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merge=lfs -text +preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf filter=lfs diff=lfs merge=lfs -text +preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/README.md b/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..552c773d6b84f7302c8d6e0b385b0da055062692 --- /dev/null +++ b/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/README.md @@ -0,0 +1,18 @@ +# [Equality and rigidity in the spacetime Penrose inequality](paper.pdf) + +**Author:** OpenAI + +**Date:** September 27, 2026 + +## Citation + +```bibtex +@misc{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026, + author = {{OpenAI}}, + title = {{Equality and rigidity in the spacetime Penrose inequality}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf}{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/build/three/06-global-classification.tex b/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/build/three/06-global-classification.tex new file mode 100644 index 0000000000000000000000000000000000000000..6cfa4a46c45d4dbf0def8f1fa5543acf3b2f4791 --- /dev/null +++ b/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/build/three/06-global-classification.tex @@ -0,0 +1,730 @@ +\section{Global classification and recovery through the horizon} +\label{q3:global:section} + +We continue with the finite-component class of +Theorem~\ref{q3:intro:rigidity}, with no prescribed boundary genera. +Starting from the complete static base and end estimates just proved, +we compactify one end of the double at its actual regularity and retain +all possible other ends while proving zero-mass rigidity. We then exclude the +interior null set. This order preserves the information needed to +recover both the original metric and second fundamental form. + +\begin{lemma}[The two conformal metrics] +\label{q3:static:conformal-completion} +Define +\[ +h_\pm=\left(\frac{1\pm\lambda}{2}\right)^4h +\quad\text{on }\mathcal E. +\] +Both are scalar flat. The plus end satisfies +\(h_+=\delta+O_k(r^{-1-\epsilon})\), hence has zero ADM mass. +The minus end compactifies by one point to a nondegenerate +\(C^{1,\epsilon}\cap W^{2,p}_{\mathrm{loc}}\) metric for some +\(p>3\), with scalar curvature zero as a distribution at that point. +\end{lemma} + +\begin{proof} +Scalar flatness follows from +Equation~\eqref{q3:static:static-equations} and the three-dimensional +conformal scalar-curvature formula, since \(1\pm\lambda\) are harmonic. +In terms of the preceding variables, +\begin{equation} +\label{q3:static:conformal-gamma} +h_+=\cosh^4(U/2)\gamma,\qquad +h_-=\sinh^4(U/2)\gamma. +\end{equation} +The first factor is \(1+O_k(r^{-2})\), proving the assertion for +\(h_+\). Its ADM flux is \(O(r^{-\epsilon})\), so its mass is zero. + +For the minus end invert \(x=y/|y|^2\), write \(\rho=|y|\), and +put \(R(y)=I-2y\otimes y/\rho^2\). +The inversion differential is \(\rho^{-2}R(y)\), with +\(R(y)^TR(y)=I\). +Equation~\eqref{q3:static:gamma-expansion} gives +\[ +I^*\gamma=\rho^{-4} +\bigl(I+O_k(\rho^{1+\epsilon})\bigr), +\qquad +U(I(y))=-M\rho+(b_1\cdot y)\rho + +O_k(\rho^{2+\epsilon}). +\] +Derivatives of the angular matrix \(R(y)\) cost the corresponding +powers of \(\rho^{-1}\), so the displayed metric estimate retains +the symbol bounds. Dividing the expansion of \(\sinh(U/2)\) by +\(\rho\), and using \(\epsilon<1\), gives +\begin{equation} +\label{q3:static:inverted-minus} +I^*h_-=\left(\frac M2\right)^4 +\left[ +\left(1-\frac{4b_1\cdot y}{M}\right)I + +O_k(\rho^{1+\epsilon}) +\right]. +\end{equation} +In particular, its only first-order term is linear and scalar. + +For the remainder \(E(y)\), the bounds +\[ +|E|\leq C\rho^{1+\epsilon},\quad +|DE|\leq C\rho^\epsilon,\quad +|D^2E|\leq C\rho^{\epsilon-1} +\] +show that \(E(0)=DE(0)=0\) extends it as \(C^{1,\epsilon}\). +To see the Hölder estimate between two points, use the gradient +bound when their separation is comparable to their radii; otherwise +integrate the Hessian along the short segment in a common annulus. +Moreover choose +\[ +30,p>3\), is scalar flat as a distribution, +and has an end on which it is smooth with +\[ +\mathfrak g=\delta+O_2(r^{-1-\epsilon}),\qquad \epsilon>0. +\] +The other ends can be arbitrary. +Then \((W,\mathfrak g)\) is isometric to Euclidean space. +The asserted regularity is sufficient at a compactified point and +at a reflected hypersurface. +\end{lemma} + +\begin{proof} +An orientable three-manifold is spin. +Choose a spin structure and use a smooth background metric to +identify its spinor bundle with that for \(\mathfrak g\). +The positive square-root change of orthonormal frames has the +stated metric regularity. +The spin connection consequently has continuous locally bounded +coefficients, with weak first derivatives in \(L^p_{\mathrm{loc}}\). +Write \(\nabla^S\) for that connection, \(D\) for the Dirac operator, +and \(c\) for Clifford multiplication. + +The weak Lichnerowicz formula has the following integrated form for +every compactly supported \(H^1\) spinor \(\phi\): +\begin{equation} +\label{q3:static:lichnerowicz} +\|D\phi\|_2^2=\|\nabla^S\phi\|_2^2. +\end{equation} +Here and below norms and integrals are for \(\mathfrak g\). +To check the formula at the indicated regularity, first use a smooth +compactly supported spinor. Approximate the metric on its support +in \(C^1\) and \(W^{2,p}\) by smooth metrics, using the same +background-bundle identification. +The smooth Lichnerowicz formula passes to the limit: connection +coefficients converge uniformly and scalar curvatures converge +in \(L^p\). +The limiting scalar term is zero by the assumed distributional +scalar flatness and the \(L^p\) curvature expression. +Density then proves Equation~\eqref{q3:static:lichnerowicz} for compact +\(H^1\) spinors. +Thus no integration surface is retained around the compactified point. + +We next establish the coercivity needed to complete compact spinors. +For a compactly supported scalar function \(f\) on the AE end, +one-dimensional integration by parts along coordinate rays gives +\begin{equation} +\label{q3:static:radial-hardy} +\int_{r_0}^\infty f^2\dd r +\leq4\int_{r_0}^\infty r^2|\partial_rf|^2\dd r. +\end{equation} +The inner boundary term has the favorable sign. +Apply this to \(f=|\phi|\) and integrate over angles. +Metric comparability and +\(|\dd|\phi||\leq|\nabla^S\phi|\) give +\begin{equation} +\label{q3:static:spin-hardy} +\int_{\mathrm{AE}}r^{-2}|\phi|^2\dd V_{\mathfrak g} +\leq C\|\nabla^S\phi\|_2^2. +\end{equation} +For any fixed compact \(K\subset W\), connect it to a fixed end +annulus by a relatively compact connected domain. +Poincaré's inequality with the norm on that annulus retained gives +\[ +\||\phi|\|_{L^2(K)} +\leq C_K\left( +\|\dd|\phi|\|_{L^2(\text{domain})} + +\|\phi\|_{L^2(\text{annulus})}\right). +\] +For example, this version follows from the ordinary Poincaré +inequality after subtracting the mean; the observed annulus +controls that mean because it has positive measure. +Equation~\eqref{q3:static:spin-hardy} controls its last term. +We have proved +\begin{equation} +\label{q3:static:local-coercivity} +\|\phi\|_{L^2(K)}\leq C_K\|\nabla^S\phi\|_2. +\end{equation} +Connectedness is the only global input in this propagation. + +Let \(\mathcal H\) be the completion of compactly supported smooth +spinors in the norm \(\|\nabla^S\phi\|_2\). +Equation~\eqref{q3:static:local-coercivity} embeds \(\mathcal H\) +in \(L^2_{\mathrm{loc}}\), and the locally bounded connection +then embeds it in \(H^1_{\mathrm{loc}}\). +Both \(\nabla^S:\mathcal H\to L^2\) and +\(D:\mathcal H\to L^2\) are well-defined, and +Equation~\eqref{q3:static:lichnerowicz} makes the latter an isometry. +Its range is therefore closed. +Equation~\eqref{q3:static:spin-hardy} also passes to this completion. +The local constants may depend on the chosen compact set; no uniform +estimate over the remaining ends is required. + +Fix the ordinary orthonormalized coordinate frame on the distinguished +end and a smooth cutoff \(\theta\), equal to one near infinity and +zero before a slightly smaller end neighborhood. +For each constant spinor \(z\in\mathbb C^2\), set +\(\psi_0=\theta z\). +Because the connection there is \(O(r^{-2-\epsilon})\), +\(\nabla^S\psi_0,D\psi_0\in L^2\). +Orthogonally project \(D\psi_0\) onto the closed subspace +\(D\mathcal H\) and choose \(\eta\in\mathcal H\) such that +\[ +D\eta=-\operatorname{proj}_{D\mathcal H}(D\psi_0). +\] +Set \(\psi=\psi_0+\eta\), \(w=D\psi\). +Then \(w\in L^2\) and +\[ +\int\langle w,D\phi\rangle\dd V_{\mathfrak g}=0 +\quad\text{for every compact smooth }\phi. +\] +Thus \(Dw=0\) weakly. +Local elliptic regularity for a first-order elliptic operator with +locally Lipschitz principal coefficients and bounded lower-order +coefficients gives \(w\in H^1_{\mathrm{loc}}\). +This regularity can also be seen by freezing the principal +coefficients on small coordinate balls: the constant-coefficient +Dirac estimate controls one derivative in \(L^2\), the oscillation +is absorbed, and commutators of mollification are bounded by the +local Lipschitz norm. Apply the estimate to regularized \(w\) +and pass to a weak limit. +The metric in the lemma has more than this coefficient regularity. + +Completeness supplies compactly supported Lipschitz distance cutoffs +\(\chi_R\), equal to one on the radius-\(R\) ball and with +\(|\dd\chi_R|\leq C/R\). +Indeed a complete locally compact Riemannian length space is proper, +and a piecewise linear cutoff of its distance has these properties. +Equation~\eqref{q3:static:lichnerowicz} applies to \(\chi_Rw\), and +the weak equation gives +\[ +\|\nabla^S(\chi_Rw)\|_2^2 +=\|D(\chi_Rw)\|_2^2 +=\|c(\dd\chi_R)w\|_2^2 +\leq \frac C{R^2}\|w\|_2^2. +\] +On every fixed compact set the cutoff eventually equals one. +It follows that \(\nabla^S w=0\). +A parallel spinor has constant length on the connected manifold; +the distinguished end has infinite volume, so \(w\in L^2\) +forces \(w=0\). +No condition at another end entered this argument. + +It remains to show that \(\psi\), rather than just \(D\psi\), is +parallel. Define the continuous sesquilinear form +\[ +\mathcal B(a,b)= +\int\left(\langle\nabla^Sa,\nabla^Sb\rangle +-\langle Da,Db\rangle\right)\dd V_{\mathfrak g} +\] +on fields for which the indicated derivatives are in \(L^2\). +For compact \(\phi\), the polarized Lichnerowicz identity, +with one slot compactly supported, gives +\(\mathcal B(\psi_0,\phi)=0\). +Approximate \(\eta\) in \(\mathcal H\) by compact spinors. +These limits use the global derivative norms; no boundary flux of +\(\eta\) at another end is invoked. +Continuity and Equation~\eqref{q3:static:lichnerowicz} imply +\[ +\mathcal B(\psi_0,\eta)=0,\qquad +\mathcal B(\eta,\eta)=0,\qquad +\mathcal B(\psi,\psi)=\mathcal B(\psi_0,\psi_0). +\] +The last expression is zero. To see this directly, integrate the +Lichnerowicz identity for \(\psi_0\) up to a large end sphere. +There is no contribution at another end because \(\psi_0\) +vanishes there. On that sphere \(\psi_0=O(1)\) and +\(\nabla^S\psi_0,D\psi_0=O(r^{-2-\epsilon})\); +the boundary integral is \(O(r^{-\epsilon})\) and tends to zero. +This is also the zero ADM boundary term of the spin proof. +Since \(D\psi=w=0\), we conclude that +\(\|\nabla^S\psi\|_2^2=0\). + +The parallel equation gives each resulting spinor a +\(C^{1,\beta}_{\mathrm{loc}}\) representative for some \(\beta>0\). +Indeed, \(H^1\subset L^6\) and the locally bounded connection first +give \(W^{1,6}\), hence local H\"older continuity; the locally H\"older +connection coefficients then make its first derivatives H\"older. +Write \(\psi_z\) for this parallel spinor. The fixed cutoff, +orthogonal projection, and inverse of \(D:\mathcal H\to D\mathcal H\) +make \(z\mapsto\psi_z\) complex linear. +For any point \(p\in W\), consider +\[ + E_p:\mathbb C^2\longrightarrow\mathbb S_p, + \qquad E_p(z)=\psi_z(p), +\] +where \(\mathbb S_p\) is the complex spinor fiber. +If \(E_p(z)=0\), parallelism and connectedness give \(\psi_z=0\) +everywhere. Then \(\eta=-\theta z\), which for \(z\ne0\) has infinite +\(r^{-2}\)-weighted \(L^2\) norm on the distinguished end, +contrary to Equation~\eqref{q3:static:spin-hardy}. +Thus \(E_p\) is injective for every \(p\), and it is an isomorphism +because the complex spinor rank is two. The images of a basis of +\(\mathbb C^2\) form a global parallel spinor frame. + +Its Hermitian Gram matrix is constant and positive definite; +a constant change of basis makes the frame orthonormal. +The weak curvature equations annihilate this frame, so the spin +curvature vanishes. Faithfulness of the spin representation on +\(\mathfrak{spin}(3)\) gives zero Riemann curvature. +Moreover, Clifford multiplication identifies the real tangent bundle +with the traceless skew-Hermitian endomorphisms of the spinor bundle. +Three fixed orthonormal matrices in this parallel spinor frame, +with matrix inner product \(-\tfrac12\operatorname{tr}(AB)\), +therefore give a global parallel orthonormal tangent frame. +The dual coframe is closed by torsion-freeness and integrates +locally to coordinates in which the metric is exactly Euclidean. +This also removes any apparent metric singularity at the +compactified point. + +Finally, completeness makes the universal flat cover Euclidean +\(\R^3\). The global parallel frame makes its deck transformations +translations. A nontrivial discrete translation group has rank +at least one, and a quotient by it has volume growth at most +quadratic. The distinguished AE end gives a cubic lower bound: +radial paths in that end put a Euclidean annulus of radius +comparable to \(R\) inside a metric ball of radius \(CR\). +The two volume bounds are incompatible. +The deck group is therefore trivial, proving the lemma. +\end{proof} + +\begin{lemma}[The infinity component has no interior missing boundary] +\label{q3:static:no-internal-null} +The component \(\mathcal E\) is all of \(\Omega\). +In particular \(N>0\) everywhere in the open exterior and the +stationary metric is vacuum there. +\end{lemma} + +\begin{proof} +The complete base of Lemma~\ref{q3:static:base-completion} reaches every +component of $S$. Take its complete two-copy double, use $h_+$ on +the plus copy and $h_-$ on the minus copy, and identify corresponding +boundary points. The signed reflected lapse expresses the joined +metric as +\[ +q=\left(\frac{1+\lambda_{\mathrm{signed}}}{2}\right)^4h_{\mathrm d}. +\] +It is smooth at each positive-lapse join and $C^2$ at each zero-lapse +join. The matched second jets make its scalar curvature zero across +every join, as well as on the two open sides. Compactify the unique +AF end of the minus copy by +Lemma~\ref{q3:static:conformal-completion}. + +We verify completeness of this conformal manifold without placing any +restriction on the number or regularity of the possible internal-null +ends. On the plus copy the factor $(1+\lambda)/2$ is at least $1/2$. +Fix a sufficiently large original AF tail. On its complement the +continuous function $\sqrt N$ is defined on a compact subset of the +original closed exterior, equals zero on $S$ and the internal zero +set, and is strictly less than one elsewhere by +Lemma~\ref{q3:static:base-completion}. Its maximum on this compact set +is therefore some $a_0<1$. Thus on the entire corresponding minus +remainder $(1-\lambda)/2\geq(1-a_0)/2>0$ uniformly. On the double +away from the minus AF tail the conformal metric is consequently +bounded below by a fixed positive multiple of the complete doubled +base metric. All joins are already attached. A finite-length escape +can therefore only run down the minus AF tail, and its proved +nondegenerate one-point compactification supplies its limit. More +explicitly, truncate that tail at a large sphere. The other side of +the sphere is complete with compact boundary by the uniform lower +bound, and the compactified tail is a compact metric region with +that same boundary; their gluing is complete. + +The conformal manifold is connected, orientable, and without boundary. +It has local $C^{1,\alpha}\cap W^{2,p}$ regularity for some +$\alpha>0$ and $p>3$: the finitely many joins are at least $C^2$, +and the sole compactification point has the regularity established +in Lemma~\ref{q3:static:conformal-completion}. Its scalar curvature is +zero distributionally, including at that point, and its distinguished +plus end has the fast decay required by +Lemma~\ref{q3:static:spin-rigidity}. That lemma makes this entire +manifold Euclidean space. + +Choose a sufficiently large sphere in the distinguished plus end. +Under the Euclidean isometry it is a compact embedded sphere. Its +end tail is the unbounded component of the complement: paths from +large end radii to that sphere have lengths tending to infinity, +and the tail has precisely that sphere as frontier. The other +component is bounded and has compact closure. A sequence in +$\mathcal E$ approaching an interior point with $N=0$ lies outside +the chosen end tail and escapes every compact set of the complete +conformal manifold. Indeed it cannot converge at an ordinary plus +point by continuity of $N$, cannot converge on any joined component +because every original boundary collar has no interior zero, and +cannot converge at a point in the open minus copy or its compactified +end. Such a sequence is impossible in the compact Euclidean +complement. Thus $\mathcal E$ has no interior boundary in $\Omega$. + +It is a nonempty open and closed subset of connected $\Omega$, and +hence equals $\Omega$. Vacuum now follows from +Proposition~\ref{q3:variation:adjoint}, since $N>0$ everywhere in +$\Omega$. +\end{proof} + +\begin{lemma}[Equality forces a single spherical boundary component] +\label{q3:static:one-boundary-component} +In the Euclidean conformal double, every original component $S_i$ +is a round sphere of radius $a=(8\kappa)^{-1}$ and has original area +$|S_i|_g=16\pi m^2=|S|_g$. Consequently $\ell=1$. +\end{lemma} + +\begin{proof} +The completed conformal double in the proof of +Lemma~\ref{q3:static:no-internal-null} is Euclidean. In particular its +plus metric $q=h_+$ is flat up to every original boundary component. +Put $\phi=(1+\lambda)/2$. On each $S_i$, +Lemma~\ref{q3:static:base-completion} gives +\[ +\phi=\tfrac12,\qquad +\partial_{\nu_h}\log\phi=\kappa,\qquad +\mathrm{II}_h=0,\qquad h|_{TS_i}=g|_{TS_i}. +\] +The conformal second-fundamental-form formula, with normal pointing +into the plus side, is therefore +\[ +\mathrm{II}_q + =\phi^2\bigl(\mathrm{II}_h+ + 2\partial_{\nu_h}\log\phi\,h|_{TS_i}\bigr) + =8\kappa\,q|_{TS_i}, + \qquad q|_{TS_i}=\tfrac1{16}g|_{TS_i}. +\] +Let $F_i$ be the Euclidean position vector on $S_i$ and $\nu_i$ +its unit normal into the plus side. Its shape operator is +$8\kappa I$, so the tangential differential of $F_i-a\nu_i$ is +zero for $a=(8\kappa)^{-1}$. Connectedness makes $F_i-a\nu_i$ +a constant point $c_i$. Thus the image lies on the sphere of radius +$a$ centered at $c_i$, with $\nu_i$ its outward radial normal. +The image is open in that sphere by local invertibility and closed +by compactness, hence is the whole sphere. The Euclidean embedding +therefore identifies $S_i$ diffeomorphically with it. +Since $q|_{TS_i}=g|_{TS_i}/16$, its area satisfies +\begin{equation} +\label{q3:static:individual-boundary-area} +|S_i|_g=16|S_i|_q=64\pi a^2 + =\frac{\pi}{\kappa^2}=16\pi m^2=|S|_g. +\end{equation} +Summing over the finite family yields +$|S|_g=\sum_{i=1}^{\ell}|S_i|_g=\ell|S|_g$. The boundary is nonempty +and $|S|_g>0$, hence $\ell=1$. In particular no connectedness or spherical +topology assumption was needed before this step. +\end{proof} + +\begin{lemma}[Identification of the base and its mass] +\label{q3:static:schwarzschild-base} +The regular completion of \((\Omega,h,\lambda)\) is the +Schwarzschild static exterior +\begin{equation} +\label{q3:static:isotropic-base} +h=\left(1+\frac a\rho\right)^4\delta,\qquad +\lambda=\frac{1-a/\rho}{1+a/\rho},\qquad +\rho\geq a, +\end{equation} +where \(a=m/2\). +The completion is homeomorphic to the original \(\Mext\), +and \(S\) is identified diffeomorphically with its horizon sphere +in the regular base boundary structure. +\end{lemma} + +\begin{proof} +Lemma~\ref{q3:static:one-boundary-component} identifies the boundary +in the Euclidean conformal double as one round sphere of radius +$a=(8\kappa)^{-1}$. The plus side is its outside: its normal is the +outward radial normal, its entire boundary is this sphere, and it +contains the distinguished unbounded end. Center Euclidean +coordinates at this sphere and write $\rho$ for their radius. + +In the resulting Euclidean coordinates, write +\(\psi=2/(1+\lambda)\), so \(h=\psi^4\delta\). +The scalar-flat conformal equation gives \(\Delta_\delta\psi=0\) +outside the sphere, with \(\psi=2\) on it and \(\psi\to1\) +at infinity. The function \(1+a/\rho\) has the same data. +Their difference vanishes by the exterior maximum principle. +This proves Equation~\eqref{q3:static:isotropic-base}. +Its Schwarzschild mass is $2a=(4\kappa)^{-1}=m$, since +$\kappa=1/(4m)$. In particular $a=m/2$. + +The regular base collar has the same underlying boundary topology +as the original collar, even when its coordinate is +\(\lambda=\sqrt N\) instead of the original defining function \(N\). +Together with the unchanged interior this identifies its completion +homeomorphically with \(\Mext\). +An interior base isometry extends uniquely to the metric completions. +On the regular boundary structures it is smooth: its boundary +restriction preserves induced lengths, hence is a smooth boundary +isometry, and normal geodesic coordinates extend it smoothly to +a collar. +Thus the completion has the claimed global topology, and +\(\Omega\cong(a,\infty)\times S^2\) is simply connected. +\end{proof} + +\subsection{The time graph and the Kruskal attachment} +\label{q3:static:embedding-subsection} + +\begin{proof}[Proof of Proposition~\ref{q3:static:classification}] +Lemma~\ref{q3:static:one-boundary-component} proves $\ell=1$ and the +spherical topology of $S$. Lemma~\ref{q3:static:no-internal-null} proves +\(N>0\) on \(\Omega\), and +Lemma~\ref{q3:static:schwarzschild-base} gives the global static +base with mass \(m\). +For the reconstruction write \(M=m\), so \(\kappa=1/(4M)\). +Simple connectivity and Lemma~\ref{q3:static:base-completion} +give a global smooth function \(H\) with +\begin{equation} +\label{q3:static:time-primitive} +\dd H=-A. +\end{equation} +In static Schwarzschild coordinates use time \(T=t+H(x)\). +Equations~\eqref{q3:static:base-definitions} and +\eqref{q3:static:time-primitive} give the exact identity +\begin{equation} +\label{q3:static:reconstructed-metric} +-N\dd T^2+h +=-u^2\dd t^2+g_{ij}(\dd x^i+X^i\dd t) + (\dd x^j+X^j\dd t). +\end{equation} +The graph \(T=H(x)\), corresponding to \(t=0\), therefore induces +the original metric \(g\). +The stationary future unit normal is \(n=(\partial_t-X)/u\). +With the second-fundamental-form convention of the theorem, +\begin{equation} +\label{q3:static:reconstructed-k} +K_{\mathrm{graph}} +=\frac1{2u}\bigl(\partial_tg-\mathcal L_Xg\bigr) +=-\frac1{2u}\mathcal L_Xg +=K. +\end{equation} +This calculation recovers the original \(K\), including a +non-time-symmetric slice. + +We next establish smoothness in the original boundary structure. +If \(u|_S=0\), Equation~\eqref{q3:static:double-norm} gives +\[ +A=\frac{Y^\flat}{a^2-s^2|Y|_g^2}. +\] +This is smooth in the original collar. +The primitive \(H\) therefore extends smoothly and finitely to \(S\). +For instance, fix its values on one interior collar section and +integrate its smooth normal derivative to the boundary. +Tangential derivatives of this extension agree with those prescribed +by \(\dd H=-A\), by continuity from the interior. + +If \(u|_S>0\), the numerator +\(X^\flat-(2\kappa)^{-1}\dd N\) vanishes as a one-form on \(S\), +by Equation~\eqref{q3:static:simple-norm}. +Smooth division by the defining function \(N\) gives +\begin{equation} +\label{q3:static:log-time} +A=\frac1{2\kappa}\dd\log N+\beta,\qquad +H=-\frac1{2\kappa}\log N+B, +\end{equation} +where \(\beta\) and \(B\) are smooth in the original collar. +The smoothness of \(B\) follows by applying the same normal-integration +argument to its differential \(-\beta\). +In this case \(H\to+\infty\) at the boundary, so static time itself +is not a regular attachment coordinate. + +Let \(r\) be the Schwarzschild area radius. On the whole exterior +\[ +N=1-\frac{2M}{r},\qquad +r=\frac{2M}{1-N}. +\] +Normalize the tortoise coordinate and Kruskal coordinates by +\begin{equation} +\label{q3:static:kruskal-definitions} +\begin{aligned} +r_*&=r+2M\log\left(\frac r{2M}-1\right),\\ +\mathsf U&=-e^{-\kappa(T-r_*)},& +\mathsf V&= e^{\kappa(T+r_*)}. +\end{aligned} +\end{equation} +On the right exterior \(\mathsf U<0,\mathsf V>0\), and +the Schwarzschild metric extends as +\[ +\mathbf g_M +=-\frac{32M^3}{r}e^{-r/(2M)} + \dd\mathsf U\,\dd\mathsf V+r^2\dd\omega^2. +\] +An exact useful form of Equation~\eqref{q3:static:kruskal-definitions} +is +\begin{equation} +\label{q3:static:kruskal-factor} +\mathsf U=-Q(N)\sqrt N\,e^{-\kappa T},\qquad +\mathsf V= Q(N)\sqrt N\,e^{\kappa T},\qquad +Q(N)=\frac{\exp\!\bigl(1/[2(1-N)]\bigr)}{\sqrt{1-N}}. +\end{equation} +The function \(Q\) is smooth and positive near zero, with +\(Q(0)=\sqrt e\). + +If \(u|_S=0\), both \(H\) and +\(\sqrt N=s\sqrt{a^2-s^2|Y|^2}\) are smooth. +Substitution into Equation~\eqref{q3:static:kruskal-factor} gives +\(\mathsf U=\mathsf V=0\) on \(S\) and +\[ +\partial_s\mathsf U|_S +=-\sqrt e\,\kappa e^{-\kappa H|_S}<0,\qquad +\partial_s\mathsf V|_S +=\sqrt e\,\kappa e^{\kappa H|_S}>0. +\] +This is a smooth attachment at the bifurcation sphere. +If \(u|_S>0\), Equation~\eqref{q3:static:log-time} instead gives +\begin{equation} +\label{q3:static:future-attachment} +\mathsf U=-Q(N)N e^{-\kappa B},\qquad +\mathsf V=Q(N)e^{\kappa B}. +\end{equation} +These are smooth in the original collar; +\(\mathsf U\) vanishes simply and +\(\mathsf V|_S=\sqrt e\,e^{\kappa B|_S}>0\). +Thus the attachment is on the future horizon, with each generator +met once after the angular identification. + +The angular regularity requires a further check when +\(u|_S>0\), because the regular base coordinate was \(\lambda\), +not the original \(N\). +On the smooth reflected collar in +Equation~\eqref{q3:static:even-base}, normal projection onto its fixed +boundary is invariant under \(\lambda\mapsto-\lambda\). +This follows from uniqueness of the normal geodesics and reflection +invariance of the metric. +Schwarzschild angular coordinates are constant along these normal +geodesics, and their boundary values are the smooth round-sphere +identification from Lemma~\ref{q3:static:schwarzschild-base}. +They are therefore smooth even functions of \(\lambda\). +A smooth even function, with smooth tangential parameters, is a +smooth function of \(N=\lambda^2\) for \(N\geq0\): +all odd Taylor coefficients vanish, and repeated application of +\((2\lambda)^{-1}\partial_\lambda\) to the Taylor remainder gives +continuous derivatives of every order at zero. +This proves the required angular smoothness in the original collar. +In the zero boundary-lapse case the original collar is already a +smooth base collar, so its completed base isometry supplies the +angular extension directly. + +We have constructed a smooth map \(\iota:\Mext\to\) Schwarzschild. +Equation~\eqref{q3:static:reconstructed-metric} extends to \(S\) +by continuity, hence \(\iota^*\mathbf g_M=g\) there as well. +The positive definiteness of \(g\) proves that \(\dd\iota\) is +injective at every boundary point. +Interior injectivity follows from the global base isometry; +boundary injectivity follows from the sphere identification. +The boundary image, at \(r=2M\), is disjoint from the open image, +where \(r>2M\). +The map is proper: a compact subset of Schwarzschild has bounded +area radius, and its inverse image is closed in the compact base +region corresponding to +\([2M,R]\times S^2\). +Thus the injective immersion is an embedding. + +The future normal extends smoothly even where \(u=0\). +To avoid taking a limit of the expression involving \(u^{-1}\), +choose the smooth future timelike Kruskal vector +\(Z=\partial_{\mathsf U}+\partial_{\mathsf V}\) near the boundary. +Let \(Z^\top\) be its tangential projection along the embedded +spacelike hypersurface and set +\[ +\widehat n= +\frac{Z-Z^\top} +{\sqrt{-\mathbf g_M(Z-Z^\top,Z-Z^\top)}}. +\] +The denominator is positive, and this is a smooth future unit +normal. On the open exterior it agrees with the normal used in +Equation~\eqref{q3:static:reconstructed-k}. +Its second fundamental form is smooth and equals \(K\) in the +interior, so equality extends to \(S\). + +There is also a smooth spacelike extension of the image through +its boundary. In the positive boundary-lapse case, use +\((\mathsf U,\omega)\) as hypersurface coordinates; its normal +derivative is nonzero by +Equation~\eqref{q3:static:future-attachment}. +The remaining null coordinate is a smooth graph function and can +be extended across \(\mathsf U=0\). +In the zero boundary-lapse case use +\((\chi,\omega)\), where +\(\chi=(\mathsf V-\mathsf U)/2\); the displayed normal derivatives +give \(\partial_s\chi>0\). +Again extend the remaining graph function across zero. +Smooth one-sided graph functions extend across a smooth boundary, +and compactness of \(S\), together with openness of the spacelike +condition, preserves spacelikeness on a sufficiently short collar. +This extension makes no assertion about the original data behind +\(S\). + +Finally, the exact inverse-metric identity gives +\begin{equation} +\label{q3:static:end-slope} +|\dd H|_h^2 +=\left|\frac{X^\flat}{N}\right|_C^2 +=\frac{|X|_g^2}{u^2N} +\longrightarrow\frac{|b|^2}{(b^0)^2}<1. +\end{equation} +Choose a constant \(a_0<1\) larger than the limiting slope norm. +Along radial rays in the Schwarzschild base, uniformly in angle, +\[ +|H(r,\omega)| +\leq C+a_0\int_R^r +\frac{\dd\rho}{\sqrt{1-2M/\rho}} +=a_0r+O(\log r). +\] +Since \(r_*=r+O(\log r)\), it follows that +\[ +H-r_*\longrightarrow-\infty,\qquad +H+r_*\longrightarrow+\infty. +\] +This is approach to the chosen Schwarzschild spatial infinity. +It allows a nonzero asymptotic tilt and does not impose \(P=0\) +or \(K=0\). +All conclusions of Proposition~\ref{q3:static:classification} follow. +\end{proof} + +Propositions~\ref{q3:variation:adjoint} and~\ref{q3:static:classification} +prove Theorem~\ref{q3:intro:rigidity}. They also prove +Theorem~\ref{q3:intro:connected-rigidity}: the connected-case argument +in Lemma~\ref{q3:variation:area-support} supplies the same localized +multiplier identity, and all subsequent adjoint, static, and +reconstruction steps use that identity rather than +partial-coincidence outermostness. diff --git a/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf b/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9977434b9bda3ae6bf8657b84e6736dfba00a45c --- /dev/null +++ b/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4f1aab59ba484eb8a903daed582b30ffe17bbd42392e233213a30abb3aadc710 +size 1274586 diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/README.md b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..be8e3681b6bac347162c67aed0135ebb5e7f9477 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality](sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026, + author = {{OpenAI}}, + title = {{Equality cases in the sharp one-dimensional matrix Lieb--Thirring inequality}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf}{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/main.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..9913872fbf2d05a80dfa12323ca042e0ff2fbd77 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/main.tex @@ -0,0 +1,29 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{Equality cases in the sharp one-dimensional\newline matrix Lieb--Thirring inequality} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +We classify all equality cases in the sharp one-dimensional Lieb--Thirring +inequality for every finite matrix size and $1/2<\gamma<3/2$. For measurable +Hermitian positive semidefinite potentials $W$ with +$\int_\R\tr(W^{\gamma+1/2})<\infty$, equality holds precisely for direct sums, +in one constant unitary basis, of scalar one-bound-state solitons and zero +channels. The nonzero solitons may have independent scales and centers. +\end{abstract} +\setcounter{tocdepth}{1} +\tableofcontents +\clearpage +\input{sections/introduction} +\input{sections/preliminaries} +\input{sections/action} +\input{sections/spectral-bound} +\input{sections/compactness} +\input{sections/rigidity} +\appendix +\input{sections/appendix-trace} +\input{sections/appendix-paths} +\input{references} +\end{document} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/preamble.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..435957f7e3ad9f443f80f2504c52d07ae59932b6 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/preamble.tex @@ -0,0 +1,28 @@ +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} +\usepackage{graphicx,microtype,xcolor,tikz} +\usetikzlibrary{arrows.meta,positioning,fit,calc} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black,pdftitle={Equality cases in the sharp one-dimensional matrix Lieb--Thirring inequality},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{\C}{\mathbb C} +\newcommand{\HS}{\mathrm{HS}} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\sech}{sech} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\op}{op} +\numberwithin{equation}{section} +\allowdisplaybreaks[2] +\setlength{\emergencystretch}{1.5em} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/references.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..66d32b2eb2b9c655f06e2f9a67e1134b35b6963b --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/references.tex @@ -0,0 +1,90 @@ +\begin{thebibliography}{99} +\raggedright +\bibitem{AizenmanLieb1978} +Michael Aizenman and Elliott H. Lieb, +\emph{On semi-classical bounds for eigenvalues of Schr\"odinger operators}, +Physics Letters A \textbf{66} (1978), no.~6, 427--429. +\href{https://doi.org/10.1016/0375-9601(78)90385-7}{doi:10.1016/0375-9601(78)90385-7}. + +\bibitem{BenguriaLoss2000} +Rafael Benguria and Michael Loss, +\emph{A simple proof of a theorem of Laptev and Weidl}, +Mathematical Research Letters \textbf{7} (2000), no.~2--3, 195--203. +\href{https://doi.org/10.4310/MRL.2000.v7.n2.a5}{doi:10.4310/MRL.2000.v7.n2.a5}. + +\bibitem{FrankGontierLewin2025} +Rupert L. Frank, David Gontier and Mathieu Lewin, +\emph{Optimizers for the finite-rank Lieb--Thirring inequality}, +American Journal of Mathematics \textbf{147} (2025), no.~2, 503--560. +\href{https://doi.org/10.1353/ajm.2025.a954649}{doi:10.1353/ajm.2025.a954649}. + +\bibitem{HLW2000} +Dirk Hundertmark, Ari Laptev and Timo Weidl, +\emph{New bounds on the Lieb--Thirring constants}, +Inventiones Mathematicae \textbf{140} (2000), no.~3, 693--704. +\href{https://doi.org/10.1007/s002220000077}{doi:10.1007/s002220000077}. + +\bibitem{HLT1998} +Dirk Hundertmark, Elliott H. Lieb and Lawrence E. Thomas, +\emph{A sharp bound for an eigenvalue moment of the one-dimensional +Schr\"odinger operator}, +Advances in Theoretical and Mathematical Physics \textbf{2} (1998), no.~4, +719--731. +\href{https://doi.org/10.4310/ATMP.1998.v2.n4.a2}{doi:10.4310/ATMP.1998.v2.n4.a2}. + +\bibitem{Keller1961} +Joseph B. Keller, +\emph{Lower bounds and isoperimetric inequalities for eigenvalues of the +Schr\"odinger equation}, +Journal of Mathematical Physics \textbf{2} (1961), no.~2, 262--266. +\href{https://doi.org/10.1063/1.1703708}{doi:10.1063/1.1703708}. + +\bibitem{LaptevWeidl2000} +Ari Laptev and Timo Weidl, +\emph{Sharp Lieb--Thirring inequalities in high dimensions}, +Acta Mathematica \textbf{184} (2000), no.~1, 87--111. +\href{https://doi.org/10.1007/BF02392782}{doi:10.1007/BF02392782}. + +\bibitem{LiebThirring1975} +Elliott H. Lieb and Walter E. Thirring, +\emph{Bound for the kinetic energy of fermions which proves the stability +of matter}, +Physical Review Letters \textbf{35} (1975), no.~11, 687--689; +erratum, \textbf{35} (1975), 1116. +\href{https://doi.org/10.1103/PhysRevLett.35.687}{doi:10.1103/PhysRevLett.35.687}. + +\bibitem{LiebThirring1976} +Elliott H. Lieb and Walter E. Thirring, +\emph{Inequalities for the moments of the eigenvalues of the Schr\"odinger +Hamiltonian and their relation to Sobolev inequalities}, +in \emph{Studies in Mathematical Physics: Essays in Honor of Valentine +Bargmann}, Elliott H. Lieb, Barry Simon and Arthur S. Wightman (eds.), +Princeton University Press, Princeton, NJ, 1976, 269--303. + +\bibitem{Milnor1965} +John W. Milnor, +\emph{Topology from the Differentiable Viewpoint}, +University Press of Virginia, Charlottesville, VA, 1965. + +\bibitem{MLT} +OpenAI, +\emph{Sharp one-dimensional Lieb--Thirring inequalities for matrix potentials}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf}{OAI:Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026}, +2026. + +\bibitem{ReadSchulz2026} +Larry Read and Marvin R. Schulz, +\emph{The sharp one-dimensional Lieb--Thirring inequality for the sum +of eigenvalues}, +\href{https://arxiv.org/abs/2609.10478v2}{arXiv:2609.10478v2}, +September 14, 2026. + +\bibitem{Sard1942} +Arthur Sard, +\emph{The measure of the critical values of differentiable maps}, +Bulletin of the American Mathematical Society \textbf{48} (1942), +883--890. +\href{https://doi.org/10.1090/S0002-9904-1942-07811-6}{doi:10.1090/S0002-9904-1942-07811-6}. + +\end{thebibliography} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/action.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/action.tex new file mode 100644 index 0000000000000000000000000000000000000000..c9d67289222b4643ebdfd4a4c91144f951a612d5 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/action.tex @@ -0,0 +1,328 @@ +\section{An action estimate that retains the equality defects} +\label{act:section} + +The finite-interval argument for the sharp inequality can also measure +failure of its equality relations. We retain the first-order square +and a quantitative part of the matrix trace deficit. The coefficient +of this second defect will be independent of the number of rows. This +uniformity allows us to exhaust an eigenvalue list before knowing that +it is finite. + +The construction adapts the field, connecting paths, and action +identity of \cite[Sections~3--6]{MLT}. Their analytic foundations are +proved in Appendices~\ref{app:trace} and~\ref{app:paths}; the additional +estimate below extracts operator-norm rigidity from the trace deficit. + +Fix positive integers $n,d$, a bounded interval +$\Omega=(x_-,x_+)$, and +$K=\operatorname{diag}(\ell_1,\ldots,\ell_n)$ with +$0<\ell_i\leq\kappa$. For +$A\in H^1_0(\Omega;\R^{n\times d})$, set $\rho=AA^{\mathsf T}$ +and define +\begin{equation}\label{act:energy} + \mathcal E_K(A)=\int_\Omega + \bigl(\|A'\|_{\HS}^2+\tr(K^2\rho)-\tr(\rho^q)\bigr)\,dx. +\end{equation} +For pointwise matrix values $A\in\R^{n\times d}$ and +$B\in\Sym_n$, put +\begin{equation}\label{act:defect} + \Phi_K(A,B)=\min\left\{1, + \|[B,K^2]\|_{\op}^2 + +\|K^2-B^2-(AA^{\mathsf T})^r\|_{\op}^2\right\}. +\end{equation} +Thus $\Phi_K=0$ expresses two algebraic relations, while +$A'-BA=0$ is a first-order differential relation. + +\begin{theorem}[Quantitative finite-interval action estimate] +\label{act:quantitative-action} +Let $0<\sigma<1$ and $\kappa>0$. There is a constant +$c_*=c_*(\sigma,\kappa)>0$ with the following property. +For $n,d,\Omega,K$ as above, suppose +$U\in H^1_0(\Omega;\R^{n\times d})$ satisfies +\[ + \int_\Omega UU^{\mathsf T}\,dx=I_n. +\] +For every $\eta>0$ there exist a real lower triangular matrix +$C\in\R^{n\times n}$ and a path +$B\in C^1(\overline\Omega;\Sym_n)$ such that +\[ + B(x_-)=K,\qquad B(x_+)=-K,\qquad + \sup_{x\in\overline\Omega}\|B(x)\|_{\op}\leq\kappa, +\] +and, for $A=CU$, +\begin{equation}\label{act:bound} + \mathcal E_K(A)\geq b_\sigma\sum_{i=1}^n\ell_i^{2\gamma} + +\int_\Omega\left(\|A'-BA\|_{\HS}^2 + +c_*\Phi_K(A,B)\right)\,dx-\eta. +\end{equation} +In particular, $c_*$ is independent of $n,d,\Omega,U$, and the +smallest diagonal entry of $K$. +\end{theorem} + +\subsection{Coercivity at the matrix-field constraint} + +The algebraic relations $\Phi_K(A,B)=0$ require $[B,K^2]=0$ +and $\rho=(K^2-B^2)^\sigma$. For $0<\delta\leq1$, we use a +locally Lipschitz field $M_\delta:\Sym_n\to\Sym_n$ extending +the commuting rule +\[ + M_\delta(B)=(K^2-B^2+\delta I)^\sigma-\delta^\sigma I + \quad\text{if }[B,K^2]=0\text{ and }K^2-B^2\geq0. +\] +Thus the constraint $M_\delta(B)=\rho$ is a regularized form of +the algebraic equality relation in this case. The extension to +arbitrary symmetric $B$ has a $C^1$ primitive +$J_\delta:\Sym_n\to\R$ satisfying +\begin{equation}\label{act:primitive} + dJ_\delta(B)[H]=-\tr(M_\delta(B)H). +\end{equation} +Proposition~\ref{field:properties} constructs these maps and proves +their required properties. Along a path $B$, the derivative identity +converts $-\tr(M_\delta(B)B')$ into the total derivative of +$J_\delta(B)$, which will supply the boundary term in the action +identity. +Its endpoint difference is explicit: +\begin{align} + J_\delta(-K)-J_\delta(K) + &=\sum_{i=1}^n\int_{-\ell_i}^{\ell_i} + \bigl((\ell_i^2-s^2+\delta)^\sigma-\delta^\sigma\bigr)\,ds + \notag\\ + &\longrightarrow b_\sigma\sum_{i=1}^n\ell_i^{2\gamma} + \qquad(\delta\downarrow0). + \label{act:endpoints} +\end{align} +The limit follows by dominated convergence and +$2\sigma+1=2\gamma$. + +Define the scalar function +\begin{equation}\label{act:F} + F(A,B)=\tr(\rho^q)-\tr\bigl((K^2-B^2)\rho\bigr), + \qquad \rho=AA^{\mathsf T}. +\end{equation} +The following lemma bounds this nonlinear term while keeping the +algebraic defect. The constraint in its hypothesis will later be +enforced by a penalty. + +\begin{lemma}\label{act:constraint} +The constant $c_*(\sigma,\kappa)>0$ can be chosen so that, whenever +$0<\delta\leq1$, $\|B\|_{\op}\leq\kappa$, and +$M_\delta(B)=AA^{\mathsf T}$, one has +\begin{equation}\label{act:constraint-bound} + F(A,B)+c_*\Phi_K(A,B)\leq E_{n,\delta}, + \qquad + E_{n,\delta} + =n\bigl(\delta\kappa^{2\sigma} + +\delta^\sigma(\kappa^2+\delta)\bigr) + +\delta^{2\sigma}. +\end{equation} +\end{lemma} + +\begin{proof} +Write $\rho=M_\delta(B)\geq0$. Proposition~\ref{field:positive} +gives +\begin{equation}\label{act:positive-symbol} + Q_s:=s^2I-2sB+K^2\geq0\quad(s\in\R), + \qquad 0\leq\rho\leq\kappa^{2\sigma}I. +\end{equation} +Put $V=K^2-B^2+\delta I$ and $\beta=\sigma-\tfrac12$. +With the positive normalization constant $c_\sigma$ specified in +\eqref{trace:normalization}, define +\begin{equation}\label{act:RN} + \begin{split} + R(t)&=\frac1\pi\int_\R + \bigl(Q_s+(\delta+t)I\bigr)^{-1}\,ds,\qquad t>0,\\ + N&=c_\sigma\int_0^\infty + t^\beta\bigl(t^{-1/2}I-R(t)\bigr)\,dt + =\rho+\delta^\sigma I. + \end{split} +\end{equation} +The last identity is also part of Proposition~\ref{field:positive}. +Since $(sI-B)^2+V=Q_s+\delta I\geq\delta I$ and $N\geq0$, +Theorem~\ref{trace:deficit} applies. It supplies +$L(t)=R(t)^{-1}>0$ and Hermitian matrices $Z(t)$ satisfying +\begin{equation}\label{act:LZ} + V+tI=L^2+Z^2+i[B,Z],\qquad + LZ+ZL+i[B,L]=0, +\end{equation} +and the exact identity +\begin{align} + \Delta&:=\tr(NV)-\tr(N^q)\notag\\ + &=c_\sigma\int_0^\infty t^\beta + \left(\tr(R(t)Z(t)^2) + +q\bigl\|R(t)^{-1/2} + -R(t)^{1/2}(tI+N^r)^{1/2}\bigr\|_{\HS}^2\right)\,dt + \geq0. + \label{act:trace-deficit} +\end{align} +The matrices in \eqref{act:LZ} may be complex Hermitian even though +$A$ and $B$ are real; $i$ denotes the imaginary unit. + +We extract a dimension-independent bound from the part of the +integral with $1\leq t\leq2$. On this interval there are constants +$a_\kappa,A_\kappa>0$ such that +\begin{equation}\label{act:R-bounds} + a_\kappa I\leq R(t)\leq A_\kappa I. +\end{equation} +For the upper bound, use $Q_s\geq0$ on a bounded interval in $s$, +and $Q_s\geq\tfrac12s^2I$ for $|s|\geq\max\{1,4\kappa\}$. +For the lower bound, restrict the integral to $|s|\leq1$, where +$Q_s+(\delta+t)I\leq(4+2\kappa+\kappa^2)I$. +These estimates depend on $\kappa$ alone. Moreover +$\|N\|_{\op}\leq\kappa^{2\sigma}+1$, so +$H_t=(tI+N^r)^{1/2}$ and $L(t)$ are bounded in operator norm by +constants depending only on $\sigma,\kappa$. + +The two integrands in \eqref{act:trace-deficit} control +$\|Z(t)\|_{\HS}^2$ and $\|L(t)-H_t\|_{\HS}^2$: +indeed $\tr(RZ^2)=\tr(ZRZ)$ and +\[ + R^{-1/2}-R^{1/2}H_t=R^{1/2}(L-H_t). +\] +It follows by averaging over $[1,2]$ that at some $t$ in this +interval, +\begin{equation}\label{act:small-squares} + \|Z(t)\|_{\op}+\|L(t)-H_t\|_{\op} + \leq C_1\sqrt\Delta, +\end{equation} +where $C_1$ depends only on $\sigma,\kappa$. +When $\Delta=0$, the same conclusion follows by taking a point +where both nonnegative integrands vanish. + +Suppose first that $\Delta\leq1$, and use this value of $t$. +The first identity in \eqref{act:LZ} gives +\[ + V-N^r=(L^2-H_t^2)+Z^2+i[B,Z]. +\] +The boundedness of $L,H_t,B$, together with +\eqref{act:small-squares}, bounds the norm of this expression by +$C_2\sqrt\Delta$. To control the commutator, the second identity +in \eqref{act:LZ} first yields +$\|[B,L]\|_{\op}\leq2\|L\|_{\op}\|Z\|_{\op}$. +Then +\[ + [B,V]=[B,L^2]+[B,Z^2]+i[B,[B,Z]] +\] +has the same bound. Thus +\begin{equation}\label{act:V-coercivity} + \|V-N^r\|_{\op}+\|[B,V]\|_{\op} + \leq C_3\sqrt\Delta. +\end{equation} +Here and below the constants in this proof depend only on +$\sigma,\kappa$. + +The matrices $N=\rho+\delta^\sigma I$ and $\rho$ commute. +Because $r>1$, the scalar function $y\mapsto y^r$ is Lipschitz on +$[0,\kappa^{2\sigma}+1]$. Consequently +\[ + \|N^r-\rho^r\|_{\op}\leq C_4\delta^\sigma. +\] +Since $[B,V]=[B,K^2]$ and $\delta\leq\delta^\sigma$, +\eqref{act:V-coercivity} implies +\[ + \|[B,K^2]\|_{\op}^2 + +\|K^2-B^2-\rho^r\|_{\op}^2 + \leq C_5(\Delta+\delta^{2\sigma}). +\] +Choose $c_*>0$ sufficiently small, also with $c_*\leq1$. +For $\Delta\leq1$ the preceding estimate gives +\begin{equation}\label{act:coercivity} + c_*\Phi_K(A,B)\leq\Delta+\delta^{2\sigma}. +\end{equation} +For $\Delta>1$ this follows directly from $\Phi_K\leq1$. +This proves the required uniform control of the algebraic defect. + +It remains to relate $\Delta$ to $F$. Expanding $N$ and $V$ gives +\[ + F(A,B)=-\Delta+\tr(\rho^q)-\tr(N^q) + +\delta\tr\rho+\delta^\sigma\tr V. +\] +The first two powers are ordered because $N=\rho+\delta^\sigma I$. +Also $\tr\rho\leq n\kappa^{2\sigma}$ and +$V\leq(\kappa^2+\delta)I$. Adding \eqref{act:coercivity} now +proves \eqref{act:constraint-bound}. +\end{proof} + +\subsection{The action identity and the penalty limit} + +\begin{proof}[Proof of Theorem~\ref{act:quantitative-action}] +Fix $0<\delta\leq1$. Proposition~\ref{path:connecting} supplies, +for every $0<\varepsilon\leq1$, a lower triangular matrix +$C_\varepsilon$ and a path $B_\varepsilon$ with the prescribed +endpoints and operator-norm bound such that, for +$A_\varepsilon=C_\varepsilon U$, +\begin{equation}\label{act:path} + \varepsilon B_\varepsilon' + =M_\delta(B_\varepsilon) + -A_\varepsilon A_\varepsilon^{\mathsf T}. +\end{equation} +For fixed $\delta,K,\Omega,U$, the matrices $C_\varepsilon$ are +bounded uniformly in $0<\varepsilon\leq1$. + +Temporarily suppress the subscript $\varepsilon$ and write +\[ + u_\delta(A,B) + =\|M_\delta(B)-AA^{\mathsf T}\|_{\HS}^2. +\] +Since $A\in H^1_0$ and $B\in C^1$, the product and chain rules +for absolutely continuous functions, \eqref{act:primitive}, and +\eqref{act:path} yield almost everywhere +\begin{align*} + \bigl(J_\delta(B)+\tr(A^{\mathsf T}BA)\bigr)' + &=2\tr((A')^{\mathsf T}BA) + -\tr\bigl((M_\delta(B)-AA^{\mathsf T})B'\bigr)\\ + &=2\tr((A')^{\mathsf T}BA)-\varepsilon^{-1}u_\delta(A,B). +\end{align*} +Complete the square in the action and integrate this identity. +Because $A$ vanishes at both endpoints, the result is the exact +formula +\begin{align} + \mathcal E_K(A) + &=J_\delta(-K)-J_\delta(K)\notag\\ + &\quad+\int_\Omega\left( + \|A'-BA\|_{\HS}^2-F(A,B) + +\varepsilon^{-1}u_\delta(A,B)\right)\,dx. + \label{act:identity} +\end{align} + +We now use the penalty to pass from the constraint estimate in +Lemma~\ref{act:constraint} to all matrix values on these paths. +The continuous representative of $U$ is bounded on +$\overline\Omega$. The uniform bound on $C_\varepsilon$ and the +bound $\|B_\varepsilon\|_{\op}\leq\kappa$ therefore place all pairs +$(A_\varepsilon(x),B_\varepsilon(x))$ in one compact set +$\mathcal K_\delta\subset\R^{n\times d}\times\Sym_n$ on which +$\|B\|_{\op}\leq\kappa$. On this set $F$, $\Phi_K$, and +$u_\delta$ are continuous. Lemma~\ref{act:constraint} implies +\begin{equation}\label{act:penalty} + \limsup_{\varepsilon\downarrow0} + \sup_{(A,B)\in\mathcal K_\delta} + \bigl(F(A,B)+c_*\Phi_K(A,B)-\varepsilon^{-1}u_\delta(A,B)\bigr) + \leq E_{n,\delta}. +\end{equation} +Indeed, if the left side exceeded $E_{n,\delta}$, some sequence +$\varepsilon_j\downarrow0$ and corresponding pairs in the compact +set would make the expression at least $E_{n,\delta}+\tau$ for +some $\tau>0$. Boundedness of $F+c_*\Phi_K$ would force +$u_\delta=O(\varepsilon_j)$ along that sequence. A convergent +subsequence would then give a limiting pair with $u_\delta=0$ and +$F+c_*\Phi_K\geq E_{n,\delta}+\tau$, contradicting +Lemma~\ref{act:constraint}. + +For any $\tau>0$, choose $\varepsilon$ small enough that the +supremum in \eqref{act:penalty} is at most $E_{n,\delta}+\tau$. +Substitution into \eqref{act:identity} gives +\begin{align*} + \mathcal E_K(A_\varepsilon) + \geq{}&J_\delta(-K)-J_\delta(K) + -|\Omega|(E_{n,\delta}+\tau)\\ + &+\int_\Omega\left(\|A_\varepsilon'-B_\varepsilon A_\varepsilon\|_{\HS}^2 + +c_*\Phi_K(A_\varepsilon,B_\varepsilon)\right)\,dx. +\end{align*} +Finally, first choose $\delta>0$ so small that the endpoint error +in \eqref{act:endpoints} and $|\Omega|E_{n,\delta}$ together are +less than $\eta/2$. Next choose $\tau>0$ with +$|\Omega|\tau<\eta/2$, and then choose the required $\varepsilon$. +These choices prove \eqref{act:bound}. All compactness arguments +were made after fixing $\delta$; no bound uniform in $\delta$ is +needed. +\end{proof} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-paths.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-paths.tex new file mode 100644 index 0000000000000000000000000000000000000000..00a762b7d3427136e0cb65841936260cb6d0abca --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-paths.tex @@ -0,0 +1,465 @@ +\section{The regularized field and connecting paths} +\label{app:paths} + +The finite-interval action argument requires a symmetric matrix path +from $K$ to $-K$ whose derivative measures the difference between a +matrix field and $AA^{\mathsf T}$. We construct both the field and the +path here. These are the constructions of \cite[Sections~4--5]{MLT}; +we include their proofs to make the analytic input to the equality +argument self-contained. The positivity statement below also +identifies the resolvent to which Theorem~\ref{trace:deficit} applies. + +Throughout this appendix, +$K=\operatorname{diag}(k_1,\ldots,k_n)$ with $k_i>0$, +$\kappa=\max_i k_i$, and $\Sym_n$ denotes the real symmetric matrices. +We use $\beta=\sigma-\tfrac12$ and the constant $c_\sigma$ from +\eqref{trace:normalization}. + +\subsection{A field with a scalar boundary integral} + +For $\delta>0$, define on $y\geq0$ +\begin{equation}\label{field:f} + f_\delta(y)=\frac{c_\sigma}{\pi}\int_0^\infty t^\beta + \left(\frac1{t+\delta}-\frac1{t+\delta+y}\right)\,dt, +\end{equation} +and extend to $y<0$ by the tangent line $f_\delta'(0)y$. +The resulting function is $C^1$, strictly increasing, and concave. +Indeed, for $y\geq0$, +\begin{equation}\label{field:f-derivative} + f_\delta'(y)=a_\sigma(y+\delta)^{\beta-1}, + \qquad + a_\sigma=\frac{c_\sigma}{\pi} + \int_0^\infty\frac{u^\beta}{(1+u)^2}\,du>0. +\end{equation} +Both integrals converge since $-\tfrac12<\beta<\tfrac12$. +The extension allows the functional calculus below even when its +matrix argument has negative eigenvalues. + +Set +\begin{equation}\label{field:definition} + \begin{split} + Q_s(B)&=s^2I-2sB+K^2,\\ + M_\delta(B)&=\lim_{R\to\infty}\int_{-R}^{R} + \bigl(f_\delta(Q_s(B))-f_\delta(s^2)I\bigr)\,ds, + \qquad B\in\Sym_n, + \end{split} +\end{equation} +where the symmetric cutoff cancels the leading term +$-2s f_\delta'(s^2)B$, which is odd in $s$. +The scalar counterpart is +\begin{equation}\label{field:mu} + \mu_\delta(z)=\int_\R + \bigl(f_\delta(s^2+z)-f_\delta(s^2)\bigr)\,ds, + \qquad z\in\R. +\end{equation} + +\begin{proposition}[Regularized field]\label{field:properties} +The limit in \eqref{field:definition} exists locally uniformly, and +$M_\delta:\Sym_n\to\Sym_n$ is locally Lipschitz. There is a +$C^1$ function $J_\delta:\Sym_n\to\R$, normalized by +$J_\delta(0)=0$, such that +\begin{equation}\label{field:primitive} + dJ_\delta(B)[H]=-\tr(M_\delta(B)H). +\end{equation} +The scalar function $\mu_\delta$ is locally Lipschitz and strictly +increasing, with $\mu_\delta(0)=0$, and +\begin{equation}\label{field:mu-positive} + \mu_\delta(z)=(z+\delta)^\sigma-\delta^\sigma + \qquad(z\geq0). +\end{equation} +For every real unit vector $e$, and for every standard basis vector +$e_i$, respectively, +\begin{align} + e^{\mathsf T}M_\delta(B)e + &\leq\mu_\delta\bigl(e^{\mathsf T}K^2e + -(e^{\mathsf T}Be)^2\bigr), + \label{field:direction}\\ + Be_i=ye_i\quad&\Longrightarrow\quad + M_\delta(B)e_i=\mu_\delta(k_i^2-y^2)e_i. + \label{field:coordinate} +\end{align} +For each diagonal sign matrix $S$, +\begin{equation}\label{field:equivariance} + M_\delta(SBS)=SM_\delta(B)S. +\end{equation} +Finally, +\begin{equation}\label{field:endpoints} + \begin{split} + J_\delta(-K)-J_\delta(K) + &=\sum_{i=1}^n\int_{-k_i}^{k_i}\mu_\delta(k_i^2-s^2)\,ds\\ + &\longrightarrow b_\sigma\sum_{i=1}^n k_i^{2\sigma+1} + \qquad(\delta\downarrow0). + \end{split} +\end{equation} +\end{proposition} + +\begin{proof} +We first justify the matrix integral, including its local Lipschitz +dependence. If a scalar function $h$ is Lipschitz on an interval +containing the spectra of Hermitian matrices $X,Y$, then +\begin{equation}\label{field:HS-calculus} + \|h(X)-h(Y)\|_{\HS}\leq\operatorname{Lip}(h)\|X-Y\|_{\HS}. +\end{equation} +To see this, use eigenbases $e_i$ of $X$ and $v_j$ of $Y$. +The mixed matrix entries of the two differences are +$(h(\lambda_i)-h(\nu_j))\langle e_i,v_j\rangle$ and +$(\lambda_i-\nu_j)\langle e_i,v_j\rangle$; square the scalar +Lipschitz inequality and sum over $i,j$. + +On a bounded set of $B$'s the spectrum of +$E_s(B)=-2sB+K^2$ is contained in $[-a|s|,a|s|]$ for all sufficiently +large $|s|$. For +\[ + h_s(y)=f_\delta(s^2+y)-f_\delta(s^2)-f_\delta'(s^2)y, +\] +Equation~\eqref{field:f-derivative} gives +$\sup_{|y|\leq a|s|}|h_s'(y)|=O(|s|^{2\beta-3})$. +It follows from \eqref{field:HS-calculus} that +\[ + f_\delta(Q_s(B))-f_\delta(s^2)I + =-2s f_\delta'(s^2)B+T_s(B), +\] +where both $\|T_s(B)\|_{\HS}$ and the local Lipschitz constant of +$T_s$ are $O(|s|^{2\beta-2})$. For the first bound, include +$f_\delta'(s^2)K^2$ in $T_s$; for the second, apply +\eqref{field:HS-calculus} to $h_s(E_s(B))$ and use +$E_s(B_1)-E_s(B_2)=-2s(B_1-B_2)$. +Because $2\beta-2<-1$, these remainders are integrable at infinity. +The odd term cancels at each symmetric cutoff. On bounded +$s$-intervals, \eqref{field:HS-calculus} applies directly to +$f_\delta$. This proves the asserted convergence and regularity. + +For the primitive, choose a scalar antiderivative +$g_\delta'=f_\delta$. Trace differentiation gives +$d\tr(g_\delta(X))[H]=\tr(f_\delta(X)H)$. +This formula follows for polynomials by cyclicity and for +$g_\delta$ by approximation in $C^1$ on compact spectral intervals. +For $s\ne0$, the function +\[ + B\longmapsto + \frac{\tr(g_\delta(Q_s(B))-g_\delta(K^2))}{2s} + +f_\delta(s^2)\tr B +\] +has differential +$-\tr((f_\delta(Q_s(B))-f_\delta(s^2)I)H)$. +At $s=0$ that differential is constant in $B$ and also has a +primitive. Thus the negative of every finite-cutoff field has +zero integral around each closed piecewise smooth curve in +$\Sym_n$. Local uniform convergence passes this property to +$-M_\delta$. Path integration, starting at zero, defines +$J_\delta$ and proves \eqref{field:primitive}. + +For bounded $z$, the integrand in \eqref{field:mu} and its +Lipschitz constant in $z$ are $O(|s|^{2\beta-2})$ at infinity. +This proves absolute convergence and local Lipschitz continuity +of $\mu_\delta$. Strict increase and $\mu_\delta(0)=0$ follow +from the same properties of $f_\delta$. For $z\geq0$, +Tonelli's theorem applied to \eqref{field:f} gives +\[ + \mu_\delta(z)=c_\sigma\int_0^\infty t^\beta + \bigl((t+\delta)^{-1/2}-(t+\delta+z)^{-1/2}\bigr)\,dt. +\] +The normalization \eqref{trace:normalization}, after scaling, +evaluates this as \eqref{field:mu-positive}. + +Translations in the variable $s$ remain legitimate with a +symmetric cutoff. More precisely, for fixed $a,z\in\R$, +\begin{equation}\label{field:shift} + \lim_{R\to\infty}\int_{-R}^{R} + \bigl(f_\delta((s-a)^2+z)-f_\delta(s^2)\bigr)\,ds + =\mu_\delta(z). +\end{equation} +Indeed $F_z(s)=f_\delta(s^2+z)$ is even and +$F_z'(s)=O(|s|^{2\beta-1})\to0$ at infinity. Pairing the two +short intervals produced by a translation bounds the difference +between the shifted and unshifted cutoff integrals by +\[ + |a|^2\sup_{R-|a|\leq u\leq R+|a|}|F_z'(u)|, +\] +which tends to zero. + +Scalar concavity in an eigenbasis of $Q_s(B)$ now gives +\[ + e^{\mathsf T}f_\delta(Q_s(B))e + \leq f_\delta(e^{\mathsf T}Q_s(B)e). +\] +The scalar argument on the right is +$(s-e^{\mathsf T}Be)^2+e^{\mathsf T}K^2e-(e^{\mathsf T}Be)^2$. +Integration and \eqref{field:shift} prove \eqref{field:direction}. +If $Be_i=ye_i$, functional calculus gives equality in the +corresponding coordinate, proving \eqref{field:coordinate}. +Since $SK^2S=K^2$, conjugation inside the defining integral proves +\eqref{field:equivariance}. + +On diagonal matrices, \eqref{field:primitive} and +\eqref{field:coordinate} read +\[ + dJ_\delta(\operatorname{diag}(b_1,\ldots,b_n)) + =-\sum_i\mu_\delta(k_i^2-b_i^2)\,db_i. +\] +Integrating from $K$ to $-K$ gives the first equality in +\eqref{field:endpoints}. For $z\geq0$, +$0\leq\mu_\delta(z)\leq z^\sigma$ and +$\mu_\delta(z)\to z^\sigma$. Dominated convergence and the +substitution $s=k_i u$ give its stated limit. +\end{proof} + +The field is defined on all symmetric matrices, but the action +argument compares it with a positive matrix $AA^{\mathsf T}$. +At such values its directional bound forces positivity of the +entire quadratic symbol $Q_s(B)$. This is the condition needed +for the trace deficit formula. + +\begin{proposition}[Positive values of the field]\label{field:positive} +If $M=M_\delta(B)\geq0$, then +\begin{equation}\label{field:positive-bounds} + Q_s(B)\geq0\quad(s\in\R),\qquad + 0\leq M\leq\kappa^{2\sigma}I. +\end{equation} +Set $V=K^2-B^2+\delta I$ and use $B,V$ in +\eqref{trace:resolvents} and \eqref{trace:N}. Then the symbol +$(sI-B)^2+V=Q_s(B)+\delta I$ is bounded below by $\delta I$, and +\begin{equation}\label{field:N} + N=M+\delta^\sigma I. +\end{equation} +In particular, the hypotheses of Theorem~\ref{trace:deficit} hold. +\end{proposition} + +\begin{proof} +By \eqref{field:direction}, positivity of $M$ and strict increase +of $\mu_\delta$ imply +$e^{\mathsf T}K^2e\geq(e^{\mathsf T}Be)^2$ for every real unit +vector $e$. Completing the scalar square gives +\[ + e^{\mathsf T}Q_s(B)e=(s-e^{\mathsf T}Be)^2 + +e^{\mathsf T}K^2e-(e^{\mathsf T}Be)^2\geq0. +\] +Thus $Q_s(B)\geq0$, also as a Hermitian matrix on $\C^n$. +Moreover, \eqref{field:direction} gives +$M\leq\mu_\delta(\kappa^2)I\leq\kappa^{2\sigma}I$, using +$(a+b)^\sigma\leq a^\sigma+b^\sigma$. + +Write $Y_s=(sI-B)^2$. Diagonalization of $B$ and +\eqref{field:shift} allow replacement of the scalar subtraction +$f_\delta(s^2)I$ in \eqref{field:definition} by $f_\delta(Y_s)$. +Since both $Q_s(B)$ and $Y_s$ are positive, their difference under +$f_\delta$ is, by \eqref{field:f}, +\[ + \frac{c_\sigma}{\pi}\int_0^\infty t^\beta + \left((Y_s+(t+\delta)I)^{-1} + -(Q_s(B)+(t+\delta)I)^{-1}\right)\,dt. +\] +For fixed $B,\delta$, both matrices being inverted are bounded +below by $c(1+t+s^2)I$. Their difference is $K^2-B^2$, so the +resolvent identity bounds the norm of the integrand by +$C t^\beta(1+t+s^2)^{-2}$. Integrating first in $s$ yields +$C't^\beta(1+t)^{-3/2}$, which is integrable on $(0,\infty)$. +Fubini's theorem is therefore applicable and gives +\[ + M=c_\sigma\int_0^\infty t^\beta + \bigl((t+\delta)^{-1/2}I-R(t)\bigr)\,dt. +\] +Subtracting this expression from \eqref{trace:N} and scaling +\eqref{trace:normalization} gives \eqref{field:N}. +\end{proof} + +\subsection{Choosing the lower triangular coefficient matrix} + +Let $\mathcal L_n$ be the real lower triangular $n\times n$ +matrices. Their dimension equals that of $\Sym_n$. This dimension +match allows us to prescribe a symmetric terminal value by choosing +$C\in\mathcal L_n$. At a simple initial problem there is one choice +up to signs of the rows. Continuation preserves the parity of this +number, and hence produces the required terminal value. + +\begin{proposition}[Connecting path]\label{path:connecting} +Let $\Omega=(x_-,x_+)$ be bounded and let +$U\in H^1_0(\Omega;\R^{n\times d})$, where $n,d\geq1$, satisfy +\[ + \int_\Omega UU^{\mathsf T}\,dx=I_n. +\] +For every $\delta,\varepsilon>0$ there are +$C_\varepsilon\in\mathcal L_n$ and +$B_\varepsilon\in C^1(\overline\Omega;\Sym_n)$ such that, with +$A_\varepsilon=C_\varepsilon U$, +\begin{equation}\label{path:ode} + \varepsilon B_\varepsilon' + =M_\delta(B_\varepsilon)-A_\varepsilon A_\varepsilon^{\mathsf T}, + \qquad B_\varepsilon(x_-)=K,\quad B_\varepsilon(x_+)=-K. +\end{equation} +They satisfy +\begin{equation}\label{path:bounds} + \sup_{x\in\overline\Omega}\|B_\varepsilon(x)\|_{\op}\leq\kappa, + \qquad + \sup_{0<\varepsilon\leq1}\|C_\varepsilon\|_{\HS}<\infty, +\end{equation} +where the second bound is for fixed $\delta,K,\Omega$. +\end{proposition} + +\begin{proof} +Use the continuous representative of $U$. Choose a smooth, +nonnegative, compactly supported cutoff $\chi$ on $\Sym_n$, equal +to one on $\{B:\|B\|_{\op}\leq\kappa\}$ and invariant under +conjugation by diagonal sign matrices. Averaging a cutoff over +the finite sign group produces such a function. The truncated +field $\widehat M=\chi M_\delta$ is bounded and globally Lipschitz. + +Fix $\varepsilon>0$. For $C\in\mathcal L_n$ and +$\theta\in[0,1]$, the equation +\begin{equation}\label{path:homotopy} + \varepsilon B'=\theta\widehat M(B)-(CU)(CU)^{\mathsf T}, + \qquad B(x_-)=K +\end{equation} +has a unique $C^1$ solution on $\overline\Omega$. Define the +continuous endpoint map +\[ + \Phi(C,\theta)=B(x_+)+K\in\Sym_n. +\] +Let $\mathcal G$ be the group of diagonal sign matrices, acting +by $C\mapsto SC$ on $\mathcal L_n$ and by conjugation on +$\Sym_n$. Equivariance of the field and uniqueness of the ODE give +\begin{equation}\label{path:equivariance} + \Phi(SC,\theta)=S\Phi(C,\theta)S. +\end{equation} + +We record the three facts needed for continuation: compactness +of the zero set, freeness of the sign action there, and one regular +orbit of zeros at $\theta=0$. +At a zero, integration of \eqref{path:homotopy} gives +\begin{equation}\label{path:C-identity} + CC^{\mathsf T}=2\varepsilon K + +\theta\int_\Omega\widehat M(B(x))\,dx. +\end{equation} +Consequently all zeros lie in a bounded set, and their joint set +in $\mathcal L_n\times[0,1]$ is compact. For +$0<\varepsilon\leq1$, this calculation gives the uniform bound +\begin{equation}\label{path:C-bound} + \|C\|_{\HS}^2\leq2\tr K + +n|\Omega|\sup_{B\in\Sym_n}\|\widehat M(B)\|_{\op}. +\end{equation} + +No row of $C$ can vanish at a zero. Otherwise the reflection +changing only row $i$ would fix $C$; ODE uniqueness and +\eqref{path:equivariance} would force $Be_i=y e_i$ throughout the +interval. Equation~\eqref{field:coordinate} would then give +\[ + \varepsilon y'=\theta\chi(B(x))\mu_\delta(k_i^2-y^2), + \qquad y(x_-)=k_i. +\] +The coefficient $\chi(B(x))$ is continuous and the right side +is locally Lipschitz in $y$. Uniqueness forces $y\equiv k_i$, +contradicting $y(x_+)=-k_i$. Thus the action of $\mathcal G$ +on the zero set is free. + +At $\theta=0$ one has +\begin{equation}\label{path:initial-map} + \Phi(C,0)=2K-\varepsilon^{-1}CC^{\mathsf T}. +\end{equation} +Its lower triangular zeros are exactly +$C=\operatorname{diag}(d_1,\ldots,d_n)$ with +$d_i=\pm\sqrt{2\varepsilon k_i}$. Indeed, the first row of +$CC^{\mathsf T}=2\varepsilon K$ fixes $|d_1|$ and forces all +other entries of the first column to vanish, and induction does +the same for subsequent columns. These $2^n$ zeros form one +sign orbit. At each, the derivative in a lower triangular +direction $H$ has diagonal entries +$-2\varepsilon^{-1}d_i H_{ii}$ and lower off-diagonal entries +$-\varepsilon^{-1}d_j H_{ij}$ for $i>j$. It is therefore an +isomorphism $\mathcal L_n\to\Sym_n$. + +We next justify the continuation using only smooth approximation +and the regular-value argument for parity +\cite[Sections~2--4]{Milnor1965}. Suppose there were no zero at +$\theta=1$. Compactness and freeness give a bounded invariant +open set $O\subset\mathcal L_n$ containing the projection of +every zero, whose closure lies where all rows are nonzero. +There is a positive lower bound for $\|\Phi\|_{\HS}$ on +\begin{equation}\label{path:gap-set} + (\partial O\times[0,1])\ \cup\ (\overline O\times\{1\}). +\end{equation} +Reparametrize $\theta$ by a smooth map equal to zero near zero +and equal to one at one. The resulting map $F$ agrees with +\eqref{path:initial-map} on an initial collar and retains the gap +on \eqref{path:gap-set}. + +Extend $F$ constantly beyond the two ends of the parameter +interval, smooth by convolution, and splice to +\eqref{path:initial-map} in a smaller initial collar. +Averaging the approximation $\Psi_0$ in the form +\[ + \Psi(C,\theta)=\frac1{|\mathcal G|} + \sum_{S\in\mathcal G}S\Psi_0(SC,\theta)S +\] +gives a smooth equivariant approximation $\Psi$ that still equals +the initial map on that collar. It can be made uniformly close +enough on $\overline O\times[0,1]$ to preserve the positive gap. + +Freeness permits an equivariant perturbation with arbitrary target +directions. At each $C_0\in\overline O$, choose a ball whose +distinct sign translates are disjoint, and a smooth bump $b$ +supported in this ball with $b(C_0)=1$. For $H\in\Sym_n$ the map +\[ + V_H(C)=\sum_{S\in\mathcal G}b(SC)SHS +\] +is equivariant and satisfies $V_H(C_0)=H$. +Choose a basis of target matrices and then a finite collection of +such neighborhoods covering $\overline O$. This gives finitely +many smooth equivariant maps $V_1,\ldots,V_N$ whose values span +$\Sym_n$ at every point of $\overline O$. + +Let $h\geq0$ be smooth, vanish near zero, and be positive outside +the collar where $\Psi$ is the initial map. Choose it so that +every zero of $h$ lies within that collar, and set +\[ + \Psi_\alpha(C,\theta) + =\Psi(C,\theta)+h(\theta)\sum_{j=1}^{N}\alpha_jV_j(C). +\] +For $\alpha$ in a sufficiently small open ball, the gap on +\eqref{path:gap-set} persists. At a zero of this family with +$0<\theta<1$, its full differential in $(C,\theta,\alpha)$ +is onto: when $h(\theta)>0$ the parameter directions span the +target, and when $h(\theta)=0$ the derivative in $C$ is the +isomorphism computed for \eqref{path:initial-map}. +Its interior zero set is consequently a smooth manifold. +Sard's theorem \cite{Sard1942}, applied to projection of this +manifold onto the parameter ball, gives arbitrarily small +$\alpha$ for which zero is a regular value of +$(C,\theta)\mapsto\Psi_\alpha(C,\theta)$. +For completeness, if the full derivative is $(L_1,L_2)$ in +the $(C,\theta)$ and $\alpha$ directions, regularity of that +projection means that each $w$ admits $v$ with +$L_1v+L_2w=0$. Hence $\operatorname{ran}L_2\subset +\operatorname{ran}L_1$, and surjectivity of $(L_1,L_2)$ implies +surjectivity of $L_1$. + +For such an $\alpha$, the zero set in $O\times[0,1]$ is a +compact smooth one-dimensional manifold with boundary. The gap +excludes boundary zeros except at $\theta=0$, where the initial +collar gives exactly the $2^n$ regular zeros above. The free +sign action has a quotient that is again a compact +one-dimensional manifold with boundary: a neighborhood disjoint +from its nontrivial translates supplies the local interval or +half-interval chart. All $2^n$ boundary points form one orbit, +so the quotient has exactly one boundary point. This is +impossible, since a compact one-dimensional manifold is a finite +union of circles and closed intervals. Therefore the original +map $\Phi(\cdot,1)$ has a zero. + +It remains to remove the cutoff. For the path at this zero and +any fixed real unit vector $e$, put $y=e^{\mathsf T}Be$. +Whenever $|y|>\kappa$, Equation~\eqref{field:direction} yields +\[ + e^{\mathsf T}M_\delta(B)e + \leq\mu_\delta(e^{\mathsf T}K^2e-y^2)<0. +\] +Since $\chi\geq0$ and $(CU)(CU)^{\mathsf T}\geq0$, the ODE +implies $y'\leq0$ there. Both endpoint values lie in +$[-\kappa,\kappa]$. An excursion above $\kappa$ cannot start +from $\kappa$ while $y$ is nonincreasing; an excursion below +$-\kappa$ cannot return to $-\kappa$ while $y$ is nonincreasing. +The two endpoint conditions therefore rule out both excursions. +This holds for every $e$, so $\|B(x)\|_{\op}\leq\kappa$. +Consequently $\chi(B)=1$, giving \eqref{path:ode}, and +\eqref{path:C-bound} supplies the remaining assertion in +\eqref{path:bounds}. +\end{proof} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-trace.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-trace.tex new file mode 100644 index 0000000000000000000000000000000000000000..2984b450c4b407bb72c976396cd309a344b53883 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-trace.tex @@ -0,0 +1,287 @@ +\section{The exact integrated trace deficit}\label{app:trace} + +The matrix comparison needed in the action argument has an exact +remainder. We give the proof from \cite[Section~3]{MLT}, retaining +both nonnegative terms. Together they control the commutator and +the algebraic equality relation used in the action argument. +Throughout this appendix, $0<\sigma<1$, $\beta=\sigma-\tfrac12$, and +$r=1/\sigma$, $p=1+\sigma$, $q=1+r$. +Define $c_\sigma>0$ by +\begin{equation}\label{trace:normalization} + c_\sigma^{-1} + =\int_0^\infty t^\beta + \bigl(t^{-1/2}-(t+1)^{-1/2}\bigr)\,dt. +\end{equation} +The integrand is $O(t^{\sigma-1})$ at zero and +$O(t^{\sigma-2})$ at infinity. Scaling therefore gives +\begin{equation}\label{trace:scaling} + c_\sigma\int_0^\infty t^\beta + \bigl(t^{-1/2}-(t+y)^{-1/2}\bigr)\,dt=y^\sigma, + \qquad y\geq0. +\end{equation} + +\begin{theorem}[Exact trace deficit]\label{trace:deficit} +Let $n\geq1$ and let $B,V\in\C^{n\times n}$ be Hermitian. +Suppose that +\begin{equation}\label{trace:symbol} + Y_s=(sI-B)^2,\qquad X_s=Y_s+V\geq bI + \quad(s\in\R) +\end{equation} +for some $b>0$. Set +\begin{equation}\label{trace:resolvents} + R(t)=\frac1\pi\int_\R(X_s+tI)^{-1}\,ds, + \qquad P(t)=t^{-1/2}I-R(t),\qquad t>0, +\end{equation} +and +\begin{equation}\label{trace:N} + N=c_\sigma\int_0^\infty t^\beta P(t)\,dt. +\end{equation} +These integrals converge absolutely in matrix norm, and $R(t)>0$. +There are continuous Hermitian matrices $L(t)=R(t)^{-1}>0$ and +$Z(t)$ satisfying +\begin{equation}\label{trace:LZ} + V+tI=L^2+Z^2+i[B,Z],\qquad + LZ+ZL+i[B,L]=0. +\end{equation} +If $N\geq0$, then +\begin{equation}\label{trace:deficit-identity} + \begin{split} + \tr(NV)-\tr(N^q) + =c_\sigma\int_0^\infty t^\beta\Bigl( + \tr(RZ^2) + +q\bigl\|R^{-1/2}-R^{1/2}(tI+N^r)^{1/2}\bigr\|_{\HS}^2 + \Bigr)\,dt. + \end{split} +\end{equation} +In particular, $\tr(NV)\geq\tr(N^q)$. +\end{theorem} + +The hypothesis concerns the full symbol $X_s$; the perturbation +$V$ itself need not be positive. The proof first recovers $L$ and +$Z$ from solutions on the two half-lines, then evaluates a logarithmic +integral, and finally completes a matrix square. + +\subsection{Half-line boundary maps} + +Quadratic growth in $s$, together with \eqref{trace:symbol}, gives +an integrable bound for $X_s^{-1}$. Hence $R(t)>0$ and +$R(t)=O(1)$ as $t\downarrow0$. Diagonalization of $B$ gives +\[ + \frac1\pi\int_\R(Y_s+tI)^{-1}\,ds=t^{-1/2}I. +\] +For $t\geq1$, both $X_s+tI$ and $Y_s+tI$ dominate +$c(s^2+t)I$, with $c>0$ independent of $s,t$. +The resolvent identity bounds their inverse difference by +$C(s^2+t)^{-2}$, and therefore +\begin{equation}\label{trace:P-bounds} + \|P(t)\|_{\op}= + \begin{cases} + O(t^{-1/2}),&t\downarrow0,\\ + O(t^{-3/2}),&t\to\infty. + \end{cases} +\end{equation} +These bounds prove the asserted convergence of $N$. + +Fix $t>0$ and put $\nabla=\partial_x-iB$. We shall use the +equation +\begin{equation}\label{trace:ode} + (-\nabla^2+V+tI)\psi=0. +\end{equation} +Matrix Riccati equations also appear in the commutation method for +Schr\"odinger inequalities \cite{BenguriaLoss2000}. Here their role +is to compute the integrated resolvent through boundary data. + +For each $v\in\C^n$ there is a unique $H^1$ solution of +\eqref{trace:ode} on either half-line with $\psi(0)=v$. +Indeed, the sesquilinear form +\[ + a_t(\phi,\psi)=\int + \bigl(\langle\nabla\phi,\nabla\psi\rangle + +\langle\phi,(V+tI)\psi\rangle\bigr)\,dx, +\] +with inner products linear in the second variable, is coercive on +$H^1(\R;\C^n)$: its Fourier symbol $X_s+tI$ dominates +$c_t(1+s^2)I$. Zero extension gives coercivity on the trace-zero +subspace of either half-line. Correcting any $H^1$ extension of +$v$ by the Riesz representation theorem gives the unique weak +solution. The equation implies $\psi''\in L^2$, so this solution +lies in $H^2$; both $\psi$ and $\psi'$ vanish at the infinite end. + +Define the right and left boundary matrices $S,T$ by +\begin{equation}\label{trace:boundary} + \nabla\psi(0+)=-Sv,\qquad \nabla\psi(0-)=Tv. +\end{equation} +Integration by parts gives +$a_t(\psi_v,\psi_w)=\langle v,Sw\rangle$ on the right and +$a_t(\psi_v,\psi_w)=\langle v,Tw\rangle$ on the left. Thus +$S,T$ are Hermitian. Translation and uniqueness show that +$\nabla\psi=-S\psi$ throughout the right half-line and +$\nabla\psi=T\psi$ throughout the left half-line. +Substituting these relations into \eqref{trace:ode}, and allowing +arbitrary boundary data, gives +\begin{equation}\label{trace:riccati} + V+tI=S^2+i[B,S]=T^2-i[B,T]. +\end{equation} + +Glue the two solutions with boundary value $v$ at zero. +The resulting continuous function has derivative jump +$-(S+T)v$, and hence satisfies +\[ + (-\nabla^2+V+tI)\psi=\delta_0(S+T)v. +\] +Fourier inversion, with forward kernel $e^{-isx}$, gives +\[ + v=\frac1{2\pi}\int_\R(X_s+tI)^{-1}\,ds\,(S+T)v + =\tfrac12 R(t)(S+T)v. +\] +The inverse symbol is $O(s^{-2})$, so its Fourier inverse is +continuous and this evaluation at zero is justified. Therefore +\[ + L:=\tfrac12(S+T)=R^{-1}>0,\qquad Z:=\tfrac12(S-T) +\] +satisfy \eqref{trace:LZ}, by adding and subtracting +\eqref{trace:riccati}. The second equation in \eqref{trace:LZ} +determines $Z$ uniquely from $L$, since $L>0$; in an eigenbasis +of $L$, the map $Z\mapsto LZ+ZL$ multiplies its $(j,k)$ entry +by the positive number $L_{jj}+L_{kk}$. This also proves the +continuity of $Z(t)$, because $R(t)$, and hence $L(t)$, is continuous. + +The diagonal entries of that same equation give $Z_{jj}=0$ +in an eigenbasis of $L$, so $\tr Z=0$. Multiplication on both +sides by $R=L^{-1}$ gives +$i[R,B]=-(ZR+RZ)$. Cyclicity of the trace now yields +\begin{equation}\label{trace:RV} + \tr\bigl(R(V+tI)\bigr)=\tr L-\tr(RZ^2). +\end{equation} +In particular, the correction $\tr(RZ^2)=\tr(ZRZ)$ is nonnegative. +We will also need the location of the spectrum of $B+iS$. +The right solutions obey $\psi'=i(B+iS)\psi$. +For an eigenvector with eigenvalue $z$, the solution is $e^{izx}v$; +its square integrability forces $\operatorname{Im}z>0$. + +\subsection{A logarithmic integral} + +The next step converts the boundary matrices into a scalar identity +that can be integrated in $t$. Define +\begin{equation}\label{trace:F} + F(t)=\frac{\tr V}{\sqrt t} + -\frac1\pi\int_\R + \log\frac{\det(X_s+tI)}{\det(Y_s+tI)}\,ds. +\end{equation} +The determinants are positive, and the logarithmic ratio is +$O(|s|^{-2})$ at infinity. Its derivative in $t$ is the trace +of the resolvent difference, with an integrable bound locally +uniform for $t>0$. Thus the integral converges absolutely and +$F$ is continuously differentiable. + +The first identity in \eqref{trace:riccati} factors the symbol as +\[ + X_s+tI=(sI-B+iS)(sI-B-iS). +\] +If $u_j+iv_j$ are the eigenvalues of $B+iS$, counted with +algebraic multiplicity, then $v_j>0$ and +\[ + \det(X_s+tI)=\prod_{j=1}^n\bigl((s-u_j)^2+v_j^2\bigr). +\] +The denominator in \eqref{trace:F} has the same form, with centers +the eigenvalues of $B$ and widths $\sqrt t$. +Translations of the individual logarithms change their +symmetric-cutoff integrals by $o(1)$: for +$h(s)=\log(s^2+v^2)$, +\[ + \frac{d}{da}\int_{-M}^M h(s-a)\,ds + =h(M+a)-h(M-a)\longrightarrow0 +\] +uniformly for bounded $a$. After centering the factors, the identity +\[ + \int_\R\log\frac{s^2+v^2}{s^2+w^2}\,ds=2\pi(v-w), + \qquad v,w>0, +\] +follows by differentiation in $v$ and integration from $w$ to $v$. +Since $\sum_jv_j=\tr S=\tr L$, we obtain +\begin{equation}\label{trace:F-evaluation} + F(t)=\frac{\tr V}{\sqrt t}-2\tr(L-\sqrt t I). +\end{equation} + +Set +\begin{equation}\label{trace:G} + G(t)=\tr(L-2\sqrt t I+tR). +\end{equation} +Both functions are nonnegative. For $G$, apply +$\lambda-2\sqrt t+t/\lambda\geq0$ to the positive eigenvalues +$\lambda$ of $L$. For $F$, take the trace of \eqref{trace:LZ} +and use \eqref{trace:F-evaluation} to obtain +\[ + F(t)=t^{-1/2}\tr\bigl((L-\sqrt t I)^2+Z^2\bigr)\geq0. +\] +Equations \eqref{trace:RV} and \eqref{trace:F-evaluation} give +the identity +\begin{equation}\label{trace:FG} + \tr(PV)=F+G+\tr(RZ^2). +\end{equation} +Every term on the right is nonnegative. The bounds +\eqref{trace:P-bounds} therefore show that $F$, $G$, and +$\tr(RZ^2)$ are integrable against $t^\beta\,dt$, and that +$t^{\beta+1}F(t)$ vanishes at both endpoints. + +Differentiating \eqref{trace:F} and using +\eqref{trace:F-evaluation}--\eqref{trace:G} gives +\[ + F'(t)=-\frac{\tr V}{2t^{3/2}}+\tr P(t), + \qquad tF'(t)=-\tfrac12F(t)-G(t). +\] +Integration by parts, with the endpoint limits just proved, yields +$\int_0^\infty t^\beta G\,dt + =\sigma\int_0^\infty t^\beta F\,dt$. +Consequently, integration of \eqref{trace:FG} gives +\begin{equation}\label{trace:weighted} + \tr(NV)=q c_\sigma\int_0^\infty t^\beta G(t)\,dt + +c_\sigma\int_0^\infty t^\beta\tr(RZ^2)\,dt. +\end{equation} +It remains to express the first integral as $\tr(N^q)/q$ plus +a nonnegative remainder. + +\subsection{The noncommutative square} + +For a positive semidefinite matrix $D$, let +\[ + F_0(t,D)=\frac{\tr D}{\sqrt t} + -2\tr\bigl((tI+D)^{1/2}-\sqrt t I\bigr). +\] +For a scalar eigenvalue $d\geq0$, the corresponding summand is +$\int_0^d(t^{-1/2}-(t+y)^{-1/2})\,dy$. +Tonelli's theorem and \eqref{trace:scaling} therefore imply +\begin{equation}\label{trace:F0} + c_\sigma\int_0^\infty t^\beta F_0(t,D)\,dt + =\frac1p\tr(D^p). +\end{equation} +Now expand the Hilbert--Schmidt square +\begin{align} + \mathcal D_D(t) + &:=\bigl\|R^{-1/2}-R^{1/2}(tI+D)^{1/2}\bigr\|_{\HS}^2 + \notag\\ + &=\tr\bigl(R^{-1}+R(tI+D)-2(tI+D)^{1/2}\bigr) + \label{trace:square}\\ + &=G(t)-\tr(P(t)D)+F_0(t,D).\notag +\end{align} +No commutation is used: the cross terms cancel adjacent factors +$R^{-1/2}R^{1/2}$, and cyclicity changes the remaining quadratic +term into $\tr(R(tI+D))$. +All three terms in the last line are integrable by +\eqref{trace:P-bounds}, \eqref{trace:FG}, and \eqref{trace:F0}. +Thus +\begin{equation}\label{trace:duality} + c_\sigma\int_0^\infty t^\beta G(t)\,dt + =\tr(ND)-\frac1p\tr(D^p) + +c_\sigma\int_0^\infty t^\beta\mathcal D_D(t)\,dt. +\end{equation} +When $N\geq0$, choose $D=N^r$. Since $rp=q=1+r$, +the first two terms on the right equal $\tr(N^q)/q$. +Substitution into \eqref{trace:weighted} proves +\eqref{trace:deficit-identity} and completes the proof of +Theorem~\ref{trace:deficit}. + +If $B$ and $V$ commute, the symbol hypothesis implies $V\geq bI$. +Simultaneous diagonalization then gives +$R(t)=(tI+V)^{-1/2}$, $N=V^\sigma$, and $Z=0$. +Both remainders vanish, as required by scalar equality. diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/compactness.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/compactness.tex new file mode 100644 index 0000000000000000000000000000000000000000..ad2398e350ccf148bbe16dfbe14afae3c50140e4 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/compactness.tex @@ -0,0 +1,379 @@ +\section{From equality to pointwise matrix relations} +\label{comp:section} + +We now apply Theorem~\ref{act:quantitative-action} to finite collections of +negative eigenfunctions. Equality forces each of its nonnegative errors to +vanish. The main issue is that the number of eigenfunctions may grow without +bound, whereas the potential has only $m$ channels. We will use this fixed +number of columns to control the tails of the resulting matrices. + +Throughout this section, $W$ is real symmetric, $W\geq0$ almost everywhere, +and $0<\int_{\R}\tr(W^p)<\infty$. Suppose that equality holds, in the form +\begin{equation} + b_\sigma\sum_{i\in\mathcal I}k_i^{2\gamma} + =D_\sigma\int_{\R}\tr(W^p)\,dx, + \label{comp:equality} +\end{equation} +where $-k_i^2$ are the negative eigenvalues of $H_W$, repeated according to +multiplicity, and +\[ + k_1\geq k_2\geq\cdots>0. +\] +By Lemma~\ref{pre:spectral}, the index set $\mathcal I$ is a nonempty finite +initial segment of $\mathbb N$ or all of $\mathbb N$, each eigenvalue has +finite multiplicity, and $k_i\to0$ in the infinite case. Fix real +orthonormal eigenfunctions $\varphi_i\in H^1(\R;\R^m)$ and regard them as +row vectors. Write +\[ + \mathscr H=\ell^2(\mathcal I;\R),\qquad + \Lambda=\operatorname{diag}(k_i:i\in\mathcal I),\qquad \kappa=k_1. +\] +For operators between real Hilbert spaces, the superscript $T$ will also +denote the adjoint. + +\begin{proposition}[Pointwise relations at equality] +\label{comp:relations} +Under these assumptions, there are rows $a_i\in H^1(\R;\R^m)$, +$i\in\mathcal I$, each a linear combination of eigenfunctions with +eigenvalue $-k_i^2$, such that +\begin{equation} + -a_i''-a_iW=-k_i^2a_i + \quad\text{in distributions.} + \label{comp:row-eigenfunction} +\end{equation} +The rows $a_i$ are allowed to vanish. On a common set of full measure, they +define a Hilbert--Schmidt operator $A(x):\R^m\to\mathscr H$, and there exists +a bounded self-adjoint operator $B$ on $\mathscr H$ with +$\|B\|_{\op}\leq\kappa$ such that +\begin{equation} + [B,\Lambda^2]=0,\qquad + \Lambda^2-B^2=(AA^T)^r,\qquad + a_i'(x)=(BA)_{i,\cdot}\quad(i\in\mathcal I). + \label{comp:limit-relations} +\end{equation} +Moreover, +\begin{equation} + W(x)=q\bigl(A(x)^TA(x)\bigr)^r + \quad\text{almost everywhere.} + \label{comp:potential-limit} +\end{equation} +The operator $B$ is asserted to exist separately at each such point $x$; +no measurable choice is required. +\end{proposition} + +We divide the proof into the construction of a sequence whose errors +vanish, convergence of every fixed row, and control of the remaining rows. + +\subsection{Finite spectral approximations and vanishing errors} +\label{comp:approximation} + +Let $n_j=j$ when $\mathcal I=\mathbb N$, and let $n_j=|\mathcal I|$ when +$\mathcal I$ is finite. Choose $0<\xi_j<1/j$ so small that +\begin{equation} + \ell_{j,i}:=(k_i^2-\xi_j)^{1/2}>0\quad(1\leq i\leq n_j), + \qquad + \sum_{i\leq n_j}\bigl(k_i^{2\gamma}-\ell_{j,i}^{2\gamma}\bigr)<1/j. + \label{comp:common-shift} +\end{equation} +Set $K_j=\operatorname{diag}(\ell_{j,1},\ldots,\ell_{j,n_j})$. +The same shift $\xi_j$ is used for every eigenvalue: this leaves the +differences of their squares unchanged. + +The approximation leading to \eqref{bound:trial} can be made arbitrarily +accurate in $H^1$. Apply it to the first $n_j$ eigenfunctions, choosing +the $H^1$ errors below $1/j$ and the form-matrix error below $\xi_j$. +This gives real, smooth, compactly supported functions +$u_{j,1},\ldots,u_{j,n_j}$, orthonormal in $L^2(\R;\R^m)$, such that +\begin{equation} + \|u_{j,i}-\varphi_i\|_{H^1}<1/j, + \qquad + \bigl(h_W[u_{j,i},u_{j,l}]\bigr)_{i,l=1}^{n_j}\leq-K_j^2. + \label{comp:form-approximation} +\end{equation} +Here $h_W[\cdot,\cdot]$ is the polarized quadratic form, and the second +inequality is the ordering of real symmetric matrices. + +Let $U_j$ be the matrix with rows $u_{j,i}$, and choose a bounded interval +$\Omega_j$ containing their supports in its interior. Apply +Theorem~\ref{act:quantitative-action} with $K=K_j$, $d=m$, $\kappa=k_1$, +and $\eta=1/j$. It gives a lower triangular matrix $C_j$, a symmetric +path $B_j$ with $\|B_j\|_{\op}\leq\kappa$, and $A_j=C_jU_j$. +Extend $A_j$ by zero outside $\Omega_j$ and extend $B_j$ by $K_j$ to the +left and by $-K_j$ to the right. All the errors in that theorem vanish +outside $\Omega_j$, so their integrals may be taken over $\R$. + +The lower triangular form of $C_j$ gives a useful additional error. Put +$Z_j=A_j^TA_j$, an $m\times m$ positive semidefinite matrix. By +\eqref{comp:form-approximation}, +\begin{equation} + \mathcal E_{K_j}(A_j) + \leq\int_{\R}\bigl(\tr(WZ_j)-\tr(Z_j^q)\bigr)\,dx-T_j, + \qquad + T_j=\sum_{l\leq i\leq n_j}(C_j)_{il}^{\,2}(k_l^2-k_i^2)\geq0. + \label{comp:triangular-gap} +\end{equation} +Indeed, the $i$th row of $A_j$ is $\sum_{l\leq i}(C_j)_{il}u_{j,l}$. +Its quadratic form, plus $\ell_{j,i}^2$ times its squared $L^2$ norm, is +at most +\[ + \sum_{l\leq i}(C_j)_{il}^{\,2} + (\ell_{j,i}^2-\ell_{j,l}^2) + =-\sum_{l\leq i}(C_j)_{il}^{\,2}(k_l^2-k_i^2). +\] +Summing over $i$ proves \eqref{comp:triangular-gap}, since $A_jA_j^T$ and +$Z_j$ have the same nonzero eigenvalues, with multiplicity. The ordering +of the $k_i$ makes every summand of $T_j$ nonnegative. + +Recall the nonnegative Young deficit from Lemma~\ref{pre:young}: +\begin{equation} + Y(W,Z)=D_\sigma\tr(W^p)-\tr(WZ)+\tr(Z^q)\geq0, + \qquad + Y(W,Z)=0\ \Longleftrightarrow\ W=qZ^r. + \label{comp:young-deficit} +\end{equation} +Combining Theorem~\ref{act:quantitative-action}, +\eqref{comp:equality}, and \eqref{comp:triangular-gap} yields +\begin{align} + 0\leq T_j+\int_{\R}\Bigl( + Y(W,Z_j)+\|A_j'-B_jA_j\|_{\HS}^2 + +c_*\Phi_{K_j}(A_j,B_j)\Bigr)\,dx + &\leq d_j,\label{comp:vanishing-errors}\\ + d_j:=b_\sigma\left( + \sum_{i\in\mathcal I}k_i^{2\gamma} + -\sum_{i\leq n_j}\ell_{j,i}^{2\gamma}\right)+1/j + &\longrightarrow0.\notag +\end{align} +Here $c_*>0$ is fixed: its independence of $n_j$ is essential. Thus all +three integral errors tend to zero, as does $T_j$. + +We will also need a uniform bound on $A_j$. The coercive estimate +\eqref{pre:young-coercive} gives +\begin{equation} + Y(W,Z_j)\geq\frac12\tr(Z_j^q) + -(2^{p-1}-1)D_\sigma\tr(W^p). + \label{comp:coercivity} +\end{equation} +Since $Z_j$ has size $m$, scalar H\"older's inequality gives +\[ + \|A_j\|_{\HS}^{2q}=(\tr Z_j)^q + \leq m^{q-1}\tr(Z_j^q). +\] +Equations~\eqref{comp:vanishing-errors} and \eqref{comp:coercivity} +therefore bound $A_j$ uniformly in $L^{2q}(\R)$ with the +Hilbert--Schmidt norm, independently of its number of rows. + +\subsection{Convergence of each fixed row} +\label{comp:rows} + +For fixed $i\in\mathcal I$, the coefficients $(C_j)_{il}$, $l\leq i$, +are bounded as $j\to\infty$. To see this, choose a bounded interval $J_i$ +so large that the restricted Gram matrix +\[ + \left(\int_{J_i}\varphi_l(x)\varphi_s(x)^T\,dx + \right)_{l,s=1}^{i}\geq\frac34 I_i, +\] +where $I_i$ denotes the $i\times i$ identity matrix. +Such an interval exists because the full Gram matrix is the identity. +By \eqref{comp:form-approximation}, the corresponding restricted Gram +matrix of the $u_{j,l}$ is at least one half of the identity for all +sufficiently large $j$. Hence +\[ + \frac12\sum_{l\leq i}(C_j)_{il}^{\,2} + \leq\int_{J_i}|(A_j)_{i,\cdot}|^2\,dx + \leq |J_i|^{1-1/q} + \left(\int_{J_i}\|A_j\|_{\HS}^{2q}\,dx\right)^{1/q}. +\] +The last expression is uniformly bounded. Lower triangularity is used +here to keep the number of coefficients in a fixed row finite. + +A diagonal subsequence now makes every coefficient $(C_j)_{il}$ converge +to a limit $c_{il}$, for $l\leq i$. From +$T_j\to0$ and \eqref{comp:triangular-gap}, +\begin{equation} + c_{il}=0\quad\text{whenever }k_l>k_i. + \label{comp:gap-suppression} +\end{equation} +Define +\[ + a_i=\sum_{l\leq i}c_{il}\varphi_l. +\] +The sum is finite, and \eqref{comp:form-approximation} implies +\begin{equation} + (A_j)_{i,\cdot}\longrightarrow a_i + \quad\text{in }H^1(\R;\R^m) + \quad\text{for every fixed }i. + \label{comp:row-convergence} +\end{equation} +The ordering of the eigenvalues and \eqref{comp:gap-suppression} show +that only eigenfunctions with eigenvalue $-k_i^2$ occur in $a_i$. +This proves \eqref{comp:row-eigenfunction}, including the possibility +$a_i=0$. The eigenfunction equations hold distributionally by testing +the form equations against smooth compactly supported functions; the +potential is locally integrable and the rows are locally bounded. + +Choose the continuous, locally absolutely continuous representatives +of these rows. The Sobolev embedding $H^1(\R)\subset C_b(\R)$ shows +that \eqref{comp:row-convergence} also gives pointwise convergence at +every $x$. Passing to a further subsequence, we may arrange that the +integrals of the nonnegative integrand in +\eqref{comp:vanishing-errors} are summable, and that, for every fixed +$i$, the squared $L^2$ errors of the derivatives in +\eqref{comp:row-convergence} are summable. A diagonal choice achieves +both conditions, by making the first $j$ available derivative errors and the +integral error less than $2^{-j}$ along the chosen subsequence. +Consequently, on a common set of full measure, +\begin{equation} + \begin{gathered} + Y(W,Z_j)\to0,\qquad + \|A_j'-B_jA_j\|_{\HS}\to0,\qquad + \Phi_{K_j}(A_j,B_j)\to0,\\ + (A_j')_{i,\cdot}(x)\to a_i'(x) + \quad\text{for every }i\in\mathcal I. + \end{gathered} + \label{comp:pointwise-errors} +\end{equation} +We also exclude the null set where $W$ is not a finite positive +semidefinite matrix. It remains to show that convergence of the +individual rows is strong enough to pass to the products in these +relations. + +\subsection{Tail control in the row-index space} +\label{comp:tails} + +Pad $A_j$, $B_j$, and $K_j$ with zeros in the unused coordinates of +$\mathscr H$. Thus $A_j:\R^m\to\mathscr H$, and $B_j,K_j$ are +self-adjoint finite-rank operators on $\mathscr H$. Since the omitted +$k_i$ tend to zero, +\begin{equation} + \|K_j^2-\Lambda^2\|_{\op} + \leq\max\{\xi_j,\sup_{i>n_j}k_i^2\}\longrightarrow0, + \label{comp:diagonal-limit} +\end{equation} +where a supremum over no indices is zero. + +Fix a point $x$ in the full-measure set from +\eqref{comp:pointwise-errors} and suppress $x$ for the moment. Write +$\rho_j=A_jA_j^T$. The definition of $\Phi$ and its convergence to zero +give +\begin{equation} + \|[B_j,K_j^2]\|_{\op}\to0, + \qquad e_j:=\|K_j^2-B_j^2-\rho_j^r\|_{\op}\to0. + \label{comp:pointwise-algebra-error} +\end{equation} +For $h\geq1$, let $P_h$ be the orthogonal projection onto the first $h$ +coordinates of $\mathscr H$, with all coordinates included if $h$ exceeds +$|\mathcal I|$. Set $Q_h=I-P_h$ and +$s_h=\sup_{i>h}k_i^2$, again with the empty supremum equal to zero. +Then $s_h\to0$. For every unit vector $v$ in the range of $Q_h$, +\begin{equation} + \|B_jv\|^2+\langle v,\rho_j^rv\rangle\leq s_h+e_j. + \label{comp:tail-quadratic} +\end{equation} +Both terms on the left are nonnegative. Jensen's inequality for the +spectral measure of $\rho_j$, using $r>1$, implies +\[ + \langle v,\rho_jv\rangle^r\leq\langle v,\rho_j^rv\rangle. +\] +It follows that +$\|Q_h\rho_jQ_h\|_{\op}\leq(s_h+e_j)^{1/r}$. +Although $\mathscr H$ may be infinite dimensional, $Q_h\rho_jQ_h$ +has rank at most $m$, because $A_j$ has exactly $m$ columns. Its trace +is therefore at most $m$ times its operator norm. We obtain the two +tail bounds +\begin{equation} + \boxed{\quad + \|Q_hA_j\|_{\HS}^2\leq m(s_h+e_j)^{1/r}, + \qquad + \|B_jQ_h\|_{\op}\leq(s_h+e_j)^{1/2}. + \quad} + \label{comp:tail-bounds} +\end{equation} +The first estimate converts decay of the eigenvalue parameters into +decay of the total mass of all omitted rows. Its factor is the fixed +physical dimension $m$, rather than the number of retained +eigenfunctions; Figure~\ref{comp:tail-figure} depicts this distinction. + +\begin{figure}[tb] + \centering + \begin{tikzpicture}[font=\small,>=Latex] + \fill[black!5] (0,2.05) rectangle (2.7,3.65); + \fill[blue!7] (0,0) rectangle (2.7,2.05); + \draw[thick] (0,0) rectangle (2.7,3.65); + \draw[thick] (0,2.05)--(2.7,2.05); + \draw[<->] (0,4.05)--node[above]{$m$ columns, fixed}(2.7,4.05); + \node at (1.35,2.83) {$P_hA_j$}; + \node at (1.35,1.28) {$Q_hA_j$}; + \node at (1.35,0.55) {$\vdots$}; + \node[anchor=east,align=right] at (-0.2,2.83) + {first $h$\\rows}; + \node[anchor=east,align=right] at (-0.2,1.08) + {all remaining\\rows}; + \draw[->] (2.9,2.83)--(3.6,2.83); + \node[anchor=west,align=left] at (3.75,2.83) + {Finite-dimensional convergence\\for each fixed $h$}; + \draw[->] (2.9,1.08)--(3.6,1.08); + \node[anchor=west,align=left] at (3.75,1.08) + {$\operatorname{rank}(Q_hA_j)\leq m$\\[4pt] + $\|Q_hA_j\|_{\HS}^2\leq m(s_h+e_j)^{1/r}$}; + \end{tikzpicture} + \caption{The number of rows can grow with the number of negative + eigenfunctions, but the number of physical channels remains $m$. + After fixing the first $h$ rows, the identity + $K_j^2\approx B_j^2+(A_jA_j^T)^r$ bounds the operator norm in the + remaining rows. The rank bound converts it into a Hilbert--Schmidt + bound, which tends to zero as first $j\to\infty$ and then $h\to\infty$. + The drawing is schematic; all matrices are padded by zero rows in + $\mathscr H$.} + \label{comp:tail-figure} +\end{figure} + +For each fixed $h$, the first $h$ rows of $A_j$ converge by +\eqref{comp:row-convergence}. The first bound in +\eqref{comp:tail-bounds} makes the remaining rows uniformly small by +choosing $h$ large and then $j$ sufficiently large. Thus $A_j$ is Cauchy in +Hilbert--Schmidt norm and converges to an operator $A:\R^m\to\mathscr H$ +whose rows are precisely $a_i(x)$: +\begin{equation} + A_j\longrightarrow A\quad\text{in Hilbert--Schmidt norm.} + \label{comp:HS-limit} +\end{equation} +This argument applies to the whole subsequence already selected and +does not require a further choice depending on $x$. + +For $B_j$ we need only pointwise compactness. Its finite compressions +$P_hB_jP_h$ are bounded in finite-dimensional spaces, so a diagonal +subsequence makes them converge for every $h$. Self-adjointness and +the second estimate in \eqref{comp:tail-bounds} give +\[ + \|B_j-P_hB_jP_h\|_{\op} + \leq2\|B_jQ_h\|_{\op} + \leq2(s_h+e_j)^{1/2}. +\] +The chosen subsequence consequently converges in operator norm to a +self-adjoint $B$ with $\|B\|_{\op}\leq\kappa$. This subsequence, and +hence $B$, may depend on the point $x$. + +We can now pass to all the required products. Hilbert--Schmidt +convergence of $A_j$ implies operator norm convergence of +$A_jA_j^T$ and of $A_j^TA_j$. Continuous functional calculus for +nonnegative operators then gives convergence of their $r$th powers. +Together with \eqref{comp:diagonal-limit} and +\eqref{comp:pointwise-algebra-error}, this proves +\[ + [B,\Lambda^2]=0,\qquad + \Lambda^2-B^2=(AA^T)^r. +\] +Also $B_jA_j\to BA$ in Hilbert--Schmidt norm. Taking its $i$th row +and using both the derivative convergence and the first-order error +in \eqref{comp:pointwise-errors} gives +$a_i'(x)=(BA)_{i,\cdot}$ for every $i$. Finally, continuity of the +finite-dimensional Young deficit and +$Y(W,A_j^TA_j)\to0$ yield +$Y(W,A^TA)=0$. Lemma~\ref{pre:young} gives +$W=q(A^TA)^r$. + +We have proved these statements at every point of one common set of +full measure. The construction selects no function $x\mapsto B(x)$: +only the existence of a suitable operator at each such point enters +the next section. This completes the proof of +Proposition~\ref{comp:relations}. diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/introduction.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..5676bd290b60fb89bb8bebe974b1743fc0c89962 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/introduction.tex @@ -0,0 +1,161 @@ +\section{Introduction}\label{intro:section} + +Lieb--Thirring inequalities compare the binding energy of a Schr\"odinger +operator with an integral of its potential. Equality asks how the potential +must arrange that binding. In one dimension, a scalar one-bound-state +optimizer has a hyperbolic-secant profile. For a matrix potential, several +such profiles can occupy orthogonal internal channels. The issue is whether +equality also permits channels that rotate with position, or several bound +states that interact within the same channel. We prove that throughout +$1/2<\gamma<3/2$ every equality case consists of independent scalar profiles +in one fixed basis. + +Let $m\ge1$ be finite, let $1/2<\gamma<3/2$, and set +$p=\gamma+1/2$. For a measurable Hermitian positive semidefinite function +$W:\R\to\C^{m\times m}$ satisfying +\begin{equation}\label{intro:class} + \int_\R\tr(W(x)^p)\,dx<\infty, +\end{equation} +let $H_W=-d^2/dx^2-W$ be the operator associated with +\begin{equation}\label{intro:form} + h_W[\psi]=\int_\R\|\psi'\|^2 + -\int_\R\langle\psi,W\psi\rangle, + \qquad \psi\in H^1(\R;\C^m). +\end{equation} +Lemma~\ref{pre:spectral} proves that the form is closed and bounded below +and that the negative spectrum is discrete, with possible accumulation +only at zero. We write +\[ + \Tr(H_W)_-^\gamma=\sum_{\lambda_j(H_W)<0}|\lambda_j(H_W)|^\gamma, +\] +counting multiplicities. The finite-dimensional trace $\tr$ is unnormalized; +matrix powers are defined by spectral functional calculus. + +\begin{theorem}[Equality classification]\label{thm:main} +For every $W$ satisfying~\eqref{intro:class}, +\begin{equation}\label{intro:sharp} + \Tr(H_W)_-^\gamma\le C_\gamma\int_\R\tr(W^p), + \qquad + C_\gamma=\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2} + \frac{\Gamma(\gamma+1)}{\sqrt\pi\,\Gamma(\gamma+3/2)}. +\end{equation} +Put $r=(\gamma-1/2)^{-1}$. Equality holds if and only if there are +$k\in\{0,\ldots,m\}$, a constant unitary matrix $U$, positive numbers +$a_1,\ldots,a_k$, and real numbers $x_1,\ldots,x_k$ such that almost +everywhere +\begin{equation}\label{intro:classification} + W(x)=U\diag\bigl(w_1(x),\ldots,w_k(x),0,\ldots,0\bigr)U^*, + \quad + w_j(x)=(r+1)a_j^2\sech^2\!\bigl(ra_j(x-x_j)\bigr). +\end{equation} +The case $k=0$ means $W=0$. Each nonzero channel has exactly one negative +eigenvalue, $-a_j^2$, and therefore every extremal potential has at most +$m$ negative eigenvalues, counted with multiplicity. +\end{theorem} + +The inequality in~\eqref{intro:sharp} is the sharp matrix inequality +of~\cite[Theorem~1.1]{MLT}. Our contribution is its equality classification. +We retain the analytic constructions in the proof, so that the inequality +and its equality conditions are established here under the same measurable +potential hypothesis. The scales and centers in~\eqref{intro:classification} +are independent; repeated scales and identical profiles are allowed. + +\subsection{History and related results} +Lieb and Thirring introduced spectral moment inequalities in their work +on the kinetic energy of fermions and stability of matter +\cite{LiebThirring1975,LiebThirring1976}. Their sharp constants distinguish +the regimes in which binding is best concentrated in a single state or +distributed among many states. Keller's optimization of the lowest +Schr\"odinger eigenvalue~\cite{Keller1961} supplies the one-state problem +underlying the scalar profiles in~\eqref{intro:classification}; their +explicit form appears in~\cite[Section~4(B), Equations~(4.19)--(4.20)]{LiebThirring1976}. +The present question concerns equality in the sum of all negative +eigenvalue moments, with a matrix potential that need not commute with +itself at different points. + +The sharp scalar endpoint $\gamma=1/2$ was proved by +Hundertmark, Lieb and Thomas~\cite{HLT1998}; Hundertmark, Laptev and +Weidl~\cite[Theorem~3.1]{HLW2000} extended it to operator-valued +potentials. The semiclassical constant +is sharp for $\gamma\ge3/2$; the lifting of moments of Aizenman and +Lieb~\cite{AizenmanLieb1978} and the operator-valued inequalities of +Laptev and Weidl~\cite{LaptevWeidl2000} are central to this development. +Benguria and Loss~\cite{BenguriaLoss2000} gave a commutation proof of the +matrix result at $\gamma=3/2$. More recently, Read and +Schulz~\cite{ReadSchulz2026} proved the sharp scalar inequality at +$\gamma=1$, with constant $4/(3\sqrt3\pi)$, and its operator-valued +extension. The matrix companion~\cite{MLT} proves the sharp one-state +constant for the whole open interval considered here. + +The endpoint $\gamma=3/2$ has a different equality structure. In the +scalar finite-rank problem, Frank, Gontier and Lewin +\cite[Theorem~5]{FrankGontierLewin2025} identify the optimizers as KdV +multisolitons. In particular, several bound states can occur in a single +scalar channel at that endpoint. The conclusion of +Theorem~\ref{thm:main}, one bound state per nonzero channel, uses the +strict inequality $r>1$ and is asserted only for the open interval. + +\subsection{Quantitative defects and fixed channels} +The proof begins with the finite-interval matrix action method +of~\cite{MLT}. For orthonormal real trial functions approximating finitely +many negative eigenfunctions, place the functions in the rows of $U(x)$ +and choose a lower triangular matrix $C$. The rows of $A=CU$ remain in the +successive trial spans. +If $K$ is the diagonal matrix of positive square roots of the trial +binding energies, the action involves $A'$, $K^2AA^T$, and a power of +the density $AA^T$. A symmetric matrix path $B(x)$ connects $K$ to +$-K$ and gives a lower bound for that action. + +For equality, the lower bound must retain more information. Besides the +square $\|A'-BA\|_{\HS}^2$, we retain a term controlling +\[ + [B,K^2]\quad\hbox{and}\quad K^2-B^2-(AA^T)^r. +\] +Its coefficient depends on the largest binding energy and the exponent, +but not on the number of trial functions. This uniformity is the first +essential point: an extremal potential is not known in advance to have +finitely many negative eigenvalues. The integrated trace identity of the +matrix companion supplies the two nonnegative squares from which this +estimate follows. + +The second point is compactness with a fixed number of columns. As the +trial list grows, $A$ has more rows but still only $m$ columns. Its +density therefore has rank at most $m$. The retained defect bounds the +tail rows by the small binding energies in that tail. This yields strong +pointwise convergence even for an infinite eigenvalue list and gives +exact algebraic relations in the limit. Those relations separate the +rows with different energies. A differential inequality then makes +their row spaces independent of position; their mutual orthogonality +leaves at most $m$ nonzero directions. On each resulting finite block, +a matrix Riccati equation has a fixed diagonalizing basis and determines +the profiles in~\eqref{intro:classification}. + +Section~\ref{pre:section} gives the spectral and convexity preliminaries. +Section~\ref{act:section} proves the quantitative action estimate, and +Section~\ref{bound:section} recovers the sharp spectral bound. +Section~\ref{comp:section} passes to exact relations for an equality +case. Section~\ref{rig:section} proves that the channels are fixed, +solves the Riccati equation, and treats complex potentials and the +converse. Appendices~\ref{app:trace} and~\ref{app:paths} prove the trace +identity, regularized field construction, and connecting-path theorem +used by the action estimate. + +\subsection{Notation} +Throughout the proof, +\begin{equation}\label{intro:parameters} + \sigma=\gamma-\tfrac12\in(0,1),\qquad + p=1+\sigma,\qquad r=\sigma^{-1},\qquad q=r+1, +\end{equation} +and +\begin{equation}\label{intro:constants} + D_\sigma=\frac{\sigma^\sigma}{(1+\sigma)^{1+\sigma}},\qquad + b_\sigma=\int_{-1}^1(1-s^2)^\sigma\,ds,\qquad + C_\gamma=\frac{D_\sigma}{b_\sigma}. +\end{equation} +The last identity follows from the beta integral. We use the operator +norm $\|\cdot\|_{\op}$ and the Hilbert--Schmidt norm +$\|A\|_{\HS}^2=\tr(A^*A)$. The symbol $T$ denotes transpose of a real +matrix, and $\Sym_n$ is the space of real symmetric $n\times n$ matrices +with inner product $\tr(BH)$. The commutator is $[B,H]=BH-HB$. +The identity matrix, or identity operator, has the size determined by +its context. diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/preliminaries.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..62a3a895710f3d4fa7e5706b11bd15fc595fb6bf --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/preliminaries.tex @@ -0,0 +1,83 @@ +\section{Spectral and convexity preliminaries}\label{pre:section} + +Two elementary facts let us work with measurable potentials and identify +the equality condition in the potential estimate. The first is the form +argument used in the matrix inequality~\cite[Lemma~7.1]{MLT}; we give its +proof to fix the precise regularity available below. + +\begin{lemma}\label{pre:spectral} +Let $p>1$ and let $W:\R\to\C^{m\times m}$ be measurable, Hermitian and +positive semidefinite, with $\int_\R\tr(W^p)<\infty$. +The form $h_W$ on $H^1(\R;\C^m)$ is closed and bounded below. +Multiplication by $W^{1/2}$ is compact from $H^1$ to $L^2$. +The negative spectrum of $H_W$ consists of isolated eigenvalues of finite +multiplicity, with possible accumulation only at zero. Its eigenfunctions +belong to $H^1$ and satisfy $-\psi''-W\psi=\lambda\psi$ distributionally. +\end{lemma} + +\begin{proof} +Set $w=\|W\|_{\op}$. Then $w^p\le\tr(W^p)$, so $w\in L^p$. +The one-dimensional Sobolev estimate, applied to the norm of a vector +function, gives $\|\psi\|_\infty^2\le2\|\psi\|_2\|\psi'\|_2$. +H\"older's inequality, interpolation and Young's inequality imply +\begin{equation}\label{pre:form-bound} + \int_\R\langle\psi,W\psi\rangle + \le 2^{1/p}\|w\|_p\|\psi'\|_2^{1/p}\|\psi\|_2^{2-1/p} + \le \varepsilon\|\psi'\|_2^2+C_\varepsilon\|\psi\|_2^2. +\end{equation} +A sufficiently shifted form norm is therefore equivalent to the $H^1$ +norm. This proves closedness and semiboundedness, and polarization gives +continuity of the potential form on $H^1$. + +Write $T\psi=W^{1/2}\psi$. The truncated multiplier +$T_{R,L}=\mathbf1_{\{|x|\le R,\,w\le L\}}T$ is compact by the compact +embedding of $H^1([-R,R])$ into $L^2([-R,R])$. Moreover, +\[ + \|(T-T_{R,L})\psi\|_2^2 + \le C\|w\mathbf1_{\{|x|>R\}\cup\{w>L\}}\|_p\|\psi\|_{H^1}^2. +\] +The coefficient tends to zero, so $T$ is compact. +For each $a>0$, an infinite-dimensional spectral subspace in +$(-\infty,-a]$ would contain an $L^2$-orthonormal sequence bounded in +$H^1$ by~\eqref{pre:form-bound}. A subsequence converges weakly to zero +in $H^1$, hence its images under $T$ converge strongly to zero. This +contradicts +$-a\ge h_W[\psi]\ge-\|T\psi\|_2^2$ for unit vectors in that subspace. +Thus every such subspace is finite-dimensional. Finally, the weak form +equation tested against smooth compactly supported functions is the stated +distributional equation; its potential term is locally integrable because +$W\in L^p_{\mathrm{loc}}$ and $\psi$ is locally bounded. +\end{proof} + +\begin{lemma}[Trace Young inequality and its equality condition]\label{pre:young} +For positive semidefinite matrices $W,Z$ of the same finite size, set +\begin{equation}\label{pre:young-defect} + Y(W,Z)=D_\sigma\tr(W^p)-\tr(WZ)+\tr(Z^q). +\end{equation} +Then $Y(W,Z)\ge0$, with equality if and only if $W=qZ^r$. +Also, +\begin{equation}\label{pre:young-coercive} + Y(W,Z)\ge\tfrac12\tr(Z^q) + -(2^{p-1}-1)D_\sigma\tr(W^p). +\end{equation} +\end{lemma} + +\begin{proof} +Scalar maximization over $z\ge0$ gives +$yz-z^q\le D_\sigma y^p$ for $y\ge0$, with equality precisely when +$y=qz^r$. In an orthonormal eigenbasis $(v_j)$ of $Z$, let $z_j$ be its +eigenvalues and put $y_j=\langle v_j,Wv_j\rangle$. Strict convexity of +$t^p$ on $[0,\infty)$ gives +\[ + \tr(WZ)-\tr(Z^q) + \le D_\sigma\sum_j y_j^p + \le D_\sigma\sum_j\langle v_j,W^pv_j\rangle. +\] +Equality in the second inequality holds exactly when each $v_j$ belongs +to an eigenspace of $W$. Equality in the first then requires +$Wv_j=qz_j^rv_j$ for every $j$. This proves both directions of the +equality assertion, including zero eigenvalues. Applying the same +inequality to $2W,Z$ gives +$\tr(WZ)\le\tfrac12\tr(Z^q)+2^{p-1}D_\sigma\tr(W^p)$, +which is~\eqref{pre:young-coercive}. +\end{proof} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/rigidity.tex b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/rigidity.tex new file mode 100644 index 0000000000000000000000000000000000000000..7a8a501d673f3ad31ac7ad91645685f5b4e9ebc4 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/rigidity.tex @@ -0,0 +1,293 @@ +\section{Fixed channels and the equality profiles} +\label{rig:section} + +The pointwise relations of Proposition~\ref{comp:relations} still involve +an auxiliary operator $B$ that may depend on the point without a prescribed +measurable choice. We now extract constant subspaces from those relations. +On each nonzero subspace, $B$ becomes a uniquely determined matrix and +satisfies an ordinary differential equation. This will identify every +real equality potential; realification and a scalar calculation will then +complete the proof of Theorem~\ref{thm:main}. + +\subsection{The channel subspaces are constant} + +Assume first that $W$ is real and is a nonzero equality potential. Use the +rows $a_i$, numbers $k_i$, and operator $\Lambda=\operatorname{diag}(k_i)$ +from Proposition~\ref{comp:relations}. For each distinct value $\alpha>0$ +in the list $(k_i)$, let $n_\alpha$ be its finite multiplicity, and let +$G_\alpha$ be the $n_\alpha\times m$ matrix whose rows are the $a_i$ with +$k_i=\alpha$. Each $G_\alpha$ has entries in $H^1(\R)$, and we use their +continuous representatives. + +At every point in the common full-measure set of that proposition, choose +one of the operators $B$ it supplies. Since $B$ commutes with $\Lambda^2$, +it preserves each of its eigenspaces. Moreover, +\[ + AA^T=(\Lambda^2-B^2)^{1/r} +\] +commutes with both $B$ and $\Lambda^2$. Writing $B_\alpha$ for the block +on the $\alpha^2$-eigenspace therefore gives +\begin{align} + G_\alpha G_\beta^T&=0 &&(\alpha\ne\beta), + \label{rig:orthogonal-blocks}\\ + G_\alpha'&=B_\alpha G_\alpha, + & [B_\alpha,G_\alpha G_\alpha^T]&=0, + \label{rig:block-first-order}\\ + \alpha^2I-B_\alpha^2&=(G_\alpha G_\alpha^T)^r, + &\|B_\alpha\|_{\op}&\le\kappa. + \label{rig:block-square} +\end{align} +These statements hold almost everywhere and require no compatibility +between the choices of $B$ at different points. + +\begin{lemma}\label{rig:fixed-right-spaces} +For every $\alpha$, the subspace +\[ + \mathcal R_\alpha=(\ker G_\alpha(x))^\perp\subset\R^m +\] +is independent of $x\in\R$. The nonzero spaces $\mathcal R_\alpha$ are +mutually orthogonal, and +\begin{equation}\label{rig:dimension-bound} + \sum_\alpha\dim\mathcal R_\alpha\le m. +\end{equation} +In particular, only finitely many $G_\alpha$ are nonzero. The potential +$W$ has a continuous representative, equal to zero on the orthogonal +complement of these spaces and equal to +$q(G_\alpha^TG_\alpha)^r$ on $\mathcal R_\alpha$. With this +representative, every $G_\alpha$ is $C^2$. +\end{lemma} + +\begin{proof} +Fix $v\in\R^m$. Equation~\eqref{rig:block-first-order} implies +\[ + \|(G_\alpha v)'(x)\|\le\kappa\|G_\alpha(x)v\| + \quad\text{for almost every }x. +\] +The function $G_\alpha v$ is locally absolutely continuous. If it +vanishes at $x_0$, integration and Gronwall's inequality, first to the +right and then to the left of $x_0$, show that it vanishes everywhere. +Consequently $\ker G_\alpha(x)$ is independent of $x$, including points +outside the original full-measure set. This proves the first assertion. + +The products in \eqref{rig:orthogonal-blocks} are continuous, so their +almost-everywhere vanishing implies their vanishing everywhere. Thus the +constant row spaces $\mathcal R_\alpha$ are mutually orthogonal. Any +finite collection has total dimension at most $m$, proving +\eqref{rig:dimension-bound} and finiteness of the nonzero blocks. + +On the full-measure set from Proposition~\ref{comp:relations}, +$A^TA=\sum_\alpha G_\alpha^TG_\alpha$. Each summand acts on its own +orthogonal space $\mathcal R_\alpha$, and only finitely many summands +are nonzero. Equation~\eqref{comp:potential-limit} therefore yields +\begin{equation}\label{rig:continuous-potential} + W=q\sum_\alpha(G_\alpha^TG_\alpha)^r + \quad\text{almost everywhere}. +\end{equation} +The right-hand side is continuous and has the asserted restrictions; +we henceforth use it as the representative of $W$. The row eigenfunction +equation \eqref{comp:row-eigenfunction} now reads +\begin{equation}\label{rig:block-second-order} + G_\alpha''=\alpha^2G_\alpha-G_\alpha W + \quad\text{in distributions}. +\end{equation} +Its right-hand side is continuous. Integrating this identity twice shows +that the continuous representative of $G_\alpha$ is $C^2$. +\end{proof} + +The row spaces already split the physical space $\R^m$ into finitely +many constant channels. To determine the potential within one of them, +we also need a constant range for $G_\alpha$ in the eigenfunction index +space $\R^{n_\alpha}$. + +\begin{lemma}\label{rig:fixed-left-spaces} +For each $\alpha$ with $\ell:=\dim\mathcal R_\alpha>0$, the range of +$G_\alpha(x)$ in $\R^{n_\alpha}$ is independent of $x$. In constant +orthonormal bases of this range and of $\mathcal R_\alpha$, the +restriction of $G_\alpha$ is an invertible $C^2$ matrix +$D:\R\to\R^{\ell\times\ell}$. The matrix +$B_0=D'D^{-1}$ is $C^1$ and satisfies, everywhere on $\R$, +\begin{equation}\label{rig:active-identities} + B_0=B_0^T,\qquad + \alpha^2I-B_0^2=(DD^T)^r>0,\qquad + B_0'=r(B_0^2-\alpha^2I). +\end{equation} +\end{lemma} + +\begin{proof} +Choose a constant orthonormal basis of $\mathcal R_\alpha$, and let +$E(x)$ be the matrix of the restriction of $G_\alpha(x)$ to this space. +It has full column rank $\ell$ at every point, since the kernel of +$G_\alpha$ is constant. The projection onto its range is +\[ + P(x)=E(x)(E(x)^TE(x))^{-1}E(x)^T. +\] +The commutator in \eqref{rig:block-first-order} shows that $B_\alpha$ +preserves the range of $G_\alpha G_\alpha^T$, which is precisely the +range of $E$. Hence $E'=B_\alpha E$ has its columns in this range +almost everywhere. Differentiating the projection gives +\begin{align*} + P'={}&(I-P)E'(E^TE)^{-1}E^T\\ + &+E(E^TE)^{-1}(E')^T(I-P)=0 + \quad\text{almost everywhere}. +\end{align*} +Since $P$ is $C^1$, it is constant. + +Choose a constant orthonormal basis of this range as well. The resulting +square matrix $D$ is invertible everywhere and $C^2$. Almost everywhere, +$D'D^{-1}$ is the restriction of $B_\alpha$ to its invariant range in +these coordinates. Equations~\eqref{rig:block-first-order} and +\eqref{rig:block-square} give the symmetry and square identity in +\eqref{rig:active-identities} almost everywhere; continuity extends +them to every point. Positivity is strict because $D$ is invertible. + +On $\mathcal R_\alpha$, Equation~\eqref{rig:continuous-potential} +identifies the potential with $q(D^TD)^r$. Equation~ +\eqref{rig:block-second-order} consequently becomes +\[ + D''=\alpha^2D-qD(D^TD)^r + =\alpha^2D-q(DD^T)^rD. +\] +The second equality follows, for example, by a singular-value +decomposition. Differentiating $B_0=D'D^{-1}$ and using $q=r+1$ +now gives +\[ + B_0'=D''D^{-1}-B_0^2 + =\alpha^2I-q(\alpha^2I-B_0^2)-B_0^2 + =r(B_0^2-\alpha^2I), +\] +as claimed. +\end{proof} + +\subsection{The matrix Riccati equation separates the profiles} + +Fix a nonzero block and the matrices of Lemma~\ref{rig:fixed-left-spaces}. +Diagonalize the symmetric matrix $B_0(x_*)$ at one point by a constant +orthogonal change of range coordinates. The right-hand side of its +Riccati equation is a polynomial in $B_0$. Uniqueness for this matrix +ordinary differential equation therefore preserves the diagonal form. +More explicitly, the strict inequality in \eqref{rig:active-identities} +places each initial diagonal entry in $(-\alpha,\alpha)$. There is a +unique $y_j\in\R$ such that the scalar solution with that initial value is +\begin{equation}\label{rig:tanh} + b_j(x)=-\alpha\tanh\bigl(r\alpha(x-y_j)\bigr), + \qquad 1\le j\le\ell. +\end{equation} +The diagonal matrix with these entries solves the same initial-value +problem as $B_0$ on all of $\R$, so uniqueness gives equality everywhere. + +Equation~\eqref{rig:active-identities} implies that $DD^T$ is diagonal +in these coordinates. Since $D'=\operatorname{diag}(b_j)D$, each row +of $D$ is a positive scalar multiple of its value at $x_*$. None of +these rows is zero, and their directions are mutually orthogonal because +$DD^T$ is diagonal. Taking their normalized directions as a constant +orthonormal basis of $\mathcal R_\alpha$ makes $D$ diagonal with positive +entries. The potential in this fixed basis is therefore diagonal, with +entries +\begin{equation}\label{rig:profiles} + q\bigl(\alpha^2-b_j(x)^2\bigr) + =(r+1)\alpha^2\operatorname{sech}^2\bigl(r\alpha(x-y_j)\bigr). +\end{equation} +Combining these bases over the finitely many nonzero spaces +$\mathcal R_\alpha$, and adding a basis of their orthogonal complement, +proves the required necessity for real $W$. There are at most $m$ +nonzero profiles by \eqref{rig:dimension-bound}. The zero potential +has the same description with no nonzero profile. + +\subsection{Complex Hermitian potentials} + +For a complex Hermitian potential $W$, take the realification +$\widetilde W$ from \eqref{bound:realification}. The proof of +Proposition~\ref{bound:sharp} shows that it is equivalent to +$W\oplus\overline W$ by a constant unitary matrix and that both the trace-power +integral and the negative spectral moment are doubled. Thus equality +for $W$ implies equality for the real potential $\widetilde W$. + +The real result provides a constant unitary matrix $V$ such that +\[ + W(x)\oplus\overline W(x) + =V\operatorname{diag}(f_1(x),\ldots,f_{2m}(x))V^* + \quad\text{almost everywhere}, +\] +where each $f_j$ is either a profile of the form \eqref{rig:profiles} or +zero. Let $P$ be the projection onto the first summand $\C^m$ and put +$\widehat P=V^*PV$. Since $P$ commutes with $W\oplus\overline W$, +\[ + (f_i(x)-f_j(x))\widehat P_{ij}=0 + \quad\text{almost everywhere}. +\] +Consequently $\widehat P_{ij}=0$ whenever the functions $f_i,f_j$ are +not equal almost everywhere. Partition the indices into groups of +identical functions, including the possible zero function. The range +of $\widehat P$ splits as the orthogonal sum of its intersections with +these coordinate groups. On each group the diagonal potential is a +scalar multiple of the identity, so any constant orthonormal basis of +that intersection diagonalizes its restriction. Transporting these +bases by $V$ to the first summand gives a constant unitary basis of +$\C^m$ in which $W$ has the asserted profiles. This proves necessity +for complex Hermitian potentials. + +\subsection{Each listed potential attains equality} + +It remains to verify attainment, including the exact spectral moment, +without appealing to a classification of scalar optimizers. Fix $a>0$ +and $x_0\in\R$, and set +\[ + V(x)=(r+1)a^2\operatorname{sech}^2\bigl(ra(x-x_0)\bigr), + \qquad H=-\partial_x^2-V. +\] +The first-order factorization used in the commutation argument +of~\cite[Section~II]{BenguriaLoss2000} also explains why this profile has +only one negative eigenvalue. Put +\[ + w(x)=a\tanh\bigl(ra(x-x_0)\bigr), + \qquad Q=\partial_x+w,\qquad \operatorname{Dom}Q=H^1(\R). +\] +Then $Q^*=-\partial_x+w$ on $H^1(\R)$. Since $w$ and $w'$ are bounded, +both product domains are $H^2(\R)$, and direct differentiation yields +\begin{align} + Q^*Q&=H+a^2, + \label{rig:factorization}\\ + QQ^*&=-\partial_x^2+a^2 + +(r-1)a^2\operatorname{sech}^2\bigl(ra(x-x_0)\bigr) + \ge a^2. + \label{rig:partner} +\end{align} +Here the last inequality uses $r>1$. The equation $Q\psi=0$ has the +one-dimensional $L^2$ solution space +\[ + \ker Q= + \operatorname{span}\left\{ + \operatorname{sech}^{1/r}\bigl(ra(x-x_0)\bigr) + \right\}. +\] +Thus $-a^2$ is a simple eigenvalue of $H$, and +\eqref{rig:factorization} excludes eigenvalues below it. If +$H\phi=\lambda\phi$ with $-a^2<\lambda<0$, put +$\mu=\lambda+a^2$ and $g=Q\phi$. Then +$\|g\|_2^2=\mu\|\phi\|_2^2>0$. Moreover, +$Q^*Q\phi=\mu\phi\in\operatorname{Dom}Q$, so +$g\in\operatorname{Dom}(QQ^*)$ and $QQ^*g=\mu g$. This contradicts +\eqref{rig:partner}, since $\mu0$, and real orthonormal eigenfunctions $\varphi_i$. +For $0<\xi0$. Let $\eta\downarrow0$, then $\xi\downarrow0$, +and finally exhaust the negative eigenvalues. The case of an empty +negative spectrum is immediate. + +For complex Hermitian $W=E+iF$, with $E,F$ real, its realification is +\begin{equation}\label{bound:realification} + \widetilde W=\begin{pmatrix}E&-F\\F&E\end{pmatrix}. +\end{equation} +The constant unitary matrix +$2^{-1/2}\left(\begin{smallmatrix}I&iI\\I&-iI\end{smallmatrix}\right)$ +conjugates $\widetilde W$ to $W\oplus\overline W$. Hence $\widetilde W$ +is real symmetric and positive semidefinite, +$\tr(\widetilde W^p)=2\tr(W^p)$, and +$\Tr(H_{\widetilde W})_-^\gamma=2\Tr(H_W)_-^\gamma$. +Here conjugation identifies the spectra of $H_W$ and $H_{\overline W}$. +The real inequality, divided by two, proves the result. +\end{proof} diff --git a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c3964859737c802cd8273ebc0ce98f036eb84842 --- /dev/null +++ b/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e88f3118d513e3133e672b809b58c6c7be43fcf09e802302ac0b569eeccd3298 +size 467633 diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/README.md b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..6724a3c22271b9146e1ab596bac000d5dfdb2a82 --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](paper.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026, + author = {{OpenAI}}, + title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex new file mode 100644 index 0000000000000000000000000000000000000000..cfa53ce62b28c247651ec04cc2534ca1db41d4d3 --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex @@ -0,0 +1,139 @@ +\section{Adelic formulation and choices}\label{subsec:adelic-packet} + +The model of Lemma~\ref{lem:packet-model} also identifies the measure obtained from adelic +torus Haar probability. We give the identification for arbitrary local +type, so the lattice formulation includes all finite lattice and real +diagonalization choices in the usual adelic formulation. + +Let $\mathbb A_f$ be the finite adeles of $\Q$, let +$\widehat\Z=\prod_\ell\Z_\ell$, and put +$\mathbb A_\Q=\R\times\mathbb A_f$ and +$\mathbb A_K=K\otimes_\Q\mathbb A_\Q$. +Write $\widehat R=R\otimes\widehat\Z$. +Choose a rational basis $b_1,\ldots,b_n$ of $K$, and let +$\rho(\alpha)$ denote multiplication by $\alpha$ on row coefficient +vectors. It embeds the torus +\[ + \mathbf T_K=(\operatorname{Res}_{K/\Q}\mathbb G_m)/\mathbb G_m + \quad\hbox{in}\quad G=\operatorname{PGL}_{n/\Q}. +\] +The embedding matrix $B_\sigma=(\sigma_j(b_i))_{i,j}$ satisfies +\begin{equation}\label{eq:regular-diagonalization} + \rho(\alpha)B_\sigma + =B_\sigma\operatorname{diag}(\sigma_1(\alpha),\ldots,\sigma_n(\alpha)). +\end{equation} +For $g_f\in G(\mathbb A_f)$, choose a lift to +$\operatorname{GL}_n(\mathbb A_f)$ and set +\[ + \widehat M(g_f)=\widehat\Z^{\,n}g_f^{-1},\qquad + M(g_f)=K\cap\widehat M(g_f), +\] +using the chosen rational basis to identify coefficient vectors with +elements of $K$. This is a full lattice, and its completion is +$\widehat M(g_f)$: choose an integer $d\ge1$ with +$d\widehat\Z^{\,n}\subset\widehat M(g_f) +\subset d^{-1}\widehat\Z^{\,n}$, and patch in the finite quotient +$d^{-1}\Z^n/d\Z^n$. Changing the projective lift multiplies it by a +scalar finite idele. Every such idele is a rational scalar times an +element of $\widehat\Z^\times$; the latter preserves every completed +full lattice. Consequently this change only rescales $M(g_f)$ by +a rational number. + +Choose $g_\infty\in G(\R)$ with +$g_\infty^{-1}\mathbf T_K(\R)^\circ g_\infty=A_n$, using the +odd-degree identification of $G(\R)$ with $\SL_n(\R)$. +For $g=(g_\infty,g_f)$, the adelic packet is the projection of +$\mathbf T_K(\Q)\backslash\mathbf T_K(\mathbb A_\Q)g$ to +$G(\Q)\backslash G(\mathbb A_\Q)/G(\widehat\Z)$, equipped with +the pushforward of torus Haar probability. + +\begin{proposition}\label{prop:adelic-packet} +The adelic torus quotient is compact. Its packet and canonical +probability agree with $\mathcal P_{K,M(g_f),\sigma'}$ and +$\mu_{K,M(g_f),\sigma'}$ for an ordering $\sigma'$ of the real +embeddings. Every prescribed local homothety type occurs this way. +Changing the rational basis or the projective lifts does not change +the packet; changing the allowed real diagonalizer only reorders the +embeddings. Thus Theorem~\ref{thm:main} includes every such adelic choice. +\end{proposition} + +\begin{proof} +Write $M=M(g_f)$ and $\widehat M=\widehat M(g_f)$. +First identify the ambient quotient with $X_n$. Given a finite matrix +lift $g_f$, choose a basis matrix $c\in\operatorname{GL}_n(\Q)$ +for $\Q^n\cap\widehat\Z^{\,n}g_f^{-1}$. Then +$cg_f\in\operatorname{GL}_n(\widehat\Z)$, and the corresponding +real lattice is $\Z^ncg_\infty$, normalized to covolume one. +Changing this basis or either matrix lift has no effect on that +lattice. This constructs the usual quotient identification directly. +For odd $n$, the signed real $n$th root gives +$[h]\mapsto(\det h)^{-1/n}h$, identifying +$G(\R)$ with $\SL_n(\R)$ and $G(\Z)$ with $\SL_n(\Z)$. + +Multiplication matrices modulo scalars give +\begin{equation}\label{eq:idelic-torus-quotient} + \mathbf T_K(\Q)\backslash\mathbf T_K(\mathbb A_\Q) + \simeq K^\times\backslash\mathbb A_K^\times/\mathbb A_\Q^\times. +\end{equation} +Here the pointwise lifts can also be seen by elementary linear algebra: +over any ground field $F_0$ containing $\Q$, a matrix whose projective +class lies in $\mathbf T_K$ becomes a scalar multiple of a multiplication +operator over an algebraic closure. It therefore commutes exactly with +the regular representation already over $F_0$. The centralizer of that +representation is the algebra of multiplication operators, so the matrix +belongs to $(K\otimes_\Q F_0)^\times$, with scalar ambiguity $F_0^\times$. +Finite lifts may be chosen integral at almost all primes. + +A finite idele $t_f$ acts on the completed lattice by +$\widehat M\mapsto t_f^{-1}\widehat M$. Hence its translates are +exactly the prescribed local homothety type. The stabilizer of +$\widehat M$ among finite ideles is +\[ + \{z:z\widehat M=\widehat M\} + =(\cO(M)\otimes\widehat\Z)^\times: +\] +equality holds precisely when both $z$ and $z^{-1}$ preserve +$\widehat M$. In the projective torus the stabilizer is the image +of this group, since a representative that preserves the lattice +up to a scalar idele can be rescaled to preserve it exactly. +The unit subgroup is compact open in the finite ideles and has finite +index in $\widehat R^\times$; at each prime not dividing $q$ it equals +$R_\ell^\times$. Its projective image is compact open as well. +No invertibility of $M$ over $\cO(M)$ is used. + +To describe the quotient and its measure, first divide +\eqref{eq:idelic-torus-quotient} by the image of +$\widehat R^\times$. Fractional-ideal factorization and +$\mathbb A_\Q^\times +=\Q^\times\R^\times\widehat\Z^\times$ +give one real fiber $E\backslash C$ for each ordinary ideal class. +These fibers are compact by the unit theorem, and the omitted +finite-unit group is compact. Thus the torus quotient is compact. +At the stabilizer of $\widehat M$, the finite fibers instead retain +the translates $\mathcal N(I)$. The same decomposition becomes +\[ + \coprod_{[I]}(\mathcal N(I)\times C)/E, +\] +with exactly the action in Lemma~\ref{lem:packet-model}. +Haar measure pushes to uniform counting measure on each finite +unit orbit and to the stated logarithmic Haar measure on $C$. +After normalization it is therefore precisely the probability in +that lemma. For $g_\infty=[B_\sigma]$, +\eqref{eq:regular-diagonalization} identifies the resulting real +lattices with its displayed map. This proves both the packet +identification and the equality of measures, including the orbit +volume weights. + +Finally, every full $M$ is obtained by choosing a basis matrix for +its completed lattice and taking its inverse as $g_f$. +Any other allowed real diagonalizer has the form $[B_\sigma dP]$, +where $d$ is diagonal and $P$ is a permutation matrix, because it +must permute the joint eigenspaces. The diagonal factor translates +the real torus and permutes its sign components, preserving its +Haar probability; the permutation reorders the embeddings. +If a rational change of basis has matrix $c$, then +$B'_\sigma=cB_\sigma$ and $\rho'=c\rho c^{-1}$, while the same +packet data have representatives $g'_\infty=cg_\infty$ and +$g'_f=cg_f$. Thus the torus is conjugated and the translated packet +is left-multiplied by $c$, which disappears in the ambient rational quotient. +\end{proof} diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/analytic.tex b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/analytic.tex new file mode 100644 index 0000000000000000000000000000000000000000..89cfc305090627de217f4f24b0656fe4134955dd --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/analytic.tex @@ -0,0 +1,189 @@ +\section{Residue-relative weighted ideal counts}\label{sec:analytic} + +Write +\[ + a_K(m)=\#\{\mathfrak a\subseteq R:\N\mathfrak a=m\}, + \qquad \kappaK=\operatorname*{Res}_{s=1}\zeta_K(s)>0. +\] +We first retain the residue in the analytic estimates. All constants +subscripted by $n$ are uniform over the fields under consideration. + +\begin{lemma}\label{lem:residue} +For $D$ sufficiently large in terms of $n$, +\begin{equation}\label{eq:residue} + h\zeta_K(1+h)\asymp_n\kappaK + \quad\left(00$, every nontrivial zero +$\rho$ satisfies $|1-\rho|\ge c_n/L$. + +The real logarithmic derivative of the completed zeta function gives +\[ + S_K(\sigma):=\sum_\rho\operatorname{Re}\frac1{\sigma-\rho} + =\frac1\sigma+\frac1{\sigma-1}+\frac L2 + +\frac{\zeta_K'}{\zeta_K}(\sigma)+G_n(\sigma),\qquad \sigma>1, +\] +where zeros are counted with multiplicity and +\[ + G_n(\sigma)=\frac n2 + \left(\frac{\Gamma'(\sigma/2)}{\Gamma(\sigma/2)}-\log\pi\right). +\] +This is the Hadamard identity in +\cite[Equation~(9)]{Stark1974}. The zero terms are nonnegative, +$G_n$ is bounded on $[1,2]$, and the Euler product gives +$\zeta_K'/\zeta_K\le0$ there. Thus, at $\sigma_0=1+L^{-1}$, +$S_K(\sigma_0)\ll_n L$. + +This bound persists uniformly for $1<\sigma\le1+4/L$. +Indeed, for $\rho=\beta+i\gamma$, set +$X=L(1-\beta)$, $Y=L\gamma$, and $U=L(\sigma-1)$. +Then $X\ge0$, $00$, +integration over an interval of length at most $4/L$ proves the +comparison in \eqref{eq:residue}. At $\sigma_0$, the inequality +$\zeta_K(\sigma_0)\ge1$ also gives $\kappaK\gg_n L^{-1}$. + +For completeness, only finitely many fields of this fixed degree +have discriminant below the cutoff. One can see this directly here: +nonzero vectors of $D^{-1/(2n)}\sigma(R)$ have absolute coordinate +product at least $D^{-1/2}$, so Mahler's criterion puts these lattices +in a fixed compact set. Local basis lifts and a finite cover give +bases of $R$ with uniformly bounded conjugates. A basis element +outside $\Q$ generates $K$, since $n$ is prime; its monic integral +minimal polynomial therefore has bounded coefficients. There are +only finitely many such polynomials. The simple pole of each of +these finitely many zeta functions now gives the remaining claims +uniformly for sufficiently small $h>0$. +\end{proof} + +We also use the following uniform form of Shiu's theorem +\cite{Shiu1980,Pollack2020}: if $b$ is nonnegative and multiplicative +and $b(m)\le d_n(m)$, where $d_n$ is the $n$-fold divisor function, +then +\begin{equation}\label{eq:shiu} + \sum_{u-vu}} + \frac{a_K(\ell)}\ell + \ll_n\frac{\log(2q)}{Q^{5/8}}\ll_n1. +\] +The first error is bounded by a constant times +$\sum_\ell\ell^{-2}$. Removing the primes of $S$ thus gives +\begin{equation}\label{eq:coprime-euler} + \exp\left(\sum_{\ell\le u}\frac{b(\ell)}\ell\right) + \ll_n\frac{\kappaK\log u}{Z_S}. +\end{equation} +For $Q$ sufficiently large at fixed $\eta$, the length +$v=y/d\ge\eta u$ is at least $u^{1/2}$. +Applying \eqref{eq:shiu}, then \eqref{eq:coprime-euler} and +\eqref{eq:shell-sum}, bounds the contribution of these $d$ by +\[ + C_n\frac{\kappaK y}{Z_S} + \sum_{d(\mathbf k)\le Q^{1/8}}\frac{P(\mathbf k)}{d(\mathbf k)} + \le C_n\frac{\kappaK y}{q}. +\] + +For $d>Q^{1/8}$, terms with $d>x$ vanish because the inner sum +in \eqref{eq:weighted-split} then contains no positive integer. +For the remaining terms, $x/d\ge1$, and the elementary bound +$\sum_{m\le z}d_n(m)\le z(1+\log z)^{n-1}$ for $z\ge1$ +follows by summing reciprocals in the first $n-1$ factors. +Using \eqref{eq:weight-tail}, with $s=1/(2n)$, the remaining +contribution to \eqref{eq:weighted-split} is at most +\[ + C_n\frac{x}{q}Q^{-s/16}(1+\log x)^{n-1} + =o\left(\frac{\kappaK x}{q}\right). +\] +This is uniform in the stated range: Lemma~\ref{lem:residue} +gives $\kappaK^{-1}\ll_n\log(3D)$, and $D\le Q^2$. +At fixed $A_0,\eta$, increase the threshold until this term is at +most $\kappaK\eta x/q\le\kappaK y/q$. The implied constant has +not changed with $\eta$, proving the proposition. +\end{proof} diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/entropy.tex b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/entropy.tex new file mode 100644 index 0000000000000000000000000000000000000000..fb95e71664c82c1a75253e33fd7920a066697a52 --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/entropy.tex @@ -0,0 +1,167 @@ +\section{Entropy and identification of the limit}\label{sec:entropy} + +The ball estimate has exponent $n$, whereas $A_n$ has dimension $n-1$. +We follow the ordinary-ball strategy of +\cite[Section~2.7.1]{ELMV2011}. We retain the estimate for every invariant +probability dominated by a finite multiple of the limit, so it forces +positive entropy after restriction to any positive-weight collection of +ergodic components. Measure classification then identifies the limit. + +For an invariant probability $\nu$, write $h_\nu(a)$ for the +measure-theoretic entropy of the map $x\mapsto xa$. +Fix distinct real numbers $\lambda_1,\ldots,\lambda_n$ with sum zero and put +\[ + a(t)=\operatorname{diag}(e^{\lambda_1t},\ldots,e^{\lambda_nt}), + \qquad \lambda=\min_{i\ne j}|\lambda_i-\lambda_j|>0. +\] + +\begin{proposition}[Entropy from ordinary balls]\label{prop:entropy-from-balls} +Let $n\ge3$ and let $\mu$ be an $A_n$-invariant probability on $X_n$. +Suppose that for every compact $\Omega\subset X_n$ there are +$C_\Omega,r_\Omega>0$ such that +\[ + \mu(xB(r))\le C_\Omega r^n + \qquad(x\in\Omega,\ 00$, then +\[ + \xi=\frac1w\int_{Z_0}\nu_z\,\dd\tau(z) +\] +would be an $A_n$-invariant probability with $\xi\le w^{-1}\mu$. +Equation~\eqref{eq:entropy-decomposition}, applied to this restricted +decomposition, would give $h_\xi(T)=0$, contradicting +Proposition~\ref{prop:entropy-from-balls}. +\end{proof} + +\paragraph{Measure classification.} +Einsiedler--Katok--Lindenstrauss +\cite[Theorem~1.3 and Corollary~1.4]{EKL2006} prove that, for prime +$n\ge3$, an $A_n$-invariant and $A_n$-ergodic probability on +$\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)$ is Haar whenever +some diagonal one-parameter subgroup has positive entropy. +Their action is on the left. The map +$\Gamma g\mapsto g^{-1}\Gamma$, where $\Gamma=\operatorname{SL}_n(\Z)$, +changes our right action by $a(t)$ into the left action by $a(-t)$. +Since entropy is unchanged by inversion of an invertible transformation, +their result applies to our convention without any additional hypothesis. + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Let $\mu_i$ be the packet probabilities for a sequence in the theorem. +Equation~\eqref{eq:discriminant-scale} gives $Q_i\to\infty$. +By Proposition~\ref{prop:tightness}, this sequence is tight and every +weak subsequential limit $\mu$ is an $A_n$-invariant probability. +Proposition~\ref{prop:ball-bound} supplies the hypotheses of +Proposition~\ref{prop:entropy-from-balls}. +Lemma~\ref{lem:entropy-components} shows that almost every +$A_n$-ergodic component of $\mu$ has positive entropy for $a(1)$. +As $n$ is prime, the classification just stated makes every such +component equal to $m_n$. Integrating the components gives $\mu=m_n$. + +Thus every convergent subsequence has the same probability limit. +Together with tightness, this proves $\mu_i\to m_n$ and excludes +loss of mass. All preceding estimates were for each prescribed local +homothety packet separately, so no averaging over different local +module types is used. +\end{proof} diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/geometry.tex b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..04a59b535f1271acdf698e499b9bb5deed3dd8e0 --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/geometry.tex @@ -0,0 +1,150 @@ +\section{Nonescape and ordinary ball bounds}\label{sec:geometry} + +We apply the weighted count in two ways. Cubes about zero control +the cusp; small boxes away from the coordinate hyperplanes control +ordinary neighborhoods in the lattice space. + +\begin{proposition}\label{prop:tightness} +Every sequence of the packet probabilities with $Q\to\infty$ +is tight. More precisely, for $0<\theta<1$, +\begin{equation}\label{eq:cusp} + \limsup_{Q\to\infty}\mu_* + \{\Lambda:\exists\,0\ne v\in\Lambda,\ \|v\|_\infty<\theta\} + \le C_n\theta^n. +\end{equation} +Every subsequential probability limit is $A_n$-invariant. +\end{proposition} + +\begin{proof} +For $f=\one_{[-\theta,\theta]^n}$, the first $n-1$ logarithms +in \eqref{eq:product-fiber} are at most $\log\theta$. +The last coordinate bounds the sum of their gaps from +$\log\theta$ by $\log(\theta^n/z)$. Thus +\[ + \cV_f(z)=\frac{2^n}{(n-1)!} + \bigl(\max\{0,\log(\theta^n/z)\}\bigr)^{n-1}. +\] +Put $X=\theta^nQ$ and $U=Q^{3/4}$. Up to an $n$-dependent +constant, Proposition~\ref{prop:unfolding} reduces the first +moment to +\[ + \frac1{\sqrt D\,\kappaK} + \sum_{\N\mathfrak a\le X} + P(\mathfrak a)\left(\log\frac X{\N\mathfrak a}\right)^{n-1}. +\] +If $X<1$, the sum is empty. Otherwise its logarithmic factor is +at most $(\log Q)^{n-1}$, since $\theta<1$. The terms with +$\N\mathfrak a\le\min(U,X)$ contribute at most +$C_nQ^{-1/4}(\log Q)^{n-1}$, by applying +Proposition~\ref{prop:weighted-count} at $x=y=U$. +When $X>U$, define the truncated cumulative sum +\[ + A_U(t)=\sum_{U<\N\mathfrak a\le t}P(\mathfrak a), + \qquad U\le t\le X. +\] +Writing each logarithmic power as an integral and applying Tonelli gives +\begin{align*} + \sum_{U<\N\mathfrak a\le X} + P(\mathfrak a)\left(\log\frac X{\N\mathfrak a}\right)^{n-1} + &= (n-1)\int_U^X A_U(t) + \left(\log\frac Xt\right)^{n-2}\frac{dt}{t} \\ + &\ll_n \frac{\kappaK}{q} + \int_U^X\left(\log\frac Xt\right)^{n-2}\dd t + \ll_n \frac{\kappaK}{q}X. +\end{align*} +Here the same proposition at $x=y=t$ bounds $A_U(t)$, and the +last inequality follows from +$\int_0^1(\log(1/v))^{n-2}\dd v<\infty$. +There is no lower boundary term, since $A_U(U)=0$, and the +logarithmic weight vanishes at $X$. If $X\le U$, this part is absent. +Dividing by $\sqrt D\,\kappaK$ gives $O_n(\theta^n)$. +Since $E_f\ge1$ whenever the displayed cusp event occurs, +this proves \eqref{eq:cusp}. + +Mahler's compactness criterion \cite[Section~3, Theorem~2]{Mahler1946} says that the +complement of this cusp event is compact. Given a desired +mass error, choose $\theta$ first and then $Q$ sufficiently large. +The finitely many earlier packet supports are compact by +Lemma~\ref{lem:packet-model}, so the whole sequence is tight. +The lattice space is locally compact and second countable; +weak compactness for tight probabilities gives probability +subsequential limits. Invariance passes to them by testing +against continuous compactly supported functions. +\end{proof} + +For $r>0$, let +\[ + B(r)=\{h\in\SL_n(\R): + \max_{i,j}|h_{ij}-\delta_{ij}|0$ such that +\begin{equation}\label{eq:ball-bound} + \mu(xB(r))\le C_\Omega r^n + \qquad(x\in\Omega,\ 04Cr_0$ on $V$. +Use Proposition~\ref{prop:weighted-count} with +\[ + x=(z_w+2Cr)Q,\qquad y=4CrQ. +\] +This slightly enlarged interval contains every contributing +ideal norm. Uniformly in $w$, for sufficiently large $Q$ it has +$Q^{3/4}\le x\le A_0Q$ with a fixed $A_0\ge1$, and +$\eta x\le y\le x$ with $\eta=4Cr/A_0\le1$, after decreasing +$r_0$ if necessary. For every fixed $r\in(0,r_0)$, +Proposition~\ref{prop:unfolding} therefore gives, eventually, +\begin{equation}\label{eq:vector-box} + \int E_{\one_{w+[-b_0r,b_0r]^n}}\dd\mu_* + \ll_{V,b_0,n} + \frac{r^{n-1}}{\sqrt D\,\kappaK} + \frac{\kappaK rQ}{q}=r^n . +\end{equation} +Here $Q=q\sqrt D$. The implicit constant is independent of $r$. The threshold for $Q$ +may depend on $r$, which is all that will be needed. + +We next choose a suitable lattice vector at each $x\in\Omega$. +Every full lattice has a vector with no zero coordinate: +for any basis, its combination with coefficients +$(1,N,\ldots,N^{n-1})$ has coordinates that are nonzero +polynomials in $N$, so an integer $N$ can avoid their finitely +many roots. Near each center choose a continuous local lift to basis +matrices, and continue the chosen vector using its fixed integer +basis coefficients. On a sufficiently small neighborhood with compact +closure in that chart, its coordinates remain bounded and bounded away +from zero. A finite cover of $\Omega$ by such neighborhoods therefore +gives choices $w_x$ lying in one compact +$V\subset(\R^\times)^n$. This pointwise selection need not be +continuous or measurable. + +For $h$ in $B(r)$, each coordinate of $w_xh-w_x$ has absolute +value at most $r\sum_i|(w_x)_i|$. Thus one fixed $b_0$ makes +every lattice in $xB(r)$ contain a nonzero vector in +$w_x+[-b_0r,b_0r]^n$. Equation~\eqref{eq:vector-box} bounds +the packet mass of $xB(r)$ by $C_\Omega r^n$. +This set is open, so Portmanteau gives +\[ + \mu(xB(r))\le\liminf_i\mu_i(xB(r))\le C_\Omega r^n. +\] +The argument holds for every fixed $x$ and $r$ with the same +constant on $\Omega$. It proves \eqref{eq:ball-bound} for +all $00$, +\begin{equation}\label{eq:weight-moment} + \sum_{\mathbf k}P(\mathbf k)d(\mathbf k)^{s-1} + \ll_{n,\eps}\frac{q^\eps}{q}. +\end{equation} +In particular, +\begin{equation}\label{eq:weight-tail} + \sum_{d(\mathbf k)>Q^{1/8}}\frac{P(\mathbf k)}{d(\mathbf k)} + \ll_n \frac1q Q^{-s/16}. +\end{equation} +\end{lemma} + +\begin{proof} +Normalize additive Haar measure on $R_\ell$ to have mass one. +The valuation shell $\mathbf k_\ell$ has mass +$Z_\ell^{-1}/d_\ell(\mathbf k_\ell)$. Its conditional law is +invariant under the transitive unit action, so $P_\ell$ is +exactly the fraction of this shell lying in $L_\ell$. +Since $L_\ell$ has mass $1/q_\ell$, summing over shells and +then multiplying over $S$ proves \eqref{eq:shell-sum}. +Elements with a zero component have additive measure zero. + +Choose an additive-Haar uniform element of $L_\ell$, and denote +its component valuations by $v_{\mathfrak p}$. The condition +$R_\ell L_\ell=R_\ell$ implies that the projection to every +component contains a unit. Modulo the $k$th power of its maximal +ideal, the projection therefore contains all $\Z_\ell$ multiples +of that unit, at least $\ell^{\lceil k/e_{\mathfrak p}\rceil}$ +elements. Uniform measure on $L_\ell$ induces uniform measure +on this finite additive image. Consequently +\begin{equation}\label{eq:valuation-tail} + \Pr(v_{\mathfrak p}\ge k) + \le\ell^{-\lceil k/e_{\mathfrak p}\rceil}\qquad(k\ge1). +\end{equation} +This argument does not require the component projections to be +independent, or even to equal their full integer rings. + +Put $j=\max_{\mathfrak p\mid\ell} +\lceil v_{\mathfrak p}/e_{\mathfrak p}\rceil$. Then +$d_\ell(\mathbf v_\ell)\le\ell^{nj}$. If $j\ge a\ge1$, some +$v_{\mathfrak p}$ is at least $e_{\mathfrak p}(a-1)+1$. +Equation~\eqref{eq:valuation-tail} and a union bound give +$\Pr(j\ge a)\le n\ell^{-a}$. Hence +\begin{equation}\label{eq:local-moment} + \mathbb E_{L_\ell}d_\ell(\mathbf v_\ell)^s + \le1+n\sum_{a\ge1}\ell^{(ns-1)a} + \le1+C_n\ell^{-1/2}. +\end{equation} +Using the shell masses once more yields the exact factorization +\[ + \sum_{\mathbf k}P(\mathbf k)d(\mathbf k)^{s-1} + =\frac1q\prod_{\ell\in S} + \left(Z_\ell\,\mathbb E_{L_\ell}d_\ell(\mathbf v_\ell)^s\right). +\] +Each factor is at most +$(1-\ell^{-1})^{-n}(1+C_n\ell^{-1/2})$. For fixed $\eps>0$ +this is at most $\ell^\eps$ at all sufficiently large primes; +the finitely many remaining primes contribute a constant +depending only on $n,\eps$. As $\prod_{\ell\in S}\ell\le q$, +this proves \eqref{eq:weight-moment}. Finally, take +$\eps=s/16$ and use $q\le Q$ to obtain +\[ + \sum_{d>Q^{1/8}}\frac{P(\mathbf k)}d + \le Q^{-s/8}\sum_{\mathbf k}P(\mathbf k)d^{s-1} + \ll_n q^{-1}Q^{-s/16}. + \qedhere +\] +\end{proof} diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/packets.tex b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/packets.tex new file mode 100644 index 0000000000000000000000000000000000000000..716c00425869a814bb397665ddb531331b08759b --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/packets.tex @@ -0,0 +1,215 @@ +\section{The packet as a weighted ideal sum}\label{sec:packets} + +We first identify the packet measure exactly, including its signs and +orbit volumes. Throughout the proof, $K$ is totally real of the fixed +prime degree $n$, and +\[ + R=\cO_K,\quad E=R^\times,\quad D=|\Disc(K)|,\quad + \kappaK=\operatorname*{Res}_{s=1}\zeta_K(s). +\] +The absolute norm $\N I$ of a nonzero fractional $R$-ideal is extended +multiplicatively from integral ideals. Thus +$\operatorname{covol}(\sigma(I))=\sqrt D\,\N I$. +We use standard ideal theory and the unit theorem +\cite[Chapters~3--5]{Milne2020}, together with the analytic +class-number formula \cite[Theorem~19.12]{Sutherland2021}. +The classical packet interpretation is discussed in +\cite[Sections~2 and~5]{ELMV2011}. + +\subsection{Saturation and local choices} + +Let $J_0=RM$, $q=[J_0:M]$, and $Q=q\sqrt D$. For a rational prime +$\ell$, write $(J_0)_\ell=j_\ell R_\ell$, and put +$L_\ell=j_\ell^{-1}M_\ell$. Then +\begin{equation}\label{eq:local-normalization} + L_\ell\subset R_\ell,\qquad R_\ell L_\ell=R_\ell,\qquad + [R_\ell:L_\ell]=q_\ell,\qquad q=\prod_\ell q_\ell. +\end{equation} +Set $S=\{\ell:\ell\mid q\}$; off $S$, $L_\ell=R_\ell$. +Since $qJ_0\subset M$, one has $qR\subset\cO(M)$. +Conversely, a multiplier of $M$ has an integral endomorphism matrix +on $M$ and hence is an algebraic integer. It follows that +\begin{equation}\label{eq:discriminant-scale} + qR\subset\cO(M)\subset R,\qquad + D(M)\le Dq^{2n}\le Q^{2n}. +\end{equation} +Only the last implication, $D(M)\to\infty\Rightarrow Q\to\infty$, +will be needed. + +These quantities are constant within the prescribed local type. +Indeed, if $M'_\ell=c_\ell M_\ell$, multiplication by $c_\ell$ +identifies $(RM)_\ell/M_\ell$ with $(RM')_\ell/M'_\ell$. +It also identifies the local multiplier orders, since multiplication +in $K\otimes_\Q\Q_\ell$ is commutative. The multiplier order commutes +with completion: in a $\Z$-basis of $M$, its defining condition is +that the matrix of multiplication have integral entries, and the +same condition over $\Z_\ell$ defines the local order. Thus $q$, +$Q$, $\cO(M)$, and $D(M)$ are unchanged by local homothety. +Different local types can nevertheless have the same multiplier order. + +Choose one fractional ideal $I$ in each ordinary ideal class of $R$. +Let $\mathcal N(I)$ be the set of lattices $N$ locally homothetic to $M$ +with $RN=I$. If $I_\ell=i_\ell R_\ell$, its possible completions are +exactly +\begin{equation}\label{eq:local-choices} + N_\ell=i_\ell z_\ell L_\ell,\qquad z_\ell\in R_\ell^\times. +\end{equation} +Indeed, taking maximal-order spans in a local homothety forces the +remaining multiplier to be a unit. Conversely, these local lattices +lie between $qI_\ell$ and $I_\ell$ and patch uniquely by the primary +decomposition of $I/qI$. Thus $\mathcal N(I)$ is finite, +$[I:N]=q$ for all its members, and its cardinality +$T=|\mathcal N(I)|$ is independent of $I$. The label count $T$ need +not equal the saturation index $q$. Local unit Haar probability induces +uniform probability on each finite set of local choices, so the choices at different +primes are independent. Every global homothety class in the +packet occurs in these sets, by moving its maximal-order span +to the chosen ideal-class representative. + +\subsection{Real parameters and measure} + +Let +\[ + C=\{t\in(\R^\times)^n:\textstyle\prod_j|t_j|=1\}. +\] +Give $C$ counting measure on its $2^n$ sign components and measure +$du_1\cdots du_{n-1}$ on the coordinates $u_j=\log|t_j|$. +The embedding of $E$ is a discrete cocompact subgroup of $C$. +If $F$ is a measurable fundamental domain, the unit theorem and +the totally real case of the analytic class-number formula +\cite[Theorem~19.12]{Sutherland2021} give +\begin{equation}\label{eq:unit-volume} + \vol(F)=2^{n-1}R_K,\qquad + h_K\vol(F)=\sqrt D\,\kappaK, +\end{equation} +where $h_K$ is the ordinary class number and $R_K$ the regulator +in deleted logarithmic coordinates. The factor $2^{n-1}$ comes +from the $2^n$ signs and the logarithmic kernel $\{1,-1\}$. +In the class-number formula this is the factor $2^n/w_K$, since +the roots of unity in a totally real field are exactly $\{1,-1\}$. + +\begin{lemma}\label{lem:packet-model} +The maps +\[ + (N,t)\longmapsto (Q\N I)^{-1/n}\sigma(N)t + \qquad (N\in\mathcal N(I),\ t\in C) +\] +induce a bijection from the disjoint union of the quotients by +\[ + \epsilon:(N,t)\longmapsto + (\epsilon N,\sigma(\epsilon)^{-1}t),\qquad \epsilon\in E, +\] +onto the union of the packet orbits. +Counting measure on the $N$ labels times the stated measure on $C$ +descends to the sum of the orbit Haar measures. In particular, +the packet probability is obtained by averaging uniformly over $I$ +and $\mathcal N(I)$ and integrating over $F$ with measure +$dt/\vol(F)$. +\end{lemma} + +\begin{proof} +Surjectivity follows from \eqref{eq:local-choices}. A global +homothety changes a normalized embedded lattice by an element +of $C$. Since $n$ is odd, each pair of opposite sign patterns has +exactly one representative in $W_n$; a common negative sign +does not change a lattice. + +For injectivity, suppose two displayed normalized lattices coincide. +Absorb the ratio of the normalizing scalars into the real diagonal +map carrying $\sigma(N)$ onto $\sigma(N')$. Applying this map +to one nonzero $\alpha\in N$ gives $\sigma(\beta)$ for some +$\beta\in N'$. None of the coordinates of $\sigma(\alpha)$ +vanishes, so the map is multiplication by $\sigma(\beta/\alpha)$. +Thus $N'=bN$ for $b\in K^\times$. Their maximal-order spans +represent the same ideal class and therefore equal the same +chosen $I$. Hence $bI=I$, so $b\in E$, and the parameters differ +by exactly the stated action. + +Combine an $N$ label with a sign component of $C$. Its stabilizer in $E$ +is the totally positive subgroup of +$\{\epsilon\in E:\epsilon N=N\}$, and has finite index in $E$. +It acts faithfully by logarithmic translations. By injectivity, +these are all periods of the corresponding $A_n$-orbit. +Each orbit of labels therefore contributes exactly one compact +$A_n$-orbit, with precisely its prescribed Haar volume. +This proves both finiteness and the measure assertion. +One may equally use $\mathcal N(I)\times F$ as a fundamental +domain, since the $E$-action preserves counting measure times $dt$. +The total measure of these fundamental domains is +$h_KT\vol(F)$. Dividing by this quantity gives precisely the +uniform averages over ideal classes and finite labels, followed +by integration against $dt/\vol(F)$. +\end{proof} + +\subsection{Containment probabilities and unfolding} + +Write $\mathfrak p\mid\ell$ for the primes of $R$ over $\ell$, +with ramification indices $e_{\mathfrak p}$ and residue degrees +$f_{\mathfrak p}$. Normalize $v_{\mathfrak p}$ by a uniformizer, +so $v_{\mathfrak p}(\ell)=e_{\mathfrak p}$ and +$\sum_{\mathfrak p\mid\ell}e_{\mathfrak p}f_{\mathfrak p}=n$. +For a tuple $\mathbf k_\ell=(k_{\mathfrak p})_{\mathfrak p\mid\ell}$ +of nonnegative integers, define +\[ + P_\ell(\mathbf k_\ell) + =\Pr_{z\in R_\ell^\times}\{x\in zL_\ell\}, +\] +where $x$ has those component valuations and the probability is +normalized unit Haar measure. The definition is independent of $x$ +because the units act transitively on each valuation shell. +For an integral nonzero $R$-ideal $\mathfrak a$, set +\[ + P(\mathfrak a)=\prod_{\ell\in S} + P_\ell((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid\ell}). +\] +For $\alpha\in I\setminus\{0\}$, the probability that a uniform +$N\in\mathcal N(I)$ contains $\alpha$ is +$P((\alpha)I^{-1})$. + +For a nonnegative Borel function $f$ on $\R^n$, put +$E_f(\Lambda)=\sum_{0\ne v\in\Lambda}f(v)$. +The product-fiber integral below integrates $f$ over the $C$-orbit +with absolute coordinate product $z$, using logarithmic Haar measure +and summing over all sign components: +\begin{equation}\label{eq:product-fiber} + \cV_f(z)=\sum_{\omega\in\{\pm1\}^n} + \int_{\R^{n-1}} + f\left(\omega_1e^{u_1},\ldots,\omega_{n-1}e^{u_{n-1}}, + \omega_n z e^{-\sum_{j0. +\end{equation} + +\begin{proposition}\label{prop:unfolding} +Writing $\mu_*=\mu_{K,M,\sigma}$, one has +\begin{equation}\label{eq:unfolding} + \int_{X_n}E_f\dd\mu_* + =\frac1{\sqrt D\,\kappaK} + \sum_{\mathfrak a\subset R,\ \mathfrak a\ne0} + P(\mathfrak a)\cV_f(\N\mathfrak a/Q). +\end{equation} +The identity holds for nonnegative extended integrals. +\end{proposition} + +\begin{proof} +Lemma~\ref{lem:packet-model} expresses the left side as +\[ + \frac1{h_K\vol(F)} + \sum_I\int_F\sum_{\alpha\in I\setminus\{0\}} + P((\alpha)I^{-1}) + f((Q\N I)^{-1/n}\sigma(\alpha)t)\dd t. +\] +The map $\alpha\mapsto(\alpha)I^{-1}$ runs over the integral +ideals in the inverse class of $I$, each fiber being one free +$E$-orbit. Summing that orbit and integrating over $F$ unfolds +to an integral over $C$, by Tonelli. The absolute coordinate +product of the normalized vector is +$\N((\alpha)I^{-1})/Q$. Translating the first $n-1$ logarithms +and relabeling the signs identifies the $C$-integral with +\eqref{eq:product-fiber}, with Jacobian one. +Every integral ideal occurs in exactly one inverse class. +Equation~\eqref{eq:unit-volume} supplies the coefficient in +\eqref{eq:unfolding}. +\end{proof} + +When $M$ is an $R$-ideal, $q=1$ and $P(\mathfrak a)=1$. +Thus \eqref{eq:unfolding} recovers the maximal-order formula. diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..a6c1ef2081882a4f141b6b9569da289b05d80102 --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex @@ -0,0 +1,247 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage{amsmath,amssymb,amsthm} +\usepackage[margin=1in]{geometry} +\usepackage{microtype} +\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} +\hypersetup{ + pdftitle={Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}, + pdfauthor={OpenAI}, + pdfsubject={Full-packet equidistribution for arbitrary local module types}, + pdfkeywords={torus packets, equidistribution, prime degree, nonmaximal orders, entropy} +} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\Q}{\mathbb Q} +\DeclareMathOperator{\N}{N} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\Disc}{Disc} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\diag}{diag} +\newcommand{\kappaK}{\kappa_K} +\newcommand{\dd}{\,d} +\newcommand{\one}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\newcommand{\cO}{\mathcal O} +\newcommand{\cV}{\mathcal V} +\numberwithin{equation}{section} +\title{Equidistribution of Prime-Degree Torus Packets\\ +with Arbitrary Local Type} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the packet form of the higher-dimensional Duke equidistribution +problem for totally real fields of any fixed prime degree at least five, +allowing arbitrary local homothety types of full lattices. As the +multiplier-order discriminant tends to infinity, the volume-weighted packet +measures converge to Haar probability measure, with no escape of mass. +\end{abstract} + +\section{Introduction}\label{sec:intro} + +Compact diagonal orbits in the space of unimodular lattices encode +ideal classes and units of totally real number fields. The higher-dimensional +Duke equidistribution problem asks whether arithmetically complete families +of these orbits become uniformly distributed as their discriminants grow. +The grouping matters: one local homothety type specifies a complete packet, +whereas a multiplier order alone can combine several packets. We prove the +prime-degree packet statement for every such local type. + +Fix a prime $n\ge5$, and write +\[ + X_n=\SL_n(\Z)\backslash\SL_n(\R),\qquad + A_n=\{\diag(e^{u_1},\ldots,e^{u_n}):\textstyle\sum_j u_j=0\}. +\] +We regard $X_n$ as the space of covolume-one row lattices, with the +right action of $A_n$, and let $m_n$ be its $\SL_n(\R)$-invariant +probability measure. Haar measure on $A_n$ is +$du_1\cdots du_{n-1}$ in the displayed logarithmic coordinates. + +Let $K$ be a totally real field of degree $n$, let +$\sigma=(\sigma_1,\ldots,\sigma_n)$ order its real embeddings, and let +$M\subset K$ be a free rank-$n$ $\Z$-submodule spanning $K$ over $\Q$. +We call such an $M$ a full lattice. Its multiplier order and discriminant +are +\[ + \cO(M)=\{\alpha\in K:\alpha M\subset M\},\qquad + D(M)=|\Disc(K)|[\cO_K:\cO(M)]^2. +\] +Two such lattices are \emph{locally homothetic} if +$M'_\ell=c_\ell M_\ell$ for every rational prime $\ell$, where +$c_\ell\in(K\otimes_\Q\Q_\ell)^\times$ and subscripts denote completion. +Choose representatives $M'$ for their $K^\times$-homothety classes +within the local type of $M$, and normalize +\[ + \Lambda_{M',\sigma} + =\operatorname{covol}(\sigma(M'))^{-1/n}\sigma(M'). +\] +The packet $\mathcal P_{K,M,\sigma}$ consists of the distinct orbits +$\Lambda_{M',\sigma}wA_n$, with +\[ + w\in W_n=\{\diag(\epsilon_1,\ldots,\epsilon_n): + \epsilon_j\in\{\pm1\},\ \textstyle\prod_j\epsilon_j=1\}. +\] +These orbits are compact and their number is finite; the +parametrization in Section~\ref{sec:packets} also proves these facts. +For an orbit $O$, let $\nu_O$ be its invariant probability and +$\vol(O)$ its volume for the specified Haar measure. Set +\begin{equation}\label{eq:packet-probability} + \mu_{K,M,\sigma} + =\frac{\sum_{O\in\mathcal P_{K,M,\sigma}}\vol(O)\nu_O} + {\sum_{O\in\mathcal P_{K,M,\sigma}}\vol(O)}. +\end{equation} + +\begin{theorem}\label{thm:main} +For every fixed prime $n\ge5$ and every sequence +$(K_i,M_i,\sigma_i)$ as above with \mbox{$D(M_i)\to\infty$}, +\[ + \mu_{K_i,M_i,\sigma_i}\longrightarrow m_n. +\] +The convergence is weak convergence of probability measures. In +particular, it holds against every continuous compactly supported +function, and no mass escapes into the cusp. +\end{theorem} + +Theorem~\ref{thm:main} permits nonmaximal orders of unbounded index +and arbitrary local module types. The field discriminants may remain +bounded or grow independently of those indices. The statement is +separate for each local homothety type; it makes no assertion about an +individual orbit, a convergence rate, or composite degrees. + +\subsection*{The maximal-order specialization} + +Take $M=\cO_K$. Every lattice in its local type is a fractional +$\cO_K$-ideal, and its homothety class is an ordinary ideal class. +Thus the packet is obtained by embedding one representative of each +ideal class, normalizing its covolume to one, and taking the distinct +sign translates of its positive diagonal orbit. For every sequence of +totally real fields of degree $n$ with $|\Disc(K)|\to\infty$, the +volume-weighted measures of these packets converge to $m_n$ by +Theorem~\ref{thm:main}. This construction also explains why an individual +ideal class is not the object being averaged. + +For general $M$, the same maximal-order ideal classes remain the global +parameters, but each carries a finite set of local module choices. +Proposition~\ref{prop:adelic-packet} in Appendix~\ref{subsec:adelic-packet} +identifies this construction with the +pushforward of adelic torus Haar probability, independently of the +rational basis and finite-lattice presentation. The theorem therefore +includes all these presentations of the maximal-order specialization, +as well as arbitrary noninvertible modules over nonmaximal orders. + +\subsection*{Context and prior work} + +Linnik's ergodic approach connected arithmetic distribution problems +with dynamics \cite{Linnik1968}. Duke proved equidistribution of the +length-weighted closed geodesics associated with fundamental +real-quadratic discriminants on the modular surface +\cite[Theorem~1]{Duke1988}. In higher rank, Einsiedler, Lindenstrauss, +Michel, and Venkatesh developed the relationship between arithmetic +separation and entropy for periodic torus orbits \cite{ELMV2009}. +Their cubic equidistribution theorem combines subconvexity, local +analysis, and measure classification +\cite[Theorem~1.4]{ELMV2011}. Their packet construction already +distinguishes local homothety from the coarser condition of having the +same multiplier order \cite[Section~5.5]{ELMV2011}. + +The higher-prime-degree extension described in +\cite[Section~1.6.3]{ELMV2011} is conditional on subconvexity. +Khayutin subsequently obtained entropy bounds using arithmetic +separation and Galois invariants, under maximal-order and +two-transitivity hypotheses \cite[Theorem~1.1]{Khayutin2019}. +Lemke Oliver, Thorner, and Zaman proved equidistribution for sequences +of orders in totally real fields of each fixed prime degree at least five, +outside a small +exceptional family of fields \cite[Theorem~2.3]{LOTZ2024}. +Their result allows nonmaximal orders; the exceptional family has at +most $O_{n,\eps}(X^\eps)$ fields of discriminant at most $X$. +Theorem~\ref{thm:main} treats every field and every full-lattice local +type in the stated prime degrees. It gives an unconditional affirmative +answer to this packet form of the higher-dimensional Duke problem, +including sequences in a fixed field with increasing order index. + +\subsection*{The arithmetic ingredient} + +The proof follows the counting-to-rigidity strategy of +Einsiedler--Lindenstrauss--Michel--Venkatesh +\cite[Section~2.7.1]{ELMV2011}: packet averages of lattice-vector +counts control the cusp and small ordinary balls, and these bounds +feed into positive-entropy rigidity. Their cubic argument obtains +asymptotics for these averages using global and local subconvexity +\cite[Sections~2.7.2--2.7.3]{ELMV2011}. For the same rigidity step, +we obtain the needed upper bounds by retaining the Dedekind-zeta +residue in the unfolded ideal count. Stark's exceptional-zero descent +\cite{Stark1974}, together with Shiu's theorem in Pollack's uniform +formulation \cite{Shiu1980,Pollack2020}, supplies the required bounds +without a subconvexity hypothesis. Odd degree excludes the quadratic +subfields that occur in Stark's exceptional-zero alternative. + +For an arbitrary prescribed local type, we keep maximal-order ideal +counting but insert local containment probabilities: at each prime, +the weight records the probability that a vector belongs to a randomly +chosen unit translate of the prescribed module. +The essential estimate is a small positive moment of the local norm, +uniform over full local modules whose maximal-order span is the full +ring. Its proof uses a unit in each component projection, not +independence of the components or invertibility of the module. +The unit-mass argument in \cite[Lemma~9.7]{ELMV2011} is an antecedent; +here we prove the exact shell normalization and the positive moment +for arbitrary local modules with this full-span condition. + +The exact unfolding belongs to the classical Hecke method, presented +in the packet setting in \cite[Section~10]{ELMV2011}. The dynamical +conclusion uses the two-sided tube criterion of +\cite[Corollary~3.3]{ELMV2009} and the prime-degree measure +classification of Einsiedler, Katok, and Lindenstrauss +\cite[Corollary~1.4]{EKL2006}. We give the residue estimate, +local-module bounds, and the passage to entropy in almost every +ergodic component below. + +\subsection*{Proof outline} + +Put $R=\cO_K$, $D=|\Disc(K)|$, and +$q=[RM:M]$. The natural counting scale is $Q=q\sqrt D$, and +$D(M)\le Q^{2n}$. Section~\ref{sec:packets} expresses the packet +mean of a lattice-vector count as a maximal-order ideal sum with +local weights. Section~\ref{sec:local} proves an exact total-mass +identity and a power-saving tail for these weights. +Section~\ref{sec:analytic} combines them with a residue-relative +zeta estimate and Shiu's theorem. Removing the Euler factors at +primes dividing $q$ produces precisely the factor needed to cancel +the local normalization. + +The resulting count controls both short lattice vectors and small +boxes about vectors with nonzero coordinates. Section~\ref{sec:geometry} +deduces tightness and an $O(r^n)$ bound for the mass of ordinary balls +in any limit measure. Since $A_n$ has dimension $n-1$, this bound +forces positive entropy in almost every $A_n$-ergodic component. +Section~\ref{sec:entropy} gives this implication and applies the +prime-degree measure-classification theorem. In that step it is +essential to prove positive entropy for every invariant probability +dominated by a finite multiple of the limit, not only for the limit +itself: this rules out a positive-weight family of zero-entropy +components. + +\input{packets} +\input{local} +\input{analytic} +\input{geometry} +\input{entropy} + +\appendix +\input{adelic} + +{\small +\bibliographystyle{plain} +\bibliography{references} +} +\end{document} diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..3e1610d9e1d9bb6d604b5edcfae4d1de4cfe888f --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib @@ -0,0 +1,123 @@ +@book{Linnik1968, + author = {Linnik, Yurij V.}, + title = {Ergodic Properties of Algebraic Fields}, + series = {Ergebnisse der Mathematik und ihrer Grenzgebiete}, + volume = {45}, publisher = {Springer}, address = {Berlin--Heidelberg}, + year = {1968}, + url = {https://link.springer.com/book/9783642866333} +} + +@article{Duke1988, + author = {Duke, William}, + title = {Hyperbolic distribution problems and half-integral weight {Maass} forms}, + journal = {Inventiones Mathematicae}, + volume = {92}, number = {1}, year = {1988}, pages = {73--90}, + doi = {10.1007/BF01393993} +} + +@article{ELMV2011, + author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay}, + title = {Distribution of periodic torus orbits and {Duke}'s theorem for cubic fields}, + journal = {Annals of Mathematics}, + volume = {173}, number = {2}, year = {2011}, pages = {815--885}, + doi = {10.4007/annals.2011.173.2.5} +} + +@article{ELMV2009, + author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay}, + title = {Distribution of periodic torus orbits on homogeneous spaces}, + journal = {Duke Mathematical Journal}, + volume = {148}, number = {1}, year = {2009}, pages = {119--174}, + doi = {10.1215/00127094-2009-023} +} + +@article{EKL2006, + author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon}, + title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture}, + journal = {Annals of Mathematics}, + volume = {164}, number = {2}, year = {2006}, pages = {513--560}, + doi = {10.4007/annals.2006.164.513} +} + +@article{LOTZ2024, + author = {Lemke Oliver, Robert J. and Thorner, Jesse and Zaman, Asif}, + title = {An approximate form of {Artin}'s holomorphy conjecture and non-vanishing of {Artin} {$L$}-functions}, + journal = {Inventiones Mathematicae}, + volume = {235}, number = {3}, year = {2024}, pages = {893--971}, + doi = {10.1007/s00222-023-01232-2}, + eprint = {2012.14422}, archivePrefix = {arXiv}, primaryClass = {math.NT} +} + +@article{Khayutin2019, + author = {Khayutin, Ilya}, + title = {Arithmetic of double torus quotients and the distribution of periodic torus orbits}, + journal = {Duke Mathematical Journal}, + volume = {168}, number = {12}, year = {2019}, pages = {2365--2432}, + doi = {10.1215/00127094-2019-0016}, + eprint = {1510.08481}, archivePrefix = {arXiv}, primaryClass = {math.NT} +} + +@article{Stark1974, + author = {Stark, H. M.}, + title = {Some effective cases of the {Brauer--Siegel} theorem}, + journal = {Inventiones Mathematicae}, + volume = {23}, number = {2}, year = {1974}, pages = {135--152}, + doi = {10.1007/BF01405166} +} + +@article{Shiu1980, + author = {Shiu, Peter}, + title = {A {Brun--Titchmarsh} theorem for multiplicative functions}, + journal = {Journal f\"ur die reine und angewandte Mathematik}, + volume = {313}, year = {1980}, pages = {161--170}, + doi = {10.1515/crll.1980.313.161} +} + +@article{Pollack2020, + author = {Pollack, Paul}, + title = {Nonnegative multiplicative functions on sifted sets, and the square roots of {$-1$} modulo shifted primes}, + journal = {Glasgow Mathematical Journal}, + volume = {62}, number = {1}, year = {2020}, pages = {187--199}, + doi = {10.1017/S0017089519000041}, + note = {\href{https://www.pollack-math.net/brunmultiplicative4.pdf}{Corrected author version}}, + url = {https://www.pollack-math.net/brunmultiplicative4.pdf} +} + +@misc{Milne2020, + author = {Milne, James S.}, + title = {Algebraic Number Theory}, + year = {2020}, note = {Version 3.08}, + url = {https://www.jmilne.org/math/CourseNotes/ANT.pdf} +} + +@misc{Sutherland2021, + author = {Sutherland, Andrew V.}, + title = {The Analytic Class Number Formula}, + year = {2021}, + note = {MIT 18.785, Fall 2021, Lecture 19, November 15}, + url = {https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_lec19.pdf} +} + +@article{Mahler1946, + author = {Mahler, Kurt}, + title = {On lattice points in {$n$}-dimensional star bodies. {I}. {Existence} theorems}, + journal = {Proceedings of the Royal Society of London. Series A}, + volume = {187}, number = {1009}, year = {1946}, pages = {151--187}, + doi = {10.1098/rspa.1946.0072} +} + +@article{Rokhlin1967, + author = {Rokhlin, V. A.}, + title = {Lectures on the entropy theory of measure-preserving transformations}, + journal = {Russian Mathematical Surveys}, + volume = {22}, number = {5}, year = {1967}, pages = {1--52}, + doi = {10.1070/RM1967v022n05ABEH001224} +} + +@article{GreschonigSchmidt2000, + author = {Greschonig, Gernot and Schmidt, Klaus}, + title = {Ergodic decomposition of quasi-invariant probability measures}, + journal = {Colloquium Mathematicum}, + volume = {84/85}, number = {2}, year = {2000}, pages = {495--514}, + doi = {10.4064/cm-84/85-2-495-514} +} diff --git a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ebc9ac7f1cf60fd10ad883a4db1c06db72465cc2 --- /dev/null +++ b/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7e6a0188abd45ae4520d4592c773d264e029751e3ef6fb2623388aee8b0f56c0 +size 404189 diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/README.md b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..af925bc8549cc72a54f11ecdd50dfdde4d7f7878 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [Equidistribution of primitive quartic torus packets for arbitrary orders](quartic-torus-packets.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026, + author = {{OpenAI}}, + title = {{Equidistribution of primitive quartic torus packets for arbitrary orders}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/additive.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/additive.tex new file mode 100644 index 0000000000000000000000000000000000000000..db026e4b7fdf540e286b2ddda5454fb08d3673dd --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/additive.tex @@ -0,0 +1,861 @@ +\section{Additive distribution of the conductor weights} +\label{sec:additive} + +The factors \(f_p^*\) introduced in Section~\ref{sec:exterior} retain the +local lattice condition of an order at the primes in \(S\). We now average their +product on the real trace plane. The estimate will also permit a residue +class at a small modulus prime to \(S\); this is the form needed to average +the factors at the remaining primes in Section~\ref{sec:sieve}. + +Fix a nonzero integer \(m\). In the three real embeddings of \(F\), write +\(z=(z_1,z_2,z_3)\), so that \(\Tr z=z_1+z_2+z_3\). Choose two distinct +indices \(i,j\), a fixed nonnegative function +\(\Phi\in C_c^\infty(\R^2)\) which is at least one on \([-1,1]^2\), and a +fixed nonnegative \(\chi\in C_c^\infty(\R)\) with +\[ + \chi(m)=1,\qquad + \supp\chi\subset(m-\tfrac13,m+\tfrac13). +\] +For \(c=(c_i,c_j)\in\R^2\) and positive \(r_i,r_j\), put +\begin{equation} + \phi(z)= + \Phi\left(\frac{z_i-c_i}{r_i},\frac{z_j-c_j}{r_j}\right)\chi(\Tr z), + \qquad + A_\phi=r_i r_j\int_{\R^2}\Phi(u,v)\,du\,dv . + \label{eq:add-box} +\end{equation} +Thus \(A_\phi\) is the integral of \(\phi\) on \(\Tr z=m\), using \(z_i,z_j\) +as coordinates on that plane. Fix \(r_*>0\), keep \(r_i,r_j\le r_*\), and +require the supports of their planar factors to lie in a fixed +bounded region. The supports of all the resulting \(\phi\)'s then lie in a +fixed compact subset \(\Omega_\infty\) of \(\R^3\) on which \(\Tr z\) stays +away from zero. All constants below may depend on \(m,\Phi,\chi\), the +upper bound for the scales, and this bounded region. They do not depend on +the locations \(c\) or on the scales within their specified range. + +For \(p\notin S\), the algebra \(F\otimes\Q_p\) is unramified and its +integral trace map \(R_{F,p}\to\Z_p\) is onto. The affine set +\[ + \mathcal T_{m,p}=\{z\in R_{F,p}:\Tr z=m\} +\] +is consequently a translate of a free rank-two direct summand of \(R_{F,p}\). +A residue class on this set modulo \(p^a\) means a translate of +\(p^a\ker(\Tr:R_{F,p}\to\Z_p)\). Equivalently, if \(z_p^0\in\mathcal T_{m,p}\) +represents the class, its members satisfy \(z-z_p^0\in p^aR_{F,p}\). +There are \(p^{2a}\) such classes. + +\begin{proposition}[Additive distribution] +\label{prop:additive-equidistribution} +There are absolute constants \(\tau,\theta>0\) with the following property. +Let \(K,\OO,F\) be as in Section~\ref{sec:exterior}, put +\(X=[\OO_K:\OO]\sqrt{|\Disc(K)|}\), and take \(S\) as in that section for +the fixed nonzero integer \(m\). In Equation~\eqref{eq:add-box}, choose any +two real partition coordinates and scales +\[ + X^{-\tau}\le r_i,r_j\le r_* +\] +whose planar supports stay in the fixed bounded region above. Let +\(1\le l\le X^\theta\) be prime to every prime in \(S\). For each \(p\mid l\) +choose any \(z_p^0\in\mathcal T_{m,p}\), specifying a class modulo +\(p^{v_p(l)}\). Then +\begin{equation} + \sum_{\substack{z\in F,\ \Tr z=m\\ + z\in R_{F,p}\ (p\notin S)\\ + z-z_p^0\in p^{v_p(l)}R_{F,p}\ (p\mid l)}} + \phi(z)\prod_{p\in S}f_p^*(z) + = + A_\phi\,l^{-2}\prod_{p\in S}d_p(m) + \bigl(1+O(X^{-\theta})\bigr). + \label{eq:add-equidistribution} +\end{equation} +Here \(d_p\) is the trace density of Lemma~\ref{lem:fourier-scalar-density}. +The error is uniform over all the fields, orders, ordered real embeddings, +permitted box locations and scales, moduli \(l\), and chosen residue classes. +The implied constant and the lower threshold for \(X\) depend only on the +fixed real data listed above and on \(m\). +\end{proposition} + +\subsection{Finite expansion and Poisson summation} + +We first assemble the local inverse transforms into a finite global +expansion. This is the point at which the fact that the cubic algebra is a +field is essential. Let \(\mathfrak d_F\) be the different of \(F\), and +write \(\mathfrak d_p=\mathfrak d_F R_{F,p}\) for its localization. +For a choice of denominator shells at the primes of \(S\), write +\[ + \mathfrak h=\prod_{p\in S}\mathfrak h_p,\qquad + H=\prod_{p\in S}H_p=\Nm\mathfrak h,\qquad + G_0=\prod_{p\in S}p^{g_p}. +\] +Local ideals here and below are identified with global ideals supported at +the corresponding primes. The functions \(K_{p,\mathfrak h_p}\) and their +boundary sets \(\partial_{\mathfrak h_p}(z)\) are those of +Lemma~\ref{lem:fourier-shell-estimates}. Set +\[ + \mathscr L_{\mathfrak h}=\{p\in S:g_p>0\text{ or }p\le5\}. +\] + +\begin{lemma}[Global bounds for denominator profiles] +\label{lem:add-profile-bounds} +For every profile one has +\begin{equation} + G_0\le [R_F:C_F]\le H G_0. + \label{eq:add-profile-indices} +\end{equation} +Suppose that \(z\in F\) is integral outside \(S\), its real image lies in +\(\Omega_\infty\), and +\(\prod_{p\in S}K_{p,\mathfrak h_p}(z)\ne0\). Define +\[ + T_\partial=T_\partial(\mathfrak h,z) + =\prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\ + \mathfrak q\in\partial_{\mathfrak h_p}(z)}} + \Nm\mathfrak q , +\] +where an empty product is one. There is a constant depending only on +\(\Omega_\infty\) such that +\begin{align} + H&\ll D_FG_0^{24}T_\partial, + &T_\partial&\le\left(\prod_{p\in S}p\right)^3, + \label{eq:add-profile-norm}\\ + \left|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\right| + &\ll 8^{|S|}T_\partial^{-1/2}. + \label{eq:add-profile-inverse} +\end{align} +All such \(z\) lie in a fractional \(R_F\)-ideal +\(\mathfrak a_{\mathfrak h}\), independent of \(z\), with +\begin{equation} + \Nm(\mathfrak a_{\mathfrak h})^{-1} + \le 30^{12}D_FG_0^{24}. + \label{eq:add-profile-support} +\end{equation} + +Set \(C_3=371\). For all sufficiently large \(X\), depending only on the +fixed real data and \(m\), a profile contributing at any such \(z\) has +\(H\le X^{C_3}\). For these \(X\) and every \(z\in F\) integral outside +\(S\) with real image in \(\Omega_\infty\), +\begin{equation} + \prod_{p\in S}f_p^*(z) + =\sum_{\substack{\mathfrak h=(\mathfrak h_p)_{p\in S}\\H\le X^{C_3}}} + \prod_{p\in S}K_{p,\mathfrak h_p}(z). + \label{eq:add-profile-expansion} +\end{equation} +The number of profiles in this sum is \(O_\eta(X^\eta)\) for every fixed +\(\eta>0\), uniformly in the arithmetic data. +\end{lemma} + +\begin{proof} +Let \(M_p=\mathfrak h_p\cap C_{F,p}\). The additive lattice identity +\[ + [R_{F,p}:C_{F,p}] + =[R_{F,p}:C_{F,p}+\mathfrak h_p]\,[\mathfrak h_p:M_p] +\] +has first factor between \(1\) and +\([R_{F,p}:\mathfrak h_p]=H_p\). The index of \(C_F\) is supported on \(S\) +by Lemma~\ref{lem:ext-order-scale}; multiplying gives +Equation~\eqref{eq:add-profile-indices}. + +At \(p\in\mathscr L_{\mathfrak h}\), the local support is contained in +\(p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_p^{-1}\). +At every other \(p\in S\), the support loses at most one power of each +boundary prime \(\mathfrak q\), and loses no power in the other components. +It follows that a supporting \(z\) belongs to +\[ + \mathfrak b_{\mathfrak h,z} + =\mathfrak h\mathfrak d_F^{-1} + \prod_{p\in\mathscr L_{\mathfrak h}} + (pR_F)^{-4(g_p+1)} + \prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\ + \mathfrak q\in\partial_{\mathfrak h_p}(z)}} + \mathfrak q^{-1}. +\] +The real trace condition excludes \(z=0\). Since \(F\) is a field, its norm +is nonzero, and containment of the principal ideal \((z)\) in +\(\mathfrak b_{\mathfrak h,z}\) gives +\[ + |\Nm_{F/\Q}z| + =\Nm(\mathfrak b_{\mathfrak h,z}) + [\mathfrak b_{\mathfrak h,z}:(z)] + \ge \frac{H} + {D_FT_\partial\prod_{p\in\mathscr L_{\mathfrak h}}p^{12(g_p+1)}}. +\] +The left side is bounded above on \(\Omega_\infty\). For \(g_p>0\) we have +\(g_p+1\le2g_p\); the only additional factors, for \(g_p=0\), are at +\(2,3,5\). Hence +\[ + \prod_{p\in\mathscr L_{\mathfrak h}}p^{12(g_p+1)} + \le30^{12}G_0^{24}. +\] +This proves the first inequality of Equation~\eqref{eq:add-profile-norm}. +Each boundary prime occurs once, and +\(\sum_{\mathfrak q\mid p}f_{\mathfrak q}\le +\sum_{\mathfrak q\mid p}e_{\mathfrak q}f_{\mathfrak q}=3\); +this proves the bound for \(T_\partial\). Equation~\eqref{eq:add-profile-inverse} +is the product of the boundary bounds of +Lemma~\ref{lem:fourier-shell-estimates}; its absolute implied constant +accounts for the three small primes. + +Dropping \(\mathfrak h_p\) in the first support condition enlarges that +support. In the second, the exponent \(v_{\mathfrak q}(\mathfrak h_p)-1\) +at a boundary is nonnegative. Thus every supporting +point belongs to the ideal +\[ + \mathfrak a_{\mathfrak h} + =\mathfrak d_F^{-1} + \prod_{p\in\mathscr L_{\mathfrak h}} + (pR_F)^{-4(g_p+1)} , +\] +which gives Equation~\eqref{eq:add-profile-support}. + +Lemma~\ref{lem:ext-order-scale} gives +\(\Delta_F\ll X^{28}\), +\(D_F\le\Delta_F\), and +\([R_F:C_F]\le\Delta_F^{1/2}\). The definition of \(S\) gives +\(\prod_{p\in S}p\ll_m X^2\). Equations +\(\eqref{eq:add-profile-indices}\)--\(\eqref{eq:add-profile-norm}\) therefore +give \(H\ll X^{28+24\cdot14+6}=X^{370}\). The fixed implied constant is +absorbed by \(X^{371}\) for sufficiently large \(X\). + +A nonzero field element has nonzero image in every component of +\(F\otimes\Q_p\). Lemma~\ref{lem:fourier-inverse-expansion} therefore +identifies the density \(f_p^*(z)\) with its pointwise finite sum of inverse +shells at all the present \(z\). The norm bound just proved excludes every +product with \(H>X^{371}\), uniformly in \(z\), and proves +Equation~\eqref{eq:add-profile-expansion}. Finally, for any fixed \(u>0\), +Rankin's inequality bounds the number of profiles with \(H\le X^{C_3}\) by +\[ + X^{C_3u}\prod_{p\in S}(1-p^{-u})^{-3}. +\] +There are at most three prime-ideal components over each \(p\). The radical +bound on \(S\) implies \(|S|=O(\log X/\log\log X)\), since the product of +\(k\) distinct primes is at least \(k!\). The displayed Euler product is +then \(X^{o(1)}\) for fixed \(u\). Taking \(u\) arbitrarily small proves +the asserted bound for every \(\eta>0\). +\end{proof} + +We apply this expansion before imposing Poisson summation. For \(p\notin S\) +let +\[ + \eta_{p,l}(z)= + \begin{cases} + \mathbf 1_{z_p^0+p^{v_p(l)}R_{F,p}}(z),&p\mid l,\\ + \mathbf 1_{R_{F,p}}(z),&p\nmid l . + \end{cases} +\] +Let \(\eta_{\mathfrak h,l}\) be the finite-adelic test function whose local +factor is \(K_{p,\mathfrak h_p}\) at \(p\in S\) and \(\eta_{p,l}\) elsewhere. +By Lemma~\ref{lem:fourier-shell-estimates}, every \(K_{p,\mathfrak h_p}\) +is supported on elements of integral trace. The factors outside \(S\) are +supported in \(R_{F,p}\) and also have integral trace. Thus any \(z\in F\) +with \(\eta_{\mathfrak h,l}(z)\ne0\) has rational trace integral at every +finite prime, hence an integer. Only \(m\) meets \(\supp\chi\). The finite +profile expansion in Lemma~\ref{lem:add-profile-bounds} consequently shows +that the left side of +Equation~\eqref{eq:add-equidistribution}, denoted by \(\Sigma\), equals +\begin{equation} + \Sigma + =\sum_{\substack{\mathfrak h\\H\le X^{C_3}}} + S_{\mathfrak h},\qquad + S_{\mathfrak h} + =\sum_{z\in F}\phi(z)\eta_{\mathfrak h,l}(z). + \label{eq:add-original-profiles} +\end{equation} +Here \(C_3=371\) is fixed. The preceding lemma makes the profile interchange +finite. Each inner sum also has finitely many terms, since its finite +conditions place \(z\) in a fractional ideal and its real support is compact. + +In the local results of Section~\ref{sec:local-fourier}, choose the finite +components of the standard character of \(\mathbb A_\Q/\Q\); they all have +conductor \(\Z_p\). Compose this global character with the field trace. +At a finite place the self-dual measure satisfies +\(\vol(R_{F,p})=|D_F|_p^{1/2}\), while at infinity it is Lebesgue measure +in the three embeddings. The real covolume of \(R_F\) is \(\sqrt{D_F}\) +and \(\prod_p\vol(R_{F,p})=D_F^{-1/2}\). Thus the diagonal copy of \(F\) +has covolume one for these measures. Write +\[ + \widehat\phi(A)=\int_{\R^3}\phi(z)\psi_\infty(\Tr(Az))\,dz +\] +for the real transform. Field Poisson summation in this +normalization \cite[Section~4.2, Lemma~4.2.4]{Tate1967} gives +\begin{equation} + S_{\mathfrak h} + =\sum_{A\in F}\widehat\phi(A) + \widehat\eta_{\mathfrak h,l}(A),\qquad + \widehat\eta_{\mathfrak h,l}(A) + =\prod_{p\in S}\gamma_{p,\mathfrak h_p}(A) + \prod_{p\notin S}\widehat\eta_{p,l}(A). + \label{eq:add-poisson} +\end{equation} +Here \(\gamma_{p,\mathfrak h_p}\) is the shell restriction of \(\gamma_p\), +extended by zero, and hats use the chosen character. Each finite test is +Schwartz--Bruhat, so the formula is also ordinary lattice Poisson summation +with congruence classes. At \(p\notin S\), trace duality is unimodular and +\[ + \widehat\eta_{p,l}(A) + =p^{-3v_p(l)} + \psi_p(\Tr(Az_p^0)) + \mathbf 1_{p^{-v_p(l)}R_{F,p}}(A) + \quad (p\mid l); +\] +for \(p\nmid l\) it is \(\mathbf 1_{R_{F,p}}\). +In particular every such transform is bounded in absolute value by one. +The finite Fourier factors in Equation~\eqref{eq:add-poisson} are supported +on the fractional ideal +\begin{equation} + \Gamma_{l,\mathfrak h}=l^{-1}\mathfrak h^{-1},\qquad P=l^3H, + \qquad + \Nm(\Gamma_{l,\mathfrak h})=P^{-1}. + \label{eq:add-frequency-lattice} +\end{equation} + +The following elementary geometry of this ideal will be used both to count +frequencies and to express the local congruences in global integer +coordinates. + +\begin{lemma}[Lattice bounds for a cubic ideal] +\label{lem:add-ideal-lattice} +Let \(\mathfrak a\) be a fractional ideal of a totally real cubic field of +discriminant \(D_F\), and write \(\Nm(\mathfrak a)=P^{-1}\) with \(P>0\). +Let \(B_Y=\{x\in\R^3:\|x\|\le Y\}\). +In embedding coordinates it has covolume \(\sqrt{D_F}/P\), first Euclidean +minimum \(\gg P^{-1/3}\), and, for \(Y>0\), +\begin{equation} + \#\{A\in\mathfrak a:\|A\|\le Y\} + \ll 1+YP^{1/3}+Y^2P^{2/3}+\frac{Y^3P}{\sqrt{D_F}} . + \label{eq:add-lattice-count} +\end{equation} +There is a \(\Z\)-basis \(b_1,b_2,b_3\) of \(\mathfrak a\) in which every +\(A=\sum n_i b_i\) with \(\|A\|\le Y\) satisfies +\[ + |n_i|\ll YP^{1/3}\quad(1\le i\le3). +\] +All constants here are absolute. +\end{lemma} + +\begin{proof} +The covolume formula follows from the discriminant. For \(0\ne A\in +\mathfrak a\), containment \((A)\subset\mathfrak a\) implies +\(|\Nm_{F/\Q}A|\ge \Nm\mathfrak a=P^{-1}\). The product inequality in +three real coordinates gives +\(|\Nm A|\le(\|A\|/\sqrt3)^3\), proving the first-minimum bound. + +Let \(\lambda_1\le\lambda_2\le\lambda_3\) be the Euclidean successive +minima. A lattice in fixed dimension has a basis of bounded orthogonality +defect, by Mahler's basis theorem \cite[Section~2, Theorem~1]{Mahler1946}. +For such a basis, order its lengths increasingly and write its defect as +\(\omega=\prod_i\|b_i\|/\covol(\mathfrak a)\ll1\). +Cramer's rule and Hadamard's inequality bound the norm of the \(i\)-th +coordinate functional by \(\omega/\|b_i\|\). Since +\(\|b_i\|\ge\lambda_i\), counting its possible integer coordinates gives +\[ + \#(\mathfrak a\cap B_Y)\ll + \prod_{i=1}^3(1+Y/\lambda_i). +\] +Minkowski's second theorem for the Euclidean ball, in the formulation of +\cite[Theorem~4.3]{Widmer2010}, gives +\(\lambda_1\lambda_2\lambda_3\asymp\covol(\mathfrak a)\). +Expanding the last product, using \(\lambda_2\ge\lambda_1\), and inserting +the first-minimum and covolume bounds proves +Equation~\eqref{eq:add-lattice-count}. The same coordinate functional +bound with \(\|b_i\|\ge\lambda_1\) proves the final assertion. +\end{proof} + +\subsection{The nonscalar contribution} + +Scalar frequencies \(A\in\Q\subset F\) in +Equation~\eqref{eq:add-poisson} record only the trace. The main work is to +show that the remaining frequencies contribute an absolute power saving. +We prove that estimate before evaluating the scalar terms. + +\begin{lemma}[Nonscalar contribution] +\label{lem:add-nonscalar} +There are absolute \(\tau,\theta,\epsilon_F>0\), with +\(\epsilon_F>2\tau+3\theta\), such that for \(C_3=371\) and the data of +Proposition~\ref{prop:additive-equidistribution} with these +\(\tau,\theta\), +\begin{equation} + \sum_{\substack{\mathfrak h\\H\le X^{C_3}}} + \left| + \sum_{A\in F\setminus\Q} + \widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A) + \right| + \ll X^{-\epsilon_F}. + \label{eq:add-nonscalar} +\end{equation} +The constants have the same uniformity as in that proposition. +\end{lemma} + +\begin{proof} +We use \(\lambda=1/3\). Lemma~\ref{lem:ext-order-scale}, which gives +\(\Delta_F\ge X^{1/2}/2\), implies +\begin{equation} + \Delta_F\ge X^\lambda + \label{eq:add-order-lower} +\end{equation} +for all sufficiently large \(X\). + +After bounding the real Fourier tail, we split according to the field +discriminant. If \(D_F\) is large, comparing it with the discriminant of +the order generated by a retained nonscalar frequency after clearing its +denominators forces \(H\) large. Otherwise the lower bound for +\(\Delta_F\) makes the index \([R_F:C_F]\) large. For profiles with small +\(G_0\), this forces \(H\) large; inverse support and boundary decay then +bound the possibly signed original sum \(S_{\mathfrak h}\). Poisson +expresses its full nonscalar Fourier sum as \(S_{\mathfrak h}\) minus the +separately estimated scalar series. For profiles with large \(G_0\), the +inverse depths attached to each frequency and its chosen shifts provide +either sufficient pointwise decay or congruences; in the latter range a +further discriminant comparison supplies the required denominator size. + +The positive exponents +\(d,e,e_1,e_2,w,\theta,\tau\) below will be fixed at the end of the proof; +put \(Y=X^w\) and \(s=w+\theta\). + +First the frequencies outside \(\|A\|\le Y\) can be discarded. For every +fixed integer \(r\ge0\), integration by parts in the fixed real coordinates +gives, uniformly over the box family, +\begin{equation} + |\widehat\phi(A)|\ll_r + X^{r\tau}(1+\|A\|)^{-r}. + \label{eq:add-real-decay} +\end{equation} +Indeed each derivative of order \(r\) costs at most a constant times +\(X^{r\tau}\), and all supports are in one compact set. +Lemma~\ref{lem:fourier-shell-estimates} and the bounds for \(p\notin S\) give +\(|\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}\). +The first-minimum bound also gives the rough count +\(\#(\Gamma_{l,\mathfrak h}\cap B_R)\ll(1+RP^{1/3})^3\). +For \(R\ge Y\ge1\) this is \(O(R^3P)\), since \(P=l^3H\ge1\). +Dyadic summation of Equation~\eqref{eq:add-real-decay}, for \(r>3\), yields +\begin{equation} + \sum_{\substack{A\in F\\\|A\|>Y}} + |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)| + \ll_r X^{r\tau}l^3Y^{3-r} + \le X^{r\tau+3\theta+w(3-r)}\,O_r(1). + \label{eq:add-real-tail} +\end{equation} +This also proves absolute convergence in Equation~\eqref{eq:add-poisson}. +The profile count costs only \(X^{o(1)}\). When bounding Fourier terms +directly, we may therefore restrict to the ball and use the uniform bound +\(|\widehat\phi|\ll1\) there. + +Suppose first that \(D_F\ge X^d\). A nonscalar \(A\) generates \(F\), since +the field has degree three. Moreover \(HR_F\subset\mathfrak h\): the +integer \(H=|R_F/\mathfrak h|\) annihilates this finite group. Thus +\(lHA\in R_F\) for \(A\in\Gamma_{l,\mathfrak h}\). The discriminant of the +order \(\Z[lHA]\) is a positive integer multiple of \(D_F\), whereas the +product of the squared differences of its three real conjugates is +\(O((lHY)^6)\). Existence of a nonscalar frequency in the ball therefore +implies +\[ + D_F\ll(lHY)^6,\qquad H\gg X^{d/6-s}. +\] +Apply Equation~\eqref{eq:add-lattice-count} to +\(\Gamma_{l,\mathfrak h}\) and multiply it by \(H^{-1}\). The four resulting +terms are bounded by constants times +\begin{equation} + X^{-d/6+s} + +X^{-d/9+5s/3} + +X^{-d/18+7s/3} + +X^{-d/2+3s}. + \label{eq:add-large-field} +\end{equation} +All have a fixed power saving once \(s0\), uniformly in the arithmetic data; it includes factors +\(8^{|S|}\). It follows that +\begin{equation} + |S_{\mathfrak h}| + \ll X^{-\lambda/6+(3/2)d+(73/2)e+o(1)}. + \label{eq:add-small-index-profile} +\end{equation} +The use of the original sum has gained the boundary factor from the inverse +transform. It remains to check that its scalar portion is small as well, +so that this bounds the nonscalar portion. + +If a rational scalar has denominator \(p^k\) at \(p\), its denominator +ideal in \(R_{F,p}\) is \(p^kR_{F,p}\); its component exponents are +\(e_{\mathfrak q}k\), so its norm is \(p^{3k}\), including at ramified +primes. Thus a scalar in this profile has its \(S\)-part denominator +\(t=H^{1/3}\), or the profile contains no scalar. Its denominator away +from \(S\) divides \(l\), so there are +\(O(1+YlH^{1/3})\) such rationals in the ball. Their absolute contribution +is at most +\begin{equation} + O\bigl(H^{-1}+YlH^{-2/3}\bigr). + \label{eq:add-small-index-scalars} +\end{equation} +By \(H\ge X^{\lambda/3-e}\) this also saves a power for the choices below. +Equation~\eqref{eq:add-poisson} expresses the full dual sum as +\(S_{\mathfrak h}\). Subtracting its scalar frequencies in the ball and +using Equation~\eqref{eq:add-real-tail} for the scalar tail therefore bounds +the full nonscalar sum for these profiles by +Equations~\eqref{eq:add-small-index-profile}--\eqref{eq:add-small-index-scalars} +and the tail. No positivity of an individual inverse shell is needed. + +It remains to treat \(D_FX^e\). For each frequency in +the denominator shell of the profile choose at each \(p\in S\) one of the shifts \(s_p\) of +Lemma~\ref{lem:fourier-shift}, and write +\[ + \alpha_p=A+s_p,\qquad \beta_p=\alpha_p^{-1},\qquad + J=\prod_{p\in S}p^{j_p}, +\] +where \(j_p\) is the least nonnegative integer with +\(p^{j_p}\beta_p\in C_{F,p}\). All sums involving \(J\) below are over +frequencies in this profile. The pointwise local estimate now reads +\[ + |\widehat\eta_{\mathfrak h,l}(A)| + \ll H^{-1}J^{-1/2}. +\] +For frequencies with \(J\ge X^{e_2}\), the rough lattice count immediately +gives +\begin{equation} + \sum_{\substack{\|A\|\le Y\\J\ge X^{e_2}}} + |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)| + \ll \frac{(1+YP^{1/3})^3}{H}\,X^{-e_2/2} + \ll X^{3s-e_2/2}. + \label{eq:add-large-j} +\end{equation} +We used \(H,l,Y\ge1\). This leaves \(J5\), and +differs from this by at most three at \(2,3,5\). +For \(p\notin S\), \(lA\) is integral and \(C_{F,p}=R_{F,p}\) is unramified, +so \(v_p(\operatorname{disc}(A))\ge-6v_p(l)\). +Combining these valuation bounds for the nonzero rational +\(\operatorname{disc}(A)\), with only an absolute factor from \(2,3,5\), +gives +\begin{equation} + Y^6\gg |\operatorname{disc}(A)| + \gg \Delta_F\,J^{-6}H^{-4}l^{-6}. + \label{eq:add-frequency-discriminant} +\end{equation} +The first inequality uses \(\|A\|\le Y\). +Thus for \(J5\), the shift lemma makes +\(x\) a unit and \(\beta_p=i_px^{-1}\in\mathfrak h_p\). The condition +\(p^{j_p}\beta_p\in C_{F,p}\) implies +\begin{equation} + \ell_p(x^{-1})\equiv0\pmod {p^k} + \quad\text{for every }1\le k\le(g_p'-j_p)_+. + \label{eq:add-inverse-congruence} +\end{equation} +Inversion permutes the units in the finite algebra +\(R_{F,p}/p^kR_{F,p}\). A primitive linear form has \(p^{2k}\) zeros among +the \(p^{3k}\) additive residues. Thus the unit condition and +Equation~\eqref{eq:add-inverse-congruence} retain at most the fraction +\(p^{-k}\) of all residues modulo \(p^k\). Nilpotents in the finite algebra +do not affect this argument. + +For \(p=2,3,5\), the shift lemma gives \(x\in R_{F,p}\) and +\(x^{-1}\in p^{-1}R_{F,p}\). Put \(y=p x^{-1}\). Then +\(p^{j_p}i_py=p(p^{j_p}\beta_p)\) still lies in +\(\mathfrak h_p\cap C_{F,p}\), and the congruence in +Equation~\eqref{eq:add-inverse-congruence} holds for \(y\) in place of +\(x^{-1}\). On the permitted locus the inverse identities +\[ + y(x)-y(x')=p x^{-1}(x'-x)(x')^{-1},\qquad + x-x'=p^{-1}x\bigl(y(x')-y(x)\bigr)x' +\] +show respectively that \(x\bmod p^{k+1}\) determines \(y\bmod p^k\), and +that a fixed residue of \(y\bmod p^k\) confines \(x\) to one residue modulo +\(p^{k-1}\). The permitted locus has \(v_{\mathfrak q}(x)=0\) on the pole +components and \(0\le v_{\mathfrak q}(x)\le e_{\mathfrak q}\) on the other +components. These conditions are determined modulo \(p^2\), since a +change in \(p^2R_{F,p}\) has component valuation at least \(2e_{\mathfrak q}\). +At modulus \(p^{k+1}\), each of the at most \(p^{2k}\) residues in +the linear kernel therefore allows at most \(p^6\) residues of \(x\). +The retained fraction is at most \(p^{3-k}\), hence \(O(p^{-k})\) with an +absolute constant at these three primes. This proves the required +congruence fraction with only one extra power of a small prime in its +modulus. + +Write \(a_p=(g_p'-j_p)_+\). The available congruence powers satisfy +\begin{equation} + \prod_{p\in S}p^{a_p} + \ge G_0^{1/3}/J + > X^{e/3-e_2}. + \label{eq:add-available-congruences} +\end{equation} +We will choose \(e/3-e_2>2e_1\). If every prime with \(a_p>0\) is at most +\(X^{e_1}\), multiply available factors \(p\), one at a time, until their +product \(M_1\) first reaches \(X^{e_1/2}\). Equation +\(\eqref{eq:add-available-congruences}\) ensures this is possible, and +\[ + X^{e_1/2}\le M_1\le X^{3e_1/2}X^{e_1}\); for large \(X\) +it exceeds \(5\). Use its congruence modulo \(p\). Set +\(B=R_{F,p}/pR_{F,p}\), a three-dimensional commutative \(\mathbf F_p\)-algebra, +possibly with nilpotents, and reduce \(\ell_p\) modulo \(p\). Choose an +\(\mathbf F_p\)-basis of \(B\) and let \(x\) be the universal element of +\(B\otimes_{\mathbf F_p}\mathbf F_p[X_1,X_2,X_3]\) in this basis. +Let \(M_x\) be its multiplication matrix and put +\(\mathcal N_B(x)=\det M_x\). This is a nonzero polynomial of degree three, +since \(\mathcal N_B(1)=1\). On the unit open set \(\mathcal N_B\ne0\), +\[ + \ell_p(x^{-1})=\frac{P_\ell(x)}{\mathcal N_B(x)},\qquad + P_\ell(x)=\ell_p\bigl(\operatorname{adj}(M_x)1_B\bigr), +\] +where \(P_\ell\) has degree at most two. This numerator is nonzero. +Indeed the affine space parametrizing elements of the vector space \(B\) +has coordinate ring \(\mathbf F_p[X_1,X_2,X_3]\), which is a domain even +when multiplication in \(B\) has nilpotents. The unit open set is dense, +and inversion is an automorphism of that open set. The nonzero linear +polynomial \(\ell_p\) stays nonzero on the open set and after composition +with inversion. The displayed formula, or injectivity of localization at +\(\mathcal N_B\), then forces \(P_\ell\ne0\). +The affine \(\operatorname{GL}_3(\mathbf F_p)\) change from global lattice +coordinates preserves nonvanishing. + +Let \(N_L=2\lfloor L\rfloor+1\asymp L\) be the number of possible integers in +one coordinate of the cube. A nonzero polynomial of bounded degree has +\(O(N_L^2)\) zeros on a product of three residue intervals of length \(N_L2e_1,\quad 2e_1<\lambda/24,\quad + s+e_2<\lambda/12. +\] +The saving in Equation~\eqref{eq:add-small-index-profile} is at least +\[ + \lambda/6-(3/2)d-(73/2)e + \ge (1/6-0.038)\lambda_0, +\] +and Equation~\eqref{eq:add-small-index-scalars} has a still larger margin +than \(\epsilon_F\), since \(e\le\lambda/1000\). +All four exponents in Equation~\eqref{eq:add-large-field} also have margins +at least \(29d/900>\epsilon_F\), by \(s\epsilon_F; +\] +the congruence margins \(e_1/2-3s\), \(e_1-3s\), and +\(\lambda/24-3s\) are larger. With \(r=100\), +the exponent in Equation~\eqref{eq:add-real-tail} is +\[ + r\tau+3\theta+w(3-r)=-\frac{9597}{10000}e_2. +\] +All \(X^{o(1)}\) costs from profiles, shifts, and exponent lists can thus be +absorbed while retaining \(\epsilon_F=e_2/4\). This proves +Equation~\eqref{eq:add-nonscalar}. Finally +\(2\tau+3\theta=e_2/2000<\epsilon_F\), as asserted. +\end{proof} + +\subsection{The scalar term and completion} + +\begin{proof}[Proof of Proposition~\ref{prop:additive-equidistribution}] +Use the constants in Equation~\eqref{eq:add-parameters}. +Equations~\eqref{eq:add-original-profiles} and \(\eqref{eq:add-poisson}\), +together with Lemma~\ref{lem:add-nonscalar}, leave the scalar frequencies +with absolute error \(O(X^{-\epsilon_F})\). We first remove the profile +cutoff from those frequencies. + +For \(a\in\Q\), let \(t=\prod_{p\in S}p^{\max(0,-v_p(a))}\) be its +\(S\)-part denominator, with \(t=1\) for \(a=0\). As noted above, its +unique profile has \(H=t^3\). +Lemma~\ref{lem:fourier-scalar-density} gives the exact product +\(\prod_{p\in S}\gamma_p(a)=t^{-3}\), including at \(2,3,5\). +The other denominator divides \(l\), so all such scalars with a given \(t\) +lie in \((tl)^{-1}\Z\). The change of real variables +\((z_i,z_j,z_k)\mapsto(z_i,z_j,\Tr z)\) has absolute Jacobian one. +Writing \(\mathbf 1=(1,1,1)\), it gives the exact identity +\[ + \widehat\phi(a\mathbf 1)=A_\phi\,\widehat\chi(a). +\] +For any fixed \(r>1\), rapid decay of the fixed function \(\chi\) implies +\(\sum_{n\in\Z}|\widehat\chi(n/(tl))|\ll_r tl\). +The scalars omitted by \(H\le X^{C_3}\) therefore contribute at most +\begin{equation} + \sum_{t>X^{C_3/3}}t^{-3} + \sum_{n\in\Z}|\widehat\phi((n/(tl))\mathbf 1)| + \ll A_\phi\,l\,X^{-C_3/3}. + \label{eq:add-scalar-tail} +\end{equation} +We have enlarged the set of \(t\)'s to all positive integers, which is an +upper bound. With \(C_3=371\), this is \(O(X^{-\epsilon_F})\), since +\(A_\phi\ll1\), \(l\le X^\theta\), and \(371/3-\theta>\epsilon_F\). +The same calculation without the cutoff proves +absolute convergence of the unrestricted scalar series. + +That series is one-dimensional Poisson summation for the trace marginals. +At infinity the marginal is \(A_\phi\chi(t)\). At \(p\in S\) it is +\(d_p(t)\mathbf 1_{\Z_p}(t)\). At \(p\mid l\), the onto integral trace map +and two kernel coordinates show that the indicator +\(z_p^0+p^{v_p(l)}R_{F,p}\) has trace density +\[ + p^{-2v_p(l)}\mathbf 1_{m+p^{v_p(l)}\Z_p}(t) +\] +with respect to Haar measure of mass one on \(\Z_p\). At every other +finite prime the marginal is \(\mathbf 1_{\Z_p}\). +Their Fourier transforms at a rational \(a\) are exactly the finite factors +in Equation~\eqref{eq:add-poisson} evaluated at the scalar \(a\), because +\(\Tr(az)=a\Tr z\). + +For completeness, this one-dimensional application of Poisson does not +require a smoothness assumption on \(d_p\). Its Fourier coefficients on +\(\Q_p/\Z_p\) of denominator \(p^k\) are \(p^{-3k}\), and there are +\(p^k-p^{k-1}\) such characters. Their absolute sum is finite. +Truncating each expansion gives a compactly supported locally constant +density on \(\Z_p\), to which Poisson applies. The densities converge +uniformly by Lemma~\ref{lem:fourier-scalar-density}, and the preceding +absolute scalar bound permits passage to the limit on the Fourier side. +On the original side, the finite marginals force the rational trace to be +an integer. Only \(m\) meets the real cutoff. Hence the unrestricted +scalar contribution is precisely +\[ + A_\phi\,l^{-2}\prod_{p\in S}d_p(m). +\] + +Lemma~\ref{lem:fourier-scalar-density} bounds the product of \(d_p(m)\) +above and below by positive absolute constants. Also +\(A_\phi\gg X^{-2\tau}\) by Equation~\eqref{eq:add-box}, and +\(l^{-2}\ge X^{-2\theta}\). Dividing the combined absolute error +\(O(X^{-\epsilon_F})\) by the main term therefore costs at most +\(O(X^{2\tau+2\theta})\). Since +\(\epsilon_F>2\tau+3\theta\), the relative error is \(O(X^{-\theta})\). +Every estimate used fixed real derivative and support bounds and constants +independent of the fields, orders, residues, and box locations and scales. +This proves Equation~\eqref{eq:add-equidistribution} with its stated +uniformity. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/conclusion.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/conclusion.tex new file mode 100644 index 0000000000000000000000000000000000000000..c30548f42dcc877b707dd46e9219da125c4efc2a --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/conclusion.tex @@ -0,0 +1,287 @@ +\section{Exclusion of the obstruction}\label{sec:conclusion} + +We apply Theorem~\ref{thm:exterior-count} to the exterior planes in +Proposition~\ref{prop:rig-obstruction}. Four small exterior coordinates +give two small signed products, whose range in the trace plane has area +of order $r^4$. The integral along a diagonal orbit contributes +logarithmic factors; we will show that they are summable at this scale. + +\subsection{The exterior vector integral} + +Fix a partition $b_1=ij|kl$ of $\{1,2,3,4\}$, and denote the other two +partitions by $b_2,b_3$. For $m\in\Z\setminus\{0\}$, $L>1$, and +$00$ in absolute value, the +logarithm of the chosen entry lies in an interval of length +$\log_+(h^2/|z_b|)$. Therefore, for a raw vector with the required +trace after the sign change, its integral over $(u_1,u_2,u_3)$ is +$\frac12\mathcal L_{L,r}(z)$. + +The $U^+$-orbit sum of the weights, including the sum over $I$, is exactly +$M(z)$ by Equation~\eqref{eq:ext-multiplicity}. Equation +\eqref{eq:end-signed-products} allows raw trace $m$ for the eight positive +determinant signs and raw trace $-m$ for the other eight. Multiplying +eight by the Jacobian factor $1/2$ gives the factor $4$ in +Equation~\eqref{eq:end-unfold}. The unit $-1$, although it acts trivially +on exterior vectors, has already been accounted for in the index $d_U$. +\end{proof} + +\begin{proposition}[Small exterior boxes]\label{prop:end-tube} +For every fixed $m\in\Z\setminus\{0\}$ and $L>1$, there are constants +$r_0>0$ and $C_{m,L}>0$ with the following property. For every sequence +of the fields, orders, and embedding orderings in Theorem~\ref{thm:main} +with $X_i\to\infty$, and every fixed $00$, Lemma~\ref{lem:ext-support} gives +$|z_b|\gg_{\Omega_t}X^{-84}$ for each $b$. Hence the last two logarithms +in Equation~\eqref{eq:end-logarithms} are $O(\log X)$, uniformly for +$04X^{-\tau}$, that width exceeds $2X^{-\tau}$. Hence every +dyadic rectangle meeting the remaining region satisfies the minimum +side lengths of Theorem~\ref{thm:exterior-count}; any boundary rectangle +may be included in full. Its area is $O(r^4 2^{-a-b})$, while the last +two factors of Equation~\eqref{eq:end-logarithms} are at most +$(a+1)\log2$ and $(b+1)\log2$. Theorem~\ref{thm:exterior-count} gives +\[ + \frac1{\sqrt D\,\kappa} + \sum_{z\text{ in that rectangle}} M(z) + \ll_{m,L} r^4 2^{-a-b}. +\] +The sum over the four choices of signs and over all $a,b$ is therefore +at most +\[ + C_{m,L}r^4 + \sum_{a,b\ge0}(a+1)(b+1)2^{-a-b} + \ll_{m,L}r^4. +\] +All constants are independent of $r$: the rectangles $\Omega_t$ and the +uniform rectangle constant were fixed before choosing $r$. The two +strips tend to zero for each fixed $r$, and $4/d_U\le2$ in +Equation~\eqref{eq:end-unfold}. These observations prove +Equation~\eqref{eq:end-tube} for both traces and all three partitions. +\end{proof} + +\subsection{Identification of every probability limit} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +For the sequence in that theorem, +$X_i=\sqrt{|\Disc(\OO_i)|}$ tends to infinity. By +Proposition~\ref{prop:ord-tightness}, the packet probabilities are tight, +and every subsequence has a further weakly convergent subsequence with +an $A_4$-invariant probability limit $\mu$. +Proposition~\ref{prop:ord-balls} gives the local ball bound required by +Corollary~\ref{cor:rig-positive-components}. Hence almost every component in +the $A_4$-ergodic decomposition of $\mu$ has positive entropy for a +diagonal flow. Proposition~\ref{prop:rig-obstruction} reduces the +identification of these components to the exclusion of its three +exterior-plane events. + +Fix a partition $b_1$, a nonzero integer $m$, and $L>1$. The event +\[ + E_{m,L,r}=\{\Lambda\in X_4:N_{m,L,r}(\Lambda)\ge1\} +\] +is open by the observation following Equation~\eqref{eq:end-vector-box}. + +The open-set inequality for weak convergence and +Proposition~\ref{prop:end-tube} now give, along the convergent subsequence, +\[ + \mu(E_{m,L,r}) + \le\liminf_i\mu_{\OO_i,\sigma_i}(E_{m,L,r}) + \le\limsup_i\int N_{m,L,r}\,d\mu_{\OO_i,\sigma_i} + \le C_{m,L}r^4 +\] +for every sufficiently small fixed $r>0$. In particular, +\[ + \mu\left(\bigcap_{n\ge2}E_{m,L,1/n}\right) + \le \inf_{\substack{n\ge2\\1/n0$, one +compact set carrying at least $1-\epsilon$ of every sufficiently late +packet probability. This is the required no-escape assertion. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/exterior.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/exterior.tex new file mode 100644 index 0000000000000000000000000000000000000000..9de8933716a8f7a7a0abea76cf225c16a55ab048 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/exterior.tex @@ -0,0 +1,1144 @@ +\section{The cubic resolvent and exterior multiplicities} +\label{sec:exterior} + +The obstruction in Section~\ref{sec:rigidity} is detected by an exterior +vector concentrated in one pair of opposite coordinates. We shall count +such vectors through their three signed opposite products. After packet +normalization these products form a triple in a cubic field, and +concentration near one exterior plane confines +two of them to a small rectangle. Our goal is a uniform count in that +rectangle. We first define the count and reduce its multiplicities to local +densities. We then construct the integral quadratic structure that will +control those densities in the next two sections. + +\subsection{The three products and the counting problem} + +Let \(\widetilde K\subset\R\) be the normal closure of \(K\), and let +\(G=\Gal(\widetilde K/\Q)\) act on the four ordered embeddings of \(K\). +This permutation action is primitive: a nontrivial block would give a field +strictly between \(\Q\) and \(K\). A transitive subgroup of \(S_4\) has +order \(4,8,12\), or \(24\). A subgroup of order \(4\) or \(8\) is contained +in a Sylow \(2\)-subgroup, which preserves a partition into two pairs. +It follows that +\[ +G=\mathfrak A_4\quad\hbox{or}\quad G=S_4. +\] +Both groups act transitively on the six two-element subsets of +\(\{1,2,3,4\}\) and on the set of pair partitions +\[ +\mathcal B=\{12|34,\ 13|24,\ 14|23\}. +\] +Define +\begin{equation} + F=\left\{(z_b)_{b\in\mathcal B}\in\widetilde K^{\mathcal B}: + g(z_b)=z_{gb}\ \text{for all }g\in G,\ b\in\mathcal B\right\}. + \label{eq:ext-resolvent} +\end{equation} +Evaluation at any partition identifies this algebra with the fixed field +of its stabilizer. Thus \(F\) is a totally real cubic field, and its three +real embeddings are the coordinates in Equation~\eqref{eq:ext-resolvent}. +In particular, \(\Tr_{F/\Q}z=\sum_b z_b\). This is the classical cubic +resolvent from the action on pair partitions; for its relation to cubic +resolvent rings, see \cite[Section~2.3]{Bhargava2004III}. We write +\(R_F=\OO_F\) and \(D_F=|\Disc(F)|\). + +Put +\[ + V=\bigwedge\nolimits^2_{\Q}K,\qquad + L_V=\bigwedge\nolimits^2_{\Z}\OO. +\] +For \(w=\sum_r x_r\wedge y_r\in V\), its embedding coordinates are +\[ + w_{ij}=\sum_r + \bigl(\sigma_i(x_r)\sigma_j(y_r)-\sigma_j(x_r)\sigma_i(y_r)\bigr) + \quad(i0\quad\Longrightarrow\quad + z\in F^\times,\qquad \Tr_{F/\Q}z\in\Z. + \label{eq:ext-trace-support} +\end{equation} + +For \(m\in\Z\setminus\{0\}\), put +\[ + \mathcal H_m=\{(y_b)_{b\in\mathcal B}\in\R^{\mathcal B}: + \textstyle\sum_b y_b=m\}. +\] +A rectangle in \(\mathcal H_m\) means the inverse image of a product of +two intervals under any two distinct partition coordinates. Its area +is the product of those interval lengths. The following estimate is +the arithmetic input needed to exclude the obstruction. + +\begin{theorem}[Planar exterior count] +\label{thm:exterior-count} +There is an absolute \(\tau>0\) with the following property. Fix +\(m\in\Z\setminus\{0\}\) and a bounded set +\(\Omega\subset\mathcal H_m\). For every primitive totally real quartic +field \(K\), every order \(\OO\subset K\), every ordering of its real +embeddings, and every chosen system of fractional \(R\)-ideal +representatives for the ordinary classes, let \(M\) be defined by +Equation~\eqref{eq:ext-multiplicity}. +For all sufficiently large \(X\), every rectangle +\(\mathcal R\subset\Omega\) whose two side lengths are at least +\(X^{-\tau}\) satisfies +\begin{equation} + \sum_{\substack{z\in F,\ \Tr z=m\\z\in\mathcal R}}M(z) + \ll_{m,\Omega}\kappa\sqrt D\,\operatorname{area}(\mathcal R). + \label{eq:ext-planar-count} +\end{equation} +The implied constant and the threshold for \(X\) are uniform in the +field, the order, the embedding ordering, the ideal representative system, +the position of the rectangle, and its side lengths and aspect ratio. +\end{theorem} + +Here and below we identify \(z\in F\) with its real triple. To see the +scale of this statement, suppose \(Q(\widehat w)=m\) and the four +coordinates outside \(12|34\) have absolute value less than \(r\). +The signed products of \(\widehat w\) then have the form +\((m-y_2-y_3,y_2,y_3)\) with \(|y_2|,|y_3|0\). +The fraction is independent of the representative of the coset. The sum of +these fractions is independent of the reference. For an empty fiber +we declare this sum to be zero. + +A scalar \(c\in\Z_p^\times\) preserves \(L_{V,p}\). The pushforward of +the product Haar probability on \(R_p^\times\times\Z_p^\times\) is +Haar probability on \(\mathscr U_p\). Hence the representative constructed +from \((I,w)\) occupies a coset \(\mathfrak c\) for which +\(P_p(I,w)=\alpha_p(\mathfrak c;z(I,w))\). + +We now measure the sum over all local cosets. Fix an additive character +\(\psi_p\) of \(\Q_p\) with conductor \(\Z_p\), and let \(dz\) be the +self-dual Haar measure on \(F_p\) for the trace pairing. Thus +\begin{equation} + \vol_{dz}(R_{F,p})=|D_F|_p^{1/2}. + \label{eq:ext-self-dual-measure} +\end{equation} + +For every \(p\), consider the law of the random variable +\begin{equation} + u\Psi(x),\qquad x\ \text{uniform additive in }L_{V,p},\qquad + u\ \text{uniform in }\Z_p^\times, + \label{eq:ext-density-law} +\end{equation} +with the two choices independent. The next lemma proves that this law +has a density with respect to \(dz\); we denote its locally constant +version on \(F_p^\times\) by \(f_p^*(z)\). Only the densities for +\(p\in S\) will enter the multiplicity bound. + +\begin{lemma}[Local density and disintegration] +\label{lem:ext-density} +The law in Equation~\eqref{eq:ext-density-law} has a density, with a +locally constant version on \(F_p^\times\). For every \(z\in F_p^\times\), +only finitely many cosets are accepted, and their fractions satisfy the +exact identity +\begin{equation} + \sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p} + \alpha_p(\mathfrak c;z) + = + |D|_p^{-1/2} + \frac{1-p^{-1}} + {\displaystyle\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R} + (1-\Nm(\mathfrak p)^{-1})} + f_p^*(z). + \label{eq:ext-density-disintegration} +\end{equation} +The sum and the density are both zero if \(\mathscr F_p(z)\) is empty. +The identity holds at every prime, including primes dividing \(q\) and +the small ramified primes. +\end{lemma} + +\begin{proof} +Let \(\Phi_p(x,u)=u\Psi(x)\). If \(\Phi_p(x,u)\in F_p^\times\), all six +split coordinates of \(x\) are nonzero. After splitting, the derivative +in \(x\) is the derivative of three products on three disjoint +two-dimensional planes. It has rank three, and therefore has rank +three over \(\Q_p\). The map is a submersion there. The remaining +input set is a finite union of proper algebraic zero loci and has +additive measure zero. Local change of variables gives an absolutely +continuous pushforward on the submersion locus, hence a density for +the full law. To obtain a locally constant version at a fixed +\(z\in F_p^\times\), cover its inverse image in the compact set +\(L_{V,p}\times\Z_p^\times\) by finitely many submersion charts, +refined to disjoint compact open pieces. The compact complement has +image avoiding a sufficiently small neighborhood of \(z\). +On that neighborhood each chart contributes a locally constant +density. This also treats an empty inverse image. + +Give \(V_p\) its additive measure for which \(L_{V,p}\) has mass one, +before restricting to that lattice. We first compute the fiber form for +the product input measure after disintegration against \(dz\), and then +the mass of \(\mathscr U_p\) for that form. In a \(\Q_p\) submersion chart, +the relative absolute four-form is obtained by evaluating the input top +form on four fiber basis vectors and three lifts of a target basis, then +dividing by the \(dz\)-form on that target basis. We may compute this +identity after extension to a splitting field, still evaluating on \(\Q_p\) tangent +vectors and using the fixed extension of \(|p|_p=p^{-1}\). We use the +same interpretation for the scalar quotient below. + +Over a splitting field, the exterior embedding matrix has determinant +\(\delta_{\OO}^{\,3}\). Consequently its measure form in the six +embedding coordinates is +\[ + |\delta_{\OO}|_p^{-3}\left|\bigwedge_{i0. + \end{cases} +\] +The first line follows by projecting onto the constrained factors +while a factor with \(r_{\mathfrak q}=0\) supplies the trace. +The second uses \(p\nmid m\). It also covers the case that \(F_p\) +is a field. + +Expanding \(b_{\mathfrak q}(j)=\sum_{r=0}^j\epsilon_{\mathfrak q}^r\) +and summing the absolutely convergent geometric series gives +\begin{equation} + \E_p b_p= + \frac{1-(\prod_{\mathfrak q}\epsilon_{\mathfrak q})t^3} + {\prod_{\mathfrak q}(1-\epsilon_{\mathfrak q}t^{f_{\mathfrak q}})}. + \label{eq:ext-geometric-mean} +\end{equation} +The five Frobenius types give the following data. A pair +\((f,\epsilon)\) in the second column is attached to one prime of \(F\). +\[ +\begin{array}{c|c|c|c} + \text{cycle type on four labels} + & (f_{\mathfrak q},\epsilon_{\mathfrak q}) + & \prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}}) + & a_K(p)-1\\ \hline + 1^4 &(1,+),(1,+),(1,+)&(1-t)^4&3\\ + 2\,1^2 &(1,+),(2,+)&(1-t)^2(1-t^2)&1\\ + 2^2 &(1,+),(1,-),(1,-)&(1-t^2)^2&-1\\ + 3\,1 &(3,+)&(1-t)(1-t^3)&0\\ + 4 &(1,-),(2,-)&1-t^4&-1 +\end{array} +\] +In every row \(\prod_{\mathfrak q}\epsilon_{\mathfrak q}=1\) and +\[ + \prod_{\mathfrak q}(1-\epsilon_{\mathfrak q}t^{f_{\mathfrak q}}) + =\frac{\prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})}{1-t}. +\] +Substitution in Equation~\eqref{eq:ext-geometric-mean} proves the exact +mean. Its linear term is the number of degree-one primes of \(K\) +minus one, which is \(a_K(p)-1\); the five fixed rows give the uniform +error term. + +If \(F_p\) is a field, \(n_p\ge1\) would make its trace divisible by +\(p\), and is impossible. Otherwise the uniform projection above +gives +\(\P_p(v_{\mathfrak q}(z)\ge k)=p^{-f_{\mathfrak q}k}\). +Taking the union over at most three factors proves +Equation~\eqref{eq:ext-unramified-tail}. Finally, the trace map is +onto, so the affine trace lattice has two integral coordinates. +Projection to each unramified factor modulo \(p^{k+1}\) is determined +by these coordinates modulo \(p^{k+1}\). The valuations truncated +at \(k+1\), and hence the asserted threshold and finite-depth +functions, have the stated congruence dependence. This remains true +on the component-vanishing set with the convention above. +\end{proof} + +\begin{proposition}[Multiplicities from local densities] +\label{prop:ext-multiplicity-majorant} +For \(z\in F\) with \(\Tr z=m\ne0\), a positive value of \(M(z)\) +requires \(z\) to be integral outside \(S\). For such integral \(z\), +\begin{equation} + M(z)\le32 + \prod_{p\in S} + \left( + |D|_p^{-1/2} + \frac{1-p^{-1}} + {\displaystyle\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R} + (1-\Nm(\mathfrak p)^{-1})} + f_p^*(z)\right) + \prod_{p\notin S}b_p(z). + \label{eq:ext-local-majorant} +\end{equation} +All products in this assertion are finite at every contributing \(z\). +\end{proposition} + +\begin{proof} +The field \(F\) and the condition \(\Tr z=m\ne0\) imply +\(z\in F^\times\). At every prime the set of accepted cosets is +finite by Lemma~\ref{lem:ext-density}. At \(p\notin S\), +Lemma~\ref{lem:ext-unramified-data} proves the integrality requirement +and bounds the local sum by \(b_p(z)\). The set +\[ + E_z=S\cup\{p:z\notin R_{F,p}^\times\} +\] +is finite, since a nonzero global \(z\) is a unit at all but finitely +many primes. At \(p\notin E_z\), \(b_p=1\), and each coset fraction is +\(0\) or \(1\), so there is at most one accepted coset, of fraction +\(1\). If some prime has no accepted coset, then \(M(z)=0\) and the +claimed inequality is immediate. Otherwise the accepted coset is +unique with fraction \(1\) outside \(E_z\), and each prime inside the +finite set \(E_z\) has finitely many accepted cosets. Thus there are +only finitely many assignments of accepted cosets, and all local sums +outside \(E_z\) equal \(1\). + +Choose local references. Equation~\eqref{eq:ext-unit-membership} +identifies the weight of a global pair at each prime with its coset +fraction \(\alpha_p\). By Lemma~\ref{lem:ext-global-cosets}, at most +\(32\) global pairs occupy any assignment of accepted cosets. +Summing over the finite set of assignments, including ones not +realized globally, bounds \(M(z)\) by \(32\) times the product of the +local sums of \(\alpha_p\). Use +Equation~\eqref{eq:ext-density-disintegration} at \(p\in S\) and +Equation~\eqref{eq:ext-unramified-cosets} outside \(S\). +This proves Equation~\eqref{eq:ext-local-majorant} and finiteness. +\end{proof} + +The unramified mean in Equation~\eqref{eq:ext-unramified-mean} has the +Euler factor needed for the residue \(\kappa\). The remaining problem +in Theorem~\ref{thm:exterior-count} is to average the densities +\(f_p^*\) for \(p\in S\) before inserting these unramified factors. +For a frequency \(\alpha\in F_p\), testing the law \(u\Psi(x)\) against +the character \(z\mapsto\psi_p(\Tr_{F_p/\Q_p}(\alpha z))\) produces the +quadratic phase \(u\Tr_{F_p/\Q_p}(\alpha\Psi(x))\). +We express these forms through a single unimodular bilinear form and an +action of \(F\). The elements of \(F\) whose operators preserve \(L_V\) +will then form the order that controls the local Fourier analysis. + +\subsection{The integral quadratic structure and its order} + +Let \(B_V\) be the polarization of \(Q_V\) without division by two: +\[ + B_V(x,y)=Q_V(x+y)-Q_V(x)-Q_V(y). +\] +Over a splitting field, write \(V_b\) for the plane spanned by the +two coordinate axes belonging to the partition \(b\), the same +coordinate planes used over \(\R\) in Section~\ref{sec:rigidity}. +For an equivariant triple \(\alpha=(\alpha_b)\in F\), let \(\alpha_b\) +scale both coordinate axes of \(V_b\). + +\begin{lemma}[Self-adjoint quadratic structure] +\label{lem:ext-quadratic-structure} +The plane-scalar action descends to a rational action +\(\rho:F\to\End_{\Q}(V)\) for which \(V\) has rank two over \(F\). +The form \(Q_V\) in an exterior \(\Z\)-basis of \(L_V\) is the three +split pairs in Equation~\eqref{eq:ext-integral-form}; in particular +\(B_V\) is unimodular on \(L_V\). The action \(\rho\) is self-adjoint +for \(B_V\), and for \(\alpha\in F\), +\begin{align} + Q_\alpha(x):=\Tr_{F/\Q}(\alpha\Psi(x)) + &=\tfrac12 B_V(\rho(\alpha)x,x), \label{eq:ext-polar-operator}\\ + B_\alpha(x,y)&=B_V(\rho(\alpha)x,y), \notag\\ + \det_{\Q}\rho(\alpha)&=\Nm_{F/\Q}(\alpha)^2,\qquad + \Psi(\rho(\alpha)x)=\alpha^2\Psi(x). \label{eq:ext-rank-two} +\end{align} +Here \(B_\alpha\) is the polarization of \(Q_\alpha\). +The same assertions hold after base change to every \(\Q_p\). +\end{lemma} + +\begin{proof} +Galois permutes the planes \(V_b\), including possible alternating +signs on their coordinate axes, and permutes the scalars \(\alpha_b\) +in the same way. The plane-scalar action therefore commutes with +Galois and descends. After splitting there are three planes of +dimension two, proving the rank assertion. Equation~\eqref{eq:ext-integral-form} +shows that the matrix of \(B_V\) pairs the three opposite pairs by +unit off-diagonal entries, with the displayed signs, so it is +unimodular. On each plane, multiplying both coordinates by +\(\alpha_b\) is self-adjoint for its split pairing. The contribution +of that plane to \(\frac12B_V(\rho(\alpha)x,x)\) is +\(\alpha_b\Psi_b(x)\), which proves +Equation~\eqref{eq:ext-polar-operator}. Polarization gives the next +identity. The determinant and the last quadratic identity follow +by multiplying, respectively, the two equal eigenvalues and the +two coordinate scalars on each plane. All these are rational +identities, hence persist under base change. +\end{proof} + +Define the stabilizer order and its discriminant by +\begin{equation} + C_F=\{\alpha\in F:\rho(\alpha)L_V\subset L_V\},\qquad + \Delta_F=\Disc(C_F). + \label{eq:ext-stabilizer-order} +\end{equation} +It is a ring containing \(1\) and a full \(\Z\)-lattice: this follows +from the rationality of \(\rho\) by clearing denominators on a basis +of \(F\). It is contained in \(R_F\), since an \(\alpha\) whose +matrix on \(L_V\) is integral is a root of that integral monic +characteristic polynomial. Thus it is an order. +The stabilizer condition is a finite list of matrix-entry integrality +conditions in a lattice basis, so it commutes with completion: +\begin{equation} + C_{F,p}=C_F\otimes_\Z\Z_p + =\{\alpha\in F_p:\rho(\alpha)L_{V,p}\subset L_{V,p}\}. + \label{eq:ext-local-stabilizer} +\end{equation} +The following comparison ensures that its discriminant tends to infinity +with the discriminant of \(\OO\), even when the field \(K\) stays fixed. + +\begin{lemma}[Scale of the stabilizer order] +\label{lem:ext-order-scale} +There is an absolute constant \(C\) such that +\begin{equation} + \frac12 X^{1/2}\le + \Delta_F=D_F[R_F:C_F]^2\le C X^{28}. + \label{eq:ext-order-scale} +\end{equation} +More precisely, with \(d=\Disc(\OO)=X^2\), +\[ + d^2R_F\subset C_F\subset R_F,\qquad D_F\ll D^2. +\] +In particular \(C_{F,p}=R_{F,p}\) for \(p\nmid Dq\). +\end{lemma} + +\begin{proof} +Let \(E=\bigwedge^2 A_{\OO}\) be the exterior embedding matrix. +Its entries are integral in every local splitting ring, and +\(\det E=\delta_{\OO}^{\,3}\). Since \(d=\delta_{\OO}^{\,2}\), +\[ + d^2 E^{-1}=\delta_{\OO}\operatorname{adj}(E) +\] +also has integral entries. For \(\alpha\in R_{F,p}\), its +plane-scalar matrix over the splitting ring is integral. +Consequently +\(d^2\rho(\alpha)=d^2E^{-1}\diag(\alpha_b,\alpha_b)E\) +is integral over the splitting ring. Its entries +are in \(\Q_p\), so they lie in \(\Z_p\). This proves +\(d^2R_F\subset C_F\) and \([R_F:C_F]\le d^6\). + +If \(p\nmid D\), the etale algebra \(R_p\) splits over an unramified +extension of \(\Z_p\); its splitting field and cubic resolvent are +unramified. Hence every prime dividing \(D_F\) divides \(D\). +For \(p>3\), a cubic algebra is tamely ramified and its discriminant +exponent is at most \(2\). At \(2\) and \(3\), the finitely many +local extensions of degree at most three have bounded discriminant +exponents \cite[Proposition~7.64 and Remark~7.65]{MilneANT}. +It follows that \(D_F\ll D^2\). The discriminant-index formula now +gives +\[ + \Delta_F=D_F[R_F:C_F]^2\ll D^2d^{12}\le X^{28}. +\] +The containment \(d^2R_F\subset C_F\) also proves equality of the +local orders away from \(Dq\). + +For the reverse inequality, we first explain why a projection +denominator controls \(\Disc(\OO)\). Write \(\operatorname{tr}\) for +matrix trace, put \(\mathscr E=\End_\Z(\OO)\), and let +\(\mathscr W_K\subset\End_\Q(K)\) be the rational subspace of +multiplication matrices. Its intersection +\(\mathscr L=\mathscr W_K\cap\mathscr E\) is saturated. It is +multiplication by \(\OO\): if \(\alpha\OO\subset\OO\), then +\(\alpha=\alpha\cdot1\in\OO\), and the converse follows because +\(\OO\) is a ring. The pairing +\((T,S)\mapsto\operatorname{tr}(TS)\) is unimodular on +\(\mathscr E\), since its matrix units are dual. Its restriction +to \(\mathscr W_K\) is the nondegenerate field trace pairing. +Let \(\mathcal E_K\) be the orthogonal projection onto +\(\mathscr W_K\). + +For the restricted pairing one has +\begin{equation} + \mathcal E_K(\mathscr E)=\mathscr L^\vee. + \label{eq:ext-projected-dual} +\end{equation} +One inclusion follows by pairing a projected integral matrix with +\(\mathscr L\). Conversely, an integer functional on the primitive +sublattice \(\mathscr L\) extends to one on \(\mathscr E\). +Unimodularity represents the extension by an element of +\(\mathscr E\); its orthogonal projection represents the original +functional on \(\mathscr L\). Nondegeneracy identifies that +projection with the prescribed element of \(\mathscr L^\vee\). +Consequently, if a positive integer \(N\) satisfies +\(N\mathcal E_K(\mathscr E)\subset\mathscr E\), then +\[ + \mathscr L^\vee\subset N^{-1}\mathscr L,\qquad + X^2=\Disc(\OO)=[\mathscr L^\vee:\mathscr L]\le N^4. +\] +It remains to produce the denominator \(N=2\Delta_F\). + +Choose trace-dual bases \(a_i\) of \(C_F\) and \(b_i\) of \(C_F^\vee\), +where +\[ + C_F^\vee=\{\beta\in F:\Tr_{F/\Q}(\beta C_F)\subset\Z\}. +\] +The adjugate of the trace Gram matrix gives +\(\Delta_F b_i\in C_F\). Define on \(\End_\Q(V)\) +\begin{equation} + \mathcal E_F(T)=\sum_{i=1}^3\rho(a_i)T\rho(b_i). + \label{eq:ext-centralizer-projection} +\end{equation} +For two embeddings \(\sigma,\tau\) of \(F\), the dual-basis identity is +\[ + \sum_i\sigma(a_i)\tau(b_i)= + \begin{cases}1,&\sigma=\tau,\\0,&\sigma\ne\tau.\end{cases} +\] +Thus, after splitting, \(\mathcal E_F\) extracts exactly the three +diagonal blocks \(\End(V_b)\). It is the orthogonal projection +for the matrix trace pairing, and +\begin{equation} + \Delta_F\mathcal E_F(\End_\Z(L_V))\subset\End_\Z(L_V). + \label{eq:ext-centralizer-denominator} +\end{equation} +This denominator conclusion uses the trace-dual lattice only; it +does not require that lattice to be an invertible \(C_F\)-ideal. + +We transfer this projection to matrices on \(K\) explicitly. +Let +\[ + j(T)(v\wedge w)=Tv\wedge w+v\wedge Tw + \qquad(T\in\End_\Q(K)). +\] +There is a natural contraction +\(\mathfrak c:\End_\Q(\bigwedge^2K)\to\End_\Q(K)\). +For clarity, in a basis \(e_1,\ldots,e_4\) write +\[ + S(e_j\wedge e_l)=\sum_{i0\ \Longrightarrow\ z\in(C_F^\sharp)^\vee. +\] +The following stronger containment is used to average the majorant in +Proposition~\ref{prop:ext-multiplicity-majorant}: if \(z\in F^\times\) is +integral outside \(S\) and \(\prod_{p\in S}f_p^*(z)>0\), then +\(z\in(C_F^\sharp)^\vee\). + +Fix \(m\ne0\) and a bounded +\(\Omega\subset\mathcal H_m\). Every +\(z\in(C_F^\sharp)^\vee\cap\Omega\) satisfies +\begin{align} + &\Delta_F^\sharp z\in R_F,\qquad + 1\le|\Nm_{F/\Q}(\Delta_F^\sharp z)|\ll_\Omega X^{84}, + \label{eq:ext-norm-bound}\\ + &\prod_{p\notin S}p^{n_p(z)} + \le|\Nm_{F/\Q}(\Delta_F^\sharp z)|,\qquad + |z_b|\gg_\Omega X^{-84}\quad(b\in\mathcal B). + \label{eq:ext-depth-coordinate-bound} +\end{align} +The constants are uniform in all arithmetic data and in the point +of \(\Omega\). +\end{lemma} + +\begin{proof} +Since \(C_F\) is an order, \(C_F\cap\Q=\Z\), so \(1\) is primitive +in its lattice. A basis \(1,a_2,a_3\) of \(C_F\) gives the basis +\(1,2a_2,2a_3\) of \(C_F^\sharp\). This proves the index and +discriminant assertions in Equation~\eqref{eq:ext-sharp-discriminant}. +The functional taking \(1\) to \(1\) and the other two basis vectors +to \(0\) is represented in the trace dual. Hence trace from that +dual is onto \(\Z\). Lemma~\ref{lem:ext-order-scale}, and \(2\in S\), +give \(C_{F,p}^\sharp=R_{F,p}\) outside \(S\). + +The two integrality identities preceding the lemma show that the +law in Equation~\eqref{eq:ext-density-law} is supported on +\((C_{F,p}^\sharp)^\vee\). For a contributing global pair, use its local +representation \(z=u\Psi(x)\) and Equation~\eqref{eq:ext-fiber-action} +to translate \(x\) into \(L_{V,p}\), adjusting \(u\) by the inverse +norm of that unit. This proves local trace-dual membership +at every prime. Intersecting the local lattices proves the stated +global support. For the second support assertion, the density factors +give this membership at \(p\in S\). At \(p\notin S\), the order is +\(R_{F,p}\), whose trace pairing is unimodular because \(F_p\) is +unramified; thus \(R_{F,p}^\vee=R_{F,p}\), and the assumed integrality +gives membership there as well. + +The adjugate of the trace Gram matrix of \(C_F^\sharp\) gives +\(\Delta_F^\sharp(C_F^\sharp)^\vee\subset C_F^\sharp\subset R_F\). +If \(z\in\Omega\), then \(\Tr z=m\ne0\), so \(z\ne0\); since \(F\) +is a field, all its embeddings are nonzero. The norm +\(\Nm(\Delta_F^\sharp z)\) is consequently a nonzero integer. The +boundedness of \(\Omega\) and \(\Delta_F^\sharp\ll X^{28}\) give +\[ + 1\le|\Nm(\Delta_F^\sharp z)| + =(\Delta_F^\sharp)^3\prod_{b\in\mathcal B}|z_b| + \ll_\Omega X^{84}. +\] +At \(p\notin S\), \(z\) is integral and the local algebra is +unramified. Hence +\[ + n_p(z)\le\sum_{\mathfrak q\mid p} + f_{\mathfrak q}v_{\mathfrak q}(z) + \le v_p\bigl(\Nm(\Delta_F^\sharp z)\bigr). +\] +The product of these inequalities gives the depth bound because +\(\Nm(\Delta_F^\sharp z)\) is an integer. Finally, if the coordinates of +\(\Omega\) have absolute value at most \(B\), with \(B\ge1\), then +for each \(b\) +\[ + |z_b|=\frac{|\Nm(\Delta_F^\sharp z)|} + {(\Delta_F^\sharp)^3\prod_{b'\ne b}|z_{b'}|} + \ge\frac{1}{(\Delta_F^\sharp)^3 B^2}\gg_\Omega X^{-84}. +\] +This proves Equation~\eqref{eq:ext-depth-coordinate-bound}. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/introduction.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..2b9af98d05f97ff60baa79bba97a938e33ee9af2 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/introduction.tex @@ -0,0 +1,253 @@ +\section{Introduction}\label{sec:introduction} + +Ideal classes in a totally real number field give compact orbits of a +diagonal group on a space of lattices. A natural arithmetic question is +whether the finite collections of orbits obtained in this way become +uniformly distributed as their discriminants grow. We prove this for the +complete packets attached to invertible ideals of every order in a primitive +totally real quartic field. The order may have arbitrarily large index in +the ring of integers. + +Let +\[ + X_4=\mathrm{SL}_4(\Z)\backslash\mathrm{SL}_4(\R) +\] +be the space of covolume-one row lattices in $\R^4$, and let $m_4$ be its +invariant probability measure. Put +\[ + H_4=\{u\in\R^4:u_1+u_2+u_3+u_4=0\},\qquad + A_4=\{\diag(e^{u_1},e^{u_2},e^{u_3},e^{u_4}):u\in H_4\}. +\] +The group $A_4$ acts on $X_4$ by right multiplication. Its Haar measure +is $du_1\,du_2\,du_3$, with $u_4=-u_1-u_2-u_3$. + +A quartic field $K$ is \emph{primitive} if it has no field strictly between +$\Q$ and $K$. Let $K$ be primitive and totally real, let $\OO\subseteq\OO_K$ +be an order, and fix an ordering $\sigma=(\sigma_1,\ldots,\sigma_4)$ of its +real embeddings. Here an order is a full-rank subring containing $1$. +A fractional $\OO$-ideal $I$ is \emph{proper} if its multiplier order +$\{a\in K:aI\subseteq I\}$ is $\OO$, and it is \emph{invertible} if +$IJ=\OO$ for some fractional $\OO$-ideal $J$. The ordinary ideal class +group $\Pic(\OO)$ consists of proper invertible fractional $\OO$-ideals +modulo $I\sim aI$ for $a\in K^\times$. + +For such an ideal put +\begin{equation}\label{eq:intro-lattice} + \Lambda_{I,\sigma} + =\covol(\sigma(I))^{-1/4}\sigma(I)\in X_4. +\end{equation} +A covolume-one lattice is represented here by any positively oriented +basis; two such bases differ by an element of $\mathrm{SL}_4(\Z)$. +Let $W_{\mathrm{all}}$ be the sixteen diagonal matrices with entries in +$\{1,-1\}$. Choose one ideal in each ordinary class and form the finite +set of distinct orbits +\[ + \mathcal P_{\OO,\sigma} + =\{\Lambda_{I,\sigma}wA_4:I\text{ a class representative},\ + w\in W_{\mathrm{all}}\}. +\] +A matrix in $W_{\mathrm{all}}$ acts on the underlying lattice, so a sign +matrix of determinant $-1$ also gives a point of $X_4$. Using all signs +makes the set independent of the chosen ordinary ideal representatives. +Every orbit is compact by the unit theorem; Section~\ref{sec:packets} +proves these assertions and describes the associated measure. + +For an orbit $\mathcal Y\in\mathcal P_{\OO,\sigma}$, let $\nu_{\mathcal Y}$ +be its invariant probability and let $\vol(\mathcal Y)$ be its volume for +the specified Haar measure. Define +\begin{equation}\label{eq:intro-measure} + \mu_{\OO,\sigma} + =\frac{\displaystyle\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}} + \vol(\mathcal Y)\nu_{\mathcal Y}} + {\displaystyle\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}} + \vol(\mathcal Y)}. +\end{equation} +Each distinct orbit occurs once in these sums. + +\begin{theorem}\label{thm:main} +Let $(K_i,\OO_i,\sigma_i)$ be any sequence in which $K_i$ is a primitive +totally real quartic field, $\OO_i$ is an order in $K_i$, and $\sigma_i$ is +an ordering of its real embeddings. If $|\Disc(\OO_i)|\to\infty$, then +\[ + \int_{X_4} f\,d\mu_{\OO_i,\sigma_i} + \longrightarrow \int_{X_4} f\,dm_4 + \qquad\text{for every }f\in C_c(X_4). +\] +There is no escape of mass: for every $\varepsilon>0$ there is a compact +$\mathcal K\subset X_4$ such that +$\mu_{\OO_i,\sigma_i}(\mathcal K)\ge1-\varepsilon$ for all sufficiently +large $i$. +\end{theorem} + +In particular, the field may remain fixed while the order index tends to +infinity. The packet in Theorem~\ref{thm:main} uses every class in +$\Pic(\OO)$, so its ideals are locally free of rank one over the specified +order. The proof uses this complete class average in both of its counting +arguments. + +\subsection{History and context} + +For real quadratic fields, compact diagonal orbits are closed geodesics on +the modular surface. Linnik's ergodic method and Skubenko's theorem for +positive discriminants established arithmetic distribution under the +condition $\left(\frac d p\right)=1$ for the discriminant $d$ at a fixed +odd prime $p$ \cite{Linnik1968,Skubenko1962}. +Using Iwaniec's estimates for half-integral weight forms \cite{Iwaniec1987}, +Duke removed that restriction for the geodesic collections associated with +positive fundamental discriminants \cite[Theorem~1]{Duke1988}. +Chelluri extended the corresponding statement to the unit tangent bundle, +as recorded in \cite[Section~1.2, p.~253]{ELMV2012}. +Einsiedler, Lindenstrauss, Michel, and Venkatesh later gave a proof using +entropy for all positive nonsquare order discriminants +\cite[Theorem~2.3]{ELMV2012}. + +In higher degree the diagonal group has higher rank, and the arithmetic +packets lie in spaces of higher-dimensional lattices. Einsiedler, +Lindenstrauss, Michel, and Venkatesh developed the discriminant and volume +framework connecting separation of periodic torus orbits to entropy +\cite{ELMV2009}. Their cubic theorem proves equidistribution of packets +that include those arising from nonmaximal totally real cubic orders +\cite[Theorem~1.4]{ELMV2011}. The local homothety class of a lattice is +part of their packet data; a multiplier order alone need not specify it +\cite[Section~5.5]{ELMV2011}. Their higher-dimensional discussion also +identifies the possibility of proper intermediate homogeneous limits +\cite[Section~1.6.3]{ELMV2011}. The theorem of Einsiedler, Katok, and +Lindenstrauss makes a diagonal-ergodic probability with positive entropy +homogeneous \cite[Theorem~1.3]{EKL2006}; in prime dimension at least three +its homogeneous conclusion is already Haar \cite[Corollary~1.4]{EKL2006}. + +For each fixed prime $p\ge5$, Lemke Oliver, Thorner, and Zaman proved +equidistribution for the orbit collections attached to orders in totally +real degree-$p$ fields as the order discriminant grows, outside an +exceptional set containing $O_{p,\varepsilon}(Q^\varepsilon)$ fields of +field discriminant at most $Q$ \cite[Theorem~2.3]{LOTZ2024}. +The separate preprint \cite[Theorem~1.1]{PrimeDegreePackets} treats every +packet in each fixed prime degree at least five and permits arbitrary local module types. +We adapt its ordinary-vector method in the argument below. +For quartic packets, Khayutin obtained entropy bounds for probability +weak-$*$ limits of maximal-order packet measures with two-transitive Galois +groups, including the primitive quartic types $\mathfrak A_4$ and $S_4$ +\cite[Theorem~1.1]{Khayutin2019Tori}. Wieser and Yang obtain entropy bounds +for certain maximal-order quartic packets whose fields have a quadratic +subfield, under further hypotheses \cite{WieserYang2026}. The present +argument must establish tightness and positive entropy in almost every +diagonal-ergodic component for arbitrary orders, and then exclude the +proper homogeneous limits allowed in degree four. + +The arithmetic use of a sieve to rule out intermediate algebraic limits +has a precedent in Khayutin's work on joint distributions of CM points +\cite{Khayutin2019CM}. Local Fourier analysis for nonmaximal orders already +appears in the cubic work of Einsiedler, Lindenstrauss, Michel, and Venkatesh +\cite[Proposition~9.9]{ELMV2011}. Here the object being transformed is a +specific quadratic product map on an exterior lattice. The resulting +estimate controls arbitrary conductor depth in a cubic-resolvent count, +before the remaining multiplicative weights are inserted by a sieve. + +\subsection{The two counts and the conductor estimate} + +Write +\[ + \begin{gathered} + R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\\ + X=q\sqrt D=\sqrt{|\Disc(\OO)|},\qquad + \kappa=\Res_{s=1}\zeta_K(s). + \end{gathered} +\] +We also write $a_K(n)$ for the number of nonzero integral $R$-ideals of +norm $n$. Both counting arguments preserve the factor $\sqrt D\,\kappa$ +that normalizes the packet after its finite local labels are averaged. + +Sections~\ref{sec:packets} and~\ref{sec:ordinary} treat ordinary lattice +vectors. An invertible $\OO$-ideal is locally a unit translate of $\OO$ +inside a maximal-order ideal. Averaging those translates gives an exact +unfolding into a weighted sum over integral $R$-ideals. A local moment +bound controls the order index, while Stark's exceptional-zero result and +the absence of a quadratic subfield retain the residue $\kappa$ in the +ideal count. Counting short vectors proves tightness. Counting vectors +in boxes away from the coordinate hyperplanes gives an $O(r^4)$ bound for +radius-$r$ neighborhoods in $X_4$, uniformly on each compact set of centers +in every probability limit. For a diagonal flow with distinct weights, +a two-sided tube is covered by $O(r^{-3})$ such neighborhoods with centers +in a fixed compact enlargement, where $r$ decreases exponentially with the +tube length. The resulting exponential decay also holds for normalized +averages over sets of $A_4$-ergodic components with positive measure, +because they are dominated by a multiple of the limit measure. This forces +positive entropy in almost every component. + +Section~\ref{sec:rigidity} applies measure classification and identifies +the remaining obstruction. A proper positive-entropy component has two +coordinate blocks of dimension two. Each lattice in its supporting orbit +then has a vector in its exterior square supported on the two opposite +coordinates $ij$ and $kl$, with both coefficients nonzero. There are three +choices of the partition $ij|kl$. + +For an exterior vector $v=(v_{ij})_{i0$. +An upper bound $X^{o(1)}$ means $O_\eta(X^\eta)$ for every fixed $\eta>0$. +Unless stated otherwise, constants are uniform in $K$, $\OO$, and the +ordering of the embeddings. In the exterior estimates a nonzero trace $m$ +and a bounded real region are fixed before $X$ tends to infinity; permitted +dependence on them is stated in the relevant results. diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/local-fourier.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/local-fourier.tex new file mode 100644 index 0000000000000000000000000000000000000000..247d4d39e256f03957613cee4e9da1a5aee00229 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/local-fourier.tex @@ -0,0 +1,680 @@ +\section{Local Fourier estimates}\label{sec:local-fourier} + +The exterior multiplicity bound in Section~\ref{sec:exterior} uses an +additive density at each prime in $S$. We now estimate its Fourier transform. +Scalar frequencies detect only the trace and have an exact, order-independent +value. For the remaining frequencies we record both decay and the support of +the inverse transform on a fixed denominator shell. The support is needed +because the denominator at a prime dividing the order index can be arbitrarily +large. + +\subsection{Quadratic data and the trace marginal} + +Fix a rational prime $p$. We first specify the local hypotheses used in the +argument. Let +\[ + E=\prod_{\nu=1}^r E_\nu,\qquad R_E=\prod_{\nu=1}^r R_\nu +\] +be a cubic \'etale $\Q_p$-algebra and its maximal order. Here $E_\nu$ is a +field, $R_\nu$ is its integer ring, and $r\le3$. Write +$\mathfrak q_\nu=(\pi_\nu)$ for the maximal ideal of $R_\nu$, +$v_\nu(\pi_\nu)=1$, $e_\nu=v_\nu(p)$, and +$q_\nu=|R_\nu/\mathfrak q_\nu|=p^{f_\nu}$. Thus +\[ + \sum_\nu e_\nu f_\nu=3, + \qquad |\Nm_{E_\nu/\Q_p}a|_p=q_\nu^{-v_\nu(a)}. +\] +Let $\mathfrak d_\nu=\mathfrak q_\nu^{d_\nu}$ be the different of +$E_\nu/\Q_p$, and put $\mathfrak d_E=\prod_\nu\mathfrak d_\nu$. + +Let $W$ be a free $E$-module of rank two, with its unital action denoted by +$\rho:E\longrightarrow\End_{\Q_p}(W)$. Suppose that $B$ is a nondegenerate +symmetric $\Q_p$-bilinear form satisfying +\[ + B(\rho(a)x,y)=B(x,\rho(a)y)\qquad(a\in E,\ x,y\in W). +\] +Let $L$ be a full $\Z_p$-lattice in $W$ which is self-dual for $B$: +\[ + L=\{y\in W:B(y,L)\subset\Z_p\}. +\] +Assume also that $Q_1(x)=B(x,x)/2$ is integral on $L$. For the scalar +calculation we shall use the stronger property that in some integral +coordinates on $L$, +\begin{equation}\label{eq:fourier-split-form} + Q_1(x_1,y_1,x_2,y_2,x_3,y_3) + =x_1y_1-x_2y_2+x_3y_3. +\end{equation} +The division by $2$ takes place in $\Q_p$. Equation~\eqref{eq:fourier-split-form} +asserts the required integrality also when $p=2$. The shell estimates below +use only $Q_1(L)\subset\Z_p$; the exact split expression is used for the +scalar marginal. + +For $a\in E$, set $Q_a(x)=B(\rho(a)x,x)/2$. The nondegenerate trace pairing +on $E$ determines a unique quadratic map $\Psi_E:W\longrightarrow E$ by +\begin{equation}\label{eq:fourier-polar-identity} + \Tr_{E/\Q_p}(a\Psi_E(x))=Q_a(x) + =\frac12 B(\rho(a)x,x)\qquad(a\in E). +\end{equation} +Define the stabilizer order and the auxiliary order +\[ + \mathcal C_L=\{a\in E:\rho(a)L\subset L\}, + \qquad \mathcal C_L^\sharp=\Z_p+2\mathcal C_L. +\] +Indeed $\mathcal C_L$ is an order in $R_E$. It is a subring containing $1$, +and a sufficiently large power of $p$ times $R_E$ preserves $L$. Conversely, +an element preserving $L$ satisfies the monic characteristic polynomial of +its integral matrix on $L$. The action is faithful, so the element is +integral over $\Z_p$ and belongs to $R_E$. + +In the application, $E=F_p$, $W=V_p$, $L=L_{V,p}$, $B=B_V$, and +$\mathcal C_L=C_{F,p}$. Lemma~\ref{lem:ext-quadratic-structure} supplies +all the stated hypotheses and identifies $\Psi_E$ with the base change of +$\Psi$. We use the abstract notation temporarily to include every product +algebra and every ramification type in one argument. + +Choose an additive character $\psi_p$ of $\Q_p$ with conductor $\Z_p$. +We give $W$ the self-dual Haar measure for $(x,y)\mapsto\psi_p(B(x,y))$; +it gives $L$ mass $1$. On $E$ use the self-dual measure for +$(A,z)\mapsto\psi_p(\Tr_{E/\Q_p}(Az))$, denoted by $dz$ or $dA$ according +to the variable. If $\mathfrak a$ is a fractional $R_E$-ideal, its +annihilator for this pairing is +$\mathfrak a^\perp=\mathfrak a^{-1}\mathfrak d_E^{-1}$. In particular +\begin{equation}\label{eq:fourier-self-dual-volume} + \vol(R_E)=\prod_\nu q_\nu^{-d_\nu/2}\le1. +\end{equation} +To see the normalization, $R_E^\perp=\mathfrak d_E^{-1}$ and +$\vol(R_E)\vol(R_E^\perp)=1$, whereas +$[\mathfrak d_E^{-1}:R_E]=\prod_\nu q_\nu^{d_\nu}$. +For $E=F_p$, Equation~\eqref{eq:fourier-self-dual-volume} reads +$\vol(R_{F,p})=|D_F|_p^{1/2}$. + +Let $\mu_p$ be the law of +\begin{equation}\label{eq:fourier-density-law} + u\Psi_E(x),\qquad x\in L\text{ with additive probability}, + \qquad u\in\Z_p^\times\text{ with Haar probability}, +\end{equation} +with $x$ and $u$ independent. This is the law defining $f_p^*$ in +Lemma~\ref{lem:ext-density}. The following regularity fact specifies the +version of that density which we shall invert pointwise. + +\begin{lemma}\label{lem:fourier-density} +The measure $\mu_p$ has a density $f_p^*$ with respect to $dz$. It has a +finite locally constant version at every $z=(z_\nu)\in E$ with +$z_\nu\ne0$ for all $\nu$. This version vanishes outside +\[ + (\mathcal C_L^\sharp)^\vee + =\{z\in E:\Tr_{E/\Q_p}(z\mathcal C_L^\sharp)\subset\Z_p\}. +\] +In particular $\Tr_{E/\Q_p}(u\Psi_E(x))\in\Z_p$ on the domain of +Equation~\eqref{eq:fourier-density-law}. +\end{lemma} + +\begin{proof} +Define an $E$-valued bilinear form $b_E$ by +\[ + \Tr_{E/\Q_p}(a b_E(x,y))=B(\rho(a)x,y)\qquad(a\in E). +\] +Self-adjointness of the action makes $b_E$ symmetric and $E$-bilinear. +It is nondegenerate because $B$ is nondegenerate. Equation~\eqref{eq:fourier-polar-identity} +then gives $\Psi_E(x)=b_E(x,x)/2$ and +$d\Psi_{E,x}(h)=b_E(x,h)$. On the $\nu$-th field factor, a nonzero +component of $\Psi_E(x)$ implies that the component $x_\nu$ of $x$ is +nonzero. The nondegenerate form on the rank-two $E_\nu$-space makes +$h_\nu\mapsto b_E(x_\nu,h_\nu)$ a nonzero, hence onto, $E_\nu$-linear +functional. It follows that $\Psi_E$ is a $\Q_p$-submersion at every +preimage of an output with nonzero components. Multiplication by a unit +$u$ is invertible on $E$, so the map $(x,u)\mapsto u\Psi_E(x)$ is a +submersion at the same points. + +For each $\nu$ the polynomial $\Psi_{E,\nu}$ is nonzero: a nondegenerate +symmetric form in characteristic zero has a nonzero quadratic form. Choose +a nonzero $\Q_p$-coordinate polynomial of this component. Its zero set in +the open lattice $L$ has Haar measure zero, by the usual one-variable root +bound followed by induction and Fubini. Off the union of +these null sets, the local submersion theorem gives absolute continuity of +the pushforward. For a fixed output with nonzero components, its preimage +in the compact set $L\times\Z_p^\times$ is compact. Cover this preimage +by finitely many compact-open submersion charts, refined to disjoint +compact-open pieces within those charts. The image of the remaining compact +set avoids a sufficiently small neighborhood of the output. After shrinking +that neighborhood, integration in the fiber coordinates of the pieces gives +a finite constant density on it. This proves the asserted +pointwise version, including value zero when the neighborhood misses the +image. + +Finally, for $c\in\mathcal C_L$ and $r\in\Z_p$, +\[ + \Tr_{E/\Q_p}((r+2c)\Psi_E(x)) + =rQ_1(x)+B(\rho(c)x,x)\in\Z_p\qquad(x\in L). +\] +Multiplication by $u\in\Z_p^\times$ preserves this inclusion. The image +of the compact domain is therefore contained in the displayed trace dual, +which proves the support and trace assertions. +\end{proof} + +We use the Fourier convention +\[ + \widehat f(A)=\int_E f(z)\psi_p(\Tr_{E/\Q_p}(Az))\,dz, + \qquad + \check g(z)=\int_E g(A)\psi_p(-\Tr_{E/\Q_p}(Az))\,dA. +\] +The transform of the measure in Equation~\eqref{eq:fourier-density-law} is +\begin{equation}\label{eq:fourier-gamma} + \gamma_p(A)=\widehat{f_p^*}(A) + =\E_{u\in\Z_p^\times}\int_L\psi_p(uQ_A(x))\,dx. +\end{equation} + +\begin{lemma}[The scalar trace density]\label{lem:fourier-scalar-density} +Suppose that the integral scalar form is exactly +$Q_1=x_1y_1-x_2y_2+x_3y_3$ as in +Equation~\eqref{eq:fourier-split-form}. Regard $a\in\Q_p$ as a scalar in +$E$. In the quartic application this scalar form is $Q_1=Q_V$. Then, for +every prime $p$, including $p=2$, +\begin{equation}\label{eq:fourier-scalar-coefficients} + \gamma_p(a)= + \begin{cases} + 1,&a\in\Z_p,\\ + p^{-3k},&v_p(a)=-k<0. + \end{cases} +\end{equation} +The trace of the law in Equation~\eqref{eq:fourier-density-law} has density +$d_p(t)\mathbf1_{\Z_p}(t)$ for additive measure giving $\Z_p$ mass $1$. +This density is continuous on $\Z_p$ and satisfies +\begin{equation}\label{eq:fourier-scalar-density-bound} + |d_p(t)-1|\le\frac1{p(p+1)}\qquad(t\in\Z_p). +\end{equation} +In particular $5/6\le d_p(t)\le7/6$, and for every finite set of primes +$S$ and every choice $t_p\in\Z_p$, +\[ + \prod_{p\in S}d_p(t_p)\asymp1 +\] +with absolute upper and lower constants. +\end{lemma} + +\begin{proof} +For fixed $u\in\Z_p^\times$ and $v_p(a)=-k<0$, integration in the second +coordinate gives +\[ + \int_{\Z_p^2}\psi_p(ua xy)\,dx\,dy + =\int_{\Z_p}\mathbf1_{\{ua x\in\Z_p\}}\,dx=p^{-k}. +\] +For $a\in\Z_p$ the integral is $1$. The sign of $a$ has no effect, so the +three independent pairs in Equation~\eqref{eq:fourier-split-form} give +Equation~\eqref{eq:fourier-scalar-coefficients}, even before averaging $u$. +This calculation uses the integral split expression and does not divide by +$2$ in $\Z_p$. + +The scalar coefficient is the Fourier coefficient of the trace law, since +$\Tr(a z)=a\Tr(z)$. That law is supported on $\Z_p$ by +Lemma~\ref{lem:fourier-density}. The character group of $\Z_p$ is +$\Q_p/\Z_p$, and it has $p^k-p^{k-1}$ characters of exact denominator +$p^k$. Hence its nonconstant Fourier coefficients have total absolute sum +\[ + \sum_{k\ge1}(p^k-p^{k-1})p^{-3k} + =\frac{1-p^{-1}}{p^2-1}=\frac1{p(p+1)}. +\] +Fourier inversion on the compact group $\Z_p$ is therefore uniformly +convergent and gives the continuous density and +Equation~\eqref{eq:fourier-scalar-density-bound}. The lower bound is +positive at every prime, and $\sum_p1/(p(p+1))<\infty$. Taking logarithms +of the upper and lower bounds proves the last assertion. +\end{proof} + +\subsection{Denominator shells and quadratic transformation} + +For $A=(A_\nu)\in E$, its denominator ideal is +\[ + \mathfrak h_p(A)=\prod_\nu\mathfrak q_\nu^{h_\nu(A)},\qquad + h_\nu(A)=\max\{0,-v_\nu(A_\nu)\}, +\] +where $v_\nu(0)=+\infty$. For an integral $R_E$-ideal +$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, its shell is thus +\[ + \mathscr S_{\mathfrak h_p} + =\prod_{h_\nu=0}R_\nu\ \times\! + \prod_{h_\nu>0}\pi_\nu^{-h_\nu}R_\nu^\times. +\] +The integral components of a shell may be zero. Define +\begin{equation}\label{eq:fourier-shell-indices} + H_p=\Nm(\mathfrak h_p)=\prod_\nu q_\nu^{h_\nu},\qquad + p^{g_p}=[\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L]. +\end{equation} +The shell is a compact open subset of $\mathfrak h_p^{-1}$. We shall shift +its frequencies by integral scalars so that the quadratic form is +nondegenerate, without changing the integral in +Equation~\eqref{eq:fourier-gamma}. + +\begin{lemma}[A scalar shift on a shell]\label{lem:fourier-shift} +Put $\iota_p=1$ for $p\le5$ and $\iota_p=0$ for $p>5$. For every +$A\in\mathscr S_{\mathfrak h_p}$ there is an $s\in\{0,1,2,3\}$ such that +$\alpha=A+s\in E^\times$ and +\[ + v_\nu(\alpha_\nu)=-h_\nu\quad(h_\nu>0),\qquad + 0\le v_\nu(\alpha_\nu)\le e_\nu\iota_p\quad(h_\nu=0). +\] +Writing $\beta=\alpha^{-1}$, one has +\begin{equation}\label{eq:fourier-shift-bounds} + \beta\in p^{-\iota_p}\mathfrak h_p, + \qquad p^{-3\iota_p}H_p + \le |\Nm_{E/\Q_p}\alpha|_p\le H_p. +\end{equation} +In particular the norm is exactly $H_p$ when $p>5$. If +\begin{equation}\label{eq:fourier-j-definition} + j_p=j_p(A,s)=\min\{j\ge0:p^j\beta\in\mathcal C_L\}, +\end{equation} +then $j_p$ is finite and $j_p\le g_p+\iota_p$. Finally, for any generator +$i\in E^\times$ of the ideal $\mathfrak h_p$, the element $x=i(A+s)$ is in +$R_E$, is a unit of $R_E$ if $p>5$, and satisfies +$x^{-1}\in p^{-\iota_p}R_E$. +\end{lemma} + +\begin{proof} +An integral scalar shift leaves every pole valuation $-h_\nu<0$ unchanged. +For an integral component and $p>5$, the condition that $A_\nu+s$ fail to be +a unit forbids at most one residue $s\in\mathbf F_p$: it forbids none if +the residue of $-A_\nu$ does not belong to the scalar subfield. There are +at most three components, and $0,1,2,3$ are distinct modulo $p$. Some choice +therefore makes all integral components units. + +For $p\le5$, normalize the extension of $v_p$ by $v_p(p)=1$, so that on +$E_\nu$ it is $v_\nu/e_\nu$. Distinct elements $s,s'$ of the list satisfy +$v_p(s-s')\le1$. The ultrametric inequality shows that for a fixed integral +component at most one candidate can satisfy $v_p(A_\nu+s)>1$. Again there +are at most three components, so a candidate remains for which all their +valuations are at most $1$. Thus $A_\nu+s$ is nonzero in every integral +component. + +The bounds on $\beta$ follow componentwise. In the norm, the possible +loss from the integral components is at most +$p^{-\iota_p\sum_\nu e_\nu f_\nu}=p^{-3\iota_p}$, proving +Equation~\eqref{eq:fourier-shift-bounds}. The group +$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)$ has order $p^{g_p}$ and is +killed by $p^{g_p}$. Thus +\begin{equation}\label{eq:fourier-order-containment} + p^{g_p}\mathfrak h_p\subset\mathcal C_L. +\end{equation} +Multiplying the containment for $\beta$ by $p^{g_p+\iota_p}$ proves the +claim about $j_p$. Finally $v_\nu(i_\nu)=h_\nu$. The components of +$i\alpha$ consequently have valuation zero on the pole factors and valuation +between zero and $e_\nu\iota_p$ on the others. This proves the assertions +about $x$ and $x^{-1}$. +\end{proof} + +For the shift in Lemma~\ref{lem:fourier-shift}, +$Q_{A+s}=Q_A+sQ_1$ and $Q_1(L)\subset\Z_p$. Thus for every fixed unit $u$ +\begin{equation}\label{eq:fourier-scalar-shift-invariance} + \int_L\psi_p(uQ_{A+s}(x))\,dx + =\int_L\psi_p(uQ_A(x))\,dx. +\end{equation} +The following quadratic transformation turns the norm of $A+s$ into decay, +and leaves an integral involving its inverse. + +\begin{lemma}[Quadratic Fourier transformation]\label{lem:fourier-quadratic-transform} +Let $W_0$ be a finite-dimensional $\Q_p$-space with a nondegenerate symmetric +form $B_0$, and let $L_0\subset W_0$ be a full $\Z_p$-lattice self-dual for +$B_0$. Use the self-dual Haar +measure, which gives $L_0$ mass $1$. For an invertible $B_0$-self-adjoint +operator $P$, put $q_P(x)=B_0(Px,x)/2$. There is a phase $\omega(P)$ with +$|\omega(P)|=1$ such that +\begin{equation}\label{eq:fourier-general-quadratic} + \int_{L_0}\psi_p(q_P(x))\,dx + =\omega(P)|\det P|_p^{-1/2} + \int_{L_0}\psi_p(-q_{P^{-1}}(x))\,dx. +\end{equation} +The normalized phase is invariant under linear isometry of the quadratic +form: $\omega(T^*PT)=\omega(P)$ for every invertible $T$, where $T^*$ is +the $B_0$-adjoint. + +For the local data above, $\alpha\in E^\times$, $\beta=\alpha^{-1}$ and +$u\in\Z_p^\times$, this gives +\begin{equation}\label{eq:fourier-quadratic-transform} + \int_L\psi_p(uQ_\alpha(x))\,dx + =\omega_p(u,\alpha)|\Nm_{E/\Q_p}\alpha|_p^{-1} + \int_L\psi_p(-u^{-1}Q_\beta(x))\,dx, + \qquad |\omega_p(u,\alpha)|=1. +\end{equation} +For every $r\in E^\times$ one has +$\omega_p(u,\alpha r^2)=\omega_p(u,\alpha)$. +All assertions hold at $p=2$. +\end{lemma} + +\begin{proof} +We use Weil's Fourier formula for a quadratic character +\cite[Chapter~I, \S14, Theorem~2, p.~161]{Weil1964}. +We record its normalization for the transform +\[ + \mathcal F_{B_0}h(y)=\int h(x)\psi_p(B_0(x,y))\,dx. +\] +Completing the square gives +\[ + q_P(x)+B_0(x,y)=q_P(x+P^{-1}y)-q_{P^{-1}}(y). +\] +Let $B_N=p^{-N}L_0$ and $c_N=\int_{B_N}\psi_p(q_P(x))\,dx$. On any fixed +compact set of $y$, translation by $P^{-1}y$ preserves $B_N$ for all +sufficiently large $N$. The transform of +$\mathbf1_{B_N}\psi_p(q_P)$ on that set is therefore +$c_N\psi_p(-q_{P^{-1}})$. If this identity is paired with a +Schwartz--Bruhat function, its left side stabilizes with $N$, since the +Fourier transform of the test function has compact support. Choosing a +test function whose integral against $\psi_p(-q_{P^{-1}})$ is nonzero shows +that $c_N$ stabilizes as well. This proves the distributional shape of the +formula. + +Differencing the absolute square of $c_N$ and integrating in one variable +gives +\[ + |c_N|^2=\vol(B_N) + \int_{B_N\cap P^{-1}B_N^\perp}\psi_p(q_P(h))\,dh. +\] +For sufficiently large $N$, the intersection is $P^{-1}p^NL_0$ and +$q_P$ is integral on it. The right side then equals +$\vol(p^{-N}L_0)\vol(P^{-1}p^NL_0)=|\det P|_p^{-1}$. +Hence the stabilized constant is $\omega(P)|\det P|_p^{-1/2}$ with unit +phase. The transform of $\mathbf1_{L_0}$ is $\mathbf1_{L_0}$, so pairing +the distribution identity with this function gives +Equation~\eqref{eq:fourier-general-quadratic}. + +Under $q_{T^*PT}=q_P\circ T$, changing variables multiplies the Gaussian +constant by $|\det T|_p^{-1}$ and its normalizing determinant square root by +$|\det T|_p$. Their product, the phase, is unchanged; this is also the +isometry invariance in Weil~\cite[Chapter~II, \S25, p.~173]{Weil1964}. +For $P=u\rho(\alpha)$ its inverse is $u^{-1}\rho(\beta)$. Rank two over +$E$ gives $\det\rho(\alpha)=\Nm_{E/\Q_p}(\alpha)^2$, and $|u|_p=1$. +This proves Equation~\eqref{eq:fourier-quadratic-transform}. Taking +$T=\rho(r)$ proves the last phase identity. The polar operator of $q_P$ is +$P$, because its polarization is $B_0(Px,y)$. In particular this proof +never assumes that $2$ is a unit of $\Z_p$. +\end{proof} + +\subsection{The inverse transform on a shell} + +Extend the restriction of $\gamma_p$ to a shell by zero, and denote its +inverse transform by +\begin{equation}\label{eq:fourier-shell-transform} + \gamma_{p,\mathfrak h_p}(A) + =\mathbf1_{\mathscr S_{\mathfrak h_p}}(A)\gamma_p(A), + \qquad + K_{p,\mathfrak h_p}(z) + =\int_E\gamma_{p,\mathfrak h_p}(A) + \psi_p(-\Tr_{E/\Q_p}(Az))\,dA. +\end{equation} +The next lemma gives the estimates for these functions which will be used +in Poisson summation. Its quantitative support assertions separate the case where +the denominator ideal is contained in the stabilizer order. + +\begin{lemma}[Intrinsic shell estimates]\label{lem:fourier-shell-estimates} +Let $E$ be the cubic algebra above, let $W$ have rank two over $E$ with +self-adjoint action for a nondegenerate symmetric form $B$, and let $L$ be +self-dual for $B$, with $Q_1(L)\subset\Z_p$. Use the stabilizer order +$\mathcal C_L=\{a\in E:\rho(a)L\subset L\}$ and $\Psi_E,\gamma_p$ from +Equations~\eqref{eq:fourier-polar-identity} and \eqref{eq:fourier-gamma}. +Fix an integral ideal +$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, and use +$H_p,g_p$ from Equation~\eqref{eq:fourier-shell-indices}. +The zero-extended function $\gamma_{p,\mathfrak h_p}$ and its inverse +$K_{p,\mathfrak h_p}$ are Schwartz--Bruhat functions. Moreover, +\[ + \supp K_{p,\mathfrak h_p}\subset(\mathcal C_L^\sharp)^\vee + \subset\{z\in E:\Tr_{E/\Q_p}z\in\Z_p\}. +\] +For +$A\in\mathscr S_{\mathfrak h_p}$ choose $s$ and $j_p=j_p(A,s)$ as in +Lemma~\ref{lem:fourier-shift}. Then +\begin{equation}\label{eq:fourier-shell-decay} + |\gamma_p(A)|\le p^{3\iota_p}H_p^{-1}p^{-j_p/2}, + \qquad + \|K_{p,\mathfrak h_p}\|_\infty + \le p^{3\iota_p}\vol(R_E)\le p^{3\iota_p}. +\end{equation} +Here $\iota_p=\mathbf1_{\{p\le5\}}$, so the first constant is $1$ at +every $p>5$ and is at most $125$ at the three remaining primes. + +If $g_p>0$ or $p\le5$, then +\begin{equation}\label{eq:fourier-inverse-support} + \supp K_{p,\mathfrak h_p} + \subset p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_E^{-1}. +\end{equation} +If $g_p=0$ and $p>5$, then support instead requires, for each $\nu$, +\begin{equation}\label{eq:fourier-boundary-support} + z_\nu\in + \begin{cases} + \mathfrak d_\nu^{-1},&h_\nu=0,\\ + \mathfrak q_\nu^{h_\nu-1}\mathfrak d_\nu^{-1},&h_\nu>0. + \end{cases} +\end{equation} +For a $z$ satisfying these support conditions, define its set of boundary +primes by +\[ + \partial_{\mathfrak h_p}(z) + =\{\mathfrak q_\nu:h_\nu>0, + \ z_\nu\notin\mathfrak q_\nu^{h_\nu}\mathfrak d_\nu^{-1}\}. +\] +In this case +\begin{equation}\label{eq:fourier-boundary-decay} + |K_{p,\mathfrak h_p}(z)| + \le 8\vol(R_E) + \prod_{\mathfrak q_\nu\in\partial_{\mathfrak h_p}(z)}q_\nu^{-1/2}. +\end{equation} +Every constant is uniform in the lattice, the order $\mathcal C_L$, the +shell, and the field factors and their ramification. +\end{lemma} + +\begin{proof} +For $c\in\mathcal C_L^\sharp$, Equation~\eqref{eq:fourier-polar-identity} +shows that $Q_c(L)\subset\Z_p$. Consequently $\gamma_p(A+c)=\gamma_p(A)$. +Each denominator shell is invariant under translation by $R_E$: adding an +integral element preserves every negative component valuation and preserves +the set of integral components. Since $\mathcal C_L^\sharp\subset R_E$, +the zero-extended shell restriction retains this period. Its inverse is +therefore supported in the annihilator $(\mathcal C_L^\sharp)^\vee$, whose +elements have integral trace because $1\in\mathcal C_L^\sharp$. +The order $\mathcal C_L^\sharp$ is an open lattice, and the shell is compact +open. Thus the zero extension in Equation~\eqref{eq:fourier-shell-transform} +is Schwartz--Bruhat, as is its Fourier inverse. + +For a fixed unit $u$ put +$I_\beta(u)=\int_L\psi_p(-u^{-1}Q_\beta(x))\,dx$. +Polarization and additive orthogonality on the self-dual lattice give +\[ + |I_\beta(u)|^2 + =\int_{\substack{h\in L\\\rho(\beta)h\in L}} + \psi_p(-u^{-1}Q_\beta(h))\,dh. +\] +Indeed, in the difference $Q_\beta(y+h)-Q_\beta(y)$ the term linear in $y$ +is $B(\rho(\beta)h,y)$, whose integral is the indicator of +$\rho(\beta)h\in L$. The absolute value of the last display is at most the +relative mass of $L\cap\rho(\beta)^{-1}L$. If +$p^{a_1},\ldots,p^{a_6}$ are the Smith elementary divisors of +$\rho(\beta)$ on $L$, then +\[ + j_p=\max\{0,-\min_i a_i\},\qquad + [L:L\cap\rho(\beta)^{-1}L] + =p^{\sum_i\max(0,-a_i)}\ge p^{j_p}. +\] +The equality for $j_p$ is exactly the definition of $\mathcal C_L$. +It follows that $|I_\beta(u)|\le p^{-j_p/2}$. Apply +Equations~\eqref{eq:fourier-scalar-shift-invariance} and +\eqref{eq:fourier-quadratic-transform}, then use +Equation~\eqref{eq:fourier-shift-bounds} and average over $u$. This proves +the first bound in Equation~\eqref{eq:fourier-shell-decay}. The shell is +contained in $\mathfrak h_p^{-1}$, which has volume +$H_p\vol(R_E)$. Integration of the first bound, with $p^{-j_p/2}\le1$, +proves the second. + +We next prove Equation~\eqref{eq:fourier-inverse-support}. Put +$N=4(g_p+1)$ and take $t\in p^N\mathfrak h_p^{-1}$. For $A$ in the shell +choose $s$ as in Lemma~\ref{lem:fourier-shift}, and write +$\alpha=A+s$, $\beta=\alpha^{-1}$. The component valuations of $t$ are +at least $-h_\nu+e_\nu N$. Translation by $t$ therefore preserves the +shell, and the same $s$ still has the valuation properties of that lemma. +Set $\zeta=t\beta$. Equation~\eqref{eq:fourier-shift-bounds} gives +\[ + \zeta\in p^{N-\iota_p}R_E,\qquad + \alpha+t=\alpha(1+\zeta),\qquad + (\alpha+t)^{-1}-\beta + =-\frac{t\beta^2}{1+\zeta} + \in p^{N-2\iota_p}\mathfrak h_p. +\] +Here $1+\zeta$ is a unit. It is a square unit in each factor when $p$ is +odd, by Hensel's lemma on principal units. When $p=2$, one has +$\zeta\in8R_E$. In the $\nu$-th field the Hensel inequality for +$Y^2-(1+\zeta_\nu)$ at $Y=1$ is +$v_\nu(\zeta_\nu)\ge3e_\nu>2e_\nu=2v_\nu(2)$, so it is a square unit +there as well, including for ramified dyadic factors. + +Moreover $N-2\iota_p\ge g_p+v_p(2)$. Equation~\eqref{eq:fourier-order-containment} +therefore places the inverse difference in $2\mathcal C_L$. Its $Q$-value +on $L$ is integral by Equation~\eqref{eq:fourier-polar-identity}. +In Equation~\eqref{eq:fourier-quadratic-transform}, the inverse integral is +unchanged by this difference. The norm is unchanged by the unit $1+\zeta$, +and the phase is unchanged because that unit is a square. The original +integral, for each fixed $u$, is consequently unchanged under $A\mapsto A+t$. +So is its unit average. Translation by $p^N\mathfrak h_p^{-1}$ preserves +the shell as a set, hence preserves its zero-extended function. The inverse +transform is supported on the annihilator of this group, namely +$p^{-N}\mathfrak h_p\mathfrak d_E^{-1}$. This proves +Equation~\eqref{eq:fourier-inverse-support} with the absolute exponent $4$. + +It remains to prove the sharper assertions for $g_p=0$ and $p>5$. The +equality $g_p=0$ says $\mathfrak h_p\subset\mathcal C_L$. For each factor +with $h_\nu=0$, its component idempotent $1_\nu$ and its whole component +ring $1_\nu R_\nu$ lie in $\mathfrak h_p$. The self-adjoint projections +$\rho(1_\nu)$ preserve $L$ and split off their lattices orthogonally. +The complementary projection $1-\sum_{h_\nu=0}\rho(1_\nu)$ also preserves +$L$. Each resulting summand is self-dual, because the splitting is +orthogonal and $L$ is self-dual. + +On an $h_\nu=0$ summand, an integral $A_\nu$ preserves its lattice, so +$Q_{A_\nu}$ is integral there since $2$ is a unit. That factor of the +integral in Equation~\eqref{eq:fourier-gamma} is $1$. On the combined +summand of all $h_\nu>0$, the frequency $A_+$ is invertible. Its inverse, +extended by zero on the other factors, lies in +$\mathfrak h_p\subset\mathcal C_L$. Its inverse quadratic integral in +Equation~\eqref{eq:fourier-general-quadratic} is therefore also $1$. +On that summand the square root of the polar determinant has absolute value +$H_p$, because each positive factor still has rank two. Applying the formula +therefore shows that the remaining frequency function is $H_p^{-1}$ times a +unit-average of Gaussian phases. Its absolute value is at most $H_p^{-1}$. +If there are no positive +$h_\nu$, this conclusion has $H_p=1$ and the empty summand contributes $1$. + +Let $\chi_\nu$ be the quadratic character of +$(R_\nu/\mathfrak q_\nu)^\times$ for a factor with $h_\nu>0$. At odd +residue characteristic every principal unit is a square, so unit square +classes are exactly the two residue square classes. Phase invariance in +Lemma~\ref{lem:fourier-quadratic-transform} shows that the bounded unit-average +just obtained depends only on the tuple of these classes. This use of phase +invariance concerns the ambient quadratic form; a unit need not preserve +the lattice. Fourier expansion on the product of the two-element +square-class groups expresses the shell function as a sum of at most +$2^{|\{\nu:h_\nu>0\}|}\le8$ terms of the form +\[ + c_{\boldsymbol\varepsilon}H_p^{-1} + \prod_{h_\nu=0}\mathbf1_{R_\nu}(A_\nu) + \prod_{h_\nu>0} + \mathbf1_{\pi_\nu^{-h_\nu}R_\nu^\times}(A_\nu) + \chi_\nu(\overline{\pi_\nu^{h_\nu}A_\nu})^{\varepsilon_\nu}, + \qquad |c_{\boldsymbol\varepsilon}|\le1, +\] +where $\varepsilon_\nu\in\{0,1\}$. The bound for the coefficients follows +from the normalized finite Fourier expansion of a function of absolute +value at most $1$. It also covers any coupling of components caused by +the common average over $u$. + +We compute the inverse of each factor. Put +$\psi_\nu=\psi_p\circ\Tr_{E_\nu/\Q_p}$ and use the component self-dual +measure. For $h_\nu=0$, the integral over $R_\nu$ is +$\vol(R_\nu)\mathbf1_{\mathfrak d_\nu^{-1}}(z_\nu)$. +For $h=h_\nu>0$, the normalized inverse of the trivial unit character is +\[ + q_\nu^{-h}\int_{\pi_\nu^{-h}R_\nu^\times} + \psi_\nu(-Az)\,dA + =\vol(R_\nu)\left( + \mathbf1_{\mathfrak q_\nu^h\mathfrak d_\nu^{-1}}(z) + -q_\nu^{-1}\mathbf1_{\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}}(z) + \right). +\] +This follows by subtracting the integral over $\pi_\nu^{-h+1}R_\nu$ from +that over $\pi_\nu^{-h}R_\nu$ and using their annihilators. For the quadratic +unit character, its function of $a=\pi_\nu^hA$ is invariant under addition +by $\mathfrak q_\nu$. This forces the normalized inverse to vanish outside +$\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}$. It also vanishes on +$\mathfrak q_\nu^h\mathfrak d_\nu^{-1}$, where the additive character on +$R_\nu$ is trivial and the sum of $\chi_\nu$ is zero. At the intervening +boundary the additive character descends to a nontrivial character $\eta$ +of the residue field $k_\nu$, and the normalized integral is +$\vol(R_\nu)q_\nu^{-1}$ times its quadratic Gauss sum. + +For completeness, if $\chi$ is the quadratic character of a finite field +$k$ of odd order $q$ and $\eta$ is a nontrivial additive character, then +\[ + \left|\sum_{a\in k^\times}\chi(a)\eta(a)\right|^2 + =\sum_{t\in k^\times}\chi(t) + \sum_{b\in k^\times}\eta((t-1)b) + =(q-1)-\sum_{\substack{t\in k^\times\\t\ne1}}\chi(t)=q. +\] +Thus the quadratic factor has magnitude +$\vol(R_\nu)q_\nu^{-1/2}$ at the boundary. The trivial factor there has +the smaller magnitude $\vol(R_\nu)q_\nu^{-1}$. Multiplying these +component estimates, using $H_p=\prod_{h_\nu>0}q_\nu^{h_\nu}$ and +$\prod_\nu\vol(R_\nu)=\vol(R_E)$, and summing at most eight terms proves +Equations~\eqref{eq:fourier-boundary-support} and +\eqref{eq:fourier-boundary-decay}. The annihilator computation retains the +different in every ramified factor. +\end{proof} + +The density need not be locally constant on the component-vanishing locus. +For the nonvanishing outputs that occur in the global field, the shell +inverses nevertheless give a finite pointwise expansion. + +\begin{lemma}[Pointwise inverse expansion]\label{lem:fourier-inverse-expansion} +If $z=(z_\nu)\in E$ has $z_\nu\ne0$ for every $\nu$, then +\begin{equation}\label{eq:fourier-inverse-expansion} + f_p^*(z)=\sum_{\mathfrak h_p}K_{p,\mathfrak h_p}(z), +\end{equation} +where the sum is over integral $R_E$-ideals and only finitely many summands +are nonzero. +\end{lemma} + +\begin{proof} +Let $B_n=p^{-n}R_E$. Fubini and additive orthogonality give +\[ + \int_{B_n}\gamma_p(A)\psi_p(-\Tr_{E/\Q_p}(Az))\,dA + =\vol(B_n)\,\mu_p(z+p^n\mathfrak d_E^{-1}). +\] +The self-dual normalizations give +\[ + B_n^\perp=p^n\mathfrak d_E^{-1},\qquad + \vol(B_n)\vol(B_n^\perp)=1. +\] +Thus the right side is the average density on that neighborhood. By +Lemma~\ref{lem:fourier-density} it equals $f_p^*(z)$ for sufficiently large +$n$. The ball $B_n$ is the disjoint union of the finitely many shells with +$h_\nu\le ne_\nu$. + +There can in fact be only finitely many nonzero shell inverses at this $z$. +The injection +$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)\hookrightarrow R_E/\mathcal C_L$ +gives $g_p\le v_p([R_E:\mathcal C_L])$. In +Equation~\eqref{eq:fourier-inverse-support}, a nonzero value requires +\[ + h_\nu\le v_\nu(z_\nu)+d_\nu+4e_\nu(g_p+1). +\] +In Equation~\eqref{eq:fourier-boundary-support}, it requires +$h_\nu\le v_\nu(z_\nu)+d_\nu+1$ whenever $h_\nu>0$. +All right sides are bounded for this $z$, because its component valuations +are finite. The shell sum therefore stabilizes, and the preceding ball +inversion proves Equation~\eqref{eq:fourier-inverse-expansion}. +\end{proof} + +The exact scalar coefficients in Lemma~\ref{lem:fourier-scalar-density} +supply the trace marginal in the next section. The nonscalar estimate uses +the pointwise decay in Equation~\eqref{eq:fourier-shell-decay}, the inverse +supports in Equations~\eqref{eq:fourier-inverse-support} and +\eqref{eq:fourier-boundary-support}, and the boundary bound in +Equation~\eqref{eq:fourier-boundary-decay}. In the case $g_p=0$ and $p>5$, +the last upper bound contains a factor $q_\nu^{-1/2}$ for each actual boundary +component where the support permits one fewer power of $\mathfrak q_\nu$. +Once the real boxes are specified, Section~\ref{sec:additive} combines the +supports with the nonzero field norm to restrict the denominator profiles in +Poisson summation. diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/macros.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/macros.tex new file mode 100644 index 0000000000000000000000000000000000000000..35d2387ec65e7abca3063af80247e3dab4a45f4c --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/macros.tex @@ -0,0 +1,30 @@ +\newcommand{\Q}{\mathbb Q} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\C}{\mathbb C} +\newcommand{\E}{\mathbb E} +\renewcommand{\P}{\mathbb P} +\newcommand{\OO}{\mathcal O} +\newcommand{\eps}{\varepsilon} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\Nm}{N} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\Disc}{Disc} +\DeclareMathOperator{\Res}{Res} +\DeclareMathOperator{\covol}{covol} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\id}{id} +\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} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/main.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..49af2730f7a9c6bc7b53f7e8c8b35a68534070ee --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/main.tex @@ -0,0 +1,30 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools} +\usepackage[mathscr]{eucal} +\usepackage[margin=1.05in]{geometry} +\usepackage{microtype,graphicx,xcolor,booktabs} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={Equidistribution of primitive quartic torus packets for arbitrary orders},pdfauthor={OpenAI}} +\input{macros} +\title{Equidistribution of primitive quartic torus packets\\for arbitrary orders} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +Let $K$ range over totally real quartic fields with no proper intermediate field, and let $\OO$ be any order in $K$. We prove that the packets of periodic diagonal orbits attached to invertible $\OO$-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as $|\Disc(\OO)|$ tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index. +\end{abstract} +\tableofcontents +\input{introduction} +\input{packets} +\input{ordinary} +\input{rigidity} +\input{exterior} +\input{local-fourier} +\input{additive} +\input{sieve} +\input{conclusion} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/ordinary.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/ordinary.tex new file mode 100644 index 0000000000000000000000000000000000000000..79e7610264678c8ad17eb26dadafbf47459737fe --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/ordinary.tex @@ -0,0 +1,771 @@ +\section{Ordinary vectors and probability limits}\label{sec:ordinary} + +We first control vectors in the lattices themselves. A count near the +origin will prevent escape into the cusp, and a count near a vector with +no zero coordinate will bound the mass of small neighborhoods in \(X_4\). +The latter bound is the input to the entropy argument in +Section~\ref{sec:rigidity}. We follow the method of +\cite[Sections~2--5]{PrimeDegreePackets}, proving the estimates with +constants uniform over primitive totally real quartic fields, their orders, +and the orderings of their embeddings. The absence of a quadratic +subfield supplies the residue estimate needed in this degree. + +\subsection{Containment probabilities and unfolding} + +For a rational prime \(p\), write +\[ + R_p=\prod_{\mathfrak p\mid p}R_{\mathfrak p}. +\] +Here \(R_{\mathfrak p}\) is the integer ring of the completion +\(K_{\mathfrak p}\). Let \(v_{\mathfrak p}\) be its integer valuation, +normalized to be \(1\) on a uniformizer, and put +\[ + e_{\mathfrak p}=v_{\mathfrak p}(p),\qquad + f_{\mathfrak p}=[R/\mathfrak p:\Z/p\Z]. +\] +Thus \(\Nm\mathfrak p=p^{f_{\mathfrak p}}\) and +\(\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}=4\); +see \cite[Theorem~3.34]{MilneANT}. For \(p\mid q\) and a tuple +\(\boldsymbol l_p=(l_{\mathfrak p})_{\mathfrak p\mid p}\) of +nonnegative integers, call the set of \(x\in R_p\) with +\(v_{\mathfrak p}(x_{\mathfrak p})=l_{\mathfrak p}\) in every component +the valuation shell \(\boldsymbol l_p\). Choose \(x\) in this shell +and define +\begin{equation}\label{eq:ord-local-probability} + P_p(\boldsymbol l_p) + =\P_{t_p\in R_p^\times}\{x\in t_p\OO_p\}, +\end{equation} +where the probability is normalized unit Haar measure. The definition +is independent of \(x\): the unit group acts transitively on each set +with specified component valuations. + +For a nonzero integral \(R\)-ideal \(\mathfrak a\), set +\begin{equation}\label{eq:ord-ideal-weight} + P(\mathfrak a)= + \prod_{p\mid q} + P_p\bigl((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid p}\bigr). +\end{equation} +An empty product, including the case \(q=1\), is \(1\). If +\(0\ne b\in I\), its image has no zero local component and +\((b)I^{-1}\) is integral. In the coordinates +\(I_p=\xi_pR_p\), the valuations of \(\xi_p^{-1}b\) are those of +\((b)I^{-1}\). Lemma~\ref{lem:pkt-local-labels} and the independence +of the local choices therefore give +\begin{equation}\label{eq:ord-membership} + \P_{L\in\mathscr L_{\OO}(I)}\{b\in L\} + =P((b)I^{-1}). +\end{equation} + +For a nonnegative Borel function \(f\) on \(\R^4\), write +\[ + E_f(\Lambda)=\sum_{0\ne x\in\Lambda}f(x). +\] +This is a Borel function on \(X_4\): in a local basis chart it is a +countable sum of the Borel functions obtained by evaluating \(f\) at +the integer linear combinations of that basis. +For \(z>0\), define the integral over the surface with absolute +coordinate product \(z\) by +\begin{equation}\label{eq:ord-fiber} + \mathcal V_f(z)= + \sum_{\omega\in\{\pm1\}^4} + \int_{\R^3} + f\bigl(\omega_1e^{u_1},\omega_2e^{u_2},\omega_3e^{u_3}, + \omega_4z e^{-u_1-u_2-u_3}\bigr)\,du_1\,du_2\,du_3 . +\end{equation} +The signs and logarithmic measure are exactly those of \(T_\infty\). + +\begin{proposition}[Ordinary vector unfolding]\label{prop:ord-unfolding} +For every nonnegative Borel \(f\) on \(\R^4\), +\begin{equation}\label{eq:ord-unfolding} + \int_{X_4}E_f\,d\mu_* + =\frac1{\sqrt D\,\kappa} + \sum_{\substack{\mathfrak a\subset R\\\mathfrak a\ne0}} + P(\mathfrak a)\, + \mathcal V_f\bigl(\Nm\mathfrak a/X\bigr). +\end{equation} +Both sides are allowed to be infinite. +\end{proposition} + +\begin{proof} +This is the unfolding of \cite[Proposition~2.2]{PrimeDegreePackets} +with the packet normalization of Proposition~\ref{prop:pkt-model}. +Average the vector sum over \(L\) in +Equation~\eqref{eq:pkt-integration}, and use +Equation~\eqref{eq:ord-membership}. The factor \(N_{\OO}\) cancels, +and the left side becomes +\[ + \frac1{h_K\vol(\mathscr F_U)} + \sum_I\int_{\mathscr F_U} + \sum_{0\ne b\in I}P((b)I^{-1})f(c_I\sigma(b)v)\,dv . +\] +For fixed \(I\), the map \(b\mapsto(b)I^{-1}\) takes \(I\setminus\{0\}\) +onto the integral ideals in the inverse class of \(I\). Each fiber +is one free \(U\)-orbit. Indeed, two generators of the same principal +ideal differ by a unique unit. The coefficient \(P((b)I^{-1})\) is +constant on that orbit. Summing its unit translates and integrating +over \(\mathscr F_U\) gives +\[ + \int_{\mathscr F_U}\sum_{\epsilon\in U} + f(c_I\sigma(\epsilon b)v)\,dv + =\int_{T_\infty}f(c_I\sigma(b)v)\,dv . +\] +The translates tile \(T_\infty\), including its sign components. +All changes in the order of integration and summation are justified +by Tonelli's theorem. + +The absolute product of the coordinates of \(c_I\sigma(b)\) is +\[ + c_I^4|\Nm_{K/\Q}(b)| + =\frac{\Nm((b)I^{-1})}{X}. +\] +Translation of the first three logarithmic coordinates and relabeling +of the signs therefore identify the last integral with +\(\mathcal V_f(\Nm((b)I^{-1})/X)\), with Jacobian one. Each integral +ideal belongs to the inverse class of exactly one chosen \(I\). +Equation~\eqref{eq:pkt-unit-volume} now gives the coefficient in +Equation~\eqref{eq:ord-unfolding}. +\end{proof} + +\subsection{The mass and tail of the local weights} + +The ideal sum in Equation~\eqref{eq:ord-unfolding} need not have simple +coefficients at primes dividing \(q\). We will use its exact total +mass over valuation shells and a positive moment of their norms. +Both statements are the order case of +\cite[Lemma~3.1]{PrimeDegreePackets}; the argument below explains why +they are uniform even when the local order has large index. + +For \(p\mid q\), put +\[ + q_p=[R_p:\OO_p],\qquad + N_p(\boldsymbol l_p)=p^{\sum_{\mathfrak p\mid p} + f_{\mathfrak p}l_{\mathfrak p}}, + \qquad + Z_p=\prod_{\mathfrak p\mid p}(1-p^{-f_{\mathfrak p}})^{-1}. +\] +For a collection \(\boldsymbol l=(\boldsymbol l_p)_{p\mid q}\), write +\[ + d(\boldsymbol l)=\prod_{p\mid q}N_p(\boldsymbol l_p),\qquad + P(\boldsymbol l)=\prod_{p\mid q}P_p(\boldsymbol l_p),\qquad + Z_q=\prod_{p\mid q}Z_p . +\] +Thus \(d(\boldsymbol l)\) is the norm of the part of an integral ideal +supported above primes dividing \(q\), and \(\prod_{p\mid q}q_p=q\). + +\begin{lemma}[Local mass and moment]\label{lem:ord-shells} +With the preceding notation, +\begin{equation}\label{eq:ord-shell-mass} + \sum_{\boldsymbol l}\frac{P(\boldsymbol l)}{d(\boldsymbol l)} + =\frac{Z_q}{q}. +\end{equation} +For every \(\eta>0\) there is a constant \(C_\eta\), independent of +\(K,\OO\), such that +\begin{equation}\label{eq:ord-shell-moment} + \sum_{\boldsymbol l} + P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8} + \le C_\eta\,\frac{q^\eta}{q}. +\end{equation} +In particular there is an absolute constant \(C\) for which +\begin{equation}\label{eq:ord-shell-tail} + \sum_{d(\boldsymbol l)>X^{1/8}} + \frac{P(\boldsymbol l)}{d(\boldsymbol l)} + \le \frac Cq X^{-1/128}. +\end{equation} +\end{lemma} + +\begin{proof} +Normalize additive Haar measure \(m_p\) on \(R_p\) by \(m_p(R_p)=1\). +In a component \(R_{\mathfrak p}\), the set of elements of valuation +\(l_{\mathfrak p}\) has measure +\((1-p^{-f_{\mathfrak p}})p^{-f_{\mathfrak p}l_{\mathfrak p}}\). +Consequently the shell \(\boldsymbol l_p\) in \(R_p\) has measure +\begin{equation}\label{eq:ord-shell-measure} + m_p(\text{shell }\boldsymbol l_p) + =\frac1{Z_p\,N_p(\boldsymbol l_p)}. +\end{equation} +Its conditional additive measure is invariant under the transitive +unit action. It is therefore the image of unit Haar probability. +By Equation~\eqref{eq:ord-local-probability}, +\(P_p(\boldsymbol l_p)\) is exactly the fraction of this shell lying +in \(\OO_p\). The elements with a zero component have measure zero, +and \(m_p(\OO_p)=q_p^{-1}\). Summing +Equation~\eqref{eq:ord-shell-measure} over the intersections with +\(\OO_p\) gives +\[ + \sum_{\boldsymbol l_p} + \frac{P_p(\boldsymbol l_p)}{N_p(\boldsymbol l_p)} + =\frac{Z_p}{q_p}. +\] +Multiplication over \(p\mid q\) proves +Equation~\eqref{eq:ord-shell-mass}. + +For the moment, choose \(x\) with normalized additive Haar probability +on \(\OO_p\), and let \(v_{\mathfrak p}=v_{\mathfrak p}(x_{\mathfrak p})\). +The projection \(H_{\mathfrak p}\) of \(\OO_p\) into +\(R_{\mathfrak p}\) contains the embedded copy of \(\Z_p\), because +\(1\in\OO_p\). Its induced probability is normalized Haar on +\(H_{\mathfrak p}\). For an integer \(k\ge1\), the image of this +copy of \(\Z_p\) modulo the ideal of valuation \(k\) has +\(p^{\lceil k/e_{\mathfrak p}\rceil}\) elements. Hence the image of +\(H_{\mathfrak p}\) has at least that many elements, and +\begin{equation}\label{eq:ord-valuation-tail} + \P_{\OO_p}\{v_{\mathfrak p}\ge k\} + \le p^{-\lceil k/e_{\mathfrak p}\rceil}. +\end{equation} +This uses only the size of the additive image of the projection; it +does not require the projections to be independent. + +Set +\[ + j=\max_{\mathfrak p\mid p} + \left\lceil\frac{v_{\mathfrak p}}{e_{\mathfrak p}}\right\rceil. +\] +If \(j\ge a\ge1\), then some \(v_{\mathfrak p}\) is at least +\(e_{\mathfrak p}(a-1)+1\). Equation~\eqref{eq:ord-valuation-tail} +and a union bound over at most four components give +\(\P_{\OO_p}\{j\ge a\}\le4p^{-a}\). On the other hand, +\[ + N_p(x)=p^{\sum_{\mathfrak p\mid p}f_{\mathfrak p}v_{\mathfrak p}} + \le p^{j\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}} + =p^{4j}. +\] +These assertions hold outside the null sets where a component vanishes. +The tail formula for the expectation of \(p^{j/2}\) now yields +\begin{align} + \E_{\OO_p}N_p(x)^{1/8} + &\le \E_{\OO_p}p^{j/2} \notag\\ + &=1+\sum_{a\ge1} + (p^{a/2}-p^{(a-1)/2})\P_{\OO_p}\{j\ge a\} + \le 1+4p^{-1/2}. + \label{eq:ord-local-moment} +\end{align} + +Applying Equation~\eqref{eq:ord-shell-measure} once more gives the +exact factorization +\[ + \sum_{\boldsymbol l} + P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8} + =\frac1q\prod_{p\mid q} + \left(Z_p\,\E_{\OO_p}N_p(x)^{1/8}\right). +\] +Each factor is at most +\((1-p^{-1})^{-4}(1+4p^{-1/2})\). For any fixed \(\eta>0\), this is +at most \(p^\eta\) at all sufficiently large primes. The product of +the factors at the finitely many remaining primes is bounded by a +constant depending only on \(\eta\). Since +\(\prod_{p\mid q}p\le q\), this proves +Equation~\eqref{eq:ord-shell-moment}. Finally, take \(\eta=1/128\) +there and use \(q\le X\): +\[ + \sum_{d(\boldsymbol l)>X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)} + \le X^{-1/64} + \sum_{\boldsymbol l}P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8} + \le \frac Cq X^{-1/128}. +\] +This proves Equation~\eqref{eq:ord-shell-tail}. +\end{proof} + +\subsection{Ideal counts relative to the residue} + +The shell identity supplies the factor \(q^{-1}\). To use it in +Equation~\eqref{eq:ord-unfolding}, the count of the remaining ideal +part must retain the factor \(\kappa\). Recall +\[ + a_K(n)=\#\{\mathfrak a\subset R:\mathfrak a\ne0,\ \Nm\mathfrak a=n\}, + \qquad n\ge1, +\] +and let \(d_4\) be the fourfold divisor function. Unique factorization +of ideals makes \(a_K\) multiplicative. At every rational prime, +including a ramified one, its local generating series is +\[ + \sum_{j\ge0}a_K(p^j)T^j + =\prod_{\mathfrak p\mid p}(1-T^{f_{\mathfrak p}})^{-1}. +\] +Each factor is coefficientwise at most \((1-T)^{-1}\), and there are +at most four factors. Hence \(a_K(n)\le d_4(n)\) for every \(n\); +in particular \(a_K(p)\le4\). + +\begin{lemma}[Residue and prime sums]\label{lem:ord-residue} +There are absolute constants \(D_0>1\), \(h_0>0\), \(Y_0>1\), and +\(C\ge1\) with the following properties for every primitive totally +real quartic field \(K\). +\begin{enumerate} +\item If \(D\ge D_0\), then +\begin{equation}\label{eq:ord-residue-comparison} + C^{-1}\kappa\le h\zeta_K(1+h)\le C\kappa + \qquad(01\), the logarithmic derivative of the completed zeta +function gives +\begin{equation}\label{eq:ord-zero-sum} +\begin{aligned} + S_K(s):=\sum_\rho\Re\frac1{s-\rho} + &=\frac1s+\frac1{s-1}+\frac L2+ + \frac{\zeta_K'}{\zeta_K}(s)+G(s),\\ + G(s)&=2\left(\frac{\Gamma'(s/2)}{\Gamma(s/2)}-\log\pi\right); +\end{aligned} +\end{equation} +see \cite[Equation~(9)]{Stark1974}. The real zero sum converges and +has nonnegative terms, since \(\Re\rho\le1\). At \(s_0=1+L^{-1}\), +the Euler product gives \(\zeta_K'/\zeta_K(s_0)\le0\). The function +\(G\) is bounded on \([1,2]\), so for sufficiently large \(D\), +\[ + 0\le S_K(s_0)\ll L. +\] +This estimate persists for all \(10\). Integrating its logarithmic derivative across an +interval of length at most \(4/L\) proves +Equation~\eqref{eq:ord-residue-comparison}, with an absolute constant. +At \(s_0\), the inequality \(\zeta_K(s_0)\ge1\) then gives +\(\kappa\gg L^{-1}\). + +There are only finitely many fields with \(D0). +\] +The theorem's constant and threshold depend only on these growth +bounds and on the fixed parameter. Its strict length condition +\(u^\beta1\). +This gives Equation~\eqref{eq:ord-shiu}, including both endpoints. +\end{proof} + +\begin{proposition}[Weighted ideal intervals]\label{prop:ord-weighted-intervals} +There is an absolute constant \(C\) such that, for each fixed +\(A\ge1\) and \(c>0\), there is a threshold \(X_0(A,c)\) with the +following property. For every \(K,\OO,\sigma\) under consideration +with \(X\ge X_0(A,c)\), and all real \(x,y\) satisfying +\[ + X^{3/4}\le x\le AX,\qquad cx\le y\le x, +\] +one has +\begin{equation}\label{eq:ord-weighted-interval} + \sum_{\substack{\mathfrak a\subset R,\ \mathfrak a\ne0\\ + x-y<\Nm\mathfrak a\le x}}P(\mathfrak a) + \le C\,\frac{\kappa y}{q}. +\end{equation} +The constant \(C\) is independent of \(A,c\); the threshold may +depend on them. +\end{proposition} + +\begin{proof} +The range is empty for \(c>1\), so suppose \(0u}}\frac{a_K(p)}p + \ll\frac{\log(2q)}u\ll1 . +\end{equation} +For the first identity, expand +\(\log Z_p=\sum_{\mathfrak p\mid p}\sum_{j\ge1}p^{-jf_{\mathfrak p}}/j\). +The terms of order \(p^{-1}\) count exactly the primes of residue +degree one, and the remaining terms are \(O(p^{-2})\), uniformly +in the splitting type. Their sum over rational primes is bounded. +For the second estimate, use \(a_K(p)\le4\), at most +\(\log(2q)/\log2\) primes dividing \(q\), and +\(u\ge X^{5/8}\), \(q\le X\). +Consequently Equations~\eqref{eq:ord-prime-sum} and +\eqref{eq:ord-euler-removal} give +\[ + \exp\left(\sum_{p\le u}\frac{b(p)}p\right) + \ll \frac{\kappa\log u}{Z_q}. +\] +Lemma~\ref{lem:ord-shiu} and +Equation~\eqref{eq:ord-shell-mass} now bound the contribution of +these tuples to Equation~\eqref{eq:ord-ideal-split} by +\[ + C\frac{\kappa y}{Z_q} + \sum_{d(\boldsymbol l)\le X^{1/8}} + \frac{P(\boldsymbol l)}{d(\boldsymbol l)} + \le C\frac{\kappa y}{q}. +\] + +It remains to control tuples with \(d>X^{1/8}\). Those with \(d>x\) +contribute nothing. For \(z\ge1\), summing the first three factors +of the divisor convolution gives +\[ + \sum_{n\le z}d_4(n) + =\sum_{a_1a_2a_3a_4\le z}1 + \le z\left(\sum_{a\le z}\frac1a\right)^3 + \le z(1+\log z)^3 . +\] +Use this with \(z=x/d\), and then +Equation~\eqref{eq:ord-shell-tail}. The remaining contribution is +at most +\[ + Cx(1+\log x)^3 + \sum_{d(\boldsymbol l)>X^{1/8}} + \frac{P(\boldsymbol l)}{d(\boldsymbol l)} + \le C\frac{x}{q}X^{-1/128}(1+\log x)^3 . +\] +For fixed \(A\), once \(X\ge A\) the condition \(x\le AX\) gives +\(\log x\le2\log X\). By +Equation~\eqref{eq:ord-residue-lower} and \(D\le X^2\), the ratio of +this last bound to \(\kappa x/q\) is at most +\[ + C X^{-1/128}(\log(3X))^4=o(1). +\] +This bound is independent of the field and the order. Increase the +threshold in terms of \(A,c\) until the ratio is at most \(c\). +The tail is then at most \(\kappa cx/q\le\kappa y/q\). The constant +in the resulting bound has not changed with \(A\) or \(c\), proving +the proposition. +\end{proof} + +\subsection{Short vectors and small neighborhoods} + +We now apply the weighted interval bound to the two vector counts +announced at the start of the section. If \(f_r=\mathbf 1_{[-r,r]^4}\), direct integration +in Equation~\eqref{eq:ord-fiber} gives +\begin{equation}\label{eq:ord-cube-fiber} + \mathcal V_{f_r}(z)=\frac{16}{6} + \bigl(\log_+(r^4/z)\bigr)^3 . +\end{equation} +Indeed, in a fixed sign component put \(s_i=\log r-u_i\) for +\(1\le i\le3\). The cube conditions become \(s_i\ge0\) and +\(\sum_{i=1}^3s_i\le\log(r^4/z)\), a simplex of the displayed volume +divided by sixteen. + +\begin{proposition}[Tightness]\label{prop:ord-tightness} +There is an absolute constant \(C\) such that, for every fixed +\(00\), there is a compact \(\mathcal K_\epsilon\subset X_4\) with +\(\mu_i(\mathcal K_\epsilon)\ge1-\epsilon\) for all sufficiently large \(i\). +Such a sequence is tight, and every weak subsequential limit is an +\(A_4\)-invariant probability on \(X_4\). +\end{proposition} + +\begin{proof} +The count \(E_{f_r}\) dominates the indicator in +Equation~\eqref{eq:ord-cusp}. Put \(T=r^4X\) and \(T_0=X^{3/4}\). +By Equations~\eqref{eq:ord-unfolding} and \eqref{eq:ord-cube-fiber}, +it suffices to bound +\begin{equation}\label{eq:ord-cube-sum} + \frac1{\sqrt D\,\kappa} + \sum_{\Nm\mathfrak a\le T} + P(\mathfrak a) + \bigl(\log(T/\Nm\mathfrak a)\bigr)^3 . +\end{equation} +The sum is empty when \(T<1\). Otherwise its logarithmic factor is +at most \((\log X)^3\), since \(\Nm\mathfrak a\ge1\) and \(r<1\). +Apply Proposition~\ref{prop:ord-weighted-intervals} at +\(x=y=T_0\), with \(A=c=1\). The part with +\(\Nm\mathfrak a\le\min\{T,T_0\}\) is at most +\[ + C\frac{\kappa T_0}{q}(\log X)^3. +\] +After division by \(\sqrt D\,\kappa=X\kappa/q\), this contributes +\(O(X^{-1/4}(\log X)^3)=o(1)\). + +If \(T>T_0\), put +\[ + A_{T_0}(t)=\sum_{T_0<\Nm\mathfrak a\le t}P(\mathfrak a) + \qquad(T_0\le t\le T). +\] +The same interval estimate at \(x=y=t\) gives +\(A_{T_0}(t)\le C\kappa t/q\), since \(T\le X\). Writing each +logarithmic cube as an integral and using Tonelli yields +\begin{align*} + \sum_{T_0<\Nm\mathfrak a\le T} + P(\mathfrak a)\bigl(\log(T/\Nm\mathfrak a)\bigr)^3 + &=3\int_{T_0}^T A_{T_0}(t) + \bigl(\log(T/t)\bigr)^2\frac{dt}{t}\\ + &\le C\frac{\kappa}{q} + \int_0^T\bigl(\log(T/t)\bigr)^2\,dt + \le C\frac{\kappa T}{q}. +\end{align*} +The last integral is finite after the substitution \(t=Ts\). +This part of Equation~\eqref{eq:ord-cube-sum} is therefore \(O(r^4)\). +If \(T\le T_0\), it is absent. The constants used at \(A=c=1\) are +absolute, so the resulting estimate proves +Equation~\eqref{eq:ord-cusp}. This is the compactness argument of +\cite[Proposition~5.1]{PrimeDegreePackets}, now using the quartic +weighted interval bound. + +Mahler's compactness criterion \cite[Section~3, Theorem~2]{Mahler1946} +says that +\[ + \mathcal K_r=\{\Lambda\in X_4:\|v\|_\infty\ge r + \text{ for all }0\ne v\in\Lambda\} +\] +is compact. Choose \(r\) so that \(Cr^4<\epsilon/2\). +Equation~\eqref{eq:ord-cusp} then gives +\(\mu_i(\mathcal K_r)\ge1-\epsilon\) for all sufficiently large \(i\). +Each of the finitely many preceding packet measures is supported on a +finite union of compact orbits, by Proposition~\ref{prop:pkt-model}. +Enlarging the compact set to include those supports proves tightness +of the entire sequence. The space \(X_4\) is locally compact, +second countable, and metrizable; the weak compactness theorem for +tight probabilities therefore gives probability subsequential limits. +For \(a\in A_4\) and \(f\in C_c(X_4)\), the equality +\[ + \int f(xa)\,d\mu_i(x)=\int f(x)\,d\mu_i(x) +\] +passes to a weak limit, since both test functions are continuous with +compact support. This proves \(A_4\)-invariance. +\end{proof} + +\begin{lemma}[Boxes away from the coordinate hyperplanes]\label{lem:ord-boxes} +Let \(V_0\subset(\R^\times)^4\) be compact, and let \(b>0\). +There are constants \(C,r_0>0\), depending only on \(V_0,b\), such +that for every fixed \(00\) and \(t_+<\infty\) be the minimum and maximum of \(t_v\) +on \(V_0\), and shrink \(r_0\) so that \(6C_1r_00\), after decreasing \(r_0\) if +necessary so that \(c\le1\). + +Proposition~\ref{prop:ord-weighted-intervals} applies for every fixed +\(r\), with this \(c\) and the fixed \(A\). Its constant is independent +of \(c\), so the sum of the relevant ideal weights is at most +\(C\kappa rX/q\), with a constant independent of \(r\). Combining +this with Equations~\eqref{eq:ord-unfolding} and +\eqref{eq:ord-box-fiber}, and using \(\sqrt D=X/q\), proves +Equation~\eqref{eq:ord-vector-box}. Only the threshold for \(X\) +depends on the fixed radius. +\end{proof} + +For \(r>0\), define the open identity neighborhood +\begin{equation}\label{eq:ord-ball-neighborhood} + B(r)=\{h\in\mathrm{SL}_4(\R): + \max_{i,j}|h_{ij}-\delta_{ij}|0\) such that +\begin{equation}\label{eq:ord-balls} + \mu(xB(r))\le C_\Omega r^4 + \qquad(x\in\Omega,\ 04M\). If \(y=xh\) with \(h\in B(r)\), then +\(v_xh\in\Lambda_y\), and +\[ + |(v_xh-v_x)_j| + \le \sum_i |(v_x)_i|\,|h_{ij}-\delta_{ij}|0\) and all sufficiently large \(i\), +\[ + \mu_i(xB(r)) + \le \int E_{\mathbf 1_{v_x+[-br,br]^4}}\,d\mu_i + \le C_\Omega r^4 + \qquad(x\in\Omega). +\] +The constants are common to all \(x\) and \(r\) in the stated ranges; +the index after which this estimate holds may depend on \(r\). +The set \(xB(r)\) is open in \(X_4\). For each fixed \(x,r\), +Portmanteau's inequality therefore yields +\[ + \mu(xB(r))\le\liminf_{i\to\infty}\mu_i(xB(r)) + \le C_\Omega r^4. +\] +This proves the bound for every \(r\) with the same constant. It uses +no interchange between a shrinking radius and the packet limit. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/packets.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/packets.tex new file mode 100644 index 0000000000000000000000000000000000000000..3288df2cad4ff93057b2e1b337ce7f81105c3258 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/packets.tex @@ -0,0 +1,298 @@ +\section{Packets from arbitrary orders}\label{sec:packets} + +Fix a primitive totally real quartic field \(K\), an order \(\OO\subset K\), +and an ordering \(\sigma=(\sigma_1,\ldots,\sigma_4)\) of its real embeddings, +as in Theorem~\ref{thm:main}. We describe the packet using finitely many +local lattice choices above each ideal class of the maximal order. This +description lets us average a containment condition at each finite prime +before performing an integral at the real place. The construction is +adapted from the packet model in +\cite[Section~2, Lemma~2.1]{PrimeDegreePackets}; we give the argument with +all sixteen sign patterns in degree four. + +Throughout the paper put +\begin{equation}\label{eq:pkt-scale} + R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\qquad + X=q\sqrt D,\qquad \kappa=\Res_{s=1}\zeta_K(s). +\end{equation} +In particular, \(|\Disc(\OO)|=q^2D=X^2\). Absolute ideal norms are +extended multiplicatively to fractional \(R\)-ideals. Thus +\begin{equation}\label{eq:pkt-covolume} + \covol(\sigma(I))=\sqrt D\,\Nm I +\end{equation} +for every nonzero fractional \(R\)-ideal \(I\): this is the defining +discriminant identity for \(I=R\), follows by index for integral \(I\), +and then follows for fractional \(I\) after multiplication by an integer. +For a full \(\Z\)-lattice \(J\subset K\), write +\(J_p=J\otimes_\Z\Z_p\), and put +\[ + K_p=K\otimes_\Q\Q_p,\qquad R_p=R\otimes_\Z\Z_p,\qquad + \OO_p=\OO\otimes_\Z\Z_p . +\] +Here \(K_p\) is a product of finite extensions of \(\Q_p\), and \(R_p\) +is the product of their integer rings. We use the usual arithmetic of +fractional ideals in the Dedekind domain \(R\); see +\cite[Theorems~3.7, 3.20, and~3.29]{MilneANT}. + +\subsection{The finite local choices} + +Choose one fractional \(R\)-ideal \(I\) in each ordinary ideal class of +\(R\). Write \(h_K\) for the number of these classes. At every prime +choose a generator +\[ + I_p=\xi_p R_p,\qquad \xi_p\in K_p^\times. +\] +Such a generator exists because each component of \(R_p\) is a discrete +valuation ring. Let \(\mathscr L_{\OO}(I)\) be the set of full lattices +\(L\subset I\) whose completions have the form +\begin{equation}\label{eq:pkt-local-choices} + L_p=\xi_p t_p\OO_p,\qquad t_p\in R_p^\times + \quad\text{for every }p. +\end{equation} +Replacing \(\xi_p\) by another generator merely changes \(t_p\) by a +unit, so this set does not depend on the chosen generators. + +\begin{lemma}\label{lem:pkt-local-labels} +The set \(\mathscr L_{\OO}(I)\) is finite, and its cardinality is +\begin{equation}\label{eq:pkt-label-count} + N_{\OO}=\prod_{p\mid q}[R_p^\times:\OO_p^\times], +\end{equation} +independently of \(I\). Its members satisfy +\[ + [I:L]=q,\qquad RL=I, +\] +and are proper invertible fractional \(\OO\)-ideals. Every ordinary +class of proper invertible fractional \(\OO\)-ideals is represented, +up to multiplication by \(K^\times\), by a member of one of the sets +\(\mathscr L_{\OO}(I)\). Uniform probability on +\(\mathscr L_{\OO}(I)\) is induced by independent normalized Haar +probabilities for \(t_p\in R_p^\times\). +\end{lemma} + +\begin{proof} +The additive group \(R/\OO\) has order \(q\), so \(qR\subset\OO\). +For every \(p\) and every unit \(t_p\), it follows that +\[ + qI_p=\xi_p t_p qR_p\subset \xi_p t_p\OO_p\subset I_p. +\] +At \(p\nmid q\) the middle lattice is \(I_p\). At the remaining +primes its image is a submodule of the finite module \(I_p/qI_p\). +The primary decomposition of \(I/qI\) identifies its \(p\)-primary +part with \(I_p/qI_p\). Thus a choice of these local submodules +patches to a unique subgroup of \(I/qI\), and its inverse image in +\(I\) is the unique lattice with the prescribed completions. + +The stabilizer of \(\OO_p\) under multiplication by \(R_p^\times\) +is exactly \(\OO_p^\times\). Indeed, \(t_p\OO_p=\OO_p\) implies +that both \(t_p\) and \(t_p^{-1}\) belong to \(\OO_p\), by applying +the equality and its inverse to \(1\). The converse is immediate. +The local choices therefore form the finite homogeneous space +\(R_p^\times/\OO_p^\times\). Its invariant probability is uniform +and is the image of normalized unit Haar measure. Patching over the +finitely many primes dividing \(q\) proves +Equation~\eqref{eq:pkt-label-count} and the assertion about +probabilities. The index of the lattice in +Equation~\eqref{eq:pkt-local-choices} is +\([R_p:\OO_p]\), and its \(R_p\)-span is \(I_p\) because \(1\in\OO_p\). +Taking products of the indices and comparing all completions gives +\([I:L]=q\) and \(RL=I\). + +We record why the local condition describes exactly invertible ideals +of the order. A full fractional \(\OO\)-ideal \(J\) is invertible +if and only if +\begin{equation}\label{eq:pkt-local-principality} + J_p=\eta_p\OO_p\quad\text{for some }\eta_p\in K_p^\times + \quad\text{at every }p. +\end{equation} +For the forward implication, complete an equality +\(JJ^{-1}=\OO\). The resulting equality over \(\OO_p\) expresses +\(1\) as a finite sum \(\sum_i a_i b_i\), with \(a_i\in J_p\) and +\(b_i\in J_p^{-1}\). At each maximal ideal \(\mathfrak m\) of the +semilocal ring \(\OO_p\), some \(a_i b_i\) is a unit after localization, +and that \(a_i\) generates \((J_p)_{\mathfrak m}\). To obtain one +generator for all the maximal ideals, choose such an \(a_{\mathfrak m}\) +for each \(\mathfrak m\), and use the Chinese remainder theorem +\cite[Theorem~1.14]{MilneANT} to +choose \(c_{\mathfrak m}\in\OO_p\) congruent to \(1\) at +\(\mathfrak m\) and to \(0\) at every other maximal ideal. Then +\(\eta_p=\sum_{\mathfrak m}c_{\mathfrak m}a_{\mathfrak m}\) generates +\((J_p)_{\mathfrak m}\) modulo \(\mathfrak m\) at each \(\mathfrak m\); +Nakayama's lemma \cite[Lemma~1.9]{MilneANT} shows that it generates each localization. Hence it +generates \(J_p\). Since \(J_p\) spans \(K_p\) over \(\Q_p\), +\(\eta_p\) has no zero component and lies in \(K_p^\times\). + +Conversely, define +\[ + J^{-1}=\{a\in K:aJ\subset\OO\}. +\] +This definition commutes with completion: after choosing \(\Z\)-bases +of \(J\) and \(\OO\), the containment is a finite collection of +integrality conditions on the coordinates of multiplication by \(a\), +and the same conditions over \(\Z_p\) describe the completed lattice. +Under Equation~\eqref{eq:pkt-local-principality} we therefore have +\((J^{-1})_p=\eta_p^{-1}\OO_p\). Consequently +\((JJ^{-1})_p=\OO_p\) for every \(p\), which implies \(JJ^{-1}=\OO\). +This proves the equivalence. It also proves properness: if \(aJ\subset J\) +for an invertible \(J\), multiplication by \(J^{-1}\) gives +\(a\OO\subset\OO\), and hence \(a\in\OO\). Thus its multiplier order +is \(\OO\). + +The lattices in Equation~\eqref{eq:pkt-local-choices} are stable under +\(\OO_p\) at every prime, so comparison of completions shows that +\(\OO L\subset L\). They are therefore fractional \(\OO\)-ideals +satisfying Equation~\eqref{eq:pkt-local-principality}. In the other +direction, let \(J\) be an invertible fractional \(\OO\)-ideal. Its \(R\)-span +\(RJ\) is a fractional \(R\)-ideal. Multiplying \(J\) by an element +of \(K^\times\), we may arrange \(RJ=I\) for the chosen representative +of this maximal-order class. If \(J_p=\eta_p\OO_p\), then +\(\eta_p R_p=I_p=\xi_pR_p\), so \(\eta_p=\xi_p t_p\) for a unit +\(t_p\in R_p^\times\). Thus the resulting \(J\) belongs to +\(\mathscr L_{\OO}(I)\), as required. +\end{proof} + +\subsection{Real parameters, identifications, and volume} + +Let \(U=R^\times\), and let \(U^+\) be its totally positive subgroup. +Put +\[ + T_\infty=\{v=(v_1,\ldots,v_4)\in(\R^\times)^4: + \textstyle\prod_i|v_i|=1\}. +\] +We give \(T_\infty\) counting measure on its sixteen sign components +and measure \(du_1\,du_2\,du_3\) on the logarithms +\(u_i=\log|v_i|\), with \(u_4=-u_1-u_2-u_3\). Denote this measure +by \(dv\). The unit theorem +\cite[Theorems~5.1 and~5.9]{MilneANT} says that +\[ + \ell(U)=\{(\log|\sigma_i(\epsilon)|)_{i=1}^4:\epsilon\in U\} +\] +is a full lattice in the hyperplane \(\sum_i u_i=0\). Its +covolume for the deleted-coordinate measure is the ordinary regulator +\(R_K\). Its kernel on \(U\) consists of \(1\) and \(-1\), since +these are the roots of unity in a totally real field. + +Let \(\mathscr F_U\) be a measurable fundamental domain for the embedded +group \(\sigma(U)\) in \(T_\infty\). Reducing logarithms modulo +\(\ell(U)\) leaves sixteen sign choices, on which the remaining kernel +\(\{1,-1\}\) identifies pairs of opposite signs. It follows that +\(\vol(\mathscr F_U)=8R_K\). The analytic class-number formula in +signature \((4,0)\), with two roots of unity, gives +\begin{equation}\label{eq:pkt-unit-volume} + h_K\vol(\mathscr F_U)=8h_KR_K=\sqrt D\,\kappa; +\end{equation} +see \cite[Chapter~V, Theorem~2.4(a)]{MilneCFT}. + +For \(L\in\mathscr L_{\OO}(I)\), set +\[ + c_I=(X\Nm I)^{-1/4}. +\] +Equations~\eqref{eq:pkt-covolume} and \([I:L]=q\) show that +\(c_I\sigma(L)v\) has covolume one for every \(v\in T_\infty\). +This notation denotes the corresponding row lattice, and thus a point +of \(X_4\), after choosing a basis of determinant one. This convention +also applies when the diagonal \(v\) has negative determinant. + +\begin{proposition}[Packet model and volume]\label{prop:pkt-model} +The maps +\[ + (L,v)\longmapsto c_I\sigma(L)v + \qquad + (L\in\mathscr L_{\OO}(I),\ v\in T_\infty) +\] +give a bijection from the disjoint union over the chosen \(I\) of the +quotients by +\begin{equation}\label{eq:pkt-unit-action} + \epsilon:(L,v)\longmapsto + (\epsilon L,\sigma(\epsilon)^{-1}v),\qquad \epsilon\in U, +\end{equation} +onto \(\bigcup_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\mathcal Y\subset X_4\). +Counting measure on the \(L\)'s times \(dv\) descends to the sum of +the Haar measures of those orbits. In particular, +\begin{equation}\label{eq:pkt-total-volume} + \sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\vol(\mathcal Y) + =N_{\OO}\sqrt D\,\kappa . +\end{equation} +Writing \(\mu_*=\mu_{\OO,\sigma}\), for every nonnegative Borel +function \(\Phi\) on \(X_4\) one has +\begin{equation}\label{eq:pkt-integration} + \int_{X_4}\Phi\,d\mu_* + =\frac{1}{h_KN_{\OO}\vol(\mathscr F_U)} + \sum_I\sum_{L\in\mathscr L_{\OO}(I)} + \int_{\mathscr F_U}\Phi(c_I\sigma(L)v)\,dv . +\end{equation} +The identity permits infinite values on both sides. +\end{proposition} + +\begin{proof} +The action in Equation~\eqref{eq:pkt-unit-action} preserves the +parameter set, since \(\epsilon I=I\) for \(\epsilon\in U\), and +preserves its image lattice. Lemma~\ref{lem:pkt-local-labels} shows +that every ordinary invertible \(\OO\)-ideal class occurs. If two +ideals differ by multiplication by \(\alpha\in K^\times\), their +normalized embedded lattices differ by the diagonal with coordinates +\(\sigma_i(\alpha)/|\Nm_{K/\Q}(\alpha)|^{1/4}\), an element of +\(T_\infty\). Finally \(T_\infty\) is the union of all sixteen +sign translates of \(A_4\). These facts prove surjectivity. + +Suppose two displayed normalized lattices coincide. After absorbing +the ratio of their positive scalars, this equality gives a diagonal +map carrying \(\sigma(L)\) onto \(\sigma(L')\). Choose \(0\ne b\in L\) +and let \(b'\in L'\) be its image. None of the four coordinates of +\(\sigma(b)\) is zero. The diagonal entries must therefore be +\(\sigma_i(b')/\sigma_i(b)=\sigma_i(b'/b)\). The map is multiplication +by \(\alpha=b'/b\), and \(L'=\alpha L\). Taking \(R\)-spans gives +\(I'=\alpha I\). The chosen representatives then force \(I'=I\) +and \(\alpha I=I\), so \(\alpha\in U\). The scalar normalizations +are now equal, and the two parameters differ by exactly +Equation~\eqref{eq:pkt-unit-action}. This proves injectivity, including +all identifications involving signs. Equality of covolume-one row +lattices is the same as equality in \(X_4\): any two positively oriented +bases of the same lattice differ by an element of \(\mathrm{SL}_4(\Z)\). + +It remains to check that this identification gives the stipulated +orbit weights. Write +\(\OO^{\times,+}=\OO^\times\cap U^+\). The \(U\)-stabilizer of a +label \(L\) is \(\OO^\times\), by its multiplier order. The stabilizer +of \(L\) together with a sign pattern is therefore +\(\OO^{\times,+}\). This subgroup has finite index in \(U\). +For \(q=1\) this follows directly from \(\OO=R\). For \(q>1\), +the kernel of \(U\longrightarrow(R/qR)^\times\) lies in +\(\OO^\times\), since both a unit congruent to \(1\) modulo \(qR\) +and its inverse belong to \(\OO\), and imposing positive signs costs +at most a further factor of sixteen. The same diagonal-identification +argument shows that all periods of the corresponding \(A_4\)-orbit +are precisely \(\ell(\OO^{\times,+})\). They form a full lattice, +so the orbit is compact. + +Combining a label with a sign produces \(16N_{\OO}\) finite choices +above each \(I\). Each \(U\)-orbit of these choices has +\([U:\OO^{\times,+}]\) elements and parametrizes one \(A_4\)-orbit. +On that orbit the quotient of logarithmic Lebesgue measure is its +Haar measure, with volume +\[ + \covol\bigl(\ell(\OO^{\times,+})\bigr) + =[\ell(U):\ell(\OO^{\times,+})]R_K + =\frac{[U:\OO^{\times,+}]}{2}R_K . +\] +The last equality uses \(\ker(\ell|_U)=\{1,-1\}\) and +\(-1\notin\OO^{\times,+}\). Consequently the number of distinct +orbits times their common volume is +\[ + \frac{16h_KN_{\OO}}{[U:\OO^{\times,+}]} + \cdot \frac{[U:\OO^{\times,+}]}2 R_K + =8h_KN_{\OO}R_K. +\] +This proves both the assertion about the descended measure and +Equation~\eqref{eq:pkt-total-volume}, by +Equation~\eqref{eq:pkt-unit-volume}. + +One can also take \(\mathscr L_{\OO}(I)\times\mathscr F_U\) as a +fundamental domain for the diagonal action in +Equation~\eqref{eq:pkt-unit-action}: reducing \(v\) into +\(\mathscr F_U\) transports \(L\) by the same unique unit, outside +the usual null boundary set. Counting measure times \(dv\) is +invariant under this action. These domains have total measure +\(h_KN_{\OO}\vol(\mathscr F_U)\), equal to the total volume just +computed. Dividing their integral by that quantity proves +Equation~\eqref{eq:pkt-integration}. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/references.bib b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..1eebf1542f938881727621d34870c0740966b942 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/references.bib @@ -0,0 +1,311 @@ +@book{Linnik1968, + author = {Linnik, Yu. V.}, + title = {Ergodic Properties of Algebraic Fields}, + series = {Ergebnisse der Mathematik und ihrer Grenzgebiete}, + volume = {45}, + publisher = {Springer-Verlag}, + address = {Berlin--Heidelberg--New York}, + year = {1968}, + note = {Translated from the Russian by M. S. 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Helv.}, + year = {2026}, + note = {Published online January 22, 2026}, + doi = {10.4171/CMH/613} +} + +@article{Stark1974, + author = {Stark, Harold M.}, + title = {Some effective cases of the {Brauer--Siegel} theorem}, + journal = {Inventiones Mathematicae}, + volume = {23}, + number = {2}, + year = {1974}, + pages = {135--152}, + doi = {10.1007/BF01405166} +} + +@article{Pollack2020, + author = {Pollack, Paul}, + title = {Nonnegative multiplicative functions on sifted sets, and the square roots of {$-1$} modulo shifted primes}, + journal = {Glasgow Mathematical Journal}, + volume = {62}, + number = {1}, + year = {2020}, + pages = {187--199}, + doi = {10.1017/S0017089519000041}, + note = {Corrected author version: \url{https://www.pollack-math.net/brunmultiplicative4.pdf}}, + url = {https://www.pollack-math.net/brunmultiplicative4.pdf} +} + +@article{LindenstraussWeiss2001, + author = {Lindenstrauss, Elon and Weiss, Barak}, + title = {On sets invariant under the action of the diagonal group}, + journal = {Ergodic Theory and Dynamical Systems}, + volume = {21}, + number = {5}, + year = {2001}, + pages = {1481--1500}, + doi = {10.1017/S0143385701001717} +} + +@article{Shiu1980, + author = {Shiu, Peter}, + title = {A {Brun--Titchmarsh} theorem for multiplicative functions}, + journal = {Journal f{\"u}r die reine und angewandte Mathematik}, + volume = {313}, + year = {1980}, + pages = {161--170}, + doi = {10.1515/crll.1980.313.161} +} + +@article{Selberg1947, + author = {Selberg, Atle}, + title = {On an elementary method in the theory of primes}, + journal = {Norske Videnskabers Selskabs Forhandlinger, Trondheim}, + volume = {19}, + number = {18}, + year = {1947}, + pages = {64--67} +} + +@article{Rankin1938, + author = {Rankin, Robert 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} +} + +@book{Koukoulopoulos2019, + author = {Koukoulopoulos, Dimitris}, + title = {The Distribution of Prime Numbers}, + series = {Graduate Studies in Mathematics}, + volume = {203}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {2019}, + isbn = {978-1-4704-4754-0}, + url = {https://dms.umontreal.ca/~koukoulo/documents/publications/primes.pdf} +} + +@article{ELMV2009, + author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay}, + title = {Distribution of periodic torus orbits on homogeneous spaces}, + journal = {Duke Mathematical Journal}, + volume = {148}, + number = {1}, + year = {2009}, + pages = {119--174}, + doi = {10.1215/00127094-2009-023}, + url = {https://people.math.ethz.ch/~einsiedl/ELMV1-final-DMJ.pdf} +} + +@article{EKL2006, + author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon}, + title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {164}, + number = {2}, + year = {2006}, + pages = {513--560}, + doi = {10.4007/annals.2006.164.513}, + url = {https://annals.math.princeton.edu/2006/164-2/p04} +} + +@article{Rokhlin1967, + author = {Rokhlin, V. A.}, + title = {Lectures on the entropy theory of measure-preserving transformations}, + journal = {Russian Mathematical Surveys}, + volume = {22}, + number = {5}, + year = {1967}, + pages = {1--52}, + doi = {10.1070/RM1967v022n05ABEH001224} +} + +@article{Mahler1946, + author = {Mahler, Kurt}, + title = {On lattice points in {$n$}-dimensional star bodies. {I}. {Existence} theorems}, + journal = {Proceedings of the Royal Society of London. Series A}, + volume = {187}, + number = {1009}, + year = {1946}, + pages = {151--187}, + doi = {10.1098/rspa.1946.0072}, + url = {https://carmamaths.org/resources/mahler/docs/090.pdf} +} + +@misc{MilneANT, + author = {Milne, James S.}, + title = {Algebraic Number Theory}, + year = {2020}, + note = {Version 3.08}, + url = {https://www.jmilne.org/math/CourseNotes/ANT.pdf} +} + +@article{Bhargava2004III, + author = {Bhargava, Manjul}, + title = {Higher composition laws {III}: The parametrization of quartic rings}, + journal = {Annals of Mathematics}, + volume = {159}, + number = {3}, + year = {2004}, + pages = {1329--1360}, + doi = {10.4007/annals.2004.159.1329} +} + +@misc{PrimeDegreePackets, + author = {{OpenAI}}, + title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}}, + year = {2026} +} + +@misc{MilneCFT, + author = {Milne, James S.}, + title = {Class Field Theory}, + year = {2020}, + note = {Version 4.03}, + url = {https://www.jmilne.org/math/CourseNotes/CFT.pdf} +} + +@article{Weil1964, + author = {Weil, Andr{\'e}}, + title = {Sur certains groupes d'op{\'e}rateurs unitaires}, + journal = {Acta Mathematica}, + volume = {111}, + year = {1964}, + pages = {143--211}, + doi = {10.1007/BF02391012} +} + +@incollection{Tate1967, + author = {Tate, John T.}, + title = {Fourier analysis in number fields and {Hecke}'s zeta-functions}, + booktitle = {Algebraic Number Theory}, + editor = {Cassels, J. W. S. and Fr{\"o}hlich, Albrecht}, + publisher = {Academic Press}, + address = {London}, + year = {1967}, + pages = {305--347} +} + +@article{Widmer2010, + author = {Widmer, Martin}, + title = {Counting primitive points of bounded height}, + journal = {Transactions of the American Mathematical Society}, + volume = {362}, + number = {9}, + year = {2010}, + pages = {4793--4829}, + doi = {10.1090/S0002-9947-10-05173-1} +} + +@article{GreschonigSchmidt2000, + author = {Greschonig, Gernot and Schmidt, Klaus}, + title = {Ergodic decomposition of quasi-invariant probability measures}, + journal = {Colloquium Mathematicum}, + volume = {84/85}, + number = {2}, + year = {2000}, + pages = {495--514}, + doi = {10.4064/cm-84/85-2-495-514} +} diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/rigidity.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/rigidity.tex new file mode 100644 index 0000000000000000000000000000000000000000..05e36e817344699a6ccb3484ccaf9794abe097af --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/rigidity.tex @@ -0,0 +1,498 @@ +\section{The homogeneous obstruction in degree four}\label{sec:rigidity} + +For a sequence with \(X\to\infty\), Proposition~\ref{prop:ord-tightness} +shows that every weak subsequential limit of the packet measures is an +\(A_4\)-invariant probability. Proposition~\ref{prop:ord-balls} +supplies the compact ball estimate in Equation~\eqref{eq:ord-balls}. +We first turn that estimate +into positive entropy in almost every \(A_4\)-ergodic component. +Measure classification then makes those components homogeneous. +In degree four, the proper homogeneous components that remain have two +coordinate blocks of dimension two. We will show that each of them +carries an exterior-square lattice vector in one of three fixed planes. +The arithmetic part of the proof will exclude these three events. + +\subsection{Entropy from the compact ball estimate} + +Fix distinct real numbers \(\lambda_1,\ldots,\lambda_4\) with sum zero, +and set +\[ + a(t)=\diag(e^{\lambda_1t},\ldots,e^{\lambda_4t}), + \qquad + \lambda_a=\min_{i\ne j}|\lambda_i-\lambda_j|>0. +\] +For an \(a(\R)\)-invariant probability \(\xi\), let \(h_\xi(a(1))\) +denote the measure-theoretic entropy of right multiplication by \(a(1)\). +The argument below is the degree-four specialization of the +ordinary-ball argument in +\cite[Proposition~6.1 and Lemma~6.2]{PrimeDegreePackets}, which in turn +follows the strategy of \cite[Section~2.7.1]{ELMV2011}. +We include the argument because its statement for every dominated +invariant probability is needed when passing to ergodic components. + +Here is the entropy input. For an open identity neighborhood \(B\) +in \(\mathrm{SL}_4(\R)\), write +\[ + B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t). +\] +The following is the specialization to \(X_4\) of the +Einsiedler--Lindenstrauss--Michel--Venkatesh tube criterion +\cite[Corollary~3.3 and Equation~(3.4)]{ELMV2009}. The flow \(a(t)\) +is \(\R\)-diagonalizable, as required there. + +\begin{theorem}[Entropy criterion for two-sided tubes] +\label{thm:rig-tube-criterion} +Let \(\xi_i\) and \(\xi\) be \(a(\R)\)-invariant Borel probabilities +on \(X_4\), with \(\xi_i\) converging weak-* to \(\xi\). +Suppose that \(t_i>0\) tend to infinity and that \(\eta>0\). +Assume that for every compact \(\Omega\subset X_4\) there are an open +identity neighborhood \(B\subset\mathrm{SL}_4(\R)\) and a constant +\(C_\Omega\) such that +\[ + \xi_i\bigl(xB^{(-t_i,t_i)}\bigr) + \le C_\Omega e^{-2\eta t_i} + \qquad(x\in\Omega,\ i\ge1). +\] +Then \(h_\xi(a(1))\ge\eta\). +\end{theorem} + +In this criterion \(B\) may depend on \(\Omega\), and the +probabilities need not be ergodic under the flow. The requirement +that the weak-* limit is a probability is part of the theorem. + +\begin{proposition}[Entropy for dominated probabilities] +\label{prop:rig-dominated-entropy} +Let \(\mu\) be an \(A_4\)-invariant probability on \(X_4\) satisfying +the compact ball bound in Equation~\eqref{eq:ord-balls} for every compact set of +centers. If \(\xi\) is an \(A_4\)-invariant probability with +\(\xi\le c\mu\) for some finite \(c\), then +\[ + h_\xi(a(1))\ge \lambda_a/3. +\] +\end{proposition} + +\begin{proof} +Use the entrywise neighborhoods \(B(r)\) defined in +Equation~\eqref{eq:ord-ball-neighborhood}. Fix \(00\), then +\[ + \xi=\frac1w\int_{Z_0}\nu_z\,d\tau(z) +\] +would be an \(A_4\)-invariant probability with \(\xi\le w^{-1}\mu\). +The restricted form of Equation~\eqref{eq:rig-entropy-decomposition} +would give \(h_\xi(T)=0\), contrary to +Proposition~\ref{prop:rig-dominated-entropy}. +\end{proof} + +\subsection{Finite-volume subgroups containing the diagonal group} + +We next use the measure-classification theorem of +Einsiedler--Katok--Lindenstrauss +\cite[Theorem~1.3; definition in Conjecture~1.1]{EKL2006}. +We state the exact specialization that supplies the homogeneous +measures needed here. + +\begin{theorem}[Positive-entropy measure classification] +\label{thm:rig-ekl-classification} +For \(n\ge3\), let \(A_n\) be the group of positive diagonal matrices +in \(\mathrm{SL}_n(\R)\). Let \(\nu\) be an \(A_n\)-invariant and +\(A_n\)-ergodic probability on +\(\mathrm{SL}_n(\R)/\mathrm{SL}_n(\Z)\), with the left action. +If \(h_\nu(a_0(1))>0\) for some one-parameter subgroup +\(a_0:\R\to A_n\), then there are a +closed connected subgroup \(L\subset\mathrm{SL}_n(\R)\) containing +\(A_n\) and a point \(x\) such that \(Lx\) is closed and \(\nu\) is +the \(L\)-invariant probability on \(Lx\). +\end{theorem} + +No ergodicity for the chosen one-parameter subgroup is assumed. +For \(n=4\), inversion +\(\mathrm{SL}_4(\Z)g\mapsto g^{-1}\mathrm{SL}_4(\Z)\) +turns right multiplication by \(a(t)\) into left multiplication by +\(a(-t)\), and entropy is unchanged upon inverting an invertible +transformation. Thus, on our right quotient, the theorem says that +a positive-entropy \(A_4\)-ergodic probability is invariant +probability on an orbit \(\Lambda J\), where \(J\) is closed and +connected, contains \(A_4\), and the stabilizer of \(\Lambda\) is a +lattice in \(J\). + +For a partition \(\mathcal I=\{I_1,\ldots,I_s\}\) of +\(\{1,2,3,4\}\) into nonempty subsets, define +\begin{equation}\label{eq:rig-block-group} + J_{\mathcal I} + =\left\{g\in\mathrm{SL}_4(\R): + \begin{array}{l} + g_{ij}=0\text{ if \(i,j\) belong to different \(I_v\)},\\ + \det(g|_{\R^{I_v}})>0\text{ for every }v + \end{array}\right\}. +\end{equation} +This is the identity component of the corresponding block-diagonal +group. Lindenstrauss and Weiss identified the equal-block phenomenon +for orbit closures containing a compact diagonal orbit +\cite[Theorem~1.3]{LindenstraussWeiss2001}; their result also enters the +proof of \cite[Corollary~1.4]{EKL2006}. The next lemma gives a direct +finite-volume argument and records the rational structure of the blocks. + +\begin{lemma}[Equal blocks and a rational commutant] +\label{lem:rig-equal-blocks} +Let \(J\) be a closed connected subgroup of \(\mathrm{SL}_4(\R)\) +containing \(A_4\), and let \(\Lambda\in X_4\). Suppose that the +stabilizer \(\Gamma_\Lambda=\{j\in J:\Lambda j=\Lambda\}\) is a +lattice in \(J\). +Then \(J=J_{\mathcal I}\) for a partition \(\mathcal I\) whose blocks +all have the same size. + +More precisely, put \(W_\Q=\Q\Lambda\) and +\(W_\R=W_\Q\otimes_\Q\R=\R^4\). There is a totally real number field +\(E\subset\End_\Q(W_\Q)\), of degree \(s=|\mathcal I|\), whose scalar +extension to \(\R\) is exactly the commutant of \(J\) in +\(\End_\R(W_\R)\). The coordinate blocks \(\R^{I_v}\) are the real +eigenspaces of \(E\) on \(W_\R\). +\end{lemma} + +\begin{proof} +Let \(\mathfrak j\) be the Lie algebra of \(J\). +It contains the trace-zero diagonal algebra and is invariant under +diagonal conjugation. The other weight spaces in +\(\mathfrak{sl}_4(\R)\) are the distinct one-dimensional spaces +\(\R E_{ij}\), \(i\ne j\), where \(E_{ij}\) is a matrix unit. +An invariant subspace is a sum of these weight spaces, so +\(\mathfrak j\) consists of the diagonal algebra and a selection of +the root spaces. + +The group \(J\) is unimodular. +One way to see this standard fact is to use the quotient +criterion for Haar measures: existence of an invariant quotient +measure and triviality of the modular character of the discrete +group \(\Gamma_\Lambda\) imply that the modular character of \(J\) +is trivial on \(\Gamma_\Lambda\). It therefore descends to +\(\Gamma_\Lambda\backslash J\). Its pushforward of invariant +probability would be invariant under translations by its image. +A nontrivial positive dilation admits no invariant probability on +\(\R_{>0}\), as its disjoint dilates of a fundamental interval all +have the same mass. Hence the modular character is trivial on \(J\). + +Draw a directed edge \(i\to j\) when \(E_{ij}\in\mathfrak j\). +The trace of the adjoint action of +\(\diag(u_1,\ldots,u_4)\), with \(\sum_i u_i=0\), on +\(\mathfrak j\) is the sum of the selected roots \(u_i-u_j\). +Unimodularity makes this sum zero. Its coefficient at \(u_i\) is +the outgoing degree of \(i\) minus its incoming degree. These +coefficients sum to zero, and a linear form vanishing on +\(\sum_i u_i=0\) has all coefficients equal. Thus every vertex +has equal outgoing and incoming degrees. + +Every edge now lies within a strongly connected component. +Otherwise the finite directed acyclic graph of components would +have a source with an outgoing edge; summing degree balance over +that source would give a positive outgoing degree and zero incoming +degree. For distinct indices, the bracket identity +\([E_{ij},E_{jk}]=E_{ik}\) makes the edge relation transitive. +Each strongly connected component is consequently a complete +directed graph. These components form a partition \(\mathcal I\), +and \(\mathfrak j\) is the Lie algebra of \(J_{\mathcal I}\). +Connected Lie subgroups with the same Lie algebra agree, giving +\(J=J_{\mathcal I}\). + +We use \(\Gamma_\Lambda\) to recover a rational structure on the +block scalars. Its elements preserve \(\Lambda\), hence act +rationally on \(W_\Q\). Let +\[ + \mathcal C_\Q + =\{M\in\End_\Q(W_\Q):M\gamma=\gamma M + \text{ for every }\gamma\in\Gamma_\Lambda\}. +\] +In a lattice basis the commutation equations have integer +coefficients. A finite subcollection of these linear equations +defines the same solution space, by finite dimensionality. +It follows that \(\mathcal C_\Q\otimes_\Q\R\) is the real +commutant of \(\Gamma_\Lambda\). + +We claim that this is also the real commutant of \(J\). +Let \(M\) be a real endomorphism commuting with \(\Gamma_\Lambda\). +The continuous map +\[ + \Gamma_\Lambda\backslash J\longrightarrow\End_\R(W_\R), + \qquad \Gamma_\Lambda j\longmapsto j^{-1}Mj +\] +is well-defined. Its pushforward of invariant probability is +invariant under conjugation by \(A_4\). Each off-diagonal +coordinate is multiplied by a nontrivial positive dilation under +some element of \(A_4\). A probability on \(\R\) invariant under +such a dilation is supported at zero: the annuli +\(\{x:r^k\le |x|1\), +are disjoint dilates covering \(\R\setminus\{0\}\). +Thus every off-diagonal coordinate of \(j^{-1}Mj\) is zero almost +everywhere. Invariant probability on the transitive space +\(\Gamma_\Lambda\backslash J\) has full support; continuity makes +\(j^{-1}Mj\) diagonal for every \(j\in J\). +In particular \(M\) is diagonal. Conjugating by +\(1+tE_{ij}\) for \(i,j\) in the same block forces its \(i\)-th +and \(j\)-th diagonal entries to agree. Therefore \(M\) is scalar +on each block, which is exactly the condition that it commute with +\(J\). The reverse inclusion is immediate, proving the claim. + +It follows that \(\mathcal C_\Q\otimes_\Q\R\simeq\R^s\). +In particular, \(\mathcal C_\Q\) is commutative and has no +nonzero nilpotents. The structure theorem for finite reduced +commutative algebras over \(\Q\) expresses it as a product of +number fields, all totally real because its real scalar extension +has no complex factor. + +This product has only one factor. If +\(e\in\mathcal C_\Q\) were an idempotent other than \(0\) and \(1\), +then \(W=e(W_\R)\) would be a nonempty proper union of coordinate +blocks, rational relative to \(\Lambda\). Every +\(\gamma\in\Gamma_\Lambda\) preserves the full lattice +\(\Lambda\cap W\) in \(W\), so +\(\lvert\det(\gamma|_W)\rvert=1\). +The continuous character +\[ + \chi:J\longrightarrow\R_{>0},\qquad + \chi(j)=\lvert\det(j|_W)\rvert +\] +is therefore trivial on \(\Gamma_\Lambda\). +It is surjective: if \(k=\dim W\), a diagonal matrix may scale the +coordinates in \(W\) by \(e^{(4-k)t}\) and those in its complement +by \(e^{-kt}\). Its total determinant is one and its determinant +on \(W\) is \(e^{k(4-k)t}\). +The character \(\chi\) would consequently push the invariant +probability on \(\Gamma_\Lambda\backslash J\) to a probability on +\(\R_{>0}\) invariant under all multiplicative translations. +The disjoint-interval argument above rules this out. +Thus \(\mathcal C_\Q\) is a field, which we denote by \(E\). + +Since \(E\otimes_\Q\R\simeq\R^s\), its degree is \(s\). +The space \(W_\Q\) is a vector space over \(E\), and +\[ + W_\Q\otimes_\Q\R + \simeq\bigoplus_{\tau:E\hookrightarrow\R} + W_\Q\otimes_{E,\tau}\R. +\] +Each summand has dimension \(\dim_E W_\Q=4/s\). +These summands are the eigenspaces of the block-scalar algebra, +namely the coordinate blocks. The blocks therefore have equal size. +\end{proof} + +For a positive-entropy homogeneous probability the lemma leaves +\(J=\mathrm{SL}_4(\R)\) or two blocks of size two. +Indeed, the only other equal-block possibility is four singleton +blocks, giving \(J=A_4\). A lattice in \(A_4\simeq\R^3\) has +compact quotient, and translations on this torus have zero entropy: +for a flat metric they are isometries, so the iterated Bowen metrics +equal the original metric and the topological entropy is zero; +the variational principle bounds measure entropy by topological entropy. + +For a possible two-block limit orbit \(\Lambda J\), the quadratic field +\(E\subset\End_\Q(\Q\Lambda)\) has real scalar extension equal to +the commutant of \(J\) in \(\End_\R(\R^4)\). +The construction supplies no embedding of \(E\) into the packet fields +\(K_i\), so their primitivity does not by itself exclude this orbit. +We will exclude it by proving that packet limits give zero mass to the +exterior obstruction below. + +\subsection{The three exterior obstruction events} + +Let \(e_1,\ldots,e_4\) be the coordinate basis of \(\R^4\). +For \(w\in\bigwedge^2\R^4\), write \(w_{ij}\), \(i0\) for some +one-parameter subgroup \(a_0:\R\to A_4\). +Then either \(\nu=m_4\), or there exists +\(b\in\mathcal B\) such that \(\nu(\mathcal E_b)=1\). +In the latter case every lattice \(\Lambda'\) in the homogeneous orbit carrying +\(\nu\) has a vector in \(\bigwedge^2\Lambda'\cap V_b\) with both +coordinates nonzero and with nonzero integer \(Q\)-value. +\end{proposition} + +\begin{proof} +Theorem~\ref{thm:rig-ekl-classification}, in the right-quotient +form explained above, represents \(\nu\) as invariant probability +on \(\Lambda J\) for a connected closed subgroup \(J\) containing +\(A_4\). By Lemma~\ref{lem:rig-equal-blocks} and the +positive-entropy assumption, either \(J=\mathrm{SL}_4(\R)\), which +gives \(\nu=m_4\), or \(J\) has two coordinate blocks of size two. + +In the second case write the blocks as \(\{i,j\}\) and \(\{k,l\}\), +and put \(b=ij|kl\). The commutant field \(E\) in +Lemma~\ref{lem:rig-equal-blocks} is quadratic. Choose a nonzero +\(\alpha\in E\) with \(\Tr_{E/\Q}(\alpha)=0\). +Its two scalar values on the blocks are \(c\) and \(-c\), with +\(c\ne0\). The rational endomorphism \(\alpha\) of +\(W_\Q=\Q\Lambda\) induces a rational endomorphism +\[ + D_\alpha:\bigwedge\nolimits^2 W_\Q\longrightarrow + \bigwedge\nolimits^2 W_\Q,\qquad + D_\alpha(v\wedge w)=\alpha(v)\wedge w+v\wedge\alpha(w). +\] +On the exterior square of the first block it acts by \(2c\), and +on that of the second by \(-2c\). It vanishes on the cross-block +summand. Its real image is therefore exactly +\[ + \bigwedge\nolimits^2\R^{\{i,j\}} + \oplus \bigwedge\nolimits^2\R^{\{k,l\}}=V_b. +\] +Because \(D_\alpha\) is rational, \(V_b\) is a rational subspace +relative to \(\bigwedge^2 W_\Q\). + +It follows that \(V_b\cap\bigwedge^2\Lambda\) is a rank-two lattice +in \(V_b\). It contains a vector off both coordinate axes: take +two independent lattice vectors; if neither is already off the +axes, their sum is. For this vector \(Q(w)\ne0\), and +Equation~\eqref{eq:rig-Q-integral} makes its value a nonzero integer. +Every element of \(J\) preserves each of the two exterior lines, +multiplying its coefficient by the nonzero determinant of the +corresponding block. The translated vector therefore has both +coordinates nonzero at every lattice of \(\Lambda J\). +This proves \(\nu(\mathcal E_b)=1\). +\end{proof} + +Corollary~\ref{cor:rig-positive-components} and +Proposition~\ref{prop:rig-obstruction} reduce identification of a +packet probability limit \(\mu\) to proving +\(\mu(\mathcal E_b)=0\) for all three \(b\in\mathcal B\). +Equation~\eqref{eq:rig-Q-integral} lets us test those events one +nonzero integer \(Q\)-value at a time. The next sections establish +the exterior-vector estimate used for this exclusion. diff --git a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/sieve.tex b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/sieve.tex new file mode 100644 index 0000000000000000000000000000000000000000..87c4075f1f14483107c65404242074be926e272c --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/sieve.tex @@ -0,0 +1,574 @@ +\section{Averaging the unramified factors}\label{sec:sieve} + +Proposition~\ref{prop:additive-equidistribution} distributes the +nonnegative mass $\phi(z)\prod_{p\in S}f_p^*(z)$ among residue classes at +primes outside $S$. To prove Theorem~\ref{thm:exterior-count}, we must +insert the remaining +factors $b_p(z)$ while retaining their local means. Some of those means are +less than one, so the argument must also keep track of the probability that +a prime does not divide any component of $z$. We first obtain a weighted +sieve estimate with this feature and then combine its local means with the +Euler factors in Proposition~\ref{prop:ext-multiplicity-majorant}. + +\subsection{Residue functions and the local bounds} + +Fix $m\in\Z\setminus\{0\}$ and a bounded region of the real trace-$m$ +plane. Throughout the sieve argument, $\phi$ is one of the nonnegative +smooth box functions in Proposition~\ref{prop:additive-equidistribution}, +with its planar support in this region. Its two scales lie between +$X^{-\tau}$ and +fixed constants. Define a measure on $F$ by +\begin{equation}\label{eq:sieve-measure} + \nu_\phi + =\sum_{\substack{z\in F,\ \Tr z=m\\z\text{ integral outside }S}} + \phi(z)\prod_{p\in S}f_p^*(z)\,\delta_z, + \qquad + V_\phi=A_\phi\prod_{p\in S}d_p(m). +\end{equation} +Here $\delta_z$ is unit point mass at $z$. All its coefficients are +nonnegative, and Proposition~\ref{prop:additive-equidistribution} with +modulus one gives a finite total mass. +Lemma~\ref{lem:fourier-scalar-density} gives +\begin{equation}\label{eq:sieve-volume} + V_\phi\asymp A_\phi +\end{equation} +uniformly in $S$: for large primes the logarithms of $d_p(m)=1+O(p^{-2})$ +have an absolutely convergent sum, and the finitely many small-prime +densities have uniform positive upper and lower bounds. + +For a positive integer $l$ prime to every prime of $S$, the integral +trace-$m$ coset has $l^2$ residue classes modulo $l$ in its two integral +coordinates. Write $\E_l W$ for the uniform mean of a function $W$ on +these classes, and also write $W(z)$ for its pullback to points integral +outside $S$. + +\begin{lemma}[Nonnegative residue functions]\label{lem:sieve-residue-function} +With $\phi$ and $V_\phi$ as in Equation~\eqref{eq:sieve-measure}, let +$1\le l\le X^\theta$ be an integer prime to every prime of $S$. +For every nonnegative +function $W$ on the residue classes of the integral trace-$m$ coset +modulo $l$, +\begin{equation}\label{eq:sieve-residue-function} + \int W\,d\nu_\phi + =V_\phi\E_l W+O\bigl(V_\phi X^{-\theta}\E_l W\bigr). +\end{equation} +The implied constant is uniform in the fields, orders, embedding +orderings, permitted box locations and scales, $l$, and $W$. It may +depend on $m$, the fixed bounded region, and the fixed smooth box function. +\end{lemma} + +\begin{proof} +For each residue class $a$, Equation~\eqref{eq:add-equidistribution} gives +\[ + \nu_\phi(a)=V_\phi l^{-2}(1+\epsilon_a), + \qquad |\epsilon_a|\le C_{\mathrm{add}}X^{-\theta}, +\] +with the same $C_{\mathrm{add}}$ for every $a$ and $l$ in the stated range. +Multiply by +$W(a)$ and sum. Since $W(a)\ge0$, the absolute value of the summed error +is at most +$C_{\mathrm{add}}V_\phi X^{-\theta}l^{-2}\sum_a W(a) + =C_{\mathrm{add}}V_\phi X^{-\theta}\E_l W$. +This also covers $\E_l W=0$. In particular the error retains the local +mean even when that mean is very small. +\end{proof} + +We recall the information about the functions to be inserted. For +$p\notin S$, let $\E_p$ and $\P_p$ denote expectation and probability for +normalized additive Haar measure on +$\{z\in R_{F,p}:\Tr z=m\}$. On $R_{F,p}\cap F_p^\times$, recall that +$n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z)$ is the largest +component valuation, with $v_{\mathfrak q}(p)=1$; see +Equation~\eqref{eq:ext-unramified-factor}. On the null set where a +component vanishes, keep the convention $n_p=\infty$, $b_p=0$. +Lemma~\ref{lem:ext-unramified-data} permits one fixed integer $C\ge1$ +such that, on the nonvanishing locus, +\begin{align} + &b_p\ge0,\qquad b_p=1\ \text{if }n_p=0,\qquad + b_p\le C(1+n_p)^C,\label{eq:sieve-local-weight}\\ + &\P_p(n_p\ge j)\le Cp^{-j}\quad(j\ge1),\qquad + \vartheta_p:=\P_p(n_p\ge1)\le C/p,\quad + \vartheta_p\le\tfrac12. \label{eq:sieve-local-tail} +\end{align} +The event $\{n_p\ge j\}$ and the function +$b_p\mathbf 1_{\{n_p=j\}}$ are determined by the two integral coordinates +modulo $p^{C(j+1)}$ for every integer $j\ge1$. They have extensions +defined by congruence classes on the whole local coset, agreeing with their values at every +global point. A prime with $n_p(z)\ge1$ will be called active at $z$. +In particular, +\begin{equation}\label{eq:sieve-mean-lower} + \E_p b_p\ge1-\vartheta_p\ge\tfrac12. +\end{equation} + +Lemma~\ref{lem:ext-support} places the support of $\nu_\phi$ in the +trace-$m$ coset of $(C_F^\sharp)^\vee$. On this support all $n_p(z)$, +$p\notin S$, are finite. The lemma's norm and depth conclusions bound +the product of the depth factors by $O(X^{84})$ in our fixed bounded real region. Thus, +for a fixed $B>84$ and all sufficiently large $X$, +\begin{equation}\label{eq:sieve-depth-product} + \prod_{p\notin S}p^{n_p(z)}\le X^B + \qquad\bigl(\nu_\phi(\{z\})>0\bigr). +\end{equation} + +Two consequences will be used repeatedly. There is a fixed $A_0>1$ such +that $b_p(z)\le A_0^{n_p(z)}$. This follows from +Equation~\eqref{eq:sieve-local-weight} for $n_p\ge1$, since +$\sup_{j\ge1}(C(1+j)^C)^{1/j}<\infty$, and from $b_p=1$ at depth zero. +Also, with +\[ + \mathcal W(z)=\prod_{p\notin S}b_p(z), +\] +the product is finite on the support of $\nu_\phi$ and satisfies +\begin{equation}\label{eq:sieve-pointwise-weight} + \mathcal W(z)\ll_\epsilon X^\epsilon + \qquad(\epsilon>0). +\end{equation} +Indeed, for any $t>0$, the inequality +$C(1+j)^C\le p^{tj}$ holds for all $j\ge1$ once $p$ is sufficiently large. +For each of the remaining finitely many primes it holds with a constant +factor independent of $j$. Multiplying these inequalities and using +Equation~\eqref{eq:sieve-depth-product}, with $t=\epsilon/B$, proves +Equation~\eqref{eq:sieve-pointwise-weight}. All constants here are uniform +in the arithmetic data and in the permitted boxes. + +\begin{proposition}[Weighted averaging]\label{prop:sieve-averaging} +For the measure $\nu_\phi$ in Equation~\eqref{eq:sieve-measure}, there is +a fixed $\delta>0$ such that +\begin{equation}\label{eq:sieve-average} + \int\mathcal W(z)\,d\nu_\phi(z) + \ll V_\phi\prod_{\substack{p\le X^\delta\\p\notin S}}\E_p b_p. +\end{equation} +The choice of $\delta$ and the implied constant are independent of the +field, order, embedding ordering, and the permitted box location and +scales. They may depend on $m$, the fixed bounded real region, and the +fixed smooth box function. +\end{proposition} + +The proof first counts bounded products of prescribed positive depths, +while requiring inactivity at any other chosen small primes. The +inactivity condition preserves the factors in +Equation~\eqref{eq:sieve-mean-lower}. We then bound the contribution of two +exceptional sets of points and truncate the remaining depth products within +the modulus range of +Lemma~\ref{lem:sieve-residue-function}. + +\subsection{Prescribed depths and inactive primes} + +Write +$\mathcal P_y=\{p\le y:p\notin S\}$. Chebyshev's estimate +$\sum_{p\le y}(\log p)/p\ll\log y$ and +Equation~\eqref{eq:sieve-local-tail} give a fixed $C_*\ge1$ such that +\[ + \sum_{p\in\mathcal P_y}\vartheta_p\log p\le C_*\log y + \qquad(y\ge2). +\] +Choose fixed positive parameters satisfying +\begin{equation}\label{eq:sieve-parameters} + 20(C+1)\eta<\theta,\qquad + 0<\delta<\min\{\eta/4,\eta/(4C_*)\}, + \qquad T_0=X^\eta,\quad Y=X^\delta. +\end{equation} +Here $T_0$ truncates the squarefree Selberg divisors and is the stopping +threshold for depth products, whose crossing prefixes may reach $T_0^2$; +$Y$ bounds the small primes outside $S$ handled by the sieve. +For every $\mathcal I\subset\mathcal P_Y$, Mertens' estimate and +$-\log(1-\vartheta_p)\le2\vartheta_p\le2C/p$ give, for a fixed $C_1$, +\begin{equation}\label{eq:sieve-inactivity-lower} + \prod_{p\in\mathcal I}(1-\vartheta_p) + \gg(\log X)^{-C_1}. +\end{equation} +The bound is uniform in the subset $\mathcal I$ and in $S$. + +\begin{lemma}[Prescribed depths and inactive primes]\label{lem:sieve-assignment} +Let $\mathcal A,\mathcal I$ be disjoint subsets of $\mathcal P_Y$, and +assign an integer $j_p\ge1$ to each $p\in\mathcal A$. Suppose that +$\prod_{p\in\mathcal A}p^{j_p}\le T_0^2$. Then +\begin{equation}\label{eq:sieve-assignment} + \int + \prod_{p\in\mathcal A} + b_p\mathbf 1_{\{n_p=j_p\}} + \prod_{p\in\mathcal I}\mathbf 1_{\{n_p=0\}}\, + d\nu_\phi + \ll V_\phi + \prod_{p\in\mathcal A}\E_p + \bigl(b_p\mathbf 1_{\{n_p=j_p\}}\bigr) + \prod_{p\in\mathcal I}(1-\vartheta_p). +\end{equation} +The constant is uniform in the two subsets and all their assigned depths. +\end{lemma} + +\begin{proof} +We use the Selberg upper-bound square +\cite{Selberg1947}; see also \cite[Chapter~21]{Koukoulopoulos2019}. +Its calculation is included because the relative error in +Lemma~\ref{lem:sieve-residue-function} must preserve the assigned means. +If $\vartheta_p=0$, the activity event at $p$ is a union of residue classes +of Haar measure zero, hence is empty on global points. Such a prime can +be removed from $\mathcal I$. For the remaining primes put +\[ + \vartheta(d)=\prod_{p\mid d}\vartheta_p,\qquad + h(d)=\prod_{p\mid d}\frac{\vartheta_p}{1-\vartheta_p} +\] +for squarefree $d$ supported on $\mathcal I$. Both functions take value +one at $d=1$. Let $\mathcal D$ be the set of those integers with +$d\le T_0$, and put $\mathcal H=\sum_{d\in\mathcal D}h(d)$. + +For coefficients $\lambda_d$ supported on $\mathcal D$ with +$\lambda_1=1$, the square +\[ + \left(\sum_{d\in\mathcal D}\lambda_d + \prod_{p\mid d}\mathbf 1_{\{n_p\ge1\}}\right)^2 +\] +majorizes the indicator that every prime of $\mathcal I$ is inactive. +Its expectation in the independent product of the local probability +spaces is +\begin{equation}\label{eq:sieve-quadratic} + \sum_{d,e\in\mathcal D}\lambda_d\lambda_e\vartheta([d,e]) + =\sum_{l\in\mathcal D}\frac1{h(l)} + \left(\sum_{\substack{d\in\mathcal D\\l\mid d}} + \lambda_d\vartheta(d)\right)^2. +\end{equation} +Here $[d,e]$ denotes the least common multiple. The identity follows +by expanding the right side and using +$\vartheta([d,e]) + =\vartheta(d)\vartheta(e)\sum_{l\mid(d,e)}h(l)^{-1}$. + +The set $\mathcal D$ is closed under divisors. Finite M\"obius inversion +therefore permits the choice +\[ + \sum_{\substack{d\in\mathcal D\\l\mid d}}\lambda_d\vartheta(d) + =\frac{\mu_{\mathrm{Mob}}(l)h(l)}{\mathcal H}, + \qquad l\in\mathcal D, +\] +where $\mu_{\mathrm{Mob}}$ is the M\"obius function. Explicitly, +\[ + \lambda_d= + \frac{\mu_{\mathrm{Mob}}(d)h(d)} + {\vartheta(d)\mathcal H} + \sum_{\substack{r\le T_0/d\\(r,d)=1\\dr\in\mathcal D}}h(r). +\] +Consequently $\lambda_1=1$, the value of +Equation~\eqref{eq:sieve-quadratic} is $\mathcal H^{-1}$, and +\begin{equation}\label{eq:sieve-coefficients} + |\lambda_d| + \le\frac{h(d)}{\vartheta(d)} + =\prod_{p\mid d}(1-\vartheta_p)^{-1} + \le2^{\omega(d)}\le T_0. +\end{equation} + +To estimate $\mathcal H$, give every squarefree integer supported on +$\mathcal I$ probability proportional to $h(d)$, without the bound +$d\le T_0$. In this finite product distribution, prime $p$ is selected +with probability $\vartheta_p$. Thus +\[ + \E\log d=\sum_{p\in\mathcal I}\vartheta_p\log p + \le C_*\log Y<\tfrac14\log T_0. +\] +Markov's inequality shows that at least half the mass lies at $d\le T_0$. +It follows that +\begin{equation}\label{eq:sieve-selberg-main} + \mathcal H\ge\tfrac12\prod_{p\in\mathcal I}(1+h(p)), + \qquad + \mathcal H^{-1}\le2\prod_{p\in\mathcal I}(1-\vartheta_p). +\end{equation} + +It remains to average this square against $\nu_\phi$ together with the +assigned weights. Write +\[ + A_{\mathcal A}= + \prod_{p\in\mathcal A}\E_p + \bigl(b_p\mathbf 1_{\{n_p=j_p\}}\bigr). +\] +The assigned weight is defined modulo +\[ + \prod_{p\in\mathcal A}p^{C(j_p+1)} + \le\left(\prod_{p\in\mathcal A}p^{j_p}\right)^{2C} + \le T_0^{4C}. +\] +For a pair $d,e\in\mathcal D$, its activity indicators require only the +additional modulus $[d,e]^{2C}\le T_0^{4C}$. The two prime sets are +disjoint, so the total modulus is at most +\begin{equation}\label{eq:sieve-modulus-budget} + T_0^{8C}\le X^\theta +\end{equation} +and is prime to every prime of $S$. The function before multiplication +by $\lambda_d\lambda_e$ is nonnegative, and its residue mean is +$A_{\mathcal A}\vartheta([d,e])$ by the Chinese remainder theorem. +Lemma~\ref{lem:sieve-residue-function} therefore gives the corresponding +main term $V_\phi A_{\mathcal A}\vartheta([d,e])$ and an error of absolute +value at most +$C V_\phi X^{-\theta}A_{\mathcal A}\vartheta([d,e])$. +There are at most $T_0^2$ pairs and, by +Equation~\eqref{eq:sieve-coefficients}, each coefficient product has +absolute value at most $T_0^2$. Summing the errors absolutely costs at +most +\begin{equation}\label{eq:sieve-square-error} + C V_\phi A_{\mathcal A}X^{-\theta}T_0^4. +\end{equation} +This estimate keeps $A_{\mathcal A}$ even if it is very small. +The main term is $V_\phi A_{\mathcal A}\mathcal H^{-1}$ by +Equation~\eqref{eq:sieve-quadratic}. Finally, +$X^{-\theta}T_0^4=o((\log X)^{-C_1})$ by +Equation~\eqref{eq:sieve-parameters}. Equations +\eqref{eq:sieve-inactivity-lower} and +\eqref{eq:sieve-selberg-main} absorb +Equation~\eqref{eq:sieve-square-error} and prove the lemma. +\end{proof} + +\subsection{Truncating and summing the depths} + +\begin{proof}[Proof of Proposition~\ref{prop:sieve-averaging}] +Put +$P_Y=\prod_{p\in\mathcal P_Y}\E_p b_p$. Equations +\eqref{eq:sieve-mean-lower} and +\eqref{eq:sieve-inactivity-lower} imply +\begin{equation}\label{eq:sieve-product-lower} + P_Y\gg(\log X)^{-C_1}. +\end{equation} +First observe that primes exceeding $Y$ cost only a fixed factor. +Equations~\eqref{eq:sieve-depth-product} and +\eqref{eq:sieve-pointwise-weight}, together with $b_p\le A_0^{n_p}$, give +\begin{equation}\label{eq:sieve-large-primes} + \prod_{\substack{p>Y\\p\notin S}}b_p(z) + \le A_0^{\sum_{p>Y,\ p\notin S}n_p(z)} + \le \exp(B\log A_0/\delta). +\end{equation} + +Consider the points for which +$\prod_{p\in\mathcal P_Y}p^{n_p(z)}\le T_0$. +Assign all their active primes in $\mathcal P_Y$ and require inactivity +at the other primes in $\mathcal P_Y$. These assignments satisfy +Lemma~\ref{lem:sieve-assignment}. After applying it, we may enlarge the +sum of its nonnegative upper bounds by allowing every subset and every +positive depth. The resulting product is +\[ + V_\phi\prod_{p\in\mathcal P_Y} + \left(1-\vartheta_p+ + \sum_{j\ge1}\E_p(b_p\mathbf 1_{\{n_p=j\}})\right) + =V_\phi P_Y. +\] +Together with Equation~\eqref{eq:sieve-large-primes}, this bounds the +contribution from these points. + +For a remaining point, the product through $\mathcal P_Y$ exceeds $T_0$. +Order its active primes in $\mathcal P_Y$ increasingly and stop at the +first prefix whose product exceeds $T_0$. If every individual factor +$p^{n_p(z)}$ for $p\in\mathcal P_Y$ is at most $T_0$, the prefix has product in +$(T_0,T_0^2]$, since its preceding product and its last factor are both +at most $T_0$. If also +$\prod_{p\in\mathcal P_{y_0}}p^{n_p(z)}\le T_0$, for a fixed cutoff +$y_0>4$ to be chosen below independently of $X$, its last prime exceeds +$y_0$. The next two estimates allow these restrictions at negligible +weighted cost. + +First, if $p^{n_p(z)}>T_0$ for $p\in\mathcal P_Y$, then +$n_p(z)\ge J_p$, where +$J_p=\lfloor\log T_0/\log p\rfloor+1$. The threshold is determined modulo +\[ + p^{C(J_p+1)}\le T_0^C Y^{2C}\le X^\theta. +\] +Lemma~\ref{lem:sieve-residue-function} and +Equation~\eqref{eq:sieve-local-tail} bound its $\nu_\phi$-measure by +$O(V_\phi p^{-J_p})=O(V_\phi/T_0)$. Summing over $p\in\mathcal P_Y$ +gives $O(V_\phi Y/T_0)$. By Equation~\eqref{eq:sieve-pointwise-weight}, +the contribution after multiplication by $\mathcal W$ is +$O_\epsilon(V_\phi X^{\epsilon+\delta-\eta})$. +Choose $\epsilon<\eta-\delta$. This is smaller than $V_\phi P_Y$ by a +power of $X$, in view of Equation~\eqref{eq:sieve-product-lower}. + +We also need the analogous removal for a fixed nonempty finite set +$\mathcal Q$ of primes, even though some of them may belong to the varying +set $S$. Put $N=|\mathcal Q|$. If +$\prod_{p\in\mathcal Q\setminus S}p^{n_p(z)}>T_0$, some prime in +$\mathcal Q\setminus S$ has $p^{n_p(z)}>T_0^{1/N}$. Put +$J_{p,\mathcal Q}=\lfloor\log T_0/(N\log p)\rfloor+1$. The event +$n_p\ge J_{p,\mathcal Q}$ is defined modulo at most +$T_0^{C/N}p^{2C}\le X^\theta$ for large $X$, and its local probability +is at most $Cp^{-J_{p,\mathcal Q}}\le C T_0^{-1/N}$. +Lemma~\ref{lem:sieve-residue-function} therefore bounds the measure of +the union by +$O_{\mathcal Q}(V_\phi T_0^{-1/N})$. After multiplication by +Equation~\eqref{eq:sieve-pointwise-weight} with $\epsilon<\eta/N$, its +weighted contribution again saves a power of $X$ relative to $V_\phi$. +If $\mathcal Q\setminus S$ is empty the event is empty. Thus this +removal is uniform in the possible intersections of $\mathcal Q$ with $S$. + +Choose a fixed $M_0>0$ and then a fixed $y_0>4$ such that +\begin{equation}\label{eq:sieve-crossing-parameters} + \eta M_0>2B\log A_0+1,\qquad M_0/\log y_0<\tfrac14. +\end{equation} +Apply the finite-set removal with $\mathcal Q=\{p:p\le y_0\}$. +These choices precede the choice of the small $\epsilon$ used in that +removal, and all are independent of $X$. + +The retained first-crossing prefixes therefore have product in +$(T_0,T_0^2]$ and last prime greater than $y_0$. Group these assignments +by the interval containing their last prime: +\begin{equation}\label{eq:sieve-buckets} + \sqrt yy_0$ occur. Every unassigned active prime is greater +than the last assigned prime and hence greater than $\sqrt y$. The total +weight of those primes, including primes beyond $Y$, is therefore at most +\begin{equation}\label{eq:sieve-unassigned} + \exp\left(\frac{2B\log A_0}{\log y}\log X\right). +\end{equation} +For each assignment in the group, require inactivity at its unassigned +primes in $\mathcal P_{\sqrt y}$ and apply +Lemma~\ref{lem:sieve-assignment}. We use Rankin's power majorant +\cite{Rankin1938}: since the assigned product exceeds $T_0$, its upper +bound can be multiplied by +\[ + T_0^{-s}\prod_{\text{assigned }p}p^{s j_p}\ge1, + \qquad s=M_0/\log y. +\] +Now enlarge the sum of these nonnegative upper bounds to all subsets and +all positive depths through $y$. This enlarges only a numerical sum after +the lemma has been applied to the genuine prefixes; it invokes no residue +estimate for the newly admitted products above $T_0^2$. + +For $p\in\mathcal P_{\sqrt y}$, the option of no assignment has weight +$1-\vartheta_p$ and the assigned options have total weight +$\sum_{j\ge1}\E_p(b_p\mathbf 1_{\{n_p=j\}})p^{sj}$. Their sum is +$\E_p(b_p p^{s n_p})$. For $\sqrt yy_0$. By +Equation~\eqref{eq:sieve-crossing-parameters}, $s<1/4$. The total +contribution of depths $j\ge2$ to $\E_p(b_p p^{s n_p})$ is at most +\[ + C^2\sum_{j\ge2}(1+j)^C p^{-(1-s)j}\ll_C p^{-3/2}. +\] +At depth one, the increase over $\E_p b_p$ is bounded by +\[ + \frac{C'}p(p^s-1) + \ll_{M_0}\frac{\log p}{p\log y} + \qquad(p\le y), +\] +because $0\le s\log p\le M_0$. Chebyshev's estimate shows that these +increases have bounded sum over $p\le y$. Mertens' estimate also gives +$\sum_{\substack{\sqrt y={Stealth[length=2mm]}, + patch/.style={draw=black!75,line width=.55pt,fill=black!4}, + selected/.style={draw=blue!65!black,line width=.55pt,fill=blue!16}, + coarse/.style={draw=black!70,dashed,line width=.65pt}, + fine/.style={draw=black!35,line width=.25pt}, + bins/.style={draw=black!70,densely dotted,line width=.5pt}, + note/.style={align=left,font=\footnotesize}, + leader/.style={draw=black!65,line width=.4pt,->} +] +\path[use as bounding box] (0,-5.15) rectangle (14.2,4.35); + +\node[anchor=west,font=\small\bfseries] at (0,4.1) + {(a) Contract $W$: identify the $U$ factor and prove independence}; +\node[align=center,font=\footnotesize] at (2.35,3.67) + {one $q$-cell; full $w,u$ ranges}; +\draw[patch] (.65,1.3) rectangle (4.05,3.35); +\draw[selected] (2.15,1.3) rectangle (2.75,3.35); +\node[font=\footnotesize,anchor=north] at (2.45,1.19) {$w$-cell}; +\node[anchor=east] at (.51,3.2) {$u$}; +\node[anchor=north] at (3.9,1.19) {$w$}; + +\draw[->,line width=.8pt] (4.65,2.35) -- (7.1,2.35); +\node[above] at (5.87,2.45) {$a_0^{t}$, $t\asymp T$}; +\node[below,font=\footnotesize,align=center] at (5.87,2.18) + {$W$ contracts; $U$ is fixed\\$T\prec F$}; + +\draw[coarse] (7.95,1.15) rectangle (10.15,3.5); +\draw[patch] (8.55,1.3) rectangle (9.55,3.35); +\draw[selected] (8.991,1.3) rectangle (9.168,3.35); +\foreach \x in {8.6,8.7,8.8,8.9,9.0,9.1,9.2,9.3,9.4,9.5} + \draw[fine] (\x,1.3) -- (\x,3.35); +\foreach \y in {1.78,2.31,2.84} + \draw[bins] (7.95,\y) -- (10.15,\y); +\node[font=\footnotesize] at (9.05,3.73) {output $U$ chart}; +\node[note,anchor=west] at (10.58,3.18) + {One coarse transverse cell\\contains the whole patch.}; +\draw[leader] (10.5,3.15) -- (10.17,3.15); +\node[note,anchor=west] at (10.58,2.26) + {Fine transverse cells\\resolve the $w$-label.}; +\draw[leader] (10.5,2.24) -- (9.28,2.24); +\node[note,anchor=west] at (10.58,1.42) {Common $U$ bins}; +\draw[leader] (10.5,1.48) -- (10.04,1.78); +\node[align=center,font=\footnotesize] at (7.1,.5) + {Mesh comparison: revealing the $w$-cell changes the $U$-bin law by $o(1)$ in sampling mean.}; + +\node[anchor=west,font=\small\bfseries] at (0,-.22) + {(b) Expand a $W$ tile: identify the remaining $W$ factor}; +\node[align=center,font=\footnotesize] at (2.35,-.86) + {one stage-$k$ tile\\shape $\Delta^k\mathcal F$}; +\draw[patch] (2.0,-3.7) rectangle (2.7,-1.55); +\draw[selected] (2.0,-2.68) rectangle (2.7,-2.43); +\foreach \x in {2.175,2.35,2.525} + \draw[bins] (\x,-3.7) -- (\x,-1.55); +\node[anchor=east] at (1.86,-1.72) {$u$}; +\node[anchor=north] at (2.65,-3.81) {$w$}; +\node[note,anchor=east] at (1.72,-2.57) {$u$-cell\\depth $S$}; +\draw[leader] (1.8,-2.57) -- (1.98,-2.57); + +\draw[->,line width=.8pt] (3.65,-2.57) -- (6.5,-2.57); +\node[above] at (5.08,-2.45) {$a_0^{-(k+m_0)}$}; +\node[below,font=\footnotesize,align=center] at (5.08,-2.73) + {$W$ expands; $U$ is fixed\\$S\prec T\prec F$, $k\asymp T$}; + +\node[align=center,font=\footnotesize] at (8.63,-.88) + {fixed output shape\\$\Delta^{-m_0}\mathcal F$}; +\draw[patch] (7.05,-3.7) rectangle (10.2,-1.55); +\draw[selected] (7.05,-2.68) rectangle (10.2,-2.43); +\foreach \x in {7.8375,8.625,9.4125} + \draw[bins] (\x,-3.7) -- (\x,-1.55); +\draw[coarse] (6.9,-2.92) rectangle (10.35,-2.2); +\foreach \y in {-2.63,-2.58,-2.53,-2.48} + \draw[fine] (7.05,\y) -- (10.2,\y); +\node[note,anchor=west] at (10.58,-1.94) + {Only the $u$-subpatch fits\\in one coarse transverse\\cell for the $W$ chart.}; +\draw[leader] (10.5,-2.02) -- (10.35,-2.2); +\node[note,anchor=west] at (10.58,-3.18) + {Fine transverse cells\\resolve the $u$-label.}; +\draw[leader] (10.5,-3.05) -- (9.72,-2.6); +\node[align=center,font=\footnotesize] at (8.63,-4.04) + {Stage-$(k+j)$ subtiles resolve $W$ tests.}; +\node[align=center,font=\footnotesize,text width=14cm] at (7.1,-4.78) + {Information comparison: an $O(S)$ budget over $T$ stages makes the $u$-label change\\ + the descendant-tile law by $o(1)$ in sampling mean, at a suitable stage $k\asymp T$.}; +\end{tikzpicture} +\caption{The two different comparisons in the product rule. Each input picture +fixes a $q$-cell at depth $F$; the coordinate $q$ is suppressed. Dashed outlines +denote coarse output transverse cells; thin solid lines denote fine transverse +cells. Dotted lines denote fixed $U$ bins in (a) and stage-$(k+j)$ $W$ subtiles in (b). +Shading marks a labelled subpatch, not a density. In (a), contraction permits +one coarse transverse-cell comparison for the whole patch and its subpatches. +In (b), expansion leaves the full $U$ interval broad. The full patch is compared +with $V$ data and the thin $u$-subpatch with $W$ data; the information estimate +compares their descendant-tile laws and hence their resolved $W$ tests. All widths and +tile shapes are schematic: $W$ can have several dimensions and its tiles need +not be rectangles. The resulting factor identifications are projective.} +\label{fig:product-mechanism} +\end{figure} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/main.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..f91dbcf65386f72c752379e8a7292791f414bd45 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/main.tex @@ -0,0 +1,88 @@ +\documentclass[11pt]{article} +\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem,booktabs} +\usepackage{xcolor,tikz} +\usetikzlibrary{arrows.meta,positioning,calc,patterns,decorations.pathreplacing} +\usepackage[numbers,sort&compress]{natbib} +\usepackage[hyphens]{url} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black, + urlcolor=blue!45!black,pdfauthor={OpenAI}, + pdftitle={Equidistribution of Primitive Sextic Torus Packets}} +\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{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\Q}{\mathbb Q} +\newcommand{\C}{\mathbb C} +\newcommand{\E}{\mathbb E} +\newcommand{\Prb}{\mathbb P} +\newcommand{\one}{\mathbf 1} +\newcommand{\dd}{\mathop{}\!d} +\newcommand{\idtext}[1]{\begingroup\urlstyle{same}\nolinkurl{#1}\endgroup} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\Ad}{Ad} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\covol}{covol} +\DeclareMathOperator{\Disc}{Disc} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\Res}{Res} +\DeclareMathOperator{\Fix}{Fix} +\DeclareMathOperator{\Mat}{Mat} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\rank}{rank} +\setlist[enumerate]{leftmargin=*,itemsep=4pt} +\setlength{\emergencystretch}{2em} +\setcounter{tocdepth}{1} +\title{Equidistribution of Primitive Sextic Torus Packets} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that the complete ideal-class packets of maximal orders in +totally real sextic fields with no proper intermediate fields become +equidistributed, with no escape of mass, in the space of unimodular +lattices as their field discriminants tend to infinity. The limit is +Haar probability measure. Each packet includes all coordinate-sign +translates and is weighted by diagonal-orbit volume. +\end{abstract} +\tableofcontents +\input{sections/introduction} +\input{sections/arithmetic} +\input{sections/homogeneous} +\input{sections/conditioning} +\input{sections/product} +\input{sections/activity} +\input{sections/entropy} +\input{sections/combination} +\input{sections/assembly} +\begingroup +\small\raggedright +\setlength{\bibsep}{3pt} +\bibliographystyle{plainnat} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/references.bib b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..ac326c352d624141140a07ae1e0512f9b38c85e9 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/references.bib @@ -0,0 +1,198 @@ +@book{Linnik1968, + author = {Linnik, Yu. V.}, + title = {Ergodic Properties of Algebraic Fields}, + series = {Ergebnisse der Mathematik und ihrer Grenzgebiete}, + volume = {45}, + publisher = {Springer-Verlag}, + address = {Berlin--Heidelberg--New York}, + year = {1968}, + note = {Translated from the Russian by M. S. Keane} +} + +@article{Duke1988, + author = {Duke, William}, + title = {Hyperbolic distribution problems and half-integral weight {Maass} forms}, + journal = {Invent. Math.}, + volume = {92}, + number = {1}, + year = {1988}, + pages = {73--90}, + doi = {10.1007/BF01393993} +} + +@article{ELMV2009, + author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay}, + title = {Distribution of periodic torus orbits on homogeneous spaces}, + journal = {Duke Math. J.}, + volume = {148}, + number = {1}, + year = {2009}, + pages = {119--174}, + doi = {10.1215/00127094-2009-023} +} + +@article{ELMV2011, + author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay}, + title = {Distribution of periodic torus orbits and {Duke}'s theorem for cubic fields}, + journal = {Ann. of Math. (2)}, + volume = {173}, + number = {2}, + year = {2011}, + pages = {815--885}, + doi = {10.4007/annals.2011.173.2.5} +} + +@article{ELMV2012, + author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay}, + title = {The distribution of closed geodesics on the modular surface, and {Duke}'s theorem}, + journal = {Enseign. Math. (2)}, + volume = {58}, + number = {3--4}, + year = {2012}, + pages = {249--313}, + doi = {10.4171/LEM/58-3-2} +} + +@article{EKL2006, + author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon}, + title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture}, + journal = {Ann. of Math. (2)}, + volume = {164}, + number = {2}, + year = {2006}, + pages = {513--560}, + doi = {10.4007/annals.2006.164.513} +} + +@article{Khayutin2019, + author = {Khayutin, Ilya}, + title = {Arithmetic of double torus quotients and the distribution of periodic torus orbits}, + journal = {Duke Math. J.}, + volume = {168}, + number = {12}, + year = {2019}, + pages = {2365--2432}, + doi = {10.1215/00127094-2019-0016} +} + +@article{WieserYang2026, + author = {Wieser, Andreas and Yang, Pengyu}, + title = {A uniform {Linnik} basic lemma and entropy bounds}, + journal = {Comment. Math. Helv.}, + year = {2026}, + note = {Published online January 22, 2026}, + doi = {10.4171/CMH/613} +} + +@article{Stark1974, + author = {Stark, Harold M.}, + title = {Some effective cases of the {Brauer--Siegel} theorem}, + journal = {Invent. Math.}, + volume = {23}, + number = {2}, + year = {1974}, + pages = {135--152}, + doi = {10.1007/BF01405166} +} + +@article{Shiu1980, + author = {Shiu, Peter}, + title = {A {Brun--Titchmarsh} theorem for multiplicative functions}, + journal = {J. Reine Angew. Math.}, + volume = {313}, + year = {1980}, + pages = {161--170}, + doi = {10.1515/crll.1980.313.161} +} + +@article{Pollack2020, + author = {Pollack, Paul}, + title = {Nonnegative multiplicative functions on sifted sets, and the square roots of {$-1$} modulo shifted primes}, + journal = {Glasg. Math. J.}, + volume = {62}, + number = {1}, + year = {2020}, + pages = {187--199}, + doi = {10.1017/S0017089519000041}, + note = {Corrected author version available at \url{https://www.pollack-math.net/brunmultiplicative4.pdf}} +} + +@book{Cassels1959, + author = {Cassels, J. W. S.}, + title = {An Introduction to the Geometry of Numbers}, + series = {Grundlehren der mathematischen Wissenschaften}, + volume = {99}, + publisher = {Springer-Verlag}, + address = {Berlin--G{\"o}ttingen--Heidelberg}, + year = {1959} +} + +@article{EK2003, + author = {Einsiedler, Manfred and Katok, Anatole}, + title = {Invariant measures on {$G/\Gamma$} for split simple {Lie} groups {$G$}}, + journal = {Comm. Pure Appl. Math.}, + volume = {56}, + number = {8}, + pages = {1184--1221}, + year = {2003}, + doi = {10.1002/cpa.10092} +} + +@article{EK2005, + author = {Einsiedler, Manfred and Katok, Anatole}, + title = {Rigidity of measures---the high entropy case and non-commuting foliations}, + journal = {Israel J. Math.}, + volume = {148}, + pages = {169--238}, + year = {2005}, + doi = {10.1007/BF02775436} +} + +@incollection{EL2010, + author = {Einsiedler, Manfred and Lindenstrauss, Elon}, + title = {Diagonal actions on locally homogeneous spaces}, + booktitle = {Homogeneous Flows, Moduli Spaces and Arithmetic}, + series = {Clay Mathematics Proceedings}, + volume = {10}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {2010}, + pages = {155--241} +} + +@article{Rokhlin1967, + author = {Rokhlin, V. A.}, + title = {Lectures on the entropy theory of measure-preserving transformations}, + journal = {Russian Math. Surveys}, + volume = {22}, + number = {5}, + year = {1967}, + pages = {1--52}, + doi = {10.1070/RM1967v022n05ABEH001224} +} + +@book{CoverThomas2006, + author = {Cover, Thomas M. and Thomas, Joy A.}, + title = {Elements of Information Theory}, + edition = {Second}, + publisher = {Wiley-Interscience}, + address = {Hoboken, NJ}, + year = {2006}, + doi = {10.1002/047174882X} +} + +@misc{PrimeDegreePackets, + author = {{OpenAI}}, + title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}}, + year = {2026} +} + +@misc{QuarticPackets, + author = {{OpenAI}}, + title = {{Equidistribution of primitive quartic torus packets for arbitrary orders}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026}}, + year = {2026} +} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/activity.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/activity.tex new file mode 100644 index 0000000000000000000000000000000000000000..12b0cb5b18b00c8929128251c1a042e03179532d --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/activity.tex @@ -0,0 +1,289 @@ +\section{Active roots and their support}\label{sec:activity} + +The product rule turns the plaque measures into a directed graph at each +augmented state. We first show that its arrows compose and describe the +support of the positive-root plaques. We then realize this graph by +measurable indicators on the original spaces, retaining the long-wall +comparison needed for the entropy argument. + +Write $\mathsf P$ for the law of the limiting augmented state +$\omega=(x,(P^B)_B)$. For $j\ne k$, define +\[ + a_{jk}(\omega) + =\mathbf 1_{\{m^{U_{jk}}_\omega\text{ is not supported on }\{1\}\}}. +\] +We call $jk$ \emph{active} when $a_{jk}=1$. This definition does not depend +on the representative of the projective measure. It is measurable: +after normalizing by a positive integral against a bump near the identity, +activity is detected by a countable family of nonnegative continuous +tests supported away from the identity. All assertions below are made +on a common set of full $\mathsf P$-measure on which the comparison and +product rules hold. + +\begin{proposition}[Composition and positive-root support] +\label{prop:activity} +Almost surely, activity is transitively closed: +\[ + a_{pq}(\omega)a_{qr}(\omega) + \bigl(1-a_{pr}(\omega)\bigr)=0 + \qquad(p,q,r\text{ distinct}). +\] +Let $b\in L$ be regular, let $V$ be the pattern group of all +$b$-positive roots, and let $V_\omega^+$ be the subgroup generated by +the active $b$-positive root groups. These active roots form a pattern, +and +\[ + \supp m^V_\omega\subset V_\omega^+. +\] +Both conclusions hold simultaneously for every regular $b$. +\end{proposition} + +\begin{proof} +First observe that Theorem~\ref{thm:product} factors the measure of any +pattern group into its one-dimensional root measures, in a suitable +order. Indeed, the cone generated by its roots is pointed. Choose a +root $\alpha$ on an exposed extremal ray. No other root in the pattern +is proportional to $\alpha$, so a diagonal logarithm can vanish on +$\alpha$ and be strictly negative on every remaining root. Removing +$\alpha$ leaves a normal pattern subgroup $W$: a sum of two roots in +the original pattern cannot equal $\alpha$, since extremality would +force both summands onto its ray. Thus the hypotheses of +Theorem~\ref{thm:product} hold for $V=WU_\alpha$. Iterate, allowing +$W=\{1\}$ at the last step. + +To prove transitivity, fix distinct $p,q,r$ and consider the +three-root Heisenberg group generated by $U_{pq},U_{qr},U_{pr}$. +Suppose that $a_{pr}=0$. Factoring first with $U_{pq}$ as the final +factor, and then with $U_{qr}$ as the final factor, gives two +descriptions of the same projective measure. Write +$u_{jk}(t)=1+tE_{jk}$. Since the $U_{pr}$ factor is a point mass, the +first description is supported on the matrices +\[ + u_{qr}(t)u_{pq}(s)=1+sE_{pq}+tE_{qr}, +\] +whereas the other is the pushforward of the product of the two root +measures under +\[ + u_{pq}(s)u_{qr}(t) + =1+sE_{pq}+tE_{qr}+stE_{pr}. +\] +If both roots were active, each root measure would give positive mass +to some compact parameter set disjoint from zero. Their product would +then give positive mass to matrices with nonzero $pr$ entry, contrary +to the first description. The projective identities apply on compact +ranges containing these sets, so all masses used here are finite. +This proves transitivity. + +For the support assertion, use the successive factorization already +proved. Every inactive root factor is a point mass at the identity; +every other factor is supported on its own active root group. The +product is therefore supported on the subgroup generated by the active +roots. Transitivity says exactly that the active positive roots are +closed under root addition, so this subgroup is a pattern group and is +closed. There are only finitely many positive-root patterns, which +also gives the simultaneous assertion. +\end{proof} + +The entropy argument will use plaques enlarged by the diagonal group. +The following consequence retains the activity at the sampled state. + +\begin{lemma}[Support on plaques enlarged by $A$]\label{lem:va-support} +With $V$ and $V_\omega^+$ as in Proposition~\ref{prop:activity}, the +limiting $VA$-plaque probability, written in displacement coordinates +from its sampled point, is supported on $V_\omega^+A$ within its chart. +\end{lemma} + +\begin{proof} +Use a product box with coordinates $\psi(q)va$, with product coordinate +domains. Local $A$-invariance gives, for each $i$, a product +decomposition +\[ + d\mu_i(q,v,a)=d\nu_i(q,v)\,da +\] +inside the box, where $da$ is Haar measure restricted to its coordinate +box and the normalization is absorbed into $\nu_i$. For $VA$ plaques +the transverse coordinate is $q$; for $V$ plaques it is $(q,a)$. +Choose a common $q$ grid and supplement it by an $a$ grid for the latter +conditioning. Conditional on a $q$ cell, the $VA$ probability is the +product of a probability on $v$ and normalized Haar measure on $a$. +Conditioning also on an $a$ cell leaves the $v$ probability unchanged. +These are exact identities before taking a limit. The interior-depth +comparisons permit these compatible grids in the defining data, and +the identities pass to the joint limit by continuous tests and the +null-boundary convention. + +At a sample $x=\psi(q_s)v_sa_s$, the $V$ displacements corresponding to +varying $v$ at fixed $a_s$ are +\[ + a_s^{-1}v_s^{-1}va_s. +\] +By Proposition~\ref{prop:activity}, their probability is supported on +$V_\omega^+$. A general $VA$ displacement is +\[ + a_s^{-1}v_s^{-1}va + =\bigl(a_s^{-1}v_s^{-1}va_s\bigr)\bigl(a_s^{-1}a\bigr), +\] +and hence belongs to $V_\omega^+A$ almost surely. The argument applies +on inner chart boxes; exhausting them gives the statement on the whole +plaque box, whose boundary has zero probability. +\end{proof} + +\begin{lemma}[Indicators along the sequence]\label{lem:activity-indicators} +Fix an interior scale $R_i$ allowed in the long-wall comparison of +Proposition~\ref{prop:plaque-comparison}, with ratio-separated depths on +both sides of it inside $(l_{-,i},l_{+,i})$. There exist measurable +functions $g_{jk,i}:X\to\{0,1\}$ such that +\[ + \operatorname{Law}_{\mu_i}\bigl(Z_i(x),(g_{jk,i}(x))_{j\ne k}\bigr) + \ \Longrightarrow\ + \operatorname{Law}_{\mathsf P} + \bigl(\omega,(a_{jk}(\omega))_{j\ne k}\bigr). +\] +They satisfy asymptotic transitivity: +\[ + \int_X g_{pq,i}(x)g_{qr,i}(x) + \bigl(1-g_{pr,i}(x)\bigr)\,d\mu_i(x)=o(1) + \qquad(p,q,r\text{ distinct}), +\] +and, for every fixed $C<\infty$, +\begin{equation}\label{eq:wall-invariance} + \sup_{\substack{b\in\ker\beta_{jk}\\ \|b\|\le CR_i}} + \int_X\bigl|g_{jk,i}(xa(b))-g_{jk,i}(x)\bigr|\,d\mu_i(x) + \longrightarrow0. +\end{equation} +\end{lemma} + +\begin{proof} +Let $\mathcal S$ be the separable metric state space used for $Z_i$, +and put $A_{jk}=\{\omega:a_{jk}(\omega)=1\}$. Choose +$\epsilon_r\downarrow0$. For each root and each $r$, there is a +$\mathsf P$-continuity set $D_{jk,r}\subset\mathcal S$ such that +\[ + \mathsf P(D_{jk,r}\mathbin\triangle A_{jk}) + <\epsilon_r. +\] +For completeness, use a countable family of continuous coordinate +tests generating the Borel sets of $\mathcal S$. For each test choose +a countable dense family of thresholds outside the atoms of its +distribution under $\mathsf P$. +The finite Boolean combinations of the resulting threshold sets have +null boundary and approximate every measurable set in +$\mathsf P$-measure. Thus the approximations may be made using +finitely many data tests at each stage. + +For fixed $r$, set +$g^{(r)}_{jk,i}(x)=\mathbf 1_{D_{jk,r}}(Z_i(x))$. +Weak convergence of the states and the null boundaries give joint +convergence of the states and these finitely many indicators. Since +the exact indicators are transitive, the limiting transitivity defect +for any triple is at most $3\epsilon_r$. + +We next prove the uniform wall estimate at this fixed approximation +level. Fix a root, $C$, and any sequence +$b_i\in\ker\beta_{jk}$ with $\|b_i\|\le CR_i$. Both marginals of +\[ + \bigl(Z_i(x),Z_i(xa(b_i))\bigr) +\] +have the ordinary state law, because $\mu_i$ is $A$-invariant. +Consequently these paired laws are tight. In every subsequential +limit their two marginals are $\mathsf P$, and +Proposition~\ref{prop:plaque-comparison} identifies the two projective +$U_{jk}$ measures. Their exact activity indicators are therefore +equal. Applying the continuity-set approximation separately in the +two marginals shows that every such limit has +\[ + \mathbb E\bigl| + \mathbf 1_{D_{jk,r}}(\omega) + -\mathbf 1_{D_{jk,r}}(\omega')\bigr| + \le 2\epsilon_r. +\] +The boundary of the tested event is contained in the union of the two +marginal boundaries, so the corresponding prelimit expectations +converge along this subsequence. Since the sequence $b_i$ was +arbitrary, a sequence chosen within a vanishing error of the supremum +proves +\[ + \limsup_i\ \sup_{\substack{b\in\ker\beta_{jk}\\\|b\|\le CR_i}} + \int_X\bigl|g^{(r)}_{jk,i}(xa(b))-g^{(r)}_{jk,i}(x)\bigr|\,d\mu_i(x) + \le2\epsilon_r. +\] +This argument uses limits of paired states even when $b_i$ diverges. + +Finally choose $r=r(i)\to\infty$ sufficiently slowly. At stage $r$ +impose the preceding estimates for all roots, all triples, and all +integers $1\le C\le r$, together with the first $r$ tests of a +convergence-determining family for the joint state and indicator law. +The limiting laws at fixed $r$ differ from the exact activity law by +at most the sum of the finitely many approximation errors. A +diagonal choice therefore gives the asserted joint convergence, +vanishing transitivity defects, and \eqref{eq:wall-invariance} for +$g_{jk,i}=g^{(r(i))}_{jk,i}$. Bounds for integral $C$ imply the bounds +for every fixed finite $C$. +\end{proof} + +We conclude with the relation between activity and pieces having +countable intersections with root plaques. +It is this relation that will exclude proper homogeneous components +once all roots have been shown to be active. + +\begin{lemma}[Inactivity on pieces with countable plaque sections]\label{lem:transverse-inactive} +Let $S\subset X$ be a Borel set whose intersection with each local +$U_{jk}$ plaque is countable. Then +\[ + a_{jk}(\omega)=0 + \quad\text{for $\mathsf P$-almost every state with }x\in S. +\] +In particular, this applies to the fixed-point pieces of +Lemma~\ref{lem:fixed-pieces} for roots crossing distinct eigenvalues: +each such piece meets a sufficiently small root plaque in at most one +point. It also applies to countable unions of these pieces. +\end{lemma} + +\begin{proof} +Work first in a $U_{jk}$ flow box with coordinates $(q,h)$ and plaque +probability $P^B$. For each $q$, the section +\[ + S_q=\{h:\psi(q)h\in S\} +\] +is countable. The sampling identity for this single chart states +that the conditional law of the sampled $h$, given $(q,P^B)$, is +$P^B$. A probability measure assigns mass zero to those points of a +countable set which are not its atoms. Consequently +\[ + \mathsf P\{x\in S\cap B,\ P^B(\{h\})=0\}=0. +\] +No conditioning on the other plaque probabilities is used here. +Projective agreement in the chart identifies an atom at the sampled +$h$ with an atom at the identity of $m^{U_{jk}}_\omega$. A countable +flow-box cover therefore shows that this root measure has an atom at +the identity almost surely over $S$. + +It remains to show that, under the full state law, an atom at the +identity forces the entire root measure to be supported there. +Identify $U_{jk}$ with $\R$ by $t\mapsto1+tE_{jk}$. Choose a fixed +$b\in L$ with $\lambda=e^{\beta_{jk}(b)}>1$. Fixed-motion equivariance +in Proposition~\ref{prop:plaque-comparison}, together with +$A$-invariance of the prelimit sampling law, says that the projective +law of $m^{U_{jk}}_\omega$ is invariant under the pushforward by +$t\mapsto\lambda t$. + +The event that this measure has an atom at zero is itself invariant +under dilation. If that event has positive probability, condition +the projective-measure law on it and normalize each measure $m$ by +its atom: $\nu=m/m(\{0\})$. This is a measurable normalization of a +locally finite measure, with $\nu(\{0\})=1$, and its law remains +dilation-invariant. For every compact interval $K$ disjoint from +zero, +\[ + (t\mapsto\lambda^n t)_*\nu(K) + =\nu(\lambda^{-n}K)\longrightarrow0. +\] +Indeed, $\lambda^{-n}K$ lies in a shrinking punctured neighborhood of +zero, whose mass tends to zero by local finiteness. Dilation +invariance makes each quantity in this display have the same +distribution as $\nu(K)$. Almost-sure convergence to zero therefore +forces $\nu(K)=0$ almost surely. A countable family of such intervals +covers $\R\setminus\{0\}$, and hence $\nu=\delta_0$ almost surely. +The atom event of probability zero is harmless. Combining this +conclusion with the first part proves the lemma. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/arithmetic.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/arithmetic.tex new file mode 100644 index 0000000000000000000000000000000000000000..614bf395966f0e512f35304697fe32d149492ab8 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/arithmetic.tex @@ -0,0 +1,677 @@ +\section{Arithmetic estimates for the packets}\label{sec:arithmetic} + +This section establishes the three estimates on which the dynamical +argument rests: uniform control of the cusp, a dimension bound for +probability limits, and exponential decay on two-sided diagonal tubes. +The first two follow by counting ordinary lattice vectors. For the +third, we count an ordinary vector together with a dual vector; their +coordinatewise product lies in the codifferent and can be counted in a +thin slice. + +For the volume, unfolding, and ordinary-vector estimates, we adapt +the method of \cite[Sections~2--6]{PrimeDegreePackets} and +\cite[Section~3]{QuarticPackets}. We reproduce the needed arguments +in the packet and Haar normalizations of this paper. + +Throughout this section, $K$ is a totally real field of degree six with +no proper intermediate field, and $\sigma$ is an ordering of its real +embeddings. Write +\[ + D=|\Disc(K)|,\qquad Q=D^{1/2},\qquad + \kappa=\mathop{\rm Res}_{s=1}\zeta_K(s),\qquad + d_K(l)=\#\{\mathfrak b\subset\mathcal O_K:N\mathfrak b=l\}. +\] +All ideals counted here are nonzero. For a sequence of fields and +embedding orderings, we use the corresponding notation +\[ + D_i=|\Disc(K_i)|,\qquad Q_i=D_i^{1/2},\qquad + \kappa_i=\mathop{\rm Res}_{s=1}\zeta_{K_i}(s),\qquad + \mu_i=\mu_{K_i,\sigma_i}. +\] +Unless another dependence is indicated, constants below are absolute +for this fixed degree. In particular, they are independent of the +field and of the ordering of its embeddings. + +\subsection{Volume and unfolding} + +We first make the sign convention in the packet definition explicit. +The space $X=\SL_6(\Z)\backslash\SL_6(\R)$ parametrizes ordinary +unimodular lattices: a lattice has a positively oriented basis, and +any two such bases differ by an element of $\SL_6(\Z)$. Consequently +a diagonal sign matrix acts on lattices even when its determinant is +$-1$. The resulting lattice again represents a point of $X$ by choosing +a positively oriented basis. There is no additional orientation +coordinate. + +Let $h_K$ and $R_K$ denote the ordinary class number and regulator, +respectively. Our convention for $R_K$ is the covolume of the absolute +logarithms of the units after deleting the sixth coordinate. Put +\[ + U=\mathcal O_K^\times,\qquad + U^+=\{\epsilon\in U:\sigma_j(\epsilon)>0\ (1\le j\le6)\}, + \qquad s=[U:U^+]. +\] +For a nonnegative Borel function $f$ on $\R^6$, define its lattice sum by +\[ + \widehat f(\Lambda)=\sum_{v\in\Lambda\setminus\{0\}}f(v). +\] +For $p>0$, define a logarithmic fiber integral +\begin{align} + V_f(p) + =\sum_{\epsilon\in\{\pm1\}^6}\int_{\R^5} + f\bigl(\epsilon_1e^{u_1},\ldots,\epsilon_5e^{u_5}, + \epsilon_6p e^{-u_1-\cdots-u_5}\bigr) + \,\dd u_1\cdots\dd u_5 . \label{eq:fiber-integral} +\end{align} +Thus $V_f(p)$ integrates over the set on which the absolute product of +the six coordinates is $p$, with the Haar normalization used for $A$. + +\begin{proposition}[Packet volume and vector unfolding] +\label{prop:packet-volume} +Every orbit in $\mathcal P_{K,\sigma}$ has $A$-volume $(s/2)R_K$, +and there are $h_K2^6/s$ distinct orbits. In particular, the total +packet volume is $Q\kappa$. For every nonnegative Borel function $f$, +\begin{equation} + \int_X\widehat f\,\dd\mu_{K,\sigma} + =\frac1{Q\kappa}\sum_{l\ge1}d_K(l)V_f(l/Q), + \label{eq:vector-unfolding} +\end{equation} +where both sides are allowed to be infinite. +\end{proposition} + +\begin{proof} +The only roots of unity in a totally real field are $1$ and $-1$. +The kernel of the absolute logarithmic map on $U$ therefore has +order two. Its image on $U^+$ has index $s/2$ in its image on $U$. +The unit theorem now gives covolume $(s/2)R_K$ for the positive-unit +logarithms in the coordinates $u_1,\ldots,u_5$. + +We check that these are exactly the stabilizers and identifications +in the packet. Suppose a real diagonal map sends $\sigma(I)$ onto +$\sigma(J)$. Choose $0\ne d\in I$, and let its image be $\sigma(e)$ +with $e\in J$. All embeddings of $d$ are nonzero, so the diagonal +entries of the map are $\sigma_j(e/d)$. It is multiplication by +$e/d\in K^\times$, and hence $J=(e/d)I$. This observation also +applies to normalized lattices after their positive covolume factors +have been undone. It follows that different ordinary ideal classes +give disjoint packet orbits. For a fixed ideal, the diagonal +automorphisms are its units, because $(I:I)=\mathcal O_K$. + +The stabilizer in $A$ is consequently the image of $U^+$. +Two sign translates belong to the same $A$-orbit exactly when their +signs differ by a unit signature. The signature image has cardinality +$s$, giving $2^6/s$ orbits for each ordinary class and the stated orbit +volume. The analytic class number formula, in this normalization, is +\[ + Q\kappa=2^5h_KR_K. +\] +It agrees with the product of the orbit number and orbit volume. + +For completeness, let $\mathcal F$ be a fundamental domain for the +positive-unit logarithms in the traceless diagonal algebra. Summing +over ordinary ideal-class representatives and all sign matrices +counts each orbit exactly $s$ times. Thus for any nonnegative $F$ on +$X$, +\begin{equation} + \int_XF\,\dd\mu_{K,\sigma} + =\frac1{sQ\kappa}\sum_{[I]}\sum_w + \int_{\mathcal F}F(\Lambda_{I,\sigma}w a(u))\,\dd u. + \label{eq:packet-integral} +\end{equation} +Here $\dd u=\dd u_1\cdots\dd u_5$ and the sixth coordinate is minus +their sum. + +Apply this identity to $F=\widehat f$. A nonzero $d\in I$ determines +the integral ideal +\[ + \mathfrak b=(d)I^{-1}, + \qquad + \prod_{j=1}^6 + \left|\covol(\sigma(I))^{-1/6}\sigma_j(d)\right| + =\frac{N\mathfrak b}{Q}, +\] +because $\covol(\sigma(I))=QNI$. The ideal $\mathfrak b$ +determines the class $[I]=[\mathfrak b]^{-1}$, and every integral +$\mathfrak b$ occurs. For that ideal, the possible $d$ form a single +$U$-orbit, or $s$ orbits under $U^+$. Summing a positive-unit orbit +and integrating over $\mathcal F$ unfolds to the entire traceless +diagonal algebra. Multiplication by a representative of $U/U^+$ +translates the absolute logarithms and permutes the signs. After the +sum over $w$, each of these $s$ orbits therefore contributes +$V_f(N\mathfrak b/Q)$. Their number cancels the factor $s$ in +\eqref{eq:packet-integral}, proving \eqref{eq:vector-unfolding}. +All rearrangements are justified by nonnegativity. +\end{proof} + +\subsection{Uniform counts of ideals} + +The normalization by $\kappa$ in \eqref{eq:vector-unfolding} makes it +important to keep this same factor in upper bounds for ideal counts. +We obtain it from a zero-free neighborhood of $1$, followed by Shiu's +estimate for nonnegative multiplicative functions. + +\begin{lemma}[Uniform ideal estimates]\label{lem:ideal-estimate} +For all sufficiently large discriminants $D$ in the present family, +\begin{align} + \kappa&\gg (\log D)^{-1}, \label{eq:residue-lower}\\ + \sum_{p\le D}\frac{d_K(p)}p + &\le \log(\kappa\log D)+O(1), \label{eq:prime-sum}\\ + \sum_{x-y0$, +\begin{equation} + |1-\rho|\ge \frac{c_0}{\log D} + \quad\text{for every nontrivial zero $\rho$ of $\zeta_K$}. + \label{eq:zero-exclusion} +\end{equation} +One can take $c_0=1/2880$. This use of Stark's result does not require +$K/\Q$ to be normal. + +Set $s_0=1+1/\log D$. The logarithmic derivative of the completed +Dedekind zeta function gives, for real $10). +\] +For any fixed $0<\beta<1/2$, the conclusion is +\begin{equation} + \sum_{x-y0$ and $c>0$ such that, for all sufficiently large $D$, +\[ + \mu_{K,\sigma}\{x:m(x)\le r\}\le Cr^c + \qquad(0r^6). +\] +Since the occurrence of a short vector is bounded by its count, +\eqref{eq:vector-unfolding} gives +\begin{equation} + \mu_{K,\sigma}\{m\le r\} + \ll\frac1{Q\kappa} + \sum_{l\le X_r}d_K(l) + \bigl(1+\log(X_r/l)\bigr)^5. \label{eq:cusp-sum} +\end{equation} +The sum is empty if $X_r<1$. + +Suppose first that $r\ge D^{-1/48}$, so $X_r\ge D^{3/8}$. +For $l\ge D^{1/4}$, divide the range of summation into intervals +$(2^{-j-1}X_r,2^{-j}X_r]$. Whenever such an interval contains one +of these $l$, its upper endpoint is at least $D^{1/4}$ and at most +$Q$. Lemma~\ref{lem:ideal-estimate}, with interval length half the +upper endpoint, bounds its ideal count by +$O(\kappa 2^{-j}X_r)$. Its weight in \eqref{eq:cusp-sum} is +$O((j+1)^5)$. The total contribution of these terms before division +by $Q\kappa$ is therefore at most +\[ + C\kappa X_r\sum_{j\ge0}2^{-j}(j+1)^5\ll\kappa X_r. +\] +For the remaining terms, the elementary divisor estimate +$\sum_{l\le Y}d_6(l)\ll Y(\log(2Y))^5$ gives an upper bound +$CD^{1/4}(\log D)^{10}$ before normalization. In view of +\eqref{eq:residue-lower}, its normalized contribution is at most +\[ + CD^{-1/4}(\log D)^{11}\ll r^6; +\] +the last inequality follows from $r^6\ge D^{-1/8}$. +This proves an $O(r^6)$ bound in the present range. + +Now let $r0$. The bound just proved +therefore gives tightness. Any weak limit has mass one, and +$A$-invariance passes to the limit because right translation by each +fixed element of $A$ is continuous and preserves compact supports. +\end{proof} + +Let $B(r)$ denote an open matrix ball about the identity in $G$, for +$r$ small; all fixed smooth local metrics give equivalent estimates. + +\begin{proposition}[Ordinary balls in a probability limit] +\label{prop:ball} +Suppose $D_i\to\infty$ and $\mu_i$ converges weakly to a probability +$\mu$. For every compact $\Omega\subset X$, there are constants +$C_\Omega,r_\Omega>0$ such that +\[ + \mu(xB(r))\le C_\Omega r^6 + \qquad(x\in\Omega,\ 00$. For all sufficiently large $i$, uniformly in the finitely +many charts and in $x\in\Omega$, the ideal norms in $Q_iJ_{x,r}$ +can be covered by an interval with upper endpoint comparable to $Q_i$ +and length at most $C_4Q_ir$. We may choose that length comparable +to $Q_ir$, so it is at least the square root of its upper endpoint +for large $i$. The upper endpoint lies between $D_i^{1/4}$ and +$D_i$. Lemma~\ref{lem:ideal-estimate} and +\eqref{eq:vector-unfolding} therefore give +\[ + \mu_i(xB(r)) + \le\int_X\widehat f_{x,r}\,\dd\mu_i + \le\frac{C_3r^5}{Q_i\kappa_i} + \sum_{l\in Q_iJ_{x,r}}d_{K_i}(l) + \le C_5r^6. +\] +The constants are independent of $x$ and $r$ in the stated range; +the threshold for $i$ may depend on $r$. Since $xB(r)$ is open, +the open-set inequality for weak convergence gives the same bound +for $\mu(xB(r))$. This proves the assertion for each $x$ and $r$ +with the uniform constants supplied by the finite cover. +\end{proof} + +\subsection{Trace slices of the codifferent} + +The vector estimate has now supplied tightness and the ordinary-ball +bound. To control diagonal tubes, we need to count a quantity which +is nearly preserved by block-diagonal matrices. For an ordinary +vector $v$ and a dual vector $y$, that quantity is a partial sum of +the products $v_jy_j$. The following lattice estimate makes such +partial sums effective arithmetic coordinates. + +Let $\mathfrak D_K$ be the different of $K$, so +\[ + \mathfrak D_K^{-1} + =\{z\in K:\Tr_{K/\Q}(z\mathcal O_K)\subset\Z\}. +\] +Give $\R^6$ its Euclidean inner product, let +$\mathbf e=(1,\ldots,1)$, and put +\[ + H=\mathbf e^\perp, + \qquad \mathcal L_K=\sigma(\mathfrak D_K^{-1})\cap H. +\] + +\begin{lemma}[Small cells in a trace slice]\label{lem:trace-slice} +The lattice $\mathcal L_K$ has Euclidean covolume $\sqrt6/Q$ in +$H$ and admits a fundamental parallelotope of diameter +$O(D^{-1/30})$. Let $S$ be a nonempty proper subset of +$\{1,\ldots,6\}$. For fixed $M>0$, uniformly in $m\in\Z$ and +intervals $J\subset\R$ of length $\delta>0$, +\begin{equation} + \#\left\{z\in\mathfrak D_K^{-1}: + \Tr_{K/\Q}z=m, + \ |\sigma_j(z)|\le M\ (1\le j\le6), + \ \sum_{j\in S}\sigma_j(z)\in J\right\} + \ll_M Q\bigl(\delta+D^{-1/30}\bigr). + \label{eq:trace-slice-count} +\end{equation} +\end{lemma} + +\begin{proof} +Since $1$ is primitive in the free abelian group $\mathcal O_K$, +choose an integral basis $\alpha_1=1,\alpha_2,\ldots,\alpha_6$. +Let $\beta_1,\ldots,\beta_6$ be its trace-dual basis, so +$\Tr_{K/\Q}(\alpha_j\beta_k)=\delta_{jk}$. By the definition of +the codifferent, these $\beta_k$ form a basis of +$\mathfrak D_K^{-1}$. In particular, +$\Tr_{K/\Q}\beta_k=\delta_{1k}$, which proves that the trace map +onto $\Z$ is surjective and that +$\sigma(\beta_2),\ldots,\sigma(\beta_6)$ form a basis of +$\mathcal L_K$. For $2\le j,k\le6$, orthogonal projection gives +\[ + \bigl\langle\operatorname{proj}_H\sigma(\alpha_j), + \sigma(\beta_k)\bigr\rangle=\delta_{jk}. +\] +These projected vectors are therefore the dual basis inside $H$. +Since $\operatorname{proj}_H\sigma(\alpha_1)=0$, we obtain +\begin{equation} + \mathcal L_K^*=\operatorname{proj}_H\sigma(\mathcal O_K). + \label{eq:projected-dual} +\end{equation} + +Take a nonzero vector in the right side of +\eqref{eq:projected-dual}, arising from $\alpha\in\mathcal O_K$. +The vector is nonzero precisely when $\alpha\notin\Q$. +Primitivity of the field implies $\Q(\alpha)=K$, so the order +$\Z[\alpha]$ has discriminant at least $D$. Differences of +embedding coordinates are unchanged by orthogonal projection to +$H$. It follows that +\[ + D\le |\Disc(\Z[\alpha])| + =\prod_{j0$, because the first five +absolute logarithms range over a fixed bounded set. Repeating +the unfolding of Proposition~\ref{prop:packet-volume}, with the +additional restriction $\mathfrak b\mid(z)\mathfrak D_K$, gives +\begin{equation} + \int_X\#\{\text{admissible pairs with product }z\} + \,\dd\mu_{K,\sigma} + \le\frac1{Q\kappa} + \sum_{\mathfrak b\mid(z)\mathfrak D_K} + V_f(N\mathfrak b/Q) + \ll\frac1{Q\kappa} + \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\}. + \label{eq:pair-unfolding} +\end{equation} +Here we have dropped the restrictions on the dual vector after +fixing $z$, which can only increase the count. The divisibility +restriction is unchanged by multiplication of $d$ by a unit, so +the same unfolding and sign multiplicities apply. + +The embedding bound on $z$ implies +$N((z)\mathfrak D_K)=D|N_{K/\Q}z|\ll D$. +For an integral ideal of norm $N$, its number of ideal divisors is +at most $d(N)^6$, where $d$ is the ordinary divisor function: over +each rational prime there are at most six prime ideals, and every +one of their exponents is at most the exponent of that prime in +$N$. The classical divisor bound consequently gives, uniformly +for our allowed $z$, +\begin{equation} + \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\} + \le\exp\left(C\frac{\log D}{\log\log D}\right). + \label{eq:ideal-divisors} +\end{equation} + +Lemma~\ref{lem:trace-slice} bounds the number of allowed products +by $O(Q(e^{-t}+D^{-1/30}))$. Uniformly for $t\le T_{+,i}$, +\[ + D_i^{-1/30}=o(e^{-t}), +\] +because $T_{+,i}/\log D_i\to0$. The product count is therefore +$O(Q_i e^{-t})$. The event defining the tube is bounded by the +number of admissible pairs, so summing \eqref{eq:pair-unfolding} +over these products and using \eqref{eq:ideal-divisors} gives +\[ + \mu_i\bigl(x(a(tb)Oa(-tb)\cap a(-tb)Oa(tb))\bigr) + \ll\kappa_i^{-1} + \exp\left(-t+C\frac{\log D_i}{\log\log D_i}\right). +\] +Finally, \eqref{eq:residue-lower} and $t\ge T_{-,i}$ show that +the logarithm of the factor multiplying $e^{-t}$ is $o(t)$, +uniformly throughout the time range. Enlarging the discriminant +threshold makes the last display at most $e^{-t/3}$. +The same choices work for finitely many cuts and diagonal elements +by taking the smallest of their identity neighborhoods and the +largest of their thresholds. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/assembly.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/assembly.tex new file mode 100644 index 0000000000000000000000000000000000000000..733d16ca86726beea6dbc3a66383812f75e33500 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/assembly.tex @@ -0,0 +1,36 @@ +\section{Proof of the equidistribution theorem}\label{sec:assembly} + +We now return to the packets of Theorem~\ref{thm:main}. +Proposition~\ref{prop:cusp} and Mahler's compactness criterion give +tightness. Take an arbitrary subsequence with a weak probability limit +$\mu$. It is $A$-invariant, and Proposition~\ref{prop:ball} gives +the ordinary-ball estimate required by +Proposition~\ref{prop:positive-entropy}. Theorem~\ref{thm:ekl} +therefore makes almost every $A$-ergodic component of $\mu$ +homogeneous. + +Apply the intermediate-conditioning construction to this subsequence, +passing further to a subsequence wherever necessary. The augmented +limiting state still projects to $\mu$ on $X$. +Proposition~\ref{prop:bowen} supplies the tube estimate throughout +the required time interval. Hence +Theorem~\ref{thm:full-activity} makes every root active almost surely +in this state. + +If proper homogeneous components had positive total weight, +Lemma~\ref{lem:fixed-pieces} would place positive $\mu$-mass on the +countable union of their fixed-point pieces. At least one piece would +have positive mass. Its fixing diagonal is nonidentity, so it has +two unequal diagonal entries. The corresponding root plaques meet +that piece in at most countably many points. By +Lemma~\ref{lem:transverse-inactive}, that root is inactive almost +surely on states projecting to the piece, a contradiction. +Thus almost every component is the $G$-invariant probability, and +$\mu=m_6$. + +Every weakly convergent subsequence supplied by tightness has the same +limit. It follows that the original sequence converges to $m_6$ on +$C_c(X_6)$. Proposition~\ref{prop:cusp} already gives the compact sets +required for the nonescape assertion. The estimates are uniform in the +embedding ordering, so the conclusion holds for the arbitrary choices +in Theorem~\ref{thm:main}. diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/combination.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/combination.tex new file mode 100644 index 0000000000000000000000000000000000000000..f4f82f087a47a8933ca893e26caaa8109195b258 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/combination.tex @@ -0,0 +1,462 @@ +\section{Combining the root orientations}\label{sec:combination} + +Proposition~\ref{prop:segment-activity} guarantees activity somewhere on +every long segment across each oriented cut. It does not yet assert that +the roots are active at the same point. The wall invariance of the root +indicators supplies the missing relation: on the scale $R_i$, an indicator +can be read as a function of the value of its root alone. Transitivity +then becomes an approximate addition rule for functions on the real line. +We first obtain activity in every real interval and then use two-edge +cycles to show that the inactive sets have measure zero. + +\begin{theorem}[Full activity]\label{thm:full-activity} +For every ordered pair $j\ne k$ in $\{1,\ldots,6\}$, +\begin{equation}\label{eq:full-activity} + \int_X g_{jk,i}\,\dd\mu_i\longrightarrow1. +\end{equation} +Consequently every root is active almost surely in the limiting state law. +\end{theorem} + +\subsection{Functions of a root coordinate} + +For each root $\beta$, fix $b_\beta\in L$ such that +$\beta(b_\beta)=1$. If $x$ is sampled with law $\mu_i$, define the random +measurable functions +\begin{equation}\label{eq:root-line-functions} + G_{\beta,i}^{x}(s)=g_{\beta,i}\bigl(xa(R_i s b_\beta)\bigr), + \qquad s\in\R. +\end{equation} +They take values in $\{0,1\}$. Regard them as elements of the compact +metrizable space +\[ + \mathcal B=\{f\in L^\infty_{\mathrm{loc}}(\R):0\leq f\leq1\}, +\] +with weak-* convergence on every bounded interval. More explicitly, +one may use a countable dense set of $L^1$ test functions on each +$[-q,q]$, $q\in\N$, to metrize this topology. The finite product +$\mathcal B^{30}$ is also compact and metrizable. After passing to a +subsequence, the laws of the arrays in \eqref{eq:root-line-functions} +converge to a law of functions $(F_\beta)_\beta$ in this space. +We also write $G_{jk,i}^{x}=G_{\beta_{jk},i}^{x}$ and +$F_{jk}=F_{\beta_{jk}}$. + +\begin{lemma}\label{lem:root-line-laws} +For any such convergent subsequence, the limiting law and its prelimit +arrays have the following properties. +\begin{enumerate} + \item Almost surely, for every $v_0\in L$ with rational coordinates, + every oriented cut $J$, and every positive rational $\tau$, + \begin{equation}\label{eq:positive-cut-integral} + \sum_{\beta\in\mathcal E_J}\int_0^\tau + F_\beta\bigl(\beta(v_0+t b_J)\bigr)\,\dd t>0. + \end{equation} + \item For distinct $j,k,\ell$, and every bounded rectangle + $B\subset\R^2$, the nonnegative defects satisfy + \begin{equation}\label{eq:line-transitivity-defect} + \E_{x\sim\mu_i}\int_B + G_{jk,i}^{x}(s)G_{k\ell,i}^{x}(t) + \bigl(1-G_{j\ell,i}^{x}(s+t)\bigr)\,\dd s\,\dd t + \longrightarrow0. + \end{equation} +\end{enumerate} +\end{lemma} + +\begin{proof} +For $v$ in a fixed bounded subset of $L$, the vector +$v-\beta(v)b_\beta$ belongs to $\ker\beta$ and remains bounded. +Wall invariance \eqref{eq:wall-invariance}, followed by invariance of +$\mu_i$ under the common translation, gives +\begin{equation}\label{eq:common-translate-comparison} + \sup_{v\text{ in a fixed bounded set}} + \int_X\left| + G_{\beta,i}^{x}\bigl(\beta(v)\bigr) + -g_{\beta,i}\bigl(xa(R_i v)\bigr) + \right|\,\dd\mu_i(x)\longrightarrow0. +\end{equation} + +Fix the parameters in assertion~1. The left side of +\eqref{eq:positive-cut-integral} is a continuous function of the +weak-* array, because $\beta(b_J)\ne0$ for every crossing root. +By \eqref{eq:common-translate-comparison}, its prelimit differs in +$L^1(\mu_i)$ by $o(1)$ from +\[ + Y_i(x)=\sum_{\beta\in\mathcal E_J}\int_0^\tau + g_{\beta,i}\bigl(xa(R_i(v_0+t b_J))\bigr)\,\dd t. +\] +Suppose that the limiting observable were zero on a set of probability +$p>0$. Convergence in law and nonnegativity allow us to choose thresholds +$\varepsilon_i\downarrow0$ and events $E_i$ with +$\mu_i(E_i)\geq p/2$ such that the original prelimit observable is at +most $\varepsilon_i$ on $E_i$. To see this directly, first choose a +fixed sufficiently small continuity threshold, use convergence in law, +and then decrease the threshold along a diagonal subsequence. +The normalized restrictions $\mu_i|_{E_i}/\mu_i(E_i)$ consequently have +$Y_i$ expectation tending to zero. Translate these probabilities by +$a(R_i v_0)$ and call the results $\xi_i$. They satisfy +$\xi_i\leq(2/p)\mu_i$, whereas the change of variable $u=R_i t$ +gives +\[ + \frac1{R_i}\int_0^{\tau R_i}\int_X + A_{J,i}\bigl(xa(u b_J)\bigr)\,\dd\xi_i(x)\,\dd u + \longrightarrow0. +\] +This contradicts Proposition~\ref{prop:segment-activity}. The parameters +in assertion~1 range over a countable set, so the assertion holds +simultaneously almost surely. + +For assertion~2, the roots $\beta_{jk}$ and $\beta_{k\ell}$ are linearly +independent. Choose $v(s,t)$ linearly in $(s,t)$ with +\[ + \beta_{jk}(v(s,t))=s,\qquad + \beta_{k\ell}(v(s,t))=t. +\] +It follows that $\beta_{j\ell}(v(s,t))=s+t$. On a bounded rectangle, +\eqref{eq:common-translate-comparison} permits replacing all three +functions in the defect by their indicators at the common point +$xa(R_i v(s,t))$, with total expected error $o(1)$. The expected defect +there is $o(1)$ by the approximate transitivity in +Lemma~\ref{lem:activity-indicators}; its distribution is unchanged by +the translation. Integrating over the rectangle proves the claim. +\end{proof} + +The transitivity defect is nonlinear in the functions, so we do not pass +it directly through weak-* convergence. Instead we select deterministic +arrays for which the defects themselves tend to zero. + +\begin{lemma}\label{lem:deterministic-root-arrays} +If \eqref{eq:full-activity} fails for some root, there is a subsequence +and a deterministic choice of arrays $(G_{\beta,i})_\beta$, each taken +from \eqref{eq:root-line-functions}, with the following properties: +\begin{enumerate} + \item $G_{\beta,i}\to F_\beta$ weak-* on bounded intervals, and the + deterministic limiting functions satisfy + \eqref{eq:positive-cut-integral} for all its stated parameters; + \item every defect in \eqref{eq:line-transitivity-defect}, without its + expectation, tends to zero on every bounded rectangle; + \item for some root $\beta_0$ and some bounded interval $I$ of positive + length, + \begin{equation}\label{eq:limiting-deficiency} + \int_I(1-F_{\beta_0}(s))\,\dd s>0. + \end{equation} +\end{enumerate} +\end{lemma} + +\begin{proof} +Pass to a subsequence on which +$\int_X(1-g_{\beta_0,i})\,\dd\mu_i\geq\varepsilon$ for a fixed +$\varepsilon>0$. Invariance of $\mu_i$ and Fubini's theorem imply, for +any fixed interval $I$, +\[ + \E\int_I(1-G_{\beta_0,i}^{x}(s))\,\dd s + =|I|\int_X(1-g_{\beta_0,i})\,\dd\mu_i + \geq\varepsilon|I|. +\] +The integral is a bounded weak-* continuous observable, so the same +inequality holds for the limiting law. This law therefore gives positive +probability to a positive deficiency. Choose a point $F$ in its support +that has a positive deficiency and satisfies assertion~1 of +Lemma~\ref{lem:root-line-laws}. + +For each integer $q$, choose a neighborhood $U_q$ of $F$ of diameter +less than $1/q$. It has positive limiting probability, and thus the +prelimit probabilities of $U_q$ are bounded below by a positive number +for all sufficiently large indices. There are finitely many root +compositions. By \eqref{eq:line-transitivity-defect}, the sum of their +defects on $[-q,q]^2$ has expectation tending to zero. Choose an index +so large that, by Markov's inequality, the probability that this sum +exceeds $1/q$ is less than half the probability of $U_q$. There is +therefore a sampled array in $U_q$ with total defect at most $1/q$. +Make these choices with increasing indices. They converge to $F$ and +have vanishing defects on every bounded rectangle, by nonnegativity +and containment in some $[-q,q]^2$. +\end{proof} + +\subsection{From cut activity to every interval} + +For the rest of the argument, fix deterministic arrays as in +Lemma~\ref{lem:deterministic-root-arrays}. The labels are still +$\{0,1\}$-valued; their weak-* limits may initially take any value in +$[0,1]$. + +\begin{lemma}\label{lem:positive-root-intervals} +For every root $\beta$ and every nonempty open interval $I\subset\R$, +\[ + \int_I F_\beta(s)\,\dd s>0. +\] +\end{lemma} + +\begin{proof} +Fix a root $\beta_{jk}$ and a desired interval $I$; replacing it by a +bounded nonempty open subinterval only strengthens what we must prove. +Choose a rational +$v_0\in L$ so that $\beta_{jk}(v_0)$ lies in $I$. Choose $r>0$ small +enough that +\[ + [\beta_{jk}(v_0)-6r,\ \beta_{jk}(v_0)+6r]\subset I. +\] +For every root $\alpha$, put +$I_\alpha=(\alpha(v_0)-r,\alpha(v_0)+r)$. Draw the directed edge +$p\to q$ if $\int_{I_{\beta_{pq}}}F_{pq}>0$. + +This directed graph is strongly connected. Indeed, for any nonempty +proper subset $J$, choose a positive rational $\tau$ sufficiently small +that $\alpha(v_0+t b_J)\in I_\alpha$ for every +$\alpha\in\mathcal E_J$ and $0\leq t\leq\tau$. +Equation~\eqref{eq:positive-cut-integral} then supplies an edge from +$J$ to its complement. A finite directed graph with an outgoing edge +across every nontrivial cut is strongly connected: otherwise the set of +vertices reachable from a suitable vertex is a nonempty proper set +with no outgoing edge. + +Take a simple directed path +$j=p_0,p_1,\ldots,p_m=k$, where $m\leq5$. Write +$\alpha_h=\beta_{p_{h-1}p_h}$ and $I_h=I_{\alpha_h}$. Weak-* +convergence gives positive lower bounds, for sufficiently large $i$, on +\[ + \int_{I_h}G_{\alpha_h,i}(s)\,\dd s, + \qquad 1\leq h\leq m. +\] +The product of these integrals is consequently bounded below by a +positive constant. Interpret it as the measure of tuples +$(s_1,\ldots,s_m)\in\prod_h I_h$ on which all the path labels equal +one. + +Except for a set of tuple measure tending to zero, successive composition +of these labels gives +\[ + G_{p_0p_h,i}(s_1+\cdots+s_h)=1, + \qquad 1\leq h\leq m. +\] +Here is the integration detail. At step $h$, the failure of this +implication, if all earlier implications held, is bounded by the +transitivity defect for $p_0,p_{h-1},p_h$. The path is simple, so these +are distinct vertices. Integration in $s_1,\ldots,s_{h-1}$ pushes +Lebesgue measure on bounded intervals to a measure of bounded density +in their sum. Multiplying by previous indicator restrictions can only +decrease this measure. The defect integral on a fixed bounded +rectangle therefore bounds the failure measure by a fixed multiple of +an $o(1)$ quantity. Summing over the finitely many steps proves the +assertion. If $m=1$, no composition is necessary. + +The final sum $s_1+\cdots+s_m$ lies within $mr$ of +$\alpha_1(v_0)+\cdots+\alpha_m(v_0)=\beta_{jk}(v_0)$, hence in $I$. +Its pushforward density is bounded above by a fixed constant. Since +a positive measure of tuples has final label one, we obtain +$\liminf_i\int_I G_{jk,i}>0$. Weak-* convergence proves the lemma. +\end{proof} + +We have now located positive activity in every interval for every root. +To eliminate a positive inactive set, we will translate an active set +by sums of parameters from two opposite auxiliary roots. Composing +successively with those two roots returns to the original root while +adding their two parameters. Interval positivity supplies parameter +sets of positive measure on both sides of zero. By mixing normalized +Lebesgue measure on these sets, we will obtain bounded densities of +mean zero, bounded support, and variance bounded away from zero. +The sum of two independent parameters with these densities describes +one cycle increment. For a fixed bounded interval +$I$, the argument needs a fixed number of independent cycle increments +to have density bounded below on $I-I$: this makes every part of the +active set interact with every part of the inactive set inside $I$. +The next lemma gives that lower bound uniformly over the densities +that arise. + +\subsection{A uniform convolution estimate} + +\begin{lemma}\label{lem:convolution} +Fix $B,M,v,L>0$. Consider the probability densities $p$ on $\R$ such +that +\[ + \supp p\subset[-B,B],\qquad 0\leq p\leq M, + \qquad \int x p(x)\,\dd x=0, + \qquad \int x^2p(x)\,\dd x\geq v. +\] +There are an integer $m\geq2$ and $c>0$, depending only on +$B,M,v,L$, such that +\[ + p^{*m}(x)\geq c\qquad(|x|\leq L) +\] +for every such density. The inequality uses the continuous representative +of the convolution density. +\end{lemma} + +\begin{proof} +If the family is empty there is nothing to prove. Otherwise write +$\sigma_p^2=\int x^2p(x)\,\dd x$, so +$v\leq\sigma_p^2\leq B^2$, and let +\[ + \phi_p(t)=\int e^{itx}p(x)\,\dd x. +\] +Bounded support and the zero mean give the uniform expansion +\begin{equation}\label{eq:uniform-characteristic-expansion} + \phi_p(t)=1-\frac12\sigma_p^2t^2+O(B^3|t|^3). +\end{equation} +Applying the same expansion to +$|\phi_p(t)|^2$, the characteristic function of the difference of two +independent variables with density $p$, shows that for a sufficiently +small $\delta>0$, depending only on $B,v$, +\begin{equation}\label{eq:small-frequency-domination} + |\phi_p(t)|\leq e^{-vt^2/4}\qquad(|t|\leq\delta). +\end{equation} + +There is also a uniform gap below one outside this neighborhood. For +$|t|\geq\delta$, an angle $\theta$, and $0\delta\sqrt m$, \eqref{eq:large-frequency-gap} and Plancherel give +\[ + \int_{|u|>\delta\sqrt m}|\phi_p(u/\sqrt m)|^m\,\dd u + \leq2\pi M\sqrt m\,\rho^{m-2}\longrightarrow0. +\] +The Gaussian tail there also tends to zero uniformly. These three +estimates prove \eqref{eq:uniform-local-clt}. + +For $|x|\leq L$ and all sufficiently large $m$, the Gaussian term in +\eqref{eq:uniform-local-clt} is at least +$1/(2\sqrt{2\pi}B)$, uniformly in $p$. Increase $m$ so that the +error is at most $1/(4\sqrt{2\pi}B)$. Then +\[ + p^{*m}(x)\geq\frac1{4\sqrt{2\pi}B\sqrt m} + \quad(|x|\leq L), +\] +which is the required conclusion. +\end{proof} + +\subsection{Cycles remove the inactive sets} + +\begin{proof}[Proof of Theorem~\ref{thm:full-activity}] +Suppose the theorem fails, and take the deterministic arrays and limiting +functions from Lemma~\ref{lem:deterministic-root-arrays}. Fix an edge +$j\to k$, and choose $\ell$ distinct from $j,k$. +Lemma~\ref{lem:positive-root-intervals} supplies, for each of the roots +$\ell j$ and $j\ell$, positive limiting integral on each of the +intervals $(-2,-1)$ and $(1,2)$. Accordingly, for all sufficiently +large $i$, each of the four sets where the relevant $G$ equals one +has length bounded below by a positive constant. + +For each auxiliary root, normalize Lebesgue measure on its two sets +to probability densities $p_i^-$ and $p_i^+$. Their means belong to +$[-2,-1]$ and $[1,2]$, respectively. There is a convex combination +\[ + p_i=\lambda_i p_i^-+(1-\lambda_i)p_i^+ +\] +with mean zero. The coefficients can be chosen in $[1/3,2/3]$ by +solving the mean-zero equation. Thus these densities have support in +$[-2,2]$, a common finite upper bound, and variance at least one. +They are supported entirely where the corresponding auxiliary +indicator is one. Denote the density for $\ell j$ by $p_i$ and that +for $j\ell$ by $q_i$. + +Starting from $G_{jk,i}(t)=1$, composition with a parameter $s$ in the +support of $p_i$ gives the edge $\ell k$, and composition with a +parameter $s'$ in the support of $q_i$ returns to $jk$: +\[ + \ell j+jk=\ell k,\qquad j\ell+\ell k=jk. +\] +Both are root compositions with three distinct vertices. Their defect +estimates imply, for every fixed bounded interval $I$, +\begin{equation}\label{eq:one-cycle-error} + \int_I\int_\R\int_\R + G_{jk,i}(t)\bigl(1-G_{jk,i}(t+s+s')\bigr) + p_i(s)q_i(s')\,\dd s\,\dd s'\,\dd t\longrightarrow0. +\end{equation} +Indeed, except on null sets of parameters, both auxiliary labels are +one. The indicator in \eqref{eq:one-cycle-error} is then bounded by +the sum of the two composition defects. For the first defect, +integrate out $s'$ and use the uniform density bound on $p_i$. +For the second, the variable $t+s$ has a bounded density on a fixed +bounded interval when $t\in I$ is integrated with Lebesgue measure +and $s$ with $p_i$. The bound on $q_i$ and the defect estimate for +$(j\ell,\ell k)$ finish the verification. + +Put $r_i=p_i*q_i$. These cycle-increment densities have support in +$[-4,4]$, a common density bound, mean zero, and variance at least two. +We can iterate \eqref{eq:one-cycle-error} any fixed number $m$ of times +to get +\begin{equation}\label{eq:repeated-cycle-error} + \int_I\int_\R + G_{jk,i}(t)\bigl(1-G_{jk,i}(t+u)\bigr) + r_i^{*m}(u)\,\dd u\,\dd t\longrightarrow0. +\end{equation} +For clarity, along a sequence of $m$ increments, a path starting with +label one and ending with label zero has a first transition from one +to zero. Bound its indicator by the sum of the indicators of these +transitions. The intermediate positions lie in +$I+[-4m,4m]$. After integrating all preceding increments, their +distributions have densities bounded by one relative to Lebesgue +measure on that larger interval, because the initial $t$ is integrated +over $I$ and all increments have probability laws. Each term is +therefore bounded by the one-cycle error on the larger interval. +There are only $m$ terms, proving \eqref{eq:repeated-cycle-error}. + +Fix a bounded interval $I$ of positive length and apply +Lemma~\ref{lem:convolution} to the $r_i$, with $L$ large enough that +$I-I\subset[-L,L]$. Choose its fixed power $m$ and lower bound +$c>0$. In \eqref{eq:repeated-cycle-error}, retain only the pairs +$t,t+u\in I$. We obtain +\[ + c\left(\int_I G_{jk,i}(t)\,\dd t\right) + \left(\int_I(1-G_{jk,i}(t))\,\dd t\right) + \longrightarrow0. +\] +The first factor has positive lower limit by +Lemma~\ref{lem:positive-root-intervals}. Hence +$\int_I(1-G_{jk,i})\to0$ for every such interval. Weak-* convergence +shows that $F_{jk}=1$ almost everywhere. The argument applies to every +root, contradicting the deficiency +\eqref{eq:limiting-deficiency}. Thus \eqref{eq:full-activity} holds. + +By Lemma~\ref{lem:activity-indicators}, the sequence indicators converge +jointly in law to the activity indicators of the limiting state. Since +each has expectation tending to one, every limiting root indicator +equals one almost surely. There are only thirty roots, so this holds +for all of them simultaneously. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/conditioning.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/conditioning.tex new file mode 100644 index 0000000000000000000000000000000000000000..7f737d87eade5db8c920ad315472c5eda138988a --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/conditioning.tex @@ -0,0 +1,664 @@ +\section{Conditional probabilities at intermediate resolutions} +\label{sec:conditioning} + +We now pass from the measures $\mu_i$ to conditional probabilities along +unipotent plaques. The conditioning must be performed before taking the +limit: exact conditional measures of a compact diagonal orbit along a +unipotent group contain no information about its distribution at the +scales of the arithmetic estimate. We instead condition on small +\emph{transverse cells}. The freedom to choose their depth over a broad +range will make the resulting data compatible with changes of charts and +with certain diagonal motions whose length tends to infinity. + +Throughout this section, $G=\SL_n(\R)$, $X=\SL_n(\Z)\backslash G$, and +$n=6$ in the application. We assume only that $\mu_i$ are $A$-invariant +probability measures converging weakly to an $A$-invariant probability +$\mu$, and that the available times satisfy +\[ + T_{-,i}\longrightarrow\infty, + \qquad T_{+,i}/T_{-,i}\longrightarrow\infty. +\] +The particular arithmetic values of these times will be used later. +All subsequences taken below preserve these assumptions. + +\subsection{Plaque charts and their extensions} + +Write $L$ for the traceless diagonal algebra and $a(b)=\exp(b)$ for +$b\in L$. For $j\ne k$, set +\[ + \beta_{jk}(b)=b_k-b_j, + \qquad U_{jk}=\{1+tE_{jk}:t\in\R\}. +\] +These signs correspond to the right action on $X$: the displacement $u$ +at $x$ becomes $a(-b)ua(b)$ at $xa(b)$, and its $jk$ parameter is +multiplied by $e^{\beta_{jk}(b)}$. + +A \emph{pattern} is a set of roots positive on some element of $L$ and +closed under root addition whenever the sum is a root. Its +\emph{pattern group} $V$ consists of the unipotent matrices with arbitrary +entries in those positions and zero entries elsewhere. After permuting +coordinates it is a triangular unipotent group. All these groups are +normalized by $A$, and there are only finitely many of them. We also use +the groups $VA$, but only in local charts. + +For any such group $H$, a plaque chart is an injective smooth map +\begin{equation} + \Phi:Q\times B\longrightarrow X, + \qquad \Phi(q,h)=\psi(q)h, + \label{eq:plaque-chart} +\end{equation} +where $Q$ is an open box in a complementary transverse manifold and +$B$ is an open bounded coordinate box in $H$. The map extends smoothly +and injectively to a slightly larger domain. The sets with $q$ fixed +are its plaques. Smooth reparametrizations of $h$, allowed to depend +smoothly on $q$, describe the same plaques and will also be used. A +smaller chart with closure in the displayed chart supplies a fixed +margin from its boundary. + +The comparison of probabilities on different bounded portions of a +plaque requires charts extending beyond a prescribed displacement +range. The following elementary recurrence observation supplies them. + +\begin{lemma}[No periods in a pattern group]\label{lem:no-periods} +Let $\nu$ be an $A$-invariant probability on $X$ and let $V$ be a +pattern group. For $\nu$-almost every $x$, +\[ + xv=x,\quad v\in V \quad\Longrightarrow\quad v=1. +\] +At every such $x$, the image $xK$ of any prescribed compact +displacement range $K\subset V$ is contained in an embedded plaque +with a transverse neighborhood. Consequently, for any compact displacement range in $V$ +and any $\epsilon>0$, finitely many inner plaque charts cover a set of +$\nu$-measure at least $1-\epsilon$, and their outer charts contain that +range about every point of the corresponding inner charts. +\end{lemma} + +\begin{proof} +Choose $b\in L$ such that $a(-tb)va(tb)\to1$ as $t\to\infty$ for every +$v\in V$. At a point recurrent under $a(b)$, a relation $xv=x$ gives +\[ + xa(kb)\,a(-kb)va(kb)=xa(kb). +\] +A subsequence of these points returns to a compact neighborhood of +$x$. Stabilizers of points in that neighborhood contain no nonidentity +element in a fixed neighborhood of $1$: this follows from the +discreteness of the lattice and local quotient charts. The displayed +relation is therefore impossible for $v\ne1$. Poincar\'e recurrence +applies at almost every point. + +The orbit map $v\mapsto xv$ is then an injective immersion. Its +restriction to a compact set is an embedding. Take a slightly larger +compact coordinate domain in $V$ and use a complementary transverse +slice. Compactness permits shrinking this slice until its product +with the prescribed plaque domain is injective. Indeed, a failure +along slices shrinking to $x$ would give either two distinct points +of the limiting embedded plaque with the same image, or contradict +local injectivity near one of its points. Shrinking once more provides +inner charts with margins. A countable subcover and continuity from +below give the finite-cover assertion. +\end{proof} + +Fix a countable stock of plaque charts and inner charts with the +following properties. It is a basis wherever local charts are needed; +for pattern groups it also supplies the extensions in +Lemma~\ref{lem:no-periods}, for each compact displacement range in a +countable exhaustion. It includes subboxes, fixed subdivisions of the +plaque coordinates into arbitrarily fine bins, and the simultaneous +coordinate usages described next. Chart and bin boundaries have +$\mu$-measure zero. Such choices are possible by varying coordinate +endpoints; for each coordinate only countably many levels can have +positive measure. One may start with multiplication charts and choose +countably many centers, transversals, and domain sizes dense among +the choices with compact closure and an injective enlargement. + +Two simultaneous usages will matter. If $V=WU$, where $U$ is a root +group, $W$ is a normal pattern subgroup, and the patterns are disjoint, +include charts +\[ + (q,w,u)\longmapsto\psi(q)wu. +\] +Their $V$, $U$, and $W$ transverse coordinates are, respectively, +$q$, $(q,w)$, and $(q,u)$. Normality gives the last assertion; the +$w$ coordinate is a smoothly reparametrized $W$ coordinate when $u$ +is fixed. Also include charts $(q,v,a)\mapsto\psi(q)va$ for $VA$, with +the simultaneous $V$ transverse coordinate $(q,a)$. All chart bounds, +plaque extensions, bin sizes, and fixed margins in a comparison are +chosen before $i$ tends to infinity. + +\subsection{A broad range of almost unchanged conditionals} + +In a chart $\mathcal B=\Phi(Q\times B)$, restrict $\mu_i$ to +$\mathcal B$. Partition $Q$ by a shifted dyadic grid of side $2^{-l}$. +For a point $x=\Phi(q,h)$ in the chart, let +\[ + P^{\mathcal B}_{i,l}(x) + =\text{the conditional distribution of $h$ given the grid cell of $q$}. +\] +The conditioning uses the restricted measure. Zero-mass columns may +be assigned any probability. An expectation involving this chart +means integration over $\mathcal B$ with the unnormalized measure +$\mu_i|_{\mathcal B}$; equivalently, one can normalize that measure and +then multiply the expectation by $\mu_i(\mathcal B)$. + +For positive sequences $r_i,s_i$, write $r\prec s$ if $r_i/s_i\to0$. +An interior depth will mean a sequence $d_i$ with +$l_-\prec d\prec l_+$, where the endpoints are supplied by the next +lemma. Indices $i$ on scales will often be suppressed. + +\begin{lemma}[A common plateau]\label{lem:plateau} +After passage to a subsequence there are integer depths $l_{-,i}$ and +$l_{+,i}$ such that +\begin{equation}\label{eq:plateau-room} + T_-\prec l_-\prec l_+\prec T_+. +\end{equation} +For every chart and every fixed finite plaque-bin partition in the +countable stock, the expected $\ell^1$ distance between the +conditional bin-probability vectors at depths $l_-$ and $l_+$ tends +to zero, averaged also over a uniform shift of the nested dyadic +grid. The same conclusion holds between any two intervening depths +for that grid. + +The shifts can be selected so that these conclusions hold for every +fixed chart and fixed bin partition and so that random-grid boundary +estimates hold at the two endpoint depths and at any prescribed +countable collection of interior depth sequences. When coordinate +usages share transverse labels, +their grids may share the corresponding shifts. At interior depths, +conditional bin probabilities are unchanged in expected $\ell^1$ +distance by using auxiliary shifted grids selected with the same +boundary convention. +\end{lemma} + +\begin{proof} +Fix a chart, a finite bin partition, and a grid shift. If $p_l$ is the +conditional bin-probability vector, its coordinates form bounded +martingales in the increasing transverse grid sigma-fields. Thus +\[ + \int_{\mathcal B}\|p_v-p_u\|_2^2\,\dd\mu_i + =\int_{\mathcal B}\|p_v\|_2^2\,\dd\mu_i + -\int_{\mathcal B}\|p_u\|_2^2\,\dd\mu_i, + \qquad u\le v. +\] +The sum of these differences over disjoint depth intervals is at most +$1$. This remains true after averaging in the shift. Enumerate all +fixed chart/bin choices and assign them positive summable weights. +Their weighted sum has the same bounded total variation. + +Put $L_i=\log(T_{+,i}/T_{-,i})$. Divide the interval of logarithmic +depths into $m_i$ consecutive intervals, where $m_i\to\infty$ and +$m_i=o(L_i)$. Discard its first and last intervals. Some remaining +interval has weighted martingale cost $O(m_i^{-1})$. Its endpoints, +rounded to integers, give $l_-$ and $l_+$. The discarded end intervals +and the length $L_i/m_i\to\infty$ give all three ratios in +\eqref{eq:plateau-room}. Each fixed chart/bin choice has cost tending +to zero; Cauchy--Schwarz gives the asserted $\ell^1$ conclusion. +Orthogonality of martingale increments gives it for intervening +depths as well. + +For the boundary assertion, fix a point $q$ in a Euclidean transverse +coordinate domain. Under a uniform grid shift, the probability that +$q$ is within $\rho 2^{-l}$ of a depth-$l$ grid wall is $O(\rho)$, +uniformly in $q,l$, and the sampling measure. Integrating first in +the shift gives the same estimate for any measure on that domain. +Choose $\rho_i\downarrow0$ slowly. Markov's inequality, followed by +a diagonal choice over the chart/bin and depth lists, selects shifts +with both the conditional-probability estimates and boundary masses +tending to zero. Common labels can use common shift variables: +each marginal shift remains uniform, which is all the argument uses. + +Here is the comparison with an auxiliary grid at an interior depth +$d$. Outside negligible boundary mass, one of its cells lies inside +one depth-$l_-$ cell and is a union of depth-$l_+$ cells, apart from +the latter cells meeting its walls. The relevant relative collar +widths are $O(2^{l_--d})$ and $O(2^{d-l_+})$. Both tend to zero much +faster than the chosen $\rho_i$. Summing the martingale bin errors +over the contained fine cells compares the auxiliary conditional +law to the coarse one. The discarded mass changes the weighted +conditional $\ell^1$ error by at most twice that mass. This proves +the last assertion. The same argument compares any two of the +selected grids by sandwiching them between these endpoint grids. +\end{proof} + +We make the selection convention precise for later use. All defining +charts, simultaneous coordinate usages, fixed bin subdivisions, and +depth sequences belong to countable lists. A diagonal selection +enforces their estimates for each fixed member; it does not assert +uniformity over the whole infinite list at a finite index $i$. +Within $[l_-,l_+]$ any fixed finite number of scales can be chosen +with successive ratios tending to infinity and with room at both +ends. For instance, use distinct fixed-fraction interpolations on +the logarithmic depth interval. Different arguments may reuse such +scales. Auxiliary grids chosen for one comparison need not define +the data, because Lemma~\ref{lem:plateau} compares their fixed-bin +laws with those of the defining grids. Such a grid may also be +chosen later at an interior depth not in the original list; it +does not change the augmented states defined below. We always let $\rho_i$ decrease +slowly enough to dominate the exponentially small relative collar +widths arising from these scale separations. None of these conventions +requires absolute continuity. + +\subsection{The augmented probability space} + +Choose a defining interior depth for every chart and selected grids +as above. Record at $x$ its conditional plaque probability in each +chart containing it, and an isolated dummy symbol for every other +chart. Write $Z_i(x)$ for the resulting tuple together with $x$. +Each plaque probability is regarded as a probability on the compact +closure of its coordinate domain. The countable product of these +compact spaces, together with the $X$ coordinate, is a separable +metrizable space. Tightness of $\mu_i$ therefore permits passage to +a subsequence for which the laws of $Z_i(x)$ converge. Denote the +limiting state by +\[ + \omega=\bigl(x,(P^{\mathcal B})_{\mathcal B}\bigr). +\] +Its $x$ marginal is $\mu$. Fixed-bin probabilities pass to the limit: +the averaged plaque probability in a chart is its plaque-coordinate +marginal, so the null-boundary choices imply that the limiting +probabilities assign zero mass to the chosen bin boundaries almost +surely. + +\begin{lemma}[Sampling one plaque probability]\label{lem:sampling} +In any fixed chart, conditional on its transverse coordinate $q$ +and its own limiting probability $P^{\mathcal B}$, the sampled +plaque coordinate $h$ has law $P^{\mathcal B}$. In particular, +$h\in\supp P^{\mathcal B}$ almost surely on that chart. This assertion +conditions on this one probability, rather than on the entire +augmented state. +\end{lemma} + +\begin{proof} +Let $q_i^0$ be a representative of the cell containing $q$. At the +sequence level, $P^{\mathcal B}_{i}$ is a function of that cell. +Consequently, for bounded continuous tests $f$ and $F$, +\[ + \int_{\mathcal B} f(h)F(q_i^0,P^{\mathcal B}_{i})\,\dd\mu_i + =\int_{\mathcal B}P^{\mathcal B}_{i}(f) + F(q_i^0,P^{\mathcal B}_{i})\,\dd\mu_i. +\] +On compact inner boxes, replacing $q_i^0$ by $q$ incurs an error +tending to zero. Let $i\to\infty$ and then exhaust the chart by +inner boxes. Its boundary has zero limiting mass. This proves the +conditional-law identity. The standard support assertion for a +sample from a probability measure follows, for example, by using a +countable basis of the plaque domain. +\end{proof} + +The remaining task in this section is to show that probabilities +recorded in different charts describe the same measure along a +plaque, up to normalization. We first isolate the estimate that +allows many small transverse columns to be regrouped without losing +control of the error. + +\subsection{Aggregating fine columns into patches} + +In an output chart use the defining nested grid at depths $l_-$ and +$l_+$. Write $E$ for a coarse cell and $e$ for a fine cell contained +in $E$. Fix finitely many plaque bins $J_j$, and set +\[ + M_e=\mu_i(\text{$e$ column}),\qquad + M_{e,j}=\mu_i(\text{$e$ column with }h\in J_j),\qquad + P_E(j)=\Prb_i(h\in J_j\mid q\in E). +\] +Here and below the chart restriction is understood. The plateau +estimate is precisely +\begin{equation}\label{eq:mass-error} + \delta_i:=\sum_E\sum_{e\subset E}\sum_j + |M_{e,j}-M_eP_E(j)|\longrightarrow0. +\end{equation} + +\begin{lemma}[Comparison on trimmed patches]\label{lem:patch-comparison} +In the preceding chart, let $p$ range over pairwise disjoint +measurable patches, each lying transversely in one coarse cell +$E(p)$. Suppose a subset of each patch has the product form +\[ + p^{\mathrm{in}}= + \bigcup_{e\in I_p}\bigl(e\times B_p\bigr), + \qquad B_p=\bigcup_{j\in J(p)}J_j, +\] +and the sum of the discarded masses $\mu_i(p\setminus p^{\mathrm{in}})$ +is at most $\epsilon$. The collections $I_p,J(p)$ may depend on +$i,p$, and the number of patches need not be bounded. If $L_p$ is +the plaque-bin law conditional on $p$, then, with errors weighted by +$\mu_i(p)$, +\[ + L_p\quad\hbox{agrees with}\quad + \frac{\one_{B_p}P_{E(p)}}{P_{E(p)}(B_p)} + \quad\text{to total expected }\ell^1\text{ error } + O(\epsilon)+o(1). +\] +An arbitrary law is used if the denominator vanishes. The +unnormalized version is +\begin{equation}\label{eq:patch-unnormalized} + \sum_p\mu_i(p) + \left\|P_{E(p)}(B_p)L_p- + \one_{B_p}P_{E(p)}\right\|_1 + \le O(\epsilon)+o(1). +\end{equation} +If patches and a disjoint refinement into subpatches admit such trims with the same +coarse cell and the same plaque-bin union, their conditional bin +laws agree with one another with the corresponding weighted error. + +These assertions also hold for bounded continuous tests, up to their +uniform continuity error on the fixed bins. A test may depend on +the sampled point, provided its norm and continuity modulus in the +tested plaque variable are uniform. At the output sample, +$P_{E(p)}$ may be replaced in fixed-bin expressions by the recorded +data probability, with an additional error tending to zero. +\end{lemma} + +\begin{proof} +For a trimmed patch, sum the errors in \eqref{eq:mass-error} over +$e\in I_p$ and $j\in J(p)$. The resulting comparison is between +its unnormalized bin-mass vector and +\[ + \left(\sum_{e\in I_p}M_e\right) + \one_{B_p}P_{E(p)}. +\] +Every pair $(e,j)$ occurs in at most one trimmed patch, since those +sets are disjoint. Hence the total unnormalized error is at most +$\delta_i$. Restoring discarded mass adds at most $\epsilon$. +For two nonnegative finite vectors $v,w$, comparison after +normalization, weighted by $\|v\|_1$, costs at most +$2\|v-w\|_1$; this follows by adding and subtracting +$w/\|v\|_1$. The same inequality covers zero mass by an arbitrary +choice of normalized vector. It proves the first assertion and, +on multiplying by $P_{E(p)}(B_p)\le1$, proves +\eqref{eq:patch-unnormalized}. + +Apply the same argument separately to patches and subpatches and +use their common restricted vector $P_E|_{B_p}$ to obtain their +comparison. Approximating a continuous test by its values on bins +gives the remaining test assertion. The bound is in $\ell^1$, so it +is uniform over all choices of bounded bin values, including values +depending on the sample. Finally, the plateau controls the expected +fixed-bin distance between the coarse conditional vector and the +recorded vector. Restricting to disjoint patches does not increase +that expectation. +\end{proof} + +In applications the bins and charts are fixed first. Their sizes +may then be chosen sufficiently small to achieve any prescribed +limsup error, after which $i$ tends to infinity. Thus ``arbitrarily +small error'' below means an actual bound by a prescribed +$\epsilon>0$ in this order of choices. It is not a claim about +uniform convergence over all subdivisions. + +\subsection{Projective agreement and diagonal motions} + +Center the probability in a $V$ chart at its sampled point: if the +sample has group coordinate $h_s$, the group displacement of $h$ is +$h_s^{-1}h$. For a smoothly reparametrized chart, use the corresponding +group elements in this formula. Two nonzero measures on a common +domain agree \emph{projectively} if one is a positive scalar multiple +of the other there. + +\begin{proposition}[Comparison of plaque data]\label{prop:plaque-comparison} +For the augmented limit constructed above the following assertions +hold. +\begin{enumerate} +\item For every pattern group $V$, its centered probabilities in +different charts agree projectively on their common plaque sheet +near the sampled point. They extend to a measurable nonzero +projective locally finite measure $m^V_\omega$ on $V$, with +$1\in\supp m^V_\omega$. The chart probabilities represent its +restrictions on all bounded displacement ranges contained in their +plaque domains. +\item Fix $b\in L$. In every subsequential joint limit of +\[ + \bigl(Z_i(x),Z_i(xa(b))\bigr),\qquad x\sim\mu_i, +\] +the two $V$ measures are related projectively by the pushforward +under $v\mapsto a(-b)va(b)$. +\item Fix a root $\beta$ and an interior time scale $R$ with room +for interior depths on both sides. For every fixed $C<\infty$ and +every sequence +\[ + b_i\in\ker\beta,\qquad \|b_i\|\le CR_i, +\] +the measures $m^{U_\beta}$ in every subsequential joint limit of +$\bigl(Z_i(x),Z_i(xa(b_i))\bigr)$ agree projectively. This assertion +holds for every such sequence $b_i$; the defining grids and state +construction are unchanged. +\end{enumerate} +\end{proposition} + +\begin{proof} +We prove one local comparison that covers all three assertions. The +motion is either the identity, a fixed $a(b)$, or $a(b_i)$ with +$b_i\in\ker\beta$ and $V=U_\beta$. Denote its induced conjugation on +$V$ by $c$. In the first two cases $c$ is fixed; in the third case +$c$ is the identity. + +We display the geometry in group-element plaque coordinates. For +a reparametrized chart, write its original coordinate as $t\in B_0$ +and its group element as $h=r_q(t)$, where the maps $r_q$ and their +inverses are smooth on the fixed extended chart. Its group-coordinate +domain is $B(q)=r_q(B_0)$; for a multiplication chart $B(q)$ is +constant. The conditional probabilities and bins remain in the +original fixed coordinates. When converting a probability to group +coordinates at a sample, use $r_{q_s}$. Across a depth-$F$ cell this +conversion differs from the actual $r_q$ by $O(2^{-F})$, which has +vanishing effect on every continuous test considered below. + +In the long-motion case choose an input +transverse depth $F$ with +\begin{equation}\label{eq:comparison-scales} + l_-\prec R\prec F\prec l_+, +\end{equation} +For a fixed motion, it suffices to take $l_-\prec F\prec l_+$; +all factors $\exp(O(R))$ below are then bounded constants. +The depth-$F$ input grid may be an auxiliary grid selected for this +comparison. Thus $R$ and $F$ need not belong to the countable list +used to define the states; Lemma~\ref{lem:plateau} compares the +auxiliary laws back to the recorded ones. Its boundary selection +uses $\mu_i$ and the scale bound, and does not depend on the +particular $b_i$ within that bound. +The input patches are the depth-$F$ transverse cells times the full +plaque-coordinate domain. Their conditional laws are the input +probabilities at depth $F$. + +\smallskip\noindent +\emph{Fitting the images into output charts.} +Choose input inner charts and fixed displacement test ranges first. +An input chart may have any fixed plaque extension needed for these +ranges. By Lemma~\ref{lem:no-periods}, output inner charts can be +chosen whose outer charts accommodate the entire image plaque +range, with a margin. A finite such cover omits arbitrarily small +limiting mass. The output distribution is again $\mu_i$, so the +same is true for the sequence. If an image patch meets one of these +inner charts, its whole image fits in the corresponding outer chart +for all sufficiently large $i$. This last assertion follows from +the transverse diameter estimate below. Assign such patches to +output charts, or work with each applicable pair of charts; the +finite number of pairs only multiplies error bounds by a fixed +constant. + +More precisely, let $B(q)$ and $B'(q')$ be the full input and output +domains in group coordinates, and let $(q_s,h_s),(q'_s,h'_s)$ be the +two samples. The output extension is required to contain +\begin{equation}\label{eq:full-plaque-margin} + h'_s\,c\bigl(h_s^{-1}\overline{B(q_s)}\bigr)\subset B'(q'_s) + \quad\text{with a positive boundary margin}. +\end{equation} +This condition concerns the entire input plaque domain. It is an +open condition on the paired sample coordinates: the domains are +smooth images of fixed compact closures. Their possible centered +ranges lie in a fixed compact subset of $V$, so the extension lemma +supplies output covers with this property. + +In the two charts the map has the form +\begin{equation}\label{eq:plaque-transition} + (q,h)\longmapsto\bigl(\theta_i(q),\,d_i(q)c(h)\bigr). +\end{equation} +The first coordinate is independent of $h$, because the motion +normalizes $V$. The factor $d_i(q)$ is a left translation in $V$. +It remains in a fixed compact set on the comparisons in question: +both plaque-coordinate ranges and $c$ are fixed. For the varying +motion, right multiplication and its inverse have differential norm +at most $\exp(O(R))$ in a left invariant metric on $G$ and the +induced quotient metric. Fixed chart changes have bounded +differentials. Consequently both $\theta_i(q)$ and $d_i(q)$ vary over an +input cell by +\begin{equation}\label{eq:transverse-image-width} +O\bigl(2^{-F}\exp(O(R))\bigr)=o(2^{-l_-}). +\end{equation} +For reparametrized charts, the input domains $B(q)$ also vary by +$O(2^{-F})$ over the cell, and their admitted output ranges vary by +$O(2^{-F}\exp(O(R)))$. The fixed boundary margins absorb these +variations. Thus the same estimate applies to those charts. +The same bounds hold for pullbacks of paths in the extended charts. +They justify following one plaque sheet throughout its prescribed +range, rather than including a different sheet at a return of the +orbit. They also justify the fitting assertion: vary $q$ along the +short path in its input cell and then follow the fixed plaque range +inside the output extension. + +On one such patch, $\theta_i$ is injective. To see this, the admitted +output plaque ranges have a common open subset, by the margin and +the small variation of $d_i(q)$. If two transverse inputs had the +same image, one could choose the same output plaque point in this +common subset, contradicting injectivity of the motion and of the +input chart. This observation permits product trims in output +coordinates, even if many distinct input patches return to the +same output box. + +\smallskip\noindent +\emph{Trimming to fine columns and fixed plaque bins.} +Discard image patches meeting a coarse transverse wall. By +\eqref{eq:transverse-image-width}, every point of such a patch is in +a collar whose relative width tends to zero. The grid convention, +applied to the output distribution $\mu_i$, makes the discarded +mass $o(1)$. Each remaining patch lies transversely in one coarse +cell $E$. + +Take fine transverse cells whose closures lie in its projected +transverse range, and plaque bins admitted over the whole range. +Their products form an inner set as in +Lemma~\ref{lem:patch-comparison}. Here are the two loss estimates. +First, a path varying the transverse output coordinate through one +fine cell pulls back a distance +\begin{equation}\label{eq:fine-pullback-width} + O\bigl(2^{-l_+}\exp(O(R))\bigr)=o(2^{-F}). +\end{equation} +Away from the input cell walls and a fixed small collar of the +input chart boundary, it therefore stays in the input transverse +range and in the chart. Input grid-wall loss is $o(1)$ by the +selected shifts; chart-boundary loss can be made arbitrarily small +by its null limiting mass. This verifies that almost all sampled +fine transverse cells are admitted. + +Second, on each plaque the transformation in +\eqref{eq:plaque-transition} has bounded distortion, uniformly in +$i$, and its dependence on $q$ tends to zero over the patch. A +sample away from a small collar of the input plaque boundary thus +lies in an output plaque bin admitted over the whole transverse +range, provided the fixed bins are chosen sufficiently fine. The +collar has arbitrarily small sampling mass, again by the fixed +null-boundary choice. For a reparametrized chart these collars are +pulled back to its original fixed coordinate boundaries, which +have zero limiting mass. Continuous tests expressed in the fixed +coordinates pull back with uniform continuity moduli on the compact +chart ranges. +This gives arbitrarily small total trim loss. +No bound on the number of patches is involved. + +\smallskip\noindent +\emph{Comparing the conditional laws.} +The image of $\mu_i$ restricted to an input patch is exactly +$\mu_i$ restricted to its image, by $A$ invariance (or trivially for +the identity motion). Apply Lemma~\ref{lem:patch-comparison} to its +trim. The input patch law, converted by +\eqref{eq:plaque-transition}, agrees with $P_E$ restricted to the +admitted bins, up to arbitrarily small weighted error. + +To express this without dividing by a possibly small admitted +mass, take two continuous displacement tests $f,\phi$ supported +with margin inside the input plaque range. Their output bin +approximations are wholly inside the admitted bins. The +unnormalized comparison says that each input test integral, +multiplied by $P_E(B_p)$, agrees with the corresponding output +$P_E$ integral. Cross-multiplication therefore gives +\begin{equation}\label{eq:projective-cross-products} + P_{\mathrm{in}}(f)P_{\mathrm{out}}(\phi\circ c^{-1}) + -P_{\mathrm{in}}(\phi)P_{\mathrm{out}}(f\circ c^{-1}) + \longrightarrow0 +\end{equation} +in probability on the compared samples. All integrals here are in +displacement coordinates centered at the respective samples. +The small $q$ variation changes the tests by a uniformly vanishing +amount; sample-dependent centering is permitted by the uniform +test assertion in Lemma~\ref{lem:patch-comparison}. The input +depth-$F$ law can be replaced by its recorded probability by +Lemma~\ref{lem:plateau}; the output coarse law can be replaced by +the output recorded probability by the same lemma. This proves +\eqref{eq:projective-cross-products} for the data that actually +define the states. + +For completeness, the errors just used are localized as follows. +Fix a pair of charts and open conditions placing the samples in +inner boxes, requiring the full-plaque margin +\eqref{eq:full-plaque-margin}, and placing the displacement test +supports inside the input plaque range with positive margins. +The patch fits and trims hold for all sequence samples satisfying +these conditions, uniformly for +the chosen motion bound. Trim accuracy may be made arbitrarily +good independently of the fixed cover or these margin conditions. +Thus the cross-product discrepancy tends to zero in probability +on each such event. Exhaust by stricter margins and use continuous +tests on compact inner boxes. Every joint distributional limit +satisfies the corresponding exact identity there, since its tests +and coordinate changes are continuous. Countably many charts, +margin conditions, and tests suffice. + +\smallskip\noindent +\emph{Gluing along one plaque.} +Apply this result first to the identity motion. Compare any chart +at the sample with a chart extending far enough to contain its +bounded plaque domain. By Lemma~\ref{lem:sampling}, each probability +has positive mass on every neighborhood of the sampled point. +Choose a nonnegative continuous test supported in a common such +neighborhood and positive at that point. Its integral is positive +for both probabilities. The cross-product identities then give +projective agreement for every continuous test with compact support +in the first plaque domain. + +To compare two bounded domains, use a single extension containing +both, on the same plaque sheet. Exhaust $V$ by compact connected +coordinate ranges containing $1$ in their interiors. Successive +extensions agree after normalization near $1$, and so define a +locally finite measure on the union. Local finiteness follows +because each compact set lies in one extension carrying a finite +chart probability with a finite positive normalization factor. +The identity lies in its support by the sampling lemma. This +construction does not require chaining through intermediate support +points: each comparison uses a chart containing both the identity +neighborhood and the entire displacement range in question. +Using countably many tests and choosing normalizing bumps from a +countable collection makes the resulting projective measure +measurable. + +\smallskip\noindent +\emph{Equivariance, including the long motion.} +For a fixed diagonal motion, the same cross-product identities +give pushforward by $c(v)=a(-b)va(b)$, first on bounded test ranges +and then globally by the extensions just constructed. For the +varying motion in $\ker\beta$, $c$ is the identity, and all estimates +above are uniform for $\|b_i\|\le CR_i$: the only growing chart cost +is $\exp(O(R_i))$, dominated by +\eqref{eq:comparison-scales}. In a paired-state limit the second +base point need not be a fixed function of the first. This causes +no difficulty. The local identities hold on every paired open +margin event, and these events exhaust probability one because +both marginals admit the required extended charts. Their exact +identities therefore hold in every such limit. Since the argument +applies to an arbitrary sequence $b_i$ under the displayed bound, +it proves the last assertion with the stated quantifiers. +\end{proof} + +The measures $m^V_\omega$ retain information obtained by conditioning +the original sequence, while having the chart consistency and +diagonal comparison properties of ordinary plaque measures. The +next step is to relate the data for different pattern groups; +that requires a product argument, rather than a further change of +coordinates. diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/entropy.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/entropy.tex new file mode 100644 index 0000000000000000000000000000000000000000..c9eceb9fdd6d87f06f7fe62d3c81c1e93e9dd33a --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/entropy.tex @@ -0,0 +1,456 @@ +\section{Activity along long diagonal segments}\label{sec:segment-entropy} + +The root probabilities constructed above detect variation within a plaque at +the intermediate resolutions. We now use the arithmetic tube estimate to +show that this variation cannot disappear across a cut for a positive +proportion of starting points over a long diagonal segment. A positive +average under $\mu_i$ alone could coexist with zero activity on a set of +starting points of positive measure. If $\mu_i(E_i)\geq p>0$, the +normalized restrictions $\xi_i=\mu_i|_{E_i}/\mu_i(E_i)$ satisfy +$\xi_i\leq p^{-1}\mu_i$. Proving the estimate for every such dominated +law excludes these positive-mass exceptional sets. In +Section~\ref{sec:combination}, this stronger conclusion will give +almost-sure segment positivity for the limiting functions. + +An \emph{oriented cut} is a nonempty proper subset +$J\subset\{1,\ldots,6\}$, with the roots directed from $J$ to its complement: +\[ + \mathcal E_J=\{\beta_{jk}:j\in J,\ k\notin J\}. +\] +The complementary subset gives the opposite orientation. Fix a small +$\delta>0$. For each such $J$, choose a regular $b_J\in L$ such that +\begin{equation}\label{eq:cut-direction} + \beta(b_J)\geq1\quad(\beta\in\mathcal E_J),\qquad + |(b_J)_k-(b_J)_j|\leq\delta + \quad(j,k\text{ on the same side of }J). +\end{equation} +These choices can be made in a fixed bounded subset of $L$: put the two +sides at two distinct levels, perturb the entries within each side, and +subtract their common average. There are only finitely many cuts. +Write $\eta=1/3$ for the exponential constant in +Proposition~\ref{prop:bowen}; it is independent of $\delta$. +Write $g_{\beta,i}=g_{jk,i}$ when $\beta=\beta_{jk}$, and put +\[ + A_{J,i}(x)=\sum_{\beta\in\mathcal E_J}g_{\beta,i}(x). +\] +Choose the common interior scale $R_i$ as in +\eqref{eq:wall-invariance}, leaving separated interior depths both below and +above it. In particular, $T_{-,i}\prec R_i\prec T_{+,i}$. + +\begin{proposition}[Activity on every long segment]\label{prop:segment-activity} +For $\delta$ sufficiently small, the preceding choices have the following +property. For every oriented cut $J$, every $C<\infty$, every $\tau>0$, +and every sequence of probabilities $\xi_i$ on $X$ satisfying +$\xi_i\leq C\mu_i$, +\begin{equation}\label{eq:segment-activity} + \liminf_{i\to\infty}\frac1{R_i} + \int_0^{\tau R_i}\int_X + A_{J,i}\bigl(xa(tb_J)\bigr)\,\dd\xi_i(x)\,\dd t>0. +\end{equation} +The same assertion holds after passage to any subsequence. +\end{proposition} + +The proof compares two bounds for the entropy of orbit names. The tube +estimate gives a positive entropy cost per unit time. If the roots crossing +the cut are inactive, the plaque probabilities vary only in directions +whose expansion rates are at most $\delta$. Their names then have a much +smaller coding cost. This use of arithmetic separation to bound the entropy +of diagonal orbit names belongs to the method of +\cite{ELMV2009}; here the conditioning at intermediate resolutions and the +arbitrary dominated starting probabilities require the additional details +below. We use Shannon entropy with natural logarithms, including its chain +rule and the elementary finite-alphabet error bound; see +\cite[Chapter~2]{CoverThomas2006}. + +\subsection{Partitions and the cusp} + +For a countable partition $\mathcal P$ and a probability $\nu$, write +\[ + H_\nu(\mathcal P)=-\sum_{P\in\mathcal P}\nu(P)\log\nu(P), + \qquad 0\log0=0. +\] +Fix a regular $b\in L$ and put $T(x)=xa(b)$. The notation +$\mathcal P_{[r,s]}=\bigvee_{t=r}^sT^{-t}\mathcal P$ records the symbols at +the integer times from $r$ through $s$. Let $V$ be the pattern group of +the $b$-positive roots. + +\begin{lemma}[A partition for consistent orbit names]\label{lem:entropy-partition} +Given a sufficiently small identity neighborhood $O\subset G$, there is a +countable Borel partition $\mathcal P$ of $X$ with the following properties. +\begin{enumerate} + \item Every atom has $\mu$-null boundary and lies in an inner part of a + relatively compact $VA$ plaque chart from the fixed stock. The chart + extends beyond the atom with a margin. + \item If $x,y$ have the same symbols at times $0,\ldots,N$, there are + relative lifts $h_t\in O$, $0\leq t\leq N$, such that + \[ + ya(tb)=xa(tb)h_t, + \qquad h_{t+1}=a(-b)h_ta(b). + \] + \item For every fixed $C$, the entropies $H_\nu(\mathcal P)$ are bounded + uniformly for all sufficiently large $i$ and all probabilities + $\nu\leq C\mu_i$. Moreover, there are finite coarsenings + $\mathcal P^{(q)}$, obtained by keeping finitely many atoms and combining + the others into one atom, for which + \begin{equation}\label{eq:partition-entropy-tails} + \lim_{q\to\infty}\limsup_{i\to\infty} + \sup_{\nu\leq C\mu_i} + H_\nu(\mathcal P\mid\mathcal P^{(q)})=0. + \end{equation} +\end{enumerate} +\end{lemma} + +\begin{proof} +Let $m(x)$ denote the shortest-vector length. Divide $X$ into layers on +which $m(x)$ is comparable to $e^{-k}$, for integers $k\geq0$, using a +first layer that also contains the thick part. Layer boundaries may be +chosen $\mu$-null. Successive minima and lattice basis reduction give a +representative matrix $g$ for each point of the $k$th layer such that +\[ + \|g\|+\|g^{-1}\|\leq e^{C_1(1+k)}. +\] +Consequently the injectivity radius for relative group coordinates on that +layer is at least $e^{-C_2(1+k)}$. Indeed, if a nonidentity integral matrix +$\gamma$ gave a smaller return, conjugating the return by $g$ would give +$\|\gamma-1\|<1$, which is impossible. + +We impose a quantitative margin condition on these charts. Write $d_G$ +for the fixed left-invariant Riemannian distance on $G$. In a fixed +identity neighborhood of $G$, adapted product coordinates give a smooth +retraction $\pi$ onto $VA$, with $\pi(h)=h$ for $h\in VA$ and +\[ + d_G(h,\pi(h))\leq C_{\pi}\dist_G(h,VA), + \qquad d_G(1,\pi(h))\leq C_{\pi}d_G(1,h). +\] +After shrinking the neighborhood, $1$ and $\pi(h)$ are joined inside +$VA$ by a path of length at most $C_{\pi}d_G(1,h)$; one may use the +local exponential coordinates of $VA$. + +Choose outer $VA$ product charts in the $k$th layer of radius comparable +to $r_k=e^{-C_3(1+k)}$, with $C_3$ sufficiently large for injectivity, +and inner boxes whose relative-coordinate distance from the outer +boundary is at least a fixed multiple of $r_k$. Use translates of a +fixed smooth group-coordinate chart, so that its local product and +metric comparison constants are uniform. Cover the layer by still +smaller symbol boxes inside these inner boxes. Require their +relative-group diameter to be at most $c_*r_k$, where $c_*>0$ is small +enough compared with both the inner-to-outer margin and the radius of +the domain of $\pi$. In particular, whenever $x,y=xh$ are in one +symbol box, the whole path from $x$ to $x\pi(h)$ just described stays +inside its outer chart. It follows that $x\pi(h)$ is on the same +chart plaque as $x$, rather than on another local sheet of that orbit. + +The number of symbol boxes needed is at most $e^{C_4(1+k)}$. +Indeed, subdivide the bounded-matrix representatives at an exponentially +small mesh, and refine by the fixed factor $c_*$ when necessary. These +operations preserve an exponential bound in $k$. Passing from the cover +to a partition by successive differences does not increase the number +of pieces or their diameters. The charts and their cuts can be chosen +from the stock with $\mu$-null boundaries: perturb centers within a small +fixed fraction of the covering radius and choose each coordinate width +in a fixed comparable interval, avoiding the at most countably many +widths that give a boundary of positive measure. These choices preserve +the exponential bounds on sizes and on the number of boxes, as well as +the inner-chart margins. Thus each partition atom remains +inside a chart with a margin, although it need not itself be a coordinate +rectangle. + +The shortest-vector length changes by at most a fixed multiplicative factor +under $T$. We can therefore choose the boxes so small that both a relative +lift in one symbol and its conjugate by $a(b)$ lie in an injective +neighborhood at the next symbol. Uniqueness in that neighborhood identifies +the latter lift with the lift chosen there. Induction gives the consistency +in assertion~2. Decreasing all box sizes by a common fixed factor ensures +that every lift lies in the prescribed $O$. + +By Proposition~\ref{prop:cusp}, a probability $\nu\leq C\mu_i$ assigns the +$k$th layer mass at most $C_5e^{-c k}$, uniformly for sufficiently large +$i$. If that mass is $p_k$ and there are at most $M_k$ symbols in the +layer, their entropy is bounded by +\[ + p_k\log M_k-p_k\log p_k. +\] +Since $\log M_k=O(1+k)$ and $p_k\leq C_5e^{-ck}$, the sum of these bounds +converges uniformly, and its tail tends to zero uniformly. Keeping all +symbols from the first finitely many layers proves both entropy assertions. +\end{proof} + +\subsection{What an inactive cut implies inside a cell} + +Fix $J$, put $b=b_J$, and let $V_s$ be the pattern of positive roots whose +endpoints lie on the same side of $J$. Set $H_s=V_sA$. The cross-root +group, denoted $Z_J$, is normal in $VA$, and each element of $VA$ has a +unique factorization +\[ + h=zd,\qquad z\in Z_J,\quad d\in H_s. +\] +In a product chart $x=\psi(q)zd$, the $H_s$ plaques are precisely the sets +with $(q,z)$ fixed. Lemma~\ref{lem:va-support} says that at a limiting +state whose crossing-root indicators all vanish, the $VA$ probability +is supported on the $H_s$ plaque through the sampled point. + +We need this support assertion for conditional probabilities of a dominated +measure, rather than only for $\mu_i$. The following consequence records +the required change of measure. + +\begin{lemma}\label{lem:inactive-cell-concentration} +Fix finitely many atoms $S$ of $\mathcal P$, with their $VA$ charts, and an +interior transverse depth $l_i\prec R_i$. Suppose that probabilities +$\nu_i\leq C\mu_i$ satisfy $\int A_{J,i}\,\dd\nu_i\to0$. Condition on +one of these symbols $S$ and on its transverse $q$ cell. Draw two points +independently from the resulting conditional probability of $\nu_i$. +Then their $z$ coordinates differ by a quantity tending to zero in +probability, averaged with the probabilities of the conditioning events. + +The conclusion also holds on average over any finite list of such +probabilities, with arbitrary probability weights, if their weighted mean +of $\int A_{J,i}\,\dd\nu_i$ tends to zero. +\end{lemma} + +\begin{proof} +First sample $x$ under $\nu_i$, and draw the second plaque coordinate from +the $\mu_i$ probability conditioned only on its transverse cell in the +whole chart. The depth-comparison rules allow this conditional probability +to be replaced, in the limit, by the chart probability in the state data. +Any joint limit of the state with $x$ sampled under $\nu_i$ is dominated by +$C$ times the usual state law. Its crossing-root indicators vanish almost +surely, since their nonnegative sum has expectation tending to zero. +Lemma~\ref{lem:va-support} therefore puts the second point on the same +$H_s$ plaque as the sample. Its $z$ coordinate equals the sampled $z$. +The transverse mesh tends to zero; continuous coordinate tests on the +fixed compact chart boxes now show that the prelimit $z$ difference tends +to zero in probability. This argument uses the probability attached to +one chart and its statewise support. It does not condition the sampled +point on the other plaque probabilities in the state. + +To change the second draw to the stated $\nu_i$ conditional law, fix +$\varepsilon>0$ and discard events $E=S\cap\{q\text{ in a given cell}\}$ +such that +\[ + \nu_i(E)<\varepsilon\mu_i(\{q\text{ in that cell}\}\cap\text{chart}). +\] +Their total $\nu_i$ mass is at most $\varepsilon$ times the number of +symbols under consideration. On every remaining event the conditional +$\nu_i$ law has density at most $C/\varepsilon$ with respect to the +$\mu_i$ chart-cell law. The preceding convergence thus remains valid for +the second draw. Let $i\to\infty$ and then $\varepsilon\to0$. + +For the averaged statement, first observe a uniform sequential consequence +of the first assertion: no sequence of dominated probabilities can have +indicator expectation tending to zero while the asserted pair error stays +bounded away from zero. Let $a_i\to0$ be the weighted mean of the +indicator expectations in a list. Discard members whose expectation is +greater than $\sqrt{a_i+i^{-1}}$. Their total weight is at most +$a_i/\sqrt{a_i+i^{-1}}\to0$ by Markov's inequality. On the remaining +members, the pair errors tend uniformly to zero by the sequential +consequence: otherwise a sequence of members violating it could be +selected. This proves the averaged assertion. +\end{proof} + +\subsection{The lower and upper entropy bounds} + +\begin{proof}[Proof of Proposition~\ref{prop:segment-activity}] +Suppose, along a subsequence, that the expression in +\eqref{eq:segment-activity} tends to zero. Choose even integers +$N_i\sim\tau R_i$ with $N_i+2<\tau R_i$. Averaging a starting shift +$s\in[0,1]$ shows that we may replace $\xi_i$ by its translate through +some $s_i\in[0,1]$ so that +\begin{equation}\label{eq:vanishing-discrete-activity} + \frac1{N_i+1}\sum_{t=0}^{N_i} + \int A_{J,i}(T^t x)\,\dd\xi_i(x)\longrightarrow0. +\end{equation} +All these translated probabilities still satisfy $\xi_i\leq C\mu_i$. +More generally, every time-$t$ distribution +$\nu_{i,t}=(T^t)_*\xi_i$ has the same domination bound. + +Choose a compact set $K$ with $\nu_{i,N_i/2}(K)\geq1-\varepsilon_0$ for +all sufficiently large $i$, where $\varepsilon_0>0$ will be small. This +follows from the uniform cusp estimate and domination. Take the +neighborhood $O$ in Lemma~\ref{lem:entropy-partition} small enough to use +Proposition~\ref{prop:bowen} uniformly on $K$. +If a name atom $B\in\mathcal P_{[0,N_i]}$ meets $T^{-N_i/2}K$, choose a +reference point in that intersection. Consistency of the relative lifts +puts every midpoint of $B$ in the tube about its reference midpoint with +time $N_i/2$. Since $T_{-,i}\prec R_i\prec T_{+,i}$, this time is in the +arithmetic range for all sufficiently large $i$. Hence +\[ + \xi_i(B)\leq C e^{-\eta N_i/2}. +\] +Such atoms have total $\xi_i$ mass at least $1-\varepsilon_0$. +Integrating the negative logarithm of the atom mass gives, after decreasing +$\varepsilon_0$ and taking $i$ large, +\begin{equation}\label{eq:name-entropy-lower} + \frac1{N_i}H_{\xi_i}(\mathcal P_{[0,N_i]})\geq\frac\eta3. +\end{equation} + +We next derive the incompatible upper bound. Fix a small target error +$\varepsilon>0$. First choose a finite coarsening $\mathcal P^0$ so that +the conditional entropy in \eqref{eq:partition-entropy-tails} is as small +as required below. Let $q_0$ be its alphabet size. Choose also a finite +collection $\mathcal S$ of original symbols whose total complement has +arbitrarily small probability under every $\nu\leq C\mu_i$. The symbols +in $\mathcal S$ need not be the symbols retained in $\mathcal P^0$. + +The boundary of $\mathcal P^0$ is a finite union of compact $\mu$-null +sets. We can choose a fixed distance accuracy $\rho>0$ so that its +$2\rho$ neighborhood has arbitrarily small probability under every +$\nu\leq C\mu_i$ for all sufficiently large $i$. To justify this +uniformly, take decreasing closed neighborhoods of these boundaries; +their $\mu$ masses tend to zero, and the weak convergence upper bound +applies to each fixed neighborhood. All subsequent orbit predictions +will be within distance $\rho$ of the actual point. Incorrect symbols +can then occur only near these boundaries, or on the exceptions specified +below. + +Choose transverse grids in the charts for $\mathcal S$ at an interior +depth $l_i\prec R_i$. Let $M$ be a large integer, fixed while +$i\to\infty$. Its size will be chosen after the charts and $\rho$; +the concentration tolerances used below may then depend on $M$. +Split the name into blocks of $M$ future symbols given the +past through the block's initial time. We may omit an initial +$t_i=o(N_i)$ symbols and a final remainder of length less than $M$: +their entropy cost divided by $N_i$ tends to zero by the uniform +single-symbol entropy bound. + +Here is why the extra transverse-cell label costs only a bounded amount +at every remaining block. In a past-name atom at time $t$, the negative +root entries of its relative lift are $O(e^{-c_b t})$, where +$c_b=\min_{\beta(b)>0}\beta(b)>0$. Triangular coordinates in a small +group neighborhood therefore put the lift at distance +$O(e^{-c_b t})$ from $VA$. Let $x,y=xh$ be two current points in one +past-name atom, and use the retraction $\pi$ from the partition +construction. The quantitative diameter condition ensures that +$x\pi(h)$ lies on the same plaque of the current outer chart as $x$. +Thus $q(x\pi(h))=q(x)$. Lipschitz continuity of the transverse +coordinate on that chart gives +\[ + \begin{aligned} + |q(y)-q(x)| + &=|q(xh)-q(x\pi(h))|\\ + &\leq C\,d_G(h,\pi(h)) + \leq C'\dist_G(h,VA) + \leq C''e^{-c_b t}. + \end{aligned} +\] +The constants are uniform over the fixed finite collection of current +charts. Choose +$t_i$ to be a sufficiently large fixed multiple of $l_i$; then +$e^{-c_b t}<2^{-l_i}$ for $t\geq t_i$, and $t_i=o(N_i)$. +Consequently the possible transverse coordinates meet only a bounded +number $B_0$ of neighboring cells. Given the past, their label has +entropy at most $\log B_0$. For current symbols outside $\mathcal S$ +use a single dummy label. + +Let $\mathcal Q_i$ record this cell label in the current symbol, with +the dummy label outside $\mathcal S$, and write +$\mathcal F_t=\mathcal P_{[0,t]}$. After telling the cell label, we may +discard the rest of the past when upper-bounding the entropy of the next +coarsened block. Precisely, the chain rule and monotonicity under +conditioning give +\begin{equation}\label{eq:coarse-block-entropy} + \begin{aligned} + &H_{\xi_i}\bigl((\mathcal P^0)_{[t+1,t+M]}\mid\mathcal F_t\bigr)\\ + &\quad\leq + H_{\xi_i}\bigl(T^{-t}\mathcal Q_i\mid\mathcal F_t\bigr) + +H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]} + \mid\mathcal P\vee\mathcal Q_i\bigr)\\ + &\quad\leq\log B_0+ + H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]} + \mid\mathcal P\vee\mathcal Q_i\bigr). + \end{aligned} +\end{equation} +The conditioning in the last entropy is just the current symbol and +its cell. By +\eqref{eq:vanishing-discrete-activity}, the mean crossing-root activity at +the block starts tends to zero. Indeed their nonnegative sum is bounded +by the full sum in that equation, and normalizing by the number of blocks +loses at most the fixed factor $M$. The averaged form of +Lemma~\ref{lem:inactive-cell-concentration} applies to their distributions. +For each symbol-cell event we may therefore choose a $z$ center so that, +apart from a probability tending to zero on average, $z$ lies within any +prescribed fixed tolerance of that center. Such a center exists by +averaging the two-draw error and choosing one of the points as center. + +The $d$ coordinates lie in a fixed compact part of $H_s$ for each of the +finitely many charts. Every root occurring in $H_s$ has expansion rate +at most $\delta$, and $A$ commutes with the motion. A mesh in these +coordinates, of accuracy a fixed multiple of $\rho e^{-\delta M}$, +therefore predicts the trajectory through the next $M$ steps to the +required distance accuracy, provided the $z$ and transverse coordinates +are sufficiently close to their centers. The number of mesh points is +at most +\begin{equation}\label{eq:slow-direction-mesh} + C_0\exp(C_6\delta M). +\end{equation} +Here $C_6$ depends only on the dimension, while $C_0$ depends on the fixed +charts and $\rho$, not on $M$. This follows by using finitely many smooth +coordinate meshes on the compact $d$ ranges: conjugation expands every +entry of a group-relative error by at most $e^{\delta M}$. Left +invariance of the metric transfers the resulting relative-group error +to an error on $X$. + +The cross-root center tolerance may need to be exponentially small in +$M$, because cross-root directions expand faster. This causes no +additional mesh labels: $M$ is fixed before applying the concentration +lemma, so any such positive tolerance is allowed. The transverse width +also tends to zero for each fixed $M$. Thus $C_0$ in +\eqref{eq:slow-direction-mesh} remains independent of $M$. + +Encode the next $M$ symbols of $\mathcal P^0$ by a mesh label, with a +single additional label for an omitted chart or an outlier, and then by +corrections to the predicted symbols. Let $e_{i,t}$ be the average +probability of an incorrect prediction over these $M$ positions, with +the starting point sampled under $\nu_{i,t}$. Put +$h_2(e)=-e\log e-(1-e)\log(1-e)$, with the usual zero conventions. +The mesh and correction costs give +\begin{equation}\label{eq:mesh-correction-entropy} + \begin{aligned} + &H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]} + \mid\mathcal P\vee\mathcal Q_i\bigr)\\ + &\quad\leq\log\bigl(C_0e^{C_6\delta M}+1\bigr) + +M\bigl(h_2(e_{i,t})+e_{i,t}\log q_0\bigr). + \end{aligned} +\end{equation} +For completeness, reveal the error indicator at each position and, on an +error, reveal the actual symbol; subadditivity and concavity of $h_2$ +give this bound also when the error probabilities differ across positions +and blocks. The chart omissions and boundary neighborhoods were chosen +with arbitrarily small probabilities. The additional concentration +errors tend to zero on average. Consequently the mean of $e_{i,t}$ +over the block starts can be made as small as required; concavity of +$h_2$ gives the same small correction bound after averaging blocks. + +Finally, the original block is recovered from its coarsened block at +the following cost: +\begin{equation}\label{eq:restore-block-entropy} + \begin{aligned} + &H_{\xi_i}\bigl(\mathcal P_{[t+1,t+M]}\mid\mathcal F_t\bigr)\\ + &\quad\leq + H_{\xi_i}\bigl((\mathcal P^0)_{[t+1,t+M]}\mid\mathcal F_t\bigr) + +\sum_{r=1}^M H_{\nu_{i,t+r}}(\mathcal P\mid\mathcal P^0). + \end{aligned} +\end{equation} +Indeed, conditional on the entire coarsened block, revealing each +original symbol costs at most its one-step conditional entropy given +its own coarsened symbol. Every summand on the right is controlled by +\eqref{eq:partition-entropy-tails}, because +$\nu_{i,t+r}\leq C\mu_i$. + +Sum \eqref{eq:restore-block-entropy} over the blocks and use +\eqref{eq:coarse-block-entropy} and \eqref{eq:mesh-correction-entropy}. +The chain rule, together with the negligible initial and final pieces, +now gives +\begin{equation}\label{eq:name-entropy-upper} + \limsup_{i\to\infty}\frac1{N_i} + H_{\xi_i}(\mathcal P_{[0,N_i]}) + \leq C_6\delta+\varepsilon+ + \frac{\log B_0+\log(C_0+1)}{M}. +\end{equation} +All constants in the last numerator were fixed before $M$ was chosen. +Choose $\delta$ first so that $C_6\delta<\eta/12$, then choose the +truncation, charts and distance accuracy to make $\varepsilon<\eta/12$, +and finally choose $M$ so large that the last term is less than +$\eta/12$. Letting $i\to\infty$ contradicts +\eqref{eq:name-entropy-lower}. This proves the proposition, including its +subsequence formulation. +\end{proof} diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/homogeneous.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/homogeneous.tex new file mode 100644 index 0000000000000000000000000000000000000000..bea7214644a84363422f9add785d199b861d2585 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/homogeneous.tex @@ -0,0 +1,268 @@ +\section{Homogeneous limits and their fixed points}\label{sec:homogeneous} + +The estimates of Section~\ref{sec:arithmetic} give probability limits and +place a restriction on their local concentration. We first use this +restriction to obtain positive entropy in almost every ergodic component. +Measure classification then reduces the problem to excluding proper block +orbits. The useful feature of those orbits is concrete: every point on one +is fixed by a nonidentity diagonal matrix with algebraic entries. + +\subsection{Positive entropy in every positive-weight collection of components} + +For an invariant probability $\nu$ and an invertible measure-preserving map +$T$, write $h_\nu(T)$ for its measure-theoretic entropy. Fix +$b\in L$ with distinct coordinates and put +\[ + \lambda=\min_{j\ne k}|b_j-b_k|>0,\qquad T(x)=xa(b). +\] +The next statement is the ordinary-ball argument of +\cite[Proposition~6.1 and Lemma~6.2]{PrimeDegreePackets}, included here +with its proof. It explains why a ball estimate for a measure can control +its ergodic components without passing that estimate to each individual +component. + +\begin{proposition}\label{prop:positive-entropy} +Let $\mu$ be an $A$-invariant probability on $X$. Suppose that, for every +compact $\Omega\subset X$, there are $C_\Omega,r_\Omega>0$ such that +\begin{equation}\label{eq:ordinary-ball-hypothesis} + \mu(xB(r))\le C_\Omega r^n + \quad(x\in\Omega,\ 00$ and consider the two-sided +tube +\[ + \mathcal T_t=a(tb)B(\epsilon)a(-tb) + \cap a(-tb)B(\epsilon)a(tb). +\] +If $h\in\mathcal T_t$ and $r=e^{-\lambda t}$, then +\[ + |h_{jj}-1|<\epsilon,\qquad |h_{jk}|\le\epsilon r\quad(j\ne k). +\] +Every nondiagonal term in the determinant expansion contains at least two +off-diagonal entries. Hence $\prod_jh_{jj}=1+O(r^2)$. The matrix +\[ + d(h)=\diag\left(h_{11},\ldots,h_{n-1,n-1}, + \Bigl(\prod_{j0$, their normalized average +$\xi$ would satisfy $\xi\le w^{-1}\mu$ and $h_\xi(T)=0$. This +contradicts the first part of the proposition. +\end{proof} + +\subsection{The algebraic alternatives} + +We use the following measure-classification theorem in its full-degree +form. + +\begin{theorem}[Einsiedler--Katok--Lindenstrauss]\label{thm:ekl} +For $n\ge3$, an $A$-invariant, $A$-ergodic probability on +$\SL_n(\Z)\backslash\SL_n(\R)$ that has positive entropy for some +diagonal one-parameter subgroup is the invariant probability on a closed +finite-volume orbit $xJ$, where $J$ is a closed connected subgroup +containing $A$. +\end{theorem} + +This is \cite[Theorem~1.3]{EKL2006}, with the definition of an algebraic +measure given there. Their left-action convention is transferred to ours +by $\Gamma g\mapsto g^{-1}\Gamma$; this replaces $a(b)$ by $a(-b)$, +which has the same entropy. The Haar conclusion for prime $n$ in +\cite[Corollary~1.4]{EKL2006} does not identify the sextic alternatives. +We shall distinguish them using their fixed points. + +\begin{lemma}\label{lem:block-groups} +Let $J$ be a connected closed subgroup of $G$ containing $A$, and suppose +$xJ$ carries a finite nonzero $J$-invariant measure. There is a partition +$\{1,\ldots,n\}=B_1\sqcup\cdots\sqcup B_d$ such that $J$ is the +identity component of the determinant-one block-diagonal group for this +partition. +\end{lemma} + +\begin{proof} +The Lie algebra $\mathfrak j$ contains $L$ and is invariant under +$\Ad(A)$. Since the off-diagonal weight spaces are one-dimensional, +$\mathfrak j$ is the direct sum of $L$ and a collection of root lines +$\R E_{jk}$. Draw the edge $j\to k$ when this line occurs. The bracket +$[E_{ij},E_{jk}]=E_{ik}$ makes the graph transitively closed whenever +$i\ne k$. + +A Lie group with a lattice is unimodular. Applying the vanishing of its +modular character to $A$ shows that +\[ + \sum_{j\to k}(b_j-b_k)=0\qquad(b\in L). +\] +Thus each vertex has equal indegree and outdegree. Every strongly +connected class is complete by transitivity. The graph of these classes +is acyclic. If it had an edge, a source class of an edge-containing +component would have strictly more outgoing than incoming edges in the +original graph, contradicting the sum of the degree balances over that +class. Hence there are no edges between classes. +The resulting Lie algebra is precisely the block-diagonal one. A +connected Lie subgroup is determined by its Lie algebra, giving the +assertion. +\end{proof} + +The next lemma supplies a countable obstruction to each proper orbit. +It uses finite volume twice: first to identify a rational commutant and +then to exclude its idempotents. + +\begin{lemma}\label{lem:fixed-pieces} +Every proper orbit $xJ$ in Theorem~\ref{thm:ekl} is contained in +$\Fix(a)=\{y\in X:ya=y\}$ for some nonidentity $a\in A$ whose entries +are algebraic numbers. There are only countably many possible matrices +$a$. For each such $a$, its fixed set is covered by countably many +smooth pieces. Whenever $a_{jj}\ne a_{kk}$, the direction of +$U_{jk}=\{1+tE_{jk}:t\in\R\}$ is nowhere tangent to these pieces, +and each piece meets a sufficiently small $U_{jk}$ plaque in at most +one point. +\end{lemma} + +\begin{proof} +Write $x=\Gamma g$ and +$\Delta=g^{-1}\Gamma g\cap J$. This is a lattice in $J$. +For the rational linear algebra in this proof, use the auxiliary column +lattice $\Lambda=g^{-1}\Z^n$, which $\Delta$ preserves, and its rational +span $\Lambda_{\Q}$. This convention is separate from the row-lattice +realization of the point $x$. +We first show that the real matrix commutant of $\Delta$ equals the +commutant of $J$. +For any matrix $C$ commuting with $\Delta$, the map +\[ + \Delta\backslash J\longrightarrow\Mat_n(\R),\qquad + \Delta j\longmapsto j^{-1}Cj +\] +is well-defined. It sends normalized Haar measure to a probability +invariant under conjugation by $A$. Choose a regular element of $A$. +Every off-diagonal matrix coordinate has a nonzero weight under its +conjugation. Recurrence of an invariant probability forces all such +coordinates to vanish almost surely: a nonzero coordinate expands +geometrically in one of the two time directions and cannot recur to a +bounded set. The map is continuous and Haar measure has full support, +so $j^{-1}Cj$ is diagonal for every $j\in J$. + +Taking $j=1$ shows that $C$ is diagonal. Conjugating by a root subgroup +within one of the blocks of Lemma~\ref{lem:block-groups} forces the two +corresponding diagonal entries of $C$ to agree. Consequently the +commutant of $\Delta$ is exactly the algebra of block scalars, namely +the commutant of $J$. + +Define +\[ + F=\{C\in\End_{\Q}(\Lambda_{\Q}):C\delta=\delta C + \text{ for all }\delta\in\Delta\}. +\] +In the lattice coordinates supplied by $g$, these commutation relations +are rational linear equations, since $g\Delta g^{-1}\subset\Gamma$. +Only finitely many of these equations are needed in the finite-dimensional +matrix space. Thus $F$ has real extension +isomorphic to $\R^d$, where $d$ is the number of blocks. In particular +$F$ is a commutative, reduced, finite-dimensional $\Q$-algebra, a product +of totally real fields. + +Suppose $F$ had a nontrivial idempotent. Its image in real coordinates +would select a nonempty proper union of blocks and hence a subspace $V$ +defined over the rational lattice. The group $\Delta$ preserves the +full lattice in $V$ and its inverse does too. Thus +\[ + \chi(j)=|\det(j|_V)| +\] +is a character of $J$ trivial on $\Delta$. It is nontrivial on $A$, +because both the selected and the unselected union of blocks are +nonempty. It is therefore onto $\R_{>0}$. Pushing the finite invariant +measure on $\Delta\backslash J$ forward by $\chi$ would give a finite +nonzero multiplicatively invariant measure on $\R_{>0}$, an +impossibility. Hence $F$ is a field. + +The representation of $F$ on the rational lattice space has some +dimension $m$ over $F$. After extending scalars to $\R$, each embedding +of $F$ has multiplicity $m$, so all blocks have size $m$ and $dm=n$. +The elements of $F$ carrying $\Lambda$ into itself form an order. +Indeed, in lattice coordinates this multiplier set is represented by +$(gFg^{-1})\cap\Mat_n(\Z)$, hence is discrete. Clearing the denominators of a +rational basis gives a full-rank submodule. Thus the multiplier set is +a finitely generated group of rank $d$, and it is a subring containing $1$. +Dirichlet's unit theorem for this order gives unit rank $d-1$. + +If $J\ne G$, then $d>1$. Choose a unit of infinite order and take an even +power so that every real embedding is positive. Its action on the +lattice is an automorphism with determinant +$N_{F/\Q}(u)^m=1$. In real coordinates it is a matrix $a\in A$, +constant on each block and therefore central in $J$. It fixes $x$, +and centrality gives $(xj)a=(xa)j=xj$ for every $j\in J$. +Its entries are conjugates of an algebraic unit. The collection of +possible such diagonal matrices is countable. + +Finally, fixing $a$ and lifting to $G$, the fixed-point equation is +\[ + ga g^{-1}=\gamma\qquad\text{for some }\gamma\in\Gamma. +\] +For each $\gamma$, the nonempty solution set is a coset of the +centralizer of $a$. It is smooth, with tangent root directions exactly +those satisfying $a_{jj}=a_{kk}$. Restricting to countably many +injective local quotient charts gives the asserted pieces in $X$. +If $a_{jj}\ne a_{kk}$, two points of one local $U_{jk}$ plaque cannot +solve the same lifted equation: their relative element $1+tE_{jk}$ +would commute with $a$, forcing $t=0$. This proves the last assertion. +\end{proof} + +For $n=6$, the equality $dm=6$ leaves only block sizes $3+3$, +$2+2+2$, or $1+1+1+1+1+1$ for a proper finite-volume orbit. +The last case is $J=A$, whose translation action on a compact quotient +has zero entropy. Thus the positive-entropy alternatives that must be +excluded have two blocks of size three or three blocks of size two. + +By Proposition~\ref{prop:ball}, Proposition~\ref{prop:positive-entropy}, +and Theorem~\ref{thm:ekl}, almost every ergodic component of any packet +limit is one of these homogeneous probabilities. To complete the proof, +it will suffice to show that such a limit assigns no mass to the +countable fixed-point pieces in Lemma~\ref{lem:fixed-pieces}. We now +construct conditional measures that can detect those pieces while +retaining the arithmetic information at the growing times of +Proposition~\ref{prop:bowen}. diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/introduction.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..4ce645cfd3968647e45d4f92a6668974e24f7272 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/introduction.tex @@ -0,0 +1,237 @@ +\section{Introduction}\label{sec:introduction} + +The real embeddings of a totally real number field turn its ideals into +lattices. Multiplication by positive diagonal matrices changes the relative +sizes of the embedding coordinates while preserving covolume. The units +make each resulting diagonal orbit compact, and the ideal classes assemble +these orbits into a finite packet. The distribution of such packets connects +the arithmetic of number fields with the dynamics of higher-rank diagonal +groups. We prove equidistribution for totally real sextic fields having no +proper intermediate field. + +\subsection{Packets and the main theorem} + +Let +\[ + X_6=\SL_6(\Z)\backslash\SL_6(\R),\qquad + A_6=\{\diag(e^{u_1},\ldots,e^{u_6}):\textstyle\sum_j u_j=0\}. +\] +We regard $X_6$ as the space of covolume-one lattices in $\R^6$, with +matrices acting on row vectors on the right. Let $m_6$ denote its +$\SL_6(\R)$-invariant probability. We normalize Haar measure on $A_6$ +as $\dd u_1\cdots\dd u_5$, with $u_6=-\sum_{j=1}^5u_j$. + +A degree-six number field $K$ is \emph{primitive} if there is no field +$F$ with $\Q\subsetneq F\subsetneq K$. Suppose $K$ is totally real, +choose an ordering $\sigma=(\sigma_1,\ldots,\sigma_6)$ of its real +embeddings, and write $\mathcal O_K$ for its ring of integers. For a +nonzero fractional $\mathcal O_K$-ideal $I$, set +\[ + \sigma(I)=\{(\sigma_1(d),\ldots,\sigma_6(d)):d\in I\},\qquad + \Lambda_{I,\sigma}=\covol(\sigma(I))^{-1/6}\sigma(I). +\] +Choose one representative of each ordinary ideal class. Let +$\mathcal P_{K,\sigma}$ be the set of distinct compact orbits +\[ + \Lambda_{I,\sigma}wA_6, + \qquad w=\diag(\epsilon_1,\ldots,\epsilon_6),\quad + \epsilon_j\in\{1,-1\}. +\] +Here every sign matrix acts on the underlying lattice, including those +of determinant $-1$. Dirichlet's unit theorem gives compactness of the +orbits. Including all signs makes the packet independent of the chosen +ordinary ideal-class representatives. + +For $\mathcal O\in\mathcal P_{K,\sigma}$, let $m_{\mathcal O}$ be +its invariant probability and $\vol_A(\mathcal O)$ its volume for the +specified Haar measure. The packet probability is +\begin{equation}\label{eq:packet-definition} + \mu_{K,\sigma}= + \frac{\displaystyle\sum_{\mathcal O\in\mathcal P_{K,\sigma}} + \vol_A(\mathcal O)m_{\mathcal O}} + {\displaystyle\sum_{\mathcal O\in\mathcal P_{K,\sigma}} + \vol_A(\mathcal O)}. +\end{equation} + +\begin{theorem}\label{thm:main} +Let $K_i$ be totally real primitive fields of degree six, with +$|\Disc(K_i)|\to\infty$, and choose any ordering $\sigma_i$ of the +embeddings of each field. Then +\[ + \int_{X_6} f\,\dd\mu_{K_i,\sigma_i} + \longrightarrow\int_{X_6} f\,\dd m_6 + \qquad(f\in C_c(X_6)). +\] +There is no escape of mass: for every $\epsilon>0$ there is a compact +$C\subset X_6$ such that +$\mu_{K_i,\sigma_i}(C)\ge1-\epsilon$ for all sufficiently large $i$. +\end{theorem} + +The theorem concerns each complete packet attached to a field. It allows +the fields and their embedding orderings to vary freely under the stated +hypotheses. The arithmetic role of primitivity is to make every +nonrational integer of $K_i$ generate the full field; this will yield +uniform separation in the dual lattice used to count thin tubes. + +\subsection{History and the composite-degree obstruction} + +For real quadratic fields, the corresponding packets project to packets +of closed geodesics on the modular surface. Linnik's ergodic method, +including Skubenko's treatment of the one-sheeted hyperboloid, gave +equidistribution under an auxiliary splitting condition +\cite[Chapter~VI]{Linnik1968}; see also the historical statements in +\cite[Theorems~1.1--1.2]{ELMV2012}. Duke removed this condition for +fundamental discriminants by analytic methods +\cite[Theorem~1]{Duke1988}. Einsiedler, Lindenstrauss, Michel, and +Venkatesh later gave a dynamical proof formulated for all positive +nonsquare order discriminants~\cite[Theorem~2.3]{ELMV2012}. +In higher degree the compact diagonal orbits have dimension greater +than one, and the dynamics of the full diagonal group provides +additional rigidity. + +Einsiedler, Katok, and Lindenstrauss proved that, on +$\SL_d(\Z)\backslash\SL_d(\R)$ for $d\ge3$, an ergodic probability +invariant under the full positive diagonal group and having positive +entropy for a diagonal flow is algebraic~\cite[Theorem~1.3]{EKL2006}. +Einsiedler, Lindenstrauss, Michel, +and Venkatesh developed the connection between arithmetic separation of +periodic torus orbits and entropy of their limiting measures +\cite{ELMV2009}. Their subsequent cubic equidistribution theorem +combined measure classification, subconvexity, and local harmonic +analysis~\cite{ELMV2011}. These works distinguish two tasks that remain +central here: establishing nonescape and entropy, and identifying the +algebraic components of a limit. + +The second task is particularly important in composite degree +\cite[Section~1.6.3]{ELMV2011}. In prime +degree, the positive-entropy classification gives Haar measure directly +\cite[Corollary~1.4]{EKL2006}. In degree six there are proper block +subgroups containing the full diagonal group whose closed orbits can +carry finite invariant measures. Primitivity of the fields in the +sequence does not, by itself, rule out concentration near such orbits. +The proof must retain enough information from the sequence to exclude +these limiting components. + +Subsequent work has sharpened the entropy estimates in higher degree. +Khayutin obtained bounds for maximal-order packets whose Galois closures +act two-transitively on the embeddings~\cite[Theorem~1.1]{Khayutin2019}. +Wieser and Yang obtained entropy bounds for quartic packets of maximal +type with quadratic subfields, under a fixed split-place condition and +additional discriminant-growth hypotheses~\cite{WieserYang2026}. +These results clarify the arithmetic information available in composite +degree; identifying every component of a limit remains a separate step +in the argument here. + +The arithmetic preparation adapts the ordinary-vector method used for +prime-degree packets~\cite[Sections~2--6]{PrimeDegreePackets} and primitive +quartic packets~\cite[Section~3]{QuarticPackets}. We give the needed estimates in full. +They use Stark's effective zero-free information +\cite{Stark1974} and the short-interval bounds of Shiu and Pollack +\cite{Shiu1980,Pollack2020}. The quartic work treats its additional +obstruction using exterior-square arithmetic. Here an ordinary vector +and a dual vector produce a codifferent element, and primitivity gives +the separation needed to estimate its partial traces. + +\subsection{The intermediate-resolution argument} + +The main additional argument concerns conditional measures that retain +information at growing, but separated, resolutions. We describe its +purpose before introducing the construction. + +Let $\mu_i=\mu_{K_i,\sigma_i}$. Ordinary-vector estimates prove +tightness and give every probability limit $\mu$ an $O(r^6)$ bound +for the mass of radius-$r$ balls, uniformly when their centers range +over a compact set. Since $A_6$ has dimension five, this bound +forces positive entropy in almost every $A_6$-ergodic component. +Theorem~\ref{thm:ekl} then makes those components homogeneous. +Every proper component is supported on points fixed by a nonidentity +diagonal matrix with algebraic entries, so only a countable family of +fixed-point sets must be excluded. + +The arithmetic estimate used for this exclusion concerns two-sided +tubes, with $O$ a sufficiently small identity neighborhood, +\[ + x\bigl(a(tb)Oa(-tb)\cap a(-tb)Oa(tb)\bigr), + \qquad a(b)=\diag(e^{b_1},\ldots,e^{b_6}),\quad\sum_jb_j=0. +\] +If the coordinates of $b$ are separated across a prescribed cut of the +indices, ordinary and dual vectors bound the packet mass of such a +tube by $e^{-\eta t}$ for a fixed $\eta>0$. The estimate holds throughout a growing interval +$[T_{-,i},T_{+,i}]$ with $T_{+,i}/T_{-,i}\to\infty$. +The breadth of this interval is used throughout the proof. + +For $j\ne k$, let $U_{jk}=\{1+sE_{jk}:s\in\R\}$. +Exact conditional measures of a finite packet along these root groups +are supported at a single point. They therefore cannot express the +spread detected by the arithmetic tube bound. Instead, in a local box +$(q,h)$ for a unipotent group, we condition the plaque coordinate $h$ +on a small cell of the transverse coordinate $q$. We choose logarithmic +cell depths inside the growing interval so that the conditional +probabilities change negligibly across a much wider internal range. +In their joint limits with the sampled lattice, the plaque point still +has the recorded probability as its conditional law, given the transverse +coordinate and that one probability. This sampling identity will detect +atoms on sets meeting each root plaque in only countably many points. +The limiting probabilities also retain their behavior under suitable +long diagonal motions. + +Product structure for genuine leafwise measures and the use of +noncommuting root foliations are basic ingredients of the higher-rank +entropy method of Einsiedler and Katok +\cite[Proposition~8.3]{EK2003}~\cite[Theorems~8.4--8.5]{EK2005}; +see also~\cite[Corollary~8.8 and Theorem~9.8]{EL2010}. The point here is +to prove the corresponding structure for the conditional probabilities +formed before taking the limit. If a diagonal element fixes one root +group $U$ and contracts a complementary unipotent subgroup $W$ generated +by root groups and normal in $WU$, +the limiting measure on $WU$ is the product of the limiting measures +on $W$ and $U$. Contraction identifies the $U$ factor. A separate +conditional-information argument shows that adding a fine $U$-cell label +barely changes the law of any fixed further subdivision of a typical +$W$ tile. Expanding that tile turns this comparison into one for bounded +$W$ tests and identifies the $W$ factor. Both arguments are needed, and neither +follows by simply disintegrating the weak limit. + +Call $j\to k$ active when the resulting $U_{jk}$ measure has support +beyond the identity. The product rule makes activity transitive under +composition of roots. It also permits an entropy argument: if crossing +roots are inactive along a positive proportion of long trajectories, +then their names can be encoded using only the slowly expanding +directions inside the two sides of a cut. This contradicts the tube +bound. Long motions in the diagonal wall $b_j=b_k$ fixing $U_{jk}$ leave +its activity unchanged. Combining the resulting one-dimensional functions +of the differences $b_k-b_j$ upgrades activity along trajectories to full +activity at almost every limiting state. A proper homogeneous +fixed-point piece forces an inactive root and is therefore excluded. + +The construction separates the arithmetic estimate from this last +argument. Its potentially reusable ingredients are the comparison of +conditional probabilities across a broad range of transverse depths, +their product rule, and the passage from entropy bounds for long segments +starting in any fixed positive proportion of a packet to full root activity. + +Section~\ref{sec:arithmetic} proves the arithmetic bounds, and +Section~\ref{sec:homogeneous} identifies the homogeneous alternatives. +Sections~\ref{sec:conditioning} and~\ref{sec:product} construct and +factor the intermediate conditional measures. +Section~\ref{sec:activity} records their root-group consequences. +Sections~\ref{sec:segment-entropy} and~\ref{sec:combination} prove the +long-segment and full-activity assertions, respectively, and +Section~\ref{sec:assembly} concludes the proof. + +\subsection{Conventions} + +For the remainder of the paper write $n=6$, +$G=\SL_n(\R)$, $\Gamma=\SL_n(\Z)$, $X=\Gamma\backslash G$, +and $A=A_n$. Let $L=\{b\in\R^n:\sum_j b_j=0\}$, identified with the +traceless diagonal matrices, and +$a(b)=\diag(e^{b_1},\ldots,e^{b_n})$. We use the right action +throughout. Thus motion by $a(b)$ changes a relative displacement $h$ +to $a(-b)ha(b)$. +Matrix neighborhoods of the identity and smooth metrics are interchangeable +on fixed compact sets. When a metric on all of $X$ is needed, we use +the quotient of a left-invariant Riemannian metric on $G$. +The notation $B(r)$ denotes a sufficiently small identity neighborhood +of radius $r$. Implicit constants may depend on fixed compact sets, +charts, and $n$, but not on the varying field. All passages to +subsequences preserve the probability limit under consideration. diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/product.tex b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/product.tex new file mode 100644 index 0000000000000000000000000000000000000000..2825b9c4cfb01d709de0638d5a134534c74bf495 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/product.tex @@ -0,0 +1,842 @@ +\section{A product rule for the plaque probabilities}\label{sec:product} + +The projective measures constructed in the preceding section retain +information obtained by conditioning the measures \(\mu_i\) at depths +tending to infinity. We now prove a product rule for these measures. +The factors in the rule belong to the \emph{same augmented state}; this +point will allow us to compose root directions in the next section. + +Product decompositions for genuine leafwise conditional measures were +developed by Einsiedler and Katok +\citep[Proposition~8.3]{EK2003} +\citep[Theorems~8.4--8.5]{EK2005}; see also +Einsiedler and Lindenstrauss +\citep[Corollary~8.8 and Theorem~9.8]{EL2010}. +The use of a diagonal element that contracts one factor and fixes +another follows this established method. Here the measures are the +augmented-state data obtained by conditioning before weak convergence. +We prove the product identity for those data using the preceding +mesh comparisons; the cited leafwise theorems are not applied to them. + +We continue to use the notation \(p\prec q\) for \(p_i/q_i\to0\). +Every scale used below lies in the plateau +\([l_{-,i},l_{+,i}]\), with room for the displayed separations and for +additional separations at both ends. Expectations in a chart mean +integration against the restriction of \(\mu_i\) to that chart; thus +all error estimates are weighted by the mass of the chart. A chart of +zero limiting mass can be omitted. + +\begin{theorem}[Product rule]\label{thm:product} +Let \(V\) and \(W\) be pattern groups, let \(U=U_\alpha\) be a +one-dimensional root group, and suppose that multiplication gives a +coordinate decomposition +\[ + V=WU,\qquad W\mathrel{\triangleleft}V. +\] +Suppose also that some \(a_0\in A\) fixes \(U\) by conjugation and that +the automorphism +\[ + \Delta(w)=a_0^{-1}wa_0 +\] +strictly contracts \(W\). Then, for almost every augmented state +\(\omega\), +\begin{equation}\label{eq:product-rule} + m^V_\omega + \ \propto\ + ( (w,u)\longmapsto wu)_* + \bigl(m^W_\omega\otimes m^U_\omega\bigr). +\end{equation} +Here \(\propto\) denotes equality up to multiplication by a positive +constant. +\end{theorem} + +The case \(W=\{1\}\) follows from the plaque comparisons. Henceforth +\(W\ne\{1\}\). The root constraints defining a contracting logarithm +are rational linear equalities and strict inequalities. We can +therefore choose \(a_0\), and rescale its logarithm, so that in the +matrix-entry coordinates of \(W\), +\begin{equation}\label{eq:integer-contraction} + \Delta(w)_{jk}=2^{-d_{jk}}w_{jk}, + \qquad d_{jk}\in\N,\quad d_{jk}>0. +\end{equation} +Whenever \(jk\) and \(k\ell\) compose, their weights add: +\(d_{j\ell}=d_{jk}+d_{k\ell}\). +This is also the compatibility that makes \(\Delta\) an automorphism. + +We use the simultaneous charts in the stock of the preceding section: +\begin{equation}\label{eq:simultaneous-product-chart} + (q,w,u)\longmapsto\psi(q)wu +\end{equation} +on product domains. The transverse coordinates are \(q\) for \(V\), +\((q,w)\) for \(U\), and \((q,u)\) for \(W\). The last assertion uses +normality of \(W\); its plaque parametrization may be a smooth +reparametrization of the group coordinate. Compatible transverse +grids are used when the same \(q\) occurs in two conditionings. + +The proof has two parts. Contraction of \(W\) shows that the \(U\) +coordinate has the same law in the \(V\) and \(U\) data, independently +of the \(W\) coordinate. Identifying the remaining factor requires a +different argument. A depth-\(S\) \(U\)-cell label contains only +\(O(S)\) information. When \(S\prec T\), revealing this label can +appreciably change the law of a fixed-length further \(W\)-tile +refinement, conditional on its parent tile, at only a vanishing +proportion of \(T\) successive stages. Expanding a selected parent +tile then turns this refinement comparison into a comparison of +bounded \(W\) tests. + +\subsection{The factor fixed by the contraction} + +\begin{lemma}\label{lem:u-factor} +In each chart \eqref{eq:simultaneous-product-chart}, the limiting +\(V\)-plaque probability is a product in the coordinates \(w,u\). +Its \(u\) marginal agrees with the \(U\)-plaque probability at the +sampled state. Consequently there is a projective locally finite +measure \(\nu^W_\omega\) on \(W\) such that +\begin{equation}\label{eq:provisional-product} + m^V_\omega + \ \propto\ + ((w,u)\longmapsto wu)_* + (\nu^W_\omega\otimes m^U_\omega). +\end{equation} +\end{lemma} + +\begin{proof} +Choose interior scales \(l_-\prec T\prec F\prec l_+\), and apply +the diagonal motion \(a_0^{\lfloor T\rfloor}\). +In one fixed input chart, a patch specifies a \(q\)-cell at depth +\(F\), with the whole \(w\) and \(u\) ranges. A subpatch also +specifies a \(w\)-cell at depth \(F\). Denote these two labels by +\(Q_F\) and \(W_F\). + +Fix an arbitrarily small permitted coverage loss. Take finitely +many output \(U\)-charts, with inner boxes and sufficiently large +plaque extensions, covering all but that loss of output probability. +The law of the output point is still \(\mu_i\). An input patch whose +image meets one of these inner boxes lies in the corresponding +extended chart, for all sufficiently large \(i\): its \(U\) range +has bounded fixed size, its \(W\) displacements shrink by +\(O(e^{-cT})\), and its \(q\) variation contributes at most +\[ + O(2^{-F}e^{CT}) +\] +for fixed constants \(c,C>0\). Assign such patches to output charts +and omit the others. The omitted mass is bounded by the coverage +loss. + +In an assigned \(U\)-chart, the transverse image of a patch has +diameter +\[ + O(e^{-cT}+2^{-F}e^{CT}) + =o(2^{-l_-}). +\] +The transverse projection is independent of \(u\). Its restriction +to the input \((q,w)\) range is injective: the chart has a common +open \(U\) range for all these inputs, and two equal transverse +coordinates would otherwise give overlapping images of two +different points of the injective input chart. In the output +plaque coordinate, the input \(u\) interval is translated by a +parameter varying by \(o(1)\) across the patch. Thus patches and +their subpatches admit a common collection of inner \(U\) bins. + +Here are the trimming details needed for +Lemma~\ref{lem:patch-comparison}. A patch meeting a coarse output +wall has its sampled point in a collar of relative width \(o(1)\); +the grid convention makes the total mass of these patches \(o(1)\). +Fine output transverse cells can be pulled back at cost at most +\(e^{CT}\). Since \(l_+\succ F\), the pullback variation is +negligible compared with the input \(q\)- and \(w\)-meshes. +Away from their grid walls, the whole fine cell therefore stays +inside the prescribed input labels. Fixed small collars of the +input chart boundary have arbitrarily small mass. After removing +these collars, sufficiently fine fixed output \(U\) bins lie wholly +in the admitted plaque interval, except for arbitrarily small +sampling mass. One may first trim each subpatch into products of +fine transverse cells and these common plaque bins, and take their +union as a trim of the patch. Both levels of the comparison use the +same coarse output cell and the same plaque bins. + +The patch/subpatch conclusion of +Lemma~\ref{lem:patch-comparison} now gives, for every continuous +function \(f\) on the closure of the input \(u\) interval, +\begin{equation}\label{eq:u-independence-prelimit} + \int + \left| + \E_{\mu_i}\bigl(f(u)\mid Q_F,W_F\bigr) + -\E_{\mu_i}\bigl(f(u)\mid Q_F\bigr) + \right|\,\dd\mu_i + \longrightarrow0. +\end{equation} +To obtain the limit zero, first make the bin and chart-collar losses +arbitrarily small for the fixed finite cover, and then make the +coverage loss arbitrarily small. The common translation of the +\(u\) coordinate causes no extra error, because its variation tends +to zero and \(f\) is uniformly continuous. + +The plateau comparison replaces the displayed depths by the depths +defining the state data. It follows that the \(u\) marginal of the +\(V\) data is the \(U\) probability at the sample. For independence, +let \(g\) be a bounded continuous function of \(w\), with modulus +of continuity \(\omega_g\). Approximate it by a constant +function \(g_F\) on each \(W_F\)-cell. Conditional expectation, +followed by the triangle inequality, gives +\[ +\begin{split} + &\int\left| + \E_{\mu_i}(f(u)g(w)\mid Q_F) + -\E_{\mu_i}(f(u)\mid Q_F)\E_{\mu_i}(g(w)\mid Q_F) + \right|\,\dd\mu_i\\ + &\quad\le + \|g\|_\infty + \int\left| + \E_{\mu_i}(f(u)\mid Q_F,W_F) + -\E_{\mu_i}(f(u)\mid Q_F) + \right|\,\dd\mu_i + +2\|f\|_\infty\omega_g(C2^{-F}). +\end{split} +\] +The right side tends to zero by +\eqref{eq:u-independence-prelimit}. Passing to the state limit +therefore gives +\[ + P^V(f(u)g(w))=P^V(f(u))P^V(g(w)) +\] +where \(P^V\) denotes the limiting \(V\)-chart probability. +A countable dense family of continuous +tests proves that the limiting probability is a product. + +For a sampled plaque point with coordinates \(w_s,u_s\), the group +displacement from that point to \(wu\) has \(W,U\) coordinates +\begin{equation}\label{eq:product-displacement} + \left(\Ad_{u_s^{-1}}(w_s^{-1}w),\ u_s^{-1}u\right). +\end{equation} +This changes the two coordinates separately, so the product +assertion holds in displacement coordinates as well. +Proposition~\ref{prop:plaque-comparison} identifies overlapping +chart probabilities projectively. Apply this comparison on +product ranges containing the identity and exhausting \(W\times +U\); charts with the required plaque extensions are available. +The resulting \(W\) marginals agree projectively and define +\(\nu^W_\omega\). This proves \eqref{eq:provisional-product}. +\end{proof} + +It remains to prove that \(\nu^W_\omega=m^W_\omega\) projectively. +The contraction just used does not shrink the \(U\) direction, so +interchanging the names of the two factors would not prove this. +We instead compare small \(W\) pieces before a reverse diagonal +motion with bounded \(W\) ranges after that motion. + +\subsection{Nested tiles and their information cost} + +We first construct subdivisions of \(W\) compatible with \(\Delta\). +Let \(\Gamma_W\) be the lattice of pattern matrices with integral +entries. Haar measure on \(W\) is Lebesgue measure in its matrix +entries, with \(\Gamma_W\) of covolume one. + +\begin{lemma}[Tiles adapted to the contraction]\label{lem:tiles} +There is a compact set \(\mathcal F\subset W\), of Haar measure one +and with Haar-null boundary, such that the translates +\(\gamma\mathcal F\), \(\gamma\in\Gamma_W\), tile \(W\) with disjoint +interiors. Moreover, +\[ + \mathcal F + =\bigcup_{d\in\mathcal D}\Delta(d)\Delta(\mathcal F), + \qquad + \mathcal D= + \{\gamma\in\Gamma_W:0\le \gamma_{jk}<2^{d_{jk}} + \text{ for each pattern entry }jk\}, +\] +where \(d_{jk}\) is the contraction weight in +\eqref{eq:integer-contraction}. +The union is disjoint off boundaries. + +For every integer \(k\ge0\), the tiles +\[ + \gamma\Delta^k\mathcal F,\qquad + \gamma\in\Delta^k\Gamma_W, +\] +form a subdivision, and these subdivisions are nested as \(k\) +increases. They may all be shifted by the same left translation +\(g\). If \(g\) is distributed according to Haar probability on +\(W/\Gamma_W\), then, for every fixed \(w\in W\), its coordinate +in its shifted level-\(k\) tile has uniform Haar distribution on +\(\mathcal F\), after rescaling by \(\Delta^{-k}\). +In particular, the probability of lying in a fixed +\(\rho\)-collar of \(\partial\mathcal F\) in these coordinates +tends to zero as \(\rho\downarrow0\), uniformly in \(k\) and \(w\). +\end{lemma} + +\begin{proof} +For this proof write \(d_\alpha\) for the positive integral weight +of a pattern entry \(\alpha\), and put \(B_\alpha=2^{d_\alpha}\). +The digit set consists of integral pattern matrices \(d\) whose +\(\alpha\) entry belongs to \(\{0,\ldots,B_\alpha-1\}\). +Define +\begin{equation}\label{eq:tile-digits} + \mathcal F= + \left\{ + \prod_{\nu=1}^{\infty}\Delta^\nu(d_\nu): + d_\nu\in\mathcal D + \right\}, +\end{equation} +with factors ordered by increasing \(\nu\). +Contraction and the finite nilpotent multiplication formulas imply +uniform convergence of these products. More explicitly, in any +entry the product is a finite sum of convergent series of products +of digit entries, with positive geometrically decaying weights. +Thus \(\mathcal F\) is compact. + +Order the pattern entries by increasing root height after putting +the pattern in upper triangular form. In an entry \(\alpha\), the +coordinate in \eqref{eq:tile-digits} is +\[ + \sum_{\nu\ge1}B_\alpha^{-\nu}(d_\nu)_\alpha + +C_\alpha, +\] +where \(C_\alpha\) is determined entirely by digits in entries of +smaller height. At height one, this is the usual base-\(B_\alpha\) +expansion of a number in \([0,1]\). Inductively, once the +lower-height digits have been specified outside their null sets +of ambiguous expansions, the possible \(\alpha\) coordinates +form the interval \([C_\alpha,C_\alpha+1]\), with a unique digit +expansion except at its countably many base endpoints. + +Left multiplication by an integral pattern matrix has the same +triangular form: the \(\alpha\) coordinate changes by its integral +\(\alpha\) entry plus a function of already chosen lower-height +entries. Reduce entries in this order. At each step exactly one +integer places the remaining coordinate in an interval of length +one, except at an endpoint, and its base expansion is then unique +almost everywhere. Fubini's theorem at each of the finitely many +entries proves that almost every point of \(W\) belongs to exactly +one translate \(\gamma\mathcal F\). It also proves that +\(\mathcal F\) has Haar measure one. + +The translates are locally finite, because \(\mathcal F\) is +compact and \(\Gamma_W\) is discrete. Their union is consequently +closed, and the almost-everywhere covering just proved implies +that they cover all of \(W\). Splitting the first digit from +\eqref{eq:tile-digits} gives +\begin{equation}\label{eq:tile-subdivision} + \mathcal F + =\bigcup_{d\in\mathcal D} + \Delta(d)\Delta(\mathcal F). +\end{equation} +Triangular reduction also shows that \(\mathcal D\) represents the +right cosets of \(\Delta^{-1}\Gamma_W\) in \(\Gamma_W\): +\[ + \Gamma_W= + \bigsqcup_{d\in\mathcal D} + (\Delta^{-1}\Gamma_W)d. +\] +Applying \(\Delta\) shows that the origins in the subdivision of +all level-zero tiles are exactly +\[ + \Delta\Gamma_W + =\bigsqcup_{d\in\mathcal D}\Gamma_W\Delta(d). +\] +Iteration proves the asserted level-\(k\) subdivision and nesting. + +For completeness, these compact tiles have the regularity needed +for boundary trimming. The closed translates cover \(W\), so the +Baire category theorem gives nonempty interior to \(\mathcal F\). +Every point of \(\mathcal F\) lies in a descendant tile in the +iterated subdivision \eqref{eq:tile-subdivision}. Such descendants +have nonempty interior and diameters tending to zero. Therefore +\(\mathcal F\) is the closure of its interior. An interior +intersection of two translates would then give a positive-volume +overlap, contradicting almost-everywhere uniqueness. A boundary +point of one translate belongs to another: otherwise local +finiteness would give a neighborhood covered only by the first +translate, making that point interior. Thus the boundary is +contained in the union of the pairwise overlaps, which is null. +Boundary assignments can be chosen measurably and consistently +through the nested subdivisions. + +It remains to check the law of a shifted tile coordinate; this +check matters because the sampling measure on \(W\) need not have +a density. Since the weights are positive integers, +\(\Gamma_W\subset\Delta^k\Gamma_W\). +The projection +\[ + W/\Gamma_W\longrightarrow W/\Delta^k\Gamma_W +\] +sends Haar probability to Haar probability. Inversion and right +multiplication by \(w\) send the latter distribution to invariant +volume on the left quotient +\(\Delta^k\Gamma_W\backslash W\). +The representative of \(g^{-1}w\) in \(\Delta^k\mathcal F\) +therefore has normalized Haar distribution on that tile, and +rescaling gives Haar measure on \(\mathcal F\). +This is precisely the coordinate of \(w\) in its tile +\(g\gamma\Delta^k\mathcal F\). +The null boundary implies the collar assertion by continuity from +above. Fubini's theorem permits the same conclusion after +sampling \(w\) from any probability measure. +\end{proof} + +For finite labels we write \(H\) for Shannon entropy, with natural +logarithms, and +\[ + I(Y;Z\mid X)=H(Y\mid X)-H(Y\mid X,Z) +\] +for conditional mutual information. Tile labels restricted to a +fixed chart are finite at every fixed level. + +\begin{lemma}[Information in a transverse label]\label{lem:stage-information} +Fix a chart \eqref{eq:simultaneous-product-chart} and integers +\(F,S\ge1\). Let \(Q_F\) be its depth-\(F\) \(q\)-cell label, +and let \(U_S\) be its depth-\(S\) \(u\)-cell label. +Let \(K_k\) be the level-\(k\) tile label of \(w\), with any common +shift as in Lemma~\ref{lem:tiles}, restricted to the chart. +For every pair of integers \(T\ge j\ge1\), +\begin{equation}\label{eq:stage-information} + \sum_{k=0}^{T-j} + I_{\mu_i}(K_{k+j};U_S\mid Q_F,K_k) + \le Cj(1+S). +\end{equation} +Here the measure in the chart is normalized for defining entropy; +its mass is restored when errors are integrated over the chart. +The constant \(C\) depends only on the fixed coordinate domain, +and is independent of \(i,F,S,T,j\), the sampling measure, and +the tile shift. +Consequently, along any sequence with \(T\to\infty\) and +\(S/T\to0\), for every fixed \(j\), averaged over +\(T/2\le k\le3T/4\), revealing \(U_S\) changes the conditional +law of \(K_{k+j}\), given \((Q_F,K_k)\), by \(o(1)\) in expected +\(\ell^1\) distance. The same assertions hold after averaging +over shifts. +\end{lemma} + +\begin{proof} +The number of \(u\)-cells meeting its fixed bounded interval is +\(O(2^S)\), so \(H(U_S)\le C(1+S)\). +Because the tile labels are nested, the chain rule gives +\[ + \sum_{k=0}^{L-1} + I(K_{k+1};U_S\mid Q_F,K_k) + = I(K_L;U_S\mid Q_F,K_0) + \le H(U_S). +\] +For a jump of length \(j\), expand its mutual information into +these one-stage increments. Each increment occurs at most \(j\) +times in the sum over starting stages, proving +\eqref{eq:stage-information}. Pinsker's inequality and +Cauchy--Schwarz bound the average conditional \(\ell^1\) error. +For \(T\ge4j\), write +\(\mathcal K_T=\{k\in\Z:T/2\le k\le3T/4\}\). The bound is +\[ + \begin{split} + &\frac1{|\mathcal K_T|}\sum_{k\in\mathcal K_T} + \E\bigl\| + \mathcal L(K_{k+j}\mid Q_F,K_k,U_S) + -\mathcal L(K_{k+j}\mid Q_F,K_k) + \bigr\|_1\\ + &\hspace{25mm}\le C\sqrt{\frac{j(1+S)}{T}}. + \end{split} +\] +This tends to zero for fixed \(j\). The entropy bound is uniform +in the shift, so it can also be averaged over shifts. +\end{proof} + +\subsection{Identification of the contracted factor} + +We now compare \(\nu^W_\omega\) with \(m^W_\omega\). Choose +interior scales +\begin{equation}\label{eq:w-factor-scales} + l_-\prec S\prec T\prec F\prec l_+. +\end{equation} +Fix nonnegative functions \(f,\phi\in C_c(W)\), bounded by one, +with \(\phi(1)>0\). It suffices to show +\begin{equation}\label{eq:w-ratio-target} + \frac{\nu^W_\omega(f)}{\nu^W_\omega(\phi)} + = + \frac{m^W_\omega(f)}{m^W_\omega(\phi)} +\end{equation} +almost surely. Identity support makes the denominators positive. +A countable determining family of such tests will then give +projective equality. + +The comparison is made at the output of the motion +\begin{equation}\label{eq:reverse-motion} + a_0^{-(k+m_0)}, +\end{equation} +where \(T/2\le k\le3T/4\) and \(m_0\) is a fixed positive integer. +A level-\(k\) input tile has, relative to its translated origin, +the fixed output shape \(\Delta^{-m_0}\mathcal F\). +The role of \(m_0\) is to make this shape contain the supports of +the tests about most sampled points. The role of the much finer +level \(k+j\), with \(j\) fixed after \(m_0\), is to resolve those +tests at output. + +The two distributions to be compared can now be specified before +the geometric details. Inside one input chart, a parent patch has +label \(p=(Q_F,K_k)\), leaving the whole \(u\) interval available; +a subpatch has label \(p'=(Q_F,K_k,U_S)\). +Lemma~\ref{lem:stage-information} compares +\begin{equation}\label{eq:w-two-tile-laws} + \mathcal L_{\mu_i}(K_{k+j}\mid Q_F,K_k) + \quad\text{and}\quad + \mathcal L_{\mu_i}(K_{k+j}\mid Q_F,K_k,U_S). +\end{equation} +The measure here is restricted to the input chart. +After \eqref{eq:reverse-motion}, the parent \(W\) tile has fixed +shape \(\Delta^{-m_0}\mathcal F\), and its descendants have shape +\(\Delta^{j-m_0}\mathcal F\). Increasing the fixed \(j\) therefore +allows \eqref{eq:w-two-tile-laws} to compare bounded continuous +\(W\) tests on that output range. + +In a simultaneous output chart with coordinates \((q',w',u')\), +the transverse coordinate of the whole patch is \(q'\); +for its thin \(u\)-subpatch it is \((q',u')\). +The geometry and plateau estimate below will consequently compare +the whole-patch test law with a restriction of the \(V\) data, +and the subpatch test law with a restriction of the \(W\) data. +These comparisons have normalization factors and retain an inner +\(u'\)-range in the \(V\) data. We will first compare their cross +products, then use Lemma~\ref{lem:u-factor} to remove that \(u'\) +restriction, and finally control the denominators. This is how +the two conditional tile laws will identify the \(W\) factor +without presupposing the desired projective equality. + +Figure~\ref{fig:product-mechanism} contrasts these two comparisons: +contraction determines the \(U\) factor, whereas expansion must be +combined with the information estimate to identify the \(W\) factor. +\input{figures/product-mechanism} + +\paragraph{Coverage and the order of choices.} +Fix a permitted exceptional probability \(\epsilon>0\). +Take finitely many input charts +\eqref{eq:simultaneous-product-chart}, with inner boxes and fixed +margins, covering all but a sufficiently small multiple of +\(\epsilon\). Estimates can be added over these charts; their +finite multiplicity only changes constants. +Use the patches and subpatches just described, with whole +level-\(k\) \(w\)-tiles. Tiles meeting +the fixed inner boxes lie inside the outer input \(w\) ranges for +large \(i\), because their diameters tend to zero. + +First choose a tile-coordinate collar thin enough that its +expected mass is much smaller than the permitted coverage loss. +Lemma~\ref{lem:tiles} does this uniformly in \(k\) and in the input +distribution. Outside this collar the sample has a fixed +positive margin in \(\mathcal F\). Choose \(m_0\) so large that +every displacement in a fixed compact set containing +\(\supp f\cup\supp\phi\) and the identity lies, with margin, in +the output tile about such a sample. +Indeed, pull such a displacement back to the input tile +coordinate. It is contracted by \(\Delta^{m_0}\); the +conjugation coming from the input \(u\) coordinate is uniformly +bounded, and commutes with \(\Delta\), since \(a_0\) fixes \(U\). +Thus the pulled-back displacement fits inside the chosen margin +for sufficiently large fixed \(m_0\). +The input inner \(u\) intervals also give a fixed \(U\) +neighborhood about the sampled output point. + +Now take finitely many output charts of the same simultaneous +form, with inner boxes and \(V\)-extensions large enough for the +images of these patches. Such a finite cover loses an arbitrarily +small further probability. For fixed \(m_0\), all displacements +from a sample to points in its output patch lie in a fixed compact +range: use the tile origin as base in \(W\), a fixed base in \(U\), +and the fixed shape \(\Delta^{-m_0}\mathcal F\). +The \(q\) variation after \eqref{eq:reverse-motion} is +\(O(2^{-F}e^{CT})\). +Hence a patch meeting an output inner box fits in its extended +chart with margin. We can assign these patches to the finitely +many output charts, or perform the estimates for each applicable +input--output pair. We keep only samples with the indicated +inner margins. + +These choices fix the chart cover, the output tile extent, and +the test-range margins. They do \emph{not} yet fix the accuracy +of bin comparisons. Afterwards one may make a second, +arbitrarily thinner tile-boundary exclusion, refine the fixed +output bins, and increase the fixed integer \(j\). These later +choices improve errors without changing the earlier coverage +requirements. + +All these requirements can be met simultaneously with the +information estimate. Initially average both over shifts and +over \(k\in[T/2,3T/4]\). The first collar was chosen to have +expected loss small enough that its required coverage bound +holds with high probability. Any later, finer collar can have +arbitrarily small expected loss. For every fixed \(j\), +Lemma~\ref{lem:stage-information} gives an error tending to zero +in this joint average. Markov's inequality therefore selects +common shifts and \(k=k_i\) satisfying the coverage bounds and +the desired comparison-error bounds. The transverse grids in +\(q,u\) may be fixed using the grid convention of the preceding +section. Every scale still satisfies +\eqref{eq:w-factor-scales}. + +\paragraph{Geometry of the output comparisons.} +Write the coordinates of an assigned output chart as +\((q',w',u')\). For fixed input \(q\), its patch image is obtained +by left multiplication in \(V\) from +\[ + \Delta^{-m_0}\mathcal F + \ \times\ (\text{input }u\text{ interval}). +\] +Left multiplication by \(w_0u_0\) sends \(wu\) to +\begin{equation}\label{eq:separate-left-translation} + \bigl(w_0\Ad_{u_0}(w)\bigr)\,(u_0u). +\end{equation} +Thus the two plaque coordinates change separately. The +parameters of this bounded left multiplication vary by +\(O(2^{-F}e^{CT})\) across the input \(q\)-cell. +The \(V\)-transverse diameter in \(q'\) is bounded by the same +quantity. It is smaller than \(2^{-CS}\) for every fixed \(C\) +when \(i\) is large. +For a subpatch, the \(W\)-transverse coordinates are +\((q',u')\); their diameter is +\[ + O(2^{-S}+2^{-F}e^{CT}) + =o(2^{-l_-}). +\] +Discard patches or subpatch comparisons crossing a coarse +output transverse wall. Their sampling mass is \(o(1)\). + +We apply Lemma~\ref{lem:patch-comparison} to \(V\) data on the +patches and \(W\) data on the subpatches. The needed inner +products are as follows. For a patch take collections of +\(w'\) and \(u'\) bins in product form; for each of its subpatches +use the same inner collection of \(w'\) bins. These collections +are admitted over the entire corresponding transverse +projection, by \eqref{eq:separate-left-translation} and the +vanishing variation of its parameters. + +We verify that these inner products lose as little sampling +mass as desired. The extended charts describe one plaque sheet +throughout each of the ranges involved. The transverse maps +are injective on a patch or subpatch: their images have a common +open plaque range, and injectivity follows from injectivity of +the original chart and the diagonal motion. Vary a fine output +transverse coordinate across its cell and pull the path back. +Its input length is at most \(e^{CT}\) times the fine mesh side. +Because \(F,S\prec l_+\), this path stays in its input \(q\)- and +\(u\)-cells except at grid-wall collars of mass \(o(1)\). +Fixed input chart-boundary collars have arbitrarily small mass. +In tile coordinates the same pullback costs at most \(e^{C'T}\), +so it stays in the input tile away from any fixed thin collar +of that tile's boundary. The second collar chosen above makes +this loss arbitrarily small. + +Finally, away from these collars, the sample has a definite +margin in its admitted \(w'\) and \(u'\) ranges. For \(w'\) the +margin depends on \(m_0\), the fixed charts, and the tile +collar thickness; for \(u'\) it follows from the fixed interval +margin. Sufficiently fine fixed plaque bins are then wholly +admitted. This proves the required plaque-bin trim estimate. +The collections can retain the supports of \(f,\phi\) about +the covered samples and a fixed \(U\) neighborhood about their +\(u'\) coordinates. Thus the comparison errors can be made +arbitrarily small after the original coverage and range choices. + +\paragraph{Comparison of the \(W\) tests.} +At a sampled output point, let \(P_V\) and \(P_W\) denote its +finite-\(i\) chart probabilities for \(V\) and \(W\). +Interpret \(f,\phi\) in the output \(w'\) coordinate using the +displacement formula \eqref{eq:product-displacement}. Their +parameters may depend on the sample, but their norms and +continuity moduli are uniform on the finitely many chart ranges. +Let \(B_u\) denote the union of the inner \(u'\) bins for its +patch. We claim that, on the covered samples, +\begin{equation}\label{eq:w-cross-product} + P_V(f\,1_{B_u})P_W(\phi) + -P_V(\phi\,1_{B_u})P_W(f) +\end{equation} +has arbitrarily small expectation of its absolute value in the +limit \(i\to\infty\). + +Here is the mass calculation. Let \(A_p\) be the law of the +output plaque coordinates when sampling the input patch \(p\), +and let \(A_{p'}\) be the corresponding law for its subpatch. +The unnormalized form of Lemma~\ref{lem:patch-comparison}, +first for \(V\) and then for \(W\), gives +\[ + \begin{aligned} + c_p A_p(f)&\simeq P_V(f\,1_{B_u}),& + c_p A_p(\phi)&\simeq P_V(\phi\,1_{B_u}),\\ + c_{p'} A_{p'}(f)&\simeq P_W(f),& + c_{p'} A_{p'}(\phi)&\simeq P_W(\phi). + \end{aligned} +\] +Let \(e_f(x),e_\phi(x),e'_f(x),e'_\phi(x)\) denote the absolute +errors in these four comparisons at the sampled point \(x\). +The factors \(c_p,c_{p'}\) are the probabilities of the admitted +plaque-bin ranges in the respective coarse output laws, hence +belong to \([0,1]\). Each \(\simeq\) means an absolute error whose +sum weighted by the relevant patch masses is arbitrarily small. +The tests are supported in the common inner \(w'\) range, so +there is no additional \(w'\)-restriction in the right-hand +integrals. Replacing coarse laws by the data laws uses the +plateau comparison. These estimates remain true for the +sample-dependent test parameters by the uniform-test clause +of Lemma~\ref{lem:patch-comparison}. + +At input, Lemma~\ref{lem:stage-information} compares the law of +the level-\((k+j)\) tile in \(p\) with its law in \(p'\). +At output these finer tiles have relative shapes +\(\Delta^{j-m_0}\mathcal F\). Once \(m_0\) and the finite chart +ranges are fixed, taking the fixed integer \(j\) large makes +their diameters as small as prescribed. The conversion to +\(w'\) coordinates has uniformly bounded distortion and is +independent of \(u\), by +\eqref{eq:separate-left-translation}. +Let \(\eta_i(x)\) be the \(\ell^1\) distance between the two +conditional level-\((k+j)\) tile laws for the patch and the +subpatch containing \(x\). There is a uniform bound +\[ + \sigma_{i,j} + \le C\,\diam(\Delta^{j-m_0}\mathcal F)+O(2^{-F}e^{CT}) +\] +for the variation of \(w'\) within one such tile, including the +variation across its \(q\)-cell. If \(h\) is either \(f\) or +\(\phi\), and \(\omega_h\) is a common continuity modulus for its +chart-coordinate versions, then +\begin{equation}\label{eq:tile-test-comparison} + |A_p(h)-A_{p'}(h)| + \le \eta_i(x)+2\omega_h(\sigma_{i,j}). +\end{equation} +Indeed, replace \(h\) by one value on each finer tile. The two +replacements cost at most \(\omega_h(\sigma_{i,j})\) each, and +the remaining finite-label difference is bounded by \(\eta_i(x)\). +This also proves the assertion for tests based at the sampled +point, since the bound is uniform over the test coefficients. +The expected value of \(\eta_i\) tends to zero for the selected +shifts and stages, by Lemma~\ref{lem:stage-information}. + +Multiply the four mass comparisons and cancel by cross +multiplication, without dividing by \(c_p\) or \(c_{p'}\). +The result is \eqref{eq:w-cross-product}: the remaining expression +is bounded by +\[ + \begin{split} + &e_f(x)+e_\phi(x)+e'_f(x)+e'_\phi(x)\\ + &\hspace{15mm} + +2\eta_i(x) + +2\omega_f(\sigma_{i,j})+2\omega_\phi(\sigma_{i,j}). + \end{split} +\] +For this inequality, the term left after the four mass +comparisons is +\(c_pc_{p'}(A_p(f)A_{p'}(\phi)-A_p(\phi)A_{p'}(f))\); +apply \eqref{eq:tile-test-comparison} and use that all factors +are bounded by one. +Integrating over the covered samples bounds +the first four terms by the total trim and bin-comparison +errors, the fifth by the stage-information error, and the last +two by their uniform moduli. Choose the trim accuracy and fixed +bins first, then choose the fixed \(j\) sufficiently large, and +finally let \(i\) tend to infinity. +This proves the claimed integrated bound for +\eqref{eq:w-cross-product}. +The number of patches may grow with \(i\); the absolute mass estimate +\eqref{eq:mass-error} charges each fine-cell/bin pair at most +once within a fixed chart comparison. + +\paragraph{Removing the \(U\) restriction.} +By Lemma~\ref{lem:u-factor}, in the limiting \(V\) data the +\(w'\) and \(u'\) coordinates are independent. Consequently, +for the finite-\(i\) probabilities, +\[ + P_V(f1_{B_u})-P_V(f)P_V(B_u)\longrightarrow0 +\] +in probability on the charts under consideration, and the same +holds with \(\phi\). To justify the varying \(B_u\), its bins +come from one fixed finite subdivision, and every bin boundary +has zero mass almost surely in the limiting data. The asserted +convergence therefore holds simultaneously for all subsets of +that finite bin collection. The displacement tests depend +continuously on the sample on the fixed inner chart ranges. +Output sampling has law \(\mu_i\), so the limiting independence +applies there even though the patches were chosen at input. +If several input charts are used, their finite multiplicity +only multiplies the error bound. + +Thus \eqref{eq:w-cross-product} becomes +\begin{equation}\label{eq:w-cross-product-reduced} + P_V(B_u) + \bigl(P_V(f)P_W(\phi)-P_V(\phi)P_W(f)\bigr) + \simeq0. +\end{equation} + +\paragraph{Denominators and passage to the projective limit.} +We now explain why \eqref{eq:w-cross-product-reduced} gives +\eqref{eq:w-ratio-target}. On each of the fixed covered chart +ranges, \(B_u\) contains a fixed \(U\) neighborhood about the +sample. Choose a continuous bump supported in that neighborhood +and positive at the sampled \(u'\) coordinate. Its integral in +the limiting \(V\) probability is positive by identity support. +The same support property gives positivity of the limiting +integrals \(P_V(\phi)\) and \(P_W(\phi)\). +For any further prescribed exceptional probability there is, +therefore, a constant \(\kappa>0\) such that, for all large \(i\), +\begin{equation}\label{eq:w-denominators} + P_V(B_u)\ge\kappa,\qquad + P_V(\phi)\ge\kappa,\qquad + P_W(\phi)\ge\kappa +\end{equation} +outside that exception on the covered samples. +This follows from convergence of the joint data in the finitely +many charts, using the fixed continuous bumps. + +The order of choices is essential. Fix the coverage, \(m_0\), +the output charts, and their test-range margins first. Then +choose \(\kappa\) from these fixed bumps. Only afterwards refine +the bin approximations and tile collars used for comparison, +and enlarge \(j\), to make the error in +\eqref{eq:w-cross-product-reduced} as small as required relative +to \(\kappa^3\). The coverage collar and the fixed neighborhoods +giving the lower bounds need not change. Division in +\eqref{eq:w-cross-product-reduced} is now legitimate and yields +arbitrarily small difference between +\[ + \frac{P_V(f)}{P_V(\phi)} + \quad\hbox{and}\quad + \frac{P_W(f)}{P_W(\phi)} +\] +outside an arbitrarily small exceptional probability. +Explicitly, let \(\mathcal G_i\) be the covered set on which +\eqref{eq:w-denominators} holds, and denote the expression on the +left of \eqref{eq:w-cross-product-reduced} by \(D_i(x)\). +For every \(\delta>0\), +\[ + \mu_i\left(\mathcal G_i\cap + \left\{\left|\frac{P_V(f)}{P_V(\phi)} + -\frac{P_W(f)}{P_W(\phi)}\right|>\delta\right\}\right) + \le \frac{1}{\delta\kappa^3} + \int_{\mathcal G_i}|D_i(x)|\,\dd\mu_i(x). +\] +The right side can be made arbitrarily small with the previously +fixed coverage and denominators. + +To see that this proves the assertion for the limiting +projective measures, suppose otherwise. A countable choice of +tests and charts would then give a reference output chart and +a positive-probability set on which the two ratios differ by +more than some fixed positive number. Restrict this set so +that the denominators have a positive lower bound and the +supports of the tests, together with a \(U\) neighborhood, have +inner margins in the chart. Continuous displacement integrals +and the open-set inequality preserve a positive probability +of discrepancy in the sequence data. + +Apply the preceding coverage construction with exceptional +probability much smaller than this discrepancy probability. +On samples covered by both the reference chart and an assigned +output chart, Proposition~\ref{prop:plaque-comparison} and +Lemma~\ref{lem:u-factor} identify their limiting ratios. +This also identifies their finite-\(i\) ratios up to an error +tending to zero outside a further small set: first impose +closed inner margins and lower bounds on the denominators, +then use joint convergence of the finitely many chart data. +The arbitrarily accurate comparison just proved in the +assigned charts contradicts persistence of the discrepancy +in the reference chart. + +We have proved \eqref{eq:w-ratio-target}. Taking countably many +tests and then exhausting compact \(W\) ranges gives +\(\nu^W_\omega\propto m^W_\omega\) almost surely. +Together with \eqref{eq:provisional-product}, this completes +the proof of Theorem~\ref{thm:product}. diff --git a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b13faaca758d31dbede73484692528ffaabd0c92 --- /dev/null +++ b/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3d96820596471fc3099f9d73a92d1bce10b991416925374a9d57b36597e41828 +size 712636 diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/README.md b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..ce67d7ba70c0243948ea293dc96be22f837fc7db --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case](paper.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026, + author = {{OpenAI}}, + title = {{Equivariant Jiang--Su Stability for Amenable Actions in the Unital Stably Finite Case}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf}{OAI:Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/paper.tex b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..d485d64f549ce5cfa66b71f3a6859f4b05b91104 --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/paper.tex @@ -0,0 +1,53 @@ +\documentclass[11pt,a4paper]{article} +\usepackage[margin=28mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{xcolor} +\usepackage[colorlinks=true,linkcolor=blue!55!black,citecolor=blue!55!black,urlcolor=blue!55!black]{hyperref} +\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{\Zalg}{\mathcal Z} +\newcommand{\N}{\mathbb N} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\T}{\operatorname{T}} +\newcommand{\Tr}{\operatorname{Tr}} +\newcommand{\HS}{\mathrm{HS}} +\newcommand{\Aut}{\operatorname{Aut}} +\newcommand{\Ad}{\operatorname{Ad}} +\newcommand{\U}{\mathcal U} +\newcommand{\eps}{\varepsilon} +\newcommand{\iu}{\mathrm{i}} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\dist}{dist} +\title{Equivariant Jiang--Su Stability\\for Amenable Actions\\in the Unital Stably Finite Case} +\author{OpenAI} +\date{October 5, 2026} +\hypersetup{pdfauthor={OpenAI},pdftitle={Equivariant Jiang-Su Stability for Amenable Actions in the Unital Stably Finite Case}} +\begin{document} +\maketitle +\begin{abstract} +Every action of a countable discrete amenable group on a simple, +separable, unital, infinite-dimensional, nuclear, stably finite complex +\(\mathrm C^*\)-algebra that is already Jiang--Su stable absorbs the +trivial action on the Jiang--Su algebra up to cocycle conjugacy. +No restriction is imposed on the action on the tracial-state simplex. +This proves the unital, stably finite case of Szab\'o's Conjecture~A +on automatic equivariant Jiang--Su stability. +\end{abstract} +\input{sections/introduction} +\input{sections/preliminaries} +\input{sections/rows} +\input{sections/gaussian} +\input{sections/splittings} +\input{references} +\end{document} diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/references.tex b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..4bd56b8eba597ea09e9ae7a216d21917ddd32412 --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/references.tex @@ -0,0 +1,36 @@ +\begin{samepage} +\begin{thebibliography}{99} +\raggedright + +\bibitem{Haagerup2014} +U.~Haagerup, +\emph{Quasitraces on exact \(C^*\)-algebras are traces}, +C.~R. Math. Rep. Acad. Sci. Canada \textbf{36} (2014), no.~2--3, +67--92. +\href{https://arxiv.org/abs/1403.7653}{arXiv:1403.7653}. + +\bibitem{SzaboSI2021} +G.~Szab\'o, +\emph{Equivariant property (SI) revisited}, +Anal. PDE \textbf{14} (2021), no.~4, 1199--1232. +\href{https://doi.org/10.2140/apde.2021.14.1199} +{doi:10.2140/apde.2021.14.1199}. +\href{https://arxiv.org/abs/1904.10897v4}{arXiv:1904.10897v4}. + +\bibitem{SzaboWouters2025} +G.~Szab\'o and L.~Wouters, +\emph{Equivariant property Gamma and the tracial local-to-global principle +for \(C^*\)-dynamics}, +Anal. PDE \textbf{18} (2025), no.~6, 1385--1432. +\href{https://doi.org/10.2140/apde.2025.18.1385} +{doi:10.2140/apde.2025.18.1385}. +\href{https://arxiv.org/abs/2301.12846v3}{arXiv:2301.12846v3}. + +\bibitem{TomsWinter2007} +A.~S. Toms and W.~Winter, +\emph{Strongly self-absorbing \(C^*\)-algebras}, +Trans. Amer. Math. Soc. \textbf{359} (2007), no.~8, 3999--4029. +\href{https://arxiv.org/abs/math/0502211v2}{arXiv:math/0502211v2}. + +\end{thebibliography} +\end{samepage} diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/gaussian.tex b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/gaussian.tex new file mode 100644 index 0000000000000000000000000000000000000000..b15929e67fc7de54e8d9a19314099d6f9cd18099 --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/gaussian.tex @@ -0,0 +1,457 @@ +\section{Gaussian quadratics over F{\o}lner sets} +\label{sec:gaussian} + +We retain the algebra \(C\), its invariant commuting orbit algebras +\(C_i\), and the variables \(x_{i,g}\), \(z_{N,g}\), and \(k_{N;g,h}\) +from Section~\ref{sec:rows}. In particular, \(\T(C)\) is nonempty, +the \(x_{i,g}\) are uniformly bounded by \(M\), and +Proposition~\ref{prop:joint-clt} applies to arbitrary sequences of +extreme traces. All traces of finite-dimensional matrices in this +section are unnormalized. For a matrix \(B\), we write +\(\|B\|_{\HS}=(\Tr(B^*B))^{1/2}\) and +\(\|B\|_1=\Tr|B|\). + +\begin{proposition}[Uniform approximately invariant unitaries]\label{prop:uniform-unitaries} +For every finite set \(H\subset G\), every \(\eps>0\), and every +integer \(m\geq1\), there are \(t>0\) and a self-adjoint element +\(Q\in C\) such that \(V=\exp(\iu tQ)\) satisfies +\begin{align} + \sup_{\varphi\in\T(C)} + \|\bar\alpha_h(V)-V\|_{2,\varphi} + &<\eps &&(h\in H), \label{eq:uniform-unitary-invariance}\\ + \sup_{\varphi\in\T(C)}|\varphi(V^\ell)| + &<\eps &&(1\leq\ell\leq m). + \label{eq:uniform-unitary-moments} +\end{align} +\end{proposition} + +We first record the matrix and probability estimates used in the +proof. + +\begin{lemma}[Square-root perturbation bound]\label{lem:square-root-trace} +If \(S,T\) are positive matrices of the same size, then +\[ + \|S^{1/2}-T^{1/2}\|_{\HS}^{\,2}\leq\|S-T\|_1. +\] +\end{lemma} + +\begin{proof} +Set \(a=S^{1/2}\), \(b=T^{1/2}\), \(d=a-b\), and +\(u=\operatorname{sign}(d)\), with \(\operatorname{sign}(0)=0\). +Since \(a^2-b^2=ad+db\), cyclicity of the trace gives +\[ + \Tr\bigl(u(a^2-b^2)\bigr) + =\Tr\bigl(|d|(a+b)\bigr). +\] +In a basis diagonalizing \(d\), positivity gives +\(a_{jj}+b_{jj}\geq|a_{jj}-b_{jj}|=|d_j|\). Thus +\[ + \Tr\bigl(|d|(a+b)\bigr) + =\sum_j |d_j|(a_{jj}+b_{jj}) + \geq\sum_j d_j^2=\Tr(d^2). +\] +On the other hand, \(u\) is a contraction, so trace-norm duality +gives +\[ + \Tr\bigl(u(a^2-b^2)\bigr)\leq\Tr|a^2-b^2|. +\] +These two inequalities prove the assertion. +\end{proof} + +\begin{lemma}[Gaussian quadratic estimates]\label{lem:gaussian-quadratic} +Let \(W\) be a standard Gaussian vector in a finite-dimensional +real Euclidean space, and let \(B\) be a real symmetric operator +on that space. Put +\[ + q_B=\langle W,BW\rangle-\Tr(B). +\] +Then +\begin{equation}\label{eq:quadratic-variance-characteristic} + \|q_B\|_{L^2}=\sqrt{2}\,\|B\|_{\HS}, + \qquad + \bigl|\mathbb E\exp(\iu s q_B)\bigr| + \leq\bigl(1+4s^2\|B\|_{\HS}^2\bigr)^{-1/4} + \quad(s\in\R). +\end{equation} +Its law has an exponential moment of its absolute value and is +determined by its moments. +\end{lemma} + +\begin{proof} +Orthogonal diagonalization of \(B\) gives +\[ + q_B\ \overset{\mathrm{law}}{=} + \sum_{j=1}^d\lambda_j(U_j^2-1), +\] +where \(\lambda_j\) are its eigenvalues and \(U_j\) are independent +standard real normal variables. The summands are centered and +have variances \(2\lambda_j^2\), proving the first formula. +The one-dimensional Gaussian integral gives +\[ + \bigl|\mathbb E\exp(\iu s q_B)\bigr| + =\prod_{j=1}^d(1+4s^2\lambda_j^2)^{-1/4}. +\] +For nonnegative numbers \(a_j\), one has +\(\prod_j(1+a_j)\geq1+\sum_j a_j\). Applying this inequality with +\(a_j=4s^2\lambda_j^2\) proves the second formula in +Equation~\eqref{eq:quadratic-variance-characteristic}. + +Choose \(c>0\) so that \(2c\max_j|\lambda_j|<1\); if \(B=0\), +any \(c>0\) will do. The pointwise estimate +\[ + \exp\!\left(c\left|\sum_j\lambda_j(U_j^2-1)\right|\right) + \leq + \exp\!\left(c\sum_j|\lambda_j|\right) + \prod_j\exp(c|\lambda_j|U_j^2) +\] +and the Gaussian integral show that +\(\mathbb E\exp(c|q_B|)<\infty\). + +For completeness, let \(\nu\) be another probability measure with +the same moments as \(q_B\). By the nonnegative power series for +\(\cosh\) and monotone convergence, +\[ + \int_{\R}\cosh(cx)\,d\nu(x) + =\sum_{r=0}^{\infty} + \frac{c^{2r}}{(2r)!}\mathbb E(q_B^{2r}) + =\mathbb E\cosh(cq_B)<\infty. +\] +Since \(e^{c|x|}\leq2\cosh(cx)\), the measure \(\nu\) also has an +exponential moment. The characteristic functions of \(\nu\) and +of \(q_B\) extend analytically to the strip +\(\{z\in\C:|\operatorname{Im}z|d\). Moment convergence supplies a constant +\(C_r<\infty\) such that +\(\int |x|^{2r}\,d\mu_j(x)\leq C_r\) for all \(j\). Consequently +\begin{equation}\label{eq:spectral-moment-tail} + \int_{|x|>K}|x|^d\,d\mu_j(x) + \leq C_r K^{d-2r} + \qquad(K>0). +\end{equation} + +Consider any weakly convergent subsequence, with limit \(\mu\). +Lower semicontinuity gives +\(\int |x|^{2r}\,d\mu(x)\leq C_r\), so the same tail bound holds +for \(\mu\). Multiply \(x^d\) by a continuous cutoff which equals +one on \([-K,K]\) and zero outside \([-2K,2K]\). Weak convergence +applies to this bounded continuous function. Letting \(K\) tend +to infinity and using Equation~\eqref{eq:spectral-moment-tail} shows that +\[ + \int x^d\,d\mu(x)=\mathbb E(q_B^d). +\] +This holds for every \(d\). Lemma~\ref{lem:gaussian-quadratic} +therefore identifies \(\mu\) with the law of \(q_B\). +Tightness now shows that the whole sequence \(\mu_j\) converges +weakly to that law: every subsequence has a weakly convergent +further subsequence, and all such limits have just been +identified. This is precisely the claimed convergence for \(f\). +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:uniform-unitaries}] +\textbf{Step 1. Choice of parameters.} +Fix \(H,\eps,m\) as in the statement. Choose \(t>0\) so that +\begin{equation}\label{eq:gaussian-t-choice} + (1+t^2)^{-1/4}<\eps/4. +\end{equation} +Next choose \(\delta>0\) and \(0<\eta<1/2\) so that +\begin{equation}\label{eq:gaussian-error-choice} + t\sqrt{2}\,(\sqrt{\delta}+2\eta)<\eps/4. +\end{equation} +Amenability gives a nonempty finite \(F\subset G\) satisfying +\[ + |hF\mathbin{\triangle}F|<\delta|F| + \qquad(h\in H). +\] +Set +\[ + n=|F|, + \qquad J=F\cup\bigcup_{h\in H}hF. +\] +Finally, choose a real polynomial +\begin{equation}\label{eq:square-root-polynomial} + p(s)=\sum_{r=1}^{D}c_rs^r, + \qquad + \sup_{0\leq s\leq1}|p(s)-\sqrt{s}|<\eta/\sqrt n. +\end{equation} +Such a polynomial exists: first approximate \(\sqrt{s}\) on +\([0,1]\) by a real polynomial to within +\(\eta/(2\sqrt n)\), then subtract its value at zero. +Thus \(t,\delta,\eta,F,n,J\), and \(p\), including its degree +and coefficients, are all fixed before \(N\) tends to infinity. +None is chosen using a trace or a limiting covariance. + +\textbf{Step 2. Uniform Gaussian comparison.} +Let \(R=(R_{g,g'})_{g,g'\in J}\) be any real positive semidefinite +matrix with diagonal one. Choose unit vectors \(v_g\) with Gram +matrix \(R\), spanning a finite-dimensional real Euclidean space +\(L\), and let \(W\) be standard Gaussian in \(L\). Write +\(\xi_g=\langle v_g,W\rangle\). For \(E=F\) or \(E=hF\), define +\[ + P_E=\frac1n\sum_{g\in E}v_gv_g^{\mathsf T}, + \qquad + B_E=p(P_E), + \qquad + q_E=\langle W,B_EW\rangle-\Tr(B_E). +\] +The operator \(P_E\) is positive with trace one, so its spectrum +lies in \([0,1]\); its rank is at most \(n\). +Because \(p(0)=0\), the operator +\(p(P_E)-P_E^{1/2}\) vanishes on the kernel of \(P_E\). +The scalar approximation in Equation~\eqref{eq:square-root-polynomial} +therefore gives +\begin{equation}\label{eq:gaussian-rank-control} + \|B_E-P_E^{1/2}\|_{\HS}^2 + \leq \operatorname{rank}(P_E)\,\eta^2/n + \leq\eta^2, + \qquad + \|P_E^{1/2}\|_{\HS}^2=\Tr(P_E)=1. +\end{equation} +In particular, +\begin{equation}\label{eq:gaussian-nondegeneracy} + 1-\eta\leq\|B_E\|_{\HS}\leq1+\eta. +\end{equation} + +The summands indexed by \(F\cap hF\) cancel when subtracting +\(P_F\) and \(P_{hF}\). Each remaining \(v_gv_g^{\mathsf T}\) +has trace norm one. Hence Lemma~\ref{lem:square-root-trace} +implies +\[ + \|P_F^{1/2}-P_{hF}^{1/2}\|_{\HS}^2 + \leq\|P_F-P_{hF}\|_1 + \leq\frac{|F\mathbin{\triangle}hF|}{n}<\delta. +\] +Combining this with Equation~\eqref{eq:gaussian-rank-control} yields +\[ + \|B_F-B_{hF}\|_{\HS}\leq\sqrt{\delta}+2\eta. +\] +Both forms \(q_F,q_{hF}\) are evaluated on the same \(W\), so +their difference is the centered quadratic associated with +\(B_{hF}-B_F\). Lemma~\ref{lem:gaussian-quadratic} gives +\begin{equation}\label{eq:gaussian-translation-bound} + \|q_{hF}-q_F\|_{L^2} + \leq\sqrt{2}\,(\sqrt{\delta}+2\eta). +\end{equation} +Also, Equation~\eqref{eq:gaussian-nondegeneracy} and \(\eta<1/2\) +give, for every integer \(\ell\geq1\), +\begin{equation}\label{eq:gaussian-fourier-bound} + \bigl|\mathbb E\exp(\iu\ell t q_F)\bigr| + \leq + \bigl(1+4\ell^2t^2\|B_F\|_{\HS}^2\bigr)^{-1/4} + \leq(1+t^2)^{-1/4}. +\end{equation} +All these bounds hold for every such \(R\), including singular +matrices. No invariance of \(R\) under translation of its indices +has been used. + +\textbf{Step 3. Equivariant polynomial prescription.} +For \(r\geq1\), direct multiplication of the rank-one summands +gives +\[ + P_E^r + =\frac1{n^r} + \sum_{g_1,\ldots,g_r\in E} + R_{g_1,g_2}\cdots R_{g_{r-1},g_r} + v_{g_1}v_{g_r}^{\mathsf T}. +\] +Taking its quadratic form and its trace, respectively, shows that +\begin{equation}\label{eq:gaussian-covariance-polynomial} + q_E= + \sum_{r=1}^{D}\frac{c_r}{n^r} + \sum_{g_1,\ldots,g_r\in E} + R_{g_1,g_2}\cdots R_{g_{r-1},g_r} + \bigl(\xi_{g_1}\xi_{g_r}-R_{g_r,g_1}\bigr). +\end{equation} +The product of covariance entries is interpreted as \(1\) when +\(r=1\). + +Use the same prescription in \(C\) by setting +\begin{align} + \widetilde Q_{N,E} + &= + \sum_{r=1}^{D}\frac{c_r}{n^r} + \sum_{g_1,\ldots,g_r\in E} + k_{N;g_1,g_2}\cdots k_{N;g_{r-1},g_r} + \bigl(z_{N,g_1}z_{N,g_r}-k_{N;g_r,g_1}\bigr), + \label{eq:empirical-quadratic-polynomial}\\ + Q_{N,E} + &=\tfrac12\bigl(\widetilde Q_{N,E} + +\widetilde Q_{N,E}^{\,*}\bigr). + \label{eq:empirical-quadratic-selfadjoint} +\end{align} +Multiplication in Equation~\eqref{eq:empirical-quadratic-polynomial} is +in the displayed order. The entries of the empirical covariance +array need satisfy no matrix positivity condition: this is a +finite polynomial in elements of \(C\), whereas square-root +functional calculus was applied only to \(P_E\) in the Gaussian +comparison. + +The exact relabeling in Equation~\eqref{eq:empirical-equivariance} carries +each tuple in the sum for \(F\) to its left translate in the sum +for \(hF\). It preserves the order of +all factors, and taking self-adjoint parts commutes with +\(\bar\alpha_h\). Consequently +\begin{equation}\label{eq:quadratic-exact-equivariance} + Q_{N,hF}=\bar\alpha_h(Q_{N,F}) + \qquad(h\in H,\ N\geq1). +\end{equation} + +\textbf{Step 4. Transfer of moments and scalar laws.} +Consider now any sequence \(N_j\to\infty\) and extreme traces +\(\varphi_j\in\T(C)\) such that +\[ + \varphi_j(k_{N_j;g,g'})\longrightarrow R_{g,g'} + \qquad(g,g'\in J). +\] +Proposition~\ref{prop:joint-clt} gives convergence of every +ordered mixed moment of the empirical variables. Any moment of +the polynomials in +Equations~\eqref{eq:empirical-quadratic-polynomial}--\eqref{eq:empirical-quadratic-selfadjoint} +is a finite linear combination of such moments. Since the +limiting variables commute and +Equation~\eqref{eq:gaussian-covariance-polynomial} is real-valued, it +follows that the \(Q_{N_j,E}\) converge jointly in moments under +\(\varphi_j\) to the \(q_E\). In particular, +\begin{equation}\label{eq:quadratic-difference-limit} + \|Q_{N_j,hF}-Q_{N_j,F}\|_{2,\varphi_j} + \longrightarrow \|q_{hF}-q_F\|_{L^2}. +\end{equation} +Lemma~\ref{lem:quadratic-moment-transfer} also gives +\begin{equation}\label{eq:quadratic-exponential-limit} + \varphi_j\bigl(\exp(\iu s Q_{N_j,F})\bigr) + \longrightarrow\mathbb E\exp(\iu s q_F) + \qquad(s\in\R). +\end{equation} +This use of scalar spectral laws is pertinent because the +operator norms of the \(Q_{N,E}\) may grow with \(N\). +Indeed, the elementary estimates +\(\|k_{N;g,g'}\|\leq M^2\) and +\(\|z_{N,g}\|\leq\sqrt N\,M\) yield only +\[ + \|Q_{N,E}\|\leq(N+1)\sum_{r=1}^{D}|c_r|M^{2r}. +\] +Convergence of the scalar moments, together with the tail +estimate Equation~\eqref{eq:spectral-moment-tail}, is sufficient for +Equation~\eqref{eq:quadratic-exponential-limit}. + +\textbf{Step 5. Uniformity over all traces.} +Set \(V_N=\exp(\iu tQ_{N,F})\). For any self-adjoint \(X,Y\) +in a unital \(\mathrm{C}^*\)-algebra and any tracial state +\(\varphi\), differentiation gives +\[ + \frac{d}{ds} + \bigl(e^{\iu(1-s)tX}e^{\iu stY}\bigr) + =e^{\iu(1-s)tX}\,\iu t(Y-X)\,e^{\iu stY}. +\] +The tracial \(2\)-seminorm is unchanged by left and right +unitary multiplication. Integration and the triangle inequality +therefore give +\begin{equation}\label{eq:tracial-duhamel} + \|e^{\iu tX}-e^{\iu tY}\|_{2,\varphi} + \leq t\|X-Y\|_{2,\varphi}. +\end{equation} +In view of Equations~\eqref{eq:quadratic-exact-equivariance}, +\eqref{eq:quadratic-difference-limit}, and +\eqref{eq:gaussian-translation-bound}, every sequence of traces +and indices with convergent covariance as above satisfies +\begin{align} + \limsup_{j\to\infty} + \|\bar\alpha_h(V_{N_j})-V_{N_j}\|_{2,\varphi_j} + &\leq t\sqrt{2}\,(\sqrt{\delta}+2\eta)<\eps/4, + \label{eq:sequential-invariance-estimate}\\ + \lim_{j\to\infty}|\varphi_j(V_{N_j}^{\ell})| + &= + \bigl|\mathbb E\exp(\iu\ell t q_F)\bigr| + \leq(1+t^2)^{-1/4}<\eps/4. + \label{eq:sequential-fourier-estimate} +\end{align} +Here \(h\in H\) and \(1\leq\ell\leq m\), and the strict +inequalities are Equations~\eqref{eq:gaussian-error-choice} and +\eqref{eq:gaussian-t-choice}. + +It follows that, for all sufficiently large \(N\), every extreme +trace \(\varphi\in\T(C)\) satisfies +\begin{equation}\label{eq:extreme-uniform-estimates} + \|\bar\alpha_h(V_N)-V_N\|_{2,\varphi}<\eps/2 + \quad(h\in H),\qquad + |\varphi(V_N^\ell)|<\eps/2 + \quad(1\leq\ell\leq m). +\end{equation} +Indeed, otherwise choose \(N_j\to\infty\) and an extreme trace +\(\varphi_j\) for which one of these finitely many tests fails. +Passing to a subsequence fixes the failed test. The matrices +\[ + \bigl(\varphi_j(k_{N_j;g,g'})\bigr)_{g,g'\in J} +\] +belong to the compact set of real positive semidefinite matrices +with diagonal one. A further subsequence has a limiting +covariance \(R\), and +Equation~\eqref{eq:sequential-invariance-estimate} or +Equation~\eqref{eq:sequential-fourier-estimate} contradicts the failure +at tolerance \(\eps/2\). +Only the finite covariance matrices were extracted; no +compactness of the extreme trace boundary is needed. Nor is a +convergence rate uniform in \(R\) required: each putative +sequence of failures supplies one limiting \(R\), to which the +preceding moment argument applies. Moment determinacy is used +after that \(R\) has been fixed, so no common exponential-moment +radius over all covariance matrices is required. + +Fix an \(N\) for which Equation~\eqref{eq:extreme-uniform-estimates} +holds. For each \(h\in H\), the squared seminorm +\[ + \varphi\longmapsto + \varphi\bigl((\bar\alpha_h(V_N)-V_N)^* + (\bar\alpha_h(V_N)-V_N)\bigr) +\] +is weak-star continuous and affine on \(\T(C)\). For each +\(1\leq\ell\leq m\), the map +\(\varphi\mapsto|\varphi(V_N^\ell)|\) is weak-star continuous +and convex. The sublevel sets with bounds \(\eps^2/4\) and +\(\eps/2\), respectively, are closed and convex and contain +all extreme traces. By the Krein--Milman Theorem they contain +\(\T(C)\). Thus the suprema in +Equations~\eqref{eq:uniform-unitary-invariance} and +\eqref{eq:uniform-unitary-moments} are at most \(\eps/2\), +and in particular strictly less than \(\eps\). +Taking \(Q=Q_{N,F}\) and \(V=V_N\) completes the proof. +\end{proof} diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/introduction.tex b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..d2f8e484e44cf6b9618491aa5553f0cabb310ff6 --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/introduction.tex @@ -0,0 +1,93 @@ +\section{Introduction} + +Jiang--Su stability is a regularity property of a \(\mathrm C^*\)-algebra +that also has a dynamical counterpart: an action should absorb the +trivial action on the Jiang--Su algebra up to cocycle conjugacy. +The theorem below establishes this implication for all amenable actions +in the simple, nuclear, finite unital setting. Throughout, tensor products +are minimal, and \(\Zalg\) denotes the Jiang--Su algebra. + +\begin{theorem}\label{thm:main} +Let \(A\) be a simple, separable, unital, infinite-dimensional, nuclear, +stably finite complex \(\mathrm C^*\)-algebra satisfying +\(A\cong A\otimes\Zalg\). Let \(G\) be a countable discrete amenable group +and let \(\alpha:G\to\Aut(A)\) be any action. Put +\[ + B=A\otimes\Zalg,\qquad + \beta_g=\alpha_g\otimes\operatorname{id}_{\Zalg}. +\] +There are a unital \(*\)-isomorphism \(\Phi:A\to B\) and unitaries +\(u_g\in B\), \(g\in G\), such that +\begin{align} + u_e&=1,& u_{gh}&=u_g\beta_g(u_h),\label{eq:cocycle}\\ + \Phi(\alpha_g(a))&=u_g\beta_g(\Phi(a))u_g^* + &&(g\in G,\ a\in A).\label{eq:conjugacy} +\end{align} +\end{theorem} + +Here stable finiteness means that every isometry in every matrix algebra +over \(A\) is unitary. Amenability is used in its F{\o}lner form: for +every finite \(H\subset G\) and \(\delta>0\), some nonempty finite +\(F\subset G\) satisfies +\(\lvert hF\mathbin{\triangle}F\rvert<\delta\lvert F\rvert\) for \(h\in H\). +Theorem~\ref{thm:main} imposes no outerness, Rokhlin, or UCT assumption, +and no condition on the extreme tracial boundary or on trace orbits. +In particular, it includes finite groups and trivial actions. + +\subsection{Context and scope} + +Szab\'o formulated automatic equivariant Jiang--Su stability for +amenable actions on separable simple nuclear Jiang--Su-stable algebras +as Conjecture~A of \cite{SzaboSI2021}. +Theorem~\ref{thm:main} resolves its unital, stably finite case. +The original formulation also includes nonunital algebras, which lie +beyond the scope of the present theorem. Earlier affirmative cases +include finite algebras with finitely many rays of extremal traces +and very weak comparison \cite[Theorem~C]{SzaboSI2021}. + +Szab\'o and Wouters proved that equivariant uniform property Gamma +characterizes equivariant Jiang--Su stability in the setting used here +\cite[Corollary~C]{SzaboWouters2025}. Their theorem supplies the +absorption step of the proof. Our task is to construct the required +invariant central trace splittings for an arbitrary action. +These splittings are uniform over the entire trace space, even when +the action moves traces. + +\subsection{Construction and proof structure} + +The technical argument has three parts. +First, tensorial absorption supplies countably many commuting +\emph{orbit algebras} inside the uniform tracial central sequence algebra. +Under every extreme trace on the algebra they generate, distinct orbit +algebras are independent. This intrinsic independence is useful because +the traces under consideration need not extend to the ambient algebra. +We then prove a joint central limit statement for normalized sums and +their empirical covariances, allowing the extreme trace to vary with +the number of rows. + +Second, the covariance is incorporated into a quadratic Gaussian +construction. For a F{\o}lner set \(F\), its Gram vectors define a +positive operator \(P_F\) of trace one. Taking a polynomial approximation +to \(P_F^{1/2}\) gives a quadratic form whose variance stays bounded +away from zero and whose change under translation is small. +This square-root normalization handles covariance matrices of arbitrary +rank, including highly correlated ones. Empirical covariances turn the +same formula into exactly equivariant noncommutative polynomials. +Compactness of the finite covariance space then promotes the scalar +moment limits to estimates uniform over every tracial state. + +Finally, weighting traces by elements of \(A\) yields a central invariant +unitary with Haar distribution relative to \(A\). Two explicit +diagonal selections give this unitary and then equal trace projections. +The construction of the rows appears in Section~\ref{sec:rows}, the +uniform unitary estimates in Section~\ref{sec:gaussian}, and the +diagonal selections and absorption argument in Section~\ref{sec:splittings}. +All parts of this construction are proved below; the final absorption +theorem is stated separately in Theorem~\ref{thm:absorption}. + +Several ingredients can be used separately under their stated hypotheses: +intrinsic independence for commuting rows +(Lemma~\ref{lem:factor-independence}), moment estimates under varying +tracial states (Proposition~\ref{prop:joint-clt}), and convergence of +scalar spectral laws against bounded continuous functions without +uniform operator-norm bounds (Lemma~\ref{lem:quadratic-moment-transfer}). diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/preliminaries.tex b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..03ce61f2e1b442c8b2d2116f50d633cff4175a74 --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/preliminaries.tex @@ -0,0 +1,74 @@ +\section{Traces and the absorption criterion}\label{sec:preliminaries} + +For a unital \(\mathrm C^*\)-algebra \(D\), write \(\T(D)\) for its +tracial-state space. For \(\tau\in\T(D)\), set +\[ + \|x\|_{2,\tau}=\tau(x^*x)^{1/2}. +\] +Under the assumptions of Theorem~\ref{thm:main}, \(\T(A)\) is a +nonempty compact convex set. Indeed, nuclearity implies exactness, +and a unital stably finite exact algebra has a tracial state; +moreover its \(2\)-quasitraces are traces +\cite[Theorem~5.11 and Corollary~5.12]{Haagerup2014}. +Every tracial state on the simple algebra \(A\) is faithful: +its null space \(\{a:\tau(a^*a)=0\}\) is a closed two-sided ideal +that does not contain \(1\). + +Fix a free ultrafilter \(\omega\) on \(\N=\{1,2,\ldots\}\). Define +\begin{equation}\label{eq:ultrapower} + \|a\|_{2,u}=\sup_{\tau\in\T(A)}\|a\|_{2,\tau},\qquad + A^\omega=\ell^\infty(\N,A)/I_\omega,\qquad + I_\omega=\{(a_j):\lim_{j\to\omega}\|a_j\|_{2,u}=0\}. +\end{equation} +The inequalities +\(\|ab\|_{2,u}\leq\|a\|\|b\|_{2,u}\) and +\(\|ba\|_{2,u}\leq\|a\|\|b\|_{2,u}\) show that \(I_\omega\) +is a closed two-sided ideal. Faithfulness of traces embeds \(A\) +as the constant sequences in \(A^\omega\). +For any sequence \((\tau_j)\) in \(\T(A)\), the formula +\[ + \tau([(a_j)])=\lim_{j\to\omega}\tau_j(a_j) +\] +defines a tracial state on \(A^\omega\), since +\(\lvert\tau_j(a_j)\rvert\leq\|a_j\|_{2,\tau_j}\). +These are the \emph{limit traces}, denoted \(\T_\omega(A)\). +We do not identify them with all tracial states on \(A^\omega\). + +Automorphisms permute \(\T(A)\) and therefore preserve +\(\|\cdot\|_{2,u}\). Applying \(\alpha_g\) to representative sequences +defines an action \(\bar\alpha:G\curvearrowright A^\omega\). +It preserves \(A^\omega\cap A'\). The quotient in +Equation~\eqref{eq:ultrapower} is also the quotient of the norm +ultrapower by its uniform trace-kernel ideal, since norm-null +sequences belong to \(I_\omega\). + +\begin{definition}\label{def:gamma} +The action \(\alpha\) has \emph{equivariant uniform property Gamma} +if, for every integer \(k\geq2\), there are pairwise orthogonal +projections \(p_1,\ldots,p_k\in(A^\omega\cap A')^{\bar\alpha}\) with +\begin{equation}\label{eq:gamma} + \tau(ap_i)=\frac{\tau(a)}{k} + \qquad(a\in A,\ \tau\in\T_\omega(A),\ 1\leq i\leq k). +\end{equation} +\end{definition} + +This is the unital simple formulation in +\cite[Definition~A]{SzaboWouters2025}. +Our construction also gives \(\sum_i p_i=1\). +We use the following established theorem. + +\begin{theorem}[Szab\'o--Wouters]\label{thm:absorption} +Let a countable discrete amenable group act on a separable, unital, +simple, nuclear, \(\Zalg\)-stable \(\mathrm C^*\)-algebra with +nonempty tracial-state space. If the action has equivariant uniform +property Gamma, it is cocycle conjugate to its tensor product with +the trivial action on \(\Zalg\). +\end{theorem} + +Theorem~\ref{thm:absorption} is the implication of +\cite[Corollary~C]{SzaboWouters2025} needed here. +Thus it remains to construct the projections required by Definition~\ref{def:gamma}. +The next two sections work inside a separable subalgebra \(C\) of +\(A^\omega\cap A'\). There we use \emph{all} traces in \(\T(C)\); +this will allow weighted restrictions of limit traces in +Section~\ref{sec:splittings}. diff --git a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/rows.tex b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/rows.tex new file mode 100644 index 0000000000000000000000000000000000000000..fdd5de343c1ad5598447f5b249175ce93125b609 --- /dev/null +++ b/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/build/sections/rows.tex @@ -0,0 +1,399 @@ +\section{Commuting orbit rows and joint moments} +\label{sec:rows} + +Throughout this section, \(A\), \(G\), and \(\alpha\) satisfy the +hypotheses of Theorem~\ref{thm:main}. Only countability of \(G\) is +needed for the constructions in this section. + +\begin{proposition}[Commuting orbit rows] +\label{prop:rows} +There are unital embeddings +\(\iota_i\colon\Zalg\to A^\omega\), \(i\geq 1\), for which the unital +orbit algebras +\[ + C_i=C^*\bigl(\bar\alpha_g(\iota_i(\Zalg)):g\in G\bigr) +\] +commute with \(A\) and commute pairwise. In particular, +\[ + C=C^*\Bigl(\bigcup_{i\geq 1}C_i\Bigr) + \subseteq A^\omega\cap A' +\] +is unital, separable, and \(\bar\alpha\)-invariant, and +\(\T(C)\neq\varnothing\). + +Moreover, there are self-adjoint elements \(x_{i,g}\in C_i\), +\(i\geq 1\), \(g\in G\), and a constant \(M<\infty\), such that +\begin{equation} +\label{eq:row-normalization} + \|x_{i,g}\|\leq M,\qquad + \varphi(x_{i,g})=0,\qquad + \varphi(x_{i,g}^2)=1 + \quad\bigl(\varphi\in\T(C)\bigr), +\end{equation} +and +\begin{equation} +\label{eq:row-equivariance} + \bar\alpha_h(x_{i,g})=x_{i,hg} + \qquad(i\geq1,\ g,h\in G). +\end{equation} +\end{proposition} + +\begin{proof} +The Jiang--Su algebra is simple, unital, uniquely tracial, and +isomorphic to its infinite tensor power; see +\cite[Example~1.14(iv) and Corollary~1.11]{TomsWinter2007}. +Write \(\rho\) for its unique tracial state. Since \(A\) is +\(\Zalg\)-stable, fix an identification +\[ + A\cong A\otimes\bigotimes_{l=1}^{\infty}\Zalg. +\] +We use the increasing finite tensor stages in this identification. + +We first show that every norm-separable \(C^*\)-subalgebra +\(D\subseteq A^\omega\) admits a unital embedding +\begin{equation} +\label{eq:relative-tail-embedding} + \Zalg\longrightarrow A^\omega\cap D'. +\end{equation} +Choose a norm-dense sequence \((d_s)_{s\geq1}\) in \(D\), and choose +bounded representative sequences +\(d_s=[(d_{s,j})_{j\geq1}]\). +For each \(j\), choose a finite tensor stage containing elements +within \(1/j\) in norm of +\(d_{1,j},\ldots,d_{j,j}\). Let +\(\theta_j\colon\Zalg\to A\) be the embedding into a tensor factor +strictly after this stage. For \(j\geq s\) and \(z\in\Zalg\), +\[ + \|[\theta_j(z),d_{s,j}]\|\leq \frac{2\|z\|}{j}. +\] +The coordinatewise homomorphisms therefore induce a unital +\(*\)-homomorphism \(z\mapsto[(\theta_j(z))_j]\) into +\(A^\omega\cap D'\). It is injective because \(\Zalg\) is simple +and the unit of \(A^\omega\) is nonzero. This proves +Equation~\eqref{eq:relative-tail-embedding}. + +Set \(D_0=A\). Inductively, suppose that \(D_{i-1}\) is a separable, +unital, \(\bar\alpha\)-invariant subalgebra of \(A^\omega\) containing +\(A\) and the preceding orbit algebras. Apply +Equation~\eqref{eq:relative-tail-embedding} to obtain +\(\iota_i(\Zalg)\subseteq D_{i-1}'\), define \(C_i\) as in the +statement, and put +\[ + D_i=C^*(D_{i-1},C_i). +\] +For \(b\in\iota_i(\Zalg)\), \(d\in D_{i-1}\), and \(g\in G\), +\[ + [\bar\alpha_g(b),d] + =\bar\alpha_g\bigl([b,\bar\alpha_{g^{-1}}(d)]\bigr)=0, +\] +since \(D_{i-1}\) is invariant. Thus \(C_i\) commutes with +\(D_{i-1}\). Countability of \(G\) preserves separability at each +stage. This gives the required algebras and their commutation +properties. A limit trace on \(A^\omega\) restricts to a tracial +state on \(C\), so \(\T(C)\) is nonempty. + +The trace \(\rho\) is faithful by simplicity of \(\Zalg\). +Choose a non-scalar self-adjoint element \(c\in\Zalg\). Then +\(\rho((c-\rho(c)1)^2)>0\), so rescaling \(c-\rho(c)1\) gives +a self-adjoint \(b\in\Zalg\) with +\(\rho(b)=0\) and \(\rho(b^2)=1\). Set +\[ + x_{i,g}=\bar\alpha_g(\iota_i(b)),\qquad M=\|b\|. +\] +Every \(\varphi\in\T(C)\) restricts along each translated unital +embedding of \(\Zalg\) to \(\rho\). This proves +Equation~\eqref{eq:row-normalization}, and +Equation~\eqref{eq:row-equivariance} follows from the action law. +\end{proof} + +The following independence statement is intrinsic to the algebra +generated by the rows. In particular, it applies to traces of \(C\) +that do not extend to the ambient ultrapower. + +\begin{lemma}[Independence at extreme traces] +\label{lem:factor-independence} +Let \(B\) be a unital \(C^*\)-algebra generated by pairwise commuting +unital \(C^*\)-subalgebras \(B_i\), \(i\geq1\). If +\(\varphi\in\partial_{\mathrm e}\T(B)\), then +\[ + \varphi(b_1\cdots b_s)=\prod_{j=1}^s\varphi(b_j) +\] +whenever \(b_j\in B_{i_j}\) and the indices \(i_1,\ldots,i_s\) +are distinct. +\end{lemma} + +\begin{proof} +Let \((H_\varphi,\pi,\Omega)\) be the tracial GNS representation +of \(\varphi\), and put +\[ + \mathcal M=\pi(B)'',\qquad + \mathcal M_i=\pi(B_i)''. +\] +We first record the relevant properties of this representation, +allowing \(\varphi\) to have a nonzero kernel on \(B\). +On the dense subspace of vectors \([a]\), \(a\in B\), right +multiplication \(R_b[a]=[ab]\) is bounded with norm at most +\(\|b\|\), by traciality. It commutes with \(\pi(B)\), while +the vectors \(R_b\Omega=[b]\) are dense. If +\(T\in\mathcal M\) satisfies \(T\Omega=0\), then +\[ + TR_b\Omega=R_bT\Omega=0\qquad(b\in B), +\] +so \(T=0\). Thus \(\Omega\) is separating for \(\mathcal M\). +The normal vector state +\(\tau(T)=\langle T\Omega,\Omega\rangle\) is consequently +faithful: for \(T\geq0\), \(\tau(T)=0\) implies +\(T^{1/2}\Omega=0\). Its traciality extends from \(\pi(B)\) +to \(\mathcal M\) by separate ultraweak continuity. Hence +\(\mathcal M\) is a finite von Neumann algebra with faithful +normal tracial state \(\tau\). + +Extremality of \(\varphi\) implies that \(\mathcal M\) is a factor. +Indeed, a nontrivial central projection \(p\in\mathcal M\) would +give the convex decomposition +\[ + \varphi(a) + =\tau(p)\frac{\tau(p\pi(a))}{\tau(p)} + +\tau(1-p)\frac{\tau((1-p)\pi(a))}{\tau(1-p)}. +\] +The two normalized functionals are tracial states on \(B\). They are distinct: +if, for instance, the first equalled \(\varphi\), equality of +the corresponding normal functionals would extend from +\(\pi(B)\) to \(\mathcal M\), and evaluation at \(p\) would +give \(1=\tau(p)\). This contradicts +\(0<\tau(p)<1\). + +The algebras \(\mathcal M_i\) commute and generate \(\mathcal M\). +Every element of the center of \(\mathcal M_i\) therefore +commutes with all of \(\mathcal M\), so +\[ + Z(\mathcal M_i)\subseteq Z(\mathcal M)=\C1. +\] +The restriction of \(\tau\) makes each \(\mathcal M_i\) a finite +factor with a faithful normal tracial state. + +Fix \(i\), and let \(y\geq0\) belong to the von Neumann algebra +generated by the other \(\mathcal M_j\)'s. It commutes with +\(\mathcal M_i\). The functional +\[ + \psi_y(x)=\tau(y^{1/2}xy^{1/2})=\tau(xy), + \qquad x\in\mathcal M_i, +\] +is normal and positive: sandwiching by \(y^{1/2}\) preserves +suprema of bounded increasing nets of positive operators. +Commutation of \(y\) with \(\mathcal M_i\) makes it tracial, +and \(0\leq\psi_y\leq\|y\|\tau\). +The Radon--Nikodym Theorem for normal functionals on a finite +von Neumann algebra therefore gives a density +\(h\in\mathcal M_i\), \(0\leq h\leq\|y\|1\), with +\(\psi_y(x)=\tau(hx)\). Invariance of \(\psi_y\) under unitary +conjugation forces \(h\) to commute with every unitary in +\(\mathcal M_i\). Hence \(h\) is central and thus scalar. +Evaluating at the unit identifies that scalar as \(\tau(y)\). +It follows that +\[ + \tau(xy)=\tau(x)\tau(y). +\] +Linearity extends this identity to arbitrary \(y\) in the +algebra generated by the other rows. Applying it successively +proves the required product formula. +\end{proof} + +We now fix the algebras and elements from +Proposition~\ref{prop:rows}. For \(N\geq1\) and \(g,h\in G\), set +\begin{equation} +\label{eq:empirical-variables} + z_{N,g}=\frac1{\sqrt N}\sum_{i=1}^N x_{i,g}, + \qquad + k_{N;g,h}=\frac1N\sum_{i=1}^N + \frac{x_{i,g}x_{i,h}+x_{i,h}x_{i,g}}2. +\end{equation} +The last two subscripts of \(k_{N;g,h}\) are covariance indices. +These elements are self-adjoint, \(k_{N;g,h}=k_{N;h,g}\), and +\begin{equation} +\label{eq:empirical-equivariance} + \bar\alpha_l(z_{N,g})=z_{N,lg}, + \qquad + \bar\alpha_l(k_{N;g,h})=k_{N;lg,lh} + \quad(l,g,h\in G). +\end{equation} + +\begin{proposition}[Joint ordered moment limit] +\label{prop:joint-clt} +Let \(J\subseteq G\) be nonempty and finite. Let \(N_j\to\infty\) +be a sequence of positive integers, and let +\(\varphi_j\in\partial_{\mathrm e}\T(C)\). Suppose that +\begin{equation} +\label{eq:covariance-convergence} + \varphi_j(k_{N_j;g,h})\longrightarrow R_{g,h} + \qquad(g,h\in J). +\end{equation} +Then \(R=(R_{g,h})_{g,h\in J}\) is a real positive semidefinite +matrix with diagonal entries one. +Choose unit vectors \(v_g\), \(g\in J\), spanning a +finite-dimensional real Euclidean space \(L\), with +\(\langle v_g,v_h\rangle=R_{g,h}\). If \(W\) is a standard +Gaussian vector in \(L\) and +\(\xi_g=\langle v_g,W\rangle\), then +\[ + \bigl((z_{N_j,g})_{g\in J},(k_{N_j;g,h})_{g,h\in J}\bigr) + \longrightarrow + \bigl((\xi_g)_{g\in J},(R_{g,h})_{g,h\in J}\bigr) +\] +in all ordered joint moments under \(\varphi_j\). +Equivalently, for every noncommutative \(*\)-polynomial \(P\) +in these self-adjoint variables, +\begin{equation} +\label{eq:joint-polynomial-limit} + \varphi_j\!\left( + P\bigl((z_{N_j,g})_g,(k_{N_j;g,h})_{g,h}\bigr)\right) + \longrightarrow + \mathbb E\,P\bigl((\xi_g)_g,(R_{g,h})_{g,h}\bigr). +\end{equation} +For every sequence \((N_j,\varphi_j)\) as above, a subsequence +satisfying Equation~\eqref{eq:covariance-convergence} exists. +\end{proposition} + +\begin{proof} +For a tracial state \(\varphi\) and self-adjoint \(x,y\), +\[ + \varphi(xy)=\varphi(yx) + =\overline{\varphi(xy)}. +\] +Thus all the second moments under consideration are real +and symmetric. For real scalars \(a_g\), +\[ + \sum_{g,h\in J}a_ga_h\varphi_j(k_{N_j;g,h}) + = + \frac1{N_j}\sum_{i=1}^{N_j} + \varphi_j\!\left(\left(\sum_{g\in J}a_gx_{i,g}\right)^2\right) + \geq0. +\] +The diagonal entries are one by +Equation~\eqref{eq:row-normalization}. Cauchy--Schwarz bounds every +entry in absolute value by one. Consequently these covariance +matrices lie in a compact set of real symmetric matrices; +their limits have the asserted properties, and the final +subsequence assertion follows. A Gram representation of +the positive semidefinite matrix \(R\) supplies \(L\) and +the vectors \(v_g\). + +The empty word is immediate. We first consider a word of +length \(d\geq1\) in the fluctuation +variables. At one term of the given sequence write +\(N=N_j\) and \(\varphi=\varphi_j\). Expanding the sums gives +\begin{equation} +\label{eq:fluctuation-expansion} + \varphi(z_{N,g_1}\cdots z_{N,g_d}) + =N^{-d/2} + \sum_{i_1,\ldots,i_d=1}^N + \varphi(x_{i_1,g_1}\cdots x_{i_d,g_d}). +\end{equation} +Partition the positions \(1,\ldots,d\) according to equality +of their row labels \(i_a\). Since distinct rows commute, +we may group the factors belonging to each row, preserving +their order within that row. Lemma~\ref{lem:factor-independence} +then factors the expectation over the distinct rows. +A singleton block gives zero by +Equation~\eqref{eq:row-normalization}. Each remaining summand has +absolute value at most \(M^d\). + +A partition with \(b\) blocks has at most \(N^b\) choices of +distinct row labels. Its contribution to +Equation~\eqref{eq:fluctuation-expansion} is therefore bounded by +\(M^dN^{b-d/2}\). Among partitions without singleton blocks, +\(b\leq\lfloor d/2\rfloor\). It follows that odd moments +tend to zero, and, when \(d=2q\), only partitions into \(q\) +pairs can have a nonzero limit. + +Fix such a pairing, writing its pairs as +\((a_l,b_l)\), \(1\leq l\leq q\), with \(a_l0\), its normalization is therefore an element of +\(\T(C)\); if \(\tau(a)=0\), the functional is zero. +Proposition~\ref{prop:uniform-unitaries} thus gives, for its unitary \(V\), +\begin{equation}\label{eq:weighted-small-moments} + |\tau(aV^\ell)|<\eps\,\tau(a)\leq\eps + \quad + \bigl(0\leq a\leq1,\ \tau\in\T_\omega(A),\ + 1\leq\ell\leq m,\ \tau(a)>0\bigr), +\end{equation} +with zero left side when \(\tau(a)=0\). +The approximate invariance estimates apply to limit traces by +restriction to \(C\). + +\begin{proposition}[A weighted Haar unitary]\label{prop:haar} +There is a unitary \(w\in(A^\omega\cap A')^{\bar\alpha}\) such that, +for every \(a\in A\), \(\tau\in\T_\omega(A)\), and +\(f\in C(\mathbb T)\), +\begin{equation}\label{eq:haar} + \tau(af(w))=\tau(a)\int_{\mathbb T}f\,d\mu, +\end{equation} +where \(\mathbb T=\{z\in\C:|z|=1\}\) and \(\mu\) is normalized +Haar measure. +\end{proposition} + +\begin{proof} +Choose increasing finite sets \(H_m\subset G\) whose union is \(G\), +and a sequence \(a_1,a_2,\ldots\) dense in the positive unit ball of +\(A\), with \(a_1=1\). Apply +Proposition~\ref{prop:uniform-unitaries} with +\((H,\eps,m)=(H_m,1/m,m)\), and denote its exponential unitary by +\(V^{(m)}\in C\). +It has a representative \((v^{(m)}_j)_j\) consisting of unitaries +in \(A\): lift its self-adjoint exponent to a bounded self-adjoint +sequence and exponentiate coordinatewise. The bounds for these +self-adjoint lifts may depend on \(m\), whereas the unitary +representatives always have norm one. + +Since \(V^{(m)}\) commutes with \(A\), Lemma~\ref{lem:coordinate}, +Equation~\eqref{eq:weighted-small-moments}, and the invariance +estimates allow a coordinate \(j(m)\geq m\) at which +\(w_m=v^{(m)}_{j(m)}\) satisfies +\begin{align} + \|[w_m,a_r]\|_{2,u}&<2/m &&(1\leq r\leq m),\label{eq:first-diagonal}\\ + \|\alpha_h(w_m)-w_m\|_{2,u}&<2/m &&(h\in H_m),\notag\\ + \sup_{\sigma\in\T(A)}|\sigma(a_r w_m^\ell)| + &<2/m &&(1\leq r,\ell\leq m).\notag +\end{align} +For the last line, apply Equation~\eqref{eq:coordinate-trace} to +the class \(a_r(V^{(m)})^\ell\). All coordinate tests are finite in +number, and their suprema already range over all traces. + +The class \(w=[(w_m)]\) is a unitary. The first two lines of +Equation~\eqref{eq:first-diagonal}, density, and +\(\|[w_m,a-a_r]\|_{2,u}\leq2\|a-a_r\|\) show that it is central +relative to \(A\) and fixed by \(\bar\alpha\). +The last line, first for every \(a_r\), then by norm density for +positive contractions, and then by linearity for \(a\in A\), gives +\[ + \tau(aw^\ell)=0 + \qquad(a\in A,\ \tau\in\T_\omega(A),\ \ell\geq1). +\] +The same holds for negative powers, because +\(\tau(aw^{-\ell})=\overline{\tau(a^*w^\ell)}\). +The zeroth power has trace \(\tau(a)\). +Thus Equation~\eqref{eq:haar} holds for trigonometric polynomials. +Uniform approximation on \(\mathbb T\) and +\(|\tau(a f(w))|\leq\|a\|\|f\|_\infty\) prove it for every continuous +\(f\). +\end{proof} + +\subsection{Equal trace projections} + +\begin{proposition}[Equal trace projections]\label{prop:gamma} +The action \(\alpha\) has equivariant uniform property Gamma. +The projections in Definition~\ref{def:gamma} can be chosen to sum to \(1\). +\end{proposition} + +\begin{proof} +Fix \(k\geq2\) and let \(w=[(w_j)]\) be the unitary of +Proposition~\ref{prop:haar}, represented by unitaries \(w_j\in A\). +Partition \(\mathbb T\) into \(k\) half-open arcs \(I_1,\ldots,I_k\) +of Haar measure \(1/k\), and write \(\chi_i=1_{I_i}\). +Choose continuous functions \(f_i^{(r)}:\mathbb T\to[0,1]\) satisfying +\[ + e_r:=\max_{1\leq i\leq k}\|f_i^{(r)}-\chi_i\|_{L^2(\mu)} + \longrightarrow0. +\] +For example, replace each indicator near its two endpoints by linear +ramps, with the total lengths of the transition arcs tending to zero. +For this proof fix \(r\), and write \(f_i=f_i^{(r)}\) and \(e=e_r\). +Pointwise bounds and the triangle inequality imply +\begin{align} + \|f_i^2-f_i\|_{L^2(\mu)}&\leq e,\label{eq:arc-bounds}\\ + \|f_i f_j\|_{L^2(\mu)}&\leq2e &&(i\ne j),\notag\\ + \Big\|\sum_i f_i-1\Big\|_{L^2(\mu)}&\leq ke,\notag\\ + \Big|\int_{\mathbb T} f_i\,d\mu-\frac1k\Big|&\leq e.\notag +\end{align} +For the first inequality, use +\(|f_i^2-f_i|\leq|f_i-\chi_i|\) almost everywhere. +For the second, expand +\(f_i f_j-\chi_i\chi_j=(f_i-\chi_i)f_j+\chi_i(f_j-\chi_j)\) +and use \(\chi_i\chi_j=0\) almost everywhere. +The remaining inequalities follow from \(\sum_i\chi_i=1\) almost +everywhere and the \(L^1\)--\(L^2\) inequality for a probability measure. + +Set \(y_i^{(r)}=f_i^{(r)}(w)\). +These are central, invariant positive contractions, since continuous +functional calculus respects commutation and automorphisms. +Equation~\eqref{eq:haar} with \(a=1\), applied to the squared modulus +of each continuous function in Equation~\eqref{eq:arc-bounds}, +gives the corresponding \(2\)-norm bounds for \(y_i^{(r)}\), +uniformly over \(\tau\in\T_\omega(A)\). +For every positive contraction \(a\in A\), it also gives +\begin{equation}\label{eq:arc-weighted} + \left|\tau(a y_i^{(r)})-\frac{\tau(a)}k\right| + =\tau(a)\left|\int_{\mathbb T}f_i^{(r)}\,d\mu-\frac1k\right| + \leq e_r . +\end{equation} + +Represent \(y_i^{(r)}\) by \(f_i^{(r)}(w_j)\). +Use \(H_r\) and \(a_1,\ldots,a_r\) from the proof of +Proposition~\ref{prop:haar}. +For this fixed \(r\), Lemma~\ref{lem:coordinate} gives a coordinate +\(j(r)\geq r\) such that the positive contractions +\(y_{i,r}=f_i^{(r)}(w_{j(r)})\) satisfy, simultaneously for all +indicated indices, +\begin{align} + \|y_{i,r}^2-y_{i,r}\|_{2,u}&0\) such +that, for all sufficiently small \(\eps>0\), every center with +\(r(x)>K\eps\) has an embedded flat disk of radius \(4\eps\). +Near a fixed cone point a disk of radius \(c r(x)\) is embedded for +some \(c>0\); away from the ends, compactness supplies the assertion. +Also, for each fixed \(L\), +\begin{equation}\label{eq:area} + \operatorname{Area}\{r\le L\eps\}=O_L(\eps^2). +\end{equation} + +Choose a nonnegative smooth radial kernel of mass one in the unit disk, +denoted by \(\rho\), and put +\(\rho_\eps(y)=\eps^{-2}\rho(y/\eps)\). +At a center with \(r>K\eps\), define +\[ + (S_\eps u)(x,\theta) + =\int_{\mathbb R^2}\rho_\eps(y)u(x-y,\theta)\,dy +\] +in an embedded flat disk, keeping the direction parallel. The formula +uses ordinary Euclidean area inside that chart. Radiality makes it +independent of the oriented flat chart. Call \(\{r>K\eps\}\) the +safe region. When a norm is taken over a base set, its inverse image +in \(SM\) is understood. + +\begin{lemma}\label{lem:smoothing} +The operators just defined have the following properties. +\begin{enumerate} +\item On the safe region, \(S_\eps\) and \(\eps\nabla S_\eps\) are +uniformly bounded from \(L^2(SM)\) to \(L^2\). They commute locally +with \(X,Y\), including on distributions, and +\(\norm{S_\eps u}_\infty\le\norm{u}_\infty\). +\item For every fixed \(u\in L^2(SM)\), +\begin{equation}\label{eq:small-derivative} + \eps\norm{Y S_\eps u}_{L^2(r>K\eps)}\longrightarrow0. +\end{equation} +On compact regular sets, \(S_\eps u\to u\) locally in \(L^2\). +\item Fix \(L>K+4\). If \(g\in L^2(SM)\) vanishes on +\(\{r0\), put \(T=s/\eps\), and fix \(L>K+4\). +This time scale keeps the discarded mass of order \(s\), while the +endpoint error will vanish as \(\eps\to0\) at fixed \(s\). +Let \(G_\eps\) be the set of regular states whose trajectories satisfy +\(r(\Psi_t z)>L\eps\) throughout \(0\le t\le T\). +The distance function is \(1\)-Lipschitz along a unit-speed path. +A time mesh of spacing at most \(\eps\), with at most +\(T/\eps+2\) points, detects every failure of this condition inside +\(\{r\le(L+1)\eps\}\). Measure preservation and \eqref{eq:area} +therefore give +\begin{equation}\label{eq:bad-measure} + \nu(SM\setminus G_\eps)\le C(T\eps+\eps^2) + =C(s+\eps^2). +\end{equation} +Null sets of singular states may be ignored. + +Set \(a_\eps=f\ind_{G_\eps}\) and form the negative-time average +\begin{equation}\label{eq:average} + g_\eps=\frac1T\int_0^T a_\eps\circ\Psi_{-t}\dd t. +\end{equation} +It has absolute value at most \(H\). If an integrand is nonzero at +\(z\), then \(z\) lies at time \(t\) on a good segment starting +at \(\Psi_{-t}z\). Consequently \(g_\eps=0\) on +\(\{rK\eps)}=o(1) + \qquad(\eps\to0,\ s\text{ fixed}), +\] +by \eqref{eq:small-derivative}. Here \(U\) is fixed. + +Put \(w_\eps=S_\eps g_\eps\), with the supported extension of +\Cref{lem:smoothing}. Its adjoint identity converts +\eqref{eq:segment-identity} into +\begin{equation}\label{eq:pairing-small} + \ip{B}{Yw_\eps}=o(1). +\end{equation} +The same lemma and \eqref{eq:average-bounds} give +\[ + \norm{\nabla w_\eps}_2\le CH/\eps,\qquad + \norm{Xw_\eps}_2\le CH/T. +\] +It has compact spatial support in \(M\). Applying +\Cref{lem:energy} with \(E=M\) yields +\begin{equation}\label{eq:fixed-s-bound} + \norm{\nabla(w_\eps)_j}_2^2\le CH^2/s\qquad(j\in\Z). +\end{equation} + +\medskip\noindent\emph{The first limit: fixed \(s\).} +Take a sequence \(\eps\to0\) and a subsequence along which +\(w_\eps\) converges weakly in \(L^2\) to \(g^{(s)}\). +The closed convex set \(\{|u|\le H\}\subset L^2\) is weakly closed, +so \(\norm{g^{(s)}}_\infty\le H\). +For a fixed smooth compactly supported test field \(\varphi\), +symmetry of convolution gives, for sufficiently small \(\eps\), +\[ + \ip{w_\eps-g_\eps}{\varphi} + =\ip{g_\eps}{S_\eps\varphi-\varphi}\longrightarrow0. +\] +The uniform \(L^2\) bound and density of these tests show that +\(g_\eps\) has the same weak \(L^2\) limit. Therefore +\begin{equation}\label{eq:limit-function} + Xg^{(s)}=0,\qquad + \norm{g^{(s)}-f}_2\le CH\sqrt{s}. +\end{equation} +The first equality follows either from \eqref{eq:average-bounds} or +the bound for \(Xw_\eps\); the second follows by weak lower +semicontinuity. + +Only coefficient derivatives at modes \(m\) and \(m-2\) contribute +to \eqref{eq:pairing-small}, because \(B\) has mode \(m-1\). +For fixed \(s\), \eqref{eq:fixed-s-bound} allows a further subsequence +on which both these gradients converge weakly in their global \(L^2\) +spaces. Testing in regular charts identifies the limits as the +gradients of the corresponding coefficients of \(g^{(s)}\). +Since \(B\) is fixed in \(L^2\), passing to the limit gives +\begin{equation}\label{eq:first-limit} + \ip{B}{P_{m-1}Yg^{(s)}}=0. +\end{equation} +Here this projected derivative means the combination of the two +coefficient derivatives in \eqref{eq:modes}; no full \(L^2\) bound +on \(Yg^{(s)}\) is asserted. + +\medskip\noindent\emph{The second limit: \(s\to0\).} +We do not use the deteriorating bound \eqref{eq:fixed-s-bound} for +this limit. Instead, \(g^{(s)}\) is bounded by \(H\) and solves the +exact local equation in \eqref{eq:limit-function}, so +\Cref{lem:bounded-modes} gives +\[ + \norm{\nabla(g^{(s)})_j}_2\le CH, +\] +uniformly in \(s\). Choose \(s_n\downarrow0\) and the limits just +constructed for each \(s_n\). By \eqref{eq:limit-function}, they +converge strongly to \(f\) in \(L^2\). Their relevant gradients +are bounded in \(L^2\), and every weak subsequential limit is +identified locally with the corresponding gradient of \(f\). +Thus \eqref{eq:first-limit} implies +\[ + \ip{B}{P_{m-1}Yf}=0. +\] +The equation \(Xf=0\) in mode \(m-1\) says +\(\pa f_{m-2}=-\pb f_m\). By \eqref{eq:modes}, +\[ + P_{m-1}Yf=-2iB,\qquad + 0=\ip{B}{-2iB}=2i\norm{B}_2^2. +\] +Hence \(\pb f_m=0\). Since \(m\) was arbitrary, the recurrence +also gives \(\pa f_j=0\) for every \(j\), proving the proposition. +\end{proof} + +\section{Holonomy and the conclusion}\label{sec:conclusion} + +\begin{proof}[Proof of \Cref{thm:main}] +Relabel the vertices so that \(\alpha/\pi\) is irrational. +Let \(A\) be a measurable invariant set as in the theorem, and lift +its indicator to a bounded invariant function \(f\) on \(SM\). +By \Cref{prop:rigidity}, each coefficient has zero flat gradient. +In every regular chart it therefore has a constant representative, +and these constants match under the unitary transitions. Each +coefficient is a parallel section. + +Continuation around the vertex of angle \(\alpha\) has holonomy +rotation by \(-2\alpha\) modulo \(2\pi\), as follows by developing +its punctured cone of angle \(2\alpha\). A parallel coefficient of +mode \(j\) must consequently be fixed by the phase +\(e^{2ij\alpha}\). If \(j\ne0\), irrationality of \(\alpha/\pi\) +makes that phase different from one, so the coefficient vanishes. +The zeroth coefficient is constant on the connected surface \(M\). +Fourier completeness shows that \(f\) is constant almost everywhere. +As an indicator with the same mass as \(A\), it has integral zero +or one. Thus \(\mu(A)\in\{0,1\}\). +\end{proof} + +\begin{remark} +The precise angle condition is visible in the last step. All the analytic +arguments apply to the double of a fixed nondegenerate triangle. A +single irrational angle is enough to eliminate every nonzero parallel +mode, which is why \Cref{thm:main} includes triangles with one rational +angle as well as triangles with three irrational angles. Since the +three angles sum to \(\pi\), exactly one irrational angle and two +rational angles cannot occur. The constants in the analytic estimates +are allowed to depend on the triangle. +\end{remark} + +\section{Quantum ergodicity and nodal domains}\label{sec:spectral} +The full-flow theorem has spectral consequences for each fixed triangle +in its scope. The following statement keeps the boundary-condition +dependence of the Cauchy data and their limiting measure explicit. + +\begin{corollary}\label{cor:quantum} +Let \(Q\) satisfy the hypotheses of Theorem~\ref{thm:main}, and fix +either Dirichlet (\(B=D\)) or Neumann (\(B=N\)) boundary conditions. +Let \((u_j)_{j\ge1}\) be any complete orthonormal eigenbasis of +\(L^2(Q)\) for the corresponding Laplacian, with +\[ + -\Delta_B u_j=\lambda_j^2u_j,\qquad + 0\le\lambda_1\le\lambda_2\le\cdots, +\] +where frequencies are listed with multiplicity. A set +\(S\subset\mathbb N\) has density one if +\(\#(S\cap\{1,\ldots,N\})/N\to1\). +Then the following conclusions hold. +\begin{enumerate} +\item There is a density-one set \(S_{\mathrm{int}}\) such that, for +every classical order-zero pseudodifferential operator \(A\) whose +Schwartz kernel is compactly supported in \(Q\times Q\), +\[ + \lim_{\substack{j\to\infty\\j\in S_{\mathrm{int}}}} + \langle Au_j,u_j\rangle_{L^2(Q)} + =\int_{Q\times S^1}\sigma_0(A)\dd\mu. +\] +Here the Euclidean metric identifies unit covectors with \(Q\times S^1\), +and \(\sigma_0(A)\) is restricted to them. +\item Let \(\Gamma\) be the regular boundary, the union of the three open sides. +For \(\lambda_j>0\), let \(h_j=\lambda_j^{-1}\) and define the Cauchy +data on the open sides by +\[ + u_j^b=\begin{cases} + h_j\,\partial_\nu u_j|_\Gamma,&B=D,\\ + u_j|_\Gamma,&B=N, + \end{cases} +\] +where \(\nu\) is the outward unit normal. Write +\(B^*\Gamma=\{(s,\eta)\in T^*\Gamma:|\eta|<1\}\), using arc length +\(s\) and signed tangential covector \(\eta\) on each side. There is +a density-one set \(S_\partial\subset\{j:\lambda_j>0\}\) such that, +for every \(a\in C_c^\infty(T^*\Gamma)\) and a properly supported order-zero +semiclassical quantization \(A_h=\operatorname{Op}_h(a)\), +\[ + \lim_{\substack{j\to\infty\\j\in S_\partial}} + \langle A_{h_j}u_j^b,u_j^b\rangle_{L^2(\Gamma)} + =\frac{2}{\pi\operatorname{Area}(Q)} + \int_{B^*\Gamma}a(s,\eta)w_B(\eta)\dd s\,d\eta, +\] +where +\[ + w_D(\eta)=\sqrt{1-\eta^2},\qquad + w_N(\eta)=(1-\eta^2)^{-1/2}. +\] +The Cauchy data are not further normalized to have unit boundary norm. +\item If the eigenbasis is real-valued, there is a density-one set +\(S_{\mathrm{nod}}\) such that the number of nodal domains of \(u_j\), +the connected components of \(Q\setminus\{u_j=0\}\), tends to infinity +as \(\lambda_j\to\infty\) with \(j\in S_{\mathrm{nod}}\). +\end{enumerate} +\end{corollary} + +\begin{proof} +Place \(\overline Q\) in a sufficiently large flat torus. Its boundary +is Lipschitz, with three smooth open sides and a finite codimension-two +singular set; incident sides meet transversely. Thus the triangle meets +the piecewise-smooth hypotheses in \cite[Section~2]{Burq}, +\cite[Definition~3.1]{HZ}, and \cite[Definition~1.16]{Hezari}, including +the last reference's metric-extension condition. +Theorem~\ref{thm:main} supplies the required billiard-flow ergodicity. +The first conclusion is the interior quantum-ergodicity theorem of +Zelditch and Zworski \cite{ZZ}, in the Dirichlet and Neumann formulation +recalled in \cite[Theorem~2 and Remark~2.1]{Burq}. + +For the boundary conclusion, discard the null sets of vertex hits and +tangential collisions. The collision section is \(B^*\Gamma\), with +invariant measure \(ds\,d\eta\). Boundary-flight coordinates identify +the flow, up to null sets, with the suspension of its collision map +under the positive free-flight time \(\tau\), and Liouville measure is +proportional to \(ds\,d\eta\,dt\) for \(0\le t<\tau\). +The roof \(\tau\) is positive almost everywhere and bounded by the +diameter of \(Q\). Hence an invariant collision set of positive measure +with positive-measure complement would suspend to a nontrivial invariant +flow set, contradicting Theorem~\ref{thm:main}. The collision map is +therefore ergodic. Hassell--Zelditch's boundary theorem +\cite[Theorem~1.1, equations~(1.3)--(1.5), and Section~7]{HZ} now gives the second +conclusion: its prefactor is +\(4/(\operatorname{vol}(S^1)\operatorname{Area}(Q)) +=2/(\pi\operatorname{Area}(Q))\). +Their inward normal instead of our outward normal changes the Dirichlet +data by a sign and leaves the quadratic pairing unchanged. + +Finally, Hezari's nodal-domain theorem for two-dimensional ergodic +billiards with piecewise-smooth boundary +\cite[Theorem~1.6]{Hezari} gives the third conclusion for real +eigenfunctions. The finitely many zero-frequency terms have no effect on +density. The three density-one sets may be replaced by their finite +intersection when a simultaneous subsequence is desired. +\end{proof} + +The subsequences may depend on the fixed triangle, boundary condition, +and eigenbasis. Zero-density exceptional subsequences remain possible: +these conclusions assert neither quantum unique ergodicity nor an +equidistribution or nodal-count rate, mixing, or a spectral-statistics +law. + + +\begingroup +\raggedright +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/build/source/references.bib b/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/build/source/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..13d8fd0d1ad55c7324462b4ec6636bce9e61f240 --- /dev/null +++ b/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/build/source/references.bib @@ -0,0 +1,164 @@ +@article{ZK, + author = {Zemlyakov, A. N. and Katok, A. B.}, + title = {Topological transitivity of billiards in polygons}, + journal = {Mathematical Notes}, + volume = {18}, + number = {2}, + pages = {760--764}, + year = {1975}, + note = {\href{https://doi.org/10.1007/BF01818045}{doi:10.1007/BF01818045}}, + doi = {10.1007/BF01818045}, + url = {https://www.mathnet.ru/eng/mzm7652} +} + +@article{KMS, + author = {Kerckhoff, Steven and Masur, Howard and Smillie, John}, + title = {Ergodicity of billiard flows and quadratic differentials}, + journal = {Annals of Mathematics}, + series = {2}, + volume = {124}, + number = {2}, + pages = {293--311}, + year = {1986}, + note = {\href{https://doi.org/10.2307/1971280}{doi:10.2307/1971280}}, + doi = {10.2307/1971280}, + url = {https://annals.math.princeton.edu/1986/124-2/p04} +} + +@incollection{MT, + author = {Masur, Howard and Tabachnikov, Serge}, + title = {Rational billiards and flat structures}, + booktitle = {Handbook of Dynamical Systems}, + editor = {Hasselblatt, Boris and Katok, Anatole}, + volume = {1A}, + pages = {1015--1089}, + publisher = {North-Holland}, + address = {Amsterdam}, + year = {2002}, + doi = {10.1016/S1874-575X(02)80015-7}, + url = {https://math.uchicago.edu/~masur/handbook1.pdf}, + note = {Author version dated December 16, 1999, + \url{https://math.uchicago.edu/~masur/handbook1.pdf}. + \href{https://doi.org/10.1016/S1874-575X(02)80015-7}{doi:10.1016/S1874-575X(02)80015-7}} +} + +@article{Vorobets, + author = {Vorobets, Ya. 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year = {2014}, + note = {\href{https://doi.org/10.1103/PhysRevE.89.042918}{doi:10.1103/PhysRevE.89.042918}}, + doi = {10.1103/PhysRevE.89.042918}, + url = {https://doi.org/10.1103/PhysRevE.89.042918}, + eprint = {1309.7617}, + archivePrefix = {arXiv} +} + +@article{ZZ, + author = {Zelditch, Steven and Zworski, Maciej}, + title = {Ergodicity of eigenfunctions for ergodic billiards}, + journal = {Communications in Mathematical Physics}, + volume = {175}, + number = {3}, + pages = {673--682}, + year = {1996}, + note = {\href{https://doi.org/10.1007/BF02099513}{doi:10.1007/BF02099513}}, + doi = {10.1007/BF02099513}, + url = {https://doi.org/10.1007/BF02099513} +} + +@misc{Burq, + author = {Burq, Nicolas}, + title = {Quantum ergodicity of boundary values of eigenfunctions: A control theory approach}, + year = {2003}, + eprint = {math/0301349v1}, + archivePrefix = {arXiv}, + primaryClass = {math.AP}, + url = {https://arxiv.org/abs/math/0301349v1}, + note = {\href{https://arxiv.org/abs/math/0301349v1}{arXiv:math/0301349v1}, January 30, 2003} +} + +@article{HZ, + author = {Hassell, Andrew and Zelditch, Steve}, + title = {Quantum ergodicity of boundary values of eigenfunctions}, + journal = {Communications in Mathematical Physics}, + volume = {248}, + number = {1}, + pages = {119--168}, + year = {2004}, + doi = {10.1007/s00220-004-1070-2}, + eprint = {math/0211140v4}, + archivePrefix = {arXiv}, + primaryClass = {math.SP}, + url = {https://arxiv.org/abs/math/0211140v4}, + note = {\href{https://doi.org/10.1007/s00220-004-1070-2}{doi:10.1007/s00220-004-1070-2}. + Cited numbering follows \href{https://arxiv.org/abs/math/0211140v4}{arXiv:math/0211140v4}, November 24, 2003} +} + +@misc{Hezari, + author = {Hezari, Hamid}, + title = {Quantum ergodicity and {$L^p$} norms of restrictions of eigenfunctions}, + year = {2016}, + eprint = {1606.08066v1}, + archivePrefix = {arXiv}, + primaryClass = {math.AP}, + url = {https://arxiv.org/abs/1606.08066v1}, + note = {\href{https://arxiv.org/abs/1606.08066v1}{arXiv:1606.08066v1}, June 26, 2016} +} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/README.md b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..d318f24a835ca1b2e768a84d67701d2721e8e549 --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/README.md @@ -0,0 +1,18 @@ +# [Essential Singularity of the Correlation Length in the Planar XY Model](paper.pdf) + +**Author:** OpenAI + +**Date:** October 5, 2026 + +## Citation + +~~~bibtex +@misc{OAI:Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026, + author = {{OpenAI}}, + title = {{Essential Singularity of the Correlation Length in the Planar XY Model}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf}{OAI:Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026}}, + year = {2026} +} +~~~ + diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/figures/flow-profile.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/figures/flow-profile.tex new file mode 100644 index 0000000000000000000000000000000000000000..da49e240cdc9575652437e7c96d45c51bebef2de --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/figures/flow-profile.tex @@ -0,0 +1,36 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\small] +\begin{scope}[xshift=0cm,yshift=1.9cm,x=1.42cm,y=.39cm] + \draw[->] (0,-4.2)--(0,4.7) node[above] {$X/k,\ Y/k$}; + \draw[->] (0,0)--(3.42,0) node[right] {$kw$}; + \draw[densely dashed,gray] (3.14159,-4.1)--(3.14159,4.3); + \node[below right] at (3.14159,0) {$\pi$}; + \node[below] at (1.5708,0) {$\pi/2$}; + \draw[blue!65!black,very thick,domain=.26:2.88,samples=100] + plot (\x,{cos(\x*180/pi)/sin(\x*180/pi)}); + \draw[orange!75!black,very thick,domain=.26:2.88,samples=100] + plot (\x,{1/sin(\x*180/pi)}); + \node[blue!65!black,fill=white,inner sep=1pt] at (2.2,-2.8) {$\cot(kw)$}; + \node[orange!75!black,fill=white,inner sep=1pt] at (1.75,2.5) {$\csc(kw)$}; + \draw[->,blue!65!black,thick] (1.17,.43)--(1.34,.235); + \node[align=center] at (1.57,-5.8) {Limiting scalar trajectory}; +\end{scope} +\begin{scope}[xshift=7.2cm,yshift=.3cm,x=1.45cm,y=2.8cm] + \draw[->] (0,0)--(3.4,0) node[right] {$w$}; + \draw[->] (0,0)--(0,1.35) node[above] {$\mathcal A(w)$}; + \draw[densely dashed,gray] (1.85,0)--(1.85,1.15); + \node[below] at (1.85,0) {$T=\pi/k$}; + \draw[blue!65!black,very thick] (.06,1)--(1.85,1); + \draw[blue!65!black,very thick] (1.85,.015)--(3.1,.015); + \draw[blue!65!black,fill=white,thick] (1.85,1) circle[x radius=.045,y radius=.023]; + \draw[blue!65!black,fill=white,thick] (1.85,.015) circle[x radius=.045,y radius=.023]; + \node[left] at (0,1) {$8\pi$}; + \node[above,align=center,font=\footnotesize] at (.87,.09) {Positive pinned\\covariance}; + \node[above,align=center,font=\footnotesize] at (2.66,.35) {Vanishing\\height variance}; + \node[align=center] at (1.55,-.23) {Limiting height coefficient}; +\end{scope} +\end{tikzpicture} +\caption{The two limits that determine the correlation length. Left: the explicit rescaled trajectory, with $k=\sqrt{3\mathcal P}$, has its outgoing pole at $w=T$. Right: the finite-scale free-height coefficient has a step at the same value. No value at the jump is asserted. Positive covariance below $T$ and circulation estimates above $T$ give the two bounds on $\delta\log_L\xi(b)$.} +\label{fig:flow-profile} +\end{figure} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/figures/mass-geometry.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/figures/mass-geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..469a93501afa2664edca2b677e2cc1bef00029b5 --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/figures/mass-geometry.tex @@ -0,0 +1,39 @@ +\begin{figure}[tb] +\centering +\begin{tikzpicture}[>=Latex,font=\small] +\begin{scope}[xshift=2.3cm,yshift=2.3cm] + \fill[gray!18] (-2,-2) rectangle (2,2); + \fill[white] (-1.5,-1.5) rectangle (1.5,1.5); + \draw[gray!70] (-2,-2) rectangle (2,2); + \draw[gray!70] (-1.5,-1.5) rectangle (1.5,1.5); + \node at (0,1.75) {$h=0$}; + \node[anchor=north east] at (1.48,1.48) {$D$}; + \fill[blue!18] (-1.03,-.38) rectangle (-.53,.38); + \fill[orange!22] (.53,-.38) rectangle (1.03,.38); + \node at (-.78,0) {$f$}; + \node at (.78,0) {$g$}; + \draw[blue!65!black,thick] (0,0) ellipse[x radius=1.27cm,y radius=.82cm]; + \draw[->,blue!65!black,thick] (-.36,.786)--(-.1,.818); + \draw[<->] (-.53,-1.13)--(.53,-1.13) node[midway,below] {$c n$}; + \node[align=center] at (0,-2.5) {Pinned covariance below $T$}; +\end{scope} +\begin{scope}[xshift=8.6cm,yshift=2.3cm] + \fill[blue!12,even odd rule] (0,0) circle (1.68) (0,0) circle (1.4); + \fill[blue!12,even odd rule] (0,0) circle (.42) (0,0) circle (.35); + \draw[blue!55!black] (0,0) circle (1.4); + \draw[blue!55!black] (0,0) circle (1.68); + \draw[blue!55!black] (0,0) circle (.35); + \draw[blue!55!black] (0,0) circle (.42); + \fill (0,0) circle (.035) node[below left] {$0$}; + \fill (2.6,0) circle (.035) node[right] {$x$}; + \draw[dashed,gray] (.45,0)--(2.53,0); + \node at (.9,1.05) {$4R$}; + \node[below right,font=\footnotesize] at (.17,-.42) {$R$}; + \draw[->,blue!60!black] (-.33,.86) arc[start angle=111,end angle=145,radius=.92]; + \node[align=center] at (0,-2.08) {$\delta\log_LR_i\in[u,v]\subset(T,w')$}; + \node[align=center] at (0,-2.5) {Circulation tilts above $T$}; +\end{scope} +\end{tikzpicture} +\caption{Two finite geometries for the mass comparison. Left: a primary pin region of positive width surrounds $D$. A loop contributing to the covariance of two separated density smears must have diameter at least their separation. The displayed loop is schematic. Right: geometrically spaced annuli surround one spin insertion and remain inside the separation scale of the other. Only two annuli are drawn. Their logarithmic scales lie in an interval where the height coefficient vanishes, so a fixed circulation tilt has arbitrarily small fluctuation cost on each annulus.} +\label{fig:mass-geometry} +\end{figure} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/main.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..63d6d79574ba28675885b13e95a9d5de502d482f --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/main.tex @@ -0,0 +1,66 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,calc} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,breaklinks=true]{hyperref} +\hypersetup{pdftitle={Essential Singularity of the Correlation Length in the Planar XY Model},pdfauthor={OpenAI},pdfsubject={Near-critical correlation length of the square-lattice XY model}} +\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{assumption}[theorem]{Assumption} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Pbb}{\mathbb P} +\newcommand{\one}{\mathbf 1} +\newcommand{\bc}{b_c} +\newcommand{\eps}{\varepsilon} +\newcommand{\mathscr}[1]{\mathcal{#1}} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Cov}{Cov} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\Lip}{Lip} +\DeclareMathOperator{\diag}{diag} +\DeclareMathOperator{\Tr}{Tr} +\newcommand{\articleid}[1]{\begingroup\small\ttfamily\hyphenchar\font=45\relax #1\endgroup} +\setlist{topsep=5pt,itemsep=3pt,parsep=0pt} +\title{Essential Singularity of the Correlation Length\\in the Planar XY Model} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the Berezinskii--Kosterlitz--Thouless essential singularity for the +correlation length of the nearest-neighbor cosine XY model on the square +lattice. Let \(m(b)\) be the mass obtained by taking first the free-box +thermodynamic limit and then the separation limit along a coordinate axis. +As the inverse temperature \(b\) approaches \(b_c\) from the massive side, +\[ +\sqrt{b_c-b}\log\frac{1}{m(b)} +\longrightarrow A_{\mathrm{XY}}, +\] +where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant. +\end{abstract} +\input{sections/introduction} +\input{sections/finite} +\input{sections/entry} +\input{sections/trajectory} +\input{sections/profile} +\input{sections/mass} +\input{references} +\end{document} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/references.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..07d986853ca7b9946aa5f8d213e1565e18ec200f --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/references.tex @@ -0,0 +1,120 @@ +\begin{thebibliography}{99} +\raggedright + +\bibitem{AHPS} +M.~Aizenman, M.~Harel, R.~Peled, and J.~Shapiro, +Depinning in integer-restricted Gaussian fields and BKT phases of +two-component spin models, +preprint, revised May 27, 2022. +\href{https://arxiv.org/abs/2110.09498v3}{arXiv:2110.09498v3}. + +\bibitem{BPR1} +R.~Bauerschmidt, J.~Park, and P.-F.~Rodriguez, +The Discrete Gaussian model, I. 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Classical systems, +\emph{Soviet Physics JETP} \textbf{32} (1971), 493--500. +\href{https://www.jetp.ras.ru/cgi-bin/dn/e_032_03_0493.pdf}{English translation}. + +\bibitem{DurandLammers} +T.~Durand and P.~Lammers, +The BKT transition and surface tension differentiability, +preprint, May 27, 2026. +\href{https://arxiv.org/abs/2605.28473v1}{arXiv:2605.28473v1}. + +\bibitem{Falco} +P.~Falco, +Critical exponents of the two dimensional Coulomb gas at the +Berezinskii--Kosterlitz--Thouless transition, +preprint, revised November 25, 2013. +\href{https://arxiv.org/abs/1311.2237v2}{arXiv:1311.2237v2}. + +\bibitem{FS} +J.~Fr\"ohlich and T.~Spencer, +The Kosterlitz--Thouless transition in two-dimensional Abelian spin systems +and the Coulomb gas, +\emph{Communications in Mathematical Physics} \textbf{81} (1981), 527--602. +\href{https://doi.org/10.1007/BF01208273}{doi:10.1007/BF01208273}. + +\bibitem{Ginibre} +J.~Ginibre, +General formulation of Griffiths' inequalities, +\emph{Communications in Mathematical Physics} \textbf{16} (1970), 310--328. +\href{https://doi.org/10.1007/BF01646537}{doi:10.1007/BF01646537}. + +\bibitem{Kosterlitz} +J.~M.~Kosterlitz, +The critical properties of the two-dimensional XY model, +\emph{Journal of Physics C: Solid State Physics} \textbf{7} (1974), 1046--1060. +\href{https://doi.org/10.1088/0022-3719/7/6/005}{doi:10.1088/0022-3719/7/6/005}. + +\bibitem{KT} +J.~M.~Kosterlitz and D.~J.~Thouless, +Ordering, metastability and phase transitions in two-dimensional systems, +\emph{Journal of Physics C: Solid State Physics} \textbf{6} (1973), 1181--1203. +\href{https://doi.org/10.1088/0022-3719/6/7/010}{doi:10.1088/0022-3719/6/7/010}. + +\bibitem{Lammers} +P.~Lammers, +Bijecting the BKT transition, +preprint, revised April 18, 2023. +\href{https://arxiv.org/abs/2301.06905v2}{arXiv:2301.06905v2}. + +\bibitem{Lieb} +E.~H.~Lieb, +A refinement of Simon's correlation inequality, +\emph{Communications in Mathematical Physics} \textbf{77} (1980), 127--135. +\href{https://doi.org/10.1007/BF01982712}{doi:10.1007/BF01982712}. + +\bibitem{Lupu} +T.~Lupu, +From loop clusters and random interlacements to the free field, +\emph{The Annals of Probability} \textbf{44} (2016), 2117--2146. +\href{https://doi.org/10.1214/15-AOP1019}{doi:10.1214/15-AOP1019}. +Author's version: +\href{https://arxiv.org/abs/1402.0298v7}{arXiv:1402.0298v7}. + +\bibitem{X} +OpenAI, +The critical correlation exponent of the planar XY model, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf}{OAI:The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026}, +2026. + +\bibitem{F} +OpenAI, +The critical logarithmic correction for the planar XY model, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf}{OAI:The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026}, +2026. + +\bibitem{RSD} +O.~Regev and N.~Stephens-Davidowitz, +An inequality for Gaussians on lattices, +\emph{SIAM Journal on Discrete Mathematics} \textbf{31} (2017), 749--757. +\href{https://doi.org/10.1137/15M1052226}{doi:10.1137/15M1052226}. + +\bibitem{Rivasseau} +V.~Rivasseau, +Lieb's correlation inequality for plane rotors, +\emph{Communications in Mathematical Physics} \textbf{77} (1980), 145--147. +\href{https://doi.org/10.1007/BF01982714}{doi:10.1007/BF01982714}. + +\bibitem{vEL} +D.~van Engelenburg and M.~Lis, +An elementary proof of phase transition in the planar XY model, +\emph{Communications in Mathematical Physics} \textbf{399} (2023), 85--104. +\href{https://doi.org/10.1007/s00220-022-04550-3}{doi:10.1007/s00220-022-04550-3}. + +\end{thebibliography} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/entry.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/entry.tex new file mode 100644 index 0000000000000000000000000000000000000000..af818c8b549064f5e8eb79102ad96a8416ca2b97 --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/entry.tex @@ -0,0 +1,862 @@ +\section{Analytic entry and the activity coordinates}\label{sec:entry} + +We now connect the finite height laws to the small-activity map of +\cite{F}. The map itself acts on analytic functions and is independent +of the temperature of the physical input. The point requiring a new +argument is initialization: the critical construction must remain small +on a parameter disk large enough to differentiate with respect to $b$. +We prove this on a disk of radius proportional to $s^{-2}$, where $s$ is +the observation-cell side. All constants depending on fixed geometric +parameters are chosen before $s$ tends to infinity. + +\subsection{The reference field and the initialization norm} + +The critical free-height coefficient is $a=8\pi$ by +\cite[Section~6]{X}. Fix $\alpha=\sqrt a$ and $\omega=2\pi/\alpha$. +The reference field has covariance $A^+$, so that $\alpha$ times the +reference field has height covariance $aA^+$. We keep this reference, +including $\alpha$, fixed when $b$ varies. + +Partition the primary graph into whole square cells of side $s$, let +$B_s$ average over each cell, and introduce the observations +\begin{equation}\label{eq:observations} + \eta=B_sh+e, +\end{equation} +where the components of $e$ are independent standard Gaussians. On a +retained graph $G$, let $Z_{G,b}(\eta)$ be the height partition sum with +penalty $\exp(-\|B_sh-\eta\|^2/2)$, and define the fixed reference energy +\begin{equation}\label{eq:observation-energy} + E_G(\eta)=\min_{v\in\R^{V(G)}} + \{a^{-1}\|\nabla v\|_G^2+\|B_sv-\eta\|^2\}. +\end{equation} +Each component constant is free in this minimization. For a retained +cell face $f=(Q,Q')$, put $w_f=\eta_{Q'}-\eta_Q$ and use the entire +cutoff +\begin{equation}\label{eq:entry-cutoff} + \chi_s(z)= + \frac{\int_{-s}^s e^{-(z-t)^2/2}\,dt}{\int_{-s}^se^{-t^2/2}\,dt}, + \qquad + \widehat Z_{G,b}(\eta)=Z_{G,b}(\eta)\prod_f\chi_s(w_f). +\end{equation} +A deleted face loses both its primary bonds and its cutoff factor. + +Here are the reference split and norm used in \cite{F}. For an integer +$u\ge1$, set +\[ + P_u(\cos t)=\left(\frac{\sin(ut/2)}{u\sin(t/2)}\right)^{64}, + \qquad \mathsf P_u=P_u(1-A/64), +\] +with removable values understood, and put +\begin{equation}\label{eq:reference-split} + C_{1$, the activity norm is +\begin{equation}\label{eq:entry-activity-norm} + \|K\|_{u,s}=\sup_{X,\phi} + A_w^{|X|}\frac{|K(X)|_{u,\phi}} + {\mathscr W_u^\kappa(X,\phi)e^{V_u(X,\phi)}}. +\end{equation} +Write $\mathscr A^{\mathrm{bulk}}_{u,s}$ for the complex space of +activity families satisfying the support and symmetry conditions above +and having finite norm \eqref{eq:entry-activity-norm}. Every activity +ball and activity variation below is taken in this space or its +compatible-torus counterpart. The ordinary regulator is kept exactly +as in the cited construction; its shell averaging and translation estimates, together with the +explicit observation regulator, are part of the imported local-map +bounds below. Norms with different observation sides $s$ need not be +identified. + +The cutoff gives a strict large-field reserve. Indeed, adjacent-cell +averages have a primary flow representation with coefficient bound +$C/s$ and squared cost $C$. Applying +\eqref{eq:bounded-flow-mgf} with tilts of size at most a fixed multiple +of $s$, then using the Gaussian tails of $\chi_s$, gives uniformly for +$b$ in a compact neighborhood of $\bc$ +\begin{equation}\label{eq:entry-large-field} + e^{E_G(\eta)/2}\frac{\widehat Z_{G,b}(\eta)}{Z_{G,b}(0)} + \le e^{(1-c_*)E_G(\eta)/2},\qquad \eta\in\R^{\mathcal Q(G)}, +\end{equation} +for some fixed $c_*>0$ on every retained topology used in the +construction. More explicitly, transporting each site straight to its +translate in a neighboring cell gives flows $\Phi_f$ with +\[ + \max_e\left|\sum_ft_f\Phi_f(e)\right|\le Cs^{-1}\max_f|t_f|, + \qquad \sum_e\left|\sum_ft_f\Phi_f(e)\right|^2\le C\sum_ft_f^2. +\] +The tilted affine comparison applies also at prescribed observations; +releasing the height and noise equalities gives a bound +$\exp(C\sum_ft_f^2)$. Choose +$t_f$ proportional to $\operatorname{sgn}(w_f)\min(|w_f|,s)$. +For $|w_f|>s$ the cutoff supplies the remaining quadratic decay. +Finally $E_G(\eta)\le C\sum_fw_f^2$, by cell interpolation. +These are the steps in \cite[Lemma~4.4 and Equation~(57)]{F}; the only physical transform +used there is $\exp\{b(\cosh(2\pi t)-1)\}$, whose constants are uniform +on the chosen $b$ interval. + +Choose fixed parameters with strict inequalities +\begin{equation}\label{eq:entry-reserves} + 1-c_*0$, the fixed choices above can be +made so that there exist $c_{R_0}>0$ and $s_0<\infty$ with the following +property. For every admissible $s\ge s_0$, the ordinary plane and +compatible-torus entry activities $K_s(b)$ are Banach analytic on +\begin{equation}\label{eq:entry-parameter-disk} + |b-\bc|0$, depending only on $R_0$, so that its exponent is +at most $-cR_0l(\pi)/2$, after enlarging $R_0$ if needed. Thus the +small pattern bound holds on the reduced complex parameter disk. +For patterns containing only links, both boundary estimates already +have the common link gains. + +The fixed field disk has radius exceeding the polarization radius in +\eqref{eq:entry-point-norm}. Cauchy's formula and polarization +therefore sum all field derivatives with that norm. Connected +summation over the complete occupied covers, with the gains just +proved, is the same absolutely convergent sum as in +\cite[Theorem~4.6]{F}. Its norm is at most $Ce^{-cR_0}$, uniformly in +$s$, the absolute entry index, and compatible volume. The same bound +on compact parameter disks makes the sum Banach analytic. Absorb +$\vartheta$ into $c_{R_0}$ and choose $R_0$ to reach the prescribed +$\rho$. Parameter Cauchy estimates give \eqref{eq:entry-cauchy}. +The occupied covers retain every field dependency, as in the critical +construction. The covariant placement and color rules, reference +Gaussian averaging, and field-negation identities are unchanged when +only the physical weights and centered scalar normalizers depend on +$b$. Thus \eqref{eq:bulk-activity-symmetries} and the prescribed collar +rules hold throughout the parameter disk. The exact finite algebra +and real positive normalizers give the partition representation for +real $b$, and the same local formulas give the copying identities. +\end{proof} + +\subsection{The map at fixed reference} + +The initialization puts the entire activity into the remainder. At a +later side $L^j$, separate two singleton functions on each regular block: +\[ + e_B^0(\phi)=\frac12\sum_{v\in B}\sum_{e=\pm e_1,\pm e_2} + |\nabla_e\phi(v)|^2, + \qquad + c_B^\alpha(\phi)=L^{-2j}\sum_{v\in B}\cos(\alpha\phi_v). +\] +Their coefficients are denoted by $t_j/2$ and $z_j$; the remaining +activity is $\mathcal R_j$. Block covariance makes these coefficients +independent of the block. Field negation excludes the sine singleton, +and square symmetry makes the retained neutral gradient quadratic +isotropic. These are the two scalar directions in the bulk weak map +of \cite[Section~5.1]{F}. The exact map integrates the shell +$\Gamma_{L^j}$, removes designated field-independent factors, and +reexpresses the result in the same subset partition algebra. The +following interface collects the activity estimates used in the rest +of the paper. + +\begin{proposition}[Activity map]\label{prop:map} +For all sufficiently large absolute indices $j$, the map on +$(t,z,\mathcal R)\in\mathbb C^2\times +\mathscr A^{\mathrm{bulk}}_{L^j,s}$, with the occupied-support and +designated-localization conventions of \cite[Proposition~4.1]{F}, +preserves the bulk symmetry class and is analytic on a fixed ball +$|t|+|z|+\|\mathcal R\|_{L^j,s}<\rho_{\mathrm{map}}$. +Its linear part is +\begin{equation}\label{eq:map-triangular} + (t,z,\mathcal R)\longmapsto + (t+P_j^t\mathcal R,\lambda_jz+P_j^z\mathcal R, + \mathsf C_j\mathcal R), + \quad \|\mathsf C_j\|\le\theta_0<1, + \quad \lambda_j=1+O(e^{-cj}), +\end{equation} +with uniformly bounded rows $P_j^t,P_j^z$. The nonlinear part and its +Taylor derivatives have quadratic bounds with uniform constants. + +There are analytic normal coordinates $(u,Z,Q)$, uniform in the entry +side, such that the pure Gaussian curve is $(u,0,0)$ and the map fixes +$u$ on that curve. With $b_*=2\log L$ and a constant $H>0$, define +\begin{equation}\label{eq:balanced-coordinates} + x=b_*u,\qquad y=\sqrt{b_*H}\,Z,\qquad W=(x,y). +\end{equation} +The coordinate changes obey +\begin{equation}\label{eq:coordinate-comparison} + t=u+O(u^2+\|Q\|),\qquad + z=(1+O(e^{-cj}))Z+O(u^2+\|Q\|),\qquad + \mathcal R=O(u^2)+Q, +\end{equation} +with the differentiated analytic bounds of these orders. In a fixed +small tube, +\begin{equation}\label{eq:normal-map} + \begin{split} + W_+&=W+(-y^2,-xy)+\mathcal E_j(W,Q),\\ + |\mathcal E_j(W,Q)| + &\le C\{j^{-2}|W|^2+|W|^3+(|W|+\|Q\|)\|Q\|\},\\ + \|Q_+\|&\le\theta\|Q\|+C|W|^2,\qquad \theta<1. + \end{split} +\end{equation} +The error is an analytic function with the corresponding Taylor +bounds on all fixed derivatives. + +On a separately initialized compatible torus with a fixed range buffer, +the two singleton coefficients copy the plane coefficients through its +stop. The torus remainder has its own contraction and analytic +quadratic bound. Its derivative at zero has no singleton forcing. +All constants are uniform in the stop, the entry side, and their +associated norms. +\end{proposition} + +\begin{proof} +The exact activity map and its uniform norm bounds are +\cite[Proposition~4.1]{F}; the triangular form is +\cite[Equation~(93)]{F}. We recall the coordinate construction to +specify the derivative bounds being used. Put +$\ell_j=\prod_{h\ge j}\lambda_h=1+O(e^{-cj})$ and +$\mathsf C_{h:j}=\mathsf C_{h-1}\cdots\mathsf C_j$. The balanced +linear coordinates are +\[ + T=t+\sum_{h\ge j}P_h^t\mathsf C_{h:j}\mathcal R, + \qquad + Z=\ell_jz+\sum_{h\ge j}\ell_{h+1}P_h^z\mathsf C_{h:j}\mathcal R. +\] +The series converge geometrically. At the chosen entry, initialize an +exact Gaussian interaction $\exp(d\sum_Be_B^0/2)$ and iterate the +same map. Its balanced gradient is $T_j^g(d)=d+O(d^2)$ and its +remainder is $\mathcal R_j^g(d)=O(d^2)$, uniformly on a common complex +disk. Define $u=(T_j^g)^{-1}(T)$ and +$Q=\mathcal R-\mathcal R_j^g(u)$, as in +\cite[Equations~(95), (99), and (101)]{F}. This gives +\eqref{eq:coordinate-comparison}; in particular its linear remainder +rows are bounded, while derivatives of its quadratic scalar terms are +$O(|u|)$. + +The parity estimates of \cite[Lemma~5.2]{F} give leading scalar terms +$-H_jZ^2$ and $-b_juZ$, with no nonconstant term on the Gaussian curve. +Their remainders have the differentiated orders used in +\eqref{eq:normal-map}. Moreover +\cite[Lemma~5.3]{F} gives +$|H_j-H|+|b_j-b_*|\le C(1+j)^{-2}$. +Rescaling by \eqref{eq:balanced-coordinates} yields the displayed +map. Terms with a small coefficient multiplying $Q$ are absorbed into +a slightly larger contraction constant. The copying assertion and the +separate torus contraction are part of the same local-map statement; +no equality between plane and torus remainders is required. +\end{proof} + +The triangular contraction also has a useful elementary consequence. +In equivalent scalar and remainder norms, the derivative of the full +map on a sufficiently small tube has norm at most $1+\gamma$, for any +prescribed $\gamma>0$. To see this, first use the balanced coordinates, +whose scalar linear rows contain no remainder, and choose a product +norm with the fixed remainder contraction. At zero the operator norm +is at most one. Uniform analytic derivative bounds then give the +claim on a small tube. A torus remainder driven by these scalar +coordinates satisfies the corresponding growth bound by convolution. + +\subsection{Finite torus observables and comparison of observation grids} + +Fix a sufficiently large dyadic range buffer $D_0$ and, for an absolute +stop $j$, use the torus of side +\begin{equation}\label{eq:stopping-tori} + n_j=D_0L^j. +\end{equation} +Each such torus is initialized separately at side $qR_0s$. It has +$D_0^2$ blocks at its stop and its singletons copy the plane history. +The physical cutoff initialization requires $n_j\ge sP_0$. +Define the physical first cosine and its noisy-cell approximation by +\begin{equation}\label{eq:physical-cosine} + \begin{split} + H_n(h)&=n^{-2}\sum_{v\in(\Z/n\Z)^2}h_v\cos(2\pi v_1/n),\\ + X_{n,s}&=\langle\eta,f_{n,s}\rangle, + \qquad f_{n,s}(Q)=(s/n)^2\cos(2\pi x_{Q,1}/n), + \end{split} +\end{equation} +where $x_Q$ is the arithmetic center of the primary sites in $Q$. +The origins of all grids agree. Both real tests annihilate constants. +The phase test is $e^{i\bar\eta}$, where $\bar\eta$ is the uniform +mean of the cell observations, interpreted modulo $2\pi$. +Set $\mathfrak g=1/(8\pi^2)$ and let +\[ + v^0_{n,s}=\langle f_{n,s},(I+aB_sA^+B_s^*)f_{n,s}\rangle +\] +be the observation variance in the fixed Gaussian reference. + +\begin{proposition}[Analytic torus observables]\label{prop:readouts} +On a fixed small complex ball of terminal activities, the cutoff +variance correction and the cutoff phase expectation extend to analytic +functions of the activity coordinates. If +$\nu_j=|t_j|+|z_j|+\|\mathcal R^{\rm tor}_{j|j}\|$, they satisfy +\begin{equation}\label{eq:torus-readouts} + \begin{split} + \widehat\Var(X_{n_j,s})-v^0_{n_j,s} + &=\kappa_{n_j,s}t_j+O(\|\mathcal R^{\rm tor}_{j|j}\|+\nu_j^2),\\ + \widehat\E e^{i\bar\eta} + &=p_{j,s}z_j+O(\|\mathcal R^{\rm tor}_{j|j}\|+\nu_j^2). + \end{split} +\end{equation} +All fixed activity derivatives have the corresponding Taylor bounds, +uniformly in entry and stop. The multipliers are positive, bounded +above and below, and +\begin{equation}\label{eq:readout-multipliers} + v^0_{n,s}=a\mathfrak g+O(s/n+n^{-2}),\qquad + \kappa_{n,s}=a\mathfrak g+O(s/n+n^{-2}). +\end{equation} +At a common absolute stop, the multipliers $p_{j,s}$ for different +entries agree except for their mean-observation-noise factor; their +relative difference is $O((s_1/n)^2+(s_2/n)^2)$. +\end{proposition} + +\begin{proof} +These are the terminal analytic functionals of +\cite[Equations~(97)--(98), (116)--(119), and (130)]{F}. +Their analyticity does not require a scaling limit for the physical +input. Indeed, on the fixed $D_0^2$-block grid the Gaussian and common-mean +integral of the terminal subset sum is $1+O(D_0^2\nu_j)$, uniformly +on a fixed complex source disk. +Choose the activity ball so that this error is less than $1/2$. +Both normalized numerator and denominator are then analytic and the +denominator is nonzero. + +Completing the Gaussian source $wX_{n,s}$ before entry averaging gives +the factor $e^{v^0_{n,s}w^2/2}$ and translates the interaction by a +microscopic cosine $F(w)=w\gamma_{n,s}\cos(2\pi v_1/n)$. +Writing $\beta_{n,s}$ for the cell-average cosine factor, +\[ + B_s\cos(2\pi v_1/n)=\beta_{n,s}\cos(2\pi x_{Q,1}/n),\qquad + \gamma_{n,s}=\frac{2v^0_{n,s}}{\alpha\beta_{n,s}},\qquad + \beta_{n,s}=1+O((s/n)^2). +\] +The linear gradient singleton contributes +$t_j\langle F,AF\rangle/2$ to the logarithm; mean integration removes +the first harmonic at linear order. Since +$\langle\cos(2\pi v_1/n),A\cos(2\pi v_1/n)\rangle +=2n^2\sin^2(\pi/n)$, this gives the first line and +\eqref{eq:readout-multipliers}. The remainder costs its norm and all +higher terms cost $O(\nu_j^2)$. + +For the phase insertion the common-mean integral instead selects the +fundamental singleton. Its coefficient is +\[ + p_{j,s}=\frac{D_0^2}{2} + \exp\{\tau_{j,s}/2-aD_{{\rm tail},j}/2\}>0, +\] +where $D_{{\rm tail},j}$ is the diagonal of the retained mean-free +covariance and $\tau_{j,s}$ is the variance, in height units, of the +integrated high mean including the observation noise. Both are bounded +on the fixed terminal geometry. At the same absolute stop the high +covariances telescope to the same constant-mode continuation. The only +entry-dependent contribution to $\tau_{j,s}$ is the noise variance +$(s/n)^2$. This proves the second assertion and the comparison of +multipliers. Cauchy estimates on a smaller activity ball give all +stated derivatives. +\end{proof} + +The next two lemmas permit comparison through these observables without +identifying the activity spaces of distinct observation grids. + +\begin{lemma}[Cutoff error and its derivative]\label{lem:cutoff-derivative} +For real $b$ in the interval of Proposition~\ref{prop:entry}, and +$n\ge sP_0$ with $\log n=o(s^2)$, replacing the original torus +expectation by its normalized cutoff expectation changes the variance +of $X_{n,s}$ and the phase expectation by at most +\begin{equation}\label{eq:cutoff-derivative-error} + Cn^Ce^{-cs^2}. +\end{equation} +The same bound holds for their first $b$ derivatives and for divided +differences between any two real parameters in that interval. +\end{lemma} + +\begin{proof} +The cell-flow proof of \eqref{eq:entry-large-field} also bounds the +probability that a prescribed face difference exceeds $s/2$ by +$Ce^{-cs^2}$, uniformly in the compact $b$ interval. On its complement, +$1-\chi_s$ has that bound. Summing over faces and applying H\"older +with the uniform reference moments gives the unnormalized loss for the +phase and the first two moments of $X_{n,s}$, as in +\cite[Lemma~4.4]{F}. The cutoff mass is consequently $1-o(1)$ after +normalization by the uncut law. + +To differentiate a normalized expectation, insert the Poisson score +$N/b$. Lemma~\ref{lem:factorial} bounds every fixed moment of $N$ by a +fixed power of $n$, uniformly on the parameter interval. Apply +H\"older to the product of this insertion, the observable, and the +cutoff loss. The face tail retains a bound $e^{-c's^2}$, while the +other factors cost a power of $n$. Differentiating the cutoff +normalizer has the same estimate because it is bounded away from zero. +This proves \eqref{eq:cutoff-derivative-error} for derivatives. +Integrating the derivative bound proves the divided-difference version. +\end{proof} + +\begin{lemma}[Observation-grid comparison]\label{lem:grid-comparison} +After dividing the phase expectation by its deterministic mean-noise +factor, it is exactly $\E e^{i\bar h}$, independently of $s$. +After subtracting the independent observation-noise variance and dividing +by $\beta_{n,s}^2$, the variance of $X_{n,s}$ differs from +$\Var H_n$ by $O((s/n)^4)$. Its first $b$ derivative, and its divided +differences over a compact real parameter interval, differ by at most +\begin{equation}\label{eq:grid-derivative-error} + Cn^2(s/n)^4. +\end{equation} +The estimates are uniform for compatible grids with $n/s$ sufficiently +large. Cutoff observables incur the additional error in +\eqref{eq:cutoff-derivative-error}. +\end{lemma} + +\begin{proof} +The spatial mean of $B_sh$ is exactly the microscopic mean. The +independent mean noise has variance $(s/n)^2$, proving the phase +identity. The profile $B_s^*f_{n,s}$ projects onto the microscopic +first cosine with coefficient $\beta_{n,s}$ and has no first sine +component, by reflection around arithmetic cell centers. The remaining +Fourier modes have frequency-index distance at least $cn/s$ from zero. +Their total squared coefficient norm is at most +$Cn^{-2}(s/n)^2$, whereas $A^+$ restricted to these modes has norm at +most $Cs^2$. Their reference cost is therefore $O((s/n)^4)$. +Translation invariance of the physical torus law makes distinct Fourier +modes exactly orthogonal. Thus the variance is the first-cosine +variance plus the alias variance, without a cross term. The reference +bound controls the latter variance. Lemma~\ref{lem:response}, applied +to the alias test, controls its derivative by its reference cost times +$|E|=2n^2$. The factor $\beta_{n,s}^{-2}$ is uniformly bounded and +independent of $b$. Integration gives the divided-difference estimate. +\end{proof} + +\subsection{The critical histories and late entry} + +We finish the interface by recording exactly which critical information +is used later. A history means the plane sequence initialized wholly +in its remainder and iterated with Proposition~\ref{prop:map}, together +with the separately initialized tori at its valid stops. + +\begin{proposition}[Critical histories]\label{prop:critical} +The fixed parameters and entry threshold can be chosen so that there +is a family $\mathcal H^{(k)}$ at $b=\bc$, with admissible +$s_k\asymp2^k$ and entry indices $j_0(k)=O(k)$, having the following +properties. For every fixed $00$ there. Rescaling by +\eqref{eq:balanced-coordinates} gives \eqref{eq:critical-asymptotic}. +The plane remainder bound is part of the same proposition and its odd +cone estimate. For a larger torus, its separate contraction is driven +by the copied scalar sequence. Contracting from an earlier fractional +range suppresses its initial contribution exponentially and sums an +$O(j^{-2})$ forcing, giving the asserted bound at every intermediate +stage. + +For the last assertion, the physical input to critical confinement is +\cite[Lemma~5.4]{F}. It gives the critical torus Gaussian first-cosine +limit and factorization with the mean phase. For any fixed tolerance, +first $P_0$ and then $s$ can be chosen so that the variance and mixed +phase discrepancies are below that tolerance for every compatible +$n\ge sP_0$. No rate of convergence in $n$ is asserted or needed. +The mixed phase functional has a nonzero multiplier on $z$, as shown +in \cite[Equations~(118)--(119)]{F}. The variance multiplier in +Proposition~\ref{prop:readouts} is nonzero as well. + +Suppose there were a first exit from a tube of radius $\rho$. +Before that exit both plane and torus remainder contractions give +\[ + \|\mathcal R_j\|+\|\mathcal R^{\rm tor}_{j|j}\| + \le C\theta^{j-j_0}\rho_{\rm ent}+C\rho^2, +\] +where $\rho_{\rm ent}$ is the entry norm. At the candidate exit the +bounded map leaves all states inside a fixed multiple of the tube, so +the terminal observables are still defined. The variance and mixed +phase identities then bound the two singleton coefficients by +$C(\rho_{\rm ent}+\rho^2+\delta_{\rm phys}+\delta_{\rm cut})$. +Here $\delta_{\rm phys}$ is the uniform critical discrepancy just +described. The number of initial indices with $n_j0$ and a compact set $K$ of complex observation vectors on a +fixed finite list of observation shapes. There are constants $c,C>0$ +such that, for $\eta\in K$ and $|b-b_0|M\le c$, +\begin{equation}\label{eq:finite-tv} + \frac{\|\mu_b^\eta-\mu_{b_0}^\eta\|_{\rm TV}}{Z_{b_0}^0} + \le C|b-b_0|M. +\end{equation} +Here total variation for complex weights means the sum of their absolute +values. The constants can be uniform over any prescribed finite family +of graph topologies and observation shapes. In particular, each +centered normalizer is nonzero on a disk of radius $c/M$ about $b_0$. +\end{lemma} + +\begin{proof} +The parameter likelihood in the augmented variables is exactly +\begin{equation}\label{eq:poisson-likelihood} + \exp\{-(b-b_0)M\}(b/b_0)^N. +\end{equation} +Expand the second factor in falling factorials. For real observations, +Lemma~\ref{lem:factorial} bounds the absolute sum of its positive-order +terms, divided by $Z_{b_0}^0$, by +\[ + \sum_{\ell\ge1}\frac{|b-b_0|^\ell}{\ell!b_0^\ell}(b_0M)^\ell + =e^{|b-b_0|M}-1. +\] +The scalar exponential in \eqref{eq:poisson-likelihood} has the same +bound. For $\eta=\eta_1+i\eta_2$, the absolute observation penalty is +its real-$\eta_1$ penalty times $e^{\|\eta_2\|^2/2}$, uniformly bounded +on $K$. This proves \eqref{eq:finite-tv}. At zero observation it gives +$|Z_b^0/Z_{b_0}^0-1|<1/2$ after decreasing $c$, hence nonvanishing. +All sums are analytic on compact parameter sets by the same expansion. +\end{proof} + +\subsection{A strict response bound} + +The upper bound in Lemma~\ref{lem:reference-cost} alone does not imply +that changing $b$ moves the long-scale height coefficient. The following +inequality supplies the needed strict response using only finite graphs. + +\begin{lemma}[Temperature response]\label{lem:response} +Let $G$ be a finite connected graph with the centered Bessel height law, +and let $f\ne0$ have coefficient sum zero. Set +$X=\langle h,f\rangle$ and $G(f)=\langle f,A_G^+f\rangle$. +For $b$ in a compact interval $I\subset(0,\infty)$, +\begin{equation}\label{eq:temperature-response} + \frac{\Var_b(X)^2}{4\pi^2b^2G(f)} + \le \partial_b\Var_b(X) + \le C_I|E|G(f). +\end{equation} +In particular, these inequalities hold on every zero-winding periodic +torus, with constants independent of its size. +\end{lemma} + +\begin{proof} +Let $C_b$ be the covariance operator on mean-zero tests. For each +oriented edge $e$, removing one positive Poisson unit gives the shifted +height weight +\[ + \prod_{d\in E}p_b\bigl(\nabla_dh/(2\pi)-\one_{\{d=e\}}\bigr). +\] +Let $r_e$ be its partition sum divided by the centered sum, and $v_e$ +its mean of $X$. Reversing $h$ identifies the negative-removal law with +its reflection, so its partition ratio is $r_e$ and its mean is $-v_e$. +By Lemma~\ref{lem:comparisons}, $00$, the free-boundary measure is +\begin{equation}\label{eq:spin-law} + \frac{1}{Z_{R,b}}\exp\!\left(b\sum_{\{u,v\}\in E_R} + \cos(\theta_u-\theta_v)\right) + \prod_{u\in\Lambda_R}\frac{d\theta_u}{2\pi}, + \qquad \theta_u\in\R/(2\pi\Z). +\end{equation} +There are no exterior edges. Write $e_1=(1,0)$ and define +\begin{equation}\label{eq:mass-definition} + C_b(r)=\lim_{R\to\infty}\E^{\mathrm{free}}_{R,b} + \cos(\theta_0-\theta_{re_1}),\qquad + m(b)=\lim_{r\to\infty}-\frac1r\log C_b(r),\qquad + b_c=\inf\{b>0:m(b)=0\}. +\end{equation} +The thermodynamic limit precedes the limit in $r$. We use the standard existence of these limits and the facts $0T). +\end{equation} +Figure~\ref{fig:flow-profile} relates the limiting flow to this height profile. + +Below $T$, positive pinned-height covariance requires loops of macroscopic diameter and hence a correlation length of at least the corresponding exponential scale. Above $T$, the vanishing height coefficient makes fixed circulation tilts arbitrarily inexpensive. A stack of annuli gives arbitrarily strong polynomial decay at those scales; the Lieb--Rivasseau finite-size criterion then gives an upper bound on $\xi(b)$. Both bounds have the same limiting logarithmic scale. In particular, +\begin{equation}\label{eq:amplitude} + A_{\mathrm{XY}}=(\log L)\frac{\pi}{\sqrt{3\mathcal P}}. +\end{equation} +Although the intermediate coordinates depend on the chosen block construction, the left side is determined by \eqref{eq:spin-law} and \eqref{eq:mass-definition}. + +Section~\ref{sec:finite} develops the finite-graph estimates. Section~\ref{sec:entry} states the analytic map inputs and proves the temperature extension. Section~\ref{sec:trajectory} establishes \eqref{eq:intro-response} and the passage time. Section~\ref{sec:profile} identifies the moving-scale height profile, and Section~\ref{sec:mass} completes the transfer to the mass. +\input{figures/flow-profile} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/mass.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/mass.tex new file mode 100644 index 0000000000000000000000000000000000000000..0e377f7c63d769f0cb8f56b113df4714839dce5e --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/mass.tex @@ -0,0 +1,610 @@ +\section{From the height profile to the spin mass}\label{sec:mass} + +The height profile identifies a transition at the logarithmic scale +\(T\). We now show that the correlation length has this same scale. +Positive pinned-height covariance below \(T\) prevents a substantially +shorter correlation length. Above \(T\), small free-height fluctuations +allow a circulation estimate with an arbitrarily large fixed tilt; +a finite-size correlation inequality then gives the opposite bound. +Both arguments concern the free-box thermodynamic correlation used in +Theorem~\ref{thm:main}. + +Throughout this section, fix a sequence \(b_k< b_c\) tending to \(b_c\), +put \(\delta_k=\sqrt{b_c-b_k}\), and pass to a subsequence furnished by +Proposition~\ref{prop:profile-step}. On this subsequence the height +profile satisfies +\begin{equation}\label{eq:mass-profile} + \mathcal A(w)=a\quad(0T),\qquad a=8\pi. +\end{equation} +All finite-graph estimates below are uniform when \(b\) belongs to a +fixed compact interval \(I\subset(0,\infty)\) containing the tail of +\((b_k)\). We will prove +\begin{equation}\label{eq:mass-scale-goal} + \delta_k\log_L\frac1{m(b_k)}\longrightarrow T. +\end{equation} + +\subsection{Correlation inequalities with the prescribed mass} + +For a finite graph \(G\), write +\[ + C_{G,b}(u,v)=\left\langle e^{i(\theta_u-\theta_v)}\right\rangle_{G,b}, + \qquad + C^{(2)}_{G,b}(u,v)= + \left\langle e^{2i(\theta_u-\theta_v)}\right\rangle_{G,b}. +\] +Here every edge has the cosine interaction with parameter \(b\), and +the boundary is free. Reflection of all angles makes these quantities +real, and their Fourier expansions make them nonnegative. We use +\(C_b(x)\) for the infinite-volume free correlation between \(0\) and +\(x\in\Z^2\), so \(C_b(r)=C_b(re_1)\). + +\begin{lemma}\label{lem:mass-correlation-bounds} +For every \(b>0\) and \(x\in\Z^2\), +\begin{equation}\label{eq:mass-pointwise} + C_b(x)\le e^{-m(b)\|x\|_\infty}. +\end{equation} +If \(G\) is a finite subgraph of \(\Z^2\), then +\begin{equation}\label{eq:mass-charge-two} + 0\le C^{(2)}_{G,b}(u,v)\le4C_{G,b}(u,v) + \le4C_b(v-u). +\end{equation} +\end{lemma} + +\begin{proof} +The Ginibre inequalities \cite{Ginibre} imply positive correlation +for cosine characters and monotonicity of two-point correlations +under increasing ferromagnetic edge couplings. In particular, +finite free correlations are bounded above by their free-exhaustion +limits; see also \cite[Lemma~16]{vEL}. These limits are translation +invariant: translating any finite graph gives another finite graph +in the same monotone exhaustion. + +We first record supermultiplicativity. For three vertices \(u,v,z\), +Ginibre's inequality gives +\[ + \left\langle + \cos(\theta_u-\theta_v)\cos(\theta_z-\theta_v) + \right\rangle + \ge C_{G,b}(u,v)C_{G,b}(z,v). +\] +The expectation of +\(\sin(\theta_u-\theta_v)\sin(\theta_z-\theta_v)\) is nonnegative. +To see this, condition on \(\theta_v\), rotate it to zero, and then +condition on all cosine coordinates. Writing a sine coordinate +as its absolute value times its sign, the remaining signs have a +ferromagnetic zero-field Ising distribution: the sign coupling on +an edge \(\{x,y\}\) is +\(b|\sin\theta_x\sin\theta_y|\ge0\). +The sign at \(v\), whose sine is zero, is irrelevant. +Their two-point sign expectations are therefore nonnegative. +The cosine addition formula now proves +\[ + C_{G,b}(u,z)\ge C_{G,b}(u,v)C_{G,b}(v,z). +\] +After free exhaustion this becomes +\(C_b(x+y)\ge C_b(x)C_b(y)\). +For every positive integer \(n\), +\(C_b(jne_1)\ge C_b(ne_1)^j\). Taking \(j\to\infty\) in the +definition of the axis mass yields +\(C_b(ne_1)\le e^{-m(b)n}\). +If \(x=(p,q)\) and \(|p|\ge|q|\), reflection symmetry and +supermultiplicativity give +\[ + C_b(x)^2=C_b(p,q)C_b(p,-q)\le C_b(2p,0) + \le e^{-2m(b)|p|}. +\] +Interchanging coordinates treats the other case and proves +\eqref{eq:mass-pointwise} with the mass specified in the statement +of the main theorem. + +For the charge-two estimate, fix an axis and abbreviate +\(s_x=\sin\theta_x\), \(c_x=\cos\theta_x\). +Rotation invariance gives +\[ + C_{G,b}(u,v)=2\langle s_us_v\rangle_{G,b},\qquad + C^{(2)}_{G,b}(u,v)=8\langle c_uc_vs_us_v\rangle_{G,b}. +\] +The same conditional sign argument gives +\(\langle s_us_v\mid(c_x)_{x\in G}\rangle\ge0\). +Since \(c_uc_v\le1\), the second identity is at most +\(8\langle s_us_v\rangle_{G,b}=4C_{G,b}(u,v)\). +Edge monotonicity finishes \eqref{eq:mass-charge-two}. +\end{proof} + +\subsection{Pinned covariance gives a lower bound on length} + +We use the random-loop representation of the XY current due to +van Engelenburg and Lis \cite[Section~3]{vEL}. We describe its +normalization because the height boundary condition and the loop +orientation are important here. +For each oriented edge of a finite graph \(G\), let +\(n_{uv}\in\Z_{\ge0}\) be a current. A current with zero divergence at +every vertex has weight +\begin{equation}\label{eq:mass-current-weight} + \prod_{\{u,v\}\in E(G)} + \frac{(b/2)^{n_{uv}+n_{vu}}}{n_{uv}!\,n_{vu}!}. +\end{equation} +Equivalently, take independent Poisson variables of mean \(b/2\) +on the oriented edges and condition on zero divergence. +For an unoriented edge put \(N_e=n_{uv}+n_{vu}\), label its +\(N_e\) copies, and assign an orientation to each copy. +Each labeled assignment receives edge weight +\((b/2)^{N_e}/N_e!\). +At a vertex with \(\ell_v\) incoming and \(\ell_v\) outgoing +half-edges, pair them by a uniformly chosen bijection. +The resulting directed loops have weight +\begin{equation}\label{eq:mass-loop-weight} + \prod_{e\in E(G)}\frac{(b/2)^{N_e}}{N_e!} + \prod_{v\in V(G)}\frac1{\ell_v!}. +\end{equation} +Indeed, summing orientations introduces the binomial factor +\(N_e!/(n_{uv}!n_{vu}!)\), and summing pairings cancels the vertex +factor. Thus this is a probability extension of the normalized +current law \eqref{eq:mass-current-weight}. + +Let \(\mathbf P_{G,b}\) denote this loop law. Given the undirected +paired loops, their orientations are independent and fair. +The reason is that \(\ell_v=\deg(v)/2\) is fixed by that paired +geometry, so every choice of loop orientations has the same weight +in \eqref{eq:mass-loop-weight}. For a planar rectangle, current +integration with exterior face height zero gives +\begin{equation}\label{eq:mass-winding-height} + h(z)=2\pi\sum_\gamma W_\gamma(z), +\end{equation} +where \(W_\gamma(z)\) is the oriented winding number of the loop +\(\gamma\) around the bounded face \(z\) +\cite[Proposition~7]{vEL}. The factor \(2\pi\) accounts for our +height spacing. + +We will also use the following consequence of the loop switching +identity \cite[Lemma~8]{vEL}: +\begin{equation}\label{eq:mass-loop-connection} + \mathbf P_{G,b}\{\text{a loop visits both }u\text{ and }v\} + \le2C^{(2)}_{G,b}(u,v). +\end{equation} +In the notation of that lemma, let \(m_{uv}\) be the number of +directed pieces of loops from \(u\) to \(v\) with no internal visit +to either endpoint. A loop visiting both endpoints implies +\(m_{uv}\ge1\), and the identity is +\(C^{(2)}_{G,b}(u,v)=\mathbf E_{G,b}[m_{uv}/(m_{uv}+1)]\). +Thus \eqref{eq:mass-loop-connection} applies to precisely the +normalized loop law above. + +\begin{lemma}[Separated height covariance]\label{lem:mass-loop-covariance} +Fix \(d,M,c_1>0\) and a compact \(I\subset(0,\infty)\). +Let \(G_n\) be a free spin rectangle contained in a square of side +\(c_1n\). On its bounded faces take the dual height law with +exterior face height zero. Let +\[ + F_n=\sum_z f_{n,z}h(z),\qquad + H_n=\sum_z g_{n,z}h(z), +\] +where \(\sum_z|f_{n,z}|,\sum_z|g_{n,z}|\le M\), and the two +supports are separated horizontally by at least \(dn\). +There are constants \(c,C>0\), depending only on the displayed +fixed parameters, such that, for \(b\in I\) and sufficiently +large \(n\), +\begin{equation}\label{eq:mass-covariance-decay} + |\Cov(F_n,H_n)|\le + Cn^{12}e^{-c n m(b)}+Ce^{-c n^4\log n}. +\end{equation} +\end{lemma} + +\begin{proof} +Condition on the undirected paired geometry. Choose one reference +orientation for each loop and write its random orientation as +\(\sigma_\gamma\in\{-1,1\}\). Equation~\eqref{eq:mass-winding-height} +gives +\[ + F_n=\sum_\gamma\sigma_\gamma F_n(\gamma),\qquad + H_n=\sum_\gamma\sigma_\gamma H_n(\gamma), +\] +where +\(F_n(\gamma)=2\pi\sum_zf_{n,z}W_\gamma^{\rm ref}(z)\), +and similarly for \(H_n(\gamma)\). +The conditional means vanish, and the conditional covariance is +\(\sum_\gamma F_n(\gamma)H_n(\gamma)\). +If one summand is nonzero, the loop has nonzero winding around +at least one face in each support. Each such face lies in the +horizontal range of the loop. Consequently this loop has +horizontal diameter at least \(dn-O(1)\). + +Let \(N=\sum_eN_e\) be the total number of traversals, and let +\(\mathcal B_n\) be the event that some loop has horizontal diameter +at least \(dn/2\). A loop of length \(|\gamma|\) has winding +number at most \(|\gamma|\) in absolute value. Hence +\[ + \left|\sum_\gamma F_n(\gamma)H_n(\gamma)\right| + \le (2\pi M)^2\sum_\gamma|\gamma|^2\,\mathbf1_{\mathcal B_n} + \le (2\pi M)^2N^2\mathbf1_{\mathcal B_n}. +\] +There are \(O(n^4)\) ordered pairs of vertices in \(G_n\). +A loop counted by \(\mathcal B_n\) visits a pair separated by at +least \(dn/2\). Using \eqref{eq:mass-loop-connection} and +Lemma~\ref{lem:mass-correlation-bounds}, we obtain +\[ + \mathbf P_{G_n,b}(\mathcal B_n) + \le C n^4 e^{-c n m(b)}. +\] +On \(N\le n^4\), the preceding conditional-covariance bound is +therefore at most \(C n^{12}e^{-c n m(b)}\) after expectation. + +It remains to bound the weighted tail of \(N\). Before the +zero-divergence conditioning, \(N\) is Poisson with mean +\(\mu=b|E(G_n)|\le C_I n^2\). The probability of that conditioning +event is at least the probability of the zero configuration, +namely \(e^{-\mu}\). Thus +\[ + \mathbf E_{G_n,b}[N^2\mathbf1_{\{N>n^4\}}] + \le e^{\mu}\, + \E_{\operatorname{Pois}(\mu)}[N^2\mathbf1_{\{N>n^4\}}] + \le Ce^{-c n^4\log n}. +\] +For the last estimate one may use the Poisson exponential moment +with parameter \(\log n\): its logarithm is +\(\mu(n-1)=O(n^3)\), whereas the tail factor is +\(e^{-n^4\log n}\); the polynomial insertion changes only a +polynomial prefactor. Pairings do not change the marginal law of +\(N\). This proves \eqref{eq:mass-covariance-decay}. +\end{proof} + +We apply the lemma to the nondegenerate pinned limits supplied by +Proposition~\ref{prop:gaussian-profile}. Let \(D=(0,1)^2\), take a +larger fixed rectangle \(U\), and pin every primary height in a +positive-width rim \(p=U\setminus D\), with the usual admissible +rounding at mesh \(n^{-1}\). Choose the unpinned vertices to be +exactly the \(n\)-by-\(n\) grid of bounded faces of the spin rectangle +\(\{0,\ldots,n\}^2\). This is only a translation and an +\(O(n^{-1})\) rounding of the primary mesh. + +The resulting interior height law is exactly the dual law of that +\emph{free} spin rectangle. All pin--pin edge factors cancel from +the normalized law. Every edge from an unpinned face to the pinned +rim becomes an edge to the exterior face; their multiplicities are +retained. In particular a corner face has two such exterior edges. +There are no fixed boundary spins or additional period constraints. +This agrees with free-boundary duality +\cite[Lemma~2.1]{X}. + +Fix smooth nonzero functions \(f,g\ge0\), compactly supported in two +interior boxes in \(D\) with positive horizontal separation, and +put +\[ + F_n=n^{-2}\sum_z f(z/n)h(z),\qquad + H_n=n^{-2}\sum_z g(z/n)h(z). +\] +The harmless translations of face centers are included in the +argument of the sampled functions. If +\(\delta_k\log_L n_k\to w\in(0,T)\), +Proposition~\ref{prop:gaussian-profile} gives +\begin{equation}\label{eq:mass-positive-pinned} + \Cov_{b_k,p}(F_{n_k},H_{n_k}) + \longrightarrow a\int_D\int_Df(z)G_D(z,z')g(z')\,dz\,dz'>0, +\end{equation} +where \(G_D\) is the Dirichlet Green kernel on \(D\). +The strict inequality follows from its positivity in the connected +square. These smears have nonzero total mass. Their convergence +uses the grounded inverse-form part of +Proposition~\ref{prop:gaussian-profile}: the primary zero rim fixes +the constant, and the approximation cost is the grounded cost on +\(D\). No mean-zero condition is imposed on \(f\) or \(g\). + +\begin{proposition}\label{prop:mass-lower-length} +Along the extracted sequence, +\begin{equation}\label{eq:mass-liminf} + \liminf_k\delta_k\log_L\frac1{m(b_k)}\ge T. +\end{equation} +\end{proposition} + +\begin{proof} +If the assertion failed, a further subsequence and numbers +\(00\) would satisfy +\[ + \delta_k\log_L\frac1{m(b_k)}\le w-2\eta. +\] +Take \(n_k=\lfloor L^{w/\delta_k}\rfloor\). Then, for all large +\(k\), +\[ + n_km(b_k)\ge L^{\eta/\delta_k} + \ge n_k^{\eta/(2w)}. +\] +The coefficients of \(F_{n_k}\) and \(H_{n_k}\) have uniformly +bounded absolute sums. Lemma~\ref{lem:mass-loop-covariance} makes +their covariance tend to zero, because an exponential in a positive +power of \(n_k\) dominates the polynomial in +\eqref{eq:mass-covariance-decay}. This contradicts +\eqref{eq:mass-positive-pinned}. +\end{proof} + +\subsection{Zero coefficient on an interval gives strong decay} + +We next use the part of the profile above \(T\). The argument is a +finite-scale version of the circulation proof of +\cite[Proposition~2.6]{X}. Its relevant input is especially simple: +on a compact interval of logarithmic scales with zero coefficient, +the transforms of bounded affine tests on free squares tend +uniformly to one. This is +Corollary~\ref{cor:profile-zero-uniform}; it follows from vanishing +affine variance and the uniform reference exponential moments. +The tilt below is fixed before the limit in \(k\). + +\begin{proposition}\label{prop:mass-polynomial} +Fix \(w'>T\), positive integers \(r_k\) satisfying +\(\delta_k\log_Lr_k\to w'\), and constants \(00\), all sufficiently large \(k\) satisfy +\begin{equation}\label{eq:mass-polynomial} + \sup_{c_0r_k\le\|x\|_\infty\le C_0r_k}C_{b_k}(x) + \le r_k^{-K}. +\end{equation} +\end{proposition} + +\begin{proof} +Choose \(T0. +\end{equation} +For \(R\in\mathcal R_k\), the annuli +\(R<|z|<2R\) about the source at \(0\) are disjoint. +Their maximal radius is \(o(r_k)\); therefore they avoid the other +source \(x\), uniformly over the displacements in the proposition. +Figure~\ref{fig:mass-geometry} compares this annular geometry with +the pinned geometry used for the lower bound. +\input{figures/mass-geometry} + +Choose a nonnegative, nonzero smooth radial function \(\chi\) +supported in \((1,2)\), and let +\[ + p(z)=\chi(|z|)\nabla\arg z, + \qquad + D=\int_{\R^2}p(z)\cdot\nabla\arg z\,dz>0. +\] +This is a smooth divergence-free form, compactly supported in the +unit annulus. Choose its radial stream function \(\psi\), constant +on each complementary component, with \(\nabla^\perp\psi=p\). +At scale \(R\), sample \(\psi_R(z)=\psi(z/R)\) on the faces of +the primary height graph and take its rotated discrete difference. +Denote the resulting edge form by \(\eta_R\), choosing its sign +to agree with the positive source. Thus, on an edge with midpoint +\(z_e\), its corresponding coordinate is +\(R^{-1}p(z_e/R)+O(R^{-2})\). +It has zero discrete divergence, +\begin{equation}\label{eq:mass-annulus-form} + \max_e|\eta_R(e)|\le C/R,\qquad + \sum_e|\eta_R(e)|^2\le C. +\end{equation} +For any affine height difference \(\nabla h+\alpha\) representing +the unit spin twist at \(0,x\), discrete summation by parts gives +\begin{equation}\label{eq:mass-deterministic-circulation} + \sum_e\eta_R(e)(\nabla h+\alpha)_e=D_R, + \qquad D_R\longrightarrow D>0. +\end{equation} +Indeed the gradient part pairs to zero. The remaining pairing is +the difference of the stream-function values at the two source +faces, multiplied by their circulations \(\pm2\pi\). +The stream function is constant near each source, so there is no +dependence on the height configuration or on the path chosen for +\(\alpha\). The convergence in +\eqref{eq:mass-deterministic-circulation} also follows directly by +Riemann sums; its error is uniform here because the second source +lies outside every annulus. Fix \(d_{\rm circ}>0\) so that \(D_R\ge d_{\rm circ}\) +for all sufficiently large radii. + +We make the partition-ratio estimate at fixed \(k,x\) in a finite +free spin box large enough to contain all these annuli in its +interior. Write \(Z_\alpha\) and \(Z_0\) for the twisted and +centered height sums, both with exterior face height zero. +Free-boundary duality gives \(C_{G,b_k}(0,x)=Z_\alpha/Z_0\). +For any fixed \(\lambda>0\), insertion of the deterministic tilt +from \eqref{eq:mass-deterministic-circulation} gives the exact identity +\begin{equation}\label{eq:mass-tilted-ratio} + e^{\lambda\sum_{R\in\mathcal R_k}D_R}C_{G,b_k}(0,x) + =\frac{Z_\alpha\left( + e^{\lambda\sum_R(\eta_R,\nabla h+\alpha)}\right)}{Z_0}. +\end{equation} +Here \(Z_\alpha(F)\) means the unnormalized twisted sum with +insertion \(F\). We will bound the right side independently of +the outer spin box. + +Fix a small relative cell size \(t>0\). Cover the support of each +\(\eta_R\) by square primary cells of side +\(m_R=\lfloor tR\rfloor\), with disjoint vertex sets, leaving +individual edges between cells as seam increments. The support +of \(\chi\) has positive distance from the annular boundary. +Thus for sufficiently small fixed \(t\), every cell meeting this +support lies inside the annulus, avoids both sources, and is +simply connected. There are \(O(t^{-2})\) such cells at each +radius. Edges not in a retained cell and carrying a nonzero tilt +number \(O_t(R)\). + +Release the endpoint constraints between these cells and the seam +increments, and release all remaining constraints outside the +cells. The tilted affine-to-centered comparison +\cite[Lemma~2.2(ii)]{X} bounds the normalized expression in +\eqref{eq:mass-tilted-ratio} by the product of the cell transforms +and the independent whole-edge seam transforms. Untilted factors +on the rest of the graph have affine-to-centered ratio at most +one. On a retained cell the connection is an integral gradient, +\(\alpha=\nabla\kappa\); reindexing \(h'=h+\kappa\) removes it +from both the weight and the statistic being tilted. Accordingly +the factor for that cell is the centered free transform +\[ + \E_{Q,b_k}^{\rm free} + \exp\{\lambda(\eta_R,\nabla h)_Q\}. +\] +In particular, this gauge change introduces no additional real +exponential factor. Released constant modes are taken modulo +common translations; the edge-difference tests annihilate every +such constant. Equivalently, one can retain vanishing quadratic +penalties on these constants in both sectors and cancel their +identical factors before removing them. No outer-box normalizer +is present in the resulting bound. + +We now estimate the factors for one annulus. The total squared +coefficient cost of its seam increments is \(O_t(R^{-1})\) by +\eqref{eq:mass-annulus-form}. The Bessel increment transform is +\(\exp\{b_k(\cosh(2\pi s)-1)\}\), so their combined logarithmic +cost is \(O_{t,\lambda}(R^{-1})\), uniformly in \(k\). +On each retained square replace \(\eta_R\) by its value at a +chosen point of the square. Smoothness of \(p\) makes the total +squared cost of the coefficient error over all cells +\(O(t^2)+o_R(1)\). The bounded-flow estimate \eqref{eq:bounded-flow-mgf} and +H\"older's inequality with exponent two therefore give +\begin{align} + &\sum_Q\log\E_{Q,b_k}^{\rm free} + e^{\lambda(\eta_R,\nabla h)_Q} + \notag\\[-2pt] + &\qquad\le + \frac12\sum_Q\log\E_{Q,b_k}^{\rm free} + e^{2\lambda(\bar\eta_{R,Q},\nabla h)_Q} + +C_\lambda t^2+o_{t,\lambda,R}(1), + \label{eq:mass-affine-cell-reduction} +\end{align} +where \(\bar\eta_{R,Q}\) is the constant coefficient on \(Q\). +The error coefficients multiplied by the fixed tilt are uniformly +bounded, as required by \eqref{eq:bounded-flow-mgf}. That estimate +bounds each error transform by its squared coefficient cost. +Summing the costs gives the total \(C_\lambda t^2\) in +\eqref{eq:mass-affine-cell-reduction}. + +Each constant test in the sum is a linear combination of the two +square affine observations defining \(a_{m_R}(b_k)\). Its +coefficient vector has size at most \(C_\lambda m_R/R\le C_\lambda t\). +Translations of the square do not change its law. Moreover, +\[ + \delta_k\log_Lm_R + =\delta_k\log_LR+\delta_k\log_Lt+o(1). +\] +Thus these scales lie in a slightly enlarged compact subinterval +of \((T,w')\) for all large \(k\). By +Corollary~\ref{cor:profile-zero-uniform}, the transforms of these +affine tests tend uniformly to one, uniformly also in their +bounded coefficient vectors. Since the number of cells is uniformly bounded +once \(t\) is fixed, their total logarithmic cost is \(o_k(1)\) +uniformly over \(R\in\mathcal R_k\). +The reference estimates and \(\min\mathcal R_k\to\infty\) also +make all seam and discretization errors uniform. We obtain, +for every fixed \(\lambda,t\), the per-annulus bound +\begin{equation}\label{eq:mass-annulus-cost} + \log(\text{product of its transforms}) + \le C_\lambda t^2+o_k(1). +\end{equation} + +Choose \(\lambda\) first, and then \(t\) so small that +\(C_\lambda t^2\le\lambda d_{\rm circ}/4\). For all sufficiently large +\(k\), the uniform error in \eqref{eq:mass-annulus-cost} is at most +\(\lambda d_{\rm circ}/4\). Equations~\eqref{eq:mass-tilted-ratio} and +\eqref{eq:mass-annulus-count} imply +\[ + C_{G,b_k}(0,x) + \le\exp\{-\lambda d_{\rm circ}N_k/2\}. +\] +This estimate holds uniformly in every sufficiently large outer +spin box, at the fixed values of \(b_k,r_k,x\). We may therefore +first take the prescribed free-box thermodynamic limit. The +resulting estimate for \(C_{b_k}(x)\) is uniform in the indicated +range of \(x\). Finally choose the fixed \(\lambda\) large enough +that \(\lambda d_{\rm circ}q/4>K\). Equation~\eqref{eq:mass-annulus-count} +then gives \eqref{eq:mass-polynomial}. +\end{proof} + +\subsection{A finite-size criterion and completion of the proof} + +The Lieb--Rivasseau inequality for plane rotators +\cite{Lieb,Rivasseau}, in the form stated in +\cite[Lemma~17]{vEL}, says that for a finite subgraph \(H\subset G\), +\(u\in H\), and \(v\notin H\), +\begin{equation}\label{eq:mass-lieb-rivasseau} + C_{G,b}(u,v)\le\sum_{z\in\partial H} + C_{H,b}(u,z)C_{G,b}(z,v). +\end{equation} +Here \(\partial H\) is the set of vertices of \(H\) adjacent to a +vertex outside \(H\); the first factor is the correlation with +free boundary on \(H\). We give the iteration so its conclusion +refers to exactly the axis mass of this paper. + +\begin{lemma}\label{lem:mass-finite-size} +Let \(H_r=[-r,r]^2\cap\Z^2\), and put +\[ + S_{r,b}=\sum_{z\in\partial H_r}C_{H_r,b}(0,z). +\] +If \(S_{r,b}\le1/2\), then \(m(b)\ge(\log2)/r\). +\end{lemma} + +\begin{proof} +At fixed \(b,r\), take the free-box thermodynamic limit in +\eqref{eq:mass-lieb-rivasseau}. For a target \(se_1\), apply the +inequality successively in translates of \(H_r\) centered at the +boundary point reached in the previous step. Each step moves the +center by at most \(r\) in the supremum norm. Hence at least +\(\lfloor s/r\rfloor-1\) iterations are possible before the target +can belong to the next translated square. Translation invariance +and \(C_b\le1\) give +\[ + C_b(se_1)\le + S_{r,b}^{\lfloor s/r\rfloor-1}. +\] +Taking \(-s^{-1}\log\) and then \(s\to\infty\) proves the claim. +\end{proof} + +\begin{proposition}\label{prop:mass-upper-length} +Along the extracted sequence, +\begin{equation}\label{eq:mass-limsup} + \limsup_k\delta_k\log_L\frac1{m(b_k)}\le T. +\end{equation} +\end{proposition} + +\begin{proof} +Fix \(w'>T\), and set \(r_k=\lfloor L^{w'/\delta_k}\rfloor\). +Proposition~\ref{prop:mass-polynomial}, with any fixed \(K>1\), +and finite-to-infinite monotonicity imply +\[ + S_{r_k,b_k}\le\sum_{z\in\partial H_{r_k}}C_{b_k}(z) + \le8r_k\,r_k^{-K}<\frac12 +\] +for all sufficiently large \(k\). Thus +Lemma~\ref{lem:mass-finite-size} yields +\(m(b_k)\ge(\log2)/r_k\). Consequently +\[ + \limsup_k\delta_k\log_L\frac1{m(b_k)} + \le\lim_k\delta_k\log_L\frac{r_k}{\log2}=w'. +\] +Letting \(w'\downarrow T\) proves \eqref{eq:mass-limsup}. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Start with any sequence \(b_k0$ of $\mathcal A$, every integer sequence $m_k$ satisfying +$\delta_k\log_L m_k\to w$ obeys +\begin{equation}\label{eq:profile-limit} + a_{m_k}(b_k)\longrightarrow\mathcal A(w). +\end{equation} +One may choose $\mathcal A$ to be right continuous. +\end{proposition} + +\begin{proof} +For positive rational $q$, set $m_k(q)=\lfloor L^{q/\delta_k}\rfloor$. +A diagonal subsequence gives a limit of $a_{m_k(q)}(b_k)$ for every +such $q$. If $qw,\ q\in\mathbb Q}\alpha(q). +\] +This is nonincreasing and right continuous. Values at rational jump +points need not be preserved; only continuity points will be used. + +For $q_-Y$ are nonnegative because +the centered second differences of $F$ decrease with their center; +the two outer-face coefficients have positive leading terms. +Exceptional diagonal-strip coefficients again have total transport +cost $O(n_k^{-1})$. We obtain saturation of +\begin{equation}\label{eq:profile-fold} + (F(x_1)-F(x_2+t))\one_{\{x_1>x_2+t\}}. +\end{equation} +Every smooth $F$ is a difference of two functions with those strict +properties, by adding a sufficiently large cubic of negative third +derivative and then a sufficiently large positive linear function. +Thus \eqref{eq:profile-fold} holds for arbitrary smooth $F$. +Rounding a reflection line changes the squared gradient cost by +$O(n_k^{-1})$. + +Given smooth $g$, choose a bounded interval $I$ containing all +$x_1-x_2$ in the target rectangle. Choose a smooth family $F_t$, +compactly supported in $I$ after slightly enlarging $I$, with +$F_t'(x)=-g_{12}(x,x-t)$ whenever $(x,x-t)$ lies in that rectangle. +Integrate \eqref{eq:profile-fold} over $t\in I$. Indeed, differentiating first in +$x_1$ and then in $x_2$ gives the distributional identity +\[ + \partial_{12}\int_I + (F_t(x_1)-F_t(x_2+t))\one_{\{x_1>x_2+t\}}\,dt + =-F'_{x_1-x_2}(x_1)=g_{12}(x_1,x_2). +\] +The difference from $g$ is a sum of one-variable functions. Riemann +sums converge in gradient energy, including across the moving +reflection lines. Linearity and energy approximation therefore prove +saturation for every smooth $g$. + +The finite centered comparison gives +$\log\E e^X\ge\Var(X)/2$, providing the Laplace lower bound. +For the upper bound, cut into cells of fixed macroscopic side $r$. +A retained smooth-gradient test $X$ on one cell has reference-cost +square root $O(r)$. The normalized exponential moment estimate gives, +for each fixed $p>1$, +\[ + \log\E e^{pX}=\tfrac12p^2\Var(X)+O_p(r^3), + \qquad \E(|X|^3e^{p|X|})=O_p(r^3), +\] +with constants uniform in $k$. H\"older's inequality and the same +reference bound remove the seam errors. There are $O(r^{-2})$ cells. +First send $k\to\infty$, then $r\downarrow0$, and finally +$p\downarrow1$. This proves \eqref{eq:profile-free-gradient}. +Apply it to real linear combinations to obtain joint Gaussian limits; +the exponential bounds supply convergence of moments. + +For densities, approximate by finite sums of Neumann eigenfunctions +and use their smooth inverse-Laplacian shifts. The discrete Neumann +form approximation and Poincar\'e inequality bound the reference cost +of the error by its squared scaled $L^2$ norm. Using the actual +rectangle endpoints before passing to their limits handles the +rounding. Finally move a face probability to a parallel strip of +width $r$. Parallel normal flows cost $O(r+n_k^{-1})$; the strip is a +density smear. Let $k\to\infty$ and then $r\downarrow0$. +\end{proof} + +\subsection{Fractional means} + +For a free graph and a profile $s$ of integer total, write +$\phi(s)=\E e^{i(h,s)}$. It is independent of the representative of +$h$ modulo a common $2\pi$ translation. Finite comparison gives +$0<\phi(s)\le1$. If $v$ has zero total, the logarithmic Hessian bound +and the fact that $\log\phi\le0$ imply +\begin{equation}\label{eq:profile-phase-transfer} + \left|\sqrt{-\log\phi(s+v)}-\sqrt{-\log\phi(s)}\right| + \le\sqrt{\Var(h,v)/2}. +\end{equation} +Indeed, for all real $t$ let $F(t)=\log\phi(s+tv)$ and +$V=\Var(h,v)$. The inequalities $F''\ge-V$ and $F\le0$ give +$|F'|^2\le-2VF$ by optimizing the quadratic lower bound for $F$; +integration gives \eqref{eq:profile-phase-transfer}, as in +\cite[Equation~(2.2)]{X}. + +\begin{lemma}[Uniform fractional means]\label{lem:profile-phase} +Under the hypotheses of Lemma~\ref{lem:profile-free}, suppose $a_*>0$. +The fractional mean modulo $2\pi$ converges to uniform measure, +independently of the Gaussian observations in that lemma. For every +fixed nonzero integer $l$ and every zero-total profile $v_k$ of bounded +reference cost, +\begin{equation}\label{eq:profile-phase-zero} + \E\exp\{i(h,lu_{U_k}+v_k)\}\longrightarrow0, +\end{equation} +where $u_{U_k}$ is uniform probability on $U_k$. +\end{lemma} + +\begin{proof} +We describe the soft joining and replica argument of +\cite[Theorem~2.5, Equations~(2.10)--(2.13)]{X}, keeping +its lattice and geometric limits separate. Start with a square of side +$m=m_k$ on the chosen logarithmic scale. Let $u_m$ be its volume +probability, $\rho_m$ its right-face probability, and set +\[ + X=(h,A_{B_m}(x_1/m)),\qquad Y=(h,\rho_m-u_m). +\] +Lemma~\ref{lem:profile-free} gives joint Gaussian limits, with +$\Var X\to a_*$ and $\Cov(X,Y)\to a_*/2$. +Suppose along a subsequence that $\phi(lu_m)$ is bounded away from +zero for a nonzero integer $l$. Uniform exponential bounds provide a +further subsequence on which +\[ + F_m(t,v)=\E e^{i(h,lu_m)+itX+ivY} +\] +converges locally with all derivatives. On real arguments the limit +$F$ is strictly positive by \eqref{eq:profile-phase-transfer}. + +The replica identity is a finite quadratic-lattice identity. Let +$F_{m,N}$ denote the preceding transform with each primary edge +replaced by $N$ Gaussian chain bonds and the observations evaluated +at the primary endpoints. For two independent fields under this +subdivided law put $S=h_1+h_2$, $D=h_1-h_2$. +Choose both representatives to vanish at the same reference vertex. +At every vertex of the subdivided graph, $S\equiv D$ modulo +$4\pi\Z$. Conditional on this full common parity class, $S$ and $D$ +are independent and identically distributed. If $V_{ij}$ is the conditional second moment +of $S_i,S_j$, for linear observations indexed by $i,j$, then +\begin{equation}\label{eq:profile-replica} + \E V_{ij}=2\Cov(h_i,h_j),\qquad + \Cov(V_{ij},V_{rs})=2\kappa(h_i,h_j,h_r,h_s), +\end{equation} +where all moments in this identity are in the subdivided law and +$\kappa$ is its fourth joint cumulant. This follows by expanding +$\E S_iS_jD_rD_s$ in the original two copies. Differentiating +$F_{m,N}$ in its observation variables gives +\[ + \E S_iS_j e^{i(l(h_1-h_2,u_m)+tX(D)+vY(D))} + =-2(F_{m,N}F_{m,N,ij}-F_{m,N,i}F_{m,N,j}). +\] +In the conditional expectation on the left, replace $V_{ij}$ by its +mean. The error is at most $\sqrt{\Var V_{ij}}$. First take the chain +limit on each fixed primary graph. Then the Gaussian moment limits +in Lemma~\ref{lem:profile-free} make the needed fourth cumulants zero. +For $ij=XX,XY$ this yields +\[ + (\log F)_{tt}=-a_*,\qquad (\log F)_{tv}=-a_*/2. +\] +Reflection gives $F_t(0,0)=0$. Consequently +\begin{equation}\label{eq:profile-phase-transform} + F(t,v)=F(0,v)\exp\{-a_*t^2/2-a_*tv/2\}. +\end{equation} + +Softly join two horizontal squares as follows. Keep independent +seam increments with the Bessel law and put +$H_2=(h_1,\rho_R)-(h_2,\rho_L)-\bar\eta$, +where $\bar\eta$ is their mean with the joining orientation. +Penalize this mismatch by $e^{-H_2^2/2}$. Sum relative cell translations; quotient by their +common translation. Adding exact seam matching recovers the joined +rectangle. The soft variance of the sum of the internal affine tests +is bounded above by twice the free square variance and below by its +variance in the joined rectangle. The joined full affine test differs +from this sum by a seam flow of squared cost $O(m^{-1})$. +Lemma~\ref{lem:profile-free} therefore makes the soft variance tend to +$2a_*$. + +The comparisons just used with summed translations can be obtained +from ordinary finite comparisons without reweighting the original +free cells. First confine each cell mean by an auxiliary precision +$\zeta\sum_i\bar h_i^2$. For each representative mean $\mu$, +its translation sum is +$\sum_{q\in\Z}e^{-\zeta(\mu+2\pi q)^2/2}$. +After division by its Gaussian integral scalar, this tends uniformly +to one as $\zeta\downarrow0$. After adding the soft penalty, divide +out the common-translation scalar; the relative-translation Gaussian +sum is uniformly bounded in its representative-dependent shift. +The finite precision comparisons at $\zeta>0$ therefore pass to +$\zeta=0$ by dominated convergence, using the uniform internal +moments. This justifies both variance comparisons with the original +free-cell product law. + +Poisson summation of the relative translations expresses the soft +characteristic transform as a positive sum of integer sectors, with +weights $e^{-q^2/2}\chi_m(q)$; here $0<\chi_m(q)\le1$ is the +independent seam characteristic. Explicitly, +$\chi_m(q)=\exp\{mb(\cos(2\pi q/m)-1)\}$, so +$\chi_m(q)\to1$ for fixed $q$, uniformly in the temperatures here. +To specify the sector functions, write +$F_m^{(q)}(t,v)=\E e^{iq(h,u_m)+itX+ivY}$. If $\rho_R,\rho_L$ +are the two vertical face probabilities, reflection and charge +reversal give +\[ + \psi_q(t)=\phi(q\rho_R+tA(x_1/m)) + \phi(-q\rho_L+tA(x_1/m)) + =F_m^{(q)}(t,q)^2. +\] +Each sector satisfies $(\log\psi_q)''\ge-2\Var X$. The logarithmic Hessian of a positive +sum is the mean of the sector logarithmic Hessians plus the variance +of the sector logarithmic first derivatives. Thus the soft variance +is at most twice the free variance minus this latter, nonnegative +variance. Differentiation under the sector sum is justified uniformly: +with $V=2\Var X$, +\[ + \psi_q| (\log\psi_q)'|^2 + \le2V(-\psi_q\log\psi_q)\le2V/e, +\] +and raw second derivatives are bounded by $V$. The Gaussian weights +are summable. + +At $t=0$ the normalizing constant is +$Z_m=\sum_qe^{-q^2/2}\chi_m(q)\psi_q(0)$, and +$1\le Z_m\le\sum_qe^{-q^2/2}$ because the zero sector is one. +Its normalized weight $p_0$ is therefore bounded below and its +logarithmic derivative is zero. The sector $q=l$ also has normalized +weight $p_l$ bounded below: \eqref{eq:profile-phase-transfer} moves $lu_m$ to a +seam face at bounded cost. By reflection, charge reversal and +\eqref{eq:profile-phase-transform}, its logarithmic derivative tends +to $-a_*l$, up to orientation. If $D_q$ denotes the sector +logarithmic derivative, their contribution bounds the variance of +derivatives below by $p_0p_l(D_l-D_0)^2$. Its liminf is strictly +positive, contradicting convergence of the +soft variance to $2a_*$. Hence $\phi(lu_m)\to0$. +Equation~\eqref{eq:profile-phase-transfer} then proves +\eqref{eq:profile-phase-zero} for square profiles of bounded cost. + +For a fixed rectangle, enclose it in a fixed-multiple square. Regard +its mean phase as a phase on the square. The difference from the +square mean has bounded reference cost, so its square coefficient +tends to zero. Cutting the complement decreases characteristic +coefficients and leaves the rectangle phase on one free component; +hence the rectangle coefficient also tends to zero. Bounded-cost +phase transfers inside the rectangle give the stated generality. +Mixed real characteristic functions with the mean-free observations +now vanish whenever $l\ne0$, and for $l=0$ they have the Gaussian +limit. Fourier approximation on the circle proves joint independence. +Uniform exponential moments permit passage with any fixed real tilt +and polynomial insertion in the mean-free observations. +\end{proof} + +\subsection{Primary pins at a moving parameter} + +We next transfer the free limits to a hard pin. This requires more +than convergence of finitely many unconditioned observations: a pin +specifies every primary height in a region. We use the band and cable +argument of \cite[Section~3]{X}, stating its finite estimates explicitly +and verifying the limits with the parameter varying along the sequence. +At a positive continuity value of the coefficient profile, the free +and phase assumptions below are supplied by +Lemmas~\ref{lem:profile-free} and~\ref{lem:profile-phase}. + +\begin{lemma}[Moving-parameter pinned limits]\label{lem:profile-pins} +Let \(b_k\) lie in a compact interval \(J\subset(0,\infty)\), and let +\(n_k\to\infty\). Suppose that, on every fixed nondegenerate rectangle +with mesh \(n_k^{-1}\), the free Bessel height laws at \(b_k\) have +Gaussian Laplace and moment limits for mean-free density smears and +energy-approximable weak Laplacian tests, with coefficient \(a_*>0\). +Suppose also that the fractional mean modulo \(2\pi\) converges jointly +with those tests to an independent uniform variable. + +Let \(U\) be a fixed rectangle and let \(p\subset U\) be a nonempty +finite union of aligned positive-area boxes. Assume that the resulting +pieces have nondegenerate faces and positive-width passages; bounded +lattice rounding is allowed. Pin every primary vertex of \(p\) to zero. +Define +\[ + \mathcal H_p=\{v\in H^1(U):v=0\text{ almost everywhere on }p\}, + \qquad \mathcal E_U(v,w)=\int_U\nabla v\cdot\nabla w, +\] +and let \(G_p\) be the inverse of this form: for a bounded density \(f\), +\(G_pf\in\mathcal H_p\) satisfies +\(\mathcal E_U(G_pf,v)=\int_Ufv\) for \(v\in\mathcal H_p\). +For every fixed finite collection of bounded piecewise continuous +densities \(f_1,\ldots,f_d\), their primary smears converge jointly, +with all Laplace transforms and polynomial moments, to the centered +Gaussian vector with covariance +\begin{equation}\label{eq:profile-pinned-covariance} + a_*\int_U f_iG_pf_j. +\end{equation} +The densities need not have zero integral. +\end{lemma} + +\begin{proof} +Write \(n=n_k\) and \(b=b_k\) in this proof, and index primary vertices +by \(x\in nU\cap\Z^2\), with the permitted boundary rounding. +For a continuum density \(f\), put \(f_n(x)=n^{-2}f(x/n)\); +equivalent cell quadratures give the same limits. All discrete +pairings use ordinary sums: \((h,f_n)=\sum_xh_xf_n(x)\), whereas +\(g_n(x)=g(x/n)\). Probability averages, including cell averages, +use weights equal to the reciprocal of their full primary counts. +The unit-edge form is denoted by \(A_{U,n}\); its energy tends to +\(\mathcal E_U\). All unspecified finite-graph constants below are +uniform for \(b\in J\). We first prove the assertion for +\[ + X_{g,n}=(h,A_{U,n}g_n), +\] +where \(g\) is smooth on the rectangle, may meet its free walls, and +vanishes in a neighborhood of \(p\). The final approximation will use +the form grounded on \(p\) to reach arbitrary densities. + +Fix a closed neighborhood \(V\) of \(\supp g\), separated from \(p\), +so that the primary support of \(A_{U,n}g_n\) lies in \(V\) for all +sufficiently large \(k\). Choose a positive-width band \(K\) +separating \(p\) from \(V\), with positive gaps on both sides. +Such a band can be chosen as a finite union of rectangular strips, clipped at the free +walls. Partition it into rectangular cells \(d\) of side comparable +to \(\ell\), with uniformly bounded aspect ratios. If \(h_d\) is the +probability average in \(d\), set +\begin{equation}\label{eq:profile-band} + W_{\ell,\lambda}(h) + =\exp\left\{-\frac\lambda2\sum_d|d|h_d^2\right\}, + \qquad + \widehat W_{n,\ell,\lambda} + =\frac{W_{\ell,\lambda}}{\mathcal P_n(W_{\ell,\lambda})}. +\end{equation} +Here \(\mathcal P_n\) is the sigma-finite free measure obtained by +normalizing the law modulo common \(2\pi\Z\)-translations and then +summing all translations. The band confines its constant, so the +denominator in \eqref{eq:profile-band} is finite and positive. We write +\(\E_W\) for integration against \(\widehat W\,d\mathcal P_n\). +Throughout the proof the order is +\begin{equation}\label{eq:profile-band-order} + k\to\infty,\qquad \ell\downarrow0, + \qquad \lambda\to\infty. +\end{equation} +In particular, every band partition and strength is fixed before the +moving-parameter mesh limit. + +We will prove that these successive band limits give variance +\(a_*\mathcal E_U(g,g)\) for \(X_{g,n}\), and that this value is +no larger than \(\liminf_k\Var_pX_{g,n}\). The free variance upper +bound will then give equality. The comparison is the main task. +The finite exploration argument below controls the linear tilts +introduced by the band and proves that exploration from \(p\) reaches +\(V\) with vanishing probability. When it does not reach \(V\), +the exposed test is zero and the centered posterior variance is at +most the hard-pin variance; the bounded band fourth moment controls +the contribution when it does reach \(V\). We finish by squeezing +Laplace transforms and using the grounded form to approximate density +smears. + +\emph{Band limits.} +At a fixed partition and strength, the assumed joint free limits turn +translation summation into integration of the constant against +\(dc/(2\pi)\). This passage also holds with every fixed polynomial +insertion. Indeed, if \(S\) is the probability average over \(U\) and +\(\bar h_K=|K|^{-1}\sum_d|d|h_d\), then +\[ + W_{\ell,\lambda}\le + e^{-\lambda|K|\bar h_K^2/2}, + \qquad S-\bar h_K\text{ has bounded primary reference cost}. +\] +The reference-cost moment and fixed-linear-exponential bounds, uniform +on \(J\), control this difference; the displayed Gaussian factor +controls the sum over translations. They give uniform integrability +of the normalizing constants and polynomial insertions. The limiting +law is therefore the Gaussian with precision +\(a_*^{-1}\mathcal E_U+\lambda\sum_d|d|v_d^2\). + +The subsequent cell refinement is a statement about these Gaussian +forms. Integrating their constant first leaves the covariance +\(C_\ell\) of the cell projection of the mean-free Neumann field, +after subtracting its \(K\)-average. The inverse Neumann Laplacian +on a rectangle is Hilbert--Schmidt, so +\(\sup_\ell\|C_\ell\|_{\mathrm{HS}}<\infty\). For \(q\ge1\), +if \(t_j\) are the eigenvalues of \(\lambda C_\ell\), the determinant +factor in the normalized \(q\)-th weight moment satisfies +\[ + \frac12\sum_j\{q\log(1+t_j)-\log(1+qt_j)\} + \le C_q\sum_jt_j^2. +\] +The constant integral is a bounded positive factor at fixed \(\lambda\). +Poincar\'e's inequality, with the constant controlled by the band, +also bounds every moment of \(S\) in the resulting Gaussian law. +Consequently, for every fixed \(q\ge1\), \(r\ge0\), and \(\lambda>0\), +\begin{equation}\label{eq:profile-band-density} + \limsup_{\ell\downarrow0}\limsup_{k\to\infty} + \mathcal P_n\bigl(\widehat W^q(1+|S|)^r\bigr)<\infty. +\end{equation} +Strong convergence of cell projections and the ordinary form +approximation give the band covariance +\begin{equation}\label{eq:profile-band-Gaussian} + T_\lambda^{-1},\qquad + T_\lambda=-\Delta_U/a_*+\lambda\one_K. +\end{equation} +These are the ingredients of the band-density argument in +\cite[Lemma~3.3]{X}, now deduced from the assumed sequence limits. +Since \(g\) vanishes near \(K\), +\(T_\lambda(a_*g)=-\Delta_Ug\) in the weak Neumann sense. Thus the +limiting variance of \(X_{g,n}\) is \(a_*\mathcal E_U(g,g)\). +Its limiting fourth moment is bounded uniformly for +\(\lambda\ge\lambda_0>0\), by Gaussian covariance monotonicity. + +\emph{Finite cable estimates.} +Metric-graph Gaussian sign clusters are developed in +\cite[Section~3, Lemma~3.2]{Lupu}; the Brownian-bridge extension of +integer-restricted Gaussian fields is described in +\cite[Section~7]{AHPS}. We use this construction on the finite +Gaussian-chain approximants. +We record the finite inputs used to remove the band and impose the +primary pin. Replace each primary edge by \(N\) lattice-Gaussian +bonds with increment weight \(q_N^{j^2}\), where \(q_N=b/(2N)\), +and attach the corresponding independent Brownian bridges conditional +on their endpoint heights. Let \(\mathcal P_{n,N}\) be the resulting +unweighted sigma-finite cable measure, and let \(\E_{W,N}\) and +\(\Pbb_{W,N}\) denote its normalized band law; write +\(\widehat W_N=W/\mathcal P_{n,N}(W)\). A cable bond is open +when its bridge does not hit zero. Under the unweighted cable measure, +conditional on the magnitudes and the open graph, the nonzero clusters +have independent fair signs. Exploration stops +a bridge at its first zero or at its next vertex. The centered +unvisited law has additional zero pins and increased precision on +the stopped segments. These finite disintegration and comparison +facts, together with the estimates below, are those of +\cite[Lemmas~2.2, 3.4, and~3.5]{X}. +All cluster quantities in the following argument refer to these +finite-\(N\) measures. Subdivision is removed by a +\(\limsup_{N\to\infty}\) at each fixed primary graph, before taking +\(k\to\infty\). Primary observables and their band normalizers +have actual fixed-graph limits, equal to those of the Bessel law. +No limit of the cable clusters is required. A remainder denoted by +\(o_k(1)\) in a cable estimate means that its absolute value has +\(\limsup_{k\to\infty}\limsup_{N\to\infty}\) equal to zero at the +fixed auxiliary parameters then under consideration. + +The dependence on \(b\) is controlled by the whole-chain estimate +\begin{equation}\label{eq:profile-chain-mgf} + \log\frac{\sum_{j\in\Z}q_N^{j^2}e^{2\pi tj}} + {\sum_{j\in\Z}q_N^{j^2}} + \le \frac{C_{J,R}t^2}{N},\qquad |t|\le R, +\end{equation} +for all sufficiently large \(N\), uniformly on \(J\). +This is \cite[Equation~(3.5)]{X} with its parameter dependence made explicit. +Grouping opposite integers proves this directly: the numerator +increment is bounded by a constant times +\(t^2\sum_{j\ge1}q_N^{j^2}j^2e^{2\pi Rj}=O_{J,R}(t^2/N)\). +A primary flow extended through all \(N\) bonds therefore has a +uniform moment cost. Additional zero pins and increased precision +decrease this centered cost. No bare resistance of the subdivided +network is used. + +For clarity, write \(\mathcal R(f)\) for the minimum squared +unit-edge flow cost of a mean-free primary test \(f\). After exposing +a cluster of scaled diameter at least \(r>0\), the full-component +unvisited mean has grounded reference cost at most \(V_r\). If +\(\mathcal D\) is the collection of band cells touched by an exposed +path and \(\ell0\), some \(M<\infty\) satisfies +\begin{equation}\label{eq:profile-damping} + \mathcal P_{n,N}\left((1+|S|)^{-M} + \one_{\{t^21\}} + (1+h_i^2+M_C(i)^2) + \le \sqrt{Q_BE_2}\,(1+h_i^2+Q_i). +\end{equation} +For \(\zeta_C\le1\), the corresponding errors are bounded by +\(\sqrt{Q_B}(1+\sqrt{Q_i})(\sqrt{E_2}+E_2)\). +H\"older, the weighted moments, and \eqref{eq:profile-tilt-moments} +make both expectations vanish at fixed \(\lambda\) after the mesh +and cell limits. For exploration from a filled region the same +conclusion follows directly by discarding \(\zeta>1\) using +\eqref{eq:profile-tilt-moments} and the actual band moments. +The finite tilt estimates used here are +\cite[Equations~(3.12)--(3.14)]{X}; the argument above identifies +the sequence limits of all their moment inputs. + +\emph{Vanishing reach.} +It remains to show that exploration from \(p\) is unlikely to cross +the band and reach \(V\). Choose finite small-cell coverings +\(i\) and \(B\) on the starting and target sides, respectively, +with fixed positive gaps from the band, the starting region, and +the target region. They cover separator strips, +so any crossing cluster meets a cell on each side. The radius +\(\rho\) of these cells is fixed before the band refinement. +Choose the macroscopic threshold \(r>0\) below the fixed separations +that a cluster joining the starting region, either comparison strip, +the band, or the target must span. Every cluster contributing to +the cross terms below then has diameter at least \(r\), and the +previous moment and tilt estimates apply with this one threshold. +Path anchoring bounds the remaining centered variance of an average +in a reached cell by a constant \(C_0\) independent of \(\rho\). +On the other hand, a logarithmic trial function supported away from +the band gives the continuum band variance lower bound +\[ + c a_*\log(1/\rho)-C. +\] +The trial pays no band mass, so this bound is uniform in \(\lambda\). +At a free wall it is restricted to the rectangle, which is permitted +for the Neumann form. Since \(a_*>0\), choose \(\rho\) so small +that the prior variance exceeds \(C_0+2c_0\) for a fixed \(c_0>0\). +The band limits then give a gap at least \(c_0\) in the discrete +laws, for each fixed strength and sufficiently fine partition and mesh. +This is the only step that requires a positive height coefficient; +it uses \(a_*>0\) in the limiting band law. The finite estimates +require only \(b\in J\), with no assumption on \(a(b_k)\). + +There are two useful consequences of this gap. Write a target +average as exposed plus unvisited height. The centered posterior +variance is at most the prior variance and loses at least \(c_0\) +if the cell is reached. Restoring its tilt has the vanishing +errors just proved. Fair-sign resampling of the exposed term gives +\begin{equation}\label{eq:profile-exterior-deficit} + c_0\Pbb_{W,N}(B\text{ reached}) + \le\E_{W,N}\sum_{C:C\cap p\ne\varnothing}M_C(B)^2+o(1). +\end{equation} +Here and below \(o(1)\) vanishes in the first two limits at fixed +\(\lambda\). A second comparison starts by biasing with \(|h_x|\), +for \(x\in B\), and exposing its cluster. The finite size-bias +inequality +\[ + \E[X^2|Y|]\ge\E X^2\,\E|Y| +\] +holds for real linear tests under centered band precision. It follows +from the minimal characteristic logarithmic Hessian in the finite +comparison rules: integrate +\(\E[X^2\cos(tY)]\le\E X^2\,\E\cos(tY)\) against +\(2\,dt/(\pi t^2)\) after subtracting the identity at zero. +Consequently the pre-exposure second moment at \(i\) is at least +its unbiased variance. The completed-cluster bias cancels from +the conditional complement law. Apply the same variance deficit, +multiply by \(\E|h_x|\), and average \(x\) over \(B\). This yields +\begin{equation}\label{eq:profile-biased-deficit} + c_0\E_{W,N}\sum_{C:C\cap i\ne\varnothing}M_C(B) + \le\E_{W,N}\sum_C M_C(B)M_C(i)^2+o(1). +\end{equation} +Only clusters spanning the fixed separation contribute to the +exposed or tilted terms. They have a fixed positive diameter, +so \eqref{eq:profile-large-tilts} justifies their summed errors. +Equations \eqref{eq:profile-exterior-deficit} and +\eqref{eq:profile-biased-deficit} are the two finite deficit +comparisons in \cite[Equations~(3.16)--(3.17), +proof of Proposition~3.8]{X}. + +As \(\lambda\to\infty\), the inverse forms \(T_\lambda^{-1}\) +decrease to the inverse with a hard zero band on \(K\). This follows +from weak energy compactness and recovery by functions zero on \(K\). +The band separates \(B\) from \(i\), so +\((B,T_\lambda^{-1}i)\to0\). By +\eqref{eq:profile-cluster-cross}, the limiting expectation of +\(\sum_CM_C(B)M_C(i)\) therefore vanishes. On +\(\max_CM_C(i)\le R\), the right side of +\eqref{eq:profile-biased-deficit} is at most \(R\) times this cross +sum. Its remaining tail tends to zero as \(R\to\infty\), uniformly +after the first two limits and toward large \(\lambda\), because +\[ + \sum_CM_C(B)M_C(i)^2\le\sqrt{Q_B}\,Q_i +\] +and the \(Q\)'s have arbitrary fixed moments uniformly in that +order of limits. Thus the first-mass sum on the left of +\eqref{eq:profile-biased-deficit} vanishes. Every cluster joining +\(p\) to \(B\) visits some cell \(i\). Truncating +\(\max_CM_C(B)\) in \eqref{eq:profile-exterior-deficit} and applying +these first-mass bounds, with the same moment control of its tail, +proves +\begin{equation}\label{eq:profile-vanishing-reach} + \lim_{\lambda\to\infty}\limsup_{\ell\downarrow0} + \limsup_{k\to\infty}\limsup_{N\to\infty} + \Pbb_{W,N}\bigl(\text{exploration from }p + \text{ reaches }V\bigr)=0. +\end{equation} + +\emph{Pinned tests and the grounded approximation.} +On a finite subdivided graph, explore every cluster meeting the +primary pin region. If it does +not reach \(V\), the exposed part of \(X_{g,n}\) is zero. +The centered posterior has the pin \(p\), possibly further zero +pins and increased segment precision, and a nonnegative restricted +band form. Its variance is at most the variance of \(X_{g,n}\) +under the primary hard-pin law on that subdivided graph, by finite +precision comparison. At fixed primary graph, this latter variance +converges as \(N\to\infty\) to the Bessel hard-pin variance. +The averaged tilt errors vanish as above. On failure, +Cauchy--Schwarz bounds the actual second-moment contribution by +the fourth-moment root times the square root of the failure +probability. The band fourth moment is uniformly bounded in the +successive limits, whereas \eqref{eq:profile-vanishing-reach} +makes that probability vanish. Hence +\[ + \liminf_{k\to\infty}\Var_p X_{g,n} + \ge a_*\mathcal E_U(g,g). +\] +The free variance bound gives the opposite limsup. For every real +\(t\), the finite lower bound for the logarithmic moment generating +function and the upper bound obtained by releasing the pin give +\[ + \frac{t^2}{2}\Var_p X_{g,n} + \le \log\E_p e^{tX_{g,n}} + \le \log\E_{\rm free}e^{tX_{g,n}}. +\] +The free release is permitted because \(A_{U,n}g_n\) has total zero. +The two endpoints converge to \(a_*t^2\mathcal E_U(g,g)/2\). +Applying this to finite linear combinations gives the joint +Gaussian Laplace limit for these tests; uniform primary exponential +moments give their polynomial moment limits. Weak free-face terms +may be approximated first by bounded density smears in primary +reference cost, with the gap from \(p\) and \(K\) preserved. + +Finally let \(f\) be any bounded piecewise continuous density, +and set \(v=G_pf\). Smooth shifts \(g_j\) vanishing near \(p\), +and allowed to meet the free walls, are dense in \(\mathcal H_p\) +in energy. Choose \(\mathcal E_U(v-g_j,v-g_j)\to0\). +The ordinary mixed-form approximation on these rectangular +geometries, as stated in \cite[Section~3.1 and proof of +Theorem~3.1]{X}, gives +\begin{equation}\label{eq:profile-grounded-approximation} + \lim_{k\to\infty} + \mathcal R_{p,n}\bigl(f_n-A_{U,n}g_{j,n}\bigr) + =\mathcal E_U(v-g_j,v-g_j), +\end{equation} +where \(\mathcal R_{p,n}\) is the squared dual norm of the unit-edge +form grounded on \(p\). To see the precise form of this approximation, +let \(A_{p,n}\) be the restriction to unpinned coordinates and put +\(v_n=A_{p,n}^{-1}f_n\). The left side before its limit is +\[ + (f_n,A_{p,n}^{-1}f_n)-2(f_n,g_{j,n}) + +(g_{j,n},A_{U,n}g_{j,n}). +\] +Mixed inverse-form convergence, quadrature, and smooth energy +consistency give respectively +\(\mathcal E_U(v,v)\), \(\mathcal E_U(v,g_j)\), and +\(\mathcal E_U(g_j,g_j)\), proving +\eqref{eq:profile-grounded-approximation}. Values of a test on pinned vertices are +immaterial in this norm. Equivalently, a flow representing its +unpinned coefficients may discharge into the pinned vertices. +This is why \eqref{eq:profile-grounded-approximation} imposes no +zero-total condition on \(f\). + +Uniform reference-cost bounds make the difference between the two +smears in \eqref{eq:profile-grounded-approximation} tend to zero +in every fixed moment, first as \(k\to\infty\) and then +\(j\to\infty\). Their fixed linear exponential bounds give the +same passage for Laplace transforms, by Cauchy--Schwarz applied to +\(e^{t(h,f_n)}-e^{tX_{g_j,n}}\). The limiting variance is +\(a_*\mathcal E_U(v,v)=a_*\int_UfG_pf\). +Apply the argument to every finite linear combination of the +\(f_i\)'s and use polarization to obtain +\eqref{eq:profile-pinned-covariance}, with the asserted joint +Laplace and moment convergence. +\end{proof} + +Two choices of pin will be used below. If \(D\) is an inner rectangle +and \(p=U\setminus D\) is its positive-width rectangular rim, then +\(\mathcal H_p\) identifies +with \(H^1_0(D)\) by zero extension, and +\eqref{eq:profile-pinned-covariance} is the Dirichlet covariance +\(a_*G_D\). If a positive-width primary collar surrounds a compactly +supported shift inside an unwrapped torus rectangle, zero pinning +that collar separates its interior law from the exterior. The lemma +therefore applies to that interior law as well. + + +\subsection{Torus limits and uniform zero-coefficient estimates} + +\begin{proposition}[Gaussian laws at continuity scales]\label{prop:gaussian-profile} +Let $w$ be a continuity point of the extracted profile, let +$\delta_k\log_Ln_k\to w$, and put $a_*=\mathcal A(w)$. +The free rectangular limits are those in +Lemma~\ref{lem:profile-free}; if $a_*>0$, their fractional means have +the conclusion of Lemma~\ref{lem:profile-phase}, and their fixed +primary pinned geometries have the Gaussian Laplace and moment limits +of Lemma~\ref{lem:profile-pins}. In particular, on a square interior +surrounded by a fixed positive-width zero rim, the covariance of +bounded piecewise continuous density smears converges to +$a_*(-\Delta_D)^{-1}$, with +Dirichlet boundary on that square. + +On the zero-winding height torus of side $n_k$, every smooth periodic +shift $g$ satisfies +\begin{equation}\label{eq:profile-torus} + \Var(h,A_{\mathrm{tor}}g_k)\longrightarrow + a_*\int_{\mathbb T^2}|\nabla g|^2. +\end{equation} +Thus the area-weighted first cosine observation in +\eqref{eq:physical-cosine} has limiting variance +$a_*\mathfrak g$, where $\mathfrak g=1/(8\pi^2)$. +\end{proposition} + +\begin{proof} +The free and pinned assertions have been proved. For the torus upper +bound, cut into small fixed rectangles, restrict the gradient test to +each piece, and use \eqref{eq:profile-partial-upper}. The omitted seams +and the gradient approximations have uniformly vanishing reference +cost. Therefore the limsup in \eqref{eq:profile-torus} is at most its +right side. This already proves the assertion when $a_*=0$. + +For $a_*>0$, first suppose that $g$ is supported in an unwrapped +rectangle strictly inside one torus chart. Add primary zero pins on a +collar of fixed positive width surrounding its support inside that +chart. Centered precision comparison decreases variance. The collar +separates the interior law from the torus exterior, and +Lemma~\ref{lem:profile-pins} gives the interior variance limit. +Indeed the torus test equals the grounded interior test there, and +$(-\Delta_D)^{-1}(-\Delta g)=g$ because $g$ vanishes near the +collar. Its variance is therefore $a_*\int|\nabla g|^2$, matching +the upper bound. For general $g$, +use a finite smooth partition of unity to write $g$ as a sum of shifts +supported in such charts. On each joint covariance subsequence the +upper-bound defect is a positive semidefinite quadratic form. Since +its diagonal vanishes on each summand, its cross terms with those +summands vanish as well. It therefore vanishes on their sum. + +For the cosine density $f(x)=\cos(2\pi x_1)$, its inverse torus +Laplacian is $g=f/(4\pi^2)$ and +$\int|\nabla g|^2=1/(8\pi^2)$. The discrete eigenvalue converges to +$4\pi^2$ after multiplication by $n_k^2$. The normalizing correction +has vanishing reference cost, which proves the last assertion. +\end{proof} + +\begin{corollary}[Uniform affine transforms on a zero interval] +\label{cor:profile-zero-uniform} +Suppose $I=[u,v]\subset(0,\infty)$ consists of continuity points of +$\mathcal A$ and $\mathcal A=0$ on $I$. Fix a compact set +$K\subset\R^2$ and $\lambda\in\R$. For a free square of side $m$ +write $X_{m,p}=(h,A_{B_m}(p\cdot x/m))$. Then +\begin{equation}\label{eq:profile-zero-uniform} + \sup_{\substack{m\in\N:\ \delta_k\log_Lm\in I\\p\in K}} + \left|\E_{B_m,b_k}^{\mathrm{free}}e^{\lambda X_{m,p}}-1\right| + \longrightarrow0. +\end{equation} +The assertion is unchanged by translating the squares or multiplying +their sides by factors in a fixed compact subset of $(0,\infty)$, +provided their resulting logarithmic scales stay in a fixed +zero-profile neighborhood of $I$. +\end{corollary} + +\begin{proof} +If $a_m(b_k)$ failed to tend to zero uniformly on the stated set, +there would be offending $m_k$ and a subsequence with +$\delta_k\log_Lm_k\to w\in I$. This contradicts +\eqref{eq:profile-limit}. Square symmetry gives +$\Var X_{m,p}=a_m(b_k)|p|^2$, uniformly small for $p\in K$. +The reference exponential bound at $\pm2\lambda$ supplies uniform +integrability of $e^{\lambda X_{m,p}}$. Since $X_{m,p}\to0$ in +probability uniformly over these choices, the expectations tend to +one uniformly. Translation does not change the law. The extra side +factors change $\delta_k\log_Lm$ by $o(1)$; compactness gives the +last statement. +\end{proof} + +\subsection{Excluding intermediate positive coefficients} + +The fixed-temperature coefficient gap in \cite[Theorem~2.5]{X} +is proved by growth +of a face characteristic coefficient. For a moving temperature, a +single growth comparison cannot be iterated at arbitrarily many +scales. We instead integrate those comparisons over a compact +logarithmic interval. The initial characteristic coefficient must +therefore have a quantitative lower bound on this interval. + +Let $\rho_m^{\mathrm{sym}}$ be one quarter of the sum of the four +face probabilities of $B_m$, and define +\begin{equation}\label{eq:profile-z} + z_m(b)=\E_{B_m,b}^{\mathrm{free}} + e^{i(h,\rho_m^{\mathrm{sym}})}. +\end{equation} +It is strictly positive and at most one. + +\begin{lemma}[Relative growth of the face phase]\label{lem:profile-ratio} +There is $c>0$, independent of sufficiently large fixed integer $G$, +with the following property. If $w$ is a continuity point and +$\mathcal A(w)=a_*>0$, then along every sequence +$\delta_k\log_Lm_k\to w$, +\begin{equation}\label{eq:profile-ratio} + \liminf_k\frac{z_{Gm_k}(b_k)}{z_{m_k}(b_k)} + \ge cG^{2-a_* /(4\pi)}. +\end{equation} +For each fixed such $G$ there is also $c(G)>0$ such that +\begin{equation}\label{eq:profile-ratio-rough} + z_{Gm}(b)/z_m(b)\ge c(G) +\end{equation} +for all sufficiently large $m$ and all $b$ in the fixed compact +temperature interval. +\end{lemma} + +\begin{proof} +We use the finite soft joining formula +\cite[Equation~(2.11)]{X}. Tile $B_{Gm}$ by $G^2$ free +$m$-squares. The outer-face phase has cell totals $\xi_i$, supported +on boundary cells, with $\sum_i\xi_i=1$ and +$\sum_i\xi_i^2\le C/G$. On the space of neutral cell totals choose a +flow right inverse $P$ on the neighboring-cell graph, and let +$\rho_{ij}$ be the probability on the face of cell $i$ adjoining +cell $j$. Define a vector +$\mathsf H$ in that neutral Euclidean space by +\[ + (\mathsf H,d)=\sum_{\langle i,j\rangle}P(d)_{ij} + \bigl((h_i,\rho_{ij})-(h_j,\rho_{ji}) + -\bar\eta_{ij}\bigr), +\] +where $\bar\eta_{ij}$ is the mean of the independent Bessel seam +increments with the same orientation. Translating cells by constants +changes $\mathsf H$ by the neutral projection of those constants. Penalize by +$e^{-|\mathsf H|^2/2}$ and sum relative translations, quotienting by their +common translation. The auxiliary-confinement argument in the proof +of Lemma~\ref{lem:profile-phase} justifies the finite precision +comparisons for these sums. +Poisson summation gives the positive numerator +\begin{equation}\label{eq:profile-soft-sum} + \sum_{\substack{q_i\in\Z\\\sum_iq_i=1}} + e^{-|q-\xi|^2/2}\, + \prod_i\phi_{m,b}\left(s_i+ + \sum_{j\sim i}P(q-\xi)_{ij}\rho_{ij}\right) + \chi_m(P(q-\xi)), +\end{equation} +up to a common scalar. Here $s_i$ is the restriction of the outer +phase to cell $i$, $\rho_{ij}$ is its seam-face probability, and +$\chi_m$ is the characteristic function of the independent seam +averages. The denominator is the corresponding sum with $s=0$ and +total zero. This follows by Fourier transforming the Gaussian penalty +on the neutral space. The reciprocal of the projected cell-translation +lattice is the neutral integer lattice. Exact seam matching adds centered precision, so the joined +coefficient is at least this soft coefficient. + +For fixed $G$, the denominator tends to one along the sequence in +\eqref{eq:profile-ratio}. Its zero sector is one. Every nonzero +sector contains a nonzero integer cell total and tends to zero by +Lemma~\ref{lem:profile-phase}; the Gaussian weights permit dominated +convergence. Each fixed-flux seam characteristic tends to one because +the seam averages have variance $O(m^{-1})$. + +Retain the sectors $q=e_{i_0}$ with $i_0$ at distance at least $G/4$ +from the boundary. The radial flow of \cite[Equation~(2.14)]{X} has outward face +fluxes $H_{i,\pm j}$, constant outer flux $1/(4G)$, and cell +divergence $e_{i_0}$. Set +\[ + D_{ij}=\tfrac12(H_{i,+j}-H_{i,-j}),\qquad + S_{ij}=H_{i,+j}+H_{i,-j}. +\] +The unit-cell radial field, corrected to have constant boundary flux, +gives +\begin{equation}\label{eq:profile-radial-cost} + \sum_{i,j}D_{ij}^2\le\frac1{2\pi}\log G+C, + \qquad \sum_{i,j}S_{ij}^2\le C. +\end{equation} +One may obtain it by integrating +$(x-x_0)/(2\pi|x-x_0|^2)+E(x)$ across cell faces, where the Neumann +boundary correction has $|E|\le C/G$ and $|\nabla E|\le C/G^2$. +The radial energy is $(2\pi)^{-1}\log G+O(1)$, and the face-sum +errors are summable. The constants are uniform for the retained +central cells. Prescribe $P(e_{i_0}-\xi)$ to be these flows for all +retained $i_0$ and extend linearly. This is possible because the +vectors $e_{i_0}-\xi$ are linearly independent, their central +coordinates being distinct. + +In each cell expand the logarithm of its characteristic coefficient +about the symmetric four-face profile of the same total. Square +symmetry makes the linear term vanish on every zero-total face +perturbation. The logarithmic Hessian bound thus gives the product +lower bound +\begin{equation}\label{eq:profile-relative-lower} + z_m(b)\exp\left\{-\tfrac12\sum_i\Var(h,Y_i)\right\}, +\end{equation} +where $Y_i$ is the difference from that symmetric profile. Since +$q_i=S_{i1}+S_{i2}$, its decomposition is explicitly +\begin{align*} + Y_i={}&\sum_{j=1}^2D_{ij}(\rho_{i,+j}-\rho_{i,-j})\\ + &+\frac{S_{i1}-S_{i2}}4 + (\rho_{i,+1}+\rho_{i,-1}-\rho_{i,+2}-\rho_{i,-2}). +\end{align*} +Square reflections make these three components mutually orthogonal. +The two odd face differences have variance exactly $a_m(b)$; the even +component has reference cost at most $C(S_{i1}^2+S_{i2}^2)$. Lemma~\ref{lem:profile-free} and +the reference bound give +\[ + \limsup_k\sum_i\Var(h,Y_i) + \le a_*\sum_{i,j}D_{ij}^2+C\sum_{i,j}S_{ij}^2. +\] +The Gaussian sector weight is bounded below because +$|e_{i_0}-\xi|$ is bounded. There are at least $c_0G^2$ such sectors. +Combining these facts with \eqref{eq:profile-radial-cost} proves +\eqref{eq:profile-ratio}. The additive energy constant and the even +component constant are uniform, since $a_*\le a$ and the reference +bounds are uniform. Therefore its prefactor $c$ does not depend on +$G$ or on the positive continuity value. + +For \eqref{eq:profile-ratio-rough}, keep $G$ fixed and retain just one +central sector. Bound every variance in +\eqref{eq:profile-relative-lower} by its reference cost. The soft +denominator is at most the fixed finite-dimensional Gaussian lattice +sum, since all its characteristic factors are at most one. The +selected seam factor has a uniform positive lower bound: a mean over +$m$ independent Bessel increments at fixed flux $q$ has +characteristic function +$\exp\{bm(\cos(2\pi q/m)-1)\}$, which is bounded below uniformly +in large $m$ and in the compact temperature interval. There are only +finitely many seams. These bounds prove \eqref{eq:profile-ratio-rough}. +Crucially, \eqref{eq:profile-relative-lower} retains the factor +$z_m(b)$ regardless of how small it is. +\end{proof} + +\begin{lemma}[Critical lower bound for the face phase] +\label{lem:profile-subexponential} +For the fixed buffer $D_0$ and block ratio $L$, +\begin{equation}\label{eq:profile-subexponential} + -\log z_{D_0L^j}(\bc)=o(j),\qquad j\to\infty. +\end{equation} +Moreover $z_m(b)\ge z_m(\bc)$ whenever $b\le\bc$. +\end{lemma} + +\begin{proof} +At $\bc$, \cite[Equation~(2.15)]{X}, the fixed-temperature version +of \eqref{eq:profile-ratio}, has coefficient $a=8\pi$. Taking $c_1$ to be the smaller of $1$ and half the prefactor in that +liminf bound, we obtain a constant independent of sufficiently large +fixed $r$ such that +for $G=L^r$, +\[ + z_{Gm}(\bc)\ge c_1z_m(\bc) +\] +for all $m$ beyond a threshold depending on $r$. Fix $r$ and iterate +on each residue class of $j$ modulo $r$. The finitely many initial +coefficients are strictly positive. Hence +\[ + \liminf_{j\to\infty}j^{-1}\log z_{D_0L^j}(\bc) + \ge r^{-1}\log c_1. +\] +Now let $r\to\infty$ and use $z_m\le1$ to obtain +\eqref{eq:profile-subexponential}. Characteristic coefficients +decrease with $b$ by the chain precision comparison, giving the last +claim. The lower bound is therefore uniform for all $b\le\bc$. +\end{proof} + +\begin{proposition}[The profile has one jump, at the passage time] +\label{prop:profile-step} +The profile extracted in Proposition~\ref{prop:profile} satisfies +\begin{equation}\label{eq:profile-step} + \mathcal A(w)=a\quad(0T), + \qquad T=\frac{\pi}{\sqrt{3\mathcal P}}. +\end{equation} +In particular all points of these two open intervals are continuity +points. No assertion about its value at the single point $T$ is +needed. +\end{proposition} + +\begin{proof} +First we exclude a continuity value in $(0,a)$. If such a value +exists, monotonicity and continuity give a compact interval +$[u,v]\subset(0,\infty)$ of positive length and $\eta>0$ such that +all its continuity values lie in $[\eta,a-\eta]$. Choose a fixed +integer stride $r$ so large that, for $G=L^r$, +$cG^{2-(a-\eta)/(4\pi)}>2$, where $c$ is from +Lemma~\ref{lem:profile-ratio}. + +On one residue class modulo $r$ take consecutive indices $j$ for +which $\delta_k j$ runs across $[u,v]$, and define on their cells +of width $\delta_kr$ the step function +\[ + F_k(t)=\log\frac{z_{D_0L^{j+r}}(b_k)}{z_{D_0L^j}(b_k)}. +\] +Choose the endpoints of this union of cells to converge to $u,v$. +At almost every $t\in(u,v)$ the profile is continuous. The starting +cell size has logarithmic scale tending to $t$, so +\eqref{eq:profile-ratio} gives $\liminf_k F_k(t)\ge\log2$. +Equation~\eqref{eq:profile-ratio-rough} gives the common lower bound +$F_k\ge\log c(G)$. Extend by zero outside the chosen cells in a +fixed enclosing interval. Fatou's lemma therefore gives +\begin{equation}\label{eq:profile-fatou} + \liminf_k\int F_k(t)\,dt\ge(v-u)\log2>0. +\end{equation} +On the other hand, the integral telescopes exactly: +\[ + \int F_k(t)\,dt + =\delta_kr\bigl[\log z_{D_0L^{j_+(k)}}(b_k) + -\log z_{D_0L^{j_-(k)}}(b_k)\bigr]. +\] +Since $z\le1$, Lemma~\ref{lem:profile-subexponential} bounds its +limsup by +$\limsup_k\delta_kr\,o(j_-(k))=0$; here +$\delta_kj_-(k)\to u>0$. This contradicts +\eqref{eq:profile-fatou}. Thus every continuity value belongs to +$\{0,a\}$. + +For $0T$ also had value $a$. Monotonicity and the bound +$\mathcal A\le a$ would make the profile identically $a$ on a +neighborhood of $T$, so $T$ would be a continuity point. The moving +exit sizes $n_{j_e}=D_0L^{j_e}$ in Proposition~\ref{prop:passage} +have $\delta_kj_e\to T$. Proposition~\ref{prop:gaussian-profile} +would give cosine variance tending to $a\mathfrak g$ there, +contradicting the strictly smaller exit limsup. Every continuity +point above $T$ therefore has value zero. Finally a monotone function +whose values on the continuity points of an open interval are a +single constant is that constant everywhere on the interval. +This proves \eqref{eq:profile-step}. +\end{proof} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/trajectory.tex b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/trajectory.tex new file mode 100644 index 0000000000000000000000000000000000000000..463ec52f15aae961f87493157177523bb3d7ee7f --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/build/sections/trajectory.tex @@ -0,0 +1,769 @@ +\section{Temperature response and the passage through the small-activity region} +\label{sec:trajectory} + +The activity estimates now determine the scale at which the height +variance first departs from its critical value. The essential point is +to determine how the physical temperature crosses the critical +trajectory. We first prove a nonzero linear response in its growing +direction, then follow a real temperature perturbation until it reaches +a fixed small exit level. All coordinates and Gaussian reference +parameters in this section are those of Proposition~\ref{prop:map}; +only the input activities depend on the physical inverse temperature +\(b\). The comparison of changing entries follows the physical-readout +method of \cite[Lemma~5.7 and the proof of Proposition~5.1]{F}; +here we apply it also to temperature derivatives and secants. + +Write \(\mathcal H^{(k)}\) for the critical families in +Proposition~\ref{prop:critical}, with observation side +\(s_k\asymp2^k\). The index \(j\) is always the absolute block index. +In particular the corresponding physical torus has side +\(n_j=D_0L^j\), even when the entry side changes. We use +\[ + W_j=(x_j,y_j),\qquad + x_j=b_*u_j,\qquad y_j=\sqrt{b_*H}\,Z_j, + \qquad b_*=2\log L. +\] +The symbol \(y_j\) here denotes the harmonic coordinate, rather than +an observation field. A superscript \(\mathrm{tor}\) on a remainder +always refers to the separately initialized torus at its specified +stop. + +We fix the order of the smallness choices. Enlarge the remainder +contraction constant in Proposition~\ref{prop:map} slightly to a +number \(\theta_1<1\), and choose \(r\in(\theta_1,1)\). Choose a +constant \(C_*\) so large that +\begin{equation}\label{eq:traj-parameter-choice} + C_*\log L>1000,\qquad + \frac2{C_*}<\frac{-\log r}{100}. +\end{equation} +Next choose \(\gamma>0\) with +\[ + \gamma C_*<\frac1{100},\qquad + \gamma<\frac{-\log r}{100}. +\] +Choose a sufficiently small activity tube, so that the derivative of +the plane map in balanced coordinates has norm at most +\(e^\gamma\) at every point of that tube. This is possible because +its linearization is the identity on the two scalar coordinates and +a contraction on the remainder. The nonlinear remainder terms are +absorbed while keeping their contraction at most \(\theta_1\). +Finally choose the entry parameter and its lower scale threshold as +in Propositions~\ref{prop:entry} and~\ref{prop:critical}, for this +tube. Every choice in this paragraph is fixed independently of +\(b\). Changing the equivalent coordinate norms only changes fixed +constants in the estimates below. + +\subsection{The derivative of the critical trajectory} + +\begin{proposition}[Nonzero temperature response]\label{prop:response} +For the fixed choices above there is a number +\(\mathcal P\in(0,\infty)\) such that, with +\[ + V_j^{(k)}= + \left.\frac{\partial W_j^{(k)}(b)}{\partial b}\right|_{b=\bc}, + \qquad q_u=(1,-\tfrac12), +\] +we have +\begin{equation}\label{eq:traj-response-limit} + \sup_{\frac12 2^k\le j\le\frac52 2^k} + \left|\frac{V_j^{(k)}}j-\mathcal Pq_u\right| + \longrightarrow0. +\end{equation} +The plane transverse remainder and every separately initialized +torus remainder, at an index in this range behind a valid stop in +the critical history, have temperature derivative bounded by +\(C|V_j^{(k)}|/j+e^{-cj}\). +\end{proposition} + +\begin{proof} +We omit the history superscript until we compare two entries. The +entry analyticity in Proposition~\ref{prop:entry} gives an initial +derivative bound \(Cs_k^2\). Propagation in the chosen tube bounds +all plane derivatives by \(Cs_k^2e^{\gamma j}\); contraction of a +torus remainder driven by the same scalars gives the weaker common +bound \(Cs_k^2e^{2\gamma j}\). These estimates are used only to +remove the initial remainder contribution. + +On a slightly larger fixed fractional interval than that in the +statement, Proposition~\ref{prop:critical} gives +\[ + W_j=\frac{(1,1)}j+O\!\left(\frac{\log j}{j^2}\right),\qquad + \|Q_j\|+\|\mathcal R_{j|h}^{\mathrm{tor}}\|=O(j^{-2}). +\] +Here \(h\ge j\) is any valid stop within the history's horizon. +Differentiating the map of Proposition~\ref{prop:map} therefore gives +\begin{equation}\label{eq:traj-linear-response} + V_{j+1}=A_jV_j+B_j\partial_b Q_j, + \qquad + A_j=I+\frac{\mathsf M}{j} + +O\!\left(\frac{\log j}{j^2}\right), + \qquad \|B_j\|\le\frac Cj, + \qquad + \mathsf M=\begin{pmatrix}0&-2\\-1&-1\end{pmatrix}. +\end{equation} +For a fixed torus stop \(h\), put +\[ + r_{j|h}=\|\partial_bQ_j\| + +\|\partial_b\mathcal R_{j|h}^{\mathrm{tor}}\|. +\] +The analytic remainder estimates, the absence of linear forcing +from the scalar inputs into the unbalanced torus remainder, and the +coordinate changes in Proposition~\ref{prop:map} imply +\begin{equation}\label{eq:traj-derivative-contraction} + r_{j+1|h}\le\theta_1r_{j|h}+\frac Cj|V_j|. +\end{equation} +Terms with a small coefficient multiplying either remainder +derivative have been included in \(\theta_1\). The constants are +independent of \(h\). + +We give the backward estimate that makes this contraction useful. +Fix \(j\) in the asserted interval and take +\(l=\lfloor j/2\rfloor\). For large \(k\), the interval +\([l,j]\) lies inside the slightly enlarged fractional range on +which the preceding bounds hold. Define +\[ + S_v=\max_{l\le i\le j}r^{j-i}|V_i|, + \qquad + S_q=\max_{l\le i\le j}r^{j-i}r_{i|h}. +\] +Iterating \eqref{eq:traj-derivative-contraction}, and using +\(i\asymp j\) on this interval, gives +\begin{equation}\label{eq:traj-weighted-forward} + S_q\le\frac CjS_v+Cr^{j-l}r_{l|h}. +\end{equation} +For large \(j\), the matrices \(A_i\) are invertible and +\(\|A_i^{-1}\|\le1+C/j\) in a fixed equivalent scalar norm. +Solving \eqref{eq:traj-linear-response} backward from \(j\), +multiplying at index \(i\) by \(r^{j-i}\), and summing the +geometric factors \(r^{v-i}\) over intermediate indices +\(i\le v0. +\end{equation} +This lower bound is the step that will exclude a purely decaying +solution of the linearized recurrence. + +To compare two entries, define the normalized diagnostic vector +\begin{equation}\label{eq:traj-normalized-diagnostic} + \mathscr O_{j,s}(b)= + \begin{pmatrix} + \bigl(\widehat\Var_b X_{n_j,s} + -\|f_{n_j,s}\|_2^2\bigr)/\beta_{n_j,s}^2\\[2pt] + e^{(s/n_j)^2/2}\widehat\E_b e^{i\bar\eta} + \end{pmatrix}. +\end{equation} +The subtracted term is the independent observation-noise variance +in the uncut law. Lemma~\ref{lem:grid-comparison} compares this +vector with the same physical pair +\((\Var_b^{\mathrm{tor}}H_{n_j}, +\E_b^{\mathrm{tor}}e^{i\bar h})\) for either entry. +At zero activity, its linear scalar coefficient in the +\(W_j,Q_j\) coordinates is +\[ + \operatorname{diag}\left( + \frac{\kappa_{n_j,s}}{b_*\beta_{n_j,s}^2}, + \frac{e^{(s/n_j)^2/2}p_{j,s}\ell_j^{-1}}{\sqrt{b_*H}}\right). +\] +Here \(\ell_j=1+O(e^{-cj})\) is the balancing factor in the +proof of Proposition~\ref{prop:map}. The linear remainder rows +are controlled by \eqref{eq:traj-slaving}. +Its constant background is independent of \(b\). Differentiating +and using \eqref{eq:traj-slaving} gives +\begin{equation}\label{eq:traj-response-readout} + D_jV_j+O(|V_j|/j)+O(e^{-cj}), + \qquad + D_j=\operatorname{diag}\left( + \frac{a\mathfrak g}{b_*}, + \frac{p_j}{\sqrt{b_*H}}\right). +\end{equation} +The numbers \(p_j\) and their inverses are uniformly bounded and +positive. At a common absolute stop this diagonal matrix may be +chosen identically for both entries: the difference of the actual +multipliers is exponentially small and is included in the error. +The variance comparison error is bounded by +\(Cn_j^2(s/n_j)^4\); the phase comparison is exact after removal +of the mean noise, apart from the cutoff error. All these errors +are exponentially small when the entries being compared have +\(s\asymp j\). Since both readouts describe the same finite +physical torus, \eqref{eq:traj-response-readout} and +\eqref{eq:traj-response-nonvanishing} give +\begin{equation}\label{eq:traj-response-overlap} + |V_j^{(k)}-V_j^{(k')}| + \le \frac Cj\bigl(|V_j^{(k)}|+|V_j^{(k')}|\bigr) +\end{equation} +whenever the index lies in the fixed interior ranges for both +histories. In particular their norms are comparable. Only two +physical scalar readouts were compared; no subtraction of +remainders belonging to different activity norms is involved. + +Define a single sequence by taking \(V_j=V_j^{(k)}\) for +\(2^k\le j<2^{k+1}\). At an ordinary step, +\eqref{eq:traj-linear-response} and \eqref{eq:traj-slaving} show +that this sequence obeys +\[ + V_{j+1}=\left(I+\frac{\mathsf M}{j}+E_j\right)V_j, + \qquad \|E_j\|\le C\frac{\log j}{j^2}. +\] +Indeed an error vector of norm at most \(\eta|V_j|\) can be +written as a matrix of norm at most \(\eta\) applied to \(V_j\). +The nonvanishing bound permits this representation also for the +exponentially small errors. At a change of entry there is, by +\eqref{eq:traj-response-overlap}, an additional matrix error of +norm \(C/j\). + +The matrix \(\mathsf M\) has eigenvectors +\(q_u=(1,-1/2)\) and \(q_s=(1,1)\), with eigenvalues \(1\) and +\(-2\), respectively. Let \((U_k,S_k)\) be the coordinates of +\(V_{2^k}\) in this basis. Telescoping products over a dyadic +interval gives +\begin{equation}\label{eq:traj-epoch} + \begin{pmatrix}U_{k+1}\\S_{k+1}\end{pmatrix} + =\left\{ + \begin{pmatrix}2&0\\0&1/4\end{pmatrix}+\mathsf E_k + \right\} + \begin{pmatrix}U_k\\S_k\end{pmatrix}, + \qquad \|\mathsf E_k\|\le Ck2^{-k}. +\end{equation} +For completeness, the unperturbed growing product is exactly +\(\prod_{j=n}^{2n-1}(1+1/j)=2\), and the decaying product is +\(\prod_{j=n}^{2n-1}(1-2/j)=1/4+O(n^{-1})\). +All partial products within one dyadic interval and their inverses +are bounded. Summing \(C\log j/j^2\) over the interval, together +with its one \(C/j\) change of entry, proves the stated bound on +\(\mathsf E_k\). + +For all sufficiently large \(k\), the cone +\(\{|S_k|\le|U_k|\}\) is invariant under +\eqref{eq:traj-epoch}. If a sequence never enters this cone, +then \(|U_k|<|S_k|\) at every sufficiently large index, and +\eqref{eq:traj-epoch} gives +\(|S_{k+1}|\le\tfrac12|S_k|\). Both coordinates would tend to +zero, contrary to \eqref{eq:traj-response-nonvanishing}. Thus the +sequence enters the cone. Thereafter +\[ + U_{k+1}=2U_k(1+\eta_k),\qquad + |\eta_k|\le Ck2^{-k}. +\] +The factors \(1+\eta_k\) are positive on the tail and their +logarithms are absolutely summable. It follows that +\(U_k/2^k\) converges to a finite nonzero number \(\mathcal P\). +The second row of \eqref{eq:traj-epoch}, divided by its first, +gives +\[ + \frac{|S_{k+1}|}{|U_{k+1}|} + \le \rho\frac{|S_k|}{|U_k|}+Ck2^{-k} + \quad\text{for some }\rho<1, +\] +so this ratio tends to zero. The same product estimates on partial +dyadic intervals show that \(V_j/j\to\mathcal Pq_u\) at every +intermediate index. Applying \eqref{eq:traj-response-overlap} +then gives the uniform assertion for either history on its interior +range. Finally the first component of +\eqref{eq:traj-response-readout} has positive physical response. +Its leading term would be negative and of order \(j\) if +\(\mathcal P<0\). Hence \(\mathcal P>0\). +\end{proof} + +\subsection{Matching a late entry to the critical trajectory} + +We will use one observation side much larger than the absolute index +at which the perturbation is first measured. The next lemma supplies +the critical estimates for that history. Its proof compares finite +physical diagnostics, so no rate of convergence of a critical height +scaling limit is needed. + +\begin{lemma}[Critical estimates after a logarithmic initial interval] +\label{lem:traj-late-critical} +Let \(s\to\infty\) through admissible observation sides, put +\(J=\lfloor C_*\log s\rfloor\), and let \(K(s)\ge J\) satisfy +\(K(s)=o(s^2)\). For the history with observation side \(s\), at +\(b=\bc\), uniformly for \(J/2\le i\le K(s)\), +\begin{equation}\label{eq:traj-late-critical} + W_i=\frac{(1,1)}i+O\!\left(\frac{\log i}{i^2}\right), + \qquad + \|Q_i\|+\|\mathcal R_i\|=O(i^{-2}). +\end{equation} +Every separately initialized torus with a valid stop +\(h\in[i,K(s)]\) also has +\(\|\mathcal R_{i|h}^{\mathrm{tor}}\|=O(i^{-2})\). +\end{lemma} + +\begin{proof} +The extended critical confinement in Proposition~\ref{prop:critical} +keeps all these states in the chosen small tube. Put +\(i_0=\lceil J/100\rceil\). At every \(i\ge i_0\), compare the +history with a canonical history \(\mathcal H^{(k)}\) for which +\(2^k\le i<2^{k+1}\). Both are exact descriptions of the same +physical torus of side \(n_i\). The first choice in +\eqref{eq:traj-parameter-choice} ensures +\(s/n_i\le e^{-ci}\) throughout this range. The observation +sampling errors and multiplier differences are consequently +exponentially small. The cutoff error is uniformly at most +\[ + C\exp\{CK(s)-cs^2\}, +\] +and is also exponentially small in \(i\), since \(K(s)=o(s^2)\). +The canonical history has the bounds of +Proposition~\ref{prop:critical}. + +Write \(d_i=|t_i|+|z_i|\), and let \(R_i\) be the maximum of +\(\|\mathcal R_i\|\) and the remainder norms at stage \(i\) +of all the separate tori with stops in \([i,K(s)]\). The two +nonzero scalar multipliers in Proposition~\ref{prop:readouts} give +\begin{equation}\label{eq:traj-late-readout} + d_i\le\frac Ci+CR_i+Cd_i^2. +\end{equation} +In writing this inequality we absorbed remainder squares and mixed +terms using the uniform small tube. Remainder contraction, with +copied scalar forcing also on each torus, gives +\begin{equation}\label{eq:traj-late-contraction} + R_{i+1}\le\theta_1R_i+Cd_i^2. +\end{equation} +The family of tori in the maximum only decreases when \(i\) +increases, so the same inequality holds for the maximum itself. + +Absorb \(Cd_i^2\) into the left side of +\eqref{eq:traj-late-readout}. Then +\(d_i\le C/i+CR_i\), and, since \(R_i\) is uniformly small, +\eqref{eq:traj-late-contraction} becomes +\[ + R_{i+1}\le\theta_2R_i+\frac C{i^2} + \quad\text{for a fixed }\theta_2<1. +\] +Iteration from \(i_0\) shows that +\(R_i\le C\theta_2^{i-i_0}+C/i^2\). For \(i\ge J/2\) the +first term is at most \(C/i^2\), uniformly for all later \(i\), +when \(s\) is large. Thus \(R_i=O(i^{-2})\) and +\(d_i=O(i^{-1})\). + +Repeat the readout comparison with these improved bounds. Each +nonlinear and remainder error is now \(O(i^{-2})\). Consequently +the two unbalanced scalar coordinates agree with those of the +canonical history to \(O(i^{-2})\). Each coordinate change to +\(W,Q\) differs from its linear part by \(O(i^{-2})\), by +Proposition~\ref{prop:map}. The canonical critical estimates +therefore give \eqref{eq:traj-late-critical}. +\end{proof} + +\subsection{A temperature secant at the matching index} + +Let +\[ + \eps=\bc-b>0,\qquad \delta=\sqrt\eps, + \qquad s\asymp\eps^{-1/3},\qquad + J=\lfloor C_*\log s\rfloor, +\] +where \(s\) is chosen admissibly. Consecutive admissible sides have +a fixed ratio, so the two-sided bound implicit in \(\asymp\) +is independent of \(\eps\). These choices give +\begin{equation}\label{eq:traj-scale-slack} + \eps s^2\longrightarrow0, + \qquad \delta^{-1}=o(s^2), + \qquad J=o(\delta^{-1}). +\end{equation} +The first inequality places \(b\) inside the analytic entry disk; +the second keeps every stop of order \(\delta^{-1}\) within the +cutoff range. + +\begin{lemma}[Matching the temperature secant]\label{lem:traj-secant} +For the history with observation side \(s\), +\begin{equation}\label{eq:traj-secant-match} + \frac{W_J(\bc-\eps)-W_J(\bc)}{-\eps} + =J\bigl(\mathcal Pq_u+o(1)\bigr), + \qquad + \left\|\frac{Q_J(\bc-\eps)-Q_J(\bc)}{\eps}\right\|\le C. +\end{equation} +The same uniform bound holds for the secant of each unbalanced torus +remainder at stage \(J\) behind a stop of order \(\delta^{-1}\). +\end{lemma} + +\begin{proof} +Entry analyticity and propagation in the small tube first give, +through stage \(J\), the crude bound +\begin{equation}\label{eq:traj-crude-proximity} + \text{distance between the states at \(\bc-\eps\) and \(\bc\)} + \le C\eps s^2e^{2\gamma J}. +\end{equation} +This statement uses each history's own norm. The torus bound follows +by contracting its remainder with the same scalar inputs. Since +\(\eps\asymp s^{-3}\) and \(2\gamma C_*<1\), the right side +decays as a negative power of \(s\); in particular it is smaller +than every fixed negative power of \(J\). Induction with a fixed +margin to the boundary of the tube justifies the propagation up to +\(J\). + +Denote the scalar secants by \(\widetilde V_i\), with denominator +\(-\eps\). On \([J/2,J]\), Lemma~\ref{lem:traj-late-critical} +and \eqref{eq:traj-crude-proximity} allow us to integrate the +analytic derivatives of the map along the segment between its two +input states. This gives the same form of scalar and remainder +inequalities as \eqref{eq:traj-linear-response} and +\eqref{eq:traj-derivative-contraction}. The extra coefficients +from \eqref{eq:traj-crude-proximity} are exponentially small in +\(J\), and hence fit inside their displayed errors. The crude +bound for any initial remainder secant is \(Cs^2e^{2\gamma J}\). +The weighted estimates \eqref{eq:traj-weighted-forward}-- +\eqref{eq:traj-weighted-backward}, now on a fixed-fraction interval +ending at \(J\), therefore give +\begin{equation}\label{eq:traj-secant-slaving} + \widetilde r_J\le\frac CJ|\widetilde V_J|+e^{-cJ}. +\end{equation} +Indeed the homogeneous contribution is bounded by a constant times +\(r^{c_1J}s^2e^{2\gamma J}\). The choices of \(C_*\) and +\(\gamma\) make its exponent strictly negative; taking, for +example, an interval of length \(J/4\) within \([J/2,J]\) +suffices. The estimate is uniform for the torus remainders behind +later stops. + +To identify this secant, choose a canonical history with +\(s_k\asymp J\) and \(J\) in its fixed interior range. In that +history we can compare the secant directly with its derivative. +For all stages through \(J\), analytic continuation in the +parameter is valid and bounded on a disk of radius +\[ + \rho_J=cJ^{-2}e^{-2\gamma J} +\] +about \(\bc\). To see this, the entry displacement is bounded +by \(Cs_k^2\rho_J\), and its amplification through \(J\) +is bounded by \(Ce^{\gamma J}\); the resulting displacement +tends to zero. The same argument and contraction control the +separate torus states. Cauchy's estimate bounds second parameter +derivatives of the stopped states by \(C\rho_J^{-2}\). +Since \(\log\eps=-3J/C_*+O(1)\), +\begin{equation}\label{eq:traj-cauchy-error} + \eps\rho_J^{-2} + \le CJ^4\exp\{(-3/C_*+4\gamma)J\} + \longrightarrow0. +\end{equation} +Thus the canonical scalar secant is +\(V_J^{(k)}+o(1)\), with analogous absolute control of its +remainder secants. This is the only use of a second temperature +derivative; it is taken in the history with entry side comparable +to \(J\). + +Compare the two histories through their physical readout secants. +Lemma~\ref{lem:grid-comparison} and +Lemma~\ref{lem:cutoff-derivative} apply uniformly on the real +parameter segment \([\bc-\eps,\bc]\), so their first-derivative +bounds apply also to the divided differences. Taylor expansion of +\(\mathscr O_{J,s}\) from \eqref{eq:traj-normalized-diagnostic} +on the segment between each history's two temperature states, +followed by \eqref{eq:traj-secant-slaving}, gives +\begin{equation}\label{eq:traj-secant-comparison} + D_J\widetilde V_J + =D_JV_J^{(k)} + +O\!\left( + \frac{|\widetilde V_J|+|V_J^{(k)}|}{J}\right)+o(1), +\end{equation} +with the same uniformly invertible diagonal matrix as in +\eqref{eq:traj-response-readout}. The multipliers and observation +errors are exponentially small because the first condition in +\eqref{eq:traj-parameter-choice} also gives +\(n_J^2(s/n_J)^4\le e^{-cJ}\). + +Proposition~\ref{prop:response} gives \(|V_J^{(k)}|=O(J)\). +Absorbing \(C|\widetilde V_J|/J\) in +\eqref{eq:traj-secant-comparison} shows +\(|\widetilde V_J|=O(J)\), then +\(\widetilde V_J-V_J^{(k)}=O(1)+o(1)\). +Proposition~\ref{prop:response} and +\eqref{eq:traj-secant-slaving} prove all the assertions. +\end{proof} + +\subsection{The nonlinear passage time} + +We now follow the real perturbed trajectory beyond the matching +index. The result includes exactly the two physical variance facts +needed in the following section. Recall the microscopic torus +cosine \(H_n\) from \eqref{eq:physical-cosine}, and +\(\mathfrak g=1/(8\pi^2)\). + +\begin{proposition}[Passage time and the variance diagnostic] +\label{prop:passage} +Put +\[ + k_0=\sqrt{3\mathcal P},\qquad T=\frac\pi{k_0}. +\] +For \(b=\bc-\eps\), use an admissible observation side +\(s\asymp\eps^{-1/3}\). On every compact subinterval of +\((0,T)\), the scalar trajectory satisfies, uniformly for +\(w=\delta j\) in that subinterval, +\begin{equation}\label{eq:traj-scaled-limit} + \frac{x_j(b)}\delta\longrightarrow k_0\cot(k_0w), + \qquad + \frac{y_j(b)}\delta\longrightarrow k_0\csc(k_0w), + \qquad \|Q_j(b)\|=O(\delta^2). +\end{equation} +In particular, for every sequence \(\eps\downarrow0\) and +\(\delta j\to w\in(0,T)\), +\begin{equation}\label{eq:traj-before-variance} + \Var_{b,n_j}^{\mathrm{tor}}H_{n_j} + \longrightarrow a\mathfrak g. +\end{equation} +For every sufficiently small fixed \(d>0\), there are exit indices +\(j_e=j_e(\eps,d)\) such that +\begin{equation}\label{eq:traj-exit} + \delta j_e\longrightarrow T,\qquad + x_{j_e}=-d+O(d^2),\qquad y_{j_e}=O(d),\qquad + \|Q_{j_e}\|+\|\mathcal R_{j_e|j_e}^{\mathrm{tor}}\|=O(d^2). +\end{equation} +All states through the exit lie in the small-activity map domain. +For some \(c_d>0\), +\begin{equation}\label{eq:traj-exit-variance} + \limsup_{\eps\downarrow0} + \Var_{b,n_{j_e}}^{\mathrm{tor}}H_{n_{j_e}} + \le a\mathfrak g-c_d. +\end{equation} +The torus law in these assertions is the original periodic Bessel +height law in the zero-winding sector, with its common height +translation factored out. +\end{proposition} + +\begin{proof} +Set +\(\Delta W_i=W_i(\bc-\eps)-W_i(\bc)\) and define +\(\Delta Q_i\) similarly. We first prove that, for a sufficiently +small fixed \(\eta>0\), uniformly on +\(J\le i\le\eta/\delta\), +\begin{equation}\label{eq:traj-early-perturbation} + \Delta W_i=-\eps i\bigl(\mathcal Pq_u+O(\eta^2)+o(1)\bigr), + \qquad \|\Delta Q_i\|\le C\eps. +\end{equation} +Here the constant in \(O(\eta^2)\) is independent of sufficiently +small \(\eta\), and \(o(1)\to0\) as \(\eps\to0\). +The critical history has the estimates of +Lemma~\ref{lem:traj-late-critical} through every fixed multiple of +\(\delta^{-1}\), by \eqref{eq:traj-scale-slack}. + +Bootstrap the estimate \(|\Delta W_i|\le C_1\eps i\). +The differentiated remainder bounds in +Proposition~\ref{prop:map} give +\[ + \|\Delta Q_{i+1}\| + \le\theta_1\|\Delta Q_i\| + +C_{C_1}\bigl(i^{-1}\eps i+(\eps i)^2\bigr). +\] +The second term is \(O_{C_1}(\eps)\) on the indicated range. +Lemma~\ref{lem:traj-secant} and contraction therefore give +\(\|\Delta Q_i\|\le C_{C_1}\eps\). +Subtract the two scalar recurrences and linearize their leading +quadratic terms about the critical trajectory. The result is +\begin{equation}\label{eq:traj-difference-recurrence} + \Delta W_{i+1} + =\left(I+\frac{\mathsf M}{i}\right)\Delta W_i+e_i, + \qquad + |e_i|\le C_{C_1}\left( + \frac{\eps\log i}{i}+\frac\eps i+\eps^2i^2\right). +\end{equation} +The first error uses the \(O(\log i/i^2)\) critical correction +and the differentiated cubic terms. The second uses +\(\|\Delta Q_i\|=O(\eps)\) and the \(O(i^{-1})\) scalar +remainder row. The last is the quadratic term in \(\Delta W_i\). + +In the eigenbasis of \(\mathsf M\), the unperturbed products from +\(h\) to \(i\) have norm at most \(Ci/h\), and send \(q_u\) +to \((i/h)q_u\) exactly. Variation of constants in +\eqref{eq:traj-difference-recurrence} and +Lemma~\ref{lem:traj-secant} therefore gives, after division by +\(\eps i\), a total error bounded by +\[ + o(1)+C_{C_1}\sum_{h\ge J}\frac{\log h+1}{h^2} + +C_{C_1}\eps\sum_{h\le i}h + \le o(1)+C_{C_1}\left(\frac{\log J}{J}+\eta^2\right). +\] +Choose \(C_1\) larger than the leading bound, then choose +\(\eta\) small. The improved inequality closes the bootstrap. +The trajectory stays within the chosen tube throughout this +argument, by the same bound and the critical estimates. +This proves \eqref{eq:traj-early-perturbation}. The unbalanced +remainder on every torus behind a stop of order \(\delta^{-1}\) +satisfies its corresponding bounds by the same contraction with +copied scalars. + +For the continuous-scale limit, linearly interpolate +\(x_j/\delta,y_j/\delta\) at times \(w=\delta j\). +On every compact subinterval of \((0,\eta]\), +\eqref{eq:traj-early-perturbation} and the critical estimates bound +these interpolants uniformly and give \(Q_j=O(\delta^2)\). +The map in Proposition~\ref{prop:map} then gives bounded slopes and, +uniformly on such a subinterval, the limiting equations +\begin{equation}\label{eq:traj-ode} + X'=-Y^2,\qquad Y'=-XY. +\end{equation} +Indeed the scalar map error is \(O(\delta^3)\) there, while a +step has length \(\delta\) and the scalar coordinates were divided +by \(\delta\). Its contribution to the slope tends to zero. +Compactness and integration of the discrete equations therefore +show that every subsequential limit solves +\eqref{eq:traj-ode}. + +Applying \eqref{eq:traj-early-perturbation} with arbitrarily small +\(\eta\) determines its behavior at the left endpoint: +\begin{equation}\label{eq:traj-left-asymptotic} + (X(w),Y(w)) + =\frac{(1,1)}w + -w\bigl(\mathcal Pq_u+O(w^2)\bigr) + \qquad(w\downarrow0). +\end{equation} +The equations preserve \(X^2-Y^2\), whose value is +\(-3\mathcal P\) by \eqref{eq:traj-left-asymptotic}. +Thus \(X'=-(X^2+3\mathcal P)\). Its pole at zero and the +positive sign of \(Y\) near zero give uniquely +\[ + X(w)=k_0\cot(k_0w),\qquad Y(w)=k_0\csc(k_0w). +\] +Every subsequential limit is the same, so convergence holds on +compact subintervals of \((0,\eta]\). + +We extend it to each compact subinterval of \((0,T)\). +Starting from any fixed time in \((0,\eta)\), choose a bound +strictly larger than the supremum of the displayed solution on +the target compact interval. While the discrete scalars divided +by \(\delta\) remain within this bound, remainder contraction +gives \(\|Q_j\|=O(\delta^2)\), so the scalar error in the Euler +approximation tends uniformly to zero. The quadratic vector field +is Lipschitz on this bounded set. The discrete Gronwall inequality +then gives uniform convergence up to the end of the interval and +excludes a first exit from the larger bound. This proves +\eqref{eq:traj-scaled-limit}, including the analogous torus +remainder bounds behind these stops. + +It remains to control the final part of the trajectory, where the +scaled solution diverges but the unscaled activity is still small. +Fix a time \(w_0\in(T/2,T)\). Starting at +\(j_{\mathrm{start}}=\lfloor w_0/\delta\rfloor\), let \(j_e\) be the first +index with \(-x_{j_e}\ge d\), provided such an index exists. +We prove its existence and the estimates in +\eqref{eq:traj-exit} using a cone on which both scalar coordinates +have a definite direction of motion. + +Take a large fixed \(M>k_0\) and let \(w_M\in(T/2,T)\) be the +outgoing time at which \(Y(w_M)=M\). At an index with +\(\delta j\to w_M\), compact convergence gives +\[ + y_j=(M+o(1))\delta, + \qquad -x_j/y_j=1+O(M^{-2})+o(1). +\] +The remainder estimate at this index can be made +\begin{equation}\label{eq:traj-cone-initial} + \|Q_j\|+\|\mathcal R_{j|h}^{\mathrm{tor}}\| + \le K M^2\delta^2 +\end{equation} +with \(K\) independent of large fixed \(M\). Here \(h\) can +be any later stop up to a fixed sufficiently large multiple of +\(\delta^{-1}\). To verify the independence, use the fixed +scaled interval on which the limiting outgoing \(Y\) increases +from \(M/2\) to \(M\). Throughout it \(|W|\le CM\delta\) +for all sufficiently small \(\delta\), with an absolute \(C\), +and its number of discrete steps tends to infinity. Contraction +bounds the forced remainder by \(CM^2\delta^2\) and removes +the contribution from the earlier state. The threshold for +\(\delta\) may depend on \(M\), as is appropriate when \(M\) +is fixed before taking \(\delta\to0\). + +For sufficiently small coordinates consider +\begin{equation}\label{eq:traj-outgoing-cone} + y>0,\qquad \frac12\le q:=-x/y\le2, + \qquad \|Q\|+\|\mathcal R^{\mathrm{tor}}\|\le K_1y^2. +\end{equation} +Choose \(K_1\) sufficiently large for the remainder contraction. +In this cone the scalar map gives +\begin{align} + y_+&=y\{1+qy+O_{K_1}(y^2+j^{-2}y)\}, + \label{eq:traj-y-growth}\\ + q_+-q&=y(1-q^2)+O_{K_1}(y^2+j^{-2}y). + \label{eq:traj-ratio-growth} +\end{align} +At each endpoint of \([1/2,2]\) the leading term in +\eqref{eq:traj-ratio-growth} points inward. It also points +uniformly inward on fixed small neighborhoods of the endpoints. +Away from those neighborhoods a sufficiently small step cannot +cross either endpoint. Thus the ratio condition is preserved +for small \(y\) and large \(j\). Equation~\eqref{eq:traj-y-growth} +gives constants \(0C'/(1-\theta_1)\), and use \(y_+\ge y\). +This applies to the plane remainder and separately to every +stopped torus under consideration. + +For large fixed \(M\), \eqref{eq:traj-cone-initial} places the +trajectory in this cone. Choose \(d\) small enough that the cone +estimates hold until the first crossing of \(-x=d\), including +that crossing. Before it, \(y\le2d\), and reciprocal summation +gives +\[ + \frac1{y_i}-\frac1{y_{i+1}} + =\frac{y_{i+1}-y_i}{y_iy_{i+1}}\ge c'>0. +\] +It follows that the exit occurs within \(C/(M\delta)\) further +steps: otherwise the last inequality would force \(1/y_i<0\). +At the crossing, the change of \(x\) is \(O(y^2)=O(d^2)\). +The cone and remainder bounds therefore give the last three +assertions in \eqref{eq:traj-exit}. + +Compact convergence on \([w_0,T-\zeta]\), for any fixed +\(\zeta>0\), shows that no such exit occurs there when +\(\delta\) is small. Hence \(\liminf\delta j_e\ge T\). +The preceding upper bound gives +\[ + \limsup_{\delta\downarrow0}\delta j_e\le w_M+C/M. +\] +Since \(w_M\uparrow T\) as \(M\to\infty\), this proves +\(\delta j_e\to T\). Throughout this argument all unscaled +coordinates remain small; no activity map is used beyond its +domain. The fixed horizon of stopped tori can be taken larger +than \((T+1)/\delta\), and the exit lies before it for small +\(\delta\). + +Finally apply the variance readout of +Proposition~\ref{prop:readouts}. At a stop with +\(\delta j\to w\in(0,T)\), the scalar activity is \(O(\delta)\) +and the remainder is \(O(\delta^2)\), so the variance converges +to \(a\mathfrak g\). At the exit, the coordinate change gives +\[ + t_{j_e}=-\frac d{b_*}+O(d^2),\qquad + \|\mathcal R_{j_e|j_e}^{\mathrm{tor}}\|+\nu_{j_e}^2=O(d^2). +\] +Here \(\nu_{j_e}\) is the full stopped activity size. The +readout multiplier tends to \(a\mathfrak g\), and therefore +\[ + \Var_{b,n_{j_e}}^{\mathrm{tor}}H_{n_{j_e}} + =a\mathfrak g-\frac{a\mathfrak g}{b_*}d + +O(d^2)+o_{\eps\downarrow0}(1). +\] +To pass from the cutoff observation to \(H_n\), remove the +independent noise and use the sampling comparison and cutoff +bound. Their errors tend to zero both before and at the exit, +since \(\log n_j=O(\delta^{-1})=o(s^2)\) and \(s/n_j\to0\). +Taking \(d\) sufficiently small gives +\eqref{eq:traj-exit-variance}, for example with +\(c_d=a\mathfrak g d/(2b_*)\) after decreasing its allowed range. +\end{proof} diff --git a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b6f57ec90323a6b2059032810a67001b122c3be9 --- /dev/null +++ b/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5630748347a3657af3940a9429b440b420fcc31b6fdc96beedb348ae0b68f5a2 +size 686498 diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/README.md b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..52ac3c556912730a3dd31aca2f303291f0c8156d --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [Étale covers with a prescribed exterior sheet](main.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026, + author = {{OpenAI}}, + title = {{\'{E}tale covers with a prescribed exterior sheet}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf}{OAI:Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/figures/path-to-cycle.tex b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/figures/path-to-cycle.tex new file mode 100644 index 0000000000000000000000000000000000000000..b366cbfc2081c70b4de2aa70c9871f544a747ddf --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/figures/path-to-cycle.tex @@ -0,0 +1,16 @@ +\begin{tikzpicture}[line width=0.65pt, every node/.style={font=\small}] + \draw (0,0) -- (2,0); + \fill (0,0) circle (1.8pt) node[above left] {$a$}; + \fill (2,0) circle (1.8pt) node[above right] {$b$}; + \node at (1,-0.7) {$P_i$, with one edge}; + + \draw[->] (4.2,0) -- (2.8,0); + \node at (3.5,0.4) {$\ell=3$}; + + \draw (5,0) -- (7,0); + \draw (5,0) .. controls (5.6,0.85) and (6.4,0.85) .. (7,0); + \draw (5,0) .. controls (5.6,-0.85) and (6.4,-0.85) .. (7,0); + \fill (5,0) circle (1.8pt) node[left] {$a'$}; + \fill (7,0) circle (1.8pt) node[right] {$b'$}; + \node at (6,-1.1) {inverse subgraph}; +\end{tikzpicture} diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/main.tex b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..c479076c3337714ac2a295a4948a82f4723d8697 --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/main.tex @@ -0,0 +1,46 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[a4paper,margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,amscd,mathtools} +\usepackage{tikz} +\usepackage{microtype} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={Étale covers with a prescribed exterior sheet},pdfauthor={OpenAI}} +\usepackage{enumitem} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\DeclareMathOperator{\Gal}{Gal} +\title{\'{E}tale covers with a prescribed exterior sheet} +\author{OpenAI} +\date{September 24, 2026} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\begin{document} +\maketitle +\begin{abstract} +Let $X$ be a smooth proper hyperbolic curve over an algebraic closure +of a $p$-adic local field. Given finitely many disjoint small open disks, +we construct one connected finite \'etale cover with a sheet isomorphic +to their entire exterior and with degree divisible by $p$ on every +connected component above each disk. +\end{abstract} + +\tableofcontents + +\input{sections/01-statement} +\input{sections/02-cycles} +\input{sections/03-characters} +\input{sections/04-quotient} +\begingroup +\raggedright +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/references.bib b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..489f5c1c1c870956bd8896226f435b59903d70a6 --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/references.bib @@ -0,0 +1,93 @@ +@misc{MochizukiTsujimura, + author = {Mochizuki, Shinichi and Tsujimura, Shota}, + title = {Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for {$p$}-adic Hyperbolic Curves}, + howpublished = {RIMS Preprint 1974}, + year = {2024}, + note = {Revision of March 30, 2024. \href{https://www.kurims.kyoto-u.ac.jp/~stsuji/Resolution%20of%20Nonsingularities%20for%20Arbitrary%20Hyperbolic%20Curves.pdf}{Authors' revised preprint}} +} + +@article{TamagawaRNS, + author = {Tamagawa, Akio}, + title = {Resolution of Nonsingularities of Families of Curves}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + volume = {40}, + number = {4}, + year = {2004}, + pages = {1291--1336}, + doi = {10.2977/PRIMS/1145475448} +} + +@article{LepageRNS, + author = {Lepage, Emmanuel}, + title = {Resolution of Nonsingularities for {Mumford} Curves}, + journal = {Publications of the Research Institute for Mathematical Sciences}, + volume = {49}, + number = {4}, + year = {2013}, + pages = {861--891}, + doi = {10.4171/PRIMS/122} +} + +@book{SGA1, + editor = {Grothendieck, Alexander}, + title = {Rev{\^e}tements {\'e}tales et groupe fondamental ({SGA}~1)}, + series = {Documents Math{\'e}matiques}, + volume = {3}, + publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France}, + address = {Paris}, + year = {2003}, + note = {Recomposed and annotated edition of the 1971 original. \url{https://arxiv.org/abs/math/0206203v2}} +} + +@article{GAGA, + author = {Poineau, J{\'e}r{\^o}me}, + title = {Raccord sur les espaces de {Berkovich}}, + journal = {Algebra \& Number Theory}, + volume = {4}, + number = {3}, + year = {2010}, + pages = {297--334}, + doi = {10.2140/ant.2010.4.297}, + note = {Appendix A, Theorem A.1. \url{https://msp.org/ant/2010/4-3/ant-v4-n3-p03-s.pdf}} +} + +@misc{TemkinAnalytic, + author = {Temkin, Michael}, + title = {Introduction to {Berkovich} analytic spaces}, + year = {2010}, + howpublished = {Lecture notes, Jussieu summer school}, + note = {Section 5.1, \url{https://berkovich-2010.imj-prg.fr/lecture-notes/Temkin-lecture-notes.pdf}} +} + +@misc{MilneAV, + author = {Milne, James S.}, + title = {Abelian Varieties}, + year = {2022}, + note = {Author version, January 2, 2022, of the chapter in Arithmetic Geometry (1986), pp. 103--150. Theorem 8.2 and Remark 8.4. \url{https://www.jmilne.org/math/xnotes/AVs.pdf}} +} + +@article{DeJongCoverings, + author = {de Jong, A. J.}, + title = {\'{E}tale fundamental groups of non-{Archimedean} analytic spaces}, + journal = {Compositio Mathematica}, + volume = {97}, + number = {1--2}, + year = {1995}, + pages = {89--118}, + note = {\url{https://www.numdam.org/item/CM_1995__97_1-2_89_0/}} +} + + +@incollection{BakerPayneRabinoff, + author = {Baker, Matthew and Payne, Sam and Rabinoff, Joseph}, + title = {On the structure of nonarchimedean analytic curves}, + booktitle = {Tropical and Non-Archimedean Geometry}, + series = {Contemporary Mathematics}, + volume = {605}, + publisher = {American Mathematical Society}, + address = {Providence, RI}, + year = {2013}, + pages = {93--121}, + doi = {10.1090/conm/605/12113}, + note = {Preprint version: arXiv:1404.0279v1, \url{https://arxiv.org/abs/1404.0279v1}} +} diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/01-statement.tex b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/01-statement.tex new file mode 100644 index 0000000000000000000000000000000000000000..cb83a84dde0e6612676501b6397e8f67442d71ff --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/01-statement.tex @@ -0,0 +1,83 @@ +\section{The simultaneous covering statement} + +Fix a prime $p$, a finite extension $k/\mathbb Q_p$, an algebraic +closure $\bar k$, and its completion $\Omega=\widehat{\bar k}$. +All analytic spaces below are Berkovich spaces over $\Omega$. +For a curve $X/\bar k$ write $X^{\rm an}=(X_\Omega)^{\rm an}$. +The additive valuation is normalized by $v(p)=1$, so that +$v(\Omega^\times)=\mathbb Q$. + +Let $X/\bar k$ be smooth, proper and connected, of genus at least two. +Choose distinct points $b_1,\ldots,b_d\in X(\bar k)$, where $d\ge1$, +and rational local parameters $t_i$ at these points. Suppose there +are pairwise disjoint open disks $B_i\subset X^{\rm an}$ on which +$t_i$ identifies $B_i$ with $\{|T|0. +\] +It therefore contains an embedded circle, entirely above $V_i$. + +\smallskip\noindent +\emph{Galois refinement preserves the cycles.} +Deck transformations of $Z_0/X$ extend to its stable model. Take the +product over that model of all conjugates of the cyclic cover. After +a finite constant extension, choose a geometric connected component +of its generic fibre and its open-and-closed closure in the normal +product model. On geometric generic fibres this corresponds to the +intersection of the conjugate normal index-$\ell$ subgroups of +$\pi_1(Z_0)$, which is normal in $\pi_1(X)$. The resulting cover +$Z\to X$ is therefore Galois. The chosen component is finite \'etale +and surjective over the model of $Z_1$. + +Above a circle with $q$ edges and $q$ vertices, a degree-$e$ finite +\'etale model cover has $eq$ edges and at most $eq$ vertices. Its +nonempty inverse graph consequently satisfies $b_1=E-V+c\ge1$. +The source model is stable by the preceding argument, so its skeleton +contains an embedded circle above each $V_i$, as required. +\end{proof} diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/03-characters.tex b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/03-characters.tex new file mode 100644 index 0000000000000000000000000000000000000000..f374813972c03784874857ecce17c19e8293a3e9 --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/03-characters.tex @@ -0,0 +1,65 @@ +\section{Graph characters supported above the disks} + +Fix $Z$ as in Proposition~\ref{prop:cycles}, and write +$\tau:Z^{\rm an}\to\Gamma$ for the retraction to its stable skeleton. +Put $U_Z=Z^{\rm an}\times_{X^{\rm an}}U$. + +\begin{lemma}\label{lem:graph-character} +For every $i$ there is a connected cyclic finite \'etale cover of $Z$ +of degree $p$ which is trivial over all of $U_Z$ and whose restriction +to at least one component above $V_i$ is connected. +\end{lemma} +\begin{proof} +Choose an embedded circle $C_i\subset\Gamma$ lying above $V_i$, +and select one of its edges. We will construct a cyclic graph cover +by gluing sheets only across a cut in this edge. To make its pullback +split over all of $U_Z$, the cut must avoid not just $U_Z$ but its +image under the retraction. + +Choose a point $\eta_i$ strictly inside the selected edge at an +irrational position for the normalized valuation. Vertex positions +and edge lengths are rational, so such a point exists. It is of +type three and satisfies +\[ + \tau^{-1}(\eta_i)=\{\eta_i\}. +\] +Indeed side disks attach only at type-two points, whose coordinates +on this edge have valuation in $\mathbb Q$. Since $\eta_i$ lies +above $V_i$, its singleton retraction fibre misses $U_Z$. Thus +$\tau(U_Z)$ avoids $\eta_i$. + +Orient the selected edge and cut it open at $\eta_i$, giving two +new endpoints $\eta_i^-$ and $\eta_i^+$ on its incoming and outgoing +halves. Take $p$ copies of the cut graph, labelled by +$a\in\mathbb F_p$, and identify $\eta_i^-$ in copy $a$ with +$\eta_i^+$ in copy $a+1$. This gives a cyclic graph cover of degree +$p$. Away from $\eta_i$ it is the disjoint union of the $p$ labelled +copies of $\Gamma\setminus\{\eta_i\}$. Traversing $C_i$ in the +selected orientation sends the sheet label $a$ to $a+1$, so the +cover of $C_i$ is connected. + +Pull this graph cover back by $\tau$. It splits over the entire +$U_Z$ because $\tau(U_Z)$ misses the cut. On the component above +$V_i$ containing $C_i$, the loop $C_i$ permutes all $p$ sheets +transitively. The restriction to that component, and hence the +pulled-back cover itself, is connected. + +It remains to realize this topological cover algebraically. A finite +topological covering of a Berkovich curve obtains an analytic structure +by pulling back the structure sheaf on its locally trivial sheets +\cite[Lemma~2.6]{DeJongCoverings}. The local product +algebras glue by their sheet permutations, giving a coherent finite +\'etale analytic algebra. Proper GAGA algebraizes this algebra, +its unit and multiplication, with their identities preserved by +full faithfulness \cite[Appendix~A, Theorem~A.1(ii)]{GAGA}; +\'etaleness is detected after analytification +\cite[Fact~5.1.5]{TemkinAnalytic}. The deck action also algebraizes. +Invariance under the algebraically closed extension $\bar k\subset\Omega$ +descends this cover and its action to $\bar k$ +\cite[Expos\'e~X, Corollary~1.8]{SGA1}. The resulting cyclic finite +\'etale cover has the two restriction properties just established. +\end{proof} + +We have obtained the local nontriviality without sacrificing exterior +splitting. To make it hold on every disk component after one quotient, +we assemble the characters equivariantly. diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/04-quotient.tex b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/04-quotient.tex new file mode 100644 index 0000000000000000000000000000000000000000..5e87df3600303e1cf6280c40c093fd0641e673f1 --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/build/sections/04-quotient.tex @@ -0,0 +1,81 @@ +\section{The stabilizer quotient} + +\begin{proof}[Proof of Theorem~\ref{thm:exterior}] +Take the covers of Lemma~\ref{lem:graph-character} for all $i$, +their conjugates under $\Gal(Z/X)$, and a connected component of their +fibre product over $Z$. Call the resulting curve $T$. The intersection +of their kernels in $\pi_1(Z)$ is normal in $\pi_1(X)$, so $T\to X$ +is connected, finite \'etale and Galois. Put +\[ + G=\Gal(T/X),\qquad P=\Gal(T/Z). +\] +The group $P$ is elementary abelian of exponent $p$, since it embeds +in the product of the cyclic deck groups. All conjugate covers split +over $U_Z$: each deck transformation of $Z/X$ preserves $U_Z$. +Their composite $P$-cover therefore splits over $U_Z$ as well. + +Let $H\subset G$ be the stabilizer of a connected component of the +inverse image of $U$ in $T^{\rm an}$. Splitting over $U_Z$ gives +\begin{equation}\label{eq:disjoint} + H\cap P=1. +\end{equation} +Indeed $P$ permutes freely the trivial sheets above each component +of $U_Z$. + +For a fixed $i$, choose a component $W$ above $V_i$ in $Z$ on +which the cyclic cover from Lemma~\ref{lem:graph-character} is +connected. Let $A$ be a component above $W$ in $T$, and let +$B\subset G$ be its stabilizer. The stabilizer for $T\to Z$ is +$B\cap P$, so the same fibre calculation as in +Lemma~\ref{lem:components} gives $\deg(A/W)=|B\cap P|$. +The map $A\to W$ factors through the connected restriction of +the chosen cyclic cover. The map to that restriction is surjective: +it is finite \'etale, and its nonzero locally constant rank is +positive everywhere on the connected target. Consequently +$p$ divides $|B\cap P|$. +Lemma~\ref{lem:components} gives transitivity of $G$ on all components +over $V_i$, and $P$ is normal in $G$. Consequently +\begin{equation}\label{eq:nontrivial} + B\cap P\ne1 + \quad\text{for every such component stabilizer }B. +\end{equation} + +Set $Y=T/H$. The two quotients fit into the commutative square +\[ +\begin{CD} + T @>{\text{quotient by }P}>> Z\\ + @V{\text{quotient by }H}VV @VVV\\ + Y @>>> X . +\end{CD} +\] +The curve $Y$ is smooth, proper and connected, and the induced map +$g:Y\to X$ is finite \'etale. We use one component calculation for +both required assertions. Let $D$ be $U$ or one of the $V_i$, and +let $A$ be a component of its inverse image in $T$, with stabilizer +$B\subset G$. The inverse image in $T$ of the image of $A$ in $Y$ +is the union of its $H$-translates. This union is open and closed +in the inverse image of $D$, so that image is a connected component +of $g^{-1}(D)$. An +$H$-translate of $A$ is either $A$ or disjoint from it; hence the +image is $A/(B\cap H)$. By Lemma~\ref{lem:components}, its degree +over $D$ is +\[ + [B:B\cap H]. +\] + +For the chosen exterior component, $B=H$, so its image has degree +one over $U$. A finite \'etale map of degree one is an isomorphism, +which proves assertion~\textup{(i)}. + +Now take any component over $V_i$ in $Y$, and choose a component +above it in $T$ with stabilizer $B$. Put $N=B\cap P$. +By~\eqref{eq:nontrivial}, $N$ is a nontrivial normal $p$-subgroup +of $B$. By~\eqref{eq:disjoint}, $N\cap(B\cap H)=1$. Hence +$B\cap H$ injects into $B/N$, and +\[ + [B:B\cap H] + =|N|\,[B/N:\operatorname{im}(B\cap H)] +\] +is divisible by $p$. This proves assertion~\textup{(ii)} for every +component above every fixed disk and completes the proof. +\end{proof} diff --git a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c4975e96422637722288a139d89e977da0a12f3f --- /dev/null +++ b/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:713430a73f1530f5d89cd38050c45a6bc2ccc458878a191917bb37b451b2f26a +size 319576 diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/README.md b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..2cd5f23ffe746f2abf89666a3a6e2341aaaab52b --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/README.md @@ -0,0 +1,17 @@ +# [Eventual categoricity for abstract elementary classes](paper.pdf) + +**Author:** OpenAI +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026, + author = {{OpenAI}}, + title = {{Eventual categoricity for abstract elementary classes}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf}{OAI:Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/figures/proof-route.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/figures/proof-route.tex new file mode 100644 index 0000000000000000000000000000000000000000..d5ebc250bbe36f23ab4c309f68fd41918401ff97 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/figures/proof-route.tex @@ -0,0 +1,39 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[ + node distance=4mm and 8mm, + box/.style={draw=linkblue!70,rounded corners=2pt,align=center, + text width=5.4cm,minimum height=10mm,inner sep=5pt,font=\small}, + closure/.style={box,dashed,text width=4.2cm}, + arr/.style={-{Stealth[length=2mm]},draw=linkblue!80,thick}] +\node[box] (orders) {Unbounded categoricity\\Order presentations and orbit reversal}; +\node[box,below=of orders] (diagrams) {Single-model diagrams\\Uniform testing and compact placement}; +\node[box,below=of diagrams] (calculus) {Working schemes\\Free extension and relative isolation\\Extensions controlled by a mark}; +\node[box,below=of calculus] (point) {Minimal type\\Fixed pointed models}; +\node[box,below=of point] (marked) {Finite marked systems\\Exact boundary lifting\\at fixed depth}; +\node[box,below=of marked] (growth) {No proper extension with unchanged points\\Basis size equals model size on a tail}; +\node[box,below=of growth] (systems) {Geometric systems and relative bases\\Exact lifting for each finite shape}; +\node[box,below=of systems] (tail) {Induction through larger cardinalities\\Categoricity on a tail}; +\foreach \a/\b in {orders/diagrams,diagrams/calculus,calculus/point, + point/marked,growth/systems} + \draw[arr] (\a) -- (\b); +\node[closure,right=of marked] (markedclosure) + {Common marked level\\All finite marked depths}; +\draw[arr,dashed] (marked.east) -- (markedclosure.west); +\draw[arr,dashed] (markedclosure.south) |- (growth.east); +\node[closure,right=of systems] (geometricclosure) + {Common geometric level\\All finite shapes\\Above the no-growth threshold}; +\draw[arr,dashed] (systems.east) -- (geometricclosure.west); +\draw[arr,dashed] (geometricclosure.south) |- (tail.east); +\end{tikzpicture} +\caption{The qualitative proof, ordered by its principal dependencies. +The two dashed boxes are separate applications of +Section~\ref{close:section-uniform}. The first makes all finite marked +depths available for the no-growth argument over the fixed point. The +second is chosen after that argument +and supplies the starting cardinal for the final transfer. These +classes of common levels need not contain a tail. The small-code +reduction then gives the uniform threshold of +Theorem~\ref{main:uniform}.} +\label{fig:proof-route} +\end{figure} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/main.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..e4c4e47a7ed322d0a759c1c7289b0d0b000706c5 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/main.tex @@ -0,0 +1,89 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem,booktabs,longtable,tabularx,needspace,etoolbox} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,fit,calc,shapes.geometric,decorations.pathreplacing} +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{25,65,105} +\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref} +\usepackage{bookmark} +\usepackage[capitalise,noabbrev,nameinlink]{cleveref} +\hypersetup{pdftitle={Eventual categoricity for abstract elementary classes},pdfauthor={OpenAI},pdfsubject={A proof of Shelah's eventual categoricity conjecture in ZFC}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{claim}[theorem]{Claim} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{notation}[theorem]{Notation} +\newtheorem{convention}[theorem]{Convention} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{example}[theorem]{Example} +\forcsvlist{\listadd\resultenvironments}{theorem,lemma,proposition,corollary,claim,definition,notation,convention,remark,example} +\newcommand{\preserveresultheading}[1]{\BeforeBeginEnvironment{#1}{\Needspace{5\baselineskip}}} +\forlistloop{\preserveresultheading}{\resultenvironments} +\newcommand{\preserveresulttype}[1]{% + \AtBeginEnvironment{#1}{\crefalias{theorem}{#1}}} +\forlistloop{\preserveresulttype}{\resultenvironments} +\pretocmd{\subsection}{\Needspace{9\baselineskip}}{}{\errmessage{Subsection spacing hook failed}} +\newcommand{\K}{K} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\DeclareMathOperator{\LS}{LS} +\DeclareMathOperator{\Cat}{Cat} +\DeclareMathOperator{\tp}{tp} +\DeclareMathOperator{\cl}{cl} +\DeclareMathOperator{\cf}{cf} +\DeclareMathOperator{\ci}{ci} +\DeclareMathOperator*{\colim}{colim} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\dom}{dom} +\DeclareMathOperator{\ran}{ran} +\DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\otp}{otp} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\Lim}{Lim} +\DeclareMathOperator{\Ord}{Ord} +\setcounter{tocdepth}{2} +\setcounter{secnumdepth}{3} +\setlist{topsep=5pt,itemsep=3pt} +\setlength{\emergencystretch}{1.5em} +\title{Eventual categoricity for abstract elementary classes} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove Shelah's eventual categoricity conjecture for abstract elementary +classes in ZFC. For each infinite bound on the L\"owenheim--Skolem number +there is a uniform threshold such that categoricity in any one cardinal +at or above that threshold implies categoricity in every cardinal at or +above the same threshold. +\end{abstract} +\tableofcontents +\clearpage +\input{sections/01-introduction} +\input{sections/02-reduction-orders} +\input{sections/03-reversal} +\input{sections/04-diagrams} +\input{sections/05-working-schemes} +\input{sections/06-calculus} +\input{sections/07-isolation} +\input{sections/08-marked-increments} +\input{sections/09-marked-boundaries} +\input{sections/10-geometry} +\input{sections/11-geometric-systems} +\input{sections/12-geometric-boundaries} +\input{sections/13-uniform-closure} +\input{sections/14-cardinal-transfer} +\bibliographystyle{alpha-linked} +\bibliography{references} +\end{document} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/references.bib b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..2c42f5393d95b7a0484cd49440844c0e9da0a0a8 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/references.bib @@ -0,0 +1,207 @@ +@book{Shelah2009, + author = {Shelah, Saharon}, + title = {Classification Theory for Abstract Elementary Classes}, + series = {Studies in Logic (London)}, + volume = {18}, + publisher = {College Publications}, + address = {London}, + year = {2009}, + isbn = {978-1-904987-71-0}, + url = {https://shelah.logic.at/papers/h/} +} + +@misc{BoneyVasey2015, + author = {Boney, Will and Vasey, Sebastien}, + title = {Categoricity and Infinitary Logics}, + year = {2015}, + howpublished = {\url{https://arxiv.org/abs/1508.03316v2}}, + eprint = {1508.03316v2}, + archivePrefix = {arXiv}, + primaryClass = {math.LO}, + note = {Preprint, revised 16 October 2015, 9 pp.} +} + +@article{ShelahVasey2024, + author = {Shelah, Saharon and Vasey, Sebastien}, + title = {Categoricity and Multidimensional Diagrams}, + journal = {Journal of the European Mathematical Society}, + volume = {26}, + number = {7}, + pages = {2301--2372}, + year = {2024}, + doi = {10.4171/JEMS/1477}, + url = {https://ems.press/journals/jems/articles/14297812} +} + +@article{VaseyUniversalI, + author = {Vasey, Sebastien}, + title = {{Shelah}'s Eventual Categoricity Conjecture in Universal Classes: {Part I}}, + journal = {Annals of Pure and Applied Logic}, + volume = {168}, + number = {9}, + pages = {1609--1642}, + year = {2017}, + doi = {10.1016/j.apal.2017.03.003}, + url = {https://doi.org/10.1016/j.apal.2017.03.003} +} + +@article{VaseyUniversalII, + author = {Vasey, Sebastien}, + title = {{Shelah}'s Eventual Categoricity Conjecture in Universal Classes: {Part II}}, + journal = {Selecta Mathematica. New Series}, + volume = {23}, + number = {2}, + pages = {1469--1506}, + year = {2017}, + doi = {10.1007/s00029-016-0296-0}, + url = {https://doi.org/10.1007/s00029-016-0296-0} +} + +@article{VaseyTamePrimes, + author = {Vasey, Sebastien}, + title = {{Shelah}'s Eventual Categoricity Conjecture in Tame Abstract Elementary Classes with Primes}, + journal = {Mathematical Logic Quarterly}, + volume = {64}, + number = {1--2}, + pages = {25--36}, + year = {2018}, + doi = {10.1002/malq.201500068}, + url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.201500068} +} + +@article{GrossbergVanDieren2006, + author = {Grossberg, Rami and VanDieren, Monica}, + title = {Categoricity from One Successor Cardinal in Tame Abstract Elementary Classes}, + journal = {Journal of Mathematical Logic}, + volume = {6}, + number = {2}, + pages = {181--201}, + year = {2006}, + doi = {10.1142/S0219061306000554} +} + +@article{VaseySpectrum, + author = {Vasey, Sebastien}, + title = {The Categoricity Spectrum of Large Abstract Elementary Classes}, + journal = {Selecta Mathematica. New Series}, + volume = {25}, + number = {5}, + eid = {65}, + year = {2019}, + note = {Article 65, 51 pp.}, + doi = {10.1007/s00029-019-0511-x}, + url = {https://doi.org/10.1007/s00029-019-0511-x} +} + +@misc{Espindola2019, + author = {Esp{\'i}ndola, Christian}, + title = {A Proof of {Shelah}'s Eventual Categoricity Conjecture and an Extension to Accessible Categories with Directed Colimits}, + year = {2022}, + howpublished = {\url{https://arxiv.org/abs/1906.09169v7}}, + eprint = {1906.09169v7}, + archivePrefix = {arXiv}, + primaryClass = {math.LO}, + note = {Preprint, version 7, 13 April 2022; first posted 21 June 2019} +} + +@misc{Espindola2023, + author = {Esp{\'i}ndola, Christian}, + title = {A Complete Classification of Categoricity Spectra of Accessible Categories with Directed Colimits}, + year = {2023}, + howpublished = {\url{https://arxiv.org/abs/2301.13167v1}}, + eprint = {2301.13167v1}, + archivePrefix = {arXiv}, + primaryClass = {math.LO}, + note = {Preprint, version 1, 30 January 2023} +} + +@article{Morley1965, + author = {Morley, Michael}, + title = {Categoricity in Power}, + journal = {Transactions of the American Mathematical Society}, + volume = {114}, number = {2}, pages = {514--538}, year = {1965}, + doi = {10.1090/S0002-9947-1965-0175782-0} +} +@article{BaldwinLachlan1971, + author = {Baldwin, J. T. and Lachlan, A. H.}, + title = {On Strongly Minimal Sets}, + journal = {The Journal of Symbolic Logic}, + volume = {36}, number = {1}, pages = {79--96}, year = {1971}, + doi = {10.2307/2271517} +} +@article{ErdosRado1956, + author = {Erd{\H o}s, Paul and Rado, Richard}, + title = {A Partition Calculus in Set Theory}, + journal = {Bulletin of the American Mathematical Society}, + volume = {62}, number = {5}, pages = {427--489}, year = {1956}, + doi = {10.1090/S0002-9904-1956-10036-0} +} +@article{Fodor1956, + author = {Fodor, G.}, + title = {{Eine Bemerkung zur Theorie der regressiven Funktionen}}, + journal = {Acta Scientiarum Mathematicarum (Szeged)}, + volume = {17}, pages = {139--142}, year = {1956}, + url = {https://acta.hu/download.phtml?id=537} +} + +@article{EhrenfeuchtMostowski1956, + author = {Ehrenfeucht, Andrzej and Mostowski, Andrzej}, + title = {Models of axiomatic theories admitting automorphisms}, + journal = {Fundamenta Mathematicae}, + volume = {43}, year = {1956}, pages = {50--68}, + doi = {10.4064/fm-43-1-50-68} +} +@article{Shelah1983B, + author = {Shelah, Saharon}, + title = {Classification theory for nonelementary classes. {I}. The number of + uncountable models of {$\psi\in L_{\omega_1,\omega}$}. {Part B}}, + journal = {Israel Journal of Mathematics}, + volume = {46}, number = {4}, year = {1983}, pages = {241--273}, + doi = {10.1007/BF02762887}, + url = {https://shelah.logic.at/papers/87b/} +} + +@misc{OpenAI2026CHThreshold, + author = {{OpenAI}}, + title = {{A CH obstruction to a prescribed categoricity threshold}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf}{OAI:A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026}}, + year = {2026} +} + + +@incollection{BoneyVasey2017Survey, + author = {Boney, Will and Vasey, Sebastien}, + title = {A survey on tame abstract elementary classes}, + booktitle = {Beyond First Order Model Theory}, + editor = {Iovino, Jos{\'e}}, + publisher = {CRC Press}, + pages = {353--427}, + year = {2017}, + eprint = {1512.00060}, + archivePrefix = {arXiv}, + primaryClass = {math.LO}, + note = {Preprint: \url{https://arxiv.org/abs/1512.00060v4}} +} + +@article{MakkaiShelah1990, + author = {Makkai, Michael and Shelah, Saharon}, + title = {Categoricity of theories in {$L_{\kappa\omega}$}, + with {$\kappa$} a compact cardinal}, + journal = {Annals of Pure and Applied Logic}, + volume = {47}, + number = {1}, + pages = {41--97}, + year = {1990}, + doi = {10.1016/0168-0072(90)90016-U} +} +@article{Boney2014, + author = {Boney, Will}, + title = {Tameness from Large Cardinal Axioms}, + journal = {The Journal of Symbolic Logic}, + volume = {79}, + number = {4}, + pages = {1092--1119}, + year = {2014}, + doi = {10.1017/jsl.2014.30} +} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/01-introduction.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..6c1140977719729fad75324f05f97562f37309ac --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/01-introduction.tex @@ -0,0 +1,308 @@ +\section{Introduction}\label{main:introduction} + +Categoricity asks whether cardinality determines a model up to isomorphism. +For abstract elementary classes, the eventual categoricity conjecture predicts +that uniqueness in one sufficiently large cardinal forces uniqueness in every +cardinal above a uniform threshold. We prove this prediction in ZFC, with the +threshold depending only on a bound for the L\"owenheim--Skolem number. + +\subsection{The statement} + +We specify the axioms, including the union clause used throughout the proof. +All structures have sets as their universes, and all languages are set-sized +and finitary. We write $|M|$ for the universe of a structure and $\|M\|$ for +its cardinality. + +\begin{definition}[Abstract elementary class]\label{main:aec} +An \emph{abstract elementary class} (AEC) is a pair $K=(\mathcal C,\le_K)$, +where $\mathcal C$ is a class of structures in a fixed language $\tau$ and +$\le_K$ is a partial order satisfying the following conditions. +\begin{enumerate}[label=\textup{(\arabic*)}] +\item The class is closed under isomorphism. If $M\le_K N$, then $M$ is a +$\tau$-substructure of $N$. If $f:N\cong N'$ and $M\le_K N$, then +$f[M]\le_K N'$. +\item \emph{Coherence:} if $M_0$ is a $\tau$-substructure of $M_1$, and +$M_0\le_K M_2$ and $M_1\le_K M_2$, then $M_0\le_K M_1$. +\item For every nonzero ordinal $\delta$ and increasing chain +$(M_i:i<\delta)$ under $\le_K$, its union $U$ belongs to $\mathcal C$ and +$M_i\le_K U$ for all $i<\delta$. If also $M_i\le_K N$ for every $i$, then +$U\le_K N$. +\item There is an infinite cardinal $\ell\ge |\tau|+\aleph_0$ such that, +for every $M\in\mathcal C$ and $A\subseteq |M|$, there is $N\le_K M$ with +$A\subseteq |N|$ and $\|N\|\le |A|+\ell$. +\end{enumerate} +The least possible $\ell$ is denoted $\LS(K)$. A \emph{strong map} is an +isomorphism onto a strong substructure of its codomain. The class is +\emph{categorical in $\kappa$} if it has a model of cardinality $\kappa$ and +all its models of that cardinality are isomorphic. +\end{definition} + +The chain axioms equivalently give unions and the common-upper-bound clause +for nonempty set-indexed directed systems of strong inclusions. We also use +the corresponding direct limits of coherent strong maps, obtained by +renaming their injective images. No intersection axiom is included. + +\begin{theorem}[Uniform eventual categoricity]\label{main:uniform} +For every infinite cardinal $\lambda$ there is a cardinal +$\mu(\lambda)\ge\lambda$ such that the following holds for every AEC $K$ +with $\LS(K)\le\lambda$: +\[ + \bigl(\exists\kappa\ge\mu(\lambda)\quad K\text{ is categorical in }\kappa\bigr) + \quad\Longrightarrow\quad + \bigl(\forall\kappa'\ge\mu(\lambda)\quad + K\text{ is categorical in }\kappa'\bigr). +\] +The same threshold is used in the premise and the conclusion. +\end{theorem} + +In particular, Theorem~\ref{main:uniform} resolves Shelah's eventual +categoricity conjecture positively. No amalgamation, joint embedding, +tameness, no-maximal-models, or arbitrarily-large-models hypothesis is +assumed. The starting categoricity cardinal may be a successor, a regular +limit, or a singular cardinal. The theorem asserts the existence of a +threshold, without prescribing a numerical bound for it. + +On the resulting categorical tail, every model strongly embeds into every +model of at least its cardinality. Indeed, the L\"owenheim--Skolem axiom +supplies a strong submodel of the smaller cardinality, and categoricity +identifies it with the given model. Taking a larger target gives proper +strong extensions and a common target for any two models. Thus the tail +has joint embedding and no maximal models. + +It is useful to separate the uniform quantifier from the main structural +argument. + +\begin{theorem}[The qualitative form]\label{main:qualitative} +An AEC categorical in unboundedly many cardinals is categorical on a tail of +the cardinals. +\end{theorem} + +Here ``unbounded'' means unbounded in the class of all cardinals, and a tail +contains every cardinal above some fixed cardinal. The set-coding argument +in Proposition~\ref{found:uniform-reduction} proves the equivalence of the +two theorems. Once that reduction is made, arbitrarily large models are +available. The order construction in Theorem~\ref{found:order-template} +then supplies actual models at every sufficiently large cardinal; subsequent +categoricity arguments concern both existence and uniqueness. + +The uniform theorem applies in particular to sentences of +$L_{\omega_1,\omega}$ in countable languages, where countable conjunctions +and disjunctions are allowed but quantifier strings are finite. +For such a sentence $\varphi$, choose a countable fragment containing it +and order its models by elementarity for that fragment. This is an AEC +with L\"owenheim--Skolem number $\aleph_0$ +\cite[Example~2.0.2]{BoneyVasey2017Survey}. The models and their +isomorphisms are unchanged, so the same $\mu(\aleph_0)$ works for all +these sentences. The prescribed $\beth_{\omega_1}$ threshold in the +classical infinitary formulation remains a separate question +\cite[Conjecture~1.1]{BoneyVasey2015}. + +\subsection{History and scope} + +Morley's categoricity theorem gives the countable first-order prototype: +a countable theory categorical in one uncountable cardinal is categorical +in every uncountable cardinal \cite{Morley1965}. Its structural approach +connects uniqueness of models with stability and dimension. The geometry +of strongly minimal sets, developed by Baldwin and Lachlan +\cite[Sections~1--2]{BaldwinLachlan1971}, provides a central antecedent for +classification by bases. In an AEC, elementary compactness and the +first-order description of types are not part of the axioms. The +corresponding amalgamation, independence, and geometry therefore require +separate constructions. + +Shelah's eventual categoricity conjecture asks for an eventual dichotomy +between categoricity and noncategoricity; see +\cite[Conjecture~N.4.2]{Shelah2009} and \cite{BoneyVasey2015}. +The equivalent formulation using one threshold depending on an LS bound +is discussed in \cite[Remark~1.3]{VaseyUniversalI}. Our set-coding +argument makes that reduction explicit: it treats bounded categoricity +spectra as well as classes categorical at unboundedly many cardinals. +An explicit numerical threshold is a further question, separate from the +existential threshold in Theorem~\ref{main:uniform}; compare +\cite[Conjecture~N.4.3(1)(b)]{Shelah2009}. +Under CH, the companion \cite[Theorem~1.1]{OpenAI2026CHThreshold} +constructs an AEC $K$ with $\LS(K)=\aleph_0$ that is categorical on a +higher tail but not at $\beth_{(2^{\aleph_0})^+}=\beth_{\omega_2}$, +ruling out this prescribed endpoint for arbitrary AECs. + +Substantial special cases have been established. Makkai and Shelah +proved transfer from sufficiently large successor categoricity for +$L_{\kappa,\omega}$-theories when $\kappa$ is strongly compact +\cite[Conclusion~5.1]{MakkaiShelah1990}. Grossberg and VanDieren +obtained upward categoricity transfer from one sufficiently large successor +cardinal for tame AECs with amalgamation, joint embedding, and arbitrarily +large models \cite[Theorem~5.2]{GrossbergVanDieren2006}. +Boney subsequently derived tameness from strongly compact cardinals and, +combining it with the Grossberg--VanDieren transfer, established eventual +categoricity for arbitrary AECs with a successor starting cardinal under +a proper class of strongly compact cardinals \cite{Boney2014}. +These results explain the importance of removing restrictions on the +starting cardinal as well as structural hypotheses on the class. + +Universal classes admit +a ZFC transfer without assuming amalgamation +\cite[Theorem~7.3]{VaseyUniversalII}. Tameness, amalgamation, and primes +give another transfer \cite[Theorem~3.8]{VaseyTamePrimes}. +Shelah and Vasey prove a ZFC categoricity theorem for excellent AECs and +obtain broader results using strongly compact cardinals, or the weak +generalized continuum hypothesis (WGCH), that is, +$2^\kappa<2^{\kappa^+}$ for every infinite cardinal $\kappa$, together +with amalgamation +\cite[Theorem~14.2 and Corollaries~14.7, 14.10]{ShelahVasey2024}. +Under WGCH, amalgamation, and arbitrarily large models, Vasey also gives +a classification of categoricity spectra +\cite[Corollary~9.7]{VaseySpectrum}. These hypotheses supply substantial +structure. Here the amalgamation, bounded-tuple calculus, and geometry +needed for the working schemes are constructed at selected levels from +the AEC axioms and unbounded categoricity. + +The proof uses several established methods in forms that are proved +locally. Order presentations originate in the work of Ehrenfeucht and +Mostowski \cite{EhrenfeuchtMostowski1956}; their AEC formulation and +functorial maps are developed in +\cite[IV.0.8--IV.0.9 and IV.0.12]{Shelah2009}. We retain a finite-support +presentation throughout the proof. Its extraction uses classical +partition-calculus methods \cite{ErdosRado1956}, while the later +isolation argument uses Fodor's pressing-down method \cite{Fodor1956}. +Stationary independence and local character are central to Shelah's good +frames \cite[II.2.1]{Shelah2009}. Sections~\ref{calc:section-calculus} +and~\ref{iso:section-isolation} establish the bounded-tuple calculus, +relative isolation, and dominated extensions required for the systems +used here. + +A version-sensitive issue matters for the order argument. Boney and +Vasey \cite[Fact~1.4 and the discussion of Claim~1.5, pp.~3--4]{BoneyVasey2015} +identify a gap in the unrestricted syntactic-characterization claim of +Shelah's Chapter~IV and withdraw an earlier attempted repair. Their +valid variants with exponentiation or cofinality assumptions do not +cover all the starting cardinals of Theorem~\ref{main:uniform}. We prove +the required reversal of strong maps directly in +Theorem~\ref{found:reversal}. + +Finite systems and dimension-raising arguments have a long history in +nonelementary categoricity. Shelah's existence and uniqueness methods +\cite[Sections~3--5]{Shelah1983B} are antecedents of the passage from +one additional dimension to a larger cardinal. The modern treatment in +\cite[Definition~8.16 and Lemma~11.6]{ShelahVasey2024} distinguishes +amalgamated completion from strong uniqueness onto a prescribed target. +Both properties are needed below and are proved separately. Our final +cardinal induction uses exact boundary maps for systems whose basis +pieces may have different cardinalities. + +Esp\'indola has announced eventual categoricity for arbitrary AECs and +further accessible-category results \cite{Espindola2019}, and subsequently +an explicit-threshold categoricity theorem \cite[Theorem~4.1]{Espindola2023}. +These are separate public claims. The argument here constructs its +working-scheme calculus, isolation, marked and geometric boundary lifting, +and common-level closure directly. None of these interfaces is imported +from those preprints. + +\subsection{Proof overview} + +By the set-coding reduction in Section~\ref{found:section-reduction-orders}, +it suffices to fix one AEC $\K$ categorical in unboundedly many cardinals +and prove categoricity on a tail. The proof first builds structure at +selected cardinals, called working levels, and only at the end transfers +it to every larger cardinal. A tuple is bounded at a level $\mu$ when +its length is strictly less than $\mu$. + +\paragraph{From orders to comparison of tuples.} +An order presentation $E$ assigns a model to a linear order: each element +is represented by a label and finitely many indices, and an order embedding +induces a strong map by substituting indices. Section~\ref{found:section-reversal} +uses specially chosen orders to control the automorphism orbits of short +tuples. A small subpresentation can be aligned with a larger presentation +so that all its short tuples have matching orbit records. An elimination +chain then proves that a strong map can be reversed on any prescribed +short tuple. The small model supplies the alignment; the larger model +accommodates the enumeration of the small model's tuples. This avoids an +exponentiation hypothesis at the categoricity cardinal. + +In Section~\ref{lev:section-diagrams}, the resulting equivalence classes +of tuples under strong maps stabilize as the categoricity cardinal +increases. These are +the comparison diagrams used later. Finite supports give two further +tools: a bound on the parameter sets needed to distinguish comparisons, +and simultaneous placement of coherent bounded prescriptions. Together +they yield amalgamation and homogeneity for bounded tuples at suitable +countably cofinal closure levels. + +\paragraph{A calculus that can be applied to systems.} +The same tools will be needed for finite systems of models, not only for +individual models. Section~\ref{lev:section-working-schemes} records their +precise hypotheses as a working scheme: a finite system shape, specified +arrows, one finite-support presentation, and compatible tuple comparisons. +Sections~\ref{calc:section-calculus} and~\ref{iso:section-isolation} derive +the calculus from this package. A type over a base is determined by +identification of tuples in an amalgam over that base. Its distinguished +free extensions are computed by long approximation sequences already in +the base. The isolation construction gives extensions controlled by +a specified mark: the full type of each additional bounded tuple, together +with the mark, is determined by the mark's type and a bounded fragment of +the base. Such extensions have the primeness and uniqueness needed for +iteration. + +\paragraph{Marked systems detect growth of a geometry.} +Section~\ref{mark:section-increments} shows that adjoining a finite mark +and an extension controlled in this way again gives a working scheme. +Iterating produces a model at each subset of a finite axis set, with +commuting strong maps. Its boundary consists of the proper-subset +vertices and their maps. Section~\ref{iter:section-boundaries} first +completes a compatible boundary map into a common extension, then extends +a boundary isomorphism onto the prescribed target. The return-tree +argument keeps full maps coherent along individual branches but retains +only boundary maps in the common accumulator. This distinction lets the +orbit bound control joint comparisons across different faces, including +their actual equalities. + +Section~\ref{geom:section-geometry} chooses a minimal singleton type, +one with a unique nonalgebraic extension over each relevant base. Its +realizations form the points of a pregeometry, with independence defined +by finite products of the type. We retain a distinguished strong copy of +its base model $M_0$, with a fixed diagram for its enumeration, in the +later models; these are the pointed models used below. Exact lifting for all finite marked +systems at one common level rules out a proper strong extension with no +new points. A downward construction then extends this no-growth +conclusion to arbitrary larger cardinalities. Thus the geometry detects +every proper extension, and its basis cardinality equals model +cardinality above the resulting threshold. + +\paragraph{From geometric systems to a categorical tail.} +Sections~\ref{sys:section-geometric-systems} and~\ref{exact:section-boundaries} +organize bases into finitely many pieces assigned to vertices. Tiled +order presentations give systems with these pieces; a paired induction +establishes their working packages and classification by prescribed +basis maps. A second boundary argument gives exact lifting for every +finite geometric shape at suitable levels. + +The two finite-dimensional applications require separate common-level +choices. Section~\ref{close:section-uniform} supplies both: fixed set codes +for all finite constructions and their bound requests produce a level +where every finite depth is available. The geometric application is made +after the marked application has supplied the no-growth threshold. +Finally, Section~\ref{tail:section-transfer} starts from a common geometric +level and inducts on the maximum cardinality of the basis pieces, +simultaneously for all finite shapes. One additional axis records the +previously constructed isomorphism at each extension step; coherent +unions handle limits. The zero-axis case gives pointed categoricity. +The L\"owenheim--Skolem axiom and categoricity at one fixed smaller +cardinal place the chosen base pattern in every sufficiently large model. +Forgetting that base gives the qualitative theorem, and the set-code +reduction gives the uniform theorem. + +Figure~\ref{fig:proof-route} summarizes the proof and the two uses of +common-level closure. + +\input{figures/proof-route} + +\subsection{Conventions} + +All size variables are cardinals unless explicitly called ordinals. +An assertion about bounded tuples at a level $\mu$ concerns tuples of +length strictly below $\mu$. Eligible working levels have countable +cofinality and are strong limits closed under the displayed bounds. +Whenever smaller adequate levels are needed cofinally below a working +level, that requirement is stated and included in the closure bookkeeping. +Automorphisms of an index order always mean automorphisms of its pure +linear order. Charts, auxiliary hulls, and basis choices are not added to +the structures whose isomorphisms are being compared. diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/02-reduction-orders.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/02-reduction-orders.tex new file mode 100644 index 0000000000000000000000000000000000000000..61f2dbf9c113ff7b5d1b851df3572eace7b20f92 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/02-reduction-orders.tex @@ -0,0 +1,387 @@ +\section{Set codes and models indexed by orders} +\label{found:section-reduction-orders} + +We first separate the uniformity over AECs from the transfer argument for +one AEC. All directed families below are nonempty and set-indexed. We +begin by deriving the directed forms of the chain axioms in +Definition~\ref{main:aec}. + +\begin{lemma}[Directed unions and coherent direct limits] +\label{found:directed-colimits} +If a nonempty set of models is directed under strong inclusion, its +union belongs to $\K$ and every member is strong in the union. If +every member is strong in a common model $N$, its union is strong in +$N$. More generally, a coherent diagram of strong maps indexed by a +nonempty directed partial order has a direct limit in $\K$, with +strong canonical maps. Every compatible family of strong maps from +the diagram into a model induces a strong map from the direct limit. +\end{lemma} + +\begin{proof} +The union is a well-defined $\tau$-structure: finitely many +arguments are contained in one family member, and any two +choices of such a member agree after passing to a common +strong upper member. Here finitarity is used for functions +and relations alike. +We prove the union and common-upper-bound assertions together by +induction on the cardinality of the family. A finite directed +family has a member containing every other member, so its union is +that member. A countable directed family has a cofinal increasing +sequence: enumerate the family and successively choose a member +above the previous choice and the next enumerated member. The +chain axioms apply to this sequence, including their +common-upper-bound clause. + +Suppose the family is indexed by a set $I$ of uncountable +cardinality $\kappa$, and the two assertions hold for smaller +families. Choose an operation assigning to each pair of indices +an index of a strong upper bound. Express $I$ as the union of an +increasing continuous sequence $(I_\alpha)_{\alpha<\cf(\kappa)}$ +of nonempty subsets of size less than $\kappa$, each closed under +that operation. To obtain this sequence, enumerate $I$ by +$\kappa$, use a cofinal sequence of ordinals below $\kappa$ +to specify the required initial segments, close each stage +under the binary operation by countably many iterations, and +take unions at limit stages. A union of fewer than +$\cf(\kappa)$ sets of size less than $\kappa$ still has size +less than $\kappa$; countable closure preserves the size of +each infinite stage. This construction also works when +$\kappa$ is a successor cardinal, using $\kappa$ many stages. + +Let $U_\alpha$ be the union of the family on $I_\alpha$. The +induction hypothesis gives $U_\alpha\in\K$. For $\alpha<\beta$, +every member indexed by $I_\alpha$ is strong in $U_\beta$. +The inductive common-upper-bound assertion therefore gives +$U_\alpha\leq_\K U_\beta$. The chain axioms now give the +required union over $I$. If the original family is strong in +$N$, induction first gives $U_\alpha\leq_\K N$ for every +$\alpha$, and the chain common-upper-bound clause gives the +same conclusion for their union. This completes the induction. + +For a diagram $(M_i,f_{ij})$ indexed by a directed partial order, +form the quotient of the disjoint union of its underlying sets +by +\[ +(i,x)\sim(j,y) +\quad\Longleftrightarrow\quad +(\exists k\geq i,j)\ f_{ik}(x)=f_{jk}(y). +\] +Directedness and coherence make this an equivalence relation. +The canonical map from each $M_i$ is injective. Interpret any +finitary function or relation in a common later model containing +representatives of its arguments; coherence and the fact that +the maps are structure embeddings make this interpretation +well-defined. The images of the $M_i$ are models forming a +directed family of strong inclusions. The first assertion puts +their union, which is the quotient just constructed, in $\K$ +and makes all canonical maps strong. + +Finally, compatible strong maps $g_i:M_i\longrightarrow N$ +induce a map on that quotient. It is injective: if two images +agree, compare their representatives in a common later $M_k$ +and use injectivity of $g_k$. The same argument gives a +structure embedding. Its image is the directed union of the +strong submodels $g_i[M_i]$ of $N$, so the +common-upper-bound assertion makes the induced map strong. +The quotient construction also gives uniqueness, and hence +the stated direct-limit property. +\end{proof} + +\begin{lemma}[Recovery from small models and pairs] +\label{found:small-code} +Fix an infinite cardinal $\lambda$ and a finitary language $\tau$ of +cardinality at most $\lambda$. An AEC in $\tau$ with +$\LS(\K)\leq\lambda$ is determined by its models of cardinality at most +$\lambda$ and its strong inclusions between such models, both taken up to +isomorphism. These data have a set code, and the recovered class and +strong-submodel relation are uniformly first-order definable in set +theory from that code. +\end{lemma} + +\begin{proof} +Represent small structures on ordinals of cardinality at most $\lambda$. +Represent a small pair $M\subseteq N$ by such a representative for $N$ +together with the distinguished subset that is the domain of $M$. +The possible structures and pairs form sets. A code specifies which +structure types and which pair types are allowed. Testing whether a +given small structure or pair has an allowed type is a set-theoretic +condition: quantify over the relevant isomorphisms. + +For an actual $M\in\K$, let +\[ +\mathcal D(M)=\{A\leq_\K M:\|A\|\leq\lambda\}. +\] +This is a set. It is nonempty by the LS axiom applied to the empty set, +and its union is $M$ by the LS axiom applied to singletons. If +$A,B\in\mathcal D(M)$, the LS axiom inside $M$ supplies a small strong +submodel containing $|A|\cup|B|$. Coherence makes both $A$ and $B$ strong +in that submodel. Thus $\mathcal D(M)$ is directed. + +Conversely, suppose a structure $M$ is the union of a nonempty directed +family of small structures whose membership and pair inclusions are +allowed by the code of $\K$. These are actual members and strong +inclusions of $\K$, so the directed-union axiom gives $M\in\K$. +Consequently the following condition recovers membership, including for +structures not already assumed to belong to $\K$: +\begin{quote} +$M$ is a union of a nonempty directed family of allowed small models, +with all its directed inclusions allowed by the small-pair code. +\end{quote} + +To recover the strong relation, require a directed presentation +$N=\bigcup\mathcal D$ of this kind and a nonempty directed subfamily +$\mathcal E\subseteq\mathcal D$ with $M=\bigcup\mathcal E$. +If $M\leq_\K N$, use $\mathcal D=\mathcal D(N)$ and +$\mathcal E=\mathcal D(M)$; transitivity makes the second a subfamily +of the first. In the reverse direction, every member of $\mathcal E$ +is strong in $N$ by the union clause for $\mathcal D$. The +common-upper-bound clause for $\mathcal E$ then gives $M\leq_\K N$. + +Each presentation can be quantified over as a set of substructures of +the structure being decoded. All the preceding conditions are thus +first-order set-theoretic formulas with the small code as parameter. +They recover both the class and the order, as asserted. +\end{proof} + +The equivalence below is the usual Hanf-number reduction; compare +\cite[Remark~1.3]{VaseyUniversalI}. We spell out the set coding to make +the uniform quantifier explicit. + +\begin{proposition}[The uniform reduction] +\label{found:uniform-reduction} +The uniform assertion in Theorem~\ref{main:uniform} is equivalent to +the following qualitative assertion: +\begin{quote} +Every AEC categorical in unboundedly many cardinals is categorical in +all cardinals above some bound depending on that AEC. +\end{quote} +\end{proposition} + +\begin{proof} +At a fixed LS bound $\lambda$, the languages can be renamed into a +fixed set of symbols with specified finite arities. By +Lemma~\ref{found:small-code}, the possible AECs in those languages are +determined by members of a set of small codes. A code is valid exactly +when its decoded class and relation satisfy the AEC axioms with LS +bound $\lambda$. This is a first-order condition on the code, so the +valid codes form a set. + +Assume the qualitative assertion. For each valid code $c$, its +categoricity spectrum is either bounded or unbounded. In the bounded +case choose the least infinite cardinal $b_c$ strictly above every +categoricity cardinal. In the unbounded case choose the least infinite +cardinal $b_c$ such that the decoded AEC is categorical in every +cardinal at least $b_c$. These choices are definable, and the second +choice exists by the qualitative assertion. Replacement bounds the +set of all $b_c$. Choose a cardinal $\mu(\lambda)\geq\lambda$ strictly +above that set of bounds. If an AEC with LS number at most $\lambda$ +is categorical at some $\kappa\geq\mu(\lambda)$, its code cannot be +in the bounded-spectrum case. It is therefore categorical at every +$\kappa'\geq b_c$, in particular at every +$\kappa'\geq\mu(\lambda)$. This is exactly the same threshold in +the premise and conclusion. + +Conversely, for an AEC categorical unboundedly often, apply the uniform +assertion with $\lambda=\LS(\K)$. Some categoricity cardinal crosses +the resulting threshold, and the asserted tail follows. +\end{proof} + +\begin{remark} +Lemma~\ref{found:small-code} also supplies the usual ZFC interpretation +of quantification over AECs: one quantifies over their set codes and +uses the decoded class predicates. No choice of representatives from +a proper class and no class choice axiom is needed. Below we fix one +AEC categorical unboundedly often and write $\ell=\LS(\K)$. It has +arbitrarily large models because categoricity includes existence. +\end{remark} + +\subsection{A finite-arity partition argument} + +The following end-homogeneous argument belongs to classical partition +calculus; compare Erd\H{o}s and Rado +\cite[Theorem~39 and its corollary]{ErdosRado1956}. We prove the +finite-arity form and bounds used by the order-template construction. + +\begin{lemma} +\label{found:partition} +For infinite cardinals $\xi,\nu$ and a positive integer $k$, there is +an infinite cardinal $R_k(\xi,\nu)$ such that every coloring of the +$k$-element subsets of an ordinal of cardinality at least +$R_k(\xi,\nu)$ into $\xi$ colors has a homogeneous subset of +cardinality $\nu$. Finitely many specified arities can be made +homogeneous simultaneously by enlarging the initial cardinal. +\end{lemma} + +\begin{proof} +For $k=1$, a set of cardinality greater than $\xi\cdot\nu$ cannot be +partitioned into $\xi$ sets all of cardinality less than $\nu$. +Suppose the assertion is known for $k$. Choose an infinite initial +ordinal $\alpha$ large enough for that $k$-ary assertion with target +$\nu$, and put $b=\xi^{|\alpha|}$. Consider a coloring $c$ of the +$(k+1)$-element subsets of an ordinal $\Omega$ with $|\Omega|>b$. + +Construct a tree of residual subsets of $\Omega$ through height +$\alpha$. Its root is $\Omega$. At a nonempty node, name and remove +its least point $x$. The previously named points on its branch, +together with $x$, form an increasing sequence. Partition the +remaining residual set according to all values +\[ +u\longmapsto c(s\cup\{u\}), +\qquad +s\text{ a $k$-subset of that sequence}. +\] +The nonempty parts are the children. At a limit height, a branch has +as its residual set the intersection of the earlier residual sets; +retain the node precisely when this intersection is nonempty. + +At each node there are at most $|\alpha|$ coordinates to record and +hence at most $b$ children. The total number of possible nodes +through height $\alpha$ is at most +$b^{|\alpha|}=b$. This includes the limit nodes, which are specified +by their branches. At most $b$ points are removed as node labels. +Choose $u\in\Omega$ which is never removed. At every height it +belongs to one residual set; at limit heights it belongs to the +intersection on that branch. Consequently it determines a branch +with $\alpha$ named points and nonempty remainder. + +For every $k$-subset $s$ of these named points, all later named points +give the same value of $c(s\cup\{x\})$: that value was fixed by the +partition immediately after the last point of $s$ was named. +Therefore $c$ on increasing $(k+1)$-tuples from the branch depends +only on their first $k$ entries. Apply the $k$-ary induction +hypothesis to this induced coloring on the branch. Its homogeneous +$\nu$-subset is homogeneous for $c$ as well. For example, the +recursion just proved allows +$R_{k+1}(\xi,\nu)=(\xi^{R_k(\xi,\nu)})^+$ after increasing the +initial $R_1$ as needed. + +For a finite list of arities, work backward to choose a bound large +enough for the last desired thinning after all the earlier ones. +Successive thinning preserves homogeneity already obtained. This +proves the final assertion. +\end{proof} + +\subsection{A fixed presentation on every order} + +The construction is an Ehrenfeucht--Mostowski presentation +\cite{EhrenfeuchtMostowski1956}; for its AEC form compare +\cite[Definition~IV.0.8 and Claims~IV.0.9, IV.0.12]{Shelah2009}. +We include the finite-support construction in the label notation used below. + +\begin{theorem}[Order presentation] +\label{found:order-template} +Suppose $\K$ has arbitrarily large models and $\LS(\K)=\ell$. +There are at most $\ell$ labels, each of positive finite arity, and a +functor $E$ from nonempty linear orders and order embeddings to +$\K$ and strong maps, with the following properties. +\begin{enumerate} +\item Every element of $E(I)$ is a label evaluated on an increasing +finite list from $I$. Equality and all structure relations and +function values are determined by the labels and the order/equality +pattern of the finitely many indices involved. +\item $E(I)$ contains distinct distinguished elements $(a_i)_{i\in I}$. +An order embedding $f:I\longrightarrow J$ sends $a_i$ to $a_{f(i)}$, +and its action on any labeled element is position substitution. +\item For every finite nonempty $s\subseteq I$, the elements labeled +on $s$ form a strong submodel $E_s(I)$ of cardinality at most $\ell$. +For $s\subseteq t$ these submodels are strongly included, and +$E(I)=\bigcup_s E_s(I)$. +\item $|I|\leq\|E(I)\|\leq |I|+\ell$. In particular, for every +infinite $\kappa\geq\ell$, $\K$ has a model of cardinality exactly +$\kappa$. +\end{enumerate} +Here a subpresentation is identified with its image under the +canonical strong map when an inclusion of underlying universes is +convenient. +\end{theorem} + +\begin{proof} +Take an arbitrarily large $M\in\K$ and distinct elements +$(a_i)_{i\in A}$ indexed by a large ordinal $A$. Recursively on the +positive finite size of $s\subseteq A$, choose $M_s\leq_\K M$ of +cardinality at most $\ell$ containing the displayed elements on $s$ +and every $M_t$ for nonempty $t\subsetneq s$. There are only finitely +many such $t$, so the LS axiom applies with bound $\ell$. Coherence +gives $M_t\leq_\K M_s$. + +Choose a surjective enumeration +$(m_{s,\eta})_{\eta<\ell}$ of each $M_s$, allowing repetitions. If +$s=\{i_0<\cdots0},\quad \rk(v),\rk(w)0$. +It sends the first displayed vector to the second. Combining this +map with translations sends any ordered pair of distinct points to +any other such pair. Restricting gives the interval assertion. + +Finally, an increasing map on ordinal positions preserves the +greatest differing position of two integer vectors. The induced +map on these integer vectors preserves the greatest differing +position of rational vectors as well. This proves the embedding +claim. Expanding the finitely many integer-vector positions used +by a rational vector gives the asserted finite code and comparison +rule. +\end{proof} + +\begin{lemma}[Balance of cuts] +\label{found:balanced-cuts} +Every interior nonprincipal cut of $\Gamma_\delta$, and every such +cut of $H(\delta)$, is balanced. +\end{lemma} + +\begin{proof} +First consider a cut $(L,R)$ of $\Gamma_\delta$. Among the ranks of +$r-l$, with $l\in L$ and $r\in R$, choose the least ordinal +$\alpha$. Fix one crossing pair of that rank and its common head +above $\alpha$. In that head, each fiber with a fixed integer +coefficient at $\alpha$ lies wholly on one side: a crossing pair +within a fiber would have smaller rank. The resulting cut in the +integer coefficients separates two adjacent integers. A point +with a head above the fixed head would exceed the fixed upper +point, and a point with a head below it would precede the fixed +lower point. The two bordering fibers are therefore cofinal +and coinitial toward the original cut, and their tails are +copies of $\Z^{(\alpha)}$. Their end +characters agree by negation in that tail group. The case +$\alpha=0$ would give a last lower and a first upper point, and +is excluded. This proves balance in $\Gamma_\delta$. + +Now let $(L,R)$ be an interior nonprincipal cut of $H=H(\delta)$, +and put +\[ +T=\{\rk(r-l):l\in L,\ r\in R\}\subseteq\Gamma_\delta. +\] +This is a final segment: from a crossing pair of rank $g$, subtract +a positive basis vector at any $h>g$ from its lower point. The new +pair still crosses and has rank $h$. + +Suppose first that $T$ has a least member $g$. Fix a crossing head +above $g$. Every fiber with a fixed rational coefficient at $g$ +lies on just one side of the cut. Otherwise a crossing difference +would have rank below $g$. The induced cut of $\Q$ is nonempty +and proper. If neither coefficient side has a nearest rational, +countable rational approximations are cofinal and coinitial in +the original cut. If the lower coefficients have a greatest +member, the entire fiber at that coefficient lies on the lower +side and is cofinal there. Its tail, supported below $g$, has +greatest available rank $g-e_0$, and the vectors +$n u_{g-e_0}$ and $-n u_{g-e_0}$ show that both its end characters +are $\aleph_0$. On the upper side, rational coefficients decreasing +to the greatest lower coefficient give coinitiality $\aleph_0$. +If instead the upper coefficients have a least member, the +reversed calculation gives cofinality and coinitiality +$\aleph_0$. These cases exhaust the rational cuts, so the original +cut is balanced. + +Suppose that $T$ has no least member, and let +$\epsilon=\ci(T)$. Put $S=\Gamma_\delta\setminus T$ and let +$H_S$ be the subgroup supported on $S$. We prove that the lower +character is $\epsilon$ by separating two cases. + +Assume there is $l\in L$ for which +\[ +\{\rk(r-l):r\in R\} +\quad\text{is coinitial in }T. +\] +The coset $l+H_S$ is contained in $L$, since a crossing within it +would have rank in $S$. It is cofinal in $L$. In fact, if a point +$l'\in L$ lay above the entire coset, $l'-l$ would have positive +rank $h\in T$. Choose $r\in R$ with $\rk(r-l)l'$. The set $S$ is nonempty, since +otherwise the cofinal coset would be the singleton $\{l\}$. +It has no greatest member: if $s$ were greatest, $s+e_0$ would +be the least member of $T$. Balance in $\Gamma_\delta$, applied +to $(S,T)$, gives $\cf(S)=\epsilon$. The cofinality of $H_S$ at +its upper end equals $\cf(S)$. A cofinal set of ranks gives +cofinal positive basis vectors, while fewer ranks are bounded +and cannot give a cofinal family of vectors. Hence +$\cf(L)=\epsilon$ in this case. + +In the other case, each $l\in L$ has a bound $t_l\in T$ strictly +below all ranks $\rk(r-l)$, $r\in R$. Choose crossing pairs +$(l_i,r_i)$, $i<\epsilon$, whose ranks are coinitial in $T$. +For fixed $l\in L$, choose $i$ with +$\rk(r_i-l_i)\leq t_l$. If $l_i\leq l$, then +$0\theta$, each realized +record in $E(H(\delta))$ determines exactly one such model orbit. +\end{lemma} + +\begin{proof} +The finitely many positions belonging to each tuple entry can +be indexed in $\theta\times\omega$. Choices of arities and +labels number at most $2^{\ell+\theta}$. The flattened +order/equality diagrams number at most $2^\theta$; their sets +of cuts have size at most $2^\theta$, and occupancy subsets +number at most $2^{2^\theta}$. The displayed generous bound +therefore bounds a universal record set. + +For a fixed $\delta$ with $\cf(\delta)>\theta$, equal records +give an order automorphism matching the flattened lists by +Theorem~\ref{found:order-orbits}. Applying $E$ gives an +automorphism of the model matching the tuples. Hence a +realized record determines one model orbit. Every tuple has +a labeled presentation, so the number of orbits is bounded +by the number of records. + +For any $\kappa\geq\max(\ell,\theta^+)$, the ordinal +$\delta=\kappa+\theta^+$, with ordinal addition here, has +cardinality $\kappa$ and cofinality $\theta^+$. The model +$E(H(\delta))$ has cardinality $\kappa$ and is therefore a +copy of the categoricity model. This proves the bound without +any condition on $\cf(\kappa)$. +\end{proof} + +Choose a model $C_\kappa$ at each categoricity size needed in a +given set-sized construction. On its $\theta$-tuple orbits put +$p\longrightarrow r$ when some strong self-map sends a +representative of $p$ to a representative of $r$. This relation +is reflexive and transitive: compose maps after using an +automorphism to align the intermediate representatives. +Choose an injection of the orbit set into the fixed cardinal +$B=B(\ell,\theta)$, once for each $\kappa$. These are its +\emph{orbit names}. For any copy of $C_\kappa$, a tuple's +name is independent of the isomorphism used to $C_\kappa$, +because two such isomorphisms differ by an automorphism. + +For a suitable $\delta$ of size $\kappa$, let $F_{\kappa,\delta}$ +be the partial function from $\mathcal R_\theta$ to $B$ sending +each realized record to its orbit name. The \emph{profile} +of this presentation consists of its named arrow relation +and $F_{\kappa,\delta}$. The profiles belong to a fixed set +of cardinality at most $2^B$; in particular this bound is +independent of $\kappa$ and $\delta$. + +\subsection{Preserving every short record in a small subpresentation} + +\begin{lemma}[Closure of ordinal supports] +\label{found:ordinal-support-closure} +Let $\theta^+\leq a$ be infinite cardinals and let $\beta$ be +an ordinal of cofinality greater than $a$. Every subset +$S\subseteq\beta$ of size at most $a$ is contained in a set +$T_0\subseteq\beta$ of size at most $a$ with the following +property: for every $P\subseteq T_0$ of size at most $\theta$ +and every finite tuple from $\beta$, there is a tuple from +$T_0$ having the same order/equality diagram over $P$. +Moreover $T_0$ is bounded in $\beta$ and has cofinality +greater than $\theta$ in its induced order. +\end{lemma} + +\begin{proof} +Construct increasing sets $S_i\subseteq\beta$ for $i<\theta^+$, +starting with $S$. At a successor stage consider all strict +slots over the entire well-ordered set $S_i$. There are at +most $|S_i|+1\leq a$ of them. In each finite slot add every +point; in each infinite slot add a countably infinite subset. +Together with $S_i$ this realizes every finite order/equality +pattern over $S_i$ which occurs in $\beta$. At limits take +unions. The cardinality stays at most $a$ throughout. + +Set $T_0=\bigcup_{i<\theta^+}S_i$. Every set +$P\subseteq T_0$ of size at most $\theta$ is contained in +some $S_i$, by regularity of $\theta^+$. For a given finite +tuple, keep any coordinates already in $S_i$ and match its +other coordinates in their respective slots using the next +stage. This proves the asserted property over $P$. +Since $|T_0|\leq a<\cf(\beta)$, $T_0$ is bounded in +$\beta$. Each $S_i$ is bounded too, and its uppermost slot +is infinite. The next stage therefore contains points above +all of $S_i$. Every at-most-$\theta$ subset of $T_0$ is +bounded in this way, proving $\cf(T_0)>\theta$. +\end{proof} + +\begin{lemma}[Appending a long indecomposable segment] +\label{found:append-segment} +Under the hypotheses of Lemma~\ref{found:ordinal-support-closure}, +suppose $\beta$ is additively indecomposable. Let $T_0$ be as +there, and let $\delta<\beta$ be additively indecomposable, +with $\otp(T_0)<\delta$ and $\cf(\delta)>\theta$. +Place a consecutive segment $J$ of order type $\delta$ +strictly after $T_0$ in $\beta$, and set $T=T_0\cup J$. +Then $\otp(T)=\delta$, and $T$ has the same finite-pattern +matching property over its at-most-$\theta$ subsets. +Consequently $H(T)\subseteq H(\beta)$ computes correctly +the occupancies of every strict slot over any sequence of +at most $\theta$ of its points. +\end{lemma} + +\begin{proof} +Choose $\alpha<\beta$ above $T_0$. Indecomposability of +$\beta$ gives $\alpha+\delta<\beta$, so +$J=[\alpha,\alpha+\delta)$ is available. Indecomposability +of $\delta$ gives $\otp(T_0)+\delta=\delta$. + +Fix a parameter subset $P\subseteq T$ of size at most +$\theta$ and a finite tuple in $\beta$. Match the +coordinates below $\alpha$ into $T_0$ over $P\cap T_0$ +using Lemma~\ref{found:ordinal-support-closure}. Keep +coordinates in $J$ fixed. The parameters in $J$, together +with these finitely many retained coordinates, are bounded +in $J$ because $\cf(\delta)>\theta$. Match all coordinates +at or beyond the end of $J$ by suitably many distinct +points of $J$ above that bound, retaining any repetitions. +The three portions stay in their correct mutual order, +and this gives the required finite tuple over $P$. + +Here $\Gamma_T=\Z^{(T)}$ is identified with its image in +$\Gamma_\beta$, and $H(T)=\Q^{(\Gamma_T)}$ with its image +in $H(\beta)$. A sequence of at most $\theta$ points of +$H(T)$ uses at most $\theta$ ordinal positions in its +finite codes. A witness in $H(\beta)$ to an occupied +strict slot uses only finitely many additional ordinal +positions. Match those into $T$ over all the parameter +positions and retain the integer and rational +coefficients. Lemma~\ref{found:order-basic} makes every +comparison with the sequence unchanged. The resulting +point of $H(T)$ witnesses that slot. The reverse +implication follows from inclusion. +\end{proof} + +\begin{proposition}[Alignment of model orbits] +\label{found:alignment} +Fix the record set and orbit naming at an infinite length +$\theta$. Let $a<\mu$ be categoricity cardinals with +$a\geq\max(\ell,\theta^+)$. Suppose the following data +have one common profile: +\begin{enumerate} +\item a presentation $E(H(\beta))$ of $C_\mu$, where +$\beta$ is additively indecomposable, $|\beta|=\mu$, +and $\cf(\beta)>a$; +\item presentations $E(H(\delta))$ of $C_a$ for a +cofinal subset of $a^+$ of additively indecomposable +ordinals $\delta$ of size $a$ and cofinality greater +than $\theta$. +\end{enumerate} +Then in any copy $N$ of $C_\mu$, every subset of size +at most $a$ is contained in $A\leq_\K N$ of size $a$ +such that the orbit name of \emph{every} $\theta$-tuple +in $A$, computed as a tuple of $C_a$, is the same as +its name computed in $N$. +\end{proposition} + +\begin{proof} +Identify $N$ with the presentation in the first item. +Represent the requested set by labeled elements and +collect the ordinal positions in the finite codes of +their indices. There are at most $a$ positions. Apply +Lemma~\ref{found:ordinal-support-closure} to obtain +$T_0$. Its order type is below $a^+$. Choose a +profile occurrence $\delta$ from the second item above +$\otp(T_0)$. Since $a^+\leq\cf(\beta)\leq\beta$ and +$\delta\eta$ with $\kappa\leq\rho<\kappa^+$ and +$\cf(\rho)=\tau$, for example by appending an ordinal +segment of type $\tau$ beyond $\eta+\kappa$. +Set $\delta=\omega^\rho$. It is additively +indecomposable. A cofinal sequence of exponents gives +a cofinal sequence in $\omega^\rho$ by continuity of +exponentiation. Fewer than $\cf(\rho)$ ordinals below +$\omega^\rho$ have all their Cantor-normal-form +exponents bounded below $\rho$, and are therefore +bounded in $\omega^\rho$. This proves +$\cf(\delta)=\cf(\rho)=\tau$. +Every ordinal below $\omega^\rho$ +has a finite Cantor normal form with exponents below +$\rho$. Thus $|\delta|\leq\max(\aleph_0,|\rho|)=\kappa$. +Since $\delta\geq\rho\geq\kappa$ and $\rho>\eta$, +we have $|\delta|=\kappa$, $\delta<\kappa^+$, and +$\delta>\eta$. +\end{proof} + +\begin{theorem}[Eventual reversal at categoricity cardinals] +\label{found:reversal} +For each infinite cardinal $\theta$ there is a cardinal +$b_\K(\theta)$ such that, whenever +$\kappa\geq b_\K(\theta)$ is a categoricity cardinal, +$M\in\K$ has size $\kappa$, $f:M\longrightarrow M$ +is strong, and $\bar a$ is a tuple of length at most +$\theta$ in $M$, there is a strong +$g:M\longrightarrow M$ with +\[ +g(f(a_i))=a_i +\quad\text{for every coordinate }i. +\] +The bound concerns all sufficiently large categoricity +cardinals, of arbitrary cofinality. No equality of the +form $\kappa^\theta=\kappa$ is assumed. +\end{theorem} + +\begin{proof} +It suffices to treat length exactly $\theta$; pad a +nonempty shorter tuple by repetitions, and use any +strong self-map for the empty tuple. In a categoricity +model, the asserted reversal is equivalent to symmetry +of the arrow relation on its $\theta$-tuple orbits. +Indeed a reverse orbit arrow can be precomposed and +postcomposed with automorphisms to match the particular +image tuple and original tuple. The converse follows +by applying reversal to a map witnessing an arrow. + +Suppose that symmetry fails at unboundedly many +categoricity cardinals. Let $P$ be an infinite cardinal +bounding the profile palette; for example take $P=2^B$. +Choose a strictly increasing list of failing +categoricity cardinals $(\kappa_i)_{iP,\qquad +\kappa_i>\kappa_j^\theta\quad(ja. +\] +Their common named arrow relation has $p\longrightarrow r$ +but no $r\longrightarrow p$. Both names occur in +both sizes. Proposition~\ref{found:alignment} applies +inside every $\mu$-sized model, using any of its +cofinal profile occurrences. + +We construct $\theta^+$ rounds of an increasing strong +chain of $\mu$-sized models. At the start of round +$i<\theta^+$, choose an aligned $a$-sized submodel +$A_i$ containing all previous $A_j$. This is possible +because their union has cardinality at most $a$. +They are all strong in the current large model, and +coherence will make $A_j\leq_\K A_i$ for $j\theta$. After +round $i$ it cannot reach $p$ in a large model. +At the next aligned small model its small orbit +has the same name, and therefore cannot reach $p$ +in the common relation. The inclusion of that small +model into $A$ gives an arrow from this earlier +small orbit to its orbit in $A$. If the latter +could reach $p$, transitivity would contradict the +former exclusion. Thus no tuple in $A$ can have +an orbit reaching $p$. + +But $A\cong C_a$, so the orbit named $p$ occurs in +$A$; it reaches itself by the identity map. This +contradiction proves that the failing categoricity +cardinals are bounded. A cardinal above their bound +is $b_\K(\theta)$, and the first paragraph gives +the claimed reversals, including all shorter lengths. +\end{proof} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/04-diagrams.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/04-diagrams.tex new file mode 100644 index 0000000000000000000000000000000000000000..bce9ffbc665f2c2c239b5d45d1dd0c91b4586ae9 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/04-diagrams.tex @@ -0,0 +1,614 @@ +\section{Stable diagrams and amalgamation at closed levels} +\label{lev:section-diagrams} + +Throughout this section, \(\K\) is the AEC fixed after +Proposition~\ref{found:uniform-reduction}: it is categorical in unboundedly +many cardinals, and \(\ell=\LS(\K)\). We use the order presentation \(E\) +of Theorem~\ref{found:order-template}. A tuple may have any set as its +index set. Comparisons between tuples always specify the identification +of their index sets. Reindexing by an ordinal, padding, and repetitions +do not change any of the arguments below. + +The goal is to turn bounded-tuple reversal into amalgamation at selected +levels. We first stabilize tuple comparisons across categoricity sizes, +then use the order presentation to test and realize those comparisons +over an entire base. Amalgamation will be a conclusion of this process. + +\subsection{Stabilization of strong-map classes} + +Fix an infinite cardinal \(\sigma\). At a categoricity cardinal +\(\kappa>\ell+\sigma\) where Theorem~\ref{found:reversal} holds for +\(\sigma\)-tuples, put +\[ + (M,a)\equiv^\sigma_\kappa(N,b) + \quad\Longleftrightarrow\quad + \text{some strong map }f:M\longrightarrow N\text{ satisfies }f(a)=b. +\] +Here \(\|M\|=\|N\|=\kappa\), and the lengths of the displayed +tuples are at most \(\sigma\). +This is an equivalence relation. Reflexivity is immediate. Categoricity +allows one to regard all the maps as maps between copies of one model; +reversal then gives symmetry, and composition gives transitivity. +Using structures with universe \(\kappa\), or one fixed representative +and its tuples, its classes form a set. Their number, for all the tuple +lengths under consideration, is bounded by the fixed cardinal supplied +by Lemma~\ref{found:orbit-bound}. Denote an infinite such bound by +\(B_\sigma\). It depends on \(\ell,\sigma\), not on \(\kappa\). + +\begin{lemma}[Canonical classes above one categoricity size] +\label{lev:canonical-classes} +For \(\|N\|\ge\kappa\), a tuple \(a\) in \(N\) has a well-defined +\(\equiv^\sigma_\kappa\)-class: compute it in any strong +\(\kappa\)-submodel containing \(a\). Strong embeddings between models +of size at least \(\kappa\) preserve this class. +\end{lemma} + +\begin{proof} +The downward LS axiom supplies the submodel, after adding +\(\kappa\) many elements to the requested set if necessary. If +\(M_0,M_1\le_\K N\) are two such submodels, apply LS to +\(|M_0|\cup|M_1|\). It gives \(M_2\le_\K N\) of size \(\kappa\) +containing both. Coherence gives \(M_i\le_\K M_2\). The two occurrences +of \(a\) therefore have the same class. For a strong embedding +\(f:N\to N'\), the image of a witnessing submodel is a strong +\(\kappa\)-submodel of \(N'\), so the same observation proves +preservation. All these comparisons take place inside a given ambient +model; no amalgamation property is used. +\end{proof} + +\begin{proposition}[Stable diagrams] +\label{lev:stabilization} +There is a definable, nondecreasing cardinal function \(t\), with +\(t(\sigma)>\ell+\sigma\) a categoricity cardinal for every infinite +\(\sigma\), having the following properties. +\begin{enumerate} +\item Tuples of length at most \(\sigma\) in models of size at least +\(t(\sigma)\) have canonical diagrams. Equality of diagrams is +preserved by every strong embedding between such models, and there +are at most \(B_\sigma\) diagrams at these lengths. +\item Diagrams include atomic diagrams, respect reindexing and +restriction, and are compatible when the bound \(\sigma\) is +increased. We denote equality by \(D(a;M)=D(b;N)\), suppressing the +bound when it is understood. +\item If \(\kappa\ge t(\sigma)\) is a categoricity cardinal, equality +of diagrams between tuples in \(\kappa\)-models is implemented by +a strong map between those models. +\item A possible diagram of a pair of tuples \((a,c)\), of total +length at most \(\sigma\), can be realized over any tuple with the +diagram of \(a\) in any categoricity model of size at least +\(t(\sigma)\). ``Possible'' means that the diagram occurs in a +model in the domain of its definition. +\end{enumerate} +\end{proposition} + +\begin{proof} +If \(\gamma\ge\kappa\) are categoricity cardinals above the reversal +threshold, Lemma~\ref{lev:canonical-classes} associates a +\(\kappa\)-class to every \(\gamma\)-class. This gives a projection +\(\pi_{\gamma\kappa}\), and these projections compose. They are +surjective. Indeed, a \(\kappa\)-model is isomorphic to \(E(I)\) for +\(|I|=\kappa\); extend \(I\) to an order of size \(\gamma\) and use +the strong order map. Every tuple class in the smaller model then +occurs in the larger one. This argument supplies the needed extension +without a no-maximal-model assumption. + +These projections are eventually all bijective. Otherwise, construct +a sequence of categoricity cardinals with more than \(B_\sigma\) +successive noninjective projections, and choose one final +categoricity cardinal \(\rho\) above the entire sequence. For each +noninjective projection, surjectivity supplies a pair of +\(\rho\)-classes which agree when projected to the earlier stage and +differ when projected to the later stage. Pairs selected at distinct +stages are distinct: after a pair has separated it remains separated +under every later refinement. This gives more than +\(B_\sigma\) pairs from a set of size at most \(B_\sigma\), a +contradiction. The construction uses only a set-length sequence of +cardinals and choices from sets. + +Choose a categoricity cardinal \(c_\sigma\) beyond this stabilization +and reversal threshold. The canonical \(c_\sigma\)-class defines +the desired diagram on models of size at least \(c_\sigma\). +The bijective projections show that its equality can equally be +computed at every later categoricity cardinal. If two tuple bounds +are used, compare them at a categoricity cardinal above both +thresholds. Restriction of strong maps then gives the same +restricted equivalence. Atomic formulas and their negations are +preserved because strong maps are structure embeddings. + +For definiteness, thresholds can be chosen as least cardinals having +the stated eventual properties. Choose \(t(\sigma)\) to be the +least categoricity cardinal strictly above \(\ell+\sigma\) and +above all the thresholds \(c_\tau\) for infinite \(\tau\le\sigma\). +This is a nondecreasing definable function. The preceding arguments +prove the first three assertions. + +For the fourth, first put a realization of \((a,c)\) into a strong +\(c_\sigma\)-submodel. If the originally chosen ambient size is +smaller than a desired target categoricity size, this categorical +submodel can be extended to that size by the order presentation. +If the original ambient size is larger, LS gives a submodel of the +desired size containing the realization. Thus the joint diagram +has a realization in a model of the target categoricity size. +Apply the third assertion to the first tuple and the specified tuple +in the target. The resulting strong map transports the whole +realization as required. +\end{proof} + +\begin{proposition}[Categoricity at countably cofinal closure points] +\label{lev:closure-categoricity} +Suppose \(\zeta>\ell\) is an infinite cardinal of countable +cofinality and +\[ + t(\sigma)<\zeta\qquad\text{for every infinite }\sigma<\zeta. +\] +Then \(\K\) is categorical in \(\zeta\). Moreover any diagram match +between bounded tuples in two \(\zeta\)-models extends to an +isomorphism between those models. Here and throughout the paper, +bounded at a level means of cardinality strictly less than that level. +\end{proposition} + +\begin{proof} +Existence follows from \(E(I)\) for \(|I|=\zeta\). Let \(M,N\) +have size \(\zeta\), and choose countable increasing exhaustions of +their underlying sets by sets of size less than \(\zeta\). +Start with the prescribed bounded diagram match, or with the empty +match. At a forward step, enumerate together the current source +tuple and the next part of the exhaustion of \(M\). Its length and +the length of the current target tuple are bounded by an infinite +\(\sigma<\zeta\). Put the data on each side inside a strong +submodel of size \(t(\sigma)<\zeta\). Their current tuples have +the same diagram. Proposition~\ref{lev:stabilization} gives a +strong map of these submodels matching those tuples. Its values +on the new entries extend the match, and the enlarged tuples +again have equal diagrams. At the next step interchange \(M,N\). + +Only finitely many previously chosen bounded tuples are used at +each step, so their combined length stays below \(\zeta\). +The union of the countable sequence of partial matches is a +bijection \(|M|\to|N|\). Every finite tuple and every value of a +finitary function occur at some step, where the atomic diagram +is preserved. The bijection is therefore a structure isomorphism. +\end{proof} + +The closure cardinals in Proposition~\ref{lev:closure-categoricity} +are unbounded. For example, starting above any specified cardinal, +choose a strictly increasing sequence \((\alpha_n:n<\omega)\) +such that \(\alpha_{n+1}\) exceeds all \(t(\sigma)\) for +\(\sigma\le\alpha_n\). Its supremum has the required closure and +countable cofinality. Including \(2^{\alpha_n}\) in these requirements +makes the supremum a strong limit. We will use these stronger +closure points below. + +\subsection{Natural diagrams and interchange of blocks} + +\begin{definition}[Natural and placed diagrams] +\label{lev:natural-diagrams} +Consider a set-sized list of terms in an order presentation. Each +term consists of a label and an increasing finite list of positions. +Extend the order to size at least the threshold for the list and +compute its diagram there. The result is its \emph{natural diagram}. +If \(A=E(I)\), a \emph{placement} of an additional tuple \(d\) +over \(A\) is a labeled representation in an order extension of +\(I\), retaining the positions and chosen representations of the +parameters in \(A\). Its \emph{placed comparison over \(A\)} is +the natural diagram of \((d,a_A)\), where \(a_A\) is a fixed +enumeration of \(A\). +\end{definition} + +The natural diagram is independent of the extension used. Indeed, +linear orders amalgamate over a common suborder: for each cut of +the suborder, place the remaining portions of the two orders in +that cut while preserving each portion's order. Extending the +amalgam further if necessary gives a common sufficiently large +presentation. Strong order maps and +Proposition~\ref{lev:stabilization} identify the two diagrams. +The same argument shows that only the labels and the joint +order/equality pattern of their supports matter. All diagrams +with parameters use matching parameter enumerations. One may +instead list all labels on the base order, allowing repetitions: +restriction gives an enumeration of the elements, and reinserting +the repetitions is forced by their equalities. + +A placed comparison over a whole \(\mu\)-model is evaluated in a +sufficiently large presentation. It is not, by this definition, +an assertion that the original \(\mu\)-model itself lies above the +threshold for diagrams of length \(\mu\). + +\begin{lemma}[Orbits over parameters] +\label{lev:parameter-orbits} +Let \(\sigma\) be infinite and suppose +\[ + \chi^\sigma=\chi, + \qquad \chi\ge \ell+2^{2^\sigma}. +\] +At every sufficiently large categoricity cardinal, there are at +most \(\chi\) automorphism orbits of \(\sigma\)-tuples over any +specified set of at most \(\chi\) parameters. +\end{lemma} + +\begin{proof} +Present the categoricity model as \(E(H(\delta))\), choosing +\(|\delta|\) to be its cardinality and \(\cf(\delta)>\chi\). +This is possible at every sufficiently large cardinal, including +singular cardinals. Fix term representations of the parameters, +and let \(S\subseteq H(\delta)\) be their support set. Its size +is at most \(\chi\). Every member of \(S\) is coded by a finite +list of ordinals and integer and rational coefficients. The union +of its ordinal supports consequently also has size at most +\(\chi\). + +For one further position, its finite ordinal support has at most +\(\chi\) possible order/equality patterns over these old ordinal +supports. The discrete coefficients contribute only countably +many choices for each finite code. Such a pattern determines +all comparisons of the new position with \(S\). For at most +\(\sigma\) new positions, include their mutual comparisons, +their identifications with old positions, and the term labels. +There are at most \(\chi^\sigma=\chi\) choices so far. + +To apply Theorem~\ref{found:order-orbits}, also record which strict +slots over the old and new positions are realized in +\(H(\delta)\). The old support \(S\) is fixed. A slot over \(S\) +not containing a new position is unchanged. At most \(\sigma\) +old slots contain new positions, and inside each there are at +most \(2^\sigma\) cuts of the new support to test. The new flags +therefore have at most \(2^{2^\sigma}\le\chi\) possible values. +The full record has at most \(\chi\) possibilities. + +The combined list of old and new positions has size at most +\(\chi<\cf(\delta)\). Theorem~\ref{found:order-orbits} extends +equality of full records to an order automorphism fixing \(S\) +pointwise. Its induced automorphism of the presentation fixes +the parameters and matches the tuples. Thus the number of +structure-automorphism orbits is at most the number of records. +\end{proof} + +\begin{lemma}[Disjoint block transpositions] +\label{lev:block-transpositions} +Take two set-sized families of labeled terms. In disjoint convex +regions of their joint support order, suppose each region consists +of two adjacent blocks, one used only by the first family and one +only by the second. Interchange the blocks in every such region, +retaining their internal orders and all positions outside the +regions. The joint natural diagram is unchanged. +\end{lemma} + +\begin{proof} +Suppose the original and modified diagrams were different. Let +\(\sigma\) bound the two lists, their supports, and the shared +positions. Choose \(\chi\ge\ell+2^{2^\sigma}\) with +\(\chi^\sigma=\chi\), for instance by taking a sufficiently +large power of two. + +In each region repeat the pair of blocks along an arbitrary +linear order \(L\). If the first family's block was on the left, +use the order \(L\) for the repetitions; if it was on the right, +use the reverse of \(L\). Write \(d_i\) for the first-family +tuple using the blocks at index \(i\), and \(e_j\) for the +second-family tuple using the blocks at index \(j\); retain all +the shared positions outside the regions. For \(ij\) +it is the modified one. Comparisons within either family and +across distinct regions do not change. + +Let \(\epsilon\) be the least infinite cardinal satisfying +\(2^\epsilon>\chi\). The binary tree \(2^{<\epsilon}\) has +at most \(\chi\) nodes: for \(\alpha<\epsilon\) each level has +size at most \(\chi\), and \(\epsilon\le\chi\). Give it +lexicographically ordered divider points, one between the two +successor subtrees at each node. Explicitly, encode branch +digits by \(-1,1\), and encode the divider at a node \(s\) +by the sequence \(s\) followed by \(0\), using +\(-1<0<1\) in lexicographic comparison. Adjoin one point for every +branch in \(2^\epsilon\). Distinct branch points determine +distinct strict cuts over the divider points. Use this as the +order \(L\). + +The tuples \(e_j\) at the divider indices, together with the +fixed shared data, use at most \(\chi\) parameters. If two +branch-indexed tuples \(d_i\) were in the same automorphism +orbit over these parameters, their diagrams with every divider +tuple would agree. A divider between the two branches gives +the two different diagrams, a contradiction. Hence there are +more than \(\chi\) such orbits. Pad the constructed order to a +sufficiently large categoricity size and apply +Lemma~\ref{lev:parameter-orbits}. This is impossible. +\end{proof} + +\subsection{Bounded tests over a well-ordered presentation} + +The next elementary set-theoretic observation will also be used for +the more elaborate layouts of later sections. + +\begin{lemma}[Ordinal-vector witnesses] +\label{lev:ordinal-vectors} +Let \(C\) be a set, and let \(U\) be a definable class of functions +\(C\to\Ord\), with fixed set parameters allowed in its definition. +There is a set \(U_0\subseteq U\) such that each \(u\in U\) +dominates some \(v\in U_0\) coordinatewise. +\end{lemma} + +\begin{proof} +The assertion is immediate if \(C\) is empty or \(U\) is empty. +Well-order \(C\), and suppose no such set exists. For every +ordinal \(\beta\), the set \(U\cap V_\beta\) fails to generate +\(U\) upwards. There is therefore an element of \(U\) which +dominates none of its members. Let \(h(\beta)>\beta\) be the +least ordinal such that one such witness lies in \(V_{h(\beta)}\). +This is a definable ordinal function. + +Take an ordinal length \(\lambda\) satisfying the partition +relation for pairs, \(|C|\) colors, and an infinite homogeneous +set, supplied by Lemma~\ref{found:partition}. Define increasing +ordinals \(\beta_i\), for \(i\le\lambda\), recursively, with +\(\beta_{i+1}>h(\beta_i)\) and with limit values above all earlier +values. Choice for a set-indexed collection of nonempty subsets +of \(V_{\beta_\lambda}\) now selects witnesses \(u_i\) which +dominate no member of \(U\cap V_{\beta_i}\), for \(i<\lambda\). +For \(iu_j(c)\). Color the pair \(i\ell+\sigma\) satisfy \(\chi^\sigma=\chi\). Suppose that +for every \(X\subseteq A\) of size at most \(\chi\) a possible +joint diagram \(q_X(d,X)\) is prescribed, where \(d\) has length +at most \(\sigma\). Assume these prescriptions restrict +coherently and that their parameter diagrams are the natural +ones of \(X\). Then one placement of \(d\) over \(I\) realizes +all the prescriptions. +\end{proposition} + +\begin{proof} +Fix one term representation per parameter in \(A\). Write +\(S_X\subseteq I\) for the supports of \(X\). For one set +\(X\), its prescription has a placement. To see this, extend +\(I\) to a sufficiently large categoricity size and use +Proposition~\ref{lev:stabilization} to realize the possible +joint diagram over the actual parameter tuple. The realization +has term representations in that presentation. + +A support profile over \(X\) records the labels for \(d\), +its formal support order \(T_0\) of size at most \(\sigma\), +and comparisons and coincidences with \(S_X\). A well-order +of size at most \(\chi\) has at most \(\chi\) cuts. Hence +there are at most \(\chi^\sigma=\chi\) profiles. The positions +of \(T_0\) in one strict slot over \(S_X\) form a cluster. +Each profile has a continuation over \(I\) which leaves +every such cluster intact: insert the cluster at a single +cut of \(I\) extending its specified cut over \(S_X\). + +Call a hit of a new parameter support on \(T_0\), or the +separation of two positions in one cluster by a new parameter +support, a new event. For \(X'\supseteq X\), all continuations +of a profile with no new event over \(S_{X'}\) have the same +natural diagram over \(X'\). Within each old slot, the whole +cluster can only change its position across a block of new +parameter positions. The slots are disjoint, so +Lemma~\ref{lev:block-transpositions} applies simultaneously. + +Suppose no placement solves all the prescriptions. Start with +any \(X_0\) of size at most \(\chi\). Given \(X_i\), choose +for every profile over \(X_i\) a continuation with no new +event over all of \(I\). It fails a prescription on some set +of at most \(\chi\) parameters. Adjoin one such failing set +for each profile to obtain \(X_{i+1}\). At limits take unions, +and carry out \(\sigma^+\) stages. Every set constructed, +including \(X_* = \bigcup_{i<\sigma^+}X_i\), has size at most +\(\chi\), because \(\sigma^+\le\chi\). + +Take a placement realizing \(q_{X_*}\). At each stage its +profile over \(X_i\) was considered. If it had no new event +between \(S_{X_i}\) and \(S_{X_{i+1}}\), the preceding +no-event comparison would identify its diagram on the +selected failing set with that of the chosen continuation. +This contradicts its realization of \(q_{X_*}\). Thus it +has a new event at every stage. A support position can be +newly hit only once, and a pair can be newly separated only +once. There are at most \(\sigma+\sigma^2=\sigma\) such +events. They cannot occur at \(\sigma^+\) different stages. +\end{proof} + +\begin{lemma}[Unique lifts of bounded diagrams] +\label{lev:placed-lifts} +Let \(\mu>\ell\) be a strong-limit cardinal of countable +cofinality strictly closed below itself under \(t\) and \(g\). +Let \(A\le_\K B\) have size \(\mu\), and identify +\(A=E(I)\) for a well-order \(I\) of size \(\mu\). For a +tuple \(d\) in \(B\) of length at most \(\sigma<\mu\), the +diagrams +\[ + D(d,X;B),\qquad X\subseteq A,\quad |X|<\mu, +\] +lift to one placed comparison over all of \(A\), uniquely +as a comparison diagram. Restricting a lifted tuple gives +the lift of its restricted diagram. +\end{lemma} + +\begin{proof} +The displayed bounded diagrams are defined, since their +thresholds lie below \(\mu\), and they agree on the parameters +with the natural diagrams. For every sufficiently large +\(\chi<\mu\) with \(\chi^\sigma=\chi\), +Proposition~\ref{lev:compact-placement} gives a placement +realizing the tests of size at most \(\chi\). Such \(\chi\) +are cofinal below \(\mu\): use \(\chi=2^\alpha\) for +\(\sigma\le\alpha<\mu\), enlarging \(\alpha\) as necessary. + +Once \(\chi\ge g(\sigma)\), any two resulting placements +have the same full comparison by Proposition~\ref{lev:testing}. +Fix one of them. Every bounded test is covered by a larger +choice of \(\chi\), whose placement has that same full +comparison. The fixed placement consequently realizes every +bounded test. The same argument gives uniqueness. A +restriction realizes the restricted tests and hence, by +uniqueness, is their lift. +\end{proof} + +\subsection{Amalgamation and type counts} + +\begin{theorem}[Amalgamation at closed levels] +\label{lev:amalgamation} +There is an unbounded class of strong-limit cardinals \(\mu\) +of countable cofinality with the following properties. +\begin{enumerate} +\item \(\K\) is categorical in \(\mu\), and any two +\(\mu\)-sized strong extensions of a \(\mu\)-model have a +strong amalgam of size \(\mu\). +\item If \(A\le_\K B_0,B_1\) all have size \(\mu\), and +\(d_i\in B_i\) are tuples of the same length at most +\(\sigma<\mu\), there is such an amalgam identifying +\(d_0\) with \(d_1\) if and only if +\[ + D(d_0,X;B_0)=D(d_1,X;B_1) + \quad\text{for every }X\subseteq A, + \quad |X|\le g(\sigma). +\] +\item There are at most \(\mu\) singleton types over each +\(\mu\)-model, where type means identification in an amalgam +over that model. +\end{enumerate} +One may take every sufficiently large strong-limit cardinal +of countable cofinality strictly closed under \(t\) and \(g\). +\end{theorem} + +\begin{proof} +Such levels are unbounded by the countable closure construction +following Proposition~\ref{lev:closure-categoricity}. +Categoricity at them has already been proved. Identify a +given base with \(A=E(I)\), where \(I\) is a well-order of +size \(\mu\). Extend \(I\) to an order \(J\) whose cardinality +\(\gamma\) is a categoricity cardinal at least \(t(\mu)\), +and let \(G=E(J)\). Thus \(A\) is strongly included in its +specified position in \(G\). + +We show that every \(\mu\)-sized strong extension \(B\) of +\(A\) embeds strongly into \(G\) over \(A\). Choose +\(M_n\le_\K B\), increasing with union \(B\), with +\(\|M_n\|<\mu\). This follows by LS from a countable +exhaustion of \(B\), containing the preceding submodel at +each step; coherence makes the chain strong. Choose compatible +enumerations \(d_n\) of these models, so that each earlier +list is a sublist of the next. + +Lemma~\ref{lev:placed-lifts} assigns coherent placed comparisons +to \((d_n,A)\). Each is realized over \(A\) in \(G\), and +the realizations can retain the previous assignments. More +explicitly, realize the corresponding term pattern in an +order extension of size \(\gamma\). Its parameter tuple +\(A\), and then its tuple \((A,d_{n-1})\), have the same +diagram as the already assigned ones. Their length is at +most \(\mu\), so Proposition~\ref{lev:stabilization} at +\(\gamma\ge t(\mu)\) supplies the strong map realizing the +new list while fixing these previous data. + +It remains to verify strongness of the copied small models; +atomic agreement by itself would not suffice. Fix \(n\), +and bound the length of \(d_n\) by an infinite +\(\sigma_n<\mu\). Choose the categoricity cardinal +\(\rho=t(\sigma_n)<\mu\). LS puts the original enumeration +inside \(B_n'\le_\K B\) of size \(\rho\), and the copied +enumeration inside \(G_n'\le_\K G\) of that size. Coherence +gives \(M_n\le_\K B_n'\). The copied and original lists +have the same bounded diagram; Proposition~\ref{lev:stabilization} +gives a strong map \(B_n'\to G_n'\) with exactly the prescribed +values on \(d_n\). Its restriction shows that the copied +\(M_n\) is a strong submodel of \(G\). + +The copied models are nested, and coherence makes their +inclusions strong. Their union is a strong submodel of \(G\) +by the chain and common-upper-bound axioms. It is the image +of an embedding of \(B\). Every element of \(A\) eventually +occurs in some \(d_n\), and its equality with that parameter +is part of the lifted comparison. The embedding therefore +fixes \(A\). Performing this construction for \(B_0,B_1\) +in the same \(G\) gives an amalgam. LS supplies a strong +\(\mu\)-submodel of \(G\) containing both images; coherence +makes it a strong amalgam of the required size. + +For the second assertion, any identifying amalgam preserves +the displayed bounded diagrams, giving necessity. Conversely, +Proposition~\ref{lev:testing} identifies the placed lifts +of \(d_0,d_1\) over \(A\). Realize this common lift once +in \(G\). On each side start the preceding countable +construction with the specified tuple already assigned, +and choose every small-model enumeration to contain that +tuple. Homogeneity over \(A\) and the old assignments works +exactly as above. The two resulting embeddings identify the +tuples, and the same LS shrinking applies. + +For a singleton, a placement uses one label on a finite +support. Relative to the well-order \(I\), each support +position is either an old position or lies in one of its +\(|I|+1\) cuts. Together with the finitely many mutual +comparisons and the label, these data have at most +\(\mu\) possibilities. They determine the full natural +comparison. The second assertion identifies exactly the +types with equal comparison, proving the bound. +\end{proof} + +Amalgamation has thus been obtained at the stated levels from the +AEC axioms and unbounded categoricity. We have not assumed that +all models of intermediate cardinalities amalgamate. The next +section records precisely the level-by-level properties that +will be required of later categories of finite systems. diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/05-working-schemes.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/05-working-schemes.tex new file mode 100644 index 0000000000000000000000000000000000000000..5c9db5bc01ee95dae7558aec9179a21d47ec1141 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/05-working-schemes.tex @@ -0,0 +1,782 @@ +\section{Working schemes and uniform comparison of diagrams} +\label{lev:section-working-schemes} + +We isolate the conditions used when the preceding constructions are applied +to finite systems of models. The definition includes the hypotheses that +must be checked for a new system; it does not infer them merely from +categoricity of its vertices. + +\subsection{Systems and the working package} + +\begin{convention}[Closure below a level] +\label{lev:level-convention} +An eligible level is an infinite cardinal belonging to the class of +cardinals on which the particular construction under consideration has +been established. Except when explicitly discussing preliminary levels, +we use strong limit levels of countable cofinality. A list is +\emph{bounded at $\mu$} if its cardinality is strictly less than $\mu$. +If $u$ is a cardinal-valued function, strict closure under $u$ means +\[ + u(\theta)<\mu + \qquad\text{for every infinite cardinal }\theta<\mu. +\] +Bounds may include fixed parameters, may be increased to infinite +cardinals, and may be replaced by monotone majorants. +\end{convention} + +A system has finitely many vertices, each a $\K$-model, and specified +maps between specified vertices. We code it as a many-sorted structure, +with one sort for each occurrence of a vertex and function symbols for +the specified maps. An element occurring in two vertices therefore +has two occurrences in the code, related by the specified map when +appropriate. Distinguished lists are coded by names. A system +embedding consists of commuting embeddings on its vertices, preserving +these names. An \emph{arrow} is such an embedding whose vertex maps are +strong, together with any additional condition stated for the scheme. +The geometric application will have one such additional condition. +No condition on intersections of images is implicit in this definition. +An isomorphism means a structural isomorphism onto the entire system. + +\begin{definition}[Working package] +\label{lev:working-package} +Fix an unbounded definable class $S$ of eligible levels, a finite system +shape, and a set-sized finite-support term template $F$. A +\emph{working scheme} consists of classes $\mathcal C_\mu$, for $\mu\in S$, +arrows between their objects, and comparisons $D$ of bounded lists, +with the following properties. Every vertex of an object in +$\mathcal C_\mu$ has cardinality $\mu$. +\begin{enumerate} +\item \emph{Structural arrows and unions.} +Arrows include all structural isomorphisms, compose, and are invariant +under isomorphisms. They are coherent: if $X$ is a subsystem of $Y$, +and both inclusions into $Z$ are arrows, then $X\to Y$ is an arrow. +For an increasing nonempty chain of objects at level $\mu$, whose +union has total cardinality $\mu$, the union is an object at that level +and the inclusions into it are arrows. If all members of the chain +map by the inclusions into one object $Z$ at an eligible level +$\nu\geq\mu$, then the inclusion of the union into $Z$ is an arrow. + +\item \emph{Presentations and mixed downward closure.} +For every well-order $I$ of eligible cardinality $\mu$, $F(I)$ is an +object of $\mathcal C_\mu$. Its elements are represented by labels +applied to finite increasing lists from $I$; equality, the symbols of +the system structure, and the specified maps are evaluated from the +labels and the finite order and equality patterns. Substitution of +positions along an order embedding induces an arrow, including when +the source and target levels differ. Every object at a level is +isomorphic to a well-ordered presentation. If $\nu\leq\mu$ are eligible +and $A$ is a set of at most $\nu$ vertex occurrences in an object of +level $\mu$, there is an arrow inclusion of an object of level $\nu$ +containing $A$. Fixed distinguished data are included in this request. + +\item \emph{Diagrams, homogeneity, and saturation.} +For each index set of bounded cardinality there is an equivalence +relation, denoted by equality of $D$-diagrams, on the corresponding +lists in all eligible objects large enough for that index set. +It includes the atomic diagram of the entire coded system, respects +restrictions and reindexing, and sends every list to a $D$-equal list +under every arrow, including mixed-level arrows. At each level, +two bounded lists have the same diagram if and only if a structural +isomorphism between their objects takes one list to the other. +The number of diagrams at a fixed arity has a set-sized bound +independent of the level. A possible joint diagram of bounded length +extending the diagram of an actual bounded list is realized over that +list at its level. Here ``possible'' includes a diagram witnessed at +any larger eligible level, provided the total requested length is +bounded at the receiving level. On term lists diagrams are natural: +they depend only on labels and the order and equality pattern of all +positions used, including in comparisons between eligible levels. + +\item \emph{Exhaustion criterion for arrows.} +Suppose a map is obtained from compatible diagram matches on increasing +bounded lists which exhaust its domain object. Include in these lists, +for every vertex $V$, nested enumerations of strong submodels +$V_n\leq_\K V$ of sizes below the source level, with union $V$. +The image of each such enumeration must enumerate a strong submodel +of the corresponding target vertex. Under these conditions the map +is an arrow. Component strongness follows from the chain union and +common-upper-bound axioms of $\K$; an additional arrow condition, when +present, must be proved for this construction separately. +\end{enumerate} +The empty list is included in the third condition, and has one +diagram. Consequently all objects at any one eligible level are +isomorphic. +\end{definition} + +The mixed downward assertion in the second condition usually needs no +separate construction. In a chart for the large object, choose finite +supports for the requested occurrences. Their union has size at most +$\nu$; enlarge it to a suborder of size $\nu$ and use the presentation +arrow. If an already given small object is also to be included, put +supports for all its occurrences into the request and use coherence. + +The assertion about images of small strong submodels in the fourth +condition is also a precise obligation, rather than a claim that +arbitrary structural embeddings are strong. At one level it follows +when the bounded enumeration is carried by a diagram match, since that +match is implemented by an isomorphism of the containing objects. +For a mixed-level use, first place the smaller containing object in a +large presentation by an order arrow. Diagram preservation and an +isomorphism at the receiving level then show that the matched small +submodel has strong image. Finally, the union of these strong images +is strong in the target by the common-upper-bound clause. This proves +the componentwise part of the criterion in precisely the situations +in which it will be used. + +\subsection{Refining the eligible levels} +\label{lev:refining-levels} + +The later scheme constructions refine the class of levels on which the working +package is available. The following hypothesis ensures that the required +refinements still leave an unbounded supply of levels. + +Here and below every bound being closed under is given by a fixed +set-theoretic definition with set parameters. To see the elementary +closure construction explicitly, start above a prescribed cardinal and +choose strictly increasing infinite cardinals $\kappa_n$ such that +\[ + \kappa_{n+1}>2^{\kappa_n} + \quad\text{and}\quad + \kappa_{n+1}> + \sup\{u_i(\theta):\theta\leq\kappa_n,\ i\in A\}, +\] +where $A$ is the fixed set of functions to be accommodated. Replacement +makes the displayed supremum a set-sized ordinal. The cardinal +$\mu=\sup_{n<\omega}\kappa_n$ has countable cofinality, is a strong limit, +and is strictly closed under every $u_i$. + +There are two distinct uses of this observation. If a theorem proves +its conclusion at every such closure point, it establishes eligibility +of $\mu$. If eligibility is already restricted to an unbounded class +$S$, closing under the least member of $S$ above $\theta$ ensures only +that $S\cap\mu$ is unbounded in $\mu$; it does not by itself prove +$\mu\in S$. Membership at the limiting cardinal must come from the +relevant theorem or from an already established closure property of +$S$. In particular, if $S$ is unbounded and closed under suprema of +strictly increasing countable sequences, we may choose the $\kappa_n$ +inside $S$ in the construction above. The resulting $\mu$ then belongs +to $S$ and has cofinally many smaller members of $S$. This is the +meaning of requiring sufficiently many smaller eligible levels; an +arbitrary unbounded class of cardinals is not silently assumed to have +this property. + +\begin{convention}[Supply of eligible levels] +\label{lev:eligible-subclass} +For the generic scheme refinements below, we use the following +\emph{eligible-subclass hypothesis} whenever an unbounded class of +refined levels is asserted. The input eligible class $S$ must contain +a fixed definable unbounded subclass $T$ closed under suprema of +strictly increasing countable sequences. In addition to the input +scheme's individual conditions, this is a hypothesis on the supply +of eligible levels; arbitrary unboundedness of $S$ does not imply it. +\end{convention} + +This hypothesis persists through every finite list of the stated +refinements. For a fixed definable bound function, choose successive +members of $T$ beyond all its values on the ordinals below the previous +member, and take their countable supremum. This gives arbitrarily +large members of $T$ strictly closed under that bound. The intersection +with the bound-closure class is still closed under strictly increasing +countable suprema. A threshold only removes an initial segment. +For a limit-point refinement of a current class containing such a $T$, +use $\{\mu\in T:\sup(T\cap\mu)=\mu>0\}$; it is unbounded and has the same countable closure, +and is contained in the required limit-point class. Repeat for the +finite refinement list, including the finitely many bound functions +then required. Thus an unbounded subclass with the same property +remains available, including for larger auxiliary-level choices. + +\paragraph{Advance notation for common levels.} +\label{lev:closure-preview} +For a definable class $A$ of ordinals, write +\[ + \Lim(A)=\{\mu:\sup(A\cap\mu)=\mu>0\}. +\] +Here is the hierarchy that will organize the two applications of +Section~\ref{close:section-uniform}. That section constructs one fixed +bound majorant $\Phi$ for each application. Its initial class $G_0$ +consists of the strong limit cardinals $\mu$ of countable cofinality with +$\Phi(\alpha)<\mu$ for every ordinal $\alpha<\mu$; subsequent classes are +\[ + G_{j+1}=G_j\cap\Lim(G_j)\quad(j<\omega), + \qquad G_\omega=\bigcap_{j<\omega}G_j. +\] +This is advance notation: the definition of $\Phi$ and the proof that +each $G_j$ is unbounded and closed under strictly increasing countable +suprema are given in that section. A member of $G_{j+1}$ is thus both +a member of $G_j$ and a limit of smaller members of $G_j$. + +In those applications, the input class contains $G_j$ for some finite +$j$. This subclass serves as $T$, and the finite refinements retain a +later $G_k$. These subclasses supply the required levels; they are not inserted as +additional class parameters into the definitions of bounds. Those +definitions, given in Section~\ref{close:section-uniform}, use fixed +formulas for the required working assertions and for the finite +refinements already described. Without the eligible-subclass hypothesis, the generic +proofs assert their conclusions at levels satisfying their displayed +bound and limit-point requirements, but do not assert that those +levels are unbounded. + +\subsection{Directed unions and rich orders} + +\begin{lemma}[Directed unions and padded presentations] +\label{lev:directed-unions} +Assume the structural union and presentation conditions of +Definition~\ref{lev:working-package} at a level $\mu$. +Every nonempty directed family of inclusion subobjects at that level, +whose union has total cardinality $\mu$, has an object as its union, +with arrow inclusions and the common-upper-bound conclusion, including +into an eligible larger level. + +Consequently $F(I)$ is an object whenever $I$ has eligible cardinality +$\mu$ and contains a well-ordered subset of cardinality $\mu$. +We call such an order \emph{padded}. Order embeddings between padded +orders induce arrows, also between different eligible levels. +If the diagram conditions of the package hold, diagrams on bounded +term lists in padded orders are natural as well. +\end{lemma} + +\begin{proof} +We first prove the directed assertion by induction on the cardinality +of an indexing set. A finite directed family has a member containing +all its members. A countable directed family has a cofinal increasing +sequence, obtained by successively choosing an upper bound for the +previous choice and the next enumerated member. The chain assertion +gives its union and its common-upper-bound property. + +For an uncountable indexing cardinal $\tau$, fix a binary operation on +the indexing set selecting an upper bound for each pair. Write the +indexing set as an increasing union of sets of size less than $\tau$, +and close each under this operation. This closure adds only countably +many stages and preserves the bound of being smaller than $\tau$. +The sets can be chosen nested, and their union is the whole indexing +set. By the inductive assertion each corresponding smaller directed +family has an object as its union. That union still has cardinality +$\mu$: it contains a member of cardinality $\mu$ and is contained in +the given total union of cardinality $\mu$. The common-upper-bound +part of the inductive assertion makes these unions a chain of arrows. +The chain assertion now gives the total union. Applying the same +common-upper-bound clauses to a specified ambient object proves the +last assertion, including the mixed-level version. + +Fix a well-ordered subset $J_0\subseteq I$ of size $\mu$. Let +$\mathcal J$ consist of all well-ordered subsets of $I$ of size $\mu$. +A union of two well-ordered subsets is well-ordered: in any nonempty +subset take the smaller of the minima supplied by its two nonempty +intersections. Thus $\mathcal J$ is directed under inclusion. Every +finite subset of $I$ is contained in a member, namely its union with +$J_0$. The finite-support description of the template identifies +\[ + F(I)=\bigcup_{J\in\mathcal J}F(J). +\] +The union has cardinality $\mu$, and the directed assertion applies. +For an order embedding $I\to I'$, first restrict to well-ordered +suborders of the source. Their images are well-ordered; if necessary, +enlarge each image by a padding subset in the target. The known order +arrows and the common-upper-bound assertion give the induced map on +the total union as an arrow. + +Finally, two occurrences of the same bounded support pattern in +orders can be identified by amalgamating the orders over that +suborder. Such an amalgamation is obtained cut by cut: retain the +common suborder and put the points from each of the two orders in +their prescribed cuts, keeping their own orders within each cut. +Append a well-order of sufficiently large eligible cardinality to +make the amalgam padded. The two order arrows identify the term +lists there. Diagram preservation proves their naturalness. +\end{proof} + +When the chain assertion for a newly defined scheme is still being +proved, the lemma is used only for an underlying scheme whose chain +assertion has already been established. Directed unions are not used +to presume the assertion under proof. + +\begin{lemma}[Rich padded orders] +\label{lev:rich-orders} +Let $\eta$ be infinite. There is a dense linear order $Q_\eta$ without +endpoints of cardinality $2^\eta$ such that every cut given by at most +$\eta$ parameters on either side is filled. Equivalently, every +extension of an order pattern with at most $\eta$ old and at most +$\eta$ added positions can be realized over its old positions. +If $\mu$ is a strong limit cardinal and $\eta<\mu$, there is a padded +order with these properties of cardinality $\mu$. No restriction on +$\cf(\mu)$ is needed for this assertion. +\end{lemma} + +\begin{proof} +Put $\xi=2^\eta$, so that $\xi^\eta=\xi$ and $\eta^+\leq\xi$. +Starting with an order of size $\xi$, repeatedly fill every cut +specified by disjoint sets $A|F|+\beta$. The diagram of $c$ has a witness +at $\alpha$. If its original witness is at a larger level, use +saturation at $\alpha$ with the empty tuple as base. If its witness +is at a smaller level, first extend that object to a presentation +at a level at least $\alpha$ using an order arrow. This gives a +choice of $\alpha$ uniform over all diagrams of the specified arity. +Choose a well-ordered chart for this new witness and fix +representations of $c$. Let $J$ be the union of their finite supports, with its inherited +well-order. In the presentations $F(J+H)$ name the terms representing +$c$ on $J$. Regard this as a template in the variable order $H$: +any term using positions from both parts becomes a label recording +its original label and its finitely many fixed positions from $J$, +applied to its remaining positions from $H$. Terms supported wholly +on $J$ are labels of arity zero. The new collection of labels has +cardinality at most $|F|+|J|+\aleph_0$ and is fixed independently of +the level. Order maps on $H$, extended by the identity on $J$, +induce arrows preserving the names. + +Naturality gives the prescribed diagram on the named list in every +sufficiently large such presentation. Homogeneity for $c$ shows +that every object with such a named realization is isomorphic to one +of these presentations. +Define its diagram on a further list $a$ to be the old diagram on +the combined list $(c,a)$. The old package gives exactly restriction +compatibility, the uniform bound, preservation, homogeneity, and +saturation for these diagrams. The same support construction gives +mixed downward closure, now including the named terms. Chain unions +and the exhaustion criterion retain the names, so they are inherited. + +For the final assertion suppose a sublist enumerates +$M\leq_\K V$ in the original witness, which may be at a level +larger than $\alpha$. Embed that original system and the new +level-$\alpha$ witness into objects of a common larger eligible +level. Mixed preservation and homogeneity there give an +isomorphism matching their named lists. The original copy of $M$ +has strong image in that large vertex. This image is exactly the +substructure enumerated inside the embedded level-$\alpha$ vertex, +so coherence makes it strong in that vertex as well. It is a +$\K$-model because its complete atomic enumeration is isomorphic +to $M$. Thus strong naming already holds in the uniformly small +witness. Above $\alpha$, order arrows and the same homogeneity +argument preserve it. The argument applies to every named +vertex model and specified map. +Thus the new presentations satisfy the intended strong naming +condition, not just the atomic diagram of the enumeration. +\end{proof} + +\subsection{Two routes to working diagrams} + +The next statement records the exact diagram input that may be used +for a new scheme. In its first route the role of +Theorem~\ref{found:reversal} is an explicit hypothesis; it is not +assumed for a new category without proof. + +\begin{proposition}[Obtaining the diagram clause] +\label{lev:scheme-diagram-routes} +Suppose a finite-system scheme has the structural and presentation +conditions of Definition~\ref{lev:working-package}, and is categorical +at its eligible levels. For an unbounded supply after refinement, +assume the eligible-subclass hypothesis of +Convention~\ref{lev:eligible-subclass}. Either of the following inputs +supplies the diagram clause, after the indicated closure refinements. +\begin{enumerate} +\item For each infinite $\sigma$, at all sufficiently large eligible +levels, the relation of being sent to another list by an arrow is +reversible on lists of cardinality at most $\sigma$, and their orbit +sets have a bound depending only on $\sigma$ and the fixed scheme. +In this case take stabilization bounds and require enough smaller +levels below the receiving level for the bounded arities in use. + +\item Same-level arrows preserve isomorphism orbits of bounded lists, +and comparisons of such orbits in well-ordered charts are independent +of the eligible level. More precisely, for each arity, equivalence +of two term patterns by an isomorphism in their respective charts is +the same relation on patterns whenever both patterns are bounded at +the levels under comparison. +\end{enumerate} +The componentwise exhaustion condition follows whenever arrows have +only componentwise strongness as their additional structural content. +For any other arrow condition its exhaustion assertion remains a +separate obligation. +\end{proposition} + +\begin{proof} +For the first route, fix an arity bound $\sigma$. At a sufficiently +large eligible cardinal $\kappa$, arrow comparison on lists of this +arity is reflexive and transitive, and reversal makes it symmetric. +A list in a larger object can be put into a level-$\kappa$ subobject. +Two choices of that subobject are included in a third by mixed +downward closure and coherence. Their arrow classes therefore agree. +This defines the level-$\kappa$ comparison in all larger objects and +shows that it is preserved under mixed arrows. + +If $\gamma\geq\kappa$ is another sufficiently large eligible cardinal, +there is consequently a projection from the $\gamma$ comparison +classes to the $\kappa$ comparison classes. This projection is onto: +an object at level $\kappa$ embeds into one at level $\gamma$ by +identifying it with a chart and using an order arrow. The projections +compose. Let $b$ bound the numbers of classes. If strict refinement +continued arbitrarily high, recursively choose $b^+$ many successive +strict refinements and then an eligible cardinal above all the +chosen levels. At each strict refinement some pair of classes at +this final level has equal images before the refinement and unequal +images after it. A fixed pair can witness this only once, since +unequal images cannot merge in a refinement. There are at most +$b^2=b$ pairs, a contradiction. Thus the comparisons stabilize. + +Their stabilized equality is canonical, and comparisons at different +arities commute with restriction: raise the cardinal beyond the +stabilization thresholds for both arities and restrict an arrow +implementing the longer comparison. Atomic diagrams are preserved +by arrows. A prescribed bounded extension is realized over an actual +list in a sufficiently high same-level object by placing a witness +and the actual list into objects of one common eligible size and +using the reversible comparison to map the witness list to the actual +one. This gives the joint extension as well. + +At a receiving level $\mu$ of countable cofinality, close below $\mu$ +under these thresholds and under choices of a sufficiently large +smaller eligible cardinal for each arity. Enumerate each of two +objects by countably many nested bounded lists. To extend a match, +choose a smaller eligible cardinal containing all the old and newly +requested entries and above their stabilization thresholds. Put the +data into subobjects at that cardinal and implement the equal +stabilized diagrams by an arrow in the desired direction. Alternate +the direction of extension and include the next enumeration list on +each side. The union is a bijection preserving and reflecting every +atomic formula, hence an isomorphism of systems. Starting with a +given equal-diagram tuple makes it part of that isomorphism. The +same smaller-level construction realizes a possible bounded extension +diagram over an actual tuple. Order arrows and order amalgamation +give naturality. These arguments use both eligibility of $\mu$ and +the availability of smaller eligible cardinals, as distinguished in +Section~\ref{lev:refining-levels}. + +For the second route, instances of the same well-ordered pattern +have the same orbit. Indeed two well-orders amalgamate over a common +suborder into a well-order: in each cut of the common well-order +concatenate the two remaining well-ordered pieces, then retain the +common points between successive cuts. Enlarge to an eligible size +if necessary. The induced order arrows and same-level orbit +preservation identify the two instances. The assumed independence +of orbit comparison from the level therefore defines diagrams on +well-ordered patterns. The set of such patterns at a fixed arity +gives a uniform bound. + +For a mixed arrow from level $\nu$ to level $\mu$, include the image +of its whole source in a level-$\nu$ subpresentation of the target. +Coherence factors the arrow through that subpresentation. Same-level +orbit preservation computes the comparison there, and the assumed +agreement of chart comparisons computes the same diagram in the +target. This proves mixed preservation. For saturation, take a +well-ordered support pattern witnessing the desired joint diagram +in a larger object. Its support has cardinality below the receiving +level, so it embeds into a well-order of that level. The old subtuple +has the diagram of the prescribed actual tuple; an isomorphism at +the receiving level carries the new realization over that tuple. +The orbit hypothesis itself supplies homogeneity. Naturality on +padded orders follows from Lemma~\ref{lev:directed-unions}. + +In either route, diagrams on bounded enumerations of small vertex +submodels have strong images by homogeneity, with the preliminary +order embedding in the mixed-level case. The AEC union axioms give +the asserted componentwise exhaustion condition. +\end{proof} + +\subsection{A set-theoretic code for diagram equality} + +The next construction supplies one definition of comparison that can +be used uniformly with a finite descriptor for the scheme. It does +not define truth for a varying family of proper-class predicates. + +A code for a finite-support template consists of its set-sized +finitary language, sorts and specified-map symbols, its labels and +their finite arities, and tables evaluating the function symbols, +equality, and relations on finite ordered support patterns. A +function table supplies a label and sublist for its output within +the union of its inputs' supports; its arity-zero case is included. +Adjoin labels of arity zero for closed terms if necessary. This +does not increase the infinite bound on the language and label set. +Fixed names, including +the labels of arity zero in Proposition~\ref{lev:named-package}, are +part of this code. The usual consistency requirements say that the +evaluated equality is an equivalence relation and a congruence, that +the function tables are total and correctly sorted, and that the +evaluations agree under passage to larger finite support +patterns. These are tests on sets. A coherent code constructs a +structure $F(I)$ on any set-sized order $I$ by taking the quotient +of the set of labeled finite lists by the evaluated equality. This +construction makes sense whether or not $F(I)$ is an object of a +working scheme at that size. + +For an arity bound $\sigma$, a \emph{pattern} records an indexed list +of labels, finite increasing support lists for them, and the order +on the union of these supports. Use canonical support universes +of cardinality at most $\sigma$ and index sets of cardinality at most +$\sigma$. These codes form a set, denoted by $\mathcal P_F(\sigma)$. +Two codes representing the same terms need not be identified in +advance. The comparison below handles such duplicate representations. +When names are represented by reserved support positions instead of +labels of arity zero, include their fixed support in every pattern +and every support map; the test bounds must then also dominate its +cardinality. + +\begin{proposition}[Uniform coding of working diagrams] +\label{lev:diagram-coding} +There is a fixed first-order set-theoretic definition $D_F$ of an +equivalence relation on term patterns, uniform in a coherent +finite-support template code $F$. There is a definable cardinal +bound $b_F(\sigma)$ such that, on $\mathcal P_F(\sigma)$, a single +back-and-forth test with bound at least $b_F(\sigma)$ computes $D_F$. +These definitions do not require the system axioms at the cardinals +used for the tests. + +Suppose, in addition, that $F$ presents a working scheme at unboundedly +many eligible strong limit levels of countable cofinality. At every +eligible level $\mu>|F|$ strictly closed under $b_F$, the working +diagram of a bounded term list is exactly its $D_F$-comparison class. +The same definition therefore computes the diagrams of every such +scheme, including the schemes with fixed named data. +\end{proposition} + +\begin{proof} +Fix an infinite $\eta$ at least the cardinality of the template code +and of the patterns to be tested. For two set structures $A,B$ in +its language, form the set of all pairs of tuples of the same index +set of cardinality at most $\eta$. Index sets can be taken to be +ordinals below $\eta^+$, reindexing concatenations when necessary. +Let $R_0$ consist of the pairs with the same atomic diagram. Having +defined $R_\alpha$, retain in $R_{\alpha+1}$ just those pairs +$(a,b)\in R_\alpha$ such that every added tuple of cardinality at +most $\eta$ on either side has a matching added tuple on the other +side with the extended pair in $R_\alpha$. At a limit take the +intersection of all preceding relations. + +This decreasing recursion stabilizes. If its state set has +cardinality $\tau$, there cannot be a strict decrease at every +successor stage below $\tau^+$, since each strict decrease removes +a previously unremoved state. Equality at one successor stage +gives a fixed point. Write $R_\infty$ for the stabilized relation. +In particular, from a pair in $R_\infty$ every allowed extension has +a matching extension still in $R_\infty$; quantification over all +ranks is not being exchanged with an existential choice without +this fixed-point argument. Reflexivity and symmetry follow from +the construction. Transitivity follows by composing matches: +the composition of two relations having the atomic and extension +properties again has those properties, and is contained in the +largest such relation $R_\infty$. This also proves invariance under +isomorphisms. + +Use orders satisfying Lemma~\ref{lev:rich-orders}, and realize the +two indicated patterns in their term structures. Declare +$p\equiv_{F,\eta}q$ when their tuple pair belongs to $R_\infty$. +Every support pattern of size at most $\eta$ embeds into such an +order, by the successive insertion assertion in that lemma. + +The definition is independent of the rich orders and of the +realizations of an individual pattern. To prove this, relate two +term tuples when their representations are carried to each other +by an order isomorphism between support sets of size at most $\eta$. +This relation has atomic matching by the definition of the template. +An added tuple has at most $\eta$ new support positions. Order +richness extends the support map to these positions on the other +side, in either direction. Thus this relation has the extension +property and is contained in $R_\infty$. Replacing an instance of +either pattern by another instance and using transitivity proves +independence for the comparison of two possibly different patterns. + +If $\eta'\geq\eta$, orders rich for $\eta'$ are also rich for $\eta$, +and their back-and-forth test with the larger moves implies the test +with the smaller moves. Independence of the chosen orders gives +\[ + \equiv_{F,\eta'}\ \subseteq\ \equiv_{F,\eta} +\] +on patterns in the common domain. Define $D_F$ to be agreement at +every sufficiently large test bound $\eta$. + +Here is both uniform definability and the promised single bound. +In defining the test at $\eta$ it is enough to use rich orders of +cardinality $2^\eta$ on a fixed standard universe, whose existence +was proved in Lemma~\ref{lev:rich-orders}. All their order relations, +the realizations of the patterns, the associated term structures, +and the states of the recursion are sets. The recursion up to the +successor cardinal of its state set defines $R_\infty$ by an ordinary +first-order set-theoretic formula. Independence means that one may +quantify over the rich orders and realizations in that set, without +making a uniform class choice of an order. + +For each pair in $\mathcal P_F(\sigma)^2$ which fails some test, +take its least failing infinite cardinal above the fixed input bounds. +This is a uniquely defined ordinal when it exists. Replacement +collects these witnesses into a set. Choose $b_F(\sigma)$ above +their supremum, above $\sigma$, and above the template and naming +bounds. Monotonicity proves +\[ + p\mathrel{D_F}q + \quad\Longleftrightarrow\quad + p\equiv_{F,b_F(\sigma)}q, + \qquad p,q\in\mathcal P_F(\sigma). +\] +The same holds at every larger test bound. Taking suprema over +smaller arities makes $b_F$ monotone if desired. Notice that the +cardinality of the set of patterns justifies taking the set of +separation witnesses; it does not assert an a priori numerical +upper bound on their ranks. A code failing the coherence tests may, +if a total definition on all set codes is desired, be assigned the +discrete comparison and a fixed default bound. This convention has +no mathematical role for a coherent template. + +We now compare with the working diagrams. First suppose two bounded +patterns have equal working diagrams. Fix any test bound $\eta$. +Choose an eligible level large enough for the patterns and $\eta$, +and use a rich padded order of that level from +Lemma~\ref{lev:rich-orders}. Lemma~\ref{lev:directed-unions} makes +its term structure an object of the scheme. Naturalness transfers +the equal working diagrams to realizations of the two patterns +there. The package gives an isomorphism taking one realization to +the other. It passes the back-and-forth test at $\eta$. Independence +of rich orders now gives $p\equiv_{F,\eta}q$. Since $\eta$ was +arbitrary, equal working diagrams imply $D_F$-equivalence. + +Conversely fix an eligible $\mu$ as in the statement and two objects +of that level with bounded lists whose patterns are $D_F$-equivalent. +Choose countable nested bounded enumerations exhausting both objects. +We construct compatible matches containing these enumerations, +maintaining $D_F$-equivalence of their term patterns. Suppose the +current lists have been matched, and a next bounded list on one side +is requested. Let an infinite $\rho<\mu$ bound the cardinality of the combined +old and requested lists. Choose +\[ + b_F(\rho)\leq\eta<\mu. +\] +This is possible by strict closure. At the same level $\mu$, form +rich padded presentations for this test. Realize the current +patterns there. Naturality and homogeneity for the working diagrams +give isomorphisms from the current objects to these presentations +taking their current lists to the chosen realizations. + +Transport the requested list through the first isomorphism. The +current pair passes the test at $\eta$, so the fixed-point extension +property supplies a matching list in the other rich presentation. +Choose finite-support term representations for the extended lists. +Their patterns pass the $\eta$-test; because +$\eta\geq b_F(\rho)$, they are $D_F$-equivalent. Transport the +matching list back to the actual target object. The isomorphisms +used in this transport preserve working diagrams, which were +already shown to imply $D_F$-equivalence. Transitivity therefore +gives the required $D_F$-equivalence for the extended actual lists, +independently of their chosen representations. + +Repeat with the sides reversed and with the next lists in the two +enumerations. All stages have cardinality below $\mu$; only their +countable union exhausts the objects. Atomic matching ensures that +the compatible assignments define a well-defined injective map, and +the alternating exhaustion makes it onto every sort. Every atomic +instance involves finitely many entries and occurs at one stage, +so the resulting bijection is a structural isomorphism. It carries +the original list to the other list. The working package therefore +identifies their working diagrams, as required. +\end{proof} + +\begin{remark}[What the code does and does not establish] +\label{lev:coding-interface} +The code $F$, its finite shape, and its fixed naming data are set +parameters. The recursion in Proposition~\ref{lev:diagram-coding} +is a recursion on a set of states of set structures. Its definition +is the same formula for every such parameter, including templates +arising after a finite number of later constructions. It therefore +introduces no satisfaction predicate for arbitrary class definitions. +The bound $b_F(\sigma)$ depends on the full fixed descriptor; names +or labels of unbounded size cannot be omitted from that input. + +The identification of $D_F$-equivalence with equality of working diagrams +uses an already established working package, including its actual +isomorphism assertion. It supplies a +uniform way to express that package in subsequent closure tests; it +is not a substitute for proving homogeneity, chain closure, or an +additional arrow condition for a new scheme. The later simultaneous +closure argument must still prove that its descriptors and its bound +recipes are uniformly set-theoretically definable and that their +requested bounds exist on relevant inputs. +\end{remark} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/06-calculus.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/06-calculus.tex new file mode 100644 index 0000000000000000000000000000000000000000..80b8af231796f19cc12c1ec7a4e9a10061d48562 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/06-calculus.tex @@ -0,0 +1,893 @@ +\section{The one-dimensional calculus} +\label{calc:section-calculus} + +Throughout this Section, $\mathcal C$ is a scheme satisfying the working +package of Definition~\ref{lev:working-package}. Its fixed presentation is +denoted by $F$, and $b$ is an infinite cardinal bounding the set of labels +and the finite amount of structural data in the scheme. We use the +occurrence sorts of that Definition: a tuple in several vertices is a +tuple of tagged occurrences, and a map of systems commutes with every +specified structural map. No choice of a hull or a basis is part of a +diagram. A level is an eligible strong limit cardinal of countable +cofinality. All further closure requirements below are requirements on +functions of bounded arities and fixed scheme data. +For the supply of unbounded refinements and larger auxiliary levels, +we use the eligible-subclass hypothesis of +Convention~\ref{lev:eligible-subclass}: the input class contains a fixed +definable unbounded subclass closed under suprema of strictly increasing +countable sequences. This subclass is used only to supply levels; +the bounds below depend on fixed scheme data and bounded arities. + +Stationary independence and local character are central to the theory of +good frames; see \cite[II.2.1]{Shelah2009}. The assertions below are +proved from the working package for bounded tuples and finite systems. +Their role is to supply the independence and descent properties needed +for the next isolation construction. + +A \emph{query} is an enumerated parameter set $Z$ together with the +diagram of the displayed tuple and that enumeration. Repetitions in an +enumeration are allowed. Saying that an answer is correct on $Z$ means +equality of the whole joint diagram, rather than equality only on finite +subtuples. At a level $\mu$, every tuple or query called bounded has +cardinality less than $\mu$. + +\subsection{Testing, placement, and amalgamation} + +\begin{lemma}[Parameter orbits for a scheme] +\label{calc:parameter-orbits} +Fix an infinite $\sigma$. If +$\chi>b+\sigma$ and $\chi^\sigma=\chi$, then, at every sufficiently closed +level greater than $\chi$, there are at most $\chi$ diagrams of +$\sigma$-tuples over a fixed parameter tuple of length at most $\chi$. +Moreover, natural diagrams are unchanged by simultaneous disjoint block +transpositions: within each of a family of disjoint convex regions, +interchange an adjoining block belonging exclusively to the first tuple +family with one belonging exclusively to the second, retaining each +block's internal order. +\end{lemma} + +\begin{proof} +Present the parameter object on a well-order. After choosing one term +representation of each parameter, its supporting positions form a +well-ordered set of size at most $\chi$. There are at most $\chi$ slots +and positions relative to this set. Labels, comparisons and coincidences +for at most $\sigma$ additional positions therefore have at most +$\chi^\sigma=\chi$ possibilities. Equal profiles give equal natural +diagrams. The diagram-isomorphism clause of the working package turns +this into the asserted orbit bound over the actual parameters. + +Here are the hypotheses needed for the block argument in +Lemma~\ref{lev:block-transpositions}. We have just proved its parameter +orbit bound. Repeating the two blocks along an arbitrary order is +permitted after padding: the scheme has the union and common-upper-bound +clauses, so Lemma~\ref{lev:directed-unions} supplies the padded +presentations and their order arrows. Naturalness compares the two +patterns at a common higher eligible level. If the transposition changed +a diagram, the repeated patterns would distinguish cuts by whether the +first block preceded or followed a parameter block. A binary tree of +least height $\epsilon$ with $2^\epsilon>\chi$ has at most $\chi$ +nonterminal nodes and more than $\chi$ branches. Divider points at those +nodes give at most $\chi$ parameter blocks and more than $\chi$ different +diagrams over them. The preceding orbit bound gives a contradiction. +The repetition uses only a set of positions, so a higher eligible level +can contain the entire test. This proves the assertion also when there +are boundedly many simultaneous transpositions. +\end{proof} + +\begin{definition}[Natural parameter layouts] +\label{calc:natural-layout} +A natural parameter layout consists of finitely many well-ordered +segments, a fixed finite set of colors and their prescribed order, and +specified families of terms on specified collections of those positions. +Its parameters are all the designated terms; vertex sorts may be among +the designations. The data must have the following monotonicity +property: after freezing bounded extra supports, increasing the ordinal +lengths of the remaining slots gives an embedding of layouts that +preserves the designated parameter terms. Bounded frozen parameters +are permitted and count towards the input arity. The layout, labels and +color conventions are fixed set data. + +The whole object $F(H)$, with $H$ well-ordered, is a natural layout. +Unions of designated vertex sorts generated on specified segments are +further examples. An arbitrary subset of the parameters in an unrelated +target chart is not asserted to be a natural layout. +\end{definition} + +\begin{proposition}[Uniform testing for natural layouts] +\label{calc:testing} +\label{calc:layout-testing} +For each fixed natural layout there is a monotone cardinal function +$g(\sigma)$ with the following property. Two placed comparison types of +tuples of length at most $\sigma$ over that layout are equal if and only +if they agree on parameter queries of size at most $g(\sigma)$. +The bound is independent of the eligible level. The extra tuple supports +may lie in a non-well-ordered extension of the layout. The assertion is +uniform when the fixed layout and frozen bounded data range over a fixed +set of descriptors of bounded size. +\end{proposition} + +\begin{proof} +Amalgamate the orders containing the two placements over the parameter +order. Freeze the labels of the two tuples, their combined ordered +support $T$, the coincidences with old positions, and the colors. There +are set-many such frozen patterns at the prescribed bound. The +unfrozen well-ordered parameter positions are described by a vector of +ordinal lengths, one coordinate for each strict slot and each prescribed +segment. There are set-many coordinates, even when $T$ itself is not +well-ordered. By the monotonicity requirement in +Definition~\ref{calc:natural-layout}, a pointwise increase embeds the old +layout and its designated terms into the new one. Unequal comparisons +remain unequal because diagrams are preserved by the induced arrows. + +For a fixed frozen pattern, apply +Lemma~\ref{lev:ordinal-vectors} to the upward closure, among all ordinal +vectors, of its valid unequal configurations. It gives a set of +generators, which can be chosen from valid unequal configurations. Each +generator has a set-sized parameter order. Take a cardinal above the +sizes of the terms on all these orders, then above the resulting bounds +for all the frozen patterns. An unequal configuration contains one of +the generating configurations and is consequently witnessed on at most +that many parameters. Increasing the bound monotonically gives $g$. +This is precisely the ordinal-vector proof of +Proposition~\ref{lev:testing}; its required ingredients here are natural +diagrams, their preservation, and the explicit monotonicity of the +layout, all verified above. Taking the same supremum over a set of +descriptors proves the last assertion. +\end{proof} + +\begin{lemma}[Compact placement for a scheme] +\label{calc:compact-placement} +Let $P$ be any chosen parameter set in a well-ordered presentation, with +one fixed term representation for each parameter. Let $v$ have length +at most $\sigma$, and let +\[ + \chi>b+\sigma,\qquad \chi^\sigma=\chi. +\] +Suppose that coherent diagrams for $(v,X)$ are prescribed for every +$X\subseteq P$ of size at most $\chi$, and that each individual +prescription has a placement. At a sufficiently closed eligible level +containing the data, one order enlargement realizes all these +prescriptions simultaneously. The same assertion holds with bounded +frozen data, after adding its size to $\sigma$. + +For a padded order that need not be well-ordered, it suffices instead to +have locally possible coherent prescriptions on every bounded subset +and closure under the profile-count growth used in the proof. The +conclusion then holds simultaneously at any one specified bounded test +size. All the bounds are uniform in the fixed scheme and layout data. +\end{lemma} + +\begin{proof} +For $X\subseteq P$, let $S_X$ be the union of its chosen supports. A +profile records the labels of $v$, its formal ordered support, and all +comparisons and coincidences with $S_X$. If $|X|\le\chi$, there are at +most $\chi$ profiles. In each strict slot over $S_X$, all the new +positions there form one cluster. A profile has a continuation over the +whole old order with no new event: place each cluster intact into a +single strict slot of the old order extending its slot over $S_X$. +Order enlargement supplies positions if necessary. A new event on +enlarging $X$ means a new hit of an old position or a separation of two +previously unseparated new positions by an old position. + +Two continuations with no new event over $S_Y$, for $Y\supseteq X$, +have the same diagram over $Y$. Indeed, their differences consist of +moving whole clusters past blocks of parameter supports. The regions +belong to distinct slots, so the block transpositions in +Lemma~\ref{calc:parameter-orbits} apply simultaneously. + +If simultaneous realization fails, choose $X_0$ of size at most $\chi$. +For each profile over $X_i$, choose a no-new-event continuation and one +prescribed query where it fails. Adjoin those queries to $X_i$ to obtain +$X_{i+1}$; at limits take unions. There are at most $\chi$ profiles, +each chosen query has size at most $\chi$, and $\sigma^+\le\chi$. +Thus every $X_i$, including $X_{\sigma^+}$, has size at most $\chi$. +Choose a placement realizing the prescription on $X_{\sigma^+}$. +At every earlier step this placement must have a new event, since +otherwise the no-event comparison would transfer the chosen failure to +it. There are at most $\sigma$ possible hits and at most $\sigma$ +pairs of new positions. Each event can occur for the first time only +once. This contradicts the $\sigma^+$ steps. + +In a non-well-ordered chart, the number of profiles over $X_i$ is at most +$2^{|X_i|+\sigma+b}$. Starting with the desired test size, use this as +the next cardinal bound and include a failed query of at most that +original test size for every profile. The larger sets $X_i$ have +prescriptions because those were supplied on all bounded subsets. +More explicitly, start with an infinite $\beta_0$ above the desired +test size and set $\beta_{i+1}=2^{\beta_i+\sigma+b}$, taking suprema +at limits. Require the single bound +$\sup_{i\le\sigma^+}\beta_i$ to be strictly below the level. +This is closure under the whole indicated recursion, rather than +merely closure under each individual exponentiation. It keeps the +final set bounded even though the level has countable cofinality. +Its prescription +is available by hypothesis, and the identical event count is a +contradiction. This is a set recursion depending only on the starting +arity and fixed data. +\end{proof} + +For later reference, fix a convenient explicit threshold for the whole +object layout. Increase $g$ so that $g(\sigma)\ge b+\sigma$, and put +\begin{equation} +\label{calc:test-threshold} + c(\sigma,\tau)=2^{g(\sigma)+b+\sigma+\tau}, + \qquad c(\sigma)=c(\sigma,\aleph_0). +\end{equation} +These cardinals exceed the displayed arities and satisfy +$c(\sigma,\tau)^\sigma=c(\sigma,\tau)$. We close levels under these +functions and their successors, as well as under the diagram and rich +order bounds at the resulting arities. For a different fixed natural +layout, use its testing function in the same formula. + +\begin{lemma}[Placed comparisons of actual tuples] +\label{calc:placed-comparisons} +Suppose $A\to N$ is an arrow between same-level objects and $v$ is a +bounded tuple in $N$. Its diagrams over bounded subsets of $A$ have a +placed comparison over a well-ordered presentation of $A$. This +comparison is unique as a diagram over all of $A$. Two such actual +tuples have equal comparisons if and only if their diagrams agree over +queries in $A$ of size at most $g(|v|+\aleph_0)$. +The conclusion also holds over any natural parameter layout for which +the same coherent local diagrams are given. +\end{lemma} + +\begin{proof} +A single bounded request is possible in a presentation: the parameter +diagram is preserved by the arrow, and saturation realizes its desired +extension over the actual parameters in a level object. Apply +Lemma~\ref{calc:compact-placement} at +$\chi=c(|v|+\aleph_0,\tau)$, for any desired bounded query size $\tau$. +Two outputs agree on all tests of size at most $g(|v|+\aleph_0)$, and +Proposition~\ref{calc:testing} identifies their full placed comparisons. +For any bounded query, choose $\tau$ large enough to include it. The +unique comparison therefore computes every actual bounded diagram. +This argument does not require an amalgamation relation on types. +\end{proof} + +\begin{theorem}[Amalgamation and types for a scheme] +\label{calc:scheme-amalgamation} +At sufficiently closed levels, $\mathcal C$ has amalgamation. If +$A\to B_0,B_1$ are arrows and $v_i$ are equally indexed bounded tuples +in $B_i$, then an amalgam over $A$ identifying $v_0$ and $v_1$ exists +if and only if their diagrams agree on all parameter queries in $A$ of +size at most $g(|v_i|+\aleph_0)$. + +Consequently $\tp(v/A;B)$, defined by identification in an amalgam over +$A$, is an equivalence class. It is determined by these bounded +diagrams and has an order placement. Restriction of a type means +restriction of its actual tuple and parameter diagrams. +\end{theorem} + +\begin{proof} +Write arrows as inclusions by taking renamed copies. We prove the +stronger construction with a prescribed identification of bounded +tuples; omit the tuples for ordinary amalgamation. Let the target +initially be $N_0=B_1$. Choose increasing bounded lists $d_n$ exhausting +$B_0$, starting with $d_0=v_0$ (or the empty list when no identification +is prescribed). In each vertex also choose an +increasing countable exhaustion by strong submodels of cardinality less +than the level, and include their enumerations in the $d_n$'s. Such +exhaustions exist by the component LS property, coherence and +$\cf(\mu)=\omega$. + +We construct targets $N_n$ and compatible images of $d_n$. The invariant +is equality of all bounded joint diagrams of $(d_n,A)$ with their +original diagrams in $B_0$. For the prescribed initial tuple this is +the hypothesis and Lemma~\ref{calc:placed-comparisons}; the empty initial +tuple is automatic. + +Suppose an image of $d_n$ has been chosen. In a well-ordered chart of +$N_n$, regard $A$ and this whole image as the parameter set $P$. +For a bounded subset of $P$, its required joint diagram with $d_{n+1}$ +is read in $B_0$, interpreting the old image as $d_n$. This prescription +is well-defined on repetitions and coincidences by the invariant. +It is locally possible: equality on the old parameters and saturation +give a realization in $N_n$. Choose a test cardinal of the form +\eqref{calc:test-threshold}, large enough also for the length of $d_n$, +and apply Lemma~\ref{calc:compact-placement}. The resulting tuple lies +in an order enlargement $N_{n+1}$ and retains the old assignments. + +To verify the invariant, use a natural well-ordered chart of the +\emph{source} $A$. Both the desired tuple and the newly placed tuple +have comparisons there by Lemma~\ref{calc:placed-comparisons}. Their +agreement on the chosen tests includes the testing bound for the joint +arity, so Proposition~\ref{calc:testing} makes their comparisons equal. +They therefore agree on every bounded query in $A$. We have not +applied uniform layout testing to the possibly unnatural image of $A$ +in the target chart; compact placement alone was used on that image. + +Take the union of the countable target chain. Its total size is $\mu$, +so the union and upper-bound clauses of the package give an object and +an arrow from $B_1$. The assignments define a map from $B_0$ into this +union. They match diagrams on increasing lists that exhaust $B_0$ and +include the component strong-submodel exhaustions. The image of each +such small submodel is strong: the matching diagram of its enumeration +is implemented by an isomorphism between its original containing +object and the target object. The arrow criterion +of the package consequently makes this map an arrow, including any +extra basis condition already checked for the current scheme. The +initial tuples have the same image. Necessity of diagram equality +follows from preservation in any amalgam. + +Amalgamation makes the identification relation transitive: amalgamate +two witnesses over their common intervening object. Reflexivity and +symmetry follow from identity maps and interchange of the two legs. +The equivalence and testing descriptions now follow from the proved +criterion and Lemma~\ref{calc:placed-comparisons}. +\end{proof} + +\begin{corollary}[Realization into a larger level] +\label{calc:mixed-realization} +Let $\nu<\mu$ be sufficiently closed eligible levels. If $A\to B$ are +$\nu$-objects and $A\to N$ is an arrow into a $\mu$-object, then there +is an arrow $B\to N$ extending the given arrow on $A$. One may prescribe +the image of a tuple of length less than $\nu$ whose diagrams over +queries in $A$ below $\nu$ already match. Types of tuples of length +less than $\nu$ over $A$ obtained in a larger object are actual +$\nu$-types, and the mixed diagram comparisons at those arities agree. +\end{corollary} + +\begin{proof} +Use the countable exhaustion in the preceding proof, with all source +lists and testing thresholds below $\nu$. Perform the compact +placement in a chart of $N$. The placed tuple, all of $A$, and the +previously assigned list together have cardinality at most $\nu$, which +is bounded at $\mu$. Saturation at $\mu$ therefore realizes their +whole joint diagram back in the actual $N$, fixing the existing +parameters. Continue there, without enlarging the final target. +For component strongness, first place the source object at level $\mu$ +by a presentation arrow. Mixed preservation and an isomorphism of +$\mu$-objects implementing the matching enumeration show that every +small source strong submodel has strong image in $N$. The exhaustion +criterion now gives the arrow. + +To interpret a tuple of length less than $\nu$ over $A$ inside a larger +object, take a $\nu$-subobject containing $A$ and the tuple, using mixed +LS. Coherence makes it an extension of $A$. Mixed preservation +identifies its diagrams with those in the larger object. The testing +criterion shows independence of the chosen subobject. This also +justifies the source-level type comparisons used during the construction. +\end{proof} + +\subsection{Approximations and stationary free extensions} + +Amalgamation has made types over a whole object available. We now choose +their free extensions by means of sequences in the base: on each bounded +parameter query, almost all entries of the sequence give the same answer. +The next two results construct these sequences and show that their answers +determine an actual type, independently of the choices. + +\begin{definition}[Approximation sequence] +An approximation sequence for $p=\tp(v/A)$ at query size $\tau$ is an +indexed family $I=(v_i:i<\kappa)$ in $A$, each $v_i$ having the same +index set as $v$, such that the concatenation of the family is bounded +and, for every $Z\subseteq A$ of size at most $\tau$, +\begin{equation} +\label{calc:exception-estimate} + \bigl|\{i<\kappa:D(v_i,Z)\ne p\mathord\upharpoonright Z\}\bigr| + \le |Z|+\aleph_0. +\end{equation} +Here $p\mathord\upharpoonright Z$ denotes its joint diagram, and a +sequence is always indexed: different indices may carry the same tuple. +We use the same terminology for all projections of the displayed tuple. +Whenever $\kappa>|Z|+\aleph_0$, the unique value assumed outside at most +$|Z|+\aleph_0$ indices is called its \emph{majority value} on $Z$. +\end{definition} + +\begin{lemma}[Uniform majority approximations] +\label{calc:approximations} +Let $|v|\le\sigma$ and let $\tau\ge\aleph_0$ be bounded. At +sufficiently closed levels there are approximation sequences for +$p=\tp(v/A)$ at size $\tau$, of every prescribed bounded infinite length +$\kappa$. They satisfy \eqref{calc:exception-estimate} simultaneously +for all subtuples of $v$. The construction has bounds depending only +on $b,\sigma,\tau,\kappa$, rather than on the level or the type. + +In particular, take $\tau=c(\sigma)$ and +$\kappa=c(\sigma)^+$. The concatenated sequence then has a uniform +bounded length depending only on $\sigma$ and the fixed scheme. Its +diagram is a code for $p$ in the precise transport sense proved in +Theorem~\ref{calc:free-extension} below. +\end{lemma} + +\begin{proof} +By the rich-order and padding constructions of +Lemmas~\ref{lev:rich-orders} and~\ref{lev:directed-unions}, present $A$ +on a padded dense order sufficiently saturated for every support size +used below. The presentation is an object of the current scheme; +categoricity in the package gives an isomorphism with $A$. +Lemma~\ref{calc:placed-comparisons}, or its compact-placement proof in +the padded chart, places $p$ correctly at size $\tau$. + +Keep every extra supporting position that belongs to the old chart. +Partition the other supporting positions into clusters according to +their strict cut over the entire old order. For every pair of distinct +cuts choose an old position separating them. Include the shared +supporting positions as well. This gives a bounded anchor set of size +at most $\sigma+\aleph_0$. Each unshared cluster now belongs to a +different strict slot over the anchors. + +In each such anchor slot, choose $\kappa$ pairwise disjoint convex +regions in the old order and, in the $i$th region, put a copy of the +corresponding cluster with its internal order. The prescribed +richness supplies these patterns; its required arity is bounded in +terms of $\sigma+\kappa$. Use the original labels and the unchanged +shared positions to obtain $v_i\in A$. + +Choose a query $Z$ and one finite support for each of its elements. +If these supports avoid every region used for the $i$th copies, then +each copied cluster and the original cluster can differ relative to +the query only by passing whole blocks of parameter positions. There +are no hits or separations inside a cluster. Their anchor slots are +disjoint, so Lemma~\ref{calc:parameter-orbits} gives +$D(v_i,Z)=p\mathord\upharpoonright Z$. Every supporting position of +$Z$ belongs to at most one of the regions, and therefore excludes at +most one index $i$. Finitarity of supports gives at most +$|Z|+\aleph_0$ excluded indices. The same set of good indices works +for every subtuple of $v$. Only bounded support patterns, the fixed +labels, rich orders at bounded arities, and the compact-placement +bounds were used. This proves the uniformity assertions. +\end{proof} + +\begin{theorem}[Stationary free extension] +\label{calc:free-extension} +Let $A\to B$ be an arrow at a sufficiently closed level and let +$p=\tp(v/A)$, where $|v|\le\sigma$. There is a specified actual type +$p^B$ over $B$, called its \emph{free lift}, extending $p$. At every +desired query size whose approximation and testing costs are bounded at +the originating level, it is computed by transporting sufficiently long +approximations in $A$. At one level this includes every bounded query +size; in a mixed-level use the costs must remain below the level of $A$. +It has the following properties. +\begin{enumerate} +\item The answer is independent of the approximation sequence, its +length, and the auxiliary test size used in its construction. +\item Free lifts commute with restriction of the tuple, are invariant +under isomorphisms, and satisfy +$(p^B)^C=p^C$ whenever $A\to B\to C$. +\item If $A$ has an eligible size $\nu$ and $B$ has a larger eligible +size, the construction still works for $|v|<\nu$, with its construction +thresholds and approximation lists below $\nu$. The restriction to +$A$ is the actual $\nu$-type of +Corollary~\ref{calc:mixed-realization}. +\item A sequence from Lemma~\ref{calc:approximations} at the fixed +threshold $c(\sigma)$ is bounded code data. If an equally indexed +sequence in any sufficiently closed eligible object has the same +diagram, it determines a unique corresponding type there by the same +majority prescription. Arrows carrying one such sequence to another +carry the corresponding type to its free lift. In particular an +isomorphism carrying the code sequence transports the type. +\end{enumerate} +We say that an actual type over $B$ is \emph{free from $A$} when it is +the free lift of its restriction to $A$. +\end{theorem} + +\begin{proof} +First fix a query bound $\tau$ and take an approximation sequence +$I=(v_i:i<\kappa)$ in $A$ with $\kappa>\tau+\aleph_0$. Given a +query $Z$ in $B$ of size at most $\tau$, saturation moves it into $A$ +over the entire sequence: there is $Z_0\subseteq A$, with the same +enumeration, such that +\begin{equation} +\label{calc:query-move} + D(I,Z)=D(I,Z_0). +\end{equation} +The total length $|I|+|Z|$ is bounded. For mixed levels it is bounded +at the originating level, and the mixed saturation clause applies. +Define the answer on $Z$ to be the majority of $D(v_i,Z)$, equivalently +the answer $p\mathord\upharpoonright Z_0$. Equation +\eqref{calc:query-move} preserves every entry of the sequence, so this +definition is independent of the chosen $Z_0$ and retains the same +exception estimate. + +If $I'$ is another sufficiently long approximation sequence, move $Z$ +into $A$ over $I\cup I'$. On that moved query both majorities compute +the original $p$. Thus the answers agree. This proof requires +saturation over the concatenation of the sequences; it does not +require either approximation estimate to hold for a query containing +that concatenation. The averaging queries still have size at most +$\tau$. The identical argument compares different query bounds on +their common domain. It also proves coherence under restrictions and +reindexing of queries. + +We next obtain an actual type. Use +$\tau=c(\sigma)$ and $\kappa=\tau^+$. In a well-ordered chart of $B$ +prescribe the majority answers on every query of size at most $\tau$. +For each single query these are realized by some $v_i\in A\subseteq B$, +since fewer than $\kappa$ indices are exceptional. Thus they have +local placements. Lemma~\ref{calc:compact-placement} gives a placement +realizing all of them in an order enlargement of $B$. This is an +actual tuple in an extension object, and +Theorem~\ref{calc:scheme-amalgamation} makes its type unique, since +$\tau\ge g(\sigma)$. This type is $p^B$. Its restriction to $A$ +equals $p$ by the same testing criterion. + +For a larger desired bounded query size $\tau'$, repeat with a power +closed bound at least $c(\sigma,\tau')$. The comparison of sequences +just proved identifies the resulting type with $p^B$ on the initial +testing threshold and hence everywhere. In a mixed-level use, this +additional direct majority evaluation is asserted only when its costs +remain below the originating level. The type itself is already +actual over $B$, and its diagrams at larger arities are those of this +actual type; no averaging inside a smaller model at unbounded arities +is being asserted. + +Restriction to a subtuple commutes with every majority evaluation. +Choose the threshold large enough for both arities and use testing to +obtain equality of types. Isomorphism invariance follows by applying +the isomorphism to the sequences and their query diagrams. +For transitivity, let $J\subseteq B$ be a canonical approximation +sequence for $p^B$ at a sufficient fixed threshold, and retain an +original sequence $I\subseteq A$ at that threshold. The latter +computes $p^B$ on queries in $B$ by the construction. Move a query in +$C$ into $B$ over $I\cup J$. The two sequences give the same answer +there, because each computes $p^B$. The first also computes $p^C$, +and the second computes $(p^B)^C$. Testing proves transitivity. +All these sequence lengths can be chosen as the uniform bounds in +Lemma~\ref{calc:approximations}, below the smallest originating level. +This proves the mixed-level assertions as well. + +Finally, suppose $J\subseteq B$ has the same whole diagram as a fixed +code $I\subseteq A$ at threshold $c(\sigma)$. For each query $Z$ in +$B$ of that size, saturation realizes $D(J,Z)$ over $I$ in $A$. +The resulting majority is therefore well-defined, with the same +exception estimate. For each query one of the entries of $J$ realizes +the answer in $B$. Coherence, compact placement and testing give an +actual unique type, as in the preceding construction. Moving queries +over code sequences shows that an arrow carrying $J$ into a larger +object gives exactly its free lift there. The code length and every +threshold used here depend only on $\sigma$ and the fixed scheme, +which proves the final assertion. +\end{proof} + +\begin{lemma}[An average criterion for descent] +\label{calc:average-descent} +Let $A\to B$ have eligible sizes $\nu\le\mu$, and let $q=\tp(d/B)$ +be actual, with $|d|=\sigma<\nu$. Choose +$\chi\ge c(\sigma)$ below $\nu$. Suppose that a bounded sequence +$I=(d_i:i<\kappa)$ in $A$, with $\chi+\aleph_0<\kappa<\nu$, has +majority value $q\mathord\upharpoonright Z$ on every query +$Z\subseteq B$ of size at most $\chi$, with at most +$|Z|+\aleph_0$ exceptions. Then $q$ is the free lift of its actual +$\nu$-restriction to $A$. +\end{lemma} + +\begin{proof} +Let $p=q\mathord\upharpoonright A$ and choose a canonical sequence +$J\subseteq A$ approximating $p$ at size $\chi$. For a query +$Z\subseteq B$ of that size, mixed saturation moves it to $Z_0$ in +$A$ over the whole $I\cup J$. The total length is less than $\nu$. +The majority from $I$ on $Z$ computes $q$, and on $Z_0$ computes $p$. +These answers agree by diagram preservation. The majority from $J$ +on $Z_0$ also computes $p$, and on $Z$ computes $p^B$. +Hence $q$ and $p^B$ agree on all $\chi$-queries. Apply the testing +criterion. The same proof works for an arbitrary longer indexed +family whenever a bounded subsequence of the displayed length retains +the exception estimate. +\end{proof} + +\subsection{Products and local character} + +\begin{proposition}[Products of bounded types] +\label{calc:products} +For bounded types $p(x),q(y)$ over $A$, realize $p$ in an extension +$B\ge A$ and then realize $q^B$ in a further extension. The joint +type of $(x,y)$ over $A$, denoted $p\otimes q$, is independent of all +choices. Binary products are symmetric, with the indicated +permutation of coordinates. Finite products associate, are +permutation invariant, and commute with coordinate projections. +Furthermore +\begin{equation} +\label{calc:free-product-formula} + (p\otimes q)^B=p^B\otimes q^B. +\end{equation} +These assertions hold across eligible levels when all originating +arities and uniform testing costs are bounded at the smallest level. + +If $x/B$ is free from $A$ and $x,d$ have product type over $B$, then +$x$ and $(d,e)$ have product type over $A$ for every bounded tuple +$e$ from $B$. +\end{proposition} + +\begin{proof} +Two realizations of $p$ can be identified over $A$ by amalgamation. +In a common model containing their witnessing extensions, the free +lifts of $q$ agree by stationarity and restriction. A second +type-identifying amalgamation identifies the realizations of $q$. +This proves independence of the choices in the definition. + +We give the cardinal calculation for symmetry. Fix a bounded query +$Z$ in $A$, and put +$e=|Z|+|x|+|y|+\aleph_0$. Choose approximation sequences +$(a_i:i<\kappa)$ for $p$ and $(b_j:j<\lambda)$ for $q$, sufficient +for queries of size $e$, where +\[ + \kappa=e^+,\qquad \lambda=\kappa^+. +\] +All these cardinals lie below a sufficiently closed level. +For the array $M_{ij}=D(a_i,b_j,Z)$, first realize $x$ and freely +realize $y$. The resulting answer $R$ is obtained by first taking +the majority in $i$ and then in $j$. Except at at most $e$ columns, +the column majority is $R$, and in each such column at most $e$ +entries differ from $R$. Reversing the realization order gives an +answer $S$: except at at most $e$ rows, its row majority is $S$, +with at most $e$ exceptions in each such row. The bounds concern +the size of $(x,Z)$ or $(y,Z)$, not the lengths of the sequences. + +There are $\kappa$ good rows. Their exceptional columns together +number at most $\kappa e=\kappa$. Since $\lambda>\kappa$, choose +a column outside this set and outside the at most $e$ bad columns. +Every good-row entry in that column equals $S$, whereas all but at +most $e$ entries equal $R$. As $\kappa>e$, some entry has both +values. Thus $R=S$. This explains why two equally sized sequences +would not suffice for the argument. Testing gives symmetry of +the joint types. + +For \eqref{calc:free-product-formula}, take the two sequences just +used in $A$ and a sequence $K\subseteq A$ approximating the actual +old product. Choose their accuracy at a common sufficient test +threshold. Given a query $Z$ in the enlarged base $B$, move it into +$A$ over these three whole sequences. Both successive free +evaluations over $B$ are the iterated majority of the same array +$D(a_i,b_j,Z)$. Transitivity permits the original sequence for the +second factor to be used after the first factor has been realized. +The query movement identifies this array with the old array in $A$, +where it computes $p\otimes q$. The sequence $K$ evaluates that +old product and computes its free lift on the original query. +The answers therefore agree. Testing proves the formula. All the +lists fixed during the query movement are bounded, including in the +mixed case. + +Now realize $x,y,z$ successively along a chain of free lifts from +$A$. The first two form $p\otimes q$, and $z$ is free over a model +containing them, so the joint type is $(p\otimes q)\otimes r$. +Over a model containing $x$, the last two have type +$q^B\otimes r^B=(q\otimes r)^B$, by the formula just proved. +Thus the same joint type is $p\otimes(q\otimes r)$. This proves +associativity. Projection follows from the projection property of +free lifts in each successive realization. Associativity and the +binary symmetry permit every permutation of a finite list of factors. + +For the final assertion, use symmetry and a type-identifying amalgam +to realize the given joint type by first putting $d$ in a model +containing $B$ and then realizing $x$ freely over it from $B$. +Since $x/B$ is itself free from $A$, transitivity makes this last +realization free from $A$. The tuple $(d,e)$ is in the intervening +model, so the definition of product gives the assertion. The amalgam +identifies the full original tuple over $B$, and hence retains its +actual joint type over $A$. +\end{proof} + +\begin{proposition}[Local character and continuity] +\label{calc:local-character} +Let $(A_i:i<\delta)$ be an increasing chain of objects at one eligible +level $\mu$, whose union $A$ has size $\mu$, and let $\delta$ be a +nonzero limit ordinal. If $p=\tp(v/A)$ and either $v$ is finite or +$|v|<\cf(\delta)$, then $p$ is free from some $A_i$. +If, in addition, every restriction $\tp(v/A_i)$ is free from a fixed +base $X\leq A_0$, then $p$ is free from $X$. +\end{proposition} + +\begin{proof} +Pass to a cofinal chain indexed by the regular cardinal +$\gamma=\cf(\delta)$. Choose a sufficient testing size +$\chi\ge c(|v|+\aleph_0)$ below $\mu$. If $\gamma$ is larger than +the size of an approximation sequence for $p$ at that threshold, +the whole sequence lies in one stage. Its exception estimate is +still correct on all queries in $A$, so +Lemma~\ref{calc:average-descent}, at the same level, applies. + +In the other case $\gamma<\mu$. Choose a regular +$\kappa<\mu$ above $\gamma,\chi,|v|$ and the needed bounds, and +take $\kappa$ approximations inside $A$. Every whole approximation +tuple belongs to some stage: this uses finiteness, or the hypothesis +$|v|<\gamma$. If every stage contained fewer than $\kappa$ of the +tuples, regularity of $\kappa$ and $\gamma<\kappa$ would make their +union contain fewer than $\kappa$ tuples. Thus one stage contains +$\kappa$ of them. Pass to those indices. The exception estimate +survives unchanged, and Lemma~\ref{calc:average-descent} again +identifies $p$ as the free lift from that stage. + +For the final assertion, local character makes $p$ free from some +$A_i$. Its restriction there is free from $X$ by hypothesis. +Transitivity in Theorem~\ref{calc:free-extension} makes $p$ free +from $X$. +\end{proof} + +\subsection{Universal strips and approximate retractions} + +The remaining construction provides maps from an extension back into its +base which fix each individual base element at all but finitely many +indices. It +will allow isolation to descend from a larger level without requiring a +bounded tuple to lie in one stage of a countable exhaustion. + +\begin{lemma}[Closure of a bounded support set] +\label{calc:support-closure} +Let $J\subseteq J^*$ be linear orders, let $V$ be a convex strip in +$J^*$, and put $J_0=J\setminus V$. Allow any fixed finite coloring +of positions in $J_0$, including their side of $V$. Let +$T^*\subseteq J^*$ and $S_0\subseteq J_0$ be bounded sets. Given +an infinite $\chi$, choose an infinite $\lambda$ with +\[ + \lambda\ge\chi+|T^*|+|S_0|, + \qquad \kappa=2^\lambda. +\] +There is $S\subseteq J_0$, of size at most $\kappa$, containing +$S_0\cup(T^*\cap J_0)$, with this property: for every +$R\subseteq S$ of size at most $\chi$ and every indexed tuple $Q$ +of at most $\chi$ positions from $J_0$, there is an equally indexed +tuple $Q'$ in $S$ having the same order, equality and color pattern +over $R\cup T^*$. In particular all positions named by $R$ are +fixed whenever they occur in $Q$. + +At a strong limit level with the displayed input bounded, $S$ is +bounded. The assertion needs no well-order hypothesis on $J$. +\end{lemma} + +\begin{proof} +Build $S_\alpha$ for $\alpha<\chi^+$, starting with the required +set. At a successor stage, for every $R\subseteq S_\alpha$ of +size at most $\chi$, and every order, equality and color pattern of +at most $\chi$ further positions over $R\cup T^*$ that is actually +realized in $J_0$, choose one realizing tuple and adjoin its positions. +Use unions at limits. If $|S_\alpha|\le\kappa$, the number of +choices of $R$ is at most $\kappa^\chi=\kappa$, and the number of +patterns for each $R$ is at most $2^\lambda=\kappa$. Each chosen +tuple adds at most $\chi$ positions. Consequently every stage and +their union have size at most $\kappa$. + +For $R$ contained in the final union, regularity of $\chi^+$ puts +all of $R$ into a single stage. The actual tuple $Q$ witnesses that +its pattern was among the requirements processed at the next stage. +Its chosen witness is the required $Q'$. Equality with elements of +$R$ and $T^*$ is part of the pattern, so it is preserved literally. +This also covers arbitrary enumerations and repetitions in $Q$. +\end{proof} + +\begin{proposition}[Universality of a fresh strip] +\label{calc:universal-strip} +Let $\mathcal C$ already satisfy the working package. In addition to +its fixed template $F$, one may use a presentation $G$ with the +following properties: it has a fixed set of finite-support labels; +every presentation $G(J)$ used below, for an order of size $\mu$, is +an object of $\mathcal C$ at level $\mu$; order embeddings give +$\mathcal C$-arrows; and the diagrams of its term lists are natural, +including in the order enlargements and padded orders used. Assume +also that these labels represent every element of $G(J)$. +Require $\mu$ to exceed the cardinality of the label set of $G$ and +to be closed under the bounds below with that cardinal included in +their fixed input data. + +Put $A=G(J)$. Insert into $J$ a fresh convex strip of size $\mu$ +with countably many disjoint convex substrips, each of size $\mu$, +and denote the resulting object by $A^+$. Then every same-level +extension of $A$ embeds into $A^+$ over $A$. In particular this +applies to $G=F$ and to well-ordered presentations. No separate +union or categoricity assertion for a category of $G$-presentations +is required. +\end{proposition} + +\begin{proof} +We first realize a bounded type $p(v/A)$ using just one fresh +substrip $V$. Choose a testing threshold $\chi\ge c(|v|+\aleph_0)$ +large enough also for $|v|$ and the number of labels of $G$. +Take a sequence $I$ in $A$ approximating $p$ at query size $\chi$. +Represent its entries by $G$-terms and collect their old supports in +$S_0\subseteq J$. Apply Lemma~\ref{calc:support-closure}, with +the side of $V$ recorded, to obtain a bounded $S\subseteq J$. +Its closure permits every at-most-$\chi$ old query pattern to move +into $S$ over any at-most-$\chi$ subset of $S$. The approximation +supports, which may be larger than $\chi$, can instead be included +as the fixed set $T^*$ in that Lemma. Thus all of them are fixed +during every such movement. + +Let $P_S$ be the set of all $G$-terms on $S$. It is bounded, since +$G$ has a bounded set of finite-support labels. The natural +parameter diagram of $P_S$ is the same in $A$ and in $G(S\cup V)$. +The latter is a level object because $|V|=\mu$. Saturation there +realizes the desired whole joint diagram of $(v,P_S)$, giving a +tuple $w$ whose supporting positions belong to $S\cup V$. +Only at most $|v|+\aleph_0\le\chi$ of its supporting positions +are old; call them $R\subseteq S$. + +For any query $Z$ in $A$ of size at most $\chi$, move its chosen +support tuple into $S$ over $R$ and the fixed approximation +supports, preserving the side of $V$. Let $Z'$ be the resulting +term tuple. All comparisons with the new supporting positions +of $w$ in $V$ are forced by that side. Naturalness therefore gives +\[ + D(w,Z)=D(w,Z'),\qquad D(I,Z)=D(I,Z'). +\] +The majority from $I$ computes $p$ on both queries. The chosen +realization of $(v,P_S)$ gives $D(w,Z')=p\mathord\upharpoonright Z'$. +It follows that $D(w,Z)=p\mathord\upharpoonright Z$. +Testing of actual types, using a natural $F$-chart of $A$ if $G$ +differs from $F$, proves $\tp(w/A)=p$. Naturality of $G$ was used +only for the displayed support movements; the testing and saturation +are those already proved in $\mathcal C$. + +Now let $B\ge A$ be arbitrary and choose increasing bounded lists +$d_n$ exhausting it, including countable strong-submodel +exhaustions in every vertex. Use the first substrip to realize the +type of $d_0$ over $A$. At subsequent step $n$, the target base consists +of $A$ and the whole previously used substrips. Suppose $d_n$ has +already been assigned there with its correct type over $A$. +Theorem~\ref{calc:scheme-amalgamation} amalgamates $B$ with this +target base over $A$, identifying that entire assigned tuple. +In this amalgam, $d_{n+1}$ has an actual bounded type over the +target base. Apply the one-substrip construction to the next fresh +substrip. Since its type includes equality with the already +assigned coordinates, the new assignment extends the previous one. +The final union of the assignments lands in $A^+$. The arrow +criterion applies to the included strong-submodel exhaustions, so +it is an arrow $B\to A^+$ over $A$. +\end{proof} + +\begin{corollary}[Approximate retractions] +\label{calc:retractions} +For same-level $A\le B$ there are arrows +$(r_i:B\to A:i<\mu)$ such that, for every $a\in A$, +\[ + \bigl|\{i<\mu:r_i(a)\ne a\}\bigr|<\aleph_0. +\] +The maps are not asserted to fix all of $A$ at once. +\end{corollary} + +\begin{proof} +Present $A$ by $F$ on a well-order consisting of $\mu$ disjoint +convex blocks $J_i$, each of cardinality $\mu$ and itself containing +countably many convex substrips of that size. The whole order still +has cardinality $\mu$, and the working package identifies its +presentation with $A$. Let $A_i$ be the presentation on the +complement of $J_i$. It is a level subobject of $A$. +Proposition~\ref{calc:universal-strip}, with $J_i$ as the inserted +strip, embeds $B$ into $A$ over $A_i$; call the arrow $r_i$. +Choose one finite supporting set for $a$. It meets only finitely +many $J_i$. For every other index, $a\in A_i$ and $r_i(a)=a$. +\end{proof} + +\begin{lemma}[Descent through approximate retractions] +\label{calc:retraction-descent} +Let $A\le U$ be objects at an eligible level $\mu$, and let +$A'\le U'$ be objects at an eligible level $\nu\le\mu$, with +compatible arrows $A'\le A$ and $U'\le U$. Suppose there are +arrows $(r_i:U\to A:i<\nu)$ such that every $a\in A$ is fixed +for all but finitely many $i$, and $r_i[U']\subseteq A'$ for all +$i<\nu$. Then for every tuple $d$ in $U'$ of length less than +$\nu$, its actual type over $A$ is the free lift of +$\tp(d/A';U')$. All comparisons are comparisons of the working +diagrams, including mixed-level diagrams at arities bounded at $\nu$. +\end{lemma} + +\begin{proof} +Coherence makes $r_i\mathord\upharpoonright U':U'\to A'$ an +arrow: its image and $A'$ are subobjects of $A$, and its image is +contained in $A'$. Put $\sigma=|d|+\aleph_0$, choose +$\chi\ge c(\sigma)$ below $\nu$, and choose +$\chi^+\le\kappa<\nu$. Retain any $\kappa$ indices and form +$I=(r_i(d))$ on them. This is a bounded concatenated list in $A'$. + +For a query $Z\subseteq A$ of size at most $\chi$, at most +$|Z|+\aleph_0$ indices fail to fix every element of $Z$. At every +other index, arrow preservation gives +\[ + D(r_i(d),Z)=D(r_i(d),r_i(Z))=D(d,Z). +\] +Thus $I$ has the required majority estimate for the actual type +over $A$. On queries in $A'$ the same identity is a comparison at +level $\nu$, using the restricted arrows. Mixed preservation +identifies it with the larger-level identity. + +Apply Lemma~\ref{calc:average-descent} to $A'\le A$. Its +restriction type is precisely $\tp(d/A';U')$, by +Corollary~\ref{calc:mixed-realization}. The conclusion is therefore +the stated free lifting of the actual type, including tuples that +span a chosen resolution of $U'$. +\end{proof} + +The preceding calculus will be applied only after the working package +has been established for the relevant scheme. Its approximation, +testing and support-closure bounds are functions of fixed scheme data +and bounded input arities. This is the uniformity needed when the +schemes are iterated at finite depths. diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/07-isolation.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/07-isolation.tex new file mode 100644 index 0000000000000000000000000000000000000000..d140a33d17dd1ad7ef5a7187e1d9826a1fd41536 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/07-isolation.tex @@ -0,0 +1,724 @@ +\section{Isolation and dominated extensions} +\label{iso:section-isolation} + +Throughout this section, $\mathcal C$ is a working scheme in the sense of +Definition~\ref{lev:working-package}. We use the refinements of the eligible +levels required in Section~\ref{calc:section-calculus}. All objects and +extensions in a displayed type are at one eligible level, unless a change of +level is stated. We retain the eligible-subclass hypothesis of +Convention~\ref{lev:eligible-subclass}; every larger auxiliary sufficiently +closed level below is chosen in its unbounded refined subclass. +A type over an object at a smaller eligible level is computed +in a section of that level; Corollary~\ref{calc:mixed-realization} makes this +independent of the section. A tuple is bounded when its cardinality is less +than the level of its base. We write $p^{\mathrm{fr}}_B$ for the free lift +$p^B$ of Theorem~\ref{calc:free-extension} when its role needs emphasis. + +We will repeatedly use the following convention concerning bounds. A request +involving a tuple of cardinality $\sigma$ is first enlarged to an infinite +arity at least $\sigma$. The testing, compact-placement, and approximation +bounds of Section~\ref{calc:section-calculus} are then applied at that arity. +There are only finitely many such enlargements in each argument below. A +source level is required to be strictly closed under those bound functions. +Thus, when a tuple is bounded at a smaller level, every test used to determine +its type may be taken below that smaller level, even if the receiving object +is at a much larger eligible level. This does not assert saturation over an +unbounded parameter set. + +Our goal is an extension controlled by a given mark, with an embedding +into every ambient object realizing that mark's type. Relative isolation +will give this control on bounded tuples. The main existence argument +first treats one bound on tuple length and then obtains all bounded +lengths in a single extension. + +\subsection{Relative isolation} + +\begin{definition}\label{iso:relative-isolation} +Let $q=\tp(u,c/A)$ and $p=\tp(u/A)$, where the concatenation $(u,c)$ is +bounded. We say that $q$ is \emph{isolated relative to $u$} if there is a +bounded set $Z\subseteq A$ such that, for every $q'=\tp(u',c'/A)$, +\[ + \tp(u'/A)=p\quad\text{and}\quad + D(u',c',Z)=D(u,c,Z) + \quad\Longrightarrow\quad q'=q. +\] +Here and below a parameter set in a diagram is supplied with a fixed +enumeration, and the same enumeration is used on both sides. We call $Z$ an +\emph{isolating fragment}. A projection retains the entire tuple $u$ unless +explicitly stated otherwise. +\end{definition} + +\begin{lemma}\label{iso:free-isolation}\label{iso:upward-transfer} +The following hold at sufficiently closed eligible levels. +\begin{enumerate} +\item A projection of a relatively isolated type is relatively isolated. +\item Suppose $A\leq B$, $q=\tp(u,c/A)$ is isolated relative to $u$, and an +actual coupling over $B$ satisfies +\[ + \tp(u/B)=\tp(u/A)^{\mathrm{fr}}_B. +\] +Then +\[ + \tp(u,c/B)=q^{\mathrm{fr}}_B. +\] +Moreover this type is isolated relative to $u$, using the same isolating +fragment as $q$. +\end{enumerate} +The second assertion also holds when $A$ is at a smaller eligible level than +$B$, provided $(u,c)$ and the required uniform bounds are below the level of +$A$. +\end{lemma} + +\begin{proof} +For the first assertion, write $c=(c_0,c_1)$ and let $Z$ isolate $q$. +Suppose $(u',c'_0)$ has the prescribed marginal on $u$ and the same diagram +as $(u,c_0)$ over $Z$. In a model containing $A,u',c'_0$, bounded saturation +extends this diagram by a tuple $c'_1$ so that +\[ + D(u',c'_0,c'_1,Z)=D(u,c_0,c_1,Z). +\] +Isolation identifies the full type over $A$, and its projection identifies +$\tp(u',c'_0/A)$. + +For the second assertion, fix an isolating fragment $Z$. It is enough to +compare the actual joint type over $B$ with $q^{\mathrm{fr}}_B$ on the +uniform testing arity for $(u,c)$. Enlarge that arity to include $Z$ and +choose approximation lists in $A$ for both $q$ and $p=\tp(u/A)$, long enough +to evaluate all the comparisons just specified. Let $t$ be a test tuple +from $B$ at this enlarged arity. Bounded saturation moves $t$ to a tuple +$t'$ in $A$, preserving its diagram over $Z$ and both approximation lists. + +Choose a realization $u_0$ of $p$ over $A$. The free evaluation of $p$ gives +\[ + D(u,t,Z)=D(u_0,t',Z). +\] +Saturation over the bounded list $(u_0,t',Z)$ supplies $c_0$ such that +\[ + D(u_0,c_0,t',Z)=D(u,c,t,Z). +\] +In particular $(u_0,c_0)$ agrees with $q$ on $Z$ and has marginal $p$. +Isolation gives $\tp(u_0,c_0/A)=q$. Consequently the actual answer to the +query $t$ is the answer that $q$ gives to $t'$. Because the query was moved +over the approximation list for $q$, that is also the free evaluation of +$q$ on $t$. The testing theorem identifies the types over $B$. + +Now let a competing type over $B$ have the same marginal on $u$ and agree +with the displayed type on $Z$. Its restriction to $A$ is $q$. The argument +just given identifies it with $q^{\mathrm{fr}}_B$, proving isolation there. +In the mixed-level case, perform these comparisons at the uniform test +bound below the source level. Mixed realization supplies the required +sections, and the same argument applies to those diagrams. +\end{proof} + +\begin{proposition}\label{iso:alternative} +There is a bound function $h$, depending only on the fixed working scheme, +with the following property. Put +$\sigma=|u,c|+\aleph_0$. At sufficiently closed eligible levels, precisely +one of the following alternatives holds for $q=\tp(u,c/A)$: +\begin{enumerate} +\item $q$ is isolated relative to $u$ by a fragment of size at most +$h(\sigma)$; +\item there are an extension $A\leq B$ at the same level and an extension +$q_B$ of $q$ such that +\[ + q_B\ne q^{\mathrm{fr}}_B, + \qquad + q_B\mathbin{\upharpoonright}u + =\tp(u/A)^{\mathrm{fr}}_B. +\] +\end{enumerate} +In particular, every relatively isolated type has an isolating fragment +bounded by $h(\sigma)$, uniformly in the level and in the type. +\end{proposition} + +\begin{proof} +Choose an infinite $\chi$ large enough for the testing theorem at arity +$\sigma$, for the approximation lists for $q$ and its marginal $p$, and +for the finitely many joint comparisons made below. These choices are +uniform in $\sigma$. Increasing $\chi$ if necessary, all the supports in +the chosen approximation lists have cardinality at most $\chi$. + +Present $A=F(I)$ on a well-order of the level cardinality $\mu$. We may +choose $I$ to have more than $\sigma$ disjoint consecutive blocks, each +of order type and cardinality $\mu$: this is still a well-order of +cardinality $\mu$, and all such presentations are objects at this level. +Let $J_0$ consist of the supports of the approximation lists, together +with an initial segment of order type $\chi^+$ in every candidate block. +Its size has a bound depending only on $\sigma$. Let $Z_0$ be the set of +all term values on $J_0$. Since the label set is fixed, the same is true +of $|Z_0|$. + +If $Z_0$ isolates $q$, we have the first alternative. Suppose it does not. +Choose $q^1\ne q^0=q$ with the same marginal $p$ and the same diagram on +$Z_0$. By Lemma~\ref{calc:compact-placement}, give both types placements +over the displayed presentation of $A$, and amalgamate their support +orders over $I$. The combined new support has cardinality at most +$\sigma$. Each new support position either hits the interior of one +candidate block or determines an interior cut of at most one such block. +We can therefore choose a candidate block $J$ whose interior contains +neither a hit nor a cut of either placement. + +Set $A_*=F(J_0\cup J)\leq A$. This object has cardinality $\mu$. The +restrictions of $q^0$ and $q^1$ to $A_*$ are equal. To check this, take a +test at the fixed sufficient arity. Move every position of the test +lying in $J$ into the initial $\chi^+$ part of $J$, preserving the order +and equality pattern of the test. This can be done above any positions +of that initial part which need to remain in the pattern: fewer than +$\chi^+$ positions have been specified. Leave the positions outside +$J$ unchanged. No position of either candidate separates the interior +of $J$, so the move preserves the natural diagram with either candidate. +The moved query uses only $J_0$ and therefore has the same answer for +$q^0$ and $q^1$. The testing theorem proves equality over $A_*$. + +The approximation lists for $q^0$ and $p$ lie in $A_*$. Their majority +evaluations show that $q^0$ and $p$ over $A$ are free from $A_*$. Also, +an isomorphism $f:A_*\simeq A$ may be chosen to fix those bounded lists, +by bounded diagram homogeneity. The transport assertion for free types +then gives +\[ + f\bigl(q^0\mathbin{\upharpoonright}A_*\bigr)=q. +\] +Transport the inclusion $A_*\leq A$ along $f$, viewing its larger object +as an extension $B$ of the original base $A$. Transport $q^1$ as well. +Its restriction is $q$ because the two restrictions to $A_*$ agree. Its +marginal is the free lift of $p$. Its joint type is not free, since the +unique free lift before transport was $q^0$, distinct from $q^1$. +This is the second alternative. + +Choose $h(\sigma)$ to bound the size of $Z_0$ for these uniform choices. +Lemma~\ref{iso:free-isolation} excludes the second alternative when $q$ +is isolated, even if an initially given isolating fragment was larger. +This proves both exclusivity and the asserted uniform bound. +\end{proof} + +\subsection{Primeness and descent of isolation} + +\begin{definition}\label{iso:all-dominated-definition} +Let $X\leq Y$ be same-level objects and let $b$ be a bounded tuple in +$Y$. We say that $Y$ is \emph{all-dominated over $X+b$} if, for every +bounded tuple $c$ from $Y$, the type $\tp(b,c/X)$ is isolated relative +to $b$. +\end{definition} + +The quantifier over all bounded tuples is part of this definition. It is +not enough to verify isolation for individual entries, or for the tuples +in one chosen countable exhaustion of $Y$. + +\begin{proposition}\label{iso:prime} +Let $Y$ be all-dominated over $X+b$ at a sufficiently closed eligible +level $\mu$. +\begin{enumerate} +\item If an extension $N$ of $X$ contains a tuple $b'$ of the same type +over $X$, there is an arrow $Y\longrightarrow N$ over $X$ sending $b$ +to $b'$. The target may be at a larger eligible level. A prescribed +bounded tuple match can be retained whenever its joint type with $b$ +over $X$ already agrees. +\item Two all-dominated objects over the same data $X+b$ are isomorphic +over those data. The isomorphism may retain a prescribed bounded tuple +match of the kind just described. +\item In an actual coupling, if a bounded tuple $x$ has product with +$b$ over $X$, then $\tp(x/Y)$ is free from $X$. +\end{enumerate} +\end{proposition} + +\begin{proof} +Choose increasing bounded lists $c_n$, $n<\omega$, exhausting $Y$; +include $b$, the prescribed initial data, and exhaustions of every vertex +by smaller strong submodels. Suppose the image of $c_n$ has already been +chosen with the correct full type over $X$. Let $Z_{n+1}\subseteq X$ +isolate $\tp(b,c_{n+1}/X)$. The old image and the source list have the +same diagram over $Z_{n+1}$. Saturation in $N$ extends this bounded match +to $(b,c_{n+1},Z_{n+1})$. Relative isolation makes its full type over $X$ +correct. Taking the union gives a map on all of $Y$ fixing $X$: whenever +an entry of $X$ occurs in a list, its equality with the corresponding +parameter is part of its full type. The strong-submodel exhaustions and +the map criterion in the working package show that this map is an arrow. +This proves the first assertion, also into a larger target because every +saturation request is bounded at the source level. + +For the second assertion, perform the same construction alternately on +the two sides, at each step including the next list in the chosen +exhaustion on the side to be covered. Both objects have isolation for +the enlarged bounded list. The construction therefore continues in both +directions and its union is an onto structural isomorphism. + +For the third assertion, use symmetry and the definition of product to +choose a model $W\geq X$ containing $x$ over which $b$ is free from $X$. +Type-identifying amalgamation brings $W$ and the actual coupling into a +common extension, identifying $x$ and $b$ over $X$. For every bounded +$c$ in $Y$, Lemma~\ref{iso:free-isolation} says that $(b,c)/W$ is free +from $X$. Hence $x$ and $(b,c)$ have product over $X$. Symmetry and +projection give the free evaluation of $x$ on every bounded test $c$ +from $Y$. The testing theorem identifies $\tp(x/Y)$ with the free lift +of $\tp(x/X)$. +\end{proof} + +\begin{lemma}\label{iso:downward-transfer} +Suppose $A\leq B$ are objects at eligible levels $\nu\leq\mu$, +$|u,c|<\nu$, and the actual type +\[ + q_B=\tp(u,c/B) +\] +is the free lift of $q_A=\tp(u,c/A)$. If $q_B$ is isolated relative to +$u$, then $q_A$ is isolated relative to $u$. Here $\nu$ is sufficiently +closed for the uniform bounds at $|u,c|+\aleph_0$. +\end{lemma} + +\begin{proof} +By Proposition~\ref{iso:alternative}, choose a large-base isolating +fragment $W\subseteq B$ with +\[ + |W|\leq h(|u,c|+\aleph_0)<\nu. +\] +Let $w$ enumerate $W$. The type $\tp(w/A)$ is a small-level type, +computed by taking a level-$\nu$ section containing $A$ and $w$. +Choose a sufficiently long bounded approximation list $(w_i)_{i<\tau}$ +for this type inside $A$, where $\tau<\nu$. Enlarge its testing bound +to include the arities of $w$ and $(u,c)$ and choose $\tau$ larger than +the two exception bounds that will occur below. Put +\[ + Z=\bigcup_{i<\tau}\ran(w_i)\subseteq A. +\] +This is a bounded fragment. + +Let $q'_A$ have the same marginal on $u$ as $q_A$ and agree with it over +$Z$. Lift $q'_A$ freely to $B$. The marginal on $u$ then agrees with +the marginal of $q_B$. In either free coupling, symmetry identifies the +joint diagram with $w$ by the majority evaluation of $\tp(w/A)$ against +the candidate tuple. Thus all but the prescribed bounded number of +$i<\tau$ give the diagram of that candidate with $w$. Choose one index +outside the two exception sets. The candidates agree on $w_i$, because +$w_i$ lies in $Z$, so their free lifts agree on $w$. Isolation of $q_B$ +identifies the free lifts, and restriction identifies $q'_A$ with $q_A$. +Every approximation and comparison here has size below $\nu$; no +approximation to the whole of $B$ is used. +\end{proof} + +\subsection{Existence with a prescribed bound on tuple length} + +We first isolate the continuity requirement on the marked marginal. This +will let us identify exactly where the proof for a finite mark must be +supplemented for a long independent list. + +\begin{definition}\label{iso:marginal-continuity} +A bounded marginal type $p$ satisfies the \emph{continuity condition used +here} if the following holds at every sufficiently closed eligible level +to which $p$ is freely lifted. Suppose +\[ + X_0\leq X_1\leq\cdots\leq X_i\leq\cdots\quad(i<\delta) +\] +is a continuous chain at that level, its union is an object at that level, +and a fixed realization $b$ in a coupled extension satisfies +\[ + \tp(b/X_i)=\tp(b/X_0)^{\mathrm{fr}}_{X_i} + \qquad(i<\delta). +\] +Then the same equality holds over $\bigcup_{i<\delta}X_i$. +The type at $X_0$ is required to be the appropriate free lift of $p$. +\end{definition} + +Every finite-tuple marginal has this property. At a limit, local character +for the finite tuple gives freeness from some earlier stage, and +transitivity gives freeness from $X_0$. This is a direct application of +Proposition~\ref{calc:local-character}, including when the chain has small +cofinality. + +\begin{proposition}\label{iso:finite-existence} +Let $\lambda$ be a sufficiently closed eligible level, let $X$ be an +object there, and let $p=\tp(b/X)$ be a finite-tuple type. For every +infinite $\theta<\lambda$ there is a same-level extension $X\leq U$ +containing a realization of $p$ such that +\[ + \tp(b,c/X)\text{ is isolated relative to }b + \quad\text{for every }c\in U\text{ with }|c|\leq\theta. +\] +More generally the assertion holds for a bounded mark $b$ satisfying +Definition~\ref{iso:marginal-continuity}, with $|b|\leq\theta$. +The closure requirements at the output level are uniform functions of +bounded arities; they do not require $\theta$ to have been fixed before +the level was chosen. +\end{proposition} + +\begin{proof} +We prove the more general statement. Fix a bounded approximation list +$I_p$ in $X$ which determines $p$ at a sufficient testing arity. Choose +a much larger eligible level $\kappa$ and a cardinal $\rho$ such that +\[ + \lambda<\rho=\rho^\lambda, + \qquad \rho^+<\kappa. +\] +Require $\rho$ to dominate the approximation and small-hull costs at +arity $\lambda$, and choose $\kappa$ after those set-sized bounds. Such +choices exist: first choose a sufficiently large power $2^\xi$ with +$\xi\geq\lambda$ for $\rho$, and then choose an eligible $\kappa$ +above $\rho^+$ and all remaining bounds. Extend $X$ by a presentation +arrow to a level-$\kappa$ object $X_0$, and choose $Y_0\geq X_0$ +containing a realization $b$ of the free lift of $p$. + +We construct continuous pairs +\[ + X_i\leq Y_i\qquad(i\leq\rho^+) +\] +at level $\kappa$, with a single fixed $b$ in all the upper objects. +The invariant on the marked marginal is +\begin{equation}\label{iso:matrix-marginal} + \tp(b/X_i)=\tp(b/X_0)^{\mathrm{fr}}_{X_i}. +\end{equation} +For $i<\rho^+$, we also keep increasing continuous sets +\[ + R_i\subseteq X_i,\qquad S_i\subseteq Y_i,\qquad + R_i\subseteq S_i,\qquad |R_i|,|S_i|\leq\rho, +\] +with $I_p\subseteq R_i$ and $b\subseteq S_i$. They are bookkeeping +sets, not additional structure on the objects. + +Here are the two closure requirements at a stage $i$ with +$\cf(i)>\lambda$. +\begin{enumerate} +\item Any specified at-most-$\lambda$ sets of base and top parameters +from $R_i,S_i$ are contained in a pair $A\leq V$ of level-$\lambda$ +subobjects of $X_i\leq Y_i$, with $A\subseteq R_i$ and +$V\subseteq S_i$. +\item Every tuple $d$ of length at most $\lambda$ from $S_i$ has a +sequence of at least $\lambda$ copies inside $R_i$ computing its +actual type over $X_i$ on all tests of size less than $\lambda$. +For every subtuple and each such parameter test $Z$, all but at most +$|Z|+\aleph_0$ of the copies give its joint $D$-diagram. +\end{enumerate} + +We explain how to arrange the requirements; in particular, they do not +amount to assuming a small saturated submodel. At successor stages, +process all at-most-$\lambda$ lists in the current parameter sets. +There are at most $\rho^\lambda=\rho$ of them. Mixed +L\"owenheim--Skolem supplies the requested level-$\lambda$ pairs. +Lemma~\ref{calc:approximations} supplies copies of each top list in the +current base, at the test sizes below $\lambda$ and with the displayed +exception estimate. Their total number of coordinates is at most +$\rho$. Close the sets under these operations for $\lambda^+$ steps; +the size is still $\rho$. If new elements are added later, repeat these +successor closures. At a limit $i$ of cofinality greater than $\lambda$, +every requested list was contained at an earlier stage. Local character +applied to that list, whose length is at most $\lambda<\cf(i)$, says +that its actual type over $X_i$ is free from some earlier stage. +Approximations placed in a later intervening successor base still +compute it over $X_i$ by the free-extension theorem. Their copies lie +in $R_i$. The same cofinality argument places every small-hull request +in an earlier closure. This proves both requirements at the indicated +limits. + +At any such stage, if some list $d$ of length at most $\lambda$ from +$S_i$ makes $\tp(b,d/X_i)$ nonisolated relative to $b$, apply +Proposition~\ref{iso:alternative}. It gives a same-level extension of +$X_i$ in which this joint type has a nonfree extension while the +$b$-marginal remains free. By type-identifying amalgamation, extend +$Y_i$ into a common $Y_{i+1}$, identifying the old $(b,d)$ with that +realization. The new base is the extension just chosen. Transitivity +preserves~\eqref{iso:matrix-marginal}. At other stages use compatible +extensions and the bookkeeping closures. At limits use the working +scheme's unions; Definition~\ref{iso:marginal-continuity} preserves +the marginal invariant. All these unions have size $\kappa$, since +$\rho^+<\kappa$. + +This process must reach an indicated stage at which every list in the +second closure requirement is isolated jointly with $b$. Suppose not, +and choose a failing list $d_i$ at every stage with $\cf(i)>\lambda$. +The set of these stages is stationary in the regular cardinal +$\rho^+$. Indeed, in any closed unbounded subset of $\rho^+$ take a +continuous increasing sequence of length $\lambda^+$ and its +supremum; the supremum is still below $\rho^+$, is in the closed +set, and has cofinality $\lambda^+$. + +Because $S_i$ is continuous and $|d_i|\leq\lambda<\cf(i)$, there is +$j(i)\nu$ and lift the type to a level-$\lambda$ base by a +presentation arrow. Apply Proposition~\ref{iso:finite-existence} at +level $\lambda$ with $\theta=\nu$. We obtain a pair $A\leq U$ of +level $\lambda$ such that every tuple of length at most $\nu$ in $U$, +jointly with $b$, has isolation relative to $b$ over $A$. The original +anchors may be retained in $A$ throughout that construction. + +Choose $\nu$ distinct indices from the family in +Corollary~\ref{calc:retractions}, and write the corresponding +approximate retractions as +\[ + r_j:U\longrightarrow A\qquad(j<\nu) +\] +such that each $a\in A$ is fixed by all but finitely many of them. +Use mixed L\"owenheim--Skolem to choose level-$\nu$ subobjects +$A'\leq U'$ of $A\leq U$ which contain $I_p,b$ in their indicated +places and satisfy +\begin{equation}\label{iso:retraction-closure} + r_j[U']\subseteq A'\qquad(j<\nu). +\end{equation} +Here is the closure construction. Start with level-$\nu$ sections +containing the desired entries. From the current top of size $\nu$, +adjoin to the next base all its images under the $\nu$ selected maps; +there are at most $\nu\cdot\nu=\nu$ coordinates. Enlarge the next +top to contain that base and the old top. Iterate countably many +times and take unions. Same-level chain closure and the common upper +bound clause into $A,U$ give the pair and +\eqref{iso:retraction-closure}. + +Fix any bounded tuple $d$ in $U'$. The tuple $(b,d)$ is still bounded +at $\nu$. The maps in~\eqref{iso:retraction-closure} place all its +images in $A'$. For a test $Z\subseteq A$, all but at most +$|Z|+\aleph_0$ of the maps fix $Z$ pointwise, and hence for all those +indices +\[ + D(r_j(b,d),Z)=D(b,d,Z). +\] +At a fixed testing arity below $\nu$, choose a bounded subsequence of +indices longer than the exception bound. Lemma~\ref{calc:retraction-descent} +gives +\[ + \tp(b,d/A)=\tp(b,d/A')^{\mathrm{fr}}_A. +\] +The type on the left is isolated, by the length-$\nu$ assertion at +level $\lambda$. Its uniform fragment bound for the actual arity +$|b,d|<\nu$ is below $\nu$. Downward transfer gives isolation of +$\tp(b,d/A')$ relative to $b$. + +This argument applies separately to every bounded $d\subseteq U'$; +it makes no assumption that $d$ lies in one stage of the countable +construction. Thus $U'$ is all-dominated over $A'+b$. The retained +anchors and bounded diagram homogeneity identify its marginal with +the original $p$ by an isomorphism $A'\simeq X$. Transport gives the +required object over $X$. Its embedding into an actual ambient is +Proposition~\ref{iso:prime}. +\end{proof} + +\subsection{Independent lists and coherent witnesses} + +Finite marks already have all-dominated extensions. To obtain the same +conclusion for a bounded independent list of finite marks, we must first +show that its finite product data determine one whole type and that this +type persists along the actual coupled chains used above. + +\begin{definition}\label{iso:independent-definition} +A list $(b_i)_{i<\delta}$ of finite tuples is +\emph{product-independent over $A$} if each finite sublist has the +product of its marginal types over $A$. +A \emph{witnessing chain} is a continuous chain $(M_i)_{i\leq\delta}$ +of objects, with $M_0=A$, such that $b_i$ lies in $M_{i+1}$ and +\[ + \tp(b_i/M_i)=\tp(b_i/A)^{\mathrm{fr}}_{M_i}. +\] +It is an \emph{all-dominated-increment chain} if, in addition, +$M_{i+1}$ is all-dominated over $M_i+b_i$. In uses at level $\mu$, +the chain has total size $\mu$, in particular $|\delta|\leq\mu$. +\end{definition} + +\begin{lemma}\label{iso:finite-domination} +Suppose $(M_i)_{i\leq\delta}$ is an all-dominated-increment chain over +$A$ for finite tuples $(b_i)_{i<\delta}$. If a finite tuple $x$ in an +actual coupling has product over $A$ with every finite selection of +the accumulated $b_i$, then $\tp(x/M_\delta)$ is free from $A$. +\end{lemma} + +\begin{proof} +Induct on the stage. At a successor $i+1$, the finite tuple $(x,b_i)$ +has product with every finite selection of earlier marks, by the +finite product rules. The induction hypothesis applied to this tuple +says that its joint type over $M_i$ is free from $A$. Products lift +freely by Proposition~\ref{calc:products}; hence $x$ and $b_i$ still have +product over $M_i$. Proposition~\ref{iso:prime}, applied to the +all-dominated increment $M_{i+1}$, makes $x/M_{i+1}$ free from $M_i$. +Transitivity makes it free from $A$. + +At a limit $j$, local character for the finite tuple $x$ gives a +proper stage $i\chi$. Cantor's Theorem gives +$\epsilon\le\chi$, and minimality gives $2^{|i|}\le\chi$ +for every $i<\epsilon$. Consequently +\[ + \left|{}^{<\epsilon}2\right|\le\chi, + \qquad 2^\epsilon\le 2^\chi<\mu, +\] +where $\mu$ is the common level. Every accumulator and branch +world below will have size $\mu$, and the total number of +amalgamations remains below $\mu$. + +We construct branch worlds $W_s\in\mathcal C_m$ for +$s\in{}^{\le\epsilon}2$, carrying tuples $f_s$, with +coherent full arrows along each branch. Their proper +boundaries have compatible arrows +\[ + q_s:\partial W_s\longrightarrow\partial A +\] +into an increasing common accumulator $A\in\mathcal C_m$. +The accumulator may be enlarged at each operation; existing +boundary arrows are always composed with its enlargement +arrow. The invariant requires coherence of these boundary +arrows with the branch arrows. It does not require full maps +of all the branch worlds into the accumulator to be retained. + +Start with $W_\varnothing=X^-$, its tuple $f$, and the +identity boundary map into $A=X^-$. At a node $s$, diagram +preservation along the branch shows that +$D(f_s)=D(f)$. Bounded homogeneity supplies an isomorphism +\[ + k_s:X^-\overset{\cong}{\longrightarrow}W_s, + \qquad k_s(f)=f_s. +\] +There is no coherence requirement on the choices of $k_s$. +Choose the minus child to be a copy of $Y^-$ attached over +$W_s$ by $k_s^{-1}$, and the plus child to be a copy of +$Y^+$ attached by $e k_s^{-1}$. These choices give the full +branch arrows. In each child, $f$ denotes the image of $f_s$; +the corresponding upper boundary lists have diagrams +$\delta_0$ and $\delta_1$ with it. Transporting $h$ gives +a boundary arrow +\[ + h_s:\partial W_{s0}\longrightarrow\partial W_{s1} +\] +over $\partial W_s$, matching the two copies of $v$. + +Apply the already proved completion theorem in dimension $m$ +to $q_s$. After enlarging the accumulator, this gives a full +map $W_s\to A_0$ extending $q_s$. Ordinary amalgamation in +$\mathcal C_m$ then puts the plus child in an enlargement +$A_1$ over this full map. Let $G_1:W_{s1}\to A_1$ be the +resulting arrow. Retain its boundary restriction as $q_{s1}$, +and define +\[ + q_{s0}=q_{s1}h_s. +\] +Both restrict to the preceding $q_s$, after enlargement, and +the two boundary copies of $v$ have one common image $v_s$ +in $A_1$. The temporary full maps to $A_0$ and $A_1$ are +no longer part of the invariant. In particular the minus +child has not been required to send its copy of $f$ to the +temporary image of $f_s$ in the accumulator. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[>=Latex, every node/.style={font=\small}, + world/.style={draw,rounded corners,inner sep=5pt}, + boundary/.style={draw,inner sep=4pt}] + \node[world] (w) at (0,2.4) {$W_s,f_s$}; + \node[world] (w0) at (-2,1.1) {$W_{s0},f_{s0}$}; + \node[world] (w1) at (2,1.1) {$W_{s1},f_{s1}$}; + \node[boundary] (b0) at (-2,-.1) {$\partial W_{s0}$}; + \node[boundary] (b1) at (2,-.1) {$\partial W_{s1}$}; + \node[world] (a) at (0,-1.7) {$A_1\ni v_s$}; + \draw[->] (w) -- (w0); + \draw[->] (w) -- (w1); + \draw (w0) -- node[left] {boundary} (b0); + \draw (w1) -- node[right] {boundary} (b1); + \draw[->] (b0) -- node[above] {$h_s$} (b1); + \draw[->,dashed] (b0) -- node[left] {$q_{s0}$} (a); + \draw[->,dashed] (b1) -- node[right] {$q_{s1}$} (a); + \draw[->,dotted,rounded corners=6pt] (w1.east) -- (4.2,1.1) + -- node[right] {$G_1$} (4.2,-1.7) -- (a.east); +\end{tikzpicture} +\caption{A return-tree split. Solid arrows between worlds are +retained along their separate branches. Dashed boundary maps +persist in the common accumulator and agree on the test $v_s$. +Each dashed arrow denotes a compatible family of maps into the +proper projections of $A_1$. +The dotted full map is used to choose the plus boundary map +and need not be retained at later splits. Full maps of all +leaf worlds are installed only after the splitting tests have +been fixed.} +\label{iter:fig-return-tree} +\end{figure} + +Process the nodes of each level successively. At a limit +height take the union of the worlds along each branch and +the union of the accumulators constructed so far. Their +sizes are still $\mu$. The boundary arrows into the union +accumulator exist by the chain and common-upper-bound clauses +in every proper projection. Full branch arrows and the +carried tuples also pass to the unions. This establishes +the invariant through height $\epsilon$. + +At that height, apply lower-dimensional completion to each +leaf boundary map, successively enlarging the accumulator. +This supplies a full map of each $W_\eta$, for +$\eta\in{}^\epsilon2$, extending its retained boundary +map. There are fewer than $\mu$ leaves, so the final union +is still at level $\mu$. If two leaves first differ after +a node $s$, their images of $f$ have diagrams $\delta_0$ +and $\delta_1$ over the common tuple $v_s$. Indeed the +corresponding copy of $v$ was in that child's boundary, +remained there along that branch, and its boundary image was never +changed except by composition with an accumulator arrow. +Thus the final images of $f$ occupy more than $\chi$ +orbits over the entries of $(v_s:s\in{}^{<\epsilon}2)$. +This parameter list has size at most $\chi$. The orbit +bound of Lemma~\ref{calc:parameter-orbits} gives the +contradiction. The claimed equality of types of $v$ follows. + +To pass from this type equality to the full pair diagrams, retain +an arbitrary bounded tuple $a$ from the lower face $X^-$. +The marginal of $b^+$ over +$X^+$ is the prescribed free lift from $e(X^-)$. +Relative isolation of the source type of $(b^-,v)$ and +Lemma~\ref{iso:free-isolation} therefore identify the actual type +of $(b^+,h(v))$ over $X^+$ with the free lift of that source type. + +Choose sufficiently long bounded approximation lists in $X^-$ +computing the source type. Their images under $e$ compute its +free lift over $X^+$ by Theorem~\ref{calc:free-extension}. +Base homogeneity gives an isomorphism $X^-\cong X^+$ matching +$a$ and these lists with their images under $e$, since $e$ +preserves their joint bounded diagram. The transport assertion +for the approximation lists says that this isomorphism carries +the whole source type to the type of $(b^+,h(v))$. +Lemma~\ref{mark:pair-isomorphism} now extends it to an isomorphism +of the marked pairs matching $(a,v)$ with $(e(a),h(v))$. +Every bounded boundary list is of this form. Thus its full +$\mathcal C_n$-diagram is preserved, including cross-face equalities. + +We finish the induction by deriving completion from this +diagram equality. Present the source $J$ on a well-order. +Its boundary union is a natural collection of the +designated term sorts in that presentation. Choose nested +bounded lists $d_i$ exhausting $J$, including increasing +strong submodel enumerations in every vertex. We construct +extensions $K_i$ of $K$ and compatible assignments of +$d_i$ into $K_i$ satisfying +\[ + D_J(d_i,u)=D_{K_i}(d_i',h(u)) + \quad\text{for every bounded boundary list }u. + \tag{C} +\] +Here and subsequently $h$ is composed with the current +extension arrow. At the start the assertion for the empty +list is exactly the boundary diagram equality just proved. + +For the next list prescribe the source joint diagrams on +the parameter set consisting of the boundary images and +the previous assignment. Each bounded local request, +including the whole previous assigned list, is possible +by saturation: the parameter diagram agrees by $(\mathrm C)$. +This also checks repetitions and equalities between old +entries and boundary entries, so the requests define one +coherent prescription. Apply compact placement in a +well-ordered chart of $K_i$, keeping these previous +entries fixed. Work at one bounded cardinal large enough +for the profile count, the new tuple, and the testing +bound for the source boundary layout with the previous +tuple included. The resulting placement lies in a +level-$\mu$ extension $K_{i+1}$. + +To check $(\mathrm C)$ beyond that initial testing cardinal, +pull the candidate diagrams back along $h$ to the natural +boundary layout of $J$. Their local possibility there +follows from the boundary diagram equality and saturation. +Lemma~\ref{calc:compact-placement} compactifies these +requests over this source layout, and +Proposition~\ref{calc:testing} says that agreement at the chosen +testing bound determines all bounded comparisons. +The actual images of the boundary inside $K_i$ have not +been assumed to form a natural layout. All uses of that +hypothesis are on the source layout after pullback. + +Continue through $\omega$ steps and take the union of the +$K_i$. The working-package criterion, with the included +strong submodel enumerations, makes the union assignment +an arrow $J\to K^*$. Whenever a boundary entry is first +included in $d_i$, its equality test against that boundary +entry forces its assigned image to be $h$ of that entry. +The union arrow therefore extends the entire boundary +map. The map from $K$ is the chain inclusion. This proves +completion and closes the induction. +\end{proof} + +\subsection{A bounded support for boundary tests} + +We have extended boundary maps into common extensions. Exact lifting +must instead land onto the given target. The next lemma supplies bounded +data that control a tuple's boundary comparisons at a fixed bounded test +arity. Together with uniform testing, this permits back-and-forth to +remain inside the two given systems. + +\begin{lemma}[Fixed parameters for a boundary test function] +\label{iter:boundary-fixed-parameters} +Fix $n<\omega$. At further sufficiently closed levels, +let $J\in\mathcal C_n$, let $z$ be a bounded tuple of +$J$, and let $\sigma$ be a bounded infinite test arity. +There is a bounded list $a$ in $\partial J$ such that +every boundary automorphism $\alpha$ fixing $a$ pointwise +satisfies +\[ + D_J(z,u)=D_J(z,\alpha(u)) + \quad\text{for every boundary tuple }u + \text{ of length at most }\sigma. + \tag{F} +\] +A boundary automorphism means a compatible family of +isomorphisms of all proper projections; a full +automorphism of $J$ is not assumed. +\end{lemma} + +\begin{proof} +The assertion is immediate for the empty boundary. +Suppose it fails otherwise. Choose a bounded infinite +$\chi$ with $\chi^{|z|+\aleph_0}=\chi$, bounding +$\sigma$ and the orbit-counting and label costs. Let +$\epsilon$ be the least cardinal with $2^\epsilon>\chi$. +Choose a smaller eligible level $\nu$ such that +\[ + |z|+\sigma+2^\chi<\nu<\mu, +\] +where $\mu$ is the level of $J$, with all required testing bounds +below $\nu$ and with +Theorem~\ref{iter:boundary-completion} available at $\nu$. +The further closure requirement on the level of $J$ +ensures this choice. It is a finite refinement for this +fixed $n$. + +Fix a well-ordered presentation $J=F_n(I)$. Construct +continuous subpresentations $J_i$ of size $\nu$ for +$i\le\epsilon$, beginning with $z$ in $J_0$. +At step $i$, the boundary of $J_i$ is a bounded parameter +set in $J$. By the assumed failure, a boundary +automorphism $\alpha_i$ of $J$ fixes it pointwise and +changes the value of the function at some tuple $u_i$ +of length at most $\sigma$. Thus +\[ + D_J(z,u_i)\ne D_J(z,\alpha_i(u_i)). + \tag{S} +\] +We choose $J_{i+1}$ containing both tests so that +$\alpha_i$ restricts to an automorphism of its boundary. + +Here is the support closure for this choice. Begin with +the positions for $J_i$, $z$, $u_i$, and +$\alpha_i(u_i)$, padding to size $\nu$. For every element +of every proper projected subpresentation currently +generated, add positions for its image and its inverse +image under the corresponding face map of $\alpha_i$. +There are finitely many faces and $\nu$ elements in their +union, and each representation uses finitely many +positions. Repeat this operation countably many times. +The resulting position set still has size $\nu$. +Every element generated at its union appeared at a +finite stage, so both its image and inverse image are +generated in the union. Hence all restricted face maps +are onto and remain compatible with the specified +inclusions. They give the asserted boundary +automorphism of $J_{i+1}$. At a limit use the union of +the position sets. Mixed diagram preservation transfers +$(\mathrm S)$ to $J_{i+1}$, since $|z|+\sigma<\nu$. + +Now build a binary tree of boundary embeddings +\[ + q_s:\partial J_i\longrightarrow\partial A, + \qquad s\in{}^i2, +\] +where $A$ is an increasing accumulator of size $\nu$. +Start with $A=J_0$ and the identity map on $\partial J_0$. +At a successor node first apply completion to $q_s$ +on $J_i$ and then amalgamate $J_{i+1}$ over that full +copy of $J_i$. This yields an arrow +$G_s:J_{i+1}\to A'$ whose boundary extends $q_s$. +Define the two next boundary embeddings by +\[ + q_{s0}=\partial G_s, + \qquad + q_{s1}=(\partial G_s)\alpha_i^{-1}. +\] +They restrict to the same old $q_s$, because +$\alpha_i$ fixes $\partial J_i$ pointwise. The common +parameter test $G_s(u_i)$ is the image of $u_i$ on +branch $s0$ and of $\alpha_i(u_i)$ on branch $s1$. +Only these boundary embeddings are retained for later +steps; the full map $G_s$ is temporary. + +At limits the continuity of the $J_i$ and the +common-upper-bound clauses give the boundary maps into +the union accumulator. At height $\epsilon$, complete +the boundary maps to full maps of $J_\epsilon$, one +leaf at a time. The full maps at the leaves carry $z$. +If two leaves first separate at $s\in{}^i2$, their +images of $z$ have the two distinct diagrams from +$(\mathrm S)$ over the one retained test $G_s(u_i)$. +This follows from preservation by their final full maps +and from coherence of their boundary maps with the +chosen maps at the split. + +There are at most $\chi$ nonterminal nodes, hence at +most $\chi$ parameters in all the splitting tests. +There are $2^\epsilon>\chi$ distinguishable leaf +images of $z$. All leaf completions and all unions +remain at level $\nu$, since $2^\chi<\nu$. +Lemma~\ref{calc:parameter-orbits} again contradicts +this number of orbits over the tests. Thus the bounded +fixing list $a$ exists. +\end{proof} + +\begin{theorem}[Exact boundary lifting] +\label{iter:exact-boundary} +For each finite $n$, at further sufficiently closed +levels every boundary isomorphism +$h:\partial J\overset{\cong}{\longrightarrow}\partial K$ +between $\mathcal C_n$-objects at the same level +extends to an isomorphism $J\cong K$ onto the given +target. All the prescribed marks and bounded named +data are preserved. +\end{theorem} + +\begin{proof} +We build a back-and-forth sequence of matches +$a\mapsto a'$ of bounded tuples, maintaining +\[ + D_J(a,u)=D_K(a',h(u)) + \quad\text{for every bounded boundary tuple }u. + \tag{BF} +\] +For the empty tuple this is +Lemma~\ref{iter:boundary-diagrams}. Suppose that a +match satisfying $(\mathrm{BF})$ has been chosen, and let $z$ +be any bounded enlargement of $a$ in $J$. + +Take a bounded infinite $\sigma$ at least the uniform +testing bound for a tuple of length $|z|$ over the +natural boundary layout of a $\mathcal C_n$-object. +Choose fixing data $c$ for $z,\sigma$ in $\partial J$ +by Lemma~\ref{iter:boundary-fixed-parameters}. +The maintained equality with the particular boundary +list $c$ gives +\[ + D_J(a,c)=D_K(a',h(c)). +\] +Working-package homogeneity supplies a full +isomorphism $H:J\cong K$ carrying $a$ to $a'$ and +$c$ to $h(c)$. Put +$\alpha=(\partial H)^{-1}h$. This is a boundary +automorphism of $J$ fixing $c$ pointwise. Therefore, +for every boundary tuple $u$ of length at most +$\sigma$, +\[ + \begin{split} + D_K(H(z),h(u)) + &=D_J(z,\alpha(u))\\ + &=D_J(z,u). + \end{split} + \tag{BT} +\] + +We justify explicitly the passage from these tests +to all bounded boundary tests. By +Theorem~\ref{iter:boundary-completion}, embed $J$ +into a same-level extension $K^*$ of $K$ by an arrow +$j$ extending $h$. Both $j(z)$ and $H(z)$ are now +tuples in $K^*$, and $(\mathrm{BT})$ says that their diagrams +agree over boundary tests from $K$ of size at most +$\sigma$. The boundary of $K$, in a well-ordered +chart of $K$, is a natural collection of term sorts. +The types of these two tuples over $K$ have order +placements by the one-dimensional calculus. +Proposition~\ref{calc:testing} restricted to this natural +boundary layout therefore gives equality over every +bounded boundary test. Preservation by $j$ then +shows +\[ + D_K(H(z),h(u))=D_J(z,u) +\] +for all bounded $u$. Thus $z\mapsto H(z)$ extends +the previous match and satisfies $(\mathrm{BF})$. + +Apply this extension step alternately to $J,K$ with +$h$, and to $K,J$ with $h^{-1}$. Their level has +countable cofinality, so countably many bounded +enlargements can exhaust both systems. In each +vertex choose increasing smaller strong submodels +whose enumerations are included in these tuples. +The union assignment is onto in every component +and is an arrow by the working-package map +criterion; the reverse construction gives its +inverse arrow. If an entry lies on the boundary, +its equality test with that boundary entry in +$(\mathrm{BF})$ forces its image to be exactly the image +under $h$. The resulting isomorphism therefore +extends all of $h$, as required. +\end{proof} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/10-geometry.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/10-geometry.tex new file mode 100644 index 0000000000000000000000000000000000000000..b1fb37fd0e7dbba8653182f30b7582957184d673 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/10-geometry.tex @@ -0,0 +1,607 @@ +\section{A minimal type and its geometry} +\label{geom:section-geometry} + +We continue with the fixed AEC $\K$ that is categorical in unboundedly +many cardinals. Throughout this Section, a level is eligible for the +single-model working scheme and for the refinements of the calculus +that are explicitly invoked. The first level chosen below is denoted +by $\lambda_0$. The level used for all finite marked constructions will +be a later cardinal $\lambda$; these two choices serve different purposes. + +\begin{lemma}[Preservation of nonalgebraicity] +\label{geom:nonalgebraic} +Let $p$ be a nonalgebraic singleton type over an object $A$ of a working +scheme. Its free lift over an extension $B$ is nonalgebraic. The same +assertion holds for a mixed arrow whenever the input arity and the +testing bounds lie below the smaller level. +\end{lemma} + +\begin{proof} +Suppose that the free lift were realized by $c\in B$. Include equality +with $c$ among its tests. To evaluate that test, the free-extension +construction moves $c$ into $A$, over enough of the chosen +approximations for $p$. Its value is then the value of equality with an +element of $A$. Every such equality has value false for $p$, whereas +equality with its alleged realization $c$ has value true. This +contradicts the evaluation rule in Theorem~\ref{calc:free-extension}. +The mixed-level construction uses the same rule with the bounds taken +below the input level. +\end{proof} + +\begin{proposition}[A minimal type] +\label{geom:minimal-type} +There is a sufficiently closed single-model level $\lambda_0$, a model +$M_0\in\K$ of size $\lambda_0$, and a nonalgebraic singleton type +$p_0$ over $M_0$ such that, over every $M_0\leq_\K A\in\K$ of size +$\lambda_0$, there is exactly one nonalgebraic extension of $p_0$. +This extension is its free lift. +\end{proposition} + +\begin{proof} +Choose $\lambda_0$ with the single-model working package, the +stationary calculus, the all-dominated hull construction, and the +singleton type bound of Theorem~\ref{lev:amalgamation}(3). +Thus there are at most $\lambda_0$ singleton +types over a model of that size. We may require at the outset strict +closure under all the fixed bounds used below for finite tuples and +their bounded tests. + +There is a nonalgebraic singleton type at this level. For example, +extend an order presentation by a fresh skeleton point. That point is +outside the old model: if it were a labeled term on finitely many old +indices, adding a second distinct skeleton point with the same order +pattern over those indices would make both equal that term, contrary +to the distinctness of the skeleton. + +Suppose that no nonalgebraic singleton type at this level is minimal in +the stated sense. A nonalgebraic type always has a nonalgebraic free +extension by Lemma~\ref{geom:nonalgebraic}. Consequently every such +type has two distinct nonalgebraic extensions over some model of size +$\lambda_0$. + +Let $\epsilon$ be the least infinite cardinal with +$2^\epsilon>\lambda_0$. Then $\epsilon\leq\lambda_0$ and +$2^{|i|}\leq\lambda_0$ for $i<\epsilon$. Construct a continuous chain +$(A_i:i\leq\epsilon)$ of models of size $\lambda_0$ and types +$p_\eta$ over $A_i$, for $\eta\in{}^i2$, as follows. At a successor +step, split every current $p_\eta$ over a model extending $A_i$. +There are at most $\lambda_0$ such models. Successive amalgamation +over $A_i$, followed by unions, puts all of them into one next base +$A_{i+1}$ of size $\lambda_0$. Freely extend each of the two split +types to that base. Their restrictions to the splitting model are +different, so they remain different, and +Lemma~\ref{geom:nonalgebraic} keeps them nonalgebraic. + +Along each branch retain a witness model and the same distinguished +realization. The required successor map exists by amalgamation with +identification of realizations having the same type over the preceding +base. At a limit take the unions of the common bases and, separately +on each branch, of its coherent witness models. The realization is +outside every base stage, so it is outside the union base. All these +unions have size $\lambda_0$ because the construction has length at +most $\lambda_0$. At the final stage distinct branches have different +types over $A_\epsilon$, witnessed at their first splitting stage. +This gives more than $\lambda_0$ singleton types over $A_\epsilon$, +contrary to the type bound. A minimal $p_0$ therefore exists. Its +unique nonalgebraic extensions are free by existence of free lifts. +\end{proof} + +The passage from a minimal type to closure and dimension follows the +geometric pattern of first-order categoricity developed through strongly +minimal sets; see Baldwin and Lachlan \cite[Sections~1--2]{BaldwinLachlan1971}. +Here the required containment, exchange, and change-of-base properties +are established directly for the AEC and its locally constructed types. + +\subsection{Finite configurations over the chosen model} + +\begin{definition} +\label{geom:points} +For $M_0\leq_\K M$ with $\|M\|\geq\lambda_0$, let $P(M)$ be the +set of realizations of $p_0$ in $M$. More explicitly, $a\in P(M)$ +means that in a strong section $N\leq_\K M$ of size $\lambda_0$ +containing $M_0\cup\{a\}$, the type $\tp(a/M_0)$ is $p_0$. +A finite ordered list of distinct points in $P(M)$ is independent if +its type over $M_0$, computed in such a section, is the product of the +corresponding copies of $p_0$. +\end{definition} + +These definitions do not depend on the section. Two sections lie in +a third section of size $\lambda_0$, by the L\"owenheim--Skolem axiom +and coherence. The embeddings into that section preserve their types +over $M_0$. Products are permutation invariant, so independence is a +property of the underlying finite set. Strong embeddings over $M_0$ +preserve and reflect both membership in $P$ and finite independence. + +We give a precise meaning to the containment test used in constructing +closure. Fix a finite configuration $F\cup\{x\}\subseteq P(M)$ and +a size-$\lambda_0$ section $N$ containing it and $M_0$. Say that $F$ +\emph{forces} $x$ if, whenever $f:N\to W$ is a strong embedding fixing +$M_0$, with $W$ of size $\lambda_0$, and +$M_0\leq_\K D\leq_\K W$ has size $\lambda_0$ and contains $f[F]$, +one has $f(x)\in D$. Thus the test permits a common extension of the +finite configuration and the proposed model; it does not presume that +intersections of strong models are strong models. + +The test is invariant under the type of the entire finite +configuration over $M_0$. Indeed, an identification of two copies of +that type is witnessed in an amalgam. Any further configuration +witnessing failure of forced containment can be amalgamated with that +identification over the first witness model. Its avoiding submodel +and the omitted point remain disjoint at that point because the maps +are injective. This transports the failure to the second copy. The +reverse transport is identical with the copies interchanged. + +\begin{lemma}[The containment test] +\label{geom:containment-test} +For a finite independent set $S\subseteq P(M)$ and $x\in P(M)$, +$S$ fails to force $x$ if and only if $S\cup\{x\}$ is independent +with $x\notin S$. +\end{lemma} + +\begin{proof} +First suppose that a coupling contains a model $D$ over $M_0$ and $S$ +but omits $x$. The type of $x$ over $D$ is a nonalgebraic extension of +$p_0$. By Proposition~\ref{geom:minimal-type} it is the free lift of +$p_0$. Restricting its parameters to $S$ shows, by +Proposition~\ref{calc:products}, that $x$ has product type with $S$ over +$M_0$. + +Conversely suppose that $S,x$ has product type. Use +Theorem~\ref{iso:all-lengths} to obtain a model all-dominated over +$M_0+S$, and amalgamate it with the actual finite configuration, +identifying the copy of $S$. The domination conclusion of +Proposition~\ref{iso:prime} makes $x$ free over this model from $M_0$. +It is outside the model by Lemma~\ref{geom:nonalgebraic}. The same +amalgamation can retain any further specified finite entries of the +original configuration. This is a coupling witnessing that $S$ does +not force $x$. +\end{proof} + +\begin{theorem}[The geometry of the minimal type] +\label{geom:pregeometry} +On $P(M)$, for every $M_0\leq_\K M$ of size at least $\lambda_0$, +the preceding finite independence is the independence relation of a +finitary pregeometry. Its closure, denoted by $\cl_M$, has the +following properties. +\begin{enumerate} +\item For finite $F\subseteq P(M)$ and any maximal independent + $S\subseteq F$, the set $\cl_M(F)$ consists exactly of those + points that do not independently extend $S$. Equivalently, + $\cl_M(F)$ is the set of points forced by $F$. +\item Strong embeddings over $M_0$ preserve and reflect finite + closure. If $M_0\leq_\K A\leq_\K M$, then + $P(A)$ is relatively closed in $P(M)$. +\item For finite $F$, $|\cl_M(F)|\leq\lambda_0$. For every infinite + $F\subseteq P(M)$, + \[ + |\cl_M(F)|\leq |F|+\lambda_0. + \] +\end{enumerate} +In particular bases exist, all bases have the same cardinality, and +each point of the span of a basis has finite basis support. +\end{theorem} + +\begin{proof} +Choose a maximal independent subset $S$ of a finite $F$. Every +$y\in F\setminus S$ fails to independently extend $S$, so +Lemma~\ref{geom:containment-test} says that every model containing $S$ +in the actual configuration must contain all of $F$. A point that +fails to independently extend $S$ is therefore forced by $F$. +If $x$ independently extends $S$, the avoiding dominated model from +Lemma~\ref{geom:containment-test} can be adjoined while retaining +$F,x$. It contains $F$ by the preceding observation and omits $x$. +Thus $F$ does not force $x$. This proves the first assertion and its +independence of the choice of $S$. + +Define closure on arbitrary subsets by taking the union of closures +of finite subsets. Extensivity and monotonicity hold directly from +the containment test. To verify idempotence on finite configurations, +suppose that finitely many $y_j$ are in $\cl_M(F)$ and that $x$ +independently extends a maximal independent $S\subseteq F$. The +avoiding model over $S$ can be coupled with the whole finite +configuration $F,(y_j),x$. It contains $F$ and every $y_j$, because +all these points are forced by $S$, and it omits $x$. Hence the +$y_j$ cannot force $x$. Finite character now yields idempotence for +all subsets. + +For exchange, it suffices by finite character to take finite $F$ and +points $x,y$ with +\[ +y\in\cl_M(F\cup\{x\})\setminus\cl_M(F). +\] +If $x\in\cl_M(F)$, idempotence would imply $y\in\cl_M(F)$, so +$x\notin\cl_M(F)$. Let $S$ be a maximal independent subset of $F$. +Then $S,x$ and $S,y$ are independent. Since every member of $F$ is +already forced by $S$, the set $S\cup\{x\}$ is maximal independent +in $F\cup\{x\}$. The displayed membership says that $S,x,y$ is +dependent. Permutation invariance of products makes $S,y,x$ +dependent as well. Applying the first assertion with $S\cup\{y\}$ +gives $x\in\cl_M(F\cup\{y\})$. This is exchange. + +Preservation and reflection follow from the invariant finite-type +definition and the containment test. For relative closedness, take +finite $F\subseteq P(A)$ and $x\in\cl_M(F)$. Choose +$D\leq_\K A$ of size $\lambda_0$ containing $M_0\cup F$, and then +choose a section of $M$ containing $D\cup\{x\}$. Forced containment +gives $x\in D\subseteq A$. The finite closure of $F$ lies in this +same $D$, proving the finite cardinal bound. There are $|F|$ finite +subsets of an infinite $F$, so finite character proves the displayed +bound. + +For completeness, a maximal independent subset of a set can be +constructed by well-ordering it and retaining a point precisely when +it is outside the span of the points retained earlier. Finite +character makes the resulting set independent, and maximality makes +it spanning. If an independent set is spanned by a finite set of +$n$ points, it has at most $n$ points: successively insert its points +into a spanning list, using exchange to remove one of the original +points whenever a new independent point is inserted. After $n$ +insertions no further independent point is in the span. This also +proves equality of finite basis sizes. + +For infinite bases $H,J$ of the same closed set, choose for each +$j\in J$ a finite support $S_j\subseteq H$. Then +$\cl(J)\subseteq\cl(\bigcup_{j\in J}S_j)$. Independence of $H$ +implies $H\subseteq\bigcup_{j\in J}S_j$, because a member of $H$ +cannot belong to the span of the other members. Hence +$|H|\leq|J|$; interchanging the bases gives the reverse inequality. +The finite argument excludes the possibility of one basis finite +and the other infinite. Finally, for a closed $C$, the operator +$U\mapsto\cl(C\cup U)\setminus C$ on $P(M)\setminus C$ satisfies +the same closure and exchange axioms. Finite character follows by +discarding all but the finitely many entries of $U$ in a dependence +witness. Applying the preceding constructions to this operator +gives relative bases and their finite supports. +\end{proof} + +\subsection{Pointed levels and change of base} + +Choose an enumeration $\bar m_0$ of $M_0$ and place it as a strong +model in a sufficiently large well-ordered presentation of $\K$. +Such a placement exists by the order construction and categoricity at +$\lambda_0$. Fix the stabilized diagram $\Delta$ of this entire +enumeration. The initial supporting positions of this placement will +be held fixed in later order presentations. + +\begin{proposition}[Pointed working levels] +\label{geom:pointed-levels} +At all sufficiently closed levels $\mu$ above the bounds for +$|\bar m_0|=\lambda_0$, models with a distinguished strong copy of +$M_0$ whose enumeration has diagram $\Delta$ form a categorical +working scheme. Arrows preserve the indicated copy pointwise. +For a pointed model $B$ at such a level, $p_0$ has a unique +nonalgebraic extension $p^B$ over $B$; it is its free lift. This family +lifts freely along pointed arrows, including the permitted mixed +arrows. Naming further bounded data of a fixed diagram within the +base does not change type computation over the whole base. +\end{proposition} + +\begin{proof} +The enumeration is a bounded tuple at $\mu$. Bounded-diagram +homogeneity in Proposition~\ref{lev:closure-categoricity} carries it +between any two instances of $\Delta$ by an isomorphism of the entire +models. The pointed version of the working package follows by +reserving its supporting positions and including the enumeration in +every relevant diagram, as in +Proposition~\ref{lev:named-package}. + +Existence of $p^B$ is mixed free extension of $p_0$, and +Lemma~\ref{geom:nonalgebraic} gives nonalgebraicity. To prove +uniqueness, take two nonalgebraic extensions over $B$. A distinguishing +test for singleton types may be chosen with a fixed bound below +$\lambda_0$. For any such parameter set $Z\subseteq B$, choose +$M_0\leq_\K A\leq_\K B$ of size $\lambda_0$ containing $Z$. +Both types restrict to nonalgebraic extensions of $p_0$ over $A$ and +are equal there by Proposition~\ref{geom:minimal-type}. Their +diagrams on $Z$ are consequently equal. The bounded testing theorem +gives equality over $B$. Uniqueness and transitivity of free +extension prove the assertion for pointed arrows. + +All distinguished data here already belong to the parameter model. +An amalgam over that model fixes them, and an identification of types +over that model respects them. Thus adding them as constants does +not change this type computation. +\end{proof} + +\begin{proposition}[Change of base for the geometry] +\label{geom:base-change} +Let $B$ be a pointed model at a sufficiently closed level much larger +than $\lambda_0$. A finite list of points in an extension of $B$ is +independent over the closed set $P(B)$ in the geometry of +Theorem~\ref{geom:pregeometry} if and only if its type over $B$ is the +product of the corresponding copies of $p^B$. +\end{proposition} + +\begin{proof} +A product can be witnessed by successive free realizations in models +containing all preceding entries. Each realization is outside that +model by Lemma~\ref{geom:nonalgebraic}. Its point set is relatively +closed, so the realization is outside the geometric span of $P(B)$ +and the earlier entries. This proves one implication. + +For the converse let $x$ be a finite list independent over $P(B)$. +Fix a parameter test $Z\subseteq B$ large enough to be used in the +uniform comparison of the type of $x$ over $B$, but with +$|Z|<\lambda_0$. Choose a regular infinite $\theta<\lambda_0$ +with $|Z|<\theta$. We may and do choose $\lambda_0$ closed enough +that the comparisons and local-character bounds used here fit below +it. + +Inside $B$ choose lists $x_i$, $i<\theta$, so that every +$D^\K(x_i,Z)$ equals $D^\K(x,Z)$ and all the $x_i$ together are +independent over $M_0$. To make the successor choice, include in the +bounded match the entire enumeration of $M_0$, $Z$, and all earlier +copies. The original $x$, being independent over $P(B)$, is independent +from those earlier copies. Saturation realizes its joint diagram +inside $B$. Diagram equality is witnessed by an isomorphism, so it +preserves every finite independence comparison over $M_0$. The total +parameters at every choice are bounded at the level of $B$. + +Take a strong section $N\leq_\K B$ of size $\lambda_0$ containing +$M_0$, $Z$, and all the copies. Their finite sublists have product +type over $M_0$. The independent-list and finite-domination +conclusions of Theorem~\ref{iso:independent-lists} give a continuous +witness chain $(A_i:i\leq\theta)$ inside $N$, starting at $M_0$, in +which $x_i$ is free over $A_i$ from $M_0$ and is included in +$A_{i+1}$. One may keep the chain inside $N$: at each step primeness +embeds the required all-dominated increment into $N$ over its already +placed base and the specified next list. At limits finite domination +keeps the next finite list free, and unions remain strong in $N$. + +Since $\cf(\theta)>|Z|$, local character applies to $\tp(Z/A_\theta)$. +It is free from some $A_i$. For a later $j<\theta$, its restriction +to $A_{j+1}$ is free over $A_j$, so $Z$ and $x_j$ have product over +$A_j$. The type of $x_j$ over $A_j$ is free from $M_0$. +The base-change identity in Proposition~\ref{calc:products} therefore +says that $x_j$ has product with $Z$ over $M_0$. In particular +$D^\K(x_j,Z)$ is determined by $p_0$ and $\tp(Z/M_0)$. + +A known product of the $p^B$ has this same joint diagram with $Z$: +products of free lifts are free lifts of products, and one may +restrict the resulting type to $M_0$ together with $Z$. Since +$D^\K(x,Z)=D^\K(x_j,Z)$, the two candidate types over $B$ agree on +every test of the uniform comparison size. Bounded testing makes +their full types equal. This proves the converse. +\end{proof} + +\subsection{Eliminating proper extensions with unchanged geometry} + +The obstruction considered here is related to Vaughtian pairs: proper +extensions of the same cardinality with no new realizations of a fixed +nonalgebraic type; compare \cite{GrossbergVanDieren2006}, where the type +is based on a limit model. Here finite marked systems and exact boundary +lifting eliminate the obstruction in the pointed setting. + +The next use of the marked construction requires a type prescription +that can be chosen from a fixed set before choosing the common level. +We record the uniformity explicitly. + +\begin{lemma}[Anchoring a singleton type] +\label{geom:anchored-type} +\label{geom:root-prescription} +There is a fixed cardinal $\beta$, depending on the single-model +template, its singleton comparison bound, and the fixed point, with +the following property. If $A$ is a sufficiently closed pointed +model and $r$ is a nonalgebraic singleton type over $A$, a list +$c\subseteq A$ of length at most $\beta$ can be named so that: +\begin{enumerate} +\item the joint stabilized diagram of $c,\bar m_0$, together with a + fixed majority prescription, defines a unique nonalgebraic + singleton type $r^B$ over every sufficiently closed model $B$ + having that named diagram; +\item the prescription at $A$ is $r$, and the family $r^B$ lifts + freely along arrows preserving the names, including across + eligible levels; +\item the prescription occurs at arbitrarily large eligible + single-model levels. +\end{enumerate} +The possible descriptors obtained this way form a set, with their +length bound independent of $\|A\|$. +\end{lemma} + +\begin{proof} +Fix a sufficiently large uniform testing arity $\tau$ for singleton +types, including the bound needed to apply compact placement to those +tests. The approximation construction for +Theorem~\ref{calc:free-extension} gives a list $c=(c_i:i<\beta')$ of +approximations in $A$ that computes $r$ by majority on all these +tests. Its length and all ancillary approximation data can be bounded +by one cardinal $\beta$, independently of the level. Choose the +length larger than the exception bound for the tests; increase +$\beta$ also to include the fixed point data. The prescription is +the value attained by all but the permitted number of entries in +$D^\K(c_i,z)$, for each test $z$ of arity at most $\tau$. + +Freely lift the actual $r$ to arbitrarily large eligible models +containing $A$. Query matching over the fixed approximation list +shows that these lifts compute exactly the same majority +prescription. They are nonalgebraic by +Lemma~\ref{geom:nonalgebraic}. For another sufficiently closed +pointed $B$ with the same named diagram, embed $B$ into such a larger +instance, respecting the names. This is possible by bounded-diagram +homogeneity and mixed placement. Restrict the lifted type to that +copy of $B$. It is nonalgebraic, since its realization is outside the +larger parameter model, and query matching makes its bounded answers +the prescribed ones. + +Any two types having these answers are equal by the singleton +comparison bound. Thus the construction is independent of the +larger instance and the chosen embedding. A free lift along an arrow +preserving the names has the same prescribed answers: the +approximation evaluation is transported by matching each query over +the named list inside the old base. Uniqueness therefore identifies +it with the prescribed receiving type. At $A$ the answers are those +of $r$ by the choice of $c$. + +The list lengths have a fixed set bound. Their stabilized diagrams, +the fixed testing arity, and the majority rule range over a set of +descriptors. Only descriptors realized at arbitrarily high eligible +single-model levels are needed; the preceding construction proves +that the descriptor extracted from an actual $r$ has this property. +The named diagram itself can be witnessed at one fixed eligible +level large enough for arity $\beta$, by saturation for diagrams from +larger levels. Proposition~\ref{lev:named-package} therefore applies +with a lower threshold depending on $\beta$ and the original working +scheme, rather than on $\|A\|$. In particular, extracting the names +from a model of size $\lambda$ does not require moving above +$\lambda$ in order to use the pointed working scheme at that level. +\end{proof} + +\begin{theorem}[No proper pair with unchanged points at one level] +\label{geom:no-growth} +There is a sufficiently large cardinal $\lambda>\lambda_0$ such that +there is no proper strong inclusion $A<_\K B$ of pointed models of +size $\lambda$ with $P(A)=P(B)$. +\end{theorem} + +\begin{proof} +Take the set of anchored descriptors from +Lemma~\ref{geom:anchored-type}. Choose $\lambda$ at a common level +for all finite marked constructions for these descriptors, using +Theorem~\ref{close:common-levels}. Require also all the relevant +single-model and pointed bounds, and $\lambda>\lambda_0+\beta$. +This is a choice uniform in the possible descriptor. In particular +the choice does not depend on a pair that might later be assumed to +exist. + +Suppose toward a contradiction that $A<_\K B$ is such a pair, and +choose $a\in B\setminus A$. Apply +Lemma~\ref{geom:anchored-type} to $r=\tp(a/A)$ and name its list +$c$. We work in the pointed scheme also naming this list, with its +fixed joint diagram. Write $\mathcal C_n$ for its $n$-fold marked +iteration with the root prescription $r$, as in +Theorem~\ref{iter:finite-systems}. + +Every one-step all-dominated increment of this scheme has no new +$P$-points. Indeed, pointed categoricity carries its base, including +the named list $c$, onto $A$. Transporting the witness $A<_\K B$ +back gives an extension with a realization of the required $r$ and +unchanged $P$. Primeness embeds the all-dominated increment into +that extension over its base and mark. Since strong maps preserve +and reflect $P$ and fix the base, the increment has exactly the base +points. + +It follows by induction that the $P$-points in the top of every +$\mathcal C_n$ object are precisely those in its root. For the +induction step, its lower and upper layers are $\mathcal C_{n-1}$ +objects and hence each has the points of its own root. Projection +onto the new axis is a valid one-step marked increment by +Theorem~\ref{iter:finite-systems}. The two roots therefore have the +same points, proving the assertion for the entire object. + +We next construct coherent finite systems. For each $n<\omega$ +choose an object $J_n\in\mathcal C_n$. For every +$s\subseteq\{1,\dots,n\}$ choose an isomorphism +\[ +e_{n,s}:J_{|s|}\longrightarrow P_sJ_n +\] +such that these maps commute with all further face projections, using +the increasing identifications of finite index sets. Here the full +face map is the identity. To make these choices, induct first on +$n$ and then on the size of a proper subset $s$. For a proper subset +$u$ of the axes of $J_{|s|}$, let $t$ be the corresponding subset of +$s$. The boundary map for $e_{n,s}$ on this face is +$e_{n,t}e_{|s|,u}^{-1}$, with the indicated index identifications. +The already chosen maps make these boundary prescriptions compatible. +Theorem~\ref{iter:exact-boundary} extends them to the desired +isomorphism. At dimension zero it is simply pointed categoricity. +This constructs every $e_{n,s}$ with the asserted commutation. + +For the finite subsets $s$ of any ordinal $I$, take a tagged copy of +the top vertex of $J_{|s|}$. If $s\subseteq t$, use the top component +of $e_{|t|,u}$ followed by the inclusion of that face into the top, +where $u$ records the positions of $s$ in the increasing list of $t$. +The preceding commutation gives a directed system of strong maps. +All its roots are copies of the same actual $J_0$, and no vertex has +more $P$-points than that root. + +The marks corresponding to different indices of $I$ are distinct in +this directed system. It suffices to inspect a two-axis object. Its +new mark lies in the root of its upper one-axis layer, whereas the +old mark is the nonalgebraic mark of that layer and is outside its +root. Face isomorphisms preserve the marks, so the distinction +persists under all transition maps. + +Take $I$ of cardinality $\mu>\lambda$, where $\mu$ is a sufficiently +closed single-model pointed level. The ordinary directed colimit in +the AEC exists: a coherent diagram of injective maps is realized as +a diagram of inclusions in its direct limit, and the directed-union +axioms apply. There are $\mu$ finite subsets of $I$, each contributing +at most $\lambda$ elements, and there are $\mu$ distinct marks. Thus +the colimit $N$ has size exactly $\mu$. Every element of $N$ comes +from one finite vertex. Preservation and reflection of $P$ give +\[ +P(N)=P(J_0),\qquad |P(N)|\leq\lambda. +\] +The enumeration of its distinguished $M_0$ retains the fixed +stabilized diagram $\Delta$ under the strong colimit embeddings. + +On the other hand, there is a pointed model of size $\mu$ with more +than $\lambda$ points. Start with any pointed model of that size. +Successively realize a nonalgebraic free extension of $p_0$ outside +the current model, for $\lambda^+$ steps. The receiving models and +their unions may all be kept of size $\mu$. The new realizations +are distinct and belong to $P$, so the union has at least +$\lambda^+$ points. Pointed categoricity at $\mu$ would make this +model isomorphic to $N$ over $M_0$, a contradiction. + +Only the stationary prescription $r$ was lifted to higher levels. +The assumed witness with unchanged $P$ was used solely at +$\lambda$. The large colimit used the full AEC directed-union +axioms, rather than a same-level closure assertion for a marked +scheme. +\end{proof} + +\begin{corollary}[No-growth and rank on the entire tail] +\label{geom:tail-no-growth} +Increase the threshold $\lambda$ if necessary to include the bounds +for the fixed point diagram. In arbitrary cardinalities, if +$A\leq_\K B$ are pointed models with $\|A\|\geq\lambda$ and +$P(A)=P(B)$, then $A=B$. +Moreover, for every pointed model $M$ with $\|M\|>\lambda$, +\[ +|P(M)|=\rk(P(M))=\|M\|. +\] +The point sets of strong submodels are relatively closed, and +relative bases and finite basis supports are available as in +Theorem~\ref{geom:pregeometry}; no eligibility condition is imposed +on these arbitrary cardinalities. +\end{corollary} + +\begin{proof} +Suppose that $a\in B\setminus A$ and $P(A)=P(B)$. Construct +increasing strong submodels $A_i\leq_\K A$ and $B_i\leq_\K B$ of +size $\lambda$, for $i<\omega$, all containing the distinguished +copy of $M_0$, so that +\[ +A_i\leq_\K B_i,\qquad a\in B_i,\qquad +P(B_i)\subseteq A_{i+1}. +\] +For the successor choice, $P(B_i)$ has size at most $\lambda$ and +is contained in $A$, so the L\"owenheim--Skolem axiom places it, +together with $A_i$, in the next $A_{i+1}$. Choose $B_{i+1}$ to +contain $A_{i+1}\cup B_i\cup\{a\}$. Coherence makes all the +required inclusions strong. The two unions $A_\omega\leq_\K +B_\omega$ have size $\lambda$. The point diagram restricts +correctly to them, and +\[ +P(B_\omega)=\bigcup_{i<\omega}P(B_i) + \subseteq A_\omega. +\] +Thus their point sets are equal, while $a\in B_\omega\setminus +A_\omega$. This contradicts Theorem~\ref{geom:no-growth}. + +Now let $\|M\|=\kappa>\lambda$. If $|P(M)|<\kappa$, choose a +strong submodel containing $M_0\cup P(M)$ of size at most +$|P(M)|+\lambda_0$. Enlarge it inside $M$, if necessary, to a +strong submodel of size at least $\lambda$ and still less than +$\kappa$. It has the same $P$ as $M$, contrary to the assertion +just proved. Hence $|P(M)|=\kappa$. If $H$ is a basis of $P(M)$, +the closure bound gives +\[ +\kappa=|P(M)|=|\cl_M(H)|\leq |H|+\lambda_0. +\] +Since $\kappa>\lambda\geq\lambda_0$, it follows that +$|H|=\kappa$. The final assertions were established in +Theorem~\ref{geom:pregeometry} independently of eligibility. +\end{proof} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/11-geometric-systems.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/11-geometric-systems.tex new file mode 100644 index 0000000000000000000000000000000000000000..f5ee8d3053c7e00a5e24609703c018b68b9f46e5 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/11-geometric-systems.tex @@ -0,0 +1,905 @@ +\section{Geometric systems and classification by bases} +\label{sys:section-geometric-systems} + +Throughout this section the distinguished model $M_0$, its fixed pointed +pattern, and the pregeometry $P$ are those of +Proposition~\ref{geom:pointed-levels} and +Theorem~\ref{geom:pregeometry}. +All model sizes exceed the threshold of +Corollary~\ref{geom:tail-no-growth}. Thus a strong inclusion of pointed +models with equal $P$-sets is equality. At an eligible uniform level +$\mu$, all vertices have size $\mu$. A statement about unions of +objects at that level concerns chains whose total size is $\mu$, as in +the working conditions of Definition~\ref{lev:working-package}. + +We first prove structural and presentation assertions without assuming +categoricity of systems. The classification argument then proceeds by +the number of axes. Its first assertion extends a root isomorphism and +prescribed bijections of the other basis pieces; this gives categoricity +for the current shape. That categoricity supplies the working package +used to prove the second assertion: prescribed bijections of the basis +pieces added to a fixed system extend to an isomorphism over that system. +By splitting off the last axis, the second assertion gives the first at +the next dimension. We establish the conditional package before carrying +out this paired induction. + +\subsection{Shapes, relative bases, and arrows} + +\begin{definition}[Admissible shapes and geometric systems] +\label{sys:shapes} +Let $A$ be a finite linearly ordered set of axes. An admissible family +$L\subseteq\mathcal P(A)$ of \emph{active labels} is defined recursively. +On no axes the family is $\{\varnothing\}$. If $k$ is a new last axis, +choose an upward closed subset $Q$ of the previously constructed family +$L$, and put +\[ + L'=L\cup\{s\cup\{k\}:s\in Q\}. +\] +Here upward closed means that $s\in Q$, $s\subseteq t$, and $t\in L$ +imply $t\in Q$. + +A geometric system of shape $L$ is a family +$(X_t:t\subseteq A)$ of pointed models in a common top model, with +$X_t\leq_\K X_u$ when $t\subseteq u$, satisfying the following condition. +There is a partition $(H_s:s\in L)$ of a basis of $P(X_A)$ such that +$H_s\subseteq P(X_s)$ and +\begin{equation} +\label{sys:vertex-span} + P(X_t)=\cl_{P(X_A)}\left(\bigcup_{s\in L,\ s\subseteq t}H_s\right) + \qquad(t\subseteq A). +\end{equation} +At uniform level $\mu$ we require $|H_s|=\mu$ for every $s\in L$. +The resulting scheme is denoted $\mathcal G_L$. +The basis partition witnesses the definition; it is not named in the +system structure. + +For a vertex $t$, write +\[ + D_t(X)=\cl_{P(X_A)}\left(\bigcup_{r\subsetneq t}P(X_r)\right). +\] +For $t=\varnothing$ this is the closure of the empty set in the +pregeometry. An arrow $X\to Y$ is a commuting family of pointed strong +embeddings for which a witnessing partition for $X$ extends to a +witnessing partition for $Y$. Equivalently, after identifying $X$ with +its image, each finite tuple from $P(X_t)$ independent over $D_t(X)$ +remains independent over $D_t(Y)$. +\end{definition} + +The equivalence in the last sentence, including its independence from +the initially chosen partition, will be proved below. All closures and +independence assertions are computed in a common containing pointed +model. Theorem~\ref{geom:pregeometry} makes this convention +independent of the chosen common strong extension. +The notation $\cl_{P(X_A)}$ means the closure $\cl_{X_A}$ defined +there; its subscript emphasizes the pregeometry in which it is +computed. We omit that subscript when the ambient model is clear. + +\begin{lemma}[Combinatorics of shapes] +\label{sys:shape-combinatorics} +An admissible family contains $\varnothing$ and is closed under unions. +Its restriction to any subset of the axes is admissible, with the +inherited order. Consequently every $t\subseteq A$ has a largest +active subset +\[ + a_L(t)=\bigcup\{s\in L:s\subseteq t\}. +\] +In a geometric system, $X_t=X_{a_L(t)}$ whenever $t$ is inactive. +\end{lemma} + +\begin{proof} +The union assertion follows by induction on the axes. Unions of old +labels are old. If a union involves $s\cup\{k\}$ with $s\in Q$, its +old part is an active label containing $s$, and therefore belongs to +$Q$. This also handles the union of two labels containing $k$. +Restriction to a set of axes replaces $L$ and $Q$, at every retained +step, by their restrictions. The restricted $Q$ is upward closed in +the restricted $L$. Steps on discarded axes make no contribution. +This proves the restriction assertion. + +The displayed union is an active label, since the family is finite and +contains $\varnothing$. The collections of active labels below $t$ +and below $a_L(t)$ agree. Equation~\eqref{sys:vertex-span} therefore +gives equality of their $P$-sets. The inclusion +$X_{a_L(t)}\leq_\K X_t$ and +Corollary~\ref{geom:tail-no-growth} give equality of the models. +\end{proof} + +\begin{remark} +Union closure alone would not imply the recursive last-axis +description. For example, $\{\varnothing,\{1,2\}\}$ is union closed, +but its two layers on splitting off axis $2$ have different active +families. We use the recursive definition, including the upward +closed set $Q$, when identifying a system with an arrow between +systems on fewer axes. +\end{remark} + +\begin{lemma}[Replacement and extension of relative bases] +\label{sys:relative-bases} +Suppose $X$ is a geometric system. For each active $s$, choose any +basis $B_s$ of $P(X_s)$ over $D_s(X)$. Then the $B_s$ form a +witnessing partition for $X$. + +A componentwise strong system embedding is an arrow precisely when +it preserves the relative independence in +Definition~\ref{sys:shapes}. For an arrow, every witnessing partition +of its domain extends to one of its codomain. +\end{lemma} + +\begin{proof} +Fix a witnessing partition $(H_s:s\in L)$, and enumerate $L$ in an +order extending inclusion. For active $s$, +\[ + D_s(X)=\cl\left(\bigcup_{u\in L,\ u\subsetneq s}H_u\right). +\] +When the piece at $s$ is replaced, previously processed labels not +below $s$ are independent, together with the original piece at $s$, +over this lower span: they are parts of a single independent set. +Replacing $H_s$ by any basis of the same flat over its lower span +preserves both independence and the span of the union. Explicitly, +let $C$ be the current pieces below $s$. Then +$\cl(C\cup B_s)=\cl(C\cup H_s)=P(X_s)$, and $C\cup B_s$ is +independent. The remaining current pieces were independent over +$\cl(C\cup H_s)$, since the full current partition was independent. +They remain independent over the equal flat $\cl(C\cup B_s)$. +Thus the new full partition is independent and has the same span. +Applying this replacement at the finitely many active labels gives the +first assertion and Equation~\eqref{sys:vertex-span} for the new +partition. + +If pieces extend along an embedding, any finite independence +calculation in a vertex over its lower span is the same calculation +using the corresponding pieces in the larger independent set. +Relative independence is therefore preserved. + +Conversely, let $(B_s)$ be an arbitrary witnessing partition of the +domain, viewed as a subsystem of $Y$. Preservation implies that +$B_s$ is independent over $D_s(Y)$. Extend it to a relative basis +$B_s\cup C_s$ of $P(Y_s)$ over $D_s(Y)$. The first assertion, applied +to $Y$, says that these choices assemble into a witnessing partition +of $Y$. This proves both the equivalence and the final assertion. +\end{proof} + +\begin{lemma}[Structural properties and the last-axis description] +\label{sys:structural} +Arrows compose and satisfy coherence. Same-level chains of total +size the level have unions in $\mathcal G_L$, with the common-upper- +bound clause also for arrows into a larger eligible level. + +Restriction to any subset of axes preserves systems and arrows. +If $L'=L\cup\{s\cup\{k\}:s\in Q\}$, splitting off the last axis +identifies a system of shape $L'$ with an arrow $X\to Y$ in +$\mathcal G_L$ such that the complementary piece at $s$ has size +$\mu$ for $s\in Q$ and is empty for $s\notin Q$. +\end{lemma} + +\begin{proof} +Composition follows from the relative independence criterion. +For coherence, suppose $X\subseteq Y$, and both embed by arrows in +$Z$. Independence over $D_t(X)$ implies independence over $D_t(Z)$, +and hence over its subset $D_t(Y)$. Componentwise coherence is the +AEC coherence axiom. + +For a continuous chain, extend witnessing pieces successively using +Lemma~\ref{sys:relative-bases}, and take their unions at limit stages. +Every finite independent sublist and every finite dependence witness +occurs at a stage. The resulting union pieces are independent and +span every vertex of the componentwise union. Each contains $\mu$ +elements from the first stage and has size at most $\mu$, so all +pieces still have size $\mu$. For a chain which is not initially +continuous, insert its earlier unions: the preceding construction +applies recursively, and the common-upper-bound argument in the next +paragraph supplies its connecting arrows. + +If each stage maps by an arrow into $Z$, a finite tuple independent +over the lower span of the union is independent over the lower span +at any stage containing it. Its image is independent over $D_t(Z)$. +The componentwise AEC common-upper-bound axiom supplies the strong +maps, so the union map is an arrow. This argument also permits a +larger level for $Z$. + +On restricting to axes $B\subseteq A$, retain the pieces whose labels +are contained in $B$. They are a basis of $P(X_B)$ and satisfy the +required span identities. For a retained vertex all its proper +subsets are still retained; its relative independence test is +therefore exactly the old test. + +Finally let $Z$ have shape $L'$, and put $X_s=Z_s$ and +$Y_s=Z_{s\cup\{k\}}$. A partition for $Y$ is obtained by combining +$H_s$ with $H_{s\cup\{k\}}$ when $s\in Q$, and using $H_s$ alone +otherwise. These choices extend the pieces of $X$, and their +complements have the asserted sizes. Conversely, extend pieces of +$X$ to pieces of $Y$ along such an arrow, give the old pieces labels +$s$ and their complements labels $s\cup\{k\}$, and use $X$ and $Y$ +as the two layers. The span identities give a system of shape $L'$. +Inactive vertices cause no exception, by +Lemma~\ref{sys:shape-combinatorics}. +\end{proof} + +For later use, preservation of relative independence also preserves +finite rank calculations modulo the lower span. For finite tuples +$a,c$ in a vertex, the condition that $a$ be independent over +$D_t(X)\cup c$ is determined by the ranks, modulo $D_t(X)$, of +sublists of $a,c$. These ranks are determined by which sublists are +independent. Thus arrows preserve this condition as well. + +\subsection{Order presentations and their rank} + +\begin{proposition}[Tiled presentations] +\label{sys:tiling} +Let $L$ be admissible and let $\mu$ be sufficiently large for the +pointed pregeometry and the no-proper-pair conclusion. The order +template $E$ gives a presentation in $\mathcal G_L$ as follows. +Choose a fixed initial order carrying the whole enumeration of $M_0$. +For each active label $t$, choose a tile $T_t$, preceded by a buffer +$B_t$, and concatenate these finitely many pairs after the initial +order, in any fixed order. Every tile and buffer is a dense linear +order without endpoints of size $\mu$. In vertex $s$ retain the +initial order, all buffers, and just the tiles with labels $t\subseteq +s$. + +These presentations have the following properties. +\begin{enumerate} +\item Each active relative piece has rank $\mu$. +\item Order embeddings of the corresponding tiles and buffers, +fixing the initial pointed data, induce arrows, including between +different eligible levels. +\item A presentation has an extension by an arrow gaining $\mu$ +independent complementary coordinates at every active label +simultaneously. +\item Tiles may be chosen with $\mu$ pairwise disjoint convex +substrips, each of size $\mu$. Further insertions of such substrips +preserve the preceding assertions and the naturalness of the +underlying $\K$-diagrams. +\end{enumerate} +No categoricity assumption about $\mathcal G_L$ is used here. +\end{proposition} + +\begin{proof} +Let $I_s$ be the indicated suborder and set $X_s=E(I_s)$, using the +literal embeddings supplied by Theorem~\ref{found:order-template}. +All $I_s$ have size $\mu$, since they contain all buffers. The +models have size $\mu$ and are strongly included according to their +labels. The initial order was chosen to contain the supports of the +fixed enumeration of $M_0$. Its unchanged term pattern carries the +same pointed model in every vertex, and coherence gives the required +strong inclusion of $M_0$. + +Choose a relative basis $H_s$ of $P(X_s)$ over the union of its +proper lower $P$-sets for every active $s$. We first prove that the +indexed family of these choices is independent; this also proves +that the pieces are disjoint. Suppose there is a finite circuit +of labeled entries, allowing a two-entry equality dependence when +two differently labeled entries coincide. Choose a maximal label $s$ represented among its +entries. Retain the finite term supports of all entries with label +$s$, and indeed retain every support position in a tile available +at $s$. In each unavailable tile replace all support positions by +distinct positions in its preceding buffer, beyond all the finitely +many retained positions in that buffer. Such positions exist because +the buffer has no greatest point and is dense. No retained position +lies in that unavailable tile. Consequently this replacement +preserves the order and equality pattern of all circuit supports and +fixes the whole initial pointed order. + +The labels and this order pattern preserve the pointed finite +geometry comparisons. One can formulate this either using the +natural $\K$-diagrams or by taking the term model on the fixed initial +order and the finite supports and applying its two pointed strong +embeddings into the original presentation. In the latter formulation +Theorem~\ref{geom:pregeometry} preserves and reflects the +independence relation. + +An entry with label $t\ne s$ in the circuit has no label properly +containing $s$, by maximality. Thus $s\nsubseteq t$. After the +replacement its term uses only common buffers, initial data, and +tiles with labels contained in $s\cap t$. Its image belongs to +$P(X_{s\cap t})\subseteq D_s(X)$. The entries with label $s$ have +not moved. The preserved circuit makes a sublist of $H_s$ dependent +over $D_s(X)$, a contradiction. The union is therefore independent. +It spans every vertex by induction on the labels: its relative basis +spans that vertex over lower vertices, which have already been +spanned. If a vertex is inactive, the order retained there is exactly +the order retained at its largest active subset. This proves the +decomposition without first assuming the system axioms. + +We give the fresh-skeleton observation needed for the rank assertion. +If $J\subseteq I$ and $i\in I\setminus J$, the skeleton entry at $i$ +does not belong to $E(J)$. Otherwise it equals a labeled term on a +finite tuple of positions from $J$. Enlarge the order by a second +position $i'$ having the same order pattern as $i$ relative to that +finite tuple. Literal pattern preservation would make both +distinguished skeleton entries equal to that term. This contradicts +their required distinctness. The enlargement need only duplicate +the cut relative to the finite tuple, not the cut relative to all of +$J$. + +For an active $t$, omit $T_t$ from $I_t$. The resulting order still +has size $\mu$, contains the pointed data and all proper lower +vertices, and presents a strong submodel. Enumerate $\mu$ points of +$T_t$ and adjoin them one at a time. The fresh-skeleton observation +makes each successor inclusion proper. All these models have size +$\mu$, because the buffers were retained. By +Corollary~\ref{geom:tail-no-growth}, each successor adds a point of +$P$. Choose one such point at each successor. Relative closedness +of $P$ in strong submodels makes these choices independent over the +initial omitted-tile model. There are $\mu$ choices, so the rank over +the proper lower vertices is at least $\mu$. It is at most $\mu$ +by the size of $X_t$. This proves the first assertion, including the +root case $t=\varnothing$. + +Next consider componentwise order embeddings into enlarged tiles and +buffers, identifying the source with its image. Suppose a finite +tuple $a$ in $P(X_s)$, independent over $D_s(X)$, becomes dependent +over $D_s(Y)$. Dependence has a finite witness consisting of points +from proper lower vertices of $Y$. Fix the finitely many old +positions supporting $a$. Move every other position needed for the +witness into its corresponding old tile or buffer, preserving its +order relative to the fixed positions. Density and absence of +endpoints of the old order suffice for each finite pattern; it is +not necessary that the old order be dense in the enlarged order. +The initial pointed order remains fixed. Each witness remains in +its designated proper lower vertex, now in $X$, and its finite +dependence diagram is preserved. This contradicts independence of +$a$ over $D_s(X)$. Lemma~\ref{sys:relative-bases} therefore gives an +arrow. The proof did not require equal source and target sizes. + +To obtain simultaneous gains, enlarge all tiles with $\mu$ fresh +points and, if desired, enlarge the buffers. For a particular active +$t$, begin the strict-growth argument over the order containing the +old $t$-tile, all enlarged proper-lower tiles, all enlarged buffers, +and the initial data. This base contains $X_t$ and all proper lower +vertices of $Y$. Adjoining the fresh points of the $t$-tile yields +$\mu$ independent points over that larger base. Hence the +complementary rank at $t$ is $\mu$. The preceding arrow assertion +allows any old pieces to extend, and +Lemma~\ref{sys:relative-bases} combines the relative choices into +one partition. This proves the simultaneous assertion. + +Finally a lexicographic sum, indexed by a linear order of size +$\mu$, of dense endpoint-free orders of size $\mu$ is still dense +and endpoint-free and has size $\mu$. Its summands are the desired +convex substrips. In particular, for a presentation indexed by a +well-order $H$, one can use $H\times\Q$ and reserve intervals of +$H$ of order type and cardinality $\mu$. Taking $H$ of order type +$\mu\cdot\mu$ supplies $\mu$ disjoint such intervals. Additional +summands or strips do not change any of the preceding arguments. +Underlying $\K$-diagrams remain natural because the entire system +was obtained by designated term subcollections of the single +template $E$. +\end{proof} + +\subsection{Obtaining the working package from categoricity} + +\begin{proposition}[Conditional working package] +\label{sys:working-package} +Fix an admissible shape $L$. Suppose that $\mathcal G_L$ is +categorical at every level in an unbounded class of eligible levels +closed sufficiently for the preceding constructions. Assume also +the eligible-subclass hypothesis of Convention~\ref{lev:eligible-subclass}. +On an unbounded further class of sufficiently closed levels, it satisfies the full working +package of Definition~\ref{lev:working-package}, with its actual +system isomorphisms and its arrows from Definition~\ref{sys:shapes}. +The resulting diagrams, saturation, and calculus include mixed +arrows between eligible levels, and the output class retains the +eligible-subclass hypothesis. + +The closure requirements are uniform in the finite shape and the +fixed pointed template. They use only the smaller and larger input +levels specified by the general working-package construction. +\end{proposition} + +\begin{proof} +The structural conditions are Lemma~\ref{sys:structural}. +Proposition~\ref{sys:tiling} supplies term presentations and mixed +presentation arrows before categoricity. By the assumed +categoricity, every object at an input level is isomorphic to its +presentation. + +We check the extra relative-basis condition throughout the reversal +argument. In each tile and buffer use the order $H(\delta)$ from +Theorem~\ref{found:order-orbits}. For a tuple of length $\theta$, record +its term labels, the fixed initial positions it uses, and, separately +in each of the finitely many variable orders, the order and +realized-cut record of its support positions. If +$\theta<\cf(\delta)$, equal records are carried to one another by +order automorphisms of the respective tiles and buffers. Their +product fixes the initial data and induces an automorphism of the +whole system. The set of possible records has a bound depending +only on $\theta$, the finite shape, and the template. Hence the +system orbits at that arity have a uniform set bound. + +Number these orbits and record the arrows between them, together +with the partial map from support records to orbit names. This is +again a bounded palette. If reversal fails unboundedly often, +choose repeated profiles at eligible levels $a<\mu$ exactly as in +Theorem~\ref{found:reversal}, with $a^\theta\leq\mu$ and the required +large cofinalities for the orders. Apply the ordinal-position +shrinking of Proposition~\ref{found:alignment} simultaneously to the +finitely many tiles and buffers. Given at most $a$ parameters in +the large system, the resulting small presentation contains them, +has size $a$ in every tile and buffer, and computes the support +records of all its $\theta$-tuples correctly. Its inclusion is an +actual arrow by Proposition~\ref{sys:tiling}. Thus it is an aligned +small geometric system, not merely a componentwise subsystem. + +For completeness, the contradiction uses two orbit names $p,r$ with +$p\to r$ and $r\not\to p$. Repeatedly choose aligned small copies, +each containing the previous one, and process their at most +$a^\theta$ tuples in the large level. Whenever a tuple can still be +sent to $p$, send it onward to $r$. From that point it cannot reach +$p$. Coherence makes the successive small copies a chain of +arrows. After $\theta^+$ rounds every $\theta$-tuple of their union +occurred at a proper round. Both unions exist by +Lemma~\ref{sys:structural}: the small union has total size $a$, and +the large construction has total size $\mu$. Alignment says that +none of the tuples of the small union can reach $p$. Categoricity +of that union at $a$ says that its orbit $p$ is realized, a +contradiction. This proves eventual reversal at each fixed arity. + +Mixed L\"owenheim--Skolem closure also has the required form. +In a presentation, close the supports of a prescribed small set +under finitely many intermediate and exterior positions in every +tile and buffer, repeat countably many times, and pad each order +to the chosen smaller eligible size. The resulting dense +endpoint-free suborders give a small presentation containing the +set. The initial pointed order is retained in full. Its inclusion +is an arrow by Proposition~\ref{sys:tiling}. Transport through a +presentation of an arbitrary large object gives mixed containment; +coherence supplies comparisons between such subobjects. + +We now apply the first route of +Proposition~\ref{lev:scheme-diagram-routes}. Its structural +hypothesis is Lemma~\ref{sys:structural}; its presentation and mixed +downward-closure hypotheses are Proposition~\ref{sys:tiling} and +the preceding paragraph. Categoricity is the input assumption, +and the required arity-by-arity reversal and uniform orbit bounds +were proved above. Thus the proposition applies to the actual +system arrows, including their relative-basis condition. + +In particular, the canonical comparison at a smaller input level +is computed in a small geometric subobject. Two such sections +are contained in a common small section by mixed downward closure, +and coherence gives its arrows, so the comparison is well defined. +The projections between comparisons at two input levels are onto +because the smaller system is a tiled presentation and extends by +the mixed order arrows of Proposition~\ref{sys:tiling}. The bounded +palette therefore stabilizes, exactly as in the first route of +Proposition~\ref{lev:scheme-diagram-routes}. + +Choose receiving levels which remain eligible, are strictly closed +under the resulting arity bounds, and have sufficiently large +smaller input levels below them for every bounded request. +These are the closure refinements described in +Section~\ref{lev:refining-levels}, supplied simultaneously in +Theorem~\ref{close:common-levels}; closure under the next input +level is not being used to infer eligibility by itself. At such a +level, countable alternating matching in smaller input-level +sections gives actual system isomorphisms on equal bounded +diagrams, and realizes possible bounded joint diagrams over their +given subtuples. Thus diagrams, homogeneity, and saturation have +the system interpretation required here, rather than merely their +componentwise $\K$ interpretation. Order embeddings and order +amalgamation make the diagrams natural on term tuples, so +Proposition~\ref{lev:diagram-coding} also applies. + +It remains to check the additional exhaustion condition, which +Proposition~\ref{lev:scheme-diagram-routes} explicitly leaves as a +separate obligation. A finite tuple independent modulo the +proper lower vertices stays so when its target diagram agrees with +a tuple placed along an actual presentation arrow: the comparison +is realized by a system isomorphism, and such an isomorphism +preserves relative independence. Every tuple in an exhaustive +sequence is handled at a stage containing it. The finite +relative-independence criterion in Lemma~\ref{sys:relative-bases} +then supplies the additional arrow condition. Componentwise +strongness is the conclusion of the same proposition's exhaustion +argument. This completes the working-package verification. + +All refinements used here are the bounds for the original reversal, +alignment, diagram stabilization, and saturation arguments, applied +to finitely many orders and the fixed initial data. None introduces +an amalgamation assumption on the original AEC; the amalgamation +used below is the conclusion of the working package and +Theorem~\ref{calc:scheme-amalgamation}. +\end{proof} + +\subsection{Fresh coordinates and the two induction assertions} + +Fix pieces $(H_s^X:s\in L)$ in a system $X$. +For an arrow $X\to Y$, extend them to pieces +\[ + H_s^Y=H_s^X\mathbin{\dot\cup}C_s^Y. +\] +The sets $C_s^Y$ are its \emph{complementary pieces}. Their sizes +may be any cardinals at most $\mu$, including zero, even though all +the total pieces have size $\mu$. + +\begin{definition}[Joint freshness] +\label{sys:fresh} +A list of coordinates in an extension $Y$ of $X$, with individual +active labels, is jointly fresh over $X$ if it is a sublist of the +complementary pieces for some such extension of the pieces of $X$. +At a single label $s$, the corresponding condition is independence +over +\[ + \cl\bigl(P(X_s)\cup D_s(Y)\bigr). +\] +For several labels these relative conditions assemble into joint +freshness by Lemma~\ref{sys:relative-bases}. Coordinates are +regarded as occurrences in their designated vertex sorts. +\end{definition} + +Freshness is preserved by arrows of extensions. Indeed every +failure or success of finite independence over the displayed flat +is determined by finite rank calculations using parameters from +$P(X_s)$ and the lower span. The observation following +Lemma~\ref{sys:structural} preserves those calculations. This also +shows that freshness is an invariant of the type over $X$ once the +working package is available. + +\begin{theorem}[Classification with prescribed basis maps] +\label{sys:basis-classification} +For each finite admissible shape $L$, at all sufficiently closed +uniform levels $\mu$, the following assertions hold. +\begin{enumerate} +\item If $X,Y\in\mathcal G_L$, an isomorphism +$e_\varnothing:X_\varnothing\simeq Y_\varnothing$ and arbitrary +bijections between the chosen nonroot pieces of $X$ and $Y$ extend +to a system isomorphism $X\simeq Y$. +In particular $\mathcal G_L$ is categorical at these levels. +\item Suppose $X\to Y$ and $X\to Z$ are arrows in +$\mathcal G_L$. Choose pieces of $X$, extend them to pieces of +$Y$ and $Z$, and prescribe a bijection +$C_s^Y\simeq C_s^Z$ for every active label $s$. +Then there is a system isomorphism $Y\simeq_X Z$ realizing every +prescribed bijection. The complementary cardinalities may be +less than $\mu$, and zero is allowed. +\end{enumerate} +The level requirements are uniform over the finitely many shapes +at any fixed number of axes. +\end{theorem} + +\begin{proof} +We prove the two assertions in order, by the number of axes. +On no axes, the first assertion is exactly the assertion that a +prescribed isomorphism of the root models is an isomorphism of the +systems. Existence of such a root isomorphism, hence categoricity +of the zero-axis scheme, is +Proposition~\ref{geom:pointed-levels}. + +Suppose the second assertion is known for all shapes on fewer +axes. For a shape with last-axis description $L'$, apply the first +assertion for $L$ to its lower layers and then the second assertion +for $L$ to its upper layers. The pieces added at labels in $Q$ +are exactly the pieces bearing labels $s\cup\{k\}$ in $L'$. +Prescribe their given bijections, and prescribe the empty maps at +the other labels. This produces the desired isomorphism of the +two-layer systems by Lemma~\ref{sys:structural}. +If an auxiliary root basis has been chosen, replace the target +root basis by its image under the prescribed root isomorphism; +Lemma~\ref{sys:relative-bases} permits that replacement without +changing any prescribed nonroot piece. Thus the first assertion +at the next number of axes follows from the preceding second +assertion. + +It remains to prove the second assertion for a fixed shape $L$ +whose first assertion has now been established. Refine the +levels using Proposition~\ref{sys:working-package}. All the +one-dimensional calculus and dominated-hull results are then +available for $\mathcal G_L$. The rest of the proof is given in +four claims. + +\begin{claim}[Uniqueness of finite fresh types] +\label{sys:finite-fresh-types} +For a fixed base $X$ and a finite ordered list of active labels, +all jointly fresh tuples with those labels have the same type +over $X$. +\end{claim} + +\begin{proof} +At zero axes this is the base-change calculation of +Proposition~\ref{geom:base-change}. Types in the pointed scheme +over a whole pointed base are the underlying $\K$-types: an +amalgam or type identification over that base already fixes the +point. All objects at the level have full $P$-rank by +Corollary~\ref{geom:tail-no-growth}. + +At a positive number of axes suppose two finite jointly fresh +tuples $b^0,b^1$ have different types over $X$. Bounded testing +gives a bounded tuple $f$ from $X$ such that +\[ + D(f,b^0)\ne D(f,b^1). +\] +Choose a cardinal $\chi<\mu$ large enough for the orbit bound at +arity $|f|$, for the finite tuple lengths, and for the template +labels, with $\chi^{|f|+\aleph_0}=\chi$. Let $\epsilon$ be the +least cardinal for which $2^\epsilon>\chi$. Then +$\epsilon\leq\chi$, there are at most $\chi$ nodes at all proper +levels of the binary tree of height $\epsilon$, and +$2^\epsilon\leq2^\chi<\mu$. + +Build a tree of extensions of $X$ at level $\mu$. At a node $V$, +the arrow from $X$ preserves the diagram of $f$. Bounded-tuple +homogeneity therefore supplies an isomorphism $X\simeq V$ +fixing $f$. Transport the two witness extensions along this +isomorphism to form the children of $V$. Their marked tuples +are fresh over $V$, and their diagrams with $f$ have the two +specified different values. At limit nodes take unions along +the branch. Extend chosen basis pieces at every step. Hence +along every branch all marked coordinates, at their respective +labels, lie in complementary pieces over the initial $X$. +The total size along a branch is $\mu$, so all unions are +allowed. Notice that the return isomorphism fixing $f$ uses +diagram homogeneity, not categoricity alone. + +At every leaf make a further rich extension by +Proposition~\ref{sys:tiling}, so that the root has complementary +rank $\mu$ over the original $X_\varnothing$. There is a common +system $U$ with the same property: present any system and make a +rich extension, transporting it over $X$ using categoricity. +For each level $\eta<\epsilon$ reserve in $U$ a finite tuple +$c_\eta$ of coordinates with the same labels as the marked +tuple at that level. Choose all of these coordinates from +complementary pieces over $X$. Their total number is at most +$\chi$. + +For a leaf $W$, its root marked coordinates and those in $U$ +are independent over $P(X_\varnothing)$. Both full root +complements have size $\mu$, and the prescribed match uses +fewer than $\mu$ coordinates. Extend that match to a bijection +of root complements. The already proved second assertion for +zero axes gives +\[ + W_\varnothing\simeq_{X_\varnothing}U_\varnothing +\] +carrying all root marked coordinates correctly. At nonroot +labels extend the prescribed matches of marked coordinates to +bijections of the full pieces, whose sizes are $\mu$. The +first assertion for the present shape extends these maps and +the root isomorphism to an isomorphism $W\simeq U$. +The old nonroot pieces of $X$ need not be fixed in this step. + +Let $f_W$ be the image of $f$ in $U$. Two different branches +first separate at a level $\eta$; preservation of diagrams +along their arrows gives different values of +$D(f_W,c_\eta)$ after the two leaf isomorphisms. Thus the +$2^\epsilon$ images of $f$ lie in different orbits over the +fixed set of at most $\chi$ reserved coordinates. The +parameter-orbit bound of Lemma~\ref{calc:parameter-orbits}, +applied to the working scheme, is $\chi$ at +these arities, a contradiction. +\end{proof} + +For each active $s$, let $p_s^X$ denote the unique fresh singleton +type given by the claim. Such types exist by the rich extensions +of Proposition~\ref{sys:tiling} and categoricity. Their realizations +lie outside the base vertex, so they are nonalgebraic. + +\begin{claim}[Free lifts preserve freshness] +\label{sys:free-fresh} +The free lift of $p_s^X$ along an arrow $X\to V$ is fresh over $V$. +Finite jointly fresh tuples have the product of their singleton +types. Bounded jointly fresh lists have a unique whole type +specified by their labels and admit all-dominated hulls. +\end{claim} + +\begin{proof} +Let $a$ realize the free lift in an extension $Y$ of $V$. +If $a$ is not fresh, finite character gives a finite tuple +$c\subseteq P(V_s)$ with +\[ + a\in\cl\bigl(c\cup D_s(Y)\bigr). +\] +This condition is invariant under system isomorphisms and is +therefore determined by the bounded diagram of $(a,c)$ in $Y$. +Its witnesses in lower vertices need not lie in $V$; the +isomorphism-invariant condition itself is what the diagram +records. + +Put $m=|c|+1$, and choose an $m$-tuple $b$ jointly fresh over +$X$, all at label $s$. Every singleton marginal is $p_s^X$ by +Claim~\ref{sys:finite-fresh-types}. Take a sufficiently long +bounded family of approximations, inside $X$, for +$\tp(b/X)$, as in Lemma~\ref{calc:approximations}. +By the support-copy construction in that Lemma, every +approximating tuple has the same diagram as +$b$ without parameters, and is consequently independent +modulo $D_s(X)$. In the free lift to $V$, each coordinate has +the same marginal as $a$. For each coordinate, the majority +estimate gives the diagram of $(a,c)$ outside a uniformly +bounded set of approximation indices. Choose the family +longer than the union of these finitely many exceptional sets. +One approximating tuple $(a_1,\ldots,a_m)$ is independent +modulo $D_s(X)$ and has, coordinate by coordinate, the required +diagram with $c$ in $V$. + +Since $X\to V$ is an arrow, this tuple remains independent +modulo $D_s(V)$. The comparison just obtained is an equality of +$\mathcal G_L$-diagrams, +\[ + D_{\mathcal G_L}(a,c;Y)=D_{\mathcal G_L}(a_i,c;V). +\] +The established working package therefore gives a system isomorphism +$F_i:Y\cong V$ taking $a$ to $a_i$ and fixing the displayed tuple $c$. +It carries every lower vertex onto its counterpart and preserves the +pointed closure, so $F_i[D_s(Y)]=D_s(V)$. Transporting the dependence +witness gives $a_i\in\cl(c\cup D_s(V))$ for every $i$. The latter +flat has relative rank at most $|c|$, whereas the tuple has +relative rank $m>|c|$. This contradiction proves that the +free lift is fresh. + +Realize a finite product successively by free lifts along a +chain of extensions. The assertion just proved puts each +new coordinate in a complementary piece over the preceding +system. Extending pieces along the chain makes the whole +tuple jointly fresh over $X$. Claim~\ref{sys:finite-fresh-types} +then says that every jointly fresh finite tuple has this +product type. A bounded jointly fresh list is consequently +product-independent in the sense of +Theorem~\ref{iso:independent-lists}. That theorem gives +its unique whole type and its witnessing free chains; its +all-dominated-hull conclusion gives existence over the list. +\end{proof} + +We now know the types of fresh coordinate lists and have all-dominated +hulls over them. To extend a partially prescribed basis map, we also +need to cover an arbitrary bounded tuple using only boundedly many +new coordinates. The next claim provides this size control, including +when the target basis has already been chosen. + +\begin{claim}[Bounded rank cost and chosen coordinates] +\label{sys:rank-cost} +Let $X\to Y$ be an arrow. A finite tuple $d$ of $Y$ is +contained in a subextension $X\to U\to Y$ whose complementary +pieces have finite sizes. For an arbitrary bounded tuple +$d$, their sizes can be bounded by +$\max(|d|,\aleph_0)$. + +If specified pieces of $Y$ extend those of $X$, the subextension +containing a bounded $d$ can be chosen with all its +complementary pieces subsets of the specified complementary +pieces of $Y$, still of bounded total size. +\end{claim} + +\begin{proof} +For finite $d$, take an all-dominated object over $X+d$ and +embed it into $Y$ by Proposition~\ref{iso:prime}. Call its +image $U$. Suppose one complementary piece of $U$ is +infinite. Choose a sequence $(a_i:i<\omega)$ from that piece. +It is jointly fresh and hence product-independent over $X$. +The witnessing-chain assertion of +Theorem~\ref{iso:independent-lists} retains this +configuration in a chain +\[ + X=V_0\longrightarrow V_1\longrightarrow\cdots + \longrightarrow V_\omega, +\] +where $a_i$ is free over $V_i$ from $X$ and belongs to +$V_{i+1}$. We may construct this chain inside the actual $U$. At each stage, +finite domination makes the actual next coordinate $a_i$ free over +$V_i$ from $X$, since its products with the preceding coordinates +remain those of the original list. Choose an all-dominated increment +over $V_i+a_i$ and use primeness to embed it into $U$, fixing this +base and this coordinate. At limits take unions inside $U$, using +the common-upper-bound clause. Thus $V_\omega\leq U$, and the tuple +$d$ and every $a_i$ retain their actual positions in $U$. Equivalently, +the witnessing-chain amalgamation may retain all of these data in a +common extension. + +The tuple $d$ is finite, so Proposition~\ref{calc:local-character} +at the limit of +cofinality $\omega$ makes the actual type $\tp(d/V_\omega;U)$ free from +some $V_n$. Hence $d$ and $a_n$ have product over $V_n$. +The base-change rule of Proposition~\ref{calc:products}, +together with the freeness +of $a_n$ over $V_n$ from $X$, gives that $a_n$ and $d$ have +product over $X$. Domination of $U$ over $X+d$ now makes +$\tp(a_n/U)$ free from $X$. This is impossible by +Claim~\ref{sys:free-fresh}: the free lift of its fresh +singleton marginal must be fresh over $U$, whereas +$a_n\in P(U_s)$. Therefore every complementary piece of +$U$ is finite. + +For a bounded list of cardinality $\tau$, include its entries +successively by this finite construction, with the preceding +subextension as base, and take unions at limit stages. +There are finitely many labels. At most $\tau$ finite +increments therefore add at most +$\kappa=\max(\tau,\aleph_0)<\mu$ coordinates in total. +All objects continue to have size $\mu$, and their unions have +total size $\mu$. This proves the bounded version. + +For the final assertion fix the target partition +$H_s^X\mathbin{\dot\cup}C_s^Y$. Starting with a bounded-rank +subextension containing $d$, repeat the following construction +countably many times. Choose relative bases for the current +subextension over $X$. Each gained basis element in vertex +$t$ has a finite support in the fixed target basis, consisting +of old coordinates and coordinates of $C_s^Y$ with +$s\subseteq t$. Request all the latter supporting +coordinates in their own vertices, and include them in the +next subextension using the bounded construction just proved. +At every step at most $\kappa$ coordinates are requested and +the total new rank remains at most $\kappa$. + +In the union let $C_s$ consist of the requested coordinates +of label $s$. These sets are independent, together with the +pieces of $X$, because they are subsets of the target +partition. They span every union vertex over its old pieces: +an element occurs at a stage, is spanned there by the gained +pieces and the old pieces, and the finite target supports of +those gained pieces were requested at the next stage. +Conversely the closed $P$-set of each union vertex contains +all the indicated coordinates below its label. Thus the +span identities hold in both directions. The $C_s$ are +precisely complementary pieces of the union, have total +size at most $\kappa$, and are subsets of the prescribed +$C_s^Y$. +\end{proof} + +\begin{claim}[Primeness over bounded complementary pieces] +\label{sys:bounded-prime} +If $X\to U$ has a bounded list $b$ enumerating all its +complementary pieces, then $U$ is all-dominated over $X+b$. +It embeds over $X$ into any extension realizing a prescribed +matching of these coordinates. +\end{claim} + +\begin{proof} +The list $b$ is product-independent by +Claim~\ref{sys:free-fresh}. Take an all-dominated object over +$X+b$ and embed it into $U$ over these data. Its image $V$ +contains in each vertex the old $P$-set and all the +complementary coordinates with labels below that vertex. +Relative closedness and the span identity imply +$P(V_t)=P(U_t)$ for every $t$. The no-proper-pair conclusion +then gives $V_t=U_t$ for every vertex. Thus $V=U$, proving +the domination assertion. + +A prescribed matching of $b$ into another extension gives +the same whole type over $X$, by +Claim~\ref{sys:free-fresh}. Primeness of the all-dominated +object therefore gives the desired arrow fixing $X$ and +realizing that matching. +\end{proof} + +We finish the second assertion of the theorem. Identify the two +bases with $X$, and keep the prescribed bijections of all +complementary pieces. Choose countable exhaustions of the two +targets by bounded lists; these exist because the eligible level +$\mu$ has cofinality $\omega$ and there are only finitely many +vertex sorts. Construct alternating isomorphisms between +subextensions of the two targets. Initially use $\id_X$. +Maintain the invariant that the complementary pieces in each +current subextension are subsets of the specified target pieces, +and that the current isomorphism agrees with their prescribed +bijections. + +At a forth step, apply Claim~\ref{sys:rank-cost} over the current +subextension on the first side to include the next bounded list +and to use only remaining specified coordinates. Its new +complementary list is bounded. The corresponding coordinates +on the second side are jointly fresh over its current +subextension. Claim~\ref{sys:bounded-prime}, transported by the +current isomorphism, embeds the enlarged subextension into the +second target over that isomorphism and realizes their +prescribed matching. Its image is the next subextension there. +The span identities show that its complementary pieces are +exactly the specified images of the new pieces, so the invariant +is preserved. A back step applies the same construction with +the roles interchanged. + +The union of these countably many coherent isomorphisms covers +both targets, since every list in both exhaustions is eventually +included. It is therefore a system isomorphism over $X$. +It realizes all prescribed basis bijections: every basis element +is eventually in the domain, and the invariant determines its +image. This proves the second assertion, with no lower bound +on complementary cardinalities. + +The induction is complete. At each fixed finite number of axes +there are only finitely many active-label families. Taking the +maximum of their finitely many bounds gives the stated +uniformity. Simultaneous closure through all finite numbers of +axes is addressed separately in +Theorem~\ref{close:common-levels}. +\end{proof} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/12-geometric-boundaries.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/12-geometric-boundaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..56a9b975471ac95204c16a4a3844fc7b50e15b4a --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/12-geometric-boundaries.tex @@ -0,0 +1,514 @@ +\section{Exact isomorphisms of geometric systems} +\label{exact:section-boundaries} + +We continue with the pointed pregeometry of +Theorem~\ref{geom:pregeometry}, above the threshold of +Corollary~\ref{geom:tail-no-growth}. A shape and its active labels have the +meaning of Definition~\ref{sys:shapes}. Unless otherwise stated, every +active basis piece in this section has cardinality equal to the uniform +level~$\mu$. The bases remain auxiliary choices. + +\begin{definition}[A prescribed boundary isomorphism] +\label{exact:boundary-data} +Let $a$ be a finite ordered set of axes, and let +$\mathcal X^-=(X_s^-:s\subseteq a)$ and +$\mathcal X^+=(X_s^+:s\subseteq a)$ be systems of the same shape. +A prescribed boundary isomorphism is a family +\[ + h_s:X_s^-\longrightarrow X_s^+\qquad(s\subsetneq a) +\] +of pointed isomorphisms such that +$h_t\mathbin{\upharpoonright}X_s^-=h_s$ whenever +$s\subseteq t\subsetneq a$. This condition refers to the displayed +inclusions. It does not add an assumption about intersections of two +incomparable vertices inside the top. +\end{definition} + +A list on the boundary can consequently be specified by assigning a +proper face to each of its occurrences. Before consistency on actual +intersections has been proved, its prescribed image means the list +obtained by applying the corresponding $h_s$ to each occurrence. +In particular, the assertion that two occurrences name the same element +is one of the atomic comparisons that must be preserved. + +\begin{theorem}[Exact geometric boundary lifting] +\label{exact:geometric-boundary} +For every finite allowed shape, at all sufficiently closed uniform +levels, every prescribed boundary isomorphism between two systems of +that shape extends to a pointed isomorphism of their tops. The top +isomorphism, together with its prescribed restrictions, is an +isomorphism of the systems. + +The refinements of the eligible levels needed for a fixed shape use +only the one-dimensional bounds for its working scheme, the bounds +for its tiling layouts, and the corresponding assertions for shapes +on fewer axes. In particular these refinements have the uniformity +specified in Section~\ref{close:section-uniform}. +\end{theorem} + +We prove the theorem by induction on the number of axes. At zero axes +it is pointed categoricity. The induction at a positive number of +axes has two parts: preserving the joint diagrams of the entire +boundary, and realizing successive extensions of a partial match +inside the given target. The first part supplies, among other +things, the missing compatibility on actual intersections. + +\begin{lemma}[Joint boundary comparisons] +\label{exact:joint-boundary} +Assume exact geometric boundary lifting for all allowed shapes on +fewer than $n$ axes. At a sufficiently closed uniform level, a +prescribed boundary isomorphism between two $n$-axis systems preserves +the bounded $\K$-diagram of every occurrence-tagged boundary list, +computed in the respective top models. +\end{lemma} + +\begin{proof} +If the top label is inactive, let $u$ be the largest active label. +It is a proper subset of the axis set, and the top vertex is $X_u$. +Every vertex is the vertex at its largest active lower label, which +is contained in $u$. Thus the map at $u$ already gives all the +comparisons in question. We may assume the top label is active. + +Write the axis set as $a\cup\{k\}$, with $k$ last and $|a|=n-1$. +Splitting off $k$ writes the two systems as +\[ + X^-\longrightarrow Y^- ,\qquad X^+\longrightarrow Y^+ +\] +in the scheme $\mathcal G_L$ on $a$. +Its categoricity is supplied by +Theorem~\ref{sys:basis-classification}; after refining the levels, +Proposition~\ref{sys:working-package} supplies its working package. +The arrows add rank $\mu$ at the labels in a fixed upward closed +$Q\subseteq L$ and add no rank at the other labels. Call such a +step a step of shape~$Q$. The boundary prescription gives a full +system isomorphism $e:X^-\simeq X^+$ and compatible isomorphisms +\[ + h_s:Y_s^-\simeq Y_s^+\qquad(s\subsetneq a) +\] +whose restrictions to $X_s^-$ agree with~$e$. + +If joint comparison fails, choose bounded lists $f$ from $X^-$ and +$d^-$ from the proper vertices of $Y^-$ witnessing failure, and put +$d^+=h(d^-)$. All occurrences from the lower layer may be included +in $f$; all the remaining occurrences lie in these upper proper +vertices. The comparisons of +\begin{equation} + (f,d^-)\text{ in }Y^- + \quad\text{and}\quad + (e(f),d^+)\text{ in }Y^+ + \label{exact:opposite-comparisons} +\end{equation} +differ. The notation in \eqref{exact:opposite-comparisons} includes +the indicated vertex occurrences. If there is no upper test $d^-$, +the base isomorphism and the strong arrows already give equality, +so this case cannot witness failure. + +A system isomorphism matching these lists would in particular be a +$\K$-isomorphism of the tops matching them. Hence the working +$\mathcal G_L$-diagrams of the lists in +\eqref{exact:opposite-comparisons} also differ. Choose an infinite +cardinal $\sigma<\mu$ bounding the lengths of the lists. Enlarge a +cardinal $\chi<\mu$ so that $\chi^\sigma=\chi$, it bounds the +relevant labels and parameter-orbit constants, so that +Lemma~\ref{calc:parameter-orbits} gives at most $\chi$ orbits of +$\sigma$-lists over $\chi$ parameters. Choose $\mu$ closed far enough that all +the following tree costs are below it. Let $\epsilon$ be the least +infinite cardinal with $2^\epsilon>\chi$. There are at most $\chi$ +nodes below height $\epsilon$, and $2^\epsilon<\mu$ leaves. + +We construct a binary tree of branch worlds $C_\eta$ in +$\mathcal G_L$, for $\eta\in{}^{\le\epsilon}2$, and a single +increasing reference chain $(R_i:i\le\epsilon)$ in that scheme. +All these objects have level~$\mu$. A branch world carries a copy +$f_\eta$ of~$f$. For every node $\eta$ of length $i$ the invariant +also includes a prescribed isomorphism +\begin{equation} + b_\eta:\partial C_\eta\simeq\partial R_i. + \label{exact:reference-invariant} +\end{equation} +Along a branch, these boundary maps commute with the inclusions into +later branch and reference worlds. At each splitting node, the two +new copies of its upper test $d$ have the same image in the next +reference boundary. Those images will be retained at all later +stages. + +Start with $C_\varnothing=R_0=X^-$ and the identity boundary map. +Suppose the objects at level $i$ have been constructed. Choose a +reference step $R_i\to R_{i+1}$ of shape~$Q$. Such a step exists: +categoricity of $\mathcal G_L$ identifies $R_i$ with $X^-$, and we +transport the given extension $X^-\to Y^-$ along that +identification. + +For each $\eta\in{}^i2$, an isomorphism $X^-\simeq C_\eta$ can +be chosen to take $f$ to $f_\eta$. Indeed arrows along the branch +preserve its bounded diagram, and equality of bounded diagrams in +the working scheme is implemented by an isomorphism. Transport +the minus and plus extensions along this return isomorphism, using +$e$ on the plus base. They give children $C_{\eta^\frown 0}$ and +$C_{\eta^\frown 1}$ over $C_\eta$, with opposite comparisons to +$f_\eta$ and with the given identification between their proper +upper vertices. On their lower proper vertices that identification +is the identity of the current branch world. + +It remains to extend \eqref{exact:reference-invariant} to the +children; merely choosing arbitrary isomorphisms of the new worlds +would not suffice. First do this for the plus child. Process all +proper subsets $s\subsetneq a$ in increasing cardinality. Consider +the restriction of the plus step to the axes $s\cup\{k\}$ and the +corresponding restriction of $R_i\to R_{i+1}$. On the base +vertex, the prescribed map is $(b_\eta)_s$. On every other proper +vertex it is either a restriction of that map or one of the upper +maps already constructed at a proper subset of~$s$. These maps +commute on all indicated subfaces by construction. The restriction +has at most $n-1$ axes, so the induction hypothesis lifts this +boundary prescription onto its upper vertex. This supplies the +map at $s$, whether that vertex is active or repeated from a +smaller active one. The resulting family is the required plus +boundary map. Compose it with the given minus-to-plus boundary +identification to obtain the minus boundary map. Both extend +$b_\eta$, and the two copies of the test have equal reference +images. + +At a limit $i\le\epsilon$, take unions along each branch and in +the reference chain. The lower-dimensional working scheme has +the chain and common-upper-bound properties, and these unions have +size $\mu$. The coherent boundary maps have unions which are +onto the reference boundary: each previous reference vertex was +already the image of the corresponding branch vertex. Thus the +invariant persists. No assertion that a limit inclusion is itself +a step of shape~$Q$ is needed. At its next successor stage a fresh +step of that shape is available over the union by categoricity. + +At a leaf, apply the induction hypothesis once more, this time to +the $(n-1)$-axis worlds $C_\eta$ and $R_\epsilon$, with the +boundary prescription $b_\eta$. Obtain a full isomorphism +$j_\eta:C_\eta\simeq R_\epsilon$. Let $A$ be the list of all +reference images of the splitting tests. Its length is at most +$\chi$, since there were at most $\chi$ splitting nodes and each +test had length at most~$\sigma$. + +For two different leaves, use their first splitting node. The +associated test has the same image in $A$ on both branches, whereas +its comparison with the images of $f$ is opposite. Arrow +preservation of diagrams carries this distinction to the leaves, +and $j_\eta$ preserves it in $R_\epsilon$. Consequently the +$2^\epsilon$ lists $j_\eta(f_\eta)$ lie in different orbits over +$A$. This contradicts the bound~$\chi$. Joint boundary +comparison follows. +\end{proof} + +\begin{remark} +The proof just given applies to equalities between occurrences from +incomparable faces. It therefore proves that the prescribed maps +have a well-defined injective union on the actual union of the +boundary vertices. Applying the same observation to the inverse +family proves that this union is a bijection between the two +boundary unions. Equality of actual intersections was not an +input to the argument. +\end{remark} + +The boundary prescription now respects actual intersections and all +bounded joint diagrams. The remaining task is to extend a bounded match +inside the prescribed target. We first establish testing for its tiled +boundary layout, then use a strip avoided by all boundary supports to +realize the resulting comparison without enlarging that target. + +\begin{lemma}[Testing on rationally tiled parameter orders] +\label{exact:tiled-testing} +Fix the finite-support order template $E$ and its natural stabilized +$\K$-diagrams. Fix a nonempty set-sized initial order $J_0$, finitely many tile +labels $i\chi$. These operations cover +the chain lengths and cardinal costs in the domination and return-tree +arguments. + +\paragraph{Placed comparison and syntax tests.} +For a specified permitted layout and tuple arity, the comparison request +asks for a bound on the size of a parameter test witnessing inequality +between two placed comparisons. More explicitly, quantify over an +eligible level, its indicated order layout, the two candidate term lists +in order enlargements, and the parameters generated by the designated +parts. If their comparisons differ on some bounded parameter list, +require that they differ on one of size at most the proposed bound. +Inequality is evaluated in a larger eligible padded presentation, +using isomorphism to a well-ordered chart, or equivalently by its $D_F$ +comparison where the chart clause has been obtained. All the objects +and positions involved in a particular test are sets. Universal +quantification over the level cardinal is an ordinary first-order +quantifier. + +The permitted layouts are the whole-model layout, designated vertex +parts, and finitely segmented or tiled layouts with the prescribed +order and colors. Their codes belong to the descriptor set. The +ordinary and rationally tiled ordinal-vector arguments of +\Cref{calc:testing,exact:tiled-testing} prove that this request has a +uniform witness. The syntax request asks for a test size at which the fixed +$D_F$ comparison on patterns of a specified size is already determined. +The set of such patterns and the set-structure back-and-forth +construction in \Cref{lev:diagram-coding} prove existence. Neither +request mentions a truth predicate for $\mathcal C_n$. + +For the rationally tiled layouts $\sum_i(H_i\times\mathbb Q)$ used +in the geometric application, the comparison request includes the +bounded frozen data of Lemma~\ref{exact:tiled-testing}. +That lemma supplies its witness also for candidates in arbitrary +enlargements of the underlying $E$-order. The exceptional fibers, +their frozen positions, and the set of cut coordinates have bounds +in the input arities and fixed descriptor; no new dependence on the +level or the finite depth is introduced. + +\paragraph{Approximation, compactification, and isolation bounds.} +The approximation and compactification constructions take a comparison +bound and a profile bound at the specified arity, increase the lengths +to dominate the exception counts, and perform the bounded power-set +iteration just described. Thus their sizes are fixed compositions of +the first two requests and cardinal operations. The isolating-fragment +bound is the size of the initial supports, the bounded initial parts of +the candidate blocks, and the terms on those supports in the proof of +\Cref{iso:alternative}. These again use the same comparison and +cardinal requests. The construction of all-dominated objects uses +larger input ordinals in these requests, mixed LS, and the indicated +successor-length recursions; it introduces no new form of predicate +depending on the marked depth. The existence assertions used here are +\Cref{iso:finite-existence,iso:all-lengths,iso:independent-lists}, with the mixed interface just verified. + +\paragraph{Eventual orbit thresholds.} +The remaining request asks for the first threshold above which a +specified reversal, orbit-preservation, or stabilization assertion holds +at the eligible levels and the indicated arity. The assertion itself +uses the standardized object sets, arrows, and orbit relation defined +above. Thus the threshold test has the form +\[ + \forall\mu\bigl(S(\mu)\land\mu>\beta + \ \Longrightarrow\ R(d,\alpha,\mu)\bigr), +\] +or its two-level version, with $R$ one of these fixed orbit assertions. +The reversal and stabilization arguments prove the existence of the +threshold when the current categoricity and the previous required +conditions hold on the unbounded eligible class. The use for geometric +shapes has exactly those hypotheses by +\Cref{sys:working-package}. Passing from these thresholds to +diagram homogeneity also requires smaller eligible levels cofinal +below the final level; that requirement is recorded separately as a +limit-point refinement. + +We check that this list accounts for a whole generic step. In a marked +step, extraction uses partition bounds, the underlying comparison and +isolation bounds, and arbitrarily large samples. It chooses a template +from the set established in +Proposition~\ref{close:uniform-definitions}; see +\Cref{mark:template}. After that choice, the two-part comparison of +\Cref{mark:two-part-testing}, used in \Cref{mark:chain-closure}, +is the placed comparison request with +this template as parameter. The downward isolation argument requires +the same bounds and sufficiently closed smaller levels. Orbit +comparison and saturation use the transport argument; the passage to +the unary chart clause uses the syntax request. Projection, completion, +and exact boundary lifting then use the ordinary comparison requests, +the prior unary conditions, and the tree costs; exact lifting may require +one further supply of cofinal smaller levels. These are respectively +the constructions of \Cref{iter:projection-types,iter:boundary-completion,iter:exact-boundary}. + +For geometry, first use the current categoricity condition and prior +basis conditions as unary inputs to the eventual orbit request, then +require smaller eligible levels and the chart syntax bounds. Basis +lifting and exact boundary lifting use the same one-dimensional and +tiled-layout requests and tree costs, by +\Cref{sys:basis-classification,exact:geometric-boundary}. A simultaneous projection calculation +or a maximum over finitely many lower descriptors is expressed by +bounded quantification over that finite descriptor. It does not require +successively inserting their formulas into one another. + +Within each of these generic proofs there are finitely many successive +uses of closure and of cofinal smaller levels. Label those positions +in the proof in their order. The predicate for a position is the initial +adequacy predicate conjoined with the earlier stated threshold, +closure, and limit-point conditions. Substituting these descriptions +at the finitely many positions gives a fixed formula for each request. +The number of positions is attached to the generic proof, not to the +finite axis count; finite lower-scheme lists are handled by the bounded +quantification just described. This produces the asserted finite +register with $S$ eliminated in favor of its uniform defining formula. + +\paragraph{The eligible predicates in a marked step.} +The following substitution order also previews the common-level induction +at the end of this section. References here to retaining $G_j$ use the +\hyperref[lev:closure-preview]{advance notation in Section~\ref*{lev:section-working-schemes}}; +the actual classes and +their absorption property are constructed below, after the register is +fixed. They supply levels for that induction, not parameters for the +requests being defined here. + +Let $S_0$ be the +unary conjunction of the full individual working conditions for the +input scheme and the projection and stationary-prescription conditions +already established for the finitely many lower descriptors. The mixed +interface supplies their mixed working conditions. First refine $S_0$ +by the testing, approximation, isolation, and domination bounds of +Sections~\ref{calc:section-calculus} and~\ref{iso:section-isolation}; +call this class $S_1$. On its sufficiently closed members the actual +constructions give the samples used in Lemma~\ref{mark:template}. +The induction maintains that $S_1$ is unbounded by retaining a $G_j$; +unboundedness of an arbitrary class alone would not justify this step. + +After selecting $F^b$, let $R(\mu)$ be the conjunction of $S_1(\mu)$, +the individual presentation flag for $F^b$, and the natural-interpretation +flags for its base and upper values in the old scheme. Evaluate the +testing request for the layout $U_H\cup A_{H'}$ on this preliminary +class $R$, using the old scheme's $D_F$. Lemma~\ref{mark:template} +supplies the individual validity and natural interpretations; +Lemma~\ref{mark:two-part-testing} reduces the comparison to the old +working scheme, where the ordinal-vector argument of +Proposition~\ref{calc:testing} supplies its uniform bound. + +Now refine $R$ by that testing bound and its placement and arity costs; +call the resulting class $S_2$. Thus $S_2$ is the output of this +refinement, not the eligible predicate in its own defining request. +The preliminary predicate $R$ uses only the earlier costs and the two +specified kinds of flags. Neither the request nor its witness uses +a $D_{F^b}$ chart clause or chain closure for the pair scheme. + +At members of $S_2\cap\Lim(S_2)$ closed under the same arity costs, +Lemma~\ref{mark:chain-closure} proves pair-chain closure, including +bounded tuples spanning the chain. The actual pre-closure orbit +argument and this chain assertion then give +Theorem~\ref{mark:package}; only now impose the syntax bound for +$D_{F^b}$ and request its chart clause. Projection correspondence and +amalgamated boundary completion follow after the fixed comparison and +tree-cost refinements in Section~\ref{iter:section-boundaries}. +The fixed-parameter boundary argument and exact lifting may add one +further limit-point refinement to obtain smaller levels where +completion already holds. Each displayed eligible predicate is thus +a finite conjunction of unary flags, previously listed bound closures, +and limit-point conditions. Substitution follows this displayed order; +no predicate for a previously chosen $G_j$ or closure function is +inserted into a bound request. More generally the auxiliary subclass +$T$ in the eligible-level convention is used only to ensure an +unbounded supply of levels through these finite refinements. It is +not an additional parameter of any request. In the present induction +$T=G_j$ supplies this existence assertion, while each request still +uses the unary adequacy predicate and the finitely expanded refinements +just displayed. + +\paragraph{The eligible predicates in a geometric step.} +The first assertion of Theorem~\ref{sys:basis-classification} for the +current shape follows from the previously proved assertions for its +finitely many lower shapes. Start with the unary conjunction of this +current categoricity assertion and the needed lower-shape assertions. +The structural and tiled-presentation proofs give the actual arrows, +coherence, union conditions, and mixed containment used by +Proposition~\ref{sys:working-package}. Its finite-tile alignment and +orbit calculation prove eventual reversal on this class. The first +route in Proposition~\ref{lev:scheme-diagram-routes} then proves +stabilization; impose those arity bounds and a limit-point refinement +within the current eligible class. Actual diagram homogeneity is +proved at the retained members before the $D_F$ syntax bound is imposed. + +The four subsequent claims in the proof of +Theorem~\ref{sys:basis-classification} use this established package, +the preceding one-dimensional bounds, and finite tree costs. They +prove the relative-basis assertion at the retained levels. +For exact boundary lifting, include the already proved exact assertions +for all proper restricted shapes in the initial unary conjunction. +The tree and unoccupied-strip proofs of +Theorem~\ref{exact:geometric-boundary} add only the indicated layout, +support-closure, and tree-cost refinements. Increasing the number of +axes changes the finite list of lower descriptors and the ordinal +size inputs; it does not add a new kind of eligible predicate or a new +position in this generic substitution order. + +Finally, the witness assertions are precisely the uniform bounds proved +in the cited constructions with their stated hypotheses. An irrelevant +descriptor, or a stage whose hypotheses fail, may give a request with +no witness. Its value $0$ makes a total definable ordinal function and +has no mathematical force. At a used stage we invoke the preceding +construction to establish existence before applying the bound. In +particular, totalization supplies neither an adequate template nor a +missing lifting or testing witness. +\end{proof} + +\subsection{The common majorant and its limit refinements} + +We now construct the hierarchy used in the preceding previews. The +register and its finite substitutions have been fixed independently of +this construction; the common majorant bounds their values, while the +limit refinements supply smaller eligible levels cofinally below a new +level. + +Choose a cardinal $\vartheta$ above the hereditary sizes of the normalized +descriptor set and all fixed parameters. This is one cutoff, chosen +before any finite-depth induction; it includes all candidate templates, +not just those already selected. For $\alpha\in\Ord$, take the supremum +of the values of all the functions in the expanded finite register, +for every descriptor and for all ordinal inputs at most $\alpha$. +Finite lists of size inputs are included by taking all their coordinates +at most $\alpha$. Each such collection is a set, and each request has +been totalized, so Replacement bounds its values. We may therefore +choose a fixed, monotone definable ordinal function $\Phi$ such that +\[ + \Phi(\alpha)>\vartheta,\qquad \Phi(\alpha)>\alpha, + \qquad B_q(d,\vec\beta)<\Phi(\alpha) + \quad(\max\vec\beta\le\alpha). +\] +Include the initial single-model requirements in this finite register. +The formulas defining its members have already been fixed. No partial +truth construction, and no bound on a variable formula's complexity, +is being used here. + +For a definable class $A$ of ordinals write +\[ + \Lim(A)=\{\lambda:\sup(A\cap\lambda)=\lambda>0\}, + \qquad + \operatorname{Cl}(f)= + \{\lambda:\forall\alpha<\lambda\ f(\alpha)<\lambda\}. +\] +Multiple ordinal inputs can be replaced by their maximum in the closure +condition. Define +\[ +\begin{split} +G_0&=\{\mu:\mu\text{ is a strong limit cardinal}, + \cf(\mu)=\omega,\ \mu\in\operatorname{Cl}(\Phi)\},\\ +G_{j+1}&=G_j\cap\Lim(G_j)\qquad(j<\omega),\qquad +G_\omega=\bigcap_{j<\omega}G_j. +\end{split} +\] +Because $\Phi(0)>\vartheta$, every member of every $G_j$ exceeds the +common parameter cutoff. This detail cannot be replaced by discarding +an arbitrary initial segment after each new template choice. + +The hierarchy itself has a uniform definition. For an ordinal $\lambda$, +form $A_0=G_0\cap(\lambda+1)$ by Separation and then, by finite recursion +on subsets of $\lambda+1$, put +\[ + A_{i+1}=\{\beta\in A_i:\beta>0\text{ and } + \sup(A_i\cap\beta)=\beta\}. +\] +The assertion $\lambda\in A_j$ is one formula with $j,\lambda$ as inputs, +and is equivalent to $\lambda\in G_j$. Quantifying over $j\in\omega$ +defines $G_\omega$. In particular the countable constructions below +can choose least witnesses using this formula and Replacement. + +\begin{lemma}[Closure and absorption] +\label{close:absorption} +Each $G_j$ is unbounded and closed under suprema of strictly increasing +countable sequences of its members. The class $G_\omega$ is unbounded. +If a current eligible class $S$ contains $G_j$, and a finite sequence of +refinements consists of the bound closures and thresholds in the fixed +register and restrictions to limit points of earlier eligible classes, +then the resulting class contains $G_{j+m}$ for some finite $m$. +Here every used bound must have a witness under its stated hypotheses. +\end{lemma} + +\begin{proof} +Given any starting ordinal, choose strictly increasing infinite cardinals +$(\kappa_n)_{n<\omega}$ above it with +\[ + \kappa_{n+1}> + \max\bigl(2^{\kappa_n}, + \sup\{\Phi(\alpha):\alpha<\kappa_n\}\bigr). +\] +The supremum $\kappa$ of this sequence is a cardinal of cofinality +$\omega$. For every cardinal $\lambda<\kappa$, some $\kappa_n$ exceeds +$\lambda$, and then $2^\lambda\le2^{\kappa_n}<\kappa$. For every +$\alpha<\kappa$, the same choice gives +$\Phi(\alpha)<\kappa_{n+1}<\kappa$. Thus $\kappa\in G_0$. +If instead the $\kappa_n$ are an increasing sequence in $G_0$, their +supremum is again a countable-cofinality strong limit cardinal and is +strictly $\Phi$-closed by these same two tests. This proves the two +assertions for $G_0$. + +Suppose they hold for $G_j$. Above any starting ordinal choose a strictly +increasing countable sequence in $G_j$. Its supremum belongs to $G_j$ +by closure and to $\Lim(G_j)$ because the sequence is cofinal there. +This proves unboundedness of $G_{j+1}$. A countable increasing sequence +in $G_{j+1}$ has its supremum in $G_j$, and its own members witness that +this supremum belongs to $\Lim(G_j)$. Hence $G_{j+1}$ has the asserted +closure as well. + +Choose $\lambda_n\in G_n$, strictly increasing and above an arbitrary +prescribed ordinal. For each fixed $j$, its tail lies in $G_j$ because +the classes are nested. The supremum belongs to $G_j$ by its countable +closure. It consequently belongs to $G_\omega$, proving unboundedness. + +For absorption, maintain that the current eligible class contains a +particular $G_k$. A request value on inputs below a member $\mu\in G_k$ +is less than $\Phi(\alpha)$ for a suitable $\alpha<\mu$, and hence less +than $\mu$. Thus a bound-closure refinement retains all of $G_k$. +A constant threshold is included as a request with input $0$ and has +the same property. If the next refinement demands cofinally many +members of the current class below the new level, retain $G_{k+1}$: +its members are limit points of $G_k$ and thus of the current class. +If several earlier eligible classes are mentioned, the maintained +index can first be increased to the maximum of their finitely many +indices. Induction through the finite refinement list proves the +claim, with $m$ bounded by its number of limit-point refinements. +\end{proof} + +\begin{theorem}[Common levels for all finite depths] +\label{close:common-levels} +With the fixed data of either application and the constructions proved +in the preceding sections, there is an unbounded class of +countable-cofinality strong limit cardinals at which all its finite-depth +conclusions hold simultaneously. In the marked application this +includes the working packages, projection and type correspondences, +amalgamated completion, and exact boundary lifting for every +$\mathcal C_n$, uniformly over the root prescription codes realized +at arbitrarily large eligible single-model levels. +In the geometric application it includes categoricity, the working +packages, basis lifting, and exact boundary lifting for every finite +shape. One such common level may be chosen above any specified set of +fixed cardinal thresholds. +\end{theorem} + +\begin{proof} +Use the fixed register and $\Phi$ just constructed. The initial +single-model results hold on $G_0$ after the initial requirements have +been included. Suppose the assertions required as input to a finite +induction step hold throughout $G_j$. Take $S$ to be the class defined +by their unary adequacy formula, with the current descriptor. It +contains $G_j$. Lemma~\ref{close:mixed-interface} supplies the mixed +conditions for that step. Lemma~\ref{close:recipe-register} identifies +its bound requests and gives the existence of each relevant witness +at its place in the proof. Lemma~\ref{close:absorption} then retains +some $G_{j'}$, with $j'<\omega$, through all its refinements. + +When a marked step extracts a new template, its code lies in the fixed +descriptor set by Proposition~\ref{close:uniform-definitions}. The +remaining requests with that code as parameter were already majorized +by $\Phi$. Thus making the choice introduces no new closure function +or cutoff. For a given finite shape, the finitely many previously +required lower shapes can first be brought to one $G_j$ by taking the +largest of their indices. The same observation applies to the finite +projection lists in the marked argument. The two ordinary finite +inductions therefore give, for each finite descriptor, all its required +conclusions on $G_j$ for some finite $j$. + +In the marked application only root codes consistent at arbitrarily +large eligible single-model levels are used. A code obtained from an +actual pair at a sufficiently closed starting cardinal has that property: +freely lift its type with the named approximation data to arbitrarily +large single-model levels. The majority rule is preserved there, and +restriction from such a level gives the same stationary prescription +at every sufficiently closed smaller single-model level with that +named pattern. This is the prescription argument in +Lemma~\ref{geom:root-prescription}, prior to the use of the finite marked +systems. It requires no persistence of a no-growth statement about +$P$ at larger levels. The fixed code set and majorant include every +such possible input code, so the common-level construction can precede +the choice of the actual pair. + +Every member of $G_\omega$ belongs to $G_j$ for every finite $j$. It consequently +satisfies the conclusion for every finite descriptor used in the +application, although the index $j$ and the selected template may depend +on that descriptor. Lemma~\ref{close:absorption} makes $G_\omega$ +unbounded, so choose a member above the desired thresholds. All +constructions in this argument are ordinary set recursions, Separation, +Replacement, and finite induction with uniform formulas. This completes +the simultaneous closure justification in ZFC. +\end{proof} diff --git a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/14-cardinal-transfer.tex b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/14-cardinal-transfer.tex new file mode 100644 index 0000000000000000000000000000000000000000..7a8f4925a781ec74b8fc05e232d21e7c7c2b04d1 --- /dev/null +++ b/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/build/sections/14-cardinal-transfer.tex @@ -0,0 +1,432 @@ +\section{Transfer to all larger cardinals} +\label{tail:section-transfer} + +The common levels of Theorem~\ref{close:common-levels} provide one +cardinal at which all finite geometric shapes have exact boundary +lifting. This section uses that single cardinal as the base of an +induction through the cardinals. The levels supplied by the closure +construction themselves need not contain a tail. + +The trade between one extra dimension and an increase in cardinality +goes back to Shelah's multidimensional existence and uniqueness methods +\cite[Theorems~5.1--5.2]{Shelah1983B}; compare their later formulation in +\cite[Lemma~11.6]{ShelahVasey2024}. The argument below +is given for the exact boundary maps and cardinal assignments used here. + +Fix a common uniform level $\gamma$ from +Theorem~\ref{close:common-levels}, for all the geometric shapes and +their exact boundary assertions. Choose it strictly above the +no-growth threshold $\lambda$ of +Corollary~\ref{geom:tail-no-growth}, and above all fixed point and +single-model thresholds. In particular $\gamma>\lambda_0$, the +cardinality of the distinguished model~$M_0$. In every pointed +model of cardinality at least $\gamma$, model cardinality equals +the cardinality of a basis of its point geometry. + +\begin{definition}[Prescribed piece cardinalities] +\label{tail:piece-cardinalities} +Let $L$ be an allowed family on a finite axis set $a$. A cardinal +assignment on $L$ is a family of cardinals +\[ + (\kappa_s:s\in L),\qquad + \gamma\leq\kappa_s\leq\kappa_t\quad(s\subseteq t\text{ in }L). +\] +A system with this assignment satisfies the span and inclusion +conditions of Definition~\ref{sys:shapes}, with a witnessing basis +partition $(H_s:s\in L)$ such that $|H_s|=\kappa_s$. +The bases are not named. A prescribed boundary isomorphism has +the meaning of Definition~\ref{exact:boundary-data}. +\end{definition} + +For $t\subseteq a$ let $a_L(t)$ be its largest active subset. +The span identities and monotonicity imply +\begin{equation} + \rk P(X_t)=\max\{\kappa_s:s\in L,\ s\subseteq t\} + =\kappa_{a_L(t)}. + \label{tail:vertex-size} +\end{equation} +All these ranks are at least $\gamma$, so +Corollary~\ref{geom:tail-no-growth} shows that the same formula +gives $\|X_t\|$. It also shows that an inactive vertex equals +$X_{a_L(t)}$. Lemma~\ref{sys:relative-bases} applies without a +uniform-cardinality requirement: its proof only uses the finite +shape and the basis axioms of the pregeometry. + +\begin{lemma}[Choosing subcubes with prescribed supports] +\label{tail:support-subcubes} +Suppose two systems $\mathcal X^-$ and $\mathcal X^+$ have the +same cardinal assignment, and prescribed isomorphisms $h_t$ on +their proper faces. Suppose their top is active and put +\[ + \kappa=\max_{s\in L}\kappa_s,\qquad + Q=\{s\in L:\kappa_s=\kappa\}. +\] +One can choose witnessing global basis partitions +$(H_s^-)$ and $(H_s^+)$ such that +\begin{equation} + h_s[H_s^-]=H_s^+\qquad(s\in L,\ s\subsetneq a). + \label{tail:proper-basis-choice} +\end{equation} +Fix such partitions. Let $\chi$ be a cardinal with +$\gamma\leq\chi<\kappa$ which bounds all $\kappa_s<\kappa$. +Given at most $\chi$ requested elements in the finitely many +vertices on both sides, there are subsystems $\mathcal U^\pm$ +inside $\mathcal X^\pm$ with the following properties: +\begin{enumerate} +\item Each requested element belongs to its indicated vertex. + Every vertex whose size is strictly below $\kappa$ is + retained in full. +\item Their witnessing pieces are subsets + $I_s^\pm\subseteq H_s^\pm$. They equal $H_s^\pm$ when + $s\notin Q$, and have cardinality $\chi$ when $s\in Q$. +\item On every proper face, + $h_t[U_t^-]=U_t^+$. +\end{enumerate} +If compatible subsystems have already been chosen whose witnessing +pieces are subsets of these fixed partitions $(H_s^\pm)$, and whose +total vertex sizes are at most $\chi$, their entire vertices can be +included among the requests. Moreover, at every $s\in Q$ one can +require exactly $\chi$ additional basis coordinates beyond those +old selected pieces, on both sides. All resulting inclusions +preserve the relative independence condition for system arrows. +\end{lemma} + +\begin{proof} +Choose the pieces on the minus side. For each proper active $s$, +the isomorphism $h_s$ takes its lower span onto the corresponding +lower span on the plus side, since it agrees with the prescribed +maps on every lower vertex. Thus $h_s[H_s^-]$ is a relative basis +at~$s$. By Lemma~\ref{sys:relative-bases}, these relative bases +assemble in the plus system. Complete them by a relative basis +at its top. Their cardinalities have the required values because +relative rank is invariant under isomorphism. This gives the +claimed partitions. + +We construct the desired subsystems by a countable closure +process, on both sides and at all the finitely many vertices. +Begin with the requested sets, the fixed point, and any specified +old subsystems. At every growing active label $s\in Q$, also +request $\chi$ basis coordinates. In the extension version +choose these outside the old selected piece. There are enough +coordinates since $|H_s^\pm|=\kappa$ and the old piece has size +at most $\chi<\kappa$. At a proper label the choices may first +be made on the minus side and transported by $h_s$; these are +still fresh on the plus side. At the top make the choices on +both sides separately. Retain all nongrowing vertices in full. + +At each finite stage choose, by the L\"owenheim--Skolem axiom, +strong submodels of the ambient vertices containing all requests +so far. Their sizes are at most $\chi$: the LS bound, point, +retained smaller vertices, and requested data all fit within that +cardinal. At the next stage make the following further requests. +\begin{enumerate} +\item For each inclusion of labels $s\subseteq t$, put all + elements of the currently chosen model at $s$ into the + requests at~$t$. +\item For every point obtained in a chosen vertex, choose a finite + support for it from the union of the global basis pieces + below that vertex. Request each supporting coordinate in + the vertex of its own active label. +\item At each proper face, request the image under $h_t$ of every + element currently included on the minus side, and the + inverse image of every element currently included on the + plus side. +\end{enumerate} +Also retain all previous requests and chosen models. These +operations make at most $\chi$ requests at each stage. Each +ambient point set has the given basis partition, so the supports +in the second operation exist and lie in the indicated vertices. +For a retained smaller vertex all supporting labels are retained +as well: if $s\subseteq t$ and $\kappa_t<\kappa$, monotonicity +gives $\kappa_s<\kappa$. Thus no operation requires enlarging a +retained vertex beyond its given ambient model. + +Take the unions of the chosen models through the countable +process. Coherence makes all successive and vertex inclusions +strong, and the AEC chain axioms give strong limit models +$U_t^\pm\leq_\K X_t^\pm$. On a proper face the image and +inverse-image requests give exactly $h_t[U_t^-]=U_t^+$. +This construction used the maps one face at a time; it did not +assume that their union on possible extra intersections was +already a function. + +Put $I_s^\pm=H_s^\pm\cap P(U_s^\pm)$. Every point of +$P(U_t^\pm)$ occurs in a finite-stage model, and its supporting +coordinates were subsequently placed at their own labels. +Conversely, the point set of $U_t^\pm$ is relatively closed in +the ambient model and contains all selected coordinates below +$t$. Therefore +\[ + P(U_t^\pm)=\cl\left(\bigcup_{s\in L,\ s\subseteq t} + I_s^\pm\right). +\] +The selected pieces are subsets of the single global independent +basis and hence are independent together. They supply precisely +the required decomposition. Nongrowing pieces and vertices were +retained in full. At a growing label at least $\chi$ coordinates +were requested and its model has size at most $\chi$, so its +piece has size exactly $\chi$. In the extension version, at +least $\chi$ fresh coordinates were requested; at most $\chi$ +coordinates altogether were added. The additional piece +therefore has cardinality exactly~$\chi$. + +Finally, the selected subsets increase when old subsystems are +included. Being subsets of the same global basis proves the +relative independence requirement for their inclusions, using +the criterion of Lemma~\ref{sys:relative-bases}. +\end{proof} + +The subcube lemma makes compatible smaller systems available, but an +arbitrary isomorphism between the next two subcubes need not extend the +previous one. The cardinal induction resolves this by adding an axis: +the old full isomorphism becomes part of a boundary prescription at a +strictly smaller maximum cardinal. This is why the induction must cover +all finite shapes simultaneously. + +\begin{theorem}[Exact lifting at every larger cardinal] +\label{tail:all-cardinals} +For every finite allowed shape and every monotone assignment +$(\kappa_s:s\in L)$ of cardinals at least $\gamma$, any prescribed +boundary isomorphism between two systems with that assignment +extends to an isomorphism of their tops. +\end{theorem} + +\begin{proof} +We induct on the cardinal +$\kappa=\max_{s\in L}\kappa_s$, proving the assertion +simultaneously for all finite shapes and all assignments with +that maximum. For $\kappa=\gamma$ all pieces have size +$\gamma$, so Theorem~\ref{exact:geometric-boundary} applies at +the chosen common level. Suppose $\kappa>\gamma$ and the +assertion is known at every smaller maximum. + +As before, an inactive top equals a proper active vertex and its +prescribed map already gives the conclusion. Assume the top is +active. Keep all smaller pieces and their vertices in full. +There are only finitely many of them, so their cardinalities +have a common bound below~$\kappa$. Choose the global basis +partitions of Lemma~\ref{tail:support-subcubes}. + +Let $\delta=\cf(\kappa)$. Choose an increasing sequence +$(\alpha_i:i<\delta)$ of ordinals below $\kappa$, cofinal in +$\kappa$, and fix enumerations of every growing vertex on each +side by~$\kappa$. We construct continuous chains of subsystems +\[ + (\mathcal U_i^-:i<\delta),\qquad + (\mathcal U_i^+:i<\delta), +\] +and compatible full system isomorphisms +$f_i:\mathcal U_i^-\simeq\mathcal U_i^+$. +Every proper-face restriction of $f_i$ is the given $h_t$ on +that subface. Their pieces are increasing subsets of the +chosen global partitions. Nongrowing vertices are retained +in full; all growing pieces at stage $i$ have the same +cardinality $\rho_i<\kappa$, at least~$\gamma$. + +For the initial stage, choose $\chi<\kappa$ bounding $\gamma$, +the retained pieces, and the sizes of the requested initial +enumeration segments. Apply Lemma~\ref{tail:support-subcubes} +on both sides. This gives systems with the same piece sizes +and maximum below $\kappa$, whose proper-face maps are the +restrictions of the given prescription. The induction +hypothesis at that maximum supplies~$f_0$. + +Suppose $\mathcal U_i^\pm$ and $f_i$ are given. Choose a +cardinal $\chi<\kappa$ which bounds $\gamma$, all retained +piece sizes, all old vertex sizes, and the next requested +enumeration segments. The size checks justifying this choice +at every proper stage are given below. Apply the extension +part of Lemma~\ref{tail:support-subcubes}. It produces +$\mathcal U_{i+1}^\pm$ containing the old systems and the +requested segments, with exactly $\chi$ new independent +coordinates at every label in +\[ + Q=\{s\in L:\kappa_s=\kappa\}, +\] +and none at the other labels. The set $Q$ is upward closed: +if $s\in Q$ and $s\subseteq t\in L$, then +$\kappa\leq\kappa_t\leq\kappa$, so $t\in Q$. + +Add a new last axis $k$. Regard +$\mathcal U_i^\pm\to\mathcal U_{i+1}^\pm$ as the two layers +of systems with active labels +\[ + L'=L\cup\{s\cup\{k\}:s\in Q\}. +\] +The old pieces have their previous cardinalities and the +additional pieces all have cardinality~$\chi$. These are +valid decompositions: the old and added coordinate sets are +disjoint subsets of the fixed global basis, and their unions +span the new vertices. The cardinal assignment on $L'$ is +monotone. Comparisons among old labels use the old +monotonicity, comparisons among new labels use equality to +$\chi$, and an old label below a new one has size at most +$\chi$ by its choice. No new label is contained in an old +one. All assigned cardinals are at least $\gamma$ and the +maximum is still strictly below~$\kappa$. + +Prescribe the boundary maps of these enlarged shapes by $f_i$ +on the entire old cube and by the given $h_t$ on every enlarged +proper face of the new cube. They commute on all indicated +subfaces: on an old proper face the two prescriptions already +agree by the invariant for~$f_i$. The cardinal induction +hypothesis applies to this shape as well, since it is +simultaneous in the number of axes and its maximum decreased. +It yields an isomorphism of the new tops extending $f_i$ and +all the enlarged proper-face maps. This is the required +$f_{i+1}$. + +At a nonzero limit $i<\delta$, take componentwise unions and +put $f_i=\bigcup_{j=Latex,font=\small] + \begin{scope} + \clip (0,0) ellipse (4.1 and 1.55); + \fill[black!4] (-4.2,-1.6) rectangle (4.2,0); + \fill[black!9] (-4.2,0) rectangle (4.2,1.6); + \end{scope} + \draw (0,0) ellipse (4.1 and 1.55); + \draw[thick] (-4.1,0) -- (4.1,0); + \node[fill=white,inner sep=3pt] at (0,0) + {$F=Q\cap\{\operatorname{Im}z_1=0\}$}; + \node at (0,0.82) {$G$}; + \node at (0,-0.82) {$f^\sharp$ on $Q_-$}; + \node at (-3.6,1.38) {$Q$}; + \draw[->] (4.75,-1.35) -- (4.75,1.35); + \node[above] at (4.75,1.35) {$\operatorname{Im}z_1$}; +\end{tikzpicture} +\caption{A schematic real section of the single-face enlargement; +passive variables are suppressed. The lower region carries $f^\sharp$ +and the upper region carries $G$. The input maps are compared only on +a slightly enlarged real face $F$, with no prescribed complex collar +and no bound for $G$ away from the interface.} +\label{fig:real-interface} +\end{figure} diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/paper.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..646d6d3e12d2bc4346b9d9830fa95bc21c2dd723 --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/paper.tex @@ -0,0 +1,68 @@ +\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{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,patterns} +\usepackage{hyperref} +\hypersetup{colorlinks=true,linkcolor=black,citecolor=black,urlcolor=blue, + pdftitle={Every complex K3 surface is Oka},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{remark}[theorem]{Remark} +\newtheorem{example}[theorem]{Example} +\newcommand{\C}{\mathbb C} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\Q}{\mathbb Q} +\newcommand{\PP}{\mathbb P} +\newcommand{\B}{\mathbb B} +\newcommand{\dd}{\,\mathrm d} +\DeclareMathOperator{\ImPart}{Im} +\DeclareMathOperator{\RePart}{Re} +\DeclareMathOperator{\NS}{NS} +\DeclareMathOperator{\SO}{SO} +\DeclareMathOperator{\SU}{SU} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Pic}{Pic} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\id}{id} +\setlist{itemsep=3pt,topsep=5pt} +\numberwithin{equation}{section} +\setlength{\emergencystretch}{2em} +\title{Every complex K3 surface is Oka} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that every complex K3 surface $X$ is Oka, resolving the K3 Oka +conjecture. Equivalently, for every $m\geq1$, every holomorphic map to $X$ +from a neighborhood of a compact convex set in $\C^m$ can be approximated +uniformly on that set by entire maps $\C^m\to X$. +\end{abstract} +{\small\tableofcontents} + +\input{sections/introduction} + +\input{sections/local} +\input{sections/gluing} +\input{sections/globalization} +\input{sections/projective} +\input{sections/sewing} +\input{sections/degeneration} +\input{sections/rational} +\input{sections/dynamics} +\input{sections/dense} + +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/references.bib b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..05aff996e117b18643dca8fdf06db651bba4445a --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/references.bib @@ -0,0 +1,181 @@ +@book{Forstneric, + author={Forstneri{\v c}, Franc}, + title={Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis}, + edition={second}, series={Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge}, + volume={56}, publisher={Springer}, year={2017}, doi={10.1007/978-3-319-61058-0} +} +@article{ForstnericCAP, + author={Forstneri{\v c}, Franc}, + title={Runge approximation on convex sets implies the {Oka} property}, + journal={Annals of Mathematics}, volume={163}, number={2}, pages={689--707}, + year={2006}, doi={10.4007/annals.2006.163.689} +} +@article{ForstnericLarussonSurfaces, + author={Forstneri{\v c}, Franc and L{\'a}russon, Finnur}, + title={Holomorphic flexibility properties of compact complex surfaces}, + journal={International Mathematics Research Notices}, + volume={2014}, number={13}, pages={3714--3734}, year={2014}, + doi={10.1093/imrn/rnt044}, note={arXiv:1207.4838v3}, + url={https://arxiv.org/abs/1207.4838v3} +} +@article{ForstnericLarussonSurvey, + author={Forstneri{\v c}, Franc and L{\'a}russon, Finnur}, + title={Survey of {Oka} theory}, + journal={New York Journal of Mathematics}, volume={17a}, + pages={11--38}, year={2011}, + url={https://nyjm.albany.edu/j/2011/17a-2.html} +} +@article{ForstnericLarussonElliptic, + author={Forstneri{\v c}, Franc and L{\'a}russon, Finnur}, + title={Every projective {Oka} manifold is elliptic}, + journal={Mathematical Research Letters}, volume={33}, number={1}, + pages={297--312}, year={2026}, doi={10.4310/MRL.260503003231}, + note={arXiv:2502.20028v6, revised May 5, 2026. + \href{https://arxiv.org/pdf/2502.20028v6}{Manuscript PDF}} +} +@misc{GallegoGonzalezPurnaprajna, + author={Gallego, Francisco Javier and Gonz{\'a}lez, Miguel and Purnaprajna, Bangere P.}, + title={{K3} double structures on {Enriques} surfaces and their smoothings}, + year={2006}, note={arXiv:math/0604629v2}, + url={https://arxiv.org/abs/math/0604629v2} +} +@misc{OpenAIGSS, + author={{OpenAI}}, + title={{Global Spherical Shells on Minimal Surfaces of Class VII}}, + howpublished={OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf}{OAI:Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026}}, + year={2026} +} +@book{Hormander, + author={H{\"o}rmander, Lars}, title={An Introduction to Complex Analysis in Several Variables}, + edition={third}, publisher={North-Holland}, year={1990} +} +@article{Siu, + author={Siu, Yum-Tong}, title={Every {Stein} subvariety admits a {Stein} neighborhood}, + journal={Inventiones Mathematicae}, volume={38}, pages={89--100}, year={1976}, + doi={10.1007/BF01390170} +} +@article{Grauert, + author={Grauert, Hans}, + title={Ein {Theorem} der analytischen {Garbentheorie} und die {Modulr\"aume} komplexer {Strukturen}}, + journal={Publications Math{\'e}matiques de l'IH{\'E}S}, volume={5}, pages={5--64}, + year={1960}, doi={10.1007/BF02684746} +} +@article{ChenGounelas, + author={Chen, Xi and Gounelas, Frank}, title={Curves of maximal moduli on {K3} surfaces}, + journal={Forum of Mathematics, Sigma}, volume={10}, year={2022}, + note={Paper No. e36}, doi={10.1017/fms.2022.24} +} +@article{AlarconForstneric, + author={Alarc{\'o}n, Antonio and Forstneri{\v c}, Franc}, title={{Oka}-1 manifolds}, + journal={Mathematische Zeitschrift}, volume={311}, year={2025}, + note={Paper No. 33}, doi={10.1007/s00209-025-03833-4} +} +@book{Huybrechts, + author={Huybrechts, Daniel}, title={Lectures on {K3} Surfaces}, + series={Cambridge Studies in Advanced Mathematics}, volume={158}, + publisher={Cambridge University Press}, year={2016}, doi={10.1017/CBO9781316594193} +} +@misc{XieZhao, + author={Xie, Song-Yan and Zhao, Shengyuan}, title={Periods of {Oka K3} surfaces are dense}, + year={2026}, note={arXiv:2608.24392v2, revised September 17, 2026}, url={https://arxiv.org/abs/2608.24392v2} +} +@misc{VerbitskyErratum, + author={Verbitsky, Misha}, + title={Ergodic complex structures on hyperk{\"a}hler manifolds: an erratum}, + year={2017}, note={arXiv:1708.05802v1}, url={https://arxiv.org/abs/1708.05802v1} +} +@article{KamenovaLuVerbitsky, + author={Kamenova, Ljudmila and Lu, Steven and Verbitsky, Misha}, + title={Kobayashi pseudometric on hyperk{\"a}hler manifolds}, + journal={Journal of the London Mathematical Society}, + volume={90}, number={2}, pages={436--450}, year={2014}, + doi={10.1112/jlms/jdu038} +} +@article{Ratner, + author={Ratner, Marina}, + title={Raghunathan's topological conjecture and distributions of unipotent flows}, + journal={Duke Mathematical Journal}, volume={63}, number={1}, pages={235--280}, + year={1991}, doi={10.1215/S0012-7094-91-06311-8} +} +@article{BorelDensity, + author={Borel, Armand}, title={Density properties for certain subgroups of semi-simple groups without compact components}, + journal={Annals of Mathematics}, volume={72}, number={1}, pages={179--188}, year={1960}, + doi={10.2307/1970150} +} +@article{BorelHarishChandra, + author={Borel, Armand and Harish-Chandra}, title={Arithmetic subgroups of algebraic groups}, + journal={Annals of Mathematics}, volume={75}, number={3}, pages={485--535}, year={1962}, + doi={10.2307/1970210} +} +@article{Eichler, + author={Gritsenko, Valeri and Hulek, Klaus and Sankaran, Gregory K.}, + title={Abelianisation of orthogonal groups and the fundamental group of modular varieties}, + journal={Journal of Algebra}, volume={322}, number={2}, pages={463--478}, + year={2009}, doi={10.1016/j.jalgebra.2009.01.037}, note={Proposition 3.3(i)} +} +@article{BuzzardLu, + author={Buzzard, Gregery T. and Lu, Steven S. Y.}, + title={Algebraic surfaces holomorphically dominable by {$\mathbb{C}^2$}}, + journal={Inventiones Mathematicae}, volume={139}, number={3}, + pages={617--659}, year={2000}, doi={10.1007/s002220050021} +} +@article{Campana, + author={Campana, Fr{\'e}d{\'e}ric}, + title={Orbifolds, special varieties and classification theory}, + journal={Annales de l'Institut Fourier}, volume={54}, number={3}, + pages={499--630}, year={2004}, doi={10.5802/aif.2027} +} +@article{CCR, + author={Cadorel, Beno{\^\i}t and Campana, Fr{\'e}d{\'e}ric and Rousseau, Erwan}, + title={Hyperbolicity and specialness of symmetric powers}, + journal={Journal de l'{\'E}cole polytechnique---Math{\'e}matiques}, + volume={9}, pages={381--430}, year={2022}, doi={10.5802/jep.185} +} +@article{ChenGounelasLiedtke, + author={Chen, Xi and Gounelas, Frank and Liedtke, Christian}, + title={Curves on {K3} surfaces}, + journal={Duke Mathematical Journal}, volume={171}, number={16}, + pages={3283--3362}, year={2022}, doi={10.1215/00127094-2021-0107} +} + +@article{ForstnericSplitting, + author={Forstneri{\v c}, Franc}, + title={Noncritical holomorphic functions on {Stein} manifolds}, + journal={Acta Mathematica}, volume={191}, pages={143--189}, year={2003}, + doi={10.1007/BF02392963} +} + +@article{GromovOka, + author={Gromov, Mikhael}, + title={{Oka}'s principle for holomorphic sections of elliptic bundles}, + journal={Journal of the American Mathematical Society}, + volume={2}, number={4}, pages={851--897}, year={1989}, + doi={10.1090/S0894-0347-1989-1001851-9} +} + +@article{HormanderLTwo, + author={H{\"o}rmander, Lars}, + title={{$L^2$} estimates and existence theorems for the $\bar\partial$ operator}, + journal={Acta Mathematica}, volume={113}, pages={89--152}, year={1965}, + doi={10.1007/BF02391775} +} + +@article{ForstnericInterpolation, + author={Forstneri{\v c}, Franc}, + title={Extending holomorphic mappings from subvarieties in {Stein} manifolds}, + journal={Annales de l'Institut Fourier}, volume={55}, number={3}, + pages={733--751}, year={2005}, doi={10.5802/aif.2112} +} +@article{ArtebaniSarti, + author={Artebani, Michela and Sarti, Alessandra}, + title={Non-symplectic automorphisms of order 3 on {K3} surfaces}, + journal={Mathematische Annalen}, volume={342}, pages={903--921}, + year={2008}, doi={10.1007/s00208-008-0260-1} +} +@misc{ChenLiHodge, + author={Chen, Gui-Qiang G. and Li, Siran}, + title={Global weak rigidity of the {Gauss--Codazzi--Ricci} equations and isometric immersions of {Riemannian} manifolds with lower regularity}, + year={2017}, note={arXiv:1607.06862v2}, + url={https://arxiv.org/abs/1607.06862v2} +} diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/degeneration.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/degeneration.tex new file mode 100644 index 0000000000000000000000000000000000000000..f19bfad16cefae407db86d69b1b39b64f511556d --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/degeneration.tex @@ -0,0 +1,561 @@ +\section{An open region of K3 periods} +\label{deg:section} + +Theorem~\ref{sew:chain} permits the centers of successive annuli to +move. We now construct a degeneration for which every possible new +center admits another block. This is the reason an infinite chain +can be continued after the exactness adjustments. + +\begin{proposition}\label{deg:open} +There is a nonempty open subset of the full marked K3 period domain +such that every surface with period in this subset has property $L$ +at every point. +\end{proposition} + +The openness here is unrestricted: the nearby surfaces need not be +projective. We shall first produce one smooth quartic with a finite +collection of buffered chart constructions, then deform all those +constructions to its full local deformation space. + +\subsection{The semistable model and its regular pencils} + +Choose two general smooth quadrics $Q_+,Q_-\subset\mathbb P^3$. +Their intersection $E$ is a smooth elliptic quartic. For a general +quartic form $G$ consider the family +\begin{equation}\label{deg:quartic-family} + Q_+Q_-=\alpha G. +\end{equation} +Here the same symbols $Q_+,Q_-$ denote their quadratic equations. +The sixteen zeros $Z$ of $G|_E$ are simple. Near such a zero the +total space has equation $xy=\alpha t$, up to holomorphic changes +of coordinates. Blowing up one component, locally the ideal +$(x,\alpha)$, is a small resolution. For example, in one chart +write $\alpha=xa$; the strict transform has $y=at$ and the map to +the parameter disc is $\alpha=xa$. In the other chart write +$x=\alpha b$; the equation becomes $t=by$, with smooth coordinates +$(\alpha,b,y)$ and projection $\alpha$. Thus the resolved total +space is smooth and the central fiber has normal crossings. + +Its two components, with a suitable choice of labels, are a quadric +$S^+$ and the blowup $S^-$ of a quadric at $Z$; they meet along the +strict transform of $E$, again denoted $E$. The resolution is +crepant because it is small and the original hypersurface is +Gorenstein. Adjunction for the quartic family supplies a nowhere-zero +relative dualizing form. On $S^\pm$ it becomes a logarithmic +symplectic form $\omega_\pm$ with simple pole along $E$ and opposite +nonzero residues. In a normal-crossing chart we have +\begin{equation}\label{deg:neck-form} + uv=\alpha,\qquad + \omega_\alpha=k(u,v,t)\,\frac{du}{u}\wedge dt, +\end{equation} +where $k$ is holomorphic and nowhere zero and $t$ is a coordinate +along $E$. In particular $k(0,0,t)dt$ is its residue on the +$u$-component; the residue on the $v$-component is its negative. +For small nonzero $\alpha$, the fibers of +Equation~\eqref{deg:quartic-family} are smooth quartic K3 surfaces. + +Call a fiber of a base-point-free pencil on $S^+$ or $S^-$ +\emph{regular} if it is a smooth rational curve, the pencil map is +smooth along it, and it meets $E$ transversely at two distinct +points. We use the two ruling pencils, pulled back on $S^-$, and +on $S^-$ also the pencils of $(1,1)$ curves through two distinct +points of $Z$, with their two base points subtracted. + +\begin{lemma}\label{deg:regular-pencils} +The quadrics and $G$ can be chosen so that: +\begin{enumerate} + \item Through every point of $E$ on each component there is a + regular fiber of one of these finitely many pencils. + \item The union of their regular fibers contains the complement + of a proper algebraic subset of each component. + \item Every regular fiber has, away from $E$, a transverse + intersection with a regular fiber of another pencil. +\end{enumerate} +\end{lemma} + +\begin{proof} +On a plain quadric, identify the two rulings with the projections +of $\mathbb P^1\times\mathbb P^1$. Their restrictions to $E$ have +degree two. They cannot both ramify at one point: otherwise the +differential of the inclusion of the smooth curve $E$ in the +quadric would vanish. This proves the first assertion on $S^+$. + +For the blown-up quadric, a ruling fiber is additionally forbidden +when it passes through a point of $Z$. Choose $Z$ generally so +that its points avoid both ruling ramification sets, and so +that a forbidden level of one ruling and a forbidden level of the +other have a common point on $E$ only at a point of $Z$. To check +that these are proper conditions, write the two projections as +$\pi_1,\pi_2$. Their ramification sets are disjoint. Conditions +involving one ramification set and one $\pi_i(Z)$ exclude only +finitely many values for the corresponding point of $Z$. For +distinct $z_i,z_j$, the condition that the grid point with +coordinates $(\pi_1(z_i),\pi_2(z_j))$ lie on $E$ is a proper +condition on the pair. For $i=j$ the grid point is $z_i$ itself. +Thus, away from $Z$, at least one ruling remains regular. + +Fix $z_i\in Z$. Choose another center $z_j$ and take the plane +through the tangent line to $E$ at $z_i$ and through $z_j$. For +general choices its intersection with the quadric is a smooth +$(1,1)$ curve. Its divisor on $E$ is +\[ + 2z_i+z_j+z_k, +\] +with $z_k$ different from $z_i,z_j$, and it passes through no other +point of $Z$. Its strict transform is a member of the pencil of +$(1,1)$ curves through $z_i,z_j$. The chord through these two +points is not a ruling line; hence the pencil has exactly these +two simple base points, which are resolved by the blowups. An +order-two contact at $z_i$ becomes a simple transverse +intersection with the strict transform of $E$. The simple +intersection at $z_j$ disappears, and the intersection at $z_k$ +remains. The selected strict transform is therefore a regular +fiber through the point of $E$ over $z_i$. + +All the generic requirements just used are simultaneous finite +proper conditions. More explicitly, on the elliptic curve the +residual $z_k$ is determined by the hyperplane divisor class and +$2z_i+z_j$; excluding coincidences or another center imposes a +proper condition on at most three of the centers. The other +requirements involve at most two centers and fixed branch data. +The divisor $Z$ ranges over the complete degree-sixteen linear +system $|\mathcal O_E(4)|$. Indeed the restriction of quartic +forms is surjective, as follows from the Koszul resolution for +the intersection of two quadrics. In an ordered general divisor +of this degree any specified fewer than sixteen points vary +freely; the remaining points satisfy the one elliptic sum +condition. Hence the finitely many proper conditions above can +all be avoided. + +For each of the finitely many pencils, only finitely many members +fail to be smooth or meet $E$ nontransversely. Their union is a +proper algebraic set, proving the second assertion. Finally, on +any regular rational fiber, at least one of the two ruling +projections is nonconstant. This is immediate for a ruling fiber +by using the other ruling; for a $(1,1)$ fiber both projections are +nonconstant. Away from finitely many points of the fiber the +corresponding ruling member is regular and the projection has +nonzero derivative. Such an intersection is transverse and can +be chosen off $E$. +\end{proof} + +We now make finite compact choices, each with an open buffer. +Let $G_{\rm reg}$ be the union of \emph{all} regular fibers from +the listed pencils. Its complement is contained in a proper +algebraic set, and $G_{\rm reg}$ contains a neighborhood of $E$. +By the first assertion of Lemma~\ref{deg:regular-pencils}, compact +subfamilies of regular fibers supply entries at every point of +$E$. By its third assertion each required family can, after +restriction, be equipped with a transverse partner at an +interior point, with a positive transversality margin in local +charts. Compactness supplies finitely many such choices near +$E$. + +For each point of the algebraic exceptional set away from $E$, +choose a coordinate line not contained in that set and a small +radius whose boundary avoids it. The boundary then lies in +$G_{\rm reg}$. Small translations and tilts retain this boundary +containment, so the disc construction supplies a neighborhood of +possible centers. Points of $G_{\rm reg}$ themselves have regular +coordinate neighborhoods. On the compact region outside a +chosen neighborhood of $E$, finitely many of these two kinds of +center neighborhoods suffice. Only after choosing these finitely +many discs do we add finitely many regular patches covering their +compact boundaries, along with the corresponding regular +families and transverse partners. Thus every retained datum +belongs to a finite buffered collection. The algebraic +exceptional set is used before the finite selection. These +disc choices will be used with +Corollary~\ref{loc:disc} after constructing the strips. + +\subsection{Core and neck coordinates} + +\begin{lemma}\label{deg:core-coordinates} +On a small parameter disc of any compact regular family there are +coordinates $(z,p)\in\mathbb P^1\times\Delta_r$ in which the two +ends are $z=0,\infty$ and +\[ + \omega_\pm=\frac{1}{2\pi i}\frac{dz}{z}\wedge dp. +\] +Either end can be designated as the entrance. With entrance at +$\infty$, exit at $0$, and $z=c\exp(2\pi i w)$, this reads +$\omega_\pm=dw\wedge dp$. The parameter $p$ is a local affine +coordinate for either end position on $E$, read using its signed +residue differential. +\end{lemma} + +\begin{proof} +Over a small disc the smooth proper family of rational fibers is +a holomorphic $\mathbb P^1$ bundle and is trivial there. One can +see this by choosing a local section, taking its relative +degree-one line bundle, and using sections and base change to +identify the family with a projective bundle. The two distinct +end sections can then be put at zero and infinity. + +In a parameter $s$ on this disc write the logarithmic form as +$f(z,s)dz\wedge ds$. For each $s$, the one-form +$f(z,s)dz$ on $\mathbb P^1$ has only simple poles at zero and +infinity. It is therefore $b(s)dz/z$. Nondegeneracy makes $b$ +nonzero. Set $dp=2\pi i b(s)ds$. + +Let $t_+(s)$ and $t_-(s)$ be the exit and entrance positions on +$E$, and let $\rho$ be the residue differential on this +component. Taking residues gives the exact identities +\[ + dp=2\pi i\,t_+^*\rho=-2\pi i\,t_-^*\rho. +\] +They also follow from the constancy of the elliptic sum of the +two intersection points. Since both ends are transverse, +$t_+$ and $t_-$ are locally invertible. Thus $p$ moves each +end freely and the stated residue interpretation follows. +\end{proof} + +In a fixed normal-crossing chart at an end, its nonzero normal +coordinate on the corresponding central component has the +uniform expansions +\begin{align} + u&=k_-(p)z^{-1}\bigl(1+O(z^{-1})\bigr) + &&\text{at the entrance}, + \label{deg:entrance}\\ + u&=k_+(p)z\bigl(1+O(z)\bigr) + &&\text{at the exit}. + \label{deg:exit} +\end{align} +The coordinate along $E$ differs from its end value by +$O(z^{-1})$ or $O(z)$ respectively. The functions $k_\pm$ +are nonzero, with uniformly bounded logarithmic derivatives on +the selected smaller parameter discs. These estimates, with +any fixed number of derivatives in logarithmic and parameter +coordinates, are uniform over the finite buffered atlas. + +Choose a small positive normal cutoff $d$. Truncate a core at +its two ends where $|u|$ is comparable to $d$, retaining fixed +buffers in logarithmic height. In the coordinate $w$ of +Lemma~\ref{deg:core-coordinates}, this is a cylinder of length +tending to infinity as $d\to0$, uniformly over the compact +families. For every fixed $d$ it is a relatively compact annulus +times a disc in the smooth part of the central fiber. +Lemma~\ref{loc:deformation} deforms this map into nearby smooth +fibers on a slightly larger annulus and parameter disc. Its +symplectic normalization gives maps with pullback form +$dw\wedge dp$ converging to the central maps on these fixed +domains. This uses only finite-domain deformations, and can be +done simultaneously for the finite raw atlas. Restrictions and +recentering of its larger parameter discs then supply all the +moving centers used below. + +The connecting neck has a more direct description. In +Equation~\eqref{deg:neck-form} put +\[ + u=c\exp(2\pi i w),\qquad v=\alpha/u, +\] +and allow it to run from $|u|\asymp d$ to $|v|\asymp d$, +with fixed additional logarithmic buffers. Normalize its +transverse coordinate by +\begin{equation}\label{deg:neck-normalization} + p(w,t)=\int_{t_0}^t2\pi i\,k(u,\alpha/u,s)\,ds. +\end{equation} +This gives $dw\wedge dp=\omega_\alpha$. On this entire neck, +including its buffers, both $|u|$ and $|v|$ are at most $Cd$. +Consequently +\[ + p(w,t)=\int_{t_0}^t2\pi i\,k(0,0,s)\,ds+O(d). +\] +The estimate is uniform in neck length, and holds with any fixed +number of derivatives on smaller parameter discs and logarithmic +buffers. Indeed each $w$ derivative differentiates $u$ or $v$ +by a fixed constant multiple, and both remain $O(d)$; parameter +derivatives are bounded on the fixed $t$-patches. The derivative +in $t$ is bounded away from zero, so the normalization is +uniformly invertible there. + +\begin{figure}[ht] +\centering +\begin{tikzpicture}[x=0.93cm,y=0.85cm,>=Latex,font=\small] + \fill[black!7] (0,0) rectangle (2.8,1.05); + \fill[black!3] (2.4,0) rectangle (8.6,1.05); + \fill[black!7] (8.2,0) rectangle (11,1.05); + \draw (0,0) rectangle (2.8,1.05); + \draw (2.4,0) rectangle (8.6,1.05); + \draw (8.2,0) rectangle (11,1.05); + \draw[dashed] (2.6,-0.2) -- (2.6,1.35); + \draw[dashed] (8.4,-0.2) -- (8.4,1.35); + \node at (1.2,0.53) {$S^+$ core}; + \node at (5.5,0.72) {neck: $uv=\alpha$}; + \node at (9.8,0.53) {$S^-$ core}; + \draw[->] (3.5,0.25) -- (7.5,0.25); + \node[above] at (2.6,1.35) {$|u|=d$}; + \node[above] at (8.4,1.35) {$|v|=d$}; + \node[below] at (5.5,-0.1) {$Y$ increases; $|u|$ decreases and $|v|$ increases}; +\end{tikzpicture} +\caption{The core--neck joins in logarithmic height, not to scale. +The overlap widths are fixed. Once $d$ is fixed, the neck length +grows as $\alpha\to0$, while its transverse width stays bounded +below. After sewing, the same parameter $p$ labels a leaf across +both joins.} +\label{deg:neck-figure} +\end{figure} + +Figure~\ref{deg:neck-figure} separates the fixed-width overlaps from +the part of the cylinder whose length grows. Only the former +enter the local transition estimates. + +\begin{lemma}[Buffered seams]\label{deg:seams} +There are fixed positive parameter and seam widths and a fixed +bound on translation derivatives with the following property. +Choose $d$ sufficiently small, then $\alpha\neq0$ sufficiently +small depending on $d$. At matching nominal end positions the +normalized core and neck charts have degree-one symplectic +transitions of the form +\[ + F_j=J_j\circ(\operatorname{id}+E_j),\qquad + J_j(w,p)=(w+a_j(p),p), +\] +on buffered seams. After choosing long-coordinate origins and +cut heights, they satisfy the domain hypotheses of +Theorem~\ref{sew:chain}, apart from exactness, and +\[ + \sup_j\|E_j\|\leq C d+\epsilon(d,\alpha), + \qquad \epsilon(d,\alpha)\longrightarrow0 + \quad(\alpha\to0, d\text{ fixed}). +\] +The widths and the bound for $a_j'$ can be chosen before $d$. +\end{lemma} + +\begin{proof} +At an exit seam Equation~\eqref{deg:exit} has leading part +$u=k_+(p)z$. Taking a local logarithm of the nonzero coefficient +gives a change $w_{\rm neck}=w_{\rm core}+a(p)$. Its derivative +is bounded by the logarithmic derivative of $k_+$. Its transverse +change is exactly the identity at leading order: both parameters +are primitives of the same signed residue differential, with +the same constant at the matching nominal end. + +At the entrance into the next component, the leading relation is +$v=k_-(p')/z'$ and $uv=\alpha$. Hence $z'$ is a nonzero +holomorphic multiple of $u$, so again $w'=w+a(p)$, with the same +orientation. The sign of the residue changes between the two +components, and the use of $z'=\infty$ at the entrance changes +it once more. The transverse primitives therefore again agree +as germs. The potentially large constant $\log\alpha$ affects +the translation itself, not its derivative in $p$. + +The actual seam maps differ from these leading maps by $O(d)$ +from the expansions and Equation~\eqref{deg:neck-normalization}, +plus the core deformation error $\epsilon(d,\alpha)$. These +estimates are in $(\log u,t)$ or $(\log v,t)$ coordinates. The +leading changes are uniformly invertible there. The inverse +function theorem with the reserved buffers therefore produces +the actual inverse branches throughout the seams. Local branches +agree by uniqueness near the specified leading branch, including +after going once around the period. Their lifts have degree one, +and they are symplectic because both charts pull back the same +surface form to $dw\wedge dp$. + +For completeness, the cuts can be prepared without imposing +incompatible origins at the two ends of a block. Work first in +raw coordinates. Enlarge each core range to +$cd<|z|<1/(cd)$ with a fixed sufficiently small $c>0$, and add +fixed logarithmic buffers to every neck. At a seam choose the +two matching cut heights by its leading formula at the nominal +center. Each raw block has independent room for the choice of +its two cuts. Then translate its single long-coordinate origin +so that its entrance is at height zero; its exit defines $L_j$. +For small $d$, and then small $\alpha$, all core and neck lengths +are at least one. Shrink the fixed transverse discs so that +the variation of $\operatorname{Im}a_j$ about its central value +is less than $h/8$. The logarithmic derivative bounds show that +this shrinking is independent of $d$ and $\alpha$. The two +cut heights then give precisely +$|\operatorname{Im}a_j+L_j|0$, smaller than a fixed fraction of the +available end-position width. The selected core charts cover +the region from a fixed multiple of $d$ up to $d_*$, after +shrinking their end-position patches. Their sewing corrections +are uniformly small in the same logarithmic coordinates. +The neck chart covers below that multiple of $d$ and overlaps +the core region by its extra logarithmic buffers. The identical +argument applies at the $v$-cut. A finite collection of the +smaller end-position patches covers $E$. + +This argument does not ask for a lower bound on an ordinary +metric chart radius at $|u|=d$. All inverse estimates there +are in logarithmic and end-position coordinates, where the +leading maps and the buffers have uniform bounds. One first +chooses the position widths and the needed error bound for +coverage and for the core crossings, and then chooses $d$. +The fixed $d_*$ can be reduced after the widths are chosen; +finally take $d$ much smaller than $d_*$. On the remaining +compact region outside these end neighborhoods, the preceding +finite collection of regular patches and coordinate discs +applies. If the end neighborhoods have been reduced in choosing +$d_*$, the intervening closed annular region is already covered +by the same regular core families; it is a fixed compact region +before $d$ is chosen. Properness of the semistable family ensures that +these core and neck regions cover every nearby smooth fiber. + +At each reserved coordinate disc away from $E$, use the +submersion coordinates of the total space to transport its +local coordinate construction to the smooth fibers. Its +boundary, and the boundaries of sufficiently nearby translated +or tilted discs, remain in the regular patches where $L$ was +just proved. Corollary~\ref{loc:disc} gives $L$ throughout the +remaining smaller coordinate neighborhoods. Thus $L$ holds +at every point of the selected smooth quartic. + +Now fix the small nonzero $\alpha$ used above. Every raw core +and neck chart is a map from a \emph{finite} annulus times a +disc, with margins. There are finitely many raw charts; +all the moving versions are restrictions and translations of +these larger charts. Apply Lemma~\ref{loc:deformation} in the +full local deformation family of this smooth K3, choosing a +nearby holomorphic symplectic form and normalizing each chart. +They deform simultaneously with arbitrarily small additional +error. Their seam inverse branches persist: on each compact +buffered seam they are the nearby branches of the original +local inverse, and uniqueness makes those branches agree +throughout the seam and around its period circle. Thus all +the domain, derivative, period-adjustment, coverage, and +transversality estimates persist in a sufficiently small +unrestricted deformation neighborhood. This argument does +not deform the infinitely many corrected charts individually; +it deforms the finite raw atlas and repeats its uniformly +controlled construction. + +The order of choices is worth recording because it prevents +any dependence of the allowable error on the length of the +neck: +\begin{enumerate} + \item Choose the finite compact regular families, partner + crossings, end-position patches, disc boundaries, + parameter widths and logarithmic buffers. + \item Choose an allowable sewing and coverage error for + these data; the derivative bounds in the leading + shears have already been fixed. + \item Choose $d$ so that the $O(d)$ errors are below that + allowance, and then choose a nonzero $\alpha$ so that + all fixed-$d$ deformation errors are below it. + \item Choose the full deformation neighborhood of that + fixed smooth quartic so that its additional errors + remain within the same allowance. +\end{enumerate} + +K3 deformation theory gives a smooth local deformation space, +and the marked period map is a local isomorphism by local +Torelli; see \cite{Huybrechts}. Its image of this last +neighborhood is a nonempty open subset of the full period +domain. Any other marked K3 with the same period has an +isomorphic underlying surface: signs and reflections in +algebraic roots match the K\"ahler chambers while fixing the +period, and global Torelli then applies \cite{Huybrechts}. +Consequently every surface with a period in this open subset +has $L$ everywhere, proving +Proposition~\ref{deg:open}. Theorem~\ref{ana:cap} consequently +gives convex approximation for each of these surfaces. diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/dense.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/dense.tex new file mode 100644 index 0000000000000000000000000000000000000000..14caa9d71050ba5de9bcf422430b4fd0e4f3db60 --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/dense.tex @@ -0,0 +1,164 @@ +\section{Interpolation and further consequences} +\label{sec:dense-entire} + +\subsection{Interpolation and dense immersions} + +We now obtain approximation and interpolation from open Riemann +surfaces and derive the dense entire immersion in +Corollary~\ref{cor:dense}. An \emph{open Riemann surface} is a +connected noncompact Riemann surface. A compact subset is +\emph{Runge} if its complement has no relatively compact component. +The notation $j_a^k$ records the value and derivatives through order +$k$ in local coordinates, and $d_S$ denotes any fixed distance +inducing the topology of $S$. + +\begin{corollary}\label{cor:oka1} +Let $S$ be a complex K3 surface, $R$ an open +Riemann surface, $K\subset R$ a compact Runge set, and +$A=\{a_j:j\geq1\}\subset R$ a closed discrete set of distinct points. +Let $k_j\geq1$ be integers. For every continuous map $h:R\to S$ +holomorphic near $K\cup A$ and every $\epsilon>0$, there is a +holomorphic map $f:R\to S$, homotopic to $h$, satisfying +\[ + \sup_{z\in K}d_S(f(z),h(z))<\epsilon, + \qquad j_{a_j}^{k_j}f=j_{a_j}^{k_j}h\quad(j\geq1). +\] +Finite or empty interpolation sets are also allowed. If each +prescribed first derivative is nonzero, $f$ can be chosen to be an +immersion. +\end{corollary} + +\begin{proof} +Theorem~\ref{thm:main} and the convex approximation characterization +\cite[Theorem~0.1]{ForstnericCAP} imply that $S$ is Oka. Every Oka manifold is +Oka-1, which is precisely the approximation and interpolation +property stated here. The CAP-to-interpolation implication is +established in \cite[Corollary~1.3]{ForstnericInterpolation}; for the +Oka-1 formulation with individually prescribed jet orders, see \cite[Definition~1.1 and the following +discussion]{AlarconForstneric}. Since $\dim_{\C}S=2$, the same +interpolating map can be chosen to be an immersion by +\cite[Corollary~2.10]{AlarconForstneric} when all prescribed first +derivatives are nonzero. +\end{proof} + +\begin{proof}[Proof of Corollary~\ref{cor:dense}] +Choose a countable dense sequence $(x_j)_{j\geq1}$ in the ordinary +topology of $S$, set $x_0=x$, and choose nonzero vectors +$v_j\in T_{x_j}S$, with $v_0=v$. At the distinct source points +$a_j=3j$, choose holomorphic germs $g_j:(\C,a_j)\to(S,x_j)$ +with $g'_j(a_j)=v_j$. Explicitly, if $\kappa_j$ is a chart taking +$x_j$ to the origin in a ball in $\C^2$, put +\[ + g_j(z)=\kappa_j^{-1}\bigl((z-a_j)d\kappa_j(v_j)\bigr). +\] +Choose $00$, the global spherical shell theorem +\cite[Theorem~1.1]{OpenAIGSS} therefore applies, so the surface is a +Kato surface. The class-VII decomposition recalled in +\cite[Introduction]{ForstnericLarussonSurfaces} now makes the list +exhaustive: Hopf and Inoue surfaces for $b_2=0$, and Enoki, +Inoue--Hirzebruch, and intermediate surfaces for $b_2>0$. +Theorem~4 of \cite{ForstnericLarussonSurfaces} shows that exactly the +Hopf and Enoki cases are Oka. +\end{proof} + +The classification assertion is restricted to minimal surfaces; no +claim is made here for arbitrary blowups or for properly elliptic +surfaces. + +A complex manifold $Y$ is \emph{elliptic} in Gromov's sense if it has +a global dominating holomorphic spray: a holomorphic vector bundle +$E\to Y$ and a holomorphic map $s:E\to Y$ such that $s(0_y)=y$ +and the fiber differential $d(s|_{E_y})_{0_y}:E_y\to T_yY$ is +surjective for every $y\in Y$ \cite{GromovOka}. + +\begin{corollary}[Global holomorphic sprays]\label{cor:global-spray} +Every projective complex K3 surface and every complex Enriques surface +is elliptic in Gromov's sense. +\end{corollary} + +\begin{proof} +Projective complex K3 surfaces are Oka by Theorem~\ref{thm:main}. +Complex Enriques surfaces are projective +\cite[Section~2]{GallegoGonzalezPurnaprajna} and are Oka by +Corollary~\ref{cor:surface-classification}. +The conclusion follows from Forstneri\v c and L\'arusson's theorem +that every projective Oka manifold is elliptic +\cite[Theorem~1.1]{ForstnericLarussonElliptic}. +\end{proof} + +This last implication requires projectivity; no assertion about a +global dominating spray on a nonprojective K3 surface is made here. diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/dynamics.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/dynamics.tex new file mode 100644 index 0000000000000000000000000000000000000000..467d42b63cae90b616fb512d10e4c07fc8aa0f14 --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/dynamics.tex @@ -0,0 +1,171 @@ +\section{Period dynamics and the conclusion} +\label{dyn:section} + +We now pass from the two open regions constructed above to every +K3 surface. The distinction between no rational direction, one +rational direction, and a rational plane is essential. In +particular, density of an Oka locus alone would not justify this +step. The corrected period-orbit analysis is due to Verbitsky +\cite{VerbitskyErratum}; the recent dense-period application in +\cite{XieZhao} uses the same broad strategy of geometric flexibility, arithmetic +period dynamics, and Torelli theory. Here the two preceding geometric +constructions supply open regions for both nonrational orbit types. +We give the particular group argument needed for this transfer. + +Write $V=\Lambda\otimes\R$, and let +$G=\operatorname{SO}_0(V)$ be the identity component. Passing to +finite-index subgroups, the integral isometry group gives an +arithmetic lattice $\Gamma\subset G$ by the arithmetic lattice +theorem \cite{BorelHarishChandra}. + +\begin{lemma}\label{dyn:orbits} +Let $P\in\mathcal D$. +\begin{enumerate} +\item If $P\cap(\Lambda\otimes\Q)=\{0\}$, its integral-isometry +orbit is dense in $\mathcal D$. +\item If $P\cap(\Lambda\otimes\Q)=\Q v$ for a primitive integral +vector $v$, its orbit under integral isometries fixing $v$ is +dense in $\mathcal D_v$. +\end{enumerate} +\end{lemma} + +\begin{proof} +We first work with an ordered orthonormal positive frame of $P$. +Its connected pointwise stabilizer is +$H=\operatorname{SO}_0(1,19)$. This group is generated by +one-parameter unipotent subgroups. Ratner's orbit closure +theorem \cite{Ratner}, applied to $H\cdot e\Gamma$ in +$G/\Gamma$, gives +\[ + \overline{H\cdot e\Gamma}=S\cdot e\Gamma +\] +for a connected closed subgroup $H\subset S\subset G$ such +that $S\cap\Gamma$ is a lattice in $S$. Thus $H$ is the +pointwise stabilizer of the actual plane $P$ throughout, and the lattice +matrices in every intermediate group are rational in the +original lattice coordinates. + +Put $W=P^\perp$. As a module for +$\mathfrak h=\mathfrak{so}(W)$, the orthogonal block-matrix +decomposition is +\[ + \mathfrak{so}(V) + =\mathfrak h\oplus\mathfrak{so}(P)\oplus(W\otimes P). +\] +The standard real representation $W$ is absolutely irreducible. +Thus an intermediate Lie algebra obtains its mixed part by +choosing a subspace $A\subset P$, with mixed module $W\otimes A$. +If $\dim A=2$, brackets give the full Lie algebra. If +$\dim A=1$, they give +$\mathfrak{so}(W\oplus A)$, the stabilizer of the positive line +$A^\perp\cap P$. Including the nonzero rotation algebra of $P$ +in this case generates the missing mixed summand and hence the +full algebra. If $A=0$, the only possibilities are +$\mathfrak h$ and +$\mathfrak h\oplus\mathfrak{so}(P)$. + +The proper alternatives force rational directions in $P$. +For the stabilizer of one positive line, Borel density +\cite{BorelDensity} makes its lattice Zariski dense in that +stabilizer. As the lattice consists of rational matrices, its +Zariski closure is defined over $\Q$. Its fixed line is therefore +rational. The same reasoning for $H$ makes its fixed plane +rational. + +There is also no difficulty from the compact rotation factor in +the remaining algebra. Up to finite isogeny the corresponding +connected group is +$\operatorname{SO}_0(W)\times\operatorname{SO}(P)$. +Projection to the first factor is proper because its kernel is +compact. Thus the image of the lattice meets each compact set +in only finitely many points and is discrete. The finite +invariant measure on the original quotient pushes forward to +a finite invariant measure on the projected quotient, so the +image is a lattice there. Borel density implies that the +Lie algebra of the Zariski closure of the original lattice +surjects onto $\mathfrak h$. Its derived algebra is exactly +$\mathfrak h$: it is contained there because the other factor +is commutative, and it maps onto $\mathfrak h$ because that +algebra is simple. The Zariski closure and its derived algebra +are defined over $\Q$, so the fixed space of the derived +algebra, namely $P$, is rational. Here the rationality of the +Zariski closure of rational matrices follows directly from the +vanishing equations: in each bounded degree their linear +conditions have rational coefficients and hence a rational +basis of solutions. + +If $P$ has no rational direction, none of these proper +alternatives is possible. Ratner's closure is consequently the +whole homogeneous space. Equivalently $H\Gamma$ is dense in +$G$. Inversion makes $\Gamma H$ dense as well, and projection +to $G/H$ makes the $\Gamma$-orbit of the chosen frame dense in +frame space. +The projection from frame space to $\mathcal D$ proves the first +assertion. The use of the frame stabilizer $H$, rather than +$H\times\operatorname{SO}(2)$, is what permits the application +of Ratner's theorem. + +For the second assertion, restrict to $v^\perp$, of signature +$(2,19)$. The integral stabilizer of $v$ is arithmetic and gives +a lattice in $G_v=\operatorname{SO}_0(v^\perp)$. More explicitly, +the subgroup of integral isometries of the orthogonal lattice +acting trivially on its discriminant group extends to the full +K3 lattice while fixing $v$ \cite[Chapter~14, Section~2]{Huybrechts}, +so the integral stabilizer has +finite index in the relevant orthogonal arithmetic group. + +An oriented plane containing $v$ is equivalent to a positive +unit vector $u\in v^\perp$: choose the sign of $u$ using the +given plane orientation. The connected stabilizer of $u$ is +again $H=\operatorname{SO}_0(1,19)$. This time +\[ + \mathfrak{so}(2,19)=\mathfrak h\oplus W +\] +as an $\mathfrak h$-module, so irreducibility shows that $H$ is +maximal at Lie algebra level. Ratner's alternatives are a closed +$H$-orbit or a dense orbit. In the closed case Borel density +makes the line $\R u$ rational in the rational quadratic space +$v^\perp$. This would give a second rational direction in $P$, +contrary to the hypothesis. The orbit is therefore dense. +The connected group $G_v$ is transitive on positive unit +vectors, so the resulting density is throughout $\mathcal D_v$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Choose a marking of $X$ and let $P\in\mathcal D$ be its period. +The rational dimension of $P\cap(\Lambda\otimes\Q)$ is zero, +one, or two. We show in each case that $X$ has $L$ everywhere. + +If it is zero, Lemma~\ref{dyn:orbits} makes the marked period +orbit dense in $\mathcal D$. It meets the nonempty open subset +provided by Proposition~\ref{deg:open}. Re-marking does not change +the complex surface. Moreover K3 surfaces with the same marked +period are isomorphic after their markings are forgotten: signs +and reflections in algebraic roots match their K\"ahler +chambers, and global Torelli then supplies an isomorphism. +Thus the all-point property $L$ in that open region holds on $X$. + +If the rational dimension is one, let $v$ be a primitive +integral generator of that direction. It is positive because +$P$ is positive. The second part of Lemma~\ref{dyn:orbits} +allows us to re-mark within the stabilizer of $v$ so that the +period lies in the relatively open set $\mathcal O_v$ of +Theorem~\ref{rat:open}. The same Torelli argument gives $L$ +everywhere on $X$. Notice that this step uses an open subset +inside the fixed-v locus itself. + +If the rational dimension is two, $P$ is a rational plane. +Its rational orthogonal complement has signature $(1,19)$. +The positive cone is open and nonempty, so it contains a +rational vector, which can be multiplied by an integer to give +a positive integral $(1,1)$-class. The K3 projectivity criterion +implies that $X$ is projective. Proposition~\ref{prj:strips} +therefore gives $L$ at every point of $X$. + +Theorem~\ref{ana:cap} now applies to $X$, proving the stated +approximation statement with all its quantifiers. The estimates +there are compact-uniform estimates for maps into $X$. +On the compact target, any two smooth Hermitian distances are +uniformly equivalent, so the conclusion holds for the +arbitrary metric specified in the statement. +\end{proof} diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/globalization.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/globalization.tex new file mode 100644 index 0000000000000000000000000000000000000000..99e4a46f0f81de6ba0df5e8fd6fbaa00b1c1d8c9 --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/globalization.tex @@ -0,0 +1,646 @@ +\section{From polydiscs to convex approximation}\label{sec:globalization} + +We now have approximation on polydiscs in every dimension. Convex +sets need not fit a polydisc inside the original map domain, so this +alone does not prove CAP. The required enlargement has the form +\[ + Q_-=Q\cap\{\operatorname{Im}z_1\leq0\}\subset Q, +\] +where $Q$ is compact and convex. Given a map near $Q_-$, we will +first approximate it on the joining real face by an entire map, +with control of tangential derivatives. That map supplies values on +the upper side. Its accuracy need not persist on any fixed complex +collar of the face, so the second step is a gluing estimate measured +on the face itself. Repeating this enlargement across the faces of +a polytope will prove Theorem~\ref{ana:cap}. + +The induction must retain arbitrary extra polydisc factors: they +include both genuine source parameters and the auxiliary variables +of the chart sprays used to glue maps. + +For \(r\geq0\), denote by \({\cal A}_r\) the following statement: +for every compact convex \(C\subset\C^r\), every extra closed +polydisc \(P\), and every map holomorphic near \(C\times P\), +there are entire maps approximating it uniformly on \(C\times P\). +All the approximations in \({\cal A}_r\) may also be required in +any fixed finite number of derivatives on a slightly smaller +compact product. To justify this last formulation, first +thicken \(C,P\) within the given domain, approximate uniformly +there, and use chart-spray coordinates and Cauchy estimates on +the smaller product. This uses no new approximation assertion. + +\subsection{Holomorphic approximation with a real parameter} + +The slices of a convex real face vary with its real coordinate. +We will join approximations on neighboring slices by a smooth +real-parameter path. The next lemma makes such a path holomorphic +near the real interval; the following lemma then makes it entire. + +\begin{lemma}[Complexifying a real parameter]\label{glue:interval} +Let $I\subset\R$ be a compact interval and let $C\subset\C^k$ be a +compact convex set. Suppose $H_s(q)$ is a smooth family, for $s$ in a +neighborhood of $I$, of holomorphic maps on a common neighborhood of +$C$, with values in a complex surface. Smoothness is understood on a +common neighborhood with all finite derivatives under consideration. +For every finite $j$ and every $\epsilon>0$, after an arbitrarily small +loss of the specified compact margins, $H$ can be approximated in +$C^j(I\times C)$ by a map holomorphic on a neighborhood of $I\times C$ +in $\C\times\C^k$. The size of this complex neighborhood may depend on +the approximant. Additional closed polydisc factors can be included in +$C$, with fixed larger neighborhoods. +\end{lemma} + +\begin{proof} +We give the gluing detail to avoid assuming a holomorphic extension of +the original smooth family. By Lemma~\ref{loc:spray}, a finite cover of +$I$ by short intervals admits fixed graph charts +$E_i(q,v)$ over the corresponding maps. First choose a smooth family of +small chart sprays $H_s^\sharp(q,v)$ centered at $H_s(q)$, holomorphic in +$(q,v)$, on a common small fiber ball. This can be done explicitly along +the interval. In one fixed chart, translate its fiber coordinates to +the center representing $H_s$. When changing charts at a division point +$s_0$, express the preceding spray in the new chart and take its fiber +displacement at $s_0$. On a short overlap interpolate this displacement +to the fixed displacement at $s_0$, leaving the moving center unchanged. +The two displacements are as close in the first fiber derivative as +desired when the overlap is short. The derivatives therefore remain +invertible. Finitely many such operations and a smaller common fiber +ball give the asserted smooth spray family. + +In each division interval write this family in one fixed graph chart. +Approximate its coordinate functions by polynomials in $s$ with +coefficients holomorphic in $(q,v)$, retaining any prescribed finite +number of derivatives, and interpolate its endpoint jets through order +$M$. This is ordinary approximation in a Banach space of holomorphic +functions on a smaller fixed neighborhood: approximate a sufficiently +high real derivative uniformly, integrate, and make a finite Hermite +correction at the endpoints. The correction tends to zero with the +initial approximation error. The target sprays obtained on adjacent +intervals consequently have identical $s$-jets through order $M$ at the +division point. + +Now, after the finitely many polynomials have been fixed, choose a small +complex thickness $\delta>0$ around the real intervals. Extend each +interval slightly past its endpoints, so adjacent rectangles overlap +within distance $O(\delta)$ of their common division point. Their +spray transition differs from the identity by $O(\delta^{M+1})$ there, +uniformly on a fixed smaller fiber ball and parameter compactum. This +is Taylor's formula applied to the holomorphic coordinate transitions; +the constants are fixed before $\delta$ is chosen. + +Additive splitting on these finitely many thin planar rectangles costs +at most $C\delta^{-1}$. Indeed, use cutoffs across the vertical overlap +bands, whose first derivatives are $O(\delta^{-1})$, and the +Cauchy--Green operator on a bounded planar set. For simultaneous +splitting, each cut contributes a pair of corrections on the entire +portion to its left and right; that contribution has zero jump at every +other cut. The number of cuts is fixed. The source variables $(q,v)$ +are passive for this operator, so no parameter radius is lost. + +For clarity, write the consecutive transitions as +$\gamma_i=\operatorname{id}+c_i$, with +$\|c_i\|+\|D_vc_i\|\le A\delta^{M+1}$ on a fixed smaller fiber ball. +Let $R$ be the simultaneous right inverse, of norm at most $K/\delta$, +for the jump operator $(a_i)\mapsto(a_{i+1}-a_i)$. +The central maps are identified by the fixed-point equation +\[ + (a_i)=R\bigl((c_i(\,\cdot\,,a_i))_i\bigr). +\] +It preserves the ball of radius $2KA\delta^M$ and is contractive there +when $KA\delta^M<1/2$. The corrections are therefore $O(\delta^M)$. +Substituting them into the corresponding sprays gives central maps +agreeing on overlaps. Cauchy estimates on rectangles with margins +comparable to $\delta$ bound their correction derivatives through order +$j$ by $O(\delta^{M-j})$. Derivatives in the other variables use their +fixed margins. Take $M>j+2$ and then $\delta$ small. Together with the +initial polynomial accuracy this gives the prescribed $C^j$ estimate. +\end{proof} +\begin{lemma}\label{ana:parameter-disc} +Assume \({\cal A}_r\). Let \(I\subset\R\) be a compact interval, +\(C\subset\C^r\) compact and convex, and \(P\) a closed polydisc. +A map holomorphic near \(I\times C\times P\) can be approximated +there, with any prescribed finite derivative accuracy and smaller +positive margins, by entire maps of all the displayed complex +variables. +\end{lemma} + +\begin{proof} +First shrink any unused margins so that the given map \(H\) is +defined on \(V\times U\), where \(V\subset\C\) is a simply +connected bounded neighborhood of \(I\), and \(U\) contains a +slightly thickened compact product \(C\times P\). +A thin open rectangle is a possible \(V\). +Let \(\phi:V\to\Delta\) be a Riemann map. Choose \(0<\rho<1\) +so that \(\phi(I)\Subset\Delta_\rho\). +The map +\[ + (\zeta,y,u)\longmapsto H(\phi^{-1}(\zeta),y,u) +\] +is defined near +\(\overline\Delta_\rho\times C\times P\), with slightly smaller +positive margins. Apply \({\cal A}_r\), using the \(\zeta\) +disc as an extra passive factor. This gives an entire map +\(E(\zeta,y,u)\) with arbitrarily good derivative approximation +on a smaller product still containing \(\phi(I)\times C\times P\). +Consequently \(E(\phi(s),y,u)\) approximates \(H(s,y,u)\) with +the required derivatives along \(I\times C\times P\). + +Choose a closed convex neighborhood \(I'\) of \(I\) contained in +\(V\). Scalar polynomial approximation, with Cauchy estimates +after a further harmless shrink, approximates \(\phi\) and any +specified finite number of its derivatives on \(I'\) by a +polynomial \(p\). At this stage \(E\) is fixed. On the compact +sets in question all its derivatives up to the required order +are bounded. The chain rule therefore shows that +\(E(p(s),y,u)\) is as close to \(E(\phi(s),y,u)\) as desired. +The map \(E(p(s),y,u)\) is entire in every variable. +\end{proof} + +\subsection{Approximation on a convex real face} + +\begin{lemma}\label{ana:face} +Assume \({\cal A}_{n-1}\), where \(n\geq1\), and +Proposition~\ref{ana:polydisc}. Let +\(F\subset\R\times\C^{n-1}\) be compact and convex, and let \(P\) +be a closed polydisc. Every holomorphic map near \(F\times P\) +can be approximated on \(F\times P\) by entire maps on +\(\C^n\times\C^{\dim P}\), with any fixed finite derivative +accuracy along the real face and smaller positive margins. +\end{lemma} + +\begin{proof} +Write the variables as \((t,y,u)\), with \(t\) real on the face. +Thicken \(F\) slightly within \(\R\times\C^{n-1}\), and thicken +\(P\), so that all eventual estimates take place strictly inside +the initial domain. Denote the thickened face again by \(F\). +Its projection is a compact interval \(I\), and write +\(F_t=\{y:(t,y)\in F\}\). + +We first specify the finite cover used below. For each \(t_0\in I\), +choose a compact convex neighborhood \(C_{t_0}\) of \(F_{t_0}\) +in the \(y\) variables and an open complex disc \(D_{t_0}\) about \(t_0\) +such that the given map \(f\) is defined near +\[ + \overline D_{t_0}\times C_{t_0}\times P, +\] +including some fixed smaller margins. These choices are possible +by compactness of \(F_{t_0}\times P\). After reducing the real +interval about \(t_0\), it has +\[ + F_t\Subset\operatorname{int}C_{t_0} +\] +uniformly on that interval. Indeed, otherwise a convergent +sequence \((t_\nu,y_\nu)\in F\), with \(t_\nu\to t_0\), would +contradict \(F_{t_0}\subset\operatorname{int}C_{t_0}\). +Choose a finite subdivision of \(I\), with small overlaps at +its division points, subordinate to these intervals. +Denote the corresponding products by \(D_i\times C_i\times P\). +All sets can be chosen a little larger than the sets used in +the subsequent estimates. + +At a division point \(b\), both adjacent \(C_i\)'s contain +\(F_b\) in their interiors. Choose compact convex sets \(C^-\) +and \(C\) with nonempty interiors such that +\[ + F_b\subset\operatorname{int}C^-, + \qquad C^-\Subset\operatorname{int}C, + \qquad C\Subset\operatorname{int}(C_i\cap C_{i+1}). +\] +By the preceding compactness argument, after reducing a complex +disc \(D_b\) about \(b\), the sets \(F_t\), for real \(t\) near +\(b\), lie strictly inside \(C^-\), and +\[ + \overline D_b\times C\times P + \Subset (D_i\times\operatorname{int}C_i\times + \operatorname{int}P^+) + \cap(D_{i+1}\times\operatorname{int}C_{i+1}\times + \operatorname{int}P^+) +\] +for an available larger passive polydisc \(P^+\). +All transition intervals and their cutoff functions are fixed now. +Also fix a large \(y\)-polydisc \(B\) containing all the \(C_i\) +and all face projections, with a positive margin. + +By \({\cal A}_{n-1}\), applied with the \(t\) disc and \(u\) +polydisc passive, choose entire maps \(g_i(t,y,u)\) arbitrarily +close to \(f\), with any fixed finite derivative accuracy, on +\(\overline D_i\times C_i\times P\). +More explicitly, apply the approximation assertion on slightly +enlarged closed products still in the given domain and use +Cauchy estimates on the displayed products. +We join two adjacent \(g_i\)'s over +one transition, with control on its required face slices. + +On a slightly enlarged \(D_b\times C\times P\), a chart spray +over \(f\), from Lemma~\ref{loc:spray}, expresses the two maps as +\[ + g_i=f^\sharp(\cdot,a_i),\qquad + g_{i+1}=f^\sharp(\cdot,a_{i+1}), +\] +where \(a_i,a_{i+1}\) and their required derivatives are as small +as desired. Choose a smooth function \(\chi:[0,1]\to[0,1]\) +equal to \(0\) near \(0\) and \(1\) near \(1\), and set +\[ + f_s=f^\sharp\bigl(\cdot,(1-\chi(s))a_i+\chi(s)a_{i+1}\bigr). +\] +This is a smooth \(s\)-family, holomorphic in \((t,y,u)\). +It equals the entire map \(g_i\) near \(s=0\) and \(g_{i+1}\) +near \(s=1\); on these subintervals use those entire definitions +on all \(y\). It is arbitrarily close to \(f\) on the local +product, and its positive \(s\)-derivatives up to any fixed +order are arbitrarily small there. The constants here depend +only on the previously fixed local chart and cutoffs. + +The path $f_s$ is defined only over the small convex set $C$, except +near its endpoints, where it equals an entire map. We need a path +defined over the common large polydisc $B$, still close to $f_s$ on +$C^-$, and equal to the prescribed endpoint maps on all of $B$. +Direct interpolation in the chart over $f$ cannot provide this, +because that chart is not defined over $B$. + +We obtain the larger domain by expanding the $y$ variable only +where the path already equals an entire endpoint map. After entire +approximation on the fixed small set $C$, we undo that expansion. +Choose \(d_*\in\operatorname{int}C\). For some \(L\geq1\), +\begin{equation}\label{ana:eq:inverse-scaling} + d_*+L^{-1}(B-d_*)\Subset\operatorname{int}C. +\end{equation} +Choose a smooth positive function \(\lambda:[0,1]\to[1,L]\) +which equals \(1\) outside the endpoint intervals where \(f_s\) +is an entire endpoint map, and equals \(L\) on smaller endpoint +intervals. All of these choices are fixed. The family +\[ + \widetilde f_s(t,y,u) + =f_s\bigl(t,d_*+\lambda(s)(y-d_*),u\bigr), + \qquad y\in C, +\] +is defined on a common neighborhood of +\([0,1]\times\overline D_b\times C\times P\). +This assertion uses entirety precisely where \(\lambda>1\); +where only the local definition of \(f_s\) is available, +\(\lambda=1\). +Apply Lemma~\ref{glue:interval} to make this family jointly +holomorphic near that compact product, with arbitrarily good +finite derivative accuracy. Next apply +Lemma~\ref{ana:parameter-disc}, with \({\cal A}_{n-1}\) and +passive factors \(D_b,P\), to obtain an entire approximant +\(E(s,t,y,u)\). + +For real \(s\), put +\[ + A_s(t,y,u)= + E\bigl(s,t,d_*+\lambda(s)^{-1}(y-d_*),u\bigr). +\] +It is smooth in \(s\) and holomorphic in \((t,y,u)\) on the +whole prescribed large product. On \(C'=C^-\Subset +\operatorname{int}C\), which contains all required overlap slices, +the inverse scaling stays in a fixed compact subset of +\(\operatorname{int}C\): by convexity it lies in the convex hull +of \(C'\cup\{d_*\}\). The two scalings cancel exactly: +\[ + \widetilde f_s\bigl(t,d_*+\lambda(s)^{-1}(y-d_*),u\bigr) + =f_s(t,y,u). +\] +Thus approximating $\widetilde f_s$ on that fixed compact subset +makes \(A_s\) arbitrarily close to \(f_s\), with the required +derivatives, on these slices. +On the smaller endpoint intervals, \eqref{ana:eq:inverse-scaling} +gives the stronger statement that \(A_s\) is arbitrarily close +to \(g_i\), respectively \(g_{i+1}\), on the entire large +product \(\overline D_b\times B\times P\). +The chain rule costs only finite constants, since \(\lambda\) +and all scaling factors have already been fixed. + +We may arrange exact endpoint joins. A chart spray over +\(g_i\) exists on a slightly larger version of that large +product. On a smaller initial \(s\)-interval, write +\(A_s=g_i^\sharp(\cdot,\beta_s)\). Here \(\beta_s\), including +all required derivatives, is as small as desired. +Replace \(\beta_s\) by \(\kappa(s)\beta_s\), where \(\kappa\) +is \(0\) near \(0\) and \(1\) before leaving this endpoint +interval. Do the analogous replacement near \(1\). +The repaired path, denoted \(H_s\), is exactly \(g_i\) near +one endpoint and exactly \(g_{i+1}\) near the other, and retains +the desired approximation on the overlap slices. It need +only be holomorphic on a neighborhood of the fixed large +product, which the chart construction provides. + +There is no circular smallness requirement in this procedure. +First choose the local \(g_i\)'s sufficiently accurate that +the interpolated \(f_s\) has the desired small error after the +fixed real transition cutoff. Then, with \(g_i,\lambda,B\) +and the endpoint chart sprays fixed, choose the complexification +and entire approximation of \(\widetilde f_s\) as accurate as +needed to overcome all their finite derivative and chart +constants. Finally perform the exact endpoint repairs. + +Substitute a fixed smooth cutoff \(s=s(t)\) across the real +transition interval. The result is a smooth real \(t\)-family, +holomorphic in \((y,u)\) on the large product, which joins the +two local entire maps and is as close to \(f\) as prescribed, +with all required tangential derivatives, on the face slices. +Doing this at the finitely many disjoint transition intervals +gives one such family on all of \(I\). Extend it smoothly +past the endpoints of \(I\) using the first and last entire +maps \(g_i\), to which it already agrees on endpoint intervals. + +Apply Lemma~\ref{glue:interval} once more, now to this real +\(t\)-family on the fixed large polydisc in \((y,u)\). +The result is jointly holomorphic near +\(I\times B\times P\) and retains the prescribed accuracy on +the face. Proposition~\ref{ana:polydisc} is \({\cal A}_0\); +thus Lemma~\ref{ana:parameter-disc} with \(r=0\) converts this +last family into an entire map in \((t,y,u)\), still with the +required accuracy. +When \(n=1\), there are no \(y\) variables and the scaling +step is omitted; the same argument, or just this last +complexification and \({\cal A}_0\) step, applies. +\end{proof} + +\subsection{Gluing across a real interface} + +Lemma~\ref{ana:face} provides an entire upper-side map close to the +old map on the real face. Unlike the buffered overlap in +Lemma~\ref{glue:open}, the complex neighborhood on which this +accuracy persists may depend on the approximant. We therefore solve +the additive jump problem using only its H\"older trace. The +nonlinear correction will use the same quadratic iteration as +open-overlap gluing, now in this trace norm. + +Fix $0<\alpha<1$. For a compact convex set $Q\subset\C^n$, write +\[ + Q_r=\{z:\operatorname{dist}(z,Q)0\}. +\] +We always start with $r>0$, including when $Q$ has empty interior. +An auxiliary variable $v$ ranges over a ball $B_b\subset\C^N$. +All functions under consideration are holomorphic in $v$. All norms +below are uniform in $v$; the H\"older seminorm is taken in the +base variables only. Traces on $M_r$ are holomorphic in +$z'=(z_2,\ldots,z_n)$ on every available slice. For a side function set +\[ + \|a\|_{r,b,*}=\sup_{Q_r^\pm\times B_b}|a| + +\sup_{v\in B_b}\|a(\,\cdot\,,v)|_{M_r}\|_{C^\alpha}. +\] +The H\"older norm on an open bounded set means the supremum norm plus +the supremum of all difference quotients in that set. The constants in +this section may depend on a bounded range of $r$, the dimensions, +$\alpha$, and the fixed joining geometry, but not on the input functions. + +\begin{lemma}[Additive splitting at a real interface]\label{glue:linear-trace} +There are $C>0$ and an integer $p\geq1$ with the following property. +For $00$. +An auxiliary variable $v$ ranges over a ball $B_b\subset\C^N$; +the splitting operators below act only on $z$ and preserve +holomorphic dependence on $v$. + +Suppose two open sets +$A,B$ cover a neighborhood of $\overline{Q_r}$ and have a fixed buffered +overlap: there is a smooth cutoff equal to zero near $A\setminus B$ and +one near $B\setminus A$, whose transition lies inside $A\cap B$ with a +positive collar. The collar is understood relative to the total domain; +after shrinking $Q_r$ its width has a positive lower bound depending +only on the fixed geometry. For a bounded holomorphic jump $u$, +the functions $\chi u$ and $(\chi-1)u$, extended by zero where their +respective multipliers vanish, have the same $\bar\partial$ derivative. +Fix $00$ fixed. +After any prescribed positive loss of base and fiber margins, the two +sprays admit fiber reparametrizations close to the identity which make +them agree. Evaluating at $v=0$ gives a glued holomorphic map. It is +arbitrarily close to the original map on each compactum where the +corresponding input spray is fixed, or has a uniform modulus of +continuity in the fiber variable. No such uniform target estimate is +asserted on a side whose spray varies without this control. +\end{lemma} + +\begin{proof} +Use the additive splitting to find $a,b$ with $b-a=c$. Shrink the +base and fiber domains by $h$. The splitting bounds and Cauchy +estimates in $v$ imply, with a fixed integer $p$ enlarged as necessary, +\[ + \|a\|+\|b\|\le Ch^{-p}e,\qquad e=\|c\|, +\] +including their needed fiber derivatives on intermediate smaller balls. +For $Ch^{-p}e$ sufficiently small relative to these margins, +$\operatorname{id}+b$ has a fiber inverse on a smaller ball, uniformly +in the base. The new transition is +\[ + (\operatorname{id}+b)^{-1}\circ\gamma\circ(\operatorname{id}+a) + =\operatorname{id}+d. +\] +The equation determining $d$ is +\[ + d= c(v+a)-c(v)-\{b(v+d)-b(v)\}. +\] +The first difference is quadratic in $e$, up to powers of $h^{-1}$. +The operation in braces is contractive on a small supremum-norm ball, +by the first fiber derivative bound. It follows that +\begin{equation}\label{eq:glue-quadratic} + \|d\|\le Ch^{-p}e^2. +\end{equation} +These estimates involve a fixed finite number of derivatives. Thus one +common exponent $p$ works throughout the iteration; no derivatives of +increasing order are requested. + +Choose losses $h_i=h_0 2^{-i}$ whose sum fits within the prescribed +margins, leaving intermediate buffers at each step. With +$A=Ch_0^{-p}$, Equation~\eqref{eq:glue-quadratic} reads +$e_{i+1}\le A2^{pi}e_i^2$. Put $E_i=A2^{p(i+1)}e_i$. Then +$E_{i+1}\le E_i^2$. Choosing $E_0<1/2$ gives doubly exponential decay. +In particular all sums $\sum_i h_i^{-q}e_i$ converge for every fixed +$q$, and are as small as desired with $e_0$. + +The successive fiber reparametrizations are composed on the nested +domains. Their displacements fit within the reserved buffers and their +fiber derivatives have summable deviations from the identity. The +compositions therefore converge uniformly. Their limiting transitions +are the identity. +Composing the two original sprays with these limiting side maps gives +equal maps on the overlap. +The target approximation assertion follows from the stated continuity +control and the smallness of the limiting fiber corrections. +\end{proof} + +\subsection{A change of coordinates with a thin parameter range} + +Enlarging one coordinate of a polydisc directly leaves an entire +transverse polydisc over each boundary point; its image need not fit +one witness neighborhood for $L$. We instead deform the source by a +polynomial automorphism. It will be nearly the identity on the old +polydisc, while the rest of the enlarged source has coordinates +$(s,\xi)$ with $s$ planar and $\xi$ bounded. The following estimate +constructs these coordinates on inverse images of the unit polydisc. +The factors $d^{-1}$ in their formula will then make the $x$-image +thin as $\xi$ varies with $s$ fixed on the joining annulus. + +\begin{lemma}\label{ana:concentration} +Fix \(m\geq2\), numbers +\[ + 1r_1\}\), all \(x_j\) are nonzero and +\begin{equation}\label{ana:eq:ratio-bound} + \left|\frac{H_d(x)_j}{a x_{j-1}}\right|\leq q + \qquad(2\leq j\leq m). +\end{equation} +Consequently the holomorphic functions +\[ + \xi_j(x)=\log\left(\frac{P_j(x_j)}{a x_{j-1}}\right), + \qquad 2\leq j\leq m, +\] +using the logarithm on \(\{|\zeta-1|<1\}\), satisfy +\[ + |\xi_j(x)|\leq B:=-\log(1-q). +\] +Here the assertions about holomorphicity are on any open inverse +image of the unit polydisc; all the estimates also hold on its +closure. With +\[ + D=d^{m-1},\qquad + \rho_1=1,\qquad \rho_j=r_j(a\rho_{j-1})^{1/d}, +\] +there is a single-valued holomorphic function \(s\) on this end such +that +\begin{equation}\label{ana:eq:parameterization} + x_j=\rho_j s^{d^{m-j}} + \exp\left(\sum_{k=2}^j\frac{\xi_k}{d^{j-k+1}}\right), + \qquad 1\leq j\leq m. +\end{equation} +For \(j=1\), the sum is empty and this says \(x_1=s^D\). +\end{lemma} + +\begin{proof} +Put \(r_*=r_1\). Choose \(S>\max(2,r_*)\), and then \(d_0\geq2\) +large enough that +\[ + (M/r_*)^d\geq4M\qquad(M\geq S,\ d\geq d_0). +\] +For example, it suffices to impose this at \(M=S,d=d_0\), since +\((M/r_*)^d/M\) increases in both variables on this range. +The cone +\[ + {\cal E}=\{u:\max_{j\geq2}|u_j|>\max(S,|u_1|)\} +\] +is forward invariant under \(H_d\). Indeed, if +\(M=\max_{j\geq2}|u_j|\) and \(|u_k|=M\), then +\[ + |H_d(u)_k|\geq(M/r_*)^d-aM\geq3M, + \qquad |H_d(u)_1|\leq aM. +\] +The transverse maximum therefore grows by at least a factor \(3\). +In particular, no forward orbit that ends in +\(\overline\Delta_1^{\,m}\) can enter \({\cal E}\). + +Choose \(C\) larger than +\[ + 1,\quad 4a,\quad 16a^2r_*^2,\quad 4S. +\] +If \(A=\max_{j\geq2}|P_j(x_j)|>C\max(1,|x_1|)\), then +\[ + \max_{j\geq2}|x_j|\leq r_* A^{1/d}\leq r_*\sqrt A. +\] +For an index \(k\) attaining \(A\), both possible terms +\(a|x_{k-1}|\), namely \(a|x_1|\) when \(k=2\) and a transverse +coordinate otherwise, are at most \(A/4\). Hence +\[ + |H_d(x)_k|\geq3A/4,\qquad + |H_d(x)_1|\leq a r_*\sqrt A\leq A/4. +\] +Thus \(H_d(x)\in{\cal E}\), a contradiction. This proves +\eqref{ana:eq:power-bound}; the same argument applies at every +nonfinal point of the orbit. At its final point the bound is +immediate. + +Write \(y=H_d(x)\). Applying the power bound at \(y\), or using +\(\|y\|_\infty\leq1\) if \(y\) is the final point, gives +\begin{equation}\label{ana:eq:next-coordinate} + |y_j|\leq r_j\bigl(C\max(1,a|x_m|)\bigr)^{1/d} + \qquad(j\geq2). +\end{equation} +Let \(L_j=\log^+|x_j|\), \(M=L_m\), and +\(A_1=\log(r_*\sqrt C+a)\). From +\(P_j(x_j)=y_j+a x_{j-1}\) and +\eqref{ana:eq:next-coordinate}, +\[ + |P_j(x_j)| + \leq (r_*\sqrt C+a)\exp(L_{j-1}+M/d). +\] +Whether or not \(|x_j|\) exceeds \(1\), this implies +\begin{equation}\label{ana:eq:log-recursion} + L_j\leq \log r_j+A_1/d+L_{j-1}/d+M/d^2. +\end{equation} +Put \(A_2=\log r_*+A_1/2\). Iterating +\eqref{ana:eq:log-recursion} to \(j=m\) yields +\[ + M\leq 2A_2+d^{-(m-1)}L_1+ + M\sum_{i=0}^{m-2}d^{-2-i} + \leq 2A_2+d^{-(m-1)}L_1+\tfrac12M. +\] +For \(|x_1|\geq r_1\), define +\[ + t=(|x_1|/r_1)^{1/D}\geq1,\qquad + A_3=4A_2+2\log r_1. +\] +It follows that +\[ + M\leq A_3+2\log t,\qquad + |y_j|\leq r_j\exp((\log C+A_3)/d)t^{2/d}. +\] + +Choose a sufficiently small \(\delta>0\) so that +\[ + q:=\max_{2\leq j\leq m} + \frac{r_j+\delta}{a(r_{j-1}-\delta)}<1, + \qquad r_j-\delta>0. +\] +Increase \(d_0\), if necessary, so that for all \(d\geq d_0\) +\[ + r_j\exp((\log C+A_3)/d)\leq r_j+\delta,\qquad + r_j\bigl((1-q)a(r_{j-1}-\delta)\bigr)^{1/d} + \geq r_j-\delta . +\] +We prove successively that +\[ + |x_j|\geq(r_j-\delta)t^{d^{m-j}}\quad(1\leq j\leq m). +\] +The assertion for \(j=1\) follows from the definition of \(t\). +If it holds for \(j-1\), then +\[ + \frac{|y_j|}{a|x_{j-1}|} + \leq + \frac{r_j+\delta}{a(r_{j-1}-\delta)} + t^{\,2/d-d^{m-j+1}}\leq q. +\] +Therefore +\[ + |P_j(x_j)|\geq(1-q)a|x_{j-1}|, +\] +and taking the \(d\)-th root proves the next assertion. +This also proves \eqref{ana:eq:ratio-bound}. +Since \(P_j/(a x_{j-1})=1+y_j/(a x_{j-1})\), the indicated +logarithm is well defined; its power series gives the bound +\(|\xi_j|\leq\sum_{\nu\geq1}q^\nu/\nu=B\). + +There is no root choice in the definition of \(s\). Define it by +\[ + s=\frac{x_m}{\rho_m} + \exp\left(-\sum_{k=2}^m\frac{\xi_k}{d^{m-k+1}}\right). +\] +The equations +\[ + x_j^d=r_j^d a x_{j-1}\exp(\xi_j) +\] +then prove \eqref{ana:eq:parameterization} backwards from \(j=m\) +to \(j=1\), using the definition of the positive real numbers +\(\rho_j\). +\end{proof} + +\begin{proposition}\label{ana:polydisc} +Under the hypotheses of Theorem~\ref{ana:cap}, a map holomorphic near +a closed polydisc in \(\C^m\) can be approximated uniformly there by +entire maps \(\C^m\to Y\). More precisely, a map near +\(\overline\Delta_1^{\,m}\) can be approximated there by maps +holomorphic near \(\overline\Delta_R^{\,m}\), for every finite +\(R>1\). +\end{proposition} + +\begin{proof} +First prove the finite-radius assertion. When \(m=1\), the +one-dimensional boundary circle is covered by finitely many of +the neighborhoods in \(L\), and Proposition~\ref{loc:planar} +applies with no passive variables. + +Suppose \(m\geq2\). Let \(f\) be the initial map and let +\(\varepsilon>0\). Choose \(b>1\) with \(f\) holomorphic near +\(\overline\Delta_b^{\,m}\). Choose the radii \(r_j\), the number +\(a\), and \(\ell>0\) so that Lemma~\ref{ana:concentration} applies +and +\[ + 1R\), and fix \(N\geq1\) so that \(a^N R_+<1\). +Set +\[ + h_d=H_d^{-N}L^N,\qquad \Omega=\Delta_{R_+}^{\,m}. +\] +Thus \(h_d(\overline\Omega)\subset E_{d,N}\). + +On some fixed neighborhood of \(\overline\Delta_1^{\,m}\), +\begin{equation}\label{ana:eq:near-identity} + h_d\longrightarrow\operatorname{id}\quad(d\longrightarrow\infty). +\end{equation} +Here is a quantitative justification. Choose +\(10\). Its vertical differential is invertible and +its vertical maps are injective after reducing \(\beta\). +The value \(b'\) and this spray are fixed before \(d\) is increased. + +Write \(\Phi_d(s,\xi)\) for the right side of +\eqref{ana:eq:parameterization}, now allowing \(\xi\) to vary +independently. Use, for example, the three parameter polydiscs +\[ + \Xi_i=\{\xi\in\C^{m-1}:|\xi_j|\leq B+i\},\qquad i=1,2,3. +\] +Choose fixed numbers +\[ + r_1j), +\] +Integrating along line segments in the convex polydisc \(\Xi_3\) +and using \(\sum_{\nu\geq1}d^{-\nu}\leq2/d\) proves the diameter +bound with the displayed \(A_5\). +In particular \(\Phi_d\) takes the indicated product into +\(\Delta_{b'}^{\,m}\) for large \(d\). + +The compact set \(f(\overline\Delta_{b'}^{\,m})\) has a finite cover +by neighborhoods on which \(L\) is available. A Lebesgue number +for a slightly smaller such cover, uniform continuity of \(f\), +and \eqref{ana:eq:thin-range} show the following. After increasing +\(d\) and decreasing the fixed spray radius, all the values +\[ + f^\sharp(\Phi_d(s,\xi),v),\qquad \xi\in\Xi_3,\quad + v\in\overline\Delta_{\beta}^{\,2}, +\] +belong to one of these neighborhoods for each fixed \(s\) on the +outer circle \(|s|^D=c_3\). The chosen neighborhood may depend +on \(s\). The same increase of \(d\) ensures that +\[ + \sup_{\overline\Delta_1^{\,m}}d_Y(f(h_d(z)),f(z)) + <\varepsilon/2,\qquad + |h_d(z)_1|c_1\}, +\] +consider the maps with the common auxiliary variable \(v\) +\[ + F_A(z,v)=f^\sharp(h(z),v),\qquad + F_{B'}(z,v)=G(s(h(z)),\xi(h(z)),v). +\] +The first is defined on all of \(A\): by +\eqref{ana:eq:power-bound}, its transverse coordinates satisfy +\[ + |h(z)_j|\leq r_j(Cc_2)^{1/d}0$, there exists a holomorphic map +$F:\C^m\to X$ such that +\[ + \sup_{z\in K}d_X(F(z),f(z))<\epsilon. +\] +\end{theorem} + +Theorem~\ref{thm:main} answers the K3 question in +\cite[Section~8, Problem~C]{ForstnericLarussonSurvey} and proves the +positive conjecture formulated explicitly in +\cite[Introduction]{XieZhao}. Its scope includes nonprojective +surfaces and imposes no condition on the Picard rank or existence of +an elliptic fibration. + +An \emph{entire curve} is a holomorphic map from $\C$, and it is an +\emph{immersion} if its derivative never +vanishes. The approximation and interpolation theory gives the +following concrete consequence, proved in Section~\ref{sec:dense-entire}. + +\begin{corollary}\label{cor:dense} +Let $S$ be a complex K3 surface. For every $x\in S$ +and every nonzero vector $v\in T_xS$, there is a holomorphic immersion +$f:\C\to S$ such that +\[ + f(0)=x,\qquad f'(0)=v,\qquad \overline{f(\C)}=S, +\] +where closure is taken in the ordinary complex topology. If $S$ is +projective, then $f(\C)$ is Zariski dense. +\end{corollary} + +Here the tangent vector, not just its line, is prescribed. For projective +$S$, the density conclusion confirms the K3 case of Campana's +prediction of Zariski-dense entire curves on special projective +manifolds \cite{Campana}. Here specialness means the absence of a +meromorphic fibration whose positive-dimensional base, endowed with +the orbifold structure induced by the multiple fibers, is of general +type. +See \cite[Conjecture~1.3]{CCR} for the prediction and +\cite[Proposition~8.9]{CCR} for the specialness of projective varieties +with trivial canonical bundle. The ordinary density and prescribed +first jet are additional conclusions. + +Section~\ref{sec:surface-consequences} derives the Oka property for +Enriques surfaces and hence for all minimal compact complex surfaces +of Kodaira dimension zero. It also obtains global dominating sprays +on projective K3 and Enriques surfaces. A separate class-VII +classification there combines earlier Oka results with the global +spherical shell theorem of the companion manuscript \cite{OpenAIGSS}; +that theorem is not an input to the K3 argument. + +\subsection{Earlier results and the remaining obstacles} + +The use of holomorphic sprays and approximation to realize topological +flexibility is central to Gromov's Oka principle for elliptic bundles +\cite{GromovOka}. Forstneri\v c's convex approximation characterization +\cite[Theorem~0.1]{ForstnericCAP} makes CAP an effective criterion for +this flexibility, while his interpolation theorem +\cite[Corollary~1.3]{ForstnericInterpolation} links it to prescribed +holomorphic data on subvarieties. Our proof uses local chart sprays +and splitting arguments in this tradition, with the uniform estimates +needed for the strip and convex-face constructions made explicit. + +Holomorphic flexibility on K3 surfaces has been studied through entire +curves, domination by affine space, and approximation. Buzzard and Lu +proved that elliptic and Kummer K3 surfaces are holomorphically +dominable by $\C^2$ \cite[Propositions~4.4--4.5]{BuzzardLu}. +Dominability means the existence of a map $\C^2\to X$ whose +differential is surjective somewhere. Forstneri\v c and L\'arusson +strengthened this for every Kummer surface to \emph{strong +dominability}: such a map can pass through each prescribed target +point with surjective differential +\cite[Corollary~3]{ForstnericLarussonSurfaces}. These properties +concern individual maps; CAP asks for approximation of every local +map in every source dimension. + +For arbitrary complex K3 surfaces, Kamenova, Lu and Verbitsky proved +that the Kobayashi pseudodistance vanishes +\cite[Corollary~2.2]{KamenovaLuVerbitsky}; the period-dynamics argument +is corrected in \cite[Section~4]{VerbitskyErratum}. This removes an +intrinsic obstruction to holomorphic flexibility, but does not +provide approximation or interpolation. + +Alarc\'on and Forstneri\v c introduced the Oka-1 property and established +it for Kummer surfaces and elliptic K3 surfaces +\cite[Proposition~8.4 and Corollary~8.6]{AlarconForstneric}. +Oka-1 concerns approximation and discrete jet interpolation from open +Riemann surfaces, within prescribed continuous homotopy classes. Its +full formulation is recalled below. +The source dimension distinction matters: the one-dimensional +property alone does not give Theorem~\ref{thm:main}. + +For projective K3 geometry, Chen, Gounelas and Liedtke developed +existence results for curves of prescribed geometric genus +\cite{ChenGounelasLiedtke}. Chen and Gounelas then proved that every +projective K3 surface contains genus-one curves of unbounded +self-intersection whose smooth normalizations vary in moduli +\cite[Theorem~A]{ChenGounelas}. We use two of their families with +different ample degrees. Their complete genus-one fibers supply +holomorphic flows for all complex times, and the difference in degrees +forces their tangent directions to be generically independent. + +More recently, Xie and Zhao proved that smooth $(2,2,2)$ hypersurfaces +in $(\PP^1)^3$ are Oka and established dense $G_\delta$ sets of Oka +K3 periods and corresponding local deformation parameters +\cite[Theorems~A, B, and~E]{XieZhao}. Their work combines geometric +constructions of flexibility with arithmetic dynamics of periods and +Torelli theory. We use the same period-theoretic framework, including +the corrected orbit alternatives of Verbitsky +\cite{VerbitskyErratum}, with different geometric constructions. +A dense collection of Oka surfaces does not by itself settle the +property for each surface: the transfer must address every possible +rational subspace of its period plane. + +Two analytic obstacles arise even before that transfer. Local +approximation in one variable must retain arbitrary holomorphic +parameters and then imply approximation on arbitrary convex sets. +Moreover, the geometric constructions produce long chains of annuli; +errors at their joins must be corrected without exhausting a fixed +positive transverse width. We handle the first issue by a polynomial +change of source coordinates and gluing estimates measured on real +faces. We handle the second by exact symplectic sewing with bounds +independent of the chain lengths. + +\subsection{The proof and its organization} + +The analytic and geometric parts meet at a local operation, denoted +$L$. It enlarges a planar disc across an attached disc, for maps with +image in a sufficiently small target neighborhood, while retaining +any prescribed smaller polydisc of passive holomorphic parameters. +Section~\ref{sec:local} defines $L$ precisely and obtains it from two +strip maps, one defined on $\C\times\Delta$ and the other on +$\Delta\times\C$, with transverse complete directions. Aligning +their coordinate axes exactly makes the overlap error arbitrarily +small. A bounded additive splitting and a contraction correct it. +The same section propagates $L$ across the center of a small disc +whose boundary already lies in the locus where $L$ holds. + +Sections~\ref{sec:gluing} and~\ref{sec:globalization} prove that $L$ +at every point implies CAP in every dimension +(Theorem~\ref{ana:cap}). A polynomial automorphism concentrates the +part of an enlarged polydisc outside the original domain into one +long coordinate and bounded transverse coordinates. The planar +operation with passive parameters then gives approximation on +polydiscs. To pass to an arbitrary convex set, we remove the defining +real faces of a containing polytope one at a time. The necessary +gluing estimate uses H\"older accuracy on each real face, without +requiring a fixed complex thickness on which that accuracy holds. + +Section~\ref{sec:projective} constructs the two complete directions +on projective K3 surfaces using the Chen--Gounelas families. Propagation +across their algebraic exceptional set gives $L$ everywhere. This +handles one case of the final period argument and also exhibits a +direct source of the required local geometry. + +The nonprojective cases require strips on open regions of period +space. Section~\ref{sew:section} first establishes the uniform sewing +theorem. On an annulus, a closed holomorphic one-form can have a +nonzero period. Exactness of the symplectic transition removes this +obstruction: the transverse constant Fourier coefficient becomes +quadratic in the error. The other Fourier modes admit a splitting +whose bounds are independent of the number and lengths of the +annuli. A quadratic iteration then retains a positive transverse +radius along an infinite chain. + +To describe the geometric transfer, fix the K3 lattice $\Lambda$, +the integral second-cohomology lattice with its intersection form. +A \emph{marking} identifies $H^2(X,\Z)$ with $\Lambda$. The real +and imaginary parts of a nonzero holomorphic two-form span an +oriented positive two-plane, the \emph{period} of the marked surface. +Its rational directions are the vectors in its intersection with +$\Lambda\otimes\Q$. + +Section~\ref{deg:section} constructs a nonempty open region in the +full period domain on which every surface has $L$ everywhere. It +smooths a union of two quadrics and sews annuli along their common +elliptic curve. The chart estimates persist under unrestricted small +K3 deformations. Section~\ref{rat:section} constructs a second kind +of region: for each fixed positive integral vector $v$, a relatively +open set among the period planes containing $v$. Here an isotrivial +genus-one fibration supplies the central model. Conservation of +imaginary symplectic action keeps all recentered annuli away from +singular fibers. Two independent cycles supply distinct strip leaves +after deformation and hence the transverse directions needed for $L$. + +Finally, Section~\ref{dyn:section} applies Ratner's orbit theorem and +Verbitsky's corrected rational-direction analysis. Periods with no +rational direction have orbits meeting the first open region. Periods +with exactly one rational direction have orbits, under its stabilizer, +meeting the corresponding second region. Rational period planes give +projective K3 surfaces, already treated above. Torelli theory +transfers $L$ along these orbits, and Theorem~\ref{ana:cap} finishes +the proof. Section~\ref{sec:dense-entire} derives interpolation, +Corollary~\ref{cor:dense}, and the further surface consequences. + +\paragraph{Conventions and standard inputs.} +Maps on a compact set are always defined on a neighborhood of that set. +Approximation may lose any prescribed positive amount of an auxiliary +domain margin. A strip means $\C\times\Delta_b$, with no injectivity +requirement unless stated; local regularity is specified at each use. +The analytic background consists of Stein theory, scalar approximation +and the $L^2$ $\bar\partial$ theorem +\cite{Forstneric,Hormander,Siu}. We recall the needed K3 period, +Torelli, and nef-pencil results at their applications +\cite[Chapters~2, 6--8, and~11]{Huybrechts}. The arithmetic inputs +are likewise stated where used. The local approximation, sewing, +geometric constructions, and reductions between them are proved here. diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/local.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/local.tex new file mode 100644 index 0000000000000000000000000000000000000000..7510424ee504dfe978bcf35404c11d2c67b1e04d --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/local.tex @@ -0,0 +1,553 @@ +\section{Local approximation from complete directions}\label{sec:local} + +The immediate goal is an operation that enlarges one planar source +variable while retaining arbitrary passive holomorphic parameters. +Two transverse complete strip maps provide this operation; a second +construction propagates it across points not initially covered by +strips. These are the inputs to the convex approximation argument +of Section~\ref{sec:globalization}. + +Write $\Delta_r=\{z\in\C:|z|0$ so small that +$5\rho0$ there is a polynomial $h_2$ satisfying +\[ +\|h_2-g_2\|_{D\times\overline P_*}<\eta,\qquad +\|h_2\|_{A\times\overline P_*}<\eta. +\] +Let $h=(h_1,h_2)$ and $C=U_A\cap U_B$. On $U_B\times P_*$, +$T=T_0(h_1)$ is holomorphic and boundedly invertible. +We seek corrections satisfying +\[ +\tau(h+\alpha)=h+\beta,\qquad +\alpha=R_Au,\quad\beta=T R_Bu. +\] +By Lemma~\ref{loc:splitting}, this is the fixed-point equation +\[ +u=\mathcal T(u):= +R_Au+T^{-1}\bigl(h-\tau(h+R_Au)\bigr) +\] +in $H^\infty(C\times P_*,\C^2)$. + +On the overlap $|h_1|<\rho/2$ and $|h_2|<\eta$. +Bounded first and second derivatives of $\tau$, and its exact +axis identity, give a constant $M\ge1$ with +\[ +\|T\|,\|T^{-1}\|\le M,\quad +\|\tau(h)-h\|\le M\eta,\quad +\|D\tau(h+\alpha)-T\|\le M(\eta+\|\alpha\|). +\] +These constants do not depend on the degrees or uncontrolled +coordinates of the polynomials. If $\kappa\ge1$ bounds both +splitting operators and $L_0=2M^2+1$, then on $\|u\|\le L_0\eta$, +\[ +\|\mathcal T(0)\|\le M^2\eta,\qquad +\|D\mathcal T(u)\|\le +M^2\kappa(1+\kappa L_0)\eta. +\] +For small $\eta$ the last bound is at most $1/2$, the ball maps +strictly into itself, and all transition evaluations remain in +$\Delta_{3\rho}^2$. The contraction theorem gives +\[ +\|\alpha\|\le\kappa L_0\eta,\qquad +\|\beta\|\le M\kappa L_0\eta. +\] + +On $U_A\times P_*$ only the second coordinate of $h+\alpha$ has +to be small, and on $U_B\times P_*$ only the first coordinate +of $h+\beta$ has to be small. Consequently +$\sigma_1(h+\alpha)$ and $\sigma_2(h+\beta)$ are defined on +these full domains and agree on their overlap. On $D\times P_*$, +the second formula is evaluated close to $g$ in $Q_c$ and +approximates $f$ as $\epsilon_1,\eta\to0$. Since the union of the +planar domains contains $D'$ and $P_0\Subset P_*$, this proves $L$. + +For the finite-length version choose $V$, $g$, and the two +polynomials using the limiting chart at $(0,c)$. Convergence on +slightly larger compact coordinate neighborhoods gives uniform +first and second derivative bounds, uniform inverse-chart domains, +and thus uniform constants in the contraction argument. +Exact axis agreement gives the same $O(\eta)$ initial error for +every pair. The polynomials are bounded on the fixed bounded +planar pieces and $\overline P_*$, so only a finite range of either +long coordinate is used. Choose a sufficiently late pair to contain +these ranges and make its chart near $(0,c)$ sufficiently close to +the limiting chart. The construction then approximates the original +data with the requested accuracy. +\end{proof} + +\begin{remark}\label{loc:alignment} +Two strip maps locally biholomorphic at chosen meeting preimages, +with transverse central curves there, can be put in the form of +Proposition~\ref{loc:criterion}. Locally lift the +first central curve through the inverse of the second strip, writing +the lift as $(a(z),t(z))$, where its bounded coordinate is $a$. +Replace the second strip by +$\widetilde\sigma_2(z,w)=\sigma_2(a(z),t(z)+w)$. +Transversality says $a'(0)\ne0$, and time zero gives exact axis +agreement. If the original second central curve also passes through +$x$ and its strip is regular there, some finite $c$ gives a regular +chart $\widetilde\sigma_2(0,c)=x$. +For finite-length families whose central germs converge to transverse +germs through a common point, the implicit function theorem first +gives nearby intersection points and meeting preimages. Translate +the preimages to the origin and apply the same construction. +Convergence and transversality give a uniformly positive +bounded-coordinate radius; the translations and the function $t$ +lose only a bounded amount of long-coordinate length. Thus this +construction also supplies the aligned families in the finite-length +part of the proposition. +\end{remark} + +\subsection{Sprays, deformation, and planar gluing} + +The strip criterion provides approximation when all input values lie +in one witness neighborhood. To apply it successively to a map whose +image meets several such neighborhoods, we need coordinates around +the graph of the map and a way to join close maps in those coordinates. +The next three lemmas supply exactly these tools. The use of sprays +and their gluing follows the framework of Gromov \cite{GromovOka} +and Forstneri\v c \cite{Forstneric}; here the sprays are constructed +only near the graph on a relatively compact Stein source. + +\begin{lemma}[Local chart sprays]\label{loc:spray} +Let $f:S\to X$ be holomorphic on a Stein manifold, let +$K\Subset S$ be compact, and assume $f^*TX$ is holomorphically +trivial on $S$. After shrinking the source neighborhood there +is a holomorphic map $f^\sharp(z,v)$ for $z$ near $K$ and +$v\in\Delta_r^2$, with $f^\sharp(z,0)=f(z)$, such that +$v\mapsto f^\sharp(z,v)$ is injective with invertible differential. +The hypothesis on $f^*TX$ holds for a contractible Stein source +and for a planar domain times a polydisc. +\end{lemma} + +\begin{proof} +The graph of $f$ is a closed Stein submanifold of $S\times X$ and +has a Stein neighborhood by the Stein neighborhood theorem +\cite{Siu}. On that neighborhood consider the vertical tangent +bundle for projection to $S$. A holomorphic frame along the graph +extends to holomorphic vertical vector fields by Cartan's +Theorem~B. Compose their small-time flows. The resulting map has +the prescribed value and a frame as its vertical differential +at $v=0$. Compactness and the holomorphic inverse function theorem +give a common radius on a neighborhood of $K$ and injectivity +in each parameter fiber. +For the last assertion, a planar domain has the homotopy type of +a one-dimensional CW complex, so every complex vector bundle on +its product with a polydisc is topologically trivial. The +Oka--Grauert principle makes it holomorphically trivial. +The same argument applies to a contractible Stein source +\cite{Forstneric}. +\end{proof} + +\begin{lemma}[Deforming a map on a relatively compact Stein domain] +\label{loc:deformation} +Let $\pi:\mathcal X\to B$ be a holomorphic family, and let +$f:S\to\mathcal X_{b_0}$ be holomorphic from a Stein manifold +with image in the submersion locus of $\pi$. On every relatively +compact source domain there is a holomorphic family of maps +$f_b$ into $\mathcal X_b$ for $b$ near $b_0$, with $f_{b_0}=f$. +\end{lemma} + +\begin{proof} +Work in local coordinates on $B$ at $b_0$. The graph of $f$ has +a Stein neighborhood in $S$ times the submersion locus. On this +neighborhood the surjection from tangent vectors vertical over $S$ +to the pulled-back base tangent bundle admits holomorphic lifts of +the base coordinate vector fields: its kernel is coherent, and +Cartan's Theorem~B gives surjectivity on global sections. Flow the +lifts successively for the small coordinate displacements of $b$. +On a relatively compact source set a common time radius is +available. The flows are vertical over $S$, and their projections +to $B$ are the desired coordinate translations. Their composition +is the required family of maps. +\end{proof} + +\begin{lemma}[Planar spray gluing]\label{loc:gluing} +Suppose two holomorphic sprays over bounded planar domains $U_A,U_B$ and +a parameter polydisc are uniformly close on their overlap, with +a positive margin in an auxiliary spray polydisc. Suppose the +first spray is a chart spray there with uniform inverse-chart +margins, and the closed planar fringes +are positively separated. If the closeness is sufficiently small, +their central maps can be glued after arbitrarily small +holomorphic corrections in the auxiliary variables. The glued +map is arbitrarily close to the first central map on any prescribed +compact subset of its domain on which that spray is fixed, or has +a uniform modulus of continuity in its auxiliary variables. +\end{lemma} + +\begin{proof} +Write the sprays as $F_A$ and $F_B$. Compact chart margins give +the transition +$\gamma(s,t,v)=F_A(s,t,\cdot)^{-1}(F_B(s,t,v)) +=v+E(s,t,v)$ on the overlap, with both +$\|E\|$ and $\|D_vE\|$ small on a smaller auxiliary polydisc. +The derivative estimate follows from the original closeness by +Cauchy estimates on the larger polydisc. Equality of the resulting +maps asks for $a=\gamma(b)$. +Use $a=R_Au$, $b=R_Bu$. Then +\[ +u=E(s,t,R_Bu) +\] +is a contraction on a small ball in the overlap $H^\infty$ space. +Lemma~\ref{loc:splitting} supplies its uniform bounds and preserves +holomorphic dependence on the passive parameters. The resulting +corrections remain in the auxiliary polydisc and are as small +as the original mismatch. Uniform continuity on the prescribed +compact sets gives the asserted approximation. +\end{proof} + +The domains in this lemma may be thin neighborhoods of compact +planar sets. In applications below the cutoff is chosen first on +a fixed overlap band. Its derivative bound, and the diameter bound +in Lemma~\ref{loc:splitting}, do not increase when the domains are +subsequently narrowed away from that band. + +\subsection{One planar enlargement with passive parameters} + +\begin{proposition}\label{loc:planar} +Let $K$ be a closed circular disc of radius $r$, or a closed +circular annulus with outer radius $r$, and let $f$ be holomorphic +near $K\times P$, where $P$ is a closed polydisc. Suppose that for +every $\zeta$ on the outer circle there is a coordinate witness +neighborhood $V_\zeta$ for $L$ containing $f(\{\zeta\}\times P)$. +For every $R>r$ and $P_0\Subset\operatorname{int}P$, one can +approximate $f$ uniformly on $K\times P_0$ by a map holomorphic +near the enlarged disc, respectively annulus with the same inner +radius and outer radius $R$, times $P_0$. +\end{proposition} + +\begin{proof} +We give a finite construction with explicit approximation margins. +Choose a compact intermediate parameter polydisc strictly between +$P_0$ and $P$. Compactness gives a partition of the outer circle +into finitely many closed arcs, with endpoint margins, such that +all values over each slightly enlarged arc and the intermediate +parameter set lie in one coordinate witness neighborhood $V_j$. +At the common endpoint of consecutive arcs the compact set of +values lies in $V_j\cap V_{j+1}$ with a positive target margin. +Take a slightly larger outer radius than $R$ during the construction. + +\emph{Attach radial spokes.} +At each endpoint choose a small filled polar rectangle crossing +the old outer circle, on which $f$ and a chart spray from +Lemma~\ref{loc:spray} have their values in both adjacent target +charts. The rectangles can be chosen mutually disjoint. Attach +to each rectangle a radial segment reaching the larger outer +radius. In one of its target charts, continue the coordinate +spray constantly along the segment from its attachment point on +the outer edge of the rectangle. This defines continuous +coordinate functions on the filled rectangle and attached segment, +holomorphic on its interior and holomorphic in the passive and +auxiliary parameters. + +The planar compact set has connected complement. Mergelyan +approximation, with parameters, approximates these coordinate +functions by polynomials. Here the parameter assertion follows +directly by taking finite Taylor sums on a slightly larger +parameter polydisc and applying scalar Mergelyan to their +coefficients. Consequently the approximation can be made uniform +with an auxiliary-parameter margin, and Cauchy estimates preserve +the required spray derivatives. On the radial segment its values +remain close to the endpoint values for all parameters. A thin +neighborhood of each segment therefore also has its values in +both adjacent $V_j$'s. + +Glue this extension to the old chart spray across a radial band +inside each original rectangle, using Lemma~\ref{loc:gluing}. +More explicitly, cut the union domain at a radius slightly larger +than $r$, still inside the old map domain. At the cut it consists +only of the disjoint rectangles. A radial cutoff with a fixed +transition width gives separated fringes and a bounded splitting +constant. The tubes around the longer portions of the spokes may +then be narrowed without changing that constant. Gluing errors +can be as small as desired. This produces a map near $K$ and all +spokes, close to $f$ on $K$, with all spoke values still in the +two adjacent target charts. + +\emph{Fill the sectors.} +List the finitely many closed polar sectors between successive +spokes. At a given step let $K_{\rm old}$ be the union of the +original compact set, all spokes, and the sectors already filled. +The next sector $S$ meets $K_{\rm old}$ exactly on its bottom +arc and its two radial sides. Along these three sides all map +values, with a parameter margin, belong to the chosen $V_j$. +This is true initially by the spoke construction, and it remains +true after each sufficiently accurate gluing, because the sides +of every unfilled sector already belong to $K_{\rm old}$. + +Choose a thin closed U-shaped band $D$ about these three sides, +inside the current map domain, and a slightly enlarged filled +sector $D'$ containing $S$ in its interior, so that $D\subset D'$ +is a special pair. To see this elementary geometry, straighten +the sector to a rectangle: $D$ is a thick band around its bottom +and two side edges, and the missing central rectangle attaches +along the U-shaped portion of its boundary. The bands and +protrusions can be made as thin as needed. +All values near $D$ lie in $V_j$. Apply $L$ to the restriction +of a chart spray over the current map, regarding its small +auxiliary variables as additional passive parameters. After any +prescribed small parameter loss this gives a spray on $D'$, +arbitrarily close to the old spray on $D$. + +We spell out the overlap used to glue. Fix an open band +$W\Subset\operatorname{int}D$ containing $K_{\rm old}\cap S$. +Using a slightly smaller band first, the compact remainders on +opposite sides are positively separated. A smooth cutoff can +therefore be zero near $K_{\rm old}\setminus W$ and one near +$S\setminus W$. On a sufficiently thin neighborhood of +$K_{\rm old}\cup S$, its sublevel and superlevel sets give two +open domains with separated fringes and overlap contained in $W$. +They may be chosen so that any point of $K_{\rm old}$ in the new +side also lies in the approximation band. The old spray is +defined on the first domain, the approximating spray on the +second, and they are close on their overlap with an auxiliary +margin. Lemma~\ref{loc:gluing} produces the required new map, +close to the old map on all of $K_{\rm old}$. + +There are only finitely many spokes and sectors. Fix in advance +a finite nested sequence of parameter polydiscs between the +intermediate one and $P_0$. At each step choose errors smaller +than the remaining compact target-chart margins and allocate a +summable finite part of the requested final error. These choices +preserve every unfilled sector's joining sides. No chart condition +on the outer boundary of a newly filled sector is required. +At the end we have a map on a neighborhood of the desired +enlarged compact set times $P_0$, with the stated approximation. +\end{proof} + +\begin{corollary}[Propagation across a small disc]\label{loc:disc} +Suppose a coordinate neighborhood contains a closed affine +complex-line disc centered at $x$, with its boundary in the open +locus where $L$ holds. Then $L$ holds at $x$. +In particular, if the complement of that locus is contained +locally in a proper analytic subset, $L$ holds everywhere. +\end{corollary} + +\begin{proof} +Slightly tilt the line in two different directions through $x$. +Openness and compactness of the original boundary imply that the +two tilted discs still have boundary in the good locus, and their +tangent directions at $x$ are transverse. Thicken each disc in a +complementary coordinate by a sufficiently small fixed parameter +disc. At every boundary point the values for all parameters then +lie in one witness neighborhood; finitely many boundary patches +give one common positive transverse radius. + +For each integer $n$, apply Proposition~\ref{loc:planar} once to +each thickened disc, enlarging its long coordinate to radius +$n$, retaining a fixed smaller transverse radius, and making its +error on the initial product less than $1/n$. The resulting +finite strips converge near their centers to the original +coordinate charts. Their limiting long directions are transverse. +Remark~\ref{loc:alignment} and the finite-length case of +Proposition~\ref{loc:criterion} imply $L$ at $x$. + +If a proper analytic subset $E$ contains the exceptional points +in a coordinate neighborhood of $x$, choose a complex line +through $x$ not contained in $E$. Its intersection with $E$ is +discrete near $x$, so a sufficiently small circle on that line +avoids $E$. The first assertion applies. +\end{proof} + +\begin{remark} +The proof also records a useful uniform version: finitely many +coordinate discs whose boundaries lie in fixed compact subsets of +the good locus retain the required boundary containment after +sufficiently small deformations of the coordinate charts. +All arguments in this section concern one planar variable with +passive parameters. No approximation theorem for arbitrary +multidimensional source compacta has been used. +\end{remark} diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/projective.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/projective.tex new file mode 100644 index 0000000000000000000000000000000000000000..d8ec0f1a342c1c6b2fd3ab247bb785982693bab8 --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/projective.tex @@ -0,0 +1,269 @@ +\section{The projective case}\label{sec:projective} + +We construct two complete directions from families of genus-one +curves. The curves on the K3 surface need not be smooth: the +complete flows take place on their smooth normalizations. + +\begin{proposition}\label{prj:strips} +Let $S$ be a smooth projective complex K3 surface. There is a +nonempty Zariski open subset $U\subset S$ such that, for every +$x\in U$, there are $b>0$ and holomorphic maps +\[ +\sigma_1:\C\times\Delta_b\longrightarrow S,\qquad +\sigma_2:\Delta_b\times\C\longrightarrow S +\] +which are locally biholomorphic at the origin and satisfy +\[ +\sigma_1(0,0)=\sigma_2(0,0)=x,\qquad +\sigma_1(z,0)=\sigma_2(z,0)\quad (|z|0$. Thus all fiber +maps are nonconstant and finite onto their reduced integral +images. + +Consider the graph map +\[ +Q=(p,q):Y\longrightarrow T\times S. +\] +It is proper, by factoring through the closed graph in +$Y\times S$ followed by the proper map $p\times\mathrm{id}_S$, +and it has finite fibers. The proper quasi-finite criterion +therefore makes it finite. Its reduced image $Z$ is an integral +surface. +Choose a smooth point $z_0$ of $C$ and its unique preimage +$y_0\in Y_{t_0}$. The fiber $Q^{-1}(t_0,z_0)$ is a singleton. +Moreover $dQ_{y_0}$ is injective: a vector in its kernel is +vertical for $p$, and the normalization is an isomorphism near +$z_0$, so $dq$ is injective on that vertical tangent line. + +The holomorphic constant-rank theorem embeds a small +neighborhood $W$ of $y_0$ in $T\times S$. Properness makes +$Q(Y\setminus W)$ closed in the analytic topology. Shrink a neighborhood +of $(t_0,z_0)$ to avoid that closed set and remain inside the +embedding chart. Thus $Q$ is an isomorphism over a nonempty +analytic open subset of $Z$. That subset meets the dense +Zariski open locus where the finite map is etale of its generic +degree. The degree is therefore one. + +Since $Y$ is normal, $Q$ is the normalization of $Z$. Its +nonisomorphism locus $N\subset Z$ is closed of dimension at +most one. Remove the finitely many parameters of the vertical +curve components of $N$. Every horizontal component of $N$ +has finite intersection with every remaining fiber, so the +irreducible support +$|Z_t|=q_t(Y_t)$ is not contained in $N$. The restriction $q_t$ +is consequently finite and birational onto its reduced image +$C_t$, hence is its normalization. The projection formula gives +\[ +H\cdot C_t=\deg(q_t^*H)=H\cdot C. +\] + +If the closure of $q(Y)$ were a curve, each retained fiber map +would normalize that same integral curve. Their smooth source +curves would all be isomorphic, contrary to nonconstant moduli. +Thus $q$ dominates $S$. Source and target have dimension two, +so the induced function-field extension is finite and, in +characteristic zero, separable. + +Here the desired finite etale locus must be justified even +though $q$ itself is not assumed proper. Choose affine opens +$V_0=\operatorname{Spec}B\subset Y$ and +$U_0=\operatorname{Spec}A\subset S$ with $q(V_0)\subset U_0$. +Every one of finitely many $A$-algebra generators of $B$ is +algebraic over $\operatorname{Frac}A$. Invert a common nonzero +element of $A$ to make their monic equations integral; the +resulting affine map is finite. Remove also the closure of +$q(Y\setminus V_0)$, of dimension at most one, and shrink to +the etale locus. This leaves a nonempty Zariski open $V$ on +which the full inverse image is the finite etale piece. +It is surjective because it is finite and dominant. +\end{proof} + +Normalize the parameter curves supplied by Chen--Gounelas, +retaining a point above each specified normalization fiber, +and pull back the families. The bases are then smooth +irreducible curves; smoothness, properness, and nonconstant +moduli persist. Fix a very ample divisor $H$ on $S$. The +Hodge index inequality +\[ +(H\cdot C_n)^2\ge H^2 C_n^2 +\] +shows that the ample degrees are unbounded. +Choose two distinct degrees $d_1,d_2$ and apply +Lemma~\ref{prj:prepare}. We obtain two families +\[ +p_i:Y_i\longrightarrow T_i,\qquad q_i:Y_i\longrightarrow S +\quad(i=1,2) +\] +whose fiber maps normalize integral curves of degree $d_i$. +Choose a common nonempty open $V\subset S$ on which +$Y_i^V=q_i^{-1}(V)\to V$ are both finite etale and surjective. +Only their inverse sheets are restricted to $V$: the maps +$q_i$ remain defined on all of $Y_i$. + +\begin{lemma}\label{prj:transverse} +There is a nonempty Zariski open $U\subset V$ such that any +two lifts $y_i\in Y_i^V$ of a point $x\in U$ have distinct +transported vertical tangent lines +\[ +dq_1\bigl(\ker(dp_1)_{y_1}\bigr) +\ne dq_2\bigl(\ker(dp_2)_{y_2}\bigr) +\quad\text{in }T_xS. +\] +The closure in $S$ of the exceptional locus has dimension +at most one. +\end{lemma} + +\begin{proof} +Form the finite etale cover +\[ +W=Y_1^V\times_VY_2^V\xrightarrow{\,Q\,}V,\qquad +\tau_i=p_i\circ\operatorname{pr}_i. +\] +Every irreducible component of the smooth surface $W$ +dominates $V$, since a finite etale map is both open and closed. +The maps $\tau_i$ are submersions. Their algebraic rank-drop +locus +\[ +I=\{d\tau_1\wedge d\tau_2=0\} +\] +records equality of the transported vertical directions. + +No component of $W$ can be contained in $I$. If one were, +identify a small analytic neighborhood in it with an open +subset of $S$ using $Q$. In a coordinate $s=\tau_1$, +dependence of the differentials makes $\tau_2$ depend only on +$s$. A local $\tau_1$-fiber would then lie in a local +$\tau_2$-fiber. Its nonconstant image germ in $S$ lies in an +integral curve from each family. Two distinct integral curves +on a projective surface have finite intersection, so the +curves must coincide. This contradicts their distinct +$H$-degrees. + +Consequently $\dim I\le1$. Since $Q$ is finite, $Q(I)$ is +closed in $V$ and has dimension at most one; its algebraic +closure in $S$ has the same dimension bound. Taking +\[ +U=S\setminus\bigl((S\setminus V)\cup\overline{Q(I)}^{\,S}\bigr) +\] +proves all assertions. +\end{proof} + +\begin{lemma}[Complete vertical flows]\label{prj:flow} +Let $p:Y\to\Delta$ be a proper holomorphic submersion with +connected genus-one fibers. After shrinking the base disc, +there is a nowhere-zero holomorphic vertical vector field +whose flow is jointly holomorphic for all complex times. +Its restriction to the central fiber may be prescribed as +any nonzero holomorphic vector field there. +\end{lemma} + +\begin{proof} +For every fiber $C$, the space $H^0(C,T_C)$ is one-dimensional +and its nonzero elements have no zeros. The relative tangent +bundle is locally free and flat over the base. Grauert's +cohomology and base-change theorem \cite[\S~7, Satz~5]{Grauert} makes +$p_*T_{Y/\Delta}$ a holomorphic line bundle with these spaces +as fibers. The evaluation morphism +\[ +p^*(p_*T_{Y/\Delta})\longrightarrow T_{Y/\Delta} +\] +is an isomorphism on every fiber, hence everywhere. +A local frame gives the vertical field, and scaling it +prescribes its central restriction. + +Take a smaller closed base disc. Its full inverse image is +compact. Local holomorphic flow existence and a finite cover +of this compact set give a common time radius $\epsilon>0$. +The flow preserves fibers, so successive small-time flows +remain in the same compact inverse image. Composing the +time-$z/M$ flow $M$ times defines a holomorphic flow for +$|z|0$, set +\[ + \mathcal D_v=\{P\in\mathcal D:v\in P\}. +\] +All open subsets of $\mathcal D_v$ below have its relative real topology. +We use the K3 period, Torelli, and lattice theorems in their usual +unpolarized form; a reference for these results and the facts about +genus-one pencils used below is +\cite[Chapters~2, 6--8, and~11]{Huybrechts}. + +\begin{theorem}\label{rat:open} +For every primitive positive integral vector $v\in\Lambda$, there is a +nonempty relatively open subset $\mathcal O_v\subset\mathcal D_v$ such +that every marked K3 surface with period in $\mathcal O_v$ has property +$L$ at every point. +\end{theorem} + +On a marked K3 surface whose period contains $v$, the holomorphic +symplectic form has a unique normalization satisfying +\begin{equation}\label{rat:normalization} + [\operatorname{Im}\sigma]=v. +\end{equation} +Indeed the real and imaginary parts of a nonzero holomorphic two-form +are orthogonal and of equal positive square, and multiplication by a +nonzero complex scalar gives all oriented orthogonal frames of its +period plane. This normalization varies smoothly on $\mathcal D_v$. + +We first produce a convenient central surface, then construct strips +on its nearby deformations satisfying \eqref{rat:normalization}. + +\subsection{An isotrivial center for every positive integral vector} + +\begin{lemma}\label{rat:center} +For every primitive $v\in\Lambda$ of square $2d>0$, there is a marked +projective K3 surface $X_0$ whose period contains $v$ and which admits +a genus-one fibration +\[ + \pi:X_0\longrightarrow\mathbb P^1 +\] +with all smooth fibers of $j$-invariant zero. Its primitive fiber class +$e\in\Lambda$ satisfies $(e,v)=0$. +\end{lemma} + +\begin{proof} +We begin with an auxiliary elliptic K3 surface with section, given by +the Weierstrass equation +\[ + y^2=x^3+a(t), +\] +where $a$ is a section of $\mathcal O_{\mathbb P^1}(12)$ with twelve +simple zeros, and the fundamental line bundle is +$\mathcal O_{\mathbb P^1}(2)$. The canonical bundle formula gives +trivial canonical bundle, and +$R^1\pi_*\mathcal O=\mathcal O_{\mathbb P^1}(-2)$ gives irregularity +zero. The simple zeros give cuspidal fibers with smooth total space. +Thus the smooth Weierstrass surface is a K3 surface. These global +Weierstrass and direct-image formulas are recalled in +\cite[Chapter~11, Sections~1--2]{Huybrechts}. + +Let $g$ be the automorphism which multiplies $x$ by a primitive cube +root of unity. Its fixed locus is the disjoint union of the zero +section and the double cover $x=0$ of the base. The latter is branched +at twelve points and has genus five. The fixed locus has Euler +characteristic $2+(2-10)=-6$. The topological Lefschetz formula, in the order-three K3 formulation +of \cite[proof of Theorem~2.2]{ArtebaniSarti}, gives +\[ + \operatorname{tr}(g\mid H^2)=-8. +\] +If $r$ is the invariant rank, the other eigenvalues occur in conjugate +pairs of primitive cube roots, so +$r-(22-r)/2=-8$, whence $r=2$. The fiber and zero section generate a +unimodular hyperbolic plane in this invariant lattice; consequently +the invariant lattice is exactly $U$. Its orthogonal complement is +even unimodular of signature $(2,18)$ and contains two hyperbolic +planes. + +That complement contains a primitive vector of square $2d$. +Eichler's criterion, applied to the unimodular K3 lattice with its +two hyperbolic summands, says that primitive vectors with the same +square are isometric \cite{Eichler}. We may therefore transport the +preceding lattice isometry so that the specified vector $v$ belongs +to its noninvariant complement. There +$1+g+g^2=0$, and thus +\[ + (v,v)=(gv,gv)=2d,\qquad (v,gv)=-d. +\] +The plane $P_0=\operatorname{span}_{\R}\{v,gv\}$ is positive. Choose +either of its two orientations, and realize it as a marked K3 period +by period surjectivity. The invariant $U$ is algebraic at this period +and contains a positive integral class, so the resulting surface is +projective. + +Choose a primitive isotropic generator $e$ of this $U$, with sign +chosen so that its represented class is effective. We keep the abstract +lattice data $g,U,e$ fixed and change the marking by reflections in +algebraic $(-2)$-classes so that the class represented by $e$ becomes +nef. The reflections fix $P_0$ pointwise, hence fix $v$; all preceding +lattice relations are unchanged. +For completeness, if an effective isotropic class has negative +intersection with an irreducible curve, that curve is a $(-2)$-curve; +reflection in it reduces the positive integral degree against a +fixed ample class. Repetition terminates in the nef cone. The +primitive nef isotropic pencil theorem then gives a base-point-free +complete pencil $|e|$ with +\[ + h^0(\mathcal O(e))=2,\qquad H^1(\mathcal O(e))=0. +\] +It defines a genus-one fibration on our center. + +We must show that this particular pencil is isotrivial. The +$g$-invariant period planes of the chosen eigenvalue form a complex +ball: they are the positive lines in one of the two nonreal +eigenspaces of $g$ on $U^\perp\otimes\C$. Take a small ball through +$P_0$, realized by a marked local K3 family. At a generic point its +algebraic lattice is precisely $U$. To see this, each integral class +outside $U$ imposes a proper linear Hodge condition; such a condition +cannot vanish on the entire eigenspace because the two conjugate +eigenspaces span $U^\perp\otimes\C$. Their countable union has empty +interior. + +The line bundle with class $e$ extends to this small family. One way +to verify the extension is to take a contractible Stein base. The +topological class extends to the total space and has zero image in +$H^2(\mathcal O)$: by the Leray spectral sequence and Stein +vanishing this group is the space of sections of +$R^2\pi_*\mathcal O$, and the class vanishes fiberwise because it is +of type $(1,1)$. The exponential sequence supplies the line bundle. +The vanishing of $H^1(\mathcal O(e))$ and cohomology and base change +extend its two sections. Their having no common zero is open by +properness. Thus the genus-one pencils persist near $P_0$. + +At every nearby generic period, $g$ acts identically on the Picard +lattice, hence fixes an ample class. Global Torelli realizes it as +an automorphism of order three. Suppose its induced action on the +pencil base were nontrivial. All its fixed points would then lie +over the two fixed points of that base action. Every fixed curve +would be vertical. Since an irreducible vertical curve $C$ satisfies +$(C,e)=0$, the Hodge index theorem gives $C^2\leq0$ and adjunction +gives $p_a(C)\leq1$. The fixed locus of a finite-order automorphism +of a smooth surface is a disjoint union of smooth curves and +isolated points. It would therefore have nonnegative Euler +characteristic, contrary to the value $-6$ determined by the +cohomological action. + +The base action is consequently trivial. The multiplier on a +smooth fiber's holomorphic differential equals the primitive +cube-root multiplier on the surface form. A smooth elliptic curve +with such an automorphism has $j=0$. Finally fix any smooth member +of the central pencil. It remains smooth in the extended pencil +over a neighborhood in the deformation base, and its $j$-invariant +varies continuously there. The nearby generic values are zero, so +the central value is zero as well. This proves the assertion for +every smooth member at $P_0$. The relation $(e,v)=0$ holds because +$e$ belongs to the original invariant lattice. +\end{proof} + +\subsection{Actions on the smooth fibers} + +Fix the surface of Lemma~\ref{rat:center} and the normalized form +$\sigma_0$. Write $S\subset\mathbb P^1$ for the finite set of +singular values. The linear monodromy of a constant-modulus family +of elliptic curves is finite. Choose a connected finite cover of +$\mathbb P^1\setminus S$ on which that monodromy is trivial, and +compactify it to a smooth compact curve $B$. Denote the resulting +punctured curve by $B^\circ$. A fixed oriented basis +$\gamma_1,\gamma_2$ now identifies the homology of all smooth +fibers over $B^\circ$. + +The base change need not possess a global section. Locally it has +holomorphic sections and product torus charts. For any primitive +cycle $\gamma$ there are cylinder coordinates, with the chosen +cycle represented by the period $1$, in which +\begin{equation}\label{rat:cylinder-form} + \sigma_0=dw\wedge dp,\qquad dp=\eta_\gamma. +\end{equation} +Here $\eta_\gamma$ is a nowhere-zero holomorphic one-form on the +corresponding base patch. Indeed, in a local product chart the +coefficient of the surface form is constant along each compact +elliptic fiber. The period normalization makes these base forms +agree on overlaps. They therefore define a global one-form +$\eta_\gamma$ on $B^\circ$. For the two basis cycles the forms +are in a constant nonreal ratio. + +If $l$ is an oriented smooth loop in a K3 surface with normalized +form $\sigma$, define its action by +\begin{equation}\label{rat:action} + H_\sigma(l)=\int_C\operatorname{Im}\sigma\pmod{\Z}, + \qquad \partial C=l. +\end{equation} +The chain $C$ exists because $H_1(X,\Z)=0$, and the value is +independent of $C$ because $[\operatorname{Im}\sigma]=v$ is +integral. On $X_0$, representatives of a fixed homology class in a +smooth elliptic fiber give the same action, since the restriction +of $\sigma_0$ to that fiber is zero. Let +\[ + h_\gamma:B^\circ\longrightarrow\R/\Z +\] +be this action function. Orientations in +\eqref{rat:cylinder-form} may be chosen so that +$dh_\gamma=\operatorname{Im}\eta_\gamma$; changing all these signs +would make no difference below. Actions are additive in the cycle. + +\begin{lemma}\label{rat:safe} +The forms $\eta_\gamma$ and functions $h_\gamma$ extend to $B$. +The map +\[ + h=(h_{\gamma_1},h_{\gamma_2}):B\longrightarrow(\R/\Z)^2 +\] +has finite fibers. For each primitive $\gamma$, put +\[ + F_\gamma=h_\gamma(B\setminus B^\circ). +\] +Outside a finite subset of $B$, at least two of the three cycles +$\gamma_1,\gamma_2,\gamma_1+\gamma_2$ have action outside their +respective finite sets $F_\gamma$. Any two of these cycles form an +integral homology basis. +\end{lemma} + +\begin{proof} +Integration over the elliptic fibers in local product charts +expresses the pullback symplectic volume as a fixed positive +constant times $i\eta_\gamma\wedge\overline{\eta_\gamma}$. +The base change has finite degree and $X_0$ is compact, so this +one-form has finite $L^2$ norm on $B^\circ$. In a punctured local +coordinate, a holomorphic one-form with finite $L^2$ norm has no +negative Laurent coefficients. It thus extends holomorphically +across each puncture. A local primitive of the extended form +also extends the circle-valued action function. + +The holomorphic forms $\eta_{\gamma_1}$ and +$\eta_{\gamma_2}$ have constant nonreal ratio. Consequently an +invertible real linear map identifies the pair of their imaginary +parts with a complex-linear differential. Endow $(\R/\Z)^2$ with this +constant complex structure. Then $h$ is a nonconstant holomorphic +map from the compact Riemann surface $B$ to an elliptic curve. +It has finite fibers. + +A point without two safe choices satisfies two forbidden-level +conditions. Each pair of the integral linear functions +$h_{\gamma_1},h_{\gamma_2},h_{\gamma_1}+h_{\gamma_2}$ is an +automorphism of the real two-torus. Thus any two such conditions +restrict $h$ to a finite set of torus values. The inverse image is +finite by the preceding paragraph. +\end{proof} + +We call a cycle safe at a point if its action avoids $F_\gamma$. +Every closed set of safe action values has compact inverse image +in $B^\circ$. This elementary compactness statement, rather than +a bound on the number of cylinder blocks, will control all the +successive center adjustments. + +\subsection{Uniform strips after deformation} + +Take a marked local deformation of $X_0$, identify its underlying +smooth fibers, and restrict its parameters to $\mathcal D_v$. +Let $X_t$ and $\sigma_t$ denote the fibers and normalized forms. +The genus-one fibration is used only on $X_0$; it is not asserted +to survive on $X_t$. + +\begin{lemma}\label{rat:strips} +Fix compact families of regular torus charts on $X_0$, each +equipped with a safe primitive cycle, and suppose their actions +are separated from the corresponding forbidden values by a +positive margin. Prescribe any finite number of period cells in +these initial data. For all sufficiently small $t\in\mathcal D_v$, +each datum gives a symplectic immersed strip on $X_t$, periodic in +its long coordinate with period $1$, of positive transverse width. +On the prescribed initial cells its chart is arbitrarily close in +$C^1$ to the original cylinder chart. All widths and closeness +requirements can be chosen uniformly over the given compact data. +\end{lemma} + +\begin{proof} +The centers may move after each exactness correction. We therefore +prepare the compact region in which the construction will continue +before choosing the charts. Let $\rho>0$ be a common lower bound +for the distance of the specified initial actions from their +forbidden sets. For every cycle used in the initial data, put +\[ + K_\gamma=\{b\in B: + \operatorname{dist}(h_\gamma(b),F_\gamma)\geq\rho/4\}. +\] +This set is compact in $B^\circ$. Choose finitely many +product-torus charts covering it, with larger buffered domains +still lying in $B^\circ$. All subsequent restrictions and +recentering will use this finite atlas. We shall prove that the +updated centers remain a fixed distance inside the prepared region. + +In a chart for a chosen cycle, the torus has constant lattice +$\Z+\tau\Z$, where $\operatorname{Im}\tau>0$ after choosing a +positive complementary cycle. Unroll this complementary period +while retaining $w$ modulo $\Z$. We obtain finite cylinders of +any prescribed fixed length. Starting sections can be chosen at +any fiber phase, so the chart families cover all phases, not just +a particular section. + +At matching nominal base positions, successive model cylinders +are identified on their overlaps by +\begin{equation}\label{rat:model-transition} + (w,p)\longmapsto(w+a(p),p). +\end{equation} +The functions $a$ incorporate both changes of local section and +any required number of complementary periods. The latter are +constant, because the modulus is constant. On smaller members of +the finite buffered atlas, the derivatives of the remaining +translation functions have uniform bounds. Choose long block +cores containing the prescribed cells and leave several period +cells as buffers at each end. At a joint, integral complementary +translations let the two cut heights agree at the center with a +bounded adjustment inside these buffers. Both cuts of each raw +chart may be chosen independently. After translating the long +origins, these charts have the block and overlap geometry required +by Theorem~\ref{sew:chain}. Shrinking the transverse radius makes +the variations of their cut heights uniformly smaller than the +overlap margins. + +Each cylinder times its transverse disc is Stein. The local +deformation argument used earlier therefore deforms these maps +on their finite buffered domains to the nearby fibers. Although +a cylinder map is usually not injective, it is an immersion, and +on each overlap its local inverse branch close to +\eqref{rat:model-transition} is unique. The branches consequently +agree around the cylinder, including its period identification. +The resulting transition is a degree-one periodic lift. The +pulled-back forms can be normalized to $dw\wedge dp$ by the +transverse integration used in Theorem~\ref{sew:chain}. +All these constructions are uniform on the finite atlas, and +their errors tend to zero with $t$. + +We next compare the action of an individual deformed block with +that of its model, before any blocks have been joined. Let $b$ be +the nominal base position of a freshly centered block, and let +$l_b$ be the image of a period loop on its zero slice. Uniformly +over the prepared charts and their allowed recenterings, +\begin{equation}\label{rat:action-comparison} + \operatorname{dist}_{\R/\Z} + \bigl(h_\gamma(b),H_{\sigma_t}(l_b)\bigr)<\varepsilon_t, + \qquad \varepsilon_t\longrightarrow0. +\end{equation} +To verify this, under the fixed smooth identification the real forms +$\operatorname{Im}\sigma_t-\operatorname{Im}\sigma_0$ are exact +and tend to zero smoothly. Fix a Riemannian metric on the underlying compact manifold and let +$G$ be its Hodge Green operator. For the exact two-form +$\eta_t=\operatorname{Im}\sigma_t-\operatorname{Im}\sigma_0$, put +$\beta_t=d^*G\eta_t$. Its harmonic projection vanishes and +$dG\eta_t=Gd\eta_t=0$, so Hodge decomposition gives $d\beta_t=\eta_t$. +Elliptic regularity makes $\beta_t$ tend to zero smoothly; +see \cite[Theorem~3.2 and Equation~(3.26)]{ChenLiHodge} for this +Hodge decomposition and Green-operator formulation. Integration of +these primitives around the model loops, together with the +small cylinders between nearby model and actual loops, proves +\eqref{rat:action-comparison}. The loop families are compact: +the base positions and fiber phases range over compact sets, and +the number of wraps in a block is fixed. The estimate depends on +the comparison of this block with its own model, not on any +previous choices of centers. + +Starting from any chosen block, extend the chain in both +directions. At a joint initially use a chart with matching nominal +base position. Its actual symplectic transition is close to +\eqref{rat:model-transition}. Its period defect is the constant +\[ + \int_{\R/\Z}(P\,dW-p\,dw). +\] +Moving this candidate's transverse center by the appropriately +signed defect makes the period exactly zero. Retain this +translated actual chart as the next block, so the exact overlap +just obtained is preserved. Move its nominal center by the same +amount in the model $p$ coordinate; that updated position is used +only when selecting the following candidate block. The adjustment +is small, and comparison with the model of the newly centered +chart retains its original error bound. It remains to justify +that the nominal positions never approach a puncture. + +For this purpose use the action \eqref{rat:action} on the actual +surface. Its value on the period loop of the zero slice is +constant within a block. It is also constant between adjacent +blocks. Indeed the primitive of $dw\wedge dp$ is $-p\,dw$; +the change of action between the image of a zero-slice loop and +the next zero-slice loop is the imaginary part, up to sign, of +the displayed period defect. The image loop winds once in the +next cylinder. Vanishing of the defect proves equality of its +action with that of the next zero slice. Denote this common +circle value by $c$. + +Thus the single-block estimate \eqref{rat:action-comparison} reads +$\operatorname{dist}_{\R/\Z}(h_\gamma(b),c)<\varepsilon_t$ +at every stage. We can now verify that the next center stays in +the prepared region. Require the action error and the change in +nominal action caused by a single center adjustment to be less +than $\rho/8$. For an initial block the conserved value $c$ has +distance greater than $7\rho/8$ from $F_\gamma$. Inductively, +a next nominal center may first be chosen at the preceding one; +the small adjustment keeps it inside the prepared region. The +exact action comparison then improves its distance from the +forbidden set to greater than $3\rho/4$. This restores a fixed +margin for the following step. The same argument applies +backwards. Thus the entire doubly infinite chain is available +with common transverse and overlap margins. + +We have obtained exact symplectic transitions with uniformly +small errors and the block geometry required by +Theorem~\ref{sew:chain}. That theorem supplies the periodic strip +and arbitrarily small corrections on every prescribed compact +part of the initial block. Estimates on slightly larger domains +give the asserted $C^1$ closeness. The transverse width has a +uniform positive lower bound after the fixed margin losses. +\end{proof} + +\subsection{Two cycles cannot give the same leaf} + +\begin{lemma}\label{rat:two-periods} +In the setting of Lemma~\ref{rat:strips}, suppose two initial +strips are obtained from independent primitive cycles of one +central smooth torus. One can choose a finite required initial block +length and a positive $C^1$ error tolerance from the central torus +and its two cycles, before restricting the deformation parameter, +with the following property. For strips meeting these bounds, if +\begin{equation}\label{rat:destroy-fiber} + ([\sigma_t],e)\ne0, +\end{equation} +then $L$ holds at every point of the smaller tube covered by both +initial strip charts. +\end{lemma} + +\begin{proof} +Fix such a point $x$ and take one leaf of each strip through it, +using sheets whose preimages lie inside the buffered initial +blocks. Identify a smooth tube of the central torus with a disc +times that torus, and let $q$ be its smooth projection to the +torus. On a sufficiently large finite box in the universal plane +cover of the torus, the lifted projections of these two leaf +parametrizations are $C^1$ close to invertible affine maps. +The prescribed cylinder lengths may be chosen to include this +box and every lattice translate used below, with a fixed positive +buffer. These choices are made on the central model; the required +$C^1$ tolerance is then fixed on these finite buffered domains. +A small $C^1$ perturbation of an invertible affine map +is injective on a slightly smaller convex box, and its image +contains a further smaller box. We may therefore invert the +two projections and express the two leaf pieces as graphs over +one common convex box $Q$ in the torus cover. Choose this box to +contain a fundamental parallelogram for the two cycles and buffers +beyond every edge. The finite block lengths were chosen large +enough to accommodate these translates and buffers. + +Each graph is periodic under its own primitive cycle wherever +both points and the required buffers belong to $Q$. To check +this exact assertion, shifting $w$ by $1$ leaves its strip map +unchanged. The lift of its torus projection changes by an +integral lattice vector; closeness to the original chart forces +that vector to be exactly the chosen cycle. Inversion and +uniqueness in the buffered box give the claimed graph equality. + +Suppose the two holomorphic curve germs at $x$ were equal. The +set where the graph germs agree is open in $Q$. It is also +closed. At a limit of such points, both graph images have the +same limiting point. In small immersion charts at that point, +the two embedded holomorphic curve germs have accumulating +intersections, so the identity theorem makes them equal. +The use of the smooth projection causes no difficulty here: +it only supplies graph coordinates, while the identity theorem +is applied in holomorphic surface charts. Connectedness of $Q$ +now makes the graphs agree throughout $Q$. + +The common graph is +periodic under both edge identifications. It consequently +defines a compact smooth torus mapped into the tube whose local +image is a holomorphic curve. Let $m$ be the index of the +sublattice generated by the two cycles in the full period +lattice. Contraction of the disc component shows that the +represented integral homology class in $X_t$ is $m e$, with +$m>0$; in the application from Lemma~\ref{rat:safe}, $m=1$. +The pullback of $\sigma_t$ to a holomorphic curve is zero, +contradicting +\eqref{rat:destroy-fiber}. The two leaf germs are therefore +distinct. + +They need not be transverse at $x$. Take a local foliation chart +for the second strip. Its transverse coordinate restricts to a +nonconstant holomorphic function on the first leaf, since the +two germs are distinct. Its derivative is nonzero at points +arbitrarily close to $x$. At such a point the first leaf meets +a member of the second strip family transversely. Apply +Remark~\ref{loc:alignment} at this intersection, assigning the +first strip, whose leaf contains $x$, to the role of $\sigma_2$ +and the transverse partner to the role of $\sigma_1$. +Proposition~\ref{loc:criterion} then gives $L$ at $x$ by the +permitted displacement along the $\sigma_2$ leaf. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{rat:open}] +By Lemma~\ref{rat:safe}, all central fibers except finitely many +have two independent safe cycle choices. Let $A\subset X_0$ be +the union of the excluded fibers and the singular fibers. This +is a proper analytic subset of dimension at most one. + +We make a finite compact preparation before deforming. At every +point of $X_0\setminus A$, choose smaller regular torus patches +with two safe cycles and larger buffered patches. At a point of +$A$, choose a small holomorphic coordinate line disc through the +point whose boundary avoids $A$. Such a disc exists because a +generic line is not contained in $A$ and meets it discretely. +Two sufficiently small different tilts give transverse discs +with boundaries still outside $A$. Their centers may vary in a +smaller coordinate neighborhood. Compactness of $X_0$ permits +a finite choice of these center neighborhoods and regular +patches. Enlarge the compact regular data to include all the +disc boundaries. Every selected safe action then has a positive +common margin from its finite forbidden set. + +Apply Lemma~\ref{rat:strips} to these finitely prepared data, +with blocks long enough for Lemma~\ref{rat:two-periods}. +On all sufficiently small deformations in $\mathcal D_v$, the +initial charts still cover fixed smaller tube patches, and the +coordinate-disc boundaries remain in those patches. If +\eqref{rat:destroy-fiber} holds, Lemma~\ref{rat:two-periods} +gives $L$ at every point of the required regular patches. +Corollary~\ref{loc:disc} gives $L$ in the reserved center +neighborhoods. Thus $X_t$ has $L$ everywhere. + +Finally \eqref{rat:destroy-fiber} holds on a nonempty relatively +open subset arbitrarily close to $P_0$ in $\mathcal D_v$. +Indeed $(e,v)=0$, but the functional $(e,\cdot)$ does not vanish +identically on the positive vectors of $v^\perp$. Varying the +other normalized positive direction therefore breaks the +equation $(e,[\sigma_t])=0$. Local Torelli identifies our small +deformation neighborhood with a period neighborhood, giving +the desired nonempty relatively open set $\mathcal O_v$. +For any other marked K3 with the same period, global Torelli +after matching K\"ahler chambers identifies the underlying +surface, so the property has the stated period-theoretic scope. +\end{proof} diff --git a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/sewing.tex b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/sewing.tex new file mode 100644 index 0000000000000000000000000000000000000000..9e3690869e77393ea3633b861c4e481846f81722 --- /dev/null +++ b/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/build/sections/sewing.tex @@ -0,0 +1,436 @@ +\section{Exact symplectic sewing of annular chains} +\label{sew:section} + +This section isolates the analytic construction needed in the geometric +arguments. The domains are cylinders, not discs. Consequently, a closed +holomorphic one-form on an overlap need not be exact. Its period is the +obstruction that must be removed before the iteration can converge. + +Put $\mathcal C=\C/\Z$, and write $Y=\operatorname{Im}w$ on +$\mathcal C$. We use the form +\[ + \omega_0=dw\wedge dp +\] +on products of $\mathcal C$ with a parameter disc. A symplectic map +$F=(W,P)$ between such products is called \emph{exact} if +\[ + F^*(P\,dW)-p\,dw +\] +is an exact holomorphic one-form. Since a cylinder times a disc has first +homology $\Z$, this is equivalent to the vanishing of its integral around +one positively oriented period circle. All lifts below have degree one: +$W(w+1,p)=W(w,p)+1$ and $P(w+1,p)=P(w,p)$. + +\begin{theorem}[Annular-chain sewing]\label{sew:chain} +Let $M$ be a complex surface with a nowhere-zero holomorphic two-form +$\omega$. Fix $00$, and $A<\infty$. For each $j\in\Z$, +let $L_j\geq1$ and let +\[ + \phi_j:\{ -2h0$ and $e_*>0$, depending only on $h,r,A$ and the indicated margins, +such that, if $eL_j-s$. Thus $S_j$ lies in +$D_j$, and its translate lies in $D_{j+1}$. The other two inequalities +have a fixed buffer because $L_j\geq1$ and $s<1/8$. + +If $s$ and $\rho$ are each decreased by $\delta$, then the smaller +$D_j$ admits coordinate Cauchy discs of radius at least +$c\delta/(1+A)$ in the larger $D_j$. The same assertion holds for +seams and for their translated copies. This follows directly from +\[ + |a_j(p')-a_j(p)|\leq A|p'-p|. +\] +In particular, all fixed-order Cauchy bounds cost only a fixed power +of $\delta^{-1}$, independently of $j$ and $L_j$. In the estimates +below constants may depend on $h,r,A$ and on fixed fractions of the +initial margins; we take $0<\delta<1$. + +\begin{lemma}[Removal of the transverse zero mode]\label{sew:linearize} +Suppose $E=(w+u,p+v)$ is exact symplectic on a seam and +$\|(u,v)\|\leq e$. After a margin loss $\delta$ there are a +holomorphic function $u_0(p)$ and a periodic mean-zero function $H$ +such that +\[ + (u,v)=(u_0,0)+X_H+R,\qquad + X_H=(H_p,-H_w),\qquad + \|H\|\leq Ce,\quad \|R\|\leq C\delta^{-2}e^2. +\] +Here $u_0$ is the zero Fourier coefficient of $u$. +\end{lemma} + +\begin{proof} +Symplecticity and exactness respectively give +\begin{align} + u_w+v_p&=-u_wv_p+u_pv_w,\label{sew:symplectic-error}\\ + 0&=\int_0^1 v(w,p)(1+u_w(w,p))\,dw. + \label{sew:period-error} +\end{align} +The second identity follows by expanding +$(p+v)d(w+u)-p\,dw$ on a horizontal period circle; the integral of +$p u_w$ is zero. Consequently its mean $v_0(p)$ satisfies +\[ + |v_0(p)|\leq C\delta^{-1}e^2. +\] +Take the unique mean-zero periodic primitive satisfying +\[ + H_w=-v+v_0. +\] +Integration along one horizontal period, followed by subtraction of +the mean, bounds $H$ by $Ce$. This primitive is holomorphic in $p$. +Equation~\eqref{sew:symplectic-error} gives +\[ + \partial_w(u-H_p)=-u_wv_p+u_pv_w-v_0'(p). +\] +The right side has zero mean and is $O(\delta^{-2}e^2)$ after one +additional fixed fraction of the margin loss. Integrating its +mean-zero primitive bounds $(u-H_p)-u_0$ by the same quantity. +The other component of $R$ is $v_0$. Rescaling the allocation of +$\delta$ among these finitely many steps proves the stated estimate. +\end{proof} + +The role of exactness is now visible. Without +Equation~\eqref{sew:period-error}, the first-order constant term in +$v$ need not vanish. It cannot be absorbed into a translation that +preserves the common transverse coordinate. + +\subsection{A uniformly bounded Fourier splitting} + +\begin{lemma}[Splitting along the whole chain]\label{sew:split} +Suppose the periodic functions $H_j$ have mean zero, are holomorphic +on $S_j(s,\rho)$, and have norm at most $e$. Assume +\[ + |\operatorname{Im}a_j+L_j|\leq\kappa<1/4. +\] +There are periodic holomorphic functions $G_j$ on +$D_j(s-\delta,\rho)$ satisfying +\begin{equation}\label{sew:split-equation} + G_j-G_{j+1}\circ J_j=-H_j. +\end{equation} +on the smaller seams. Their bounds, on domains with the appropriate +additional losses, are +\[ + \|G_j\|\leq C\delta^{-1}e,\qquad + \|X_{G_j}\|\leq C\delta^{-2}e,\qquad + \|DX_{G_j}\|\leq C\delta^{-3}e. +\] +All constants are uniform in the block lengths and the number of +blocks. +\end{lemma} + +\begin{proof} +Set $T_j=-H_j$ and write +\[ + T_j(w,p)=\sum_{n\neq0}t_{j,n}(p)e^{2\pi i n w}. +\] +The coefficients are holomorphic in $p$. Denote the negative and +positive parts by $T_j^-$ and $T_j^+$. Starting from $(w_j,p)$, +continue the coordinates by +\[ + w_{s+1}=w_s+a_s(p) +\] +in either direction, and define +\begin{equation}\label{sew:fourier-sum} + G_j(w_j,p)=\sum_{l\geq j}T_l^-(w_l,p) + -\sum_{l0$ and take $\delta_n=d_0 2^{-n}$. Choose $d_0$ so that +\[ + c_0\sum_{n\geq0}\delta_n + <\tfrac14\min(s_0,\rho_0). +\] +At stage $n$ reduce both margins by $c_0\delta_n$. Thus their +limits remain positive. Set $B=C d_0^{-q}$, increasing the uniform +constant in Equation~\eqref{sew:newton-bound} once if needed. If +$e_n$ bounds the normalized errors, then +\[ + e_{n+1}\leq B2^{qn}e_n^2. +\] +For $x_n=B2^{q(n+1)}e_n$ this becomes $x_{n+1}\leq x_n^2$. +Consequently, if $x_0\leq\theta<1$, then +\[ + e_n\leq B^{-1}2^{-q(n+1)}\theta^{2^n}. +\] +Every series $\sum_n\delta_n^{-k}e_n$, with fixed $k$, converges +and tends to zero as $e_0\to0$. Taking $e_0$ still smaller ensures +all the margin conditions above, keeps +$|\operatorname{Im}a_j^{(n)}+L_j|0\), optimized narrowness \(\ell\)}; +\node[branch,below=10mm of path] (finite) {\(0<\ell<\infty\)\\ + wide intervals or a horizontal model\\ + Sections~\ref{sec:finite-narrowness}--\ref{sec:critical-rates}}; +\node[branch,left=of finite] (iso) {\(\ell=0\)\\ + polynomial potential;\\ + restoration to degree two\\ + Section~\ref{sec:isotropic}}; +\node[branch,right=of finite] (unbounded) {\(\ell=\infty\)\\ + full-time localization and scalar projection\\ + Section~\ref{sec:unbounded-narrowness}}; +\node[box,below=11mm of finite] (candidate) {Improvement on every substantial residual\\ + greedy coverage and scalar-sheet matching}; +\node[box,below=of candidate] (atlas) {A chart system with recoverable labels in the original problem\\ + smaller terminal time cost or, for finite \(\ell\), excess narrowness}; +\draw[flow] (model)--(path); +\draw[flow] (path.south)--(finite.north); +\draw[flow] (path.south) -- ++(0,-4mm) -| (iso.north); +\draw[flow] (path.south) -- ++(0,-4mm) -| (unbounded.north); +\draw[flow] (finite.south)--(candidate.north); +\draw[flow] (iso.south) |- ([yshift=4mm]candidate.north); +\draw[flow] (unbounded.south) |- ([yshift=4mm]candidate.north); +\draw[flow] (candidate)--(atlas); +\end{tikzpicture} diff --git a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/paper.tex b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..68cc2974f85d66d5d91ae6a9d74e68a6d65864c8 --- /dev/null +++ b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/paper.tex @@ -0,0 +1,78 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{enumitem} +\usepackage{booktabs,longtable,array} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,fit,decorations.pathreplacing} +\usepackage{needspace} +\usepackage[numbers,sort&compress]{natbib} +\usepackage{xurl} +\usepackage[hidelinks]{hyperref} +\usepackage[capitalise,nameinlink,noabbrev]{cleveref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{assumption}[theorem]{Assumption} +\newtheorem{notation}[theorem]{Notation} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{claim}[theorem]{Claim} +\numberwithin{equation}{section} +\newcommand{\R}{\mathbb R} +\newcommand{\C}{\mathbb C} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\E}{\mathbb E} +\newcommand{\Pp}{\mathbb P} +\newcommand{\eps}{\varepsilon} +\newcommand{\1}{\mathbf 1} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\Span}{span} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Lip}{Lip} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Cov}{Cov} +\setlist{topsep=4pt,itemsep=2pt,parsep=0pt} +\allowdisplaybreaks[2] +\hypersetup{pdftitle={Every Four-Dimensional Kakeya Set Has Full Hausdorff Dimension}, + pdfauthor={OpenAI},pdfsubject={The unrestricted four-dimensional Kakeya Hausdorff dimension theorem}} +\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi +\ifdefined\pdftrailerid\pdftrailerid{}\fi +\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi +\title{Every Four-Dimensional Kakeya Set\\Has Full Hausdorff Dimension} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: +every subset of \(\R^4\) containing a unit line segment in every +direction has Hausdorff dimension four. No compactness or regularity +assumption is imposed on the set or its witnessing line family. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-framework} +\input{sections/03-comparison} +\input{sections/04-critical-paths} +\input{sections/05-scalar-tools} +\input{sections/06-isotropic} +\input{sections/06a-retained-edge} +\input{sections/07-finite-narrowness} +\input{sections/08-critical-rates} +\input{sections/09-unbounded-narrowness} +\input{sections/10-consequences} +\setlength{\bibsep}{6pt plus 1pt minus 1pt} +\bibliographystyle{plainnat} +\bibliography{references} +\end{document} diff --git a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/references.bib b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..b83a5adef27448c6b61ff134791da3cf96b8246e --- /dev/null +++ b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/references.bib @@ -0,0 +1,440 @@ +@misc{ThreeDimensional, + author = {{OpenAI}}, + title = {{The Kakeya maximal conjecture in three dimensions}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf}{OAI:The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026}}, + year = {2026}, + note = {Lemma~2.3} +} + +@article{WangZahlSticky, + author = {Wang, Hong and Zahl, Joshua}, + title = {Sticky {Kakeya} sets and the sticky {Kakeya} conjecture}, + journal = {Journal of the American Mathematical Society}, + volume = {39}, + number = {2}, + pages = {515--585}, + year = {2026}, + doi = {10.1090/jams/1067}, + eprint = {2210.09581}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Theorem numbering refers to arXiv:2210.09581v2, 7 October 2025}, + url = {https://arxiv.org/abs/2210.09581v2} +} + +@article{CarberyValdimarsson, + author = {Carbery, Anthony and Valdimarsson, Stef{\'a}n Ingi}, + title = {The endpoint multilinear {Kakeya} theorem via the {Borsuk--Ulam} theorem}, + journal = {Journal of Functional Analysis}, + volume = {264}, + number = {7}, + pages = {1643--1663}, + year = {2013}, + doi = {10.1016/j.jfa.2013.01.012}, + eprint = {1205.6371}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Theorem 1 on page 2 of arXiv:1205.6371v1, 29 May 2012}, + url = {https://arxiv.org/abs/1205.6371v1} +} + +@misc{OrponenDeSaxceShmerkin, + author = {Orponen, Tuomas and de Saxc{\'e}, Nicolas and Shmerkin, Pablo}, + title = {On the {Fourier} decay of multiplicative convolutions}, + year = {2024}, + howpublished = {arXiv:2309.03068v2}, + eprint = {2309.03068}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 2, 27 February 2024; Theorem 1.5}, + url = {https://arxiv.org/abs/2309.03068v2} +} + +@book{TaoVu2006, + author = {Tao, Terence and Vu, Van H.}, + title = {Additive Combinatorics}, + series = {Cambridge Studies in Advanced Mathematics}, + volume = {105}, + publisher = {Cambridge University Press}, + address = {Cambridge}, + year = {2006}, + doi = {10.1017/CBO9780511755149}, + note = {Theorem 2.29 and Exercise 6.4.10; see the author's \href{https://www.math.ucla.edu/~tao/preprints/books.html}{errata} for the corrected fifth-power loss} +} + +@article{KatzZahlPlanebrush, + author = {Katz, Nets Hawk and Zahl, Joshua}, + title = {A {Kakeya} maximal function estimate in four dimensions using planebrushes}, + journal = {Revista Matem{\'a}tica Iberoamericana}, + volume = {37}, + number = {1}, + pages = {317--359}, + year = {2021}, + doi = {10.4171/RMI/1219}, + eprint = {1902.00989}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 3, 7 October 2025; Theorem 2}, + url = {https://arxiv.org/abs/1902.00989v3} +} + +@article{RaiChoudhuriSticky, + author = {Rai Choudhuri, Mukul}, + title = {An improved bound on the {Hausdorff} dimension of sticky {Kakeya} sets in {$\mathbb{R}^4$}}, + journal = {Revista Matem{\'a}tica Iberoamericana}, + volume = {42}, + number = {4}, + pages = {1227--1256}, + year = {2026}, + doi = {10.4171/RMI/1621}, + url = {https://doi.org/10.4171/RMI/1621} +} + +@misc{WangZakharovSticky, + author = {Wang, Hong and Zakharov, Dmitrii}, + title = {{Kakeya} and {Furstenberg} problems for sticky sets of tubes}, + year = {2026}, + howpublished = {arXiv:2609.22035v1}, + eprint = {2609.22035}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 1, 18 September 2026; Theorem 1.2}, + url = {https://arxiv.org/abs/2609.22035v1} +} + +@book{BasuPollackRoy, + author = {Basu, Saugata and Pollack, Richard and Roy, Marie-Fran{\c{c}}oise}, + title = {Algorithms in Real Algebraic Geometry}, + edition = {2nd}, + series = {Algorithms and Computation in Mathematics}, + volume = {10}, + publisher = {Springer}, + address = {Berlin, Heidelberg}, + year = {2006}, + doi = {10.1007/3-540-33099-2}, + isbn = {978-3-540-33098-1}, + url = {https://doi.org/10.1007/3-540-33099-2} +} + +@misc{WangZahlThreeDimensional, + author = {Wang, Hong and Zahl, Joshua}, + title = {Volume estimates for unions of convex sets, and the {Kakeya} set conjecture in three dimensions}, + year = {2025}, + howpublished = {arXiv:2502.17655v1}, + eprint = {2502.17655}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 1, 24 February 2025}, + url = {https://arxiv.org/abs/2502.17655v1} +} + +@misc{GuthWangZahlStreamlined, + author = {Guth, Larry and Wang, Hong and Zahl, Joshua}, + title = {A streamlined proof of the {Kakeya} set conjecture in {$\mathbb{R}^3$}}, + year = {2026}, + howpublished = {arXiv:2601.14411v1}, + eprint = {2601.14411}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 1, 20 January 2026}, + url = {https://arxiv.org/abs/2601.14411v1} +} + +@article{Besicovitch1928, + author = {Besicovitch, A. S.}, + title = {On {Kakeya}'s problem and a similar one}, + journal = {Mathematische Zeitschrift}, + volume = {27}, + pages = {312--320}, + year = {1928}, + doi = {10.1007/BF01171101}, + url = {https://link.springer.com/article/10.1007/BF01171101} +} + +@article{Davies1971, + author = {Davies, Roy O.}, + title = {Some remarks on the {Kakeya} problem}, + journal = {Proceedings of the Cambridge Philosophical Society}, + volume = {69}, + number = {3}, + pages = {417--421}, + year = {1971}, + doi = {10.1017/S0305004100046867}, + url = {https://doi.org/10.1017/S0305004100046867} +} + +@article{Wolff1995, + author = {Wolff, Thomas H.}, + title = {An improved bound for {Kakeya} type maximal functions}, + journal = {Revista Matem{\'a}tica Iberoamericana}, + volume = {11}, + number = {3}, + pages = {651--674}, + year = {1995}, + doi = {10.4171/RMI/188}, + url = {https://ems.press/journals/rmi/articles/5248} +} + +@misc{GaoLiuXi2025, + author = {Gao, Chuanwei and Liu, Diankun and Xi, Yakun}, + title = {Curved {Kakeya} sets and {Nikodym} problems on manifolds}, + year = {2025}, + howpublished = {arXiv:2503.11574v1}, + eprint = {2503.11574}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + doi = {10.48550/arXiv.2503.11574}, + note = {Version 1, 14 March 2025; Definition 1.3, Conjecture 1.4, and Theorem 1.5; Definitions 1.7--1.11, Theorem 1.12, Corollary 4.2, and Section 4.2}, + url = {https://arxiv.org/abs/2503.11574v1} +} + +@article{GuthZahlPolynomialWolff, + author = {Guth, Larry and Zahl, Joshua}, + title = {Polynomial {Wolff} axioms and {Kakeya}-type estimates in {$\mathbb{R}^4$}}, + journal = {Proceedings of the London Mathematical Society}, + volume = {117}, + number = {1}, + pages = {192--220}, + year = {2018}, + doi = {10.1112/plms.12138}, + eprint = {1701.07045}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Corrected version: arXiv:1701.07045v3, 19 April 2019}, + url = {https://arxiv.org/abs/1701.07045v3} +} + +@article{ZahlSeveri, + author = {Zahl, Joshua}, + title = {A discretized {Severi}-type theorem with applications to harmonic analysis}, + journal = {Geometric and Functional Analysis}, + volume = {28}, + pages = {1131--1181}, + year = {2018}, + doi = {10.1007/s00039-018-0455-x}, + eprint = {1801.05106}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + url = {https://arxiv.org/abs/1801.05106v3} +} + +@article{KatzRogersPolynomialWolff, + author = {Katz, Nets Hawk and Rogers, Keith M.}, + title = {On the polynomial {Wolff} axioms}, + journal = {Geometric and Functional Analysis}, + volume = {28}, + number = {6}, + pages = {1706--1716}, + year = {2018}, + doi = {10.1007/s00039-018-0466-7}, + eprint = {1802.09094}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + url = {https://arxiv.org/abs/1802.09094v3} +} + +@article{RoytwarfYomdin1997, + author = {Roytwarf, N. and Yomdin, Y.}, + title = {{Bernstein} classes}, + journal = {Annales de l'Institut Fourier}, + volume = {47}, + number = {3}, + pages = {825--858}, + year = {1997}, + doi = {10.5802/aif.1582}, + url = {https://doi.org/10.5802/aif.1582} +} + +@book{CoverThomas2006, + author = {Cover, Thomas M. and Thomas, Joy A.}, + title = {Elements of Information Theory}, + edition = {2nd}, + publisher = {Wiley}, + year = {2006}, + doi = {10.1002/047174882X}, + url = {https://doi.org/10.1002/047174882X} +} + +@article{SudakovSzemerediVu2005, + author = {Sudakov, Benny and Szemer{\'e}di, Endre and Vu, Van H.}, + title = {On a question of {Erd\H{o}s} and {Moser}}, + journal = {Duke Mathematical Journal}, + volume = {129}, + number = {1}, + pages = {129--155}, + year = {2005}, + doi = {10.1215/S0012-7094-04-12915-X}, + url = {https://doi.org/10.1215/S0012-7094-04-12915-X} +} + +@article{BennettCarberyTao2006, + author = {Bennett, Jonathan and Carbery, Anthony and Tao, Terence}, + title = {On the multilinear restriction and {Kakeya} conjectures}, + journal = {Acta Mathematica}, + volume = {196}, + number = {2}, + pages = {261--302}, + year = {2006}, + doi = {10.1007/s11511-006-0006-4}, + url = {https://doi.org/10.1007/s11511-006-0006-4} +} + +@article{Guth2010, + author = {Guth, Larry}, + title = {The endpoint case of the {Bennett--Carbery--Tao} multilinear {Kakeya} conjecture}, + journal = {Acta Mathematica}, + volume = {205}, + number = {2}, + pages = {263--286}, + year = {2010}, + doi = {10.1007/s11511-010-0055-6}, + url = {https://doi.org/10.1007/s11511-010-0055-6} +} + +@article{BourgainGuth2011, + author = {Bourgain, Jean and Guth, Larry}, + title = {Bounds on oscillatory integral operators based on multilinear estimates}, + journal = {Geometric and Functional Analysis}, + volume = {21}, + number = {6}, + pages = {1239--1295}, + year = {2011}, + doi = {10.1007/s00039-011-0140-9}, + url = {https://doi.org/10.1007/s00039-011-0140-9} +} + +@article{GuoWangZhangDichotomy, + author = {Guo, Shaoming and Wang, Hong and Zhang, Ruixiang}, + title = {A dichotomy for {H{\"o}rmander}-type oscillatory integral operators}, + journal = {Inventiones mathematicae}, + volume = {238}, + number = {2}, + pages = {503--584}, + year = {2024}, + doi = {10.1007/s00222-024-01288-8}, + eprint = {2210.05851}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {The invariant formulation is Theorem 2.1 in arXiv:2210.05851v2, 3 February 2023}, + url = {https://arxiv.org/abs/2210.05851v2} +} + +@article{NadjimzadahBourgain, + author = {Nadjimzadah, Arian}, + title = {{Bourgain}'s Condition, Sticky {Kakeya}, and New Examples}, + journal = {Geometric and Functional Analysis}, + volume = {36}, + pages = {1131--1173}, + year = {2026}, + doi = {10.1007/s00039-026-00750-4}, + eprint = {2511.10918}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Example 1.24 and Proposition 1.25; arXiv:2511.10918v4, 18 August 2026}, + url = {https://arxiv.org/abs/2511.10918v4} +} + +@article{Bourgain1999, + author = {Bourgain, Jean}, + title = {On the dimension of {Kakeya} sets and related maximal inequalities}, + journal = {Geometric and Functional Analysis}, + volume = {9}, + number = {2}, + pages = {256--282}, + year = {1999}, + doi = {10.1007/s000390050087}, + url = {https://link.springer.com/article/10.1007/s000390050087} +} + +@article{KatzLabaTao2000, + author = {Katz, Nets Hawk and {\L}aba, Izabella and Tao, Terence}, + title = {An improved bound on the {Minkowski} dimension of {Besicovitch} sets in {$\mathbb{R}^3$}}, + journal = {Annals of Mathematics}, + volume = {152}, + number = {2}, + pages = {383--446}, + year = {2000}, + doi = {10.2307/2661389}, + eprint = {math/9903166}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + url = {https://arxiv.org/abs/math/9903166v2} +} + +@article{KatzTaoNewBounds, + author = {Katz, Nets Hawk and Tao, Terence}, + title = {New bounds for {Kakeya} problems}, + journal = {Journal d'Analyse Math{\'e}matique}, + volume = {87}, + pages = {231--263}, + year = {2002}, + doi = {10.1007/BF02868476}, + eprint = {math/0102135}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + url = {https://arxiv.org/abs/math/0102135} +} + +@misc{KeletiMatheEquivalences, + author = {Keleti, Tam{\'a}s and M{\'a}th{\'e}, Andr{\'a}s}, + title = {Equivalences between different forms of the {Kakeya} + conjecture and duality of {Hausdorff} and packing + dimensions for additive complements}, + year = {2023}, + eprint = {2203.15731}, + archivePrefix = {arXiv}, + primaryClass = {math.MG}, + note = {Version 3, 17 July 2023}, + url = {https://arxiv.org/abs/2203.15731v3} +} + +@misc{NadjimzadahLifting, + author = {Nadjimzadah, Arian}, + title = {Lifting curved {Kakeya} sets to linear {Kakeya} sets}, + year = {2026}, + eprint = {2609.06299}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 1, 5 September 2026}, + url = {https://arxiv.org/abs/2609.06299v1} +} + +@article{LabaTaoMediumDimension, + author = {{\L}aba, Izabella and Tao, Terence}, + title = {An improved bound for the {Minkowski} dimension of {Besicovitch} sets in medium dimension}, + journal = {Geometric and Functional Analysis}, + volume = {11}, + pages = {773--806}, + year = {2001}, + doi = {10.1007/PL00001685}, + eprint = {math/0004015}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Theorem 1.3 in arXiv:math/0004015v1}, + url = {https://arxiv.org/abs/math/0004015v1} +} + +@misc{BorgesEtAlRestrictionKakeya, + author = {Borges, Tainara and Chan, Tiklung and Chen, Mingfeng and Liu, Diankun and Xi, Yakun and Zhan, Yufei}, + title = {Restriction and {Kakeya} maximal estimates in {$\mathbb{R}^4$}}, + year = {2025}, + howpublished = {arXiv:2511.22824v1}, + eprint = {2511.22824}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 1, 28 November 2025; Theorem 1.2}, + url = {https://arxiv.org/abs/2511.22824v1} +} + +@misc{ChristQuasiextremals, + author = {Christ, Michael}, + title = {Quasiextremals for a {Radon}-like transform}, + year = {2011}, + howpublished = {arXiv:1106.0722v1}, + eprint = {1106.0722}, + archivePrefix = {arXiv}, + primaryClass = {math.CA}, + note = {Version 1, 3 June 2011}, + url = {https://arxiv.org/abs/1106.0722v1} +} diff --git a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/sections/01-introduction.tex b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..47261a37c5661198bf8114c47dc8fce6324c7a6c --- /dev/null +++ b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/sections/01-introduction.tex @@ -0,0 +1,328 @@ +\section{Introduction}\label{sec:introduction} + +A \emph{Kakeya set} in \(\R^n\) is a set containing a unit line segment +in every direction. The Hausdorff-dimension Kakeya conjecture asserts +that every such set has Hausdorff dimension \(n\). We prove its +four-dimensional case. + +\begin{theorem}\label{thm:main} +Let \(K\subset\R^4\). Suppose that for every \(e\in S^3\) there is a point +\(a_e\in\R^4\) such that +\[ + \{a_e+te:0\leq t\leq1\}\subset K. +\] +Then \(\dim_H K=4\). +\end{theorem} + +Theorem~\ref{thm:main} resolves the Hausdorff-dimension Kakeya conjecture +positively in four dimensions. Neither \(K\) nor the choice of its +witnessing segments is required to be measurable. In particular, no +packing-dimension or stickiness hypothesis is imposed on the family +of witnessing lines. + +For arbitrary sets, we use the covering definition +\[ + \mathcal H_\delta^s(K) + =\inf\left\{\sum_i(\diam U_i)^s: + K\subset\bigcup_i U_i,\quad \diam U_i\leq\delta\right\}, + \qquad + \mathcal H^s(K)=\lim_{\delta\downarrow0}\mathcal H_\delta^s(K), +\] +where the covers are countable, and +\(\dim_H K=\inf\{s:\mathcal H^s(K)=0\}\). +This definition requires no measurability assumption. The reduction +in Section~\ref{sec:framework} uses finite selections of witnessing +segments and outer-measure estimates in direction space. + +Related dimension statements and geometric consequences are developed +in Section~\ref{sec:consequences}. Besides the packing- and +bounded-set Minkowski-dimension implications, these include a lower +bound in higher dimensions by orthogonal projection, arbitrary +direction sets and extension of segments to lines, and Nikodym and +curved-Kakeya applications. The latter implications use the transfer +results of \citet{KeletiMatheEquivalences}, +\citet{GaoLiuXi2025}, and \citet{NadjimzadahLifting}, with their +distinct hypotheses stated there. + +\subsection{Context and input estimates} + +The Kakeya problem connects the geometry of line segments with +estimates for oscillatory operators. The classical needle problem +asks how small a planar region can be if a unit segment is to turn +continuously within it and return with its endpoints reversed; the +area can be arbitrarily small \citep{Besicovitch1928,Davies1971}. +A central difficulty in the dimension problem is that many segments +can overlap substantially even when their directions are well +distributed. Besicovitch's constructions show that a Kakeya set can +have Lebesgue measure zero, whereas Davies proved full Hausdorff +dimension in the plane \citep{Besicovitch1928,Davies1971}. + +For bounded Kakeya sets, the upper Minkowski formulation asks for +dimension \(n\), measured by covering numbers at one common scale; +the Hausdorff formulation allows covers with varying diameters. +For fixed ambient dimension \(n\ge2\), let \(T_\delta(a,e)\) be the +cylinder centered at \(a\in\R^n\), with axis direction +\(e\in S^{n-1}\), length one, and transverse radius \(\delta\). +Define the maximal operator by +\(K_\delta f(e)=\sup_{a\in\R^n}|T_\delta(a,e)|^{-1} +\int_{T_\delta(a,e)}|f(x)|\,dx\). +The stronger Kakeya maximal conjecture asks that, for every +\(\varepsilon>0\), there be a constant \(C_{\varepsilon,n}\) such that +\[ + \|K_\delta f\|_{L^n(S^{n-1})} + \leq C_{\varepsilon,n}\delta^{-\varepsilon} + \|f\|_{L^n(\R^n)} +\] +for all \(f\in L^n(\R^n)\) and \(0<\delta<1\) +\citep{KatzTaoNewBounds}. For partial maximal estimates, the +dimension parameter records the Hausdorff lower bound supplied by +that estimate. Theorem~\ref{thm:main} proves the four-dimensional +Hausdorff assertion. + +Wolff's bound \(\dim_H K\ge(n+2)/2\) gives dimension at least three +in \(\R^4\). His geometric method studies the hairbrush of a tube: +the collection of tubes meeting it +\citep{Wolff1995,GuthZahlPolynomialWolff}. +Bourgain connected line incidences in separated slices to sumset +and difference-set estimates, using Gowers's quantitative form of +the Balog--Szemer\'edi theorem \citep{Bourgain1999}. +Katz and Tao developed sums--differences iterations for +higher-dimensional bounds \citep{KatzTaoNewBounds}. +In three dimensions, Katz, \L{}aba, and Tao combined multiscale +clustering and local planar structure with the arithmetic method +to improve the upper Minkowski lower bound beyond \(5/2\) +\citep{KatzLabaTao2000}. +\L{}aba and Tao then used multiscale geometric structure to obtain +upper Minkowski dimension strictly greater than \((n+2)/2\) for +\(n\ge4\), in particular greater than three in four dimensions +\citep[Theorem~1.3]{LabaTaoMediumDimension}. + +The later four-dimensional Hausdorff improvements addressed +concentration near algebraic sets. For compact Kakeya sets, +the polynomial Wolff estimates of Guth and Zahl, together with the +algebraic direction bounds of Zahl and Katz and Rogers, give the +lower bound \(3+1/40\) +\citep{GuthZahlPolynomialWolff,ZahlSeveri,KatzRogersPolynomialWolff}. +The polynomial Wolff axioms restrict concentration near low-degree +algebraic sets; the accompanying multilinear estimates exploit +quantitative linear independence of tube directions +\citep{GuthZahlPolynomialWolff}. +Katz and Zahl's planebrush organizes tubes around a common plane +and combines this geometry with Guth and Zahl's trilinear estimate. +It gives Hausdorff dimension at least \(3.059\), while their distinct +maximal-function estimate has dimension parameter at least \(3.049\) +\citep{KatzZahlPlanebrush}. +Borges, Chan, Chen, Liu, Xi, and Zhan subsequently improved the +maximal parameter to +\((159+\sqrt{145})/56\approx3.0543\), combining the planebrush +with a planar two-ends Furstenberg estimate +\citep[Theorem~1.2]{BorgesEtAlRestrictionKakeya}. +This improves the maximal estimate while leaving the larger +Hausdorff bound \(3.059\) as a separate conclusion. + +Rai Choudhuri proved the bound \(13/4\) for compact sticky sets, +which admit a witnessing line family of packing dimension three +\citep{RaiChoudhuriSticky}. Wang and Zakharov have since obtained a +full-dimensional union estimate for almost AD-regular sticky tube +families satisfying convex Wolff axioms and a dense-shading condition +\citep{WangZakharovSticky}; those hypotheses do not cover arbitrary +Kakeya sets. In three dimensions, Wang and Zahl proved the Kakeya +set conjecture \citep{WangZahlThreeDimensional}, and Guth, Wang, +and Zahl subsequently streamlined the proof +\citep{GuthWangZahlStreamlined}. + +The three-dimensional estimate used in our reduction is the weighted +full-time plank bound of \citet[Lemma~2.3]{ThreeDimensional}, recorded +as \Cref{lem:weighted-planks}. It is proved there by weighted convex +clustering from the union estimate of +\citet[Theorem~1.1]{GuthWangZahlStreamlined}. It supplies spatial +boxes with enough incidence mass to support the quadratic local +fits described below. +The further analytic inputs have distinct roles. Determinant +multilinear Kakeya \citep{CarberyValdimarsson} controls configurations +of independent velocities. The restricted-triple dot-product +theorem of \citet{WangZahlSticky} and the multiplicative-convolution +estimate of \citet{OrponenDeSaxceShmerkin} provide planar rigidity. +The latter is applied in \Cref{re:retained-edge-rigidity} to construct +scalar coefficients with small interval probabilities. +That step also uses a graph Balog--Szemer\'edi--Gowers argument and +fixed-iterate sumset calculus +\citep{SudakovSzemerediVu2005,TaoVu2006}. The graph argument retains +a substantial set of original incidence edges, which are needed +to fit the resulting planar structure back to the trajectories. + +\subsection{Mechanisms of the proof} + +The proof uses quadratic local fits to capture line concentration +that persists across scales. Its basic object is a \emph{chart}: +a collection of line pieces in a moving spatial box and a time +interval, on which one quadratic polynomial test stays small. +Each chart also carries a mass requirement, measured using selected +time-line pairs. This prevents a fit to an exceptional collection of +negligible weight from counting as progress. Normalizing inside a +chart produces another problem of the same kind, with its time +interval rescaled to unit length. Charts can therefore be nested. + +The two quantities to balance are the thickness of the quadratic +relation and the length of the line pieces on which it holds. +Longer intervals at a prescribed thickness give better charts. +The reduction in Section~\ref{sec:framework} turns a hypothetical +Hausdorff deficit into a weighted line problem with two competing +properties: refining to terminal depth loses less than one power of +density per unit depth, yet every substantial terminal chart system must use +intervals of a definite shrinking length. The proof rules out this +combination. + +More precisely, write the lines as +\((t,y_0+tV,w_0+tW)\), where \(y\in\R^2\). +An observation retains exact \((t,y)\) and bins the remaining +coordinate \(w\); increasing the depth refines this bin. +At resolution \(N_0\to\infty\), the selected incidence density in a +bin has a common typical size \(n(r)=N_0^{-f(r)}\). +Here density is measured per \(dt\,dy\) and per hidden bin within +each component of the weighted mixture. The counterexample gives +conditional velocity nonconcentration and +\[ + f(L)-f(r)\le d(L-r)\qquad(0\le r\le L),\qquad d<1, +\] +where \(L\) is the terminal depth. +A chart is called \emph{true} when its incidence mass, divided by +its interval length, is at least this density times its spatial +volume, up to subpower factors. +A chart system's coverage is the selected incidence mass, and mass +\(N_0^{-o(1)}\) is called substantial. +Writing a common interval length as \(N_0^{-h+o(1)}\), its +\emph{time cost} is \(h\). +The \emph{horizon} \(H(E)\) is the infimum of these costs over true +terminal systems of substantial coverage in the weighted problem +\(E\), allowing further selections and subsequences. +The reduction yields \(H(E)>0\). + +The exponent below one makes a comparison principle available. +Local graph representations of the polynomial tests give scalar +sheets, which are analytic branches of polynomial equations. +The density bound and the lower mass bounds of retained entries +supply lists of size at most \(\delta^{-d+o(1)}\) at accuracy +\(\delta\), for a fixed \(d<1\). +Under the derivative and sampling bounds in +\Cref{lem:two-color-matching}, two such lists cannot frequently +agree in value while keeping their rescaled derivative data apart. +Alternating fresh conditional samples of sheets and times turns a +persistent derivative discrepancy into a polynomial endpoint map +whose image has more volume than the density bounds allow. +Successful paths remain unnormalized submeasures, so this volume +argument retains their dependence on the preceding choices. +The comparison both lengthens terminal charts and joins the local +approximations constructed later. + +To locate a forbidden improvement, we follow nested charts in a +nearly extremal weighted problem. Their relative density and time +profiles become linear, with time rate \(k>0\). +A further extremal choice controls how much thinner the smaller +spatial axis can become than the time scale; the larger width is +comparable to that scale up to subpower factors. +Call the optimized rate of this \emph{narrowness} \(\ell\). +The choices are arranged so that a definite improvement in time, +or an additional gain in narrowness at fixed time when \(\ell\) +is finite, contradicts extremality after completion and return to +the original problem. Section~\ref{sec:critical-paths} gives the +precise order of these choices and the resulting critical paths. + +There are three geometric regimes. When \(\ell=0\), enough +separated velocities remain to determine an approximate polynomial +field. Conditional scalar estimates first give polynomial fits +along individual lines. Multilinear Kakeya and interpolation combine +them, and switching between lines makes the field approximately +conservative, producing a potential. +The potential can initially have degree higher than two. The +obstruction has a specific form: the broad directions may lie near +a conic. The proof identifies the surviving cubic and quartic terms +and uses the existing hidden coordinate to encode them, on the +retained trajectories, in a quadratic correction to the chart test. +This restores the degree required for an admissible chart +(\Cref{prop:degree-restoration}). +The same argument with one spatial coordinate also gives zero +horizon for an auxiliary scalar model; that result supplies the +scalar projections used in the other regimes. + +When \(0<\ell<\infty\), projection onto the scalar model gives an +alternative. Sufficiently wide projected intervals already yield an +improvement by comparison. Otherwise planar rigidity forces the +trajectories into a horizontal model with equation \(Y'=-Z+tv\). +Its natural boxes have horizontal size \(H\) and transverse size +\(H^2\). The noncommuting horizontal directions force the relevant +scalar sheets to be nearly affine. +At the critical time rate \(2k=1\), a path with three refreshed +line segments fills enough three-dimensional base volume to improve +the fit. Away from that rate the aim is to recover wide projected +intervals. When \(2k<1\), this first requires repeated scalar +estimates that improve the resolution of trajectory parameters over +the whole normalized time interval. + +When \(\ell=\infty\), two nested chart labels instead localize +trajectories over the whole normalized time interval. +Normalization can introduce large shifts in the density exponents, +but these cancel in the relative density ratios used by the +argument. A final scalar projection gives charts whose time cost +tends to zero, contradicting the positive cost inherited from the +counterexample. + +Each regime initially produces local candidates on substantial +residual portions of the weighted family. To obtain a contradiction, +these fits must become chart assignments with enough total coverage +and with finite lists of labels recoverable from the original +observations. In the finite-narrowness cases, greedy coverage and +the small-list comparison give this passage through +\Cref{prop:candidate-upgrade}. The unbounded-narrowness construction +recovers coverage and finite label lists after its full-time +normalization. In each case, the improved chart system is lifted +to the original weighted problem, where it contradicts the extremal +choice. The comparison, the finite-narrowness conversion, and the +conditional higher-jet argument (\Cref{prop:higher-jets}) are stated +separately because their hypotheses do not require a low-entropy +original line family. + +\begin{figure}[t] + \centering + \input{figures/proof-map} + \caption{The contradiction from a hypothetical Kakeya counterexample. + A critical weighted path has one of three narrowness regimes. + Each produces improvements on substantial residuals, which are + assembled into a chart system with recoverable labels in the + original problem. In the middle branch, \(2k=1\) uses the three-leg + switch; \(2k\ne1\) returns to wide intervals, with a bootstrap + when \(2k<1\). The scalar version of the isotropic argument supplies + the projection theorem used in the other two branches.} + \label{fig:proof-map} +\end{figure} + +\subsection{Organization and scale conventions} + +Section~\ref{sec:framework} gives the weighted models, chart +normalization, and reduction from an arbitrary Kakeya set. +Section~\ref{sec:comparison} proves the scalar-sheet comparison and +terminal extension. Section~\ref{sec:critical-paths} constructs the +critical paths and proves the criterion that turns local candidates +into forbidden improvements. + +Section~\ref{sec:scalar-tools} develops the conditional scalar +estimates, and Section~\ref{sec:isotropic} treats the isotropic +case and its auxiliary scalar model. +Section~\ref{sec:retained-edge-tools} proves the planar rigidity +tools and fits their conclusions to the original incidence edges. +Sections~\ref{sec:finite-narrowness} and~\ref{sec:critical-rates} +use these results to treat finite positive narrowness and its +critical time regimes. +Section~\ref{sec:unbounded-narrowness} treats unbounded narrowness +and completes the proof of Theorem~\ref{thm:main}. +Section~\ref{sec:consequences} develops the dimensional and geometric +consequences. + +Two scale conventions are essential when these steps are combined. +Section~\ref{sec:critical-paths} distinguishes the original resolution +\(N_0\) from the local tangent resolution \(N\): returned chart +systems must satisfy mass and list bounds subpower in \(N_0\), even +when the local analysis only supplies bounds subpower in \(N\). +Section~\ref{sec:framework} distinguishes current incidence laws +from earlier witness laws. A witness floor used for counting is +paired with a density cap for that same law. diff --git a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/sections/02-framework.tex b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/sections/02-framework.tex new file mode 100644 index 0000000000000000000000000000000000000000..2cb57ede9ff053e9b3deadab92bfa544c10beb2f --- /dev/null +++ b/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/build/sections/02-framework.tex @@ -0,0 +1,1652 @@ +\section{Weighted models and the reduction from arbitrary Kakeya sets} +\label{sec:framework} + +We encode a line family by a probability law on trajectories and a +selection of trajectory--time incidences. A local fit confines the +base spatial projection of each whole trajectory piece to a moving +box and keeps a quadratic polynomial small along that piece. Its mass is measured on the +selected incidences. +Theorem~\ref{thm:model-reduction} turns a hypothetical Kakeya +counterexample into a weighted problem in which fitting at the final +thickness requires a positive power cost in time localization. +The infimum of these costs is the horizon. + +We first develop the observations and density estimates used to measure +these fits. We then define the charts, show how their priors and densities +transform under normalization, and prove the reduction from an arbitrary +Kakeya set. All subpower factors in this section refer to one exact +sequence with resolution tending to infinity; later sequences of exact +problems will be introduced separately. + +\subsection{Exact problems and observations} + +The model has two versions. Version \(2\) retains the lines in +\(\R^4\); version \(1\) is the scalar model used after projection. +In each version a prior samples a trajectory independently of time, +and a separate event specifies which trajectory--time incidences +are retained. + +\begin{definition}[Exact problems]\label{def:exact-problem} +An exact problem is a sequence indexed by a real parameter +\(N_0\to\infty\), with a fixed length \(00\) and a polynomial +\(P_*(t,y,w)\) of total degree at most two. Writing +\(X(t)=P_*(t,y(t),w(t))\), the native bounds are +\begin{align} + \sup_{[0,1]}|X|&\le K_0,& + \sup_{[0,1]}|\partial_wP_*|&\le K_0B, + \label{a01:native-bounds}\\ + |\partial_wP_*|&\ge B/K_0 + \quad\hbox{at selected incidences},& + |\partial_{ww}P_*|&\le K_0B^2. + \notag +\end{align} +Tests in this version are arbitrary polynomials \(P(t,y,w)\) of +total degree at most two. +\end{enumerate} +In both versions \(X\) is a quadratic polynomial along each trajectory; +its coefficients are subpower bounded by interpolation on \([0,1]\). + +An incidence selection is an event \(E_\gamma\) in trajectory--time +space, and +\begin{equation} + d\mu_\gamma(l,t)=\1_{E_\gamma}(l,t)\,d\nu_\gamma(l)\,dt, + \qquad + \mu=\sum_\gamma\omega_\gamma\mu_\gamma,\qquad + \mu(\mathrm{all})\ge K_0^{-1}. + \label{a01:incidence-law} +\end{equation} +Thus \(\mu_\gamma\) is not normalized to probability. +A further selection restricts this event and must retain total mass +at least an inverse subpower. +\end{definition} + +Here and below polynomial bounds are \(N_0^M\) for a finite exponent +\(M\) fixed throughout the exact sequence. The exponent may be +arbitrarily large and may change when a new exact problem is constructed. +We impose the following finiteness convention. + +\begin{definition}[Tempered data]\label{a01:tempered} +The numbers of worlds, specified parameter ranges and coefficients, +and \(B,B^{-1}\), are polynomially bounded. In the full trajectory +coefficient coordinates, each prior has piecewise constant Lebesgue +density, polynomially bounded on polynomially many semialgebraic +pieces. Trajectory and incidence predicates use polynomially many +polynomial conditions of bounded individual degree, with arbitrary +Boolean combinations. The number of variables in each geometric +test is bounded. Real constants in these conditions have no height +restriction. + +Exact-level selections, labels, cells, and frames obey such uniform +bounds. Chart widths and time lengths have polynomially bounded +reciprocals. Changes of trajectory coordinates are invertible and +have polynomially controlled constant Jacobian. We permit +polynomially many copies of coefficient space, with an internal +discrete trajectory index. This index is not a world or an observed +point feature; all assignments concern indexed trajectories. + +Measurable arguments may be used to locate a finite family of boxes +or polynomial tests. The output is encoded by those constants and +geometric tests, not by a description of the intermediate measurable +search. +\end{definition} + +Worlds with extremely small success fractions can be removed. +After forming chart worlds, charts with extremely small conditional +prior mass can also be removed: there are only polynomially many +charts, and the selected measure is bounded by prior times time. +The threshold can be a sufficiently small reciprocal polynomial. +These observations preserve the convention in +\Cref{a01:tempered}. + +Use nested half-open dyadic grids, rounding requested widths upwards +within a factor of two. The observation at depth \(r\) is +\begin{equation} + p_r=(\gamma,t,y(t),[w(t)]_{a_r/B}). + \label{a01:observation} +\end{equation} +Within a world its density is with respect to \(dt\,dy\) and counting +measure on the hidden bin. Such densities exist: at fixed \(t\), +replacing \(y_0\) by \(y(t)=y_0+tV\) has Jacobian one. + +A density is \emph{typically} \(n=N_0^{-f}\) if, for every fixed +\(\tau>0\), the selected incidences where it is outside +\([N_0^{-\tau}n,N_0^\tau n]\) have total weighted mass at most +\(N_0^{-c(\tau)}\), eventually, for some \(c(\tau)>0\). +One-sided typical bounds have the corresponding meaning. +A problem is homogeneous for a family of observations if each +observation has a common exponent, independent of the world and +fixed along the exact sequence. Unless otherwise stated the +observations are \(p_r\), \(0\le r\le L\), and their typical densities +are denoted \(n(r)=N_0^{-f(r)}\). + +\subsection{Density selection and homogeneous profiles} + +The selected observation densities need not initially have uniform +sizes. We will restrict to incidences on which their logarithmic sizes +stabilize. Restrictions and finite refinements obey a simple mass +estimate; a finite-table approximation will make the subsequent density +selections compatible with the tempered data. + +\begin{lemma}[Restriction, refinement, and caps] +\label{a01:density-operations} +Let \(g\) be the unnormalized density of an observation and \(g'\) +the density after restriction. For \(0<\epsilon<1\), +\begin{equation} + \int_{\{g'<\epsilon g\}}g'\le + \epsilon\int g. + \label{a01:restriction-integral} +\end{equation} +If an observation is refined by a discrete label with at most \(D\) +possible values on the retained events, the total refined mass at +entries whose density is less than \(\epsilon g\) is at most +\(\epsilon D\int g\). +These statements apply within conditional worlds and may be +summed with the original world weights. + +Consequently further inverse-subpower restrictions, and refinements +by subpower lists, preserve typical logarithmic densities. +Upper density bounds may be integrated over arbitrary base regions +and summed over discrete entries after the exceptional observations +have been removed. +\end{lemma} + +\begin{proof} +Restriction gives \(0\le g'\le g\), and integrating the defining +inequality on \(\{g'<\epsilon g\}\) proves +\Cref{a01:restriction-integral}. For a refinement with densities +\(g'_c\), sum the inequalities \(g'_c<\epsilon g\) over at most \(D\) +labels at each old observation and then integrate. +Take \(\epsilon=N_0^{-\tau}\), and absorb a subpower \(D\) or a +subpower normalization loss into a smaller positive power. + +For clarity about exceptional sets, write the pushed-forward +measure as a good part plus an omitted part, with densities +\(g_{\mathrm{good}},g_{\mathrm{bad}}\). +On the new states where \(g_{\mathrm{bad}}>g_{\mathrm{good}}\), +their total mass is at most \(2\int g_{\mathrm{bad}}\). +On the other states the total density is at most twice the good +density. This transfers a ceiling proved for the restricted good +measure to a typical ceiling for the full resulting measure. +It does not justify summing an unremoved exceptional portion +against small conditional probabilities. +\end{proof} + +A datum is \emph{known by \(p_s\)} if, after permitted pruning, it +belongs to a deterministic list of size at most \(K_0\) depending +only on \(p_s\). List knowledge, rather than a choice of one +representative, is the convention throughout. +The exact base coordinates in an observation vary continuously. +To choose labels or density ranges while preserving temperedness, we +approximate their joint law on a sufficiently fine finite base mesh. +The next lemma controls the error uniformly over subsequent choices. + +\begin{lemma}[Joint base flattening]\label{a01:flattening} +Fix tempered discrete labels and predicates of polynomial total +count. Given a fixed \(T>0\), there is a fixed finite \(D\) such that +uniformly redistributing the base \((t,y)\) inside its mesh cells +of width \(\rho\sim N_0^{-D}\), while retaining the discrete entries, +changes the joint measure in total variation by \(O(N_0^{-T})\). +Thus a posterior list selector for these entries can be replaced, +with that error in success, by a cell-constant tempered list table. +\end{lemma} + +\begin{proof} +Use coordinates \((t,y,V)\) and the remaining trajectory +coefficients, extending all densities by zero outside their +domains. Holding the other variables fixed, each polynomial +condition has a bounded number of changes on a base-coordinate +axis, unless it is identically zero there; the latter condition +has no changes. There are polynomially many conditions. +The weighted density and the discrete labels therefore change +across only polynomially many boundaries on each such fiber. +Their heights and the ranges of the remaining variables are +polynomially bounded. Integration first on the fiber and then in +the other variables shows that a base translation of size at most +\(\rho\) changes the joint measure by at most +\(N_0^C\rho\), with \(C\) fixed by the tempered bounds. +Sum over the finitely many base axes and over internal index copies. + +Within-cell averaging has the same bound, up to a dimensional +constant: write its \(L^1\) difference from the original density +as the integral of pairwise differences of density values in the +same cell, then compare these with translations of length at most +a constant times \(\rho\). Choosing \(D>C+T+1\) proves the assertion. +All labels are included in this argument jointly. + +A measurable selector has the same success up to total variation +error on the flattened law. On that law the posterior of the +discrete entries is constant inside each cell with its conditioning +bins fixed; a constant maximizing list can be chosen there. +There are polynomially many such cells and entries. The resulting +lookup table is tempered, and its success transfers back by the +same total variation estimate. +\end{proof} + +When two grids overlap with at most \(K_0\) members at fixed exact +base and other conditioning data, their common refinement has the +same typical exponent by \Cref{a01:density-operations}. +For a one-way upper bound it suffices to sum the good measure over +the corresponding covering entries. + +\begin{lemma}[Homogenization]\label{a01:homogenization} +After a tempered further selection of inverse-subpower mass and a +subsequence, an exact problem has a homogeneous profile \(f(r)\) +on every prescribed compact depth interval. It is nondecreasing +and \(1\)-Lipschitz. Jointly, one may homogenize the observation +\((p_r,[V]_{a_u})\) on compact nonnegative \((r,u)\)-ranges, obtaining +a function \(f(r,u)\) that is nondecreasing in each variable and +has Lipschitz constants \(1\) in \(r\) and \(j\) in \(u\). +Moreover \(f(r,0)=f(r)\). +\end{lemma} + +\begin{proof} +Choose a countable dense depth grid containing all required +endpoints. At stage \(N_0\), test a finite initial part of this +grid, increasing its size sufficiently slowly. +By \Cref{a01:flattening}, use a common fixed polynomial base mesh +fine enough for all the observations and for a power-small +total variation error. The tested hidden and velocity grids +are nested and lie in fixed compact depth ranges, so their joint +entries are determined by the finest tested bins. Their total +number, including base cells, is polynomial. + +Polynomial bounds give a polynomial ceiling for each flattened +density. Densities below \(N_0^{-C}\), for a sufficiently large +fixed \(C\), have power-small total mass, by summing over the +polynomial ranges and entries. Quantize the remaining logarithmic +densities in intervals of width \(\epsilon_{N_0}\to0\). +If \(m_{N_0}\) is the number of tested observations, choose the +rates so slowly that the number of density-type lists is +\[ + \bigl(O(\epsilon_{N_0}^{-1})\bigr)^{m_{N_0}}=N_0^{o(1)}. +\] +One list carries inverse-subpower mass. Restrict to it. +This restriction is a cell table testing the joint discrete +observations and is tempered. Its upper bounds persist, and +\Cref{a01:density-operations} shows that its densities lose a +fixed power relative to the preselection values only on +power-small mass. + +The total variation estimate transfers these conclusions to +the exact base densities. Indeed, where two densities differ +by a large multiplicative factor, their \(L^1\) difference +controls the mass under the larger density there. +Pass to a subsequence on which the type exponents converge at +every fixed tested depth. + +Refinement from depth \(r\) to \(r'\ge r\) has at most +\(K_0N_0^{r'-r}\) hidden-bin choices, and refinement from velocity +depth \(u\) to \(u'\ge u\) has at most +\(K_0N_0^{j(u'-u)}\) choices. Monotonicity of densities under +refinement and \Cref{a01:density-operations} imply +\begin{align*} + 0&\le f(r',u)-f(r,u)\le r'-r,\\ + 0&\le f(r,u')-f(r,u)\le j(u'-u). +\end{align*} +These inequalities extend the profiles uniquely to all +intermediate depths; nesting squeezes their typical densities +between nearby tested ones. Since \(|V|\le K_0\), appending +the unit velocity bin costs only \(K_0\) choices, proving +\(f(r,0)=f(r)\). +\end{proof} + +Other fixed logarithmic quantities with polynomial upper and +reciprocal bounds may be homogenized by the same argument. +In subsequent cap estimates, fixed tolerances on fixed finite +grids are sent to zero sufficiently slowly, and the grids +increase sufficiently slowly, that the exceptional mass is +negligible compared with the particular inverse-subpower +selection in use. This convention concerns the present exact +\(N_0\)-sequence. + +The class used in the contradiction imposes two further bounds. +The first limits how rapidly the typical density can decrease +as the hidden observation is refined toward the terminal depth. +The second limits velocity concentration within a terminal +observation. + +\begin{definition}[The cap class]\label{a01:cap-class} +Fix \(S,\eta>0\). A homogeneous exact problem belongs to \(C(d)\), +\(d<1\), if +\begin{equation} + f(L)-f(r)\le d(L-r)\qquad(0\le r\le L), + \label{a01:endpoint-cap} +\end{equation} +and, for \(0