\) such that the following holds with +\begin{equation}\label{syn:cutoff} + r_*=\exp\!\left[-\frac{C_0(d+\log\Lambda)}{d-1-u-k}\right]\in(0,1). +\end{equation} +For every full-row-rank \(k\times d\) matrix \(A\), put \(a=\norm A_{\op}\) +and define +\begin{align} + U_A(b)={}&(2\pi)^{-k/2}\ind_{\{\norm b\le a\}} + \left[1+a^k\int g_A(v)\ind_{\{\norm{v-b}\le2a\}}\right. \notag\\[-2mm] + &\hspace{42mm}\left. + {}\cdot\min\{(ar_*)^{-k},\norm{v-b}^{-k}\}\,dv\right]. + \label{syn:majorant} +\end{align} +At \(v=b\), the second entry in the minimum is interpreted as \(+\infty\). +If \(Ax=b\) and \(L\subset\ker A\), then +\begin{equation}\label{syn:kernel-bound} + (2\pi)^{-k/2}\int\delta(s)^{-k}\,P(ds)\le U_A(b). +\end{equation} +Writing \(V_k\) for the volume of the Euclidean unit ball in \(\R^k\), +the same function satisfies +\begin{align} + \int U_A(b)\,db + &\le (2\pi)^{-k/2}V_k a^k[2+k\log(2/r_*)], + \label{syn:majorant-mass}\\ + \E_{A\sim\gamma_{k,d}}\log\int U_A(b)\,db + &\le C\bigl[d+\log(1+\log\Lambda)\bigr]. + \label{syn:majorant-log} +\end{align} +The logarithm in the last display is absolutely integrable, and +\(\E_A\log(1+\int U_A)<\infty\). +\end{lemma} + +\begin{proof} +At the fixed tuple, the restriction of \(B\) to \(L\) has rank \(u\) +almost surely. Choose an orthonormal \(m\times q\) matrix \(N\) whose +columns span \((BL)^\perp\), using only \(BD_{\mathbf x}\). Gram--Schmidt +with a fixed ordering gives a measurable choice on the full-rank set. +With an independent standard Gaussian \(k\times q\) matrix \(H\), the +required conditional law of \(F\) is realized by +\begin{equation}\label{syn:synthetic-realization} + F=HN^{\mathsf T}. +\end{equation} +The choice of \(N\) uses only \(B|_L\). The Gaussian restriction +\(B|_{L^\perp}\) is independent of \(B|_L\). Write \(s-x=v+w\) with +\(v\in L\) and \(w\in L^\perp\). Then \(N^{\mathsf T}Bv=0\), while, +conditionally on \(B|_L\), the vector \(N^{\mathsf T}Bw\) has law +\(N(0,\norm w^2I_q)\). Therefore, conditionally on \(H\) and \(B|_L\), +\begin{equation}\label{syn:covariance} + FB(s-x)\sim N(0,\delta(s)^2HH^{\mathsf T}). +\end{equation} +This conditional calculation accounts for the dependence of \(F\) on \(B\). + +For \(\eta>0\), Tonelli's theorem and \eqref{syn:covariance} give +\[ + \E_{B,F}\int\varphi_{\eta,k}(FB(s-x))\,P(ds) + =(2\pi)^{-k/2}\int + \E_H\det(\delta(s)^2HH^{\mathsf T}+\eta^2I_k)^{-1/2}\,P(ds). +\] +The integrand on the right increases as \(\eta\downarrow0\). The tube +bound \eqref{eq:affine-tube} gives \(\delta(s)>0\) for \(P\)-almost every +\(s\); it also gives a finite inverse \(k\)th moment, since +\(d-1-u>k\). The matrix \(H\) has full row rank almost surely. +Monotone convergence on the right and Fatou's lemma on the left, along +\(\eta=1/j\), prove the first inequality in \eqref{syn:synthetic-bound}. + +For completeness, successive orthogonal residuals of the rows of \(H\) +have independent squared lengths +\[ + V_i\sim\chi^2_{q-i+1},\qquad + \det(HH^{\mathsf T})=\prod_{i=1}^kV_i. +\] +Indeed, conditionally on the preceding rows, the new row is standard +Gaussian on a complement of dimension \(q-i+1\); this conditional law is +constant, which also proves independence of the residual lengths. This is +the elementary Gaussian orthogonal reduction described in +Edelman and Rao~\cite[Section~5]{EdelmanRao2005}; the rectangular product +used here follows from the displayed conditioning. Direct integration of +the chi-squared density gives \(\E V^{-1}=1/(p-2)\) for +\(V\sim\chi_p^2\), \(p>2\). Cauchy--Schwarz hence gives +\[ + a_q=\prod_{i=1}^k\E V_i^{-1/2} + \le\prod_{i=1}^k(q-i-1)^{-1/2} + \le(q-k-1)^{-k/2}. +\] + +We next use the actual observations. If \(Ax=b\) and \(L\subset\ker A\), +then, for \(v=P_L(s-x)\), +\begin{equation}\label{syn:actual-kernel} + \norm{As-b}=\norm{A(s-x-v)}\le a\,\delta(s). +\end{equation} +Let \(\alpha=d-1-u\). The tube bound, now applied to \(x+L\), gives +\[ + P\{\delta\le t\}\le\Lambda C^d t^\alpha,\qquad 0r_*\}\), +\eqref{syn:actual-kernel} gives both +\(\delta^{-k}\le r_*^{-k}\) and +\(\delta^{-k}\le a^k\norm{As-b}^{-k}\). +The pushforward of \(P\) under \(A\) has density \(g_A\), and both \(As\) +and \(b\) have norm at most \(a\). Integration of these two bounds gives +\eqref{syn:kernel-bound} with the two support cutoffs in +\eqref{syn:majorant}. + +The remaining singularity has the critical exponent \(k\) in dimension +\(k\). Radial integration gives the exact mass +\begin{equation}\label{syn:critical-mass} + \int_{\{\norm w\le2a\}} + \min\{(ar_*)^{-k},\norm w^{-k}\}\,dw + =V_k[1+k\log(2/r_*)]. +\end{equation} +The ball of radius \(ar_*\) contributes \(V_k\), and the annulus contributes +\(kV_k\int_{ar_*}^{2a}dt/t\). Dropping the outer cutoff in +\eqref{syn:majorant} only for its convolution term, and using +\(\int g_A=1\), proves \eqref{syn:majorant-mass}. + +We record the scale in this bound before taking its logarithm. Integrating +the standard Gaussian density over the ball of radius \(\sqrt k\) gives +\((2\pi)^{-k/2}V_k\le(e/k)^{k/2}\). Also +\(\E\norm A_{\op}\le C\sqrt d\): choose \(1/4\)-nets on the two unit +spheres of cardinalities at most \(9^k\) and \(9^d\), bound the operator +norm by twice the maximum of the scalar bilinear forms on the nets, and +integrate the union bound for their standard Gaussian tails. Jensen's +inequality gives +\[ + \E\log\bigl[(2\pi)^{-k/2}V_k\norm A_{\op}^k\bigr] + \le\frac k2(1-\log k)+k\log(C\sqrt d)\le Cd. +\] +The cutoff \eqref{syn:cutoff} and \eqref{syn:dimension-margins} give +\[ + \log[2+k\log(2/r_*)] + \le C+\log d+\log(1+\log\Lambda). +\] +These estimates prove \eqref{syn:majorant-log}. Finally, +\(\log\norm A_{\op}\) is integrable below because the norm dominates the +absolute value of one standard Gaussian entry. Retaining the first term +of \(U_A\) bounds its total mass below by +\((2\pi)^{-k/2}V_k\norm A_{\op}^k\). +Together with \eqref{syn:majorant-mass}, these observations prove the two +remaining integrability assertions. +\end{proof} + +\subsection{Integrability and cancellation of the correlation costs} + +The majorant now controls the synthetic density at the actual label. +Before subtracting the two correlation bounds, we verify integrability +of the comparison logarithms. This uses both the finite actual +correlation and the normalization of \(R_B\). + +\begin{lemma}[Integrability of the projected comparison]\label{syn:logs} +In the replica experiment \(Q\), enlarged by the independent \(B\) and +the conditional \(F\) above, +\[ + \E|\log w(\mathbf Y)|+\E|\log\Xi| + +\E|\log R_B(\mathbf Y)|<\infty. +\] +If \(G\) is a standard Gaussian \(m\times u\) matrix, then +\begin{align} + \E\log w(\mathbf Y) + &=\E\log w(\mathbf X)-\frac{k}{2}\E\log\det(G^{\mathsf T}G), + \label{syn:gram-log}\\ + \frac12\E\log\det(G^{\mathsf T}G)&\le\frac u2\log m. + \label{syn:gram-upper} +\end{align} +Moreover, +\begin{equation}\label{syn:correlation-lower} + \mathcal C_Y + \ge\E\log R_B(\mathbf Y) + \ge\E\log w(\mathbf Y)-u\,\E\log\Xi. +\end{equation} +\end{lemma} + +\begin{proof} +At a fixed full-affine-rank \(\mathbf X\), write +\(D_{\mathbf X}=UR\), where \(U\) has orthonormal columns. Since \(B\) +is independent, \(BU\) is a standard Gaussian \(m\times u\) matrix. +Consequently +\[ + \Delta(\mathbf Y)\stackrel{\mathrm{law}}{=} + \Delta(\mathbf X)\det(G^{\mathsf T}G)^{1/2} + \quad\text{conditionally on }\mathbf X. +\] +The logarithm of \(\Delta(\mathbf X)\) is integrable by +Lemma~\ref{lem:replica-logs}. Gaussian column residuals factor +\(\det(G^{\mathsf T}G)\) into independent chi-squared variables of degrees +\(m,m-1,\ldots,m-u+1\). Their absolute logarithms are integrable from +their densities at zero and infinity. This proves +\eqref{syn:gram-log} and the asserted integrability of \(\log w(\mathbf Y)\). +Jensen applied to each factor gives \eqref{syn:gram-upper}. + +The actual equal-label experiment has \(L_{\mathbf X}\subset\ker A\). +Conditionally on \((A,b,\mathbf X)\), the matrix \(B\) remains independent, +and the conditional law of \(F\) is +\(\gamma_{k,m}^{B L_{\mathbf X}}\). Lemma~\ref{syn:synthetic} therefore +gives +\begin{equation}\label{syn:conditional-mean} + \E[\Xi\mid A,b,\mathbf X]\le a_qU_A(b). +\end{equation} +It follows that \(\E\log(1+\Xi)<\infty\). Here are the details needed for +the possible small values of \(g_A(b)\). Conditional Jensen reduces the +claim to integrability of \(\log(1+U_A(b))\). On the actual finite positive +set for \(g_A\), +\[ + \log(1+U_A(b)) + \le\log(1+g_A(b))+\log(1+U_A(b)/g_A(b)). +\] +The first term is integrable by Lemma~\ref{lem:replica-logs}. For the second, +average over \(b\) with density \(g_A\) and apply Jensen to obtain at most +\(\log(1+\int U_A)\). Lemma~\ref{syn:synthetic} makes its average finite. +Thus \((\log\Xi)^+\) is integrable. + +Let the product reference first draw \(B\) and then draw \(\ell\) +independent points from \(P^B\). The divergence of the actual joint law +from this reference is \(\mathcal C_Y<\infty\), and \(R_B\) integrates to one under this +reference. Equation~\eqref{eq:entropy-positive-log} gives +\begin{equation}\label{syn:positive-comparison-log} + \E(\log R_B(\mathbf Y))^+ + \le\E\log(1+R_B(\mathbf Y))\le\mathcal C_Y+\log2<\infty. +\end{equation} +Conditionally on \(B,\mathbf Y\), Equation +\eqref{syn:comparison-density} has a finite positive right side, so +\(\E_F\Xi^{-u}<\infty\). Hence +\(\E_F(\log\Xi)^-<\infty\) on almost every such fiber; also +\(\E_F(\log\Xi)^+<\infty\) there by the preceding joint bound. +Conditional Jensen is now legitimate and gives +\[ + \log R_B(\mathbf Y) + \ge\log w(\mathbf Y)-u\,\E_F\log\Xi. +\] +Rearranging this inequality, +\[ + u\,\E_F(\log\Xi)^- + \le(\log R_B(\mathbf Y))^+ + +|\log w(\mathbf Y)|+u\,\E_F(\log\Xi)^+. +\] +Its right side has finite expectation, proving joint integrability of +\((\log\Xi)^-\). The same Jensen lower bound gives +\[ + (\log R_B(\mathbf Y))^- + \le|\log w(\mathbf Y)|+u\,\E_F|\log\Xi|, +\] +so the negative comparison logarithm is integrable as well. +Finally use the normalized test \(R_B\) in \eqref{eq:entropy-test}. +All logarithms are now integrable, and that test together with the +conditional Jensen bound proves \eqref{syn:correlation-lower}. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{syn:block}] +The preceding lemma permits subtraction of all likelihood and correlation +terms. Conditional Jensen in \eqref{syn:conditional-mean} gives +\[ + \E[\log\Xi\mid A,b,\mathbf X]\le\log a_q+\log U_A(b). +\] +The last logarithm is integrable under the actual label law: \(U_A(b)\) +is bounded below there by \((2\pi)^{-k/2}\), and its positive logarithm +was controlled in the proof of Lemma~\ref{syn:logs}. At fixed \(A\), +Jensen with \(b\sim g_A\) gives +\[ + \int g_A(b)\log\frac{U_A(b)}{g_A(b)}\,db + \le\log\int U_A(b)\,db. +\] +Together with \eqref{syn:synthetic-bound} and \eqref{syn:majorant-log}, +this yields +\begin{equation}\label{syn:likelihood-difference} + \E\log\Xi-\E\log g_A(b) + \le-\frac k2\log(q-k-1) + +C[d+\log(1+\log\Lambda)]. +\end{equation} +Combine \eqref{syn:correlation-upper}, \eqref{syn:gram-log}, +\eqref{syn:gram-upper}, \eqref{syn:correlation-lower}, and +\eqref{syn:likelihood-difference}. The result is +\begin{align} + \mathcal C_X-\mathcal C_Y + &\le\frac{ku}{2}\log\frac{m}{q-k-1} + +Cu[d+\log(1+\log\Lambda)] \notag\\ + &\le C\ell[d+\log(1+\log\Lambda)]. + \label{syn:correlation-loss} +\end{align} +The first term is \(O(k u)\) by \eqref{syn:dimension-margins}. +Thus the scale from the projected Gaussian determinant cancels the +inverse-determinant scale in the synthetic likelihood. The critical +kernel was integrated before taking a logarithm, leaving only +\(\log(1+\log\Lambda)\). + +Apply Lemma~\ref{lem:replica-information} with \(t=\ell\), +the representative \(S=X_1\), message \(W=z\), and map +\(\phi_B(x)=Bx\). Every replica has the required signal--message pair law, +\(B\) is independent of the whole tuple and message, and +\(\mathcal C_X<\infty\). Its conclusion and \(H(z)\le\log K\) give +\[ + \ell I(X_1;z\mid B,BX_1) + \le\log K+\mathcal C_X-\mathcal C_Y. +\] +The joint law of \((B,X_1,z)\) is the product of the \(B\)-law and the +original \((X,z)\)-law. Substitution of \eqref{syn:correlation-loss} +therefore proves \eqref{syn:block-bound}. +\end{proof} + +\subsection{Iteration along the full state history} + +\begin{proposition}[The synthetic route to the sample bound]\label{syn:streaming} +For every nonnegative integer-valued \(M(d)=o(d^2)\) and every sequence +\(0<\epsilon(d)\le1/10\), consider learners in +Definition~\ref{def:streaming-model} with deterministic horizon \(T(d)\) +and signal \(X\sim\sigma_d\). If their angular +success probability, averaged over the uniform signal, rows, and learner +randomness, is at least \(2/3\), then +\[ + T(d)\ge c\,d\log(1/\epsilon(d)) +\] +for all sufficiently large \(d\), with an absolute \(c>0\). The threshold +may depend on the memory sequence. +\end{proposition} + +\begin{proof} +Apply Lemma~\ref{lem:padding} with \(p=2/3\) and +\(p'=s_0=16/25=0.64\) to the learner in +Definition~\ref{def:streaming-model}. This gives deterministic Borel +state-based rules and data-independent initialization, with uniform-prior +success at least \(s_0\); all remaining randomness is fixed. +Pre-generate all \(T\) rows, including rows after an early stop. + +Put \(L_\epsilon=\log(1/\epsilon)\). It suffices to consider dimensions +for which \(T\le dL_\epsilon\). At other dimensions the claimed lower +bound already holds. Lemma~\ref{lem:early-clock}, applied with +\(p_0=s_0\), gives +\begin{equation}\label{syn:output-count} + s_0\le(T+1)2^M\epsilon^{d-1},\qquad + L_\epsilon=o(d),\qquad T=o(d^2). +\end{equation} +Pad first to time \(T\) and then +to the least multiple of \(k\) at least \(T\), using the stopping-index +tags and ignored fresh pairs of Lemma~\ref{lem:padding}. Its exact +per-index state count gives +\begin{equation}\label{syn:padded-width} + K=(T+2)2^M,\qquad \log K=o(d^2). +\end{equation} + +Since \(s_0>3/5\), Lemma~\ref{lem:raw-sample-bound} gives +\(T>\lfloor d/2\rfloor\) for all sufficiently large \(d\). We use the +weaker consequence +\begin{equation}\label{syn:linear-endpoint} + T>d/4. +\end{equation} + +Let \(p=\lceil T/k\rceil\). Write \(H_i\) for the tuple of states at the +ends of the first \(i\) blocks, with \(H_0\) constant. Use one independent +\(B\sim\gamma_{m,d}\) and \(Y=BX\) throughout. If a history +\(h\) has probability \(p_h>0\), Lemma~\ref{lem:conditional-priors} +gives its conditional signal density +\[ + \frac{\Prob\{H_{i-1}=h\mid X=x\}}{p_h}\le\frac1{p_h} + \quad\text{relative to }\sigma_d. +\] +The next block matrix is independent of \((X,H_{i-1})\). Given \(h\), +the initial state is fixed, and the next state is a measurable \(K\)-valued +function of the new block and its exact labels. The side matrix remains +independent of this conditional experiment. Proposition~\ref{syn:block} +therefore applies with \(\Lambda=1/p_h\). Averaging it over histories gives +\begin{align} + I(X;H_i\mid H_{i-1},B,Y) + &\le\frac{\log K}{\ell} + +C\left[d+\sum_h p_h\log(1+\log(1/p_h))\right] \notag\\ + &\le\frac{\log K}{\ell} + +C[d+\log(1+(i-1)\log K)]. + \label{syn:history-step} +\end{align} +The last line uses concavity of \(t\mapsto\log(1+t)\) and +\(H(H_{i-1})\le(i-1)\log K\). In particular it applies the bounded-prior +block theorem before averaging its density cost. Since +\(p=o(d)\), \(\ell\ge d/16\), and \(\log K=o(d^2)\), the right side is +at most \(Cd\), with one absolute \(C\), for all sufficiently large \(d\). +The chain rule then gives +\begin{equation}\label{syn:history-information} + I(X;H_p\mid B,Y)\le Cdp\le C(T+d). +\end{equation} +The postponed deterministic output is a function of the final tagged state, +and therefore of \(H_p\). + +To obtain the matching endpoint, use Lemma~\ref{lem:residual-information} +with \(S=X\), \(V=H_p\), \(G=B\), and \(r_0=1/4\). +The output is a function of \(H_p\), and \(B\) is independent of the +entire signal--history experiment. Moreover, \(2\epsilon<1/4\) and +\(m\le d/2\le d-2\). For the residual radius +\(\rho_B=\norm{P_{\ker B}X}\), \eqref{eq:residual-markov} gives +\[ + \Prob\{\rho_B<1/4\} + \le\frac{m/d}{1-1/16}\le\frac8{15}. +\] +Thus the lemma applies with success minus radius-failure probability at +least \(s_0-8/15=8/75\), and yields +\begin{align} + I(X;H_p\mid B,Y) + &\ge\frac8{75}(d-m-1)\log\frac1{8\epsilon}-\log2 \notag\\ + &\ge c_1dL_\epsilon + \label{syn:history-lower} +\end{align} +for all sufficiently large \(d\), with an absolute \(c_1>0\). Indeed, +\[ + \log\frac1{8\epsilon} + \ge\left(1-\frac{\log8}{\log10}\right)L_\epsilon, +\] +and \(d-m-1\ge d/2-1\). + +By \eqref{syn:linear-endpoint}, \(T+d<5T\). Combining +\eqref{syn:history-information} and \eqref{syn:history-lower} therefore +gives \(T\ge c_2dL_\epsilon\) with an absolute \(c_2>0\). The constants +are absolute; the eventual threshold comes only from \(M(d)=o(d^2)\). +All replicas and auxiliary projections have served solely to bound +information. The learner's observations remain the prescribed fresh +Gaussian rows and exact labels. +\end{proof} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/07-incidence.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/07-incidence.tex new file mode 100644 index 0000000000000000000000000000000000000000..864f2dc9e5b62043dab45ba9105ec0a47f9c8700 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/07-incidence.tex @@ -0,0 +1,979 @@ +\section{Stiefel incidence and conditional block information} +\label{sec:incidence} + +This comparison bounds conditional information directly by the relative +entropy of two laws on tuples of signals. Both laws make the signals share +an exact projection. The first uses the actual observation block; the +second obtains its shared projection from an orthonormal frame inside the +coordinates of the independent side projection. Given the separate side +projections of all the signals, the second law leaves them conditionally +independent with their original fiber laws. This product structure makes +the divergence of the two laws an upper bound for the information in the +message, multiplied by the number of signals. + +The geometric work is to express both laws relative to one incidence +measure. We first construct the laws and prove their information +comparison. We then compute their exact Jacobians, transfer density +exceptions before division, and estimate the resulting likelihood ratio. + +Write \(\sigma=\sigma_d\), and let \(v_q\) denote the volume of the +unit ball in \(\mathbb R^q\). For the conditional chain rule on standard +Borel spaces, see +Austin~\cite[Sections~3.3--3.4]{Austin2020}. We use, for all sufficiently +large \(d\), +\begin{equation}\label{inc:dimensions} + n=k=\lfloor d/16\rfloor,\qquad + \ell=k-1,\qquad p=\lfloor d/4\rfloor . +\end{equation} +Thus \(\ell\ge2\), while \(p-n-\ell\) and \(d-n-\ell\) are at least an +absolute positive multiple of \(d\). + +\begin{proposition}[Information in one block]\label{inc:block} +Let \(\mu=f\sigma\) be a probability measure with a Borel density +\(0\le f\le F<\infty\), where \(F\ge1\). Draw \(S\sim\mu\) and standard +Gaussian matrices \(A\in\mathbb R^{n\times d}\) and +\(B\in\mathbb R^{p\times d}\), all independently. For an integer +\(w\ge1\), a message \(J\) takes values in a fixed \(w\)-point alphabet +(unused symbols are allowed) and is drawn by an arbitrary measurable +probability kernel from +\((A,AS)\); conditional on \((A,AS)\), its randomness is independent of +\((S,B)\). There is an absolute constant \(C\) such that +\begin{equation}\label{inc:block-bound} + I_\mu(S;J\mid B,BS) + \le \frac{\log w}{k}+Cd+C\log(1+\log F). +\end{equation} +\end{proposition} + +The dependence on \(F\) is doubly logarithmic. This permits us later to +condition on an entire finite history: a history of probability \(a\) +changes the uniform prior to a density bounded by \(1/a\). + +\subsection{Two probability laws on compatible signals} +\label{inc:laws} + +We use the spherical projection versions from +\eqref{eq:finite-tilt}, with prior \(\mu=f\sigma\). Recall them locally: +for a full-row-rank \(q\times d\) matrix \(L\), with +\(1\le q\le d-2\), and an interior label \(t\), let +\(\sigma_{L,t}\) be uniform probability on the spherical fiber \(Ls=t\), +and let \(p_L^0\) be the uniform-sphere projection density in +\eqref{eq:spherical-projection-density}. Then +\begin{equation}\label{inc:recalled-fibers} + p_L(t)=p_L^0(t)\int f\,d\sigma_{L,t},\qquad + \mu_{L,t}=\frac{f\,\sigma_{L,t}}{\int f\,d\sigma_{L,t}} + \quad\text{when }p_L(t)>0. +\end{equation} +Set \(p_L=0\) outside the interior projection ellipsoid and fix one +measurable probability kernel on unused fibers. These jointly measurable +versions specify all the conditional draws below. Their geometric +finite-measure form will enter only when we compute the two likelihoods. + +Write \(\phi_j(A,y)\) for the probabilities of the message kernel. +For a tuple \(\mathbf s=(s_1,\ldots,s_k)\), put +\[ + U=[s_2-s_1,\ldots,s_k-s_1]\in\mathbb R^{d\times\ell}. +\] +The notation \(U^\perp\) means the orthogonal complement of the column +space of \(U\), and \(P_{U^\perp}\) is its orthogonal projector. +Let \(\mathcal V_{n,h}\) be the manifold of \(n\times h\) matrices with +orthonormal rows, with the metric induced by the Euclidean matrix norm. +Write \(\upsilon_{n,h}\) for normalized Riemannian volume on it. +This is the Euclidean Stiefel metric discussed by Edelman, Arias, and +Smith~\cite[Section~2.2.1]{EdelmanAriasSmith1998}; we compute its volume +factor below. + +We use an auxiliary \(C\in\mathcal V_{n,p}\) and set \(D=CB\). +For a full-rank \(p\times\ell\) matrix \(W\), write \(\upsilon_W\) for +uniform frames whose rows lie in \((\operatorname{col}W)^\perp\). +Write \(\gamma_{r,d}\) for standard Gaussian measure on +\(\mathbb R^{r\times d}\), and \(\gamma_U\) for the law of \(n\) +independent \(N(0,P_{U^\perp})\) rows. All these kernels are measurable +on their rank domains. For example, projecting a Gaussian matrix onto +\((\operatorname{col}W)^\perp\) and taking its orthonormal row polar +factor gives \(\upsilon_W\); orthogonal invariance identifies this with +normalized induced volume. We may assign arbitrary values on rank +exceptions. + +Define \(Q\) by the following experiment. +\begin{enumerate} +\item Draw \(A\sim\gamma_{n,d}\) and \(s_1\sim\mu\) independently. + Put \(y=As_1\). Draw \(J\) from \(\phi(A,y)\) and draw + \(s_2,\ldots,s_k\) independently from \(\mu_{A,y}\), independently + of \(J\) conditional on \((A,y)\). +\item Draw \(B\sim\gamma_{p,d}\) independently. Given \((\mathbf s,B)\), + draw \(C\sim\upsilon_{BU}\). +\end{enumerate} +Then \(AU=0\). Conditional on \((A,y)\), all \(k\) signals are +independent with law \(\mu_{A,y}\), which is absolutely continuous on +a positive-radius sphere of dimension \(d-n-1\). Each new point +therefore avoids the affine span of its predecessors; hence \(U\) has +rank \(\ell\) almost surely. The independent Gaussian \(B\) makes +\(BU\) have rank \(\ell\) almost surely, so \(\upsilon_{BU}\) is +available because \(p-\ell\ge n\). Each marginal +\begin{equation}\label{inc:q-marginals} + \mathcal L_Q(s_i,J,B,Bs_i)=\mathcal L_\mu(S,J,B,BS) + \qquad(1\le i\le k) +\end{equation} +is the law in Proposition~\ref{inc:block}. + +The matrix \(D=CB\) has a useful conditional law under \(Q\): +\begin{equation}\label{inc:conditional-D} + \mathcal L_Q(D\mid \mathbf s,A,y,J) + =N(0,P_{U^\perp})^{\otimes n}. +\end{equation} +To verify this, split each row of \(B\) between +\(\operatorname{col}U\) and \(U^\perp\). The second component is +independent of \(BU\). Given the tuple, the kernel for \(C\) depends +on \(B\) only through \(BU\); its orthonormal rows send the independent +Gaussian second component to \(n\) independent standard Gaussian rows +in \(U^\perp\). The first component is annihilated by \(C\). + +Define \(R\) in the other order. First draw independent +\(B\sim\gamma_{p,d}\) and \(C\sim\upsilon_{n,p}\); then \(D=CB\) is +standard Gaussian. Draw \(s_1\sim\mu\) independently and, conditional +on \((D,Ds_1)\), draw \(s_2,\ldots,s_k\) independently from +\(\mu_{D,Ds_1}\). Draw \(J\) uniformly from its \(w\) possible values, +independently, and finally draw \(A\sim\gamma_U\). +Again \(U\) has rank \(\ell\). Conditional on \((C,D)\), the other +\(p-n\) rows in an orthogonal row rotation of \(B\) are independent +standard Gaussian rows and were not used to draw the tuple. Since +\(p-n\ge\ell\), they make \(BU\) have rank \(\ell\) almost surely. +In these experiments \(A\) and \(D\) also have row rank \(n\) almost +surely: each is either standard Gaussian or has conditional Gaussian +support \(U^\perp\), of dimension \(d-\ell\ge n\). + +\begin{lemma}[Product conditioning]\label{inc:product} +Put \(Z_i=Bs_i\). After \(A,C\) are discarded under \(R\), +\begin{equation}\label{inc:product-law} + \mathcal L_R(\mathbf s\mid B,Z_1,\ldots,Z_k,J) + =\bigotimes_{i=1}^k\mu_{B,Z_i}. +\end{equation} +Consequently, +\begin{equation}\label{inc:entropy-lower} + \operatorname{KL}(Q\|R)\ge kI_\mu(S;J\mid B,BS). +\end{equation} +\end{lemma} + +\begin{proof} +Retain \(B,C\) temporarily and put \(t=Ds_1\). Conditional on +\((B,C,t)\), the signals are independent with law \(\mu_{D,t}\). +Disintegrating this law further according to \(z=Bs\) gives +\(\mu_{B,z}\): the earlier equation \(Ds=t\) is just \(Cz=t\), +already determined by \(z\). Formally, disintegrate the law +of \(z=Bs\) according to \(Cz\), and compose its conditional kernel +with \(\mu_{B,z}\); the resulting kernel is a conditional law given +\(Ds=t\). These identities hold at the needed values because each +individual signal has marginal law \(\mu\) given \((B,C)\). +Conditioning the independent signals on their separate \(Z_i\)'s +therefore yields the product in \eqref{inc:product-law}. That product +does not depend on \(C,t\), so discarding them leaves it unchanged. +The independent uniform \(J\) and the later kernel for \(A\) do not +change the marginal under consideration. + +Data processing and the conditional chain rule give +\[ + \operatorname{KL}(Q\|R)\ge + \mathbb E_Q\operatorname{KL}\!\left( + Q(\mathbf s\mid B,Z_1,\ldots,Z_k,J) + \,\middle\|\,\bigotimes_{i=1}^k\mu_{B,Z_i}\right). +\] +This is valid as an extended inequality; absolute continuity will be +established below. Relative entropy to a product is at least the sum +of the relative entropies of the marginals. For each such summand, +convexity of relative entropy permits us, after averaging, to discard +the other \(Z_j\)'s from its first conditional law. The result is at +least +\[ + \mathbb E_Q\operatorname{KL}\bigl( + Q(s_i\mid B,Z_i,J)\,\|\,\mu_{B,Z_i}\bigr) + =I_\mu(S;J\mid B,BS) +\] +by \eqref{inc:q-marginals}. Sum over \(i\). +\end{proof} + +The product conditional law has reduced the block information bound to +one task: bound \(\operatorname{KL}(Q\|R)\) by \(\log w\) plus the +geometric and prior-density costs. We next put both laws on a common +incidence measure. Its exact densities will make that divergence +computable, after exceptional sets have been removed before division. + +\subsection{The common measure and its density versions} +\label{inc:common-measure}\label{inc:fibers} + +To compare the two laws, we express their conditional probabilities as +normalized finite measures on the same incidence set. The underlying +geometric formula is the regular-submersion case of coarea; see +Federer~\cite[Definition~2.10 and Theorem~3.1, pp.~423 and 426--427]{Federer1959}. +The following calculation specifies the induced fiber volume and the +probability normalization used here. + +\begin{lemma}[Spherical projection measure]\label{inc:projection} +Suppose \(L\in\mathbb R^{q\times d}\) has full row rank and +\(1\le q\le d-2\). In the interior +\[ + E_L=\{t:t^{\mathsf T}(LL^{\mathsf T})^{-1}t<1\} +\] +define a finite measure on \(\{s\in\mathbb S^{d-1}:Ls=t\}\) by +\begin{equation}\label{inc:fiber-measure} + \lambda_{L,t}(ds)= + \frac{f(s)}{|\mathbb S^{d-1}|\,j_L(s)} + \,d\operatorname{vol}_{d-1-q}(s),\qquad + j_L(s)=\det\!\bigl(L(I-ss^{\mathsf T})L^{\mathsf T}\bigr)^{1/2}. +\end{equation} +Set \(\lambda_{L,t}=0\) outside \(E_L\). Its mass and normalization +are exactly the versions recalled in \eqref{inc:recalled-fibers}: +\[ + p_L(t)=\lambda_{L,t}(\mathbb S^{d-1}),\qquad + \mu_{L,t}=\lambda_{L,t}/p_L(t)\quad\text{where }p_L(t)>0. +\] +The finite-measure kernel is jointly measurable in \((L,t)\). Moreover, +\begin{equation}\label{inc:ellipsoid-bound} + V_L\|p_L\|_\infty\le Fd^d,\qquad + V_L=|E_L|=v_q\det(LL^{\mathsf T})^{1/2}. +\end{equation} +\end{lemma} + +\begin{proof} +The tangent space at \(s\) is \(s^\perp\). The Gram matrix of the +differential \(L:s^\perp\to\mathbb R^q\) is +\(L(I-ss^{\mathsf T})L^{\mathsf T}\), proving the formula for its normal +Jacobian. With +\(P_L=L^{\mathsf T}(LL^{\mathsf T})^{-1}L\), the determinant lemma also gives +\[ + j_L(s)=\det(LL^{\mathsf T})^{1/2}\sqrt{1-\|P_Ls\|^2}. +\] +The differential is onto except when \(s\) belongs to the row space of +\(L\); that sphere has zero \((d-1)\)-dimensional surface measure. +Applying change of variables on coordinate patches of its complement, +and then exhausting the complement, gives +\[ + \int h(s)\,d\mu(s) + =\int_{\mathbb R^q}\int h(s)\,\lambda_{L,t}(ds)\,dt + \quad(h\ge0). +\] +To identify these measures with the versions already used, put +\(s_0=L^{\mathsf T}(LL^{\mathsf T})^{-1}t\) and +\(r=\sqrt{1-\|s_0\|^2}\). An interior fiber is +\(s_0+r\mathbb S(\ker L)\), where \(\mathbb S(\ker L)\) is the unit +sphere of \(\ker L\), and \(j_L\) is constant there. Its induced +volume is \(|\mathbb S^{d-q-1}|r^{d-q-1}\). Dividing this volume by +\(|\mathbb S^{d-1}|j_L\) gives precisely the uniform projection density +\(p_L^0(t)\) in \eqref{eq:spherical-projection-density}. Consequently, +as finite measures on every interior fiber, +\[ + \lambda_{L,t}=p_L^0(t)f\,\sigma_{L,t}. +\] +Taking mass and then normalizing gives \eqref{inc:recalled-fibers} +pointwise on its positive set. A uniform +direction in \(\ker L\) can be generated by projecting a standard +Gaussian vector onto that subspace and normalizing. The projection +matrix is a measurable function of full-rank \(L\); integrating the +bounded Borel function \(f(s_0+r\theta)\) against this kernel therefore +gives measurable \(\lambda_{L,t}\), \(p_L(t)\), and \(\mu_{L,t}\), +with the fixed convention on unused fibers from \eqref{inc:recalled-fibers}. + +For orthonormal rows and \(f=1\), the displayed fiber formula gives a +density proportional to +\((1-\|t\|^2)^{(d-q-2)/2}\) on the unit ball. Its normalizing integral +is at least \(\tfrac12v_qd^{-q/2}\): restrict to the ball of radius +\(d^{-1/2}\) and use +\((1-1/d)^{(d-q-2)/2}\ge1/2\). +The maximum of the density times \(v_q\) is consequently at most +\(2d^{q/2}\le d^d\). The bound \(f\le F\) multiplies this estimate by +at most \(F\). Finally, an invertible change from orthonormal +row coordinates to \(L\) divides the density by +\(\det(LL^{\mathsf T})^{1/2}\) and multiplies the ellipsoid volume by +the same factor. This proves \eqref{inc:ellipsoid-bound}; in particular +our chosen \(p_L\) is finite everywhere. +\end{proof} + + +\begin{lemma}[Interchange on a regular incidence set]\label{inc:swap} +Let \(X,Z\) be second-countable Riemannian manifolds, and let +\(G:X\times Z\to\mathbb R^a\) be smooth. On a relatively open subset +\(\Omega\) of \(G^{-1}(0)\), suppose both partial differentials +\(d_xG\) and \(d_zG\) are onto. Write \(J_xG,J_zG\) for their normal +Jacobians. For every nonnegative measurable \(h\) on \(\Omega\), +\begin{align} + &\int_Z\int_{\{x:(x,z)\in\Omega\}} + \frac{h(x,z)}{J_xG(x,z)} + \,d\operatorname{vol}_{\dim X-a}(x)\,d\operatorname{vol}_Z(z) + \notag\\ + &\qquad= + \int_X\int_{\{z:(x,z)\in\Omega\}} + \frac{h(x,z)}{J_zG(x,z)} + \,d\operatorname{vol}_{\dim Z-a}(z)\,d\operatorname{vol}_X(x). + \label{inc:swap-identity} +\end{align} +The fiber volumes are induced volumes, and the equality holds for +nonnegative integrals even when their common value is infinite. +\end{lemma} + +\begin{proof} +Near a point of \(\Omega\), choose coordinates +\(x=(x',x'')\), \(z=(z',z'')\) so that \(x''\) and \(z''\) each have +\(a\) coordinates and both \(G_{x''}\) and \(G_{z''}\) are invertible. +Let \(u_X(x)\,dx\), \(u_Z(z)\,dz\) be the volume forms in these +coordinates. Changing variables from \((x',x'')\) to \((x',G)\) +shows that the induced \(x\)-fiber volume divided by \(J_xG\) is +\[ + \frac{u_X(x)}{|\det G_{x''}|}\,dx'. +\] +Indeed, the tangent-volume factor and the normal-volume factor of this +change of variables are respectively the induced fiber volume and the +reciprocal normal Jacobian. Thus the left integral has local density +\(u_Xu_Z/|\det G_{x''}|\) in \((x',z',z'')\); the right integral has +density \(u_Xu_Z/|\det G_{z''}|\) in \((x',z',x'')\). +At fixed \((x',z')\), implicit differentiation gives +\[ + \frac{dx''}{dz''}=-G_{x''}^{-1}G_{z''},\qquad + \left|\det\frac{dx''}{dz''}\right| + =\frac{|\det G_{z''}|}{|\det G_{x''}|}. +\] +The two local integrals agree. Split \(\Omega\) into measurable pieces +subordinate to a countable coordinate cover and sum using Tonelli's +theorem. This also proves the assertion for infinite integrals. +\end{proof} + +Let \(\mathsf H\) be the probability law that draws +\(\mathbf s\sim\mu^{\otimes k}\), then independent +\(B\sim\gamma_{p,d}\), and, conditional on these, independently draws +\(A\sim\gamma_U\) and \(C\sim\upsilon_{BU}\). Include an independent +uniform \(J\). Independent points from \(\mu\) give +\(\operatorname{rank}U=\ell\) almost surely. The same Gaussian +decomposition as in \eqref{inc:conditional-D} then gives full ranks for +\(BU,A,D\) under \(\mathsf H\). + +\begin{lemma}[Exact undivided identities and probability densities] +\label{inc:densities} +Define the following marginal probability measures of \(\mathsf H\): +\[ + \begin{split} + \mathsf H_A(d\mathbf s,dA) + &=\mu^{\otimes k}(d\mathbf s)\,\gamma_U(dA),\\ + \mathsf H_C(d\mathbf s,dB,dC) + &=\mu^{\otimes k}(d\mathbf s)\,\gamma_{p,d}(dB)\,\upsilon_{BU}(dC). + \end{split} +\] +Using the unnormalized measures of \eqref{inc:fiber-measure}, define +nonnegative measures on the corresponding exact incidence sets by +\begin{align} + \mathsf M_A(d\mathbf s,dA) + &=\mu(ds_1)\gamma_{n,d}(dA) + \prod_{i=2}^k\lambda_{A,As_1}(ds_i),\label{inc:MA}\\ + \mathsf M_C(d\mathbf s,dB,dC) + &=\mu(ds_1)\gamma_{p,d}(dB)\upsilon_{n,p}(dC) + \prod_{i=2}^k\lambda_{D,Ds_1}(ds_i),\qquad D=CB.\label{inc:MC} +\end{align} +Values at row-rank failures may be set to zero. The weights on the +right below are assigned arbitrary finite values on their Gram-rank +null sets. These measures satisfy the exact identities +\begin{align} + \mathsf M_A + &=(2\pi)^{-n\ell/2}\det(U^{\mathsf T}U)^{-n/2}\,\mathsf H_A, + \label{inc:undivided-A}\\ + \mathsf M_C + &=\kappa\det(U^{\mathsf T}B^{\mathsf T}BU)^{-n/2}\,\mathsf H_C, + \qquad + \kappa=\frac{|\mathcal V_{n,p-\ell}|}{|\mathcal V_{n,p}|}. + \label{inc:undivided-C} +\end{align} +These are equalities of nonnegative measures. + +For \(L=A,D\), the values \(p_L(Ls_1)\) are positive and finite +\(\mathsf H\)-almost surely and equal the shrinking-ball limits +\[ + p_L(Ls_1)=\lim_{j\to\infty} + \frac{\mu\{z:\|Lz-Ls_1\|<2^{-j}\}}{v_n2^{-jn}}. +\] +On this full-measure set, the densities of \(Q\) and \(R\) with respect +to \(\mathsf H\) are +\begin{align} + q_*&=w\phi_J(A,As_1)(2\pi)^{-n\ell/2} + \det(U^{\mathsf T}U)^{-n/2}p_A(As_1)^{-\ell}, + \label{inc:q-density}\\ + r_*&=\kappa\det(U^{\mathsf T}B^{\mathsf T}BU)^{-n/2} + p_D(Ds_1)^{-\ell}.\label{inc:r-density} +\end{align} +In particular \(Q\ll R\). +\end{lemma} + +\begin{proof} +We first establish the identities without dividing by any projection +density. In \(\mathsf M_A\), integrating the tuple variables gives the +exact \((s_1,A)\)-marginal +\begin{equation}\label{inc:MA-marginal} + \mu(ds_1)\gamma_{n,d}(dA)\,p_A(As_1)^\ell. +\end{equation} +For fixed full-rank \(A\), this weight is finite by +\eqref{inc:ellipsoid-bound}. Where it is positive, +\(\lambda_{A,As_1}^{\otimes\ell} +=p_A(As_1)^\ell\mu_{A,As_1}^{\otimes\ell}\). +The rank argument for the \(Q\) experiment therefore shows that +\(\operatorname{rank}U=\ell\) outside an \(\mathsf M_A\)-null set. +At zero weight the fiber product has zero mass. The same reasoning in +\(\mathsf M_C\) gives the exact \((s_1,B,C)\)-marginal +\begin{equation}\label{inc:MC-marginal} + \mu(ds_1)\gamma_{p,d}(dB)\upsilon_{n,p}(dC)\,p_D(Ds_1)^\ell, +\end{equation} +and the rank argument for \(R\) removes the loci where \(U\) or \(BU\) +has rank less than \(\ell\). Multiplication by the finite weight in +\eqref{inc:MC-marginal} preserves these native null sets. + +On either incidence set, the spherical partial differentials are +regular on the remaining locus. Indeed, if some \(s_i\) belonged to +the row space of \(A\), then its row-space projection would have norm +one. The equations \(As_j=As_i\) would give every \(s_j\) that same +row-space projection, and their unit norms would force \(s_j=s_i\) +for all \(j\), contradicting \(\operatorname{rank}U=\ell\). +The same argument applies to \(D\). The Gaussian rank assertions +above also remove row-rank failures. Thus removing the complements of +the full-rank regular loci changes neither \(\mathsf M_A,\mathsf M_C\) +nor the corresponding \(\mathsf H_A,\mathsf H_C\). + +Fix \(s_1\) and apply Lemma~\ref{inc:swap} to +\[ + x=(s_2,\ldots,s_k),\qquad z=A,\qquad G(x,z)=AU. +\] +For fixed \(A\), the target has \(\ell\) independent \(n\)-coordinate +blocks, so the partial normal Jacobian is +\(\prod_{i=2}^k j_A(s_i)\). These are precisely the denominators in +\(\prod_{i=2}^k\lambda_{A,As_1}(ds_i)\). +For fixed tuple, the differential is \(\Delta A\mapsto\Delta A\,U\). +It consists of \(n\) row maps with Gram matrix \(U^{\mathsf T}U\); +its normal Jacobian is \(\det(U^{\mathsf T}U)^{n/2}\). +The \(A\)-fiber has rows in \(U^\perp\). Restricting the ambient +standard Gaussian density to this subspace and then expressing it +relative to the normalized Gaussian \(\gamma_U\) contributes +\((2\pi)^{-n\ell/2}\). This proves \eqref{inc:undivided-A} on the +regular locus, and hence exactly after the null loci just proved are +removed. In particular, integrating over a restricted regular tuple +fiber gives at most \(p_A(As_1)^\ell\), and gives equality for native +almost every \((s_1,A)\). + +For the second identity, fix \((s_1,B)\) and use +\(G(x,C)=CBU\). Put \(W=BU\). The tuple partial normal Jacobian is +\(\prod_{i=2}^k j_D(s_i)\). At \(CW=0\), +\[ + T_C\mathcal V_{n,p} + =\{\Delta:\Delta C^{\mathsf T}+C\Delta^{\mathsf T}=0\}. +\] +Variations whose rows belong to \(\operatorname{col}W\) lie in this +tangent space, since \(C W=0\). They form the orthogonal complement +of the tangent space to the constrained frame fiber. On this normal +space the differential is \(\Delta\mapsto\Delta W\), with normal +Jacobian +\(\det(W^{\mathsf T}W)^{n/2}\). There is no additional metric factor +in these normal directions. The constrained fiber, with its induced +metric, is isometric to \(\mathcal V_{n,p-\ell}\). Replacing its +unnormalized volume by \(\upsilon_W\) leaves the volume ratio \(\kappa\). +Lemma~\ref{inc:swap} now proves \eqref{inc:undivided-C}, again exactly +after removal of native rank-null sets. The corresponding restricted +tuple mass is at most \(p_D(Ds_1)^\ell\) and equals it for native almost +every \((s_1,B,C)\). + +We next transfer the density exceptions using these undivided measures. +For a full-rank \(n\times d\) matrix \(L\), define +\[ + b_j(L,t)= + \frac{\mu\{z:\|Lz-t\|<2^{-j}\}}{v_n2^{-jn}}, +\] +and let \(\mathcal E(L,s)\) be the condition that either +\(p_L(Ls)\notin(0,\infty)\), or the limit of \(b_j(L,Ls)\) does not +exist, or that limit differs from \(p_L(Ls)\). This is a measurable +condition. Set +\[ + \mathcal E_A=\{(\mathbf s,A):\mathcal E(A,s_1)\},\qquad + \mathcal E_D=\{(\mathbf s,B,C):\mathcal E(CB,s_1)\}. +\] +For every fixed full-rank \(L\), Lebesgue differentiation and the +density formula imply \(\mu\{s:\mathcal E(L,s)\}=0\). +The exact native marginals \eqref{inc:MA-marginal} and +\eqref{inc:MC-marginal}, whose density weights are finite for each +fixed native projection, therefore give +\[ + \mathsf M_A(\mathcal E_A)=0,\qquad + \mathsf M_C(\mathcal E_D)=0. +\] +On the right sides of \eqref{inc:undivided-A} and +\eqref{inc:undivided-C}, the multiplying Gram weights are strictly +positive and finite almost surely. If \(h\) denotes either weight, +then on \(\{1/m\le h\le m\}\) the equality +\(\int_{\mathcal E}h\,d\mathsf H_\bullet=0\) gives +\(\mathsf H_\bullet(\mathcal E\cap\{1/m\le h\le m\})=0\). +Letting \(m\to\infty\) proves that \(\mathcal E_A\) is +\(\mathsf H_A\)-null and \(\mathcal E_D\) is \(\mathsf H_C\)-null. +This argument is valid even if an undivided measure has infinite total +mass. Adding the unused probability kernels gives both assertions +under \(\mathsf H\). + +Only now replace each unnormalized fiber measure by its normalized +conditional law. The \(Q\) experiment divides +\eqref{inc:undivided-A} by \(p_A(As_1)^\ell\), then attaches the same +kernels for \(B,C\) and the message density \(w\phi_J(A,As_1)\) relative +to uniform \(J\). This gives \eqref{inc:q-density}. The \(R\) +experiment divides \eqref{inc:undivided-C} by \(p_D(Ds_1)^\ell\) and +attaches the unchanged \(\gamma_U\) kernel for \(A\) and uniform \(J\), +giving \eqref{inc:r-density}. The divisions are legitimate on the +full-measure set just proved. Since \(r_*>0\) and is finite there, +\(Q\ll R\). +\end{proof} + +\begin{lemma}[The induced Stiefel volume]\label{inc:stiefel} +For \(1\le n\le h\), with the induced Euclidean matrix metric, +\begin{equation}\label{inc:stiefel-volume} + |\mathcal V_{n,h}|= + 2^{n(n-1)/4}\prod_{j=1}^n|\mathbb S^{h-j}|. +\end{equation} +Consequently, for \eqref{inc:dimensions}, +\begin{equation}\label{inc:kappa} + \kappa=\pi^{-n\ell/2}\prod_{j=1}^n + \frac{\Gamma((p-j+1)/2)}{\Gamma((p-j+1-\ell)/2)} + \ge\left(\frac{p-n+1-\ell}{2\pi}\right)^{n\ell/2}. +\end{equation} +\end{lemma} + +\begin{proof} +For \(n=1\), the frame manifold is \(\mathbb S^{h-1}\), giving the base +case of \eqref{inc:stiefel-volume}. For \(n\ge2\), the map from +\(\mathcal V_{n,h}\) to its first row in +\(\mathbb S^{h-1}\) has constant normal Jacobian +\(2^{-(n-1)/2}\). To compute it, move the frame by an orthogonal +transformation to the first \(n\) coordinate rows. Tangent matrices +then consist of a skew-symmetric block in the first \(n\) columns and +an arbitrary remaining block. Each of the \(n-1\) skew coordinates +affecting the first row appears twice in the squared matrix norm, so +the corresponding singular value of the first-row map is \(2^{-1/2}\). +The other first-row coordinates have singular value one. The fiber +is isometric to \(\mathcal V_{n-1,h-1}\). +Slicing thus gives +\[ + |\mathcal V_{n,h}|= + 2^{(n-1)/2}|\mathbb S^{h-1}|\,|\mathcal V_{n-1,h-1}|. +\] +Iteration proves \eqref{inc:stiefel-volume}. The metric factor depends +only on \(n\), so it cancels in the ratio defining \(\kappa\). +The sphere-area formula +\(|\mathbb S^{h-1}|=2\pi^{h/2}/\Gamma(h/2)\) gives the product in +\eqref{inc:kappa}. + +For \(a>0\) and \(\ell/2\ge1\), a gamma random variable \(X\) of shape +\(a\) and scale one satisfies +\(\Gamma(a+\ell/2)/\Gamma(a)=\mathbb E X^{\ell/2} +\ge(\mathbb E X)^{\ell/2}=a^{\ell/2}\) by Jensen's inequality. +Apply this with \(a=(p-j+1-\ell)/2\) and then use +\(a\ge(p-n+1-\ell)/2\). +\end{proof} + +\subsection{An inverse kernel at two scales} +\label{inc:inverse-section} + +The density ratio in Lemma~\ref{inc:densities} involves +\(p_D(Ds_1)/p_A(As_1)\). It has no useful pointwise bound. We control +its positive logarithm instead, using the fact that \(A\) annihilates +the same difference space that governs the conditional Gaussian law +of \(D\). For \(u>0\), write \(\log^+u=\max\{0,\log u\}\), and set +\(\log^+0=0\). + +\begin{lemma}[Two-scale inverse kernel]\label{inc:kernel} +Under \(Q\), define +\[ + \begin{split} + K&=\int + \operatorname{dist}(z-s_1,\operatorname{col}U)^{-n}\,d\mu(z),\\ + p_A^*(t)&=\sup_{\rho>0} + \frac{\mu\{z:\|Az-t\|<\rho\}}{v_n\rho^n},\qquad + K_0=v_n\|A\|_{\mathrm{op}}^n p_A^*(As_1). + \end{split} +\] +Then \(K_0\ge2^{-n}\), and +\begin{equation}\label{inc:kernel-bound} + K\le K_0L_F,\qquad + L_F=1+\frac{n}{d-\ell-n}+n\log(2/\tau),\qquad + \tau=\left(\frac{2^{-n}}{Fd^d}\right)^{1/(d-\ell-n)}. +\end{equation} +In particular, +\(\log L_F\le C\log d+C\log(1+\log F)\), and \(K<\infty\). +Moreover, +\begin{equation}\label{inc:conditional-density} + \mathbb E_Q[p_D(Ds_1)\mid\mathbf s,A,y,J] + \le (2\pi)^{-n/2}K . +\end{equation} +\end{lemma} + +\begin{proof} +For fixed \(z\), \eqref{inc:conditional-D} makes the coordinates of +\(D(z-s_1)\) independent centered Gaussians with variance +\(\operatorname{dist}(z-s_1,\operatorname{col}U)^2\). Hence +\[ + \frac{\Pr_Q\{\|D(z-s_1)\|<\rho\mid\mathbf s,A,y,J\}}{v_n\rho^n} + \le (2\pi)^{-n/2} + \operatorname{dist}(z-s_1,\operatorname{col}U)^{-n}. +\] +The right side is interpreted as \(+\infty\) when the distance is zero. +Integrate in \(z\) using Tonelli's theorem and then apply conditional +Fatou along \(\rho=2^{-j}\). Lemma~\ref{inc:densities} and +\(Q\ll\mathsf H\) identify the almost-sure limit of the ball averages +with \(p_D(Ds_1)\). This proves \eqref{inc:conditional-density} +without evaluating an arbitrary density version at a dependent point. + +Let \(H(t)\) be the \(\mu\)-probability that the distance in \(K\) is +at most \(t\). Since \(AU=0\), +\[ + \operatorname{dist}(z-s_1,\operatorname{col}U)\le t + \quad\Longrightarrow\quad + \|A(z-s_1)\|\le\|A\|_{\mathrm{op}}t. +\] +The projection law has a Lebesgue density, so projection-ball +boundaries have zero mass. The definition of \(p_A^*\) therefore gives +\(H(t)\le K_0t^n\). All these distances are at most two; taking +\(t\downarrow2\) from above also gives \(K_0\ge2^{-n}\). + +A second estimate uses the projection onto \(U^\perp\), of dimension +\(d-\ell\le d-2\). Applying Lemma~\ref{inc:projection} to a matrix +with orthonormal rows for this projection bounds its density by +\(Fd^d/v_{d-\ell}\). A ball of radius \(t\) centered at the projection +of \(s_1\) consequently has probability at most +\(Fd^dt^{d-\ell}\). Thus +\begin{equation}\label{inc:two-scales} + H(t)\le\min\{K_0t^n,\;Fd^dt^{d-\ell}\}. +\end{equation} +The exponent \(d-\ell\) uses the bounded projection density and the +condition \(\ell\ge2\). + +The layer-cake formula, including the possibility of an initially +infinite integral, is +\[ + K=2^{-n}+n\int_0^2H(t)t^{-n-1}\,dt. +\] +Here \(d-\ell-n>0\) and \(\tau\le1\). Use the second estimate in +\eqref{inc:two-scales} up to \(\tau\) and the first thereafter: +\[ + \begin{split} + K&\le2^{-n} + +\frac{nFd^d\tau^{d-\ell-n}}{d-\ell-n} + +nK_0\log(2/\tau)\\ + &\le K_0\left(1+\frac{n}{d-\ell-n}+n\log(2/\tau)\right). + \end{split} +\] +The last inequality uses +\(Fd^d\tau^{d-\ell-n}=2^{-n}\le K_0\). Finally, +\[ + \log(1/\tau) + =\frac{n\log2+\log F+d\log d}{d-\ell-n}. +\] +The dimension ratios in \eqref{inc:dimensions} give +\(L_F\le C(1+d\log d+\log F)\). Taking its logarithm gives the +stated bound and finiteness of the inverse-distance integral \(K\). +\end{proof} + +\begin{lemma}[Maximal average at an actual label]\label{inc:maximal} +For every full-rank \(A\), +\begin{equation}\label{inc:maximal-bound} + \int\log\frac{p_A^*(As)}{p_A(As)}\,d\mu(s) + \le Cd+C\log(1+\log F). +\end{equation} +The integrand is nonnegative almost surely. +\end{lemma} + +\begin{proof} +Lebesgue differentiation gives \(p_A^*(t)\ge p_A(t)\) for +\(A_\#\mu\)-almost every label. We prove the particular +centered maximal inequality we need. Its one-dimensional ancestry is +the maximal-function theory of Hardy and +Littlewood~\cite[Section~III]{HardyLittlewood1930}; for the dimension +cost of customary covering arguments, see Stein and +Str\"omberg~\cite[p.~259]{SteinStromberg1983}. + +For \(a>0\), a compact subset of \(\{t:p_A^*(t)>a\}\) has a finite +cover by balls centered in that set, each of projection mass greater +than \(a\) times its volume. Select disjoint balls in decreasing order +of radius. Every discarded ball is contained in the triple of a +selected ball, since it meets a selected ball of at least its radius. +The selected masses sum to at most one, so the compact set has volume +at most \(3^n/a\). Inner regularity gives +\begin{equation}\label{inc:weak-maximal} + |\{t:p_A^*(t)>a\}|\le3^n/a\qquad(a>0). +\end{equation} +The function is measurable: the supremum may be restricted to positive +rational radii, because a density gives zero mass to ball boundaries. + +Let \(E_A\) be the projection ellipsoid and \(V_A=|E_A|\). +Since \(p_A^*\le\|p_A\|_\infty\), integration of level sets, using +the bound \(V_A\) up to level \(1/V_A\) and +\eqref{inc:weak-maximal} above that level, gives +\[ + \int_{E_A}p_A^*(t)\,dt + \le1+3^n\log^+(V_A\|p_A\|_\infty) + \le1+3^n(\log F+d\log d). +\] +Jensen's inequality with respect to the probability density \(p_A\) +then yields +\[ + \int p_A(t)\log\frac{p_A^*(t)}{p_A(t)}\,dt + \le\log\int_{\{p_A>0\}}p_A^*(t)\,dt + \le\log\int_{E_A}p_A^*(t)\,dt. +\] +Values on \(\{p_A=0\}\) do not enter the integral. The preceding +estimate and \(n\asymp d\) prove \eqref{inc:maximal-bound}. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{inc:block}] +On the full-measure set in Lemma~\ref{inc:densities}, the log likelihood +ratio is the sum of the message term \(\log(w\phi_J(A,As_1))\), the term +\begin{equation}\label{inc:gram-term} + -\frac{n\ell}{2}\log(2\pi)-\log\kappa+ + \frac n2\log + \frac{\det(U^{\mathsf T}B^{\mathsf T}BU)}{\det(U^{\mathsf T}U)}, +\end{equation} +and \(\ell\log(p_D(Ds_1)/p_A(As_1))\). +The message term is at most \(\log w\). + +Under \(Q\), \(B\) is independent of \(U\). Conditional on \(U\), put +\(V=U(U^{\mathsf T}U)^{-1/2}\). Then \(G=BV\) is a standard Gaussian +\(p\times\ell\) matrix and +\[ + \frac{\det(U^{\mathsf T}B^{\mathsf T}BU)}{\det(U^{\mathsf T}U)} + =\det(G^{\mathsf T}G). +\] +The lower bound in \eqref{inc:kappa} and the determinant--trace +inequality bound \eqref{inc:gram-term} above by +\[ + \frac{n\ell}{2}\log + \frac{\operatorname{tr}(G^{\mathsf T}G)} + {\ell(p-n+1-\ell)}. +\] +Its positive part has expectation \(O(n\ell)\). Indeed, +\(\mathbb E\operatorname{tr}(G^{\mathsf T}G)=p\ell\), the ratio +\(p/(p-n+1-\ell)\) is bounded, and +\(\mathbb E\log^+X\le\log(1+\mathbb E X)\) for \(X\ge0\). + +For the remaining ratio, use +\(\log^+u\le\log(1+u)\), conditional Jensen, and +\eqref{inc:conditional-density} and \eqref{inc:kernel-bound}. Since +\(p_A^*(As_1)/p_A(As_1)\ge1\) almost surely, this gives +\begin{align} + &\mathbb E_Q\!\left[ + \log^+\frac{p_D(Ds_1)}{p_A(As_1)} + \,\middle|\,\mathbf s,A,y,J\right]\notag\\ + &\quad\le \log2+ + \log^+\!\bigl((2\pi)^{-n/2}v_n\|A\|_{\mathrm{op}}^n\bigr) + +\log L_F+\log\frac{p_A^*(As_1)}{p_A(As_1)}. + \label{inc:log-ratio-conditional} +\end{align} +The Gaussian operator term has expectation \(O(n)\). Here are the +elementary estimates. A \(1/4\)-net of the unit sphere in +\(\mathbb R^h\) has size at most \(9^h\), by packing disjoint radius +\(1/8\) balls. Nets in the two dimensions bound \(\|A\|_{\mathrm{op}}\) +by twice the maximum of at most \(9^{n+d}\) absolute standard Gaussian +bilinear forms. A Gaussian tail bound gives +\(\mathbb E\|A\|_{\mathrm{op}}\le C\sqrt d\). +Also, integrating the Gaussian density over the ball of radius +\(\sqrt n\) gives \(v_n\le(2\pi e/n)^{n/2}\). Therefore +\[ + \log^+\!\bigl((2\pi)^{-n/2}v_n\|A\|_{\mathrm{op}}^n\bigr) + \le n\log(1+\sqrt{e/n}\,\|A\|_{\mathrm{op}}), +\] +whose expectation is \(O(n)\) by Jensen and \(n\asymp d\). + +The \((A,s_1)\)-marginal under \(Q\) is the original independent +Gaussian/prior draw. Averaging \eqref{inc:log-ratio-conditional} and +using Lemmas~\ref{inc:kernel} and~\ref{inc:maximal} consequently gives +\begin{equation}\label{inc:log-ratio} + \mathbb E_Q\log^+\frac{p_D(Ds_1)}{p_A(As_1)} + \le Cd+C\log(1+\log F). +\end{equation} +All positive parts in the log likelihood ratio are now integrable. +Since \(Q\ll R\), its negative part is integrable as well: with +\(h=dQ/dR\), +\(\int_{\{h<1\}}-h\log h\,dR\le1/e\). +Thus no undefined subtraction occurs, and +\[ + \operatorname{KL}(Q\|R) + \le\log w+Ckd+Ck\log(1+\log F). +\] +Combine this with \eqref{inc:entropy-lower} and divide by \(k\). +\end{proof} + +\subsection{Finite histories and the precision endpoint} +\label{inc:history-section} + +We state the consequence first for average success under the uniform +spherical prior, using the exact finite-state randomized model in +Definition~\ref{def:streaming-model}. + +\begin{theorem}[Average-prior precision bound]\label{inc:precision} +There is a universal \(c>0\) with the following property. Let +\(M=M(d)\) be a nonnegative integer with \(M(d)=o(d^2)\). +For all sufficiently large \(d\), with the +threshold allowed to depend on the rate \(M(d)/d^2\to0\), suppose +\(S\sim\sigma\) and a learner in Definition~\ref{def:streaming-model} stops by a +deterministic integer horizon \(T\ge0\). If +\[ + \Pr\{\angle(\widehat S,S)\le\epsilon\}\ge2/3, + \qquad 0<\epsilon\le1/10, +\] +then +\begin{equation}\label{inc:precision-bound} + T\ge c\,d\log(1/\epsilon). +\end{equation} +The bound permits arbitrary computation within a transition. +\end{theorem} + +For the proof of Theorem~\ref{inc:precision} and its auxiliary lemmas +below, apply the fixed-shared-seed part of Lemma~\ref{lem:padding} to the +stated learner in Definition~\ref{def:streaming-model}. We work in the +resulting Borel experiment with data-independent initialization and +uniform-prior success at least \(2/3\), retaining fresh transition and +output randomness. This is the standing reduction for this subsection. + +\begin{lemma}[Short-horizon reduction]\label{inc:short} +Put \(L_\epsilon=\log(1/\epsilon)\). Pad stopping to \(T\) by retaining +the stopped state and its stopping index. At most +\begin{equation}\label{inc:width} + w=(T+2)2^M +\end{equation} +padded states suffice. Under the hypotheses of +Theorem~\ref{inc:precision}, if \(T0\). In the learner model above, success +probability at least \(2/3\) is also impossible for \(T\le d/4\), +even if all observations are retained. +\end{lemma} + +\begin{proof} +Apply Lemma~\ref{lem:residual-information} with \(V=\widehat S\), the +identity output kernel, \(b=p\), and \(r_0=1/2\). For the residual radius +\(R=\|P_{\ker B}S\|\), Lemma~\ref{lem:residual-sphere} gives +\[ + \Pr\{R<1/2\}\le\frac{4p}{3d}\le\frac13. +\] +Success at least \(2/3\) therefore leaves event mass at least \(1/3\), +and \eqref{eq:residual-information} gives the first inequality in +\eqref{inc:fiber-bound}. The second follows from +\[ + \log\frac1{4\epsilon} + \ge\left(1-\frac{\log4}{\log10}\right)L_\epsilon, +\] +because \(d-p-1\) is an absolute positive fraction of \(d\); the +subtractive \(\log2\) is absorbed for large \(d\). + +For the final assertion, grant the learner all pre-generated observations +through its horizon, including ignored rows after stopping. Its output +is a kernel of those data and independent randomness. Lemma~\ref{lem:raw-sample-bound} +shows that success at least \(3/5\), and hence at least \(2/3\), already +requires \(T>\lfloor d/2\rfloor\) for large \(d\). In particular, +\(T>d/4\). That lemma includes the deterministic zero-row residual +sphere, so the assertion covers \(T=0\) as well. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{inc:precision}] +If \(T\ge dL_\epsilon\), the desired bound is immediate. Otherwise +apply Lemmas~\ref{inc:short} and~\ref{inc:history}. The terminal output +is a kernel of the final padded state, hence of \(V_b\); conditional +data processing and Lemma~\ref{inc:fiber-precision} give +\[ + Cbd\ge I(S;V_b\mid B,BS) + \ge I(S;\widehat S\mid B,BS) + \ge c_1dL_\epsilon. +\] +Thus \(b\ge c_0L_\epsilon\) for an absolute \(c_0>0\). +If \(L_\epsilon\ge2/c_0\), then +\[ + T>n(b-1)\ge(c_0/2)nL_\epsilon\ge c\,dL_\epsilon +\] +after adjusting an absolute constant. If \(L_\epsilon<2/c_0\), the +separate conclusion \(T>d/4\) of Lemma~\ref{inc:fiber-precision} +gives the same order after another absolute adjustment. This proves +\eqref{inc:precision-bound} uniformly over the allowed \(\epsilon\). +\end{proof} + +A guarantee of success probability at least \(2/3\) for every fixed +unit signal implies the uniform-prior average guarantee, so it has the +same consequence. The argument uses fresh exact labels, a finite +persistent state alphabet, and a deterministic horizon; its additional +projections and tuple resampling belong only to the analysis. diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/08-streaming.tex b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/08-streaming.tex new file mode 100644 index 0000000000000000000000000000000000000000..107f57db146834eb064ef0956db4964bfeaba8f4 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/build/sections/08-streaming.tex @@ -0,0 +1,220 @@ +\section{Current-state recurrences and the streaming bound} +\label{sec:streaming-consequences} + +We now complete the streaming applications of the Gaussian determinant +test and the distance-bin test through two current-state iterations. +Both use one independent side matrix throughout the stream. They condition +on the present padded state, apply a complete bounded-prior block bound in +that conditional experiment, and only then average over state values. +Thus their information upper bounds never subtract two possibly infinite +averages of conditional total correlations. + +\subsection{The Gaussian replica clock} + +\begin{proposition}[Current-state iteration of Gaussian replicas] +\label{prop:gaussian-streaming} +Let $M(d)=o(d^2)$ be nonnegative and integer-valued, and let +$0<\epsilon(d)\le1/10$. A learner in +Definition~\ref{def:streaming-model} whose uniform-prior angular success is +at least $3/5$ satisfies +\[ + T(d)\ge c\,d\log(1/\epsilon(d)) +\] +for an absolute $c>0$ and all sufficiently large $d$. The threshold may +depend on the memory sequence. +\end{proposition} + +\begin{proof} +Use the first step of Lemma~\ref{lem:padding} to choose a fixed shared-seed +conditional experiment with success at least $3/5$ and then take its Borel +replacement. Its initialization is independent of the signal and all rows; +fresh transition and output randomness remains in the kernels. This +selection uses the original jointly measurable experiment and does not +require a jointly measurable family of Borel replacements. + +Put $\beta=\log(1/\epsilon)$. If $T>d\beta$, the conclusion is immediate. +On the remaining branch, Lemmas~\ref{lem:padding} and +\ref{lem:early-clock} give padded states of entropy at most +\begin{equation}\label{eq:gaussian-clock-width} + h_T=M\log2+\log(T+2)=o(d^2), + \qquad \beta=o(d),\qquad T=o(d^2). +\end{equation} +Let +\[ + k=2\lfloor d/16\rfloor,\qquad t=k+1,\qquad + \ell=4k,\qquad n=\lceil T/k\rceil. +\] +Append ignored fresh samples so that the padded stream has $n$ complete +blocks. Let $V_i$ be the padded state at the end of block $i$, with $V_0$ +the data-independent initial padded state. The latter includes a possible +stopping decision at index zero. Draw a single standard Gaussian +$\ell\times d$ matrix $B$ independently of the entire selected learner +experiment and all appended rows, and put $Z=BS$. + +Consider a positive-probability value $V_i=v$, and put +$p_v=\Prob\{V_i=v\}$. By Lemma~\ref{lem:conditional-priors}, its conditional +signal law has a Borel density $f_v$ satisfying +\[ + f_v\le L_v:=1/p_v + \quad\hbox{relative to }\sigma_d,\qquad + \sum_v p_v\log L_v=H(V_i). +\] +The next block matrix is independent of the pair $(S,V_i)$: the state uses +only the signal, completed rows, and learner randomness, and the new rows +are fresh. Given $v$, the next state is a finite kernel of the new matrix +$A$ and its exact label $AS$. The side matrix $B$ remains independent of +this entire conditional signal, block, and message experiment. Therefore +Theorem~\ref{thm:gaussian-block} applies with $L=L_v$ and with message +$V_{i+1}$. + +Apply that numerical bound at each $v$ before averaging. Concavity of +$x\mapsto\log(2+x)$ and the entropy bound give +\begin{align} + I(S;V_{i+1}\mid V_i,B,Z) + &\le \frac{H(V_{i+1}\mid V_i)}{t} + +Cd+C\sum_vp_v\log(2+\log L_v) \notag\\ + &\le \frac{h_T}{t}+Cd+C\log(2+H(V_i)) \notag\\ + &\le \frac{h_T}{t}+Cd+C\log(2+h_T). + \label{eq:gaussian-current-step} +\end{align} +All conditional uses have bounded priors and finite conditional replica +correlation, as established in the block theorem. Since $t\asymp d$ and +$h_T=o(d^2)$, the last line is at most $C_1d$ for all sufficiently large +$d$, with one absolute $C_1$. + +Write $\mathcal I(V)=I(S;V\mid B,Z)$. The chain rule and data processing +give +\[ + \mathcal I(V_{i+1}) + \le \mathcal I(V_i)+I(S;V_{i+1}\mid V_i,B,Z). +\] +Here $\mathcal I(V_0)=0$ by the independence of initialization and the +index-zero decision from $(S,B,Z)$. Iterating +\eqref{eq:gaussian-current-step} yields +\begin{equation}\label{eq:gaussian-information-clock} + \mathcal I(V_n)\le C_1dn. +\end{equation} + +The side row count satisfies $\ell=8\lfloor d/16\rfloor\le d/2$. +The postponed output is a kernel of $V_n$ with fresh randomness, so +\eqref{eq:endpoint-gaussian} gives +\begin{equation}\label{eq:gaussian-streaming-endpoint} + \mathcal I(V_n)\ge + \tfrac12(d-\ell-1)\log\frac1{5\epsilon}-\log2 + \ge c_1d\beta +\end{equation} +for large $d$. Lemma~\ref{lem:raw-sample-bound} also gives +$T>\lfloor d/2\rfloor$. In particular $T>0$, and for $d\ge32$ the +inequality $k\ge d/16$ gives the explicit clock comparison +\[ + dn\le d(T/k+1)\le16T+d\le18T. +\] +Combining this with \eqref{eq:gaussian-information-clock} and +\eqref{eq:gaussian-streaming-endpoint} proves the proposition. +\end{proof} + +\subsection{The distance-bin clock} + +\begin{proposition}[Current-state iteration of distance bins] +\label{prop:distance-streaming} +Under the hypotheses of Theorem~\ref{thm:streaming}, the distance-bin +block estimate \eqref{eq:distance-block} also gives +$T(d)\ge c\,d\log(1/\epsilon(d))$ for an absolute $c>0$ and all +sufficiently large $d$. +\end{proposition} + +\begin{proof} +Apply Lemma~\ref{lem:padding} with $p=2/3$ and $p'=3/5$. It first fixes a +shared seed using the original jointly measurable experiment, then chooses +Borel versions in that conditional experiment and fixes its independent +initialization and remaining fresh randomness. We obtain deterministic +Borel state-based rules and an output with uniform-prior success at least +$3/5$. Pad stopping and postpone this output. The final output is a +function of the final tagged state. + +For this route keep the convenient clock bound +\begin{equation}\label{eq:distance-clock-width} + \mathfrak m_T=M\log2+\log(T+3). +\end{equation} +Indeed, the exact count in Lemma~\ref{lem:padding} is $(T+2)2^M$, so the +entropy of every padded state is at most $h_T\le\mathfrak m_T$. +Use +\[ + k=\lfloor d/16\rfloor,\qquad q=\lfloor d/8\rfloor,\qquad + p=\lfloor d/2\rfloor,\qquad n=\lceil T/k\rceil, +\] +and append ignored fresh rows through the last block. Let $V_0,\ldots,V_n$ +be the block-end states, starting with the data-independent tagged initial +state. Independently of the entire deterministic learner experiment and +all rows, draw one $p\times d$ standard Gaussian matrix $B$. It is used +only for analysis. Put $Z=BS$ and +$\mathcal D(V)=I(S;V\mid B,Z)$. + +For a block from $V$ to $W$, condition on a value $v$ with probability +$p_v>0$. By Lemma~\ref{lem:conditional-priors}, the conditional signal density is bounded by +$\exp(b_v)$, where $b_v=\log(1/p_v)$. The next standard Gaussian matrix +$A$ is independent of $(S,V)$, the ending state is a function of +$(v,A,AS)$, and $B$ remains independent of the entire conditional +experiment. Apply Theorem~\ref{thm:distance-block} at this fixed $v$ and +then average. Since $\sum_vp_vb_v=H(V)$, +\begin{align} + I(S;W\mid V,B,Z) + &\le \frac{H(W\mid V)}q+C\left(d+\frac{H(V)}d\right) \notag\\ + &\le C_2\left(d+\frac{\mathfrak m_T}{d}\right). + \label{eq:distance-current-step} +\end{align} +The last inequality uses $q\ge d/16$ for large $d$. In this calculation +the prior cost $b_v/d$ is averaged linearly; no entire state history is +retained. The chain rule gives +$\mathcal D(W)-\mathcal D(V)\le I(S;W\mid V,B,Z)$, and +$\mathcal D(V_0)=0$. Consequently +\begin{equation}\label{eq:distance-information-clock} + \mathcal D(V_n)\le C_2 n\left(d+\frac{\mathfrak m_T}{d}\right). +\end{equation} + +For the exact side row count $p=\lfloor d/2\rfloor$, the independent +residual-sphere endpoint \eqref{eq:endpoint-bins} gives +\begin{equation}\label{eq:distance-streaming-endpoint} + \mathcal D(V_n)\ge + \tfrac12(d-p-1)\log\frac1{4\epsilon}-\log2 + \ge c_2d\beta,\qquad \beta=\log(1/\epsilon), +\end{equation} +for sufficiently large $d$. This is the direct $90/d$ radius estimate +and the event of mass at least $1/2$ used in the distance-bin route. +The corresponding raw-data bound in Lemma~\ref{lem:raw-sample-bound} +also yields $T>\lfloor d/2\rfloor$, equivalently $T>d/2$ for integer $T$. +The zero-row experiment in that lemma has residual radius one. + +If $T\ge d\beta$, the asserted lower bound is immediate. Otherwise +$\mathcal D(V_n)\le H(V_n)\le\mathfrak m_T$ and +\eqref{eq:distance-streaming-endpoint} give the route's clock inequality +\[ + c_2d\beta\le M\log2+\log(T+3) + \le M\log2+\log(d\beta+3). +\] +Since $d\beta+3\le(d+3)(\beta+1)$ and +$\log(1+\beta)\le\beta$, this implies, for large $d$, +\begin{equation}\label{eq:distance-short-clock} + \beta\le \frac{C(M+\log d+1)}d=o(d). +\end{equation} +Thus $Td/2$ gives +\[ + dn\le32T+d\le34T. +\] +Comparing the resulting upper bound $\mathcal D(V_n)\le C_4T$ with +\eqref{eq:distance-streaming-endpoint} proves the proposition. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:streaming}] +Its uniform-prior success $2/3$ is at least the $3/5$ required by +Proposition~\ref{prop:gaussian-streaming}, which proves the theorem. +Proposition~\ref{prop:distance-streaming} gives a second proof using +the linear entropy cost. The later sections give three further proofs: +the Haar current-state argument in Proposition~\ref{haar:streaming}, and +the synthetic and incidence full-history arguments in +Proposition~\ref{syn:streaming} and Theorem~\ref{inc:precision}. +Each uses the same learner model, while +retaining its own side law, conditional experiment, and clock estimate. +\end{proof} diff --git a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9bf750d24128510d51058bc47be6963efe32a310 --- /dev/null +++ b/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:135be898557ce66eca6ac31cf62bc40967ab32fc0a30b41ffb9c1c20b063e6ea +size 749646 diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/README.md b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..ac2a058bb369202f58d7f08446d9078d81ae7872 --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [Power-law violations of Yau's nodal upper bound](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026, + author = {{OpenAI}}, + title = {{Power-law violations of Yau's nodal upper bound}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf}{OAI:Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/main.tex b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..5eb54e929514a03490d75bfd8b7aaae9b3278fe0 --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/main.tex @@ -0,0 +1,89 @@ +\documentclass[11pt]{article} +\newcommand{\DoNotLoadEpstopdf}{} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\pdfglyphtounicode{tfm:lmsy10/mapsto}{007C} +\pdfglyphtounicode{tfm:lmex10/parenleftbig}{0028} +\pdfglyphtounicode{tfm:lmex10/parenrightbig}{0029} +\pdfglyphtounicode{tfm:lmex10/vextendsingle}{23D0} +\pdfglyphtounicode{tfm:lmex10/parenleftBig}{0028} +\pdfglyphtounicode{tfm:lmex10/parenrightBig}{0029} +\pdfglyphtounicode{tfm:lmex10/parenleftbigg}{0028} +\pdfglyphtounicode{tfm:lmex10/parenrightbigg}{0029} +\pdfglyphtounicode{tfm:lmex10/braceleftbigg}{007B} +\pdfglyphtounicode{tfm:lmex10/bracerightbigg}{007D} +\pdfglyphtounicode{tfm:lmex10/parenleftBigg}{0028} +\pdfglyphtounicode{tfm:lmex10/parenrightBigg}{0029} +\pdfglyphtounicode{tfm:lmex10/summationtext}{2211} +\pdfglyphtounicode{tfm:lmex10/producttext}{220F} +\pdfglyphtounicode{tfm:lmex10/integraltext}{222B} +\pdfglyphtounicode{tfm:lmex10/summationdisplay}{2211} +\pdfglyphtounicode{tfm:lmex10/integraldisplay}{222B} +\pdfglyphtounicode{tfm:lmex10/hatwide}{0302} +\pdfglyphtounicode{tfm:lmex10/tildewide}{0303} +\pdfglyphtounicode{tfm:lmex10/bracketleftBig}{005B} +\pdfglyphtounicode{tfm:lmex10/bracketrightBig}{005D} +\pdfglyphtounicode{tfm:lmex10/radicalbig}{221A} +\pdfglyphtounicode{tfm:lmex10/radicalBig}{221A} +\pdfglyphtounicode{tfm:lmex10/parenlefttp}{239B} +\pdfglyphtounicode{tfm:lmex10/parenleftex}{239C} +\pdfglyphtounicode{tfm:lmex10/parenleftbt}{239D} +\pdfglyphtounicode{tfm:lmex10/parenrighttp}{239E} +\pdfglyphtounicode{tfm:lmex10/parenrightex}{239F} +\pdfglyphtounicode{tfm:lmex10/parenrightbt}{23A0} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage[margin=1.05in]{geometry} +\usepackage{enumitem} +\PassOptionsToPackage{hyphens}{url} +\usepackage[hidelinks]{hyperref} +\hypersetup{ + pdftitle={Power-law violations of Yau's nodal upper bound}, + pdfauthor={OpenAI}, + pdfsubject={A fixed smooth metric with a positive power of excess nodal volume} +} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\C}{\mathbb C} +\newcommand{\E}{\mathbb E} +\newcommand{\Pp}{\mathbb P} +\newcommand{\dd}{\mathrm d} +\newcommand{\Hh}{\mathcal H} +\newcommand{\Z}{\mathcal Z} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\diver}{div} +\DeclareMathOperator{\tr}{tr} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\allowdisplaybreaks[1] +\setlist[enumerate]{itemsep=3pt,topsep=5pt} +\setlist[itemize]{itemsep=3pt,topsep=5pt} +\emergencystretch=1em +\title{Power-law violations of\\Yau's nodal upper bound} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We construct a smooth Riemannian metric on $S^4\times S^1$ and a sequence +of real Laplace eigenfunctions whose nodal four-volume grows faster than +$\lambda^{1/2+\epsilon_0}$ for one fixed $\epsilon_0>0$. +This disproves the smooth upper-bound assertion in Yau's nodal-set +conjecture and the proposed bound with an arbitrarily small positive +power loss. +\end{abstract} +\input{sections/introduction} +\input{sections/framework} +\input{sections/envelopes} +\input{sections/waves} +\input{sections/global} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/references.bib b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..877014288c1c9ca742fbccb384f73a5151367c70 --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/references.bib @@ -0,0 +1,143 @@ +@incollection{Yau1982, + author = {Yau, Shing-Tung}, + title = {Problem section}, + booktitle = {Seminar on Differential Geometry}, + editor = {Yau, Shing-Tung}, + series = {Annals of Mathematics Studies}, + volume = {102}, + publisher = {Princeton University Press}, + address = {Princeton, NJ}, + year = {1982}, + pages = {669--706}, + doi = {10.1515/9781400881918-035}, + url = {https://doi.org/10.1515/9781400881918-035} +} + +@article{DonnellyFefferman1988, + author = {Donnelly, Harold and Fefferman, Charles}, + title = {Nodal sets of eigenfunctions on {Riemannian} manifolds}, + journal = {Inventiones Mathematicae}, + volume = {93}, + number = {1}, + year = {1988}, + pages = {161--183}, + doi = {10.1007/BF01393691}, + url = {https://doi.org/10.1007/BF01393691} +} + +@article{LogunovUpper2018, + author = {Logunov, Alexander}, + title = {Nodal sets of {Laplace} eigenfunctions: polynomial upper estimates of the {Hausdorff} measure}, + journal = {Annals of Mathematics (2)}, + volume = {187}, + number = {1}, + year = {2018}, + pages = {221--239}, + doi = {10.4007/annals.2018.187.1.4}, + url = {https://annals.math.princeton.edu/2018/187-1/p04} +} + +@article{LogunovLower2018, + author = {Logunov, Alexander}, + title = {Nodal sets of {Laplace} eigenfunctions: proof of {Nadirashvili}'s conjecture and of the lower bound in {Yau}'s conjecture}, + journal = {Annals of Mathematics (2)}, + volume = {187}, + number = {1}, + year = {2018}, + pages = {241--262}, + doi = {10.4007/annals.2018.187.1.5}, + url = {https://annals.math.princeton.edu/2018/187-1/p05} +} + +@article{Hezari2023, + author = {Hezari, Hamid}, + title = {Upper bounds on the size of nodal sets for {Gevrey} and quasianalytic {Riemannian} manifolds}, + journal = {Communications in Mathematical Physics}, + volume = {404}, + number = {3}, + pages = {1341--1359}, + year = {2023}, + doi = {10.1007/s00220-023-04853-z}, + url = {https://doi.org/10.1007/s00220-023-04853-z} +} + +@article{EncisoPeraltaSalas2015, + author = {Enciso, Alberto and Peralta-Salas, Daniel}, + title = {Eigenfunctions with prescribed nodal sets}, + journal = {Journal of Differential Geometry}, + volume = {101}, + number = {2}, + pages = {197--211}, + year = {2015}, + doi = {10.4310/jdg/1442364650}, + url = {https://doi.org/10.4310/jdg/1442364650} +} + +@incollection{EncisoPeraltaSalasTorres2018, + author = {Enciso, Alberto and Peralta-Salas, Daniel and Torres de Lizaur, Francisco}, + title = {High-energy eigenfunctions of the {Laplacian} on the torus and the sphere with nodal sets of complicated topology}, + booktitle = {Nonlinear Partial Differential Equations for Future Applications}, + editor = {Koike, Shigeaki and Kozono, Hideo and Ogawa, Takayoshi and Sakaguchi, Shigeru}, + series = {Springer Proceedings in Mathematics \& Statistics}, + volume = {346}, + publisher = {Springer}, + address = {Singapore}, + year = {2021}, + pages = {245--261}, + doi = {10.1007/978-981-33-4822-6_7}, + url = {https://arxiv.org/abs/1810.09277v1} +} + +@article{Aronszajn1957, + author = {Aronszajn, N.}, + title = {A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order}, + journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es (9)}, + volume = {36}, + year = {1957}, + pages = {235--249}, + url = {https://sites.math.washington.edu/~blwilson/Nodal/Aronszajn.pdf} +} + +@article{DenckerSjostrandZworski2004, + author = {Dencker, Nils and Sj{\"o}strand, Johannes and Zworski, Maciej}, + title = {Pseudospectra of semiclassical {\mbox{(pseudo-)}} differential operators}, + journal = {Communications on Pure and Applied Mathematics}, + volume = {57}, + number = {3}, + year = {2004}, + pages = {384--415}, + doi = {10.1002/cpa.20004}, + url = {https://arxiv.org/abs/math/0301242v1}, + note = {Section locators follow arXiv:math/0301242v1} +} + +@article{CharalambousLu2015, + author = {Charalambous, Nelia and Lu, Zhiqin}, + title = {Heat kernel estimates and the essential spectrum on weighted manifolds}, + journal = {Journal of Geometric Analysis}, + volume = {25}, + year = {2015}, + pages = {536--563}, + doi = {10.1007/s12220-013-9438-1}, + url = {https://doi.org/10.1007/s12220-013-9438-1} +} + +@book{Evans2010, + author = {Evans, Lawrence C.}, + title = {Partial Differential Equations}, + edition = {Second}, + series = {Graduate Studies in Mathematics}, + volume = {19}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {2010}, + url = {https://bookstore.ams.org/gsm-19-r} +} + +@misc{OpenAINodalQualitative2026, + author = {{OpenAI}}, + title = {{Smooth counterexamples to Yau's nodal upper bound in dimensions three and four}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf}{OAI:Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026}}, + year = {2026} +} diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/envelopes.tex b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/envelopes.tex new file mode 100644 index 0000000000000000000000000000000000000000..dee80fdb5a2060cf4d5328af4de969d37e6498cb --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/envelopes.tex @@ -0,0 +1,659 @@ +\section{Envelopes with a fixed power gain}\label{sec:envelopes} + +We construct envelopes whose gradient integral grows geometrically under +iteration. The part of each parent cube available for the next iteration +will have more gradient integral than the entire parent had before the +change. All scales and gain factors are fixed before any finite +approximation order is prescribed. + +Throughout this Section, tensor norms in Hessian inequalities are operator +norms for $G$. Coordinate derivatives and integrals refer to fixed smooth +coordinates. The notation $p,Q,J_h$ is that of +\eqref{eq:hessian-test}. + +\subsection{A seed and the preparation condition} + +\begin{lemma}\label{lem:seed} +There are a number $a_*>0$, a smooth real function $h_0$ on $B$, a +$C^1$ neighborhood $\mathcal G$ of $g_0$ among smooth metrics, and a +coordinate chart $\Omega\subset\{t>1/2\}$ with open sets +$U_0\Subset V\Subset\Omega$ such that the following assertions hold +uniformly for $G\in\mathcal G$: +\begin{enumerate}[label=\textup{(\roman*)}] + \item $h_0=-a_*t$ on $\{t\le1/2\}$; + \item $\max_e J_{h_0}(e)\ge c_0>0$ on $\{t\ge1/4\}$; + \item on $U_0$, the Hessian of $h_0$ is positive definite with a + uniform positive lower bound, and + $01/2\}$, two Hessian eigenvalues therefore +have a uniform positive lower bound and the gradient is arbitrarily +small. In round normal coordinates $\xi$ at $x_*$, add +$\varepsilon|\xi|^2/2$ times a fixed smooth cutoff equal to one near +$\xi=0$. Choose $\varepsilon>0$ sufficiently small that this is a +$C^2$-small perturbation on its support. The resulting function $h_0$ +has positive definite Hessian at $x_*$. + +Here is why the cutoff region retains the required test. Shrinking its +support and then the perturbation if necessary, the perturbed Hessian +has two eigenvalues at least $c a_*$, its norm is at most $C a_*$, +and $|\nabla_{g_0}h_0|$ is sufficiently small that +$C a_*|\nabla_{g_0}h_0|^21/2\}$ in a fixed +coordinate chart, and shrink $\mathcal G$ once more. All the stated +constants can then be chosen uniformly. Notice that $U_0$ is chosen +away from the critical point $x_*$. +\end{proof} + +We use the coordinates of $\Omega$ for all corrugations. They may be +taken on a neighborhood of its closure, so metric and inverse metric +have common ellipticity and $C^1$ bounds there. Fix positive constants +$C_{\mathrm p}$ and $c_{\mathrm p}$. A smooth function $h$ is +\emph{prepared at scale $S$} on a cube $P$ of side $\nu S$, +$00,\qquad + |\nabla^2h|\le C_{\mathrm p}|\nabla h|/S,\\ + \text{at each point there is a two-plane } + E\subset\ker\dd h\text{ such that } + J_h(e)\ge c_{\mathrm p}|\nabla h|/S + \quad(e\in E,\ |e|=1). + \end{gathered} +\end{equation} +The plane $E$ need not vary continuously. Cubes are parallel to the +fixed coordinate axes, and the inequalities are required on their +closures. + +The two Hessian conditions serve different parts of the next corrugation. +The Hessian norm bound controls the inherited Hessian and the variation +of the parent gradient. When the gradient changes by a small relative +amount, we can rotate the favorable two-plane slightly into the +hyperplane perpendicular to the new gradient. One further linear +tangency constraint then leaves a direction with a positive test. +On the region selected for repetition, the modification supplies two +favorable directions again. This is why the preparation condition +asks for a two-plane even though the envelope ultimately needs only one +positive test at each point. + +By Lemma~\ref{lem:seed}, for all sufficiently large +$C_{\mathrm p}$ and sufficiently small $c_{\mathrm p}>0$, every +sufficiently small cube in $U_0$ is prepared for $h_0$ at scale $1$, +uniformly over $G\in\mathcal G$. + +\subsection{A repeatable local corrugation} + +\begin{lemma}\label{lem:corrugation} +Under fixed ellipticity and $C^1$ metric bounds in the coordinate +patch, there are constants $C_{\mathrm p}^*>0$, +$c_{\mathrm p}^*>0$ and $\zeta>0$ with the following property. +For any $C_{\mathrm p}\ge C_{\mathrm p}^*$ and +$00$, once and for all, such that every +prepared cube $P$ at scale $S\le1$ admits a smooth modification +$v=h+w$, with $w$ supported in the interior of $P$, satisfying: +\begin{enumerate}[label=\textup{(\roman*)}] + \item for fixed constants $0<\alpha\le1\le\beta$ and $c>0$, + \[ + \alpha|\nabla h|\le|\nabla v|\le\beta|\nabla h|, + \qquad + \max_eJ_v(e)\ge c|\nabla h|/S + \quad\hbox{on }P; + \] + \item writing $\ell=\nu^2S$, some cubes in the partition of $P$ + into cubes of side $\nu\ell$ are prepared for $v$ at scale $\ell$ + with the same constants in \eqref{eq:preparation}, and their union + $\mathcal R_P$ satisfies + \begin{equation}\label{eq:local-mass-gain} + \int_{\mathcal R_P}|\nabla v|\,\dd x + \ge(1+\zeta)\int_P|\nabla h|\,\dd x; + \end{equation} + \item if $s=|\nabla h(x_P)|$ at the center $x_P$ of $P$, then for + every integer $j\ge0$, + \begin{equation}\label{eq:corrugation-derivatives} + \|D^jw\|_{L^\infty(P)}\le C_j s\ell^{1-j}. + \end{equation} +\end{enumerate} +All constants are uniform over prepared functions and cubes and over +the stated class of metrics. They may depend on the fixed pattern, +preparation constants, and final choice of $\nu$. No derivative of +the input $h$ beyond its prepared second-derivative bound is used. +The pattern and $\zeta$ are chosen before the preparation constants +and the final $\nu$; increasing $C_{\mathrm p}$ or decreasing +$c_{\mathrm p}$ only requires taking $\nu$ smaller. +\end{lemma} + +\begin{proof} +The pattern will depend on three variables transverse to the parent +gradient frozen at its center. Radial wells in these variables have +two positive tangential Hessian directions; a small amplitude controls +their potentially negative radial contribution to $J_v$. We first +choose the wells and smaller concentric balls that will supply the +repeatable gain. We then place this pattern in the parent and verify +the Hessian test, including where the parent cutoff varies. + +\paragraph{The periodic wells.} +Choose a smooth function $F$ on $[0,1]$, positive on $(0,1)$, equal +to $v$ near $v=0$, and flat at $v=1$. Inside a unit cube in $\R^3$ +take finitely many balls with pairwise disjoint closures and radii +$R_i>0$. On a ball, at radius $r$ from its center, set +\[ + \psi(r)=-\int_r^{R_i}F(q/R_i)\,\dd q,\qquad + f(r)=F(r/R_i),\qquad u_0(r)=f(r)/r. +\] +Extend $\psi$ by zero outside the balls and periodically to $\R^3$. +It is smooth: near a center it is a constant plus $r^2/(2R_i)$, +and it is flat at the outer boundary. The extended value of $u_0$ +at the center is $1/R_i$. Write $\sigma$ for the radial unit vector. +Then +\begin{equation}\label{eq:well-identities} + \nabla\psi=f\sigma,\qquad + D^2\psi=f'\sigma\otimes\sigma + +u_0(I-\sigma\otimes\sigma),\qquad + f^2|f'|\le C u_0,\quad f^2\le C u_0. +\end{equation} +The first constant in these inequalities is independent of the radii, +because, for $v=r/R_i$, +\[ + \frac{f^2|f'|}{u_0}=vF(v)|F'(v)|. +\] +The second follows from $f^2/u_0=R_i vF(v)$ and the bounded radii. +Set $f,f',u_0$ equal to zero outside the balls. These coefficients +are continuous, and +\begin{equation}\label{eq:well-vanishing} + u_0=0\quad\Longrightarrow\quad f=f'=\psi=0. +\end{equation} +At a center $f'=u_0$, so any choice of $\sigma$ gives the same +Hessian in \eqref{eq:well-identities}; outside the balls choose +$\sigma$ arbitrarily as well. + +Choose a small amplitude $a>0$. Its smallness will use only $F$ and +the radius-independent first bound in \eqref{eq:well-identities}. +In particular it can be chosen before the packing. The mean gradient +factor on any one well is +\[ + \mathcal W(a)=3\int_0^1v^2\sqrt{1+a^2F(v)^2}\,\dd v>1. +\] +Put $\eta_0=(\mathcal W(a)-1)/8$. Choose the finite packing with +volume fraction $\varpi$ such that +\[ + \varpi\mathcal W(a)>1+6\eta_0. +\] +Such a packing exists. To see this without requiring an infinite +packing in the construction, fill a fixed positive fraction of any +remaining open set by finitely many closed grid cubes lying inside +it, and put a ball with strictly smaller radius than the inradius +in each cube. This captures a fixed positive fraction of the +remaining volume. Finitely many repetitions make the uncovered +volume as small as desired, with all ball closures still disjoint. + +Choose $\eta>0$ so small that, on the core balls +$r<(1-\eta)R_i$, the weighted integral per period satisfies +\[ + C_{\mathrm{core}} + :=\sum_i\int_{r<(1-\eta)R_i}\sqrt{1+a^2f^2}\,\dd z' + >\varpi\mathcal W(a)-\eta_0>1+5\eta_0. +\] +Choose an inner cube fraction $q\in(0,1)$ such that +\begin{equation}\label{eq:repeat-slack} + qC_{\mathrm{core}}>C_{\mathrm{core}}-\eta_0>1+4\eta_0. +\end{equation} +Here $q$ is a four-dimensional volume fraction. Fix a smooth cutoff +on the unit parent cube which equals one on the concentric inner +cube of fraction $q$, is supported in the parent interior, and is +flat wherever it is zero. Products of one-dimensional plateau +cutoffs give such a function. The wells, cores, and cutoff are now +fixed. All their derivative bounds are finite, although no bound +uniform in derivative order is asserted. + +\paragraph{Normalization on a prepared parent.} +Take affine coordinates $z=(z_1,z')$ centered at $x_P$, making the +metric Euclidean at the center and +$\nabla h(x_P)=s\partial_{z_1}$. The linear maps and their inverses +have uniform bounds; the image of $P$ is a parallelepiped of uniformly +bounded distortion. The coordinate identity relating $D(\dd h)$ +to the covariant Hessian and the bounded connection gives +\[ + |D(\dd h)|\le C|\dd h|/S +\] +on $P$, since $S\le1$. Integration along segments of length +$O(\nu S)$ yields relative upper and lower bounds for the gradient +and then +\begin{equation}\label{eq:parent-normalization} + \dd h/s=\dd z_1+O(\nu),\qquad + (G_{ij})_z=I+O(\nu),\qquad + |\nabla h|/s=1+O(\nu). +\end{equation} +The constants here may depend on the preparation constants. Only +the bounds already specified in the lemma enter them. + +Rescale the fixed parent cutoff to $P$ and call it $\chi$. Set +\begin{equation}\label{eq:corrugation-formula} + v=h+s\ell a\chi\psi(z'/\ell),\qquad \ell=\nu^2S. +\end{equation} +In the following estimates the radial coefficients are evaluated at +$z'/\ell$. A cutoff derivative at its parent scale is denoted +$D_{\mathrm{sc}}\chi$; thus an ordinary derivative of $\chi$ is +$(\nu S)^{-1}D_{\mathrm{sc}}\chi$. The bounded affine changes only +alter the fixed bounds for these scaled derivatives. Since +$\ell/(\nu S)=\nu$, Equation~\eqref{eq:corrugation-formula} gives +\begin{equation}\label{eq:new-gradient} + \dd v/s=\dd z_1+a\chi f\sigma\cdot\dd z'+O(\nu). +\end{equation} +This proves the gradient comparison in \textup{(i)} for fixed +$\alpha,\beta$, when $\nu$ is sufficiently small. + +\paragraph{The new Hessian contribution.} +Write +\[ + D=\frac{\nabla v}{\sqrt{1+|\nabla v|^2}},\qquad + \mathcal F=\{e:\dd v(e)=0,\ \sigma\cdot e'=0\}, +\] +where $e'$ denotes the static transverse components in the affine +coordinates. The two defining linear conditions are independent +by \eqref{eq:new-gradient}, so $\mathcal F$ is two-dimensional. +For a metric unit vector $e\in\mathcal F$, +\begin{equation}\label{eq:test-components} + |e'|^2\ge\tfrac12,\qquad + |\sigma\cdot D'|\le C_0a\chi f+C\nu. +\end{equation} +Indeed $e_1=O(\nu)$ follows from the two linear conditions. The +second estimate follows from \eqref{eq:new-gradient} and +$G^{ij}=\delta_{ij}+O(\nu)$, using a denominator at least a fixed +multiple of $s$. We may take $C_0$ absolute by first taking $\nu$ +small for the fixed data. These arguments are uniform even when +$s$ is arbitrarily small or large. + +Let $\Pi_\sigma$ be Euclidean projection onto $\sigma^\perp$ in +the transverse space. The leading contribution of $v-h$ to its +Hessian test on $D,e$, divided by $s/\ell$, is +\[ + a\chi\left[ + f'(\sigma\cdot D')^2+ + u_0\bigl(|\Pi_\sigma D'|^2+|e'|^2\bigr)\right]. +\] +By \eqref{eq:test-components}, this is at least +\[ + a\chi\left[ + \tfrac12u_0-|f'|\bigl(2C_0^2a^2\chi^2 f^2+C\nu^2\bigr) + \right]. +\] +At the initial choice of amplitude, impose +$2C_0^2a^2\sup_v vF(v)|F'(v)|\le1/4$. +The main term is then at least +$a\chi u_0/4-Ca\nu^2\chi|f'|$. + +The mixed cutoff terms on $e,e$ vanish because +$\sigma\cdot e'=0$. On $D,D$ their size is bounded by +\[ + C\nu f|D_{\mathrm{sc}}\chi|(\chi f+\nu). +\] +The pure cutoff terms have size at most +$C\nu^2|\psi||D_{\mathrm{sc}}^2\chi|$. +Finally, replacing the coordinate Hessian of $v-h$ by its covariant +Hessian gives an error of size +\[ + C\ell(\chi f+\nu) +\] +in the same units, because the connection is uniformly bounded. + +We verify uniformity of these errors also where the favorable +coefficient vanishes. Set $X=\chi u_0$. On a fixed compact +parameter space, if continuous functions $X,b\ge0$ satisfy +$b=0$ on $\{X=0\}$, then +\begin{equation}\label{eq:vanishing-coefficient} + \sup\frac{\nu^2b}{X+\nu^2}\longrightarrow0 + \qquad(\nu\longrightarrow0). +\end{equation} +For $X\ge\tau$ the quotient is at most +$\nu^2\sup b/\tau$; for $X<\tau$ it is at most +$\sup_{X<\tau}b$, which tends to zero as $\tau\downarrow0$. +The compact parameter space here consists of cutoff position, +period position, and bounded affine frames. Taking its closure is +harmless. Transformed cutoff derivatives are continuous on this +space and have uniform bounds. + +Apply \eqref{eq:vanishing-coefficient} to +$b=\chi|f'|$, $b=f|D_{\mathrm{sc}}\chi|$, and +$b=|\psi||D_{\mathrm{sc}}^2\chi|$. Their vanishing follows from +\eqref{eq:well-vanishing} and the flatness of $\chi$ on its zero +set. The other mixed term satisfies +$\nu\chi f^2|D_{\mathrm{sc}}\chi|\le C\nu X$. +Moreover, $\chi f\le C\sqrt X$ and $\ell\le\nu^2$ give +\[ + \frac{\ell(\chi f+\nu)}{X+\nu^2} + \le C\frac{\nu^2\sqrt X}{X+\nu^2}+\nu + \le C'\nu. +\] +We have proved, uniformly in all prepared parents and allowed tests, +\begin{equation}\label{eq:corrugation-hessian-error} + \frac{\ell}{s}\left[ + \nabla^2(v-h)(D,D)+\nabla^2(v-h)(e,e)\right] + \ge c_a X-\varepsilon(\nu)(X+\nu^2), + \qquad \varepsilon(\nu)\longrightarrow0, +\end{equation} +for a fixed $c_a>0$. No positive lower bound on a nonzero value of +$\chi$ or $u_0$ has been used. + +\paragraph{Retaining one direction everywhere.} +The old Hessian has norm $O(\nu^2)$ after division by $s/\ell$. +Thus the preceding estimate compares a favorable new contribution of +size $X$ with inherited terms of size $\nu^2$. When $X$ dominates +$\nu^2$, the new favorable directions suffice. When $X=O(\nu^2)$, +the gradients are close enough to carry an old favorable direction to +the new tangent space. To obtain one uniform lower bound, consider +any sequence $\nu\to0$ of admissible configurations. +If $X/\nu^2\to\infty$ along a subsequence, the positive term in +\eqref{eq:corrugation-hessian-error} dominates the old Hessian in +every unit direction of $\mathcal F$. + +If instead $X/\nu^2$ stays bounded, then +$\chi f\le C\sqrt X=O(\nu)$, so +$|\nabla v-\nabla h|\le\varepsilon_1|\nabla h|$ with +$\varepsilon_1=O(\nu)$. This also makes their normalized gradients +close uniformly for unbounded slopes. In fact, for +$T(p)=p/\sqrt{1+|p|^2}$ and $|q-p|\le\varepsilon_1|p|$, +$\varepsilon_1\le1/2$, the segment from $p$ to $q$ stays outside +the ball of radius $|p|/2$. Therefore +\begin{equation}\label{eq:normalized-gradient-stability} + |T(q)-T(p)| + \le\frac{\varepsilon_1|p|}{\sqrt{1+|p|^2/4}} + \le2\varepsilon_1. +\end{equation} +There is a metric orthogonal rotation $R=I+O(\varepsilon_1)$ taking +$\ker\dd h$ to $\ker\dd v$. Rotate the old good two-plane $E$ +in \eqref{eq:preparation} by $R$. Inside the three-dimensional +space $\ker\dd v$, the planes $RE$ and $\mathcal F$ intersect in +dimension at least +\[ + 2+2-3=1. +\] +For a unit vector $e$ in their intersection, compare the old test on +$p/Q,R^{-1}e$ with the old Hessian evaluated on $D,e$. +Equation~\eqref{eq:normalized-gradient-stability} shows that the +change costs $o(\nu^2)$ in the normalized units. Thus the old +contribution is at least $\tfrac12c_{\mathrm p}\nu^2$ for small +$\nu$. The new contribution in +\eqref{eq:corrugation-hessian-error} is bounded below by +$-o(\nu^2)$ in this bounded-ratio regime. + +These alternatives prove the uniform margin +$c_{\mathrm p}\nu^2/8$ in the normalized units for some test when +$\nu$ is sufficiently small. Indeed, a contrary +sequence has a subsequence on which $X/\nu^2$ is bounded or tends +to infinity, and both alternatives just excluded failure of that +margin. Multiplication by $s/\ell$ and +\eqref{eq:parent-normalization} prove the Hessian assertion in +\textup{(i)}. This is where base dimension four is used: in +dimension $d$ the corresponding intersection lower bound is $d-3$. + +\paragraph{The region prepared for repetition.} +Inside the inner parent cube where $\chi=1$, keep the points for +which $z'/\ell$ lies in a core ball. On this region $u_0$ has a +fixed positive lower bound. Equations +\eqref{eq:corrugation-hessian-error} and +\eqref{eq:parent-normalization} imply, for every unit +$e\in\mathcal F$, +\[ + J_v(e)\ge c' s/\ell, + \qquad + |\nabla^2v|\le(C'+O(C_{\mathrm p}\nu^2))s/\ell. +\] +The leading constants depend only on the fixed pattern and the +uniform metric bounds. Gradient comparison now permits fixed +thresholds $C_{\mathrm p}^*$ and $c_{\mathrm p}^*$ such that, for +$C_{\mathrm p}\ge C_{\mathrm p}^*$ and +$00$. This proves +\eqref{eq:local-mass-gain}; taking $\nu$ smaller still preserves +this value of $\zeta$. + +\paragraph{Derivatives and the order of choices.} +Differentiating \eqref{eq:corrugation-formula} in the fixed +coordinates gives \eqref{eq:corrugation-derivatives}. In a term +with $q$ derivatives on $\chi$, the scale factor is +\[ + s\ell(\nu S)^{-q}\ell^{-(j-q)} + =s\ell^{1-j}\nu^q\le s\ell^{1-j}. +\] +All derivatives of the pattern and all powers of the bounded affine +matrix enter the fixed constant $C_j$. The center slope and the +affine matrix are constants in the expression being differentiated. + +To summarize the choices within this proof, first fix $F$ and its +small amplitude $a$, then the finite packing, cores, and parent +cutoff with the slack in \eqref{eq:repeat-slack}. Choose the +preparation constants next, large and small respectively, and +finally choose a reciprocal $\nu$ satisfying the local positivity, +repeat preparation, and mass-error requirements. This proves the +stated uniformity and completes the lemma. +\end{proof} + +\subsection{Iteration and a common exponent for every finite order} + +\begin{proposition}\label{prop:envelope} +After fixing the data of Lemma~\ref{lem:seed} and possibly shrinking +$\mathcal G$, there are common constants $\gamma>0$ and $c>0$ +with the following property. For every smooth $G\in\mathcal G$ +and all sufficiently large $k$, there is a smooth real envelope +$h=h_k$ on $B$ such that $h=-a_*t+c_k$ on $\{t\le1/2\}$ and +\begin{equation}\label{eq:envelope-bounds} + \begin{gathered} + \|h\|_{C^j}\le C_j k^{\delta(j+1)} + \quad\text{for every fixed integer }j\ge0,\\ + \max_{\substack{|e|_G=1\\\dd h(e)=0}}J_h(e) + \ge c k^{-\delta}\quad\text{on }\{t\ge1/4\}, + \qquad + \int_V\sqrt{1+|\nabla_Gh|_G^2}\,\dd x\ge c k^\gamma. + \end{gathered} +\end{equation} +Here $\delta=10^{-6}$, and the chart $\Omega$ and region +$V\Subset\Omega$ are fixed. All corrugations are supported in +one fixed initial cube compactly contained in $V$. +The constants $c_k$ may be any bounded prescribed real numbers. +The exponent $\gamma$ and all construction parameters are +independent of derivative orders, subsequent accuracy requirements, +and round gluing collars. Constants $C_j$ may depend on the order, +the fixed smooth metric and coordinate systems, and the bound for +$|c_k|$; no uniform bound as $j\to\infty$ is asserted. +\end{proposition} + +\begin{proof} +Choose $C_{\mathrm p}$ large enough and $c_{\mathrm p}>0$ small +enough for both Lemma~\ref{lem:corrugation} and the seed preparation +following Lemma~\ref{lem:seed}. Choose the final $\nu$ sufficiently +small for the former lemma and so that a fixed cube $P_0$ of side +$\nu$ has closure in $U_0$. All these choices are uniform for the +metrics in $\mathcal G$. Put $r=\nu^2$. + +Start with the prepared family $\mathcal P_0=\{P_0\}$ for $h_0$. +At generation $i\ge1$, apply Lemma~\ref{lem:corrugation} on every +cube of $\mathcal P_{i-1}$, using parent scale $S_i=r^{i-1}$, +and call the resulting function $h_i$. The selected child cubes +form $\mathcal P_i$; their sides are $\nu r^i$ and they are +prepared at scale $r^i$. Every generation consists of finitely +many cubes with disjoint interiors. Each perturbation is supported +strictly inside its parent, so each $h_i$ is smooth, including across +all cube boundaries. Nothing outside $P_0$ changes. + +Let +\[ + I_i=\sum_{P\in\mathcal P_i}\int_P|\nabla h_i|\,\dd x. +\] +Equation~\eqref{eq:local-mass-gain} gives +\begin{equation}\label{eq:iterated-mass} + I_i\ge(1+\zeta)I_{i-1},\qquad + \int_V\sqrt{1+|\nabla h_i|^2}\,\dd x + \ge I_i\ge I_0(1+\zeta)^i. +\end{equation} +The initial mass $I_0$ has a positive uniform lower bound by +Lemma~\ref{lem:seed} and the fixed volume of $P_0$. + +We give the derivative accounting for all orders with one growth +constant. Let $\beta\ge1$ and $0<\alpha\le1$ be the gradient +comparison factors of Lemma~\ref{lem:corrugation}. The largest +gradient after $i$ generations is at most a fixed multiple of +$\beta^i$. On $P_0$ every point has gradient at least +$s_{\min}\alpha^i$, since a point is changed at most once per generation. +At generation $i$, the center slope of each parent is therefore +at most $C\beta^{i-1}$. By +\eqref{eq:corrugation-derivatives}, disjointness within that +generation gives +\begin{equation}\label{eq:generation-derivatives} + \|D^j(h_i-h_{i-1})\|_{L^\infty} + \le C_j\beta^{i-1}r^{i(1-j)}. +\end{equation} +There is no factor counting cubes in this supremum norm estimate. +There is also no product of higher pattern derivative constants +over generations: the increments are added, and their individual +formulas involve frozen slopes and affine matrices. Passing to +the fixed finite atlas only adds fixed chain-rule constants and +lower-order derivatives, which satisfy the same upper bound. +Summation yields +\begin{align} + \|h_m\|_{C^j} + &\le C_j\left(1+\sum_{i=1}^m + \beta^{i-1}r^{i(1-j)}\right)\notag\\ + &\le C'_j\exp\left( + [\log\beta+(j-1)_+|\log r|+1]m\right) + \le C'_j e^{C_*(j+1)m}, + \label{eq:all-order-envelope-growth} +\end{align} +where $(j-1)_+=\max\{j-1,0\}$ and one finite $C_*$ works for +every $j\ge0$. + +The lower Hessian margin has the same exponential control. +At a point outside $P_0$, the seed margin remains unchanged. At a +point in $P_0$, consider the last generation $i\le m$ that changes +the function there. If no such generation occurs, use the seed +margin. Otherwise Lemma~\ref{lem:corrugation} bounds its retained +margin below by +\[ + c\,\frac{s_{\min}\alpha^{i-1}}{r^{i-1}} + \ge c s_{\min}\alpha^m. +\] +Later perturbations vanish near that point, or all their jets there +vanish at their support boundaries, so this margin persists. By +increasing $C_*$ if necessary, +\begin{equation}\label{eq:iterated-hessian-margin} + \max_eJ_{h_m}(e)\ge c e^{-C_*m} + \quad\text{on }\{t\ge1/4\}. +\end{equation} +This argument requires no nonzero-gradient assumption outside +$P_0$; critical points there are covered by the seed lemma. + +Fix $\theta>0$ so small that $C_*\theta\le\delta$, and for each +$k\ge2$ take +\[ + m=\lfloor\theta\log k\rfloor, + \qquad \gamma=\theta\log(1+\zeta)>0, + \qquad h=h_m+c_k. +\] +Equations \eqref{eq:all-order-envelope-growth} and +\eqref{eq:iterated-hessian-margin} give the first two estimates in +\eqref{eq:envelope-bounds}. Equation~\eqref{eq:iterated-mass} +gives the third, since +$(1+\zeta)^m\ge(1+\zeta)^{-1}k^\gamma$. +Adding bounded constants affects only the zeroth-order bound, and +$h=-a_*t+c_k$ on $t\le1/2$ because both the seed perturbation and +all corrugations lie above that set. + +For clarity about subsequent uses, the constants determining +$C_*$, $\theta$, and $\gamma$ use only the fixed pattern, the +seed, and common $C^1$ metric bounds. Higher derivatives of an +individual input metric may enter constants in a later smooth +coordinate change, but cannot alter these exponents. Indeed, for +any fixed such coordinate change, an order-$j$ derivative of the +pullback is a finite sum of derivatives of $h$ of orders at most +$j$ multiplied by fixed smooth coefficients. Its growth is still +bounded by $C_jk^{\delta(j+1)}$. Each finite order may have its +own constant and frequency threshold. This establishes the common +exponent for arbitrarily high finite accuracy and completes the +proof. +\end{proof} diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/framework.tex b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/framework.tex new file mode 100644 index 0000000000000000000000000000000000000000..88c81e38d58ec8e08c8cbe41e9a87aae0dd6d403 --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/framework.tex @@ -0,0 +1,156 @@ +\section{Framework and the quantitative step}\label{sec:framework} + +\subsection{The weighted base operator} + +Let $B=S^4\subset\R^5$, with round metric $g_0$ and round Riemannian +density $\mu$. Write a point of $B$ as $(x_1,x_2,y)$, where $y\in\R^3$, +and set $t=|y|^2$. A \emph{positive pair} consists of a smooth positive +definite symmetric contravariant tensor $A$ and a smooth function +$\rho>0$. Define a metric $G$ and a differential operator $L$ by +\begin{equation}\label{eq:weighted-operator} + G^{-1}=\rho^{-1}A,\qquad + Lf=\rho^{-1}\diver_\mu(A\,\dd f). +\end{equation} +Thus $-L$ is symmetric and nonnegative in $L^2(B,\rho\,\dd\mu)$, with +quadratic form +\[ + \mathfrak a(f,f)=\int_B A(\dd f,\dd f)\,\dd\mu . +\] +The tensor and density will be changed independently. Their conversion +to an ordinary metric occurs only at the end of the proof. + +Fix a finite smooth atlas with compact coordinate subpatches covering +$B$; all $C^r$ norms of functions and tensors may be taken in that atlas. +For a pair, the norm means the sum of the two component norms. +Equivalent fixed choices change constants but do not change the +assertions. All pairs used below lie in one sufficiently small $C^1$ +neighborhood of $(g_0^{-1},1)$, within the space of smooth positive +pairs. Accordingly $G$ and $G^{-1}$ have common $C^1$ bounds and common +ellipticity constants. Higher derivative bounds need only be finite for +each fixed input pair. + +Gradients, Hessians, lengths, and orthogonality in envelope and phase +estimates refer to $G$ unless another metric is displayed. Integration +in the elliptic operator refers to $\mu$; integration in a coordinate +corrugation chart refers to Lebesgue measure $\dd x$. +For a real smooth function $h$, set +\begin{equation}\label{eq:hessian-test} + \begin{split} + p&=\nabla_G h,\qquad Q=(1+|p|_G^2)^{1/2},\\ + J_h(e)&=(\nabla_G^2h)(p/Q,p/Q)+(\nabla_G^2h)(e,e), + \qquad |e|_G=1,\quad \dd h(e)=0 . + \end{split} +\end{equation} +At $p=0$ every unit vector is an admissible test. +The positivity conditions imposed below on $J_h$ supply a gap between +the envelope Hessian and the real part of a complex phase Hessian. +Lemma~\ref{lem:phase-hessian} and Proposition~\ref{prop:packets} use +this gap in the construction and localization of the packets. +Complex inner products +and tensor evaluations in the phase equations are extended +bilinearly; estimates of the size of a complex vector or matrix use +the ordinary Hermitian norm. + +We fix once and for all +\[ + \delta=10^{-6}. +\] +The numerical exponents below leave generous margins; we do not optimize +the final positive gain. +Positive constants written $c,C,C_r$ may change between occurrences. +Any additional dependencies are specified where their uniformity +matters. In particular, a finite approximation order is fixed before a +frequency tends to infinity. + +\subsection{A round region and sign certificates} + +For an integer $N>1$, let +\begin{equation}\label{eq:round-beam} + k=\sqrt{N(N+3)},\qquad + u_*=\operatorname{Re}(x_1+i x_2)^N . +\end{equation} +The Euclidean polynomial in \eqref{eq:round-beam} is homogeneous of +degree $N$ and harmonic in $\R^5$. The polar-coordinate formula +\[ + \Delta_{\R^5} + =\partial_r^2+\frac4r\partial_r+\frac1{r^2}\Delta_{g_0} +\] +therefore gives $-\Delta_{g_0}u_*=k^2u_*$ on $B$. +Where $t<1$, the angular coordinate $\vartheta$ in the +$(x_1,x_2)$-plane also gives +\[ + u_*=(1-t)^{N/2}\cos(N\vartheta). +\] +The round beam will be retained in part of a region where the input +pair is round. This region serves two purposes: it supplies an exact +solution to which the packets can be glued, and it leaves a patch where +a later tensor perturbation can make the eigenvalue simple while +preserving that solution. The construction uses both properties before +passing to the limiting operator. + +The envelope construction fixes a coordinate chart +$\Omega\subset\{t>1/2\}$ and an open region $V$ whose closure is compact +in $\Omega$. The initial corrugation cube is compactly contained in +$V$. Euclidean balls centered in $V$ with radii tending uniformly to +zero are consequently contained in $\Omega$. + +\begin{definition}\label{def:certificate} +A \emph{sign certificate} for a continuous real function $u$ in +$\Omega$ is a finite family of pairs $(D_i^-,D_i^+)$ of congruent +closed parallel three-dimensional Euclidean disks such that: +\begin{enumerate}[label=\textup{(\roman*)}] + \item each pair is the two ends of a straight closed cylinder + compactly contained in a Euclidean coordinate ball; + \item these coordinate balls are pairwise disjoint; + \item $u$ has strictly opposite signs on $D_i^-$ and $D_i^+$. +\end{enumerate} +The \emph{size} of the certificate is +$\sum_i\Hh_{\mathrm{Euc}}^3(D_i^+)$. +\end{definition} + +The signs in Definition~\ref{def:certificate} can be reversed on each +pair. Since the disks are compact and finite in number, a certificate +persists under sufficiently small uniform perturbations of $u$. +It is also preserved by multiplying $u$ by any nonzero real constant. +Every segment joining corresponding points of $D_i^-$ and $D_i^+$ +contains a zero. This elementary projection property will provide the +Hausdorff-measure bound for the final metric. + +\subsection{The one-step construction} + +\begin{proposition}\label{prop:step} +There exist a $C^1$ neighborhood $\mathcal U$ of the round pair +$(g_0^{-1},1)$ among smooth positive pairs, a chart $\Omega$ as above, +and a number $\gamma>0$ with the following property. +Fix a smooth pair $(A,\rho)\in\mathcal U$, numbers +\[ + \frac14\le l_-0$, there exist $c>0$ and +$N_0$ such that for every integer $N\ge N_0$, with $k$ as in +\eqref{eq:round-beam}, there are a smooth positive pair +$(\widehat A,\widehat\rho)\in\mathcal U$ and a nonzero real smooth +function $u$ satisfying: +\begin{enumerate}[label=\textup{(\roman*)}] + \item $\|(\widehat A-A,\widehat\rho-\rho)\|_{C^r}<\eta$, and + $(\widehat A,\widehat\rho)=(g_0^{-1},1)$ on $\{t0$ supplied by Proposition~\ref{prop:envelope}. Shrink the +neighborhood of positive pairs, if necessary, so that its associated +metrics belong to the neighborhood required there. Throughout a single +step the input pair is fixed. Constants may depend on this pair, on a +fixed positive collar width, and on fixed finite approximation orders. +Their frequency exponents, when specified below, do not have these +dependencies. + +\subsection{A weighted Gaussian construction} + +Fix $l_-,l_+$ as in Proposition~\ref{prop:step}, put $w=l_+-l_->0$, +and choose +\[ + s_i=l_-+iw/5\quad(1\le i\le4). +\] +For the allowed frequency $k=\sqrt{N(N+3)}$, define +\[ + b_N(t)=\frac{N}{2k}\log(1-t),\qquad t<1. +\] +Apply Proposition~\ref{prop:envelope} and choose the additive constant +in $h$ so that $h(s_2)=b_N(s_2)$. Here the notation $h(s_2)$ is +unambiguous because $h=-a_*t+\text{constant}$ on $t\le1/2$. +The constants added in this way are bounded, since +$s_2\in[1/4,1/2]$ and $N/k\le1$. For the fixed sufficiently small +seed slope $a_*$ and all sufficiently large $N$, +\[ + \frac{\dd}{\dd t}(h-b_N) + =-a_*+\frac{N}{2k(1-t)}\ge c_0>0 + \qquad(0\le t\le1/2), +\] +where $c_0$ can be fixed independently of the collar. Consequently, +\begin{equation}\label{eq:gluing-gaps} + h\le b_N-cw\quad(t\le s_1),\qquad + h\ge b_N+cw\quad(s_3\le t\le s_4) +\end{equation} +for a fixed $c>0$. + +Choose a smooth cutoff $0\le\beta\le1$ equal to one for $t\le s_3$ +and zero for $t\ge s_4$. Set +\begin{equation}\label{eq:gluing-weight} + W=e^{kh}+\beta(t)e^{kb_N(t)}. +\end{equation} +The second summand is extended by zero outside $t0$, +\begin{equation}\label{eq:beam-weighted-bounds} + \begin{split} + |D^q(\beta u_*)|&\le C_q k^qW,\\ + |D^q(L+k^2)(\beta u_*)|&\le C_{q,M}k^{-M}W,\\ + |\dd W|/W&\le Ck^2 . + \end{split} +\end{equation} +All derivatives here and below are in the fixed finite coordinate +systems. To justify the first bound, work on $t\le s_4<1/2$ and use +$u_*=(1-t)^{N/2}\cos(N\vartheta)$ in finitely many angular charts. +Each derivative costs at most a fixed constant times $k$. In the +cutoff region the exponential gap absorbs the cutoff derivatives. +The second bound follows in the same way from the commutator with +$\beta$. Finally, $|\dd h|\le Ck^{2\delta}$ and the same gap control +the derivative of \eqref{eq:gluing-weight}, giving its last bound. +For a fixed collar, all derivatives of $\beta$ are fixed constants; +more quantitatively they may be bounded by $C_qw^{-q}$. + +Choose a net in $\{t\ge s_1\}$ of mesh at most a fixed small multiple +of $k^{-10}$, with at most $Ck^{40}$ centers. At every center take the +fixed finite family of Proposition~\ref{prop:packets}. Denote the +resulting collection, counting packets with their center and family +index, by $\mathcal W_k$. Its cardinality is at most $Ck^{40}$. +For each $v\in\mathcal W_k$ take two independent standard real +Gaussian variables $g_v^{(1)},g_v^{(2)}$, all independent, and set +\begin{equation}\label{eq:gaussian-sum} + u=\beta u_*+ + \sum_{v\in\mathcal W_k} + \bigl(g_v^{(1)}\operatorname{Re}v+ + g_v^{(2)}\operatorname{Im}v\bigr). +\end{equation} +Let $\mathcal B_k$ be the event that all these coefficients have +absolute value at most $k$. A Gaussian tail estimate gives +\begin{equation}\label{eq:coefficient-event} + \Pp(\mathcal B_k^c)\le Ck^{40}e^{-k^2/2}\longrightarrow0. +\end{equation} + +Given a finite integer $J\ge2$ and a number $M>0$, choose the packet +orders so that their residuals through order $J$ decay at least as +$k^{-M-42}e^{kh}$. This choice is made before $k$ is taken large. +The packet estimates, \eqref{eq:beam-weighted-bounds}, and the +cardinality bound imply, on $\mathcal B_k$, +\begin{equation}\label{eq:sum-weighted-bounds} + \begin{split} + |D^qu|&\le C_q k^{2q+41}W\qquad(0\le q\le J),\\ + |D^q\mathfrak r|&\le C k^{-M}W\qquad(0\le q\le J),\\ + \mathfrak r&=\diver_\mu(A\,\dd u)+k^2\rho u . + \end{split} +\end{equation} +The multiplication by the fixed smooth $\rho$ is included in the +second estimate. In particular, increasing $J$ or $M$ does not change +the exponent $2q+41$ in the first estimate. + +Every packet is supported within distance $Ck^{-1/3-100\delta}$ of +its center. The function $t$ has bounded differential, so for a fixed +$C$ all packets vanish on +\begin{equation}\label{eq:untouched-region} + tk^{50}$, then +$\beta=1$ for large $k$: on its transition region +\eqref{eq:gluing-gaps} instead gives $W/e^{kh}\le1+e^{-cw k}$. +In the uncut region, the cosine formula and the angular derivative +give +\[ + |u_*|+|\dd u_*|_{g_0}\ge c e^{kb_N}. +\] +On $\mathcal B_k$, the packet value and first derivative together are +bounded by $Ck^{43}e^{kh}$. The beam therefore dominates their sum +on this region and gives a lower bound $cW$ for the first jet. +The first inequality in \eqref{eq:gluing-gaps} puts all of $t0$, independent of +$k$, such that for all sufficiently large allowed frequencies, +the probability that $u$ has a sign certificate in $\Omega$ of size +at least $c_*k^{1+\gamma}$ is at least $p_*$. The certificate can be +chosen from pairs of disks fixed deterministically before sampling +the Gaussian coefficients. +\end{lemma} + +\begin{proof} +For $x\in V$, put +\begin{equation}\label{eq:sign-radius} + R_x=(kQ_x)^{-1},\qquad Q_x=(1+|\nabla_G h(x)|_G^2)^{1/2}. +\end{equation} +All balls below are Euclidean coordinate balls. Their radii are at +most $k^{-1}$, so fixed enlargements lie in a common relatively +compact subset of $\Omega$ for large $k$. The envelope bounds give +$|DQ|\le Ck^{4\delta}$. If two candidate balls intersect, their +centers are at distance at most $2/k$, and their $Q$ values differ +by at most $Ck^{-1+4\delta}$. Since $Q\ge1$, the corresponding +radii are uniformly comparable. + +At each fixed $k$ the candidate radii have a positive lower bound. +Choose a finite maximal disjoint family of open balls +$\{B(x,R_x):x\in\mathcal X_k\}$, with centers in $V$. +Uniformly bounded enlargements of these balls cover $V$, and $Q$ is +comparable to $Q_x$ in each enlargement. Thus +\begin{equation}\label{eq:weighted-packing} + \int_V kQ(y)\,\dd y + \le C\sum_{x\in\mathcal X_k}kQ_xR_x^4 + =C\sum_{x\in\mathcal X_k}R_x^3, + \qquad + \sum_{x\in\mathcal X_k}R_x^3\ge c k^{1+\gamma}. +\end{equation} +The last inequality uses \eqref{eq:envelope-bounds}. + +Fix one ball. The beam vanishes on its radius-two enlargement for +large $k$, because $\overline V\subset\{t>1/2\}$ and $s_4<1/2$. +Define +\[ + S_x^2=\sum_{v\in\mathcal W_k}|v(x)|^2, + \qquad U_x(Y)=u(x+R_xY)/S_x,\qquad |Y|\le2. +\] +In expressions differentiated with respect to $Y$, a packet $v$ +means its composition $v(x+R_xY)$. +Lemma~\ref{lem:scaled-packets} applies to the entire packet +collection, including packets centered outside $V$ that meet this +ball. It gives $S_x\ge ce^{kh(x)}$; uniformly bounded sums of +normalized squared packet derivatives through every fixed order; +and, for every non-tail packet, +\begin{equation}\label{eq:local-imaginary-lower} + \left|\operatorname{Im}\frac{\dd_Yv}{v}(0)\right|\ge c. +\end{equation} +Here a non-tail has its cutoff identically one on the scaled ball +and nonzero value at the center. For tails, the lemma supplies +absolute exponential bounds, even if $v(x)=0$. Polynomially many +tails consequently contribute $o(1)$ to the normalized squared +value and derivative sums. In particular, +\begin{equation}\label{eq:non-tail-mass} + S_x^{-2}\sum_{v\ \mathrm{non\text{-}tail}}|v(x)|^2=1-o(1). +\end{equation} +All these statements are uniform over $x\in\mathcal X_k$. + +The value $U_x(0)$ is a standard real Gaussian. We verify that some +directional derivative has a uniformly positive conditional +variance given this value. Let $z$ be the coefficient-space vector +whose coordinates for packet $v$ are +$(\operatorname{Re}v(x),\operatorname{Im}v(x))$, and let $T$ be the +matrix whose corresponding columns are the four-dimensional real +derivative vectors $\dd_Y\operatorname{Re}v(0)$ and +$\dd_Y\operatorname{Im}v(0)$. Thus $|z|^2=S_x^2$. If $g$ is the +vector of independent standard Gaussian coefficients, the value and +gradient are $z^{\mathsf t}g/S_x$ and $Tg/S_x$. The conditional +gradient covariance is +\begin{equation}\label{eq:conditional-covariance} + S_x^{-2}T\left(I-\frac{zz^{\mathsf t}}{S_x^2}\right)T^{\mathsf t}. +\end{equation} +For each non-tail packet define the unit coefficient vector $c_v$ +supported on its two coordinates, with entries +\[ + \frac{(-\operatorname{Im}v(x),\operatorname{Re}v(x))}{|v(x)|}. +\] +These vectors are mutually orthonormal and perpendicular to $z$. +The orthogonal projection in \eqref{eq:conditional-covariance} +therefore dominates $\sum_v c_vc_v^{\mathsf t}$ as a nonnegative +quadratic form. Moreover +\[ + Tc_v=|v(x)|\operatorname{Im}\frac{\dd_Yv}{v}(0). +\] +Equations~\eqref{eq:local-imaginary-lower} and +\eqref{eq:non-tail-mass} show that the trace of +\eqref{eq:conditional-covariance} is at least a fixed $c>0$. +There is consequently a deterministic unit direction $e_x$ in +$\R^4$ such that the derivative +$Z_x=\partial_{e_x}U_x(0)$, conditioned on $U_x(0)$, has variance +between fixed positive constants. Its unconditional variance is +bounded above by the deterministic squared-derivative estimates. +Those estimates also bound the regression coefficient +$\operatorname{Cov}(Z_x,U_x(0))$. + +For clarity, the needed uniform control of the local $C^2$ norm +follows from finitely many derivatives. The deterministic estimates +give +\[ + \sup_{|Y|\le2}\E|D_Y^qU_x(Y)|^2\le C_q + \qquad(0\le q\le5). +\] +Integration and the Sobolev inequality on concentric balls in +$\R^4$ yield +\begin{equation}\label{eq:local-c2-moment} + \E\lVert U_x\rVert_{C^2(\overline{B(0,1)})}^2 + \le C\E\lVert U_x\rVert_{H^5(B(0,2))}^2\le C. +\end{equation} +We choose the $C^2$ norm to control the Euclidean gradient norm and +Hessian operator norm, changing only the fixed constant in this +inequality. The Sobolev embedding used here is valid since +$5>2+4/2$; see \cite[Sections~5.6--5.7]{Evans2010}. + +Let $a'=1/4$ and fix $00$. + +On that intersection Taylor's formula at $\pm a'e_x/H$ gives +opposite signs, with absolute values at least $a'/(2H)$: +the linear term has magnitude at least $a'/H$, the center value +has magnitude at most $c'/H$, and the quadratic error is at most +$a'^2/(2H)$. The gradient bound preserves these signs on the +parallel transverse disks of radius +\[ + \sigma_* = a'/(4H^2) +\] +centered at those two points. The whole closed cylinder between +the disks is compactly contained in the unit ball. The direction +$e_x$, centers and radii are deterministic functions of the packet +data. After scaling by $R_x$, they provide a potential certificate +pair inside $B(x,R_x)$. + +Let $F_k$ be the fraction of successful balls weighted by $R_x^3$. +No independence among these successes is claimed or needed. +Since $0\le F_k\le1$ and $\E F_k\ge\pi_*$, +\begin{equation}\label{eq:weighted-success} + \Pp(F_k\ge\pi_*/2)\ge\frac{\pi_*}{2-\pi_*}>0. +\end{equation} +The Euclidean three-area of one successful disk is +$\omega_3\sigma_*^3R_x^3$, where $\omega_3$ is the unit-ball +volume in $\R^3$. On the event in \eqref{eq:weighted-success}, +\eqref{eq:weighted-packing} therefore gives a certificate of size +at least $c_*k^{1+\gamma}$. Its cylinders lie in disjoint open +balls, as required by Definition~\ref{def:certificate}. This proves +the lemma with $p_* =\pi_* /(2-\pi_*)$. +\end{proof} + +Lemmas~\ref{lem:noncancellation} and~\ref{lem:signs} can be realized +simultaneously. For each fixed choice of the orders, first fix the +positive number $p_*$ in Lemma~\ref{lem:signs}, and then take $k$ +so large that the failure probability in +\eqref{eq:noncancellation} is less than $p_*/2$. The simultaneous +event has probability at least $p_*/2$. Thus, for every sufficiently +large allowed $k$, there is a real smooth $u$ satisfying +\eqref{eq:sum-weighted-bounds}, the pointwise lower bound in +\eqref{eq:noncancellation}, and a certificate of size +$c_*k^{1+\gamma}$. This probability subtraction is valid even if +$p_*$ is extremely small. The probability bounds above were obtained +without conditioning on $\mathcal B_k$; only deterministic derivative +estimates and interpolation used that event. + +\subsection{Exact coefficient correction} + +\begin{lemma}\label{lem:correction} +Let $(A,\rho)$ be a fixed smooth positive pair, let $u$ be real and +smooth, and let $W$ be a smooth positive weight. Suppose that +\[ + |u|+|\dd u|_{g_0}\ge k^{-500}W. +\] +Fix an integer $R\ge0$. Suppose that +\[ + |D^qu|\le C_qk^{2q+41}W\quad(0\le q\le R+2),\qquad + |D^q\mathfrak r|\le C_qk^{-M}W\quad(0\le q\le R+1), +\] +where $\mathfrak r=\diver_\mu(A\,\dd u)+k^2\rho u$. +There are smooth real corrections $(\delta A,\delta\rho)$ canceling +the eigenfunction residual exactly, and +\begin{equation}\label{eq:correction-size} + \lVert(\delta A,\delta\rho)\rVert_{C^R} + \le C_R k^{-M+2129+1088R}. +\end{equation} +The corrections vanish on every open set where $\mathfrak r=0$. +In particular, choosing $M\ge2131+1088R$ makes them tend to zero +in $C^R$ as $k\to\infty$. For sufficiently small corrections the +output pair is positive and $u$ is its exact eigenfunction at $k^2$. +\end{lemma} + +\begin{proof} +Write $I=g_0^{-1}$ and define globally +\begin{equation}\label{eq:correction-formula} + \begin{split} + \mathcal D&=u^2+I(\dd u,\dd u),\qquad F=-\mathfrak r/\mathcal D,\\ + \delta A&=FuI,\qquad + \delta\rho=k^{-2}\bigl(Fu-\diver_\mu(FI\,\dd u)\bigr). + \end{split} +\end{equation} +The lower bound on the first jet gives +$\mathcal D\ge\tfrac12 k^{-1000}W^2>0$, so these are smooth. +The product rule gives the exact identity +\begin{align*} + \diver_\mu(\delta A\,\dd u)+k^2\delta\rho\,u + &=u\diver_\mu(FI\,\dd u)+F I(\dd u,\dd u)\\ + &\quad+Fu^2-u\diver_\mu(FI\,\dd u)\\ + &=F\mathcal D=-\mathfrak r. +\end{align*} +Thus $\diver_\mu((A+\delta A)\dd u) ++k^2(\rho+\delta\rho)u=0$ identically. + +We include finite derivative bounds to specify the accuracy needed +in this step. For $0\le q\le R+1$, the hypotheses imply +\[ + |D^q\mathcal D|\le C_q k^{2q+86}W^2. +\] +Every term in a derivative of order $q$ of $\mathcal D^{-1}$ is a +constant times +$\mathcal D^{-m-1}\prod_{i=1}^m D^{q_i}\mathcal D$, where +$q_i\ge1$, $\sum_iq_i=q$, and $m\le q$; the order-zero case is +included separately. Hence +\[ + |D^q(\mathcal D^{-1})| + \le C_q k^{1000+2q+1086q}W^{-2} + =C_q k^{1000+1088q}W^{-2}. +\] +By the product rule, +\begin{equation}\label{eq:F-derivatives} + |D^qF|\le C_qk^{-M+1000+1088q}W^{-1} + \qquad(0\le q\le R+1). +\end{equation} +No derivatives of $W^{-1}$ are required for these estimates: the +displayed weights bound each differentiated factor before the +factors are multiplied. + +Multiplying \eqref{eq:F-derivatives} by derivatives of $u$ gives +$|D^q(Fu)|\le C_qk^{-M+1041+1088q}$. Replacing $u$ by $I\dd u$ +gives the bound $C_qk^{-M+1043+1088q}$, with the derivatives of +the fixed tensor $I$ absorbed in the constant. The divergence in +\eqref{eq:correction-formula} costs one further derivative, and +the density correction has the factor $k^{-2}$. These bounds give +\eqref{eq:correction-size}; its exponent for the density is +$1043+1088(R+1)-2=2129+1088R$. +The cancellation of the two weights is what permits arbitrarily +small coefficient changes even where the original function is +exponentially small. + +If $\mathfrak r=0$ on an open set, then $F=0$ there and both +corrections vanish. Finally, the cone of positive pairs is open +in $C^0$ on the compact base, so sufficiently small corrections +preserve positivity. +\end{proof} + +\subsection{Simplicity on a protected round patch} + +\begin{lemma}\label{lem:simplicity} +Suppose a smooth positive pair has a real eigenfunction $u$ at +$k^2=N(N+3)$, with $N>1$. Suppose there is an open region on which +the pair equals the round pair and $u=u_*$, containing a point with +$01$ and $t<1$, it follows +that $b''(u)=0$. Substitution back into +\eqref{eq:simplicity-composition} eliminates the constant term of +$b$, so $f=cu$ on $O_0$. + +The difference $f-cu$ satisfies a second-order elliptic equation +with smooth real positive definite principal coefficients and +smooth lower-order coefficients. Aronszajn's unique continuation +theorem applies: its local hypotheses allow principal coefficients +of class $C^2$ with Lipschitz second derivatives and bounded +lower-order coefficients, and smooth solutions satisfy its +solution hypotheses \cite[pp.~235--236]{Aronszajn1957}. Vanishing +on an open patch gives infinite-order vanishing there. On the +connected base the difference is therefore identically zero, +proving the assertion about the kernel. + +For $\tau>0$ replace $A$ by $A+\tau E$. The eigenfunction $u$ and +its eigenvalue remain exact because $E\dd u=0$ identically. We +claim that $k^2$ is simple for every sufficiently small positive +$\tau$. Otherwise there would be $\tau_m\downarrow0$ and real +eigenfunctions $f_m$ at $k^2$, normalized in $L^2(\rho\,\dd\mu)$ +and orthogonal to $u$. Their energy identity is +\[ + \int_B(A+\tau_m E)(\dd f_m,\dd f_m)\,\dd\mu=k^2, +\] +so uniform ellipticity bounds them in $H^1$. A subsequence +converges weakly in $H^1$ and strongly in $L^2$ to a normalized +function $f_0$ orthogonal to $u$. Passing to the weak equation +shows $f_0\in\mathcal E$; elliptic regularity makes it smooth. +For any fixed $\phi\in\mathcal E$, testing the two eigenfunction +equations against one another cancels the original form and gives +\[ + \int_B E(\dd f_m,\dd\phi)\,\dd\mu=0. +\] +Weak convergence passes this fixed bilinear identity to the limit. +Now choose $\phi=f_0$ to see that $f_0$ is in the kernel just +identified, contradicting its normalization and orthogonality. +This proves the claim. The tensor $E$ is fixed before choosing +$\tau$; hence $\tau E$ can meet every prescribed finite norm +tolerance, while remaining supported in $O$. +\end{proof} + +\subsection{Proof of the quantitative step} + +\begin{proof}[Proof of Proposition~\ref{prop:step}] +Fix the input pair, the collar, $r$ and $\eta$. Choose +$R\ge\max\{r,1\}$ and a smaller positive tolerance $\eta'$ so +that total $C^R$ movement less than $\eta'$ implies the requested +$C^r$ movement and keeps the pair positive and in $\mathcal U$. +Such a choice exists because the fixed input belongs to the open +set $\mathcal U$. + +Choose $J=R+2$ and $M\ge2131+1088R$, and then choose the finite +packet orders giving \eqref{eq:sum-weighted-bounds}. All these +choices precede frequency selection. Lemmas~\ref{lem:noncancellation} +and~\ref{lem:signs} give, for every sufficiently large allowed $k$, +a sample with the required first-jet bound, residual bounds and a +certificate of size $c_*k^{1+\gamma}$. Lemma~\ref{lem:correction} +changes the pair by at most $C_Rk^{-2}$ in $C^R$ while preserving +that same function. Choose $k$ large enough for this change to be +less than $\eta'/2$. + +We describe explicitly why a narrow collar only changes this +frequency threshold. Packet supports avoid the region below +$s_1-Ck^{-1/3-100\delta}$. Requiring +\begin{equation}\label{eq:collar-thresholds} + Ck^{-1/3-100\delta}(M'+P)\log k+\log C_q(w) +\end{equation} +protects a fixed subcollar and makes any required cutoff residual +$C_q(w)k^Pe^{-cw k}$ smaller than $k^{-M'}$. Only finitely many +orders $q$ and finite targets $M'$ are needed at this step. +For every already fixed $w>0$, all inequalities in +\eqref{eq:collar-thresholds} hold for every sufficiently large $k$. +In particular the exact corrections vanish below $s_1-w/100$, +where the coefficients remain round and $u=u_*$. + +In the subcollar +\[ + l_-+w/200$ below its simplicity threshold and so small that +$\lVert\tau E\rVert_{C^R}<\eta'/2$. This tensor can have large +finite derivatives depending on $k$; the subsequent scalar choice +of $\tau$ absorbs them. It preserves $u$, its exact eigenvalue and +all current signs exactly. Its support lies in $t>l_-$, so the +final pair is round on $t0$ such that every smooth positive pair +$\mathcal P'$ satisfying +\[ + \lVert\mathcal P'-\mathcal P\rVert_{C^s}\le\varepsilon +\] +has a simple eigenvalue in $I$ and a real eigenfunction with the +same certificate disks. Thus a closed ball defined by one finite +norm can be contained in a persistence neighborhood. +\end{lemma} + +\begin{proof} +We recall the spectral and regularity facts used here. The form +\[ + \int_B A(\dd f,\dd v)\,\dd\mu+ + \int_B\rho fv\,\dd\mu +\] +is coercive on $H^1(B)$ and defines the inverse of $-L+1$ in +$L^2(\rho\,\dd\mu)$. The compact embedding +$H^1\hookrightarrow L^2$ makes this inverse compact, and it is +selfadjoint. The spectral theorem and elliptic regularity give a +discrete nonnegative spectrum with finite multiplicities and smooth +real eigenfunctions. Its ordered eigenvalues obey the min--max +principle with Rayleigh quotient +\begin{equation}\label{eq:weighted-rayleigh} + \frac{\int_B A(\dd f,\dd f)\,\dd\mu} + {\int_B\rho f^2\,\dd\mu}. +\end{equation} + +Uniform multiplicative comparisons of nearby positive tensors and +densities compare \eqref{eq:weighted-rayleigh} for every $f$ and +therefore imply convergence of each ordered eigenvalue under +$C^0$ convergence of pairs. Choose a smaller interval about +$\lambda$, with closure in $I$, separating it from its neighboring +ordered eigenvalues. Their convergence implies that for sufficiently +close pairs the corresponding ordered eigenvalue is still simple +and lies in $I$. + +We also need uniform convergence of a corresponding normalized +eigenfunction, up to sign. Here only finitely many coefficient +derivatives are needed. For a fixed smooth elliptic $L$ on the +closed base, the local interior estimates, a finite partition of +unity and interpolation give +\begin{equation}\label{eq:elliptic-persistence-estimate} + \lVert f\rVert_{H^{a+2}} + \le C_a\bigl(\lVert Lf\rVert_{H^a}+\lVert f\rVert_{L^2}\bigr) + \qquad(a=0,1,2,\ldots). +\end{equation} +These are the usual interior elliptic estimates; the Sobolev and +compactness statements used here are described in +\cite[Sections~5.6--5.7 and~6.3]{Evans2010}. A sufficiently small +finite coefficient norm makes $L'-L$ small as an operator +$H^{a+2}\to H^a$ for each of finitely many specified $a$. Its +contribution in \eqref{eq:elliptic-persistence-estimate} can then +be absorbed on the left. Applying the resulting estimates at +$a=0,2$, together with the eigenfunction equation, gives a uniform +$H^4$ bound for eigenfunctions whose eigenvalues stay in the +chosen bounded interval and whose $L^2$ norms are bounded. +For example, requiring sufficiently small $C^4$ changes of the +pair is more than enough for these finitely many operator bounds: +the principal coefficients of $L$ involve $A,\rho$ without +derivatives, and its first-order coefficients involve at most +their first derivatives. + +Suppose that pairs converge in such a finite norm to $\mathcal P$. +Normalize their corresponding real eigenfunctions $f_m$ by +$\int\rho_m f_m^2\,\dd\mu=1$. Positivity bounds their ordinary +$L^2$ norms, so the preceding argument bounds them in $H^4$. +Rellich compactness and Sobolev embedding in dimension four give +a subsequence converging in $C^0$ (for example, first converge in +$H^3$, which embeds in $C^0$). Passing to the weak equation shows +that the limit is a normalized eigenfunction at $\lambda$ for +$\mathcal P$. By simplicity it equals the prescribed normalized +eigenfunction or its negative. + +The finite collection of strict signs on compact disks has a +positive minimum absolute margin after this normalization. Hence +all sufficiently close such eigenfunctions have the same opposite +signs on the same disks, allowing an overall change of sign. If +no sufficiently small finite-norm neighborhood had this property, +a sequence converging in that norm would contradict the preceding +compactness argument. Finally, decrease its radius to put a closed +finite-norm ball inside the neighborhood. This proves the lemma, +including the assertion for varying densities. +\end{proof} + +\subsection{One smooth limiting pair and the ordinary metric} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Use monotone fixed coordinate norms for pairs. Choose $b>0$ so +that the closed $C^1$ ball of radius $b$ about +$\mathcal P_0=(g_0^{-1},1)$ consists of positive pairs in +$\mathcal U$. Fix +\[ + l^{(j)}=\frac14+2^{-j-2}\quad(j\ge0),\qquad + \epsilon_0=\gamma/4>0. +\] +We construct pairs $\mathcal P_j=(A_j,\rho_j)$, frequencies $k_j$, +and certificates $\mathcal C_j$ inductively. Every output is round +on $t0$. +At stage $j$, these numbers are already fixed for $i2k_{j-1}$ when $j>1$, and also so large that +\[ + c_jk_j^{\gamma/2}>j. +\] +The output certificate then satisfies +\begin{equation}\label{eq:stage-certificate-size} + |\mathcal C_j|>j k_j^{1+\gamma/2}. +\end{equation} +Only after this output has been constructed do we select its own +$s_j,\varepsilon_j$ by Lemma~\ref{lem:persistence}. They constrain +subsequent stages, so no current frequency choice depends +circularly on its own future persistence radius. + +The increments are summable in every fixed $C^r$ norm: for all +large $j$, $r_j\ge r$ and their $C^{r_j}$ norms are less than +$2^{-j}$. Therefore the pairs converge to one smooth pair +$\mathcal P_\infty=(A_\infty,\rho_\infty)$. Their total $C^1$ +movement from the round pair is less than +$b\sum_{j\ge1}2^{-j-1}=b/2$, so the limit remains positive and +belongs to $\mathcal U$. For each fixed $i$, the same budgets give +\[ + \lVert\mathcal P_\infty-\mathcal P_i\rVert_{C^{s_i}} + \le\sum_{j>i}2^{-(j-i+1)}\varepsilon_i + =\varepsilon_i/2. +\] +The limit lies in every selected persistence neighborhood. Hence +there are real smooth eigenfunctions $v_i$ for the limiting base +operator, with the same certificate disks $\mathcal C_i$, and +\begin{equation}\label{eq:limiting-eigenvalue-intervals} + -L_\infty v_i=\lambda_i v_i,\qquad + k_i^2/2<\lambda_i<2k_i^2. +\end{equation} +The frequency separation makes these eigenvalue intervals disjoint +and tending to infinity. The functions $v_i$ are nonzero by their +normalization, or just by their strict signs. + +Set $G_\infty^{-1}=A_\infty/\rho_\infty$ and let +\[ + q=\frac{\rho_\infty\,\dd\mu}{\dd\operatorname{vol}_{G_\infty}}. +\] +This is a smooth positive function, the ratio of two smooth +positive base densities. We use the standard warped-product +realization of a weighted Laplacian +\cite[Section~4, equation~(2)]{CharalambousLu2015}, with a circle +as the fiber. On the fixed manifold $M=B\times S^1$ +define +\begin{equation}\label{eq:circle-metric} + g=G_\infty+q^2\dd\varphi^2, +\end{equation} +where $\dd\varphi^2$ is the standard unit-circle metric. The +manifold is smooth, closed and connected, of dimension five, and +\[ + \dd\operatorname{vol}_g + =q\,\dd\operatorname{vol}_{G_\infty}\,\dd\varphi + =\rho_\infty\,\dd\mu\,\dd\varphi. +\] +For a function $f$ pulled back from $B$, the coordinate divergence +formula gives +\begin{equation}\label{eq:circle-operator} + \Delta_g f + =\frac{1}{\rho_\infty\mu} + \partial_a\bigl(\rho_\infty\mu + G_\infty^{ab}\partial_b f\bigr) + =\rho_\infty^{-1}\diver_\mu(A_\infty\dd f) + =L_\infty f. +\end{equation} +Here $\mu$ in the coordinate expression denotes its positive +coordinate density. Thus the pullbacks $u_i$ of $v_i$ are exact +real Laplace--Beltrami eigenfunctions with eigenvalues +$\lambda_i$. + +It remains to convert their certificates into the claimed measure +bound. Fix a closed interval $I_0$ of positive length in a circle +coordinate chart. All certificate cylinders are in one fixed +compact subset of $\Omega$: their centers lie in $V\Subset\Omega$ +and their radii tend uniformly to zero, with the initial +frequencies chosen large enough for the same inclusion. On the +product of this compact set with $I_0$, coordinate and Riemannian +distances have fixed Lipschitz comparison constants. One may see +this directly by extending the coordinate functions smoothly with +cutoffs to the compact manifold; their gradients are then +uniformly bounded in the single fixed metric $g$. + +For one certificate cylinder $C_i^{(a)}$, let $D_i^{(a)}$ be one +of its transverse disks. The opposite endpoint signs imply that +every longitudinal segment through the disk contains a zero of +$v_i$. Therefore the subset +\[ + \Z(u_i)\cap(C_i^{(a)}\times I_0) +\] +projects onto $D_i^{(a)}\times I_0$ by coordinate orthogonal +projection in the base and the circle coordinate. The Lipschitz +constant of every such projection is bounded by one fixed $K$, +independent of the disk's radius and orientation. The defining +covering inequality for Hausdorff measure under a Lipschitz map +gives +\[ + \Hh_g^4\bigl(\Z(u_i)\cap(C_i^{(a)}\times I_0)\bigr) + \ge K^{-4}\Hh_{\mathrm{Euc}}^4(D_i^{(a)}\times I_0) + =K^{-4}|I_0|\Hh_{\mathrm{Euc}}^3(D_i^{(a)}). +\] +The closed cylinders lie in disjoint open balls. Their indicated +nodal subsets are consequently pairwise disjoint Borel sets, so +their Hausdorff measures add. Using +\eqref{eq:stage-certificate-size}, we obtain one constant $c>0$ +belonging to the final metric such that +\begin{equation}\label{eq:final-nodal-lower} + \Hh_g^4(\Z(u_i))\ge c|\mathcal C_i| + > c i k_i^{1+\gamma/2}. +\end{equation} +This argument uses continuity and signs only, without any +regularity assumption on the nodal set. + +Finally, \eqref{eq:limiting-eigenvalue-intervals} and +$\epsilon_0=\gamma/4$ imply +\[ + \lambda_i^{1/2+\epsilon_0} + \le2^{1/2+\epsilon_0}k_i^{1+\gamma/2}. +\] +Combining this with \eqref{eq:final-nodal-lower} yields +\[ + \frac{\Hh_g^4(\Z(u_i))}{\lambda_i^{1/2+\epsilon_0}} + \ge c\,2^{-1/2-\epsilon_0}i\longrightarrow\infty. +\] +Both $g$ and $\epsilon_0$ are fixed throughout this sequence, +which proves Theorem~\ref{thm:main}. +\end{proof} + +\begin{corollary} +For every integer $n>5$ there is a smooth closed connected +$n$-dimensional Riemannian manifold and a fixed $\epsilon_0>0$ +with the same divergence of nodal measure divided by +$\lambda^{1/2+\epsilon_0}$. +\end{corollary} + +\begin{proof} +Take the product of \eqref{eq:circle-metric} with any fixed smooth +closed connected manifold of dimension $n-5$, and pull back the +same eigenfunctions. The product Laplacian preserves their +eigenvalues. Multiply each of the projection certificates above +by a fixed compact coordinate box of positive $(n-5)$-volume in +the extra factor. The same Lipschitz projection argument, now in +dimension $n-1$, gives a fixed positive multiple of the preceding +certificate lower bound. The denominator and its fixed exponent +are unchanged, so the ratio still tends to infinity. +\end{proof} diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/introduction.tex b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..602476f97ba4616ce1f9b9dc3f862d25f6588ecb --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,141 @@ +\section{Introduction}\label{sec:introduction} + +Let $(M^n,g)$ be a smooth closed connected Riemannian manifold. We use +$\Delta_g=\diver_g\nabla_g$ and write +\[ + -\Delta_g u=\lambda u,\qquad + \Z(u)=\{x\in M:u(x)=0\}. +\] +Yau posed the nodal-size question in his problem collection +\cite{Yau1982}. In its smooth closed-manifold formulation, Yau's +nodal-set conjecture predicts that the $(n-1)$-dimensional +Hausdorff measure of $\Z(u)$ is bounded above and below by positive +constant multiples of $\sqrt\lambda$, with the constants depending only +on the fixed manifold and metric \cite[p.~162]{DonnellyFefferman1988}. +The upper assertion is required to +hold for every nonzero real eigenfunction, including all eigenfunctions +in a multiple eigenspace. + +We disprove that upper assertion in the smooth category. In fact, the +assertion that one has an upper bound with every positive power loss, +\begin{equation}\label{eq:near-sharp-target} + \Hh_g^{n-1}(\Z(u)) + \le C(M,g,\epsilon)\lambda^{1/2+\epsilon} + \qquad\text{for every }\epsilon>0, +\end{equation} +fails. + +\begin{theorem}\label{thm:main} +There exist a smooth Riemannian metric $g$ on $S^4\times S^1$, a number +$\epsilon_0>0$, and nonzero real smooth functions $u_j$ with positive +eigenvalues $\lambda_j$ such that +\[ + -\Delta_g u_j=\lambda_j u_j,\qquad + \lambda_j\longrightarrow\infty,\qquad + \frac{\Hh_g^4(\Z(u_j))} + {\lambda_j^{1/2+\epsilon_0}}\longrightarrow\infty. +\] +The metric and $\epsilon_0$ are fixed throughout the sequence. +\end{theorem} + +In particular, \eqref{eq:near-sharp-target} fails for every +$0<\epsilon\le\epsilon_0$ for the metric of Theorem~\ref{thm:main}. + +The theorem concerns ordinary Laplace--Beltrami eigenfunctions on one +closed connected manifold. Taking a product with a fixed closed manifold +gives the same conclusion in each dimension greater than five. The +construction makes no assertion about the corresponding upper bound in +dimensions three or four. + +In dimensions three and four, a companion paper constructs fixed smooth +metrics and real eigenfunction sequences with $\lambda\to\infty$ for which +the nodal measure divided by $\sqrt\lambda$ tends to infinity, without +asserting a fixed positive power-law excess +\cite[Theorems~1.1 and~1.2]{OpenAINodalQualitative2026}. + +\paragraph{Background.} +Donnelly and Fefferman proved the sharp two-sided nodal estimate for +real-analytic metrics \cite[Theorem~1.2]{DonnellyFefferman1988}. +For smooth metrics, Logunov proved the sharp lower bound +\cite[Theorem~1.1]{LogunovLower2018} and, in dimensions at least three, +a polynomial upper bound with a dimension-dependent exponent +\cite[Theorem~6.2]{LogunovUpper2018}. +Regularity assumptions between analyticity and smoothness also yield +stronger upper estimates: Hezari proved a $\lambda^{s/2}$ bound in +Gevrey class $s\ge1$, and a $\sqrt\lambda\log(\lambda+e)$ bound for a +specified quasianalytic class +\cite{Hezari2023}. +Those hypotheses do not cover arbitrary smooth metrics. +Theorem~\ref{thm:main} disproves the smooth upper-bound part of Yau's +conjecture and even the relaxed estimate \eqref{eq:near-sharp-target}; +the lower-bound results remain compatible with it. + +There is a complementary tradition of prescribing nodal geometry. +In dimensions at least three, Enciso and Peralta-Salas realize a smooth +closed connected orientable separating hypersurface as the nodal set of +a first nonconstant eigenfunction by choosing a metric +\cite{EncisoPeraltaSalas2015}. At high energy, Enciso, Peralta-Salas, +and Torres de Lizaur realize prescribed stable nodal components, up to +ambient diffeomorphism, on round spheres and flat tori, with a +localization restriction in the torus case +\cite{EncisoPeraltaSalasTorres2018}. Their construction already gives +arbitrarily high frequencies on fixed analytic metrics. The distinction +here is power-law excess in total nodal measure, which is not +furnished by those prescription theorems. + +\paragraph{The quantitative mechanism.} +We first work with a symmetric elliptic operator on the four-sphere +whose positive principal tensor and positive density can be varied +independently. A local corrugation increases the integral of the +gradient of a logarithmic envelope. More specifically, the part of a +parent cube prepared for another corrugation has more new gradient +integral than the entire old parent cube. Repetition at a fixed scale +ratio therefore gives a multiplicative gain. The derivatives of order +$r$ grow at most exponentially in $(r+1)$ times the number of +generations. Let $k$ denote the construction frequency, with target +eigenvalue $k^2$. Taking a fixed small multiple of $\log k$ generations +produces a smooth real envelope $h=h_k$ on the base and a fixed +coordinate region $V$ such that +\[ + \int_V\sqrt{1+|\nabla h|^2}\,\dd x\ge c k^\gamma +\] +for one fixed $\gamma>0$, while keeping every fixed derivative of $h$ +within a controlled power of $k$. + +Finite Taylor solutions of the complex eikonal and transport equations +then give localized packets beneath $e^{kh}$. They achieve any prescribed +finite residual accuracy while preserving fixed derivative exponents. +A real Gaussian combination has a quantitatively nonvanishing first +jet everywhere and opposite signs on many pairs of small disks. +The first property permits an explicit simultaneous correction of the +principal tensor and density to make the function an exact +eigenfunction. The second gives stable lower bounds for nodal measure +without a regularity assumption on the zero set. + +The distinction between constants and exponents is essential here. +At a given stage, constants and frequency thresholds may depend +arbitrarily badly on the required finite coefficient accuracy. +The exponent $\gamma$ does not; nor do the chart or coefficient +neighborhood. The positive constant in the nodal lower bound and the frequency threshold +may depend on the fixed input pair, collar, finite norm, and tolerance. +Consequently the construction can +meet every finite collection of earlier spectral-persistence tolerances +and still retain a fixed power gain. A summable iteration gives one +smooth limiting pair. Adding a warped circle realizes its weighted +operator as an ordinary Laplace--Beltrami operator. + +These steps also give reusable local facts: a corrugation with a +repeatable integral gain, a phase-Hessian construction valid at zero +real slope, a polynomial first-jet noncancellation argument, and an +exact tensor--density correction. Each is proved below. In particular, +the argument does not appeal to a qualitative existence theorem with an +uncontrolled frequency threshold. + +\paragraph{Organization.} +Section~\ref{sec:framework} states the quantitative one-step construction +and fixes the geometric notation. +Section~\ref{sec:envelopes} proves the envelope estimates. +Section~\ref{sec:waves} constructs and estimates the local packets. +Section~\ref{sec:global} produces exact eigenfunctions, proves the +one-step construction, and passes to the fixed metric in +Theorem~\ref{thm:main}. diff --git a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/waves.tex b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/waves.tex new file mode 100644 index 0000000000000000000000000000000000000000..2c276d51d5ef801e72f269511e04921f849aa22d --- /dev/null +++ b/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/build/sections/waves.tex @@ -0,0 +1,708 @@ +\section{Local packets below an envelope}\label{sec:waves} + +Throughout this Section the smooth positive pair \((A,\rho)\), and hence +\(G\) and \(L\), is fixed. The parameter \(k\) tends to infinity, and +\(h=h_k\) satisfies \eqref{eq:envelope-bounds}. Constants may depend on +the fixed pair, the constants in those bounds, and finitely many +construction orders. They are independent of \(k\) and of the packet +center. A lower threshold for \(k\) may depend on the same data. + +The phase, transport amplitudes, and cutoff follow the complex WKB +construction of localized quasimodes; compare +\cite[Section~3]{DenckerSjostrandZworski2004}. Here the envelope varies +with the frequency. We therefore construct finite Taylor solutions and +track their dependence on its derivatives and Hessian margin, keeping +the support, derivative, and real-jet exponents fixed as the residual +accuracy increases. + +We use complex bilinear extensions of real quadratic forms. When the +norm of a complex vector is used instead, it is the Hermitian norm. +In ordinary coordinate derivative estimates, a fixed finite atlas +with compactly contained subcharts is understood. + +\subsection{The phase Hessian} + +We first explain how the envelope test \(J_h\) enters the phase equation. +At a center \(z\), use \(G\)-normal coordinates and write +\(p=\nabla_Gh(z)\), \(Q=(1+|p|^2)^{1/2}\), and +\(H=\nabla_G^2h(z)\). For a unit vector \(v\perp p\), the complex +vector +\[ + P=p+iQv +\] +satisfies \(P\cdot P=-1\). A phase \(\phi\) with +\(\phi(z)=h(z)\) and \(\dd\phi(z)=P\) therefore satisfies the +constant term of the eikonal equation +\(G^{ij}\partial_i\phi\,\partial_j\phi=-1\). Its first-order terms +require \(SP=0\), where \(S=\partial^2\phi(z)\): the first metric +derivatives vanish in normal coordinates. A positive gap +\(H-\operatorname{Re}S\) then places \(\operatorname{Re}\phi\) below +\(h\) to quadratic order and supplies localization. + +Write \(p=|p|e_1\) and \(b=|p|/Q\), choosing any unit +\(e_1\perp v\) when \(p=0\). For a symmetric matrix \(S=U+iD\), +the identity \(S(be_1+iv)=0\) forces +\[ + b^2U(e_1,e_1)+U(v,v)=0. +\] +Consequently, the corresponding test on the required gap is +\[ + b^2(H-U)(e_1,e_1)+(H-U)(v,v) + =b^2H(e_1,e_1)+H(v,v)=J_h(v). +\] +This explains the positivity condition imposed on the envelope. The +following lemma constructs a compatible Hessian with a quantitative +gap, including when the real phase gradient vanishes. + +\begin{lemma}[A dominated characteristic Hessian]\label{lem:phase-hessian} +Let \(e_1,v\) be orthonormal vectors in \(\R^4\), let \(0\le b<1\), and +let \(H\) be a real symmetric matrix such that +\[ + |H|\le k^{4\delta},\qquad + J'=b^2H(e_1,e_1)+H(v,v)\ge k^{-2\delta}. +\] +For all sufficiently large \(k\), there are real symmetric matrices +\(U,D\) satisfying +\begin{equation}\label{eq:phase-hessian} + (U+iD)(b e_1+i v)=0,\qquad + H-U\ge k^{-3\delta}I,\qquad |U+iD|\le k^{20\delta}. +\end{equation} +The threshold can be chosen independently of \(H,e_1,v,b\) under these +hypotheses. +\end{lemma} + +\begin{proof} +For \(b>0\), the first identity in \eqref{eq:phase-hessian} is equivalent +to +\[ + Dv=bUe_1,\qquad De_1=-Uv/b. +\] +The two prescribed columns are compatible with symmetry precisely when +\begin{equation}\label{eq:waves-column-compatibility} + b^2U(e_1,e_1)+U(v,v)=0. +\end{equation} + +First suppose that \(b\ge k^{-12\delta}\), and set +\[ + c_0=\frac{J'}{1+b^2},\qquad U=H-c_0I. +\] +Then \eqref{eq:waves-column-compatibility} holds. To give the extension +of \(D\) explicitly, write its blocks relative to +\(\R e_1\oplus\R v\oplus E\), where \(E=\{e_1,v\}^{\perp}\). +Denote the corresponding blocks of \(U\) by +\(u_{11},u_{12},u_{1E},u_{2E},\ldots\). Take +\[ + D= + \begin{pmatrix} + -u_{12}/b & b u_{11} & -u_{2E}/b\\ + b u_{11} & b u_{12} & b u_{1E}\\ + -u_{E2}/b & b u_{E1} & 0 + \end{pmatrix}. +\] +The identity \(u_{22}=-b^2u_{11}\) verifies both required columns. +Moreover, +\[ + H-U=c_0I\ge\tfrac12k^{-2\delta}I,\qquad + |U|\le Ck^{4\delta},\qquad |D|\le Ck^{16\delta}. +\] +These imply \eqref{eq:phase-hessian} for large \(k\). + +Now suppose that \(0\le b0\). Finite +construction orders can be chosen, before taking \(k\) large, so that +every center \(z\) with \(t(z)\ge1/4\) has a family \(\mathcal W_z\) of +four smooth complex packets with the following properties. + +Each packet is supported in an ordinary coordinate ball about \(z\) +of radius at most \(Ck^{-1/3-100\delta}\). For every fixed +nonnegative integer \(j\), +\begin{equation}\label{eq:packet-derivatives} + |D_x^j w(x)|\le C_j k^{2j}e^{kh(x)}. +\end{equation} +For the prescribed orders, +\begin{equation}\label{eq:packet-residual} + |D_x^j(L+k^2)w(x)|\le C e^{kh(x)}k^{-M}, + \qquad 0\le j\le J. +\end{equation} +The center jets and the nearby jets satisfy +\begin{equation}\label{eq:jet-span} + \sigma_{\min}(\mathcal J_z(z))\ge c k^{-3/4},\qquad + \sigma_{\min}(\mathcal J_z(x))\ge k^{-1} + \quad\text{if }\operatorname{dist}(x,z)\le k^{-10}, +\end{equation} +for sufficiently large \(k\). Any fixed smooth reference distance +may be used here. The same conclusion holds if the latter radius +is multiplied by a fixed constant. + +The exponents in the support, derivative, and jet bounds are +independent of \(J,M\). Constants and lower frequency thresholds may +depend on the chosen finite orders. +\end{proposition} + +\begin{proof} +We divide the construction into the selection of jets, finite +polynomial equations, and localization estimates. + +\paragraph{Normal coordinates and the linear jets.} +For each center choose a \(G\)-orthonormal frame and use normal +coordinates \(\xi\) in that frame. Put +\[ + p=\nabla h(z),\qquad Q=Q_z,\qquad H=\nabla^2h(z). +\] +In these coordinates the tangent and cotangent components agree at +the center. The envelope bounds, with a little slack in their +exponents, give +\begin{equation}\label{eq:waves-center-bounds} + Q\le k^{3\delta},\qquad |H|\le k^{4\delta} +\end{equation} +for large \(k\). + +If \(p\ne0\), set \(e_1=p/|p|\), and choose a favorable unit test \(e_2\) +from \eqref{eq:envelope-bounds}. If \(p=0\), first choose such an +\(e_2\), then choose any unit \(e_1\perp e_2\). +Complete to an orthonormal basis \(e_1,e_2,e_3,e_4\), and put +\[ + \eta_k=k^{-9\delta},\qquad + v_2=e_2,\quad + v_3=\frac{e_2+\eta_k e_3}{\sqrt{1+\eta_k^2}},\quad + v_4=\frac{e_2+\eta_k e_4}{\sqrt{1+\eta_k^2}}. +\] +These three vectors span \(e_1^\perp\) with least spanning singular +value at least \(c\eta_k\). + +Use the four pairs +\[ + (p',v)=(p,v_2),(p,v_3),(p,v_4), + \bigl(p+Qk^{-3/4}e_1,v_2\bigr). +\] +For each pair define +\[ + Q'=\sqrt{1+|p'|^2},\qquad P=p'+iQ'v,\qquad b=|p'|/Q'. +\] +Then \(p'=|p'|e_1\), \(P\cdot P=-1\), and +\begin{equation}\label{eq:waves-imaginary-size} + 1\le Q'/Q\le1+k^{-3/4}. +\end{equation} +Changing \(e_2\) to a \(v_j\) changes its Hessian value by at most +\(Ck^{-5\delta}\). If \(a=|p|\), the function \(a^2/(1+a^2)\) +changes by at most \(2k^{-3/4}\) when \(a\) is increased by +\(Qk^{-3/4}\). Indeed its derivative at \(s\ge a\) is at most +\(2(1+a^2)^{-3/2}\). The additional error in the Hessian test is +at most \(Ck^{-3/4+4\delta}\). Both errors are negligible compared +with the original \(ck^{-\delta}\) margin. Consequently +\[ + b^2H(e_1,e_1)+H(v,v)\ge k^{-2\delta} +\] +for every selected pair. Apply Lemma~\ref{lem:phase-hessian} to obtain +\(U,D\). Multiplying its characteristic identity by \(Q'\) gives +\((U+iD)P=0\). + +\paragraph{Frozen charts and scaling.} +The choice of frame may depend on \(k\), but each packet uses one +fixed chart when spatial derivatives are taken. More explicitly, if +\(E\) is its chosen frame, the chart map is +\[ + \Psi_{z,E}(\xi)=\exp_z\left(\sum_i\xi_iE_i\right). +\] +For each fixed derivative order \(q\), the derivatives of these maps +and their inverses are uniformly bounded over all centers and +orthonormal frames on sufficiently small common coordinate balls. +This follows from smoothness and compactness of the orthonormal +frame bundle of the fixed metric. In particular, the chain rule +gives +\[ + |D_\xi^q(h\circ\Psi_{z,E})| + \le C_q(G)\sum_{\ell=1}^q\|h\|_{C^\ell} + \le C'_q k^{\delta(q+1)}. +\] +There are no spatial derivatives of the rule used to select \(E\). + +Set +\begin{equation}\label{eq:waves-scaling} + r_0=k^{-100\delta},\qquad X=\xi/r_0,\qquad + \kappa=kr_0Q,\qquad + h_{\rm sc}(X)=\frac{h(\Psi_{z,E}(r_0X))-h(z)}{r_0Q}. +\end{equation} +For every fixed \(q\ge2\), +\begin{equation}\label{eq:waves-scaled-envelope} + |D_X^q h_{\rm sc}| + \le C_q Q^{-1}k^{\delta(101-99q)}. +\end{equation} +All these derivatives are bounded independently of \(k,z\). The +constant may depend on \(q\) and on high derivatives of the fixed +metric. + +In normal coordinates write +\(L=g^{ij}(\xi)\partial_{\xi_i\xi_j} ++\beta^i(\xi)\partial_{\xi_i}\). +After multiplication by \(r_0^2\), the operator \(L+k^2\) becomes +\[ + \mathcal L+\kappa^2/Q^2,\qquad + \mathcal L=g^{ij}(r_0X)\partial_{X_iX_j} + +r_0\beta^i(r_0X)\partial_{X_i}. +\] +Its coefficients have uniformly bounded derivatives of every fixed +order on a fixed \(X\)-ball. The normal-coordinate identities are +\(g^{ij}(0)=\delta_{ij}\) and +\(\partial_{\xi_\ell}g^{ij}(0)=0\). + +\paragraph{Finite eikonal and transport equations.} +Choose an integer \(T\ge3\), to be specified below. We construct a +complex polynomial \(\Phi\) with +\[ + \Phi(0)=0,\qquad + \partial_X\Phi(0)=P/Q,\qquad + \partial_X^2\Phi(0)=r_0(U+iD)/Q +\] +such that +\[ + E_\Phi(X) + =g^{ij}(r_0X)\partial_i\Phi\,\partial_j\Phi+Q^{-2} +\] +vanishes to order \(T\) at zero. Here \emph{vanishes to order \(T\)} +means that all derivatives of order less than \(T\) vanish. +The constant equation follows from \(P\cdot P=-1\), and the linear +equations follow from \((U+iD)P=0\) and the vanishing first derivatives +of the metric. + +At homogeneous degree \(m\ge2\), the as-yet unspecified homogeneous +term \(\Phi_{m+1}\) enters the equation through +\(2(P/Q)\cdot\partial\,\Phi_{m+1}\). All other terms have already +been determined. The Hermitian norm of \(P/Q\) lies between \(1\) +and \(3\) for large \(k\), by \eqref{eq:waves-imaginary-size}. +Lemma~\ref{lem:polynomial-inverse} therefore solves each equation +with a uniformly bounded right inverse. Solving through degree +\(T-1\) gives the desired polynomial. Its coefficients are bounded +for each fixed \(T\): the prescribed Hessian is bounded by +\(k^{-80\delta}\), the coefficient jets of \(\mathcal L\) are bounded, +and only finitely many bounded algebraic operations are used. + +Define +\[ + \mathcal T a + =2g^{ij}(r_0X)(\partial_i\Phi)(\partial_ja) + +(\mathcal L\Phi)a. +\] +Fix an integer \(s\ge0\). Choose polynomials \(a_0,\ldots,a_s\) such +that +\[ + a_0(0)=1,\qquad a_j(0)=0\quad(j\ge1), +\] +and each of +\[ + \mathcal Ta_0,\qquad + \mathcal Ta_j+\mathcal La_{j-1}\quad(1\le j\le s) +\] +vanishes to order \(T\) at zero. To see this directly, solve +\(\mathcal Ta=f\) by homogeneous degrees. The new amplitude term +of degree \(m+1\) enters the degree-\(m\) equation through +\(2(P/Q)\cdot\partial\); the remaining terms use only coefficients +already chosen. Lemma~\ref{lem:polynomial-inverse} again applies. +First construct the entire \(a_0\), then \(a_1\), and so on. +Thus the second derivative in \(\mathcal La_{j-1}\) causes no circular +dependence. Constants at zero are free, as stipulated above. +For fixed \(T,s\), all polynomial coefficients are bounded uniformly +in \(k,z\). No bound uniform as \(T,s\to\infty\) is asserted or needed. + +Choose once a real smooth cutoff \(\omega\), equal to one on +\(\{|X|\le1\}\), supported in \(\{|X|<2\}\), and taking values in +\([0,1]\). Define +\begin{equation}\label{eq:packet-definition} + \begin{split} + a(X)&=\sum_{j=0}^s\kappa^{-j}a_j(X),\\ + w(\Psi_{z,E}(r_0X)) + &=e^{kh(z)}e^{\kappa\Phi(X)}a(X)\omega(k^{1/3}X), + \end{split} +\end{equation} +and extend \(w\) by zero. For large \(k\), its support is compactly +contained in the normal chart, so this extension is smooth and has +the asserted support radius. On \(|X|\le2k^{-1/3}\), +\begin{equation}\label{eq:waves-amplitude} + a(X)=1+O(k^{-1/3}),\qquad |a(X)|\ge\tfrac12, +\end{equation} +where the constant depends on \(T,s\). This follows from the chosen +constant terms and \(\kappa\ge1\). + +\paragraph{Localization.} +The linear part of +\(\operatorname{Re}\Phi-h_{\rm sc}\) is either zero or +\(k^{-3/4}e_1\cdot X\). Lemma~\ref{lem:phase-hessian}, +\eqref{eq:waves-center-bounds}, and +\eqref{eq:waves-scaled-envelope} imply +\begin{equation}\label{eq:waves-localization} + \operatorname{Re}\Phi(X)-h_{\rm sc}(X) + \le k^{-3/4}|X|-c k^{-106\delta}|X|^2+C|X|^3. +\end{equation} +The third derivative bounds used here are uniform for each fixed +construction order. On the packet support the cubic term can be +absorbed into half the quadratic deficit, since +\(k^{-1/3+106\delta}\to0\). +The maximum of the remaining linear term minus the quadratic term +is \(O(k^{-3/2+106\delta})\). Also +\[ + k^{1-100\delta}\le\kappa\le k. +\] +Consequently +\[ + |e^{kh(z)+\kappa\Phi(X)}| + \le 2e^{kh(\Psi_{z,E}(r_0X))} + \quad (|X|\le2k^{-1/3}) +\] +for large \(k\). + +On the smaller annulus +\(\tfrac14k^{-1/3}\le|X|\le2k^{-1/3}\), the negative exponent has +magnitude at least \(ck^{1/3-206\delta}\), whereas the positive linear +exponent is at most \(Ck^{-1/12}\). Since +\(1/3-206\delta>1/4\), a larger threshold gives +\begin{equation}\label{eq:waves-exponential-tail} + |e^{kh(z)+\kappa\Phi(X)}| + \le e^{-2k^{1/4}}e^{kh(\Psi_{z,E}(r_0X))} + \quad\left(\tfrac14k^{-1/3}\le|X|\le2k^{-1/3}\right). +\end{equation} + +\paragraph{Residual and derivative estimates.} +For the uncut expression, direct differentiation gives +\[ + \begin{split} + e^{-\kappa\Phi} + (\mathcal L+\kappa^2/Q^2)(e^{\kappa\Phi}a) + ={}&\kappa^2E_\Phi a+\kappa\mathcal Ta_0\\ + &+\sum_{j=1}^s\kappa^{1-j} + (\mathcal Ta_j+\mathcal La_{j-1}) + +\kappa^{-s}\mathcal La_s. + \end{split} +\] +Taylor's theorem bounds a remainder vanishing to order \(T\), after +\(q\le J\le T\) derivatives in \(X\), by \(C|X|^{T-q}\). +On the support this is at most \(Ck^{-(T-q)/3}\). +Each ordinary derivative of such a remainder therefore has a cost +at most \(k^{1/3+100\delta}\), relative to its zeroth-order Taylor +bound. A derivative falling on the exponential, amplitude, or cutoff +has a cost at most a fixed constant times +\[ + r_0^{-1}(1+\kappa+k^{1/3})\le Ck^{1+3\delta}3(M+200\delta+2J+3),\qquad + s>\frac{M+200\delta+2J+1}{1-100\delta}. +\] +These choices are made before the frequency threshold. +Increasing that threshold absorbs all their finite constants and +proves \eqref{eq:packet-residual}. The same derivative calculation +without the operator factor proves \eqref{eq:packet-derivatives} for +every fixed \(j\), with exponent \(2j\) independent of \(T,s\). + +\paragraph{Real jet spanning.} +At the center, \(a(0)=1\) and the cutoff is constant, so +\[ + \mathbf j_z w(z)=\left(1,\frac{P}{Q}\right)+O(\kappa^{-1}). +\] +The error is bounded by \(Ck^{-1+100\delta}\). +Before this error, the real parts contain the two vectors +\[ + (1,(|p|/Q)e_1),\qquad + (1,(|p|/Q+k^{-3/4})e_1). +\] +Their coefficients are bounded, and their least spanning singular +value on the value--\(e_1\) plane is at least \(ck^{-3/4}\). +The imaginary parts span the orthogonal \(e_1^\perp\) slope space +with least singular value at least \(ck^{-9\delta}\). +Together their least spanning singular value is at least +\(ck^{-3/4}\). Since +\(k^{-1+100\delta}=o(k^{-3/4})\), the actual center jets retain this +bound, with a smaller constant. The argument applies without +change at \(p=0\). + +Finally, \eqref{eq:packet-derivatives} through order two and the +first envelope derivative bound give +\[ + |D_x(\mathbf j_z w(x))|\le Ck^3 +\] +near the center. In differentiating this expression \(Q_z\) remains +fixed, while differentiating \(e^{-kh(x)}\) costs +\(k|D_xh|\le Ck^{1+2\delta}\). +A displacement of \(O(k^{-10})\) therefore changes the jet matrix +by \(O(k^{-7})\). Uniformly bounded changes between ordinary and +normal coordinates preserve these estimates. +This is negligible relative to \(k^{-3/4}\), and proves the second +inequality of \eqref{eq:jet-span}, after increasing the threshold. +\end{proof} + +\subsection{Deterministic estimates at the local oscillation scale} + +The next Lemma supplies estimates for an arbitrary polynomial-sized +collection of the preceding packets. It includes packets whose +centers lie outside the region in which signs will be measured. +Use the fixed chart \(\Omega\subset\{t>1/2\}\) and the open set +\(V\Subset\Omega\) from the preceding Sections. Points of \(\Omega\) +are identified with their coordinate vectors. + +\begin{lemma}[Packets on a wavelength ball]\label{lem:scaled-packets} +Fix the finite construction orders in Proposition~\ref{prop:packets} +and a fixed finite number \(K_1\ge0\). Let \(\mathcal W\) be the collection of +all four packets from each of finitely many centers in +\(\{t\ge1/4\}\), with \(|\mathcal W|\le Ck^{K_1}\). +Suppose that for every \(x\in V\) some center is within +\(C_0k^{-10}\) of \(x\), where \(C_0\) is fixed. Set +\[ + R_x=(kQ_x)^{-1},\qquad + S_x^2=\sum_{w\in\mathcal W}|w(x)|^2,\qquad + w_x(Y)=w(x+R_xY),\quad |Y|\le2. +\] +The coordinate ball in this definition lies in \(\Omega\) for large +\(k\). + +Consider the packets whose supports meet this closed ball. Their +centered scaled coordinates \(X_z\) are defined on the whole ball. +Partition them into +\[ + \mathcal N_x=\{w:|X_z(x)|<\tfrac12k^{-1/3}\},\qquad + \mathcal T_x=\{w:|X_z(x)|\ge\tfrac12k^{-1/3}\}. +\] +For every fixed nonnegative integer \(j\), uniformly in \(x\in V\), +\begin{equation}\label{eq:scaled-packet-bounds} + \begin{aligned} + S_x&\ge c e^{kh(x)},\\ + \sup_{|Y|\le2}|D_Y^j w_x(Y)| + &\le C_j|w(x)| &&(w\in\mathcal N_x),\\ + \sup_{|Y|\le2}|D_Y^j w_x(Y)| + &\le C_j e^{-k^{1/4}}e^{kh(x)} + &&(w\in\mathcal T_x). + \end{aligned} +\end{equation} +Every packet in \(\mathcal N_x\) has cutoff identically one and +nonzero amplitude on the ball. Moreover, +\begin{equation}\label{eq:imaginary-gradient} + \left|\operatorname{Im} + \left(\frac{\dd_Yw_x}{w_x}\right)(0)\right|\ge c + \qquad(w\in\mathcal N_x). +\end{equation} +In particular, +\begin{equation}\label{eq:waves-square-sums} + \begin{aligned} + \frac1{S_x^2}\sum_{w\in\mathcal W} + \sup_{|Y|\le2}|D_Y^jw_x(Y)|^2&\le C_j,\\ + \frac1{S_x^2}\sum_{w\in\mathcal N_x}|w(x)|^2&=1-o(1). + \end{aligned} +\end{equation} +The constants and thresholds may depend on the fixed construction +orders and on \(K_1,C_0\), but not on \(k,x\). +\end{lemma} + +\begin{proof} +The bounds on the first two derivatives of \(h\) imply +\[ + |D_xQ|\le Ck^{4\delta} +\] +in the fixed chart. A packet centered at \(z\) that meets the ball +about \(x\) has +\[ + |z-x|\le Ck^{-1/3-100\delta}+C/k. +\] +For large \(k\), such a \(z\), and the whole ball, belong to a +compactly contained part of \(\Omega\). Normal coordinates centered +at \(z\) are defined throughout the ball. The preceding estimates +give +\[ + |Q_z-Q_x| + \le Ck^{-1/3-96\delta}+Ck^{-1+4\delta}=o(1). +\] +In particular, \(Q_z/Q_x\) is bounded above and below independently +of the packet. The same comparison holds for \(Q\) throughout the +ball. Since \(|\dd h|\le CQ\) in the coordinate chart, +\begin{equation}\label{eq:waves-local-weight} + k|h(x+R_xY)-h(x)|\le C,\qquad |Y|\le2. +\end{equation} + +The nearest center has +\(|X_z(x)|=O(k^{-10+100\delta})\). +The center formula and its nearby-jet estimate in the proof of +Proposition~\ref{prop:packets} show that one of its packets has +\(|w(x)|\ge c e^{kh(x)}\). This proves the first line of +\eqref{eq:scaled-packet-bounds}. + +Write \(\kappa_z=kr_0Q_z\). For every fixed \(q\ge1\), the chain +rule and normal-chart bounds give +\[ + |D_Y^qX_z|\le C_qr_0^{-1}R_x^q,\qquad + \kappa_z|D_Y^qX_z| + \le C_q\,\frac{Q_z}{Q_x}(kQ_x)^{1-q}\le C_q. +\] +For \(q=1\) there is also a lower bound: the matrix +\begin{equation}\label{eq:waves-scaled-jacobian} + \kappa_z D_YX_z=(Q_z/Q_x)D_x\xi_z +\end{equation} +has inverse bounded independently of \(k,x,z\). +In particular, +\[ + |X_z(x+R_xY)-X_z(x)| + \le Ck^{-1+100\delta}/Q_x=o(k^{-1/3}). +\] + +Suppose first that \(w\in\mathcal T_x\). +At every point where its cutoff or any derivative of the cutoff can +contribute, +\[ + \tfrac14k^{-1/3}\le |X_z|\le2k^{-1/3}. +\] +Apply \eqref{eq:waves-exponential-tail}. Each fixed positive +\(Y\)-derivative of \(\kappa_z\Phi(X_z)\) is bounded: in a term +containing \(m\) derivatives of \(X_z\), the displayed derivative +bounds leave a factor \(\kappa_z^{1-m}\le1\). +Amplitude derivatives are bounded as well. +Derivatives of \(\omega(k^{1/3}X_z)\) are bounded because +\(k^{1/3}/\kappa_z\le k^{-2/3+100\delta}\le1\). +Thus differentiation introduces only fixed constants at each fixed +order. Together with \eqref{eq:waves-local-weight}, this proves +the tail bound in \eqref{eq:scaled-packet-bounds}. +This argument uses an absolute estimate and is valid even if +\(w(x)=0\). + +For \(w\in\mathcal N_x\), the displacement estimate shows that +\(|X_z|0$ and let $U$ be a prescribed smooth divergence-free velocity +on $[0,\infty)\times\mathbb R^3$, with $U(0)=0$. On each finite interval, +assume $U\in CH^2\cap C^1L^2$ and $U,\nabla U$ are bounded. Put +$f=\mathcal F_\nu[U]$. Then $(U,0)$ is a solution. Its velocity is unique +in each of the following separately stated comparison classes, with the +same force and initial data: +\begin{enumerate}[label=(\roman*),itemsep=2pt] +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v$, and a +pressure representative $p\in CH^1$. +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, +and an $H^1$ pressure representative at each time, with no required time +continuity in that pressure norm. +\item Strong solutions with $v\in CH^4\cap C^1H^2$ and distributional +$\nabla p\in CL^2$. +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, +and a pressure representative $p\in CL^2$. +\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$, +without an additional integrability or normalization condition on pressure. +\end{enumerate} +The pressure gradient is unique. An $L^2$ pressure representative, when +specified, is zero; otherwise pressure is determined up to a function of +time. The reference material flow is a smooth diffeomorphism at every +finite time and exists for every finite forward time. + +On a flat torus $\T_L^3$, the same conclusion holds in the classical class +where $v,v_t$, spatial derivatives through order two, and $p,\nabla p$ are +continuous on finite closed cylinders, with periodic mean-zero pressure. +No Sobolev conditions at infinity are involved in that assertion. +\end{lemma} +\begin{proof} +Substitution proves existence of the prescribed solution. Set $w=v-U$. +The difference equation is +\[ + w_t+(v\cdot\nabla)w+(w\cdot\nabla)U + =\nu\Delta w-\nabla p,\qquad \operatorname{div}w=0. +\] +The stated velocity regularity implies $w\in C^1L^2\cap CH^2$, so +$\frac d{dt}\|w\|_2^2=2\langle w,w_t\rangle$ at each time. In case +(iii), the stronger assumptions imply these inclusions. They also imply +bounded velocity and spatial derivatives through order two: for $j\le2$, +Cauchy--Schwarz in Fourier space uses the finite integral +$\int_{\mathbb R^3}|\xi|^{2j}(1+|\xi|^2)^{-4}\,d\xi$. + +For the transport term, insert a cutoff $\chi_R$ that equals one on the +ball of radius $R$, is supported in the ball of radius $2R$, and satisfies +$|\nabla\chi_R|\le C/R$. Its boundary error is at most +$CR^{-1}\|v\|_\infty\|w\|_2^2$, which tends to zero. Diffusion is +integrated by parts in $H^2$; alternatively its cutoff error is bounded by +$CR^{-1}\|w\|_2\|\nabla w\|_2$. The deformation term is bounded by +$\|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2$. + +The same pressure cancellation applies in all five classes. At each time, +the equation gives the following equality of distributional gradients in +$L^2$: +\[ + \nabla p=f-v_t-(v\cdot\nabla)v+\nu\Delta v\in L^2, + \qquad \|(v\cdot\nabla)v\|_2\le\|v\|_\infty\|\nabla v\|_2. +\] +The velocity assumptions in every class give all terms on the right in +$L^2$; the identical estimate for $U$ and its stated time and space +regularity give $f\in L^2$. Since a distributional gradient is curl free, +Lemma~\ref{lem:b-gradient} gives $\langle\nabla p,w\rangle=0$. +This cancellation neither chooses an $L^2$ pressure representative nor +uses continuity of pressure in a norm. + +In every case the resulting pointwise energy inequality is +\[ + \frac12\frac d{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2 + \le \|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2. +\] +The coefficient is bounded on each finite interval and $w(0)=0$, so an +integrating factor gives $w=0$. The pressure has already disappeared before +time integration; the proof imposes no unlisted temporal pressure norm. +Substitution then gives $\nabla p=0$. A spatially constant $L^2$ function +on $\mathbb R^3$ is zero, which proves the normalization assertion. + +On the torus every integration by parts is periodic and has no boundary +term. The specified classical regularity justifies norm differentiation, +diffusion, and pressure cancellation. The same energy inequality gives +uniqueness and the mean-zero normalization fixes pressure. In either domain, +the smooth reference velocity and its bounded spatial derivative give +unique material trajectories; bounded speed prevents finite-time escape. +Solving backwards on a finite interval gives the inverse flow, and smooth +ODE dependence gives its smoothness. +\end{proof} + +Cases (i)--(v) identify the precise hypotheses used by the applications; +their presence in a common lemma does not replace a theorem's declared +comparison class by a different one. In particular, a pressure norm is not +silently imposed in a gradient-only application. The proof is the classical +energy comparison for prescribed smooth solutions \cite[Section~18]{Leray1934}, +with its noncompact pressure pairings made explicit. + +For the flow $\Phi_t$, differentiation of its trajectory equation gives +\[ + U(t)=(\partial_t\Phi_t)\circ\Phi_t^{-1},\qquad + \partial_{tt}\Phi_t(a)= + (f-\nabla p+\nu\Delta U)(t,\Phi_t(a)). +\] +These identities describe the material form of the same constructed solution; +they do not add an independent existence claim. + +\begin{lemma}[Physical scaling]\label{lem:p2-chart} +For $a,b>0$, $x=x_0+ay$, and $s=bt$, let +$U(t,x)=abV(bt,(x-x_0)/a)$. Then trajectories are transformed by this change +of variables, incompressibility is preserved, and at the prescribed physical +viscosity $\nu$, +\[ + \mathcal F_\nu[U](t,x)=ab^2\left[ + V_s+(V\cdot\nabla_y)V-\frac{\nu}{a^2b}\Delta_yV + \right](bt,(x-x_0)/a). +\] +\end{lemma} +\begin{proof} +The trajectory equation gives $\dot x=abV$; differentiating in space gives +$\operatorname{div}_xU=b\operatorname{div}_yV$. The three residual terms +scale by $ab^2,ab^2$, and $b/a$, respectively, proving the formula. +\end{proof} +Thus a side-$10$ torus is kept as a physical side-$10$ torus. If a chart +is rescaled, the residual is recomputed with the fixed $\nu$; the equation +is not identified with a differently normalized viscosity. + +\begin{proposition}[Projection on a flat torus]\label{prop:projection-l2} +An effective smooth mean-zero force $g$ on $\T_L^3$ has an effective +mean-zero potential $\phi$ satisfying $\Delta\phi=\operatorname{div}g$. +Replacing $(g,p)$ by $(g-\nabla\phi,p-\phi)$ preserves velocity and makes +the force solenoidal. Uniform mixed-derivative bounds, temporal $L^2$ +bounds of their spatial suprema, and separately imposed time periodicity, +eventual stationarity, or orderwise decay bounds are preserved. +\end{proposition} +\begin{proof} +In frequencies $\xi_k=2\pi k/L$, set +$\widehat\phi(k)=-i\xi_k\cdot\widehat g(k)/|\xi_k|^2$ for $k\ne0$ +and zero for $k=0$. The multiplier from $g$ to $\nabla\phi$ is +$\xi_k\xi_k^{\mathsf T}/|\xi_k|^2$. For a spatial derivative order $r$, +integrate each Fourier coefficient $r+5$ times in a coordinate with largest +$|k_i|$. There are $O(n^2)$ lattice points on the shell $\|k\|_\infty=n$, +so the derivative series and its tail converge absolutely, with +\[ + \|\partial_t^h\partial_x^\alpha\nabla\phi\|_\infty + \le C_{\alpha,L}\max_i + \|\partial_t^h\partial_{x_i}^{|\alpha|+5}g\|_\infty. +\] +This bound is pointwise in time, and proves each asserted norm or weighted +time estimate. Derivative bounds give effective Fourier tails; coefficient +integrals can be computed by Riemann sums with derivative error estimates. +The formula gives the Poisson identity, divergence cancellation, and the +pressure sign by substitution. Linearity preserves the time symmetries. +\end{proof} +Projection generally changes pressure. It supplies no compact-support +conclusion on Euclidean space and is distinct from the direct zero-pressure +solenoidal shear construction. + +\paragraph{Effective flat profiles.} +For later cutoff formulas put +\begin{equation}\label{eq:p2-step} + \rho(s)=\begin{cases}e^{-1/s},&s>0,\\0,&s\le0,\end{cases} + \qquad \sigma(s)=\frac{\rho(s)}{\rho(s)+\rho(1-s)}. +\end{equation} +Positive-side derivatives of $\rho$ are polynomials in $1/s$ times +$e^{-1/s}$ and extend flatly at zero. The estimate +$v^me^{-v}\le(m+k)!v^{-k}$ gives effective error bounds near the seam, +and $\rho(s)+\rho(1-s)\ge e^{-2}$. Products, translations, rescalings, +and derivatives therefore yield effective cutoffs with all requested +derivative bounds. Periodization uses zero collars; at approximate real +arguments a finite superset of possible translates is evaluated. No real +equality test at a seam or execution of the encoded computation is used. diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/autonomous.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/autonomous.tex new file mode 100644 index 0000000000000000000000000000000000000000..0c217e9f36071e7b89efb377411afbed8681808a --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/autonomous.tex @@ -0,0 +1,338 @@ +\section{An autonomous processor with a spatial clock} +\label{sec:sa-autonomous} +The preceding constructions repeat a time-dependent pulse. We now make +that repetition part of an autonomous divergence-free field: a third +coordinate runs through the phases of a planar Hamiltonian motion. +The observer is restricted to a quiet clock window, so transient transport +paths cannot give false detections. An onto scalar time change then starts +the fluid from rest and makes its force decay. + +\begin{theorem}\label{thm:sa-autonomous} +At fixed positive computable viscosity, a deterministic one-tape machine +and finite input effectively determine a smooth mean-zero general force +on the unit torus such that, for every $k\ge0$ and spatial multi-index +$\beta$, +\begin{equation}\label{eq:sa-autonomous-force-class} + \|\partial_t^k\partial_x^\beta f(t)\|_\infty + \le C_{k,\beta}(1+t)^{-1}. +\end{equation} +Its zero-data Navier--Stokes solution is globally smooth and unique in +the classical periodic comparison class, with uniformly bounded kinetic +energy and mean-zero pressure. For the fixed label and open set +\begin{equation}\label{eq:sa-autonomous-event} + y_*=(1/2,1/2,1/4),\qquad + \mathcal O=\{(x_1,x_2,z):1/20 +\] +with rational $\lambda_i$ and the respective centers, and require the +compatibility condition +\[ + \widehat U_i=\widehat c_i+ + \diag(\lambda_i,\lambda_i^{-1})(U_i-c_i). +\] +There is an effective smooth compactly supported planar Hamiltonian +field $V(\xi,X)$, with time support in $(1/4,3/4)$, whose unit-time +map has this formula on each $U_i$ and a positive neighborhood. +The construction gives a rational bound on its spatial support and all +controlled rectangle paths. +\end{lemma} +\begin{proof} +Choose a positive rational margin $\epsilon$ at most one quarter of a +positive coordinate-separating gap for each pair in each family; for +empty pair lists take $\epsilon=1$. The enlarged families stay disjoint. +Choose rational $D_0$ larger than $\epsilon$ plus every source or target +half-side. Take an integer $G\ge1$ so that within each family the +scaled centers $Gc_i$ and $G\widehat c_i$ are more than $10D_0$ apart +in the sup norm. Choose parking centers $e_i$ more than $10D_0$ apart +from one another and from all these scaled centers. A sufficiently +distant row with spacing $20D_0$ does this effectively. + +Use affine paths +\begin{equation}\label{eq:sa-autonomous-moving-box} + X\mapsto C_i(\xi)+\diag(a_i(\xi),a_i(\xi)^{-1})(X-c_i). +\end{equation} +First translate centers simultaneously $c_i\to Gc_i$, keeping shapes +fixed; any coordinate separation can only increase. Next move one +rectangle at a time to $e_i$. At those parking centers change $a_i$ +from $1$ to $\lambda_i$. Each half-side stays between its endpoint +values and hence below $D_0-\epsilon$. Move the target-shaped rectangles +one at a time to $G\widehat c_i$, then translate centers simultaneously +down to $\widehat c_i$. In this last phase the target separations stay +valid down to scale one. +During source evacuation, occupied centers are scaled source centers or +parking centers. During target delivery, every source has already been +evacuated, and occupied centers are parking centers or scaled target +centers. Each such mixed family has separation greater than $10D_0$ by +the choices above. Thus arbitrary intersections between the original +source and target families never place two occupied rectangles at the +same location. The obstacle family for each transfer consists of every +other currently occupied rectangle, wherever it is in that phase. + +For a one-at-a-time transfer, put a closed square of sup radius $4D_0$ +around every stationary center. These obstacles are disjoint, and the +transfer endpoints are outside them. Follow the straight segment, replacing +any portion inside one obstacle by a boundary detour in a fixed orientation. +Entry and exit points solve rational linear inequalities, so all waypoints +are rational. A tangent contact, a corner contact without entry, or a segment already +on the boundary needs no detour: it already has the required clearance. +A segment crossing the interior has rational first and last intersection +points, found by clipping its rational parameter interval against the +four rational side inequalities. A fixed clockwise boundary arc between +those points uses only rational corners. The obstacles are separated, +so each detour misses every other obstacle. The moving center remains at sup distance at least +$4D_0$ from all stationary centers; both enlarged rectangles have radius +less than $D_0$, proving separation throughout. + +Schedule the finitely many submotions in successive equal slots inside +$[1/3,2/3]$, using a flat step for each segment and scale change. This +smooths time without changing any path or clearance. Let $\chi_i$ +equal one on a positive collar of the moving rectangle and be supported +in its $\epsilon$ enlargement. With $Y=X-C_i$, set +\begin{equation}\label{eq:sa-autonomous-hamiltonian} + \mathcal H=\sum_i\chi_i\left( + \dot C_{i1}Y_2-\dot C_{i2}Y_1 + +\frac{\dot a_i}{a_i}Y_1Y_2\right),\qquad + V=(\partial_{X_2}\mathcal H,-\partial_{X_1}\mathcal H). +\end{equation} +The supports are disjoint. On rectangle $i$ the field is +$\dot C_i+\diag(\dot a_i/a_i,-\dot a_i/a_i)(X-C_i)$, so +\eqref{eq:sa-autonomous-moving-box} solves its ODE exactly. +The same formula holds on a positive initial neighborhood by the plateau +and finite affine norm bounds. Smooth ODE uniqueness identifies the +motion. A rational support bound follows from the finite endpoint and +waypoint list plus $D_0+1$. Every operation is effective finite geometry +and the fixed flat step, proving the lemma. +\end{proof} + +Apply this lemma to \eqref{eq:sa-autonomous-branches}. With +$I_\eta=h_\eta([0,1])$, the normalized target rectangles for moves +$0,1,-1$ are, respectively, $[0,1]\times I_\beta$, +$I_\beta\times[0,1]$, and $[0,1]\times h_\gamma(I_\beta)$. +Their side lengths are exactly those of their source rectangles multiplied +by $(1,1)$, $(K^{-1},K)$, and $(K,K^{-1})$. Adding the respective state +offsets verifies the compatibility equation for every branch. +The branch list is +nonempty because of the fresh initial state. Choose rational $M>1$ so +that its support, controlled paths, and all state squares lie in $(-M,M)^2$. +We now have one complete planar transition pulse with its own proof of +area preservation; no third-dimensional strain has been used. + +\subsection{The spatial clock and the observation window} +\label{subsec:sa-autonomous-clock} +Set $\kappa=1/(16M)$ and $T(X)=(1/2,1/2)+\kappa X$. +All relevant geometry lies after scaling in $(7/16,9/16)^2$. +Periodically extend +\[ + \widetilde V(x_1,x_2,z)=\kappa V(z,T^{-1}(x_1,x_2)). +\] +Its known zero collars at the horizontal boundaries and at $z=0,1$ +make this smooth. It has horizontal divergence zero; each horizontal +component has integral zero in $(x_1,x_2)$ since it is a scaled derivative +of the compact Hamiltonian. + +For an explicit vertical clock choose a flat step $\theta$ that is zero +on $(-\infty,1/3]$ and one on $[2/3,\infty)$. Put +\[ + \psi(s)=(1-\theta(16(s-1/8))) + (1-\theta(16(-s-1/8))),\qquad + b(x_1,x_2)=\psi(x_1-1/2)\psi(x_2-1/2), +\] +periodize $b$, and set $w(x_1,x_2)=b(x_1,x_2)-b(x_1-1/2,x_2)$. +The bump is one on $[3/8,5/8]^2$ and supported in +$(5/16,11/16)^2$. Its half-period translate has disjoint support, +so $w=1$ on all scaled state squares and controlled paths, and $\int w=0$. +The autonomous field +\begin{equation}\label{eq:sa-autonomous-W} + W=(\widetilde V_1,\widetilde V_2,w) +\end{equation} +is divergence free and mean zero. Write $\Psi_\tau$ for its global +smooth flow on the torus. + +From $(T(X),1/4)$ with $X$ in a source rectangle, the clock increases +at speed one and the horizontal coordinates follow its prescribed path +with planar phase $\xi=1/4+\tau$. After that pulse, the horizontal target +stays fixed while the clock passes from $z=0$ to $z=1/4$. The entire +path has $w=1$, so this description solves the autonomous ODE. Thus at +unit times the map performs one primitive transition and returns the clock +to $1/4$. If $X_j$ is the primitive configuration code, induction gives +\begin{equation}\label{eq:sa-autonomous-samples} + \Psi_j(y_*)=(T(X_j),1/4) +\end{equation} +through halt or indefinitely in a nonhalting run. +Along these controlled paths, $z=1/4+\tau\pmod1$. Hence for every +positive such $j$, in the window +\begin{equation}\label{eq:sa-autonomous-window} + j-1/4<\tau0$ for $t>0$ and +$S(t)\to\infty$, this clock is continuous and strictly increasing from +zero onto $[0,\infty)$. Every finite autonomous event and every sample +$S^{-1}(j)$ therefore occur at finite physical time. The established +window equivalence proves the theorem. + +All finite tables, initial geometric tails, rectangle offsets, clearances, +waypoints, and scaling constants are effective. The Hamiltonian cutoff +uses only rescaled flat steps and positive rational margins, and its +periodization has known zero collars. The remaining force operations +are finite differentiation and the rational profile \eqref{eq:sa-autonomous-profile}. +The input affects the initial-state translation in +\eqref{eq:sa-autonomous-offsets}; no list of executed states is used. +At every finite time the material map is a smooth diffeomorphism, with +$u=\partial_tF\circ F^{-1}$ and +$\partial_t^2F=(\nu\Delta u-\nabla p+f)\circ F$, as follows by + differentiating its trajectory equation. diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/compact-reciprocal.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/compact-reciprocal.tex new file mode 100644 index 0000000000000000000000000000000000000000..4a1e050e6e23bfbdb7c48dc27ec9b733401c4974 --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/compact-reciprocal.tex @@ -0,0 +1,210 @@ +\section{Two compact realizations of the reciprocal recorder} +\label{subsec:p2:rapid-periodic} +The preceding area-changing constructions use the third coordinate to +compensate a planar determinant. For the ordinary one-head recorder, +the planar maps already have determinant one. The third coordinate can +then serve only to separate instructions during transport. We give two +compact Euclidean realizations: one uses four stationary fields in +succession, and the other follows entire moving boxes in three phases. +Their observer estimates will follow directly from the horizontal +motion, without a vertical restriction on the detector. + +Use the frontier recorder of Lemma~\ref{lem:sb-frontier} throughout this +section. Its frontier starts at absolute cell two; every other history +cell is blank and every return mark is zero. Its fixed incoming moves +and distinguishing written symbols will separate the target rectangles. +At checkpoint $k$ the records occupy $2,\ldots,k+1$, the frontier is at +$k+2$, and transition $k+1$ takes $2(k+2-h_{k+1})+4$ instructions. +These properties hold for the same entire partial table as in that lemma. + +Choose the combined alphabet $\mathcal A$ of the recorder, of size $m$. +Give its letters the even digits $g_a=0,2,\ldots,2m-2$, with the fully +blank combined letter first. We will use either $B=2m$ or $B=2m+1$. +For a tape $v$ whose head is at its relative index zero, set +\[ + x=\sum_{j\ge1}g_{v_{-j}}B^{-j},\qquad + y=\sum_{j\ge0}g_{v_j}B^{-j-1},\qquad + I_a=[g_a/B,(g_a+1)/B]. +\] +These are the gapped codes of Section~\ref{subsec:sb-codes}. +A state $s$ is placed at $(o_s+x,y,0)$ for a specified offset $o_s$. +If a rule reads $a$, writes $b$, moves by $d$, and enters $s'$, its +coordinate map relative to the source and target offsets is +\begin{equation}\label{eq:p2:rapid-rectangle-rules} +\begin{array}{c|c|c} + d&\text{source rectangle}&(x',y')\\\hline + 1 &[0,1]\times I_a&((g_b+x)/B,\ By-g_a)\\ + 0 &[0,1]\times I_a&(x,\ y+(g_b-g_a)/B)\\ + -1&I_\ell\times I_a& + (Bx-g_\ell,\ (g_\ell+(g_b+By-g_a)/B)/B). +\end{array} +\end{equation} +For the last row there is one branch for each $\ell\in\mathcal A$. +The target rectangles are respectively $I_b\times[0,1]$, +$[0,1]\times I_b$, and $[0,1]\times(g_\ell/B+I_b/B)$. +These formulas follow by removing the scanned digit and pushing the +written digit. They apply to every point of the displayed closed +rectangles. Determinism separates the sources; the fixed incoming +displacement and distinguishing written symbol separate the targets. +For a left move the target pair $(\ell,b)$ supplies that distinction. +Every linear part is $\operatorname{diag}(\lambda,\lambda^{-1})$. + +The code also retains absolute, rather than only head-relative, +configuration information at the machine checkpoints. Count the records +immediately to the left of the frontier to recover $k$. The frontier +has absolute position $k+2$; its location in the decoded tape therefore +recovers the absolute head position. The work track then recovers the +original tape. This use of history records is the retained-history +version of the reversible-computation idea of Bennett~\cite{Bennett1973}; +the positional affine representation follows the generalized-shift +viewpoint of Moore~\cite{Moore1990,Moore1991}. + +\begin{proposition}[Two compact periodic processors] +\label{prop:p2:rapid-periodic} +For every deterministic machine and finite input, and every fixed +computable viscosity $\nu>0$, there are effective smooth velocities +$U$ and forces $f$ on $\mathbb R^3$ with the following properties. +Both have all mixed derivatives bounded, support in a compact spatial +set independent of time, and period one for $t\ge1$; $U(0)=0$. +The solution with pressure zero is unique in the class +\[ + u\in C([0,T];H^2)\cap C^1([0,T];L^2),\qquad + p\in C([0,T];H^1),\qquad + \sup_{[0,T]\times\mathbb R^3}(|u|+|\nabla u|)<\infty +\] +for every finite $T$, and its kinetic energy is uniformly bounded. +The following are two separate constructions with their stated fixed +particle events: +\begin{enumerate} +\item Four successive stationary localized fields, with initial label +$(-1,0,0)$ and detector $\{x_1>0\}$. +\item Three successive phases of moving full-box fields, with initial +label $(0,0,0)$ and detector \mbox{$\{x_1<-1\}$}. +\end{enumerate} +In each case the particle enters the detector at some real time if and +only if the machine halts. The repeated part of the prescription depends +only on the machine; the input appears only in a one-time loading pulse. +\end{proposition} +\begin{proof} +Let $R_i,R_i'$ be the planar source and target rectangles in +\eqref{eq:p2:rapid-rectangle-rules}, with centers $c_i,c_i'$ and first +scale $\lambda_i$. All their side lengths are at most one. If there are +no instructions, use the zero repeated field and the same loader described +below. Empty pairwise margin tests are omitted for a singleton family. + +For the first realization choose $B=2m$. Put terminal offsets at +$4,7,10,\ldots$ and all other offsets at $-4,-7,-10,\ldots$. +Choose private heights $h_i=4i$. On four successive quarters of a unit +period, perform the following autonomous motions, each with flat progress +from zero to one: +\begin{enumerate} +\item Translate upward by $h_i e_3$, with a cutoff equal to one near +$R_i\times[0,h_i]$. +\item At height $h_i$, use the Hamiltonian vector potential +\[ + (0,0,\chi_i\log(\lambda_i) + (X_1-c_{i1})(X_2-c_{i2})). +\] +Its curl on the plateau is +$\log(\lambda_i)(X_1-c_{i1},-(X_2-c_{i2}),0)$. +\item Translate horizontally by $c_i'-c_i$ at height $h_i$. +\item Translate downward by $h_i e_3$ near $R_i'\times[0,h_i]$. +\end{enumerate} +A translation by $w$ is localized by +$\operatorname{curl}(\chi(w\times X)/2)$. +In the first and last quarters, choose cutoff collars below one third +of the smallest positive horizontal separation in the source and target +families, respectively. In the middle quarters, collars of thickness +one in height do not meet because the heights differ by four. For the +stretch, choose the horizontal plateau to include +$c_i+[-1,1]^2$. For translation, include the bounding rectangle of +$[c_i,c_i']$ enlarged by one in each horizontal direction. + +More explicitly, with $\vartheta(s)=\sigma(2s-1/2)$ and +$\vartheta_j(s)=\vartheta(4s-j+1)$, multiply the sum of the stationary +fields for quarter $j$ by $\vartheta_j'(s)$. The tracked rectangle rises, +scales by $(\lambda_i^{\vartheta_2},\lambda_i^{-\vartheta_2})$, +translates its center along $[c_i,c_i']$, and descends. Each intermediate +half-width is at most $1/2$, so the entire rectangle remains on the +specified plateaus. The positive collars and finite affine norms also +give exactness on a positive initial neighborhood. If both endpoint +states are nonterminal, both first center coordinates are at most +$-3$, and the whole tracked path has negative first coordinate. + +For the second realization choose $B=2m+1$, nonterminal offsets +$3,6,9,\ldots$, and terminal offsets $-3,-6,-9,\ldots$. +Thicken each rectangle by $[-1,1]$ in height, and write $c_i^0,c_i^1$ +for the three-dimensional endpoint centers. Each pair within the +source family, and within the target family, has a horizontal coordinate +gap at least $B^{-2}$. Use height $6i$ and collar +$\epsilon=1/(4B^2)$. In three successive phases the prescribed affine +paths have center and linear part +\[ +\begin{array}{c|c|c} + &c_i(s)&L_i(s)\\\hline + \text{lift}&c_i^0+6i\Theta e_3&I\\ + \text{transfer}&(1-\Theta)c_i^0+\Theta c_i^1+6i e_3& + \operatorname{diag}(\lambda_i^\Theta,\lambda_i^{-\Theta},1)\\ + \text{lower}&c_i^1+6i(1-\Theta)e_3& + \operatorname{diag}(\lambda_i,\lambda_i^{-1},1), +\end{array} +\] +where $\Theta$ is $\vartheta$ rescaled to the active third. +The proof of Lemma~\ref{lem:sb-boxes} applies to these exact data: +horizontal gaps separate the collars during lift and lowering, while +height gaps six separate vertical half-widths $1+\epsilon$ during +transfer. The localized field is explicitly +\begin{equation}\label{eq:p2:rapid-moving-box} + \sum_i\operatorname{curl}_X\left[ + \chi_i\left\{\frac{\dot c_i\times(X-c_i)}2+ + \frac{(\dot L_iL_i^{-1}(X-c_i))\times(X-c_i)}3\right\}\right]. +\end{equation} +The trace of $\dot L_iL_i^{-1}$ is zero. The curl identity in +Lemma~\ref{lem:sa-curl-motion} therefore gives the prescribed affine +velocity near each moving box, and ODE uniqueness gives its exact +full-box motion. A branch with two nonterminal endpoint states has +first center coordinate at least three and half-width at most $1/2$; +hence its entire path has first coordinate at least $5/2$. + +It remains to initialize and to pass from the finite maps to the fluid +event. The initial combined tape is blank except at finitely many cells, +so its initial code $Z_0$ is rational. During $[0,1]$, a localized curl +translation with flat progress carries the fixed label to $Z_0$ along +the straight segment. A compact cutoff contains the whole segment with +a positive collar. Repeat the chosen instruction pulse on every later +unit interval. The time collars make all joins smooth, and only +finitely many spatial supports occur. Thus $U$ has the support and +derivative bounds in the statement. Set +\[ + f=\partial_tU+(U\cdot\nabla)U-\nu\Delta U. +\] +The residual realization and comparison argument of +Section~\ref{sec:model} gives the asserted solution and uniqueness; +fixed support and bounded $U$ give bounded kinetic energy. The formula +can also be returned as $f=f^{(0)}+\nu f^{(1)}$, with +$f^{(0)}=U_t+(U\cdot\nabla)U$ and $f^{(1)}=-\Delta U$ effective +independently of $\nu$. Evaluation for an arbitrary fixed real viscosity +is relative to that parameter. + +At each integer $1+j$ through a halt, or at every such integer in a +nonhalting run, the particle is the exact code after $j$ auxiliary +instructions. Lemma~\ref{lem:sb-frontier} supplies finite positive +instruction counts between the original machine checkpoints. In the +first realization, a nonhalting loading segment has negative first +coordinate, and all subsequent nonterminal branches stay negative; +a terminal code has positive first coordinate. In the second, +nonhalting loading has nonnegative first coordinate and later +nonterminal paths stay above $5/2$ in that coordinate, while a terminal +code has first coordinate at most $-2$. An initially halted machine is +detected at the end of loading. Thus the exclusions apply at every real +time, and a halting run enters the detector no later than its terminal +checkpoint. No exclusion is needed during a final transition into a +terminal state. + +All rectangles, cutoffs, scales and finite branch fields are computed +from the finite table. Positive rational scales have effective logarithms +and exponentials. Smooth switching and derivative bounds permit +evaluation at time joins without equality tests. The finite repeated +field acts on all branch domains at every period; neither its definition +nor its evaluation uses a predicted sequence of machine configurations. +\end{proof} diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/first-simulation.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/first-simulation.tex new file mode 100644 index 0000000000000000000000000000000000000000..03a313aac1650f82708918a4e7873a6b9a720c0e --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/first-simulation.tex @@ -0,0 +1,98 @@ +\section{The first initialized simulation}\label{sec:sa-histories} +We now use the moving-sheet construction to prove the first result. +Records interspersed among work symbols retain the information erased by +a machine instruction. The processor below first supplies its exact +initialized simulation and the stronger separation of its full rule +cylinders. We then place those cylinders and check the fixed observer +during loading and throughout every subsequent motion. + +\subsection{History interspersed among work symbols}\label{subsec:sa-interleaved} +Let $\Gamma$ be the work alphabet, $Q$ the states, and $H$ the halt set. +Adjoin an origin bit to each work symbol, preserving it under writes; +initially only cell zero has bit one. Write $A=\Gamma\times\{0,1\}$ and +$b_0=(\square,0)$. For each nonhalt instruction +$\widehat\delta(q,a)=(q',b,d)$, $d\in\{0,+,-\}$, and incoming mode +$h\in\{0,+,-\}$, introduce a distinct record $g_{q,h,a}$. The records +form an alphabet $G$ disjoint from $A$. Main controls are $M(q,h)$; +$C_+(q),C_-(q)$ finish a head movement. + +\begin{lemma}[Interspersed-history simulation]\label{lem:sa-interleaved} +Starting at $M(q_0,0)$ with left stack $b_0^\infty$ and right stack the +origin-tagged input followed by blanks, this table simulates every ordinary +step in finitely many positive rule applications. Erasing records at a +main control gives the work tape, head, and state. The origin bit gives +absolute tape positions. The full source cylinders are pairwise disjoint, +as are the full image cylinders. The initialized run reaches a halting +main control exactly when the machine halts. +\end{lemma} +\begin{proof} +At a main control, erasing records makes $R$ the head-and-right tape and +$L$ the reversed left tape. The first row writes $b$ and saves the lost +information in $g$. A stay is complete. For a right move, $b$ has been +put on $L$ and $C_+$ transfers the intervening records until the next +work symbol is exposed. For a left move, $C_-$ transfers the intervening +records from $L$ and finally its first work symbol; that symbol becomes +the new head immediately before $b$ after erasure. At finite rule time +there are finitely many records and infinitely many work cells in each +direction, so every scan finishes. Writes preserve the origin bit. + +The complete rule list is +\begin{equation}\label{eq:sa-interleaved-table} +\begin{aligned} + (M(q,h);\epsilon,a)&\to + \begin{cases} + (M(q',0);g,b),&d=0,\\ + (C_+(q');bg,\epsilon),&d=+,\\ + (C_-(q');\epsilon,bg),&d=-, + \end{cases}\quad g=g_{q,h,a},\\ + (C_+(q);\epsilon,g)&\to(C_+(q);g,\epsilon),&g\in G,\\ + (C_+(q);\epsilon,c)&\to(M(q,+);\epsilon,c),&c\in A,\\ + (C_-(q);g,\epsilon)&\to(C_-(q);\epsilon,g),&g\in G,\\ + (C_-(q);c,\epsilon)&\to(M(q,-);\epsilon,c),&c\in A. +\end{aligned} +\end{equation} +No rule leaves a main control with $q\in H$. + +Source separation uses the tested top symbol on $R$ for main and $C_+$ +controls, and on $L$ for $C_-$. Into $C_+$ the left prefix starts with a +work symbol in a main entry and a record in a loop; the record names +separate the main entries. Into $C_-$ the same distinction is on $R$. +Into $M(q,0)$ the first left record identifies the incoming instruction; +into $M(q,+)$ or $M(q,-)$ the first right work symbol identifies the exit. +Thus every image has a unique inverse on its full cylinder. The simulation +invariant proves the halt equivalence, including an initial halt. +\end{proof} + +We now prove Theorem~\ref{thm:sa-interleaved}. + +\begin{proof} +For $K=|A\sqcup G|$, use digits $1,3,\ldots,2K-1$ and base $B=2K+2$. +Enumerate all $m$ controls by $i=1,\ldots,m$, put $\ell=1/[16(m+1)]$, +and use the code +\[ + (\beta_i+\ell E(L),1/4+\ell E(R),1/4),\qquad + \beta_i=2i\ell+\begin{cases}5/8,&\text{halting},\\1/4,&\text{otherwise}. + \end{cases} +\] +Nonhalt intervals lie in $(1/4,3/8)$ and halt intervals in $(5/8,3/4)$. +All prefixes in \eqref{eq:sa-interleaved-table} have length at most two; +Lemma~\ref{lem:sa-radix} therefore gives source and target gaps at least +$\ell/B^2$. Let $N$ be the number of rules. Apply Lemma~\ref{lem:sa-sheet-routing}, with collar at most +$\min(\ell/(4B^2),1/[16(N+1)],1/16)$, to their full closed rectangles. +Its normal scale compensates the unequal total prefix lengths. +The initial code is rational. A localized curl translation of radius +$1/32$ loads it along the segment from the fixed label during $[0,1]$. +Repeat the unit rule pulse thereafter. All pulses have zero collars +inside the cube and are curls, giving a smooth mean-zero velocity $U$ +with $U(0)=0$. Lemma~\ref{lem:sa-realization} supplies the force. + +At time $1+k$ induction gives the code after $k$ rules until the first +halt. A halt is in the observed arc, including at $t=1$ if initial. +For a nonhalting run the loader stays below $1/2$, and every subsequent +horizontal coordinate is constant during vertical motion and between its +two nonhalt endpoints during the middle stage. Thus the observer is +avoided at every time. At rule time $k$ the raw stacks are blank beyond +$\max(1,|w|)+2k$ symbols, so this code retains the whole configuration, +not merely the event bit. +\end{proof} + diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/full-boxes.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/full-boxes.tex new file mode 100644 index 0000000000000000000000000000000000000000..4666c6bd828b6557a6c70955f818b53f66a114eb --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/full-boxes.tex @@ -0,0 +1,152 @@ +\section{Prescribing a positive thickness}\label{sec:sheets-boxes} +We shall also need boxes of a prescribed positive thickness, for which +an unbounded choice of layer heights is convenient. +\begin{lemma}[Exponential scaling of full boxes] +\label{lem:sa-full-box-layers}\label{lem:sb-boxes} +Suppose $R_i^\pm\subset\mathbb R^2$, $1\le i\le N$, $N\ge1$, +are separately separated nondegenerate rational closed rectangles. +Let $F_i$ be their positive diagonal affine maps, and explicitly require +$F_i(R_i^-)=R_i^+$. Their factors $\lambda_{i1},\lambda_{i2}$ are +positive rational numbers, given by the corresponding rational side-length +ratios. Put +$\lambda_{i3}=(\lambda_{i1}\lambda_{i2})^{-1}$, and choose any positive +rational source half-heights $h_i^-$. Define $h_i^+=\lambda_{i3}h_i^-$. +There is an effective compactly supported divergence-free unit-time pulse +whose map on each full box $R_i^-\times[-h_i^-,h_i^-]$ is the specified +horizontal map together with $z\mapsto\lambda_{i3}z$. It maps that box +to $R_i^+\times[-h_i^+,h_i^+]$, and the formula is exact on a positive +three-dimensional neighborhood. Its centers interpolate linearly in the +horizontal coordinates, while every half-side is at most the larger of +its endpoint half-sides. The pulse vanishes near both time endpoints. +\end{lemma} +\begin{proof} +Let $Z=\max_{i,\pm}h_i^\pm$. Choose a positive rational collar +$0<\eta\le1/4$ no larger than one quarter of each within-family horizontal +coordinate-separating gap. If a family has only one member, there is no +pairwise condition. Assign heights $z_i=4(Z+1)i$. More generally, the +same proof permits any heights whose pairwise gaps exceed $2Z+2\eta$. +With successive flat switches $s_1,s_2,s_3$, lift, deform, and lower using +\[ + C_i=((1-s_2)c_i^-+s_2c_i^+,z_i(s_1-s_3)),\qquad + D_i=\diag(\lambda_{i1}^{s_2},\lambda_{i2}^{s_2},\lambda_{i3}^{s_2}). +\] +The middle-stage vertical half-size is at most $Z$, because every +positive geometric interpolation stays between its two endpoint values. +During ascent and descent, the fixed source and target projections, +respectively, keep the $\eta$ enlargements disjoint. During deformation +the height condition separates those enlargements. Product cutoffs equal +to one on the $\eta/2$ enlargement and supported in the $\eta$ +enlargement therefore meet Lemma~\ref{lem:sa-curl-motion}. +The determinant of $D_i$ is one. That lemma gives the exact affine path +on every full box and a positive initial neighborhood. Only finitely +many bounded paths occur, so the support lies in a fixed compact set. +Logarithms and exponentials of positive rational factors are effective. + +This proof permits different initial thicknesses because it uses only +the finite maximum of all endpoint half-heights. In the special case +$h_i^-=1$ one has $Z=\max_i(1,\lambda_{i3})$ and the stated default +heights are $4(Z+1)i$. Unlike the linear-width sheet path, this +interpolation does not make each individual horizontal trajectory a +convex combination of its endpoints; the center and half-width bounds +are the assertions used below. +\end{proof} + +\subsection{Records attached to individual work cells}\label{subsec:sa-cell-records} +A related recorder keeps a word of history immediately after each work +cell. This changes the actual rule cylinders: a stay writes its record +on the right, and a right move transfers its newly written history in a +separate scan. It is not a renaming of \eqref{eq:sa-interleaved-table}. + +Use an origin-tagged work alphabet $\Sigma$, moves $D=\{0,+,-\}$, and +an incoming tag $e\in D\sqcup\{*\}$. Extend halt states by idle +instructions. The controls are $C(q,e),F(p),G(p)$; the distinct records +are $\gamma(q,e,a)$, all outside $\Sigma$; write $\mathcal G$ for +their alphabet. Initialize at $C(q_0,*)$ with stacks $\square^\infty$ and +$w\square^\infty$, where $w$ includes the permanent origin tag. +At a main control with head at $j$, its invariant is +\[ + L=\operatorname{rev}(a_{j-1}W_{j-1}) + \operatorname{rev}(a_{j-2}W_{j-2})\cdots, + \qquad R=a_jW_j\,a_{j+1}W_{j+1}\cdots, +\] +where every $W_i$ is a finite record word, initially empty. For an instruction writing $b$ and retaining its record $g$, the main +update changes $a_jW_j$ to $bgW_j$. The $F$ scan transfers the whole record word $gW_j$ +before exposing the next work cell; the $G$ scan transfers the previous +cell's reversed record and then its work symbol. Thus every instruction +finishes in finitely many positive steps and gives the stated invariant. +Erasing records and using the origin tag recovers the original tape. + +If $\delta(q,a)=(p,b,d)$ and $g=\gamma(q,e,a)$, +use +\begin{equation}\label{eq:sa-cell-records-3} +\begin{array}{ll} + d=0:&(C(q,e);\epsilon,a)\to(C(p,0);\epsilon,bg),\\ + d=+:&(C(q,e);\epsilon,a)\to(F(p);b,g),\\ + d=-:&(C(q,e);\epsilon,a)\to(G(p);\epsilon,bg). +\end{array} +\end{equation} +For each record $g'$ and work symbol $a'$, add +\begin{equation}\label{eq:sa-cell-records-4} +\begin{array}{ll} + (F(p);\epsilon,g')\to(F(p);g',\epsilon),& + (F(p);\epsilon,a')\to(C(p,+);\epsilon,a'),\\ + (G(p);g',\epsilon)\to(G(p);\epsilon,g'),& + (G(p);a',\epsilon)\to(C(p,-);\epsilon,a'). +\end{array} +\end{equation} + + +The sources are separated by their tested top symbols. Image cylinders +into $C(p,0)$ are separated by their right prefixes $bg$, and those into +$C(p,+)$ or $C(p,-)$ by the exiting work symbol. No rule enters $C(p,*)$. +Into $F(p)$ a main entry has work first on $L$ and its identifying record +first on $R$, whereas a loop has a record first on $L$. Into $G(p)$ a +main entry has $bg$ first on $R$, whereas a loop has a record first there. +These distinctions recover the incoming rule for all tails. + +\begin{theorem}[Input shifts and periodic full-box motion] +\label{thm:sa-cell-records} +This recorder gives an effective smooth general force on $\mathbb R^3$ +with fixed compact spatial support, period one from $t=0$, and all mixed +derivatives bounded. Its zero-data solution is globally smooth with +uniformly bounded energy. The particle starting at the origin has first +coordinate greater than $2$ at some time exactly when the machine halts. +Uniqueness holds for smooth velocities in $C_tH^2\cap C_t^1L^2$ with +velocity and gradient bounded on finite intervals, and pressure in $H^1$ +modulo constants at each time. Pressure is determined modulo a function +of time. +\end{theorem} +\begin{proof} +Take odd digits and base $B=2|\Sigma\sqcup\mathcal G|+1$, where +$\mathcal G$ is the record alphabet. Put +$l_0=E(\square^\infty)$ and $r_0=E(w\square^\infty)$. +Give $C(q_0,*)$ offset zero, other nontrigger controls distinct offsets +$-4,-8,\ldots$, and trigger controls +$C(q,d)$ with $q$ halting and $d\in D$ offsets $4,8,\ldots$. +Encode by +\begin{equation}\label{eq:sa-cell-records-5} + (o_s-l_0+E(L),-r_0+E(R),0). +\end{equation} +The initial code is exactly zero. For every rule use its two prefix +rectangles, with scales +$\lambda_1=B^{|v|-|v'|}$, $\lambda_2=B^{|w|-|w'|}$ and +$\lambda_3=(\lambda_1\lambda_2)^{-1}$. +Lemma~\ref{lem:sa-full-box-layers} acts on the entire box +$R^-\times[-1,1]$, sending it to $R^+\times[-\lambda_3,\lambda_3]$. +The encoded central sheet returns to the same plane $z=0$ on successive +periods; the full box thickness generally changes. One may take margin $B^{-m_*}/10$, +where $m_*$ is the maximum prefix length, and heights $4(Z+1)i$ as in +that lemma. Periodize the resulting pulse from time zero; the time +collars give $U(0)=0$. Lemma~\ref{lem:sa-realization} and its pointwise +$H^1$ pressure refinement give the equation and uniqueness. + +The code induction now begins at $t=0$. A reached halt has a trigger +code with first coordinate at least $4-l_0>2$. An initial halt reaches +one after its first idle instruction, changing tag $*$ to $0$. +For a nonhalting run, every used source and target has first-coordinate +center at most $1$ and half-width at most $1/2$. The exponential box +path has convexly interpolated centers and half-width at most the larger +endpoint half-width; hence it stays below $3/2<2$ throughout. Input +shifts affect the finite spatial prescription, and no loader is required. +\end{proof} + diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/introduction.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..16459546ccfb18331b07f50b5f123b932460ccb1 --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/introduction.tex @@ -0,0 +1,154 @@ +\section{Introduction} +A finite machine instruction reads a short part of a configuration and +changes it. A material flow, by contrast, must be invertible at every +finite time. Two questions therefore arise when a particle is to carry a +computation: where is the information erased by an instruction retained, +and how is the resulting injective update made compatible with +incompressibility? We study these questions for finite prefix instructions +and their smooth three-dimensional realizations. + +A stack is an infinite word whose first letter is its accessible end. +A prefix instruction has the form +\[ + (s;v,w)\longmapsto(s';v',w'), +\] +meaning that it sends $(s,vL,wR)$ to $(s',v'L,w'R)$ for every pair of +unchanged tails $L,R$. Here $s,s'$ belong to a finite set of control +states, and all four prefixes are finite words over a finite alphabet. +A positional code turns this operation into a positive diagonal affine +map between two rectangles. The source rectangles must be separately +disjoint, and so must the target rectangles. Intersections between a +source rectangle and a target rectangle are allowed. These conditions +are stronger than injectivity along one initialized computation. + +The planar map need not preserve area. Appending one history letter, +for example, contracts one stack coordinate without expanding the +other. We realize such instructions in two ways. A compensating normal +strain transports the entire coded rectangle and a three-dimensional +neighborhood. Alternatively, a compressible planar motion is placed in +an invariant plane of an incompressible field. The latter preserves the +prescribed planar dynamics without prescribing an affine map off the +plane. Both methods require control of the complete continuous path: +correctness at integer sampling times alone does not establish a fixed +particle observation. + +\subsection{The fluid problem and a first result} +Let $\mathbb T^3=(\mathbb R/\mathbb Z)^3$ with its flat metric, and fix a +positive computable viscosity $\nu$. For an arbitrary fixed positive real +$\nu$, the same formulas and conclusions hold, and numerical evaluation +is relative to that coefficient. We use +\begin{equation}\label{eq:sheets-ns} + u_t+(u\cdot\nabla)u-\nu\Delta u+\nabla p=f, + \qquad \operatorname{div}u=0,\qquad u(0)=0. +\end{equation} +An instance is a deterministic machine with finite state set, finite work +alphabet with a blank, an initial state, a specified halt set, and a +finite input written on an otherwise blank two-sided tape. Its head +starts at cell zero. Missing transitions may be interpreted as halting. +For a fixed initial label $a$, the material position satisfies +$X'(t)=u(t,X(t))$ and $X(0)=a$. The observation asks whether +$X(t)\in O$ for some $t\ge0$, where $O$ is a fixed open set. + +A finite effective force prescription is a finite program that evaluates +$f$ and any requested mixed derivative at computably specified arguments +to any positive rational error, and supplies the derivative bounds used +on compact sets. The program describes every instruction branch. It does +not generate a machine execution and replay that execution as a force. +On a torus our classical comparison class requires $u,u_t$, spatial +derivatives of $u$ through order two, and $p,\nabla p$ to be continuous +on every finite closed time cylinder. Velocity and pressure are periodic, +and pressure has spatial mean zero. The prescribed solutions themselves +are smooth. + +\input{principal-theorem.tex} + +This first result is proved with history records interspersed among work +letters. Its geometric map changes planar area, so a third scale restores +volume. Later constructions change the history arrangement or initialization +when a different property is wanted: compact Euclidean support, a bounded +detector, periodicity from the initial time, or a spatial clock. The +comparison classes, initial labels, detectors and temporal conclusions are +stated separately for these alternatives. + +\subsection{Antecedents and the role of the present constructions} +The logical obstruction comes from Turing's undecidable symbol-printing +problem~\cite[Section~8]{Turing1936}. Modify such a machine to halt at +the first printing of the designated symbol, and redirect any earlier +ordinary stop into an infinite dummy loop. It then halts exactly when +the designated symbol would have been printed, giving the halting +formulation used here. Retaining information lost by an irreversible instruction was +discussed by Landauer~\cite[Section~3]{Landauer1961} and implemented by +Bennett's history recording~\cite[Eqs.~(9)--(11)]{Bennett1973}. Our +histories remain stored. We prove the local tables and their inverses +explicitly; the cleanup and complexity statements of Bennett's reversible +simulation are not used. + +Moore's generalized shifts connect finite symbolic instructions with +positional codes and piecewise affine dynamics +\cite{Moore1990,Moore1991}. The smooth realization and suspension for +invertible generalized shifts in +\cite[Section~6, Theorem~12 and its corollary]{Moore1991} are direct +antecedents of the present viewpoint. We need additional statements on +full closed rectangles, effective derivative bounds, incompressibility, +zero initial velocity, and observation at every intermediate real time. +These are established by the explicit motions below. The classical +fragility of positional tape storage also explains why an exact event +should not be read as a uniform finite-precision tolerance. + +Cardona, Miranda, Peralta-Salas and Presas constructed Turing-complete +stationary Euler particle flows on a three-sphere with a chosen metric +\cite{CardonaMirandaPeraltaSalasPresas2021}. Their area-preserving +extension of generalized shifts uses contraction, transport and expansion +of separated blocks followed by a relative area correction +\cite[Proposition~5.1]{CardonaMirandaPeraltaSalasPresas2021}. +That construction is an important predecessor of the planar routing used +here. Our explicit normal compensation and invariant-plane formulas give +the prescribed flat three-dimensional motions directly; their full +neighborhood and detector properties are proved at the point of use. +Cardona, Miranda and Peralta-Salas also constructed Turing-complete +stationary Euler flows for the standard Euclidean metric on $\R^3$, +with computation on a noncompact invariant plane and infinite kinetic +energy \cite[Theorem~1 and the discussion following it]{CardonaMirandaPeraltaSalas2023Beltrami}. + +Universality is also known with viscosity. Dyhr, Gonz\'alez-Prieto, Miranda +and Peralta-Salas construct unforced stationary Navier--Stokes particle +universality on suitable compact three-manifolds after deforming the metric, +using Hodge viscosity and a harmonic velocity +\cite[Theorem~A]{DyhrGonzalezPrietoMirandaPeraltaSalas2026}. +The fixed flat domains, prescribed external force, and zero initial +velocity in the present paper specify a different fluid experiment. +Our results concern these constructed solutions and exact material events; +they make no assertion about arbitrary-data regularity. + +The companion \emph{Finite Instructions and Solenoidal Shear Flows} +\cite{OpenAIShear2026} gives a complete initialized construction with +reciprocal planar maps and directly solenoidal force at pressure zero. +We use its full-domain recorder in one later specialization, with an +explicit identification of all guards. The general residual forces below +need not be solenoidal. When a torus force is projected, the accompanying +pressure changes; that operation is kept distinct from the direct shear +identity. + +\subsection{Proof strategy and reading order} +The first application has four steps. Gapped radix intervals turn complete +prefix cylinders into separated closed rectangles. A local curl formula +then implements an affine volume-preserving motion on a neighborhood. +Private heights separate the instruction rectangles while a reciprocal +normal strain corrects their planar area changes. Finally, a history +invariant identifies the computation, a loading segment places the fixed +particle at its initial code, and the horizontal interpolation proves +the detector equivalence throughout every instruction period. + +Section~\ref{sec:model} records the residual realization and the exact +analytic comparison classes. Section~\ref{sec:sa-sheets} develops the +radix and moving-sheet tools. Section~\ref{sec:sa-histories} completes +Theorem~\ref{thm:sa-interleaved}. Sections~\ref{sec:sheets-boxes} +and~\ref{sec:sheets-planes} then explain the additional requirements of +full boxes and invariant-plane routing. Sections~\ref{sec:sa-separator-histories}--\ref{sec:sa-marked-five-stage} +organize the resulting processors by where they retain history: next to +a separator, on a tape track, or in a separate stack. +Sections~\ref{sec:sb-prefix}--\ref{sec:sb-events} treat further +history placements, boot instructions and fixed observers. +Section~\ref{subsec:p2:rapid-periodic} gives two compact reciprocal +realizations. The final section replaces a temporal program by a spatial +clock and verifies its separate decay and phase-window observation. diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/main.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..c5212058b62d1be43ee3ed76fe173ba11f01b55c --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/main.tex @@ -0,0 +1,56 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,enumitem,microtype,longtable,array,tikz} +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\usepackage[margin=1in]{geometry} +\usepackage[colorlinks=true,linkcolor=blue,urlcolor=blue,citecolor=blue]{hyperref} +\hypersetup{pdftitle={Prefix Instructions and Incompressible Flows},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\T}{\mathbb T} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\R}{\mathbb R} +\newcommand{\Q}{\mathbb Q} +\newcommand{\diag}{\operatorname{diag}} +\newcommand{\supp}{\operatorname{supp}} +\newcommand{\divg}{\operatorname{div}} +\newcommand{\id}{\operatorname{id}} +\title{Prefix Instructions and Incompressible Flows} +\author{OpenAI} +\date{September 27, 2026} +\begin{document} +\maketitle +\begin{abstract} +Finite prefix instructions may change area and erase information. We give +explicit history processors and smooth incompressible motions that retain +that information and realize every instruction on its full domain. A first +application assigns each machine and finite input a smooth mean-zero force +on the flat unit three-torus, at any fixed positive computable viscosity. +The solution starts from rest; a fixed particle enters a fixed open strip +exactly when the machine halts. The force repeats with period one after an +initial loading interval. We then prove alternative realizations using +full boxes, normal compensation, invariant planes, and a spatial clock. +The constructions specify their initialization, comparison class, derivative +bounds, and continuous-time observation. Exact formulas and effective +cutoffs provide finite descriptions of the fields and all their derivatives. +\end{abstract} +\input{introduction.tex} +\input{analysis.tex} +\input{sheet-geometry.tex} +\input{first-simulation.tex} +\input{full-boxes.tex} +\input{planes-and-histories.tex} +\input{prefix-realizations.tex} +\input{compact-reciprocal.tex} +\input{autonomous.tex} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/planes-and-histories.tex b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/planes-and-histories.tex new file mode 100644 index 0000000000000000000000000000000000000000..f205bf2bd0ce9ebe06e7923212e6693f53cf9bcd --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/build/planes-and-histories.tex @@ -0,0 +1,1083 @@ +\section{Planar routing and invariant coding planes}\label{sec:sheets-planes} +The contraction, parking and expansion strategy has an area-preserving predecessor in \cite[Proposition~5.1]{CardonaMirandaPeraltaSalasPresas2021}. The proof below gives the needed planar motion explicitly; the three-dimensional incompressibility is supplied separately by the invariant-plane lift. +\begin{lemma}[Planar routing through parking positions]\label{lem:sa-planar-routing} +Let $D$ be an open convex rational rectangle. Let $B_i$ and $C_i$ be +two separately disjoint finite families of nondegenerate rational closed +rectangles compactly contained in $D$. An effective smooth planar field, +supported in $(0,1)\times D$, realizes their matched positive diagonal +affine maps on positive neighborhoods of all $B_i$. If $G\subset D$ is +an open convex rational rectangle, every pair $B_i,C_i\Subset G$ can have +its entire controlled motion confined to $G$. +\end{lemma} +\begin{proof} +Take $G=D$ if no smaller safe rectangle is specified. +For an empty family use zero. Otherwise choose distinct rational +parking points in $G$ avoiding all source and +target centers. Move centers first from the sources to their parking +points, one at a time, and then from parking to their targets, one at a +time. Each destination is unoccupied. For a move $p\to q$, a two-segment +path $p\to a\to q$ avoids the finite set of stationary centers if $a$ +avoids the lines joining an endpoint to one of those centers. Choose +$a$ rational in $G$ when both endpoints are there, and in $D$ otherwise. +To make this choice effective, take a rational parabola segment in the +chosen open rectangle: every forbidden line meets it at at most two +points, so finitely many rational samples suffice. Convexity keeps the +segments inside the chosen rectangle. + +All segment-to-stationary-center squared distances and boundary +clearances have positive rational lower bounds. Choose a rational +$\epsilon>0$ smaller than every endpoint half-side, with $10\epsilon$ +below every relevant distance and boundary clearance. Shrink each +source rectangle to the square of half-side $\epsilon$ about its center, +translate these squares along the chosen segments in order, and expand +inside the target rectangles. During contraction the source cutoffs have pairwise disjoint supports; +during expansion the target cutoffs do. The moving square's support +of half-side $2\epsilon$ misses each stationary square. During a shrink/expansion about $c$ with +positive half-widths $v_a(t)$, use a cutoff times +\[ + \left(\frac{\dot v_1}{v_1}(x_1-c_1), + \frac{\dot v_2}{v_2}(x_2-c_2)\right); +\] +during a translation use a moving plateau times $\dot c$. +The exact path is $c(t)+\diag(v_1(t),v_2(t))y$, $y\in[-1,1]^2$. +Its endpoint factors are the ratios of the target to source half-sides. +Flat schedules on finitely many successive slots make a smooth pulse. +Shrinking and expanding stay in the endpoint rectangles; the prescribed +path clearances prove the extra confinement in $G$. + +Widths may be interpolated exponentially or linearly. With padded +plateaus, either choice gives an effective positive initial neighborhood +by the finite products of affine norms. A cutoff that equals one only +on the closed moving rectangle still proves the same formula there, +including its boundary, but by itself asserts no exterior neighborhood. +We shall state explicitly when such a closed-rectangle implementation +is being retained. The empty-family case uses the zero field. +\end{proof} + +\begin{lemma}[Invariant-plane lift]\label{lem:sa-invariant-lift} +Let $b(t,r)$ be an effective smooth planar field, periodic on $\mathbb T^2$ +or compactly supported in $\mathbb R^2$, with bounded mixed derivatives. +Let $h$ be smooth effective with bounded derivatives, $h(z_0)=0$ and +$h'(z_0)=1$. Then +\begin{equation}\label{eq:sa-lift} + U(t,r,z)=(h'(z)b_1,h'(z)b_2,-h(z)\operatorname{div}_r b) +\end{equation} +is divergence free and carries exactly the planar flow on $z=z_0$. +The choice $h(z)=\sin(2\pi z)/(2\pi)$ for $z_0=0$ gives a mean-zero +unit-torus field. The choice $h=(z-z_0)\chi(z)$, with $\chi=1$ near +$z_0$ and compact support, gives compact support on $\mathbb R^3$ when +$b$ is compactly supported, or a mean-zero torus field when $h$ is +supported in one vertical chart. Time periods and mixed-derivative +bounds are preserved. +\end{lemma} +\begin{proof} +The divergence is $h'\operatorname{div}b-h'\operatorname{div}b=0$. +On the indicated plane $U=(b,0)$, so ODE uniqueness gives its invariance. +For the sine choice both vertical factors integrate to zero. For compact +$h$, $\int h'=0$ and $\int\operatorname{div}b=0$ give the three means. +All remaining assertions follow by differentiation of finite products. +\end{proof} + +\section{Histories above and below separators} +\label{sec:sa-separator-histories} +A separator lets one stack contain both work symbols and a finite history +without mixing their roles. Storing history below the separator makes +the scan length depend on the updated work word. Storing it above the +separator makes the scan traverse the accumulated history instead. This +choice changes the instruction count and the unused image cylinders +available for initialization. + +\subsection{History below a delimiter}\label{subsec:sa-delimiter} +A delimiter lets a finite left work word sit above its history. A marker +on the right remembers how far the subsequent return scan must go. +Use work alphabet $\Gamma$ and normalize to one halt state $q_h$. +Introduce $\#,\star$ and records +$h_i$. For a right or stay instruction $\delta(q,a)=(q',b,d)$ let +$i=(q,a)$; for a left instruction let $i=(q,a,c)$ with +$c\in\Gamma\sqcup\{\#\}$. Write $q^+(i)=q'$. + +\begin{lemma}[Delimiter-history simulation]\label{lem:sa-delimiter} +Starting at $(N_{q_0},\#\square^\infty,w\square^\infty)$, normal +configurations have form $(N_q,L\#H\square^\infty,R)$, where $L,H$ +are finite, $R$ is eventually blank, and the represented left tape is +$L\square^\infty$. One machine instruction takes exactly $2|L'|+3$ +rules, where $L'$ is the updated finite left work word. Full source +and image cylinders are separately disjoint, and the initialized run +reaches $N_{q_h}$ exactly when the machine halts. +\end{lemma} +\begin{proof} +For $R=aR_*$ the first rule gives the updated pair +\[ + (L',R')=\begin{cases} + (bL,R_*),&d=+,\\ + (L,bR_*),&d=0,\\ + (L_*,cbR_*),&d=-,\ L=cL_*,\\ + (\epsilon,\square bR_*),&d=-,\ L=\epsilon. + \end{cases} +\] +It also places $\star$ before $R'$. The $D_i$ scan transfers $|L'|$ +work symbols until $\#$ is exposed, inserts $h_i$ below it, and enters +$U_{q^+(i)}$. Exactly $|L'|$ reverse transfers restore $L'$, and the +marker-removal rule finishes the step. This proves the count and tape +invariant, including the new blank at the left boundary. Record +displacements recover the absolute head position. + +The complete rules are +\begin{equation}\label{eq:sa-delimiter-table} +\begin{aligned} + (N_q;\epsilon,a)&\to(D_i;b,\star),&&d=+,\\ + (N_q;\epsilon,a)&\to(D_i;\epsilon,\star b),&&d=0,\\ + (N_q;c,a)&\to(D_i;\epsilon,\star cb),&&d=-,\ c\in\Gamma,\\ + (N_q;\#,a)&\to(D_i;\#,\star\square b),&&d=-,\ c=\#,\\ + (D_i;c,\epsilon)&\to(D_i;\epsilon,c),&&c\in\Gamma,\\ + (D_i;\#,\epsilon)&\to(U_{q^+(i)};\#h_i,\epsilon),\\ + (U_q;\epsilon,c)&\to(U_q;c,\epsilon),&&c\in\Gamma,\\ + (U_q;\epsilon,\star)&\to(N_q;\epsilon,\epsilon). +\end{aligned} +\end{equation} +No normal rule leaves $N_{q_h}$. + +The tested top symbols separate sources, with an extra left-top test +for left moves. Into $D_i$ a normal entry starts with $\star$ on the +right and a loop with an ordinary letter. Into $U_q$ a history entry +starts with $\#h_i$ on the left and a loop with an ordinary letter. +Distinct records identify distinct entries. Into $N_q$ there is just +its marker-removal rule. These are inverse checks for arbitrary tails, +so induction gives all assertions including initial halting. +\end{proof} + +Three useful choices of observation follow from this same delimiter +mechanism. One uses its table directly; a second changes its coordinates; +a third separates the work update and marker push into two genuine rules. +This last modification changes the sample times even though contracting +those two rules recovers the original table. + +\begin{theorem}[Delimiter processors on invariant planes] +\label{thm:sa-delimiter}\label{thm:sa-delimiter-origin} +\label{thm:sa-left-stack-plane} +For every machine and finite input, each row below gives an effective +smooth mean-zero general force on the unit torus, periodic of period one +for $t\ge1$, with all mixed derivatives bounded. Its zero-data solution +is unique in the classical periodic class and has uniformly bounded +energy. The stated fixed particle enters the stated fixed open set +exactly when the machine halts. +\begin{center}\small +\begin{tabular}{lll} +Processor & Initial label & Open set\\\hline +Delimiter & $(1/2,1/2,0)$ & $(2/3,5/6)^2\times\mathbb T$\\ +Delimiter, origin coordinates & $(0,0,0)$ & $\{x:2/30$. +\end{theorem} +\begin{proof} +For $m$ letters use digits $2,4,\ldots,2m$ and base $K=2m+2$. +Set $a_S=-E(\sharp\square^\infty)$, give halt main states distinct +offsets $4,8,\ldots$, and other states offsets $-4,-8,\ldots$. +With $b_0=-E(\square^\infty)$, encode by +$(a_s+E(L),b_0+E(R),0)$, so the fixed initial code is zero. +Lemma~\ref{lem:sa-radix} gives separated source and target rectangles. +Thicken to $R_i^-\times[-1,1]$ and apply +Lemma~\ref{lem:sa-full-box-layers} with scales +$K^{|v|-|v'|},K^{|w|-|w'|}$ and their reciprocal product. +For example take the three switches $\sigma(8t-(2j-1))$, $j=1,2,3$, +and collar one quarter of the minimum of $1$ and the rectangle gaps. +The field is zero for $t\le1/8$ and $t\ge6/8$ within a period. +Its periodization is smooth from zero, so +Lemma~\ref{lem:sa-realization} applies. + +Induction at integer times follows the initialized rule sequence. +A halt code has first coordinate at least $4$. Every nonhalting used +branch has endpoint centers at most $1$ and widths at most $1$; +the exponential-width bound gives first coordinate at most $3/2$ +throughout. The force therefore has the required all-time event. +The complete input-insertion branch is included in every period, and +its actual one-time use follows from the symbolic dynamics. +\end{proof} + +\subsection{A finite work window on each side}\label{subsec:sa-delimiter-history} +The preceding delimiter processor leaves the right tape infinite. If both +work words are finite, crossing either end must instead insert a new blank. +This changes the guards of the right-move rules and introduces a distinct +marker-push state. Here the work alphabet is $\Gamma$ and the sole halt +state is denoted by $h$. + +At a ready control $A_q$ use stacks +$L\#H\square^\infty$ and $R\#\square^\infty$, where $L,H$ are finite +and $R$ is a nonempty work word. Initially $L=H=\epsilon$ and $R=w$, +except that empty input is replaced by the one-letter word $\square$. +Denote this initial right work word by $R_{\rm in}$. +For $\delta(q,a)=(q',b,d)$ the work prefix replacements are +\[ +\begin{array}{c|c|c|l} + d&(v,w)&(v',w')&\text{condition}\\\hline + 0&(\epsilon,a)&(\epsilon,b)&\\ + +&(\epsilon,ac)&(b,c)&c\in\Gamma\\ + +&(\epsilon,a\#)&(b,\square\#)&\\ + -&(c,a)&(\epsilon,cb)&c\in\Gamma\\ + -&(\#,a)&(\#,\square b)& +\end{array} +\] +Give every alternative its own index $r$, control $P_r$, and record +$\rho_r$. Its state change is $A_q\to P_r$ and $q'(r)=q'$. +Add a marker $J$ and controls $S_r,T_q$, where $q$ runs through every +machine state, including $h$. The rules are +\[ +\begin{array}{ll} + (P_r;\epsilon,\epsilon)\to(S_r;\epsilon,J),\\ + (S_r;c,\epsilon)\to(S_r;\epsilon,c),&c\in\Gamma,\\ + (S_r;\#,\epsilon)\to(T_{q'(r)};\#\rho_r,\epsilon),\\ + (T_q;\epsilon,c)\to(T_q;c,\epsilon),&c\in\Gamma,\\ + (T_q;\epsilon,J)\to(A_q;\epsilon,\epsilon). +\end{array} +\] +The full alphabet is $\Gamma_{\rm full}=\Gamma\sqcup\{\#,J\} +\sqcup\{\rho_r\}_r$, with all added symbols distinct. +There is no work rule from $A_h$. +The work rule updates the finite tape interval, adjoining a blank exactly +when a move crosses one of its ends. After pushing $J$, the $S_r$ scan +transfers the updated left word, inserts $\rho_r$ below its delimiter, +and the $T_q$ scan restores it and removes $J$. Hence one step takes +$4+2|L'|$ rules and leaves $R'$ nonempty. All finite scans terminate. +Sources at $A_q$ are separated by the scanned symbol and the displayed +additional boundary test. Each $P_r$ has one incoming rule. Images +into $S_r$ have right top $J$ or an ordinary letter; images into $T_q$ +have left prefix $\#\rho_r$ or an ordinary letter. Each $A_q$ has one +marker-removal entry. These observations prove source and image +separation for arbitrary tails, as well as the initialized halt equivalence. + +\begin{theorem}[Finite windows and a fixed slab]\label{thm:sa-delimiter-history} +This processor gives an effective smooth mean-zero force on the unit +torus, periodic of period one for $t\ge1$, with every mixed derivative +bounded. Its zero-data solution is globally smooth, unique in the +classical periodic class, and has uniformly bounded energy. The particle +from $(1/4,1/4,1/4)$ enters $\{x:1/2h_n$. The compute rule marks $h_n$; the +outgoing scan reaches $T_n$, records the old data, advances the pointer, +and returns from $T_n-1$. Every return-scan cell is unmarked until the +marked one, and its right neighbor is never the new pointer. Thus each +guard holds. Clearing the mark and moving by $d$ gives the next machine +configuration, with $h_n+d\le T_n0$. The empty list uses zero. + +Let $E_0,y_*$ be the rational initial planar codes. On the torus load +from $a=(1/16,1/16)$ by the spatially constant field +$\dot\eta(t)(E_0-a)$, with $\eta$ flat from zero to one in +$[1/4,3/4]$. Follow by the repeated rule pulse and lift with +$h(z)=\sin(2\pi z)/(2\pi)$. In Euclidean space load the origin along +$\theta(t)y_*$ with a compact moving cutoff equal to one on a square +of half-side one and supported in its enlargement by one. Follow by +the repeated planar pulse and lift with $h(z)=z\chi(z)$, where +$\chi=1$ on $[-1,1]$ and is supported in $(-2,2)$. +Lemma~\ref{lem:sa-invariant-lift} gives the exact planar motions and +Lemma~\ref{lem:sa-realization} gives the forces. Its bounded-velocity +comparison refinement is exactly the second row's uniqueness class. + +At time one both particles are their initial codes. Subsequent rule +samples encode the run. In the torus case the $n$th compute visit is +at $1+\sum_{j0$&$t=0$\\ +\multicolumn{4}{l}{\emph{Reserved-loader sheets}}\\ +Left reserved sheets&$\mathbb T^3$, $(1/2,1/2,1/4)$&$(1/8,1/4)$&$t=0$\\ +Right reserved sheets&$\mathbb T^3$, $(1/2,1/8,1/4)$&$(3/4,15/16)$&$t=0$\\ +\multicolumn{4}{l}{\emph{Invariant planes}}\\ +Frontier sine plane&$\mathbb T^3$, $(1/2,1/4,0)$&$(5/8,15/16)$&$t=1$\\ +Previous-tag plane&$\mathbb T^3$, $(1/8,1/4,0)$&$(2/3,5/6)$&$t=0$\\ +Shifted-start plane&$\mathbb T^3$, $(3/4,3/4,1/2)$&$(0,1/3)$&$t=0$\\ +Separator sine plane&$\mathbb T^3$, $(1/4,1/2,0)$&$(1/2,1)$&$t=1$\\ +Two-cell-frontier plane&$\mathbb T^3$, $(1/8,1/8,0)$&$(2/3,7/8)$&$t=1$\\ +Zero-blank plane&$\mathbb T^3$, $(1/4,1/2,1/2)$&$(3/4,7/8)$&$t=1$ +\end{tabular} +\end{center} +For the rows beginning at $t=1$, the repeated program depends only on the machine table, while the input appears in the loading pulse. The rows beginning at zero incorporate loading into the repeated prefix table, reserve a disjoint loader image, or place the initial code at the fixed label as specified in the proof. No force in this theorem is required to be solenoidal. The separator sine-plane row additionally permits an effective mean-zero solenoidal force with the same velocity and event. For arbitrary fixed real $\nu>0$, the same algebraic formulas and conclusions hold, with numerical evaluation relative to that parameter. +\end{theorem} + +\subsection{Logging after an update} +\label{subsec:sb-postlog} +The first processor keeps a finite left tape word above a delimiter and hides the history below it. It handles a left move beyond that word by exposing an implicit blank. + +\begin{lemma}[Finite left word and a subsequent history sweep]\label{lem:sb-postlog} +The delimiter-history table of Lemma~\ref{lem:sa-delimiter}, with +$(N_q,D_i,U_q,\star)$ renamed $(P_q,T_i,U_q,\diamond)$, applies to any finite halt set $H$ by omitting its normal rules at every $q\in H$. It simulates one machine transition in exactly $2|L'|+3$ instructions, with $L'$ the updated finite left word, and has separately disjoint full source and image cylinders. Starting with an empty left word, $|L_j|\le j$ after $j$ machine transitions. +\end{lemma} +\begin{proof} +The displayed renaming is bijective and changes no write, displacement or +top-letter guard in \eqref{eq:sa-delimiter-table}; it therefore conjugates +the partial maps on all their cylinders. Allowing several halt states +only omits their normal rules. The inverse distinctions at transfer, +history-insertion and restoration states are unchanged, so the full-domain +proof of Lemma~\ref{lem:sa-delimiter} applies to this extension. +That lemma also gives the initialized simulation and the exact count. +The finite left word increases in length by at most one per ordinary +transition, hence $|L_j|\le j$. Displacement records recover the absolute +head position. It lies in $[-j,j]$ after $j$ transitions, and every +nonblank work cell lies in $[-j,|w|+j]$: each step writes at the old head +and moves by at most one. These bounds recover the finite tape content +in addition to the halting event. +\end{proof} + +The bottom history tail may be $\#^\infty$, $\square^\infty$, or a +reserved letter $h_*\square^\infty$. None changes a rule: the history +scan stops at the exposed delimiter. More precisely, on the invariant +sets reached from these starts, replace the bottom tail while retaining +the finite record list. This replacement commutes with every instruction. +The following applications choose the tail according to their loading +geometry; the symbolic inverse and count remain those just proved. + +\subsection{Logging the previous tag before an update} +\label{subsec:sb-prelog} +A second table incorporates loading into the repeated program and logs the previous tag before executing the next transition. This changes both the location of history and the exact instruction count. + +\begin{lemma}[Previous-tag processor]\label{lem:sb-prelog} +There is an injective partial prefix processor starting from a distinguished state $I$ with both stacks $\square^\infty$. One instruction loads the input. At a nonhalting main state with $k$ stored left cells, the next machine transition takes exactly $2k+4$ instructions. +\end{lemma} +\begin{proof} +Give each update case of Lemma~\ref{lem:sb-postlog} its own tag $\lambda$, and add a reserved tag $\bot$. Use distinct letters $h_\lambda$, markers $\#,\diamond$, and states $B(q,\lambda)$, $U(q,\lambda)$, $V(q),D(q)$, with the last three types needed only for nonhalting $q$. + +At a main state the stacks are $L\#Z,R$. The first rule inserts $\diamond$, the $k=|L|$ transfers and $k$ restores leave the tape unchanged, insert $h_\lambda$ into $Z$, and remove $\diamond$. The final update is exactly the four-case tape update proved above. The instruction count is $1+k+1+k+1+1=2k+4$. Write $\lambda_0=\bot$ and let $\lambda_j$ be the $j$th update tag for $j\ge1$. After $j\ge0$ transitions the current tag is $\lambda_j$ and the history is $Z=h_{\lambda_{j-1}}\cdots h_{\lambda_0}\square^\infty$, with an empty recorded prefix when $j=0$. Thus the loaded configuration has tag $\bot$ and blank history, while for $j\ge1$ the recorded prefix ends at $h_\bot$. This recovers the displacement history and proves the configuration and halting correspondence, including an initially halted machine after the one loading instruction. + +The complete rule list begins +\begin{align*} + (I;\epsilon,\epsilon)&\mapsto(B(q_0,\bot);\#,w),\\ + (B(q,\lambda);\epsilon,\epsilon)&\mapsto(U(q,\lambda);\epsilon,\diamond),\quad q\notin H,\\ + (U(q,\lambda);c,\epsilon)&\mapsto(U(q,\lambda);\epsilon,c),\quad c\in A,\\ + (U(q,\lambda);\#,\epsilon)&\mapsto(V(q);\#h_\lambda,\epsilon),\\ + (V(q);\epsilon,c)&\mapsto(V(q);c,\epsilon),\quad c\in A,\\ + (V(q);\epsilon,\diamond)&\mapsto(D(q);\epsilon,\epsilon). +\end{align*} +For a transition $\delta(q,a)=(q',b,d)$ the rules from $D(q)$ are +\[ +\begin{array}{c|c|c} +1&(D(q);\epsilon,a)&(B(q',\lambda);b,\epsilon)\\ +0&(D(q);\epsilon,a)&(B(q',\lambda);\epsilon,b)\\ +-1, c\in A&(D(q);c,a)&(B(q',\lambda);\epsilon,cb)\\ +-1, c=\#&(D(q);\#,a)&(B(q',\lambda);\#,\square b). +\end{array} +\] +There are no outgoing rules from halting $B(q,\lambda)$. +The tags make every incoming branch to a $B$-state unique, including the loader, whose tag is $\bot$. Images at $U$ are separated by right tops $\diamond$ versus $c\in A$; images at $V$ by left prefixes $\#h_\lambda$ versus $c\in A$; into $D(q)$ there is one rule. Source separation is given by the displayed top-letter tests. Consequently both cylinder families are disjoint. +\end{proof} + +\subsection{A finite right word extended by a blank} +\label{subsec:sb-rightword} +The previous processors store only finitely many left cells. The following one instead keeps the left half-tape infinite and the right tape window finite. Its scan explicitly appends a blank before returning. + +\begin{lemma}[Right-word processor]\label{lem:sb-rightword} +An effective injective prefix table represents a machine configuration by +\[ + (q,L,r\#h\square^\infty),\qquad L\in A^{\mathbb N},\quad r\in A^*,\quad |r|\ge1, +\] +where $r$ starts at the head and all unlisted cells to its right are blank. If the local write and move leave a word $v$ before $\#$, one simulated transition uses exactly $2|v|+3$ instructions and replaces the right stack by $v\square\#\tau_eh\square^\infty$. +\end{lemma} +\begin{proof} +Index $e=(q,\ell,a)$ for every nonhalting $q$ and $\ell,a\in A$, and write $(q_e,b_e,d_e)=\delta(q,a)$. Add $\#,\diamond,\tau_e$ and states $F_e,J_q$. The first rule has source $(q;\ell,a)$ and target +\[ + \begin{cases} + (F_e;\diamond b_e\ell,\epsilon),&d_e=1,\\ + (F_e;\diamond,\ell b_e),&d_e=-1,\\ + (F_e;\diamond\ell,b_e),&d_e=0. + \end{cases} +\] +Add +\begin{align*} + (F_e;\epsilon,c)&\mapsto(F_e;c,\epsilon)&&(c\in A),\\ + (F_e;\epsilon,\#)&\mapsto(J_{q_e};\epsilon,\square\#\tau_e),\\ + (J_q;c,d)&\mapsto(J_q;\epsilon,cd)&&(c,d\in A),\\ + (J_q;\diamond,\epsilon)&\mapsto(q;\epsilon,\epsilon). +\end{align*} +All source tests are disjoint. Into $F_e$ the entry starts the left stack with $\diamond$, whereas each loop starts it with a distinct tape letter. Into $J_q$ the right prefixes $\square\#\tau_e$ are separated from one another by their tags and from every $cd$ by the second letter; the $cd$ themselves distinguish loop images. There is one incoming rule to $q$. + +Writing $r=ar_1$, the word $v$ is respectively $r_1$, $\ell b_er_1$, or $b_er_1$. The left stack below the inserted $\diamond$ is exactly the updated infinite left tape. The $F_e$ loop moves the finite word $v$ onto that stack in reverse order. The delimiter rule adds a blank and records $e$. The $J_{q_e}$ loop restores $v$ while its second-stack first letter remains in $A$, and its exit removes $\diamond$. There are $1+|v|+1+|v|+1$ rules. In particular the possibly empty $v$ after a right move causes no undefined rule: the inserted blank supplies the next scanned cell. Start with $L=\square^\infty$, $r=w\square$, and empty history. Induction proves the claimed tape interpretation and exact halting equivalence. History length gives the number of simulated transitions and their displacements. After $k$ transitions the head lies in $[-k,k]$ and no nonblank tape cell lies outside $[-k,|w|+k]$. Reading the finite right word up to $\#$ and the first $2k$ letters of $L$ therefore recovers all possibly nonblank tape cells: a left offset greater than $2k$ has absolute index less than $-k$. The record list supplies the absolute head index. +\end{proof} + +\subsection{Tagged transfer without a temporary right delimiter} +\label{subsec:sb-tagged} +\begin{lemma}[Tagged left-word sweep]\label{lem:sb-tagged} +There is an injective prefix processor whose update sweeps the finite left word through a disjoint tagged alphabet. Its exact transition count is $2m+3$, where $m$ is the updated left-word length. +\end{lemma} +\begin{proof} +Use main labels $M_q$, branch labels $D_e$, restore labels $C_q$, a delimiter $\#$, distinct records $k_e$, and a disjoint tagged copy $\bar A=\{\bar a:a\in A\}$. + +At $M_q$ the stacks are $L\# Z,R$, starting at $\#^\infty,w\square^\infty$. Each main rule performs precisely the head-relative tape update; for a right move the lookahead $c$ is retained and becomes current. The $m$ updated left letters are transferred with tags, the record is inserted below $\#$, and the $m$ tagged letters are restored and untagged. The untagged right top then triggers the exit. This gives $2m+3$ instructions and the next main configuration. Every scan is finite, and no untagged tape data are confused with the tagged scan segment. The same induction as in Lemma~\ref{lem:sb-postlog} proves the halting equivalence and reconstruction of head displacement. + +A right-moving transition is split by its next right letter: +\[ + (M_q;\epsilon,ac)\mapsto(D_e;b,c),\qquad c\in A. +\] +For a left move use $(M_q;\alpha,a)\mapsto(D_e;\epsilon,\alpha b)$ for $\alpha\in A$, and $(M_q;\#,a)\mapsto(D_e;\#,\square b)$. For a stay use $(M_q;\epsilon,a)\mapsto(D_e;\epsilon,b)$. Every displayed case has its own $e$, with destination $q'(e)$. Complete the table by +\begin{align*} + (D_e;\alpha,\epsilon)&\mapsto(D_e;\epsilon,\bar\alpha),\quad\alpha\in A,\\ + (D_e;\#,\epsilon)&\mapsto(C_{q'(e)};\#k_e,\epsilon),\\ + (C_q;\epsilon,\bar\alpha)&\mapsto(C_q;\alpha,\epsilon),\quad\alpha\in A,\\ + (C_q;\epsilon,a)&\mapsto(M_q;\epsilon,a),\quad a\in A. +\end{align*} +Sources separate by their indicated letters and the right lookahead. Into $D_e$, the main-rule image has an untagged right top while loop images have distinct tagged tops. Into $C_q$, the left prefixes $\#k_e$ separate entries from loops with untagged left tops. The incoming rules to $M_q$ preserve distinct right tops $a$. These facts prove full-cylinder injectivity. +\end{proof} + +\subsection{A finite tape window and a boot instruction} +\label{subsec:sb-window} +This processor stores a finite tape window in one stack and its history in the other. Its fixed initial code has both stack coordinates equal to zero, which will make the entire velocity periodic from the initial time. + +\begin{lemma}[Window processor]\label{lem:sb-window} +There is an effective injective prefix table with a boot state $s_*$ and main states $N_q$. From $(s_*,E^\infty,E^\infty)$ it loads the input in one instruction. At a main configuration +\[ + (N_q,\ell Ja rE^\infty,\sigma g_*E^\infty) +\] +it represents the finite tape window $\ell ar$, with head at $a$; both words $\ell,r$ are in left-to-right order and all cells outside the window are blank. One machine transition takes $3+|\ell|+|\ell_{\rm new}|$ instructions. +\end{lemma} +\begin{proof} +Use fresh markers $J,E$, a history base $g_*$, and records $g_\tau$. A tag $\tau$ specifies $(q,a,d,c)$ for a moving rule, where $c\in A$ is the neighboring tape letter or $c=\perp$ denotes a window edge; a stay has tag $(q,a,0,\perp)$. Use scan states $S_q$ for nonhalting $q$ and restore states $D_\tau$. With $w_0=w$ for nonempty $w$ and $w_0=\square$ otherwise, the boot and scan rules are +\begin{align*} + (s_*;\epsilon,\epsilon)&\mapsto(N_{q_0};Jw_0E,g_*),\\ + (N_q;\epsilon,h)&\mapsto(S_q;\epsilon,h),\quad q\notin H,\ h\in\Lambda,\\ + (S_q;c,\epsilon)&\mapsto(S_q;\epsilon,c),\quad c\in A, +\end{align*} +where $\Lambda=\{g_*\}\cup\{g_\tau\}$. For $\delta(q,a)=(q',b,d)$ the update rules are +\[ +\begin{array}{c|c|c} +0&(S_q;Ja,\epsilon)&(D_{(q,a,0,\perp)};Jb,\epsilon)\\ +1, c\in A&(S_q;Jac,\epsilon)&(D_{(q,a,1,c)};Jc,b)\\ +1,\ \text{edge}&(S_q;JaE,\epsilon)&(D_{(q,a,1,\perp)};J\square E,b)\\ +-1, c\in A&(S_q;Ja,c)&(D_{(q,a,-1,c)};Jcb,\epsilon)\\ +-1,\ \text{edge}&(S_q;Ja,h)&(D_{(q,a,-1,\perp)};J\square b,h),\quad h\in\Lambda. +\end{array} +\] +Finally include $(D_\tau;\epsilon,c)\mapsto(D_\tau;c,\epsilon)$ for $c\in A$ and $(D_\tau;\epsilon,h)\mapsto(N_{q'_\tau};\epsilon,g_\tau h)$ for $h\in\Lambda$. + +The domains separate by history tops at $N$, by tape tops versus $J$ at $S$, by the tested current letter and neighbor, and by second-stack tops at $D$. Into $S_q$, history tops distinguish entry from tape-letter scan images. Into $D_\tau$, first tops $J$ distinguish updates from restore loops; the only multiple update entries sharing one tag are the left-edge variants, whose second tops $h$ distinguish their images. Into a main state the prefixes $g_\tau h$ distinguish every restore image; the boot image instead starts with $g_*$. Hence both full cylinder families are disjoint. + +Entering $S_q$ and transferring $\ell$ leaves $Ja rE^\infty$ and puts $\operatorname{rev}(\ell)$ above the old history. A stay replaces $a$ by $b$. A right move pushes $b$ onto this reversed left segment and takes the next right cell as current, inserting $\square$ if at the edge. A left move pops the nearest left cell, or uses a new blank if there is none, and puts $b$ at the front of the right segment. The $D_\tau$ loops restore the new left segment in its original order, and the exit records the tag. Counting entry, the $|\ell|$ scan moves, the update, $|\ell_{\rm new}|$ restore moves, and the exit proves the formula. This also proves the invariant and that every simulated step is finite. The tag list determines absolute head position. An initially halted machine is detected immediately after boot; otherwise halting and nonhalting follow by induction. +\end{proof} + +\subsection{Two finite-window history programs} +\label{subsec:sb-frontier} +The next two processors use an ordinary tape with data, history, and marker tracks. They differ in how the right history frontier advances. Retaining this distinction records two finite local implementations of the same tape computation. + +\begin{lemma}[Frontier moved by a separate instruction]\label{lem:sb-frontier}\label{lem:p2:rapid-history} +There is a partial one-head machine with alphabet +$\Sigma=A\times\{\mathtt e,\mathtt F,\lambda_r:r\in\mathcal R\}\times\{0,1\}$, +where $\mathcal R=(Q\setminus H)\times A$, and states +$B_q,N_q,L_q,A_r,R_r$. With the history frontier initially at absolute cell $2$, it simulates transition number $n+1$ in +\[ + 2(2+n-h_{n+1})+4 +\] +auxiliary steps, where $h_{n+1}$ is the new machine head position. For every target state, all incoming rules have the same direction and the written combined symbol distinguishes their old state and scanned symbol. +\end{lemma} +\begin{proof} +This is the recorder of \cite[Lemma~2.1]{OpenAIShear2026} with initial frontier +$r_0=2$, under the following bijections of control states and history +letters: +\[ + (S_q,A_r,R_r,F_q,L_q)\mapsto(B_q,A_r,R_r,N_q,L_q),\qquad + (E,P,[r])\mapsto(\mathtt e,\mathtt F,\lambda_r). +\] +The work alphabet and marker bit are unchanged. In rows one, two, three, +six, and seven the history symbol is required to differ from the +frontier; row four reads the frontier, and row five reads an empty +history cell. These are exactly the guards in \cite[Lemma~2.1]{OpenAIShear2026}. +The writes and displacements also agree row by row. Thus these +bijections conjugate the entire partial transition, not only its +initialized orbit. Its full-domain inverse, destination-determined +incoming directions, and distinguishing written symbols transfer. + +The initialization sets every history cell to $\mathtt e$ except +for $\mathtt F$ at absolute cell $2$, and sets every marker bit to +zero. The checkpoint invariant of \cite[Lemma~2.1]{OpenAIShear2026} gives records +at $2,\ldots,n+1$, frontier $2+n$, and original work head $h_n$. +Its count $2(r_0+n-h_{n+1})+4$ is the asserted count. The same invariant +includes an initially halted machine. +\end{proof} + +To convert this table to two stacks, list cells to the left nearest first and cells from the head rightwards. For an auxiliary rule scanning $g$, writing $g'$, and moving $d$, use prefix pairs +\[ +\begin{array}{c|c|c} +d&\text{source}&\text{target}\\\hline +1&(\epsilon,g)&(g',\epsilon)\\ +0&(\epsilon,g)&(\epsilon,g')\\ +-1&(e,g)&(\epsilon,eg'),\quad e\in\Sigma. +\end{array} +\] +Determinism separates the sources. Within one target state the direction is fixed; the written symbol separates targets in the left coordinate for $d=1$, in the right coordinate for $d=0$, and in the two-letter right prefix $(e,g')$ for $d=-1$. Thus this is again an injective full-cylinder table. + +\begin{lemma}[Two-cell frontier update]\label{lem:sb-neighbor} +There is an injective partial update on a centered tape with alphabet +$\Sigma=A\times(\{\bot,\#\}\sqcup\mathcal R)\times\{0,1\}$ and labels +$S_q,L_q,P_r,R_r$. At the $n$th main checkpoint its frontier is $e=2+n$, and the next machine transition takes exactly $2(e-h_{n+1})+2$ updates. Each branch can be specified by the three source cells $-1,0,1$ and has a whole product prefix cylinder as its image. +\end{lemma} +\begin{proof} +Write a tape cell as $(a_j,g_j,m_j)$ with indices relative to the head. After the indicated edits a displacement $d$ recenters by $s'_j=\widetilde s_{j+d}$. + +Initially the auxiliary head is at absolute cell $0$ in $S_{q_0}$, all marker bits are zero, the history symbol is $\#$ at absolute cell $2$ and $\bot$ everywhere else, and the work track is the input followed by blanks on the rest of the tape. At checkpoint $n$, the records occupy cells $2,\ldots,n+1$, the frontier is at $e=2+n$, and the other history cells are empty. This holds at initialization. After the work update the new head $h'0$. Take $\eta=g/4$ and +$Z_i=(2H_3+2)i$, using successive flat switches on +$(1/8,1/4)$, $(3/8,1/2)$ and $(3/4,7/8)$. +Endpoint gaps lose only $2\eta\le g/2$, while a height gap exceeds +$2H_3+2\eta$. Thus these particular margins and heights satisfy every +separation and plateau condition of that proof. The centers interpolate +horizontally and the half-widths interpolate geometrically, so its +observer bounds apply without any change of physical coordinates. + +The normal scale has an explicit symbolic meaning. For an unscaled state interval $[o_s,o_s+1]$ and a prefix rule, the horizontal factors are +\[ + \lambda_1=B^{|\alpha|-|\alpha'|},\qquad + \lambda_2=B^{|\beta|-|\beta'|},\qquad + \lambda_3=(\lambda_1\lambda_2)^{-1}. +\] +Source thickness $[-1,1]$ and target thickness $[-\lambda_3,\lambda_3]$ therefore satisfy Lemma~\ref{lem:sb-boxes}. A code lies at height zero at every instruction boundary, so it lies in the next source box regardless of the previous target thickness. No invariant three-dimensional neighborhood of the code set is asserted. + +The same proof permits the following two concrete choices of collars +and stage times. These choices change the interpolating field, while +preserving its proved endpoint map and observer estimate. For the odd-digit left-word code let $m_*$ be the largest prefix length and take relative padding $\eta_*=B^{-m_*}/4$. Enlarge every moving box by a factor $1+\eta_*$. Endpoint horizontal widths are at most $1$, so a separating gap loses at most $\eta_*$ and stays positive. The heights +\[ + Z_i=4(1+\eta_*)\max_j\{1,\lambda_{j3}\}\,i +\] +separate the middle-stage vertical extents. Use three successive ramps $r_j(t)=r(3t-j+1)$, where +$r(v)=\rho(v-1/4)/(\rho(v-1/4)+\rho(3/4-v))$ and $\rho$ is the flat function in \eqref{eq:p2-step}. These are exactly the three lift, change, and lower stages of the proof, with geometric shape interpolation. For the boot-window code use instead $\gamma=B^{-m_*}$, additive padding $\gamma/4$, and private heights $4(R+1)i$, $R=\max(1,\max_i\lambda_{i3})$, on three equal stages with progress $\sigma(3v-1)$. Both choices meet the same disjoint-neighborhood proof, including every closed-box boundary point. + +\begin{lemma}[Storage and delivery for a bounded observer]\label{lem:sb-storage} +Suppose the horizontal source and target rectangles for a prefix table lie in state bands $[o_s,o_s+1]\times[0,1]$, where the unique halting band has $o_h=0$ and every other band has $o_s\ge3$. Give source boxes half-height +\[ + h=\frac1{1+\sum_j\lambda_{j3}} +\] +and target boxes half-height $h\lambda_{j3}$. An effective divergence-free pulse implements their determinant-one affine maps in $7N$ slots. Every trajectory from a box whose source and target bands are nonhalting avoids +$\mathcal O=(-1,2)\times(-1,2)\times(-1,1)$ throughout the pulse. +\end{lemma} +\begin{proof} +If $N=0$, use the zero pulse. Assume henceforth that $N>0$. +Every source vertical half-width is $h\le1$, and every target half-width +is $h\lambda_{j3}<1$; horizontal half-widths are at most $1/2$. +Thus every endpoint half-width is at most one. Put storage centers $m_j=(-10-5j,0,0)$, horizontal storage squares of half-side one at these centers, and travel height $Z=6$. The source projections together with these storage squares form a disjoint family, and so do the target projections together with the storage squares. Choose a rational margin $0<\eta<1/2$ so their respective enlargements remain disjoint. + +Use $3N$ consecutive slots to take each source box in turn vertically up by $6$, horizontally to its storage center at height $6$, and down. A localized translation is the curl of +\[ + \tfrac12\chi(x-c(t))\,[c'(t)\times(x-c(t))], +\] +where $\chi=1$ on a positive neighborhood of the moving box and is supported in its $\eta$ enlargement. For a half-width vector $r$, write $\chi_{r,\eta}$ for the product +cutoff equal to one on the box of half-widths $r+\eta/2$ and supported +in the box of half-widths $r+\eta$. Use another $N$ slots to apply each +diagonal scaling at its storage center, by the curl of +\[ + \tfrac13\chi_{(1,1,1),\eta}(x-m_j) + [\theta'(t)G_j(x-m_j)\times(x-m_j)],\qquad + G_j=\operatorname{diag}(\log\lambda_{j1},\log\lambda_{j2},\log\lambda_{j3}). +\] +Here $\operatorname{tr}G_j=0$ and $\theta$ is a flat progression from $0$ to $1$ on that slot. The exact scaling path is $m_j+\exp(\theta G_j)(x-m_j)$; all its half-widths stay at most one, inside the cutoff plateau. Finally use $3N$ slots to deliver the scaled boxes, one at a time, to their target centers by the reverse type of high route. + +During each vertical leg the projected support is separated from every stationary box by the relevant source/storage or storage/target family. During a horizontal leg the support is above $6-1-\eta$, while stationary boxes lie in $[-1,1]$ in height. Scaling supports are disjoint at the storage centers. Hence the other tracked boxes remain stationary at every slot; induction proves the precise composition on each complete source box and on a positive neighborhood, by the curl-motion lemma and the finite positive margins. Source-target intersections cause no conflict because all sources are stored before any target is occupied. + +A nonhalting tracked point has first coordinate at least $3$ in source and target footprints, and less than $-1$ in storage footprints. Its vertical legs and scaling stay over these footprints. Horizontal legs have height at least $5$. Each possibility excludes $\mathcal O$, proving all-time avoidance. +\end{proof} + + +\begin{figure}[ht] +\centering +\begin{tikzpicture}[x=0.78cm,y=0.70cm,>=stealth] + \draw[fill=gray!15] (4,0) rectangle (6,1.4); + \node at (5,.65) {$\mathcal O$}; + \draw[fill=blue!12] (.5,-.15) rectangle (1.5,.15); + \node[below] at (1,-.25) {storage}; + \draw[fill=orange!15] (8.5,-.15) rectangle (9.5,.15); + \node[below] at (9,-.25) {source or target}; + \draw[->,thick] (9,.2)--(9,3)--(1,3)--(1,.2); + \node[above] at (5,3) {travel above the detector}; + \draw[dashed] (-.1,2.5)--(10,2.5); + \node[right] at (10,2.5) {$z>1$}; +\end{tikzpicture} +\caption{Storage routing for a bounded detector, with the second +horizontal coordinate suppressed. All sources are evacuated before +any target is occupied. Scaling takes place at the storage positions; +horizontal travel takes place at height six. The drawing is schematic: +the storage and endpoint footprints are separated by the explicit +margins in Lemma~\ref{lem:sb-storage}.} +\label{fig:sb-storage} +\end{figure} + +\subsection{Sheet and planar alternatives} +\label{subsec:sb-sheets} +When only a planar sheet carries the code, its vertical thickness can +be chosen after the horizontal motion. This allows a stronger path +statement: every horizontal coordinate of a tracked point can be made +a convex combination of its own endpoints. The widths in the two +horizontal directions need neither agree nor have equal products. For source and target horizontal centers $a_i,b_i$ and half-widths $r_{ik}^0,r_{ik}^1$, interpolate the center and both half-widths by the same parameter $s$: +\[ + c_{i,\mathrm{hor}}=(1-s)a_i+sb_i,\qquad + r_{ik}=(1-s)r_{ik}^0+sr_{ik}^1. +\] +Take starting height $z_0=1/4$ and private heights $z_i=1/2+i/(4(N+1))$. Lift, interpolate, and lower on disjoint time stages. A point of fixed fractional horizontal coordinates follows the convex combination of its own two endpoints. The affine generator is +\[ + c_i'(t)+\operatorname{diag}\left(\frac{r'_{i1}}{r_{i1}}, + \frac{r'_{i2}}{r_{i2}},-\frac{r'_{i1}}{r_{i1}}-\frac{r'_{i2}}{r_{i2}}\right)(x-c_i(t)). +\] +This is exactly the reciprocal-normal-strain construction in Lemma~\ref{lem:sa-sheet-routing}, applied inside the open unit cube to the stated rectangles. To verify its chart hypothesis for rectangles elsewhere than $[1/4,3/4]^2$, take one tenth of the minimum of $1$, the actual endpoint chart margins, the horizontal sup-norm gaps in the two endpoint families, and the private-height gaps. The expanded plates then remain inside the cube and are separated by footprints during lift and lowering, and by height during interpolation. The proof of that lemma consequently applies without a coordinate or viscosity normalization. + +An invariant-plane realization solves a different problem. We first +allow the planar field to compress the rectangles to small squares, +route the squares through parking positions, and expand at the targets. +The lift will cancel its divergence in the normal direction. +For planar routing we use Lemma~\ref{lem:sa-planar-routing} with $D=(0,1)^2$ and the safe rectangle $G=(0,1/2)\times(0,1)$, or the right half of $D$ as specified below. There are several explicit choices of the same finite path construction. They preserve the full-rectangle affine maps but have their own smooth paths: +\begin{enumerate} +\item For the frontier sine plane, use initial contraction on $[0,1/4]$, $4N$ successive segment intervals on $[1/4,3/4]$, and expansion on $[3/4,1]$. The contraction factors are $1-s+s\epsilon/r_{ik}$, and the expansion factors $1-s+sr'_{ik}/\epsilon$. To avoid $l$ forbidden lines, select a vertex among $2l+3$ rational points $(a,1/4+a^2)$, $1/81/3$. +\item For the separator sine plane, use moving centers $c_i$ and half-size matrices $M_i$ throughout all stages, with linear interpolation during shrink and expansion. Relative padded boxes $c_i+\lambda M_i[-1,1]^2$ are disjoint, and the field is +\[ + \sum_i\psi(M_i^{-1}(X-c_i)) + [c_i'+M_i'M_i^{-1}(X-c_i)]. +\] +Here $\lambda=1+d_0/4$ with $d_0$ a positive endpoint margin/gap bound. If $d_1$ bounds required boundary margins and $d_2\le1$ bounds the positive squared center separations along all planned segments, choose +$\epsilon=\tfrac12\min(m_0,d_1/\lambda,d_2/(4\lambda))$, with $m_0$ the minimum half-side. Then padded tiny squares do not intersect, since their intersection would force center distance at most $2\sqrt2\lambda\epsilon0$, the product +\[ + \sigma\!\left(\frac{x-a+\rho}{\rho/2}\right) + \sigma\!\left(\frac{b+\rho-x}{\rho/2}\right) +\] +equals one on its $\rho/2$ enlargement and is supported in its $\rho$ +enlargement. Products in the coordinates give box cutoffs. The effective +flatness estimates in Section~\ref{sec:model} justify evaluation of all +mixed derivatives across their seams; no equality test at a seam is used. + +\begin{lemma}[Localized affine motions]\label{lem:sa-curl-motion} +Let $c(t)\in\mathbb R^3$ and a positive diagonal matrix $D(t)$ be smooth +effective functions on a compact interval, with $D(t_0)=I$ and +$\det D(t)=1$. Put $A=\dot D D^{-1}$. Let $K_0$ be a rectangular box, allowing +zero thickness in a coordinate. If an effective smooth compactly supported +cutoff $\chi$ equals one on a +positive collar of the moving box $K_t=c(t)+D(t)(K_0-c(t_0))$, then +\begin{equation}\label{eq:sa-curl} + V(t,x)=\nabla\times\left\{\chi(t,x) + \left[\tfrac12\dot c\times(x-c) + +\tfrac13(A(x-c))\times(x-c)\right]\right\} +\end{equation} +is divergence free and its trajectory from $x_0\in K_0$ is +\begin{equation}\label{eq:sa-affine-path} + x(t)=c(t)+D(t)(x_0-c(t_0)). +\end{equation} +The formula holds on an effective positive neighborhood of $K_0$. +Disjoint moving supports allow finitely many such motions to be summed. +\end{lemma} +\begin{proof} +Differentiating $\det D=1$ gives $\operatorname{tr}A=0$. With $Y=x-c$, +\[ + \nabla\times(\dot c\times Y)=2\dot c,\qquad + \nabla\times((AY)\times Y)=3AY. +\] +The second identity follows by expanding the curl and using +$\operatorname{div}(AY)=0$. Thus on the plateau $V=\dot c+AY$, and +\eqref{eq:sa-affine-path} solves its ODE. If $M\ge1$ bounds $\|D\|$ and +the collar width is $\eta$, an initial perturbation smaller than +$\eta/(2M)$ remains on that plateau. For a finite sequence, decrease +this radius by the product of the preceding affine norms. Smooth ODE +uniqueness proves the assertion, and disjoint supports prevent interference. +\end{proof} + +\begin{lemma}[Transport of area-changing sheets]\label{lem:sa-sheet-routing} +Let $B_i$ and $C_i$, $1\le i\le N$, be two separately positively +separated finite families of nondegenerate rational closed rectangles +in $(0,1)^2$. Let $F_i:B_i\to C_i$ be the positive diagonal affine +map, and let $z_0\in(0,1)$ be rational. There is an effective +smooth divergence-free field supported in $(0,1)\times(0,1)^3$ whose +time-one map agrees on a positive three-dimensional neighborhood of each +$B_i\times\{z_0\}$ with a diagonal affine map restricting to the +map $(x,z_0)\mapsto(F_i(x),z_0)$, with normal scale +reciprocal to its planar area multiplier. On the +sheet, each horizontal coordinate is constant during ascent and descent +and is a convex combination of its two endpoint coordinates during the +middle stage. The same construction applies in a larger bounded +Euclidean region after translations and a choice of separated heights. +\end{lemma} +\begin{proof} +For $N=0$ take zero. Write $p_i,q_i$ for the centers and $w_i^0,w_i^1$ +for side-length vectors. In three successive stages, lift to +$z_i=1/2+i/[4(N+1)]$, transform there, and lower to $z_0$. During the +middle stage, with a flat progress parameter $s\in[0,1]$, put +\[ + p_i(s)=(1-s)p_i+sq_i,\quad w_i(s)=(1-s)w_i^0+sw_i^1, +\] +\[ + D_i(s)=\diag\left(\frac{w_{i1}(s)}{w_{i1}^0}, + \frac{w_{i2}(s)}{w_{i2}^0}, + \frac{w_{i1}^0w_{i2}^0}{w_{i1}(s)w_{i2}(s)}\right). +\] +All scales are positive and their product is one. Give the sheets a +sufficiently small rational thickness. The reciprocal scale is bounded +by $\max_i[\min(1,w_{i1}^1/w_{i1}^0) +\min(1,w_{i2}^1/w_{i2}^0)]^{-1}$, so an effective positive thickness +keeps the middle-stage boxes apart vertically. During lifting use the +source separation, and during lowering the target separation. Choose a +rational collar smaller than a quarter of the corresponding coordinate +gaps and boundary clearances. Lemma~\ref{lem:sa-curl-motion} now supplies +the three local motions on disjoint supports, including a positive initial +neighborhood. A sheet point has zero normal displacement relative to its +center. Its fixed fractional coordinate in the interpolated interval +gives $(1-s)x_a+sF_i(x)_a$ in each horizontal direction. Flat stage +collars finish the construction. The larger-region version uses exactly +the same finite clearance argument. +\end{proof} + +\subsection{Closing the fluid argument}\label{subsec:sa-fluid-assembly} +\begin{lemma}[Realization and comparison]\label{lem:sa-realization} +Let $U$ be an effective smooth divergence-free field, $U(0)=0$, on the +unit torus or with fixed compact spatial support on $\mathbb R^3$. +Assume its mixed derivatives are bounded on each finite time interval. +Then +\begin{equation}\label{eq:sa-residual} + f=\partial_tU+(U\cdot\nabla)U-\nu\Delta U +\end{equation} +is an effective prescribed force and $(u,p)=(U,0)$ is a global solution. +Its velocity is unique in the applicable comparison classes of +Lemma~\ref{lem:b-comparison}; pressure is determined with exactly the +normalization or additive time function specified there. +Uniform mixed-derivative bounds, a time period, fixed compact support, +and spatial mean zero pass from $U$ to $f$ when imposed on $U$. +The material flow is defined for every finite time. +\end{lemma} +\begin{proof} +The compactly supported field satisfies the whole-space velocity class +$C_tH^2\cap C_t^1L^2$ and has bounded velocity and gradient; zero pressure +satisfies $C_tH^1$. Thus Lemma~\ref{lem:p2-realization} applies in either +domain. Derivatives of \eqref{eq:sa-residual} are finite sums of products +of derivatives of $U$, proving the stated bounds and support. For mean +zero use $(U\cdot\nabla)U=\operatorname{div}(U\otimes U)$ and integrate. +The effective cutoff formulas and differentiation give the force program. +Periodic evaluation uses a finite superset of potentially active translates, +including adjoining intervals at a seam. +\end{proof} + +Two refinements of the whole-space comparison will be used explicitly. +Lemma~\ref{lem:b-comparison}(ii) permits pressure in $H^1$ modulo spatial +constants at each time, without continuity into that norm, when the +competing velocity retains $C_tH^2\cap C_t^1L^2$ and finite-interval +bounds on velocity and gradient. Its case (i) instead permits omission +of the competing-gradient bound when pressure belongs to $C_tH^1$. +The compactly supported reference fields below satisfy the hypotheses +of both statements. We retain each theorem's declared comparison class +when applying the corresponding case. + +For each construction below, the finite rule list gives all spatial +branches before a trajectory is chosen. The choice of initial coordinates +or a loading motion supplies the exact rational starting code; a loading +rule inside the repeated program is included in the induction at integer +times. We then check the entire path of every used nonhalting branch +against its observer. This last check is essential: a safe pair of +endpoints alone need not give a safe interpolation. Bounds on the +finitely many pulse types give uniform kinetic energy on the torus, +or under fixed compact support in Euclidean space. No simulation trace +enters a force prescription. diff --git a/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf new file mode 100644 index 0000000000000000000000000000000000000000..534d7705e25270077810ccc252a7dc72247a6bbb --- /dev/null +++ b/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8cd581420ec9967ac33296ba4f91342cb275bd63db72d0db630139381d3aa59e +size 702942 diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/README.md b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..d46cefe74cd0a415616360d763b3bbef414e8395 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [Prescribed large-volume charges on threefolds with trivial canonical bundle](paper.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026, + author = {{OpenAI}}, + title = {{Prescribed large-volume charges on threefolds with trivial canonical bundle}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf}{OAI:Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/01-introduction.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..b23b3c7a8dc4cd4281e933a8a24c8c1c1cbeef2a --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/01-introduction.tex @@ -0,0 +1,404 @@ +\section{Introduction}\label{sec:introduction} + +A central charge assigns a complex number to each object of a derived +category. A stability condition organizes the objects by its argument and +requires every object to have a finite filtration by semistable factors. +Bridgeland introduced this structure and its deformation theory +\cite{Bridgeland2007}. On a projective threefold, the expected large-volume +charges are exponential expressions in the Chern character. The +prescribed-charge problem asks whether these particular expressions admit +a compatible heart, finite semistable filtrations, and a support bound +controlling the numerical classes of all semistable objects. + +We construct stability conditions with the exact ordinary and +square-root-Todd large-volume charges on every smooth projective complex +threefold with trivial canonical bundle. One volume threshold works on an +open set of real twists and real ample directions. The support property +holds on the full numerical Grothendieck group, and the ordinary charge +has the standard double-tilt heart. An independent argument proves strong +tilt inequalities on every smooth projective complex threefold: a fixed +rational correction works at every volume, and the correction vanishes +beyond a threshold independent of the object and of the twist along a +fixed polarization. + +All varieties are over $\CC$, and all Chern characters and intersection +products are numerical. Write $N^1(X)_{\RR}$ for the space of numerical +real divisor classes, $\operatorname{Amp}(X)$ for its ample cone, and +$N_1(X)_{\QQ}$ for the space of numerical rational curve classes. +A threefold is smooth, connected and projective. +The assertions for a disjoint union follow by applying the results to +its finitely many connected components and taking the largest constants. +Write $\Db(X)=\Db\Coh(X)$ and +$\ch^B(E)=e^{-B}\ch(E)$ for a real numerical divisor class $B$. + +\subsection{Prescribed charges and the full numerical group} + +The Euler pairing is +$\chi(E,F)=\sum_j(-1)^j\dim\Ext^j(E,F)$, and we use +\[ + \Knum(X)=K_0(X)/\{v:\chi(v,w)=0\text{ for every }w\in K_0(X)\}. +\] +A numerical stability condition consists of a charge homomorphism +$Z:\Knum(X)\to\CC$ and a locally finite slicing $\mathcal P$ of +$\Db(X)$. A slicing gives the semistable objects of each real phase, +with $Z(E)=m(E)e^{i\pi\phi}$, $m(E)>0$, on $\mathcal P(\phi)$, +the shift and Hom-vanishing rules, and finite Harder--Narasimhan +filtrations. Local finiteness requires the categories of sufficiently +short phase intervals to have finite length. We require the support +property of Kontsevich--Soibelman +\cite[Sections~1.2 and~2.1]{KontsevichSoibelman2008} on the full real space +$\Knum(X)_{\RR}$: for a norm there is $C>0$ such that +$\|[E]\|\leq C|Z(E)|$ for every semistable $E$. +Its heart is $\mathcal P(0,1]$. We call a stability condition geometric +if all point sheaves $\OO_x$ are stable of phase $1$. + +We specify the tilt conventions needed to state the heart in our theorem. +For a real ample class $H$, a real divisor class $B$, and a coherent sheaf +of positive rank put +\[ + \mu_{H,B}(F)=\frac{H^2\ch_1^B(F)}{H^3\ch_0(F)}, +\] +and give a nonzero torsion sheaf slope $+\infty$. +The slope Harder--Narasimhan filtration has semistable factors of strictly +decreasing slopes; write $\mu^+$ and $\mu^-$ for its largest and smallest +slopes. The slope torsion pair and its tilt are +\[ + \mathcal T_{H,B}=\{F:\mu^-_{H,B}(F)>0\},\qquad + \mathcal F_{H,B}=\{F:\mu^+_{H,B}(F)\leq0\},\qquad + \mathcal B_{H,B}=\langle\mathcal F_{H,B}[1],\mathcal T_{H,B}\rangle, +\] +with the zero sheaf in both subcategories. Here a heart is the abelian +category specified by a bounded $t$-structure, $[1]$ is the cohomological +shift, and brackets denote extension closure. Thus an object of +$\mathcal B_{H,B}$ has ordinary cohomology only in degrees $-1,0$, in the +indicated two subcategories. + +For real $B$, real ample $H$ and $t>0$, the ordinary physical charge is +\begin{equation}\label{eq:intro-physical} + Z_{B,tH}(E)= + -\ch_3^B(E)+\frac{t^2}{2}H^2\ch_1^B(E) + +i\left(tH\ch_2^B(E)-\frac{t^3}{6}H^3\ch_0(E)\right). +\end{equation} +The first tilt is unchanged if the polarization $H$ is replaced by +$tH$. On this first tilt, use the second slope +\[ + \nu_{B,tH}(E)= + \frac{tH\ch_2^B(E)-t^3H^3\ch_0(E)/6}{t^2H^2\ch_1^B(E)}. +\] +Its denominator-zero value is $+\infty$. An object is semistable for +this slope if every nonzero proper subobject in $\mathcal B_{H,B}$ has +slope at most that of the quotient; stability uses a strict inequality. +The Harder--Narasimhan property holds for these real ample parameters +\cite[Section~2]{BMS2016}. Let $\mathcal T'_{B,tH}$ +have strictly positive smallest Harder--Narasimhan slope, and +$\mathcal F'_{B,tH}$ have nonpositive largest slope. The standard +double-tilt heart is +\[ + \mathcal A_{B,tH}= + \langle\mathcal F'_{B,tH}[1],\mathcal T'_{B,tH}\rangle. +\] +These definitions use the same strict-positive/nonpositive boundary +at both tilts. + +\begin{theorem}[Exact large-volume charges on a real sector] +\label{thm:geometric-sector} +Let $X$ be a smooth projective complex threefold with +$K_X\simeq\OO_X$. For any $B_*\in N^1(X)_{\RR}$ and +$H_*\in\operatorname{Amp}(X)$ there are open neighborhoods $V$ of +$B_*$ and $W$ of $H_*$ and one $T>0$ such that, for all +$B\in V$, $H\in W$ and $t\geq T$, the following hold. +\begin{enumerate}[label=\textup{(\roman*)}] +\item $(Z_{B,tH},\mathcal A_{B,tH})$ is a geometric numerical +stability condition with support on $\Knum(X)_{\RR}$. +\item There is a geometric numerical stability condition, with support +on the same full numerical space, whose charge is exactly +\[ + Z^{\mathrm{LV}}_{B,tH}(E) + =-\int_Xe^{-B-itH}\ch(E)\sqrt{\td(X)}, + \qquad \sqrt{\td(X)}=1+\frac{c_2(X)}{24}. +\] +Its heart is the one constructed in Section~\ref{sec:ambient}. +\end{enumerate} +\end{theorem} + +Only triviality of $K_X$ is assumed; the additional vanishings +$H^1(X,\OO_X)=H^2(X,\OO_X)=0$ in the strict Calabi--Yau convention +are unnecessary. The quantifiers give one threshold for both charges +throughout the same real sector. The two hearts are specified separately: +the ordinary charge uses the double tilt defined above, while the +Todd-corrected charge uses the heart constructed in +Section~\ref{sec:ambient}. + +The ordinary and square-root-Todd charges have large-volume antecedents in +Bayer's polynomial stability conditions +\cite[Theorem~3.2.2 and Section~4]{BayerPolynomial2009}. Polynomial stability +orders phases asymptotically as the volume tends to infinity. Our theorem +constructs a numerical Bridgeland stability condition at each finite +volume in the stated sector. The passage to finite volume must also +control the charge kernel: on a threefold of Picard rank greater than +one, the full numerical group contains classes invisible to the four +intersection degrees along a single polarization. + +\subsection{Independent uniform tilt inequalities} + +The numerical problem concerns the third Chern character, which is absent +from the classical Bogomolov--Gieseker inequality. Bayer--Macr\`i--Toda +introduced the double-tilt construction and proposed a third-character +inequality that would make it a source of stability +conditions~\cite{BMT2014}. Bayer--Macr\`i--Stellari established that +inequality for abelian threefolds and Calabi--Yau threefolds of abelian +type, and developed the discriminants, deformation and wall framework used +below~\cite{BMS2016}. For this part fix an ample integral divisor $H$ +and a rational numerical divisor $B_0$. + +For $a>0$, $b\in\RR$ and $B=B_0+bH$, the tilt slope is +\begin{equation}\label{eq:intro-tilt} + \nu_{a,b}(E)= + \frac{H\ch_2^B(E)-\tfrac12a^2H^3\ch_0(E)} + {H^2\ch_1^B(E)}, +\end{equation} +with value $+\infty$ when the denominator is zero. +We use the same subobject--quotient comparison for tilt stability. +At finite slope, semistability is equivalently the comparison with the +slope of the object itself. The treatment of zero-dimensional quotients +and the finite factor filtrations needed below is given in +Section~\ref{sec:tilt}. + +\begin{theorem}[Uniform strong inequality]\label{thm:uniform-inequality} +Let $X$ be a smooth projective complex threefold, $H$ an ample integral +divisor, and $B_0\in N^1(X)_{\QQ}$. +There are constants $k\in\QQ_{\geq0}$ and $R_*>0$, depending only on +$(X,H,B_0)$, such that every finite-slope $\nu_{a,b}$-semistable +$E\in\mathcal B_{H,B_0+bH}$ with $\nu_{a,b}(E)=0$ satisfies +\begin{equation}\label{eq:intro-corrected} + \ch_3^{B_0+bH}(E) + \leq \left(\frac{a^2}{6}+k\right)H^2\ch_1^{B_0+bH}(E) + \qquad(a>0,\ b\in\RR). +\end{equation} +For the same fixed data, the stronger bound +\begin{equation}\label{eq:intro-tail} + \ch_3^{B_0+bH}(E) + \leq \frac{a^2}{6}H^2\ch_1^{B_0+bH}(E) +\end{equation} +holds whenever $a>R_*$, for every $b\in\RR$. +\end{theorem} + +There is no canonical-bundle hypothesis in this theorem. Both constants +are independent of the rank and Chern character of $E$. The curve class +$\Gamma=kH^2\in N_1(X)_{\QQ}$ gives the correction +$\Gamma\cdot\ch_1^{B_0+bH}(E)$ and satisfies $\Gamma\cdot H\geq0$. + +Bernardara--Macr\`i--Schmidt--Zhao proved a corrected inequality of this +kind for Fano threefolds polarized by a divisor proportional to $-K_X$, +and asked for such a curve-class correction +on every polarized threefold +\cite[Theorem~1.1 and Question~2.4]{BMSZ2017}. +Martinez--Schmidt stated the version allowing a fixed rational +transverse twist \cite[Question~2.6]{MartinezSchmidt2019}. +Theorem~\ref{thm:uniform-inequality} answers these corrected-inequality +questions affirmatively and also gives a volume threshold beyond which +the correction can be removed. + +The uncorrected inequality at every parameter is false. Schmidt's +blow-up example~\cite[Theorem~3.1]{Schmidt2017} and the contracted-divisor +and Weierstrass examples of Martinez--Schmidt +\cite[Theorem~1.1]{MartinezSchmidt2019} give explicit violations. +Section~\ref{sec:counterexamples} computes their bounded volume ranges +and proves a uniform bound for semistable sheaves of arbitrary rank on +the reduced exceptional surface. These comparisons retain both parts of +Theorem~\ref{thm:uniform-inequality}: the corrected all-volume statement +and the uncorrected tail. + +The physical volume and the tilt variable satisfy $t=\sqrt3a$. +Consequently the strong bound \eqref{eq:intro-tail} has coefficient +$t^2/18$, whereas positivity of the ordinary charge +\eqref{eq:intro-physical} on zero-slope objects supplies the weaker +coefficient $t^2/2$. The proof of the strong inequality is independent of +the construction of the charges and their full numerical support. + +The uncorrected tail also gives an all-slope quadratic formulation. +For $v=(v_0,v_1,v_2,v_3)$ define +\begin{equation}\label{eq:intro-quadratic} + Q_{\alpha,\beta}(v)= + \alpha(v_1^2-2v_0v_2) + +\beta(3v_0v_3-v_1v_2)+2v_2^2-3v_1v_3. +\end{equation} +\begin{corollary}[A parabolic region with trivial correction] +\label{cor:quadratic-tail} +For every polarized smooth projective threefold $(X,H)$ there is $R_*>0$ +such that, if +\[ + \alpha>f(\beta):=\frac{\beta^2+R_*^2}{2}, +\] +then $Q_{\alpha,\beta}(v_H(E))\geq0$ for every nonzero semistable object +in $\mathcal B_{H,\beta H}$ for the slope +\[ + \frac{v_2-\beta v_1+(\beta^2-\alpha)v_0}{v_1-\beta v_0}, + \qquad + v_H(E)=(H^3\ch_0(E),H^2\ch_1(E),H\ch_2(E),\ch_3(E)), +\] +again interpreted as $+\infty$ at zero denominator. +\end{corollary} +The function $f$ is continuous. This is the generalized quadratic +formulation with trivial correction: $v_H$ uses the ordinary, +unmodified Chern character. The slope in the corollary +is exactly~\eqref{eq:intro-tilt} with +$a=\sqrt{2\alpha-\beta^2}$ and $b=\beta$; Section~\ref{sec:wall} +proves the all-slope implication, including zero denominator. + +\subsection{Previous constructions} + +General existence of stability conditions on projective varieties is +supplied by Li's construction~\cite{Li2026} and the framework of +Li--Liu--Liu--Macr\`i--Perry--Stellari--Zhao~\cite{LiEtAl2026}. +Cheng establishes full numerical support and general nonemptiness +in~\cite{Cheng2026}. Thus nonemptiness alone is already known in the +setting of Theorem~\ref{thm:geometric-sector}. Our construction uses the +product, restriction and deformation methods of these works to prescribe +the two charges and the real sector. +The product-embedding route to full numerical support in +Section~\ref{sec:ambient} adapts the construction of +Giovenzana--Robotis--Rota--Zuliani~\cite{GiovenzanaRobotisRotaZuliani2026}, +distinct from the quantitative product inputs cited there. + +There is also a preceding large-volume construction with a different +character. Cheng--Feyzbakhsh identify a large-volume double-tilt heart for +the character $\td(N_{X/\mathbb P^n})^{-1}\ch$, where $N_{X/\mathbb P^n}$ +is the normal bundle of their projective embedding +\cite[Section~3]{ChengFeyzbakhsh2026}. A reparameterization expresses +that heart as an ordinary double tilt while leaving a curve-class +correction in its charge. +The real-parameter extension discussed in their Section~3.3 follows from +\cite[Theorem~4.1, Proposition~4.5 and Theorem~6.2]{LiEtAl2026}. +Combining the mass--Hom estimate +\cite[Theorem~7.5]{LiEtAl2026} with Cheng's full-support criterion +\cite[Theorem~2.1]{Cheng2026} gives full numerical support for this +fixed-polarization real family, pointwise in its parameters. +Theorem~\ref{thm:geometric-sector} prescribes the ordinary and +square-root-Todd charges on an open set of real twists and ample +directions with one volume threshold. Its heart comparison adapts the +surface-section method of Cheng--Feyzbakhsh to the ordinary character +and positive weighted sections. + +Restriction to lower-dimensional varieties has also led to strong tilt +inequalities. Feyzbakhsh--Koseki--Liu--Rekuski derive a corrected tilt +inequality from an improved sheaf Bogomolov--Gieseker bound, and obtain +that bound on families of Calabi--Yau threefolds from Brill--Noether +estimates on curves inside surface sections +\cite[Theorems~1.1, 1.3 and~1.4]{FeyzbakhshEtAl2025}. +Our numerical argument likewise uses successive restrictions, but its +input is a rank-uniform cohomology estimate on surfaces. A Frobenius +limit then gives the inequality at one fixed positive volume on an +arbitrary smooth projective threefold. The wall argument builds on the +discriminant-descent method of +\cite[Lemma~2.7]{BMSZ2017}; the fixed-volume defect estimate is what +makes its resulting threshold uniform in the twist and the object. + +\subsection{Two independent proof strategies} + +We prove the prescribed-charge theorem first, in +Sections~\ref{sec:ambient}--\ref{sec:heart}, and then establish the +strong inequalities in +Sections~\ref{sec:tilt}--\ref{sec:wall}. Figure~\ref{fig:proof} shows +the two arguments. Their distinction is mathematical: controlling four +Chern-character degrees along one polarization does not itself control +the full numerical group or provide a common threshold when the ample +direction varies. + +For the charge construction, choose very ample line bundles whose +positive span contains the ample directions under consideration. They +give an embedding $i:X\hookrightarrow Y$ into a product of projective +spaces. In dimension three, products of divisor classes span all +numerical Chern-character classes; choosing the line bundles to span +$N^1(X)_{\RR}$ therefore makes $i_*$ injective on the numerical +Grothendieck group. Ambient support will then control every numerical +class on $X$. + +The ambient construction must also permit a deformation which is uniform +at large volume. Liu's product-with-a-curve +construction~\cite{LiuProduct2021}, the positive-genus product theory +of~\cite{ProductsCurves2025}, the support induction +of~\cite{LiEtAl2026}, and Cheng's two-block +construction~\cite{Cheng2026} supply its starting framework. We prove +a weighted support estimate for finitely many projective blocks after +rescaling each Chern-character coordinate by its power of the volume. +In these coordinates the support constant is independent of volume, and +every fixed positive-degree correction to the exponential charge tends +to zero as the volume grows. This permits the ambient deformation +throughout an open set of twists and ample directions. A fixed polynomial +correction cancels the Todd factors in Grothendieck--Riemann--Roch, so +restriction gives exactly the chosen character on $X$. + +This construction already gives full numerical support and stable point +sheaves. For the ordinary charge, compatible restrictions to smooth +surfaces identify its heart. The induced surface hearts control the +outer cohomology sheaves; positive weights combine their slope bounds +for a real ample direction. A separate argument at the real axis fixes +the zero-slope boundary of the second tilt. The Todd-charge construction +uses the geometric heart obtained by restriction. + +The independent numerical proof is concentrated at one fixed volume $A$. +For a slope-zero object $E$ at this volume, put +$d=H^2\ch_1^B(E)/H^3$ and $r=\ch_0(E)$. The tilt discriminant +inequality gives $d\geq A|r|$. We prove +\[ + \frac{\ch_3^B(E)}{H^3}-\frac{A^2d}{6}\le C(d-A|r|) +\] +with $C$ independent of $E$ and $b$. The error $e=d-A|r|$ can vanish +even at large rank, so the estimates must remain proportional to this +error without an additional rank term. Truncating the ordinary slope +filtrations separates sheaf factors in a narrow slope interval from a +residual complex whose rank and first two intersection degrees are +bounded by $e$. +Langer's restriction inequality controls the resulting slope variance +\cite{Langer2004,Langer2004Addendum}. Hermitian--Einstein theory and the +Bochner identity~\cite{Donaldson1985,UhlenbeckYau1986,DemaillyCADG}, +together with a matrix spectral estimate of Cwikel--Lieb--Rozenblum type +\cite{Frank2014}, control the residual cohomology in terms of $e$. +For each fixed object, we reduce finitely many +sheaves and maps to positive characteristic. Frobenius pullback and +Riemann--Roch turn the cohomology bounds into the displayed inequality +on $X$; the errors depending on the chosen object vanish in the limit. +Transport along zero-slope branches, with strict discriminant descent at +walls, gives the corrected global bound and the uncorrected tail. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[ + box/.style={draw,rounded corners=2pt,align=center,font=\small, + text width=4.4cm,minimum height=10mm,inner sep=5pt}, + arrow/.style={-{Stealth[length=2mm]},thick}] +\node[box] (ambient) {Weighted ambient stability\\and support}; +\node[box,below=7mm of ambient] (deform) {Exact charge deformation\\and restriction to $X$}; +\node[box,below=7mm of deform] (support) {Full numerical support\\and stable points for both charges}; +\node[box,below=7mm of support] (heart) {Ordinary charge: surface comparison\\and the double-tilt heart}; +\node[box,right=15mm of ambient] (analysis) {Rank-uniform restriction\\and cohomology}; +\node[above=3mm of ambient,align=center,font=\small] {Prescribed charges: $K_X\simeq\OO_X$}; +\node[above=3mm of analysis,align=center,font=\small] {Inequalities: arbitrary threefold}; +\node[box,below=7mm of analysis] (apex) {Fixed-volume defect bound}; +\node[box,below=7mm of apex] (wall) {Wall transport}; +\node[box,below=7mm of wall] (bounds) {Corrected all-volume inequality\\and uncorrected tail}; +\draw[arrow] (ambient)--(deform); +\draw[arrow] (deform)--(support); +\draw[arrow] (support)--(heart); +\draw[arrow] (analysis)--(apex); +\draw[arrow] (apex)--(wall); +\draw[arrow] (wall)--(bounds); +\end{tikzpicture} +\caption{The independent categorical and numerical arguments. Full support +is obtained in Section~\ref{sec:ambient}, before the ordinary-heart +comparison in Section~\ref{sec:heart}. The numerical argument begins +with the tilt tools of Section~\ref{sec:tilt} and proves the inequalities +in Sections~\ref{sec:analysis}--\ref{sec:wall}. The relation $t=\sqrt3a$ +compares volume conventions.} +\label{fig:proof} +\end{figure} + +For the numerical argument, Section~\ref{sec:tilt} supplies the circle +and duality tools, Sections~\ref{sec:analysis} and~\ref{sec:apex} +establish the fixed-volume bound, and Section~\ref{sec:wall} proves +Theorem~\ref{thm:uniform-inequality} and +Corollary~\ref{cor:quadratic-tail}. +Section~\ref{sec:counterexamples} compares the inequalities with +counterexamples and bounds arbitrary-rank sheaves on reduced exceptional +surfaces. diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/02-tilt.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/02-tilt.tex new file mode 100644 index 0000000000000000000000000000000000000000..9d3444536b99bf3a5330b2044ad4e429d2f8620f --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/02-tilt.tex @@ -0,0 +1,366 @@ +\section{Tilt stability and numerical transport} +\label{sec:tilt} + +We now begin the independent proof of +Theorem~\ref{thm:uniform-inequality}, which applies to arbitrary smooth +projective threefolds along a fixed integral ample direction. The +inequality will first be proved at one volume, then transported to all +positive volumes. This section supplies the tilt conventions and +deformation facts needed for that transport. Objects with zero reduced +charge must be retained: they affect both finite factor filtrations and +duality at real parameters. + +\subsection{Characters, slopes, and the first tilt} + +Throughout this section, $X$ is a smooth projective complex threefold, +$H$ is an integral ample divisor class, and $h=H^3$. All intersections +and Chern characters are numerical. For a real divisor class $T$, set +$\ch^T(E)=e^{-T}\ch(E)$ and use the normalized coordinates +\begin{equation}\label{eq:normalized-characters} +\begin{gathered} + r(E)=\ch_0(E),\qquad c_T(E)=\ch_1^T(E),\\ + d_T(E)=\frac{H^2c_T(E)}h,\qquad + w_T(E)=\frac{H\ch_2^T(E)}h,\qquad + z_T(E)=\frac{\ch_3^T(E)}h. +\end{gathered} +\end{equation} +The subscript $0$ is reserved for the untwisted character. Fix +$B_0\in N^1(X)_{\QQ}$ and put $B=B_0+bH$ for $b\in\RR$. +We abbreviate the twist $B_0+bH$ by a symbolic subscript $b$; +when evaluating at $b=0$, we write the subscript $B_0$ explicitly. +The exponential formula gives +\begin{align} + d_b&=d_{B_0}-br,& + w_b&=w_{B_0}-b d_{B_0}+\tfrac12b^2r,\label{eq:twist-two}\\ + z_b&=z_{B_0}-b w_{B_0}+\tfrac12b^2d_{B_0}-\tfrac16b^3r. + \label{eq:twist-three} +\end{align} + +For a positive-rank coherent sheaf define +$\mu_{H,B}=d_B/r$, and give every nonzero torsion sheaf slope +$+\infty$. Let $\mu^+$ and $\mu^-$ be the largest and smallest slopes +in the slope Harder--Narasimhan filtration. The torsion pair and its tilt +are +\begin{equation}\label{eq:first-tilt} +\begin{gathered} + \mathcal T_{H,B}=\{U:\mu^-_{H,B}(U)>0\},\qquad + \mathcal F_{H,B}=\{V:\mu^+_{H,B}(V)\le0\},\\ + \mathcal B_{H,B}=\langle\mathcal F_{H,B}[1],\mathcal T_{H,B}\rangle, +\end{gathered} +\end{equation} +where the zero sheaf belongs to both parts. This is the heart of a +bounded $t$-structure. For $a>0$ put +\begin{equation}\label{eq:tilt-numerator} + n_{a,b}=w_b-\tfrac12a^2r,\qquad + \nu_{a,b}(E)= + \begin{cases}n_{a,b}(E)/d_b(E),&d_b(E)>0,\\ + +\infty,&d_b(E)=0. + \end{cases} +\end{equation} +For a general real twist $B$, the same formula defines +$\nu_{a,B}$ using $d_B,w_B$ and $n=w_B-a^2r/2$; the notation +$\nu_{a,b}$ specifies the line $B=B_0+bH$. +An object is tilt-stable, respectively tilt-semistable, if every proper +nonzero subobject $F\subset E$ in $\mathcal B_{H,B}$ satisfies +$\nu(F)<\nu(E/F)$, respectively $\nu(F)\le\nu(E/F)$. +This is the convention of \cite[Definition~2.3]{BMS2016}. +The physical parameter $\omega=\sqrt3aH$ used in +\cite{BMT2014,BMS2016} multiplies our finite slope by $1/(\sqrt3a)$, +so defines the same stable and semistable objects. + +Here are two harmless reductions for the inequality. Write +$q=H^2B_0/h$ and replace $(B_0,b)$ by $(B_0-qH,b+q)$; the twist +$B$ is unchanged, and the new rational $B_0$ satisfies $H^2B_0=0$. +If $H'=sH$ is very ample, use $a'=a/s$ and $b'=b/s$. +The hearts agree and $\nu'_{a',b'}=\nu_{a,b}/s$. An estimate with +correction $k'(H')^2$ becomes an estimate with correction +$s^2k'H^2$, and a threshold $a'>R'$ becomes $a>sR'$. +We may therefore assume $H$ very ample and $H^2B_0=0$ while proving +the estimates. Constants used with duality are chosen for both $B_0$ +and $-B_0$. + +\subsection{Discriminants and finite-slope filtrations} + +The Hodge index theorem gives a useful norm on $N^1(X)_{\RR}$. +Define +\begin{equation}\label{eq:divisor-norm} + \langle C,D\rangle=\frac{HCD}{h},\qquad + C_\perp=C-\langle H,C\rangle H,\qquad + \|C\|^2=\langle H,C\rangle^2-\langle C_\perp,C_\perp\rangle. +\end{equation} +The pairing is negative definite on $H^\perp$, so this is a positive +definite norm. For a positive-rank torsion-free sheaf $G$ put +\begin{equation}\label{eq:sheaf-discriminants} +\begin{gathered} + \mu(G)=\frac{d_0(G)}{r(G)},\qquad + \xi_G=\left\langle\frac{c_0(G)}{r(G)},\frac{c_0(G)}{r(G)}\right\rangle + -\frac{2w_0(G)}{r(G)},\\ + \delta_G^T=\left(\frac{d_T(G)}{r(G)}\right)^2 + -\frac{2w_T(G)}{r(G)}. +\end{gathered} +\end{equation} +Classical Bogomolov--Gieseker gives $\xi_G\ge0$ for slope-semistable +$G$. Twist invariance of the full discriminant and the orthogonal +decomposition in \eqref{eq:divisor-norm} give +\begin{equation}\label{eq:xi-delta} + \delta_G^T=\xi_G-\left\langle + \left(\frac{c_0(G)}{r(G)}-T\right)_\perp, + \left(\frac{c_0(G)}{r(G)}-T\right)_\perp\right\rangle\ge\xi_G. +\end{equation} +Slope filtrations of torsion-free sheaves may be refined to slope-stable +torsion-free factors: take saturated equal-slope subsheaves inside a +semistable factor and induct on rank. + +For arbitrary derived objects the projected discriminant is +\begin{equation}\label{eq:projected-discriminant} + \Delta(E)=d_b(E)^2-2r(E)w_b(E) + =d_{B_0}(E)^2-2r(E)w_{B_0}(E). +\end{equation} +Its independence of $b$ follows from \eqref{eq:twist-two}. +We use the following established tilt-stability input for a \emph{fixed} +integral ample class $H$. There are Harder--Narasimhan filtrations for +all $a>0$ and real $B$; tilt-stability is open in $(a,B)$; the extremal +phases of a fixed derived object vary continuously; and every +tilt-semistable object satisfies +\begin{equation}\label{eq:tilt-discriminants} + \Delta\ge0,\qquad + \langle c_B,c_B\rangle-2rw_B+C_Hd_B^2\ge0 +\end{equation} +for a constant $C_H\ge0$ depending only on $X,H$. +These are \cite[Theorem~3.5 and Propositions~B.2 and~B.5]{BMS2016}, +with volume factors absorbed into $C_H$. The continuous reduced charge +is $d_B+i(w_B-a^2r/2)$, with heart phase interval +$(-1/2,1/2]$ and phase $1/2$ assigned to its zero-charge objects. +No assertion about varying the direction of $H$ is used here; +\cite[Remark~B.6(b)]{BMS2016} expressly distinguishes that question. + +\begin{lemma}\label{lem:tilt-zero-denominator} +An object $E\in\mathcal B_{H,B}$ has $d_B(E)\ge0$. +Write $n=w_B-a^2r/2$. +If $d_B(E)=0$, then $n(E)\ge0$, with equality exactly when +$E$ is a zero-dimensional sheaf, including the zero object. +For a finite-slope object $E$, semistability is equivalent to the +condition that every proper nonzero subobject has positive denominator +and slope at most $\nu(E)$. Every finite-slope quotient of a +finite-slope semistable object has slope at least $\nu(E)$. +\end{lemma} + +\begin{proof} +Write $V=\mathcal H^{-1}(E)$ and $U=\mathcal H^0(E)$ for ordinary +cohomology. The defining torsion pair gives $d_B(V)\le0$ and +$d_B(U)\ge0$, whence $d_B(E)=d_B(U)-d_B(V)\ge0$. +If it is zero, a nonzero $V$ is torsion-free and slope-semistable of +slope zero, whereas $U$ has dimension at most one. Bogomolov--Gieseker +and Hodge index give $w_B(V)\le0$; effectiveness of the curve cycle +gives $w_B(U)\ge0$. Thus +\[ + n(E)=w_B(U)-w_B(V)+\tfrac12a^2r(V)\ge0. +\] +Equality forces $V=0$ and the curve cycle of $U$ to vanish, so $U$ is +zero-dimensional. The converse is immediate. + +In a short exact sequence $0\to F\to E\to G\to0$ with $d(E)>0$, +a subobject with $d(F)=0$ has slope $+\infty$ while $d(G)>0$. +It is forbidden by semistability. If both denominators are positive, +additivity makes $\nu(E)$ their weighted average, proving the desired +equivalences. If $d(G)=0$, then $n(G)\ge0$, so +$\nu(F)\le\nu(E)$ and comparison with $\nu(G)=+\infty$ is automatic. +\end{proof} + +Strict comparison with the quotient is significant: a zero-dimensional +quotient does not destroy stability in our convention, although it +gives equality between the slopes of the subobject and the object. +An object stable for the stricter subobject--object convention is +semistable in ours, and is therefore covered by all the inequalities +below. + +\begin{lemma}\label{lem:tilt-support-bound} +For fixed $a>0$ there is a constant $C_a$, depending only on $a,X,H$, +such that every tilt-semistable object, at any real twist $B$, satisfies +\begin{equation}\label{eq:tilt-support-bound} + |r|+|w_B|+\|c_B\|\le C_a(d_B+|n|),\qquad n=w_B-a^2r/2. +\end{equation} +The constants can be bounded uniformly when $a$ ranges in a compact +subinterval of $(0,\infty)$. +\end{lemma} + +\begin{proof} +The first inequality of \eqref{eq:tilt-discriminants} gives +$a^2r^2+2rn\le d^2$, and hence +$|r|\le 2|n|/a^2+d/a$. The equality $w=n+a^2r/2$ bounds $|w|$. +Finally the second inequality gives +\[ + \|c_B\|^2=2d^2-\langle c_B,c_B\rangle + \le(2+C_H)d^2-2rw, +\] +which proves the norm bound. These expressions also prove uniformity +on compact $a$-intervals. +\end{proof} + +This estimate controls rank, the full first character, and one degree +of the second character. It is not a support statement on the full +numerical Grothendieck group. Its immediate use is to make a +finite-slope factorization terminate even when $b$ is irrational. + +\begin{lemma}\label{lem:finite-tilt-factors} +Every nonzero finite-slope tilt-semistable object has a finite +filtration in $\mathcal B_{H,B}$ whose factors are tilt-stable of the +same finite slope. Extensions of finite-slope semistable objects of +the same slope are semistable of that slope. +\end{lemma} + +\begin{proof} +Let $\nu(E)=u\in\RR$. If $E$ is not stable, there is a proper +nonzero subobject $F$ with $\nu(F)=\nu(E/F)=u$ and positive +denominators. The object $F$ is semistable by +Lemma~\ref{lem:tilt-zero-denominator}. To obtain a splitting into +two semistable objects, we may have to absorb a zero-dimensional +subobject of the quotient back into $F$. Put $G=E/F$ and take its +Harder--Narasimhan filtration. Its last factor is a quotient of $E$, +so, if its slope is finite, that slope is at least $u$. The other +finite slopes are no smaller. There must be a finite factor since +$d(G)>0$. Additivity of $n-ud$, together with the nonnegativity of +$n$ on infinite-slope factors, now forces all finite slopes to equal +$u$ and all infinite-slope factors to have $d=n=0$. +Thus $G$ is either semistable or has a zero-dimensional initial +subobject $T$ followed by a semistable quotient of slope $u$. +Replace $F$ by the preimage of $T$. It is still a proper +equal-slope subobject of $E$, hence semistable, and its quotient is +now semistable as well. + +Repeated splitting cannot produce arbitrarily many nonzero factors. +For a semistable factor of slope $u$, Lemma~\ref{lem:tilt-support-bound} +gives $|r|\le C_a(1+|u|)d$. If $r\ne0$, integrality of rank +therefore bounds $d$ below by $1/(C_a(1+|u|))$. +If $r=0$, then $d=H^2\ch_1/h$ is positive and belongs to +$h^{-1}\ZZ$, so $d\ge1/h$. Each factor has denominator at least +the smaller of these two positive constants, and their denominators +sum to $d(E)$. A finite number of splits consequently gives stable +factors. + +For the extension assertion, intersect a subobject with the first +term and take its image in the last term. A nonzero intersection or +image has positive denominator and slope at most $u$ by semistability. +Their sum has the same properties, proving the subobject criterion. +\end{proof} + +\subsection{Semistability on the numerical semicircle} + +The next observation brings a nonzero finite slope to slope zero. +It also supplies the side points used in the fixed-apex argument. +It extends the calculation of \cite[Lemma~4.3]{BMS2016} to a fixed +transverse twist $B_0$. + +\begin{lemma}\label{lem:tilt-semicircle} +Suppose $E$ is tilt-semistable of finite slope $u$ at $(a,b)$. +Put $c=b+u$ and $R=\sqrt{a^2+u^2}$. Then $E$ is +tilt-semistable at every point +\begin{equation}\label{eq:tilt-semicircle} + b'=c+x,\qquad a'=\sqrt{R^2-x^2},\qquad -R0$. +On the whole open arc, therefore, +\[ + d_{b'}=d_c-xr>0,\qquad + n_{a',b'}=-x(d_c-xr). +\] + +Consider the set of points on the arc where $E$ is semistable in +the corresponding heart. It contains the original point. It is +closed within the arc: the formal charge has positive real part, +and continuity of the largest and smallest phases makes both +converge to its finite phase. In particular, this argument also +ensures membership in the limiting heart. +It is open within the arc as well. At a semistable point take the +finite stable filtration from Lemma~\ref{lem:finite-tilt-factors}. +Every factor has the same slope there and thus the same $c,R$ in +\eqref{eq:tilt-semicircle}. Openness of stability keeps all these +finitely many factors stable in nearby hearts, with their common +slope $-x$. Their fixed triangles are short exact sequences in +those hearts, and their extensions are semistable by the preceding +lemma. Connectedness of the arc proves the assertion. +\end{proof} + +\subsection{Duality, including the point defect} + +To treat negative rank we use the shifted derived dual +$\mathbb D(E)=R\mathcal Hom(E,\OO_X)[1]$. + +\begin{lemma}\label{lem:tilt-duality} +Let $a>0$, let $B$ be real, and let $E\in\mathcal B_{H,B}$ +be tilt-semistable of finite slope $u$. There are a zero-dimensional +sheaf $T_0$ and a distinguished triangle +\begin{equation}\label{eq:tilt-dual-triangle} + \widetilde E\longrightarrow\mathbb D(E)\longrightarrow T_0[-1] + \longrightarrow\widetilde E[1], +\end{equation} +where $\widetilde E\in\mathcal B_{H,-B}$ is tilt-semistable of +slope $-u$. At twists $B$ and $-B$ respectively their coordinates +are related by +\begin{equation}\label{eq:tilt-dual-characters} + (r,d,w,z)(\widetilde E) + =\bigl(-r(E),d_B(E),-w_B(E),z_B(E)+\operatorname{length}(T_0)/h\bigr). +\end{equation} +Moreover $\widetilde E=\tau^{\le0}\mathbb D(E)$ for the ordinary +cohomological truncation. +\end{lemma} + +\begin{proof} +For rational $\omega=\sqrt3aH$ and rational $B$, +\cite[Proposition~5.1.3(b)]{BMT2014} gives the triangle and +semistability directly: its hypothesis is finite maximal tilt slope, +which holds for a finite-slope semistable object. In that triangle +$\widetilde E$ has ordinary cohomology in degrees $-1,0$, whereas +$T_0[-1]$ is in degree $1$. Hence $\widetilde E$ is the stated +ordinary truncation; it depends only on $E$. + +First suppose $E$ is stable at real parameters. By openness it is +stable at a sequence of nearby parameters with rational $\omega,B$. +The rational triangles all have the same truncation +$\tau^{\le0}\mathbb D(E)$ and the same sheaf +$\mathcal H^1(\mathbb D(E))$. Their mirror finite phases converge +to the phase of slope $-u$, with the denominator still positive. +Continuity of extremal phases therefore gives semistability of this +fixed truncation in $\mathcal B_{H,-B}$. + +For a semistable $E$, use Lemma~\ref{lem:finite-tilt-factors} and +induct on its number of stable factors. For an extension +$0\to E_1\to E\to E_2\to0$, duality reverses the triangle. +We must show that its degree-zero cohomology in the mirror tilt heart +is semistable and that its degree-one cohomology is zero-dimensional. +The cohomology sequence is +\[ + 0\longrightarrow\widetilde E_2\longrightarrow + \mathcal H^0_{\mathcal B_{H,-B}}(\mathbb D(E)) + \longrightarrow\widetilde E_1\longrightarrow T_{0,2} + \longrightarrow\mathcal H^1_{\mathcal B_{H,-B}}(\mathbb D(E)) + \longrightarrow T_{0,1}\longrightarrow0, +\] +and all other tilt cohomology vanishes. Subobjects and quotients +of a zero-dimensional sheaf in this heart are zero-dimensional: +additivity first forces both denominators to vanish, then both +nonnegative numerators to vanish, and +Lemma~\ref{lem:tilt-zero-denominator} applies. +Consequently the kernel of $\widetilde E_1\to T_{0,2}$ has the +same positive denominator and the same slope as $\widetilde E_1$; +it is semistable by the subobject criterion. The degree-zero term +is an extension of it by $\widetilde E_2$, and is semistable of +slope $-u$. The degree-one term is zero-dimensional. This proves +\eqref{eq:tilt-dual-triangle}, including its truncation description, +for the extension. + +Finally the shifted dual has characters $(-r,c_B,-\ch_2^B,\ch_3^B)$ +at twist $-B$. The term $T_0[-1]$ has third character +$-\operatorname{length}(T_0)$, so taking characters of the triangle +gives \eqref{eq:tilt-dual-characters}. +\end{proof} diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/03-analysis.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/03-analysis.tex new file mode 100644 index 0000000000000000000000000000000000000000..f1695bad19877c2f174af52235966859b64aa241 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/03-analysis.tex @@ -0,0 +1,600 @@ +\section{Restriction and cohomology bounds} +\label{sec:analysis} + +The estimates in this section convert numerical control of slope factors +into control of sections and first cohomology. The constants depend on the +polarized variety and on specified twist ranges, but not on the ranks of +the sheaves. This distinction is essential: the first-cohomology estimate +will be proportional to a discriminant error, which can vanish even when +the rank is large. + +Throughout this section, $H$ is very ample and $h=H^3$. On a smooth +surface $S\in|H|$, we use +\[ + d_T(G)=\frac{H|_S\,\ch_1^T(G)}h, + \qquad w_T(G)=\frac{\ch_2^T(G)}h, + \qquad \mu_T(G)=\frac{d_T(G)}{\rk G}. +\] +For divisor classes on $S$, put $\langle C,D\rangle=CD/h$ and use the +norm +\[ + \|C\|^2=\langle H,C\rangle^2-\langle C_\perp,C_\perp\rangle, + \qquad C_\perp=C-\langle H,C\rangle H. +\] +Here and below $H$ also denotes its restriction. The definitions of +$\xi_G$ and $\delta_G^T$ consequently have the same formulas as on $X$. +In particular, for a positive-rank sheaf they give the identity +\begin{equation}\label{eq:surface-norm-discriminant} + \mu_T(G)^2+\delta_G^T + =\left\|\frac{c_1(G)}{\rk G}-T\right\|^2+\xi_G. +\end{equation} +On a smooth curve $C_m\in|mH|_S$, normalize the slope as +$\mu(G)=\deg(G)/(mh\rk G)$. This makes the slope of an unmodified +restriction equal to the slope before restriction. + +The first-cohomology estimate will concern stable surface sheaves +whose normalized first character is close to a negative scalar class. +For the fixed rational twist $B_0$, write +\[ + T=B_0|_S+(b-A)H,\qquad |b|\le1. +\] +For sufficiently large $A$, our target is to bound first cohomology +at fixed twists by sums of +\[ + r(G)\left(\left\|\frac{c_1(G)}{r(G)}-T\right\|^2+\xi_G\right). +\] +This quantity can vanish even when the total rank is large. The +negative scalar part of $T$ will give positive central curvature on +the dual bundle; its trace-free curvature will then control the +remaining harmonic forms. A separate estimate for sections uses the +rank of a subsheaf's evaluation map. We first choose surfaces and +curves with uniform geometry, so that both estimates have constants +independent of the sheaves and their ranks. + +\subsection{Restriction and the choice of smooth sections} + +\begin{lemma}[Restriction variance]\label{lem:restriction-variance} +Let $G$ be a slope-semistable torsion-free sheaf of rank $r>0$ on $X$. +For a very general smooth $S\in|H|$, restrict $G$ and refine its slope +Harder--Narasimhan filtration into slope-stable torsion-free factors +$G_i$, with ranks $r_i$ and normalized slopes $\mu_i$. Then +\begin{equation}\label{eq:restriction-variance} + \sum_i r_i\bigl(\mu_i-\mu(G)\bigr)^2\le r\xi_G. +\end{equation} +The same assertion holds for a slope-semistable torsion-free sheaf on +$S$, restricted to a very general smooth $C_m\in|mH|_S$, for every +positive integer $m$. + +More generally, restrict a finite filtration with semistable +torsion-free factors and refine each restricted factor separately. +From $X$ to $S$, for any fixed divisor twist $T$ on $X$, restricted +to $S$, the increases in +$\sum_i r_i\mu_T(G_i)^2$ and in $\sum_i r_i\delta_{G_i}^T$ are equal +and lie between zero and the original sum $\sum_G r(G)\xi_G$. +From $S$ to $C_m$, the same upper bound holds for the increase of the +second slope moment, using any fixed divisor twist on $S$ and its +restriction to $C_m$. The concatenated filtration need not be the +Harder--Narasimhan filtration of the whole restriction. +\end{lemma} + +\begin{proof} +Write +\[ + \operatorname{Disc}(G) + =2r c_2(G)-(r-1)c_1(G)^2=c_1(G)^2-2r\ch_2(G). +\] +The characteristic-zero semistable case of Langer's restriction +inequality \cite[Theorem~3.1]{Langer2004} has zero extremal-slope +term. On $X$ it gives, for raw restricted slopes $s_i=h\mu_i$, +\[ + \sum_{i0$, depending only on $X,H$, such that every +smooth $S\in|H|$ satisfies +\begin{equation}\label{eq:surface-fixed-m} + h^0(S,\OO_S(mH))=\chi(\OO_S(mH)), + \qquad \mu(K_S)-m\le-1, +\end{equation} +and $H^0(X,\OO_X(mH))\to H^0(S,\OO_S(mH))$ is surjective. +One can choose relatively compact analytic neighborhoods of a smooth +transverse pair of divisors in $|H|\times|mH|$ such that the resulting +surfaces and curves, equipped with the induced Fubini--Study metrics, +have uniform smooth local geometry. + +Within these neighborhoods one may impose, for each application, the +very-general restriction conditions of +Lemma~\ref{lem:restriction-variance} and exactness of any specified +finite collection of restricted sequences. Restrictions of a +specified finite collection of torsion-free sheaves may also be +required to remain torsion-free. For a specified bounded complex, +one may require its derived restriction to have ordinary cohomology +equal to the restrictions of its ordinary cohomology sheaves. +\end{lemma} + +\begin{proof} +Choose $m$ so large that $(m-1)H-K_X$ is ample, +$H^1(X,\OO_X((m-1)H))=0$, and $m\ge\mu(K_S)+1$. +The last slope is independent of $S$, by adjunction +$K_S=(K_X+H)|_S$. Kodaira vanishing on $S$ +\cite[Chapter~VII, Theorem~3.3]{DemaillyCADG} gives the first equality +in \eqref{eq:surface-fixed-m}, and the restriction sequence on $X$ +gives surjectivity on sections. + +Bertini's theorem supplies a smooth pair meeting transversely. +Shrink an analytic neighborhood of that pair so that its closure is +compact and all intersections remain smooth. The universal +surfaces and curves over this closure are smooth proper families. +Local trivializations on a finite cover give uniform bounds for +the metrics, their derivatives, coordinate comparisons, and the +partitions of unity used below. Surjectivity on sections allows the +curve on each chosen surface to vary in an analytic open subset of +its complete linear system. + +For a fixed coherent sheaf, a general defining section avoids all +its associated points, and is thus a nonzerodivisor. Applying this +to the quotients in a finite collection of exact sequences preserves +their exactness under restriction. To ensure torsion-freeness, +embed a torsion-free sheaf into a vector bundle and require the +defining section to be a nonzerodivisor on its cokernel. Restriction +then embeds the restricted sheaf into the restricted bundle. Such +an embedding exists because a sufficiently positive twist of the +dual is globally generated and the original sheaf injects into its +double dual. Repeating the argument on the surface gives the curve +assertion. Applied to the ordinary cohomology sheaves of a bounded +complex, the nonzerodivisor condition kills the first Tor terms in +the restriction spectral sequence, giving the asserted cohomology +identification. + +These are finitely many nonempty Zariski-open conditions at each +stage. A very-general condition removes at most countably many +proper algebraic closed subsets. Each such subset has empty +analytic interior, so the Baire theorem ensures a choice in every +one of our nonempty analytic neighborhoods. The geometric +constants were fixed before this choice and do not depend on the +sheaves being restricted. +\end{proof} + +\subsection{Sections of subsheaves} + +We first bound sections using a scalar heat estimate. The rank of a +subsheaf enters only through the rank of its evaluation map; the +ambient bundle may have arbitrarily large rank. + +\begin{lemma}[Uniform heat estimates]\label{lem:uniform-heat} +Let $Y$ range over the compact families of curves or surfaces in +Lemma~\ref{lem:uniform-sections}, with real dimension $n$. Write +$\Delta_{\mathrm{sc}}$ for the nonnegative scalar Laplacian. Its heat +kernel satisfies +\[ + 0\le k_s(x,y)\le C s^{-n/2}\qquad(00), +\end{equation} +with a constant independent of the bundle, its rank, and its +connection. +\end{lemma} + +\begin{proof} +The uniform charts and partitions of unity give the Nash inequality +\begin{equation}\label{eq:nash-uniform} + \|f\|_2^{2+4/n} + \le C\bigl(\|df\|_2^2+\|f\|_2^2\bigr)\|f\|_1^{4/n}. +\end{equation} +For completeness, in a Euclidean chart split the Fourier integral +at frequency $R$: its low-frequency part is bounded by +$CR^n\|f\|_1^2$, and its high-frequency part by +$R^{-2}\|df\|_2^2$. Optimize in $R$, and use a fixed finite +partition of unity. Derivatives of the partition produce the +additional $\|f\|_2^2$ term. All comparison constants are uniform +over the chosen families. + +Set $L_{\mathrm{sc}}=\Delta_{\mathrm{sc}}+1$ and +$f_s=e^{-sL_{\mathrm{sc}}}f$. +The scalar semigroup is positivity preserving and contracts $L^1$. +For $y(s)=\|f_s\|_2^2$, \eqref{eq:nash-uniform} and +$y'=-2\langle L_{\mathrm{sc}}f_s,f_s\rangle$ imply +\[ + y'\le-c\|f\|_1^{-4/n}y^{1+2/n}. +\] +Integration gives $\|e^{-sL_{\mathrm{sc}}}\|_{1\to2}\le Cs^{-n/4}$. +Selfadjointness and composition give +$\|e^{-sL_{\mathrm{sc}}}\|_{1\to\infty}\le Cs^{-n/2}$ for every $s>0$. +Multiplication by $e^s$ gives the stated estimate for $k_s$ when +$s\le1$. + +For a section-valued heat solution, Kato's inequality and the +maximum principle give +\[ + |e^{-sL}v|(x)\le(e^{-sL_{\mathrm{sc}}}|v|)(x). +\] +One obtains the differential inequality by differentiating +$(|v|^2+\varepsilon^2)^{1/2}$ and then letting +$\varepsilon\downarrow0$; compatibility of the connection with the +metric is the only bundle input. Approximate a vector-valued point +mass at $y$ in this inequality. This bounds the operator norm of +$K_s(x,y)$ by the scalar kernel of $e^{-sL_{\mathrm{sc}}}$, and proves +\eqref{eq:connection-heat}. +\end{proof} + +\begin{lemma}[Sections of a subsheaf]\label{lem:subsheaf-sections} +Let $Y$ be one of the curves or surfaces in +Lemma~\ref{lem:uniform-sections}, let $G$ be a slope-stable +torsion-free sheaf on $Y$, and let $J\subset G$ have rank $q$. +Then +\begin{equation}\label{eq:subsheaf-sections} + h^0(Y,J)\le Cq\bigl(1+\mu(G)_+\bigr)^{\dim Y}, + \qquad x_+=\max\{x,0\}, +\end{equation} +where $C$ depends only on the chosen geometric families. The same +bound applies to subsheaves of a stable dual bundle tensored with +a line bundle, using the slope of that twisted bundle. +\end{lemma} + +\begin{proof} +On a curve $G$ is locally free. On a surface its reflexive hull +$W=G^{**}$ is locally free and slope-stable: intersecting a +subsheaf of $W$ with $G$ preserves rank and first Chern class, since +$W/G$ is zero-dimensional. The Hermitian--Einstein theorem +\cite{Donaldson1985,UhlenbeckYau1986} therefore gives a metric on +$W$ for which $i\Lambda F_W$ is the scalar determined by +$\mu(G)$. Its proportionality constant is fixed by the dimension +and our volume normalization. + +For a holomorphic section $s$ of $W$, the Bochner identity gives +\[ + \Delta_{\mathrm{sc}}|s|^2\le C\mu(G)_+|s|^2. +\] +Scalar heat comparison consequently yields +$|s(x)|^2\le e^{C u\mu(G)_+}(e^{-u\Delta_{\mathrm{sc}}}|s|^2)(x)$. +Take $u=(1+\mu(G)_+)^{-1}$ and apply +Lemma~\ref{lem:uniform-heat}. The evaluation operator, with the +$L^2$ norm on $H^0(Y,W)$, satisfies +\begin{equation}\label{eq:evaluation-operator} + \|\operatorname{ev}_x\|_{\mathrm{op}}^2 + \le C\bigl(1+\mu(G)_+\bigr)^{\dim Y}. +\end{equation} +Give $H^0(Y,J)\subset H^0(Y,W)$ the induced $L^2$ norm and choose +an orthonormal basis $s_1,\ldots,s_N$. Outside a proper analytic +subset, their evaluations lie in a subspace of dimension at most +$q$. Thus +\[ + \sum_{j=1}^N|s_j(x)|^2 + =\|\operatorname{ev}_x|_{H^0(J)}\|_{\mathrm{HS}}^2 + \le q\|\operatorname{ev}_x\|_{\mathrm{op}}^2. +\] +Integration proves \eqref{eq:subsheaf-sections}; the volumes are +uniformly bounded. The case $q=0$ means $J=0$. Dualization and +tensoring with a line preserve slope stability, so the same proof +gives the last assertion. +\end{proof} + +\subsection{A matrix spectral estimate on surfaces} + +To control first cohomology, a bound proportional to rank would +not suffice. We instead count harmonic forms using the square of +the Hilbert--Schmidt norm of the trace-free curvature. The following +spectral estimate is a four-dimensional, bundle-valued +Cwikel--Lieb--Rozenblum estimate; compare +\cite[Theorem~3.2 and Lemma~3.4]{Frank2014}. We give the +heat-semigroup proof, retaining the matrix trace to keep its constants +independent of rank. + +\begin{lemma}[Spectral dimension bound]\label{lem:matrix-spectral} +Let $S$ be one of the surfaces in Lemma~\ref{lem:uniform-sections}. +Let $\mathcal V$ be a Hermitian vector bundle with metric connection, +$L=\nabla^*\nabla+1$, and let $P$ be a bounded measurable +nonnegative selfadjoint endomorphism of $\mathcal V$. +If a finite-dimensional subspace $U$ of the form domain of $L$ +satisfies +\begin{equation}\label{eq:spectral-form-hypothesis} + \langle Lu,u\rangle\le\langle Pu,u\rangle\qquad(u\in U), +\end{equation} +then +\begin{equation}\label{eq:matrix-spectral-bound} + \dim U\le C\int_S\operatorname{tr}(P^2). +\end{equation} +The constant is independent of $\mathcal V$, its rank, $\nabla$, +and $P$. +\end{lemma} + +\begin{proof} +The compact positive operator $L^{-1/2}PL^{-1/2}$ has at least +$\dim U$ eigenvalues greater than or equal to $1$: apply min--max +to the subspace $L^{1/2}U$ and +\eqref{eq:spectral-form-hypothesis}. Its nonzero eigenvalues, +with multiplicity, are those of $P^{1/2}L^{-1}P^{1/2}$. +Define an operator with an additional time variable by +\[ + (\mathcal Bv)(s)=e^{-sL/2}P^{1/2}v, + \qquad + \mathcal B:L^2(S,\mathcal V)\longrightarrow + L^2((0,\infty)\times S,\mathcal V). +\] +Integration of the heat semigroup gives +$\mathcal B^*\mathcal B=P^{1/2}L^{-1}P^{1/2}$. + +Fix $0<\eta<1$ and split $\mathcal B=\mathcal B_{\le}+\mathcal B_>$ +by inserting the fiberwise spectral projections +$\mathbf1_{sP\le\eta}$ and $\mathbf1_{sP>\eta}$ immediately after +$P^{1/2}$. Heat contraction gives +\[ + \|\mathcal B_{\le}v\|^2 + \le\int_0^\infty\langle P\mathbf1_{sP\le\eta}v,v\rangle\,ds + \le\eta\|v\|^2. +\] +Indeed, each positive eigenvalue $p$ contributes +$\int_0^{\eta/p}p\,ds=\eta$, and a zero eigenvalue contributes zero. + +Let $K_s$ be the kernel of $e^{-sL}$. The semigroup identity and +Lemma~\ref{lem:uniform-heat}, in real dimension four, imply +\begin{align*} + \|\mathcal B_>\|_{\mathrm{HS}}^2 + &=\int_0^\infty\int_S + \operatorname{tr}\bigl(P\mathbf1_{sP>\eta}K_s(x,x)\bigr)\,dx\,ds\\ + &\le C\int_S\int_0^\infty + s^{-2}\operatorname{tr}\bigl(P\mathbf1_{sP>\eta}\bigr)\,ds\,dx\\ + &=C\eta^{-1}\int_S\operatorname{tr}(P^2). +\end{align*} +The last identity follows eigenvalue by eigenvalue from +$\int_{\eta/p}^\infty s^{-2}p\,ds=p^2/\eta$. +All integrands are nonnegative, so Tonelli's theorem applies. + +On the spectral subspace where $\mathcal B^*\mathcal B\ge1$, +the triangle inequality gives +$\|\mathcal B_>v\|\ge(1-\sqrt\eta)\|v\|$. +Apply this to an orthonormal basis of that subspace and sum. +Taking $\eta=1/4$ proves \eqref{eq:matrix-spectral-bound}. +\end{proof} + +In the application below, $P$ will be built from trace-free curvature. +The matrix trace in \eqref{eq:matrix-spectral-bound} will therefore +bound the dimension of harmonic forms by curvature energy, with no +additional term proportional to the bundle rank. + +\subsection{First cohomology near a negative scalar class} + +Fix the rational class $B_0$ from the threefold, and restrict it to +the chosen surfaces. The next proposition concerns sheaves whose +normalized first Chern classes are close to +$B_0+(b-A)H$. A sufficiently negative scalar part makes the central +curvature component of the dual bundle positive; the trace-free curvature measures the +failure of the resulting vanishing theorem. + +\begin{proposition}[Surface first cohomology]\label{prop:negative-surface-cohomology} +Fix a real number $M$. There is a number $A_0$, depending only on +$X,H,B_0,M$ and the chosen smooth neighborhoods, with the following +property. For $A\ge A_0$ and a finite set +$I\subset\{t\in\ZZ:t\le M\}$, there is a constant $C$ such that, +for every chosen surface $S$, every $b\in[-1,1]$, every finite +collection of slope-stable torsion-free sheaves $G_i$ on $S$, and +every $t\in I$, +\begin{equation}\label{eq:negative-surface-cohomology} + \sum_i h^1(S,G_i(tH)) + \le C\sum_i r_i\left( + \left\|\frac{c_1(G_i)}{r_i}-T\right\|^2+\xi_{G_i}\right), + \qquad T=B_0|_S+(b-A)H. +\end{equation} +The constant is independent of the collection and its ranks. +The same upper bound holds for $h^1(S,G(tH))$ if the $G_i$ are +the factors of a filtration of $G$. +In particular, a bound $C_0e$ for the sum on the right gives a +bound $CC_0e$ for the first cohomology. +\end{proposition} + +The order of constants matters in the next section. The threshold +$A_0$ uses only the upper twist bound $M$. Once $A$ and the finite +set $I$ are fixed, the cohomology constant $C$ may depend on both; +it remains independent of the sheaves, their ranks, and $b$. + +\begin{proof} +We prove the bound for one factor $G$, writing +$r=\rk G$, $C_1^{\mathrm{num}}=c_1(G)/r$, and +$\rho=\|C_1^{\mathrm{num}}-T\|$. We separate $\rho\ge1$, +where the numerical error already controls rank, from $\rho<1$, +where the curvature argument is needed. + +Suppose first that $\rho\ge1$. The class $T$ is uniformly bounded +once $A$ is fixed, so $1+\|C_1^{\mathrm{num}}\|^2\le C_A\rho^2$. +Lemma~\ref{lem:subsheaf-sections}, applied to $G(tH)$, gives +$h^0(G(tH))\le Cr(1+\|C_1^{\mathrm{num}}\|^2+t^2)$. +Put $W=G^{**}$. Serre duality identifies $H^2(G(tH))^*$ with +$\Hom(G,K_S(-tH))=H^0(W^*\otimes K_S(-tH))$; the equality follows +by extending across the finite support of $W/G$. +The dual version of Lemma~\ref{lem:subsheaf-sections} gives the +same bound for $h^2$. + +Surface Riemann--Roch reads +\[ + \chi(G(tH)) + =r\chi(\OO_S)+\frac{hr}{2} + \left(\langle C_1^{\mathrm{num}}+tH, + C_1^{\mathrm{num}}+tH\rangle-\xi_G\right) + -\frac r2(C_1^{\mathrm{num}}+tH)K_S. +\] +The Hodge norm controls both the intersection form and pairing +with $K_S$. Consequently +$|\chi(G(tH))|\le Cr(1+\|C_1^{\mathrm{num}}\|^2+t^2+\xi_G)$. +Since $I$ is finite, $h^1=h^0+h^2-\chi$ is at most +$Cr(\rho^2+\xi_G)$, as required. + +Now assume $\rho<1$. The hull sequence +$0\to G\to W\to Q\to0$ has $Q$ zero-dimensional and +\begin{equation}\label{eq:hull-length} + \xi_G-\xi_W=\frac{2\operatorname{length}Q}{hr}, + \qquad + \operatorname{length}Q\le\frac{hr\xi_G}{2}. +\end{equation} +Here $W$ is stable, as in the proof of +Lemma~\ref{lem:subsheaf-sections}, and $\xi_W\ge0$ by the +classical Bogomolov--Gieseker inequality. The long exact +cohomology sequence gives +$h^1(G(tH))\le h^1(W(tH))+\operatorname{length}Q$. +It remains to bound $h^1(W(tH))$ by $Cr\xi_W$. + +Equip $W$ with its Hermitian--Einstein metric. Write its Chern +curvature as +\[ + F_W=F_0+\frac{\operatorname{tr}F_W}{r}\operatorname{id}_W, + \qquad + \vartheta=\frac{i}{2\pi r}\operatorname{tr}F_W. +\] +The real $(1,1)$-form $\vartheta$ is closed and has constant +contraction, hence is harmonic. It represents +$C_1^{\mathrm{num}}$. The Hodge star formula on primitive +$(1,1)$-forms identifies its $L^2$ norm with a fixed multiple of +the divisor norm. Uniform elliptic estimates on our compact +smooth families therefore give a uniform bound for the supremum +norm of a harmonic representative in terms of this divisor norm. +In particular, since $\rho<1$ and $|b|\le1$, +\begin{equation}\label{eq:central-curvature-positive} + -\vartheta-t\omega_S\ge(A-M-C_1)\omega_S + \qquad(t\le M). +\end{equation} +The fixed constant $C_1$ bounds the harmonic representatives of +$B_0|_S+bH$ and the unit ball of errors. Here $\omega_S$ is the +induced Kähler form, and the line bundle $\OO_S(H)$ has curvature +$2\pi\omega_S$. Choose $A_0$ so that the right side is strictly +positive with a fixed lower bound. This choice uses only the +upper endpoint $M$, not the lower endpoint of $I$. + +The trace-free curvature $F_0$ is primitive. Chern--Weil and the +primitive Hodge star formula give +\begin{equation}\label{eq:tracefree-energy} + \int_S|F_0|_{\mathrm{HS}}^2\le C\frac{\operatorname{Disc}(W)}r + =Chr\xi_W. +\end{equation} +Indeed +$\operatorname{Disc}(W)=-r\int_S +\operatorname{tr}((iF_0/2\pi)\wedge(iF_0/2\pi))$, +and the last integrand is the negative squared norm because +$F_0$ is primitive. The fixed conversion constant depends on the +curvature convention, not on $r$. + +By Serre duality and the Dolbeault theorem, $h^1(W(tH))$ is the +dimension of the space $U$ of harmonic $(2,1)$-forms with values in +$W^*(-tH)$. On this bundle the Bochner--Kodaira--Nakano identity +\cite[Chapter~VII, Corollary~1.3]{DemaillyCADG} is +\begin{equation}\label{eq:nakano-identity} + \Delta''=\Delta'+[iF,\Lambda]. +\end{equation} +For the central curvature, \eqref{eq:central-curvature-positive} +makes the commutator on $(2,1)$-forms at least $\lambda I$ for a +fixed $\lambda>0$. To see the sign directly, diagonalize the +central $(1,1)$-form in a unitary frame. On top holomorphic +degree, its commutator is the sum of the positive eigenvalues +whose antiholomorphic indices occur; there is one such index +here. Denote the remaining selfadjoint curvature endomorphism by +$K_0$. Wedge and contraction act on fixed-dimensional form +spaces, so +\begin{equation}\label{eq:curvature-endomorphisms} + |K_0|_{\mathrm{HS}}\le C|F_0|_{\mathrm{HS}}. +\end{equation} + +We explain explicitly how to retain an elliptic operator in +\eqref{eq:nakano-identity}. Let $\nabla$ be the total metric +connection on $(2,1)$-forms with these bundle coefficients. +For a pure-type form the terms in $\partial u+\bar\partial u$ +have different types, as do their adjoints. Hence the quadratic +form of the connection de Rham Laplacian is the sum of the forms +of $\Delta'$ and $\Delta''$. On $U$, the latter vanishes, so the +de Rham form equals the $\Delta'$ form. The Weitzenböck formula, +restricted to this type, consequently gives +\begin{equation}\label{eq:weitzenbock-on-harmonics} + \langle\Delta'u,u\rangle + \ge\|\nabla u\|_2^2-C_2\|u\|_2^2+\langle R_0u,u\rangle + \qquad(u\in U), +\end{equation} +where $R_0$ is selfadjoint and +$|R_0|_{\mathrm{HS}}\le C|F_0|_{\mathrm{HS}}$. +The bounded term contains the tangent curvature and the central +bundle curvature; its bound $C_2$ is uniform for fixed $A,I$. +The formula follows by anticommuting wedge and contraction in +$d_\nabla$ and $d_\nabla^*$: the second-order part is +$\nabla^*\nabla$, and the remaining terms contract the tangent +and bundle curvature. Only the fixed form dimension enters +the contraction constants. + +Choose +$0<\varepsilon\le\min\{1,\lambda/(C_2+1)\}$. +On $U$, retain $\varepsilon\Delta'$ in +\eqref{eq:nakano-identity}, discard the nonnegative remainder, +and use \eqref{eq:weitzenbock-on-harmonics}. This gives +\[ + \varepsilon\|\nabla u\|_2^2+ + (\lambda-\varepsilon C_2)\|u\|_2^2 + \le\langle(|K_0|+\varepsilon|R_0|)u,u\rangle. +\] +Our choice ensures $\lambda-\varepsilon C_2\ge\varepsilon$. +Thus, with spectral absolute values of selfadjoint matrices, +\[ + L=\nabla^*\nabla+1,\qquad + P=\varepsilon^{-1}|K_0|+|R_0|, + \qquad + \langle Lu,u\rangle\le\langle Pu,u\rangle\quad(u\in U). +\] +Moreover, +\begin{equation}\label{eq:potential-trace} + \operatorname{tr}(P^2) + \le2\varepsilon^{-2}|K_0|_{\mathrm{HS}}^2 + +2|R_0|_{\mathrm{HS}}^2 + \le C|F_0|_{\mathrm{HS}}^2. +\end{equation} +This uses the Hilbert--Schmidt norm of the matrices, without +replacing them by their operator norms times the identity. + +Lemma~\ref{lem:matrix-spectral}, followed by +\eqref{eq:tracefree-energy}, now proves +$h^1(W(tH))=\dim U\le Cr\xi_W$. +Together with \eqref{eq:hull-length}, this gives the required bound +for the factors with $\rho<1$ as well. Sum over the factors. +For an extension $0\to G'\to G\to G''\to0$, the cohomology +sequence gives $h^1(G(tH))\le h^1(G'(tH))+h^1(G''(tH))$; +induction proves the final filtration assertion. +\end{proof} + +The combination of Lemma~\ref{lem:restriction-variance} and +Proposition~\ref{prop:negative-surface-cohomology} is the point of +the section. Restriction increases the relevant numerical sums by +at most a discriminant contribution, while the cohomology bound +uses those sums without adding a bare rank term. The fixed-apex +argument can therefore apply these estimates even when its +numerical error is zero. diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/04-apex.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/04-apex.tex new file mode 100644 index 0000000000000000000000000000000000000000..9da9f01b91f8c65ab003bc2518f6fceea08035e6 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/04-apex.tex @@ -0,0 +1,918 @@ +\section{A uniform estimate at one volume}\label{sec:apex} + +The preceding surface estimates control cohomology in terms of a +discriminant, without a factor depending on the rank. We now use them to +prove the following estimate at one fixed value of the volume parameter. +The factor on its right measures the distance from the equality case of +the first-tilt discriminant inequality. + +\begin{proposition}\label{prop:apex} +Let $X$ be a smooth projective complex threefold, let $H$ be very ample, +and let $B_0$ be a rational divisor class with $H^2B_0=0$. +There are constants $A>0$ and $C\geq0$, depending only on $X,H,B_0$, +such that, for every $b\in\RR$ and every finite-slope +$\nu_{A,b}$-semistable object $E\in\mathcal B_{H,B_0+bH}$ +with $\nu_{A,b}(E)=0$, +\begin{equation}\label{eq:apex-estimate} + z_B(E)-\frac{A^2}{6}d_B(E) + \leq C\bigl(d_B(E)-A|r(E)|\bigr),\qquad B=B_0+bH. +\end{equation} +The same constants can be chosen for $B_0$ and $-B_0$. +\end{proposition} + +Here $h=H^3$ and $d_B,w_B,z_B$ are normalized by $h$ as in +Section~\ref{sec:tilt}. In particular, $d_B\geq A|r|$ at slope zero. +We first choose all the constants from the fixed geometry. For an +individual $E$, we then separate a narrow range of sheaf slopes from a +remainder whose cohomology is controlled by +\[ + e=d_B(E)-A|r(E)|. +\] +Reduction to characteristic $p$ and Riemann--Roch convert these +cohomology estimates into \eqref{eq:apex-estimate}. The remainder +contributes at most $Cp^3e$. The narrow sheaf tail can have large +rank even when $e=0$; two restriction sequences will instead bound +its sections by a cubic expression in its slopes, producing the +coefficient $A^2/6$. Only errors that vanish after division by +$p^3$ will be allowed to depend on $E$. +We write $F(t)=F\otimes\OO_X(tH)$, also for complexes, and write +$h^i(F)=\dim\HH^i(X,F)$. + +\subsection{Constants chosen before the object} + +Tensoring by $\OO_X(kH)$ identifies the parameters $b$ and $b+k$ +and preserves all the $B$-twisted characters. It therefore suffices +to treat $|b|\leq1$. Choose a positive integer $q$ and a divisor $D$ +representing $qB_0$. On a smooth reduction of $X$ in characteristic +$p$, let $F$ denote absolute Frobenius and put +\begin{equation}\label{eq:rounding-lines} + L_p=\OO_X\bigl(-\lfloor p/q\rceil D-\lfloor pb\rceil H\bigr), + \qquad L'_p=K_X\otimes L_p^{-1}. +\end{equation} +Either nearest integer can be chosen at a tie. Thus +$c_1(L_p)/p\longrightarrow-B$. +We will estimate Frobenius sections through sections at two fixed +twists on the original object. The following presentation supplies +those twists; its consequence immediately below explains their roles. + +\begin{lemma}\label{lem:frobenius-presentation} +There are positive integers $j,j_2$ and a constant $C_0$, determined +by $X,H,D$, with the following property. After choosing and shrinking +a model of this fixed geometry over a finitely generated integral +subring of $\CC$, every smooth geometric fiber in characteristic $p$ +and every $|b|\leq1$ satisfy the following assertion, for either +$L=L_p$ or $L=L'_p$: +\[ + G_{p,L}=F_*(K_X^{1-p}\otimes L^{-1}) +\] +has a presentation +\begin{equation}\label{eq:frobenius-presentation} + 0\longrightarrow K_{p,L}\longrightarrow\OO_X(-j)^{N_{p,L}} + \longrightarrow G_{p,L}\longrightarrow0, + \qquad N_{p,L}\leq C_0p^3, +\end{equation} +where $K_{p,L}(j_2)$ is generated by its global sections. +\end{lemma} + +\begin{proof} +For each line bundle in the fixed list +$\OO_X(\pm D),\OO_X(\pm H),K_X^{\pm1}$, choose three successive +surjections from sums of negative powers of $\OO_X(H)$, with locally +free kernels. They exist by global generation after twisting; a +surjection onto a locally free sheaf has locally free kernel. +Spread these finitely many sequences and a regularity bound for +$\OO_X$ to a model, and shrink so that they remain exact on the +fibers. + +There is consequently an integer $v_0$, independent of the fiber, +such that tensoring by any one line bundle in the list increases by +at most $v_0$ a threshold above which all higher cohomology vanishes. +Indeed, tensor its three-step presentation by the original line +bundle and use the cohomology sequences. The three dimension shifts +end in cohomological degree greater than three. If the finitely many +twists in the presentation are bounded by $v_0$, every intervening +sum of lines has vanishing higher cohomology above the shifted +threshold. Starting with the fixed bound for $\OO_X$ proves the +claim by induction on the number of tensor factors. + +The line $K_X^{1-p}\otimes L^{-1}$ is a product of at most $c_0p+c_1$ +members of this fixed list, uniformly for $|b|\leq1$ and for both +choices of $L$. It therefore has no higher cohomology after twisting +by $sH$ whenever $s\geq c_2p+c_3$. Projection formula gives +\[ + H^i\bigl(G_{p,L}(j-i)\bigr) + =H^i\bigl(K_X^{1-p}\otimes L^{-1}(p(j-i))\bigr)=0 + \quad(1\leq i\leq3) +\] +for a fixed sufficiently large $j$. Castelnuovo--Mumford regularity +then gives the evaluation surjection in +\eqref{eq:frobenius-presentation}, using a basis of +$H^0(G_{p,L}(j))$, and gives surjectivity of multiplication of these +sections at every higher twist. + +The cohomology sequence for its kernel gives a uniform regularity +bound $j_2$. For degrees at least two use regularity of $G_{p,L}$ +and of $\OO_X(-j)$; in degree one use the just-proved multiplication +surjectivity. Increasing $j_2$ makes the kernel generated at $j_2$. +Finally, the higher cohomology of +$K_X^{1-p}\otimes L^{-1}(pj)$ vanishes, so +\[ + N_{p,L}=\chi\bigl(K_X^{1-p}\otimes L^{-1}(pj)\bigr). +\] +This Euler characteristic is constant under specialization for +each of the indicated line bundles. On the complex fiber, +Riemann--Roch is a polynomial of degree at most three in exponents +bounded by a fixed multiple of $p$. This proves $N_{p,L}\leq C_0p^3$. +All the choices preceded any choice of a tilt object. +\end{proof} + +\begin{corollary}[From fixed twists to Frobenius sections] +\label{cor:frobenius-section-transfer} +On a smooth geometric fiber in the preceding lemma, let $P_1$ be a +perfect complex with ordinary cohomology in degrees $[-1,0]$. +If +\[ + H^0\bigl(\mathcal H^{-1}(P_1)(j_2)\bigr)=0, +\] +then, for either $L=L_p$ or $L=L'_p$, +\begin{equation}\label{eq:frobenius-section-transfer} + h^0(F^*P_1\otimes L) + \leq C_0p^3h^0(P_1(j)). +\end{equation} +The twists $j,j_2$ and the constant $C_0$ are those chosen from the +fixed geometry in Lemma~\ref{lem:frobenius-presentation}. +\end{corollary} + +\begin{proof} +Finite duality and projection formula give +\begin{equation}\label{eq:frobenius-hom-identity} + h^0(F^*P_1\otimes L)=\dim\Hom(G_{p,L},P_1). +\end{equation} +Indeed $F^!P_1=F^*P_1\otimes K_X^{1-p}$, so adjunction applied +to $K_X^{1-p}\otimes L^{-1}$ proves the identity. +The semilinearity of absolute Frobenius does not change these +dimensions. Apply \eqref{eq:frobenius-presentation}. The obstruction +to injectivity of +\[ + \Hom(G_{p,L},P_1)\longrightarrow + \Hom(\OO_X(-j)^{N_{p,L}},P_1) +\] +is the image of +\[ + \Hom(K_{p,L}[1],P_1) + =\Hom(K_{p,L},\mathcal H^{-1}(P_1)). +\] +The equality uses the ordinary cohomological range of $P_1$. +Generation of $K_{p,L}(j_2)$ and the assumed vanishing make this +group zero. The Hom map is therefore injective, and its target +has dimension $N_{p,L}h^0(P_1(j))\leq C_0p^3h^0(P_1(j))$. +\end{proof} + +Thus the characteristic-zero estimates we shall need are a section +bound at $j$ and a vanishing for the negative cohomology sheaf at +$j_2$. They will pass to the chosen fibers by semicontinuity. +We now choose the remaining constants so that both estimates can +be proved uniformly in the tilt object. + +Fix $m$ and the compact smooth neighborhoods of surfaces and curves +from Lemma~\ref{lem:uniform-sections}. In particular, +\begin{equation}\label{eq:apex-m-choice} + h^0(\OO_S(m))=\chi(\OO_S(m)),\qquad + \mu(K_S)-m\leq-1. +\end{equation} +Put $M=\max(j,j_2)$, choose $c>M+2$, and then choose $A$ so large that +Proposition~\ref{prop:negative-surface-cohomology} applies for upper +twist $M$ and both $B_0,-B_0$, and +\begin{equation}\label{eq:apex-A-choice} + A>c,\qquad A>j_2+2,\qquad A\geq1,\qquad + A^2-c^2\geq\delta_{\rm line}:=-HB_0^2/h. +\end{equation} +The number $\delta_{\rm line}$ is nonnegative by the Hodge index +theorem. We shall choose one further integer $m_0$ below, depending +only on these constants, and put $t_0=-m_0$. Thus $A$ does not +depend on the negative endpoint $t_0$: the negative-surface estimate +requires an upper bound on the twists for its positivity, and its +constant may subsequently depend on the chosen finite range. + +\subsection{The two decompositions at slope zero} + +Fix a finite-slope semistable $E$ at $(A,b)$ with $|b|\leq1$ and +$\nu_{A,b}(E)=0$. Suppress $B=B_0+bH$ in numerical subscripts when +there is no ambiguity. Write +\[ + V=\mathcal H^{-1}(E),\qquad U=\mathcal H^0(E),\qquad + \bar U=U/U_{\rm tor}. +\] +The sheaf $V$ is torsion-free. All the ordinary slope factors of +$V$ have slope at most $b-A$, and all those of $\bar U$ have slope +at least $b+A$. To prove the first assertion, the highest slope +factor $V_1[1]$ is a subobject of $E$ in the first-tilt heart. +Finite-slope semistability excludes a denominator-zero subobject +and gives +\[ + w_B(V_1)\geq A^2r(V_1)/2. +\] +Classical Bogomolov--Gieseker gives +$2w_B(V_1)/r(V_1)\leq(\mu(V_1)-b)^2$; the first torsion pair gives +$\mu(V_1)\leq b$. Together they give $\mu(V_1)\leq b-A$. +For the other assertion use the lowest slope factor of $\bar U$, +which is a quotient of $E$, and the same computation with +$\mu>b$. + +Throughout this section, $O(e)$ denotes an absolute value bounded +by a fixed constant times $e=d-A|r|$. Constants in this notation +may depend on the choices above, but not on $E$, its rank, or $b$. +Rank and character bounds for a complex do not mean bounds for the +length of its zero-dimensional cohomology. + +For nonnegative rank, we retain a quotient with slopes within one +unit of $b+A$ and bound sections of the residual complex by $e$. +For nonpositive rank, we retain a shifted subsheaf with slopes +within one unit of $b-A$; its surface first cohomology will allow us to +bound all sections of $E$ by $e$. This second bound will also be +applied to the shifted dual of a nonnegative-rank object, controlling +its degree-two cohomology. + +Suppose first that $r(E)\geq0$. On refined ordinary slope +filtrations the equality $e=d-Ar$ reads +\begin{equation}\label{eq:positive-excess} + e=\sum_{\bar U}r_i(\mu_i-b-A) + +d(U_{\rm tor})+\sum_Vr_i(b-\mu_i+A). +\end{equation} +Every term is nonnegative. Let $Q$ be the slope tail quotient of +$\bar U$ whose slopes lie in $[b+A,b+A+1]$, and let $P$ be the kernel +of the epimorphism $E\to Q$ in the tilt heart: +\begin{equation}\label{eq:positive-residual-triangle} + P\longrightarrow E\longrightarrow Q. +\end{equation} +Empty parts mean zero sheaves. The negative cohomology of $P$ is +$V$, and its degree-zero cohomology is an extension of the slope +prefix with slopes $>b+A+1$ by $U_{\rm tor}$. The corresponding +sheaf sequences, shifted where appropriate, are exact in the tilt +heart because their torsion-free terms stay in the indicated halves +of the torsion pair. + +Equation~\eqref{eq:positive-excess} gives +\begin{equation}\label{eq:residual-character-bound} + d(P),\ |r(P)|,\ r(\mathcal H^{-1}(P)),\ + r(\mathcal H^0(P))=O(e). +\end{equation} +For example, on the high prefix the positive number +$\mu_i-b-A>1$ controls both $r_i$ and $r_i(\mu_i-b)$; on $V$ the +number $b-\mu_i+A\geq2A$ does the same. The torsion degree is at +most $e$. Quotient comparison and the classical discriminant +inequality on the factors of $Q$ give +\begin{equation}\label{eq:positive-tail-moments} + \frac{A^2r(Q)}2\leq w_B(Q) + \leq\frac{A^2r(Q)}2+O(e),\qquad + \sum_Qr_i\delta_i^B=O(e). +\end{equation} +Indeed, write $\mu_i-b=A+s_i$, where $0\leq s_i\leq1$ and +$\sum r_is_i\leq e$. Then +$\sum r_i(\mu_i-b)^2=A^2r(Q)+O(e)$, and subtract $2w_B(Q)$. +Additivity with $w_B(E)=A^2r(E)/2$ now also gives $w_B(P)=O(e)$. + +For $r(E)\leq0$ use instead +\begin{equation}\label{eq:negative-excess} + e=\sum_Vr_i(b-A-\mu_i) + +\sum_{\bar U}r_i(\mu_i-b+A)+d(U_{\rm tor}). +\end{equation} +Let $V_0$ be the slope prefix of $V$ with slopes in +$[b-A-1,b-A]$ and form the exact triangle in the tilt heart +\begin{equation}\label{eq:negative-residual-triangle} + V_0[1]\longrightarrow E\longrightarrow P. +\end{equation} +The same estimates give \eqref{eq:residual-character-bound} and +$w_B(P)=O(e)$. More explicitly, subobject comparison applied to +$V_0[1]$ gives +\[ + A^2r(V_0)/2\leq w_B(V_0)\leq A^2r(V_0)/2+O(e). +\] +Put $T=B-AH$. If $\mu_i-b=-A-s_i$ with $0\leq s_i\leq1$, +then $\sum r_is_i\leq e$ and +\[ + \sum_{V_0}r_i\mu_T^2\leq e,\qquad + w_T(V_0)=w_B(V_0)+A d_B(V_0)+A^2r(V_0)/2\geq-Ae. +\] +Since $\delta_i^T=\mu_T^2-2w_T(G_i)/r_i$, it follows that +\begin{equation}\label{eq:negative-tail-energy} + \sum_{V_0}r_i(\mu_T^2+\delta_i^T)=O(e). +\end{equation} +After very general restriction to a surface in the fixed smooth +neighborhood and refinement into stable factors, this bound remains +valid by Lemma~\ref{lem:restriction-variance}. For each such factor, +\[ + \mu_T^2+\delta^T=\|c_0/r-T\|^2+\xi. +\] +Consequently Proposition~\ref{prop:negative-surface-cohomology} +implies, for every integer $t_00$. A nonzero proper tilt quotient $I$ fits into +$0\to K\to L\to I\to0$ in the heart, with $K$ a sheaf and +\[ + 0\longrightarrow\mathcal H^{-1}(I)\longrightarrow K + \longrightarrow L\longrightarrow\mathcal H^0(I)\longrightarrow0. +\] +The image in $L$ is nonzero: otherwise $K=\mathcal H^{-1}(I)$ +belongs to both halves of the slope torsion pair and is zero, so +the quotient is the whole line. The image therefore has rank one. +It follows that $K$ is torsion-free, all its slopes lie in $(b',\ell]$, +and $r(I)=-r(\mathcal H^{-1}(I))$. Classical Bogomolov--Gieseker +on the slope factors gives $w_{B_0+b'H}(K)\leq u d(K)/2$. +Since $r(L)=1$, $d(L)=u$ and +$2w(L)=u^2-\delta_{\rm line}$, subtraction gives +\begin{equation}\label{eq:line-quotient-numerator} + n(I)\geq\frac{u d(I)-\delta_{\rm line} + +a^2r(\mathcal H^{-1}(I))}{2}. +\end{equation} +If $a^2\geq\delta_{\rm line}$, every finite quotient slope is at +least $u/2$ when $r(I)<0$, and at least +$u/2-h\delta_{\rm line}/2$ when $r(I)=0$. +For the latter assertion, $d(I)>0$ lies in $h^{-1}\ZZ$, so +$d(I)\geq1/h$. The whole line has slope +\begin{equation}\label{eq:line-whole-slope} + \nu_{a,b'}(L)=\frac u2-\frac{a^2+\delta_{\rm line}}{2u}. +\end{equation} + +Choose the positive integer $m_0$ so large that +\[ + m_0-1>h\delta_{\rm line},\qquad + (m_0-1)^2>A^2+\delta_{\rm line}. +\] +At $(A,b)$ all finite quotient slopes of $\OO_X(m_0H)$ are then +strictly positive. A nonzero map from this line to $E$ would have +an image that is both such a quotient and a subobject of the +slope-zero semistable $E$. A denominator-zero image is also +excluded by finite-slope semistability. Thus, with $t_0=-m_0$, +\begin{equation}\label{eq:initial-section-vanishing} + h^0(E(t_0))=0,\qquad + h^0(P(t_0))=0\quad\hbox{in the positive-rank decomposition}. +\end{equation} + +For the remaining estimates work at the side point +\[ + b_s=b-c,\qquad a_s=\sqrt{A^2-c^2},\qquad B_s=B_0+b_sH. +\] +Lemma~\ref{lem:tilt-semicircle} shows that $E$ is semistable there +with slope $c$. +Both triangles above remain short exact sequences in its tilt +heart: the negative sheaf slopes are at most $b-Ab-c$. The numerical quantities +$d(P),r(P),w(P)$ are still $O(e)$ after this bounded twist change. +For any integer $t_0\leq t\leq M$, the line $\OO_X(-tH)$ has +$u=-t-b_s>1$. Equations~\eqref{eq:line-quotient-numerator} and +\eqref{eq:line-whole-slope} provide a single lower bound $-C_1$ for +all its finite tilt quotient slopes, uniformly in this finite range. + +\begin{lemma}\label{lem:restriction-image} +Let $E\in\mathcal B_{H,B_0+bH}$ be finite-slope +$\nu_{A,b}$-semistable with $\nu_{A,b}(E)=0$ and $|b|\leq1$. +Put $B=B_0+bH$ and $e=d_B(E)-A|r(E)|$. Let $P$ be the residual term in +\eqref{eq:positive-residual-triangle} when $r(E)\geq0$, or in +\eqref{eq:negative-residual-triangle} when $r(E)\leq0$. +Let +$W\subset\Hom(\OO_X(-tH),P)$ be any finite-dimensional subspace, +where $t_0\mu(K_S)-m$ has no homomorphism to $K_S(-m)$. +For all the other factors, \eqref{eq:apex-m-choice} gives +$\mu_i\leq-1$, so their total rank, as well as their second slope +moment, is $O(e)$. Serre duality and +Lemma~\ref{lem:subsheaf-sections} applied to their dual hulls, +twisted by $K_S(-m)$, bound their $h^2$ by $O(e)$. +Surface Riemann--Roch and +\eqref{eq:evaluation-image-numerics} give +\[ + \chi\bigl(K'|_S(t+m)\bigr) + =r(K')\chi(\OO_S(m))+O(e). +\] +Indeed the error consists of $w_0(K'(t))$ and the intersection of +$c_0(K'(t))$ with the fixed class $mH-K_S/2$; here +$K_S=(K_X+H)|_S$. Consequently +\begin{equation}\label{eq:kernel-many-sections} + h^0\bigl(K'|_S(t+m)\bigr) + \geq r(K')h^0(\OO_S(m))-O(e). +\end{equation} + +Restrict the triangle $K'\to\OO_X(-tH)^N\to I$ and twist by +$(t+m)H$. The kernel on $H^0$ from $K'|_S(t+m)$ has dimension at +most +$h^{-1}(I|_S(t+m))=h^0(V_I|_S(t+m))$. +Equations~\eqref{eq:evaluation-negative-sections} and +\eqref{eq:kernel-many-sections}, with $r(K')=N+O(e)$, therefore +give +\[ + \begin{aligned} + &\dim\im\!\left( + H^0(K'|_S(t+m))\longrightarrow H^0(\OO_S(m))^N + \right)\\ + &\hspace{25mm}\geq N h^0(\OO_S(m))-O(e). + \end{aligned} +\] +The source of the next map has dimension $Nh^0(\OO_S(m))$. +Subtracting the displayed lower bound shows that the image of +\[ + H^0(\OO_S(m))^N\longrightarrow\HH^0(I|_S(t+m)) +\] +has dimension $O(e)$. This is the cancellation which removes +the unrestricted dimension $N$ from the estimate. + +It remains to remove the auxiliary twist $m$. +Multiplication by the section defining +$C_m$ gives a map from $\HH^0(I|_S(t))$ to this latter group. +Its kernel has dimension at most +$h^{-1}(I|_{C_m}(t+m))$, which is again $O(e)$ by +\eqref{eq:evaluation-negative-sections}. The image of +$H^0(\OO_S)^N$ in $\HH^0(I|_S(t))$ consequently has dimension +$O(e)$. Composing with $I\to P$ proves the assertion for $W$. +All equalities involving $h^{-1}$ use derived restriction with +nonzerodivisors on the ordinary cohomology sheaves; no exactness +of a tilt-image operation under restriction was assumed. +\end{proof} + +\begin{corollary}\label{cor:residual-sections} +Let $E\in\mathcal B_{H,B_0+bH}$ be finite-slope +$\nu_{A,b}$-semistable with $\nu_{A,b}(E)=0$ and $|b|\leq1$. +Put $B=B_0+bH$ and $e=d_B(E)-A|r(E)|$. For $r(E)\geq0$, take $P$ from +\eqref{eq:positive-residual-triangle} and put +$V_P=\mathcal H^{-1}(P)$. For $r(E)\leq0$, use the decomposition +\eqref{eq:negative-residual-triangle} and put $V=\mathcal H^{-1}(E)$. +There is a constant $C_3$, independent of $E,b$, such that +\begin{equation}\label{eq:residual-sections} + \begin{aligned} + h^0(P(j))&\leq C_3e &&\text{for the positive decomposition},\\ + h^0(E(j))&\leq C_3e &&\text{for the negative decomposition}. + \end{aligned} +\end{equation} +Moreover +\begin{equation}\label{eq:negative-cohomology-vanishing} + \begin{aligned} + H^0(V_P(j_2))&=0&&\text{in the positive decomposition},\\ + H^0(V(j_2))&=0&&\text{in the negative decomposition}. + \end{aligned} +\end{equation} +\end{corollary} + +\begin{proof} +For the positive decomposition, apply +Lemma~\ref{lem:restriction-image} to all of $H^0(P(t))$ and the +triangle $P(t-1)\to P(t)\to P|_S(t)$. +The increase in $h^0$ is at most the dimension of the restriction image, +at most $C_2e$. Telescope over the fixed set of integers +$t_0\mu(G)$, contradicting semistability. +Here the Frobenius twist of the algebraically closed base field +is an automorphism and does not change the slope assertion. +Thus no adjacent gap is greater than $c_Y$. There are at most +$r$ factors, so the maximum slope is at most the mean plus +$c_Y(r-1)$, as claimed. +\end{proof} + +The localizations above concern finitely many fixed sheaves, after +$E$ has been chosen. The resulting base still has geometric closed +fibers in unbounded prime characteristics: its constructible image +in $\Spec\ZZ$ contains the generic point and hence a nonempty open +set. Very-generality was used only to choose the original complex +divisors; it is not asserted for their reductions. + +\subsection{The tail and the Frobenius limit} + +Suppose $r(E)\geq0$ and use +\eqref{eq:positive-residual-triangle}. For every stable ordinary +slope factor $G$ of $Q$, choose a very general smooth +$S\in|H|$ and a very general $C\in|H|_S$, imposing all required +exactness conditions. Refine $G|_S$ into stable factors $G_i$ +and their restrictions to $C$ into stable factors $G_{ij}$. +These curves, which serve only in the present algebraic argument, +need not belong to the neighborhoods used for the analytic +estimates. Lemma~\ref{lem:restriction-variance} twice gives, +with $\mu=\mu(G)$, +\begin{equation}\label{eq:twice-restricted-variance} + \sum_{i,j}r_{ij}(\mu_{ij}-\mu)^2 + \leq3r(G)\delta_G^B. +\end{equation} +Indeed the first variance is at most $r(G)\delta_G^B$. +The sum of $r_i\delta_i^B$ on $S$ equals $r(G)\delta_G^B$ plus +that variance, so is at most $2r(G)\delta_G^B$; apply the +restriction estimate once more and add the two variances. +The weighted means are preserved at each step. Spread this +finite data and apply Lemma~\ref{lem:fixed-sheaf-spreading} to +$G,G_i,G_{ij}$. + +Since $H^2D=0$, the normalized slope shift of $L_p$ on $X,S,C$ is +$-pb+O(1)$, with a bounded rounding error. Frobenius commutes with +these restrictions, and its flatness preserves their exact +filtrations. In the following estimates an $O_G$ constant may +depend on all the fixed data chosen for $G$, but not on $p$ or on +the fiber. In particular the ranks in the Frobenius slope error +are now fixed, so they may enter these constants. They must not +enter the coefficients which survive division by $p^3$. +Lemma~\ref{lem:frobenius-slope-gap} gives a vanishing +twist +\[ + k_X=-p(\mu-b)+O_G(1),\qquad + H^0(F^*G\otimes L_p(k_X))=0, +\] +where the bounded integer adjustment is chosen to make the maximum +slope negative. Telescope from $k_X$ to zero using the surface +restriction sequence. At each integer $k$ in this interval, +bound the surface sections by those of its factors $G_i$. +For each such factor telescope on $S$ from its own vanishing +twist $-p(\mu_i-b)+O_G(1)$ up to $k$, if $k$ exceeds that twist. + +On $C$, the $H$-degree is $h$. Taking successive jets at a point +until the maximum ordinary degree slope is negative, and using +\eqref{eq:frobenius-slope-gap}, gives for every factor +\begin{equation}\label{eq:curve-ramp-bound} + h^0(F^*G_{ij}\otimes L_p(l)) + \leq h r_{ij}\bigl(p(\mu_{ij}-b)+l\bigr)_++O_G(1). +\end{equation} +The error includes the integer rounding and at most a bounded +number of point jets, each of dimension $r_{ij}$; all ranks here +are fixed before $p$ varies. Each surface telescope has only +$O_G(p)$ terms in the range under consideration. Summing the +ramp in \eqref{eq:curve-ramp-bound}, and enlarging the sum down +to all its positive terms when necessary, gives +\begin{align} + h^0(F^*G|_S\otimes L_p(k)) + &\leq\frac{hp^2}{2}\sum_{i,j}r_{ij} + (\mu_{ij}-b+k/p)_+^2+O_G(p)\notag\\ + &\leq\frac{hp^2}{2}\bigl[ + r(G)(\mu-b+k/p)_+^2+3r(G)\delta_G^B\bigr]+O_G(p). + \label{eq:surface-square-bound} +\end{align} +For the second inequality use +$x_+^2\leq y_+^2+2y_+(x-y)+(x-y)^2$ and the vanishing of the +weighted first displacement from the mean, followed by +\eqref{eq:twice-restricted-variance}. The ramp sum differs from +the corresponding half-square by $O_G(p)$, uniformly for the +indices used above. + +The outer telescope has length +$p(\mu-b)+O_G(1)\leq(A+1)p+O_G(1)$. Summing +\eqref{eq:surface-square-bound} and using the Riemann sum for +the square on $[-(\mu-b),0]$ gives the following bound. +Here and below, limits are taken along geometric fibers of the chosen +model, after all the specified localizations, with characteristics +$p$ tending to infinity: +\begin{equation}\label{eq:tail-factor-frobenius} + \limsup_{p\to\infty}p^{-3}h^0(F^*G\otimes L_p) + \leq\frac{h r(G)}6(\mu-b)^3 + +C_Ah r(G)\delta_G^B. +\end{equation} +The summed $O_G(p)$ errors are $O_G(p^2)$ and vanish in this +limit. One may take a fixed multiple of $A+1$ for $C_A$; +it does not depend on the restriction sheaves or their ranks. +Summing over the finite filtration of $Q$, using +\eqref{eq:positive-tail-moments}, gives +\begin{equation}\label{eq:tail-frobenius} + \limsup_{p\to\infty}p^{-3}h^0(F^*Q\otimes L_p) + \leq\frac{hA^2}{6}d_B(Q)+O(e). +\end{equation} +To verify its leading term, write $\mu-b=A+s$, $0\leq s\leq1$. +Then $(A+s)^3-A^2(A+s)\leq C_As$, and +$\sum r_is_i\leq e$ by \eqref{eq:positive-excess}. + +It remains to bound the residual contribution and the positive +even cohomology group in degree two. Let $P_1$ be the positive +residual $P$ or a nonpositive-rank object treated by the negative +decomposition. It has ordinary cohomology in degrees $[-1,0]$, +and \eqref{eq:negative-cohomology-vanishing} supplies the vanishing +required in Corollary~\ref{cor:frobenius-section-transfer}. +That corollary and \eqref{eq:residual-sections} give, for either +rounding line $L$, +\begin{equation}\label{eq:residual-frobenius} + h^0(F^*P_1\otimes L) + \leq N_{p,L}h^0(P_1(j))\leq C_4p^3e. +\end{equation} +The characteristic-zero section bounds pass to these fibers by +the finite semicontinuity conditions already imposed. + +Apply Lemma~\ref{lem:tilt-duality} to $E$: +\begin{equation}\label{eq:apex-duality-triangle} + \widetilde E\longrightarrow R\mathcal Hom(E,\OO_X)[1] + \longrightarrow T_0[-1], +\end{equation} +where $T_0$ is zero-dimensional and $\widetilde E$ is finite-slope +semistable of slope zero at $(A,-B)$. Its rank is $-r(E)$, +its denominator is $d(E)$, its excess is $e$, and +$z_{-B}(\widetilde E)=z_B(E)+\operatorname{length}(T_0)/h$. +Include this triangle and the negative decomposition for +$\widetilde E$ in the fixed spreading data. Serre duality, +flatness of $F$, and \eqref{eq:apex-duality-triangle} give +\begin{equation}\label{eq:degree-two-frobenius} + h^2(F^*E\otimes L_p) + =h^0(F^*\widetilde E\otimes L'_p)\leq C_4p^3e. +\end{equation} +For clarity, Serre duality first identifies the dual of the +degree-two group with degree zero of +$F^*R\mathcal Hom(E,\OO_X)[1]\otimes L'_p$. +The term $F^*T_0[-1]\otimes L'_p$ has zero hypercohomology in +degrees $-1$ and $0$, so it does not alter that group. + +The complex $F^*E\otimes L_p$ has hypercohomology in degrees +$[-1,3]$. The only even degrees in this interval are zero and +two, whence +\[ + \chi(F^*E\otimes L_p) + \leq h^0(F^*E\otimes L_p)+h^2(F^*E\otimes L_p). +\] +The degree-zero group is bounded above by the sum of the +corresponding groups for $P$ and $Q$ in +\eqref{eq:positive-residual-triangle}. Equations +\eqref{eq:tail-frobenius}, \eqref{eq:residual-frobenius} and +\eqref{eq:degree-two-frobenius} therefore imply +\begin{equation}\label{eq:euler-frobenius-bound} + \limsup_{p\to\infty}p^{-3}\chi(F^*E\otimes L_p) + \leq\frac{hA^2}{6}d_B(Q)+C_5e. +\end{equation} + +Riemann--Roch identifies the limit on the left: +\begin{equation}\label{eq:rr-frobenius-limit} + \lim_{p\to\infty}p^{-3}\chi(F^*E\otimes L_p)=h z_B(E). +\end{equation} +Here is a way to compare varying characteristics without +identifying their cohomology rings. On a smooth fiber, +$\ch_i(F^*E)=p^i\ch_i(E)$ by the splitting principle. The same +formula holds for the Adams class $\psi^p[E]$ in vector-bundle +$K$-theory. Riemann--Roch thus identifies the Euler characteristic +in \eqref{eq:rr-frobenius-limit} with that of +$\psi^p[E]\otimes L_p$. This class is defined by exterior-power +operations on a fixed perfect presentation and commutes with +specialization. Its Euler characteristic is constant from the +model to the complex fiber. On that fiber, write +$c_1(L_p)=-pB+R_p$, where $R_p$ remains in a fixed bounded set of +divisor classes. The degree-three leading term of +\[ + \int_X\left(\sum_{i=0}^3p^i\ch_i(E)\right) + e^{-pB+R_p}\td(X) +\] +is $p^3\ch_3^B(E)$; all other terms are $O_E(p^2)$. +This proves \eqref{eq:rr-frobenius-limit}, without a condition on +$K_X$. + +Combining \eqref{eq:euler-frobenius-bound} and +\eqref{eq:rr-frobenius-limit}, and using $d_B(Q)\leq d_B(E)$, +proves \eqref{eq:apex-estimate} for $r(E)\geq0$. +The surviving constants came only from the fixed regularity, +surface estimates and the interval $[A,A+1]$; every constant +depending on the individual spread sheaves occurred in an error +of order at most $p^2$ and disappeared after division by $p^3$. +This establishes the asserted order of quantifiers. + +If $r(E)<0$, apply the just-proved case to $\widetilde E$ in +\eqref{eq:apex-duality-triangle}, which has positive rank. +Its $d$ and $e$ agree with those of $E$, and its third character +is at least $z_B(E)$. The estimate for $\widetilde E$ therefore +implies the estimate for $E$. Our constants were chosen for both +signs of $B_0$, so this use of the mirror parameters costs nothing. +Finally undo the integral tensor normalization of $b$. +This completes the proof of Proposition~\ref{prop:apex}. diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/05-wall.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/05-wall.tex new file mode 100644 index 0000000000000000000000000000000000000000..9004df189544d8bca06c9181f62e4a61fe8eaabf --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/05-wall.tex @@ -0,0 +1,241 @@ +\section{From one apex to uniform inequalities} +\label{sec:wall} + +The fixed-apex estimate now supplies the global inequality. The +mechanism is numerical: a differential inequality preserves a strict +violation as the volume parameter moves toward the apex. If an object +becomes strictly semistable, one continues with a violating stable +factor. A discrete discriminant makes this process finite. +Transport along a zero-slope hyperbola and induction on the +discriminant appear in \cite[Section~5]{BMS2016} and, for a corrected +inequality, \cite[Lemma~2.7]{BMSZ2017}. Here the terminal estimate is +at a fixed positive volume $A$. The comparison functions below turn +that estimate into an all-volume correction and an uncorrected tail. + +\subsection{A transport principle} + +At slope zero define the defect and the nonnegative comparison quantity +\begin{equation}\label{eq:wall-defect} + D_a(E)=z_b(E)-\frac{a^2d_b(E)}6,\qquad + S_a(E)=\frac{\Delta(E)}{d_b(E)} + =d_b(E)-\frac{a^2r(E)^2}{d_b(E)}. +\end{equation} +They are defined whenever $d_b(E)>0$ and $w_b(E)=a^2r(E)/2$. +In particular $0\le S_a\le d_b$ for a semistable object. + +\begin{proposition}[Transport from a fixed apex]\label{prop:apex-transport} +Fix a smooth projective complex threefold $X$, an integral ample class +$H$, and $B_0\in N^1(X)_{\QQ}$. Suppose that there are $A>0$ and +$C\ge0$ such that every finite-slope tilt-semistable object of slope +zero at $(A,b)$, for every $b\in\RR$, satisfies +\begin{equation}\label{eq:wall-apex-input} + D_A(E)\le C S_A(E). +\end{equation} +Then, with +\begin{equation}\label{eq:wall-constants} + R_0=A+\frac{3C}{A},\qquad + k_0=\max\left\{C+\frac{A^2}{6},\frac{CR_0}{A}\right\}, +\end{equation} +every finite-slope tilt-semistable object of slope zero at $(a,b)$ +satisfies $D_a(E)\le k_0d_b(E)$ for all $a>0,b\in\RR$, and +$D_a(E)\le0$ whenever $a>R_0$. +\end{proposition} + +\begin{proof} +We first compute along the slope-zero curve of a fixed class. Its +positive-denominator branch is +\begin{equation}\label{eq:zero-branch} + d(a)=\sqrt{\Delta+a^2r^2},\qquad + b(a)=\frac{d_{B_0}-d(a)}r\quad(r\ne0). +\end{equation} +For rank zero, $d$ and $b$ remain constant. If the branch starts at +a semistable object, then $\Delta\ge0$, and its denominator stays +positive at every positive $a$. Indeed $d(a)\ge a|r|$ when +$r\ne0$, whereas $d$ is a positive constant when $r=0$. +The identities $dz_b/db=-w_b$ and $w_b=a^2r/2$ give +\begin{equation}\label{eq:wall-derivatives} + D_a'=-\frac a3S_a,\qquad + S_a'=-\frac{ar^2}{d(a)^2}S_a. +\end{equation} +For example, when $r\ne0$ one has $d'=ar^2/d$, +$b'=-ar/d$, and $z'=a^3r^2/(2d)$; these give both equalities +directly. The rank-zero case follows from the definitions. +For a differentiable nonnegative function $k$, therefore, +\begin{equation}\label{eq:wall-comparison-derivative} + (D_a-k(a)S_a)'= + S_a\left(-\frac a3-k'(a)+k(a)\frac{ar^2}{d(a)^2}\right). +\end{equation} + +There are three comparisons. For the corrected bound below $A$, +take $k(a)=C+(A^2-a^2)/6$ and move upward toward +$A$. The right side of \eqref{eq:wall-comparison-derivative} is +$k(a)ar^2S_a/d(a)^2\ge0$, so a strict violation persists upward. +To rule out positive defect at a starting value $R>R_0$, take +$k(a)=a(R-a)/3$ on $[A,R]$ and move downward. Since +$ar^2/d(a)^2\le1/a$, +\[ + -\frac a3-k'(a)+k(a)\frac{ar^2}{d(a)^2} + \le-\frac a3-k'(a)+\frac{k(a)}a=0. +\] +A strict violation again persists in the direction of movement. +Here $k(R)=0$ and $k(A)=A(R-A)/3>C$. +For the corrected bound in the remaining interval, on any +interval $[A,R]$ use $k(a)=Ca/A$ and move +downward. The same upper bound for the last term gives a derivative +at most $-aS_a/3\le0$, and $k(A)=C$. + +Each comparison preserves a strict violation while the class remains +stable. We next show how to continue through a change of stability, +with a strict decrease of a discrete discriminant at each replacement. +For an equal-slope-zero filtration with stable factors $E_i$, +all $d_i$ are positive and $w_i=a^2r_i/2$. Additivity and +weighted Cauchy--Schwarz give +\begin{equation}\label{eq:wall-additivity} + D_a(E)=\sum_iD_a(E_i),\qquad + S_a(E)=\sum_i d_i-a^2\frac{(\sum_i r_i)^2}{\sum_i d_i} + \ge\sum_iS_a(E_i). +\end{equation} +Thus a strict inequality $D_a(E)>k(a)S_a(E)$, with $k(a)\ge0$, +passes to at least one stable factor. Such a factorization is finite +by Lemma~\ref{lem:finite-tilt-factors}. + +Start with a violating stable factor and follow its branch +\eqref{eq:zero-branch} toward $A$ as long as it remains stable. +Stability is open. At an endpoint in the positive $a$-interval, +continuity of extremal phases gives semistability, since the formal +denominator remains positive. The strict violation persists at the +endpoint: along the movement $D-kS$ is at least its initial positive +value. If the endpoint is $A$, this contradicts +\eqref{eq:wall-apex-input}. Otherwise the endpoint is nonstable, +since stability there would extend the interval, and we split again +and select a violating stable factor. + +At every such genuine wall, the discriminant of each factor is +strictly smaller than that of its parent. Indeed, +\begin{equation}\label{eq:wall-discriminant-split} + \Delta(E)=\sum_i\Delta(E_i)+ + 2\sum_{iR_0$. +The third proves $D_a\le(Ca/A)S_a$ for $A\le a\le R_0$. +Finally use $S_a\le d_b$ and \eqref{eq:wall-constants}. +\end{proof} + +\subsection{The corrected inequality and the uncorrected tail} + +\begin{proof}[Proof of Theorem~\ref{thm:uniform-inequality}] +First use the very-ample and transverse-twist reductions of +Section~\ref{sec:tilt}. Proposition~\ref{prop:apex} supplies +$A>0$ and $C\ge0$, chosen independently of $E$ and $b$, such that +\[ + D_A(E)\le C\bigl(d_B(E)-A|r(E)|\bigr). +\] +At slope zero put $e=d_B-A|r|$. Since $d_B\ge A|r|$, +\[ + S_A=\frac{(d_B-A|r|)(d_B+A|r|)}{d_B},\qquad + e\le S_A\le2e. +\] +The hypothesis of Proposition~\ref{prop:apex-transport} follows. +Multiplying its conclusion by $h$ gives +\[ + \ch_3^B(E)\le\frac{a^2}{6}H^2\ch_1^B(E) + +k_0H^2\ch_1^B(E), +\] +and gives the same inequality without the correction for $a>R_0$. +Undo the polarization rescaling as described in Section~\ref{sec:tilt}. +This gives constants $k\ge0$ and $R_*>0$ for the original +$X,H,B_0$, uniform in both $E$ and $b$. + +Enlarge $k$ to a rational nonnegative number. The curve class +$\Gamma=kH^2\in N_1(X)_{\QQ}$ then satisfies +$\Gamma\cdot H=kh\ge0$, and the correction is exactly +$\Gamma\cdot\ch_1^B(E)$. This proves the corrected statement +with its required rational class, as well as the independent +uncorrected large-volume threshold. No canonical-bundle hypothesis +has entered either argument. +\end{proof} + +\subsection{The quadratic inequality at every slope} + +We finish by expressing the uncorrected tail in the alternative +coordinates used for generalized Bogomolov--Gieseker inequalities. +Here $B_0=0$ and +\[ + v(E)=(v_0,v_1,v_2,v_3) + =(H^3\ch_0(E),H^2\ch_1(E),H\ch_2(E),\ch_3(E)). +\] +For $\alpha>\beta^2/2$, the first tilt is +$\mathcal B_{H,\beta H}$ and its slope is +\begin{equation}\label{eq:alternative-slope} + \widehat\nu_{\alpha,\beta}(E)= + \frac{v_2(E)-\beta v_1(E)+(\beta^2-\alpha)v_0(E)} + {v_1(E)-\beta v_0(E)}, +\end{equation} +with value $+\infty$ at zero denominator. Define +\begin{equation}\label{eq:quadratic-form} + Q_{\alpha,\beta}(v)= + \alpha(v_1^2-2v_0v_2)+\beta(3v_0v_3-v_1v_2) + +2v_2^2-3v_1v_3. +\end{equation} +Thus no correction of any character is being incorporated into $v$. + +\begin{proof}[Proof of Corollary~\ref{cor:quadratic-tail}] +Take the threshold of Theorem~\ref{thm:uniform-inequality} with +$B_0=0$. Set $b=\beta$ and $a=\sqrt{2\alpha-b^2}$. +The numerator in \eqref{eq:alternative-slope} is +$h(w_b-a^2r/2)$ and its denominator is $hd_b$, so its slope is +exactly $\nu_{a,b}$. Expanding \eqref{eq:quadratic-form} with +\eqref{eq:twist-two}--\eqref{eq:twist-three} gives +\begin{equation}\label{eq:quadratic-twisted} + \frac{Q_{\alpha,b}(v(E))}{h^2} + =\frac{a^2\Delta(E)}2+2w_b(E)^2-3d_b(E)z_b(E). +\end{equation} +If $d_b(E)=0$, this is nonnegative by +\eqref{eq:tilt-discriminants}; it includes zero-dimensional objects. + +Otherwise put $u=\nu_{a,b}(E)$, $c=b+u$, and +$R=\sqrt{a^2+u^2}$. Lemma~\ref{lem:tilt-semicircle} makes $E$ +semistable of slope zero at $(R,c)$, with $d_c>0$. +Using $w_b=ud_b+a^2r/2$, +\[ + d_c=d_b-ur,\qquad + z_c=z_b-uw_b+\tfrac12u^2d_b-\tfrac16u^3r +\] +in \eqref{eq:quadratic-twisted} gives the exact identity +\begin{equation}\label{eq:quadratic-zero-slope-identity} + \frac{Q_{\alpha,b}(v(E))}{h^2} + =-3d_b(E)\left(z_c(E)-\frac{R^2d_c(E)}6\right). +\end{equation} +Since $\alpha>f(b)$ implies $R\ge a>R_*$, the expression in +parentheses is nonpositive by the uncorrected tail. This proves the +claim. The passage uses the entire semistability arc and the +inequality $R\ge a$; an all-parameter equivalence alone would not +justify a conclusion on this truncated region. +\end{proof} diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/06-ambient.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/06-ambient.tex new file mode 100644 index 0000000000000000000000000000000000000000..f272b9d65019a911765d2eaef7305e723885a657 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/06-ambient.tex @@ -0,0 +1,695 @@ +\section{Prescribed charges on the full numerical lattice} +\label{sec:ambient} + +We begin the proof of Theorem~\ref{thm:geometric-sector} by constructing +both prescribed charges with full numerical support. This argument is +independent of the strong tilt inequalities. Throughout the section, +$X$ is a smooth projective complex threefold with $K_X\simeq\OO_X$. + +The construction takes place first on a product of projective spaces. +We choose an embedding that retains every numerical class of $X$, prove +a support estimate adapted to increasing volume on the product, and use +that estimate to deform its charge. A fixed polynomial correction then +makes Grothendieck--Riemann--Roch restrict the deformed charge to the +chosen charge on $X$. The same construction on smooth surface sections +will later identify the ordinary heart. +We use numerical Grothendieck groups in the Euler-pairing sense: +\[ + \Knum(X)=K_0(X)/\{v:\chi(v,u)=0\text{ for every }u\in K_0(X)\}. +\] +The left and right radicals coincide by Serre duality, so either side of +the Euler pairing may be used in the adjunction argument below. +All real vector spaces associated with these groups are finite-dimensional. +We use Bridgeland's stability conditions and locally finite slicings +\cite{Bridgeland2007}. A stability condition has \emph{full numerical support} if, for a norm on +$\Knum(X)_\RR$, there is a constant $C$ such that +$\|[E]\|\le C|Z(E)|$ for every semistable object $E$. +The constant may depend on the stability condition. This is equivalent to +the quadratic-form definition +\cite[Definition~1.1]{BayerDeformation2019}: one may take +$C^2|Z(v)|^2-\|v\|^2$, which is negative definite on $\ker Z$. + +\subsection{An embedding which retains every numerical class} + +The product-projective embedding and full-support construction here adapt +the approach of Giovenzana--Robotis--Rota--Zuliani +\cite{GiovenzanaRobotisRotaZuliani2026}. The numerical injectivity argument +is given below; the quantitative product inputs and the prescribed-charge, +real-sector refinements are treated separately in the subsequent subsections. + +\begin{lemma}\label{lem:full-numerical-embedding} +Let $X$ be a smooth projective threefold. Rational Chern characters identify +its numerical Grothendieck group with +\begin{equation}\label{eq:full-numerical-lattice} + \Knum(X)_\QQ\simeq + \QQ\oplus N^1(X)_\QQ\oplus N_1(X)_\QQ\oplus\QQ. +\end{equation} +Products of numerical divisor classes span this space. Given a real ample +class $H_*$, there are very ample line bundles $L_1,\ldots,L_q$ whose classes +form a basis of $N^1(X)_\RR$ and positive numbers $s_1,\ldots,s_q$ with +$H_*=\sum_hs_hL_h$. For the product embedding +\begin{equation}\label{eq:product-embedding} + i:X\hookrightarrow Y=\prod_{h=1}^q\mathbb P^{d_h}, + \qquad d_h=h^0(X,L_h)-1, +\end{equation} +the pushforward $i_*:\Knum(X)_\RR\to\Knum(Y)_\RR$ is injective. +\end{lemma} + +\begin{proof} +The rational Chern character identifies the Grothendieck group with the +rational Chow group +\cite[Tags 0FDI, 0FEW, 0FB2 and 0FB5]{StacksProject}. In Riemann--Roch the Euler pairing is the intersection +pairing after applying the invertible operations of dualizing a class and +multiplying it by $\td(X)$. Its radical therefore gives precisely the +numerical quotient of the Chow group. In dimensions zero and three this +quotient is $\QQ$; in the other two dimensions it is the divisor--curve +pairing in \eqref{eq:full-numerical-lattice}. + +For a rational ample class $A$, the Hodge index theorem makes +$(D,D')\mapsto ADD'$ nondegenerate on $N^1(X)_\QQ$; see +\cite[Theorem~6.4 and Corollary~6.5]{GrebRossToma2016} for its +real-ample formulation. The divisor--curve +pairing is nondegenerate by numerical equivalence, so multiplication by $A$ +is an isomorphism $N^1(X)_\QQ\to N_1(X)_\QQ$. Divisors, their products with +$A$, and $A^3$ consequently span every summand of +\eqref{eq:full-numerical-lattice}. + +Choose an affine slice transverse to the ray of $H_*$ and a rational simplex +in its ample cone whose interior contains that ray. Its $q=\rho(X)$ +vertices are linearly independent. Taking sufficiently large integral +multiples makes them very ample without changing their positive cone. This +also explains the rank-one case, where a single very ample class suffices. + +Pushforward is defined on numerical groups by adjunction with perfect +complexes. If $i_*v=0$ numerically, the projection formula gives +\[ + \chi(i^*W,v)=\chi(W,i_*v)=0\qquad(W\in K_0(Y)). +\] +Chern characters of line bundles on $Y$ span its polynomial Chow ring, and +their restrictions span \eqref{eq:full-numerical-lattice}. Nondegeneracy of +the Euler pairing gives $v=0$. The spanning argument here uses dimension +three; it makes no analogous claim in arbitrary dimension. +\end{proof} + +\subsection{Scaled support on elliptic products} + +The next estimate is the quantitative reason that the ambient charge can +be corrected uniformly at large volume. Its coordinates rescale a +codimension-$j$ character by the expected power $R^{n-j}$. In these +coordinates a fixed positive-codimension correction of the exponential +charge will tend to zero as $R\to\infty$. A support bound independent +of $R$ then turns this coordinate estimate into a bound on every +semistable object. + +We first establish the support bound on an elliptic product. The product +construction and the uniqueness needed to compare its different orders +of induction are external inputs. The estimate below makes their volume +scaling explicit. + +\begin{lemma}[Scaled support on an elliptic product] +\label{lem:elliptic-scaled-support} +Let $C$ be an elliptic curve, let $n\ge1$, and fix positive rational +numbers $\rho_1,\ldots,\rho_n$. For rational $R\ge1$, put +$w_r=R\rho_r$ and let $H_r$ be the point-divisor class from the $r$th +factor of $C^n$. The product stability condition with charge +\[ + Z_{\mathbf w}(u)=-\int_{C^n}e^{-i\sum_rw_rH_r}\ch(u) +\] +has support on the coordinate lattice +\[ + x_I(u)=H_I\ch_{n-|I|}(u),\qquad + H_I=\prod_{r\in I}H_r,\qquad I\subset\{1,\ldots,n\}. +\] +More precisely, set $y_I=(\prod_{r\in I}w_r)x_I$. In these coordinates, +\[ + Z_{\mathbf w}(u)=\widehat Z_n(y(u)),\qquad + \widehat Z_n(y)=-\sum_I(-i)^{|I|}y_I. +\] +For a fixed Euclidean norm there is $C_n>0$, independent of $R$, such that +\[ + \|y(u)\|\le C_n|Z_{\mathbf w}(u)| + \qquad(u\text{ the class of a semistable object}). +\] +The constants can be chosen uniformly when the positive ratios $\rho_r$ +vary in a compact set. Point sheaves are stable of phase $1$. +\end{lemma} + +\begin{proof} +We use the product-with-a-curve construction of +\cite[Lemma~5.7 and Remark~5.8]{LiuProduct2021}, the positive-genus +product construction of \cite[Theorem~4.5]{ProductsCurves2025}, and +the support induction of \cite[Lemma~6.6]{LiEtAl2026}. +For $n=1$, the usual slope stability on $C$ has charge +$-x_{\varnothing}+iw_1x_{\{1\}}$ and the assertion is immediate. + +Suppose that the scaled support bound has been proved in dimension +$n-1$. Omitting factor $r$ splits the coordinates into $x',x''$, +where $x'$ consists of those containing $r$, with that index removed. +The product construction has charge +\[ + Z_{n-1}(x'')-iw_rZ_{n-1}(x'). +\] +It initially gives support with respect to its own quotient lattice. +Every choice of omitted factor gives the same exponential charge and +the same phase $1$ for point sheaves. The uniqueness theorem +\cite[Theorem~3.11]{ProductsCurves2025} therefore identifies the +slicings: that theorem permits the two support lattices to differ. +We may consequently test all the product support forms on the same +semistable objects, before proving support on the full coordinate +lattice. + +Write $M_{\widehat r}$ for scaling the remaining coordinates by their +products of weights. The scaled coordinates split as +\[ + y'=w_rM_{\widehat r}x',\qquad y''=M_{\widehat r}x''. +\] +The lower-dimensional charge in scaled coordinates is a fixed linear +map $\widehat Z$. Increase its support constant $C_{n-1}$ if needed +and choose the support form +\begin{equation}\label{eq:elliptic-support-form} + q_{n-1}(v)=C_{n-1}^2|\widehat Z(v)|^2-\|v\|^2. +\end{equation} +It is nonnegative on lower-dimensional semistable classes. Crucially, +\[ + q_{n-1}(v)\le C_{n-1}^2|\widehat Z(v)|^2 + \quad\text{for every }v, +\] +not merely for semistable classes. This is the norm form used in the +product support estimate. + +For clarity, that estimate has the following coefficient condition. +Write $Z_{n-1}(x')=\alpha+i\beta$ and +$Z_{n-1}(x'')=\gamma+i\delta$. With equal positive product parameters +$s=t=w_r$, the form +\[ + \beta\gamma-\alpha\delta + +\eta_r q_{n-1}(M_{\widehat r}x') +\] +is nonnegative on product-semistable classes when +$0\le\eta_r\le w_r/C_{n-1}^2$ +\cite[Lemma~5.7]{LiuProduct2021}; this is the substitution in +\cite[Lemma~6.6]{LiEtAl2026}. +Choose $0<\lambda_r0$. There is a continuous family of +numerical stability conditions $(U_R,\mathcal R_R)$ for $R\ge1$ with +\[ + U_R(u)=-\int_Ye^{-iR\sum_hs_h\xi_h}\ch(u). +\] +All point sheaves are stable of phase $1$. Write +$\ch(u)=\sum_{\mathbf j}u_{\mathbf j}\xi^{\mathbf j}$, where +$0\le j_h\le d_h$, and set +\begin{equation}\label{eq:ambient-scaled-coordinates} + (M_Ru)_{\mathbf j}= + u_{\mathbf j}\prod_h(Rs_h)^{d_h-j_h}. +\end{equation} +For a fixed Euclidean norm there is $D>0$, independent of $R$, such that +\begin{equation}\label{eq:ambient-support} + \|M_Ru\|\le D|U_R(u)|\qquad + (u\text{ the class of an }\mathcal R_R\text{-semistable object}). +\end{equation} +For every $\mathbf e\in\ZZ^q$, +\begin{equation}\label{eq:ambient-twist-distance} + d(\mathcal R_R,\mathcal R_R\otimes\OO_Y(\mathbf e)) + \le\sum_h\frac{d_h|e_h|}{\pi Rs_h}. +\end{equation} +\end{proposition} + +\begin{proof} +Cheng proves the two-block construction and phase estimate in +\cite[Proposition~4.3 and Remark~4.4]{Cheng2026}. We apply the same +quotient construction block by block, using +Lemma~\ref{lem:elliptic-scaled-support} to control all coordinates +uniformly in the volume. + +First take $R$ and the positive weights $s_h$ rational. Put +$n=\sum_hd_h$ and give +each of the $d_h$ elliptic factors in block $h$ the weight +$w_r=2Rs_h$. The lemma supplies the starting stability condition and +its scaled support bound. Only the coordinate lattice of the elliptic +power is used here; full numerical support will follow on $Y$ from the +explicit pullback calculation below. + +First apply \cite[Proposition~5.1]{LiEtAl2026} to the stability condition +just constructed on the coordinate lattice. It supplies the descent +by $(\ZZ/2)^n$ to $(\mathbb P^1)^n$ and the Bayer property with $k=0$ +for every effective line bundle $\OO(a_1,\ldots,a_n)$, with all $a_r\ge0$. +We then take the symmetric quotients one block at a time. + +Here is the filtration used at a symmetric step. For +$Q=(\mathbb P^1)^d\to\mathbb P^d$, there is a finite filtration of +$\OO_{Q\times_{\mathbb P^d}Q}$ whose successive quotients are +\[ + p_2^*\mathcal L_j^{-1}\otimes\OO_{\Gamma(g_j)}, + \qquad g_j\in S_d,\qquad + \mathcal L_j=\OO(a_{j1},\ldots,a_{jd}),\quad a_{jr}\ge0, +\] +where $p_2$ is the second projection and $\Gamma(g_j)$ is the graph +of the permutation $g_j$ +\cite[Definition~3.19 and Example~3.20(3)]{LiEtAl2026}. +At a block stage the map is $f:Q\times M\to\mathbb P^d\times M$, +where $M$ is the product of the other blocks, and its fiber square is +$(Q\times_{\mathbb P^d}Q)\times M$. +Flat pullback to this product preserves the filtration and its exact +sequences. Its graph automorphisms are $g_j\times\operatorname{id}_M$, +and its line bundles remain pulled back from $Q$. + +The finite-flat descent criterion \cite[Proposition~3.21]{LiEtAl2026} +therefore requires precisely invariance under these permutations and +the Bayer properties with $k=0$ for these effective line bundles. +Indeed, for a phase-$\phi$ object $F$, the factors of $f^*f_*F$ are +$\mathcal L_j^{-1}\otimes g_{j*}F$; the two properties put their largest +phases at most $\phi$. On the elliptic +product, signs and permutations within each equal-weight block preserve +the charge and point phases, so the uniqueness theorem used above gives +their invariance. Block permutations normalize the full sign group; +their invariance descends through the completed involution quotient. +For subsequent block quotients, the required properties pass to the +remaining blocks as follows +\cite[Lemma~3.9 and Remark~3.14(3)]{LiEtAl2026}. +At each quotient $f:V\to V'$, pullback computes extremal phases for the +induced slicing $\mathcal S'=f_\#\mathcal S$. If a line bundle $L$ comes +from an unquotiented block, then +\[ + \phi^-_{\mathcal S'}(F\otimes L[k]) + =\phi^-_{\mathcal S}(f^*F\otimes f^*L[k]). +\] +The Bayer property for $f^*L$ therefore descends to $L$. An automorphism +of another block commutes with $f$, so its invariance descends by the +same pullback description. Thus every remaining quotient has +the required Bayer properties and invariance, for any finite number of +blocks. + +For completeness, let $g$ be the resulting map from the elliptic power to +$Y$. Pullback satisfies +\[ + g^*\xi_h^{j_h}=2^{j_h}j_h! + \sum_{|J_h|=j_h}H_{J_h}, + \qquad \deg g=2^n\prod_hd_h!. +\] +If $|I\cap\text{block }h|=d_h-j_h$, then the scaled $I$-coordinate of +$g^*u$ is +\begin{equation}\label{eq:block-coordinate-comparison} + \Bigl(\prod_{r\in I}w_r\Bigr) + H_I\ch_{n-|I|}(g^*u) + =2^n\Bigl(\prod_hj_h!\Bigr)(M_Ru)_{\mathbf j}. +\end{equation} +The fixed multiplicities are $\prod_h\binom{d_h}{j_h}$. Pullback is thus +injective on the complete Chern-character lattice of $Y$, with norm +comparison constants independent of $R$. Semistability pulls back, and +$Z_{\mathbf w}(g^*u)=\deg(g)U_R(u)$. This proves \eqref{eq:ambient-support}. + +For the phase estimate, vary the real tensor parameter on each elliptic +factor successively. The one-factor argument in the proof of +\cite[Theorem~2.4]{ChengFeyzbakhsh2026}, used in +\cite[Remark~4.4]{Cheng2026}, gives its absolute variation divided by +$\pi w_r$. A twist $e_h\xi_h$ pulls back with coefficient $2e_h$ on the +$d_h$ factors in block $h$; its factor $2$ cancels the factor $2$ in $w_r$. +Adding the variations gives \eqref{eq:ambient-twist-distance}. +Finally the deformation and multiplication-isogeny argument in the proof of +\cite[Proposition~4.3]{Cheng2026} extends the rational-weight construction +continuously to positive real weights. It applies factor by factor: the +scaled support forms just constructed remain fixed on compact positive +weight sets, the exponential charges vary continuously, and uniqueness on +overlapping deformation neighborhoods identifies the lifts. The closed +support inequalities and the phase bounds persist. This also extends $R$ +to all real $R\ge1$. Point stability in this extension follows separately +from \cite[Theorem~4.1(1)]{LiEtAl2026} on the elliptic power and preservation +of geometricity under finite pushforward +\cite[Remark~3.11]{LiEtAl2026}. Concretely, for the finite faithfully flat +quotient $g$, the sheaf $g^*\OO_y$ has a finite filtration by point sheaves, +all stable of phase $1$. A phase-one subobject and quotient of $\OO_y$ +pull back to objects whose Jordan--H\"older factors are among these points, +so both pullbacks are coherent finite-length sheaves. Faithful flatness +descends this cohomological concentration. The original subobject and +quotient are therefore sheaves, and simplicity of $\OO_y$ rules out a +proper nonzero subobject. Continuity and the charge value $-1$ normalize +the point phase to $1$. +\end{proof} + +\subsection{Geometric slicings and the restriction criterion} + +We now prepare the restriction step. The first consequence of point +stability below gives the cohomological bounds needed for a truncated +resolution. The second will keep points stable during deformation in +all ambient charges. Both will also be used to identify the induced +heart. Write +$\mathcal R(I)$ for the extension-closed category generated by semistable +objects with phases in an interval $I$. + +\begin{lemma}[Cohomological bounds from points]\label{lem:geometric-range} +Let $V$ be a smooth projective variety of dimension $n\ge2$, and let +$\mathcal R$ be a locally finite slicing in which every point sheaf is +stable of phase $1$. Then +\begin{align} + \mathcal R(0,1]&\subset D^{[-n+1,0]}(V), + &\Coh(V)&\subset\mathcal R(1-n,1],\label{eq:geometric-range}\\ + D^{\le-n}(V)&\subset\mathcal R(>0), + &\mathcal R(>0)&\subset D^{\le0}(V).\label{eq:geometric-aisles} +\end{align} +The sheaf $\mathcal H^{-n+1}(E)$ is torsion-free for +$E\in\mathcal R(0,1]$. Moreover, $\mathcal H^0(E)$ is zero-dimensional +for $E\in\mathcal R(1)$. +\end{lemma} + +\begin{proof} +This is the point-object argument of \cite[Lemma~10.1]{BridgelandK3}; we +include it to specify the dimension and boundary statements being used. +Let $E$ be stable with phase $0<\phi\le1$ and not a point of phase $1$. +Phase vanishing gives $\Hom(E,\OO_x[j])=0$ for $j<0$. +For $j\ge n$, Serre duality identifies this space with +$\Hom(\OO_x,E[n-j])^*$, which is zero by phases. At $j=n,\phi=1$, use +nonisomorphism of the two stable objects. The canonical bundle causes no +change, since its restriction to a point is trivial. A minimal free +complex at each closed point consequently has nonzero terms only in +degrees $[-n+1,0]$. Its bottom cohomology is a subsheaf of a locally free +sheaf and hence is torsion-free. At phase $1$ we also have +$\Hom(E,\OO_x)=0$ for every $x$, so $\mathcal H^0(E)=0$. +Point sheaves themselves satisfy the stated cohomological bounds. Finite +Jordan--Hölder filtrations and extensions give these assertions for the whole +heart; in phase $1$ its zeroth cohomology is an extension of sheaves supported +at finitely many points. + +A coherent sheaf cannot receive a nonzero morphism from an object whose +phase is greater than $1$, by the cohomological bound just proved and a +shift. It cannot map nontrivially to an object of phase at most $1-n$: +such an object is in $D^{\ge1}$, including the endpoint by the phase-one +observation. Applied to its extreme Harder--Narasimhan factors, these +vanishings prove $\Coh(V)\subset\mathcal R(1-n,1]$. Shifting this +inclusion and the heart bound proves \eqref{eq:geometric-aisles}. +\end{proof} + +\begin{lemma}[Stability from an open set of charges]\label{lem:primitive-point} +Let a stability condition have support on a finite-rank lattice $\Lambda$. +Suppose that an object $E$ has primitive class in $\Lambda$ and remains +semistable throughout a neighborhood which is open in all central charges +$\Hom(\Lambda,\CC)$. Then $E$ is stable. +\end{lemma} + +\begin{proof} +If $E$ is strictly semistable, take a first stable subobject $G$ in a finite +Jordan--Hölder filtration. Its class cannot be real proportional to $[E]$. +Indeed their aligned charges would make $[G]=c[E]$ with $00)\subset D^{\le n_2}(Y), + \qquad N\ge n_2-n_1+1. + \end{gathered} +\end{equation} +An untruncated cofiltration has no last numerical requirement. All tensor +products in this criterion are derived. The conclusion is a slicing +\begin{equation}\label{eq:induced-slicing} + f^\sharp\mathcal R(\phi)= + \{F\in\Db(V):f_*F\in\mathcal R(\phi)\}, +\end{equation} +with charge $U\circ f_*$ and support on the ambient lattice through $f_*$. +In particular it provides the required Harder--Narasimhan filtrations. +No surjectivity assumption is imposed on $f$. Smoothness makes the bounded +finite-Tor-dimension formulation applicable here. We shall verify the +cofiltration and every bound in \eqref{eq:restriction-hypotheses} before +using the criterion. + +\subsection{Correcting the charge before restriction} + +Fix real classes $B_*,H_*$ with $H_*$ ample, and choose the embedding in +Lemma~\ref{lem:full-numerical-embedding}. We write $L_h$ also for its first +Chern class, so $i^*\xi_h=L_h$ and $H_*=\sum_hs_hL_h$ with $s_h>0$. +Consider separately the factors +\begin{equation}\label{eq:two-character-factors} + \Theta=1\qquad\text{and}\qquad + \Theta=\sqrt{\td(X)}=1+\frac{c_2(X)}{24}. +\end{equation} +The assumption on the canonical bundle gives +$\td(X)=1+c_2(X)/12$ in the degrees relevant on $X$. By +Lemma~\ref{lem:full-numerical-embedding}, choose a degree-two polynomial +$\delta$ in the $\xi_h$ with $i^*\delta=c_2(X)$ numerically. A suitable +polynomial on $Y$ with constant term one is +\[ + \gamma=\begin{cases} + \td(Y)(1-\delta/12),&\Theta=1,\\ + \td(Y)(1-\delta/24),&\Theta=\sqrt{\td(X)}. + \end{cases} +\] +Products are taken in the finite-dimensional numerical Chow ring. Thus +\begin{equation}\label{eq:gamma-lift} + i^*\bigl(\gamma/\td(Y)\bigr)\td(X)=\Theta. +\end{equation} +In particular the two choices remove different Todd contributions. + +Write $B=\sum_hb_hL_h$ and $H=\sum_h\ell_hL_h$. Deform the ambient charge +to +\begin{equation}\label{eq:deformed-ambient-charge} + U_t^{B,H}(u)=-\int_Y + e^{-\sum_hb_h\xi_h-it\sum_h\ell_h\xi_h}\ch(u)\gamma. +\end{equation} +Grothendieck--Riemann--Roch and \eqref{eq:gamma-lift} give the exact identity +\begin{equation}\label{eq:exact-restricted-charge} + U_t^{B,H}(i_*E)=-\int_Xe^{-B-itH}\ch(E)\Theta. +\end{equation} +Numerical equality in \eqref{eq:gamma-lift} suffices because only +intersection degrees occur. For $\Theta=1$ this is exactly $Z_{B,tH}$; +for the other choice it is exactly $Z^{\mathrm{LV}}_{B,tH}$. + +\begin{lemma}\label{lem:uniform-ambient-deformation} +For every $0<\epsilon<1/8$, there are open neighborhoods $V\ni B_*$ and +$W\ni H_*$ in the real divisor and ample spaces, and $T\ge1$, such that for all +$B\in V$, $H\in W$, and $t\ge T$, the charge +\eqref{eq:deformed-ambient-charge} lifts to a stability condition with +slicing at distance less than $\epsilon$ from +$\mathcal R_t$. The construction is available in an open neighborhood of +that charge in the full space $\Hom(\Knum(Y),\CC)$. +\end{lemma} + +\begin{proof} +Take $V$ bounded in the $b_h$ coordinates, and restrict $W$ by +$|\ell_h/s_h-1|<\eta$. In the scaled coordinates +\eqref{eq:ambient-scaled-coordinates}, expansion of the two exponentials +gives +\begin{equation}\label{eq:uniform-deformation-estimate} + \bigl\|(U_t^{B,H}-U_t)\circ M_t^{-1}\bigr\| + \le C_1\eta+C_2t^{-1}. +\end{equation} +To see the uniformity, the coefficient of $u_{\mathbf j}$ in the +uncorrected exponential has degree $n-|\mathbf j|$ in $t$. Varying the +positive coefficients $s_h$ to $\ell_h$ changes its ratio to the weight of +$M_t$ by $O(\eta)$. A codimension-$d>0$ term of +$e^{-\sum b_h\xi_h}\gamma-1$ lowers that degree by $d$ and hence contributes +$O(t^{-d})$. There are finitely many monomials, and their coefficients are +bounded on the chosen neighborhoods. This proves +\eqref{eq:uniform-deformation-estimate} independently of the object. + +Combine it with \eqref{eq:ambient-support}. Choose $\eta$ first and then +$T$ so that +\[ + D(C_1\eta+C_2/T)<\sin(\pi\epsilon). +\] +To check the support hypothesis of the deformation theorem explicitly, put +\[ + Q_t(u)=D^2|U_t(u)|^2-\|M_tu\|^2. +\] +It is nonnegative on the original semistable classes. If $U'(u)=0$ and +$\|(U'-U_t)M_t^{-1}\|\le\delta<1/D$, then +\[ + Q_t(u)\le(D^2\delta^2-1)\|M_tu\|^2<0\qquad(u\ne0). +\] +Thus $Q_t$ is negative definite on every charge kernel along the straight +deformation, and throughout the indicated open charge region. +The support-property deformation theorem +\cite[Theorem~1.2]{BayerDeformation2019} lifts the straight path while +preserving $Q_t$. On its nonnegative cone, +$\|M_tu\|\le D|U_t(u)|$, so the relative charge change is less than +$\sin(\pi\epsilon)$. The quantitative phase estimate +\cite[Lemma~2.9]{BayerDeformation2019} then makes the slicing distance +less than $\epsilon$ as long as it is less than $1/4$. It is zero at the start; +continuity along the path and $\epsilon<1/8$ prevent a first exit from +that $1/4$ neighborhood. This proves the asserted strict distance bound. +The charge bound is strict, so for each fixed $t$ it also holds on an open +neighborhood in \emph{all} ambient numerical charges. This is the space +on which we deform, before applying restriction. +\end{proof} + +\begin{proposition}\label{prop:geometric-prescribed-charges} +For either factor in \eqref{eq:two-character-factors}, the choices in +Lemma~\ref{lem:uniform-ambient-deformation} can be made so that every +$B\in V,H\in W,t\ge T$ gives a locally finite geometric stability +condition on $X$, with charge \eqref{eq:exact-restricted-charge} and full +numerical support. Every point sheaf is stable of phase $1$. +For $\Theta=1$, the construction also gives compatible geometric slicings +on every smooth section $S_h\in|L_h|$, with Harder--Narasimhan phases +computed by pushforward to $Y$. +\end{proposition} + +\begin{proof} +Set $A=\OO_Y(1,\ldots,1)$. Choose a locally free resolution +$P^\bullet\to i_*\OO_X$ with $P^0=\OO_Y$ and with each $P^{-k}$ a +finite sum of line bundles $A^{-b}$ for $k\ge1$. Such a resolution is +obtained by repeatedly generating the successive coherent kernels after +sufficiently positive twists. Fix $N>\dim Y+1$ and truncate after the +terms in degrees $0,\ldots,-(N-1)$. The remaining kernel is shifted by +$[N]$, so the cofiltration remainder lies in $D^{\le-N}(Y)$. + +For a defining ambient linear form of a smooth section $S_h$, first form +the cone of the untruncated map +$P^\bullet(-\xi_h)\to P^\bullet$. It resolves +$(i\circ j_h)_*\OO_{S_h}$, and its term in degree $-k$ is +\[ + P^{-k}\oplus P^{1-k}(-\xi_h),\qquad P^1=0. +\] +Truncate each cone resolution after its terms in degrees +$0,\ldots,-(N-1)$. Its remaining kernel is again shifted by $[N]$, so +its cofiltration remainder also lies in $D^{\le-N}(Y)$. +The line bundles in all these terms are independent of the chosen smooth +section. Include all their summands, in particular the mixed twists +$A^{-b}\otimes\OO_Y(-\xi_h)$, together with $A$ and $\OO_Y(\xi_h)$, +in one finite list. This list is fixed before the parameters $B,H,t$ and +the sections are chosen. + +Let $\mathcal R$ be a lift from +Lemma~\ref{lem:uniform-ambient-deformation}. Tensoring is an isometry for +slicing distance, so \eqref{eq:ambient-twist-distance} gives, after +increasing the single threshold $T$, +\begin{equation}\label{eq:deformed-twist-distance} + d(\mathcal R,\mathcal R\otimes L)<1 +\end{equation} +for every line bundle $L$ in the finite list, uniformly in the claimed +sector. The triangle inequality bounds the left side by +$2\epsilon+O(t^{-1})$. The same strict inequalities hold in a full open +neighborhood of each lifted charge. + +In particular tensoring by $A^{-1}[1]$ has the Bayer property. +\cite[Proposition~3.27]{LiEtAl2026} makes every point sheaf on $Y$ +semistable throughout that open neighborhood. Its numerical class is +primitive, because $\chi(\OO_Y,\OO_y)=1$. +Lemma~\ref{lem:primitive-point} upgrades semistability to stability. +At the charge \eqref{eq:deformed-ambient-charge} its value is $-1$, and +closeness to $\mathcal R_t$ fixes its phase to be $1$. + +We can now apply Lemma~\ref{lem:geometric-range} on $Y$. +It supplies \eqref{eq:restriction-hypotheses} with +$n_1=-\dim Y$ and $n_2=0$. Each resolution term $L[k]$, $k\ge1$, has +the Bayer property: \eqref{eq:deformed-twist-distance} says that tensoring +by $L$ lowers no extremal phase by as much as one, and the shift $[k]$ +restores the required lower bound. The initial $\OO_Y$ term is the +identity. The chosen $N$ exceeds the truncation bound. Thus the +finite-map criterion applies to the closed immersion $i$ and, when needed, +to every $i\circ j_h$. It supplies slicings with the indicated charges +and Harder--Narasimhan filtrations. Definition +\eqref{eq:induced-slicing} and uniqueness of those filtrations show that +pushforward computes their extremal phases. + +Pushforward is exact for these hearts and detects zero objects. A proper +destabilizing subobject of a point would therefore give one for the stable +point on $Y$. The induced points are stable of phase $1$. +If $E$ is semistable on $X$, the ambient support inequality and the +injectivity in Lemma~\ref{lem:full-numerical-embedding} give +\[ + \|[E]\|\le C\|i_*[E]\|\le C'|U_t^{B,H}(i_*E)|. +\] +This is support on the full Euler numerical group of $X$. For each +auxiliary surface the ambient lattice through pushforward suffices; no +injectivity assertion for its full numerical group is needed. +Finally the support bound on a finite-rank lattice gives local finiteness: +in a sufficiently short phase interval, the projection of a nonzero +semistable charge onto its middle ray has a fixed positive lower bound. +Additivity of this projection bounds the length of any chain of strict +subobjects or quotients of a fixed object in that interval. The induced +slicings are thus locally finite, as asserted. +\end{proof} + +Apply the construction to both choices of $\Theta$, intersect the two +neighborhoods $V$ and $W$, and take the larger of their thresholds. The +same real sector and one threshold therefore work for both charges. +The proposition completes the prescribed Todd-charge assertion of +Theorem~\ref{thm:geometric-sector}. It also supplies the exact ordinary +charge and full numerical support. Identifying its heart requires the +additional argument in the next section. Only $K_X\simeq\OO_X$ has been +used; no vanishing of $H^1(X,\OO_X)$ or $H^2(X,\OO_X)$ is required. diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/07-heart.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/07-heart.tex new file mode 100644 index 0000000000000000000000000000000000000000..b930cd55421bcf8ccfc08fd91969c63263b9695e --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/07-heart.tex @@ -0,0 +1,378 @@ +\section{The ordinary double tilt throughout the real sector} +\label{sec:heart} + +For the factor $\Theta=1$, we now identify the heart furnished by +Proposition~\ref{prop:geometric-prescribed-charges}. The method of comparing +hearts through smooth surface sections follows Cheng--Feyzbakhsh +\cite[Section~3]{ChengFeyzbakhsh2026}. Here the charge has already been +corrected to the ordinary Chern character, and positive weighted sections +allow the polarization to be real. The argument below proves the phase +boundaries directly. + +Fix $B\in V,H\in W,t\ge T$ from the preceding construction. Let +$\mathcal P$ be the induced slicing on $X$, with charge $Z_{B,tH}$, and put +$\mathcal C=\mathcal P(0,1]$. As before, +$H=\sum_h\ell_hL_h$ with every $\ell_h>0$. All restrictions of complexes +in this section are derived restrictions. + +\subsection{The geometric heart on a surface} + +\begin{lemma}\label{lem:surface-geometric-heart} +Let $S$ be a smooth projective surface and $(Z,\mathcal R)$ a locally finite +geometric stability condition, normalized so that point sheaves have phase +$1$. Suppose +\[ + \Im Z(G)=\omega\bigl(c_1(G)-\beta\rk(G)\bigr), +\] +where $\omega$ is a real ample divisor class and $\beta$ a real divisor +class. Then +\[ + \mathcal R(0,1]=\langle\mathcal F_{\omega,\beta}[1], + \mathcal T_{\omega,\beta}\rangle, +\] +where $\mathcal T_{\omega,\beta}$ has strictly positive slope factors +(including torsion) and $\mathcal F_{\omega,\beta}$ has nonpositive +slope factors. +\end{lemma} + +\begin{proof} +Lemma~\ref{lem:geometric-range} gives +$\mathcal R(0,1]\subset D^{[-1,0]}(S)$ and +$\Coh(S)\subset\mathcal R(-1,1]$. The comparison of these two bounded +$t$-structures makes $\mathcal R(0,1]$ a tilt of $\Coh(S)$, with torsion +pair +\[ + \mathcal T_0=\Coh(S)\cap\mathcal R(0,1],\qquad + \mathcal F_0=\Coh(S)\cap\mathcal R(-1,0]. +\] +For instance their torsion sequence is obtained by truncating a sheaf at +phase $0$; the cohomological range ensures that both pieces are sheaves. + +If $G\in\mathcal T_0$, every quotient sheaf also belongs to +$\mathcal T_0$ and has nonnegative imaginary charge. A nonzero +positive-rank quotient cannot have imaginary charge zero: it would lie in +$\mathcal R(1)$, whereas Lemma~\ref{lem:geometric-range} makes its zeroth +cohomology zero-dimensional. Thus all positive-rank quotients of $G$ have +strictly positive slope. Applying this to its last slope +Harder--Narasimhan quotient proves +$\mathcal T_0\subset\mathcal T_{\omega,\beta}$. +If $F\in\mathcal F_0$, then $F[1]\in\mathcal R(0,1]$, so $F$ is +torsion-free by the same lemma. Every subsheaf remains in $\mathcal F_0$ +and has nonpositive imaginary charge. Its maximal-slope factor therefore +has slope at most zero, giving +$\mathcal F_0\subset\mathcal F_{\omega,\beta}$. + +These inclusions identify the torsion pairs. Indeed the torsion sequence +for either pair, applied to an object in the corresponding part of the +other, leaves a quotient or subobject in the intersection of its two +orthogonal parts, hence zero. This proof includes slope zero; it does not +use a generic-parameter argument. +\end{proof} + +Take a smooth section $j_h:S_h\hookrightarrow X$ in $|L_h|$ and its +compatible slicing $\mathcal P_h$. Grothendieck--Riemann--Roch for this +divisor embedding, whose normal line bundle is $L_h|_{S_h}$, gives +\begin{equation}\label{eq:surface-induced-charge} + Z_h(G)=-\int_{S_h}e^{-B|_{S_h}-itH|_{S_h}}\ch(G) + \frac{1-e^{-L_h|_{S_h}}}{L_h|_{S_h}}. +\end{equation} +Since $(1-e^{-L})/L=1-L/2+L^2/6$ on a surface, its imaginary part is +\begin{equation}\label{eq:surface-imaginary-charge} + \Im Z_h(G)=tH|_{S_h}\bigl(c_1(G) + -(B|_{S_h}+L_h|_{S_h}/2)\rk(G)\bigr). +\end{equation} +Lemma~\ref{lem:surface-geometric-heart} identifies +\begin{equation}\label{eq:section-surface-heart} + \mathcal P_h(0,1]=\langle\mathcal F_h[1],\mathcal T_h\rangle, +\end{equation} +where the surface slope uses $H|_{S_h}$ and twist +$B|_{S_h}+L_h|_{S_h}/2$. Only the imaginary part of the induced charge +is needed for this identification; its real part need not be the +ordinary surface charge. + +\subsection{Weighted sections give strict slope bounds} + +The section-phase comparison of +\cite[Lemma~3.1]{ChengFeyzbakhsh2026} applies using the line-twist +estimate \eqref{eq:deformed-twist-distance}. The two section triangles give, for every nonzero restricted object, +\begin{align} + \phi^-_{\mathcal P_h}(E|_{S_h}) + &\ge\phi^-_{\mathcal P}(E),\label{eq:section-lower-phase}\\ + \phi^+_{\mathcal P_h}(E(L_h)|_{S_h}) + &\le\phi^+_{\mathcal P}(E)+1.\label{eq:section-upper-phase} +\end{align} +To verify the first inequality, push the triangle +$E(-L_h)\to E\to j_{h,*}(E|_{S_h})$ to $Y$. +Its last term is an extension of $E$ and $E(-L_h)[1]$, and the latter +has smallest phase at least that of $E$ by the strict twist-distance bound. +For the second, $j_{h,*}(E(L_h)|_{S_h})$ is an extension of $E(L_h)$ and +$E[1]$, both of largest phase at most $\phi^+_{\mathcal P}(E)+1$. +Pushforward computes the induced phases, proving the assertions. + +For a positive-rank sheaf, use the unnormalized slope +\[ + m_{H,B}(F)=\frac{H^2(c_1(F)-B\rk F)}{\rk F}; +\] +its signs agree with the normalized slopes defining +$\mathcal B_{H,B}$. Superscripts $+$ and $-$ denote the largest and +smallest Harder--Narasimhan slopes, with torsion assigned slope $+\infty$. +The next estimate adapts the argument of +\cite[Proposition~3.3]{ChengFeyzbakhsh2026} +by combining the sections with positive real weights. Put +\begin{equation}\label{eq:weighted-slope-gap} + c=\frac12\sum_h\ell_hHL_h^2>0. +\end{equation} + +\begin{lemma}\label{lem:heart-cohomology-slopes} +Every $E\in\mathcal C$ has cohomology in degrees $[-2,0]$, its sheaf +$\mathcal H^{-2}(E)$ is torsion-free, and +\begin{equation}\label{eq:heart-sign-bounds} + m^+_{H,B}(\mathcal H^{-2}(E))\le-c, + \qquad m^-_{H,B}(\mathcal H^0(E))>c. +\end{equation} +A bound for a zero sheaf is vacuous. +\end{lemma} + +\begin{proof} +The range and torsion-freeness follow from +Lemma~\ref{lem:geometric-range}. Write $F=\mathcal H^{-2}(E)$ and +$G=\mathcal H^0(E)$. For each $h$, choose a general smooth $S_h$ which +is a nonzerodivisor on the finitely many cohomology sheaves and the +subobjects and quotients used below. Those sheaves and sequences then +restrict without Tor. Such choices are available inside the complete +very ample linear system. + +By \eqref{eq:section-upper-phase}, $E(L_h)|_{S_h}$ has largest phase at +most $2$. The surface tilt \eqref{eq:section-surface-heart}, applied to +its lowest ordinary cohomology, implies +$F(L_h)|_{S_h}\in\mathcal F_h$. Similarly +\eqref{eq:section-lower-phase} places $E|_{S_h}$ in +$\mathcal P_h(>0)$, and the highest ordinary cohomology of the surface +tilt gives $G|_{S_h}\in\mathcal T_h$. These conclusions can also be +read directly from the aisles: $\mathcal P_h(\le2)$ has its degree $-2$ +cohomology in $\mathcal F_h$, while $\mathcal P_h(>0)$ has its degree +$0$ cohomology in $\mathcal T_h$. + +If $F\ne0$, restrict its maximal-slope subsheaf $F'\subset F$. +The class $\mathcal F_h$ is closed under subsheaves, so the shifted twist +in the definition of the surface slope gives +\[ + \frac{HL_h(c_1(F')-B\rk F')}{\rk F'} + \le-\frac12HL_h^2. +\] +If $G$ has a positive-rank last slope quotient $G''$, quotient-closure of +$\mathcal T_h$ gives +\[ + \frac{HL_h(c_1(G'')-B\rk G'')}{\rk G''} + >\frac12HL_h^2. +\] +For a torsion sheaf $G$ its minimum slope is $+\infty$ instead. +Multiply the displayed inequalities by the positive $\ell_h$ and sum. +Since $\sum_h\ell_hL_h=H$, they are exactly +\eqref{eq:heart-sign-bounds}. Positivity of the weights is what permits +this argument for a real ample $H$. +\end{proof} + +\subsection{Comparison with the first tilt} + +Write $\mathcal B=\mathcal B_{H,B}$ for the ordinary first tilt. The slope +bounds just obtained place it between two adjacent translates of the +constructed heart. + +\begin{lemma}\label{lem:first-tilt-phase-range} +We have +\begin{equation}\label{eq:first-tilt-phase-range} + \mathcal B\subset\mathcal P(-1,1]. +\end{equation} +Consequently +\[ + \mathcal T_0=\mathcal B\cap\mathcal P(0,1],\qquad + \mathcal F_0=\mathcal B\cap\mathcal P(-1,0] +\] +is a torsion pair in $\mathcal B$, and +$\mathcal C=\langle\mathcal F_0[1],\mathcal T_0\rangle$. +\end{lemma} + +\begin{proof} +Let $D\in\mathcal T_{H,B}$ be a sheaf with $m^-_{H,B}(D)>0$. +Lemma~\ref{lem:geometric-range} places all its phases in $(-2,1]$. +If a last Harder--Narasimhan factor $G$ had phase in $(-2,-1]$, then +$G[2]\in\mathcal P(0,1]$. It follows that $G\in D^{[0,2]}$ and +$m^+_{H,B}(\mathcal H^0(G))\le-c$. The canonical nonzero map $D\to G$ +is tested on its degree-zero cohomology: +\[ + \Hom(D,G)=\Hom(D,\mathcal H^0(G))=0. +\] +Here the equality follows from $G\in D^{\ge0}$ and standard truncation; +the vanishing is the slope comparison, including the case of torsion $D$ +since the target is torsion-free. This contradiction excludes that phase +interval and gives $D\in\mathcal P(-1,1]$. + +Now let $J\in\mathcal F_{H,B}$, so $J$ is torsion-free with +$m^+_{H,B}(J)\le0$. The coherent-sheaf range gives +$J[1]\in\mathcal P(-1,2]$. If a first Harder--Narasimhan factor $G$ +had phase in $(1,2]$, then $G[-1]\in\mathcal P(0,1]$, +$G\in D^{[-3,-1]}$, and +$m^-_{H,B}(\mathcal H^{-1}(G))>c$. +The canonical nonzero map $G\to J[1]$ is tested on its top cohomology: +\[ + \Hom(G,J[1])=\Hom(\mathcal H^{-1}(G),J)=0, +\] +again by truncation and slope. Thus $J[1]\in\mathcal P(-1,1]$. +Extensions prove \eqref{eq:first-tilt-phase-range}. + +The final assertion is the comparison theorem for two bounded hearts one +of which lies in two adjacent degrees of the other. Explicitly, truncate +an object of $\mathcal B$ at phase $0$ in $\mathcal P$. +The inclusion \eqref{eq:first-tilt-phase-range} and its corresponding +inclusions of aisles make both truncations objects of $\mathcal B$; +they give its torsion sequence with the two stated intersections. +Tilting that sequence recovers $\mathcal P(0,1]$. +\end{proof} + +\subsection{The real-axis boundary and the second tilt} + +The constructed heart is now a tilt of $\mathcal B$. It remains to +identify the torsion pair, including the distinction between positive +and zero second slope. The following observation, following the boundary argument of +\cite[Lemma~3.6]{ChengFeyzbakhsh2026}, controls precisely those objects. + +\begin{lemma}\label{lem:phase-one-first-tilt} +An object of $\mathcal B\cap\mathcal P(1)$ is a zero-dimensional sheaf. +\end{lemma} + +\begin{proof} +Let $E$ belong to the stated intersection. Its ordinary cohomology is +$J=\mathcal H^{-1}(E)\in\mathcal F_{H,B}$ and +$U=\mathcal H^0(E)\in\mathcal T_{H,B}$. +Lemma~\ref{lem:geometric-range} makes $U$ zero-dimensional. +Choose the smooth sections $S_h$ to avoid its finite support and to +restrict $J$ without Tor. If $J\ne0$, it is torsion-free of positive +rank, and $E|_{S_h}=J|_{S_h}[1]$. +Inequality \eqref{eq:section-lower-phase} gives +$\phi^-_{\mathcal P_h}(J|_{S_h})\ge0$. +Its phases as a coherent sheaf are also at most $1$, by +Lemma~\ref{lem:geometric-range}. Hence +$\Im Z_h(J|_{S_h})\ge0$ and \eqref{eq:surface-imaginary-charge} gives +\[ + \frac{HL_h(c_1(J)-B\rk J)}{\rk J}\ge\frac12HL_h^2 + \quad\text{for each }h. +\] +The positive weighted sum says $m_{H,B}(J)\ge c>0$, contrary to +$m^+_{H,B}(J)\le0$. Therefore $J=0$ and $E=U$. +\end{proof} + +We use the physical second slope from the introduction, writing +$\nu^{\mathrm{phys}}_{B,tH}=\nu_{B,tH}$ to distinguish it from +$\nu_{a,b}$: +\begin{equation}\label{eq:physical-second-slope} + \nu^{\mathrm{phys}}_{B,tH}(E)= + \frac{tH\ch_2^B(E)-t^3H^3\ch_0(E)/6} + {t^2H^2\ch_1^B(E)}, +\end{equation} +with value $+\infty$ at a zero denominator. The tilt +Harder--Narasimhan property for each fixed real ample class $tH$ and real +$B$ is the input of \cite[Section~2]{BMS2016}; here no continuity as the +ample direction varies is required. Its Harder--Narasimhan filtration +defines the torsion pair $(\mathcal T',\mathcal F')$ in +$\mathcal B$ with strictly positive slopes in $\mathcal T'$ and +nonpositive slopes in $\mathcal F'$. Multiplication by a positive +constant does not affect this pair. The claimed double tilt is +$\mathcal A_{B,tH}=\langle\mathcal F'[1],\mathcal T'\rangle$. + +\begin{lemma}\label{lem:physical-zero-denominator} +For $E\in\mathcal B$ the denominator in +\eqref{eq:physical-second-slope} is nonnegative. If it vanishes, then +$\Im Z_{B,tH}(E)\ge0$, with equality only for a zero-dimensional sheaf. +\end{lemma} + +\begin{proof} +Let $J=\mathcal H^{-1}(E)$ and $U=\mathcal H^0(E)$. +The first torsion pair makes each contribution to +$H^2\ch_1^B(E)=H^2\ch_1^B(U)-H^2\ch_1^B(J)$ nonnegative. +If the sum is zero, $J$, when nonzero, is slope-semistable of slope zero, +and $U$ has dimension at most one. Indeed positive-rank quotients in the +positive torsion part have strictly positive degree, and a nonzero +effective divisorial cycle has positive intersection with $H^2$. +For real $H$, the K\"ahler Bogomolov--Gieseker inequality follows from +\cite[Theorem~1.1, Equation~(1.6)]{LiZhangZhang2017}, applied with zero +Higgs field to the reflexive hull $J^{**}$ when $J\ne0$. +This hull is semistable for the same $H$: intersecting a subsheaf of +$J^{**}$ with $J$ preserves its rank and first Chern class. +The quotient $Q=J^{**}/J$ has codimension at least two, and, writing +$r=\rk J$, additivity gives +\[ + c_1(J)^2-2r\ch_2(J) + =c_1(J^{**})^2-2r\ch_2(J^{**})+2r\ch_2(Q), +\] +where $\ch_2(Q)$ is an effective curve cycle, so the inequality descends +to $J$. +The same argument on a surface uses an effective zero-cycle correction; +neither application changes the real polarization. +Classical Bogomolov--Gieseker and the real-ample Hodge index theorem +\cite[Theorem~6.4]{GrebRossToma2016}, applied to +$H^2\ch_1^B(J)=0$, give +$H\ch_2^B(J)\le0$. The curve cycle of $U$ is effective. Consequently +\[ + \Im Z_{B,tH}(E)= + tH\ch_2^B(U)-tH\ch_2^B(J)+\frac{t^3H^3}{6}\rk J\ge0. +\] +If $J\ne0$, the last term is strictly positive. If $J=0$ and $U$ has a +nonzero one-dimensional part, its curve degree is strictly positive. +Thus equality is possible exactly when $E=U$ is zero-dimensional. +\end{proof} + +\begin{proposition}\label{prop:exact-double-tilt-heart} +For the factor $\Theta=1$ throughout the real sector of +Proposition~\ref{prop:geometric-prescribed-charges}, +\[ + \mathcal P(0,1]=\mathcal A_{B,tH}. +\] +\end{proposition} + +\begin{proof} +Compare $(\mathcal T',\mathcal F')$ with the torsion pair +$(\mathcal T_0,\mathcal F_0)$ of +Lemma~\ref{lem:first-tilt-phase-range}. +For $D\in\mathcal T'$, take its latter torsion sequence +$0\to D_1\to D\to D_2\to0$. +A nonzero $D_2\in\mathcal F_0$ has $\Im Z(D_2)\le0$. +Its denominator cannot vanish: Lemma~\ref{lem:physical-zero-denominator} +would make it zero-dimensional, whereas zero-dimensional sheaves are in +$\mathcal P(1)$ and $\mathcal F_0\subset\mathcal P(-1,0]$. +Thus $D_2$ has positive denominator and +$\nu^{\mathrm{phys}}_{B,tH}(D_2)\le0$, contradicting the strictly positive +slopes of every nonzero quotient of $D$. Hence +$\mathcal T'\subset\mathcal T_0$. + +For $D\in\mathcal F'$, consider $D_1$ in the same torsion sequence. +A nonzero subobject of $D$ has second slope at most zero, so $D_1$ has +positive denominator and $\Im Z(D_1)\le0$. +But $D_1\in\mathcal T_0\subset\mathcal P(0,1]$ makes its imaginary +charge nonnegative. It must therefore vanish, and $D_1\in\mathcal P(1)$. +Lemma~\ref{lem:phase-one-first-tilt} makes it zero-dimensional, whose second +slope is $+\infty$, a contradiction. Thus +$\mathcal F'\subset\mathcal F_0$. +The two inclusions identify the torsion pairs, and their tilts coincide. +\end{proof} + +Proposition~\ref{prop:exact-double-tilt-heart}, together with +Proposition~\ref{prop:geometric-prescribed-charges}, proves the ordinary +charge assertion of Theorem~\ref{thm:geometric-sector}, including its exact +heart, full numerical support, local finiteness, stable points, and one +threshold for an open real sector. The equality of hearts was proved only +for $\Theta=1$. The Todd-charge construction retains its geometric heart +from the preceding section. + +Finally, the coefficient in the strong inequality should be kept distinct +from the weaker sign consequence of this construction. If $E\ne0$ is +first-tilt semistable of finite physical slope zero, then +$E[1]\in\mathcal A_{B,tH}$ and its nonzero real charge is negative. Thus +\[ + \ch_3^B(E)<\frac{t^2}{2}H^2\ch_1^B(E). +\] +With $t=\sqrt3a$, the strong inequality has coefficient $t^2/18$ instead. +The categorical argument supplies the real-sector and full-support +conclusions; the sharper coefficient will be proved independently in +Sections~\ref{sec:apex} and~\ref{sec:wall} by the fixed-volume estimate +and wall transport. diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/08-counterexamples.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/08-counterexamples.tex new file mode 100644 index 0000000000000000000000000000000000000000..09c6693f98a1b2e01c0797ebeb34b7f935d261b1 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/08-counterexamples.tex @@ -0,0 +1,190 @@ +\section{Comparison with the known counterexamples} +\label{sec:counterexamples} + +The uncorrected strong inequality fails at small volume, even on +Calabi--Yau threefolds. We record the parameters of two established +counterexamples and then give a uniform estimate for sheaves on their +exceptional surfaces. Throughout this section, the volume parameter is +\(a\) in the ordinary tilt slope \(\nu_{a,b}\); the physical volume is +\(t=\sqrt3a\). + +\subsection{A line bundle and a contracted divisor} + +Let \(X=\operatorname{Bl}_p\mathbb P^3\), let \(L\) be the pulled-back +hyperplane class, and let \(E\) be the exceptional divisor. Fix +\(H=2L-E\) and \(B_0=0\). The intersection identities +\(L^3=E^3=1\), \(LE=0\), and \(H^3=7\) give +\[ + \bigl(H^3\ch_0,H^2\ch_1,H\ch_2,\ch_3\bigr)(\OO_X(L)) + =\left(7,4,1,\frac16\right). +\] +For the conventional quadratic expression +\[ + \mathcal Q_{a,b}(F) + =a^2\bigl((H^2\ch_1(F))^2-2H^3\ch_0(F)H\ch_2(F)\bigr) + +4(H\ch_2^{bH}(F))^2 + -6H^2\ch_1^{bH}(F)\ch_3^{bH}(F), +\] +direct expansion yields +\begin{equation} + \mathcal Q_{a,b}(\OO_X(L)) + =2\left(a^2+\left(b-\frac14\right)^2-\frac1{16}\right). + \label{eq:schmidt-disk} +\end{equation} +Schmidt proves that actual tilt-stable points occur in the negative +region of \eqref{eq:schmidt-disk} +\cite[Theorem~3.1]{Schmidt2017}. All these violations have \(a<1/4\). +On the positive-denominator zero-slope branch one has, more explicitly, +\[ + a^2=b^2-\frac87b+\frac27, + \qquad b<\frac{4-\sqrt2}{7}, + \qquad + \ch_3^{bH}(\OO_X(L)) + -\frac{a^2}{6}H^2\ch_1^{bH}(\OO_X(L)) + =\frac{4b-1}{42}. +\] +These formulas use the same ordinary characters and normalization as +the strong inequality. + +For an integral effective divisor \(D\subset X\), still with \(B_0=0\), +the identity \(\ch(\OO_D)=1-e^{-D}\) shows that its unique zero-slope +line and its defect there are +\begin{align} + b_0&=-\frac{HD^2}{2H^2D},\notag\\ + \ch_3^{b_0H}(\OO_D) + -\frac{a^2}{6}H^2D + &=\frac{D^3}{6} + -\frac{(HD^2)^2}{8H^2D} + -\frac{a^2}{6}H^2D. + \label{eq:divisor-counterexample} +\end{align} +The wall-radius argument in +\cite[Lemma~3.1 and its proof]{MartinezSchmidt2019} gives semistability +on this line for \(a\ge H^2D/(2H^3)\), and stability for the strict +inequality in our subobject-versus-quotient convention. Indeed, every +positive-rank wall has radius at most this bound, while a proper +rank-zero subobject of \(\OO_D\) is a subsheaf with quotient supported +in dimension at most one, hence of infinite tilt slope. + +Suppose now that \(D\) contracts to a point, \(-D\) is relatively +ample, and \(H=ML-D\), where \(L\) is the pullback of an ample +divisor and \(M\) is sufficiently large. Put \(s=D^3>0\). +Since \(LD=0\), +one has \(H^2D=s\) and \(HD^2=-s\). Formula +\eqref{eq:divisor-counterexample} becomes +\begin{equation} + b_0=\frac12, + \qquad + \ch_3^{b_0H}(\OO_D)-\frac{a^2}{6}H^2D + =s\left(\frac1{24}-\frac{a^2}{6}\right). + \label{eq:contracted-defect} +\end{equation} +Thus, when \(H^3>s\), the interval +\(s/(2H^3)0\), and the uncorrected strong +inequality for every \(a\ge1/2\). Both constants are independent of +\(F\), its rank, and \(b\). +\end{lemma} + +\begin{proof} +Write \(\ell=H|_D\), \(s=\ell^2=D^3>0\), +\(\rho=\rk(F)\), and \(C=B_0|_D\). Put +\[ + u=\ch_1^C(F),\qquad v=\ch_2^C(F),\qquad + \mu=\frac{\ell u}{\rho s},\qquad + \delta=\mu^2-\frac{2v}{\rho s}. +\] +Every exact sequence of sheaves on \(D\) pushes forward to an exact +sequence of torsion sheaves in the first-tilt heart. Their tilt +slopes differ from their \(C\)-twisted normalized surface slopes by +the common constant \(1/2-b\). Twisting shifts all surface slopes +equally, so tilt semistability of \(i_*F\) implies slope semistability +of \(F\) for \(\ell\). +The classical surface Bogomolov--Gieseker inequality +\cite[Theorem~3.1.4]{BMT2014} and the Hodge index theorem give +\[ + u^2-2\rho v\ge0, + \qquad (\ell u)^2\ge s u^2, + \qquad \delta\ge0. +\] +The first expression is unchanged by twisting the Chern character +by \(C\), so these conclusions hold for the specified \(B_0\). + +The normal line bundle has first Chern class \(-\ell\), and hence +\(\td(N_{D/X})^{-1}=1+\ell/2+\ell^2/6\). +Grothendieck--Riemann--Roch now gives +\[ + H^2\ch_1^B(i_*F)=\rho s, + \qquad + H\ch_2^B(i_*F)=\rho s(\mu+1/2-b). +\] +Since the threefold rank of \(i_*F\) is zero, its zero-slope condition +is \(b=\mu+1/2\). The degree-three term of the same formula yields +\[ + \frac{\ch_3^B(i_*F)}{\rho s} + =\frac{v}{\rho s}+\frac\mu2+\frac16 + -b(\mu+1/2)+\frac{b^2}{2} + =\frac1{24}-\frac\delta2\le\frac1{24}. +\] +This proves \eqref{eq:exceptional-uniform}; the two strong bounds +follow by subtracting \(a^2/6\). +\end{proof} + +For the Weierstrass examples of +\cite[Section~3.2 and Appendix~A]{MartinezSchmidt2019}, let +\(p:X\to S\) be a Weierstrass elliptic fibration from a smooth +Calabi--Yau threefold to a del Pezzo surface, and let \(\Sigma\) +be its section. +Since \(K_X\simeq\OO_X\), adjunction gives +\(\Sigma|_\Sigma=K_S\). For the ample polarization +\(H=q\Sigma-(1+q)p^*K_S\), with a positive integer \(q\), one has +\(H|_\Sigma=-K_S\). Thus +Lemma~\ref{lem:exceptional-surface-bound} applies in arbitrary rank +and after every line-bundle twist on \(\Sigma\) whose pushforward is +tilt-semistable. In particular, +the known Calabi--Yau counterexample has the defect +\eqref{eq:contracted-defect}, with \(s=K_S^2\), and its violating +volume range is bounded by \(a<1/2\). + +\begin{remark} +\label{rem:nonreduced-multiple} +Replacing \(D\) by \(nD\) in the character calculation does not +produce a stable counterexample at arbitrarily large volume. Under +the hypotheses of Lemma~\ref{lem:exceptional-surface-bound}, the exact +sequence +\[ + 0\longrightarrow\OO_D(-(n-1)D)\longrightarrow\OO_{nD} + \longrightarrow\OO_{(n-1)D}\longrightarrow0 +\] +lies in every first-tilt heart. For \(B_0=0\) and \(b=n/2\), its +three slopes are \((n-1)/2\), \(0\), and \(-1/2\), respectively. +Thus \(\OO_{nD}\) is unstable for every \(a>0\) when \(n>1\). +The uniform estimate above concerns sheaves on the reduced divisor; +its proof uses surface semistability, rather than the character of +an arbitrary multiple. +\end{remark} diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/main.tex b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..0b81a62d86a88575e9446e0eec2ffe5831e445f3 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/main.tex @@ -0,0 +1,99 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=27mm]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype,booktabs,array,longtable,enumitem} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage[unicode,hidelinks]{hyperref} +\hypersetup{pdftitle={Prescribed large-volume charges on threefolds with trivial canonical bundle},pdfauthor={OpenAI}} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\input{glyphtounicode} +\pdfgentounicode=1 +\pdfglyphtounicode{parenleftbig}{0028} +\pdfglyphtounicode{parenrightbig}{0029} +\pdfglyphtounicode{bracketleftbig}{005B} +\pdfglyphtounicode{bracketrightbig}{005D} +\pdfglyphtounicode{parenleftBig}{0028} +\pdfglyphtounicode{parenrightBig}{0029} +\pdfglyphtounicode{parenleftbigg}{0028} +\pdfglyphtounicode{parenrightbigg}{0029} +\pdfglyphtounicode{angbracketleftbigg}{27E8} +\pdfglyphtounicode{angbracketrightbigg}{27E9} +\pdfglyphtounicode{parenleftBigg}{0028} +\pdfglyphtounicode{parenrightBigg}{0029} +\pdfglyphtounicode{braceleftBigg}{007B} +\pdfglyphtounicode{bracerightBigg}{007D} +\pdfglyphtounicode{angbracketleftBigg}{27E8} +\pdfglyphtounicode{angbracketrightBigg}{27E9} +\pdfglyphtounicode{summationtext}{2211} +\pdfglyphtounicode{producttext}{220F} +\pdfglyphtounicode{integraltext}{222B} +\pdfglyphtounicode{summationdisplay}{2211} +\pdfglyphtounicode{productdisplay}{220F} +\pdfglyphtounicode{integraldisplay}{222B} +\pdfglyphtounicode{hatwide}{0302} +\pdfglyphtounicode{tildewide}{0303} +\pdfglyphtounicode{radicalbig}{221A} +\pdfglyphtounicode{radicalBig}{221A} +\pdfglyphtounicode{vextenddouble}{20E6} +\setlength{\emergencystretch}{2em} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\DeclareMathOperator{\ch}{ch} +\DeclareMathOperator{\td}{td} +\DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Ext}{Ext} +\DeclareMathOperator{\Coh}{Coh} +\DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\supp}{Supp} +\newcommand{\OO}{\mathcal O} +\newcommand{\CC}{\mathbb C} +\newcommand{\RR}{\mathbb R} +\newcommand{\QQ}{\mathbb Q} +\newcommand{\ZZ}{\mathbb Z} +\newcommand{\HH}{\mathbb H} +\newcommand{\Db}{D^{\mathrm b}} +\newcommand{\Knum}{K_{\mathrm{num}}} +\title{Prescribed large-volume charges on threefolds with trivial canonical bundle} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +Let $X$ be a smooth projective complex threefold with trivial canonical +bundle. We construct numerical Bridgeland stability conditions with the exact +ordinary and square-root-Todd central charges at every sufficiently large +volume. One volume threshold works on an open set of real twists and ample +directions. The resulting stability conditions have the support property on +the full numerical Grothendieck group and stable point sheaves. Separately, on +every smooth projective complex threefold we prove a strong tilt inequality +above a volume threshold uniform in the object and twist along a fixed +polarization. +\end{abstract} +\tableofcontents +\input{01-introduction} +\input{06-ambient} +\input{07-heart} +\input{02-tilt} +\input{03-analysis} +\input{04-apex} +\input{05-wall} +\input{08-counterexamples} +\bibliographystyle{plain} +\bibliography{refs} +\end{document} diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/refs.bib b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/refs.bib new file mode 100644 index 0000000000000000000000000000000000000000..ce99122301e4b9ebf0f395c0d83eaecb3667647d --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/build/refs.bib @@ -0,0 +1,338 @@ +@article{BMT2014, + author = {Bayer, Arend and Macr{\`i}, Emanuele and Toda, Yukinobu}, + title = {{Bridgeland} stability conditions on threefolds {I}: {Bogomolov--Gieseker} type inequalities}, + journal = {J. 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Accessed 2 October 2026; + Tags 0FDI, 0FEW, 0FB2, 0FB5, 0FET} +} + +@article{Frank2014, + author = {Frank, Rupert L.}, + title = {Cwikel's theorem and the {CLR} inequality}, + journal = {J. Spectr. Theory}, + volume = {4}, + number = {1}, + year = {2014}, + pages = {1--21}, + doi = {10.4171/JST/59}, + url = {https://ems.press/content/serial-article-files/33504} +} + +@misc{GiovenzanaRobotisRotaZuliani2026, + author = {Giovenzana, Franco and Robotis, Antonios-Alexandros and Rota, Franco and Zuliani, Vanja}, + title = {Stability conditions supported on {Lefschetz} classes}, + year = {2026}, + note = {arXiv:2608.16013v1, 17 August 2026}, + url = {https://arxiv.org/abs/2608.16013v1} +} diff --git a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1eb7046f67e6c3048c73af7f95a190a13fb67c27 --- /dev/null +++ b/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:84ce978a670157043e5a8bc24583d2eb290cecf379a9f1ec495b6084e8c3f3f2 +size 625204 diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/README.md b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..b573d9796c7d099d5ac98df109d7405a24a627bd --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/README.md @@ -0,0 +1,18 @@ +# [Prime Predecessors with an Even Number of Prime Factors](paper.pdf) + +**Author:** OpenAI + +**Date:** September 17, 2026 + +## Citation + +```bibtex +@misc{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026, + author = {{OpenAI}}, + title = {{Prime Predecessors with an Even Number of Prime Factors}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf}{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/main.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..837740bf086f56d9c20fba6b3c7ff20ff53b581d --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/main.tex @@ -0,0 +1,141 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage[margin=1in]{geometry} +\usepackage{enumitem,microtype,booktabs,array} +\usepackage[colorlinks=true,linkcolor=black,citecolor=blue,urlcolor=blue]{hyperref} +\hypersetup{pdftitle={Prime Predecessors with an Even Number of Prime Factors},pdfauthor={OpenAI}} +\newcommand{\1}{\mathbf{1}} +\newcommand{\e}{\mathrm{e}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Pcal}{\mathcal{P}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\Li}{Li} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\cond}{cond} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\setlist[enumerate]{itemsep=3pt,topsep=5pt} +\allowdisplaybreaks[1] + +\title{Prime Predecessors with an Even Number of Prime Factors} +\author{OpenAI} +\date{September 17, 2026} + +\begin{document} +\maketitle + +\begin{abstract} +We prove that there are infinitely many primes $p$ for which $p-1$ is +squarefree and has an even number of prime factors. Equivalently, +there are infinitely many primes $p$ with $\mu(p-1)=1$. +\end{abstract} + +\section{Introduction}\label{sec:introduction} + +Let $\Omega(n)$ be the number of prime factors of $n$, counted with +multiplicity, and let $\omega(n)$ count its distinct prime factors. +We prove the following affirmative answer to the parity question for +predecessors of primes. + +\begin{theorem}\label{thm:main} +There are infinitely many primes $p$ such that $p-1$ is squarefree +and $\Omega(p-1)$ is even. In particular, there are infinitely many +such primes under either convention for counting prime factors. +\end{theorem} + +The squarefreeness condition is useful rather than cosmetic: it removes +the ambiguity between the two counting conventions. Writing $p=2u+1$, +we will construct nonnegative weights on odd integers $u$ for which +$\Omega(u)$ is odd. These weights have enough bilinear distribution +to detect prime values of $2u+1$, and their nonsquarefree part is +negligible at the scale of the detected prime mass. + +\paragraph{Relation to earlier work.} +Prime-factor parity is a classical obstruction in sieve theory: +congruence information alone need not distinguish the two Liouville +signs. The role of additional bilinear information is illustrated by +the asymptotic sieve of Friedlander and Iwaniec +\cite[Section~1]{FriedlanderIwaniec}. Work on smooth shifted primes, +including Baker and Harman~\cite{BakerHarman} and +Lichtman~\cite{Lichtman}, provides related context for imposing +multiplicative restrictions on a prime's predecessor. The parity +condition here is a different restriction; we do not deduce it from +smoothness alone. + +Our structural inputs are the dilation-graph transference, ideal-kernel, +weighted-sieve, and proxy estimates of the companion paper +\emph{Weighted dilation graphs, smooth shifted primes and totient +fibers}~\cite{SmoothShifted}, hereafter S. Their hypotheses and the +conclusions needed here are stated where they enter the proof. +We do not reprove those results. + +S's shifted-correlation argument uses a rough integer in a designated +long factor slot and unsigned marked weights. Neither smoothness of +the predecessor nor that unsigned statement gives the parity conclusion +of Theorem~\ref{thm:main}. The new work is to permit a prime in the long +slot and the same group-supported Liouville sign at both endpoints. +We prove both extensions here. + +\paragraph{The two analytic extensions.} +First, a logarithmic-phase estimate gives arbitrary fixed logarithmic +cancellation in a prime Dirichlet polynomial of positive-power length +at all sufficiently large logarithmic heights up to $x^2$. A +quantitative power-sum iteration supplies the weak high-height +zero-free strip needed for this estimate. The growing degree and all +its constants are kept explicit enough for the application. + +Second, the sign of a shared dilation occurs twice in a lifted +correlation and cancels by complete multiplicativity. This observation +allows the graph reduction to retain a parity-sensitive weight. +The comparison terms are controlled by local Fourier energy: a +small-prime factor restricts difficult frequencies to a sparse set, +the new prime estimate treats its large frequencies, and a prescribed +character--Mellin discrepancy treats its small frequencies. The +resulting correlation estimate implies a Type II theorem for the +nonnegative parity-selecting weight. + +The common-sign cancellation and the prime-slot estimate are stated +separately, so their use is not tied to the final sieve. In the +extraction argument, ordinary congruence distribution handles the +initial sieve; the stronger Type II statement justifies replacing +prime and roughness indicators by bounded local-density proxies. +An upper bound for balanced semiprimes then leaves positive prime mass. + +\paragraph{Organization.} +Section~\ref{sec:preliminaries} fixes conventions and records the +classical inputs. Section~\ref{lp:section} proves the prime-polynomial +estimate, Lemma~\ref{lp:prime-polynomial}, and +Section~\ref{sh:section} proves the twisted correlation, +Theorem~\ref{sh:correlation}. Section~\ref{wt:section} constructs the +weights. Their Type II and congruence distribution are +Theorems~\ref{ti:distribution} and~\ref{cg:distribution}, proved in +Sections~\ref{ti:section} and~\ref{cg:section}. +Section~\ref{ex:section} extracts the required primes. +All auxiliary constants are fixed before $x$ tends to +infinity; in particular, the final proxy precision is chosen only +after the required discrepancy exponent. We make this order explicit +at the steps where it matters. + +\input{sections/01-preliminaries} +\input{sections/02-long-prime} +\input{sections/03-shifted} +\input{sections/04-weights} +\input{sections/05-type-ii} +\input{sections/06-congruences} +\input{sections/07-extraction} +\input{references} + +\end{document} diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/references.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..2a9ed5641d3673c24b787a5e1f449ad6cbca053a --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/references.tex @@ -0,0 +1,39 @@ +\begingroup +\raggedright +\begin{thebibliography}{9} + +\bibitem{BakerHarman} +R.~C. Baker and G.~Harman, +\emph{Shifted primes without large prime factors}, +Acta Arith. \textbf{83} (1998), no.~4, 331--361. +\href{https://doi.org/10.4064/aa-83-4-331-361}{doi:10.4064/aa-83-4-331-361}. + +\bibitem{FriedlanderIwaniec} +J.~Friedlander and H.~Iwaniec, +\emph{Asymptotic sieve for primes}, +Ann. of Math. (2) \textbf{148} (1998), no.~3, 1041--1065. +\href{https://doi.org/10.2307/121035}{doi:10.2307/121035}; +\href{https://arxiv.org/abs/math/9811186v1}{arXiv:math/9811186v1}. + +\bibitem{Lichtman} +J.~D. Lichtman, +\emph{Primes in arithmetic progressions to large moduli, and shifted +primes without large prime factors}, +\href{https://arxiv.org/abs/2211.09641v1}{arXiv:2211.09641v1}, +14 November 2022. + +\bibitem{SmoothShifted} +OpenAI, +\emph{Weighted dilation graphs, smooth shifted primes and totient fibers}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf}{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026}, +2026. + +\bibitem{Wooley2012} +T.~D. Wooley, +\emph{Vinogradov's mean value theorem via efficient congruencing}, +Ann. of Math. (2) \textbf{175} (2012), 1575--1627. +\href{https://doi.org/10.4007/annals.2012.175.3.12}{doi:10.4007/annals.2012.175.3.12}. + +\end{thebibliography} +\endgroup diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/01-preliminaries.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/01-preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..ff9bef0b5a1e1351960623bd29260ef06dcef235 --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/01-preliminaries.tex @@ -0,0 +1,150 @@ +\section{Conventions and classical estimates}\label{sec:preliminaries} + +Throughout the proof, +\[ + x\longrightarrow\infty,\qquad L=\log x,\qquad T=\log L, + \qquad W=\exp(\sqrt L),\qquad \e(t)=\exp(2\pi i t). +\] +A logarithmic power always has a fixed exponent, chosen before $x$ +tends to infinity. The implied constants may depend on all previously +fixed parameters. No effective threshold for $x$ is asserted. +The letters $p$ and $q$ in explicitly indicated prime sums range over +primes; general factorization variables are positive integers. +We put $P^-(1)=\infty$, where $P^-(n)$ is the least prime factor of +$n>1$, and write $\tau(n)$ for its divisor function. The notation +$n\asymp Y$ restricts $n$ to a fixed constant enlargement of a dyad +$[Y,2Y)$. All such enlargements remain bounded independently of $x$. +Dirichlet characters are extended by zero on nonunits. + +We use the following classical estimates in the forms recorded in +S~\cite[Section~2]{SmoothShifted}. This explicit list fixes their +uniformity; it does not require any distribution theorem beyond the +stated ranges. + +\begin{proposition}[Prime estimates]\label{pre:primes} +The prime number theorem holds with an error smaller than every fixed +negative logarithmic power. Mertens' estimates give +\[ + \sum_{p\le y}\frac1p=\log\log y+O(1),\qquad + \prod_{p\le y}\left(1-\frac1p\right) + \sim\frac{e^{-\gamma_E}}{\log y}, +\] +where $\gamma_E$ is Euler's constant. For fixed $A,C>0$, uniformly +for $r\le(\log y)^C$ and $(a,r)=1$, the Siegel--Walfisz estimate is +\[ + \#\{p\le y:p\equiv a\pmod r\} + =\frac{\Li(y)}{\varphi(r)} + +O_{A,C}\left(\frac{y}{(\log y)^A}\right). +\] +Its constants need not be effective. +\end{proposition} + +All short intervals used with the prime number theorem have either a +fixed logarithmic-length interpretation in a fixed positive-power +band, or relative length a fixed negative power of $L$; the absolute +error above is made sufficiently small before taking their +differences. We never assume a relative prime asymptotic on every +arbitrarily short interval. + +\begin{proposition}[Large sieve and moment bounds]\label{pre:moments} +For arbitrary coefficients supported on the integers of an interval +of length $H$, +\[ + \sum_{r\le R}\frac r{\varphi(r)} + \sum_{\chi\bmod r}^{*} + \left|\sum_n a_n\chi(n)\right|^2 + \ll (H+1+R^2)\sum_n|a_n|^2. +\] +The asterisk denotes primitive characters. For every fixed integer +$k\ge0$ there is $C_k$ such that +\[ + \sum_{n\le Y}\tau(n)^k\ll_k Y(\log(2Y))^{C_k},\qquad + \sum_{n\le Y}\frac{\tau(n)^k}{n} + \ll_k(\log(2Y))^{C_k}. +\] +If $P(t)=\sum_{n\le Y}c_n n^{it}$, then on any real interval $I$ +of length $H$, +\[ + \int_I|P(t)|^2\,dt + \ll (H+Y\log(2Y))\sum_n|c_n|^2. +\] +For a set $\mathcal T\subset I$ with mutual spacing at least one, +\[ + \sum_{t\in\mathcal T}|P(t)|^2 + \ll (H+1+Y\log(2Y))(\log(2Y))^2\sum_n|c_n|^2. +\] +The constants in the last two estimates are absolute, independently +of how the coefficients were formed. +\end{proposition} + +These are Theorem~2.3 and Lemmas~2.5--2.6 of S. In particular, the +fixed-moment estimate applies to a convolution of a fixed number of +logarithmically bounded sequences, giving coefficient-square mass +$UL^{O(1)}$ on a dyad of size $U$. With reciprocal-index +normalization the bound is $L^{O(1)}/U$. For the one growing power +used later, we instead prove a factorial coefficient bound and use +the coefficient-independent mean-square inequality. + +\begin{proposition}[Derivative tests]\label{pre:derivatives} +Let $f$ be real-valued on an interval containing $H$ consecutive +integers. If $f'$ is monotone and remains in +$[j+\lambda,j+1-\lambda]$ for an integer $j$ and +$0<\lambda\le1/2$, then +\[ + \sum_n\e(f(n))\ll\lambda^{-1}. +\] +If instead $f$ is twice continuously differentiable and +$\lambda\le|f''|\le C\lambda$ throughout the interval, then +\[ + \sum_n\e(f(n))\ll_C H\sqrt\lambda+\lambda^{-1/2}. +\] +Both estimates apply on every subinterval satisfying their hypotheses. +\end{proposition} + +This is the form of S, Lemma~2.7. The distance from integers in the +first test is part of the hypothesis, not merely a lower bound on +the absolute derivative. + +We will repeatedly separate smooth functions in logarithmic +coordinates. The following elementary version of S, Lemma~2.8, +explains the order in which frequency cutoffs and accuracies may be +chosen. + +\begin{lemma}[Smooth separation]\label{pre:separation} +Let $F$ be smooth and supported on a fixed bounded box in +$\R^b$, with $b$ fixed. Suppose all derivatives of order $j$ are +bounded by $C_jL^{C(j+1)}$. Fourier inversion separates its variables +with integrated absolute coefficient mass $L^{O(1)}$. There is a +fixed exponent $C'>0$, depending on $b,C$, such that truncating each +frequency at $L^{C'}$ makes an error $O_A(L^{-A})$ for every fixed +$A>0$. The exponent $C'$ is fixed before $A$. +\end{lemma} + +\begin{proof} +Integration by parts gives, for each fixed $j$, an integrable tail +bounded by a constant times +$L^{C(j+1)}(1+|\xi|)^{-j}$. Using a fixed $j>b$ first bounds the +full integral by a fixed logarithmic power. Outside +$|\xi|\ge L^{C'}$ the same calculation gives +\[ + O_j\bigl(L^{C(j+1)-C'(j-b)}\bigr). +\] +Fix $C'>C$ and then increase $j$ for any prescribed $A$. +The Fourier phase factors into one-dimensional phases. Applied to +normalized logarithmic coordinates, these are multiplicative +power twists. +\end{proof} + +The fixed-dimension condition in Lemma~\ref{pre:separation} will not +be used with a growing-dimensional box. In the allocation argument +there are $O(T)$ separate one-dimensional transitions, each with a +fixed Fourier integral norm. Their product costs only +$\exp(O(T))=L^{O(1)}$; a telescoping estimate controls their combined +tails. + +The weighted block sieve and local-density proxies are stated in the +prime-extraction section, where their particular coefficient bounds +and precision dependence are needed. The generic dilation-graph +results are likewise stated at their application in the shifted +correlation section. These are the only nonclassical imported +ingredients of the proof. diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/02-long-prime.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/02-long-prime.tex new file mode 100644 index 0000000000000000000000000000000000000000..4357c3892b542531779e7aa1a6a371681c9e8dd7 --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/02-long-prime.tex @@ -0,0 +1,579 @@ +\section{A long prime polynomial}\label{lp:section} + +The correlation argument will require cancellation in a prime polynomial at +frequencies as large as $x^2$. We prove the required estimate here, including +the dependence on a growing degree in the elementary mean-value argument. +Throughout this section, $L=\log x$ and $T=\log L$. + +\begin{lemma}[Long prime polynomial]\label{lp:prime-polynomial} +Fix $0<\tau<\eta<1$, $C>0$, and $A>0$. There is a constant +$B_0=B_0(\tau,\eta,C,A)$ such that, uniformly for +\[ + \frac{x^\tau}{2}\le N\le x^\eta,\qquad + q\le L^C,\qquad L^{B_0}\le |t|\le x^2, +\] +every Dirichlet character $\chi$ modulo $q$ and every interval +$I\subset[N,2N]$ satisfy +\begin{equation}\label{lp:prime-estimate} + \left|\sum_{p\in I}\chi(p)p^{-1+it}\right| + \ll_{\tau,\eta,C,A} L^{-A}. +\end{equation} +\end{lemma} + +The proof proceeds from a quantitative power-sum estimate to cancellation +for a logarithmic phase, then to a high-height zero-free strip, and finally +to primes by Mellin inversion. +We first establish a power-sum estimate with sufficient uniformity in its +degree. For integers $k\ge2$, $s\ge1$, and $M\ge1$, let $J_{s,k}(M)$ count +the solutions of +\[ + \sum_{i=1}^s u_i^j=\sum_{i=1}^s v_i^j + \quad(1\le j\le k),\qquad 1\le u_i,v_i\le M. +\] +Writing $K=k(k+1)/2$ and +\[ + f(\boldsymbol\alpha)=\sum_{n=1}^M + \e\!\left(\sum_{j=1}^k\alpha_j n^j\right), +\] +orthogonality gives +$J_{s,k}(M)=\int_{[0,1]^k}|f(\boldsymbol\alpha)|^{2s} +\,d\boldsymbol\alpha$. + +The argument uses the classical Vinogradov mean-value method in +Linnik's $p$-adic form; see Wooley~\cite[Section~2, pp.~1583--1585]{Wooley2012} +for an exposition. We derive the required degree-uniform quantitative +estimate below, without invoking the modern efficient-congruencing +theorem of that paper. + +\begin{lemma}[A quantitative power-sum bound]\label{lp:mean-value} +There are absolute constants $C_1,C_2>0$ such that, for every integer +$k\ge2$, some integer $s$ with $k\le s\le C_1k^4$ satisfies +\begin{equation}\label{lp:mean-value-bound} + J_{s,k}(M)\le \exp(C_1k^{C_2})M^{2s-K+1/100} + \qquad(M\ge1). +\end{equation} +\end{lemma} + +\begin{proof} +We iterate estimates +\begin{equation}\label{lp:inductive-bound} + J_{s,k}(M)\le C_s M^{E_s},\qquad + E_s=2s-K+\epsilon_s, +\end{equation} +starting with $s=k$, $C_k=1$, and $\epsilon_k=K$. The iteration sends +$s$ to $s+k$ and $\epsilon_s$ to $(1-1/k)\epsilon_s$. + +Choose a sufficiently large absolute constant $A_1$. If +$M^{1/k}k$. There are at most $k!$ +orderings. The Jacobian of the power sums has determinant +\[ + k!\prod_{i0$. +Thus the asserted iteration holds. Its constants can be chosen with +\[ + \log C_{s+k}\le \log(1+C_s) + +O\!\left((s+k)\log(2k)+k\log(s+k)+k^3\log(2k)\right), +\] +including \eqref{lp:small-M-cost}. + +After $O(k\log(2k))$ steps, +$\epsilon_s=K(1-1/k)^{(s-k)/k}\le1/100$. +Then $s=O(k^2\log(2k))\le C_1k^4$, and summing the displayed costs +gives $\log C_s\le C_1k^{C_2}$ for absolute constants. +Increasing the exponent from $\epsilon_s$ to $1/100$ proves the Lemma. +\end{proof} + +\begin{lemma}[A logarithmic phase on progressions]\label{lp:log-phase} +Fix $C>0$. There is an absolute constant $C_3>0$ such that, for +sufficiently large $x$ in terms of $C$, the following holds uniformly: +\[ + \exp(L/T^2)\le N'\le2x^5,\qquad q\le L^C,\qquad + \exp(L/(2T^2))\le |v|\le4x^3. +\] +For every residue $a\pmod q$ and every interval $I\subset[N',2N']$, +\begin{equation}\label{lp:progression-estimate} + \left|\sum_{\substack{n\in I\\n\equiv a\pmod q}}n^{iv}\right| + \ll \frac{N'}q\exp(-L/T^{C_3}). +\end{equation} +\end{lemma} + +\begin{proof} +Put $H=N'/q$ and $y=\log|v|/\log H$. Then +\begin{equation}\label{lp:phase-scales} + \log H\ge L/T^2-CT\gg L/T^2,\qquad y\ll T^2. +\end{equation} +Writing $n=q(b+a/q)$, with $0\le a\frac15. +\] +Counting integer points proves \eqref{lp:occupancy}. + +Let $s$ be supplied by Lemma~\ref{lp:mean-value}. We also need the +following bound for suprema over boxes $Q$: +\begin{equation}\label{lp:box-supremum} + \sum_Q\sup_{\boldsymbol\alpha\in Q}|S(\boldsymbol\alpha)|^{2s} + \le\exp(O(sk+k))M^KJ_{s,k}(M). +\end{equation} +To prove it, rescale each box to $[0,1]^k$. Iterating the +one-dimensional fundamental theorem of calculus gives, for any smooth +function $F$ on that cube, +\[ + \sup|F|\le\sum_{\boldsymbol\epsilon\in\{0,1\}^k} + \int_{[0,1]^k}|\partial^{\boldsymbol\epsilon}F|. +\] +H\"older's inequality bounds the $2s$th power of the right-hand side +by $2^{k(2s-1)}$ times the sum of the corresponding $2s$th moments. +For the rescaled $S$, every mixed derivative has coefficients bounded +in modulus by $(2\pi)^k$: each differentiation in coordinate $j$ +introduces the factor $2\pi i h^j/M^j$. +On summing over the boxes, change of variables contributes $M^K$. +Orthogonality then bounds the full-torus moment of every such derivative +by $(2\pi)^{2sk}J_{s,k}(M)$. This proves +\eqref{lp:box-supremum}, with constants controlled at the growing degree. + +Apply H\"older's inequality to the sum in +\eqref{lp:shifted-phase}, and use \eqref{lp:occupancy}, +\eqref{lp:box-supremum}, and \eqref{lp:mean-value-bound}. Since +$|\mathcal I|\ll H$, the result is +\begin{equation}\label{lp:phase-final-bound} + \frac1M\sum_{b\in\mathcal I}|S(\boldsymbol\alpha(b))| + \ll H\left(\exp(O(k^{C_4}))H^{-1/5}M^{1/100}\right)^{1/(2s)} +\end{equation} +for an absolute constant $C_4$. +We have $k\ll T^2$ and $s\ll T^8$, whereas +\[ + -\frac15\log H+\frac1{100}\log M + \le-\frac{77}{400}\log H. +\] +Every fixed power of $T$ is $o(L/T^2)$. Thus the right-hand side +of \eqref{lp:phase-final-bound} is at most +$H\exp(-c_1L/T^{10})$ for some absolute $c_1>0$, once $x$ is +sufficiently large. The errors in \eqref{lp:shifted-phase} are smaller. +Increasing a fixed exponent $C_3>10$ absorbs $c_1$ and proves the Lemma. +\end{proof} + +We write $D(s,\chi)$ for the Dirichlet $L$-function, to distinguish it +from $L=\log x$. Characters need not be primitive. Periodicity gives +$\sum_{n\le u}\chi(n)=c_\chi u+O(q)$, where +$c_\chi=q^{-1}\sum_{a\bmod q}\chi(a)$. Partial summation therefore +continues $D(s,\chi)$ meromorphically to $\operatorname{Re}s>0$, with +only the possible simple pole at $s=1$, and gives, for +$\sigma=\operatorname{Re}s$ in a fixed compact subinterval of $(0,\infty)$, +\begin{equation}\label{lp:L-tail} + D(s,\chi)=\sum_{n\le Y}\frac{\chi(n)}{n^s} + +\frac{c_\chi Y^{1-s}}{s-1} + +O\bigl(q(1+|s|)Y^{-\sigma}\bigr). +\end{equation} + +\begin{lemma}[A bound near the line $\operatorname{Re}s=1$] +\label{lp:L-upper} +Fix $C>0$ and put $r_*=T^4/L$. Uniformly for $q\le L^C$, +\begin{equation}\label{lp:L-upper-bound} + \log|D(\sigma+iv,\chi)|\ll_C T^2 + \quad\left(|\sigma-1|\le10r_*,\quad + \frac12\le|v|\le3x^3\right). +\end{equation} +\end{lemma} + +\begin{proof} +First suppose $|v|\le\exp(L/(2T^2))$, and use +$Y=\exp(2L/T^2)$ in \eqref{lp:L-tail}. Absolute summation gives +\[ + \sum_{n\le Y}n^{-\sigma} + \ll(1+\log Y)\max(1,Y^{1-\sigma})\le\exp(O(T^2)). +\] +Since $|s-1|\ge1/2$, the pole term has the same bound. The error is +at most +\[ + \exp\left(-\frac{3L}{2T^2}+O(T^2+CT)\right), +\] +and hence is negligible. + +For $\exp(L/(2T^2))<|v|\le3x^3$, take $Y=x^5$. +The terms with $n\le\exp(L/T^2)$ again contribute +$\exp(O(T^2))$ in absolute value. On a dyadic interval above this +threshold, sum \eqref{lp:progression-estimate}, with phase $-v$, over +the residue classes modulo $q$, including their character coefficients. +The factors $q$ and $1/q$ cancel. Partial summation with $n^{-\sigma}$ +bounds that dyad by +\[ + \exp(-L/T^{C_3})\exp(O(T^4)). +\] +There are $O(L)$ dyads, and the same bound applies to a final partial +dyad. Their total is negligible, since $L/T^{C_3}$ exceeds every +fixed power of $T$. Finally, the pole term and remainder in +\eqref{lp:L-tail} are bounded respectively by +\[ + \exp\left(-\frac{L}{2T^2}+O(T^4)\right),\qquad + \exp\bigl(-2L+O(T^4+CT)\bigr). +\] +This proves \eqref{lp:L-upper-bound}. +\end{proof} + +\begin{lemma}[Local logarithmic derivative]\label{lp:local-log-derivative} +Fix $C>0$, let $q\le L^C$, and put +$s_0=1+r_*+iv$, where $1\le|v|\le\tfrac52x^3$. +There is a radius $R$ with $4r_*\le R\le5r_*$, with no zero on +its boundary, for which +\begin{equation}\label{lp:local-log-derivative-formula} + \frac{D'}{D}(s,\chi) + =\sum_{|\rho-s_0|0$ there is $c=c(C)>0$ such that every character +of modulus at most $L^C$ has no zero in +\begin{equation}\label{lp:zero-free-region} + \operatorname{Re}s\ge1-cT^2/L,\qquad + 1\le|\operatorname{Im}s|\le x^3. +\end{equation} +Moreover, +\begin{equation}\label{lp:log-derivative-bound} + \left|\frac{D'}D(s,\chi)\right|\ll_C L + \quad\left(1-\frac{cT^2}{2L}\le\operatorname{Re}s\le1+\frac1L, + \quad2\le|\operatorname{Im}s|\le\frac{x^3}{2}\right). +\end{equation} +\end{lemma} + +\begin{proof} +For $\sigma>1$, the Euler products and the inequality +$3+4\cos\theta+\cos(2\theta)=2(1+\cos\theta)^2\ge0$ give +\begin{equation}\label{lp:trigonometric-positivity} + 0\le-3\frac{\zeta'}\zeta(\sigma) + -4\operatorname{Re}\frac{D'}D(\sigma+iv,\chi) + -\operatorname{Re}\frac{D'}D(\sigma+2iv,\chi^2). +\end{equation} +Indeed this follows term by term in the absolutely convergent +prime-power expansions; primes dividing the modulus contribute only +the positive zeta term. Also +$-\zeta'/\zeta(\sigma)=1/(\sigma-1)+O(1)$ near $1$. + +Suppose $\rho=\beta+iv$ were a zero in +\eqref{lp:zero-free-region}, and set +$\sigma=1+20cT^2/L$. Apply +Lemma~\ref{lp:local-log-derivative} at heights $v$ and $2v$. +The evaluation points lie in the respective inner disks, and $\rho$ +lies in the first zero sum, because $T^2/L=o(r_*)$. +No zero has real part greater than $1$, by the absolutely convergent +Euler product. All terms in the zero sums therefore contribute +nonpositively to the negative real logarithmic derivatives. Retaining +the term at $\rho$, and using +$0<\sigma-\beta\le21cT^2/L$, bounds the right-hand side of +\eqref{lp:trigonometric-positivity} by +\[ + \left(\frac{3}{20c}-\frac{4}{21c}+O_C(1)\right)\frac L{T^2} + =\left(-\frac{17}{420c}+O_C(1)\right)\frac L{T^2}. +\] +Here the errors are $O_C(T^2/r_*)=O_C(L/T^2)$. +Choosing a sufficiently small fixed $c>0$ gives a contradiction. + +For $s$ in the region of \eqref{lp:log-derivative-bound}, apply the +local formula centered at $1+r_*+i\operatorname{Im}s$. +Every zero in its sum has absolute imaginary part between $1$ and +$x^3$, since $2\le|\operatorname{Im}s|\le x^3/2$ and $r_*=o(1)$. +By \eqref{lp:zero-free-region}, its horizontal distance from $s$ is +at least $cT^2/(2L)$. The $O_C(T^2)$ zero terms and the local error +therefore total $O_C(L)$, proving +\eqref{lp:log-derivative-bound}. +\end{proof} + +\begin{proof}[Proof of Lemma~\ref{lp:prime-polynomial}] +We first establish the analogous bound with the von Mangoldt weight. +Let $\delta=L^{-A-4}$. Smooth the indicator of $I/N\subset[1,2]$ +by convolution with a nonnegative smooth kernel of width $\delta$. +This gives $0\le g\le1$, supported in $[1/2,3]$, whose difference +from the indicator is supported within $O(\delta)$ of its endpoints, +and with $\|g^{(j)}\|_\infty\ll_j\delta^{-j}$. +The construction also applies when $I$ is shorter than $\delta N$. +Since $\Lambda(n)\le\log n$, the error in replacing the interval +by $g(n/N)$ is +\begin{equation}\label{lp:smoothing-error} + O\!\left((\delta N+1)\frac{\log(3N)}N\right) + =O(L^{-A-2}), +\end{equation} +uniformly in $I$. + +Define the Mellin transform by +\[ + \widehat g(z)=\int_0^\infty g(w)w^z\,\frac{dw}{w}. +\] +On every fixed bounded real-part strip, integration by parts gives +\begin{equation}\label{lp:mellin-decay} + |\widehat g(u+iv)|\ll_j + L^{(j+1)(A+5)}(1+|v|)^{-j}. +\end{equation} +Also $|\widehat g(u+iv)|\ll1$ there, by absolute integration. +Mellin inversion and the absolutely convergent logarithmic derivative +on $\operatorname{Re}z=1/L$ yield +\begin{equation}\label{lp:mellin-inversion} + \sum_n\Lambda(n)\chi(n)n^{-1+it}g(n/N) + =\frac1{2\pi i}\int_{1/L-i\infty}^{1/L+i\infty} + \widehat g(z)N^z + \left(-\frac{D'}D(1-it+z,\chi)\right)dz. +\end{equation} +On this full line, absolute convergence gives +\[ + \left|\frac{D'}D(1+1/L+i u,\chi)\right| + \le\sum_n\frac{\Lambda(n)}{n^{1+1/L}} + =-\frac{\zeta'}\zeta(1+1/L)\ll L, +\] +at every height $u$. + +Choose a fixed $C_5>A+6$ and put $H_0=L^{C_5}$. +Then choose a fixed integration-by-parts order $j$ large enough that +\eqref{lp:mellin-decay} makes the two tails with +$|\operatorname{Im}z|>H_0$ in \eqref{lp:mellin-inversion} +$O(L^{-A-2})$. Explicitly their bound is +\[ + O_j\!\left(L^{1+(j+1)(A+5)}H_0^{1-j}\right), +\] +since $N^{1/L}\ll1$. Fix $B_0>C_5+2$. +For $L^{B_0}\le|t|\le x^2$, every point in the rectangle +\[ + -\frac{cT^2}{2L}\le\operatorname{Re}z\le\frac1L, + \qquad |\operatorname{Im}z|\le H_0 +\] +has +\begin{equation}\label{lp:contour-heights} + 2\le|-t+\operatorname{Im}z| + \le x^2+H_0<\frac{x^3}{2} +\end{equation} +for large $x$. Thus Lemma~\ref{lp:zero-free} applies throughout +the rectangle. There are no zeros or poles of the logarithmic +derivative inside it; in particular, the possible principal-character +pole at $z=it$ is outside it. + +Shift the truncated contour to +$\operatorname{Re}z=-cT^2/(2L)$. +The horizontal edges are $O(L^{-A-2})$, by +\eqref{lp:log-derivative-bound} and \eqref{lp:mellin-decay}, increasing +the already fixed order $j$ if necessary. +On the new vertical segment, +\[ + |N^z|=\exp\left(-\frac{cT^2\log N}{2L}\right) + \ll\exp(-c\tau T^2/2). +\] +Its remaining factors and length cost at most a fixed power of $L$. +Since $\exp(-c\tau T^2/2)$ is smaller than every prescribed fixed +power of $L^{-1}$, the shifted integral is $O(L^{-A-2})$. +Together with \eqref{lp:smoothing-error}, this proves, uniformly for +all subintervals $I\subset[N,2N]$, +\begin{equation}\label{lp:mangoldt-estimate} + \sum_{n\in I}\Lambda(n)\chi(n)n^{-1+it}\ll L^{-A-2}. +\end{equation} + +Prime powers of exponent at least two contribute in absolute value at +most $O(N^{-1/2}(\log(3N))^2)$: there are +$O(\sqrt N\log(3N))$ possible bases and exponents with +$N\le p^a\le2N$, and every summand has size $O(\log(3N)/N)$. +This is smaller than every fixed power of $L^{-1}$ in the present +range. Removing them from \eqref{lp:mangoldt-estimate} gives the +same uniform bound for +$\sum_{p\in I}(\log p)\chi(p)p^{-1+it}$. +Finally, partial summation against $1/\log u$, whose value and total +variation on $[N,2N]$ are $O_{\tau}(1/L)$, removes the logarithmic +weight and proves \eqref{lp:prime-estimate}. +\end{proof} diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/03-shifted.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/03-shifted.tex new file mode 100644 index 0000000000000000000000000000000000000000..c1d5b043b349b9c2eb818eb860723afb1cfd97b4 --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/03-shifted.tex @@ -0,0 +1,800 @@ +\section{A prime-slot shifted correlation with a common parity twist} +\label{sh:section} + +We prove the shifted-correlation estimate needed below. The graph +transference and residual ideal operator are imported from +\cite{SmoothShifted}; the replacement of a long rough-integer factor by +a prime factor, and the common parity twist, are proved here. + +\subsection{Groups, marked weights, and the assertion} + +Fix +\[ + 00, + \qquad 0W$, and the modulus of $\chi_0$ and +$|\sigma_0|,|\sigma_1|$ are at most $L^C$. The constants in this +section may depend on these fixed bounds. The word invariant refers +to the identity $F(vn)=F(n)$ whenever $v_*=1$. + +\begin{theorem}[Twisted prime-slot correlation]\label{sh:correlation} +Let $F,G$ have the form \eqref{sh:core}, with possibly different +coefficients, characters, intervals, and frequencies, subject to fixed +bounds as above. For every fixed $D_0>0$ there is a fixed $B>0$ with +the following property. Suppose that the sequence $\alpha$ in $F$ +satisfies +\begin{equation}\label{sh:discrepancy} + \left|\sum_{m\in I}\alpha_m\chi(m)m^{-1+it}\right|\leq L^{-B} + \quad\left( + \begin{array}{l} + I\subset[1,x^2]\text{ an interval},\\ + \operatorname{mod}(\chi)\leq L^B,\quad |t|\leq L^B + \end{array}\right). +\end{equation} +There is no discrepancy requirement on $G$. Uniformly for +$xL^{-C}\leq Z\leq xL^C$, integers $0<|k|\leq L^C$, +$|a_1|,|a_2|\leq L^C$, and smooth functions $\psi$ supported in a +fixed compact subinterval of $(0,\infty)$ with +$\|\psi^{(j)}\|_\infty\ll_j L^{C(j+1)}$, one has +\begin{equation}\label{sh:correlation-bound} + \sum_{\substack{z>0\\z+k>0}}\frac{\psi(z/Z)}z + z^{ia_1}(z+k)^{ia_2}\overline{F(z)}G(z+k) + W_\ell(z)W_\ell(z+k)\varepsilon(z)\varepsilon(z+k) + \ll L^{-D_0}. +\end{equation} +The exponent $B$ depends only on $D_0$ and the displayed fixed data. +\end{theorem} + +The proof uses a residual operator made up of one raw term and signed +comparison terms. Transference makes the residual pairing small; after +the shared dilation is removed, the raw term is the correlation in +\eqref{sh:correlation-bound}, because the common sign cancels. +Each comparison has two independent large free products, allowing a +Fourier estimate. We will bound these comparisons and subtract them +from the residual pairing. + +\subsection{The graph inputs} + +We give the operator definitions to specify their normalizations. +For a sufficiently large fixed integer $J$, put $M=J+\ell$ and +$\mu_g(p)=1/(pV_g)$. A pattern $\nu$ specifies +$C_g\subset\{1,\ldots,J\}$ with $C_g=\varnothing$ in small groups +and $|C_g|\leq m_0$ in big groups. Put $t_g=\ell+|C_g|$. +The first $J$ slots outside $C_g$ are shared between the endpoints. +To form the dilation $D$, take their labels and independent auxiliary +labels of law $\mu_g$ in the slots of $C_g$. Thus $D$ contains $J$ +labels from each group, counted with multiplicity. +The coefficient $K_\nu$ may depend only on the ordered big-group +source and target labels and the auxiliary big-group free labels. + +A physical state at $n\in\mathbb Z$ consists of $M$ ordered, +distinct divisors from each group. Each state has mass +$\prod_gV_g^{-M}$, with counting measure in $n$. For a row +transition from $n$ to $n'=n+kD$, retain the shared labels and sum +over all physical target lists with coefficient $\prod_gV_g^{-t_g}$. +Auxiliary free labels are forbidden from both endpoint lists; they +may coincide with one another. The row multiplier is +\begin{equation}\label{sh:physical-multiplier} + K_\nu D^{i\zeta} + q_0^{(\omega_P(n)-Ms_P)/2}q_0^{(\omega_P(n')-Ms_P)/2}. +\end{equation} +The operator $\mathcal A_\zeta$ is the sum of these row operations, +averaged over independent permutations of the $M$ slots in every +group at each endpoint. In particular the joint normalization of +the two lists in group $g$ is $V_g^{-M-t_g}$. + +The ideal operator acts on the probability space of the ordered +big-group lists, with independent coordinates of law $\mu_g$; +repetitions are permitted. It retains the same shared slots and +samples both the unshared target slots and the auxiliary slots +independently. If $D_{\mathcal B}$ is the big-group part of $D$, +its multiplier is +$K_\nu D_{\mathcal B}^{i\zeta}\e(\Theta D_{\mathcal B})$. +The sum, symmetrized at both ends, is $\mathcal T_\zeta(\Theta)$. + +\begin{proposition}[Transference input]\label{sh:transference} +The following are the Local Transference Theorem, its Endpoint Pairing +Corollary, and the Absolute Physical Bounds Lemma of +\cite[Section~3, Theorem~3.5, Corollary~3.11, and Lemma~3.4]{SmoothShifted}. +Assume $\sum_\nu\sup|K_\nu|\leq L^{C_1}$, where $m_0,C_1$ are +fixed independently of $J$. For every fixed $E>0$ there is an +$E_{\rm id}>0$, depending only on $E$ and the group data, such that +\[ + \sup_{\Theta\in\mathbb R} + \|\mathcal T_\zeta(\Theta)\|_{2\to2}\leq L^{-E_{\rm id}} +\] +implies the following conclusion for all sufficiently large fixed +$J$. Let $I=[X,2X)$ with $x^\gamma\leq X\leq x^{1/\gamma}$ for +some fixed $\gamma>0$. If $f$ is supported on positions in $I$, +is independent of the chosen marks, and +\[ + \sup|f|\leq\exp(O(\sqrt L)),\qquad + \|f\|,\|g\|\leq X^{1/2}L^{C_3}, +\] +then +\begin{equation}\label{sh:pairing} + |\langle f,\mathcal A_\zeta g\rangle| + \ll XL^{-E+2C_3}. +\end{equation} +The lower bound for $J$ may depend on $m_0,C_1$; the threshold for +$x$ may also depend on $J$, the fixed exponent bounding $|k|$, and +$\gamma$. There is no additional restriction on $\zeta$ beyond the +ideal norm hypothesis. The operator obtained by taking absolute +values of all transition coefficients has row and column sums at +most $L^{C_{\rm abs}}$, with $C_{\rm abs}$ independent of $J$. +\end{proposition} + +\begin{proposition}[Residual ideal family and comparison kernels] +\label{sh:ideal} +The following are the Residual Ideal Operator Theorem and Comparison +Kernel Lemma of \cite[Section~4, Theorem~4.1 and Lemma~4.3]{SmoothShifted}. +For every prescribed $E_{\rm id}>0$ and fixed $C_4\geq0$, there are +fixed $m_0,J_0,C_1$ such that, for each fixed $J\geq J_0$, a family +consisting of the raw pattern $C_g=\varnothing$, $K_0=1$, and signed +comparison patterns satisfies +\begin{equation}\label{sh:ideal-bound} + \sum_\nu\sup|K_\nu|\leq L^{C_1},\qquad + \sup_{\substack{\Theta\in\mathbb R\\|\zeta|\leq L^{C_4}}} + \|\mathcal T_\zeta(\Theta)\|_{2\to2} + \leq L^{-E_{\rm id}}. +\end{equation} +The family is independent of the individual values of $\Theta,\zeta$, +and all its defining parameters are independent of $J$. + +Split the big groups into two blocks. Every comparison frees +nonempty sets of probes $A,B$ in the respective blocks and has +coefficient +\begin{equation}\label{sh:comparison-kernel-product} + -(-1)^{|A|+|B|}\mathcal K_A(D_A,X_A)\mathcal K_B(D_B,X_B). +\end{equation} +Here $D_A,D_B$ are the free products, $X_A$ is the corresponding +target-mark product, and $X_B$ the corresponding source-mark product. +No shared label or final $\ell$ mark enters these kernels. For a +slot set $\Lambda$, their exact form is +\begin{equation}\label{sh:kernel} + \mathcal K_\Lambda(D,X) + =\sum_{j:\nu_{\Lambda j}\geq\xi} + \frac{\1_{\log D\in I_j}\1_{\log X\in I_j}}{\nu_{\Lambda j}} + \sum_{\substack{\chi\ {\rm primitive}\\ + \operatorname{cond}(\chi)\leq Q}} + \overline{\chi(D)}\chi(X), + \quad I_j=[jh,(j+1)h), +\end{equation} +where $\nu_{\Lambda j}$ is the probability of the indicated cell +under the independent slot laws. The quantities $Q,h^{-1},\xi^{-1}$ +are fixed powers of $L$, with $h\leq1$. There are at most $Q^2$ +characters and $O_{m_0}(1+TL^d/h)$ nonempty cells, and +\[ + \sup|\mathcal K_\Lambda|\leq Q^2\xi^{-1},\qquad + \sup_X\mathbb E_D|\mathcal K_\Lambda(D,X)|\leq Q^2. +\] +More precisely, the parameters can be chosen, independently of $J$, +so that for any fixed $A_0>0$, any tuple tests $f(X_B),g(X_A)$, +and every $\theta\in\mathbb R$, $|\zeta|\leq L^{C_4}$, +\begin{align} + \Big|\mathbb E\,\overline{f(X_B)}g(X_A) + \big[&(X_AX_B)^{i\zeta}\e(\theta X_AX_B)\notag\\ + &-\mathcal K_A(D_A,X_A)\mathcal K_B(D_B,X_B) + (D_AD_B)^{i\zeta}\e(\theta D_AD_B)\big]\Big| + \leq L^{-A_0}\|f\|_2\|g\|_2. + \label{sh:comparison-contract} +\end{align} +All four label tuples in this expectation are independent. The tests +need not be functions only of their products. Expanding the two +kernels in \eqref{sh:comparison-kernel-product}, including all +patterns, has total coefficient mass and number of terms bounded by +fixed powers of $L$. +\end{proposition} + +These are precisely the graph inputs we use. In particular, +\eqref{sh:kernel} consists of two global character and cell expansions, +not one expansion for each group. + +\subsection{Endpoint norms and removal of the shared dilation} + +We now prove Theorem~\ref{sh:correlation} using these contracts. +Write $\tau(v)$ for the divisor-counting function. The pointwise +estimate +\begin{equation}\label{sh:core-divisor} + |F(n)|\ll L^{C'}\tau(n_*)^2 +\end{equation} +follows by counting the three-factor decompositions in +\eqref{sh:core}; the same estimate holds for $G$. +On physical states the function +\[ + F(n)\varepsilon(n)q_0^{(\omega_P(n)-Ms_P)/2} +\] +is independent of the chosen marks, and its damping is at most one. +For a fixed physical list of product $P$, one has $n_*\mid n/P$. +Consequently Proposition~\ref{pre:moments} gives, uniformly on a fixed +enlargement of a position dyad, +\[ + \sum_{\substack{n\asymp X\\P\mid n}}|F(n)|^2 + \ll L^{C'}\sum_{v\ll X/P}\tau(v)^4 + \ll (X/P)L^{C''}. +\] +The normalized reciprocal sum of the admissible lists is at most +\[ + \prod_g V_g^{-M} + \sum_{\text{lists}}\frac1P + \leq\prod_g\left(\sum_{p\in\mathcal P_g}\frac1{pV_g}\right)^M=1. +\] +Here $P\leq\exp(O_J(TL^d))=x^{o(1)}$, so $X/P$ is in the +ordinary divisor-moment range. We have proved +\begin{equation}\label{sh:endpoint-norm} + \|F\varepsilon q_0^{(\omega_P-Ms_P)/2}\|_{n\asymp X} + \ll X^{1/2}L^{C_3}, +\end{equation} +with $C_3$ independent of $J$ and of the subsequent kernel choices. +Bounded smooth factors do not affect this independence. Moreover, +both $m$ and $p$ in \eqref{sh:core} are $W$-rough for large $x$. +There are at most $O(\sqrt L)$ prime factors above $W$ in +$n\asymp x^{1+o(1)}$, counted with multiplicity. Assigning each such +factor to $m,p$, or the remaining factor shows +\begin{equation}\label{sh:endpoint-sup} + |F(n)|\leq L^{C'}3^{O(\sqrt L)}=\exp(O(\sqrt L)). +\end{equation} +This proves every endpoint hypothesis of Proposition~\ref{sh:transference}. + +Consider the physical pairing with extra cutoff, harmonic measure, +and phases +\begin{equation}\label{sh:lifted-pairing} + \frac{\psi(n/(ZD))}{n} + D^{-i(a_1+a_2)}n^{ia_1}(n')^{ia_2} + \overline{F(n)}G(n')\varepsilon(n)\varepsilon(n') + q_0^{(\omega_P(n)-Ms_P)/2}q_0^{(\omega_P(n')-Ms_P)/2}. +\end{equation} +The row operation itself contributes the other two half-damping +factors in \eqref{sh:physical-multiplier}. Summation over states, +auxiliary draws, and patterns is understood in +\eqref{sh:lifted-pairing}, with the row coefficient $K_\nu$. + +Here $n\asymp ZD=x^{1+o(1)}$ and $|n'-n|=x^{o(1)}$. Split +$n$ among $O(L)$ dyads $[X,2X)$ and restrict $n'$ to a fixed +enlargement of each dyad. As in Lemma~\ref{pre:separation}, Fourier inversion of +$u\mapsto\psi(e^u)$ separates the cutoff into powers of $n$ and +$D$; put the bounded factor $X/n$ in the first vector and divide +the pairing by $X$. The total Fourier integral mass is a fixed +power of $L$, independent of $J$. + +We specify a useful quantifier here. If the logarithmic cutoff +derivatives are bounded by $O_j(L^{C_\psi(j+1)})$, fix an exponent +$F_0>C_\psi+2$ before forming the ideal family. Integration by +parts gives, for each fixed $N$, +\[ + \int_{|v|>L^{F_0}}|\widehat{\psi\circ\exp}(v)|\,dv + \ll_N L^{C_\psi(N+1)-F_0(N-1)}. +\] +Thus this same cutoff exponent permits arbitrary saving by increasing +$N$. Choose $C_4$ to contain $a_1+a_2$ and all frequencies in the +truncated integral. By \eqref{sh:endpoint-norm} and +\eqref{sh:endpoint-sup}, choose $E$ large enough for the desired +saving after the fixed Fourier and dyadic costs; then choose +$E_{\rm id}$, the residual family, and finally $J$. The Fourier +tail is bounded using the absolute row and column bounds of +Proposition~\ref{sh:transference}; its accuracy can be increased +after the family and $J$ have been fixed without changing $F_0$ or +$C_4$. It follows that the sum of the residual pairings is +$O(L^{-D})$ for any prescribed fixed $D$. + +We next identify each individual pairing. Write $D=D_RD_C$, +where $D_R$ is the shared product and $D_C$ the free product, and put +$n=D_Rw$, $n'=D_Rw'$. Then $w'=w+kD_C$. Away from overlaps of +shared labels with one another or with the divisors of $w,w'$, the +number of shared labels in group $g$ is $M-t_g$, and +\[ + \omega_g(D_Rw)-M=\omega_g(w)-t_g. +\] +The full damping therefore leaves exactly the two damping factors +of $W_{\mathbf t}(w)W_{\mathbf t}(w')$. The identity +\[ + V_g^{-M-t_g}=V_g^{-(M-t_g)}V_g^{-2t_g} +\] +and $1/n=1/(D_Rw)$ turn the shared labels into independent draws +of law $\mu_g$, leaving the normalizations for the two unshared +marked lists. The cores are unchanged. The sign identity is exact, +even when overlaps occur: +\begin{equation}\label{sh:shared-sign} + \varepsilon(D_Rw)\varepsilon(D_Rw') + =\varepsilon(D_R)^2\varepsilon(w)\varepsilon(w') + =\varepsilon(w)\varepsilon(w'). +\end{equation} +The phases and cutoff in \eqref{sh:lifted-pairing} become +\[ + \psi(w/(ZD_C))\,(w/D_C)^{ia_1}(w'/D_C)^{ia_2}/(D_Rw). +\] +Since the outer vectors were mark-independent, the initial +symmetrizations do not alter this computation. After summing the +shared labels, the pairing for pattern $\nu$ is, up to negligible +error, +\begin{align} + \mathbb E_{\rm free}\sum_{\substack{w>0\\w'=w+kD_C>0}} + &\frac{\psi(w/(ZD_C))}{w} + (w/D_C)^{ia_1}(w'/D_C)^{ia_2} + \overline{F(w)}G(w')\varepsilon(w)\varepsilon(w')\notag\\ + &\hspace{10mm}\cdot W_{\mathbf t}(w)W_{\mathbf t}(w') + \mathbb E_{\rm marks}K_\nu. + \label{sh:stripped} +\end{align} +The final expectation means averaging over the unshared marked lists. + +For completeness, all discarded restrictions have superpolynomial +logarithmic saving. Conditional on $w,w'$, the free labels, and +earlier shared draws, at most $O_J(L)$ prime values are excluded +from a shared draw. Every atom is $O(\exp(-L^a))$. When bounding +actual overlapping configurations, unshared physical marks still +divide $w,w'$; ignoring damping changes their weight by at most +$q_0^{-2Ms_P}=L^{O_J(1)}$. Equations \eqref{sh:weight-bound} and +\eqref{sh:core-divisor}, followed by Cauchy--Schwarz and fixed divisor +moments on the two translated intervals, bound the remaining harmonic +sum by $L^{O_J(1)}$. Shared overlaps thus cost +$L^{O_J(1)}\exp(-L^a)$. + +If an auxiliary free label $p'$ belongs to an unshared endpoint +list, then $p'\mid D_C$ and hence $p'\mid w,w'$. Set $w=p'y$. +The new shift $kD_C/p'$ is integral, the invariant cores are +unchanged, and the harmonic measure supplies $1/p'$. Applying the +same divisor moments at scale $ZD_C/p'$ bounds this contribution +by $L^{O_J(1)}/p'$. There are $O_J(T)$ free slots and +$p'\geq\exp(L^c)$. This bounds all free exclusions. After summing +the fixed-power coefficient costs, the error in +\eqref{sh:stripped} is $O_A(L^{-A})$ for every fixed $A$. +For the raw pattern, $D_C=1$, $\mathbf t=(\ell,\ldots,\ell)$, +and \eqref{sh:stripped} is exactly the left side of +\eqref{sh:correlation-bound}. It remains to bound the comparisons. + +\subsection{Comparison multipliers and minor arcs} + +Expand the two kernels in a comparison using \eqref{sh:kernel}. +Their four factors are bounded weights on $D_A,D_B$ and on the +big unshared marks at the respective endpoints, times a total +coefficient cost $L^{O(1)}$. The free products are restricted to +logarithmic cells of width at most one. Let their lower endpoints +be $U_1,U_2$, put $H'=U_1U_2$, and set $Y=ZH'$. Then +\begin{equation}\label{sh:comparison-scales} + U_1,U_2\geq\exp(L^c-O(1)),\quad + H'\leq\exp(O(TL^d)),\quad Y=x^{1+o(1)}. +\end{equation} +The positions $w,w'$ in \eqref{sh:stripped} are comparable to $Y$. +Logarithmic Fourier separation of the smooth cutoff and phases +therefore reduces it, with total cost $L^{O(1)}/Y$, to expressions +\begin{equation}\label{sh:comparison-convolution} + \mathbb E\,b_1(D_A)b_2(D_B) + \sum_w\overline{\mathcal U(w)}\mathcal V(w+kD_AD_B), + \qquad |b_1|,|b_2|\leq1. +\end{equation} +An endpoint here is its core times $\varepsilon$, a power twist of +fixed log-power frequency, $W_{\mathbf t}^{\mathbf b}$ with +$\mathbf b$ depending only on big marks, and a smooth size cutoff +at $Y$. Fixed divisor moments give +\begin{equation}\label{sh:comparison-norms} + \sum_w|\mathcal U(w)|^2+\sum_w|\mathcal V(w)|^2 + \ll YL^{C_5}. +\end{equation} +They also control the Fourier-separation tails to arbitrary accuracy. + +The Fourier multiplier of \eqref{sh:comparison-convolution} is +\[ + \mathfrak m(\theta)=\mathbb E\,b_1(D_A)b_2(D_B) + \e(k\theta D_AD_B), + \qquad |\mathfrak m(\theta)|\leq1. +\] +For a selected product $D$ with at most $m_0$ slots per group, +unique factorization gives +\begin{equation}\label{sh:product-atoms} + \mathbb P(D=n)=\frac1n\prod_g + \frac{k_g!}{V_g^{k_g}\prod_{p\in\mathcal P_g}v_p(n)!} + \leq\frac{L^{C_6}}n. +\end{equation} +Thus a bounded test collapsed onto product values $n\asymp U_i$ +has squared coefficient norm $O(L^{C_6}/U_i)$. + +We use the integer bilinear estimate of +\cite[Section~4, Lemma~4.2]{SmoothShifted}: for coefficients on +intervals of lengths $O(U),O(V)$, and +$\alpha=u/r+\beta$ with $(u,r)=1$, $|\beta|\leq r^{-2}$, +\[ + \left|\sum_{n,t}a_nb_t\e(\alpha nt)\right| + \ll\|a\|_2\|b\|_2 + [U+(1+V/r)\{U+r\log(2r)\}]^{1/2}. +\] +Arbitrary missing coefficients and translated containing intervals +are allowed. Together with \eqref{sh:product-atoms}, this bounds +$|\mathfrak m(\theta)|$ by +\begin{equation}\label{sh:minor-bound} + L^{C_7}\left(\frac1{U_2}+\frac1r+ + \frac{\log(2r)}{U_1}+ + \frac{r\log(2r)}{H'}\right)^{1/2} +\end{equation} +whenever $r$ is a Dirichlet-approximation denominator for $k\theta$. + +Choose such an approximation with maximum denominator +$\lfloor H'/L^{C'}\rfloor$. If $r>L^{C'}$, all terms in +\eqref{sh:minor-bound} save an arbitrarily large fixed log power +when $C'$ is sufficiently large, by \eqref{sh:comparison-scales}. +If $r\leq L^{C'}$, then $\theta$ belongs, after accounting for +$|k|\leq L^C$, to an arc +\begin{equation}\label{sh:arcs} + \left|\theta-\frac hr\right|_{\mathbb R/\mathbb Z} + \leq\frac{L^{C_{\rm arc}}}{H'}, + \qquad 1\leq r\leq L^{C_{\rm arc}},\quad h\bmod r, +\end{equation} +for a sufficiently large fixed $C_{\rm arc}$. The letters $h,r$ +in \eqref{sh:arcs} denote the resulting rational center, rather +than necessarily the original approximation. There are a fixed +logarithmic power of arcs. Parseval and +\eqref{sh:comparison-norms} dispose of their complement. On the +arcs, Cauchy--Schwarz shows that it suffices to prove, for every +fixed $A>0$ and every arc center $h/r$, +\begin{equation}\label{sh:arc-target} + \int_{|\beta|\leq1/H} + |\widehat{\mathcal U}(h/r+\beta)|^2\,d\beta + \ll_A YL^{-A},\qquad H=H'/L^{C_{\rm arc}}. +\end{equation} +Here $\widehat{\mathcal U}(\theta)=\sum_w\mathcal U(w)\e(\theta w)$, +$H\to\infty$, and $H=x^{o(1)}$. + +\subsection{The endpoint on a major arc} + +Choose a small group $g_s$ and a designated one of its +$\ell\geq1$ marks. Because the mark weight $\mathbf b$ uses only +big marks, removing this prime gives, unless a prime in $g_s$ +divides $w$ twice, +\begin{equation}\label{sh:small-mark-extraction} + W_{\mathbf t}^{\mathbf b}(w) + =\frac1{V_{g_s}}\sum_{\substack{p_s\mid w\\p_s\in\mathcal P_{g_s}}} + W_{\mathbf t-\mathbf e_{g_s}}^{\mathbf b}(w/p_s). +\end{equation} +The equality includes damping: removing $p_s$ decreases both +$\omega_P$ and $\sum_g t_g$ by one. On the excluded set both +weights are bounded by fixed log powers. Since $w_*\mid w/p_s^2$ +on a multiple of $p_s^2$, divisor moments show that the endpoint +error has squared norm +\begin{equation}\label{sh:small-square-error} + \ll YL^{C_8}\sum_{p_s\in\mathcal P_{g_s}}p_s^{-2} + \ll YL^{C_8}\exp(-L^a). +\end{equation} +Its Fourier energy is negligible by Parseval. + +The factors $m,p$ in \eqref{sh:core} contain no group prime. +After \eqref{sh:small-mark-extraction}, write $w=mp_sp r'$. +Complete multiplicativity gives +$\varepsilon(w)=\varepsilon(p_s)\varepsilon(r')$; the remaining +marked weight and $c_{r'_*}$ are functions of $r'$ alone. They +have a fixed log-power bound, so may be absorbed into one residual +coefficient. + +To remove the rational center, fix $d'=(r',r)$ and put $q=r/d'$. +The first three factors $m,p_s,p$ are units modulo $r$ for all +large $x$, and $r'/d'$ is a unit modulo $q$. On units the exact +character expansion is +\begin{equation}\label{sh:rational-characters} + \e(hv/q)=\sum_{\chi\bmod q}c_{h,q}(\chi)\chi(v),\qquad + c_{h,q}(\chi)=\frac1{\varphi(q)} + \sum_{v\bmod q}^{*}\e(hv/q)\overline{\chi(v)}, + \quad |c_{h,q}(\chi)|\leq1. +\end{equation} +Apply this with $v=mp_sp(r'/d')$. The gcd restriction and +$\chi(r'/d')$ go into the residual coefficient. Summing over +$d'$ and all characters costs a fixed power of $L$ and includes +the principal and imprimitive characters. The product of the new +character on $m$ with $\chi_0$ is a character of fixed log-power +modulus. It is therefore enough to bound the energy at zero of +\begin{equation}\label{sh:four-factors} + a_w=\psi_1(w/Y)w^{iv_1} + \sum_{mp_sp r'=w}\alpha_m\chi_m(m)m^{i\sigma_0} + b_s(p_s)\1_{p\in I_p,\ p\ {\rm prime}} + \chi_p(p)p^{i\sigma_1}d_{r'}. +\end{equation} +Here $p_s\in\mathcal P_{g_s}$, $|b_s|\ll1$, +$|d_{r'}|\leq L^{C_9}$, and all characters, frequencies and +smooth-cutoff derivatives have fixed log-power bounds. The sign +on $p_s$ and $1/V_{g_s}$ are included in $b_s$. In particular, +\begin{equation}\label{sh:a-norm} + \sum_w|a_w|^2\ll YL^{C_{10}}. +\end{equation} + +\subsection{Local Fourier energy and Mellin polynomials} + +We include the scale conversion from +\cite[Section~5, equation~(5.16)]{SmoothShifted} with its proof, +as its normalization is useful here. + +\begin{lemma}[Local Mellin energy]\label{sh:mellin-energy} +Let $a_w$ be supported on $w\asymp Y$, where $Y=x^{1+o(1)}$, +and let $H=x^{o(1)}$ tend to infinity. For every fixed $j\geq1$, +\begin{equation}\label{sh:mellin-inequality} + \int_{|\beta|\leq1/H}|\widehat a(\beta)|^2\,d\beta + \ll_j \frac1Y\int_{\mathbb R}(1+H|t|/Y)^{-j} + \left|\sum_w a_ww^{it}\right|^2dt + +\left(\frac HY\right)^2\sum_w|a_w|^2. +\end{equation} +\end{lemma} + +\begin{proof} +Take a smooth bump $K$ of integral one, supported sufficiently near +zero that its Fourier transform, with convention +$\widehat K(\xi)=\int K(u)\e(-\xi u)\,du$, has modulus at least +$1/2$ on $[-1,1]$. Plancherel gives +\[ + \int_{|\beta|\leq1/H}|\widehat a(\beta)|^2\,d\beta + \ll H^{-2}\int_{\mathbb R} + \left|\sum_w a_wK((w-v)/H)\right|^2dv. +\] +Only $v\asymp Y$ contributes. Keep this restriction when replacing +the kernel by $K(v\log(w/v)/H)$. On the union of their supports, +$|w-v|\ll H$, and their arguments differ by $O(H/Y)$. +Cauchy--Schwarz over the $O(H)$ possible integers $w$, followed +by integration over the $O(H)$ centers for each $w$, bounds the +normalized squared error by $(H/Y)^2\sum_w|a_w|^2$. + +Put $h_0=H/Y$, $v=Y\exp(u)$, and $P_a(t)=\sum_w a_ww^{it}$. +Fourier inversion for the new kernel gives +\[ + \sum_w a_wK(v\log(w/v)/H) + =\frac{h_0}{2\pi}\int_{\mathbb R} + e^{-u}\widehat K(h_0e^{-u}t/(2\pi))v^{-it}P_a(t)\,dt. +\] +Insert a fixed smooth compact cutoff in $u$ equal to one on the +required range. The Fourier transform in $u$ of the resulting +amplitude $e^{-u}\widehat K(h_0e^{-u}t/(2\pi))$ is bounded by +\[ + C_j(1+|s|)^{-2}(1+h_0|t|)^{-j}, +\] +by two integrations by parts and the Schwartz bounds for +$\widehat K$. Expand that amplitude in its $u$-Fourier transform, +apply Minkowski in $s$ and Plancherel in $u$, and use $dv\ll Y\,du$. +The normalization is $H^{-2}Yh_0^2=Y^{-1}$. Increasing the decay +order if necessary proves \eqref{sh:mellin-inequality}. +\end{proof} + +Apply this lemma to \eqref{sh:four-factors}. The error term is +$O_A(YL^{-A})$ for every fixed $A$, by \eqref{sh:a-norm}. +Dirichlet-polynomial mean squares imply, on every dyadic time scale +$R\geq1$, +\begin{equation}\label{sh:mellin-tail-ms} + Y^{-2}\int_{R\leq|t|\leq2R}|P_a(t)|^2dt + \ll(1+R/Y)L^{C_{11}}. +\end{equation} +Consequently, with $T_0=LY/H$, the part $|t|>T_0$ of the first +term in \eqref{sh:mellin-inequality} is at most +\[ + YL^{C_{11}}\sum_{r\geq0} + (2^rL)^{-j}\left(1+\frac{2^rL}{H}\right) + \ll_j YL^{C_{11}-j}(1+L/H). +\] +It is $O_A(YL^{-A})$ on taking $j$ large enough. + +Split the four variables in \eqref{sh:four-factors} into dyads. +There are $O(L^4)$ relevant boxes, and their product scales are +comparable to $Y$. Write +\[ + \frac{P_a(t)}Y + =\sum_w\frac1w\big((w/Y)\psi_1(w/Y)\big)w^{i(t+v_1)} + \sum_{mp_sp r'=w}(\text{the four coefficients}). +\] +Fourier-separate the parenthesized function in $\log(w/Y)$. On +every box the fourfold collapsed reciprocal coefficients have +squared norm $\ll L^{C_{12}}/Y$, by a fixed divisor moment. +Hence centered mean squares give a bound $L^{C_{13}}$ for the +integral of their squared polynomial on any translate of +$[-T_0,T_0]$. Minkowski therefore bounds a separating-frequency +tail by a fixed log power times the square of the $L^1$ mass of +that tail. Integration by parts makes it arbitrarily small at a +fixed log-power cutoff. This proves the claimed separation also +for unbounded tail frequencies. Retained shifts enlarge the +time interval to at most $[-2T_0,2T_0]$, since $Y/H$ exceeds every +fixed log power. + +After absorbing these shifts into $t$ and the fixed frequencies, +it suffices to prove, for arbitrary fixed $A'>0$, +\begin{equation}\label{sh:polynomial-target} + \int_{|t|\leq2LY/H}|P_s(t)P_l(t)\mathcal R(t)|^2dt + \ll L^{-A'}, +\end{equation} +where +\begin{align*} + P_s(t)&=\sum_{p_s\asymp P}b_s(p_s)p_s^{-1+it},\\ + P_l(t)&=\sum_{\substack{p\asymp P',\ p\in I_p\\p\ {\rm prime}}} + \chi_p(p)p^{-1+i\sigma_1+it},\\ + \mathcal R(t)&= + \left(\sum_{m\asymp M_1}\alpha_m\chi_m(m)m^{-1+i\sigma_0+it}\right) + \left(\sum_{r'\asymp R_1}d_{r'}(r')^{-1+it}\right). +\end{align*} +The small primes retain their group restriction, and each dyad is +intersected with its original interval. Their sizes satisfy +\begin{equation}\label{sh:polynomial-scales} + \tfrac12\exp(L^a)\leq P\leq\exp(L^b),\qquad + x^\tau/2\leq P'\leq x^\eta,\qquad PM_1P'R_1\asymp Y. +\end{equation} +For any subproduct of these four polynomials, collapsed as +$\sum_{n\asymp U}c_nn^{it}$, a fixed divisor moment gives +\begin{equation}\label{sh:collapsed-norm} + \sum_n|c_n|^2\ll L^{C_{14}}/U. +\end{equation} +Only four convolution factors are involved: all remaining marked +weights have already entered the single bounded coefficient $d_{r'}$. + +\subsection{Exceptional times and completion of the proof} + +First consider $|P_s(t)|\leq L^{-A_s}$. The polynomial +$P_l\mathcal R$ has size $Y/P$, and +\[ + \frac{Y/P}{2LY/H}=\frac{H}{2LP}\longrightarrow\infty +\] +by $bL^{-A_s}\}$. We establish +\begin{equation}\label{sh:exceptional-count} + |\mathcal J|\leq + \exp\big(O(L^b+L^{1-a}\log L)\big)=Y^{o(1)}. +\end{equation} +Put $j_0=\lfloor\log Y/\log(2P)\rfloor$. Unique factorization +shows directly that the squared coefficient norm of $P_s^{j_0}$ +is at most +\begin{equation}\label{sh:factorial-norm} + j_0!\left(\sum_{p_s\asymp P}|b_s(p_s)|^2p_s^{-2}\right)^{j_0} + \leq j_0!(C/P)^{j_0}. +\end{equation} +Indeed for each multiset of $j_0$ primes, one of the two multinomial +coefficients in its squared coefficient is at most $j_0!$. +This argument remains valid when $j_0$ grows; no growing-order +divisor-moment estimate is being used. + +Select a point with $|P_s|>L^{-A_s}$ in each occupied interval. +Split the intervals into three classes according to their integer +index modulo three, so the chosen points in a class are separated +by at least one. The indices of $P_s^{j_0}$ are at most $Y$, and +the time range is contained in $[-Y,Y]$ for large $x$. The +separated-point mean-square inequality and \eqref{sh:factorial-norm} +give +\[ + |\mathcal J|\ll YL^{C_{15}}j_0! + (CL^{2A_s}/P)^{j_0}. +\] +Since $YP^{-j_0}\leq2P\,2^{j_0}$, +$\log P\leq L^b$, and $j_0=O(L^{1-a})$, this proves +\eqref{sh:exceptional-count}. + +We next prove the sparse mean-square estimate +\begin{equation}\label{sh:sparse-supremum} + \sum_{I\in\mathcal J}\sup_{t\in I}|\mathcal R(t)|^2 + \ll L^{C_{16}}. +\end{equation} +Collapse $\mathcal R(t)=\sum_{n\asymp U}c_nn^{it}$, where +\[ + U=M_1R_1\asymp Y/(PP')\geq Y^{1-\eta-o(1)}>Y^{0.6}, + \qquad \sum_n|c_n|^2\ll L^{C_{14}}/U. +\] +Choose a maximizing point on each closed unit interval and again +split into three separated classes. The Gram matrix of the +evaluation vectors $(n^{it_i})_{n\asymp U}$ has entries +$\sum_{n\asymp U}n^{i(t_i-t_j)}$. For +$1\leq|\Delta|\leq c_*U$, with a sufficiently small fixed $c_*>0$, +the derivative of $\Delta\log n/(2\pi)$ is monotone, has magnitude +comparable to $|\Delta|/U$, and is bounded away from nonzero +integers. The first derivative test of +Proposition~\ref{pre:derivatives} gives $O(U/|\Delta|)$. +For $c_*U\leq|\Delta|\ll Y$, the second derivative test gives +\[ + O\big(\sqrt{|\Delta|}+U/\sqrt{|\Delta|}\big)=O(Y^{1/2}). +\] +The diagonal entry is $O(U)$. Thus all entries are bounded by +\[ + O\left(\frac{U}{1+|t_i-t_j|}+Y^{1/2}\right). +\] +Separation and \eqref{sh:exceptional-count} bound an absolute row +sum by +\[ + O\big(U\log Y+|\mathcal J|Y^{1/2}\big)=O(U\log Y). +\] +The operator norm is bounded by this row sum. Multiplying by +$\sum|c_n|^2$ proves \eqref{sh:sparse-supremum}. + +On the exceptional intervals, $|P_s(t)|\ll1$. Prescribe a +sufficiently large saving for $P_l$, and apply +Lemma~\ref{lp:prime-polynomial}. Choose $B_1$ larger than its +frequency threshold and the fixed exponent bounding $|\sigma_1|$. +For $|t|\geq L^{B_1}$ in the present range, the lemma applies to +$t+\sigma_1$: it is above the required logarithmic threshold and +\[ + |t+\sigma_1|\leq2LY/H+|\sigma_1|B_1+C_{17}+A'$, makes this last integral $O(L^{-A'})$. +This proves \eqref{sh:polynomial-target}, hence +\eqref{sh:arc-target}. + +The major-arc estimate, the minor-arc estimate, and Parseval now +bound every comparison \eqref{sh:stripped} by an arbitrarily large +negative log power, after paying the fixed number of arcs, kernel +expansions, and Fourier integrals. Their sum has the same bound. +Subtracting it from the controlled residual pairing leaves the raw +pattern, which is \eqref{sh:correlation-bound}. This completes +the proof of Theorem~\ref{sh:correlation}. + +\begin{remark}[Order of choices]\label{sh:choice-order} +The endpoint norm exponent and the original Fourier cutoff exponent +are fixed from the input data. Then choose the physical accuracy +$E$, ideal accuracy $E_{\rm id}$, ideal frequency range $C_4$, all +kernel parameters, and finally a sufficiently large fixed $J$. +Once this family is fixed, choose the minor-arc and major-arc +accuracies, the prime-polynomial accuracy and threshold $B_1$, and +last the discrepancy exponent $B$. Any stronger Fourier-tail +saving required by these choices is obtained by more integrations +by parts at the already fixed cutoff exponent. No kernel parameter +or frequency range has to be changed after choosing $J$. +\end{remark} diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/04-weights.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/04-weights.tex new file mode 100644 index 0000000000000000000000000000000000000000..b65802358dc1d5fe9fe32e240f9ee19990b0b670 --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/04-weights.tex @@ -0,0 +1,299 @@ +\section{Candidate weights and their mass}\label{wt:section} + +The ordinary bands $Q_j$ give an even number of prime slots and allow +the candidate to be split near a prescribed size. The smaller groups +$\mathcal P_g$ supply the graph marks and the odd parity restriction. +Separating the two families keeps these roles independent. + +We now fix the prime groups used in the remainder of the proof. All +constants in this section depend only on the following fixed geometry +and on the function $\Psi$. In particular, they are independent of the +parameters $\kappa,b_1$ and of the proxy precision chosen later. + +Choose a fixed even integer $r_0$ sufficiently large that +\[ + \frac{(r_0-1)c_1}{2}\ge c_2+1, + \qquad c_2s'<\frac14, +\] +where +\begin{equation}\label{wt:geometry} + c_1=1,\quad c_2=\frac65,\quad c_3=\frac32,\quad c_4=\frac95, + \qquad s'=\frac1{r_0(c_1+c_2)},\qquad s_j=s'2^{-j}L. +\end{equation} +Let $j_x$ be the largest nonnegative integer with $c_1s_{j_x}\ge20T$, +and let $Q_j$ consist of the primes with +$c_1s_j\le\log p\le c_2s_j$, for $0\le j\le j_x$. +For each $j$ such that $[c_3s_j,c_4s_j]$ is contained in either +$[L^{1/10},L^{1/5}]$ or $[L^{3/10},L^{2/5}]$, take the primes whose +logarithms belong to $[c_3s_j,c_4s_j]$ as one group $\mathcal P_g$. +As before, $\mathcal P$ is the union of these groups and $n_*$ is the +part of $n$ supported outside $\mathcal P$. + +These intervals are mutually disjoint, including between the $Q_j$ +and the $\mathcal P_g$: indeed $c_20\quad\Longrightarrow\quad + \Omega(u)\ \hbox{is odd},\qquad \Omega(2u)\ \hbox{is even}. +\end{equation} + +\subsection{Harmonic measures and the group tail} + +Set +\begin{equation}\label{wt:harmonic-masses} + H_Q=\sum_v\frac{a(v)}v + =\prod_{j=0}^{j_x}\frac{V_{Q,j}^{r_0}}{r_0!}, + \qquad + H_P=\sum_{r:\,r_*=1}\frac{W_1(r)}r. +\end{equation} +Since $j_x+1=O(T)$, Equation~\eqref{wt:band-masses} gives +$L^{-C}\le H_Q\le L^C$ for a fixed $C$. +For one group introduce +\[ + Z_g(q)=\prod_{p\in\mathcal P_g}\left(1+\frac q{p-1}\right). +\] +The factor of $H_P$ associated with this group is +\begin{equation}\label{wt:group-mass} + H_g=\frac{Z_g'(q_0)}{V_g} + =\frac{Z_g(q_0)}{V_g} + \sum_{p\in\mathcal P_g}\frac1{p-1+q_0}. +\end{equation} +This follows by expanding the Euler product: each occurring prime has +harmonic mass $\sum_{k\ge1}p^{-k}=1/(p-1)$, and differentiation chooses +one of the distinct prime divisors as its mark. Because $q_0\le1$, the +sum on the right of Equation~\eqref{wt:group-mass} is at least $V_g$. +The same sum is bounded above, and $Z_g(q_0)$ is bounded above, by +constants uniform in $g$. Hence +\begin{equation}\label{wt:HP-bounds} + 1\le Z_g(q_0)\le H_g\le C, + \qquad + H_P=\prod_gH_g,\qquad 1\le H_P\le L^C, + \qquad + \prod_{p\in\mathcal P}\left(1+\frac{q_0}{p-1}\right)\le H_P. +\end{equation} + +Normalize $a(v)/v$ and $W_1(r)/r$ by $H_Q$ and $H_P$, respectively, +and draw the two parts independently. The $Q$-measure can equivalently +be sampled by drawing every ordered slot independently with law +$1/(pV_{Q,j})$ in band $j$. The group measure is a product of the +normalized measures belonging to Equation~\eqref{wt:group-mass}. +Write $V$ and $R$ for these two random integers and $U=VR$. + +In one group the unnormalized harmonic mass of total multiplicity one +is exactly $V_g/V_g=1$. The contribution to multiplicity two from +two distinct primes is +\[ + \frac{2q_0}{V_g}\sum_{p0$. +Conditioning on all groups except one proves +\begin{equation}\label{wt:odd-mass} + \mathbb P\bigl(\mathcal E(R)=-1\bigr)\ge\delta_0. +\end{equation} +The bound is independent of the number of groups. + +\begin{lemma}[Tail of the group part]\label{wt:tail} +There is a fixed $C$ such that, for each fixed $\sigma>0$, +\[ + \sum_{\substack{r>x^\sigma\\r_*=1}}\frac{W_1(r)}r + \le L^C\exp(-\sigma L^{3/5}) +\] +for all sufficiently large $x$. +\end{lemma} + +\begin{proof} +Put $\delta=L^{-2/5}$. The mass in one group after inserting $r^\delta$ +is +\[ + \prod_{p\in\mathcal P_g} + \left(1+\frac{q_0}{p^{1-\delta}-1}\right) + \frac1{V_g}\sum_{p\in\mathcal P_g} + \frac1{p^{1-\delta}-1+q_0}. +\] +Every group prime satisfies $p^\delta\le e$, so this expression is +bounded above by a fixed constant, using Equation~\eqref{wt:band-masses}. +Multiplying over $O(T)$ groups gives +$\sum_{r_* =1}W_1(r)r^{-1+\delta}\le L^C$. +For $r>x^\sigma$, the extra factor is at least +$x^{\sigma\delta}=\exp(\sigma L^{3/5})$, proving the assertion. +\end{proof} + +\subsection{Localization at the target size} + +\begin{lemma}[Total candidate mass]\label{wt:mass} +There is a fixed $C_A$ such that +\[ + X_A\asymp\frac{x}{L}H_QH_P, + \qquad X_A\gg xL^{-C_A}. +\] +The implied constants and $C_A$ depend only on the fixed geometry and +on $\Psi$. +\end{lemma} + +\begin{proof} +The exact probability identity is +\begin{equation}\label{wt:expectation} + X_A=H_QH_P\, + \mathbb E\left[U\Psi(U/x)\, + \1_{\{\mathcal E(R)=-1\}}\right]. +\end{equation} +Leave one ordered slot in $Q_0$ free and condition on every other +variable. On the support of $\Psi(U/x)$, its logarithm must lie in an +interval of length $\log2$. Uniformly in the location of such an +interval, its normalized harmonic prime mass in $Q_0$ is $O(1/L)$. +For example, after intersecting with $Q_0$, the interval is contained +in $[y,2y]$ with $\log y\asymp L$, and the prime number theorem bounds +the reciprocal prime sum there by $O(1/L)$. +Since $U\le2x$ on the effective support, this proves the upper bound. + +For the lower bound let +\[ + m_j=\frac{c_1+c_2}{2}s_j, + \qquad \Delta=(c_2-c_1)s'L. +\] +The infinite midpoint sum satisfies +$r_0\sum_{j\ge0}m_j=L$, by the choice of $s'$. +Choose a fixed integer $J_0$ such that +\[ + \sum_{j>J_0}r_0c_2s_j<\Delta/24. +\] +For sufficiently large $x$ we have $j_x>J_0$. Restrict all slots in +$0\le j\le J_0$, except the free slot in $Q_0$, to fixed small +neighborhoods of their respective midpoints, choosing the widths so +that their total logarithmic deviation is less than $\Delta/24$. +There are only finitely many restrictions, each with harmonic +probability bounded below by a positive constant by the prime number +theorem. Their joint probability is therefore bounded below. + +Independently, require $\mathcal E(R)=-1$ and +$\log R\le\Delta/12$. Equations~\eqref{wt:HP-bounds} and +\eqref{wt:odd-mass}, together with Lemma~\ref{wt:tail} applied to the +fixed positive number $\Delta/(12L)$, show that these two requirements +have probability at least $\delta_0/2$ for sufficiently large $x$. +No restrictions are needed on the remaining $Q$-slots. Their total +logarithm and the infinite midpoint sum for their bands both lie +between zero and $\Delta/24$. Consequently, if $B$ is the logarithm +of the product of all variables except the free prime, then on the +event just described +\[ + \left|B-(L-m_0)\right|<\Delta/6. +\] + +Choose $1<\alpha<\beta<2$ with +$\Psi(t)\ge c_\Psi>0$ throughout $[\alpha,\beta]$. +For the free prime it is enough to require +\[ + \log p\in[L+\log\alpha-B,\ L+\log\beta-B]. +\] +This interval has the fixed positive length $\log(\beta/\alpha)$. +Its endpoints lie strictly inside $[c_1s_0,c_2s_0]$ with distance +$\gg\Delta$ from the boundary, since $m_0$ is its midpoint and +$\Delta$ its full width. The prime number theorem and partial +summation give harmonic mass $\gg1/L$ for the interval, uniformly in +the conditioned variables. On it $U\ge\alpha x$ and +$\Psi(U/x)\ge c_\Psi$. Equation~\eqref{wt:expectation} now gives the +lower bound. Finally $H_Q\ge L^{-C}$ and $H_P\ge1$ imply the stated +fixed-power lower bound for $X_A$. +\end{proof} + +\subsection{Squarefree predecessors} + +\begin{lemma}[Squarefree loss]\label{wt:squarefree} +The mass of nonsquarefree predecessors satisfies +\[ + \sum_{\substack{u\ge1\\2u\ {\rm not\ squarefree}}} + A(u)\Psi(u/x)=o(X_A/L). +\] +\end{lemma} + +\begin{proof} +In one $Q$-band the probability that two given slots coincide is +\[ + \sum_{p\in Q_j}\frac1{p^2V_{Q,j}^2} + \le\frac{L^{-20}}{V_{Q,j}}\ll L^{-20}. +\] +The number of pairs of slots within bands is $O(T)$, so the +probability of any repeated $Q$-prime is $O(TL^{-20})$. + +For completeness, fix $p\in\mathcal P_g$ and write +$Z_{g,p}(q)=\prod_{q'\in\mathcal P_g,\ q'\ne p} +(1+q/(q'-1))$. The unnormalized mass of integers in this group that +are divisible by $p$ is +\[ + \frac{Z_{g,p}(q_0)+q_0Z_{g,p}'(q_0)}{V_g(p-1)}\ll\frac1p. +\] +The bound follows from the uniform group harmonic masses; division +by $H_g\ge1$ preserves it. Conditional on the set of distinct prime +divisors, the exponent of an occurring prime has distribution +proportional to $p^{-k}$, $k\ge1$. The conditional probability that +its exponent is at least two is therefore exactly $1/p$. +Thus the probability of a square from the group part is +\[ + \ll\sum_{p\in\mathcal P}\frac1{p^2} + \le e^{-L^{1/10}}\sum_gV_g + \ll T e^{-L^{1/10}}. +\] +All bands and groups are disjoint, and all their primes are odd. +It follows that $2U$ is not squarefree with probability +$O(TL^{-20})$ under the unrestricted harmonic measure. +Using Equation~\eqref{wt:expectation} and dropping parity and size +localization gives the upper bound +\[ + 2x\|\Psi\|_\infty H_QH_P\,O(TL^{-20}) + \ll xH_QH_PT L^{-20}. +\] +By Lemma~\ref{wt:mass}, its ratio to $X_A/L$ is +$O(TL^{-18})=o(1)$, as required. +\end{proof} diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/05-type-ii.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/05-type-ii.tex new file mode 100644 index 0000000000000000000000000000000000000000..49ee57b34c5a9e6c03f5d0b5675554d2e1bb58dd --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/05-type-ii.tex @@ -0,0 +1,551 @@ +\section{Bilinear distribution}\label{ti:section} + +We retain the groups, ordinary prime bands, and weights of the preceding +section. In particular, every ordinary slot contains a prime, there are +$r_0$ labelled slots in each $Q_j$, and +\[ + s_j=s'2^{-j}L,\qquad 20T\le s_{j_x}<40T,\qquad + \frac{(r_0-1)c_1}{2}\ge c_2+1. +\] +The smooth allocation and change of variables below follow the Type II +method of \cite{SmoothShifted}. We give the argument for the present +all-prime weight and its parity twist, using +Theorem~\ref{sh:correlation} for the resulting correlations. + +\begin{theorem}[Type II distribution]\label{ti:distribution} +Fix $D_*,b_*,C>0$. Let $U,V$ be dyadic scales such that +\[ + UV\asymp x,\qquad x^{b_*}\le U,V\le x^{1-b_*}, +\] +and let $K\subset[U,2U)$ and $J\subset[V,2V)$ be intervals. Suppose +$|\alpha_m|,|\beta_n|\le L^C$ and $\alpha_m=0$ unless +$P^-(m)>W$. There is a fixed $B$, depending only on these fixed data +and on the fixed weight and cutoff data, for which the following holds. +Assume that the sequence $\alpha_m\1_{m\in K}$ satisfies +\eqref{sh:discrepancy}; explicitly, assume +\begin{equation}\label{ti:actual-discrepancy} + \left|\sum_{m\in I}\alpha_m\1_{m\in K} + \chi(m)m^{-1+it}\right|\le L^{-B} + \quad\left( + \begin{array}{c} + I\subset[1,x^2]\text{ an interval},\\ + \operatorname{mod}(\chi)\le L^B,\quad |t|\le L^B + \end{array}\right). +\end{equation} +Then +\begin{equation}\label{ti:bound} + \sum_{\substack{m\in K,\ n\in J,\ u\ge1\\mn=2u+1}} + \alpha_m\beta_n A(u)\Psi(u/x) + \ll xL^{-D_*}. +\end{equation} +The estimate is uniform in the intervals and coefficient sequences +satisfying these assumptions. No discrepancy or smoothness assumption is +imposed on $\beta$. +\end{theorem} + +\begin{proof} +Extend the actually restricted sequences by zero, and write them again +as $\alpha$ and $\beta$. Thus $\alpha$ in every subsequent convolution +is the sequence appearing in \eqref{ti:actual-discrepancy}. By +$A=A_0(1-\mathcal E)/2$, it suffices to prove the assertion with +$A_0(u)\varepsilon(u)$, for each of the two choices +\[ + \varepsilon=1\quad\hbox{or}\quad + \varepsilon=\mathcal E. +\] +The choice will be the same at the two endpoints of each correlation. + +\paragraph{Removing a large group-prime part.} +Choose a fixed band index $j_{**}$ so large that +\begin{equation}\label{ti:designated-range} + \tau=c_1s'2^{-j_{**}}>0,\qquad + \eta=c_2s'2^{-j_{**}}<\min(b_*/4,1/4). +\end{equation} +For sufficiently large $x$, this band is present. Designate one of its +labelled slots, whose prime therefore lies in $[x^\tau,x^\eta]$. +This merely names an existing slot in the weight. + +Put $u_P=u/u_*$. We first discard $u_P>x^{b_*/4}$. For $u\asymp x$, +the part of $2u+1$ supported on primes exceeding $W$ has at most +$O(L/\log W)=O(\sqrt L)$ prime factors with multiplicity. Its number +of divisors is consequently at most $\exp(O(\sqrt L))$, since +$a+1\le2^a$ for $a\ge1$. This bounds the possible rough divisors $m$. +The tail bound in Lemma~\ref{wt:tail}, with $d=2/5$, gives +\begin{align} + &\sum_{\substack{mn=2u+1,\ u\asymp x\\u_P>x^{b_*/4}}} + |\alpha_m\beta_n|A_0(u)|\Psi(u/x)|\notag\\ + &\qquad\ll + xL^{O(1)}\exp(O(\sqrt L)) + \left(\sum_v\frac{a(v)}v\right) + \sum_{\substack{r>x^{b_*/4}\\r_*=1}}\frac{W_1(r)}r + \ll xL^{O(1)} + \exp\left(-\frac{b_*}{4}L^{1-d}+O(\sqrt L)\right). + \label{ti:large-group-error} +\end{align} +To obtain the first inequality, write $u=vr$, use $vr\asymp x$ to +bound $1\ll x/(vr)$, and then drop the product restriction. +The harmonic ordinary-slot mass is $H_Q=L^{O(1)}$. Since $1-d>1/2$, +\eqref{ti:large-group-error} is $O_H(xL^{-H})$ for every fixed $H$. +The same conclusion will hold whenever we reinstate these discarded +tuples with a multiplier of absolute value at most one. + +\paragraph{A smooth allocation with bounded residual.} +Set +\begin{equation}\label{ti:E} + E=UL^{-K_0}, +\end{equation} +where the fixed positive constant $K_0$ will be chosen below. Put the +entire group-prime part and the designated prime into $h$. Process all +remaining ordinary slots by increasing band index, in a fixed order +within each band, assigning each slot either to $e$ or to $h$. +Write these slots as $p_1,\ldots,p_N$, with band indices $j(i)$; +here $N=r_0(j_x+1)-1=O(T)$. + +Fix a smooth function $\gamma:\mathbb R\to[0,1]$ equal to zero on +$(-\infty,0]$ and to one on $[1,\infty)$. At slot $i$, with residual +target $R_i$, assign the prime to $e$ with weight +\[ + \gamma\left(\frac{R_i-\log p_i}{s_{j(i)}}\right) +\] +and to $h$ with the complementary weight. Start with +$R_1=\log E$, and subtract $\log p_i$ from the residual precisely +when the slot is assigned to $e$. + +For a tuple surviving the preceding deletion, the product withheld +from allocation is at most $x^{b_*/2}$. Since $u\asymp x$ and +$U\le x^{1-b_*}$, its available logarithms satisfy, for large $x$, +\[ + 0<\log E\le\sum_{i=1}^N\log p_i. +\] +Every branch of positive weight has the invariant +\begin{equation}\label{ti:allocation-invariant} + 0\le R_i\le\sum_{t=i}^N\log p_t+(c_2+1)s_{j_x} + \qquad(1\le i\le N+1). +\end{equation} +Indeed, taking a slot requires $R_i>\log p_i$, so preserves both +nonnegativity and the upper bound after subtracting that logarithm. +Skipping a slot in band $j$ requires +\[ + R_i<\log p_i+s_j\le(c_2+1)s_j. +\] +If $j40(c_2+1)$. Every surviving branch has +\begin{equation}\label{ti:e-range} + EL^{-C_s}\le e\le E. +\end{equation} +This choice of $C_s$ is independent of $K_0$. + +Choose a fixed smooth $\rho:\mathbb R\to[0,1]$ of compact support +which equals one throughout the possible argument interval, and put +\[ + f_1=\rho\gamma,\qquad f_0=\rho(1-\gamma). +\] +The allocation is unchanged on the retained tuples. On any tuple the +sum of its full branch weights is at most one, since +$f_0+f_1=\rho\le1$ at every node of the choice tree. We may consequently +impose \eqref{ti:e-range} and reinstate all discarded tuples, with the +same negligible error \eqref{ti:large-group-error}. + +\paragraph{Separation and its uniform cost.} +Use the Fourier convention +\[ + \widehat f(\xi)=\int_{\mathbb R}f(t)e^{-i\xi t}\,dt, + \qquad f(t)=\int_{\mathbb R}\widehat f(\xi)e^{i\xi t} + \frac{d\xi}{2\pi}, +\] +and set +\[ + C_F=\max\left(1, + \int|\widehat f_0(\xi)|\frac{d\xi}{2\pi}, + \int|\widehat f_1(\xi)|\frac{d\xi}{2\pi}\right). +\] +For a fixed branch $\delta=(\delta_1,\ldots,\delta_N)\in\{0,1\}^N$, +Fourier inversion of its $N$ transitions has total variation at most +$C_F^N$. Summing over the branches costs at most +\begin{equation}\label{ti:fourier-cost} + (2C_F)^N\le L^{C_f} +\end{equation} +for a fixed exponent, increased below to include the fixed cutoff +$\Psi$. In particular this exponent is independent of $K_0$. +Truncating each transition variable at $|\xi_i|\le L$ has total error +at most +\begin{equation}\label{ti:fourier-tail} + 2^N N C_M C_F^{N-1}L^{-M} +\end{equation} +for any fixed $M$, by the union bound on the complementary integration +regions and the rapid decay of a single Fourier factor. The constant +$C_M$ occurs only once in each term of this estimate. Increasing $M$ +therefore does not increase the exponent arising from $C_F^N$. + +To see the separated phases explicitly, use +$R_i=\log E-\sum_{tt}\frac{\xi_i}{s_{j(i)}}. +\end{equation} +The dependence on $E$ is entirely in the scalar of modulus one. +Since $\sum_i s_{j(i)}^{-1}=O(T^{-1})$, the retained frequencies +satisfy $|\sigma_t|=O(L/T)$. Fourier inversion of the fixed smooth +function $t\mapsto\Psi(e^t)$ likewise separates $\Psi(eh/x)$, +with bounded integral norm; truncation at a fixed power of $L$ has +arbitrary logarithmic accuracy. Its contribution is a factor +$h^{i\varrho}$, a factor $e^{i\varrho}$ absorbed into the coefficient +of $e$, and a scalar of modulus one. + +All these errors can be summed before Cauchy--Schwarz. The original +$m,n$ restrictions still imply $u\asymp x$, and the unmodified +factorization mass is bounded by +\[ + \sum_{\substack{mn=2u+1\\u\asymp x}} + |\alpha_m\beta_n|A_0(u) + \ll L^{O(1)}\sum_{v\asymp x}\tau(v) + \ll xL^{O(1)}. +\] +Here and below $\tau$ is the divisor function. The same estimate +applies to the labelled-slot sum with its original factorial +normalization. Combining it with \eqref{ti:fourier-tail} proves an +error $O_H(xL^{-H})$ for arbitrary fixed $H$, without changing the +fixed exponent in \eqref{ti:fourier-cost}. + +Consequently the required sum is, up to these errors, a sum and +integral of total variation at most $L^{C_f}$ of expressions +\begin{equation}\label{ti:separated-bilinear} + \begin{split} + B'&=\sum_{e,n}a_e(e)\beta_n + \sum_{mn=2eh+1}\alpha_m a_h(h),\\ + a_h(h)&=h^{i\varrho}\varepsilon(h)W_1(h)H(h_*),\qquad + H(t)=\sum_{pr=t}\1_{p\in Q_{j_{**}}}p^{i\sigma}c_r. + \end{split} +\end{equation} +The original $m,n$ restrictions are in $\alpha,\beta$, and the +$e$ range \eqref{ti:e-range} is in $a_e$. The prime in $H$ is the +designated prime. Its phase can be taken to be zero at this stage; +we allow a bounded-power frequency $\sigma$ in the notation. All +phases in \eqref{ti:separated-bilinear} have fixed logarithmic-power +bounds. + +Both $a_e$ and $c_r$ are bounded coefficients. In fact, for a fixed +branch and a fixed product, disjointness of the bands determines the +prime multiset in every band. There are at most $r_0!$ assignments to +the relevant labelled slots of any one band, even when primes repeat. +Distributing the original factorial normalizations between the two +coefficients only improves the resulting bound +\begin{equation}\label{ti:coefficient-bounds} + |a_e(e)|,|c_r|\le(r_0!)^{j_x+1}\le L^{C_a}, + \qquad |a_h(h)|\le L^{C_h}\tau(h_*). +\end{equation} +The last bound comes from summing over the designated prime divisor +of $h_*$. The exponents $C_a,C_h$ are fixed independently of $K_0$ +and of the Fourier error accuracy. Only ordinary-band primes were +assigned to $e$, so its removal leaves the marked group weight and +the sign exactly on $h$, as used in \eqref{ti:separated-bilinear}. + +\paragraph{Cauchy--Schwarz and the diagonal.} +Choose bounded nonnegative smooth majorants $\eta_e,\eta_n$, equal +to at least one on the $e,n$ ranges. We can arrange +\[ + \operatorname{supp}\eta_e\subset[\tfrac12L^{-C_s},2], + \qquad \operatorname{supp}\eta_n\subset[1/2,3], + \qquad + \|\eta_e^{(j)}\|_\infty+\|\eta_n^{(j)}\|_\infty + \ll_j L^{C_{\eta}(j+1)}, +\] +where $C_{\eta}$ depends only on $C_s$. There are $O(EV)$ pairs +$(e,n)$ in the original ranges. Hence \eqref{ti:coefficient-bounds} +and Cauchy--Schwarz give +\begin{equation}\label{ti:cauchy} + |B'|^2\ll EVL^{C_p}\mathcal N,\qquad + \mathcal N=\sum_{e,n}\eta_e(e/E)\eta_n(n/V) + \left|\sum_{mn=2eh+1}\alpha_m a_h(h)\right|^2. +\end{equation} +Here $C_p$ is fixed independently of $K_0$. + +The diagonal $m=m'$ in the square forces $h=h'$. Collecting by +$v=mn$ and then using \eqref{ti:coefficient-bounds} gives +\begin{align} + \mathcal N_{\mathrm{diag}} + &\ll L^{C_d}\sum_{\substack{v\asymp x\\v\ {\rm odd}}} + \tau(v)\sum_{e\mid(v-1)/2} + \tau\left(\frac{v-1}{2e}\right)^2\notag\\ + &\le L^{C_d}\sum_{v\asymp x}\tau(v)\tau(v-1)^3 + \ll xL^{C_d'}. + \label{ti:diagonal} +\end{align} +The final estimate follows from Cauchy--Schwarz and the fixed second +and sixth moments of $\tau$ in Proposition~\ref{pre:moments}. +Enlarging the $e$ divisor set in this +calculation makes $C_d'$ independent of $K_0$. Since +$EV\asymp xL^{-K_0}$, the diagonal contribution to the right side +of \eqref{ti:cauchy} is at most +$x^2L^{-K_0+C_p+C_d'}$. We now fix +\begin{equation}\label{ti:K-choice} + K_0>2(D_*+C_f+1)+C_p+C_d'. +\end{equation} +It remains to obtain an arbitrarily strong logarithmic saving for +the off-diagonal part of $\mathcal N$ with this fixed $K_0$. + +\paragraph{The off-diagonal change of variables.} +A pair of representations in the expanded square satisfies +\[ + mn=2eh+1,\qquad m'n=2eh'+1. +\] +The first identity gives $(n,2e)=1$. Subtracting the two identities +therefore gives a unique integer $k$ with +\begin{equation}\label{ti:determinant-forward} + m'-m=2ek,\qquad h'-h=kn,\qquad + z=m'h,\qquad z+k=mh'. +\end{equation} +Indeed $mh'-m'h=k(mn-2eh)=k$. On the off-diagonal $k\ne0$, and +the lower support bound for $e$ gives +\begin{equation}\label{ti:k-range} + 0<|k|\ll L^{K_0+C_s}. +\end{equation} + +For completeness, this parametrization is reversible for both signs +of $k$. Start from positive factorizations +$z=m'h$, $z+k=mh'$, require +$m'\equiv m\pmod{2|k|}$, and put +\begin{equation}\label{ti:determinant-reverse} + e=\frac{m'-m}{2k}>0,\qquad + n=\frac{2eh+1}{m}. +\end{equation} +The first quantity is an integer. The determinant identity implies +\[ + m(h'-h)=k(2eh+1). +\] +Since $P^-(m)>W$ and $k$ has the fixed logarithmic-power bound +\eqref{ti:k-range}, we have $(m,k)=1$ for large $x$. Thus $n$ is +a positive integer and $h'-h=kn$. Substituting $m'=m+2ek$ recovers +$m'n=2eh'+1$. Every reconstructed variable is unique. Applying the +size cutoffs in \eqref{ti:cauchy} therefore gives an exact bijection, +with no additional multiplicity or divisibility condition. + +Both $m,m'$ are odd and coprime to $k$. With $q_k=2|k|$, their +congruence is imposed exactly by +\begin{equation}\label{ti:character-orthogonality} + \1_{m'\equiv m\ ({\rm mod}\ q_k)} + =\frac1{\varphi(q_k)}\sum_{\chi\ ({\rm mod}\ q_k)} + \chi(m)\overline{\chi(m')}. +\end{equation} +The normalized character sum has total coefficient mass one. + +\paragraph{The pulled-back cutoffs.} +On the supports above, $h\asymp x/e$ and $m'\asymp U$. Thus +$z\asymp Ux/e$ lies between fixed constant multiples of +$xL^{K_0}$ and $xL^{K_0+C_s}$. Insert a smooth dyadic partition +in $z$, retaining an outer cutoff $\psi_Z(z/Z)$ in every piece. +Only $O(1+T)$ dyads are needed, and all have $Z=xL^{O(1)}$. + +For a fixed $k,Z$, put +\[ + t_1=\log(m/U),\quad t_2=\log(m'/U),\quad t_3=\log(z/Z). +\] +The arguments of the two majorants are exactly +\begin{equation}\label{ti:pulled-cutoffs} + \begin{split} + \frac eE + &=\frac{m'-m}{2kE} + =\frac{U}{2kE}(e^{t_2}-e^{t_1}),\\ + \frac nV + &=\frac1V\left(\frac{(m'-m)z}{kmm'}+\frac1m\right)\\ + &=\frac{Z}{kUV}e^{t_3}(e^{-t_1}-e^{-t_2}) + +\frac1{UV}e^{-t_1}. + \end{split} +\end{equation} +On a fixed compact box in these coordinates, every log-coordinate +derivative of the first expression is +$O_j(U/(|k|E))$, and every derivative of the second is +\[ + O_j\left(\frac{Z}{|k|UV}+\frac1{UV}\right). +\] +All these quantities are bounded by fixed powers of $L$ after +\eqref{ti:K-choice}. The chain rule and the derivative bounds for +$\eta_e,\eta_n$ therefore give, for each multi-index $\nu$, +\begin{equation}\label{ti:pulled-derivatives} + \|\partial^\nu b_{k,Z}\|_\infty + \ll_\nu L^{C_0(|\nu|+1)} +\end{equation} +for the pulled-back product of majorants, multiplied by fixed smooth +buffers in $t_1,t_2,t_3$. Choose these buffers to equal one on the +original $m,m'$ dyads and on the support of the outer $z$ cutoff. +This places $b_{k,Z}$ on a fixed compact box and leaves the sequence +$\alpha$ unchanged. The majorants vanish in neighborhoods of +nonpositive $e/E$ and $n/V$, so extending the product by zero across +those regions is smooth. Positivity in +\eqref{ti:determinant-reverse} is thereby imposed by the same cutoffs. + +This is the fixed-dimensional setting of Lemma~\ref{pre:separation}. +Choose a fixed $C_8$ larger than the slope required in +\eqref{ti:pulled-derivatives}, and put $P=L^{C_8}$. Repeated +integration by parts in these three coordinates gives +\[ + |\widehat b_{k,Z}(\boldsymbol\xi)| + \ll_j(1+|\boldsymbol\xi|/P)^{-j},\qquad + \int_{\mathbb R^3}|\widehat b_{k,Z}(\boldsymbol\xi)| + \,d\boldsymbol\xi\ll P^3, +\] +and +\begin{equation}\label{ti:post-fourier-tail} + \int_{|\boldsymbol\xi|>PL}|\widehat b_{k,Z}(\boldsymbol\xi)| + \,d\boldsymbol\xi\ll_j P^3L^{3-j}. +\end{equation} +Consequently the cutoffs separate into phases +$m^{i\xi_1}(m')^{i\xi_2}z^{i\xi_3}$ at fixed logarithmic-power +cost and with fixed logarithmic-power frequencies. Scalars involving +$U$ and $Z$ have modulus one. + +To justify the tails absolutely, an endpoint convolution is bounded +by $L^{O(1)}\tau(v_*)^2$: there are at most +$\tau_3(v_*)\le\tau(v_*)^2$ triples $mpr=v_*$. Including the bounded +group weights and dropping the congruence, Cauchy--Schwarz and the +fixed fourth divisor moment in Proposition~\ref{pre:moments} give +\begin{equation}\label{ti:absolute-correlation} + L^{O(1)}\sum_{z\asymp Z}\tau(z)^2\tau(z+k)^2 + \ll ZL^{O(1)}. +\end{equation} +Here $|k|\ll L^{O(1)}=o(Z)$ and both positions are positive. +This estimate controls \eqref{ti:post-fourier-tail}, after summing +over all $k$ and $Z$, to arbitrary logarithmic accuracy by increasing +$j$. The fixed exponent $C_8$ need not increase with that accuracy. + +\paragraph{Identifying the two endpoints.} +Fix one character in \eqref{ti:character-orthogonality} and one +retained Fourier term. Before the character and Fourier factors, the +coefficient from the expanded square is +\[ + \alpha_m\overline{\alpha_{m'}} + a_h(h)\overline{a_h(h')}. +\] +Define invariant cores, for positive integers $v$, by +\begin{equation}\label{ti:endpoint-cores} + \begin{split} + F_0(v)&=\sum_{mpr=v_*}\alpha_m\chi(m) + m^{i(\varrho-\xi_2)} + \1_{p\in Q_{j_{**}}}p^{-i\sigma}\overline{c_r},\\ + G_0(v)&=\sum_{mpr=v_*}\alpha_m\chi(m) + m^{i(\xi_1+\varrho)} + \1_{p\in Q_{j_{**}}}p^{-i\sigma}\overline{c_r}. + \end{split} +\end{equation} +Both cores contain the original restricted sequence $\alpha$. +All group primes are below $W$, whereas $m,m'$ have no prime factor +at most $W$. Therefore +\begin{equation}\label{ti:group-identities} + \begin{gathered} + z_*=m'h_*,\qquad (z+k)_*=mh'_*,\\ + W_1(z)=W_1(h),\quad W_1(z+k)=W_1(h'),\qquad + \varepsilon(z)=\varepsilon(h),\quad + \varepsilon(z+k)=\varepsilon(h'). + \end{gathered} +\end{equation} +Using also $h=z/m'$ and $h'=(z+k)/m$, direct expansion shows that +the separated sum is precisely +\begin{equation}\label{ti:separated-correlation} + \sum_{\substack{z>0\\z+k>0}} + \psi_Z(z/Z)z^{i(\varrho+\xi_3)}(z+k)^{-i\varrho} + \overline{F_0(z)}G_0(z+k) + W_1(z)W_1(z+k)\varepsilon(z)\varepsilon(z+k). +\end{equation} +For example, conjugating the first core supplies +$\overline{\alpha_{m'}}\overline{\chi(m')} +(m')^{i(\xi_2-\varrho)}H(h_*)$, while the second core supplies +$\alpha_m\chi(m)m^{i(\xi_1+\varrho)}\overline{H(h'_*)}$. +The remaining powers in \eqref{ti:separated-correlation} give exactly +$h^{i\varrho}(h')^{-i\varrho}z^{i\xi_3}$. +Thus the discrepancy-bearing first core has the required coefficient +$\alpha$, with its conjugation occurring outside that core. + +Every hypothesis of Theorem~\ref{sh:correlation} is now satisfied. +The designated slot is a prime in the fixed range +\eqref{ti:designated-range}; the residual coefficients are bounded +by \eqref{ti:coefficient-bounds}; the character modulus $2|k|$ and +all frequencies have fixed logarithmic-power bounds; and the two +endpoints have the same allowed sign. Finally, set +\[ + \widetilde\psi_Z(t)=t\psi_Z(t). +\] +Then $\psi_Z(z/Z)=(Z/z)\widetilde\psi_Z(z/Z)$. Consequently +\eqref{ti:separated-correlation} is exactly $Z$ times a harmonic +correlation covered by that theorem. For any prescribed $D_0$ it is +therefore +\begin{equation}\label{ti:correlation-bound} + O(ZL^{-D_0}), +\end{equation} +provided the discrepancy exponent $B$ is chosen sufficiently large +after all the fixed bounds and $D_0$. + +\paragraph{Completing the parameter choices.} +Here is an explicit ordering of the choices. First fix the weight +geometry, $b_*,D_*,C$, the designated band, the allocation functions, +and the cutoff. These determine the exponents +\[ + C_s,\ C_a,\ C_h,\ C_f,\ C_p,\ C_d'. +\] +Fix $K_0$ by +\eqref{ti:K-choice}. This fixes the derivative and frequency bounds +after Cauchy--Schwarz, the Fourier scale $P$, and exponents +$K_k,K_z,C_{\rm sep}$ such that there are at most $L^{K_k}$ shifts, +$Z\le xL^{K_z}$, and the total post-Cauchy separation cost, including +the dyads, is at most $L^{C_{\rm sep}}$. These exponents are all fixed. + +Choose $A_{\rm off}>0$ so that +\[ + -K_0+C_p-A_{\rm off}<-2(D_*+C_f+1), +\] +and then choose +\[ + D_0>A_{\rm off}+K_k+K_z+C_{\rm sep}+2. +\] +Choose the differentiation orders in the Fourier tail estimates +large enough for the same error target. +Theorem~\ref{sh:correlation} now supplies a sufficiently large fixed $B$. +Summing \eqref{ti:correlation-bound} over shifts, dyads, characters, +and the retained Fourier integrals gives +\[ + |\mathcal N_{\rm off}| + \ll xL^{-A_{\rm off}}. +\] +Together with \eqref{ti:diagonal}, \eqref{ti:K-choice}, and +\eqref{ti:cauchy}, this proves +$|B'|\ll xL^{-D_*-C_f-1}$. The total pre-Cauchy variation +$L^{C_f}$ and the arbitrarily small previously discarded errors +prove \eqref{ti:bound} for each choice of $\varepsilon$, and hence +for $A$. +\end{proof} + +\begin{remark}[Precision in subsequent proxy replacements] +\label{ti:proxy-order} +Theorem~\ref{ti:distribution} permits the actual interval restriction +to be included in $\alpha$, and it asks only for boundedness of the +other coefficient. Thus it can be applied a second time after +exchanging the two factors, even if the first coefficient has already +been replaced by a nonsmooth bounded proxy. On $O(L)$ contributing +factor dyads, its total error is $O(xL^{-D_*+1})$. The lower bound +$X_A\gg xL^{-C_A}$ in Lemma~\ref{wt:mass} makes this +$o(X_A/L)$ whenever $D_*>C_A+2$. + +In particular, if a later proxy construction has coefficients bounded +by one independently of its logarithmic cell precision $S_1$, use +that coefficient bound when making all the choices above. First fix +$K_0$, the shifted-correlation accuracy, and finally the discrepancy +exponent $B$; only afterwards choose the proxy precision +$S_1>2B+3$. Neither $C_f$ nor the coefficient or diagonal exponents +depends on $S_1$. Increasing $S_1$ therefore changes only how large +$x$ must be and does not reopen any earlier choice. +\end{remark} diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/06-congruences.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/06-congruences.tex new file mode 100644 index 0000000000000000000000000000000000000000..c6ec62011352eea453f35a1928dff7bdf6eb9e44 --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/06-congruences.tex @@ -0,0 +1,207 @@ +\section{Distribution in congruence classes}\label{cg:section} + +For an odd squarefree integer $d$ define +\begin{equation}\label{cg:remainder} + r(d)=\sum_{\substack{u\ge1\\2u+1\equiv0\pmod d}} + A(u)\Psi(u/x)-\frac{X_A}{\varphi(d)}. +\end{equation} +The following estimate supplies the distribution needed by the sieve. +Its proof uses only the prime slot in $Q_0$, the weight bounds already +proved, and the classical Siegel--Walfisz and multiplicative large +sieve estimates in +\cite[Section ``Notation and preliminary estimates'']{SmoothShifted}. + +\begin{theorem}[Congruence distribution]\label{cg:distribution} +For every fixed $0<\vartheta<1/2$ and $D>0$, +\[ + \sum_{\substack{d\le x^\vartheta\\d\ {\rm odd\ and\ squarefree}}} + |r(d)|\ll_{\vartheta,D}xL^{-D}+X_A L^{-18}. +\] +The constants may depend on the fixed weight geometry and $\Psi$, +but not on any later proxy precision. No effective bound on the +threshold for $x$ is asserted. +\end{theorem} + +\begin{proof} +Write $Q=x^\vartheta$ and $w(u)=A(u)\Psi(u/x)$. The congruence in +Equation~\eqref{cg:remainder} is the single unit class +$a_d\equiv-2^{-1}\pmod d$. Character orthogonality gives the exact +identity +\begin{equation}\label{cg:characters} + r(d)=\frac1{\varphi(d)} + \sum_{\substack{\chi\bmod d\\\chi\ne\chi_0}} + \overline{\chi(a_d)}\sum_u w(u)\chi(u) + -\frac1{\varphi(d)}\sum_{(u,d)>1}w(u). +\end{equation} +We first bound the correction in the second term. Mertens' estimate +implies +\begin{equation}\label{cg:totient-sum} + \sum_{h\le Q}\frac{\mu^2(h)}{\varphi(h)} + \le\prod_{p\le Q}\left(1+\frac1{p-1}\right)\ll L. +\end{equation} +For a fixed $u$ in the effective support of $w$, we have +$x\le u\le2x$ and every prime divisor of $u$ is at least $L^{20}$. +Writing a squarefree $d$ divisible by $p$ as $ph$, and then using a +union bound, gives +\[ + \sum_{\substack{d\le Q\\d\ {\rm squarefree}\\(d,u)>1}} + \frac1{\varphi(d)} + \le\sum_{p\mid u}\frac1{p-1} + \sum_{h\le Q}\frac{\mu^2(h)}{\varphi(h)} + \ll L\sum_{p\mid u}\frac1{p-1}\ll L^{-18}. +\] +For the last inequality, $u$ has at most +$\log(2x)/(20T)$ distinct prime divisors. Since $w\ge0$, the total +principal correction is therefore $O(X_A L^{-18})$. + +To treat the nonprincipal characters, delete one designated ordered +slot of $Q_0$ from the definition of $a$. Retain the original divisor +$(r_0!)^{j_x+1}$ in the resulting coefficient, denoted by $a^-(v)$. +If the residual exponents are $e_p$, then +\[ + a^-(v)=\frac1{r_0}\prod_{j=0}^{j_x}\prod_{p\in Q_j}\frac1{e_p!} + \le\frac1{r_0}, +\] +with $r_0-1$ slots in $Q_0$ and $r_0$ slots in every other band, and +$a^-(v)=0$ off this support. Partitioning ordered tuples by the +deleted prime gives, including when primes repeat, +\[ + a(v)=\sum_{\substack{p\in Q_0\\p\mid v}}a^-(v/p). +\] +The $Q_0$-primes lie outside $\mathcal P$, so removing that slot +changes neither $W_1$ nor $\mathcal E$. Consequently +\begin{equation}\label{cg:prime-convolution} + A(u)=\sum_{\substack{pr=u\\p\in Q_0}}b(r), + \qquad + b(r)=W_1(r)\frac{1-\mathcal E(r)}2a^-(r_*), + \qquad |b(r)|\le L^{B_0}, +\end{equation} +where $B_0$ is fixed by the weight geometry. + +Split $p,r$ into dyads of sizes $P,R$. Only $O(L)$ pairs of dyads +contribute, and on these $PR\asymp x$. Since +$x^{c_1s'}\le p\le x^{c_2s'}$, there is a fixed $c>0$, depending +only on Equation~\eqref{wt:geometry}, such that +\begin{equation}\label{cg:factor-ranges} + P,R\ge x^c +\end{equation} +for every contributing pair and all sufficiently large $x$. +For example any $c<\min(c_1s',1-c_2s')$ suffices. + +Every character modulo an odd squarefree $d$ is induced by a unique +primitive character $\chi_f$ of conductor $f\mid d$. Write $d=fh$; +then $(f,h)=1$ and $\varphi(d)=\varphi(f)\varphi(h)$. For such an +induced character, +\[ + \chi(pr)=\chi_f(p)\chi_f(r) + \1_{\{(p,h)=1\}}\1_{\{(r,h)=1\}}. +\] +The nonprincipal characters have $f>1$; there is no multiplicity in +this parametrization. After applying the triangle inequality in +Equation~\eqref{cg:characters}, it remains to bound sums of the form +\begin{equation}\label{cg:primitive-sum} + \begin{split} + \sum_{\substack{h\le Q\\h\ {\rm odd\ and\ squarefree}}} + \frac1{\varphi(h)} + \sum_{\substack{1L^K$. Put $\psi(v)=\Psi(e^v)$ and +$\widehat\psi(t)=\int_{\mathbb R}\psi(v)e^{-itv}\,dv$. +Fourier inversion gives +\[ + \Psi(pr/x)=\frac1{2\pi}\int_{\mathbb R} + \widehat\psi(t)x^{-it}p^{it}r^{it}\,dt, + \qquad \int_{\mathbb R}|\widehat\psi(t)|\,dt\ll_\Psi1. +\] +There is no frequency truncation error. For fixed $h,t$, define +\[ + \mathcal P_h(\chi,t)= + \sum_{\substack{p\in Q_0,\ p\asymp P\\(p,h)=1}}\chi(p)p^{it}, + \qquad + \mathcal R_h(\chi,t)= + \sum_{\substack{r\asymp R\\(r,h)=1}}b(r)\chi(r)r^{it}. +\] +Their coefficient squared norms are at most $O(P)$ and +$O(RL^{2B_0})$, respectively, uniformly in $h,t$. On a conductor +dyad $FL^K$, so their lower endpoints are at least $L^K/2$. +In particular the summation over $h$ costs a logarithmic factor, +not a factor $Q$. Taking $K>B_2+D+2$ makes +Equation~\eqref{cg:large-conductors} $O(xL^{-D})$, because +$\vartheta<1/2$. + +It remains to consider $1B_3+K+D+2$. Equation~\eqref{cg:small-conductors} is then +$O(xL^{-D})$. Together with the large-conductor estimate and the +principal correction, this proves the theorem. +\end{proof} diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/07-extraction.tex b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/07-extraction.tex new file mode 100644 index 0000000000000000000000000000000000000000..c502440dc9e099bcd1483700e7deb06e00cc1de9 --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/build/sections/07-extraction.tex @@ -0,0 +1,677 @@ +\section{Prime extraction}\label{ex:section} + +Put $N_u=2u+1$ and $a_x(u)=A(u)\Psi(u/x)$, so that +$a_x(u)\ge0$ and $X_A=\sum_u a_x(u)$. Throughout this section the +band geometry, the even integer $r_0$, the damping $q_0=1/2$, and +$\Psi$ are the fixed data of the weight construction. In particular, +Lemma~\ref{wt:mass} gives +\begin{equation}\label{ex:mass-reminder} + X_A\asymp\frac{x}{L}H_QH_P,\qquad X_A\gg xL^{-C_A},\qquad H_P\ge1. +\end{equation} +We use the Type II and congruence distribution results proved above, +together with the following explicitly stated sieve and proxy inputs. + +\subsection{Sieve and proxy inputs} + +\begin{lemma}[Block sieve, imported]\label{ex:block-sieve} +Consider finitely many objects with nonnegative weights and, at some +primes $p\le z$, designated bad conditions. Suppose that for every +squarefree product $d$ of these primes, including $d=1$, the weight +on which all conditions indexed by $p\mid d$ hold is +$Xg(d)+r(d)$. Here $X\ge0$, $g$ is multiplicative, and, for fixed +$\eta_0>0$ and $C$, one has +\[ + 0\le g(p)\le1-\eta_0,\qquad + \sum_{w1). +\] +For every sufficiently large even integer $h$, the weight avoiding +all the bad conditions equals +\begin{equation}\label{ex:block-formula} + X\prod_{p\le z}(1-g(p))(1+O(e^{-h})) + +O\!\left(\sum_{\substack{d\le z^{4h+2}\\d\text{ squarefree}}} + |r(d)|\right). +\end{equation} +The products and sums use only the designated primes. Constants +depend only on the two density bounds, and are uniform when $h$ +grows. For $h=2$ there is an upper bound by a fixed constant times +the displayed main term, plus the same remainder sum. + +The upper and lower inclusion--exclusion polynomials have +coefficients of absolute value at most one, supported on squarefree +$d\le z^{4h+2}$. The upper polynomial is nonnegative. Its overcount +is bounded by the nonnegative difference between the upper and +lower polynomials; that difference also has coefficients of absolute +value at most one and the same support bound. +\end{lemma} + +This is the Block sieve, Lemma~2.9 of \cite{SmoothShifted}. Its +coefficient bound, its growing-$h$ uniformity, and its fixed-depth +upper bound will each be used below. No sieve assertion about the +present weight is included in the import: its required remainder +estimates come from Theorem~\ref{cg:distribution} or from the explicit +box count below. + +Define +\begin{equation}\label{ex:euler-products} + \begin{split} + V(y)&=\prod_{p\le y}(1-1/p),\qquad + V_1(y)=\prod_{3\le p\le y}(1-1/(p-1)),\\ + \mathfrak S&=2\prod_{p\ge3}\left(1-\frac1{(p-1)^2}\right)>0. + \end{split} +\end{equation} +Mertens' estimate and the convergent quotient product give +\begin{equation}\label{ex:mertens} + V(y)\sim\frac{e^{-\gamma_E}}{\log y},\qquad + V_1(y)\sim\mathfrak S V(y), +\end{equation} +where $\gamma_E$ is Euler's constant. + +Here are the proxy estimates we use. For the moment let +$0<\kappa<1/50$ and $0x^\gamma}. +\] +For a fixed precision $S_1>0$, partition the logarithmic axis into +cells of width $\Delta_L=L^{-S_1}$. On every full cell +$C_Y=(Y,e^{\Delta_L}Y]$ meeting the ambient interval, put +\begin{equation}\label{ex:proxy-definition} + c_I(C_Y)= + \frac{\sum_{m\in C_Y}I(m)} + {\#\{m\in C_Y:P^-(m)>W\}},\qquad + B_I(m)=c_I(C_Y)\1_{P^-(m)>W}\quad(m\in C_Y). +\end{equation} +Even if a later factor range cuts a cell, both counts in this +definition are taken on the full cell. Range restrictions are +applied only after the ratio has been defined. + +\begin{lemma}[Proxy estimates, imported]\label{ex:proxies} +For sufficiently large $x$, the denominators in +\eqref{ex:proxy-definition} are positive and $0\le c_I\le1$. +For every fixed $B>0$, every fixed $S_1>2B+3$, and every interval +$K$ inside a dyad in the stated factor ranges, the sequence +\[ + \alpha_m=(I(m)-B_I(m))\1_{m\in K} +\] +vanishes outside $P^-(m)>W$, has absolute value at most one, and +satisfies +\begin{equation}\label{ex:proxy-discrepancy} + \left|\sum_{m\in J}\alpha_m\chi(m)m^{-1+it}\right|\le L^{-B} +\end{equation} +for every interval $J\subset[1,x^2]$, every Dirichlet character +of modulus at most $L^B$, and every $|t|\le L^B$. + +Write +\[ + \mathcal B_\kappa=[x^{1/2-2\kappa},x^{1/2+2\kappa}]. +\] +There is an absolute constant $C_{\mathrm{pr}}$ such that +\begin{equation}\label{ex:prime-density} + c_{\mathrm{pr}}(m)\le\frac{C_{\mathrm{pr}}}{LV(W)} + \qquad(m\in\mathcal B_\kappa). +\end{equation} +The constant is independent of $\kappa$ and $S_1$; the threshold +for $x$ may depend on all fixed parameters. + +For $w>\gamma>0$, set +\begin{equation}\label{ex:rough-density} + D_\gamma(w)=\frac1w+ + \sum_{l\ge2}\frac1{l!} + \int_{\substack{t_i\ge\gamma\ (1\le i\gamma>0$, including the +thresholds $w=l\gamma$. If $mn=N_u$ on the support of $\Psi(u/x)$ +and $\log n/L\in(\gamma,\gamma']$, then +\begin{equation}\label{ex:rough-proxy-bound} + c_\gamma(m)\le + \frac{\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t)+o(1)}{LV(W)}. +\end{equation} +Here the parameters are restricted to a compact subset of +$w>\gamma>0$, as holds for the bins under consideration. For fixed +sufficiently small $b>0$, +\begin{align} + \int_b^{1/2}D_\alpha(1-\alpha)\frac{d\alpha}{\alpha} + &=D_b(1)-1,\label{ex:density-integral}\\ + D_b(1)&\le\frac{e^{-\gamma_E}}b(1+O(e^{-c/b})) + \label{ex:density-upper} +\end{align} +with an absolute $c>0$. The endpoint of the integral is interpreted +by its left limit. +\end{lemma} + +The discrepancy assertion and the density assertions are, +respectively, Lemmas~7.3 and~7.4 of \cite{SmoothShifted}, in the +section ``Extracting primes with smooth predecessors.'' These +results concern ordinary integer cells; their hypotheses contain +no candidate weight. In particular they apply to the present $A$. +The exact bound $c_I\le1$ also follows directly because each test +selects a subset of the $W$-rough integers in its cell. Thus the +coefficient bounds of both $B_I$ and $I-B_I$ do not depend on the +precision $S_1$. In \eqref{ex:rough-proxy-bound}, +\[ + \frac{\log m}{L}=1-\frac{\log n}{L}+O(1/L),\qquad + \frac{\log m}{L}-\gamma\ge2\kappa+O(1/L), +\] +which verifies the required separation from $w=\gamma$. Every +little-oh assertion in the proxy estimates is taken only after its +parameters have been fixed. + +\subsection{A uniform positive pair bound} + +We first prove an upper bound that uses only the original weight +geometry. This establishes the constant needed to choose $\kappa$ +without referring to $b_1$, the Type II discrepancy exponent, or +the eventual proxy precision. + +Choose a sufficiently large fixed integer $j'$ and a fixed $\theta$ +in the gap +\begin{equation}\label{ex:micro-choice} + c_2s'2^{-j'}<\theta0$ and a compatible +choice of ordered micro entries and marks. The band gap in +\eqref{ex:micro-choice} implies that all non-$\mathcal P$ prime +factors of $u$ below $x^\theta$ are among these micro entries, +whereas every macro prime exceeds $x^\theta$. All group primes +are below $x^\theta$ for sufficiently large $x$. The only random +survival conditions are therefore at the distinct unmarked +group-prime divisors of $u$. There are exactly +$\omega_P(u)-s_P$ of them, so the survival probability is +$q_0^{\omega_P(u)-s_P}$, even when marked or unmarked primes +occur to higher powers. + +Summing one mark per group with normalization $\prod_g V_g^{-1}$ +recovers the factor $W_1(u)$. For a fixed prime multiset in any +band, the number of ordered $r_0$-tuples divided by $r_0!$ is +$1/\prod_p v_p!\le1$. Summing the normalized compatible micro +lists retains their exact contribution to $a(u_*)$; dropping the +remaining macro-list contribution can only increase it. Thus +the compatible assignments already dominate $A_0(u)$. +Incompatible assignments contribute nonnegative terms, proving +\eqref{ex:positive-majorant}. +\end{proof} + +\begin{lemma}[Positive dyadic pair estimate]\label{ex:box} +There is a constant $C_{\mathrm{box}}$, depending only on the +fixed weight data and the choices in \eqref{ex:micro-choice}, such +that, on every dyadic box with $M_1M_2\asymp x$ and +$M_i\ll x^{0.54}$, +\begin{equation}\label{ex:box-bound} + \sum_{\substack{M_1\le m<2M_1,\ M_2\le n<2M_2\\ + mn=N_u,\ P^-(m),P^-(n)>W}} + a_x(u)\le C_{\mathrm{box}}X_AV(W)^2. +\end{equation} +The fixed constants in the size comparisons can be taken to cover +all boxes meeting $mn=2u+1$ with $u\in[x,2x]$. +\end{lemma} + +\begin{proof} +Use Lemma~\ref{ex:mask} and bound $\Psi$ by its fixed supremum. +Fix an assignment and a mask. Since $D_1$ is odd, $D_1\mid u$ +implies $mn\equiv1\pmod{D_1}$. We count all pairs in the box with +this congruence, dropping both the parity of $mn$ and the remaining +support restriction on $u$. + +For every odd prime $l\le x^\theta$ not dividing $D_1$, impose +the bad condition $mn\equiv0\pmod l$ when $l\le W$, and the bad +condition $mn\equiv1\pmod l$ when $l$ is banned. There are +$2l-1$ pairs of residues with product zero and $l-1$ with product +one; the two sets are disjoint. Hence the local number and density +of bad residue pairs are +\begin{equation}\label{ex:local-density} + \nu(l)=(2l-1)\1_{l\le W}+(l-1)\1_{l\text{ banned}}, + \qquad g(l)=\frac{\nu(l)}{l^2}. +\end{equation} +In particular $g(l)\le7/9$ and $g(l)\le3/l$. These give uniform +density hypotheses for Lemma~\ref{ex:block-sieve}, independent of +the assignment and mask. + +There are exactly $\varphi(D_1)$ residue pairs with product one +modulo $D_1$: choose the first unit and then its unique inverse. +This statement does not require $D_1$ to be squarefree. For a +squarefree $d$ built from the sieving primes, the Chinese remainder +theorem gives exactly $\varphi(D_1)\nu(d)$ pairs modulo $D_1d$, +where $\nu$ is multiplicative. Each pair of classes contributes +\[ + \frac{M_1M_2}{(D_1d)^2} + +O\left(1+\frac{M_1+M_2}{D_1d}\right). +\] +Thus the base mass is +$M_1M_2\varphi(D_1)/D_1^2$, with local density $g(d)$. +Since $\nu(d)\le d\tau(d)^2$, the lattice remainders satisfy +\[ + |r_a(d)|\ll\tau(d)^2\{M_1+M_2+D_1d\}. +\] +Use the $h=2$ upper bound in Lemma~\ref{ex:block-sieve}, whose +level is $D=(x^\theta)^{10}$. Fixed divisor-moment bounds give +\begin{equation}\label{ex:box-error} + \sum_{d\le D}|r_a(d)| + \ll L^{C_0}\{(M_1+M_2)D+D_1D^2\} +\end{equation} +for a fixed $C_0$. This includes $d=1$ and is uniform in the mask. + +We next compare the sieve product with Mertens products. For +$l\le W$ banned by the mask, its factor is +$1-3/l+2/l^2$; for an allowed $l\le W$ it is $(1-1/l)^2$. +For $WW}}a_x(u) + \le C_1(\kappa L+1)X_AV(W)^2. +\end{equation} +There are $O(\kappa L+1)$ first-factor dyads because the interval +has logarithmic length $4\kappa L$, and only $O(1)$ second-factor +dyads per first-factor dyad because $mn\in[2x+1,4x+1]$. +Enlarging partially intersected boxes is legitimate by positivity. +For $\kappa<1/50$ all their sides are $O(x^{0.54})$. +Consequently $C_1$ is independent of $\kappa,b_1,S_1$. +Fix, at this point, the constant +\begin{equation}\label{ex:balanced-constant} + C_{\mathrm{bal}}=C_{\mathrm{pr}}^2C_1, +\end{equation} +enlarging it if necessary to absorb the fixed dyadic conventions. + +\subsection{Parameter choice and test replacement} + +Choose $0<\kappa<1/50$ so small that +\begin{equation}\label{ex:kappa-choice} + C_{\mathrm{bal}}\kappa<\mathfrak S/4. +\end{equation} +Next choose $b_1>0$ sufficiently small, with $b_1<1/100$, to +satisfy the presieve and subtraction conditions below. The +constants in those conditions do not depend on a later proxy +precision. Set +\[ + \vartheta=\frac12-\frac\kappa2,\qquad b_* = b_1/3. +\] +By continuity in Lemma~\ref{ex:proxies}, a sufficiently fine fixed +partition of $(b_1,1/2-\kappa]$ satisfies +\begin{equation}\label{ex:mesh} + \sum_{(\gamma,\gamma']} + \sup_{\gamma\le t\le\gamma'}D_\gamma(1-t) + \log(\gamma'/\gamma) + \le D_{b_1}(1)-1+\frac1{10}. +\end{equation} +In fact these sums converge to +$\int_{b_1}^{1/2-\kappa}D_\alpha(1-\alpha)\,d\alpha/\alpha$: +on this fixed compact range, replacing $\gamma$ and $t$ by the +same point changes the continuous integrand uniformly by a +quantity tending to zero with the mesh. Positivity and +\eqref{ex:density-integral} then give \eqref{ex:mesh}. + +Prescribe a Type II saving $D_*>C_A+4$ and a congruence saving +$D_{\mathrm{cg}}>C_A+4$. Apply Theorem~\ref{ti:distribution} with +$b_*$ and coefficient bound one, obtaining its required fixed +discrepancy exponent $B$. Only now fix $S_1>2B+3$ and define the +proxies by \eqref{ex:proxy-definition}. All these parameters, +including any prime number theorem accuracies in the proxy inputs, +are fixed before $x$ tends to infinity. + +\begin{lemma}[Replacement on the required factor ranges] +\label{ex:replacement} +In all factor sums used below, a test $I$ on either factor of +$mn=N_u$ can be replaced by $B_I$ with total error $o(X_A/L)$, +provided the coefficient on the other factor has absolute value at +most one. This holds also when that other coefficient is a proxy. +\end{lemma} + +\begin{proof} +For the unbalanced sums, the prime factor $n$ is in +$(x^{b_1},x^{1/2-\kappa}]$, and the complementary factor lies +between fixed positive multiples of $x^{1/2+\kappa}$ and +$x^{1-b_1}$. For the balanced sums both factors are in +$\mathcal B_\kappa$. Thus every contributing factor dyad, after +intersecting with its actual interval, has scale between +$x^{b_1/3}$ and $x^{1-b_1/3}$ for sufficiently large $x$. +The actual factor ranges also lie inside the ambient proxy +interval. The fixed gaps in these exponent inequalities absorb +all dyadic endpoint constants. + +On a dyad use $(I-B_I)\1_K$ as the first sequence in +Theorem~\ref{ti:distribution}, where $K$ is the actual summation +interval. Lemma~\ref{ex:proxies} supplies its rough support, +coefficient bound, and discrepancy. Exchange the factor names +when the test is on the second factor. The theorem requires only +boundedness of the companion sequence, so a nonsmooth proxy is +permitted. There are $O(L)$ contributing dyadic pairs for each +of the fixed finitely many tests. The total error is therefore +$O(xL^{-D_*+1})=o(X_A/L)$ by \eqref{ex:mass-reminder}. +The full-cell normalization is retained when $K$ cuts a cell. +In particular, neither $B$ nor the coefficient exponent used in +the Type II theorem depends on $S_1$. +\end{proof} + +\subsection{Presieving and unbalanced composites} + +For odd squarefree $d$ write, as in the congruence theorem, +\[ + r(d)=\sum_{d\mid N_u}a_x(u)-\frac{X_A}{\varphi(d)}. +\] +Theorem~\ref{cg:distribution}, with the already fixed +$\vartheta,D_{\mathrm{cg}}$, gives +\begin{equation}\label{ex:remainder-sum} + \sum_{\substack{d\le x^\vartheta\\d\text{ odd and squarefree}}} + |r(d)| + \ll xL^{-D_{\mathrm{cg}}}+X_AL^{-18}=o(X_A/L). +\end{equation} + +\begin{lemma}[Initial sifted mass]\label{ex:presieve} +For sufficiently small fixed $b_1$, +\begin{equation}\label{ex:presieve-bound} + \sum_{P^-(N_u)>x^{b_1}}a_x(u) + \ge\frac{\mathfrak S X_A}{L} + \left\{\frac{e^{-\gamma_E}}{b_1} + (1-O(e^{-c/b_1}))+o(1)\right\}. +\end{equation} +\end{lemma} + +\begin{proof} +Apply Lemma~\ref{ex:block-sieve} to the nonnegative weights +$a_x(u)$, with bad condition $p\mid N_u$ at odd primes +$p\le z=x^{b_1}$. Its density is $g(p)=1/(p-1)$, at most $1/2$; +the sums of these densities over $(w,w^2]$ are bounded. Take +\[ + h=2\left\lfloor\frac1{40b_1}\right\rfloor. +\] +Making $b_1$ small ensures that this even integer is sufficiently +large for the two-sided sieve. Its level satisfies +\[ + b_1(4h+2)\le\frac15+2b_1<\vartheta. +\] +Thus \eqref{ex:remainder-sum} supplies the full remainder sum. +Since $N_u$ is odd, avoiding the specified odd primes is exactly +$P^-(N_u)>x^{b_1}$. Finally use \eqref{ex:mertens} and +$e^{-h}=O(e^{-c/b_1})$ in \eqref{ex:block-formula}. +\end{proof} + +\begin{lemma}[Conditional small-prime sieve]\label{ex:conditional} +For each fixed bin $(\gamma,\gamma']$ of the chosen partition, +\begin{equation}\label{ex:conditional-sieve} + \sum_{\substack{x^\gammaW}}a_x(u) + =X_AV_1(W)\{\log(\gamma'/\gamma)+o(1)\}. +\end{equation} +\end{lemma} + +\begin{proof} +Fix such a prime $n$. For each odd squarefree $d$ supported on +primes at most $W$, one has $n>W$ for sufficiently large $x$, and +\[ + \sum_{nd\mid N_u}a_x(u) + =\frac{X_A}{(n-1)\varphi(d)}+r(nd). +\] +Apply Lemma~\ref{ex:block-sieve} on the objects with $n\mid N_u$, +using base mass $X_A/(n-1)$, local density $1/(p-1)$, and +$h=2\lceil T^2\rceil$. Its stated growing-$h$ uniformity applies; +the level for $d$ is +\[ + D_W=W^{4h+2}=\exp(O(\sqrt L\,T^2))=x^{o(1)}. +\] +Every product $nd$ used is at most +$x^{1/2-\kappa+o(1)}x^\gamma$. This includes a repeated occurrence of +$n$. Its weight is therefore bounded above by the sum over +$mn=N_u$ of $I_\gamma(m)\1_{x^\gammaW$, $P^-(m)>W$ is equivalent to $P^-(N_u)>W$; no +coprimality between $m$ and $n$ is needed. Lemma~\ref{ex:conditional} +and \eqref{ex:mertens} bound the weight removed for this bin by +\[ + \frac{\mathfrak S X_A}{L} + \left\{\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t) + \log(\gamma'/\gamma)+o(1)\right\}, +\] +apart from the total $o(X_A/L)$ replacement error. Summing over +the fixed mesh and using \eqref{ex:mesh} bounds the total removed +weight by +\[ + \frac{\mathfrak S X_A}{L} + \{D_{b_1}(1)-1+1/10+o(1)\}. +\] +Subtract this from Lemma~\ref{ex:presieve}. By +\eqref{ex:density-upper}, the coefficient of +$\mathfrak S X_A/L$ left over is at least +\[ + 1-\frac1{10}-O(b_1^{-1}e^{-c/b_1})+o(1). +\] +Choose $b_1$ so small that this is greater than $1/2$ for +sufficiently large $x$. This choice also satisfies the initial +sieve conditions. The surviving presieved values are exactly +among the two classes in the assertion, so their weight proves +\eqref{ex:surviving-mass}. +\end{proof} + +\subsection{Balanced composites and the conclusion} + +\begin{lemma}[Balanced composite bound]\label{ex:balanced} +With the constant fixed in \eqref{ex:balanced-constant}, the weight +of composite $N_u$ with $P^-(N_u)>x^{1/2-\kappa}$ is at most +\begin{equation}\label{ex:balanced-bound} + C_{\mathrm{bal}}(\kappa+1/L)\frac{X_A}{L}+o(X_A/L). +\end{equation} +\end{lemma} + +\begin{proof} +Such a composite has exactly two prime factors counted with +multiplicity: three would give +$N_u>x^{3(1/2-\kappa)}>4x+1$. Both factors lie in +$\mathcal B_\kappa$ for sufficiently large $x$, since the larger +one is at most $(4x+1)x^{-1/2+\kappa}W}}a_x(u). +\end{equation} +Apply \eqref{ex:all-balanced-pairs} and +\eqref{ex:balanced-constant}, and add the $o(X_A/L)$ replacement +error. This proves \eqref{ex:balanced-bound}. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Combining Lemmas~\ref{ex:unbalanced} and~\ref{ex:balanced} gives +\[ + \sum_{N_u\text{ prime}}a_x(u) + \ge\left(\frac{\mathfrak S}{2} + -C_{\mathrm{bal}}\kappa-o(1)\right)\frac{X_A}{L} + \gg\frac{X_A}{L}, +\] +by \eqref{ex:kappa-choice}. Lemma~\ref{wt:squarefree} removes only +$o(X_A/L)$ from this mass. +Consequently there is a supported $u$ with $2u+1$ prime and +$2u$ squarefree. + +For completeness, the support condition counts prime factors with +their multiplicities. The ordinary part $u_*$ has +$r_0(j_x+1)$ prime factors, an even number because $r_0$ is even. +The factor $(1-\mathcal E(u))/2$ in $A(u)$ imposes an odd total +multiplicity on the group-prime part. Hence $\Omega(u)$ is odd. +Every supported $u$ is odd, so +$\Omega(2u)=1+\Omega(u)$ is even. On the retained squarefree +predecessors this is also an even number of distinct prime +factors. Finally $u\in[x,2x]$ wherever $a_x(u)>0$; the resulting +primes exceed $x$ for arbitrarily large $x$, proving infinitude. +\end{proof} + +The order of choices is intrinsic to the argument: the original +weight data and the micro-band choices determine +$C_{\mathrm{box}}$ and $C_{\mathrm{bal}}$; then come $\kappa$, +$b_1$, the finite mesh, the distribution savings and their +discrepancy exponent, and finally $S_1$. All thresholds for $x$ +are imposed afterwards. In particular the absolute prime-proxy +constant and the coefficient bound one, rather than any +precision-dependent regularity, are what make this order valid. diff --git a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..04fc532a68726620bf6a9bd1b2dcd14572af481d --- /dev/null +++ b/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8ee2d8f79198dd5f09c5fbf71e309a63493d1c089772499614ee58dc8575eae9 +size 599500 diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/README.md b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..c97336dae8abeecc46f39e13bb51211779ddb058 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/README.md @@ -0,0 +1,18 @@ +# [Primitive roots for every admissible integer base](primitive-roots-all-integer-bases.pdf) + +**Author:** OpenAI + +**Date:** October 4, 2026 + +## Citation + +```bibtex +@misc{OAI:Primitive-roots-for-every-admissible-integer-base-October-4-2026, + author = {{OpenAI}}, + title = {{Primitive roots for every admissible integer base}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf}{OAI:Primitive-roots-for-every-admissible-integer-base-October-4-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/main.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..d33302d7dc15e0b564ae1fee2ecff014d252217f --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/main.tex @@ -0,0 +1,91 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[a4paper,margin=30mm]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} +\ifdefined\pdfmapfile\pdfmapfile{+rsfs.map}\fi +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs} +\usepackage{xcolor} +\usepackage{graphicx} +\usepackage{tikz} +\usepackage{hyperref} +\usepackage[capitalise,nameinlink,noabbrev]{cleveref} +\hypersetup{ + pdftitle={Primitive roots for every admissible integer base}, + pdfauthor={OpenAI}, + colorlinks=true, + linkcolor=blue!45!black, + citecolor=blue!45!black, + urlcolor=blue!45!black +} + +\numberwithin{equation}{section} +\theoremstyle{plain} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Li}{Li} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Res}{Res} +\newcommand{\Norm}{\mathrm{N}} +\newcommand{\one}{\mathbf{1}} +\newcommand{\1}{\mathbf{1}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\dd}{\,\mathrm{d}} +\newcommand{\eps}{\varepsilon} +\newcommand{\Adele}{\mathbb{A}} +\newcommand{\GL}{\mathrm{GL}} +\newcommand{\PGL}{\mathrm{PGL}} +\newcommand{\SL}{\mathrm{SL}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Fq}{\mathbb{F}} +\newcommand{\e}{\mathrm{e}} +\newcommand{\im}{\operatorname{Im}} +\newcommand{\re}{\operatorname{Re}} +\newcommand{\wh}{\widehat} +\newcommand{\ov}{\overline} +\newcommand{\sfr}{\mathrm{sf}} +\setlist[enumerate]{leftmargin=*,itemsep=3pt} +\setlist[itemize]{leftmargin=*,itemsep=3pt} +\allowdisplaybreaks[2] +\emergencystretch=2em + +\title{Primitive roots for every admissible integer base} +\author{OpenAI} +\date{October 4, 2026} + +\begin{document} +\maketitle +\input{sections/00-abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/01a-reduction} +\input{sections/02-arithmetic} +\input{sections/02a-coefficient-family} +\input{sections/03-theta} +\input{sections/04-reflection} +\input{sections/05-poisson} +\input{sections/06-zero-free} +\input{sections/07-splitting} +\input{sections/08-type-ii} +\input{sections/11-sieve} +\input{sections/12-construction} +\appendix +\input{sections/10-whittaker-calculation} +\input{references} +\end{document} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/references.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..a5ae613858b131e24ec8d3696c4cb74f07848803 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/references.tex @@ -0,0 +1,209 @@ +\phantomsection +\addcontentsline{toc}{section}{References} +\begin{thebibliography}{99} +\raggedright +\setlength{\itemsep}{2pt plus 1pt minus 1pt} + +\bibitem{ArtinHasse08} +E.~Artin and H.~Hasse, +\emph{Emil Artin und Helmut Hasse: Die Korrespondenz 1923--1934}, +edited and commented by G.~Frei and P.~Roquette, with F.~Lemmermeyer, +Universit\"atsverlag G\"ottingen, 2008. +\href{https://univerlag.uni-goettingen.de/handle/3/isbn-978-3-940344-50-2}{Publisher's edition}. + +\bibitem{BaierYoung10} +S.~Baier and M.~P.~Young, +Mean values with cubic characters, +\emph{J. 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Sbornik] N.S.} \textbf{2(44)} (1937), no.~6, +1239--1246 (Russian, with a German summary). +\href{https://www.mathnet.ru/eng/sm5649}{MathNet record and original text}. + + +\bibitem{DunnRadziwill24} +A.~Dunn and M.~Radziwi\l{}\l{}, +Bias in cubic Gauss sums: Patterson's conjecture, +\emph{Ann. of Math.} (2) \textbf{200} (2024), no.~3, 967--1057. +\href{https://doi.org/10.4007/annals.2024.200.3.3}{doi:10.4007/annals.2024.200.3.3}. +Author version: +\href{https://arxiv.org/abs/2109.07463v3}{arXiv:2109.07463v3}. + +\bibitem{FordHalberstam2000} +K.~Ford and H.~Halberstam, +The Brun--Hooley sieve, +\emph{J. Number Theory} \textbf{81} (2000), no.~2, 335--350. +\href{https://doi.org/10.1006/jnth.1999.2479}{doi:10.1006/jnth.1999.2479}. + + +\bibitem{Goldfeld68} +M.~Goldfeld, +Artin's conjecture on the average, +\emph{Mathematika} \textbf{15} (1968), no.~2, 223--226. +\href{https://doi.org/10.1112/S0025579300002606}{doi:10.1112/S0025579300002606}. + +\bibitem{GoldmakherLouvel13} +L.~Goldmakher and B.~Louvel, +A quadratic large sieve inequality over number fields, +\emph{Math. Proc. Cambridge Philos. Soc.} \textbf{154} (2013), +no.~2, 193--212. +\href{https://doi.org/10.1017/S0305004112000370}{doi:10.1017/S0305004112000370}. + +\bibitem{GuptaMurty84} +R.~Gupta and M.~R.~Murty, +A remark on Artin's conjecture, +\emph{Invent. Math.} \textbf{78} (1984), no.~1, 127--130. +\href{https://doi.org/10.1007/BF01388719}{doi:10.1007/BF01388719}. + +\bibitem{DedekindZeros} +E.~Hasanalizade, Q.~Shen, and P.-J.~Wong, +Counting zeros of Dedekind zeta functions, +\emph{Math. Comp.} \textbf{91} (2022), no.~333, 277--293. +\href{https://doi.org/10.1090/mcom/3665}{doi:10.1090/mcom/3665}. + +\bibitem{HeathBrown86} +D.~R.~Heath-Brown, +Artin's conjecture for primitive roots, +\emph{Quart. J. Math. Oxford} (2) \textbf{37} (1986), +no.~1, 27--38. +\href{https://doi.org/10.1093/qmath/37.1.27}{doi:10.1093/qmath/37.1.27}. + +\bibitem{HeathBrown95} +D.~R.~Heath-Brown, +A mean value estimate for real character sums, +\emph{Acta Arith.} \textbf{72} (1995), no.~3, 235--275. +\href{https://doi.org/10.4064/aa-72-3-235-275}{doi:10.4064/aa-72-3-235-275}. + +\bibitem{Hooley67} +C.~Hooley, +On Artin's conjecture, +\emph{J. Reine Angew. Math.} \textbf{225} (1967), 209--220. +\href{https://doi.org/10.1515/crll.1967.225.209}{doi:10.1515/crll.1967.225.209}. + +\bibitem{HoshiKanai22} +A.~Hoshi and K.~Kanai, +Davenport and Hasse's theorems and lifts of multiplication matrices of +Gaussian periods, +\emph{Finite Fields Appl.} \textbf{84} (2022), article~102101. +\href{https://doi.org/10.1016/j.ffa.2022.102101}{doi:10.1016/j.ffa.2022.102101}. + +\bibitem{KP84} +D.~A.~Kazhdan and S.~J.~Patterson, +Metaplectic forms, +\emph{Publ. Math. Inst. Hautes \'Etudes Sci.} \textbf{59} (1984), 35--142. +\href{https://doi.org/10.1007/BF02698770}{doi:10.1007/BF02698770}. + +\bibitem{KP85Erratum} +D.~A.~Kazhdan and S.~J.~Patterson, +Corrections to \emph{Metaplectic forms}, +\emph{Publ. Math. Inst. Hautes \'Etudes Sci.} \textbf{62} (1985), 203. +\href{https://doi.org/10.1007/BF02698809}{doi:10.1007/BF02698809}. + +\bibitem{KlurmanShparlinskiTeravainen25} +O.~Klurman, I.~E.~Shparlinski, and J.~Ter\"av\"ainen, +On Artin's conjecture on average and short character sums, +\emph{Bull. Lond. Math. Soc.} \textbf{57} (2025), no.~8, 2429--2443. +\href{https://doi.org/10.1112/blms.70103}{doi:10.1112/blms.70103}. + +\bibitem{Kubota67} +T.~Kubota, +Topological covering of \(\mathrm{SL}(2)\) over a local field, +\emph{J. Math. Soc. Japan} \textbf{19} (1967), no.~1, 114--121. +\href{https://doi.org/10.2969/jmsj/01910114}{doi:10.2969/jmsj/01910114}. + +\bibitem{Neukirch99} +J.~Neukirch, +\emph{Algebraic Number Theory}, +translated by N.~Schappacher, +Grundlehren der mathematischen Wissenschaften, vol.~322, +Springer, Berlin, 1999. +\href{https://doi.org/10.1007/978-3-662-03983-0}{doi:10.1007/978-3-662-03983-0}. + +\bibitem{BlockPaper} +OpenAI, +Weighted dilation graphs, smooth shifted primes and totient fibers, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf}{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026}, +2026. Lemmas~2.9 and~7.4. + +\bibitem{PoissonDirichlet} +OpenAI, +The Poisson--Dirichlet law for prime predecessors, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf}{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026}, +2026. Theorem~3.1 and Sections~3--5. + +\bibitem{PrimePredecessors} +OpenAI, +Prime Predecessors with an Even Number of Prime Factors, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf}{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}, +2026. Section~3, especially Lemmas~3.1 and~3.3. + +\bibitem{QuasiRH} +OpenAI, +The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane +$\mathrm{Re}(s)>7/8$, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf}{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026}, +2026. Part~I, Sections~5--7. + +\bibitem{Patterson77} +S.~J.~Patterson, +A cubic analogue of the theta series, +\emph{J. Reine Angew. Math.} \textbf{296} (1977), 125--161. +\href{https://doi.org/10.1515/crll.1977.296.125}{doi:10.1515/crll.1977.296.125}. + +\bibitem{Patterson77II} +S.~J.~Patterson, +A cubic analogue of the theta series. II, +\emph{J. Reine Angew. Math.} \textbf{296} (1977), 217--220. +\href{https://doi.org/10.1515/crll.1977.296.217}{doi:10.1515/crll.1977.296.217}. + +\bibitem{Stephens69} +P.~J.~Stephens, +An average result for Artin's conjecture, +\emph{Mathematika} \textbf{16} (1969), no.~2, 178--188. +\href{https://doi.org/10.1112/S0025579300008159}{doi:10.1112/S0025579300008159}. + +\bibitem{Tate67} +J.~T.~Tate, +Fourier analysis in number fields and Hecke's zeta-functions, +in \emph{Algebraic Number Theory}, edited by J.~W.~S.~Cassels and +A.~Fr\"ohlich, Academic Press, London, 1967, 305--347. +\href{https://math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf}{Scan of the volume}. + +\bibitem{ThornerZaman19} +J.~Thorner and A.~Zaman, +A unified and improved Chebotarev density theorem, +\emph{Algebra Number Theory} \textbf{13} (2019), no.~5, 1039--1068. +\href{https://doi.org/10.2140/ant.2019.13.1039}{doi:10.2140/ant.2019.13.1039}. + +\bibitem{Vinogradov65} +A.~I.~Vinogradov, +On the density hypothesis for Dirichlet \(L\)-series, +\emph{Izv. Akad. Nauk SSSR Ser. Mat.} \textbf{29} (1965), +no.~4, 903--934 (Russian). +\href{https://www.mathnet.ru/eng/im3080}{MathNet record and original text}. +Correction, \emph{ibid.} \textbf{30} (1966), no.~3, 719--720; +\href{https://www.mathnet.ru/eng/im2856}{MathNet record and correction}. + +\bibitem{Weil64} +A.~Weil, +Sur certains groupes d'op\'erateurs unitaires, +\emph{Acta Math.} \textbf{111} (1964), 143--211. +\href{https://doi.org/10.1007/BF02391012}{doi:10.1007/BF02391012}. + +\end{thebibliography} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/00-abstract.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/00-abstract.tex new file mode 100644 index 0000000000000000000000000000000000000000..92009a09fe55023715d1fe1e156f3e358a8273ca --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/00-abstract.tex @@ -0,0 +1,6 @@ +\begin{abstract} +We prove the infinitude assertion in Artin's primitive root conjecture: +for every integer $a$ that is neither $-1$ nor a square, there are +at least $c_a x/(\log x)^2$ primes in $(x,2x)$ with primitive root $a$, +for some $c_a>0$ and every sufficiently large $x$. +\end{abstract} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/01-introduction.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..d3ed0c86e7524058a601145a4863acdb89d576e7 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/01-introduction.tex @@ -0,0 +1,219 @@ +\section{Introduction}\label{sec:introduction} + +Let $a$ be an integer and let $p$ be a prime not dividing $a$. +The multiplicative order $\ord_p(a)$ is the least positive integer $k$ +such that $a^k\equiv1\pmod p$. The residue class of $a$ is a primitive +root modulo $p$ if $\ord_p(a)=p-1$. Call $a$ \emph{admissible} if +$a\ne-1$ and $a$ is not the square of an integer. These are necessary +conditions for primitive-root behavior at infinitely many primes: +the order of $-1$ is at most $2$, and a nonzero square modulo an odd +prime has order dividing $(p-1)/2$. + +Artin's primitive root conjecture, formulated in 1927, asserts that +every admissible integer is a primitive root modulo infinitely many +primes; its quantitative form predicts a positive-density asymptotic +\cite[Section~17.2]{ArtinHasse08}. We prove the following lower bound. + +\begin{theorem}\label{thm:primitive-root}\label{thm:main} +For every admissible integer $a$, there are constants $c_a>0$ and +$x_a\ge2$ such that, for every real $x\ge x_a$, +\[ + \#\{p\text{ prime}:x1-10^{-6}. +\] +\end{theorem} + +The principal character is included, with its pole at $s=1$ permitted. +The width is common to all fields and characters in the statement, +with no conductor or height cutoff. In the proof, $F$ and $\eta$ are +fixed before the auxiliary data are chosen. Constants and the points +from which estimates hold may depend on them; the numerical saving +in the final Mellin argument does not. This distinction is what +allows a uniform splitting estimate for a growing family of fields. + +\subsection{Historical context and the two obstructions} + +Hooley proved Artin's predicted asymptotic under the Riemann hypothesis +for the Dedekind zeta functions of the Kummer fields +$\Q(\mu_n,a^{1/n})$, with $n$ squarefree~\cite{Hooley67}. +These fields encode the obstructions to maximal order. For a prime +$q$ and $p\nmid aq$, the conditions +\[ + p\equiv1\pmod q,\qquad a^{(p-1)/q}\equiv1\pmod p +\] +say that $p$ splits completely in $\Q(\mu_q,a^{1/q})$. +They hold precisely when $q$ divides the index +$(p-1)/\ord_p(a)$. Hooley's method counts and sums such splitting +conditions. Here a restricted factorization of $p-1$ confines the +possible index divisors to ranges where they can all be removed. + +Classical unconditional results allow a finite choice of bases. +Gupta and Murty constructed a set of thirteen integers from three +distinct primes, at least one of which is a primitive root modulo +infinitely many primes~\cite{GuptaMurty84}. Heath-Brown proved that +at most two prime bases fail the infinitude assertion, so at least +one of $2,3,5$ satisfies it~\cite{HeathBrown86}. +Averaging over the base gives another approach. Goldfeld and Stephens +established early average theorems~\cite{Goldfeld68,Stephens69}; +Klurman, Shparlinski, and Ter\"av\"ainen obtained the expected +asymptotic for almost all integers in a range as short as +$1\le a\le\exp((\log\log x)^2)$~\cite{KlurmanShparlinskiTeravainen25}. +The exceptional set may depend on $x$, so these average results +do not by themselves settle a prescribed base. + +Two difficulties must be addressed for a fixed base. One is analytic: +the splitting fields vary with the possible index divisor. A standard +simultaneous zero-free region for Hecke characters associated with +an abelian extension has logarithmic width depending on the field, +conductors, and height, and permits at most one simple real +exceptional zero~\cite[Theorem~3.1]{ThornerZaman19}. +The common width in Theorem~\ref{thm:hecke-zero-free} instead gives +a common power saving. The other difficulty is the prime construction: +we need many primes for which the possible index divisors fall in +the range controlled by that saving. We now describe the two parts +of the proof and the points where they meet. + +\subsection{The Hecke zero-free argument} + +Fix a cyclotomic field $F$ containing $\mu_{12}$ and a finite-order +Hecke character $\eta$. The analytic construction uses cubic theta +coefficients under sextic Kummer twists. Patterson's cubic theta +series over $\Q(\sqrt{-3})$ is the model for the Gauss coefficients +and their transformations~\cite{Patterson77,Patterson77II}; the +comparison formulas are recorded by Dunn and Radziwi\l{}\l{} +\cite[Section~5.1 and Appendix~A]{DunnRadziwill24}. +The companion manuscript \emph{The Quasi-Riemann Hypothesis} +develops a coefficient reflection and an additive probe over that +field while retaining unrestricted cube factors +\cite[Part~I, Sections~5--7]{QuasiRH}. Its zero-free theorem treats +Dirichlet $L$-functions and finite-order Hecke $L$-functions over +$\Q(\sqrt{-3})$. We construct the required mechanisms over each +fixed $F$, using the cubic theta representation of Kazhdan and +Patterson and a normalized global Whittaker factorization consistent +with their correction~\cite{KP84,KP85Erratum}. + +Two local calculations govern the comparison. We choose generators +for which sextic reciprocity has no supplementary factor. In a +reflected interaction between a prime occurring to valuation one +in the varying twist and a varying squarefree coefficient prime, +the sextic exponents are $1$ and $-4$. Their sum is $3$ modulo $6$, +so the interaction is quadratic. The quadratic large sieve of +Heath-Brown, in its number-field form due to Goldmakher and Louvel, +then controls the reflected moment +\cite{HeathBrown95,GoldmakherLouvel13}. + +On the Poisson side, a relative-norm formula identifies the cube of +the cubic Gauss phase with the negative of a unitary Hecke-character +value. A finite-order correction gives the target character $\eta$. +At a prime ideal $\mathfrak p$ outside a fixed excluded set, the +first squarefree theta term at the principal frequency is +$-\eta(\mathfrak p)(\Norm\mathfrak p)^{-\xi}$, where $\xi$ is +the Mellin variable for the coefficient norm. Together with the +constant term, it supplies the local factor of the reciprocal +target $L$-function. The remaining local terms give a holomorphic, +nonvanishing correction in the region used at the end of the proof. + +Reflection and Poisson summation thus estimate the same additive +average in two ways. A conductor large sieve and zero detection +control the nonprincipal frequencies. The resulting power saving +for the principal inverse Mellin integral continues its Mellin +transform across the line $\re s=1$ and excludes zeros of the +target function. The use of ideal norms keeps the numerical +exponents in this argument independent of the degree of $F$. +Sections~\ref{sec:arithmetic}--\ref{sec:zero-free} carry out this +proof; Appendix~\ref{app:whittaker} provides the theta normalization. + +\subsection{Controlled predecessors in a fixed progression} + +The construction produces many primes of the form +\[ + p=crQ+1,\qquad c\in\{2,4\},\qquad Q>x^{0.9}\text{ prime}, +\] +where every prime divisor of $r$ lies between +$\exp((\log x)^{0.1})$ and $\exp((\log x)^{0.3})$. +We also prescribe a residue class on which the Legendre symbol +$(a/p)$ equals $-1$. This excludes $2$ as an index divisor. +A nontrivial remaining index has a prime divisor in $rQ$: +the factor $Q$ leads to a small elementary exceptional set, and +the factors of $r$ fall within the uniform splitting estimate. + +The construction starts with weighted integers $d=crQ+1$ of this +shape. A sieve removes small prime factors of $d$. For a remaining +composite $d=mn$, with $n$ its least prime factor, a marked bilinear +estimate replaces the restriction on the cofactor $m$ by an explicit +density and a weaker condition on its small prime factors. +Buchstab's least-prime-factor identity leaves a positive margin +between the initial sifted mass and the mass removed in this +comparison. A separate upper bound makes the contribution from two +prime factors close to $\sqrt x$ smaller than that margin. + +The marked prime estimate is an input from +\emph{The Poisson--Dirichlet law for the prime factors of $p-1$} +\cite[Theorem~3.1]{PoissonDirichlet}. We extract its scalar coefficient +criterion and verify the full character and frequency estimates for +the rough-factor comparison. The weights require $d-1$ to have a +prime divisor in each of several prescribed ranges; these choices +are the marks. A bounded multiplier in the bilinear estimate may +depend on the quotient left after every power of these primes has +been removed. A congruence condition on $d=mn$ need not have this +invariance. We therefore expand $mn\equiv u\pmod M$ in Dirichlet +characters and separate it into factors $\lambda(m)\lambda(n)$ +in the two scalar coefficient sequences. Fixed twists preserve +the required long character-sum bound. + +The resulting fixed-progression construction is stated separately +in Proposition~\ref{prop:controlled}. It applies to any fixed data +$M,c,u$ satisfying its local coprimality conditions, independently +of the primitive-root application. Its proof uses +Bombieri--Vinogradov distribution, a Brun--Hooley block sieve, +and the continuous rough-number density +\cite{Bombieri65,Vinogradov65,FordHalberstam2000,Buchstab1937}. +The precise elementary sieve and density forms are adapted from +\cite[Lemmas~2.9 and~7.4]{BlockPaper}, with proofs included here. + +\subsection{Organization and notation} + +Section~\ref{sec:reduction} states the controlled-predecessor and +splitting estimates and combines them to prove +Theorem~\ref{thm:primitive-root}. Sections~\ref{sec:arithmetic}--\ref{sec:zero-free} +prove Theorem~\ref{thm:hecke-zero-free} for a fixed arbitrary $F$ +and $\eta$, independently of the integer base. The analytic scale +parameters in those sections are local to that proof. +Section~\ref{sec:splitting} returns to the fixed integer $a$ and +derives the uniform splitting bound. Section~\ref{sec:type-ii} +proves the marked coefficient and rough-factor estimates; +Section~\ref{sec:sieve} supplies the elementary sieve facts; +Section~\ref{sec:prime-construction} proves the fixed-progression +construction. Appendix~\ref{app:whittaker} verifies the selected +Kazhdan--Patterson specialization and Whittaker normalization. + +Throughout, $\log$ is the natural logarithm, and $L=\log x$ in the +prime-counting arguments. The symbols $\ll$ and +$O$ have their usual meaning; subscripts record permitted dependence +when it matters. Auxiliary parameters are fixed before the relevant +asymptotic variable tends to infinity. Uniformity in those parameters +is stated explicitly. The general representation-theoretic, +class-field, Hecke-continuation, and classical sieve theorems are +cited at their points of use. The marked Type II argument also uses +the companion papers cited in the bibliography. diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/01a-reduction.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/01a-reduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..260e5612b0f4e3b70a870d7dc28001abb24a16ee --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/01a-reduction.tex @@ -0,0 +1,216 @@ +\section{Reduction to controlled predecessors and complete splitting} +\label{sec:reduction} + +For a prime $p\nmid a$, failure of $a$ to generate $\mathbb F_p^\times$ +means that some prime divides the index +\[ + i_p(a)=\frac{p-1}{\ord_p(a)}. +\] +We construct primes for which the possible divisors of this index belong +to two ranges. A large divisor forces $p$ to divide one of a short list +of integers $a^j-1$; the remaining divisors are controlled by complete +splitting in Kummer fields. This is the index-divisor reduction underlying +Hooley's work \cite[Sections~4--7]{Hooley67}. + +The first proposition supplies the primes, with a fixed congruence that +will force quadratic nonresiduosity. + +\begin{proposition}[Primes with controlled predecessors] +\label{prop:controlled}\label{prop:prime-reservoir} +Let $M$ be a fixed positive multiple of $8$, let $c\in\{2,4\}$, and let +$u$ be an integer satisfying +\begin{equation} + (u,M)=1,\qquad c\mid u-1,\qquad + \left(\frac{u-1}{c},\frac Mc\right)=1. + \label{alg:progression-conditions} +\end{equation} +For every sufficiently large real $x$, there are +$\gg_{M,c,u}x/L^2$ distinct primes $p\in(x,2x)$ satisfying +\begin{equation} + p\equiv u\pmod M,\qquad p-1=crQ, + \qquad Q>x^{0.9}\ \text{prime}, + \label{eq:controlled-predecessor} +\end{equation} +where $r$ is a positive integer all of whose prime divisors lie in +\[ + \bigl(\exp(L^{0.1}),\exp(L^{0.3})\bigr). +\] +\end{proposition} + +We prove Proposition~\ref{prop:controlled} in +Section~\ref{sec:construction}. For the second input, write $\mu_m$ for +the group of $m$th roots of unity. For a nonzero integer $a$ and a prime +$q$, put +\begin{equation} + E_q=\Q(\mu_q),\qquad K_q=\Q(\mu_q,a^{1/q}). + \label{alg:kummer-fields} +\end{equation} +The choice of the root $a^{1/q}$ does not affect $K_q$. + +\begin{proposition}[Uniform bound for complete splitting] +\label{prop:splitting} +Fix an integer $a$ with $|a|>1$, and put $\delta_0=10^{-6}$. +For all sufficiently large real $x$ and all primes $q$ sufficiently +large in terms of $a$, with $q\le\exp(L^{0.3})$, one has +\begin{equation} + \#\{p\text{ prime}:x1$. Choose $M,c,u$ by +Lemma~\ref{alg:quadratic-progression}, and take the +$\gg_a x/L^2$ primes supplied by Proposition~\ref{prop:controlled}. +For each such prime $p$, Euler's criterion gives +$a^{(p-1)/2}\equiv-1\pmod p$. Consequently $i_p(a)$ is odd: an even +index would make $\ord_p(a)$ divide $(p-1)/2$. + +If $i_p(a)>1$, choose a prime divisor $q$ of this index. Since +$p-1=crQ$ and $c$ is a power of $2$, we have $q\mid rQ$. Moreover +$\ord_p(a)\mid(p-1)/q$, so \eqref{eq:kummer-frobenius-test} holds. +We bound the number of such primes $p$ in the two possible ranges of $q$. + +If $q=Q$, then +\[ + 1\le\frac{p-1}{Q}<2x^{0.1}. +\] +Thus $p$ divides the positive integer +\[ + B_a(x)=\prod_{1\le j\le2x^{0.1}}|a^j-1|. +\] +No factor vanishes because $|a|>1$, and +\[ + \log B_a(x) + \le\sum_{1\le j\le2x^{0.1}}\bigl(j\log|a|+\log2\bigr) + \ll_a x^{0.2}. +\] +Every distinct prime divisor $p>x$ contributes at least $L$ to this +logarithm. The number of primes lost in this case is therefore +$O_a(x^{0.2}/L)=o(x/L^2)$. + +If $q\mid r$, then +$\exp(L^{0.1})\exp(L^{0.1})}\frac1{m(m-1)} + +x^{1-\delta_0}\exp(L^{0.3})\notag\\ + &\ll_a \frac{x}{L}\exp(-L^{0.1}) + +x\exp(-\delta_0L+L^{0.3}) + =o(x/L^2) + \label{alg:exceptional-small-divisors} +\end{align} +primes, where the first sum is over integers. The first estimate uses +\eqref{eq:uniform-splitting} for each possible $q$; the second uses the +telescoping sum of $1/(m(m-1))$. + +Subtracting these two losses leaves $\gg_a x/L^2$ primes with +$i_p(a)=1$. All choices depend only on $a$, and both estimates hold for +every sufficiently large real $x$. This proves the theorem. +\end{proof} + +For $a=2$, the theorem gives $\gg x/L^2$ primes in $(x,2x)$ with +primitive root $2$, with absolute constants. This case can use the +particularly simple choice $(M,c,u)=(8,4,5)$ in +Proposition~\ref{prop:controlled}: the predecessors are $4rQ$, and +$p\equiv5\pmod8$ gives $(2/p)=-1$ directly. + +\begin{remark} +Negative squares are covered by the row $d_0=-1$ in the progression +lemma. Odd perfect powers also require no separate exclusion. If +$a=b^h$ with fixed odd $h>1$, every prime divisor of $h$ is eventually +smaller than every prime factor of $rQ$; hence $(h,p-1)=1$ for the +constructed primes. This explains directly how the controlled +predecessor removes the obstruction from the fixed exponent. +\end{remark} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/02-arithmetic.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/02-arithmetic.tex new file mode 100644 index 0000000000000000000000000000000000000000..f867118933bcbca858a62a3d7b495d894d7a40be --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/02-arithmetic.tex @@ -0,0 +1,625 @@ +\section{S-integers, residue symbols, and Gauss sums}\label{sec:arithmetic} + +The analytic argument will sum over ideals while also using their generators +in local fields. We first choose generators for which reciprocity has no +supplementary factor. We then identify the two Hecke characters which account +for the phases of the cubic and sextic Gauss sums. + +Fix a cyclotomic field \(F\) containing \(\mu_{12}\), and put +\([F:\Q]=2r\). At a finite place \(v\), normalize the absolute value by +\(|\varpi_v|_v=(\Norm v)^{-1}\); at a complex place use the square of the +ordinary modulus. These are the absolute values in the product formula. +Let +\[ + \psi=\prod_v\psi_v:\Adele_F/F\longrightarrow \C^\times +\] +be the standard trace additive character. We also fix a finite-order Hecke +character \(\eta\). Choose a finite set \(S\) containing the complex places, +all places above \(6\), the finite conductor of \(\eta\), and every place at +which \(\mathcal O_v\) is not self-dual for the pairing +\(\psi_v(xy)\). The theta normalization in \Cref{prop:theta-input} +and the local Euler-product estimate in \Cref{prop:euler-product} require +two additional fixed finite sets of excluded places. Include both sets in +\(S\). Their existence is established at those points, independently of +the summation variables and of the arithmetic choices below. We may +include any further fixed finite set of places. +Enlarging \(S\) by finitely many primes which generate the ideal class group, +we arrange that the ring of S-integers +\[ + R=\mathcal O_S=\{a\in F:v_{\mathfrak p}(a)\geq0 + \text{ for every }\mathfrak p\notin S\} +\] +is principal. This is the final choice of \(S\), and all arithmetic data +below are constructed for it before the coefficient family is defined. +All assertions in this section concern this fixed \(F\) and \(S\); their +auxiliary constants may depend on them. + +Write \(S_f\) for the finite places in \(S\), \(s_f=|S_f|\), and +\[ + U_S=R^\times,\qquad F_S=\prod_{v\in S}F_v,\qquad + |y|_S=\prod_{v\in S}|y_v|_v,\qquad + F_S^1=\{y\in F_S^\times:|y|_S=1\}. +\] +An \emph{exterior ideal} is a nonzero integral ideal of \(R\), equivalently +an ideal supported at primes outside \(S\). Its norm is +\(q_{\mathfrak a}=\Norm\mathfrak a=|R/\mathfrak a|\), extended +multiplicatively to fractional ideals. Roman letters will denote generators +of the corresponding fraktur ideals. For \(a\in F^\times\), put +\(q_a=|a|_S\). Finally, on the additive group \(F_S\) set +\[ + \e(x)=\prod_{v\in S}\psi_v(x_v), + \qquad + \widehat f(y)=\int_{F_S}f(x)\e(-xy)\,dx, +\] +where \(dx\) is the product of the self-dual local additive measures. + +\begin{lemma}[The S-integer lattice]\label{lem:s-lattice} +The diagonal copy of \(R\) is a discrete cocompact subgroup of \(F_S\), is +its own annihilator for the pairing \(\e(xy)\), and has covolume \(1\). +For every Bruhat--Schwartz function \(f\) on \(F_S\), +\[ + \sum_{m\in R}f(m)=\sum_{\nu\in R}\widehat f(\nu). +\] +If \(\mathfrak a=aR\) is an exterior fractional ideal, then +\[ + q_a=q_{\mathfrak a},\qquad + (aR)^\perp=a^{-1}R,\qquad + \sum_{m\in aR}f(m) + =q_{\mathfrak a}^{-1}\sum_{\nu\in a^{-1}R}\widehat f(\nu). +\] +\end{lemma} + +\begin{proof} +Put \(K^S=\prod_{v\notin S}\mathcal O_v\), embedded in the adeles with zero +S-components. At every place outside \(S\), +the defining conductor condition says that \(\mathcal O_v\) is its own +annihilator. Its self-dual measure is consequently \(1\), and +\(\widehat{\one_{K^S}}=\one_{K^S}\). Adelic additive duality identifies the +annihilator of \(F\) in \(\Adele_F\) with \(F\), and the additive quotient +\(\Adele_F/F\) has volume \(1\); see +\cite[Chapter~II, \S29]{Weil64}. Strong approximation identifies +\[ + F_S/R\simeq \Adele_F/(F+K^S). +\] +The quotient is compact because \(\Adele_F/F\) is compact. To check +discreteness, choose a compact neighborhood \(C\) of zero in \(F_S\). +The intersection \(F\cap(C\times K^S)\) is finite because diagonal \(F\) +is discrete and closed in the adeles. Shrinking \(C\) excludes the finitely +many nonzero S-components in this intersection, so zero is isolated in +\(R\). Taking annihilators in the last quotient gives +\[ + (F+K^S)^\perp=F\cap(F_S\times K^S)=R, +\] +with \(R\) diagonally embedded, so its annihilator in \(F_S\) is indeed \(R\). +The covolume of a lattice and that of its annihilator are reciprocal under +self-dual measure; hence this covolume is \(1\). + +More directly, apply adelic Poisson summation +\cite[Chapter~III, \S41, equation~(39)]{Weil64} to +\(f\otimes\one_{K^S}\). Only the elements of \(R\) occur on either side, +and the exterior factor is unchanged by Fourier transform. This gives the +displayed Poisson identity with no scalar factor. Multiplication by \(a\) +has additive modulus \(|a|_S\), so scaling this identity gives the formula +for \(aR\). The product formula gives +\[ + |a|_S=\prod_{v\notin S}|a|_v^{-1} + =\Norm(aR)=q_{\mathfrak a}, +\] +which finishes the proof. +\end{proof} + +We use Hilbert symbols to remove the supplementary factor in sextic +reciprocity. Choose the local reciprocity map \(\operatorname{Art}_v\) in +which a uniformizer acts as arithmetic Frobenius on an unramified extension. +For \(n\in\{2,3,6\}\), set +\[ + (a,b)_{n,v} + =\frac{\operatorname{Art}_v(b)(a^{1/n})}{a^{1/n}}. +\] +This orders the two arguments oppositely to the convention in +\cite[Chapter~V, \S3]{Neukirch99}. Thus, at a place of residue +characteristic prime to \(n\), for a unit \(u\) and a uniformizer \(\varpi\), +\[ + (u,\varpi)_{n,v}\equiv u^{(\Norm v-1)/n}\pmod v. +\] +The value is read in \(\mu_n\subset F\). On +\[ + \mathcal V=F_S^\times/F_S^{\times6} +\] +define \(H_S(x,y)=\prod_{v\in S}(x_v,y_v)_{6,v}\), and let \(E\) be the image +of \(U_S\) in \(\mathcal V\). For \(T>0\), let \(\ell_T\in F_S^\times\) +have finite components \(1\) and component \(T^{1/(2r)}\) at every complex +place. Then \(|\ell_T|_S=T\). + +\begin{lemma}[Generators with fixed power classes]\label{lem:balanced-generators} +The pairing \(H_S\) on \(\mathcal V\) is nondegenerate and alternating. +The map \(U_S/U_S^6\to\mathcal V\) is injective, its image \(E\) satisfies +\(E^\perp=E\), and there is an isotropic subgroup \(B\) such that +\(\mathcal V=E\oplus B\). + +Every exterior fractional ideal \(\mathfrak a\) has a generator \(a\) whose +class \([a]\) in \(\mathcal V\) belongs to \(B\). Any two such generators +differ by an element of \(U_S^6\). They can be chosen so that +\[ + a/\ell_{q_{\mathfrak a}}\in\mathcal K_{\mathrm{bal}} +\] +for one fixed compact subset \(\mathcal K_{\mathrm{bal}}\subset F_S^1\), independent of +\(\mathfrak a\). +\end{lemma} + +\begin{proof} +For \(\ell=2,3\), put \(\mathcal V_\ell=F_S^\times/F_S^{\times\ell}\). +Local Kummer theory and local reciprocity give a nondegenerate Hilbert +pairing on this group \cite[Chapter~V, Proposition~(3.2)]{Neukirch99}. +At a finite place \(v\), +\[ + \dim_{\Fq_\ell}F_v^\times/F_v^{\times\ell} + =2+\one_{v\mid\ell}[F_v:\Q_\ell]. +\] +Indeed the valuation contributes one dimension. When \(v\nmid\ell\), the +residue units contribute one dimension and the principal units are +\(\ell\)-divisible. When \(v\mid\ell\), the principal units have a +\(\Z_\ell\)-part of rank \([F_v:\Q_\ell]\) and an \(\ell\)-power +root-of-unity part contributing one more dimension. The latter is present +because \(F\) contains \(\mu_\ell\). This is also the local power-class +formula \cite[Chapter~II, Corollary~(5.8)]{Neukirch99}. +Complex power-class groups are trivial. +All places above \(\ell\) lie in \(S\), so their local degrees sum to \(2r\) +and +\begin{equation} + \dim_{\Fq_\ell}\mathcal V_\ell=2s_f+2r.\label{eq:local-power-dimension} +\end{equation} +The S-unit theorem \cite[Chapter~I, Corollary~(11.7)]{Neukirch99} gives +\(U_S\simeq\mu(F)\times\Z^{r+s_f-1}\). Since \(\mu(F)\) is cyclic and +contains \(\mu_\ell\), +\begin{equation} + \dim_{\Fq_\ell}U_S/U_S^\ell=r+s_f.\label{eq:unit-power-dimension} +\end{equation} + +We next check that the latter group injects into \(\mathcal V_\ell\). +Suppose that \(u\in U_S\) is an \(\ell\)-th power in every \(F_v\) with +\(v\in S\). The Kummer extension \(F(u^{1/\ell})/F\) is cyclic and split +at \(S\). At every exterior prime \(u\) is a unit and \(\ell\) is +invertible, so this extension is unramified there. It is consequently an +everywhere unramified abelian extension in which every S-prime splits. +The Hilbert class-field correspondence, together with the quotient +description of the S-class group, identifies such extensions with quotients +of the class group of \(R\), which is trivial +\cite[Chapter~I, Proposition~(11.6); Chapter~VI, Proposition~(6.9) and +Theorem~(7.3)]{Neukirch99}. Hence +\(u=t^\ell\) for \(t\in F\). Its exterior valuations show \(t\in U_S\), +proving the injection. + +For two S-units, every exterior Hilbert symbol is \(1\), since both arguments +are units and the residue characteristic is prime to \(6\). Hilbert +reciprocity \cite[Chapter~VI, Theorem~(8.1)]{Neukirch99} therefore makes +their S-product \(1\). The Hilbert pairing is +alternating here: \(-1=\zeta_{12}^6\), and the diagonal identity +\((x,x)_{n,v}=(x,-1)_{n,v}\) makes its diagonal trivial for \(n=2,3,6\). +By \Cref{eq:local-power-dimension} and \Cref{eq:unit-power-dimension}, the +injected unit image in \(\mathcal V_\ell\) is an isotropic subspace of half +the dimension. Nondegeneracy implies that it is self-orthogonal. +Symplectic linear algebra supplies an isotropic complementary subspace for +each \(\ell\). The Chinese remainder isomorphism +\(\mathcal V\simeq\mathcal V_2\oplus\mathcal V_3\), under which the two +pairings are the third and second powers of \(H_S\), combines these +complements into \(B\). It also gives the asserted injection and +self-orthogonality for exponent \(6\). + +Let \(\mathfrak a=a_0R\). Decompose \([a_0]=e b\) in \(E\oplus B\), and +choose an S-unit representing \(e^{-1}\). Its product with \(a_0\) is a +generator \(a\) with \([a]\in B\). If \(a'\) is another, then \(a'/a\) is +an S-unit whose class lies in \(E\cap B\). It is therefore locally a sixth +power at \(S\), and the injection already proved says \(a'/a\in U_S^6\). + +Finally \(a/\ell_{q_{\mathfrak a}}\) belongs to \(F_S^1\), by +\Cref{lem:s-lattice}. To see the relevant compactness explicitly, the +norm-one idele class group is compact +\cite[Chapter~VI, Theorem~(1.6)]{Neukirch99}. The subgroup +\(F_S^1\times\prod_{v\notin S}\mathcal O_v^\times\) is open in the +norm-one ideles and meets diagonal \(F^\times\) in \(U_S\). Its quotient +by \(U_S\) is therefore an open, hence closed, subgroup of that compact +idele class group. Its projection onto \(F_S^1/U_S\) is surjective, so +\(F_S^1/U_S\) is compact. Since \(U_S^6\) has finite index in \(U_S\), +\(F_S^1/U_S^6\) is compact as well. Choosing a compact set which maps onto +this quotient and multiplying \(a\) by a suitable element of \(U_S^6\) +proves the last assertion. +\end{proof} + +Call a generator with class in \(B\) \emph{allowed}, and call it +\emph{balanced} when it satisfies the compactness condition in +\Cref{lem:balanced-generators}. We use balanced representatives for +estimates involving S-components. In multiplicative identities we may +instead use the product of the allowed generators of the prime factors: +the product remains allowed and differs from the selected representative by +an element of \(U_S^6\). + +For an exterior prime \(\mathfrak p\), the reduction of \(\mu_6\) into +\((R/\mathfrak p)^\times\) is injective. Define the sextic residue character +by +\[ + \chi_{\mathfrak p}(x)\equiv x^{(q_{\mathfrak p}-1)/6}\pmod{\mathfrak p} + \quad (x\not\equiv0\pmod{\mathfrak p}),\qquad + \chi_{\mathfrak p}(0)=0. +\] +For every integer \(j\), the notation \(\chi_{\mathfrak p}(x)^j\) means the +usual character power on units and means \(0\) on nonunits, also when +\(j\equiv0\pmod6\). If +\(\mathfrak a=\prod_{\mathfrak p}\mathfrak p^{k_{\mathfrak p}}\), put +\[ + \chi_{\mathfrak a}(x)= + \prod_{\mathfrak p\mid\mathfrak a} + \chi_{\mathfrak p}(x)^{k_{\mathfrak p}}, + \qquad \chi_R(x)=1. +\] +The same zero convention applies to powers of \(\chi_{\mathfrak a}\). + +For distinct exterior primes with allowed generators \(p,p'\), global +reciprocity gives the exact equality +\begin{equation} + \chi_{\mathfrak p}(p')=\chi_{\mathfrak p'}(p). +\label{eq:sextic-reciprocity} +\end{equation} +Indeed the only possibly nontrivial exterior symbols in the product formula +for \((p,p')_6\) are +\((p,p')_{6,\mathfrak p}=\chi_{\mathfrak p}(p')^{-1}\) and +\((p,p')_{6,\mathfrak p'}=\chi_{\mathfrak p'}(p)\). The S-product is \(1\) +because \(B\) is isotropic. If \(m\in R\setminus\{0\}\) and +\(\mathfrak b=(m)R\), write \(m=\epsilon b\) +with \(b\) allowed and \(\epsilon\in U_S\). Multiplying +\Cref{eq:sextic-reciprocity} over prime factors gives, for an allowed +generator \(a\) of \(\mathfrak a\), +\begin{equation} + \chi_{\mathfrak a}(m) + =\chi_{\mathfrak a}(\epsilon)\chi_{\mathfrak b}(a). +\label{eq:generator-reciprocity} +\end{equation} +This includes the case \((\mathfrak a,\mathfrak b)\ne R\): both sides are +then zero. For an S-unit \(\epsilon\), the same product formula gives +\begin{equation} + \chi_{\mathfrak a}(\epsilon)=H_S(\epsilon,a)^{-1}. +\label{eq:unit-symbol} +\end{equation} +Thus its dependence on \(\mathfrak a\) is through the finite class \([a]\). + +\begin{lemma}[Kummer and class characters]\label{lem:kummer-characters} +For \(u\in R\setminus\{0\}\), the ideal function +\[ + \chi_\bullet(u)(\mathfrak a)=\chi_{\mathfrak a}(u) +\] +is induced away from \(S\) and the prime divisors of \((u)R\) by a +finite-order Hecke character. Its values at those prime divisors are set +equal to zero. If \(\mathfrak f_u\) is the conductor of the inducing +primitive character, then +\begin{equation} + \Norm\mathfrak f_u\le C_{F,S}\Norm\rad((u)R) + \le C_{F,S}q_u.\label{eq:kummer-conductor} +\end{equation} +The same bound, with a changed constant, holds after twisting by the fixed +\(\eta\). The inducing characters associated to \(u\) and \(v\) agree if +and only if \(u/v\in F^{\times6}\). + +Every character of the finite group \(B\), evaluated on allowed generators, +extends to a finite-order Hecke character unramified outside \(S\). + +The following parametrization will index the nonzero Poisson frequencies; +its character distinctness and conductor bound will be used in zero detection. +Choose a complete set of representatives \(\mathcal U\) modulo \(U_S^6\) among the nonzero +S-integers whose exterior valuations belong to \(\{0,1,\ldots,5\}\), with +\(1\in\mathcal U\). For allowed generators \(r\), the map +\begin{equation} + (u,\mathfrak r)\longmapsto [u r^6] + \quad\text{from }\mathcal U\times\{\text{exterior ideals}\} + \quad\text{to }(R\setminus\{0\})/U_S^6\label{eq:frequency-decomposition} +\end{equation} +is a bijection. Among \(u\in\mathcal U\), only \(u=1\) induces the +principal character, and distinct \(u\)'s induce distinct characters, also +after the fixed twist by \(\eta\). +\end{lemma} + +\begin{proof} +Because \(\mu_6\subset F\), Kummer theory +\cite[Chapter~IV, Theorem~(3.3) and Corollary~(3.6)]{Neukirch99} +identifies \(u\in F^\times\) with +the character +\[ + \kappa_u(\sigma)=\frac{\sigma(u^{1/6})}{u^{1/6}} + \quad\text{of }\Gal(\overline F/F), +\] +whose kernel as a function of \(u\) is exactly \(F^{\times6}\). +Global class-field theory identifies this finite abelian character with a +finite-order Hecke character +\cite[Chapter~VI, Theorem~(5.5) and Proposition~(5.6)]{Neukirch99}. +At an exterior prime +\(\mathfrak p\nmid(u)R\), arithmetic Frobenius satisfies +\[ + \kappa_u(\operatorname{Frob}_{\mathfrak p}) + \equiv (u^{1/6})^{q_{\mathfrak p}-1} + =u^{(q_{\mathfrak p}-1)/6} + =\chi_{\mathfrak p}(u)\pmod{\mathfrak p}. +\] +Reduction is injective on \(\mu_6\), so these are equal as roots of unity. +This proves the asserted identification on good primes. + +At an exterior place, the principal units are sixth powers by Hensel's +lemma. Hence the associated local Hilbert character is trivial on principal +units, giving conductor exponent at most one. The tame Hilbert formula +\cite[Chapter~V, Proposition~(3.4)]{Neukirch99} shows that the pairing of +two units is \(1\), so if \(u\) is a local unit the character is unramified. +At a place in \(S\), it depends only on the class of \(u\) in +the finite group \(F_v^\times/F_v^{\times6}\); the finitely many resulting +local conductors have a fixed maximum. This proves +\Cref{eq:kummer-conductor}. A fixed twist supported in \(S\) only changes +that maximum. Equality of two inducing Hecke characters is equality of +their Kummer characters, since unramified Frobenius elements determine a +finite abelian character. The Kummer kernel gives the stated criterion. +Setting extra prime values to zero merely omits Euler factors and does not +alter the inducing character. + +For a character \(\theta_B\) of \(B\), extend it to \(\mathcal V=E\oplus B\) +by \(\theta_S(eb)=\theta_B(b)\). This is a continuous finite-order character +of \(F_S^\times\) trivial on \(U_S\). If \(\mathfrak a=aR\), define +\(\theta(\mathfrak a)=\theta_S(a)\). It is independent of the generator +and multiplicative. At every exterior place take the unramified local +character sending a uniformizer to \(\theta(\mathfrak p)\), and at \(S\) +take \(\theta_S^{-1}\). Their product is trivial on diagonal \(F^\times\) +by construction and is the required Hecke character. + +To prove \Cref{eq:frequency-decomposition}, divide each exterior valuation +of a nonzero \(h\) by \(6\). This determines an ideal \(\mathfrak r\) and +the exterior valuations of an S-integer \(u_0=h/r^6\) in +\(\{0,\ldots,5\}\). Replacing \(u_0\) by its representative in +\(\mathcal U\) changes \(u_0r^6\) only by an element of \(U_S^6\). +Conversely, the exterior valuations of \([ur^6]\) determine +\(\mathfrak r\) and those of \(u\); equality of two such classes then says +that the two representatives \(u\) differ by \(U_S^6\). Changing an +allowed \(r\) by \(U_S^6\) does not affect its image in the quotient. +This proves bijectivity. + +If \(u\in\mathcal U\) induces the principal character, the first part gives +\(u=t^6\) in \(F\). Its exterior valuations are divisible by \(6\) and +between \(0\) and \(5\), hence are all zero; the same is true for the +exterior valuations of \(t\). Thus \(u\in U_S^6\), giving \(u=1\). +If \(u,v\in\mathcal U\) induce the same character, the differences of their +exterior valuations are divisible by \(6\) and lie in \([-5,5]\), so vanish. +Their quotient is then a sixth power of an S-unit, and \(u=v\). +The fixed twist by \(\eta\) cancels from an equality of twisted characters. +\end{proof} + +We now determine the phases of the Gauss sums. If \(\mathfrak c\) is a +squarefree exterior ideal with allowed generator \(c\), define +\begin{equation} + \gamma_j(\mathfrak c) + =q_{\mathfrak c}^{-1/2} + \sum_{x\bmod\mathfrak c}\chi_{\mathfrak c}(x)^j\e(x/c), + \qquad \gamma_j(R)=1.\label{eq:normalized-gauss} +\end{equation} +The additive character on \(R/\mathfrak c\) here is primitive by +\Cref{lem:s-lattice}. A change \(c\mapsto\epsilon^6c\), with +\(\epsilon\in U_S\), changes the additive argument by a sixth power and is +absorbed by a change of residue variable; hence the sum is independent of +the allowed generator. At an exterior prime, the adelic product formula +gives +\[ + \e(x/p)=\psi_{\mathfrak p}(-x/p). +\] +The reduction of \(\mu_{12}\) is injective there, so +\(q_{\mathfrak p}\equiv1\pmod{12}\) and +\(\chi_{\mathfrak p}(-1)=1\). The minus sign can therefore be suppressed +in all these prime Gauss sums. + +\begin{lemma}[The Gauss correction characters]\label{lem:gauss-characters} +There are a unitary Hecke character \(\Lambda\), unramified outside \(S\), +and a finite-order Hecke character \(G\), unramified outside \(S\), such that +for every exterior prime \(\mathfrak p\), +\begin{equation} + \gamma_2(\mathfrak p)^3=-\Lambda(\mathfrak p),\qquad + G(\mathfrak p)=\overline{\chi_{\mathfrak p}(4)}\, + \gamma_3(\mathfrak p),\qquad + \gamma_1(\mathfrak p)\gamma_2(\mathfrak p) + =-\Lambda(\mathfrak p)G(\mathfrak p). +\label{eq:gauss-corrections} +\end{equation} +At each complex place the infinity type of \(\Lambda\) is purely angular +of exponent \(1\) or \(-1\), in the ordinary angular coordinate. +The values \(\gamma_3(\mathfrak p)\) are signs and depend multiplicatively +on the class of an allowed \(p\) in \(B\). + +For coprime squarefree exterior ideals, +\begin{equation} + \gamma_j(\mathfrak c_1\mathfrak c_2) + =\chi_{\mathfrak c_1}(c_2)^j + \chi_{\mathfrak c_2}(c_1)^j + \gamma_j(\mathfrak c_1)\gamma_j(\mathfrak c_2). +\label{eq:gauss-crt} +\end{equation} +\end{lemma} + +\begin{proof} +First work over a finite field \(k\) of size \(Q\), with a nontrivial +additive character \(\psi_k\), a character \(\chi\) of exact order \(6\), and +\(\rho=\chi^2\). Write +\[ + G_j=\sum_{x\in k}\chi(x)^j\psi_k(x),\qquad + J(\alpha,\beta)=\sum_{x\in k}\alpha(x)\beta(1-x). +\] +For nontrivial characters the usual Gauss inversion and Gauss--Jacobi +identities give +\begin{equation} + G_2G_4=Q,\qquad G_2^2=J(\rho,\rho)G_4,\qquad + (Q^{-1/2}G_2)^3=Q^{-1/2}J(\rho,\rho).\label{eq:cubic-jacobi} +\end{equation} +The first equality uses \(\rho(-1)=1\). Changing \(\psi_k(x)\) to +\(\psi_k(ax)\) multiplies \(G_j\) by \(\overline{\chi(a)}^{\,j}\), so the +cube in \Cref{eq:cubic-jacobi} is independent of the additive character. + +Let \(E_0=\Q(\omega)\), where \(\omega^2+\omega+1=0\). Every ideal of +\(\Z[\omega]\) prime to \(3\) has a unique generator congruent to \(1\) +modulo \(3\); denote it by \(\pi_+(\mathfrak a)\). This follows from class +number one and the bijection of the six units with +\((\Z[\omega]/3)^\times\), and is the convention used in +\cite[\S2.1]{BaierYoung10}. For a prime \(\mathfrak q\nmid6\) with residue +size \(Q\), take +\(\rho(x)\equiv x^{(Q-1)/3}\pmod{\mathfrak q}\). Reduction of the Jacobi +sum gives +\[ + J(\rho,\rho)\equiv + \sum_{x\in\Fq_Q}x^{(Q-1)/3}(1-x)^{(Q-1)/3}=0 + \pmod{\mathfrak q}, +\] +since every monomial has degree strictly between \(0\) and \(Q-1\). +The Gauss identities also give \(|J(\rho,\rho)|^2=Q\). +To determine the unit, put \(t=1-\omega\); the ideals \((t^2)\) and \((3)\) +are equal. Writing \(\rho(x)=\omega^{k(x)}\), with \(k(x)\in\{0,1,2\}\), +and expanding modulo \(t^2\) yields +\[ + J(\rho,\rho)\equiv Q-2 + -t\sum_{x\ne0,1}\bigl(k(x)+k(1-x)\bigr) + \equiv Q-2-2(Q-1)t\equiv-1\pmod3. +\] +Here each value of \(\rho\) occurs \((Q-1)/3\) times on +\(\Fq_Q^\times\), and \(Q\equiv1\pmod3\). Divisibility by +\(\mathfrak q\), the norm equality, and this congruence show +\begin{equation} + J(\rho,\rho)=-\pi_+(\mathfrak q). +\label{eq:eisenstein-jacobi} +\end{equation} +This is also the prime case of the classical cubic Gauss identity +\(g(n)^3=\mu(n)\Norm(n)n\) in +\cite[\S2.2, equation~(11) in arXiv:0804.2233v4]{BaierYoung10}. + +Define on ideals of \(E_0\) prime to \(3\) +\[ + \Lambda_{E_0}(\mathfrak a) + =\frac{\pi_+(\mathfrak a)}{|\pi_+(\mathfrak a)|}. +\] +The generators congruent to \(1\pmod3\) multiply, so this is a +multiplicative ideal character. On a principal ideal generated by +\(a\equiv1\pmod3\) its value is \(a/|a|\); thus it is the unitary Hecke +character of finite conductor dividing \((3)\) and angular infinity type +of exponent \(1\), in ideal-character convention. Set +\(\Lambda=\Lambda_{E_0}\circ\Norm_{F/E_0}\). +If \(\mathfrak p\) lies above \(\mathfrak q\) with residue degree \(f\), +the power definition of the residue character shows that the cubic +character at \(\mathfrak p\) is the lift of that at \(\mathfrak q\) by +the residue-field norm. The Davenport--Hasse Jacobi lifting formula says +\[ + J_{\Fq_{Q^f}}(\rho\circ\Norm,\rho\circ\Norm) + =(-1)^{f-1}J_{\Fq_Q}(\rho,\rho)^f; +\] +its hypotheses hold because \(\rho\), \(\rho\), and \(\rho^2\) are all +nontrivial \cite[Theorem~1.1]{HoshiKanai22}. By +\Cref{eq:eisenstein-jacobi}, the right side is +\(-\pi_+(\mathfrak q)^f\). Since +\(\Norm_{F/E_0}\mathfrak p=\mathfrak q^f\), +\Cref{eq:cubic-jacobi} proves +\(\gamma_2(\mathfrak p)^3=-\Lambda(\mathfrak p)\). +The local norm at infinity is a product of complex coordinates or their +conjugates. Consequently its angular exponent at each complex place of +\(F\) is \(1\) or \(-1\), with no radial factor. Pullback by norm is +unramified wherever \(\Lambda_{E_0}\) is unramified, so its finite conductor +is supported in \(S\). + +For the quadratic sum we use one global additive character to relate the +local signs. For \(a\in F_v^\times\), define the normalized Weil index +\[ + \mathfrak w_v(a)= + \frac{\operatorname{WeilIndex}_{\psi_v}(a x^2)} + {\operatorname{WeilIndex}_{\psi_v}(x^2)}. +\] +It depends only on the square class of \(a\). Weil's product formula and +the Hilbert-symbol identity are +\begin{equation} + \prod_v\mathfrak w_v(a)=1\quad(a\in F^\times),\qquad + \mathfrak w_v(a)\mathfrak w_v(b) + =(a,b)_{2,v}\mathfrak w_v(ab); +\label{eq:weil-index} +\end{equation} +see \cite[Chapter~II, \S28 and Proposition~5 in \S30]{Weil64}. +At an odd conductor-zero place, the local Gauss integral for the Weil index +\cite[Chapter~II, \S27]{Weil64} gives \(\mathfrak w_v(u)=1\) for a unit \(u\) +and +\[ + \mathfrak w_{\mathfrak p}(p^{-1}) + =q_{\mathfrak p}^{-1/2} + \sum_{x\bmod\mathfrak p}\psi_{\mathfrak p}(x^2/p) + =q_{\mathfrak p}^{-1/2} + \sum_{x\bmod\mathfrak p} + \chi_{\mathfrak p}(x)^3\psi_{\mathfrak p}(x/p) + =\gamma_3(\mathfrak p). +\] +For the second equality, the number of square roots of \(x\) is +\(1+\chi_{\mathfrak p}(x)^3\), and the sum of a nontrivial additive +character is zero. The last equality uses +\(\chi_{\mathfrak p}(-1)=1\). + +Put \(\mathfrak w_S=\prod_{v\in S}\mathfrak w_v\). Applying the first +identity of \Cref{eq:weil-index} to \(p^{-1}\), all exterior places other +than \(\mathfrak p\) contribute \(1\), and hence +\[ + \gamma_3(\mathfrak p)=\mathfrak w_S(p^{-1})^{-1}. +\] +On \(B\) the multiplicative defect in the second identity is +\[ + \prod_{v\in S}(a,b)_{2,v}=H_S(a,b)^3=1. +\] +Thus \([a]\mapsto\mathfrak w_S(a^{-1})^{-1}\) is a character of \(B\) +factoring through square classes. It takes values in \(\{\pm1\}\); +equivalently Gauss inversion gives +\(\gamma_3(\mathfrak p)^2=\chi_{\mathfrak p}(-1)^3=1\). +Extend this B-character by \Cref{lem:kummer-characters}. Since \(4\) is an +S-unit, \(\chi_\bullet(4)\) is also a finite-order Hecke character unramified +outside \(S\). Their product, with \(\chi_\bullet(4)\) conjugated, defines +\(G\) and gives its displayed prime value. + +It remains to check the sextic duplication factor. In the finite-field +notation above put \(\phi=\chi^3\). Completing the square and counting +square roots gives +\[ + J(\chi,\chi) + =\overline{\chi(4)}\sum_{t\in k}\chi(1-t^2) + =\overline{\chi(4)}J(\phi,\chi). +\] +In the second equality the part without \(\phi\) vanishes, since \(\chi\) +is nontrivial. Applying the Gauss--Jacobi identity to both sides yields +\[ + G_1G_4=\overline{\chi(4)}G_2G_3. +\] +Together with \(G_2G_4=Q\), this gives +\[ + \gamma_1(\mathfrak p)\gamma_2(\mathfrak p) + =\overline{\chi_{\mathfrak p}(4)}\, + \gamma_3(\mathfrak p)\gamma_2(\mathfrak p)^3 + =-\Lambda(\mathfrak p)G(\mathfrak p). +\] + +Finally use the allowed product \(c_1c_2\) to evaluate the sum at +\(\mathfrak c_1\mathfrak c_2\). Chinese remaindering writes its additive +character as the product of the characters modulo \(\mathfrak c_i\), with +their arguments multiplied respectively by \(c_2^{-1}\) and \(c_1^{-1}\). +Changing the two residue variables contributes +\(\chi_{\mathfrak c_1}(c_2)^j\chi_{\mathfrak c_2}(c_1)^j\). +This proves \Cref{eq:gauss-crt}, including \(j\equiv0\pmod6\) with the +specified zero convention. +\end{proof} + +For later use set +\[ + \vartheta=\overline\Lambda\,\overline G\,\eta. +\] +Its finite conductor is supported in \(S\). If \(\vartheta_v\) denote its +idele-class components, define +\[ + L_S(y)=\prod_{v\in S}\vartheta_v(y_v)^{-1}. +\] +The global character is trivial on \(F^\times\), so for every exterior +ideal with allowed generator \(a\), +\begin{equation} + \vartheta(\mathfrak a)=L_S(a),\qquad L_S(\epsilon)=1 + \quad(\epsilon\in U_S).\label{eq:vartheta-s} +\end{equation} +This is a smooth unitary character of \(F_S^\times\). Finite-order +characters are trivial on the connected group \(\C^\times\); hence the +infinity components contributed by \(G\) and \(\eta\) are trivial. +Those of \(L_S\) are therefore purely angular of exponent \(1\) or \(-1\) +at each complex place. diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/02a-coefficient-family.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/02a-coefficient-family.tex new file mode 100644 index 0000000000000000000000000000000000000000..f2c84a0da5e4bf302633d3b77ca109bd39ae35d2 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/02a-coefficient-family.tex @@ -0,0 +1,179 @@ +\section{The coefficient family and analytic comparison} +\label{sec:coefficient-family} +\label{subsec:coefficient-family} + +The two analytic evaluations use the same family of coefficient sums. We +define that family and state the second moment proved by the theta and +reflection arguments, then explain how the moment enters the zero-free +argument. + +Retain the field \(F\), the final set \(S\), the ring \(R=\mathcal O_S\), +and the generator convention fixed in \Cref{sec:arithmetic}. Unless +explicitly stated otherwise, all ideals in this section are integral +exterior ideals. A roman letter +representing an ideal is always an allowed \(B\)-generator; it need not be +balanced unless this is stated. In particular, products of allowed prime +generators are permitted. The row variable \(m\) below ranges over all of +\(R\). + +For \(\re s>1\), write +\(\zeta_F^S(s)=\sum_{\mathfrak a}q_{\mathfrak a}^{-s}\) for the Dedekind +zeta function with its factors at \(S\) omitted, and set +\[ + L_F^S(s,\eta)=L^S(s,\eta):=\prod_{\mathfrak p\notin S} + (1-\eta(\mathfrak p)q_{\mathfrak p}^{-s})^{-1}. +\] +The latter notation also denotes the usual meromorphic continuation of +this incomplete Hecke \(L\)-function; a principal pole is permitted. All +primes dividing the conductor of \(\eta\) already lie in \(S\), so there +are no further omitted exterior factors for this target character. +For \(t>0\) put +\begin{equation}\label{eq:reflection-gaussian} + P_G(t)=\frac{1}{2\sqrt\pi}\exp\!\left(-\frac{(\log t)^2}{4}\right), + \qquad + \int_0^\infty P_G(t)t^\xi\,\frac{\dd t}{t}=e^{\xi^2} + \quad(\xi\in\C). +\end{equation} +The transform identity follows by putting \(v=\log t\) and completing the +square, first for real \(\xi\) and then by entire continuation. Write +\(\vartheta=\overline\Lambda\,\overline G\,\eta\). For \(m\in R\) +and \(Z>0\) define +\begin{equation}\label{eq:reflection-B-definition} + B_m(Z)= + \sum_{\substack{\mathfrak c\ \mathrm{squarefree}\\ \mathfrak n}} + \frac{\gamma_2(\mathfrak c)}{q_{\mathfrak c}^{1/2}q_{\mathfrak n}} + \vartheta(\mathfrak A)\chi_{\mathfrak A}(m) + P_G\!\left(\frac{q_{\mathfrak A}}{Z}\right), + \qquad + \mathfrak A=\mathfrak c\mathfrak n^3,\quad A=c n^3. +\end{equation} +Here \(c,n,A\) are allowed generators, and \(\mathfrak n\) is unrestricted, +so it may share primes with \(\mathfrak c\). +Every residue-character power has the zero extension on nonunits fixed in +\Cref{sec:arithmetic}, including powers with exponent zero modulo six. +The sum is independent of the permitted generator choices. It converges +absolutely: \(|\gamma_2(\mathfrak c)|=1\) for squarefree \(\mathfrak c\), +and for any \(\sigma>1/2\) the bound \(P_G(t)\ll_\sigma t^{-\sigma}\) +reduces the absolute sum, after a factor \(O_\sigma(Z^\sigma)\), to +\begin{equation}\label{eq:reflection-original-dirichlet-majorant} + \sum_{\mathfrak c,\mathfrak n} + q_{\mathfrak c}^{-1/2-\sigma}q_{\mathfrak n}^{-1-3\sigma} + \leq \zeta_F^S(\sigma+1/2)\zeta_F^S(3\sigma+1)<\infty. +\end{equation} +At \(m=0\), the mask kills every term with \(\mathfrak A\ne R\), so +\(B_0(Z)=P_G(1/Z)\). The factor +\(\gamma_2(\mathfrak c)/(q_{\mathfrak c}^{1/2}q_{\mathfrak n})\) will be +identified in \Cref{sec:theta} as the exterior coefficient supplied by the +normalized local theta formula. + +The comparison uses rows at scale \(XY\), where +\begin{equation}\label{eq:reflection-scales} +\begin{gathered} + a=\frac9{10},\qquad b=\frac{133}{1000},\qquad + M_0=a+b=\frac{1033}{1000},\\ + h_0=1-a=\frac1{10},\qquad X=Z^a,\quad Y=Z^b. +\end{gathered} +\end{equation} +Recall the norm section \(\ell_T\) fixed in \Cref{sec:arithmetic}, for which +\(|\ell_T|_S=T\). If \(\mathscr K\subset F_S^\times\) is compact, set +\[ + \mathscr B_Z(\mathscr K)=R\cap\ell_{Z^{M_0}}\mathscr K. +\] +Thus every local absolute value of \(\ell_{Z^{M_0}}^{-1}m\) is bounded +above and away from zero on this set. These are the compact balanced +annular row sets used below. For \(m\ne0\) decompose its exterior ideal +uniquely as +\[ + (m)_S=\mathfrak h(m)\mathfrak d(m), + \qquad + \mathfrak h(m)=\prod_{v_{\mathfrak p}(m)\geq2} + \mathfrak p^{v_{\mathfrak p}(m)}, + \qquad + \mathfrak d(m)=\prod_{v_{\mathfrak p}(m)=1}\mathfrak p. +\] +The ideal \(\mathfrak d(m)\) is squarefree and coprime to \(\mathfrak h(m)\); +every prime exponent of \(\mathfrak h(m)\) is at least two. We call such an +ideal powerful. + +The theta and reflection arguments in \Cref{sec:theta,sec:reflection} +prove the following estimate. + +\begin{proposition}[Reflected second moment]\label{prop:reflection} +Let \(F\) be a fixed cyclotomic field containing \(\mu_{12}\), let \(S\) and the +arithmetic data be as above, and fix a compact subset +\(\mathscr K\subset F_S^\times\). For every \(\epsilon>0\), every \(Z\geq2\), +and all dyadic lengths \(P,D\geq1\), +\begin{equation}\label{eq:reflection-dyadic-mass} + \sum_{\substack{m\in\mathscr B_Z(\mathscr K)\\ + P\leq q_{\mathfrak h(m)}<2P\\ + D\leq q_{\mathfrak d(m)}<2D}} + |B_m(Z)|^2 + \ll_{F,S,\eta,\mathscr K,\epsilon} + Z^\epsilon P^{1/2}\left(D+\frac{D^2P}{Z}\right). +\end{equation} +In particular, +\begin{equation}\label{eq:reflection-total-mass} + \sum_{m\in\mathscr B_Z(\mathscr K)}|B_m(Z)|^2 + \ll_{F,S,\eta,\mathscr K,\epsilon} Z^{2M_0-1+\epsilon}. +\end{equation} +The estimate is uniform if \(\mathscr K\) is multiplied by an element of any +fixed compact subset of the norm-one group of \(F_S^\times\). +\end{proposition} + +Here is the analytic comparison proved in \Cref{sec:poisson,sec:zero-free}. +The first of those sections constructs a scalar +\(I_Z=I(Z^a,Z^b,Z)\) by pairing \(B_m(Z)\), on a fixed compact annular +row set of scale \(XY\), with a normalized sum of +\(\chi_{\mathfrak s}(d)\e(-md/s)\), where +\(q_{\mathfrak s}\asymp Y\) and \(d\in(R/\mathfrak s)^\times\). +It specifies the weights and the average over \(F_S^1/U_S^6\). +The whole additive factor has squared row mass \(O(Y^{-1})\), uniformly in +that average. Hence \Cref{eq:reflection-total-mass} and Cauchy--Schwarz give, +for every \(\epsilon>0\), +\[ + I_Z\ll Z^{(2M_0-1-b)/2+\epsilon} + =Z^{933/2000+\epsilon}. +\] +The constants here and below may depend on the fixed \(F,S,\eta\) and the +fixed weights, as well as on any displayed loss. + +Poisson summation gives a second evaluation. Its nonzero frequencies are +grouped by \(u\in\mathcal U\) through \Cref{eq:frequency-decomposition}. +Only the principal class \(u=1\), the class of sixth powers, contributes +both numerator poles in the auxiliary Mellin variables: \(w=1\) and +\(z=1/6\). The Euler calculation constructs a specific principal correction +\(\mathcal H_\eta(s)\) by specializing the correction for \(u=1\) at +\((w,z)=(1,1/6)\). It proves that \(\mathcal H_\eta\) is holomorphic and +nowhere zero on an open neighborhood of the closed half-plane +\(\re s\ge .998\), and is bounded on that closed half-plane uniformly in +height. The double residue equals \(c_0f_\eta(Z)\), where \(c_0\ne0\) and +\[ + f_\eta(Z)=\frac1{2\pi i}\int_{(2)} + Z^{s-8/15}e^{(s-5/6)^2} + \frac{\mathcal H_\eta(s)}{L^S(s,\eta)}\,ds + \qquad(Z>0). +\] +The exponent left by the two residues is +\(a/2+s-1+h_0/6=s-8/15\), whose value at \(s=1\) is \(7/15\). + +The zero-detection argument proves, for sufficiently large \(Z\), +\[ + I_Z=c_0f_\eta(Z)+O(Z^{7/15-.0004}). +\] +Since \(933/2000=7/15-1/6000\), this identity and the preceding bound, +with the stated losses, give \(f_\eta(Z)\ll Z^{7/15-1/20000}\). +Moving the original \(s\)-line in the inverse Mellin integral to the right +gives \(f_\eta(Z)=O_A(Z^A)\) for every \(A>0\) and \(01-1/20000\). Fourier inversion on \(\re s=2\), followed by +meromorphic continuation, identifies it throughout that half-plane with +\[ + e^{(s-5/6)^2}\frac{\mathcal H_\eta(s)}{L^S(s,\eta)}. +\] +Its numerator is holomorphic and nowhere zero there, so \(L^S(s,\eta)\) +has no zero there; a principal pole only gives a zero of the reciprocal. +The numerical saving \(1/20000\) is absolute; the constants and thresholds +may depend on \(F,S,\eta\). The later +sections supply the residue calculation and the contour and Fourier +justifications. diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/03-theta.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/03-theta.tex new file mode 100644 index 0000000000000000000000000000000000000000..d1fb734792b74e137d10b68624949ff75dc4d0c9 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/03-theta.tex @@ -0,0 +1,775 @@ +\section{The normalized cubic theta input and its radial profiles} +\label{sec:theta} + +To prove the reflected moment, we first realize the exterior coefficient +\(\gamma_2(\mathfrak c)/(q_{\mathfrak c}^{1/2}q_{\mathfrak n})\) +in the family \(B_m\). The cubic theta representation used here has +one-dimensional local Whittaker spaces. Its selected central datum, +together with normalized induction, gives the factor \(q_{\mathfrak p}^{-1}\) +for a triple valuation step. Compatible rational, torus, root, and compact +sections fix the unit factor at valuation \(e\) as +\(\chi_{\mathfrak p}(u)^{2e}\). We state this normalized input first and +then identify its exterior coefficient. The verification of the +Kazhdan--Patterson specialization is in \Cref{app:whittaker}. + +Throughout this section \(F\) is the fixed cyclotomic field from the +preceding section, and \(\Adele_F\) is its adele ring. In particular, +\(\mu_{12}\subset F\). We use the standard trace character +\(\psi:F\backslash\Adele_F\longrightarrow\C^\times\), and choose the +self-dual additive Haar measures, for which +\(F\backslash\Adele_F\) has volume one. Write +\[ + n(x)=\begin{pmatrix}1&x\\0&1\end{pmatrix},\qquad + \bar n(x)=\begin{pmatrix}1&0\\x&1\end{pmatrix},\qquad + a(y)=\begin{pmatrix}y&0\\0&1\end{pmatrix},\qquad + w=\begin{pmatrix}0&1\\1&0\end{pmatrix}. +\] +The same notation will denote the specified lifts below. A Whittaker +functional with character \(\psi_v\) means a linear functional \(\lambda\) +satisfying +\(\lambda(\pi(n(x))\varphi)=\psi_v(x)\lambda(\varphi)\). + +\subsection{The automorphic input and its exterior coefficients} + +We use the determinant-modified cover of Kazhdan and Patterson with +parameter \(c=2\), with the ordinary torus section and the canonical root +splittings. The embedding \(\iota:\mu_3(F)\hookrightarrow\C^\times\) +is oriented as in the exterior cocycle formula below. If \(a_v(y)\) +denotes the chosen torus lift, define the effective scalar +\(\mathcal C_v(x,y)\) by +\[ + \pi(a_v(x))\pi(a_v(y)) + =\mathcal C_v(x,y)\pi(a_v(xy)) +\] +in a genuine representation \(\pi\). Thus this notation includes the +chosen embedding of the central \(\mu_3\). + +\begin{proposition}[Cubic theta input]\label{prop:theta-input} +For the fixed field \(F\), there is a threefold central cover +\[ + 1\longrightarrow\mu_3 + \longrightarrow\widetilde{\PGL}_2(\Adele_F) + \longrightarrow\PGL_2(\Adele_F)\longrightarrow1 +\] +which splits over \(\PGL_2(F)\), and an irreducible genuine automorphic +representation +\(\Theta=\bigotimes'_v\Theta_v\) on this cover, with the following +properties. + +\begin{enumerate}[label=\textup{(\roman*)}] +\item \emph{Global representation and Fourier expansion.} + The cover is the quotient of the \(c=2\) cover + \(\widetilde{\GL}_2^{(2)}\) by the ordinary split scalar subgroup. + The inducing character of \(\Theta_v\), on the center of the covering + torus before this quotient, is + \begin{equation}\label{eq:theta-central-datum} + \omega_v\!\left(\zeta\,s_v(\operatorname{diag}(x^3,y^3))s_v(zI)\right) + =\iota(\zeta)\left|x^3/y^3\right|_v^{1/6}. + \end{equation} + Here \(s_v\) is the ordinary torus section. The normalized inducing + norm exponents are \(1/6,-1/6\). The local exceptional representation is the irreducible + quotient for these exponents and the irreducible subrepresentation for + their Weyl conjugates. + Every \(\Theta_v\) has a one-dimensional Whittaker space. + The global functional + \[ + \lambda(\varphi)= + \int_{F\backslash\Adele_F}\theta_\varphi(n(x)) + \overline{\psi(x)}\,\dd x + \] + on the automorphic realization \(\varphi\mapsto\theta_\varphi\) is + nonzero. The representation is spherical at all but finitely many + finite places. + There is a preliminary pure tensor \(\varphi^\circ\) on which + \(\lambda\) is nonzero. Its finitely many nondistinguished finite + places lie in the final fixed \(S\), along with the archimedean places, + the places over \(6\), and the places where \(\psi_v\) has nonzero conductor. + + For a pure tensor + \(\varphi=\varphi_S\otimes\bigotimes_{v\notin S}\varphi_v\), + with the distinguished spherical vector at almost every \(v\notin S\), + the global Whittaker function factors as + \begin{equation}\label{eq:theta-normalized-factorization} + \mathcal W_\varphi(g) + :=\lambda(\pi(g)\varphi) + =\mathcal W_{S,\varphi_S}(g_S) + \prod_{v\notin S}W_{v,\varphi_v}(g_v). + \end{equation} + Each exterior spherical functional is normalized at one, so only + finitely many factors in this product differ from one for adelic \(g\). + The functional at \(S\) factors at its complex places; at its finite + places it may be retained as a finite sum of products of local Whittaker + functionals. For a smooth automorphic vector, + \begin{equation}\label{eq:theta-fourier-expansion} + \theta_\varphi(g)=\theta_{\varphi,N}(g) + +\sum_{\nu\in F^\times}\mathcal W_\varphi(a(\nu)g),\qquad + \theta_{\varphi,N}(g) + =\int_{F\backslash\Adele_F}\theta_\varphi(n(x)g)\,\dd x . + \end{equation} + This expansion converges absolutely and locally uniformly on compact + sets of \(g\). + +\item \emph{Exact lifts and exterior local values.} + On the quotient the chosen lifts satisfy + \begin{equation}\label{eq:theta-lift-identities} + \begin{gathered} + n(x)a(y)=a(y)n(x/y),\qquad + wa(y)w^{-1}=a(y^{-1}),\\ + wn(H)=n(H^{-1})a(-H^{-2})\bar n(H^{-1}). + \end{gathered} + \end{equation} + In the last identity \(H\ne0\). At an exterior prime, when \(H^{-1}\) + is integral the last factor is compact, and a compact diagonal may absorb + the minus sign. + + For every exterior prime \(\mathfrak p\), choose a uniformizer + \(\varpi\), put \(q_{\mathfrak p}=\Norm\mathfrak p\), and use the + compact splitting on \(\PGL_2(\mathcal O_{\mathfrak p})\). + The orientation of \(\iota\), equivalently that of the Hilbert-symbol + convention, can be fixed so that the scalar by which the torus + cocycle acts in a genuine representation is + \begin{equation}\label{eq:theta-effective-cocycle} + \mathcal C_{\mathfrak p}(\varpi,u)=\chi_{\mathfrak p}(u)^{-2} + \qquad(u\in\mathcal O_{\mathfrak p}^{\times}). + \end{equation} + The compact splitting agrees with the chosen sections on unit + diagonals, Weyl elements, and integral upper and lower root elements. + Define + \[ + g_{2,\mathfrak p} + =q_{\mathfrak p}^{-1/2} + \sum_{x\bmod\mathfrak p} + \chi_{\mathfrak p}(x)^2\psi_{\mathfrak p}(x/\varpi). + \] + Then \(|g_{2,\mathfrak p}|=1\). If \(W_{\mathfrak p}\) is the + spherical Whittaker function normalized by \(W_{\mathfrak p}(1)=1\), + then, for \(u\in\mathcal O_{\mathfrak p}^{\times}\), + \begin{equation}\label{eq:theta-local-values} + W_{\mathfrak p}(a(\varpi^e u)) + =\chi_{\mathfrak p}(u)^{2e} + \begin{cases} + 0,&e<0,\\ + q_{\mathfrak p}^{-k},&e=3k,\quad k\ge0,\\ + q_{\mathfrak p}^{-k-1/2}g_{2,\mathfrak p}, + &e=3k+1,\quad k\ge0,\\ + 0,&e=3k+2,\quad k\ge0. + \end{cases} + \end{equation} + In particular a shift by three nonnegative valuations multiplies the + value by \(q_{\mathfrak p}^{-1}\). + +\item \emph{Complex local behavior.} + At a complex place, \(\Theta_v\) is the ordinary spherical + \(\PGL_2(\C)\) principal series with normalized characters + \((|\cdot|_v^{1/6},|\cdot|_v^{-1/6})\), where \(|y|_v\) is the square + of the ordinary modulus. Its spherical Whittaker function is a + nonzero constant times + \begin{equation}\label{eq:theta-complex-bessel} + |y|_v^{1/2}K_{1/3}\bigl(c_v|y|_v^{1/2}\bigr),\qquad c_v>0. + \end{equation} + The constant term of the automorphic theta function belongs to the + Weyl-conjugate torus induction. In particular its dependence on a + complex diagonal \(a(y)\) is radial, also when the vector on its right + has been acted on by a fixed compact element. +\end{enumerate} +\end{proposition} + +In \eqref{eq:theta-normalized-factorization}, the condition +\(W_{\mathfrak p}(1)=1\) fixes every exterior spherical scalar; the +remaining normalization belongs to the \(S\)-functional. +\Cref{app:whittaker} derives this product from the nonzero global +functional and local uniqueness, including the correction to the original +global formula. The noncompact norm +integrations below require the separate absolute bounds in +\Cref{lem:sheet-profile}. + +\begin{corollary}[The exterior theta coefficient] +\label{cor:theta-exterior-coefficient} +At every exterior prime use an allowed prime generator as the uniformizer +in \Cref{prop:theta-input}. Let \(A\in R\setminus\{0\}\) be an allowed +generator of an integral exterior ideal \(\mathfrak A\). Then +\begin{equation}\label{eq:theta-exterior-coefficient} + \prod_{\mathfrak p\notin S}W_{\mathfrak p}(a(A)) + = + \begin{cases} + \displaystyle + \frac{\gamma_2(\mathfrak c)} + {q_{\mathfrak c}^{1/2}q_{\mathfrak n}}, + &\mathfrak A=\mathfrak c\mathfrak n^3,\quad + \mathfrak c\ \text{squarefree},\\[2mm] + 0,&\text{otherwise}. + \end{cases} +\end{equation} +In the first case \(\mathfrak c,\mathfrak n\) are uniquely determined; +they need not be coprime. +\end{corollary} + +\begin{proof} +The local support in \eqref{eq:theta-local-values} permits precisely the +valuations \(3k\) and \(3k+1\). They give the unique decomposition +\(\mathfrak A=\mathfrak c\mathfrak n^3\), with \(\mathfrak c\) squarefree, +and magnitudes \(q_{\mathfrak p}^{-k}\) and +\(q_{\mathfrak p}^{-k-1/2}\), respectively. Use allowed product +generators \(A=cn^3\); any other allowed \(A\) differs by an element of +\(U_S^6\), which has no effect on the local values. At a prime +\(\mathfrak p\mid\mathfrak c\), the local unit factor is +\(\chi_{\mathfrak p}(c/p)^2\). The unit from \(n^3\) has exponent +divisible by six in the symbol, including when \(\mathfrak p\mid\mathfrak n\), +and disappears. The normalized prime sum \(g_{2,\mathfrak p}\) equals +\(\gamma_2(\mathfrak p)\): the local additive sign from +\Cref{eq:normalized-gauss} has no effect because +\(\chi_{\mathfrak p}(-1)=1\). The product of the unit factors and these +prime sums is \(\gamma_2(\mathfrak c)\) by \Cref{eq:gauss-crt}. +Multiplying the magnitudes gives \eqref{eq:theta-exterior-coefficient}. +\end{proof} + +Thus theta supplies the exterior factor in \(B_m\). The ideal character +will enter through an \(S\)-average, and finite upper-unipotent translations +will insert the moving sextic twist. The chosen \(S\)-Whittaker function +first supplies a radial profile; an exact radial average will replace it +by \(P_G\). We prepare those \(S\)-components next. + +The arithmetic power classes also describe the torus factor at \(S\). +Put \(\mathcal C_S=\prod_{v\in S}\mathcal C_v\). At an exterior prime, +\Cref{eq:theta-effective-cocycle} and the tame Hilbert-symbol formula give +\(\mathcal C_{\mathfrak p}(x,y)=(x,y)_{6,\mathfrak p}^{2}\) for all +local \(x,y\). For rational \(a,b\), the product formulas for the chosen +torus lifts and for the Hilbert symbol therefore imply +\[ + \mathcal C_S(a,b) + =\prod_{\mathfrak p\notin S}\mathcal C_{\mathfrak p}(a,b)^{-1} + =H_S(a,b)^2. +\] +Both sides are continuous on \(F_S^\times\times F_S^\times\), so weak +approximation gives +\begin{equation}\label{eq:reflection-S-cocycle} + \mathcal C_S(x,y)=H_S(x,y)^2 + \qquad(x,y\in F_S^\times). +\end{equation} +In particular this factor depends only on the classes in the fixed finite +group \(\mathcal V=F_S^\times/F_S^{\times6}\). The local sections here +are the same ones used by the rational splitting. + +\subsection{Bounds and a prescribed finite Whittaker function} + +For \(v\mid\infty\), define the derivative in the logarithm of the norm by +\[ + \mathcal D_v f(y) + =\left.\frac{\dd}{\dd t}f(e^{t/2}y)\right|_{t=0}. +\] +The factor \(1/2\) makes +\(|e^{t/2}y|_v=e^t|y|_v\). + +We give both the Bessel calculation and bounds for the fixed rotations +of the vectors that we will use. Choose complex coordinates in which +\(\psi_v(z)=e^{i c_v\operatorname{Re}z}\), with \(c_v>0\); a harmless +rotation and a change of sign give this form. Put \(\varrho=|y|_v^{1/2}\). +In the dominant normalized principal series set +\(\nu_0=1/2+1/6=2/3\). Its spherical section, restricted to +\(wn(x)a(y)\), is a constant times +\[ + \varrho^{2\nu_0}(\varrho^2+|x|_{\mathrm{ord}}^2)^{-2\nu_0}, +\] +where \(|x|_{\mathrm{ord}}\) is the ordinary complex modulus. The +Jacquet integral is absolutely convergent because \(2\nu_0>1\). +Using the Laplace representation of the negative power and the +two-dimensional Gaussian Fourier integral gives +\begin{align*} + \varrho^{2\nu_0}\int_{\R^2} + \frac{e^{-ic_v x_1}}{(\varrho^2+x_1^2+x_2^2)^{2\nu_0}} + \,\dd x_1\dd x_2 + &=\frac{\pi \varrho^{2\nu_0}}{\Gamma(2\nu_0)} + \int_0^\infty t^{2\nu_0-2} + e^{-t\varrho^2-c_v^2/(4t)}\,\dd t\\ + &=c'_v \varrho K_{2\nu_0-1}(c_v\varrho) + =c'_v \varrho K_{1/3}(c_v\varrho),\qquad c'_v\ne0. +\end{align*} +The Laplace and Gaussian integrals may be interchanged absolutely: +after integrating the Gaussian without its phase the integrand at +zero is \(O(t^{2\nu_0-2})\), with exponent greater than \(-1\). +This proves \eqref{eq:theta-complex-bessel}. + +For an upper root translate, Whittaker equivariance gives +\begin{equation}\label{eq:theta-lie-multiplier} + W_{\pi(n(z))\varphi}(a(y))=\psi_v(yz)W_\varphi(a(y)). +\end{equation} +Suitable complex linear combinations of the two real Lie derivatives +in \(z\) therefore multiply the spherical function by \(y\) and by +\(\bar y\), respectively. Write +\(\omega_m(y)=(y/|y|_v^{1/2})^m\) for \(m\in\{1,-1\}\). +After multiplying each derivative vector by a nonzero constant, its +Whittaker function is +\begin{equation}\label{eq:theta-complex-derivative} + W_{v,m}(a(y)) + =|y|_v\,\omega_m(y)K_{1/3}(c_v|y|_v^{1/2}). +\end{equation} + +Here are uniform bounds for every vector in a fixed finite list +consisting of these derivative vectors and their fixed compact +rotations. A rotation of a first Lie derivative of the spherical +vector is another first Lie derivative, since the spherical vector +is compact invariant and +\(\pi(k)\dd\pi(X)=\dd\pi(\operatorname{Ad}(k)X)\pi(k)\). +For a fixed real Lie algebra element \(X\), differentiate the +spherical section using +\[ + f^\circ(g)= + \left(\frac{|\det g|_{\mathrm{ord}}} + {|g_{21}|_{\mathrm{ord}}^2+|g_{22}|_{\mathrm{ord}}^2}\right)^{2\nu_0}. +\] +On \(wn(x)a(y)\), its first Lie derivative and any number of +\(\mathcal D_v\)-derivatives are the preceding spherical integrand +times a finite sum of bounded homogeneous rational functions of +\((y,x)\). More precisely, for every norm derivative order \(b\) +and every \(x_1\)-derivative order \(h\), the resulting derivative is +bounded by +\[ + C_{b,h} \varrho^{2\nu_0} + (\varrho^2+|x|_{\mathrm{ord}}^2)^{-2\nu_0-h/2}. +\] +This follows by differentiating the displayed quotient: each +derivative adds a polynomial numerator of the same degree as its +denominator, and an \(x_1\)-derivative lowers the homogeneous degree +by one. The integral of the bound with \(h=0\) is +\(O(\varrho^{2-2\nu_0})=O(\varrho^{2/3})\). For \(\varrho\ge1\), integrating by parts +\(h\) times against \(e^{-ic_vx_1}\) instead gives +\(O_{b,h}(\varrho^{2/3-h})\). These integrable derivative bounds justify the +integrations by parts by cutoff and limit. Consequently, for every +integer \(j\ge0\) and every \(A>0\), the fixed list satisfies +\begin{equation}\label{eq:theta-complex-bounds} + |\mathcal D_v^j W(a(y))| + \le C_{A,j}\min\bigl(|y|_v^{1/3},|y|_v^{-A}\bigr). +\end{equation} + +At a finite place the corresponding elementary bounds hold for every +fixed smooth vector. To be explicit, let +\(W_\varphi(y)=\lambda_v(\pi_v(a(y))\varphi)\). There is an additive +open subgroup \(J\subset F_v\) for which \(n(J)\) fixes \(\varphi\). +The identity +\[ + W_\varphi(y)=\psi_v(yx)W_\varphi(y)\qquad(x\in J) +\] +forces \(W_\varphi(y)=0\) when \(|y|_v\) is sufficiently large. +For the other end, put \(Q_v=\#(\mathcal O_v/\mathfrak p_v)\) and +\(z_v=a(\varpi_v^3)\). This element is central in the covering +torus. By the two-cell Jacquet calculation +\cite[Proposition~I.2.1, p.~61, and Theorem~I.2.9(e), p.~72]{KP84}, the two +constituents of the unnormalized ordinary Jacquet module of the +principal series have \(z_v\)-eigenvalues of absolute values +\[ + Q_v^{-1}\quad\hbox{and}\quad Q_v^{-2}. +\] +These are the exponents \(1/2-1/6\) and \(1/2+1/6\) on a triple +valuation step. Exactness of the ordinary Jacquet functor gives the +same possible eigenvalues on the exceptional subquotient. Hence a +polynomial all of whose roots have those absolute values annihilates +the action of \(z_v\) on this Jacquet module. + +For completeness, this gives a recurrence on the Whittaker values +themselves. If \(P\) is such an annihilating polynomial, then +\[ + P(\pi_v(z_v))\varphi + =\sum_{i=1}^h\bigl(\pi_v(n(x_i))-1\bigr)\varphi_i +\] +for finitely many \(x_i,\varphi_i\). Apply +\(\lambda_v\pi_v(a(y))\). For all sufficiently small \(|y|_v\) the +right side is zero because every \(\psi_v(yx_i)\) is one. Cube +cocycles are trivial, so the left side is the recurrence \(P(E)W_\varphi(y)\), +where \(E W_\varphi(y)=W_\varphi(y\varpi_v^3)\). +On each valuation progression its solutions are sums of a polynomial +in the valuation times the powers of the roots of \(P\). +The unit variable ranges over only finitely many classes after +restricting to a sufficiently small unit subgroup which fixes the +vector and consists of cubes. It follows that, for every fixed +\(\delta<1/3\), +\begin{equation}\label{eq:theta-finite-bounds} + W_\varphi(y)=0\quad(|y|_v>C_\varphi),\qquad + |W_\varphi(y)|\le C_{\varphi,\delta}|y|_v^\delta + \quad(|y|_v\le1). +\end{equation} +All constants in this paragraph may depend on the place, the vector, +and the functional. For a fixed finite list one may choose a common +positive \(\delta\). The local twisted Jacquet quotients of +\(\Theta_v\) are finite dimensional by the quotient argument in +\Cref{subsec:theta-global-normalization}. Thus the same conclusions +hold term by term for the finite sum of products at \(S\) in +\Cref{prop:theta-input}. + +\begin{lemma}[Finite torus cutoff]\label{lem:finite-whittaker-cutoff} +Let \(S_f\) be a finite set of nonarchimedean places and let +\(\lambda_f\ne0\) be a Whittaker functional with character +\(\prod_{v\in S_f}\psi_v\) on a smooth representation of the product +of the local covers. For a vector \(\varphi\), put +\[ + W_\varphi(y)=\lambda_f\!\left(\pi\bigl((a(y_v))_{v\in S_f}\bigr) + \varphi\right), + \qquad y\in\prod_{v\in S_f}F_v^\times. +\] +Given any open subgroup \(H_f\) of the product of the unit groups, +there are a compact open product subgroup \(U\subset H_f\) and a +smooth vector \(\varphi_U\) for which +\[ + W_{\varphi_U}(y)=\one_U(y) + \qquad\left(y\in\prod_{v\in S_f}F_v^\times\right). +\] +\end{lemma} + +\begin{proof} +Choose a vector with \(\lambda_f(\varphi)\ne0\). The map +\(y\mapsto W_\varphi(y)\) is locally constant near one. Choose a +product of sufficiently small groups \(1+\mathfrak p_v^{m_v}\), +contained in \(H_f\), on which it is the constant +\(\lambda_f(\varphi)\); call that product \(U\). As a subset of the +additive product \(\prod F_v\), \(U\) is compact and open. +By additive Fourier inversion choose a Schwartz--Bruhat function +\(\phi\) with +\[ + \int_{\prod F_v}\phi(t)\prod_v\psi_v(y_vt_v)\,\dd t=\one_U(y). +\] +The vector \(\varphi'=\int\phi(t)\pi(n(t))\varphi\,\dd t\) is a +well-defined smooth vector: its integrand is locally constant on the +compact support of \(\phi\), so the integral is a finite linear +combination of vectors. The split root relation gives +\[ + W_{\varphi'}(y) + =W_\varphi(y)\int\phi(t)\prod_v\psi_v(y_vt_v)\,\dd t + =W_\varphi(y)\one_U(y). +\] +Divide \(\varphi'\) by \(\lambda_f(\varphi)\). +\end{proof} + +Here is the choice at \(S\) that will be used later. Write +\(S_f=S\setminus S_\infty\). Let \(\mathscr L\) be any fixed finite +family of smooth unitary characters \(\mathcal L_S\) of +\(F_S^\times\), each trivial on \(U_S\), radially trivial at infinity, +and with angular component \(\omega_{m_v}\), \(m_v\in\{1,-1\}\), at +each complex place. This permits the arithmetic character \(L_S\) +and its finitely many finite-order twists used below. Factor the +nonzero global functional at its complex places, leaving a nonzero +functional at the finite places of \(S\). Apply +\Cref{lem:finite-whittaker-cutoff} with an open subgroup contained in +the sixth powers and in the intersection of the kernels of the finite +components of all \(\mathcal L_S\in\mathscr L\). We obtain one product +\(U=\prod_{v\in S_f}U_v\) and one finite vector. For each +\(\mathcal L_S\), choose at each complex place the derivative of its +angular type. The functions in \eqref{eq:theta-complex-derivative} +are nonzero at one, and their normalization gives an \(S\)-Whittaker +function +\begin{equation}\label{eq:theta-chosen-S-function} + W_{S,\mathcal L_S}(a(y))=\one_U(y_f) + \prod_{v\mid\infty}|y_v|_v\,\omega_{m_v}(y_v) + K_{1/3}(c_v|y_v|_v^{1/2}). +\end{equation} +In particular \(\mathcal L_S(y)^{-1}W_{S,\mathcal L_S}(a(y))\) is the +product of the same radial Bessel factors and \(\one_U(y_f)\) for +every member of the family. The radial profile and its Gaussian +averaging kernel below may therefore be chosen in common for this +finite family. + +The complex constant term is radial by \Cref{prop:theta-input}, whereas +\(\mathcal L_S^{-1}\) has a nontrivial angular component. We will use this +distinction in \Cref{sec:reflection}, after the actual weighted theta sums +have been shown to descend to the quotient on which they are integrated. + +\subsection{Integration on norm sheets} + +Recall that \(r=\#S_\infty=[F:\Q]/2\), and define +\[ + X_u=\{y\in F_S^\times:|y|_S=u\}\qquad(u>0). +\] +For \(d\in F_S^\times\), put \(q_d=|d|_S\). Multiplication by \(d\) +sends \(X_l\) to \(X_{q_dl}\), and inversion sends \(X_l\) to \(X_{1/l}\). +The forward and reflected theta terms will therefore lead to sheet +integrals at norms \(q_dl\) and \(q_d/l\), respectively. We now bound +such one-variable profiles, including absolute envelopes that will justify +the ideal sums and the radial average. The two signed orbit substitutions +will be proved with their actual integrands in \Cref{sec:reflection}. + +There is at least one complex place. Use the norm section \(\ell_u\) from +\Cref{sec:arithmetic}; it satisfies \(|\ell_u|_S=u\) and +\(\ell_u\ell_v=\ell_{uv}\). Choose multiplicative Haar measures and Haar +measure on \(X_1\) so that +\begin{equation}\label{eq:sheet-disintegration} + \int_{F_S^\times}f(y)\,\dd^\times y + =\int_0^\infty\int_{X_u}f(y)\,\dd^\times_u y\,\frac{\dd u}{u}, +\end{equation} +where the measure on \(X_u\) is the translate by \(\ell_u\) of that +on \(X_1\). At a complex place we take angle measure of total mass +one, so its multiplicative measure is a fixed constant times +\(\dd t/t\) times angle measure, with \(t=|y_v|_v\). +At a finite place every valuation shell has the same multiplicative +measure. + +\begin{lemma}[Norm-sheet profiles]\label{lem:sheet-profile} +Let \(\mathcal I\) be a finite index set. For every +\(\kappa\in\mathcal I\), let \(f_\kappa\) be a finite sum of products +\[ + f_\kappa(y)=\sum_{\ell=1}^{L_\kappa} + \prod_{v\in S}f_{\kappa,\ell,v}(y_v), + \qquad y\in F_S^\times. +\] +Assume that for one \(\delta>0\) all these local factors satisfy the +following conditions. + +\begin{enumerate}[label=\textup{(\alph*)}] +\item At a finite \(v\), \(f_{\kappa,\ell,v}\) is locally constant, + vanishes for \(|y_v|_v>C_v\), and is + \(O(|y_v|_v^\delta)\) for \(|y_v|_v\le1\). +\item At a complex \(v\), the factor is smooth on \(F_v^\times\) and, for + every \(j\ge0\) and \(A>0\), + \[ + |\mathcal D_v^j f_{\kappa,\ell,v}(y_v)| + \le C_{A,j}\min(|y_v|_v^\delta,|y_v|_v^{-A}). + \] +\end{enumerate} +All constants may depend on the finite family. Define +\[ + H_\kappa(u)=\int_{X_u}f_\kappa(y)\,\dd^\times_u y,\qquad + D=u\frac{\dd}{\dd u}. +\] +Fix one complex place \(v_0\), and define the nonnegative absolute +derivative profile +\[ + H^{\mathrm{abs}}_{\kappa,j}(u) + =\sum_{\ell=1}^{L_\kappa}\int_{X_u} + |\mathcal D_{v_0}^j f_{\kappa,\ell,v_0}(y_{v_0})| + \prod_{v\ne v_0}|f_{\kappa,\ell,v}(y_v)|\,\dd^\times_u y. +\] +Then \(H_\kappa\) is smooth, and for every \(j\ge0\) and +\(\sigma>-\delta\), +\begin{equation}\label{eq:sheet-weighted-bound} + |D^jH_\kappa(u)|\le H^{\mathrm{abs}}_{\kappa,j}(u) + \le C_{\sigma,j}u^{-\sigma}\qquad(u>0). +\end{equation} +In particular, for any \(0<\alpha<\delta\) and every \(A>0\), +\begin{equation}\label{eq:sheet-two-ended-bound} + |D^jH_\kappa(u)|+H^{\mathrm{abs}}_{\kappa,j}(u) + \le C_{A,j}\min(u^\alpha,u^{-A}). +\end{equation} +The conclusions are unchanged if each \(f_\kappa\) is multiplied by +a fixed smooth unitary character of \(F_S^\times\) and by a bounded function +of its class in the finite group \(\mathcal V\). + +Let \(\mathcal K:(0,\infty)\to\C\) have every absolute +moment +\[ + M_\beta(\mathcal K)=\int_0^\infty|\mathcal K(T)|T^\beta + \,\frac{\dd T}{T}<\infty + \qquad(\beta\in\R). +\] +Both convolutions +\[ + \int_0^\infty\mathcal K(T)H_\kappa(uT)\,\frac{\dd T}{T}, + \qquad + \Phi_\kappa(u):=\int_0^\infty\mathcal K(T)H_\kappa(u/T) + \,\frac{\dd T}{T} +\] +and their norm derivatives obey \eqref{eq:sheet-two-ended-bound}, +with constants depending also on the indicated moments. For every +\(j\), the two absolute majorants +\(\int|\mathcal K(T)|H^{\mathrm{abs}}_{\kappa,j}(uT^{\pm1})\,\dd T/T\) +satisfy the same two-ended size bound. +For any \(x_i>0\), complex \(a_i\), indices +\(\kappa_i\in\mathcal I\), \(\tau\in\{1,-1\}\), and +\(\sigma>-\delta\), one has the absolute bound +\begin{equation}\label{eq:sheet-absolute-sum} +\begin{split} + &\int_0^\infty|\mathcal K(T)| + \sum_i|a_i|H^{\mathrm{abs}}_{\kappa_i,0}(u x_iT^\tau) + \,\frac{\dd T}{T}\\ + &\hspace{12mm}\le + C_{\sigma,0}u^{-\sigma}M_{-\tau\sigma}(\mathcal K) + \sum_i|a_i|x_i^{-\sigma}. +\end{split} +\end{equation} +In particular the sum and integral may be interchanged if the final +series is finite. +\end{lemma} + +\begin{proof} +For a finite place, let \(A_v(Q_v^{-k})\) be the supremum of the +absolute local factor on its \(k\)-th valuation shell. Condition +(a) implies, for every \(\sigma>-\delta\), +\[ + \sum_{k\in\Z} A_v(Q_v^{-k})Q_v^{-\sigma k}<\infty: +\] +there are only finitely many negative \(k\), and the positive tail +is geometric with ratio \(Q_v^{-(\delta+\sigma)}\). +At a complex place condition (b) likewise gives +\[ + \int_{F_v^\times}|f_v(y)|\,|y|_v^\sigma\,\dd^\times y<\infty, + \qquad + \sup_{y\in F_v^\times}|y|_v^\sigma + |\mathcal D_v^j f_v(y)|<\infty. +\] +These statements remain valid for all factors in the fixed finite +family. + +Use \(t_{v_0}=|y_{v_0}|_{v_0}\) as the dependent norm in +\eqref{eq:sheet-disintegration}. On \(X_u\), +\[ + t_{v_0}=u\prod_{v\ne v_0}|y_v|_v^{-1}, +\] +and the sheet measure is the product of the angle measure at \(v_0\) +and the multiplicative measures at the other places, up to its fixed +normalization. Multiplying the absolute integrand by +\(u^\sigma=\prod_v|y_v|_v^\sigma\), take the weighted supremum of the +\(v_0\)-factor and the weighted integrals of all the others. This +gives \eqref{eq:sheet-weighted-bound}, including its absolute +version. On this parametrization \(D\) differentiates only the +\(v_0\)-factor and is exactly \(\mathcal D_{v_0}\). +The same weighted majorants for each derivative justify +differentiation under the integral on compact \(u\)-intervals. +Taking \(\sigma=-\alpha\) and then \(\sigma=A\) proves +\eqref{eq:sheet-two-ended-bound}. + +At a complex place the sixth-power quotient is trivial, and at a +finite place it is finite. Thus a bounded function of +\(\mathcal V\) is a finite linear combination of products of local +class indicators. A unitary character factors over the finite +product \(S\); at a complex place its norm derivatives only introduce +fixed constants. Multiplying by these functions preserves the +hypotheses after splitting into a finite family. + +For the convolutions, apply the bound with \(\sigma=-\alpha\) or +\(\sigma=A\) under the \(T\)-integral. For example the two bounds +for the reflected convolution have additional factors +\(M_{-\alpha}(\mathcal K)\) and \(M_A(\mathcal K)\). +The corresponding absolute derivative profiles dominate +differentiation on compact \(u\)-intervals. Finally apply +\eqref{eq:sheet-weighted-bound} to each summand in +\eqref{eq:sheet-absolute-sum}; Tonelli and the definition of the +moment give the displayed bound. +\end{proof} + +The lemma also gives the smooth separation on norm dyads that we +will use. If \(\rho\) is a fixed smooth compactly supported +function on \(\R\), integration by parts twice in its ordinary +Fourier transform gives +\begin{equation}\label{eq:sheet-fourier-separation} + \left\|\widehat{\rho(t)\Phi_\kappa(u e^t)}\right\|_{L^1(\R)} + \le C_{A,\rho}\min(u^\alpha,u^{-A}). +\end{equation} +Indeed the \(L^1\) norm is bounded by a constant times the suprema +of the function and its derivatives of orders one and two; +these derivatives are controlled by +\eqref{eq:sheet-two-ended-bound}. Fourier inversion with +\(t=\log x-\log y\), for \(x,y\) in fixed ratio intervals, therefore +separates the two norm variables with bounded \(L^1\) cost and +retains any chosen large-\(u\) power saving. + +\subsection{Exact radial smoothing} + +For the chosen \(S\)-function \eqref{eq:theta-chosen-S-function}, +the angular characters cancel against \(\mathcal L_S^{-1}\). Define its +radial profile by +\begin{equation}\label{eq:theta-V-profile} + V(u)=\int_{X_u}\one_U(y_f) + \prod_{v\mid\infty}|y_v|_v + K_{1/3}(c_v|y_v|_v^{1/2})\,\dd^\times_u y. +\end{equation} +It is a positive, nonzero profile covered by +\Cref{lem:sheet-profile}. In this subsection the Mellin convention is +\(\widehat f(s)=\int_0^\infty f(u)u^s\,\dd u/u\). +The unreflected theta average will contain \(V(q_{\mathfrak A}l)\). +At \(l=T/Z\), the next lemma averages this profile to exactly +\(P_G(q_{\mathfrak A}/Z)\). After the Weyl operation the norm variable +is inverted, and the same kernel instead produces the controlled profiles +\(\int\mathcal K(T)H_\kappa(u/T)\,\dd T/T\). + +\begin{lemma}[Exact Gaussian smoothing]\label{lem:gaussian-smoothing} +For \(V\) in \eqref{eq:theta-V-profile}, there are constants +\(C_0>0\) and \(A_0>0\) such that +\begin{equation}\label{eq:theta-V-mellin} + \widehat V(s) + =C_0 A_0^s + \bigl[\Gamma(s+5/6)\Gamma(s+7/6)\bigr]^r + \qquad(\re s>-5/6). +\end{equation} +There is a smooth function \(\mathcal K:(0,\infty)\to\C\) such that, +for every \(N>0\) and \(j\ge0\), +\begin{equation}\label{eq:theta-kernel-bounds} + \left|(T\partial_T)^j\mathcal K(T)\right| + \le C_{N,j}\min(T^N,T^{-N}), +\end{equation} +and, for every \(u>0\), +\begin{equation}\label{eq:theta-exact-gaussian} + \int_0^\infty\mathcal K(T)V(uT)\,\frac{\dd T}{T} + =P_G(u)=\frac{1}{2\sqrt{\pi}} + \exp\!\left(-\frac{(\log u)^2}{4}\right). +\end{equation} +Its Mellin transform satisfies the exact sign convention +\begin{equation}\label{eq:theta-kernel-mellin} + \widehat{\mathcal K}(-s) + =\frac{e^{s^2}}{\widehat V(s)}. +\end{equation} +The quotient on the right is interpreted by its entire continuation. +For every fixed finite family of rotated \(S\)-profiles \(H_\kappa\) +covered by \Cref{lem:sheet-profile}, the functions +\(\Phi_\kappa(u)=\int\mathcal K(T)H_\kappa(u/T)\,\dd T/T\) +obey the signed and absolute bounds of that lemma. +\end{lemma} + +\begin{proof} +The disintegration \eqref{eq:sheet-disintegration} and Tonelli give +the Mellin transform of \(V\) as the product of its local Mellin +integrals. Each finite factor is the positive volume of \(U_v\), +because \(|y_v|_v=1\) there. At a complex place put \(t=|y_v|_v\). +The classical Bessel integral, or the same Laplace integral used +above, gives +\begin{align} + \int_0^\infty t^{s+1}K_{1/3}(c_v\sqrt t)\,\frac{\dd t}{t} + &=2^{2s+1}c_v^{-2s-2} + \Gamma(s+5/6)\Gamma(s+7/6), + \qquad \re s>-5/6. \label{eq:theta-bessel-mellin} +\end{align} +The angle integrals only change the positive measure constant. +Multiplication over the \(r\) complex places proves +\eqref{eq:theta-V-mellin}. + +Set +\[ + \mathcal R_V(s)=\frac{e^{s^2}}{\widehat V(s)} + =C_0^{-1}A_0^{-s}e^{s^2} + \bigl[\Gamma(s+5/6)\Gamma(s+7/6)\bigr]^{-r}. +\] +The reciprocal gamma function is entire, so \(\mathcal R_V\) is +entire. On each fixed closed vertical strip, Stirling's formula +implies +\[ + |\mathcal R_V(\sigma+it)| + \le C(1+|t|)^C e^{-t^2+\pi r|t|}. +\] +The same bound with a different polynomial holds after +multiplication by any power of \(s\). Hence +\begin{equation}\label{eq:theta-kernel-definition} + \mathcal K(T)=\frac{1}{2\pi i}\int_{(\sigma)} + \mathcal R_V(s)T^s\,\dd s +\end{equation} +converges absolutely and is independent of the real line +\(\sigma\): the horizontal sides of a rectangle tend to zero by the +Gaussian bound. Differentiation in \(\log T\) inserts a power of +\(s\). Moving the line to \(N\) for \(00\), absolute +Mellin Fubini is valid because +\[ + \int_0^\infty|\mathcal K(T)|T^{-\sigma}\frac{\dd T}{T}<\infty, + \qquad + \int_0^\infty V(u)u^\sigma\frac{\dd u}{u}<\infty. +\] +It follows that the Mellin transform of the left side of +\eqref{eq:theta-exact-gaussian} is +\(\widehat V(s)\widehat{\mathcal K}(-s)=e^{s^2}\). +The function on the right of \eqref{eq:theta-exact-gaussian} has +the same transform, by the elementary Gaussian integral. +Mellin inversion proves the identity. The final assertion follows +from the convolution part of \Cref{lem:sheet-profile}. +\end{proof} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/04-reflection.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/04-reflection.tex new file mode 100644 index 0000000000000000000000000000000000000000..c02dfdd3c0b4f10595e1a1668f01d79815da79ce --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/04-reflection.tex @@ -0,0 +1,1228 @@ +\section{Reflection and the quadratic large sieve}\label{sec:reflection} + +We prove the second-moment estimate in \Cref{prop:reflection}. Following +the reflection method of \cite[Part~I, Sections~5--7]{QuasiRH}, we realize +\(B_m\) as a theta average and apply the rational Weyl element. At a +valuation-one row prime the reflected coefficient contributes a first power +of the sextic symbol; a squarefree column prime contributes power \(-4\) +in the same pairing. Their product has exponent \(1-4\equiv3\pmod6\), +so a quadratic large sieve applies. The reflected norm gives a column +length of order \(D^2P/Z\), and the rapid decay of the norm profile controls +the remaining terms. + +\subsection{An automorphic expression for \texorpdfstring{\(B_m\)}{B m}} + +All implied constants in this section may depend on the fixed field, \(S\), +the target character \(\eta\), the compact set \(\mathscr K\), and the +chosen finite family of local theta data. + +We first replace elements by ideal rows. Choose a balanced allowed generator +\(m_0\) of each exterior ideal \(\mathfrak m\). For every nonzero +\(m\in R\) generating \(\mathfrak m\), write +\(m=\varepsilon_m m_0\) with \(\varepsilon_m\in U_S\). When +\(m\in\mathscr B_Z(\mathscr K)\), the balanced shapes of \(m_0\), the +fixed shape set \(\mathscr K\), and +\(q_{\mathfrak m}\asymp_{\mathscr K}Z^{M_0}\) put \(\varepsilon_m\) in a +fixed compact subset of the norm-one group. The \(S\)-unit theorem makes +\(U_S\) discrete there, so one ideal is generated by only boundedly many +elements of the row set. Put +\[ + \Gamma=U_S^6,\qquad U_6=U_S/\Gamma,\qquad + \upsilon_m=\varepsilon_m\Gamma\in U_6. +\] +The group \(U_6\) is finite. The value of \(B_m\) depends on the unit +\(\varepsilon_m\) only through \(\upsilon_m\), which is fixed when the row +\(m\) is fixed. + +For a nonzero row let \(j_{\mathfrak p}=v_{\mathfrak p}(\mathfrak m)\bmod6\) +in \(\{0,\ldots,5\}\). For an allowed generator \(A\) of an integral +exterior ideal \(\mathfrak A\), reciprocity gives +\begin{equation}\label{eq:reflection-moving-twist} + \chi_{\mathfrak A}(m) + =\chi_{\mathfrak A}(\varepsilon_m) + \prod_{\mathfrak p\mid\mathfrak m} + \chi_{\mathfrak p}(A)^{j_{\mathfrak p}}. +\end{equation} +For coprime ideals this is the multiplicative reciprocity law from the +arithmetic section. At a shared prime both sides vanish by zero extension, +including when \(j_{\mathfrak p}=0\); the zeroth power there is the unit +indicator. + +Define +\(\lambda_{\varepsilon_m}(y)=H_S(\varepsilon_m,y)^{-1}\) on \(F_S^\times\). +By \Cref{eq:unit-symbol}, its value at an allowed generator \(A\) is +\(\chi_{\mathfrak A}(\varepsilon_m)\), and isotropy of \(E\) makes it +trivial on \(U_S\). It depends only on \(\upsilon_m\). With the character +\(L_S\) from \Cref{eq:vartheta-s}, put +\(L_{\upsilon_m}=L_S\lambda_{\varepsilon_m}\). Then +\begin{equation}\label{eq:reflection-S-character} + L_{\upsilon_m}(A) + =\vartheta(\mathfrak A)\chi_{\mathfrak A}(\varepsilon_m) + \qquad\text{for every allowed generator }A. +\end{equation} +These characters form a fixed finite family. They are smooth, unitary, +and trivial on \(U_S\); their radial components at infinity are trivial, +and their angular component at each complex place has power \(1\) or \(-1\). + +We insert the moving twists by finite upper-unipotent translations. At an +exterior prime \(\mathfrak p\), write +\(k_{\mathfrak p}=\mathcal O_F/\mathfrak p\) and use its allowed generator +\(p\) as a local uniformizer. For \(1\le j\le5\) put +\[ + \tau_{\mathfrak p,j} + =\sum_{v\in k_{\mathfrak p}^{\times}} + \chi_{\mathfrak p}(v)^{-j}\psi_{\mathfrak p}(v/p). +\] +The character is nontrivial, so +\(\lvert\tau_{\mathfrak p,j}\rvert=q_{\mathfrak p}^{1/2}\). Define +\begin{equation}\label{eq:reflection-Fourier-mask} + \alpha_{\mathfrak p,j}(v)= + \begin{cases} + \tau_{\mathfrak p,j}^{-1}\chi_{\mathfrak p}(v)^{-j}, + &1\le j\le5,\ v\ne0,\\ + 0, &1\le j\le5,\ v=0,\\ + 1-q_{\mathfrak p}^{-1},&j=0,\ v=0,\\ + -q_{\mathfrak p}^{-1},&j=0,\ v\ne0. + \end{cases} +\end{equation} +For every \(t\in\mathcal O_{\mathfrak p}\), +\begin{equation}\label{eq:reflection-mask-identity} + \sum_{v\in k_{\mathfrak p}}\alpha_{\mathfrak p,j}(v) + \psi_{\mathfrak p}(tv/p) + =\chi_{\mathfrak p}(t)^j. +\end{equation} +For \(j\ne0\), substitution by \(t^{-1}\) proves this for a unit \(t\), +and both sides vanish for a nonunit. For \(j=0\), the left side is +\(1-q_{\mathfrak p}^{-1}\sum_v\psi_{\mathfrak p}(tv/p)\), the unit indicator. +Thus the formula includes the masked zeroth power. + +For each \(\upsilon\in U_6\), choose the theta vector in +\Cref{prop:theta-input} whose finite \(S\)-Whittaker function is the +indicator of the common small product subgroup contained in sixth powers +and in \(\ker L_\upsilon\), and whose complex angular type is that of +\(L_\upsilon\). Denote its automorphic function by \(\theta_\upsilon\). +The cutoff and the normalizations are those of +\Cref{eq:theta-chosen-S-function}; these choices form a fixed finite family. +For the remainder of this subsection fix a nonzero \(m\), put +\(\upsilon=\upsilon_m\), and write \(L=L_\upsilon\). + +For \(y\in F_S^\times\) and digits +\(\boldsymbol v=(v_{\mathfrak p})_{\mathfrak p\mid\mathfrak m}\), let +\(g(y,\boldsymbol v)\) have component \(a(y)\) at \(S\), component +\(n(v_{\mathfrak p}/p)\) at \(\mathfrak p\mid\mathfrak m\), and identity +at the other exterior places. Use the lifts in +\Cref{eq:theta-lift-identities}, choose representatives of the residue +digits, and set +\[ + \boldsymbol\alpha_m(\boldsymbol v) + =\prod_{\mathfrak p\mid\mathfrak m} + \alpha_{\mathfrak p,j_{\mathfrak p}}(v_{\mathfrak p}), + \qquad w_+=1,\quad w_-=w. +\] +The combined digit sums that we shall integrate are +\begin{equation}\label{eq:reflection-combined-digits} +\begin{split} + \mathscr D_m^\pm(y) + &=\sum_{\boldsymbol v}\boldsymbol\alpha_m(\boldsymbol v) + \theta_\upsilon(w_\pm g(y,\boldsymbol v)),\\ + \mathscr D_{m,N}^\pm(y) + &=\sum_{\boldsymbol v}\boldsymbol\alpha_m(\boldsymbol v) + \theta_{\upsilon,N}(w_\pm g(y,\boldsymbol v)). +\end{split} +\end{equation} +Rational automorphy gives \(\mathscr D_m^+=\mathscr D_m^-\). We next +show that all four sums descend under \(\Gamma\). + +For \(\gamma\in\Gamma\), choose representatives +\(v'_{\mathfrak p}\equiv\gamma v_{\mathfrak p}\pmod{\mathfrak p}\), and put +\[ + u_{\mathfrak p} + =(\gamma v_{\mathfrak p}-v'_{\mathfrak p})/p\in\mathcal O_{\mathfrak p}. +\] +Let \(k_S=1\), +\(k_{\mathfrak p}=n(u_{\mathfrak p})a(\gamma)\) for +\(\mathfrak p\mid\mathfrak m\), and \(k_{\mathfrak q}=a(\gamma)\) at the +other exterior places. Because \(\gamma\) is an exterior unit, these are +compact lifts fixing the original exterior spherical vector. At \(S\) +the torus factors involving \(\gamma\) are trivial, since it is a sixth +power. At a row prime the exact root relation gives +\[ + n(v'_{\mathfrak p}/p)n(u_{\mathfrak p})a(\gamma) + =a(\gamma)n(v_{\mathfrak p}/p). +\] +The rational diagonal and Weyl lifts therefore satisfy +\begin{equation}\label{eq:reflection-quotient-lifts} +\begin{split} + g(\gamma y,\boldsymbol v') + &=a(\gamma)g(y,\boldsymbol v)k^{-1},\\ + w g(\gamma y,\boldsymbol v') + &=a(\gamma^{-1})w g(y,\boldsymbol v)k^{-1}. +\end{split} +\end{equation} +The second identity has the same compact factor at the far right. + +The upper constant term has the required invariance at these right +arguments as well. For \(q\in F^\times\), any adelic \(h\), and any such +exterior compact \(k\), the root relation and the product formula give +\begin{equation}\label{eq:reflection-constant-term-descent} +\begin{split} + \theta_{\upsilon,N}(a(q)hk^{-1}) + &=\int_{F\backslash\Adele_F} + \theta_\upsilon(a(q)n(x/q)hk^{-1})\,\dd x\\ + &=\theta_{\upsilon,N}(h). +\end{split} +\end{equation} +Indeed \(n(x)a(q)=a(q)n(x/q)\); rational automorphy removes \(a(q)\), +and right invariance removes \(k^{-1}\) at the far right. The substitution +\(x\mapsto x/q\) preserves Haar measure on \(F\backslash\Adele_F\). +This argument applies in particular when \(h_{\mathfrak p}=wn(v/p)\). +Finally \(\chi_{\mathfrak p}(\gamma)=1\), so +\(\alpha_{\mathfrak p,j}(\gamma v)=\alpha_{\mathfrak p,j}(v)\), including +the zero digit. Reindexing by \(\boldsymbol v'=\gamma\boldsymbol v\) in +\Cref{eq:reflection-quotient-lifts,eq:reflection-constant-term-descent} +proves the asserted descent of the combined sums before and after \(w\). + +Recall \(X_l=\{y\in F_S^\times:|y|_S=l\}\) and the sheet measures from +\Cref{eq:sheet-disintegration}. The quotient \(X_l/\Gamma\) is compact; +choose a measurable fundamental set \(\Omega_1\subset X_1\) and put +\(\Omega_l=\ell_l\Omega_1\), using the norm section from that disintegration. +Every integral over \(X_l/\Gamma\) means the integral over \(\Omega_l\) +with the restricted sheet measure, equivalently the quotient measure +induced using counting measure on \(\Gamma\). The factors +\(\one_B([y])\) and \(L(y)^{-1}\) also descend, since +\([\gamma]=1\) in \(\mathcal V\) and \(L(\gamma)=1\). The resulting +integrands are integrable on the quotient: the digit sums are finite and +the factors are smooth or locally constant. + +Both combined constant terms have zero weighted integral. To see this, +rotate \(y\) by a complex number \(z\) of modulus one at one complex place. +In the plus sum this +acts by the left diagonal \(a(z)\); in the minus sum it acts by \(a(z^{-1})\). +The constant term belongs to the torus induction in +\Cref{prop:theta-input}. Its complex inducing law is radial for every +right argument, so both sums are unchanged, including at the exterior +arguments \(wn(v/p)\). The class condition is unchanged because +\(\C^\times\) is sixth-divisible, while \(L(y)^{-1}\) transforms by a +nontrivial angular character. Haar invariance therefore gives +\begin{equation}\label{eq:reflection-angular-cancellation} + \int_{X_l/\Gamma}\one_B([y])L(y)^{-1} + \mathscr D_{m,N}^\pm(y)\,\dd_l^\times y=0. +\end{equation} +We may now define the quotient average +\begin{equation}\label{eq:reflection-theta-average} +\begin{split} + \mathcal T_m(l) + &=\int_{X_l/\Gamma}\one_B([y])L(y)^{-1} + \mathscr D_m^+(y)\,\dd_l^\times y\\ + &=\int_{X_l/\Gamma}\one_B([y])L(y)^{-1} + \mathscr D_m^-(y)\,\dd_l^\times y. +\end{split} +\end{equation} + +\begin{lemma}[Theta representation of the smoothed sum] +\label{lem:reflection-theta-representation} +Let \(V\) and \(\mathcal K\) be the common unrotated profile and Gaussian +averaging weight from \Cref{lem:gaussian-smoothing}. For \(l,Z>0\), +\begin{align} + \mathcal T_m(l) + &=\sum_{\substack{\mathfrak c\ \mathrm{squarefree}\\\mathfrak n}} + \frac{\gamma_2(\mathfrak c)}{q_{\mathfrak c}^{1/2}q_{\mathfrak n}} + L(A)\prod_{\mathfrak p\mid\mathfrak m} + \chi_{\mathfrak p}(A)^{j_{\mathfrak p}} + V(q_{\mathfrak A}l), \label{eq:reflection-unreflected}\\ + B_m(Z) + &=\int_0^\infty\mathcal K(T)\mathcal T_m(T/Z)\,\frac{\dd T}{T}. + \label{eq:reflection-exact-average} +\end{align} +Here \(\mathfrak A=\mathfrak c\mathfrak n^3\) and \(A=cn^3\) is an +allowed product generator. The ideal sums are over integral exterior +ideals. Both formulas converge absolutely after the sheet integrations, +and the second converges absolutely after the \(T\)-integration. +\end{lemma} + +\begin{proof} +Use the Fourier expansion and normalized factorization in +\Cref{eq:theta-fourier-expansion,eq:theta-normalized-factorization} in the +plus average. The constant term vanishes by +\Cref{eq:reflection-angular-cancellation}. At a nonconstant rational +frequency \(\nu\), a row-prime factor is +\[ + W_{\mathfrak p}(a(\nu)n(v/p)) + =\psi_{\mathfrak p}(\nu v/p)W_{\mathfrak p}(a(\nu)). +\] +Spherical support forces \(\nu\in R\); summing the digits with +\Cref{eq:reflection-mask-identity} supplies +\(\prod_{\mathfrak p\mid\mathfrak m}\chi_{\mathfrak p}(\nu)^{j_{\mathfrak p}}\). + +Write \(f_\upsilon(v)\) for the chosen unrotated \(S\)-Whittaker function. +The \(S\)-factor of the frequency is +\(\mathcal C_S(\nu,y)f_\upsilon(\nu y)\). Its cutoff has +\([\nu y]=1\) in \(\mathcal V\). Together with \([y]\in B\), this gives +\([\nu]\in B\). By alternation and \Cref{eq:reflection-S-cocycle}, +\[ + \mathcal C_S(\nu,y) + =H_S(\nu,\nu^{-1})^2H_S(\nu,\nu y)^2=1 +\] +on that support. If \(A\) is an allowed generator of the exterior ideal +of \(\nu\), then \(\nu/A\in U_S\). The direct sum \(E\oplus B\) and the +injection \(U_6\longrightarrow E\) now imply \(\nu=\gamma A\) for +\(\gamma\in\Gamma\). The grouped exterior factor is unchanged by this +multiplication, by the local unit rule and +\(\chi_{\mathfrak p}(\gamma)=1\). + +We justify the orbit substitution with an absolute integral first. +The sets \(\gamma\Omega_l\) partition \(X_l\) up to null sets. Tonelli, +multiplicative Haar invariance, and \Cref{eq:sheet-weighted-bound} give +for \(\sigma>1/2\) +\begin{equation}\label{eq:reflection-forward-absolute} +\begin{split} + \sum_{\gamma\in\Gamma}\int_{\Omega_l} + |f_\upsilon(\gamma Ay)|\,\dd_l^\times y + &=\int_{X_{q_{\mathfrak A}l}}|f_\upsilon(v)|\, + \dd_{q_{\mathfrak A}l}^\times v\\ + &\ll_\sigma(q_{\mathfrak A}l)^{-\sigma}. +\end{split} +\end{equation} +The signed substitution \(v=\gamma Ay\) is therefore legitimate. Since +\([y]=[A]^{-1}[v]\) and \(L(y)^{-1}=L(A)L(v)^{-1}\), it gives +\begin{equation}\label{eq:reflection-forward-unfolding} +\begin{split} + &\sum_{\gamma\in\Gamma}\int_{\Omega_l} + \one_B([y])L(y)^{-1}f_\upsilon(\gamma Ay)\,\dd_l^\times y\\ + &\quad=L(A)\int_{X_{q_{\mathfrak A}l}} + \one_B([A]^{-1}[v])L(v)^{-1}f_\upsilon(v)\, + \dd_{q_{\mathfrak A}l}^\times v\\ + &\quad=L(A)V(q_{\mathfrak A}l). +\end{split} +\end{equation} +For the last equality, \([v]=1\) on the cutoff, \([A]\in B\), and the +angular characters cancel as in \Cref{eq:theta-V-profile}. Thus the +integral depends on \(A\) only through its norm. Multiplicative Haar +measure introduces no norm Jacobian. + +By \Cref{cor:theta-exterior-coefficient}, the exterior spherical +coefficient vanishes unless \(\mathfrak A=\mathfrak c\mathfrak n^3\) +with \(\mathfrak c\) squarefree, and then equals +\(\gamma_2(\mathfrak c)/(q_{\mathfrak c}^{1/2}q_{\mathfrak n})\). +This includes primes shared by \(\mathfrak c\) and \(\mathfrak n\). +Together with \Cref{eq:reflection-forward-unfolding} it proves +\Cref{eq:reflection-unreflected}. + +Here are the absolute bounds for the exchanges. Summing +\Cref{eq:reflection-forward-absolute} with the exterior magnitudes is at +most \(l^{-\sigma}\) times the convergent expression in +\Cref{eq:reflection-original-dirichlet-majorant}. Before evaluating +the finite digit sums, their total absolute mass is at most +\(q_{\mathfrak p}^{1/2}\) for \(j\ne0\) and is less than \(2\) for \(j=0\). +For a fixed row their finite product multiplies the same spherical +majorant, which justifies that earlier exchange. Finally +\Cref{lem:gaussian-smoothing} gives every absolute moment of \(\mathcal K\). +The averaged absolute sum is bounded by a constant times +\[ + Z^\sigma M_{-\sigma}(\mathcal K) +\] +times the finite sum in \Cref{eq:reflection-original-dirichlet-majorant}. +We may therefore apply \Cref{eq:theta-exact-gaussian} term by term. +\Cref{eq:reflection-moving-twist,eq:reflection-S-character} and the +definition of \(B_m\) give \Cref{eq:reflection-exact-average}. +\end{proof} + +\subsection{The local Weyl calculation} + +The second expression in \Cref{eq:reflection-theta-average} is now the +starting point. At \(S\), \(wa(y)=a(y^{-1})w\) replaces the vector by a +fixed rotation and the argument by \(y^{-1}\). Its constant term has +already been removed in \Cref{eq:reflection-angular-cancellation}. +At an exterior place with zero or no digit, \(w\) is compact. We compute +the effect of each nonzero digit. + +\begin{lemma}[Local effect of reflection]\label{lem:reflection-local-Weyl} +Let \(\mathfrak p\) be an exterior prime and put +\(W_e=W_{\mathfrak p}(a(p^e))\). For \(j\in\{0,\ldots,5\}\), a unit \(t\), +and an integer \(e\), define +\[ + \mathcal R_{\mathfrak p,j,e}(t) + =\sum_{v\in k_{\mathfrak p}^{\times}}\alpha_{\mathfrak p,j}(v) + W_{\mathfrak p}(a(p^{e-2}t)w n(v/p)). +\] +It vanishes for \(e<0\). For \(e\ge0\), +\begin{equation}\label{eq:reflection-local-formula} + \mathcal R_{\mathfrak p,j,e}(t) + =W_e\chi_{\mathfrak p}(t)^{4+2e} + \sum_{v\in k_{\mathfrak p}^{\times}} + \alpha_{\mathfrak p,j}(v)\chi_{\mathfrak p}(v)^4 + \psi_{\mathfrak p}(p^{e-1}t/v). +\end{equation} +For \(j=1\), it vanishes for \(e>0\), while +\begin{equation}\label{eq:reflection-valuation-one} +\begin{split} + \mathcal R_{\mathfrak p,1,0}(t) + &=\omega_{\mathfrak p}\chi_{\mathfrak p}(t), + \qquad |\omega_{\mathfrak p}|=1,\\ + \omega_{\mathfrak p} + &=\frac{\sum_{x\ne0}\chi_{\mathfrak p}(x)^3\psi_{\mathfrak p}(x/p)} + {\tau_{\mathfrak p,1}}. +\end{split} +\end{equation} +For all \(j\), its magnitude at \(e=0\) is at most \(1\), and for \(e>0\) +is at most \(q_{\mathfrak p}^{1/2}|W_e|\). More precisely, for \(e\ge0\) +put \(s_{\mathfrak p,j,e}=\mathcal R_{\mathfrak p,j,e}(1)\) and choose +the exponents modulo \(6\) by +\begin{equation}\label{eq:reflection-local-monomial} + b_{j,0}\equiv8-j,\qquad b_{j,e}\equiv4+2e\ (e>0),\qquad + \mathcal R_{\mathfrak p,j,e}(t) + =s_{\mathfrak p,j,e}\chi_{\mathfrak p}(t)^{b_{j,e}}. +\end{equation} +For \(e>0\), \(s_{\mathfrak p,j,e}=0\) unless \(j=4\), and it also +vanishes for \(e\equiv2\pmod3\). The zero-digit term is available only +for \(j=0\); at a frequency \(\nu\) it is +\((1-q_{\mathfrak p}^{-1})W_{\mathfrak p}(a(\nu))\), with ordinary +spherical support. +\end{lemma} + +\begin{proof} +For a unit digit \(v\), the exact lift identity in +\Cref{eq:theta-lift-identities}, with \(H=v/p\), is +\[ + w n(H)=n(H^{-1})a(H^{-2})k_v,\qquad + k_v=a(-1)\bar n(p/v) +\] +with \(k_v\) in the spherical compact group. Only the diagonal minus +sign has been absorbed in \(k_v\); its symbols are trivial because +\(F\) contains \(\mu_{12}\). Commuting the upper root past +\(a(p^{e-2}t)\) gives the positive additive argument +\(\psi_{\mathfrak p}(p^{e-1}t/v)\). The effective torus law gives, for +integers \(e_1,e_2\) and units \(u,v\), +\[ + \mathcal C_{\mathfrak p}(p^{e_1}u,p^{e_2}v) + =\chi_{\mathfrak p}(u)^{2e_2}\chi_{\mathfrak p}(v)^{-2e_1}. +\] +The unit--unit and uniformizer--uniformizer factors are trivial. Thus +torus multiplication contributes +\[ + \mathcal C_{\mathfrak p}(p^{e-2}t,p^2v^{-2}) + =\chi_{\mathfrak p}(t)^4\chi_{\mathfrak p}(v)^{4(e-2)}. +\] +The remaining spherical argument is \(p^etv^{-2}\). It is zero for +\(e<0\); otherwise \Cref{eq:theta-local-values} contributes +\(\chi_{\mathfrak p}(t)^{2e}\chi_{\mathfrak p}(v)^{-4e}W_e\). +Their product proves \Cref{eq:reflection-local-formula}. + +For \(e>0\) the additive character is trivial. Unit-character +orthogonality makes the digit sum zero unless \(j=4\), including the +case \(j=0\). For every \(j\) its absolute value is at most +\(q_{\mathfrak p}^{1/2}\): use +\(|\alpha_{\mathfrak p,j}(v)|=q_{\mathfrak p}^{-1/2}\) for \(j\ne0\) +and \(q_{\mathfrak p}^{-1}\) for \(j=0\). Its unit dependence is the +power \(4+2e\), and \(W_e=0\) for \(e\equiv2\pmod3\). + +For \(e=0\), substitute \(x=t/v\). If \(j\ne0\), this leaves the power +\(\chi_{\mathfrak p}(t)^{8-j}\), divided by \(\tau_{\mathfrak p,j}\), +and the Gauss sum of \(\chi_{\mathfrak p}^{j-4}\) against +\(\psi_{\mathfrak p}(x/p)\). That sum has magnitude +\(q_{\mathfrak p}^{1/2}\), except for \(j=4\), when it has magnitude \(1\). +For \(j=1\) its character is quadratic and \(8-j\equiv1\pmod6\), +giving \Cref{eq:reflection-valuation-one}. For \(j=0\), the same +substitution leaves \(\chi_{\mathfrak p}(t)^8\) times a nontrivial +Gauss sum divided by \(-q_{\mathfrak p}\), again of magnitude at most one. +These observations prove \Cref{eq:reflection-local-monomial} and the +remaining bounds. Finally, compactness of \(w\) gives the zero-digit +assertion by spherical right invariance. +\end{proof} + +For a row \(\mathfrak m=\mathfrak h\mathfrak d\), call a prime active +when its digit is nonzero. Every prime of \(\mathfrak d\) is active, +because there \(j_{\mathfrak p}=1\). For \(\mathfrak p\mid\mathfrak h\) +put \(j_{\mathfrak p}=v_{\mathfrak p}(\mathfrak h)\bmod6\). The exact +set of possible active products in \(\mathfrak h\) is +\begin{equation}\label{eq:reflection-active-set} + \mathcal F(\mathfrak h)= + \left\{\mathfrak f\ \mathrm{squarefree}: + \mathfrak f\mid\rad(\mathfrak h),\ + \mathfrak p\mid\mathfrak f\ \text{if }\ + \mathfrak p\mid\mathfrak h,\ j_{\mathfrak p}\ne0\right\}. +\end{equation} +Only a prime with \(j_{\mathfrak p}=0\) can be inactive. For +\(\mathfrak f\in\mathcal F(\mathfrak h)\), its exact inactive scalar is +\[ + I_{\mathfrak h,\mathfrak f} + =\prod_{\substack{\mathfrak p\mid\mathfrak h\\ + \mathfrak p\nmid\mathfrak f}} + (1-q_{\mathfrak p}^{-1}), +\] +and powerfulness gives +\begin{equation}\label{eq:reflection-active-norm} + \mathfrak f\mid\rad(\mathfrak h),\qquad + q_{\mathfrak f}^2\le q_{\mathfrak h}. +\end{equation} +Let \(\mathcal A(\mathfrak f)\) be the integral exterior ideals +\(\mathfrak a'_{\mathfrak f}\) supported on \(\mathfrak f\), with every +exponent congruent to \(0\) or \(1\) modulo \(3\); set +\(\mathcal A(1)=\{1\}\). This is a convenient common support: the actual +coefficient \(s_{\mathfrak p,j_{\mathfrak p},e}\) remains zero in the +locally vanishing cases of \Cref{lem:reflection-local-Weyl}. + +The local support shows that every frequency with a nonzero grouped +exterior coefficient occurs among +\begin{equation}\label{eq:reflection-frequency} + \nu=\varepsilon\frac{a'_{\mathfrak f}c'(n')^3}{(df)^2}, + \qquad + \mathfrak a'=\mathfrak a'_{\mathfrak f}\mathfrak c'(\mathfrak n')^3, + \qquad \varepsilon\in U_S, +\end{equation} +where \(\mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f)\), +\(\mathfrak c'\) is squarefree, and +\((\mathfrak c'\mathfrak n',\mathfrak d\mathfrak f)=1\). +The displayed elements \(d,f,a'_{\mathfrak f},c',n'\) are allowed +generators. The ideals \(\mathfrak c'\) and \(\mathfrak n'\) may share +primes, and primes of \(\mathfrak h\) omitted from \(\mathfrak f\) may +occur in either. The valuation-one formula forces numerator exponent +zero at each prime of \(\mathfrak d\). Away from +\(\mathfrak d\mathfrak f\), the valuations \(3k\) and \(3k+1\) +determine \(\mathfrak c'\) and \(\mathfrak n'\) uniquely. The remaining +ambiguity is the displayed \(S\)-unit, since allowed product generators +are unique modulo \(\Gamma\). + +For later absolute estimates define +\begin{equation}\label{eq:reflection-frozen-weight} + \beta_{\mathfrak p}(e)= + \begin{cases} + 1,&e=0,\\ + q_{\mathfrak p}^{-k},&e=1+3k,\ k\ge0,\\ + q_{\mathfrak p}^{1/2-k},&e=3k,\ k\ge1,\\ + 0,&\text{otherwise}, + \end{cases} + \qquad + \beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f}) + =\prod_{\mathfrak p\mid\mathfrak f} + \beta_{\mathfrak p}(v_{\mathfrak p}(\mathfrak a'_{\mathfrak f})). +\end{equation} +The local lemma and the spherical magnitudes give +\(|s_{\mathfrak p,j,e}|\le\beta_{\mathfrak p}(e)\). The other exterior +magnitudes together are at most +\(q_{\mathfrak c'}^{-1/2}q_{\mathfrak n'}^{-1}\). + +\subsection{The full reflected expansion} + +We now assemble these local factors. The \(S\)-integrals form a fixed +family of one-variable profiles. Let +\(f_{w,\upsilon}(v)\) be the \(S\)-Whittaker function of the chosen vector +after the fixed Weyl rotation. Choose once a presentation +\[ + f_{w,\upsilon}(v) + =\sum_{\ell=1}^{J_\upsilon}\prod_{a\in S} + f_{\upsilon,\ell,a}(v_a). +\] +The finite \(S\)-functional and the fixed cutoff vector provide such +presentations by \Cref{prop:theta-input,eq:theta-chosen-S-function}; +their lengths are bounded over +the finite row-class family. All fixed scalar coefficients are included +in the displayed local factors. Write \(H_S\) also for its induced +pairing on \(\mathcal V\). For \(\kappa\in\mathcal V\), define +\begin{equation}\label{eq:reflection-profiles} +\begin{split} + H_{\upsilon,\kappa,\ell}(x) + &=\int_{X_x}\one_B(\kappa[v]^{-1})H_S(\kappa,[v])^2L_\upsilon(v)\\ + &\hspace{25mm}\times + \prod_{a\in S}f_{\upsilon,\ell,a}(v_a)\,\dd_x^\times v,\\ + \Phi_{\upsilon,\kappa,\ell}(x) + &=\int_0^\infty\mathcal K(T)H_{\upsilon,\kappa,\ell}(x/T) + \,\frac{\dd T}{T},\\ + \Phi_{\upsilon,\kappa}(x) + &=\sum_{\ell=1}^{J_\upsilon}\Phi_{\upsilon,\kappa,\ell}(x). +\end{split} +\end{equation} +The class multiplier is bounded and depends only on \(\mathcal V\), by +\Cref{eq:reflection-S-cocycle}. The local bounds for the fixed rotated +vectors and \Cref{lem:sheet-profile,lem:gaussian-smoothing} therefore give +one \(\alpha>0\) such that, for every \(N>0\) and integer \(j\ge0\), +\begin{equation}\label{eq:reflection-profile-bound} + |(x\partial_x)^j\Phi_{\upsilon,\kappa}(x)| + \ll_{N,j}\min(x^\alpha,x^{-N})\qquad(x>0). +\end{equation} +The constants are uniform in \(\upsilon\in U_6\) and +\(\kappa\in\mathcal V\). This is a fixed finite family: the functional +and tensor terms have been combined in \(\Phi_{\upsilon,\kappa}\), while +the bounded class function stays inside its integral. In particular +those fixed coefficients will not enter the local majorant +\(\beta_{\mathfrak f}\). + +Choose representatives \(\varepsilon_u\) for the separate frequency-unit +classes \(u\in U_6\). For +\[ + \mathfrak t=(\mathfrak h,\mathfrak f,\mathfrak a'_{\mathfrak f}, + \mathfrak n',u),\qquad + \mathfrak f\in\mathcal F(\mathfrak h),\quad + \mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f),\quad + (\mathfrak n',\mathfrak f)=1, +\] +and class slices \(s_d,s_c\in B\), put +\begin{equation}\label{eq:reflection-profile-class} + \kappa_6(\mathfrak t;s_d,s_c) + =[\varepsilon_u][a'_{\mathfrak f}]s_c[n']^3 + s_d^{-2}[f]^{-2}\in\mathcal V. +\end{equation} +This is a selected class, rather than a further summation index. For +\(\mathfrak p\mid\mathfrak f\), abbreviate +\[ + e_{\mathfrak p}=v_{\mathfrak p}(\mathfrak a'_{\mathfrak f}),\qquad + s_{\mathfrak p}=s_{\mathfrak p,j_{\mathfrak p},e_{\mathfrak p}},\qquad + b_{\mathfrak p}=b_{j_{\mathfrak p},e_{\mathfrak p}}. +\] +With \(\upsilon,\mathfrak t,s_d,s_c\) fixed, the next three factors organize the +dependence on the two varying ideals: \(\omega\) depends on neither, +\(\rho\) depends on the row \(\mathfrak d\) alone, and \(\lambda\) depends +on the column \(\mathfrak c'\) alone. All three are independent of \(Z\). +The quadratic symbol and norm profile will contain the remaining joint +dependence on \(\mathfrak d\) and \(\mathfrak c'\). +For \(L=L_\upsilon\) and \(\varepsilon=\varepsilon_u\), define +\begin{equation}\label{eq:reflection-frozen-factor} +\begin{split} + \omega_{\upsilon,\mathfrak t} + ={}&I_{\mathfrak h,\mathfrak f} + L(a'_{\mathfrak f})^{-1}L(n')^{-3}L(f)^2\\ + &{}\times\prod_{\mathfrak p\mid\mathfrak f} + s_{\mathfrak p} + \chi_{\mathfrak p}\!\left( + \varepsilon\frac{a'_{\mathfrak f}}{p^{e_{\mathfrak p}}} + (n')^3(f/p)^{-2}\right)^{b_{\mathfrak p}}. +\end{split} +\end{equation} +For a squarefree \(\mathfrak d\) with +\((\mathfrak d,\mathfrak h\mathfrak n')=1\) and \([d]=s_d\), put +\begin{equation}\label{eq:reflection-row-factor} +\begin{split} + \rho_{\upsilon,\mathfrak t,s_d}(\mathfrak d) + ={}&L(d)^2\\ + &{}\times\prod_{\mathfrak p\mid\mathfrak d} + \omega_{\mathfrak p} + \chi_{\mathfrak p}\!\left( + \varepsilon a'_{\mathfrak f}(n')^3(d/p)^{-2}f^{-2}\right)\\ + &{}\times\prod_{\mathfrak p\mid\mathfrak f} + \chi_{\mathfrak p}(d)^{-2b_{\mathfrak p}}, +\end{split} +\end{equation} +and put it equal to zero otherwise. For a squarefree \(\mathfrak c'\) +with \((\mathfrak c',\mathfrak f)=1\) and \([c']=s_c\), put +\begin{equation}\label{eq:reflection-column-factor} +\begin{split} + \lambda_{\upsilon,\mathfrak t,s_c}(\mathfrak c') + ={}&L(c')^{-1} + \prod_{\mathfrak p\mid\mathfrak f} + \chi_{\mathfrak p}(c')^{b_{\mathfrak p}}\\ + &{}\times\prod_{\mathfrak q\mid\mathfrak c'} + g_{2,\mathfrak q} + \chi_{\mathfrak q}\!\left( + \varepsilon a'_{\mathfrak f}(c'/q)f^{-2}\right)^2, +\end{split} +\end{equation} +and put it equal to zero otherwise. All character arguments in these +formulas are units at their primes on the stated supports. Thus negative +character powers are well-defined. These definitions give one coherent +row function and one coherent column function for each frozen tuple and +class slice. + +\begin{lemma}[The reflected expansion]\label{lem:reflection-quadratic-pairing} +For every nonzero \(m\in R\) and \(Z>0\), let +\((m)_S=\mathfrak h\mathfrak d\) and \(\upsilon=\upsilon_m\). Then +\begin{equation}\label{eq:reflection-global-expansion} +\begin{split} + B_m(Z) + ={}&\sum_{\mathfrak f\in\mathcal F(\mathfrak h)} + \sum_{\substack{\mathfrak a'_{\mathfrak f}\in\mathcal A(\mathfrak f)\\ + \mathfrak n'\\(\mathfrak n',\mathfrak f)=1}} + \sum_{\substack{u\in U_6\\s_d,s_c\in B}} + \frac{\omega_{\upsilon,\mathfrak t}}{q_{\mathfrak n'}}\\ + &{}\times\rho_{\upsilon,\mathfrak t,s_d}(\mathfrak d) + \Biggl[\sum_{\mathfrak c'\ \mathrm{squarefree}} + \frac{\lambda_{\upsilon,\mathfrak t,s_c}(\mathfrak c')} + {q_{\mathfrak c'}^{1/2}} + \chi_{\mathfrak d}(c')^3\\ + &{}\times\Phi_{\upsilon,\kappa_6(\mathfrak t;s_d,s_c)} + \!\left(\frac{Zq_{\mathfrak a'_{\mathfrak f}} + q_{\mathfrak c'}q_{\mathfrak n'}^3} + {q_{\mathfrak d}^2q_{\mathfrak f}^2}\right)\Biggr]. +\end{split} +\end{equation} +All ideal sums are over integral exterior ideals, and the expansion is +absolutely convergent. The coefficients satisfy +\[ + |\omega_{\upsilon,\mathfrak t}| + \le\beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f}),\qquad + |\rho_{\upsilon,\mathfrak t,s_d}|\le1,\qquad + |\lambda_{\upsilon,\mathfrak t,s_c}|\le1. +\] +The row-unit class \(\upsilon\) is fixed; +\(u\) is the independently summed frequency-unit class. +\end{lemma} + +\begin{proof} +We use the minus average in \Cref{eq:reflection-theta-average} and its +nonconstant Fourier terms. We first supply the absolute bound needed +before unfolding any signed unit orbit. Define +\[ + F^{\mathrm{abs}}_{w,\upsilon}(v) + =\sum_{\ell=1}^{J_\upsilon} + \prod_{a\in S}|f_{\upsilon,\ell,a}(v_a)|,\qquad + H^{\mathrm{abs}}_{w,\upsilon}(x) + =\int_{X_x}F^{\mathrm{abs}}_{w,\upsilon}(v)\,\dd_x^\times v. +\] +This is an absolute profile of \Cref{lem:sheet-profile}; in particular +\(H^{\mathrm{abs}}_{w,\upsilon}(x)\ll_\sigma x^{-\sigma}\) for +\(\sigma>1/2\), uniformly over the fixed row-class family. For any +\(\nu_0\in F^\times\), the sets \(\gamma\Omega_l^{-1}\) partition +\(X_{1/l}\). Tonelli and multiplicative Haar invariance give +\begin{equation}\label{eq:reflection-reflected-absolute} +\begin{split} + \sum_{\gamma\in\Gamma}\int_{\Omega_l} + F^{\mathrm{abs}}_{w,\upsilon}(\gamma\nu_0/y)\,\dd_l^\times y + &=H^{\mathrm{abs}}_{w,\upsilon}(|\nu_0|_S/l)\\ + &\ll_\sigma(|\nu_0|_S/l)^{-\sigma}. +\end{split} +\end{equation} + +For a fixed row, this also justifies the exchanges before evaluating the +digits. At one nonzero digit, the absolute Whittaker factor in the proof of +\Cref{lem:reflection-local-Weyl} is \(|W_e|\) at numerator valuation +\(e\ge0\). Summing the absolute digit weights costs at most +\(q_{\mathfrak p}^{1/2}\) for \(j\ne0\) and is less than \(2\) for \(j=0\). +For \(\sigma>1/2\), the weighted spherical series is +\[ + \sum_{e\ge0}|W_e|q_{\mathfrak p}^{-\sigma e} + =\frac{1+q_{\mathfrak p}^{-1/2-\sigma}} + {1-q_{\mathfrak p}^{-1-3\sigma}}<\infty. +\] +The denominator shift \(e-2\) multiplies this by +\(q_{\mathfrak p}^{2\sigma}\). There are only finitely many row primes, +so their resulting product is finite for the fixed row. Here the numerator +valuations at the primes of \(\mathfrak d\) are still summed. The product +of ordinary spherical series at the other primes is bounded by +\(\zeta_F^S(\sigma+1/2)\zeta_F^S(3\sigma+1)\). Partitioning the units +into the finitely many \(U_6\)-classes and applying +\Cref{eq:reflection-reflected-absolute} bounds the absolute sheet sum. +The \(T\)-average then uses \(M_\sigma(\mathcal K)<\infty\). +Thus these preliminary exchanges are valid for each fixed row. The +uniform estimates below use the coefficients after the finite digit sums +have been evaluated. + +Partition the zero and nonzero digits by +\(\mathfrak f\in\mathcal F(\mathfrak h)\), apply +\Cref{lem:reflection-local-Weyl}, and parameterize the frequencies by +\Cref{eq:reflection-frequency}. For the fixed representative +\(\varepsilon=\varepsilon_u\), write +\[ + \nu=\gamma\nu_0,\qquad + \nu_0=\varepsilon\frac{a'_{\mathfrak f}c'(n')^3}{(df)^2}, + \qquad \gamma\in\Gamma. +\] +The valuations and then the frequency-unit class give a unique such +parameterization on the stated support. At +\(\mathfrak p\mid\mathfrak d\), the local unit of \(\nu_0=p^{-2}t\) is +\[ + t=\varepsilon a'_{\mathfrak f}c'(n')^3(d/p)^{-2}f^{-2}. +\] +At \(\mathfrak p\mid\mathfrak f\), the local unit of +\(\nu_0=p^{e_{\mathfrak p}-2}t\) is +\[ + t=\varepsilon(a'_{\mathfrak f}/p^{e_{\mathfrak p}}) + c'(n')^3d^{-2}(f/p)^{-2}. +\] +At \(\mathfrak q\mid\mathfrak c'\), put +\(k=v_{\mathfrak q}(\mathfrak n')\). The spherical valuation is \(1+3k\), +and its unit exponent is \(2\) modulo \(6\). The cube unit from +\((n'/q^k)^3\) disappears from the symbol. A prime of \(\mathfrak n'\) +outside \(\mathfrak c'\) has unit exponent divisible by \(6\). +Consequently the exact grouped exterior factor on +\((\mathfrak d,\mathfrak h\mathfrak n')= + (\mathfrak c',\mathfrak f)=(\mathfrak d,\mathfrak c')=1\) is +\begin{equation}\label{eq:reflection-exterior-factorization} +\begin{split} + \mathcal E_{\mathfrak t}(\mathfrak d,\mathfrak c') + ={}&\frac{I_{\mathfrak h,\mathfrak f}} + {q_{\mathfrak c'}^{1/2}q_{\mathfrak n'}}\\ + &{}\times\prod_{\mathfrak p\mid\mathfrak d} + \omega_{\mathfrak p} + \chi_{\mathfrak p}\!\left( + \varepsilon a'_{\mathfrak f}c'(n')^3(d/p)^{-2}f^{-2}\right)\\ + &{}\times\prod_{\mathfrak p\mid\mathfrak f} + s_{\mathfrak p} + \chi_{\mathfrak p}\!\left( + \varepsilon(a'_{\mathfrak f}/p^{e_{\mathfrak p}}) + c'(n')^3d^{-2}(f/p)^{-2}\right)^{b_{\mathfrak p}}\\ + &{}\times\prod_{\mathfrak q\mid\mathfrak c'} + g_{2,\mathfrak q} + \chi_{\mathfrak q}\!\left( + \varepsilon a'_{\mathfrak f}(c'/q)(df)^{-2}\right)^2 . +\end{split} +\end{equation} +The factor \(I_{\mathfrak h,\mathfrak f}\) is present whether or not an +inactive prime occurs in \(\mathfrak c'\) or \(\mathfrak n'\). Thus the +formula includes their overlap at such a prime as well as every other +\(\mathfrak c'\)--\(\mathfrak n'\) overlap. The local zeros +\(s_{\mathfrak p}=0\) keep the broader active numerator support exact. + +We identify the only factor involving both row and column variables. +At a pair \(\mathfrak p\mid\mathfrak d\), +\(\mathfrak q\mid\mathfrak c'\), the row factor supplies +\(\chi_{\mathfrak p}(q)\), and the denominator \(d^{-2}\) in the +spherical column factor supplies \(\chi_{\mathfrak q}(p)^{-4}\). +Allowed-generator reciprocity gives +\begin{equation}\label{eq:reflection-one-minus-four} + \chi_{\mathfrak p}(q)\chi_{\mathfrak q}(p)^{-4} + =\chi_{\mathfrak p}(q)^{1-4} + =\chi_{\mathfrak p}(q)^3. +\end{equation} +Multiplying over the pairs gives \(\chi_{\mathfrak d}(c')^3\). +There is no further pairing from a prime occurring only in +\(\mathfrak n'\), since its unit exponent is divisible by \(6\). +The separated \(S\)-character is +\[ + L(\nu_0)^{-1} + =L(a'_{\mathfrak f})^{-1}L(c')^{-1}L(n')^{-3}L(d)^2L(f)^2, +\] +because \(L(\varepsilon)=1\). Comparing +\Cref{eq:reflection-exterior-factorization} with +\Cref{eq:reflection-frozen-factor,eq:reflection-row-factor,eq:reflection-column-factor} +therefore gives, on the indicated class slices, +\begin{equation}\label{eq:reflection-factor-separation} + \mathcal E_{\mathfrak t}(\mathfrak d,\mathfrak c')L(\nu_0)^{-1} + =\frac{\omega_{\upsilon,\mathfrak t} + \rho_{\upsilon,\mathfrak t,s_d}(\mathfrak d) + \lambda_{\upsilon,\mathfrak t,s_c}(\mathfrak c')} + {q_{\mathfrak c'}^{1/2}q_{\mathfrak n'}} + \chi_{\mathfrak d}(c')^3. +\end{equation} +The local bounds prove the three coefficient bounds in the lemma. +The zero extension of the quadratic symbol makes the right side zero +when \(\mathfrak d\) and \(\mathfrak c'\) meet, as the valuation-one +support requires. A prime shared by \(\mathfrak d\) and \(\mathfrak n'\) +must instead be removed by the separate row support of \(\rho\). +Together with \((\mathfrak n',\mathfrak f)=1\) and the column support of +\(\lambda\), these conditions reproduce exactly +\((\mathfrak c'\mathfrak n',\mathfrak d\mathfrak f)=1\). + +It remains to evaluate the \(S\)-integral. The grouped digit sums +\(\mathcal R_{\mathfrak p,j,e}(t)\) are unchanged under \(t\mapsto\gamma t\) +for \(\gamma\in\Gamma\), by \Cref{eq:reflection-local-monomial}. +The same is true of each ordinary spherical factor and of the zero-digit +factor, since \(\chi_{\mathfrak p}(\gamma)=1\). This includes active +\(j=0\) terms and the zero digit. Thus the grouped exterior coefficient +may be taken outside the signed \(\Gamma\)-orbit. That orbit is absolutely +dominated by \Cref{eq:reflection-reflected-absolute}. + +The reflected \(S\)-factor is +\(\mathcal C_S(\gamma\nu_0,y^{-1})f_{w,\upsilon}(\gamma\nu_0/y)\). +In the legitimate substitution \(v=\gamma\nu_0/y\), +\([y]=[\nu_0][v]^{-1}\) and +\(L(y)^{-1}=L(\nu_0)^{-1}L(v)\). Alternation and +\Cref{eq:reflection-S-cocycle} give +\[ + \mathcal C_S(\gamma\nu_0,y^{-1}) + =H_S(\gamma\nu_0,v/(\gamma\nu_0))^2 + =H_S(\nu_0,v)^2. +\] +We obtain the exact identity +\begin{equation}\label{eq:reflection-S-unfolding} +\begin{split} + &\sum_{\gamma\in\Gamma}\int_{\Omega_l} + \one_B([y])L(y)^{-1}\mathcal C_S(\gamma\nu_0,y^{-1}) + f_{w,\upsilon}(\gamma\nu_0/y)\,\dd_l^\times y\\ + &\quad=L(\nu_0)^{-1}\int_{X_{|\nu_0|_S/l}} + \one_B([\nu_0][v]^{-1})H_S(\nu_0,v)^2L(v)f_{w,\upsilon}(v) + \,\dd_{|\nu_0|_S/l}^\times v . +\end{split} +\end{equation} +The sheet norm is +\[ + \frac{|\nu_0|_S}{l} + =\frac{q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3} + {q_{\mathfrak d}^2q_{\mathfrak f}^2l}; +\] +there is no norm Jacobian. Inside the integral \(\nu_0\) occurs only +through its class in \(\mathcal V\). On the supports of the two class +indicators this class is exactly +\(\kappa_6(\mathfrak t;s_d,s_c)\). Thus no continuous shape of the row +or column generator remains inside the profile. + +Insert the two class partitions, use +\Cref{eq:reflection-factor-separation}, and split the fixed \(S\)-function +into its \(\ell\)-terms. Putting \(l=T/Z\) in +\Cref{eq:reflection-exact-average} produces precisely the combined profile +in \Cref{eq:reflection-profiles}, and hence +\Cref{eq:reflection-global-expansion}. For a fixed row class and fixed +ideal data \((\mathfrak h,\mathfrak f,\mathfrak a'_{\mathfrak f},\mathfrak n')\), +there are only \(|U_6||B|^2\) choices of the frequency unit and two slices. +After those choices, the class +\(\kappa_6\) is determined and there is one combined profile. The active +sets and the ideal sums are the explicit outer sums in the expansion. + +We finish by verifying the uniform absolute bound after grouping. For +\(\sigma\ge0\), +\begin{equation}\label{eq:reflection-frozen-Dirichlet} + E_{\mathfrak p}(\sigma) + :=\sum_{e\ge0}\beta_{\mathfrak p}(e)q_{\mathfrak p}^{-\sigma e} + =1+\frac{q_{\mathfrak p}^{-\sigma} + +q_{\mathfrak p}^{-1/2-3\sigma}} + {1-q_{\mathfrak p}^{-1-3\sigma}} + \le 3+\sqrt2. +\end{equation} +In particular, for every \(\delta_1>0\), +\(\prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma) + \ll_{F,\delta_1}q_{\mathfrak f}^{\delta_1}\), uniformly for +\(\sigma\ge0\). Absorb the finitely many primes of norm below +\((3+\sqrt2)^{1/\delta_1}\) into the constant and bound each remaining +factor by \(q_{\mathfrak p}^{\delta_1}\). + +Put \(Q=q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3\). +Dropping squarefreeness and the unfrozen coprimality restrictions in the +nonnegative majorant gives +\begin{equation}\label{eq:reflection-reflected-Dirichlet} +\begin{split} + \mathcal D_{\mathfrak f}(\sigma) + &:=\sum_{\mathfrak a'_{\mathfrak f},\mathfrak c',\mathfrak n'} + \beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f}) + q_{\mathfrak c'}^{-1/2}q_{\mathfrak n'}^{-1}Q^{-\sigma}\\ + &\le\zeta_F^S(\sigma+1/2)\zeta_F^S(3\sigma+1) + \prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma) + <\infty\qquad(\sigma>1/2). +\end{split} +\end{equation} +The frozen ideal still ranges over \(\mathcal A(\mathfrak f)\). +For \(t_{\mathfrak d}=q_{\mathfrak d}^2q_{\mathfrak f}^2/Z\), the +absolute profile above dominates every class multiplier and yields +\begin{equation}\label{eq:reflection-reflected-Fubini} +\begin{split} + &\int_0^\infty|\mathcal K(T)| + \sum_{\mathfrak a'_{\mathfrak f},\mathfrak c',\mathfrak n'} + \beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f}) + q_{\mathfrak c'}^{-1/2}q_{\mathfrak n'}^{-1} + H^{\mathrm{abs}}_{w,\upsilon}\!\left( + \frac{Q}{t_{\mathfrak d}T}\right)\frac{\dd T}{T}\\ + &\hspace{2cm}\ll_\sigma + t_{\mathfrak d}^{\sigma}M_\sigma(\mathcal K) + \mathcal D_{\mathfrak f}(\sigma)<\infty. +\end{split} +\end{equation} +This justifies all exchanges in the grouped expansion. Its constants +are uniform in the ideal data, apart from the displayed product of local +majorants. Since \(\mathcal F(\mathfrak h)\) is finite for each fixed row, +the full expansion is absolutely convergent. +\end{proof} + +The case \(m=0\) remains the separate identity +\(B_0(Z)=P_G(1/Z)\) from the coefficient-family section. + +The expansion also displays the length used in the second moment. On dyads +\(P\le q_{\mathfrak h}<2P\) and \(D\le q_{\mathfrak d}<2D\), put +\begin{equation}\label{eq:reflection-column-cutoff} +\begin{aligned} + Q&=q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3, + &\qquad + t_{\mathfrak d}&=\frac{q_{\mathfrak d}^2q_{\mathfrak f}^2}{Z},\\ + \mathcal L&=\frac{8D^2P}{Z}, + & R_0&=Z^\delta\mathcal L, +\end{aligned} +\end{equation} +where \(\delta>0\) will be chosen small. By +\Cref{eq:reflection-active-norm}, \(t_{\mathfrak d}\le\mathcal L\). +We shall retain \(Q\le R_0\), which in particular gives +\(q_{\mathfrak c'}\le R_0\). If \(R_0<1\), this range is empty because +\(Q\) is the norm of an integral ideal. The tail estimate below justifies +this truncation for every dyad, including that case. + +\subsection{The quadratic large sieve and the moment estimate} + +We use the quadratic large sieve established over \(\Q\) by Heath-Brown +and extended to Hecke families by Goldmakher--Louvel +\cite{HeathBrown95,GoldmakherLouvel13}. We state the latter result in the +form needed for \Cref{eq:reflection-global-expansion} and verify its family +hypotheses for the present generator convention. + +\begin{lemma}[Quadratic large sieve for the exterior symbols] +\label{lem:reflection-quadratic-sieve} +For $C,D\geq1$ and complex coefficients $z_{\mathfrak c}$ supported on +squarefree exterior ideals with $q_{\mathfrak c}\leq C$, one has, for every +$\epsilon>0$, +\begin{equation}\label{eq:reflection-quadratic-sieve} + \sum_{\substack{\mathfrak d\ \mathrm{squarefree}\\q_{\mathfrak d}\leq D}} + \left|\sum_{\substack{\mathfrak c\ \mathrm{squarefree}\\q_{\mathfrak c}\leq C}} + z_{\mathfrak c}\chi_{\mathfrak d}(c)^3\right|^2 + \ll_{F,S,\epsilon}(C+D)(CD)^\epsilon + \sum_{\mathfrak c}|z_{\mathfrak c}|^2. +\end{equation} +Arbitrary restrictions on the rows and separate restrictions or unitary +factors on the columns are allowed by deleting rows and changing the +coefficients. +\end{lemma} + +\begin{proof} +We verify Definition~1 of Goldmakher--Louvel before applying their +Theorem~1.1 \cite{GoldmakherLouvel13}. Choose an ordinary integral ideal +$\mathfrak q_S$ supported on the finite places of $S$, with a positive +exponent $m_v$ at each such place large enough that +$1+\mathfrak p_v^{m_v}\mathcal O_v\subset F_v^{\times2}$. +Every local quadratic character at $v$ is trivial on this subgroup, so its +conductor exponent is at most $m_v$. Exterior ideals are exactly the +ordinary ideals prime to $\mathfrak q_S$, with the same norms. + +For a squarefree exterior ideal $\mathfrak c$, let $\Xi_{\mathfrak c}$ be +the primitive Hecke character of the possibly trivial quadratic Kummer extension +$F(\sqrt c)/F$. It is independent of the allowed generator, since two +such generators differ by a sixth power of an $S$-unit. At an exterior +prime its local conductor exponent is one if the prime divides +$\mathfrak c$ and zero otherwise: at the odd residue characteristics under +consideration an odd valuation gives a tamely ramified quadratic extension, +whereas a unit gives an unramified or trivial extension. Thus +\[ + \operatorname{cond}(\Xi_{\mathfrak c}) + =\mathfrak c\mathfrak s_{\mathfrak c}, + \qquad \mathfrak s_{\mathfrak c}\mid\mathfrak q_S. +\] +Its infinite type is trivial because all infinite places of $F$ are complex. +On an exterior ideal $\mathfrak d$, +\[ + \Xi_{\mathfrak c}(\mathfrak d)=\chi_{\mathfrak d}(c)^3. +\] +This includes the zero at a shared prime, which is in the conductor of +$\Xi_{\mathfrak c}$. These are the primitive global characters, not a +character with additional artificially deleted Euler factors. + +Let $B_2$ be the image of $B$ in $F_S^\times/F_S^{\times2}$ and set +$\mathrm{cl}_2(\mathfrak c)=[c]_{S,2}\in B_2$. Allowed generators +multiply modulo $U_S^6$, so this extends to a homomorphism on the group of +all fractional exterior ideals. It has finite image. It is also ray-class +data for $\mathfrak q_S$: for a principal exterior ideal $(x)$ its value is +the $B_2$-projection of $[x]_{S,2}$ in the direct sum induced by $E\oplus B$; +if $x\equiv1\pmod{\mathfrak q_S}$, all its local classes at $S$ are squares, +so this projection is trivial. + +For coprime squarefree exterior ideals, the exact reciprocity of allowed +generators gives +\[ + \Xi_{\mathfrak c}(\mathfrak d) + =\chi_{\mathfrak d}(c)^3 + =\chi_{\mathfrak c}(d)^3 + =\Xi_{\mathfrak d}(\mathfrak c). +\] +Thus the finite-class reciprocity factor in that definition may be taken +identically one. If $\mathfrak c_1,\mathfrak c_2$ are coprime squarefree exterior ideals and have the +same $\mathrm{cl}_2$, their quotient $c_1/c_2$ is a square at every place +of $S$. The primitive character inducing +$\Xi_{\mathfrak c_1}\overline{\Xi_{\mathfrak c_2}}$ is consequently +trivial there. At every prime of $\mathfrak c_1\mathfrak c_2$ it is tamely +ramified with conductor exponent one, and at every other exterior prime it +is unramified. Its \emph{global} conductor is therefore exactly +$\mathfrak c_1\mathfrak c_2$. The product here is understood as the +primitive inducing character, with the convention specified in +\cite[\S2]{GoldmakherLouvel13}. This verifies the conductor-product axiom, +including the part at $S$. + +We have thus obtained a quadratic Hecke family in the precise sense of +\cite[Definition~1]{GoldmakherLouvel13}. Its Theorem~1.1 gives +\Cref{eq:reflection-quadratic-sieve}. Apply that theorem to the full +family and then delete unwanted rows or set unwanted coefficients to zero; +a single class slice need not itself be a Hecke family. +\end{proof} + +\begin{proof}[Proof of \Cref{prop:reflection}] +The initial element-to-ideal reduction lets us fix one row-unit class +\(\upsilon\in U_6\) and sum over ideal rows at a bounded cost. +Fix dyads $P\leq q_{\mathfrak h}<2P$ and +$D\leq q_{\mathfrak d}<2D$ which meet the row set. They satisfy +\begin{equation}\label{eq:reflection-PD-range} + PD\ll_{\mathscr K}Z^{M_0},\qquad P,D\geq1. +\end{equation} +There are $O_F(P^{1/2})$ powerful ideals on the first dyad. In fact every +powerful ideal has a unique expression $\mathfrak r^2\mathfrak s^3$ with +$\mathfrak s$ squarefree (choose its exponent as the parity of each prime +exponent). The fixed-field bound for the number of ideals of norm at most +$t$ is $O_F(t)$, hence +\begin{equation}\label{eq:reflection-powerful-count} + \#\{\mathfrak h\text{ powerful}:q_{\mathfrak h}<2P\} + \ll_F P^{1/2}\sum_{\mathfrak s}q_{\mathfrak s}^{-3/2} + \ll_F P^{1/2}. +\end{equation} + +Fix \(\mathfrak h\), and use \(Q,t_{\mathfrak d},\mathcal L,R_0\) from +\Cref{eq:reflection-column-cutoff}. For each +\(\mathfrak f\in\mathcal F(\mathfrak h)\) and +\(\mathbf s=(u,s_d,s_c)\in U_6\times B^2\), split its partial sum in +\Cref{eq:reflection-global-expansion} according to \(Q\le R_0\) or +\(Q>R_0\). Denote these sums, which still include +\(\mathfrak a'_{\mathfrak f},\mathfrak n',\mathfrak c'\), by +\(\operatorname{Ret}_{\mathfrak f,\mathbf s,\mathfrak d}\) and +\(\operatorname{Tail}_{\mathfrak f,\mathbf s,\mathfrak d}\). +Sum them over the admitted \(\mathfrak f\) and the finite \(\mathbf s\) +to define \(\operatorname{Ret}_{\mathfrak h,\mathfrak d}\) and +\(\operatorname{Tail}_{\mathfrak h,\mathfrak d}\). Thus, on a row with +class \(\upsilon\), +\[ + B_m(Z)=\operatorname{Ret}_{\mathfrak h,\mathfrak d} + +\operatorname{Tail}_{\mathfrak h,\mathfrak d}. +\] +Set both parts to zero outside the squarefree rows +\(\mathfrak d\) coprime to \(\mathfrak h\). The threshold \(R_0\) is +common to the dyad and independent of the individual row. When \(R_0<1\) +the retained part is empty, so no large sieve at a length below one is +required. + +We first bound the discarded terms, uniformly in this case as well. Put +$\mathcal J=\max(Z^\delta,\mathcal L^{-1})$, and fix $N>\sigma>1/2$. +For $Q>R_0$ we have both $Q/t_{\mathfrak d}>Z^\delta$ and +$Q/t_{\mathfrak d}\geq\mathcal L^{-1}$, because $Q\geq1$. +The profile estimate therefore gives +\[ + \one_{Q>R_0}|\Phi_{\upsilon,\kappa}(Q/t_{\mathfrak d})| + \ll_N \mathcal J^{-(N-\sigma)}(t_{\mathfrak d}/Q)^\sigma. +\] +Summing absolutely with \Cref{eq:reflection-reflected-Dirichlet}, for one +active product and one fixed \(\mathbf s\), yields +\begin{equation}\label{eq:reflection-tail-bound} + |\operatorname{Tail}_{\mathfrak f,\mathbf s,\mathfrak d}| + \ll_{N,\sigma,F} + \mathcal L^\sigma\mathcal J^{-(N-\sigma)} + \zeta_F^S(\sigma+1/2)\zeta_F^S(3\sigma+1) + \prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma). +\end{equation} +This one estimate includes all column, cube, and frozen-exponent tails. +If $R_0<1$, then $\mathcal J=\mathcal L^{-1}$ and the corresponding +entire partial sum is $O(\mathcal L^N\prod E_{\mathfrak p}(\sigma))$. + +For every $\delta_1>0$, \Cref{eq:reflection-frozen-Dirichlet} also gives +\begin{equation}\label{eq:reflection-active-choice-sum} + \sum_{\mathfrak f\mid\rad(\mathfrak h)} + \prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(\sigma) + =\prod_{\mathfrak p\mid\mathfrak h}(1+E_{\mathfrak p}(\sigma)) + \ll_{F,\delta_1}q_{\mathfrak h}^{\delta_1} + \qquad(\sigma\geq0). +\end{equation} +The same finite-small-prime argument proves this inequality; including all +divisors only adds active choices that may actually be forbidden. Sum +\Cref{eq:reflection-tail-bound} over them and over the fixed finite class +list, square, and use $O_F(D)$ for the number of row ideals on the second +dyad and \Cref{eq:reflection-powerful-count}. The total squared tail is +\begin{equation}\label{eq:reflection-tail-mass} + \ll_{N,\sigma,F,\delta_1} + P^{1/2+2\delta_1}D\, + \mathcal L^{2\sigma}\mathcal J^{-2(N-\sigma)}. +\end{equation} +Here \Cref{eq:reflection-PD-range} implies +$P\ll Z^{M_0}$ and $\mathcal L\ll Z^{2M_0-1}$, while +$\mathcal J\geq Z^\delta$. Given any $A_0>0$, choosing $N$ with +\[ + 2\delta(N-\sigma)>A_0+2M_0\delta_1+2\sigma(2M_0-1) +\] +makes \Cref{eq:reflection-tail-mass} +$O(Z^{-A_0}P^{1/2}D)$. This proves the promised absolute domination of +the discarded range, including all dyads for which $R_0<1$. + +It remains to bound the retained terms when \(R_0\ge1\). Fix +\(\mathfrak f,\mathbf s,\mathfrak a'_{\mathfrak f},\mathfrak n'\). +Then \(\kappa=\kappa_6(\mathfrak t;s_d,s_c)\) is fixed, while +\(\rho_{\upsilon,\mathfrak t,s_d}\) and +\(\lambda_{\upsilon,\mathfrak t,s_c}\) are coherent functions of all rows +and columns in the two slices. Split \(\mathfrak c'\) into dyads +\(C\le q_{\mathfrak c'}<2C\), \(C\ge1\). Put +\[ + v_0=\frac{Zq_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak n'}^3C} + {D^2q_{\mathfrak f}^2},\qquad + x=\log(q_{\mathfrak c'}/C)-2\log(q_{\mathfrak d}/D). +\] +Then $x\in[-2\log2,\log2]$, and the profile in +\Cref{eq:reflection-global-expansion} is $\Phi_{\upsilon,\kappa}(v_0e^x)$. +Choose a fixed $\varphi\in C_c^\infty(\R)$ equal to one on that interval +and set $g_{v_0}(x)=\varphi(x)\Phi_{\upsilon,\kappa}(v_0e^x)$. Applying +\Cref{eq:sheet-fourier-separation} with this cutoff gives +\begin{equation}\label{eq:reflection-Fourier-separation} + \int_\R|\widehat g_{v_0}(\tau)|\,\dd\tau + \ll_N\min(v_0^\alpha,v_0^{-N})\ll_N1. +\end{equation} +Fourier inversion now separates the norm factors as +$(q_{\mathfrak c'}/C)^{i\tau}(q_{\mathfrak d}/D)^{-2i\tau}$ with the +bounded integral cost in \Cref{eq:reflection-Fourier-separation}. +The retained indicator +$\one_{q_{\mathfrak a'_{\mathfrak f}}q_{\mathfrak c'}q_{\mathfrak n'}^3\leq R_0}$ +is a column coefficient after the two frozen ideals are fixed. + +Use Minkowski's inequality for this Fourier integral and apply +\Cref{lem:reflection-quadratic-sieve} to each $\tau$. Separate row factors +have modulus at most one, and the column square mass is +\[ + \sum_{C\leq q_{\mathfrak c'}<2C}q_{\mathfrak c'}^{-1}\ll_F1 +\] +by ideal counting. For any $\delta_2>0$, the row $\ell^2$ norm of this +one column dyad, before the factor +$\beta_{\mathfrak f}(\mathfrak a'_{\mathfrak f})/q_{\mathfrak n'}$, is +therefore at most +\begin{equation}\label{eq:reflection-one-column-dyad} + \ll_{N,\delta_2}(D+C)^{1/2}(DC)^{\delta_2/2} + \min(v_0^\alpha,v_0^{-N}). +\end{equation} +The restrictions $(\mathfrak d,\mathfrak h\mathfrak n')=1$ merely delete +rows. The zero extension of the symbol already treats shared row and +column primes, so no joint restriction was removed in applying the sieve. + +For the retained range, $C\leq R_0$ and +$q_{\mathfrak n'}\leq R_0^{1/3}$. There are $O(\log(2R_0))$ such column +dyads, and partial summation of the fixed-field ideal count gives +\[ + \sum_{q_{\mathfrak n'}\leq R_0^{1/3}}q_{\mathfrak n'}^{-1} + \ll_F\log(2R_0). +\] +Drop the final minimum in \Cref{eq:reflection-one-column-dyad} and use +Minkowski successively for the column dyads, $\mathfrak n'$, and +$\mathfrak a'_{\mathfrak f}$. +\Cref{eq:reflection-frozen-Dirichlet} at $\sigma=0$ shows that their +retained row norm, for fixed \(\mathfrak h,\mathfrak f,\mathbf s\), is +\[ + \ll_{\delta_2} + \left(\prod_{\mathfrak p\mid\mathfrak f}E_{\mathfrak p}(0)\right) + \log^2(2R_0)(D+R_0)^{1/2}(DR_0)^{\delta_2/2}. +\] +Sum these norms over the admitted \(\mathfrak f\) and the finite +\(\mathbf s\), using \Cref{eq:reflection-active-choice-sum} for the former. +Square the resulting estimate for each fixed \(\mathfrak h\), and only +then sum the squares over \(\mathfrak h\). Since each row has a unique powerful part, no triangle +inequality across different powerful parts is necessary. +\Cref{eq:reflection-powerful-count} yields +\begin{equation}\label{eq:reflection-retained-mass} + \sum_{\substack{\mathfrak h\ \mathrm{powerful}\\P\leq q_{\mathfrak h}<2P}} + \sum_{D\leq q_{\mathfrak d}<2D} + |\operatorname{Ret}_{\mathfrak h,\mathfrak d}|^2 + \ll_{\delta_1,\delta_2} + P^{1/2+2\delta_1}\log^4(2R_0)(D+R_0)(DR_0)^{\delta_2}. +\end{equation} +All the finite unit and $S$-class choices have only multiplied this by a +fixed constant. + +We make the power losses explicit. On the range +\Cref{eq:reflection-PD-range}, +\[ + D+R_0\ll Z^\delta\left(D+\frac{D^2P}{Z}\right),\qquad + DR_0\ll Z^{3M_0-1+\delta},\qquad P\ll Z^{M_0}. +\] +For any $\delta_3>0$, $\log^4(2R_0)\ll_{\delta_3}Z^{\delta_3}$ because +$1\leq R_0\ll Z^{2M_0-1+\delta}$. Choose the positive auxiliaries so that +\[ + \delta+2M_0\delta_1+(3M_0-1+\delta)\delta_2+\delta_3<\epsilon. +\] +Then \Cref{eq:reflection-retained-mass} is bounded by the right side of +\Cref{eq:reflection-dyadic-mass}. The retained mass is zero for $R_0<1$, +and \Cref{eq:reflection-tail-mass} is smaller than that right side in all +cases after $N$ is chosen as above. Adding the two parts and restoring the +bounded element multiplicities proves \Cref{eq:reflection-dyadic-mass}. + +Finally, for every nonempty dyad, $P\geq1$ and +\Cref{eq:reflection-PD-range} give +\[ + P^{1/2}D\leq PD\ll Z^{M_0}\leq Z^{2M_0-1},\qquad + P^{1/2}\frac{D^2P}{Z} + =\frac{(PD)^2}{ZP^{1/2}}\ll Z^{2M_0-1}, +\] +where the first comparison uses $M_0>1$. There are only $O((\log Z)^2)$ +possible dyads. Applying the dyadic estimate with a smaller preliminary +loss absorbs this factor and proves \Cref{eq:reflection-total-mass}. +Multiplying $\mathscr K$ by any element of a fixed compact norm-one set +only enlarges the fixed compact shape set used in the initial multiplicity +bound, which proves the asserted uniformity. +\end{proof} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/05-poisson.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/05-poisson.tex new file mode 100644 index 0000000000000000000000000000000000000000..c8526f74b281d37fda034c17195c4bb5afd5f79b --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/05-poisson.tex @@ -0,0 +1,765 @@ +\section{The Poisson probe and its Euler product}\label{sec:poisson} + +We insert the theta sum \(B_m(Z)\) of +\Cref{eq:reflection-B-definition} into an additive average. The reflected +second moment bounds this average from above. Poisson summation will give a +second expression whose principal frequency contains the reciprocal of the +target \(L\)-function. The use of an auxiliary additive polynomial while +retaining the unrestricted cube factors follows the method of +\cite[Part~I, \S\S6--7]{QuasiRH}; the calculation here is over the fixed field +\(F\) and uses its sextic characters. + +Keep the field \(F\), the set \(S\), the target \(\eta\), and the allowed +generators fixed as in \Cref{sec:arithmetic}. In particular +\(\vartheta=\overline\Lambda\,\overline G\,\eta\), and every character power +is zero on nonunits, including the zeroth power. The scale notation +\(a,b,M_0,h_0,X,Y\) is fixed in \Cref{eq:reflection-scales}. +All implied constants in this and the next section may depend on the fixed +\(F,S,\eta\) and the fixed weights. Exponents displayed as numerical +constants do not have that dependence. An arbitrary positive power loss +may always be chosen smaller before an estimate is applied. + +\subsection{Weights and the estimate from reflection} + +Choose a nonzero nonnegative function \(W_1\in C_c^\infty((0,\infty))\), +and set +\[ + \widehat W_1(w)=\int_0^\infty W_1(t)t^w\,\frac{dt}{t}. +\] +Choose a tensor Bruhat--Schwartz function \(W_0\) on \(F_S\), compactly +supported in \(F_S^\times\), such that +\begin{equation} + \widehat W_0(0)=0,\qquad + M(1/6)\ne0,\qquad + M(z)=\int_{F_S^\times}\widehat W_0(y)|y|_S^z\,d^\times y . +\label{eq:poisson-Fourier-weight} +\end{equation} +The condition \(\widehat W_0(0)=0\) will remove the zero additive +frequency after Poisson summation. The condition \(M(1/6)\ne0\) will +ensure that the weight does not kill the residue at \(z=1/6\) in that +expansion. These are separate from the support condition on \(W_0\), +which removes the original row \(m=0\) from the probe. +Here the Fourier sign is the one fixed in \Cref{sec:arithmetic}. We +explain the choice and the analytic properties that will be needed. At +each place the local zeta functional equation gives, for \(00\), each local integral of +\(|\widehat W_v(y)||y|_v^\sigma\) is finite. At a finite place this follows +by summing the geometric series of shells near zero and using compact +support at infinity; at a complex place it follows from boundedness at +zero and Schwartz decay at infinity. The same assertions hold after any +number of derivatives in a complex logarithmic norm. Thus \(M\) is +holomorphic for \(\re z>0\), and integration by parts in that norm gives, +for every compact interval \(J\subset(0,\infty)\) and every \(A>0\), +\begin{equation} + |M(\sigma+i\tau)|\ll_{J,A}(1+|\tau|)^{-A} + \quad(\sigma\in J). +\label{eq:poisson-M-decay} +\end{equation} +The analogous estimate holds for \(\widehat W_1\) on every fixed real +strip. + +For an exterior ideal \(\mathfrak s\), with an allowed generator \(s\), put +\[ + g_{\mathfrak s}(k) + =\sum_{d\bmod\mathfrak s}\chi_{\mathfrak s}(d)\e(kd/s). +\] +The value is independent of the allowed generator: a change by +\(\epsilon^6\), \(\epsilon\in U_S\), is absorbed by the same change of +the residue variable. Use product multiplicative Haar measures for which +the norm decomposition of \(F_S^\times\) has the form +\[ + d^\times y=\frac{dT}{T}\,dt,\qquad y=t\ell_T,\quad t\in F_S^1. +\] +The measure on \(F_S^1/U_S^6\) below is the quotient measure induced by +\(dt\). This choice also fixes the normalization in \(M\). +Define +\begin{equation} + \begin{split} + I(X,Y,Z)=\int_{F_S^1/U_S^6} + \frac1Y\sum_{\mathfrak s}W_1(q_{\mathfrak s}/Y) + \frac1{\sqrt{q_{\mathfrak s}X}}\sum_{m\in R} + q_{\mathfrak s}^{-1/2}g_{\mathfrak s}(-m) + W_0\!\left(\frac{m}{t_0\ell_{q_{\mathfrak s}X}}\right) + B_m(Z)\,dt_0 . + \end{split} +\label{eq:poisson-probe} +\end{equation} +All undecorated ideal sums in this section are over nonzero integral +exterior ideals. The summand for \(m=0\) vanishes because \(W_0\) is +supported in \(F_S^\times\). For \(\epsilon\in U_S^6\), +\(\chi_{\mathfrak A}(\epsilon)=\chi_{\mathfrak s}(\epsilon)=1\); +changing \(m\) to \(\epsilon m\) shows that the integrand descends to the +quotient. That quotient is compact by +\Cref{lem:balanced-generators}, and we use representatives in a fixed +bounded subset of \(F_S^1\). For each such representative the \(m\)-sum +has finite support. The Gaussian weight in \(B_m\) makes its ideal sum +absolutely convergent. + +We first record the additive estimate in the form also used later for +conductor characters. If \(\mathfrak s=R\), its one residue class is +understood to give the single fraction \(0\). + +\begin{lemma}[Separated additive fractions]\label{lem:additive-sieve} +Let \(K\) be a fixed compact subset of \(F_S\), and let \(N,V\ge2\). +Choose balanced generators \(s\) for the exterior ideals with +\(q_{\mathfrak s}\le V\). Let \(\mathcal P\) be any subset of the distinct +points +\[ + \left\{\frac d s+R: + q_{\mathfrak s}\le V,\ d\in(R/\mathfrak s)^\times\right\} + \subset F_S/R. +\] +For any complex \(b_\alpha\), +\begin{equation} + \sum_{m\in R\cap\ell_NK} + \left|\sum_{\alpha\in\mathcal P}b_\alpha\e(m\alpha)\right|^2 + \ll_{F,S,K}(N+V^2)\sum_{\alpha\in\mathcal P}|b_\alpha|^2 . +\label{eq:additive-sieve} +\end{equation} +The dual inequality, with the sums over \(m\) and \(\alpha\) interchanged, +has the same constant. +\end{lemma} + +\begin{proof} +Reduced fractions with distinct denominator ideals are distinct modulo +\(R\). Indeed, at a prime dividing its denominator the valuation of a +reduced fraction is the negative of the denominator valuation; at any +other exterior prime it is nonnegative. If the denominator valuations +differ, their difference therefore has negative valuation there. +If all agree, congruence modulo \(R\) says the residue classes agree. +For distinct fractions and any \(r\in R\), the numerator in +\[ + \frac d s-\frac {d'}{s'}+r + =\frac{ds'-d's+rss'}{ss'} +\] +is a nonzero element of \(R\). The norm identity of +\Cref{lem:s-lattice} gives +\begin{equation} + \left|\frac d s-\frac {d'}{s'}+r\right|_S + \ge (q_{\mathfrak s}q_{\mathfrak s'})^{-1}\ge V^{-2}. +\label{eq:additive-separation} +\end{equation} +For \(r\in R\setminus\{0\}\) it also gives \(|r|_S\ge1\). + +Put \(d_F=[F:\Q]\) and +\(\delta=c\min(N^{-1/d_F},V^{-2/d_F})\), with \(c>0\) fixed sufficiently +small. There is a nonnegative Bruhat--Schwartz function \(\Phi\) which is +at least one on \(\ell_NK\), has +\(\widehat\Phi(0)\ll\delta^{-d_F}\), and whose Fourier transform is supported +in a product of fixed finite-place additive subgroups and complex disks +of ordinary radius \(C\delta\). Here is a direct construction. At each +complex place take the inverse transform of a smooth bump +supported in a disk, evaluated at \(\delta y_v\). Its value stays +bounded below when \(\delta|y_v|\) is sufficiently small. At finite +places use the inverse transform of the indicator of a sufficiently +small compact open subgroup, whose annihilator contains the finite +projection of \(K\). The product, multiplied by a fixed constant and +squared in modulus, majorizes the box. Fourier support follows from +convolution, and a change of scale gives integral \(O(\delta^{-d_F})\). +Shrinking \(c\) makes this support contain no translate by \(R\) of a +difference of distinct points in \(\mathcal P\), by +\Cref{eq:additive-separation}: the product of the complex squared absolute +values on that support is \(O(\delta^{d_F})\), and its finite-place product +is bounded by a fixed constant. It contains no nonzero point of \(R\) +for the same reason. + +Expand the square after majorizing it by \(\Phi\), and apply Poisson +summation on \(R\). For \(\alpha,\beta\in\mathcal P\), +\[ + \sum_{m\in R}\Phi(m)\e(m(\alpha-\beta)) + =\sum_{r\in R}\widehat\Phi(r-\alpha+\beta) + =\one_{\alpha=\beta}\widehat\Phi(0). +\] +Since \(\delta^{-d_F}\ll N+V^2\), this proves +\Cref{eq:additive-sieve}. Equality of the operator norms of a finite +matrix and its adjoint gives the dual assertion. +\end{proof} + +\begin{proposition}\label{prop:probe-low} +For the weights above, for every \(\epsilon>0\) and \(Z\ge2\), +\begin{equation} + I(Z^a,Z^b,Z)\ll_{F,S,\eta,W_0,W_1,\epsilon} + Z^{(2M_0-1-b)/2+\epsilon} + =Z^{933/2000+\epsilon}. +\label{eq:probe-low} +\end{equation} +\end{proposition} + +\begin{proof} +Write \(v=q_{\mathfrak s}/Y\) and +\(y=m/(t_0\ell_{XY})\). For \(v\) in the support of \(W_1\), the support +of \(W_0(y/\ell_v)\) puts \(y\) in one fixed compact annular set. +On this set the smooth function +\[ + v^{-1/2}W_1(v)W_0(y/\ell_v) +\] +has a Fourier series in \(\log v\) whose coefficients \(c_j(y)\) +satisfy \(\sum_j\|c_j\|_\infty<\infty\), uniformly in \(y\). +To see this, take a compact interval containing the logarithmic support, +extend by zero to a slightly larger period, and integrate twice by parts. +All the required derivatives are bounded uniformly on the compact +\(y\)-set. Thus we can separate this weight from the additive sums with +a bounded total coefficient cost. + +For each Fourier term, expansion of \(g_{\mathfrak s}\) gives a +coefficient of modulus \(Y^{-1}q_{\mathfrak s}^{-1/2}\) at each reduced +fraction \(d/s\), with \(q_{\mathfrak s}\asymp Y\). Their total squared +mass is at most +\[ + Y^{-2}\sum_{q_{\mathfrak s}\asymp Y} + q_{\mathfrak s}^{-1}|(R/\mathfrak s)^\times| + \ll_F Y^{-1}, +\] +using the fixed-field ideal count. Take \(N=XY\) and \(V\ll Y\) in +\Cref{lem:additive-sieve}. The bounded set of representatives \(t_0\) +only enlarges the fixed compact row box. By the triangle inequality for +the Fourier series, the squared \(\ell^2\) norm in \(m\) of the whole +additive factor in \Cref{eq:poisson-probe}, including +\((q_{\mathfrak s}X)^{-1/2}\), is +\[ + \ll (XY)^{-1}\frac{XY+Y^2}{Y}\ll Y^{-1}. +\] +The last inequality uses \(b0\), the local integrability used above gives an absolute +sheet majorant \(O_\sigma(T^{-\sigma})\), by using one complex weighted +supremum and the other weighted local integrals. Taking \(\sigma>1\) +sums the \(q_u^{-\sigma}q_{\mathfrak r}^{-6\sigma}\) majorant, while +the Gaussian handles the ideal \(\mathfrak A\); this permits Tonelli's +theorem in the orbit sum. The zero frequency contributes +nothing because \(\widehat W_0(0)=0\). + +By the measure disintegration, the Mellin transform of \(V_0\) is \(M\). +The same weighted derivative majorants in the complex norm variables +justify differentiation of the sheet integral in \(\log T\), so +\(V_0(T)\) is smooth for \(T>0\). Its absolute weighted integrability +and \Cref{eq:poisson-M-decay} then give pointwise Mellin inversion by +Fourier inversion in \(\log T\) on each line \(\re z>0\). +By \Cref{eq:frequency-decomposition}, write the orbits uniquely as +\(h=ur^6\), where \(u\in\mathcal U\) and \(\mathfrak r\) is an exterior +ideal with allowed generator \(r\). The principal row \(u=1\) consists +of the surviving nonzero sixth-power frequencies \(h=r^6\). Hence the +precise Poisson expression is +\begin{equation} + \begin{split} + I(X,Y,Z) + ={}&\frac{X^{1/2}}Y + \sum_{\substack{\mathfrak c\ {\rm squarefree}\\ + \mathfrak n,\mathfrak s}} + \frac{\gamma_2(\mathfrak c)\vartheta(\mathfrak A)} + {q_{\mathfrak c}^{1/2}q_{\mathfrak n}q_{\mathfrak A}}\, + P_G(q_{\mathfrak A}/Z)W_1(q_{\mathfrak s}/Y)\\ + &{}\times + \sum_{u\in\mathcal U}\sum_{\mathfrak r} + F(s,A,ur^6) + V_0\!\left(\frac{Xq_uq_{\mathfrak r}^6}{q_{\mathfrak A}}\right), + \qquad \mathfrak A=\mathfrak c\mathfrak n^3. + \end{split} +\label{eq:Poisson-orbits} +\end{equation} +Here \(P_G\) is the Gaussian of \Cref{eq:reflection-gaussian}. + +Use Mellin inversion for \(P_G,W_1,V_0\), with transforms +\(e^{t^2},\widehat W_1(w),M(z)\), respectively. The factors involving +\(\mathfrak A\) are +\(q_{\mathfrak A}^{-1}q_{\mathfrak A}^{-t}q_{\mathfrak A}^{z}\). +Set \(t=\xi+z-1\). We obtain +\begin{equation} + \begin{split} + I(X,Y,Z)=\frac1{(2\pi i)^3} + \int_{(4)}\int_{(4)}\int_{(2)} + &X^{1/2-z}Z^{\xi+z-1}Y^{w-1} + e^{(\xi+z-1)^2}M(z)\widehat W_1(w)\\ + &{}\times\sum_{u\in\mathcal U}q_u^{-z} + \mathcal F_u(\xi,w,z)\,dz\,dw\,d\xi , + \end{split} +\label{eq:poisson-mellin} +\end{equation} +where the innermost line is \(\re z=2\), the middle line is +\(\re w=4\), and the outer line is \(\re\xi=4\), and +\begin{equation} + \mathcal F_u(\xi,w,z)= + \sum_{\substack{\mathfrak c\ {\rm squarefree}\\ + \mathfrak n,\mathfrak s,\mathfrak r}} + \gamma_2(\mathfrak c)\vartheta(\mathfrak A)F(s,A,ur^6) + q_{\mathfrak c}^{-1/2-\xi}q_{\mathfrak n}^{-1-3\xi} + q_{\mathfrak s}^{-w}q_{\mathfrak r}^{-6z}, + \qquad \mathfrak A=\mathfrak c\mathfrak n^3. +\label{eq:calF-definition} +\end{equation} +All interchanges used for \Cref{eq:poisson-mellin} are in absolute +convergence. For example, +\(|F(s,A,h)|\le q_{\mathfrak s}q_{\mathfrak A}\), +\(|\gamma_2(\mathfrak c)|=1\), and the ideal count, including the finitely +many S-unit classes for \(u\), give absolute convergence on these lines. +The factors \(M(z)\) and \(\widehat W_1(w)\) have arbitrary vertical +polynomial decay, while the Gaussian has Gaussian decay in +\(\im(\xi+z)\). Changing height variables to +\((\im(\xi+z),\im z,\im w)\) proves absolute integrability as well. + +\subsection{The local correction} + +We extend the notation \(L^S(s,\eta)\) from +\Cref{sec:coefficient-family} to a Hecke character on exterior primes: +the factors at \(S\) are omitted, and any additional zero prime values +are retained in the Euler product. +Thus, for example, +\[ + L^S(w,\chi_\bullet(u)) + =\prod_{\mathfrak p\notin S} + (1-\chi_{\mathfrak p}(u)q_{\mathfrak p}^{-w})^{-1} + \quad(\re w>1). +\] +As before, +\(\zeta_F^S(s)=\prod_{\mathfrak p\notin S}(1-q_{\mathfrak p}^{-s})^{-1}\). + +\begin{proposition}\label{prop:euler-product} +For every \(u\in\mathcal U\), the series in +\Cref{eq:calF-definition} has the Euler factorization +\begin{equation} + \mathcal F_u(\xi,w,z)= + \frac{\zeta_F^S(6z)L^S(w,\chi_\bullet(u))} + {L^S(\xi,\eta\overline{\chi_\bullet(u)})} + \mathcal H_u(\xi,w,z) +\label{eq:probe-euler-product} +\end{equation} +initially in absolute convergence. The correction \(\mathcal H_u\) is +holomorphic in open neighborhoods of both real-part regions +\begin{equation} + \begin{array}{lll} + \re\xi\ge .998,&\re w\ge .003,&\re z\ge .4,\\ + \re\xi\ge .998,&\re w\ge .95,&\re z\ge .16. + \end{array} +\label{eq:euler-regions} +\end{equation} +For every \(\epsilon>0\) it satisfies +\[ + |\mathcal H_u(\xi,w,z)|\ll_{F,S,\eta,\epsilon}q_u^\epsilon +\] +there, uniformly in all heights. If \(S\) contains every prime of norm +at most a sufficiently large fixed constant, then +\(\mathcal H_1\) is bounded and nowhere zero on both neighborhoods. +\end{proposition} + +The two regions serve different contour movements in the next section. +The first permits the small and intermediate nonprincipal rows to place +\(w\) at \(.003\) and \(z\) at \(.4\), while \(\xi\) can reach \(.998\) +once the reciprocal denominator has been controlled. The second contains +the principal numerator poles \(w=1\), \(z=1/6\) and the nearby lines +\(w=.95\), \(z=.16\) used to isolate their double residue. The +nowhere-zero conclusion for \(u=1\) will ensure that its correction does +not cancel a singularity of \(1/L^S(\xi,\eta)\). + +\begin{proof} +We first factor the finite expression, keeping the units that Chinese +remaindering introduces. Use products of allowed prime generators for +\(s,A,r\). They differ from any other allowed choices by sixth powers +of S-units, which have no effect on the expressions. At an exterior +prime \(\mathfrak p\), with generator \(p\) and norm \(Q=q_{\mathfrak p}\), +write +\[ + \begin{gathered} + t=v_{\mathfrak p}(\mathfrak A)=e_0+3l,\quad + e_0=v_{\mathfrak p}(\mathfrak c)\in\{0,1\},\quad + l=v_{\mathfrak p}(\mathfrak n),\quad k=v_{\mathfrak p}(\mathfrak s),\\ + j=v_{\mathfrak p}(u),\quad m'=v_{\mathfrak p}(\mathfrak r), + \quad J=j+6m'. + \end{gathered} +\] +Write \(s=p^ks_p\), \(A=p^tA_p\), and \(ur^6=p^Jh_p\); the subscripts +mean that the full \(p\)-power has been omitted. These three subscripted +quantities are units at \(\mathfrak p\). Put +\(\psi_{\mathfrak p}^{\circ}(x)=\psi_{\mathfrak p}(-x)\). +For \(x\in F\), the global additive product gives +\(\e(x)=\prod_{\mathfrak p\notin S}\psi_{\mathfrak p}^{\circ}(x)\), +with only finitely many nontrivial factors. Expand the inner Gauss sum +in \Cref{eq:finite-Poisson-factor} and apply the Chinese remainder +decomposition to each residue ring. Before changing variables, the local +factor of \(F(s,A,ur^6)\) is +\[ + \sum_{\substack{d\bmod p^k\\p^Jh_p\equiv p^tA_p d\pmod{p^k}}} + \chi_{\mathfrak p}(d)^k + \sum_{v\bmod p^t}\chi_{\mathfrak p}(v)^t + \psi_{\mathfrak p}^{\circ}\!\left( + \frac{(p^Jh_p-p^tA_p d)v}{p^{k+t}s_pA_p}\right). +\] +Here an absent modulus \(p^0\) has one residue and character value \(1\). +For a positive modulus the character retains its unit mask, even when +its exponent is divisible by six. Now make the changes +\[ + d=(h_p/A_p)d',\qquad v=(s_pA_p/h_p)v'. +\] +They are bijections of the relevant residue rings. The congruence becomes +\(p^J\equiv p^t d'\pmod{p^k}\), and the local additive argument becomes +\((p^J-p^td')v'/p^{k+t}\). The unit multiplier removed by these changes +is exactly +\begin{equation} + \chi_{\mathfrak p}(h_p)^{k-t} + \chi_{\mathfrak p}(s_p)^t + \chi_{\mathfrak p}(A_p)^{t-k}. +\label{eq:Euler-CRT-unit} +\end{equation} +There are no hidden coprimality assumptions here: these changes only use +the units left after omitting the full \(p\)-powers. Moreover +\(\chi_{\mathfrak p}(h_p)=\chi_{\mathfrak p}(u/p^j)\), because the +remaining factors of \(r^6\) are sixth powers. + +For completeness define the stripped local sum by +\begin{equation} + C_{t,k}(J)= + \sum_{\substack{d\bmod p^k\\p^J\equiv p^td\pmod{p^k}}} + \chi_{\mathfrak p}(d)^k + \sum_{v\bmod p^t}\chi_{\mathfrak p}(v)^t + \psi_{\mathfrak p}^{\circ} + \left(\frac{(p^J-p^td)v}{p^{k+t}}\right). +\label{eq:C-local-definition} +\end{equation} +When \(k=0\) the outer modulus is one, its character is one, and its +single residue may be taken to be zero. The analogous convention applies +to \(t=0\) in the inner sum. When \(k>0\) or \(t>0\), the corresponding +character is supported on units even if its exponent is divisible by +six. These conventions distinguish the absent modulus from the zeroth +power character on a nontrivial modulus. + +We check explicitly that the remaining factors in different primes +separate. Expand \Cref{eq:Euler-CRT-unit} using the product generators, +and consider an unordered pair of distinct primes \(\mathfrak p,\mathfrak +p'\). In their common symbol +\(\chi_{\mathfrak p}(p')=\chi_{\mathfrak p'}(p)\), the contributions from +\(\chi_{\mathfrak p}(s_p)^t\) and +\(\chi_{\mathfrak p}(A_p)^{t-k}\), together with the analogous factors at +\(\mathfrak p'\), have exponent +\[ + tk'+t'k+(t-k)t'+(t'-k')t=2tt'. +\] +In particular all \(s\)-\(A\) interactions cancel. By +\Cref{eq:gauss-crt}, the Chinese remainder factors in +\(\gamma_2(\mathfrak c)\) add \(4e_0e_0'\). Their total is +\[ + 2(e_0+3l)(e_0'+3l')+4e_0e_0'\equiv0\pmod6. +\] +Thus all cross-prime terms cancel, including when both moduli have powers +of the same primes; those shared powers were retained inside +\Cref{eq:C-local-definition}. The \(h_p\)-factors remain precisely as +characters of \(u/p^j\). If +\(\theta_{\mathfrak p}=\chi_{\mathfrak p}(u/p^j)\in\mu_6\), the exact +local factor is consequently +\begin{equation} + \begin{split} + \mathcal F_{u,\mathfrak p}(\xi,w,z) + =\sum_{\substack{e_0=0,1\\l,k,m'\ge0}} + &\gamma_2(\mathfrak p)^{e_0}\vartheta(\mathfrak p)^{e_0+3l} + \theta_{\mathfrak p}^{\,k-e_0-3l}C_{e_0+3l,k}(j+6m')\\ + &{}\times + Q^{-e_0(1/2+\xi)-l(1+3\xi)-kw-6m'z}. + \end{split} +\label{eq:exact-Euler-factor} +\end{equation} +It follows initially by absolute convergence that +\(\mathcal F_u=\prod_{\mathfrak p\notin S}\mathcal F_{u,\mathfrak p}\). + +Here are explicit evaluations of every \(C_{t,k}(J)\). Put +\[ + T_v(y)=\sum_{x\bmod p}\chi_{\mathfrak p}(x)^v + \psi_{\mathfrak p}^{\circ}(yx/p), + \qquad \tau_v=T_v(1). +\] +The unit mask is retained for \(v\equiv0\pmod6\). Thus +\[ + T_v(y)= + \begin{cases} + \overline{\chi_{\mathfrak p}(y)}^{\,v}\tau_v,&p\nmid y,\\ + Q-1,&p\mid y,\ v\equiv0\pmod6,\\ + 0,&p\mid y,\ v\not\equiv0\pmod6, + \end{cases} + \qquad + |\tau_v|=\sqrt Q\ (v\not\equiv0\!\!\pmod6),\quad \tau_0=-1. +\] +For \(t=0\) the value is \(C_{0,0}(J)=1\), and +\(C_{0,k}(J)=\one_{J=0}\) for \(k>0\). For \(t>0\) it is +\begin{equation} + \begin{split} + C_{t,0}(J)&=\one_{J\ge t-1}Q^{t-1}T_t(p^{J-t+1}),\\ + C_{t,1}(J)&=\one_{J\ge t}\tau_1Q^{t-1}T_{t-1}(p^{J-t}),\\ + C_{t,k}(J)&=0\qquad(k\ge2). + \end{split} +\label{eq:C-local-values} +\end{equation} +For the first formula sum the \(Q^{t-1}\) lifts of a residue \(v\bmod p\) +in the inner sum; they cancel unless \(J\ge t-1\). For \(k=1\) the +congruence first requires \(J\ge1\). Sum \(d\bmod p\) before \(v\). +Because \(v\) is a unit, the sum is +\(\tau_1\overline{\chi_{\mathfrak p}(v)}\); +\(\chi_{\mathfrak p}(-1)=1\) removes the sign. The remaining sum over +lifts of \(v\) is zero unless \(J\ge t\), and then gives the second +formula. For \(k\ge2\), varying \(d\) by \(p^{k-1}\lambda\), \(\lambda\bmod p\), +preserves the congruence and its unit character. For every unit \(v\) +the phase varies by \(\psi_{\mathfrak p}^{\circ}(-\lambda v/p)\), whose sum in +\(\lambda\) is zero. This proves the last formula. In particular, for \(t>0\), +\begin{equation} + |C_{t,0}(J)|\le Q^t,\qquad + |C_{t,1}(J)|\le Q^{t+1/2}, +\label{eq:C-local-bounds} +\end{equation} +with the support restrictions visible in \Cref{eq:C-local-values}. + +Suppose first that \(\mathfrak p\nmid(u)R\), so \(j=0\). Set +\[ + V=Q^{-6z},\quad + W=\chi_{\mathfrak p}(u)Q^{-w},\quad + D_1=\eta(\mathfrak p)\overline{\chi_{\mathfrak p}(u)}Q^{-\xi}, + \quad + E_1=\vartheta(\mathfrak p)^6Q^{4-6\xi}V. +\] +The \(t=0\) values give +\((1-V)^{-1}+W/(1-W)\): a positive \(k\) requires \(J=0\). +The first squarefree theta term has +\(e_0=1\), \(l=k=m'=0\), hence \(t=1\), \(J=0\), and +\(C_{1,0}(0)=\tau_1\). Its contribution is +\[ + \begin{split} + &\gamma_2(\mathfrak p)\vartheta(\mathfrak p) + \overline{\chi_{\mathfrak p}(u)}Q^{-1/2-\xi}\tau_1\\ + &\qquad=\gamma_1(\mathfrak p)\gamma_2(\mathfrak p) + \vartheta(\mathfrak p)\overline{\chi_{\mathfrak p}(u)}Q^{-\xi} + =-D_1. + \end{split} +\] +Indeed \(\tau_1=\sqrt Q\,\gamma_1(\mathfrak p)\) in the chosen additive +sign, and \Cref{eq:gauss-corrections} gives +\(\gamma_1\gamma_2\vartheta=-\eta\). This negative term is the first +nonconstant term of the reciprocal local \(L\)-factor \(1-D_1\); it is +the reason that \(L^S(\xi,\eta\overline{\chi_\bullet(u)})\) occurs in the +denominator of \Cref{eq:probe-euler-product}. We extract +\((1-V)^{-1}(1-W)^{-1}(1-D_1)\), whose first two factors are those of +\(\zeta_F^S(6z)\) and \(L^S(w,\chi_\bullet(u))\). The mixed terms and +the higher theta terms will be included in the correction. + +For \(t>0\) and \(J=6m'\), the formulas show that only +\(t\equiv0,1\pmod6\) occur. For \(t=6d\), \(d\ge1\), the \(k=0\) sum is +\[ + E_1^d\frac{1-Q^{-1}}{1-V}, +\] +and the \(k=1\) term is \(E_1^dW\). For the latter, the only possible +valuation is \(J=t\), and \(\tau_1\tau_5=Q\). For \(t=6d+1\), \(d\ge0\), +the \(k=0\) term is \(-D_1E_1^d\), extending the first squarefree term +just computed. Finally its \(k=1\) sum is +\[ + -(Q-1)D_1E_1^d\,\frac{WV}{1-V}. +\] +Here \(J>t\) and \(T_0(p^{J-t})=Q-1\); the first possible extra +\(m'\) gives the displayed \(V\). Summing in \(d\) proves the closed +formula +\begin{equation} + \begin{split} + \mathcal F_{u,\mathfrak p} + ={}&\frac1{1-V}+\frac W{1-W} + +\frac{E_1}{1-E_1} + \left(\frac{1-Q^{-1}}{1-V}+W\right)\\ + &-\frac{D_1}{1-E_1} + \left(1+\frac{(Q-1)WV}{1-V}\right). + \end{split} +\label{eq:Euler-good-exact} +\end{equation} +This derivation also explains the vanishing of the allowed theta exponents +\(t\equiv3,4\pmod6\): the possible boundary values of \(J\) in +\Cref{eq:C-local-values} have the wrong residue modulo six, and the +remaining \(T\)-sum is a nontrivial character sum at zero. + +Define at every prime +\[ + \mathcal H_{u,\mathfrak p} + =(1-V)\frac{1-W}{1-D_1}\mathcal F_{u,\mathfrak p}, +\] +where \(W=D_1=0\) at \(\mathfrak p\mid(u)R\), in accordance with the +zero extension. We verify normal convergence on explicit open +neighborhoods: in each line of \Cref{eq:euler-regions} decrease all three +lower endpoints by \(10^{-5}\), and use the resulting strict inequalities. +On these neighborhoods all of \(V,W,D_1,E_1\) have modulus bounded away +from one once \(Q\) exceeds a fixed constant. Subtracting \(1\) after +multiplying \Cref{eq:Euler-good-exact} by +\((1-V)(1-W)/(1-D_1)\) gives +\begin{equation} + |\mathcal H_{u,\mathfrak p}-1| + \ll |VW|+|D_1V|+|D_1W|+Q|D_1WV|+|E_1| + \ll Q^{-1-\delta},\qquad \delta=\frac1{4000}, +\label{eq:Euler-good-error} +\end{equation} +uniformly for \(\mathfrak p\nmid(u)R\). For clarity, before the +\(10^{-5}\) enlargement the weakest of the first three terms is +\(|D_1W|\le Q^{-1.001}\), in the first region. The exponents in the +last two terms are at most +\[ + 1-\re\xi-\re w-6\re z\le-1.908,\qquad + 4-6\re\xi-6\re z\le-2.948 +\] +on the union of the two regions. The enlargement leaves respectively +the margins \(0.00098\), \(0.90792\), and \(1.94788\) beyond exponent +\(-1\), which suffice for the stated \(\delta\). More explicitly, +the specialization at \(E_1=0\) satisfies +\[ + \left.\mathcal H_{u,\mathfrak p}\right|_{E_1=0}-1 + =\frac{D_1(W+V-WV)-WV-(Q-1)D_1(1-W)WV}{1-D_1}. +\] +This accounts for the \(Q|D_1WV|\) term as well as the first three +terms in \Cref{eq:Euler-good-error}. The difference from this +specialization is \(O(|E_1|)\) on the stated neighborhoods: the remaining +denominators are bounded, and \(Q|D_1WV|<1\) once \(Q\) exceeds the fixed +cutoff. + +At \(\mathfrak p\mid(u)R\), \(1\le j\le5\). The \(t=0\) contribution +is \((1-V)^{-1}\). The term \(t=1,k=0\) vanishes, since \(J\ge1\) and +\(T_1(p^J)=0\). By \Cref{eq:C-local-bounds}, the sum for \(t=1,k=1\) +has modulus at most +\[ + \frac{Q^{1-\re\xi-\re w}}{1-|V|}. +\] +Its exponent is at most \(-0.001\) on the first region and at most +\(-0.948\) on the second. For \(l\ge1\), before the factor \(V^{m'}\), +the bound for the summand with \(e_0\in\{0,1\}\) is at most a constant +times \(Q\) to the power +\begin{equation} + (2-3\re\xi)l+(1/2-\re\xi)e_0 + +\max(0,1/2-\re w). +\label{eq:Euler-bad-exponents} +\end{equation} +This is at most \(-0.497\) for \(l=1,e_0=0\) on the first region and +decreases by at least \(0.994\) for each further \(l\); the second region +has a larger saving. The \(10^{-5}\) enlargement still leaves a saving +greater than \(0.49\) in this sum, and leaves \(0.00098\) in the +\(t=1,k=1\) term. The geometric sums in \(l,m'\), followed by +multiplication by \(1-V\), therefore give +\begin{equation} + \mathcal H_{u,\mathfrak p}=1+O(Q^{-\delta}) + \quad(\mathfrak p\mid(u)R),\qquad \delta=\frac1{4000}, +\label{eq:Euler-bad-error} +\end{equation} +on the same neighborhoods. These estimates prove absolute local +convergence there as well. + +The product of the factors in \Cref{eq:Euler-good-error} converges +normally, since \(\sum_{\mathfrak p}q_{\mathfrak p}^{-1-\delta}<\infty\). +The remaining primes are precisely the divisors of \((u)R\). For every +\(\epsilon>0\), +\[ + \prod_{\mathfrak p\mid(u)R}(1+Cq_{\mathfrak p}^{-\delta}) + \ll_{F,S,\epsilon}q_u^\epsilon: +\] +for sufficiently large \(Q\) each local factor is at most \(Q^\epsilon\), +and the finitely many smaller primes contribute a fixed constant. +Consequently \(\mathcal H_u=\prod_{\mathfrak p\notin S} +\mathcal H_{u,\mathfrak p}\) is holomorphic and obeys the asserted bound. +Multiplication of its defining local factors gives +\Cref{eq:probe-euler-product} in the initial region and then as a +meromorphic identity on either neighborhood. + +For \(u=1\) every prime satisfies \Cref{eq:Euler-good-error}. The local +estimate determines a fixed norm cutoff beyond which its uniform bound +is smaller than \(1/2\). Choose \(S\) to contain all primes up to that +cutoff from the outset, as permitted in \Cref{sec:arithmetic}. Every remaining +\(\mathcal H_{1,\mathfrak p}\) is nonzero, and its logarithm defined near +\(1\) has an absolutely and uniformly convergent sum. Its exponential is +\(\mathcal H_1\), which is therefore bounded and nowhere zero on both +neighborhoods. This proves the proposition. +\end{proof} + +For the remainder of the proof, write +\[ + \mathcal H_\eta(s):=\mathcal H_1(s,1,1/6). +\] +This is the principal specialization announced in +\Cref{sec:coefficient-family}. Since \((w,z)=(1,1/6)\) lies in the +second region of \Cref{eq:euler-regions}, \(\mathcal H_\eta\) is +holomorphic, bounded, and nowhere zero on an open neighborhood of the +closed half-plane \(\re s\ge .998\), uniformly in height. diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/06-zero-free.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/06-zero-free.tex new file mode 100644 index 0000000000000000000000000000000000000000..589df3a177b3961e83ff3b8f6fadcba6ebb2f64a --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/06-zero-free.tex @@ -0,0 +1,764 @@ +\section{Zero detection and Mellin continuation}\label{sec:zero-free} + +We now prove \Cref{thm:hecke-zero-free} for the fixed but arbitrary +\(F\) and \(\eta\). The factorization in +\Cref{prop:euler-product} singles out the frequency \(u=1\): it is the +only row whose numerator \(L\)-function has a pole. We shall bound the +other rows by a conductor large sieve, isolate the two residues of that +principal row, and compare them with \Cref{prop:probe-low}. The resulting +bound for one Mellin integral will exclude zeros of the target function. + +\subsection{A conductor large sieve and a zero detector} + +Let \(\omega_u\) denote the primitive Hecke character inducing +\(\eta\overline{\chi_\bullet(u)}\). By +\Cref{lem:kummer-characters}, on a dyad \(U\le q_u<2U\), +\begin{equation} + \Norm\mathfrak f_{\omega_u}\ll_{F,S,\eta}U, + \qquad + \#\{u\in\mathcal U:U\le q_u<2U\}\ll_{F,S}U . +\label{eq:row-conductor-count} +\end{equation} +The second bound follows from the ideal count and the finite number of +S-unit classes modulo sixth powers. The characters \(\omega_u\) in this +family are distinct: equality cancels the fixed twist and forces \(u/v\) +to be a sixth power in \(F\); the selected exterior valuations then agree, +and the quotient is in \(U_S^6\). Their local components at \(S\) run +through a fixed finite set, since the Kummer part depends on +\(F_v^\times/F_v^{\times6}\). The principal numerator occurs only for +\(u=1\). The primitive denominator \(\omega_u\) may be principal for +a different row, so it will be treated separately in zero detection. + +We give the conductor estimate required for that detection. In the next +lemma and in the detector, ideals without the adjective exterior are +integral ideals of \(\mathcal O_F\), and \(q_{\mathfrak n}=\Norm\mathfrak n\) +also for those ideals. + +\begin{lemma}[Primitive conductor large sieve]\label{lem:conductor-sieve} +Fix \(C_0>0\) and a finite set \(\mathscr L_S\) of possible local components at \(S\). +Let \(\mathscr X(U)\) be any family of distinct primitive finite-order +Hecke characters of \(F\) whose conductor norms are at most \(C_0U\) +and whose components at \(S\) belong to that set. For \(N,U\ge2\), +any complex coefficients \(b_{\mathfrak n}\), and every \(\epsilon>0\), +\begin{equation} + \sum_{\omega\in\mathscr X(U)} + \left|\sum_{q_{\mathfrak n}\le N} + b_{\mathfrak n}\omega(\mathfrak n)\right|^2 + \ll_{F,S,C_0,\mathscr L_S,\epsilon} + (N+U^2)(NU)^\epsilon + \sum_{q_{\mathfrak n}\le N}|b_{\mathfrak n}|^2 . +\label{eq:conductor-sieve} +\end{equation} +As usual a primitive character is extended by zero on ideals meeting +its conductor. +\end{lemma} + +\begin{proof} +First restrict the coefficient ideals to one exterior norm dyad and the +characters to one choice of their components at \(S\). If +\(\mathfrak q\) is the exterior part of the conductor, principality of +\(R\) and the global character relation write the value at +\(\mathfrak n=nR\), for its selected balanced generator \(n\), as +\[ + \omega(\mathfrak n)=\lambda_S(n)\lambda_{\mathfrak q}(n). +\] +Here \(\lambda_S\) is a common fixed unitary function of the +S-components, and \(\lambda_{\mathfrak q}\) is a primitive residue +character modulo \(\mathfrak q\). The orientation of the local +characters is absorbed in these two symbols. The formula includes zero +on nonunits. It follows directly by evaluating the idele character at +the rational element \(n\): the unramified exterior components give +\(\omega(\mathfrak n)\), leaving the ramified exterior components and +the components in \(S\). In particular, for this fixed S-choice the +residue character determines \(\omega\). + +Let \(q\) be a balanced generator of \(\mathfrak q\). By +\Cref{lem:s-lattice}, \((n,d)\mapsto\e(nd/q)\) is a perfect additive +pairing on \(R/\mathfrak q\). Put +\[ + \tau(\overline{\lambda_{\mathfrak q}}) + =\sum_{d\bmod\mathfrak q} + \overline{\lambda_{\mathfrak q}(d)}\,\e(d/q). +\] +For a primitive residue character, this Gauss sum has modulus +\(q_{\mathfrak q}^{1/2}\), and +\[ + \lambda_{\mathfrak q}(n)= + \frac1{\tau(\overline{\lambda_{\mathfrak q}})} + \sum_{d\bmod\mathfrak q} + \overline{\lambda_{\mathfrak q}(d)}\,\e(nd/q). +\] +For a unit \(n\) this follows by a change of residue variable. For a +nonunit the Gauss sum is zero: locally at a prime dividing \(n\) the +additive character factors through a proper quotient of the conductor, +and multiplication by a unit in the kernel of that quotient on which +the primitive multiplicative character is nontrivial forces the sum to +vanish. Finite additive Parseval now gives +\[ + \sum_{n\bmod\mathfrak q} + \left|\sum_{d\bmod\mathfrak q} + \overline{\lambda_{\mathfrak q}(d)}\e(nd/q)\right|^2 + =q_{\mathfrak q}|(R/\mathfrak q)^\times|. +\] +The left side is +\(|(R/\mathfrak q)^\times|\,|\tau(\overline{\lambda_{\mathfrak q}})|^2\), +which proves the asserted modulus. Put +\[ + T_{d/q}=\sum_{\mathfrak n}b_{\mathfrak n}\lambda_S(n)\e(nd/q). +\] +Sum the square of the preceding Gauss expansion over the characters of +conductor \(\mathfrak q\). Extending that sum to all residue characters +and using character orthogonality bounds it by +\[ + \frac{|(R/\mathfrak q)^\times|}{q_{\mathfrak q}} + \sum_{d\in(R/\mathfrak q)^\times}|T_{d/q}|^2 + \le \sum_{d\in(R/\mathfrak q)^\times}|T_{d/q}|^2 . +\] +The same assertion holds for the unit conductor with its single residue. +Sum now over \(q_{\mathfrak q}\le C_0U\). +The fractions are distinct reduced fractions. The balanced generators +of the coefficient ideals on a norm dyad lie in \(\ell_{N'}K\) for one +fixed compact \(K\), with \(N'\le N\), by +\Cref{lem:balanced-generators}. The dual assertion of +\Cref{lem:additive-sieve}, with \(V=\max(2,C_0U)\), bounds this sum by +\((N+U^2)\sum|b_{\mathfrak n}|^2\). + +To pass to all exterior lengths at most \(N\), split into dyads and use +Cauchy--Schwarz, at a logarithmic cost. For general \(\mathcal O_F\) +ideals write \(\mathfrak n=\mathfrak n_S\mathfrak n^S\), with the first +factor supported in \(S_f\). There are +\(O_S((1+\log N)^{|S_f|})\) possibilities for \(\mathfrak n_S\). +Within a fixed S-choice the value \(\omega(\mathfrak n_S)\) is fixed +(or zero). Cauchy--Schwarz over these possibilities and the preceding +exterior estimate introduce only powers of \(\log N\); the square masses +over the disjoint pieces sum to the original square mass. The finite +partition by S-components costs a constant. Absorbing all logarithms +into \((NU)^\epsilon\) proves \Cref{eq:conductor-sieve}. +\end{proof} + +We use standard Hecke continuation and the functional equation in a +deliberately conservative convexity bound. For a nonprincipal primitive +finite-order \(\omega\) with \(\Norm\mathfrak f_\omega\ll U\), there is +an \(A_F>0\), depending on the fixed field, such that on any of the +fixed strips below, in particular for \(.003\le\sigma\le4\), +\begin{equation} + |L_F(\sigma+it,\omega)| + \ll_{F,S,\eta,\epsilon}U^{1/2+\epsilon}(2+|t|)^{A_F}. +\label{eq:Hecke-conservative-convexity} +\end{equation} +This follows from the functional equation, the absolutely convergent +Euler product on a right line, and the Phragmén--Lindelöf principle; +the gamma factors contribute the field-dependent height power +\cite[Main Theorem~4.4.1 and \S4.5]{Tate67}. The same estimates give polynomial height bounds +on any fixed strip for \((s-1)\zeta_F(s)\). Omitting factors at a fixed set or +at primes dividing \(u\) changes a positive-real-part estimate by +\(q_u^\epsilon\): for a fixed \(\sigma_*>0\), each of +\[ + \prod_{\mathfrak p\mid(u)R}(1+q_{\mathfrak p}^{-\sigma_*}), + \qquad + \prod_{\mathfrak p\mid(u)R}(1-q_{\mathfrak p}^{-\sigma_*})^{-1} +\] +is \(O_{\epsilon,\sigma_*}(q_u^\epsilon)\), by bounding a sufficiently +large prime's factor by \(q_{\mathfrak p}^\epsilon\) and absorbing the +finitely many other primes into the constant. + +\begin{proposition}\label{prop:zero-density} +For the fixed \(F,S,\eta\) there are \(\rho>0\) and \(U_0\) with the +following property. For every \(U\ge U_0\), there is a set +\(\mathcal E(U)\subset\{u\in\mathcal U:U\le q_u<2U\}\) of cardinality +\[ + |\mathcal E(U)|\ll_{F,S,\eta}U^{1/2} +\] +which contains every row for which \(\omega_u\) is principal or +\(L_F(s,\omega_u)\) has a zero with +\(\re s\ge .996\) and \(|\im s|\le4U^\rho\). +For \(u\notin\mathcal E(U)\) and every \(\epsilon>0\), +\begin{equation} + \left|L^S(\xi,\eta\overline{\chi_\bullet(u)})^{-1}\right| + \ll_{F,S,\eta,\epsilon}U^\epsilon + \qquad + (\re\xi\ge .998,\ |\im\xi|\le2U^\rho). +\label{eq:good-reciprocal} +\end{equation} +The exponents \(.996,.998,1/2\) are absolute. The height exponent +\(\rho\), the constants, and \(U_0\) may depend on the fixed data. +\end{proposition} + +\begin{proof} +Set \(\sigma_0=.996\). For a primitive nonprincipal character \(\omega\) +in the family of \Cref{eq:row-conductor-count}, define the truncated +inverse and its product coefficients by +\[ + D_U(s,\omega)=\sum_{q_{\mathfrak d}\le U^2} + \mu_F(\mathfrak d)\omega(\mathfrak d)q_{\mathfrak d}^{-s}, + \qquad + \alpha_U(\mathfrak n) + =\sum_{\substack{\mathfrak d\mid\mathfrak n\\q_{\mathfrak d}\le U^2}} + \mu_F(\mathfrak d). +\] +Here \(\mu_F\) is the Möbius function of integral \(\mathcal O_F\) +ideals. In absolute convergence, +\[ + L_F(s,\omega)D_U(s,\omega) + =\sum_{\mathfrak n}\alpha_U(\mathfrak n) + \omega(\mathfrak n)q_{\mathfrak n}^{-s}. +\] +Complete multiplicativity with the primitive zero extension justifies +this identity also on ideals meeting the conductor. The coefficients +are independent of \(\omega\) and satisfy +\begin{equation} + \alpha_U(1)=1,\qquad + \alpha_U(\mathfrak n)=0\quad(11\), +\begin{equation} + \begin{split} + \sum_{\mathfrak n}\alpha_U(\mathfrak n)\omega(\mathfrak n) + q_{\mathfrak n}^{-\beta-i\gamma}e^{-q_{\mathfrak n}/U^{10}} + =\frac1{2\pi i}\int_{(c)} + L_F(\beta+i\gamma+v,\omega)D_U(\beta+i\gamma+v,\omega) + \Gamma(v)U^{10v}\,dv . + \end{split} +\label{eq:zero-detector} +\end{equation} +Move the \(v\)-line to \(\re v=1/2-\beta\), which lies between +\(-1/2\) and \(-.496\). The only crossed gamma pole is \(v=0\), and +its residue vanishes because \(L_F(\beta+i\gamma,\omega)=0\). +The character is nonprincipal, so its \(L\)-function has no pole in the +shift. The gamma decay and the polynomial height bounds justify the +horizontal limits. + +On the new line the fixed-field ideal count gives +\[ + |D_U(1/2+it,\omega)| + \le\sum_{q_{\mathfrak d}\le U^2}q_{\mathfrak d}^{-1/2} + \ll_F U. +\] +Use \Cref{eq:Hecke-conservative-convexity} and integrate the gamma +factor. The right side of \Cref{eq:zero-detector} is at most +\begin{equation} + \ll U^{10(1/2-\beta)+1/2+1+\epsilon}(1+|\gamma|)^{A_F} + \ll U^{-3.46+A_F\rho+\epsilon}. +\label{eq:detector-shift-bound} +\end{equation} +Choose once and for all +\[ + 0<\rho<\min\left\{\frac1{10},\frac1{A_F+1}\right\} +\] +and then take the preliminary loss small. The last bound is \(o(1)\). + +The first term on the left of \Cref{eq:zero-detector} is +\(e^{-U^{-10}}=1+o(1)\), and the intervening terms through \(U^2\) +vanish by \Cref{eq:detector-coefficients}. The portion with +\(q_{\mathfrak n}>U^{11}\) is exponentially small, uniformly in +\(\beta\ge\sigma_0\): the divisor bound and ideal count majorize it by +a fixed power of \(U\) times \(e^{-U/2}\). It follows that the sum +over \(U^2U^2/2\) and \(N\le U^{11}\), the nonlogarithmic cost in +\Cref{eq:conductor-sieve} is bounded by +\begin{equation} + (N+U^2)N^{1-2\sigma_0+\epsilon} + \ll N^{2-2\sigma_0+\epsilon} + \ll U^{.088+O(\epsilon)} . +\label{eq:density-exponent} +\end{equation} +Here \(2-2\sigma_0=.008\), and \(11(.008)=.088\); the factor +\((NU)^\epsilon\) in the large sieve is included in \(U^{O(\epsilon)}\). +Chebyshev's inequality with \Cref{eq:detecting-partial-sum}, followed by +summation over the \(O(\log U)\) dyads, gives +\[ + \#\{\omega\text{ with a detected zero}\} + \ll U^{.088+\rho+O(\epsilon)}. +\] +The choice \(\rho<.1\) leaves more than \(0.31\) before exponent \(1/2\). +Take the losses small and include the at most one principal character. +Distinctness of the \(\omega_u\)'s now gives the asserted bound for +\(\mathcal E(U)\). + +It remains to obtain the reciprocal bound from the zero-free rectangles +just proved. Let \(T=U^\rho\) and \(u\notin\mathcal E(U)\). For +\(|t|\le2T\), consider disks centered at \(2+it\) of radii +\[ + R_2=1.0035,\qquad r_2=1.003,\qquad r_0=.999,\qquad r_1=1.002. +\] +For large \(U\) the outer disk lies within +\(\re s>.996,\ |\im s|<4T\); its left edge is \(.9965\). +The nonprincipal \(L_F(s,\omega_u)\) is entire and nonzero on the disk. +It has a branch of \(\log L_F\) there agreeing at the center with the +Euler logarithm. The convexity bound gives +\(\re\log L_F=\log|L_F|\le C_F\log U\) on the outer disk, and the value +at the center is \(O_F(1)\). Borel--Carathéodory consequently bounds +\(|\log L_F|\) by \(O_F(\log U)\) on radius \(r_2\). On radius \(r_0\) +the real part is at least \(1.001\), so the absolutely convergent Euler +logarithm is \(O_F(1)\). The three-circles theorem applied to the +analytic function \(\log L_F\) now gives, on radius \(r_1\), +\[ + |\log L_F(s,\omega_u)|\ll_F(\log U)^\theta,\qquad + \theta=\frac{\log(1.002/.999)}{\log(1.003/.999)}<1. +\] +These disks reach the line \(\re s=.998\); absolute convergence covers +the larger real parts outside them. Hence for \(\re s\ge.998\) and +\(|\im s|\le2T\), +\[ + |L_F(s,\omega_u)^{-1}| + \le \exp\bigl(O_F((\log U)^\theta)\bigr) + \ll_{F,\epsilon}U^\epsilon . +\] +The estimate for omitted factors preceding the proposition gives the +same bound for the incomplete function, proving +\Cref{eq:good-reciprocal}. +\end{proof} + +\subsection{The principal component and the other frequency rows} + +Use the scales in \Cref{eq:reflection-scales}, and let +\(\sigma_\xi,\sigma_w,\sigma_z\) denote the real parts of +\(\xi,w,z\). Write \(E_*(\sigma_\xi,\sigma_w,\sigma_z)\) for the real +power of \(Z\) outside the Dirichlet factors in \Cref{eq:poisson-mellin}. +In the principal row, taking the residues at \(w=1\) and +\(z=1/6\) leaves the exponent \(C_*(\xi)\) in the second line: +\begin{equation} + \begin{split} + E_*(\sigma_\xi,\sigma_w,\sigma_z) + &=\frac a2+\sigma_\xi-1+h_0\sigma_z+b(\sigma_w-1),\\ + C_*(\xi)&=\frac a2+\xi-1+\frac{h_0}{6} + =\xi-\frac8{15},\qquad C_*(1)=\frac7{15}. + \end{split} +\label{eq:contour-exponent} +\end{equation} +Thus \(C_*(1)=7/15\) is the reference power for comparing the +principal remainders and the other frequency rows. + +We specify a decay observation that justifies the contour limits and, +later, the truncated shifts. On any fixed real strips used below, for +every \(A>0\), +\begin{equation} + \left|e^{(\xi+z-1)^2}M(z)\widehat W_1(w)\right| + \ll_A e^{-(\im\xi+\im z)^2} + (1+|\im z|)^{-A}(1+|\im w|)^{-A}. +\label{eq:triple-height-decay} +\end{equation} +The constants can depend on those strips. Away from the poles extracted +below, all numerator Hecke and zeta functions on them have polynomial height growth, and +\(\mathcal H_u\) is uniformly bounded by \(q_u^\epsilon\). +After increasing \(A\), their height powers are absorbed in +\Cref{eq:triple-height-decay}. In particular for every \(A_1>0\), +\[ + \int_{\R}e^{-(T+v)^2}(1+|v|)^{-A}\,dv + \ll_{A_1}(1+|T|)^{-A_1} +\] +with \(A\) sufficiently large. Thus integration in \(z,w\) gives +arbitrary polynomial decay in the remaining \(\xi\)-height as well. +On lines \(\re\xi>1\) the reciprocal denominator is globally bounded +by its absolutely convergent Euler product. These facts justify all +infinite contour shifts on such lines by their horizontal limits. + +Let \(I_1(Z)\) denote the contribution of \(u=1\) to the initially +absolutely convergent formula \Cref{eq:poisson-mellin}, on the scales +\((X,Y)=(Z^a,Z^b)\). Its numerator is \(\zeta_F^S(w)\). Put +\(\delta_1=10^{-7}\), \(\sigma_1=1+\delta_1\), and +\(\zeta_1=1/6+\delta_1\). Move the initial lines to +\[ + \re\xi=\sigma_1,\qquad \re w=\sigma_1,\qquad \re z=\zeta_1. +\] +These moves cross no poles: both zeta arguments remain to the right of +one, and the reciprocal denominator is in absolute convergence. +The second region of \Cref{eq:euler-regions} contains all the moves. +Now move \(w\) to \(.95\), crossing its simple pole at \(1\), and call +the remaining triple integral \(R_w(Z)\). In the \(w=1\) residue move +\(z\) to \(.16\), crossing the pole of \(\zeta_F^S(6z)\) at \(1/6\), +and call the remaining residue integral \(R_z(Z)\). Their contours are +\[ + \begin{aligned} + R_w:&\quad (\re\xi,\re w,\re z)=(\sigma_1,.95,\zeta_1),\\ + R_z:&\quad (\re\xi,w,\re z)=(\sigma_1,1,.16). + \end{aligned} +\] +The other term is the double residue. With its evaluation +\(c_0f_\eta(Z)\) below, the principal component is exactly +\[ + I_1(Z)=c_0f_\eta(Z)+R_w(Z)+R_z(Z). +\] +The powers of the two remainders relative to \(C_*(1)\) are +\[ + \begin{split} + E_*(\sigma_1,.95,\zeta_1)-C_*(1) + &=(1+h_0)\delta_1-.05b=-.00664989,\\ + E_*(\sigma_1,1,.16)-C_*(1) + &=\delta_1+h_0(.16-1/6) + =-.000666566666\ldots . + \end{split} +\] +The height bounds just given and \Cref{prop:euler-product} therefore give +\begin{equation} + \begin{aligned} + R_w(Z)&=O_{F,S,\eta,W_0,W_1}\bigl(Z^{C_*(1)-.0005}\bigr),\\ + R_z(Z)&=O_{F,S,\eta,W_0,W_1}\bigl(Z^{C_*(1)-.0005}\bigr). + \end{aligned} +\label{eq:principal-remainders} +\end{equation} + +To evaluate the double residue, write +\(R_S=\Res_{s=1}\zeta_F^S(s)>0\). It has the form +\(c_0f_\eta(Z)\), where +\begin{equation} + \begin{split} + c_0&=\frac{R_S^2}{6}M(1/6)\widehat W_1(1)\ne0,\\ + f_\eta(Z)&=\frac1{2\pi i}\int_{(2)} + Z^{C_*(\xi)}e^{(\xi-5/6)^2} + \frac{\mathcal H_\eta(\xi)}{L^S(\xi,\eta)}\,d\xi . + \end{split} +\label{eq:principal-Mellin-integral} +\end{equation} +Indeed the residue of \(\zeta_F^S(6z)\) is \(R_S/6\), and +\(\widehat W_1(1)=\int_0^\infty W_1(t)\,dt>0\). The other factor is +nonzero by \Cref{eq:poisson-Fourier-weight}. The norm measures were +fixed before the probe, so this is the exact scalar in its normalization. +The line of the double residue was initially \(\re\xi=\sigma_1\); we +moved it right to \(2\) to write \Cref{eq:principal-Mellin-integral}. +This uses only absolute convergence of \(1/L^S(\xi,\eta)\) for +\(\re\xi>1\). + +It remains to bound the rows with \(u\ne1\). Split their sum on the +initial lines into dyads \(U\le q_u<2U\), \(U=1,2,4,\ldots\), before +shifting any of them. Each dyad contains finitely many rows; the full +sum on the initial lines is absolutely convergent. Denote by +\(\mathcal J_{\mathcal C}(Z)\) the triple integral of a subset +\(\mathcal C\) of such a dyad, and set +\[ + \mathcal J_U^{\mathrm{np}}(Z) + =\mathcal J_{\{u\ne1:U\le q_u<2U\}}(Z). +\] +The numerator for each of these rows +is entire. From \Cref{eq:Hecke-conservative-convexity} and the +omitted-factor bound, +\begin{equation} + L^S(w,\chi_\bullet(u)) + \ll_\epsilon U^{1/2+\epsilon}(2+|\im w|)^{A_F} + \qquad(.003\le\re w\le4). +\label{eq:numerator-convexity} +\end{equation} +On \(\re\xi=\sigma_1\), independently of \(u\) and the height, +\[ + |L^S(\xi,\eta\overline{\chi_\bullet(u)})^{-1}| + \le\prod_{\mathfrak p\notin S}(1+q_{\mathfrak p}^{-\sigma_1}) + \le\zeta_F(\sigma_1). +\] +For each row move the initial lines to +\[ + \re\xi=\sigma_1,\qquad \re w=.003,\qquad \re z=.4. +\] +One may first move \(\xi\), then \(z\), then \(w\). +The \(z\)-line remains above \(1/6\); the numerator is entire; and the +denominator stays in absolute convergence. The first region of +\Cref{eq:euler-regions} contains the whole move. If +\(|\mathcal C|\ll U^\alpha\), the bounds above give +\begin{equation} + |\mathcal J_{\mathcal C}(Z)| + \ll_\epsilon + Z^{E_*(\sigma_1,.003,.4)} + U^{\alpha+1/2-.4+\epsilon}. +\label{eq:dyadic-contour-bound} +\end{equation} +The factor \(U^{-.4}\) is the \(q_u^{-z}\) in +\Cref{eq:poisson-mellin}. The height integral is bounded by +\Cref{eq:triple-height-decay}. + +To see how the size of the frequency determines the contours, rewrite +\Cref{eq:dyadic-contour-bound} as +\[ + |\mathcal J_{\mathcal C}(Z)| + \ll_\epsilon + Z^{a/2+\sigma_1-1+b(.003-1)} + U^{\alpha+1/2+\epsilon} + \left(\frac{Z^{h_0}}U\right)^{.4}. +\] +For a general \(z\)-line, the corresponding factor is +\((Z^{h_0}/U)^{\re z}\). Its dependence on \(\re z\) changes direction +at \(U=Z^{h_0}=Z^{.1}\): below that scale a rightward \(z\)-shift enlarges +this factor, while above it such a shift reduces the factor. We retain +the current low-\(w\) contour through \(U\le Z^{h_0+.00001}\), and use +a rightward \(z\)-shift beyond that enlarged threshold. + +The common \(Z\)-exponent on the current contour, relative to the +principal scale, is +\[ + E_*(\sigma_1,.003,.4)-C_*(1) + =10^{-7}+\frac7{300}-.132601 + =-.109267566666\ldots . +\] +For \(U\le Z^{.09}\), use \Cref{eq:dyadic-contour-bound} with +\(\alpha=1\). The \(U^{1.1}\) factor adds at most +\(.09(1.1)=.099\) to this relative exponent, so these rows are already +smaller than the principal scale. + +We use \Cref{prop:zero-density} only in the remaining interval +\begin{equation} + Z^{.09}0\) +the discarded tails on \(\re\xi=\sigma_1\) and the horizontal portions +at \(\im\xi=\pm U^\rho\) are, in total over a good dyad, at most +\begin{equation} + \ll_{A_1,\epsilon} + Z^{E_*(\sigma_1,.003,.4)} + U^{1+1/2-.4+\epsilon-\rho A_1}. +\label{eq:xi-tail-bound} +\end{equation} +On the horizontal portions the real exponent is bounded by its value +at \(\sigma_1\); the reciprocal is bounded by +\Cref{eq:good-reciprocal}. On the discarded original tails it is +bounded by absolute convergence. The convolution estimate following +\Cref{eq:triple-height-decay}, used with pointwise exponent \(A_1+2\) +before integrating the discarded \(\xi\)-tails, supplies \(U^{-\rho A_1}\) +in both cases. +Since \(U>Z^{.09}\), choosing \(A_1>1/\rho\), after \(\rho\) is fixed, +gives more than enough additional saving for these terms. + +All these comparisons leave a fixed margin after the arbitrary small +power losses. The sums over the \(O(\log Z)\) dyads in these ranges +cost only another arbitrarily small power. + +Finally consider \(U>Z^{.10001}\). Now +\(Z^{h_0}/U\le Z^{-.00001}\), so a fixed rightward \(z\)-shift suppresses +the whole dyad. Keep +\(\re\xi=\re w=4\) and move \(z\) to the fixed right line +\(\re z=R_0=350000\). The move stays in the Euler region, crosses no +pole, and uses the holomorphy of \(M\) on \(\re z>0\). +Both Hecke factors are bounded by absolute convergence on these lines, +so \Cref{prop:euler-product} and \Cref{eq:triple-height-decay} give +\[ + |\mathcal J_{\mathcal C}(Z)| + \ll_\epsilon Z^{3.849}U^{1+\epsilon} + \left(\frac{Z^{.1}}U\right)^{R_0} + =Z^{3.849+.1R_0}U^{1-R_0+\epsilon}. +\] +The dyads sum geometrically. At their lower endpoint the power of \(Z\) +is +\begin{equation} + 3.849+.1R_0+.10001(1-R_0+\epsilon) + =.44901+.10001\epsilon . +\label{eq:far-frequency-exponent} +\end{equation} +This is strictly less than \(C_*(1)-.01\) for small \(\epsilon\). +The very large \(z\)-line is fixed; its weight constants may be large, +but are independent of \(Z\) and \(U\). + +Choose the preliminary losses small enough that their total cost in the +finite frequency ranges, including the dyadic count, is less than +\(10^{-4}\) in the \(Z\)-exponent. This is possible since +\(U\le Z^{.10001}\) there. The smallest central saving in +\Cref{eq:good-numerical-margin} is larger than \(0.0012\); +\Cref{eq:xi-tail-bound} and \Cref{eq:far-frequency-exponent} have +larger savings. We have therefore proved +\begin{equation} + \sum_{j\ge0}\mathcal J_{2^j}^{\mathrm{np}}(Z) + \ll Z^{C_*(1)-.0004}. +\label{eq:nonprincipal-row-total} +\end{equation} +Each dyadic integral here is obtained from its initial absolutely +convergent integral, and the preceding bounds justify their sum. + +\subsection{Completion of the Mellin argument} + +The numerical exponents established above, before the indicated small +losses, are collected in \Cref{tab:analytic-margins}. In particular +the exponent from reflection is +\[ + \frac{2M_0-1-b}{2}=\frac{933}{2000} + =C_*(1)-\frac1{6000}. +\] +\begin{table}[ht] +\centering +\begin{tabular}{@{}lc@{}} +\toprule +Contribution & exponent relative to \(C_*(1)=7/15\)\\ +\midrule +Probe bound from reflection & \(-1/6000=-.000166666\ldots\)\\ +Principal remainder \(R_w\) & \(-.00664989\)\\ +Principal remainder \(R_z\) & \(-.000666566\ldots\)\\ +Nonprincipal rows \(U\le Z^{.09}\) & \(-.010267566\ldots\)\\ +Exceptional rows \(U\le Z^{.10001}\) & \(-.049261566\ldots\)\\ +Good central rows \(U\le Z^{.10001}\) & \(-.001256666\ldots\)\\ +Rows \(U>Z^{.10001}\), \(R_0=350000\) & \(-.017656666\ldots\)\\ +\bottomrule +\end{tabular} +\caption{The fixed exponent margins in the probe comparison. The row +bounds precede arbitrary small power losses.} +\label{tab:analytic-margins} +\end{table} + +Combining the principal decomposition \(I_1=c_0f_\eta+R_w+R_z\), +\Cref{eq:principal-remainders,eq:nonprincipal-row-total}, and +\Cref{eq:poisson-mellin} gives, for sufficiently large \(Z\), +\[ + I(Z^a,Z^b,Z)=c_0f_\eta(Z)+O(Z^{C_*(1)-.0004}). +\] +Use \Cref{prop:probe-low} with, for example, a loss \(10^{-5}\). +Since \(1/6000-10^{-5}>1/20000\) and \(c_0\ne0\), the preceding identity +implies the safe bound +\begin{equation} + f_\eta(Z)\ll_{F,S,\eta,W_0,W_1}Z^{7/15-1/20000} + \qquad(Z\text{ sufficiently large}). +\label{eq:f-large-bound} +\end{equation} + +Set +\[ + \mathscr A(\xi)=e^{(\xi-5/6)^2} + \frac{\mathcal H_\eta(\xi)}{L^S(\xi,\eta)}. +\] +For \(\re\xi>1\) this is holomorphic. In the definition of \(f_\eta\) in +\Cref{eq:principal-Mellin-integral}, its \(\xi\)-line may be moved right +from \(2\) to any fixed real part \(\sigma_R>2\). The reciprocal there is in +absolute convergence, the correction is holomorphic and bounded, and +the Gaussian gives rapid vertical decay. It follows that for every +\(A_0>0\), +\begin{equation} + f_\eta(Z)=O_{A_0}(Z^{A_0})\qquad(0A_0+8/15\). Thus this endpoint estimate has used no +zero exclusion to the left of one. + +Consider +\begin{equation} + \mathcal T(s)=\int_0^\infty f_\eta(Z)Z^{-C_*(s)}\,\frac{dZ}{Z}. +\label{eq:final-Mellin-transform} +\end{equation} +By \Cref{eq:f-large-bound,eq:f-small-bound}, it converges locally +uniformly, and hence defines a holomorphic function, for +\[ + \re s>1-\frac1{20000}. +\] +Indeed at infinity the real exponent is +\(1-1/20000-\re s<0\), and at zero +\Cref{eq:f-small-bound} allows an arbitrarily large exponent. +The same majorants with powers of \(\log Z\) justify differentiation +under the integral. + +We identify this transform on its original line without a formal contour +interchange. Put \(t=\log Z\). From +\Cref{eq:principal-Mellin-integral}, +\[ + e^{-C_*(2)t}f_\eta(e^t) + =\frac1{2\pi}\int_{\R}e^{i\tau t}\mathscr A(2+i\tau)\,d\tau . +\] +The function \(\mathscr A(2+i\tau)\) is continuous and rapidly decreasing by +the Gaussian and the bounded Euler factors. The endpoint bounds on +\(f_\eta\) make the left side integrable in \(t\). Fourier inversion +therefore gives +\begin{equation} + \mathcal T(2+i\tau)=\mathscr A(2+i\tau)\qquad(\tau\in\R). +\label{eq:Mellin-identification} +\end{equation} +For \(\re s>1-1/20000\), the second neighborhood in +\Cref{prop:euler-product} makes the numerator +\[ + e^{(s-5/6)^2}\mathcal H_\eta(s) +\] +holomorphic and nowhere zero. +The Hecke continuation makes \(\mathscr A(s)\) meromorphic there. The identity +theorem, first on \(\re s>1\) using \Cref{eq:Mellin-identification} +and then for meromorphic functions on the connected larger half-plane, +identifies \(\mathscr A\) with the holomorphic function \(\mathcal T\). +A zero of \(L^S(s,\eta)\) in that half-plane would be a pole of \(\mathscr A\), +since its numerator is nonzero, which is impossible. A pole of the +principal \(L\)-function simply makes its reciprocal vanish. + +The finite factors removed from \(L_F(s,\eta)\) are either one or +\(1-\eta(\mathfrak p)q_{\mathfrak p}^{-s}\) with +\(|\eta(\mathfrak p)|=1\); their zeros have real part zero. Thus the +same zero exclusion holds for the full \(L_F(s,\eta)\). The obtained +half-plane contains \(\re s>1-10^{-6}\). All onsets and constants in +the proof were allowed to depend on \(F,S,\eta\), but the numerical +saving \(1/20000\) is absolute. A field-dependent onset in +\Cref{eq:f-large-bound} affects only a finite portion of the integral +and does not change its half-plane of convergence. Since the initial +field and finite-order character were arbitrary in the scope of +\Cref{thm:hecke-zero-free}, this proves that theorem. diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/07-splitting.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/07-splitting.tex new file mode 100644 index 0000000000000000000000000000000000000000..ee9b3eac0decd351991b6b3f8d2b237b5df2970d --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/07-splitting.tex @@ -0,0 +1,247 @@ +\section{A uniform estimate for complete splitting} +\label{sec:splitting} + +We prove Proposition~\ref{prop:splitting} for the fields +$K_q=\Q(\mu_q,a^{1/q})$, where $a$ is a fixed integer with $|a|>1$. +Theorem~\ref{thm:hecke-zero-free}, proved in +Section~\ref{sec:zero-free}, supplies the common zero-free width +$\delta_0=10^{-6}$. We transfer that region to $\zeta_{K_q}$ and then +use a smooth explicit formula whose constants depend on the field only +through its degree and discriminant. + +\subsection{Field bounds and descent of the zero-free region} + +The field $K_q$ need not be cyclotomic. We first place it in an abelian +extension of a cyclotomic field, so that +Theorem~\ref{thm:hecke-zero-free} applies, and then descend the resulting +zero-free region to $\zeta_{K_q}$. + +\begin{lemma}\label{spl:field-bounds} +Fix an integer $a$ with $|a|>1$. For every sufficiently large prime $q$, +the degree $n_q=[K_q:\Q]$ and absolute discriminant $D_q$ satisfy +\begin{equation} + n_q=q(q-1),\qquad + D_q\le q^{q(q-2)}(q^q|a|^{q-1})^{q-1},\qquad + \log D_q+n_q\ll_a n_q\log(2q). + \label{spl:degree-discriminant} +\end{equation} +Every rational prime $p\nmid aq$ is unramified in $K_q$, and +\begin{equation} + \zeta_{K_q}(s)\ne0 + \qquad(\operatorname{Re}s>1-\delta_0,\ s\ne1). + \label{spl:kummer-zero-free} +\end{equation} +\end{lemma} + +\begin{proof} +Fix a rational prime $\ell\mid a$, and let $v_\ell(a)>0$ be its +valuation. Take $q$ large enough that $q\ne\ell$ and $q>v_\ell(a)$. +When $a=2$, the choice $\ell=2$ works for every prime $q\ge5$. +The cyclotomic discriminant formula gives +$|d_{E_q}|=q^{q-2}$, so $\ell$ is unramified in $E_q$ and the valuation +of $a$ at every prime above $\ell$ is $v_\ell(a)$. This is not divisible +by $q$, and therefore $a$ is not a $q$th power in $E_q$. + +Since $E_q$ contains $\mu_q$, the extension $K_q/E_q$ is Galois and +the map +\[ + \sigma\longmapsto\frac{\sigma(a^{1/q})}{a^{1/q}} +\] +embeds its Galois group in $\mu_q$. Its degree divides the prime $q$ +and is not one, so it equals $q$. This proves the degree formula. + +The root $a^{1/q}$ is integral, and the relative field discriminant +divides the discriminant of its power basis: +\[ + \mathfrak d_{K_q/E_q} + \mid(q^q a^{q-1})\mathcal O_{E_q}. +\] +The discriminant tower formula now gives +\[ + D_q + =|d_{E_q}|^q\operatorname{N}_{E_q/\Q}(\mathfrak d_{K_q/E_q}) + \le q^{q(q-2)}(q^q|a|^{q-1})^{q-1}. +\] +This also proves unramifiedness away from $aq$, and taking logarithms +proves the last estimate in \eqref{spl:degree-discriminant}. We use the +standard cyclotomic and relative discriminant formulas in +\cite[Chapter~I, \S10; Chapter~III, \S2]{Neukirch99}. + +To obtain \eqref{spl:kummer-zero-free}, put +\[ + F_q=\Q(\mu_{12q}),\qquad \widetilde K_q=K_qF_q. +\] +The extension $\widetilde K_q/F_q$ is cyclic, since it is obtained by +adjoining $a^{1/q}$ to a field containing $\mu_q$. The extension +$\widetilde K_q/K_q$ is also abelian: it is Galois, and restriction +embeds its Galois group in the abelian group $\operatorname{Gal}(F_q/\Q)$. +Abelian Artin factorization, with one-dimensional Artin functions +identified with primitive Hecke functions, gives +\begin{align} + \zeta_{\widetilde K_q}(s) + &=\prod_{\eta\in\widehat{\operatorname{Gal}(\widetilde K_q/F_q)}} + L_{F_q}(s,\eta),\label{spl:factorization-upstairs}\\ + \frac{\zeta_{\widetilde K_q}(s)}{\zeta_{K_q}(s)} + &=\prod_{\substack{\eta\in + \widehat{\operatorname{Gal}(\widetilde K_q/K_q)}\\\eta\ne1}} + L_{K_q}(s,\eta). + \label{spl:factorization-quotient} +\end{align} +See \cite[Chapter~VII, Corollary~(10.5), Theorem~(10.6), +and the following remarks]{Neukirch99}. +Every factor in \eqref{spl:factorization-upstairs} is covered by +Theorem~\ref{thm:hecke-zero-free}, because $F_q$ is cyclotomic and contains +$\mu_{12}$. Thus $\zeta_{\widetilde K_q}$ is zero-free in the indicated +half-plane. Every factor in \eqref{spl:factorization-quotient} is an +entire nonprincipal Hecke $L$-function. A zero of $\zeta_{K_q}$ there +would therefore be a zero of $\zeta_{\widetilde K_q}$. This proves +\eqref{spl:kummer-zero-free}. +\end{proof} + +The width in \eqref{spl:kummer-zero-free} is independent of $q$. +It remains to turn it into a prime-ideal estimate whose constants retain +only the explicit dependence on degree and discriminant in +\eqref{spl:degree-discriminant}. + +\subsection{A smooth prime-ideal estimate} + +Smoothing permits an absolutely convergent sum over zeros. The following +form of the explicit formula is useful because its error is linear in +$\log D+n$. We give the calculation with this dependence explicit. + +\begin{lemma}\label{spl:smooth-estimate} +Fix $0<\delta<1/2$ and a nonnegative function +$f\in C_c^\infty((0,\infty))$. Let $K$ be a number field of degree $n$ +and absolute discriminant $D$ whose Dedekind zeta function has no zero +in $\operatorname{Re}s>1-\delta$. For every real $x\ge2$, +\begin{equation} + \Theta_{K,f}(x) + :=\sum_{\mathfrak p}\sum_{j\ge1} + (\log\operatorname{N}\mathfrak p) + f\left(\frac{(\operatorname{N}\mathfrak p)^j}{x}\right) + \ll_f x+x^{1-\delta}(\log D+n), + \label{spl:smooth-prime-ideal-bound} +\end{equation} +where $\mathfrak p$ ranges over the nonzero prime ideals of $K$. +The implied constant is independent of $K$ and $x$. +\end{lemma} + +\begin{proof} +Let $(r_1,r_2)$ be the signature of $K$, so $n=r_1+2r_2$, and define +the Mellin transform +\[ + \Phi(s)=\int_0^\infty f(t)t^{s-1}\,dt. +\] +It is entire. Repeated integration by parts shows that, for every fixed +vertical strip and every $A>0$, +$\Phi(s)\ll_{f,A}(1+|\operatorname{Im}s|)^{-A}$ on that strip. +Mellin inversion and the absolutely convergent Euler series on +$\operatorname{Re}s=2$ give +\begin{equation} + \Theta_{K,f}(x) + =\frac1{2\pi i}\int_{(2)} + -\frac{\zeta_K'}{\zeta_K}(s)x^s\Phi(s)\,ds. + \label{spl:mellin-integral} +\end{equation} + +We shift the contour to $\operatorname{Re}s=-1/2$. +On this line the functional equation gives +\begin{equation} + \left|\frac{\zeta_K'}{\zeta_K}(s)\right| + \ll \log D+n\log(2+|\operatorname{Im}s|), + \label{spl:left-line} +\end{equation} +with an absolute implied constant. To see the dependence on $K$, the +Euler series at $1-s$, whose real part is $3/2$, is bounded by +\[ + n\sum_{p}\frac{(\log p)p^{-3/2}}{1-p^{-3/2}}\ll n. +\] +Indeed the sum of the residue degrees of primes above a rational prime +is at most $n$. The remaining terms in the functional equation are +$\log D$ and the logarithmic derivatives of +\[ + \Gamma_{\R}(s)^{r_1}\Gamma_{\C}(s)^{r_2},\qquad + \Gamma_{\R}(s)=\pi^{-s/2}\Gamma(s/2),\quad + \Gamma_{\C}(s)=2(2\pi)^{-s}\Gamma(s). +\] +On the lines involved their logarithmic derivatives are +$O(\log(2+|\operatorname{Im}s|))$, by Stirling's formula and their +fixed distance from poles. This proves \eqref{spl:left-line}; see also +\cite[Main Theorem~4.4.1 and \S4.5]{Tate67} for the functional equation. + +The poles crossed in \eqref{spl:mellin-integral} are the pole of +$\zeta_K$ at $1$, its nontrivial zeros $\rho$ counted with multiplicity, +and the zero at $0$ of +order $r_1+r_2-1$. The last order follows from the gamma factors and +the simple pole of the completed zeta function at $0$; when the order +is zero there is no residue. There are no other trivial zeros between +the two lines. The nontrivial zeros lie in $0<\operatorname{Re}\rho<1$: +the hypothesis excludes the boundary at $1$, and the functional equation +excludes nontrivial zeros on the boundary at $0$. + +For completeness, the zero count needed to justify the shift and bound +its residues is +\begin{equation} + \#\{\rho:T\le\operatorname{Im}\rho1\), write \(P^-(m)\) for its least prime factor, and +put \(P^-(1)=\infty\). Thus \(P^-(m)>y\) means that \(m\) is +\(y\)-rough. All Dirichlet characters below, including principal +characters, are extended by zero on nonunits. + +Fix an integer \(K\ge1\) and real numbers +\(0.10}\to\mathbb C\) satisfies \(F(ph)=F(h)\) for every +group prime \(p\) and every positive +\(h\) if and only if it depends only on \(h/h_{\mathcal P}\). In +particular this invariance removes every power of a group prime, not +only a selected divisor. + +Define +\begin{equation}\label{eq:marked-weight} + \begin{aligned} + V_i&=\sum_{p\in\mathcal P_i}\frac1p, + &\omega_i(h)&=\sum_{p\in\mathcal P_i}\one_{p\mid h},\\ + \omega(h)&=\sum_{i=1}^K\omega_i(h), + &\mathcal W(h)&=q^{\omega(h)-K} + \prod_{i=1}^K\frac{\omega_i(h)}{V_i}, + \qquad 00\) and \(q\in(0,1)\). Suppose +\(H_m,H_n\ge x^\delta\) and \(X=H_mH_n\asymp x\). Let \(F\) satisfy +\(|F(h)|\le1\) and \(F(ph)=F(h)\) for every prime in the groups +defined in \Cref{eq:marked-prime-groups}. Let \(\alpha_m\) be supported on an +arbitrary interval in \([H_m,2H_m]\), where it has the value +\[ + \alpha_m=m^{iv}\left(\one_{m\text{ prime}} + -\frac{\one_{P^-(m)>W}}{V(W)\log m}\right), + \qquad |v|\le L^C. +\] +Let \(\beta_n\) be supported on \(W\)-rough integers in \([H_n,2H_n]\), +with \(|\beta_n|\le L^C\). If \(K\) is sufficiently large in terms of +\(\delta,C,D_*,q\), then +\begin{equation}\label{eq:marked-type-ii} + \left|\sum_{m,n}\alpha_m\beta_n F(mn-1)\mathcal W(mn-1)\right| + \ll X L^{-D_*}. +\end{equation} +The required lower bound on \(K\) is independent of the particular +\(a_i\). The implicit constant and the threshold for \(x\) may depend +on all the fixed parameters, including the \(a_i\). +\end{theorem} + +The invariance is required for every positive \(h\), including when +\(p\mid h\). The estimate is uniform over the permitted \(F\), +coefficient intervals, and coefficients. The symbol \(X\) in this +section denotes only the product \(H_mH_n\). + +We need the same marked bound for a rough-factor discrepancy. The +following lemma isolates a sufficient condition on a scalar +coefficient: cancellation against every character of logarithmic +modulus throughout the stated frequency range. Its proof extracts +the coefficient dependence from the argument of +\cite[Theorem~3.1]{PoissonDirichlet}. + +\begin{lemma}[Coefficient criterion for the marked estimate] +\label{lem:marked-coefficient-transfer} +Fix \(\delta,C,D_*>0\) and \(q\in(0,1)\). Retain the marked data and +the hypotheses on \(F,H_m,H_n\) in \Cref{thm:marked-type-ii}, with fixed +comparison constants in \(X=H_mH_n\asymp x\). Let +\((\alpha_m)\) and \((\beta_n)\) be scalar sequences indexed by positive +integers, supported on \(W\)-rough integers in their respective dyads, +with \(\alpha\) supported on an arbitrary interval +\(I\subseteq[H_m,2H_m]\), and with +\(|\alpha_m|,|\beta_n|\le L^C\). Suppose that for every fixed +\(A_0,B,A>0\), every Dirichlet character \(\chi\) modulo +\(k\le L^{A_0}\), and every \(|t|\le2XL^B\), +\begin{equation}\label{eq:type-ii-long-coefficient-interface} + \left|\frac1{H_m}\sum_m\alpha_m\chi(m)m^{it}\right|\ll_A L^{-A}, +\end{equation} +uniformly in the permitted interval, character, and frequency. The +constant and threshold in this hypothesis may depend on \(A_0,B,A\), +the fixed setup, and all further fixed parameters defining the +coefficients; these parameters are chosen before \(x\) grows. + +Then \Cref{eq:marked-type-ii} holds when \(K\) is sufficiently large +in terms of the fixed data, independently of the particular band +exponents \(a_i\). Its implicit constant and threshold may depend on +all the fixed data, including the \(a_i\). +\end{lemma} + +\begin{proof} +We extract the assertion from the proof of +\cite[Theorem~3.1]{PoissonDirichlet}. In its marked expansion, write +\(mn-1=ah\), where \(a\) is the product of the selected primes. +Invariance gives \(F(mn-1)=F(h)\). Except when a selected prime has +its square dividing \(mn-1\), one also has +\(\omega(mn-1)-K=\omega(h)\). Replacing the damping by +\(q^{\omega(h)}\) therefore costs \(O_A(XL^{-A})\) for every fixed +\(A\). The resulting factor \(F(h)q^{\omega(h)}\) has modulus at +most one and is discarded in the following Cauchy bound. There are \(O_K(L)\) dyads +\(Y\) for the product of the selected primes. For the contribution +\(S_Y\) of one dyad, Cauchy's inequality gives +\[ + |S_Y|^2\ll\frac XY\mathcal Q_Y, +\] +where the nonnegative expanded square is +\begin{equation}\label{eq:type-ii-cauchy-square} + \mathcal Q_Y=\left(\prod_iV_i^{-2}\right) + \sum_{\substack{a,b,m,n,m',n'\\ + mn-1=ah,\ m'n'-1=bh\text{ for some }h}} + \rho(a/Y)\rho(b/Y)\alpha_m\overline{\alpha_{m'}} + \beta_n\overline{\beta_{n'}}. +\end{equation} +Here \(a,b\) each select one prime from every group, and \(\rho\) is +the fixed real smooth dyadic cutoff supported in \([1,4]\). +Pairs \(a,b\) sharing a selected prime cost \(O_A(XYL^{-A})\). +For the remaining coprime pairs, the common-\(h\) condition is +equivalent to \(t=b-a\), where \(t=bmn-am'n'\). +These reductions use only coefficient bounds and the invariance of +\(F\); see \cite[Section~3.1]{PoissonDirichlet}. + +For a fixed \(A_0>0\), let \(\mathfrak M\subset\mathbb R/\mathbb Z\) +be the union of the arcs +\[ + \operatorname{dist}_{\mathbb R/\mathbb Z}(\theta,c/k) + \le\frac{2L^{A_0}}Y,\qquad + 1\le k\le L^{A_0},\quad c\bmod k,\quad (c,k)=1. +\] +The selected-product range has \(\log Y\gg L^{0.1}\), so these arcs +are disjoint for large \(x\). +Choose the companion's fixed real \(\psi\in C_c^\infty(\mathbb R)\), +equal to one on \([-4,4]\), and define +\[ + \mathcal H_{\mathfrak M}(t;a,b) + =\psi(t/Y)\int_{\mathfrak M} + \exp\bigl(2\pi i\theta(t-b+a)\bigr)\,\dd\theta. +\] +The major form is the scalar sum +\begin{equation}\label{eq:type-ii-major-form} + \begin{split} + \mathcal Q_Y^{\mathrm{maj}} + ={}&\left(\prod_iV_i^{-2}\right) + \sum_{a,b,m,n,m',n'} + \rho(a/Y)\rho(b/Y)\\ + &\qquad{}\times\alpha_m\overline{\alpha_{m'}} + \beta_n\overline{\beta_{n'}} + \mathcal H_{\mathfrak M}(bmn-am'n';a,b), + \end{split} +\end{equation} +where \(a,b\) now run independently over products of one prime from +every group. + +The proof of the replacement conclusion in +\cite[Proposition~4.1]{PoissonDirichlet} applies to these scalar +coefficients. Its operator moment is independent of their values. +The passage from that moment to the scalar endpoint pairing uses only +\(W\)-rough support and norm and supremum bounds with logarithmic +exponents depending on \(C\), not on \(K\) +\cite[Sections~3.2--3.3]{PoissonDirichlet}. This pairing initially +has projections onto the good endpoint states of that argument. In +removing either projection, only the error from the endpoint outside +its good set is majorized absolutely: each whole scalar endpoint factor +is replaced by its logarithmic supremum bound, discarding both its +coefficient value and its support indicator before the one-edge root +replacement. The surviving unprojected pairing retains its exact +endpoint vectors. Expanding those vectors and controlling collisions +uses only size, rough support, and fixed divisor moments +\cite[Section~4.9]{PoissonDirichlet}. Thus the replacement conclusion, +including restoration of independent \(a,b\), gives +\begin{equation}\label{eq:type-ii-minor-transfer} + |\mathcal Q_Y-\mathcal Q_Y^{\mathrm{maj}}| + \ll_A XYL^{-A} +\end{equation} +for every prescribed fixed accuracy, with the choices specified below. + +The proof of the major-term estimate +\cite[Proposition~5.1]{PoissonDirichlet} separates characters only at +moduli \(k\le L^{A_0}\). The group primes and all prime divisors of +the rough variables exceed these moduli, so every variable is a unit +there. After Mellin separation its retained rectangle, for a fixed +\(B\) chosen after \(A_0\), has \(|s'|\le L^B\) and +\(|t|\le XL^B\), so both \(t\) and \(s'-t\) +lie in the range of \Cref{eq:type-ii-long-coefficient-interface}. +Transform decay and the product mean square on translated intervals +control the discarded tails using only scalar coefficient bounds and +fixed divisor moments. +The further Mellin shift enters only the one-group prime polynomial +and the bounded factor from the other groups; the sparse-frequency +estimate is uniform in this shift +\cite[Lemma~5.3]{PoissonDirichlet}. On the remaining sparse frequency +set, the only further input about +\(\alpha\) is the supremum in +\Cref{eq:type-ii-long-coefficient-interface}, in the role of +\cite[Lemma~5.2]{PoissonDirichlet}. The same hypothesis covers the +conjugated endpoint, by conjugating and replacing \(\chi,t\) by +\(\overline\chi,-t\). The major-term proof therefore gives +\begin{equation}\label{eq:type-ii-major-transfer} + |\mathcal Q_Y^{\mathrm{maj}}|\ll_D XYL^{-D} +\end{equation} +for every fixed \(D>0\). + +Choose the desired square accuracy \(D_1\) sufficiently large in terms +of \(D_*,C\), before choosing \(K\). For +\Cref{eq:type-ii-minor-transfer}, choose the moment accuracy \(E_0\) +after \(D_1,C\), then the arc exponent \(A_0\), and then \(K\), as in +\cite[Proposition~4.1]{PoissonDirichlet}. These choices do not depend +on the particular \(a_i\). The analytic accuracies in +\Cref{eq:type-ii-major-transfer} are chosen after \(A_0\), as permitted +by \Cref{eq:type-ii-long-coefficient-interface} for every fixed +accuracy. Taking \(A=D=D_1\) gives +\(\mathcal Q_Y\ll XYL^{-D_1}\). Cauchy's inequality yields +\(|S_Y|\ll XL^{-D_1/2}\). The \(O_K(L)\) dyads and all fixed +coefficient losses are absorbed by the choice of \(D_1\). The +logarithmic exponents used in that choice are independent of \(K\); +constants depending on \(K\) affect only the implicit constant and +threshold. This proves \Cref{eq:marked-type-ii}. +\end{proof} + +We now construct a coefficient satisfying this criterion for +\(x^\gamma\)-rough integers. Its comparison weight comes from the +continuous measure of products of large primes. For \(w>\gamma>0\), +define +\begin{equation}\label{eq:rough-density-definition} + \begin{split} + D_\gamma(w)&=\sum_{j\ge1}D_{\gamma,j}(w), + \qquad D_{\gamma,1}(w)=\frac1w,\\ + D_{\gamma,j}(w)&=\frac1{j!} + \int_{\substack{t_i\ge\gamma\ (1\le ix^\gamma\\\prod_i y_i\in I}} + \prod_{i=1}^j\frac{\dd y_i}{\log y_i} + =\frac1L\int_I D_{\gamma,j}(\log y/L)\,\dd y. +\end{equation} +For \(j=1\), the integrands agree because +\(D_{\gamma,1}(\log y/L)/L=1/\log y\). For \(j\ge2\), put +\(y=\prod_i y_i\), \(t_i=\log y_i/L\) for \(ix^\gamma\) differs from the weak simplex +in \Cref{eq:rough-density-definition} only on null boundaries. The +factor \(1/j!\) divides the ordered prime-product measure by the +number of permutations. This identity concerns the full domain +\(w>\gamma\); the upper exponent restriction in the next proposition +belongs to the Type II estimate, not to the density. + +The functions in \Cref{eq:rough-density-definition} are jointly +continuous on compact subsets of \(w>\gamma>0\), and, for fixed +\(\gamma\), are Lipschitz in \(w\) on each such compact interval. To +check the assertion also at \(w=j\gamma\) for \(j\ge2\), write +\(s=w-j\gamma\). For \(s\ge0\), the change of variables +\(t_i=\gamma+s y_i\) gives +\[ + D_{\gamma,j}(w)=\frac{s^{j-1}}{j!} + \int_{\substack{y_i\ge0\\\sum_{ix^\gamma} + -\frac{D_\gamma(\log m/L)}{L V(W)}\one_{P^-(m)>W} + \right),\qquad |v|\le L^C, +\end{equation} +and set \(\alpha_m=0\) outside \(I\). Then +\Cref{eq:marked-type-ii} holds for every fixed \(D_*>0\) when \(K\) is +sufficiently large in terms of the fixed data. Its required lower bound +is independent of the particular band exponents \(a_i\); the threshold +and implicit constant may depend on all the fixed data. In particular, +\(\gamma\) and the gap \(w_--\gamma\) are fixed before \(x\) grows. +\end{proposition} + +\begin{proof} +We verify \Cref{eq:type-ii-long-coefficient-interface} for this +coefficient. At low frequencies the two principal counting terms +cancel. At higher frequencies the prime-product sequence and the +\(W\)-rough comparison sequence are each small. The support and size +conditions of \Cref{lem:marked-coefficient-transfer} will then follow +directly from the coefficient formula. + +Put \(M=H_m\) and write \(R_\gamma(m)=\one_{P^-(m)>x^\gamma}\). +On this dyad a number counted by \(R_\gamma\) has at most +\(J=\lfloor w_+/\gamma\rfloor\) prime factors counted with multiplicity. +Define +\[ + T_\gamma(m)=\sum_{j=1}^{J}\frac1{j!} + \sum_{\substack{p_1,\ldots,p_j>x^\gamma\text{ prime}\\ + p_1\cdots p_j=m}}1. +\] +If \(m\) is squarefree and \(x^\gamma\)-rough, the ordered lists give +\(T_\gamma(m)=1\). If the prime multiplicities are \(e_p\), the +corresponding value is \(1/\prod_p e_p!\), which is at most one. +Thus the difference is supported on a square of a prime exceeding +\(x^\gamma\), and, uniformly for every subinterval of the dyad, +\begin{equation}\label{eq:rough-convolution-square-error} + \sum_{M\le m\le2M}|R_\gamma(m)-T_\gamma(m)| + \le\sum_{p>x^\gamma}\left\lfloor\frac{2M}{p^2}\right\rfloor + \ll Mx^{-\gamma}. +\end{equation} +The same bound holds after any character and real power twist. It is +smaller than \(M\) times every fixed negative power of \(L\). + +We next obtain the counting formula needed at low frequencies. For +every fixed \(A_1,A_0>0\), every \(k\le L^{A_0}\), every character +\(\chi\bmod k\), and every interval \(I'\subseteq[M,2M]\), we claim +\begin{equation}\label{eq:rough-convolution-low-count} + \sum_{m\in I'}R_\gamma(m)\chi(m) + =\one_{\chi\text{ principal}}\frac1L + \int_{I'}D_\gamma(\log y/L)\dd y+O(M L^{-A_1}). +\end{equation} +The constant may depend on the displayed fixed parameters and on +\(\gamma,w_-,w_+\), but the estimate is uniform in the interval and +character. + +To prove it, split every prime variable of a term of \(T_\gamma\) into +half-open dyads \([P_i,2P_i)\). All their scales satisfy +\(P_i\ge x^\gamma/2\) and \(P_i\le2M\). There are \(O(L^j)\) possible +boxes for the \(j\)-prime term. A box that meets the product interval +has +\(M/2^j\le\prod_iP_i\le2M\). On a prime dyad of scale \(P\), the +Siegel--Walfisz estimate, summed against \(\chi\) over its reduced +classes, gives for every subinterval \(J'\subseteq[P,2P]\) +\begin{equation}\label{eq:rough-slot-siegel-walfisz} + \sum_{p\in J'}\chi(p) + =\one_{\chi\text{ principal}}\int_{J'}\frac{\dd y}{\log y} + +O_{A_2}(P L^{-A_2}) +\end{equation} +with arbitrary fixed \(A_2\). Indeed \(\log P\asymp L\) with fixed +comparison constants, and one chooses the Siegel--Walfisz accuracy to +absorb the at most \(L^{A_0}\) residue classes. This is the classical +estimate in \cite[Proposition~2.1]{PrimePredecessors}. Taking +differences of its endpoint estimates gives the absolute error in +\Cref{eq:rough-slot-siegel-walfisz} even for an arbitrarily short +subinterval. The cutoff \(p>x^\gamma\) can be intersected with that +interval. For large \(x\), these primes exceed \(k\); hence the +principal character is one on them. The same orthogonality argument +covers imprimitive characters. + +Replace the prime measures in one box successively using +\Cref{eq:rough-slot-siegel-walfisz}. When all other variables are fixed, +whether at prime values or at real values in a previous replacement, the +product condition leaves an interval in the remaining prime dyad. Its +error is \(O(P_iL^{-A_2})\). The total absolute mass of each other +variable, either its prime counting measure or \(\dd y/\log y\), is +\(O(P_h)\). The error in that box is therefore \(O(M L^{-A_2})\). +There are only the fixed \(j\) replacements and \(O(L^j)\) boxes, so +their errors give \(O(M L^{-A_1})\) on choosing \(A_2\) sufficiently +large. A nonprincipal character has zero continuous main term. For a +principal character the main term of the \(j\)-prime summand is exactly +\[ + \frac1{j!}\int_{\substack{y_i>x^\gamma\\\prod_i y_i\in I'}} + \prod_{i=1}^j\frac{\dd y_i}{\log y_i}. +\] +By \Cref{eq:rough-density-pushforward}, these integrals sum to the +principal main term in \Cref{eq:rough-convolution-low-count}. +Finally \Cref{eq:rough-convolution-square-error} absorbs repeated +primes and proves the claim. + +The matching small-prime count is +\begin{equation}\label{eq:rough-small-prime-character-count} + \sum_{\substack{m\in I'\\P^-(m)>W}}\chi(m) + =\one_{\chi\text{ principal}}V(W)|I'|+O(M L^{-A_1}) +\end{equation} +for every fixed \(A_1\), with the same interval and character uniformity. +This is \cite[Lemma~2.5]{PoissonDirichlet}. Notice that all prime divisors +of \(k\le L^{A_0}\) are less than \(W\) for large \(x\), so its principal +character is one on the rough support. + +Write \(u=t+v\). Fix temporarily a constant \(B_2>C+2\). If +\(|t|\le L^{B_2}\), then \(|u|\le2L^{B_2}\). Partial summation in +\Cref{eq:rough-convolution-low-count} against \(y^{iu}\) costs at most +\(1+O(|u|)\), since this is its total variation on a dyad. Partial +summation in \Cref{eq:rough-small-prime-character-count} against +\(D_\gamma(\log y/L)y^{iu}/(LV(W))\) likewise has only a fixed +logarithmic cost: \(D_\gamma\) is Lipschitz on the fixed interval and +Mertens' theorem gives \(LV(W)\asymp L^{0.76}\). The two resulting +principal main terms are both +\[ + \frac1L\int_I D_\gamma(\log y/L)y^{iu}\dd y, +\] +while both nonprincipal main terms vanish. Choosing \(A_1\) after +\(B_2\) proves \Cref{eq:type-ii-long-coefficient-interface} with any +fixed accuracy on this low-frequency range. + +For the remaining frequencies, use the long prime polynomial estimate +\cite[Lemma~3.1]{PrimePredecessors} and the corresponding +bound for the small-prime proxy. For fixed +\(0<\tau_{\rm lp}<\theta_{\rm lp}<1\) and fixed \(A_0,A_3>0\), the +prime estimate gives a constant +\(B_0=B_0(\tau_{\rm lp},\theta_{\rm lp},A_0,A_3)\) such that +\begin{equation}\label{eq:rough-slot-high} + \left|\sum_{p\in J'}\chi(p)p^{-1+iu}\right|\ll L^{-A_3} +\end{equation} +for every interval \(J'\subseteq[P,2P]\), every character modulo +\(k\le L^{A_0}\), and +\[ + x^{\tau_{\rm lp}}/2\le P\le x^{\theta_{\rm lp}},\qquad L^{B_0}\le|u|\le x^2. +\] +Choose \(0<\tau_{\rm lp}<\gamma\) and \(w_+<\theta_{\rm lp}<1\). Every prime dyad in the +finite convolution lies in this scale range for sufficiently large \(x\). +Partial summation changes the reciprocal weight in +\Cref{eq:rough-slot-high} to give +\(\left|\sum_{p\in J'}\chi(p)p^{iu}\right|\ll P L^{-A_3}\). +In each box of the convolution, fix all but one prime; the remaining +product restriction is again an arbitrary interval. This bound in that +one slot and trivial bounds in the other slots give \(O(M L^{-A_3})\) +per box. Taking \(A_3\) to absorb the \(O(L^j)\) boxes and using +\Cref{eq:rough-convolution-square-error} proves any requested +logarithmic saving for +\(M^{-1}\sum_{m\in I}R_\gamma(m)\chi(m)m^{iu}\) throughout this range. + +For the proxy, the proof of +\cite[Lemma~5.2]{PoissonDirichlet} gives the following uniform estimate. +Put \(T_* =\exp(L/(\log L)^2)\). For every interval +\(I'\subseteq[M,2M]\), and with arbitrary fixed \(A_4\), it states +\begin{equation}\label{eq:rough-high-source} + \begin{split} + \left|\frac1{M V(W)} + \sum_{\substack{m\in I'\\P^-(m)>W}} + \frac{\chi(m)m^{iu}}{\log m}\right| + \ll L^{-A_4}+{} + \begin{cases} + L^{C_0}(|u|^{-1}+x^{-c_0}),&1\le|u|\le T_*,\\ + L^{C_0}\exp(-L/(\log L)^{C_3}),&T_*\le|u|0\) are fixed after \(A_0,\delta\), and \(C_3>0\) is +absolute; \(C_0\) does not grow with \(A_4\). The second range uses +the logarithmic-phase estimate of +\cite[Lemma~3.3]{PrimePredecessors}, valid for arbitrary residue +classes and subintervals. The first follows in the cited proof of +the long-coefficient lemma by comparison with the integral of +\(y^{iu}\) on each progression. + +For \(w=\log m/L\), the coefficient of the proxy can be written +\[ + \frac{D_\gamma(w)}{L V(W)} + =\frac{wD_\gamma(w)}{V(W)\log m}. +\] +The multiplier \(wD_\gamma(w)\) is bounded and has bounded total +variation on our fixed interval. Since \Cref{eq:rough-high-source} +holds on every subinterval, one more partial summation proves that same +bound for the proxy coefficient. Choose the requested accuracy first, +then \(A_3,A_4\) and the corresponding \(B_0\), and finally take +\[ + B_2>\max\{C+2,\ B_0+2,\ A+C_0+2\}. +\] +When \(L^{B_2}\le|t|\le2XL^B\), one has +\(|u|\asymp|t|\), \(|u|\ge L^{B_0}\), and +\(|u|\le2XL^B+L^CW\) for large \(x\), and \(D_\gamma\) is bounded +on our fixed interval. Since \(LV(W)\asymp L^{0.76}\), both parts of +\Cref{eq:rough-type-ii-coefficient} are bounded and supported on +\(W\)-rough integers. Thus the support, size, and long-coefficient +hypotheses of +\Cref{lem:marked-coefficient-transfer} all hold. Applying that lemma +proves \Cref{eq:marked-type-ii} and the proposition. +\end{proof} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/10-whittaker-calculation.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/10-whittaker-calculation.tex new file mode 100644 index 0000000000000000000000000000000000000000..7cf22626db2038f10ecb065e45500df966411422 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/10-whittaker-calculation.tex @@ -0,0 +1,495 @@ +\section{The cubic theta normalization} +\label{app:whittaker} + +This appendix supplies the Kazhdan--Patterson specialization and +normalization behind \Cref{prop:theta-input}. We first specialize their +cover, inducing datum, and global representation, and derive the normalized +Whittaker factorization. We then verify the exact lift identities and +compute the spherical Jacquet integral in the sections used in the paper. +The local calculation fixes the Gauss phase and the unit factor together, +and includes the modular factor in the triple-valuation step. The main +argument uses the resulting global Fourier expansion with normalized local +factors and the exact local rules for the Weyl calculation. + +\subsection{The selected cover and global representation} +\label{subsec:theta-specialization} + +Denote the cubic Hilbert symbol at a place \(v\) by +\((\,\cdot\,,\,\cdot\,)_{3,v}\). In the convention of Kazhdan and +Patterson, the restriction of their determinant-modified cocycle to the +diagonal torus is +\begin{equation}\label{eq:kp-torus-cocycle} + \sigma_{c,v}\bigl(\operatorname{diag}(a,b), + \operatorname{diag}(a',b')\bigr) + =(a,b')_{3,v}(ab,a'b')_{3,v}^{c}, + \qquad c\in\Z/3\Z. +\end{equation} +The root groups have their canonical splittings. These conventions are +those of \cite[Section~0.1, pp.~39--44]{KP84}. + +For the \(n\)-fold cover of \(\GL_d\), where \(d\) is the rank parameter, +the center criterion of \cite[Proposition~0.1.1, p.~42]{KP84} requires +\(z^{d-1+2dc}\in F_v^{\times n}\) for a scalar \(zI\). For \(d=2,n=3\) +the exponent is \(1+4c\), which is zero modulo three exactly when \(c=2\). +Moreover, \eqref{eq:kp-torus-cocycle} on two scalars is +\[ + \sigma_{2,v}(zI,z'I)=(z,z')_{3,v}^{\,1+4\cdot2}=1. +\] +Thus their full preimage is central and the ordinary scalar section splits. +Quotienting its image gives the asserted local covers of \(\PGL_2\). +The adelic cover and its rational splitting are obtained by the reciprocity +construction in \cite[Section~0.2]{KP84}. The ordinary scalar section +agrees with that splitting on rational scalars, so the quotient retains a +rational splitting. + +We check the datum in \eqref{eq:theta-central-datum}. A Hilbert symbol with +a cube argument is one; the cocycle on scalar pairs is also one. Consequently +the cube and scalar sections commute and multiply without any remaining +cocycle. If an element has two presentations in +\eqref{eq:theta-central-datum}, their two cube coordinates differ by a +common scalar cube, so the displayed absolute-value ratio agrees. The +formula is therefore a genuine character, trivial on the split scalar +subgroup. On a coroot cube it gives +\[ + \omega_v\bigl(s_v(\operatorname{diag}(t^3,t^{-3}))\bigr)=|t|_v. +\] +This is exactly the exceptional condition of +\cite[Section~I.2, p.~71]{KP84}. The exceptional condition alone would +only specify the square of the value on \(a(\varpi^3)\); the datum in the +main text chooses the pure absolute-value value. + +Use the idele norm in \eqref{eq:theta-central-datum} to define the global +character. A rational diagonal lying in the adelic cube torus times scalars +has a rational ratio which is a cube at every completion. If that ratio +were not a cube in \(F\), its nontrivial Kummer extension would have a +nonsplit unramified prime by Chebotarev. Thus the ratio is a rational cube. +The product formula now shows that our character is trivial on the +intersection with the rational lift. This is the compatibility hypothesis +in \cite[Section~II.1, p.~107, immediately after Lemma~II.1.1]{KP84}. +Extending by the trivial character on rational lifts gives the character +on the rational maximal abelian subgroup of that lemma. As in +\cite[Section~II.1, pp.~107--109]{KP84}, one chooses an extension on the +product of local maximal abelian subgroups which agrees with the former +character on their intersection. The character on the scalar center is +unitary. + +At finite places, the local quotient and subrepresentation assertions are +\cite[Theorem~I.2.9(a),(b), p.~72]{KP84}. At tame finite places, part (f) +of that theorem supplies the spherical vector for the normalized unramified +datum. For a tame place, the uniqueness criterion is +\[ + d=n-1,\quad 2(c+1)=0\pmod n, + \qquad\text{or}\qquad d=n +\] +by \cite[Corollary~I.3.6, p.~79]{KP84}. It applies to +\((d,n,c)=(2,3,2)\). The residue construction and restricted tensor +product are \cite[Theorem~II.2.1, p.~118]{KP84}. Since the tame local +Whittaker spaces have dimension one, +\cite[Theorem~II.2.5(a), p.~122]{KP84} gives a nonzero global Whittaker +functional, and part (b) gives local uniqueness at every place. +At the complex places this also follows directly from +\cite[Theorem~I.6.5(c), p.~105]{KP84}. The inducing spaces, their exceptional +images, and the automorphic residue all inherit the scalar action of +\(\omega\). That action is trivial, so these representations descend to the +scalar quotients. + +On the quotient torus the section of \(a(y)\) has cocycle +\((x,y)_{3,v}^{2}\). Since \(-1\) is a cube in \(F\), this is an alternating +bimultiplicative law. At a tame place it is trivial on pairs of units, and +its value on a uniformizer and a unit is the cubic residue symbol or its +inverse, with a common orientation at all places. Choose the embedding +\(\iota\) with the orientation giving \eqref{eq:theta-effective-cocycle}. +The canonical compact splitting agrees with the sections of unit diagonals, +Weyl elements, and integral upper root elements by +\cite[Proposition~0.1.3, p.~44]{KP84}. Conjugating by the Weyl element gives +the same agreement for the lower root group. +\cite[Corollary~I.3.4, p.~78]{KP84} gives relations for the coefficient +functions defining local Whittaker functionals. +\cite[Theorems~I.4.2--I.4.3, p.~87]{KP84} convert them to spherical +Whittaker values with the modular factor and give the central step. +The direct local calculation below proves +\eqref{eq:theta-local-values} and the lift identities +\eqref{eq:theta-lift-identities} in these conventions. + +At a complex place the Hilbert symbol and the cover are trivial. +\cite[Theorem~I.6.4(c), p.~104]{KP84} identifies the exceptional +representation with the ordinary principal series when \(d\le n\). +Our cube datum fixes the characters on all of \(\C^\times\), because every +complex number has a cube root. Triviality on scalars removes the determinant +character; equivalently any possible remaining finite-order continuous +character of the connected group \(\C^\times\) is trivial. This gives the +characters in part \textup{(iii)} of \Cref{prop:theta-input}. Their Bessel +function is computed in the main text from the Jacquet integral. Finally, +the two Bruhat terms in the constant term of the Eisenstein series are +described by \cite[Proposition~II.1.2, p.~110]{KP84}. At the exceptional +residue the surviving term belongs to the Weyl-conjugate induction, as in +\cite[Section~II.1, p.~114]{KP84}. Its complex inducing character, +including the modular factor, is \(|y|_v^{1/3}\); in particular it has no +angular part. + +\subsection{The normalized global Whittaker expansion} +\label{subsec:theta-global-normalization} + +Let \(\lambda\) be the nonzero global functional just obtained. The +restricted tensor product is spanned by pure tensors, so there is a pure +tensor \(\varphi^\circ\) with \(\lambda(\varphi^\circ)\ne0\). Its +nondistinguished finite places form a finite set. The final \(S\) contains +this set; the arithmetic data are constructed only after that final choice. +For \(v\notin S\), varying only the \(v\)-component of \(\varphi^\circ\) +gives a nonzero local Whittaker functional. Normalize it at +\(\varphi_v^\circ\). If other finitely many components have already been +varied, the resulting partial functional at \(v\) is still a Whittaker +functional, and local uniqueness makes it a scalar multiple of the +normalized one. Evaluating at \(\varphi_v^\circ\) identifies that scalar. +Induction on the number of varied components proves +\eqref{eq:theta-normalized-factorization}. The same argument factors the +complex components. + +At a finite product of remaining places the functional descends to the +tensor product of the local twisted Jacquet quotients. The twisted Jacquet +quotient of the principal series is finite dimensional by +\cite[Lemma~I.3.2, p.~75]{KP84}. Since \(\Theta_v\) is its quotient by +Theorem~I.2.9(b), its twisted Jacquet quotient is a quotient of that +finite-dimensional space. A functional on the finite tensor product is +therefore a finite sum of products. This is the finite-S assertion in +\Cref{prop:theta-input}. + +This normalization accounts for the correction to the original global +formula. The erratum withdraws an auxiliary spherical-functional vanishing +assertion on pp.~119--120. It replaces stabilization of the auxiliary +coefficient \(C_S\) by stabilization of \(C_ST(S)\), where in rank two +\[ + T(S)=\prod_{\substack{v\in S\\v\nmid\infty}}(1+Q_v^{-1}), + \qquad Q_v=\#(\mathcal O_v/\mathfrak p_v). +\] +The sums in Theorems~II.2.2--II.2.3 become finite-\(S\) limits with the +factor \(T(S)^{-1}\), and the same change applies on p.~130 +\cite[correction to Theorems~II.2.2--II.2.3]{KP85Erratum}. +The preceding argument begins with the nonzero automorphic functional and +normalizes each exterior local functional at the distinguished spherical +vector. It proves the product \eqref{eq:theta-normalized-factorization} +without the withdrawn assertion or an unnormalized infinite product. + +The rational splitting identifies every nonconstant Fourier coefficient +with this Whittaker function. Conjugate \(n(x)\) by the rational lift of +\(a(\nu)\) and change \(x\) to \(\nu x\); the adelic Jacobian is one. +This gives \eqref{eq:theta-fourier-expansion}. On a fixed compact set of +\(g\), smoothness at infinity and invariance under a fixed finite open +subgroup reduce the expansion along \(F\backslash\Adele_F\) to the smooth +Fourier expansion on a compact real torus. Integration by parts in +sufficiently many archimedean directions gives absolute local uniform +convergence there. + +\subsection{Sections and the spherical induced vector} + +Fix an exterior finite place \(\mathfrak p\). In this appendix write +\(\mathcal O=\mathcal O_{\mathfrak p}\), let \(\varpi\) be the chosen +uniformizer, put \(Q=q_{\mathfrak p}\), and abbreviate +\(\psi=\psi_{\mathfrak p}\), \(\chi=\chi_{\mathfrak p}\), and +\(\mathcal C=\mathcal C_{\mathfrak p}\). +Thus \(\psi\) has conductor \(\mathcal O\), additive measure gives +\(\mathcal O\) volume one, and \(\chi\) is the sextic residue +character on \(\mathcal O^\times\). In all finite sums it is +extended by zero to the zero residue, including when its exponent is +zero. The embedding of the cubic symbol has been chosen so that the +effective torus cocycle is +\begin{equation}\label{eq:app-torus-laws} + \mathcal C(\varpi,u)=\chi(u)^{-2},\quad + \mathcal C(u,\varpi)=\chi(u)^2,\quad + \mathcal C(u,u')=1,\quad \mathcal C(\varpi,\varpi)=1 + \qquad(u,u'\in\mathcal O^\times). +\end{equation} +The last two identities follow from tameness and alternation. In +particular pure uniformizer powers multiply without a cocycle. +All symbols with \(-1\) are one because \(-1\in F^{\times6}\). + +We first record explicitly why the ordinary rank-one decompositions +are valid for these lifts. See Kubota \cite[p.~114]{Kubota67} for the local +rank-one construction. In the convention of +\cite[Section~0.1, p.~41]{KP84}, the determinant-modified formula is +\[ + \sigma_c(g_1,g_2)= + \left(\frac{x(g_1g_2)}{x(g_1)}, + \frac{x(g_1g_2)}{x(g_2)\det g_1}\right)_3 + (\det g_1,\det g_2)_3^c, + \quad + x(g)= + \begin{cases} + g_{21},&g_{21}\ne0,\\ + g_{22},&g_{21}=0. + \end{cases} +\] +We take \(c=2\) and then quotient the split scalar subgroup. +Substitution in this formula gives +\[ + n(x)a(y)=a(y)n(x/y),\qquad + wa(y)=a(y^{-1})w +\] +on the quotient. Before quotienting, the second equality has the +extra scalar \(yI\). All its cocycles are symbols between powers of +\(y\) and \(-1\), hence are one. For \(H\ne0\), the matrix identity +\[ + wn(H)=n(H^{-1})\operatorname{diag}(-H^{-1},H)\bar n(H^{-1}) +\] +also lifts exactly. The product of the first two factors has cocycle +one, and multiplication by the last factor has cocycle +\((H^{-1},-H)_3=1\). The middle diagonal is +the product of the split scalar \(HI\) and \(a(-H^{-2})\), again +without a cocycle. Thus +\begin{equation}\label{eq:app-bruhat-lift} + wn(H)=n(H^{-1})a(-H^{-2})\bar n(H^{-1}) +\end{equation} +on the quotient. If \(H^{-1}\in\mathcal O\), the last factor is in +the compact splitting. The latter agrees with the sections on unit +diagonals, \(w\), and integral upper root elements by +\cite[Proposition~0.1.3, p.~44]{KP84}; conjugation by \(w\) gives +agreement on the lower root group. The diagonal \(a(-1)\) can +therefore be absorbed in a compact factor in any spherical +calculation. + +Let \(\sigma^-\) be the unramified representation of the covering +torus with central character the Weyl conjugate of +\eqref{eq:theta-central-datum}. Its norm exponent on \(a(y)\) is +\(-1/6\). Modulo scalars, the subgroup of torus elements of +valuation divisible by three is maximal abelian: the tame group +\(F_{\mathfrak p}^\times/F_{\mathfrak p}^{\times3}\) has one +valuation and one unit coordinate over \(\Z/3\Z\), and the +commutator pairing pairs those coordinates nondegenerately. +The torus construction in \cite[Section~0.3, pp.~54--55, and +Section~I.1, p.~58]{KP84} realizes \(\sigma^-\) by induction from a +character of this subgroup. Its index is three, so the unramified +Heisenberg representation has dimension three. The canonical +unramified extension is trivial on unit lifts +\cite[Section~I.1, p.~60]{KP84}; the induced basis vector supported +on that subgroup is therefore unit fixed. Call it \(e_0\), and set +\begin{equation}\label{eq:app-Heisenberg-basis} + e_j=Q^{-j/6}\sigma^-(a(\varpi^j))e_0,\qquad j\in\Z. +\end{equation} +The central element \(a(\varpi^3)\) acts by \(Q^{1/2}\), so +\(e_{j+3}=e_j\). These three vectors form a basis. From +\eqref{eq:app-torus-laws} and unit invariance one obtains +\begin{equation}\label{eq:app-torus-on-e0} + \sigma^-(a(\varpi^j u))e_0 + =Q^{j/6}\chi(u)^{2j}e_j + \qquad(j\in\Z,\ u\in\mathcal O^\times). +\end{equation} + +Use the vector-valued realization of normalized induction +\[ + I^-=\operatorname{Ind}_{\widetilde B}^{\widetilde{\PGL}_2} + (\sigma^-),\qquad + f(n(x)a(y)g)=|y|_{\mathfrak p}^{1/2} + \sigma^-(a(y))f(g). +\] +Its spherical vector \(f^\circ\) is determined by +\(f^\circ(k)=e_0\) on the compact subgroup. The total norm exponent +in this transformation law is +\(1/2-1/6=1/3\). The compact-invariant line of this normalized +unramified induction is one dimensional by +\cite[Lemma~I.1.3, p.~60]{KP84}. By +\cite[Theorem~I.2.9(a),(f), p.~72]{KP84}, the local exceptional +representation is the irreducible subrepresentation of \(I^-\) and has +a nonzero compact-invariant vector. That vector is a scalar multiple of +\(f^\circ\), so the subrepresentation contains \(f^\circ\). + +\subsection{The shells of the Jacquet integral} + +For a smooth \(f\in I^-\), define the vector Jacquet functional by +\[ + \mathscr J(f)=\lim_{L\to\infty} + \int_{\varpi^{-L}\mathcal O}f(wn(x))\overline{\psi(x)}\,\dd x. +\] +This limit stabilizes. Indeed, when \(|x|\) is sufficiently large, +the right invariance of \(f\) makes \(\bar n(x^{-1})\) fix \(f\). +The lifted Bruhat formula then expresses \(f(wn(x))\) as +\[ + |x|_{\mathfrak p}^{-1} + \sigma^-(a(-x^{-2}))f(1). +\] +Choose \(m\) so that \(1+\varpi^m\mathcal O\) consists of cubes and +its unit lifts fix the vector \(f(1)\) in the smooth torus +representation. On a shell \(x=\varpi^{-l}v\), the displayed +expression is unchanged when \(v\) is replaced by +\(vu\), \(u\in1+\varpi^m\mathcal O\): the norm is unchanged, all +cocycles with \(u\) are trivial, and \(a(u^{-2})\) fixes \(f(1)\). +Each such unit coset is an additive coset of +\(\varpi^m\mathcal O\). For \(l>m\), the character +\(\psi(\varpi^{-l}v)\) has zero integral on it. All sufficiently +large shells therefore vanish. This argument applies to every +smooth \(f\), including each of its right translates. + +If \(t\in F_{\mathfrak p}\), choose a ball large enough to contain +\(t\) and to realize the stable values for both \(f\) and +\(\pi(n(t))f\). Translation \(x\mapsto x+t\) preserves that ball +and gives +\(\mathscr J(\pi(n(t))f)=\psi(t)\mathscr J(f)\). +Thus each scalar coordinate of \(\mathscr J\) is a Whittaker functional. +This is a direct stable version, at the present inducing parameter, +of the Jacquet functionals introduced in +\cite[Section~I.3, pp.~74--75]{KP84}. + +For \(e\ge0\) put \(J_e=\mathscr J(\pi(a(\varpi^e))f^\circ)\); explicitly, +\begin{equation}\label{eq:app-Jacquet} + J_e=\int_{F_{\mathfrak p}} + f^\circ(wn(x)a(\varpi^e))\overline{\psi(x)}\,\dd x. +\end{equation} +The integral retains the stable interpretation just established. +We now evaluate it shell by shell. + +Put \(x=\varpi^eH\) and use the first two lift identities above. +The measure contributes \(Q^{-e}\), the modular factor of +\(a(\varpi^{-e})\) contributes \(Q^{e/2}\), and its torus norm +factor contributes \(Q^{-e/6}\). Thus every shell has the common +factor +\begin{equation}\label{eq:app-common-factor} + Q^{-e+e/2-e/6}=Q^{-2e/3}. +\end{equation} +If \(H\in\mathcal O\), then \(wn(H)\) belongs to the compact +subgroup, and \(\psi(\varpi^eH)=1\). This region contributes +\(e_{-e}\) inside the common factor. + +Suppose instead that \(H=\varpi^{-l}v\), with \(l\ge1\) and +\(v\in\mathcal O^\times\). In \eqref{eq:app-bruhat-lift} the last +factor is compact, while the first acts trivially in the inducing +transformation law. The modular factor of \(a(-H^{-2})\) is +\(Q^{-l}\), and its torus norm factor is \(Q^{l/3}\). Furthermore, +\[ + a(\varpi^{2l}v^{-2}) + =\mathcal C(\varpi^{2l},v^{-2})^{-1} + a(\varpi^{2l})a(v^{-2}),\qquad + \mathcal C(\varpi^{2l},v^{-2})^{-1}=\chi(v)^{-8l}. +\] +The minus sign in \(-H^{-2}\) has no effect. Multiplication by +\(a(\varpi^{-e})\) changes the basis index by \(-e\), without a +cocycle between the pure uniformizer powers. Finally +\(\dd H=Q^l\,\dd v\). It follows that this shell contributes, +inside \eqref{eq:app-common-factor}, +\begin{equation}\label{eq:app-shell} + Q^{l/3}e_{-e+2l} + \int_{\mathcal O^\times} + \chi(v)^{-8l}\overline{\psi(\varpi^{e-l}v)}\,\dd v. +\end{equation} +This accounts separately for the measure, modular, norm, and +Hilbert-symbol factors. + +Define the normalized prime Gauss sums +\[ + g_j=Q^{-1/2}\sum_{x\bmod\mathfrak p} + \chi(x)^j\psi(x/\varpi),\qquad j\in\Z. +\] +The powers in these sums retain the unit mask. Since +\(\chi(-1)=1\), replacing \(\psi(x/\varpi)\) by its complex +conjugate has no effect on \(g_j\). For \(j\not\equiv0\pmod6\), +the usual finite-field Gauss identity gives +\[ + |g_j|=1,\qquad g_jg_{-j}=\chi(-1)^j=1. +\] +In particular \(g_2g_4=1\). These identities follow, for example, +by expanding the product of the two sums, substituting \(x=ty\), +and using the additive sum in \(y\); the result is +\(Q\chi(-1)^j\) before division by \(Q\). + +Write \(I_{e,l}\) for the integral in \eqref{eq:app-shell}. The +character there is \(\chi^{-2l}\), since \(-8l\equiv-2l\pmod6\). +Its exact values are +\begin{equation}\label{eq:app-shell-integrals} + I_{e,l}= + \begin{cases} + 1-Q^{-1},&1\le l\le e,\quad 3\mid l,\\ + 0,&1\le l\le e,\quad 3\nmid l,\\ + Q^{-1/2}g_{-2l},&l=e+1,\quad 3\nmid l,\\ + -Q^{-1},&l=e+1,\quad 3\mid l,\\ + 0,&l>e+1. + \end{cases} +\end{equation} +Indeed, for \(l\le e\) the additive character is trivial, so unit +orthogonality gives the first two cases. For \(l=e+1\), integration +over the residue classes gives \(Q^{-1}\) times the corresponding +prime Gauss sum; the trivial character with its unit mask gives +\(\sum_{x\ne0}\psi(x/\varpi)=-1\). For \(l>e+1\), split each unit +class modulo \(\varpi^{l-e-1}\) into its \(Q\) lifts modulo +\(\varpi^{l-e}\). The multiplicative character is unchanged on +these lifts, whereas the additive factors sum to +\(\sum_{t\bmod\mathfrak p}\psi(t/\varpi)=0\). +This also proves the asserted stabilization of +\eqref{eq:app-Jacquet}. + +We can now sum the shells. For \(k\ge0\), the integral unit ball +and the shells \(l=3,6,\ldots,3k\) have the same basis index and +combined coefficient +\[ + 1+(1-Q^{-1})\sum_{j=1}^{k}Q^j=Q^k. +\] +For \(e=3k\), the terminal shell \(l=3k+1\) has coefficient +\(Q^{k-1/6}g_4\) and basis vector \(e_2\). For \(e=3k+1\), the +terminal shell \(l=3k+2\) has coefficient +\(Q^{k+1/6}g_2\) and basis vector \(e_0\). For \(e=3k+2\), the +terminal shell \(l=3k+3\) has coefficient \(-Q^k\) and the same +basis vector \(e_1\) as the preceding combined contribution. +Multiplying by \eqref{eq:app-common-factor} therefore gives +\begin{align} + J_{3k} + &=Q^{-k}(e_0+Q^{-1/6}g_4e_2),\notag\\ + J_{3k+1} + &=Q^{-k}(Q^{-1/2}g_2e_0+Q^{-2/3}e_2) + =Q^{-k-1/2}g_2(e_0+Q^{-1/6}g_4e_2),\notag\\ + J_{3k+2}&=0. \label{eq:app-J-values} +\end{align} +The second equality uses \(g_2g_4=1\). + +In particular \(J_0=e_0+Q^{-1/6}g_4e_2\ne0\). Choose a scalar +coordinate nonzero on \(J_0\). Its Jacquet functional is nonzero on +the exceptional subrepresentation, since that subrepresentation +contains \(f^\circ\). The local Whittaker space is one dimensional +by \cite[Corollary~I.3.6, p.~79]{KP84}. Hence every spherical Whittaker +function normalized at one has the ratios in +\eqref{eq:app-J-values}: +\begin{equation}\label{eq:app-pure-values} + W(a(\varpi^{3k}))=Q^{-k},\qquad + W(a(\varpi^{3k+1}))=Q^{-k-1/2}g_2,\qquad + W(a(\varpi^{3k+2}))=0. +\end{equation} +The middle term is +\(Q^{-k-1}\sum_x\chi(x)^2\psi(x/\varpi)\), as claimed. +This computation agrees with the central-step statement of +\cite[Theorem~I.4.3, p.~87]{KP84}: on \(a(\varpi^3)\) the Weyl-conjugate +inducing character is \(Q^{1/2}\) and the modular half-character is +\(Q^{-3/2}\), so their product is \(Q^{-1}\). + +\subsection{Negative valuations and the unit normalization} + +For \(e<0\), choose \(x\in\mathcal O\) for which +\(\psi(\varpi^e x)\ne1\). The spherical vector is fixed by +\(n(x)\), while the split root relation and Whittaker equivariance +give +\[ + W(a(\varpi^e)) + =W(a(\varpi^e)n(x)) + =\psi(\varpi^e x)W(a(\varpi^e)). +\] +Thus the value is zero. For a unit \(u\), the compact section of +\(a(u)\) fixes the spherical vector. The torus law gives +\[ + a(\varpi^e u)=\mathcal C(\varpi^e,u)^{-1}a(\varpi^e)a(u), +\] +where \(\mathcal C\) includes the scalar action on a genuine vector. +Consequently +\begin{equation}\label{eq:app-unit-rule} + W(a(\varpi^e u))=\chi(u)^{2e}W(a(\varpi^e)). +\end{equation} +Together with \eqref{eq:app-pure-values}, this proves +\eqref{eq:theta-local-values}. + +There is a useful check that fixes the orientation without referring +to a convention for the name of a cubic Gauss sum. At a tame place +the residue character is unchanged when \(\varpi\) is replaced by +\(\varpi u\). Substitution \(x=uv\) in the residue sum yields +\[ + Q^{-1}\sum_{x\bmod\mathfrak p}\chi(x)^2\psi(x/(\varpi u)) + =\chi(u)^2 Q^{-1}\sum_{v\bmod\mathfrak p} + \chi(v)^2\psi(v/\varpi). +\] +This is exactly the \(e=1\) transformation in +\eqref{eq:app-unit-rule}. Thus the Gauss phase and the unit factor +are a single compatible normalization. Changing the sign of an +additive Gauss argument is harmless because \(\chi(-1)=1\). diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/11-sieve.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/11-sieve.tex new file mode 100644 index 0000000000000000000000000000000000000000..dd08bb85f3651081d361f5b4843dd5de76d340a4 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/sections/11-sieve.tex @@ -0,0 +1,277 @@ +\section{Two elementary sieve facts}\label{sec:sieve} + +The prime construction needs two elementary facts. The first estimates +the mass left after excluding local divisibility conditions, with an +explicit bound on the moduli in the error term. The second identifies +the density removed when a least prime factor is selected. We give +proofs, following the forms in \cite[Lemmas~2.9 and~7.4]{BlockPaper}. + +\subsection{A block sieve with bounded coefficients} + +The advantage of the following form of the Brun--Hooley sieve is that +its remainders occur without divisor weights. Its underlying product +inequalities are those of Ford and Halberstam +\cite[Section~1, Lemma~1]{FordHalberstam2000}. + +\begin{lemma}[Block sieve]\label{lem:block} +Let $\mathcal A$ be a finite set with nonnegative weights, and let +$\mathcal P$ be a set of primes at most $z$, where $z\ge2$. +For each $p\in\mathcal P$, specify a bad condition on $\mathcal A$. +Suppose that, for every squarefree product $d$ of primes in +$\mathcal P$, the weight of the objects satisfying all conditions +at $p\mid d$ is +\[ + A(d)=X'g(d)+E(d), +\] +including $d=1$, where $X'\ge0$, $g(1)=1$, and $g$ is multiplicative. +Assume that, for fixed constants $\eta_0>0$ and $C_0<\infty$, +\begin{equation}\label{eq:block-hypotheses} + 0\le g(p)\le1-\eta_0, + \qquad + \sum_{\substack{v1). +\end{equation} +For every sufficiently large even integer $H$, the weight $S$ of +objects avoiding all the bad conditions satisfies +\begin{equation}\label{eq:block-sieve} + S=X'\prod_{p\in\mathcal P}(1-g(p)) + \bigl(1+O(\exp(-H))\bigr) + +O\left(\sum_{\substack{d\le z^{4H+2}\\d\mid\prod_{p\in\mathcal P}p}} + |E(d)|\right). +\end{equation} +The constants and the lower bound on $H$ depend only on +$\eta_0,C_0$. In particular, the estimate is uniform when $H$ grows. +For $H=2$ there is the upper bound +\begin{equation}\label{eq:block-sieve-upper} + S\ll_{\eta_0,C_0}X'\prod_{p\in\mathcal P}(1-g(p)) + +\sum_{\substack{d\le z^{10}\\d\mid\prod_{p\in\mathcal P}p}} + |E(d)|. +\end{equation} +\end{lemma} + +\begin{proof} +Write $b_p\in\{0,1\}$ for the indicator of the bad condition at $p$. +Use the blocks +\[ + \mathcal B_j=\{p\in\mathcal P: + z^{2^{-j-1}}0, +\] +because $-\log(1-u)\le u/\eta_0$ for $0\le u\le1-\eta_0$. +Moreover, +\[ + t_j:=\mathbb E T_j\le\frac{C_0^{h_j+1}}{(h_j+1)!}, + \qquad a_j\le\mathbb E U_j\le a_j+t_j. +\] +The factorial bound implies +\[ + R_H:=\sum_j\frac{t_j}{a_j}\ll_{\eta_0,C_0}\exp(-H) +\] +for all sufficiently large even $H$: for sufficiently large $n$, +$C_0^n/n!\le\exp(-2n)$, and $h_j=(j+1)H$. +For $H=2$, the same sum is bounded in terms of $\eta_0,C_0$. +Independence of the blocks now yields +\[ + \prod_ja_j\le\mathbb E U\le \prod_ja_j\exp(R_H), + \qquad + \mathbb E(U-B)\le \prod_ja_j R_H\exp(R_H). +\] +Thus both bounding polynomials have expectation +$\prod_ja_j(1+O(\exp(-H)))$ when $H$ is sufficiently large; +for $H=2$, the upper expectation is $O(\prod_ja_j)$. + +Finally evaluate $U$ and $B$ in the original weighted set. +Multiplicativity makes their main terms $X'\mathbb E U$ and +$X'\mathbb E B$. The coefficient and support bounds show that +each error is at most the sum of $|E(d)|$ in +\eqref{eq:block-sieve}. Squeezing $S$ between the two weighted +polynomial sums proves both assertions. +\end{proof} + +\subsection{Rough-number densities} + +Recall the density $D_\gamma(s)$ from +\eqref{eq:rough-density-definition}. Its $j$th term describes a +product of $j$ primes whose logarithms, in units of $\log x$, sum +to $s$ and are at least $\gamma$; the factor $1/j!$ normalizes the +ordered prime lists. These are Buchstab's rough-number densities, +expressed in logarithmic coordinates. The identity below is the +corresponding least-prime-factor decomposition \cite{Buchstab1937}. +The normalization and regularity were established in +Section~\ref{sec:type-ii}; we now prove the two estimates needed to +subtract composite mass. + +For the product $V(y)$ in \eqref{eq:sieve-basic-parameters}, +Mertens' theorem gives +\begin{equation}\label{eq:mertens-product} + V(y)\sim\frac{\exp(-\gamma_E)}{\log y}, +\end{equation} +where $\gamma_E$ is Euler's constant. + +\begin{lemma}[Rough-number density]\label{lem:density} +For $00$ such that, for +$00$, +put $L=\log x$, and let +\[ + R_b(x)=\#\{n\in\mathbb{N}:xx^b\}. +\] +We first claim that +\begin{equation}\label{eq:density-rough-lower} + D_b(1)\le\liminf_{x\to\infty}\frac{L}{x}R_b(x). +\end{equation} +For $j\ge2$ and fixed $\eta>0$, count ordered prime lists +$(p_1,\ldots,p_j)$ with weight $1/j!$, subject to +\[ + p_i>x^b\quad(ix^{b+\eta}$. +An integer with prime multiplicities $e_1,e_2,\ldots$ receives +total weight at most $1/\prod_i e_i!\le1$, even when prime factors +repeat. Its total number of prime factors also determines $j$. +Thus summing these counts over $j$, together with the one-prime +contribution, gives a lower bound for $R_b(x)$. + +Uniformly in the indicated range of $Q$, the prime number theorem gives +\[ + \#\{p_j:x/Q0\), and define +\begin{equation}\label{con:weight} + \begin{split} + F(h)&=\one_{\{h/h_{\mathcal P}=cQ\text{ for a prime }Q>x^{0.9}\}},\\ + w(d)&=\one_{\{d\equiv u\pmod M\}}\Psi(d/x) + F(d-1)\mathcal W(d-1)\qquad(d\ge2). + \end{split} +\end{equation} +Set \(w(1)=0\). Since \((ph)_{\mathcal P}=p h_{\mathcal P}\) for +every group prime \(p\), including when \(p\mid h\), we have +\begin{equation}\label{con:invariance} + F(ph)=F(h)\qquad(h\ge1,\ p\in\textstyle\bigcup_i\mathcal P_i). +\end{equation} +On the support of \(w\), the representation \(d=crQ+1\) is unique: +\(r=(d-1)_{\mathcal P}\), since neither \(c\) nor \(Q\) has a +group prime factor. In particular, +\begin{equation}\label{con:support} + r\le x^{0.11},\qquad \mathcal W(d-1)=\mathcal W(r),\qquad + 0\le w(d)\le B_K +\end{equation} +for a constant \(B_K\). The last assertion follows from +\(j2^{1-j}\le1\) for each integer \(j\ge1\); in fact +\(0\le\mathcal W(h)\le B_K\) for every \(h\ge1\). + +Write \(X_0=\sum_d w(d)\) for the total mass. Our aim is to prove +that the prime mass is \(\gg X_0/L\), while \(X_0\gg x/L\). +The bound \(w\le B_K\) will then give the required count of primes. +The marked estimate supplies the comparison needed to subtract +composite mass. + +Use the rough-number density $D_\gamma(s)$ from +\eqref{eq:rough-density-definition}. For fixed \(\gamma>0\) and integers +\(m>x^\gamma\), put +\begin{equation}\label{con:proxies} + R_\gamma(m)=\one_{\{P^-(m)>x^\gamma\}},\qquad + B_\gamma(m)=\frac{D_\gamma(\log m/L)}{LV(W)} + \one_{\{P^-(m)>W\}}. +\end{equation} +We will replace \(R_\gamma\) by \(B_\gamma\) inside sums weighted +by \(w(mn)\): the factor \(D_\gamma/(LV(W))\) compensates for +retaining only \(W\)-roughness. Lemma~\ref{lem:marked-coefficient-transfer} +and the character-transform bound proved in +Proposition~\ref{prop:rough-type-ii} permit this replacement after +the fixed congruence is expanded in characters. We verify that +application next. All Dirichlet characters retain the zero extension +off the units used in Section~\ref{sec:type-ii}. + +\begin{lemma}[Type~II estimate in the progression]\label{con:type-ii} +Fix $\delta>0$ and $0<\gamma0$, every character of modulus at most $L^{A_0}$, +and every $|t|\le2XL^B$, uniformly for $|v|\le L^C$ and every +interval $I$ under the present fixed exponent hypotheses. +If $\alpha$ satisfies that bound, so does +\(\alpha_m\lambda(m)\): the product \(\lambda\chi\), with its +zero extension, is a character modulo \(\operatorname{lcm}(M,k)\), +which is at most \(ML^{A_0}\le L^{A_0+1}\) for large \(x\). + +To separate the smooth factor, write +\[ + \widehat\psi(v)=\frac1{2\pi}\int_{\R}\Psi(\e^s)\e^{-ivs}\,ds. +\] +Fourier inversion yields +\[ + \Psi(mn/x)=\int_{\R}\widehat\psi(v)x^{-iv}m^{iv}n^{iv}\,dv. +\] +For \(|v|\le L\), apply Lemma~\ref{lem:marked-coefficient-transfer} +to the invariant function $F$ in \eqref{con:weight}, with +\(C=2\), \(D_*=6\), and +\[ + \alpha_m=m^{iv}\lambda(m) + (R_\gamma(m)-B_\gamma(m)),\qquad + \beta_n=b_n\lambda(n)n^{iv}. +\] +Here the formula for \(\alpha_m\) applies on \(I\), and +\(\alpha_m=0\) elsewhere. +Both sequences are \(W\)-rough and bounded by \(L^2\) for large +\(x\): \(x^\gamma>W\), \(D_\gamma\) is bounded on the fixed +exponent interval, and \(LV(W)\asymp L^{0.76}\). The finite +number of characters is fixed with \(M\). The integral of +\(|\widehat\psi|\) is finite, whereas its tail beyond \(L\) is +\(O_A(L^{-A})\) for every fixed \(A\). On that tail the bilinear +sum is \(O_K(X)\) by \eqref{con:support}. Integration proves +\eqref{con:type-ii-bound}. +\end{proof} + +All parameters in the remainder of the section are fixed before +\(x\) tends to infinity. Unless an independence is explicitly +asserted, constants, thresholds, and rates of convergence may depend +on those fixed parameters. The estimates are valid for arbitrary +fixed marks, subject to the lower bounds on \(K\) in the Type~II +applications. We will choose the parameters together at the end. + +\subsection{Mass and distribution in progressions} + +Let \(r\) range over group integers, and define +\begin{equation}\label{con:mass-definitions} + \begin{split} + J_0&=\sum_r\frac{\mathcal W(r)}r,\\ + A_r&=\sum_{\substack{Q>x^{0.9}\text{ prime}\\crQ+1\equiv u\pmod M}} + \Psi((crQ+1)/x),\qquad + X_0=\sum_d w(d)=\sum_r\mathcal W(r)A_r. + \end{split} +\end{equation} +We shall show that \(X_0\) is of order \(xJ_0/L\), and that the +family at each small \(r\) has enough distribution to apply the +block sieve. The harmonic sum makes this possible without constants +depending on the pointwise bound \(B_K\). + +\begin{lemma}[Harmonic mass]\label{con:harmonic} +For each group put +\[ + E_i(t)=\prod_{p\in\mathcal P_i}\left(1+\frac{t}{p-1}\right). +\] +The harmonic mass has the exact Euler-product expression +\begin{equation}\label{con:harmonic-identity} + \begin{aligned} + J_0&=\prod_{i=1}^K\frac{E_i'(1/2)}{V_i}\\ + &=\prod_{i=1}^K\left\{ + \frac{E_i(1/2)}{V_i} + \sum_{p\in\mathcal P_i}\frac1{p-1+1/2}\right\}. + \end{aligned} +\end{equation} +For every fixed \(\eps>0\), +\begin{equation}\label{con:harmonic-bounds} + 1\le J_0\ll_K1,\qquad + \sum_{r>x^\eps}\frac{\mathcal W(r)}r + \ll_K\exp(-\eps L^{1-a_K}). +\end{equation} +Consequently the part of \(X_0\) with \(r>x^\eps\) is +\(O_K(x\exp(-\eps L^{1-a_K}))\). +\end{lemma} + +\begin{proof} +At a prime \(p\), summing over its positive exponents contributes +\(\sum_{j\ge1}p^{-j}=1/(p-1)\). Consequently \(E_i(t)\) is the +harmonic generating function for integers supported on the \(i\)th +group, with \(t\) recording their number of distinct prime factors. +Differentiating its finite product and +evaluating at \(t=1/2\) inserts the factor +\(\omega_i(r)(1/2)^{\omega_i(r)-1}\). +The groups are disjoint, so multiplying these marked sums and +dividing by \(\prod_iV_i\) proves \eqref{con:harmonic-identity}. + +Integers with one prime, to exponent one, from each group contribute +\[ + \prod_{i=1}^K\left(V_i^{-1}\sum_{p\in\mathcal P_i}p^{-1}\right)=1 +\] +to \(J_0\). For the upper bound and tail put \(s=L^{-a_K}\). +Every group prime satisfies \(p^s\le\e^2\), so, for large \(x\), +\[ + \sum_{j\ge1}p^{-j(1-s)}\le\frac{2\e^2}{p}. +\] +Bounding the mark by \(B_K\) and multiplying the geometric series +over group primes gives +\[ + \sum_r\frac{\mathcal W(r)}{r^{1-s}}\ll_K1, +\] +since \(\sum_iV_i\ll K\). Multiplication by +\(x^{-\eps s}\) on \(r>x^\eps\) proves the tail. Finally, +\(A_r\ll x/r\) by counting integers in the interval for \(Q\) +whenever \(A_r\ne0\); otherwise the bound is immediate. +\end{proof} + +Take temporarily +\begin{equation}\label{con:small-parameters} + 0<\kappa<0.01,\qquad 0<\eps<\kappa, + \qquad 00\), +\begin{equation}\label{con:bv-sum} + \sum_{r\le x^\eps}\mathcal W(r) + \sum_{\substack{\ell\le D\\\ell\text{ odd squarefree}}}|E_r(\ell)| + \ll xJ_0L^{-A}. +\end{equation} +Moreover, writing \(A_\Psi=\int_1^2\Psi(y)\,dy\), we have +\begin{equation}\label{con:mass-asymptotic} + X_0\sim\frac{xJ_0A_\Psi}{c\varphi(M/c)L} + \qquad(x\to\infty), +\end{equation} +with all construction parameters fixed and \(J_0=J_0(x)\). +In particular, +\begin{equation}\label{con:total-mass} + X_0\asymp_{M,\Psi}\frac{xJ_0}{L}, +\end{equation} +with comparison constants independent of \(K,\kappa,\eps,b\). +\end{lemma} + +\begin{proof} +Put \(N=M/c\). For \(r\le x^\eps\), the support of \(\Psi\) +makes \(Q>x^{0.9}\) automatic. The progression for \(d\) is +equivalent to the reduced class +\[ + Q\equiv\frac{u-1}{c}\,r^{-1}\pmod N. +\] +If \((\ell,Mr)>1\), then \(A_r(\ell)=g_r(\ell)=0\). +Otherwise the additional condition is +\(Q\equiv-(cr)^{-1}\pmod\ell\), and the two classes combine to +one reduced class modulo \(N\ell\). + +We use the prime-counting form of the Bombieri--Vinogradov theorem +\cite{Bombieri65,Vinogradov65}: for fixed \(A',\eta>0\), +\begin{equation}\label{con:bv-input} + \sum_{q0, +\] +which also absorbs the fixed factor \(N\). + +Define +\[ + I_r=\frac{x}{cr}\int_1^2 + \frac{\Psi(y)}{\log((xy-1)/(cr))}\,dy. +\] +For every fixed \(0<\theta<0.1\), the support of \(\Psi\) forces +\(Q>x^{0.9}\) whenever \(r\le x^\theta\). The prime number theorem +in the fixed reduced progressions modulo \(N\), followed by partial +summation, therefore gives, for every fixed \(A>0\), +\begin{equation}\label{con:fixed-r-mass} + A_r=\frac{I_r}{\varphi(N)} + +O_{A,M,\Psi}\bigl((x/r)L^{-A}\bigr) + \qquad(r\le x^\theta). +\end{equation} +The class varies with \(r\), but \(N\) is fixed and every class is +reduced. The implied constant is independent of \(\theta\), although +the threshold for the automatic cutoff may depend on \(\theta\). +This single-modulus estimate thus holds throughout +\(0<\theta<0.1\); the restriction \(\eps<\kappa\) is needed for +the joint modulus estimate that follows. + +Partial summation in \eqref{con:bv-input} yields +\begin{equation}\label{con:smoothed-bv} + \sum_{\substack{\ell\le D\\\ell\text{ odd squarefree}}} + \left|A_r(\ell)-\frac{g_r(\ell)}{\varphi(N)}I_r\right| + \ll_{A'}\frac xrL^{-A'}. +\end{equation} +To check the smoothing uniformly, integrate the progression remainder +against the derivative of \(\Psi((crt+1)/x)\). Its total absolute +integral is \(\int|\Psi'(y)|\,dy\), independently of \(r\), and +all integration points have \(t\asymp x/(cr)\). The moduli +\(N\ell\) are distinct, and the maximum in \eqref{con:bv-input} +covers their varying residue classes. + +The term \(\ell=1\) of \eqref{con:smoothed-bv} evaluates \(A_r\). +Replacing \(I_r/\varphi(N)\) by \(A_r\) in the other terms costs +at most one logarithmic factor, because +\[ + \sum_{\ell\le D}g_r(\ell)\le\sum_{\ell\le D}\frac1{\varphi(\ell)} + \ll L. +\] +For example, expand +\(n/\varphi(n)=\sum_{d\mid n}\mu^2(d)/\varphi(d)\) and use +\(\sum_d\mu^2(d)/(d\varphi(d))<\infty\). +Choosing \(A'\) sufficiently large gives +\(\sum_{\ell\le D}|E_r(\ell)|\ll(x/r)L^{-A}\); summing with +the marks proves \eqref{con:bv-sum}. + +On the support of \(\Psi\), for \(r\le x^\eps\) the logarithm +in \(I_r\) is comparable to \(L\), with absolute comparison +constants since \(\eps<0.01\) and \(c\in\{2,4\}\). Thus +\(I_r\asymp_\Psi x/(rL)\). Summing the evaluation of \(A_r\) +and using Lemma~\ref{con:harmonic} proves \eqref{con:total-mass}. +The tail divided by \(xJ_0/L\) tends to zero since \(J_0\ge1\). +Although its threshold may depend on \(K\), the resulting comparison +constants do not. + +To obtain the precise asymptotic, fix an auxiliary exponent +\(0<\theta<\eps\), without changing the sieve parameters. +Uniformly for \(r\le x^\theta\) and \(y\) in the support of \(\Psi\), +\[ + \log((xy-1)/(cr))=L+O(\theta L+1), +\] +and hence +\[ + I_r=\frac{xA_\Psi}{crL} + \bigl(1+O(\theta)+O(L^{-1})\bigr). +\] +The constants in these relative errors are independent of \(\theta\). +Use the \(\ell=1\) term of \eqref{con:smoothed-bv} with \(A'>2\), +multiply by \(\mathcal W(r)\), and sum over \(r\le x^\theta\). +The summed progression error is \(O(xJ_0L^{-A'})\). +Lemma~\ref{con:harmonic} bounds the omitted parts of both \(X_0\) +and \(J_0\); after the harmonic tail is multiplied by \(x/L\), +both are \(o_\theta(xJ_0/L)\), since \(J_0\ge1\). Thus +\[ + \frac{c\varphi(N)L X_0}{xJ_0A_\Psi} + =1+O(\theta)+o_\theta(1). +\] +First let \(x\) tend to infinity with \(\theta\) fixed, and then +let \(\theta\) decrease to zero. This proves +\eqref{con:mass-asymptotic}. For \((M,c,u)=(8,4,5)\), it becomes +\(X_0\sim xJ_0A_\Psi/(4L)\), since \(\varphi(M/c)=\varphi(2)=1\). +\end{proof} + +To express the sieve products, define +\begin{equation}\label{con:singular-product} + \begin{split} + V_M(y)&=\prod_{\substack{22}\left(1-\frac1{(p-1)^2}\right) + \prod_{\substack{p\mid M\\p>2}} + \left(1-\frac1{p-1}\right)^{-1}>0. + \end{split} +\end{equation} +Division by \(V(y)\) and Mertens' theorem give +\begin{equation}\label{con:mertens} + V_M(y)\sim\mathfrak S_M V(y),\qquad + V(y)\sim\frac{\e^{-\gamma_E}}{\log y}, +\end{equation} +where \(\gamma_E\) is Euler's constant. The primes dividing any +supported \(r\) are large enough that their omission changes these +products negligibly. More precisely, \eqref{con:support} gives +\begin{equation}\label{con:omitted-primes} + \sum_{p\mid r}\frac1p\le0.11L^{0.9}\exp(-L^{0.1})=o(1), + \qquad + \prod_{2x^b}w(d) + \ge\frac{\mathfrak S_M X_0}{L} + \left\{\frac{\e^{-\gamma_E}}b + (1-C_s\e^{-c_s/b})+o(1)\right\}, +\end{equation} +where \(C_s,c_s>0\) are absolute; the displayed bound on \(\e^{-H}\) +permits \(c_s=1/20\), after enlarging \(C_s\). +The remainders are +\(o(X_0/L)\) by taking \(A>2\) in \eqref{con:bv-sum}. + +\begin{lemma}[Mass conditioned on a divisor]\label{con:bin-mass} +For any fixed \(b\le\gamma<\gamma'\le1/2-\kappa\), +\begin{equation}\label{con:bin-mass-formula} + \sum_{\substack{x^\gammaW}}w(d) + =\mathfrak S_M X_0 V(W) + \left(\log\frac{\gamma'}\gamma+o(1)\right). +\end{equation} +\end{lemma} + +\begin{proof} +For large \(x\), the primes \(n\) in this sum exceed \(W\) and +all group primes, and do not divide \(M\). Fix \(r\le x^\eps\) +and condition the family in \(A_r\) on \(n\mid crQ+1\). For a +squarefree product \(\ell\) of odd primes at most \(W\), the +intersection count is +\[ + A_r(n\ell)=\frac{A_r}{n-1}g_r(\ell)+E_r(n\ell). +\] +This includes \(\ell=1\), as allowed in Lemma~\ref{lem:block}. +Apply that lemma with main mass \(A_r/(n-1)\) and +\(H=2\lceil(\log L)^2\rceil\). Its relative error tends to zero, +and its remainder moduli satisfy +\[ + \ell\le W^{4H+2}=\exp(O(L^{0.24}(\log L)^2))=x^{o(1)}, + \qquad n\ell\le x^{1/2-\kappa+o(1)}0\) we have +\[ + \sum_{r\le x^\eps}\mathcal W(r)A_r + =X_0+o(X_0L^{-A}). +\] +This evaluates the contribution of \(r\le x^\eps\) with the exact +main mass \(X_0\); taking \(A\) large in \eqref{con:bv-sum} +makes the remainders \(o(X_0V(W))\). Finally, each supported \(d\) +has at most \(\log(2x)/(bL)=O(1/b)\) distinct prime divisors above +\(x^b\). Consequently the harmonic tail bounds the omitted +\(r>x^\eps\) contribution even after summing over \(n\). +This proves \eqref{con:bin-mass-formula}. +\end{proof} + +\subsection{Removing composites by their least prime factor} + +We next pass from \(S_b\) to +\[ + U_\kappa=\sum_{P^-(d)>x^{1/2-\kappa}}w(d). +\] +For a bin \((\gamma,\gamma']\subseteq(b,1/2-\kappa]\), with the +first left endpoint allowed to equal \(b\), define +\[ + T_{\gamma,\gamma'}= + \sum_{\substack{x^\gammax^\gamma}w(mn). +\] +If \(x^\gamma\gamma\). +Both dyad scales are at least \(x^{b/2}\), and their product is +comparable to \(x\). There are \(O(L)\) such pairs, since each +\(n\)-dyad allows only a bounded number of \(m\)-dyads. + +Apply Lemma~\ref{con:type-ii} in every pair with \(\delta=b/2\) +and with \(b_n\) the prime indicator restricted to its dyad and +bin. These coefficients are \(W\)-rough. The total error is +\(O(xL^{-5})=o(X_0/L)\), by \eqref{con:total-mass} and +\(J_0\ge1\). Hence +\[ + T_{\gamma,\gamma'}= + \sum_{\substack{x^\gammax^\gamma}\frac{D_\gamma(\log m/L)}{LV(W)} + \one_{\{P^-(m)>W\}}w(mn)+o(X_0/L). +\] +Joint continuity of the density on the fixed compact exponent range +gives, on the support of the summand, +\[ + D_\gamma(\log m/L) + \le\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t)+o(1). +\] +Since \(n>W\), the remaining roughness condition is precisely +\(P^-(mn)>W\). Lemma~\ref{con:bin-mass} thus gives +\begin{equation}\label{con:bin-cost} + T_{\gamma,\gamma'}\le\frac{\mathfrak S_M X_0}{L} + \left\{\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t) + \log\frac{\gamma'}\gamma+o(1)\right\}. +\end{equation} + +The density estimate in Lemma~\ref{lem:density} is normalized to +match this expression: +\begin{equation}\label{con:density-budget} + \int_b^{1/2}D_t(1-t)\frac{dt}{t}=D_b(1)-1, + \qquad + D_b(1)\le\frac{\e^{-\gamma_E}}b(1+C_d\e^{-c_d/b}) +\end{equation} +for sufficiently small fixed \(b\). On +\(b\le\gamma\le t\le1/2-\kappa\), the point \((\gamma,1-t)\) +stays a fixed distance from \(s=\gamma\). Uniform continuity and +the finite mass of \(dt/t\) therefore give a sufficiently fine +fixed partition +\[ + b=\gamma_0<\gamma_1<\cdots<\gamma_J=1/2-\kappa +\] +such that +\begin{align*} + \sum_{j=1}^J + \sup_{\gamma_{j-1}\le t\le\gamma_j}D_{\gamma_{j-1}}(1-t) + \log\frac{\gamma_j}{\gamma_{j-1}} + &\le\int_b^{1/2-\kappa}D_t(1-t)\frac{dt}{t}+0.1\\ + &\le D_b(1)-1+0.1. +\end{align*} +Subtracting \eqref{con:bin-cost} for these finitely many bins from +\eqref{con:initial-lower} proves +\begin{equation}\label{con:post-bin} + U_\kappa\ge\frac{\mathfrak S_M X_0}{L} + \left\{0.9-\frac{\e^{-\gamma_E}}b + (C_s\e^{-c_s/b}+C_d\e^{-c_d/b})+o(1)\right\}. +\end{equation} +The unit term in \eqref{con:density-budget} is what leaves a positive +mass after this subtraction. It remains to remove the composites +whose two prime factors are both close to \(\sqrt x\). + +\subsection{Composites with two nearly equal prime factors} + +A composite counted by \(U_\kappa\) has exactly two prime factors +with multiplicity: three factors greater than \(x^{1/2-\kappa}\) +would have product exceeding \(2x\). Each factor lies in +\([x^{1/2-\kappa},2x^{1/2+\kappa}]\). Cover the ordered factor +pairs by dyadic boxes +\begin{equation}\label{con:boxes} + \mathcal B=[M_1,2M_1)\times[M_2,2M_2) +\end{equation} +that meet this range in each coordinate and admit a product in +\([x,2x]\). They satisfy \(x/4\le M_1M_2\le2x\), and their +full coordinate intervals lie in \([x^{0.46},x^{0.54}]\) for large +\(x\). There are at most \(C_{\rm dyad}(\kappa L+1)\) boxes, +where \(C_{\rm dyad}\) is absolute: this bounds the number of first +coordinate dyads, and the product condition leaves only a bounded +number of second coordinate dyads for each. + +The rough-pair bound holds on a slightly wider fixed range of boxes, +which includes all the boxes just selected. +\begin{lemma}[An upper bound for rough pairs]\label{con:rough-pairs} +Let \(\mathcal B=[M_1,2M_1)\times[M_2,2M_2)\) be any dyadic box +such that +\begin{equation}\label{con:rough-box-range} + x/4\le M_1M_2\le2x,\qquad + x^{0.46}/2\le M_i\le2x^{0.54}\quad(i=1,2). +\end{equation} +Then +\begin{equation}\label{con:rough-pair-bound} + \sum_{\substack{(m,n)\in\mathcal B\\P^-(m)>W,\ P^-(n)>W}}w(mn) + \le C_1X_0V(W)^2, +\end{equation} +where \(C_1\) depends at most on \(M,\Psi\), and is independent +of \(K,\kappa,b,\eps\). +\end{lemma} + +\begin{proof} +Fix a contributing group integer \(r\le x^{0.11}\), and set +\(k_r=cr\), \(z'=x^{0.002}\). The corresponding pairs satisfy +\(mn\equiv1\pmod{k_r}\). At every odd prime \(j\le z'\) with +\(j\nmid r\), they avoid +\begin{equation}\label{con:pair-bad} + mn\equiv1\pmod j, + \qquad\text{and, if }j\le W,\qquad mn\equiv0\pmod j. +\end{equation} +Indeed, \(mn-1=k_rQ\) with \(Q>x^{0.9}>z'\), while both factors +are \(W\)-rough. We impose no further condition at primes dividing +\(k_r\) and may omit the fixed progression for this upper bound. + +For a squarefree product \(\ell\) of the designated odd primes, +put +\[ + \rho(\ell)=\prod_{j\mid\ell} + \bigl(j-1+(2j-1)\one_{\{j\le W\}}\bigr). +\] +The conditions \(mn=1\) and \(mn=0\) modulo \(j\) are disjoint +and have respectively \(j-1\) and \(2j-1\) residue pairs. There +are \(\varphi(k_r)\) pairs with product one modulo \(k_r\). +The Chinese remainder theorem therefore gives exactly +\(\varphi(k_r)\rho(\ell)\) pairs modulo \(k_r\ell\) satisfying +the base congruence and all the forbidden conditions at primes +dividing \(\ell\). Their count in the box is +\[ + X'g(\ell)+E(\ell),\qquad + X'=M_1M_2\frac{\varphi(k_r)}{k_r^2},\qquad + g(\ell)=\frac{\rho(\ell)}{\ell^2}, +\] +where +\begin{equation}\label{con:lattice-error} + E(\ell)\ll\varphi(k_r)\rho(\ell) + \left(\frac{M_1+M_2}{k_r\ell}+1\right). +\end{equation} +This formula also holds at \(\ell=1\). + +The local bounds \(g(j)\le7/9\) and \(g(j)\le3/j\) verify +the gap and block-sum hypotheses of Lemma~\ref{lem:block}, with +absolute constants. Its upper bound with \(H=2\) uses +\(\ell\le(z')^{10}=x^{0.02}\). For squarefree \(\ell\), +\[ + \rho(\ell)\le\prod_{j\mid\ell}3j\le\tau(\ell)^2\ell, +\] +where \(\tau\) is the divisor function. Since \(\varphi(k_r)\le k_r\) +and \(k_r=cr\le4r\), \eqref{con:lattice-error} gives +\[ + |E(\ell)|\ll\tau(\ell)^2(M_1+M_2+r\ell). +\] +The needed divisor sum follows elementarily. Let \(\tau_4(n)\) +count ordered decompositions of \(n\) into four positive factors. +Prime by prime, \(\tau(n)^2\le\tau_4(n)\); summing over the +first three factors therefore yields, for \(Y\ge2\), +\[ + \sum_{n\le Y}\tau(n)^2 + \le Y\left(\sum_{a\le Y}\frac1a\right)^3 + \ll Y\bigl(\log(2Y)\bigr)^3. +\] +Consequently, taking \(Y=x^{0.02}\) and using +\(M_1+M_2\le4x^{0.54}\) from \eqref{con:rough-box-range}, we obtain +\begin{equation}\label{con:lattice-sum} + \sum_{\ell\le x^{0.02}}|E(\ell)| + \ll (x^{0.56}+rx^{0.04})L^3 + =o\bigl((x/r)L^{-10}\bigr) +\end{equation} +uniformly for \(r\le x^{0.11}\). The ratios of the two displayed +power terms to \(x/r\) are at most \(x^{-0.33}\) and +\(x^{-0.74}\), respectively, and absorb the logarithmic factor. + +For \(j\le W\), the surviving sieve factor is +\[ + 1-\frac{3j-2}{j^2} + =\left(1-\frac1j\right)^3 + \left(1-\frac1{(j-1)^2}\right), +\] +and for \(W0.54\), and \(D_{0.4}(s)=1/s\). Its comparison +coefficient is therefore +\[ + B_{\rm pr}(m)=\frac{\one_{\{P^-(m)>W\}}}{V(W)\log m}. +\] +The identity +\[ + \begin{aligned} + \one_{\{m\text{ prime}\}}\one_{\{n\text{ prime}\}} + -B_{\rm pr}(m)B_{\rm pr}(n) + &=(\one_{\{m\text{ prime}\}}-B_{\rm pr}(m))\one_{\{n\text{ prime}\}}\\ + &\quad+B_{\rm pr}(m)(\one_{\{n\text{ prime}\}}-B_{\rm pr}(n)) + \end{aligned} +\] +allows two successive applications, interchanging the variables in +the second. The companion coefficient is \(W\)-rough and bounded +by one for large \(x\) in both cases. All coefficients retain +their box restrictions. The total error over the boxes is +\(O(xL^{-5})=o(X_0/L)\). + +Since \(\log m,\log n\ge0.46L\), Lemma~\ref{con:rough-pairs} +bounds the resulting sum in one box by +\((0.46)^{-2}C_1X_0/L^2\). The ordered prime-pair sum includes +prime squares and may count other semiprimes twice, which is harmless +for an upper bound. Summing the boxes gives +\begin{equation}\label{con:balanced-cost} + \sum_{\substack{P^-(d)>x^{1/2-\kappa}\\d\text{ composite}}}w(d) + \le C_2(\kappa+L^{-1})\frac{X_0}{L}+o(X_0/L), +\end{equation} +where \(C_2=(0.46)^{-2}C_1C_{\rm dyad}\) depends at most on +\(M,\Psi\), and is independent of \(K,\kappa,b,\eps\). + +\subsection{Choice of parameters and completion} + +\begin{proof}[Completion of the proof of Proposition~\ref{prop:controlled}] +Fix \(M,c,u,\Psi\) first, and hence the constants +\(\mathfrak S_M\) and \(C_2\). Choose +\(0<\kappa<0.01\) such that +\[ + C_2\kappa<\mathfrak S_M/4, +\] +and then choose \(0<\eps<\kappa\). Choose \(b>0\) sufficiently +small for the sieve and density estimates, and so that +\[ + \frac{\e^{-\gamma_E}}b + (C_s\e^{-c_s/b}+C_d\e^{-c_d/b})<0.2. +\] +Fix the partition used in the least-factor subtraction. Only +finitely many Type~II parameter sets now occur. Choose \(K\) +sufficiently large for all of them, and choose the exponents +afterward, for example +\[ + a_i=0.1+\frac{0.1i}{K+1}\qquad(1\le i\le K). +\] +This order is permitted by Lemma~\ref{lem:marked-coefficient-transfer} +and Proposition~\ref{prop:rough-type-ii}, hence by Lemma~\ref{con:type-ii}. + +Every supported prime is counted by \(U_\kappa\). +Subtracting \eqref{con:balanced-cost} from \eqref{con:post-bin}, +and then taking \(x\) sufficiently large for the fixed parameters, +gives +\[ + \frac{L}{\mathfrak S_M X_0}\sum_{d\text{ prime}}w(d) + \ge0.9-0.2-\frac14+o(1)>\frac14. +\] +By \eqref{con:total-mass}, \(J_0\ge1\), and \(w\le B_K\), +the number of distinct supported primes is therefore +\(\gg_{M,c,u}x/L^2\). Their support in \eqref{con:weight} +ensures \(xx^{0.9}\) prime. Every prime factor of +\(r\) belongs to a group in \eqref{eq:marked-prime-groups}, hence lies in +\((\exp(L^{0.1}),\exp(L^{0.3}))\) for all sufficiently large +\(x\). These are precisely the assertions of the proposition. +\end{proof} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/vendor/texmf/tex/plain/rsfs/scrload.tex b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/vendor/texmf/tex/plain/rsfs/scrload.tex new file mode 100644 index 0000000000000000000000000000000000000000..db1bf9349ffe8951e536d99d19f97bc3709ac3a4 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/build/vendor/texmf/tex/plain/rsfs/scrload.tex @@ -0,0 +1,8 @@ + +\font\tenscr=rsfs10 % +\font\sevenscr=rsfs7 % +\font\fivescr=rsfs5 % +\skewchar\tenscr='177 \skewchar\sevenscr='177 \skewchar\fivescr='177 +\newfam\scrfam \textfont\scrfam=\tenscr \scriptfont\scrfam=\sevenscr +\scriptscriptfont\scrfam=\fivescr +\def\scr{\fam\scrfam} diff --git a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a7bf0d3d837c550edc5aa95edaffc3ab7d720151 --- /dev/null +++ b/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:671ac4b3ece879f614ed8f4b782e61a07805e533f385a694fe4d7afbda3c9365 +size 891737 diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/README.md b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..5cf63933c08438b0994361f2688402848f939fd6 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [Product-projection localization and the QAC⁰ parity lower bound](paper.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026, + author = {{OpenAI}}, + title = {{Product-projection localization and the $\mathrm{QAC}^0$ parity lower bound}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf}{OAI:Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/main.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..d00b0d0ec644efa120f31f54faab963cc01a4bd4 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/main.tex @@ -0,0 +1,25 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{Product-projection localization and the\\ +\texorpdfstring{$\QAC$}{QAC0} parity lower bound} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that constant-depth quantum circuits with arbitrary one-qubit +and unbounded-arity Toffoli gates cannot compute parity with any fixed +positive worst-case advantage using polynomially many qubits. Ancillas +start in zero, only one output qubit is measured, and all final garbage +is unrestricted. This resolves Moore's parity conjecture in the +measured-output model. +\end{abstract} +\input{sections/introduction} +\input{sections/localization} +\input{sections/polynomial} +\input{sections/depth_reduction} +\input{sections/bootstrap} +\input{sections/parity} +\input{sections/consequences} +\input{references} +\end{document} diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/preamble.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..3bddecb5e444608da30422f59715fd45ff2d94e3 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/preamble.tex @@ -0,0 +1,48 @@ +\input{glyphtounicode} +\input{glyphtounicode-cmex} +\pdfgentounicode=1 +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern,microtype} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage[margin=1.05in]{geometry} +\usepackage[dvipsnames]{xcolor} +\usepackage{hyperref} +\usepackage{xurl} +\usepackage{aliascnt} +\usepackage[nameinlink,capitalise,noabbrev]{cleveref} +\hypersetup{colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue, +urlcolor=MidnightBlue,pdfauthor={OpenAI}, +pdftitle={Product-projection localization and the QAC0 parity lower bound}} +\setlength{\emergencystretch}{2em} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem} +\newaliascnt{lemma}{theorem} +\newtheorem{lemma}[lemma]{Lemma} +\aliascntresetthe{lemma} +\newaliascnt{proposition}{theorem} +\newtheorem{proposition}[proposition]{Proposition} +\aliascntresetthe{proposition} +\newaliascnt{corollary}{theorem} +\newtheorem{corollary}[corollary]{Corollary} +\aliascntresetthe{corollary} +\theoremstyle{definition} +\newaliascnt{definition}{theorem} +\newtheorem{definition}[definition]{Definition} +\aliascntresetthe{definition} +\theoremstyle{remark} +\newaliascnt{remark}{theorem} +\newtheorem{remark}[remark]{Remark} +\aliascntresetthe{remark} +\newcommand{\QAC}{\mathrm{QAC}^0} +\newcommand{\cU}{\mathcal U} +\newcommand{\cB}{\mathcal B} +\newcommand{\ket}[1]{\lvert #1\rangle} +\newcommand{\bra}[1]{\langle #1\rvert} +\newcommand{\proj}[1]{\ket{#1}\!\bra{#1}} +\newcommand{\norm}[1]{\left\lVert #1\right\rVert} +\newcommand{\E}{\mathbb E} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\parity}{PARITY} diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/references.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..732d2e6399b1bb94b98f244bf5a06ec469cbaecc --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/references.tex @@ -0,0 +1,182 @@ +\begin{thebibliography}{FFGHZ06} + +\bibitem[Ajt83]{Ajtai1983} +Mikl\'os Ajtai. +\newblock $\Sigma^1_1$-formulae on finite structures. +\newblock \emph{Annals of Pure and Applied Logic}, 24(1):1--48, 1983. +\newblock \href{https://doi.org/10.1016/0168-0072(83)90038-6}{doi:10.1016/0168-0072(83)90038-6}. + +\bibitem[ADOY25]{ADOY2025} +Anurag Anshu, Yangjing Dong, Fengning Ou, and Penghui Yao. +\newblock On the Computational Power of $\mathsf{QAC}^{0}$ with Barely +Superlinear Ancillae. +\newblock In \emph{Proceedings of the 57th Annual ACM Symposium on Theory of +Computing (STOC 2025)}, pages 1476--1487, 2025. +\newblock \href{https://doi.org/10.1145/3717823.3718189}{doi:10.1145/3717823.3718189}. +Full version: \href{https://arxiv.org/abs/2410.06499v4}{arXiv:2410.06499v4}, +21 December 2025. + +\bibitem[AM23]{AnshuMetger2023} +Anurag Anshu and Tony Metger. 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Logarithmic-Qubit State}. +\newblock 2026. \href{https://arxiv.org/abs/2609.17408v1}{arXiv:2609.17408v1}, +15 September 2026. + +\bibitem[H\aa s86]{Hastad1986} +Johan H\aa stad. +\newblock Almost optimal lower bounds for small depth circuits. +\newblock In \emph{Proceedings of the Eighteenth Annual ACM Symposium on +Theory of Computing}, pages 6--20, 1986. +\newblock \href{https://doi.org/10.1145/12130.12132}{doi:10.1145/12130.12132}. + +\bibitem[H\v S05]{HoyerSpalek2005} +Peter H\o yer and Robert \v Spalek. +\newblock Quantum Fan-out is Powerful. +\newblock \emph{Theory of Computing}, 1(5):81--103, 2005. +\newblock \href{https://doi.org/10.4086/toc.2005.v001a005}{doi:10.4086/toc.2005.v001a005}. + +\bibitem[JTVW26]{JTVW2026} +Malvika Raj Joshi, Avishay Tal, Francisca Vasconcelos, and John Wright. +\newblock Improved Lower Bounds for $\mathsf{QAC}^{0}$. +\newblock In \emph{Proceedings of the 58th Annual ACM Symposium on Theory of +Computing (STOC 2026)}, pages 2199--2209, 2026. 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+\newblock 1999. \href{https://arxiv.org/abs/quant-ph/9903046v3}{arXiv:quant-ph/9903046v3}, +17 March 1999. + +\bibitem[NPVY24]{NPVY2024} +Shivam Nadimpalli, Natalie Parham, Francisca Vasconcelos, and Henry Yuen. +\newblock On the Pauli Spectrum of $\mathsf{QAC}^{0}$. +\newblock In \emph{Proceedings of the 56th Annual ACM Symposium on Theory of +Computing (STOC 2024)}, pages 1498--1506, 2024. +\newblock \href{https://doi.org/10.1145/3618260.3649662}{doi:10.1145/3618260.3649662}. +Full version: \href{https://arxiv.org/abs/2311.09631v4}{arXiv:2311.09631v4}, +17 July 2024. + +\bibitem[O'D03]{ODonnell2003Noise} +Ryan O'Donnell. +\newblock \emph{Computational Applications of Noise Sensitivity}. +\newblock Ph.D. thesis, Massachusetts Institute of Technology, 2003. +\newblock \url{https://www.cs.cmu.edu/~odonnell/papers/thesis.pdf}. + +\bibitem[Ope26]{OpenAI2026Trajectories} +OpenAI. +\newblock \emph{Regular trajectories, pruning and quantum parity}. +\newblock OpenAI Math Release preprint 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Parity. +\newblock In \emph{12th Innovations in Theoretical Computer Science Conference +(ITCS 2021)}, volume 185 of \emph{Leibniz International Proceedings in +Informatics}, pages 32:1--32:20, 2021. +\newblock \href{https://doi.org/10.4230/LIPIcs.ITCS.2021.32}{doi:10.4230/LIPIcs.ITCS.2021.32}. +Full version: \href{https://arxiv.org/abs/2008.07470v3}{arXiv:2008.07470v3}, +30 November 2020. + +\bibitem[She09]{Sherstov2009} +Alexander A. Sherstov. +\newblock Approximate inclusion-exclusion for arbitrary symmetric functions. +\newblock \emph{Computational Complexity}, 18(2):219--247, 2009. +\newblock \href{https://web.cs.ucla.edu/~sherstov/pdf/incl-excl.pdf}{Author manuscript}. + +\bibitem[XL26]{XuLi2026Fanout} +Boyan Xu and Lvzhou Li. +\newblock \emph{Fanout Complexity of Symmetric Boolean Functions in +$\mathsf{QAC}^0$}. +\newblock 2026. \href{https://arxiv.org/abs/2609.05153v1}{arXiv:2609.05153v1}, +4 September 2026. + +\bibitem[Yao85]{Yao1985} +Andrew Chi-Chih Yao. +\newblock Separating the polynomial-time hierarchy by oracles. +\newblock In \emph{Proceedings of the 26th Annual Symposium on Foundations +of Computer Science (FOCS 1985)}, pages 1--10, 1985. +\newblock \href{https://doi.org/10.1109/SFCS.1985.49}{doi:10.1109/SFCS.1985.49}. + +\end{thebibliography} diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/bootstrap.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/bootstrap.tex new file mode 100644 index 0000000000000000000000000000000000000000..49a87cf240228074af96c0f331e1ad61e1b00952 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/bootstrap.tex @@ -0,0 +1,193 @@ +\section{The finite scale induction} +\label{sec:bootstrap} + +The preceding depth gives localization for projections with small +support. We now extend that estimate to arbitrary product projections. +The extension uses the inverse circuit at a larger mismatch scale. +The main constraint is how far that scale can move: to prove +$\cB_d(s,t)$ from $\cB_d(t,b)$, we need +$tt>a$, +they give exactly the inequalities required by the polynomial and +small-support estimates: +\begin{equation} +\label{eq:step-parameters} + 0\le s\frac{a+b}{2}. +\end{equation} +Fix \(U\in\cU_d(N)\) and counts \(M,D\), and put +\[ + P=[M=0],\qquad H=[M\ge N^t],\qquad + C=UDU^\dagger,\qquad Q=[C=0]=U[D=0]U^\dagger. +\] +Let \(p(x)=\sum_kc_kx^k\) be the polynomial in +\Cref{lem:polynomial} for \(a,b,u\). We will first control +$Hp(C)P$ by small-support localization. Squaring the desired norm +will then place the remaining high-$C$ error on $PQ$, where the +larger-scale hypothesis applies after inversion. + +Each count summand is a rank-one projection on one qubit. +Expanding \(D^k\) gives at most \(N^k\) ordered monomials, each a +product projection on at most \(k\) qubits: repeated indices collapse +by idempotence, while different indices commute. +For \(k\le\deg p\le N^u\), \Cref{lem:small-support} applies to +every such monomial with the same fixed exponents \(u,w,t\). +Since \(p(C)=Up(D)U^\dagger\), its coefficient bound gives +\begin{equation} +\label{eq:polynomial-transition} + \norm{Hp(C)P} + \le e^{-N^w}\sum_k|c_k|N^k + \le e^{N^u-N^w}. +\end{equation} +The constant term causes no difficulty: it is a multiple of \(I\), +and \(HP=0\). Also, \(\norm C\le N\) gives +\begin{equation} +\label{eq:polynomial-norm} + \norm{p(C)}\le\sum_k|c_k|N^k\le e^{N^u}. +\end{equation} + +The polynomial need not approximate \(Q\) on all of \(\mathcal H_N\). +Squaring the desired norm makes the approximation error occur as +\((Q-p(C))PQ\), rather than \((Q-p(C))P\): +\begin{align} +\norm{HQP}^2 + &=\norm{HQPQH}\notag\\ + &\le \norm{Hp(C)PQH}+\norm{H(Q-p(C))PQH}\notag\\ + &\le \norm{Hp(C)P}+\norm{(Q-p(C))PQ}. +\label{eq:square} +\end{align} +Only the leftmost \(Q\) was replaced in this decomposition. No commutation +of \(P\) with \(C\) or \(Q\) is involved. + +Split the last error at the spectral threshold \(N^b\) of \(C\). +The spectral projections of \(C\) commute with \(Q\) and \(p(C)\). +On \([Cu\), \(t>u\), and +\(a,w,t>s\). This remains true for \(s=0\), when the comparison +quantity is a fixed positive constant. Taking the square root proves +\(\cB_d(s,t)\). All sufficient lower bounds on \(N\) depend only +on the fixed parameters and \(d\), so the conclusion is uniform. +\end{proof} + +We now choose a finite chain of scales within this interval. Each step +will advance the mismatch exponent by almost the current gap $t-s$. +Keeping at least half of the initial gap for $O(1/(t-s))$ steps will +carry that exponent above one, where the tail projection vanishes. +A slack of order $(t-s)^2$ per step is small enough to preserve the gap. + +\begin{proof}[Proof of \Cref{thm:localization}] +The case \(d=0\) is \Cref{lem:base}. Fix \(d\ge1\), and assume the +theorem at depth \(d-1\) for every fixed exponent pair. +It remains to prove \(\cB_d(s,t)\) for fixed \(0\le s1\) is immediate. +Put +\[ + g=t-s,\qquad \eta=\frac{g^2}{100},\qquad + s_0=s,\quad t_0=t, +\] +and define +\begin{equation} +\label{eq:scale-recurrence} + s_{i+1}=t_i,\qquad t_{i+1}=2t_i-s_i-3\eta. +\end{equation} +Writing \(g_i=t_i-s_i\), we have +\[ + g_i=g-3i\eta,\qquad t_{i+1}-t_i=g_{i+1}. +\] +Let \(K=\lceil2/g\rceil\). Since \(0g/2\) for \(i\le K\), and +\[ + t_K=t+\sum_{i=1}^K g_i>t+Kg/2\ge t+1>1. +\] +Let \(J\le K\) be the first index for which \(t_J>1\). +This index and every exponent in the chain are independent of \(N\). +The assertion \(\cB_d(s_J,t_J)\) is trivial. + +\begin{figure}[htbp] +\centering +\[ +\underbrace{\cB_d(s_0,t_0)}_{\text{target}} +\ \Longleftarrow\ +\cB_d(s_1,t_1) +\ \Longleftarrow\ \cdots\ \Longleftarrow\ +\underbrace{\cB_d(s_J,t_J)}_{t_J>1:\ \text{trivial}} . +\] +\caption{The same-depth induction runs from right to left along a +finite chain; \(J\) is the first index with \(t_J>1\). +Each implication is \Cref{lem:backward} and also uses +the already established localization theorem at depth \(d-1\). +The inverse circuit is substituted only into the universal statement +at the next pair, after that statement has been proved.} +\label{fig:scales} +\end{figure} + +Suppose now that $\cB_d(s_{i+1},t_{i+1})$ has been proved, +where $i0$, and $0<\delta<1$. +For all sufficiently large $n$, every $n$-qubit reduced state $\rho$ +prepared from zero inputs by an allowed depth-at-most-$d$ circuit on +at most $n^c$ total qubits satisfies +\[ + \mathcal F_n(\rho)\frac1{80k}. +\] +The remaining qubits may be discarded in arbitrary final states. +\end{corollary} + +\begin{proof} +Retain all $N$ preparation qubits, including those eventually discarded, +and write $\ket\psi=U\ket{0^N}$ and $Q=\proj\psi$. Thus $Q$ is the +conjugate by $U\in\cU_d(N)$ of a product projection. For an output string +$y$, let $P_y$ fix the $n$ output qubits to $y$ and act identically on the +remaining register, and let $M_y$ count mismatches from $y$ on the outputs. +Then $P_y=[M_y=0]$ and $\norm{P_y\psi}^2=p_y$. Fix $0\frac1{8k}. +\] +All errors here are uniform for $k\le n/2$ as $k\to\infty$. +The factor four counts both complementary strings in the definition +of felinity. + +Apply the same block circuit to $\rho$, and call its new-qubit state +$\sigma$. Trace distance contracts under a unitary and partial trace. +Also, for any two states $\tau,\tau'$ with computational-basis +probabilities $q,q'$, the definition gives +\[ + |\mathcal F_\ell(\tau)-\mathcal F_\ell(\tau')| + \le4\sum_y|q_y-q'_y| + \le8\operatorname{TD}(\tau,\tau'). +\] +Thus $\operatorname{TD}(\rho,\proj{D_k^n})\le1/(80k)$ would imply +\[ + \mathcal F_\ell(\sigma)>\frac1{8k}-\frac8{80k} + =\frac1{40k}. +\] +For large $k$, $\ell\ge\sqrt k$, while $k\ge n^\delta$ gives +$n\le\ell^{2/\delta}$. The preparation of $\sigma$ therefore has fixed +depth and polynomially many total qubits as a function of $\ell$. +But $\ell^2/k\to\infty$, so its felinity exceeds $\ell^{-2}$ eventually, +contrary to the first assertion applied with target size $\ell$ and +exponent $a=2$. Since $k\ge n^\delta\to\infty$, all thresholds can be +chosen uniformly over the allowed $k$. +\end{proof} + +In particular, no such family can assign inverse-polynomial +probabilities to both a string and its complement: their two terms +contribute $4p_y p_{\bar y}$ to $\mathcal F_n(\rho)$. + +The polynomial bound in \Cref{cor:state-preparation} is measured in the +number $n$ of target qubits. Gretta, Gupta, and +Joshi~\cite[Theorem~1.1]{GGJ2026StatePreparation} show that every $n$-qubit +pure state can be prepared exactly at constant depth with $2^{O(n)}$ +ancillas, all returned to zero. + +\subsection{Scalar-output Fourier concentration} + +Gretta, Gupta, and Joshi~\cite[Conjecture~1 and Corollary~1.2]{GGJ2026} +formulate a qualitative Quantum-LMN statement and establish its +equivalence with the parity lower-bound question. Our localization +theorem gives this Fourier-concentration statement directly. +For an $n$-input circuit $C$ on $N$ total qubits with output qubit $t$, +define its expected measured sign and uniform-input Fourier coefficients by +\begin{align*} + f_C(x)&=\bra{x,0^{N-n}}C^\dagger Z_tC\ket{x,0^{N-n}}\in[-1,1],\\ + \widehat f_C(S)&=2^{-n}\sum_{x\in\{0,1\}^n} + f_C(x)(-1)^{\sum_{i\in S}x_i}. +\end{align*} +Write $W_{\ge k}[f_C]=\sum_{|S|\ge k}|\widehat f_C(S)|^2$. +This is the absolute squared Fourier tail, not a fraction of the +total weight $\mathbb E_x f_C(x)^2\le1$. + +\begin{corollary}[Qualitative scalar-output Quantum-LMN] +\label{cor:quantum-lmn} +For every fixed nonnegative integer $d$, there is a positive function +$\gamma_d$ on the positive integers, depending only on $d$, such that +$\gamma_d(k)/k^A\to\infty$ for every fixed $A>0$, and every allowed +depth-at-most-$d$ circuit $C$ on $n$ inputs and $N$ total qubits, +every designated output $t$, and every integer $1\le k\le n$ satisfy +\[ + W_{\ge k}[f_C]\le\frac{N}{\gamma_d(k)}. +\] +In particular, fix $c\ge1$, $0<\delta\le1$, and $B>0$. For all +sufficiently large $n$, uniformly over these circuits with $N\le n^c$, +\[ + W_{\ge\lceil n^\delta\rceil}[f_C]\le n^{-B}. +\] +The sufficient threshold depends only on $d,c,\delta,B$. +\end{corollary} + +\begin{proof} +Fix a circuit $C$ and let $Q=C^\dagger O_0C$, where $O_0$ projects its +output qubit onto $\ket0$. For $h\in\{0,1\}^n$, let $\xi_h$ be the +Hadamard product vector from the proof of \Cref{thm:parity}, and let +$M_h$ count mismatches from its factors on the $n$ input qubits. +For $S\subseteq[n]$, write $\mathbf 1_S$ for its indicator string. +The character calculation in \eqref{eq:parity-average}, with +$\bar h$ replaced by $h\oplus\mathbf 1_S$, gives +\[ + \widehat f_C(S) + =2\E_h\bra{\xi_{h\oplus\mathbf 1_S},0}Q\ket{\xi_h,0} + \qquad(S\ne\varnothing), +\] +since $f_C(x)=2\bra{x,0}Q\ket{x,0}-1$. +For fixed $h$, the vectors $\ket{\xi_{h\oplus\mathbf 1_S},0}$ are +orthonormal as $S$ varies, and their $M_h$ counts are exactly $|S|$. +Jensen's and Bessel's inequalities therefore yield +\begin{align} + W_{\ge k}[f_C] + &\le4\E_h\sum_{|S|\ge k} + \left|\bra{\xi_{h\oplus\mathbf 1_S},0}Q\ket{\xi_h,0}\right|^2 + \notag\\ + &\le4\sup_h\norm{[M_h\ge k]Q[M_h=0]}^2. + \label{eq:fourier-localization} +\end{align} +For any fixed $0<\sigma<\tau$, \Cref{thm:localization} bounds this +by $4e^{-2N^\sigma}$ whenever $N$ is sufficiently large and +$k\ge N^\tau$. The threshold depends only on $d,\sigma,\tau$. + +It remains to obtain one function of $k$ valid for every circuit size. +For each positive integer $k$, define +\[ + b_d(k)=\sup\frac{W_{\ge k}[f_C]}{N}, +\] +where the supremum ranges over all allowed depth-at-most-$d$ circuits, +all designated outputs, and all input and total sizes $k\le n\le N$. +Parseval's identity gives $W_{\ge k}[f_C]\le\E_x f_C(x)^2\le1$, +so $0\le b_d(k)\le1/k$. +Fix $A\ge1$. If $N\ge k^A$, then $W_{\ge k}[f_C]/N\le k^{-A}$. +For the remaining sizes $k\le N0$. +Thus the positive function +\[ + \gamma_d(k)=\frac1{b_d(k)+e^{-k}} +\] +satisfies $\gamma_d(k)/k^B\to\infty$ for every $B>0$, and the definition +of $b_d$ gives $W_{\ge k}[f_C]\le N/\gamma_d(k)$. +Finally, choose $B'>(c+B)/\delta$ in this superpolynomial growth +property to obtain the asserted cutoff consequence. +\end{proof} + +\subsection{Majority and symmetric functions} + +The reductions of Xu and Li turn the parity lower bound into a +worst-case lower bound for majority and broader symmetric families. +For a symmetric \(f_n:\{0,1\}^n\to\{0,1\}\), let \(f_{n,i}\) be its +value on inputs of Hamming weight \(i\), and define its transition radius by +\[ + \rho(f_n)=\max_{\substack{0\le i0\), or +\(\widetilde{\deg}_{1/3}(f_n)=\Omega(n^{1/2+\eta})\) for some fixed +\(\eta>0\), for all sufficiently large \(n\). For every fixed +\(\gamma>0\), no nonuniform family in the circuit model of +\Cref{thm:parity}, with fixed depth and polynomially many total qubits, +computes \(f_n\) with probability at least +\(1/2+1/(\log n)^\gamma\) on every input for all sufficiently large \(n\). +In particular, this excludes strict majority +\(\operatorname{MAJ}_n(x)=\mathbf 1\{|x|>n/2\}\), whose transition radius +is \(\lceil n/2\rceil\). +\end{corollary} + +\begin{proof} +Xu and Li's Theorem~16 and Corollaries~2--3 +\cite{XuLi2026Fanout} turn a family satisfying either hypothesis and the +stated success bound into an exact, clean implementation of fanout on +\(n\) targets, with constant entangling depth and polynomially many +ancillas. Their circuit convention~\cite[Section~2.3]{XuLi2026Fanout} +requires the measured output to be an initially zero ancilla; appending one fresh zero qubit and a CNOT from our designated +output preserves its measurement distribution, even with arbitrary +garbage. Their entangling depth \(D\) uses at most \(2D+1\) of our layers. +Thus the reduction stays within fixed depth and polynomial total qubits. +Hadamard conjugation on the fanout data wires gives the exact parity +unitary; a zero target and its final measurement contradict +\Cref{thm:parity}. Finite exceptional input lengths do not affect this +circuit-family contradiction. +\end{proof} + +There is consequently a strict separation between the Boolean families +computed with worst-case success at least \(2/3\) in our model and in the +matched model obtained by adjoining unit-cost reversible unbounded-arity +threshold gates +\[ + \ket{x,b}\longmapsto + \ket{x,b\mathbin{\oplus}\mathbf 1\{|x|\ge t\}}, + \qquad t\in\{0,\ldots,n\}, +\] +while retaining the same depth, disjoint-support, qubit, ancilla, and +output conventions. The enlarged model contains the original one and +computes strict majority exactly with one such gate at +\(t=\lfloor n/2\rfloor+1\), whereas +Corollary~\ref{cor:symmetric-functions} excludes even worst-case success \(2/3\) +for majority in the original model. + +\subsection{High average accuracy for majority} + +The preceding obstruction assumes an advantage on every input. The +high-average-accuracy consequence of Gretta, Gupta, and +Joshi~\cite[Corollary~4.6]{GGJ2026} also follows directly from our +Fourier-concentration bound: majority retains a polynomial Fourier tail +at a polynomially growing cutoff. + +\begin{corollary}[High-average-accuracy majority obstruction] +\label{cor:majority-average} +For every fixed \(\delta>0\), no nonuniform family \((C_n)\) in the +circuit model of \Cref{thm:parity}, with fixed depth and polynomially +many total qubits, satisfies +\[ + 2^{-n}\sum_{x\in\{0,1\}^n} + f_{C_n}(x)(-1)^{\operatorname{MAJ}_n(x)}\ge1-n^{-\delta} +\] +for all sufficiently large \(n\). Here \(f_{C_n}(x)\in[-1,1]\) is +the expected measured sign: the probability of output zero minus the +probability of output one. Equivalently, no such family computes strict +majority with average success at least \(1-\tfrac12 n^{-\delta}\), +where the average is over uniform inputs and the output measurement. +\end{corollary} + +\begin{proof} +Suppose such a family exists, restrict to odd $n$, and put +$g_n(x)=(-1)^{\operatorname{MAJ}_n(x)}$ and +$\delta_0=\min\{\delta,1/4\}$. Since $|f_{C_n}|\le1$ and $g_n^2=1$, +the asserted correlation gives +\[ + \|f_{C_n}-g_n\|_2^2 + \le 2\bigl(1-\mathbb E_x f_{C_n}(x)g_n(x)\bigr) + \le 2n^{-\delta_0}, +\] +where the norm uses the uniform probability measure. + +We use the following finite-size majority tail estimate: +for $\varepsilon\ge n^{-1/3}$ and sufficiently large odd $n$, +\[ + W_{\ge k}[g_n]>\varepsilon + \quad\text{whenever $k$ is odd and }1\le k\le\frac1{7\varepsilon^2}. +\] +This is \cite[Theorem~3.5.6, p.~50]{ODonnell2003Noise}. +Set $\varepsilon=4n^{-\delta_0}$ and let $k_n$ be the largest odd +integer at most $n^{2\delta_0}/112$. These parameters satisfy the +estimate's hypotheses for all sufficiently large odd $n$. +Parseval's identity and the triangle inequality for the Fourier +coefficients at degrees at least $k_n$ now give +\[ + \sqrt{W_{\ge k_n}[f_{C_n}]} + \ge \sqrt{W_{\ge k_n}[g_n]}-\|f_{C_n}-g_n\|_2 + > (2-\sqrt2)n^{-\delta_0/2}. +\] +On the other hand, $k_n\ge\lceil n^{\delta_0}\rceil$ eventually. +Applying \Cref{cor:quantum-lmn} at the latter cutoff, with +$B=\delta_0+1$, yields +$W_{\ge k_n}[f_{C_n}]\le n^{-(\delta_0+1)}$, a contradiction. +\end{proof} + +This conclusion does not assert a constant average-case gap or exclude +\(1-1/\operatorname{polylog}n\) correlation. diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/depth_reduction.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/depth_reduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..f1430f1f57c756e6d8edb83ed7588eb0d0506636 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/depth_reduction.tex @@ -0,0 +1,194 @@ +\section{Depth reduction for small-support projections} +\label{sec:depth} + +We begin the proof of \Cref{thm:localization} at depth zero. We then +show that the full localization statement at depth $d-1$ controls +projections of small support at depth $d$. All large-$N$ thresholds +in this section depend only on the displayed exponents and the depth, +not on the circuits, counts, or product projections. + +\begin{lemma}[Depth zero]\label{lem:base} +For every fixed $0\le s0$, +\begin{align*} + \norm{[M\ge r]QP}^2 + &\le \prod_{i\in I_*}(1-p_i)\,\Pr(K\ge r)\\ + &\le 2^{-r}\prod_{i\in I_*}(1-p_i)(1+p_i) + \le 2^{-r}. +\end{align*} +The middle inequality is Markov's inequality applied to $2^K$. +The same bound is immediate if the amplitude vanishes. Taking +$r=N^t$ gives +$\norm{[M\ge N^t]QP}\le\exp(-\tfrac12(\log 2)N^t)$, +which is at most $\exp(-N^s)$ for all sufficiently large $N$. +\end{proof} + +The depth-zero estimate incorporates both the chance of producing a +mismatch and the amplitude lost under projection. We next extend the +preceding-depth hypothesis from one mismatch pattern to all patterns +below a small threshold. This is the input needed to control successive +factors in the layer expansion. + +\begin{lemma}[Propagation from low counts]\label{lem:propagation} +Fix $d\ge1$ and assume $\cB_{d-1}(s,t)$ for every fixed pair +$0\le s\max\{\alpha,\sigma\}$, all sufficiently large $N$ +satisfy +\begin{equation}\label{eq:low-count-propagation} + \norm{[M\ge N^\beta]B[M\alpha,\sigma$. +\end{proof} + +We can now remove one reflection layer when the tested projection has +small support. The expansion produces three ordered factors; the +low-count estimate just proved bounds the transitions between them. + +\begin{lemma}[Small-support depth reduction]\label{lem:small-support} +Fix $d\ge1$ and assume $\cB_{d-1}(s,t)$ for every fixed pair +$0\le s\max\{u,w\}$, +all sufficiently large $N$ satisfy +\begin{equation}\label{eq:small-support} + \norm{[M\ge N^t]UAU^\dagger[M=0]} + \le\exp(-N^w) +\end{equation} +uniformly for every $U\in\cU_d(N)$, every count $M$, and +every product projection $A$ supported on at most $N^u$ qubits. +\end{lemma} + +\begin{proof} +The claim is immediate when $t>1$, so suppose $t\le1$. +Write +\[ + U=VR_1L_0,\qquad + V=(L_dR_d)\cdots(L_2R_2)L_1\in\cU_{d-1}(N), +\] +where $V=L_1$ if $d=1$, and put $A'=L_0AL_0^\dagger$. +This is a product projection with the same support as $A$, +of size $k\le N^u$. Gates of $R_1$ whose supports avoid +$\supp A'$ commute with $A'$ and cancel in $R_1A'R_1$. +Because the gates in $R_1$ have disjoint supports, at most +$k$ gates remain. + +Expand each remaining reflection $I-2P_*$ on both sides of +$A'$. Each term is a scalar multiple of +$P_{\mathrm{left}}A'P_{\mathrm{right}}$, where the two side +factors are product projections: within either side, the selected +gate projections have disjoint supports. The sum of the absolute +scalar coefficients is at most +\begin{equation}\label{eq:reflection-coefficients} + (1+2)^{2k}=9^k. +\end{equation} +Conjugation by $V$ turns each such term into a product +$B_1B_2B_3$, with +\[ + B_1=VP_{\mathrm{left}}V^\dagger,\qquad + B_2=VA'V^\dagger,\qquad + B_3=VP_{\mathrm{right}}V^\dagger. +\] +Each $B_j$ is therefore a contraction +to which \Cref{lem:propagation} applies. No commutation between +these three factors is asserted or needed. + +Choose fixed exponents +\[ + \max\{w,u\}u,w$ +absorb the entire expansion cost uniformly for $k\le N^u$. +\end{proof} diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/introduction.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..41e23169369c3cb6a99fe28115f6c23229a0eb7c --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/introduction.tex @@ -0,0 +1,240 @@ +\section{Introduction} +\label{sec:introduction} + +Parity is a basic test of whether a shallow circuit can combine +information from all of its inputs. For a computational-basis string +$x=(x_1,\ldots,x_n)\in\{0,1\}^n$, its value is +$\parity(x)=x_1\oplus\cdots\oplus x_n$. We study circuits generated +by arbitrary one-qubit unitaries and Toffoli gates with any number of +controls and one target. Gates in each parallel layer have pairwise +disjoint supports. The depth is the number of such layers. + +The circuit starts in $\ket{x}\otimes\ket0^{\otimes(N-n)}$ on $N$ +total qubits. Its answer is the result of measuring one designated output +qubit in the computational basis; the remaining registers may be +discarded in arbitrary final states, and the input need not be preserved. +For $0<\varepsilon\le1/2$, computing parity with worst-case advantage +$\varepsilon$ means giving the correct answer with probability at least +$1/2+\varepsilon$ for every $x$. The class $\QAC$ uses constant depth +and polynomially many total qubits. This also bounds the gate count: +a disjoint layer on $N$ qubits has at most $N$ nonempty gates. +The gates may depend on $n$, but not on the input string. +We use unitary circuits, with no intermediate measurements, +postselection, or primitive unbounded-fanout gate. + +\begin{theorem}[Parity lower bound] +\label{thm:parity} +Fix a nonnegative integer \(d\), a real number \(c\ge1\), and +\(0<\varepsilon\le1/2\). For all sufficiently large \(n\), no circuit +of depth at most \(d\) on \(n\) input qubits and at most \(n^c\) total +qubits, using arbitrary one-qubit unitaries and unbounded-arity Toffoli +gates, computes \(\parity(x)\) with probability at least +\(1/2+\varepsilon\) for every \(x\in\{0,1\}^n\). +Gates in each layer have disjoint supports, ancillas start in \(\ket0\), +and one output qubit is measured; the other qubits may be discarded. +\end{theorem} + +Taking \(\varepsilon=1/6\) gives the usual success threshold \(2/3\). +Thus \Cref{thm:parity} resolves positively the conjecture +\(\parity\notin\QAC\), with bounded error, polynomially many ancillas, +and only a measured output required. + +\subsection{History and significance} +In the classical class $\mathrm{AC}^0$, circuits use unbounded-fan-in +AND and OR gates and NOT gates, with unrestricted reuse of wires. +The superpolynomial constant-depth lower bounds for parity of Furst, +Saxe, and Sipser~\cite{FurstSaxeSipser1984} and, independently, +Ajtai~\cite{Ajtai1983}, were fundamental explicit circuit lower bounds. +Yao obtained exponential bounds~\cite{Yao1985}, and H\aa stad obtained +almost optimal exponential bounds through the switching +lemma~\cite{Hastad1986}. Quantum circuits introduce a different +constraint: the supports of gates in a parallel layer are disjoint, +so a wire cannot control arbitrarily many simultaneous gates. + +Moore introduced the corresponding quantum circuit classes and related +coherent parity to unbounded fanout~\cite{Moore1999}. The latter is the +unitary sending $\ket{b,y_1,\ldots,y_k}$ to +$\ket{b,y_1\oplus b,\ldots,y_k\oplus b}$. The coherent parity +transformation sends the same input to +$\ket{b\oplus\parity(y),y_1,\ldots,y_k}$, where +$y=(y_1,\ldots,y_k)$. Hadamard conjugation interchanges these two +transformations. Moore's original +formulation required ancillary qubits to return to zero; our result +also treats the more permissive measured-output model. +Green, Homer, Moore, and Pollett developed the associated quantum +counting classes and modular-gate equivalences~\cite{GreenHomerMoorePollett2002}. +The constant-depth constructions of H\o yer and \v Spalek demonstrate +how much computational power a primitive fanout operation +provides~\cite{HoyerSpalek2005}. + +Early lower bounds revealed the importance of the ancillary register. +Fang, Fenner, Green, Homer, and Zhang obtained a depth lower bound of +order $\log(n/(a+1))$ for exact, clean parity computation with $a$ +ancillary qubits~\cite[Corollary 4.6]{FangFennerGreenHomerZhang2006}; +Bera developed a different lower-bound method in the ancilla-free +setting~\cite{Bera2011}. Pad\'e, Fenner, Grier, and Thierauf ruled out +exact clean simulation at entangling depth two~\cite{PadeFennerGrierThierauf2020}. +Fenner, Grier, Pad\'e, and Thierauf subsequently removed the final-state +cleanliness requirement for exact single-output parity at that +depth~\cite[Theorem 6.1]{FennerGrierPadeThierauf2025}. +Here the quoted depth-two results count multiqubit layers, allowing +arbitrary one-qubit layers between them; we call this +\emph{entangling depth}. Rosenthal proved lower bounds for approximate +parity unitaries and related cat-like states, and gave constant-depth +approximation constructions with superpolynomial +resources~\cite{Rosenthal2021}. Approximate unitary implementation and +computation of a single measured classical bit have different output +requirements, so the distinction matters in comparing lower bounds. + +More recent work developed spectral and Fourier methods for $\QAC$. +Nadimpalli, Parham, Vasconcelos, and Yuen used the Pauli spectrum to prove +ancilla-restricted lower bounds~\cite{NPVY2024}. Anshu, Dong, Ou, and Yao +ruled out a fixed positive parity advantage with +$\widetilde O(n^{1+3^{-d}})$ ancillas at every fixed entangling depth +$d\ge1$~\cite[Corollary 4.4 and Theorem 4.6]{ADOY2025}. +Dong, Ou, and Yao increased this ancilla range to +$\widetilde O(n^{1+2^{-d}})$~\cite[Corollary 3.3]{DOY2025LinearSize}. +Here $\widetilde O$ suppresses logarithmic factors. Both results use +spectral-norm approximation of conjugated observables. + +Joshi, Tal, Vasconcelos, and Wright proved exact parity lower bounds at +entangling depth three and bounded parity correlation at entangling +depth two by $\exp(-\Omega(\sqrt n))$, without a size +restriction~\cite[Theorem 1.2 and Corollary 5.9]{JTVW2026}. Kintali's September 2026 preprint reports +correlation at most $\exp(-\Omega(n))$ at entangling depth three +and an exponential multiqubit gate lower bound for exact parity at +entangling depth four~\cite[Theorems 1--2]{Kintali2026ShallowParity}. +These results retain a resource restriction or a fixed upper bound on +entangling depth. Gretta, Gupta, and Joshi established an +equivalence between the general parity lower-bound question and Fourier +concentration~\cite{GGJ2026}, and Xu and Li extended related fanout +reductions to symmetric Boolean functions~\cite{XuLi2026Fanout}. Our theorem permits every fixed depth and +every fixed polynomial bound on the total number of qubits, with +arbitrary discarded garbage and any fixed positive worst-case advantage. + +\subsection{The localization argument} + +Unbounded gate arity is the first obstacle to a lower bound. A single +Toffoli gate can involve every input, so the number of wires in an +output's backward light cone gives no useful restriction. Our argument +instead bounds transitions from a fixed product condition to states +with many mismatches from that condition. We keep the amplitude lost +when imposing a projection, without normalizing the resulting state. + +A \emph{product projection} fixes a chosen one-qubit vector on each qubit +of a subset and acts as the identity elsewhere. Fix reference unit +vectors $v_i$ on a subset $S\subseteq\{1,\ldots,N\}$. Their +\emph{mismatch count} is +\[ + M=\sum_{i\in S}(I-\proj{v_i})_i; +\] +its eigenvalue counts the factors orthogonal to their reference vectors. +Write $[M=0]$ and $[M\ge r]$ for its zero and high-count spectral +projections. Given a shallow circuit $U$ and a product projection $A$, +our structural theorem bounds +\[ + \norm{[M\ge N^t]UAU^\dagger[M=0]}\le e^{-N^s} + \qquad(0\le sn/2\}$, where $|x|$ is the +Hamming weight, cannot be computed in this circuit model with worst-case advantage $1/(\log n)^\gamma$ +for any fixed $\gamma>0$ (\Cref{cor:symmetric-functions}). This separates +the model from the corresponding one with unbounded threshold gates. + +The state-preparation consequences rule out inverse-polynomial +measurement probabilities for both a string and its bitwise complement, +and sufficiently accurate preparation of uniform superpositions of +strings of a fixed Hamming weight $k$ with $n^\delta\le k\le n/2$ +for fixed $\delta>0$ (\Cref{cor:state-preparation}). For a single measured output, define +its expected sign as the probability of zero minus the probability of one. +Localization implies that its Fourier coefficients have total squared +weight that decays faster than every inverse power of $n$ on input +subsets of size at least $n^\delta$, for every fixed +$0<\delta\le1$ (\Cref{cor:quantum-lmn}). Finally, majority cannot be computed +with average success at least $1-\tfrac12 n^{-\delta}$ on uniformly +random inputs, for any fixed $\delta>0$ (\Cref{cor:majority-average}). +The precise quantifiers and proofs +appear after the parity proof in \Cref{sec:consequences}. + +\paragraph{Organization.} +\Cref{sec:localization} states the localization theorem and fixes the +circuit model. \Cref{sec:polynomial} proves the polynomial approximation. +\Cref{sec:depth} reduces small-support projections to the preceding +depth, and \Cref{sec:bootstrap} completes the finite scale induction. +\Cref{sec:parity} derives \Cref{thm:parity}, and +\Cref{sec:consequences} proves the consequences just described. diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/localization.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/localization.tex new file mode 100644 index 0000000000000000000000000000000000000000..688c4f2c96bf4e91a9804525306158438f6cbd98 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/localization.tex @@ -0,0 +1,98 @@ +\section{Product projections and localization} +\label{sec:localization} + +The parity proof will use localization for a single output projection. +We prove the stronger uniform statement for every product projection, +because its induction exchanges the projected and counted registers. + +All operators act on \(\mathcal H_N=(\mathbb C^2)^{\otimes N}\), and +\(\norm{\cdot}\) denotes the operator norm. Identities on unmentioned +qubits are implicit. Spectral projections are written in brackets; +for example, \([M\ge r]\) is the projection onto the sum of eigenspaces +of a self-adjoint operator \(M\) with eigenvalues at least \(r\). +All logarithms are natural. + +A \emph{count} is an operator +\begin{equation} +\label{eq:count} + M=\sum_{i\in S}\bigl(I-\proj{v_i}\bigr)_i, + \qquad S\subseteq\{1,\ldots,N\},\quad \norm{v_i}=1. +\end{equation} +The subscript indicates the qubit on which the summand acts. +The reference vector \(v_i\in\mathbb C^2\) can be chosen separately +at each qubit. The summands commute, and the spectrum of \(M\) is +contained in \(\{0,\ldots,|S|\}\). +A \emph{product projection} is a tensor product of rank-one one-qubit +projections on a subset, with identity on the complement. Its support +is the selected subset. The empty +product is \(I\). These are precisely the projections \([M=0]\). +In particular, a product projection need not have rank one on +\(\mathcal H_N\). + +We use a slightly enlarged gate set of product-state reflections, as in +\cite[Section~2 and Proposition~2.1]{Rosenthal2021}. +Let \(\cU_d(N)\) be the set of +unitaries +\begin{equation} +\label{eq:normal-form} + U=(L_dR_d)\cdots(L_1R_1)L_0, +\end{equation} +where each \(L_j\) is a product of one-qubit unitaries and each \(R_j\) +is a layer of gates \(I-2A\) with disjoint supports, with \(A\) a product +projection on the gate's qubits. Identity layers are allowed. +The class \(\cU_0(N)\) consists of products of one-qubit unitaries. + +This model contains the circuits of \Cref{thm:parity}. Indeed, if +\(C\) projects onto all controls being \(1\), a Toffoli gate equals +\[ + (I-C)\otimes I+C\otimes X + =I-2C\otimes\proj{-}, + \qquad \ket-=\frac{\ket0-\ket1}{\sqrt2}. +\] +Grouping the local gates in each parallel step gives +\eqref{eq:normal-form} with a number of reflection layers bounded by +the original depth. Discarded qubits can be retained as idle wires +without changing any later measurement probability. Here $d$ counts +reflection layers in the enlarged model; it is at most the number of +physical layers in the original circuit. The arbitrary local layers in +\eqref{eq:normal-form} are included in this definition, rather than +charged separately to $d$. + +We will also use closure under inversion at the same depth. Each +\(R_j\) is self-adjoint, since it is a product of commuting +self-adjoint reflections. Thus +\begin{equation} +\label{eq:inverse} + U^\dagger=L_0^\dagger R_1L_1^\dagger\cdots R_dL_d^\dagger + \in\cU_d(N). +\end{equation} +This is the same normal form with the layer indices reversed: the new +local layers are $L_d^\dagger,\ldots,L_0^\dagger$, in temporal order, +and the new reflection layers are $R_d,\ldots,R_1$. No factors on +overlapping supports have been commuted. + +\begin{theorem}[Localization of conjugated product projections] +\label{thm:localization} +Fix a nonnegative integer \(d\) and real numbers \(0\le s1\), it is immediate for \(N>1\), since +\([M\ge N^t]=0\). We prove the theorem by induction on \(d\), assuming +all fixed exponent pairs at the preceding depth. +The counts \(M,D\) are allowed to be different because the proof +will exchange their roles when applying \eqref{eq:inverse}. + +The next two sections supply the approximation and depth-reduction +estimates. The proof of \Cref{thm:localization} is completed in +\Cref{sec:bootstrap}. diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/parity.tex b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/parity.tex new file mode 100644 index 0000000000000000000000000000000000000000..962627737f24fe7e1b9c876c9319291e3f5b5704 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/build/sections/parity.tex @@ -0,0 +1,79 @@ +\section{From localization to parity} +\label{sec:parity} + +We finish by relating the output probability to transitions between +product states. All localization estimates have already been proved; +no assumption about coherent or clean computation is needed here. +For the broader Fourier-concentration setting, see~\cite{GGJ2026}; +the character-orthogonality identity needed here is proved directly. + +\begin{proof}[Proof of \Cref{thm:parity}] +Suppose that a circuit as in the theorem exists on \(n\) input qubits +and \(N-n\) initially zero ancillas, with \(n\le N\le n^c\). +Retain any discarded qubits as idle wires and write \(W\) for the +resulting unitary. By \Cref{sec:localization}, \(W\in\cU_d(N)\) +after padding with identity layers. +Let \(O\) project the output qubit onto \(\ket0\), and put +\[ + B=W^\dagger OW,\qquad + p_0(x)=\bra{x,0}B\ket{x,0}. +\] +The Born rule identifies \(p_0(x)\) with the probability of output +zero, irrespective of the final states of the other qubits. +Here and below the last zero denotes all initially zero ancillas. + +For \(h\in\{0,1\}^n\), define +\[ + \ket{\xi_h} + =\bigotimes_{j=1}^n + \frac{\ket0+(-1)^{h_j}\ket1}{\sqrt2}, + \qquad + \bar h=(1-h_1,\ldots,1-h_n). +\] +Let \(M_h\) be the count on the input qubits whose reference vectors +are the factors of \(\xi_h\). The vectors +\(\ket{\xi_h,0}\) and \(\ket{\xi_{\bar h},0}\) have exact counts +zero and \(n\), respectively. +Choose fixed \(0(a+b)/2$. For every sufficiently large integer +$N$, there is a real polynomial $p$ such that +\begin{equation}\label{eq:polynomial-bounds} + \deg p\le N^u, + \qquad W_N(p)\le \exp(N^u), + \qquad p(0)=1, +\end{equation} +and +\begin{equation}\label{eq:polynomial-decay} + |p(x)|\le \exp(-N^a) + \qquad\text{for every integer }1\le x\le \lceil N^b\rceil. +\end{equation} +The threshold for $N$ depends only on $a,b,u$. No restriction $b\le1$ is +imposed. +\end{lemma} + +\begin{proof} +Put +\[ + r=\lceil N^a\rceil, + \qquad m=\lceil N^b\rceil, + \qquad \ell=\left\lceil 2\sqrt{mr}\log m\right\rceil. +\] +Since $a0). +\] +In particular, $|T_j(x)|\le1$ on $[-1,1]$. Set +\begin{equation}\label{eq:polynomial-construction} + p(x)=F(x) + \frac{T_\ell\!\left((m+r-2x)/(m-r)\right)} + {T_\ell\!\left((m+r)/(m-r)\right)}. +\end{equation} +The denominator is positive, and $p(0)=1$. + +To bound the denominator, write $q=\sqrt{r/m}$ and +$h=(1+q)/(1-q)$. Then $0N$. + +Let $A=2\beta+1$. The Chebyshev recurrence and induction give +$W_N(T_j(z))\le A^j$: the inductive step uses +\[ + 2\beta A^j+A^{j-1}\le A^{j+1}, + \qquad A^2-2\beta A-1=2\beta\ge0. +\] +The initial cases follow from $W_N(T_0(z))=1$ and +$W_N(T_1(z))=\beta\le A$. Since $A\le7+4N\le11(1+N)$ and the denominator +in \Cref{eq:polynomial-construction} is at least $1$, we obtain +\[ + W_N(p)\le(1+N)^r\bigl(11(1+N)\bigr)^\ell, + \qquad \deg p\le r+\ell. +\] +Finally, with $\gamma=(a+b)/2>a$, the rounded definitions imply +\[ + r=O(N^a),\qquad + \ell=O(N^\gamma\log N),\qquad + \log W_N(p)=O\!\left(N^\gamma(\log N)^2\right). +\] +Both $r+\ell$ and $\log W_N(p)$ are $o(N^u)$ because $u>\gamma$. +All constants depend only on the fixed exponents, proving +\Cref{eq:polynomial-bounds} with the asserted uniform threshold. +\end{proof} diff --git a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..86e125f95494da09bbadfe961167e10d01fe4ba8 --- /dev/null +++ b/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:12393c9f997aa808d71792f8afbd0cd2e65cf2892725f621ef845cc87b61efcf +size 442232 diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/README.md b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..df46d9149769cf681b18d62f7aec91b43993d1bb --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/README.md @@ -0,0 +1,16 @@ +# [Projection moments, positive cap domination, and Riesz estimates on the sphere](paper.pdf) + +OpenAI · September 27, 2026 + +## Citation + +```bibtex +@misc{OAI:Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026, + author = {{OpenAI}}, + title = {{Projection moments, positive cap domination, and Riesz estimates on the sphere}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf}{OAI:Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/main.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..bbb537e2d7f11c41850bb022c5dcf3477449e916 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/main.tex @@ -0,0 +1,30 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{Projection moments, positive cap domination,\\ +and Riesz estimates on the sphere} +\author{OpenAI} +\date{September 27, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove moment estimates for exact random projections of finite measures +whose mass is controlled on Euclidean balls, and derive positive domination +by countable sums of spherical cap measures. For learners with \(M=o(d^2)\) +bits of memory, these estimates give three proofs that uniform-sphere average +success at least \(2/3\) at angular accuracy \(0<\epsilon\le1/10\) requires +\(\Omega(d\log(1/\epsilon))\) noiseless Gaussian observations. The three +proofs keep their different stopping and accuracy costs explicit. +\end{abstract} +\begingroup +\small +\tableofcontents +\endgroup +\clearpage +\input{sections/introduction} +\input{sections/geometry-model} +\input{sections/projection} +\input{sections/positive-caps} +\input{sections/riesz} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/preamble.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..b38c4f548ba13469bfe7135dd317e04f7b6bd284 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/preamble.tex @@ -0,0 +1,42 @@ +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{needspace} +\usepackage{array,enumitem,booktabs} +\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} +\usepackage[nameinlink,noabbrev]{cleveref} +\hypersetup{ + pdftitle={Projection moments, positive cap domination, and Riesz estimates on the sphere}, + pdfauthor={OpenAI}, + pdfsubject={Exact random projections and finite-state noiseless Gaussian regression}, + bookmarksopen=true, + bookmarksnumbered=true +} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Prob}{\mathbb P} +\renewcommand{\P}{\mathbb P} +\newcommand{\1}{\mathbf 1} +\newcommand{\ind}{\mathbf 1} +\newcommand{\sphere}{S^{d-1}} +\newcommand{\norm}[1]{\left\lVert #1\right\rVert} +\newcommand{\abs}[1]{\left\lvert #1\right\rvert} +\newcommand{\inner}[2]{\langle #1,#2\rangle} +\newcommand{\ip}[2]{\langle #1,#2\rangle} +\newcommand{\dist}{\operatorname{dist}} +\newcommand{\aff}{\operatorname{aff}} +\newcommand{\vol}{\operatorname{vol}} +\newcommand{\op}{\mathrm{op}} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\supp}{supp} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/references.bib b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..b861e887bc86cb3f370d021eb5888d9c682b165c --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/references.bib @@ -0,0 +1,140 @@ +@inproceedings{SSV2019, + author = {Sharan, Vatsal and Sidford, Aaron and Valiant, Gregory}, + title = {Memory-Sample Tradeoffs for Linear Regression with Small Error}, + booktitle = {Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing}, + year = {2019}, + pages = {890--901}, + note = {Full version arXiv:1904.08544v2, 10 October 2020; theorem and section locators refer to that version. \url{https://arxiv.org/abs/1904.08544v2}. DOI: \url{https://doi.org/10.1145/3313276.3316403}} +} + +@inproceedings{SteinhardtDuchi2015, + author = {Steinhardt, Jacob and Duchi, John}, + title = {Minimax Rates for Memory-Bounded Sparse Linear Regression}, + booktitle = {Proceedings of the 28th Conference on Learning Theory}, + series = {Proceedings of Machine Learning Research}, + volume = {40}, + year = {2015}, + pages = {1564--1587}, + note = {\url{https://proceedings.mlr.press/v40/Steinhardt15.html}} +} + +@inproceedings{SVW2016, + author = {Steinhardt, Jacob and Valiant, Gregory and Wager, Stefan}, + title = {Memory, Communication, and Statistical Queries}, + booktitle = {29th Annual Conference on Learning Theory}, + series = {Proceedings of Machine Learning Research}, + volume = {49}, + year = {2016}, + pages = {1490--1516}, + note = {\url{https://proceedings.mlr.press/v49/steinhardt16.html}} +} + +@inproceedings{Shamir2014, + author = {Shamir, Ohad}, + title = {Fundamental Limits of Online and Distributed Algorithms for Statistical Learning and Estimation}, + booktitle = {Advances in Neural Information Processing Systems}, + volume = {27}, + year = {2014}, + pages = {163--171}, + note = {\url{https://proceedings.neurips.cc/paper_files/paper/2014/hash/cc427d934a7f6c0663e5923f49eba531-Abstract.html}} +} + +@misc{Raz2016, + author = {Raz, Ran}, + title = {Fast Learning Requires Good Memory: A Time-Space Lower Bound for Parity Learning}, + year = {2016}, + note = {arXiv:1602.05161v1, 16 February 2016. \url{https://arxiv.org/abs/1602.05161v1}} +} + +@inproceedings{DKS2019, + author = {Dagan, Yuval and Kur, Gil and Shamir, Ohad}, + title = {Space Lower Bounds for Linear Prediction in the Streaming Model}, + booktitle = {Proceedings of the Thirty-Second Conference on Learning Theory}, + series = {Proceedings of Machine Learning Research}, + volume = {99}, + year = {2019}, + pages = {929--954}, + note = {Theorem locators refer to the published PMLR version. \url{https://proceedings.mlr.press/v99/dagan19b.html}} +} + +@article{Mattila1975, + author = {Mattila, Pertti}, + title = {Hausdorff Dimension, Orthogonal Projections and Intersections with Planes}, + journal = {Annales Academiae Scientiarum Fennicae, Series A I, Mathematica}, + volume = {1}, + number = {2}, + year = {1975}, + pages = {227--244}, + note = {DOI: \url{https://doi.org/10.5186/aasfm.1975.0110}} +} + +@article{Drury1984, + author = {Drury, S. W.}, + title = {Generalizations of {Riesz} Potentials and {$L^p$} Estimates for Certain \(k\)-Plane Transforms}, + journal = {Illinois Journal of Mathematics}, + volume = {28}, + number = {3}, + year = {1984}, + pages = {495--512} +} + +@article{FranklMaehara1990, + author = {Frankl, Peter and Maehara, Hiroshi}, + title = {Some Geometric Applications of the Beta Distribution}, + journal = {Annals of the Institute of Statistical Mathematics}, + volume = {42}, + number = {3}, + year = {1990}, + pages = {463--474}, + note = {DOI: \url{https://doi.org/10.1007/BF00049302}} +} + +@misc{SchnaubeltFA2025, + author = {Schnaubelt, Roland}, + title = {Functional Analysis: Lecture Notes of Winter Semester 2017/18}, + howpublished = {Karlsruhe Institute of Technology, author-hosted lecture notes}, + year = {2025}, + note = {Title-page version 21 July 2025; accessed 27 September 2026. \url{https://iana.math.kit.edu/downloads/iana3/schnaubelt/Skripten/fa17-skript.pdf}} +} + +@misc{FremlinTaxonomy, + author = {Fremlin, D. H.}, + title = {Measure Theory, Chapter 21: Taxonomy of Measure Spaces}, + howpublished = {University of Essex, author-hosted online development extract}, + note = {Section 211 version 20 November 2003; accessed 27 September 2026. \url{https://www1.essex.ac.uk/maths/people/fremlin/chap21.pdf}} +} + +@misc{FremlinFunctionSpaces, + author = {Fremlin, D. H.}, + title = {Measure Theory, Chapter 24: Function Spaces}, + howpublished = {University of Essex, author-hosted online development extract}, + note = {Chapter version 15 November 2013; Section 243 version 30 April 2004; accessed 27 September 2026. \url{https://www1.essex.ac.uk/maths/people/fremlin/chap24.pdf}} +} + +@book{TaoMeasure2011, + author = {Tao, Terence}, + title = {An Introduction to Measure Theory}, + publisher = {American Mathematical Society}, + series = {Graduate Studies in Mathematics}, + volume = {126}, + year = {2011}, + note = {Theorem and page locators refer to the author-hosted preliminary version; accessed 27 September 2026. \url{https://terrytao.wordpress.com/wp-content/uploads/2012/12/gsm-126-tao5-measure-book.pdf}} +} + +@misc{CalefHardin2008v1, + author = {Calef, Matthew T. and Hardin, Douglas P.}, + title = {{Riesz} \(s\)-Equilibrium Measures on \(d\)-Rectifiable Sets as \(s\) Approaches \(d\)}, + year = {2008}, + note = {arXiv:0808.3802v1, 28 August 2008; the potential definition is in Section 1, p.~2. \url{https://arxiv.org/abs/0808.3802v1}} +} + +@article{Jiang2006, + author = {Jiang, Tiefeng}, + title = {How Many Entries of a Typical Orthogonal Matrix Can Be Approximated by Independent Normals?}, + journal = {The Annals of Probability}, + volume = {34}, + number = {4}, + year = {2006}, + pages = {1497--1529}, + note = {Lemma locators refer to the published-article reprint, arXiv:math/0601457v2, 25 September 2006. \url{https://arxiv.org/abs/math/0601457v2}. DOI: \url{https://doi.org/10.1214/009117906000000205}} +} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/geometry-model.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/geometry-model.tex new file mode 100644 index 0000000000000000000000000000000000000000..65c567c94d3d3c5427c4d6d2df333c4cbad0e326 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/geometry-model.tex @@ -0,0 +1,327 @@ +\section{Spherical probabilities and exact observations} +\label{p:geometry-model} + +We collect the geometric facts used by all three estimates and then specify +the finite-state experiment to which they will be applied. The geometric +lemmas keep their normalizations visible: all measures on spheres below are +probability measures, whereas densities on Euclidean label spaces are +relative to Lebesgue measure unless another reference measure is stated. + +\subsection{Coordinates and caps} + +Write \(\sigma_d\) for uniform probability on the unit sphere +\(S^{d-1}\subset\R^d\), and write \(B(z,r)\) for the closed Euclidean ball +with center \(z\) and radius \(r\). When the ambient dimension is fixed we +abbreviate \(\sigma_d\) to \(\sigma\). The norm is Euclidean, and +\(\norm A_{\op}\) denotes the operator norm of a matrix. Constants denoted +by \(C\) are positive and absolute and may increase from one occurrence to +the next. A subscript records any permitted additional dependence. +All logarithms are natural unless a base is displayed. + +\begin{lemma}[Coordinates of a uniform direction]\label{p:coordinate-density} +Let \(D\ge2\) and \(1\le k0\) such that, +for every \(c\in S^{d-1}\) and \(00\), +\begin{equation}\label{p:ambient-cap-mass} + \sigma\bigl(B(z,r)\bigr)\le\min\{1,(2r)^n\}. +\end{equation} +An output direction independent of the signal has probability at most +\(\epsilon^n\) of angular error at most \(\epsilon\), averaged over +\(\sigma\). +\end{lemma} + +\begin{proof} +For \(r\le1\), rotate \(c\) to the north pole. A sphere point in +\(B(c,r)\) has last coordinate at least \(1-r^2/2\ge1/2\). Its surface +element as a graph over \(c^\perp\) is at most twice Lebesgue measure, +and its projection lies in the \(n\)-ball of radius \(r\). The full sphere +has area at least twice the volume of the unit \(n\)-ball, by projecting +both hemispheres. This proves the upper bound for \(r\le1\); total mass +one proves it for \(r>1\). + +For the lower bound when \(r\le1\), the angular radius +\(2\arcsin(r/2)\) is at least \(r\). In polar angle the cap mass is the +integral of \(\sin^{n-1}\theta\) over this interval divided by its +integral over \([0,\pi]\). Use +\(\sin\theta\ge2\theta/\pi\) for \(0\le\theta\le r\) and bound the +denominator by \(\pi\). Since \(n\le2^{n-1}\), the result is at least +\((r/\pi)^n\). For \(1\le r\le2\), monotonicity therefore gives the +lower bound with \(a=1/(2\pi)\). + +If an ambient ball meets the sphere, choose \(c\) in the intersection. +Its sphere portion is contained in \(B(c,2r)\), proving +\eqref{p:ambient-cap-mass}; an empty intersection is immediate. Finally, +angular error at most \(\epsilon\) implies chordal error at most +\(\epsilon\), because \(2\sin(\theta/2)\le\theta\). Apply the +sphere-centered upper bound to each fixed output and then integrate over +any independent output randomization. +\end{proof} + +\subsection{The finite-state regression experiment} + +An unknown direction \(s\in S^{d-1}\) is observed through independent +standard Gaussian features and their exact labels: +\[ + X_t\sim N(0,I_d),\qquad Y_t=\inner{X_t}{s},\qquad t=1,2,\ldots . +\] +At each sample index the learner retains a state from a set of at most +\(2^M\) values, where \(M\) is a nonnegative integer. A transition may +use the current state, the entire current pair \((X_t,Y_t)\), the sample +index, and fresh independent randomness. Computation within the transition +is unrestricted. Every piece of information depending on earlier samples +that remains available later must be encoded by the finite state. The +learner neither chooses a feature nor revisits an earlier one. + +The learner stops after at most a prescribed integer \(T\ge0\) samples. +Its stopping decisions obey the same information restriction. For fixed +rules, its output is a unit vector whose law depends only on the terminal +state and stopping index, with fresh randomness allowed. In particular, +the last observed pair influences the output only through the terminal +state. The sample index is available without being stored in the state; +the proofs below account for the information in the stopping index. + +A shared random seed may select the rules before sampling. The pair +consisting of this seed and the initial finite state is jointly independent +of the signal and the entire sequence of feature rows. The seed and initial +state may depend on each other. Fresh transition and output randomness is +independent of these variables and of initialization. Nonuniform rules may +also depend on \(d,\epsilon\), and the sample index. This is a bit-memory +model with a deterministic finite horizon. + +We impose the following measurability convention. The original experiment +is jointly measurable in the signal, pre-generated feature rows, shared +seed, initial state, and fresh randomness, including the resulting state +path, stopping index, and output. For almost every fixed shared seed, the +finite vector of transition and stopping probabilities at each fixed +state and index is measurable in the completion of +\(\gamma_d(dx)\,dy\), where \(\gamma_d=N(0,I_d)\). A rule given instead +as a completed-measurable function of the current pair and an independent +fresh seed is also allowed; Lemma~\ref{p:borel-kernels} justifies this +presentation. Each terminal output law is a probability measure on the +Borel sphere. + +For the lower bounds, let \(S\sim\sigma\) be independent of the feature +rows. We call the probability of +\(\arccos\inner{\widehat S}{S}\le\epsilon\), including all learner +randomness, the \emph{uniform-sphere average success probability}. + +\begin{theorem}[Finite-memory precision bound]\label{p:regression} +Let \(M(d)=o(d^2)\) be a sequence of nonnegative integers and let +\(0<\epsilon(d)\le1/10\). There is an absolute \(c>0\) such that, for +all sufficiently large \(d\), every learner in the experiment above whose +uniform-sphere average success satisfies +\[ + \Prob\{\arccos\inner{\widehat S}{S}\le\epsilon(d)\}\ge\frac23 +\] +must use +\[ + T(d)\ge c\,d\log\frac1{\epsilon(d)}. +\] +The dimension threshold may depend on the memory sequence. In particular, +the conclusion holds when the same success probability is required +separately for every \(s\in S^{d-1}\). +\end{theorem} + +An every-signal guarantee implies the average guarantee by integration +against \(\sigma\). All three proofs use the average guarantee itself. +The first proof also yields a uniform dependence on the memory budget. + +\begin{corollary}[Finite-dimensional memory dependence] +\label{p:finite-memory-tradeoff} +There are absolute constants \(c>0\) and \(d_0\) such that, for every +integer \(d\ge d_0\), every nonnegative integer \(M\), and every +\(0<\epsilon\le1/10\), a learner in the experiment above with +uniform-sphere average success at least \(2/3\) and deterministic finite +horizon \(T\) satisfies +\[ + T\ge \frac{c\,d\log(1/\epsilon)}{1+M/d^2}. +\] +\end{corollary} + +The proof appears at the end of Section~\ref{p:projection}. In particular, +the same precision bound holds with absolute constants throughout +\(M\le d^2\) once \(d\) exceeds an absolute threshold. + +\subsection{Measurable continuations and shared randomness} + +The block arguments will start fresh executions from specified states, +including states the learner never reaches. The next lemma supplies Borel +rules for all such continuations while preserving the actual uniform-prior +experiment. Its proof uses the marginal law of each observation pair, +so it does not require a density conditional on a selected state. + +\begin{lemma}[Borel versions and counterfactual continuations] +\label{p:borel-kernels} +Fix \(d\ge2\), finite \(T\), and finite state sets. Suppose each transition +and stopping probability, for each fixed state and index, is measurable in +the completion of the Borel sigma field of \(\R^d\times\R\) under +\(\lambda(dx,dy)=\gamma_d(dx)\,dy\), where \(\gamma_d\) is standard +Gaussian probability on \(\R^d\). The terminal output law depends only on +the terminal state and index. There are everywhere Borel probability +kernels with the same state sets and output laws that induce the same +uniform-prior joint law of signal, state path, stopping index, and output. +For the Borel learner, starting from any state and index and supplying fresh +future features defines a measurable continuation success function, whether +or not that state is reachable in the original execution. +The same conclusion holds when a rule is given before seed integration by +a finite probability vector measurable in the completion of +\(\lambda\otimes\kappa\), where \(\kappa\) is the law of an independent +fresh seed. +\end{lemma} + +\begin{proof} +First suppose the rule includes a fresh seed of law \(\kappa\). Each +completed-product measurable coordinate has a product-measurable version: +approximate it by simple functions and replace the countably many level +sets by product-measurable sets modulo null sets. Replace the resulting +finite vector by a fixed point mass wherever it fails to be a probability +vector. This repair occurs on a product-measurable null set. Integrating +against \(\kappa\) now gives a Borel probability vector in the current +pair, and Fubini shows agreement with the original integrated rule for +\(\lambda\)-almost every pair. It therefore suffices to consider the +coordinate probabilities in the first statement. + +For each fixed state and index, combine the possible transitions and stops +into a finite set of destinations, tagged as continuing or stopping. +Choose Borel versions of their coordinate probabilities by the same +simple-function construction. On the Borel null set where the versions +fail to be nonnegative or to sum to one, replace the vector by a fixed +point mass. Finitely many states and indices are involved, so all changes +lie in one Borel \(\lambda\)-null set \(Z\). + +For every nonzero feature \(x\), Lemma~\ref{p:coordinate-density} gives +a Lebesgue density for \(\inner{x}{S}\). Since a Gaussian feature is +nonzero almost surely, every pre-generated pair +\((X_t,\inner{X_t}{S})\) has law absolutely continuous with respect to +\(\lambda\). Thus each pair avoids \(Z\) almost surely. Even if the +current state depends on previous observations, the probability of being +in a particular state and seeing a pair in \(Z\) is bounded by the +marginal probability of seeing that pair in \(Z\), which is zero. + +Almost surely none of the finitely many pre-generated pairs lies in +\(Z\). Couple the original and modified finite destination kernels by +the same fresh uniform random numbers. Induction gives identical state +paths and stopping decisions; couple their unchanged terminal output laws +identically as well. This proves equality of the specified joint laws. +Finally, finite composition and integration of the Borel kernels against +independent future features and coins defines a measurable continuation +from every specified state and index, including unreachable states. +\end{proof} + +\paragraph{Convention for the three applications.} +We first condition on a shared seed for which the model assumptions hold. +The joint independence of seed and initial state from signal and feature +rows ensures that the conditional initial-state law still contains no +signal or sample information. We then use the Borel learner supplied by +Lemma~\ref{p:borel-kernels}. The lemma need only preserve the +uniform-prior law; preservation at every fixed signal is unnecessary. +All bounds below are uniform in the fixed rules and are averaged over the +shared seed at the end. The original conditional success probabilities +are measurable in that seed because the original experiment is jointly +measurable. Neither the replacement kernels nor any subsequent proof +decompositions need to be selected jointly across seeds. + +\subsection{A common full-data endpoint} + +The three moment arguments produce bounds involving \(T+d\). The next +elementary obstruction supplies the linear lower bound on \(T\) needed +to remove that additive \(d\), without using any memory restriction. + +\begin{lemma}[Antipodal full-data bound]\label{p:haar-linear} +In the experiment above, if \(\epsilon\le1/10\) and \(T\le d/8\), +even an estimator given all \(T\) features and labels has uniform-prior +success probability at most \(5/8\). +\end{lemma} + +\begin{proof} +Let \(F\) be the row span of the pre-generated feature matrix and put +\(u=P_FS\). Its dimension is \(T\) almost surely, with the evident +interpretation for \(T=0\). The labels determine \(u\). +For \(T=0\), the residual is the original uniform signal. For \(T>0\), +Lemma~\ref{p:coordinate-density}, applied after fixing the feature matrix, +shows that conditional on all the data the residual is +\(\sqrt{1-\norm u^2}\,\omega\), with \(\omega\) uniform on the unit +sphere of \(F^\perp\). Before conditioning on the labels, isotropy gives +\(\E\norm u^2=T/d\). Consequently +\(\Prob\{\norm u^2\ge1/2\}\le2T/d\). + +When \(\norm u^2<1/2\), opposite residual directions give two signals +at distance \(2\sqrt{1-\norm u^2}>\sqrt2>2\epsilon\). They have the same +conditional law, and at most one can be within chordal distance +\(\epsilon\) of a fixed output. Angular success implies this chordal +condition, so conditional success is at most \(1/2\), also after +averaging independent estimator randomness. Total success is at most +\[ + \frac12+\frac12\Prob\{\norm u^2\ge1/2\} + \le\frac12+\frac Td\le\frac58. +\] +Giving the estimator all pre-generated data also covers a learner that +stops at a prefix, with its independent shared and fresh randomness. +\end{proof} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/introduction.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..3f539a32ad9b07d2b5a788f4a184841b5a7b9c10 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/introduction.tex @@ -0,0 +1,207 @@ +\section{Introduction}\label{p:introduction} + +A linear projection can concentrate a measure even when the original +measure has no atoms. The amount of concentration depends on how much mass +can lie near the affine subspaces that a projection collapses. This paper +studies that dependence for random exact projections and uses it to analyze +what a finite memory state can retain from noiseless Gaussian equations. + +We consider three questions. The first asks for an integrated moment of a +projected density. Let \(\nu\) be a finite positive measure in \(\R^d\) +whose mass on every ball of radius \(r\) is at most \(Kr^\beta\), and let +\(P\) be a random matrix with \(k\) orthonormal rows. When \(P_\#\nu\) +has density \(g_P\), an integrated estimate controls the average over +\(P\) of \(\int g_P^q\). Such an estimate measures concentration throughout +the label space. Theorem~\ref{p:frame-moment} proves a bound with explicit +dependence on the dimension, support radius, and local-mass exponent. + +The second question asks for an actual positive dominating measure after a +finite routing decision. Let \(\sigma\) be uniform probability on +\(S^{d-1}\), and let \(\mu_B\) be \(\sigma\) conditioned on a spherical +cap \(B\). A routing rule observes a Gaussian matrix \(X\) and the exact +label \(Xs\), and selects an output \(w\) with probability +\(K_w(X,Xs)\). Its average likelihood \(f_w(s)=\E_XK_w(X,Xs)\) +weights the parent measure to \(f_w\mu_B\). We seek a positive sum of +normalized cap measures that dominates this measure, with a controlled +cost for small child radii. The positive-domination theorem in +Section~\ref{p:cap-section} constructs a countable such sum and bounds +its cost by a power of a reference routing probability. + +The third question concerns evaluation at a specified label. For a +measurable function \(h:S^{d-1}\to[0,1]\), the ambient Riesz functional +\[ + V(h)=\sup_{z\in\R^d}\int h(s)\norm{s-z}^{-d/2}\,d\sigma(s) +\] +is the supremum of the raw Riesz potentials of \(h\sigma\) over ambient +centers, and it controls mass near every center. If \(h\sigma\) is restricted +to a ball about \(z\), Section~\ref{p:riesz-section} estimates a Gaussian projected +density at \(Xz+b\) for every fixed offset \(b\). It uses one explicitly +chosen density version for all offsets. This qualification matters: +an integrated density norm is unchanged by modifications on a +matrix-dependent graph, so it cannot by itself assign or bound the values +on that graph. + +All three estimates give a precision bound for finite-state regression. +The signal is uniform on \(S^{d-1}\), and each fresh observation is a +standard Gaussian feature together with its exact inner product with the +signal. Write \(T\) for the learner's deterministic sample horizon. +In the bit-memory model of Section~\ref{p:geometry-model}, a learner +with \(M(d)=o(d^2)\) retained bits that achieves angular error at most +\(\epsilon(d)\), where \(0<\epsilon(d)\le1/10\), with average probability +at least \(2/3\) +requires +\[ + T(d)\ge c\,d\log(1/\epsilon(d)) +\] +for an absolute \(c>0\) and all sufficiently large \(d\), as stated in +Theorem~\ref{p:regression}. The dimension threshold may depend on the +memory sequence. Under the same accuracy and success requirements, the +integrated projection argument also gives the explicit tradeoff +\[ + T\ge \frac{c\,d\log(1/\epsilon)}{1+M/d^2} +\] +for every nonnegative integer \(M\) and all \(d\) above an absolute +threshold (Corollary~\ref{p:finite-memory-tradeoff}). In particular, the +same precision bound holds with absolute constants throughout \(M\le d^2\). +The underlying success-probability bounds +\eqref{p:haar-success}, \eqref{p:cap-fixed-stop-explicit}, and +Proposition~\ref{p:riesz-success} retain explicit dependence on arbitrary +\(M\). + +\subsection{Projection geometry and earlier methods} + +The local growth condition belongs to classical potential theory. +Mattila's work on Hausdorff dimension and orthogonal projections uses +inverse-distance energy averaging and compares positive-dimensional +Hausdorff measure with the existence of a measure satisfying a ball-growth +bound~\cite[Lemma~5.1, p.~236, and Theorem~5.4, p.~237]{Mattila1975}. +Mattila describes the potential-theoretic projection method as following +Kaufman's work~\cite[Introduction, p.~227]{Mattila1975}. Here the growth +measure is supplied by a hypothesis or constructed from a cap test. The quantitative estimates require successive +distances to affine spans and moments whose order grows with \(d\). + +Affine simplex volumes also have an established role in integral geometry. +Drury's affine-plane change of variables contains a simplex-volume +Jacobian~\cite[Lemma~1, pp.~497--498]{Drury1984}. In our frame calculation, +the product of successive affine heights is the volume of the associated +parallelepiped, and the \(k\) projected rows raise its inverse to the +\(k\)th power. We derive that row-Jacobian and its probability normalization +directly. The exact density of a rectangular block of a Haar orthogonal +matrix is classical; see Jiang~\cite[Lemma~2.5]{Jiang2006}. +Our derivation keeps its normalization explicit before the affine-height +change of variables. Classical beta descriptions of a uniform direction projected +onto a subspace are discussed by Frankl and +Maehara~\cite{FranklMaehara1990}; Lemma~\ref{p:coordinate-density} +gives the vector density and conditional direction law used here. + +The regression application sits within a broader study of learning with +limited retained information. Shamir's finite-message protocols include a +bounded-memory online subclass~\cite[Section~2]{Shamir2014}. +Steinhardt, Valiant, and Wager related memory, communication, and statistical +queries~\cite[Section~1.1]{SVW2016}; Raz subsequently proved a qualitative +quadratic-memory versus exponential-sample separation for parity +learning~\cite[Theorem~1]{Raz2016}. For regression, Steinhardt and Duchi +established memory-dependent minimax risk bounds in a sparse noisy +model~\cite[Section~1.1]{SteinhardtDuchi2015}. + +Sharan, Sidford, and Valiant developed a nonuniform finite-width +branching-program analysis for continuous regression. Their principal +theorem concerns independent isotropic Gaussian features with independent +uniform additive noise of half-width \(2^{-d/5}\). With at most \(d^2/4\) +bits of memory, success probability at least \(2/3\) for a uniformly +distributed unit signal, and Euclidean accuracy +\(d^{-r}\) in its stated range \(r\le O(d/\log d)\), it gives an +\(\Omega(d\log r)\) sample bound~\cite[Theorem~1]{SSV2019}. +Their projection argument expands high moments into independent copies +and uses successive orthogonalization~\cite[Section~7, Lemma~12]{SSV2019}. +That is a close methodological predecessor of our tuple calculations. +Their lemma treats one noisy interval under a global \(L^2\) density +condition. The estimates here treat exact multirow labels under local +ball or Riesz-potential conditions. Dagan, Kur, and Shamir obtained related +quadratic-space obstructions for approximate null vectors of Gaussian +rows and for empirical residual error in a different equation +model~\cite[Theorems~3 and~10]{DKS2019}. These comparisons concern +distinct statistical targets and observation laws. + +\subsection{How the three arguments work} + +For the integrated estimate, expand the \(q\)th density moment using +\(q\) points of the measure. The collision of their labels is controlled +by the projection of the \(q-1\) differences from the first point. +Successive coordinates of an orthonormal frame have ellipsoidal densities; +their determinant powers telescope, leaving one nonnegative determinant +power and an explicit \(\sqrt d\) scale. Gram--Schmidt converts the +remaining change of variables into inverse affine heights. A tube about +each preceding affine span is covered by balls, so the local mass +hypothesis integrates these heights in reverse order. This proves +Theorem~\ref{p:frame-moment}. + +The first application separates a Gaussian matrix into its positive radial +factor and an independent orthonormal frame. The routing rule may use the +whole radial factor; averaging that dependence produces weights on frame +and projected label. The projected support occupies a \(k\)-ball. +Its volume cancels the dimensional scale in the density norm and returns +the same radius exponent as the incoming local-mass bound. The backward +argument therefore preserves the ambient dimension through every block. +Section~\ref{p:projection} completes this route with local stopping-index +counting and the common full-data bound of Section~\ref{p:geometry-model}. + +For positive cap domination, the order is different. Bounded nonnegative +tests on a countable cap dictionary imply a local ball bound for a rescaled +test measure. The normalized instance of the integrated theorem bounds the +test of \(f_w\mu_B\). To turn all these inequalities into a positive +dominating measure, the proof uses a weighted sum of cap densities. +Finitely many caps occur at each scale and the weighted cost makes the +fine-scale \(L^1\) tail vanish. This gives norm compactness. Strict +separation of the closed downward convex set then produces exactly the +bounded tests already controlled. The resulting positive sums propagate +the unnormalized joint laws of signal and state, with a separate reference +probability for each parent cap. + +The specified-label argument uses \(q\) vectors anchored at \(z\), +rather than \(q-1\) differences anchored at a sampled point. Their images +under the Gaussian rows have covariance equal to the anchored Gram matrix. +The resulting ball-average estimate is uniform in every fixed offset. +Taking a single lower-limit version and applying Fatou's lemma on each +offset graph preserves that estimate for exact densities. Operator-norm +bins and radial shells then control the Riesz potential of earlier +continuations. This route retains a factor of order +\(1+\log(1/\epsilon)\) from its shell count and a global terminal clock; +its own short-run argument controls both. + +The three analytic conclusions and the costs of their regression +applications are compared below. Here \(T\) is the deterministic sample +horizon, and a stopping index is charged only where its value enters the +corresponding argument. + +\begin{center} +\small +\begin{tabular}{@{}>{\raggedright\arraybackslash}p{.18\textwidth} + >{\raggedright\arraybackslash}p{.37\textwidth} + >{\raggedright\arraybackslash}p{.39\textwidth}@{}} +\toprule +Estimate & Analytic guarantee & Regression argument \\ +\midrule +Integrated projection & A density moment integrated over the frame and +label space & Backward local-mass propagation; stopping indices counted +within each block \\[3pt] +Positive cap domination & An actual dominating sum of cap measures, with +weighted radius cost & Forward propagation of joint signal--state laws; +one final sum over stopping indices \\[3pt] +Fixed-offset density & One density version with a moment bound at +every fixed affine offset & Backward Riesz-potential propagation; global +terminal tags and a logarithmic shell count \\ +\bottomrule +\end{tabular} +\end{center} + +Section~\ref{p:geometry-model} supplies the probability normalizations, +finite-state model, Borel-kernel reduction, and common full-data endpoint. +Section~\ref{p:projection} proves the integrated estimates and their +backward application; these two sections suffice for the regression +consequence and the explicit memory tradeoff. Section~\ref{p:cap-section} +uses the normalized projection moment to construct positive dominating +measures and propagate them forward. Section~\ref{p:riesz-section} first +proves its specified-offset estimate, then develops the Riesz recurrence +and its parameter argument. The cap dual constraints test averages on +spherical caps; they are distinct from integration on affine subspheres. diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/positive-caps.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/positive-caps.tex new file mode 100644 index 0000000000000000000000000000000000000000..396770636d528e5cb433ac79f538f11482c1c40c --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/positive-caps.tex @@ -0,0 +1,454 @@ +\section{Positive domination by cap measures}\label{p:cap-section} + +This section proves the sample bound by propagating the joint law of the +signal and each memory state forward through the stream. We bound each +such law by a positive sum of uniform measures on spherical caps. A term +is charged for its mass and for a power of the reciprocal of its radius. +The main step shows that a block of observations increases the total charge +by a controlled factor. The passage from bounds on test functions to an +actual dominating measure requires compactness in the norm of $L^1$. + +We prove Theorem~\ref{p:regression} by a forward argument in the +finite-state experiment of Section~\ref{p:geometry-model}. Use that +section's fixed-seed convention and Borel kernels from +Lemma~\ref{p:borel-kernels}. All bounds below are uniform in the fixed +rules; their scalar probability conclusions are averaged over the shared +seed at the end. + +\subsection{The spherical experiment and the projection input} + +Write $\sigma$ for uniform probability on $S^{d-1}$, and set +\begin{equation}\label{p:cap-parameters} + n=d-1,\qquad k=p=\lfloor d/16\rfloor,\qquad + \alpha=\lambda=\frac n2,\qquad \theta=1-\frac1p. +\end{equation} +We take $d$ sufficiently large that $p\ge2$ and $k,p\ge d/32$. +For $c\in S^{d-1}$ and $00). +\end{equation} +For $01/4$, use the mass bound and $1\le4^\lambda t^\lambda$. +These estimates prove \eqref{p:cap-test-ball-bound}. + +\smallskip\noindent +\emph{The quantitative test estimate.} +All hypotheses of Theorem~\ref{p:frame-positive} now hold for $\nu$. +Indeed, the test, parent cap, and measure are fixed before the fresh +matrix $X$ is drawn. For every orthonormal $k$-frame $P$, the coordinate +density in Lemma~\ref{p:coordinate-density} gives $P_\#\sigma\ll dz$. +Because $h\mu_{c,r}\ll\sigma$, its projection is also absolutely +continuous, and the affine rescaling +\[ + P((s-c)/r)=(Ps-Pc)/r +\] +gives the same conclusion for $P_\#\nu$. The theorem therefore supplies +a jointly measurable density satisfying, for almost every $X$, +\[ + P_\#\nu=g_Xm_k,\qquad g_X(z)=0\text{ for }\norm z>1, + \qquad \E_X\int g_X^p\,dm_k\le\exp(C_Ddp). +\] +The density is relative to $dm_k=dz/v_k$ on the unit ball; $D$, and +hence the new constant, is absolute. Since $s=c+ru$ gives +$Xs=Xc+rHPu$, Tonelli's theorem and H\"older's inequality on the product +law of $X$ and $z$ yield +\begin{equation}\label{p:cap-dual-test} +\begin{split} + \int h f_w\,d\mu_{c,r} + &=\E_X\int K_w(X,Xc+rHz)g_X(z)\,dm_k(z)\\ + &\le\left(\E_X\int g_X^p\,dm_k\right)^{1/p} + \left(\E_X\int K_w(X,Xc+rHz)^{p'}\,dm_k\right)^{1/p'}\\ + &\le\exp(Cd)q_w^\theta,\qquad p'=p/(p-1). +\end{split} +\end{equation} +The last line uses $0\le K_w\le1$. Thus every test satisfying +\eqref{p:cap-dual-constraints} obeys \eqref{p:cap-dual-test}. + +\smallskip\noindent +\emph{Representation by positive cap sums.} +The geometric estimate is complete; the remaining step uses only the +dictionary and its test bound. If $q_w=0$, the feasible test $h=1$ shows +that $f_w\mu_{c,r}=0$, and the empty sum suffices. Otherwise set +$K=\exp(Cd)q_w^\theta>0$, with the constant from +\eqref{p:cap-dual-test}, and define in real $L^1(\sigma)$ +\[ + \mathcal C_K=\left\{\sum_{i\in I}a_i\phi_i: + a_i\ge0,\ \sum_i a_ib_i\le K\right\}. +\] +Each series converges in $L^1$, since $b_i\ge1$ and +$\norm{\phi_i}_{L^1}=1$. This convex set is norm compact. To see this, +take a sequence and choose a coefficient array for each term. A diagonal +subsequence converges coefficientwise, since $0\le a_i\le K/b_i$. +The limiting array has cost at most $K$ by Fatou's lemma. There are +finitely many indices through every level $J$, and uniformly over all +admissible arrays, +\begin{equation}\label{p:cap-compact-tail} + \left\|\sum_{j>J}\sum_{i\text{ at level }j}a_i\phi_i\right\|_{L^1} + =\sum_{j>J}\sum_{i\text{ at level }j}a_i + \le K2^{-\alpha(J+1)}. +\end{equation} +The same estimate holds for the limiting array. Finite-level convergence +followed by \eqref{p:cap-compact-tail} gives convergence in $L^1$. +This proves sequential compactness, and hence compactness in the metric +space $L^1(\sigma)$ +\cite[Section~1.3, Theorem~1.37, p.~29]{SchnaubeltFA2025}. + +Its signed downward closure +\[ + \mathcal D_K=\{u\in L^1(\sigma):u\le v\text{ a.e. for some }v\in\mathcal C_K\} +\] +is convex, contains every nonpositive $L^1$ function, and is norm closed. +In fact, if $u_m\to u$ and $u_m\le v_m\in\mathcal C_K$, compactness +provides a subsequence $v_m\to v\in\mathcal C_K$; then +\[ + \norm{(u-v)_+}_{L^1} + \le\norm{u-u_m}_{L^1}+\norm{v_m-v}_{L^1}\longrightarrow0. +\] +Put $F=f_w\phi_*$. If $F\notin\mathcal D_K$, strict separation of the +closed convex set $\mathcal D_K$ and the compact singleton $\{F\}$ +\cite[Section~5.2, Theorem~5.20(b), pp.~98--99]{SchnaubeltFA2025} +gives a continuous real functional $\ell$ with +\[ + \ell(F)>\sup_{u\in\mathcal D_K}\ell(u). +\] +A probability measure is localizable +\cite[Theorem~211L]{FremlinTaxonomy}, so real $L^1$--$L^\infty$ +duality~\cite[Theorem~243G(b)]{FremlinFunctionSpaces} represents +$\ell(u)=\int hu\,d\sigma$ for a real $h\in L^\infty(\sigma)$. +The finite supremum forces $h\ge0$: if $h\le-\delta$ on a set $E$ +of positive measure, the functions $-t\1_E\in\mathcal D_K$ would make +it infinite. Define +\[ + S_h=\sup_i\frac{\int h\phi_i\,d\sigma}{b_i},\qquad + 0\le S_h\le\norm h_{L^\infty}. +\] +Then +\begin{equation}\label{p:cap-support-functional} + \sup_{u\in\mathcal D_K}\int hu\,d\sigma=KS_h. +\end{equation} +The cost bound gives the upper inequality, and the single terms +$(K/b_i)\phi_i$ give the reverse inequality by taking the supremum over +$i$. This does not require the supremum to be attained. If $S_h=0$, +the parent term gives $\int h\phi_*\,d\sigma=0$, and $0\le f_w\le1$ +implies $\ell(F)=0$, contrary to strict separation. If $S_h>0$, the +function $h/S_h$ satisfies \eqref{p:cap-dual-constraints}, so +\eqref{p:cap-dual-test} again contradicts separation. Therefore +$F\in\mathcal D_K$. An admissible series dominates its density almost +everywhere; integrating gives the asserted measure inequality and cost +bound. Its total mass is at most $K$. +\end{proof} + +\subsection{Propagation, bounded stopping, and the accuracy endpoint} + +The preceding proposition applies to each cap in a joint-state bound. +The positivity of all measures makes the iteration linear before any +estimate is taken. + +\begin{lemma}[Cost of a forward block]\label{p:cap-forward-step} +Let $\mathcal V$ and $\mathcal W$ be finite state sets with +$\abs{\mathcal W}\le2^M$. For each $v\in\mathcal V$, let +$K_{v,w}$ be block kernels of the type in +\eqref{p:cap-reference-probabilities}, with likelihoods $f_{v,w}$. +Let $\eta_v$ be finite positive measures on the sphere and suppose +\[ + \eta_v\le\sum_i a_{v,i}\mu_{c_{v,i},r_{v,i}},\qquad + \mathcal P=\sum_{v,i}a_{v,i}(2/r_{v,i})^\alpha<\infty, +\] +where each sum is countable, $a_{v,i}\ge0$, and $00$ throughout the argument and split its first $t$ indices into +$b=\lceil t/k\rceil$ consecutive blocks of length at most $k$. At a +prefinal boundary retain only paths that have not stopped and are eligible +to continue. In the final block retain exactly the paths stopping at +$t$, with their terminal state. For a fixed starting state and a supplied +block of pairs, integrating the internal randomness gives the probability +of each retained ending state. These are Borel kernels with sum at most +one, since all omitted paths are discarded. A shorter block is padded +by independent unused Gaussian rows, which the kernels ignore. Its +likelihood for a signal $s$ is therefore $f_{v,w}(s)$ of +\eqref{p:cap-reference-probabilities}. Conditional on the fixed seed, +the fresh Gaussian feature rows in the block are jointly independent of the signal and entering state, +indeed of the signal, initialization, past rows, and past coins. Given the +signal, the past affects this likelihood only through the starting state +and the fixed index. + +At any counted boundary, let $\eta_v(E)$ be the probability that +$S\in E$ and the path is counted in state $v$. These are unnormalized +joint measures and their outgoing laws have exactly the formula in +Lemma~\ref{p:cap-forward-step}. Initially, $B(c,2)=S^{d-1}$ and +$\mu_{c,2}=\sigma$. Write $V_0$ for the initial finite state. For the +fixed shared seed $\xi$, joint initialization +independence gives +\[ + \eta_v^\xi(E)=\pi_\xi(v)\sigma(E), + \qquad \pi_\xi(v)=\Prob\{V_0=v\mid\xi\},\qquad + \sum_v\pi_\xi(v)=1. +\] +Thus the initial total cost is one before any discarded paths are removed, +and at most one afterward. +After $b$ applications of that lemma, the counted terminal measures +admit positive cap dominations of total cost at most +$[\exp(Cd)2^{M/p}]^b$. + +At each fixed terminal state and index $t$, the output is drawn from a +fixed law using fresh randomness. Apply Lemma~\ref{p:cap-terminal} to each +term of its dominating cap sum. A term of coefficient $a$ and +radius $r$ contributes at most +$\exp(Cd)(\epsilon/2)^\alpha a(2/r)^\alpha$. +Summing proves the conditional version of +\eqref{p:cap-fixed-stop-explicit}. At $t=0$, use the +initial domination and the same terminal estimate directly. The index +has been fixed throughout, so only the at most $2^M$ states at each +ending boundary were counted. No additional state is charged for the +value of $t$. + +The countable dominating sums are chosen only after fixing the shared +seed. The bound is uniform in that seed, so the common convention of +Section~\ref{p:geometry-model} allows us to average the original +conditional success probabilities. No measurable choice of cap +decompositions across shared seeds is required. +Finally, for $M(d)=o(d^2)$ we eventually have $M(d)\le d^2$. Since +$p\ge d/32$, the logarithm of the extra factor $2^{M/p}$ is then at +most $32(\log2)d$. Absorbing it into the absolute exponential constant +proves \eqref{p:cap-fixed-stop-asymptotic}. +\end{proof} + +We now sum the fixed-index estimate over all possible stopping times +and use the common full-data bound to absorb the resulting lower-order +terms. + +\begin{proof}[Proof of Theorem~\ref{p:regression} by positive cap domination] +Put $L=\log(1/\epsilon)$. + +For the given sequence $M(d)=o(d^2)$, use +\eqref{p:cap-fixed-stop-asymptotic} in all sufficiently large dimensions. +The stopping events are disjoint. Since $k\ge d/32$, $\alpha\ge d/3$, +and $\lceil t/k\rceil\le t/k+1$, summing over $0\le t\le T$ yields +\begin{equation}\label{p:cap-total-success} + \Prob\{\text{success}\} + \le(T+1)\exp(C_1d+C_2T-dL/3). +\end{equation} +Here $(\epsilon/2)^\alpha\le\exp(-dL/3)$, and $C_1,C_2$ are absolute. +If success is at least $2/3$, taking logarithms gives +\[ + \frac{dL}{3} + \le C_1d+C_2T+\log(T+1)+\log(3/2) + \le C_1d+(C_2+1)T+\log(3/2), +\] +since $\log(T+1)\le T$ for every $T\ge0$. +Lemma~\ref{p:haar-linear} also gives $T>d/8$, and therefore $T\ge1$. +Consequently +\[ + dL\le3(8C_1+C_2+2)T. +\] +This proves the claimed bound with an absolute constant, uniformly over +the full accuracy range. Only the dimension at which the memory factor +is absorbed may depend on the sequence $M(d)$. +\end{proof} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/projection.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/projection.tex new file mode 100644 index 0000000000000000000000000000000000000000..75728c5e3fa70a283ebb600b9393b993bfdd69f8 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/projection.tex @@ -0,0 +1,544 @@ +\section{Integrated moments of orthonormal projections} +\label{p:projection} + +This section answers the first question: how does a bound on the mass of +every Euclidean ball control the density after a random orthonormal +projection, when the label is integrated over its whole space? The proof +has two geometric parts. A frame calculation expresses the cost of +projecting several points through their affine Gram determinant. A covering +argument then integrates that determinant using the ball-mass bound. +We retain the dimensional factor in the density norm because it cancels +against the projected support volume in the regression application. + +For integers \(1\le k0\}}. +\end{equation} +Here \(>0\) means positive definite. In particular, this density is +bounded by \((C\sqrt d)^{km}\). + +If \(x_1,\ldots,x_{m+1}\in\R^d\) are affinely independent and +\[ + \delta_i=\dist\bigl(x_i,\aff(x_1,\ldots,x_{i-1})\bigr), + \qquad 2\le i\le m+1, +\] +then the joint Lebesgue density of the columns +\(P(x_i-x_1)\), \(2\le i\le m+1\), is bounded by +\begin{equation}\label{p:frame-difference-density} + (C\sqrt d)^{km}\prod_{i=2}^{m+1}\delta_i^{-k}. +\end{equation} +\end{lemma} + +\begin{proof} +Right rotational invariance allows us to replace the fixed columns of +\(E\) by an independent uniform orthonormal frame. Conditional on \(P\), +its projection has the same law as the first \(k\) coordinates of that +frame. Sample the full columns successively and let \(Z_j\) be the matrix +of the first \(k\) coordinates of its first \(j\) columns. Set +\[ + S_j=I_k-Z_jZ_j^{\mathsf T},\qquad S_0=I_k. +\] +Given the preceding full columns, the next column is uniform on the unit +sphere in their \(d-j\)-dimensional orthogonal complement. The coordinate +map from that complement to the first \(k\) coordinates has row Gram +matrix \(S_j\). When \(S_j>0\), multiplication by \(S_j^{-1/2}\) makes +its rows orthonormal. Lemma~\ref{p:coordinate-density} and a linear change +of variables therefore give the next projected column \(z\) conditional +density +\[ + c(d-j,k)(\det S_j)^{-1/2} + (1-z^{\mathsf T}S_j^{-1}z)^{(d-j-k-2)/2} +\] +on \(z^{\mathsf T}S_j^{-1}z<1\). This expression depends only on \(Z_j\), +so it is also the conditional density given those projected columns. +The strict inequality implies \(S_{j+1}=S_j-zz^{\mathsf T}>0\) +almost surely. The determinant identity +\[ + \det S_{j+1}=\det S_j(1-z^{\mathsf T}S_j^{-1}z) +\] +shows that multiplication of the \(m\) conditional densities cancels +every intermediate determinant. The remaining factor is +\(\det S_m^{(d-k-m-1)/2}\), proving \eqref{p:frame-density-exact}. +Its exponent is nonnegative and \(0<\det S_m\le1\). + +The normalizing constants have the stated scale. For a unit-scale gamma +variable \(G\) of shape \(a=(d-j-k)/2>0\), Cauchy--Schwarz and the gamma +recurrence give +\[ + \frac{\Gamma((d-j)/2)}{\Gamma((d-j-k)/2)} + =\E G^{k/2} + \le(\E G^k)^{1/2} + =\left(\prod_{i=0}^{k-1}(a+i)\right)^{1/2} + \le d^{k/2}. +\] +Together with the factor \(\pi^{-k/2}\), this bounds each \(c(d-j,k)\) +by \((C\sqrt d)^k\). + +For the last assertion, Gram--Schmidt writes +\([x_2-x_1,\ldots,x_{m+1}-x_1]=ER\), where \(E\) has orthonormal +columns and \(R\) is triangular with +\(\abs{\det R}=\prod_{i=2}^{m+1}\delta_i\). Right multiplication by \(R\) +on \(k\)-by-\(m\) matrices has Jacobian \(\abs{\det R}^k\), one factor +for each row. Applying this change of variables to the preceding density +proves \eqref{p:frame-difference-density}. +\end{proof} + +The product of the affine heights in \eqref{p:frame-difference-density} +is \(m!\) times the \(m\)-dimensional volume of the simplex spanned by +the points. Related simplex-volume factors occur in Drury's affine-plane +Jacobian~\cite[Lemma~1, pp.~497--498]{Drury1984}. Here the exponent and +normalization come from the displayed \(k\)-row change of variables. + +\subsection{From local mass to an exact density norm} + +The local growth hypothesis below is of Frostman type, and +inverse-distance averaging is a classical projection method; compare +Mattila~\cite[Lemma~5.1, p.~236, and Theorem~5.4, p.~237]{Mattila1975}. +We prove the quantitative affine-tube and growing-order moment bounds +needed here. + +\begin{lemma}[Affine inverse distances]\label{p:affine-inverse} +Let \(0<\beta\le d\), \(K\ge0\), and \(R>0\). Let \(\nu\) be a finite +positive Borel measure supported in \(B(z,R)\subset\R^d\) such that +\begin{equation}\label{p:all-ball-mass} + \nu(B(b,t))\le Kt^\beta + \qquad(b\in\R^d,\ t>0). +\end{equation} +If \(F\) is an affine \(j\)-plane, \(k\ge1\) is an integer, and +\(\beta-j-k\ge1\), then \(\nu(F)=0\) and +\begin{equation}\label{p:affine-inverse-bound} + \int\dist(x,F)^{-k}\,d\nu(x)\le C^d K R^{\beta-k}, +\end{equation} +provided \(j,k\le d\). +\end{lemma} + +\begin{proof} +For \(00\), continuity from above as \(t\downarrow0\) +gives \(\nu(F)=0\). Distances greater than \(R\) contribute at most +\(R^{-k}\nu(B(z,R))\le KR^{\beta-k}\). On the shell +\(2^{-h-1}R<\dist(x,F)\le2^{-h}R\), \eqref{p:affine-tube} gives a +contribution at most +\[ + 3^j2^{\beta+k}KR^{\beta-k}\,2^{-h(\beta-j-k)}. +\] +The sum over \(h\ge0\) is at most twice its first term. Since +\(\beta,j,k\le d\), the resulting factor is bounded by \(C^d\). +\end{proof} + +\begin{theorem}[Integrated orthonormal-projection moment] +\label{p:frame-moment} +Let \(d,k,q\) be integers and \(\beta\) a real number with +\(k\ge1\), \(q\ge2\), and +\begin{equation}\label{p:frame-margins} + k+q\le d,\qquad 0<\beta\le d,\qquad + \beta-k-(q-2)\ge1. +\end{equation} +Let \(\nu\) be a finite positive Borel measure supported in +\(B(z,R)\), \(R>0\), satisfying \eqref{p:all-ball-mass} for some +\(K\ge0\). Assume that \(P_\#\nu\) is absolutely continuous with +respect to Lebesgue measure for \(\vartheta_{d,k}\)-almost every \(P\). +There is a jointly measurable version \(g(P,u)\) of its density such that +\begin{equation}\label{p:frame-moment-bound} + \left(\int_{\mathcal V_{d,k}}\int_{\R^k}g(P,u)^q\,du\, + d\vartheta_{d,k}(P)\right)^{1/q} + \le C^d(C\sqrt d)^{k(1-1/q)} + K R^{\beta-k(1-1/q)}. +\end{equation} +For every \(P\), this version vanishes when \(\norm{u-Pz}>R\). +\end{theorem} + +\begin{proof} +When \(K=0\), the measure is zero and the assertion is immediate. +For \(K>0\), let \(v_k\) be the unit-ball volume and define +\[ + g_\delta(P,u)=\frac{\nu\{x:\norm{Px-u}\le\delta\}}{v_k\delta^k}. +\] +These are jointly measurable nonnegative functions. Expanding their +\(q\)th powers and using Tonelli's theorem expresses the integrated +moment through \(q\) points \(x_1,\ldots,x_q\) drawn from \(\nu\). +The intersection of the \(q\) label balls of radius \(\delta\) has +volume at most \(v_k\delta^k\); it is empty unless +\(\norm{P(x_i-x_1)}\le2\delta\) for \(i\ge2\). + +Put +\(\delta_i=\dist(x_i,\aff(x_1,\ldots,x_{i-1}))\). +Lemma~\ref{p:affine-inverse} shows that every affine plane encountered +here has zero \(\nu\)-mass: its dimension is at most \(q-2\), so the +last condition in \eqref{p:frame-margins} applies. Consequently the tuple +is affinely independent for \(\nu^{\otimes q}\)-almost every choice. +For such a tuple, Lemma~\ref{p:frame-density} applies with \(m=q-1\); +the condition \(k+q\le d\) is exactly its determinant condition. The +probability of all \(q-1\) inequalities is bounded by +\[ + (C\sqrt d)^{k(q-1)} + \prod_{i=2}^q\delta_i^{-k} + \bigl[v_k(2\delta)^k\bigr]^{q-1}. +\] +The label-ball volumes cancel those in the definition of \(g_\delta\). +Absorbing \(2^{k(q-1)}\) in \(C\), we obtain +\begin{equation}\label{p:frame-before-integration} + \int\!\!\int g_\delta^q\,du\,d\vartheta_{d,k} + \le (C\sqrt d)^{k(q-1)} + \int\prod_{i=2}^q\delta_i^{-k}\,d\nu^{\otimes q}. +\end{equation} +The degenerate tuples form a null set and can be assigned an infinite +upper bound in this nonnegative integral. + +Integrate first in \(x_q\), then in \(x_{q-1},\ldots,x_2\). +At each step Lemma~\ref{p:affine-inverse} bounds the current factor by +\(C^d K R^{\beta-k}\). The remaining first-point mass is at most +\(KR^\beta\). Hence the last integral in +\eqref{p:frame-before-integration} is at most +\[ + C^{d(q-1)}K^qR^{q\beta-k(q-1)}. +\] +This proves the \(q\)th power of \eqref{p:frame-moment-bound} uniformly +in \(\delta\). + +Define \(g(P,u)=\liminf_{j\to\infty}g_{1/j}(P,u)\). The assumed absolute +continuity and Lebesgue differentiation +\cite[Theorem~1.6.19, pp.~146--147]{TaoMeasure2011} imply that, for almost every +\(P\), this is a density of \(P_\#\nu\) for almost every \(u\). +Fatou's lemma on the product space +\cite[Corollary~1.4.47, p.~110]{TaoMeasure2011} gives +\eqref{p:frame-moment-bound}. If \(\norm{u-Pz}>R\), sufficiently small +label balls miss \(P(B(z,R))\), since \(P\) is a contraction; the +displayed version is then zero. This limiting calculation concerns the +exact projected measure. It introduces no noise into an observation. +\end{proof} + +We now spell out the two parameter choices that will be used. Their +normalizations differ, even though their frame calculation is shared. + +\begin{corollary}[All-radii projection of spherical local mass] +\label{p:frame-n30} +For sufficiently large \(d\), put +\[ + n=d-1,\qquad k=q=\lfloor d/4\rfloor,\qquad \beta=n-q. +\] +Let \(0\le f\le1\) be Borel on \(S^{d-1}\), and suppose for \(K\ge0\) that +\[ + \int_{B(b,t)}f\,d\sigma\le Kt^\beta + \qquad(b\in\R^d,\ t>0). +\] +For \(z\in\R^d\), \(R>0\), and \(E=B(z,R)\), the projection of \(\1_E f\sigma\) by +\(P\sim\vartheta_{d,k}\) has a jointly measurable Lebesgue density +\(g(P,u)\) satisfying +\begin{equation}\label{p:frame-n30-bound} + \norm g_{L^q(\vartheta_{d,k}\,du)} + \le C^d(C\sqrt d)^{k(1-1/q)} + K R^{\beta-k(1-1/q)}. +\end{equation} +\end{corollary} + +\begin{proof} +Restriction preserves the stated all-ball bound. For every \(P\), the +projection of \(\sigma\) is absolutely continuous by +Lemma~\ref{p:coordinate-density}, so the same holds for the restricted +weighted measure. The exact margins are +\[ + k+q=2\lfloor d/4\rfloor\le d,\qquad + \beta-k-(q-2)=d+1-3\lfloor d/4\rfloor\ge d/4+1. +\] +Thus Theorem~\ref{p:frame-moment} applies. This also records the +nonnegative frame-determinant exponent +\((d-k-q)/2=(d-2\lfloor d/4\rfloor)/2\). +\end{proof} + +\begin{theorem}[Normalized moment for unit-ball measures] +\label{p:frame-positive} +For sufficiently large \(d\), put +\[ + k=p=\lfloor d/16\rfloor,\qquad \lambda=(d-1)/2. +\] +Let \(X\) be a \(k\)-by-\(d\) standard Gaussian matrix and define +\[ + H=(XX^{\mathsf T})^{1/2},\qquad P=H^{-1}X +\] +on its probability-one full-row-rank event, and set \(P\) to a fixed frame +on the null complement. Let \(m_k\) be uniform +probability on the unit ball of \(\R^k\). Suppose \(\nu\) is a finite +positive Borel measure fixed independently of \(X\), of mass at most one, supported in the unit ball +of \(\R^d\), such that for a fixed \(D>0\), +\[ + \nu(B(b,t))\le D^d t^\lambda + \qquad(b\in\R^d,\ t>0). +\] +Assume \(P_\#\nu\) is absolutely continuous for almost every \(X\). +There is a jointly measurable nonnegative \(g_X(z)\) such that, for almost +every \(X\), \(P_\#\nu(dz)=g_X(z)\,dz/v_k\). This version vanishes when +\(\norm z>1\) and satisfies +\begin{equation}\label{p:frame-positive-bound} + \E_X\int g_X(z)^p\,dm_k(z)\le \exp(C_Ddp). +\end{equation} +\end{theorem} + +\begin{proof} +Right rotational invariance of \(X\) makes \(P\) have law +\(\vartheta_{d,k}\); the independence assertion about \(P,H\) used later +is proved below. The exact determinant and inverse-distance margins are +\[ + k+p=2\lfloor d/16\rfloor\le d,\qquad + \lambda-k-(p-2) + =\frac{d-1}{2}-2\lfloor d/16\rfloor+2 + \ge\frac{3d}{8}+\frac32. +\] +Apply Theorem~\ref{p:frame-moment} with \(q=p\), \(\beta=\lambda\), +\(K=D^d\), and \(R=1\). If \(\ell(P,z)\) is its Lebesgue density, +then \(g_X(z)=v_k\ell(P,z)\) is the required normalized density. Moreover +\[ + \left(\E_X\int g_X^p\,dm_k\right)^{1/p} + =v_k^{1-1/p}\norm\ell_{L^p(\vartheta_{d,k}\,dz)}. +\] +Use the unit-ball volume estimate \eqref{p:unit-ball-volume}. +Since \(k\ge d/32\) for sufficiently large \(d\), +\[ + v_k^{1-1/p}(C\sqrt d)^{k(1-1/p)}\le C^d. +\] +Theorem~\ref{p:frame-moment} now bounds the normalized \(L^p\) norm by +\(\exp(C_Dd)\), proving \eqref{p:frame-positive-bound}. The density +version and its support are inherited from that theorem. The mass-at-most-one +and absolute-continuity hypotheses have been retained explicitly. +\end{proof} + +\subsection{Radial averaging and dimension-preserving propagation} + +We next apply the all-radii estimate to the actual Gaussian data. A learner +may use both the row directions and their lengths and correlations. The +following factorization retains all of that radial information before +averaging it into the routing weights. + +\begin{lemma}[Gaussian radial factor and frame]\label{p:radial-factor} +Let \(X\) be a \(k\)-by-\(d\) standard Gaussian matrix with \(k0,\ 1\le w\le N). +\] +Let \(Q_w(X,y)\) be nonnegative Borel functions on +\(\R^{k\times d}\times\R^k\) with \(\sum_wQ_w\le1\), and define +\[ + f_{\mathrm{in}}(s)=\E_X\sum_{w=1}^N Q_w(X,Xs)f_w(s). +\] +Then +\begin{equation}\label{p:haar-block-bound} + \int_{B(z,R)}f_{\mathrm{in}}\,d\sigma + \le C^dN^{1/q}K R^\beta + \qquad(z\in\R^d,\ R>0). +\end{equation} +\end{proposition} + +\begin{proof} +Write \(X=HP\) as in Lemma~\ref{p:radial-factor} and average the full +radial dependence of the routing rule: +\[ + \tau_w(P,u)=\E_H Q_w(HP,Hu). +\] +These are measurable nonnegative weights with \(\sum_w\tau_w\le1\). +Independence gives the exact identity +\begin{equation}\label{p:haar-routing} + f_{\mathrm{in}}(s)= + \int\sum_w\tau_w(P,Ps)f_w(s)\,d\vartheta_{d,k}(P). +\end{equation} +Fix \(E=B(z,R)\), and let \(g_w(P,u)\) be the exact density in +Corollary~\ref{p:frame-n30} for \(\1_Ef_w\sigma\). After integrating +\eqref{p:haar-routing} over \(E\), pushforward integration yields +\(\int\!\int\sum_w\tau_w g_w\,du\,d\vartheta_{d,k}\). +Each \(g_w\) is supported in \(\norm{u-Pz}\le R\). +On this domain the measure with density \(\tau_w\), relative to frame +probability, label Lebesgue measure, and counting measure in \(w\), has +total mass at most \(v_kR^k\). Hölder's inequality therefore gives +\[ + \int_E f_{\mathrm{in}}\,d\sigma + \le (v_kR^k)^{1-1/q} + \left(\sum_w\int\!\!\int\tau_w g_w^q\,du\,d\vartheta_{d,k}\right)^{1/q} + \le (v_kR^k)^{1-1/q}N^{1/q}\max_w\norm{g_w}_{L^q}. +\] +Substitute \eqref{p:frame-n30-bound}. The radius powers add exactly to +\(\beta\). Also \(k\ge d/8\), so \eqref{p:unit-ball-volume} gives +\[ + v_k^{1-1/q}(C\sqrt d)^{k(1-1/q)}\le C_0^d. +\] +This proves \eqref{p:haar-block-bound}. The original \(Q_w\) could use +all entries of \(H\); only after that dependence was retained did we +average it into \(\tau_w\). +\end{proof} + +\subsection{Backward propagation of continuation success} + +We now apply Proposition~\ref{p:haar-block} to the success probability of +a fresh continuation from each state. Its local-mass exponent survives +every block, and the resulting finite-dimensional estimate retains the +full dependence on \(M\). + +\begin{proposition}[Success bound from orthonormal projections] +\label{p:haar-success-proposition} +There are absolute constants \(C\ge1\) and \(d_0\) such that the +following holds for every integer \(d\ge d_0\), every nonnegative integer +\(M\), every \(0<\epsilon\le1/10\), and every learner of deterministic +finite horizon \(T\) in Section~\ref{p:geometry-model}. Put +\[ + k=q=\lfloor d/4\rfloor,\qquad + b=\lceil T/k\rceil,\qquad N_*=(k+2)2^M. +\] +Then +\begin{equation}\label{p:haar-success} + \Prob\{\text{success}\}\le + C^d\epsilon^q\bigl[C^dN_*^{1/q}\bigr]^b. +\end{equation} +The constant and dimension threshold are independent of \(M,\epsilon,T\) +and the learner's rules. +\end{proposition} + +\begin{proof} +Use the fixed-seed and Borel-kernel convention of +Section~\ref{p:geometry-model}, and set \(n=d-1\) and \(\beta=n-q\). +For each terminal state and stopping index, let \(h(s)\) be the success +probability of its signal-independent output law. The cap estimates of +Lemma~\ref{p:cap-mass} give +\[ + \int_{B(z,R)}h\,d\sigma + \le C^d\min\{R^n,\epsilon^n\} + \le C^d\epsilon^q R^\beta + \qquad(z\in\R^d,\ R>0). +\] +The last inequality follows from \(R\le\epsilon\) or +\(R\ge\epsilon\), respectively. Thus the all-radii coefficient for +every terminal function is at most \(K_0=C^d\epsilon^q\). + +Divide the horizon into \(b\) blocks of \(k\) rows, padding the last +block with independent rows ignored by its rules. Starting at a specified +state and boundary index, define its continuation by a fresh future +execution. This definition does not condition on that state being reached. +The continuation functions are Borel and take values in \([0,1]\) by +Lemma~\ref{p:borel-kernels}. + +Within a block, a destination is either a continuing state at the next +boundary or a terminal state tagged by its local stopping index. The +possible local stopping indices are \(0,1,\ldots,k\), including an +immediate stop. There are at most \((k+1)2^M\) terminal destinations and +\(2^M\) continuing destinations, hence at most \(N_*\) in total. The +global stopping index is determined by the fixed block start and this +local index, so the terminal output retains its full allowed dependence +on time. + +Finite composition of the transition kernels produces Borel routing +probabilities \(Q_w(X,y)\) for these destinations, with +\(\sum_wQ_w\le1\). Their continuation functions \(f_w\) use only +future independent rows and coins; terminal destinations use the functions +\(h\) above. The incoming continuation therefore has precisely the form +\[ + f_{\mathrm{in}}(s)=\E_X\sum_wQ_w(X,Xs)f_w(s) +\] +from Proposition~\ref{p:haar-block}. The Gaussian block is fresh: +conditional on the fixed shared seed, its rows are jointly independent of +the signal and entering state, which depends only on initialization, +past pairs, and past coins. + +Backward induction through the blocks multiplies the all-radii +coefficient by at most \(C^dN_*^{1/q}\) per block. This factor is at +least one, so the coefficient also bounds any earlier terminal option. +Integrate the initial continuation over \(B(0,1)\) and average over the +conditional initial-state law, which is independent of the signal and +rows. Conditional success is at most the right side of +\eqref{p:haar-success}. The bound is uniform in the fixed rules, so the +shared-seed convention gives the same bound for the original learner. +When \(T=0\), the terminal bound supplies the conclusion directly. +\end{proof} + +\begin{proof}[Proof of Corollary~\ref{p:finite-memory-tradeoff} and first +proof of Theorem~\ref{p:regression}] +Put \(L=\log(1/\epsilon)\). Choose an absolute dimension threshold so +that Proposition~\ref{p:haar-success-proposition} applies and +\(k=q\ge d/8\). If success is at least \(2/3\), +\eqref{p:haar-success} implies +\[ + qL\le (b+1)d\log C + +\frac bq\bigl(M\log2+\log(k+2)\bigr) + +\log(3/2). +\] +Since \(b\le8T/d+1\) and \(\log(k+2)/q\) is bounded by an absolute +constant, this gives an absolute \(C_1\) such that +\begin{equation}\label{p:haar-memory-arithmetic} + dL\le C_1(T+d)\left(1+\frac M{d^2}\right). +\end{equation} +No bound on \(M\) was used. Lemma~\ref{p:haar-linear} forces +\(T>d/8\), and hence \(T+d<9T\). Substitution into +\eqref{p:haar-memory-arithmetic} proves +Corollary~\ref{p:finite-memory-tradeoff} with an absolute constant. + +Finally, if \(M(d)=o(d^2)\), then \(1+M(d)/d^2\le2\) for all +sufficiently large \(d\). The corollary proves +Theorem~\ref{p:regression}, with an absolute lower-bound constant and a +dimension threshold that may depend on the memory sequence. This proof +is uniform over the accuracy sequence and uses only within-block stopping +indices. +\end{proof} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/riesz.tex b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/riesz.tex new file mode 100644 index 0000000000000000000000000000000000000000..efe6925957ee186fb356dfc2038a70136095e918 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/build/sections/riesz.tex @@ -0,0 +1,568 @@ +\section{Riesz estimates at fixed affine offsets} +\label{p:riesz-section} + +An integrated density norm does not specify the value of a density on a +matrix-dependent graph of labels. We now construct a single density +version whose moment is controlled at every fixed affine offset. The +bound uses the supremum of ambient Riesz potentials of the original +measure. We first prove this analytic estimate, then use it on distance +shells to propagate finite-state continuation probabilities backward. +The shell argument retains an explicit logarithmic loss. + +Throughout this section, fix an integer $d\ge16$ and put +\begin{equation}\label{p:riesz-parameters} + k=q=\lfloor d/8\rfloor,\qquad \alpha=d/2. +\end{equation} +These choices give +\[ + q\ge2,\qquad d/16\le k=q\le d/8,\qquad + \alpha-k-q\ge1,\qquad d-1-\alpha\ge1. +\] +The matrix $X$ will always denote a $k$ by $d$ standard Gaussian matrix. +Constants denoted by $C\ge1$ are absolute and may increase between uses. + +\subsection{Ambient potentials and affine distances} + +For a measurable $h:S^{d-1}\to[0,1]$, define the supremum of its +ambient Riesz potentials by +\begin{equation}\label{p:riesz-potential} + V(h)=\sup_{z\in\R^d} + \int_{S^{d-1}}\frac{h(s)}{\norm{s-z}^{\alpha}}\,d\sigma(s). +\end{equation} +For a measure $\mu$, the raw Riesz potential is +$U_\alpha^\mu(z)=\int\norm{s-z}^{-\alpha}\,d\mu(s)$ +\cite[Section~1, p.~2]{CalefHardin2008v1}. Thus $V(h)$ is its supremum +over ambient centers for $\mu=h\sigma$; it is not the Riesz energy. +The kernel is understood as a nonnegative extended-valued function +at $s=z$. Write $B_r(z)=\{x\in\R^d:\norm{x-z}\le r\}$, and +intersect such balls with the sphere when integrating against +$\sigma$. + +\begin{lemma}[Ambient potential bounds]\label{p:riesz-potential-finite} +For every $z\in\R^d$ and $00$, +and let $F$ be an affine subspace through $z$ of dimension +$\ell\le q-1$. Then $F\cap B_\rho(z)$ has zero $h\,d\sigma$ +mass, and +\begin{equation}\label{p:riesz-distance-bound} + \int_{B_\rho(z)} + \operatorname{dist}(s,F)^{-k}h(s)\,d\sigma(s) + \le C^d V(h)\rho^{\alpha-k}. +\end{equation} +\end{lemma} + +\begin{proof} +For every ambient center $c$ and radius $r>0$, definition +\eqref{p:riesz-potential} gives +\begin{equation}\label{p:riesz-ball-control} + \int_{B_r(c)}h\,d\sigma\le V(h)r^\alpha . +\end{equation} +Fix $00$, letting $r$ decrease to zero shows that +$F\cap B_\rho(z)$ has zero mass. + +Every point of $B_\rho(z)$ is within distance $\rho$ of $F$. +Apply \eqref{p:riesz-tube} on the shells with outer distance +$r_m=\rho2^{-m}$ and bound the inverse power there by +$(2/r_m)^k$. Their sum is at most +\[ + 3^\ell2^{\alpha+k}V(h)\rho^{\alpha-k} + \sum_{m\ge0}2^{-m(\alpha-\ell-k)}. +\] +Here +$\alpha-\ell-k\ge\alpha-(q-1)-k\ge2$, +so the series is at most $2$. Since +$\ell\le d/8$, $\alpha=d/2$, and $k\le d/8$, the remaining +constant is at most $C^d$. This proves +\eqref{p:riesz-distance-bound}. +\end{proof} + +\subsection{A density at fixed affine offsets} + +The next estimate uses $q$ points anchored at the chosen center $z$. +Their images under each Gaussian row have an anchored Gram matrix as +covariance. The inverse-distance bound just proved will integrate the +negative determinant power in the joint Gaussian density. + +\begin{lemma}[Projection density at deterministic offsets] +\label{p:riesz-density} +Let $z\in\R^d$, $\rho>0$, let +$H\subset S^{d-1}\cap B_\rho(z)$ be measurable, and let +$h:S^{d-1}\to[0,1]$ be measurable. Fix these objects before drawing a +$k$ by $d$ standard Gaussian matrix $X$. The pushforward of +$h\mathbf 1_H\,d\sigma$ by $s\mapsto Xs$ admits a single jointly +Borel function +\[ + R:\R^{k\times d}\times\R^k\longrightarrow[0,\infty] +\] +such that $R(X,\cdot)$ is its Lebesgue density for every full-row-rank +$X$, and, for every fixed $b\in\R^k$, +\begin{equation}\label{p:riesz-density-bound} + \left(\E_X R(X,Xz+b)^q\right)^{1/q} + \le C^d V(h)\rho^{\alpha-k}. +\end{equation} +The same function $R$ is used for all $b$. Moreover, +\begin{equation}\label{p:riesz-density-support} + R(X,y)=0 + \quad\hbox{if}\quad + \norm{y-Xz}>\norm X_{\mathrm{op}}\rho . +\end{equation} +\end{lemma} + +\begin{proof} +Let $\mu$ be the finite Borel measure given by +$\mu(E)=\int_E h\mathbf1_H\,d\sigma$ on Borel subsets of the +sphere. For $\eta>0$, let +$B_\eta^k=\{u\in\R^k:\norm u\le\eta\}$ and let +$\operatorname{vol}_k$ denote $k$-dimensional Lebesgue measure. Define +\begin{equation}\label{p:riesz-ball-average} + R_\eta(X,y)= + \frac{1}{\operatorname{vol}_k(B_\eta^k)} + \int \mathbf1_{\{\norm{Xt-y}\le\eta\}}\,d\mu(t). +\end{equation} +The integrand is Borel in $(X,y,t)$, so the parameter integral is +Borel in $(X,y)$. We first prove +\eqref{p:riesz-density-bound} with $R_\eta$ in place of $R$, +uniformly in $\eta$. + +Expand its $q$th moment at $Xz+b$ by Tonelli's theorem, using +$t_1,\ldots,t_q$ from the $q$ copies of $\mu$. For a fixed tuple, +the columns $X(t_j-z)$ form an array whose $k$ rows are independent +centered Gaussian vectors with covariance +\[ + G=\bigl(\inner{t_i-z}{t_j-z}\bigr)_{1\le i,j\le q}. +\] +If $G$ is nonsingular, their joint density is bounded everywhere by +$(2\pi)^{-kq/2}(\det G)^{-k/2}$. Therefore the probability that +all columns belong to the radius-$\eta$ ball about $b$ is at most +this bound times $\operatorname{vol}_k(B_\eta^k)^q$. + +The singular tuples have zero $\mu^q$ mass. Indeed, at the first +linearly dependent vector $t_j-z$, the point $t_j$ lies in +$z+\operatorname{span}(t_1-z,\ldots,t_{j-1}-z)$, an affine space +through $z$ of dimension at most $q-1$. Lemma~\ref{p:riesz-distance} +assigns zero $\mu$ mass to it. The degeneracy set is measurable, +and Fubini's theorem applied at each $j$ proves the claim. For a +nonsingular tuple, Gram--Schmidt triangularizes the matrix with +columns $t_j-z$ with diagonal entries equal to the successive +orthogonal distances. Taking the determinant of its Gram matrix +gives +\begin{equation}\label{p:riesz-gram} + (\det G)^{-k/2} + =\prod_{j=1}^q + \operatorname{dist}\!\left( + t_j,z+\operatorname{span}(t_1-z,\ldots,t_{j-1}-z) + \right)^{-k}. +\end{equation} +Integrate this product first in $t_q$, then in $t_{q-1}$, and so on. +For every nonsingular prefix, Lemma~\ref{p:riesz-distance} bounds +the next integral by $C^dV(h)\rho^{\alpha-k}$. The exceptional +prefixes have zero product measure by the same degeneracy argument. +It follows that the integral of +\eqref{p:riesz-gram} is at most +$(C^dV(h)\rho^{\alpha-k})^q$. The ball volumes cancel in the +moment expansion, and the Gaussian prefactor is at most one. +Thus +\[ + \E_X R_\eta(X,Xz+b)^q + \le \bigl(C^dV(h)\rho^{\alpha-k}\bigr)^q +\] +for every $\eta>0$ and every fixed $b$. + +Now choose the function once and for all: +\begin{equation}\label{p:riesz-density-version} + R(X,y)=\liminf_{m\to\infty}R_{1/m}(X,y). +\end{equation} +It is jointly Borel. To see that it is a density, fix a full-row-rank +$X$. Set $D=(XX^{\mathsf T})^{1/2}$ and $P=D^{-1}X$, so $D$ is invertible, +$P$ has orthonormal rows, and $X=DP$. +Lemma~\ref{p:coordinate-density} makes $P_\#\sigma$ absolutely +continuous, and the invertible map $D$ preserves absolute continuity. +Since $\mu\le\sigma$, the pushforward $X_\#\mu$ is also absolutely +continuous. The Lebesgue differentiation +theorem~\cite[Theorem~1.6.19]{TaoMeasure2011}, applied to its +$L^1$ density, shows that +\eqref{p:riesz-density-version} equals that density almost +everywhere in $y$. A Gaussian $X$ has full row rank almost +surely, since each successive Gaussian row avoids the span of +the preceding rows. + +For each fixed $b$, the identity +$R(X,Xz+b)=\liminf_m R_{1/m}(X,Xz+b)$ holds by the definition of +the single function $R$. Fatou's +lemma~\cite[Corollary~1.4.47]{TaoMeasure2011} over $X$ now gives +\eqref{p:riesz-density-bound}. These are separate moment +statements for each deterministic $b$; no simultaneous +almost-sure bound in $X$ over all offsets is asserted. Finally, +if $\norm{y-Xz}>\norm X_{\mathrm{op}}\rho$, then every $Xt$ with +$t\in H$ is at distance at least +$\norm{y-Xz}-\norm X_{\mathrm{op}}\rho>0$ from $y$. The averages in +\eqref{p:riesz-ball-average} are eventually zero, proving +\eqref{p:riesz-density-support}. +\end{proof} + +The explicit choice \eqref{p:riesz-density-version} is the feature +needed for the next step. A density chosen only up to a +Lebesgue-null set in $y$ for each $X$ would not justify the moment +estimate at the matrix-dependent point $Xz+b$. +An integrated $L^q(dX\,dy)$ estimate controls a density only up to +product-null sets, while the graph $y=Xz+b$ has zero product measure. +Positive cap domination instead controls the measure of every Borel +set by a positive sum of cap measures. Neither conclusion supplies +the specified-offset moment in Lemma~\ref{p:riesz-density}; its +anchored determinant estimate and explicit limit are therefore kept +as a separate argument. + +\subsection{Continuation probabilities and the terminal potential} + +We apply the specified-offset estimate to the finite-state experiment of +Section~\ref{p:geometry-model}. Fix nonnegative integers $M,T$ and an +accuracy $0<\epsilon\le1/10$, and write +\[ + p_{\mathrm{succ}}=\Prob\{\arccos\inner{\widehat S}{S}\le\epsilon\}, + \qquad L=\log(1/\epsilon),\qquad + N=\lceil T/k\rceil,\qquad W=(T+2)2^M. +\] +The success probability averages over the uniform signal, Gaussian +features, and learner randomness. The factor $W$ will count the retained +states together with their possible terminal stopping indices. + +\begin{proposition}[Riesz success estimate]\label{p:riesz-success} +There is an absolute constant $C_0\ge1$ such that every learner just +specified satisfies +\begin{equation}\label{p:riesz-success-bound} + p_{\mathrm{succ}}\le B J^N,\qquad + B=C_0^d\epsilon^{\,d/2-1},\qquad + J=1+C_0^d(1+L)W^{1/q}. +\end{equation} +This includes $T=0$. No asymptotic assumption on $M$ is needed for +this estimate. +\end{proposition} + +We first bound the potential of each terminal output law. To control its +increase across a block, we apply Lemma~\ref{p:riesz-density} at $Xz+b$ +and then integrate over $b$. The factor $1+L$ in +\eqref{p:riesz-success-bound} counts the distance shells and remains +in the final parameter argument. + +Use the fixed-seed and Borel-kernel convention of +Section~\ref{p:geometry-model}: first fix an admissible shared seed, +under which the initial state is independent of the signal and rows, +and then apply Lemma~\ref{p:borel-kernels} to choose Borel rules. These +rules define continuations from every state. All bounds below are uniform +in the fixed rules; after proving the conditional bound, we average the +original conditional success probabilities over the seed. + +Complete unused rules and terminal laws arbitrarily, and force a halt +by $T$. After a halt, retain a tag consisting of the stopping index +and terminal state and ignore further samples. At any time there are +at most $2^M$ live states and at most $(T+1)2^M$ such tags, including +a possible immediate halt. Hence the stated bound $W$ applies. Add +independent unused Gaussian rows after $T$ to complete the last block of length $k$. Let +$\mathcal V_i$ be the state set at the boundary after block $i$, +$0\le i\le N$; thus $\lvert\mathcal V_i\rvert\le W$. + +For every $v\in\mathcal V_i$ and $s\in S^{d-1}$, let $h_i(v,s)$ be +the success probability of the remaining execution started at that +boundary state. This definition also applies to states not reached +by the original run. At $i=N$ it is the success probability of the +fixed output law associated with the terminal state. These functions +are Borel and take values in $[0,1]$: the terminal assertion follows +by integrating the Borel success indicator against the output law, +and the assertion at earlier boundaries follows inductively from +the formula below. If $X$ is a fresh $k$ by $d$ standard Gaussian +matrix, then for $1\le i\le N$ +\begin{equation}\label{p:riesz-recursion} + h_{i-1}(v,s) + =\E_X\sum_{w\in\mathcal V_i}K^{(i)}_{v,w}(X,Xs)h_i(w,s). +\end{equation} +Here $K^{(i)}_{v,w}(X,y)$ is the probability of the block transition +from $v$ to $w$ on the exact row pairs $(X,y)$. These kernels are +Borel, nonnegative, and sum to one in $w$ for every $(X,y)$. +They incorporate internal randomization and the ignored rows after a +halt. The functions $h_i(w,\cdot)$ are fixed independently of $X$. +The fresh Gaussian block is independent of the signal, initialization, +past rows, and past coins. Starting the continuation at the specified +state therefore gives exactly \eqref{p:riesz-recursion}; the state and +boundary index contain all retained sample-dependent information. + +\begin{lemma}[Terminal potential]\label{p:riesz-terminal-lemma} +Let $Q$ be a signal-independent probability law on unit-vector outputs, +and let $h(s)=Q\{u:\arccos\inner{u}{s}\le\epsilon\}$. Then +\begin{equation}\label{p:riesz-terminal} + V(h)\le C^d\epsilon^{d-1-\alpha}. +\end{equation} +\end{lemma} + +\begin{proof} +Lemma~\ref{p:cap-mass} gives $\int h\,d\sigma\le\epsilon^{d-1}$, +also for a randomized output law. For any ambient center $z$, use +$h\le1$ and \eqref{p:riesz-inner} inside $B_\epsilon(z)$. Outside +that ball, bound the kernel by $\epsilon^{-\alpha}$ and use the total +mass bound. The sum is at most $C^d\epsilon^{d-1-\alpha}$ uniformly +in $z$, proving \eqref{p:riesz-terminal}. +\end{proof} + +Set +\[ + V_i=\max_{v\in\mathcal V_i}V(h_i(v,\cdot)). +\] +The maximum is finite by Lemma~\ref{p:riesz-potential-finite}. +Lemma~\ref{p:riesz-terminal-lemma} gives $V_N\le C^d\epsilon^{d-1-\alpha}$. +We next estimate the contribution of a single distance shell in +\eqref{p:riesz-recursion}. + +\subsection{One block and the distance shells} + +\begin{lemma}[One shell across a block]\label{p:riesz-shell} +For $1\le i\le N$, $v\in\mathcal V_{i-1}$, $z\in\R^d$, and +$\rho>0$, let +$H_\rho(z)=\{s\in S^{d-1}:\rho/2<\norm{s-z}\le\rho\}$. +Then +\begin{equation}\label{p:riesz-shell-bound} + \int_{H_\rho(z)} + \frac{h_{i-1}(v,s)}{\norm{s-z}^{\alpha}}\,d\sigma(s) + \le C^d W^{1/q}V_i. +\end{equation} +\end{lemma} + +\begin{proof} +For each successor $w$, use the particular density $R_w(X,y)$ +of Lemma~\ref{p:riesz-density} for the measure +$h_i(w,\cdot)\mathbf1_{H_\rho(z)}\,d\sigma$. +Tonelli's theorem, \eqref{p:riesz-recursion}, and pushforward +integration give +\begin{equation}\label{p:riesz-shell-pushforward} +\begin{split} + \int_{H_\rho(z)} + \frac{h_{i-1}(v,s)}{\norm{s-z}^{\alpha}}\,d\sigma(s) + &\le (2/\rho)^\alpha + \E_X\int_{\R^k} + \sum_wK^{(i)}_{v,w}(X,y)R_w(X,y)\,dy\\ + &\le (2/\rho)^\alpha + \E_X\int_{\norm{y-Xz}\le\norm X_{\mathrm{op}}\rho} + \max_wR_w(X,y)\,dy . +\end{split} +\end{equation} +The pushforward identity used in the first inequality holds for every +full-row-rank $X$, hence almost surely. The second inequality uses +the probability-vector property of $K^{(i)}$ and +\eqref{p:riesz-density-support}. + +We need a deterministic integration region after averaging over +$X$. Choose a sufficiently large absolute $D_0$ and partition +the matrices into +\[ + E_1=\{\norm X_{\mathrm{op}}\le D_0\sqrt d\},\qquad + E_j=\{D_0(j-1)\sqrt d<\norm X_{\mathrm{op}}\le D_0j\sqrt d\} + \quad(j\ge2). +\] +These bins obey +\begin{equation}\label{p:riesz-bins} + \P(E_j)\le \exp\bigl(-4(j-1)^2d\bigr)\qquad(j\ge2). +\end{equation} +For completeness, take $1/4$-nets on the unit spheres of $\R^d$ +and $\R^k$. A maximal separated set has at most $9^d$ and +$9^k$ points, respectively, by the disjoint-ball volume argument. +Approximating each unit vector by its net point incurs an error +at most $\norm X_{\mathrm{op}}/2$ in a bilinear form. Hence +$\norm X_{\mathrm{op}}$ is at most twice the largest absolute +bilinear form on the two nets. Each such form is standard +Gaussian, so its elementary tail bound and a union bound give +\begin{equation}\label{p:riesz-operator-tail} + \P\{\norm X_{\mathrm{op}}>t\} + \le 2\cdot9^{d+k}\exp(-t^2/8). +\end{equation} +Putting $t=D_0(j-1)\sqrt d$ proves +\eqref{p:riesz-bins} after increasing $D_0$, since $k\le d/8$. + +For every deterministic $b\in\R^k$, H\"older's inequality and +\eqref{p:riesz-density-bound} give +\begin{equation}\label{p:riesz-bin-holder} +\begin{split} + \E_X\!\left[\mathbf1_{E_j}\max_wR_w(X,Xz+b)\right] + &\le \P(E_j)^{1-1/q} + \left(\sum_w\E_X R_w(X,Xz+b)^q\right)^{1/q}\\ + &\le \P(E_j)^{1-1/q}W^{1/q} + C^dV_i\rho^{\alpha-k}. +\end{split} +\end{equation} +No conditional version of the density lemma is being used: +H\"older's inequality pays for the bin event. For a fixed matrix +in $E_j$, translate $y=Xz+b$ in +\eqref{p:riesz-shell-pushforward}; the remaining region is +$\norm b\le D_0j\sqrt d\,\rho$. Joint measurability permits +Tonelli's theorem to integrate \eqref{p:riesz-bin-holder} over +this deterministic ball. + +The volume of that ball is at most $C^d j^k\rho^k$. Indeed, the unit-ball bound \eqref{p:unit-ball-volume} gives +$v_k\le(2\pi e/k)^{k/2}$. Since $d/k\le16$, +$v_k(D_0j\sqrt d\,\rho)^k\le C^d j^k\rho^k$. +Also, by $q\ge2$ and \eqref{p:riesz-bins}, +\[ + \sum_{j\ge1}j^k\P(E_j)^{1-1/q} + \le 1+\sum_{j\ge2} + \exp\bigl(k\log j-2(j-1)^2d\bigr) + \le 2 . +\] +For the last bound use +$k\log j\le(d/8)(j-1)^2$ for $j\ge2$ and sum the resulting +geometric upper bound. Combining the volume and bin estimates +with \eqref{p:riesz-shell-pushforward} proves +\eqref{p:riesz-shell-bound}. The powers of the shell radius +cancel exactly: +\[ + \rho^{-\alpha}\rho^{\alpha-k}\rho^k=1. +\] +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{p:riesz-success}] +Outside $B_\epsilon(z)$ use the shells +\[ + 2^m\epsilon<\norm{s-z}\le2^{m+1}\epsilon, + \qquad m=0,1,\ldots . +\] +For every ambient $z$, at most $C(1+L)$ of these meet the +sphere. If $\norm z\le3$, all distances are at most $4$, giving +at most $\lceil\log_2(4/\epsilon)\rceil$ nonempty shells. If +$\norm z>3$, the distances lie between $\norm z-1$ and +$\norm z+1$, whose ratio is less than $2$; only a bounded number +of dyadic shells can then meet that interval. + +Inside $B_\epsilon(z)$ use $h_{i-1}\le1$ and +\eqref{p:riesz-inner}. On each remaining shell use +Lemma~\ref{p:riesz-shell}. Taking the supremum over $z$ and the +maximum over $v$ gives +\begin{equation}\label{p:riesz-step} + V_{i-1}\le C^d\epsilon^{d-1-\alpha} + +C^d(1+L)W^{1/q}V_i . +\end{equation} +Choose $C_0$ large enough that the first term here and the +terminal value $V_N$ are at most $B$ in +\eqref{p:riesz-success-bound}, and that the second coefficient is +at most $J-1$. With $U_i=\max\{V_i,B\}$, equation +\eqref{p:riesz-step} gives $U_{i-1}\le J U_i$, while $U_N=B$. +Hence $V_0\le BJ^N$, also when $N=0$. + +At $z=0$ the kernel in \eqref{p:riesz-potential} equals one on +the unit sphere, so $\int h_0(v,s)\,d\sigma(s)\le V_0$ for +each initial state. For almost every fixed shared seed, averaging over its +conditional initial-state law preserves the bound, because that law is +independent of the signal and rows. Averaging the original conditional +success probabilities over the seed then +preserves it again. This proves \eqref{p:riesz-success-bound}. +\end{proof} + +\subsection{The logarithmic lower bound} + +We now prove Theorem~\ref{p:regression} from the Riesz success estimate. +The parameter argument retains the stopping-index factor in $W$ and +the shell-count factor $1+L$. Terminal-tag counting controls both in +the only case still requiring proof, namely $Td/8$; hence the additive $d$ on the right of +\eqref{p:riesz-final} is at most $8T$. +For $d\ge4$ the left coefficient is at least $d/4$, so +$T\ge c\,dL$ for an absolute $c>0$. This combines with the +case $T\ge dL$ already separated. +\end{proof} diff --git a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..38c72b52a866d62b6d29d5fbdc55f2ff7674b906 --- /dev/null +++ b/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2a212c32e3e62170f5327a686743ecf6752fe68e4993eadc5b6badfc76b8cf76 +size 508325 diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/README.md b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..d9808e6323b764a94569216fdbf9b76a36974be7 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/README.md @@ -0,0 +1,18 @@ +# [Projective Hodge lines and ordinary Iitaka subadditivity](paper.pdf) + +**Author:** OpenAI + +**Date:** September 27, 2026 + +## Citation + +```bibtex +@misc{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026, + author = {{OpenAI}}, + title = {{Projective Hodge lines and ordinary Iitaka subadditivity}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf}{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/figures/period-and-original-base.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/figures/period-and-original-base.tex new file mode 100644 index 0000000000000000000000000000000000000000..e59abdd1fcfb266cff8a91715b0ac2a8e750a262 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/figures/period-and-original-base.tex @@ -0,0 +1,14 @@ +\begin{figure}[ht] +\centering +\begin{tikzcd}[column sep=large,row sep=large] +\widetilde X \arrow[r,"g"] \arrow[dr,"f\mu"'] & +Y \arrow[d,"q"] \arrow[r,"p"] & S \\ +& Z & +\end{tikzcd} +\caption{The relative Iitaka map $g$ has Kodaira-dimension-zero +generic fibre. The original base map $q$ and the independent period +map $p$ start from the same variety $Y$. The divisor $M=p^*L$ +comes from $S$, while the desired addition inequality is measured +over $Z$.} +\label{fig:two-bases} +\end{figure} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/main.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..7005bd83d092c35b2290a1f184b659ab1223734e --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/main.tex @@ -0,0 +1,73 @@ +\documentclass[11pt]{amsart} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools} +\usepackage[margin=1.08in]{geometry} +\usepackage[expansion=false]{microtype} +\usepackage{enumitem,booktabs,array,needspace,tikz-cd} +\usepackage[hidelinks,pdfusetitle]{hyperref} +\usepackage[capitalise,noabbrev,nameinlink]{cleveref} +\hypersetup{pdftitle={Projective Hodge lines and ordinary Iitaka subadditivity},pdfauthor={OpenAI},bookmarksdepth=2} +\newcommand{\C}{\mathbf C} +\newcommand{\Q}{\mathbf Q} +\newcommand{\R}{\mathbf R} +\newcommand{\Z}{\mathbf Z} +\newcommand{\N}{\mathbf N} +\newcommand{\cO}{\mathcal O} +\newcommand{\cV}{\mathcal V} +\newcommand{\ddc}{dd^c} +\DeclareMathOperator{\Gr}{Gr} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Supp}{Supp} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\id}{id} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{example}[theorem]{Example} +\numberwithin{equation}{section} +\setcounter{tocdepth}{1} +\emergencystretch=1em +\allowdisplaybreaks[2] +\title[Projective Hodge lines and ordinary subadditivity]{Projective Hodge lines and ordinary Iitaka subadditivity} +\author{OpenAI} +\date{September 27, 2026} +\begin{document} +\begin{abstract} +We prove the ordinary Iitaka subadditivity conjecture for surjective +projective morphisms with connected fibers between smooth connected +projective varieties over algebraically closed fields of characteristic +zero. If $F$ is the geometric generic fiber of $f:X\to Z$, then +$\kappa(X)\geq\kappa(F)+\kappa(Z)$. +\end{abstract} +\maketitle +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-rational} +\input{sections/03-limits} +\input{sections/04-rankloss} +\input{sections/05-quotient} +\input{sections/06-canonical} +\input{sections/07-ordinary} +\input{sections/08-integral} +\input{sections/09-thom} +\bibliographystyle{amsplain} +\bibliography{references} +\end{document} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/references.bib b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..ec7569f9745937c9fb7ded43daa055063a98ea04 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/references.bib @@ -0,0 +1,342 @@ +@article{AmbroBoundary, + author = {Ambro, Florin}, + journal = {Journal of Differential Geometry}, + note = {Theorem numbering is cited from arXiv:math/0210271v1}, + number = {2}, + pages = {229--255}, + title = {{Shokurov}'s boundary property}, + url = {https://arxiv.org/abs/math/0210271v1}, + volume = {67}, + year = {2004} +} + +@article{AmbroModuli, + author = {Ambro, Florin}, + doi = {10.1112/S0010437X04001071}, + journal = {Compositio Mathematica}, + note = {\href{https://doi.org/10.1112/S0010437X04001071}{doi:10.1112/S0010437X04001071}; theorem numbering follows \href{https://arxiv.org/abs/math/0308143v1}{arXiv:math/0308143v1}, 14 August 2003}, + number = {2}, + pages = {385--403}, + title = {The moduli b-divisor of an lc-trivial fibration}, + url = {https://doi.org/10.1112/S0010437X04001071}, + volume = {141}, + year = {2005} +} + +@article{AndreMonodromy, + author = {Andr{\'e}, Yves}, + journal = {Compositio Mathematica}, + number = {1}, + pages = {1--24}, + title = {{Mumford--Tate} groups of mixed {Hodge} structures and the theorem of the fixed part}, + url = {https://www.numdam.org/item/CM_1992__82_1_1_0/}, + volume = {82}, + year = {1992} +} + +@misc{BFMT, + archiveprefix = {arXiv}, + author = {Bakker, Benjamin and Filipazzi, Stefano and Mauri, Mirko and Tsimerman, Jacob}, + eprint = {2508.19215}, + note = {\href{https://arxiv.org/abs/2508.19215v2}{arXiv:2508.19215v2}, 18 December 2025}, + primaryclass = {math.AG}, + title = {{Baily--Borel} compactifications of period images and the b-semiampleness conjecture}, + url = {https://arxiv.org/abs/2508.19215v2}, + year = {2025} +} + +@article{BrunebarbeCadorel, + author = {Brunebarbe, Yohan and Cadorel, Beno{\^i}t}, + doi = {10.1093/imrn/rny054}, + journal = {International Mathematics Research Notices}, + note = {Locators: \href{https://arxiv.org/abs/1707.01327v1}{arXiv:1707.01327v1}}, + number = {6}, + pages = {1601--1609}, + title = {Hyperbolicity of varieties supporting a variation of {Hodge} structure}, + url = {https://arxiv.org/abs/1707.01327v1}, + volume = {2020}, + year = {2020} +} + +@article{CKS, + author = {Cattani, Eduardo and Kaplan, Aroldo and Schmid, Wilfried}, + doi = {10.2307/1971333}, + journal = {Annals of Mathematics}, + number = {3}, + pages = {457--535}, + series = {2}, + title = {Degeneration of {Hodge} structures}, + url = {https://annals.math.princeton.edu/1986/123-3/p02}, + volume = {123}, + year = {1986} +} + +@article{FujinoMori, + author = {Fujino, Osamu and Mori, Shigefumi}, + journal = {Journal of Differential Geometry}, + number = {1}, + pages = {167--188}, + title = {A canonical bundle formula}, + url = {https://www.math.kyoto-u.ac.jp/~fujino/Canonical.pdf}, + volume = {56}, + year = {2000} +} + +@incollection{FujinoWeakPositivity, + author = {Fujino, Osamu}, + booktitle = {Algebraic Varieties and Automorphism Groups}, + doi = {10.2969/aspm/07510073}, + note = {Locators: \href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{author version 0.54, 30 June 2015}}, + pages = {73--118}, + publisher = {Mathematical Society of Japan}, + series = {Advanced Studies in Pure Mathematics}, + title = {Notes on the weak positivity theorems}, + url = {https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}, + volume = {75}, + year = {2017} +} + +@article{Hashizume, + author = {Hashizume, Kenta}, + doi = {10.3792/pjaa.96.017}, + journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences}, + note = {Theorem numbering is cited from arXiv:1902.10923v2}, + number = {10}, + pages = {87--92}, + title = {Log {Iitaka} conjecture for abundant log canonical fibrations}, + url = {https://arxiv.org/abs/1902.10923v2}, + volume = {96}, + year = {2020} +} + +@article{IitakaOriginal, + author = {Iitaka, Shigeru}, + doi = {10.2969/jmsj/02320356}, + journal = {Journal of the Mathematical Society of Japan}, + number = {2}, + pages = {356--373}, + title = {On {$D$}-dimensions of algebraic varieties}, + url = {https://www.jstage.jst.go.jp/article/jmath1948/23/2/23_2_356/_article}, + volume = {23}, + year = {1971} +} + +@article{KawamataCurves, + author = {Kawamata, Yujiro}, + doi = {10.1007/BF01404756}, + journal = {Inventiones Mathematicae}, + number = {1}, + pages = {57--71}, + title = {{Kodaira} dimension of algebraic fiber spaces over curves}, + url = {https://link.springer.com/article/10.1007/BF01404756}, + volume = {66}, + year = {1982} +} + +@article{KawamataKodaira, + author = {Kawamata, Yujiro}, + doi = {10.1515/crll.1985.363.1}, + journal = {Journal f{\"u}r die reine und angewandte Mathematik}, + pages = {1--46}, + title = {Minimal models and the {Kodaira} dimension of algebraic fiber spaces}, + url = {https://doi.org/10.1515/crll.1985.363.1}, + volume = {363}, + year = {1985} +} + +@incollection{Kollar87, + address = {Tokyo and Amsterdam}, + author = {Koll{\'a}r, J{\'a}nos}, + booktitle = {Algebraic Geometry, Sendai, 1985}, + doi = {10.2969/aspm/01010361}, + pages = {361--398}, + publisher = {Kinokuniya and North-Holland}, + series = {Advanced Studies in Pure Mathematics}, + title = {Subadditivity of the {Kodaira} dimension: fibers of general type}, + url = {https://doi.org/10.2969/aspm/01010361}, + volume = {10}, + year = {1987} +} + +@article{KovacsPatakfalvi, + author = {Kov{\'a}cs, S{\'a}ndor J. and Patakfalvi, Zsolt}, + doi = {10.1090/jams/871}, + journal = {Journal of the American Mathematical Society}, + note = {Locators: \href{https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf}{62-page author manuscript}}, + number = {4}, + pages = {959--1021}, + title = {Projectivity of the moduli space of stable log-varieties and subadditivity of log-{Kodaira} dimension}, + url = {https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf}, + volume = {30}, + year = {2017} +} + +@misc{MaeharaIitaka, + archiveprefix = {arXiv}, + author = {Maehara, Kazuhisa}, + eprint = {0910.0604}, + howpublished = {arXiv preprint}, + note = {Version 3, 16 May 2010; first posted 4 October 2009}, + title = {{Iitaka--Viehweg} conjectures {$C$} and {$C^{++}$}}, + url = {https://arxiv.org/abs/0910.0604v3}, + year = {2010} +} + +@article{Schmid, + author = {Schmid, Wilfried}, + doi = {10.1007/BF01389674}, + journal = {Inventiones Mathematicae}, + number = {3--4}, + pages = {211--319}, + title = {Variation of {Hodge} structure: The singularities of the period mapping}, + url = {https://doi.org/10.1007/BF01389674}, + volume = {22}, + year = {1973} +} + +@misc{TsujiDirectImages, + archiveprefix = {arXiv}, + author = {Tsuji, Hajime}, + eprint = {1012.0884}, + howpublished = {arXiv preprint}, + note = {Version 1, 4 December 2010}, + title = {Global generation of the direct images of relative pluricanonical systems}, + url = {https://arxiv.org/abs/1012.0884v1}, + year = {2010} +} + +@incollection{ViehwegAdditivity, + address = {Tokyo and Amsterdam}, + author = {Viehweg, Eckart}, + booktitle = {Algebraic Varieties and Analytic Varieties (Tokyo, 1981)}, + doi = {10.2969/aspm/00110329}, + pages = {329--353}, + publisher = {Kinokuniya and North-Holland}, + series = {Advanced Studies in Pure Mathematics}, + title = {Weak positivity and the additivity of the {Kodaira} dimension for certain fibre spaces}, + url = {https://esaga.uni-due.de/eckart.viehweg/ecpubl/}, + volume = {1}, + year = {1983} +} + +@article{BakkerBrunebarbeTsimerman, + author = {Bakker, Benjamin and Brunebarbe, Yohan and Tsimerman, Jacob}, + title = {o-minimal {GAGA} and a conjecture of {Griffiths}}, + journal = {Inventiones Mathematicae}, + volume = {232}, + number = {1}, + year = {2023}, + pages = {163--228}, + doi = {10.1007/s00222-022-01166-1}, + note = {Locators: arXiv:1811.12230v3, 23 October 2022}, + url = {https://arxiv.org/abs/1811.12230v3} +} + +@misc{StacksFlatten, + author = {{The Stacks Project Authors}}, + title = {The Stacks Project}, + year = {2026}, + note = {Lemma 38.31.1, Tag 081R; accessed September 26, 2026}, + url = {https://stacks.math.columbia.edu/tag/081R} +} + +@article{DeligneHodgeII, + author = {Deligne, Pierre}, + title = {Th{\'e}orie de {Hodge}. {II}}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + volume = {40}, + year = {1971}, + pages = {5--57}, + doi = {10.1007/BF02684692}, + note = {\href{https://doi.org/10.1007/BF02684692}{doi:\nolinkurl{10.1007/BF02684692}}}, + url = {https://www.numdam.org/item/PMIHES_1971__40__5_0.pdf} +} + +@incollection{CattaniKaplanDegenerating, + author = {Cattani, Eduardo and Kaplan, Aroldo}, + title = {Degenerating variations of {Hodge} structure}, + booktitle = {Th{\'e}orie de {Hodge}---Luminy, Juin 1987}, + series = {Ast{\'e}risque}, + number = {179--180}, + publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France}, + year = {1989}, + pages = {67--96}, + note = {\href{https://numdam.org/item/AST_1989__179-180__67_0/}{Numdam archive}}, + url = {https://numdam.org/item/AST_1989__179-180__67_0/} +} + +@article{BirkarIitakaSix, + author = {Birkar, Caucher}, + doi = {10.1112/S0010437X09004187}, + journal = {Compositio Mathematica}, + note = {arXiv:0806.4413, first posted 27 June 2008}, + number = {6}, + pages = {1442--1446}, + title = {The {Iitaka} conjecture {$C_{n,m}$} in dimension six}, + url = {https://arxiv.org/abs/0806.4413}, + volume = {145}, + year = {2009} +} + +@article{ChangNonvanishing, + author = {Chang, Chi-Kang}, + title = {Generalized nonvanishing conjecture and {Iitaka} conjecture}, + journal = {Mathematische Zeitschrift}, + volume = {310}, + pages = {Article 30}, + year = {2025}, + doi = {10.1007/s00209-025-03727-5}, + note = {Theorem numbering refers to arXiv:2312.04015v2, 25 March 2024}, + url = {https://arxiv.org/abs/2312.04015v2} +} + +@article{CaoPaunAbelian, + author = {Cao, Junyan and P{\u a}un, Mihai}, + doi = {10.1007/s00222-016-0672-6}, + journal = {Inventiones Mathematicae}, + note = {Locators: arXiv:1504.01095v3, submitted 3 August 2016}, + number = {1}, + pages = {345--387}, + title = {{Kodaira} dimension of algebraic fiber spaces over abelian varieties}, + url = {https://arxiv.org/abs/1504.01095v3}, + volume = {207}, + year = {2017} +} + +@article{CaoSurfaces, + author = {Cao, Junyan}, + doi = {10.14231/AG-2018-020}, + journal = {Algebraic Geometry}, + number = {6}, + pages = {728--741}, + title = {{Kodaira} dimension of algebraic fiber spaces over surfaces}, + url = {https://content.algebraicgeometry.nl/2018-6/2018-6-020.pdf}, + volume = {5}, + year = {2018} +} + +@incollection{HaconPopaSchnellAbelian, + address = {Providence, RI}, + author = {Hacon, Christopher D. and Popa, Mihnea and Schnell, Christian}, + booktitle = {Local and Global Methods in Algebraic Geometry}, + doi = {10.1090/conm/712/14346}, + note = {Locators: author manuscript dated 9 May 2017}, + pages = {143--195}, + publisher = {American Mathematical Society}, + series = {Contemporary Mathematics}, + title = {Algebraic fiber spaces over abelian varieties: around a recent theorem by {Cao} and {P{\u a}un}}, + url = {https://people.math.harvard.edu/~mpopa/papers/HPS.pdf}, + volume = {712}, + year = {2018} +} + +@article{GriffithsPeriodsIII, + author = {Griffiths, Phillip A.}, + doi = {10.1007/BF02684654}, + journal = {Publications Math{\'e}matiques de l'IH{\'E}S}, + pages = {125--180}, + title = {Periods of integrals on algebraic manifolds, {III}: Some global differential-geometric properties of the period mapping}, + url = {https://www.numdam.org/item/PMIHES_1970__38__125_0/}, + volume = {38}, + year = {1970} +} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/01-introduction.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..e88eb57f829edc7377794f79466b111b50e1c978 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/01-introduction.tex @@ -0,0 +1,268 @@ +\section{Introduction}\label{sec:introduction}\label{altord:section} + +Let $f:X\to Z$ be a surjective morphism with connected fibres between +smooth projective varieties. The pluricanonical forms on $X$ measure +its birational complexity. Iitaka's subadditivity problem asks whether +the forms on the base and on the geometric generic fibre always +contribute additively to that complexity. More precisely, the Kodaira +dimension $\kappa(X)$ is the maximum dimension of the images of the +complete linear systems $|mK_X|$, for positive integers $m$; it is +$-\infty$ if every such system is empty. If $F$ is the geometric +generic fibre of $f$, the conjecture $C_{n,m}$ asks for +\[ + \kappa(X)\geq\kappa(F)+\kappa(Z), + \qquad n=\dim X,\quad m=\dim Z. +\] +We use $(-\infty)+a=-\infty$ and $\kappa(\mathrm{point})=0$. + +The problem belongs to Iitaka's theory of dimensions of divisors +\cite{IitakaOriginal}. Its development has repeatedly connected the +growth of pluricanonical systems with positivity on the base. +Viehweg's weak-positivity method gives addition when the base is of +general type \cite{ViehwegAdditivity}. Kawamata proved addition over +curves and when the general fibre has a good minimal model +\cite{KawamataCurves,KawamataKodaira}; Koll\'ar established the case +of general-type fibres \cite{Kollar87}. The log-general-type-fibre +theorem of Kov\'acs--Patakfalvi and its formulation by Hashizume +provide the addition theorem used below +\cite{KovacsPatakfalvi,Hashizume}. + +Birkar proved the conjecture in total dimension at most six +\cite{BirkarIitakaSix}. Using canonical-bundle and nonvanishing +methods, Chang extended this to total dimension at most seven under +the additional assumption $\kappa(X)\geq0$ +\cite[Theorem~1.8]{ChangNonvanishing}. +Other advances include Cao--P\u{a}un's theorem for projective klt pairs over abelian varieties, +Hacon--Popa--Schnell's result for bases of maximal Albanese dimension, +and Cao's theorem for projective klt pairs over surfaces +\cite{CaoPaunAbelian,HaconPopaSchnellAbelian,CaoSurfaces}. +These results show how information about the base and positivity of +direct images can control the total pluricanonical system. +Tsuji's preprint states ordinary +subadditivity in Theorem~1.13, while Maehara's preprint states a +broader variation inequality in Section~7, Theorem~8 +\cite{TsujiDirectImages,MaeharaIitaka}. Those preprint claims are not +inputs to our proof. The argument here is a projective proof through +a Hodge line and a canonical bundle formula. + +\begin{theorem}[Ordinary Iitaka subadditivity]\label{thm:ordinary}\label{altord:ordinary-subadditivity} +Let $k$ be an algebraically closed field of characteristic zero, and +let $f:X\to Z$ be a surjective projective morphism with connected +fibres between smooth connected projective $k$-varieties. If $F$ is +its geometric generic fibre, then +\[ + \kappa(X)\geq\kappa(F)+\kappa(Z). +\] +\end{theorem} + +Thus the ordinary Iitaka conjecture has a positive answer in this +setting. The main step in the proof is a positivity statement for +the entire highest Hodge line of a rationally polarized variation. +We first describe that statement, then explain how the canonical +bundle formula brings it into the subadditivity problem. + +Let $Y$ be a smooth connected projective complex variety and $U$ a +dense open subset. A rationally polarized integral pure variation +$\mathbb V$ on $U$ consists of a rational local system with a +monodromy-invariant lattice, a Hodge filtration of fixed weight, and +a rational flat polarization. Suppose its highest nonzero +filtration step $F^p\mathbb V_{\mathcal O}$ is a line, so +$F^{p+1}=0$ and $\operatorname{rank}F^p=1$. On a simply connected +open set, a flat marking identifies this line with a holomorphic +map to the projective space of a fixed reference fibre. We call +this the \emph{highest-line map}. The \emph{full period map} +records every step of the Hodge filtration. + +The highest line has a rational parabolic extension across a smooth +normal-crossing compactification. This is the extension whose pullback, +after a finite level cover and resolution making local monodromy +unipotent, is the Deligne--Schmid highest line. The rational weights +retain the finite parts of the original local monodromies. + +\begin{theorem}[Projective adjoint positivity]\label{thm:adjoint}\label{altord:p:observation} +Let $Y$ be a smooth connected projective complex variety, let +$U\subset Y$ be a dense Zariski open subset, and let $\mathbb V$ +be a rationally polarized integral pure variation of Hodge structure +on $U$ whose entire highest nonzero Hodge filtration step is a line. +Let $M\in\operatorname{Pic}(Y)\otimes\Q$. Assume that, after a smooth +projective birational modification of $Y$ if necessary, there are a +smooth projective alteration $\tau:\widehat Y\to Y$ and a rational +number $c>0$ such that the pulled-back variation has unipotent +monodromy along a simple-normal-crossing boundary and +\[ + \tau^*M\sim_{\Q}cJ_{\widehat Y}, +\] +where $J_{\widehat Y}$ is its extended highest line. +After further smooth projective birational modification, there are a +surjective morphism $p:Y\to S$ with connected fibres onto a smooth +projective variety and a nef rational line $L$ on $S$ such that +\[ + M\sim_{\Q}p^*L,\qquad + K_S+jL\text{ is big for all sufficiently large integers }j. +\] +If $S$ is a point, the conclusion is $M\sim_{\Q}0$. +\end{theorem} + +The two period maps have separate jobs. The full period image provides +the projective base on which an adjoint divisor can be big. The +numerical dimension of $L$ is measured by the highest-line map, +whose rank can be smaller. The argument must therefore remove the +boundary using the rank of the line, without identifying it with +the dimension of the full period image. + +\subsection{The boundary obstruction and its resolution} + +The algebraicity theorem of Bakker--Brunebarbe--Tsimerman constructs +the full period image as an algebraic variety +\cite[Theorem~1.1]{BakkerBrunebarbeTsimerman}. At a neat level, +Brunebarbe--Cadorel's theorem gives logarithmic general type when +the full period map is generically immersive +\cite[Theorem~1.1]{BrunebarbeCadorel}. A finite quotient returns to +the original level. This introduces a second divisor to be removed: +besides the boundary with nonzero monodromy logarithm, there is +divisorial ramification of the finite quotient. Ordinary adjoint +positivity requires control of both. + +We first retain the smallest rational subvariation containing the +highest line. This is the pure transcendental-part construction +of Bakker--Filipazzi--Mauri--Tsimerman +\cite[Definition~2.9]{BFMT}. Rational minimality has a useful +consequence: a connected rational normal subgroup of monodromy +that fixes the line pointwise must be trivial. Together with +Andr\'e's monodromy normality theorem \cite[\S5]{AndreMonodromy} +and the fixed-part results of Griffiths, Deligne, and Schmid +\cite[added note to~4.2.6 and~4.2.8(i),(ii)]{DeligneHodgeII} +\cite[Theorem~7.22 and Corollary~7.23]{Schmid}, +it also detects a flat automorphism +that is scalar on all highest lines and preserves one full period. +Such an automorphism preserves every full period. + +The local analytic step concerns a family of holomorphic maps +approaching a divisor. If its limiting tangential rank equals +the full rank in the interior, then its nearby image is already +contained in the limiting image. At a nilpotent degeneration, +the limiting line is in the kernel of the monodromy logarithm; +rational minimality rules out equality of ranks. At a divisor +fixed by finite inertia, the limiting lines lie in one eigenspace. +The same rank principle and the fixed-part theorem would make +the inertia element fix the entire full period image. + +For this last contradiction we use the normalization of the +\emph{actual} period image. The finite group acts effectively on +its function field. Equality of the full periods therefore forces +the inertia element to be the identity. A further finite cover +of the image could have a nontrivial automorphism acting trivially +on every period, so connected-fibre factorization is taken only +after this argument. Sections~\ref{sec:rational}--\ref{sec:rankloss} +give the rational, analytic, and finite-quotient parts of this rank +loss in that order. + +Once the ranks drop, elementary section counting removes the +boundary and divisorial ramification. If the highest-line map has +generic rank $r$, the ample +perturbation in the interior supplies a polynomial of degree $r$ +in a large multiple of $L$, whereas the restriction polynomials +on those divisors have degree at most $r-1$. Effective divisors +exceptional over the quotient do not change its section growth. +This yields the adjoint positivity in Theorem~\ref{thm:adjoint}. + +\subsection{From the period theorem to ordinary subadditivity} + +For the nonvacuous case of Theorem~\ref{thm:ordinary}, put +$d=\kappa(F)\geq0$ and assume $\kappa(Z)\geq0$. The relative Iitaka +fibration factors a birational model of $X$ through a variety $Y$ +with $\dim(Y/Z)=d$ and Kodaira-dimension-zero generic fibre over +$Y$. The canonical bundle formula separates the singular-fibre +contribution from a nef moduli divisor: +\[ + K_{\widetilde X}\sim_{\Q} + g^*(K_Y+B+M)+R,\qquad \widetilde X\xrightarrow{g}Y\xrightarrow{q}Z. +\] +Here $B$ is an effective klt boundary, and the residual divisor $R$ +has the direct-image and exceptionality properties needed to compare +complete pluricanonical systems. This is the strategy of +Fujino--Mori's canonical bundle formula, with the birational +discriminant and moduli theory of Ambro +\cite[Theorem~4.5]{FujinoMori} +\cite{AmbroBoundary,AmbroModuli}. + +The least-index canonical root cover of the Kodaira-zero fibre +has geometric genus one. Its unique top form is therefore the +entire highest line of a rational variation, even if the character +of the cyclic covering group is not rational. The Hodge realization +of $M$ matches its extension on an alteration, not only its +restriction to the open base. These facts place $M$ within +Theorem~\ref{thm:adjoint}. The auxiliary sub-boundary in this formula +can be negative on the generic fibre; the argument uses nefness +and the Hodge realization, not a semiampleness assertion requiring +an effective generic boundary. + +The section comparison gives +\[ + \kappa(X)\geq\kappa(Y,K_Y+B+M),\qquad + (K_Y+B+M)|_{Y_{\bar\eta}}\text{ big}, +\] +where $Y_{\bar\eta}$ is the geometric generic fibre of $q$. +The period projection writes $M=p^*L$ for a morphism $p:Y\to S$. +Write $\mu:\widetilde X\to X$ for the birational modification +of the original source in the relative Iitaka construction. +Figure~\ref{fig:two-bases} separates the two maps from $Y$: their +bases need not map to one another. + +\input{figures/period-and-original-base} + +A product-map argument first gives log pluricanonical nonvanishing on a general fibre +of $p$. Fujino's twisted weak positivity then turns the big +divisor $K_S+jL-H$, for an ample $H$ and large $j$, into an +effective divisor linearly equivalent to +$K_Y+B+jM-p^*H$ \cite[Theorem~1.1]{FujinoWeakPositivity}. +Interpolating this divisor with an effective klt log canonical +class reduces the desired inequality to the established addition +theorem for log-general-type fibres +\cite[Theorem~2.11]{Hashizume}. Scalar extension of each +pluricanonical space transfers the result from $\C$ to any +algebraically closed field of characteristic zero. + +There are two further proofs of the Hodge-line boundary estimate +in the projective setting. An integral tensor replacement makes +monodromy faithful on the moving highest line; projection to a +fixed quotient by a monodromy logarithm then produces transverse +loops on which the line is constant. A second calculation obtains +the restriction-intersection bound directly from Hodge curvature: +smooth Thom forms concentrate on a boundary divisor, and a product +Poincar\'e estimate controls their limits at crossings. These +two calculations share the integral tensor replacement and the +transverse-loop criterion; the second supplies a different proof +of the boundary restriction-intersection step. These +arguments retain the same whole-top-line hypothesis but isolate +different mechanisms behind the boundary estimate. In the main +route, rational minimality and the tangential rank principle treat +the finite quotient on the actual period image directly. + +\subsection{Organization} + +Section~\ref{sec:rational} constructs the rationally minimal variation and proves the +fixed-period criterion. Section~\ref{sec:limits} compares numerical rank with +boundary limits and proves the tangential rank principle. +Section~\ref{sec:rankloss} applies these results to boundary monodromy and divisorial +ramification. Section~\ref{sec:quotient} constructs the actual period quotient and +completes the proof of adjoint positivity. Section~\ref{sec:canonical} gives the +canonical bundle data, the least-index root cover, and the +section-growth comparison. Section~\ref{sec:ordinary} proves ordinary subadditivity +and gives the scalar-extension argument in characteristic zero. +Section~\ref{p1logalt:section} proves the integral tensor replacement and the alternative +adjoint argument by transverse loops. Section~\ref{p1varhd:section} proves the +restriction-intersection estimate using Thom forms, including the +crossing calculation. + +\subsection{Conventions} +An algebraic fibre space is a surjective projective morphism with +connected fibres. All varieties in the Hodge-theoretic argument are +over $\C$. Divisors and line-bundle identities are rational unless +otherwise specified, and section comparisons are taken in degrees +clearing all denominators. We write $A\sim_{\Q}B$ when a positive +integral multiple of $A-B$ is principal. An effective rational class +means a class rationally linearly equivalent to an effective divisor. +The positive and negative parts of a rational divisor have disjoint +support. Numerical dimension is used only for nef divisors and is +defined in Section~\ref{sec:limits}. diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/02-rational.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/02-rational.tex new file mode 100644 index 0000000000000000000000000000000000000000..b851f20060455a54a5b7054fc1c064494bf504c6 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/02-rational.tex @@ -0,0 +1,169 @@ +\section{Rational minimality and fixed periods}\label{sec:rational}\label{altord:h:section} +Our objective is to recognize when a symmetry that fixes all highest +lines must fix the full periods. We first discard the rational +subvariations that do not contribute to the highest line. Rationality +is essential: the invariant spaces of rational normal monodromy +subgroups will then be Hodge substructures. + +Let $U$ be a smooth connected complex algebraic variety and let +$V=(V_{\Z},F^\bullet V_{\mathcal O},Q)$ be a polarized integral pure +variation of Hodge structure of weight $w$. Integral means that the +underlying rational local system has a monodromy-invariant lattice; +the polarization is rational and need not be integral. Write $m$ for the largest index with +$F^mV_{\mathcal O}\ne0$, and assume throughout that +$\rk F^mV_{\mathcal O}=1$. We call this piece the \emph{Hodge line}. +On a simply connected flat chart its inclusion in $V_{\mathcal O}$ +defines a holomorphic map +\[ + \ell:U_{\mathrm{loc}}\longrightarrow \mathbb P(V_{\C,u}). +\] +Changes of flat chart compose $\ell$ with a constant projective linear +transformation, so its differential rank is intrinsic. + +\subsection{The smallest rational subvariation} + +We call a lifted point \emph{Hodge-generic} if it lies outside every +proper Hodge locus of a rational flat tensor in tensor constructions on +the variation and its dual. These are countably many closed analytic +loci on the universal cover. Their complement is dense by the Baire +theorem. Only this analytic genericity is needed in the Hodge arguments. + +\begin{lemma}[Rationally minimal variation]\label{altord:h:minimal} +There is a unique smallest rational subvariation $V^{\min}\subset V_{\Q}$ +whose Hodge filtration contains $F^mV_{\mathcal O}$. At a Hodge-generic +point, its fibre is the smallest rational Hodge substructure containing +the Hodge line, and that fibre is simple. The subvariation has an induced +polarization and integral lattice. On a smooth compactification with +unipotent boundary monodromy, its extended Hodge line agrees with that +of $V$. +\end{lemma} + +\begin{proof} +In a polarizable rational Hodge structure, intersect all rational Hodge +substructures containing the specified line. A finite intersection +already attains the minimum dimension, and gives the unique smallest +one, say $W$. Semisimplicity shows that $W$ is simple: in a decomposition +into simple summands exactly one summand can have a nonzero deepest +Hodge piece, since that piece has dimension one, and that summand already +contains the line. + +Here is also the generic-fibre justification, which is useful when +passing between variations and individual Hodge structures. Trivialize +the rational local system on the connected universal cover of $U$. +For each rational endomorphism $e$, the condition that $e$ preserve every +$F^p$ is closed analytic. Choose a lifted point outside all such loci +which are proper. A rational Hodge projector onto $W$ at this point +therefore preserves the filtration everywhere on the cover. Its action +on the Hodge line is the identity everywhere: it is an idempotent on a +line bundle, and equals the identity at the chosen point. Its constant +image is thus a subvariation containing the line. At every other point +chosen in the same way, applying the argument in both directions shows +that its image is again the smallest Hodge substructure containing the +line. Uniqueness and the deck action now imply monodromy invariance, +so this subvariation descends to $U$. Any rational subvariation containing +the line contains it, by inspection at the chosen point. + +Restriction of $Q$ to a Hodge substructure is nondegenerate, and +$V^{\min}\cap V_{\Z}$ supplies the lattice. The polarization gives a +complementary subvariation whose deepest Hodge piece is zero. +Canonical extensions preserve this direct sum, proving the last claim. +This is the pure case of the transcendental-part construction of +\cite[Definition~2.9]{BFMT}. +\end{proof} + +\subsection{From a fixed highest line to a fixed full period} +A variation is \emph{minimal for its highest line} if the generic +fibre is the smallest rational Hodge substructure containing that +line. Lemma~\ref{altord:h:minimal} supplies this replacement without +changing the line. The next two lemmas explain what this minimality +buys. We use the usual Mumford--Tate group of a rational Hodge +structure: the smallest rational algebraic group through which its +Hodge homomorphism factors. + +\begin{lemma}[Normal subgroups and the deepest line] +\label{altord:p:normality} +Let $\mathbb V$ be a rationally polarized integral pure variation on a smooth +connected complex algebraic variety, minimal for its deepest Hodge line. +In a flat trivialization on the universal cover, let $G\subset\mathrm{GL}(V)$ +be connected algebraic monodromy and let $P=\operatorname{MT}(V)$ be the generic +Mumford--Tate group. If $H\subset G$ is a connected normal algebraic +subgroup defined over $\Q$, and $V^H_{\C}$ contains the deepest line at a +Hodge-generic lifted point, then $H$ is trivial. +\end{lemma} + +\begin{proof} +The monodromy and fixed-part theorems give that $G$ is semisimple and +normal in the connected reductive group $P$; see +\cite[\S5, Theorem~1 and Corollary~1]{AndreMonodromy}. Over an +algebraic closure, a connected normal subgroup of a semisimple group is +the product of a collection of its almost simple factors. Conjugation +by $P$ preserves $G$. Since $P$ is connected, it acts trivially on the +finite set of these factors, and therefore preserves $H$. + +Consequently $V^H$ is a rational $P$-subrepresentation. It is a rational +Hodge substructure at a Hodge-generic point, and it contains the deepest +Hodge line. Minimality gives $V^H=V$. The inclusion +$G\subset\mathrm{GL}(V)$ is faithful, so $H=1$. +\end{proof} + +\begin{lemma}[A flat automorphism over a fixed point] +\label{altord:p:inertia} +Let $U$ be a smooth connected complex algebraic variety with a rationally +polarized integral pure variation $\mathbb V$ minimal for its deepest Hodge line. +Suppose that an automorphism $b$ of $U$ fixes a point $e$, and that there +is a flat isomorphism of variations +\[ + \Phi_y:V_y\longrightarrow V_{b(y)}. +\] +Use the fibre $V_e$ as the flat reference space, and put $a=\Phi_e$. +If, on a nonempty open subset of the universal cover, all deepest Hodge +lines lie in a single eigenspace of $a$, then $a$ preserves the complete +Hodge filtration at every lifted point. In particular, locally at $e$ +the full lifted period maps at $y$ and $b(y)$ are equal. +\end{lemma} + +\begin{proof} +Write $\rho$ for monodromy based at $e$. Flatness and $b(e)=e$ give +\[ + a\rho(\gamma)a^{-1}=\rho(b_*\gamma) + \qquad (\gamma\in\pi_1(U,e)). +\] +Thus $a$ normalizes both algebraic monodromy and its identity component +$G$. Let $\lambda$ be the eigenvalue in the hypothesis. The condition +that the deepest line lies in $\ker(a-\lambda)$ is a holomorphic linear +condition on the universal cover, so it holds throughout that connected +cover by analytic continuation. + +Fix a Hodge-generic lifted point and let $W\subset V_{\C}$ be the complex span +of the monodromy orbit of its deepest line. This space is monodromy +invariant, hence $G$-invariant. Equivariance places every line in this +orbit among the deepest lines on the universal cover, so +$a|_W=\lambda\,\mathrm{id}_W$. +Every element $aga^{-1}g^{-1}$, with $g\in G$, therefore acts trivially +on $W$, as do all its $G$-conjugates. Let $H$ be their normal algebraic +closure in $G$. It is defined over $\Q$, since $a$ and $G$ are rational. +It is connected: the commutator map from the connected group $G$ has +connected image containing the identity, and the algebraic subgroup +generated by its conjugates is the closure of products of such +connected sets. Although $W$ need not be rational, $V^H$ is rational +and contains the chosen deepest line. +Lemma~\ref{altord:p:normality} gives $H=1$. Hence $a$ centralizes $G$. + +Pass to the connected finite \'etale cover $U_1\to U$ corresponding to +the inverse image of $G$ under monodromy, and choose $e_1$ over $e$. +Since $a$ centralizes all monodromy on $U_1$, it extends to a global flat +rational section of $\End(\mathbb V|_{U_1})$. The latter is a polarizable +pure variation of weight zero. The flat section $a$ has Hodge type +$(0,0)$ at $e_1$, since $\Phi_e$ is a Hodge isomorphism. The base +$U_1$ is a smooth algebraic variety and hence a Zariski open subset +of a compact complex variety, so the fixed-part theorem applies. +By \cite[Theorem~7.22 and Corollary~7.23]{Schmid}, $a$ has type +$(0,0)$ everywhere. Thus $a$ preserves the full filtration on the +universal cover. + +For completeness, take the lift of $b$ to the universal cover fixing a +chosen lift of $e$. Flatness of $\Phi$ gives +$F^\bullet(b\widetilde y)=aF^\bullet(\widetilde y)$ in the reference +space. The assertion just proved makes the right-hand side equal to +$F^\bullet(\widetilde y)$. +\end{proof} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/03-limits.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/03-limits.tex new file mode 100644 index 0000000000000000000000000000000000000000..8455ec53dbd5e9c32474ad69d2ebd114fdc46066 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/03-limits.tex @@ -0,0 +1,266 @@ +\section{Numerical rank and tangential limits}\label{sec:limits} +We now compare a boundary restriction of the extended highest line +with the limiting projective line map. The Hodge-theoretic input +identifies the limiting graded line. The analytic rank principle +then decides what would happen if its tangential rank did not drop. +For a nef line $J$ on a smooth projective $t$-fold we use +\[ + \nu(J)=\max\{a\in\{0,\ldots,t\}:c_1(J)^aH^{t-a}>0\}, +\] +where $H$ is ample. A line on a point has numerical dimension zero. + +Let $U$ and its variation be as in Section~\ref{sec:rational}. +Take a smooth projective compactification $T$ of $U$ with reduced +simple normal crossings boundary $D$, and suppose that all local +monodromies along $D$ are unipotent. Denote the Deligne extension by +$\overline V$ and the Schmid extension of the Hodge line by +$J=F^m\overline V$. We use the following standard inputs. + +\begin{theorem}[Extension and positivity inputs]\label{altord:h:inputs} +Under these hypotheses: +\begin{enumerate} +\item The extended Hodge filtration is by subbundles of $\overline V$. +For a morphism $g:(T',D')\to(T,D)$ of smooth projective varieties +with reduced simple normal crossings boundaries and +$g(T'\setminus D')\subset U$, the extended Hodge line of $g^*V$ +is $g^*J$. These extensions commute with direct sums and tensor products. +If the monodromy logarithms of some boundary components vanish, the +variation extends purely across those components, including their +crossings away from the remaining boundary. +\item The line $J$ is nef. It is big precisely when the top +Kodaira--Spencer map +\[ + T_U\longrightarrow + \Hom(F^mV_{\mathcal O},F^{m-1}V_{\mathcal O}/F^mV_{\mathcal O}) +\] +is generically injective, equivalently when $\ell$ is generically +immersive. +\end{enumerate} +The first assertions are the unipotent extension and nilpotent-orbit +theorems of \cite{Schmid,CKS}; functoriality is also recorded in +\cite[Lemmas~2.16 and~5.6]{BFMT}. The second assertion is +\cite[Lemma~6.17]{BakkerBrunebarbeTsimerman}. +\end{theorem} + +The equivalence in the second assertion uses Griffiths transversality: +the differential of the line map is the indicated Kodaira--Spencer +map followed by the inclusion of $F^{m-1}/F^m$ in $V_{\mathcal O}/F^m$. + +\begin{proposition}[Numerical rank]\label{altord:h:numerical-rank} +If $r=\rk_{\mathrm{gen}}d\ell$, then +\begin{equation}\label{altord:h:eq-numerical-rank} + \nu(J)=r. +\end{equation} +\end{proposition} + +\begin{proof} +Put $t=\dim T$, and fix an ample divisor $H$ on $T$. +For $1\le a\le t$, choose a sufficiently general smooth complete +intersection $C_a\subset T$ of dimension $a$, using a sufficiently +large multiple of $H$. It meets the boundary transversely. At general +points of $C_a\cap U$, the restriction of $d\ell$ to $T_{C_a}$ has +rank $\min(a,r)$. To see the required genericity, at a point where +$d\ell$ has rank $r$, the $a$-planes with this property form a nonempty +open subset of the tangent Grassmannian. A sufficiently ample linear +system prescribes the required first-order tangent conditions; Bertini +then gives the assertion for general complete intersections. +By Theorem~\ref{altord:h:inputs}, $J|_{C_a}$ is big exactly when $a\le r$. +Since it is nef, this is equivalent to +$(J|_{C_a})^a>0$, and hence to $c_1(J)^aH^{t-a}>0$. +This proves the formula, also when $r=0$; dimension zero is immediate. +\end{proof} + +\subsection{The limiting line along a divisor} + +We recall the boundary input in the form needed here. Let $E$ be a +component of $D$, let +$E^\circ=E\setminus\Supp(D-E)$, and let $N$ be its monodromy logarithm. +The monodromy weight filtration $W(N)$ is centred at $w$. +The limiting filtration induces polarized pure variations on its graded +pieces over $E^\circ$, with unipotent monodromy at the remaining boundary. +If $k$ is the unique weight carrying the deepest line, then $J|_E$ is +the Schmid extension of the top line of $\Gr^{W(N)}_k$. +These statements include compatibility at deeper strata; +see \cite[\S2.5.2--\S2.5.3 and Lemma~2.16(3)]{BFMT}. +For a single $N$, the relevant primitive summand is polarized by the +form induced by $Q(-,N^{k-w}-)$ when $k\ge w$. + +The local expression below is Schmid's nilpotent-orbit normal form +\cite[Theorem~4.12]{Schmid}; its compatibility at crossings uses +\cite{CKS}. The rank comparison is the consequence needed here. + +\begin{proposition}[Boundary line]\label{altord:h:boundary-line} +On a transverse coordinate chart $(u,v)$ about a general point of $E$, +with $E=(u=0)$, write a generator of the lifted Hodge line as +\begin{equation}\label{altord:h:nilpotent-expression} + e^{zN}a(u,v),\qquad z=\frac{\log u}{2\pi i}, +\end{equation} +where $a$ is holomorphic across $u=0$ and $a(0,v)\ne0$. +After shrinking the chart there is an integer $\ell\ge0$ such that +\begin{equation}\label{altord:h:primitive-line} + N^{\ell+1}a(0,v)=0,\qquad N^\ell a(0,v)\ne0, +\end{equation} +and the line belongs to weight $w+\ell$ in the limiting mixed Hodge +structure. Moreover, +\begin{equation}\label{altord:h:boundary-rank} + \nu(J|_E)\le + \rk_{\mathrm{gen}}d\bigl(v\longmapsto[N^\ell a(0,v)]\bigr). +\end{equation} +The map on the right is formed in the chosen flat coordinates. +If $N=0$, take $\ell=0$; the variation then extends purely near the +general point of $E$. +\end{proposition} + +\begin{proof} +Put $a_0=a(0,v)$. In the Deligne bigrading of the limiting mixed +Hodge structure, $F^m$ is a single one-dimensional summand $I^{m,q}$: +all first indices are at most $m$, and $\dim F^m=1$. +Set $\ell=m+q-w$. The primitive decomposition on weight $w+\ell$ +expresses each summand as a sum of terms +\[ + N^j P_{w+\ell+2j},\qquad j\ge\max(0,-\ell), +\] +where $P_{w+b}$ denotes the primitive part for $b\ge0$. +A contribution to Hodge type $(m,q)$ with $j>0$ would come from +type $(m+j,q+j)$, impossible because its first index exceeds $m$. +Thus $j=0$, $\ell\ge0$, and the top line is primitive. +Hard Lefschetz for the monodromy weight filtration gives nonvanishing +of its image under $N^\ell$ and vanishing under $N^{\ell+1}$ on the +associated graded. Since $N$ has bidegree $(-1,-1)$ in the Deligne +bigrading, these statements hold for $a_0$ itself: a vector in a single +Deligne summand maps injectively to its weight-graded piece. +The integer $\ell$ is constant on a sufficiently small open stratum, +so the identities hold throughout the chosen $v$-chart. + +Let $J_E^{\mathrm{gr}}$ be the extended top line of the pure variation +on $\Gr^{W(N)}_{w+\ell}$ along $E^\circ$. +The boundary extension input identifies it with $J|_E$. +Projecting $N^\ell a_0$ to $\Gr^{W(N)}_{w-\ell}$ yields the image +of this graded top line under the flat isomorphism +\[ + N^\ell:\Gr^{W(N)}_{w+\ell}\longrightarrow + \Gr^{W(N)}_{w-\ell}. +\] +Consequently the projective map of the graded line has rank no greater +than $v\mapsto[N^\ell a_0(v)]$: projection is a fixed linear map on +the flat chart, defined near the line in question. +Apply Proposition~\ref{altord:h:numerical-rank} to the graded variation on +$E^\circ$ and its smooth projective compactification $E$. +This proves \eqref{altord:h:boundary-rank}. +\end{proof} + +\subsection{A local rank principle} + +The boundary-line proposition has reduced the numerical question to a +projective differential. We next prove the local assertion that will +force a symmetry to fix the interior whenever the limiting rank is +as large as the interior rank. This analytic fact requires control of tangential derivatives +only. A sector below has the form +$S_\epsilon=\{0<|u|<\epsilon,\ \alpha<\arg u<\beta\}$ with a fixed +branch of the argument and finite opening $\beta-\alpha<2\pi$. + +\begin{lemma}[Rank and a tangential limit]\label{altord:h:rank-principle} +Let $B\subset\C^n$ be a polydisc and let +$f:S_\epsilon\times B\to M$ be a holomorphic map to a complex +manifold. Suppose $f(u,\cdot)$ converges uniformly on compact subsets +of $B$, as $u\to0$ throughout the sector, to a holomorphic map $f_0$. +Assume that the full generic differential rank of $f$ is at most $r$ +and that $\rk df_0(v_0)=r$ at some $v_0\in B$. +Then, after shrinking the sector and taking neighborhoods +$v_0\in B'\Subset B_0\subset B$, +\[ + f(S_{\epsilon'}\times B')\subset f_0(B_0), +\] +where the indicated local image of $f_0$ is an embedded complex +submanifold of dimension $r$. In particular, every fixed closed analytic +subset containing this local image also contains the nearby image of $f$. +\end{lemma} + +\begin{proof} +Use a common coordinate chart in $M$, shrinking $B$ and the sector. +Cauchy estimates give convergence of all derivatives in the $B$ +variables on smaller polydiscs. The $(r+1)$-minors of $df$ vanish +identically, so those of $df_0$ vanish as well. Thus $f_0$ has constant +rank $r$ near $v_0$. The holomorphic constant rank theorem supplies +source coordinates $v=(x,y)\in\C^r\times\C^{n-r}$ and target +coordinates in which $f_0(x,y)=(x,0)$. + +Write $f=(F,G)$ in these coordinates, with +$F(u,x,y)=x+E(u,x,y)$. On fixed smaller polydiscs, both $E$ and +$\partial_xE$ tend uniformly to zero. Choose nested $x$-balls, a +smaller ball $W$ for $w$, and a ball $Y$ for $y$. For sufficiently +small $\epsilon'$, the maps +\[ + x\longmapsto w-E(u,x,y) +\] +send a fixed closed $x$-ball into itself and have Lipschitz constant +less than $1/2$, uniformly for +$(u,w,y)\in S_{\epsilon'}\times W\times Y$. +The contraction theorem gives a unique solution +$x=\chi(u,w,y)$ of $F(u,x,y)=w$ on this common product. +The holomorphic inverse function theorem and uniqueness make $\chi$ +holomorphic in all interior parameters; moreover $\chi\to w$ +uniformly on smaller products. + +In the new source coordinates the map is +\[ + f(u,\chi(u,w,y),y)=(w,H(u,w,y)). +\] +The $w$-derivatives already have rank $r$. +The rank bound therefore forces $\partial_uH=0$ and +$\partial_yH=0$: either derivative, if nonzero, would give an +additional image vector with zero first $r$ coordinates. +Since the sector and $Y$ are connected, $H$ depends only on $w$. +Its limit as $u\to0$ is zero, hence $H=0$. +Shrinking the original source neighborhood so that its $F$-values +belong to $W$ proves the assertion. If $r=0$, the rank bound directly +makes $f$ constant on the connected source, and the same conclusion +follows. +\end{proof} + +\begin{lemma}[Nilpotent tangential convergence]\label{altord:h:nilpotent-limit} +Let $N$ be a nilpotent endomorphism of a finite-dimensional complex +vector space, and let $a(u,v)$ be holomorphic on $\Delta\times B$. +Suppose, for a fixed $\ell\ge0$, that +$N^{\ell+1}a(0,v)=0$ and $N^\ell a(0,v)\ne0$ throughout $B$. +On a sector with the above choice of $z=\log u/(2\pi i)$, +\begin{equation}\label{altord:h:eq-nilpotent-limit} + [e^{zN}a(u,v)]\longrightarrow[N^\ell a(0,v)] +\end{equation} +normally on sufficiently small $v$-charts, with all tangential +derivatives. Thus Lemma~\ref{altord:h:rank-principle} applies whenever the +rank of this limit equals an upper bound for the full rank of the +line map. Its nearby image then lies in $\mathbb P(\ker N)$. +\end{lemma} + +\begin{proof} +Choose $k$ with $N^{k+1}=0$, and write +$a(u,v)=a_0(v)+u q(u,v)$. Set $b(v)=N^\ell a_0(v)/\ell!$. +Direct expansion gives +\begin{align*} + z^{-\ell}e^{zN}a(u,v) + &=b(v)+\sum_{j<\ell}\frac{z^{j-\ell}}{j!}N^ja_0(v) + +u\sum_{j=0}^k\frac{z^{j-\ell}}{j!}N^jq(u,v). +\end{align*} +For every fixed tangential derivative, uniformly on compact subsets +of $B$, the remainder is +\[ + O(|z|^{-1})+O(|u|\,|z|^{k-\ell}); +\] +the first term is absent when $\ell=0$. +On the sector $|z|$ is comparable to $|\log|u||$, so both terms tend +to zero. Choose a linear functional nonzero on $b(v_0)$ and shrink +the $v$-chart. Dividing by that functional gives convergence with +all tangential derivatives in a fixed affine projective chart, +proving \eqref{altord:h:eq-nilpotent-limit}. Finally $Nb=0$, so the limiting +image belongs to the fixed projective linear subspace +$\mathbb P(\ker N)$, and the last assertion follows from +Lemma~\ref{altord:h:rank-principle}. +\end{proof} + +\begin{remark}\label{altord:h:scaling} +All numerical-rank statements are unchanged on replacing a Hodge line +by a positive rational multiple of its class in $\Pic(T)\otimes\Q$. +The local statements also apply with $N=0$, when the line map extends +holomorphically across the divisor. +\end{remark} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/04-rankloss.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/04-rankloss.tex new file mode 100644 index 0000000000000000000000000000000000000000..ec10f9481b1c64c0fecc74cdb1b7daf0cd7a6a1d --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/04-rankloss.tex @@ -0,0 +1,177 @@ +\section{Rank loss on the boundary and the ramification divisor} +\label{sec:rankloss} + +The preceding lemmas apply to a smooth model of a full period image. +At a neat level the variation is unipotent at the boundary. Returning +to the original level also introduces ramification divisors. We now +prove the strict numerical rank drop for both kinds of divisor. + +We specify the finite-group data because the group acting on the +variation can be larger than the effective group acting on its base. +Let $T$ be a smooth connected projective complex variety of positive +dimension, let $D_T$ be a reduced simple-normal-crossing divisor, and +put $U=T\setminus D_T$. Let $\mathbb V$ be a rationally polarized +integral pure variation on $U$, minimal for its entire highest +Hodge line, with unipotent boundary monodromy. Write $J$ for the +extended highest line. Let $D_N$ be the union of the components +of $D_T$ with nonzero monodromy logarithm, and put +$U_N=T\setminus D_N$. The variation extends purely over $U_N$ by +Theorem~\ref{altord:h:inputs}. + +Suppose a finite group $H$ acts on $T$, preserves $D_T$, and acts on +the variation by flat Hodge isomorphisms preserving its integral +lattice and rational polarization and covering that action. Set +\[ + G_f=H/\ker\bigl(H\longrightarrow\Aut(T)\bigr). +\] +Thus $G_f$ acts effectively on $T$. We retain $H$ for its action on +the variation, since its kernel on $T$ can act nontrivially on fibres. +Uniqueness of the pure extension preserves this action over $U_N$. + +Fix a connected period domain $\mathcal D$ for the rational +polarized reference space and a neat arithmetic group $\Gamma'$ of +lattice isometries containing the monodromy. Let +\[ + \varphi:U_N\longrightarrow\Gamma'\backslash\mathcal D +\] +be the full period map. Assume that the normalization $B$ of the +closure of its actual image is an algebraic variety, that +the natural rational map $T\dashrightarrow B$ is birational, and +that $\varphi$ is generically immersive. In particular, +\[ + \C(T)=\C(B). +\] +These assumptions describe the actual period-image model. The +algebraicity of $B$ is the period-image theorem of +\cite[Theorem~1.1]{BakkerBrunebarbeTsimerman}; here its construction +is an explicit input to the rank-loss statement. + +\begin{theorem}[Boundary and ramification rank loss]\label{thm:rankloss} +For the data just specified, put $r=\nu(J)$. +Then $r>0$. If $E$ is either a component of $D_N$, or a component +of $D_T$ fixed pointwise by a nonidentity element of +$G_f$, then +\[ + \nu(J|_E)0$. + +\smallskip\noindent +\emph{A component with nonzero monodromy logarithm.} +Let $E\subset D_N$ and write $N\ne0$ for its rational monodromy +logarithm. In coordinates $(u,v)$ near a general point of $E$, +with $E=(u=0)$, the lifted highest-line map has the form +\[ + [e^{zN}a(u,v)],\qquad z=\frac{\log u}{2\pi i}. +\] +Proposition~\ref{altord:h:boundary-line} provides an integer +$\ell\geq0$ with +\[ + N^{\ell+1}a(0,v)=0,\qquad N^\ell a(0,v)\ne0, + \qquad + \nu(J|_E)\leq\rk_{\rm gen}d\bigl(v\mapsto[N^\ell a(0,v)]\bigr). +\] +Lemma~\ref{altord:h:nilpotent-limit} gives its normal +tangential convergence on a sector from the interior line maps. +All interior $(r+1)$-minors vanish. Cauchy convergence of tangential +derivatives therefore bounds the rank of the limiting map by $r$. + +Suppose that $\nu(J|_E)\geq r$. The limiting map has rank exactly +$r$ at a suitable general point, and its image lies in +$\mathbb P(\ker N)$. Lemma~\ref{altord:h:rank-principle} forces all +nearby interior lines to lie in this same linear subspace. The +condition that $N$ annihilates the highest line is holomorphic, so +it continues over the connected universal cover of $U_N$. + +The one-parameter group $\exp(tN)$ belongs to $G$: it is the +connected Zariski closure of the cyclic unipotent local monodromy. +Every monodromy conjugate of this group fixes the highest line +pointwise. Let $G_N$ be the algebraic subgroup they generate. +It is connected, defined over $\Q$, and normalized by monodromy; +Zariski density makes it normal in $G$. Its invariant space contains +the highest line at a Hodge-generic point. By +Lemma~\ref{altord:p:normality}, $G_N=1$, contradicting $N\ne0$. +This proves the required inequality for components of $D_N$. + +We have controlled all divisors at which the pure variation fails +to extend. The remaining obstruction is a divisor of finite +ramification where the variation does extend. Its inertia element +can act on the rational fibre even when it fixes every point of +the divisor, so we must use the full-period conclusion of +Lemma~\ref{altord:p:inertia}. + +\smallskip\noindent +\emph{A divisor fixed by finite inertia.} +Let $b\in G_f$ be nonidentity and fix $E$ pointwise. If $E$ is +contained in $D_N$, the preceding case applies. Otherwise a general +point $e\in E$ lies in $U_N$. Choose a lift $h\in H$ of $b$. +The retained $H$-action supplies a flat Hodge isomorphism +\[ + \Phi_y:V_y\longrightarrow V_{b(y)} +\] +on $U_N$. Use $V_e$ as a flat reference space on a small simply +connected $b$-invariant neighborhood, and set $a=\Phi_e$. +If $x$ is the full lifted period map, flatness gives +\[ + x(by)=a\,x(y). +\] +The operator $a$ is a rational Hodge isometry at $e$. Because $H$ +acts on the variation and $b(e)=e$, the group-action law gives +$a^{\operatorname{ord}(h)}=1$. Since $b$ fixes $E$, every highest line +on the nearby part of $E$ is an eigenline of $a$. Its eigenvalue +is constant on a connected small patch, so these lines lie in one +eigenspace $\ker(a-\lambda)$. + +Proposition~\ref{altord:h:boundary-line} with $N=0$ bounds +$\nu(J|_E)$ by the rank of this extending line map on $E$. + +Suppose again that $\nu(J|_E)\geq r$. The rank on $E$ is at most +$r$, since all $(r+1)$-minors of the ordinary extending line map +vanish by continuation from the interior. It is therefore $r$. +Apply Lemma~\ref{altord:h:rank-principle} in a transverse coordinate +with $N=0$. A nonempty open set of lifted highest lines lies in +$\ker(a-\lambda)$. Lemma~\ref{altord:p:inertia}, applied on the +smooth algebraic variety $U_N$ with fixed point $e$, now shows +that $a$ preserves every full period. Consequently $x(by)=x(y)$. + +Projecting to $\Gamma'\backslash\mathcal D$ gives +$\varphi(by)=\varphi(y)$ on a nonempty open set. Since $B$ is the +normalization of the actual image, its normalization map is +birational and hence injective on a dense open subset of that +image. It follows that $b$ acts trivially on $\C(B)$. The chosen +model has $\C(T)=\C(B)$, so $b$ acts trivially on $T$, contrary +to its nonidentity in $G_f$. This proves the strict inequality. + +Finally, in a finite quotient in characteristic zero, ramification +at a divisorial valuation is detected by a nontrivial inertia +group, which acts identically on that divisor's function field. +Its elements therefore fix the divisor pointwise. The last +assertion follows from the case just proved. +\end{proof} + +\begin{remark} +The conclusion concerns the numerical dimension of the highest line, +while generic immersivity is a hypothesis on the full period map. +Neither the proof nor the conclusion identifies these ranks. The +function-field equality with the actual period image is used only +at the last contradiction in the finite case; it cannot be replaced +there by an arbitrary finite extension of that function field. +\end{remark} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/05-quotient.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/05-quotient.tex new file mode 100644 index 0000000000000000000000000000000000000000..8a4e1d83f7af7e6b27ebcf631e465da9b4e1fde4 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/05-quotient.tex @@ -0,0 +1,341 @@ +\section{The period quotient and adjoint bigness}\label{sec:quotient}\label{altord:sec:period} +We prove Theorem~\ref{thm:adjoint}. The full period image supplies +an algebraic base, and Section~\ref{sec:rankloss} controls the divisors +introduced by compactification and finite descent. We first show +that those numerical rank drops suffice to remove the divisors +from an adjoint class. This is a section-growth calculation; no +abundance statement for the highest line is needed. + +\subsection{Section estimates}\label{altord:sec:estimates} +\begin{lemma}\label{altord:e:volume-lemma} +Let \(T\) be a smooth projective \(t\)-fold, \(A\) a rational divisor, +and \(J\) a nef rational divisor +with \(r=\nu(J)>0\), and \(D=\sum_i d_iE_i\) an effective integral divisor +with smooth prime components. Suppose +\[ + \nu(J|_{E_i})0\) and all \(j\geq1\). +The last inequality follows from \(J^rH_1^{t-r}>0\). + +Remove \(mD\) one prime divisor at a time. Each quotient in a divisor +exact sequence is a line bundle on some \(E_i\), of the form +\[ + \mathcal O_{E_i} + \left(m(A+D+jJ)-\sum_h k_hE_h\right), + \qquad 0\leq k_h\leq md_h. +\] +There is a single ample rational divisor \(H_2\) on \(T\) for which every +such quotient has at most +\(h^0(E_i,m(H_2+jJ)|_{E_i})\) sections, in divisible degrees and for every +\(j\geq0\). To see this, choose an ample Cartier divisor \(B_0\) with a +globally generated multiple clearing the denominator of +\(B_0-(A+D)\), and ample Cartier divisors \(B_h\) such that both \(B_h\) +and \(B_h+E_h\) are globally generated. Enlarge \(B_0\) if necessary and +take \(H_2=B_0+\sum_h d_hB_h\). The difference, restricted to \(E_i\), is +\[ + m(B_0-A-D)+ + \sum_h\bigl((md_h-k_h)B_h+k_h(B_h+E_h)\bigr). +\] +It has a section nonzero on \(E_i\), by global generation and +multiplication, giving the claimed bound. + +There are \(m\sum_i d_i\) quotient terms. For fixed \(j\), +asymptotic Riemann--Roch on each \(E_i\) bounds their total contribution on +the scale \(m^t/t!\) by +\[ + t\sum_i d_i\,(H_2+jJ)^{t-1}\cdot E_i. +\] +This is \(O(j^{r-1})\), since every intersection containing at least +\(r\) factors of \(J|_{E_i}\) vanishes. Consequently +\[ + \liminf_{m\to\infty} + \frac{t!}{m^t}h^0\bigl(T,m(A+jJ)\bigr) + \ \geq\ c j^r-O(j^{r-1})>0 +\] +for all sufficiently large \(j\), proving bigness. +\end{proof} + +\begin{lemma}\label{altord:e:exceptional-pullback} +Let \(\pi:T\to S\) be a surjective generically finite morphism of smooth +projective varieties. Let \(E\geq0\) be a rational divisor on \(T\) whose +components have image of codimension at least two in \(S\). For a rational +divisor \(D\) on \(S\), the divisor \(\pi^*D+E\) is big if and only if +\(D\) is big. +\end{lemma} + +\begin{proof} +Factor \(\pi\) as \(T\xrightarrow{\rho}\bar S\xrightarrow{\tau}S\), +where \(\bar S\) is the normalization of \(S\) in \(\C(T)\). +Then \(\rho\) is birational and \(\tau\) is finite. Every component of +\(E\) is \(\rho\)-exceptional, since a finite map preserves dimensions. +Normality gives +\(\rho_*\mathcal O_T(mE)=\mathcal O_{\bar S}\) in divisible degrees: +a rational function with poles only on exceptional divisors has no +codimension-one pole downstairs. Projection formula identifies the +sections of \(m(\pi^*D+E)\) with those of \(m\tau^*D\). +Finally, bigness is invariant under surjective finite pullback. +\end{proof} + + +\subsection{Construction and descent} +\begin{proof}[Proof of Theorem~\ref{altord:p:observation}] +We may first make the birational modification allowed in the hypothesis. +Replace $\mathbb V$ by the minimal rational subvariation from +Lemma~\ref{altord:h:minimal}, and equip it with the intersection lattice and +the restricted polarization. Its deepest Hodge line and the line +$J_{\widehat Y}$ are unchanged. We continue to denote this variation +by $\mathbb V$. + +\medskip\noindent +\textbf{The actual period image and its finite quotient.} +Choose a rational reference space with lattice and polarization, and +let $\mathcal D$ be the connected period-domain component containing a +chosen lift of the periods. Let $\Gamma$ be the arithmetic group of +integral polarizing isometries preserving this component; it contains +the monodromy of $\mathbb V$. Let $\Gamma'\triangleleft\Gamma$ be a +normal neat subgroup of finite index. The kernel of +\[ + \pi_1(U)\longrightarrow\Gamma/\Gamma' +\] +defines a connected finite \'etale Galois cover $U'\to U$. Denote its +deck group by $H$; thus $H$ is the image of this homomorphism, not +necessarily all of $\Gamma/\Gamma'$. + +The period map of $\mathbb V|_{U'}$ factors algebraically through the +normalization $B$ of the closure of its image in +$\Gamma'\backslash\mathcal D$. The space $B$ is a normal +quasiprojective variety, and on a smooth dense open subset its period +realization has generically immersive differential; this is the +period-image theorem \cite[Theorem~1.1]{BakkerBrunebarbeTsimerman}. Here and below the function +field of $B$ is the function field of this normalized \emph{actual +image}. We do not replace it by its relative algebraic closure in +$\C(U')$. + +Neatness makes the arithmetic action free. The universal local system +on the quotient is +\[ + (\mathcal D\times V_{\Z})/\Gamma', +\] +with its tautological filtration. On the smooth open of $B$ this +filtration is horizontal: the dominant map $U'\to B$ is generically +submersive and its pullback is the original variation, so Griffiths +transversality holds on a dense open and hence everywhere. Thus the +restricted local system and filtration form a polarizable integral +variation. Moreover, the action of +$\Gamma/\Gamma'$ on this local system is an honest action: +\[ + [x,v]\longmapsto[\gamma x,\gamma v]. +\] +Normality of $\Gamma'$ makes the formula independent of the +representative $\gamma$. Restricting gives an action of $H$ on the +variation over $B$, covering its action on $B$. The action on $B$ is +algebraic: apply uniqueness in \cite[Theorem~1.1]{BakkerBrunebarbeTsimerman} to the period map +and its expression obtained by a deck transformation on $U'$ and the +corresponding arithmetic translation on the target, and then lift to +the normalization. Let +\[ + G_f=H/\ker(H\longrightarrow\operatorname{Aut}(B)) +\] +be the effective group on the image. The full group $H$ will still be +retained for its action on the variation. + +Choose a $G_f$-equivariant smooth projective compactification $S'$ of a +smooth dense invariant open of $B$, with snc boundary, by equivariant +compactification and resolution. All boundary monodromies of its +variation are unipotent. Indeed they are quasi-unipotent by the +monodromy theorem, and they belong to the neat group $\Gamma'$; their +root-of-unity eigenvalues must therefore all be one. Let $J'$ denote +the extended deepest Hodge line and put $L'=cJ'$. This is nef by +Theorem~\ref{altord:h:inputs}. The $H$-action extends to $J'$ by +functoriality of the unipotent Deligne and Hodge extensions. + +Let $q:S'\to Q_0=S'/G_f$ be the finite quotient. Although an element +of $\ker(H\to G_f)$ may act nontrivially on $J'$, it acts on its fibres +through finite-order characters. Taking a tensor power divisible by +$|H|$ kills the fibre action of every stabilizer in $H$, including this +kernel. That tensor power therefore descends to an actual line bundle +on $Q_0$ by finite-group descent. Consequently there is a rational +line bundle $\overline L$ on $Q_0$ with +\[ + q^*\overline L\sim_{\Q}L'. +\] +It is nef, since its pullback by the finite surjective map $q$ is nef. +Choose a smooth projective resolution $s:S_0\to Q_0$ and put +$L_0=s^*\overline L$. + +The equivariant period map on $U'$ descends to a dominant rational map +$Y\dashrightarrow Q_0$, and hence to $S_0$. Resolve its graph by a +smooth projective birational modification of $Y$, obtaining +\[ + p_0:Y\longrightarrow S_0. +\] +We claim that +\begin{equation} + M\sim_{\Q}p_0^*L_0. + \label{altord:p:line-descent} +\end{equation} +To verify this claim, choose a common smooth projective alteration $W$ +dominating the alteration in the hypothesis and the level cover, and +resolve the graph of its rational map to $S'$. Write +$\mu:W\to Y$ and $t:W\to S'$ for the resulting morphisms. Arrange +snc boundaries by further resolution, containing the inverse images of +the boundaries in question. The pulled-back variations agree on a +dense open subset. Unipotent Deligne extensions and their extended +Hodge filtrations commute with these pullbacks between smooth snc +pairs; see \cite[Lemma~5.6]{BFMT}, or the unipotent extension theory +\cite{Schmid,CKS}. The extended top line $J_W$ is consequently the +pullback of the top line on either model. The hypothesis and the +commutative maps to $Q_0$ give +\[ + \mu^*M\sim_{\Q}cJ_W + \sim_{\Q}t^*L' + \sim_{\Q}\mu^*p_0^*L_0. +\] +Pullback by a proper generically finite map is injective on rational +line-bundle classes: applying divisor pushforward, or the norm of a +principal rational function, multiplies the original class by the +degree. This proves \eqref{altord:p:line-descent}. In particular no boundary +twist has been inferred from an equality merely on the open locus. + +If $\dim S_0=0$, equation~\eqref{altord:p:line-descent} gives +$M\sim_{\Q}0$, and the theorem follows. Assume henceforth that +$\dim S_0>0$. + +\medskip\noindent +\textbf{A common equivariant model and log general type.} +Normalize $S_0$ in $\C(S')$, and take a $G_f$-equivariant smooth +projective resolution of the resulting model and its birational map +to $S'$. Denote the resulting maps by +\[ + \eta:T\longrightarrow S',\qquad + \pi:T\longrightarrow S_0. +\] +Thus $\eta$ is birational, $\pi$ is generically finite with Galois +function-field group $G_f$, and $G_f$ acts on $T$ over $S_0$. +Choose the resolution so that there is a $G_f$-invariant snc boundary +$D_T$ containing the complement of the open period locus and the +support of the Jacobian divisor of $\pi$. The pulled-back variation +has unipotent boundary monodromy. If $J_T$ is its extended top line, +then +\begin{equation} + L_T:=cJ_T\sim_{\Q}\eta^*L' + \sim_{\Q}\pi^*L_0. + \label{altord:p:T-line} +\end{equation} + +Let $D_N$ be the reduced sum of those components of $D_T$ having +nonzero monodromy logarithm. The variation extends as a pure polarized +variation over +\[ + U_N=T\setminus D_N. +\] +Indeed, near a point where all the remaining logarithms are zero, the +nilpotent-orbit extension is a pure polarized Hodge filtration; the +several-variable extension gives the same statement at crossings +\cite{Schmid,CKS}. These extensions are unique and preserve the +$H$-equivariance of the variation. The divisor $D_N$ is snc, since it +is a union of components of $D_T$. + +The full period map on $U_N$ has generically immersive differential, +because $T$ is birational to $B$. The log-general-type theorem for a +polarized variation with generically immersive period map therefore +gives +\begin{equation} + K_T+D_N\quad\text{big}. + \label{altord:p:log-big} +\end{equation} +Here the applicable theorem is \cite[Theorem~1.1]{BrunebarbeCadorel}. Its boundary is precisely +$D_N$, because the pure variation has just been extended over all the +other boundary components. + +We now have precisely the geometric data of +Theorem~\ref{thm:rankloss}: $T$ is birational to the actual normalized +period image, the full period map is generically immersive, and the +retained finite group $H$ acts on its integral polarized variation. +The effective group on $T$ is $G_f$. Minimality is preserved here: +the variation on $T$ and the original variation have the same +Hodge structures at Hodge-generic points, since the original source +dominates the period image. Consequently +\begin{equation} + r:=\nu(L_T)>0, + \qquad \nu(L_T|_E)0}), + \qquad + \operatorname{codim}_Y g(\Supp R^-)\geq2, +\end{equation} +and \(R^-\) is \(\alpha\)-exceptional. +The divisor \(M\) is the trace of the moduli b-divisor, which can be +assumed to descend to \(Y\). On a sufficiently high generically finite +base change admitting semistable reduction, its pullback is +\(\Q\)-linearly equivalent to a positive rational multiple of the +Schmid extension of the distinguished canonical Hodge eigensheaf of +the canonical root cover. +\end{theorem} + +The residual divisor and discriminant assertions, with the normalization +by the least nonzero pluricanonical index, are +\cite[\S4.4 and Theorem~4.5(i)--(v)]{FujinoMori}. +For original boundary zero, the logarithmic semistable part agrees +with the semistable part of Section~2 by +\cite[Proposition~4.7]{FujinoMori}. +The moduli interpretation follows from the discriminant-threshold +definition in \cite[Definition~4.3]{FujinoMori} and the following Ambro +results. The b-divisor and Hodge assertions use +\cite[Theorems~0.2 and~4.4(1),(3), Lemma~5.2(4)--(5), and +Propositions~5.4--5.5]{AmbroBoundary}; +see also \cite[Proposition~3.1]{AmbroModuli}. +These statements allow the auxiliary sub-boundary to have negative +coefficients. + +We record how the hypotheses of these results fit together. +In the notation of \cite[\S4.4]{FujinoMori}, applied with original boundary +zero, the crepant boundary upstairs is the negative of the effective +relative canonical divisor over the smooth original source. +Thus the choice \(\Xi=0\) satisfies the hypotheses of their +Theorem~4.5, including those of part~(v). +The coefficients of our \(B\) are their \(s_P/b\); part~(v) places them +in \([0,1)\). Their residual divisor is \(bR\). +We verify the all-degree direct-image condition from the normalization +below, so that it remains available after the final change of model. +For the exceptionality assertion, flatten before resolving the main +transform, as in \cite[\S4.4]{FujinoMori}. The inverse image of a base +subset of codimension at least two has codimension at least two in +the flat main transform; finite normalization preserves this bound. +A prime divisor of its resolution lying over that subset is therefore +exceptional over the normalized main transform. This transform maps +birationally to \(X_0\), so the prime is also \(\alpha\)-exceptional. +Apply this to every component of \(R^-\). Its small image on \(Y\) +alone would not imply exceptionality over \(X_0\). + +Here is also the relevant normalization of the associated sub-pair. +On the geometric generic fibre \(C\), let \(b\) be the least positive +integer with \(H^0(C,bK_C)\ne0\), and let \(D_C\) be the divisor of its +unique section up to scalar. The horizontal restriction is +\[ + R_C=\frac1bD_C,\qquad + \Delta_C=-R_C. +\] +The least positive integer making \(K_C+\Delta_C\) principal is +also \(b\). Indeed, \(bK_C-D_C\) is principal. Conversely, if +\(m(K_C-D_C/b)\) is principal for a positive integer \(m\), then +\(mD_C/b\) is an integral effective divisor linearly equivalent to +\(mK_C\). Thus \(H^0(C,mK_C)\ne0\), which forces \(m\geq b\). +This identifies the least-index normalization in the sub-pair +construction with the pluricanonical index used for the root cover. +A rational relative pluricanonical section first defines a divisor +\(R_0\) with this restriction. For a prime divisor \(P\subset Y\), +subtract from \(R_0\) the pullback of \(P\) multiplied by +\[ + a_P=\min_{E\mapsto P} + \frac{\operatorname{coeff}_E(R_0)} + {\operatorname{mult}_E(g^*P)}. +\] +Only finitely many \(a_P\) are nonzero. The resulting residual divisor +is nonnegative over the generic point of \(P\), with coefficient zero +on at least one component over \(P\). + +This normalization also gives the direct-image condition in every +positive degree. Let \(V\subset Y\) be a nonempty open subset, let +\(k>0\) be an integer, and let \(\varphi\) represent a section of +\(\mathcal O_{X'}(\lfloor kR^+\rfloor)\) over \(g^{-1}(V)\). +On the geometric generic fibre \(C\), its poles are bounded by +\(\lfloor kD_C/b\rfloor\), hence by an integral multiple of +\(D_C\). Since \(D_C\sim bK_C\) and \(\kappa(C)=0\), every such +function is constant on \(C\). Since \(g_*\mathcal O_{X'}=\mathcal O_Y\), +the field \(\C(Y)\) is algebraically closed in \(\C(X')\). +Thus \(\varphi=g^*\psi\) for a rational function \(\psi\) on +\(V\). For each prime divisor \(P\) meeting \(V\), +choose a component \(E\) over \(P\) on which the normalized +residual coefficient is zero. Regularity of \(\varphi\) along +\(E\) gives +\[ + 0\leq\operatorname{ord}_E(g^*\psi) + =\operatorname{mult}_E(g^*P)\operatorname{ord}_P(\psi). +\] +Thus \(\psi\) has no codimension-one pole on \(V\), and normality +of \(Y\) makes it regular. Pullbacks of regular functions give the +reverse inclusion, proving +\[ + g_*\mathcal O_{X'}(\lfloor kR^+\rfloor)=\mathcal O_Y + \qquad(k\in\Z_{>0}). +\] + +For the sub-pair \((X',-R)\) its discriminant coefficient is \(1-t_P\), +where +\[ + t_P=\sup\{t\in\R: + (X',-R+t\,g^*P)\text{ is sub-lc over the generic point of }P\}. +\] +Nonnegativity of \(R\) there implies \(t_P>0\); the component with +residual coefficient zero gives \(t_P\leq1\). +This explains the effective boundary in this normalization. + +The rank-one hypothesis for this sub-pair holds even though +\(\Delta_C\) is negative. Indeed, \(\kappa(C,R_C)=0\). +On a log resolution of \((C,-R_C)\), a section of the rounded +discrepancy sheaf pushes down to a rational function on \(C\) whose +poles are bounded by a sufficiently large integral multiple of +\(D_C\). The space of such functions has dimension one, while the +constant function is allowed. Consequently +\[ + \rk g_*\mathcal O_{X'}\bigl(\lceil\mathbf A(X',-R)\rceil\bigr)=1, +\] +where the expression is interpreted by pushing down from a log +resolution. This is the rank condition in +\cite[introductory assumptions (1)--(3)]{AmbroBoundary}. + +After choosing a model on which the moduli b-divisor descends, take +a common higher base model satisfying the flattening and +normal-crossing requirements, with boundary containing the total +transform of the original normalized discriminant support and all +exceptional base divisors. Crepantly pull back the auxiliary sub-pair +to a smooth resolved main transform, taking a log resolution that +includes its residual support and the pullbacks of the base boundary. +Apply the same minimum-coefficient normalization to this residual divisor. Its +horizontal restriction is again \(D_C/b\) for the unique +least-index pluricanonical divisor on the new geometric generic +fibre: for the induced birational morphism \(\lambda:C'\to C\), +crepant pullback gives +\[ + \lambda^*(D_C/b)+K_{C'/C}=D_{C'}/b. +\] +The preceding argument therefore restores the all-degree +direct-image condition on this final model. The normalized residual +is nonnegative over every codimension-one base point; its negative +part has base image of codimension at least two and is exceptional +over the original source by the flattening argument above. Over the +strict transform of an original base prime, its zero-coefficient +component survives and the crepant residual stays nonnegative, so +renormalization makes no further change there. Any new discriminant +support is therefore exceptional and lies in the chosen +simple-normal-crossing boundary. Its coefficients are in \([0,1)\) +by the same threshold calculation, so the discriminant is effective +and klt. + +This renormalization does not change the moduli b-divisor. Indeed, +changing a sub-boundary from \(\Delta\) to \(\Delta+g^*A\) changes +the divisor pulled back in \(K_{X'}+\Delta\) by \(A\), while the +discriminant changes by \(A\) as well, by its threshold definition. +Their difference is unchanged, and the same calculation holds on +every higher base model. Crepant birational pullback likewise +preserves the induced b-divisor. Thus the final normalized residual +retains the descended nef moduli part together with the effective +klt discriminant and the residual properties needed for the section +comparisons below. +The parabolic pullback statement and independence of the chosen root +trivialization are also given directly in +\cite[Definition-Proposition~7.1]{AmbroBoundary}. + +\begin{remark} +\label{altord:cb:no-abundance} +No abundance assertion for \(M\) is part of the input. +The stronger conclusion of +\cite[Theorem~3.3]{AmbroModuli} assumes effectivity of the boundary on +the geometric generic fibre, an assumption which need not hold for +\(-R_C\). The b-nefness and Hodge identification used above instead +come from the results of \cite{AmbroBoundary} just cited. +In particular, no good minimal model of \(C\) is required. +For the Hodge-theoretic formulation with sub-pairs, see also +\cite[Definition~6.18, Construction-Definition~6.23, +Remark~6.25, and Theorem~6.28]{BFMT}. +\end{remark} + +\subsection{The least-index root cover and its rational variation} + +\begin{lemma}[Canonical root cover] +\label{altord:cb:root-cover} +Let \(C\) be a smooth connected projective variety with +\(\kappa(C)=0\), and let \(b>0\) be the least integer for which +\(H^0(C,bK_C)\ne0\). Choose a nonzero section \(s\) of \(bK_C\) and a +nonzero rational canonical form \(\omega\). +Let \(T\) be the normalization of \(C\) in the field obtained by +adjoining a root +\[ + \xi^b=\frac{s}{\omega^b}. +\] +Then \(T\to C\) has degree \(b\), and every smooth projective +resolution \(W\) of \(T\) satisfies +\[ + \kappa(W)=0,\qquad h^0(W,K_W)=1. +\] +The unique canonical line is spanned by the form +\(\xi h^*\omega\), where \(h:W\to C\) is the induced morphism. +\end{lemma} + +\begin{proof} +Put \(n=\dim C\) and \(D_C=\operatorname{div}(s)\). +The case \(n=0\) is immediate. For \(n>0\), first note that every +nonzero pluricanonical space on \(C\) has dimension one: +two independent sections in the same degree would give a nonconstant +rational map and hence positive Kodaira dimension. + +The field \(\C(C)\) contains all roots of unity. +If \(s/\omega^b=u^p\) for a prime \(p\mid b\), the rational +\((b/p)\)-canonical form \(u\omega^{b/p}\) has \(p\)-th power \(s\). +At every prime divisor its order is \(p^{-1}\) times the nonnegative +order of \(s\), so this form has no poles. A rational section of a +line bundle on the smooth variety \(C\) with no divisorial poles is +regular. This contradicts the minimality of \(b\). +The Kummer irreducibility criterion now gives a field extension of +degree \(b\). Denote the normalization map by \(\pi:T\to C\), and a +smooth projective resolution by \(\rho:W\to T\), so \(h=\pi\rho\). + +The differential of \(h\) gives a nonzero morphism +\(h^*K_C\to K_W\) at the generic point, and a regular morphism of +line bundles everywhere. Use this morphism and its tensor powers +when pulling back canonical forms. The rational canonical form +\(\eta=\xi h^*\omega\) then satisfies +\[ + \eta^{\otimes b}=h^*s. +\] +The right side is a regular section of \(bK_W\). +Thus \(b\,\operatorname{ord}_E(\eta)\geq0\) for every prime divisor +\(E\subset W\). Normality of \(W\) again implies that \(\eta\) is +regular, and therefore \(h^0(W,K_W)\geq1\). + +We next bound the canonical divisor from above. +Choose \(K_C=\operatorname{div}(\omega)\), and on \(T\) use the +canonical Weil divisor determined by the differential pullback of +\(\omega\). At a prime divisor \(P\subset C\) outside \(\Supp D_C\), +the order of \(s/\omega^b\) is divisible by \(b\). Locally over the +discrete valuation ring at \(P\), after dividing the root by a +power of a uniformizer, the defining equation is a \(b\)-th root of +a unit. It is unramified, since the characteristic is zero. +Consequently all ramification divisors lie over \(\Supp D_C\). + +If \(Q\subset T\) lies over \(P\subset\Supp D_C\), write +\(e=e(Q/P)\) and \(d=\operatorname{ord}_P(D_C)\geq1\). +Tameness gives the ramification coefficient \(e-1\), and +\[ + e-1\leq ed=\operatorname{ord}_Q(\pi^*D_C). +\] +The canonical divisor formula for finite maps, interpreted at +codimension-one points, therefore gives +\[ + K_T\leq\pi^*(K_C+D_C). +\] +On \(W\) the same inequality holds away from the +\(\rho\)-exceptional divisors. Hence there is an effective integral +\(\rho\)-exceptional divisor \(E\) such that +\begin{equation} +\label{altord:cb:cover-upper-bound} + K_W\leq h^*(K_C+D_C)+E. +\end{equation} +This argument does not require a pullback of the Weil divisor \(K_T\). + +Set \(A=K_C+D_C\). Normality of \(T\) implies +\(\rho_*\mathcal O_W(mE)=\mathcal O_T\) for every \(m\geq0\): +a rational function allowed poles only on exceptional divisors is +regular in codimension one on \(T\), hence regular on \(T\). +Projection formula and \eqref{altord:cb:cover-upper-bound} imply +\[ + \kappa(W,K_W) + \leq \kappa(W,h^*A+E) + =\kappa(T,\pi^*A) + =\kappa(C,A). +\] +The last equality is invariance of Iitaka dimension under a finite +surjective pullback. In this situation it can be seen directly +from section rings: a homogeneous section upstairs satisfies its +monic characteristic polynomial over \(\C(C)\); its coefficients, +in degree \(jm\), are sections of \(jmA\), by the valuation bounds +for sections of \(m\pi^*A\). Thus the upstairs graded ring is +integral over the downstairs ring, and their fraction fields have +the same transcendence degree. +Finally \(D_C\sim bK_C\), so +\[ + \kappa(C,A)=\kappa(C,(b+1)K_C)=0. +\] +Together with the nonzero form \(\eta\), this proves +\(\kappa(W)=0\), and then \(h^0(W,K_W)=1\). +\end{proof} + +The underlying cover of Lemma~\ref{altord:cb:root-cover} is the +canonical root cover used in +\cite[Remark~2.6 and Theorem~3.1]{FujinoMori}. +Over a dense open subset \(U\subset Y\), a rational relative +pluricanonical section constructs this cover in families; a +projective resolution, followed by shrinking \(U\), gives a smooth +projective morphism \(h_U:\mathcal W_U\to U\). +Its geometric generic fibre has the invariants in +Lemma~\ref{altord:cb:root-cover}. Constancy of Hodge numbers in the smooth +projective family then gives geometric genus one on every fibre. +If their dimension is \(n\), then +\[ + \mathbb V=R^n(h_U)_*\Q,\qquad + \rk F^n(\mathbb V\otimes\mathcal O_U)=1. +\] +The variation has an integral lattice modulo torsion and is +polarizable, using a relative ample class and the Lefschetz +decomposition. Alternatively, its primitive middle cohomology +is polarizable and contains all of \(H^{n,0}\), since cup product +with a \((1,1)\)-class annihilates \(H^{n,0}\). +The cyclic deck transformation \(\xi\mapsto\zeta\xi\) acts on the +unique top form through \(\zeta\). Thus the distinguished +canonical eigensheaf is the entire top Hodge line of a rational +polarizable variation, even when the character itself is not +defined over \(\Q\). We do not replace the rational variation by +that character eigenspace. + +After a smooth projective alteration \(\tau:T_Y\to Y\), resolution +of the boundary, and semistable reduction in codimension one, the +identification in Theorem~\ref{altord:cb:parabolic} reads +\begin{equation} +\label{altord:cb:hodge-realization} + \tau^*M\sim_{\Q}aJ,\qquad a\in\Q_{>0}, +\end{equation} +where \(J\) is the Schmid extension of the top Hodge line of +\(\tau^*\mathbb V\), with unipotent local monodromy. +Birational changes of the resolved root cover do not change its +canonical direct image: on a common resolution, differential +pullback identifies its canonical forms. +The extension comparison is the one in +\cite[Theorem~4.4(1),(3), Lemma~5.2(5), and +Proposition~5.5]{AmbroBoundary}, rather than +an identification asserted only on \(U\). +Changing the chosen rational trivialization by a function from +the base gives the same identification after a further finite +base change extracting the corresponding root. + +\subsection{Relative Iitaka reduction and section comparisons} + +\begin{proposition}[Canonical bundle reduction] +\label{altord:cb:reduction} +Let \(f:X\to Z\) be an algebraic fibre space between smooth connected +projective varieties, and let its geometric generic fibre \(F\) have +\(d=\kappa(F)\geq0\). +There exist smooth projective varieties \(\widetilde X,Y\), +a birational morphism \(\mu:\widetilde X\to X\), and algebraic fibre +spaces +\[ + \widetilde X\xrightarrow{g}Y\xrightarrow{q}Z,\qquad + qg=f\mu, +\] +such that \(\dim Y-\dim Z=d\) and the geometric generic fibre of +\(g\) has Kodaira dimension zero. +They admit the data \(B,M,R\) of Theorem~\ref{altord:cb:parabolic}, +with \(R^-\) exceptional over \(X\). +The moduli divisor \(M\) has the realization +\eqref{altord:cb:hodge-realization} for a polarizable rational variation +whose top Hodge piece has rank one. +Writing +\[ + D=K_Y+B+M, +\] +one has +\begin{equation} +\label{altord:cb:two-comparisons} + \kappa(X)\geq\kappa(Y,D), + \qquad + D|_{Y_{\bar\eta}}\text{ is big}, +\end{equation} +where \(\bar\eta\) is a geometric generic point of \(Z\). +The same bigness holds on very general smooth fibres of \(q\). +\end{proposition} + +\begin{proof} +Over \(\C(Z)\), take a sufficiently divisible pluricanonical map +of the generic fibre and its Iitaka fibration, including the +relative algebraic closure of the image field. +The Iitaka fibration theorem gives image dimension \(d\) and +Kodaira dimension zero for the geometric generic fibre of this +fibration. Formation of its pluricanonical spaces commutes with +extension to an algebraic closure of \(\C(Z)\); thus these +statements can be checked geometrically. +Extend the rational map to projective models over \(Z\), resolve +source and target, and take the connected-fibre factorization. +This gives \(X_0\to Y_0\to Z\) with the required dimensions. +The second map has connected fibres as well: pushing +\(\mathcal O_{X_0}\) first to \(Y_0\) and then to \(Z\) gives +\(\mathcal O_Z\), because the original \(f\) and the birational +map to its smooth source have this property. + +Apply Theorem~\ref{altord:cb:parabolic} to \(X_0\to Y_0\). +All its modifications can be performed over \(Z\). +Exceptionality over \(X_0\) implies exceptionality over \(X\). +Lemma~\ref{altord:cb:root-cover} and the subsequent family construction +give the required rational variation and its top line. +The two comparisons are proved in Lemma~\ref{altord:cb:comparison} +below. +\end{proof} + +\begin{lemma}[Comparison of section growth] +\label{altord:cb:comparison} +For the factorization and canonical bundle data in +Proposition~\ref{altord:cb:reduction}, \eqref{altord:cb:two-comparisons} holds. +\end{lemma} + +\begin{proof} +The formula gives +\[ + K_{\widetilde X}+R^- + \sim_{\Q}g^*D+R^+. +\] +Write \(K_{\widetilde X}=\mu^*K_X+E_\mu\), using compatible +canonical divisors. Since \(X\) and \(\widetilde X\) are smooth, +\(E_\mu\) is effective and \(\mu\)-exceptional. +The same is true of \(E_\mu+R^-\). +For sufficiently divisible \(m\), normality and projection +formula therefore give +\[ + H^0\bigl(\widetilde X,m(K_{\widetilde X}+R^-)\bigr) + \simeq H^0(X,mK_X). +\] +Multiplication by the canonical section of \(mR^+\) injects +\(H^0(Y,mD)\) into the space on the left; here we also use +\(g_*\mathcal O_{\widetilde X}=\mathcal O_Y\). +This proves \(\kappa(Y,D)\leq\kappa(X)\). + +Now make the flat base change \(\bar\eta\to Z\). +The varieties \(\widetilde X_{\bar\eta}\) and \(Y_{\bar\eta}\) +are smooth; the former is birational to the geometric generic +fibre \(F\) of \(f\). +Restrictions of canonical divisors agree, up to linear +equivalence, with the corresponding canonical divisors of these +fibres. The restricted formula consequently implies +\[ + K_{\widetilde X_{\bar\eta}} + \leq_{\Q} + g_{\bar\eta}^*(D|_{Y_{\bar\eta}}) + +R^+|_{\widetilde X_{\bar\eta}}. +\] +Here \(\leq_{\Q}\) means that the difference is +\(\Q\)-linearly equivalent to an effective divisor. +Formation of a proper direct image commutes with this flat +base change, so \eqref{altord:cb:residual} yields, for sufficiently +divisible \(m\), +\[ + (g_{\bar\eta})_* + \mathcal O_{\widetilde X_{\bar\eta}} + (mR^+|_{\widetilde X_{\bar\eta}}) + =\mathcal O_{Y_{\bar\eta}}. +\] +Projection formula now gives +\begin{align*} + h^0(\widetilde X_{\bar\eta},mK_{\widetilde X_{\bar\eta}}) + &\leq + h^0\bigl(\widetilde X_{\bar\eta}, + m g_{\bar\eta}^*(D|_{Y_{\bar\eta}}) + +mR^+|_{\widetilde X_{\bar\eta}}\bigr)\\ + &=h^0(Y_{\bar\eta},mD|_{Y_{\bar\eta}}). +\end{align*} +It follows that +\[ + d=\kappa(F) + \leq\kappa(Y_{\bar\eta},D|_{Y_{\bar\eta}}) + \leq\dim Y_{\bar\eta}=d. +\] +Thus \(D|_{Y_{\bar\eta}}\) is big, including the +zero-dimensional case. + +For completeness, the bigness assertion spreads to a dense open +subset after restricting to the smooth locus of \(q\). +Fix a relatively ample divisor \(H\). +Bigness on the geometric generic fibre gives, for some +sufficiently divisible integer \(m\), +a nonzero section of \(mD-H\) there. Flat scalar extension of +cohomology first gives such a section over \(\C(Z)\). +Shrinking the base, the section extends to the family and remains +nonzero on its fibres. Hence \(mD|_{Y_z}\) is the sum of the +ample divisor \(H|_{Y_z}\) and an effective divisor for general +\(z\), proving the stated very-general-fibre assertion. +\end{proof} + +\begin{lemma}[Further birational modifications] +\label{altord:cb:high-model} +Suppose \(Y,B,M,D\) satisfy the conclusions of +Proposition~\ref{altord:cb:reduction}. Let \(\nu:Y'\to Y\) be a smooth +projective log resolution, and write +\[ + K_{Y'}+B^{\mathrm{cr}}=\nu^*(K_Y+B). +\] +Set \(B'=(B^{\mathrm{cr}})^+\), \(M'=\nu^*M\), and +\(D'=K_{Y'}+B'+M'\). +Then \(B'\) is effective, \((Y',B')\) is klt, \(M'\) is nef with +the same Hodge realization after pullback, and +\[ + D'=\nu^*D+(B^{\mathrm{cr}})^-. +\] +The additional divisor is effective and \(\nu\)-exceptional. +Consequently \(\kappa(Y',D')=\kappa(Y,D)\), and \(D'\) is big +on the geometric generic fibre over \(Z\). +\end{lemma} + +\begin{proof} +The nonexceptional coefficients of \(B^{\mathrm{cr}}\) are those +of the effective divisor \(B\), so its negative part is +exceptional. All coefficients of \(B^{\mathrm{cr}}\) are less +than one by the klt hypothesis. Its positive part has simple +normal-crossing support and coefficients in \([0,1)\), proving +that \((Y',B')\) is klt. +The displayed identity follows directly from the definitions. +Exceptional effective twists do not change section spaces of a +pullback to a normal base, so they do not change the Iitaka +dimension. +On the geometric generic fibre over \(Z\), the induced map is +birational and the restricted identity expresses \(D'\) as the +pullback of a big divisor plus an effective divisor. +Nefness is preserved by pullback. For the Hodge assertion, use +a common smooth alteration of \(Y'\) and the alteration in +\eqref{altord:cb:hodge-realization}; the unipotent Deligne and Schmid +extensions commute with the resulting morphisms of smooth +normal-crossing pairs. +\end{proof} + +\begin{remark}\label{altord:cb:reduced-moduli-remark} +After Lemma~\ref{altord:cb:high-model}, \(B'\) need not be described as +the discriminant of the original sub-pair. The properties used +later are precisely effectiveness and klt singularities, the +nef Hodge realization of \(M'\), and the two comparisons in +\eqref{altord:cb:two-comparisons}; all have been preserved. +In particular, since the big cone of \(Y_{\bar\eta}\) is open, +there is a rational number \(v\) with \(00\). Along a smooth fibre of \(t\), the cotangent +sequence is +\[ + 0\longrightarrow \Omega^1_{S,s}\otimes\mathcal O_{W_s} + \longrightarrow \Omega^1_W|_{W_s} + \longrightarrow \Omega^1_{W_s}\longrightarrow0. +\] +Since the last term has rank \(a\), the entire \(m\)-th exterior +power lies in the filtration term with at least \(m-a\) base +factors. Its quotient by the next term gives, after pulling back +top forms on \(Z\), a morphism +\begin{equation}\label{altord:a:cotangent-map} + h_s^*\omega_Z\longrightarrow + \bigwedge\nolimits^{m-a}\Omega^1_{S,s}\otimes\omega_{W_s}. +\end{equation} +At a general point of a general \(W_s\), the differential of \(h\) +has rank \(m\), and its restriction to the tangent space of \(W_s\) +has rank \(a\). Thus \eqref{altord:a:cotangent-map} is nonzero. A suitable +linear functional on the constant exterior-power factor yields a +nonzero morphism \(h_s^*\omega_Z\to\omega_{W_s}\). + +Choose \(n>0\) and \(0\ne\tau\in H^0(Z,nK_Z)\). +For general \(s\), the image \(h_s(W_s)\) is not contained in the +zero divisor of \(\tau\): otherwise dominance of \(h\) would fail. +The \(n\)-th tensor power of the preceding morphism therefore sends +\(h_s^*\tau\) to a nonzero section of \(nK_{W_s}\). This proves the +claim. + +Let \(G\) be a general fibre of \(r\). For general \(z\in Z\), such +fibres form a moving family covering \(Y_z\). A big divisor +restricts to a big divisor on a general member of this family: +restrict an ample-plus-effective decomposition, choosing a member +not contained in the support of the effective part. Consequently +\(D|_G\) is big. Since \(G\) maps to a point of \(S\), +\(M|_G\sim_{\Q}0\), and adjunction gives +\[ + D|_G\sim_{\Q}K_G+B|_G. +\] +For general \(s\), the morphism \(r_s:Y_s\to W_s\) has connected +fibres, \((Y_s,B|_{Y_s})\) is klt, and its general fibre has big log +canonical divisor. Theorem~\ref{altord:a:log-general-type} gives +\[ + \kappa(Y_s,K_{Y_s}+B|_{Y_s})\geq + \dim G+\kappa(W_s)\geq0. +\] +Finally choose \(s\) outside the countably many proper jumping +loci for the spaces of sections of all divisible log +pluricanonical bundles on the smooth \(p\)-fibres. Generic base +change identifies their dimensions with those on the geometric +generic fibre. Thus the last nonvanishing holds on that fibre as +well. +\end{proof} + +\subsection{Effectivity from weak positivity} + +\begin{lemma}\label{altord:a:effectivity} +Let \(p:Y\to S\) be an algebraic fibre space between smooth +projective varieties, and let \((Y,B)\) be an effective klt +\(\Q\)-pair. Suppose that the geometric generic fibre has +nonnegative log Kodaira dimension. For every big \(\Q\)-divisor +\(P\) on \(S\), the divisor +\[ + K_{Y/S}+B+p^*P +\] +is \(\Q\)-linearly equivalent to an effective divisor. +\end{lemma} + +\begin{proof} +Choose a dense open over which \(p\) is flat. Flatten \(p\) by a +projective birational modification of the base +and then resolve the base, retaining flatness of the main +transform by base change; see \cite[Lemma~38.31.1, Tag~081R]{StacksFlatten}. +Resolve the resulting main transform \(V\to S_1\), choosing the +projective resolution so that its composite \(\mu:Y_1\to Y\) is also +a log resolution of \((Y,B)\). Thus +\[ + \sigma:S_1\to S,\qquad + \mu:Y_1\to Y,\qquad + p_1:Y_1\to S_1 +\] +are compatible morphisms, \(S_1,Y_1\) are smooth projective, +\(\sigma,\mu\) are birational, and \(Y_1\to V\) is a resolution +with \(V\to S_1\) flat. +Any prime divisor of \(Y_1\) whose image in \(S_1\) has codimension +at least two is \(\mu\)-exceptional. Indeed, flatness implies that +the inverse image of such a base subset has codimension at least +two in \(V\); hence the divisor is exceptional over \(V\), and +its image in \(Y\) also has codimension at least two. + +Take the positive part \(B_1\) of the crepant boundary, so that +\((Y_1,B_1)\) is effective klt and +\begin{equation}\label{altord:a:source-correction} + K_{Y_1}+B_1=\mu^*(K_Y+B)+E,\qquad + E\geq0,\quad E\ \text{\(\mu\)-exceptional}. +\end{equation} +Generic-fibre nonvanishing and field-extension base change allow +us to choose a sufficiently divisible \(k>0\) for which +\[ + \mathcal E= + p_{1*}\mathcal O_{Y_1}\bigl(k(K_{Y_1/S_1}+B_1)\bigr) +\] +has positive rank. Theorem~\ref{altord:a:weak-positivity} applies to +\(\mathcal E\). + +Fix an ample Cartier divisor \(A\) on \(S_1\), and choose a +positive integer \(\alpha\) so large that +\[ + \sigma^*P-\frac{1}{k\alpha}A +\] +is big. For some \(\beta>0\), weak positivity makes +\[ + \bigl(\Sym^{\alpha\beta}\mathcal E\bigr)^{**} + \otimes\mathcal O_{S_1}(\beta A) +\] +generically generated. Set \(\ell=\alpha\beta\). +Let \(U\subset S_1\) be the locally free locus of the torsion-free +sheaf \(\mathcal E\); its complement has codimension at least two. +On all of \(U\), multiplication of sections gives +\[ + \Sym^\ell\mathcal E|_U\longrightarrow + p_{1*}\mathcal O_{Y_1} + \bigl(k\ell(K_{Y_1/S_1}+B_1)\bigr)|_U. +\] +This map is nonzero at the generic point: a nonzero section on the +integral generic fibre has nonzero powers. Generic generation +therefore supplies a global section of the twisted reflexive +symmetric power whose product is nonzero. Its image is a regular +section over \(p_1^{-1}(U)\) of +\[ + k\ell(K_{Y_1/S_1}+B_1)+\beta p_1^*A. +\] +Here \(U\) is the locally free locus, not merely a smaller open +where the chosen sections generate. + +The big divisor +\[ + k\ell\sigma^*P-\beta A + =k\ell\left(\sigma^*P-\frac{1}{k\alpha}A\right) +\] +has an effective sufficiently divisible multiple. Taking the +same power of the preceding section and multiplying by the +pullback of a nonzero section of that multiple gives a nonzero +section over \(p_1^{-1}(U)\) of a multiple of +\[ + D_1:=K_{Y_1/S_1}+B_1+p_1^*\sigma^*P. +\] +As a rational section on \(Y_1\), it has poles only above +\(S_1\setminus U\). Thus there are \(n>0\), a rational function +\(\varphi\ne0\), and an effective \(\mu\)-exceptional divisor \(T\) +such that +\begin{equation}\label{altord:a:section-with-poles} + \operatorname{div}(\varphi)+nD_1+T\geq0. +\end{equation} + +Choose compatible canonical divisors and put +\[ + D_0=K_{Y/S}+B+p^*P,\qquad + R=K_{S_1}-\sigma^*K_S\geq0. +\] +Equation~\eqref{altord:a:source-correction} gives the exact identity +\[ + D_1=\mu^*D_0+E-p_1^*R. +\] +Push \eqref{altord:a:section-with-poles} forward to the smooth variety +\(Y\). Since \(\mu_*E=\mu_*T=0\), we obtain +\[ + \operatorname{div}(\varphi)+nD_0 + \geq n\mu_*p_1^*R\geq0. +\] +This proves rational linear effectivity of \(D_0\). +\end{proof} + +\subsection{Completion over the complex numbers} + +\begin{proof}[Proof of Theorem~\ref{altord:ordinary-subadditivity} over \(\C\)] +If either \(\kappa(F)\) or \(\kappa(Z)\) is \(-\infty\), the +assertion follows from the convention in the statement. +Assume therefore that +\[ + d:=\kappa(F)\geq0,\qquad \kappa(Z)\geq0. +\] +Proposition~\ref{altord:cb:reduction} and Lemma~\ref{altord:cb:comparison} +provide algebraic fibre spaces +\[ + \widetilde X\longrightarrow Y\xrightarrow{q}Z +\] +and an effective klt pair \((Y,B)\), together with a nef +\(\Q\)-divisor \(M\) having the required Hodge realization. +Writing \(D=K_Y+B+M\), their conclusions include +\begin{equation}\label{altord:a:reduction} + \dim(Y/Z)=d,\qquad + \kappa(X)\geq\kappa(Y,D),\qquad + D|_{Y_{\bar\eta}}\ \text{big}, +\end{equation} +where \(Y_{\bar\eta}\) is the geometric generic \(q\)-fibre. + +Apply adjoint positivity, Theorem~\ref{altord:p:observation}, +and the compatible modifications of Lemma~\ref{altord:cb:high-model}. +We obtain an algebraic fibre space \(p:Y\to S\) with \(S\) smooth +projective and +\[ + M\sim_{\Q}p^*L,\qquad L\ \text{nef}. +\] +If \(\dim S=0\), then \(M\sim_{\Q}0\). +Theorem~\ref{altord:a:log-general-type}, applied directly to \(q\), and +\eqref{altord:a:reduction} give the result. +We may consequently suppose \(\dim S>0\). Adjoint positivity also gives +\[ + K_S+jL\ \text{big for all sufficiently large integers }j. +\] + +Openness of the big cone on \(Y_{\bar\eta}\) permits a rational +number \(v\) with \(01\) and an ample +\(\Q\)-divisor \(H\) on \(S\) such that +\[ + P_0:=K_S+jL-H +\] +is big. Lemma~\ref{altord:a:effectivity} gives +\begin{equation}\label{altord:a:effective-term} + E_1:=K_Y+B+jM-p^*H + \sim_{\Q}K_{Y/S}+B+p^*P_0 + \ \sim_{\Q}\ \text{an effective divisor}. +\end{equation} + +Choose +\[ + \lambda=\frac{1-v}{j-v}\in(0,1)\cap\Q, + \qquad \lambda j+(1-\lambda)v=1. +\] +This balances the coefficients of \(M\); to cancel the corresponding +term \(-\lambda p^*H\), put +\[ + A_S=vL+\frac{\lambda}{1-\lambda}H,\qquad + E_2=K_Y+B+p^*A_S. +\] +Then \(A_S\) is ample and +\begin{equation}\label{altord:a:interpolation} + D\sim_{\Q}\lambda E_1+(1-\lambda)E_2. +\end{equation} +Choose a sufficiently divisible integer \(N>1\) such that +\(|Np^*A_S|\) is basepoint-free. A general member +\(T_N\in|Np^*A_S|\) gives +\[ + \Gamma=\frac{1}{N}T_N\sim_{\Q}p^*A_S +\] +with \((Y,B+\Gamma)\) klt. This follows from Bertini, or after +pulling the basepoint-free system to a log resolution of +\((Y,B)\) and choosing \(N\) sufficiently large. +Thus \(E_2\sim_{\Q}K_Y+B+\Gamma\) is the log canonical class of +an effective klt pair. Its restriction to \(Y_{\bar\eta}\) is +the big divisor in \eqref{altord:a:reduced-moduli} plus the nef divisor +\(\lambda p^*H/(1-\lambda)\), and hence is big. +Theorem~\ref{altord:a:log-general-type} now yields +\[ + \kappa(Y,E_2)\geq d+\kappa(Z). +\] +In sufficiently divisible degrees, multiplication by powers of a +nonzero section in \eqref{altord:a:effective-term} embeds the corresponding +linear systems of \((1-\lambda)E_2\) into those of \(D\). +Positive rational scaling leaves Iitaka dimension unchanged, so +\[ + \kappa(X)\geq\kappa(Y,D)\geq\kappa(Y,E_2) + \geq\kappa(F)+\kappa(Z), +\] +as required. +\end{proof} + +\subsection{Extension to arbitrary characteristic-zero fields} + +\begin{lemma}\label{altord:a:field-extension} +Let \(K\subset K'\) be a field extension and let \(V\) be a smooth +projective geometrically integral variety over \(K\). +The Kodaira dimension of \(V\) is unchanged by extending scalars +to \(K'\). The same assertion holds for the geometric generic +fibre of an algebraic fibre space between smooth projective geometrically integral +varieties when the ground field is extended. +\end{lemma} + +\begin{proof} +Flat base change for proper coherent cohomology gives, in every +positive degree $n$, +\[ + H^0(V,nK_V)\otimes_K K' + \simeq H^0(V_{K'},nK_{V_{K'}}). +\] +The canonical bundle commutes with this smooth scalar extension. +Nonvanishing is therefore unchanged. When the space is nonzero, +choose a $K$-basis. The rational map defined by that basis and its +image base-change to the complete pluricanonical map over $K'$; +dimensions of finite-type schemes are unchanged by extension of +the ground field. Taking the supremum of these image dimensions +proves the first assertion, including the case $-\infty$. + +For the fibre assertion, put \(E=K(Z)\) and +\(E'=K'(Z_{K'})\), and write \(F_K\) for the generic fibre over \(E\). +The generic fibre after ground-field extension is \(F_K\times_E E'\). +Choose an algebraic closure \(\Omega\) of \(E'\) and an algebraic +closure \(\overline E\subset\Omega\) of \(E\). The two geometric +generic fibres are \(F_K\times_E\overline E\) and +\(F_K\times_E\Omega\); the latter is a scalar extension of the +former. Their Kodaira dimensions agree by the first assertion. +\end{proof} + +To complete the proof of Theorem~\ref{altord:ordinary-subadditivity}, let \(k\) be any +algebraically closed field of characteristic zero. +The varieties, their projective embeddings, and the morphism +descend to a finitely generated subfield \(K\subset k\) over +\(\Q\). Smoothness, geometric integrality and surjectivity hold +for the descended data by faithful flatness. Proper flat base +change and faithful flatness also preserve the equality +\(f_*\mathcal O_X=\mathcal O_Z\), hence the connected-fibre +condition. Choose an embedding \(K\hookrightarrow\C\). +Lemma~\ref{altord:a:field-extension} identifies all three Kodaira +dimensions in the statement with their counterparts over +\(\C\), using the geometric generic fibre for \(F\). +The complex case just proved therefore establishes the theorem +over \(k\). diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/08-integral.tex b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/08-integral.tex new file mode 100644 index 0000000000000000000000000000000000000000..34e74aeff4b6347f991ae7195687f45390421ea5 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/build/sections/08-integral.tex @@ -0,0 +1,562 @@ +\section{An integral projective proof of adjoint positivity} +\label{p1logalt:section} + +We give another proof of adjoint positivity for the whole highest Hodge +line of a rationally polarized integral pure variation. The conclusion +is the same adjoint conclusion as Theorem~\ref{thm:adjoint} for the +rational parabolic extension itself. A positive rational multiple +has the same conclusion after rescaling the large adjoint parameter. +We first replace the +variation by an integral tensor construction, then descend it through +its full period map and subtract the boundary by comparing intersection +growth. + +The tensor construction removes every monodromy element acting trivially +on the moving projective line. This makes a transverse loop detect +strict loss of the line's numerical dimension at each remaining boundary +divisor. The full period map supplies an algebraic base; the differential +of the map remembering only the highest line controls the boundary +subtraction. These two ranks need not agree. + +Let $P$ be a smooth connected projective complex variety, let $D$ be a reduced simple normal crossing +divisor, and put $P^\circ=P\setminus D$. A \emph{top Hodge line} of a rationally +polarized integral pure variation $\mathbb V$ on $P^\circ$ means +\[ + F^p\mathbb V_{\cO}\quad\text{with}\quad + F^{p+1}\mathbb V_{\cO}=0,\qquad + \rank F^p\mathbb V_{\cO}=1. +\] +Here an integral variation has a local system of finite-rank free +$\mathbb Z$-modules, and the polarization is rational and flat. The +weight and $p$ may be arbitrary integers. The line itself is a complex +Hodge subbundle; no rationality assumption is made on it. If $V$ is a flat reference +fiber, its \emph{line period map} is the equivariant holomorphic map +\[ + \ell:\widetilde{P^\circ}\longrightarrow\mathbb P(V), + \qquad x\longmapsto F^p_x. +\] +Its generic differential rank does not depend on the reference fiber. + +\begin{theorem}[Adjoint positivity]\label{p1logalt:positivity} +Let $P$ be a smooth connected projective complex variety and $D$ a +reduced simple normal crossing divisor. Let $\mathbb V$ be a rationally +polarized integral pure variation on $P\setminus D$ whose highest +nonzero Hodge filtration piece is a line, and let +$L\in\operatorname{Pic}(P)\otimes\Q$ be its rational parabolic extension. There are a projective birational +morphism $\pi:\widetilde P\to P$, with $\widetilde P$ smooth, a smooth +projective variety $Z$, a surjective morphism $h:\widetilde P\to Z$ with +connected fibers, and a nef rational divisor $A$ on $Z$ such that +\[ + \pi^*L\sim_{\Q}h^*A, + \qquad K_Z+aA\ \text{is big for every sufficiently large rational }a. +\] +Here a divisor on a point is big. Further smooth birational modifications +of the source are permitted by replacing $h$ with its composition with +the modification. +\end{theorem} + +We first specify the extension and metric facts used in the proof. + +\subsection{Extensions and numerical dimension} + +The monodromy theorem makes the local monodromies quasi-unipotent. +For unipotent local monodromy, $L$ is the top filtration subbundle in the +Deligne--Schmid extension. For quasi-unipotent monodromy, its rational +extension is characterized by the following rule: after an adapted cover +making local monodromy unipotent, its pullback is the unipotent extension, +as a rational line bundle. This convention includes the parabolic +weights; it is not an arbitrary choice of a Deligne lattice. + +\begin{lemma}[Extension facts]\label{p1logalt:extensions} +The rational extension has the following properties. +\begin{enumerate}[label=\textup{(\roman*)}] +\item It is nef and is compatible with positive tensor powers and +pullback by morphisms of smooth projective log pairs whose interiors map +to the original interior. +\item In the unipotent case, on each connected boundary stratum the local +limiting mixed Hodge structures determine a unique weight-graded piece +containing the top Hodge line. These pieces glue to a polarizable pure +variation with rank-one top filtration, whose extended top line on the +closure of the stratum is the restriction of $L$. +\item The preceding restriction statement remains valid after an adapted +cover in the quasi-unipotent case. In particular, numerical dimensions +of restrictions may be computed on a smooth adapted cover of the +stratum. +\end{enumerate} +\end{lemma} + +\begin{proof} +In the unipotent case the canonical extension and relative monodromy +weight filtration are supplied by nilpotent-orbit theory +\cite{Schmid,CKS}; precise formulations for the top +line are given in \cite[\S 2.5.3, Lemma 2.16, Lemma 5.6]{BFMT}. The +graded pieces are polarized by the primitive decomposition. + +For a boundary stratum, take the weight filtration associated with an +interior point of its monodromy cone; it is independent of that choice. +Strictness and $\rank F^p=1$ select one nonzero highest graded piece. +The global object in (ii) is this pure graded variation. To check its +gluing when the normal bundle is nontrivial, choose a local normal +coordinate $t$ and write $N$ for its monodromy logarithm. Replacing $t$ +by $t'=u(s)t$, with $u$ invertible along the stratum, changes the local +limiting filtration by +\[ + \exp\left(-\frac{\log u(s)}{2\pi\sqrt{-1}}N\right). +\] +Since $N$ lowers the monodromy weight filtration by two, this transition +acts identically on each weight graded. For several normal coordinates +the same calculation uses the sum of the commuting monodromy logarithms, +which also acts trivially on their common weight graded. The pure +graded data therefore glue. Strictness identifies the rank-one top +filtration with its unique nonzero top graded piece. At the remaining +corners, \cite[\S 2.5.3 and Lemma 2.16(3)]{BFMT} identifies the canonical +extensions on the entire smooth stratum closure, so the identification +of lines holds there as well. + +For completeness, the rational convention is compatible with these +operations as follows. In a crossing chart, take sufficiently divisible +roots of its boundary coordinates. The resulting unipotent extension +is equivariant for the finite deck group. A common positive power kills +the characters of the stabilizers on its line fibers and descends to a +line bundle in the original chart. The descents agree on overlaps: +compare them on a common root cover and use the uniqueness of the +unipotent extension. This defines a class in +$\operatorname{Pic}(P)\otimes\Q$ and proves compatibility with further +pullbacks and powers. One can perform the comparison globally on a +smooth adapted alteration; auxiliary branch divisors cause no difficulty +because the original variation extends over them. Nefness descends +under a proper surjective morphism, since every curve downstairs is +dominated by a curve upstairs. Finally, nef numerical dimension is +unchanged by a generically finite pullback, so the same construction on +strata proves (iii). +\end{proof} + +\begin{lemma}[Curvature and intersections]\label{p1logalt:rank} +The numerical dimension of the rational top Hodge line equals the +generic rank of its line period map: +\[ + \nu(L)=\rank(d\ell)_{\mathrm{gen}}. +\] +In the unipotent case, if $d=\dim P$, $\omega$ is a smooth K\"ahler form, +and $\alpha$ is the first Chern form of the Hodge metric on $L|_{P^\circ}$, +then, for $0\le k\le d$, +\begin{equation}\label{p1logalt:chern} + \int_{P^\circ}\alpha^k\wedge\omega^{d-k} + =c_1(L)^k\cdot[\omega]^{d-k}. +\end{equation} +The integrals are absolutely convergent. +\end{lemma} + +\begin{proof} +The closed-test-form version of this calculation is proved in +Lemma~\ref{p1varhd:curvature}; that proof uses only the extension facts in +Lemma~\ref{p1logalt:extensions}, not any descent or rank-drop result. +Taking the test form to be $\omega^{d-k}$ gives +Equation~\eqref{p1logalt:chern}. Griffiths' curvature formula gives +\begin{equation}\label{p1logalt:curvature-kernel} + \ker\alpha=\ker(d\ell). +\end{equation} +If $r$ is the generic differential rank, then $\alpha^{r+1}=0$ everywhere +on the open set, whereas $\alpha^r\wedge\omega^{d-r}$ is positive on a +nonempty open subset. The intersection identity and nefness therefore +give $\nu(L)=r$. For quasi-unipotent monodromy, both numerical dimension +and differential rank are unchanged by a smooth adapted generically +finite cover, and Lemma~\ref{p1logalt:extensions} identifies the pulled-back +rational line with its unipotent extension. +\end{proof} + +\subsection{Eliminating the projective monodromy kernel} + +For a variation on a connected smooth algebraic variety, we will use the +condition +\begin{equation}\label{p1logalt:faithful-line} + \gamma\ell(x)=\ell(x)\text{ for every }x + \quad\Longrightarrow\quad \gamma=\id + \quad(\gamma\text{ in the monodromy image}). +\end{equation} +This is a condition on the image of the moving line, rather than on all +of the ambient projective space. + +\begin{lemma}[An integral tensor replacement]\label{p1logalt:kernel} +Let $U$ be a smooth connected complex algebraic variety, and +let $\mathbb V$ be a rationally polarized integral pure variation on $U$ +whose highest nonzero Hodge filtration piece is a line $L^\circ$. +There are an integer $b>0$ and a rationally polarized integral pure +variation $\mathbb V_1$ on $U$ satisfying +Condition~\eqref{p1logalt:faithful-line}, whose highest nonzero Hodge +filtration piece is $(L^\circ)^{\otimes b}$. On any smooth projective +simple-normal-crossing compactification of $U$, the rational parabolic +highest line of $\mathbb V_1$ is the $b$th power of that of $\mathbb V$. +\end{lemma} + +\begin{proof} +For a general algebraic $U$, choose a dense quasi-projective open +$U_0$. The map $\pi_1(U_0)\to\pi_1(U)$ is surjective, since loops +can be moved off the complement. Thus the monodromy decompositions +below are unchanged, and a flat projector of Hodge type $(0,0)$ on +$U_0$ has that type throughout $U$ by continuity. +Write $\Gamma$ for the original monodromy image. By semisimplicity +\cite[Theorem~4.2.6 and the added-in-proof note]{DeligneHodgeII}, the complex local system is a sum of +monodromy-isotypic summands $V_j$. These are complex Hodge subvariations. To see this +precisely, the fixed-part theorem equips +$\operatorname{End}_{\Gamma}(V)$ with its Hodge algebra structure, and its +center has type $(0,0)$ by \cite[Corollary 4.2.8(i),(ii)]{DeligneHodgeII}. +Equivalently, the connected Hodge action fixes every primitive central +idempotent of this semisimple algebra. The images of those idempotents +are exactly the $V_j$. + +The top line belongs to one summand $V_{j_0}$. Retain that summand and +all its rational conjugates. Their sum is a rational subvariation: +the central idempotents are defined over a finite splitting field of the +rational commutant algebra, and the selected sum is Galois invariant. +It inherits the rational polarization and the lattice obtained by +intersecting with the original lattice. Denote it by $V_0$. None of +this construction requires the highest line, or its scalar character, +to be rational. + +Suppose $\gamma\in\Gamma$ fixes the moving line projectively everywhere. +Its projective fixed locus is the finite union of the projectivizations +of its eigenspaces. The connected line image lies in one of them, so +$\gamma$ is scalar on the linear span $S$ of that image. Equivariance +makes $S$ a nonzero $\Gamma$-subrepresentation of $V_{j_0}$. Write +$V_{j_0}=E\otimes M$, with $E$ irreducible and $\Gamma$ acting trivially +on $M$. Since $S$ contains a copy of $E$, scalar action on $S$ implies +the same scalar action on $E$ and hence on all of $V_{j_0}$. Rational +conjugation gives scalar action on every retained $V_j$. + +It follows that $\gamma|_{V_0}$ preserves the Hodge filtration at any +point. It also preserves the rational polarization and the lattice. +The stabilizer of a polarized Hodge structure in its real polarization +group is compact, since it preserves the positive definite Hodge metric. +Its intersection with the integral automorphism group is finite. +Therefore $\gamma|_{V_0}$ has finite order. For this particular +variation, one may take the exponent of that finite stabilizer at a +fixed base point. There is also a choice depending only on +$R=\rank V_0$. Indeed, an eigenvalue of an integral finite-order matrix +of rank $R$ has order $q$ with $\varphi(q)\le R$, because its cyclotomic +minimal polynomial divides the characteristic polynomial. The finite +set of such $q$ therefore gives the explicit permissible choice +\[ + b=\operatorname{lcm}\{q\geq1:\varphi(q)\leq R\}. +\] +Every scalar on every retained summand has $b$th power one. This +uniform exponent is chosen before constructing the new variation. + +Inside $V_0^{\otimes b}$ take +\[ + V_1=\bigoplus_{j\ \mathrm{retained}}V_j^{\otimes b}. +\] +The sum is taken inside $V_0^{\otimes b}$. The individual terms are +distinct direct summands in the tensor expansion, so the sum +is direct. It is Galois invariant and hence rational, and each summand +is a Hodge subvariation. Restricting the tensor polarization and +intersecting with the tensor lattice make it rationally polarized and +integral. The summand +$V_{j_0}^{\otimes b}$ has top line $(L^\circ)^{\otimes b}$; all other +summands have strictly smaller highest Hodge index. Thus this is the +unique top line of $V_1$. + +Every element considered above acts trivially on $V_1$. Conversely, if +an element of the new monodromy image fixes its moving top line, choose +a lift in $\Gamma$. Injectivity of the Veronese map +$\mathbb P(V_{j_0})\to\mathbb P(V_{j_0}^{\otimes b})$ shows that the lift +fixes the original moving line. It is therefore killed on $V_1$. +This proves Condition~\eqref{p1logalt:faithful-line}. Compatibility of extensions +with tensor powers finishes the proof. +\end{proof} + +We now descend the variation itself, using faithfulness to kill +monodromy along connected period fibres. Consequently we may +normalize the period image in the relative algebraic closure of its +function field in $\C(P)$ before estimating boundary ranks. The +finite-inertia contradiction in Section~\ref{sec:rankloss} required +the actual image; here the tensor replacement makes fibre monodromy +trivial once it fixes the moving line on an open set. + +\begin{lemma}[Full-period descent]\label{p1logalt:descent} +After applying Lemma~\ref{p1logalt:kernel} and making smooth birational +modifications of the projective models, the resulting variation +descends over a dense open subset of a smooth projective variety $Z$. +Its descended full period map is generically immersive and satisfies +Condition~\eqref{p1logalt:faithful-line}. If $A_1$ is its rational top Hodge +extension, then the original extension satisfies +\[ + b\pi^*L\sim_{\Q}h^*A_1 +\] +for a morphism $h:\widetilde P\to Z$ with connected fibers. +\end{lemma} + +\begin{proof} +Let $\mathcal D$ be the period domain for the chosen polarized lattice, +and choose an arithmetic group $\Gamma_{\mathrm{ar}}$ containing the +actual monodromy image $\Gamma$. By \cite[Theorem 1.1]{BakkerBrunebarbeTsimerman}, the full +period map factors through an algebraic dominant morphism +$P^\circ\to Y_0$, where $Y_0$ is an irreducible quasi-projective variety +and $Y_0^{\mathrm{an}}\to\Gamma_{\mathrm{ar}}\backslash\mathcal D$ is +a closed analytic immersion. Normalize $Y_0$ in the relative algebraic +closure of $\C(Y_0)$ in $\C(P)$. This extension is finite, so the +normalization is finite over $Y_0$. Its function field is relatively +algebraically closed in $\C(P)$; in characteristic zero this makes the +generic fiber geometrically connected. Shrink +source and target so that the latter is smooth and the map is a smooth +topologically locally trivial fibration with connected fibers. To +obtain this shrinking for the possibly nonproper source, stratify a +proper algebraic compactification of the map together with its source +boundary and restrict over the open base stratum. + +A local lift of the full period map along a fiber takes values in one +$\Gamma_{\mathrm{ar}}$-orbit. These orbits are discrete, including when +the quotient has finite stabilizers, so that lift is locally constant +in fiber directions. Continuation around a fiber loop therefore fixes +the lifted full period. Transport the loop to nearby fibers using the +local trivialization. In flat coordinates its actual monodromy fixes +the top line on a nonempty analytic open subset of the source. This +identity extends to its connected universal cover, and +Lemma~\ref{p1logalt:kernel} kills that monodromy. The homotopy sequence of +the fibration now factors the integral representation through the +ordinary fundamental group of the base. The filtration descends as well: on a +simply connected base chart the pulled-back flat bundle is trivial, +the filtration is constant on each connected fiber, and holomorphic +local sections of the smooth map show that the descended filtration is +holomorphic. Its transversality and polarization follow by pullback. + +The descended period map has the dimension of the original full period +image and is generically immersive. Base loops lift to the source, so +the same argument and Condition~\eqref{p1logalt:faithful-line} show that the descended +variation retains that condition. Shrinking introduces no change to +it: any monodromy element fixing the line on the smaller open fixes it +identically by analytic continuation. + +Choose a smooth projective compactification of the base, resolve its +boundary, and resolve the rational map from $P$ to this compactification. +The resulting morphism $h:\widetilde P\to Z$ has connected generic +fiber and hence connected fibers, by Stein factorization and normality +of $Z$. Enlarge the source boundary to contain the inverse image of +the chosen base boundary and resolve it. The rational extension of +the original variation is unchanged where it already extended. +Lemma~\ref{p1logalt:extensions} identifies its $b$-th power with $h^*A_1$. +\end{proof} + +\subsection{A strict rank drop at nontrivial monodromy} + +\begin{lemma}[A transverse-loop criterion]\label{p4int:transverse-loop} +Let $U=\Delta\times\Delta^{d-1}$, with coordinates $(z,s)$, and let +$U^*=\Delta^*\times\Delta^{d-1}$. Suppose that a holomorphic map +$\ell:\widetilde{U^*}\to\mathbb P(V)$ is equivariant for a linear +transformation $T\in\operatorname{GL}(V)$ under one positive turn around +$z=0$, and that $\rank(d\ell)\leq r$. Suppose that a holomorphic map +$q:U\to\mathbb C^r$ has tangential differential of rank $r$ at a point +of $z=0$, and that on $U^*$ its lift satisfies +$\ker(d\ell)\subseteq\ker(dq)$. +Then $T$ fixes every line in the image of $\ell$ on a nonempty open +subset. If $\ell$ is the restriction of a line period map on a connected +universal cover, $T$ fixes that entire line image. This includes $r=0$, +when $q$ is the map to a point. +\end{lemma} + +\begin{proof} +After shrinking around the given point, the submersion theorem provides +coordinates +\[ + (z,q_1,\ldots,q_r,u_1,\ldots,u_{d-r-1}). +\] +The coordinate $z$ is retained because $dq$ already has rank $r$ on its +zero divisor. On the punctured neighborhood, the kernel inclusion and +the rank bound imply +$\rank(d\ell)=r$ and $\ker(dq)=\ker(d\ell)$. +Fix $q$ and $u$ and make one positive turn in the $z$ coordinate. +Along every lift of that loop, the derivative of $\ell$ vanishes, so its +value is constant. Its endpoints are related by $T$, which consequently +fixes the initial line. Varying the fixed coordinates proves the claim +on an open subset. The projective equality $T\ell=\ell$ then extends by +the identity theorem to the connected universal cover. For $r=0$, the +rank bound makes $\ell$ locally constant and the same loop argument +applies with no $q$ coordinates. +\end{proof} + +\begin{lemma}[Boundary rank drop]\label{p1logalt:boundary-drop} +Let $Z$ be smooth projective, let $D$ be a reduced simple normal crossing +divisor, and let a polarizable integral pure variation on $Z\setminus D$ +have a top Hodge line satisfying Condition~\eqref{p1logalt:faithful-line}. Write $A_1$ +for its rational extension and $r=\nu(A_1)$. If $D_i$ is a component +whose transverse monodromy is not the identity, then +\[ + \nu(A_1|_{D_i})0$ with $T_s^e=1$ and set $z=t^e$. +In flat coordinates the nilpotent orbit extension supplies a holomorphic +line map $\Psi$ on the root polydisk such that +\begin{equation}\label{p1logalt:untwist} + \ell(t,s)= + \exp\left(\frac{e\log t}{2\pi\sqrt{-1}}N\right)\Psi(t,s), + \qquad + \Psi(\zeta t,s)=T_s\Psi(t,s), + \quad \zeta=\exp(2\pi\sqrt{-1}/e). +\end{equation} +These are equalities of lines; $\ell$ is evaluated on the logarithmic +cover when necessary. + +Let $W=W(N)$, centered at the weight of the variation. On the boundary +there is a unique index $k$ in which the top line has a nonzero +weight-graded image. The map $N$ is a strict morphism of limiting mixed +Hodge structures of type $(-1,-1)$. Since $F^{p+1}=0$, it gives +\begin{equation}\label{p1logalt:survives} + F^p\cap NV=N(F^{p+1})=0. +\end{equation} +Thus the limiting line survives in $C=V/NV$. + +We also need to preserve its derivative rank. The primitive decomposition +of the fixed graded local system identifies +\[ + \operatorname{Gr}^W_k V + =P_k\oplus N\operatorname{Gr}^W_{k+2}V, + \qquad + P_k\xrightarrow{\ \sim\ }\operatorname{Gr}^W_k C. +\] +Here $k$ is at least the original weight: lower graded pieces are +entirely in the image of $N$ and cannot contain the top line. The second +summand has zero $F^p$, so the graded moving top line lies in the flat +primitive summand $P_k$. Its projective derivative rank is therefore +unchanged by the displayed isomorphism. Projecting the boundary line +in $C$ further to its relevant graded piece recovers precisely this pure +line period map. By Lemmas~\ref{p1logalt:extensions} and~\ref{p1logalt:rank}, +its rank is +\[ + r_i=\nu(A_1|_{D_i}). +\] +This equality may be checked on a global adapted cover: restriction of +the pulled-back line to a component dominating $D_i$ is its generically +finite pullback, which preserves numerical dimension. + +Equation~\eqref{p1logalt:untwist} makes $\Psi(0,s)$ an eigenline of $T_s$. +Its eigenvalue $\lambda$ is constant on a sufficiently small connected +boundary neighborhood. Let $E=C_\lambda$ and let $q_0:V\to E$ be the +quotient followed by the eigenspace projection. The projective map +\[ + Q(t,s)=\mathbb P(q_0)(\Psi(t,s)) +\] +is defined near the chosen boundary point and has boundary rank at +least $r_i$. Since $N$ vanishes on $C$, it also equals +$\mathbb P(q_0)(\ell(t,s))$. Moreover $T_s$ acts by a scalar on $E$, +so $Q(\zeta t,s)=Q(t,s)$. Hence $Q$ descends holomorphically to the +original $(z,s)$-polydisk, including $z=0$. This descent uses a +projective map; a choice of a fractional scalar frame is unnecessary. + +Suppose $r_i\ge r$. Choose $r$ affine coordinates of $Q$, denoted +$q_1,\ldots,q_r$, whose differentials are independent along $D_i$. +The map $q=(q_1,\ldots,q_r)$ factors locally through $\ell$ on the +punctured chart. Its differential therefore satisfies the kernel +inclusion in Lemma~\ref{p4int:transverse-loop}. The rank bound for +$\ell$ is $r$ by Lemma~\ref{p1logalt:rank}. The transverse-loop lemma +shows that $T$ fixes the moving highest line everywhere on the connected +universal cover. Condition~\eqref{p1logalt:faithful-line} gives +$T=\id$, contrary to the choice of the divisor. If $r=0$, the same +lemma applies with no coordinate functions. If $r>d-1$, the supposed +inequality $r_i\geq r$ is already impossible. +\end{proof} + +\subsection{Removing the boundary from the adjoint class} + +\begin{lemma}[A boundary subtraction estimate]\label{p1logalt:volume} +Let $Z$ be smooth projective of positive dimension $d$, let $A_1$ be a +nef rational divisor, and let $J=\sum_iD_i$ be reduced with smooth +components. Suppose $K_Z+J$ is big and +\[ + \nu(A_1|_{D_i})0$. By Kodaira's lemma write +\[ + K_Z+J\sim_{\Q}H+E,\qquad H\text{ ample},\quad E\ge0, +\] +and set $B_a=H+aA_1$ for rational $a\ge0$. Choose one ample integral +divisor $C_0$ such that both $C_0$ and $C_0+J$ are globally generated, +and put $H'=H+C_0$. This choice is independent of $a$ and of all +section degrees below. + +For every sufficiently divisible $m>0$, filter by $kJ$ for $0\le k0\}, +\] +where $H$ is any ample class and $A$ is nef. In particular, this +numerical dimension is zero on a point. Normalize +$\ddc=\frac{i}{2\pi}\partial\bar\partial$. + +Lemma~\ref{p1logalt:extensions} supplies nefness and compatibility of +the rational extension with positive tensor powers and log-pair +pullbacks whose interiors map into the original interior. The +Chern-intersection formula below uses these extension properties, +but neither full-period descent nor a boundary rank-drop result. +For strict rank loss we also use the integral tensor replacement +of Lemma~\ref{p1logalt:kernel} and the transverse-loop criterion of +Lemma~\ref{p4int:transverse-loop}. The replacement is faithful on +the moving highest line: a monodromy element fixing that line at +every point of the connected universal cover is the identity. +The Thom-form calculation computes restriction intersections from +tangential limiting curvature. It uses the local graded Hodge data, +without identifying the boundary restriction with a globally extended +graded line as in Section~\ref{p1logalt:section}. + +\subsection{Curvature and intersection powers} + +\begin{lemma}[Chern intersections with closed test forms]\label{p1varhd:curvature} +Let $A$ be the rational parabolic highest line of an integral +polarized pure variation on the complement of a reduced +simple-normal-crossing divisor in a smooth projective variety $R$. +Then +\begin{equation}\label{p1varhd:numrank} + \nu(A)=\text{generic differential rank of its projective line map} +\end{equation} +provided $A$ is nef. In the unipotent case, if $\alpha$ is the Chern +form of its Hodge metric on $R\setminus D_R$, then for every +$0\leq j\leq n=\dim R$ and every smooth closed form $\gamma$ of type +$(n-j,n-j)$, the integral +$\int_{R\setminus D_R}\alpha^j\wedge\gamma$ is absolutely convergent +and equals $c_1(A)^j\cdot[\gamma]$. +\end{lemma} + +\begin{proof} +We may compute after a generically finite smooth alteration on +which all boundary monodromies are unipotent. To obtain one, pass +to a finite level cover with neat monodromy, compactify, and resolve. +Quasi-unipotence then gives unipotence, including for the commuting +local monodromies on the resolved boundary. The parabolic pullback +property of Lemma~\ref{p1logalt:extensions} identifies the new line +with the pullback of $A$. +Generically finite pullback preserves nef numerical dimension by +the intersection and projection formulas. It also preserves the +generic differential rank in characteristic zero. We therefore +work on the unipotent model, retaining the notation $R,A$. + +Let $\alpha$ be the semipositive Chern form of the highest-line +Hodge metric on the open part. Griffiths' curvature formula +\cite[Theorem~5.2, Formula~5.3, and Formula~6.18]{GriffithsPeriodsIII} says +that $\alpha(v,\bar v)$ is a positive fixed multiple of the squared +norm of the projective highest-line differential in direction $v$. +In particular, its rank is the rank in \eqref{p1varhd:numrank}. + +Consider a crossing chart with coordinates $z_1,\ldots,z_n$ and +boundary $z_1\cdots z_c=0$. Put +$\rho_i=-\log|z_i|^2$ and shrink the chart so that $\rho_i>e$. +The horizontal Schwarz estimate, applied separately to the +coordinate disks, gives +\begin{equation}\label{p1varhd:poincare} + 0\leq\alpha\leq C\Omega, + \qquad + \Omega= + \sum_{i=1}^{c}\frac{i\,dz_i\wedge d\bar z_i} + {|z_i|^2\rho_i^2} + +\sum_{i=c+1}^{n}i\,dz_i\wedge d\bar z_i. +\end{equation} +Here and below harmless positive normalization constants are +absorbed into $C$. The coordinate estimate follows from the +curvature of the horizontal period metric and the singular +Ahlfors--Schwarz lemma; see \cite[\S 2.1, Proposition~2.4 and +Lemma~2.5]{BrunebarbeCadorel}. Positivity and the Hermitian-form Cauchy--Schwarz +inequality control the off-diagonal entries and give the uniform +product estimate \eqref{p1varhd:poincare}, even where the period +differential is degenerate. + +By the nilpotent-orbit theorem +\cite[Theorem~4.12]{Schmid}, a nonvanishing top extension frame, +written in flat coordinates on a bounded angular sector, is +\begin{equation}\label{p1varhd:frame} + v(z)=\exp\left(\sum_{i=1}^{c} + \frac{\log z_i}{2\pi i}N_i\right)u(z), +\end{equation} +where the $N_i$ commute and $u$ is holomorphic and nonvanishing. +If $\beta$ is the curvature of a fixed smooth metric on $A$, write +$\alpha-\beta=\ddc\varphi$. The ratio of the two metrics satisfies +\begin{equation}\label{p1varhd:loglog} + |\varphi(z)|\leq C\left(1+\sum_{i=1}^{c}\log\rho_i\right). +\end{equation} +This is the logarithmic growth estimate for canonical Hodge frames; +see \cite[Theorem~5.1]{CattaniKaplanDegenerating}. Here the Schwarz +estimate also gives a direct proof of the bound needed for integration. The differential of the logarithmic Hodge norm of any +flat vector is bounded by a constant times the invariant period +metric, with a constant independent of the point by homogeneity. +Travel from a fixed point along coordinate paths in a bounded +angular sector. Their product-Poincar\'e length is bounded by +$C(1+\sum_i\log\rho_i)$, so the Hodge metric and its inverse in a +flat frame grow at most as products of powers of the $\rho_i$. +The exponential in \eqref{p1varhd:frame} and its inverse are polynomials +in the logarithms, since the $N_i$ are commuting nilpotent +operators. Finally, $u$ has ordinary norm bounded above and below +on a smaller chart. These three observations give the upper and +lower logarithmic bounds in \eqref{p1varhd:loglog}. We have therefore +obtained the two controls needed for the cutoff argument: +\begin{equation}\label{p1logalt:metric-bounds} + 0\leq\alpha\leq C\Omega, + \qquad |\varphi|\leq C\Bigl(1+\sum_i\log\rho_i\Bigr). +\end{equation} + +Chern-current representation for canonical Hodge extensions is +established in \cite[Theorem~5.2]{CattaniKaplanDegenerating}. We now +justify the highest-line intersection formula with closed test forms, +including the absence of boundary contributions. For $0\leq j\leq n$ and every +smooth closed form +$\gamma$ of type $(n-j,n-j)$, +\begin{equation}\label{p1varhd:chern} + \int_{R\setminus D_R}\alpha^j\wedge\gamma + =c_1(A)^j\cdot[\gamma]. +\end{equation} +The case $j=0$ is immediate, so assume $j>0$. Choose a smooth +function $\chi$ that is one on $(-\infty,1]$ and +zero on $[2,\infty)$, and use products of +$\chi((\log\rho_i)/M)$ to cut off the boundary. The $\rho_i$ can +be globalized using smooth metrics on the boundary divisor +bundles. Denote the resulting cutoff by $\chi_M$. Its second +derivatives are uniformly bounded in the product-Poincar\'e metric, +and their support escapes to the boundary as $M\to\infty$. + +Since all curvature forms are closed, on the open part we have +\[ + \alpha^j-\beta^j + =\ddc\left(\varphi + \sum_{k=0}^{j-1}\alpha^k\wedge\beta^{j-1-k}\right). +\] +Multiplying by $\chi_M\gamma$ and integrating by parts makes the +difference of the integrals equal to +\[ + \int\varphi\,\ddc\chi_M\wedge + \sum_{k=0}^{j-1}\alpha^k\wedge\beta^{j-1-k}\wedge\gamma. +\] +Its absolute value is bounded on escaping tails by a constant times +\[ + \left(1+\sum_i\log\rho_i\right)\Omega^n. +\] +This is integrable: in a cusp variable the relevant radial +integrals are bounded by +$\int^{\infty}(1+\log\rho)\rho^{-2}\,d\rho$. +Thus the error tends to zero. The same domination permits removal +of $\chi_M$ from the two curvature integrals, proving +\eqref{p1varhd:chern}. Rational bundles are handled by first taking an +integral tensor power and then dividing the forms and identities +by that power. + +Let $r$ be the generic rank of the projective line map. Then +$\alpha^{r+1}=0$ on the open part, whereas $\alpha^r$ is nonzero +and positive on a nonempty open subset. Apply \eqref{p1varhd:chern} +with complementary powers of an ample K\"ahler form. It gives +vanishing of the $(r+1)$st intersection power and strict positivity +of the $r$th one. The nef intersection characterization proves +\eqref{p1varhd:numrank}. +\end{proof} + +\subsection{Rank loss at a monodromy divisor} + +\begin{lemma}[Boundary rank drop by a Thom-form calculation]\label{p1varhd:boundary} +In the setting of Lemma~\ref{p1varhd:curvature}, suppose that monodromy +is faithful on the moving highest line. If $D_i$ is a boundary +component with nonidentity transverse monodromy, then +\begin{equation}\label{p1varhd:strictdrop} + \nu(A|_{D_i})<\nu(A). +\end{equation} +\end{lemma} + +\begin{proof} +Put $r=\nu(A)$. We first compute a bound for the numerical +dimension on a boundary component of a unipotent model. This +computation is unchanged on a component dominating $D_i$ under a +generically finite alteration, since restriction of the pulled-back +line is the generically finite pullback of $A|_{D_i}$. + +\paragraph{The limiting highest line.} +Near a general point of a unipotent boundary component write the +coordinates as $(t,s)$, with divisor $t=0$ and monodromy logarithm +$N$. Let $w$ be the weight and $p$ the largest filtration index. +The limiting mixed Hodge structure is polarized by $N$ +\cite[Theorem~6.16]{Schmid}; its graded structures vary as polarized +variations along the smooth boundary stratum +\cite[Proposition~2.10 and the following discussion]{CattaniKaplanDegenerating}. +The limiting $F^p$ has dimension one and $F^{p+1}=0$. Consequently +it consists of one Deligne summand $I^{p,q}$. Write +$p+q=w+l$. In the primitive decomposition for $N$, a nonzero +$N^k$-image term with $k>0$ would have to come from $F^{p+k}$, +which is zero. The line is therefore primitive, $l\geq0$, and, +for a nonzero vector $u_0(s)=u(0,s)$ spanning it, +\[ + N^lu_0(s)\ne0,\qquad N^{l+1}u_0(s)=0. +\] +Shrink the boundary stratum so that its Hodge type, and hence $l$, +is fixed. + +For a highest Hodge vector in the pure variation, the Hodge norm is +the flat polarization pairing with its conjugate, multiplied by the +fixed factor determined by its Hodge type. +Substituting the exponential frame into that pairing gives a +polynomial in $y=-\log|t|$. Primitive polarization identifies its +leading term as a positive constant times $y^l$ times the squared +norm $h_l(s)$ of the primitive graded line. Replacing $u(t,s)$ +by $u_0(s)$ introduces errors $O(|t|)$ times powers of $y$; +the same holds after tangential differentiation, by holomorphicity +of $u$. It follows, locally uniformly with tangential derivatives, +that +\begin{equation}\label{p1varhd:limitnorm} + \log\|v(t,s)\|^2 + =l\log y+\log h_l(s)+c+O(y^{-1})+O(|t|y^M) +\end{equation} +for some $M$ and a constant $c$. Thus the tangential curvature +converges to the curvature of the primitive graded highest line. +If $q_i$ denotes the generic rank of its projective differential, +the limiting tangential curvature has rank at most $q_i$. + +\paragraph{The restriction intersection and crossing points.} +We claim that +\begin{equation}\label{p1varhd:restrictionrank} + \nu(A|_{D_i})\leq q_i. +\end{equation} +The following estimate proves the claim without assuming uniform +convergence of \eqref{p1varhd:limitnorm} at crossings. Fix an ample +K\"ahler form $\omega$ and an integer $j$ with +$q_in-1$, the supposed inequality +$\nu(A|_{D_i})\geq r$ is impossible on dimensional grounds. This proves +Equation~\eqref{p1varhd:strictdrop}. +\end{proof} + + +\subsection{Connection with the integral projective argument} + +The hypotheses required for the preceding calculation arise from the +integral tensor replacement and full-period descent in +Lemmas~\ref{p1logalt:kernel} and~\ref{p1logalt:descent}. +On the resulting projective base, Lemma~\ref{p1varhd:boundary} gives the +same strict inequality as Lemma~\ref{p1logalt:boundary-drop}, with +the boundary intersection justified directly by Thom forms. The +boundary subtraction estimate of Lemma~\ref{p1logalt:volume} then +applies to the union $J$ of the divisors with nonidentity transverse +monodromy. The full-period map supplies the bigness of $K_R+J$ as in +the proof of Theorem~\ref{p1logalt:positivity}; the highest-line +rank supplies the strict decrease in the degree of the restriction +polynomials. Thus the two different period ranks retain their separate +roles in adjoint positivity. + +\begin{remark}[Tensor replacement on a restricted locus] +\label{p4int:restricted-tensor} +The same local results also give the curve-constancy part +of the highest-line properties used in such comparisons. On a smooth +complete curve, degree zero gives zero integral of the semipositive +Hodge curvature by Equation~\eqref{p1varhd:chern}, after an adapted +cover if necessary. The curvature is therefore zero, and the highest +line is projectively constant in a flat reference. If the highest line +is projectively constant on a connected smooth algebraic locus, apply +Lemma~\ref{p1logalt:kernel} to the restricted variation itself. +Use its own monodromy image and isotypic decomposition: restricting +an ambient replacement need not preserve faithfulness, because the +restricted line image may have a larger projective stabilizer. +Every monodromy element of the replacement fixes its constant highest +line, so faithfulness makes that monodromy trivial. Since the new line +is a positive tensor power of the original line, the original +highest-line character on that locus has finite image. The tensor +exponent here is allowed to depend on the restricted variation. +\end{remark} + +\begin{remark}\label{p1varhd:further-models} +The adjoint conclusion is preserved under further smooth projective +birational modification $\pi:R'\to R$, together with a compatible +modification of the source. The parabolic pullback property preserves +the line-bundle identity and nefness, while +\[ + K_{R'}+b\pi^*A + =\pi^*(K_R+bA)+E_\pi,\qquad E_\pi\geq0, +\] +preserves bigness. The calculations here concern the whole rank-one +highest step of a polarizable integral pure variation on a projective +pair; they assert no equality between highest-line rank, full-period +rank, and whole-fiber birational variation. +\end{remark} diff --git a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8ea0072aafb93b646f344024ae57bc8e07b97778 --- /dev/null +++ b/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:564a98c625b94ef479a6690bed65451fee18c47c69c23e5d9e1f4fd181deca9a +size 587812 diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/README.md b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..a8d63fb17f80ca00e155576965754383215b4be0 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/README.md @@ -0,0 +1,17 @@ +# [Projective-space bases of Lagrangian fibrations](main.pdf) + +**Author:** OpenAI +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026, + author = {{OpenAI}}, + title = {{Projective-space bases of Lagrangian fibrations}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf}{OAI:Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/main.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..901380f752c355ff73462f5264e4e350212972f4 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/main.tex @@ -0,0 +1,83 @@ +\pdftrailerid{} +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[a4paper,margin=28mm]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs,longtable} +\usepackage{graphicx,tikz} +\usepackage{placeins} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage[dvipsnames]{xcolor} +\usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue]{hyperref} +\hypersetup{pdftitle={Projective-space bases of Lagrangian fibrations},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{notation}[theorem]{Notation} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\newcommand{\C}{\mathbb C} +\newcommand{\Q}{\mathbb Q} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\F}{\mathbb F} +\newcommand{\PP}{\mathbb P} +\newcommand{\cO}{\mathcal O} +\newcommand{\cE}{\mathcal E} +\newcommand{\cN}{\mathcal N} +\newcommand{\cS}{\mathcal S} +\newcommand{\cT}{\mathcal T} +\newcommand{\cJ}{\mathcal J} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\NS}{NS} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\td}{td} +\DeclareMathOperator{\ch}{ch} +\DeclareMathOperator{\Gr}{Gr} +\DeclareMathOperator{\id}{id} +\setlist{leftmargin=*,itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{1.5em} +\title{Projective-space bases of Lagrangian fibrations} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that the normal projective base of a projective Lagrangian +fibration from a compact irreducible holomorphic symplectic Kähler manifold +is projective space. This resolves the projective-space base conjecture +for such fibrations in every dimension and deformation type. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\FloatBarrier +\input{sections/02-setup} +\input{sections/03-local-tensor} +\input{sections/04-stratum} +\input{sections/05-base-change} +\input{sections/06-euler} +\input{sections/07-deformation} +\input{sections/08-isotrivial} +\input{sections/09-second-system} +\input{sections/10-variation} +\input{sections/11-quotient} +\input{sections/12-conclusion} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/references.bib b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..02b78c83d5b3287cfa7478f48d85f1d4dfa8ad28 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/references.bib @@ -0,0 +1,627 @@ +@article{Matsushita1999, + author = {Matsushita, Daisuke}, + title = {On fibre space structures of a projective irreducible symplectic manifold}, + journal = {Topology}, + volume = {38}, + number = {1}, + year = {1999}, + pages = {79--83}, + doi = {10.1016/S0040-9383(98)00003-2}, + url = {https://arxiv.org/abs/alg-geom/9709033} +} + +@article{Matsushita2001, + author = {Matsushita, Daisuke}, + title = {Addendum: {On} fibre space structures of a projective irreducible symplectic manifold}, + journal = {Topology}, + volume = {40}, + number = {2}, + year = {2001}, + pages = {431--432}, + url = {https://arxiv.org/abs/math/9903045} +} + +@article{Matsushita2000, + author = {Matsushita, Daisuke}, + title = {Equidimensionality of {Lagrangian} fibrations on holomorphic symplectic manifolds}, + journal = {Mathematical Research Letters}, + volume = {7}, + number = {4}, + 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+@article{OttazziWarhurst2011, + author={Ottazzi, Alessandro and Warhurst, Ben}, + title={Contact and 1-quasiconformal maps on {Carnot} groups}, + journal={Journal of Lie Theory}, volume={21}, year={2011}, pages={787--811}, + url={https://www.heldermann-verlag.de/jlt/jlt21/ottala2e.pdf} +} +@article{AbramovichOlssonVistoli2008, + author={Abramovich, Dan and Olsson, Martin and Vistoli, Angelo}, + title={Tame stacks in positive characteristic}, + journal={Annales de l'Institut Fourier}, volume={58}, number={4}, + year={2008}, pages={1057--1091}, url={https://arxiv.org/abs/math/0703310} +} +@article{OlssonStarr2003, + author={Olsson, Martin and Starr, Jason}, + title={Quot functors for {Deligne--Mumford} stacks}, + journal={Communications in Algebra}, volume={31}, number={8}, + year={2003}, pages={4069--4096}, url={https://arxiv.org/abs/math/0204307}, + note={Numbering follows the arXiv author manuscript} +} +@article{Toen1999, + author={To{\"e}n, Bertrand}, + title={Th{\'e}or{\`e}mes de {Riemann--Roch} pour les champs de {Deligne--Mumford}}, + journal={{K}-Theory}, volume={18}, number={1}, year={1999}, pages={33--76}, + url={https://arxiv.org/abs/math/9803076}, + note={Numbering follows arXiv:math/9803076v2} +} +@article{PortaYu2016, + author={Porta, Mauro and Yu, Tony Yue}, + title={Higher analytic stacks and {GAGA} theorems}, + journal={Advances in Mathematics}, volume={302}, year={2016}, pages={351--409}, + url={https://arxiv.org/abs/1412.5166}, + note={Numbering follows arXiv:1412.5166v2} +} +@misc{MilanovSaito, + author={Milanov, Todor and Saito, Kyoji}, + title={Primitive Forms and Vertex Operators}, + howpublished={Undated book draft}, + url={https://member.ipmu.jp/todor.milanov/Research/pf-vo.pdf}, + note={Accessed September 23, 2026} +} +@incollection{Behrend2004, + author={Behrend, Kai}, title={Cohomology of stacks}, + booktitle={Intersection Theory and Moduli}, + series={{ICTP} Lecture Notes}, volume={XIX}, year={2004}, pages={249--294}, + publisher={Abdus Salam International Centre for Theoretical Physics}, + address={Trieste}, url={https://personal.math.ubc.ca/~behrend/CohSta-1.pdf}, + note={Based on 2002 lectures} +} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/01-introduction.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..6ef3895ff42a1432074e6d4c3699504c4747e094 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/01-introduction.tex @@ -0,0 +1,253 @@ +\section{Introduction}\label{sec:introduction} + +Let \(X\) be a compact irreducible holomorphic symplectic (IHS) Kähler +manifold of dimension \(2n\). A nontrivial fibration of \(X\) is a proper +surjective holomorphic map \(f:X\to B\) with connected fibers onto a normal +complex space, with \(0<\dim B<2n\). In the projective setting, Matsushita +proved that \(\dim B=n\) and that a smooth general fiber is an abelian +variety \cite{Matsushita1999,Matsushita2001}. His equidimensionality theorem +shows that every irreducible component of every fiber is Lagrangian once +the general fiber is Lagrangian \cite[Theorem~1]{Matsushita2000}. +For compact Kähler IHS total spaces, Greb--Lehn's Proposition~2.2 shows +that the base is projective of dimension \(n\) and that every fiber +component is Lagrangian \cite{GrebLehn2014}. The projective-space base conjecture asks +whether this normal base must be isomorphic to \(\PP^n\). + +Hwang proved the projective-space conclusion when \(X\) is projective and +\(B\) is smooth \cite[Theorem~1.2]{Hwang2008}. Greb and Lehn extended the +smooth-base theorem to compact Kähler IHS total spaces +\cite[Theorem~1.1]{GrebLehn2014}. These results reduce the base problem to +the possible singularities of \(B\). + +The conclusion is also known for the \(K3^{[n]}\), generalized Kummer, +\(\mathrm{OG6}\), and \(\mathrm{OG10}\) deformation types without assuming +the base smooth. A common nef formulation of the earlier isotropic-divisor +and rational-fibration results is recorded in +\cite[Theorem~1.3]{DebarreHuybrechtsMacriVoisin2024}: a nef line bundle \(L\) +whose first Chern class is primitive and isotropic for the +Beauville--Bogomolov--Fujiki form on such a compact IHS manifold +admits a holomorphic Lagrangian fibration \(g:X\to\PP^n\) with +\(L\simeq g^*\cO_{\PP^n}(1)\). For an existing fibration \(f:X\to B\), choose +an ample line bundle \(H\) on \(B\) and write \(f^*H\simeq L^r\) with +\(c_1(L)\) primitive. This class remains nef and isotropic, so the theorem +applies to \(L\); connected fibers identify +the graded section rings of \(H\), \(L^r\), and \(\cO_{\PP^n}(r)\). +Taking their \(\operatorname{Proj}\) identifies \(B\) with \(\PP^n\). + +In dimension four, Ou reduced the normal surface base of a projective IHS +fibration to \(\PP^2\) or the specified Fano surface +\(S^{\mathrm n}(E_8)\), with one \(E_8\) singularity and two nodal rational +anticanonical curves \cite[Theorem~1.2]{Ou2019}. Huybrechts and Chenyang Xu +excluded the second case \cite[Theorem~0.1]{HuybrechtsXu2022}. +Müller and Zheng Xu subsequently proved a local theorem: a projective +Lagrangian germ from a smooth holomorphic symplectic manifold has smooth +base if the normal analytic base has only isolated quotient singularities +\cite[Theorem~1.3]{MullerXu2026}. Their Corollary~1.4 treats such germs in +total dimension four without a singularity assumption, and their +Theorem~1.5 treats terminal threefold bases in total dimension six. +Their Example~4.1 exhibits a nonisolated quotient stratum for a noncompact +symplectic total space; excluding these strata in the compact setting +therefore requires additional global geometry. + +For projective IHS total spaces, the cohomological constraint is already +that of projective space: Shen--Yin compute the rational intersection +cohomology of \(B\) as one-dimensional in degrees \(0,2,\ldots,2n\) and +zero in odd degrees \cite[Theorem~0.4(a)]{ShenYin2022}. This does not itself establish +smoothness. Our proof first establishes smoothness and then applies Hwang's +theorem. + +\begin{theorem}\label{thm:main} +Let \(n\geq1\), and let \(X\) be a compact irreducible holomorphic symplectic +Kähler manifold of dimension \(2n\), with symplectic form \(\sigma\). +Suppose that +\[ + f:X\longrightarrow B +\] +is a projective surjective holomorphic morphism with connected fibers, +where \(B\) is a normal projective variety of dimension \(n\). +Assume that every irreducible component of every fiber has dimension +\(n\) and that the restriction of \(\sigma\) to its smooth locus is zero. +Then \(B\) is isomorphic to \(\PP^n_{\C}\). +\end{theorem} + +Here projectivity of \(f\) means the existence of a relatively ample +holomorphic line bundle. Together with projectivity of \(B\), it makes the +original \(X\) projective. Compact Kähler deformations, which can be +nonprojective, enter later in the proof. The theorem imposes no smoothness +condition on \(B\), no additional lower bound on \(b_2(X)\), and no +restriction on the deformation type. It gives a positive answer to the +projective-space base conjecture for the stated projective fibrations. + +The proof uses the companion paper \cite{StrongSYZ2026}. +Its Theorem~1.1 supplies semiampleness of a nonzero +integral nef isotropic line bundle on the given compact Kähler IHS manifold +and the associated connected Lagrangian fibration; we restate it as +Theorem~\ref{thm:strong-syz}. Its Lemma~4.2 supplies a polarized family for +a prescribed primitive ample class, with polarization a positive multiple +of that class. It also supplies a torsion-free finite-index subgroup of the +integral lattice isometries fixing the class and preserving the chosen +period-domain component. This subgroup acts freely, the quotient is smooth +and quasi-projective, and a local marked lift of the family's algebraic +period map is submersive at some point. The family monodromy lies in that subgroup. +Lemma~\ref{secsys:actual-monodromy} of this article proves that its image +has finite index there. Section~\ref{sec:second-system} +uses these inputs to construct a second Lagrangian fibration. + +Once \(B\simeq\PP^n\) has been established, Kim--Oguiso--Shinder's theorem +excludes codimension-one multiple fibers +\cite[Theorems~1.1 and~3.4]{KimOguisoShinder2026}. Kamenova--Verbitsky also +prove this conclusion over \(\PP^n\) +\cite[Theorems~1.5 and~6.7]{KamenovaVerbitsky2026}. Thus, for every prime +divisor \(D\subset B\), writing \(f^*D=\sum_i a_iE_i\) with distinct prime +divisors \(E_i\), one has \(\gcd_i a_i=1\). Equivalently, there is no +equality \(f^*D=mE\) with an integral effective divisor \(E\) and \(m>1\). +This is a consequence of the base identification. + +\subsection{Proof strategy} + +The proof has two stages. First we rule out germs that are not analytic +quotients of smooth germs by finite groups. We then eliminate the +stabilizers of the remaining quotient germs. The second stage uses both +the original fibration and a finite cover by an abelian variety constructed +from deformations that retain \(B\). This cover is available without +assuming that \(B\) has quotient singularities. + +Suppose \(B\) has a nonquotient germ. Sections~\ref{sec:local-tensor} +and~\ref{sec:stratum} use finite local covers and reflexive cotangent tensors +to find a nonempty closed normal projective stratum \(S_0\subset B\) of +dimension less than \(n\). A smooth Deligne--Mumford stack \(\cS\) with coarse +space \(S_0\) records the finite stabilizers on its local charts. +Sections~\ref{sec:base-change} and~\ref{sec:euler} prove the obstruction +\[ + \chi_{\mathrm{top}}(S_0)=0, +\] +where this is the ordinary Euler characteristic of the coarse space. +The numerical argument compares Euler characteristics after pullback to +\(X\) with alternating cotangent contributions on the local charts. +This comparison uses ordinary coefficient base change in a suitable +reduction to positive characteristic; Frobenius growth and Riemann--Roch +force the relevant values to be zero. The local tensor classification +then allows coherent coefficient modules to isolate the trivial character +of the full stabilizer. This recovers the ordinary Euler characteristic +of \(S_0\), including when some stabilizers act trivially on the stratum. + +The tensor recognition in this construction uses Deligne's criterion +\cite[Theorem~7.1]{Deligne1990}. The differential identity behind the +base-change formula uses the strictness and decomposition of projective +direct images of Hodge modules \cite{Saito1988,Saito2016}. These tools enter the +Euler obstruction; the later exclusions use the global geometry of the +original fibration and its deformations. + +Choose a very ample line bundle \(H\) on \(B\) and put \(e=c_1(f^*H)\). +Matsushita's deformation theorem gives a relatively generated line bundle +and a family of Lagrangian maps throughout the nearby locus where \(e\) +remains of type \((1,1)\) +\cite[Theorem~1.2 and Corollary~1.3]{MatsushitaDeformations2016}. +The targets in that result may vary. Section~\ref{sec:deformation} proves +that their normal Stein targets can all be identified with the original +\(B\), obtaining a map to \(B\times D\) over that entire nearby locus. +Verbitsky's degenerate-twistor construction already keeps a Lagrangian +fibration holomorphic over its original base along a complex line +\cite[Theorem~1.10]{Verbitsky2015}. Bogomolov--Déev--Verbitsky develop +this construction for closed base forms of type \((2,0)+(1,1)\) +\cite[Proposition~3.2 and Theorem~3.3]{BogomolovDeevVerbitsky2022}. +Soldatenkov--Verbitsky give a formulation for normal analytic bases and +show, for compact IHS fibrations, that these deformations are Zariski +locally trivial over the projective base and arise from twists +\cite[Section~2.2, Theorem~3.1 and Corollary~3.3]{SoldatenkovVerbitsky2025}. +The simultaneous fixed target over the full nearby \(e\)-Hodge locus is +the additional assertion needed here. + +Lemma~\ref{lem:betti-bound} gives \(b_2(X)\geq4\). If \(b_2(X)\geq5\), period +variation in the fixed-base family forces the Hodge cohomology of \(\cS\) +to be diagonal. Consequently +\[ + \chi_{\mathrm{top}}(S_0)=\sum_p h^{p,p}(\cS)>0, +\] +contradicting the obstruction. When \(b_2(X)=4\), we distinguish the +isotrivial family of polarized smooth fibers from a family with nonzero +variation. Section~\ref{sec:isotrivial} treats the isotrivial case: its +ramification and reflection arguments reduce a possible stratum to an +elliptic curve, where a connection on a negative line bundle gives the +contradiction. In the other case, Section~\ref{sec:second-system} constructs +a projective deformation \(f':X'\to B\) with a second, varying Lagrangian +fibration \(g:X'\to C\). Section~\ref{sec:variation} uses its monodromy and +the symmetric period cubic to force the same diagonal Hodge conclusion. +The cubic is the Donagi--Markman condition on the smooth polarized locus +\cite[Section~7.2, Lemmas~7.1--7.2 and Remark~7.3]{DonagiMarkman1996}; +Deligne's fixed-part theorem supplies the monodromy invariants used with it +\cite[Theorem~4.1.1(ii) and Corollary~4.1.2]{Deligne1971}. +These three cases rule out every nonquotient germ. + +Starting from the original fibration and its fixed-base family, the +second-system construction also supplies a cover in every allowed rank. +On a smooth fiber \(A\) of \(g\), the mixed Fujiki formula makes the nef +line bundle \(f'^*H|_A\) have positive top self-intersection, hence it is +ample on the abelian variety \(A\). Therefore \(f'|_A\) is finite, and +equality of dimensions makes it surjective. This is a finite cover by an +abelian variety. +After the first stage has proved that \(B\) has quotient singularities, +Section~\ref{sec:quotient} applies this cover and the original fibration. +On the canonical smooth stack over \(B\), the direct-image identities +produce a single translation vector on \(A\) whose projection to the directions +fixed by each nonidentity stabilizer element never vanishes on its fixed locus. +This use of equivariant higher direct images has a local +precedent in Müller--Xu's combination of those identities, cohomology and +base change for the flat fibration, and equivariant Lefschetz vanishing +\cite[Theorem~3.1]{MullerXu2026}. Here the finite relation associated with +the cover records pairs of points of \(A\) together with an isomorphism +between their stack images. A component containing a nonidentity +automorphism over a pair \((a,a)\) has a proper difference image in \(A\). +Choose such a component for which that image has maximal dimension. +Polarization and +ramification produce another such component with a larger difference +image, giving a contradiction. Thus \(B\) is smooth, and Hwang's theorem +completes the proof. + +Figure~\ref{fig:proof} displays the three exclusions in the first stage +and the cover available in every allowed rank. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[ + box/.style={draw=MidnightBlue!70,rounded corners=2pt, + fill=MidnightBlue!3,align=center,inner sep=6pt,font=\small}, + line/.style={semithick,draw=MidnightBlue!80}, + arr/.style={line,-{Stealth[length=2mm]}}] +\node[box,text width=11.6cm] (zero) at (0,0) + {A nonquotient germ gives a closed stratum \(S_0\) with + \(\chi_{\mathrm{top}}(S_0)=0\)\\ + (Sections~\ref{sec:local-tensor}--\ref{sec:euler})}; +\node[box,text width=3.65cm,minimum height=2cm] (large) at (-4.5,-2.3) + {\(b_2\geq5\)\\Fixed-base deformation\\and period variation\\ + (Section~\ref{sec:deformation})}; +\node[box,text width=3.65cm,minimum height=2cm] (iso) at (0,-2.3) + {\(b_2=4\), isotrivial\\Ramification and an\\elliptic connection\\ + (Section~\ref{sec:isotrivial})}; +\node[box,text width=3.65cm,minimum height=2cm] (vary) at (4.5,-2.3) + {\(b_2=4\), varying\\Fixed-base deformation\\with a second fibration\\and symmetric cubic\\ + (Sections~\ref{sec:second-system}--\ref{sec:variation})}; +\node[box,text width=5.7cm,minimum height=1.45cm] (quot) at (-3.4,-5) + {Each case excludes the stratum:\\ + \(B\) has only quotient singularities}; +\node[box,text width=5.7cm,minimum height=1.45cm] (cover) at (3.4,-5) + {Second-system construction:\\a finite cover of \(B\) by an abelian variety\\ + (Section~\ref{sec:second-system}, all \(b_2\geq4\))}; +\node[box,text width=11.6cm] (smooth) at (0,-7) + {Eliminate quotient stabilizers (Section~\ref{sec:quotient})\\ + \(B\) is smooth; Hwang's theorem gives \(B\simeq\PP^n\)}; +\draw[arr] (zero.south west)--(large.north); +\draw[arr] (zero.south)--(iso.north); +\draw[arr] (zero.south east)--(vary.north); +\draw[line] (large.south)--(-4.5,-3.75)--(0,-3.75); +\draw[line] (iso.south)--(0,-3.75); +\draw[line] (vary.south)--(4.5,-3.75)--(0,-3.75); +\draw[arr] (0,-3.75)--(quot.north); +\draw[arr] (quot.south)--(smooth.north west); +\draw[arr] (cover.south)--(smooth.north east); +\end{tikzpicture} +\caption{The first stage rules out a nonquotient germ in three cases. +The second uses both quotient singularities and a finite cover by an +abelian variety.} +\label{fig:proof} +\end{figure} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/02-setup.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/02-setup.tex new file mode 100644 index 0000000000000000000000000000000000000000..bbcaee32d01a7415c610df3810b2f9a0ea08681b --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/02-setup.tex @@ -0,0 +1,171 @@ +\section{Notation and global inputs}\label{sec:setup} + +All spaces are over $\C$ unless a reduction to positive characteristic +is specified. An irreducible holomorphic symplectic manifold is +abbreviated IHS. Write $q=q_X$ for the Beauville--Bogomolov--Fujiki +quadratic form, and $(\,\cdot\,,\,\cdot\,)$ for its associated +bilinear form. We also write $q(x,y)=(x,y)$, so that $q(x)=q(x,x)$. +We use the integral normalization; its real signature is +$(3,b_2(X)-3)$. + +Choose a very ample line bundle $H$ on $B$, and set +\[ + e=c_1(f^*H). +\] +For a coherent sheaf $M$ on a normal space, $M^{**}$ is its +reflexive hull. In particular, +$\Omega_B^{[j]}=(\bigwedge^j\Omega_B^1)^{**}$. +Reflexive tensor operations are always explicitly interpreted by +taking this hull. An associated map of a semiample line bundle is +replaced by its Stein factorization. + +\begin{proposition}[Background for the given fibration] +\label{prop:background} +In the setting of Theorem~\ref{thm:main}, the following hold. +\begin{enumerate} +\item $X$ is projective, $e\neq0$ is nef and $q(e)=0$. +The base is $\Q$-factorial and klt, has Picard number one, +and $-K_B$ is ample as a $\Q$-Cartier divisor. +A smooth fiber is an abelian variety after choosing an origin. +\item There are reflexive identifications +\[ + R^j f_*\cO_X\simeq\Omega_B^{[j]}. +\] +For every integral Weil divisor $A$ ample as a $\Q$-divisor, +\[ + H^i\!\left(B, + (\Omega_B^{[j]}\otimes\cO_B(A))^{**}\right)=0 + \quad(i>0,\ j>0). +\] +The reflexivity assertion also applies locally to the projective +equidimensional fibrations with connected fibers, smooth source \(Z\) +satisfying \(\omega_Z\simeq\cO_Z\), and normal Cohen--Macaulay +\(\Q\)-Gorenstein target that occur below. +\item The rational intersection cohomology of $B$ has the Betti +numbers of $\PP^n$, and the restriction of $H^2(X,\Q)$ to a +smooth fiber has one-dimensional image. +\item If $B$ is smooth, then $B\simeq\PP^n$. +\end{enumerate} +\end{proposition} + +\begin{proof} +Relative ampleness and projectivity of $B$ make $X$ Moishezon; +as $X$ is Kähler, it is projective. Equivalently, a sufficiently +positive twist of a relative polarization by a base polarization +is ample. The structure statements in (1) are the fibration +theorems of Matsushita +\cite{Matsushita1999,Matsushita2001}. +The pullback of $H$ is nef and nonzero. Its $(2n)$-th power is +zero because $\dim B=n$, so Fujiki's relation gives $q(e)=0$. + +For (2), Ou's Proposition~3.6 gives the reflexive cotangent +identifications for the given projective equidimensional Lagrangian +fibration, extending the symplectic differential identification of +Matsushita \cite{Matsushita2005,Ou2019}. Ou's Theorem~1.3 is the separate +local reflexivity result, and Theorem~1.4 gives the vanishing +\cite{Ou2019}. +The target hypotheses hold because klt spaces are +Cohen--Macaulay \cite[Theorem~11.1(2), Corollaries~11.9 and~11.14]{Kollar1997} +and $K_B$ is $\Q$-Cartier. +The cited reflexivity theorem has arbitrary-dimensional scope. +The higher-cohomology and fiber-restriction statements in (3) +are \cite[Theorem~0.4]{ShenYin2022}. +Statement (4) is Hwang's theorem \cite[Theorem~1.2]{Hwang2008}. +\end{proof} + +Fix once and for all degreewise global isomorphisms +\[ + \phi_j:R^j f_*\cO_X\xrightarrow{\ \sim\ }\Omega_B^{[j]} + \qquad(0\leq j\leq n), +\] +with \(\phi_0\) the unit identification \(f_*\cO_X=\cO_B\). +The later normalized-chart constructions use the restrictions of these +same maps to \(B_{\mathrm{reg}}\); Proposition~\ref{str:direct-images} +constructs their compatible reflexive extensions on the charts. + +We use intersection cohomology in Proposition~\ref{prop:background} +until rational smoothness has been proved in +Proposition~\ref{loc:rational-smoothness}, which then identifies it with +ordinary rational cohomology. + +\begin{corollary}\label{cor:tangent-rigidity} +If $n>1$, then $H^1(B,T_B)=0$. +\end{corollary} +\begin{proof} +On the smooth locus, contraction identifies +$T_B$ with $\Omega_B^{n-1}\otimes\cO_B(-K_B)$. +Both sides extend reflexively, so +\[ + T_B\simeq + (\Omega_B^{[n-1]}\otimes\cO_B(-K_B))^{**}. +\] +Apply Proposition~\ref{prop:background}(2), since $n-1>0$. +\end{proof} + +The following elementary observation also appears in +\cite[Lemma~3.1]{GrebLehn2014}. + +\begin{lemma}\label{lem:betti-bound} +One has $b_2(X)\geq4$. +\end{lemma} +\begin{proof} +Let $h$ be an ample class. Then $(h,e)>0$ and $q(e)=0$, so +$\langle h,e\rangle_{\R}$ is a hyperbolic plane. +It is perpendicular to the positive real plane generated by +$\operatorname{Re}\sigma$ and $\operatorname{Im}\sigma$. +These four independent real directions give the assertion. +\end{proof} + +For a compact Kähler manifold $Y$, write $\mathcal K_Y$ for its +Kähler cone in $H^{1,1}(Y,\R)$. + +\begin{theorem}[Strong hyperkähler SYZ, companion input] +\label{thm:strong-syz} +Let $Y$ be a compact IHS Kähler manifold of dimension $2m$. +If a holomorphic line bundle $L$ satisfies +\[ + c_1(L)\neq0,\qquad + c_1(L)\in\overline{\mathcal K_Y},\qquad + q_Y(c_1(L))=0, +\] +then a positive tensor power of $L$ is generated by its global +sections. Its associated map with connected fibers has a normal +projective base of dimension $m$. Every component of every fiber +has dimension $m$ and is Lagrangian. +\end{theorem} + +This is Theorem~1.1 of the companion \cite{StrongSYZ2026}; it has no +restriction on \(b_2(Y)\), \(m\), or deformation type. The hypothesis +\(b_2\geq5\) in that manuscript belongs to its separate finiteness +Corollary~1.2. Section~\ref{sec:second-system} applies the theorem to a +second nonzero integral nef isotropic class on an actual projective +deformation. The class is of type \((1,1)\), so the exponential sequence +supplies its holomorphic line bundle. + +\paragraph{Reduction to smoothness.} +For $n=1$, normality makes $B$ a smooth curve, and +Proposition~\ref{prop:background}(4) proves the result. +Henceforth $n>1$. It suffices to prove that $B$ is smooth. +Proposition~\ref{def:fixed-base} later supplies proper holomorphic +fibrations over this same \(B\) throughout the nearby \(e\)-Hodge locus. +The original fibration and the projective deformation carrying two +Lagrangian systems have the stronger projectivity used in their respective +arguments. + +\paragraph{The Euler obstruction.} +The first stage has a specific numerical target. If \(B\) has a +nonquotient germ, Sections~\ref{sec:local-tensor}--\ref{sec:stratum} +produce a nonempty closed irreducible normal projective stratum +\(S_0\subset B\) of dimension \(d0\). +Its finite stabilizers, including those acting trivially on the stratum, +are retained in the calculations. In characteristic \(p\), their orders +are chosen prime to \(p\). The later geometric arguments will contradict +this Euler equality when \(b_2(X)\geq5\) and in the isotrivial and varying +cases with \(b_2(X)=4\). diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/03-local-tensor.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/03-local-tensor.tex new file mode 100644 index 0000000000000000000000000000000000000000..ae0201e286072ffb8bca7babd564db67459063a7 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/03-local-tensor.tex @@ -0,0 +1,1067 @@ +\section{Finite local covers and cotangent tensors} +\label{loc:section} +\label{sec:local-tensor} + +We first establish three geometric facts about the base germs: finite +smooth domination, rational smoothness, and the existence of maximal +finite quasi-\'etale charts. The normalized source over such a chart +will also provide the direct-image and descent maps used later. +We then study reflexive cotangent tensors on one maximal chart through +a residual trace category. Its tensor group acts on the fiber spaces +of that category. The obstruction to a holomorphic connection +identifies the nontrivial irreducible summands of the cotangent fiber +representation as standard blocks for special linear and symplectic groups, +which enter the Euler calculation in +Section~\ref{sec:euler}. A finite +map between normal spaces is called \emph{quasi-\'etale} if it is +\'etale outside a subset of codimension at least two in the target. +The geometric tensor argument uses complex analytic local rings of the +maximal charts; they are excellent and henselian, with residue field +$\C$. The trace and split-lifting lemmas are stated for general normal +henselian local rings as well, so that they apply after completion. +Reflexive pullback means pullback followed by double dual. + +\begin{lemma}[Finite smooth domination] +\label{loc:smooth-domination} +Every point of $B$ has an algebraic neighborhood, after an \'etale +base change if necessary, admitting a finite surjection from a smooth +variety of dimension $n$. In particular, every analytic germ of $B$ +is finitely dominated by a smooth germ of the same dimension. +Every such germ is analytically $\Q$-factorial. +\end{lemma} + +\begin{proof} +Fix $b\in B$. Choose sufficiently positive very ample divisors on the +smooth projective variety $X$. By Bertini, a general intersection of +$n$ of them is smooth of dimension $n$. They may simultaneously be +chosen so that their intersection with each component of $f^{-1}(b)$ +is zero-dimensional and nonempty: the fiber has finitely many +components, all of dimension $n$, and successive general hyperplanes +cut each of them properly. Denote the resulting smooth intersection +by $Y$. The proper map $Y\to B$ is quasi-finite over $b$, and hence +finite over a neighborhood of $b$. Its image contains that +neighborhood, by properness and the dimension theorem. A local +branch at a point of $Y$ over $b$ gives the asserted finite map of +analytic germs. If an individual algebraic branch is required, split +the finite algebra over the henselization and descend the corresponding +idempotents to an \'etale neighborhood. + +Let $\pi:(Y,y)\to (B,b)$ be a finite dominating smooth germ, of degree +$m$, and let $D$ be a Weil divisor on $(B,b)$. Pullback of Weil +divisors under a finite map is defined by the ramification indices of +the discrete valuations at prime divisors. The divisor $\pi^*D$ is +principal on the smooth local germ, say $\operatorname{div}(g)$. +The valuation formula for the field norm gives +\[ + \operatorname{div}\bigl(\operatorname{Nm}_{\C(Y)/\C(B)}g\bigr) + =\pi_*\pi^*D=mD. +\] +Thus $mD$ is Cartier. The argument is analytic and therefore proves +the stronger local assertion needed later, rather than merely +algebraic $\Q$-factoriality. +\end{proof} + +\begin{proposition}[Rational smoothness] +\label{loc:rational-smoothness} +The space $B$ is an oriented rational homology manifold of real +dimension $2n$. Its rational intersection complex is +$\Q_B[n]$. Consequently its ordinary rational cohomology has the +Betti numbers of $\PP^n$. The local rational homology-manifold +assertion holds for every normal germ finitely dominated by a smooth +germ, in particular for the maximal charts constructed below. +\end{proposition} + +\begin{proof} +We explain the trace at a branch point. For a finite surjection +$\pi:Y\to U$ from a smooth $n$-fold to a normal connected +neighborhood, let $m_y$ be the local generic degree of the branch at +$y$. A section $a$ of $\pi_*\Q_Y$ is locally constant on $Y$. +Set +\[ + \operatorname{Tr}_{\pi}(a)(b) + =\sum_{y\in\pi^{-1}(b)}m_y a(y). +\] +This is locally constant. Indeed, choose disjoint neighborhoods of +the finitely many points above $b$. After shrinking downstairs, the +sum of the local generic degrees in each of these neighborhoods is +constant: it equals the degree over the common unbranched dense open +of the locally irreducible normal base. The value of $a$ is constant +on each chosen neighborhood. This proves the assertion and shows +that the trace is a morphism of sheaves. Its composite with the +constant-section inclusion $\Q_U\to\pi_*\Q_Y$ is multiplication by +$\deg\pi$. Therefore $\Q_U$ is a direct summand of +$\pi_*\Q_Y$. + +Proper base change and the finiteness of the fibers give +$R^j\pi_*\Q_Y=0$ for $j>0$. Applying the point costalk at $b$ to +the preceding direct summand, and using proper base change for +costalks, exhibits +\[ + H^k_{\{b\}}(U,\Q) + \quad\text{as a direct summand of}\quad + \bigoplus_{y\in\pi^{-1}(b)}H^k_{\{y\}}(Y,\Q). +\] +The latter group is zero unless $k=2n$. The top local group on the +normal, hence locally irreducible, complex analytic germ $U$ is +one-dimensional: the complex orientation gives its fundamental +class, and the top fundamental cycles of a pure analytic germ are +indexed by its local irreducible components +\cite[Section~1.4, Theorem~3.2, Section~3.3 and Lemma~4.3]{BorelHaefliger1961}. +Here fundamental classes are taken in Borel--Moore homology of +representatives and passed to local groups. Equivalently, this +follows from an analytic triangulation and the oriented top strata. +Thus the costalk of $\Q_U$ is exactly $\Q[-2n]$ at every point, +with the compatible complex orientation. This is the rational +homology-manifold assertion and identifies the intersection complex +with the shifted constant sheaf +\cite[Theorem~1.1(1)$\Leftrightarrow$(4)]{Massey2005}. +The final claim now follows from +the base-cohomology theorem recalled in +Proposition~\ref{prop:background}; the passage from intersection to +ordinary cohomology has been justified before using it. +\end{proof} + +\begin{proposition}[Maximal quasi-\'etale charts and normalized source charts] +\label{loc:maximal-charts} +Every pointed germ $(B,b)$ has a connected finite normal Galois +quasi-\'etale cover $(C,c)$ which dominates every connected finite +quasi-\'etale cover of that germ. The chart $(C,c)$ is klt, is +finitely dominated by a smooth germ, and has no nontrivial connected +finite quasi-\'etale cover. The finite diagrams can be defined over +an algebraic \'etale neighborhood of $b$. + +More generally, let $U\to B$ be an integral separated \'etale algebraic +neighborhood and let $\pi:C\to U$ be an integral finite normal +quasi-\'etale cover. Put $X_U=X\times_B U$. The base change +$X_U\times_U C$ has a unique irreducible component dominating $C$; +let $Z$ be its normalization and $h:Z\to C$ its structure map. +Then $h$ is projective and equidimensional of relative dimension $n$, +has connected fibers, and satisfies $h_*\cO_Z=\cO_C$. The finite map +$p:Z\to X_U$ is \'etale. Consequently $Z$ is smooth, the pullback of +the symplectic form on $X$ is symplectic on $Z$, its canonical bundle is +trivial, and the fibers of $h$ are Lagrangian. If $\pi$ is Galois with +group $\Gamma$, then $p$ is a $\Gamma$-torsor. Ordinary \'etale base +change on $C$ preserves projectivity, equidimensionality, connected +fibers, the equality $h_*\cO_Z=\cO_C$, and the smooth symplectic source. +The torsor in the statement belongs to the original finite chart. +\end{proposition} + +\begin{proof} +Work first over the strict henselization of the algebraic germ and +choose a finite smooth slice as in +Lemma~\ref{loc:smooth-domination}. Choose a connected local factor +of its finite algebra, with fraction field $L$ over the base field +$K$. If $K'/K$ defines a finite quasi-\'etale normal cover, the +normalization of its pullback to this slice is unramified outside a +set of codimension at least two. Finiteness preserves that +codimension bound. The source of the normalized map is normal and +the target is regular, so purity of the branch locus +\cite[Tag 0BMB]{StacksProject} makes the map \'etale everywhere. +A connected finite \'etale cover of a strictly henselian regular +local scheme is trivial. A component of the split pullback +therefore embeds $K'$ into $L$ over $K$. + +Let $M/K$ be a finite Galois closure of $L/K$, let $W$ be the +normalization of the base in $M$, and write $\Gamma$ for its Galois +group. Normalizations are finite by excellence. Let $H\lhd\Gamma$ +be the normal subgroup generated by the inertia groups of all prime +divisors of $W$. The finite quotient $C=W/H$ is unramified over +every prime divisor of the normal base and hence is quasi-\'etale. +Conversely, an intermediate extension which is quasi-\'etale has +trivial divisorial inertia; its defining subgroup contains every +conjugate of each inertia group, and hence contains $H$. Every +quasi-\'etale cover is consequently dominated by $C$. A further +quasi-\'etale cover of $C$ would give one of the base and is therefore +trivial. Applying the same split-pullback argument to $C$ shows +that it too is finitely dominated by the smooth slice. + +All finite algebras, group actions, quotient maps and their +identities descend from the strict henselization to an \'etale +neighborhood. Shrinking removes branch divisors not meeting the +marked fiber. This construction also gives the analytic maximum. +Indeed, the normalized pullback of any finite analytic +quasi-\'etale cover to the smooth germ is \'etale by analytic purity +and splits because a smooth germ is simply connected. One may +also obtain this directly: a small smooth ball minus an analytic +set of complex codimension at least two is simply connected, by +general position for disks against a stratification; its finite +cover splits there, and normalization extends that splitting +uniquely. The analytic cover is thus +an intermediate cover of the analytification of the same finite +Galois closure. Finite normalizations and finite-group quotients +commute here with analytification; the local normal components and +their degrees agree. The divisorial inertia groups through the +marked point are therefore the same in both constructions. Finally +$C$ is klt. Indeed, writing $\pi:C\to B$, we have +$K_C=\pi^*K_B$ in codimension one, and for a +divisorial valuation $E$ over $C$ restricting with ramification index +$r$ to a divisorial valuation $F$ over $B$, the canonical ramification +formula on normalized birational models gives +$A_C(E)=rA_B(F)$. The latter is positive because $B$ is klt. + +We prove the source assertions, including connectedness. The connected +fibers of $f$ and normality of $B$ give $f_*\cO_X=\cO_B$ by Stein +factorization. This equality persists after the flat base change to +$U$. Let $K=\C(U)$, $L=\C(C)$, and let $F$ be the generic fiber of +$X_U\to U$. Generic smoothness makes $F$ smooth. Proper flat field +base change gives +\[ + H^0(F_{\overline K},\cO_{F_{\overline K}})=\overline K. +\] +Thus $F_{\overline K}$ is connected. A connected smooth scheme over +an algebraically closed field is integral, since its irreducible +components are disjoint. In this setting it is an abelian variety +after an origin is chosen, by Proposition~\ref{prop:background}. +In particular $F$ is geometrically integral, so $F_L$ is integral +and normal and has $H^0(F_L,\cO_{F_L})=L$. This proves uniqueness of +the component dominating $C$ and identifies the generic fiber of its +normalization $Z$ with $F_L$. + +The normalization is finite by excellence, so $h$ is projective. +Its Stein algebra $h_*\cO_Z$ is finite over $\cO_C$ and embeds in the +generic algebra $L$: a section on the integral space $Z$ is determined +by its restriction to the generic fiber. Every such section is +integral over $\cO_C$; normality of $C$ therefore gives +$h_*\cO_Z=\cO_C$. Stein factorization now gives connected fibers. +This part of the argument used only that $C\to U$ is finite and +dominant, with $C$ normal and integral. + +Let $D\subset U$ have codimension at least two and contain the branch +locus of $\pi$. Equidimensionality of $f$ gives +\[ + \dim f_U^{-1}(D)\leq\dim D+n\leq2n-2. +\] +The map $p$ is finite over the smooth space $X_U$ and is \'etale away +from $f_U^{-1}(D)$. Purity makes it \'etale everywhere, so $Z$ is +smooth and carries the pulled-back symplectic form. Each component of +a fiber of $h$ has dimension at most $n$, because its finite image lies +in a fiber of $f_U$. The fiber dimension theorem gives the reverse +inequality. Hence $h$ is equidimensional of relative dimension $n$; +the same finite images show that its fiber components are Lagrangian. +The symplectic form trivializes the canonical bundle of $Z$. + +If $\pi$ is Galois, its deck action lifts to $Z$. Over the dense +unbranched locus the map $p$ is a deck torsor. Both sides of the torsor +map $\Gamma\times Z\to Z\times_{X_U}Z$ are finite \'etale over $Z$, +so their isomorphism on that dense locus extends over $Z$. This proves +freeness of the action also over special fibers. Subsequent \'etale +refinements of the object chart use ordinary base change of $h$ and +preserve its projectivity, equidimensionality, connected fibers and +smooth symplectic source. The torsor descent is retained from the +finite chart before that refinement. +\end{proof} + +\subsection{The trace quotient} + +Fix a maximal chart and let $(R,\mathfrak m)$ be its analytic local +ring. Put $V=\Omega_R^{[1]}$. On reflexive modules we use +$M\mathbin{\widehat\otimes}N=(M\otimes_RN)^{**}$ and +$M^\vee=\Hom_R(M,R)$. These are a symmetric rigid tensor product +and dual: on a common open set containing every point of codimension +one they are the ordinary bundle operations, and all identities +extend uniquely by reflexivity. In particular the categorical trace +is the usual generic trace, which lies in $R$ by normality. We use +the full additive, idempotent-complete tensor subcategory $\mathcal C$ +generated by $V$ and $V^\vee$. Every object in this subcategory is a vector +bundle on the smooth locus of the chart. + +There are two uses of endomorphisms below. The ordinary finite +$R$-algebra $A_M=\End_R(M)$ consists of morphisms of the module $M$. +The internal endomorphism object is +\[ + \underline{\End}(M)=M\widehat\otimes M^\vee. +\] +Its underlying reflexive module is canonically $\Hom_R(M,M)$: this is +the usual bundle identification in codimension one, extended +reflexively. We use the underline when this module is a coefficient +object in $\mathcal C$, rather than the algebra of morphisms of $M$. + +\begin{lemma}[The trace ideal is the Jacobson radical] +\label{loc:trace-radical} +Let $(R,\mathfrak m)$ be any noetherian normal henselian local +$\C$-algebra with residue field $\C$. For finite reflexive modules +$M,N$, define +\[ + \mathcal R(M,N)= + \{a\in\Hom_R(M,N):\tr(ba)\in\mathfrak m + \text{ for every }b\in\Hom_R(N,M)\}. +\] +These spaces form a tensor ideal. For every $M$, +$\mathcal R(M,M)$ is the Jacobson radical of the finite +$R$-algebra $\End_R(M)$. The quotient category is semisimple +abelian over $\C$, has finite-dimensional Hom spaces, and its +categorical dimensions are the positive integer ranks of its +nonzero objects. +\end{lemma} + +\begin{proof} +Cyclicity of trace proves stability under composition. Stability +under tensor product follows by taking a partial trace: for a map +$a:M\to N$ and a test map +$b:N\widehat\otimes P\to M\widehat\otimes P$, the trace of +$b(a\widehat\otimes\id_P)$ is the trace of $a$ composed with the +partial trace of $b$. These formulas hold for bundles in codimension +one and therefore over $R$. Duality is compatible with the same +ideal. + +Write $A=A_M=\End_R(M)$ and $J=\rad A$. The algebra $A$ is finite over +the henselian local ring $R$, so $\mathfrak m A\subset J$, +$A/J$ is a finite-dimensional semisimple $\C$-algebra, and +$J^q\subset\mathfrak m A$ for some $q$. We recall why idempotents +lift in this setting. First lift an idempotent through the nilpotent +radical of $A/\mathfrak m A$. For an ensuing lift $a\in A$ of an +idempotent modulo $\mathfrak m A$, the commutative finite algebra +$R[a]$ splits into the factors supported at $0$ and $1$ in its +closed fiber by henselianity. The resulting idempotent lifts the +given one. Induction in complementary corners lifts an entire +orthogonal system. Thus primitive idempotents of $A/J$ lift to +actual direct summands of $M$. + +For $x\in J$, all sufficiently large powers $x^j$ belong to +$\mathfrak m A$, so their traces vanish modulo $\mathfrak m$. +The generic characteristic polynomial of $x$ has coefficients in +$R$: this is true at every height-one localization, where $M$ is +free, and $R$ is the intersection of those valuation rings. If +$\lambda_1,\ldots,\lambda_r$ are the roots of its reduction over +$\C$, Newton identities give +$\sum_i\lambda_i^j=0$ for every sufficiently large $j$. +Grouping the distinct nonzero roots, a Vandermonde matrix applied to +consecutive such powers forces all their multiplicities to be zero. +There are consequently no nonzero roots, and $\tr(x)=0$ modulo +$\mathfrak m$. The same holds for $yx$ for any $y\in A$, because +$J$ is an ideal. Hence $J\subset\mathcal R(M,M)$. + +The residual trace now descends to $A/J$. On each simple block +$\operatorname{Mat}_t(\C)$ it is a scalar multiple of the ordinary +matrix trace, by cyclicity. Evaluate it on a primitive diagonal +idempotent. Its lift projects onto a nonzero reflexive direct +summand $M_0$, so the scalar is $\rk(M_0)>0$. The trace pairing is +therefore nondegenerate on every block. This proves the reverse +inclusion and the equality with $J$. + +Apply the argument to finite direct sums of objects. Their +endomorphism algebras in the quotient are semisimple matrix algebras, +and their primitive projectors lift. In particular every object is +a finite sum of simple objects, distinct simple objects have no +morphisms, and the endomorphisms of each simple object are $\C$. +This also proves that the quotient is abelian and that all its exact +sequences split. The ideal contains $\mathfrak m\Hom_R(M,N)$, +so its Hom spaces are finite-dimensional. Finally the trace of the +identity is $\rk M$, as asserted. +\end{proof} + +\begin{lemma}[Lifting split maps] +\label{loc:split-lift} +Over any ring in Lemma~\ref{loc:trace-radical}, +if a morphism of reflexive modules becomes a split injection in the +trace quotient, it is already a split injection of reflexive modules. +An isomorphism in the quotient lifts to an isomorphism. In +particular, an object is a sum of copies of the tensor unit in the +quotient if and only if its underlying module is free. +\end{lemma} + +\begin{proof} +Let $a:M\to N$ and lift a left inverse in the quotient to +$b:N\to M$. Then $ba=\id_M+r$ with $r\in\rad\End_R(M)$, +so $ba$ is invertible. Replacing $b$ by $(ba)^{-1}b$ gives an +actual left inverse. If the quotient map is an isomorphism, apply +the same argument to both composites of lifts of it and its inverse; +the lift is an isomorphism. Apply this to an isomorphism with $R^r$ +for the final statement. +\end{proof} + +Return now to the analytic maximal chart and the category $\mathcal C$ +fixed above. + +\begin{proposition}[The residual tensor group] +\label{loc:tannakian} +The residual category $\overline{\mathcal C}=\mathcal C/\mathcal R$ +admits a fiber functor $\omega$ to complex vector spaces. For a choice +of $\omega$, tensor reconstruction identifies it with +$\operatorname{Rep}_{\C}(G)$, where +$G=\Aut^\otimes(\omega)$ is a reductive algebraic subgroup of +$\operatorname{GL}(V_{\mathrm{fib}})$ and +$V_{\mathrm{fib}}=\omega(\overline V)$. This representation is +faithful and has dimension $n$. +\end{proposition} + +\begin{proof} +Lemma~\ref{loc:trace-radical} gives an abelian rigid symmetric +$\C$-linear category with $\End({\bf1})=\C$. Tensor product is +exact because every exact sequence splits. For an object of rank +$r$, its $(r+1)$st exterior power is zero: it is zero on the common +bundle locus and hence its reflexive extension is zero. Deligne's +criterion \cite[Theorem 7.1]{Deligne1990} therefore applies. The +category is tensor-generated by one object; Deligne's +Corollary~6.20 supplies a fiber functor over a finite extension of +$\C$, hence over $\C$ itself. Tensor reconstruction gives the +stated group. A tensor generator makes it an algebraic subgroup of +its general linear group, and semisimplicity of the representation +category makes it reductive in characteristic zero. The fiber +dimension equals the categorical dimension, namely $\rk V=n$. +\end{proof} + +We call $G$ the residual tensor group of the chosen fiber functor. +Its representations are the fiber spaces of $\overline{\mathcal C}$. +Finite deck groups, denoted by $\Gamma$, act on the geometric charts. +For example, if $M_{\mathrm{fib}}=\omega(\overline M)$, then +\[ + \omega\bigl(\overline{\underline{\End}(M)}\bigr) + =\End_{\C}(M_{\mathrm{fib}}),\qquad + \End_{\overline{\mathcal C}}(\overline M) + =\End_G(M_{\mathrm{fib}}). +\] +The first space is the full internal coefficient representation, with +the conjugation action; the second is its space of invariant vectors. + +\subsection{Connections before and after the trace quotient} + +For an object $M$ let $M_U$ be its vector bundle on the smooth locus +$U$ of a sufficiently small representative of the chart. Its +ordinary Atiyah class \cite{Atiyah1957} is +\[ + a(M_U)\in H^1\bigl(U,\Omega_U^1\otimes\End(M_U)\bigr). +\] +It vanishes precisely when $M_U$ has a holomorphic connection; +integrability is not part of this assertion. We take the direct +limit over shrinking representatives when discussing this class on +a germ. All cohomology in the following lemmas is this ordinary +sheaf cohomology. + +\begin{lemma}[A connection gives freeness on a maximal chart] +\label{loc:connection-free} +If $a(M_U)=0$, then $M$ is a free $R$-module. +\end{lemma} + +\begin{proof} +Pull a connection on $M_U$ back to a finite smooth dominating germ +$\pi:(Y,y)\to(C,c)$. The complement of $\pi^{-1}(U)$ has +codimension at least two. On $Y$ let +$F=(\pi^*M)^{**}$. In local smooth coordinates each coefficient +of the connection extends across that complement by the Hartogs +property of $F$. Its target $F\otimes\Omega_Y^1$ is reflexive +because $\Omega_Y^1$ is locally free. Thus the connection extends +to $F$ and still satisfies Leibniz's rule. + +Here is the elementary freeness argument, which does not require a +flat connection. For a coordinate derivation $\partial$, choose a +presentation of $F$ and lift the operator $\nabla_{\partial}$ to +the free module of generators. If $A$ is the corresponding matrix +on generators and $P$ a matrix of relations, stability of the +relations gives a matrix $Q$ with +\[ + \partial P+AP=PQ. +\] +The derivative of every minor of $P$ is therefore a linear +combination of minors of the same size. Each Fitting ideal of $F$ +is stable under every coordinate derivation. A nonzero ideal in a +smooth characteristic-zero local ring with this property is the +unit ideal: repeatedly differentiating a nonzero element of minimal +order produces an element with nonzero constant term. If $r$ is +the generic rank of $F$, its $r$th Fitting ideal is nonzero and hence +is the unit ideal. Thus $F$ is generated by $r$ elements. The +resulting surjection from a free module of rank $r$ has a torsion +kernel, which is zero because it is a submodule of a free module +over a domain. This proves that $F$ is free. + +Take a normal Galois closure $W\to C$ of $Y\to C$, with group +$\Gamma$. Its normal domain is local over the henselian germ, +so the whole group fixes the unique marked point. The +reflexive pullback $P$ of $M$ to $W$ is free: it agrees in +codimension one with the pullback of the free module $F$, and both +reflexive extensions agree. It has its natural semilinear +$\Gamma$-action. At a prime divisor, divisorial inertia acts +trivially on its restriction to that divisor, because there $P$ +is the ordinary pullback of a bundle from $C$. The same inertia +therefore acts trivially on the fiber of $P$ at the marked point. +To see the last specialization explicitly, for an inertia element +of order $m$, restrict $P$ to its fixed locus and apply the +idempotents +\[ + \frac1m\sum_{j=0}^{m-1}\zeta^{-j}g^j. +\] +Their ranks are locally constant. On the divisor all nonidentity +eigenvalues have rank zero, hence they have rank zero at its +specialization to the marked point as well. + +Since $C$ has no further quasi-\'etale cover, these inertia groups +normally generate $\Gamma$, by the construction in +Proposition~\ref{loc:maximal-charts}. The action on the marked +fiber of $P$ is consequently trivial. Lift a basis of that fiber +to sections of $P$ and average over $\Gamma$. The averaged +sections still lift the basis and hence, by Nakayama, form a basis +of the free module $P$. This basis is $\Gamma$-invariant. +The invariant basis identifies $P^\Gamma$ with $R^r$, because the +invariants of the normal cover ring are $R$. Thus $P^\Gamma$ is +reflexive. The natural map $M\to P^\Gamma$ is an isomorphism in +codimension one by ordinary finite descent for the pullback of a +bundle. Both modules are reflexive, so this isomorphism extends over +the normal germ. Hence $M=P^\Gamma$ is free, as claimed. +\end{proof} + +\begin{lemma}[Tensor tests for the ordinary Atiyah class] +\label{loc:atiyah-tests} +Let $M$ be an object, and let $t_j$ be actual sections of finitely +many functorial mixed tensor powers $T_j(M)$ of $M$ and $M^\vee$ on +the chart. Direct sums and the usual symmetric or alternating +projectors may be used; each construction carries the natural action +of every endomorphism of $M$. Put $Q=\bigoplus_jT_j(M)$ and define +\[ + \Phi_M:\underline{\End}(M)\longrightarrow Q, + \qquad A\longmapsto(A\cdot t_j)_j +\] +by the infinitesimal tensor action. The coefficient map for the +Atiyah class is the morphism in $\mathcal C$ +\begin{equation}\label{loc:psi-coefficient} + \Psi_M=\id_V\widehat\otimes\Phi_M: + V\widehat\otimes\underline{\End}(M) + \longrightarrow V\widehat\otimes Q. +\end{equation} +On the smooth locus its ordinary bundle restriction satisfies +$(\Psi_{M,U})_*a(M_U)=0$ in $H^1(U,(V\widehat\otimes Q)_U)$. +If $\overline{\Psi_M}$ is injective in $\overline{\mathcal C}$, then +$a(M_U)=0$. +\end{lemma} + +\begin{proof} +The natural tensor action defining $\Phi_M$ is a bundle map on $U$; +its reflexive extension is the displayed morphism in $\mathcal C$. +Choose local connections $\nabla_\alpha$ on $M_U$. +Their differences represent $a(M_U)$. On $T_j(M_U)$ the +corresponding differences are the infinitesimal tensor actions of +$\nabla_\alpha-\nabla_\beta$. Evaluating on the global section +$t_j$ gives the coboundary of the local sections +$\nabla_\alpha t_j$. This proves the asserted vanishing after +restricting \eqref{loc:psi-coefficient} to bundles on $U$ and taking +ordinary $H^1$. + +An injection in the semisimple quotient has a left inverse there. +Lemma~\ref{loc:split-lift} lifts it to an actual left inverse +between reflexive coefficient modules. Restriction to $U$ gives +a split map of bundles, which induces an injection on ordinary +$H^1$. Its kernel contains $a(M_U)$, so that class is zero. +The argument uses only that the individual $t_j$ are actual +sections. No algebraic identities among their images in the +quotient need hold for the chosen lifts. +\end{proof} + +\begin{proposition}[Connectedness and semisimplicity] +\label{loc:connected-semisimple} +The reductive group $G$ of Proposition~\ref{loc:tannakian} is +connected and semisimple. +\end{proposition} + +\begin{proof} +For a reductive subgroup of a general linear group, the invariant +mixed tensors have that subgroup as their simultaneous pointwise +stabilizer. Indeed, projectors onto direct summands and all maps +between tensor constructions are themselves such tensors. +Preserving them is precisely the condition to give a tensor +automorphism of the forgetful functor, and tensor reconstruction +identifies those automorphisms with the group. Differentiating, +its Lie algebra is the common kernel of the infinitesimal +tensor-action maps. Since the endomorphism space is +finite-dimensional, finitely many tensors suffice for this last +kernel. + +Suppose a representation $N$ of $G$ has finite image. Realize it +as a direct summand of tensor constructions in +$V_{\mathrm{fib}}$, and lift its projector to obtain an actual +module $M$. Choose finitely many invariant mixed tensors of $N$ +whose infinitesimal stabilizer is the zero Lie algebra. By the +definition of the quotient Hom spaces these tensors lift to actual +maps from $R$, hence to actual sections on the chart. The map +$\omega(\overline{\Phi_M})$ of Lemma~\ref{loc:atiyah-tests} has as +its kernel this infinitesimal stabilizer inside the full space +$\End_{\C}(N)$, and is therefore injective. Thus +$\overline{\Phi_M}$ is injective. Tensoring it with $\overline V$ +preserves injectivity, so $\overline{\Psi_M}$ is injective and that +lemma and Lemma~\ref{loc:connection-free} make $M$ free. +Therefore $N$ is trivial. The regular representation of the +finite component group $G/G^\circ$ is one such representation. +It is trivial only if that component group is trivial. Hence $G$ +is connected. + +A character of $G$ corresponds to a rank-one reflexive module on +the chart. By analytic $\Q$-factoriality some positive reflexive +tensor power of that module is free. Thus the character has +finite order. A connected algebraic group has no nontrivial +finite-image character. The character group of $G$ is therefore +zero. For a connected reductive group its quotient by the derived +group is a torus, whose characters detect it; that torus must be +trivial. Thus $G$ is semisimple. +\end{proof} + +\subsection{The prolongation forced by a nontrivial cotangent block} + +For a vector space $T$ and a Lie subalgebra +$\mathfrak h\subset\End(T)$, set +\[ + \mathfrak h^{(1)}= + \{A\in\Sym^2T^*\otimes T: + A(x,{-})\in\mathfrak h\text{ for every }x\in T\}. +\] +For a reflexive module $M$, let $\Sym^{[2]}M$ denote the image of the symmetric idempotent +$(1+\tau)/2$ on $M\widehat\otimes M$. It is a reflexive direct +summand and restricts to the usual symmetric square on the bundle +locus. For a block $V_i$ below with fiber +$L_i=\omega(\overline V_i)$ and $T_i=L_i^\vee$, the fiber of its +coefficient module is +\[ + \omega\bigl(\overline{\Sym^{[2]}V_i\widehat\otimes V_i^\vee}\bigr) + =\Sym^2L_i\otimes L_i^\vee + =\Sym^2T_i^*\otimes T_i. +\] +The intersection defining a prolongation will be taken in this fiber +space. +We use Lie's triangularization theorem and complete reducibility +for finite-dimensional complex representations, in the forms +\cite[Theorem 2.2(2), Corollary 2.2(2), Theorem 2.9]{Cap2016}. +The prolongation spaces belong to the Singer--Sternberg framework +\cite{SingerSternberg1965}. The infinite-type rank-one criterion is due +to Guillemin--Quillen--Sternberg \cite{GuilleminQuillenSternberg1967}; +see also the proof and attribution in +\cite[Section~2.2 and Theorem~2.3]{OttazziWarhurst2011}. +We give the prolongation argument in full, including the separate +semisimple-irreducible step from nonzero first prolongation to infinite +type. + +\begin{lemma}[One-block connection obstruction] +\label{loc:block-obstruction} +Lift a decomposition of $V_{\mathrm{fib}}$ into irreducible +$G$-modules to a decomposition $V=\bigoplus_iV_i$ of reflexive +modules. Let +\[ + \mathfrak h_i= + \mathfrak g\cap\End(V_{i,\mathrm{fib}}) + \subset\End(V_{\mathrm{fib}}), + \qquad\mathfrak g=\operatorname{Lie}(G), +\] +where the endomorphisms in this intersection are zero on every +other block. Regard $\mathfrak h_i$ on +$T_i=V_{i,\mathrm{fib}}^\vee$ by the dual action. If +$\mathfrak h_i^{(1)}=0$, then $V_i$ is free. Consequently every +nontrivial irreducible block has +$\mathfrak h_i^{(1)}\ne0$. +\end{lemma} + +\begin{proof} +Local coordinate connections on $\Omega_U^1$ are torsion-free. +Their differences show that $a(V_U)$ belongs to the direct summand +\[ + H^1\bigl(U,\Sym^2V_U\otimes V_U^\vee\bigr) + \subset H^1\bigl(U,V_U\otimes\End(V_U)\bigr). +\] +On the other hand, choose local connections preserving the actual +splitting $V_U=\bigoplus_i(V_i)_U$. The off-diagonal +endomorphism components of the same cohomology class are zero. +These two conditions imply +\begin{equation} + a(V_U)=\sum_i a_i,\qquad + a_i\in H^1\bigl(U,\Sym^2(V_i)_U\otimes(V_i)_U^\vee\bigr). + \label{loc:diagonal-atiyah} +\end{equation} +Here is the index check. Block preservation allows only +coefficients $V_a\otimes V_b\otimes V_b^\vee$. Symmetry +interchanges the first two covariant slots. If $a\ne b$, its +image is an off-diagonal coefficient +$V_b\otimes V_a\otimes V_b^\vee$, whose class is zero. +Because these are actual direct sums of bundles, the original +mixed component is also zero. This leaves precisely the terms in +\eqref{loc:diagonal-atiyah}. In particular $a_i$ is the class +$a((V_i)_U)$ under the indicated inclusion of its coefficients. + +Choose lifts $t_j$ of finitely many $G$-invariant tensors in natural +mixed powers of $V_{\mathrm{fib}}$ cutting out $\mathfrak g$ +infinitesimally. Use them to define $Q$ and $\Phi_V$ as in +Lemma~\ref{loc:atiyah-tests}. For the $i$th block set +\[ + S_i=\Sym^{[2]}V_i\widehat\otimes V_i^\vee. +\] +The symmetric inclusion in the first two slots, followed by the +diagonal inclusion of the $i$th endomorphism block, gives +\[ + \jmath_i:S_i\longrightarrow + V_i\widehat\otimes\underline{\End}(V). +\] +The required one-block coefficient map is +\begin{equation}\label{loc:kappa-coefficient} + \kappa_i=(\id_{V_i}\widehat\otimes\Phi_V)\circ\jmath_i: + S_i\longrightarrow V_i\widehat\otimes Q. +\end{equation} +On $U$ the source of this map is +$\Sym^2(V_i)_U\otimes(V_i)_U^\vee$. Projecting the first slot of +$(\Psi_{V,U})_*a(V_U)=0$ onto $(V_i)_U$ and using +\eqref{loc:diagonal-atiyah} gives +$(\kappa_{i,U})_*a_i=0$ in ordinary $H^1$. + +Write $L=V_{\mathrm{fib}}$ and $L_i=V_{i,\mathrm{fib}}$. +The kernel of $\omega(\overline{\Phi_V})$ is $\mathfrak g$ inside +$\End_{\C}(L)$. Hence the fiber kernel of +\eqref{loc:kappa-coefficient} is the explicit intersection +\begin{equation}\label{loc:kappa-kernel} + \ker\omega(\overline{\kappa_i})= + \bigl(\Sym^2L_i\otimes L_i^\vee\bigr) + \cap\bigl(L_i\otimes\mathfrak g\bigr) + \quad\text{inside }L_i\otimes\End_{\C}(L). +\end{equation} +Partial evaluation of the first covariant slot by $x\in T_i=L_i^\vee$ +is supported on the $i$th endomorphism block. It belongs to +$\mathfrak g$ precisely when it belongs to $\mathfrak h_i$. +The symmetry of the first two slots is the defining symmetry in +$\mathfrak h_i^{(1)}$ on $T_i$. Passing from the action on $L_i$ +to the dual action on $T_i$ transposes and changes sign, which leaves +this linear kernel unchanged. Thus \eqref{loc:kappa-kernel} identifies +with $\mathfrak h_i^{(1)}$. + +If the kernel is zero, faithfulness of $\omega$ makes +$\overline{\kappa_i}$ injective, hence split injective in the semisimple +quotient. Lemma~\ref{loc:split-lift} lifts a left inverse for +$\kappa_i$ between the reflexive coefficient modules. Restricting +that left inverse to $U$ makes the induced ordinary $H^1$ map injective. +Thus $a_i=0$ in ordinary $H^1$, and +Lemma~\ref{loc:connection-free} makes $V_i$ free. A free +irreducible object is the tensor unit. This proves the assertion. +\end{proof} + +\begin{lemma}[An auxiliary tensor factor kills prolongation] +\label{loc:auxiliary-factor} +Suppose $T=U\otimes W$, $\dim W>1$, and +$\mathfrak h\subset\End(U)\otimes\id_W$. Then +$\mathfrak h^{(1)}=0$. +\end{lemma} + +\begin{proof} +Write a prolongation tensor as $A(x,y)=A_x(y)$ with +$A_x\in\mathfrak h$. Fix $x=u\otimes w_1$ with $w_1\ne0$ +and choose $w_2$ independent of $w_1$. For every $v\in U$, +symmetry gives +\[ + (A_xv)\otimes w_2 + =A(u\otimes w_1,v\otimes w_2) + =(A_{v\otimes w_2}u)\otimes w_1. +\] +The two indicated tensor subspaces have zero intersection, so +$A_xv=0$ for every $v$. Thus $A_x=0$ for every pure tensor $x$, +and hence for every $x$. No basis of $W$ invariant under any +larger Lie algebra has been assumed. +\end{proof} + +\begin{lemma}[Nonzero prolongation produces a rank-one element] +\label{loc:rank-one-symbol} +Let $\mathfrak h\subset\End(T)$ be semisimple, faithful and +irreducible. If $\mathfrak h^{(1)}\ne0$, then +$\mathfrak h$ contains an endomorphism of rank one. +\end{lemma} + +\begin{proof} +Use homogeneous polynomial vector fields on the affine space $T$. +Let $\mathfrak p_{-1}=T$ be the translations, +$\mathfrak p_0=\mathfrak h$, and for $k\ge1$ set +\[ + \mathfrak p_k= + \{A\in\Sym^{k+1}T^*\otimes T: + A(x_1,\ldots,x_k,{-})\in\mathfrak h + \text{ for every }x_1,\ldots,x_k\in T\}. +\] +These spaces form a graded Lie algebra +$\mathfrak p=\bigoplus_{k\ge-1}\mathfrak p_k$ under the +vector-field bracket. Indeed, differentiating a bracket gives the +sum of brackets with derivatives. Induction on the total degree, +starting with closure of $\mathfrak h$ under brackets, proves +$[\mathfrak p_i,\mathfrak p_j]\subset\mathfrak p_{i+j}$. +The representation also gives +$[\mathfrak h,\mathfrak p_k]\subset\mathfrak p_k$ directly. + +Suppose first that $\mathfrak p$ is finite-dimensional. Its +solvable radical is graded, because it is preserved by the scaling +automorphisms of this grading. From any nonzero homogeneous +polynomial vector field in that radical, successive brackets with +translations produce a nonzero constant field. Irreducibility +under $\mathfrak h$ then puts all of $T$ in the radical. Since +$\mathfrak p_1\ne0$, the bracket +$[T,\mathfrak p_1]$ is nonzero. It lies in the intersection of +the radical with $\mathfrak h$, a solvable ideal of a semisimple +algebra, which is impossible. Thus the radical is zero. + +The grading derivation $D$, acting as $k\id$ on +$\mathfrak p_k$, must now be inner. Recall a short proof of this +fact. For any derivation $D$ of a semisimple algebra $\mathfrak a$, +nondegeneracy of the Killing form gives $z\in\mathfrak a$ such that +$E=D-\operatorname{ad}(z)$ satisfies +$\tr(E\operatorname{ad}(x))=0$ for all $x$. Then +\[ + B(Ex,y)=\tr([E,\operatorname{ad}(x)]\operatorname{ad}(y)) + =\tr(E\operatorname{ad}([x,y]))=0, +\] +so $E=0$. Apply this to $\mathfrak a=\mathfrak p$ and write +$D=\operatorname{ad}(z)$. Comparing homogeneous degrees and +using the zero center shows $z\in\mathfrak p_0=\mathfrak h$. +But $D|_T=-\id_T$. This would give a nonzero scalar in the +semisimple subalgebra $\mathfrak h\subset\mathfrak{sl}(T)$, +again impossible. Hence $\mathfrak p$ is infinite-dimensional. + +We make the symbol argument for this last alternative explicit. +Let $S=\Sym T$, and let +$\mathfrak h^\perp\subset T\otimes T^*$ be the annihilator of +$\mathfrak h\subset T^*\otimes T$. Form the finitely generated +graded $S$-module +\[ + \mathcal M=(S\otimes T^*)/ + S\mathfrak h^\perp, +\] +where the relations have degree one. Perfect pairing with +homogeneous polynomial vector fields identifies +$\mathcal M_d^*=\mathfrak p_{d-1}$ for $d\ge0$: +annihilating a relation multiplied by $d-1$ linear factors is +exactly the displayed partial-evaluation condition. Since +$\mathfrak p$ is infinite-dimensional, $\mathcal M$ is not a +finite-length module supported at the origin. Its support thus +contains a nonzero point $\xi\in T^*=\Spec(S)(\C)$. + +The fiber at $\xi$ is +\[ + T^*/\{(\xi\otimes\id)(r):r\in\mathfrak h^\perp\}, +\] +and is nonzero. Choose a nonzero vector $u\in T$ annihilating +its relation space. Then +$\langle r,u\otimes\xi\rangle=0$ for every +$r\in\mathfrak h^\perp$. Hence $u\otimes\xi\in\mathfrak h$. +Since both $u$ and $\xi$ are nonzero, this is the required +rank-one endomorphism. +\end{proof} + +\begin{lemma}[The rank-one root calculation] +\label{loc:rank-one-classification} +A faithful irreducible representation $T$ of a complex semisimple +Lie algebra containing a rank-one endomorphism is the standard +representation, or its dual, of $\mathfrak{sl}(T)$, or the +standard representation of $\mathfrak{sp}(T)$. In particular the +Lie algebra is simple. +\end{lemma} + +\begin{proof} +We give the weight argument, including the exceptional short-root +possibilities. Consider in $\PP(\mathfrak h)$ the closure of +the orbit of a rank-one operator. It consists of rank-one +operators, since the rank-at-most-one locus is closed and the zero +operator does not occur in projective space. A Borel subgroup has +a fixed point on this closure. In this particular linear setting +one can see the fixed point directly: triangularize the solvable +Borel action, choose the smallest term of an invariant complete +flag whose projectivization meets the closure, and intersect there. +This projective intersection avoids the preceding hyperplane and +is therefore also affine. It is finite; a connected group fixes +each of its points. In the adjoint representation a Borel-fixed +line is the highest-root line of one simple ideal. The highest +root is long. We have therefore found a highest-long-root +operator of rank one in one simple ideal. + +An irreducible representation of a direct sum of simple Lie +algebras is an external tensor product of their irreducible +representations. The rank of this root operator on $T$ is its +rank on the indicated factor multiplied by the dimensions of all +other factors. Rank one forces those other factors to have +dimension one. They act trivially, and faithfulness eliminates +them. Thus $\mathfrak h$ is simple. + +All long roots are Weyl-conjugate. For every long root $\alpha$, +the raising operator $e_\alpha$ has rank one. Restriction to the +root $\mathfrak{sl}_2$ is a direct sum of irreducibles; a summand +of highest weight $m$ contributes rank $m$ to $e_\alpha$. +There is consequently exactly one two-dimensional standard +summand and otherwise only trivial summands. The Cartan +subalgebra commuting with this root $\mathfrak{sl}_2$ acts on +the unique standard summand by a single character. Thus the root +reflection $s_\alpha$ interchanges exactly two full Cartan +weights, each of multiplicity one and differing by $\alpha$, +and fixes every other full weight. This assertion does not yet +exclude a zero weight of higher multiplicity. + +Consider one irreducible component of the subsystem of long roots. +Its simple reflections act by distinct transpositions on the set +of weights. Adjacent simple roots have a product of reflections +of order three, so their transpositions share exactly one vertex. +Orthogonal simple roots give commuting transpositions. Two +different simple roots cannot give the same transposition, since +the difference of the transposed weights would be both roots up +to sign. Connectedness of the diagram therefore gives a single +connected moving set of vertices. The transpositions generate +the full symmetric group on this set. + +Map the augmentation representation on these vertices to the +weight space by sending a vertex difference to the corresponding +weight difference. The map is nonzero and equivariant. The +augmentation representation of a symmetric group is irreducible, +so the map is injective. Its image is exactly the span of the +root component: consecutive vertex differences are its simple +roots, and all differences lie in that span. Conjugating a +simple-root transposition shows that every pair difference is a +root. Conversely every root in this component arises this way, +by conjugacy of its root reflections. The component is +therefore of type $A$, with the moving weights the vertices of +its standard simplex, up to their common centroid. + +Distinct long-root components have disjoint moving sets. +Their transpositions commute, so an overlap could only be an +identical pair; the difference of that pair would then belong to +two orthogonal root spans, which is impossible. The long roots +span the entire Cartan dual. Hence the centroid of a moving +simplex, being fixed by its component and by every other +component, is zero. A weight moved by no long-root component is +zero. + +The ordinary classification of reduced irreducible root systems +\cite[Theorem 3.10]{Cap2016} now leaves very few cases. +For completeness, their long-root +subsystems are +\[ +\begin{array}{c|c} + \text{root system}&\text{long-root subsystem}\\ \hline + A_\ell&A_\ell\\ + B_\ell\ (\ell\ge3)&D_\ell\\ + C_\ell&A_1^{\ell}\\ + D_\ell\ (\ell\ge4)&D_\ell\\ + E_6,E_7,E_8&E_6,E_7,E_8\\ + F_4&D_4\\ + G_2&A_2 +\end{array} +\] +Here $B_2=C_2$ and the rank-one cases are $A_1$. +For the non-simply-laced rows, this table follows directly from +the coordinate roots: the long roots of $B_\ell$ and $F_4$ are +$\pm\epsilon_i\pm\epsilon_j$, those of $C_\ell$ are +$\pm2\epsilon_i$, and the long roots of $G_2$ form a regular +hexagon. In the simply-laced cases every root is long. +Thus only $A$, $C$, $B_3$ (where $D_3=A_3$), and $G_2$ +are possible. The Weyl groups of $B_3$ and $G_2$ contain +$-\id$. In those cases the nonzero weights would be respectively +a tetrahedron or a triangle with centroid zero. Such a simplex +is not preserved by $-\id$; adjoining zero weights cannot make +it centrally symmetric. These two cases are excluded. + +For type $A_\ell$, the remaining simplex weights are the standard +weights or their negatives. The highest weight is therefore +the first or last fundamental weight. Highest-weight uniqueness +\cite[Theorem 4.3]{Cap2016} +identifies $T$ with the standard module or its dual. For type +$C_\ell$, the long roots are $\pm2\epsilon_j$; each centered +two-vertex simplex is $\{\epsilon_j,-\epsilon_j\}$. The +highest weight is $\epsilon_1$, so highest-weight uniqueness +gives the standard symplectic module. In both cases this last +uniqueness statement excludes any additional zero-weight +multiplicity. The root-system and highest-weight facts used here +are the usual finite-dimensional characteristic-zero theorems; +the rank-one and simplex deductions have been proved above. +\end{proof} + +\begin{theorem}[Structure of the residual tensor group] +\label{loc:slsp} +For a maximal chart there is a decomposition +\begin{equation} + V_{\mathrm{fib}}={\bf1}^{\oplus s}\oplus\bigoplus_{i=1}^r L_i, + \qquad + G=\prod_{i=1}^rG_i, + \qquad + G_i=\operatorname{SL}(L_i)\ \text{or}\ + \operatorname{Sp}(L_i), + \label{loc:group-product} +\end{equation} +where each $G_i$ acts by its standard representation on $L_i$ +and trivially on all other summands. There are no repeated +nontrivial irreducible summands. We place +$\operatorname{SL}_2=\operatorname{Sp}_2$ in the symplectic list. +The integer $s$ is the number of free rank-one direct summands +of $V$. +\end{theorem} + +\begin{proof} +By Proposition~\ref{loc:connected-semisimple}, +$\mathfrak g$ is semisimple. The one-block algebra +$\mathfrak h_i$ of Lemma~\ref{loc:block-obstruction} is an +ideal of $\mathfrak g$, hence semisimple. Split off its +complementary ideal. On the irreducible dual of the $i$th +block the representation is $U\otimes W$, with +$\mathfrak h_i$ acting on $U$. If $\dim W>1$, +Lemma~\ref{loc:auxiliary-factor} makes the first prolongation +zero, contrary to Lemma~\ref{loc:block-obstruction} for a +nontrivial block. Therefore $\mathfrak h_i$ acts irreducibly +there. It acts faithfully, since its elements act by zero on +every other block and the total representation is faithful. +Lemmas~\ref{loc:rank-one-symbol} and +\ref{loc:rank-one-classification} identify it with the full +special linear or symplectic algebra in its standard module. + +The ideals so obtained for distinct blocks commute, since each +is zero on all the other blocks. A remaining complementary +semisimple ideal would commute with these irreducible standard +actions. By Schur's lemma it would act by scalars on each block; +a semisimple Lie algebra has no scalar character. It also acts +trivially on the trivial blocks, so faithfulness eliminates it. +If two nontrivial blocks were isomorphic as $\mathfrak g$-modules, +an element acting trivially on one would act trivially on the +other; their one-block ideals would be zero. This again +contradicts Lemma~\ref{loc:block-obstruction}. + +It remains to check the group, rather than just its Lie algebra. +Take the simply connected covering group of the connected +semisimple group $G$. Its factors are the indicated +$\operatorname{SL}$ and $\operatorname{Sp}$ groups. Each +factor acts faithfully by its standard representation on its +own block, and trivially on the others. Thus the kernel on the +full representation is trivial, including any proposed central +or diagonal central subgroup. The covering map is therefore +an isomorphism and its image is the blockwise product in +\eqref{loc:group-product}. Finally a trivial summand in the +trace quotient is an actual free summand by +Lemma~\ref{loc:split-lift}, and conversely. This identifies $s$ +as claimed. +\end{proof} + +\begin{corollary}[A unique exterior constituent] +\label{loc:exterior-constituent} +In the notation of Theorem~\ref{loc:slsp}, suppose $r>0$ and put +$W=\bigoplus_{i=1}^rL_i$. There are a simple $G$-module $E$ and an +integer $1\leq j_0\leq\dim W$ such that, for $0\leq b\leq\dim W$, +\[ + \dim_{\C}\Hom_G(E,\Lambda^bW)= + \begin{cases}1,&b=j_0,\\0,&b\ne j_0.\end{cases} +\] +The multiplicity of $E$ in $\Lambda^{j_0}V_{\mathrm{fib}}$ is also +one. The selection of $E$ is intrinsic to the standard factors and +is preserved by permutations of the factors. +\end{corollary} + +\begin{proof} +For a standard $\operatorname{SL}_m$ block with $m>2$, select its +degree-one representation. The exterior powers in degrees +$1,\ldots,m-1$ are irreducible with distinct highest weights, while +degrees $0$ and $m$ are trivial. The selected representation therefore +occurs once and in degree one only. + +For a standard $\operatorname{Sp}_{2m}$ block $L_i$, select +\[ + P_i^m=\ker\bigl(\Lambda^mL_i\longrightarrow\Lambda^{m-2}L_i\bigr), +\] +where the arrow is symplectic contraction and negative exterior powers +are zero. The primitive decomposition +\[ + \Lambda^bL_i\simeq + \bigoplus_{a=0}^{\lfloor\min(b,2m-b)/2\rfloor} + P_i^{\min(b,2m-b)-2a} +\] +has irreducible, mutually distinct primitive terms $P_i^s$ of highest +weight $\varpi_s$ \cite[Theorem~17.5]{FultonHarris1991}. +Thus $P_i^m$ occurs once and only for $b=m$. This includes +$\operatorname{SL}_2=\operatorname{Sp}_2$. + +Take the external tensor product of the selected representations. +It is simple for the product group and occurs in $\Lambda^\bullet W$ +once, in a single degree $j_0$. In $\Lambda^{j_0}V_{\mathrm{fib}}$ +it still has multiplicity one: a positive exterior degree from the +trivial summand would require its occurrence in a smaller degree of +$W$. The factorwise rule is preserved by permutations. In the +symplectic case the kernel defining $P_i^m$ is also unchanged when +the symplectic form is rescaled, so the selection has the asserted +intrinsic meaning. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/04-stratum.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/04-stratum.tex new file mode 100644 index 0000000000000000000000000000000000000000..d72224fc2d5138be4714fff794bb293b63f02681 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/04-stratum.tex @@ -0,0 +1,789 @@ +\section{A closed stratum and compatible algebraic products}\label{sec:stratum} + +We turn the free part of the local cotangent module into a closed +geometric stratum, then construct algebraic product presentations along +it. The transition maps preserve its marked ideal at every order. +This supplies a single system in which the ambient tensor maps and the +coherent coefficients supported on finite thickenings can be compared. + +For a normal complex analytic germ $(V,v)$ put +\[ + D_v=\im\bigl(T_V{}_v\longrightarrow T_{V,v}^{\mathrm{Zar}}\bigr), + \qquad r(v)=\dim_{\C}D_v. +\] +Here the arrow evaluates a germ of a derivation at $v$, and +$T_V=\Hom(\Omega_V^1,\cO_V)$. All germs in the next two lemmas are finitely +dominated by smooth germs of the same dimension, as supplied by +Proposition~\ref{loc:maximal-charts}. + +\begin{lemma}[Evaluation and smooth factors]\label{str:evaluation} +The largest rank of a free direct summand of $\Omega_V^{[1]}{}_v$ is +$r(v)$. There is an analytic product decomposition +\begin{equation}\label{str:analytic-product} + (V,v)\simeq(\Delta^{r(v)},0)\times(T,t), +\end{equation} +where all vector fields on $(T,t)$ have zero evaluation at $t$. +Moreover, if $\beta$ is a reflexive one-form and a vector field $\xi$ +has zero evaluation at $v$, then $\beta(\xi)(v)=0$. +\end{lemma} + +\begin{proof} +We first prove the last assertion, which is stronger than the analogous +statement for ordinary differentials. A derivation has a local holomorphic +flow: after embedding the germ in a polydisc, lift its coordinate values to +an ambient vector field and solve the holomorphic differential equation. +Preservation of the defining ideal makes the flow preserve the embedded +analytic space. If the derivation evaluates to zero, its flow $\phi_z$ +fixes $v$. + +Choose a finite smooth cover $a:(Y,y)\to(V,v)$ and consider +$H(z,w)=\phi_z(a(w))$. The map $(z,w)\mapsto(z,H(z,w))$ is finite: +it is $\id_\Delta\times a$ followed by the flow automorphism of +$\Delta\times V$. Thus the inverse image of $\Delta\times V_{\mathrm{sing}}$ +has codimension at least two. The pullback of $\beta$ from +$V_{\mathrm{reg}}$ consequently extends to a holomorphic one-form +$\alpha$ on the smooth space $\Delta\times Y$. + +We claim that $\alpha$ restricts to zero on $C=\Delta\times\{y\}$. +To check this, pull a fixed finite smooth cover $a_0:Y_0\to V$ back by +$H$, normalize a component dominating $\Delta\times Y$, and resolve it. +The resulting diagram has a proper dominant map $q$: +\[ + \begin{array}{ccc} + \widetilde W&\xrightarrow{\ g\ }&Y_0\\ + {\scriptstyle q}\downarrow&&\downarrow{\scriptstyle a_0}\\ + \Delta\times Y&\xrightarrow{\ H\ }&V. + \end{array} +\] +The pullback of $\beta$ extends to a holomorphic form $\beta_0$ on +$Y_0$. On $\widetilde W$ the forms $q^*\alpha$ and $g^*\beta_0$ +agree, since they agree on a dense open. An irreducible subvariety of +$q^{-1}(C)$ dominating $C$ maps under $g$ to a point of the finite fiber +$a_0^{-1}(v)$. Restriction to a resolution of that subvariety therefore +annihilates $q^*\alpha$. Pullback of one-forms from the smooth curve +$C$ under a dominant map is injective in characteristic zero, proving +the claim. Finally, the definition of the flow gives the identity +\[ + \left.\alpha\left(\frac{\partial}{\partial z}\right)\right|_{z=0} + =a^*(\beta(\xi)). +\] +It holds first over the regular locus and then everywhere by holomorphic +extension. Evaluation at $y$, using $\alpha|_C=0$, yields +$\beta(\xi)(v)=0$. + +Suppose that $\Omega_V^{[1]}{}_v$ has a free summand with basis +$\beta_1,\ldots,\beta_s$ and dual vector fields +$\xi_1,\ldots,\xi_s$. A linear combination of their evaluations that +vanishes pairs to zero with every $\beta_j$ by the preceding paragraph. +The identities $\beta_j(\xi_i)=\delta_{ij}$ imply independence of the +evaluations. Thus $s\leq r(v)$. + +Conversely choose vector fields with $r=r(v)$ independent evaluations and +regular functions $z_1,\ldots,z_r$ with +$\xi_i(z_j)(v)=\delta_{ij}$. Cut a transverse germ by $z_1=\cdots=z_r=0$. +Successive flows of the chosen fields define a map from a polydisc times +this transverse germ to $V$. To invert it, apply the inverse flows and +solve for their times by the analytic implicit function theorem; its +Jacobian in the time variables is the identity at $v$. This argument +works in an ambient embedding and hence also on the embedded germ. +The resulting product has $r$ free cotangent summands. The transverse +factor cannot have a vector field with nonzero evaluation, since such a +field would give an additional independent evaluation on the product. +\end{proof} + +\begin{lemma}[The rank-locus dimension bound]\label{str:rank-locus} +On such a normal germ, the locus $\{v:r(v)\leq a\}$ is closed analytic +and has dimension at most $a$. +\end{lemma} + +\begin{proof} +Coherence of the sheaf of derivations, followed by evaluation on finitely +many ambient coordinates, expresses the condition by minors of a matrix +of holomorphic functions. For the dimension assertion we use the +following trace observation. If $\pi:(Y,y)\to(V,v)$ is a finite map +from a smooth germ, with this single point over $v$, and $m$ is its degree, +then a vector field $\eta$ on $Y$ induces the derivation +\begin{equation}\label{str:trace-derivation} + a\longmapsto\frac{1}{m}\operatorname{Tr}_{\C(Y)/\C(V)} + \bigl(\eta(\pi^*a)\bigr). +\end{equation} +The trace is holomorphic by integrality and normality, and the Leibniz +rule follows from linearity of trace over $\cO_V$. At $v$, the trace of a +holomorphic function on this local branch is $m$ times its value at $y$. +Consequently the evaluation of \eqref{str:trace-derivation} is exactly +$d\pi_y(\eta_y)$. A local branch can be isolated analytically even if +the original finite cover has several points over $v$. We obtain +\begin{equation}\label{str:differential-image} + d\pi_y(T_{Y,y})\subset D_v. +\end{equation} + +Let $Z$ be an irreducible subvariety of the rank locus. A component of +its reduced inverse image under a finite smooth cover dominates $Z$. +At general smooth points of this component and of $Z$, the restricted +finite map has differential rank $\dim Z$, by characteristic zero. +Equation~\eqref{str:differential-image} therefore gives $\dim Z\leq a$. +\end{proof} + +Define $s(b)$ to be the free cotangent rank on a maximal chart at $b$. +It is independent of the choices in Proposition~\ref{loc:maximal-charts}; it +is also the multiplicity of the tensor unit in Theorem~\ref{loc:slsp}. + +\begin{proposition}[The minimum stratum]\label{str:minimum} +Let $d=\min_{b\in B}s(b)$. If $d=n$, every germ of $B$ is a quotient +singularity. Otherwise the minimum locus is a nonempty closed algebraic +subset. Each of its connected components $S_0$ is irreducible, normal +and projective of dimension $d$. On a maximal chart over a point of +$S_0$, its reduced inverse image is, as a germ of a pair, +\begin{equation}\label{str:minimum-pair} + (V,S_V)\simeq(\Delta^d\times T,\Delta^d\times\{t\}). +\end{equation} +At every point of $S_V$, the vector-field evaluation space is exactly +the tangent space to $S_V$. +There is a smooth separated Deligne--Mumford stack $\cS$ with coarse +space $S_0$ whose charts are these smooth factors. Its stabilizers are +the full deck stabilizers, including those acting trivially on the +smooth factor. +\end{proposition} + +\begin{proof} +Smooth factors persist under further finite local quasi-\'etale covers. +Indeed such a cover is \'etale over the smooth locus by purity, and a +cover of $\Delta^r\times T_{\mathrm{reg}}$ is pulled back from +$T_{\mathrm{reg}}$, since the polydisc is simply connected. Normalization +extends the product across the omitted codimension-two set. More +precisely, if $c'$ is any nearby point on a finite quasi-\'etale chart +over $b'$, its pointed local branch is dominated by a maximal cover of +$(B,b')$. Each of its $r(c')$ smooth factors persists on that cover, +so $r(c')\leq s(b')$. A product with $r$ smooth directions therefore +implies $s\geq r$ throughout its image. Each pointed finite branch +dominates its base germ, so the image of a sufficiently small +neighborhood contains a neighborhood downstairs. Along the smooth factor +through its marked point, the transverse germ is unchanged; the same +description of covers shows that its maximal-cover property is unchanged +as well. Hence $s$ is lower semicontinuous in the analytic topology +and is constant along that factor. + +At a minimum point choose the product in Lemma~\ref{str:evaluation}. +The preimage of the minimum locus lies in the evaluation-rank locus +$r\leq d$: any extra smooth factor at another point of this chart +would persist on its maximal cover. The rank locus on the full chart +is the product of $\Delta^d$ with the transverse rank-zero locus $Z$. +Apply Lemma~\ref{str:rank-locus} to this full chart, which is finitely +dominated by a smooth germ. It gives $d+\dim Z\leq d$, so the germ of +$Z$ at the marked point is just that point. Conversely the whole smooth factor +has the original maximal transverse type. This proves +\eqref{str:minimum-pair} and also proves that the minimum locus is locally +analytic, rather than merely topologically closed. + +Its closedness and projectivity of $B$ make it algebraic by Chow's +theorem. Locally it is a finite quotient of the smooth orbit by its +deck stabilizer, and hence is normal. Distinct irreducible components +of a normal variety are disjoint, so a connected component is +irreducible. If $d=n$, Lemma~\ref{str:evaluation} makes every maximal +chart smooth, which is exactly the quotient-singularity assertion. +The construction of the stack, with its specified stabilizers, is +included in Proposition~\ref{str:products} below. +\end{proof} + +From now on fix such a component with $d0$. +We first prove the precise algebraization statement needed for its +neighborhood. + +\begin{lemma}[Algebraizing a marked product]\label{str:marked-artin} +Let $(V,v)$ be an algebraic complex germ, let $S\subset V$ be a reduced +closed subgerm, and suppose the analytic pair is +$(\Delta^d\times T,\Delta^d\times\{t\})$. There are an algebraic +transverse germ $(T_0,t_0)$ and common \'etale neighborhoods of pairs +\begin{equation}\label{str:etale-product-pair} + (V,S)\ \longleftarrow\ (U,S_U)\ \longrightarrow\ + (\mathbb A^d\times T_0,\mathbb A^d\times\{t_0\}). +\end{equation} +Both inverse-image ideals defining $S_U$ are equal, without truncation. +For $k\geq1$, put $A_k=\cO_{T_0,t_0}/\mathfrak m_{t_0}^k$ and let $U_k$ +be defined by the $k$th power of this common ideal. After shrinking the +displayed neighborhoods, there are compatible isomorphisms +\begin{equation}\label{str:finite-product} + U_k\simeq S_U\times\Spec A_k + \quad\text{over}\quad \mathbb A^d\times\Spec A_k. +\end{equation} +No equivariance of these product isomorphisms is asserted. +\end{lemma} + +\begin{proof} +Choose algebraic functions $z_1,\ldots,z_d$ whose restrictions give +independent cotangents on $S$ at $v$, and define $T_0$ by their vanishing. +In the given analytic product these functions may replace the polydisc +coordinates: their Jacobian in those coordinates is invertible. The +analytic implicit function theorem, with the transverse germ retained, +identifies the resulting transversal with $T$ and gives a formal +isomorphism of the algebraic pointed pairs in +\eqref{str:etale-product-pair}. + +Here is why approximation can retain the marked ideal exactly. Write +finite algebra presentations for the target product and generators +$a_1,\ldots,a_h$ and $b_1,\ldots,b_l$ for its marked ideal and the ideal +of $S$ in $V$, respectively. Besides the target's defining equations, +include unknown coefficients in the finitely many equations +\[ + \varphi(a_i)=\sum_j u_{ij}b_j, + \qquad b_j=\sum_i v_{ji}\varphi(a_i). +\] +The formal pair isomorphism supplies a solution. Artin approximation +over the henselization of the algebraic local ring produces a solution +in an \'etale neighborhood, arbitrarily close to that formal solution; +this is the polynomial-equation form of +\cite[Tags 0CAU and 0CAV]{StacksProject}. Denominators in local +presentations can be included by equations for their inverses. The +displayed equations give the two ideal inclusions exactly. + +Choose approximation modulo the square of the maximal ideal. Composing +the resulting map on completions with the original formal inverse +gives an endomorphism that is the identity on residue field and +cotangent space. It is the identity on the associated graded ring, +which is generated in degree one. Induction on the powers of the +maximal ideal and completeness show that it is an isomorphism. Thus +the second map in \eqref{str:etale-product-pair} induces an isomorphism +of completed local rings and is \'etale at the marked point; shrink +to its \'etale locus. This verifies the inverse-map issue even for a +singular transversal. + +The restriction $S_U\to\mathbb A^d$ is \'etale. Both $U_k$ and +$S_U\times\Spec A_k$ are \'etale over +$\mathbb A^d\times\Spec A_k$ and have the same reduction over +$\mathbb A^d$. Uniqueness of lifting morphisms between \'etale schemes +through a nilpotent thickening gives \eqref{str:finite-product} +\cite[Tag 025H]{StacksProject}. The same uniqueness makes the +isomorphisms compatible as $k$ varies. Notice that this last argument +uses the equality of the two ideals, not closeness of those ideals to +some finite order. +\end{proof} + +\begin{proposition}[Compatible product presentations]\label{str:products} +The following data exist along $S_0$. +\begin{enumerate} +\item There are finitely many normal finite Galois quasi-\'etale charts +$V_\alpha\to B_\alpha$, with deck groups $\Gamma_\alpha$ and with +$B_\alpha\to B$ separated and \'etale, covering $S_0$. +They are maximal at every point over $S_0$. +\item Normalized main overlaps give an \'etale groupoid along the +reduced inverse images $S_\alpha\subset V_\alpha$. Its quotient is +the stack $\cS$ of Proposition~\ref{str:minimum}. Writing $I$ for the +marked ideal on the object charts, the same finite diagrams define, +for every $k\geq1$, a proper separated Deligne--Mumford stack +$\mathcal S_k$, with $\mathcal S_1=\cS$. Its coarse space is finite +over $B$ and projective. +\item A finite family of \'etale refinements of the object charts +covers their closed strata and has the marked-product presentations of +Lemma~\ref{str:marked-artin}. All transition maps come from the +normalized ambient overlaps and preserve $I$ exactly. Their +restrictions are consequently compatible for all powers $I^k$. +\item The normalized source charts give ordinary pullback and descent +of coherent sheaves on any $\mathcal S_k$ to coherent sheaves on $X$ +supported over $S_0$. +\end{enumerate} +These assertions are encoded by finitely many algebraic schemes, +morphisms, ideals and identities. In particular they can be spread +simultaneously before choosing any order of Frobenius pullback. +\end{proposition} + +\begin{proof} +Start with a chart maximal at one point, as in +Proposition~\ref{loc:maximal-charts}. By +\eqref{str:minimum-pair} and Lemma~\ref{str:marked-artin}, its transverse +analytic type is constant along the part of $S_\alpha$ covered by one +common \'etale product neighborhood. The image is open in $S_\alpha$, +because an \'etale map is open. Every point there still has a maximal +germ, since the transverse germ and its covers have not changed. +Take the union of translates of this open neighborhood around the finite +deck orbit over the base point. The complement in $S_\alpha$ has closed +image under the finite chart map. Removing that image downstairs +arranges maximality at every point over the remaining stratum. +Quasi-compactness of $S_0$ gives the finite list in (1). Applying +Lemma~\ref{str:marked-artin} at its inverse images and using +quasi-compactness gives a finite family of product refinements as in (3). + +We construct the groupoids while keeping the full finite normalizations. +Put $V=\coprod_\alpha V_\alpha$. Choose a common dense open +$B^\circ\subset B_{\mathrm{reg}}$ over which all the charts are \'etale, +and write $V^\circ=V\times_BB^\circ$. For $a=2,3,4$, let $W_a$ be the +reduced union of the irreducible components of $V^{\times_B a}$ meeting +$(V^\circ)^{\times_{B^\circ}a}$, and let +\[ + \overline V_a=W_a^\nu +\] +be its full normalization. It is finite over the closed subscheme +$W_a\subset V^{\times_Ba}$. Its restriction over $B^\circ$ is the +ordinary product, which is \'etale over the smooth space $B^\circ$ and +therefore normal. Write $\overline R=\overline V_2$, with projections +$s,t:\overline R\to V$. + +The projections of these normalizations are \'etale at every point +over $S_0$. To see this locally, pass to a common \'etale base +neighborhood for the chosen object charts. A projection is there a +finite quasi-\'etale map followed by an \'etale map: away from the +codimension-two branch loci it is a base change of the original charts, +and finiteness preserves that codimension bound. Each pointed target +chart over $S_0$ is maximal, so every connected local finite +quasi-\'etale factor is trivial. This proves the asserted \'etaleness. + +We will also need an arrow through every pair of object points over a +point of $S_0$. Pass to the common strict henselian base germ at that +point and select the pointed local factors of the two finite charts. +By the algebraic--analytic comparison in +Proposition~\ref{loc:maximal-charts}, the maximality at these points +also gives maximality of the selected strict henselian factors. +They are therefore isomorphic over this germ. +The graph of such an isomorphism is a main branch of their product and +passes through the chosen pair. Normalization commutes with this +strict henselian base change: the base change of a finite normalization +is normal under \'etale localization and has the same generic +normalization property, and the assertion passes to the filtered +strict henselian limit. The graph branch therefore supplies a point +of $\overline R$ over the chosen pair. This proves the assertion for +the original algebraic charts. + +Let $R\subset\overline R$ be the open where both $s$ and $t$ are +\'etale. It contains the entire inverse image of $S_0$. Let +$I_\alpha$ be the reduced ideal of $S_\alpha$ and let $I$ be their +union on $V$. On the full normalization define the extended ideals +\[ + J_s=\im(s^*I\longrightarrow\cO_{\overline R}),\qquad + J_t=\im(t^*I\longrightarrow\cO_{\overline R}). +\] +Their zero sets are both the inverse image of $S_0$. On $R$ they are +equal: the two \'etale pullbacks are reduced and have that same support. +Let $V_k\subset V$ be defined by $I^k$, and define $R_k$ to be the +closed subscheme of the full $\overline R$ defined by $J_s^k$. +Its entire support lies in $R$, so +\[ + R_k=R\times_{s,V}V_k=R\times_{t,V}V_k. +\] +In particular its two maps to $V_k$ are \'etale. The map to +$V\times_BV$ factors through $V_k\times_BV_k$, because both extended +ideals vanish to order $k$ on $R_k$. The resulting map +$R_k\to V_k\times_BV_k$ is finite: it is a closed quotient of the +full finite normalization, with a factorization through that closed +subscheme. This is the finiteness used below. + +Here are the normalization comparisons needed for composition. Put +\[ + P=R\times_{t,V,s}R. +\] +It is normal, since it is \'etale over the normal object charts. +Every irreducible component of $P$ meets the open over $B^\circ$: +its image under an \'etale projection to an object component is a +nonempty open, and therefore meets the dense $V^\circ$. Over +$B^\circ$, the scheme $P^\circ$ is an open subscheme of the ordinary +triple product $(V^\circ)^{\times_{B^\circ}3}$. Hence each component +of $P$ maps dominantly onto a main component of $W_3$. Its generic +map lifts to that component of $\overline V_3$, and the lift extends +over the normal component of $P$. Indeed, each element of the +finite normalization is integral over the local target ring; its +generic pullback is integral over a normal local ring of $P$ and so +belongs to that ring. This constructs $P\to\overline V_3$. + +The three pair-forgetting maps $\overline V_3\to W_2$ lift in the +same way to maps $\rho_{01},\rho_{12},\rho_{02}:\overline V_3\to +\overline R$. Componentwise dominance here follows on $B^\circ$: +each projection of the ordinary triple product is \'etale, and its +nonempty image is open in the corresponding ordinary pair component. +Let +\[ + O_3=\rho_{01}^{-1}(R)\cap\rho_{12}^{-1}(R). +\] +It contains the entire inverse image of $S_0$, and its two adjacent +pairs give a map $O_3\to P$. Conversely, the adjacent pair maps of +$P\to\overline V_3$ agree with the two projections $P\to R$ on the +dense unbranched open, hence on $P$; the lift therefore lands in $O_3$. +This map and $O_3\to P$ are +inverse between $O_3$ and $P$. Their composites are the identity +on the componentwise dense ordinary unbranched products; equality +extends because the sources are reduced and the targets separated. +Restrict further to +$O'_3=O_3\cap\rho_{02}^{-1}(R)$, which still contains the entire +closed support. The map $\rho_{02}$ gives composition there. + +On $P$ the three extended ideals from the object coordinates are +equal, by the equality $J_s=J_t$ on each arrow. Call their common +ideal $J$. Base change by the closed immersions therefore gives the +scheme equality +\[ + R_k\times_{V_k}R_k=P/J^k. +\] +Its support is the whole inverse image of $S_0$ in $P$, hence lies in +the domain corresponding to $O'_3$. The three ideals agree there +also along $\rho_{02}$, so composition restricts to +$R_k\times_{V_k}R_k\to R_k$ for every $k$. + +For associativity use $R\times_VR\times_VR$ and $\overline V_4$. +The same argument applies component by component: the former is +normal, and its unbranched part is open in the ordinary quadruple +product and meets every component. Each component therefore dominates +a main component of $W_4$, while all pair-forgetting projections are +dominant onto their main pair components there. It identifies with the open of +$\overline V_4$ where the adjacent pairs $01,12,23$ land in $R$. +Restrict to the further open where $02,13,03$ also land in $R$. +Every one of these opens contains the entire inverse image of $S_0$. +Both parenthesizations are defined on this last open and agree on +its componentwise dense unbranched product, hence agree everywhere. +Their restrictions agree on every quotient by the common $k$th +power of the stratum ideal. Interchanging the two coordinates lifts +to the normalization $\overline R$ and gives inversion on the open +where both the original and interchanged arrow land in $R$. The +diagonal of the ordinary unbranched pair product is open and closed; +its main branch lifts the diagonal map from the normal $V$ to +$\overline R$. Restricting to the inverse image of $R$ gives the unit. +These domains contain the entire closed support, by the \'etaleness of +all pair points over $S_0$. The unit and inverse identities follow on their +support-containing domains by the same dense-open comparison and +then restrict to all $k$. Thus $R_k\rightrightarrows V_k$ is an +\'etale groupoid. Four factors suffice because associativity is the +only law involving three composable arrows. + +Since $B$ is separated, $V_k\times_BV_k$ is closed in $V_k\times V_k$. +The finiteness proved above therefore makes the diagonal of +$[V_k/R_k]$ finite. This quotient is a separated Deligne--Mumford +stack with finite inertia. Its local finite quotient presentation +can be read after base change to $B_\alpha$. The diagonal +$B_\alpha\to B_\alpha\times_BB_\alpha$ is open and closed, since +$B_\alpha\to B$ is separated and \'etale. On that component the +self-overlap is the normalization of +$V_\alpha\times_{B_\alpha}V_\alpha$, namely +$\coprod_{\gamma\in\Gamma_\alpha}V_\alpha$ with its distinct deck +labels. The existence of a main arrow through every pair of closed +object points, proved above, makes the other object charts \'etale +refinements of this presentation: the relevant arrow projections +are \'etale and surjective on the closed support, which is the +underlying support of every $V_k$. Consequently over $B_\alpha$ +the stack is +\[ + [\,(V_\alpha)_k/\Gamma_\alpha\,]. +\] +Its coarse algebra is precisely +\[ + (\cO_{V_\alpha}/I_\alpha^k)^{\Gamma_\alpha} + =\cO_{B_\alpha}/(I_\alpha^k\cap\cO_{B_\alpha}). +\] +Here the intersection is the contracted ideal in the finite chart +algebra; exactness of finite-group invariants in characteristic zero +gives the equality. These finite algebras glue \'etale locally. +They give a coarse space finite over the projective variety $B$, and +therefore projective. A quotient by a finite group is proper over +its coarse space, so the stack is proper. At $k=1$ the coarse space +is $S_0$ and the object charts are smooth, giving $\mathcal S_1=\cS$. + +The construction keeps the label of every $\gamma\in\Gamma_\alpha$ +even when it fixes the whole smooth factor. Stabilizers may increase +on closed subsets of $S_\alpha$, and remain subgroups of these fixed +finite deck groups. The ambient groupoid laws and their restrictions +therefore retain the full stabilizers on $\cS$. + +For (4), write $X_\alpha\to V_\alpha$ for the normalized source chart. +Proposition~\ref{loc:maximal-charts} makes +$X_\alpha\to X\times_BB_\alpha$ a finite \'etale +$\Gamma_\alpha$-torsor. The normalized overlaps give its ordinary +descent identifications. The ideals +$I_\alpha^k\cO_{X_\alpha}$ descend to a coherent ideal near +$f^{-1}(S_0)$; gluing it with the unit ideal off this closed set gives +a closed subscheme $X_k\subset X$. This is the descended ideal +quotient, with no identification with a fiber product over the coarse +thickening required. Put $X_{\alpha,k}=X_\alpha/I_\alpha^k\cO_{X_\alpha}$. +The local diagram +\[ + \begin{array}{ccc} + X_{\alpha,k}&\longrightarrow&(V_\alpha)_k\\ + \downarrow&&\downarrow\\ + {[X_{\alpha,k}/\Gamma_\alpha]}&\longrightarrow& + [(V_\alpha)_k/\Gamma_\alpha] + \end{array} +\] +is $2$-Cartesian. Its lower left quotient is the descended scheme +$X_k\times_BB_\alpha$, because the source action is a torsor. +These diagrams glue to a map $f_k:X_k\to\mathcal S_k$. +For a coherent sheaf $M$ on $\mathcal S_k$, the ordinary sheaves +$\cO_{X_{\alpha,k}}\otimes_{\cO_{(V_\alpha)_k}}M_\alpha$ descend to +a coherent sheaf on $X_k$, and then on $X$. Increasing $k$ does not +change this pullback, denoted $f^*M$. The displayed Cartesian +diagram is also the reason that ordinary chart base change and the +canonical higher-direct-image coefficient maps descend. After the +product refinements in (3), these constructions are their ordinary +\'etale pullbacks from the finite charts. + +All schemes, full finite normalizations, group actions, product maps, +marked ideals and maps in the diagrams with at most four factors are +of finite type. Their identities, ideal inclusions, and the +disjointness of each omitted closed complement from its closed support +are finitely many algebraic conditions. Individual marked pairs were +approximated in Lemma~\ref{str:marked-artin}; their transition maps are +the normalized algebraic maps just constructed. This proves the +asserted finite nature of the compatible data. +\end{proof} + +We write $\widehat{\mathcal B}$ for the compatible formal system +$(\mathcal S_k)_{k\geq1}$. There are two kinds of sheaves in use on this +system. First, an ambient reflexive tensor object +$\mathcal M_\alpha$ on each normal chart, with its natural \'etale +transition isomorphisms, gives the completed system +\[ + M=(M_k)_{k\geq1},\qquad + M_{\alpha,k}=\mathcal M_\alpha/I_\alpha^k\mathcal M_\alpha. +\] +In particular, for $0\leq j\leq n$ we use +\[ + \mathcal N_{j,\alpha}=\Omega_{V_\alpha}^{[j]},\qquad + N_j=(N_{j,k})_{k\geq1}. +\] +The notation $\Omega_{\widehat{\mathcal B}}^{[j]}$ will mean this +completed ambient system. It does not refer to an intrinsic reflexive +differential sheaf on the nonreduced $\mathcal S_k$. + +The ambient Hom sheaf for two such objects is +\[ + \mathcal H_{\alpha,MN} + =\mathcal Hom_{V_\alpha}(\mathcal M_\alpha,\mathcal N_\alpha). +\] +It has the natural \'etale transition maps, and we write +\[ + \widehat{\mathcal H}_{MN} + =\bigl(\mathcal H_{\alpha,MN}/I_\alpha^k + \mathcal H_{\alpha,MN}\bigr)_{k\geq1} +\] +for its completed system. On an adic complete chart this is the Hom +module of the completed ambient modules: completion is flat for +noetherian rings and Hom commutes with flat base change for finite +presentations. Composition and trace are induced by their ambient +maps. Put +\[ + N=\bigoplus_{j=0}^nN_j,\qquad + \cE=\widehat{\mathcal H}_{NN}. +\] +Thus $\cE$ is completed from the ambient endomorphism sheaf before +any quotient is taken. + +Second, a bounded-support coefficient is a coherent sheaf $M$ on some +$\mathcal S_k$, regarded at larger orders by pushforward along the +closed immersions. For such a coefficient, $N_j\otimes M$ means the ordinary +tensor product $N_{j,k}\otimes_{\cO_{\mathcal S_k}}M$ at any sufficient +order. Its pullback to $X$ is the ordinary pullback constructed in +Proposition~\ref{str:products}. This use of coefficients is separate +from the completed ambient systems. In particular the full truncated +endomorphism sheaf +\[ + \mathcal{E}nd_{\mathcal S_k}\left(\bigoplus_jN_{j,k}\right) +\] +is allowed to have maps that do not lift to ambient endomorphisms. +The natural map from $\cE/I^k\cE$ to this full sheaf may be neither +injective nor surjective. + +For ambient objects $M,N$, define their barred Hom sheaf on $\cS$ by +the residual trace pairing: +\begin{equation}\label{str:barred-hom} + \overline{\Hom}(M,N)= + \frac{\widehat{\mathcal H}_{MN}|_{\cS}} + {\ker\left[ + \widehat{\mathcal H}_{MN}|_{\cS}\longrightarrow + (\widehat{\mathcal H}_{NM}|_{\cS})^\vee,\quad + a\longmapsto\bigl(b\longmapsto\tr(ba)\bigr)\right]}. +\end{equation} +We put $\overline{\cE}=\overline{\Hom}(N,N)$. For the fixed natural +tensor expressions and structural maps used here, +Lemma~\ref{str:uniform} below shows that these are locally free and +that their fibers are the trace quotients of the corresponding local +tensor objects. A residual direct summand is then handled by the +image of its idempotent on these bundles; it is locally a direct +summand bundle as well. +Cyclicity and partial trace make the kernels in +\eqref{str:barred-hom} a tensor ideal, so composition and the indicated +tensor operations descend to these quotients. + +Coarse and chart ideals give cofinal filtrations: on a finite chart +the radical of the extended coarse ideal is $I_\alpha$, so suitable +powers contain one another by noetherianity. A finite cover makes +these comparisons uniform. + +\begin{lemma}[Uniform transverse calculations]\label{str:uniform} +On the finite family of product refinements in +Proposition~\ref{str:products}, the ambient sheaves +$\mathcal N_j=\Omega^{[j]}$, the fixed natural mixed reflexive tensor +expressions in them, and their ambient Hom and trace maps have fixed +transverse descriptions. Their completed systems at order $k$ are +obtained from fixed finite modules over $A_k$ by extension along the +smooth factor. The same holds for kernels and images of the resulting +fixed transverse maps. Residual retracts are obtained by applying +their idempotents to the resulting barred bundles. In particular the barred Hom sheaves in +\eqref{str:barred-hom} are locally free on $\cS$ and their fibers are +the local residual trace Hom spaces. + +Consequently a property of these modules and maps that, for each fixed +transverse complete ring, holds at every order $k\geq k_0$ has one +valid order on the whole stratum, provided it is preserved by flat +scalar extension and can be checked by faithfully flat completion. +The completed product descriptions persist after replacing the smooth +parameters by lifts of another coordinate system on the reduced +stratum. +\end{lemma} + +\begin{proof} +In our applications the transversal $T_0$ is normal near $t_0$. +Its analytic germ is the transverse factor of a normal product, hence +normal. Its completion is normal by excellence and is also the +completion of the algebraic local ring; faithful flatness of completion +then descends normality to that ring. Shrink $T_0$ accordingly. +On a product $\mathbb A^d\times T_0$, restriction to the smooth locus +of the normal transversal gives +\begin{equation}\label{str:cotangent-product} + \Omega^{[j]}_{\mathbb A^d\times T_0} + =\bigoplus_{a+b=j} + \operatorname{pr}_1^*\Omega^a_{\mathbb A^d} + \otimes\operatorname{pr}_2^*\Omega_{T_0}^{[b]}. +\end{equation} +Both sides are reflexive, so the equality extends across codimension +two. It remains valid under the common \'etale pullbacks. +Equation~\eqref{str:finite-product} gives the claimed description on +each whole thickened chart. For finite modules, flat base change +commutes with Hom, as follows from a finite presentation; it also +commutes with kernels and images. It therefore commutes with double +duals and the fixed reflexive tensor constructions here. Composition +and contraction are preserved, and trace can be checked on the dense +locally free locus. The trace pairing on the reduced smooth factor is +thus the scalar extension of a pairing of finite-dimensional complex +vector spaces. Its kernel and quotient are vector bundles there. +This proves the claim about barred Hom. + +We make explicit how the residual tensor category of the full maximal +chart is used in these transverse calculations. At a product point put +\[ + R=\widehat{\cO}_{T_0,t_0},\qquad + S=R[[t_1,\ldots,t_d]],\qquad + E=R^{\oplus d}\oplus\widehat{\Omega_{T_0}^{[1]}}. +\] +The completion of the full chart cotangent module is $E\otimes_RS$. +For finite reflexive tensor objects $P,Q$ generated by $E$, flat +base change gives +\[ + \Hom_S(P\otimes_RS,Q\otimes_RS) + =\Hom_R(P,Q)\otimes_RS, +\] +and also commutes with their reflexive tensor operations, composition +and trace. Reduction modulo +$\mathfrak m_S=(\mathfrak m_R,t_1,\ldots,t_d)$ gives the same trace +pairing as reduction over $R$ modulo $\mathfrak m_R$. Its kernels and +residual Hom spaces therefore agree. The same flat Hom calculation +compares the analytic local ring of the full chart with its completion. +Idempotent lifting over the henselian rings, as in +Lemmas~\ref{loc:trace-radical} and \ref{loc:split-lift}, identifies +their idempotent-complete residual tensor categories. Thus the +residual category of the full maximal chart is transported by scalar +extension to the category computed from the transverse module $E$. +Theorem~\ref{loc:slsp} and Corollary~\ref{loc:exterior-constituent} +are applied on the full chart and carried to the transverse +calculation by this identification. + +One may now choose an exponent on each of the finitely many +transversals and take their maximum. Equalities of coherent maps, +surjectivity, and specified powers annihilating a kernel descend from +faithfully flat completions. This proves precisely the asserted +uniformity for the fixed finite diagrams. + +For the last assertion, the stratum ideal in $S$ is +$\mathfrak m_RS$. If $t'_1,\ldots,t'_d$ restrict to another coordinate +system on $S/\mathfrak m_RS$, substitution $t_i\mapsto t'_i$, fixing +$R$, is an automorphism of this complete ring by the formal inverse +function theorem. It preserves the stratum ideal. Completed +differentials and their reflexive tensor maps are functorial for this +automorphism, so it transports the completed Hom, trace and ideal +data as well. +\end{proof} + +\begin{proposition}[Compatible direct images on normalized charts] +\label{str:direct-images} +Let $U\to B$ be an integral separated \'etale algebraic neighborhood, +let $\pi:C\to U$ be an integral finite normal Galois quasi-\'etale +chart, and let $h:Z\to C$ be the normalized source chart of +Proposition~\ref{loc:maximal-charts}. For the global degreewise +isomorphisms $\phi_j:R^jf_*\cO_X\to\Omega_B^{[j]}$ fixed after +Proposition~\ref{prop:background}, there are unique isomorphisms +\[ + \phi_{j,C}:R^jh_*\cO_Z\xrightarrow{\ \sim\ }\Omega_C^{[j]}, + \qquad 0\leq j\leq n, +\] +whose restrictions to $C^\circ=C\times_BB_{\mathrm{reg}}$ are the +ordinary flat pullbacks of $\phi_j$. The degree-zero map is the +unit. These isomorphisms are compatible with ordinary \'etale +refinement and with every normalized main overlap whose projections +are \'etale. In particular they give compatible ambient +identifications $\mathcal N_{j,\alpha}=R^jh_{\alpha*}\cO_{X_\alpha}$ +on the charts of Proposition~\ref{str:products}. +\end{proposition} + +\begin{proof} +The restriction $C^\circ\to B_{\mathrm{reg}}$ is \'etale. Indeed it +factors through the smooth $U\times_BB_{\mathrm{reg}}$, and purity +makes the finite quasi-\'etale factor \'etale there. Its complement +in $C$ has codimension at least two. Over $C^\circ$ the normalized +source is the ordinary base change $X\times_BC^\circ$: that base change +is smooth and normal, all its components dominate $C^\circ$ by +equidimensionality, and it has the integral generic fiber used in +Proposition~\ref{loc:maximal-charts}. Proper flat base change and +\'etale pullback of differentials therefore pull $\phi_j$ to +\[ + (R^jh_*\cO_Z)|_{C^\circ}\xrightarrow{\ \sim\ }\Omega_{C^\circ}^j. +\] +This open includes the entire inverse image of the discriminant over +the regular part of $B$; only the singular locus of $B$ has been +removed. + +The discrepancy argument in Proposition~\ref{loc:maximal-charts} applies +to any such quasi-\'etale $\pi$, so $C$ is klt. In particular it is +Cohen--Macaulay and $\Q$-Gorenstein. The same proposition gives a +projective equidimensional map $h$ with connected fibers and smooth +canonical-trivial source. Ou's local reflexivity theorem +\cite[Theorem~1.3]{Ou2019} therefore makes $R^jh_*\cO_Z$ reflexive +on $C$. Both sides of the displayed isomorphism are reflexive and +the omitted set has codimension at least two, so it extends uniquely +to $\phi_{j,C}$. At $j=0$ it is the Stein unit. + +Ordinary \'etale base change preserves both sides. On a normalized +main overlap with \'etale projections, every component meets the +inverse image of $B_{\mathrm{reg}}$, since a main component meets +the dense common unbranched open. The two pullbacks there are the +same pullback of the fixed global $\phi_j$. They consequently +agree on the entire normal overlap by reflexivity. This also proves +compatibility with deck actions and with further \'etale refinements. +\end{proof} + +We finish by recording the scope of spreading. Choose models over a +finitely generated integral subring of $\C$ for the finite diagram in +Proposition~\ref{str:products}, for the ambient coherent isomorphisms +of Proposition~\ref{str:direct-images}, and for the finite coherent +maps required in Lemma~\ref{str:uniform} at the selected orders. +After localization, the object and overlap charts retain normality, +the stated maps retain \'etaleness near the closed loci, the reduced +stratum charts are smooth, the finite source charts are smooth \'etale +torsors, and the coarse stratum is geometrically integral and projective. +Retain also a fiberwise dense common unbranched open and its finite +torsor identifications. Keep the ideal equalities and groupoid +identities as identities of the models, and invert the finitely many +deck-group orders. + +For each of the pair, triple, quadruple, unit and inverse domains used +above, retain the disjointness of its omitted closed complement from +the corresponding closed support. There are finitely many such +intersections, all with empty generic fiber, so one further localization +keeps them empty. This ensures that the quotients by every power of +the marked ideal remain in those domains after reduction. The common +cross-arrow projections also retain their coverage of the corresponding +closed supports. Their images are open by \'etaleness, and the closed +subsets of those supports omitted by the relevant unions of images have +empty generic fiber. There are finitely many such subsets, so one +further localization removes them. The common +product maps are modeled as \'etale maps of marked pairs. Their +product descriptions at every finite order then follow from formal +\'etaleness, uniformly in the model. + +For any additional fixed finite list of coherent presentations, +kernels, images and identities, generic freeness and coherent base +change retain the required calculations after further localization. +In positive characteristic we use these spread sheaves, direct-image +isomorphisms and algebraic identities. No positive-characteristic +maximal-cover or residual tensor classification is asserted. diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/05-base-change.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/05-base-change.tex new file mode 100644 index 0000000000000000000000000000000000000000..93aa8532827a5c6dc0bcde01401c72f432994a46 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/05-base-change.tex @@ -0,0 +1,1086 @@ +\section{Good reductions and ordinary base change}\label{bc:section} +\label{sec:base-change} + +We retain the finite charts and product presentations of +Proposition~\ref{str:products}. We will prove ordinary coefficient base +change on suitable reductions, simultaneously for every module on each +chart. The transverse modules used later need not be flat over the base, +so the proof will keep track of an ordinary complex and its actual chain +homotopies. + +Fix a normalized source chart and shrink its normal affine base around a +point of the stratum. The chart properties in +Proposition~\ref{loc:maximal-charts} give +\[ + h:Z\longrightarrow\Spec R,\qquad + h_*\cO_Z=\widetilde R,\qquad \dim Z=2n, +\] +where $Z$ is smooth and $h$ is projective and equidimensional of relative +dimension $n$. Fix a nowhere zero volume form +$\sigma_X\in H^0(X,\omega_X)$; its pullback $\sigma_Z$ is a frame of +$\omega_Z$. A sufficiently positive embedding over $R$, followed by a +linear projection whose center misses the chosen fiber, gives after +shrinking a finite map +\begin{equation}\label{bc:finite-projection} + \pi:Z\longrightarrow Y=\PP^n_R. +\end{equation} +Indeed the chosen sections have no common zero on that fiber, hence on +a neighborhood by properness. The resulting map is quasi-finite because +its defining line bundle is relatively ample, and is therefore finite. +Choose also a finite dominant slice +$\Spec E\to\Spec R$ whose total space is smooth, and put +\[ + \mathcal V=\pi_*\cO_Z,\qquad + E_Y=E\otimes_R\cO_Y. +\] +The finite algebras $\mathcal V$ and $E_Y$ have regular total spaces +$Z$ and $\PP^n_E$. Quasi-compactness of the stratum lets us make these +choices on a finite collection of source charts. + +For a module $E$ over a ring $R$, write $\operatorname{add}_R(E)$ for the +modules isomorphic to direct summands of finite direct sums of $E$. Our +first target on one chart is a bounded complex $C$ with terms in +$\operatorname{add}_R(E)$ and, for every $R$-module $M$, natural +identifications +\begin{equation}\label{bc:coefficient-target} + \vartheta_M^i:H^i(C\otimes_R M) + \xrightarrow{\ \sim\ }H^i(Z,h^*\widetilde M). +\end{equation} +The pullback $h^*$ and all tensor products here are ordinary. +These identifications must respect the specified coefficient maps. Namely, +for +\[ + \gamma_M^i(C):H^i(C)\otimes_RM\longrightarrow H^i(C\otimes_RM), + \qquad [z]\otimes m\longmapsto[z\otimes m], +\] +and the usual map $\beta_M^i$ obtained by multiplying a cohomology class +on $Z$ by the pulled-back section $m$, we require +\begin{equation}\label{bc:canonical-compatibility} + \vartheta_M^i\gamma_M^i(C) + =\beta_M^i(\vartheta_R^i\otimes\id_M). +\end{equation} +The remaining target in a good reduction is an actual homotopy equivalence, +locally on $\Spec R$, from $C$ to a complex with zero differential. The +following elementary observation explains why this suffices for all +ordinary modules at once. + +\begin{lemma}[The ordinary homotopy criterion]\label{bc:homotopy-criterion} +Let $C$ be a bounded complex of $R$-modules. If there are chain maps +$C\rightleftarrows D_0$ that are inverse up to chain homotopy and +$d_{D_0}=0$, then $\gamma_M^i(C)$ is an isomorphism for every $R$-module +$M$ and every $i$. If \eqref{bc:coefficient-target} and +\eqref{bc:canonical-compatibility} hold, the canonical map $\beta_M^i$ +is therefore an isomorphism. +\end{lemma} + +\begin{proof} +Tensor the two maps and their chain homotopies with $M$. They remain +homotopy inverses between $C\otimes_RM$ and $D_0\otimes_RM$. The maps +$\gamma_M^i$ are natural in chain maps, and +$\gamma_M^i(D_0)$ is the identity on $D_0^i\otimes_RM$. Naturality +under the two homotopy equivalences proves the assertion for $C$. +This uses no flatness of $M$, of $D_0$, or of the terms of $C$. +\end{proof} + +We next construct the coefficient complex, then turn a generic homotopy +splitting into the local splitting required by this criterion. + +\subsection{Finite smooth covers as coefficient modules} + +The following comparison supplies one additive generator for the finite +regular algebras above. It makes no flatness assertion over their common +normal base. + +\begin{proposition}[Comparison of finite regular algebras]\label{bc:finite-modules} +Let $L$ be a complete noetherian normal local domain, and let $A_1,A_2$ +be finite dominant local $L$-algebras that are regular local rings. +Put $d_i=\rk_L A_i$. +If these rings have positive characteristic and arise by completing local +rings of varieties over a perfect field, then +\begin{equation}\label{bc:regular-comparison} + A_1^{\oplus d_2}\simeq A_2^{\oplus d_1} + \qquad\text{as $L$-modules}. +\end{equation} +The same conclusion holds in characteristic zero when the diagram is the +completion of an algebraic finite diagram at a closed point over $\C$. +Finite products of the indicated local algebras can also be compared, +using their total generic ranks. +\end{proposition} + +\begin{proof} +Suppose first that the characteristic is $p>0$. Write $d=\dim L$ and +$\tau=\operatorname{trdeg}_k\kappa(L)$, where $k$ is the perfect ground +field. All residue fields of the $A_i$ are finite extensions of +$\kappa(L)$, so their $p$-degrees are $p^\tau$. For $q=p^a$, finite flat +Frobenius on a regular local ring, as in Kunz's theorem +\cite{Kunz1969}, gives +\begin{equation}\label{bc:frobenius-rank} + F_*^aA_i\simeq A_i^{\oplus q^{d+\tau}}. +\end{equation} +Here the module structure on the left is through $x\mapsto x^q$. +The formula follows either from a regular system of parameters and a +residue-field $q$-basis, or by taking the degree of Frobenius on the +fraction field. Completion does not change this formula. In particular, +the exponent is the same for both algebras, including at a nonclosed +point. + +The two modules in \eqref{bc:regular-comparison} have the same generic +rank. Clearing denominators in a generic isomorphism and its inverse +gives maps +\[ + u:A_1^{\oplus d_2}\longrightarrow A_2^{\oplus d_1},\qquad + v:A_2^{\oplus d_1}\longrightarrow A_1^{\oplus d_2}, + \qquad vu=r\id,\quad uv=r\id +\] +for one nonzero $r\in L$. We estimate the size of an invertible block of +$F_*^a(r)$ on $F_*^aA_i$, viewed as a free $A_i$-module. If +$\mathfrak n_i$ is the maximal ideal, its cokernel needs +\[ + \dim_{\kappa(A_i)} + \bigl(F_*^a(A_i/rA_i)/\mathfrak n_iF_*^a(A_i/rA_i)\bigr) + =q^\tau\operatorname{length}_{A_i} + A_i/(r,\mathfrak n_i^{[q]}) +\] +generators. Since $\mathfrak n_i$ has $d$ generators, +$\mathfrak n_i^{d(q-1)+1}\subset\mathfrak n_i^{[q]}$. +The Hilbert--Samuel polynomial of $A_i/rA_i$, of dimension at most $d-1$, +therefore bounds this number by $O(q^{d+\tau-1})$. The case $d=0$ is +immediate, since $r$ is a unit. Elementary invertible row and column +operations over $A_i$ now exhibit an identity block of size +$q^{d+\tau}-O(q^{d+\tau-1})$ in $F_*^a(r)$. + +Apply $F_*^a$ to $u,v$. The identity block just found, repeated $d_2$ +times, factors through $F_*^a(A_2^{\oplus d_1})$. Consequently +\[ + A_1^{\oplus d_2(q^{d+\tau}-O(q^{d+\tau-1}))} + \quad\text{is an $L$-module summand of}\quad + A_2^{\oplus d_1q^{d+\tau}}. +\] +Finite modules over the complete local ring $L$ have Krull--Schmidt +decompositions. For each indecomposable isomorphism class $T$, let +$m_i(T)$ be its multiplicity in $A_i$. The displayed summand relation, +divided by $q^{d+\tau}$ and followed by $a\to\infty$, gives +$d_2m_1(T)\le d_1m_2(T)$. Interchanging the indices gives equality. +This proves \eqref{bc:regular-comparison}. + +Here are details of the characteristic-zero passage. Henselize the +algebraic base at the chosen point. The idempotents separating the +finitely many points of a finite algebra descend over an \'{e}tale +neighborhood. Thus individual factors can be treated in a finite +algebraic diagram; after shrinking that diagram they remain regular and +dominant over a normal integral base. Let $M=A_1^{\oplus d_2}$ and +$N=A_2^{\oplus d_1}$ denote the corresponding finite modules before +completion. Choose finite generators $\phi_1,\ldots,\phi_s$ of the +ordinary module $\Hom(M,N)$. Its completion is +$\Hom(\widehat M,\widehat N)$, because $M$ is finitely presented and +completion is flat. + +An isomorphism $\widehat M\simeq\widehat N$ exists precisely when some +linear combination of the $\phi_i$ is surjective modulo the maximal +ideal. For the forward implication, reduce the coefficients of a +completed isomorphism. For the converse, Nakayama gives surjectivity, +and a surjective map of the same generic rank between torsion-free +modules has zero kernel. Existence of the required linear combination +is a finite matrix condition: at least one maximal minor of +$\sum z_i\overline\phi_i$ is a nonzero polynomial in the $z_i$. + +Spread the finite presentations, the marked point, and this Hom +calculation over a finitely generated characteristic-zero domain. +Generic freeness applied to the kernels and cokernels of the finite +presentation matrices makes the Hom calculation and its reduction at +the marked point commute with specialization. Retain normality, +regularity, ranks, and the dimensions of the two residual modules. +If all the maximal-minor polynomials vanished in characteristic zero, +their finitely many coefficients would vanish in every remaining +specialization. At any good positive-characteristic specialization, +however, the first part supplies an isomorphism of the completed +modules. This is a contradiction. A factor that splits further under +specialization causes no difficulty: the first part applies to all its +regular factors and their sum. Finally, comparisons with one fixed +factor and addition prove the assertion for finite products. +\end{proof} + +\begin{remark}\label{bc:residue-degrees} +Lengths in this comparison are always taken over the common target when they +are compared. For a finite local algebra $A/L$ and a finite-length +$A$-module $Q$, +\[ + \operatorname{length}_L Q + =[\kappa(A):\kappa(L)]\operatorname{length}_A Q. +\] +Thus tensoring \eqref{bc:regular-comparison} with any fixed finite-length +$L$-module compares the corresponding lengths with their residue-field +degrees included. Dividing two such comparisons, one for a test module +and one for a fixed reference module of nonzero length, gives the same +ratio on every regular factor. This is the form needed at generic +points of the support in the Euler calculation. +\end{remark} + +Proposition~\ref{bc:finite-modules}, applied to the completed local +factors at closed points, gives +\begin{equation}\label{bc:additive-generator} + \mathcal V\in\operatorname{add}(E_Y) + \quad\text{locally on $Y$}. +\end{equation} +To descend the assertion from completion, consider the composition +pairing +\[ + \mathcal Hom(E_Y,\mathcal V)\otimes + \mathcal Hom(\mathcal V,E_Y)\longrightarrow + \mathcal End(\mathcal V). +\] +Membership of $\id_{\mathcal V}$ in its image is exactly the existence +of a retraction through a finite sum of $E_Y$'s. Flat completion +commutes with these finite Hom modules and detects membership by the +cokernel, so the retraction exists locally before completion. A finite +open cover supplies finitely many retraction witnesses. + +Let +\[ + \mathsf H=\End_R(E),\qquad + \mathsf H_Y=\mathsf H\otimes_R\cO_Y,\qquad + \mathcal P=\mathcal Hom_Y(E_Y,\mathcal V). +\] +The right action on $\mathcal P$ is by precomposition, and $E$ is a left +$\mathsf H$-module by evaluation. Flatness of $Y/R$ identifies +$\mathsf H_Y$ with $\mathcal End_Y(E_Y)$. For a left $\mathsf H$-module +$Q$, set $Q_Y=\cO_Y\otimes_RQ$, with its induced left +$\mathsf H_Y$-action. Our ordinary sheaf tensor notation means +\[ + \mathcal P\otimes_{\mathsf H}Q + :=\mathcal P\otimes_{\mathsf H_Y}Q_Y. +\] +The same convention applies to the other right $\mathsf H_Y$-module +sheaves below. By +\eqref{bc:additive-generator}, $\mathcal P$ is locally a finite +projective right $\mathsf H_Y$-module and evaluation gives +\begin{equation}\label{bc:evaluation} + \mathcal P\otimes_{\mathsf H_Y}E_Y\simeq\mathcal V. +\end{equation} +These assertions follow first for $E_Y$ and then for sums and summands; +no general Morita equivalence or $R$-flatness of $E$ is being asserted. + +\subsection{One twist bound for all coefficient modules} + +\begin{lemma}\label{bc:uniform-twists} +Let $R$ be noetherian, let $\mathsf H$ be an algebra finite over its +central subring $R$, and let $\mathcal P$ on $Y=\PP^n_R$ be locally +finite projective as a right $\mathsf H_Y$-module. There is an integer +$m_0$ such that, for every $m\ge m_0$ and every left $\mathsf H$-module +$Q$, +\begin{align} + H^j(Y,\mathcal P(m)\otimes_{\mathsf H}Q)&=0 + &&(j>0),\label{bc:uniform-vanishing}\\ + H^0(Y,\mathcal P(m))\otimes_{\mathsf H}Q + &\xrightarrow{\ \sim\ } + H^0(Y,\mathcal P(m)\otimes_{\mathsf H}Q). + \label{bc:uniform-sections} +\end{align} +The arrow is the canonical multiplication map, and +$H^0(Y,\mathcal P(m))$ is a finite projective right $\mathsf H$-module. +The bound is independent of $Q$. +\end{lemma} + +\begin{proof} +Set $\mathcal K_0=\mathcal P$. Relative generation on projective space +constructs $n+2$ exact sequences +\begin{equation}\label{bc:finite-resolution} + 0\longrightarrow\mathcal K_{r+1}\longrightarrow\mathcal F_r + \longrightarrow\mathcal K_r\longrightarrow0, + \qquad 0\le r\le n+1, +\end{equation} +where each $\mathcal F_r$ is a finite sum of $\mathsf H_Y(-a_{rj})$. +To obtain such a surjection, choose ordinary coherent generators after +twisting and extend their maps $\mathsf H_Y$-linearly. Since +$\mathcal K_r$ is locally projective over $\mathsf H_Y$, the sequence +locally splits over that algebra. Induction shows that every kernel +$\mathcal K_{r+1}$ is again locally finite projective over +$\mathsf H_Y$. + +The sheaf $\mathsf H_Y$ is flat as a right $\mathsf H$-module: its tensor +functor is $\cO_Y\otimes_R-$, and $Y$ is $R$-flat. Therefore all the +kernels just constructed are $\mathsf H$-flat, and every sequence in +\eqref{bc:finite-resolution} stays exact after tensoring with an +arbitrary $Q$. Choose $m_0\ge a_{rj}$ for all the finitely many twists. +For $m\ge m_0$ each $\mathcal F_r(m)\otimes_{\mathsf H}Q$ is a sum of +$\cO_Y(m-a_{rj})\otimes_RQ$. The coordinate \v{C}ech calculation on +projective space gives zero higher cohomology for these sheaves for +every $R$-module $Q$; their sections are the degree $m-a_{rj}$ +homogeneous polynomials with coefficients in $Q$. + +For $j>0$, the long exact sequences consequently give +\begin{equation}\label{bc:dimension-shift} + H^j(Y,\mathcal K_r(m)\otimes_{\mathsf H}Q) + \simeq + H^{j+n+2-r}(Y,\mathcal K_{n+2}(m)\otimes_{\mathsf H}Q). +\end{equation} +The $n+1$ standard affine opens of $Y$ have affine intersections, so +quasi-coherent cohomology vanishes in degrees greater than $n$. +Thus the left side is zero for $r=0,1,2$ and $j>0$. +In particular this proves \eqref{bc:uniform-vanishing}. We have made +no assertion about the projective dimension or the low-degree +cohomology of the final kernel $\mathcal K_{n+2}$. + +The first two sequences, together with the vanishings for +$\mathcal K_1$ and $\mathcal K_2$, give +\begin{equation}\label{bc:sections-presentation} + H^0(Y,\mathcal P(m)\otimes_{\mathsf H}Q) + =\operatorname{coker}\bigl( + H^0(Y,\mathcal F_1(m))\otimes_{\mathsf H}Q + \longrightarrow + H^0(Y,\mathcal F_0(m))\otimes_{\mathsf H}Q\bigr). +\end{equation} +For $Q=\mathsf H$ this is a finite presentation of +$G_m=H^0(Y,\mathcal P(m))$ by finite free right $\mathsf H$-modules. +Right exactness of tensoring this same presentation proves +\eqref{bc:uniform-sections}, with its canonical map. +Finally, tensoring any short exact sequence of coefficient modules +with $\mathcal P(m)$ remains exact by $\mathsf H$-flatness, and taking +sections remains exact by \eqref{bc:uniform-vanishing}. +Equation~\eqref{bc:uniform-sections} says that $G_m\otimes_{\mathsf H}-$ +is exact. Hence $G_m$ is flat; being finitely presented by +\eqref{bc:sections-presentation}, it is projective. +\end{proof} + +Choose an integer $\nu\ge\max(1,m_0)$. We reserve $\nu$ for the Koszul +exponent; the powers of the stratum ideal keep their notation $I^k$. +The coordinate sections $y_0^\nu,\ldots,y_n^\nu$ have no common zero +on $Y$. Their Koszul +coresolution is the locally split exact sequence +\begin{equation}\label{bc:koszul} + 0\longrightarrow\mathcal P\longrightarrow + \mathcal P(\nu)^{\oplus(n+1)}\longrightarrow\cdots\longrightarrow + \mathcal P((n+1)\nu)\longrightarrow0. +\end{equation} +Local splitting follows by contracting the Koszul complex wherever one +of the coordinate sections is a unit. Let $K_\nu(\mathcal P)$ be the +complex of sections of its positive-twist terms, beginning in degree +zero. Lemma~\ref{bc:uniform-twists} makes it a bounded complex of finite +projective right $\mathsf H$-modules representing +$\mathbf R\Gamma(Y,\mathcal P)$. Set +\[ + C_\nu(\mathcal V)=K_\nu(\mathcal P)\otimes_{\mathsf H}E. +\] +Its terms lie in $\operatorname{add}_R(E)$ and are, explicitly, +$H^0(Y,\mathcal V(j\nu))^{\oplus\binom{n+1}{j}}$ for +$1\le j\le n+1$, in degree $j-1$. +For every $R$-module $M$, ordinary associativity and +\eqref{bc:evaluation} give +\[ + \mathcal P\otimes_{\mathsf H}(E\otimes_RM) + =\mathcal V\otimes_RM. +\] +Apply Lemma~\ref{bc:uniform-twists} with the left $\mathsf H$-module +$Q=E\otimes_RM$. The locally split sequence \eqref{bc:koszul} and the +finite projection formula for $\pi$ give natural identifications +\begin{equation}\label{bc:ordinary-complex} + C_\nu(\mathcal V)\otimes_R M + \simeq\mathbf R\Gamma(Y,\mathcal V\otimes_RM) + =\mathbf R\Gamma(Z,h^*\widetilde M). +\end{equation} +The tensor and $h^*$ in this formula are ordinary operations. No Tor +vanishing for $E$ or $M$ over $R$ has been used. + +Here is the canonical-map content of \eqref{bc:ordinary-complex}. Use +the standard affine cover of $Y$ and its \v{C}ech complex. The +augmentation from $\mathcal V\otimes_RM$ to the positive Koszul complex +gives a quasi-isomorphism of their \v{C}ech total complexes, since +\eqref{bc:koszul} is locally split before coefficients are tensored. +The maps from global sections of the positive terms to their \v{C}ech +complexes are quasi-isomorphisms by \eqref{bc:uniform-vanishing}; +\eqref{bc:uniform-sections} identifies those sections with the terms +of $C_\nu(\mathcal V)\otimes_RM$. This gives the natural zigzag meant +by \eqref{bc:ordinary-complex}. The augmentation, \v{C}ech maps, +section identifications, and finite projection formula are all natural +for maps of coefficient modules. +In particular it is natural for $R\to M$, $1\mapsto m$. On cocycles +that map sends a class to its product with the pulled-back section $m$. +Thus the induced cohomology identifications are precisely +\eqref{bc:coefficient-target} and satisfy +\eqref{bc:canonical-compatibility}. + +The same construction applies to any coherent $\mathcal W$ locally in +$\operatorname{add}(E_Y)$, using +$\mathcal Hom_Y(E_Y,\mathcal W)$ in place of $\mathcal P$ and a +sufficiently large exponent. In the explicit section description, +an $\cO_Y$-linear map of such sheaves induces an actual chain map +$C_\nu(\mathcal W)\to C_\nu(\mathcal W')$ whenever the same $\nu$ is +valid for both. These maps respect composition. + +We also need a stronger comparison between different exponents for one +sheaf. The bounded projective $\mathsf H$-complexes +$K_\nu(\mathcal P)$ represent the same object +$\mathbf R\Gamma(Y,\mathcal P)$. Full faithfulness of bounded +projective complexes in the derived category of right $\mathsf H$-modules +therefore lifts its identity to chain maps that are inverse up to +$\mathsf H$-linear chain homotopies. Tensoring these actual maps and +homotopies with $E$ gives +\begin{equation}\label{bc:power-comparison} + C_\nu(\mathcal V)\simeq C_{\nu'}(\mathcal V) + \quad(\nu,\nu'\text{ sufficiently large}) +\end{equation} +as homotopy equivalences. The maps and homotopies commute with central +multiplication by elements of $R$. We used full faithfulness over +$\mathsf H$, before tensoring with $E$, and then retained the resulting +maps and homotopies over $R$. + +Fix one such complex $C=C_{\nu_0}(\mathcal V)$ in characteristic zero. +Over $\operatorname{Frac}(R)$ it splits as a complex of vector spaces. +Its cohomology embeds as a summand of a finite zero-differential complex +whose terms are sums of $E\otimes_R\operatorname{Frac}(R)$. Clearing +denominators in the two maps and in their homotopy gives a nonzero +$g\in R$, a bounded zero-differential complex $D$ with terms finite +sums of $E$, and actual maps and a homotopy +\begin{equation}\label{bc:cleared-homotopy} + C\xrightarrow{a}D\xrightarrow{b}C, + \qquad ba-g\id_C=dH_0+H_0d. +\end{equation} +For example, clear separate denominators in the two maps, and then a +further common denominator in the scaled homotopy; rescaling one map +and $g$ gives the displayed identity. All its terms are finite +module maps. It is therefore one fixed finite datum that can be +spread out. + +The central factor will remain this same $g$ when the exponent changes. +Indeed, let $f:C\to C_{\nu'}(\mathcal V)$ and +$j:C_{\nu'}(\mathcal V)\to C$ be the homotopy inverses induced by the +$\mathsf H$-comparisons, and write +$fj-\id=dJ+Jd$. Then the actual chain maps $aj$ and $fb$ satisfy +\begin{equation}\label{bc:transported-homotopy} + (fb)(aj)-g\id_{C_{\nu'}(\mathcal V)} + =d(fH_0j+gJ)+(fH_0j+gJ)d. +\end{equation} +This follows by inserting \eqref{bc:cleared-homotopy} and using +$fgj=gfj$. It introduces no additional clearing factor. + +\subsection{The differential identity attached to a clearing factor} + +Choose a finite Noether normalization +$\mu:\Spec R\to P=\mathbb A^n_\C$, with coordinates $u_1,\ldots,u_n$, +and put +\[ + e_\mu=[\operatorname{Frac}(R):\C(u_1,\ldots,u_n)]. +\] +Choose a nonzero $\Delta_\mu\in\C[u_1,\ldots,u_n]$ such that +$R[1/\Delta_\mu]$ is finite \'{e}tale and free of rank $e_\mu$ over +$\C[u_1,\ldots,u_n][1/\Delta_\mu]$, and such that $h$ is smooth over +this locus. These are fixed finite data. In particular the retained +rank will later bound the degrees of completed local branches, including +at points outside this open set. + +Set $a=\mu h$. +The morphism $a$ is projective, with smooth source of dimension $2n$ +and smooth target of dimension $n$. Write +$du=du_1\wedge\cdots\wedge du_n$. On the dense locus where $\mu$ is +\'{e}tale and $h$ is smooth, the volume form defines the relative top form +$\eta=\sigma_Z/du$. Since $h_*\cO_Z=\cO_{\Spec R}$ and +$\omega_Z$ is trivial, the corresponding lowest Hodge sheaf is +$R\eta$, viewed on $P$. + +\begin{lemma}\label{bc:differential-identity} +For the fixed $g\ne0$ in \eqref{bc:cleared-homotopy}, on a finite +principal-open cover of $P$ there are finitely many $t_j\in R$ and +algebraic differential operators $D_j$ with +regular coefficients on the respective open sets such that +\begin{equation}\label{bc:differential-equation} + \eta=\sum_j D_j(g t_j\eta). +\end{equation} +The identity is in the middle relative de Rham module on the generic +smooth locus, but its operators are regular also at the other points +of the chosen open sets. For one integer $o_g$ fixed in characteristic +zero, they have expressions +\begin{equation}\label{bc:operator-bound} + D_j=\sum_{|\alpha|\le o_g}c_{j,\alpha}(u)\partial_u^\alpha, + \qquad + \partial_u^\alpha= + \partial_{u_1}^{\alpha_1}\cdots\partial_{u_n}^{\alpha_n}, +\end{equation} +where each $c_{j,\alpha}$ is regular on its specified open set. +\end{lemma} + +\begin{proof} +Use the constant polarizable Hodge module on $Z$ in right +$\mathcal D_Z$ notation. Its underlying module is $\omega_Z$, with +lowest filtration index $-2n$. The filtered direct-image construction +and projective strictness give +\begin{equation}\label{bc:lowest-filtration} + F_{-2n}\mathcal H^0(a_+\omega_Z)=a_*\omega_Z. +\end{equation} +To track the index, use the transfer Spencer resolution. Its term in +degree $-i$ is filtered by pieces with source filtration lowered by +$i$ and by the nonnegative order of target differential operators. +At total index $-2n$, all $i>0$ terms vanish; in degree zero only +order zero remains. The lowest filtered complex is consequently +$\mathbf Ra_*\omega_Z$. Strictness identifies its degree-zero +cohomology with the left side of \eqref{bc:lowest-filtration}. +These conventions and strictness are the ones in +\cite[Sections~B.2--B.3 and Theorem~1.4]{Saito2016}, the latter being +\cite[Theorem~5.3.1]{Saito1988}. + +The strict-support decomposition is compatible with the filtration. +The sheaf $a_*\omega_Z$ is torsion-free on $P$: multiplication by a +nonzero base function is injective on the dominant integral smooth +source. Thus its summands on proper supports are zero. After changing +from right to left modules by $\omega_P^{-1}$, the sheaf $R\eta$ lies +in the full-support part of $\mathcal H^0(a_+\omega_Z)$, which is +generically the $n$-th relative de Rham cohomology. This part is an +intersection extension of semisimple algebraic local systems, hence a +semisimple regular holonomic $\mathcal D_P$-module with no simple +constituent on a proper support. Here we use strict support +\cite[Section~1.2]{Saito2016} and algebraic semisimplicity +\cite[Proposition~4.2.5 and Theorem~4.2.6]{Deligne1971}; restricting to an arbitrary small +analytic open would not justify this semisimplicity assertion. + +Inside that semisimple module let $U$ and $U_g$ be the +$\mathcal D_P$-submodules generated by $R\eta$ and $gR\eta$. +At the generic point of $P$, the finite domain $R$ becomes the field +$\operatorname{Frac}(R)$, so $g$ is invertible and these restrictions +agree. +The quotient $U/U_g$ is supported properly. Semisimplicity makes it a +sum of constituents of the full-support module, so it is zero. +Consequently $\eta$ belongs locally to $\mathcal D_P(gR\eta)$. +Writing this membership using finitely many generators gives +\eqref{bc:differential-equation}. Quasi-compactness gives a finite +principal-open cover and a finite maximum $o_g$ for the orders. +On an open $D(f)$, a local section of $\mu_*\cO_{\Spec R}$ has the form +$r/f^m$ with $r\in R$. Absorb multiplication by $f^{-m}$ into the +operator acting on $gr\eta$; its coefficients remain regular on $D(f)$ +and its order does not increase. Thus the $t_j$ may be taken in $R$. +Algebraic differential operators on these opens of affine space have +the expressions \eqref{bc:operator-bound}; their finitely many +coefficient denominators are units on the corresponding opens. +The restriction of the identity to the +generic smooth locus involves only algebraic de Rham classes, +connections, and finite-order operations, and thus is finite algebraic +data that can be reduced after localization. +\end{proof} + +\subsection{Finite preparations and selection of the reductions} + +The preceding constructions give finite data in characteristic zero. +We first retain these data on a model. After choosing a good reduction, +we will choose its Koszul exponent and its local Cartier units there. +The coefficient modules themselves are not among the data to be spread. + +Spread the projective morphism, +the finite charts and their groupoid, the smooth source charts, the +volume form, the ideals of the strata, and the finite \'{e}tale product +presentations. Include $h,\pi,E,\mu$ on each member of the finite +chart cover, the coherent identity $h_*\cO_Z=R$, the compatible ambient +direct-image identifications of Proposition~\ref{str:direct-images}, +the finite Hom and evaluation calculations defining +$\mathsf H,\mathcal P$, the finite local retraction witnesses for +\eqref{bc:additive-generator}, the $n+2$ sequences in +\eqref{bc:finite-resolution} with local splitting witnesses, one exponent +$\nu_0$ and its complex $C_{\nu_0}$, and the maps and homotopy in +\eqref{bc:cleared-homotopy}. Include also the finite identities +\eqref{bc:differential-equation}, the expressions +\eqref{bc:operator-bound} with their denominators on the specified open +sets, and the finite operator cover itself. More explicitly, if this +cover is $P=\bigcup_\lambda D(f_\lambda)$, retain an identity +$\sum_\lambda r_\lambda f_\lambda=1$ in $\C[u_1,\ldots,u_n]$. +Its reduction keeps these principal opens covering every fiber. +Include the element $\Delta_\mu$ with the finite free \'{e}tale +rank $e_\mu$. Localizing a finitely generated integral model makes the kernels, +images, and cokernels used in these presentations flat over the model +and retains their stated base-change maps. Retain smoothness of the +source charts and slices, normality and geometric integrality of the +chart bases, dimensions, equidimensionality, and the \'{e}tale and finite +assertions on the specified neighborhoods. Retain also geometric +integrality of the smooth projective total space and smoothness of $h$ +over $D(\Delta_\mu)$. Invert the stabilizer +orders and the finitely many coefficient denominators. On every retained +fiber, $\Delta_\mu$ remains nonzero, and the algebra on +$D(\Delta_\mu)$ remains finite \'{e}tale and free of rank $e_\mu$. + +Retaining the complex $C_{\nu_0}$ includes retaining its coefficient +meaning. The spread local splitting witnesses make the same $n+2$ +sequences \eqref{bc:finite-resolution} locally split over the fiber's +$\mathsf H_Y$, with the same twists $a_{rj}$. The proof of +Lemma~\ref{bc:uniform-twists} therefore gives the old bound $m_0$ on +every retained fiber, for every coefficient module there. In +particular $\nu_0\ge m_0$ is still valid. The first two sequences give +the same finite cokernel presentation \eqref{bc:sections-presentation} +for each $H^0(Y,\mathcal P(j\nu_0))$ after specialization. Tensoring +that presentation with the residue field identifies the specialized +section module with the section module of the fiber. Include the +finitely many coordinate multiplication maps as well. Thus the +specialization of $C_{\nu_0}$ is the fiber complex +$K_{\nu_0}(\mathcal P)\otimes_{\mathsf H}E$, with +\eqref{bc:ordinary-complex} and its canonical-map compatibility. + +For the smooth projective total space retain the nowhere zero volume +form and the coherent conditions +\begin{equation}\label{bc:coherent-cup} + h^2(\cO_X)=h^{2n}(\cO_X)=1,\qquad + H^2(\cO_X)^{\otimes n}\longrightarrow H^{2n}(\cO_X) + \text{ is nonzero}. +\end{equation} +These are finite cohomological conditions, supplied originally by the +IHS Hodge decomposition. Retain also the relevant coherent and de Rham +base-change identifications. + +This preparation remains available after any fixed finite enlargement +of the algebraic and coherent data on a finite-type model, followed by +restriction to any nonempty principal open on which the required +identities and geometric properties remain valid. Thus this localization +inverts only finitely many elements. Make those choices before selecting a +characteristic-zero +closed point of the retained model over $\Q$. Its residue field is a +number field. Pull the data to its ring of integers with finitely many +elements inverted, so only finitely many further places are excluded by +the model. +The lemma below supplies arbitrarily large remaining Cartier reductions +of this number-field model. In particular the finite enlargement and +localization occur before that arithmetic choice. We use only the +retained algebraic and cohomological properties of the chosen +characteristic-zero fiber. + +\begin{lemma}\label{bc:cartier-primes} +There are arbitrarily large remaining residue characteristics, at +degree-one places of a finite extension of the number field, for which +\[ + \operatorname{Car}(F_*\sigma_X)\ne0. +\] +On the geometric special fiber, the map +\begin{equation}\label{bc:cartier-splitting} + \psi_X:F_*\cO_X\longrightarrow\cO_X, + \qquad \psi_X(z)=\operatorname{Car}(F_*(z\sigma_X))/\sigma_X +\end{equation} +satisfies $\psi_X(1)=\delta\in k^*$. +\end{lemma} + +\begin{proof} +Put $b=b_2(X)$ and choose a prime $\ell>2b$. After a finite extension, +both the residual representation on a stable lattice of +$H^2_{\mathrm{et}}(X_{\overline K},\Q_\ell)$ modulo $\ell$ and +$\mu_\ell$ are constant. At a degree-one good place of norm $p$, +geometric Frobenius has an integral trace $a_p$ satisfying +\begin{equation}\label{bc:trace-congruence} + |a_p|\le bp,\qquad a_p\equiv b\pmod\ell, + \qquad p\equiv1\pmod\ell. +\end{equation} +We use integrality and the weight-two Weil bound, and the equality of +the \'{e}tale and crystalline characteristic polynomials +\cite[Theorem~1]{KatzMessing1974}, using its formulation and the +crystalline applicability stated in +\cite[Chapter~VI, Theorem~3.9 and Section~4]{MilneTannakian2025}. + +Discard the finitely many ramified places of the number field and +the places where relative de Rham cohomology or its base change has +torsion. At a degree-one remaining place the local base is $\Z_p$. +Crystalline--de Rham comparison for the smooth proper lift identifies +the reduction of the crystalline trace with the trace of Frobenius on +$H^2_{\mathrm{dR}}(X_{\F_p})$ +\cite[Corollaries~3.8 and~3.10]{BhattDeJong2011}. +Modulo $p$, Frobenius on the de Rham complex factors as its projection +onto $\cO_X$ followed by the map taking functions to their $p$-th +powers in degree zero. On cohomology write this factorization as +$\beta\alpha$ through $H^2(\cO_X)$. The reverse composite +$\alpha\beta$ is coherent Frobenius. The identity +$\tr(\beta\alpha)=\tr(\alpha\beta)$ shows that zero coherent +Frobenius forces $p\mid a_p$. + +In that case $a_p/p$ is an integer in $[-b,b]$ congruent to $b$ modulo +$\ell$. Since $\ell>2b$, it equals $b$, so $a_p=bp$. +If coherent Frobenius vanished at all but finitely many degree-one +good places, the semisimplification of +$H^2_{\mathrm{et}}(X_{\overline K},\Q_\ell)$ would therefore have the +same character as $\Q_\ell(-1)^{\oplus b}$. Indeed degree-one places +have density one among primes of a number field; the others have norm +at least the square of the underlying rational prime and density zero. +Chebotarev for number fields and the characteristic-zero character +criterion \cite[Theorem~3.2 and Section~5.1.1.3]{Serre2014} give the +assertion: finite-quotient Frobenius density and continuity identify +the two trace functions in the projective limit. + +This contradicts Hodge--Tate comparison at a place above $\ell$: +$h^2(\cO_X)=1$ contributes a weight-zero summand, whereas all Tate +summands just found have weight one, in the convention where +$\Q_\ell(-1)$ has weight one. Comparison applies to the smooth +proper variety over that local field +\cite[Theorem~1.6 and Corollary~1.8]{Scholze2013}; de Rham representations are stable +under subquotients and their filtered dimensions are additive in +exact sequences \cite[Theorem~5.2.1(2) and Propositions~6.3.2--6.3.3]{BrinonConrad2009}, +so passing to semisimplification has not lost this obstruction. Thus coherent Frobenius is nonzero at infinitely many, +and hence arbitrarily large, good residue characteristics. + +By \eqref{bc:coherent-cup}, its $n$-fold product is nonzero on top +coherent cohomology. Serre duality identifies the transpose of this +map with Cartier trace on global top forms. The space of global top +forms is the frame line and $X$ is geometrically integral and proper, +so \eqref{bc:cartier-splitting} sends $1$ to a nonzero constant. +\end{proof} + +All subsequent positive-characteristic assertions are made in these +geometric special fibers. Choose $p$ larger than every retained +Noether-normalization degree $e_\mu$ and every differential order bound +$o_g$ on the finite chart cover. \'{E}tale compatibility of Cartier +trace gives the same constant $\delta$ on the source charts. Thus +$\psi_Z:F_*\cO_Z\to\cO_Z$ pushes down, using $h_*\cO_Z=R$, to +\[ + \psi_h:F_*R\longrightarrow R, + \qquad\psi_h(1)=\delta. +\] + +\subsection{Cartier detects every cleared factor} + +\begin{lemma}\label{bc:clearing-test} +For every closed point $b\in\Spec R$ in a retained chart, there is a +function $t$ regular near $b$ such that +\begin{equation}\label{bc:unit-test} + \psi_h(gt)(b)\ne0. +\end{equation} +\end{lemma} + +\begin{proof} +Fix $b$ and one of the principal opens in +Lemma~\ref{bc:differential-identity} containing $\mu(b)$. In this proof +$t_j,D_j$ denote its data, so the $t_j$ are regular at $b$ and the +coefficients of the $D_j$ are regular at $\mu(b)$. + +First work over $K_0=\operatorname{Frac}(R)$, where the family +$Z_0=Z\times_RK_0$ is smooth and proper. The retained \'{e}tale locus +of $\mu$ identifies $du$ with a basis of $\omega_{K_0/k}$ and gives +the relative frame $\eta=\sigma_Z/du$. Serre duality gives dual lines +\[ + V_0=H^n(Z_0,\cO_{Z_0})=K_0v,\qquad + W_0=H^0(Z_0,\omega_{Z_0/K_0})=K_0\eta,\qquad + \langle v,\eta\rangle=1. +\] +Put +\[ + Z_0^{(p)}=Z_0\times_{\Spec K_0,F_{K_0}}\Spec K_0,\qquad + V_0^{(p)}=V_0\otimes_{K_0,F_{K_0}}K_0,\qquad + W_0^{(p)}=W_0\otimes_{K_0,F_{K_0}}K_0. +\] +Flat field base change identifies these twisted vector spaces with +$H^n(Z_0^{(p)},\cO_{Z_0^{(p)}})$ and +$H^0(Z_0^{(p)},\omega_{Z_0^{(p)}/K_0})$, respectively. +Write $v^{(p)}=v\otimes1$ and $\eta^{(p)}=\eta\otimes1$. +They remain dual under the base-changed Serre pairing. +Relative Frobenius $F_{\mathrm{rel}}:Z_0\to Z_0^{(p)}$ and its +finite dual trace have the following $K_0$-linear maps on these lines: +\begin{equation}\label{bc:relative-trace-lines} + \begin{aligned} + F_{\mathrm{rel}}^*:V_0^{(p)}&\longrightarrow V_0, + &v^{(p)}&\longmapsto s\,v,\\ + \operatorname{Tr}_{F_{\mathrm{rel}}}:W_0&\longrightarrow W_0^{(p)}, + &\eta&\longmapsto s\,\eta^{(p)}. + \end{aligned} +\end{equation} +Here the second map is induced by +$F_{\mathrm{rel}*}\omega_{Z_0/K_0}\to\omega_{Z_0^{(p)}/K_0}$. +It is the Serre-dual transpose of the first, which proves that their +scalars are the same $s\in K_0$. + +Taking $p$-th powers of a \v{C}ech cocycle for $v$ gives a class +$\lambda$ in middle relative de Rham cohomology. Its projection to +$H^n(Z_0,\cO_{Z_0})$ is $F_{\mathrm{rel}}^*v^{(p)}$. Pairing +$\lambda$ with the top form $\eta$, with the order that gives Serre +evaluation, therefore gives +$\langle\lambda,\eta\rangle=s$. This distinguishes the coherent +Frobenius map in \eqref{bc:relative-trace-lines} from its de Rham lift. +The lift is horizontal: its $p$-power representatives are closed even +for absolute differentials over $k$, so the connecting operation defining +the Gauss--Manin derivative annihilates them. The de Rham pairing is +compatible with the connection. Pairing the reduced identity +\eqref{bc:differential-equation} with this horizontal class gives +\begin{equation}\label{bc:scalar-differential} + s=\sum_jD_j(gt_js). +\end{equation} +The pairing, connection, and \eqref{bc:differential-equation} on the +smooth generic family are reductions of their finite characteristic-zero +descriptions. The scalar identity \eqref{bc:scalar-differential} is +obtained in this fiber from the newly constructed horizontal class. + +Factor absolute Frobenius as +$Z_0\xrightarrow{F_{\mathrm{rel}}}Z_0^{(p)} + \xrightarrow{w}Z_0$, where $w$ is the base change of $F_{K_0}$. +This is the relative Frobenius factorization of +\cite[Tag 0CC6]{StacksProject}. +The relative trace in \eqref{bc:relative-trace-lines} sends +$m\eta$ to $sm\eta^{(p)}$. The trace for $w$, by flat base change and +the projection formula on top forms, applies the Cartier map +$\operatorname{Car}_{K_0}:F_*\omega_{K_0/k}\to\omega_{K_0/k}$ +to its remaining base form. Transitivity of finite trace therefore +gives the rational identity below; compatibility of trace with this +flat base change is \cite[Tag 0E5L]{StacksProject}. +\begin{equation}\label{bc:relative-cartier} + \psi_h(m)\,du + =\operatorname{Car}_{K_0}\bigl(F_*(s m\,du)\bigr) + \qquad(m\in K_0). +\end{equation} +We suppress $F_*$ in the input notation for $\psi_h$. The scalar +$s$ stays inside the base Cartier operator. At this point it is only +a rational element of $K_0$. + +Suppose \eqref{bc:unit-test} fails, so +$\psi_h(gR_b)\subset\mathfrak m_{R_b}$. Complete the regular base at +$\mu(b)$: +\[ + A=k[[v_1,\ldots,v_n]],\qquad v_i=u_i-u_i(\mu(b)),\qquad + K=\operatorname{Frac}(A),\qquad \mathfrak m=(v_1,\ldots,v_n). +\] +The finite completed algebra $R\otimes_{k[u]}A$ is a product of the +completed local rings at the points above $\mu(b)$. They are normal +complete local domains; keep the factor $L$ at $b$ and put +$K'=\operatorname{Frac}(L)$. After tensoring with $K$, the whole +product is finite \'{e}tale of total dimension $e_\mu$: the nonzero +$\Delta_\mu$ is invertible in $K$, and its free \'{e}tale rank was +retained. Thus its factors are finite separable fields and +\begin{equation}\label{bc:completed-degree} + [K':K]\le e_\mue_\mu,\quad p>o_g + \quad\text{on every member of the finite chart cover}. +\] +On each such reduction the following assertions hold simultaneously. +On every retained source chart, for every +$R$-module $M$ and every $i$, the canonical map +\begin{equation}\label{bc:canonical-base-change} + (R^ih_*\cO_Z)\otimes_RM\longrightarrow + R^ih_*(h^*\widetilde M) +\end{equation} +is an isomorphism, with ordinary tensor and pullback. In particular, +for every coherent module $M$ supported on any finite thickening of +$\cS$ in $\widehat{\mathcal B}$, +\begin{equation}\label{bc:ordinary-euler} + \chi(X,f^*M)=\sum_{i=0}^n(-1)^i + \chi(\widehat{\mathcal B},N_i\otimes M). +\end{equation} +One reduction works for all these $M$ and all thickening orders. +\end{proposition} + +\begin{proof} +The preceding argument makes $C_\nu(\mathcal V)$ homotopy equivalent +to a zero-differential complex near every closed point. Its +coefficient identifications satisfy +\eqref{bc:coefficient-target} and +\eqref{bc:canonical-compatibility}. Lemma~\ref{bc:homotopy-criterion} +therefore gives the canonical map \eqref{bc:canonical-base-change} for +every module on each such neighborhood. Since the base is affine, the +cohomology modules correspond to the quasi-coherent direct-image sheaves; +the map is multiplication by the pulled-back coefficient. + +The local homotopy splitting extends to a neighborhood of each closed +point: its maps involve finite modules, the local function $t$, and +the inverse of the unit $u$. These neighborhoods are chosen before +any coefficient module $M$. They cover the finite-type chart, since any nonempty +closed complement would have a closed point. Consequently the local +statement holds as a sheaf statement for every module. The maps are +compatible with the \'{e}tale overlaps and deck actions because they are +canonical multiplication maps. Proposition~\ref{str:products} +therefore descends them to every supported thickening. The ordinary +supported pullback on $X$ is exactly the one constructed there. +Taking the Euler characteristic of Leray and using the spread +chartwise direct-image isomorphisms of +Proposition~\ref{str:direct-images} gives \eqref{bc:ordinary-euler}. +Here $N_i\otimes M$ has the ordinary bounded-support meaning fixed in +Section~\ref{sec:stratum}; all supports are proper. + +The only choices required before reduction were the finite diagrams, +resolutions, one cleared homotopy per chart, and bounded differential +identities. The proof of Lemma~\ref{bc:uniform-twists} works for all +coefficient modules in each retained fiber, with the old bound for +$C_{\nu_0}$ retained by the split sequences. The exponent $\nu$, the +comparison at $p\nu$, and the local functions $t$ were constructed anew +in that fiber. No spreading of infinitely many Frobenius identities +or of the individual modules $M$ occurs. A finite chart cover and +Lemma~\ref{bc:cartier-primes} give the asserted simultaneous choice of +reductions. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/06-euler.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/06-euler.tex new file mode 100644 index 0000000000000000000000000000000000000000..8eafee3b609acdea0416152f636a00495c4438de --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/06-euler.tex @@ -0,0 +1,1045 @@ +\section{The Euler characteristic of a minimal stratum} +\label{eu:section}\label{sec:euler} + +We retain the closed stratum $S_0$, its smooth stack $\cS$, and its +formal neighbourhood $\widehat{\mathcal B}$ from +Propositions~\ref{str:minimum} and~\ref{str:products}. Thus +\[ + d=\dim\cS=\dim S_0,\qquad c=n-d>0, +\] +and $I$ is the reduced ideal of $\cS$. Stabilizers always mean the +\emph{full} stabilizers of these charts. In particular, they need not act +effectively on $\cS$. + +\begin{proposition}\label{eu:euler-zero} +If a maximal chart of $B$ is singular, then the closed minimal stratum +satisfies +\[ + \chi_{\mathrm{top}}(S_0)=0. +\] +Here $\chi_{\mathrm{top}}$ is the ordinary Euler characteristic of the coarse +space. +\end{proposition} + +We first globalize the exterior decomposition and choose a full +endomorphism quotient on one finite thickening. In characteristic zero, +a comparison at one common support point then makes a reference +Chern--Todd number vanish. After these finite data have been retained in +one good reduction, ordinary base change bounds Frobenius growth and +the reference comparison makes every product test have Euler +characteristic zero. Three ordinary dual modules finally recover the +trivial representation of every full stabilizer. + +\subsection{Exterior multiplicities and the full truncated quotient} + +Use the ambient formal systems $N_i=(N_{i,k})_{k\geq1}$ of +Section~\ref{sec:stratum}: on a chart $V_\alpha$, +\[ + N_{i,k}=\Omega_{V_\alpha}^{[i]}/I_\alpha^k\Omega_{V_\alpha}^{[i]}. +\] +They are restrictions of reflexive sheaves on the normal ambient charts, +not reflexive differentials formed on the nilpotent thickenings. Put +\[ + N=\bigoplus_{i=0}^nN_i,\qquad + \cE=\widehat{\mathcal H}_{NN}, +\] +where $\widehat{\mathcal H}_{NN}$ is the completion of ambient Hom +before quotienting by $I^k$. Flat completion and finite presentation +identify it locally with Hom between the completed ambient modules. +For the finite tensor diagram used here, write +$\overline{\Hom}(M,N)$ for the quotient of +$\widehat{\mathcal H}_{MN}|_{\cS}$ by the kernel of +\[ + \widehat{\mathcal H}_{MN}|_{\cS}\longrightarrow + (\widehat{\mathcal H}_{NM}|_{\cS})^\vee,\qquad + a\longmapsto\bigl(b\longmapsto\tr(ba)\bigr). +\] +Set +\[ + \overline{\cE}=\overline{\End}(N),\qquad + \cJ=\ker(\cE\longrightarrow\overline{\cE}). +\] +The fixed transverse product descriptions first make these barred Hom +sheaves vector bundles on $\cS$ and identify their fibers with the local +residual trace quotients. In particular the fibers of +$\overline{\cE}$ are semisimple algebras, and +\begin{equation}\label{eu:radical-power} + I\cE\subset\cJ,\qquad \cJ^\ell\subset I\cE +\end{equation} +for one $\ell$ after taking a finite cover of $\cS$. +For a coefficient $M$ supported at order $k$, the finite action uses +\[ + \im(\cJ\longrightarrow\cE/I^k\cE) + =\cJ/(\cJ\cap I^k\cE), +\] +followed by its image in $\End(N\otimes M)$. The filtrations below +use these action images. +The inclusions in~\eqref{eu:radical-power} follow from +Lemma~\ref{loc:trace-radical}, applied to the +transverse modules, and from +the compatible product presentations and uniform transverse descriptions +of Proposition~\ref{str:products} and Lemma~\ref{str:uniform}. + +\begin{lemma}\label{eu:exterior} +In the barred tensor calculus, the trivial-isotypic multiplicity bundle +of $V=N_1$ is canonically +\[ + \mathcal U=\overline{\Hom}(\cO,V)=\Omega^1_{\cS}. +\] +Let $W$ be the complementary object to this evaluation summand. There +are idempotents $p_b\in\overline{\cE}$, supported in the $N_b$ block, +whose images are $\Lambda^bW$. If $L$ is any left +$\overline{\cE}$-module, then +\begin{equation}\label{eu:block-exterior} + \mathbf1_{N_i}L\simeq + \bigoplus_{a+b=i}\Lambda^a\Omega^1_{\cS}\otimes p_bL. +\end{equation} +Moreover, there are an integer $j_0$ and an isotypic idempotent $u$ in +the $N_{j_0}$ block such that +\begin{equation}\label{eu:isolated-constituent} + u\overline{\cE}p_b= + \begin{cases} + \cO_{\cS}u,&b=j_0,\\ + 0,&b\ne j_0. + \end{cases} +\end{equation} +The identity line in this formula has trivial action under every +stabilizer. +\end{lemma} + +\begin{proof} +Ambient vector fields are tangent to the orbit and restrict onto its +tangent bundle. Their restriction kernel is the trace kernel: a vector +field vanishing at an orbit point pairs to zero there with every +reflexive one-form, whereas differentials of ordinary functions detect +every nonzero tangent vector. The perfect barred pairing between $V$ +and $V^\vee$ therefore identifies the two multiplicity bundles with +$\Omega^1_{\cS}$ and $T_{\cS}$. Evaluation and contraction give mutually +inverse inclusion and projection maps for the trivial-isotypic part. +They are intrinsic, so agree on overlaps. + +Exterior multiplication applied to $V=\mathcal U\otimes\mathbf1\oplus W$ +in this quotient category gives +$\Lambda^iV=\bigoplus_{a+b=i}\Lambda^a\mathcal U\otimes\Lambda^bW$. +Writing this identity in local dual frames gives matrices of barred +Hom maps and their inverses. Applying these matrices to a left module +proves~\eqref{eu:block-exterior}; an unbarred algebra splitting is not +needed. + +At each point, Corollary~\ref{loc:exterior-constituent} supplies a +simple constituent of $\Lambda^\bullet W$ which occurs once and in +exactly one degree $j_0$. The fixed transverse descriptions transport +this fiberwise selection, and its degree is locally constant, hence +constant on the connected $\cS$. In $\Lambda^{j_0}V$ its multiplicity +is still one: a positive exterior degree of $\mathcal U$ would require +its occurrence in a smaller degree of $W$. Its isotypic projector is +$u$, and Schur's lemma gives \eqref{eu:isolated-constituent} fiberwise. + +The selection in the corollary is intrinsic to the independent standard +factors and is unchanged by their permutations. The centers of the +semisimple block algebras are finite \'etale over $\cS$; in a product +presentation their +primitive components are constant. The invariant selection therefore +defines an algebraic idempotent on the whole component. Finally every +automorphism acts on an endomorphism algebra by conjugation and fixes +the identity of the selected simple object. This proves the assertion +about the character of its identity line, including for ineffective +stabilizers. +\end{proof} + +For a finite order $k$, ambient action gives a map +\[ + \cE/I^k\cE\longrightarrow + \End_{\cS_k}\!\left(\bigoplus_iN_{i,k}\right). +\] +This map need be neither injective nor surjective; only its image is +the ambient subalgebra of the full endomorphism algebra formed after +quotienting. The later Hom cokernel ranges over all maps of transverse +modules, including maps without an ambient lift. We therefore need +the following quotient of the full algebra, while retaining the ambient +image for the filtration-preserving projector used later. + +\begin{lemma}\label{eu:truncated-end} +For one sufficiently large integer $k$, set +\[ + \cS_k=(\widehat{\mathcal B},\cO/I^k),\quad + T_i=N_i/I^kN_i,\quad T=\bigoplus_iT_i. +\] +There is a canonical surjection +\begin{equation}\label{eu:full-end-quotient} + \End_{\cS_k}(T)\longrightarrow\overline{\cE} +\end{equation} +with nilpotent kernel, compatible with the action of $\cE$. +After specializing any smooth product parameters, the image of ambient +endomorphisms in the specialized $\End(T)$ still surjects onto the +specialized barred algebra, with nilpotent kernel. +\end{lemma} + +\begin{proof} +First let $(R,\mathfrak m)$ be a complete transverse local ring and $M$ +the corresponding finite direct sum of transverse modules. Put +\[ + E=\End_R(M),\qquad E_j=\End_R(M/\mathfrak m^jM). +\] +For every fixed $j$, the images of $E_l\to E_j$, $l\ge j$, form a +descending sequence of subspaces of a finite-dimensional vector space. +Write $D_j$ for its stable value. The maps $D_{j+1}\to D_j$ are +surjective. Indeed, for $a\in D_j$, the images in $E_{j+1}$ of its +lifts to all sufficiently large $E_l$ are nonempty descending affine +subspaces; they stabilize and give a lift in $D_{j+1}$. Recursively +choose compatible lifts. Completeness of $M$ identifies +$\varprojlim_jE_j$ with $E$, so $D_1$ is exactly the image of $E\to E_1$. +This is an all-level argument; stabilization at level one alone would +not justify the lifting assertion. + +Choose $k$ for which $\im(E_k\to E_1)=D_1$. A map in the kernel lowers +$M/\mathfrak m^kM$ into its $\mathfrak m$-multiple, so the $k$-th power +of this kernel is zero. Also $\ker(E\to D_1)\subset\rad(E)$: if $a$ +lowers $M$ into $\mathfrak mM$, then $1-ba$ is invertible for every +$b\in E$, by the convergent $\mathfrak m$-adic geometric series. +The trace-radical calculation consequently identifies +\[ + D_1/\rad(D_1)=E/\rad(E). +\] +The composite $E_k\to D_1\to D_1/\rad(D_1)$ is onto and has nilpotent +kernel. Indeed its kernel modulo the lowering kernel is the radical +of a finite-dimensional algebra and is nilpotent. Notice that no +claim has been made that every element of $E_k$ lifts to $E$; only its +residual action must lift. + +On a product presentation, finite quotients and their Hom modules are +these fixed transverse objects extended by the smooth parameter ring. +Flat extension commutes with their finite presentations and kernels. +Thus the chosen exponent works throughout that presentation. Take +the maximum of the exponents for a finite cover of $\cS$. Assertions +about the quotient, its kernel, and a fixed nilpotence exponent descend +from faithfully flat completions. + +To check canonicity, let $\mathcal A_{\mathrm{res}}$ be the common +image of the full and ambient endomorphism algebras in $\End(T/IT)$. +The original trace quotient $\cE\to\overline{\cE}$ kills the kernel +of $\cE\to\mathcal A_{\mathrm{res}}$, by the transverse calculation, +and hence factors uniquely through $\mathcal A_{\mathrm{res}}$. +Composing with $\End(T)\to\mathcal A_{\mathrm{res}}$ defines +\eqref{eu:full-end-quotient}, including on maps without ambient lifts. +Its kernel is the transverse nilpotent ideal extended along the smooth +parameters; it need not be the ordinary Jacobson radical of the full algebra +over the parameter ring. The quotient is locally a product of matrix +algebras over a reduced smooth base and has no nonzero nilpotent +two-sided ideal. Thus this kernel is intrinsically the largest +nilpotent two-sided ideal. Transition isomorphisms preserve it and +the factorization through the residual-action image, so the map glues +on the stack independently of the product coordinates. + +After setting the smooth parameters to zero, let $C$ be the image of +ambient endomorphisms in the full finite algebra $\End(T_o)$. Product +structure identifies $C$ with the transverse ambient image. It maps +onto the barred algebra because the original $E$ does. Its kernel is +its intersection with the nilpotent kernel of +\eqref{eu:full-end-quotient}. Changing the smooth parameter lifts while +keeping the transverse coordinates is a formal automorphism preserving +$I$; differential tensors transform functorially under it. This +conjugates the calculation and does not change any exponent or create +an additional lifting obstruction. +\end{proof} + +\subsection{A characteristic-zero reference number} + +For a coherent sheaf $F$ on a smooth $2n$-fold $X$ supported in +codimension at least $c$, write $[F]_c=\ch_c(F)$. This is its +codimension-$c$ support cycle, with the ordinary scheme lengths as +multiplicities; it is zero if the support has larger codimension. +This is the leading-term property of the Riemann--Roch transformation +\cite[Chapter~III, \S\S1--2]{BaumFultonMacPherson1975}. Over $\C$ we +also use $[F]_c$ for the resulting cohomology class with supports. +We now construct the reference number in characteristic zero. + +\begin{lemma}\label{eu:reference-number} +Let $F_0=f^*\cO_{\cS}$, using ordinary pullback and the descended +closed-subscheme structure on the source. Then +\begin{equation}\label{eu:reference-vanishing} + \int_X [F_0]_c\,\td_{2n-c}(X)=0. +\end{equation} +After restricting a finite coherent model to a nonempty principal +open, the same equality holds for the corresponding reference sheaf +on every geometric fiber. +\end{lemma} + +\begin{proof} +We first prove, in complex cohomology with supports, that for one +positive rational number $\rho$ +\begin{equation}\label{eu:cycle-comparison} + [F_0]_c=\rho\,f^*[S_0]. +\end{equation} +The supported class $[S_0]$ exists because $B$ is an oriented rational +homology manifold by Proposition~\ref{loc:rational-smoothness}. +Equidimensionality implies that every component of $f^{-1}(S_0)$ of +dimension $2n-c=n+d$ dominates $S_0$: one whose image has smaller +dimension has dimension at most $n+d-1$. + +Choose \emph{one} maximal chart and \emph{one} lifted component $S'$ +of its stratum. For each top-dimensional component of $f^{-1}(S_0)$, +restrict it to the chosen base neighborhood and choose a component of +its inverse image under the source torsor of +Proposition~\ref{str:products}. Its base image is a stratum lift. +The deck group is transitive on such lifts over a general coarse point, +so translating the chosen source component makes its base image $S'$. +This translation leaves its image on $X$ unchanged. Thus one chosen +lift $S'$ sees every top component, without requiring transitivity on +components of the fibers of $f$. Use the finite projection of +Section~\ref{bc:section} on this source chart, +\[ + \pi:Z\longrightarrow Y=\PP^n_V. +\] +Each chosen lifted component has dimension $n+d$ and is finite over +$S'\times\PP^n$. It therefore dominates that entire irreducible +support. We can now choose \emph{one common general closed point} +$y\in S'\times\PP^n$ at which all its reduced preimage support +branches are smooth and \'etale over the support. In analytic germs +each of these branches maps isomorphically to the support germ. + +Let $R=\widehat{\cO}_{Y,y}$ and let +$A_\alpha=\widehat{\cO}_{Z,z_\alpha}$ run through the regular local +factors above this same point. Write $r_\alpha$ for their generic ranks +over $R$, and $J\subset R$ for the support ideal. The quotient $R/J$ +is regular. If $m_\alpha$ is the scheme multiplicity of the ordinary +reference pullback on this branch, the preceding degree-one assertion +gives +\[ + m_\alpha=\rk_{R/J}(A_\alpha/JA_\alpha). +\] +Proposition~\ref{bc:finite-modules}, in characteristic zero and over +this single target ring, gives +\[ + A_\alpha^{\oplus r_\beta}\simeq + A_\beta^{\oplus r_\alpha}. +\] +Quotienting by $J$ and taking ranks yields +\begin{equation}\label{eu:common-charzero-ratio} + r_\beta m_\alpha=r_\alpha m_\beta. +\end{equation} +Thus $m_\alpha/r_\alpha$ is one common scalar. + +We compare the cohomological multiplicities by the finite local-degree +rule. For a finite local branch, the projection formula with supports, +capped with the complex orientation, multiplies the target support +class by the generic degree of the branch after pushforward. Dividing +by the degree of its reduced support map gives the coefficient of the +pulled class. Applied first to the chosen base-chart branch, this +gives $\lambda[S']$ as the pullback of $[S_0]$, with one positive +factor $\lambda$ on $S'$. The intervening \'etale base refinement +does not change that factor. Applied next to a branch of $\pi$, whose +reduced support map has degree one at the common point, the same rule +gives coefficient $r_\alpha$. Hence the coefficient of $f^*[S_0]$ +on that branch is $\lambda r_\alpha$. +Equation~\eqref{eu:common-charzero-ratio} makes the scheme-to-cohomology +ratio $m_\alpha/(\lambda r_\alpha)$ independent of the branch. +The source charts are \'etale, so both kinds of multiplicities descend +unchanged. Top support classes are determined at these generic points; +this proves~\eqref{eu:cycle-comparison} globally. + +By Propositions~\ref{loc:rational-smoothness} and~\ref{prop:background}, +ordinary $H^{2c}(B,\Q)$ is one-dimensional, and $[S_0]$ is a multiple +of $c_1(H)^c$. Its pullback +is therefore a multiple of $e^c$. For every integer $t$, the IHS +Riemann--Roch polynomial gives +\[ + \chi(X,f^*H^{\otimes t})=\operatorname{RR}_X(q(te)) + =\operatorname{RR}_X(0), +\] +since $q(e)=0$; see \cite[\S1.11]{Huybrechts1999}. +The coefficient of $t^c$ in ordinary Hirzebruch--Riemann--Roch is +$\frac1{c!}\int_Xe^c\td_{2n-c}(X)$. It vanishes for $c>0$. +Together with~\eqref{eu:cycle-comparison} this proves +\eqref{eu:reference-vanishing}. + +For completeness, specialization of this number does not use rational +smoothness of a characteristic-$p$ base. The ordinary coherent sheaf +$F_0$ on the smooth projective $X$ has a finite locally free resolution. +Include its finitely many terms, maps, kernels, the descended quotient +ideal defining $F_0$, and the identification with the ordinary pullback +in a finite coherent model. Generic flatness and coherent base change +on a nonempty principal open preserve the resolution of that quotient +on fibers and support codimension at least $c$. Thus its restriction +represents the actual reference pullback on every retained fiber. Its +Chern character below codimension $c$ vanishes and its component in +codimension $c$ is its actual support +cycle. The degree +\[ + \deg\bigl(\ch_c(F_{0,s})\td_{2n-c}(T_{X_s})\bigr) +\] +is constant: it is the degree of a fixed relative Chern polynomial in +the vector bundles of that resolution and the relative tangent bundle +on a smooth proper family. This gives the last assertion. The +argument neither keeps track of individual cohomology classes on the +reduced base nor requires a constant number of support components. +\end{proof} + +The finite characteristic-zero data needed for the Euler argument are +now defined. We first record a finite witness for the ambient +inclusion~\eqref{eu:radical-power}, which will be used at every +thickening order. On a chart put +\[ + H_\alpha=\mathcal H_{\alpha,NN},\qquad + J_{\alpha,\mathrm{alg}} + =\ker\bigl(H_\alpha\longrightarrow i_{\alpha*} + \overline{\cE}_\alpha\bigr), +\] +where $i_\alpha:S_\alpha\hookrightarrow V_\alpha$. +Exact noetherian completion identifies the completion of this coherent +ambient kernel with $\cJ$. The target modulo $I_\alpha$ is supported +on $S_\alpha$, so faithful flatness of completion at its stalks gives +the finite coherent identities +\[ + I_\alpha H_\alpha\subset J_{\alpha,\mathrm{alg}},\qquad + J_{\alpha,\mathrm{alg}}^{\otimes\ell} + \longrightarrow H_\alpha/I_\alpha H_\alpha + \quad\text{is zero}, +\] +where the last map is multiplication. Retaining these identities and +their kernels makes completion give~\eqref{eu:radical-power} on every +retained fiber at every order. + +Fix the presentations of these ambient tensor and trace maps, the +exterior idempotents, the full algebra $\End_{\cS_k}(T)$, its quotient +\eqref{eu:full-end-quotient}, and the kernels, surjections and +nilpotence identities for that full quotient at the chosen thickness +$k$. +Fix also the ordinary reference pullback, its finite locally free +resolution and relative Chern--Todd degree from +Lemma~\ref{eu:reference-number}, and coherent cohomology for the +finitely many exterior bundles of $\Omega^1_{\cS}$. + +Enlarge the common finite algebraic and coherent model of +Sections~\ref{sec:stratum}--\ref{sec:base-change} by these data. Restrict +it to a nonempty principal open on which the defining identities, +geometric hypotheses, +kernels, images, vector-bundle descriptions and coherent base-change +maps are retained. In particular finite presentation and the retained +Hom base-change maps identify the specialization of +$\End_{\cS_k}(T)$ with the \emph{full} endomorphism algebra of the +specialized $T$, including its nonlifting maps. Proposition~\ref{bc:ordinary} +then permits a characteristic-zero closed point of this retained model +and arbitrarily large good Cartier reductions of its number-field model. +Choose one such geometric reduction, of characteristic $p$, and work +over its algebraically closed residue field until the final transfer +back to $\C$. This one reduction supplies ordinary base change for +every coefficient module and every finite thickening order. The +stabilizer congruences below will be imposed on a power of this fixed +$p$; they impose no further arithmetic condition on the prime. + +\subsection{A tame Euler functional} + +The following fact works over an algebraically closed field of +characteristic prime to the stabilizer orders. Its Euler characteristic +is coherent cohomology of the stack, not integration weighted by +stabilizers. For a vector bundle $V$, we use +$\lambda_{-1}(V)=\sum_a(-1)^a[\Lambda^aV]$ in $K^0$. +For a vector bundle $G$ on a possibly disconnected stack, $\rk(G)$ +denotes the largest of its ranks on the connected components. On the +empty stack we set this rank to zero; its Euler functional is zero. + +\begin{lemma}\label{eu:tame-functional} +Let $\mathcal D$ be a smooth proper tame Deligne--Mumford stack of pure +dimension $d$ with projective coarse space. The functional +\[ + \Theta(G)=\chi\bigl(\mathcal D, + \lambda_{-1}(\Omega^1_{\mathcal D})\otimes G\bigr) +\] +on vector bundles is additive, and its additive extension to +$K^0(\mathcal D)$ depends only on pointwise full-stabilizer +representations. There is a constant $C_{\mathcal D}$ such that +$|\Theta(G)|\le C_{\mathcal D}\rk(G)$ for every vector bundle $G$. +Consequently, for any finite signed presentation +$V=\sum_i a_i[G_i]$, one has +\[ + |\Theta(V)|\le C_{\mathcal D}\sum_i|a_i|\rk(G_i). +\] +The latter bound uses the chosen presentation, not the virtual rank. +\end{lemma} + +\begin{proof} +There are finitely many connected components. The functional and its +representation dependence are componentwise; if the estimate holds +on them with constants $C_j$, summing gives the stated bound with +$C_{\mathcal D}=\sum_j C_j$ under our rank convention. We may +therefore assume $\mathcal D$ connected. + +Let $A$ be the pullback of an ample line bundle from the coarse space +and set +\[ + \Theta_m(G)=\sum_{a=0}^d(-1)^a + \chi\bigl(\mathcal D,\Lambda^a\Omega^1_{\mathcal D} + \otimes A^{-ma}\otimes G\bigr). +\] +The coarse pushforward is exact by tameness +\cite[Definition~3.1 and Theorem~3.2]{AbramovichOlssonVistoli2008}. The usual Euler polynomial +on its projective coarse space shows that $\Theta_m(G)$ is a polynomial +in $m$ of degree at most $d$. + +For $m$ sufficiently large, global sections of +$T_{\mathcal D}\otimes A^m$ evaluate onto the invariant tangent vectors +at every residual gerbe. To see the uniform assertion, push this +bundle to the coarse space, use an ample twist to generate that coherent +sheaf, and use exactness of invariants to restrict to a residual gerbe. +At a point $x$ with stabilizer $J_x$, the possible values are exactly +$(T_x\mathcal D)^{J_x}$. The locus where this dimension is at most $r$ +has dimension at most $r$. On a finite quotient chart, its stabilizer +strata lie in fixed loci of subgroups; these are smooth for tame actions +and their tangent spaces are the invariant tangent spaces. There are +finitely many relevant subgroup strata on a finite atlas. + +For a stratum of dimension $s$ with invariant tangent dimension $r$, +the incidence of sections vanishing at a point has codimension $r$ in +the space of sections over that point. Since $s\le r$, a general +section has a zero scheme $Z_m$ of dimension at most zero. At every +zero its $d$ components generate an ideal of height $d$ in a regular +local ring. They are a regular sequence, and their Koszul complex +gives +\[ + \Theta_m(G)=\chi(Z_m,G|_{Z_m}). +\] +This remains valid when $Z_m$ is empty. + +The right side is a finite sum of invariant dimensions on zero-dimensional +quotient charts. Filtering their Artin rings by powers of the maximal +ideal expresses each summand as an integral linear functional of the +fiber representation $G_x$. Its absolute value is bounded by a fixed +constant times $\dim G_x$. In particular, it depends only on these +fiber representations, not on the transition functions of $G$. +Choose $d+1$ distinct sufficiently large integers $m$. Polynomial +interpolation expresses $\Theta_0$ as a fixed rational linear +combination of the corresponding $\Theta_m$. This proves both +assertions for bundles, with one constant independent of $G$. +Additivity extends the representation dependence to virtual classes, +and the triangle inequality gives the displayed presentation bound. +The construction uses the full finite groups at zeroes and applies +without an effectiveness assumption. +\end{proof} + +Apply the lemma to $\cS$ in the fixed reduction. Call a coherent +module $M$ supported on a finite thickening a \emph{product module} if, +in completed product coordinates and after forgetting the group action, +it is a finite-length transverse module extended by the smooth parameter +ring. The parameters may depend on the chart and the test. + +For such a coefficient module, define $F^{a*}M$ by ordinary tensor +along Frobenius on the ambient charts and use its natural compatibility +with the groupoid maps to descend it. It is then regarded as a module +on a finite thickening annihilating it. In completed coordinates +$B_{\mathrm{loc}}=R_{\mathrm{tr}}[[t_1,\ldots,t_d]]$, ordinary +associativity gives +\[ + F_{B_{\mathrm{loc}}}^{a*} + (B_{\mathrm{loc}}\otimes_{R_{\mathrm{tr}}}M_{\mathrm{tr}}) + \simeq + B_{\mathrm{loc}}\otimes_{R_{\mathrm{tr}}} + F_{R_{\mathrm{tr}}}^{a*}M_{\mathrm{tr}}. +\] +Thus Frobenius pullback preserves product form without any flatness +assumption on Frobenius of the transverse ring. + +For a product module $M$, filter $N\otimes M$ by +$\cJ^s(N\otimes M)$ and denote the direct sum of its successive quotients +by $L(M)$. This filtration is finite by~\eqref{eu:radical-power}; each +quotient is killed by $I$, has a $\overline{\cE}$-action, and is a +vector bundle on $\cS$. The last assertion is the transverse product +calculation. Define +\begin{equation}\label{eu:gamma} + \gamma(M)=\sum_{b=0}^n(-1)^b[p_bL(M)]\in K^0(\cS). +\end{equation} +Ordinary base change, additivity along the filtration, and +\eqref{eu:block-exterior} give +\begin{equation}\label{eu:euler-test} + \chi(X,f^*M)= + \chi\bigl(\cS,\lambda_{-1}(\Omega^1_{\cS})\otimes\gamma(M)\bigr). +\end{equation} +Every tensor product in this formula is ordinary. In particular no +flatness in the transverse directions has been assumed. + +\begin{lemma}\label{eu:growth} +For a fixed product module $M$ on a fixed good reduction, putting +$q=p^a$ gives +\[ + \chi(X,F_X^{a*}f^*M)=O(q^c). +\] +The implicit constant may depend on the reduction and on $M$. +\end{lemma} + +\begin{proof} +If $I^bM=0$, choose generators $x_1,\ldots,x_v$ for the maximal ideal +of the transverse ring. Its Frobenius pullback is killed by +$(\mathfrak m^b)^{[q]}$, which contains +$(x_1^{bq},\ldots,x_v^{bq})$ and hence $\mathfrak m^{vbq}$. +A fixed number of generators for $M$ also generates its ordinary +Frobenius pullback. Hilbert--Samuel growth in transverse dimension $c$ +therefore bounds its length by $O(q^c)$. Tensoring with each of the +fixed $N_i$ changes this by only a fixed generator bound. The total +rank of the graded pieces of $L(F^{a*}M)$, and hence the sum of the +ranks in its expression~\eqref{eu:gamma}, is $O(q^c)$ on a finite atlas. + +The annihilating thickness just obtained grows with $q$. The +all-modules, all-orders conclusion of Proposition~\ref{bc:ordinary} +therefore matters here: it applies to each new coefficient +$F^{a*}M$ in this same reduction. The Frobenius square commutes with +ordinary pullback on the ambient charts and with their \'etale +descent, so $f^*F^{a*}M=F_X^{a*}f^*M$. Applying +\eqref{eu:euler-test} and Lemma~\ref{eu:tame-functional} proves the +bound with the original test and prime fixed. +\end{proof} + +\subsection{Support transfer and vanishing of product tests} + +\begin{lemma}\label{eu:test-number} +On a fixed good reduction, every product module $M$ satisfies +\begin{equation}\label{eu:test-leading-zero} + \int_X\ch_c(f^*M)\td_{2n-c}(X)=0. +\end{equation} +Consequently +\begin{equation}\label{eu:all-tests-zero} + \chi(X,f^*M)= + \chi\bigl(\cS,\lambda_{-1}(\Omega^1_{\cS})\otimes\gamma(M)\bigr)=0. +\end{equation} +\end{lemma} + +\begin{proof} +If $M$ has support of dimension less than $d$, its ordinary pullback +has codimension greater than $c$ by equidimensionality, and its +codimension-$c$ character is zero. Otherwise repeat the common-chart +and lifted-component choice from Lemma~\ref{eu:reference-number} in +this fixed reduction, using the retained labelled source torsor. +Translating a lift of each top component to the same $S'$ again makes +every component visible there. Now take the generic point $\eta$ of +$S'\times\PP^n$. Complete its target local ring and call it $R$; +its residue field is $K=\kappa(\eta)$. Let $A_\alpha$ be all the +regular complete local factors over it. Write +\[ + r_\alpha=\rk_RA_\alpha,\qquad + h_\alpha=[\kappa(A_\alpha):K],\qquad + m_\alpha(Q)=\operatorname{length}_{A_\alpha} + (A_\alpha\otimes_RQ) +\] +for a finite-length $R$-module $Q$. Here $Q$ is the module induced by +$M$; the reference module is $K$. The finite-regular-module identity +and length over $R$ give, for every pair of factors, +\begin{equation}\label{eu:residue-lengths} + r_\beta h_\alpha m_\alpha(Q) + =r_\alpha h_\beta m_\beta(Q),\qquad + r_\beta h_\alpha m_\alpha(K) + =r_\alpha h_\beta m_\beta(K). +\end{equation} +The second lengths are positive. Division cancels both generic ranks +and residue degrees and shows +\[ + \frac{m_\alpha(Q)}{m_\alpha(K)} + =\frac{m_\beta(Q)}{m_\beta(K)}. +\] +Proposition~\ref{bc:finite-modules} applies at these nonclosed points +with their residue-field Frobenius degrees; no perfection of $K$ is +being assumed. All top components are visible over this one generic +point, because every chosen lift dominates the support under the +finite projection. If factors split in reduction, include every new +factor in~\eqref{eu:residue-lengths}. Descent through the \'etale +source charts shows that +\[ + [f^*M]_c=\rho_M[F_0]_c +\] +as support cycles, for a scalar $\rho_M$ depending possibly on $M$ and +on the good prime. Lemma~\ref{eu:reference-number} proves +\eqref{eu:test-leading-zero}. + +It remains to explain why this one coefficient suffices. Frobenius +is flat on the smooth $X$, and Chern characters of ordinary Frobenius +pullbacks satisfy $\ch_i(F_X^{a*}F)=q^i\ch_i(F)$; one can check this +on a finite vector-bundle resolution by the splitting principle. +Hirzebruch--Riemann--Roch, applied additively to that finite resolution +\cite[\S0.1]{BaumFultonMacPherson1975}, therefore gives the polynomial +\begin{equation}\label{eu:rr-polynomial} + \chi(X,F_X^{a*}f^*M)= + \sum_{i=c}^{2n}q^i + \int_X\ch_i(f^*M)\td_{2n-i}(X),\qquad q=p^a. +\end{equation} +The growth bound of Lemma~\ref{eu:growth} kills its coefficients for +$i>c$. Equation~\eqref{eu:test-leading-zero}, proved independently, +kills the coefficient for $i=c$. Thus the polynomial is identically +zero; evaluating at $q=1$ and using~\eqref{eu:euler-test} proves +\eqref{eu:all-tests-zero}. The scalar $\rho_M$ is fixed before the +Frobenius exponent varies. Uniformity of that scalar in $M$, in the +prime, or in Frobenius pullbacks is unnecessary. +\end{proof} + +\subsection{Three ordinary tests and cancellation of stabilizer weights} + +We finish by constructing three product modules whose signed coefficient +in~\eqref{eu:gamma} has the same fiber representation as the trivial +line, up to a single sign. The reason for three modules is the following +transverse identity. Let $A$ be a transverse Artin ring at a geometric +point, with its full stabilizer action, and let $Y\to F$ be an equivariant +map of finite $A$-modules with actual kernel $Z$. For any equivariant +finite $A$-module $Q$, +left exactness of Hom gives, in the stabilizer representation ring, +\begin{equation}\label{eu:hom-cokernel-template} +\begin{aligned} + &[\Hom_A(Q,F)]-[\Hom_A(Q,Y)]+[\Hom_A(Q,Z)]\\ + &\hspace{2em}=\bigl[\operatorname{coker}\bigl( + \Hom_A(Q,Y)\longrightarrow\Hom_A(Q,F)\bigr)\bigr]. +\end{aligned} +\end{equation} +Here Hom carries its full stabilizer action, before taking invariants. +Neither surjectivity of $Y\to F$ nor projectivity of $Q$ is required. +We will choose the evaluation image so that this cokernel is the +selected barred corner. Ordinary duality will turn the three Hom +terms into the three coefficients in~\eqref{eu:gamma}; the construction +must therefore preserve the actual kernel under specialization. + +Let $\mathcal A_i$ be the inverse image of zero under the map +\begin{equation}\label{eu:right-ideal} + \Hom_{\cS_k}(T_i,T_{j_0})\longrightarrow + u\overline{\cE}\mathbf1_{N_i},\qquad a\longmapsto u\bar a. +\end{equation} +The sum of these blocks is a right ideal under precomposition in the +full endomorphism algebra. By Lemma~\ref{eu:truncated-end}, this is a +coherent ideal defined on $\cS_k$, without choosing a global lift of +$u$. + +\begin{lemma}[Stabilizers and parameters]\label{str:stabilizer-parameters} +The restriction of the ordinary ambient K\"ahler cotangent sheaf to +$\cS$ is a vector bundle faithful on every stabilizer. At a point with +stabilizer $H$, one can choose smooth parameters on the completed +stratum and lift them to the ambient local ring so that their span, +and therefore their generated ideal, is $H$-stable. These assertions +remain true for the spread diagrams in characteristics prime to all +the chart-group orders. +\end{lemma} + +\begin{proof} +In a product chart the restricted ordinary cotangent is +$\Omega^1_{S_U}\oplus +(\mathfrak m_{t_0}/\mathfrak m_{t_0}^2)\otimes\cO_{S_U}$, hence is +locally free. This is the ordinary cotangent, which records all +embedding directions, not its reflexive hull. A finite-order +automorphism of a complete local ring that is trivial on its +cotangent space and has order invertible in the residue field is +trivial: it is the identity on the associated graded, and at the first +nonzero order of its difference from the identity, iteration multiplies +that difference by the order, a contradiction. Equivalently, average +lifts of a cotangent basis and apply complete Nakayama. A deck +stabilizer acts faithfully on the ambient germ, so its cotangent +representation is faithful. + +The surjection from the ambient maximal ideal to the cotangent space +of the smooth stratum admits an $H$-equivariant linear section by +averaging. Its finite-dimensional image gives the desired lifted +functions. Their restrictions are coordinates by the formal inverse +function theorem. They can be chosen as algebraic local functions +before completion, since a finite group average of such functions is +again algebraic locally. Lemma~\ref{str:uniform} permits this change +of parameters. The arguments use only finite group averaging and +invertibility of the parameter Jacobian, and hence apply in the +indicated reductions as well. +\end{proof} + +\begin{lemma}[Rectification of product generators]\label{str:rectification} +Let $K$ be a field, let $A$ be an Artin local $K$-algebra with residue +field $K$, and put $C=A[[t_1,\ldots,t_d]]$. Fix finite $A$-modules +$E_i,F$ and submodules +$Q_i\subset\Hom_A(E_i,F)$. Given surjections +$C^{r_i}\to Q_i\otimes_A C$, form their evaluation map +\[ + \Phi:\bigoplus_i(E_i\otimes_A C)\otimes_C C^{r_i} + \longrightarrow F\otimes_A C. +\] +An automorphism of each $C^{r_i}$ makes $\Phi$ the scalar extension of +a fixed map of finite $A$-modules, with possible zero columns. Its +kernel, image and cokernel are consequently extensions of finite +$A$-modules, are flat over $K[[t_1,\ldots,t_d]]$, and specialize +exactly on setting the parameters to zero. +\end{lemma} + +\begin{proof} +Choose a minimal $A$-generating list $q_1,\ldots,q_m$ of $Q_i$. +Lift these constant elements through the given surjection. Their +residue classes in $C^{r_i}/\mathfrak m_CC^{r_i}$ are independent, +since their images form a basis of $Q_i/\mathfrak m_AQ_i$. Complete +them to a basis of the free module. The first $m$ columns are then +exactly the $q_j$; subtract suitable combinations of these columns +from the others to make the latter zero. These invertible column +operations on $C^{r_i}$ induce invertible operations on +$(E_i\otimes_A C)\otimes_C C^{r_i}$ even if $E_i$ is not $A$-flat. +Evaluation respects the same column operations, so the resulting map +has constant coefficients in the asserted sense. Flatness of +$A\to A[[t]]$ preserves its kernel, image and cokernel. A finite +$A$-module extended in this way is finite free over $K[[t]]$ as a +$K[[t]]$-module, which proves exact specialization. +\end{proof} + +When a stabilizer acts on the original objects in +Lemma~\ref{str:rectification}, the change of basis need not be +equivariant. It proves product form and exact specialization only. +The original maps and their kernels keep their given actions; their +characters are computed after specialization by an invariant parameter +ideal from Lemma~\ref{str:stabilizer-parameters}. + +When $d=0$, the parameter list is empty, the rectification is already +transverse, and the cotangent determinant used below is the trivial line. + +Choose once and for all a power $q_0=p^b$ such that +\begin{equation}\label{eu:frobenius-choice} + q_0\ge k,\qquad q_0\equiv1\pmod m + \quad\hbox{for every stabilizer order }m. +\end{equation} +There are finitely many orders and each is prime to $p$. Taking $b$ +to be a sufficiently large common multiple of the orders of $p$ in +the corresponding unit groups gives this choice. The same $q_0$ will +serve both the generating bundle and the dual modules; it is distinct +from the unbounded powers $q=p^a$ in Lemma~\ref{eu:growth}. + +\begin{lemma}\label{eu:three-modules} +There exist vector bundles $P_i$ on $\cS_k$ with surjections +$P_i\twoheadrightarrow\mathcal A_i$. For their evaluation map +\begin{equation}\label{eu:evaluation} + Y_0=\bigoplus_iT_i\otimes P_i + \xrightarrow{\ \Phi\ }T_{j_0},\qquad Z_0=\ker\Phi, +\end{equation} +the source, target, kernel, image and cokernel are product modules. +The kernel sequence remains exact after setting the smooth product +parameters to zero, including when those parameters have been chosen +equivariantly for a point stabilizer. +\end{lemma} + +\begin{proof} +By Lemma~\ref{str:stabilizer-parameters}, the ordinary ambient +cotangent restricted to $\cS$ is a vector bundle faithful on every +full stabilizer. Since $q_0\ge k$, ambient Frobenius factors through +a map $\Phi_{q_0}:\cS_k\to\cS$. Pulling back that bundle gives a +bundle on $\cS_k$ whose stabilizer representations remain faithful: +raising roots of unity of order prime to $p$ to the $q_0$-th power +is a permutation. A finite sum of +bounded tensor powers contains every irreducible representation of +each stabilizer. One elementary justification is that matrix +coefficients of a faithful representation separate elements of a +finite group, hence generate its function algebra; their products are +matrix coefficients of tensor powers. There is one bound for a +finite quotient atlas and its finitely many subgroup types. Dual +powers may equally be included. + +Let $G$ be such a generating bundle. If $\mathcal A$ is any coherent +sheaf on $\cS_k$, exact coarse pushforward and sufficiently ample coarse +twists generate the invariant Hom sheaf from $G$ to $\mathcal A$. +The evaluation maps then surject on every residual gerbe, since $G$ +contains all its simple types. Nakayama makes them surjective on +stalks. Taking suitable finite sums of twists of $G$ gives the $P_i$. +This is the generating-sheaf criterion of +\cite[Definition~5.1 and Theorem~5.2]{OlssonStarr2003}, here realized +by the faithful ambient cotangent bundle. + +For the product assertion, complete at a point and forget the action +temporarily. Put +\[ + A=R_{\mathrm{tr}}/\mathfrak m_{\mathrm{tr}}^k, + \qquad D=K[[t_1,\ldots,t_d]],\qquad C=A\otimes_KD. +\] +Every $T_i$ is $M_i\otimes_KD$ for a finite $A$-module $M_i$. +In these coordinates $u$ is a constant primitive central idempotent +in the $N_{j_0}$ block: its center is the fixed transverse center +tensored with the local ring $D$, and its idempotents are constant. +Thus the finite-dimensional transverse Hom calculation and +\eqref{eu:full-end-quotient} give +$\mathcal A_i=A_i\otimes_KD$, where +$A_i\subset\Hom_A(M_i,M_{j_0})$ is the kernel of the transverse +map $a\mapsto u\bar a$ in \eqref{eu:right-ideal}. +Trivialize $P_i$, so its surjection is $C^{r_i}\to A_i\otimes_KD$. +Apply Lemma~\ref{str:rectification} with $E_i=M_i$, $F=M_{j_0}$ +and $Q_i=A_i$. It makes the evaluation a fixed transverse map +extended by $D$ and proves that its kernel, image and cokernel are +product modules, flat over $D$, whose sequences specialize exactly +modulo $(t_1,\ldots,t_d)$. + +These basis changes establish formal product form only; they need not +be equivariant. The actual bundles, maps and kernel in +\eqref{eu:evaluation} keep their original equivariant structures. +For character calculations, choose equivariant lifts of the smooth +stratum parameters by tame averaging, as in +Lemma~\ref{str:stabilizer-parameters}. Their ideal is stable under the +stabilizer. The formal automorphism fixing the transverse ring in +Lemma~\ref{str:uniform} transports the tensor, full Hom, trace and +ideal data to these new product coordinates. Reapplying +Lemma~\ref{str:rectification} there proves flat specialization for +this stable ideal. Completion detects the +coherent local properties claimed here; it does not assert an +algebraic equivariant product splitting. +\end{proof} + +The ring of each completed chart of $\cS_k$ is an Artin ring extended +by $d$ smooth parameters, so $\cS_k$ is Cohen--Macaulay. We now +construct the global dualizing sheaf used in the tests. +The fixed Frobenius factor $\Phi_{q_0}:\cS_k\to\cS$ is representable +and finite. On a tame quotient chart Frobenius induces an isomorphism +on each finite stabilizer, which proves representability; finiteness +can then be checked on the charts. In completed product coordinates +$A[[t_1,\ldots,t_d]]$, with $A$ the transverse Artin algebra, its +structure map from the smooth parameter ring uses $t_i^{q_0}$. +This makes the source finite free over that ring. Define +$\omega_{\cS_k}$ by the $(\Phi_{q_0})_*\cO_{\cS_k}$-module +\[ + (\Phi_{q_0})_*\omega_{\cS_k} + =\Hom_{\cO_{\cS}} + ((\Phi_{q_0})_*\cO_{\cS_k},\omega_{\cS}), + \qquad (a\lambda)(b)=\lambda(ab). +\] +Finite Hom duality and the trace for Frobenius of a smooth parameter +disk identify this locally with $A^*$ tensored with the parameter +volume line, where $A^*=\Hom_K(A,K)$ over the residue field. +Hom for a finite locally free algebra commutes with \'etale base change; +the smooth trace is compatible with the same base change. Hence these +local identifications glue with their actual stabilizer actions. +This gives the dualizing sheaf and its product description. It need +not be invertible: the transverse Artin algebra need not be Gorenstein. +The pullback $\Phi_{q_0}^*\omega_{\cS}$ is an invertible line bundle. +For $T'=T_{j_0},Y_0,Z_0$ define the +\emph{ordinary} dual module +\begin{equation}\label{eu:dual-test} + M_{T'}=\Hom_{\cS_k}(T',\omega_{\cS_k}) + \otimes(\Phi_{q_0}^*\omega_{\cS})^{-1}. +\end{equation} +No derived dual is meant. Finite duality in the product coordinates +shows that these are product modules, so +Lemma~\ref{eu:test-number} applies to all three. + +\begin{lemma}\label{eu:character-extraction} +At each geometric point of $\cS$, in the representation ring of its +full stabilizer, +\begin{equation}\label{eu:trivial-fiber} + \gamma(M_{T_{j_0}})-\gamma(M_{Y_0})+\gamma(M_{Z_0}) + =(-1)^{j_0}[\mathbf1]. +\end{equation} +\end{lemma} + +\begin{proof} +Let the stabilizer be $J$, choose the equivariant smooth parameters +from Lemma~\ref{eu:three-modules}, and denote specialization modulo +them by a subscript $o$. Let $A$ be the resulting transverse Artin +ring over the algebraically closed residue field $K$, and let +$\delta=\det(\Omega^1_{\cS,o})$ be the one-dimensional cotangent +determinant representation of $J$. + +The completed finite map to the smooth parameter disk computes the +canonical module $\omega_C$ as finite Hom to the disk's canonical +module \cite[Tag 0AX0]{StacksProject}. The ring is finite free over +the parameter ring, so this dualizing complex is concentrated in its +canonical degree. The chosen parameter span is stabilizer-stable, +with determinant $\delta$, and the parameters form a regular sequence +on $C=A[[t_1,\ldots,t_d]]$. Equivariant Koszul adjunction gives +\[ + \bigl(\omega_C/(t_1,\ldots,t_d)\omega_C\bigr) + \otimes\delta^{-1} + \simeq\omega_A\simeq A^*, + \qquad A^*=\Hom_K(A,K). +\] +Thus the specialized dualizing module is $A^*\otimes_K\delta$ with its +actual stabilizer action. Product form justifies specialization of +Hom, and finite duality gives +$\Hom_A(T'_o,A^*)\simeq(T'_o)^*$ equivariantly, where +$(-)^*=\Hom_K(-,K)$. The other line in~\eqref{eu:dual-test} has +character $\delta^{-q_0}$. Consequently +\begin{equation}\label{eu:det-cancellation} + (M_{T'})_o\simeq(T'_o)^*\otimes\delta^{1-q_0} + \simeq(T'_o)^*. +\end{equation} +The last isomorphism is an assertion about the stabilizer +representation: \eqref{eu:frobenius-choice} makes +$\delta^{1-q_0}$ trivial. It does not assert that the corresponding +global line bundle is trivial. + +For each $j$, consider the image $C_j$ of the ambient $N_j$ +endomorphisms in $\End_A(T_{j,o})$. By +Lemma~\ref{eu:truncated-end}, it surjects onto the barred corner with +nilpotent kernel. This is a $J$-equivariant surjection. Exactness of +tame invariants makes $C_j^J$ surject onto the invariant barred +corner, whose idempotent $p_j$ we want to lift. The invariant kernel +is still nilpotent, so idempotent lifting gives +\[ + p'_j\in C_j^J,\qquad \overline{p'_j}=p_j. +\] +An explicit lifting proof starts with a lift $a$ and corrects +$a^2-a$ successively modulo powers of the nilpotent ideal; $2a-1$ +is invertible modulo that ideal because its square is $1$ there. +The corrections take place inside $C_j^J$. A global unbarred +projector is not asserted. + +Being in the \emph{ambient image} has an additional consequence. +Choose an ambient endomorphism inducing $p'_j$ after specialization. +It preserves every $\cJ^s(N\otimes M_{T'})$ because $\cJ$ is a +two-sided ideal. On the specialized associated graded its action is +$p_{j,o}$, since the residual specialization of this representative is +$p_{j,o}$; no assertion about its residual action before specialization +is needed. Product form makes the filtration specialize exactly. +The idempotent $p'_j$ therefore splits the specialized filtered module +with graded image $p_{j,o}L(M_{T'})_o$. Thus in the representation ring, +\begin{equation}\label{eu:fiber-hom} + \bigl[(p_jL(M_{T'}))_o\bigr] + =\bigl[(p'_jT_{j,o})\otimes_A(T'_o)^*\bigr] + =\bigl[\Hom_A(p'_jT_{j,o},T'_o)^*\bigr]. +\end{equation} +The second equality is ordinary finite-dimensional tensor--Hom +duality: dualizing its left-hand tensor gives +$\Hom_A(p'_jT_{j,o},T'_o)$. + +Put $Q_j=p'_jT_{j,o}$. Lemma~\ref{eu:three-modules} identifies +$Z_{0,o}$ with the actual kernel of the specialized evaluation +$\Phi_o:Y_{0,o}\to T_{j_0,o}$. Apply +\eqref{eu:hom-cokernel-template} with $Q=Q_j$ and this specialized +evaluation. We compute the image +of its Hom map explicitly. Each evaluation column belongs to +the right ideal~\eqref{eu:right-ideal}, and precomposing it by any +map from $Q_j$ remains in that ideal. Conversely, a right-ideal map +$Q_j\to T_{j_0,o}$ extends by zero on +$(1-p'_j)T_{j,o}$ to an element of $\mathcal A_{j,o}$. +Express that element in the generating columns of $P_{j,o}$ and +restrict to $Q_j$. These expressions lift it through $Y_{0,o}$. +This verifies equality of the image with the indicated ideal, +using all $A$-linear maps and with their genuine $J$-actions. + +The full quotient~\eqref{eu:full-end-quotient} now identifies that +cokernel with $u\overline{\cE}_op_j$. Surjectivity on this corner +follows by lifting any barred map to a full truncated map and +precomposing with $p'_j$; its kernel is exactly the inverse-image +ideal just described. Dualizing the signed identity and using +\eqref{eu:fiber-hom} leaves +$[(u\overline{\cE}_op_j)^*]$. By +\eqref{eu:isolated-constituent} this is zero unless $j=j_0$, when +it is the invariant identity line. Summing with the signs $(-1)^j$ +in~\eqref{eu:gamma} proves~\eqref{eu:trivial-fiber}. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{eu:euler-zero}] +Equation~\eqref{eu:all-tests-zero} vanishes for the three modules +\eqref{eu:dual-test}. Take their signed sum. By +Lemma~\ref{eu:character-extraction} its coefficient has at every +point the representation $(-1)^{j_0}\mathbf1$. +Lemma~\ref{eu:tame-functional} therefore gives +\begin{equation}\label{eu:cotangent-euler-zero} + \chi\bigl(\cS, + \lambda_{-1}(\Omega^1_{\cS})\bigr)=0. +\end{equation} +This coherent Euler characteristic is retained from characteristic +zero in the chosen model. Indeed the finitely many exterior bundles, +proper coarse pushforward and coherent cohomology can be spread with +base change; tame coarse pushforward is exact. Thus +\eqref{eu:cotangent-euler-zero} holds for the original complex stack. + +Finally on smooth finite quotient charts the holomorphic de Rham +complex resolves the constant sheaf, and taking invariants is exact +in characteristic zero. Hypercohomology Euler characteristics, +and proper GAGA \cite[Theorem~7.1]{PortaYu2016}, identify +\[ + \sum_{a,b}(-1)^{a+b}\dim H^b(\cS,\Omega^a_{\cS}) + =\sum_i(-1)^i\dim H^i(S_0,\C). +\] +The rational or complex cohomology of a finite stabilizer has no +positive degrees; hence passage to the coarse space in this identity +does not insert stabilizer weights. This remains true for stabilizers +acting trivially on a chart. The left side is +\eqref{eu:cotangent-euler-zero}, proving the claimed ordinary +Euler vanishing. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/07-deformation.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/07-deformation.tex new file mode 100644 index 0000000000000000000000000000000000000000..55f5136290acf099dfc784c9f4751fbccd80a066 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/07-deformation.tex @@ -0,0 +1,493 @@ +\section{Deformations with fixed base and the case \texorpdfstring{$b_2\geq5$}{b2 at least 5}} +\label{def:section} +\label{sec:deformation} + +We work in characteristic zero again and write +$f_0:X_0\to B$ for the original projective fibration, with +$e=c_1(f_0^*H)$ and symplectic form $\sigma_0$. +Matsushita's deformation theorem supplies a morphism +defined by a relatively generated line bundle on the entire nearby +$e$-Hodge locus. We prove that its normal Stein target is analytically +$B\times D$, including when $B$ is singular, by first splitting the +parameter directions locally and then using the rigidity of $B$. +The nearby manifolds need only be compact K\"ahler. + +We then follow the inverse image of the fixed closed stratum of +Proposition~\ref{str:minimum} through the family. When $b_2\geq5$, +this variation forces the Hodge cohomology of $\cS$ to be diagonal, +contradicting Proposition~\ref{eu:euler-zero}. +We first record a restriction calculation that will also control +transported fiber polarizations in Section~\ref{sec:second-system}. + +\begin{lemma}\label{def:restriction-kernel} +Let $i:F\hookrightarrow X_0$ be a smooth fiber over the smooth locus of +$f_0$. Then +\begin{equation}\label{def:kernel-equation} + \ker\bigl(i^*:H^2(X_0,\C)\longrightarrow H^2(F,\C)\bigr)=e^\perp. +\end{equation} +\end{lemma} + +\begin{proof} +The smooth fibers have one common cohomology class, and intersecting +$n$ general hyperplanes of the base gives +$e^n=\deg_H(B)[F]$. Polarizing the Fujiki relation, and using $q(e)=0$, +gives a positive constant $a$ such that +\begin{equation}\label{def:mixed-fujiki} + \int_F i^*\xi_1\cdots i^*\xi_n + =a\prod_{j=1}^n q(e,\xi_j) + \qquad(\xi_j\in H^2(X_0,\C)). +\end{equation} +Indeed, in every nonzero pairing of the $2n$ entries +$e,\ldots,e,\xi_1,\ldots,\xi_n$, each copy of $e$ must be paired +with a different $\xi_j$. Positivity of $a$ follows by taking all +$\xi_j$ to be an ample class $h$; in particular $q(e,h)>0$. + +The restrictions of $\sigma_0$ and $\bar\sigma_0$ vanish. If +$\xi\in H^{1,1}(X_0,\R)$ and $q(e,\xi)=0$, Equation~\eqref{def:mixed-fujiki} +gives +\[ + \int_F i^*\xi\,(i^*h)^{n-1}=0, + \qquad + \int_F(i^*\xi)^2(i^*h)^{n-2}=0 +\] +when $n\geq2$. The Hodge index theorem on $F$ makes the second +quadratic form negative definite on the primitive real $(1,1)$-classes, +so $i^*\xi=0$. For $n=1$, the first integral already proves this, +since $H^2(F,\R)$ is one-dimensional. Thus $e^\perp$ is contained +in the kernel. Conversely, the first equality in +Equation~\eqref{def:mixed-fujiki}, with one entry $\xi$ and all other +entries $h$, shows that every class whose restriction vanishes is +orthogonal to $e$. The Hodge decomposition and complexification give +Equation~\eqref{def:kernel-equation}. +\end{proof} + +Let $D$ be a sufficiently small polydisc about the original point in the +locus $\operatorname{Def}(X_0,e)$ on which $e$ is of type $(1,1)$. +The local Torelli theorem identifies this smooth germ with the period +hyperplane $q(e,\sigma_t)=0$. Write +$\pi:\mathcal X\to D$ for the restricted Kuranishi family, and use its +cohomology marking throughout. + +\begin{proposition}[Fixed-base deformation]\label{def:fixed-base} +After shrinking $D$, there is a proper holomorphic map +\[ + \mathfrak f:\mathcal X\longrightarrow B\times D +\] +over $D$ whose central fiber is $f_0$. Every +$f_t:X_t\to B$ has connected fibers, and every irreducible component +of every fiber is an $n$-dimensional Lagrangian subvariety. +Under the chosen marking, $c_1(f_t^*H)=e$ for every $t$. +For some $m>0$, the extension of $f_0^*H^m$ is relatively generated +and defines these fibrations after Stein factorization. +\end{proposition} + +\begin{proof} +Choose $m>0$ so that $H^m$ is very ample, and put +$L_0=f_0^*H^m$. The local universal deformation of the pair +$(X_0,L_0)$ is the restriction of the Kuranishi family to +\[ + q(c_1(L_0),\sigma_t)=m q(e,\sigma_t)=0, +\] +and carries an extension $\mathcal L$ of $L_0$; this is +\cite[Theorem~1.1]{MatsushitaDeformations2016}. Thus its restriction +to a small polydisc is exactly the family $\mathcal X\to D$ chosen +above. Theorem~1.2 and Corollary~1.3 of the same source give a locally +free sheaf $\mathcal E=\pi_*\mathcal L$ and a surjective evaluation +$\pi^*\mathcal E\to\mathcal L$. Their hypotheses are the original +Lagrangian fibration over the projective $B$ and +$L_0=f_0^*H^m$ with $H^m$ very ample; the nearby fibers may be nonprojective compact +K\"ahler manifolds. The resulting proper morphism is +\[ + \Phi:\mathcal X\longrightarrow\PP_D(\mathcal E). +\] +Here $\PP_D(\mathcal E)$ parametrizes one-dimensional quotients. +The cohomology-and-base-change argument in Theorem~1.2 identifies +$\mathcal E\otimes k(0)$ with $H^0(X_0,L_0)$. Since +$f_{0*}\cO_{X_0}=\cO_B$, projection formula identifies this last +space with $H^0(B,H^m)$. Hence $\Phi_0$ is $f_0$ followed by the +very ample embedding of $B$ defined by the complete system $|H^m|$. + +Let $\mathcal Y$ be the image of $\Phi$, with its reduced analytic +structure, and let +\[ + \mathcal X\xrightarrow{F}\mathcal B\xrightarrow{\nu} + \mathcal Y\subset\PP_D(\mathcal E) +\] +be its relative Stein factorization. Write $p:\mathcal B\to D$. +The space $\mathcal B$ is normal because $\mathcal X$ is normal, +and $p$ is proper. For each $t$, take the normal Stein factorization +of the induced map to its image, +\[ + X_t\xrightarrow{g_t}Z_t + \longrightarrow\Phi_t(X_t). +\] +Here $Z_t$ is normal and finite over a projective variety, and $g_t$ +has connected fibers. Corollary~1.3 gives a general Lagrangian fiber. +The symplectic form on the compact K\"ahler manifold $X_t$ is +$d$-closed. Thus the proper map $g_t$ satisfies +\cite[Theorem~1]{Matsushita2000}, which makes every irreducible +component of every fiber an $n$-dimensional Lagrangian subvariety. +The underlying set of each fiber of $F$ is a connected component +of a fiber of $\Phi_t$, and hence is the underlying set of a fiber +of $g_t$. In particular $F$ is equidimensional of relative +dimension $n$. This determines +its fiber dimensions before any identification of a fiber of +$\mathcal B$ with a fiberwise Stein target. + +We prove that $p$ is locally an analytic product before making any +base-change assertion for its central fiber. Fix +$b\in p^{-1}(0)$ and $x\in F^{-1}(b)$. A general codimension-$n$ +smooth slice through $x$ in $X_0$ meets $F^{-1}(b)$ only at $x$ +as a germ. Extend its local defining equations to $\mathcal X$. +The resulting slice $Y$ is smooth over $D$, and its map to the germ +$(\mathcal B,b)$ is finite and dominant. This follows from the +isolated-fiber criterion for a finite map of analytic germs and +equality of dimensions. Let $R\subset A$ be the corresponding +finite extension of local rings and let $r$ be its generic degree. +Here $R$ is normal, $A$ is regular, and the parameter functions +$t_1,\ldots,t_s$ are part of a smooth coordinate system on $Y$. + +Choose derivations $\delta_i$ of $A$ satisfying +$\delta_i(t_j)=\delta_{ij}$. Define +\begin{equation}\label{def:trace-derivations} + \partial_i(a)=\frac1r + \operatorname{Tr}_{\operatorname{Frac}A/ + \operatorname{Frac}R} + \bigl(\delta_i(a)\bigr),\qquad a\in R. +\end{equation} +Trace takes $A$ into $R$: its values are integral over $R$ and +belong to its fraction field, and $R$ is normal. Trace is +$R$-linear, so the Leibniz rule shows that $\partial_i$ is a +derivation of $R$, with $\partial_i(t_j)=\delta_{ij}$. +Holomorphic derivations of an analytic space have local flows. +For example, embed the germ in a polydisc, lift the derivation to a +vector field preserving its defining ideal, and restrict the +ambient flow. Successive flows of the $\partial_i$ give an +isomorphism of germs +\begin{equation}\label{def:base-local-product} + (\mathcal B,b)\simeq(p^{-1}(0),b)\times(D,0) +\end{equation} +over $D$. The lifts need not commute: inverse flows in the reverse +order give the inverse product map. + +Normality of $\mathcal B$ now implies normality of $p^{-1}(0)$. +The central factorization has connected first fibers and a finite +second map, and its composite is $\Phi_0$, the complete system +$|H^m|$ pulled back by $f_0$. Uniqueness of Stein factorization +therefore identifies $p^{-1}(0)$ with the original $B$. + +It remains to globalize Equation~\eqref{def:base-local-product}. +It gives an exact sequence +\[ + 0\longrightarrow T_{\mathcal B/D} + \longrightarrow T_{\mathcal B} + \longrightarrow p^*T_D\longrightarrow0. +\] +The relative tangent sheaf is flat over $D$ and restricts to $T_B$, +as is seen in the local products. By +Corollary~\ref{cor:tangent-rigidity}, $H^1(B,T_B)=0$. +Proper coherent semicontinuity consequently gives +$R^1p_*T_{\mathcal B/D}=0$ after shrinking. Also +$p_*\cO_{\mathcal B}=\cO_D$, since each fiber of $p$ is the image +of the connected $X_t$ and is therefore connected. +Pushing forward the tangent sequence and using that $D$ is Stein +therefore gives global holomorphic lifts of the coordinate vector +fields of $D$. Properness supplies one neighborhood on which +their successive flows are defined along the whole central fiber. +These flows give $\mathcal B\simeq B\times D$ over $D$. +The resulting $\mathfrak f$ has the asserted fiber properties. +The line bundle $\mathfrak f^*\operatorname{pr}_B^*H$ on +$\mathcal X$ restricts centrally to $f_0^*H$, so its fiber Chern +classes are the flat class $e$ under the chosen marking. +\end{proof} + +\begin{remark}\label{def:nonprojective-warning} +The vanishing $H^1(B,T_B)=0$ comes from the original projective +fibration. Proposition~\ref{def:fixed-base} produces proper +holomorphic fibrations on every small $e$-deformation, whose sources +may be nonprojective compact K\"ahler manifolds. Thus the next +argument uses an open set in the full period hyperplane. +Section~\ref{sec:second-system} will select a projective member by +constructing an integral K\"ahler class. +\end{remark} + +Assume that a nonquotient germ remains, and retain the closed stratum +$S_0$, its smooth stack $\cS$, and $d=\dim\cS0$. + +Fix $\alpha\in H^{p,q}(\cS)$ with $p>q$, and set +$k=d-p+1$. On every $Q_t$, +\begin{equation}\label{def:holomorphic-vanishing} + \omega_t^k\,h_t^*\alpha=0. +\end{equation} +To verify it, first note that the pullback of $\sigma_t$ by every +smooth space mapping into a fiber of $f_t$ is zero. On a resolution +of each irreducible fiber component this follows from the +Lagrangian assumption and extension of holomorphic forms. +For a map whose image is contained in a proper subvariety of that +component, resolve a component of its proper base change to that +resolution which dominates the source. Pullback of forms by a +dominant map is injective, giving the same conclusion. +Thus, at a general point where $h_t$ has constant rank, +$\omega_t$ vanishes on two vertical vectors. Each of its terms +has at least one horizontal one-form factor. Wedge multiplication +by $\omega_t^k$ consequently kills the pullback of every +holomorphic $p$-form on $\cS$, since $k+p>d$. This equality of +holomorphic bundle maps extends from the dense open set and +implies Equation~\eqref{def:holomorphic-vanishing} on Dolbeault +cohomology. + +Work over a simply connected small open subset of $D^\circ$. +Smooth proper transport makes the maps $j_t^*$ and $h_t^*$ +constant on cohomology. Accordingly +\[ + v\longmapsto (j_t^*v)^k\,h_t^*\alpha +\] +is one fixed homogeneous polynomial with values in a fixed +cohomology group. It vanishes on all period vectors over that +open set, by Equation~\eqref{def:holomorphic-vanishing}. +The quadratic form on $e^\perp_\C$ has radical $\C e$ and rank +$b_2-2\geq3$. Its null cone is therefore irreducible. Local +Torelli and Proposition~\ref{def:fixed-base} imply that the +periods just used contain an analytic open subset of its +projectivization. Such an open set is Zariski dense. Hence the +polynomial vanishes on the whole cone, including the conjugate +period of our chosen fiber. Suppressing $t$, we have obtained +\begin{equation}\label{def:antiholomorphic-vanishing} + \bar\omega^{\,k}h^*\alpha=0. +\end{equation} + +We give the contraction step with its coefficients retained. +This avoids replacing a sheaf-valued class by its exterior product +before the contractions have been applied. If $\alpha\ne0$, +Poincar\'e duality supplies +$\beta\in H^{d-p,d-q}(\cS)$ with $\alpha\beta\ne0$. +Pass to the conjugate complex structures and put +\[ + \mathcal A=h^*\Omega^1_{\overline{\cS}}, + \qquad r=d-q, + \qquad s=\bar\omega\in H^0(\bar Q,\Omega^2_{\bar Q}). +\] +Write $\rho:\mathcal A\to\Omega^1_{\bar Q}$ for differential +pullback. +In this notation +\[ + h^*\alpha\in H^p(\bar Q,\Omega^q_{\bar Q}), + \qquad + \widetilde\beta=h^*\beta + \in H^{d-p}(\bar Q,\Lambda^r\mathcal A). +\] +The second class is kept in the indicated coefficient bundle; +it has not yet been wedge-mapped to $\Omega^r_{\bar Q}$. +Since $p>q$, we have $r\geq k$. +Apply the exterior coproduct +\[ + \Delta_{k,r-k}:\Lambda^r\mathcal A + \longrightarrow\Lambda^k\mathcal A + \otimes\Lambda^{r-k}\mathcal A +\] +to its coefficients. Cup Equation~\eqref{def:antiholomorphic-vanishing} +with this class, contract the first $k$ coefficient factors by +$\overline{\mathcal H}$ from +Lemma~\ref{def:hamiltonian}, and wedge the remaining $r-k$ +factors as basic forms. All these are holomorphic sheaf maps on +$\bar Q$, so they induce well-defined operations on its sheaf +cohomology. Their output belongs to +$H^d(\bar Q,\Omega^d_{\bar Q})$. + +Here is the exterior-algebra computation of that output. If +$a_1,\ldots,a_k$ are local sections of $\mathcal A$ and $\eta$ +is a basic $q$-form, the identities +\[ + \iota_{\overline{\mathcal H}(a)}s=\rho(a), + \qquad + \iota_{\overline{\mathcal H}(a)}\rho(b)=0 +\] +give +\[ + \iota_{\overline{\mathcal H}(a_k)}\cdots + \iota_{\overline{\mathcal H}(a_1)}(s^k\wedge\eta) + =(-1)^{k(k-1)/2}k!\, + \rho(a_1)\wedge\cdots\wedge\rho(a_k)\wedge\eta. +\] +For a decomposable $r$-form the signed shuffle coproduct followed +by wedge multiplication is $\binom rk$ times the original form. +It follows, with only an overall sign depending on conventions, +that the preceding sheaf cup operation is +\begin{equation}\label{def:contraction-output} + \pm k!\binom rk\,h^*(\alpha\beta). +\end{equation} +Its input was zero, so Equation~\eqref{def:contraction-output} +forces $h^*(\alpha\beta)=0$. + +That is impossible. Choose a component $Q'$ dominating $S_0$, +a K\"ahler form $\kappa$ on $Q'$, and a K\"ahler orbifold form +$\tau$ on $\cS$. If $N=\dim Q'$, then +\[ + \int_{Q'}h^*\tau^d\wedge\kappa^{N-d}>0: +\] +the integrand is nonnegative and positive on the nonempty open +set where $h$ has rank $d$. Thus the top class of $\cS$ pulls +back nontrivially. As $\alpha\beta$ is a nonzero top class, it +cannot pull back to zero. This proves the vanishing for $p>q$; +complex conjugation proves Equation~\eqref{def:diagonal-hodge}. + +The Euler number is now +\[ + \chi_{\mathrm{top}}(S_0) + =\sum_{p=0}^d h^{p,p}(\cS)>0, +\] +because the coarse stratum is nonempty and connected. This +contradicts Proposition~\ref{eu:euler-zero}. There can therefore +be no nonquotient germ. +\end{proof} + +\begin{remark}\label{def:rank-four-boundary} +The irreducibility just used is the only place in this section where +$b_2\geq5$ is required. When $b_2=4$, the restricted quadratic form +has rank two, and its null cone is the union of two distinct linear +components. A local period branch lies in one component and its +conjugate in the other. The continuation from +Equation~\eqref{def:holomorphic-vanishing} to +Equation~\eqref{def:antiholomorphic-vanishing} then does not follow. +The two rank-four arguments below are necessary for exactly this +reason. +\end{remark} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/08-isotrivial.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/08-isotrivial.tex new file mode 100644 index 0000000000000000000000000000000000000000..febb1f2730837ecb8279228dd5a049f1ab9eec84 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/08-isotrivial.tex @@ -0,0 +1,902 @@ +\section{The isotrivial case when the second Betti number is four} +\label{iso:section} +\label{sec:isotrivial} + +\begin{proposition}\label{iso:exclusion} +Let \(f:X\to B\) be the projective Lagrangian fibration of +Section~\ref{sec:setup}. If \(b_2(X)=4\) and the polarized period map +of its smooth fibers has zero variation, then every germ of \(B\) is a +quotient singularity. +\end{proposition} + +Throughout the section we assume these hypotheses. The proof starts +with the original fibration \(f\). Its constant abelian fiber and +rational transcendental plane produce a normal variety \(T\) with +two finite maps: a Galois cover \(p:T\to B\), and a map \(a:T\to A\) +to an abelian variety. We compare their ramification indices with +the multiplicity of a divisor in a fiber of \(f\). The resulting +inequalities control both the linear monodromy and the lifting of +vector fields near a minimal stratum. This is the isotrivial cover; +the finite cover of \(B\) by an abelian variety constructed from a +second Lagrangian fibration in the next section has a different role. + +Finite-cover descriptions and the Hodge theory of +isotrivial Lagrangian systems are developed systematically by +Kim--Laza--Martin \cite[Section~2.4]{KimLazaMartin2026}; the construction +below retains the two distinct ramification maps needed for our +rank-four argument. + +\begin{lemma}\label{iso:vertical-divisor} +The Picard number of \(X\) is two, and +\[ + \mathbb T_{\Q}:=\NS(X)_{\Q}^{\perp} + \quad\text{satisfies}\quad + \mathbb T_{\Q}\otimes_{\Q}\C + =\C[\sigma]\oplus\C[\overline{\sigma}]. +\] +For every prime divisor \(D\subset B\), there is a unique prime divisor +\(F_D\subset X\) dominating \(D\). Near the generic point of \(D\), +\[ + f^*D=m_DF_D +\] +for a positive integer \(m_D\). +\end{lemma} + +\begin{proof} +Since \(h^{2,0}(X)=h^{0,2}(X)=1\), we have \(h^{1,1}(X)=2\). +An ample class \(h\) and the nonzero isotropic class \(e\) are linearly +independent, so they span \(\NS(X)_{\Q}\). Their Beauville form is +nondegenerate, and its orthogonal complement is the stated rational +plane. + +Let \(F\) be any divisor dominating \(D\). Its restriction to a general +smooth fiber is zero, whereas \(h\) restricts nontrivially and \(e\) +restricts to zero. Therefore \([F]=a e\) with \(a\in\Q\). +Intersecting with an ample class to the power \(2n-1\) shows \(a>0\). +Choose positive integers \(k,l\) with \(k[F]=l e\). +After multiplying once more if necessary, this is equality of integral +classes. Since \(H^1(X,\cO_X)=0\), the exponential sequence gives +\[ + \cO_X(kF)\simeq f^*H^{\otimes l}. +\] +The section defining \(kF\) is pulled back from a section of +\(H^{\otimes l}\), because \(f_*\cO_X=\cO_B\). +Its zero set consequently contains every component over the generic +point of \(D\). Its support is \(F\), so there is only one such +component. A local uniformizer of \(D\) now has divisor \(m_DF_D\). +\end{proof} + +\begin{lemma}\label{iso:abelian-model} +There exist a normal projective variety \(T\), a finite group \(G\), +an \(n\)-dimensional abelian variety \(P\), and an \(n\)-dimensional +abelian variety \(A\), with the following properties. +\begin{enumerate} +\item The map \(p:T\to B\) is finite Galois with group \(G\), and the +main base change of \(X\) is birational over \(T\) to \(P\times T\). +The action on the first factor is by constant affine automorphisms +preserving an ample line bundle on \(P\). +\item There is a finite surjective \(G\)-equivariant morphism +\(a:T\to A\), where \(G\) acts affinely on \(A\). +The linear actions on \(P\) and \(A\) have the same kernel. +\item Writing this kernel as \(K\), and writing \(A_l\) for the quotient +of \(A\) by the finite group of translations induced by \(K\), there +are finite morphisms +\[ + T_l:=T/K \xrightarrow{\ a_l\ } A_l, + \qquad + T_l\xrightarrow{\ p_l\ }B. +\] +The group \(G_l:=G/K\) is the Galois group of \(p_l\), and acts on +\(A_l\) with faithful linear part. +\end{enumerate} +\end{lemma} + +\begin{proof} +Choose an ample line bundle on \(X\), and use its restriction to polarize +the smooth fibers. Constancy of the polarized moduli map means that +the geometric generic fiber, after choosing an origin, is a fixed +polarized abelian variety \(P\). +The origin and a polarized isomorphism are defined over a finite +extension of \(\C(B)\): each is specified by finitely many algebraic +data over its algebraic closure. Alternatively, a sufficiently high +level structure gives this isomorphism on a dense open by the fine +moduli space of polarized abelian varieties. + +We can also make the actual line bundle constant. Once its numerical +polarization has been matched with a fixed ample bundle \(L_P\) on +\(P\), their difference lies in \(\Pic^0(P)\). +The polarization isogeny +\[ + P\longrightarrow\Pic^0(P),\qquad + x\longmapsto t_x^*L_P\otimes L_P^{-1} +\] +allows us to remove that difference by a translation after a further +finite extension. Take a Galois closure and normalize \(B\) in it. +The resulting \(T\) is projective and normal. +The descent automorphisms of \(P\) preserve \(L_P\). +The group \(\Aut(P,L_P)\) is finite: its translation subgroup is the +finite kernel of the polarization isogeny, and its origin-preserving +subgroup is a discrete subgroup of the compact unitary group of the +polarization. Thus each descent automorphism, initially over the +function field of \(T\), is a constant affine automorphism over \(\C\). +This proves the first assertion. + +Take a \(G\)-equivariant projective resolution +\(\widetilde T\to T\). +The rational map \(P\times\widetilde T\dashrightarrow X\) pulls +\(\sigma\) back to a holomorphic two-form \(\widetilde\sigma\) on +\(P\times\widetilde T\). To see regularity, resolve the graph; its +pulled-back form descends across the birational map to the smooth +product, since holomorphic differential forms are unchanged by a +proper birational modification of smooth varieties. +Choose a basis \(\alpha_1,\ldots,\alpha_n\) of \(H^0(P,\Omega_P^1)\). +The K\"unneth decomposition and the Lagrangian condition give +\begin{equation}\label{iso:cross-term} + \widetilde\sigma + =\sum_{i=1}^n\alpha_i\wedge\gamma_i+\beta, + \qquad + \gamma_i\in H^0(\widetilde T,\Omega_{\widetilde T}^1),\quad + \beta\in H^0(\widetilde T,\Omega_{\widetilde T}^2). +\end{equation} +There is no two-form term from \(P\). +At a general point the pulled-back form is nondegenerate and \(P\) +is Lagrangian; hence the \(\gamma_i\) are pointwise independent there. + +We spell out the rationality needed for an abelian quotient. +Graph pullback and projection onto the K\"unneth summand give a +morphism of rational Hodge structures +\[ + \mathbb T_{\Q}\longrightarrow + H^1(P,\Q)\otimes H^1(\widetilde T,\Q). +\] +Contract its image with \(H^1(P,\Q)^\vee\), and denote the resulting +rational subspace of \(H^1(\widetilde T,\Q)\) by \(W_{\Q}\). +By Lemma~\ref{iso:vertical-divisor} and +\eqref{iso:cross-term}, +\[ + W_{\Q}\otimes\C + =\langle\gamma_1,\ldots,\gamma_n\rangle_{\C} + \oplus + \langle\overline{\gamma_1},\ldots, + \overline{\gamma_n}\rangle_{\C}. +\] +It is a rational weight-one Hodge substructure of dimension \(2n\). +To form the corresponding Albanese quotient, take the annihilator +of \(W_{\Q}\) in +\(H_1(\operatorname{Alb}(\widetilde T),\Q)\). +It is a rational Hodge subspace, so its real span and its lattice +define a connected complex subtorus. Restriction of an ample +polarization makes this an abelian subvariety. +The quotient is an abelian variety \(A\) of dimension \(n\); +its one-forms pull back to the span of the \(\gamma_i\). +The induced morphism +\(\widetilde a:\widetilde T\to A\) is generically finite. +The subspace \(W_{\Q}\) is \(G\)-stable; the Albanese functor consequently +gives an affine action on \(A\) making \(\widetilde a\) equivariant. + +We next descend this morphism to \(T\). +Let \(C'\) be a curve contracted by \(\widetilde T\to T\). +The rational map from \(P\times C'\) to \(X\), interpreted on a resolved +graph, has image in one fiber of \(f\). +The pullback of \(\sigma\) on that graph is zero. Here the Lagrangian +condition also applies if the image lies in a smaller subvariety of a +fiber component: resolve the component, on which the pulled-back form +vanishes, and dominate the smaller subvariety by proper base change. +Restriction of \eqref{iso:cross-term} and contraction in the \(P\) +directions now show that each \(\gamma_i\) vanishes on the normalization +of \(C'\). It follows that \(\widetilde a\) is constant on every +contracted curve. It is constant on each connected resolution fiber +as well: a positive-dimensional projective image of such a fiber in +an abelian variety would contain the image of a noncontracted curve. +The fibers of \(\widetilde T\to T\) are connected and \(T\) is normal, +so this proves descent to a morphism \(a:T\to A\). + +Equivariance gives a generically finite morphism +\(\bar a:B=T/G\to A/G\). +Pull back an ample line bundle from the projective quotient \(A/G\). +The result is a nef and big line bundle on \(B\); because +\(\rho(B)=1\), it is numerically a positive multiple of an ample +class and hence is ample. Thus \(\bar a\) contracts no curve and is +finite. The commutative quotient square then shows that \(a\) is +finite too. + +Finally the cross pairing in \eqref{iso:cross-term} is perfect between +the tangent spaces of the two abelian factors. Its invariance shows +that an element has trivial linear action on \(P\) if and only if it +has trivial linear action on \(A\). Such an element acts on \(A\) +by a translation. Quotienting its finite translation image gives +\(A_l\), and equivariance gives the stated finite map from \(T/K\). +The derivative representation is unchanged by an isogeny, so the +linear action of \(G_l\) on \(A_l\) is faithful. +\end{proof} + +\begin{lemma}\label{iso:ramification} +Let \(D\subset B\) be a prime divisor, let \(E\subset T\) be a prime +divisor over \(D\), and put \(D_A=a(E)\). Write +\[ + M=e_E(p),\qquad L=e_E(a) +\] +for the ramification indices of the two finite maps, and let \(m_D\) +be the fiber multiplicity in Lemma~\ref{iso:vertical-divisor}. +Then +\begin{equation}\label{iso:multiplicity-bound} + Lm_D\le M. +\end{equation} +The linear part of a generator of the cyclic inertia is either the +identity or a pseudoreflection, on both \(A\) and \(P\). +If \(M_l,L_l\) are the analogous indices for \(p_l,a_l\), then +\begin{equation}\label{iso:linear-ramification-bound} + L_l\le M_l. +\end{equation} +\end{lemma} + +\begin{proof} +The maps and divisors being compared fit into the following diagram. +The upper square commutes as rational maps, and the lower square is +the quotient square from Lemma~\ref{iso:abelian-model}. +\begin{center} +\begin{tikzpicture}[>=Latex, every node/.style={font=\small}] + \node (prod) at (0,1.6) {\(P\times T\supset P\times E\)}; + \node (source) at (6.1,1.6) {\(X\supset F_D\)}; + \node (cover) at (0,0) {\(T\supset E\)}; + \node (base) at (6.1,0) {\(B\supset D\)}; + \node (abelian) at (0,-1.6) {\(A\supset D_A\)}; + \node (quotient) at (6.1,-1.6) {\(A/G\)}; + \draw[->,dashed] (prod) -- node[above] {\(\phi\)} (source); + \draw[->] (prod) -- node[left] {\(\operatorname{pr}_T\)} (cover); + \draw[->] (source) -- node[right,align=left] + {\(f\)\\\(\operatorname{ord}_{F_D}(f^*t)=m_D\)} (base); + \draw[->] (cover) -- node[above,align=center] + {\(p\) finite Galois\\\(e_E(p)=M\)} (base); + \draw[->] (cover) -- node[left,align=right] + {\(a\) finite\\\(e_E(a)=L\)} (abelian); + \draw[->] (base) -- node[right] {\(\bar a\)} (quotient); + \draw[->] (abelian) -- node[above] {quotient} (quotient); +\end{tikzpicture} +\end{center} +Here \(t\) is a local uniformizer of \(D\), and \(\phi\) is induced by +the birational identification over \(T\). The integers \(M\) and +\(L\) are local indices along \(E\), not the degrees of the finite +maps. They compare to the multiplicity \(m_D\) on the original +source \(X\), which may be greater than one. + +All assertions concern generic points of divisors. We may therefore +discard smaller subsets, split the unramified tangential extension, +and work with smooth transverse polydiscs. Choose a transverse +parameter \(z\) for \(E\), absorbing a unit so that \(p^*t=z^M\). +Cyclic inertia fixes \(E\) pointwise. Since \(a\) is finite and +equivariant, it fixes the divisor \(D_A\) pointwise. +A finite affine automorphism with a fixed point is conjugate by a +translation to its linear part; fixing a divisor makes its linear +part either the identity or a pseudoreflection. +The perfect cross pairing gives the same assertion on \(P\). + +Let \(\omega=\sigma^n/n!\), with an irrelevant nonzero scalar absorbed +into the choice of a translation-invariant volume form on \(P\). +In the top exterior power of \eqref{iso:cross-term}, every term must +contain all \(n\) \(P\)-directions. Thus the term \(\beta\) makes no +contribution, and on the smooth product model +\[ + \phi^*\omega=\operatorname{vol}_P\wedge a^* + \operatorname{vol}_A. +\] +Its order along \(P\times E\) is \(L-1\). More precisely, near the +generic point of \(E\), it is \(u z^{L-1}\operatorname{vol}_{\mathrm{prod}}\), +where +\(\operatorname{vol}_{\mathrm{prod}} +=\operatorname{vol}_P\wedge\operatorname{vol}_T\) +for a nowhere vanishing local top form \(\operatorname{vol}_T\) on \(T\), +and \(u\) is pulled back from a unit at the generic point of \(E\). + +Let \(v=\operatorname{ord}_{F_D}\) be the normalized valuation of +the divisor in Lemma~\ref{iso:vertical-divisor}. The product field is +the compositum \(\C(P\times T)=\C(X)\C(T)\). Geometric integrality +of the generic fiber makes this a Galois extension of \(\C(X)\) with +group \(G\). Extend \(v\) to a +normalized divisorial valuation \(w\) of the product field +\(\C(P\times T)\), choosing the extension above \(E\). Indeed, +conjugating any extension by \(G\) preserves its restriction \(v\) +and moves its base center through all divisors of \(T\) over \(D\). +Write \(r\) for the ramification index of \(w/v\). Since +\(v(t)=m_D\), +\begin{equation}\label{iso:valuation-multiplicity} + r m_D=Mw(z). +\end{equation} +The center of \(w\) on the proper product projects dominantly to \(E\). +In particular the product is smooth at the generic point of this +center, and the unit \(u\) above is a \(w\)-unit. +Denote its log discrepancy by \(A_{\mathrm{prod}}(w)\). +The volume form on the smooth \(X\) has order zero at \(v\). +The order of the same pulled-back top form can therefore be computed +in the finite extension of valuation rings and on the product model: +\begin{equation}\label{iso:top-form-valuation} + \begin{aligned} + \operatorname{ord}_w(\phi^*\omega) + &=r\,\operatorname{ord}_v(\omega)+(r-1)=r-1\\ + &=(L-1)w(z)+A_{\mathrm{prod}}(w)-1. + \end{aligned} +\end{equation} +The first line uses the different exponent \(r-1\) of a tame +extension of discrete valuation rings; the residue extension is +separable in characteristic zero. The second line uses +\(\operatorname{ord}_w(\operatorname{vol}_{\mathrm{prod}}) +=A_{\mathrm{prod}}(w)-1\). +On the smooth product, the pair defined by the smooth divisor +\(z=0\) is log canonical, so \(A_{\mathrm{prod}}(w)\ge w(z)\). +Consequently \eqref{iso:top-form-valuation} gives +\begin{equation}\label{iso:valuation-discrepancy} + r=(L-1)w(z)+A_{\mathrm{prod}}(w) + \ \ge\ Lw(z). +\end{equation} +This last inequality can also be read in local coordinates: +the pullback of a form containing \(dz\) has order at least +\(w(z)-1\). +Combining \eqref{iso:valuation-multiplicity} and +\eqref{iso:valuation-discrepancy} proves +\eqref{iso:multiplicity-bound}. + +Let \(k\) be the inertia order of \(T\to T_l\) along this divisor. +Its image on \(A\) is a group of translations fixing \(a(z=0)\), +so that image is trivial. Quotienting \(A\) by the translation +group is an isogeny and is everywhere unramified. The indices in +the two compositions consequently give +\[ + M=kM_l,\qquad L=kL_l. +\] +In particular \(L_lm_D\le M_l\), which implies +\eqref{iso:linear-ramification-bound}. +\end{proof} + +\begin{lemma}\label{iso:order-two} +Let \(g_D\) be a local inertia generator at a prime base divisor \(D\). +The linear part of \(g_D^{m_D}\) on \(P\) does not have order two. +\end{lemma} + +\begin{proof} +Set \(m=m_D\), and make the base change \(t=s^m\). +Near the generic point of \(D\), the normalized source \(X'\) of this +base change is smooth and its map to \(X\) is \'{e}tale. +Indeed \(f^*t\) has the single divisor \(mF_D\). +In the regular local rings of \(X\) one can write +\(f^*t=u h^m\), with \(u\) a unit; adjoining \(s\) and normalizing +adjoins an \(m\)-th root of the unit. This is \'{e}tale. +Equivalently, the extension is unramified in codimension one and +purity applies. The divisor \(s=0\) on \(X'\) is reduced. +These statements hold on a neighborhood of the whole fiber over +the generic point of \(D\), which is the neighborhood used below. + +Put \(\delta=\gcd(M,m)\). +On a component of the normalized compositum of the two base covers, +use a parameter \(z'\) so that +\[ + z=(z')^{m/\delta},\qquad + s=(z')^{M/\delta}. +\] +Its cyclic inertia over the rooted base has order +\[ + M'=\frac M\delta, + \qquad\text{and its generator acts on \(P\) as }g_D^m. +\] +The order of the pulled-back volume on the new product is \(L'-1\), +where +\begin{equation}\label{iso:rooted-indices} + L'=\frac{Lm}{\delta}. +\end{equation} +This follows by substituting \(z=(z')^{m/\delta}\) in +\(z^{L-1}dz\). In particular +\(01\), then +\eqref{iso:linear-ramification-bound} implies \(M_l>1\) for \(p_l\). +Its nontrivial cyclic inertia fixes \(R\) pointwise and therefore +fixes the divisor \(a_l(R)\) pointwise. +Since the linear action of \(G_l\) is faithful, a generator acts +on \(A_l\) by an affine pseudoreflection. +Consequently \(a_l(R)\) is a component of its fixed locus: +it is a translate of an abelian subvariety of codimension one. +We call such a translate an abelian hyperplane. + +After shrinking about \(z_0\), every ramification divisor that meets +the neighborhood contains \(z_0\). +Its inertia is contained in \(J\), since an automorphism fixing +the divisor pointwise fixes \(z_0\). +The tangent hyperplane of its image therefore contains \(V_J\). + +Let \(v\in V_J\). +The finite separable map \(a_l\) gives a unique meromorphic lift +\(\widetilde v\) of the constant field \(v\). +It is regular in codimension one. +At an unramified generic point this is immediate. +At a general point of a ramification divisor choose coordinates +in which \(a_l\) is +\[ + (z,y_2,\ldots,y_n)\longmapsto + (z^{L_l},y_2,\ldots,y_n). +\] +Since \(v\) is tangent to the abelian branch hyperplane, its normal +component is zero and the lift has no pole. +Normality then extends the derivation everywhere: for any regular +function \(h\), the meromorphic function \(\widetilde v(h)\) +belongs to every local height-one ring and thus to the normal +local ring itself. +Uniqueness of the lift and \(J\)-invariance of \(v\) show that +\(\widetilde v\) is \(J\)-invariant. +The germ of \(p_l\) is the quotient by \(J\), so the field descends +to that germ of \(B\). + +These descended fields have independent evaluations. +To verify this without assuming either germ smooth, center +exponential coordinates of \(A_l\) at \(a_l(z_0)\), so the finite +affine \(J\)-action becomes linear. +Choose \(J\)-invariant linear coordinate functions +\(\ell_1,\ldots,\ell_r\), dual to a basis +\(v_1,\ldots,v_r\) of \(V_J\). +The invariant functions \(a_l^*\ell_i\) descend to functions +\(u_i\) on the base germ, and the descended fields \(\xi_j\) satisfy +\[ + \xi_j(u_i)=\delta_{ij}. +\] +They therefore split \(r\) smooth directions by their local flows. +The same directions persist on the maximal quasi-\'etale chart: +the fields lift uniquely across a quasi-\'etale finite map +(check regularity in codimension one and use normality), and +the pulled-back functions retain these identities. +On \(S_0\) the evaluation rank on that chart is \(d=1\), by +Proposition~\ref{str:minimum}. Thus \(r\le1\). +\end{proof} + +\begin{lemma}\label{iso:regular-differentials} +For every local holomorphic function \(u\) on \(B\), its pulled-back +differential has an expression +\begin{equation}\label{iso:differential-coefficients} + p_l^*(du)=\sum_{i=1}^n h_i\,a_l^*\alpha_i, +\end{equation} +where the \(\alpha_i\) are a basis of translation-invariant +one-forms on \(A_l\) and the \(h_i\) are holomorphic on the +corresponding neighborhood of \(T_l\). +At a point \(z_0\) with stabilizer \(J\), the covector +\(\sum_i h_i(z_0)\alpha_i\) is \(J\)-invariant. +\end{lemma} + +\begin{proof} +The coefficients are uniquely defined meromorphic functions, since +\(a_l\) is generically \'{e}tale. +It is enough to check regularity at every prime divisor of \(T_l\). +There is nothing to prove where \(a_l\) is unramified. +At a general ramification point write \(M=M_l\), \(L=L_l\). +Use linear coordinates \((x,y_2,\ldots,y_n)\) on \(A_l\) adapted +to the affine fixed hyperplane \(x=0\). +The inertia fixes the \(y_i\) and acts on \(x\) by a root of unity. +The restrictions of \(a_l^*y_i\) are generically independent along +the divisor. They can be used as tangential coordinates on +\(T_l\), and, after absorbing a unit in the normal parameter, +\[ + a_l^*x=z^L. +\] +Choose \(z\) so that the cyclic inertia rotates it by a primitive +\(M\)-th root. The local invariant ring of this cyclic quotient +has coordinates \(z^M,y_2,\ldots,y_n\). +Thus \(p_l^*u=F(z^M,y)\) for a holomorphic function \(F\), and +\[ + d(p_l^*u) + =\frac ML z^{M-L}F_t(z^M,y)\,d(z^L) + +\sum_{i=2}^n F_{y_i}(z^M,y)\,dy_i. +\] +All coefficients are holomorphic because \(L\le M\). +The displayed coordinate covectors are constant linear combinations +of the \(\alpha_i\), so this proves the required codimension-one +regularity. Normality extends every \(h_i\). + +Expression \eqref{iso:differential-coefficients} is unique on the +function field. Its left side is \(J\)-invariant and its right +side transforms by the linear representation on the \(\alpha_i\). +Evaluation at the \(J\)-fixed point \(z_0\) therefore gives an +invariant covector, as asserted. +\end{proof} + +\begin{lemma}\label{iso:elliptic-lift} +Let \(C\) be an irreducible component of +\((p_l^{-1}(S_0))_{\mathrm{red}}\). +Then \(C\) is a smooth curve of genus one. +Its image \(E=a_l(C)\) is an elliptic translate in \(A_l\), and +\(a_l|_C:C\to E\) and \(p_l|_C:C\to S_0\) are \'{e}tale. +At every \(z\in C\), +\[ + \operatorname{Lie}(A_l)^{\operatorname{Stab}_{G_l}(z)} + =\operatorname{Lie}(E). +\] +Here \(\operatorname{Lie}(E)\) means the constant tangent direction +of the translate \(E\). +\end{lemma} + +\begin{proof} +All components \(C\) are projective curves mapping finitely onto +\(S_0\). Let \(J_0\) be their generic pointwise stabilizer. +More explicitly, discard the proper fixed loci of those finitely +many group elements that do not fix \(C\) identically. +On the remaining open the stabilizer is the subgroup \(J_0\) +fixing \(C\) pointwise. +Equivariance implies +\[ + a_l(C)\subset A_l^{J_0}. +\] +Every component of this fixed locus is a translate of an abelian +subvariety with tangent space \(V_{J_0}\). +Lemma~\ref{iso:local-fields} bounds its dimension by one, whereas +finiteness of \(a_l\) makes the image of \(C\) one-dimensional. +It follows that the image is exactly an elliptic translate \(E\), +and \(V_{J_0}=\operatorname{Lie}(E)\). + +Write \(\nu:\widehat C\to C\) for the normalization and let +\[ + b:\widehat C\to S_0,\qquad + a_C:\widehat C\to E +\] +be the two finite morphisms. +At any \(q\in\widehat C\), choose a parameter \(u\) on the smooth +curve \(S_0\) near \(b(q)\), and extend it to a local function on +\(B\). Such an extension exists because \(S_0\) has its reduced +closed-subspace structure. +Restricting \eqref{iso:differential-coefficients} gives +\begin{equation}\label{iso:curve-differential} + b^*du=\eta\,a_C^*\theta +\end{equation} +near \(q\), where \(\theta\) is a nonzero translation-invariant +one-form on \(E\) and \(\eta\) is holomorphic. +There is no assumption here that \(T_l\) is smooth along \(C\). +To justify restriction, first pull +\eqref{iso:differential-coefficients} to a resolution of \(T_l\). +Both sides are regular ordinary differentials there and agree +generically, hence everywhere. A component of the proper base +change dominating \(\widehat C\) then gives the stated equality +on \(\widehat C\), since pullback of differentials along a dominant +map to a smooth curve in characteristic zero is injective. + +Equation \eqref{iso:curve-differential} implies pointwise +\[ + \operatorname{ram}_q(b) + \ge\operatorname{ram}_q(a_C). +\] +Both targets have genus one. Riemann--Hurwitz therefore gives +the same total ramification degree, \(2g(\widehat C)-2\), for +both maps. The pointwise inequalities must all be equalities. +In particular \(\eta(q)\ne0\). + +Put \(z=\nu(q)\) and \(J=\operatorname{Stab}_{G_l}(z)\). +The covector in Lemma~\ref{iso:regular-differentials} is +\(J\)-invariant, and the nonvanishing of \(\eta(q)\) says that +its restriction to \(\operatorname{Lie}(E)\) is nonzero. +Since \(J\) contains \(J_0\), we have +\[ + V_J\subset V_{J_0} + =\operatorname{Lie}(E). +\] +A finite-dimensional representation of a finite group in +characteristic zero has equally dimensional invariant vector +and covector spaces, by the averaging projector. +Thus \(V_J\ne0\), and the displayed inclusion is equality. +This holds at every point and on every normalization branch. + +Fix a nonzero \(v_0\in\operatorname{Lie}(E)\). +Its unique meromorphic lift through \(a_l\) is holomorphic near +each point of \(C\), by Lemma~\ref{iso:local-fields}, and the +lifts agree on overlaps by uniqueness. +Call the resulting field \(\widetilde v_0\). +It has nonzero evaluation everywhere on \(C\), since +\(da_l(\widetilde v_0)=v_0\). +Locally it descends through \(p_l\) to a field on \(B\). +Its flow preserves the intrinsically defined minimum stratum +\(S_0\), and hence its lifted flow preserves the reduced inverse +image and each of its irreducible curve components. +Thus \(\widetilde v_0\) is tangent to \(C\). + +A reduced curve carrying such a nonvanishing ambient holomorphic +field is smooth: choose a function whose derivative under the +field is nonzero, and use the local flow to split off its disc +direction. An invariant reduced curve in this product is the +disc times a reduced zero-dimensional germ. +It follows that \(C=\widehat C\) is smooth. +Since \(da_l(\widetilde v_0)=v_0\ne0\), the map \(C\to E\) is +unramified. It is consequently an \'{e}tale cover of an elliptic +curve, so \(g(C)=1\). +The equality of ramification divisors already proved also makes +\(C\to S_0\) unramified. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{iso:exclusion}] +If a nonquotient germ remained, \eqref{iso:elliptic-case} and +Lemma~\ref{iso:elliptic-lift} would supply \(C\) and a nonzero +translation direction \(v_0\) as above. +Choose a positive integer \(r\) such that +\(\mathcal L=\cO_B(rK_B)\) is a line bundle, and put +\[ + \mathcal M=p_l^*\mathcal L. +\] +We construct a holomorphic connection on \(\mathcal M|_C\); +in particular we must check both its extension and its gluing. + +Near \(z\in C\), the field \(\widetilde v_0\) is invariant under +the point stabilizer and descends to a holomorphic field +\(\xi_z\) on the corresponding base germ. +On \(B_{\mathrm{reg}}\), Lie differentiation by \(\xi_z\) acts +on \(r\)-fold canonical tensors. It extends to an operator +\[ + \mathcal D_z:\mathcal L\longrightarrow\mathcal L. +\] +Indeed \(\mathcal L\) is the reflexive extension of its restriction +to \(B_{\mathrm{reg}}\), so a differentiated section extends +uniquely across the subset of codimension at least two. +The operator obeys +\[ + \mathcal D_z(us) + =\xi_z(u)s+u\mathcal D_z(s). +\] +Pull it to the chosen neighborhood in \(T_l\) using the lifted +symbol \(\widetilde v_0\): in a local pulled-back frame its rule is +\begin{equation}\label{iso:connection-operator} + \widetilde{\mathcal D}_z(h\,p_l^*s) + =\widetilde v_0(h)\,p_l^*s + +h\,p_l^*(\mathcal D_zs). +\end{equation} +The Leibniz identity and +\(\widetilde v_0(p_l^*u)=p_l^*(\xi_z u)\) +make this definition independent of the frame. + +On an overlap, both operators +\(\widetilde{\mathcal D}_z\) have the same symbol. +They agree on the dense open where \(p_l\) is \'{e}tale and +both varieties are smooth: there the differential of \(p_l\) +identifies the pulled-back canonical tensors with canonical +tensors upstairs, and both operators are Lie differentiation +by the same field \(\widetilde v_0\). +Their difference is an \(\cO_{T_l}\)-linear endomorphism of the +line bundle \(\mathcal M\); its coefficient is holomorphic and +zero on a dense open, hence is zero. +Thus the local operators glue on a neighborhood of \(C\). +This argument compares the operators upstairs and does not +require a single descended vector field on a neighborhood of +all of \(S_0\). + +Tangency of \(\widetilde v_0\) to the reduced curve \(C\) implies +that the glued operator preserves +\(\mathcal I_C\mathcal M\), by +\eqref{iso:connection-operator}. +It therefore induces a first-order operator +\[ + D:\mathcal M|_C\longrightarrow\mathcal M|_C +\] +with symbol the nowhere vanishing field +\(\widetilde v_0|_C\). +Let \(\tau\) be the holomorphic one-form on the elliptic curve +\(C\) satisfying \(\tau(\widetilde v_0)=1\). +Then +\[ + \nabla s=D(s)\otimes\tau +\] +is a holomorphic connection on \(\mathcal M|_C\). + +A line bundle with a holomorphic connection on a smooth projective +curve has degree zero, a classical connection obstruction +\cite{Atiyah1957}. One elementary proof takes a nonzero +meromorphic section \(s\): the meromorphic one-form +\(\nabla s/s\) has at each point the residue +\(\operatorname{ord}_q(s)\), and the residue theorem gives +\(\deg\operatorname{div}(s)=0\). +Consequently +\[ + \deg(\mathcal M|_C)=0. +\] +On the other hand, \(-K_B\) is ample as a \(\Q\)-Cartier divisor, +and \(p_l|_C\) is finite onto the positive-dimensional curve +\(S_0\). Hence +\[ + \deg(\mathcal M|_C) + =r\,\deg\bigl((p_l|_C)^*K_B\bigr)<0. +\] +This contradiction excludes the nonquotient germ and proves +Proposition~\ref{iso:exclusion}. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/09-second-system.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/09-second-system.tex new file mode 100644 index 0000000000000000000000000000000000000000..e9651ea9a5cbd1c6192520607921a55649a6c39d --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/09-second-system.tex @@ -0,0 +1,612 @@ +\section{A second Lagrangian system} +\label{secsys:section} +\label{sec:second-system} + +Retain the local $e$-locus $D$ and its cohomology marking from +Proposition~\ref{def:fixed-base}, and write $f_0:X_0\to B$ for the +original projective fibration. The geometric target is an actual nearby +manifold with two integral nef isotropic classes whose entire positive +interval lies in its K\"ahler cone. This gives both fibrations on one +projective manifold, with the first still mapping to the original $B$. + +\begin{proposition}[Two systems on one nearby model]\label{secsys:second-system} +Arbitrarily close to the original point of $D$, there is a member +$X'=X_t$ and an integral $(1,1)$-class $u$ such that $e$ and $u$ are +nef and +\[ + q(e)=q(u)=0,\qquad q(e,u)>0, +\] +and the whole positive interval satisfies +\begin{equation}\label{secsys:geometric-target} + \{a e+b u:a,b>0\}\subset\mathcal K_{X'}. +\end{equation} +In particular $X'$ is projective. Its fixed-base fibration +$f'=f_t:X'\to B$ admits a second projective Lagrangian fibration +$g:X'\to C$, with $C$ normal and projective. For every fiber $A$ +over the smooth locus of $g$, the restriction $f'|_A:A\to B$ is +finite and surjective. Thus $B$ has a finite cover by an abelian variety. +\end{proposition} + +The rank-four refinement proved in +Proposition~\ref{secsys:rank-four-variation} chooses $g$ with nonzero +polarized variation when the original fibration has that property. +We first construct the two classes in +\eqref{secsys:geometric-target}. + +We distinguish integral isometries from parallel-transport operators +throughout the proof. This distinction matters in applying Torelli. +Write $\Lambda=H^2(X_0,\Z)$, equipped with $q$, and fix the connected +marked deformation component containing $X_0$. Its orientation of maximal +positive subspaces is the one given by +$(\operatorname{Re}\sigma_0,\operatorname{Im}\sigma_0,\kappa)$, with +$\kappa$ K\"ahler. Markings below are anchored by parallel transport +in this component. + +\subsection{Realized positive three-planes} + +Call an oriented positive three-plane $T\subset\Lambda_{\R}$ +\emph{realized} if it is the span of the period plane and a K\"ahler +class of a marked manifold in the chosen component. The Calabi--Yau +theorem and twistor rotation imply that every compatible oriented +two-plane $P\subset T$, together with its positively oriented orthogonal +ray in $T$, is realized once $T$ is realized. We use period surjectivity +from the chosen marked component for the integral Beauville lattice +\cite[\S3.1 and Theorem~5.5]{HuybrechtsTorelli2012}, together with the +marked period identification +\cite[Corollaries~4.10 and~5.9]{HuybrechtsTorelli2012}. +The K\"ahler form of parallel-transport Torelli says that a Hodge +parallel-transport isometry carrying a K\"ahler class to a K\"ahler +class is induced by an isomorphism +\cite[Theorem~1.3(2) and \S3.2]{Markman2011}. The global Torelli input +in that proof is the preceding marked period identification. +These are the marked-component statements used below. +The twistor description is also recalled in +\cite[Section~5.1]{AmerikVerbitsky2015}. + +\begin{lemma}\label{secsys:metric-periods} +Realized positive three-planes form an open subset. In rank greater +than four, a three-plane fails to be realized precisely when it is +orthogonal to a rational negative MBM class of the marked component. +In rank four, the weaker assertion that every nonrealized three-plane +has a rational negative orthogonal line suffices and holds. +At a fixed oriented period, K\"ahler cones of marked models in the +component are either equal or disjoint. +\end{lemma} + +\begin{proof} +Given a realized pair $(P,\kappa)$, nearby planes $T'$ contain oriented +period planes $P'$ close to $P$ and orthogonal positive vectors +$\kappa'$ close to $\kappa$. Local Torelli realizes $P'$ by a small +deformation of that manifold. Openness of the relative K\"ahler cone +makes $\kappa'$ K\"ahler, proving openness. + +Recall the two parts of the MBM theorem used here: the negative rational +wall classes are invariant under deformation that keeps them of type +$(1,1)$, and the K\"ahler cone is a component of the positive cone +after removing their orthogonal hyperplanes +\cite[Theorems~1.17 and~1.19]{AmerikVerbitsky2015}. +No bound on their squares is used. A realized plane cannot be +orthogonal to such a class: that class would be of type $(1,1)$ and +orthogonal to a K\"ahler class. Conversely, suppose $T$ has no such +orthogonality. Choose a generic oriented plane $P\subset T$, so that +every rational class orthogonal to $P$ is already orthogonal to $T$. +Indeed, each rational vector not orthogonal to $T$ excludes a proper +closed subset of the sphere of planes in $T$, and there are only +countably many such vectors. Period surjectivity gives a marked +manifold with period $P$. Its rational $(1,1)$ classes include no +MBM class. The cone theorem therefore identifies its K\"ahler cone +with the whole positive component. The orthogonal ray to $P$ in $T$, +with sign fixed by the orientation, is in that component. This realizes +$T$. + +For rank four one can avoid even the MBM formulation in this step. +If the negative line $T^\perp$ is irrational, the same generic choice +of $P$ has zero Picard lattice. Its K\"ahler cone is the whole positive +component, so $T$ is realized. Thus a nonrealized plane has rational +negative normal line. + +Finally, an intersection between the K\"ahler cones of two marked models +with the same period gives a Hodge parallel-transport isometry carrying +a K\"ahler class to a K\"ahler class. Torelli makes it the isometry of +an isomorphism. The cones then agree. +\end{proof} + +Choose an integral ample class $h$ on $X_0$. A first arbitrarily small +deformation preserving both $h$ and $e$ makes the real period plane +$P_*$ rational: rational positive two-planes are dense in the rational +space $\langle e,h\rangle^\perp$. In rank four this plane is already +rational, and we take the original point. Ampleness of $h$ persists, and +Proposition~\ref{def:fixed-base} preserves the fibration of class $e$. +Denote this projective member and its fibration by $X_*,f_*$. +Set +\begin{equation}\label{secsys:four-space} + U=P_*\oplus\langle e,h\rangle_{\Q}, + \qquad \operatorname{sign}(q|_U)=(3,1). +\end{equation} +Here and below a rational plane is also used for its real span when +discussing signs and periods. The space +$P_*+\R h$ is realized. Consequently $U^\perp$ contains no MBM +class that could obstruct realization. + +Let $\mathbb H(U)$ be the hyperbolic three-space of negative lines in +$U_{\R}$. A point $[v]$ corresponds to the positive three-plane +$v^\perp\cap U_{\R}$, with its fixed orientation. Let +$J\subset\mathbb H(U)$ be the set of nonrealized planes. It is closed +by Lemma~\ref{secsys:metric-periods}. Moreover, +\begin{equation}\label{secsys:rational-bad-set} + J\subset\mathbb H(U)(\Q). +\end{equation} +For rank greater than four, project an obstructing rational MBM class +orthogonally to $U$. The projection is rational and nonzero, since +$U^\perp$ contains no such class; it is the negative normal line in +$U$. In rank four use the last assertion about nonrealized planes in +Lemma~\ref{secsys:metric-periods}. Any parallel-transport isometry +preserving $U$ preserves $J$. + +Choose the component of the negative cone whose null boundary contains +the ray $\R_{>0}e$. We call its nonzero vectors, including boundary +vectors, future vectors. Distinct future null vectors have negative +pairing. The other null line in $\langle e,h\rangle$ has a rational +future representative +\begin{equation}\label{secsys:other-cusp} + w_*=e-\frac{2q(e,h)}{q(h)}h. +\end{equation} +The negative lines in a Lorentz plane with future null boundary vectors +$v,w$ form the geodesic +$\{[s v+t w]:s,t>0\}$; we denote it by $(v,w)$. + +\subsection{Null rotations from actual algebraic families} + +\begin{lemma}\label{secsys:actual-monodromy} +Let $a\in U_{\Q}$ be nonzero and null. Every rational null rotation +fixing $a$, extended as the identity on $U^\perp$, has a positive power +that is an integral parallel-transport operator in the chosen marked +component. +\end{lemma} + +\begin{proof} +Choose $b\in U_{\Q}$ with $q(a,b)=1$ and $q(b)=0$; subtracting +$q(b_0)a/2$ from any $b_0$ paired to $1$ with $a$ gives such a vector. +Let $V=\langle a,b\rangle^\perp\cap U$, a positive rational plane. +For $x\in V_{\Q}$ the null rotation is +\begin{equation}\label{secsys:null-rotation} + \begin{split} + T_{a,x}a&=a,\\ + T_{a,x}b&=b+x-\tfrac12 q(x)a,\\ + T_{a,x}v&=v-q(v,x)a\quad(v\in V), + \end{split} + \qquad T_{a,x}|_{U^\perp}=\id. +\end{equation} +Direct pairing verifies that it preserves $q$, and substitution gives +$T_{a,x}T_{a,y}=T_{a,x+y}$. Thus $T_{a,x}^m=T_{a,mx}$. +Its matrix entries in an integral basis are polynomials in $m$ of degree +at most two with rational coefficients and constant term the identity. +For sufficiently divisible $m$, both $T_{a,mx}$ and its inverse are +integral. This accounts also for the finite gluing between the +sublattices in $U$ and $U^\perp$. + +For $x\ne0$, choose a rational positive vector +$\ell\in V\cap x^\perp$. The rotation fixes $\ell$. Replace it by +a primitive integral vector on the same line. A generic period in +$\ell^\perp$ has Picard lattice $\Z\ell$. Period surjectivity and +the cone theorem give an ample marked deformation with polarization +$\ell$ or $-\ell$; choose the ample sign. This sign does not alter +the stabilizer of $\ell$. + +We now use the prescribed-polarization conclusion of +\cite[Lemma~4.2]{StrongSYZ2026}. It applies to this primitive positive +class $\ell$, ample on a manifold in the chosen marked component. +Write $D_\ell$ for the component of its type-IV period domain containing +that ample period, and put +\[ + G_\ell(\Z)=\{g\in O(\Lambda,q):g\ell=\ell, + \ gD_\ell=D_\ell\}. +\] +The cited lemma supplies a smooth projective family of IHS manifolds +over a smooth connected quasi-projective variety $S$, with a relative +polarization of class $k\ell$ for some $k>0$ and a reference marking +obtained by parallel transport from $X_0$. It also supplies a +torsion-free finite-index subgroup $\Gamma\subset G_\ell(\Z)$ +containing the family monodromy and an algebraic period map +\[ + \Phi:S\longrightarrow Q_\ell:=\Gamma\backslash D_\ell. +\] +Here $Q_\ell$ is smooth and quasi-projective, the map +$D_\ell\to Q_\ell$ is a covering with free $\Gamma$-action, and a +local marked lift of $\Phi$ is submersive. The type-IV domain +$D_\ell$ is contractible, including in rank four, so this covering +identifies $\pi_1(Q_\ell)$ with $\Gamma$. + +The remaining finite-index assertion concerns the image of this +actual family monodromy, and we prove it here. The submersive local +lift makes the algebraic map $\Phi$ dominant. + +Choose an irreducible algebraic slice $S'\subset S$ of dimension +$\dim Q_\ell$ dominating $Q_\ell$. For example, intersect with +general hyperplanes through a smooth submersive point, transverse to +its period fiber. After normalization and restriction, this slice is +smooth and generically finite over $Q_\ell$. There is a nonempty +Zariski-open $Q^\circ\subset Q_\ell$ over which its restriction is +a nonempty finite \'{e}tale cover: generic finiteness gives a finite +open after removing the boundary in a proper compactification, and +characteristic zero allows removal of the branch locus. +Therefore its fundamental-group image has finite index in +$\pi_1(Q^\circ)$. The map +$\pi_1(Q^\circ)\to\pi_1(Q_\ell)$ is surjective, since a loop in a +complex manifold can be moved off a proper algebraic subset. +It follows that +\begin{equation}\label{secsys:finite-index-image} + [\Gamma:\Phi_*\pi_1(S)]<\infty. +\end{equation} +The period map on the universal cover of $S$, based at this reference +marking, is equivariant for the actual second-cohomology monodromy. +Because $D_\ell\to Q_\ell$ is a +covering with free action, the deck transformation induced by each +loop is exactly that cohomology operator. Hence the subgroup in +\eqref{secsys:finite-index-image} consists of actual parallel transports, +with the original marking retained. + +Our rotation fixes $\ell$; after taking an integral power, a further +power preserves $D_\ell$ and therefore lies in $G_\ell(\Z)$. +Since $\Gamma$ has finite index in $G_\ell(\Z)$, a further power +lies in $\Gamma$; by \eqref{secsys:finite-index-image}, one more +positive power lies in the family monodromy. This uses only that every element has a +positive power in any finite-index subgroup; normality of the subgroup +is unnecessary. Transport to the original reference fiber finishes +the proof. The zero parameter gives the identity. +\end{proof} + +\subsection{An integral loxodromic operator with irrational axis} + +\begin{lemma}\label{secsys:loxodromic} +There is a parallel-transport operator $\gamma$ preserving $U$ and +acting identically on $U^\perp$ such that its characteristic polynomial +on $U$ is irreducible over $\Q$. Its real eigenvalues are +$\rho,\rho^{-1}$ with $\rho>1$, and its other two eigenvalues are a +nonreal conjugate pair on the unit circle. Its attracting and repelling +future null lines $a_+,a_-$ can both be placed in any prescribed +neighborhood of $[w_*]$ in the null boundary. +\end{lemma} + +\begin{proof} +Choose a rational null partner $b$ of $e$ on the line of $w_*$ with +$q(e,b)=1$, and a rational orthogonal basis $v_1,v_2$ of their positive +perpendicular plane. Write $q(v_1)=A>0$, $q(v_2)=B>0$. +For a rational number $t\ne0$, set $x=v_1$, $y=v_1+t v_2$. +Choose $t$ so that +\begin{equation}\label{secsys:phase-choice} + \frac{2(A-Bt^2)}{A+Bt^2}\notin\Z. +\end{equation} +This is possible: the expression lies in $(-2,2)$ and assumes each +of the three possible integral values at only finitely many real $t$. +By Lemma~\ref{secsys:actual-monodromy}, for some positive integers +$m_1,m_2$ the opposite rotations $T_{e,m_1x}$ and $T_{b,m_2y}$ are +integral monodromies. + +For completeness, the spectral calculation can be performed in a real +orthonormal transverse basis without any rationality assumption on +$\sqrt A$ or $\sqrt B$. Identify +$a e+b' b+c v_1+d v_2$ with the Hermitian matrix +\[ + \begin{pmatrix} + \sqrt2 a&\sqrt A c+i\sqrt B d\\ + \sqrt A c-i\sqrt B d&-\sqrt2 b' + \end{pmatrix}. +\] +Its determinant is the negative of $q$. The action +$H\mapsto MHM^*$ of $SL_2(\C)$ induces $SO^+(3,1)$. +Formula~\eqref{secsys:null-rotation} corresponds, for the first +rotation, to an upper triangular unipotent with parameter +$-m_1\sqrt A/\sqrt2$; the opposite rotation corresponds to a lower +triangular unipotent with parameter +$m_2(\sqrt A-i t\sqrt B)/\sqrt2$. +Put +\[ + \gamma_k=T_{e,km_1x}T_{b,km_2y}. +\] +Its spin trace is +\begin{equation}\label{secsys:spin-trace} + \tau_k=2+k^2m_1m_2\zeta, + \qquad \zeta=-\tfrac12(A-i t\sqrt{AB}). +\end{equation} +Let $\lambda_k,\lambda_k^{-1}$ be its spin eigenvalues, choosing +$|\lambda_k|>1$. For large $k$, +$\lambda_k/\tau_k\to1$. The four eigenvalues on $U_{\R}$ are +\[ + |\lambda_k|^2,\quad |\lambda_k|^{-2},\quad + \lambda_k/\overline{\lambda_k},\quad + \overline{\lambda_k}/\lambda_k. +\] +The sum of the last pair converges to +$2\cos(2\arg\zeta)=2(A-Bt^2)/(A+Bt^2)$. +By \eqref{secsys:phase-choice}, it is nonintegral for all sufficiently +large $k$. In particular, this conjugate pair is nonreal and neither +eigenvalue is $1$ or $-1$. + +The characteristic polynomial on $U$ is monic with rational +coefficients, and its roots are algebraic integers because +$\gamma_k$ preserves $\Lambda$. It therefore belongs to $\Z[z]$. +It is reciprocal, has constant term $1$, and has no rational root: +a rational root would be $1$ or $-1$. If it factored into two +quadratics over $\Q$, real coefficients would force the nonreal pair +to belong to one quadratic. That monic factor has integral +coefficients, so the sum of the pair would be an integer, a +contradiction. Thus it is irreducible. Fix a sufficiently large $k$. + +Finally, a nontrivial null rotation fixing $[w_*]$ has powers converging +to $[w_*]$ on every other point of the null boundary. In boundary +coordinates with $[w_*]$ at infinity these powers are translations +$z\mapsto z+jc$, with $c\ne0$. Such a rotation has an integral +monodromy power by Lemma~\ref{secsys:actual-monodromy}. Neither +eigenline of $\gamma_k$ is rational, by irreducibility, so neither is +$[w_*]$. Conjugating $\gamma_k$ by a sufficiently high power of this +rotation brings both eigenlines as close to $[w_*]$ as desired and +preserves all the asserted properties. Call the conjugate $\gamma$. +\end{proof} + +The axis $(a_-,a_+)$ contains no rational negative line. Indeed, a +nonzero rational vector in its invariant real Lorentz plane would +have its entire $\gamma$-orbit in that plane. The rational span of +the orbit would be a nonzero proper rational invariant subspace of +$U$, contradicting irreducibility. The two geodesics $(e,a_\pm)$ +also contain no rational negative line. Otherwise such a line, +together with rational $e$, would define a rational Lorentz plane. +A rational binary quadratic form with one rational null line has +both null lines rational: if $q(e,v)\ne0$, its second null vector is +$v-q(v)e/(2q(e,v))$. The second line would be $a_\pm$, contrary +to irreducibility. Equation~\eqref{secsys:rational-bad-set} gives +\begin{equation}\label{secsys:good-limit-geodesics} + J\cap(a_-,a_+)=J\cap(e,a_+)=J\cap(e,a_-)=\varnothing. +\end{equation} + +\subsection{Uniform tails and the entire K\"ahler interval} + +For a future null line $[w]$ near $[w_*]$, choose its representative +with $q(e,w)=-1$ and put +\[ + P(e,w)=\langle e,w\rangle^\perp\cap U_{\R}, +\] +oriented near $P_*$. This is a nearby $e$-period. It gives an actual +small deformation $X_w$ of $X_*$ within $D$, with the fibration $f_w:X_w\to B$ by +Proposition~\ref{def:fixed-base}. The orthogonal projection of $h$ to +$\langle e,w\rangle$ has the form +\begin{equation}\label{secsys:projected-kahler} + h_w=\alpha(w)e-\beta w,\qquad + \alpha(w)=-q(h,w)>0,\quad \beta=q(h,e)>0. +\end{equation} +It tends to $h$ as $[w]\to[w_*]$, so it is K\"ahler on $X_w$ +after restricting to a small neighborhood. As $e$ is nef on $X_w$, +every $h_w+t e$, $t\ge0$, is K\"ahler. Its orthogonal negative +line in $\langle e,w\rangle$ is +$[(\alpha(w)+t)e+\beta w]$. On a smaller neighborhood $\mathcal V$ +of $[w_*]$, the continuous function $\alpha(w)/\beta$ is bounded +above by some $R$. Thus +\begin{equation}\label{secsys:uniform-tail} + [s e+w]\notin J\qquad + ([w]\in\mathcal V,\ q(e,w)=-1,\ s\ge R). +\end{equation} +This is the required uniformity near the rational endpoint $e$. +Choose the conjugation in Lemma~\ref{secsys:loxodromic} so that both +$a_+$ and $a_-$ belong to $\mathcal V$. + +\begin{lemma}\label{secsys:whole-geodesic} +For all sufficiently large integers $N$, +\[ + (e,\gamma^N e)\cap J=\varnothing. +\] +\end{lemma} + +\begin{proof} +Choose future eigenvectors for $a_+,a_-$ with eigenvalues +$\rho,\rho^{-1}$, respectively. The remaining real invariant +plane is positive definite and $\gamma$ acts on it by a rotation. +Since $e$ is a future null vector different from both eigenlines, +its coefficients along both future eigenvectors are positive. +Indeed, pairing its eigenspace decomposition with either eigenvector +shows the coefficient of the other to be positive. It follows that +\begin{equation}\label{secsys:endpoint-limits} + \rho^{-N}\gamma^Ne\longrightarrow v_+, + \qquad + \rho^{-N}\gamma^{-N}e\longrightarrow v_-, +\end{equation} +where $v_+,v_-$ are positive representatives of $a_+,a_-$. + +If the assertion failed, choose $N\to\infty$ and +$z_N=[s_Ne+\gamma^Ne]\in J$, with $s_N>0$. We first claim +\begin{equation}\label{secsys:balance-bounds} + s_N\rho^{-N}\longrightarrow0, + \qquad s_N\rho^N\longrightarrow\infty. +\end{equation} +To prove the first limit, normalize $\gamma^Ne$ by +$c_N=-q(e,\gamma^Ne)$; by \eqref{secsys:endpoint-limits}, +$c_N/\rho^N$ tends to a positive constant, and +$[\gamma^Ne]\to a_+\in\mathcal V$. +The tail exclusion~\eqref{secsys:uniform-tail} bounds $s_N/c_N$ +above. If a subsequence of $s_N/\rho^N$ had a positive limit, +$z_N$ would converge to an interior point of $(e,a_+)$. +Closedness of $J$ would put that point in $J$, contradicting +\eqref{secsys:good-limit-geodesics}. This proves the first limit. +Apply the same argument to +\[ + \gamma^{-N}z_N + =[s_N\gamma^{-N}e+e] + =[s_N^{-1}e+\gamma^{-N}e]\in J. +\] +The uniform tail at $e$ now uses $a_-\in\mathcal V$ and gives +$s_N^{-1}\rho^{-N}\to0$, which is the second limit. + +Choose integers $k_N$ such that +\[ + 1\le s_N\rho^{2k_N-N}<\rho^2. +\] +The two limits in \eqref{secsys:balance-bounds} imply +$k_N\to\infty$ and $N-k_N\to\infty$. +Passing to a subsequence, suppose +$s_N\rho^{2k_N-N}\to r\in[1,\rho^2]$. +Apply $\gamma^{-k_N}$ to $z_N$ and divide its representative by +$\rho^{N-k_N}$. Equation~\eqref{secsys:endpoint-limits} gives +\[ + \frac{s_N\gamma^{-k_N}e+\gamma^{N-k_N}e}{\rho^{N-k_N}} + =s_N\rho^{2k_N-N} + \frac{\gamma^{-k_N}e}{\rho^{k_N}} + +\frac{\gamma^{N-k_N}e}{\rho^{N-k_N}} + \longrightarrow r v_-+v_+. +\] +The limit has negative square, so its line is an interior point of +the axis $(a_-,a_+)$. Invariance and closedness of $J$ put that +line in $J$, contradicting \eqref{secsys:good-limit-geodesics}. +\end{proof} + +Take such an $N$ and let $X_N$ be the actual nearby $e$-deformation +with first fibration $f_N:X_N\to B$ and period +$P_N=P(e,\gamma^Ne)$, oriented near $P_*$. +The choice of $a_+$ close to $[w_*]$, followed by large $N$, makes +this deformation as small as prescribed. Put +\begin{equation}\label{secsys:second-class} + u_N=-\gamma^Ne,\qquad + \mathcal C_N=\{a e+b u_N:a,b>0\}. +\end{equation} +The minus sign is essential: $e$ and $\gamma^Ne$ are future null +vectors, so $q(e,\gamma^Ne)<0$ and $q(e,u_N)>0$. +The cone $\mathcal C_N$ is a positive component in the Lorentz plane +$\langle e,u_N\rangle$. Its positive rays correspond, by taking +orthogonal negative lines, to the geodesic $(e,\gamma^Ne)$. +Lemma~\ref{secsys:whole-geodesic} and twistor rotation show that every +ray of $\mathcal C_N$ is K\"ahler on some marked model of period $P_N$. + +All those rays are K\"ahler on $X_N$ itself. Indeed, their projectivized +interval is connected. Intersections with the K\"ahler cones of marked +models give an open cover by pairwise disjoint sets, by +Lemma~\ref{secsys:metric-periods}. Such a cover has just one nonempty +member. The class $h_w$ of \eqref{secsys:projected-kahler}, for +$[w]=[\gamma^Ne]$, belongs to this interval and is K\"ahler on $X_N$. +Thus +\begin{equation}\label{secsys:actual-cone} + \mathcal C_N\subset\mathcal K_{X_N}. +\end{equation} +In particular $e+u_N$ is an integral K\"ahler class, making $X_N$ +projective, and $u_N$ is a nonzero integral nef isotropic $(1,1)$ +class on this very manifold. It is the Chern class of a line bundle: +the exponential sequence and $H^1(X_N,\cO_{X_N})=0$ identify +$\Pic(X_N)$ with the integral $(1,1)$ classes. Applying +Theorem~\ref{thm:strong-syz}, including Stein factorization, gives +the second Lagrangian fibration $g:X_N\to C$ with normal projective +base $C$. Both $f_N$ and $g$ are projective morphisms: their graphs +are closed analytic subvarieties of projective products, hence +algebraic, and the corresponding algebraic maps are projective. + +\subsection{Finite maps from its abelian fibers} + +Let $A$ be a fiber over the smooth locus of $g$. It is an abelian +variety after choosing an origin. A positive multiple of $u_N$ is +$g^*c_1(H_C)$ for an ample line bundle $H_C$ on $C$, so the cohomology +class of $A$ is a positive rational multiple of $u_N^n$. To see the +constant, cut $C$ by $n$ general hyperplanes in a sufficiently high +multiple of $H_C$; their transverse intersection is a finite collection +of regular points, and its inverse image is the corresponding collection +of fibers. All regular fibers have the same cohomology class by +parallel transport over the connected smooth locus. + +Polarizing Fujiki gives the exact positive quantity +\begin{equation}\label{secsys:mixed-fujiki} + \binom{2n}{n}\int_{X_N}e^n u_N^n + =c_{X_N}\,2^n q(e,u_N)^n>0. +\end{equation} +Therefore the nef line bundle $f_N^*H|_A$ has positive top +self-intersection. Such a line bundle on an abelian variety is ample. +Indeed, average representatives of its class by translations. Nefness +makes the resulting invariant Hermitian form positive semidefinite, +and positive top intersection makes its determinant positive. All +its eigenvalues are then positive, so the line bundle has a positive +curvature representative and is ample. + +The morphism $f_N|_A$ has no positive-dimensional fiber, since a curve +in such a fiber would have degree zero against this ample line bundle. +It is proper, hence finite. Its image is closed of dimension +$\dim A=n=\dim B$ and therefore equals $B$. This proves all assertions +of Proposition~\ref{secsys:second-system}. In particular we have obtained +\begin{equation}\label{secsys:abelian-cover} + p:A\longrightarrow B +\end{equation} +as a finite surjective morphism from an abelian variety. + +\subsection{Nonzero variation in rank four} + +\begin{proposition}[The rank-four variation refinement] +\label{secsys:rank-four-variation} +If $b_2(X_0)=4$ and the polarized family of smooth fibers of $f_0$ +has nonzero variation, the choices in +Proposition~\ref{secsys:second-system} can be made so that $g$ also +has nonzero polarized variation. +\end{proposition} + +\begin{proof} +Nonzero variation means that, at some regular fiber, the derivative +in a base direction of its weight-one period map is nonzero. This +condition persists under small deformations of the fibration: on a +small smooth product neighborhood in the family supplied by +Proposition~\ref{def:fixed-base}, the relative Hodge bundle and that +derivative vary holomorphically, so a chosen nonzero matrix entry +stays nonzero. + +The fiber polarization can also be transported. Its class is the +restriction of an integral ample class on $X_0$. By +Lemma~\ref{def:restriction-kernel} and topological transport, the +restriction image on a nearby smooth fiber is a rational +one-dimensional space with kernel $e^\perp$. The restriction of a +K\"ahler class spans its real form, since it pairs positively with $e$, +so the transported integral class +remains of type $(1,1)$. Positivity on the smooth torus is open. +This gives a polarized family near the chosen fiber, and the same +argument applies at the conjugate fibration. + +In rank four the preliminary point was $X_*=X_0$. Complex +conjugation of its hyperk\"ahler complex structure reverses the +period-plane orientation and the K\"ahler sign. The resulting marked +manifold $\overline X_*$ has period $\overline P_*$, ample class $-h$, +and conjugate fibration +$\overline f_*:\overline X_*\to\overline B$ with fiber class $-e$. +It lies in the same marked deformation component: a half-circle of +the twistor sphere joins the complex structure to its negative. +Nonzero variation of $\overline f_*$ is equivalent to that of $f_*$. + +We choose the endpoint neighborhood in +Lemma~\ref{secsys:loxodromic} small enough for the following comparison +with this conjugate fibration. Under the actual parallel transport +$\gamma^{-N}$, the underlying plane of $P_N$ becomes +\[ + \langle \gamma^{-N}e,e\rangle^\perp\cap U_{\R}. +\] +It approaches an unoriented plane near $P_*$, since +$[\gamma^{-N}e]\to a_-$ and $a_-$ can be chosen arbitrarily close to +$[w_*]$. The transported positive cone is +\begin{equation}\label{secsys:opposite-cone} + \gamma^{-N}\mathcal C_N + =\{a\gamma^{-N}e-b e:a,b>0\}. +\end{equation} +By \eqref{secsys:other-cusp}, this cone contains a vector close to +$-h$. Preservation of the orientation of maximal positive +three-spaces therefore makes the transported oriented period close +to $\overline P_*$. + +Choose the endpoint neighborhood sufficiently small, and then $N$ +sufficiently large, that this period lies in a neighborhood where the +$(-e)$-deformation of $\overline f_*$ exists and has nonzero variation. +Let $Y_N$ be that actual small deformation of $\overline X_*$ with +period $\gamma^{-N}P_N$. It has a K\"ahler class near $-h$, in the +component \eqref{secsys:opposite-cone}. In rank four, +$H^{1,1}(X_N,\R)=\langle e,u_N\rangle_{\R}$, so +\eqref{secsys:actual-cone} says that the entire positive component +of $X_N$ is K\"ahler. The Hodge parallel-transport isometry between +$Y_N$ and $X_N$ induced by $\gamma^N$ consequently carries a +K\"ahler class to a K\"ahler class. Torelli identifies these +manifolds by an isomorphism inducing that isometry. The class $-e$ +on $Y_N$ becomes $u_N$ on $X_N$. + +Line bundles on an IHS manifold are determined by their Chern +classes. Thus the transported fibration and the fibration obtained +from $u_N$ have the same connected fibers: positive generated powers +define the same Stein factorization. It follows that $g$ has nonzero +polarized variation. The isometry in this comparison is the actual +parallel transport constructed in Lemma~\ref{secsys:actual-monodromy}. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/10-variation.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/10-variation.tex new file mode 100644 index 0000000000000000000000000000000000000000..bb82aad14755b1e1f43b551b530f0257f0a881b5 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/10-variation.tex @@ -0,0 +1,571 @@ +\section{The varying case when \texorpdfstring{$b_2=4$}{b2=4}} +\label{var:section} +\label{sec:variation} + +\begin{proposition}[The varying rank-four case]\label{var:exclusion} +Let $X$ be a projective irreducible holomorphic symplectic manifold of +dimension $2n$ with $b_2(X)=4$, and let +\[ + f:X\longrightarrow B,\qquad + g:X\longrightarrow C +\] +be projective Lagrangian fibrations with normal projective bases. +Assume that the polarized family of smooth fibers of $g$ has nonzero +variation and that, for every smooth fiber $A_t$ of $g$, the restriction +$f|_{A_t}:A_t\to B$ is finite and surjective. +Fix a connected minimum stratum $S_0\subset B$ and its smooth +Deligne--Mumford stack $\cS$ as in Propositions~\ref{str:minimum} +and~\ref{str:products}. Then +\[ + H^{a,b}(\cS)=0\quad\text{for }a\ne b,\qquad + \chi_{\mathrm{top}}(S_0)=\sum_p h^{p,p}(\cS)>0. +\] +\end{proposition} + +Proposition~\ref{secsys:rank-four-variation} supplies these two +fibrations when the original rank-four fibration has nonzero polarized +variation, with the first fibration still mapping to the original +base $B$. The positive Euler characteristic in the proposition +contradicts Proposition~\ref{eu:euler-zero} when the fixed stratum has +a singular maximal chart. + +Fix the data in the proposition, and let $C^\circ$ be the connected +smooth open of regular values of $g$. +Resolve the reduced inverse image of $S_0$ in $g^{-1}(C^\circ)$, +retaining the components which dominate $C^\circ$, and restrict to a +connected dense algebraic open $T\subset C^\circ$ over which the +resolution is smooth. The base $T$ is smooth and quasi-projective. +We obtain smooth projective families +\[ + \pi:\mathcal A=g^{-1}(T)\longrightarrow T, + \qquad \rho:\mathcal Q\longrightarrow T, +\] +with maps $j:\mathcal Q\to\mathcal A$ and $h:\mathcal Q\to\cS$. +The latter map uses the normalized maximal-chart covers, which are +\'etale over the smooth source by purity; it therefore retains all the +stabilizer data of $\cS$. Put $d=\dim S_0$. Each $Q_t$ has dimension +$d$, possibly with several components, and its map to $S_0$ is dominant and generically +finite. Indeed $A_t\to B$ is finite. The proper closed images of the +omitted nondominating components can be removed from $T$. +Put +\[ + V_\Q=R^1\pi_*\Q, + \qquad U^i_\Q=R^i\rho_*\Q, + \qquad H_1=V^{1,0}. +\] +An origin on the fibers $A_t$ is unnecessary: their holomorphic +one-forms are translation invariant and trivialize their cotangent +bundles. All variations in this section are taken over the same +algebraic open $T$. + +To prove diagonal cohomology, suppose that +$0\ne\alpha\in H^{a,b}(\cS)$ with $a>b$. We will turn its +pullbacks into a flat tensor whose exterior factor has a nonzero +all-holomorphic component. We will find a joint monodromy lowering +operator whose block from $V^{1,0}$ to $V^{0,1}$ is invertible, +whereas its action on the factor $U^b$ vanishes after $b+1$ +lowerings. Invariance of the tensor will +then force its $a$th exterior lowering to vanish, contradicting the +all-holomorphic component. The finite maps $A_t\to B$ produce the +tensor; the rank-four invariant-cycle count, together with the +nonzero variation of $g$, produces the invertible lowering operator. + +\begin{lemma}[Differentials and the top cup component] +\label{var:cup-image} +For every $t\in T$ there is a natural factorization +\begin{equation} + h_t^*\Omega^1_{\cS}\longrightarrow + j_t^*\Omega^1_{A_t}\longrightarrow\Omega^1_{Q_t}. + \label{var:factorization} +\end{equation} +If $0\ne\alpha\in H^{a,b}(\cS)$, its pullback is a nonzero global +flat section $y$ of $U^{a+b}_\C$. Moreover, for the morphism of +variations +\begin{equation} + \mathcal T:\Lambda^a V_\Q\otimes U^b_\Q\longrightarrow U^{a+b}_\Q, + \qquad v\otimes u\longmapsto j^*v\smile u, + \label{var:cup-map} +\end{equation} +we have, at every point of $T$, +\begin{equation} + y_t\in\mathcal T_t\bigl( + \Lambda^a H_{1,t}\otimes (U^b_t)^{0,b}\bigr). + \label{var:top-cup} +\end{equation} +\end{lemma} + +\begin{proof} +We first check the independence needed to define +\eqref{var:factorization}. On a maximal chart, lift a form on the +smooth orbit to an ordinary ambient differential, using the local +product presentation of Proposition~\ref{str:products}. Every tangent +vector to the smooth finite cover of this chart maps, at a point above +the orbit, into the orbit tangent space. Here is the trace argument +in this finite-map situation. Work with a pointed finite local branch +$R\subset E$ of generic degree $m$, with $E$ regular. Extend a tangent +vector upstairs to a local derivation $\delta$ of $E$. The rule +\[ + r\longmapsto m^{-1}\operatorname{Tr}_{\operatorname{Frac}(E)/ + \operatorname{Frac}(R)}(\delta r) +\] +is a derivation of $R$: its values are regular by normality, and the +Leibniz rule follows from $R$-linearity of trace. Its value at the +marked point is the differential image of the chosen tangent vector, +since all conjugates in the pointed branch have the same residue and +trace on the residue is multiplication by $m$. Evaluations of +ambient derivations are precisely the orbit tangent directions, by +Proposition~\ref{str:minimum}. + +The normalized pullback of a maximal chart to $A_t$ is smooth and +\'etale over $A_t$. Indeed the quasi-\'etale exceptional locus has +codimension at least two, as does its inverse image under the finite +map $A_t\to B$, and purity applies. Thus the preceding tangent +calculation applies there. The difference of two lifts of an orbit +form vanishes on the orbit tangent, so its pullback vanishes as an +ambient cotangent vector at every point above the orbit. Restricting +to the reduced inverse image and then pulling to its resolution kills +this difference. The resulting maps are compatible on the chart +overlaps and with the finite group actions; hence they descend to +\eqref{var:factorization}. + +The map of the entire family to the fixed stack $\cS$ makes $h^*\alpha$ +a global flat section. It is nonzero on every fiber. To see this, +choose $\beta\in H^{d-a,d-b}(\cS)$ with $\alpha\smile\beta\ne0$, using +Poincar\'e duality and the Hodge decomposition on $\cS$. Pullback on +$H^{2d}(\cS,\C)$ is nonzero: the $d$th power of a coarse ample class +has positive integral on a dominating component of $Q_t$. This also +holds for $d=0$, where the class is the unit. Since $\cS$ is connected, +its top cohomology is one-dimensional; consequently +$h_t^*(\alpha\smile\beta)\ne0$ and $h_t^*\alpha\ne0$. + +Finally, take the $a$th exterior power of +\eqref{var:factorization}, and use +\[ + j_t^*\Omega^a_{A_t} + =\Lambda^a H^0(A_t,\Omega^1_{A_t})\otimes\cO_{Q_t}. +\] +On $H^b$ the resulting sheaf maps factor Dolbeault pullback through +\[ + \Lambda^a H^0(A_t,\Omega^1_{A_t})\otimes H^b(Q_t,\cO_{Q_t}). +\] +Under the Dolbeault identification the last arrow is cup product +with the pulled-back holomorphic forms. This is exactly +\eqref{var:top-cup}. +\end{proof} + +Choose flat polarizations on the variations in +\eqref{var:cup-map}, and the induced tensor polarization on its +source. Write $\mathcal T^\dagger$ for the adjoint. This adjoint is +flat, although the Hodge metrics themselves generally vary. Indeed, +both sides of \eqref{var:cup-map} have weight $a+b$, and a morphism of +Hodge structures commutes with their Weil operators. The adjoint +for their positive Hodge metrics is therefore its transpose for the +flat polarization forms. It is a morphism of variations. +Consequently +\begin{equation} + z=\mathcal T^\dagger y + \in\Gamma(T,\Lambda^a V_\C\otimes U^b_\C) + \label{var:flat-z} +\end{equation} +is flat. Its component +\begin{equation} + z_{\mathrm{top}}\in + \Lambda^a H_1\otimes(U^b)^{0,b} + \label{var:z-top} +\end{equation} +is nonzero at every point. In fact, choose $w$ in this component +with $\mathcal T w=y$. If the orthogonal projection of $z$ to this +component were zero, then +\[ + 0=\langle z,w\rangle + =\langle y,\mathcal T w\rangle + =\langle y,y\rangle>0, +\] +a contradiction. Orthogonality here follows from the factor Hodge +decompositions of the tensor Hodge metric. + +\begin{lemma}[The simultaneous Higgs operator] +\label{var:joint-monodromy} +Let $T$ be a connected smooth quasi-projective algebraic variety, and +let $\pi:\mathcal A\to T$ and $\rho:\mathcal Q\to T$ be smooth +projective morphisms. The fibers of $\rho$ may be disconnected. +For an integer $b\ge0$, put +$V_\Q=R^1\pi_*\Q$ and $U^b_\Q=R^b\rho_*\Q$, choose flat +polarizations on these rational variations, and write $H_1=V^{1,0}$. +Assume the following fixed-part condition for every pure tensor word +in $V_\Q,U^b_\Q$ and their duals: its full-monodromy invariant subspace +is a constant Hodge structure, so every Hodge component of a global +flat invariant tensor is again globally flat. This condition holds +for the geometric systems just specified, as verified below. + +Let $G$ be the full Zariski closure of the monodromy on +$V_\Q\oplus U^b_\Q$. Then $G$ is linearly reductive, though it may +be disconnected. Its Lie algebra $\mathfrak g_\C$, transported by +conjugation as a flat subbundle of +$\End(V_\C)\oplus\End(U^b_\C)$, is a subvariation of Hodge structure +of weight zero. For every tangent vector $x\in T_tT$, +\begin{equation} + (\theta_{V,x},\theta_{U,x})\in\mathfrak g_t^{-1,1}. + \label{var:higgs-in-lie} +\end{equation} +The image +\begin{equation} + E_t=\im\bigl(\mathfrak g_t^{-1,1} + \longrightarrow\Hom(H_{1,t},\overline H_{1,t})\bigr) + \subset\Sym^2 H_{1,t}^* + \label{var:E} +\end{equation} +is a subbundle preserved by the Chern connection. In unitary frames it +satisfies +\begin{equation} + A B^* C+C B^* A\in E_t\qquad(A,B,C\in E_t), + \label{var:triple} +\end{equation} +where $B^*$ is the Hermitian adjoint matrix. +\end{lemma} + +\begin{proof} +First we verify the stated fixed-part condition from the geometric +hypotheses. Write $U=U^b$. A pure tensor word has the form +\[ + \mathbb M= + V^{\otimes r}\otimes(V^\vee)^{\otimes r'} + \otimes U^{\otimes s}\otimes(U^\vee)^{\otimes s'}. +\] +The polarizations give $V^\vee\simeq V(1)$ and +$U^\vee\simeq U(b)$, so +\[ + \mathbb M\simeq + \bigl(V^{\otimes(r+r')}\otimes U^{\otimes(s+s')}\bigr)(r'+bs'). +\] +The untwisted word is a flat Hodge direct summand of +$R^{r+r'+b(s+s')}\varpi_*\Q$ for the corresponding fiber product +$\varpi:\mathcal Y\to T$ of copies of $\pi$ and $\rho$. The canonical K\"unneth +inclusion and projection commute with monodromy and Hodge +decomposition; no algebraic K\"unneth correspondence is required. +The product remains smooth and projective when fibers of $\rho$ are +disconnected. Deligne's Theorem~4.1.1(ii) and Corollary~4.1.2 of +\cite{Deligne1971} supply invariant lifts from a nonsingular +compactification of this product and make their Hodge components +globally flat. Applying the K\"unneth projection and then the Tate +twist proves the fixed-part condition for $\mathbb M$. +It also proves it for flat Hodge direct summands such as exterior +and symmetric powers. The summand projectors decompose every tensor +construction on $V\oplus U$ into these pure words. Since $T$ is +connected, global flat sections are exactly full-monodromy invariants, +and taking the Zariski closure does not change them. + +The same geometric hypotheses give semisimplicity of +$V_\Q\oplus U^b_\Q$ by Deligne's Proposition~4.2.5 and +Theorem~4.2.6 of \cite{Deligne1971}. Both cohomology systems have +rational structures, integral lattices modulo torsion, and +polarizations on the smooth algebraic base $T$. Semisimplicity and +faithfulness imply that the unipotent radical of $G^\circ$ is trivial. +The component group $G/G^\circ$ is finite, so averaging complements +in characteristic zero makes the full, possibly disconnected group +$G$ linearly reductive. + +A linearly reductive subgroup of a general linear group is the pointwise +stabilizer of its invariant mixed tensors. One way to check the +pointwise assertion is to realize the subgroup as the stabilizer of +a line in a rational representation, realize that representation in +mixed tensors, and choose an invariant complementary subspace to the +line by linear reductivity. Its projection is an invariant tensor; fixing +that tensor preserves the line and hence belongs to the subgroup. +Applying this to the faithful representation $V\oplus U^b$, with its +invariant summand projectors included, gives +\begin{equation} + \mathfrak g_t=\{D: D\tau=0\text{ for every invariant mixed tensor } + \tau\text{ at }t\}. + \label{var:tensor-annihilator} +\end{equation} + +For a flat invariant tensor of a single Hodge type, the three parts +of its connection have different Hodge degrees. Its lowering part +therefore vanishes separately. Applying this to every flat Hodge +component just described shows that the simultaneous pair in +\eqref{var:higgs-in-lie} annihilates every tensor in +\eqref{var:tensor-annihilator}. This proves +\eqref{var:higgs-in-lie}; it is an assertion about the Lie algebra of +the pointwise stabilizer. Also, the invariant tensor spaces are +Hodge graded, so their common infinitesimal annihilator is Hodge +graded. It is flat by conjugation transport, and therefore is a +subvariation of the indicated weight-zero endomorphism variation. + +The grading-preserving part of its connection is the restriction of +the ambient Chern connection. Projection to $\End(V)$ is a morphism +of variations; its image is a subvariation, and its degree $(-1,1)$ +part is $E$. This proves constant rank and Chern parallelism. The +weight-one polarization identifies $\overline H_1$ with $H_1^*$ and +makes these lowering matrices symmetric. + +For completeness, the bracket calculation takes place in the +polarization-preserving Lie algebra. In a unitary Hodge frame a +lowering operator has block matrix +\[ + L_A=\begin{pmatrix}0&0\\ A&0\end{pmatrix} + \quad\text{on }H_1\oplus\overline H_1. +\] +The conjugate of a lift of $L_B$ in $\mathfrak g^{-1,1}$ is in +$\mathfrak g^{1,-1}$ and has upper block $B^*$, up to the same +irrelevant polarization sign for all $B$. The double bracket of +lifts of $L_A$, this upper operator, and $L_C$ has lower block +$AB^*C+CB^*A$, up to sign. Its Hodge degree is $(-1,1)$; projection +to the first factor proves \eqref{var:triple}. +\end{proof} + +Apply the lemma to the geometric systems constructed above; its +fixed-part condition was verified in the proof. The Higgs matrices +in $E$ have an additional symmetry supplied by +the Lagrangian system. Symplectic contraction gives an isomorphism +\begin{equation} + \xi:T_T\xrightarrow{\ \sim\ }H_1, + \qquad x\longmapsto[\iota_{\widetilde x}\sigma|_{A_t}], + \label{var:contraction} +\end{equation} +independent of the lift $\widetilde x$ since the fiber is Lagrangian. +Regard $\xi$ as a $V_\C$-valued one-form. It satisfies +$d_\nabla\xi=0$. Indeed its pairing with any locally transported +fiber one-cycle is the fiber integral of the closed form $\sigma$; +exterior differentiation commutes with this integration, and the +restriction of $\sigma$ to the fibers is zero. Taking the +$V^{0,1}$ component gives +\[ + \theta_{V,x}(\xi(y))=\theta_{V,y}(\xi(x)). +\] +The polarization already gives symmetry in the two matrix slots. +After the identification \eqref{var:contraction}, these two +symmetries give a cubic +\begin{equation} + c_t\in\Sym^3H_{1,t}^*,\qquad + c_t(x,-,-)=\theta_{V,\xi^{-1}(x)}\in E_t. + \label{var:cubic} +\end{equation} +This is the Donagi--Markman cubic associated with a Lagrangian system +\cite[Section~7.2, Lemmas~7.1--7.2 and Remark~7.3]{DonagiMarkman1996}; +the calculation above gives the symmetry in the conventions used here. +Nonzero variation means that this cubic is not identically zero. + +\begin{lemma}[Maximal-rank kernels and a symmetric cubic] +\label{var:matrix} +Let $H$ be a finite-dimensional Hermitian vector space and let +$E\subset\Sym^2 H^*$ be a complex subspace satisfying +\eqref{var:triple} in unitary coordinates. Suppose +$c\in\Sym^3H^*$ and $c(x,-,-)\in E$ for every $x\in H$. +If $A\in E$ has the maximum rank occurring in $E$, then +\[ + c(k,-,-)=0\qquad(k\in\ker A). +\] +\end{lemma} + +\begin{proof} +Let $r=\rk A$; the assertion is immediate if $r=0$. By unitary +symmetric diagonalization, take coordinates in which +$A=\operatorname{diag}(\lambda_1,\ldots,\lambda_r,0)$, with +$\lambda_i>0$. The triple operation gives all odd powers +$\operatorname{diag}(\lambda_i^{2m+1},0)$. Interpolation on the +finitely many nonzero singular values therefore gives +\[ + P=\operatorname{diag}(I_r,0)\in E, + \qquad H=F\oplus K,\quad K=\ker A. +\] +The linear map $Z\mapsto PP^*Z+ZP^*P$ preserves $E$ and has +eigenvalues $2,1,0$ on the $FF$, off-diagonal, and $KK$ blocks. +Its polynomial spectral projections thus separate these blocks +inside $E$. A nonzero $KK$ block would, when added to $P$, have rank +larger than $r$. Hence every member of $E$ has zero $KK$ block. +Write +\[ + \mathcal D=\left\{D: + \begin{pmatrix}D&0\\0&0\end{pmatrix}\in E\right\},\qquad + \mathcal B=\left\{B: + \begin{pmatrix}0&B\\B^{\mathsf t}&0\end{pmatrix}\in E\right\}. +\] +Here $D$ is symmetric and $B:K\to F$ is a rectangular matrix. +Triple closure, first with $(P,D,P)$ and then with +$(D,P,\left(\begin{smallmatrix}0&B\\B^{\mathsf t}&0\end{smallmatrix}\right))$, +gives +\begin{equation} + D^*\in\mathcal D,\qquad DB\in\mathcal B. + \label{var:block-stability} +\end{equation} + +For $B\in\mathcal B$, the Schur complement computes +\[ + \rk\begin{pmatrix}I_r&B\\B^{\mathsf t}&0\end{pmatrix} + =r+\rk(B^{\mathsf t}B). +\] +Maximality of $r$ implies $B^{\mathsf t}B=0$. Polarizing in $B$ +shows that, for a fixed $k\in K$, the space +\[ + W_k=\{Bk:B\in\mathcal B\}\subset F +\] +is totally isotropic for the complex symmetric form +$(u,v)\mapsto u^{\mathsf t}v$. In detail, +$B^{\mathsf t}C+C^{\mathsf t}B=0$, and evaluation on $(k,k)$ gives +$(Bk)^{\mathsf t}(Ck)=0$. Equation~\eqref{var:block-stability} +shows that $W_k$ is stable under every $D\in\mathcal D$ and every +$D^*$. + +Put $S=c(k,-,-)$. Every cubic contraction has zero $KK$ block; +by symmetry, for $f\in F$ and $k'\in K$, +\[ + c(k,f,k')=c(f,k,k')=0. +\] +Thus $S$ has only an $FF$ block, also denoted by $S$, in +$\mathcal D$. For $f\in F$, let $B_f$ be the off-diagonal block of +$c(f,-,-)$. Cubic symmetry gives $Sf=B_fk$, so +\begin{equation} + \im S\subset W_k. + \label{var:S-image} +\end{equation} +Since $S$ is symmetric and $W_k$ is isotropic, for $w\in W_k$ and +$f\in F$ we have +\[ + (Sw)^{\mathsf t}f=w^{\mathsf t}Sf=0. +\] +It follows that $S$ annihilates $W_k$. But $S^*$ preserves $W_k$ +by \eqref{var:block-stability}; together with +\eqref{var:S-image} this proves +\begin{equation} + SS^*S=0. + \label{var:positive-triple-zero} +\end{equation} +The singular values of $SS^*S$ are the cubes of those of $S$. +Hence $S=0$. This is the Hermitian positivity step; no positivity +of the complex symmetric form was asserted. +\end{proof} + +\begin{lemma}[Existence of an invertible lowering matrix] +\label{var:invertible} +Under the hypotheses of Proposition~\ref{var:exclusion}, every fiber +of the bundle $E$ attached to the families $\pi,\rho$ above contains +an invertible matrix. +\end{lemma} + +\begin{proof} +Let $r$ be the maximum rank in $E_t$. It is independent of $t$, +because Chern parallel transport is unitary and carries $E_t$ onto +$E_{t'}$ by congruence. Define intrinsically +\begin{equation} + K_t^{\mathrm{all}}= + \sum_{\substack{A\in E_t\\\rk A=r}}\ker A\subset H_{1,t}. + \label{var:all-kernels} +\end{equation} +Unitary Chern transport preserves the entire collection of these +kernels, and therefore their span. Thus \eqref{var:all-kernels} +is a global Chern-parallel subbundle of constant rank. No matrix +$P$, chosen kernel, or finite cover enters its definition. + +By Lemma~\ref{var:matrix}, the cubic vanishes when any argument lies +in $K^{\mathrm{all}}$. In particular all lowering Higgs operators +annihilate this subbundle. Write the real flat connection on $V$ as +\[ + \nabla=D+\theta+\overline\theta, +\] +with $D$ the Chern connection. On $K^{\mathrm{all}}\subset V^{1,0}$, +$\theta$ vanishes by the cubic and $\overline\theta$ vanishes by +Hodge degree. Complex conjugation in the real variation shows that +$\overline\theta$ vanishes on +$\overline{K^{\mathrm{all}}}\subset V^{0,1}$, while $\theta$ vanishes +there by degree. Consequently +\[ + K^{\mathrm{all}}\oplus\overline{K^{\mathrm{all}}} +\] +is the complexification of a real flat subbundle $L\subset V_\R$; +its flat connection is unitary. Positivity of the Hodge metric +shows that the alternating polarization is nondegenerate on $L$. +Its symplectic orthogonal $L^\perp$ is also a global flat subbundle. +These statements concern the full flat connection on $T$, and hence +full monodromy around every loop. + +We now compute the relevant full-monodromy invariant space. +The use of the one-dimensional fiber-restriction image to prove +irreducibility over $\R$ also appears in Bakker's proof of Matsushita's +variation conjecture \cite[Lemma~5]{Bakker2025}, building on Voisin's +argument. Here it rules out the specific flat summand extracted from +the maximal-rank kernels. By the +global invariant-cycle theorem, in the form of +\cite[Theorem~4.1.1(ii)]{Deligne1971} applied to +$g^{-1}(C^\circ)\to C^\circ$ with smooth projective compactification +$X$, the image of +\[ + H^2(X,\Q)\longrightarrow H^2(A_t,\Q)=\Lambda^2 V_{\Q,t} +\] +is precisely the invariant subspace. Its dimension is one: the +independent classes $\sigma$, $\overline\sigma$, and $g^*c_1(H_C)$ +restrict to zero, whereas an ample class restricts nontrivially; +here $H_C$ is ample and $b_2(X)=4$. +The inclusion $T\hookrightarrow C^\circ$ induces a surjection of +fundamental groups, because loops in a smooth complex variety can +be moved off a closed subset of positive complex codimension. +Since $V|_T$ is the restriction of the same local system, its +monodromy image is unchanged. Therefore +\begin{equation} + \dim_\R(\Lambda^2 V_{\R,t})^{\pi_1(T,t)}=1. + \label{var:one-invariant} +\end{equation} +Invariants commute with extension from $\Q$ to $\R$: they are the +common kernels of rational monodromy-minus-identity maps, and a +finite subcollection suffices in finite dimension. + +If $L$ were nonzero and proper, the polarization composed with the +two flat projections onto $L$ and $L^\perp$ would give two independent +real invariant alternating forms. The polarization identifies the +dual alternating-square representation with $\Lambda^2 V_\R$, so +this contradicts \eqref{var:one-invariant}. Rationality of $L$ is +not needed. If $L=V_\R$, the Higgs field vanishes identically, +contradicting the chosen nonzero variation of $g$ (which remains +nonzero on every dense open). Thus $L=0$. If $rb$. The flat section +$z$ in \eqref{var:flat-z} is fixed by the joint algebraic monodromy +$G$, and its component \eqref{var:z-top} is nonzero. At a point +$t\in T$, choose an invertible matrix $A\in E_t$ and a lift +$\nu\in\mathfrak g_t^{-1,1}$. Let $N_V$ be its induced action on +$\Lambda^a V_t$ and $N_U$ its action on $U^b_t$, acting on their +respective tensor factors. Invariance gives +\[ + (N_V+N_U)z_t=0. +\] +Since $U^b_t$ has Hodge types $(p,b-p)$ with $0\le p\le b$, the +operator $N_U$ satisfies $N_U^{b+1}=0$. The two tensor actions +commute, so induction gives +\begin{equation} + N_V^a z_t=(-1)^aN_U^a z_t=0, + \label{var:nilpotence} +\end{equation} +because $a>b$. + +On $V_t$, the operator $\nu$ maps $H_{1,t}$ isomorphically onto +$\overline H_{1,t}$, and kills $\overline H_{1,t}$. Consequently +$N_V^a$ kills every summand of $\Lambda^a V_t$ having fewer than +$a$ factors in $H_{1,t}$. On the whole sector with all $a$ exterior +factors holomorphic, including its $U^b_t$ factor, it is +\[ + \left.N_V^a\right|_{\Lambda^a H_{1,t}\otimes U^b_t} + =a!\,\Lambda^a A\otimes\id_{U^b_t}: + \Lambda^a H_{1,t}\otimes U^b_t + \xrightarrow{\ \sim\ } + \Lambda^a\overline H_{1,t}\otimes U^b_t. +\] +The projection of $z_t$ to this sector is nonzero, because its further +projection to $(U^b_t)^{0,b}$ is the nonzero component +\eqref{var:z-top}. The displayed map is injective and every other +exterior sector is killed, so $N_V^a z_t\ne0$, +contradicting \eqref{var:nilpotence}. Thus no $H^{a,b}(\cS)$ with +$a>b$ is nonzero; complex conjugation excludes $a0, +\] +since $h^{0,0}(\cS)=1$. Rational cohomology of the smooth stack is +that of its coarse space, including when its stabilizers are +ineffective. This proves the stated positive Euler characteristic. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/11-quotient.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/11-quotient.tex new file mode 100644 index 0000000000000000000000000000000000000000..d62546ebd9cb4076b846cae0b2a5c8a218e43a2f --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/11-quotient.tex @@ -0,0 +1,871 @@ +\section{Eliminating quotient singularities}\label{quot:section} +\label{sec:quotient} + +\begin{proposition}[Smoothness under a finite cover by an abelian variety] +\label{quot:smooth-base} +Let \(f:X\to B\) be the original projective Lagrangian fibration in +the hypotheses of Theorem~\ref{thm:main}. Suppose that \(B\) has +quotient singularities and that there is a finite surjective morphism +\[ + p:A\longrightarrow B +\] +from an abelian variety \(A\) of dimension \(n\). Then \(B\) is smooth. +\end{proposition} + +The two inputs have different roles. The original fibration supplies +the direct-image identity and the Fano property of \(B\). The abelian +variety supplies translation fields, and the Fano property will make +the ramification line bundle of the lifted cover ample. No Galois +property of \(p\) is assumed. The proof therefore retains all arrows +of its relation over the canonical stack, together with their labels. + +\subsection{The canonical stack and its labelled relation} +\label{quot:stack-field} + +Write \(\mathfrak B\) for the smooth canonical Deligne--Mumford stack +of \(B\), and \(c:\mathfrak B\to B\) for its coarse morphism. +Its local presentations are \([U/G]\), where \(U\) is smooth and the +finite group \(G\) acts faithfully and freely in codimension one. +These presentations can equivalently be obtained from the maximal +charts of Proposition~\ref{loc:maximal-charts}: their normalized overlaps +are now étale everywhere. In particular, \(\mathfrak B\) is proper, +has trivial generic stabilizer, and agrees with \(B\) over \(B_{\rm reg}\). +All fiber products of stacks below are \(2\)-fiber products. An arrow in +such a fiber product retains its isomorphism label, even when its two +coarse coordinates coincide. + +\begin{lemma}[Representable lifts]\label{quot:lifts} +The maps \(f\) and \(p\) lift to representable morphisms +\[ + \widetilde f:X\longrightarrow\mathfrak B, + \qquad \widetilde p:A\longrightarrow\mathfrak B, + \qquad c\widetilde f=f,\quad c\widetilde p=p. +\] +The first is projective and equidimensional of relative dimension +\(n\), with connected fibers. The second is finite, flat, and surjective. +\end{lemma} + +\begin{proof} +Take a local Galois chart \(U\to U/G\) as above. Its non-étale +locus on the coarse space has codimension at least two. +The inverse image of that locus has codimension at least two in \(X\), +by equidimensionality, and in \(A\), by finiteness. +Consequently, the full normalizations of the components of the +pullbacks of \(U\to U/G\) that dominate these smooth sources are +finite, by excellence, and étale over the sources, by purity of the branch locus +\cite[Tag 0BMB]{StacksProject}. Their deck actions are torsor actions: +the torsor identities hold on the dense étale locus and extend uniquely +to the finite étale covers. The torsors and their equivariant maps to +\(U\) are precisely the data defining the two lifts. +Uniqueness on normalized overlaps makes these data compatible and +gives the stated representable lifts. The normalized-source package +of Proposition~\ref{loc:maximal-charts} gives the asserted properties +of \(\widetilde f\) on each chart: its source is smooth and symplectic, +its map to \(U\) is projective with Lagrangian equidimensional fibers, +and Stein factorization +gives connected fibers and direct image \(\cO_U\). + +The pullback of \(\widetilde p\) to \(U\) is finite, since it is finite +over the ordinary finite base change \(A\times_B U\). It is a morphism +between smooth \(n\)-dimensional varieties and is flat by miracle flatness +\cite[Tag 00R4]{StacksProject}. Thus \(\widetilde p\) is finite flat, and it is +surjective because its coarse morphism is surjective. +\end{proof} + +The lifted cover defines the finite relation +\begin{equation}\label{quot:relation} + \mathcal R=A\times_{\mathfrak B}A + \ \substack{\xrightarrow{\ t\ }\\[-.6ex]\xrightarrow[\ s\ ]{\ }} + A,\qquad + \delta:\mathcal R\longrightarrow A,\qquad + \delta(\xi)=t(\xi)-s(\xi). +\end{equation} +An arrow with source \(x\) and target \(y\) is a triple +\((x,y,\eta)\), where +\(\eta:\widetilde p(x)\simeq\widetilde p(y)\) is its isomorphism label. +Both projections are finite flat. The relation is generically +étale over \(A\), hence reduced: a nilpotent section would be zero +generically and would contradict torsion-freeness over the integral +base \(A\). It is pure-dimensional of dimension \(n\), and every +irreducible component is finite and surjective over \(A\) under both +projections. +If \(\nu:Z\to\mathcal R\) is the normalization of a component followed +by inclusion, we also write \(s,t\) for the induced coordinate maps +on \(Z\), and put \(\delta_Z=\delta\circ\nu\). + +The identity section \(i:A\to\mathcal R\) is a closed immersion. +The inertia stack \(I(\mathfrak B)\) parametrizes a point of +\(\mathfrak B\) together with one of its automorphisms. Thus +\[ + \delta^{-1}(0) + =A\times_{\mathfrak B}I(\mathfrak B). +\] +In this zero-difference fiber the image of \(i\) is both open and +closed: the identity section of the finite unramified inertia is +open, and separatedness makes it closed. We call its complement +the nonidentity part. In particular, a point in that complement is +an arrow \((x,x,h)\) with \(h\ne1\); its label is retained even though +its two coordinates agree. + +Retain the ample line bundle \(H\) on \(B\) chosen in +Section~\ref{sec:setup}, and put \(P=p^*H\) on \(A\). The two +coordinate pullbacks of \(P\) to \(\mathcal R\) are isomorphic. +For an abelian subvariety \(D\subset A\), define its polarized +complement by +\begin{equation}\label{quot:orthogonal} + D^\perp + =\ker\bigl(A\xrightarrow{\lambda_P}\widehat A + \longrightarrow\widehat D\bigr)^0, + \qquad + \lambda_P(a)=t_a^*P\otimes P^{-1}. +\end{equation} +The superscript \(0\) means identity component. The nondegenerate +rational alternating form of the polarization gives +\(\dim D^\perp=n-\dim D\), \((D^\perp)^\perp=D\), and reversal of +inclusions under this operation. + +Here is the maximality argument that will eliminate nontrivial stabilizers. +If a nontrivial stabilizer existed, surjectivity of \(\widetilde p\) +would give a nonidentity arrow of difference zero. Choose, among all +relation components containing such an arrow, one with difference +image of largest dimension. We first construct one translation field +\(v\) on \(A\). For every nonidentity automorphism \(h\), the +\(h\)-fixed projection of \(d\widetilde p(v)\) will be nonzero at every +lift of the corresponding inertia point. This will force each of +these difference images to be proper +in \(A\). The equality of the two polarization pullbacks will then +make the chosen image a torsion coset \(\tau+D\). It contains zero, +so \(\tau\in D\) and the image is the subgroup \(D\) itself. Its +properness makes \(K=D^\perp\) positive-dimensional. + +Finally, ramification will supply another relation component with +differences outside \(D\). On the normalization of the chosen +component, a \(K\)-translation will move a point above nonidentity +inertia over a ramification divisor. A whole divisor through that +translated point will still map onto \(D\). We will +place that divisor and its marked point in one component of the space +of composable arrows. Composition on this component preserves the +marked nonidentity arrow and produces a difference image strictly +larger than \(D\), contradicting the choice of dimension. The next +calculation constructs the translation field needed for the first step. + +\subsection{A translation field along nonidentity inertia} + +\begin{lemma}[A \(K\)-theory identity]\label{quot:k-identity} +In \(K^0(\mathfrak B)\), +\begin{equation}\label{quot:direct-image} + \widetilde f_![\cO_X] + =\sum_{j=0}^{n}(-1)^j[\Omega^j_{\mathfrak B}] + =\lambda_{-1}(\Omega^1_{\mathfrak B}). +\end{equation} +Moreover, for some integer \(m>0\), +\begin{equation}\label{quot:nilpotence} + \bigl([T_{\mathfrak B}]-n[\cO_{\mathfrak B}]\bigr)^m + \lambda_{-1}(\Omega^1_{\mathfrak B})=0 . +\end{equation} +\end{lemma} + +\begin{proof} +Use the global isomorphisms +\(\phi_j:R^j f_*\cO_X\xrightarrow{\sim}\Omega_B^{[j]}\) +chosen once on the original base in Section~\ref{sec:setup}, with +\(\phi_0\) the unit +isomorphism. We apply the chart comparison of +Proposition~\ref{str:direct-images}. Choose a connected affine quotient +chart \(U\to U/G\) with \(U/G\to B\) an integral separated étale +neighborhood, and view \(U\to\mathfrak B\) as its atlas map. +Write \(f_U:X_U\to U\) for the normalized lift, +put \(U^0=U\times_B B_{\rm reg}\), and let \(\pi:U^0\to B_{\rm reg}\) +be the induced étale morphism. The complement of \(U^0\) has +codimension at least two. Over this open the lift is the ordinary +base change of the original fibration. Ordinary flat base change for +\(\pi\), followed by the étale cotangent identification, gives +\[ + (R^j(f_U)_*\cO_{X_U})|_{U^0} + \simeq\pi^*(R^j f_*\cO_X|_{B_{\rm reg}}) + \xrightarrow{\ \pi^*\phi_j\ }\Omega^j_{U^0}. +\] +This open includes the discriminant over the regular part of \(B\); +the flat map used here is \(\pi\), so no smoothness of the fibers is +required for the base-change step \cite[Tag 02KH]{StacksProject}. + +For \(j>0\), Ou's local reflexivity theorem applies to \(f_U\): +Proposition~\ref{loc:maximal-charts} supplies its projective, +equidimensional map with connected fibers and smooth canonical-trivial +source, and \(U\) is smooth +\cite[Theorem~1.3]{Ou2019}. Both \(R^j(f_U)_*\cO_{X_U}\) and +\(\Omega_U^j\) are therefore their reflexive extensions from \(U^0\). +The displayed isomorphism extends uniquely across the complement. +For \(j=0\) the same conclusion is the unit isomorphism from +\((f_U)_*\cO_{X_U}=\cO_U\). On a chart overlap the extended maps +agree over the inverse image of \(B_{\rm reg}\), by transitivity of +flat base change from the same \(\phi_j\). Every component of the +overlap meets this open, since it maps étale to a chart and cannot be +supported over its codimension-at-least-two complement. Uniqueness +of reflexive extension makes the maps agree everywhere. Thus they +glue to degreewise isomorphisms +\[ + R^j\widetilde f_*\cO_X\simeq\Omega^j_{\mathfrak B} + \qquad(0\le j\le n). +\] +For \(j>n\), the restriction to \(U^0\) is zero by +Proposition~\ref{prop:background}, and reflexivity gives zero on \(U\). +Thus no higher direct-image terms occur. + +These isomorphisms first give \eqref{quot:direct-image} in coherent +\(K\)-theory. In this setting coherent and vector-bundle \(K\)-theory +agree. Indeed, \(T_{\mathfrak B}\) is faithful on every stabilizer by +linearization of finite group actions. A finite direct sum +\(\mathcal G\) of bounded tensor powers of this bundle and its dual +contains every irreducible stabilizer representation occurring on a +finite system of quotient charts. Exactness of finite-group +invariants makes the evaluation +\[ + c^*c_*(\mathcal G^\vee\otimes\mathcal F)\otimes\mathcal G + \longrightarrow\mathcal F +\] +surjective for every coherent \(\mathcal F\). For sufficiently large +\(k\), the coherent sheaf +\(c_*(\mathcal G^\vee\otimes\mathcal F)\otimes H^k\) is globally +generated on \(B\). Its generators yield a surjection from a finite +sum of \(\mathcal G\otimes c^*H^{-k}\) onto \(\mathcal F\). +Iterating and using regularity of the smooth stack gives a finite +locally free resolution. This proves the claimed comparison and +allows the identity to be pulled back in vector-bundle \(K^0\), even +along an inertia morphism which is not flat. + +We spell out the nilpotence needed for +\eqref{quot:nilpotence}. If \(V\) is a vector bundle of rank \(r\) on +a smooth projective variety \(Y\), then \([V]-r\) is nilpotent in +\(K^0(Y)\). Indeed, pull back to the full flag bundle of \(V\). +This pullback is injective on \(K^0\), since pushforward of its +structure sheaf is \(\cO_Y\), with no higher direct images. +The pulled-back class is a sum of classes \([L]-1\) for line bundles. +For a globally generated line bundle \(L\), a finite set of generating +sections and its exact Koszul complex show that +\((1-[L^{-1}])^N=0\) for some \(N\). +An arbitrary line bundle is a quotient of two sufficiently positive +globally generated line bundles. Therefore its class minus one is +also nilpotent: it is an invertible multiple of a difference of two +commuting nilpotent elements. A finite sum of commuting nilpotents is +nilpotent. This proves the assertion. + +Apply it on the original projective \(X\) to +\(\widetilde f^*T_{\mathfrak B}\). +The projection formula and \eqref{quot:direct-image} give +\eqref{quot:nilpotence}, with one exponent \(m\) before any inertia +component is chosen. +\end{proof} + +\begin{lemma}[A translation field with no fixed-direction zero] +\label{quot:translation-field} +For each connected component \(S\) of the nonidentity inertia of +\(\mathfrak B\), let \(j:S\to\mathfrak B\) be its natural morphism, +let \(h\) denote the tautological finite-order automorphism, and set +\[ + d=\dim S,\qquad + W=A\times_{\mathfrak B}S,\qquad + a:W\to A,\qquad q:W\to S,\qquad E=q^*T_S . +\] +There is a translation-invariant vector field +\(v\in\operatorname{Lie}(A)\) such that, simultaneously for every such +\(S\), the section +\begin{equation}\label{quot:projected-section} + s_{v,S}\in H^0(W,E),\qquad + s_{v,S}=\operatorname{pr}_{h=1}\bigl(d\widetilde p(v)|_W\bigr), +\end{equation} +has no zero. +\end{lemma} + +\begin{proof} +An inertia component is a smooth proper stack: on a quotient chart its +presentation is a fixed locus \(U^h\) modulo the appropriate centralizer. +The fixed eigensubbundle of \(j^*T_{\mathfrak B}\) is \(T_S\). +Also, \(W\) is a scheme, since it is finite over \(A\). +The morphism \(W\to S\) is finite flat, being a base change of +\(\widetilde p\). +Thus \(W\) is projective, pure-dimensional of dimension \(d\), and +Cohen--Macaulay. These assertions hold even if \(W\) is nonreduced. + +Restrict \eqref{quot:nilpotence} to \(S\) and take the character of the +tautological automorphism. This is the inertia eigenbundle character +used in To{\"e}n's Riemann--Roch framework +\cite[Section~3.3 and Lemma~4.6]{Toen1999}; only its explicitly described +ring homomorphism and moving-factor cancellation are needed here. More explicitly, for a bundle \(V\) on +\(\mathfrak B\), decompose +\(j^*V=\bigoplus_{\zeta} V_\zeta\) into the \(h\)-eigenbundles and use +the ring homomorphism +\[ + [V]\longmapsto\sum_{\zeta}\zeta[V_\zeta] + \quad\hbox{in }K^0(S)\otimes_{\Z}\C . +\] +Pull this identity to \(W\) and take its Chern character in +\(H^{\rm even}(W,\C)\). +The degree-zero term coming from +\([T_{\mathfrak B}]-n\) is +\(\operatorname{tr}(h|T_{\mathfrak B})-n\), which is nonzero. +Indeed, the tangent action is faithful and all eigenvalues have +absolute value one; their sum can be \(n\) only if every eigenvalue is +one. This would make \(h\) the identity by finite-order linearization. +The character of each moving cotangent factor in the alternating +exterior sum has nonzero degree-zero term +\((1-\zeta)^{\operatorname{rk}V_\zeta}\), for \(\zeta\ne1\). +The eigenbundle ranks are constant on connected \(S\), so each of +these degree-zero terms is the same nonzero scalar on every connected +component of \(W\). Each full class is therefore a unit: every +positive-degree element of the cohomology of the finite-dimensional +projective scheme \(W\) is nilpotent. Cancelling them gives +\[ + \operatorname{ch} + \bigl(\lambda_{-1}(q^*\Omega^1_S)\bigr)=0. +\] +The term of degree \(2d\) on the left is \(c_d^{\rm top}(E)\), since for +Chern roots \(x_i\) of \(E\) the expression is +\(\prod_i(1-e^{-x_i})\). Hence +\begin{equation}\label{quot:fixed-chern-zero} + c_d^{\rm top}(E)=0 + \quad\hbox{in }H^{2d}(W,\C). +\end{equation} +In particular, a nonempty nonidentity inertia component of dimension +zero is impossible, as in that case this says \(1=0\). + +Consider the bundle map +\[ + \alpha:\operatorname{Lie}(A)\otimes\cO_W + =a^*T_A\longrightarrow E +\] +used in \eqref{quot:projected-section}. +For every reduced irreducible subvariety \(Y\subset W\) of dimension +\(s\), this map has rank at least \(s\) at its generic point. +In fact, \(Y\to S\) is finite onto its image and is generically +separable. On a smooth dense open of \(Y\), and after an étale chart +on \(S\), its differential consequently has rank \(s\). +The differential identity for the fiber square says +\[ + \alpha\circ d(a|_Y)=d(q|_Y), +\] +because the latter differential has values in the fixed tangent +subbundle. This proves the rank assertion. + +It follows that the locus on \(W_{\rm red}\) where +\(\operatorname{rk}\alpha\le r\) has dimension at most \(r\): +apply the preceding assertion to each of its irreducible components. +Over the stratum of rank \(r\), the incidence variety +\[ + \{(w,u)\in W_{\rm red}\times\operatorname{Lie}(A): + \alpha_w(u)=0\} +\] +has dimension at most \(r+(n-r)=n\). +There are finitely many rank strata. Thus a general \(u\) gives a +section whose zero scheme has at most zero-dimensional support. +It is a finite scheme because \(W\) is proper. + +At every such zero, the \(d\) local entries of the section are a +system of parameters in a \(d\)-dimensional Cohen--Macaulay local ring, +hence a regular sequence \cite[Tag 02JN]{StacksProject}. +Let \([W]_d=\sum_i\operatorname{length}(\cO_{W,\eta_i})[W_i]\) +be the scheme-theoretic fundamental cycle, where the \(W_i\) are the +irreducible components with generic points \(\eta_i\). The algebraic +zero-section formula gives +\[ + c_d^{\rm CH}(E)\cap[W]_d=[Z(s_{u,S})]_0 + \quad\hbox{in }\mathrm{CH}_0(W) +\] +\cite[Tag 0FA9]{StacksProject}. The complex Borel--Moore cycle class commutes with +vector-bundle Chern classes and proper pushforward +\cite[Chapter~19]{Fulton1998}. With the same generic multiplicities +in \(\operatorname{cl}_{\rm BM}([W]_d)\), this gives +\[ + \operatorname{length} Z(s_{u,S}) + =\deg\bigl(c_d^{\rm CH}(E)\cap[W]_d\bigr) + =\bigl\langle c_d^{\rm top}(E), + \operatorname{cl}_{\rm BM}([W]_d)\bigr\rangle=0, +\] +where the last equality is \eqref{quot:fixed-chern-zero}. Each +nonempty isolated zero has strictly positive length. +Thus there are no zeros. +The inertia has only finitely many connected components, so the +intersection of the corresponding nonempty open sets of choices of +\(u\) supplies the single field \(v\) in the statement. +\end{proof} + +\subsection{Difference images and translations}\label{quot:differences} + +We now use the field to control every relation component meeting the +nonidentity part of \(\delta^{-1}(0)\). The first step rules out a +dominant difference image; the polarization will then describe each +remaining image and its translation symmetry. + +\begin{lemma}[Dominant differences avoid nonidentity inertia] +\label{quot:nondominance} +If an irreducible component of \(\mathcal R\) contains a nonidentity +arrow with difference zero, its difference image is a proper +subvariety of \(A\). +\end{lemma} + +\begin{proof} +Let \(Z\) be the normalization of such a component, let \(z_0\in Z\) +map to the given arrow, and suppose that \(\delta_Z:Z\to A\) is +dominant. Take its Stein factorization +\[ + Z\xrightarrow{\beta}T\xrightarrow{\gamma}A. +\] +Here \(T\) is normal, \(\gamma\) is finite and dominant, and \(\beta\) has connected +fibers. The connected fiber through \(z_0\) lies entirely over +nonidentity arrows: its image lies in \(\delta^{-1}(0)\), where the +identity and nonidentity parts are open and closed. + +Put \(t_0=\beta(z_0)\), and take the field \(v\) of +Lemma~\ref{quot:translation-field}. +A finite dominant map to the smooth \(n\)-fold \(A\) is locally open. +For completeness, a sufficiently small local branch of \(T\) at +\(t_0\) is finite over a neighborhood of \(0\); its analytic image is +closed and has dimension \(n\), so contains a neighborhood of \(0\). +We can therefore lift a sequence +\(\exp(\varepsilon_\nu v)\), with +\(0\ne\varepsilon_\nu\to0\), to points of \(T\) tending to \(t_0\). +Lift these points farther to \(Z\). +Properness gives a subsequence tending to a point +\(z_\infty\in\beta^{-1}(t_0)\). +Its arrow is nonidentity, and the arrows in the sequence have +coordinates +\[ + (x_\nu,y_\nu),\qquad + y_\nu=x_\nu+\exp(\varepsilon_\nu v), + \qquad x_\nu,y_\nu\longrightarrow x_\infty . +\] + +Choose a local quotient chart at \(\widetilde p(x_\infty)\), a lift +\(\psi\) of \(\widetilde p\) on a small analytic neighborhood of +\(x_\infty\), and finite-group linearizing coordinates at +\(\psi(x_\infty)\). +After passing to a subsequence the labels are one fixed group element +\(h\ne1\), fixing \(\psi(x_\infty)\), and +\[ + \psi(y_\nu)=h\psi(x_\nu). +\] +Projecting to the \(h\)-fixed coordinate space yields +\[ + \operatorname{pr}_{h=1} + \bigl(\psi(x_\nu+\exp(\varepsilon_\nu v))-\psi(x_\nu)\bigr) + =0. +\] +Division by \(\varepsilon_\nu\) and passage to the limit give +\(\operatorname{pr}_{h=1}d\psi_{x_\infty}(v)=0\). +This is a zero of the section \eqref{quot:projected-section} on the +inertia component containing \(h\), contrary to +Lemma~\ref{quot:translation-field}. +\end{proof} + +\begin{lemma}[Difference images and complementary translations] +\label{quot:difference-image} +Let \(Z\) be the normalization of an irreducible component of +\(\mathcal R\), and suppose that its difference image \(D_Z\) has +dimension \(r1. +\] +Its self fiber product has, besides the identity branch, branches +\(v_1=\zeta u_1,\ v_j=u_j\) for \(\zeta^e=1,\ \zeta\ne1\). +They meet the identity along \(L\). +There are only finitely many global irreducible components of +\(\mathcal R\). Their closed intersections with \(i(L)\), excluding +the identity component, contain a dense open of \(i(L)\). +Irreducibility therefore gives a single residual component +\(\mathcal R'\ne i(A)\) containing all of \(i(L)\), including its +special points. + +Suppose, for a contradiction, that \(\delta(\mathcal R')\subset D\). +Lemma~\ref{quot:difference-image} says that this difference image is a +torsion coset. It contains zero, since \(i(L)\subset\mathcal R'\), +so it is an abelian subvariety \(D'\subset D\). +Its polarized complement \(K'=(D')^\perp\) contains \(K\). +The same lemma gives a \(K\)-action on the normalization \(Z'\) of +\(\mathcal R'\). + +Consider the closed set +\[ + F=\{x\in A:i(x)\in\mathcal R'\}. +\] +It is proper: otherwise \(\mathcal R'\) would contain \(i(A)\), +and the two irreducible \(n\)-dimensional components would be equal. +For any \(x\in F\), choose an individual preimage of \(i(x)\) in \(Z'\). +Its connected \(K\)-orbit remains over difference zero and remains +in the inverse image of the identity part of \(\delta^{-1}(0)\), +since that part is open and closed. Its image at \(k\) is therefore +precisely \(i(x+k)\). Hence \(F\) is \(K\)-invariant. +This reasoning does not require a global lift of \(i(L)\) to \(Z'\). + +The divisor \(L\subset F\) is an irreducible component of \(F\), +because \(F\) is proper in the \(n\)-fold \(A\). +A connected group preserves each irreducible component of a closed +invariant set with finitely many components. Thus \(K+L=L\). +But \(L\to A/K\) is surjective, which gives \(K+L=A\). +This contradiction proves the required assertion about +\(\delta(\mathcal R')\). +\end{proof} + +\begin{lemma}[A divisor through the marked arrow] +\label{quot:marked-divisor} +Let \(\nu:Z\to\mathcal R\) be the normalization of an irreducible +component followed by inclusion, with difference image an abelian +subvariety \(D\subsetneq A\), and put +\(K=D^\perp\). +Suppose \(z_0\in Z\) lies over a nonidentity arrow of difference zero. +For \(L\) as in Lemma~\ref{quot:ramification}, there are \(k\in K\), +\(e_0=k\cdot z_0\), and a reduced prime divisor \(E\subset Z\) +through \(e_0\), such that \(x_0:=s(e_0)=t(e_0)\in L\), +\(\nu(e_0)\) is nonidentity, and +\[ + s(E)=L,\qquad \delta_Z(E)=D . +\] +\end{lemma} + +\begin{proof} +The action of \(K\) supplied by +Lemma~\ref{quot:difference-image} preserves difference zero. +Connectedness and the open-and-closed identity partition preserve +the nonidentity condition. Since \(L\to A/K\) is surjective, a +translate of the first coordinate of \(z_0\) belongs to \(L\). +Choose such a \(k\), and put \(e_0=k\cdot z_0\) and +\(x_0=s(e_0)=t(e_0)\). This is the translated normalization point +retained below. + +The divisor \(L\) is Cartier because \(A\) is smooth. +Its local equation pulls back to a nonzero nonunit in the integral +local ring \(\cO_{Z,e_0}\). +A minimal prime over that equation has height one, by the principal +ideal theorem. Taking its closure with reduced structure gives a +prime divisor \(E\) through \(e_0\) and contained in \(s^{-1}(L)\). +The finite map \(s:Z\to A\) preserves dimension, so its closed image +\(s(E)\) has dimension \(n-1\) and equals the irreducible divisor \(L\). + +The image of the proper action morphism \(K\times E\to Z\) is closed +and irreducible. Its first-coordinate image is +\(K+L=A\). Since \(Z\to A\) is finite, this image has dimension \(n\) +and hence is all of \(Z\). As \(\delta_Z\) is \(K\)-invariant, +\[ + \delta_Z(E) + =\delta_Z(K\cdot E) + =\delta_Z(Z)=D . +\] +These are equalities of actual images; projectivity ensures that +no special point is lost by taking closures. +\end{proof} + +\begin{proof}[Completion of the proof of Proposition~\ref{quot:smooth-base}] +Apply Lemmas~\ref{quot:ramification} and~\ref{quot:marked-divisor} +to the fixed \(D,K,Z,z_*\). They provide a divisor \(L\subset A\), +a residual component \(\mathcal R'\subset\mathcal R\) containing +all of \(i(L)\), and a reduced prime divisor \(E\subset Z\) through +the translated point \(e_0\), with +\[ + \delta_Z(E)=D,\qquad + \delta(\mathcal R')\not\subset D. +\] +Write \(x_0=s(e_0)=t(e_0)\in L\) and +\(\alpha_0=\nu(e_0)\). The arrow \(\alpha_0\) is nonidentity and +has difference zero. It is this translated arrow that composition +will retain. +Form the space of composable arrows +\[ + \mathcal T + =\mathcal R'\times_{\,t,A,s} Z . +\] +The inclusion of all of \(i(L)\) in the actual, possibly nonnormal +component \(\mathcal R'\) gives a global morphism +\(i|_L:L\to\mathcal R'\): both \(L\) and \(\mathcal R'\) carry their +reduced structures. Hence the map +\[ + \iota:E\longrightarrow\mathcal T,\qquad + e\longmapsto \bigl(i(s(e)),e\bigr) +\] +is a closed embedding, the graph of this morphism over the closed +subscheme \(E\subset Z\). Since \(E\) is irreducible, one irreducible +component \(\mathcal T_0\) contains its entire image. Give +\(\mathcal T_0\) its reduced structure. Since \(E\) is reduced, +\(\iota\) factors as a closed embedding into \(\mathcal T_0\); +in particular, this component contains the retained marked point. +Figure~\ref{quot:incidence} records this component choice and the +subsequent composition. + +This same component dominates \emph{both} factors. +Indeed, \(\mathcal R'\times Z\) has pure dimension \(2n\). +The equations for \(\mathcal T\) are the pullback of the diagonal +in \(A\times A\), which is locally cut out by \(n\) equations. +The dimension inequality for an ideal with \(n\) generators implies +\(\dim\mathcal T_0\ge n\). +Both projections from \(\mathcal T\) to its factors are finite, +so \(\dim\mathcal T_0\le n\). +Its images in both irreducible \(n\)-dimensional factors are closed +of dimension \(n\), and thus are the whole factors. +The dimension estimate uses smoothness of \(A\) through the equations +of its diagonal. + +Composition of the two labelled arrows gives +\(\mu:\mathcal T_0\to\mathcal R\). +Both \(\mathcal T_0\) and \(\mathcal R\) are finite over the outer source +coordinate in \(A\), so \(\mu\) is finite. Its closed irreducible image +has dimension \(n\), since that source coordinate is surjective. +Consequently its image is an irreducible +component \(\mathcal R''\) of \(\mathcal R\). +On the embedded \(E\), composition with the identity is exactly +\(\nu(e)\), with its label. Thus \(\mathcal R''\) contains the marked +nonidentity arrow \(\alpha_0=\nu(e_0)\) over difference zero, and its +difference image contains all of \(\delta_Z(E)=D\). + +It also contains a point outside \(D\). In fact, dominance of +\(\mathcal T_0\to\mathcal R'\) allows us to choose composable arrows +\[ + z\longrightarrow x\longrightarrow y +\] +with \(x-z\notin D\), whereas \(y-x\in D\), because the second arrow +comes from \(Z\). Since \(D\) is a subgroup, +\(y-z=(y-x)+(x-z)\notin D\). +The closed irreducible difference image of \(\mathcal R''\) +therefore contains \(D\) properly, and has dimension greater than +\(r\). It still meets nonidentity inertia over zero, contradicting +the choice of \(r\). + +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[ + >=Stealth, + object/.style={font=\large, inner sep=5pt, align=center}, + explanation/.style={font=\small, align=center}, + maplabel/.style={font=\small, fill=white, inner sep=2pt}, + retained/.style={draw=blue!55!black, line width=.8pt}, + projection/.style={draw=black!75, line width=.65pt} +] + \node[object] (E) at (-4.5,0) {$E\ni e_0$}; + \node[object] (T) at (0,0) {$\mathcal T_0\ni\iota(e_0)$}; + \node[object] (Rpp) at (4.5,0) {$\mathcal R''\ni\alpha_0$}; + + \node[explanation] at (0,1.05) + {$\mathcal T_0$ is one irreducible component of + $\mathcal R'\times_{t,A,s}Z$}; + + \draw[retained,{Hooks[right]}->] (E.east) -- (T.west) + node[midway,above,maplabel] {closed embedding $\iota$} + node[midway,below,maplabel,font=\scriptsize] + {$e\longmapsto\bigl(i(s(e)),e\bigr)$}; + \draw[retained,->] (T.east) -- (Rpp.west) + node[midway,above,maplabel] {composition $\mu$} + node[midway,below,maplabel,font=\scriptsize] {onto $\mathcal R''$}; + + \node[object] (Rp) at (-2.05,-2.05) {$\mathcal R'$}; + \node[object] (Z) at (2.05,-2.05) {$Z$}; + \draw[projection,->] (T.south west) -- (Rp.north) + node[midway,left=5pt,maplabel,font=\scriptsize,align=center] + {finite,\\dominant}; + \draw[projection,->] (T.south east) -- (Z.north) + node[midway,right=5pt,maplabel,font=\scriptsize,align=center] + {finite,\\dominant}; + \node[explanation,font=\scriptsize,anchor=north] at (-2.05,-2.32) + {actual relation component}; + \node[explanation,font=\scriptsize,anchor=north] at (2.05,-2.32) + {normalization}; + + \node[explanation,text=blue!55!black] at (0,-3.08) + {$x_0=s(e_0),\qquad + \mu\bigl(\iota(e_0)\bigr)=\alpha_0=\nu(e_0)\ne i(x_0), + \qquad\delta(\alpha_0)=0$}; + + \node[object,font=\normalsize] (source) at (-3.5,-4.15) {$z$}; + \node[object,font=\normalsize] (middle) at (0,-4.15) {$x$}; + \node[object,font=\normalsize] (target) at (3.5,-4.15) {$y$}; + \draw[projection,->] (source.east) -- (middle.west) + node[midway,above,maplabel] {$\rho\in\mathcal R'$} + node[midway,below,maplabel] {$x-z$}; + \draw[projection,->] (middle.east) -- (target.west) + node[midway,above,maplabel] {$\nu(\zeta),\ \zeta\in Z$} + node[midway,below,maplabel] {$y-x$}; + \node[explanation] at (0,-5.02) + {$(\rho,\zeta)\in\mathcal T_0:\quad + \mu(\rho,\zeta):z\longrightarrow y,$\\ + $y-z=(x-z)+(y-x)$}; +\end{tikzpicture} +\caption{The component used for composition. The whole divisor +\(E\), including \(e_0\), lies in one component \(\mathcal T_0\) whose +two finite projections are dominant. Along \(E\) the first arrow is +the identity, so composition retains \(\nu(E)\) and the marked +nonidentity arrow \(\alpha_0\). The bottom row shows how the two +differences add: dominance onto \(\mathcal R'\) supplies a difference +outside \(D\), while the second difference lies in \(D\).} +\label{quot:incidence} +\end{figure} + +Thus every stabilizer is trivial. A Deligne--Mumford stack +with trivial inertia is an algebraic space, and its coarse morphism +is then an isomorphism. Therefore \(\mathfrak B=B\), and \(B\) is smooth. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/12-conclusion.tex b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/12-conclusion.tex new file mode 100644 index 0000000000000000000000000000000000000000..df7cedc90fe4f5ef34602e3ffa08bf511bbc1af3 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/build/source/sections/12-conclusion.tex @@ -0,0 +1,39 @@ +\section{Completion of the proof}\label{sec:conclusion} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +The case $n=1$ was settled in Section~\ref{sec:setup}; assume $n>1$. +By Lemma~\ref{lem:betti-bound}, $b_2(X)\geq4$. +If \(b_2(X)\geq5\), Proposition~\ref{def:large-betti} gives quotient +singularities for \(B\). If \(b_2(X)=4\) and the polarized family of +smooth fibers of the original \(f\) is isotrivial, +Proposition~\ref{iso:exclusion} gives the same conclusion. +In the remaining case, Proposition~\ref{secsys:rank-four-variation} +chooses the two systems of Proposition~\ref{secsys:second-system} on a +projective IHS deformation \(X'\), still with \(b_2(X')=4\): +projective Lagrangian fibrations \(f':X'\to B\) and \(g:X'\to C\), with +\(C\) normal and projective, such that \(g\) has nonzero polarized +variation and \(f'\) restricts to a finite surjection on every smooth +fiber of \(g\). +If \(B\) had a nonquotient germ, a maximal chart would be singular. +Propositions~\ref{str:minimum} and~\ref{str:products} would supply the +fixed stratum \((S_0,\cS)\) and its product presentations. +Proposition~\ref{eu:euler-zero}, applied to the original fibration, +would give \(\chi_{\mathrm{top}}(S_0)=0\). +For the two fibrations on \(X'\), Proposition~\ref{var:exclusion} +instead gives \(\chi_{\mathrm{top}}(S_0)>0\), a contradiction. +Thus every germ of \(B\) is a quotient singularity in this case as well. +The three cases exhaust the possibilities. + +Using the fixed-base family, Proposition~\ref{secsys:second-system} +supplies in every case a finite cover of the unchanged \(B\) by an +\(n\)-dimensional abelian variety: +restrict its first fibration to a smooth fiber of the second. +This cover, the quotient singularities just proved, and the original +projective fibration \(f:X\to B\) meet the hypotheses of +Proposition~\ref{quot:smooth-base}. Hence \(B\) is smooth. +The original $X$ is projective, $f$ is surjective with connected +positive-dimensional fibers, and $B$ is now smooth and projective. +Hwang's theorem applies with precisely these hypotheses +\cite[Theorem~1.2, in the setting of Theorem~1.1]{Hwang2008}, giving +$B\simeq\PP^n_{\C}$. +\end{proof} diff --git a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0b89729d854bf6999125914ce94b5b4482c65279 --- /dev/null +++ b/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5fdf543649f1469a097badf9e3cb603031110f8dff1dbacb85f49fcbc99c125d +size 989907 diff --git a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/README.md b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..fc93b1e2d2da82a174d40c794e0e4d65553074db --- /dev/null +++ b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/README.md @@ -0,0 +1,16 @@ +# [Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder](paper.pdf) + +OpenAI · September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026, + author = {{OpenAI}}, + title = {{Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf}{OAI:Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/paper.tex b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..23620d4dbfc0367dfa263a3753d0828e299b4415 --- /dev/null +++ b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/paper.tex @@ -0,0 +1,69 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} +\usepackage{microtype,enumitem,booktabs,array,graphicx} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,patterns,positioning,calc} +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{28,75,118} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref} +\usepackage[nameinlink,capitalise,noabbrev]{cleveref} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\T}{\mathbb T} +\newcommand{\E}{\mathbb E} +\newcommand{\PP}{\mathbb P} +\newcommand{\dd}{\,\mathrm d} +\newcommand{\one}{\mathbf 1} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\newcommand{\abs}[1]{\left\lvert#1\right\rvert} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Law}{Law} +\setlength{\emergencystretch}{1.5em} +\hypersetup{pdftitle={Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder},pdfauthor={OpenAI}} +\title{Pure-Point Spectrum for the Two-Dimensional\\Anderson Model at Every Positive Disorder} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +For each fixed positive disorder strength, we prove that the nearest-neighbor +Anderson operator on the square lattice with independent uniform site +potentials almost surely has pure-point spectral type throughout its spectrum. +This resolves the pure-point assertion of the two-dimensional Anderson +localization conjecture for the uniform single-site law. +\end{abstract} +\input{sections/introduction} +\input{sections/setup} +\input{sections/matrix} +\input{sections/resonant} +\input{sections/impedance} +\input{sections/transmission} +\input{sections/hierarchy} +\input{sections/screening} +\input{sections/vanishing} +\input{sections/bootstrap} +\input{sections/spectral} +\input{references} +\end{document} diff --git a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/references.tex b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..6a78c643480eca89278fff919b1134bde459720d --- /dev/null +++ b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/references.tex @@ -0,0 +1,113 @@ +\begin{thebibliography}{99} + +\bibitem{AbrahamsAndersonLicciardelloRamakrishnan1979} +E.~Abrahams, P.~W.~Anderson, D.~C.~Licciardello, and T.~V.~Ramakrishnan, +\emph{Scaling theory of localization: Absence of quantum diffusion in +two dimensions}, +Physical Review Letters \textbf{42} (1979), no.~10, 673--676. +\href{https://doi.org/10.1103/PhysRevLett.42.673}{doi:10.1103/PhysRevLett.42.673}. + +\bibitem{AizenmanElgartNabokoSchenkerStolz2006} +M.~Aizenman, A.~Elgart, S.~Naboko, J.~H.~Schenker, and G.~Stolz, +\emph{Moment analysis for localization in random Schr\"odinger operators}, +Inventiones Mathematicae \textbf{163} (2006), 343--413. +\href{https://doi.org/10.1007/s00222-005-0463-y}{doi:10.1007/s00222-005-0463-y}. + +\bibitem{AizenmanMolchanov1993} +M.~Aizenman and S.~Molchanov, +\emph{Localization at large disorder and at extreme energies: +An elementary derivation}, +Communications in Mathematical Physics \textbf{157} (1993), no.~2, 245--278. +\href{https://doi.org/10.1007/BF02099760}{doi:10.1007/BF02099760}. + +\bibitem{AizenmanSchenkerFriedrichHundertmark2001} +M.~Aizenman, J.~H.~Schenker, R.~M.~Friedrich, and D.~Hundertmark, +\emph{Finite-volume fractional-moment criteria for Anderson localization}, +Communications in Mathematical Physics \textbf{224} (2001), 219--253. +\href{https://doi.org/10.1007/s002200100441}{doi:10.1007/s002200100441}. + +\bibitem{Anderson1958} +P.~W.~Anderson, +\emph{Absence of diffusion in certain random lattices}, +Physical Review \textbf{109} (1958), no.~5, 1492--1505. +\href{https://doi.org/10.1103/PhysRev.109.1492}{doi:10.1103/PhysRev.109.1492}. + +\bibitem{BenedekCalderonPanzone1962} +A.~Benedek, A.~P.~Calder\'on, and R.~Panzone, +\emph{Convolution operators on Banach space valued functions}, +Proceedings of the National Academy of Sciences of the United States +of America \textbf{48} (1962), no.~3, 356--365. +\href{https://doi.org/10.1073/pnas.48.3.356}{doi:10.1073/pnas.48.3.356}. + +\bibitem{BlackDroginHernandez2025} +A.~Black, R.~Drogin, and F.~Hern\'andez, +\emph{Self-consistent equations and quantum diffusion for the Anderson model}, +arXiv:2506.06468v2 (2025). +\href{https://arxiv.org/abs/2506.06468v2}{Version 2, November 6, 2025}. + +\bibitem{CombesThomas1973} +J.~M.~Combes and L.~Thomas, +\emph{Asymptotic behaviour of eigenfunctions for multiparticle +Schr\"odinger operators}, +Communications in Mathematical Physics \textbf{34} (1973), no.~4, 251--270. +\href{https://doi.org/10.1007/BF01646473}{doi:10.1007/BF01646473}. + +\bibitem{DelyonLevySouillard1985} +F.~Delyon, Y.~L\'evy, and B.~Souillard, +\emph{Anderson localization for multi-dimensional systems at large +disorder or large energy}, +Communications in Mathematical Physics \textbf{100} (1985), no.~4, 463--470. +\href{https://doi.org/10.1007/BF01217724}{doi:10.1007/BF01217724}. + +\bibitem{DingSmart2020} +J.~Ding and C.~K.~Smart, +\emph{Localization near the edge for the Anderson Bernoulli model on the +two dimensional lattice}, +Inventiones Mathematicae \textbf{219} (2020), 467--506. +\href{https://arxiv.org/abs/1809.09041v3}{arXiv:1809.09041v3}. + +\bibitem{FMSS1985} +J.~Fr\"ohlich, F.~Martinelli, E.~Scoppola, and T.~Spencer, +\emph{Constructive proof of localization in the Anderson tight binding model}, +Communications in Mathematical Physics \textbf{101} (1985), no.~1, 21--46. +\href{https://doi.org/10.1007/BF01212355}{doi:10.1007/BF01212355}. + +\bibitem{FrohlichSpencer1983} +J.~Fr\"ohlich and T.~Spencer, +\emph{Absence of diffusion in the Anderson tight binding model for large +disorder or low energy}, +Communications in Mathematical Physics \textbf{88} (1983), no.~2, 151--184. +\href{https://doi.org/10.1007/BF01209475}{doi:10.1007/BF01209475}. + +\bibitem{Hurtado2026} +O.~Hurtado, +\emph{Localization and unique continuation for non-stationary +Schr\"odinger operators on the 2D lattice}, +Communications in Mathematical Physics \textbf{407} (2026), article~64. +\href{https://doi.org/10.1007/s00220-026-05559-8}{doi:10.1007/s00220-026-05559-8}. + +\bibitem{Simon2000} +B.~Simon, +\emph{Schr\"odinger operators in the twenty-first century}, +in \emph{Mathematical Physics 2000}, +A.~Fokas, A.~Grigoryan, T.~Kibble, and B.~Zegarlinski (eds.), +Imperial College Press, London, 2000, pp.~283--288. +\href{https://math.caltech.edu/SimonPapers/R40.pdf}{Author-hosted published version}. + +\bibitem{SimonWolff1986} +B.~Simon and T.~Wolff, +\emph{Singular continuous spectrum under rank one perturbations and +localization for random Hamiltonians}, +Communications on Pure and Applied Mathematics \textbf{39} (1986), +no.~1, 75--90. +\href{https://doi.org/10.1002/cpa.3160390105}{doi:10.1002/cpa.3160390105}. + +\bibitem{Teschl2009} +G.~Teschl, +\emph{Mathematical Methods in Quantum Mechanics: With Applications to +Schr\"odinger Operators}, +Graduate Studies in Mathematics \textbf{99}, +American Mathematical Society, Providence, RI, 2009. +\href{https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf}{Author-hosted first edition}. + +\end{thebibliography} diff --git a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/sections/bootstrap.tex b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/sections/bootstrap.tex new file mode 100644 index 0000000000000000000000000000000000000000..8a95ff038526a7ca8f585a121091b7f39e8756dc --- /dev/null +++ b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/sections/bootstrap.tex @@ -0,0 +1,579 @@ +\section{A fixed-energy fractional-moment estimate} +\label{sec:bootstrap} + +We return to the true product law and remove the hierarchy from the +operator whose decay is to be estimated. Throughout this section, +\(h>0\) and \(E\in(-4-h,4+h)\) are fixed. All choices may depend on +these two parameters. Distance means the shortest-coordinate max-norm +distance on the torus \(\T_N^2=(\Z/b_N\Z)^2\). The notation for the probe +sets below is local to this section. + +\begin{theorem}[True torus Green function]\label{boot:fractional} +Fix \(00\), and an integer +\(N_0=N_0(h,E,s)\) such that, for every \(N\ge N_0\) and every +\(x,y\in\T_N^2\), +\begin{equation}\label{boot:real-decay} + \E\abs{(H_N-E)^{-1}_{xy}}^{s/2} + \le A\exp\{-a\dist(x,y)\}. +\end{equation} +Here \(H_N\) has the true iid uniform \([-h,h]\) diagonal and periodic +nearest-neighbor hopping. Its inverse at the specified real energy +exists almost surely. No uniformity in \(E\), nor any statement about +time evolution, is part of this theorem. +\end{theorem} + +We prove the theorem by first retaining arbitrary deterministic masks of +artificial probes. At each site a mask prescribes absence, strength +\(\tau\), or strength \(\tau\vartheta(v_i)\), with at most one probe per +site. The profile \(\vartheta\) is the one fixed earlier; in particular +it equals one on \([-h/2,h/2]\), is \(C^{1,1}\), and vanishes quadratically +only at the endpoints of the potential interval. Write +\(\eta_i\) for a prescribed positive probe strength and let \(Y\) have +columns \(\sqrt{\eta_i}e_i\). In this section +\[ + G=(H_N-E-iYY^\top)^{-1},\qquad S=1+2iY^\top GY. +\] +Thus \(S\) is symmetric and unitary, and \(Y^\top GY\) is a +contraction by \Cref{mat:ward}. All exceptional algebraic singularities +below can be discarded under the continuous product law. + +The parameter order is +\begin{equation}\label{boot:order} + s,\ d_0,\ C_0,\ n,\ m,\ N, + \qquad R=\lfloor\exp(b_m^2)\rfloor,\qquad \tau=C_0/R. +\end{equation} +The barrier threshold \(d_0>0\) will be small; \(C_0\) will be large. +After they have been fixed, \(n\) and then \(m>n\) are made sufficiently +large. Finally \(N\) is sufficiently large in terms of all these +choices; all ensuing estimates hold for every larger \(N\). We always +require \(b_N>100R\) and \(\tau\le1\). + +\subsection{Filling and twisting an annulus} + +Let \(X,Y'\) be arbitrary deterministic subsets of the probe sites in +the radius-\(R\) squares centered at \(x,y\), respectively, where +\(\dist(x,y)\ge10R\). Let +\[ + C=\{i:2R\le\dist(i,x)\le4R\}. +\] +Fill every previously unoccupied site of \(C\) with a probe of strength +\(\tau\vartheta(v_i)\), keeping existing probes and their strengths. +The filled mask is again deterministic. Superscripts \({\rm old}\) and +\({\rm fill}\) distinguish the resulting scattering matrices. Identify +the probes in \(C\) with their sites and put +\[ + U=\norm{S^{\rm fill}_{CX}}_{\rm HS},\qquad + V=\norm{S^{\rm fill}_{CY'}}_{\rm HS},\qquad + D=\norm{S^{\rm fill}_{Y'X}}_{\rm HS}. +\] + +\begin{lemma}[Mixture domination]\label{boot:twist} +There is \(C_s<\infty\), independent of the masks, centers, \(R,N\), +and \(C_0\ge1\), such that +\begin{equation}\label{boot:twist-ineq} + \E\norm{S^{\rm old}_{Y'X}}_{\rm HS}^{s} + \le C_s(1+C_0)\E\bigl[D^s+V^sU^s\bigr]. +\end{equation} +As everywhere in this section, the constant may depend on the fixed +\(h,E,s\) and profile. +\end{lemma} + +\begin{proof} +Close just the newly added probes using a common Haar phase \(z\). +The closure identity in \Cref{mat:closure} and the sandwich estimate in +\Cref{mat:weak}, followed by integration of the weak \(L^1\) tail, give +\[ + \int\norm{(S_z)_{Y'X}}_{\rm HS}^{s}\,\dd z + \le C_s(D^s+V^sU^s). +\] +Here \(\dd z\) is normalized Haar measure, and the blocks linking the +new probes to \(X,Y'\) are subblocks of those defining \(U,V\). +The corresponding real closing shift at each newly occupied site is +\(w\vartheta(v_i)\), where \(w\) has density +\[ + \frac{\tau}{\pi(w^2+\tau^2)}. +\] +Only newly occupied sites are shifted, so after closing we have precisely +the old mask evaluated at these shifted true potentials. + +For \(\abs w\le c/R\), with fixed sufficiently small \(c>0\), the map +\(T_w(v)=v+w\vartheta(v)\) is a diffeomorphism of \([-h,h]\) onto +itself. If \(u=T_w(v)\), its pushforward density relative to the uniform +law is \(r_w(u)=(1+w\vartheta'(v))^{-1}\). Uniformly in \(u\), +\[ + \log r_w(u)=-w\vartheta'(u)+O(w^2). +\] +This follows from the inverse Jacobian and the Lipschitz continuity of +\(\vartheta'\). There are at most \(K R^2\) shifted sites, so at each +fixed output configuration the joint density satisfies +\[ + \log\prod_i r_w(u_i) + =-w\sum_i\vartheta'(u_i)+O(R^2w^2). +\] +Choose the half interval \([0,c/R]\) or \([-c/R,0]\) on which the linear +term is nonnegative. On this entire half interval the joint density is +at least a fixed positive constant. Its Cauchy probability is +\[ + \frac1\pi\arctan\frac c{C_0}\ge\frac{c_1}{1+C_0}. +\] +Consequently the full shifted mixture dominates +\(c_2(1+C_0)^{-1}\) times the true product law, pointwise in the output +configuration. Applying this domination to the nonnegative old-mask +block power and then averaging the Haar estimate proves the claim. +\end{proof} + +We seek a bound by an arbitrarily small multiple of \(\E V^s\). +Thus \Cref{boot:twist-ineq} leaves two tasks: attenuate \(D\) across +the filled annulus, and combine a small mean return \(U\) with a +resampling bound for \(U^sV^s\). An unweighted estimate for barrier +failure would not suffice when \(\E V^s\) is small. We therefore +control that failure also under the \(V^s\)-weighted law. + +\subsection{Two barriers under a transmission tilt} + +Call a site good when \(\vartheta(v_i)\ge d_0\). Give cost one to an +edge precisely when both endpoints are good and both radii about \(x\) +belong to \([2.2R,2.8R]\); all other edges have cost zero. Let +\(d_{\rm in}\) be the associated path-cost distance from the radius-\(2R\) +square. Independently of this definition, count good-good edges with +both radii in \([3.2R,3.8R]\), and let \(d_{\rm out}\) be the resulting +distance from \(\{i:\dist(i,x)\ge4R\}\). Integer roundings of the four +band boundaries make no difference to the constants. For a sufficiently +small absolute \(\gamma>0\), define +\begin{equation}\label{boot:barrier-event} + \mathcal B=\left\{ + \begin{array}{ll} + d_{\rm in}(i)\ge\gamma R&\text{if }\dist(i,x)\ge3R,\\ + d_{\rm out}(i)\ge\gamma R&\text{if }\dist(i,x)\le3R. + \end{array}\right\}. +\end{equation} +These two strictly buffered bands will allow separate resampling of the +inside and outside of radius \(3R\). + +\begin{lemma}[Uniform one-site estimates under tilt]\label{boot:tilted} +Suppose \(\E V^s>0\), and define the probability law +\[ + \dd\nu=\frac{V^s}{\E V^s}\,\dd\PP. +\] +For a site \(i\in C\), conditional on all other potentials, put +\(W(v_i)=V^s\). Then +\begin{equation}\label{boot:l2l1} + \bigl(\E_i W^2\bigr)^{1/2}\le K_s\E_i W, +\end{equation} +with a constant uniform in every frozen exterior and in the mask. +For \(d_0>0\) sufficiently small, chosen after \(s\) and before \(C_0\), +there are \(K,c>0\) such that +\begin{equation}\label{boot:tilted-barriers} + \PP(\mathcal B^c)+\nu(\mathcal B^c)\le K e^{-cR}, + \qquad + \E_\nu d_{\min}^{-2s}\le K R^2, + \qquad d_{\min}=\min_{i\in C}\vartheta(v_i). +\end{equation} +No independence under \(\nu\) is asserted or needed. +\end{lemma} + +\begin{proof} +Fix all variables except \(v=v_i\), and write the probe factor at \(i\) +as \(a(v)\in\{1,\vartheta(v)\}\). Schur complementation at \(i\) +expresses the other rows and the row at \(i\) of the block defining \(V\) +in the form +\[ + A'+\frac{D'}{v-i\tau a(v)-w_*},\qquad + \frac{D''\sqrt{a(v)}}{v-i\tau a(v)-w_*},\qquad \Im w_*\ge0. +\] +The coefficients are fixed matrices of compatible sizes. Passivity +gives the sign of \(\Im w_*\). The deleted physical-site system is +invertible almost surely: the imaginary part forces a kernel vector to +vanish at all remaining probe sites, so it would lie in the kernel of +the true real restriction, whose nonzero determinant polynomial +vanishes only on a product-law null set. +Writing both rows over their common denominator, their joint numerator is +\[ + N(v)=\bigl((v-i\tau a(v)-w_*)A'+D',\ D''\sqrt{a(v)}\bigr). +\] +On the plateau \(P=[-h/2,h/2]\), this is the affine vector +\(N_P(v)=(vA'+B,D'')\), where \(B=D'-(i\tau+w_*)A'\). +If \(M=\max_{v\in P}\norm{N_P(v)}_{\rm HS}\), then +\[ + \norm{A'}_{\rm HS}\le2M/h,\qquad + \norm{(B,D'')}_{\rm HS}\le M,\qquad + \sup_{[-h,h]}\norm{N(v)}_{\rm HS}\le K_hM. +\] +Moreover an interval in \(P\) of length at least \(h/4\), adjacent to +an endpoint attaining the maximum, has numerator norm at least \(M/2\). +This uses the displayed slope bound. If \(M=0\), the entire block +vanishes and the claimed inequality is immediate. + +Put \(b(v)=v-i\tau a(v)-w_*\). Its modulus is at most +\(K_h(1+\abs{w_*})\). For every \(00\), +one can choose \(n\), then \(m>n\), then a threshold for \(N\), so that +\[ + \E U^2<\epsilon +\] +for every subsequent \(N\), every deterministic mask, every center, +and every permitted subset \(X\). +\end{lemma} + +\begin{proof} +Carry the filled artificial mask through the auxiliary hierarchy. +Take a scalar voltage \(f_R\) equal to one on the radius-\(R\) square, +zero at radii at least \(2R\), and Lipschitz with constant \(K/R\). +By \Cref{screen:profiles}, at fixed \(n\), +\[ + \E\mathcal Q_{n,N}(f_R)\le Kd_n+o_{m,N;n,C_0}(1). +\] +Apply \Cref{hier:comparison} to thin to uniform stage \(m\). +The profile is supported in the radius-\(2R\) square and has Lipschitz +constant \(K/R\), while \(b_m\ll R\). Thus +\Cref{hier:charges}(ii) bounds its unnormalized charge by \(K r_n\), +where \(r_n=\sum_{l\ge n}q_lp_l\to0\), independently of \(R\). +Writing \(U_m\) for the same artificial-probe block at stage \(m\), +we obtain +\begin{equation}\label{boot:return-stage} + \E U_m^2 + \le Kd_n+o_{n\to\infty}(1)+o_{m,N;n,C_0}(1). +\end{equation} +Each probe pair contributing to \(U_m^2\) has unit squared voltage difference and is +counted in both directions in \(\mathcal Q\), canceling its factor +\(1/2\). The artificial probes are allowed in the comparison because +\(C_0/R\le\exp(-c_*b_m)\) for sufficiently large \(m\). + +Let \(\mathcal A\) be the event that no selected tile of level at least +\(m\) meets the radius-\(5R\) square. Tile counting gives +\[ + \PP(\mathcal A^c) + \le K\sum_{l\ge m}(1+R/b_l)^2 e^{-b_l^6} + =o(R^{-4}). +\] +For example each fixed polynomial loss in \(R\) is dominated by +\(\exp(-b_m^6)\), since \(\log R\sim b_m^2\). +On \(\mathcal A\), the stage-\(m\) and true filled coefficients and +probes coincide throughout this square. On \(\mathcal B\) use +\[ + \chi=(1-d_{\rm in}/(\gamma R))_+. +\] +It equals one on \(X\), vanishes outside radius \(3R\), and changes +only on shared good-good edges. Cut off stage-\(m\) fields sourced at +\(X\), and correct in the true system using +\Cref{boot:commutator,boot:cutoff-ward}. Since \(R\tau=C_0\), +\[ + U\le\left(1+\frac K{C_0d_0}\right)U_m + \quad\hbox{on }\mathcal A\cap\mathcal B. +\] +Always \(U^2\le\abs X\le K R^2\), by unitarity. The exceptional +contribution is therefore \(o(1)\), by the preceding hazard estimate +and the product-law path count in the proof of \Cref{boot:tilted}. +That count does not require \(\E V^s>0\). Now \(d_n\to0\) by +\Cref{van:zero}; choosing \(n,m,N\) in order in +\Cref{boot:return-stage} proves the assertion. The error notation means +that at fixed \(n,C_0\), any desired tolerance holds for all sufficiently +large \(m\), and then for all \(N\) above a threshold depending on \(m\). +It asserts no rate uniform in \(n\). +\end{proof} + +\subsection{Direct transmission and independent resampling} + +We next prove two pointwise estimates: +\begin{equation}\label{boot:direct} + D\le Kd_{\min}^{-1}V\quad\hbox{almost surely},\qquad + D\le K e^{-c C_0}V\quad\hbox{on }\mathcal B. +\end{equation} +For the first, use fields sourced at \(Y'\), symmetry of \(S\), and a +cutoff equal to one inside radius \(2R\), zero outside radius \(4R\), +with gradient at most \(K/R\). Its source term vanishes, while both +endpoints of every changing edge have strength at least +\(\tau d_{\min}\). \Cref{boot:commutator,boot:cutoff-ward} give +\(D\le K(R\tau d_{\min})^{-1}V\); here \(C_0\ge1\). +In particular, if \(\E V^s=0\), then \(D=0\) almost surely, and no +tilted law is required. + +For the improved estimate choose an integer +\[ + \ell=\left\lceil\frac{K_1}{\tau d_0}\right\rceil +\] +with \(K_1\) sufficiently large. For \(j\ge1\) use the cutoff +\[ + \chi_j(i)=\min\left\{1, + \frac{(d_{\rm out}(i)-(j-1)\ell)_+}{\ell}\right\}. +\] +All original sources have \(d_{\rm out}=0\). Nonzero gradients have +both endpoint distances at least \((j-1)\ell\), because cost distances +are integer valued and change by at most one across an edge. +Gradients only occur on counted good-good edges in the outer band. +For the field \(\Psi=G^{\rm fill}Y_{Y'}\), put +\[ + A_j=\norm{\one_{\{d_{\rm out}\ge j\ell\}}Y^\top\Psi}_{\rm HS}, + \qquad + B_j=\norm{\one_{C\cap\{d_{\rm out}\ge j\ell\}} + Y^\top\Psi}_{\rm HS}. +\] +The cutoff equations and Ward give +\[ + A_j\le\frac K{\ell\tau d_0}B_{j-1} + \le\tfrac12 B_{j-1},\qquad B_j\le A_j. +\] +Start with \(B_0=V/2\). On \(\mathcal B\), the set \(X\) has +\(d_{\rm out}\ge\gamma R\), so it remains beyond every one of the +\(\lfloor\gamma R/\ell\rfloor\) cutoffs. For sufficiently large fixed +\(C_0\), followed by sufficiently large \(R\), this number is at least +\(c(d_0)C_0\). The iteration proves the second estimate in +\Cref{boot:direct}. Notice that \(A_j\) includes every absorbed output, +including \(X\); the proof does not require \(X\subset C\). + +To separate the product \(V^sU^s\), enlarge the probability space with +two independent fresh families of true potentials. Let +\(A=\{i:\dist(i,x)<3R\}\). Form one comparison configuration by +retaining the original potentials on \(A^c\) and replacing those on +\(A\) by the first fresh family. Form a second by retaining the +original potentials on \(A\) and replacing those on \(A^c\) by the +second fresh family. Keep the filled deterministic mask in both +configurations, and recompute its strengths from the corresponding +potentials. Denote the first configuration's transmission to \(Y'\) +by \(V^{\rm ext}\), and the second's transmission from \(X\) by +\(U^{\rm int}\). + +On \(\mathcal B\), the cutoff +\((1-d_{\rm out}/(\gamma R))_+\) is supported in the shared exterior +\(A^c\), equals one on \(Y'\), and has all changing edges in its +buffered outer band. Applying it to the fresh-interior fields and +correcting in the original system proves +\[ + V\le\left(1+\frac K{C_0d_0}\right)V^{\rm ext}. +\] +Likewise \((1-d_{\rm in}/(\gamma R))_+\) is supported in the shared +interior \(A\) and gives +\[ + U\le\left(1+\frac K{C_0d_0}\right)U^{\rm int}. +\] +Take \(C_0\) large enough that both factors are at most two. The two +comparison configurations are independent \emph{unconditionally}: +they use disjoint parts of the original product sample and independent +fresh families. Each has the original true product distribution. +Dropping \(\one_{\mathcal B}\) before factoring therefore yields +\begin{equation}\label{boot:factorization} + \E[\one_{\mathcal B}V^sU^s] + \le 2^{2s}\E V^s\,\E U^s + \le 2^{2s}\E V^s(\E U^2)^{s/2}. +\end{equation} +This factorization makes no assertion of independence conditional on +the original barrier event. + +The exceptional event is controlled under the tilted law from +\Cref{boot:tilted}. Unitarity gives \(U\le K R\), and +\Cref{boot:direct,boot:tilted-barriers} give, respectively, +\begin{align*} + \frac{\E[\one_{\mathcal B^c}V^sU^s]}{\E V^s} + &\le K R^s\nu(\mathcal B^c),\\ + \frac{\E[\one_{\mathcal B^c}D^s]}{\E V^s} + &\le K\E_\nu[\one_{\mathcal B^c}d_{\min}^{-s}]\\ + &\le K(\E_\nu d_{\min}^{-2s})^{1/2} + \nu(\mathcal B^c)^{1/2} + \le K R e^{-cR/2}. +\end{align*} +Both quantities tend to zero uniformly over all the prescriptions. +Combining these estimates with \Cref{boot:twist}, we obtain +\begin{equation}\label{boot:one-step} + \E\norm{S^{\rm old}_{Y'X}}_{\rm HS}^s + \le K_s(1+C_0) + \left[e^{-scC_0}+(\E U^2)^{s/2}+\epsilon_R\right]\E V^s, + \qquad \epsilon_R\longrightarrow0. +\end{equation} +The constants \(K_s,c>0\) are independent of large \(C_0\). First +choose \(C_0\) to make its exponential term small, then use +\Cref{boot:return} with \(n,m,N\) in order to make the return term +small, and enlarge \(m,N\) as necessary for \(\epsilon_R\). Thus the +entire multiplier can be made arbitrarily small. These are uniform +expectation bounds for each choice of mask and centers; no simultaneous +good event over all choices is needed. + +\subsection{Contraction over all masks} + +The passage from finite-scale smallness to exponential fractional-moment +decay parallels the finite-volume criteria of +\cite{AizenmanSchenkerFriedrichHundertmark2001}. We prove the contraction +here for the probe transmission moments and deterministic masks. + +\begin{proposition}[Uniform mask contraction]\label{boot:contraction} +For fixed \(h,E,s\), parameters through \(R\) and an integer \(N_0\) +can be chosen so that, for every \(N\ge N_0\), +\[ + F_N(r)\le\tfrac12 F_N(r-4R),\qquad r\ge10R. +\] +Here \(F_N(r)\) is the supremum of +\(\E\norm{S_{Y'X}}_{\rm HS}^s\) over all deterministic artificial +masks, all subsets \(X,Y'\) lying in radius-\(R\) squares, and all +centers at distance at least \(r\). A supremum over an empty family +is zero. Consequently there are constants \(K,a_1>0\), independent +of \(N\ge N_0\), such that +\begin{equation}\label{boot:mask-decay} + F_N(r)\le K R^s e^{-a_1 r/R},\qquad r\ge0. +\end{equation} +\end{proposition} + +\begin{proof} +Partition \(C\) into at most \(J\) sets \(C_k\), each contained in a +radius-\(R\) square with center in \(C\); \(J\) is an absolute +constant. Then +\[ + V^s=\left(\sum_{k=1}^J + \norm{S^{\rm fill}_{C_kY'}}_{\rm HS}^2\right)^{s/2} + \le\sum_{k=1}^J\norm{S^{\rm fill}_{C_kY'}}_{\rm HS}^s. +\] +Each new center is at distance at least \(r-4R\) from \(y\). +The filled model is included in the same mask class. Choose the +multiplier in \Cref{boot:one-step} at most \(1/(2J)\) and take +suprema, using symmetry to interchange the two block roles when +necessary. This proves the contraction. Always +\(F_N(r)\le K R^s\) by unitarity and the rank of a radius-\(R\) input +set. Iterating in steps of \(4R\) until the argument falls below +\(10R\) proves \Cref{boot:mask-decay}. All thresholds were fixed before +taking the supremum and are valid on every sufficiently large torus. +\end{proof} + +\subsection{Removing the two endpoint probes} + +Choose a mask consisting only of two constant-strength probes +\(\tau\) at distinct sites \(x,y\). Let +\(G^0=(H_N-E)^{-1}\), and let \(g\) be its two-by-two compression to +\(\{x,y\}\). Resolvent subtraction gives the damped compression +\[ + g^{\rm damp}=(1-i\tau g)^{-1}g. +\] +For a symmetric two-by-two matrix, direct multiplication by the +adjugate shows +\[ + (g^{\rm damp})_{xy} + =\frac{g_{xy}}{\det(1-i\tau g)}. +\] +Since \(S_{xy}=2i\tau(g^{\rm damp})_{xy}\), the exact identity is +\begin{equation}\label{boot:endpoint-identity} + G^0_{xy}=\frac{S_{xy}}{2i\tau}\det(1-i\tau g). +\end{equation} + +For completeness, the moments needed to remove the determinant are +uniform in the exterior and the volume. Conditional on all other +potentials, Schur complementation gives +\[ + g=(A+\diag(v_x,v_y))^{-1} +\] +with \(A\) a fixed real symmetric matrix. Put +\(t=(v_x+v_y)/2\), \(u=(v_x-v_y)/2\). The integration square becomes +\(\abs u\le h\), \(\abs t\le h-\abs u\), with Jacobian two. +For each fixed \(u\), let \(\lambda_1(u),\lambda_2(u)\) be the real +eigenvalues of \(A+\diag(u,-u)\). If \(00\), proving +\Cref{boot:fractional}. This conclusion remains a fixed-real-energy +finite-volume estimate; the spectral passage uses it separately at +each energy. diff --git a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/sections/hierarchy.tex b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/sections/hierarchy.tex new file mode 100644 index 0000000000000000000000000000000000000000..b918e6049d7ddc94317f8875a0b4dde863ad6b6b --- /dev/null +++ b/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/build/sections/hierarchy.tex @@ -0,0 +1,611 @@ +\section{A hierarchy of probability splits and energy comparisons} +\label{sec:hierarchy} + +Throughout this section the disorder strength $h>0$ and the interior energy +$E$ are fixed. All parameters in the cell construction are fixed as well. +Constants may depend on these choices, including the corner-box exponent +chosen below, but not on the outer torus, the upper thinning level, or the +prescribed artificial-probe mask. We write +\[ + b_l=3^l,\qquad L_l=\frac{b_l-1}{2}-b, + \qquad q_l=p_l=\exp(-b_l^6). +\] +Only levels above a sufficiently large fixed $l_0$ will be used. We increase +$l_0$ whenever necessary to apply all the preceding cell estimates. + +\subsection{An iid implementation of the rare cell laws} + +On the torus of side $b_N$, choose one uniform lattice offset, independently +of the true iid potentials. For each $l_0\le l\le N$, partition the torus +into squares of side $b_l$ whose centers are congruent to that offset modulo +$b_l$. These partitions are nested. In each tile use the translated +geometry and branch law of \Cref{cell:geometry,cell:branch}, with +$L=L_l$. The word \emph{ancestor} will always refer to strict containment +in this nested family unless the contrary is stated. + +We first record the geometric feature that makes conditioning compatible +across levels. The corner triples belonging to different tiles and levels +are disjoint except when they coincide exactly. At a coincident triple +the ordering of its three sites, and the two real plane-basis vectors used +for its box coordinates, are the same. In fact, the missing corner and +its two companions have fixed displacements from the corresponding two +ends of a tile. Passing to an extreme child preserves those positions: +\[ + b_l+\frac{b_l-1}{2}-b + =\frac{b_{l+1}-1}{2}-b. +\] +Consequently, if an ancestor triple meets a descendant tile, it is one of +that descendant's own triples. Also, the set $V\cup B$ of a descendant +is contained in that of its ancestor, and every descendant center belongs +to the ancestor's $V$. The latter inclusion will ensure that all +descendant reference-potential changes are interior changes when we use +the parent-cell envelope. + +Put $K_0=10^6$. For each distinct corner triple, give its three true +potentials a single box label as follows. In each product of polynomial +pieces of $\vartheta$ lying in $(-h,h)^3$, use the fixed linear box +coordinates from \Cref{cell:branch}. Partition the resulting region into +maximal dyadic cubes whose concentric doubles lie in the region. Their +inverse images are the corner boxes. All labels are defined from the +true potentials, but we will expose only specified labels. At exact +corner coincidences we use the same label, rather than successively +refining it. Null boundaries may be assigned arbitrarily. + +Conditional on a box label, the three true variables are uniform on that +box: the original density is constant, and the coordinate change is +linear with constant nonzero Jacobian. Labels of disjoint triples are +independent. If a box has width less than $\xi$, its points lie within +$C\xi$ of the boundary of one of the finitely many piece regions. +Indeed, away from those boundary strips a dyadic cube of width comparable +to $\xi$ containing the point has its concentric double inside the region, +so a maximal cube containing it cannot be smaller. Thus +\begin{equation} + \PP(\text{width of a given triple's box}<\xi) + \le C\xi. + \label{hier:small-box} +\end{equation} +A level-$l$ tile has \emph{valid own labels} when the widths of all four +corner boxes are at least $b_l^{-K_0}$. The doubled-box requirement also +provides the polynomial clearance from support endpoints and piece +boundaries required in the cell estimates. + +Assign to every tile $U$ of level $l$ an independent selector +$\eta_U\in\{0,1\}$ of success probability $q_l$. These selectors are +independent of all true potentials. A tile is \emph{eligible} if it is +selected and has no selected strict ancestor. Eligible tiles are +pairwise disjoint, even when a selected tile later fails its branch test. +An eligible tile is \emph{allowed} if its own labels and those of every +strict ancestor are valid at their respective levels. + +For an allowed tile $U$, let $\mathcal L_U$ be the likelihood of its cell +branch law relative to the true product law with its four corner boxes +fixed. It changes the law on $V$ and the noncorner single-side +coordinates of $B$, leaves the conditional laws on the four corner +triples unchanged, and leaves all remaining sites of the tile unchanged. +Its root factor is the conditional density centered at the reference +$c(v)$ constructed from the true values on $J$. Apart from the +allowance gate, the likelihood uses only data in $U$ and its own corner +boxes. In particular it does not use ancestor labels in any other way. +The density bound of \Cref{cell:branch} gives, uniformly over valid boxes, +\[ + \mathcal L_U\le + \exp\bigl(C L_l^3+C L_l^2\log(2+L_l)\bigr). +\] +After increasing $l_0$, we have $p_l\mathcal L_U\le1$. + +Give each tile a further independent uniform variable $t_U$ on $[0,1]$. +Its acceptance indicator is +\begin{equation} + A_U=\one_{\{U\text{ eligible and allowed}\}} + \one_{\{t_U\le p_l\mathcal L_U\}}. + \label{hier:acceptance} +\end{equation} +The selectors, acceptance uniforms, and offset are auxiliary randomness; +the true potentials are sampled first from their unchanged iid law. + +\begin{proposition}[Conditional sampling] +\label{hier:sampling} +Fix the offset and all selector marks. Expose the box identities of the +eligible tiles and of their ancestors. Conditional on this information, +the unexplored variables in disjoint eligible tiles are independent. +For an allowed eligible tile $U$ of level $l$, its conditional acceptance +probability is $p_l$, and conditional on acceptance its site variables +have exactly the branch law of \Cref{cell:branch} with the indicated +corner conditioning. + +These conclusions remain valid for $U$ after fixing status decisions or +other local choices in disjoint eligible tiles, provided the exposed box +identities are retained in the conditioning. In particular, +\begin{equation} + \PP(A_U=1\mid\text{offset})\le q_l p_l. + \label{hier:incidence} +\end{equation} +The unconditional law of the true potentials remains the iid product +law. The same construction can be made prospectively by omitting all +selectors and ancestor tests above a specified cutoff level. +\end{proposition} + +\begin{proof} +The selector marks determine eligibility without using any potential. +Consider an eligible tile $U$. Any exposed triple that meets $U$ is +either one of its own triples or an ancestor triple coinciding with one +of them. To see this also for labels exposed on behalf of another +eligible tile, use nesting: that other tile is disjoint from $U$, and an +ancestor of it that meets $U$ either contains $U$ or is disjoint from its +interior. In the containing case the corner-coincidence property applies. +Thus no exposed label imposes an additional condition on the interior of +$U$ or on its single-side variables. + +The conditional true law in $U$ is consequently precisely the reference +law defining $\mathcal L_U$: the four corner triples are uniform in +their fixed boxes, and all other sites have their product laws. These +conditional laws factor across eligible tiles, since their site sets +are disjoint and the only repeated labels are identical labels on the +same triple. For an allowed tile, +\[ + \E[\mathcal L_U\mid\text{selectors and exposed labels}]=1. +\] +Integrating the independent uniform $t_U$ proves that its acceptance +probability is $p_l$. More generally, for every bounded local test +function $F$, +\[ + \E[F\one_{\{A_U=1\}}\mid\text{selectors and exposed labels}] + =p_l\E[F\mathcal L_U\mid\text{selectors and exposed labels}]. +\] +After division by $p_l$, this is exactly the branch expectation. +Local status decisions in disjoint eligible tiles multiply the +conditional product density by factors on those tiles alone, so do not +alter this conclusion. + +The event $A_U=1$ requires its own selector to be one. Conditional on +eligibility and all required valid labels, the additional acceptance +probability is $p_l$; dropping the other restrictions gives +\Cref{hier:incidence}. Finally, acceptance only attaches auxiliary +indicators to the initially sampled product field, and never resamples +or changes that field. Omitting the higher-level rules leaves the same +argument on the truncated hierarchy. +\end{proof} + +For a stage $n$, $l_0\le n\le N$, place active cell ports at precisely +the accepted roots of levels at least $n$. Use their branch strengths +and reference potentials. Every site without an active cell port has +its true real potential. Because eligible tiles are disjoint, there is +at most one cell port at a site. Denote the scattering matrix without +artificial probes by $S_{n,N}$. + +We may also prescribe at most one artificial probe per site, absent or +of strength $\tau$ or $\tau\vartheta(v_i)$. Its mask may depend on the +offset and on external randomness independent of the potentials and +branch decisions. The profile is always evaluated at the true value, +including at an active cell root. When comparing stages $n