Add preprints files (part 11)

.gitattributes

@@ -476,3 +476,78 @@ preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manif
 preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/general-weak-random-bond-ising.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/build/figures/boundary-squeeze.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/quenched-sle3-weak-random-bond-ising.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/README.md

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+# [One-sided negative sectional curvature and the holomorphic Liouville property](paper.pdf)
+
+OpenAI  
+September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026,
+  author = {{OpenAI}},
+  title = {{One-sided negative sectional curvature and the holomorphic Liouville property}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf}{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026}},
+  year = {2026}
+}
+```
+

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/03-analytic.tex

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+\section{Polynomial tests for bounded holomorphic functions}
+\label{sec:analytic}
+
+We will force the $w$-derivative of every bounded holomorphic function
+$f$ on the tube to vanish. The elementary estimate comes from a holomorphic
+disk through a point $(s_0,\zeta,w_0)$:
+\[
+ t\longmapsto(s_0,\zeta+t,w_0+i\alpha b t),\qquad |t|<L,
+ \qquad \alpha,L>0,\quad |b|=1.
+\]
+If this disk lies in the tube and $|f|\le1$, Cauchy's inequality gives
+\[
+ |f_y(s_0,\zeta,w_0)+i\alpha b f_w(s_0,\zeta,w_0)|\le L^{-1}.
+\]
+We must prevent the two derivatives from canceling at many spatial
+points. We arrange those points in root-of-unity cycles and hold $b$
+constant on each cycle. A finite Fourier average then keeps only Taylor
+degrees in one residue class. The phases will separate the prescribed
+lowest-degree part of $f_w$ from the first few surviving terms of $f_y$,
+with one phase choice valid for every polynomial of that degree.
+
+The grids constructed below provide equidistributed projective directions
+for polynomial sampling and cyclic affine lifts for this extraction.
+Their additional separation and shell estimates will be used in
+Section~\ref{sec:assembly} to localize the shear coefficients in disjoint
+spatial neighborhoods when realizing the disks. After constructing the grids
+and choosing the phases, we formulate the precise disk conditions and
+prove that repeated tests force the holomorphic Liouville property.
+
+Throughout this section, $\sigma$ denotes the unitary-invariant
+probability measure on $\mathbb P^{n-1}$.  All norms on vectors and
+derivatives are Euclidean unless indicated otherwise.  Constants may
+depend on the fixed dimension $n$; the exponents in the separation
+estimate below do not.
+
+\subsection{Polynomial grids and their affine lifts}
+
+\begin{lemma}[Homogeneous polynomial grids]
+\label{grid:polynomials}
+Fix $n\ge2$.  For every sufficiently large integer $N$ one can choose a
+system of homogeneous polynomials of degree $N$,
+\[
+ Q_N=(H_N,p_N):\mathbb C^n\longrightarrow\mathbb C^{n-1}\times\mathbb C,
+\]
+with the following properties.  These choices can be made along all
+integers $N\to\infty$, or along any prescribed sequence of degrees
+tending to infinity.
+\begin{enumerate}
+\item There is a constant $C_n$ such that
+\begin{equation}
+ \label{grid:sphere-bounds}
+ N^{-C_n}\le |Q_N(u)|\le N\qquad (|u|=1).
+\end{equation}
+\item The projective zero set of $H_N$ consists of
+$M_N=N^{n-1}$ transverse points.  A subset $\mathcal G_N$ of these
+points satisfies
+\[
+ |\mathcal G_N|=(1-o(1))M_N,
+ \qquad
+ \frac1{M_N}\sum_{\xi\in\mathcal G_N}\delta_\xi
+ \ \rightharpoonup\ \sigma.
+\]
+For every $\xi\in\mathcal G_N$ and every unit representative $u$ of
+$\xi$,
+\begin{equation}
+ \label{grid:goodness}
+ N^{-1}\le |p_N(u)|\le N,
+ \qquad
+ \operatorname{sing}_{\min}
+       \bigl(DH_N(u)|_{u^\perp}\bigr)\ge N^{-1}.
+\end{equation}
+\item Above each $\xi\in\mathcal G_N$, choose any lift $v_\xi$ such
+that $p_N(v_\xi)=1$.  All $N$ lifts
+\[
+ e^{2\pi i q/N}v_\xi,\qquad 0\le q<N,
+\]
+have radii in $[N^{-1/N},N^{1/N}]$.  With the absolute choice
+$k_*=30$, the Euclidean balls of radius $2N^{-k_*}$ about all these
+lifts are pairwise disjoint.  If $v$ is one of the lifts, then
+\begin{equation}
+ \label{grid:shell-bound}
+ |(H_N(z),p_N(z)-1)|\ge N^{-k_*-10}
+ \quad\hbox{when}\quad
+ N^{-k_*}\le |z-v|\le2N^{-k_*}.
+\end{equation}
+In a neighborhood of radius $4N^{-k_*}$ of every such lift, the first
+and second derivatives of $Q_N$ have norm $O_n(N^5)$.
+\end{enumerate}
+In particular, weak approximation to any prescribed finite collection
+of continuous test functions, with any fixed positive accuracy, can be
+required by taking $N$ sufficiently large.
+\end{lemma}
+
+\begin{proof}
+We use the standard unitary-invariant Gaussian ensemble of homogeneous
+polynomials; its covariance and conditional zero-density framework appear
+in \citet[Sections~2.2 and~2.4]{BleherShiffmanZelditch2000}. The proof
+below supplies the needed formulas and all additional quantitative grid
+properties. Randomness is used only to establish existence. A
+standard circular complex Gaussian has density
+$\pi^{-1}e^{-|z|^2}$ on $\mathbb C$.  Choose the components of $Q_N$
+independently, with
+\[
+ Q_N^{(a)}(z)=
+ \sum_{|\beta|=N}\left(\frac{N!}{\beta!}\right)^{1/2}
+       g_{a,\beta}z^\beta,
+ \qquad
+ \mathbb E\bigl[Q_N^{(a)}(z)
+       \overline{Q_N^{(a)}(u)}\bigr]=(z\cdot\bar u)^N.
+\]
+Here all $g_{a,\beta}$ are independent standard circular complex
+Gaussians.  The covariance identity is the multinomial formula.  In
+particular, at a unit vector the components of $Q_N$ are independent
+standard complex Gaussians, and the law is invariant under unitary
+changes of coordinates.
+
+We first prove \eqref{grid:sphere-bounds} with probability tending to
+one.  If $A=\sup_{|z|=1}|Q_N(z)|$, homogeneity gives
+$\sup_{|z|\le1+1/N}|Q_N(z)|\le eA$.  Cauchy's inequality on complex
+lines therefore bounds the first derivative on the unit sphere by
+$CNA$, with an absolute constant $C$.  A sphere net of mesh $c/N$,
+with a sufficiently small absolute $c$, consequently contains a point
+where the norm is at least $A/2$.  Such a net has
+$O_n(N^{2n-1})$ points.  Gaussian tails show
+\[
+ \mathbb P(A>N)\le C_n' N^{2n-1}e^{-c_n' N^2}\longrightarrow0.
+\]
+On the event $A\le N$, the Lipschitz constant on the sphere is at
+most $CN^2$.  Use a second net of mesh $cN^{-C_n-2}$.  If the norm
+somewhere on the sphere is less than $N^{-C_n}$, it is less than
+$2N^{-C_n}$ at a point of this net.  A standard complex Gaussian
+vector in $\mathbb C^n$ has probability at most $C_n' r^{2n}$ of
+lying in a ball of radius $r$.  The union bound is therefore at most
+\[
+ C_n' N^{(C_n+2)(2n-1)-2nC_n}
+   =C_n' N^{-C_n+4n-2}.
+\]
+Choose, for example, $C_n=8n$.  This proves both sphere bounds with
+probability tending to one.
+
+Put $d=n-1$.  Almost surely the projective zeros of the $d$-component
+system $H_N$ are transverse.  To see this directly, let $\mathcal A$
+be its finite-dimensional complex coefficient space.  Evaluation at
+any point of $\mathbb P^d$ maps $\mathcal A$ onto the fiber of
+$\mathcal O(N)^{\oplus d}$.  Hence the universal zero incidence
+manifold is smooth, and a coefficient vector is a regular value of its
+projection to $\mathcal A$ precisely when the corresponding section
+has only transverse zeros.  Sard's theorem gives a Lebesgue-null
+exceptional set; the Gaussian law is absolutely continuous.
+Transversality and compactness imply that the zero set is finite.
+The projective B\'ezout theorem for a zero-dimensional intersection of
+$d$ hypersurfaces of degrees $N,\ldots,N$ gives total intersection
+number $N^d$; every transverse intersection has multiplicity one
+\citep[Theorem~III-71]{EisenbudHarris2000GeometrySchemes}.  Thus the number of zeros is the deterministic
+quantity $M_N=N^d$ almost surely.
+
+We next justify the counting formulas needed for equidistribution.
+They are the one- and two-point, zero-dimensional cases of the
+correlation formula in \citet[Theorem~2.1]{BleherShiffmanZelditch2000}.
+Our subsequent off-diagonal argument proves the particular convergence
+needed here; the more general simultaneous-zero convergence theorem of
+\citet[Corollary~1.3]{Shiffman2008Convergence} supplies a broader context.
+In a holomorphic coordinate chart of $\mathbb P^d$, regard the system
+as a holomorphic map $H_a(z)$ depending linearly on its coefficients.
+Locally split those coefficients as $(a,b)$ so that
+\[
+ H_{a,b}(z)=E(z)a+R(z,b),\qquad \det E(z)\ne0.
+\]
+This is possible because evaluation is surjective.  The equation
+$H_{a,b}(z)=0$ is equivalent to
+$a=A(z,b):=-E(z)^{-1}R(z,b)$.  At a zero, with
+$J=D_zH_{a,b}(z)$,
+\[
+ D_zA=-E(z)^{-1}J.
+\]
+For fixed $b$, the change-of-variables formula with multiplicity for
+$z\mapsto A(z,b)$ thus introduces the real Jacobian
+$|\det E|^{-2}|\det J|^2$.  Integrating in $b$ against the smooth
+Gaussian coefficient density yields the one-point zero density
+\begin{equation}
+ \label{grid:one-point-density}
+ \rho_{1,N}(z)=
+ p_{H_N(z)}(0)\,
+ \mathbb E\bigl[|\det DH_N(z)|^2\mid H_N(z)=0\bigr].
+\end{equation}
+The same argument works with a bounded measurable mark multiplying
+the zero count.  It first applies to nonnegative marks, and the
+general bounded case follows by decomposition.
+
+For two distinct projective points, joint evaluation onto the two
+fibers is surjective.  Indeed, powers of linear forms that vanish at
+one point and not the other give independent prescribed values for
+each component.  Solve now for $2d$ coefficient variables and apply
+the same change of variables in the two point coordinates.  The
+point Jacobian is block diagonal, giving
+\begin{equation}
+ \label{grid:two-point-density}
+ \begin{split}
+ \rho_{2,N}(z,z')={}&p_{(H_N(z),H_N(z'))}(0,0)\\
+ &\quad\cdot
+ \mathbb E\bigl[
+   |\det DH_N(z)|^2|\det DH_N(z')|^2
+       \mid H_N(z)=H_N(z')=0\bigr].
+ \end{split}
+\end{equation}
+Partitions of unity give these formulas on projective space and on
+the complement of its diagonal, respectively.  This derivation uses
+only the coefficient density and ordinary change of variables; no
+independence of zeros has been assumed.
+
+At a point represented by a unit vector $u$, take affine tangent
+coordinates $u+\sum_{j=1}^d z_j e_j$ with $(e_j)$ orthonormal in
+$u^\perp$.  Differentiating the covariance shows that $H_N(u)$ is
+independent of $N^{-1/2}DH_N(u)|_{u^\perp}$, and that the latter is
+a $d\times d$ matrix of independent standard complex Gaussians.
+At two distinct projective points, the cross-covariances of these
+normalized value and first-derivative data tend to zero locally
+uniformly off the diagonal.  More explicitly, on a compact set of
+pairs with $|u\cdot\bar v|\le\rho<1$, they are bounded by a fixed
+power of $N$ times $\rho^{N-2}$.  This follows by differentiating
+$(u\cdot\bar v)^N$ at most once at each point.  The joint Gaussian
+covariance consequently converges uniformly there to the direct sum
+of its one-point covariances.  Conditional Gaussian covariances given
+the values zero converge as well.  Gaussian moments of the fixed
+polynomials $|\det J|^2$ and their products therefore converge.
+Equations \eqref{grid:one-point-density} and
+\eqref{grid:two-point-density} imply that the two-point density,
+divided by $M_N^2$, converges locally uniformly off the diagonal to
+the product of the normalized one-point densities.
+
+The exact total number of zeros lets us pass from off-diagonal
+decorrelation to weak convergence without estimating the two-point
+density near the diagonal. Let
+\[
+ \mu_N=\frac1{M_N}\sum_{H_N(\xi)=0}\delta_\xi.
+\]
+Unitary invariance and the deterministic count imply
+$\mathbb E\mu_N=\sigma$.  The positive measure
+$\mathbb E(\mu_N\otimes\mu_N)$ has total mass one.  Off the
+diagonal, its density converges to that of $\sigma\otimes\sigma$,
+by the preceding calculation.  This implies convergence on the
+whole product: given $\varepsilon>0$, choose a compact subset of the
+off-diagonal region having $(\sigma\otimes\sigma)$-mass greater
+than $1-\varepsilon$.  Its mass under
+$\mathbb E(\mu_N\otimes\mu_N)$ is greater than $1-2\varepsilon$
+for all sufficiently large $N$.  At most $2\varepsilon$ of the
+total mass can remain elsewhere.  Since the diagonal has zero
+$(\sigma\otimes\sigma)$-mass, there is no additional limiting
+mass there.  Thus
+\[
+ \mathbb E(\mu_N\otimes\mu_N)
+       \rightharpoonup\sigma\otimes\sigma.
+\]
+For every continuous test function, its integral against $\mu_N$
+therefore has variance tending to zero.  This proves weak
+convergence to $\sigma$ in probability.
+
+It remains to discard the bad zeros.  Use
+\eqref{grid:one-point-density} with marks.  If $G$ is a $d\times d$
+standard complex Gaussian matrix, then at a zero the derivative
+condition in \eqref{grid:goodness} fails only when
+$\operatorname{sing}_{\min}(G)<N^{-3/2}$.  Its expected discarded
+fraction is
+\[
+ \frac{\mathbb E\bigl[
+       |\det G|^2
+       \boldsymbol{1}_{\{\operatorname{sing}_{\min}(G)<N^{-3/2}\}}
+       \bigr]}{\mathbb E|\det G|^2}
+       \longrightarrow0.
+\]
+Indeed a Gaussian square matrix is nonsingular almost surely,
+$|\det G|^2$ is integrable, and dominated convergence applies.
+The independent polynomial $p_N$ has a standard complex Gaussian
+value at every unit representative.  Its two forbidden tails have
+probability
+\[
+ \mathbb P(|p_N|<N^{-1})+\mathbb P(|p_N|>N)
+   =1-e^{-N^{-2}}+e^{-N^2}\longrightarrow0.
+\]
+Markov's inequality shows that the total discarded fraction tends
+to zero in probability.  Removing that fraction does not alter weak
+convergence.  The modulus and singular-value conditions are
+unchanged on multiplying a unit representative by a phase, so they
+hold for every such representative.
+
+We may now select deterministic systems.  Take a countable uniformly
+dense family of continuous functions on projective space, and impose
+successively accurate conditions on successively longer finite
+subfamilies.  Convergence in probability, the vanishing bad
+fraction, and the probability-one transversality statement show that
+these conditions and \eqref{grid:sphere-bounds} have a nonempty
+intersection for every sufficiently large degree.  Choose one system
+from that intersection.  A diagonal choice gives all the asserted
+limits and works equally well along any prescribed sequence of
+degrees.
+
+We have obtained the projective sampling properties. It remains to
+provide disjoint neighborhoods in which Section~\ref{sec:assembly} can
+localize its shear coefficients. Fix a good direction and a lift
+$v$ with $p_N(v)=1$.
+Homogeneity and \eqref{grid:goodness} give
+$N^{-1/N}\le |v|\le N^{1/N}$.  In the splitting of the domain into
+the radial complex line and its Hermitian orthogonal complement,
+$DH_N(v)$ vanishes on the radial line, while
+$Dp_N(v)$ has radial derivative $N/|v|$.  The transverse derivative
+of $H_N$ has least singular value at least $N^{-4}$ for large $N$:
+the scaling from a unit representative costs at most one additional
+factor $N$, so this is a deliberately weaker bound than necessary.
+Homogeneity, the sphere upper bound, and Cauchy's inequality at
+scale $c/N$ give first- and second-derivative bounds $O_n(N^5)$ in
+the stated neighborhood.  At the lift itself, the transverse first
+derivatives are $O_n(N^4)$.  Block inversion therefore gives
+\[
+ \bigl\|D(H_N,p_N-1)(v)^{-1}\bigr\|\le N^9
+\]
+for all sufficiently large $N$.  Taylor's formula, uniformly for
+$|z-v|\le4N^{-30}$, now yields
+\[
+ |(H_N(z),p_N(z)-1)|
+ \ge N^{-9}|z-v|-C_n' N^5|z-v|^2
+ \ge\tfrac12 N^{-9}|z-v|.
+\]
+This proves \eqref{grid:shell-bound}.  It also excludes every other
+zero of $(H_N,p_N-1)$ within distance $4N^{-30}$, proving
+disjointness of the balls about all good lifts.
+\end{proof}
+
+\subsection{Choosing phases uniformly for a fixed Taylor order}
+
+The grids now supply many separated evaluation points. Their next role
+is linear algebraic: we choose a phase at each projective point so that
+no fixed-order polynomial jet can be hidden among the first few Taylor
+degrees surviving a root-of-unity average. The independent-phase moment
+calculation is the same quadratic-form identity used for random-phase
+trace estimates in \citet{IitakaEbisuzaki2004}.
+The additional rank and polynomial superlevel estimates give one phase
+choice uniformly for an entire coefficient space.
+
+\begin{lemma}[Phases separate low-degree jets]
+\label{grid:phases}
+Fix positive integers $k,n$, with $n\ge2$, such that
+\begin{equation}
+ \label{grid:dimension-choice}
+ \sum_{j=1}^{k-1}\frac{j^{n-1}}{(n-1)!}<\frac1{100}.
+\end{equation}
+Fix a nonnegative integer $h$.  For sufficiently large polynomial
+grids from Lemma~\ref{grid:polynomials}, choose one lift $v_\xi$
+above each good direction and let $V_N\subset\mathbb C^{\mathcal G_N}$
+be the space of evaluation vectors of sums of homogeneous
+polynomials of degrees
+\[
+ h,\ h+N,\ \ldots,\ h+(k-1)N.
+\]
+Let $\Pi_N$ be orthogonal projection onto $V_N$ for the unweighted
+Euclidean point norm.  There are phases $b_\xi$, $|b_\xi|=1$, and a
+constant $\delta_h>0$, independent of $N$ and of the chosen lift
+phases, such that
+\begin{equation}
+ \label{grid:phase-lower-bound}
+ M_N^{-1/2}
+ \left\|(1-\Pi_N)
+       \bigl(b_\xi P(v_\xi)\bigr)_{\xi\in\mathcal G_N}\right\|
+ \ge\delta_h\|P\|
+\end{equation}
+for every homogeneous polynomial $P$ of degree $h$.  Here $\|P\|$
+is any fixed Euclidean norm on its coefficient space, chosen
+independently of the grid.  All required grid accuracies and degree
+thresholds depend only on the fixed $n,k,h$ and this norm.
+\end{lemma}
+
+\begin{proof}
+Dimension gives
+\[
+ \frac{\operatorname{rank}\Pi_N}{M_N}
+ \le N^{1-n}\sum_{j=0}^{k-1}
+       \binom{h+jN+n-1}{n-1}<\frac2{100}
+\]
+for all sufficiently large $N$.  Indeed its limit is the sum in
+\eqref{grid:dimension-choice}.
+
+There is a number $t_h>0$ such that, on every sufficiently accurate
+grid, every polynomial with $\|P\|=1$ satisfies
+\begin{equation}
+ \label{grid:uniform-superlevel}
+ |P(v_\xi)|>t_h
+ \quad\hbox{at at least }0.8M_N\hbox{ good directions}.
+\end{equation}
+For completeness, a nonzero homogeneous polynomial has a
+projective zero set of $\sigma$-measure zero.  In affine charts this
+is the elementary fact that the zero set of a nonzero complex
+polynomial has Lebesgue measure zero, obtained by induction and
+Fubini's theorem.  Thus, for each norm-one polynomial $P$, some
+positive threshold has a superlevel set of measure greater than
+$0.9$.  Uniform continuity in the coefficients makes the same
+superlevel estimate, with a smaller threshold, valid in a
+neighborhood of $P$.  The coefficient unit sphere is compact, so
+finitely many such neighborhoods suffice and their thresholds have
+a positive common lower bound.  Weak convergence of the grid
+measures applied to these finitely many open superlevel sets gives
+the estimate for unit representatives, with a fixed margin in both
+threshold and counting proportion.  Since
+$|v_\xi|^h\to1$ uniformly, it gives
+\eqref{grid:uniform-superlevel}.  This proof requires only finitely
+many fixed approximation conditions for the given $h$.
+
+The projection has $0\le(\Pi_N)_{\xi\xi}\le1$ and diagonal sum
+less than $0.02M_N$.  Consequently at most $0.04M_N$ diagonal
+entries exceed $1/2$.  Choose the $b_\xi$ temporarily as independent
+uniform phases.  Writing $Q_N'=1-\Pi_N$, independence gives
+\[
+ \begin{split}
+ &\mathbb E\left[
+ M_N^{-1}\left\|Q_N'
+       \bigl(b_\xi P(v_\xi)\bigr)_\xi\right\|^2\right]\\
+ &\hspace{25mm}=
+ M_N^{-1}\sum_\xi
+       \bigl(1-(\Pi_N)_{\xi\xi}\bigr)|P(v_\xi)|^2
+ \ge0.38t_h^2
+ \end{split}
+\]
+when $\|P\|=1$.  The last inequality uses the intersection of the
+$0.8M_N$ indices in \eqref{grid:uniform-superlevel} with the
+complement of the at most $0.04M_N$ high-leverage indices.
+
+We need a single phase choice for all polynomials, rather than a
+choice depending on $P$.  Let $(e_a)$ be a fixed orthonormal basis
+of the degree-$h$ coefficient space, and put $q_a(\xi)=e_a(v_\xi)$.
+These evaluations have a common bound depending only on $n,h$ and
+the chosen norm.  The Gram matrix of the random linear map in
+\eqref{grid:phase-lower-bound} has entries
+\[
+ G_{ab}=M_N^{-1}\sum_{\xi,\eta}
+       \overline{q_a(\xi)}\,\overline{b_\xi}
+       (Q_N')_{\xi\eta}b_\eta q_b(\eta).
+\]
+Its diagonal-index terms are deterministic.  For distinct indices,
+the products $\overline{b_\xi}b_\eta$ are orthonormal in
+$L^2$ of the phase probability space: two such ordered products
+have nonzero inner product only when their ordered pairs agree.
+In particular, reversal of the pair gives expectation
+$\mathbb E(\overline{b_\xi}^{\,2}b_\eta^2)=0$.
+It follows that
+\[
+ \begin{split}
+ \mathbb E|G_{ab}-\mathbb EG_{ab}|^2
+ &\le C_{n,h}M_N^{-2}
+       \sum_{\xi\ne\eta}|(\Pi_N)_{\xi\eta}|^2\\
+ &\le C_{n,h}M_N^{-2}\operatorname{rank}\Pi_N
+   =O_{n,h}(M_N^{-1}).
+ \end{split}
+\]
+The Gram matrix has fixed finite size.  Chebyshev's inequality
+therefore implies that its operator-norm distance from its mean
+tends to zero in probability.  Its mean is bounded below by
+$0.38t_h^2$ times the identity.  A phase choice with least
+eigenvalue at least $0.19t_h^2$ exists for every sufficiently large
+grid.  Taking $\delta_h=\sqrt{0.19}\,t_h$ proves the assertion.
+All estimates used only absolute values of lift evaluations, so
+the threshold and lower bound are uniform over their arbitrary
+representative phases.
+\end{proof}
+
+\begin{remark}
+\label{grid:finite-dimension}
+For every fixed $k$, condition \eqref{grid:dimension-choice} holds
+in some sufficiently large finite dimension.  Indeed, with
+$r=n-1$, its left side is bounded by
+$(k-1)^{r+1}/r!$, which tends to zero as $r\to\infty$.
+The argument will use one such fixed dimension, not an assertion
+in every dimension.
+\end{remark}
+
+\subsection{From disk estimates to the holomorphic Liouville property}
+
+\begin{lemma}[Root-of-unity extraction]
+\label{grid:frequency}
+Let $0\le h<N$, and suppose $g$ is holomorphic on a neighborhood of
+the closed ball $|s|\le S$, with $|g|\le K$ there.  Write
+$g=\sum_{r\ge0}g_r$ for its homogeneous Taylor expansion at zero.
+If $R|v|<S$, then
+\[
+ \frac1N\sum_{q=0}^{N-1}e^{-2\pi i hq/N}
+       g(Re^{2\pi i q/N}v)
+   =\sum_{j\ge0}g_{h+jN}(Rv).
+\]
+If $R|v|/S\le e^{-0.99c/N}$ and $c\ge1$, truncating the sum after
+$j=k-1$ has error at most $C K e^{-0.99kc}$, while retaining just
+the term $g_h(Rv)$ has error at most $C K e^{-0.99c}$.  The constant
+$C$ is absolute, independent of $N,h,R,S$.
+\end{lemma}
+
+\begin{proof}
+Cauchy's inequality on the complex line through $v$ gives
+\[
+ |g_r(Rv)|\le K(R|v|/S)^r.
+\]
+The homogeneous expansion converges absolutely at the evaluation
+points.  The finite Fourier average annihilates every degree not
+congruent to $h$ modulo $N$, giving the identity.  Put
+$\theta=R|v|/S$.  The first tail is bounded by
+\[
+ K\sum_{j=k}^{\infty}\theta^{h+jN}
+   \le\frac{K e^{-0.99kc}}{1-e^{-0.99c}}.
+\]
+The case $k=1$ gives the other estimate.  The denominator is
+bounded away from zero for $c\ge1$.
+\end{proof}
+
+We now convert these finite-dimensional estimates into the analytic
+condition needed by the geometric construction. A large sheared disk
+bounds a linear combination of the $y$- and $w$-derivatives. The phases
+separate the degree-$h$ part of the latter from the former. Heights
+$D_j$ growing more slowly than the accuracy parameters $c_j$ will let
+the three-lines theorem turn smallness at those heights into vanishing.
+
+\begin{proposition}[A testing criterion for the holomorphic Liouville property]
+\label{grid:criterion}
+Fix constants $p_0,p_1\ge0$ and $m_0>p_1$.  Choose an integer $k$
+and a finite dimension $n\ge2$ so that
+\begin{equation}
+ \label{grid:exponent-choice}
+ 0.99k>p_0+p_1+1,
+ \qquad
+ \sum_{r=1}^{k-1}\frac{r^{n-1}}{(n-1)!}<0.01.
+\end{equation}
+Let $(\zeta_j,h_j)$ be a sequence of pairs from a countable dense
+subset of an open disk in $\mathbb C$ and the nonnegative integers,
+with every pair repeated infinitely often.  At stage $j$, choose
+numbers $R_j\ge1$, $c_j\ge1$, an integer $N_j>h_j$, and a grid
+from Lemma~\ref{grid:polynomials} sufficiently large and accurate
+for Lemma~\ref{grid:phases} at order $h_j$.  Set
+\[
+ S_j=R_j e^{c_j/N_j},\qquad \log N_j\le c_j/100.
+\]
+Choose heights $D_j$ and positive numbers $\alpha_j$ such that
+\[
+ \alpha_j^{-1}\le e^{p_1c_j},
+\]
+and, along the repetitions of every fixed pair,
+\begin{equation}
+ \label{grid:height-rates}
+ c_j\longrightarrow\infty,\qquad
+ D_j\longrightarrow\infty,\qquad
+ \frac{D_j}{c_j}\longrightarrow0.
+\end{equation}
+There are phases $b_{j,\xi}$, one per good projective direction
+and constant throughout its cycle of lifts, with the following
+property.
+
+Let $\ell$ be a smooth real function on $\mathbb C^n\times\mathbb C$,
+and let
+\[
+ \mathcal T=\{(s,y,w):\operatorname{Im}w>\ell(s,y)\}.
+\]
+Suppose every stage satisfies these conditions:
+\begin{enumerate}
+\item On $|s|\le S_j$,
+\begin{equation}
+ \label{grid:height-gap}
+ \ell(s,\zeta_j)\le D_j-2.
+\end{equation}
+\item For every holomorphic $f$ on $\mathcal T$ with $|f|\le1$,
+every $w$ with $\operatorname{Im}w=D_j$, and every $|s|=S_j$,
+\begin{equation}
+ \label{grid:sphere-derivative}
+ |f_y(s,\zeta_j,w)|\le e^{p_0c_j}.
+\end{equation}
+\item For every good direction $\xi$, every lift
+$v=e^{2\pi iq/N_j}v_\xi$, and every $w$ on that height line, the
+holomorphic disk
+\begin{equation}
+ \label{grid:sheared-disk}
+ t\longmapsto
+ \bigl(R_jv,\ \zeta_j+t,\ w+i\alpha_j b_{j,\xi}t\bigr),
+ \qquad |t|<e^{m_0c_j},
+\end{equation}
+is contained in $\mathcal T$.
+\end{enumerate}
+Then every bounded holomorphic function on $\mathcal T$ is constant.
+The phases depend only on the grids and their assigned jet orders,
+not on $\ell$, $f$, or $w$.
+\end{proposition}
+
+\begin{proof}
+For each stage choose the phases supplied by
+Lemma~\ref{grid:phases}.  Assign its phase to all $N_j$ lifts of
+the same direction.  We prove the assertion for a holomorphic
+function with $|f|\le1$, which suffices by constant rescaling.
+
+Fix a stage and temporarily omit its subscript.  For each $w$ with
+$\operatorname{Im}w=D$, put
+\[
+ A(s)=f_y(s,\zeta,w),\qquad I(s)=i f_w(s,\zeta,w).
+\]
+Both are holomorphic on a neighborhood of the closed spatial ball
+$|s|\le S$, by the strict height margin and compactness.  The
+maximum principle and \eqref{grid:sphere-derivative} give
+$|A|\le e^{p_0c}$ on the ball.  The unit $w$-disk centered at any
+$(s,\zeta,w)$ in this ball lies in $\mathcal T$, by
+\eqref{grid:height-gap}; Cauchy's inequality gives $|I|\le1$.
+On a sheared disk, the derivative at $t=0$ is
+$A(Rv)+\alpha b_\xi I(Rv)$.  Thus
+\begin{equation}
+ \label{grid:node-inequality}
+ |A(Rv)+\alpha b_\xi I(Rv)|\le e^{-m_0c}.
+\end{equation}
+Every estimate so far is uniform in the real part of $w$.
+
+For each good direction take the Fourier average in
+Lemma~\ref{grid:frequency} over its cycle.  Since $b_\xi$ is
+constant on that cycle, it factors out of the averaged second
+term of \eqref{grid:node-inequality}.  The lift radius bound gives
+\[
+ \frac{R|v_\xi|}{S}
+ \le\exp\left(\frac{\log N-c}{N}\right)
+ \le e^{-0.99c/N}.
+\]
+Let $a$ and $i'$ be the resulting vectors of averaged $A$ and $I$
+values, respectively.  There is a vector $a_0\in V_N$ such that
+\[
+ \|a-a_0\|_\infty\le C e^{(p_0-0.99k)c}.
+\]
+If $P$ is the homogeneous degree-$h$ Taylor term of the function
+$s\mapsto I(Rs)$, then
+\[
+ \bigl\|i'-(P(v_\xi))_\xi\bigr\|_\infty
+       \le C e^{-0.99c}.
+\]
+The averaged node inequalities still have supremum norm at most
+$e^{-m_0c}$.  Apply $1-\Pi_N$ and use $a_0\in V_N$.  Since the
+number of good directions is at most $M_N$, conversion from the
+supremum norm to $M_N^{-1/2}$ times Euclidean norm costs at most
+one.  Lemma~\ref{grid:phases} consequently gives
+\[
+ \begin{split}
+ \alpha\delta_h\|P\|
+ &\le e^{-m_0c}
+       +C e^{(p_0-0.99k)c}
+       +C\alpha e^{-0.99c},\\
+ \|P\|
+ &\le C_h\left(
+       e^{(p_1-m_0)c}
+       +e^{(p_1+p_0-0.99k)c}
+       +e^{-0.99c}\right).
+ \end{split}
+\]
+In particular, with the fixed positive number
+\[
+ \varepsilon=\min\{m_0-p_1,\ 0.99k-p_0-p_1,\ 0.99\},
+\]
+we have
+\begin{equation}
+ \label{grid:jet-decay}
+ \|P\|\le C_h e^{-\varepsilon c}.
+\end{equation}
+The constants are independent of the stage once $h$ is fixed.
+This is exactly the uniformity in Lemma~\ref{grid:phases}; the
+Fourier tail denominator is $1-e^{-0.99c}$, not
+$1-e^{-0.99c/N}$.  Since every degree-$h$ coefficient of $I(Rs)$
+is $R^h$ times the corresponding coefficient of $I(s)$ and
+$R\ge1$, \eqref{grid:jet-decay} bounds each degree-$h$ coefficient
+of $f_w(s,\zeta,w)$ by the same exponential rate.
+
+We have obtained exponential decay for every coefficient of the
+prescribed degree, uniformly along the entire horizontal $w$-line.
+It remains to bring that estimate down from the growing testing heights
+to a fixed open part of the tube.
+
+Fix now a scheduled center $\zeta$ and a multiindex $\beta$ of
+length $h$.  The coefficient
+\[
+ g_\beta(w)=\frac1{\beta!}
+       \partial_s^\beta f_w(0,\zeta,w)
+\]
+is holomorphic on the whole connected half-plane
+$\operatorname{Im}w>\ell(0,\zeta)$.  At every point of this
+half-plane a spatial neighborhood is available, so the derivative
+is well-defined and holomorphic there.  Choose once a small
+closed spatial polydisk about zero and a number $t_0$ exceeding
+the supremum of $\ell(s,\zeta)$ on that polydisk by more than two.
+Cauchy's inequalities in $s$ and on a unit $w$-disk give
+\[
+ |g_\beta(w)|\le B_\beta
+ \qquad (\operatorname{Im}w\ge t_0),
+\]
+where $B_\beta$ is independent of every testing stage and of
+$\operatorname{Re}w$.  Along the stages assigned to $(\zeta,h)$,
+\eqref{grid:jet-decay} gives
+\[
+ \sup_{\operatorname{Im}w=D_j}|g_\beta(w)|
+       \le C_h e^{-\varepsilon c_j}.
+\]
+For any fixed $t>t_0$, apply Hadamard's three-lines theorem
+\cite[Theorem~7.13]{Tropp2022} to the horizontal strip between heights
+$t_0$ and $D_j$, once $D_j>t$.
+Writing $\theta_j=(t-t_0)/(D_j-t_0)$, it yields
+\[
+ \sup_{\operatorname{Im}w=t}|g_\beta(w)|
+ \le B_\beta^{\,1-\theta_j}
+       (C_h e^{-\varepsilon c_j})^{\theta_j}.
+\]
+If $B_\beta=0$ there is nothing to prove.  Otherwise, the
+logarithm of the right side is
+\[
+ \log B_\beta+
+ \frac{t-t_0}{D_j-t_0}
+    \bigl(\log(C_h/B_\beta)-\varepsilon c_j\bigr)
+       \longrightarrow-\infty
+\]
+by \eqref{grid:height-rates}.  Hence $g_\beta$ vanishes in the
+upper half-plane and, by the identity theorem, throughout
+$\operatorname{Im}w>\ell(0,\zeta)$.  Repeating this argument for
+each order proves that all spatial Taylor coefficients of $f_w$
+at zero vanish at every scheduled center.  Dependence of the
+constants and of the preliminary height $t_0$ on the order causes
+no difficulty: each order has its own infinite sequence, and
+analytic continuation gives the same full half-plane afterwards.
+
+To pass from these statements to a common open set, choose a
+spatial ball about zero and an open subdisk of the center disk,
+both with compact closures.  Choose $L$ larger than the supremum
+of $\ell$ on their product.  On the product with
+$\operatorname{Im}w>L$, the function $f_w$ is jointly holomorphic.
+At each scheduled center in the subdisk, all of its spatial
+Taylor coefficients vanish for every such $w$.  Taylor uniqueness
+and the identity theorem on the spatial ball give vanishing on
+that ball.  Continuity in $y$ and density of the scheduled centers
+then give vanishing on the whole product.  The tube is connected:
+the coordinates
+\[
+ (s,y,\operatorname{Re}w,
+        \operatorname{Im}w-\ell(s,y))
+\]
+identify it with
+$\mathbb C^{n+1}\times\mathbb R\times(0,\infty)$.
+The several-variable identity theorem therefore gives $f_w=0$
+everywhere on $\mathcal T$.
+
+For every fixed $(s,y)$ the $w$-fiber is a connected half-plane,
+so $f$ is constant along that fiber.  It descends to a bounded
+function $F$ on $\mathbb C^{n+1}$.  This descended function is
+holomorphic: near any base point choose a single constant $w_0$
+whose imaginary part exceeds the supremum of $\ell$ on a small
+base neighborhood, and write $F(s,y)=f(s,y,w_0)$ there.  These
+local holomorphic definitions agree on overlaps by fiber
+constancy.  Liouville's theorem, applied on complex lines in
+$\mathbb C^{n+1}$, makes $F$, and therefore $f$, constant.
+\end{proof}
+
+\begin{remark}
+\label{grid:geometric-interface}
+Condition \eqref{grid:sphere-derivative} follows immediately if,
+at the stated sphere points and heights, the unsheared $y$-disk
+of radius $e^{-p_0c_j}$ lies in the tube.  Thus the criterion asks
+the geometric construction for only three quantitative facts: a
+height bound on a spatial ball, sufficiently wide unsheared disks
+on its boundary, and much wider disks in the prescribed sheared
+directions at its polynomial nodes.  The phase choices are made
+before any bounded holomorphic function is considered.
+\end{remark}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/04-fiber.tex

@@ -0,0 +1,326 @@
+\section{The fiber Legendre transform and its curvature}\label{sec:fiber}
+
+The analytic criterion leaves us a geometric task: construct the base
+potential while retaining explicit curvature margins for later changes.
+We first derive the metric and its six curvature blocks for a fixed fiber
+center. The partial Legendre transform uses, up to our fixed metric
+normalization, the fiber momentum of the circle-invariant constructions
+of Calabi and Hwang--Singer \citep{Calabi1979,HwangSinger2002}.
+We allow the radial profile to vary over the spatial base, which adds
+spatial-derivative terms to the curvature. We compute the full tensor
+before estimating these additional terms.
+
+All spatial derivatives in this section are taken with the fiber momentum
+\(\tau\) fixed, unless a different convention is stated explicitly.
+The spatial variables are \(s=(s_1,\ldots,s_n)\), and the fiber variable is
+\(y\in\mathbb C\).  Its center \(\zeta\) is constant in this section.
+Let \(q(x,s)>0\), where \(x=\log\tau\), and choose \(F\) by
+\begin{equation}\label{fiber:F}
+ F_x=q^{-1},\qquad \lim_{x\to-\infty}(F(x,s)-x)=0.
+\end{equation}
+We assume that this normalization is possible, that \(F\to+\infty\)
+as \(x\to+\infty\), and that \(q\), regarded as a function of \(\tau\),
+is smooth at zero with \(q(0,s)=1\).  All these statements are local
+uniformly in \(s\).  For smooth real functions \(B,\psi\), set
+\begin{align}
+ U(s,\tau)&=B(s)+\tau\psi(s)-\int_0^\tau F(\log v,s)\,dv,
+ \label{fiber:U}\\
+ \rho&=\log|y-\zeta|^2=F(\log\tau,s)-\psi(s),
+ \label{fiber:inverse}\\
+ \ell_{\mathrm{rad}}&=U+\tau\rho
+ =B+\int_0^\tau q(\log v,s)^{-1}\,dv.
+ \label{fiber:potential}
+\end{align}
+The last identity follows by integration by parts; the boundary term
+\(vF(\log v,s)\) tends to zero as \(v\downarrow0\).
+Since \(F_x>0\) and its endpoint limits are infinite with the indicated
+signs, \eqref{fiber:inverse} determines a unique \(\tau>0\) for every
+\(y\ne\zeta\).
+
+Hereafter \(C=-R\) denotes the negative Hermitian curvature tensor:
+\begin{equation}\label{fiber:curvature-convention}
+ C_{\alpha\bar\beta\gamma\bar\delta}
+ =\partial_\alpha\partial_{\bar\beta}g_{\gamma\bar\delta}
+ -g^{\mu\bar\nu}
+   (\partial_\alpha g_{\gamma\bar\nu})
+   (\partial_{\bar\beta}g_{\mu\bar\delta}).
+\end{equation}
+The sectional-curvature sign associated with this convention is verified
+below; in particular, positivity of the relevant contractions of \(C\)
+means nonpositive Riemannian sectional curvature.
+
+\begin{lemma}[Metric and regularity of the Legendre transform]
+\label{fiber:metric}
+Write
+\[
+ P=\tau q,\qquad V=(B_{i\bar j}),\qquad
+ G=(\psi_{i\bar j}),\qquad h=(U_{i\bar j}),\qquad
+ \mathcal N=h_\tau.
+\]
+Where \(h>0\), the complex Hessian of \(\ell_{\mathrm{rad}}\) is
+\begin{equation}\label{fiber:metric-form}
+ h_{i\bar j}\,ds_i\,d\bar s_j+
+ P\left|d\log(y-\zeta)+U_{\tau i}\,ds_i\right|^2.
+\end{equation}
+It is positive off the axis.  Its spatial Schur complement is exactly
+\(h\).  The potential and its complex Hessian extend smoothly across
+\(y=\zeta\); on that axis the Hessian in the original splitting is
+\(V\oplus e^\psi\), and is positive whenever \(V>0\).
+\end{lemma}
+
+\begin{proof}
+Use the holomorphic coordinate \(z_0=\log(y-\zeta)\), so that
+\(\rho=z_0+\bar z_0\).  Differentiating \(U_\tau+\rho=0\) gives
+\[
+ U_{\tau\tau}=-P^{-1},\qquad
+ \partial_{z_0}\tau=P,\qquad
+ \partial_{s_i}\tau=P U_{\tau i}.
+\]
+Differentiation of \(\ell=U+\tau\rho\) now gives
+\[
+ g_{0\bar0}=P,\qquad g_{i\bar0}=P U_{\tau i},\qquad
+ g_{i\bar j}=h_{i\bar j}+P U_{\tau i}U_{\tau\bar j},
+\]
+which is \eqref{fiber:metric-form}.  For regularity, write
+\(F=\log\tau+f(\tau,s)\).  The equation
+\(f_\tau=(q^{-1}-1)/\tau\), with \(f(0,s)=0\), shows that \(f\)
+is smooth.  Equation \eqref{fiber:inverse} becomes
+\(\tau e^{f(\tau,s)}=e^\psi|y-\zeta|^2\).  Its derivative with respect
+to \(\tau\) at zero is one, so the smooth implicit-function theorem
+applies.  In particular,
+\(\tau=e^\psi|y-\zeta|^2+O(|y-\zeta|^4)\).
+Substitution in \eqref{fiber:potential} proves the stated extension and
+axis metric.
+\end{proof}
+
+We use a local holomorphic change \(z_0\mapsto z_0+f(s)\) to make
+\(U_{\tau i}=0\) at a specified point.  The corresponding change of
+\(U\) is linear in \(\tau\) with a pluriharmonic spatial coefficient;
+it leaves \(h\) and \(\mathcal N\) unchanged.  The quadratic jet of
+\(f\) can also make \(U_{\tau ik}=0\) there.  We call these the tilted
+logarithmic coordinates.  The coordinate changes used to evaluate a jet
+are held fixed while that jet is differentiated.
+
+\begin{lemma}[The six curvature blocks]\label{fiber:blocks}
+In tilted logarithmic coordinates, the tensor
+\eqref{fiber:curvature-convention} has the following blocks:
+\begin{align}
+ A:=C_{0\bar0 0\bar0}
+   &=P^2P_{\tau\tau}-|\partial P|_h^2,
+ \label{fiber:blockA}\\
+ A_i:=C_{i\bar0 0\bar0}
+   &=P\bigl(\partial_iP_\tau-
+       (\mathcal N h^{-1}\partial P)_i\bigr),
+ \label{fiber:blockAi}\\
+ W_{i\bar j}:=C_{i\bar j0\bar0}
+   &=P\bigl(\partial_i\partial_{\bar j}\log P
+       +P_\tau\mathcal N_{i\bar j}
+       -P(\mathcal N h^{-1}\mathcal N)_{i\bar j}\bigr),
+ \label{fiber:blockW}\\
+ C_{i\bar0 k\bar0}&=(\nabla_h^{2,0}P)_{ik},
+ \label{fiber:blockspin}\\
+ K_{ik\bar j}:=C_{i\bar j k\bar0}
+   &=P\nabla_i^h\mathcal N_{k\bar j}
+      +P_i\mathcal N_{k\bar j}+P_k\mathcal N_{i\bar j},
+ \label{fiber:blockK}\\
+ L_{i\bar j k\bar l}:=C_{i\bar j k\bar l}
+   &=C^h_{i\bar j k\bar l}
+      +P(\mathcal N_{i\bar j}\mathcal N_{k\bar l}
+          +\mathcal N_{i\bar l}\mathcal N_{k\bar j}).
+ \label{fiber:blockL}
+\end{align}
+Here \(C^h\) and \(\nabla^h\) are the curvature with sign
+\eqref{fiber:curvature-convention} and the K\"ahler connection of the
+fixed-\(\tau\) spatial metric.  Index positions in the products use the
+indicated Hermitian metric, with the usual conjugations.
+\end{lemma}
+
+\begin{proof}
+In addition to the tilt, first choose spatial coordinates normal for
+\(h\) at the point.  On functions of \((s,\tau)\), physical coordinate
+derivatives act as
+\[
+ D_i=\partial_i+P U_{\tau i}\partial_\tau,
+ \qquad D_0=P\partial_\tau,
+\]
+and their conjugates.  At the point the complete list of cubic entries
+needed below is
+\begin{equation}\label{fiber:cubics}
+ \ell_{00\bar0}=PP_\tau,\quad
+ \ell_{i0\bar0}=P_i,\quad
+ \ell_{00\bar j}=P_{\bar j},\quad
+ \ell_{i0\bar j}=P\mathcal N_{i\bar j},\quad
+ \ell_{ik\bar0}=0,\quad \ell_{ik\bar j}=0.
+\end{equation}
+For example, the identity
+\(P^2\partial_\tau U_{\tau i}=P_i\), obtained by spatial
+differentiation of \(U_{\tau\tau}=-1/P\), and its conjugate give the
+second and third entries.
+Directly differentiating the metric entries before evaluating gives
+\begin{align*}
+ D_0D_{\bar0}P&=P(P_\tau^2+PP_{\tau\tau}),\\
+ D_iD_{\bar0}P&=P_iP_\tau+P\partial_iP_\tau,\\
+ D_iD_{\bar j}P&=P_{i\bar j}+P\mathcal N_{i\bar j}P_\tau,\\
+ D_iD_{\bar j}(PU_{\tau k})
+   &=P_i\mathcal N_{k\bar j}+P_k\mathcal N_{i\bar j}
+      +P\partial_i\mathcal N_{k\bar j},\\
+ D_iD_{\bar j}(h_{k\bar l}+PU_{\tau k}U_{\tau\bar l})
+   &=\partial_i\partial_{\bar j}h_{k\bar l}
+      +P\mathcal N_{i\bar j}\mathcal N_{k\bar l}
+      +P\mathcal N_{i\bar l}\mathcal N_{k\bar j}.
+\end{align*}
+For completeness, the fourth derivative giving the spin block has a
+cancellation which is useful later.  Put \(a_i=U_{\tau i}\).  Before
+evaluation,
+\[
+ \ell_{ik\bar0}=Pa_{ik}+P_i a_k+P_k a_i+PP_\tau a_i a_k.
+\]
+At the point \(a_i=a_{ik}=0\), while
+\[
+ (a_i)_\tau=P_i/P^2,\qquad
+ (a_{ik})_\tau=P_{ik}/P^2-2P_iP_k/P^3.
+\]
+Applying \(D_{\bar0}=P\partial_\tau\) therefore gives \(P_{ik}\):
+the two positive terms \(P_iP_k/P\) cancel the negative doubled term.
+
+Subtract the cubic contractions in
+\eqref{fiber:curvature-convention}.  For A the vertical contraction is
+\(PP_\tau^2\) and the spatial one is \(|\partial P|_h^2\).
+For \(A_i\) they are \(P_iP_\tau\) and
+\(P(\mathcal Nh^{-1}\partial P)_i\).  For \(W\) they are
+\(P_iP_{\bar j}/P\) and \(P^2(\mathcal Nh^{-1}\mathcal N)_{i\bar j}\).
+The first combines with \(P_{i\bar j}\) to give
+\(P\partial_i\partial_{\bar j}\log P\).
+For the purely spatial block the contraction is the one defining
+\(C^h\).  These calculations prove all formulas in normal spatial
+coordinates.  Without spatial normalization, the last entry of
+\eqref{fiber:cubics} is \(\partial_i h_{k\bar j}\); its contractions
+replace \(P_{ik}\) by \(\nabla^h_i\partial_kP\) and
+\(\partial_i\mathcal N_{k\bar j}\) by
+\(\nabla_i^h\mathcal N_{k\bar j}\).  This also verifies the displayed
+tensorial formulas in general spatial coordinates.
+\end{proof}
+
+The six blocks give the complete tensor, but the desired sign concerns
+real sectional curvature. The next lemma identifies the matrix tests
+corresponding to real two-planes and records exactly how much of the
+diagonal curvature can absorb each mixed block.
+
+We specify the norm convention needed to keep all subsequent constants
+independent of \(n\).  The norm of a covariant tensor is its
+multilinear operator norm on unit vectors in the stated fixed frame.
+The norm of a Hermitian test matrix is its Frobenius norm.  Thus, for a
+Hermitian matrix \(T\) of rank at most two and a covariant Hermitian form \(Q\),
+\begin{equation}\label{fiber:ranknorm}
+ |Q:T|\le\sqrt2\|Q\|\,|T|.
+\end{equation}
+Indeed diagonalize \(T\); the sum of the absolute values of its at most two
+nonzero eigenvalues is at most \(\sqrt2|T|\).
+The same decomposition bounds a curvature pairing \(C(T,T)\) by
+\(2\|C\||T|^2\).  Contractions of cubic tensors through an inverse
+metric can instead be regarded as inner products of two covectors.
+Their operator bounds therefore involve no summation factor depending
+on the dimension.
+
+\begin{lemma}[Real planes and rank-two absorption]\label{fiber:ranktwo}
+If \(C(H,H)\ge0\) for every Hermitian contravariant matrix \(H\) of rank
+at most two, the underlying real sectional curvatures are nonpositive.
+Write the spatial, cross, and fiber diagonal blocks of \(H\) as
+\(T,b_\times,t\).  The contraction has the terms
+\begin{equation}\label{fiber:contraction}
+ At^2+2t(W:T)+2W(b_\times,\bar b_\times)+L(T,T),
+\end{equation}
+and its other terms have absolute value at most
+\begin{equation}\label{fiber:otherterms}
+ 2|\nabla_h^{2,0}P(b_\times,b_\times)|
+ +4|tA_i b_\times^i|+4|(K:T)b_\times|.
+\end{equation}
+In particular the following quantitative criterion suffices.  Suppose
+\(A>0\), \(W>0\), and a nonnegative quadratic form \(\mathcal R(T)\)
+satisfies
+\begin{align}
+ L(T,T)-\frac{|W:T|^2}{.98A}&\ge\mathcal R(T),
+ \label{fiber:reserve}\\
+ |\nabla_h^{2,0}P(b,b)|&\le\tfrac14 W(b,\bar b),
+ \label{fiber:spinabs}\\
+ |A_i b^i|&\le\eta\sqrt{A W(b,\bar b)},
+ \label{fiber:Aiabs}\\
+ |(K:T)b|&\le\eta\sqrt{\mathcal R(T)W(b,\bar b)},
+ \label{fiber:Kabs}
+\end{align}
+for all such tests, where \(\eta\le1/100\).  Then
+\begin{equation}\label{fiber:retained}
+ C(H,H)\ge .01At^2+W(b_\times,\bar b_\times)
+                         +\tfrac12\mathcal R(T).
+\end{equation}
+All constants are independent of dimension.
+\end{lemma}
+
+\begin{proof}
+For real vectors with \((1,0)\) parts \(v,w\), set
+\(H=i(v\otimes\bar w-w\otimes\bar v)\).  This is Hermitian of
+rank at most two.  K\"ahler symmetry gives
+\[
+ C(H,H)=2\{C(v,\bar v,w,\bar w)
+                   -\operatorname{Re}C(v,\bar w,v,\bar w)\}.
+\]
+Expand \(R(v+\bar v,w+\bar w,w+\bar w,v+\bar v)\).  The only four
+nonzero terms are the two alternating type patterns and their
+conjugates.  In the convention in which a real vector with complex
+components \(v\) has squared length \(g_{i\bar j}v^i\bar v^j\), the complex
+bilinear extension of the real metric on a holomorphic and an
+antiholomorphic vector is one half the Hermitian pairing.  Accordingly
+the real curvature numerator is \(-\tfrac12 C(H,H)\).  This proves the
+sign assertion.
+Expanding a general block matrix \(H\) gives
+\eqref{fiber:contraction}--\eqref{fiber:otherterms}; each mixed
+conjugate pair accounts for its factor two or four.  A principal block
+\(T\) has rank at most two.
+
+For the quantitative assertion use
+\(2t(W:T)\ge-.98At^2-|W:T|^2/(.98A)\).
+The remaining diagonal coefficient is .02A.  The spin term costs at
+most \(W/2\).  The other two terms obey
+\[
+ 4\eta |t|\sqrt{AW}\le .01At^2+400\eta^2W,
+ \qquad
+ 4\eta\sqrt{\mathcal R W}\le\tfrac12\mathcal R+8\eta^2W.
+\]
+Since \(1.5-408\eta^2\ge1\) when \(\eta\le1/100\),
+\eqref{fiber:retained} follows.
+\end{proof}
+
+\begin{lemma}[Exact perturbation identity]\label{fiber:perturbation}
+In any fixed holomorphic coordinates let \(H\) denote the cubic jet viewed
+as a map from two tangent inputs to a covector, and write \(H=\Gamma g\).
+For a perturbation with \(g+\Delta g>0\), the change of the cubic
+contraction \(Hg^{-1}H^*\) is exactly
+\begin{equation}\label{fiber:perturbformula}
+ \Gamma\Delta H^*+\Delta H\Gamma^*-\Gamma\Delta g\Gamma^*
+ +(\Delta H-\Gamma\Delta g)(g+\Delta g)^{-1}
+                 (\Delta H-\Gamma\Delta g)^*.
+\end{equation}
+Consequently, if \(\|g^{-1}\Delta g\|\) in metric-unit axes is at most
+one half, the inverse of the perturbed metric is at most twice that
+of \(g\), and
+\begin{align}\label{fiber:perturbbound}
+ \|\Delta C\|\le {}&\|\Delta j^4\ell\|
+       +2\|\Gamma\|\|\Delta H\|
+       +\|\Gamma\|^2\|\Delta g\|\\
+ &+2\|g^{-1}\|
+       (\|\Delta H\|+\|\Gamma\|\|\Delta g\|)^2.
+ \notag
+\end{align}
+The same bound, with an absolute factor, controls rank-two contractions.
+\end{lemma}
+
+\begin{proof}
+Set \(Q=\Delta H-\Gamma\Delta g\).  Then
+\(H+\Delta H=\Gamma(g+\Delta g)+Q\).  Multiply out the corresponding
+cubic contraction and subtract \(\Gamma g\Gamma^*\).  The two cross
+terms and \(\Gamma\Delta g\Gamma^*\) combine to give the first three
+terms in \eqref{fiber:perturbformula}.  Taking operator norms gives
+\eqref{fiber:perturbbound}.  The inverse comparison is the elementary
+spectral estimate for \(I+g^{-1/2}\Delta g g^{-1/2}\); the final statement
+uses \eqref{fiber:ranknorm} twice.
+\end{proof}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/05-quiet.tex

@@ -0,0 +1,600 @@
+\section{A spatially varying quiet fiber profile}\label{sec:quiet}
+
+We now choose a radial fiber profile that tolerates spatial variation
+of its parameters. Its steep part gives the curvature margins needed
+near the testing disks; its logarithmic tail keeps the whole affine
+fiber available. After establishing the scalar estimates, we use the
+six-block formulas to prove a uniform curvature criterion and then
+show that it survives localized shear terms.
+
+Fix \(b_0\ge1000\) and \(K_0\ge1\); increasing \(K_0\) later is
+allowed.  Define a positive function \(u\) of \(\tau>0\) by
+\begin{equation}\label{quiet:u}
+ u=\tau\exp\left(\int_0^\tau
+       \frac{(1+v)^{-1/16}-1}{v}\,dv\right).
+\end{equation}
+It satisfies \(u/\tau\to1\) at zero and
+\(d\log u/d\log\tau=(1+\tau)^{-1/16}\).  It is increasing, is
+comparable to \(\tau\) on \((0,1]\), and tends to a finite positive
+limit at infinity: the last assertion follows by integrating
+\((1+\tau)^{-1/16}d\tau/\tau\) from 1 to infinity.
+More specifically, convexity gives
+\((1+v)^{-1/16}\ge1-v/16\), whence
+\(e^{-1/16}\tau\le u\le\tau\) for \(0<\tau\le1\).
+For a real parameter \(E\), called the severity, set
+\begin{equation}\label{quiet:q}
+ q=q_0(x,E)=(1+e^E u)^{b_0}+K_0^{-1}\log(1+\tau),
+ \qquad x=\log\tau.
+\end{equation}
+Constants in this section may depend on \(b_0,K_0\), but are independent
+of the spatial dimension.  A lower threshold for \(E\) will always be
+chosen sufficiently large for these constants.  The primitive \(F\) has
+the normalization \eqref{fiber:F}.  Because \(q-1=O_E(\tau)\) at zero,
+this normalization exists.  Because \(q\) grows only linearly in \(x\) at
+the right end, \(F\to+\infty\) there.
+
+\begin{lemma}[Scalar estimates for the profile]\label{quiet:scalar}
+Put
+\[
+ q_a=(1+e^Eu)^{b_0},\quad r=r_q=q_x/q,\quad
+ D=D_q=(q_x+q_{xx})/q=r(1+r)+r_x,\quad e_0=q_E/q.
+\]
+For all sufficiently large \(E\) and all \(\tau>0\),
+\begin{gather}
+ q\ge1,\quad q_x>0,\quad
+ D\ge r(r+\tfrac12),\quad q_{xx}/q_x\ge-\tfrac1{16},
+ \label{quiet:slopes}\\
+ 0<r\le C,\quad |e_0|+|q_{EE}/q|\le C,
+ \quad |r_E|\le Cr.
+ \label{quiet:derivatives}
+\end{gather}
+For \(\tau\le1\), the first two quantities in the middle of
+\eqref{quiet:derivatives} are also at most \(Cr\).
+For \(1\le j\le4\),
+\begin{equation}\label{quiet:Fderivatives}
+ |\partial_E^jF|+|\partial_E^j\bar F|\le C,
+ \qquad \bar F(\tau,E)=\tau^{-1}\int_0^\tau F(\log v,E)\,dv.
+\end{equation}
+For \(0\le j\le4\) and \(\tau\ge e^{-E/4}\),
+\begin{equation}\label{quiet:difference}
+ |\partial_E^j(F-\bar F)|\le C e^{-E/4},\qquad
+ |\partial_E^j(F-\bar F)|\le Cr.
+\end{equation}
+The following additional estimates will be used:
+\begin{align}
+ q_x&\ge e^E &&(e^{-E/4}\le\tau\le e^E),
+ \label{quiet:intermediate}\\
+ q&\le C(e^{b_0E}+\log(1+\tau)),\qquad
+ q_x\ge c(1+e^{b_0E}(1+\tau)^{-1/16}) &&(\tau\ge1),
+ \label{quiet:tail}\\
+ r&\ge .9 b_0(1+\tau)^{-1/16}
+              &&(1\le\tau\le e^{3b_0E}),
+ \label{quiet:middle-slope}\\
+ q&\le C,\quad q-1\le Ce^E\tau,\quad
+ r\asymp e^E\tau,\quad D\asymp e^E\tau
+              &&(0<\tau\le2e^{-E}).
+ \label{quiet:axis-slope}
+\end{align}
+In particular \(q_x\ge c\) for \(\tau\ge e^{-E}\), and
+\(q^{-1}\le Cr\) for \(\tau\ge2e^{-E}\).
+\end{lemma}
+
+\begin{proof}
+Write \(a=(1+\tau)^{-1/16}\), \(z=e^Eu\), and
+\(q_b=K_0^{-1}\log(1+\tau)\).  The logarithmic slopes of the two
+summands of \(q\) are
+\[
+ p=\frac{(q_a)_x}{q_a}=b_0a\frac z{1+z},\qquad
+ d=\frac{(q_b)_x}{q_b}
+     =\frac{\tau}{(1+\tau)\log(1+\tau)}.
+\]
+They satisfy
+\[
+ p_x/p\ge-\tfrac1{16},\qquad
+ d_x/d=\frac1{1+\tau}
+       -\frac{\tau}{(1+\tau)\log(1+\tau)}\ge-\tfrac12.
+\]
+For the second inequality use
+\(\log(1+\tau)\ge2\tau/(2+\tau)\), obtained by differentiation from
+equality at zero.  The derivative of a weighted mean of logarithmic
+slopes is the weighted mean of their derivatives plus their nonnegative
+variance.  Hence \(r_x\ge-r/2\), proving the first bound for \(D\).
+Also
+\((q_a)_{xx}/(q_a)_x=p+p_x/p\ge-1/16\) and
+\((q_b)_{xx}/(q_b)_x=(1+\tau)^{-1}>0\).  Taking the weighted mean with
+weights \((q_a)_x,(q_b)_x\) gives the other slope bound.
+
+Every positive \(E\) derivative of \(q_a\) of order at most four is bounded
+by \(Cq_a\min(1,z)\).  Differentiating \(q^{-1}\) thus gives
+\begin{equation}\label{quiet:reciprocal}
+ |\partial_E^j(q^{-1})|
+       \le C\frac{q_a}{q^2}\min(1,z),\qquad 1\le j\le4.
+\end{equation}
+Likewise \((q_a)_x\) and its \(E\) derivative have ratio bounded in
+absolute value by a profile constant, so \(|r_E|\le Cr\).
+Since \(a\ge2^{-1/16}\) for \(\tau\le1\), comparison with \(p\) gives
+\(e_0,q_{EE}/q\le Cr\) there.  The remaining bounds in
+\eqref{quiet:derivatives} follow directly from the displayed formulas.
+
+Differentiate the normalized integral for \(F\).  For \(x\le0\), use
+\(d x=d\log z/a\), where \(a\ge2^{-1/16}\), in
+\eqref{quiet:reciprocal}; the resulting absolute integral is at most
+\[
+ C\int_0^\infty\frac{\min(1,z)}{(1+z)^{b_0}}\,\frac{dz}{z}<\infty.
+\]
+For \(x\ge0\), put \(M=e^{b_0E}\).  The positive lower and upper
+bounds on \(u\) for \(\tau\ge1\) imply
+\(cM\le q_a\le CM\) and
+\(q\ge cM+x/K_0\).  The bound in \eqref{quiet:reciprocal} is therefore
+at most \(CM/(cM+x/K_0)^2\), whose integral over \([0,\infty)\)
+is uniformly bounded.  This proves \eqref{quiet:Fderivatives}, also
+for the average by integration.
+
+Integration by parts gives, including for \(j=0\),
+\begin{equation}\label{quiet:averageidentity}
+ \partial_E^j(F-\bar F)
+   =\int_0^1\partial_E^j(q^{-1})(\log\tau+\log t,E)\,dt.
+\end{equation}
+If \(\tau\ge e^{-E/4}\), split at \(t=e^{-E/4}\).  On the first
+part the integrand is uniformly bounded.  On the second part
+\(\tau t\ge e^{-E/2}\), and
+\(u(v)\ge c\min(v,1)\) implies \(e^E u(\tau t)\ge c e^{E/2}\).
+There the integrand, also for \(j=0\), is at most \(C e^{-b_0E/2}\).
+This proves the first bound in \eqref{quiet:difference}.
+For \(\tau\ge2\), split instead at \(t=\tau^{-1/2}\).
+Monotonicity and the positive finite limit of \(u\), together with
+\(\log(1+\sqrt\tau)\ge\tfrac12\log(1+\tau)\), give
+\(q(\log\sqrt\tau,E)\ge c q(\log\tau,E)\).  Thus the integral is
+at most \(C(\tau^{-1/2}+q^{-1})\).
+The estimates
+\(q\le C(M+\log\tau)\) and
+\(q_x\ge c(1+M\tau^{-1/16})\) show that
+\(q\tau^{-1/2}\le Cq_x\) and \(1\le Cq_x\); hence this last bound is
+at most \(Cr\).  On \([e^{-E/4},2]\) the explicit formula for \(p\) gives a
+positive lower bound on \(r\).  This completes \eqref{quiet:difference}.
+
+The estimates \eqref{quiet:tail} follow from the same two summands of \(q\).
+For \(1\le\tau\le e^{3b_0E}\), the logarithmic summand divided by
+\(q_a\) tends uniformly to zero as the lower threshold on \(E\) increases;
+also \(z/(1+z)\to1\) uniformly.  This proves
+\eqref{quiet:middle-slope}.  On
+\([e^{-E/4},e^E]\) one has
+\(z\ge c e^{3E/4}\) and \(a\ge c e^{-E/16}\), so
+\[
+ q_x\ge(q_a)_x=b_0az(1+z)^{b_0-1}
+     \ge c\exp((3b_0/4-1/16)E)\ge e^E.
+\]
+Finally, on \((0,2e^{-E}]\), the quantity \(z\) is comparable to
+\(e^E\tau\) and remains bounded by a profile constant.  The formulas
+for the first two \(x\) derivatives of \(q\) then give
+\eqref{quiet:axis-slope}.  Between \(2e^{-E}\) and 1 the same formulas
+give \(q_x\ge c\); for \(\tau\ge1\), use \eqref{quiet:tail}.
+These statements also prove the two final consequences.
+\end{proof}
+
+The scalar estimates make two tasks compatible: the spatial Schur
+metric remains positive, and the fiber curvature absorbs the trace
+couplings in a real-plane test. The following criterion states the
+spatial hypotheses needed for both tasks. The word \emph{quiet} refers
+to the smallness of the severity derivatives in the metric of the twist.
+
+\begin{proposition}[Quiet profile criterion]\label{quiet:profile}
+There exist positive constants \(\epsilon,\mu\) and a lower threshold
+\(E_{\min}\), depending on the fixed profile constants but not on \(n\),
+with the following property.  Let \(E\) be a smooth spatial function and
+use \(q=q_0(x,E(s))\) with constant center and no shear.  At a spatial
+point suppose, in some fixed holomorphic affine coordinates, that:
+\begin{enumerate}
+\item \(G\ge V>0\).  The operator norms of \(V,G\), their inverses,
+their first and second metric jets, and all derivatives of \(E\) of orders
+one through four are at most \(e^{\epsilon E}\).
+\item For every Hermitian contravariant matrix \(T\) of rank at most two,
+\(C^V(T,T)\ge e^{-\epsilon E}|T|^2\), with the norm in those affine
+coordinates.
+\item In \(G\)-unit axes,
+\[
+ |\partial E|_G^2\le\mu,\qquad
+ \|\partial\bar\partial E\|_G\le\mu,\qquad
+ \|\nabla_h^{2,0}E\|_G\le\mu
+ \quad\hbox{for every }\tau>0.
+\]
+\end{enumerate}
+If \(E\ge E_{\min}\), the resulting metric is positive for every \(y\)
+over that spatial point and has nonpositive real sectional curvature
+there.  Uniformly in \(\tau\),
+\begin{equation}\label{quiet:schur}
+ \mathcal N=(1+O(\mu))G,\qquad
+ h\asymp V+\tau G.
+\end{equation}
+More precisely, off the fiber axis the rank-two curvature calculation
+retains fixed positive multiples of
+\begin{equation}\label{quiet:margins}
+ At^2+W(b_\times,\bar b_\times)+P|T|_{\mathcal N}^2
+ +\boldsymbol{1}_{\{\tau\le e^{-E/4}\}}
+                          e^{-\epsilon E}|T|^2.
+\end{equation}
+Here the fiber blocks use the tilted logarithmic coordinates,
+\(|T|_{\mathcal N}\) is its Frobenius norm in \(\mathcal N\)-unit
+spatial axes, and \(\boldsymbol{1}\) denotes the indicated indicator.
+The meaning of the final hypothesis is noncircular: positivity of \(h\)
+follows from the preceding hypotheses and does not use
+\(\nabla_h^{2,0}E\).
+\end{proposition}
+
+\begin{proof}
+Differentiating \(U_\tau=\psi-F\) gives
+\begin{equation}\label{quiet:N}
+ \mathcal N=G-F_E\partial\bar\partial E
+               -F_{EE}\partial E\otimes\bar\partial E,
+ \qquad h=V+\tau\bar{\mathcal N}.
+\end{equation}
+The bar denotes averaging from 0 to \(\tau\).  The bounded derivatives
+in \eqref{quiet:Fderivatives} give
+\((1-C\mu)G\le\mathcal N,\bar{\mathcal N}\le(1+C\mu)G\).
+Choose \(\mu\) small enough that \(C\mu<1/10\).  This proves
+positivity and \eqref{quiet:schur}.
+
+We will repeatedly use finite-jet estimates of the following precise
+kind.  Differentiating \eqref{quiet:N} at most twice spatially gives
+sums with at most four derivatives of \(E\) and at most four \(E\) derivatives
+of \(F\) or \(\bar F\).  Each summand is a product of a fixed number of
+the assumed jets.  The formulas for the inverse, connection, and
+curvature add a fixed number of inverse metric factors.  Consequently
+there is an absolute finite constant \(C_*\), independent of dimension,
+such that these estimates cost at most \(C e^{C_*\epsilon E}\), with
+the factors \(1+\tau\) or \((1+\tau)^{-1}\) displayed explicitly
+below.  This conclusion follows by the product rule and
+\(h^{-1}\le C e^{\epsilon E}(1+\tau)^{-1}I\); it uses operator norms,
+not sums of tensor entries.  The same observation applies after
+normalizing spatial axes by \(\mathcal N\), because
+\(e^{-\epsilon E}I/C\le\mathcal N\le C e^{\epsilon E}I\).
+Enlarge \(C_*\) once to cover these finitely many products, and choose
+\(\epsilon>0\) so small that \(C_*\epsilon<1/100\).
+Any profile-dependent multiplicative constants are absorbed by raising
+\(E_{\min}\).
+
+In \(\mathcal N\)-unit axes define
+\begin{equation}\label{quiet:Y}
+ \mathbf Y=I-\tau\mathcal N^{1/2}h^{-1}\mathcal N^{1/2}.
+\end{equation}
+The displayed formula indicates the normalization; equivalently it is
+\(I-\tau h^{-1}\) after \(\mathcal N=I\) has been imposed in the
+fixed axes at the point.  We claim
+\begin{equation}\label{quiet:Ybounds}
+ \|\mathbf Y\|\le r/20\quad(\tau\ge2e^{-E}),\qquad
+ \mathbf Y=I+O(e^{-(1-C_*\epsilon)E})
+                         \quad(\tau\le2e^{-E}).
+\end{equation}
+Always \(\|\mathbf Y\|\le1+C\mu\), because
+\(h\ge\tau\bar{\mathcal N}\) and
+\(\bar{\mathcal N}\) is close to \(\mathcal N\).
+On \([2e^{-E},1]\), the explicit slope \(p\) in the scalar lemma yields
+\(r\ge .35b_0\) for all sufficiently large \(E\): indeed
+\(u\ge e^{-1/16}\tau\), \(z\ge2e^{-1/16}\),
+\(a\ge2^{-1/16}\), and
+\(q_a/q\ge(1+\log2/K_0)^{-1}\ge(1+\log2)^{-1}\).
+The product of these three lower bounds in
+\(p(q_a/q)=b_0a[z/(1+z)](q_a/q)\) exceeds \(.35b_0\).
+This proves the first bound on that interval.
+For \(\tau\ge1\), \eqref{quiet:difference} gives
+\(\|\bar{\mathcal N}-\mathcal N\|_{\mathcal N}\le C\mu r\).
+Set \(h_0=V+\tau\mathcal N\).  The inverse identity and
+\(h,h_0\ge c\tau\mathcal N\) show that replacing \(h\) by \(h_0\) in
+\eqref{quiet:Y} costs at most \(C\mu r\).  In the normalized axes
+\(V\le\gamma I\), where \(\gamma=(1-C\mu)^{-1}\), and hence
+\[
+ \|I-\tau h_0^{-1}\|\le\frac\gamma{\gamma+\tau}.
+\]
+This is at most \(r/40\) on \([1,e^{3b_0E}]\), by
+\eqref{quiet:middle-slope} and \(b_0\ge1000\).
+For \(\tau>e^{3b_0E}\), use \(r\ge c/q\) and
+\(q/\tau\le C(e^{b_0E}+\log(1+\tau))/\tau\); the supremum of the
+last expression on this range tends to zero with the lower threshold
+on \(E\).  Thus the same estimate holds there.  Decreasing \(\mu\) pays
+the inverse-identity error.  Finally, if \(\tau\le2e^{-E}\), then
+\(h\ge V\) and the raw size bounds give
+\(\tau\|\mathcal N^{1/2}h^{-1}\mathcal N^{1/2}\|
+ \le C\tau e^{C_*\epsilon E}\), proving the second bound.
+
+Since \(P_\tau=q(1+r)\) and
+\(P^2P_{\tau\tau}=Pq^2D\), the curvature blocks in
+Lemma~\ref{fiber:blocks} become
+\begin{align}
+ \frac W{Pq}&=rI+\mathbf Y+q^{-1}\partial\bar\partial\log q,
+ \label{quiet:Wformula}\\
+ P^{-1}\nabla_h^{2,0}P
+     &=e_0\nabla_h^{2,0}E+(q_{EE}/q)\partial E\otimes\partial E.
+ \label{quiet:spinformula}
+\end{align}
+The final term in \eqref{quiet:Wformula} and the spin form divided by
+\(Pq\) have norm at most \(C\mu r\) when \(\tau\ge2e^{-E}\),
+and at most \(C\mu\) on the remaining interval.  This follows from
+\eqref{quiet:derivatives}, the three smallness assumptions on \(E\), and
+\(q^{-1}\le Cr\) away from the exceptional interval.
+After decreasing \(\mu\),
+\begin{equation}\label{quiet:Wbounds}
+ W\asymp Pqr\,\mathcal N\quad(\tau\ge2e^{-E}),\qquad
+ W\asymp Pq(1+r)\,\mathcal N\quad(\tau\le2e^{-E}),
+\end{equation}
+and the spin inequality \eqref{fiber:spinabs} holds.
+
+For the fiber diagonal term,
+\[
+ \frac{|\partial P|_h^2}{Pq^2D}
+ \le C\mu\frac{e_0^2}{qD}\le C\mu.
+\]
+Here \(h\ge c\tau G\); the final scalar quotient is bounded using
+\(e_0\le Cr\) for \(\tau\le1\), and
+\(qD\ge cqr=cq_x\ge c\) for \(\tau\ge1\).
+Thus
+\begin{equation}\label{quiet:Abounds}
+ A=(1+O(\mu))Pq^2D>0.
+\end{equation}
+In the same normalized axes the cancellation in
+\eqref{fiber:blockAi} gives
+\[
+ |A_{\cdot}|_{\mathcal N}
+ \le CPq\sqrt\mu\bigl(|e_0|(r+\|\mathbf Y\|)+|r_E|\bigr)
+ \le CPq\sqrt\mu\,r.
+\]
+The last inequality uses \(e_0\le Cr\) on the exceptional interval.
+Combining with \eqref{quiet:Wbounds} and
+\eqref{quiet:Abounds}, the squared ratio to the dual norm determined
+by \(AW\) is at most \(C\mu\).  Consequently
+\eqref{fiber:Aiabs} holds with any fixed desired small \(\eta>0\)
+after decreasing \(\mu\).
+
+We next pay the trace coupling.  Outside the exceptional interval,
+\eqref{quiet:Ybounds} and \eqref{quiet:Wformula} imply
+\[
+ |W:T|\le Pqr\left(|\operatorname{tr}_{\mathcal N}T|
+                       +\frac{\sqrt2}{16}|T|_{\mathcal N}\right).
+\]
+By \(D\ge r^2\), its cost against .98A is at most
+\begin{equation}\label{quiet:tracecost}
+ \frac P{.98(1-C\mu)}
+ \left(|\operatorname{tr}_{\mathcal N}T|
+                       +\frac{\sqrt2}{16}|T|_{\mathcal N}\right)^2.
+\end{equation}
+For \(0\le s\le\sqrt2\),
+\((s+\sqrt2/16)^2/(1+s^2)<.8\); this can also be checked by
+differentiating, since its maximum on this interval occurs at the
+right endpoint.  Because \(T\) has rank at most two, decreasing \(\mu\)
+therefore makes \eqref{quiet:tracecost} at most
+\(.85P((\operatorname{tr}_{\mathcal N}T)^2+|T|_{\mathcal N}^2)\).
+These are exactly the positive symmetrized terms in
+\eqref{fiber:blockL}.
+On \(\tau\le2e^{-E}\), the bounds
+\(D\ge c e^E\tau\), \(q,r\le C\), and
+\eqref{quiet:Wbounds} instead give a trace cost at most
+\(Ce^{-E}|T|_{\mathcal N}^2\).
+
+To estimate the remaining \(C^h\), note from \eqref{quiet:N} that
+the difference \(h-V=\tau\bar{\mathcal N}\) and its first two jets
+are at most \(C\tau e^{C_*\epsilon E}\).
+On \(\tau\le e^{-E/4}\), the curvature formula or
+Lemma~\ref{fiber:perturbation} consequently gives
+\[
+ |C^h(T,T)-C^V(T,T)|
+       \le C\tau e^{C_*\epsilon E}|T|^2
+       \le\tfrac12e^{-\epsilon E}|T|^2.
+\]
+In the complementary range, the inverse bound and the jet bounds give
+\[
+ |C^h(T,T)|\le C(1+\tau)e^{C_*\epsilon E}|T|^2.
+\]
+This is arbitrarily small compared with \(P|T|_{\mathcal N}^2\):
+indeed, for \(\tau\ge e^{-E/4}\),
+\(\tau q_a/(1+\tau)\ge e^{2E}\) after increasing the threshold on \(E\),
+by the explicit power \(b_0\ge1000\); normalization of \(T\) costs only
+\(e^{C_*\epsilon E}\).  On the exceptional interval, its trace cost
+\(Ce^{-E}|T|_{\mathcal N}^2\) is likewise negligible compared with
+\(e^{-\epsilon E}|T|^2\).
+We have thus obtained \eqref{fiber:reserve} with
+\begin{equation}\label{quiet:Rreserve}
+ \mathcal R(T)\ge cP|T|_{\mathcal N}^2
+  +c\boldsymbol{1}_{\{\tau\le e^{-E/4}\}}
+                          e^{-\epsilon E}|T|^2,
+\end{equation}
+before paying the \(K\) coupling.
+
+For that coupling split \(K\) into \(P\nabla^h\mathcal N\) and the two
+terms containing \(P_i=Pe_0E_i\).  The latter have normalized size at
+most \(CP|e_0|\sqrt\mu\); their squared dual \(W\) norm after pairing
+with \(T\) is at most \(C\mu P|T|_{\mathcal N}^2\).  To see this directly
+outside the exceptional interval, the scalar quotient to bound is
+\(e_0^2/(qr)\); it is bounded by \eqref{quiet:derivatives} and
+\eqref{quiet:tail}.  On the exceptional interval replace \(r\) in that
+denominator by \(1+r\), which is easier.
+For \(P\nabla^h\mathcal N\) use three ranges:
+\begin{enumerate}
+\item If \(\tau\le e^{-E/4}\), then \(W\ge cPq\mathcal N\), and
+the raw jet bounds give squared dual \(W\) cost at most
+\(C\tau e^{C_*\epsilon E}|T|^2\), negligible compared with the
+second term of \eqref{quiet:Rreserve}.
+\item If \(e^{-E/4}\le\tau\le e^E\), use
+\(\|\nabla^h\mathcal N\|_{\mathcal N}\le Ce^{C_*\epsilon E}\).
+The squared dual cost is at most
+\(CPe^{C_*\epsilon E}(qr)^{-1}|T|_{\mathcal N}^2\).
+By \eqref{quiet:intermediate} this is negligible compared with
+the first term of \eqref{quiet:Rreserve}.
+\item If \(\tau>e^E\), the identity \(\nabla^h h=0\) permits
+\[
+ \nabla^h\mathcal N
+   =\nabla^h(\mathcal N-\bar{\mathcal N}-V/\tau).
+\]
+Equation \eqref{quiet:difference}, differentiated spatially once,
+and the raw bounds give a normalized norm at most
+\(C e^{-E/4+C_*\epsilon E}\).  In making this estimate the connection
+has raw norm at most \(Ce^{C_*\epsilon E}\), because \(h\) has first
+jets at most \(C(1+\tau)e^{C_*\epsilon E}\) and inverse norm at most
+\(Ce^{C_*\epsilon E}/(1+\tau)\).  Since \(qr=q_x\ge c\), the
+squared dual \(W\) cost is again negligible compared with
+\(P|T|_{\mathcal N}^2\).
+\end{enumerate}
+Fix \(\eta\le1/100\) in Lemma~\ref{fiber:ranktwo}.  Choose \(\mu\)
+small enough for the terms proportional to \(\mu\), and then
+\(E_{\min}\) large enough for the exponentially small terms, so that
+\eqref{fiber:Kabs} holds.  That lemma proves positivity of every
+rank-two contraction with the retained margins
+\eqref{quiet:margins}, and hence nonpositive real sectional curvature.
+Smoothness and positivity at the axis follow from
+Lemma~\ref{fiber:metric}; curvature extends by continuity.
+\end{proof}
+
+To realize the disks of Proposition~\ref{grid:criterion}, we need to
+add a term linear in the fiber variable whose coefficient varies over
+the spatial base. A locally constant coefficient contributes only a
+pluriharmonic potential. The issue is therefore its derivative support,
+where the estimates below must hold for arbitrarily large fiber radius.
+
+\begin{proposition}[Localized shear perturbations]\label{quiet:shear}
+Under the hypotheses of Proposition~\ref{quiet:profile}, suppose \(E\) and
+the center are constant on the spatial region under consideration.
+Assume also that the first and pure second spatial derivatives of
+\(\psi\), in the affine coordinates of that proposition, have norms
+at most \(e^{\epsilon E}\).  Add
+\[
+ \ell_{\mathrm{sh}}=\operatorname{Re}(c_{\mathrm{sh}}(s)(y-\zeta)).
+\]
+On the support of its coefficient's derivatives suppose
+\(\psi\ge-J\), where \(J\ge0\), and
+\begin{equation}\label{quiet:shear-small}
+ \|j^{\le4}c_{\mathrm{sh}}\|\le e^{-C E-J/2}.
+\end{equation}
+For a sufficiently large \(C\) depending only on the profile and the
+constants in the quiet criterion, this perturbation preserves metric
+positivity and nonpositive real sectional curvature for all \(y\).
+Moreover its metric is at least one half the unperturbed metric, and
+fixed fractions of the curvature margins remain.  The choice of \(C\) is
+independent of dimension.
+\end{proposition}
+
+\begin{proof}
+Where the coefficient is locally constant the added potential is
+pluriharmonic, so it changes no metric.  It remains to estimate the
+derivative collar in \eqref{quiet:shear-small}.  Since \(E\) is constant,
+\(P_i=0\), \(\mathcal N=G\), and \(h=V+\tau G\).
+The holomorphic tilt used in Lemma~\ref{fiber:blocks} has linear and
+quadratic coefficients bounded by \(Ce^{\epsilon E}\), because
+\(U_\tau=\psi-F\).  Its constant term may and will be zero.  Derivatives
+through order four of the exponential of this fixed quadratic
+polynomial at the point cost at most \(Ce^{C\epsilon E}\), by the
+ordinary product rule.  These are fixed-order multilinear estimates.
+
+First consider \(\tau\ge e^{-E}\) in tilted logarithmic coordinates.
+The metric inverse has norm at most \(Ce^{C\epsilon E}/\tau\).
+By \eqref{quiet:Abounds}, \eqref{quiet:Wbounds},
+\eqref{quiet:Rreserve}, and \(q_x\ge c\) on this range, the retained
+curvature on a rank-two matrix \(H\) is at least
+\begin{equation}\label{quiet:large-margin}
+ c\tau e^{-C_1E}|H|^2
+\end{equation}
+in these fixed coordinates, for a fixed \(C_1\).
+The cubic table \eqref{fiber:cubics} shows explicitly that the only
+possibly nonzero connection blocks are
+\[
+ \Gamma^0_{00}=P_\tau,\qquad
+ \Gamma^j_{i0}=P(h^{-1}G)^j_i,\qquad
+ \Gamma^k_{ij}=(\Gamma^h)^k_{ij}.
+\]
+Consequently their operator norm is at most
+\(Ce^{C_1E}q\), since \(r\) is bounded.  Allowing a larger fixed power
+of \(q\) in the following estimates would give the same result.
+
+The inequality \(q\ge1\) and the left normalization imply \(F\le x\).
+Thus
+\[
+ |y-\zeta|=e^{(F-\psi)/2}\le e^{-\psi/2}\sqrt\tau.
+\]
+After the tilt, all shear jets needed for the metric and curvature,
+of orders two, three, and four, are therefore at most
+\begin{equation}\label{quiet:shear-large-jets}
+ C\sqrt\tau\exp((-C+C_2)E),
+\end{equation}
+where \(e^{-J/2}e^{-\psi/2}\le1\) was used.
+For every fixed integer \(r_1\), the scalar estimate
+\eqref{quiet:tail}, also valid as an upper bound for small \(\tau\),
+gives
+\begin{equation}\label{quiet:log-absorption}
+ \sup_{\tau\ge e^{-E}}\frac{q^{r_1}}{\sqrt\tau}
+       \le C_{r_1}e^{C_{r_1}E}.
+\end{equation}
+Indeed \(q^{r_1}\le C_{r_1}(e^{r_1b_0E}
++\log(1+\tau)^{r_1})\); on \([e^{-E},1]\) use
+\(\tau^{-1/2}\le e^{E/2}\), and on \([1,\infty)\) the quotient
+\(\log(1+\tau)^{r_1}/\sqrt\tau\) is bounded.
+
+For clarity, this bound controls the quadratic perturbation as well
+as the linear terms.  Write \(\delta=e^{-CE}\) and absorb all other
+fixed \(E\) powers into \(e^{cE}\).  The relative metric error is bounded
+by \(\delta e^{cE}/\sqrt\tau\).  Each linear curvature error from
+\eqref{fiber:perturbbound}, divided by
+\eqref{quiet:large-margin}, is bounded by
+\(\delta e^{cE}q^{r_1}/\sqrt\tau\) for some fixed \(r_1\).
+The quadratic contraction error before division by that margin is
+at most \(\delta^2e^{cE}q^{2r_1}\); after division its bound is
+\[
+ \delta^2e^{c'E}\left(\frac{q^{r_1}}{\sqrt\tau}\right)^2.
+\]
+Equation \eqref{quiet:log-absorption} makes all these quantities
+arbitrarily small by a single sufficiently large \(C\).  It follows that
+the metric is at least half its old value and that the curvature
+errors are smaller than a fixed fraction of the retained margin,
+uniformly on this whole fiber range.
+
+For \(0<\tau\le e^{-E}\), exponentiate the tilted logarithmic
+coordinate and use the affine fiber coordinate with scale
+\(e^{-(\psi(s_0)+E)/2}\) at the fixed spatial point \(s_0\).
+Denote its value by \(z\).  The zero constant term of the tilt and
+\eqref{quiet:axis-slope} give
+\begin{equation}\label{quiet:small-z}
+ |z|^2=e^{E+F}\asymp e^E\tau.
+\end{equation}
+For example, \(|F-x|\le Ce^E\tau\) follows by integration of
+\((q^{-1}-1)d\tau/\tau\) on this range, and proves the comparison.
+The metric blocks in affine coordinates are \(h\) and \(P/|z|^2\).
+Both have inverse norm at most \(e^{C_3E}\).  The curvature tensor
+transforms tensorially: its fiber diagonal block divides by \(|z|^4\),
+its \(W\) block by \(|z|^2\), and its spatial block is unchanged.
+Since
+\[
+ A\ge c\tau^2e^E,\qquad W\ge c\tau G,\qquad
+ \mathcal R(T)\ge c e^{-\epsilon E}|T|^2,
+\]
+the retained rank-two curvature margin in affine coordinates is at
+least \(e^{-C_3E}|H|^2\), after increasing \(C_3\).
+
+The connection in the new coordinates is also bounded by
+\(e^{C_3E}\), without an apparent singularity at \(z=0\).
+Indeed the logarithmic block \(\Gamma^0_{00}=P_\tau\) becomes
+\((P_\tau-1)/z\), and
+\(P_\tau-1=O(e^E\tau)\); thus it is bounded by
+\(C\sqrt{e^E\tau}\).  The mixed spatial block becomes
+\(Ph^{-1}G/z\), bounded by \(e^{C_3E}\) using
+\eqref{quiet:small-z}.  The spatial-output, two-fiber-input block is
+zero because \(P_i=0\), and the spatial connection is already bounded.
+These checks cover all cubic entries.  The same bounds extend to zero
+by the smoothness proved in Lemma~\ref{fiber:metric}.
+
+In these affine coordinates the shear jets of orders two through four
+are bounded by
+\[
+ e^{-CE-J/2}e^{-(\psi(s_0)+E)/2}Ce^{C_4\epsilon E}
+ \le C\exp(-(C+1/2-C_4\epsilon)E).
+\]
+The estimate includes derivatives of the tilt and uses that \(|z|\) is
+bounded on this range.  Apply Lemma~\ref{fiber:perturbation}, with the
+inverse metric, connection, and reciprocal margin bounded by fixed
+powers of \(e^{C_3E}\).  Increasing the same \(C\) makes all relative
+metric errors and all curvature errors smaller than one half their
+respective retained bounds, including at the axis.  The estimates
+overlap at \(\tau=e^{-E}\), so they cover the entire fiber.
+
+Finally, all exponents just used arise from derivatives of order at
+most four, a fixed finite number of products in
+\eqref{fiber:perturbformula}, and operator norms of linear maps.
+Rank-two contractions introduce only the absolute constants in
+\eqref{fiber:ranknorm}.  None of these operations counts spatial
+indices.  This proves the asserted independence from \(n\).
+\end{proof}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/06-radial.tex

@@ -0,0 +1,273 @@
+\section{Radial spatial geometry and aligned profiles}\label{sec:radial}
+
+The quiet criterion supplies the testing regions, where $G\ge V$.
+A center change requires a different arrangement: $V$ must be close to
+$\lambda G$ with a large scalar $\lambda$. This section constructs radial
+spatial metrics that can pass between these arrangements and proves the
+curvature estimate for the aligned regime. Here \emph{aligned} means
+that their metric jets through order two are close to proportional,
+with the scalar frozen at the evaluation point.
+
+All norms of covariant jets in this section are multilinear operator norms in the stated fixed affine coordinates. The norm of a Hermitian contravariant matrix is its Frobenius norm. Constants called absolute may depend on a fixed upper bound for the first two derivatives of $\kappa$ in $\log Y$, but not on the spatial complex dimension. This convention is essential when choosing the dimension only after the geometric constants.
+
+\subsection{The radial clocks}
+
+Write $t=|s|^2$, $\nu=\log t$, and let $B$ and $\Psi$ be radial potentials. Put
+\begin{equation}\label{radial:definitions}
+ \begin{gathered}
+ a=B'(t),\qquad a_\psi=\Psi'(t)=a/\lambda,\qquad
+ u=\log a,\\
+ Y=\frac{du}{d\nu},\qquad
+ H=\frac{Y}{t a_\psi},\qquad
+ p=\frac{d\log a_\psi}{du},\qquad
+ \kappa=\frac{dY/d\nu}{Y(1+Y)}.
+ \end{gathered}
+\end{equation}
+Here a prime on a radial potential means differentiation in $t$. We use $u$ for this radial variable in this section and $u_s$ in Section~\ref{sec:assembly}. We write $V=\partial\bar\partial B$ and $G_0=\partial\bar\partial\Psi$ for their Hermitian matrices.
+
+These quantities separate the controls needed in the construction.
+The ratio $\lambda$ compares the transverse coefficients of $V$ and
+$G_0$, while $H/\lambda=Y/(ta)$ is the common scale of the spatial
+curvature components computed below. The evolution equations will give
+$d\Psi/du=1/H$ and $d\nu/du=1/Y$: large $H$ limits the change of the
+radial twist $\Psi$ during growth, and large $Y$ limits the spatial radius consumed.
+We use the first feature to prepare a center change and the second to
+keep each accelerated episode inside a finite radial interval. Capped
+growth then lets the spatial radius continue between successive tests.
+
+\begin{lemma}[Radial evolution]\label{radial:odes}
+Suppose $a,a_\psi,Y>0$, $\lambda\ge1$, and
+\begin{equation}\label{radial:kappa-assumptions}
+ \frac1{1+Y}\le\kappa\le1.1,\qquad
+ \frac{d\kappa}{d\log Y}\ge-\kappa.
+\end{equation}
+Then
+\begin{align}\label{radial:clock-equations}
+ \frac{d\nu}{du}&=\frac1Y,&
+ \frac{d\log Y}{du}&=\kappa(1+1/Y),&
+ \frac{d\log\lambda}{du}&=1-p,\\
+ \frac{d\Psi}{du}&=\frac1H,&
+ \frac{d\log H}{du}&=\kappa(1+1/Y)-p-1/Y.\notag
+\end{align}
+If $Y\ge t$ initially, then $Y\ge t$ subsequently. With smooth bounded controls $p,\kappa$ on each finite $u$ interval, positive solutions have no blowup on that interval. A cap of the form $\kappa=C_\kappa/(1+Y)$, with fixed $C_\kappa\ge1$, has no blowup at finite $\nu$ for the prescribed bounded controls.
+
+If $p=1$, then
+\begin{equation}\label{radial:H-logY}
+ \frac{d\log H}{d\log Y}=1-\frac1\kappa.
+\end{equation}
+In particular, $p=\kappa=1$ holds both $\lambda$ and $H$ constant. If $\kappa\ge1$ on an interval beginning at $u_0$, then the total increase of $\nu$ on that interval is at most $1/Y(u_0)$, even if its finite $u$ length is arbitrarily large.
+\end{lemma}
+
+\begin{proof}
+The first three identities follow by changing the independent variable from $\nu$ to $u$. Since $d\Psi/d\nu=t a_\psi$ and $H=Y/(t a_\psi)$, the fourth follows as well. Differentiating $\log H=\log Y-\nu-\log a_\psi$ gives the fifth. The lower bound on $\kappa$ implies
+\[
+ \frac{d\log(Y/t)}{d\nu}=\kappa(1+Y)-1\ge0.
+\]
+In $u$ time, $dY/du=\kappa(1+Y)\le1.1(1+Y)$, so $Y$ remains finite and positive on each finite interval. The remaining variables are obtained by integrating the displayed equations there. In a cap, $dY/d\nu=C_\kappa Y$, which gives the asserted global $\nu$ continuation. Dividing the last equation of \eqref{radial:clock-equations}, with $p=1$, by the second proves \eqref{radial:H-logY}. Finally, $\kappa\ge1$ gives $Y(u)\ge Y(u_0)e^{u-u_0}$, and integration of $d\nu/du=1/Y$ gives the last assertion.
+\end{proof}
+
+The inequalities in \eqref{radial:kappa-assumptions} allow the smooth changes used later. For example, to decrease $\kappa$ between two fixed positive values, make the change over a sufficiently long fixed interval in $\log Y$; then $d\log\kappa/d\log Y\ge-1$. To pass from $\kappa=1$ to a cap, choose a smooth function $K(Y)$ which initially equals $1+Y$, has $0\le K'\le1$, and eventually is constant, and put $\kappa=K(Y)/(1+Y)$. The change can be made on a fixed increasing-factor interval of $Y$, so its first two $\log Y$ derivatives are bounded. Moreover $1\le K\le1+Y$ and
+\[
+ Y\kappa_Y+\kappa
+ =\frac{K+Y(1+Y)K'}{(1+Y)^2}\ge0.
+\]
+Thus all inequalities persist, and $H$ is nonincreasing during this change when $p=1$.
+
+\subsection{Curvature and coordinate bounds}
+
+\begin{lemma}[Radial curvature]\label{radial:curvature}
+Under \eqref{radial:kappa-assumptions}, the negative Hermitian curvature tensor $C^V=-R^V$ satisfies
+\begin{equation}\label{radial:rank-reserve}
+ C^V(T,T)\ge c\,\kappa\frac{H}{\lambda}|T|_V^2
+ \qquad\text{if }T=T^*,\quad\operatorname{rank}T\le2,
+\end{equation}
+where $c>0$ is absolute. Here the subscript $V$ means that the matrix is expressed in $V$-unit axes.
+\end{lemma}
+
+\begin{proof}
+At $s=(\sqrt t,0,\ldots,0)$ the metric has entries
+\[
+ V_{i\bar j}=a\delta_{ij}+a'\bar s_i s_j,
+\]
+and its radial and transverse eigenvalues are $a(1+Y)$ and $a$. The unitary symmetry in the transverse variables and the K\"ahler symmetries leave only a transverse block, a radial--transverse bisectional block, and a pure radial entry. After removing the common factor $Y/(ta)=H/\lambda$, their respective values in $V$-unit axes are
+\begin{equation}\label{radial:curvature-components}
+ \delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj},\qquad
+ \kappa\delta_{kl},\qquad
+ D=\kappa\bigl(1+\kappa+Y\kappa_Y\bigr).
+\end{equation}
+For completeness, the transverse coefficient is $a'/a^2=Y/(ta)$. The radial--transverse coefficient is
+\[
+ \frac{dY/d\nu}{t a(1+Y)}=\kappa\frac{Y}{ta}.
+\]
+For the pure radial entry, differentiate the logarithm of the radial metric along the radial complex line. Since
+\[
+ \frac{d}{d\nu}\log\bigl(a(1+Y)\bigr)=Y(1+\kappa),
+\]
+its second $\nu$ derivative divided by $ta(1+Y)$ is exactly the last expression in \eqref{radial:curvature-components} times $Y/(ta)$. These calculations also follow directly by subtracting the cubic contractions in the K\"ahler curvature formula.
+
+Write a Hermitian test matrix in radial/transverse blocks as
+\[
+ T=\begin{pmatrix}r&b^*\\ b&S\end{pmatrix}.
+\]
+The contraction after removing $Y/(ta)$ is
+\[
+ Dr^2+2\kappa r\operatorname{tr}S+2\kappa|b|^2
+       +(\operatorname{tr}S)^2+|S|^2.
+\]
+The assumption on $\kappa_Y$ gives $D\ge\kappa$. Absorb the cross term with $0.9Dr^2$. Its cost is at most
+\[
+ \frac{\kappa^2}{0.9D}(\operatorname{tr}S)^2
+ \le\frac{1.1}{0.9}(\operatorname{tr}S)^2.
+\]
+Since $\operatorname{rank}S\le2$, $|\operatorname{tr}S|\le\sqrt2|S|$. The transverse reserve is therefore at least $5|S|^2/9$, in addition to $0.1Dr^2$ and $2\kappa|b|^2$. As $\kappa\le1.1$, these bound an absolute positive multiple of $\kappa(r^2+2|b|^2+|S|^2)$. This proves \eqref{radial:rank-reserve}.
+\end{proof}
+
+\begin{lemma}[Scaled affine jets]\label{radial:scaled}
+Assume \eqref{radial:kappa-assumptions} and $\lambda\ge1$.
+Suppose $Y\ge1$, $Y\ge t$, the first two derivatives of $\kappa$ in $\log Y$ are bounded by a fixed constant, and
+\begin{equation}\label{radial:p-close}
+ |p-1|+\left|\frac{dp}{d\log Y}\right|
+      +\left|\frac{d^2p}{d(\log Y)^2}\right|\le\delta,
+ \qquad 0<\delta\le\tfrac1{10}.
+\end{equation}
+At a point use $G_0$-unit affine axes divided by $\sqrt H$. In these axes $G_0=H^{-1}I$ at the point. The transverse and radial Euclidean axis lengths are
+\begin{equation}\label{radial:axis-lengths}
+ \sqrt{\frac tY},\qquad
+ \sqrt{\frac{t}{Y(1+pY)}}.
+\end{equation}
+In these fixed coordinates:
+\begin{enumerate}
+\item derivatives of $\Psi$ and $B$ in orders one through four are bounded by $C/H$ and $C\lambda/H$, respectively;
+\item metric jets through order two of $V-\lambda G_0$, with $\lambda$ frozen at its value at the point, are bounded by $C\delta\lambda/H$;
+\item derivatives of $u$ in orders one through four are bounded by $C$;
+\item if a further potential has Euclidean derivatives in orders two through four bounded by $D_e$, its Hessian and first two metric jets here are bounded by $CD_e/(a_\psi H)$.
+\end{enumerate}
+All constants are independent of dimension.
+\end{lemma}
+
+\begin{proof}
+The eigenvalues of $G_0$ are $a_\psi$ transversely and $a_\psi(1+pY)$ radially, giving \eqref{radial:axis-lengths}. In these axes $dt$ and $d^2t$ have operator norms at most $Ct/Y$, and higher derivatives of $t$ vanish. The first bound includes the fact that the only nonzero first derivative is radial; thus its axis length includes the extra factor $(1+pY)^{-1/2}$.
+
+Set a dot to mean $d/d\nu$. We have
+\[
+ \dot Y=\kappa Y(1+Y)=O(Y^2),\qquad
+ \ddot Y=O(Y^3),\qquad \dddot Y=O(Y^4).
+\]
+The last two estimates use, respectively, the first and second indicated derivatives of $\kappa$. Also $\dot p=O(\delta Y)$ and $\ddot p=O(\delta Y^2)$. Direct differentiation gives
+\begin{align*}
+ a'&=aY/t,\\
+ a''&=a\bigl(Y^2-Y+\dot Y\bigr)/t^2,\\
+ a'''&=a\bigl(Y^3-3Y^2+2Y+3(Y-1)\dot Y+\ddot Y\bigr)/t^3.
+\end{align*}
+Consequently $|a^{(j)}|\le Ca(Y/t)^j$ for $0\le j\le3$. The same formulas for $a_\psi$ use $pY$ in place of $Y$ and its derivatives. They imply the corresponding estimate for $a_\psi$, and comparison at the point, where $a=\lambda a_\psi$, improves their difference by the factor $\delta$.
+
+The chain rule for a radial potential in at most four spatial slots is a finite sum of products of its $t$ derivatives and copies of $dt,d^2t$. For a term containing its $l$th $t$ derivative, the product of these latter factors is bounded by $C(t/Y)^l$. Since $B^{(l)}=a^{(l-1)}$, every such product is at most $Cat/Y=C\lambda/H$, and the same argument gives $C/H$ for $\Psi$. The improved difference estimate proves the metric-jet assertion. For $u$, its $t$ derivatives through order four are $O((Y/t)^j)$, by the displayed bounds on $Y$ and its derivatives; the same chain rule proves the third assertion. All sums here have a number of terms bounded in terms of the derivative order, not the dimension.
+
+Finally, the largest axis length in \eqref{radial:axis-lengths} is $\sqrt{t/Y}\le1$. Applying two, three, or four such axes to the extra potential therefore gives at most $CD_e t/Y=CD_e/(a_\psi H)$.
+\end{proof}
+
+\subsection{Constant-center aligned estimates}
+
+In the remainder of this section the fiber center and severity $E$ are constant, the fiber profile is $q=q_0(x,E)$ from Proposition~\ref{quiet:profile}, and no shear is added. We use the fiber notation
+\[
+ x=\log\tau,\quad P=\tau q,\quad r=q_x/q,\quad
+ D_q=(q_x+q_{xx})/q.
+\]
+The elementary profile estimates of Lemma~\ref{quiet:scalar} give $q\ge1$, $r>0$, $D_q\ge r^2$, and, for constants allowed to depend on the fixed profile and $E$,
+\begin{equation}\label{radial:profile-bounds}
+ \begin{split}
+ &D_q\ge c_E\tau\quad(0<\tau\le1),\qquad q\le C_E\quad(0<\tau\le1),\\
+ &q/D_q\le C q^2,\qquad q\le C_E(1+\log\tau)\quad(\tau\ge1).
+ \end{split}
+\end{equation}
+These bounds also follow directly from the positive logarithmic summand of $q_0$: for $\tau\ge1$ it supplies a positive lower bound for $q_x+q_{xx}$, whereas the other summand is bounded in $\tau$ at fixed $E$.
+
+\begin{proposition}[Aligned profile]\label{radial:aligned}
+There is an absolute $\delta_0>0$ with the following property. Fix $\sigma>0$ and take the severity $E$ sufficiently large in terms of $\sigma$ and the fixed profile. Assume the hypotheses of Lemma~\ref{radial:scaled}, with $\delta\le\delta_0$, and also
+\[
+ \tfrac12\le\kappa\le1.1,\qquad H\ge\sigma,
+ \qquad G=G_0+G_e,\qquad
+ |\partial^{[j]}G_e|\le\delta/H\quad(0\le j\le2)
+\]
+in its scaled affine axes. Here $G=\partial\bar\partial\psi$ and $\partial^{[j]}$ denotes any real or complex metric jet of order $j$, measured as a multilinear operator. Suppose either
+\begin{equation}\label{radial:aligned-alternatives}
+ \lambda=1,\qquad\text{or}\qquad
+ H\ge M_E(1+\log\lambda)^2,
+\end{equation}
+where $M_E$ is sufficiently large depending only on the fixed profile and $E$. Then the constant-center fiber potential defines a positive metric with nonpositive real sectional curvature. Its spatial Schur complement satisfies $h\ge cV$.
+
+More precisely, in scaled spatial axes and the tilted logarithmic fiber coordinate of Lemma~\ref{fiber:blocks}, the contraction of its negative curvature on each Hermitian rank-at-most-two matrix with blocks $(T,b,t_f)$ is bounded below by
+\begin{equation}\label{radial:aligned-margin}
+ c\left(A t_f^2+W(b,\bar b)
+               +\frac{\lambda+\tau}{H}|T|^2
+               +P|T|_G^2\right),
+\end{equation}
+where $c>0$ can be fixed independently of dimension and the stage parameters once the stated thresholds hold. The coefficient $A$ and the form $W$ are those of Lemma~\ref{fiber:blocks}.
+\end{proposition}
+
+\begin{proof}
+Since $q$ has no spatial dependence, $\mathcal N=G$ and $h=V+\tau G$. Lemma~\ref{radial:scaled} shows that, through two metric jets, $h$ differs from the constant multiple $(\lambda+\tau)V/\lambda$ by at most $C\delta(\lambda+\tau)/H$. In particular
+\[
+ h\asymp\frac{\lambda+\tau}{H}I,\qquad
+ \Gamma^h=O(1).
+\]
+Lemma~\ref{radial:curvature} gives a covariant curvature lower bound $c\lambda|T|^2/H$ for $V$ in these scaled axes. Multiplication of a metric by a positive constant multiplies its covariant curvature by that constant. The curvature difference is at most $C\delta(\lambda+\tau)|T|^2/H$: indeed its fourth-derivative term has this bound, and its cubic contractions have the same bound by the inverse estimate $|h^{-1}|\le CH/(\lambda+\tau)$ and the first-jet bounds. Lemma~\ref{fiber:perturbation} makes this calculation explicit. Reducing $\delta_0$ therefore gives
+\begin{equation}\label{radial:h-margin}
+ C^h(T,T)\ge c\frac{\lambda+\tau}{H}|T|^2
+ \quad(\operatorname{rank}T\le2).
+\end{equation}
+
+Let $\theta=\lambda/(\lambda+\tau)$. The identity $\nabla^h(V+\tau G)=0$ implies
+\[
+ \nabla^h G=\theta\bigl(\nabla^hG-\lambda^{-1}\nabla^hV\bigr),
+ \qquad |\nabla^hG|\le C\delta\theta/H.
+\]
+The curvature table gives $A=Pq^2D_q$, vanishing spin and $A_i$ terms, and $K=P\nabla^hG$. In $G$-unit normalization it gives
+\begin{equation}\label{radial:W-normalized}
+ \frac{W}{Pq}=rI+\mathcal Y,
+ \qquad \mathcal Y=V_G(V_G+\tau I)^{-1},
+ \qquad \mathcal Y\asymp\theta I.
+\end{equation}
+Here $V_G$ denotes $V$ expressed in $G$-unit axes; its eigenvalues are $(1+O(\delta))\lambda$. In particular $W>0$. Spending a fixed small portion of $W$ on the $K$ coupling costs at most
+\[
+ C\delta^2\frac{P\theta^2}{qH(r+\theta)}|T|^2
+ \le C\delta^2\frac{\tau\theta}{H}|T|^2,
+\]
+which is an arbitrarily small portion of \eqref{radial:h-margin} after reducing $\delta_0$.
+
+It remains to pay for $2t_f(W:T)$. Spending $0.98A t_f^2$ costs $(W:T)^2/(0.98A)$. First suppose the second alternative in \eqref{radial:aligned-alternatives} holds. Split the two terms in \eqref{radial:W-normalized} using $(a+b)^2\le1.1a^2+11b^2$. The $r$ term costs at most
+\[
+ \frac{1.1}{0.98}P(\operatorname{tr}_G T)^2.
+\]
+The horizontal curvature contains the positive term
+\[
+ P\bigl((\operatorname{tr}_G T)^2+|T|_G^2\bigr).
+\]
+As $|\operatorname{tr}_G T|\le\sqrt2|T|_G$, their difference retains more than $P|T|_G^2/2$. The remaining trace cost is at most
+\begin{equation}\label{radial:trace-remainder}
+ C\frac{P\theta^2}{D_qH^2}|T|^2.
+\end{equation}
+For $\tau\le1$, \eqref{radial:profile-bounds} bounds $P/D_q$ by $C_E$. For $\tau\ge1$, it bounds \eqref{radial:trace-remainder} by
+\[
+ \frac{C_E}{H^2}\tau\theta^2(1+\log\tau)^2|T|^2
+ \le \frac{C_E(1+\log\lambda)^2}{H^2}
+             (\lambda+\tau)|T|^2.
+\]
+For the last inequality, put $z=\tau/\lambda$ and use boundedness of
+$z(1+z)^{-3}$ and $z(1+z)^{-3}(\log z)^2$ on $(0,\infty)$. Taking $M_E$ sufficiently large absorbs this cost into a small portion of \eqref{radial:h-margin}.
+
+Now suppose $\lambda=1$. The profile estimates give, for sufficiently large $E$,
+\begin{equation}\label{radial:lambda-one-slope}
+ \|\mathcal Y\|\le r/18\qquad(\tau>2e^{-E}).
+\end{equation}
+To see the uniformity of this assertion, $\|\mathcal Y\|\le(1+C\delta)/(1+\tau)$. On $2e^{-E}\le\tau\le1$, the first summand of $q_0$ gives a logarithmic slope bounded below by a large fixed multiple of one. On $1\le\tau\le e^{3b_0E}$, it gives $r\ge0.9b_0(1+\tau)^{-1/16}$ after increasing $E$. Beyond this interval the logarithmic summand gives $r\ge c/q$, which exceeds $40/(1+\tau)$ by its linear-in-$\log\tau$ upper bound and the starting size $\tau\ge e^{3b_0E}$. These statements are unchanged by taking $\delta_0$ smaller.
+
+By \eqref{radial:lambda-one-slope}, the trace expenditure outside the exceptional interval is at most
+\[
+ \frac{P}{0.98}
+       \left(|\operatorname{tr}_GT|+\frac{\sqrt2}{18}|T|_G\right)^2.
+\]
+For $0\le s\le\sqrt2$, the ratio $(s+\sqrt2/18)^2/[0.98(1+s^2)]$ is strictly less than one, by an absolute amount. Thus a fixed portion of $P|T|_G^2$ remains. On $\tau\le2e^{-E}$, the explicit left-end profile gives $q\le C$, $r\le C$, and $D_q\ge c\tau e^E$, where constants may depend on the fixed profile but not $E$. Hence the trace cost is at most $Ce^{-E}|T|^2/H^2$. Since $H\ge\sigma$, this is a small portion of $(1+\tau)|T|^2/H$ for $E$ sufficiently large in terms of $\sigma$.
+
+All cases retain fixed portions of $A$, $W$, \eqref{radial:h-margin}, and the symmetrized $PG^2$ term, proving \eqref{radial:aligned-margin}. Lemma~\ref{fiber:ranktwo} gives nonpositive real sectional curvature off the fiber axis. The metric is positive there and $h\ge cV$. Smoothness of the original affine fiber potential and its positive limiting metric extend both conclusions across the axis.
+\end{proof}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/07-centers.tex

@@ -0,0 +1,345 @@
+\section{Changing the fiber center}\label{sec:centers}
+
+A moving center is not a holomorphic change of coordinates. We therefore first widen the fibers, then estimate the actual change of potential as a perturbation in fixed holomorphic coordinates. The widening and the metric scales must be chosen in the order specified below.
+
+Throughout this section $p=\kappa=1$ and $Y\ge\max\{1,t\}$, so $\lambda,H$ are constant by Lemma~\ref{radial:odes}, and $V=\lambda G_0$ as metrics, including their jets. We use the scaled affine spatial coordinates of Lemma~\ref{radial:scaled}. The severity $E$ is constant, and no shear term is present. The old twist has the form
+\[
+ \psi_{\rm old}=\Psi+\psi_e,
+ \qquad G_e=\partial\bar\partial\psi_e,
+ \qquad |\partial^{[j]}G_e|\le\delta/H\quad(0\le j\le2),
+\]
+with $\delta$ sufficiently small as in Proposition~\ref{radial:aligned}. Suppose also that, throughout the interval being configured,
+\begin{equation}\label{center:old-data}
+ \psi_{\rm old}\le A_{\rm old},\qquad
+ |\partial^{[j]}\psi_{\rm old}|\le A_{\rm old}
+ \quad(1\le j\le4)
+\end{equation}
+in these fixed scaled axes at each point. The number $A_{\rm old}$ is known independently of the target sizes of $\lambda,H$. Additive constants already placed in the twist are included in this upper bound.
+
+\begin{proposition}[Center change]\label{center:change}
+Choose the profile constants in the following order: first the fixed exponent $b_0$ of the quiet profile; then a sufficiently large absolute constant $d_0$; and then $K_0$ sufficiently large in terms of $d_0$ and absolute constants specified in the proof. Fix $E$ and old data as in \eqref{center:old-data}. There is a width parameter $L$, chosen using only these data and the fixed profile, such that the following construction is possible.
+
+After choosing $L$, choose a sufficiently large $\lambda$, and then a sufficiently large $H$; explicitly one can require
+\begin{equation}\label{center:target-order}
+ \lambda\ge\max\{e^{3L},\Lambda_f\},
+ \qquad H\ge M_f(1+\log\lambda)^2,
+\end{equation}
+where $\Lambda_f,M_f$ are finite constants determined by the fixed profile, $E,L$, the old data, and $\delta$, but not by $\lambda,H$. Over three fixed units of the variable $\Psi$, one can change from any prescribed constant center in the unit disk to any other such center, returning to the original profile $q_0(x,E)$ at the end. The final twist differs from the old twist by a nonnegative constant, possibly very large.
+
+During the change the potential is smooth, its metric is positive and has nonpositive real sectional curvature, and the metric bounds $cV$ from below on spatial projection, for a fixed $c>0$. There is no restriction on the cost of the final constant offset. All power exponents and smallness tolerances used before choosing the spatial dimension are dimension independent.
+\end{proposition}
+
+\subsection{The profile deformation and its elementary bounds}
+
+Write $F^0_x=1/q_0$ and normalize $F^0(x)-x\to0$ at the left end. We record properties of the concrete quiet profile that will be used here. At fixed $E$,
+\begin{equation}\label{center:q-properties}
+ \begin{split}
+ &q_0\ge1,\quad (q_0)_x>0,\quad
+       (q_0)_{xx}/(q_0)_x\ge-1/16,\\
+ &|(q_0)_x|+|(q_0)_{xx}|\le C q_0,\qquad
+       q_0(x,E)\le C_E+C(1+\max(x,0)),\\
+ &\inf_{x\ge-20}(q_0)_x>0.
+ \end{split}
+\end{equation}
+Here and below a subscript $E$ permits dependence on $E$ and the already fixed profile. The last infimum can be bounded below using just the derivative of $K_0^{-1}\log(1+e^x)$. The inequality for the second derivative follows by differentiating both summands in the quiet profile. In particular $(q_0)_x+(q_0)_{xx}\ge(15/16)(q_0)_x$.
+
+Two endpoint properties are also needed. The function $Q_0(v)=q_0(\log v,E)$ is smooth down to $v=0$, with
+\begin{equation}\label{center:left-slope}
+ Q_0(0)=1,\qquad Q_0'(0)=b_0e^E+K_0^{-1}>0.
+\end{equation}
+At the other end the auxiliary function in the quiet profile tends to a positive finite limit with error $O(e^{-x/16})$, so
+\[
+ q_0(x,E)=c_E+x/K_0+O_E(e^{-x/16})\quad(x\longrightarrow\infty).
+\]
+Integration, using $1/q_0-K_0/x=O_E(x^{-2})$, gives
+\begin{equation}\label{center:primitive-asymptotic}
+ F^0(x)=K_0\log x+O_E(1),\qquad
+ \int_0^L\frac{dx}{q_0(x,E)}=K_0\log L+O_E(1).
+\end{equation}
+No uniformity in $E$ is asserted or needed in these endpoint constants; $E$ has been fixed before $L$.
+
+Take $L$ large. Choose a smooth function $\chi$ with
+\[
+ 0\le\chi\le1,\quad \operatorname{supp}\chi\subset[-20,3L],\quad
+ \chi=1\text{ on }[0,L],\quad \int\chi=2L,\quad \chi'\ge-1/16,
+\]
+and with its derivatives through any fixed needed order bounded absolutely. Such a function is obtained from a fixed rising cutoff, an adjustable plateau, and a fixed descending cutoff of length greater than $16$; adjusting the plateau gives the integral exactly. Define the increasing inverse map $m=m(x)$ by
+\begin{equation}\label{center:inverse-map}
+ x=m+L-\frac12\int_{-\infty}^m\chi(v)\,dv.
+\end{equation}
+Then
+\begin{equation}\label{center:map-bounds}
+ 1\le m'\le2,\quad m\le x,\quad
+ m=x-L\ (x\le L-20),\quad m=x\ (x\ge X:=3L).
+\end{equation}
+Its derivatives of any fixed order are bounded absolutely. Set
+\[
+ q_1(x)=q_0(m(x),E)\le q_0(x,E).
+\]
+Since $m''=\chi'(m)(m')^3/2$, putting $a=m'\in[1,2]$ gives
+\begin{align}\label{center:q1-positive}
+ (q_1)_x+(q_1)_{xx}
+ &\ge(q_0)_x(m)\bigl(a-a^2/16-a^3/32\bigr)\\
+ &\ge\tfrac34(q_0)_x(m)a=\tfrac34(q_1)_x>0.\notag
+\end{align}
+Also $|(q_1)_x|+|(q_1)_{xx}|\le Cq_1$, with constants independent of $L$. These bounds follow from \eqref{center:q-properties} and the bounds on $m',m''$.
+
+\subsection{Offsets during profile interpolation}
+
+On the first unit of $\Psi$, choose a smooth nondecreasing flat-ended cutoff $\beta=\beta(\Psi)$ from $0$ to $1$, with fixed bounds on its derivatives through order four. Define
+\begin{equation}\label{center:forward-profile}
+ q(x,\beta)=(1-\beta)q_0(x,E)+\beta q_1(x),\qquad
+ \mathfrak s=1/q,
+\end{equation}
+and let $F^\beta$ have derivative $F^\beta_x=\mathfrak s$ and left normalization $F^\beta-x\to0$. Every such primitive exists, tends to infinity on the right, and differs from $F^0$ by a constant for $x\ge X$. All spatial derivatives in the rest of this subsection are at fixed $\tau=e^x$.
+
+Use an offset $w^+(\beta)$ defined by $w^+(0)=0$ and
+\begin{equation}\label{center:offset-definition}
+ (w^+)'(\beta)=\partial_\beta F^\beta(X)
+                  +\frac{d_0}{1-\beta+1/L},\qquad
+ D(x,\beta)=(w^+)'(\beta)-\partial_\beta F^\beta(x).
+\end{equation}
+Here the prime on $w^+$ denotes differentiation in $\beta$. Since $\mathfrak s_\beta\ge0$ and vanishes for $x\ge X$,
+\begin{equation}\label{center:D-domination}
+ D\ge\frac{d_0}{1-\beta+1/L}.
+\end{equation}
+This lower bound can dominate $\sup_x|\mathfrak s_\beta|$ by any required fixed factor if $d_0$ is sufficiently large. Indeed
+\[
+ 0\le\mathfrak s_\beta=(q_0-q_1)/q^2
+ \le\min\{(1-\beta)^{-1},\sup_{x\le X}q_0(x,E)\}.
+\]
+By taking $L$ sufficiently large relative to $C_E$, the second entry is at most $CL$ with an absolute $C$. The elementary bound
+$\min\{z^{-1},CL\}\le(C+1)/(z+1/L)$ proves the assertion, including $\beta=1$ by continuity.
+
+For later restoration use, on another unit of $\Psi$, the reverse mixture $q=(1-\beta)q_1+\beta q_0$ with a fresh increasing cutoff. Now $\mathfrak s_\beta\le0$ and $\partial_\beta F^\beta\le0$. Choose a new offset $w^-(\beta)$ with $w^-(0)=0$ and constant positive derivative greater than a sufficiently large fixed multiple of $1+\sup_{x,\beta}|\mathfrak s_\beta|$. With $D=(w^-)'-\partial_\beta F^\beta$, the same domination holds. No estimate on the size of $w^-(1)$ is required.
+
+Once $L$ is fixed, all the following constants are finite and may be included in a number $C_f$: upper bounds for the offsets and their derivatives through order four; for $D$ and its first three $\beta$ derivatives; for the profile and the finite-order derivatives used below on $x\le X$; for their left-end decay coefficients; and the reciprocal lower constants in the positive estimates on that interval. To check finiteness at the noncompact left endpoint, both $q_0-1$ and $q_1-1$, with their needed derivatives, are $O_f(e^x)$. Thus $\beta$ derivatives of $\mathfrak s$ are $O_f(e^x)$, and their integrals defining derivatives of $F^\beta$ converge. In particular $C_f$ may include $e^X$ and large functions of $E,L$. It is selected before $\lambda,H$.
+
+\begin{lemma}[Curvature during interpolation]\label{center:interpolation}
+During either profile switch above, keep the center constant and set
+$\psi=\psi_{\rm old}+w^\pm(\beta(\Psi))$, also including the already frozen offset when making the reverse switch. If $\lambda$ and then $H$ satisfy sufficiently large thresholds of the form \eqref{center:target-order}, the metric is positive, $h\ge cV$, and its negative curvature has a strictly positive rank-two reserve off the fiber axis. These conclusions extend across that axis by continuity.
+\end{lemma}
+
+\begin{proof}
+A prime on $\beta$ now means differentiation in $\Psi$. Direct differentiation of $\psi-F^\beta$ gives
+\begin{equation}\label{center:N-formula}
+ \mathcal N=(1+D\beta')G_0+G_e
+       +\bigl(D\beta''+D_\beta(\beta')^2\bigr)
+                   \partial\Psi\otimes\bar\partial\Psi.
+\end{equation}
+Here $D_\beta$ is the derivative at fixed $x$; thus all dependence of the primitive on the spatial cutoff is included. The last term and its first two spatial jets are $O(C_f/H^2)$, because all positive derivatives of $\Psi$ in scaled axes are $O(1/H)$. The derivatives of the scalar $D\beta'$ beyond its value are $O(C_f/H)$. With $d_f=1+D\beta'\ge1$, these facts show
+\[
+ \mathcal N\asymp d_fG_0>0,
+ \qquad h=V+\int_0^\tau\mathcal N(v)\,dv\ge cV.
+\]
+The constants in $\asymp$ can be absolute by making $H$ large enough; the value $d_f$ itself is bounded by $C_f$.
+
+First consider $x\le X$. Since $\tau\le e^X$, the first two metric jets of $h-V$ are $O(C_f/H)$. Hence
+\begin{equation}\label{center:finite-x-h}
+ \Gamma^h=\Gamma^{G_0}+O(C_f/\lambda),\qquad
+ C^h(T,T)\ge c\lambda|T|^2/H
+ \quad(\operatorname{rank}T\le2).
+\end{equation}
+The second estimate follows from Lemma~\ref{radial:curvature}, the scaled jet estimates, and the curvature perturbation identity, taking $\lambda\gg C_f$.
+
+The mixed form in Lemma~\ref{fiber:blocks} is
+\[
+ \frac{W}{Pq}=(1+r)\mathcal N
+       -\tau\mathcal N h^{-1}\mathcal N
+       +q^{-1}\partial\bar\partial\log q.
+\]
+The last term equals
+\[
+ -\mathfrak s_\beta\beta'G_0+O(C_f/H^2).
+\]
+The spin form, divided by $Pq$, equals
+\[
+ -\mathfrak s_\beta\beta'\nabla_h^{2,0}\Psi+O(C_f/H^2).
+\]
+In fact $q_\beta/q^2=-\mathfrak s_\beta$; the remaining terms contain two first derivatives of $\Psi$. Lemma~\ref{radial:scaled} and \eqref{center:finite-x-h} give
+$|\nabla_h^{2,0}\Psi|\le C/H$ with absolute $C$. Also
+$|\tau\mathcal N h^{-1}\mathcal N|\le C_f/(\lambda H)$. Choose $d_0$ first so that the domination in \eqref{center:D-domination} is sufficiently strong in comparison with this absolute $C$, and then choose $\lambda,H$ large compared with $C_f$. It follows that
+\begin{equation}\label{center:W-spin}
+ W\ge c\frac{Pq}{H}d_f I,
+ \qquad |\nabla_h^{2,0}P(b,b)|\le\tfrac14 W(b,\bar b).
+\end{equation}
+This argument also applies to the reverse mixture, using absolute values of $\mathfrak s_\beta$.
+
+By \eqref{center:q1-positive} and convexity of the mixture,
+\[
+ D_q=(q_x+q_{xx})/q\ge c_f\min(1,\tau)\quad(x\le X).
+\]
+Left-end decay and boundedness on the rest of this interval give
+\[
+ |\partial\log P|+|\partial P_\tau|
+       \le C_f\min(1,\tau)/H.
+\]
+It follows from the curvature table that
+\begin{equation}\label{center:A-control}
+ A=(1+o(1))Pq^2D_q,\qquad
+ |A_i|\le C_f P\min(1,\tau)/H,
+\end{equation}
+where the relative error tends uniformly to zero as $\lambda,H$ increase. To verify uniformity at $\tau=0$, the subtracted term in $A$ is at most
+$C_fP^2\min(1,\tau)^2/(\lambda H)$, while the leading term is $Pq^2D_q$. The ratio tends to zero uniformly under the stated choices. Similarly \eqref{center:W-spin} and \eqref{center:A-control} make the $A_i$ coupling arbitrarily small in units $(AW)^{1/2}$.
+
+Finally $|K/P|\le C_f/H$. Spending fixed portions of $A,W$ according to Lemma~\ref{fiber:ranktwo} costs at most
+\[
+ C_f\frac{P}{qH}|T|^2\le\frac{C_f}{H}|T|^2
+ \quad\text{for the }K\text{ coupling},\qquad
+ C_f\frac{P}{D_qH^2}|T|^2\le\frac{C_f}{H^2}|T|^2
+ \quad\text{for the trace coupling}.
+\]
+In these displayed bounds $C_f$ is enlarged to include $e^X$; also $P/D_q$ is bounded on $x\le X$, including the left endpoint. Both costs are small compared with \eqref{center:finite-x-h}. Thus fixed fractions of $A$, $W$, and $\lambda|T|^2/H$ remain.
+
+Now let $x>X$. Since the two profiles coincide exactly here, $F^\beta(x)-F^0(x)$ is independent of $x$. Consequently $\mathcal N=\mathcal N_\infty$ is independent of $\tau$, and
+\begin{equation}\label{center:finite-tail}
+ h=V_f+\tau\mathcal N_\infty,
+ \qquad V_f=V+\int_0^{e^X}
+          \bigl(\mathcal N(v)-\mathcal N_\infty\bigr)\,dv.
+\end{equation}
+There is no infinite tail in this expression. Its first two spatial jets differ from those of $V$ by $O(C_f/H)$. Treating $d_f=1+D\beta'$ at the point as constant, the jets of $\mathcal N_\infty$ differ from those of $d_fG_0$ by at most $C\delta d_f/H$, after taking $H\gg C_f/\delta$; similarly those of $V_f$ differ from $\lambda G_0$ by a sufficiently small relative amount after taking $\lambda\gg C_f/\delta$.
+
+The aligned argument can now be applied with the two spatial scales $\lambda$ and $d_f$. More explicitly, let $\theta_f=\lambda/(\lambda+\tau d_f)$. The comparisons and the identity $\nabla^h(V_f+\tau\mathcal N_\infty)=0$ give
+\[
+ C^h(T,T)\ge c\frac{\lambda+\tau d_f}{H}|T|^2,
+ \qquad |\nabla^h\mathcal N_\infty|
+                   \le C\delta d_f\theta_f/H.
+\]
+The $K$ cost is bounded by $C\delta^2(\lambda+\tau d_f)|T|^2/H$. The $rI$ part of the trace cost uses the same rank-two symmetrized $P\mathcal N_\infty^2$ reserve as before. The remaining cost is at most
+\[
+ C\frac{\tau qd_f^2\theta_f^2}{D_qH^2}|T|^2
+ \le C_f\frac{(1+\log\lambda)^2(\lambda+\tau d_f)}{H^2}|T|^2.
+\]
+For the last step, use \eqref{radial:profile-bounds} and the scalar inequality from the proof of Proposition~\ref{radial:aligned} with $\lambda/d_f$ in place of $\lambda$. We have $1\le d_f\le C_f$ and may require $\lambda\ge C_f$, so the logarithm is bounded by $\log\lambda$. Taking $H\ge M_f(1+\log\lambda)^2$ with sufficiently large $M_f$ absorbs this last cost. Spin and $A_i$ vanish in this region because $q=q_0$ is spatially constant.
+
+We have proved positivity and nonpositive curvature off the axis, with fixed fractional reserves. The profiles are smooth in $\tau$ at zero, equal to one there, and have the prescribed left normalization. Thus the potential and metric are smooth in the original affine fiber coordinate, whose metric at the axis has entries $V,e^\psi$. Curvature extends by continuity. This proves the lemma.
+\end{proof}
+
+\subsection{The width gained before moving the center}
+
+At the end of the first switch freeze $\beta=1$ and the offset, and write
+$\psi=\psi_{\rm old}+w^+(1)$. Changing variables in the defining integral gives
+\begin{equation}\label{center:F1-formula}
+ F^1(x)=L+F^0(m(x))
+          -\frac12\int_{-\infty}^{m(x)}\frac{\chi(v)}{q_0(v,E)}\,dv.
+\end{equation}
+Both sides have derivative $1/q_1$ and the same left normalization. At $x=X$, $m(X)=X$, so integration of \eqref{center:offset-definition} yields
+\begin{equation}\label{center:offset-total}
+ w^+(1)=L-\frac12\int_{\mathbb R}\frac{\chi(v)}{q_0(v,E)}\,dv
+                  +d_0\log(1+L).
+\end{equation}
+In particular this offset is nonnegative: the integral is at most $\int\chi=2L$.
+
+Define the pointwise width scales
+\[
+ R_0=\exp\bigl((L-\psi)/2\bigr),\qquad
+ Z=\exp\bigl((F^1(x)-\psi)/2\bigr)=|y-\zeta|.
+\]
+Because $\chi=1$ on $[0,L]$, \eqref{center:primitive-asymptotic} and \eqref{center:offset-total} imply
+\begin{equation}\label{center:R-growth}
+ R_0\ge c_{E,A_{\rm old}}L^{K_0/4-d_0/2}.
+\end{equation}
+Choose $K_0$ so large relative to $d_0$ that $K_0/4-d_0/2>K_0/9$. Then, for $L$ sufficiently large depending on old data,
+\begin{equation}\label{center:width-bounds}
+ R_0\ge L^{K_0/9},\qquad
+ Z\ge x^{K_0/9}\quad(x\ge L-20).
+\end{equation}
+Here is the second bound in detail. At $x=L-20$, formula \eqref{center:F1-formula} gives $F^1=L+F^0(-20)$, so $Z$ is a fixed old-data multiple of $R_0$. Monotonicity of $F^1$ and \eqref{center:R-growth} prove the desired bound on $[L-20,3L]$, absorbing the fixed factor $3^{K_0/9}$ by increasing $L$. For $x\ge3L$, the integral in \eqref{center:F1-formula} has reached its total and cancels the integral in \eqref{center:offset-total}. Thus exactly
+\[
+ F^1(x)-\psi=F^0(x)-\psi_{\rm old}-d_0\log(1+L).
+\]
+The asymptotic \eqref{center:primitive-asymptotic}, together with $x\ge3L$ and the strict exponent inequality already chosen, gives the second bound on this remaining interval. These estimates are uniform over the spatial interval because only the upper bound in \eqref{center:old-data} was used.
+
+\subsection{The actual nonholomorphic center motion}
+
+The preceding calculation has produced width before any center is moved.
+We now spend that width to control the metric and curvature errors of
+moving the center. Two coordinate systems cover the fiber: logarithmic
+coordinates away from the enlarged core and a scaled affine coordinate
+on the core, including its axis.
+
+On the next unit of $\Psi$, hold $q=q_1$ and all offsets fixed, and let $\zeta$ be a smooth flat-ended convex-combination interpolation of the two prescribed centers. Its derivatives through order four as a function of $\Psi$ are bounded absolutely. By Lemma~\ref{radial:scaled}, and $H\ge1$, its spatial derivatives in scaled axes satisfy
+\begin{equation}\label{center:zeta-jets}
+ |\partial^{[j]}\zeta|\le C/H\qquad(1\le j\le4).
+\end{equation}
+At a point $s_0$, compare the moving-center potential with the potential obtained by replacing $\zeta(s)$ by the constant $\zeta(s_0)$ in a neighborhood. This is a comparison in fixed holomorphic coordinates, not a claim that $y-\zeta(s)$ is a holomorphic coordinate. All constants in the following two comparisons may depend on old data, $E$, and the fixed profile, but are independent of $L,\lambda,H$ once the stated thresholds hold.
+
+First suppose $x\ge L-20$. For the frozen center use scaled spatial axes and the tilted logarithmic fiber coordinate of Lemma~\ref{fiber:blocks}. The previous interpolation proof with cutoff constant retains fractions of $A,W$ and the horizontal curvature reserve. On this region $m\ge-20$, so \eqref{center:q-properties} and \eqref{center:q1-positive} give
+\[
+ A\ge c\tau,\qquad W\ge c\tau H^{-1}I.
+\]
+The horizontal reserve is at least $c\tau/H$: on $x\le X$ use $\lambda\ge e^X\ge\tau$ in \eqref{center:finite-x-h}, and on $x>X$ use the aligned reserve. Consequently the full rank-two curvature contraction is bounded below by $c\tau/H$ times the squared Frobenius norm of the test. The metric has the same lower scale and its inverse has norm at most $CH/\tau$. These assertions also follow directly for the metric from $h=V+\tau G$ and $P=\tau q$.
+
+The polynomial holomorphic tilt has coefficients bounded by the first two old twist derivatives; the offset is constant. In these tilted coordinates the frozen connection has norm at most $Cq$. Indeed the possibly large entries in the cubic table are $P_\tau=q+q_x=O(q)$ and $P\mathcal Nh^{-1}=O(q)$, whereas $P_i=0$, the tilted pure holomorphic second spatial derivative vanishes, and $\Gamma^h=O(1)$.
+
+Let $\Phi$ be the one-variable fiber potential as a function of
+$\rho+\psi=\log|y-\zeta|^2+\psi$. Its first derivative is $\tau$, and differentiation in this argument is $q\partial_x$. Thus its derivatives of orders one through four have size at most $C\tau q^3$. For example
+\[
+ \Phi''=\tau q,\quad
+ \Phi'''=\tau q(q+q_x),\quad
+ \Phi''''=\tau q\bigl((q+q_x)^2+q(q_x+q_{xx})\bigr).
+\]
+The fixed argument has bounded jets through order four in the chosen frame. The difference in its argument after moving the center is
+\begin{equation}\label{center:log-argument-difference}
+ \log\left|1-\frac{\zeta(s)-\zeta(s_0)}{y-\zeta(s_0)}\right|^2.
+\end{equation}
+It vanishes at the evaluation point, and each positive jet contains a derivative of $\zeta$ and a denominator of size at least $Z$. By \eqref{center:zeta-jets}, bounded tilt coefficients, and $HZ\ge1$, its jets through order four are $O(1/(HZ))$. The chain rule consequently bounds every potential-jet difference through order four by
+\begin{equation}\label{center:outer-jet-error}
+ C\frac{\tau q^3}{HZ}.
+\end{equation}
+In particular its relative metric error is at most $Cq^3/Z$. Once this is small, Lemma~\ref{fiber:perturbation}, the bound $\Gamma=O(q)$, and the inverse bound $O(H/\tau)$ show that the curvature error on unit rank-two tests is at most
+\begin{equation}\label{center:outer-curvature-error}
+ C\frac{\tau}{H}\left(\frac{q^5}{Z}
+                            +\frac{q^8}{Z^2}\right)
+ \le C\frac{\tau q^8}{HZ}.
+\end{equation}
+The first term accounts for all linear fourth-jet and cubic-contraction changes, and the second is the quadratic inverse-contraction term. Fixed-rank contraction costs only an absolute factor.
+
+For $x\ge L-20$, \eqref{center:q-properties} and $m\le x$ give $q=q_1\le Cx$ once $L$ is large relative to $C_E$. This constant is independent of $L,\lambda,H$. Choose, in addition to the earlier requirement, $K_0/9>10$. Then \eqref{center:width-bounds} makes the ratios in \eqref{center:outer-jet-error} and \eqref{center:outer-curvature-error} arbitrarily small compared with the metric and curvature reserves by increasing $L$. This choice precedes $\lambda,H$.
+
+It remains to include $x<L-20$ and the axis itself. Put $v=\tau e^{-L}$. By \eqref{center:map-bounds}, the fiber potential added to $B$ is exactly $e^L$ times the original smooth radial function, now evaluated at
+\[
+ |y-\zeta(s)|^2e^{\psi(s)-L}.
+\]
+This identity follows by changing variables in $\int_0^\tau q_1(\log t)^{-1}dt$. Use scaled spatial axes and the fixed affine fiber coordinate
+\[
+ z=\frac{y-\zeta(s_0)}{R_0(s_0)}.
+\]
+At the evaluation point $|z|^2=e^{F^0(\log v)}\asymp_E v$ for $0\le v\le e^{-20}$. The comparison is uniform down to zero because $F^0(\log v)-\log v$ is smooth there. The smooth radial function and its derivatives on this fixed compact argument interval have bounds depending only on old data and $E$.
+
+For the frozen center, \eqref{center:left-slope} gives, with positive old-data constants,
+\[
+ A\ge c_E\tau^2e^{-L},\qquad
+ W\ge c_E\tau H^{-1}I,
+\]
+throughout $0<v\le e^{-20}$. For the first inequality, the continuous positive ratio of $P^2P_{\tau\tau}$ to $\tau^2e^{-L}$ has a positive limit at zero by $Q_0'(0)>0$; positivity away from zero follows from \eqref{center:q-properties}. For $W$, the term $Pq\mathcal YG$ has a positive lower bound of this size because $\tau\le e^{L-20}$ and $\lambda\ge e^{3L}$. Divide the vertical and cross curvature blocks by $|z|^4$ and $|z|^2$, respectively, on changing from logarithmic to affine fiber coordinates. The retained vertical and cross reserves are then at least $c_Ee^L$ and $c_Ee^L/H$. The horizontal reserve $c\lambda/H$ also dominates $ce^L/H$. The metric has the same smallest scale, so its inverse is bounded by $C_EH/e^L$.
+
+Undoing the exponentiated tilt changes the affine fiber variable by $z\mapsto e^{\theta(s)}z$, where $\theta(s_0)=0$ and its coefficients are bounded by old twist derivatives. Since $|z|$ is bounded, the differential and its inverse, and the needed higher jets, have old-data bounds independent of $L,\lambda,H$. This preserves the indicated rank-two lower margin up to old-data constants. The connections in these coordinates are bounded: the logarithmic entry transforms to
+\[
+ (P_\tau-1)/z=O_E(|z|),
+\]
+and the spatial-fiber entry is bounded by
+\[
+ |P\mathcal Nh^{-1}|/|z|
+       \le C_E\tau/(\lambda|z|)\le C_E|z|,
+\]
+using $\lambda\ge e^L$. The pure vertical-to-spatial entry vanishes before the bounded tilt is undone. The remaining spatial entries are bounded by the radial jet estimates.
+
+In this affine frame the moving smooth argument, before applying the fixed smooth radial function, is
+\begin{equation}\label{center:affine-argument}
+ \left|z-\frac{\zeta(s)-\zeta(s_0)}{R_0(s_0)}\right|^2
+                         e^{\psi(s)-\psi(s_0)}.
+\end{equation}
+Its difference from the frozen argument has jets through order four bounded by $C/(HR_0)$, including at $z=0$, by \eqref{center:zeta-jets}. Therefore the potential jets change by at most $Ce^L/(HR_0)$. The inverse metric and bounded connection estimates give a curvature error at most
+\begin{equation}\label{center:inner-error}
+ C\frac{e^L}{H}\left(R_0^{-1}+R_0^{-2}\right).
+\end{equation}
+By \eqref{center:width-bounds}, this is a small fraction of the $e^L/H$ reserve for $L$ sufficiently large, independently of the later choices of $\lambda,H$. The metric error is small by the same estimate. Smoothness includes the axis, so no limiting logarithmic-coordinate assertion is being used at a singular coordinate point.
+
+We have proved that the moving-center metric remains, say, at least half the frozen-center metric and that its real sectional curvature is nonpositive. Its spatial projection bound is therefore inherited from the frozen Schur bound $h\ge cV$.
+
+\begin{proof}[Completion of Proposition~\ref{center:change}]
+Choose $d_0$ to dominate the absolute spin constants in the interpolation estimate, and choose $K_0$ so that $K_0/4-d_0/2>K_0/9$ and $K_0/9>10$. The latter explicit exponent is larger than the exponent $8$ in \eqref{center:outer-curvature-error}. For fixed old data and $E$, choose $L$ sufficiently large for \eqref{center:width-bounds} and for the moving-center metric and curvature errors to be less than the retained reserves. Next compute the finite constants $C_f$ for both switches and choose $\lambda$ and then $H$ according to \eqref{center:target-order}, with the larger thresholds needed in Lemma~\ref{center:interpolation}. This is the required noncircular order of choices.
+
+Perform the forward switch with constant old center, the center interpolation with frozen profile $q_1$ and offset, and finally the reverse switch with constant new center. Lemma~\ref{center:interpolation} controls the two switches and the preceding argument controls the actual center motion. The cutoffs are flat-ended, so all joins are smooth. At the end the profile is $q_0$, the center is the prescribed new constant, and the added offset is $w^+(1)+w^-(1)\ge0$. All profiles are at least one, have smooth left endpoints with value one, and have normalized primitives tending to infinity on the right; thus they define smooth fiber potentials for all affine fiber values throughout. The spatial projection bound follows on each part, with a fixed fractional loss during the center motion. No bound on the final constant offset is required or asserted.
+\end{proof}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/08-assembly.tex

@@ -0,0 +1,1102 @@
+\section{Assembly of the complete base}\label{sec:assembly}\label{assembly:section}
+
+The local estimates are now available. We must realize infinitely many
+analytic tests with one finite spatial dimension, preserve the earlier
+tests when adding later wells, and obtain a complete global metric.
+We choose the spatial data and join the local regimes. All tensor and
+fixed-order derivative estimates in this section use operator or
+multilinear norms in fixed affine coordinates. In particular, their
+leading exponents do not include a trace over the spatial dimension.
+
+\begin{proposition}\label{assembly:base}
+For some fixed finite integer $n$, there is a smooth strictly
+plurisubharmonic potential $\ell$ on $\mathbb C^n\times\mathbb C$
+whose K\"ahler metric is complete and has nonpositive real sectional
+curvature. The potential satisfies every hypothesis of
+Proposition~\ref{grid:criterion}, for a countable dense set of centers
+in the unit disk and every spatial Taylor order. Thus all bounded
+holomorphic functions on its half-plane tube are constant.
+\end{proposition}
+
+At each spatial point the fiber potential is recovered from
+\[
+ F(\log\tau,s)=\log|y-\zeta(s)|^2+\psi(s)
+\]
+with the left normalization of Section~\ref{sec:fiber}. The radial
+function $B$ supplies the spatial metric. The twist $\psi$ starts with
+a radial background $\Psi$: polynomial wells make selected fibers wide,
+and additional radial plurisubharmonic terms control the derivatives
+of the severity. A term
+$\operatorname{Re}(c_{\rm sh}(s)(y-\zeta(s)))$ prescribes the sheared
+disk directions. We choose these data in increasing spatial radius.
+The polynomial wells are the only later additions that extend into
+earlier regions, so their inward tails must be reserved in advance.
+
+The proof occupies this section. A cycle starts on a capped spatial
+interval, raises the severity, performs one test, and raises severity
+again on a post-test ramp. At constant severity it then accelerates the
+spatial metric, enters the aligned regimes, changes center, and restores
+the spatial parameters. Finally it increases severity while the
+normalized spatial geometry is fixed, until the severity can pay for
+the next capped interval.
+\input{figures/stage-clock}
+
+We retain the radial notation of Lemma~\ref{radial:odes}:
+\[
+ \begin{gathered}
+ t=|s|^2,\quad \nu=\log t,\quad a=B'(t),\quad u_s=\log a,\\
+ Y=\frac{du_s}{d\nu},\quad a_\psi=\Psi'(t)=a/\lambda,\quad
+ H=\frac{Y}{ta_\psi},\quad p=\frac{d\log a_\psi}{du_s}.
+ \end{gathered}
+\]
+Here $H$ is a scalar; the polynomial system below is denoted by
+$\mathcal H$. The evolution equations are
+\begin{align}
+ \frac{dY}{du_s}&=\kappa(1+Y),&
+ \frac{d\nu}{du_s}&=Y^{-1},&
+ \frac{d\log\lambda}{du_s}&=1-p,\label{assembly:clocks}\\
+ \frac{d\Psi}{du_s}&=H^{-1},&
+ \frac{d\log H}{du_s}
+ &=\kappa(1+Y^{-1})-p-Y^{-1}.
+ \label{assembly:H-clock}
+\end{align}
+The admissibility and derivative conditions on $\kappa,p$ in the radial
+lemmas will always be imposed.
+
+\subsection{Wells and the order of constants}
+
+Fix a sequence $(\zeta_j,h_j)$ in which every pair from a fixed countable
+dense subset of $\{|y|<1\}$ and $\mathbb Z_{\geq0}$ occurs infinitely
+often. The parameters of a test will be
+\begin{equation}\label{assembly:test-parameters}
+ E_*=B_*c,\qquad \eta=e^{-c},\qquad
+ N=\lfloor e^{\gamma E_*}\rfloor,\qquad
+ l=\frac N2\log(t/R^2),\qquad S=Re^{c/N}.
+\end{equation}
+The constants $B_*,\gamma$ are fixed below. At each stage $R$ is fixed
+before $c$. For a degree-$N$ system $(\mathcal H,p_N)$ from
+Lemma~\ref{grid:polynomials}, define
+\begin{equation}\label{assembly:well}
+ W_j(s)=d\log
+ \frac{|\mathcal H(s/R)|^2+|p_N(s/R)-1|^2+e^{-2c}}
+ {1+e^{-2c}}.
+\end{equation}
+The polynomial $p_N$ is unrelated to the radial control $p$.
+The logarithmic sum-of-squares formula shows that $W_j$ is smooth
+and plurisubharmonic.
+
+\begin{lemma}\label{assembly:well-bounds}
+There is an absolute exponent $C_{\rm pol}$ such that, for $t\geq1$
+and $1\leq r\leq4$,
+\begin{equation}\label{assembly:well-raw}
+ |\partial^{[r]}W_j|
+ \leq C(n,d)\exp\{C_{\rm pol}(\max(l,0)+c+\log N)\}.
+\end{equation}
+There is a constant $L_n<\infty$ such that, after increasing the degree
+threshold,
+\begin{equation}\label{assembly:well-tail}
+ |\partial^{[r]}W_j|\leq N^{L_n}
+ \qquad(1\leq r\leq4,\ l\geq L_n\log N).
+\end{equation}
+Both statements hold throughout the indicated outer ranges.
+\end{lemma}
+
+\begin{proof}
+Homogeneity and the spherical upper bound give
+$|(\mathcal H,p_N)(s/R)|\leq Ne^l$. Cauchy estimates at scale
+$c_0\sqrt t/N$ bound its derivatives of order at most four by this
+quantity times a fixed power of $N$, since $t\geq1$. In a differentiated
+logarithm there are only a fixed number of factors and denominator
+powers; the denominator in \eqref{assembly:well} is at least $e^{-2c}$.
+This proves \eqref{assembly:well-raw} with an absolute exponent;
+fixed-dimensional constants and $d$ affect only its prefactor.
+
+The spherical lower bound also gives
+$|(\mathcal H,p_N)(s/R)|\geq e^lN^{-C_n}$. Once $l$ exceeds a sufficiently
+large multiple of $\log N$, subtraction of $(0,1)$ leaves norm at least
+$e^lN^{-C_n}/2$. The factors $e^l$ in derivatives of the logarithm can
+then be canceled. The remaining costs are fixed powers of $N$, with
+exponents depending on the already fixed $n$. Increase $L_n$ to pay
+them and the degree threshold to pay the prefactor.
+\end{proof}
+
+For each stage we also introduce smooth nonnegative radial
+plurisubharmonic functions $Z_{j,\rm in}$ and $Z_{j,\rm out}$. They
+vanish before their stage and eventually become affine nondecreasing
+functions of $\nu$. The center changes add offsets that vanish before
+their change interval and are constant afterwards. We seek the final
+twist in the form
+\begin{equation}\label{assembly:twist}
+ \psi=\Psi+\sum_j W_j+\sum_j(Z_{j,\rm in}+Z_{j,\rm out})
+       +\text{center-change offsets}.
+\end{equation}
+The center is constant except on its change intervals. There is one
+fiber profile at each spatial point: it is $q_0(x,E(s))$ except during
+the coordinated switches of Proposition~\ref{center:change}.
+Shears occur only in the disjoint test collars. The recursion below
+will make the well sum converge smoothly on compact spatial sets;
+the other added terms are locally finite.
+
+The constants must provide the three outputs of
+Proposition~\ref{grid:criterion}: a fixed-height spatial ball, small
+unsheared disks on its boundary, and large prescribed sheared disks at
+the nodes. Their exponents determine $k$ and hence the dimension.
+Accordingly we first choose all geometric constants independently of
+$n$, and only then use stagewise size choices to pay dimensional
+prefactors.
+
+We give the full order of choices. First choose the profile constants,
+including $d_0,K_0$ for the center change, then $\epsilon,\mu$ for
+Proposition~\ref{quiet:profile}, the shear exponent in
+Proposition~\ref{quiet:shear}, and the tolerance $\delta$ for
+Proposition~\ref{radial:aligned}. These choices are independent of $n$.
+Use control transitions whose first two derivatives in $\log Y$ have
+absolute bounds. Let $C_{\rm rad}$ dominate the absolute exponents
+needed for radial raw jets through order four. Choose
+\begin{equation}\label{assembly:first-constants}
+ \epsilon_1\ll\epsilon/(1+C_{\rm rad}),\qquad
+ C_\uparrow\gg(1+C_{\rm rad})/\epsilon.
+\end{equation}
+Choose next
+\begin{equation}\label{assembly:second-constants}
+ m_z\geq10(C_{\rm pol}+1),\qquad
+ B_*\geq\frac{C_*(1+m_z+C_{\rm pol}+C_{\rm rad})}{\epsilon_1},
+\end{equation}
+where the absolute $C_*$ is large enough for the finitely many raw
+exponent comparisons below. Choose $C_s$ larger than the quiet shear
+severity exponent, and $C_w$ large enough to pay the shell loss and
+four derivatives of its cutoffs; $C_w>5k_*+20$ suffices for these latter
+costs after increasing the other fixed constants if necessary. Here
+$k_*$ is the absolute separation exponent of the polynomial grids.
+Choose
+\begin{equation}\label{assembly:gamma-kb}
+ 0<\gamma\ll\epsilon_1,\qquad
+ \gamma B_*\leq\min\{.005,(8C_w)^{-1}\},\qquad
+ k_b>\max\{8/\gamma,100/\epsilon\}.
+\end{equation}
+Choose $M_b$ sufficiently large for the ratios in
+Lemma~\ref{assembly:quiet-check}, and then $\sigma>0$ sufficiently small
+for that lemma and Lemma~\ref{assembly:reset}. These are choices made
+from fixed constants; in particular
+$\sigma M_bk_b^2\ll1$, and $\sigma$ pays all quiet smallness requirements
+involving $C_\uparrow$. Severity thresholds can depend on $\sigma$.
+Finally take
+\begin{equation}\label{assembly:depth}
+ d\geq8(M_bk_b+C_s+1)B_*.
+\end{equation}
+All constants so far are independent of dimension.
+
+Set
+\begin{equation}\label{assembly:test-exponents}
+ m_0=d/2,\qquad p_0=2d,\qquad
+ p_1=C_sB_*+C_wd\gamma B_*.
+\end{equation}
+Then $p_1\leq d/4<m_0$. Choose an integer $k$ with
+$.99k>p_0+p_1+1$, and only now fix a finite $n$ sufficiently large that
+\begin{equation}\label{assembly:dimension}
+ \sum_{r=1}^{k-1}\frac{r^{n-1}}{(n-1)!}<.01.
+\end{equation}
+Such a finite $n$ exists because $k$ is fixed. The constants $C_n,L_n$
+and all dimensional prefactors will be paid by the stagewise choice
+of large $c$; they do not change \eqref{assembly:test-exponents}.
+
+\subsection{Recursive convention and capped intervals}
+
+Every estimate for an already configured region is made with an
+additional allowance of one for the absolute value and spatial
+jets through order four of all future wells. Let $\nu_j^{\rm sat}$
+denote the designated endpoint of severity saturation at stage $j$,
+and let
+\[
+ K_j=\{s:|s|^2\leq e^{\nu_j^{\rm sat}}\}
+\]
+be the entire inner spatial ball ending there. The testing annulus lies
+beyond this ball. At stage $j$ we impose
+\begin{equation}\label{assembly:diagonal}
+ \sup_{s\in K_j}|\partial^{[r]}W_j(s)|\leq2^{-j},
+ \qquad 0\leq r\leq j+4.
+\end{equation}
+Each new trigger begins beyond every completed stage, so every already
+configured region lies in every later $K_j$. The omitted well sum
+therefore has absolute value and spatial jets through order four
+bounded by $\sum_j2^{-j}\leq1$ there. Its Hessian is also positive
+semidefinite. Feasibility is proved below. This convention reserves the
+entire future allowance before selecting any future coefficients. We apply
+the local all-fiber lemmas to the final twist with these allowances;
+we do not infer a uniform curvature statement from small perturbations
+on compact subsets of the total space.
+
+Initially put
+\begin{equation}\label{assembly:initial-data}
+ a=a_\psi=a_{\rm init}e^t,\quad Y=t,\quad
+ \lambda=p=1,\quad\kappa=(1+Y)^{-1},
+\end{equation}
+with $a_{\rm init}$ so large that $H\leq\sigma$, and choose $B,\Psi$
+with equal initial constants. Up to a positive scale and an additive
+constant, this initial spatial potential is the exponential radial
+example of \citet[Section~3]{Seshadri2006}. Take center $\zeta_1$ and a constant
+severity sufficiently large for the initial compact region and the
+tail allowance. The radial curvature formula is strictly positive
+on nonzero rank-two Hermitian tests at $t=0$, by its limit or by
+direct differentiation of $a_{\rm init}e^t$. Thus the quiet criterion
+also applies at the spatial origin.
+
+Before every trigger the radial data will have the form
+\begin{equation}\label{assembly:cap}
+ \lambda=p=1,\qquad
+ \kappa=\frac{C_\kappa}{1+Y},\qquad
+ 1\leq C_\kappa\leq1+Y,\qquad H\leq\sigma,
+\end{equation}
+with $C_\kappa$ constant on that interval. Then
+$dY/d\nu=C_\kappa Y$, so the cap can be continued through any
+prescribed finite interval of $\nu$ without a pole. All previously
+generated nonradial twist terms have bounded positive-order Euclidean
+jets on the entire outer range: use \eqref{assembly:well-tail} and
+the affine $Z$ tails. Hence a cap can also be continued until
+$a=e^{u_s}$ dominates any prescribed bound for those old jets.
+
+Choose the next trigger start $\nu_B$ with
+$Y_B=Y(\nu_B)\gg C_\kappa$, $e^{\nu_B}\geq1$, and severity constant
+there at a value satisfying
+\begin{equation}\label{assembly:trigger-initial}
+ E_0\geq C_\uparrow u_s(\nu_B).
+\end{equation}
+The reset below provides this inequality and any additional fixed
+threshold. Put
+\begin{equation}\label{assembly:trigger-radius}
+ \xi=Y_B(\nu-\nu_B),\qquad
+ \nu_*=\nu_B+Y_B^{-1},\qquad R=e^{\nu_*/2}.
+\end{equation}
+These data precede the choice of $c$. Retain the capped radial data
+through
+\begin{equation}\label{assembly:post-length}
+ l=T_*+4,\qquad T_*=10\{c+(1+L_n)\log N\}.
+\end{equation}
+For large $c$ this lies inside $\nu\leq\nu_B+2/Y_B$, since
+$(T_*+4)/N\to0$. Throughout this fixed continuation
+$Y\asymp Y_B$ and $u_s=u_s(\nu_B)+O(1)$, with constants allowed
+to depend on the already fixed cap data.
+
+\begin{lemma}\label{assembly:height}
+Before choosing $c$ at stage $j$, one can fix $D_j\geq j$ such that,
+for every sufficiently large permitted $c$ and every final choice of
+the reserved tails,
+\begin{equation}\label{assembly:height-bound}
+ \ell(s,\zeta_j)\leq D_j-2\qquad(|s|\leq Re^{c/N}).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+On the spatial region following the previous center-change interval,
+the center is $\zeta_j$ and $\ell(s,\zeta_j)=B(s)$, including on shear
+supports. The same identity holds on the initial cap at the first
+stage. Thus the bound there follows from the fixed cap continuation
+up to $\nu_B+2/Y_B$.
+The earlier spatial region is a fixed compact set $K$ with already
+chosen profiles and centers; future stages do not change $q(x,s)$
+on $K$. The reservation gives $|\psi-\psi_{\rm old}|\leq1$ there,
+including the current and all later wells. Both centers are in the unit disk, so
+$|\zeta_j-\zeta(s)|\leq2$. Choose
+$A>\log4+\sup_K\psi_{\rm old}+1$.
+For each $s\in K$, $F(x,s)\to\infty$ as $x\to\infty$.
+Continuity, a finite cover, and monotonicity in $x$ give a common
+finite $X$ with $F(X,s)>A$ on $K$. Inverting the fiber relation
+therefore gives $\tau\leq e^X$. Since $q\geq1$, the radial potential
+is bounded by $\sup_K B+e^X$; the fixed shear adds at most
+$2\sup_K|c_{\rm sh}|$. At center coincidences this argument is
+understood by the smooth limit $\tau=0$. These bounds are independent
+of $c$ and of the future wells. Add two and increase to at least $j$.
+\end{proof}
+
+We always require
+\begin{equation}\label{assembly:height-rate}
+ c\geq j(D_j+1)
+\end{equation}
+in addition to the finitely many current thresholds. Thus $D_j\to\infty$
+and $D_j/c_j\to0$.
+
+\subsection{The severity trigger}
+
+Choose a smooth convex nondecreasing $h_b$ on $(-\infty,1)$, zero
+for $\xi\leq0$, equal to $-\log(1-\xi)$ up to an affine function near
+$1$. To construct it, cut on the nonnegative second derivative
+$(1-\xi)^{-2}$ smoothly and integrate twice from zero.
+Use
+\begin{equation}\label{assembly:trigger}
+ E=E_0+k_bh_b(\xi),\qquad Z_{j,\rm in}=M_bk_b^2h_b(\xi)
+\end{equation}
+until severity is near $E_*$. Smoothly saturate $E$ at $E_*$ over
+a unit interval of its argument, so that $E=E_*$ by
+$k_bh_b=E_*-E_0+1$. Derivatives of the saturation function in that
+argument are bounded. Keep the unsaturated convex $Z_{j,\rm in}$
+slightly longer, and once $k_bh_b\geq E_*-E_0+2$, cut off its
+second derivative nonnegatively over half the remaining gap to
+$\xi=1$. Continue it affinely in $\nu$ thereafter. This is a smooth
+convex nondecreasing function. Such a function of $\log|s|^2$,
+zero near the origin, is a smooth radial plurisubharmonic function:
+its transverse and radial Hessian entries are its first and second
+$\nu$ derivatives divided by $t$.
+
+Near this cutoff the gap $1-\xi$ is comparable, with fixed stagewise
+factors, to $e^{-(E_*-E_0)/k_b}$. The final slope is bounded by
+\begin{equation}\label{assembly:trigger-slope}
+ C M_bk_b^2Y_Be^{(E_*-E_0)/k_b}.
+\end{equation}
+The value there is $M_bk_b(E_*-E_0+O(1+k_b))$. Multiplication
+of \eqref{assembly:trigger-slope} by the remaining gap to $\nu_*$
+costs only a fixed $O(M_bk_b^2)$, and its product with $4c/N$
+tends to zero. Thus for large $c$,
+\begin{equation}\label{assembly:trigger-value}
+ Z_{j,\rm in}\leq2M_bk_bE_*\qquad(|l|\leq2c).
+\end{equation}
+Through saturation, its derivatives and those of $E$ through order
+four have bounds given by fixed prefactors times
+$(1+Y_B)^4 e^{4(E-E_0)/k_b}$.
+The bound with $E_*$ also holds for positive-order derivatives of
+the affine tail.
+
+The well satisfies \eqref{assembly:diagonal} after increasing $c$.
+Indeed throughout severity saturation,
+\begin{equation}\label{assembly:activation-gap}
+ l\leq-C(Y_B,E_0)N e^{-E_*/k_b}.
+\end{equation}
+The absolute value on the right grows exponentially in $E_*$,
+because $\log N\sim\gamma E_*$ and $\gamma>4/k_b$.
+Homogeneity makes the polynomial evaluations on the whole inner
+ball exponentially small in this quantity. Cauchy on a collar
+of scale $R/N$ gives the same conclusion for any fixed derivative
+order, with only powers of $N$ as further costs. The logarithm in
+\eqref{assembly:well} is then close to zero with all these
+derivatives. We can therefore enforce the bound $2^{-j}$ for its
+value and derivatives through order $j+4$. This decay pays all
+fixed exponential-in-$c$ costs needed at stage $j$.
+
+\begin{lemma}\label{assembly:quiet-check}
+The trigger satisfies the hypotheses of
+Proposition~\ref{quiet:profile}, uniformly for every $\tau\geq0$.
+After saturation the same criterion applies through $l=4c$, including
+the initial part of the post-test term defined below.
+\end{lemma}
+
+\begin{proof}
+For raw estimates use Euclidean axes. Here $t\geq1$, $Y\asymp Y_B$,
+and $\log Y_B\leq C u_s(\nu_B)$; the last estimate follows by
+integrating $d\log Y/du_s=\kappa(1+Y^{-1})\leq2.2$ for $Y\geq1$.
+Differentiation of $a'=aY/t$ bounds the required radial metric jets
+by $e^{C_{\rm rad}u_s}$. Their inverses cost no more. The radial
+curvature estimate gives
+\begin{equation}\label{assembly:euclidean-curvature}
+ C^V(T,T)\geq c\kappa H|T|_V^2
+ \geq c(a/t)|T|^2,
+\end{equation}
+since $V\geq aI$ and
+$\kappa H=C_\kappa Y/((1+Y)ta)\geq c/(ta)$.
+The initial severity pays these bounds, the old data and the future
+allowance. The trigger cost $e^{4E/k_b}$ fits the quiet raw exponent.
+The current well is small before saturation. Also $G\geq V$, since
+all nonconstant extra twist terms in this interval are
+plurisubharmonic.
+
+For the covariant condition, write
+\[
+ \mathcal B(\nu)=B(e^\nu),\qquad
+ \mathcal U(\nu)=\Psi(e^\nu)+Z_{j,\rm in}(e^\nu),
+\]
+and use primes for $\nu$ derivatives within this proof. On the
+severity-varying interval,
+\begin{align}
+ \frac{\mathcal U''}{\mathcal U'},\
+ \frac{|\mathcal U'''|}{\mathcal U''},\
+ \frac{\mathcal B''}{\mathcal B'},\
+ \frac{|\mathcal B'''|}{\mathcal B''}
+ &\leq C\sqrt{\mathcal U''},
+ \label{assembly:connection-ratios}\\
+ \frac{|E'|}{\mathcal U'},\
+ \frac{|E'|}{\sqrt{\mathcal U''}},\
+ \frac{|E''|}{\mathcal U''},\
+ \frac{|E'''|}{(\mathcal U'')^{3/2}}
+ &\leq\varepsilon_b,
+ \label{assembly:small-ratios}
+\end{align}
+where $\varepsilon_b$ can be made as small as required.
+Away from the pole, $h_b$ and its relevant derivatives are bounded,
+whereas
+$\mathcal B'=Y/H$ and $\mathcal B''=Y(1+Y)/H$.
+The choice $\sigma M_bk_b^2\ll1$ makes these background terms
+dominate. Near the pole let $D_b=Y_B/(1-\xi)$. Then
+\[
+ Z_{j,\rm in}'\asymp M_bk_b^2D_b,\quad
+ Z_{j,\rm in}''\asymp M_bk_b^2D_b^2,\quad
+ |Z_{j,\rm in}'''|\leq C M_bk_b^2D_b^3,
+\]
+and $|E^{(r)}|\leq Ck_b^rD_b^r$ for $1\leq r\leq3$, including
+saturation. These prove \eqref{assembly:connection-ratios} and
+make \eqref{assembly:small-ratios} small by first enlarging $M_b$
+and then decreasing $\sigma$. The fixed intervening interval of
+$\xi$ is covered by the first estimates. We also have
+$\mathcal U''\geq1$.
+
+At fixed $\tau$, the radial part of the Schur metric is the Hessian of
+\[
+ J_h=\mathcal B+\tau(\mathcal U-\bar F(x,E)),
+\]
+where the bar denotes averaging from zero to $\tau$ in the $\tau$
+variable. Bounded severity derivatives of $\bar F$ and
+\eqref{assembly:small-ratios} show that the first two derivatives
+of $\mathcal U-\bar F$ are positively comparable with those of
+$\mathcal U$ and its third derivative is bounded by
+$C(\mathcal U'')^{3/2}$.
+Use affine axes normalized by the Hessian of $\mathcal U$, with
+radial radius $\sqrt{t/\mathcal U''}$ and transverse radii
+$\sqrt{t/\mathcal U'}$.
+At a radial point, the only nonzero holomorphic connection types
+for a radial Hessian metric are radial--radial to radial and
+radial--transverse to transverse. Differentiation gives the bounds
+\begin{equation}\label{assembly:connection-exact}
+ \frac{|J_h'''/J_h''-1|}{\sqrt{\mathcal U''}},
+ \qquad
+ \frac{|J_h''/J_h'-1|}{\sqrt{\mathcal U''}}.
+\end{equation}
+They are bounded by \eqref{assembly:connection-ratios}. No estimate
+of a radial/transverse condition number is used.
+
+Write the remaining Hessian as $\tau G_e$, with $G_e\geq0$.
+Its size and first jets in these axes are bounded: old Euclidean
+bounds are dominated by the large initial $a$, and the current
+and future wells have the reserved absolute bounds. The normalizing
+radii are at most $a^{-1/2}$ because this background is at least
+$V\geq aI$. If $h_r$ denotes the radial Schur metric, the exact
+connection difference is, with the usual index placement,
+\begin{equation}\label{assembly:connection-perturbation}
+ \Gamma^h-\Gamma^{h_r}
+ =\tau(\partial G_e-\Gamma^{h_r}G_e)(h_r+\tau G_e)^{-1}.
+\end{equation}
+Since $h_r\geq c\tau I$, $\tau h^{-1}$ is uniformly bounded for
+all $\tau>0$. Thus the full connection is uniformly bounded as well.
+
+In these axes the first derivative of $E$ has norm
+$|E'|/\sqrt{\mathcal U''}$; the complex Hessian entries are bounded
+by $|E'|/\mathcal U'$ and $|E''|/\mathcal U''$, and the only pure
+second derivative has size $|E''-E'|/\mathcal U''$.
+Consequently subtracting the connection times the first derivative
+gives a small $\nabla_h^{2,0}E$. Since the actual $G$ is at least the
+normalizing background, the same smallness holds in $G$ units.
+This proves all quiet smallness conditions uniformly in $\tau$.
+
+After saturation, severity derivatives vanish. Through $l=4c$ the
+well's raw costs are bounded by $e^{C(c+\log N)}$; the post term
+below has cost at most a fixed multiple of $N^4e^{2m_zc}$.
+Both fit $\epsilon E_*$ by \eqref{assembly:second-constants}, while
+the inverse and radial curvature estimates persist.
+\end{proof}
+
+\subsection{Testing disks and the post ramp}
+
+Choose $c$ sufficiently large that the degree in
+\eqref{assembly:test-parameters} meets the grid accuracy required for
+$h_j$, and $N>h_j$. At every good lift $v$, use a smooth cutoff
+supported in $|s-Rv|<2RN^{-k_*}$ and equal to one on
+$|s-Rv|\leq RN^{-k_*}$. Assign on the inner ball
+\begin{equation}\label{assembly:shear-amplitude}
+ c_{\rm sh}=\alpha b_v,\qquad
+ \alpha=\exp(-C_sE_*-C_wd\log N),
+\end{equation}
+and multiply by the cutoff in the collar. The phases have modulus
+one, are independently assignable by projective direction, and
+repeat on every lift of that direction. The outer balls are disjoint.
+A cutoff $\chi(|s-Rv|^2/(RN^{-k_*})^2)$ has fixed-order multilinear
+derivative bound $C_r(RN^{-k_*})^{-r}$ with absolute $C_r$.
+At most one contributes at any point, so there is no factor for
+the number of nodes or dimension.
+
+On the derivative supports, the shell estimate gives
+\begin{equation}\label{assembly:shear-lower}
+ \psi\geq-2d(k_*+11)\log N-C_{\rm old}.
+\end{equation}
+Old values are bounded on the fixed cap continuation, future values
+have the allowance, and the trigger term is nonnegative.
+These balls have $|l|\leq\log N+1$ and lie beyond severity
+saturation. First and pure second derivatives of $\psi$ also fit
+the quiet raw bounds; the extra background first-derivative cost
+is at most $a\sqrt t$. The constants $C_s,C_w$ pay respectively
+the severity exponent of Proposition~\ref{quiet:shear} and half the
+loss \eqref{assembly:shear-lower} together with the four cutoff
+derivatives. Large $c$ pays fixed old factors. Thus that proposition
+applies for all fiber radii with metric comparison loss at most
+$1/2$. Where a shear coefficient is constant, its potential term
+is pluriharmonic.
+
+At a node the new well is $-2dc+O(1)$; on $|s|=S$ it is at most
+$2d(c+\log N)+O_d(1)$. All other values on the fixed continuation
+are bounded independently of $c$, except the trigger term.
+The post term is still zero. Hence
+\begin{align}
+ \psi_{\rm node}&\leq-2dc+2M_bk_bB_*c+C_{\rm old},
+ \label{assembly:node-twist}\\
+ \psi_{\rm sphere}&\leq2d(c+\log N)+2M_bk_bB_*c+C_{\rm old,d}.
+ \label{assembly:sphere-twist}
+\end{align}
+For a small fixed profile-dependent $c_0>0$, the disk
+\begin{equation}\label{assembly:fiber-disk}
+ |y-\zeta_j|<c_0e^{-(\psi+E_*)/2}
+\end{equation}
+has $\tau\leq e^{-E_*}$ and radial potential increment at most one.
+Indeed, by the axis estimates in Lemma~\ref{quiet:scalar}, at
+$\tau=e^{-E_*}$ the normalized relation gives $F=-E_*+O(1)$ uniformly: integrate $1/q-1=O(e^{E_*}\tau)$
+up to that value of $\tau$. Monotonicity proves the disk assertion.
+
+At a node the holomorphic shear
+$w\mapsto w+i\alpha b_v(y-\zeta_j)$ has imaginary increment
+$\operatorname{Re}(\alpha b_v(y-\zeta_j))$, exactly the added
+potential. Writing $z=y-\zeta_j$, the clearance on the disk
+\eqref{assembly:fiber-disk}, over any $w$ with
+$\operatorname{Im}w=D_j$, is therefore
+\[
+ \begin{split}
+ &\operatorname{Im}(w+i\alpha b_vz)
+       -\ell(Rv,\zeta_j+z)\\
+ &\qquad=D_j-\ell_{\rm rad}(Rv,\zeta_j+z)
+       \geq D_j-B(Rv)-1\geq1.
+ \end{split}
+\]
+The last inequality uses \eqref{assembly:height-bound} on the center
+axis. Thus the shear consumes none of the height margin, and the
+sheared disk lies in the tube. Its logarithmic radius is at least
+\[
+ (d-M_bk_bB_*-B_*/2)c-C_{\rm old}
+ \geq(13d/16)c-C_{\rm old},
+\]
+which exceeds $m_0c$ for large $c$.
+On the sphere there is no shear, and Cauchy's inequality on
+\eqref{assembly:fiber-disk} bounds the transverse derivative
+of a function of modulus at most one by
+\[
+ \exp\{[d(1+\gamma B_*)+M_bk_bB_*+B_*/2]c+C_{\rm old,d}\}
+ \leq e^{p_0c}.
+\]
+Also $\alpha^{-1}\leq e^{p_1c}$.
+These are precisely the constants \eqref{assembly:test-exponents},
+with strict margins to absorb the prefactors.
+Choose the phases supplied by Proposition~\ref{grid:criterion}.
+All geometric estimates hold for arbitrary phases.
+
+For the post ramp, add a smooth convex nondecreasing function
+$Z_{j,\rm out}$ of $l$, zero through $2c$, and equal to
+$e^{m_z(l-2c)}$ up to affine terms from $2c+2$ onward.
+This is obtained by cutting on its positive second derivative and
+integrating twice. Continue through $T_*+2$, then cut off that
+second derivative nonnegatively so that the function is affine
+after $T_*+3$. Independently turn on the severity slope $B_*$
+in $l$ over $[4c,4c+1]$, and turn it off over
+$[T_*,T_*+1]$. Write $E_{\rm post}$ for the final value.
+For large $c$,
+\begin{equation}\label{assembly:post-severity}
+ E\geq B_*(c+l)/10\quad(2c\leq l\leq T_*+4),\qquad
+ E_{\rm post}\geq B_*T_*/2.
+\end{equation}
+
+\begin{lemma}\label{assembly:post}
+The quiet criterion holds throughout the post ramp.
+Beyond $l=T_*+4$, the sum of all already generated additive twist
+terms other than $\Psi$ and the constant offsets, and each such
+term individually, has Euclidean positive-order jets through order
+four bounded on the whole outer range by
+\begin{equation}\label{assembly:outer-bound}
+ D_{\rm out}=\exp(\epsilon_1E_{\rm post}/3).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Each derivative of a function of $l$ costs a fixed power of $N$.
+Thus $Z_{j,\rm out}$ has positive-order bounds through order four
+of size $C N^4e^{m_z\max(0,l-2c)}$. These and
+\eqref{assembly:well-raw} fit the raw quiet exponent by
+\eqref{assembly:second-constants} and \eqref{assembly:post-severity}.
+Other radial and old-data bounds remain fixed on the cap
+continuation. Through $l=4c$, severity is constant and the preceding
+lemma applies.
+
+Where severity varies, take $\mathcal U=\Psi+Z_{j,\rm out}$.
+Before division by $t$, its transverse and radial Hessian entries
+have sizes
+\[
+ m_zN e^{m_z(l-2c)},\qquad
+ (m_zN)^2e^{m_z(l-2c)}.
+\]
+For $l\geq4c$, $m_z(l-2c)$ exceeds
+$C_{\rm pol}(l+c+\log N)$ by a fixed positive multiple of $c+l$.
+Thus this Hessian dominates the current well and its first
+Hessian jets in normalized axes.
+The affine $Z_{j,\rm in}$ tail has coefficient at most a fixed
+factor times $e^{E_*/k_b}\ll N$; old terms and future tails are
+covered as before. Moreover
+$\mathcal U''/\mathcal U',|\mathcal U'''|/\mathcal U''\leq Cm_zN$,
+while $\sqrt{\mathcal U''}\gtrsim
+m_zN e^{m_z(l-2c)/2}$.
+Severity derivatives of orders $r\leq3$ are at most $CB_*N^r$.
+Thus the connection ratios hold and all severity ratios are small.
+Equations~\eqref{assembly:connection-exact} and
+\eqref{assembly:connection-perturbation} again prove the pure
+Hessian condition uniformly for all $\tau$. Every extra Hessian
+is positive semidefinite in this interval.
+
+Past $T_*+4$ the new well satisfies \eqref{assembly:well-tail},
+and the $Z$ terms are affine in $\nu$. Their positive-order
+Euclidean jets are bounded on the entire outer range by their
+endpoint bounds up to fixed constants. Their logarithmic costs
+are at most
+$C(1+m_z)T_*+4\log N+C_{\rm old}$ with an absolute $C$.
+The absolute constant $C_*$ in \eqref{assembly:second-constants}
+was chosen to pay this inequality too.
+Equation~\eqref{assembly:post-severity} then implies
+\eqref{assembly:outer-bound}, after increasing $c$ for the finite
+sum of old bounds, the number of generated terms, and dimensional
+prefactors. The dimension-dependent $L_n$
+multiplies $T_*$ on both sides and changes no leading comparison.
+\end{proof}
+
+\subsection{Acceleration, center change, and reset}
+
+At constant $E_{\rm post}$, increase $\kappa$ from the cap value
+to $\kappa_*=1.08$ on a fixed-factor interval of $Y$ after
+$l=T_*+4$, keeping $\lambda=p=1$. The transition can preserve
+$Y\kappa_Y\geq-\kappa$: multiply the difference between the cap
+value and $\kappa_*$ by a nondecreasing cutoff in $\log Y$.
+The upward derivative only improves that inequality.
+The $u_s$ increment is bounded in terms of the fixed prior $Y_B$,
+independently of large $c$. Keep $\kappa=\kappa_*$ until
+\begin{equation}\label{assembly:aligned-entry}
+ u_s=\epsilon_1E_{\rm post}.
+\end{equation}
+For large $c$ this endpoint lies after the transition and has
+$H\geq\sigma$. Indeed, with $p=1$,
+$d\log H/du_s=(\kappa_*-1)(1+Y^{-1})\geq.08$,
+whereas the initial values at the end of the cap have fixed
+stagewise bounds as $c\to\infty$.
+
+Until \eqref{assembly:aligned-entry}, apply the quiet criterion.
+The radial raw jets cost $e^{C_{\rm rad}u_s}$, the extra twist
+jets are bounded by \eqref{assembly:outer-bound}, and the curvature
+lower bound follows as in \eqref{assembly:euclidean-curvature}.
+Severity derivatives vanish. The choices of $\epsilon_1$ and $B_*$
+pay all these estimates. At entry $a_\psi=e^{\epsilon_1E_{\rm post}}$.
+By the scaled-axis estimate of the radial lemmas, an extra
+Euclidean Hessian with two further jets bounded by $D$ costs at
+most $CD/(a_\psi H)$ in scaled axes. Consequently at entry and
+through the later aligned intervals,
+\begin{equation}\label{assembly:aligned-extra}
+ |j^{\leq2}G_e|_{\rm scaled}\leq\delta/H.
+\end{equation}
+Here we leave a fixed fraction of the $\delta$ budget for future
+wells. The ratio $D_{\rm out}/a_\psi$ is exponentially small at
+entry and only decreases later, since $a_\psi$ increases.
+During an actual profile switch its new offset is accounted for
+by Proposition~\ref{center:change}; outside the switch, that offset
+is constant.
+
+\begin{lemma}\label{assembly:change-order}
+At constant $E_{\rm post}$ the center can be changed from $\zeta_j$
+to $\zeta_{j+1}$, and the data can then be restored to
+\[
+ \lambda=p=1,\qquad \kappa=1,\qquad H=\sigma,
+\]
+using finite clock intervals and only the aligned and center-change
+regimes. The old-twist bounds needed for choosing the profile
+shift are available independently of the targeted $\lambda,H$.
+\end{lemma}
+
+\begin{proof}
+Initially grow $H$ with $\kappa=\kappa_*$ and $p=\lambda=1$.
+Use the $\lambda=1$, $H\geq\sigma$ alternative of
+Proposition~\ref{radial:aligned}. Ultimately we will lower $p$
+slightly below one for a bump, thereby increasing $\lambda$, and
+then restore $p=1$ and return smoothly to $\kappa=1$.
+
+We first justify the order in which the targets are chosen.
+During the growth and the bump, $p\leq1$ and $\kappa=\kappa_*$,
+so $d\log H/du_s\geq.08$. Thus
+\[
+ \int_{u_{\rm entry}}^\infty \frac{du_s}{H}
+ \leq \frac{1}{.08H_{\rm entry}}.
+\]
+This bounds the total increase of $\Psi$ before the final transition
+independently of the size of either target. The transition from
+$\kappa_*$ to one takes a fixed $\log Y$ interval and leaves
+$H$ nondecreasing, adding another fixed bound to $\Psi$.
+Moreover $\kappa\geq1$ gives
+$d\log Y/du_s\geq1$, hence
+\[
+ 0\leq\nu-\nu_{\rm entry}
+ =\int \frac{du_s}{Y}\leq Y_{\rm entry}^{-1}
+\]
+through the growth and holding intervals. The old nonradial
+twists therefore have bounded values on one fixed compact
+spatial interval, independently of the targets. Their scaled
+derivatives are bounded by the fixed Euclidean bounds, since
+the scaled coordinate radii are at most one. Derivatives of
+$\Psi$ in these axes are $O(1/H)\leq O(1/\sigma)$.
+The forward switch and center move each occupy a fixed amount
+of the variable $\Psi$; these amounts can be included in the
+same bounds. Include the reserved future value and jet allowance
+as well. These are exactly old-twist bounds independent of
+$\lambda,H$, as required by Proposition~\ref{center:change}.
+
+Choose its shift $L_0$ from these bounds and $E_{\rm post}$.
+This fixes its profile constants $C_f$ and a sufficient target
+$\lambda$, and that target determines a sufficient target $H$.
+This is the order of parameter selection in
+Proposition~\ref{center:change}. The radial path reaches the targets
+in the other order: grow $H$ first until it exceeds every required
+multiple of
+\[
+ C_{E_{\rm post}}(1+\log\lambda)^2
+ \quad\hbox{and}\quad C_f(1+\log\lambda)^2.
+\]
+Choose a smooth bump of $1-p\geq0$ with small amplitude and
+small derivatives in $\log Y$, and with integral in $u_s$
+equal to the desired $\log\lambda$. Such a bump exists by
+making its middle plateau arbitrarily long; the endpoint ramps
+can have fixed small amplitude and sufficiently long fixed
+length, while the plateau length supplies the remaining integral.
+The conversion $d\log Y/du_s=\kappa(1+Y^{-1})$ and its bounded
+derivatives show that bounds through order two in $\log Y$
+follow from the same bounds in $u_s$.
+Throughout this bump $H$ increases, and the already chosen lower
+bound for $H$ pays the entire range $0\leq\log\lambda(s)\leq
+\log\lambda$. Apply the large-$H$ alternative of
+Proposition~\ref{radial:aligned}.
+
+Restore $p=1$ and lower $\kappa$ smoothly to one. The downward
+transition can be made slowly enough in $\log Y$ that
+$Y\kappa_Y\geq-\kappa$, with absolute derivative bounds.
+Since $\kappa\geq1$ here, $H$ stays large. Now $\kappa=p=1$,
+so both $H$ and $\lambda$ are constant. Perform the forward
+profile switch, the center move, and the reverse switch of
+Proposition~\ref{center:change}. Interpolate the two center
+values by a flat-ended convex combination, keeping the center
+in the unit disk. The old bounds and targets have been chosen
+in the order just proved. The changes take a bounded interval
+of $\Psi$, hence a finite interval of $u_s$ because
+$d\Psi/du_s=1/H$ and $H$ is fixed, even though this interval
+can be very long. Restore $q_0$ and retain the new constant
+offset. The center is now $\zeta_{j+1}$. The offset may be arbitrarily
+large, but it is now constant and changes no derivative of the twist. Its
+value becomes part of the old data when the next testing height is
+chosen, before the next accuracy parameter $c$.
+
+Raise $\kappa$ to $\kappa_*$ again and use a bump with
+$1<p<\kappa_*$ to bring $\log\lambda$ down to zero.
+A small positive plateau and fixed ramps give any prescribed
+finite integral of $p-1$; stop precisely when that integral
+equals the existing $\log\lambda$. Thus $\lambda$ never falls
+below one. Also
+\[
+ \frac{d\log H}{du_s}
+ =\kappa_*-p+\frac{\kappa_*-1}{Y}>0,
+\]
+so the large-$H$ aligned condition remains valid. Restore $p=1$.
+
+Finally reduce $H$ to $\sigma$. Transition $\kappa$ to $1/2$,
+hold there for an adjustable interval, then return to one from
+below. For $p=1$ the exact relation is
+\begin{equation}\label{assembly:H-descent}
+ \frac{d\log H}{d\log Y}=1-\kappa^{-1}.
+\end{equation}
+Its value is $-1$ on the middle plateau. The endpoint transitions
+have fixed finite effects on $\log H$; the initial $H$ can be
+made as large as needed. Choose the plateau length so that the
+final value is exactly $\sigma$. Throughout this descent
+$H\geq\sigma$, $\lambda=1$, and $1/2\leq\kappa\leq\kappa_*$,
+so the $\lambda=1$ aligned alternative applies. Choose the
+downward transitions slowly enough to preserve the same
+derivative inequality as above.
+
+All these intervals have finite $u_s$ or finite $\log Y$ length.
+Equations~\eqref{assembly:clocks} show that $Y$ remains finite
+on each finite $u_s$ interval and that the radius advances by
+a finite positive amount. A putative pole of an indefinitely
+continued accelerated equation is therefore never reached
+during an episode. This proves the claim.
+\end{proof}
+
+\begin{lemma}\label{assembly:reset}
+From the endpoint in Lemma~\ref{assembly:change-order}, severity
+can be increased and a new cap started so that the next trigger
+has \eqref{assembly:trigger-initial}, while the metric continues
+to satisfy the quiet criterion. The cap can be extended to
+arbitrarily large spatial radii.
+\end{lemma}
+
+\begin{proof}
+The spatial scale $u_s$ may now be enormous, and $E_{\rm post}$ need
+not dominate it. Proposition~\ref{quiet:profile} permits any fixed
+holomorphic affine frame at the evaluation point. We first use
+$V$-unit affine axes, where the curvature scale is the fixed $\sigma$
+and the relevant jets are bounded independently of $u_s$. This will
+justify the initial severity increase before arranging the size
+inequality needed for the next trigger.
+
+First hold $\kappa=p=\lambda=1$ and $H=\sigma$.
+Let $E$ increase smoothly as a function of $u_s$, starting at
+$E_{\rm post}$, with a fixed flat-ended transition to slope
+$4C_\uparrow$. Its derivatives through order four in $u_s$
+are bounded by constants depending on the fixed parameters,
+not on $\sigma$ or the possibly enormous $u_s$ at entry.
+Continue long enough that
+\begin{equation}\label{assembly:catchup}
+ E\geq2C_\uparrow u_s
+\end{equation}
+and any desired fixed threshold. The greater slope ensures
+that this occurs after a finite interval.
+
+We verify the quiet criterion even before \eqref{assembly:catchup}.
+Use $V$-unit affine axes. The scaled-jet radial estimates give
+metric jets of orders $r=1,2$ bounded by $CH^{r/2}$ in these
+axes, and scaled derivatives of $u_s$ through order four are
+bounded by an absolute constant. Therefore
+\[
+ |\partial^{[r]}E|_{V}\leq C H^{r/2},\qquad 1\leq r\leq4.
+\]
+The extras in \eqref{assembly:aligned-extra} satisfy the same
+bounds after this normalization, and $G\geq V$.
+For $\sigma$ small, the severity terms in $\mathcal N$ are
+small relative to $G$, so
+$h\asymp(1+\tau)V$, with first derivatives bounded in these
+axes by $C(1+\tau)\sqrt H$.
+Its connection is consequently $O(\sqrt H)$ uniformly in
+$\tau$. The first, complex second, and pure covariant second
+severity derivatives are small for every $\tau$.
+
+The raw curvature margin on this hold is
+$C^V(T,T)\geq c\sigma|T|^2$ in $V$-unit axes, because
+$\kappa=1$ and $H=\sigma$. All other raw data are bounded there
+by fixed constants. The previously chosen $E_{\rm post}$
+exceeds the fixed profile and $\sigma$ thresholds, so it
+pays this margin even when it is much smaller than the current
+$u_s$. Thus there is no gap before severity catches up.
+
+After \eqref{assembly:catchup}, keep the same slope while
+turning $\kappa$ into a cap. Here is an explicit admissible
+transition. If $Y_0$ is its starting clock value, set
+\[
+ A(Y)=1+Y_0+\int_{Y_0}^Y \eta_0(v/Y_0)\,dv,\qquad
+ \kappa(Y)=\frac{A(Y)}{1+Y},
+\]
+where $\eta_0$ is smooth, equals one near $1$, decreases
+nonnegatively to zero, and is zero for arguments at least two.
+For $Y\leq Y_0$ use $A=1+Y$.
+Then $1\leq A\leq1+Y$, so
+$(1+Y)^{-1}\leq\kappa\leq1$. Moreover
+\[
+ Y\kappa_Y+\kappa
+ =\frac{YA'(Y)}{1+Y}+\frac{A(Y)}{(1+Y)^2}\geq0.
+\]
+The first two derivatives of $\kappa$ in $\log Y$ have absolute
+bounds, because $Y/Y_0$ ranges over a fixed interval while
+$\eta_0$ changes. For example, writing $b=Y/(1+Y)$ and
+$D=Y\partial_Y$, one has
+$D\kappa=b(\eta_0-\kappa)$ and
+$D^2\kappa=b(1-2b)(\eta_0-\kappa)+bD\eta_0$.
+At the end $A$ is the new constant $C_\kappa$.
+Since $\kappa\leq1$ and $p=1$, $H$ is nonincreasing and
+stays at most $\sigma$.
+
+The same normalized jet and connection estimates continue
+through the transition and the cap. To check the raw curvature
+margin there, note that $Y\geq t$: it holds initially, and
+$d\log Y/d\nu=\kappa(1+Y)\geq1$ preserves it.
+Also $u_s\geq t$ after choosing the initial constant large
+enough, since $du_s/dt=Y/t\geq1$.
+Hence
+\[
+ \kappa H\geq c/(ta)\geq c e^{-2u_s}
+ \quad(t\geq1),
+\]
+and \eqref{assembly:catchup} with large $C_\uparrow$ pays the
+quiet lower bound $e^{-\epsilon E}$. The positive-order
+extras stay bounded relative to $V$ on an arbitrarily long
+cap. More explicitly, a fixed Euclidean jet bound $D_{\rm out}$
+costs at most
+$C(D_{\rm out}/a)H^{r/2}$ in $V$-unit axes for Hessian jets
+of order $r\leq2$, since $aH=Y/t\geq1$.
+The ratio only decreases as $a$ grows.
+Future wells are included in the fixed allowance until
+their own activations.
+
+Coast as far in $\nu$ as desired. Shortly before the next
+chosen trigger, turn off the severity slope smoothly over
+a fixed-length $u_s$ interval. The margin in
+\eqref{assembly:catchup} is sufficient to retain
+$E_0\geq C_\uparrow u_s(\nu_B)$ after this bounded final
+increment, by making the cap endpoint large enough.
+The existing raw jet bounds can also be absorbed by making
+that endpoint large. This produces
+\eqref{assembly:trigger-initial} and the next cap data.
+\end{proof}
+
+\subsection{Global realization and completeness}
+
+We now carry out the recursion. At each stage first extend the
+cap so that its trigger starts beyond all previous stages and
+with $\nu_B\geq j$. Fix its old-data bounds and testing height
+by Lemma~\ref{assembly:height}. Then choose $c$ large enough
+for \eqref{assembly:height-rate}, the polynomial approximation
+for $h_j$, \eqref{assembly:diagonal}, and all the finite
+thresholds in the trigger, test, post ramp, and aligned entry.
+The later finite center-change targets are chosen in the
+order proved in Lemma~\ref{assembly:change-order}.
+Lemma~\ref{assembly:reset} completes the cycle.
+Thus no clock or already configured profile is determined
+by an unknown future well.
+
+For completeness, the future allowance is valid in every
+kind of local regime. In a quiet interval, later wells have
+positive-semidefinite Hessians and bounded absolute jets;
+they preserve $G\geq V$, and the preceding connection proof
+uses exactly these bounds. In an aligned or switch interval,
+their scaled Hessian jets are bounded by
+$C/(a_\psi H)$, which fits the reserved part of
+$\delta/H$ at the large chosen $a_\psi$. In a center move,
+their values change the radius estimates by at most a fixed
+factor, and their scaled derivatives are within the old-data
+allowance included before choosing the shift. On a shear
+collar they lower the twist by at most one, already included
+in the lower bound used for its coefficient. Consequently
+the local lemmas apply to the final infinite twist for
+every fiber radius, even though curvature margins need not
+have a positive infimum over all fiber radii.
+
+The condition $\nu_B\geq j$ rules out finite accumulation of
+stages. Any compact spatial set lies before the activation
+of all sufficiently late wells. For a fixed derivative order
+$r$, those later wells satisfy
+$|\partial^{[r]}W_j|\leq2^{-j}$ once $j+4\geq r$.
+The Weierstrass criterion in every derivative order proves
+compact-smooth convergence of $\sum_jW_j$.
+The $Z$ terms and nonconstant offsets are locally finite.
+Flat-ended transitions, finite positive clock evolutions,
+and the prescribed initial data therefore give global smooth
+spatial data. On a compact spatial set the profile is smooth
+to $\tau=0$ with $q(0,s)=1$; its normalized $F-x$ is smooth
+there as well. The smooth inversion at the fiber center
+therefore gives a global smooth potential
+\[
+ \ell(s,y)=\ell_{\rm rad}(s,y)+
+ \operatorname{Re}\bigl(c_{\rm sh}(s)(y-\zeta(s))\bigr).
+\]
+The shears are absent where the center changes.
+All local curvature and positivity conclusions apply,
+including the fiber axes by their smooth limits.
+
+We spell out the metric comparison used for completeness.
+In the constant-center regimes the Schur metric is bounded
+below by a fixed positive multiple of $V$; during profile
+switches, positivity of $\mathcal N$ for all $\tau$ also
+gives $h=V+\int_0^\tau\mathcal N\,dv\geq V$.
+The shear and moving-center comparisons can each be chosen
+to lose at most a factor $1/2$. Their spatial supports are
+separated, so these losses do not multiply over infinitely
+many stages at a point. Hence a single $c_1>0$ satisfies
+\begin{equation}\label{assembly:spatial-completeness}
+ g_\ell(\dot s,\dot y)\geq c_1 V(\dot s,\dot s)
+ \geq c_1a_{\rm init}|\dot s|^2
+\end{equation}
+everywhere.
+
+A finite-length curve therefore has finite Euclidean
+spatial variation and remains in a compact spatial set.
+On such a compact set, for large $|y|$, the metric is also
+bounded below by a fixed positive multiple of the frozen
+log-block metric at the point. The tilt coefficients are
+uniformly bounded in $\tau$ there. Indeed they are spatial
+derivatives of $\psi-F$: on quiet intervals they use bounded
+$F_E$; on switch intervals they use bounded $F_\beta$, which
+is constant in $x$ past the fixed switch cutoff; and on a
+center move the profile is spatially fixed apart from the
+already bounded twist. There are only finitely many profile
+episodes on the compact set, and the well sum is smooth.
+Thus some compact-set constant $C_K$ bounds the tilt one-form.
+
+Also $q\geq1$ and $F\leq x$, so at sufficiently large $|y|$
+the fiber value satisfies $\tau\geq1$ and $P=\tau q\geq1$,
+uniformly on the compact spatial set.
+The frozen fiber differential is $dy/(y-\zeta(s))$; freezing
+means that the center has its point value in this comparison.
+The block bound and the triangle inequality imply along
+the curve, when $|y|$ is large,
+\[
+ \frac{|dy|}{|y-\zeta(s)|}
+ \leq C_K\,ds_{g_\ell}+C_K|ds|.
+\]
+Both terms on the right have finite integral by
+\eqref{assembly:spatial-completeness}. Since $|\zeta(s)|<1$,
+this bounds the total variation of $\log|y|$ outside a
+fixed disk and prevents fiber escape. Thus no divergent
+curve has finite length. Equivalently, a Cauchy sequence
+lies in a compact Euclidean coordinate set, where the
+smooth positive metric gives convergence. The base metric
+is complete.
+
+\begin{proof}[Proof of Proposition~\ref{assembly:base}]
+The recursion gives a smooth strictly plurisubharmonic
+potential on $\mathbb C^{n+1}$, with nonpositive real
+sectional curvature by the local lemmas and completeness
+by the preceding argument. Every required pair
+$(\zeta,h)$ is tested infinitely often. The stages satisfy
+$R\geq1$, $N\to\infty$, $c\to\infty$,
+$\log N\leq.005c<c/100$, $D_j\to\infty$, and
+$D_j/c_j\to0$. Lemma~\ref{assembly:height} supplies the
+height bound. The node disks, sphere derivative bound,
+and shear amplitude have the fixed exponents
+\eqref{assembly:test-exponents}. Their phases were chosen
+as permitted by Proposition~\ref{grid:criterion}, and
+their geometric realization allows all those choices.
+That proposition proves the asserted Liouville property.
+\end{proof}
+
+\begin{remark}\label{assembly:potential-unbounded}
+The potential is necessarily unbounded below, and this can also be
+seen directly from the disks. At a testing node choose
+$y-\zeta_j=-\tfrac12 e^{m_0c}\overline{b_v}$.
+The radial increment is at most one, whereas the shear contributes
+$-\tfrac12\alpha e^{m_0c}$. Thus
+\[
+ \ell(s,y)\leq D_j+1-\tfrac12e^{(m_0-p_1)c}\longrightarrow-\infty.
+\]
+Here $m_0>p_1$ and $c/(D_j+1)\to\infty$.
+In particular the elementary holomorphic functions $e^{ikw}$,
+$k>0$, are unbounded on the tube; they do not contradict its
+bounded-function conclusion.
+\end{remark}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
+Apply Lemma~\ref{tube:transfer} to the base just
+constructed. Its tube is connected, complete, and
+diffeomorphic to Euclidean space, hence simply connected.
+Its real sectional curvatures are at most $-1$ by that lemma,
+and its bounded holomorphic functions are constant by
+Proposition~\ref{assembly:base}. The complex dimension is the fixed finite
+$m=n+2$.
+
+There is no finite lower sectional-curvature bound.
+At $s=0,y=\zeta_1$, the center and profile are constant
+nearby and there is no shear. On the center axis the
+spatial curvature tensor of the base equals that of $V$:
+the added radial-fiber potential vanishes to second
+order in $y-\zeta_1$, and its cubic derivatives with two
+spatial indices and one fiber index vanish there.
+The initial radial metric has a strictly negative
+holomorphic-plane curvature in a spatial direction.
+Corollary~\ref{tube:negative-plane} therefore shows that the tube's
+sectional curvatures are unbounded below.
+This proves every assertion of Theorem~\ref{thm:main}.
+\end{proof}
+
+
+\paragraph{The two-sided question.}
+Neither this example nor the independent pinched threefold of
+\citet[Theorem~1.1]{OpenAIPinchedThreefold2026} decides whether a
+nonconstant bounded holomorphic function must exist under finite
+two-sided negative real sectional pinching. The present curvature is
+unbounded below, while the threefold retains nonconstant bounded
+base-coordinate functions.

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/stage-clock.tex

@@ -0,0 +1,43 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  stage/.style={draw=black!55,rounded corners=2pt,fill=black!3,
+    text width=3.65cm,minimum height=1.27cm,align=center,font=\small},
+  flow/.style={-{Latex[length=2mm]},thick,draw=black!65},
+  note/.style={font=\footnotesize,align=center}]
+\node[stage] (cap) at (0,0) {Cap and coast\\
+  $\lambda=1$, $H\le\sigma$};
+\node[stage] (trigger) at (4.6,0) {Severity trigger\\
+  raise $E$ to $E_*$};
+\node[stage] (test) at (9.2,0) {Polynomial test\\
+  fixed center and severity};
+\node[stage] (post) at (9.2,-2.05) {Post-test ramp\\
+  reach $E_{\rm post}$};
+\node[stage] (accelerate) at (4.6,-2.05) {Accelerate growth\\
+  enter aligned regime};
+\node[stage] (grow) at (0,-2.05) {Grow $H$, then $\lambda$\\
+  $E=E_{\rm post}$};
+\node[stage] (center) at (0,-4.1) {Widen, move, restore\\
+  hold $\kappa=p=1$};
+\node[stage] (reset) at (4.6,-4.1) {Return $\lambda$ to $1$\\
+  then lower $H$ to $\sigma$};
+\node[stage] (catchup) at (9.2,-4.1) {Severity catch-up\\
+  before the next cap};
+\draw[flow] (cap)--(trigger);
+\draw[flow] (trigger)--(test);
+\draw[flow] (test)--(post);
+\draw[flow] (post)--(accelerate);
+\draw[flow] (accelerate)--(grow);
+\draw[flow] (grow)--(center);
+\draw[flow] (center)--(reset);
+\draw[flow] (reset)--(catchup);
+\draw[flow] (catchup.south)--++(0,-.48)
+  node[below,note]{Next stage: cap and coast};
+\end{tikzpicture}
+\caption{One stage, in increasing spatial radius. The test is installed
+while the spatial growth is capped. The intervening growth and center
+change occupy finite clock intervals. After the reset, severity must
+catch up with the accumulated spatial scale before the next capped
+regime can begin. The arrows indicate order, not elapsed radius.}
+\label{fig:stage-clock}
+\end{figure}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/figures/tube.tex

@@ -0,0 +1,18 @@
+\begin{tikzpicture}[x=1.15cm,y=.85cm]
+ \fill[black!5]
+  (-3,1.8) .. controls (-2.2,.45) and (-1.05,.55) .. (0,1.05)
+  .. controls (1.05,1.55) and (2.2,.65) .. (3,1.45)
+  -- (3,4.15) -- (-3,4.15) -- cycle;
+ \draw[->] (-3.45,0) -- (3.5,0) node[right] {$z$};
+ \draw[->] (-3.35,-.1) -- (-3.35,4.4) node[above] {$v$};
+ \draw[thick]
+  (-3,1.8) .. controls (-2.2,.45) and (-1.05,.55) .. (0,1.05)
+  .. controls (1.05,1.55) and (2.2,.65) .. (3,1.45);
+ \node[anchor=north] at (1.8,.82) {$v=\ell(z)$};
+ \draw[dashed] (0,0) -- (0,1.05);
+ \draw[<->] (0,1.1) -- (0,3.05)
+  node[midway,right] {$b=v-\ell(z)$};
+ \fill (0,3.12) circle (1.6pt);
+ \node[above right] at (0,3.12) {$(z,w)$};
+ \node at (-1.75,3.55) {$\mathcal T_\ell:\ b>0$};
+\end{tikzpicture}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/main.tex

@@ -0,0 +1,59 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs,array}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit,decorations.pathreplacing}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{xcolor}
+\definecolor{linkblue}{RGB}{24,68,102}
+\PassOptionsToPackage{hyphens}{url}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
+\hypersetup{pdftitle={One-sided negative sectional curvature and the holomorphic Liouville property},pdfauthor={OpenAI}}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\setlist{topsep=5pt,itemsep=3pt,parsep=0pt}
+\allowdisplaybreaks[2]
+\emergencystretch=1.5em
+\title{One-sided negative sectional curvature\\and the holomorphic Liouville property}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct, in some sufficiently large fixed finite complex dimension
+$m$, a domain in $\mathbb C^m$ diffeomorphic to $\mathbb R^{2m}$ that
+admits a complete K\"ahler metric with real sectional curvature at most
+$-1$ and has only constant bounded holomorphic functions. Its sectional
+curvatures are unbounded below. The construction gives a negative answer
+to the one-sided bounded-holomorphic-function question.
+\end{abstract}
+\clearpage
+\begingroup
+\small
+\tableofcontents
+\endgroup
+\clearpage
+\input{01-introduction}
+\input{02-tube}
+\input{03-analytic}
+\input{04-fiber}
+\input{05-quiet}
+\input{06-radial}
+\input{07-centers}
+\input{08-assembly}
+\bibliographystyle{plainnat}
+\bibliography{refs}
+\end{document}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/build/refs.bib

@@ -0,0 +1,185 @@
+@article{WuYau2020,
+  author = {Wu, Damin and Yau, Shing-Tung},
+  title = {Invariant metrics on negatively pinched complete {K\"ahler} manifolds},
+  journal = {Journal of the American Mathematical Society},
+  volume = {33},
+  number = {1},
+  year = {2020},
+  pages = {103--133},
+  doi = {10.1090/jams/933},
+  url = {https://doi.org/10.1090/jams/933}
+}
+
+@misc{WuYauSurvey2019,
+  author = {Wu, Damin and Yau, Shing-Tung},
+  title = {Some negatively curved complex geometry},
+  year = {2019},
+  howpublished = {Author manuscript},
+  url = {https://dwu.math.uconn.edu/wp-content/uploads/sites/1743/2019/08/Wu-Yau_Some-negatively-curved-complex-geometry_ICCM_Aug-15-2019.pdf}
+}
+
+@article{WuNormal1967,
+  author = {Wu, H.},
+  title = {Normal families of holomorphic mappings},
+  journal = {Acta Mathematica},
+  volume = {119},
+  year = {1967},
+  pages = {193--233},
+  url = {https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6046-11511_2006_Article_BF02392083.pdf}
+}
+
+@article{BleherShiffmanZelditch2000,
+  author        = {Bleher, Pavel and Shiffman, Bernard and Zelditch, Steve},
+  title         = {Universality and scaling of correlations between zeros on complex manifolds},
+  journal       = {Inventiones Mathematicae},
+  volume        = {142},
+  number        = {2},
+  pages         = {351--395},
+  year          = {2000},
+  doi           = {10.1007/s002220000092},
+  url           = {https://math.jhu.edu/~shiffman/publications/44%20universality%202000.pdf},
+  eprint        = {math-ph/9904020},
+  archivePrefix = {arXiv}
+}
+
+@article{ShiffmanZelditch1999,
+  author        = {Shiffman, Bernard and Zelditch, Steve},
+  title         = {Distribution of zeros of random and quantum chaotic sections of positive line bundles},
+  journal       = {Communications in Mathematical Physics},
+  volume        = {200},
+  pages         = {661--683},
+  year          = {1999},
+  doi           = {10.1007/s002200050544},
+  url           = {https://link.springer.com/article/10.1007/s002200050544},
+  eprint        = {math/9803052},
+  archivePrefix = {arXiv}
+}
+
+@article{Shiffman2008Convergence,
+  author        = {Shiffman, Bernard},
+  title         = {Convergence of random zeros on complex manifolds},
+  journal       = {Science in China Series A: Mathematics},
+  volume        = {51},
+  number        = {4},
+  pages         = {707--720},
+  year          = {2008},
+  doi           = {10.1007/s11425-008-0060-9},
+  url           = {https://math.jhu.edu/~shiffman/publications/63%20convergence%20of%20random%20zeros%202008.pdf},
+  eprint        = {0708.2754},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.CV}
+}
+
+@book{EisenbudHarris2000GeometrySchemes,
+  author    = {Eisenbud, David and Harris, Joe},
+  title     = {The Geometry of Schemes},
+  series    = {Graduate Texts in Mathematics},
+  volume    = {197},
+  publisher = {Springer-Verlag},
+  address   = {New York},
+  year      = {2000},
+  doi       = {10.1007/b97680},
+  isbn      = {978-0-387-98638-8}
+}
+
+@article{IitakaEbisuzaki2004,
+  author        = {Iitaka, Toshiaki and Ebisuzaki, Toshikazu},
+  title         = {Random phase vector for calculating the trace of a large matrix},
+  journal       = {Physical Review E},
+  volume        = {69},
+  pages         = {057701},
+  year          = {2004},
+  doi           = {10.1103/PhysRevE.69.057701},
+  url           = {https://arxiv.org/abs/cond-mat/0401202v2},
+  eprint        = {cond-mat/0401202},
+  archivePrefix = {arXiv}
+}
+
+@article{Calabi1979,
+  author  = {Calabi, Eugenio},
+  title   = {M\'etriques k\"ahl\'eriennes et fibr\'es holomorphes},
+  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
+  series  = {4},
+  volume  = {12},
+  number  = {2},
+  pages   = {269--294},
+  year    = {1979},
+  doi     = {10.24033/asens.1367},
+  url     = {https://www.numdam.org/item/ASENS_1979_4_12_2_269_0/}
+}
+
+@article{HwangSinger2002,
+  author  = {Hwang, Andrew D. and Singer, Michael A.},
+  title   = {A momentum construction for circle-invariant {K\"ahler} metrics},
+  journal = {Transactions of the American Mathematical Society},
+  volume  = {354},
+  number  = {6},
+  pages   = {2285--2325},
+  year    = {2002},
+  doi     = {10.1090/S0002-9947-02-02965-3},
+  eprint  = {math/9811024},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DG},
+  url     = {https://arxiv.org/abs/math/9811024v1}
+}
+
+@article{ShcherbinaZhang2021,
+  author  = {Shcherbina, Nikolay and Zhang, Liyou},
+  title   = {A {Kobayashi} and {Bergman} complete domain without bounded representations},
+  journal = {Annali di Matematica Pura ed Applicata},
+  series  = {4},
+  volume  = {200},
+  pages   = {149--163},
+  year    = {2021},
+  doi     = {10.1007/s10231-020-00989-0},
+  eprint  = {2003.06728},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CV},
+  url     = {https://doi.org/10.1007/s10231-020-00989-0}
+}
+
+@misc{CaoShawWithdrawn,
+  author        = {Cao, Jianguo and Shaw, Mei-Chi},
+  title         = {Bounded holomorphic functions on negatively curved {K{\"a}hler} manifolds of dimension {$\geq 3$}},
+  year          = {2015},
+  archivePrefix = {arXiv},
+  eprint        = {1512.00368},
+  primaryClass  = {math.CV},
+  doi           = {10.48550/arXiv.1512.00368},
+  url           = {https://arxiv.org/abs/1512.00368},
+  note          = {Withdrawn in version 2, 6 February 2016}
+}
+
+
+@article{Seshadri2006,
+  author        = {Seshadri, Harish},
+  title         = {Negative sectional curvature and the product complex structure},
+  journal       = {Mathematical Research Letters},
+  volume        = {13},
+  number        = {3},
+  pages         = {495--500},
+  year          = {2006},
+  doi           = {10.4310/MRL.2006.v13.n3.a13},
+  eprint        = {math/0602289},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.DG},
+  url           = {https://arxiv.org/abs/math/0602289}
+}
+
+@misc{OpenAIPinchedThreefold2026,
+  author = {{OpenAI}},
+  title = {{A negatively pinched K{\"a}hler threefold without bounded holomorphic coordinates}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf}{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026}},
+  year = {2026}
+}
+
+@misc{Tropp2022,
+  author = {Tropp, Joel A.},
+  title = {{ACM} 204: Matrix Analysis},
+  howpublished = {Caltech CMS Lecture Notes 2022-01},
+  year = {2022},
+  doi = {10.7907/nwsv-df59},
+  url = {https://tropp.caltech.edu/notes/Tro22-Matrix-Analysis-LN.pdf},
+  note = {\href{https://tropp.caltech.edu/notes/Tro22-Matrix-Analysis-LN.pdf}{Lecture notes}, Theorem~7.13}
+}

preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:dedbbd2f24f8281c384e8927c78a54fd71d894113da70e5f702b8deb1f5bc046
+size 701177

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/README.md

@@ -0,0 +1,18 @@
+# [Open Homomorphisms of Global Solvably Closed Galois Groups](open-homomorphisms-solvably-closed-galois-groups.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026,
+  author = {{OpenAI}},
+  title = {{Open Homomorphisms of Global Solvably Closed Galois Groups}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf}{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026}},
+  year = {2026}
+}
+```
+

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/main.tex

@@ -0,0 +1,50 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{tikz-cd}
+\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
+\hypersetup{pdftitle={Open Homomorphisms of Global Solvably Closed Galois Groups},pdfauthor={OpenAI}}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\C}{\mathbb C}
+\newcommand{\F}{\mathbb F}
+\newcommand{\X}[2]{X_{#1}(#2)}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Ind}{Ind}
+\DeclareMathOperator{\Fr}{Fr}
+\DeclareMathOperator{\cor}{cor}
+\DeclareMathOperator{\res}{res}
+\DeclareMathOperator{\Ad}{Ad}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Cl}{Cl}
+\DeclareMathOperator{\Tr}{Tr}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\title{Open Homomorphisms of Global Solvably Closed Galois Groups}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/preliminaries}
+\input{sections/characters}
+\input{sections/generation}
+\input{sections/matching}
+\input{sections/identification}
+\input{references}
+\end{document}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/references.tex

@@ -0,0 +1,101 @@
+\begin{thebibliography}{99}
+
+\bibitem{bass1966}
+Hyman Bass,
+\emph{Generators and relations for cyclotomic units},
+Nagoya Math. J. \textbf{27} (1966), no.~2, 401--407.
+\href{https://doi.org/10.1017/S0027763000026258}{doi:10.1017/S0027763000026258}.
+
+\bibitem{dasgupta2023}
+Samit Dasgupta,
+\emph{Ranks of matrices of logarithms of algebraic numbers. I: The theorems of Baker and Waldschmidt--Masser},
+Essential Number Theory \textbf{2} (2023), no.~1, 93--138.
+\href{https://doi.org/10.2140/ent.2023.2.93}{doi:10.2140/ent.2023.2.93}.
+
+\bibitem{hoshi2025}
+Yuichiro Hoshi,
+\emph{Homomorphisms of global solvably closed Galois groups compatible with cyclotomic characters},
+Tohoku Math. J. (2) \textbf{77} (2025), no.~1, 17--32.
+\href{https://doi.org/10.2748/tmj.20230414}{doi:10.2748/tmj.20230414}.
+
+\bibitem{hoshi2026}
+Yuichiro Hoshi,
+\emph{A note on open homomorphisms between global solvably closed Galois groups},
+Kodai Math. J. \textbf{49} (2026), no.~1, 63--79.
+\href{https://doi.org/10.2996/kmj49104}{doi:10.2996/kmj49104}.
+
+\bibitem{laurent1991}
+Michel Laurent,
+\emph{Sur quelques r\'esultats r\'ecents de transcendance},
+Ast\'erisque \textbf{198--199--200} (1991), 209--230.
+\href{https://doi.org/10.24033/ast.93}{doi:10.24033/ast.93}.
+
+\bibitem{maosaidi2025}
+Yu Mao and Mohamed Sa\"idi,
+\emph{Open homomorphisms between $m$-step solvable Galois groups compatible with the cyclotomic characters},
+preprint, 2025.
+\href{https://arxiv.org/abs/2512.13312}{arXiv:2512.13312}.
+
+\bibitem{masser1981}
+D.~W. Masser,
+\emph{On polynomials and exponential polynomials in several complex variables},
+Invent. Math. \textbf{63} (1981), no.~1, 81--95.
+\href{https://doi.org/10.1007/BF01389194}{doi:10.1007/BF01389194}.
+
+\bibitem{neukirch1969a}
+J\"urgen Neukirch,
+\emph{Kennzeichnung der $p$-adischen und der endlichen algebraischen Zahlk\"orper},
+Invent. Math. \textbf{6} (1969), 296--314.
+\href{https://doi.org/10.1007/BF01425420}{doi:10.1007/BF01425420}.
+
+\bibitem{neukirch1969b}
+J\"urgen Neukirch,
+\emph{Kennzeichnung der endlich-algebraischen Zahlk\"orper durch die Galoisgruppe der maximal aufl\"osbaren Erweiterungen},
+J. reine angew. Math. \textbf{238} (1969), 135--147.
+\href{https://doi.org/10.1515/crll.1969.238.135}{doi:10.1515/crll.1969.238.135}.
+
+\bibitem{neukirch1977}
+J\"urgen Neukirch,
+\emph{\"Uber die absoluten Galoisgruppen algebraischer Zahlk\"orper},
+Ast\'erisque \textbf{41--42} (1977), 67--79.
+\url{https://www.numdam.org/item/AST_1977__41-42__67_0/}.
+
+\bibitem{neukirch1999}
+J\"urgen Neukirch,
+\emph{Algebraic Number Theory},
+translated by Norbert Schappacher,
+Grundlehren der mathematischen Wissenschaften, vol.~322,
+Springer-Verlag, Berlin, 1999.
+\href{https://doi.org/10.1007/978-3-662-03983-0}{doi:10.1007/978-3-662-03983-0}.
+
+\bibitem{saiditamagawa2022}
+Mohamed Sa\"idi and Akio Tamagawa,
+\emph{The $m$-step solvable anabelian geometry of number fields},
+J. reine angew. Math. \textbf{789} (2022), 153--186.
+\href{https://doi.org/10.1515/crelle-2022-0025}{doi:10.1515/crelle-2022-0025}.
+
+\bibitem{uchida1976}
+K\^oji Uchida,
+\emph{Isomorphisms of Galois groups},
+J. Math. Soc. Japan \textbf{28} (1976), no.~4, 617--620.
+\href{https://doi.org/10.2969/jmsj/02840617}{doi:10.2969/jmsj/02840617}.
+
+\bibitem{uchida1979}
+K\^oji Uchida,
+\emph{Isomorphisms of Galois groups of solvably closed Galois extensions},
+Tohoku Math. J. (2) \textbf{31} (1979), no.~3, 359--362.
+\href{https://doi.org/10.2748/tmj/1178229803}{doi:10.2748/tmj/1178229803}.
+
+\bibitem{uchida1981}
+K\^oji Uchida,
+\emph{Homomorphisms of Galois groups of solvably closed Galois extensions},
+J. Math. Soc. Japan \textbf{33} (1981), no.~4, 595--604.
+\href{https://doi.org/10.2969/jmsj/03340595}{doi:10.2969/jmsj/03340595}.
+
+\bibitem{waldschmidt1981}
+Michel Waldschmidt,
+\emph{Transcendance et exponentielles en plusieurs variables},
+Invent. Math. \textbf{63} (1981), no.~1, 97--127.
+\href{https://doi.org/10.1007/BF01389195}{doi:10.1007/BF01389195}.
+
+\end{thebibliography}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/characters.tex

@@ -0,0 +1,420 @@
+\section{Additive characters and Frobenius detection}\label{sec:characters}
+
+We will compare homomorphisms of Galois groups by evaluating additive
+$\ell$-adic characters at Frobenius elements.  The main fact needed for
+this comparison is that infinitely many primes of absolute residue
+degree one detect a character.  We first prove the logarithm-rank
+statement behind this fact, then establish the versions involving
+averaging and corestriction.  The section ends with a representation
+contained in the character space that will supply enough characters for
+the comparison.
+
+\subsection{A logarithm-rank lemma}
+
+For a finite extension $K/\Q_\ell$, normalize its absolute value by
+$|\ell|_\ell=\ell^{-1}$.  The $\ell$-adic logarithm on
+$\mathcal O_K^\times$ is the continuous homomorphism that agrees with
+the usual power series near $1$; its kernel is the finite group of
+roots of unity in $K$.  Logarithms of points in a torus are taken
+coordinatewise.
+
+For a number field $M$, class field theory expresses an additive
+character of $G_M$ as a linear form on $M\otimes_\Q\Q_\ell$.
+A zero Frobenius value at a prime $v\nmid\ell$ forces this form to
+annihilate $\log b$, where $(b)=v^h$ for some $h>0$.
+For primes of residue degree one chosen over distinct rational primes
+unramified in a Galois closure, the conjugate generators have disjoint
+ideal supports; the following lemma turns their multiplicative
+independence into full logarithmic rank.
+
+This special rank estimate admits an
+interpolation-determinant proof.  It is a special case of the
+$\ell$-adic Waldschmidt--Masser rank theorem
+\cite{waldschmidt1981,masser1981}.  We retain a proof using the
+translate estimates of that theory and Laurent's interpolation
+determinants \cite{laurent1991}; see also
+\cite[Theorems~5.1 and~5.8]{dasgupta2023}.
+
+\begin{lemma}[Logarithm rank]\label{lem:log-rank}
+Let $n,m\geq1$, let $E$ be a number field embedded in a finite
+extension $K/\Q_\ell$, and let
+\[
+ u_i=(u_{i1},\ldots,u_{in})\in(E^\times)^n,
+ \qquad 1\leq i\leq m,
+\]
+have algebraic-integer coordinates that are units in $K$.  Suppose that
+for every $a=(a_1,\ldots,a_m)\in\Z^m\setminus\{0\}$, the coordinates
+of $u^a:=\prod_{i=1}^m u_i^{a_i}$ are multiplicatively independent
+modulo roots of unity.  If $m>n(n-1)$, then
+\[
+ \operatorname{span}_K\{\log u_1,\ldots,\log u_m\}=K^n.
+\]
+\end{lemma}
+
+\begin{proof}
+The multiplicative hypothesis implies that the cyclic subgroup
+generated by every $u^a$ with $a\ne0$ is Zariski dense in
+$\mathbb G_m^n$.  Indeed, distinct Laurent monomials have distinct
+values at $u^a$.  If a Laurent polynomial vanished at all nonnegative
+powers of $u^a$, a Vandermonde matrix applied to its finitely many
+monomials would force all its coefficients to vanish.
+
+Suppose that the logarithms span a subspace of dimension $r<n$.
+The case $r=0$ contradicts the multiplicative hypothesis because the
+kernel of the logarithm on local units consists of roots of unity.
+Thus $1\leq r<n$, and in particular $n\geq2$.  We will obtain an
+algebraic determinant whose small $\ell$-adic absolute value is
+incompatible with its archimedean size.
+
+First make all choices needed for the analytic estimate.  Replacing
+all $u_i$ by the same positive integer power preserves the hypotheses
+and the logarithmic rank.  Such a power puts their coordinates in a
+neighborhood where exponential and logarithm are inverse.  The
+$\mathcal O_K$-module generated by their logarithms is free of rank
+$r$; choose a basis $b_1,\ldots,b_r\in K^n$.  After a further common
+$\ell$-power, these basis vectors have sufficiently small coordinates
+that
+\[
+ \rho:=\max_{j,t}|b_{t,j}|_\ell
+ \quad\hbox{satisfies}\quad
+ \eta:=\rho\ell^{1/(\ell-1)}<1.
+\]
+Here the basis is multiplied by the same power as the logarithms.
+Consequently there are $z_i\in\mathcal O_K^r$ such that
+\[
+ \log u_i=\sum_{t=1}^r z_{i,t}b_t.
+\]
+From now on these powered points, the number field $E$, the local
+field $K$, the basis, and $\eta$ are fixed.  In particular, none
+depends on the degree parameter introduced below.
+
+For an integer vector $\alpha=(\alpha_1,\ldots,\alpha_n)$, the
+monomial $X^\alpha$ on these points is represented by
+\[
+ f_\alpha(z)=
+ \exp\!\left(\sum_{j=1}^n\alpha_j
+                      \sum_{t=1}^r b_{t,j}z_t\right).
+\]
+It is analytic on a polydisc of radius greater than $1$.  If
+$f_\alpha(z)=\sum_{\nu\in\Z_{\geq0}^r}c_{\alpha,\nu}z^\nu$,
+the exponential series and
+$|k!|_\ell^{-1}\leq\ell^{k/(\ell-1)}$ give the uniform bound
+\begin{equation}\label{eq:log-taylor-bound}
+ |c_{\alpha,\nu}|_\ell\leq\eta^{|\nu|},
+ \qquad |\nu|=\nu_1+\cdots+\nu_r.
+\end{equation}
+The bound is independent of $\alpha$, since every integer has
+$\ell$-adic absolute value at most $1$.  Moreover, for every
+$a\in\Z_{\geq0}^m$, the point
+$z(a):=\sum_i a_i z_i$ lies in $\mathcal O_K^r$, and
+$f_\alpha(z(a))=(u^a)^\alpha$.
+
+We next show that sufficiently many of these evaluations have full
+algebraic rank.  For an integer $D\geq1$, put $A=(D+1)^n$ and choose
+an integer $S\geq1$ such that
+\begin{equation}\label{eq:log-grid-size}
+ (S+1)^m>(nD)^n,
+ \qquad S=O(D^{n/m}).
+\end{equation}
+Consider the matrix with rows indexed by
+$\alpha\in\{0,\ldots,D\}^n$, columns indexed by
+$a\in\{0,\ldots,nS\}^m$, and entries $(u^a)^\alpha$.
+We claim that its rank is $A$.
+
+Otherwise a nonzero polynomial $P$ of degree at most $D$ in each
+variable would vanish at every point $u^a$ in this grid.  Over an
+algebraic closure of $E$, define closed subsets of the torus by
+\[
+ Z_j=\{x\in\mathbb G_m^n:
+          P(u^a x)=0\ \hbox{for all }a\in\{0,\ldots,jS\}^m\},
+ \qquad 0\leq j\leq n.
+\]
+These sets are nested and nonempty, since $1\in Z_n$, whereas
+$\dim Z_0\leq n-1$.  Hence some adjacent pair $Z_j,Z_{j+1}$ has
+the same dimension $d$.  Let $V$ be a $d$-dimensional irreducible
+component of $Z_{j+1}$.  For every
+$b\in\{0,\ldots,S\}^m$, the translate $u^bV$ is contained in
+$Z_j$ and has dimension $d$, so it is an irreducible component of
+$Z_j$.  All these translates are distinct: an equality
+$u^bV=u^{b'}V$ for $b\ne b'$ would make $V$ stable under every
+power of $u^{b-b'}$.  The density proved above would then force
+$V=\mathbb G_m^n$, contrary to $V\subseteq Z_0$.
+
+For completeness, a common zero set of polynomials of total degree
+at most $\delta\geq1$ in $\mathbb A^n$ has at most $\delta^n$
+irreducible components.  One can obtain this form of the affine
+B\'ezout bound by assigning a component $Y$ the weight
+$\delta^{\dim Y}\deg Y$.  Start with $\mathbb A^n$, of weight
+$\delta^n$, and impose the equations successively.  A component
+contained in the next hypersurface retains its weight; a component
+not contained in it is cut into components of dimension one less,
+whose total degree is at most $\delta\deg Y$ by B\'ezout.  Thus
+the total weight does not increase, and removing redundant components
+can only decrease it.  The same component bound holds after
+restriction to the torus.  Applying it to $Z_j$, whose equations
+have total degree at most $nD$, contradicts
+\eqref{eq:log-grid-size}.  This proves the rank claim.
+
+Choose a nonzero $A\times A$ minor $\Delta_D$ of the evaluation
+matrix.  It is an algebraic integer in the fixed field $E$.
+Expand its rows using the convergent power series for $f_\alpha$.
+A nonzero term must involve $A$ distinct monomials in the $r$
+variables $z_1,\ldots,z_r$, since repeated monomials give repeated
+rows in the corresponding evaluation determinant.  The sum of the
+degrees of any $A$ distinct such monomials is at least
+$c_rA^{1+1/r}$ for all sufficiently large $A$: the number of
+monomials of degree at most $t$ is $\binom{t+r}{r}=O_r(t^r)$,
+so at least half of the $A$ monomials have degree bounded below by
+a positive multiple of $A^{1/r}$.  Their evaluations have absolute
+value at most $1$.  The nonarchimedean triangle inequality and
+\eqref{eq:log-taylor-bound}, first for truncated series and then by
+convergence, therefore give a constant $c>0$ such that
+\begin{equation}\label{eq:log-determinant-small}
+ \log|\Delta_D|_\ell\leq-cA^{1+1/r}.
+\end{equation}
+
+At every archimedean embedding $\tau:E\hookrightarrow\C$, an
+entry of this minor has absolute value at most $\exp(CDS)$, with
+$C$ independent of $D$.  Expanding the determinant thus gives
+\begin{equation}\label{eq:log-determinant-large}
+ \log|\tau(\Delta_D)|\leq C'ADS+A\log A.
+\end{equation}
+Let $\lambda$ be the place of $E$ induced by $E\hookrightarrow K$.
+Use absolute values at finite places extending the usual ones on
+$\Q$, and sum over all embeddings $E\hookrightarrow\C$, so that
+each complex place occurs twice.  Writing $p_v$ for the rational
+prime below a finite place $v$, the product formula reads
+\[
+ 0=\sum_{\tau:E\hookrightarrow\C}\log|\tau(\Delta_D)|
+     +\sum_{v\text{ finite}}[E_v:\Q_{p_v}]
+                            \log|\Delta_D|_v.
+\]
+Every term in the second sum is nonpositive because $\Delta_D$ is
+an algebraic integer.  Retaining its $\lambda$-term and using
+\eqref{eq:log-determinant-small}--\eqref{eq:log-determinant-large}
+would imply
+\[
+ c[E_\lambda:\Q_\ell]A^{1+1/r}
+ \leq [E:\Q](C'ADS+A\log A).
+\]
+This is impossible as $D\to\infty$: the left side has order
+$D^{n+n/r}$, whereas the right side is
+$O(D^{n+1+n/m}+D^n\log D)$, and
+\[
+ \frac nr\geq\frac n{n-1}>1+\frac nm
+\]
+by $m>n(n-1)$.  This contradiction proves the lemma.
+\end{proof}
+
+\subsection{Detection, averaging, and corestriction}
+
+For a number field $M\subset\overline{\Q}$, write
+\[
+ \X{\ell}{M}:=\Hom_{\mathrm{cont}}(G_M,\Q_\ell),
+ \qquad G_M=\Gal(\overline{\Q}/M),
+\]
+where $\Q_\ell$ is an additive group with trivial Galois action.
+Local class field theory shows that these characters are unramified
+at every finite prime $v\nmid\ell$: the local unit group has a
+pro-$p$ subgroup of finite index for $p\ne\ell$, and has no
+nonzero continuous homomorphism to $\Q_\ell$.  Thus
+$x(\Fr_v)$ is defined for $x\in\X{\ell}{M}$ and $v\nmid\ell$.
+
+If $M/\Q$ is Galois, $R=\Gal(M/\Q)$ acts by
+$(g x)(h)=x(\widetilde g^{-1}h\widetilde g)$, where
+$\widetilde g\in G_\Q$ lifts $g$.  The definition is independent
+of the lift.  For a subgroup $J\subseteq R$, write
+\[
+ E_J=\frac1{|J|}\sum_{g\in J}g
+\]
+for the projector onto the $J$-invariants.  We use
+$\res_{L/M}$ and $\cor_{L/M}$ for restriction and corestriction
+on additive characters for a finite extension $L/M$.
+
+\begin{proposition}[Frobenius detection]\label{prop:character-detection}
+Let $M$ be a number field, let $\ell$ be a rational prime, and let
+$x\in\X{\ell}{M}$.
+\begin{enumerate}
+\item[(i)] If $x(\Fr_v)=0$ at infinitely many primes $v\nmid\ell$
+of absolute residue degree one, then $x=0$.
+\item[(ii)] Suppose that $M/\Q$ is Galois, and fix
+$\delta\in\Gal(M/\Q)$.  If $x(\Fr_v)=0$ at infinitely many
+primes $v\nmid\ell$ lying over rational primes unramified in $M$
+whose Frobenius element at $v$ in $\Gal(M/\Q)$ is $\delta$, then
+$E_{\langle\delta\rangle}x=0$.
+\item[(iii)] If $\cor_{M/\Q}x\ne0$, then
+$x(\Fr_v)=0$ at only finitely many primes $v\nmid\ell$, with no
+restriction on their residue degrees.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+We first recall the class-field-theoretic description of the character
+space.  Let $V_M=M\otimes_\Q\Q_\ell$, and let $U_M\subseteq V_M$
+be the $\Q_\ell$-linear span of the logarithms of the global units,
+with logarithms taken at all places above $\ell$.  Then
+\begin{equation}\label{eq:character-class-field}
+ \X{\ell}{M}\simeq
+ \{\lambda\in\Hom_{\Q_\ell}(V_M,\Q_\ell):
+                         \lambda(U_M)=0\}.
+\end{equation}
+Indeed, pass to the inverse limit of the ray class sequences with
+moduli supported above $\ell$.  Local logarithms identify the
+rationalized local unit groups with $V_M$; the global units impose
+exactly the subspace $U_M$.  The ordinary class group and the factors
+at real places are finite, so contribute no additive
+$\Q_\ell$-characters.  This description uses the actual logarithmic
+image of the units and makes no assertion about its dimension.
+
+For $v\nmid\ell$, choose $h_v>0$ and $b_v\in\mathcal O_M$ with
+$(b_v)=v^{h_v}$.  Such a choice is possible because the ideal class
+group is finite.  Principal reciprocity, with arithmetic Frobenius,
+gives for the linear form corresponding to $x$ the identity
+\begin{equation}\label{eq:character-principal-reciprocity}
+ \lambda(\log b_v)=-h_vx(\Fr_v).
+\end{equation}
+Here $b_v$ is a unit at every place above $\ell$; all other local
+unit contributions are killed by $x$.
+
+For (i), set $n=[M:\Q]$ and choose $m>n(n-1)$ of the zero primes,
+over distinct rational primes unramified in a Galois closure $E$ of
+$M$.  Let $b_i$ be the elements just chosen for these primes, and
+let
+\[
+ u_i=(\sigma(b_i))_{\sigma:M\hookrightarrow E}\in(E^\times)^n.
+\]
+All coordinates are algebraic integers and are units at every place
+of $E$ above $\ell$.  We verify the multiplicative hypothesis of
+Lemma~\ref{lem:log-rank} by valuations.
+
+Put $G=\Gal(E/\Q)$ and $H=\Gal(E/M)$.  For a selected degree-one
+prime $v_i$ and a prime $\mathfrak p_i$ of $E$ above it, the
+decomposition group $D_i$ is contained in $H$.  In fact, the rational
+prime is unramified in $E$, and the degree-one condition says that
+$[D_i:D_i\cap H]=1$.  The support of $v_i\mathcal O_E$ is
+indexed by $H/D_i$.  Its translates corresponding to the embeddings
+$M\hookrightarrow E$, indexed by $G/H$, have supports indexed by
+the disjoint sets $gH/D_i$.  Distinct indices $i$ involve distinct
+rational primes.  It follows that the $mn$ numbers
+$\sigma(b_i)$ have pairwise disjoint nonempty supports of their
+principal ideals.  They are therefore multiplicatively independent
+modulo roots of unity.  In particular, for any nonzero
+$a\in\Z^m$, the $n$ coordinates of $u^a$ are multiplicatively
+independent modulo roots of unity.
+
+Embed $E$ in a finite extension $K/\Q_\ell$.  Under the splitting
+\[
+ (M\otimes_\Q\Q_\ell)\otimes_{\Q_\ell}K
+   \simeq\prod_{\sigma:M\hookrightarrow E}K,
+\]
+the vector $\log b_i$ becomes $\log u_i$.
+Lemma~\ref{lem:log-rank} says that these vectors span the right
+side.  Equations~\eqref{eq:character-class-field} and
+\eqref{eq:character-principal-reciprocity} therefore force
+$\lambda=0$, and hence $x=0$.
+
+For (ii), put $C=\langle\delta\rangle$, $F=M^C$, and
+$y=E_Cx$.  Since $\delta$ fixes every indicated prime $v$, averaging
+preserves its Frobenius value, so $y(\Fr_v)=0$.  The
+restriction--corestriction identity gives
+\[
+ \res_{M/F}\cor_{M/F}y=\sum_{g\in C}g y=|C|y.
+\]
+Thus $y$ is the restriction of
+$z=|C|^{-1}\cor_{M/F}y\in\X{\ell}{F}$.  At an indicated prime,
+the decomposition group of $M/\Q$ is exactly $C$.  The prime
+$u=v\cap F$ consequently has absolute residue degree one, and
+\[
+ 0=y(\Fr_v)=[\kappa(v):\kappa(u)]z(\Fr_u).
+\]
+Infinitely many distinct $u$ occur because $M/F$ is finite.
+Part~(i) gives $z=0$, and therefore $y=0$.
+
+For (iii), suppose that infinitely many zero primes exist.  Let
+$E/\Q$ be a finite Galois closure of $M$, put
+$R=\Gal(E/\Q)$, and let $y=\res_{E/M}x$.  Discarding finitely
+many primes, choose primes of $E$ above these zero primes whose
+rational primes are unramified in $E$.  Restriction multiplies a
+Frobenius value by the relative residue degree, so they are zero
+primes for $y$.  An infinite subset has one fixed Frobenius element
+$\delta\in R$.  By (ii),
+$E_{\langle\delta\rangle}y=0$, and hence $E_Ry=0$.  It follows that
+\[
+ \res_{E/\Q}\cor_{E/\Q}y=|R|E_Ry=0.
+\]
+Restriction on additive characters across a finite extension is
+injective: a character that vanishes on a subgroup of finite index
+has finite image in the torsion-free group $\Q_\ell$, and so is
+zero.  On the other hand, transitivity and the
+corestriction--restriction identity give
+\[
+ \cor_{E/\Q}y
+ =\cor_{M/\Q}\cor_{E/M}\res_{E/M}x
+ =[E:M]\cor_{M/\Q}x\ne0,
+\]
+a contradiction.
+\end{proof}
+
+\subsection{A signed representation in the character space}
+
+The class field description also controls the supply of characters
+when the number field varies.  The regular representation gives upper
+bounds for dimensions of subgroup invariants, while the signed
+representation below gives lower bounds and ensures a faithful action
+on the character space.
+
+\begin{proposition}\label{prop:signed-representation}
+Let $M/\Q$ be a finite, totally imaginary Galois extension, put
+$R=\Gal(M/\Q)$, and let $c\in R$ be a complex conjugation.
+For every rational prime $\ell$, there are $R$-equivariant embeddings
+\begin{equation}\label{eq:signed-representation}
+ W_M:=\Ind_{\langle c\rangle}^{R}(\mathrm{sign})
+ \lhook\joinrel\longrightarrow\X{\ell}{M}
+ \lhook\joinrel\longrightarrow\Q_\ell[R],
+\end{equation}
+where $\mathrm{sign}(c)=-1$.  In particular, the action of $R$ on
+$\X{\ell}{M}$ is faithful.
+\end{proposition}
+
+\begin{proof}
+By the normal basis theorem, $V_M=M\otimes_\Q\Q_\ell$ is the
+regular representation of $R$, as is its dual.  The naturality of
+\eqref{eq:character-class-field} gives the second embedding.
+
+The equivariant Dirichlet unit theorem
+\cite[Section~3]{bass1966} identifies the representation
+$\mathcal O_M^\times\otimes_\Z\Q_\ell$ with the permutation
+representation on the archimedean places minus its trivial summand.
+Since these places are indexed by $R/\langle c\rangle$, writing
+$U^{\mathrm{alg}}=\mathcal O_M^\times\otimes_\Z\Q_\ell$ gives
+\[
+ \Q_\ell\oplus U^{\mathrm{alg}}
+ \simeq\Ind_{\langle c\rangle}^{R}\mathbf1.
+\]
+The character identity follows first over the real numbers from the
+usual logarithmic unit embedding, and hence over $\Q_\ell$ by
+semisimplicity in characteristic zero.  In particular,
+$U^{\mathrm{alg}}$ is self-dual.  Its $\ell$-adic logarithmic image
+$U_M$ is a quotient of $U^{\mathrm{alg}}$; no injectivity of the
+logarithm on this tensor product is needed.
+
+The decomposition of the regular representation induced from the
+two characters of $\langle c\rangle$ is
+\[
+ \Q_\ell[R]
+ \simeq\Ind_{\langle c\rangle}^{R}\mathbf1\oplus W_M
+ \simeq\Q_\ell\oplus U^{\mathrm{alg}}\oplus W_M.
+\]
+Since $U_M^\vee$ is a subrepresentation of
+$(U^{\mathrm{alg}})^\vee\simeq U^{\mathrm{alg}}$, semisimplicity
+and $\X{\ell}{M}\simeq(V_M/U_M)^\vee$ show that
+$\X{\ell}{M}$ contains $\Q_\ell\oplus W_M$.  This proves the
+first embedding in \eqref{eq:signed-representation}.
+
+Finally, realize $W_M$ as the direct sum of the signed lines indexed
+by the left cosets of $\langle c\rangle$.  An element acting
+trivially must fix the line of the identity coset, and therefore
+belongs to $\langle c\rangle$.  The nonidentity element $c$ acts
+by $-1$ on that line.  Thus only the identity acts trivially on
+$W_M$, proving faithfulness.
+\end{proof}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/generation.tex

@@ -0,0 +1,425 @@
+\section{Normalizer extension and Frobenius generation}\label{sec:generation}
+
+Let $F$ be a number field and let
+\[
+  \alpha_0:G_F\longrightarrow P
+\]
+be a continuous open homomorphism. Our eventual aim is to identify this
+map with ordinary restriction, up to conjugation in $P$. We first enlarge
+an open part of its domain in a canonical way. For the enlarged map we
+will prove surjectivity and equality of cyclotomic logarithms at an
+infinite set of auxiliary primes. The main step is to show that images
+of local Frobenius elements generate prescribed cyclic subgroups in
+suitable finite quotients of $P$.
+
+Choose an open normal subgroup $U\trianglelefteq\mathcal G$ contained in
+$G_F$, and put
+\[
+ A=\alpha_0(U),\qquad N=\ker(\alpha_0|_U),\qquad
+ T=N_{\mathcal G}(N).
+\]
+The subgroup $A$ is open in $P$. Since $U\subset T$, the normalizer $T$
+is open and closed in $\mathcal G$; write $T=G_k$.
+
+\begin{lemma}[Extension to the normalizer]\label{lem:normalizer-extension}
+There is a unique continuous open homomorphism
+\[
+  \beta:T\longrightarrow P
+\]
+whose restriction to $U$ is $\alpha_0|_U$.
+\end{lemma}
+
+\begin{proof}
+Conjugation by $t\in T$ induces a continuous automorphism of $U/N=A$.
+By Theorem~\ref{thm:uchida-isomorphism}, applied to
+$\Q^s/(\Q^s)^A$, this automorphism is conjugation by a unique element
+$\beta(t)$ of $P$. Indeed, the implementing field automorphism of
+$\Q^s$ fixes $\Q$. Uniqueness shows both that $\beta$ is a homomorphism
+and that $\beta(u)=\alpha_0(u)$ for $u\in U$. It also shows that any
+extension of $\alpha_0|_U$ to $T$ must equal $\beta$: its values must
+induce these same conjugations on $A$.
+
+Continuity on the open subgroup $U$ implies continuity on $T$.
+Moreover, $\beta(T)$ contains $A$ and is therefore open in $P$.
+A continuous homomorphism of profinite groups is open onto its image,
+so $\beta$ is open.
+\end{proof}
+
+We keep $U,A,N,T,k$, and $\beta$ fixed for the rest of the argument.
+List the distinct conjugates of $N$ other than $N$ itself as
+\[
+ N_i=t_iNt_i^{-1}\quad(1\leq i\leq r),\qquad t_i\notin T,
+ \qquad H_i=\beta(N_i).
+\]
+There are finitely many because $U$ normalizes $N$. Normality of $U$
+in $\mathcal G$ implies that each $N_i$ lies in $U$ and is normal in
+$U$, so $H_i$ is normal in $A$.
+Each $H_i$ is nontrivial. Otherwise $t_iNt_i^{-1}\subset N$, and
+iterating this inclusion would give a descending chain returning to
+$N$: some power of $t_i$ lies in $U$ and hence normalizes $N$.
+All inclusions would be equalities, contrary to $t_i\notin T$.
+We allow $r=0$, with sums indexed by $i$ then equal to zero.
+
+\subsection{A finite quotient with enough characters}
+
+The next lemma constructs characters that distinguish a proper
+subgroup of a cyclic group while avoiding the invariant spaces attached
+to the $H_i$. The latter condition will prevent cancellation by
+conjugates of $N$ when we average a pulled-back character on the source.
+For a finite subgroup $J$ acting on a characteristic-zero vector space,
+we use the averaging operator $E_J=|J|^{-1}\sum_{g\in J}g$.
+
+\begin{lemma}[Quotient amplification]\label{lem:quotient-amplification}
+Fix an odd prime $\ell$ and a finite quotient $R_0$ of $P$. There is
+a finite Galois extension $L/\Q$ in $\Q^s$, totally imaginary, such that
+the quotient $R=\Gal(L/\Q)$ dominates $R_0$ and has the following
+property. For every $\gamma\in R$, with $C=\langle\gamma\rangle$, and
+every proper subgroup $D<C$, there is $x\in\X{\ell}{L}$ satisfying
+\begin{equation}\label{eq:separating-character}
+ x\in\X{\ell}{L}^{D},\qquad E_Cx=0,\qquad
+ x\notin\sum_{i=1}^r\X{\ell}{L}^{H_i}.
+\end{equation}
+Here each $H_i$ acts through its image in $R$.
+\end{lemma}
+
+\begin{proof}
+Choose a quotient $R_1=\Gal(L_1/\Q)$ dominating $R_0$ such that
+every $H_i$ has nontrivial image in $R_1$, the kernel of $P\to R_1$
+is contained in $A$, and $L_1$ contains $\Q(i)$. These are finitely
+many compatible conditions on an open normal subgroup of $P$.
+Put $n=|R_1|$.
+
+We will enlarge $R_1$ while keeping element orders bounded by $2n$.
+At the same time, the images of the $H_i$ and the conjugacy class of
+complex conjugation will grow. Growth of the $H_i$ makes the sum
+of their invariant spaces small. Growth of the conjugacy class
+makes the signed representation supply enough $D$-invariant
+vectors killed by $E_C$ to escape that sum.
+
+We use the regular-module auxiliary extension construction appearing
+in Uchida's proof~\cite[pp.~360--362]{uchida1979}, with quantitative
+bounds on its invariant spaces. More precisely, we construct a
+further quotient with kernel consisting of $m$ copies of the regular
+$\mathbb F_2[R_1]$-module, where $m$ will be chosen below.
+Choose $m$ distinct rational primes that split completely in $L_1$.
+For each of them, select one prime of $L_1$ above it. Approximation
+gives elements $a_1,\ldots,a_m\in L_1^\times$ such that $a_j$ has
+valuation one at its selected prime and valuation zero at every other
+prime in these $m$ sets. The $mn$ elements
+\[
+  g(a_j),\qquad g\in R_1,\quad 1\leq j\leq m,
+\]
+have independent square classes: their valuations at the selected
+sets of primes form a permutation of the identity matrix modulo two.
+Adjoin their square roots and call the resulting field $L$.
+Kummer theory gives
+\begin{equation}\label{eq:regular-kummer-kernel}
+  1\longrightarrow V\longrightarrow R=\Gal(L/\Q)
+   \longrightarrow R_1\longrightarrow1,
+  \qquad V\simeq\mathbb F_2[R_1]^m
+\end{equation}
+as $R_1$-modules. The dual module supplied by Kummer theory is again
+the regular permutation module. The field $L$ is Galois over $\Q$
+because all conjugates of the $a_j$ were included; its group is
+solvable, so $L\subset\Q^s$. No splitting of
+\eqref{eq:regular-kummer-kernel} is required.
+
+Every element of $R$ has order at most $2n$, since its $n$th power
+lies in the exponent-two group $V$. Write $A_R$ and $H_{i,R}$ for
+the images of $A$ and $H_i$ in $R$. The choice of $R_1$ gives
+$V\subset A_R$. Choose $h_i\in H_{i,R}$ with nonidentity image
+$g_i\in R_1$, of order $d_i>1$. As $H_{i,R}$ is normal in $A_R$,
+\[
+ (g_i-1)V=[h_i,V]\subset H_{i,R}.
+\]
+On $m$ regular modules the rank of $g_i-1$ is
+$mn(1-1/d_i)\geq mn/2$. Consequently
+\begin{equation}\label{eq:large-subgroups}
+  |H_{i,R}|\geq 2^{mn/2}.
+\end{equation}
+If $c\in R$ is a complex conjugation, its image in $R_1$ is a
+nontrivial involution because $L_1$ is totally imaginary.
+Conjugation by $V$ alone therefore gives at least $2^{mn/2}$ distinct
+conjugates of $c$; thus
+\begin{equation}\label{eq:large-conjugacy-class}
+  |c^R|\geq 2^{mn/2}.
+\end{equation}
+
+Set $\eta=2^{-mn/2}$ and write $X=\X{\ell}{L}$. By
+Proposition~\ref{prop:signed-representation}, $X$ embeds in the regular
+$\Q_\ell[R]$-representation and contains
+\[
+ W=\Ind_{\langle c\rangle}^{R}(\operatorname{sign}).
+\]
+The regular-representation bound and \eqref{eq:large-subgroups} give
+\begin{equation}\label{eq:forbidden-dimension}
+ \dim\left(\sum_{i=1}^rX^{H_i}\right)
+ \leq\sum_{i=1}^r\frac{|R|}{|H_{i,R}|}
+ \leq r\eta|R|.
+\end{equation}
+The character of $W$ takes the value $|R|/2$ at the identity,
+the value $-|R|/(2|c^R|)$ on $c^R$, and zero elsewhere.
+Hence, for every subgroup $J\subset R$,
+\begin{equation}\label{eq:signed-invariants}
+ \left|\dim W^J-\frac{|R|}{2|J|}\right|
+ \leq\frac{\eta|R|}{2}.
+\end{equation}
+For $D<C=\langle\gamma\rangle$, the map
+$E_C:W^D\to W^C$ is surjective. Since $|D|\leq |C|/2$ and
+$|C|\leq2n$, \eqref{eq:signed-invariants} implies
+\begin{align*}
+ \dim\ker(E_C|_{W^D})
+ &=\dim W^D-\dim W^C\\
+ &\geq\frac{|R|}{2}\left(\frac1{|D|}-\frac1{|C|}\right)
+       -\eta|R|\\
+ &\geq |R|\left(\frac1{4n}-\eta\right).
+\end{align*}
+Choose the integer $m\geq1$ so large that
+\[
+  (r+1)2^{-mn/2}<\frac1{4n}.
+\]
+This single choice works for all $\gamma$ and all proper $D<C$.
+The last dimension is then greater than
+\eqref{eq:forbidden-dimension}, so its kernel contains a vector
+outside the indicated sum. This vector gives
+\eqref{eq:separating-character}. If $C$ is trivial, there is no
+proper subgroup to consider.
+\end{proof}
+
+\subsection{Generation by local Frobenius images}
+
+We now use these characters to strengthen the local correspondence.
+Containment of a source decomposition group in a target decomposition
+group places its Frobenius image in a cyclic subgroup of a finite
+quotient. The next proposition shows that, after enlarging the
+quotient, infinitely many such images generate the entire cyclic
+subgroup.
+
+\begin{proposition}[Frobenius generation]\label{prop:frobenius-generation}
+Fix an odd prime $\ell$, a finite quotient $R_0$ of $P$, and
+$\gamma_0\in R_0$. There are a quotient
+\[
+ P\twoheadrightarrow R=\Gal(L/\Q)\twoheadrightarrow R_0
+\]
+and a lift $\gamma\in R$ of $\gamma_0$ with the following property.
+There are infinitely many $\ell$-active primes $v$ of $k$, over
+distinct rational primes $p\ne\ell$, for which compatible
+prolongations can be chosen so that:
+\begin{enumerate}
+\item the corresponding target rational prime $q\ne\ell$ is unramified
+in $L$, and the target prolongation restricts to a prime $w'$ of $L$
+whose Frobenius in $R$ is $\gamma$;
+\item the image in $R$ of an arithmetic Frobenius lift at $v$ generates
+$\langle\gamma\rangle$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+Choose $R$ by Lemma~\ref{lem:quotient-amplification} and choose any
+lift $\gamma$ of $\gamma_0$. Put $C=\langle\gamma\rangle$.
+Take an open normal subgroup $U'\trianglelefteq\mathcal G$ contained
+in $U\cap\beta^{-1}(P_L)$; for example, take the core of this open
+subgroup in $\mathcal G$. Let $F'/\Q$ be the finite Galois extension
+with $G_{F'}=U'$. The map
+\[
+ \beta|_{U'}:G_{F'}\longrightarrow P_L
+\]
+is open.
+
+Chebotarev's theorem supplies infinitely many primes $w'$ of $L$,
+over distinct rational primes $q\ne\ell$ unramified in $L$, with
+Frobenius exactly $\gamma$. Theorem~\ref{thm:local-correspondence}
+applied to $\beta|_{U'}$ pairs all but finitely many of them with
+$\ell$-active primes $v'$ of $F'$, away from $\ell$.
+Uniqueness of the target prime for an active source prime implies
+that infinitely many $v'$ occur. Since only finitely many primes of
+$F'$ lie over a given rational prime, we may pass to a subset with
+distinct source rational primes $p$, all unramified in $F'$.
+Passing to another infinite subset, we may also assume that their
+Frobenius elements in $\Gal(F'/\Q)$ are a fixed element $\delta$.
+
+Fix a source prolongation above each $v'$, and use it also for the
+prime $v=v'|_k$. The original correspondence at $v'$ can be made
+with this prolongation: changing the source prolongation conjugates
+the target group by an element of $\beta(U')\subset P_L$, which
+preserves the finite prime $w'$. Activity passes from $v'$ to $v$,
+since the relevant inertia subgroup for $U'$ is contained in that
+for $T$. Apply Theorem~\ref{thm:local-correspondence} at $v$ using
+this same source prolongation. If $w_1$ is the restriction to $L$
+of its target prolongation, then
+\[
+ \beta(D_{v'})\subset\beta(D_v)\cap P_L\subset D_{w_1}.
+\]
+Uniqueness of the target prime for $v'$ at the level of $L$ gives
+$w_1=w'$. Consequently the corresponding target rational prime
+is $q$, and the image of $D_v$ in $R$ lies in $C$.
+These changes of prolongations preserve $v'$ and $w'$, so they
+preserve the previously selected finite Frobenius elements
+$\delta$ and $\gamma$.
+
+Put $H=T/U'\subset\mathcal G/U'$ and let $f$ be the residue degree
+of $v$ over $p$. Figure~\ref{fig:generation-primes} records the
+primes involved. Here $\delta=\Fr(v'/p)$ and
+$\gamma=\Fr(w'/q)$ belong to the finite Galois groups over $\Q$.
+The symbols $\Fr_{v'}$ and $\Fr_{w'}$ used below instead denote
+local arithmetic Frobenius over $F'$ and $L$, respectively.
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzcd}[row sep=large,column sep=huge]
+ v'\text{ of }F' \arrow[d] \arrow[r,dashed] &
+ w'\text{ of }L \arrow[d] \\
+ v\text{ of }k \arrow[d,"f"'] \arrow[r,dashed] &
+ q\text{ of }\Q \\
+ p\text{ of }\Q &
+\end{tikzcd}
+\caption{Vertical arrows restrict primes; dashed arrows are the
+partial local correspondences. The residue degree of $v/p$ is $f$.
+No equality between $p$ and $q$ is assumed at this stage.}
+\label{fig:generation-primes}
+\end{figure}
+
+In the decomposition group
+$\langle\delta\rangle\subset\Gal(F'/\Q)$ we have
+\begin{equation}\label{eq:residue-degree-intersection}
+ \langle\delta\rangle\cap H=\langle\delta^f\rangle,
+ \qquad
+ f=\min\{j\geq1:\delta^j\in H\}.
+\end{equation}
+Indeed, the residue degree is the orbit size of the distinguished
+coset in the action of $\langle\delta\rangle$ on
+$\Gal(F'/\Q)/H$. This description does not require $H$ to be
+normal. In particular $f$ is constant on the chosen infinite set.
+The homomorphism $\beta$ induces $H\to R$, since
+$\beta(U')\subset P_L$; define
+\[
+ d=\beta(\delta^f)\in R.
+\]
+This notation is independent of the lift of $\delta^f$ to $T$.
+A local lift of rational Frobenius to the $f$th power is a
+Frobenius lift at $v$, so the preceding containment gives $d\in C$.
+
+We claim that $D=\langle d\rangle$ equals $C$. Suppose that $D<C$,
+and choose $x$ as in \eqref{eq:separating-character}. Pullback along
+$\beta|_{U'}$ gives a linear map
+\[
+ j:\X{\ell}{L}\longrightarrow\X{\ell}{F'},\qquad y=j(x).
+\]
+The map $j$ is injective: its defining homomorphism has open image
+in $P_L$, and an additive $\Q_\ell$-valued character vanishing on
+an open subgroup vanishes everywhere. Frobenius evaluation at $w'$
+is invariant under $\gamma$, because $\gamma$ fixes $w'$ and acts
+trivially by conjugation on its unramified local quotient. Therefore
+\[
+ x(\Fr_{w'})=(E_Cx)(\Fr_{w'})=0.
+\]
+The character $x$ is unramified at $w'$, so this equality says that
+it vanishes on the entire decomposition group there. It follows
+that $y(\Fr_{v'})=0$ for every prime in our infinite set.
+Proposition~\ref{prop:character-detection}(ii) now gives
+\begin{equation}\label{eq:source-average-zero}
+ E_{\langle\delta\rangle}y=0.
+\end{equation}
+
+To see why this contradicts the choice of $x$, write
+$a=|\langle\delta\rangle|$ and $Y=j(\X{\ell}{L})$.
+Pullback is $T$-equivariant, where the action on the target
+character space is through $\beta$. In particular $y$ is fixed
+by $N$. In the sum defining \eqref{eq:source-average-zero}, the
+powers $\delta^j$ belonging to $H$ are exactly those for which
+$f\mid j$, by \eqref{eq:residue-degree-intersection}. Each such
+power acts as the identity on $y$, because $x$ is $D$-invariant.
+If $f\nmid j$, choose a lift $t$ of $\delta^j$ to $\mathcal G$.
+Then $t\notin T$, and $\delta^jy$ is fixed by $tNt^{-1}$.
+Multiplying \eqref{eq:source-average-zero} by $a$ thus yields
+\[
+  0=\frac af\,y+\sum_{\substack{0\leq j<a\\f\nmid j}}\delta^jy,
+  \qquad
+  y\in\sum_{i=1}^r\X{\ell}{F'}^{N_i}.
+\]
+
+It remains to return this membership to the target character
+space. The finite group $U/U'$ acts on $\X{\ell}{F'}$, and its
+subspace $Y$ is stable. Semisimplicity supplies a $U$-equivariant
+projection
+\[
+  \pi:\X{\ell}{F'}\longrightarrow Y.
+\]
+Every $N_i$ lies in $U$, so $\pi$ carries $N_i$-invariants into
+$Y^{N_i}$. Equivariance and injectivity of $j$ give
+\[
+ Y^{N_i}=j\bigl(\X{\ell}{L}^{H_i}\bigr).
+\]
+Applying $\pi$ to the displayed membership and then applying
+$j^{-1}$ contradicts the last condition in
+\eqref{eq:separating-character}. Only $U$-equivariance of the
+projection is used; no normality of $T$ in $\mathcal G$ is needed.
+Thus $D=C$, proving the proposition.
+\end{proof}
+
+\subsection{Surjectivity and cyclotomic logarithms}
+
+The generation statement now gives the two properties needed in the
+next stage. The first removes any restriction on the target image.
+For the second, a congruence condition on $\ell$ forces the source
+primes supplied by the proposition to have residue degree one, so
+the character-detection theorem applies.
+
+\begin{theorem}\label{thm:normalized-map}
+The extension $\beta:G_k\to P$ is surjective. Put
+\[
+ m_0=[k:\Q]!,\qquad
+ \mathcal L=\{\ell:\ell\text{ is an odd prime and }
+                    \ell\equiv1\pmod{m_0}\}.
+\]
+The set $\mathcal L$ is infinite, and for every $\ell\in\mathcal L$,
+\begin{equation}\label{eq:cyclotomic-logarithms}
+ \log\chi_\ell\circ\beta=\log\chi_{k,\ell}
+ \quad\text{as additive characters on }G_k.
+\end{equation}
+Here $\chi_\ell$ is the $\ell$-adic cyclotomic character on $P$,
+and $\chi_{k,\ell}$ is the ordinary cyclotomic character on $G_k$.
+\end{theorem}
+
+\begin{proof}
+Let $R_0$ be any finite quotient of $P$ and let $\gamma_0\in R_0$.
+Proposition~\ref{prop:frobenius-generation} produces an element
+$d$ in the image of $\beta$ in a further quotient $R$ such that
+$\langle d\rangle=\langle\gamma\rangle$, where $\gamma$ lifts
+$\gamma_0$. Thus $\gamma$, and hence $\gamma_0$, belongs to the
+image. Since $\beta(T)$ is compact, surjectivity onto every finite
+quotient implies $\beta(T)=P$.
+
+Dirichlet's theorem on primes in arithmetic progressions shows
+that $\mathcal L$ is infinite. Fix $\ell\in\mathcal L$, take
+\[
+ R_0=\Gal(\Q(\mu_\ell)/\Q)\simeq\mathbb F_\ell^\times,
+\]
+and choose $\gamma_0$ to generate this cyclic group.
+The proposition gives infinitely many active primes $v$ of $k$
+over distinct rational primes $p\ne\ell$, for which the image of
+a Frobenius lift $\sigma_v$ acts on $\mu_\ell$ by a generator.
+Let $q$ be the corresponding target rational prime and let
+$f=[\kappa(v):\mathbb F_p]$, so $Nv=p^f$. If
+$a(\sigma_v)\in\widehat{\Z}$ is the target residue exponent,
+Lemma~\ref{lem:norm-identity} gives
+\begin{equation}\label{eq:generating-norm}
+  p^f=q^{a(\sigma_v)}\quad\text{in }\Z_\ell^\times.
+\end{equation}
+Its reduction modulo $\ell$ says that $p^f$ generates
+$\mathbb F_\ell^\times$. In particular $\gcd(f,\ell-1)=1$.
+But $1\leq f\leq[k:\Q]$, and every prime divisor of any $f>1$
+in this range divides $m_0$, hence divides $\ell-1$.
+Consequently $f=1$.
+
+Taking logarithms in \eqref{eq:generating-norm} gives
+\[
+ (\log\chi_\ell\circ\beta)(\sigma_v)
+   =\log p
+   =\log\chi_{k,\ell}(\sigma_v).
+\]
+The difference is an element of $\X{\ell}{k}$ vanishing at
+infinitely many primes of absolute residue degree one.
+Proposition~\ref{prop:character-detection}(i) proves
+\eqref{eq:cyclotomic-logarithms}.
+\end{proof}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/identification.tex

@@ -0,0 +1,102 @@
+\section{Identifying the homomorphism}\label{sec:identification}
+
+We now turn the prime correspondence into equality of group homomorphisms. The argument uses one fixed coefficient prime $r$, although the auxiliary prime witnessing activity is allowed to vary. This distinction is what makes the correspondences constructed in Section~\ref{sec:matching} sufficient.
+
+Retain the map $\beta:T=G_k\to P$, the infinite set $\mathcal L$, and the fields $B=\Q(i)$ and $K$ from Sections~\ref{sec:generation}--\ref{sec:matching}. Thus $G_K=\beta^{-1}(P_B)$, and Theorem~\ref{thm:normalized-map} gives
+\begin{equation}\label{eq:fixed-r-log}
+ \log\chi_r\circ\beta=\log\chi_{k,r}\qquad(r\in\mathcal L).
+\end{equation}
+Write $\sigma:T\to P$ for ordinary restriction.
+
+\begin{proposition}\label{prop:identify-normalized}
+There is an element $a\in P$ such that $\beta=\Ad(a)\circ\sigma$ on $T$.
+\end{proposition}
+\begin{proof}
+Fix a finite Galois extension $L/\Q$ inside $\Q^s$ containing $B$, and put $R=\Gal(L/\Q)$. Choose a finite Galois extension $F'/\Q$ containing $K$ and $L$ such that $\beta(G_{F'})\subset P_L$. This is possible by taking the core in $\mathcal G$ of the appropriate open subgroup of $T$. Fix $r\in\mathcal L$.
+
+The maps $\beta$ and $\sigma$, restricted to $G_{F'}$, give two pullbacks
+\begin{equation}\label{eq:two-pullbacks}
+ i_\beta,i_\sigma:\X rL\longrightarrow\X r{F'}.
+\end{equation}
+Both are injective: their underlying homomorphisms have open image, and an additive $\Q_r$-valued character vanishing on a finite-index subgroup is zero. For $a\in P$ let $\sigma_a=\Ad(a)\circ\sigma$ and let $i_a$ be its pullback. Since $L/\Q$ is Galois, $\sigma_a(G_{F'})\subset P_L$. The map $i_a$ depends only on $a$ modulo $P_L$, because inner automorphisms of $P_L$ act trivially on additive characters.
+
+\emph{Frobenius evaluations.}
+Let $q\ne r$ range over rational primes split completely in $F'$. By Proposition~\ref{prop:same-characteristic}, for all but finitely many such $q$ there is a prime $v$ of $K$ above $q$ whose decomposition group maps into a decomposition group at a prime $w$ of $B$ above the same $q$. Choose a prime $v'$ of $F'$ above $v$ and compatible prolongations. Then
+\[
+ \beta(D_{v'})\subset D_u\subset P_L
+\]
+for a prime $u$ of $L$ above $w$. Every prime here has absolute residue degree one, because $q$ splits completely in $F'$.
+
+Choose a lift $h$ of arithmetic Frobenius in $D_{v'}$. Let $a_q\in\widehat{\Z}$ be the residue exponent of $\beta(h)$ at $u$. Equation~\eqref{eq:fixed-r-log} gives
+\[
+ (a_q)_r\log_r q=\log_r q.
+\]
+The number $\log_r q$ is nonzero: $q\in\Z_r^\times$ is a positive rational integer larger than one and is not a root of unity. Consequently $(a_q)_r=1$. Every $x\in\X rL$ is unramified at $u$, so
+\begin{equation}\label{eq:matched-evaluation}
+ i_\beta(x)(\Fr_{v'})=x(\Fr_u).
+\end{equation}
+This calculation uses only decomposition containment from the active correspondence. It does not require that its auxiliary prime be $r$.
+
+Under ordinary restriction, the same Frobenius maps to Frobenius at another prime of $L$ above $q$. An element of $R$ carries this prime to $u$. Choose a lift $a\in P$ of that element. Then
+\[
+ i_\beta(x)(\Fr_{v'})=i_a(x)(\Fr_{v'})
+ \qquad\text{for every }x\in\X rL.
+\]
+The choice of $a$ modulo $P_L$ may initially depend on $q$. Since $R$ is finite, an infinite set of distinct $q$ uses one fixed choice. For each $x$, the difference $i_\beta(x)-i_a(x)$ therefore vanishes at infinitely many degree-one primes of $F'$. Proposition~\ref{prop:character-detection} gives
+\begin{equation}\label{eq:equal-pullbacks}
+ i_\beta=i_a.
+\end{equation}
+
+\emph{From characters to the finite group.}
+The field $F'/\Q$ is Galois, so $T$ acts on $\X r{F'}$ by conjugation. The two pullbacks intertwine this action with the respective actions on $\X rL$ induced by $\beta$ and $\sigma_a$. If $i$ denotes their common injective value in \eqref{eq:equal-pullbacks}, then for $t\in T$ and $x\in\X rL$,
+\[
+ i\bigl(\beta(t)\cdot x\bigr)
+   =t\cdot i(x)
+   =i\bigl(\sigma_a(t)\cdot x\bigr).
+\]
+The field $L$ is totally imaginary because it contains $B$. By Proposition~\ref{prop:signed-representation}, $R$ acts faithfully on $\X rL$. Hence
+\[
+ \beta(t)\bmod P_L=\sigma_a(t)\bmod P_L\qquad(t\in T).
+\]
+
+\emph{One conjugation for all finite quotients.}
+For each such $L$, let
+\[
+ C_L=\{a\in P:\ \beta(t)\equiv\Ad(a)(\sigma(t))\pmod{P_L}
+                    \text{ for every }t\in T\}.
+\]
+We have proved that $C_L$ is nonempty. It is closed, and indeed clopen, since the defining condition depends only on $a$ modulo $P_L$. If $L$ contains finitely many fields $L_1,\ldots,L_s$, then $C_L\subset\bigcap_j C_{L_j}$. Composita thus give the finite-intersection property. Compactness of $P$ supplies an element in every $C_L$. The extensions $L$ containing $B$ are cofinal among finite Galois subextensions of $\Q^s/\Q$, so this element gives the asserted equality in $P$.
+\end{proof}
+
+We remove the normalizer extension and return to the original source.
+\begin{theorem}\label{thm:prosolvable-rigidity}
+Let $F$ be a number field in $\overline{\Q}$. Every continuous open homomorphism $\varphi:G_F\to P$ is conjugate in $P$ to ordinary restriction.
+\end{theorem}
+\begin{proof}
+As in Lemma~\ref{lem:normalizer-extension}, choose an open normal subgroup $U$ of $\mathcal G$ contained in $G_F$, and extend $\varphi|_U$ to the normalizer of its kernel. Proposition~\ref{prop:identify-normalized} shows that, after one conjugation of $\varphi$, it agrees on $U$ with ordinary restriction $\sigma:G_F\to P$.
+
+For $g\in G_F$ and $u\in U$, normality of $U$ gives
+\[
+ \varphi(g)\sigma(u)\varphi(g)^{-1}
+ =\varphi(gug^{-1})
+ =\sigma(gug^{-1})
+ =\sigma(g)\sigma(u)\sigma(g)^{-1}.
+\]
+Thus $\sigma(g)^{-1}\varphi(g)$ centralizes $\sigma(U)=\varphi(U)$, an open subgroup of $P$. Lemma~\ref{lem:centralizer} makes this element trivial. The equality holds for every $g\in G_F$.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Choose copies of the given fields inside $\overline{\Q}$. By Lemma~\ref{lem:solvable-closure}, each $E_i$ contains $\Q^s$. Form the composite
+\begin{equation}\label{eq:final-composite}
+ G_{F_1}\longrightarrow G_1\xrightarrow{\alpha}G_2
+     \longrightarrow P,
+\end{equation}
+where the outer arrows are restrictions. The first is surjective. The last has open image, since its composite with $G_{F_2}\twoheadrightarrow G_2$ is ordinary restriction from an open subgroup of $\mathcal G$ to $P$. Continuous surjective homomorphisms between profinite groups are open, so \eqref{eq:final-composite} is open.
+
+By Theorem~\ref{thm:prosolvable-rigidity}, this composite is conjugate to ordinary restriction. Cyclotomic characters are unchanged by conjugation. All roots of unity lie in $\Q^s$, so the full cyclotomic characters factor through the groups in \eqref{eq:final-composite}. Surjectivity of the first arrow therefore yields
+\[
+ \chi_{E_2/F_2}\circ\alpha=\chi_{E_1/F_1}
+ \quad\text{in }\widehat{\Z}^{\times}.
+\]
+Theorem~\ref{thm:hoshi} gives an equivariant embedding $E_2\hookrightarrow E_1$. Its uniqueness is Uchida's theorem cited after Theorem~\ref{thm:hoshi}.
+\end{proof}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/introduction.tex

@@ -0,0 +1,39 @@
+\section{Introduction}\label{sec:introduction}
+
+An algebraic extension of a number field is \emph{solvably closed} if it has no nontrivial abelian algebraic extension. Algebraic closures and maximal prosolvable extensions are examples. The isomorphism theorems of Neukirch and Uchida show that Galois groups of such fields retain remarkably precise arithmetic information. The corresponding question for open homomorphisms asks whether this reconstruction also recovers field embeddings.
+
+Let $E_i/F_i$ be Galois extensions of number fields, with each $E_i$ solvably closed, and write $G_i=\Gal(E_i/F_i)$. A field embedding $j:E_2\hookrightarrow E_1$ \emph{induces} a homomorphism $\alpha:G_1\to G_2$ if
+\[
+ g\circ j=j\circ\alpha(g)\qquad(g\in G_1).
+\]
+In particular, such an embedding sends $F_2$ into $F_1$. Uchida conjectured that every continuous homomorphism with open image arises in this way \cite[p.~595]{uchida1981}. We prove the conjecture in its full form.
+
+\begin{theorem}\label{thm:main}
+Let $F_1,F_2$ be number fields, and let $E_i/F_i$ be possibly infinite solvably closed Galois extensions. Every continuous open homomorphism
+\[
+ \alpha:\Gal(E_1/F_1)\longrightarrow\Gal(E_2/F_2)
+\]
+is induced by a unique field embedding $j:E_2\hookrightarrow E_1$.
+\end{theorem}
+
+The kernel is unrestricted. The proof establishes the necessary arithmetic compatibility from openness itself.
+
+\subsection{Historical context}
+The reconstruction of number fields from their Galois groups begins with Neukirch's work on normal number fields and decomposition groups \cite{neukirch1969a,neukirch1969b}. The passage to arbitrary number fields and prescribed group isomorphisms involved Uchida, Iwasawa, and Ikeda; their contributions are recorded in the contemporary accounts \cite{uchida1976,neukirch1977}. Uchida subsequently established the isomorphism theorem for arbitrary solvably closed Galois extensions of number fields \cite{uchida1979}.
+
+In the paper formulating the homomorphism conjecture, Uchida proved it when the source number field is $\Q$, proved uniqueness in general, and established existence under a suitable local condition on decomposition groups \cite[Theorem~1, Proposition~2, and Theorem~2]{uchida1981}. His local analysis also associates primes to one another whenever a chosen pro-$\ell$ inertia group survives under the homomorphism, for an odd auxiliary prime $\ell$. For each fixed auxiliary prime $\ell$, this partial correspondence reaches all but finitely many target primes \cite[Proposition~1]{uchida1981}.
+
+A later reduction of Hoshi identifies an exact arithmetic condition for existence: an open homomorphism is induced by a field embedding if and only if it preserves the full cyclotomic character \cite[Theorem~3.4]{hoshi2025}. Hoshi's subsequent work reformulates the unrestricted conjecture in terms of radicals and normality of the field fixed by a kernel \cite[Theorem~B]{hoshi2026}. Related work on finite solvable quotients includes the isomorphism theorem of Sa\"idi and Tamagawa \cite{saiditamagawa2022} and the cyclotomic-compatible homomorphism theorem of Mao and Sa\"idi \cite{maosaidi2025}. Our proof uses the cyclotomic criterion at its endpoint. The main task is to derive that criterion from the partial local correspondences, even though their domains and exceptional sets initially depend on $\ell$.
+
+\subsection{The main ideas}
+Two arguments used below are useful beyond the final reduction. First, an additive $\ell$-adic character of the absolute Galois group of a number field is determined by its values on any infinite set of primes of absolute residue degree one. More precisely, a character vanishing at all those primes is zero. The logarithmic-rank input is a special case of the $\ell$-adic Waldschmidt--Masser theorem \cite{waldschmidt1981,masser1981,dasgupta2023}. We give a complete proof using Laurent's interpolation-determinant method \cite{laurent1991} and a translate-and-degree zero estimate on a torus. Its variants for Frobenius averages and corestriction permit us to treat primes of arbitrary residue degree as well. These statements concern the actual unit-logarithm image and require no hypothesis on its rank.
+
+Second, we refine the Kummer argument in Uchida's proof of cofinite target coverage \cite[Proposition~1]{uchida1981}. After logarithmic cyclotomic compatibility has been established, the number of target primes missed by the partial correspondence can be bounded independently of $\ell$. A missed prime supplies a Kummer class. On killing a finite cyclotomic twist, these classes lie in one character component of an $S$-unit representation. The full extension degree may grow with $\ell$, but the multiplicity of that character is bounded by the number of orbits of $S$, hence by the degree of the fixed source field. This uniformity allows us to choose $\ell$ by simultaneous Kummer conditions and force enough corresponding primes to have the same rational residue characteristic.
+
+Here is the route from these ingredients to Theorem~\ref{thm:main}. Put $P=\Gal(\Q^s/\Q)$, where $\Q^s$ is the maximal prosolvable extension of $\Q$. For a number field $F$, let $G_F$ denote its absolute Galois group. It suffices to identify every continuous open homomorphism $G_F\to P$ with ordinary restriction up to conjugation. Indeed, every solvably closed field contains $\Q^s$, and this identification will give the full cyclotomic compatibility required by Hoshi's theorem.
+
+We first restrict to a normal open subgroup of $G_{\Q}$ and extend the map to the normalizer of its kernel. Write this extension as $\beta:G_k\to P$. Following the regular-module constructions used by Uchida \cite{uchida1979}, we enlarge finite quotients of $P$ by elementary abelian kernels consisting of many copies of a regular representation. Each quotient $R=\Gal(L/\Q)$ acts on the additive character space of $G_L$. Suitable characters distinguish a cyclic subgroup generated by a target Frobenius element from each of its proper subgroups. Comparing Frobenius averages proves that $\beta$ is surjective and preserves logarithmic cyclotomic characters for an infinite congruence class of auxiliary primes.
+
+Next, pass to the preimage of $P_{\Q(i)}=\Gal(\Q^s/\Q(i))$. A sign character shows that, for almost every split target prime, only finitely many source primes can correspond to it as $\ell$ varies. The uniform Kummer bound then forces a correspondence in the same residue characteristic for almost every such target prime. Finally, use rational primes split completely in suitable finite extensions. Their additive Frobenius evaluations identify the two character pullbacks, one from $\beta$ and one from ordinary restriction. A faithful representation in the character space identifies the group maps modulo every finite quotient; compactness supplies a single conjugation.
+
+Section~\ref{sec:preliminaries} states the anabelian inputs and proves the local norm identity. Section~\ref{sec:characters} develops the character tests. Section~\ref{sec:generation} obtains surjectivity and logarithmic cyclotomic compatibility. Section~\ref{sec:matching} proves the uniform bound and matches residue characteristics. Section~\ref{sec:identification} identifies the map to $P$ and proves Theorem~\ref{thm:main}.

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/matching.tex

@@ -0,0 +1,180 @@
+\section{Matching residue characteristics}\label{sec:matching}
+
+We now show that almost every prime of $\Q(i)$ above a split rational prime has an active partner of the same residue characteristic. The auxiliary prime defining activity may vary with the pair. Two finiteness statements make this possible: almost every such target prime has only finitely many possible partners, and the number of target primes missed at a fixed auxiliary prime admits a uniform bound.
+
+Keep the surjection $\beta:G_k\to P$ and the set $\mathcal L$ from Theorem~\ref{thm:normalized-map}, and put
+\[
+ B=\Q(i),\qquad G_K=\beta^{-1}(P_B),\qquad
+ \beta_B=\beta|_{G_K}:G_K\twoheadrightarrow P_B.
+\]
+Thus $K/k$ is quadratic. An \emph{active pair at $\ell$} in this section means a source prime $v$ of $K$ active at $\ell$ for $\beta_B$, together with its corresponding prime $w$ of $B$ from Theorem~\ref{thm:local-correspondence}. In particular, both primes lie away from $\ell$. For a prime $w$ of $B$, define
+\[
+ \mathcal V_w=
+ \{v:\text{$(v,w)$ is an active pair at some $\ell\in\mathcal L$}\}.
+\]
+
+\subsection{Finite fibers and a uniform bound on missed primes}
+
+\begin{lemma}\label{lem:finite-fibers}
+For all but finitely many primes $w$ of $B$ above rational primes split in $B$, the set $\mathcal V_w$ is finite.
+\end{lemma}
+\begin{proof}
+Fix $r\in\mathcal L$. Proposition~\ref{prop:signed-representation} supplies a nonzero character $\psi\in\X{r}{B}$ on which $\Gal(B/\Q)$ acts by its nontrivial character. By Proposition~\ref{prop:character-detection}(i), $\psi(\Fr_w)$ is nonzero at all but finitely many degree-one primes $w\nmid r$. Fix such a $w$ above a split rational prime $q\ne r$, and put
+\[
+ x_0=(\log\chi_r)|_{P_B},\qquad
+ z=\frac{\psi(\Fr_w)}{\log_r q}.
+\]
+The denominator is nonzero: the kernel of the logarithm on $\Z_r^\times$ consists of roots of unity, and the positive rational integer $q$ is not a root of unity. Thus $z\ne0$, and $\psi-zx_0$ vanishes on the decomposition group at $w$, since it is unramified there and vanishes on Frobenius.
+
+Consider its pullback
+\[
+ y=\beta_B^*(\psi-zx_0)\in\X{r}{K}.
+\]
+The pullback of $\psi$ transforms by sign under $G_k/G_K$. Its corestriction to $k$ is therefore zero: restricting that corestriction back to $K$ gives the sum of the two translates, and restriction is injective for additive characters with characteristic-zero coefficients. Theorem~\ref{thm:normalized-map} identifies $\beta_B^*x_0$ with $\log\chi_{K,r}$. Consequently
+\[
+ \cor_{K/\Q}(y)
+ =-z[K:\Q]\log\chi_{\Q,r}\ne0.
+\]
+For every $v\in\mathcal V_w$ away from $r$, decomposition containment gives $y(\Fr_v)=0$, regardless of which auxiliary prime makes the pair active. Proposition~\ref{prop:character-detection}(iii) permits only finitely many such $v$. There are also only finitely many primes of $K$ above $r$, proving the assertion.
+\end{proof}
+
+For $\ell\in\mathcal L$, let $\mathcal M_\ell$ be the set of primes of $B$ which occur in no active pair at $\ell$. This includes the primes above $\ell$.
+
+We seek a bound for $\#\mathcal M_\ell$ depending only on $[K:\Q]$. The proof will place independent classes attached to missed primes in one character space of an $S$-unit group. We first record the standard cohomological identification of that group, including the completion step; see also \cite{neukirch1999}.
+
+\begin{lemma}[Unramified Kummer classes]\label{lem:kummer-unramified}
+Let $F$ be a number field, let $\ell$ be a rational prime, and let $S$ be a finite set of places of $F$ containing all places above $\ell$ and all archimedean places. Then Kummer theory induces a natural isomorphism
+\begin{equation}\label{eq:unramified-kummer}
+ H^1_{\mathrm{cont}}(G_F,\Q_\ell(1))
+ _{\text{unramified outside }S}
+ \simeq \mathcal O_{F,S}^{\times}\otimes_{\Z}\Q_\ell.
+\end{equation}
+It is equivariant for every group of field automorphisms of $F$ preserving $S$, with the natural action on cohomology combining conjugation on $G_F$ and the action on the Tate twist.
+\end{lemma}
+\begin{proof}
+Integral Kummer theory gives
+\[
+ H^1_{\mathrm{cont}}(G_F,\Z_\ell(1))
+ \simeq\varprojlim_n F^\times/(F^\times)^{\ell^n}.
+\]
+The inverse-limit assertion follows from the finite-level Kummer isomorphisms; the degree-zero groups are finite and satisfy the Mittag--Leffler condition. Every continuous rational cocycle has bounded image, since $G_F$ is compact, and hence becomes integral after multiplication by one power of $\ell$. The same observation for coboundaries gives rationalization of this integral cohomology. At a prime outside $S$, inertia acts trivially on $\Z_\ell(1)$, and its Kummer evaluation is the valuation times the tame $\ell$-adic character. These integral inertia homomorphisms are torsion-free. Thus unramifiedness after rationalization is exactly the vanishing of all integral valuation coordinates outside $S$.
+
+For completeness, put $U_S=\mathcal O_{F,S}^\times$ and let
+\[
+ H_n=\{[a]\in F^\times/(F^\times)^{\ell^n}:
+       \operatorname{ord}_v(a)\equiv0\pmod{\ell^n}
+       \text{ for every }v\notin S\}.
+\]
+Writing $\Cl_S(F)$ for the $S$-ideal class group, there is an exact sequence
+\begin{equation}\label{eq:s-unit-completion}
+ 0\longrightarrow U_S/U_S^{\ell^n}
+ \longrightarrow H_n
+ \longrightarrow \Cl_S(F)[\ell^n]
+ \longrightarrow0.
+\end{equation}
+The last map sends $[a]$ to the ideal class of
+$\operatorname{div}_S(a)/\ell^n$, where $\operatorname{div}_S$ records the valuations outside $S$. The transition from level $n+1$ to level $n$ induces multiplication by $\ell$ on these obstruction classes. Since $\Cl_S(F)$ is finite, their inverse limit is zero. It follows that the kernel of all outside-$S$ valuations in the completed multiplicative group is precisely $\varprojlim_n U_S/U_S^{\ell^n}$. The argument takes place in the full completed multiplicative group; it does not assume finite valuation support for its elements. Finally $U_S$ is finitely generated, so rationalizing its completion gives $U_S\otimes\Q_\ell$. This proves \eqref{eq:unramified-kummer}. All the constructions commute with field automorphisms preserving $S$, giving the asserted equivariance.
+\end{proof}
+
+Uchida's proof of target-side finiteness associates Kummer classes to missed primes and bounds them by a unit rank \cite[Proposition~1]{uchida1981}. We refine that argument by keeping the classes in one character space of a cyclic extension. The multiplicity of this character has a bound independent of $\ell$, even when the extension degree grows.
+
+\begin{proposition}\label{prop:uniform-missed-primes}
+For every $\ell\in\mathcal L$,
+\[
+ \#\{w\in\mathcal M_\ell:w\nmid\ell\}\le 2[K:\Q],
+ \qquad
+ \#\mathcal M_\ell\le 2[K:\Q]+2.
+\]
+\end{proposition}
+\begin{proof}
+Fix $\ell\in\mathcal L$ and any finite set $J\subset\mathcal M_\ell$ of primes away from $\ell$. For each $w\in J$, choose $a_w\in B^\times$ and $h_w>0$ with
+\[
+ (a_w)=w^{h_w}.
+\]
+Its compatible Kummer cocycles define a class
+\[
+ \kappa_w\in H^1_{\mathrm{cont}}(P_B,\Q_\ell(1)).
+\]
+They factor through $P_B$ because $\Q^s$ contains the roots of unity and all the required radicals. Valuations at the distinct primes $w$ show that these classes are linearly independent. Each is unramified at every finite prime other than $w$ and the primes above $\ell$.
+
+Write
+\[
+ \rho=\chi_\ell\circ\beta_B,\qquad
+ \epsilon=\rho\chi_{K,\ell}^{-1}.
+\]
+By Theorem~\ref{thm:normalized-map}, $\epsilon$ takes values in $\mu_{\ell-1}\subset\Q_\ell^\times$. Let $K'/K$ be the cyclic extension fixed by its kernel, and let $\Delta=\Gal(K'/K)$. Pullback along the surjection $\beta_B$ is injective on $H^1$, with pulled-back coefficient module $\Q_\ell(\rho)$. Restriction to $G_{K'}$ remains injective: its kernel is a first cohomology group of the finite group $\Delta$, which vanishes in characteristic zero. We therefore obtain independent classes
+\[
+ c_w\in H^1_{\mathrm{cont}}(G_{K'},\Q_\ell(1))
+ \qquad(w\in J).
+\]
+Under the natural $\Delta$-action on this cohomology, they all lie in the $\epsilon^{-1}$ eigenspace. Indeed the original coefficient module is $\Q_\ell(1)\otimes\epsilon$, so invariance for its twisted action becomes the eigenvalue $\epsilon^{-1}$ after restriction.
+
+We claim that every $c_w$ is unramified away from $\ell$. Suppose that it is ramified at a prime $u\nmid\ell$ of $K'$, and let $v$ be the prime below it in $K$. The cyclotomic module is trivial on $I_u$, so the restricted cocycle is a nonzero continuous homomorphism from $I_u$ to the additive group $\Q_\ell(1)$. Choose a pro-$\ell$ Sylow subgroup $J_u\subset I_u$ on which it is nonzero, and a pro-$\ell$ Sylow subgroup $J_v\subset I_v$ containing $J_u$. Then $\beta_B(J_v)\ne1$, so $v$ is active at $\ell$. Theorem~\ref{thm:local-correspondence} sends $J_v$ into inertia at a corresponding prime $t\nmid\ell$ of $B$. Since $t$ occurs in an active pair, $t\notin J$, whereas $\kappa_w$ is unramified at $t$. Inertia at $t$ acts trivially on the coefficient module, so the Kummer cocycle itself vanishes on that inertia group. Its pullback therefore vanishes on $J_u$, a contradiction. This proves the claim.
+
+Let $S$ consist of the archimedean places of $K'$ and its primes above $\ell$. By Lemma~\ref{lem:kummer-unramified}, the classes $c_w$ give independent elements of the $\epsilon^{-1}$ eigenspace in $\mathcal O_{K',S}^{\times}\otimes\Q_\ell$. It remains to bound this eigenspace. The equivariant Dirichlet $S$-unit theorem \cite[Section~3]{bass1966} gives the character of the full $S$-unit module as
+\[
+ [\mathcal O_{K',S}^{\times}\otimes\Q_\ell]
+   =[\Q_\ell[S]]-[\Q_\ell].
+\]
+Each $\Delta$-orbit in $S$ contributes at most one copy of the one-dimensional character $\epsilon^{-1}$ to the permutation representation: for a stabilizer $H\subset\Delta$, its multiplicity in $\Q_\ell[\Delta/H]$ is one if it is trivial on $H$, and zero otherwise. There are
+\[
+ \#(S/\Delta)
+ =r_1(K)+r_2(K)+\#\{v\text{ of }K:v\mid\ell\}
+ \le2[K:\Q]
+\]
+such orbits. Subtracting the trivial representation cannot increase this multiplicity. Hence the independent classes $c_w$ lie in a space of dimension at most $2[K:\Q]$, and $\#J\le2[K:\Q]$. Since $J$ was arbitrary, this bounds all missed primes away from $\ell$. There are at most two primes of $B$ above $\ell$, giving the second bound.
+\end{proof}
+
+The bound depends on $K$, but neither on $\ell$ nor on the degree of the cyclic extension $K'/K$. We can therefore choose one auxiliary prime that excludes every possible partner of several specified target primes, and contradict this bound. The following elementary Kummer construction provides that choice.
+
+\subsection{Simultaneous separation by power residues}
+
+\begin{lemma}\label{lem:power-separation}
+Let $m,d\ge1$, let $q_1,\ldots,q_n$ be distinct rational primes, and for each $j$ let $\mathcal P_j$ be a finite set of rational primes not containing $q_j$. Given any finite set of rational primes to avoid, there are infinitely many odd primes $\ell\equiv1\pmod m$ outside that set and all the displayed prime sets such that
+\[
+ p^f\bmod\ell\notin\langle q_j\bmod\ell\rangle
+ \quad
+ (1\le j\le n,\ p\in\mathcal P_j,\ 1\le f\le d).
+\]
+The sets $\mathcal P_j$ may overlap, and may contain $q_i$ for $i\ne j$.
+\end{lemma}
+\begin{proof}
+Choose distinct auxiliary primes $s_j$ larger than $m$, $d$, and every prime in the lists, and put
+\[
+ C=\Q(\zeta_{m\prod_j s_j}).
+\]
+For each $j$, the classes of the distinct rational primes in $\{q_j\}\cup\mathcal P_j$ are independent in $C^\times/(C^\times)^{s_j}$. Indeed, at any prime in those lists, the ramification index in $C/\Q$ divides $[\Q(\zeta_m):\Q]$: the other cyclotomic conductors are prime to that prime. This index is prime to $s_j$, and valuations at the separate rational primes give the asserted independence.
+
+Kummer theory therefore identifies the Galois group over $C$ of
+\[
+ E_j=C\bigl(q_j^{1/s_j},\ p^{1/s_j}\ (p\in\mathcal P_j)\bigr)
+\]
+with an elementary abelian $s_j$-group having one independent coordinate for each listed prime. Choose $\sigma_j\in\Gal(E_j/C)$ which fixes $q_j^{1/s_j}$ and acts nontrivially on every $p^{1/s_j}$ with $p\in\mathcal P_j$. The extensions $E_j/C$ are linearly disjoint, since their degrees are powers of distinct primes. Thus these automorphisms combine to an element $\sigma\in\Gal(E/C)$, where $E$ is their compositum.
+
+The extension $E/\Q$ is Galois. Moreover, conjugation in $\Gal(E/\Q)$ preserves whether each of the specified radical coordinates of $\sigma$ is trivial: a conjugation only raises its root-of-unity multiplier to a power prime to the corresponding $s_j$. Chebotarev's theorem supplies infinitely many rational primes $\ell$ with Frobenius in the conjugacy class of $\sigma$, avoiding the prescribed finite set, all listed primes, and $2$. They split completely in $C$, so $\ell\equiv1\pmod m$ and $\ell\equiv1\pmod{s_j}$. At these primes, $q_j$ is an $s_j$th power modulo $\ell$ and every $p\in\mathcal P_j$ is not.
+
+The quotient of $\F_\ell^\times$ by its $s_j$th powers has prime order $s_j$. A nontrivial class stays nontrivial on raising it to any exponent $1\le f\le d<s_j$. Every power of $q_j$, however, is an $s_j$th power. This proves the displayed separation. In particular, overlaps between different lists impose no conflict: their conditions belong to linearly disjoint Kummer extensions with different prime exponents.
+\end{proof}
+
+\begin{proposition}\label{prop:same-characteristic}
+For all but finitely many primes $w$ of $B$ above rational primes $q$ split in $B$, there exist a prime $v$ of $K$ above the same $q$ and an auxiliary prime $\ell\in\mathcal L$ for which $(v,w)$ is an active pair for $\beta_B$.
+\end{proposition}
+\begin{proof}
+Suppose that there are infinitely many exceptions. Lemma~\ref{lem:finite-fibers} allows us to choose
+\[
+ w_1,\ldots,w_n,\qquad n>2[K:\Q]+2,
+\]
+among them, above distinct split rational primes $q_1,\ldots,q_n$, with every $\mathcal V_{w_j}$ finite. Let $\mathcal P_j$ be the set of rational residue characteristics of the primes in $\mathcal V_{w_j}$. The failure of the asserted conclusion means $q_j\notin\mathcal P_j$.
+
+Apply Lemma~\ref{lem:power-separation} with $m=m_0$ and $d=[K:\Q]$. It gives $\ell\in\mathcal L$, away from all the primes involved, such that
+\begin{equation}\label{eq:norm-separation}
+ Nv\bmod\ell\notin\langle q_j\bmod\ell\rangle
+ \qquad(v\in\mathcal V_{w_j},\ 1\le j\le n),
+\end{equation}
+because $Nv=p^f$ with $p\in\mathcal P_j$ and $1\le f\le[K:\Q]$. If $w_j$ occurred in an active pair at this $\ell$, its source prime would lie in $\mathcal V_{w_j}$. Applying Lemma~\ref{lem:norm-identity} to a lift of source Frobenius and reducing modulo $\ell$ would give
+\[
+ Nv\bmod\ell\in\langle Nw_j\bmod\ell\rangle
+ =\langle q_j\bmod\ell\rangle,
+\]
+contrary to \eqref{eq:norm-separation}. All $n$ primes $w_j$ therefore belong to $\mathcal M_\ell$, contradicting Proposition~\ref{prop:uniform-missed-primes}.
+\end{proof}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/build/sections/preliminaries.tex

@@ -0,0 +1,92 @@
+\section{Galois groups and partial local correspondences}\label{sec:preliminaries}
+
+We collect the anabelian inputs and the local calculation that will connect Frobenius elements with cyclotomic characters. The substantial inputs from Uchida and Hoshi are stated explicitly; the remaining arguments use standard global and local class field theory, Kummer theory, and Chebotarev's theorem.
+
+\subsection{Conventions and solvably closed fields}
+Fix an algebraic closure $\overline{\Q}$ of $\Q$. For a number field $M\subset\overline{\Q}$ put $G_M=\Gal(\overline{\Q}/M)$, and put
+\[
+ \mathcal G=G_{\Q},\qquad
+ P=\Gal(\Q^s/\Q),
+\]
+where $\Q^s\subset\overline{\Q}$ is the maximal prosolvable extension of $\Q$. For a number field $M\subset\Q^s$ write $P_M=\Gal(\Q^s/M)$. All homomorphisms between profinite groups and all characters in this paper are continuous. An open homomorphism is a homomorphism with open image: a continuous surjection between profinite groups is a quotient map and is open.
+
+We use finite primes unless infinity is explicitly mentioned. For a finite prime $v$, its absolute norm is $Nv$, and its absolute residue degree is $[\kappa(v):\F_p]$, where $p$ is its rational residue characteristic. Frobenius is arithmetic. A chosen prolongation of $v$ determines a decomposition group $D_v$ and inertia group $I_v$; a different prolongation conjugates both. Expressions involving several such groups always use specified compatible prolongations.
+
+\begin{lemma}\label{lem:solvable-closure}
+Every solvably closed algebraic extension $E$ of $\Q$ contains $\Q^s$. The field $\Q^s$ is solvably closed. For every number field $M\subset\Q^s$, the extension $\Q^s/M$ is the maximal prosolvable extension of $M$.
+\end{lemma}
+\begin{proof}
+Since $E(\mu_n)/E$ is abelian, $E$ contains every root of unity. Adjoining an $n$th root of an element of $E$ then gives a cyclic extension, so $E$ is closed under radicals. Finite solvable Galois extensions can be built by towers of radical extensions after adjoining roots of unity; hence $\Q^s\subset E$.
+
+The same tower argument shows that $\Q^s$ contains all roots of unity and is closed under radicals. Any nontrivial finite abelian extension would have a cyclic subextension of prime degree, which Kummer theory excludes. Any nontrivial abelian algebraic extension would have a nontrivial finite abelian subextension. Thus $\Q^s$ is solvably closed.
+
+Finally, a finite solvable Galois extension of $M$ lies in a finite solvable Galois extension of $\Q$: first take a solvable normal closure of $M/\Q$ inside $\Q^s$, and then the compositum of the finitely many conjugate extensions. The group of this compositum over that normal closure embeds in a product of solvable groups. This proves the last assertion.
+\end{proof}
+
+In particular, every additive $\ell$-adic character of $G_M$ factors through $P_M$ when $M\subset\Q^s$. For any solvably closed Galois extension $E/F$, the action on roots of unity defines its cyclotomic character
+\[
+ \chi_{E/F}:\Gal(E/F)\longrightarrow\widehat{\Z}^{\times}.
+\]
+We use $\chi_{M,\ell}$ for the usual $\ell$-adic character of $G_M$, and $\chi_\ell$ for that of $P$. The $\ell$-adic logarithm is normalized to vanish on the finite torsion subgroup of $\Z_\ell^\times$.
+
+\subsection{The anabelian inputs}
+The first input permits a homomorphism to be extended from an open subgroup to the normalizer of its kernel.
+
+\begin{theorem}[Uchida]\label{thm:uchida-isomorphism}
+Let $E_i/F_i$ be solvably closed Galois extensions of number fields. Every isomorphism
+$\theta:\Gal(E_1/F_1)\xrightarrow{\sim}\Gal(E_2/F_2)$
+is induced by a unique field isomorphism $j:E_2\xrightarrow{\sim}E_1$, with
+$g j=j\theta(g)$ for all $g$.
+\end{theorem}
+This is \cite[Theorem]{uchida1979}. One consequence we shall use twice is the following.
+\begin{lemma}\label{lem:centralizer}
+The centralizer in $P$ of any open subgroup of $P$ is trivial.
+\end{lemma}
+\begin{proof}
+An open subgroup is $P_M$ for a number field $M\subset\Q^s$. An element centralizing $P_M$ is a field automorphism of $\Q^s$ implementing the identity automorphism of $P_M$. The identity field map also implements it, so uniqueness in Theorem~\ref{thm:uchida-isomorphism} gives the assertion.
+\end{proof}
+
+Here is the local input, in the precise setting in which we need it. For a homomorphism $\varphi:G_M\to P_N$, an odd prime $\ell$, and a prime $v\nmid\ell$ of $M$, say that $v$ is \emph{active at $\ell$} if $\varphi$ is nontrivial on a pro-$\ell$ Sylow subgroup of $I_v$. The condition is independent of the chosen Sylow subgroup and prolongation, since such choices are conjugate.
+
+\begin{theorem}[Uchida's partial local correspondence]\label{thm:local-correspondence}
+Let $M$ be a number field, let $N\subset\Q^s$ be a number field, and let $\varphi:G_M\to P_N$ be open. Fix an odd prime $\ell$. Every prime $v\nmid\ell$ active at $\ell$ determines a unique prime $w\nmid\ell$ of $N$ for which, with suitable prolongations,
+\[
+ \varphi(D_v)\subset D_w.
+\]
+Moreover, $\varphi$ is injective on a pro-$\ell$ Sylow subgroup of $I_v$, and sends this subgroup into $I_w$. All but finitely many primes of $N$ arise from such active primes of $M$.
+\end{theorem}
+This is the odd-prime case of the trichotomy in \cite[\S1, pp.~595--596]{uchida1981}, together with its Proposition~1. In that trichotomy, nontrivial pro-$\ell$ inertia is case~(ii); case~(i) kills that inertia, and the torsion case~(iii) requires $\ell=2$. The open-image hypothesis, rather than surjectivity onto $P_N$, suffices. We will apply the theorem after restricting to smaller open subgroups as well. The cofinite target coverage in this theorem is for a fixed $\ell$; its exceptional set is not asserted to be uniform in $\ell$.
+
+The uniqueness assertion concerns primes of the finite target field $N$. When restricting a map to smaller open subgroups, we will intersect a containing target decomposition group with $P_{N'}$ for a finite extension $N'/N$ inside $\Q^s$, and then use uniqueness at the $N'$-level. Proposition~\ref{prop:frobenius-generation} carries out this comparison with explicit source and target fields; no uniqueness of infinite prolongations is required.
+
+The final input converts cyclotomic compatibility into the required field embedding.
+\begin{theorem}[Hoshi]\label{thm:hoshi}
+Let $E_i/F_i$ be solvably closed Galois extensions of number fields, and let
+$\alpha:\Gal(E_1/F_1)\to\Gal(E_2/F_2)$ be open. There exists an equivariant field embedding $E_2\hookrightarrow E_1$ if and only if
+\[
+ \chi_{E_2/F_2}\circ\alpha=\chi_{E_1/F_1}
+ \quad\text{in }\widehat{\Z}^{\times}.
+\]
+\end{theorem}
+We use \cite[Theorem~3.4]{hoshi2025}. Uniqueness of the embedding, when it exists, is \cite[Proposition~2]{uchida1981}.
+
+\subsection{Tame inertia and norms}
+Decomposition groups in $P_M$ are the full absolute Galois groups of the corresponding local fields. Here is a justification for using the usual local ramification structure. Fix a prime of $\overline{\Q}$ above a prime of $M$, and form inside a local algebraic closure the union of the completions of finite fields between $M$ and $\Q^s$. This union contains all roots of unity and is closed under radicals: for a fixed exponent, a nonzero local element can be approximated by a global element in the field defining that completion, and their ratio can be made a power. Thus the union has no cyclic extension. Every finite Galois group over a nonarchimedean local field is solvable, by its wild inertia, tame inertia, and unramified filtration. Every closed subgroup of its prosolvable absolute Galois group is prosolvable as well; a nontrivial such group has a nontrivial finite cyclic quotient after passing through a finite solvable quotient. The union is therefore the full local algebraic closure. The usual decomposition-group identification proves the claim.
+
+For an active pair $(v,w)$ from Theorem~\ref{thm:local-correspondence}, and $g\in D_v$, let $b(g)\in\widehat{\Z}$ be its source residue exponent and let $a(g)\in\widehat{\Z}$ be the target residue exponent of $\varphi(g)$. These are measured using $Nv$- and $Nw$-Frobenius, respectively. The target residue exponent is defined on $D_v$ by composition; we do not assume that the map sends all source inertia into target inertia.
+
+\begin{lemma}\label{lem:norm-identity}
+For an active pair $(v,w)$ at an odd prime $\ell$,
+\begin{equation}\label{eq:norm-identity}
+ (Nv)^{b(g)}=(Nw)^{a(g)}\quad\text{in }\Z_\ell^\times
+ \qquad(g\in D_v).
+\end{equation}
+Powers by profinite integers are interpreted by continuous powering in the compact group $\Z_\ell^\times$.
+\end{lemma}
+\begin{proof}
+Let $p\ne\ell$ be the residue characteristic of $v$, and choose a topological generator $\tau$ of a pro-$\ell$ Sylow subgroup of $I_v$. Its image in source tame inertia generates the pro-$\ell$ factor. The tame conjugation relation shows that
+\[
+ e=g\tau g^{-1}\tau^{-(Nv)^{b(g)}}
+\]
+belongs to source wild inertia, a pro-$p$ group. In the target tame quotient, the image of $e$ also lies in the pro-$\ell$ factor of tame inertia: both $\varphi(\tau)$ and its conjugate do. Its image is therefore trivial, because it belongs to a pro-$p$ subgroup and to a pro-$\ell$ group. The nontrivial image of $\varphi(\tau)$ in the torsion-free target pro-$\ell$ tame factor then gives \eqref{eq:norm-identity} by the target tame conjugation relation. The target residue characteristic is different from $\ell$ by Theorem~\ref{thm:local-correspondence}.
+\end{proof}

preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf

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preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/README.md

@@ -0,0 +1,16 @@
+# [Optimal-order mixing of the Thorp shuffle](paper.pdf)
+
+OpenAI · September 26, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026,
+  author = {{OpenAI}},
+  title = {{Optimal-order mixing of the Thorp shuffle}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf}{OAI:Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026}},
+  year = {2026}
+}
+```
+

preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/conditional-flow.tex

@@ -0,0 +1,39 @@
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[>=Stealth, every node/.style={font=\small},
+    free/.style={circle,draw=linkblue,fill=white,minimum size=6mm},
+    seen/.style={circle,fill=black!65,minimum size=3mm,inner sep=0pt}]
+  \node at (2,2.6) {Two free positions};
+  \node[free] (a) at (0,1.7) {};
+  \node[free] (b) at (0,0.3) {};
+  \node[left=1mm of a] {$u$};
+  \node[left=1mm of b] {$v$};
+  \node[free] (c) at (4,1.7) {};
+  \node[free] (d) at (4,0.3) {};
+  \node[right=1mm of c] {$(u+v)/2$};
+  \node[right=1mm of d] {$(u+v)/2$};
+  \draw[->,linkblue] (a)--(c);
+  \draw[->,linkblue] (a)--(d);
+  \draw[->,linkblue] (b)--(c);
+  \draw[->,linkblue] (b)--(d);
+  \node at (2,-0.35) {Unrevealed fair switch};
+  \begin{scope}[xshift=7.3cm]
+    \node at (1.5,2.6) {One free position};
+    \node[free] (e) at (0,1.7) {};
+    \node[seen] (f) at (0,0.3) {};
+    \node[left=1mm of e] {$u$};
+    \node[seen] (g) at (3,1.7) {};
+    \node[free] (h) at (3,0.3) {};
+    \node[right=1mm of h] {$u$};
+    \draw[->,linkblue,thick] (e)--(h);
+    \draw[->,black!65,dashed] (f)--(g);
+    \node at (1.5,-0.35) {Switch fixed by the observed path};
+  \end{scope}
+\end{tikzpicture}
+\caption{The conditional flow on one matching edge. Open circles are free
+positions; filled circles contain an exposed card. A switch with two free
+positions averages the centered masses $u,v$. With one free position,
+the observed path fixes the switch and transports the centered mass $u$.
+The latter case produces the martingale noise before the switch is revealed.}
+\label{fig:conditional-flow}
+\end{figure}

preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/main.tex

@@ -0,0 +1,601 @@
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage{xcolor}
+\definecolor{linkblue}{RGB}{20,69,103}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
+\hypersetup{pdftitle={Optimal-order mixing of the Thorp shuffle},pdfauthor={OpenAI},bookmarksopen=true,bookmarksnumbered=true}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\title{Optimal-order mixing of the Thorp shuffle}
+\author{OpenAI}
+\date{September 26, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that the Thorp shuffle on $2^d$ cards mixes in $\Theta(d)$ complete shuffles. The full permutation law after $1600d$ shuffles converges to uniform in total variation as $d\to\infty$, uniformly over the initial deck, while a support count gives a lower bound of $2d-O(1)$.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+A single card can become uniformly distributed long before a shuffled
+deck is close to uniform. The Thorp shuffle makes this distinction
+particularly clear. Split a deck into two equal halves, pair cards that
+occupy corresponding positions, and choose the order of each pair by an
+independent fair coin. Interleaving the pairs produces the new deck.
+For a deck of size $n=2^d$, every individual card is uniform after $d$
+shuffles. The question is how much longer it takes to randomize the
+joint order of all $n$ cards.
+
+We prove that a constant multiple of $d$ shuffles suffices. Let $q_d$
+denote the law of one complete shuffle as a permutation of the
+positions, and let $U_n$ be uniform on the symmetric group $S_n$.
+Convolution is the law of composition of independent permutations, and
+\[
+ \|\mu-\nu\|_{\mathrm{TV}}
+ =\frac12\sum_g|\mu(g)-\nu(g)|.
+\]
+Write $t_{\mathrm{mix}}(d)$ for the least nonnegative integer $t$ at which
+$\|q_d^{*t}-U_n\|_{\mathrm{TV}}\le1/4$.
+
+\begin{theorem}[Optimal-order mixing]\label{thm:mixing}
+As $d\to\infty$,
+\[
+ \|q_d^{*(1600d)}-U_n\|_{\mathrm{TV}}\longrightarrow0.
+\]
+For every integer $t\ge0$,
+\[
+ \|q_d^{*t}-U_n\|_{\mathrm{TV}}
+ \ge 1-\frac{2^{tn/2}}{n!}.
+\]
+Consequently the mixing time at distance $1/4$ satisfies
+\[
+ t_{\mathrm{mix}}(d)\ge
+ \left\lceil\frac2n\log_2\frac{3n!}{4}\right\rceil
+ =2d-O(1),
+ \qquad t_{\mathrm{mix}}(d)=\Theta(d).
+\]
+\end{theorem}
+
+One step in this theorem is one complete physical shuffle. Changing the
+initial deck translates the permutation law, so its distance from
+uniform is unchanged. The upper bound is asymptotic in $d$; the
+constant $1600$ is an explicit choice in the proof, not a proposed
+sharp constant. Determining a leading constant or a cutoff remains a
+separate question.
+
+The full-law conclusion controls every statistic of the final deck and,
+in particular, the joint positions of any deterministic selection of
+cards, by contraction of total variation under a map. It also shows
+that this shuffle produces an approximately uniform permutation using
+$O(n\log n)$ independent fair bits: each of the $O(\log n)$ steps uses
+$n/2$ bits. The support bound identifies the same order of randomness
+as necessary for this construction.
+
+\subsection{History and related work}
+
+Thorp introduced the model in his 1973 study of the effects of
+nonrandom shuffling on Faro \cite[Section~3.1]{thorp}. Its simple
+pairwise rule conceals a difficult question about dependence across the
+whole deck. For dyadic decks, Morris obtained the first bound
+polynomial in $d$, namely $O(d^{44})$, using evolving sets and a
+chameleon process that tracks conditional card probabilities
+\cite{morris44}. Montenegro and Tetali sharpened this analysis to
+$O(d^{29})$ by spectral-profile and evolving-set methods
+\cite[Theorems~6.14--6.15]{montenegro-tetali2006}.
+
+A second line of attack used relative entropy. Morris related entropy
+decay to pairwise card collisions and obtained $O((\log n)^4)$
+shuffles for every even $n$ \cite{morris4}. He then proved $O(d^3)$
+for $n=2^d$ by a stronger contraction over a block of $d$ shuffles
+\cite{morris3}. Theorem~\ref{thm:mixing} attains the logarithmic order
+discussed in Jonasson's notes \cite[Section~5]{jonasson} for dyadic
+decks. The corresponding optimal-order question for arbitrary even
+deck sizes is not addressed by the cube argument here.
+
+Card marginals have also been studied for their own sake and for
+cryptography. Morris, Rogaway and Stegers analyzed the joint images of
+ordered input lists in their construction of enciphering schemes on
+small domains \cite[Theorem~1]{mrs}. Czumaj and V\"ocking proved that
+$O((\log n)^2)$ Thorp shuffles mix any fixed fraction of the cards
+strictly below one, using non-Markovian couplings and butterfly-network
+estimates \cite{CzumajVocking2014}. Our first stage needs a different,
+averaged guarantee: after $200d$ shuffles, the joint positions of a
+uniformly sampled list of $7n/8$ labels are close to uniform on average
+over the list. The passage from these averaged marginals to the full
+law is a separate part of the proof.
+
+The conditional-flow method has several predecessors. In the original
+Thorp analysis, Morris used a chameleon process to represent the law of a tagged card
+conditional on the set-valued trajectory of a collection containing
+that card; the conditional probabilities average across available pairs
+and are transported across pairs with only one available position
+\cite[Section~4, Lemma~3]{morris44}. Hoang, Morris and Rogaway later
+used conditional squared energy and a sequential total-variation
+comparison to analyze the swap-or-not shuffle
+\cite[Section~3, Theorem~3 and Lemma~4]{hmr2014}. Their uniformly random
+keys make the next partner uniform over the underlying group.
+Here the directions are prescribed cyclically. The exact identity
+retaining only one sweep of noise, together with independence of the
+two opposite halves during the intervening updates, provides the needed
+contraction.
+
+For the final passage to the group, we use complete reducibility and
+Schur orthogonality, together with the tableau description and
+hook-length formula for symmetric-group degrees
+\cite{etingof,sagan}. The inverse-degree summability needed here is a
+special case of a stronger theorem of Liebeck and Shalev
+\cite[Theorem~2.6]{liebeck-shalev2004}; we give an elementary proof of
+the required case. The final total-variation comparison is the
+finite-group Fourier method of Diaconis and Shahshahani
+\cite[Section~3, Lemma~14]{DiaconisShahshahani1981}.
+The additional step here is to turn bounds on complementary card
+marginals into operator bounds for matrix Fourier transforms, using a
+common trimming and the span of one vector's orbit under a block
+subgroup.
+
+\subsection{Proof overview}
+
+Label positions by binary strings of length $d$, the vertices of a
+cube. Undoing a deterministic cyclic rotation turns successive shuffles
+into switch layers along successive coordinate directions: the two
+positions in one switch differ only in the active bit. A sweep through
+all $d$ directions takes $d$ physical shuffles.
+Section~\ref{sec:model} gives this change of frame and proves the
+support lower bound.
+
+For the upper bound, first choose an ordered list of $7n/8$ labels
+uniformly and independently of the shuffle. Expose these cards one at
+a time. Conditional on the paths already exposed, the next card has a
+probability distribution on the remaining positions. On a switch with
+two remaining positions its masses are averaged; on a switch with
+only one remaining position its mass follows the route fixed by the
+exposed path. Subtracting the uniform distribution on the remaining
+positions gives a centered flow.
+
+A full sweep of coordinate averages annihilates every mean-zero
+vector. The actual conditional update differs from coordinate averaging
+only by noise with conditional mean zero on switches with one available
+position.
+Consequently the expected squared norm at the present time is an exact
+weighted sum of noise contributions from the preceding sweep, as
+proved in Section~\ref{sec:flow}. To control those contributions,
+Section~\ref{sec:contraction} looks back from the next active direction
+through the $d-1$ other directions. These updates act independently
+inside the two opposite halves. A flow value in one half is therefore
+independent of the availability of its partner in the other half,
+conditional on the history at the start of this interval. A random list
+leaves a positive fraction of available positions in both halves with
+high probability. This yields a scalar energy recurrence and then the
+averaged marginal estimate of Proposition~\ref{prop:averaged-marginal}
+at time $200d$.
+
+The second stage recovers the dependencies involving the omitted
+cards. Partition the labels into eight equal blocks, choosing the
+partition so that the eight complementary marginals have small total
+error. Each marginal is a distribution on cosets of the subgroup that
+permutes its omitted block. Removing a small amount of probability
+makes all eight coset densities bounded under one common law.
+Restrict an irreducible representation of $S_n$ to the product of these
+eight subgroups. Each tensor constituent has at least one factor of
+small degree. Even with arbitrary multiplicity, the orbit of a single
+vector under the corresponding block subgroup spans a space of
+controlled dimension.
+Averaging projections onto its translates then bounds each Fourier
+operator. Sections~\ref{sec:degrees} and~\ref{sec:lift} supply the degree
+estimate and this transfer; Section~\ref{sec:completion} combines eight
+independent time intervals of $200d$ shuffles and handles parity to complete the proof.
+
+The companion on random coordinate frames gives an independent
+$32800d$ proof \cite[Theorem~1.1]{companion-frames}. The
+conditional-information companion obtains bounds uniform over specified
+input lists \cite[Theorem~1.1]{companion-conditional}, while the Fourier
+companion develops general transfers from complementary coset bounds
+\cite[Theorem~1.1]{companion-transfer}. The present proof uses none of
+these results: its cyclic-coordinate estimate and eight-block transfer
+are proved here.
+
+\section{The model and the support obstruction}\label{sec:model}
+
+Let $d\ge1$ be an integer. Identify the positions with \(V=\mathbb F_2^d\), write $e_1,\ldots,e_d$ for its standard basis, and label each card by its initial position. A permutation maps a label to its current position; composition means \((gh)(x)=g(h(x))\). With
+\[
+ R(x_1,\ldots,x_d)=(x_2,\ldots,x_d,x_1),
+\]
+one physical shuffle is \(RS\), where \(S\) independently fixes or swaps each pair \(\{x,x+e_1\}\). Thus its action is
+\[
+ (x_1,u)\longmapsto (u,x_1+\xi_u),
+\]
+with independent fair bits \(\xi_u\).
+
+Let $S_1,S_2,\ldots$ be independent copies of $S$. If \(Y_t=RS_tY_{t-1}\) is the physical permutation, started from $Y_0=\mathrm{id}$, and \(X_t=R^{-t}Y_t\), then
+\[
+ X_{t+1}=Q_tX_t,\qquad
+ Q_t=R^{-t}S_{t+1}R^t.
+\]
+The \(Q_t\) are independent matching switches in the directions
+\(i_t=1+(t\bmod d)\), since $R^{-t}e_1=e_{i_t}$. A block whose length is a multiple of \(d\) ends in the physical frame. We analyze this cyclic-coordinate process, writing \(\Pi_t=X_t\).
+
+During one full sweep each coordinate of a given card is refreshed
+exactly once by a coin fair conditional on the card's earlier path.
+These refreshed bits form a uniform binary string. Restoring the
+deterministic rotation preserves uniformity, proving the one-card
+assertion in the introduction.
+
+Uniform measure is invariant under every position permutation. Starting from any fixed deck translates the law started at the identity and leaves its total-variation distance from uniform unchanged. Moreover, \(t\) shuffles have at most \(2^{tn/2}\) possible coin strings. Their support \(A\) therefore gives
+\[
+ \|q_d^{*t}-U_n\|_{\mathrm{TV}}
+ \ge q_d^{*t}(A)-U_n(A)
+ \ge1-\frac{2^{tn/2}}{n!}.
+\]
+Distance at most \(1/4\) forces \(2^{tn/2}\ge3n!/4\), proving the rounded lower bound. The inequalities \((n/e)^n\le n!\le n^n\) give its asymptotic form.
+
+For completeness, mixing exists in each fixed dimension. A full coordinate sweep has positive probability of being the identity, and of making any specified single cube-edge transposition. Edge transpositions on the connected cube generate \(S_n\). Every permutation can therefore be obtained in some number of sweeps; identity sweeps pad these representations to one common length. The kernel at that length assigns positive mass to every group element and hence dominates a positive multiple of \(U_n\). Iteration proves convergence. This finite-dimensional observation does not assert the numerical bound \(1600d\) for every \(d\).
+
+\section{A conditional flow with one sweep of memory}\label{sec:flow}
+
+Choose an ordered list \((c_1,\ldots,c_k)\) of distinct labels independently of the shuffle. We will ultimately average over uniform lists, but first fix the list. For $0\le\ell<k\le n$, expose the trajectories of its first \(\ell\) cards, and call the next card the tagged card. Let $\mathcal F_s$ be the sigma-field generated by the list and the exposed paths through time $s$. Let \(B_s\) be the positions not occupied by those \(\ell\) cards, so \(m=|B_s|=n-\ell\) is constant. Define
+\[
+ p_s(x)=\mathbb P\{\Pi_s(c_{\ell+1})=x\mid\mathcal F_s\},
+ \qquad
+ w_s(x)=p_s(x)-\frac1m\mathbf1_{B_s}(x).
+\]
+The vector \(w_s\) is supported on \(B_s\), has total sum zero, and has squared norm at most one. Averaging and transport also underlie Morris's chameleon representation \cite[Section~4, Lemma~3]{morris44}. The next lemma identifies the unexposed coins directly for labeled trajectories.
+
+\begin{samepage}
+\begin{lemma}[Path cylinders and conditional flow]\label{lem:path-cylinder}
+For every feasible specification of the exposed paths through a time $T$,
+the visited time--edge coins are fixed and all other coins remain
+independent and fair. On a matching edge with two free positions the
+conditional masses are averaged; on an edge with one free position its
+mass follows the known free route. These rules also transport the uniform
+mass on $B_s$ to that on $B_{s+1}$. For $s\le T$, conditioning on the
+full exposed paths gives the same tagged-card law at time $s$ as
+conditioning on $\mathcal F_s$.
+\end{lemma}
+\end{samepage}
+\begin{proof}
+To verify the cylinder assertion, fix any feasible specification of the exposed paths through a terminal time \(T\). The paths determine the visited time-edge pairs and the coin value at every visited pair. Conversely, prescribing these values forces the specified paths, by induction on time: the next position of an exposed card depends only on the coin of its present edge. If two exposed cards share an edge, feasibility gives the same prescribed coin. The path event is therefore a cylinder in the independent coin array. Conditional on it, unvisited coins remain independent and fair.
+
+It follows that an edge with two free positions averages their tagged-card masses. An edge with one free position transports its mass to the uniquely free output specified by the exposed card's next position. These updates also carry the uniform vector on \(B_s\) to the uniform vector on \(B_{s+1}\). Future prescribed coins involve later time coordinates of the array; they do not alter the earlier free coins. Consequently the law at time \(s\), even if initially defined by conditioning on all exposed paths through \(T\), equals this \(\mathcal F_s\)-measurable forward flow. In particular we may use its adaptedness below. Averaging and transport do not increase squared norm. The initial centered point mass has squared norm $1-1/m\le1$, which proves the stated bound on \(w_s\). Figure~\ref{fig:conditional-flow} depicts the two updates.
+\end{proof}
+
+\input{conditional-flow}
+
+Let \(A_i\) average a function across every coordinate-\(i\) pair, and write \(x^{(i)}=x+e_i\). Put
+\[
+ E_s=\mathbb E\|w_s\|_2^2,
+ \qquad
+ L_s=\sum_x w_s(x)^2\mathbf1_{\{x^{(i_s)}\notin B_s\}},
+ \qquad b_s=\mathbb E L_s.
+\]
+Here the expectation includes the initial list when it is random. Before the next exposed positions are revealed, the transport choices on edges with one free position are independent fair coins. Hence
+\begin{equation}\label{eq:h-1}
+ w_{s+1}=A_{i_s}w_s+\xi_s,
+ \qquad \mathbb E(\xi_s\mid\mathcal F_s)=0.
+\end{equation}
+On a one-free-position edge whose free endpoint is \(x\), its noise is
+\[
+ \pm\frac{w_s(x)}2(\delta_x-\delta_{x^{(i_s)}}),
+\]
+where $\delta_x$ is unit mass at $x$, and the signs are independent on distinct such edges. Other edges contribute no noise.
+
+\begin{lemma}[One-sweep energy identity]\label{lem:energy-memory}
+For every \(t\ge d\),
+\begin{equation}\label{eq:h-2}
+ E_t=\sum_{j=1}^d2^{-j}b_{t-j}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+Expand \eqref{eq:h-1} over the last \(d\) layers. The product of all \(d\) coordinate averages maps a vector to its spatial mean, so it annihilates \(w_{t-d}\). Noise terms from distinct times have zero expected cross products: the later term has conditional mean zero and the intervening averages are deterministic. At a fixed time, noises on distinct edges also have zero expected cross products, because their signs are independent conditional on the exposed past. The noise created at time \(t-j\) is followed by averaging in \(j-1\) distinct coordinates other than its own. Its two masses spread over disjoint subcubes, each of size \(2^{j-1}\). The resulting squared norm is \(2^{-j}w_{t-j}(x)^2\). Summing these contributions proves \eqref{eq:h-2}.
+\end{proof}
+
+\section{Why the recent noise contracts}\label{sec:contraction}
+
+We bound the recent-noise term \(b_s\) in terms of the energy \(E_s\), then use the resulting decay to control ordered card marginals.
+
+Fix \(s\ge d-1\), put \(a=s-d+1\), and let \(H_0,H_1\) be the two halves defined by coordinate \(i_s\). The updates from \(a\) to \(s\) use every other coordinate once and do not cross these halves. Conditional on \(\mathcal F_a\), the initial free sets \(B_a\cap H_h\) and centered flows \(w_a|_{H_h}\) are fixed, and the future switch arrays in the two halves are independent. The pair \((B_s\cap H_h,w_s|_{H_h})\) is determined by this initial state and the switch array inside \(H_h\).
+
+Every position at time \(a\) in a half is marginally uniform in that half after the intervening \(d-1\) updates: each of its remaining coordinates is refreshed by a conditionally fair coin. Summing this endpoint probability over the free positions at time \(a\) gives, for every \(y\in H_{1-h}\),
+\[
+ \mathbb P\{y\in B_s\mid\mathcal F_a\}
+ =\frac{|B_a\cap H_{1-h}|}{n/2}.
+\]
+For \(x\in H_h\), the random variables \(w_s(x)^2\) and \(\mathbf1_{\{x^{(i_s)}\in B_s\}}\) depend on opposite future arrays. Their conditional independence therefore yields
+\begin{equation}\label{eq:h-3}
+ \mathbb E\left[\sum_{x\in H_h}w_s(x)^2
+       \mathbf1_{\{x^{(i_s)}\in B_s\}}\,\middle|\,\mathcal F_a\right]
+ =\frac{|B_a\cap H_{1-h}|}{n/2}
+   \mathbb E\left[\sum_{x\in H_h}w_s(x)^2
+                        \,\middle|\,\mathcal F_a\right].
+\end{equation}
+The values of the flow and the free set within one half need not be independent.
+
+Now take a uniform random initial list, independent of the shuffle, and suppose \(m\ge n/8\). For every fixed realization of the shuffle, the image of the uniformly chosen free set is uniform among the \(m\)-subsets of \(V\). Thus \(B_a\) is unconditionally a uniform \(m\)-subset. Its count in either half has mean \(m/2\), and
+\begin{equation}\label{eq:h-4}
+ \mathbb P\{|B_a\cap H_h|<m/4\}\le e^{-m/32}.
+\end{equation}
+For completeness we give the bounded-difference exponential-moment argument of Hoeffding and Azuma \cite{hoeffding1963,azuma1967} in this setting. Expose the uniform subset in random order and use the Doob martingale for its final half-count. Changing the next draw can be coupled with the remaining completion by exchanging the two possible drawn points; the increment has absolute value at most one. A centered random variable in an interval of length two has exponential moment at most \(e^{\theta^2/2}\): for \(\psi(\theta)=\log\mathbb E e^{\theta Z}\), the second derivative is a tilted variance at most one, and integration from zero proves the bound. Conditional multiplication over the \(m\) martingale increments gives
+\(\mathbb E e^{\theta(M_m-M_0)}\le e^{m\theta^2/2}\).
+Markov's inequality and minimization in \(\theta\) give the lower-tail bound \(e^{-u^2/(2m)}\); use \(u=m/4\) to obtain \eqref{eq:h-4}.
+
+With
+\[
+ \beta=\frac1{16},\qquad \eta=2e^{-n/256},
+\]
+both half densities in \eqref{eq:h-3} are at least \(\beta\), except on an \(\mathcal F_a\)-measurable event of probability at most \(\eta\). The exceptional event may be correlated with the energy. Since the energy is at most one, \eqref{eq:h-3}, summed over the halves, nonetheless gives
+\begin{equation}\label{eq:h-5}
+ b_s\le(1-\beta)E_s+\eta.
+\end{equation}
+This uses an unconditional exceptional probability and does not claim concentration conditional on the exposed paths.
+
+Let \(\gamma=31/32=1-\beta/2\). Equations \eqref{eq:h-2} and \eqref{eq:h-5} imply
+\begin{equation}\label{eq:h-6}
+ E_t\le\gamma^{t-2d}+\frac\eta\beta
+       =\left(\frac{31}{32}\right)^{t-2d}
+          +32e^{-n/256}.
+\end{equation}
+For \(t\le2d\) the bound follows from \(E_t\le1\). For \(t>2d\), use induction in \eqref{eq:h-2}; all indices \(t-j\) are at least \(d\). The geometric contribution is at most \(\gamma^{t-2d}\), because
+\[
+ (1-\beta)\sum_{j=1}^{\infty}(2\gamma)^{-j}
+ =\frac{1-\beta}{2\gamma-1}=1.
+\]
+The constant contribution is at most
+\((1-\beta)\eta/\beta+\eta=\eta/\beta\).
+This proves \eqref{eq:h-6}.
+
+We next turn the energy estimate into a bound on the positions of an ordered list. We use the sequential comparison of Morris \cite[Section~5.3, Lemma~12]{morris-exclusion}, also given by Morris, Rogaway and Stegers \cite[Appendix~A, Lemma~2]{mrs}. It applies to any law on distinct-position tuples; $U_{\mathrm{inj}}$ denotes the uniform law on such tuples.
+\begin{lemma}[Sequential total variation]\label{lem:sequential-tv}
+For a random tuple $(Z_1,\ldots,Z_k)$ of distinct positions,
+\begin{equation}\label{eq:h-7}
+ \|\mathcal L(Z_1,\ldots,Z_k)-U_{\mathrm{inj}}\|_{\mathrm{TV}}
+ \le\sum_{j=1}^k
+  \mathbb E\left\|\mathcal L(Z_j\mid Z_1,\ldots,Z_{j-1})
+       -U_{V\setminus\{Z_1,\ldots,Z_{j-1}\}}\right\|_{\mathrm{TV}}.
+\end{equation}
+Every expectation is under the actual prefix law. Refining the conditioning in a summand can only increase its expected distance, provided the refining sigma-field contains the prefix.
+\end{lemma}
+\begin{proof}
+At the $j$th step compare the actual joint $j$-tuple with the law that
+samples its first $j-1$ coordinates from their actual distribution and
+appends a uniform available position. Their distance is exactly the
+$j$th summand. Applying this same uniform extension kernel to a uniform
+prefix gives a uniform $j$-tuple and cannot increase total variation.
+The triangle inequality and induction prove \eqref{eq:h-7}. The uniform comparison law is prefix-measurable, so the tower property and convexity of total variation prove the refinement assertion.
+\end{proof}
+
+\begin{proposition}[Averaged ordered marginals]\label{prop:averaged-marginal}
+Let \(n=2^d\), \(T=200d\), and \(1\le k\le7n/8\). Choose an ordered list \(C=(c_1,\ldots,c_k)\) uniformly among all lists of distinct labels, independently of the shuffle. Then
+\begin{equation}\label{eq:h-8}
+ \begin{aligned}
+ \mathbb E_C
+ \left\|\mathcal L\bigl((\Pi_T(c_1),\ldots,\Pi_T(c_k))\mid C\bigr)
+       -U_{\mathrm{inj}}\right\|_{\mathrm{TV}}
+ &\le\varepsilon_n,\\
+ \varepsilon_n:=\frac{n^{3/2}}2
+  \sqrt{(31/32)^{198d}+32e^{-n/256}}
+ &=O(n^{-5/2}).
+ \end{aligned}
+\end{equation}
+\end{proposition}
+\begin{proof}
+First fix \(C\) and apply Lemma~\ref{lem:sequential-tv} to \(Z_j=\Pi_T(c_j)\). In its \(j\)th summand, condition further on the paths through time \(T\) of the first \(j-1\) cards. These paths determine the prefix of endpoints, so the lemma's refinement assertion applies.
+
+Let \(w_T^{(j)}\) be the resulting centered flow, with \(\ell=j-1\) and \(m_j=n-j+1\). Its conditional distance is \(\frac12\|w_T^{(j)}\|_1\). Averaging over both the list and the shuffle, Cauchy--Schwarz gives
+\[
+ \frac12\mathbb E_{C,\mathrm{shuffle}}\|w_T^{(j)}\|_1
+ \le\frac12\sqrt{m_j\,
+       \mathbb E_{C,\mathrm{shuffle}}\|w_T^{(j)}\|_2^2}.
+\]
+Each stage leaves at least \(n/8\) free positions, so \eqref{eq:h-6} applies to the expectation under this joint law, for every \(j\le k\). Since \(m_j\le n\) and \(k\le n\), summing the last display proves the bound in \eqref{eq:h-8}. Finally, \(198\log_2(32/31)>8\) gives \((31/32)^{198d}\le n^{-8}\), which proves the stated asymptotic estimate.
+\end{proof}
+The bound averages over the input list. We will use that average to choose a single partition into eight blocks for which all eight complementary marginals have small total error.
+
+The remaining question concerns the full permutation, including dependencies involving the omitted cards. We now pass to representations of \(S_n\). Restricting a representation to the subgroups that permute labels within the eight blocks separates it into tensor factors. At least one factor is small enough that the span generated by a single vector remains controlled, even when that factor occurs with multiplicity. This converts the marginal estimates into bounds on the Fourier matrices of a slightly trimmed law. First we prove the dimension estimate needed to sum those bounds; then we carry out the trimming and orbit-span argument.
+
+\section{The representation degrees needed for the lift}\label{sec:degrees}
+
+The transfer needs two properties of symmetric-group representation degrees. A uniform bound on their reciprocal sum will control orbit dimensions for the label-block subgroups in Proposition~\ref{prop:orbit-lift}. Decay of the sum away from the trivial and sign representations will control the full-group Fourier error in Section~\ref{sec:completion}. We prove both properties here.
+
+Let \(\lambda\vdash s\) be a partition of \(s\), represented by its Young diagram, whose row lengths are the parts of \(\lambda\). Write \(f^\lambda\) for the degree of the corresponding irreducible representation of \(S_s\). A standard Young tableau fills the diagram with \(1,\ldots,s\), increasing along each row and column. We use the classical facts that \(f^\lambda\) counts these tableaux, is unchanged by transposing the diagram, and satisfies the hook-length formula
+\[
+ f^\lambda=\frac{s!}{\prod_{z\in\lambda}h(z)},
+\]
+where $h(z)$ counts the box $z$ together with the boxes to its right
+and below it. The row $(s)$ and column $(1^s)$ give the trivial and
+sign representations, respectively \cite{sagan,etingof}.
+
+\begin{lemma}[Reciprocal representation degrees]\label{lem:reciprocal-degrees}
+The reciprocal-degree sums satisfy
+\begin{equation}\label{eq:h-9}
+ C_1:=\sup_{s\ge1}\sum_{\lambda\vdash s}(f^\lambda)^{-1}<\infty,
+ \qquad
+ \sum_{\substack{\lambda\vdash s\\\lambda\notin\{(s),(1^s)\}}}(f^\lambda)^{-1}\longrightarrow0\quad(s\to\infty).
+\end{equation}
+\end{lemma}
+Stronger asymptotics for every fixed positive inverse-degree exponent
+were proved by Liebeck and Shalev \cite[Theorem~2.6]{liebeck-shalev2004}.
+The elementary proof below gives all the summability used here.
+\begin{proof}
+
+First, let \(p(u)\) be the number of partitions of \(u\). The identity
+\[
+ \sum_{v\ge0}p(v)x^v=\prod_{j\ge1}(1-x^j)^{-1}
+\]
+gives \(p(u)\le e^{3\sqrt u}\) for \(u\ge1\): evaluate at \(x=e^{-1/\sqrt u}\), when the logarithm of the product is
+\[
+ \sum_{r\ge1}\frac1{r(e^{r/\sqrt u}-1)}
+ \le\sqrt u\sum_{r\ge1}r^{-2}\le2\sqrt u,
+\]
+and multiplying by \(x^{-u}\) bounds its \(u\)th coefficient as claimed.
+
+For a diagram of size \(s\), let \(L\) be the larger of its first row and column lengths. Transpose so its first row has length \(L\), and put \(u=s-L\). If \(1\le u\le s/4\), then
+\[
+ f^\lambda\ge \binom Lu\ge(L/u)^u\ge3^u.
+\]
+For the first inequality place labels \(1,\ldots,u\) in the first \(u\) top-row boxes; choose any \(u\) labels from the remaining \(L\) labels for the lower diagram, filling it in row-major order; and fill the remaining top row increasingly. Every lower column lies under one of the first \(u\) top boxes, so all column and row inequalities hold. There are at most \(2p(u)\) shapes with this tail size, including transposes. Their reciprocal contributions are bounded by the summable sequence \(2p(u)3^{-u}\); for each fixed positive \(u\), the binomial lower bound tends to infinity with \(s\). Dominated convergence makes the whole contribution tend to zero.
+
+If \(u>s/4\) and \(L\ge s/8\), retain the top row and the first \(b=\lfloor s/16\rfloor\) boxes below it, read row by row from top to bottom and left to right. The retained boxes form a Young diagram. Every standard tableau of the retained diagram extends to the full diagram by filling the remaining boxes in the same row order with the larger labels. In the retained diagram, fix \(1,\ldots,b\) in the first \(b\) top-row boxes and choose the \(b\) lower labels from the remaining \(L\) labels, as above. Since \(L\ge2b\), this gives
+\[
+ f^\lambda\ge\binom Lb\ge2^b
+\]
+for all sufficiently large \(s\). There are at most \(e^{3\sqrt s}\) diagrams, so their reciprocal contribution tends to zero. If \(L<s/8\), every hook has length less than \(s/4\); the hook formula and \(s!\ge(s/e)^s\) give \(f^\lambda\ge(4/e)^s\), again dominating the partition count. These cases cover all shapes except the row and column. Their degrees are one, so the full sums tend to two. The finitely many initial sums are finite, which also proves the uniform bound in \eqref{eq:h-9}. At \(s=1\), the row and column coincide and are counted once.
+\end{proof}
+
+\section{From eight large marginals to the whole group}\label{sec:lift}
+
+Assume \(d\ge3\), put \(G=S_n\), and let \(\mu=q_d^{*(200d)}\). Choose a uniform ordered partition \((M_1,\ldots,M_8)\) of the labels into blocks of size \(n/8\). Each complement is a uniform subset of size \(7n/8\). Giving it an independent uniform order yields exactly the list in \eqref{eq:h-8}, and changing that order does not change the total-variation distance. Therefore some deterministic partition has
+\begin{equation}\label{eq:h-10}
+ \delta_n:=\sum_{a=1}^8
+ \|\mathcal L(\mu\text{ on }V\setminus M_a)-U_{\mathrm{inj}}\|_{\mathrm{TV}}
+ \le8\varepsilon_n=o(1).
+\end{equation}
+Fix that partition. Let \(H_a\) be the subgroup of all permutations that fix every label outside \(M_a\). Two maps \(g,g'\) agree on the complement exactly when \(g'=gh\) for some \(h\in H_a\). Thus the complementary image tuple identifies the left coset \(gH_a\), and its uniform law is the coset pushforward of uniform measure on \(G\).
+
+For each \(a\), remove every coset \(C\in G/H_a\) with \(\mu(C)>2|H_a|/|G|\), and let \(E\) be what remains after all eight removals. A bad coset satisfies
+\(\mu(C)\le2(\mu(C)-|H_a|/|G|)\).
+The sum of positive excesses is its marginal total-variation distance, so the union bound gives
+\[
+ M:=\mu(E)\ge1-2\delta_n.
+\]
+For all sufficiently large \(n\), we have \(M\ge1/2\) and may define the conditional probability law \(\nu=\mu(\cdot\mid E)\). Then
+\begin{equation}\label{eq:h-11}
+ \|\nu-\mu\|_{\mathrm{TV}}=1-M=o(1),
+ \qquad
+ \nu(gH_a)\le\frac2M\frac{|H_a|}{|G|}
+             \le4\frac{|H_a|}{|G|}.
+\end{equation}
+A coset removed for that subgroup has no retained mass; every other coset had the required original cap. This proves the same cap for all eight systems under one common law. Consequently, for every nonnegative function that is constant on the cosets of any one \(H_a\), its expectation under \(\nu\) is at most four times its uniform expectation. The next proposition applies this comparison to projections onto orbit spans.
+
+We use finite-dimensional complex Hilbert spaces. The operator norm is
+$\|B\|_{\mathrm{op}}$, and the Hilbert--Schmidt norm is
+$\|B\|_{\mathrm{HS}}^2=\operatorname{tr}(B^*B)$ with ordinary,
+unnormalized trace. Write $\widehat H_a$ for the set of equivalence classes of irreducible representations of $H_a$. For a probability measure \(\nu\) on \(G\) and a unitary representation \(\rho\), define
+\(\widehat\nu(\rho)=\sum_g\nu(g)\rho(g)\).
+\begin{proposition}[Eight-block Fourier bound]\label{prop:orbit-lift}
+Let \(n\) be a positive multiple of eight, and partition the labels of \(G=S_n\) into sets \(M_1,\ldots,M_8\) of size \(n/8\). For each \(a\), let \(H_a\) be the subgroup fixing every label outside \(M_a\). Suppose a probability measure \(\nu\) on \(G\) satisfies
+\[
+ \nu(gH_a)\le4\frac{|H_a|}{|G|}
+ \qquad(g\in G,\ 1\le a\le8).
+\]
+Then every irreducible unitary representation \(\rho\) of \(G\), of degree \(D\), satisfies
+\begin{equation}\label{eq:h-12}
+ \|\widehat\nu(\rho)\|_{\mathrm{op}}
+ \le A D^{-5/16},\qquad A=\sqrt{32C_1}.
+\end{equation}
+\end{proposition}
+\begin{proof}
+The disjoint block subgroups form \(H=H_1\times\cdots\times H_8\). The restriction of \(\rho\) to \(H\) decomposes orthogonally into spaces
+\[
+ (V_{\pi_1}\otimes\cdots\otimes V_{\pi_8})
+       \otimes\mathcal A_{\boldsymbol\pi},
+\]
+where the last space records arbitrary multiplicity and \(H\) acts trivially on it. Every occurring tensor type has product of factor degrees at most \(D\); assign its whole isotypic space to one factor whose degree is at most \(D^{1/8}\). This gives an orthogonal decomposition \(V_\rho=E_1\oplus\cdots\oplus E_8\). Each \(E_a\) is \(H\)-invariant, and all the \(H_a\)-irreducible types occurring in it have degree at most \(D^{1/8}\).
+
+Fix \(u_a\in E_a\), and let
+\(W_a=\operatorname{span}\{\rho(h)u_a:h\in H_a\}\).
+Let \(u_{a,\pi}\) be the projection of \(u_a\) onto the full \(H_a\)-isotypic component \(V_\pi\otimes\mathcal A_\pi\), with \(b=\dim\pi\). Its orbit lies in the image of
+\[
+ \operatorname{End}(V_\pi)\longrightarrow V_\pi\otimes\mathcal A_\pi,
+ \quad Q\longmapsto(Q\otimes I)u_{a,\pi}.
+\]
+Its dimension is at most \(b^2\), independently of multiplicity. Grouping all equivalent copies first, and counting each irreducible type once, \eqref{eq:h-9} gives
+\begin{equation}\label{eq:h-13}
+ \dim W_a\le\sum_{\substack{\pi\in\widehat H_a\\\dim\pi\le D^{1/8}}}
+                 (\dim\pi)^2
+ \le D^{3/8}\sum_{\pi\in\widehat H_a}(\dim\pi)^{-1}
+ \le C_1D^{3/8}.
+\end{equation}
+The uniformity of \(C_1\) applies to every block size.
+
+The space \(W_a\) is \(H_a\)-invariant, so \(\rho(g)W_a\) depends only on \(gH_a\). If \(P_W\) denotes orthogonal projection, Schur's lemma and the trace give
+\[
+ \mathbb E_{g\sim U_G}P_{\rho(g)W_a}
+       =\frac{\dim W_a}{D}I.
+\]
+Indeed this average commutes with \(\rho(G)\), and its trace is \(\dim W_a\). Applying the assumed coset cap to the nonnegative coset-constant function \(\|P_{\rho(g)W_a}z\|^2\), we obtain
+\[
+ \begin{aligned}
+ |\langle z,\widehat\nu(\rho)u_a\rangle|^2
+ &\le\mathbb E_{g\sim\nu}|\langle z,\rho(g)u_a\rangle|^2\\
+ &\le\|u_a\|^2\mathbb E_{g\sim\nu}
+                  \|P_{\rho(g)W_a}z\|^2\\
+ &\le4\frac{\dim W_a}{D}\|u_a\|^2\|z\|^2
+ \le4C_1D^{-5/8}\|u_a\|^2\|z\|^2.
+ \end{aligned}
+\]
+For \(u=\sum_a u_a\), sum these bounds after taking square roots, and use \(\sum_a\|u_a\|\le\sqrt8\|u\|\). This proves \eqref{eq:h-12}.
+\end{proof}
+
+\section{Eight convolution factors and the final comparison}\label{sec:completion}
+
+The final comparison uses the finite-group Fourier method of
+Diaconis and Shahshahani \cite[Section~3]{DiaconisShahshahani1981}.
+The shuffle law is not constant on conjugacy classes, so we retain matrix norms throughout.
+By \eqref{eq:h-11} and Proposition~\ref{prop:orbit-lift}, the nearby law
+\(\nu\) has the required Fourier operator bound. We first show that
+\(\nu^{*8}\) is close to uniform, then compare it with the law of eight
+independent intervals of \(200d\) shuffles,
+\(\mu^{*8}=q_d^{*(1600d)}\).
+
+\begin{proof}[Completion of the proof of Theorem~\ref{thm:mixing}]
+With our Fourier normalization, every probability measure \(\sigma\) on \(G\) obeys
+\begin{equation}\label{eq:h-14}
+ 4\|\sigma-U_G\|_{\mathrm{TV}}^2
+ \le\sum_{\rho\ne\mathbf1}D_\rho
+                         \|\widehat\sigma(\rho)\|_{\mathrm{HS}}^2.
+\end{equation}
+For clarity, Cauchy--Schwarz bounds the left side by
+\(|G|\sum_g|\sigma(g)-|G|^{-1}|^2\).
+The regular-character orthogonality identity gives, for every real mass function \(a\),
+\[
+ \sum_\rho D_\rho\left\|\sum_g a(g)\rho(g)\right\|_{\mathrm{HS}}^2
+ =\sum_{g,h}a(g)a(h)\sum_\rho D_\rho
+                 \operatorname{tr}(\rho(h)^*\rho(g))
+ =|G|\sum_g a(g)^2.
+\]
+Apply this to \(a=\sigma-U_G\); its trivial coefficient is zero and uniform measure has zero transform in every nontrivial irreducible. This proves \eqref{eq:h-14}, with the ordinary, unnormalized Hilbert--Schmidt norm fixed in Section~\ref{sec:lift}.
+
+Convolution multiplies Fourier matrices. The operator norm is submultiplicative and \(\|B\|_{\mathrm{HS}}\le\sqrt D\|B\|_{\mathrm{op}}\), without any normality assumption. Therefore the nonsign, nontrivial contribution for \(\sigma=\nu^{*8}\) is at most
+\[
+ \begin{aligned}
+ \sum_{\rho\ne\mathbf1,\mathrm{sgn}}D_\rho
+         \|\widehat\nu(\rho)^8\|_{\mathrm{HS}}^2
+ &\le A^{16}\sum_{\rho\ne\mathbf1,\mathrm{sgn}}
+                    D_\rho^{2-16(5/16)}\\
+ &=A^{16}\sum_{\rho\ne\mathbf1,\mathrm{sgn}}D_\rho^{-3}
+ \longrightarrow0,
+ \end{aligned}
+\]
+by \eqref{eq:h-9}, since \(D^{-3}\le D^{-1}\).
+
+The sign representation has degree one and requires a separate argument. In the final physical shuffle of a positive-time block, condition on every coin but one. Flipping that last coin changes the resulting permutation by a transposition, so the original block law \(\mu\) has zero expected sign. Since \(\nu=\mu\mathbf1_E/M\),
+\[
+ |\widehat\nu(\mathrm{sgn})|
+ =\frac1M\left|\sum_{g\notin E}\mu(g)\mathrm{sgn}(g)\right|
+ \le\frac{1-M}{M}\longrightarrow0.
+\]
+Its contribution in \eqref{eq:h-14} after eight factors also tends to zero. Thus \(\|\nu^{*8}-U_G\|_{\mathrm{TV}}\to0\).
+
+Finally, replacing independent convolution factors one at a time costs at most their individual total-variation distances. By \eqref{eq:h-11},
+\[
+ \|\mu^{*8}-U_G\|_{\mathrm{TV}}
+ \le8(1-M)+\|\nu^{*8}-U_G\|_{\mathrm{TV}}
+ \longrightarrow0.
+\]
+Each convolution factor represents \(200d\) physical shuffles, so \(\mu^{*8}=q_d^{*(1600d)}\). Translation invariance gives the same bound from every deterministic initial deck. Together with the support inequality and fixed-dimensional convergence already proved, this completes the theorem.
+\end{proof}
+
+\input{references}
+\end{document}

preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/build/references.tex

@@ -0,0 +1,152 @@
+\begin{thebibliography}{99}
+
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preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf

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preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/README.md

@@ -0,0 +1,18 @@
+# [Orbifold and logarithmic Iitaka subadditivity](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 26, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026,
+  author = {{OpenAI}},
+  title = {{Orbifold and logarithmic Iitaka subadditivity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}},
+  year = {2026}
+}
+```
+

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/figures/two-maps.tex

@@ -0,0 +1,13 @@
+\begin{figure}[ht]
+\centering
+\begin{tikzcd}[column sep=large,row sep=large]
+ X \arrow[r,"g"] & W \arrow[r,"h"] \arrow[d,"p"'] & Y\\
+ & S &
+\end{tikzcd}
+\caption{The relative Iitaka reduction factors the original fibration
+through $W$. The map $h$ retains the original base and has
+$k$-dimensional general fibers. The independent map $p$ expresses
+$M$ as the pullback of a nef line $P$ on a projective base where
+$K_S+aP$ is big. No map between $Y$ and $S$ is asserted.}
+\label{fig:iitaka-period}
+\end{figure}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/main.tex

@@ -0,0 +1,87 @@
+\documentclass[11pt]{amsart}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools}
+\usepackage[margin=1.08in]{geometry}
+\usepackage[expansion=false]{microtype}
+\newcommand{\DoNotLoadEpstopdf}{}
+\usepackage{enumitem,booktabs,longtable,array,needspace,tikz-cd}
+\usepackage[hidelinks,pdfusetitle]{hyperref}
+\usepackage[capitalise,noabbrev,nameinlink]{cleveref}
+\hypersetup{pdftitle={Orbifold and logarithmic Iitaka subadditivity},pdfauthor={OpenAI},bookmarksdepth=2}
+\newcommand{\C}{\mathbf C}
+\newcommand{\Q}{\mathbf Q}
+\newcommand{\R}{\mathbf R}
+\newcommand{\Z}{\mathbf Z}
+\newcommand{\N}{\mathbf N}
+\newcommand{\cO}{\mathcal O}
+\newcommand{\cV}{\mathcal V}
+\newcommand{\cA}{\mathcal A}
+\newcommand{\ii}{\sqrt{-1}}
+\newcommand{\ddc}{dd^c}
+\newcommand{\kbar}{\bar\kappa}
+\newcommand{\red}{\mathrm{red}}
+\newcommand{\hor}{\mathrm{hor}}
+\DeclareMathOperator{\Gr}{Gr}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\rk}{rk}
+\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\Supp}{Supp}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\vol}{vol}
+\DeclareMathOperator{\trdeg}{trdeg}
+\DeclareMathOperator{\Frac}{Frac}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\Bir}{Bir}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Pic}{Pic}
+\DeclareMathOperator{\lct}{lct}
+\DeclareMathOperator{\mult}{mult}
+\DeclareMathOperator{\id}{id}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{construction}[theorem]{Construction}
+\newtheorem{convention}[theorem]{Convention}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{example}[theorem]{Example}
+\numberwithin{equation}{section}
+\setcounter{tocdepth}{1}
+\allowdisplaybreaks[2]
+\emergencystretch=1em
+\title[Orbifold and logarithmic Iitaka subadditivity]{Orbifold and logarithmic Iitaka subadditivity}
+\author{OpenAI}
+\date{September 26, 2026}
+\begin{document}
+\begin{abstract}
+We prove Campana's orbifold Iitaka subadditivity conjecture for rational
+simple normal crossing boundaries on compact manifolds in Fujiki class
+$\mathcal C$, including coefficient one. Ordinary and logarithmic
+subadditivity follow.
+\end{abstract}
+\maketitle
+\tableofcontents
+\input{sections/introduction}
+\input{sections/models}
+\input{sections/root-comparison}
+\input{sections/adjoint-addition}
+\input{sections/period-positivity}
+\input{sections/general-type-addition}
+\input{sections/applications}
+\input{sections/reduced-root}
+\appendix
+\input{sections/normalization}
+\raggedbottom
+\input{references}
+\end{document}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.bib

@@ -0,0 +1,741 @@
+@misc{AdiprasitoLiuTemkinSemistable,
+ archiveprefix = {arXiv},
+ author = {Adiprasito, Karim and Liu, Gaku and Temkin, Michael},
+ eprint = {1810.03131},
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+ primaryclass = {math.AG},
+ title = {Semistable reduction in characteristic {0}},
+ url = {https://arxiv.org/abs/1810.03131v2},
+ year = {2019}
+}
+
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+ title = {{Shokurov}'s boundary property},
+ url = {https://arxiv.org/abs/math/0210271v1},
+ volume = {67},
+ year = {2004}
+}
+
+@article{AmbroModuli,
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+ note = {\href{https://doi.org/10.1112/S0010437X04001071}{doi:10.1112/S0010437X04001071}},
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+ url = {https://doi.org/10.1112/S0010437X04001071},
+ volume = {141},
+ year = {2005}
+}
+
+@article{AndreMonodromy,
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+ pages = {1--24},
+ title = {{Mumford--Tate} groups of mixed {Hodge} structures and the theorem of the fixed part},
+ url = {https://www.numdam.org/item/CM_1992__82_1_1_0/},
+ volume = {82},
+ year = {1992}
+}
+
+@article{BakkerMatsushita,
+ author = {Bakker, Benjamin},
+ doi = {10.1016/j.aim.2025.110554},
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+ url = {https://benjamin-bakker.github.io/matsushita.pdf},
+ volume = {482},
+ year = {2025}
+}
+
+@article{BakkerBrunebarbeTsimerman,
+  author  = {Bakker, Benjamin and Brunebarbe, Yohan and Tsimerman, Jacob},
+  title   = {o-minimal {GAGA} and a conjecture of {Griffiths}},
+  journal = {Inventiones Mathematicae},
+  volume  = {232},
+  number  = {1},
+  year    = {2023},
+  pages   = {163--228},
+  doi     = {10.1007/s00222-022-01166-1},
+  note    = {Theorem locators refer to arXiv:1811.12230v3, 23 October 2022},
+  url     = {https://arxiv.org/abs/1811.12230v3}
+}
+
+@misc{BFMT,
+ archiveprefix = {arXiv},
+ author = {Bakker, Benjamin and Filipazzi, Stefano and Mauri, Mirko and Tsimerman, Jacob},
+ eprint = {2508.19215},
+ note = {\href{https://arxiv.org/abs/2508.19215v2}{arXiv:2508.19215v2}, 18 December 2025},
+ primaryclass = {math.AG},
+ title = {{Baily--Borel} compactifications of period images and the b-semiampleness conjecture},
+ url = {https://arxiv.org/abs/2508.19215v2},
+ year = {2025}
+}
+
+@article{BakkerTsimermanAxSchanuel,
+ author = {Bakker, Benjamin and Tsimerman, Jacob},
+ doi = {10.1007/s00222-019-00863-8},
+ journal = {Inventiones Mathematicae},
+ note = {The cited locators refer to \href{https://arxiv.org/abs/1712.05088v1}{arXiv:1712.05088v1}},
+ number = {1},
+ pages = {77--94},
+ title = {The {Ax--Schanuel} conjecture for variations of {Hodge} structures},
+ url = {https://arxiv.org/abs/1712.05088v1},
+ volume = {217},
+ year = {2019}
+}
+
+@article{BierstoneMilmanFunctorial,
+ author = {Bierstone, Edward and Milman, Pierre D.},
+ doi = {10.2977/PRIMS/1210167338},
+ journal = {Publications of the Research Institute for Mathematical Sciences},
+ number = {2},
+ pages = {609--639},
+ title = {Functoriality in resolution of singularities},
+ url = {https://ems.press/journals/prims/articles/2154},
+ volume = {44},
+ year = {2008}
+}
+
+@article{BirkarIitakaSix,
+ author = {Birkar, Caucher},
+ doi = {10.1112/S0010437X09004187},
+ journal = {Compositio Mathematica},
+ note = {arXiv:0806.4413, first posted 27 June 2008},
+ number = {6},
+ pages = {1442--1446},
+ title = {The {Iitaka} conjecture {$C_{n,m}$} in dimension six},
+ url = {https://arxiv.org/abs/0806.4413},
+ volume = {145},
+ year = {2009}
+}
+
+@article{BCHM,
+ author = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
+ doi = {10.1090/S0894-0347-09-00649-3},
+ journal = {Journal of the American Mathematical Society},
+ note = {The cited locators refer to the 81-page \href{https://arxiv.org/abs/math/0610203v2}{arXiv:math/0610203v2}},
+ number = {2},
+ pages = {405--468},
+ title = {Existence of minimal models for varieties of log general type},
+ url = {https://arxiv.org/abs/math/0610203v2},
+ volume = {23},
+ year = {2010}
+}
+
+@article{BoucksomVolume,
+ author = {Boucksom, S{\'e}bastien},
+ doi = {10.1142/S0129167X02001575},
+ journal = {International Journal of Mathematics},
+ note = {Theorem locators refer to arXiv:math/0201031v1, 5 January 2002},
+ number = {10},
+ pages = {1043--1063},
+ title = {On the volume of a line bundle},
+ url = {https://arxiv.org/abs/math/0201031v1},
+ volume = {13},
+ year = {2002}
+}
+
+@article{BEGZ,
+ author = {Boucksom, S{\'e}bastien and Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed},
+ doi = {10.1007/s11511-010-0054-7},
+ journal = {Acta Mathematica},
+ pages = {199--262},
+ title = {{Monge--Amp{\`e}re} equations in big cohomology classes},
+ url = {https://www.math.univ-toulouse.fr/~guedj/fichierspdf/BEGZ10.pdf},
+ volume = {205},
+ year = {2010}
+}
+
+@article{BrunebarbeCadorel,
+ author = {Brunebarbe, Yohan and Cadorel, Beno{\^i}t},
+ doi = {10.1093/imrn/rny054},
+ journal = {International Mathematics Research Notices},
+ note = {The cited locators refer to \href{https://arxiv.org/abs/1707.01327v1}{arXiv:1707.01327v1}},
+ number = {6},
+ pages = {1601--1609},
+ title = {Hyperbolicity of varieties supporting a variation of {Hodge} structure},
+ url = {https://arxiv.org/abs/1707.01327v1},
+ volume = {2020},
+ year = {2020}
+}
+
+@article{Campana2004,
+ author = {Campana, Fr{\'e}d{\'e}ric},
+ doi = {10.5802/aif.2027},
+ journal = {Annales de l'Institut Fourier},
+ number = {3},
+ pages = {499--630},
+ title = {Orbifolds, special varieties and classification theory},
+ url = {https://www.numdam.org/item/10.5802/aif.2027/},
+ volume = {54},
+ year = {2004}
+}
+
+@article{CampanaOrbifolds,
+ author = {Campana, Fr{\'e}d{\'e}ric},
+ doi = {10.1017/S1474748010000101},
+ journal = {Journal of the Institute of Mathematics of Jussieu},
+ note = {The cited locators refer to \href{https://arxiv.org/abs/0705.0737v8}{arXiv:0705.0737v8}},
+ number = {4},
+ pages = {809--934},
+ title = {Orbifoldes g{\'e}om{\'e}triques sp{\'e}ciales et classification bim{\'e}romorphe des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes compactes},
+ url = {https://arxiv.org/abs/0705.0737v8},
+ volume = {10},
+ year = {2011}
+}
+
+@article{CaoSurfaces,
+ author = {Cao, Junyan},
+ doi = {10.14231/AG-2018-020},
+ journal = {Algebraic Geometry},
+ number = {6},
+ pages = {728--741},
+ title = {{Kodaira} dimension of algebraic fiber spaces over surfaces},
+ url = {https://content.algebraicgeometry.nl/2018-6/2018-6-020.pdf},
+ volume = {5},
+ year = {2018}
+}
+
+@article{CaoPaunAbelian,
+ author = {Cao, Junyan and P{\u a}un, Mihai},
+ doi = {10.1007/s00222-016-0672-6},
+ journal = {Inventiones Mathematicae},
+ note = {Theorem locators refer to arXiv:1504.01095v3, submitted 3 August 2016},
+ number = {1},
+ pages = {345--387},
+ title = {{Kodaira} dimension of algebraic fiber spaces over abelian varieties},
+ url = {https://arxiv.org/abs/1504.01095v3},
+ volume = {207},
+ year = {2017}
+}
+
+@article{CKS,
+ author = {Cattani, Eduardo and Kaplan, Aroldo and Schmid, Wilfried},
+ doi = {10.2307/1971333},
+ journal = {Annals of Mathematics},
+ number = {3},
+ pages = {457--535},
+ series = {2},
+ title = {Degeneration of {Hodge} structures},
+ url = {https://annals.math.princeton.edu/1986/123-3/p02},
+ volume = {123},
+ year = {1986}
+}
+
+@misc{ChenLimits,
+ archiveprefix = {arXiv},
+ author = {Chen, Qianyu},
+ eprint = {2103.03983},
+ note = {arXiv:2103.03983; the cited Lemma 8.1 is in the \href{https://qymath.github.io/Steenbrink.pdf}{61-page author manuscript}},
+ primaryclass = {math.AG},
+ title = {Limits of {Hodge} structures via holonomic {$D$}-modules},
+ url = {https://qymath.github.io/Steenbrink.pdf},
+ year = {2021}
+}
+
+@article{DeligneHodgeII,
+  author = {Deligne, Pierre},
+  title = {Th{\'e}orie de {Hodge}. {II}},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume = {40},
+  year = {1971},
+  pages = {5--57},
+  doi = {10.1007/BF02684692},
+  url = {https://www.numdam.org/item/PMIHES_1971__40__5_0.pdf}
+}
+
+@article{DemaillyRegularization,
+ author = {Demailly, Jean-Pierre},
+ journal = {Journal of Algebraic Geometry},
+ note = {The cited locator refers to the \href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}{author manuscript}},
+ number = {3},
+ pages = {361--409},
+ title = {Regularization of closed positive currents and intersection theory},
+ url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf},
+ volume = {1},
+ year = {1992}
+}
+
+@article{DemaillyPaun2004,
+ author = {Demailly, Jean-Pierre and P{\u{a}}un, Mihai},
+ doi = {10.4007/annals.2004.159.1247},
+ journal = {Annals of Mathematics},
+ number = {3},
+ pages = {1247--1274},
+ series = {2},
+ title = {Numerical characterization of the {K{\"a}hler} cone of a compact {K{\"a}hler} manifold},
+ url = {https://annals.math.princeton.edu/2004/159-3/p05},
+ volume = {159},
+ year = {2004}
+}
+
+@article{Douady,
+ author = {Douady, Adrien},
+ doi = {10.5802/aif.226},
+ journal = {Annales de l'Institut Fourier},
+ number = {1},
+ pages = {1--95},
+ title = {Le probl{\`e}me des modules pour les sous-espaces analytiques compacts d'un espace analytique donn{\'e}},
+ url = {https://www.numdam.org/item/AIF_1966__16_1_1_0/},
+ volume = {16},
+ year = {1966}
+}
+
+@article{EnokizonoHashizumeSemistable,
+ author = {Enokizono, Makoto and Hashizume, Kenta},
+ doi = {10.1090/tran/9365},
+ journal = {Transactions of the American Mathematical Society},
+ note = {The cited locators refer to \href{https://arxiv.org/abs/2404.05124v3}{arXiv:2404.05124v3}, 12 October 2024},
+ pages = {7667--7688},
+ title = {Semistable reduction for complex analytic spaces},
+ url = {https://arxiv.org/abs/2404.05124v3},
+ volume = {378},
+ year = {2025}
+}
+
+@article{FujikiDouady,
+ author = {Fujiki, Akira},
+ doi = {10.2977/PRIMS/1195189279},
+ journal = {Publications of the Research Institute for Mathematical Sciences},
+ number = {1},
+ pages = {1--52},
+ title = {Closedness of the {Douady} spaces of compact {K{\"a}hler} spaces},
+ url = {https://ems.press/content/serial-article-files/41937},
+ volume = {14},
+ year = {1978}
+}
+
+@incollection{FujinoWeakPositivity,
+ author = {Fujino, Osamu},
+ booktitle = {Algebraic Varieties and Automorphism Groups},
+ doi = {10.2969/aspm/07510073},
+ note = {The cited locators refer to \href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{author version 0.54, 30 June 2015}},
+ pages = {73--118},
+ publisher = {Mathematical Society of Japan},
+ series = {Advanced Studies in Pure Mathematics},
+ title = {Notes on the weak positivity theorems},
+ url = {https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf},
+ volume = {75},
+ year = {2017}
+}
+
+@article{FujinoAnalyticAdjunction,
+ author = {Fujino, Osamu},
+ doi = {10.3792/PJAA.100.002},
+ journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences},
+ number = {2},
+ pages = {7--11},
+ title = {Log canonical inversion of adjunction},
+ url = {https://www.math.kyoto-u.ac.jp/~fujino/on-inversion-of-adjunction-pja-final.pdf},
+ volume = {100},
+ year = {2024}
+}
+
+@article{FujinoFujisawa,
+ author = {Fujino, Osamu and Fujisawa, Taro},
+ doi = {10.4171/PRIMS/145},
+ journal = {Publications of the Research Institute for Mathematical Sciences},
+ note = {\href{https://doi.org/10.4171/PRIMS/145}{doi:10.4171/PRIMS/145}},
+ number = {4},
+ pages = {589--661},
+ title = {Variations of mixed {Hodge} structure and semipositivity theorems},
+ url = {https://ems.press/content/serial-article-files/41240},
+ volume = {50},
+ year = {2014}
+}
+
+@article{FujinoFujisawaVMHS,
+ author = {Fujino, Osamu and Fujisawa, Taro},
+ journal = {International Journal of Mathematics},
+ note = {Published article: 43 pp.; theorem locators refer to the 36-page \href{https://www.math.kyoto-u.ac.jp/~fujino/vmhs-applications8.pdf}{author version 0.55, 11 March 2025}},
+ number = {9},
+ pages = {2550020},
+ title = {Variation of mixed {Hodge} structure and its applications},
+ url = {https://www.math.kyoto-u.ac.jp/~fujino/vmhs-applications8.pdf},
+ volume = {36},
+ year = {2025}
+}
+
+@article{FujinoGongyo,
+ author = {Fujino, Osamu and Gongyo, Yoshinori},
+ doi = {10.5802/aif.2894},
+ journal = {Annales de l'Institut Fourier},
+ note = {\href{https://doi.org/10.5802/aif.2894}{doi:10.5802/aif.2894}},
+ number = {4},
+ pages = {1721--1735},
+ title = {On the moduli b-divisors of lc-trivial fibrations},
+ url = {https://www.numdam.org/item/10.5802/aif.2894.pdf},
+ volume = {64},
+ year = {2014}
+}
+
+@article{FujinoMori,
+ author = {Fujino, Osamu and Mori, Shigefumi},
+ journal = {Journal of Differential Geometry},
+ number = {1},
+ pages = {167--188},
+ title = {A canonical bundle formula},
+ url = {https://www.math.kyoto-u.ac.jp/~fujino/Canonical.pdf},
+ volume = {56},
+ year = {2000}
+}
+
+@article{GrauertDirectImages,
+ author = {Grauert, Hans},
+ doi = {10.1007/BF02684746},
+ journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+ note = {With corrigendum, Publ. Math. IH{\'E}S 16 (1963), 35--36},
+ pages = {5--64},
+ title = {Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen},
+ url = {https://www.numdam.org/item/PMIHES_1960__5__5_0/},
+ volume = {5},
+ year = {1960}
+}
+
+@article{GrauertDirectImagesCorr,
+ author = {Grauert, Hans},
+ doi = {10.1007/BF02684291},
+ journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+ pages = {35--36},
+ title = {Berichtigung: Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen},
+ url = {https://www.numdam.org/articles/10.1007/BF02684291/},
+ volume = {16},
+ year = {1963}
+}
+
+@article{GriffithsPeriodsIII,
+ author = {Griffiths, Phillip A.},
+ doi = {10.1007/BF02684654},
+ journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+ pages = {125--180},
+ title = {Periods of integrals on algebraic manifolds, {III}: Some global differential-geometric properties of the period mapping},
+ url = {https://www.numdam.org/item/PMIHES_1970__38__125_0/},
+ volume = {38},
+ year = {1970}
+}
+
+@article{GuedjZeriahiEnergy,
+ author = {Guedj, Vincent and Zeriahi, Ahmed},
+ doi = {10.1016/j.jfa.2007.04.018},
+ journal = {Journal of Functional Analysis},
+ note = {The cited locators refer to \href{https://arxiv.org/abs/math/0612630v2}{arXiv:math/0612630v2}},
+ number = {2},
+ pages = {442--482},
+ title = {The weighted {Monge--Amp{\`e}re} energy of quasiplurisubharmonic functions},
+ url = {https://arxiv.org/abs/math/0612630v2},
+ volume = {250},
+ year = {2007}
+}
+
+@incollection{HaconPopaSchnellAbelian,
+ address = {Providence, RI},
+ author = {Hacon, Christopher D. and Popa, Mihnea and Schnell, Christian},
+ booktitle = {Local and Global Methods in Algebraic Geometry},
+ doi = {10.1090/conm/712/14346},
+ note = {Theorem locators refer to the author manuscript dated 9 May 2017},
+ pages = {143--195},
+ publisher = {American Mathematical Society},
+ series = {Contemporary Mathematics},
+ title = {Algebraic fiber spaces over abelian varieties: around a recent theorem by {Cao} and {P{\u a}un}},
+ url = {https://people.math.harvard.edu/~mpopa/papers/HPS.pdf},
+ volume = {712},
+ year = {2018}
+}
+
+@article{Hashizume,
+ author = {Hashizume, Kenta},
+ doi = {10.3792/pjaa.96.017},
+ journal = {Proceedings of the Japan Academy, Series A, Mathematical Sciences},
+ note = {Theorem numbering is cited from arXiv:1902.10923v2},
+ number = {10},
+ pages = {87--92},
+ title = {Log {Iitaka} conjecture for abundant log canonical fibrations},
+ url = {https://arxiv.org/abs/1902.10923v2},
+ volume = {96},
+ year = {2020}
+}
+
+@article{HironakaFlattening,
+ author = {Hironaka, Heisuke},
+ doi = {10.2307/2373721},
+ journal = {American Journal of Mathematics},
+ number = {2},
+ pages = {503--547},
+ title = {Flattening theorem in complex-analytic geometry},
+ url = {https://doi.org/10.2307/2373721},
+ volume = {97},
+ year = {1975}
+}
+
+@article{IitakaOriginal,
+ author = {Iitaka, Shigeru},
+ doi = {10.2969/jmsj/02320356},
+ journal = {Journal of the Mathematical Society of Japan},
+ number = {2},
+ pages = {356--373},
+ title = {On {$D$}-dimensions of algebraic varieties},
+ url = {https://www.jstage.jst.go.jp/article/jmath1948/23/2/23_2_356/_article},
+ volume = {23},
+ year = {1971}
+}
+
+@incollection{IitakaLog,
+ author = {Iitaka, Shigeru},
+ booktitle = {Complex Analysis and Algebraic Geometry: A Collection of Papers Dedicated to {K. Kodaira}},
+ doi = {10.1017/CBO9780511569197.014},
+ editor = {Baily, Jr., Walter L. and Shioda, Tetsuji},
+ note = {\href{https://doi.org/10.1017/CBO9780511569197.014}{doi:10.1017/CBO9780511569197.014}},
+ publisher = {Iwanami Shoten and Cambridge University Press},
+ title = {On logarithmic {Kodaira} dimension of algebraic varieties},
+ url = {https://doi.org/10.1017/CBO9780511569197.014},
+ year = {1977}
+}
+
+@article{IitakaClassification,
+ author = {Iitaka, Shigeru},
+ doi = {10.3792/pja/1195517881},
+ journal = {Proceedings of the Japan Academy},
+ number = {3},
+ pages = {103--105},
+ title = {Classification of Algebraic Varieties},
+ url = {https://www.jstage.jst.go.jp/article/pjab1945/53/3/53_3_103/_pdf},
+ volume = {53},
+ year = {1977}
+}
+
+@article{KawamataCurves,
+ author = {Kawamata, Yujiro},
+ doi = {10.1007/BF01404756},
+ journal = {Inventiones Mathematicae},
+ number = {1},
+ pages = {57--71},
+ title = {{Kodaira} dimension of algebraic fiber spaces over curves},
+ url = {https://link.springer.com/article/10.1007/BF01404756},
+ volume = {66},
+ year = {1982}
+}
+
+@article{KawamataKodaira,
+ author = {Kawamata, Yujiro},
+ doi = {10.1515/crll.1985.363.1},
+ journal = {Journal f{\"u}r die reine und angewandte Mathematik},
+ pages = {1--46},
+ title = {Minimal models and the {Kodaira} dimension of algebraic fiber spaces},
+ url = {https://doi.org/10.1515/crll.1985.363.1},
+ volume = {363},
+ year = {1985}
+}
+
+@incollection{Kollar87,
+ address = {Tokyo and Amsterdam},
+ author = {Koll{\'a}r, J{\'a}nos},
+ booktitle = {Algebraic Geometry, Sendai, 1985},
+ doi = {10.2969/aspm/01010361},
+ pages = {361--398},
+ publisher = {Kinokuniya and North-Holland},
+ series = {Advanced Studies in Pure Mathematics},
+ title = {Subadditivity of the {Kodaira} dimension: fibers of general type},
+ url = {https://doi.org/10.2969/aspm/01010361},
+ volume = {10},
+ year = {1987}
+}
+
+@article{KovacsPatakfalvi,
+ author = {Kov{\'a}cs, S{\'a}ndor J. and Patakfalvi, Zsolt},
+ doi = {10.1090/jams/871},
+ journal = {Journal of the American Mathematical Society},
+ note = {The cited locators refer to the \href{https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf}{62-page author manuscript}},
+ number = {4},
+ pages = {959--1021},
+ title = {Projectivity of the moduli space of stable log-varieties and subadditivity of log-{Kodaira} dimension},
+ url = {https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf},
+ volume = {30},
+ year = {2017}
+}
+
+@misc{MaeharaIitaka,
+ archiveprefix = {arXiv},
+ author = {Maehara, Kazuhisa},
+ eprint = {0910.0604},
+ howpublished = {arXiv preprint},
+ note = {Version 3, 16 May 2010; first posted 4 October 2009},
+ title = {{Iitaka--Viehweg} conjectures {$C$} and {$C^{++}$}},
+ url = {https://arxiv.org/abs/0910.0604v3},
+ year = {2010}
+}
+
+@book{MilneAlgebraicGroups,
+ author = {Milne, James S.},
+ doi = {10.1017/9781316711736},
+ note = {Cited from the corrected 2022 reprint},
+ publisher = {Cambridge University Press},
+ series = {Cambridge Studies in Advanced Mathematics},
+ title = {Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field},
+ url = {https://www.jmilne.org/math/Books/iAG2022.pdf},
+ volume = {170},
+ year = {2017}
+}
+
+@article{NamikawaProjectivity,
+ archiveprefix = {arXiv},
+ author = {Namikawa, Yoshinori},
+ eprint = {math/0101019},
+ journal = {International Journal of Mathematics},
+ note = {Theorem locators refer to arXiv:math/0101019v6, 12 November 2001},
+ number = {2},
+ pages = {125--135},
+ title = {Projectivity criterion of {Moishezon} spaces and density of projective symplectic varieties},
+ url = {https://arxiv.org/abs/math/0101019v6},
+ volume = {13},
+ year = {2002}
+}
+
+@misc{IitakaVariationCompanion,
+  author = {{OpenAI}},
+  title = {{Logarithmic Kodaira dimension and whole-fiber variation}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf}{OAI:Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026}},
+  year = {2026}
+}
+
+@misc{IitakaProjectiveCompanion,
+  author = {{OpenAI}},
+  title = {{Projective Hodge lines and ordinary Iitaka subadditivity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf}{OAI:Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026}},
+  year = {2026}
+}
+
+@misc{IitakaAdditivityCompanion,
+  author = {{OpenAI}},
+  title = {{The reverse logarithmic Kodaira inequality and additivity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf}{OAI:The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026}},
+  year = {2026}
+}
+
+@misc{OpenAILogAbundance2026,
+  author = {{OpenAI}},
+  title = {{Log abundance in characteristic zero}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}},
+  year = {2026}
+}
+
+@article{PopaSchnellKahler,
+ author = {Pareschi, Giuseppe and Popa, Mihnea and Schnell, Christian},
+ doi = {10.2140/gt.2017.21.2419},
+ journal = {Geometry \& Topology},
+ pages = {2419--2460},
+ title = {{Hodge} modules on complex tori and generic vanishing for compact {K{\"a}hler} manifolds},
+ url = {https://msp.org/gt/2017/21-4/gt-v21-n4-p12-s.pdf},
+ volume = {21},
+ year = {2017}
+}
+
+@article{PatakfalviFibered,
+ author = {Patakfalvi, Zsolt},
+ doi = {10.1090/tran/6386},
+ journal = {Transactions of the American Mathematical Society},
+ number = {3},
+ pages = {1837--1869},
+ title = {Fibered stable varieties},
+ url = {https://arxiv.org/abs/1208.1787v4},
+ volume = {368},
+ year = {2016}
+}
+
+@article{Schmid,
+ author = {Schmid, Wilfried},
+ doi = {10.1007/BF01389674},
+ journal = {Inventiones Mathematicae},
+ number = {3--4},
+ pages = {211--319},
+ title = {Variation of {Hodge} structure: The singularities of the period mapping},
+ url = {https://doi.org/10.1007/BF01389674},
+ volume = {22},
+ year = {1973}
+}
+
+@article{SommesePeriodMapping,
+ author = {Sommese, Andrew John},
+ title = {On the rationality of the period mapping},
+ journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
+ series = {4},
+ volume = {5},
+ number = {4},
+ year = {1978},
+ pages = {683--717},
+ url = {https://www.numdam.org/item/ASNSP_1978_4_5_4_683_0/}
+}
+
+@misc{TsujiDirectImages,
+ archiveprefix = {arXiv},
+ author = {Tsuji, Hajime},
+ eprint = {1012.0884},
+ howpublished = {arXiv preprint},
+ note = {Version 1, 4 December 2010},
+ title = {Global generation of the direct images of relative pluricanonical systems},
+ url = {https://arxiv.org/abs/1012.0884v1},
+ year = {2010}
+}
+
+@book{UenoClassification,
+ address = {Berlin},
+ author = {Ueno, Kenji},
+ doi = {10.1007/BFb0070570},
+ publisher = {Springer},
+ series = {Lecture Notes in Mathematics},
+ title = {Classification Theory of Algebraic Varieties and Compact Complex Spaces},
+ url = {https://doi.org/10.1007/BFb0070570},
+ volume = {439},
+ year = {1975}
+}
+
+@incollection{ViehwegAdditivity,
+ address = {Tokyo and Amsterdam},
+ author = {Viehweg, Eckart},
+ booktitle = {Algebraic Varieties and Analytic Varieties (Tokyo, 1981)},
+ doi = {10.2969/aspm/00110329},
+ pages = {329--353},
+ publisher = {Kinokuniya and North-Holland},
+ series = {Advanced Studies in Pure Mathematics},
+ title = {Weak positivity and the additivity of the {Kodaira} dimension for certain fibre spaces},
+ url = {https://esaga.uni-due.de/eckart.viehweg/ecpubl/},
+ volume = {1},
+ year = {1983}
+}
+
+@misc{VilladsenHodgeKahler,
+ archiveprefix = {arXiv},
+ author = {Villadsen, Mads Bach},
+ eprint = {2401.09544},
+ note = {\href{https://arxiv.org/abs/2401.09544v1}{arXiv:2401.09544v1}, 17 January 2024},
+ primaryclass = {math.AG},
+ title = {{Hodge} modules and {K{\"a}hler} morphisms},
+ url = {https://arxiv.org/abs/2401.09544v1},
+ year = {2024}
+}
+
+@article{WangKahlerIitaka,
+ author = {Wang, Juanyong},
+ doi = {10.5802/afst.1690},
+ journal = {Annales de la Facult{\'e} des Sciences de Toulouse. Math{\'e}matiques},
+ number = {4},
+ pages = {813--897},
+ series = {6},
+ title = {On the {Iitaka} Conjecture {$C_{n,m}$} for {K{\"a}hler} Fibre Spaces},
+ url = {https://afst.centre-mersenne.org/item/10.5802/afst.1690.pdf},
+ volume = {30},
+ year = {2021}
+}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/references.tex

@@ -0,0 +1,406 @@
+\begin{thebibliography}{99}
+
+\bibitem{AdiprasitoLiuTemkinSemistable}
+Karim Adiprasito, Gaku Liu and Michael Temkin.
+\emph{\href{https://arxiv.org/abs/1810.03131v2}{Semistable reduction in characteristic {0}}}.
+Preprint, 2019.
+\href{https://arxiv.org/abs/1810.03131v2}{arXiv:1810.03131v2}, 17 June 2019; 22-page full preprint.
+
+\bibitem{AmbroBoundary}
+Florin Ambro.
+\emph{\href{https://arxiv.org/abs/math/0210271v1}{{Shokurov}'s boundary property}}.
+Journal of Differential Geometry \textbf{67} (2004), no.~2, 229--255.
+Theorem numbering is cited from arXiv:math/0210271v1.
+
+\bibitem{AmbroModuli}
+Florin Ambro.
+\emph{\href{https://doi.org/10.1112/S0010437X04001071}{The moduli b-divisor of an lc-trivial fibration}}.
+Compositio Mathematica \textbf{141} (2005), no.~2, 385--403.
+\href{https://doi.org/10.1112/S0010437X04001071}{doi:\nolinkurl{10.1112/S0010437X04001071}}.
+
+\bibitem{AndreMonodromy}
+Yves Andr{\'e}.
+\emph{\href{https://www.numdam.org/item/CM_1992__82_1_1_0/}{{Mumford--Tate} groups of mixed {Hodge} structures and the theorem of the fixed part}}.
+Compositio Mathematica \textbf{82} (1992), no.~1, 1--24.
+
+\bibitem{BakkerMatsushita}
+Benjamin Bakker.
+\emph{\href{https://benjamin-bakker.github.io/matsushita.pdf}{A short proof of a conjecture of {Matsushita}}}.
+Advances in Mathematics \textbf{482} (2025), 110554.
+\href{https://doi.org/10.1016/j.aim.2025.110554}{doi:\nolinkurl{10.1016/j.aim.2025.110554}}.
+Part A.
+
+\bibitem{BakkerBrunebarbeTsimerman}
+Benjamin Bakker, Yohan Brunebarbe and Jacob Tsimerman.
+\emph{\href{https://arxiv.org/abs/1811.12230v3}{o-minimal {GAGA} and a conjecture of {Griffiths}}}.
+Inventiones Mathematicae \textbf{232} (2023), no.~1, 163--228.
+\href{https://doi.org/10.1007/s00222-022-01166-1}{doi:\nolinkurl{10.1007/s00222-022-01166-1}}.
+Theorem locators refer to arXiv:1811.12230v3, 23 October 2022.
+
+\bibitem{BFMT}
+Benjamin Bakker, Stefano Filipazzi, Mirko Mauri and Jacob Tsimerman.
+\emph{\href{https://arxiv.org/abs/2508.19215v2}{{Baily--Borel} compactifications of period images and the b-semiampleness conjecture}}.
+Preprint, 2025.
+\href{https://arxiv.org/abs/2508.19215v2}{arXiv:2508.19215v2}, 18 December 2025.
+
+\bibitem{BakkerTsimermanAxSchanuel}
+Benjamin Bakker and Jacob Tsimerman.
+\emph{\href{https://arxiv.org/abs/1712.05088v1}{The {Ax--Schanuel} conjecture for variations of {Hodge} structures}}.
+Inventiones Mathematicae \textbf{217} (2019), no.~1, 77--94.
+\href{https://doi.org/10.1007/s00222-019-00863-8}{doi:\nolinkurl{10.1007/s00222-019-00863-8}}.
+The cited locators refer to \href{https://arxiv.org/abs/1712.05088v1}{arXiv:1712.05088v1}.
+
+\bibitem{BierstoneMilmanFunctorial}
+Edward Bierstone and Pierre D. Milman.
+\emph{\href{https://ems.press/journals/prims/articles/2154}{Functoriality in resolution of singularities}}.
+Publications of the Research Institute for Mathematical Sciences \textbf{44} (2008), no.~2, 609--639.
+\href{https://doi.org/10.2977/PRIMS/1210167338}{doi:\nolinkurl{10.2977/PRIMS/1210167338}}.
+
+\bibitem{BirkarIitakaSix}
+Caucher Birkar.
+\emph{\href{https://arxiv.org/abs/0806.4413}{The {Iitaka} conjecture {$C_{n,m}$} in dimension six}}.
+Compositio Mathematica \textbf{145} (2009), no.~6, 1442--1446.
+\href{https://doi.org/10.1112/S0010437X09004187}{doi:\nolinkurl{10.1112/S0010437X09004187}}.
+arXiv:0806.4413, first posted 27 June 2008.
+
+\bibitem{BCHM}
+Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan.
+\emph{\href{https://arxiv.org/abs/math/0610203v2}{Existence of minimal models for varieties of log general type}}.
+Journal of the American Mathematical Society \textbf{23} (2010), no.~2, 405--468.
+\href{https://doi.org/10.1090/S0894-0347-09-00649-3}{doi:\nolinkurl{10.1090/S0894-0347-09-00649-3}}.
+The cited locators refer to the 81-page \href{https://arxiv.org/abs/math/0610203v2}{arXiv:math/0610203v2}.
+
+\bibitem{BoucksomVolume}
+S{\'e}bastien Boucksom.
+\emph{\href{https://arxiv.org/abs/math/0201031v1}{On the volume of a line bundle}}.
+International Journal of Mathematics \textbf{13} (2002), no.~10, 1043--1063.
+\href{https://doi.org/10.1142/S0129167X02001575}{doi:\nolinkurl{10.1142/S0129167X02001575}}.
+Theorem locators refer to arXiv:math/0201031v1, 5 January 2002.
+
+\bibitem{BEGZ}
+S{\'e}bastien Boucksom, Philippe Eyssidieux, Vincent Guedj and Ahmed Zeriahi.
+\emph{\href{https://www.math.univ-toulouse.fr/~guedj/fichierspdf/BEGZ10.pdf}{{Monge--Amp{\`e}re} equations in big cohomology classes}}.
+Acta Mathematica \textbf{205} (2010), 199--262.
+\href{https://doi.org/10.1007/s11511-010-0054-7}{doi:\nolinkurl{10.1007/s11511-010-0054-7}}.
+
+\bibitem{BrunebarbeCadorel}
+Yohan Brunebarbe and Beno{\^i}t Cadorel.
+\emph{\href{https://arxiv.org/abs/1707.01327v1}{Hyperbolicity of varieties supporting a variation of {Hodge} structure}}.
+International Mathematics Research Notices \textbf{2020} (2020), no.~6, 1601--1609.
+\href{https://doi.org/10.1093/imrn/rny054}{doi:\nolinkurl{10.1093/imrn/rny054}}.
+The cited locators refer to \href{https://arxiv.org/abs/1707.01327v1}{arXiv:1707.01327v1}.
+
+\bibitem{Campana2004}
+Fr{\'e}d{\'e}ric Campana.
+\emph{\href{https://www.numdam.org/item/10.5802/aif.2027/}{Orbifolds, special varieties and classification theory}}.
+Annales de l'Institut Fourier \textbf{54} (2004), no.~3, 499--630.
+\href{https://doi.org/10.5802/aif.2027}{doi:\nolinkurl{10.5802/aif.2027}}.
+
+\bibitem{CampanaOrbifolds}
+Fr{\'e}d{\'e}ric Campana.
+\emph{\href{https://arxiv.org/abs/0705.0737v8}{Orbifoldes g{\'e}om{\'e}triques sp{\'e}ciales et classification bim{\'e}romorphe des vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes compactes}}.
+Journal of the Institute of Mathematics of Jussieu \textbf{10} (2011), no.~4, 809--934.
+\href{https://doi.org/10.1017/S1474748010000101}{doi:\nolinkurl{10.1017/S1474748010000101}}.
+The cited locators refer to \href{https://arxiv.org/abs/0705.0737v8}{arXiv:0705.0737v8}.
+
+\bibitem{CaoSurfaces}
+Junyan Cao.
+\emph{\href{https://content.algebraicgeometry.nl/2018-6/2018-6-020.pdf}{{Kodaira} dimension of algebraic fiber spaces over surfaces}}.
+Algebraic Geometry \textbf{5} (2018), no.~6, 728--741.
+\href{https://doi.org/10.14231/AG-2018-020}{doi:\nolinkurl{10.14231/AG-2018-020}}.
+
+\bibitem{CaoPaunAbelian}
+Junyan Cao and Mihai P{\u a}un.
+\emph{\href{https://arxiv.org/abs/1504.01095v3}{{Kodaira} dimension of algebraic fiber spaces over abelian varieties}}.
+Inventiones Mathematicae \textbf{207} (2017), no.~1, 345--387.
+\href{https://doi.org/10.1007/s00222-016-0672-6}{doi:\nolinkurl{10.1007/s00222-016-0672-6}}.
+Theorem locators refer to arXiv:1504.01095v3, submitted 3 August 2016.
+
+\bibitem{CKS}
+Eduardo Cattani, Aroldo Kaplan and Wilfried Schmid.
+\emph{\href{https://annals.math.princeton.edu/1986/123-3/p02}{Degeneration of {Hodge} structures}}.
+Annals of Mathematics (2) \textbf{123} (1986), no.~3, 457--535.
+\href{https://doi.org/10.2307/1971333}{doi:\nolinkurl{10.2307/1971333}}.
+
+\bibitem{ChenLimits}
+Qianyu Chen.
+\emph{\href{https://qymath.github.io/Steenbrink.pdf}{Limits of {Hodge} structures via holonomic {$D$}-modules}}.
+Preprint, 2021.
+arXiv:2103.03983; the cited Lemma 8.1 is in the \href{https://qymath.github.io/Steenbrink.pdf}{61-page author manuscript}.
+
+\bibitem{DeligneHodgeII}
+Pierre Deligne.
+\emph{Th\'eorie de Hodge. II}.
+Publications Math\'ematiques de l'IH\'ES \textbf{40} (1971), 5--57.
+\href{https://doi.org/10.1007/BF02684692}{doi:\nolinkurl{10.1007/BF02684692}}.
+
+\bibitem{DemaillyRegularization}
+Jean-Pierre Demailly.
+\emph{\href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}{Regularization of closed positive currents and intersection theory}}.
+Journal of Algebraic Geometry \textbf{1} (1992), no.~3, 361--409.
+The cited locator refers to the \href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/regularization.pdf}{author manuscript}.
+
+\bibitem{DemaillyPaun2004}
+Jean-Pierre Demailly and Mihai P{\u{a}}un.
+\emph{\href{https://annals.math.princeton.edu/2004/159-3/p05}{Numerical characterization of the {K{\"a}hler} cone of a compact {K{\"a}hler} manifold}}.
+Annals of Mathematics (2) \textbf{159} (2004), no.~3, 1247--1274.
+\href{https://doi.org/10.4007/annals.2004.159.1247}{doi:\nolinkurl{10.4007/annals.2004.159.1247}}.
+
+\bibitem{Douady}
+Adrien Douady.
+\emph{\href{https://www.numdam.org/item/AIF_1966__16_1_1_0/}{Le probl{\`e}me des modules pour les sous-espaces analytiques compacts d'un espace analytique donn{\'e}}}.
+Annales de l'Institut Fourier \textbf{16} (1966), no.~1, 1--95.
+\href{https://doi.org/10.5802/aif.226}{doi:\nolinkurl{10.5802/aif.226}}.
+
+\bibitem{EnokizonoHashizumeSemistable}
+Makoto Enokizono and Kenta Hashizume.
+\emph{\href{https://arxiv.org/abs/2404.05124v3}{Semistable reduction for complex analytic spaces}}.
+Transactions of the American Mathematical Society \textbf{378} (2025), 7667--7688.
+\href{https://doi.org/10.1090/tran/9365}{doi:\nolinkurl{10.1090/tran/9365}}.
+The cited locators refer to \href{https://arxiv.org/abs/2404.05124v3}{arXiv:2404.05124v3}, 12 October 2024.
+
+\bibitem{FujikiDouady}
+Akira Fujiki.
+\emph{\href{https://ems.press/content/serial-article-files/41937}{Closedness of the {Douady} spaces of compact {K{\"a}hler} spaces}}.
+Publications of the Research Institute for Mathematical Sciences \textbf{14} (1978), no.~1, 1--52.
+\href{https://doi.org/10.2977/PRIMS/1195189279}{doi:\nolinkurl{10.2977/PRIMS/1195189279}}.
+
+\bibitem{FujinoWeakPositivity}
+Osamu Fujino.
+\emph{\href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{Notes on the weak positivity theorems}}.
+In \emph{Algebraic Varieties and Automorphism Groups}.
+Advanced Studies in Pure Mathematics 75.
+Mathematical Society of Japan, 2017, pp.~73--118.
+\href{https://doi.org/10.2969/aspm/07510073}{doi:\nolinkurl{10.2969/aspm/07510073}}.
+The cited locators refer to \href{https://www.math.kyoto-u.ac.jp/~fujino/weak-posi11.pdf}{author version 0.54, 30 June 2015}.
+
+\bibitem{FujinoAnalyticAdjunction}
+Osamu Fujino.
+\emph{\href{https://www.math.kyoto-u.ac.jp/~fujino/on-inversion-of-adjunction-pja-final.pdf}{Log canonical inversion of adjunction}}.
+Proceedings of the Japan Academy, Series A, Mathematical Sciences \textbf{100} (2024), no.~2, 7--11.
+\href{https://doi.org/10.3792/PJAA.100.002}{doi:\nolinkurl{10.3792/PJAA.100.002}}.
+
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+International Journal of Mathematics \textbf{13} (2002), no.~2, 125--135.
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+{OpenAI}.
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+{OpenAI}.
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+OpenAI Math Release preprint
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+{OpenAI}.
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+
+\end{thebibliography}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/adjoint-addition.tex

@@ -0,0 +1,247 @@
+\section{The adjoint-addition principle}\label{addition:section}
+
+The section comparison reduces subadditivity to a statement about
+$K_W+T+M$. We prove that statement here from the two positivity
+inputs specified below. Its two maps have different purposes:
+$h$ is the fibration whose fiber dimension is to be added, whereas
+$p$ expresses the nef term as a pullback from a projective base.
+
+The first input is the precise adjoint-positivity theorem for the
+line produced by the construction. We use the highest-step and
+parabolic-extension conventions of Section~\ref{rankone:section}.
+
+\begin{theorem}[Adjoint-positivity input]\label{period:adjoint}
+Let $B$ be a smooth compact connected manifold in class $\mathcal C$
+and $B^\circ\subset B$ a dense Zariski open set with SNC complement.
+Let $\mathbb V$ be a pure real-polarizable variation of Hodge
+structures on $B^\circ$ with an integral ambient lattice. Suppose
+a complex direct summand of $\mathbb V_{\mathbf C}$ has a
+rank-one highest nonzero Hodge step, with parabolic rational
+extension $L$. There are a modification $\tau:B'\to B$, a
+surjective morphism $p:B'\to S$ with connected fibers to a smooth
+projective variety, and a nef rational line bundle $P$ on $S$ such that
+\begin{equation}\label{period:conclusion}
+ \tau^*L=p^*P,\qquad K_S+aP\text{ is big for some }a\in\mathbf Q_{>0}.
+\end{equation}
+Positive rational rescaling of $L$ is allowed, and the line identity
+persists under further modifications. If $S$ is a point, $\tau^*L$
+is rationally trivial.
+\end{theorem}
+
+The full proof is in Section~\ref{period:section}. The lattice belongs to the ambient
+variation: the complex summand need not itself be defined over
+$\mathbf Q$, and the real polarization need not be rational.
+
+The second input supplies addition when the fiber already has a big
+log canonical divisor. It will be proved in Section~\ref{generaltype:section}.
+
+\begin{proposition}\label{generaltype:addition}
+Let \(h\colon V\to Y\) be a surjective holomorphic map with connected fibers
+between smooth compact complex manifolds in Fujiki class \(\mathcal C\).
+Let \(T\) and \(C\) be effective rational divisors with simple normal crossing
+support and coefficients in \([0,1]\) on \(V\) and \(Y\), respectively.
+Suppose that
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item \(K_F+T_F\) is big for a very general smooth fiber \(F\) of \(h\);
+\item \(C\leq B(h,T)\);
+\item \(T_E=1\) for every prime divisor \(E\) of \(V\) whose image in \(Y\)
+has codimension at least two.
+\end{enumerate}
+Then
+\begin{equation}\label{generaltype:inequality}
+  \kappa(V,K_V+T)\ \geq\
+  \dim F+\kappa(Y,K_Y+C).
+\end{equation}
+As usual, the assertion is automatic if the last term is \(-\infty\).
+\end{proposition}
+
+
+General-type addition will give sections after a positive ample
+twist from $S$. To remove that twist, we will need one fixed section
+with a negative ample twist. The next lemma produces it from a big
+base divisor, once nonvanishing on the general $p$-fiber is known.
+It uses weak positivity with exceptional poles and
+descent to a fixed smooth reference, following the method of
+\cite[Remark~6.5(3) and Proposition~6.6]{CampanaOrbifolds}.
+We include the descent step explicitly.
+
+\begin{lemma}[Effectivity after a big base twist]
+\label{application:weak-effectivity}
+Let $p:(W,T)\to S$ be surjective, with $W$ smooth compact in
+class $\mathcal C$, $S$ smooth projective, and $T$ a rational SNC
+boundary. Suppose one fixed divisible relative log pluricanonical
+system is nonzero on very general fibers. For every big rational
+line bundle $H$ on $S$, the rational line bundle
+$K_{W/S}+T+p^*H$ has a nonzero section in some positive degree.
+\end{lemma}
+
+\begin{proof}
+Choose a K\"ahler source modification, flatten over a smooth
+projective modification $\pi:S'\to S$, and resolve the flat main
+transform. Write $\mu:W'\to W$ and $p':W'\to S'$ for the
+resulting maps. Give $W'$ the strict transform of $T$ plus the
+reduced $\mu$-exceptional divisor, denoted $T'$. Then
+\begin{equation}\label{application:source-discrepancy}
+ K_{W'}+T'=\mu^*(K_W+T)+E,
+ \qquad E\geq0\ \text{and $\mu$-exceptional}.
+\end{equation}
+Every source prime over a subset of codimension at least two in
+$S'$ is exceptional over the fixed $W$: before resolving the
+source, flatness bounds that inverse image by codimension at least
+two. Put $R=K_{S'/S}\geq0$ and $H'=\pi^*H+R$. The exact identity
+\begin{equation}\label{application:twist-cancellation}
+ K_{W'/S'}+T'+p'^*H'
+ =\mu^*(K_{W/S}+T+p^*H)+E
+\end{equation}
+will allow descent.
+
+Choose a divisible $m>0$ such that
+$\mathcal E=p'_*\mathcal O_{W'}(m(K_{W'/S'}+T'))$ has positive
+rank. Generic base change and the hypothesis provide such an $m$.
+Fujino's weak positivity theorem applies to this log canonical pair
+and its projective base \cite[Theorem~1.1]{FujinoWeakPositivity}.
+Choose an ample Cartier divisor $A_0$ and an integer $\alpha>0$
+such that $H'-A_0/(m\alpha)$ is big. For some integer $\beta>0$,
+\[
+ (\operatorname{Sym}^{\alpha\beta}\mathcal E)^{**}
+        \otimes\mathcal O_{S'}(\beta A_0)
+\]
+is generated by global sections at the generic point.
+
+On the locally free locus of $\mathcal E$, multiplication of
+relative sections gives an evaluation map to
+\[
+ \mathcal O_{W'}\bigl(m\alpha\beta(K_{W'/S'}+T')
+                         +\beta p'^*A_0\bigr).
+\]
+It is nonzero generically, because a nonzero relative section has
+nonzero powers. Generic generation supplies a global section whose
+image is nonzero. The complement of the locally free locus has
+codimension at least two. Multiply a sufficiently divisible power
+of the evaluated section by a section of a sufficiently divisible
+multiple of
+$m\alpha\beta\,p'^*(H'-A_0/(m\alpha))$.
+The latter exists by bigness. This produces a nonzero section of a
+multiple of the left side of \eqref{application:twist-cancellation}, except possibly
+at source primes above that codimension-two set.
+
+Locally, a meromorphic frame of $\mathcal E$ expresses the evaluated
+section as a meromorphic combination of products of relative
+sections. Thus the resulting form is meromorphic, and its only
+possible divisorial poles lie above that omitted codimension-two set.
+Those primes, and $E$, are exceptional over $W$. At every prime
+of the original smooth $W$, the resulting meromorphic form therefore
+has the required order for a multiple of $K_{W/S}+T+p^*H$.
+Divisorial regularity and extension across codimension two give a
+global section on $W$. If $S$ is a point, the same conclusion is
+the assumed nonvanishing.
+\end{proof}
+
+\begin{proposition}[Adjoint addition]
+\label{addition:principle}
+Let $h:W\to Y$ be a fibration between smooth compact manifolds in
+class $\mathcal C$. Let $T,C$ be rational SNC boundaries with
+$C\leq B(h,T)$, and suppose every prime of $W$ whose image has
+codimension at least two in $Y$ has coefficient one in $T$.
+Let $p:W\to S$ be a surjective morphism with connected fibers to
+a smooth projective variety. Suppose $P$ is a nef rational line
+bundle on $S$, $M=p^*P$, and $K_S+a_0P$ is big for some rational
+$a_0>0$. If $K_W+T+M$ is big on very general $h$-fibers, then
+\[
+ \kappa(W,K_W+T+M)
+ \geq\dim(W/Y)+\kappa(Y,K_Y+C).
+\]
+\end{proposition}
+
+\begin{proof}
+The assertion is automatic if the last term is $-\infty$.
+Assume it is nonnegative, put $D=K_W+T$, and write
+$k=\dim(W/Y)$. We may first replace $W$ by a K\"ahler
+modification, adding the reduced exceptional divisor to its strict
+boundary. The log section spaces, also after tensoring by the
+pullback of $M$, are unchanged. Fiberwise bigness and the
+inf-multiplicity boundary are preserved; newly exceptional primes
+have coefficient one. Thus all hypotheses remain valid.
+
+If $S$ is a point, $M$ is rationally trivial and
+\Cref{generaltype:addition} proves the proposition. Suppose $\dim S>0$,
+and choose an ample Cartier divisor $A$ on $S$. A very general
+$h$-fiber is K\"ahler and carries the big rational line
+$(D+M)|_{W_y}$, so it is projective. Openness of its big cone
+gives a rational $c$ with $0<c<1$ for which
+$(D+cM)|_{W_y}$ is still big. This choice works on very general
+fibers: choose the first fiber outside the countably many
+cohomology and evaluation-rank exceptional loci for rational $c$
+and divisible degrees. The one full-rank system just found then
+has the same rank generically. For zero-dimensional fibers this
+argument is unnecessary.
+
+For every positive rational $\eta$, $cP+\eta A$ is ample.
+Choose a sufficiently divisible large $N$ and a general member
+of the basepoint-free system $|Np^*(cP+\eta A)|$. Dividing by
+$N$ gives an effective rational divisor $Q_\eta$ such that
+\[
+ Q_\eta\sim_{\mathbf Q}cM+\eta p^*A,
+ \qquad T+Q_\eta\ \text{is an SNC boundary}.
+\]
+Bertini is applied simultaneously to the finitely many strata of
+$T$; the member contains no old boundary component, and increasing
+$N$ bounds its new coefficients by one. Existing coefficient-one
+components remain unchanged, and increasing the source boundary
+can only increase $B(h,T)$. Thus all boundary hypotheses of
+\Cref{generaltype:addition} hold for $T+Q_\eta$. Its fiber adjoint is big
+by the choice of $c$ and nefness of the added ample pullback.
+Consequently
+\begin{equation}\label{application:perturbed-bound}
+ \kappa(W,D+cM+\eta p^*A)
+       \geq k+\kappa(Y,K_Y+C)
+       \qquad(\eta\in\mathbf Q_{>0}).
+\end{equation}
+
+The systems in \eqref{application:perturbed-bound} already have the required
+image dimension. It remains to remove their ample term without
+losing these ratios of sections. In particular one such divisor has
+a nonzero section in a fixed
+positive degree. Restriction of that section to a very general
+$p$-fiber is nonzero; both pulled-back twists become trivial
+there. Hence
+\begin{equation}\label{application:p-fiber-nonvanishing}
+ \kappa(W_s,(K_W+T)|_{W_s})\geq0
+ \quad\text{for very general }s\in S.
+\end{equation}
+One fixed relative log pluricanonical direct image for
+$p:(W,T)\to S$ therefore has positive rank.
+
+Choose rational $a>\max\{1,a_0\}$. Since $P$ is nef,
+$K_S+aP$ is big. For sufficiently small rational $\lambda>0$,
+$H=K_S+aP-\lambda A$ is still big.
+Lemma~\ref{application:weak-effectivity} now gives a nonzero section of a
+positive multiple of
+\begin{equation}\label{application:negative-twist}
+ E_-=D+aM-\lambda p^*A
+     =K_{W/S}+T+p^*H.
+\end{equation}
+Put
+\[
+ \theta=\frac{1-c}{a-c},
+ \qquad \eta=\frac{(1-c)\lambda}{a-1},
+ \qquad E_+=D+cM+\eta p^*A.
+\]
+Then $0<\theta<1$, $\eta>0$, and direct calculation gives
+\begin{equation}\label{addition:cancel}
+ D+M=(1-\theta)E_++\theta E_-.
+\end{equation}
+Fix a nonzero section $e$ of $rE_-$ for one denominator-clearing
+$r>0$. For sufficiently divisible $n$, multiplication by
+$e^{n\theta/r}$ maps
+\[
+ H^0(W,n(1-\theta)E_+)
+       \lhook\joinrel\longrightarrow H^0(W,n(D+M)).
+\]
+The exponent is an integer, and the source degrees range through a
+Veronese subsequence. On the complement of the zero divisor of $e$,
+every ratio of two sections is unchanged. These systems therefore
+have the same image dimensions before and after multiplication.
+Equation~\eqref{application:perturbed-bound} for the displayed value of $\eta$
+proves the asserted lower bound.
+\end{proof}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/applications.tex

@@ -0,0 +1,441 @@
+\section{Return to the original fibration and consequences}
+\label{application:section}
+
+The previous sections supply the two positivity statements and the
+exact section comparison. We now apply them to the original fibration,
+keeping its base fixed while changing the intermediate Iitaka space.
+
+\begin{proof}[Proof of \Cref{main:orbifold}]
+If either summand on the right is \(-\infty\), the asserted inequality
+is automatic.  Suppose henceforth that both are nonnegative, and put
+\[
+ k=\kappa(F,K_F+\Delta_F),\qquad b=\kappa(f\mid\Delta).
+\]
+
+\smallskip\noindent
+\emph{Fixing the base.}
+Use Section~\ref{setup:section} and \Cref{setup:flattening} to resolve the diagram,
+take K\"ahler models, and flatten before resolving the source. Use
+strict transforms plus reduced exceptional divisors at every source
+modification. Resolve the discriminant and boundary-stratum images
+on the base, then fix this smooth K\"ahler base \(Y\). The source
+and very general fiber log section rings are unchanged, and every
+source prime mapped into codimension at least two in \(Y\) has
+coefficient one. Put
+\[
+ C=B(f,\Delta)
+\]
+on this model. It has SNC support, and the definition of the invariant
+gives
+\begin{equation}
+ \kappa(Y,K_Y+C)\geq b.
+ \label{application:invariant-bound}
+\end{equation}
+\Cref{setup:simultaneous} preserves \(C=B(f,\Delta)\) and the
+coefficient-one condition through all further changes over \(Y\).
+
+\smallskip\noindent
+\emph{The relative Iitaka map and its Hodge line.}
+Apply \Cref{application:relative-iitaka}, obtaining
+\[
+ X\xrightarrow{g}W\xrightarrow{h}Y,
+ \qquad \dim W_y=k,
+ \qquad \kappa(G,K_G+\Delta_G)=0.
+\]
+Make \(W\) K\"ahler, resolve the diagram, and fix the current smooth
+source pair \((X_0,\Delta_0)\) as reference. Flatten over \(W\)
+and resolve the generator and boundary data as required by
+\Cref{rankone:comparison}. Thus every prime mapped by \(g\) into
+codimension at least two is exceptional over \(X_0\), and on a dense
+smooth log open the relative
+systems have rank one in all sufficiently divisible degrees.
+
+Apply \Cref{rankone:comparison,period:adjoint}. Their further
+modifications remain over \(Y\); \Cref{setup:simultaneous} preserves
+the exceptional reference, while the variation and its parabolic line
+pull back from the common open. The relative Iitaka property persists
+by \Cref{application:relative-iitaka}. On the resulting common model
+we obtain a rational line \(M\), a rational SNC boundary \(T\) in
+\([0,1]\), and a surjective map to a smooth projective variety:
+\begin{equation}
+ p:W\longrightarrow S,
+ \qquad M=p^*P,
+ \qquad P\text{ nef},
+ \qquad K_S+a_0P\text{ big for some }a_0>0,
+ \label{application:period-data}
+\end{equation}
+such that
+\begin{equation}
+ B(g,\Delta)\leq T\leq1.
+ \label{application:rankone-boundary}
+\end{equation}
+Compute \(T\) by the rank-one order formula on this final model.
+Parabolic pullback and \Cref{rankone:comparison} identify the divisible
+systems of \(K_W+T+M\) with those on \((X_0,\Delta_0)\), also
+relatively over \(Y\), preserving all ratios.
+
+\smallskip\noindent
+\emph{Relative bigness and the base boundary.}
+Set \(D_W=K_W+T\).  The systems defining the relative Iitaka image
+have image dimension \(k\) on very general \(f\)-fibers. Push these systems
+into \(D_W+M\) using \Cref{rankone:comparison}.  Their ratios are
+unchanged, so the resulting systems have image dimension \(k\) on
+very general \(h\)-fibers, whose dimension is \(k\).  Consequently
+\begin{equation}
+ (D_W+M)|_{W_y}\text{ is big for very general }y.
+ \label{application:relative-big}
+\end{equation}
+Here the systems are obtained by generic base change over an open patch
+of \(Y\) and extend on the smooth \(W_y\) by the relative section
+comparison. Adjunction identifies the restricted adjoint bundles.
+
+By \Cref{application:composition},
+\begin{equation}
+ B(h,T)\geq C.
+ \label{application:composed-boundary}
+\end{equation}
+The same lemma gives coefficient one in \(T\) for every prime of
+\(W\) mapped into codimension at least two in \(Y\). Thus both
+boundary hypotheses of \Cref{generaltype:addition} hold.
+
+The hypotheses of \Cref{addition:principle} have now all been
+verified. Its conclusion is
+\begin{equation}\label{application:final-intermediate-bound}
+ \kappa(W,K_W+T+M)\geq k+\kappa(Y,K_Y+C).
+\end{equation}
+The proof of that proposition uses a single nonzero section to cancel
+the ample perturbation and preserves the ratios defining this bound.
+
+Finally, \Cref{rankone:comparison} lifts these systems by the relative
+generators, preserving their ratios and image dimensions. Untested
+\(g\)-contracted primes are exceptional over \((X_0,\Delta_0)\),
+so the forms descend and extend there. By
+\Cref{setup:log-modification}, they then descend to the original source.
+Combining \eqref{application:invariant-bound} and
+\eqref{application:final-intermediate-bound} proves
+\[
+ \kappa(X,K_X+\Delta)
+     \geq \kappa(F,K_F+\Delta_F)+\kappa(f\mid\Delta).
+\]
+
+Bigness on a point means Iitaka dimension zero, and a zero-dimensional
+connected fiber has trivial relative line. If \(Y\) is a point the
+inequality is an identity; if the relative Iitaka fiber is a point its
+Hodge line is trivial. Together with the point-period-base case above
+and the initial \(-\infty\) convention, these observations cover all
+dimensions and values of the two summands.
+\end{proof}
+
+\subsection{The logarithmic and ordinary inequalities}
+\label{spec:section}
+
+We first compare the invariant orbifold base with a prescribed
+logarithmic base.  This makes the two classical specializations of
+Theorem~\ref{main:orbifold} immediate, while retaining control of the
+base under modification.
+
+\begin{lemma}[Comparison with a logarithmic base]
+\label{spec:base-comparison}
+Let $f:(X,\Delta)\to Y$ be a fibration of smooth compact complex
+manifolds, where $\Delta$ has rational coefficients in $[0,1]$ and
+simple normal crossing support.  Let $D_Y$ be a reduced simple normal
+crossing divisor on $Y$.  Suppose that every prime component of
+$f^*D_Y$ has coefficient one in $\Delta$.  Then
+\[
+  \kappa(f\mid\Delta)\geq\kappa(Y,K_Y+D_Y).
+\]
+\end{lemma}
+
+\begin{proof}
+Choose a neat model as in Lemma~\ref{setup:flattening}, with
+$p:X'\to X$, $q:Y'\to Y$, $f':X'\to Y'$, and
+$\Delta'=p_*^{-1}\Delta+\operatorname{Exc}(p)_{\mathrm{red}}$.
+We may arrange that
+$E_Y=(q^{-1}\operatorname{Supp}D_Y)_{\mathrm{red}}$ has simple
+normal crossings.  Every prime of $X'$ dominating a component of
+$E_Y$ has coefficient one in $\Delta'$: an exceptional prime has
+coefficient one by definition, and a nonexceptional prime is the strict
+transform of a component of $f^*D_Y$.  Thus its weighted multiplicity
+is infinite, and the inf-multiplicity rule gives
+\[
+ B(f',\Delta')\geq E_Y.
+\]
+Pullback by $q$ takes logarithmic top forms with poles along $D_Y$
+to logarithmic top forms with poles along $E_Y$.  In local coordinates
+this follows by pulling back each factor $dz_i/z_i$; the same assertion
+holds for all tensor powers.  Consequently, pullback embeds every
+plurilogarithmic system on $(Y,D_Y)$ into the corresponding system of
+$K_{Y'}+B(f',\Delta')$, preserving all ratios of sections.
+The neat-model identity~\eqref{setup:neat-formula} now proves the
+claim.
+\end{proof}
+
+\begin{corollary}[Logarithmic subadditivity]
+\label{spec:logarithmic}
+Let $f:X\to Y$ be a surjective morphism with connected fibers between
+smooth connected projective complex varieties.  Let $D_X,D_Y$ be
+reduced simple normal crossing divisors, with the zero divisor allowed,
+such that
+\[
+ \operatorname{Supp}(f^*D_Y)\subseteq\operatorname{Supp}(D_X).
+\]
+For a very general fiber $F$, put $D_F=D_X|_F$.  Then
+\begin{equation}\label{spec:log-inequality}
+ \kappa(X,K_X+D_X)
+ \geq\kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y).
+\end{equation}
+The same statement holds for compact manifolds in Fujiki class
+$\mathcal C$ and holomorphic fibrations between them.
+\end{corollary}
+
+\begin{proof}
+Theorem~\ref{main:orbifold} applies to $(X,D_X)$, and
+Lemma~\ref{spec:base-comparison} bounds its base term from below.
+If either term on the right of~\eqref{spec:log-inequality} is
+$-\infty$, the assertion follows from our convention for that value.
+\end{proof}
+
+In the projective setting, this is the usual compactification form of
+the logarithmic Iitaka conjecture.  Indeed, for a smooth
+quasi-projective variety $U$ with a smooth projective compactification
+$X$ and reduced simple normal crossing boundary $D_X=X\setminus U$,
+its logarithmic Kodaira dimension is
+$\overline\kappa(U)=\kappa(X,K_X+D_X)$
+\cite{IitakaLog}.  A dominant morphism $U\to V$ between smooth
+connected quasi-projective varieties, with geometrically connected
+general fiber, admits compatible such compactifications:
+compactify the graph and resolve the graph and boundaries.  The
+Stein factor of the resulting morphism $X\to Y$ is finite
+birational over the normal variety $Y$, hence equals $Y$.
+Thus $f$ has connected fibers, and
+$\operatorname{Supp}(f^*D_Y)\subseteq\operatorname{Supp}(D_X)$.
+For very general $v\in V$, $(X_v,D_X|_{X_v})$ compactifies $U_v$.
+Thus Corollary~\ref{spec:logarithmic} gives
+$\overline\kappa(U)\geq\overline\kappa(U_v)+\overline\kappa(V)$.
+
+Ordinary subadditivity also enters the Albanese reduction of
+\cite{OpenAILogAbundance2026}. That work combines
+it with separate nonvanishing and good-model arguments to prove log
+abundance in characteristic zero.
+
+\begin{corollary}[Ordinary subadditivity in characteristic zero]
+\label{spec:ordinary}
+Let $k$ be an algebraically closed field of characteristic zero, and
+let $f:X\to Y$ be a surjective morphism with connected fibers between
+smooth connected projective $k$-varieties.  If $F$ is its geometric
+generic fiber, then
+\begin{equation}\label{spec:ordinary-inequality}
+ \kappa(X)\geq\kappa(F)+\kappa(Y).
+\end{equation}
+More generally, the logarithmic inequality of
+Corollary~\ref{spec:logarithmic}, with the same boundary hypotheses,
+holds over $k$, with the geometric generic fiber in place of a very
+general complex fiber.
+\end{corollary}
+
+\begin{proof}
+Over $\mathbb C$, the ordinary assertion follows by taking
+$D_X=D_Y=0$ in Corollary~\ref{spec:logarithmic}.  In a projective
+family over $\mathbb C$, the Iitaka dimension of the geometric
+generic fiber equals that of a very general fiber.  To see this,
+apply cohomology and base change in each positive divisible
+pluricanonical degree.  After shrinking the base separately in each
+degree, the corresponding relative rational map also has constant
+fiberwise image dimension.  Excluding the resulting countable
+union of proper closed subsets identifies all these image
+dimensions with those of the generic fiber.
+
+For the field extension, let $K\subseteq K'$ be fields and let $V$
+be a smooth projective geometrically integral $K$-variety with a
+rational divisor $D$.  For every $m$ for which $mD$ is integral,
+flat base change gives
+\[
+ H^0(V,\mathcal O_V(mD))\otimes_K K'
+   \simeq H^0(V_{K'},\mathcal O_{V_{K'}}(mD_{K'})).
+\]
+The complete linear system on the right is the scalar extension of
+the one on the left, so nonvanishing and image dimension are
+unchanged.  Hence Iitaka dimension is invariant under extension
+of the ground field.  Applying this observation to the generic
+fiber over the function field of the base proves the corresponding
+invariance for geometric generic fibers.
+
+Descend $f$, its projective embeddings, and its boundary divisors
+to a finitely generated subfield $K\subset k$ over $\mathbb Q$.
+The descended varieties are smooth and geometrically integral;
+the boundary conditions descend as well.  The equality
+$f_*\mathcal O_X=\mathcal O_Y$ descends by proper flat base change
+and faithful flatness, and therefore also holds after an embedding
+$K\hookrightarrow\mathbb C$.  Apply the complex logarithmic
+inequality to this base change.  The preceding invariance identifies
+its three Iitaka dimensions with those over $k$, proving both
+assertions.
+\end{proof}
+
+\subsection{Kodaira dimension along the core}
+\label{spec:core}
+
+A smooth compact orbifold pair $(X,\Delta)$ in class $\mathcal C$ is
+\emph{special} if it has no meromorphic fibration onto a
+positive-dimensional base of orbifold general type; here general
+type means that the invariant base dimension equals the dimension
+of the base.  Campana constructed its \emph{core}
+\[
+ c_{(X,\Delta)}:(X,\Delta)\dashrightarrow C(X,\Delta),
+\]
+whose very general orbifold fiber is special and whose orbifold base is of general
+type, unless it is a point
+\cite[Theorem~9.1]{CampanaOrbifolds}.  In particular, the core has
+a point base exactly when $(X,\Delta)$ is special. Fiber properties
+of the core and the tower below refer to Campana's stable orbifold
+fibers: they are computed on the suitable equivalent holomorphic
+representatives furnished by these constructions
+\cite[Definition~2.49 and \S7.1]{CampanaOrbifolds}.
+A neat representative $g:(Z,\Gamma)\to T$ in the sense of
+Section~\ref{setup:section} is \emph{strictly neat} if its
+inf-multiplicity boundary on the smooth base is SNC. It is
+\emph{high} if $g$ is an orbifold morphism to this base: for every
+base prime $D$ and source prime $E$ with
+$a_E=\ord_E(g^*D)>0$, including primes contracted by $g$,
+\[
+              a_E m_\Gamma(E)\geq m_{B(g,\Gamma)}(D).
+\]
+These are the model conditions of
+\cite[Definition~3.8 and Proposition~3.10]{CampanaOrbifolds}.
+The high condition is not automatic: the inf-multiplicity base tests
+divisors dominating base primes, whereas an orbifold morphism must
+also control primes contracted into higher codimension
+\cite[Remark~3.3(2)]{CampanaOrbifolds}.
+For the core and its tower we choose compatible strictly neat and
+high representatives supplied by Campana's constructions
+\cite[Proposition~3.15 and Corollary~4.14]{CampanaOrbifolds}.
+The section comparisons of Section~\ref{setup:section} do not
+assert that arbitrary model changes preserve every stable-fiber
+property.
+
+\begin{corollary}[Kodaira dimension along the core]
+\label{spec:core-dimension}
+Let $(X,\Delta)$ be a smooth compact orbifold pair in Fujiki class
+$\mathcal C$, with rational simple normal crossing boundary
+coefficients in $[0,1]$.  Let $C=C(X,\Delta)$ be its core base and
+$(G,\Delta_G)$ a very general orbifold fiber.  Then
+\begin{equation}\label{spec:core-equality}
+ \kappa(X,K_X+\Delta)
+   =\kappa(G,K_G+\Delta_G)+\dim C.
+\end{equation}
+If $\kappa(X,K_X+\Delta)\geq0$, then
+\[
+ 0\leq\dim C\leq\kappa(X,K_X+\Delta),
+\]
+and equality in the second inequality holds precisely when
+$\kappa(G,K_G+\Delta_G)=0$.
+In particular, $\kappa(X,K_X+\Delta)=0$ implies that
+$(X,\Delta)$ is special.
+\end{corollary}
+
+\begin{proof}
+Resolve the core map, using the boundary convention of
+Lemma~\ref{setup:log-modification}.  The total and fiber Iitaka
+dimensions are preserved.  Theorem~\ref{main:orbifold} gives
+the lower bound in~\eqref{spec:core-equality}, since the invariant
+base dimension is $\dim C$.
+
+For completeness, the reverse bound is the elementary upper
+addition inequality.  For each divisible $m$, let $\phi_m$ be
+the meromorphic map defined by $|m(K_X+\Delta)|$, when this
+system is nonempty.  The map $(c,\phi_m)$ has image dimension
+at most $\dim C+\kappa(G,K_G+\Delta_G)$: on a very general
+$c$-fiber its second component is given by a subspace of
+$H^0(G,m(K_G+\Delta_G))$.  Since projection from this image
+onto the image of $\phi_m$ is surjective, the same upper bound
+holds for $\dim\phi_m(X)$.  Taking the maximum over $m$ proves
+the desired inequality.  If the fiber Iitaka dimension is
+$-\infty$, restriction to a very general fiber shows that no
+global pluricanonical section can be nonzero, so both sides of
+\eqref{spec:core-equality} are $-\infty$.
+
+Finally, if the total Iitaka dimension is nonnegative, a nonzero
+section restricts nontrivially to a very general core fiber.
+Its Iitaka dimension is therefore nonnegative.  The dimension
+bound and its equality case follow from~\eqref{spec:core-equality}.
+When the total dimension is zero, the core base is a point,
+which is equivalent to specialness.
+\end{proof}
+
+These statements recover, for the core, Campana's general-type-base addition
+theorem and its consequences
+\cite[Theorems~6.3, 6.7, and~6.8]{CampanaOrbifolds}.
+The existence of the core and the implication from Kodaira
+dimension zero to specialness are established results of his
+theory.
+
+\subsection{The core tower}
+\label{spec:core-tower-section}
+
+For a smooth orbifold pair $(Z,\Theta)$, the notation
+$\kappa_+(Z,\Theta)=-\infty$ means that every meromorphic
+fibration onto a positive-dimensional base has invariant
+orbifold Kodaira dimension $-\infty$; the condition is vacuous
+for a point.  Equivalently, one may test all dominant meromorphic
+maps: in Stein factorization, the finite orbifold canonical-divisor
+inequality shows that passing to the connected-fiber factor cannot
+decrease the invariant base dimension.  Indeed, over a divisor
+with multiplicity $m_D$, a prime with ramification index $e$ and
+orbifold multiplicity $m_E$ satisfies $m_D\leq e m_E$; the
+canonical-divisor difference has coefficient
+$e/m_D-1/m_E\geq0$.
+The \emph{$\kappa$-rational quotient} is a meromorphic fibration
+$r:(Z,\Theta)\dashrightarrow R$ with $\kappa(r\mid\Theta)\geq0$
+whose very general stable orbifold fiber has $\kappa_+=-\infty$
+\cite[Definition~5.23 and Corollary~6.14]{CampanaOrbifolds}.
+Its \emph{stable orbifold base} is the inf-multiplicity base on the
+compatible representatives just described
+\cite[Proposition~3.15, Corollary~4.14, and \S7.1]{CampanaOrbifolds}.
+
+\begin{corollary}[Campana's decomposition of the core]
+\label{spec:core-tower}
+Let $(X,\Delta)$ be as in Corollary~\ref{spec:core-dimension},
+and set $n=\dim X$.  Starting from $(X,\Delta)$, successively
+take the $\kappa$-rational quotient $r$ and the
+Moishezon--Iitaka fibration $M$ of its stable orbifold base.
+At each iteration choose such compatible representatives and equip
+the new base with the stable inf-multiplicity orbifold structure
+of the composite.
+The composite after at most $n$
+iterations is the core, up to bimeromorphic equivalence of
+fibrations:
+\[
+ c_{(X,\Delta)}=(M\circ r)^n.
+\]
+The positive-dimensional very general stable orbifold fibers of
+the $r$-steps have $\kappa_+=-\infty$, and those of the
+$M$-steps have Kodaira dimension zero.  In particular, for
+a special pair this tower ends at a point.
+\end{corollary}
+
+\begin{proof}
+Theorem~\ref{main:orbifold} establishes the hypothesis of
+Campana's Corollary~6.14, which supplies the
+$\kappa$-rational quotient.  Its scope is exactly that used
+here: smooth compact class-$\mathcal C$ pairs with rational
+coefficients in $[0,1]$
+\cite[Conjecture~6.1 and Remark~6.2]{CampanaOrbifolds}.
+Its stable base has nonnegative Kodaira dimension, so $M$
+is defined and has very general orbifold fiber of Kodaira
+dimension zero.  The new smooth base remains in class
+$\mathcal C$, with rational boundary coefficients in $[0,1]$.
+Thus the construction can be repeated.
+
+Campana's Lemma~10.1 states that each composite $M\circ r$
+has special general orbifold fiber and decreases dimension
+unless its source pair is of general type.  His
+Theorems~10.2--10.3 identify the resulting stabilized
+composite with the core.  Their sole conjectural input is
+the existence of the $\kappa$-rational quotient, now supplied
+above.  Strict dimension decrease allows at most $n$
+nontrivial iterations; identity steps give the displayed
+formula.  A point is fixed throughout.
+These are precisely the decomposition and fiber assertions of
+\cite[Lemma~10.1, Theorems~10.2--10.3, and Corollary~10.4]
+{CampanaOrbifolds}.
+\end{proof}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/general-type-addition.tex

@@ -0,0 +1,685 @@
+\section{Addition with a log-general-type fiber}\label{generaltype:section}
+
+We prove subadditivity when the fiber is of log general type.  The base
+may be nonprojective: the projective positivity theorem will be applied
+to a stable family over the image of its classifying map.
+
+We prove \Cref{generaltype:addition}.
+
+The proof constructs finitely many relative pluricanonical forms whose
+ratios give a generically finite map on a general fiber.  Their poles
+must be controlled at every vertical divisor, including divisors whose
+image on the original base has codimension at least two.  Multiplying
+these forms by base pluricanonical sections then retains both the fiber
+coordinates and the base Iitaka coordinates.
+
+We first construct a projective stable family over the image of a
+classifying map.  Positivity is applied to that family.  Its sections
+are pulled back, descended through a finite cover, and tested at
+vertical valuations.  All degrees below clear the denominators of the
+relevant divisors.
+
+\subsection{Comparison with a stable family}
+
+We use the stable-family convention of
+\cite[Notation~3.7, Remark~3.8, and Definitions~3.9 and~3.11]
+{KovacsPatakfalvi}. A stable pair has semi-log-canonical singularities
+and an ample rational log canonical divisor. A stable family over a
+normal base is flat, proper and surjective with connected stable
+fibers; its boundary support avoids the generic points and the
+codimension-one singular points of every fiber. Its relative log
+canonical divisor is rational Cartier and relatively ample, so the
+family is projective. Boundaries are pulled
+back by divisorial restriction, and Cartier multiples of the relative
+log canonical divisor commute with base change between normal bases.
+Semi-log-canonicity
+allows nonnormal fibers and tests log canonicity on the normalization
+with the conductor boundary. The general fibers in our application
+are additionally Kawamata log terminal (klt): they are normal and
+all their log discrepancies are positive. This extra condition will
+be checked before applying positivity.
+A stable family has \emph{maximal variation} when its classifying map
+to the moduli space of stable pairs is generically finite onto its image.
+
+Fiberwise finite generation gives an embedding degree separately on
+each fiber.  The next lemma selects one degree on an analytic base.
+As throughout, \emph{very general} excludes a countable union of proper
+closed analytic subsets.
+
+\begin{lemma}\label{generaltype:selection}
+Let \(Y\) be an irreducible compact complex space.  For each positive integer
+\(j\), suppose that \(q_j\colon Q_j\to Y\) is a proper surjective map from an
+irreducible compact complex space and that, over a dense analytic open
+\(U_j\subset Y\), its fibers are irreducible of a fixed dimension \(r_j\).
+Let
+\[
+  \phi_j\colon Q_j\dashrightarrow Z_j
+\]
+be a meromorphic map with irreducible projective image \(Z_j\).
+Suppose that a dense open part of \(q_j^{-1}(U_j)\), called the genuine
+frame locus, meets every fiber densely and that \(\phi_j\) is holomorphic
+there.  Let \(G_j\subset Z_j\) be a countable union of algebraically
+constructible subsets.
+
+Let \(\Lambda_j\subset U_j\) be sets whose union contains a very general
+subset of \(Y\), such that for \(y\in\Lambda_j\) all genuine frames above
+\(y\) map into \(G_j\).  Then some \(\Lambda_j\) is not contained in a
+countable union of proper closed analytic subsets of \(Y\), and for
+that index one of the constructible subsets making up \(G_j\) is dense
+in \(Z_j\).  In particular \(G_j\) contains a nonempty algebraic open
+subset of \(Z_j\).
+\end{lemma}
+
+\begin{proof}
+Some \(\Lambda_j\) is not contained in a countable union of proper
+closed analytic subsets; otherwise their union would be so contained.
+Resolve the graph of \(\phi_j\) and shrink \(U_j\), if necessary.
+This discards only a proper analytic subset of \(Y\) and preserves
+the stated properties of the general fibers.
+Write \(G_j=\bigcup_iG_{ji}\).  If no \(G_{ji}\) is dense, the inverse
+images
+\[
+  E_{ji}=\phi_j^{-1}(\overline{G_{ji}})
+\]
+are proper closed analytic subsets of the resolved \(Q_j\).
+For \(y\in\Lambda_j\), the genuine frame fiber is covered by these
+subsets.  Baire's theorem implies that one \(E_{ji}\) contains a
+relatively open part of the fiber.  Irreducibility then implies
+that \(E_{ji}\) contains the whole compact fiber.
+
+The locus
+\[
+  A_{ji}=\{y\in U_j:
+            \dim(E_{ji}\cap q_j^{-1}(y))\geq r_j\}
+\]
+is analytic, by the fiber-dimension theorem for proper maps.  It is
+proper: otherwise \(E_{ji}\) would contain the general fibers and
+hence all of \(Q_j\).  The same theorem before restriction to \(U_j\)
+shows that \(A_{ji}\), together with \(Y\setminus U_j\), is contained
+in a proper closed analytic subset of \(Y\).
+This would place \(\Lambda_j\) in countably many proper closed
+analytic subsets, a contradiction.  Finally, a dense constructible
+subset of an irreducible algebraic variety contains a nonempty
+open subset.
+\end{proof}
+\begin{lemma}[Analytic comparison with a stable family]
+\label{generaltype:stable-comparison}
+Let \(h\colon V\to Y\) be a surjective holomorphic map with connected
+fibers, where \(V\) is a smooth compact Kähler manifold and \(Y\) is
+a smooth compact manifold in class \(\mathcal C\).  Let \(A\) be an
+effective rational SNC divisor on \(V\), horizontal over \(Y\), with
+coefficients strictly less than one.  Suppose \(K_F+A_F\) is big on
+very general fibers \(F\) of positive dimension.  There are
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item a proper generically finite map \(\tau\colon Y'\to Y\), with \(Y'\)
+smooth, compact, and in class \(\mathcal C\);
+\item a surjective holomorphic map \(\rho\colon Y'\to R\), with \(R\)
+smooth and projective;
+\item a projective stable family
+\(\pi\colon(\mathcal V,\mathcal A)\to R\) of maximal variation, with klt
+general fiber;
+\end{enumerate}
+such that the main component of \(V\times_Y Y'\) is bimeromorphic
+over \(Y'\) to \(\mathcal V\times_RY'\).  Over a dense analytic open,
+this meromorphic comparison is the fiberwise log canonical model map
+for \(K_F+A_F\), with the horizontal pushforward of \(A\) as boundary.
+The space \(R\) may be a point.
+\end{lemma}
+
+\begin{proof}
+The construction has three outputs.  Relative pluricanonical images
+produce algebraic parameter data even though the base is analytic.
+A classifying map then supplies a projective stable family after a
+finite base cover.  Finally, an isomorphism cover matches that family
+with the relative image; fiberwise canonical models determine the
+horizontal discrepancy signs.
+
+\emph{Relative images and frames.}
+Set \(\mathcal L=\mathcal O_V(K_{V/Y}+A)\), understood as a rational
+line bundle.  For the factorial multiples \(m\) clearing its denominator,
+the sheaves
+\[
+  \mathcal E_m=h_*\mathcal L^{\otimes m}
+\]
+are coherent. First restrict to a dense analytic open where \(h\) and
+the horizontal boundary strata are smooth, so the proper family and
+its line bundles are flat over the base. Generic constancy of the
+fiber section dimension and cohomology and base change then identify
+the fibers of \(\mathcal E_m\) with the complete systems, separately
+for each \(m\); see \cite[\S7, nos.~4--6, S\"atze~3--5]{GrauertDirectImages}
+and its corrigendum \cite{GrauertDirectImagesCorr}.
+After a modification of \(Y\),
+the strict transform of \(\mathcal E_m\), modulo torsion, is locally free.
+The evaluation map gives a relative meromorphic map into its projective
+bundle.  Resolve this map and take its proper analytic image
+\(\mathcal Z_m\).  We also take the proper images of the horizontal
+boundary components and retain their divisorial cycle parts.
+Analytic flattening makes these embedded image and cycle data flat on a
+dense analytic open, with fixed Hilbert and cycle polynomials.
+These uses of coherent direct images and analytic flattening are
+independent of any minimal-model theorem; see \cite{HironakaFlattening}.
+
+Write \(s_m=\rank\mathcal E_m\) on this open; indices with \(s_m=0\)
+are omitted.  Its projective frame bundle has a natural compactification
+\[
+ \overline Q_m=
+ \mathbb P\!\left(\Hom(\mathcal O_Y^{\,s_m},\mathcal E_m)\right).
+\]
+Here \(Y\) denotes the modification chosen for this degree; its map to
+the original base remains proper and bimeromorphic.  The locus of
+invertible matrices modulo scalars is a principal
+\(\operatorname{PGL}(s_m)\)-bundle.
+A genuine frame embeds the corresponding fiber of \(\mathcal Z_m\)
+in a fixed \(\mathbb P^{s_m-1}\).  There is therefore a holomorphic
+Hilbert--Chow parameter map on the genuine frame locus.
+The Hilbert spaces here are the algebraic Hilbert schemes of
+projective space, viewed analytically; equivalently one may use
+their Douady realization \cite{Douady}.
+
+To extend this map meromorphically, use the existing compact relative
+image and the projective linear action in local trivializations.
+The graph of that action extends over the projective matrix
+space: inversion is given by the adjugate matrix.  Apply this construction
+to the proper family \(\mathcal Z_m\) and its marked cycles, take proper
+analytic images, and flatten by strict transform.  One obtains a proper flat family
+of embedded subspaces over a modification of \(\overline Q_m\).
+Its Hilbert--Chow map is holomorphic.  The local graph constructions
+agree on the genuine frames and hence glue.  Thus the original
+parameter map is meromorphic.  After resolving it, its image is
+compact analytic in a fixed projective Hilbert--Chow space, and is
+consequently algebraic.  Denote the irreducible image by \(Z_m\).
+This construction uses a frame bundle, not a meromorphic
+trivialization of \(\mathcal E_m\).
+
+\emph{The good parameter locus.}
+In \(Z_m\), call an embedded pair \((Z,D)\) good if it is klt and stable,
+has the prescribed boundary weights, has a complete nondegenerate
+embedding, and satisfies
+\begin{equation}\label{generaltype:embedded-adjoint}
+  \mathcal O_Z(1)\simeq
+       \mathcal O_Z\bigl(m(K_Z+D)\bigr).
+\end{equation}
+One may also impose the generic Hilbert and marking data.
+This is a countable union of algebraically constructible loci.
+Here is the parameter-space justification.  Stratify the universal
+embedded families by their Hilbert polynomials, Cartier indices,
+and base-change loci.  On each such stratum,
+\Cref{generaltype:embedded-adjoint} can be imposed by parameterizing
+the map
+\[
+  \omega_Z^{[m]}\longrightarrow\mathcal O_Z(1)
+\]
+with its prescribed divisor.  These parameter spaces are of finite
+type.  The stable-pair and klt conditions are constructible there,
+and their images are constructible by Chevalley's theorem.
+This is the embedded construction used for stable log-variety
+moduli in \cite[Section~6.A, especially Propositions~6.8 and~6.11
+and Lemma~6.12]{KovacsPatakfalvi}.  In particular no assertion
+about analytic minimal models is hidden in the good locus.
+
+Every very general fiber is projective: bigness makes it Moishezon,
+and it is Kähler. We use Moishezon's projectivity criterion, which
+is also the smooth case of \cite[Theorem~6]{NamikawaProjectivity}.
+The projective klt finite-generation theorem
+\cite[Theorem~1.1 and its consequences]{BCHM} therefore gives its log
+canonical model.  Every sufficiently large divisible pluricanonical
+degree embeds that model and satisfies
+\Cref{generaltype:embedded-adjoint}.  Exclude at once the countably
+many generic base-change loci, relative-rank loci, and generic-degree
+loci for the maps constructed above.  Let \(\Lambda_m\) consist of
+the remaining parameters for which the degree-\(m\) map actually
+constructs the fiber's log canonical model.
+Their union contains a very general subset of \(Y\), and all genuine
+frames over a point of \(\Lambda_m\) give good data.
+Apply \Cref{generaltype:selection}.  We obtain one fixed \(m\) for which
+\(\Lambda_m\) is not contained in a countable union of proper analytic
+subsets and good data hold on an algebraic open of \(Z_m\).
+
+For this degree the relative image map is generically birational.
+Indeed its generic finite degree agrees with its fiberwise degree
+outside the generic-degree exceptional set already excluded.
+Choose a point of \(\Lambda_m\) there; that degree is one.
+Retain \(\Lambda_m\): its actual canonical models will determine the
+horizontal discrepancy signs at the end of the construction.
+
+\emph{The invariant classifying map.}
+Fix the dimension, the volume read from
+\Cref{generaltype:embedded-adjoint}, and a finite coefficient set
+containing the coefficients of \(A\) and closed under sums at most one.
+The good algebraic parameter locus has a classifying map to the
+projective coarse space \(M\) of stable pairs with these data.
+If an additional parameterization of the map in
+\Cref{generaltype:embedded-adjoint} was used, its moduli graph descends:
+the embedded pair determines a unique isomorphism class, so the
+closure of that graph is generically a singleton over the good
+parameter locus.  Restricting a dense algebraic open therefore gives
+the same rational classifying map on \(Z_m\).
+It is invariant under change of projective frame.
+
+Compose with the meromorphic frame parameter map.
+The resulting map \(\overline Q_m\dashrightarrow M\) is constant
+on general frame fibers.  Its graph has a proper analytic image
+in \(Y\times M\).  This image is generically a singleton over \(Y\),
+so its projection to \(Y\) is bimeromorphic.  It is the graph of
+a meromorphic map
+\[
+  \mu\colon Y\dashrightarrow M.
+\]
+Let \(M_0\) be the closure of its image; it is projective.
+
+\emph{A stable parameter cover and the matching cover.}
+Stable moduli supplies a finite parameter cover
+\(S\to M\) from a normal scheme carrying a stable family
+\cite[Corollary~6.19]{KovacsPatakfalvi}.
+The coarse space \(M\) is projective
+\cite[Corollary~7.3]{KovacsPatakfalvi}, so its finite cover \(S\)
+is projective as well. The moduli construction for the specified
+coefficient set is given in
+\cite[Definition~6.2, Proposition~6.4, and
+Proposition~6.14]{KovacsPatakfalvi}.
+Resolve \(\mu\), and normalize a component of the pullback of
+\(S\to M\) dominating \(Y\).  Let \(R_0\) be its image in \(S\).
+The restricted stable family over \(R_0\) has maximal variation,
+because \(R_0\to M_0\) is finite.
+
+At this point the projective family exists over \(R_0\).
+Equality of coarse moduli points identifies its general fibers with
+those of the original analytic image, but does not identify the
+families. We construct a further generically finite cover carrying
+such an identification.
+
+On a dense analytic open of the comparison base, embed the original
+flat image by the complete system of
+\(\mathcal O_{\mathcal Z_m}(k)\), and the pulled-back stable family
+by its \(km\)-adjoint system, for a common sufficiently large and
+divisible \(k\). The two sheaves of sections are coherent direct
+images of compact families; generic cohomology and base change make
+them locally free after shrinking. On a common compact modification
+of the base, make their torsion-free strict transforms locally free,
+resolve the evaluation maps, and flatten the embedded image families
+and their weighted boundary cycles. This gives holomorphic relative
+Hilbert--Chow data. The modification concerns the base itself, not a
+space of choices of frames.
+
+On every general good fiber, \Cref{generaltype:embedded-adjoint}
+identifies the two polarizations. Thus the projective linear maps
+identifying the two embedded varieties and their weighted boundary
+cycles form a nonempty Isom set. This set is finite, since stable
+pairs have finite automorphism groups
+\cite[Propositions~6.5 and~6.8]{KovacsPatakfalvi}; completeness and
+nondegeneracy of the embeddings exclude additional ambient
+stabilizers. At this stage we use only the embedded analytic image
+and its polarization. Its \(\Q\)-Gorenstein family structure will
+be transported from the stable family by the identification.
+
+To compactify these Isom sets, work in local trivializations of the
+two section bundles. In the product of their projective parameter
+spaces with the projective matrix space, take the algebraic closure
+of the incidence relation for invertible matrices identifying the
+embedded varieties and weighted boundary cycles, after clearing the
+fixed coefficient denominators. Pull back this closure by the two
+Hilbert--Chow maps. Equivariance under changes of both frames glues
+the local incidence spaces into a proper analytic space in the
+projective bundle of homomorphisms between the section bundles; no
+meromorphic trivialization is used. Retain its compact irreducible
+components that dominate the base and meet the invertible locus,
+and normalize them. The invertible locus has the finite Isom fibers
+just described, so the resulting cover is proper and generically
+finite. Its tautological isomorphism identifies the generic families,
+including the stable family's \(\Q\)-Gorenstein structure; its graph
+therefore extends meromorphically over the compact base.
+
+Finally resolve \(R_0\) projectively and resolve the graph of the
+map from the comparison cover to it.  Denote the resulting spaces
+by \(R\) and \(Y'\).  Stable families and their divisible relative
+adjoint line bundles commute with base change, with boundary understood
+by divisorial restriction \cite[Remark~3.8]{KovacsPatakfalvi}.
+Thus the pullback family over \(R\) remains stable; it has klt general
+fibers and maximal variation.  All covers and modifications used above stay
+in class \(\mathcal C\).
+
+On a common resolution we can now compare the source adjoint
+divisor with the pullback of the stable relative adjoint divisor,
+which is rational Cartier.  There are only finitely many horizontal
+prime coefficients in their difference.  On fibers above
+\(\Lambda_m\), away from the finitely many generic comparison
+exceptions, they are the log canonical model discrepancy coefficients.
+They are therefore nonnegative and supported on divisors exceptional
+over the stable model.  The same statements hold horizontally in
+the family.  Remove the images of the remaining vertical discrepancy
+divisors.  On the resulting dense analytic open this is the log
+canonical model comparison, with its divisorial pushforward boundary.
+This proves the asserted meromorphic comparison without relative
+analytic finite generation.  If \(M_0\) is a point, the construction
+uses the fixed stable pair and \(R\) is a point.
+\end{proof}
+
+\subsection{Relative forms and vertical valuations}
+
+For a meromorphic section \(\sigma\) of
+\(\mathcal O_V(bK_{V/Y})\), its horizontal pole bound \(bA\)
+means that \(\ord_E(\sigma)\geq-bA_E\) at every horizontal prime,
+using a local nonvanishing base volume to interpret relative forms.
+At a vertical prime over a divisor \(D\subset Y\), write
+\(\omega_D\) for a local nonvanishing logarithmic top form on the
+base with a simple pole along \(D\), and no other pole at its generic
+point.  We say that \(\sigma\) has the relative logarithmic bound at
+this prime if
+\begin{equation}\label{generaltype:relative-log-bound}
+  \ord_E\bigl(\sigma\wedge h^*\omega_D^{\,b}\bigr)\geq-b.
+\end{equation}
+The notation \(\wedge\) denotes the relative-times-base identification
+of the corresponding pluricanonical line bundles.
+
+\begin{lemma}\label{generaltype:stable-orders}
+In the situation of \Cref{generaltype:stable-comparison}, pull back a
+holomorphic section of a divisible multiple of
+\(K_{\mathcal V/R}+\mathcal A\), and compare it with relative pluriforms
+on the main component of
+\(V\times_Y Y'\).
+It satisfies the horizontal bound given by the pullback of \(A\).
+Moreover, on a common resolution it satisfies
+\Cref{generaltype:relative-log-bound} at every prime lying over a
+divisor of any smooth modification of the comparison base.
+\end{lemma}
+
+\begin{proof}
+On general fibers, pullback from the log canonical model lies in the
+original log pluricanonical system.  The effective canonical-model
+discrepancy on a common resolution therefore proves the horizontal bound.
+
+For the vertical assertion, first fix a smooth modification of the
+comparison base and a prime divisor on it.  The stable family and its
+divisible relative adjoint line bundles commute with this base change.
+Work transversely to a general point of that divisor.  The resulting
+one-parameter family is still stable.  This order of operations is
+what permits the same bound at divisors exposed only on a higher base
+model.  Its total pair with the reduced
+central fiber added is log canonical.  More explicitly, choose the
+disk with klt fibers away from its origin.  Its total space is normal:
+flatness over the smooth disk and the \(S_2\) property of the fibers
+give \(S_2\).  A codimension-one point on the central fiber is a
+generic point of one of its reduced components.  Its local ring is
+one-dimensional, and its quotient by the disk parameter is a field;
+its maximal ideal is consequently generated by that parameter, so
+this local ring is regular.  Away from the central fiber the total
+space is normal, because its fibers are klt.  Serre's criterion now
+gives normality.
+Compatibility of the relative adjoint divisor with the stable fiber
+identifies the adjunction different on the normalization of the central
+fiber with its stable boundary and conductor.  The resulting pair is
+log canonical because the central fiber is stable; see
+\cite[Lemma~2.10 and Corollary~2.11]{PatakfalviFibered} for this
+normalization-and-conductor identity.
+The analytic Cartier-divisor inversion theorem
+\cite[Theorem~1.1]{FujinoAnalyticAdjunction} consequently applies.
+The argument also applies after any finite ramified disk change.
+
+Let \(t\) be the disk parameter and let \(\xi\) denote the
+pulled-back relative section in degree \(b\).  Then
+\(\xi\wedge(dt/t)^b\) is a section of the total log canonical
+multiple with the central fiber added.  On a log resolution the
+log canonical discrepancy bound says that this form has pole
+order at most \(b\) at every vertical prime.  A further common
+resolution with the original family preserves this bound.
+This is \Cref{generaltype:relative-log-bound}.  Smooth transverse
+parameters do not change any of these orders.
+\end{proof}
+
+\begin{lemma}\label{generaltype:relative-system}
+Under the hypotheses of \Cref{generaltype:stable-comparison}, there
+is a positive integer \(b_0\) and a finite-dimensional space
+\[
+  \mathcal S\subset
+  H^0_{\mathrm{mer}}\bigl(V,\mathcal O_V(b_0K_{V/Y})\bigr)
+\]
+whose associated meromorphic map has rank \(\dim F\) on general
+\(h\)-fibers, with the following properties:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item every member has horizontal pole bound \(b_0A\);
+\item every member satisfies the relative logarithmic bound
+at primes over base divisors;
+\item the same bound may be tested after any modification of
+the base, at a divisor exposed on that modification.
+\end{enumerate}
+The same assertions hold for the systems of products of members
+of \(\mathcal S\).
+\end{lemma}
+
+\begin{proof}
+Use \Cref{generaltype:stable-comparison}.  If \(\dim R>0\), the
+stable-family bigness theorem
+\cite[Theorem~8.1 and Corollary~8.3]{KovacsPatakfalvi} applies:
+the base is normal projective, the family is stable and has
+maximal variation, and its general fiber is klt.
+Thus \(K_{\mathcal V/R}+\mathcal A\) is big.
+Choose finitely many sections of one sufficiently divisible
+multiple whose map is generically finite.  Its image has dimension
+\(\dim F+\dim R\), and its restriction to a general \(\pi\)-fiber
+has rank \(\dim F\).  If \(R\) is a point, choose such
+sections of the ample log canonical divisor of the fixed stable
+pair instead.  Pullback and the meromorphic comparison preserve these
+image dimensions and give the pole bounds of
+\Cref{generaltype:stable-orders}.
+
+Replace the comparison cover by a Galois cover and resolve the
+diagrams equivariantly.  To do this analytically, first take its Stein
+factorization over \(Y\); its finite factor has the same cover over a
+dense open set.  Over the finite étale locus, the Galois closure is a
+component of a sufficiently large fiber product of this finite factor.
+Take the corresponding compact component, normalize, and then resolve
+the graph of its map to the comparison cover.  Equivariant resolution
+gives the required diagram; write \(G\) for its finite group.
+This construction uses finite analytic maps and does not require the
+meromorphic function field of \(Y\) to have transcendence degree
+\(\dim Y\).
+
+Let \(\mathcal B\) be the graded \(\C\)-algebra generated by the
+chosen finite set of relative sections and all its \(G\)-conjugates.
+Every homogeneous element has the required horizontal and vertical
+linear pole bounds: products add the orders and sums cannot
+decrease the lower bounds.  The algebra is integral and finite over
+\(\mathcal B^G\).  Indeed each homogeneous generator satisfies its
+orbit polynomial with invariant coefficients, and invariants of
+a finite group acting on a finitely generated algebra are finitely
+generated.  The corresponding map
+\(\operatorname{Proj}\mathcal B\to\operatorname{Proj}\mathcal B^G\)
+is finite.  Before taking invariants, the system contains the sections
+pulled back from \(\mathcal V\), whose restriction to a general source
+fiber has image dimension \(\dim F\).  A finite map preserves the
+dimension of the image of that fiber, even though \(G\) may permute
+the fibers above one point of \(Y\).  Choose a Veronese degree for
+\(\mathcal B^G\) generated by finitely many elements of that single
+degree.  Their ratios therefore still have rank \(\dim F\) on general
+fibers.  Denote their pluricanonical degree by \(b_0\).
+The same finite quotient also preserves the total image dimension,
+which is at least \(\dim F+\dim R\).
+
+Use the natural identification of relative canonical bundles on the
+common smooth locus for every base change or modification.  On the
+dense étale locus, invariant relative
+tensors therefore descend to \(V\).  They extend meromorphically
+across the remaining locus by finite analytic descent.  The
+horizontal bounds descend because the comparison cover is
+generically étale along each horizontal divisor.
+
+For completeness, the vertical order calculation is exact.
+Let \(E'\) lie over \(E\), with ramification index \(e\).
+For a degree-\(b_0\) top pluriform \(\alpha\),
+\begin{equation}\label{generaltype:ramified-order}
+  \ord_{E'}(\alpha_{\mathrm{pullback}})+b_0
+       =e\bigl(\ord_E(\alpha)+b_0\bigr).
+\end{equation}
+A logarithmic top form on the base pulls back to a nonvanishing
+logarithmic top form at a prime dominating its divisor.
+Apply \Cref{generaltype:ramified-order} to the relative form
+wedged with this base logarithmic form.
+The bound upstairs is therefore equivalent to
+\Cref{generaltype:relative-log-bound} downstairs.
+For (iii), pull the chosen Galois comparison cover and its original
+sections to a common refinement over the modified base.
+Lemma~\ref{generaltype:stable-orders} applies to these same pullbacks
+at primes above the newly exposed base divisor; its bounds pass to
+their conjugates, products, and invariant combinations.  The
+ramification calculation therefore gives the bound for the pullback
+of each fixed descended member of $\mathcal S$.
+Products add the order inequalities, and taking a full product system
+is a Veronese operation on its image, so products preserve the relative
+rank as well.
+\end{proof}
+
+\begin{lemma}\label{generaltype:expose-valuation}
+Let \(h\colon V\to Y\) be a proper surjective holomorphic map
+of irreducible compact complex spaces, with \(Y\) smooth.
+If a prime divisor \(E\) of a smooth \(V\) does not dominate \(Y\),
+there is a modification of the base and a dominating main-component
+model on which the strict transform of \(E\) maps onto a prime
+divisor of the modified base.
+\end{lemma}
+
+\begin{proof}
+Apply analytic flattening to \(h\), by strict transform, and resolve
+the modified base.  Pullback preserves the flatness of the intermediate
+family, whose fibers have the generic relative dimension \(d\).
+This intermediate source is proper and bimeromorphic over \(V\).
+On a resolution dominating it, the strict transform of \(E\) remains
+a divisor, because a proper modification of the smooth \(V\) is an
+isomorphism at the generic point of \(E\).  Its image in the flat
+intermediate source has dimension \(\dim V-1\): it still dominates
+\(E\).  The fiber-dimension bound consequently gives an image on the
+new base of dimension at least
+\(\dim V-1-d=\dim Y-1\).
+That image is proper, since its projection to \(Y\) is contained in
+\(h(E)\), and properness makes it a closed analytic set.  It is
+therefore a prime divisor.  Further source resolutions retain the
+same generic image and do not change the valuation of \(E\).
+\end{proof}
+
+\subsection{Proof of the addition proposition}
+
+\begin{proof}[Proof of \Cref{generaltype:addition}]
+We may suppose \(\kappa(Y,K_Y+C)\geq0\).
+Choose a Kähler modification \(\nu\colon\widetilde V\to V\),
+resolve the boundary,
+and put
+\[
+  \widetilde T=\nu_*^{-1}T+\operatorname{Exc}(\nu)_{\mathrm{red}}.
+\]
+By \Cref{setup:log-modification,setup:modification-compatibility},
+pullback and descent identify the divisible log pluricanonical rings
+on the total spaces and on their very general fibers.  In particular,
+fiberwise bigness is preserved, and
+\(B(h\circ\nu,\widetilde T)=B(h,T)\).
+Every prime over base codimension at least two still has coefficient
+one: it is either a strict transform of such a prime or is newly
+exceptional.  Write \((V,T)\) for this Kähler model; its sections
+descend to the original pair.
+
+If \(\dim F=0\), the map is bimeromorphic.  Pull back sections
+of \(K_Y+C\).  At a prime dominating a base divisor the
+multiplicity is one and the condition \(C\leq B(h,T)\) gives
+the required pole bound.  At an exceptional prime the
+log canonical discrepancy bound for \((Y,C)\), together with
+\(T_E=1\), gives the same conclusion.  Thus pullback preserves
+all the base systems and proves the assertion.  If \(Y\) is a
+point, the assertion is simply the assumed bigness.  We may
+therefore assume both dimensions are positive.
+
+Write \(T^{\mathrm h}\) for the horizontal part of \(T\).
+Choose a rational \(\epsilon>0\) sufficiently small and set
+\[
+  A=(1-\epsilon)T^{\mathrm h}.
+\]
+Its coefficients are strictly less than one.
+To make \(K_F+A_F\) big uniformly on very general fibers, consider
+\(\epsilon=1/j\) and all denominator-clearing degrees.  Choose a fiber
+outside their countably many generic base-change and relative-rank
+exceptional sets.
+Openness of the big cone on that projective fiber gives one
+choice of \(\epsilon\) and one degree with full image dimension.
+The generic-rank choice makes the same degree have full rank
+on general fibers.  Notice that no vertical coefficient has
+been lowered, and \(C\) has not been changed.
+
+Apply \Cref{generaltype:relative-system} to \(A\), and write
+\(\sigma\) for a member of its system in degree \(b\), including
+any product degree.  Let
+\[
+  \eta\in H^0\bigl(Y,\mathcal O_Y(b(K_Y+C))\bigr).
+\]
+The relative-times-base product
+\(\sigma\wedge h^*\eta\) is a meromorphic section of
+\(\mathcal O_V(bK_V)\).  We check that its poles are bounded by
+\(bT\) at every prime.
+
+There is nothing further to check at horizontal primes, since
+\(A\leq T^{\mathrm h}\).  Suppose next that \(E\) maps onto a
+prime divisor \(D\subset Y\), and put
+\[
+  a_E=\ord_E(h^*D).
+\]
+Relative to a local nonvanishing logarithmic base volume
+\(\omega_D\), the section \(\eta\) vanishes to order at least
+\(b(1-C_D)\).  The relative logarithmic bound gives
+\begin{equation}\label{generaltype:orbifold-order}
+  \ord_E(\sigma\wedge h^*\eta)
+        \geq -b+b\,a_E(1-C_D).
+\end{equation}
+The inf-multiplicity convention is precisely
+\[
+ B(h,T)_D
+   =1-\max_{E\mapsto D}\frac{1-T_E}{a_E}.
+\]
+Consequently \(C_D\leq B(h,T)_D\) implies
+\[
+ a_E(1-C_D)\geq1-T_E,
+\]
+and \Cref{generaltype:orbifold-order} is at least \(-bT_E\).
+This uses no integrality condition on the boundary multiplicities.
+
+Finally, suppose \(\operatorname{codim}_Y h(E)\geq2\).
+Expose a base divisor \(D'\) below the strict transform of \(E\)
+by \Cref{generaltype:expose-valuation}, and resolve the diagrams.
+The pullback of \(\eta\) has at most a logarithmic pole along
+\(D'\): this is the divisorial discrepancy inequality for the
+log canonical smooth pair \((Y,C)\).
+Property (iii) of \Cref{generaltype:relative-system} therefore
+gives order at least \(-b\) at the strict transform of \(E\).
+The order is unchanged, since source modifications are isomorphisms
+at the generic point of \(E\).
+The assumption \(T_E=1\) supplies exactly this allowance.
+We have proved, at all prime divisors, that
+\begin{equation}\label{generaltype:product-sections}
+ \sigma\wedge h^*\eta
+       \in H^0\bigl(V,\mathcal O_V(b(K_V+T))\bigr).
+\end{equation}
+On a smooth space these divisorial bounds imply holomorphic
+extension of the corresponding log pluricanonical section.
+
+Choose a degree in which the complete base system has image
+dimension \(\kappa(Y,K_Y+C)\), and pass to a common multiple
+with \(b_0\) in \Cref{generaltype:relative-system}.
+The product system of relative sections still has image dimension
+\(d=\dim F\) on a general fiber.  The corresponding product of the
+chosen base system has image dimension \(\kappa(Y,K_Y+C)\), since a
+Veronese embedding preserves its image.  In this common degree write
+the nonzero relative generators as \(\sigma_i\) and the base
+generators as \(\eta_j\).
+The total system in \Cref{generaltype:product-sections} contains
+all \(\sigma_i\wedge h^*\eta_j\).
+On the open set where \(\sigma_0\) and \(\eta_0\) are nonzero,
+ratios in this system include
+\[
+  \frac{\sigma_i}{\sigma_0},
+  \qquad
+  h^*\!\left(\frac{\eta_j}{\eta_0}\right).
+\]
+Its image therefore maps onto the base Iitaka image, and over
+a general point of that image it contains the \(d\)-dimensional
+image of a general \(h\)-fiber.
+The dimension-of-fibers inequality gives image dimension at
+least \(d+\kappa(Y,K_Y+C)\).  This proves
+\Cref{generaltype:inequality} on the Kähler model.
+The initial log pluricanonical descent then proves it on the
+given smooth source.
+\end{proof}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/introduction.tex

@@ -0,0 +1,312 @@
+\section{Introduction}\label{intro:section}
+
+The Kodaira dimension measures the growth of pluricanonical forms.
+For a surjective morphism $f:X\to Y$ of smooth projective complex varieties
+with connected fibers, Iitaka's subadditivity problem asks whether
+\[
+                  \kappa(X)\geq\kappa(F)+\kappa(Y),
+\]
+where $F$ is a general fiber. The logarithmic version permits poles
+along reduced boundary divisors and therefore applies to open
+varieties. We prove a common generalization in which rational boundary
+coefficients and multiple fibers both contribute to the inequality.
+
+Iitaka's divisor-dimension and logarithmic theories provide the
+language for this question \cite{IitakaOriginal,IitakaLog}; an
+explicit logarithmic subadditivity conjecture appears in
+\cite[\S3]{IitakaClassification}.
+The lower bound requires global pluricanonical forms that retain
+the growth of both the fibers and the base. Direct-image positivity
+established central cases: Kawamata treated curve bases and fibers
+admitting good minimal models, Viehweg developed weak positivity
+and addition over a base of general type, and Koll\'ar proved
+addition for general-type fibers
+\cite{KawamataCurves,KawamataKodaira,ViehwegAdditivity,Kollar87}.
+Weak positivity controls symmetric powers after small positive
+ample twists; a general-type base provides enough canonical
+positivity to absorb them.
+
+Other advances exploit dimension or the geometry of the base:
+Birkar treated total dimension at most six, Cao--P\u{a}un and
+Hacon--Popa--Schnell treated abelian or maximal-Albanese-dimension
+bases, and Cao treated projective klt pairs over surfaces
+\cite{BirkarIitakaSix,CaoPaunAbelian,HaconPopaSchnellAbelian,CaoSurfaces}.
+Thus the established cases draw positivity from different parts of
+the fibration, rather than imposing one uniform condition on it.
+
+For logarithmic pairs, Kov\'acs--Patakfalvi proved the projective
+log-general-type-fiber case \cite[Theorem~9.6]{KovacsPatakfalvi},
+while Hashizume proved reduced-SNC subadditivity for fibers with
+abundant log canonical divisor \cite[Theorem~1.2]{Hashizume}.
+Maehara had established logarithmic addition over a base of log
+general type; Fujino gives a proof through twisted weak positivity
+\cite[Theorem~1.9]{FujinoWeakPositivity}.
+Campana proved addition over an orbifold base of general type, and
+Wang proved klt K\"ahler subadditivity over complex tori, including
+an inf-multiplicity refinement
+\cite[Theorem~6.3]{CampanaOrbifolds}
+\cite[Main Theorem~A and Theorem~E]{WangKahlerIitaka}.
+The present argument uses general-type addition only after a relative
+Iitaka reduction; the original fiber need not be of general type
+or admit a good minimal model.
+
+Earlier preprints also state general ordinary inequalities.
+Tsuji states ordinary subadditivity following a direct-image generation
+theorem \cite[Theorems~1.9 and~1.13]{TsujiDirectImages}; his
+Remark~1.14 distinguishes an unproved variation refinement.
+Maehara states an ordinary determinant inequality involving variation
+\cite[\S7, Theorem~8]{MaeharaIitaka}. This later preprint claim is
+distinct from his logarithmic general-type-base theorem above.
+Neither preprint statement is a proof input here.
+
+\subsection{The orbifold base and the main theorem}
+
+Multiple fibers carry information that the ordinary base omits.
+Campana's orbifold base records it through inf multiplicities, and
+his birational invariant gives the corresponding form of Iitaka's
+conjecture \cite{Campana2004,CampanaOrbifolds}. We specify that
+invariant before stating the theorem.
+
+\label{setup:orbifold-base}
+A fibration is a proper surjective holomorphic map with connected
+fibers. A rational SNC boundary on a smooth manifold is a finite
+sum $\Delta=\sum_i\delta_i\Delta_i$, with
+$\delta_i\in\mathbf Q\cap[0,1]$ and simple normal crossing support.
+Write $\Delta_E$ for the coefficient of a source prime $E$. Set
+\[
+ m_\Delta(E)=\frac1{1-\Delta_E},
+ \qquad \frac1{0}=\infty,
+\]
+where $\Delta_E=0$ off the boundary. If the base of
+$f:(X,\Delta)\to Y$ is smooth, define
+\begin{equation}\label{setup:inf-boundary}
+\begin{gathered}
+ m(f,\Delta;D)=\min_{E\mapsto D}
+     \bigl\{\operatorname{ord}_E(f^*D)m_\Delta(E)\bigr\},
+ \qquad
+ B(f,\Delta)=\sum_D\left(1-\frac1{m(f,\Delta;D)}\right)D.
+\end{gathered}
+\end{equation}
+Here $D$ ranges over base prime divisors, $E$ over source primes
+dominating $D$, and $1/\infty=0$. The minimum is nonempty and
+only finitely many coefficients of $B(f,\Delta)$ are nonzero.
+There is no divisibility condition on these multiplicities.
+
+For smooth sources and bases, an elementary change of model is a
+commutative diagram of fibrations
+\[
+\begin{tikzcd}
+ (X',\Delta') \arrow[r,"u"] \arrow[d,"f'"'] &
+ (X,\Delta) \arrow[d,"f"]\\
+ Y' \arrow[r,"v"'] & Y
+\end{tikzcd}
+\]
+where $u,v$ are proper bimeromorphic holomorphic maps,
+$u_*\Delta'=\Delta$, and both boundaries are rational SNC
+boundaries in $[0,1]$. We require, for every prime $D$ on $X$
+and prime $E$ on $X'$ with $t=\operatorname{ord}_E(u^*D)>0$,
+\begin{equation}\label{setup:orbifold-morphism}
+                     t\,m_{\Delta'}(E)\geq m_\Delta(D).
+\end{equation}
+We permit $\infty\geq\infty$ and impose no condition for $t=0$.
+Write $f'\sim f$ for the equivalence relation generated by these
+diagrams, allowing arrows in either direction. Define
+\begin{equation}\label{setup:invariant-base}
+ \kappa(f\mid\Delta)=
+ \inf_{f':(X',\Delta')\to Y'\sim f}
+       \kappa\bigl(Y',K_{Y'}+B(f',\Delta')\bigr).
+\end{equation}
+For a normal singular base $Y$, resolve $Y$, resolve the main
+component of the fiber product with $X$, and give the resulting
+smooth source the strict transform of $\Delta$ plus the reduced
+exceptional divisor over $X$. After resolving this boundary, apply
+the preceding definition. It is independent of the chosen resolution
+\cite[Definitions~2.1, 2.3, 3.2, 4.7, and~4.10;
+Remark~4.13 and Corollary~4.14]{CampanaOrbifolds}.
+
+For a rational line bundle $L$, $\kappa(Z,L)$ is the maximal image
+dimension of its complete linear systems in sufficiently divisible
+positive degrees, and is $-\infty$ if all these systems are empty.
+A point has Kodaira dimension zero, and
+$(-\infty)+a=-\infty$, including $a=-\infty$.
+Fujiki class $\mathcal C$ consists of compact complex spaces
+bimeromorphic to compact K\"ahler spaces.
+
+\begin{theorem}[Orbifold subadditivity]\label{main:orbifold}
+Let $X$ be a smooth compact connected complex manifold in Fujiki
+class $\mathcal C$, let $\Delta$ be a rational SNC boundary, and
+let $f:X\to Y$ be a surjective holomorphic map with connected
+fibers onto a normal compact irreducible complex space.
+For a very general smooth fiber $F$, put $\Delta_F=\Delta|_F$.
+Then
+\begin{equation}\label{main:inequality}
+ \kappa(X,K_X+\Delta)
+ \geq\kappa(F,K_F+\Delta_F)+\kappa(f\mid\Delta).
+\end{equation}
+\end{theorem}
+
+This is a positive resolution of Campana's orbifold Iitaka
+conjecture in its rational SNC, Fujiki-class-$\mathcal C$
+formulation, including coefficient one
+\cite[Conjecture~6.1 and Remark~6.2]{CampanaOrbifolds}.
+Here a very general fiber is chosen outside a countable union of
+proper closed analytic subsets, including the critical values and
+the exceptional images of boundary strata.
+
+For reduced SNC divisors $D_X,D_Y$ satisfying
+$\operatorname{Supp}(f^*D_Y)\subseteq\operatorname{Supp}(D_X)$,
+the invariant base term is at least $\kappa(Y,K_Y+D_Y)$.
+Theorem~\ref{main:orbifold} therefore gives
+\[
+ \kappa(X,K_X+D_X)
+ \geq\kappa(F,K_F+D_X|_F)+\kappa(Y,K_Y+D_Y)
+\]
+for fibrations between smooth compact class-$\mathcal C$ manifolds.
+For a smooth quasi-projective variety $U$, its logarithmic Kodaira
+dimension is $\bar\kappa(U)=\kappa(X,K_X+D_X)$ on a smooth
+projective compactification with reduced SNC boundary
+$D_X=X\setminus U$. Compatible compactifications give logarithmic
+subadditivity for dominant morphisms of smooth quasi-projective
+complex varieties with geometrically connected general fiber.
+Empty boundaries give ordinary subadditivity. The projective
+ordinary and logarithmic statements also hold over algebraically
+closed fields of characteristic zero, using geometric generic fibers.
+
+\subsection{The reduction and the two positivity inputs}
+
+The proof first replaces the original fiber by a general-type
+object without losing its complete pluricanonical systems.
+Assume the two terms on the right of Theorem~\ref{main:orbifold}
+are nonnegative, and put $k=\kappa(F,K_F+\Delta_F)$.
+A relative Iitaka map gives, after the model changes of Section~\ref{setup:section},
+\[
+ X\xrightarrow{g}W\xrightarrow{h}Y,
+ \qquad \dim W_y=k,
+ \qquad\kappa(G,K_G+\Delta_G)=0
+\]
+for a very general $g$-fiber $G$.
+
+The canonical-bundle-formula strategy separates the contribution
+of singular fibers from a moduli contribution on $W$.
+Fujino--Mori developed a higher-dimensional logarithmic formula,
+and Ambro organized its discriminant and moduli parts birationally
+\cite{FujinoMori,AmbroBoundary,AmbroModuli}.
+Our use of this strategy requires an exact comparison on the fixed
+original source. Section~\ref{setup:section} obtains it from the orders of relative
+pluriforms. A sufficiently divisible pluricanonical system on $G$
+is one-dimensional. Taking a root of its generator on a cyclic
+cover gives a differential form, and the line it spans belongs to
+the cohomology of that cover. Tensor descent and extension across
+the boundary produce a rational line bundle $M$ on $W$.
+The orders of the same form determine a boundary $T$ such that
+\[
+ B(g,\Delta)\leq T\leq1,
+ \qquad
+ H^0\bigl(X,m(K_X+\Delta)\bigr)
+ \simeq H^0\bigl(W,m(K_W+T+M)\bigr)
+\]
+in sufficiently divisible degrees. The comparison also holds
+relatively over $Y$. Thus $K_W+T+M$ is big on the
+$k$-dimensional fibers of $h$.
+
+Two inputs turn this fiberwise bigness into the required global
+lower bound. The first concerns the line $M$: the period map,
+which records how the Hodge structures of the covers vary, supplies
+a second fibration $p:W\to S$, with $S$ smooth projective, and
+a nef rational line bundle $P$ such that
+\[
+                      M=p^*P,\qquad K_S+aP\ \text{is big}
+\]
+for some positive rational $a$. The maps $h$ and $p$ have different
+jobs, as Figure~\ref{fig:iitaka-period} shows. The second input is
+log-general-type addition: a log canonical divisor that is big on
+the fibers of $h$ has Kodaira dimension at least their dimension
+plus the orbifold contribution of $Y$.
+
+\input{figures/two-maps}
+
+Section~\ref{addition:section} states both inputs precisely and proves the intervening
+addition principle. Write $D_W=K_W+T$. For a fixed rational $c<1$
+close to one, general-type addition gives sufficiently many ratios
+of sections of $D_W+cM+\eta p^*A$, where $A$ is ample on $S$
+and $\eta>0$ is rational. Weak positivity then supplies one fixed
+section of $D_W+aM-\lambda p^*A$ for suitable $a>1$ and
+$\lambda>0$. A positive rational combination cancels the ample
+terms and gives coefficient one on $M$. Multiplication by powers
+of that fixed section preserves the ratios of the first system.
+This is an exact section argument; it requires neither abundance
+of $P$ nor a limiting assertion for Kodaira dimensions.
+
+The full proofs of the inputs follow this calculation.
+Section~\ref{period:section} proves adjoint positivity of the Hodge line. It combines
+the curvature and boundary theory of Griffiths, Schmid, and
+Cattani--Kaplan--Schmid with a projective period quotient
+\cite{GriffithsPeriodsIII,Schmid,CKS}.
+The construction retains suitable rational adjoint factors of the
+ambient integral variation, associated with its monodromy, from which
+a positive power of the chosen line can be recovered. Their full
+periods give the quotient, using Villadsen's class-$\mathcal C$
+compactification theorem and the period-quotient approach of
+Bakker--Brunebarbe--Tsimerman and Sommese; curvature and a volume
+argument establish its projectivity
+\cite{BakkerBrunebarbeTsimerman,SommesePeriodMapping,VilladsenHodgeKahler}.
+
+On this quotient $S$, the retained full period map is generically
+immersive, but the map remembering only the chosen line may have
+smaller rank. Curvature identifies the latter rank with the numerical
+dimension $\nu(P)$, measured by its nonzero intersection powers.
+Let $D$ be the reduced boundary where the retained variation has
+nonidentity local monodromy. Brunebarbe--Cadorel gives $K_S+D$ big
+\cite{BrunebarbeCadorel}. Using Andr\'e's monodromy normality and
+Bakker--Tsimerman's Ax--Schanuel theorem to analyze fibers of the
+line map, the boundary argument proves
+$\nu(P|_{D_i})<\nu(P)$ for every component $D_i$ of $D$, including
+finite nonidentity monodromy
+\cite{AndreMonodromy,BakkerTsimermanAxSchanuel}.
+The boundary loss in the asymptotic section count consequently has
+lower degree in the coefficient of $P$ than the available sections
+on $S$. Lemma~\ref{period:subtract} turns this comparison into
+bigness of $K_S+aP$.
+
+Section~\ref{generaltype:section} proves log-general-type addition in class $\mathcal C$.
+Kov\'acs--Patakfalvi's stable-family positivity applies to projective
+families. We construct such an auxiliary family and descend its
+relative sections to the original class-$\mathcal C$ base.
+Fujino's weak positivity supplies the other direct-image input
+used in Section~\ref{addition:section} \cite{KovacsPatakfalvi,FujinoWeakPositivity}.
+Section~\ref{application:section} then returns once to the original fibration, applies the
+addition principle to $(W,T,M)$, and derives the ordinary,
+logarithmic, characteristic-zero, and core consequences. The core
+and the relevant specialness of Kodaira-zero fibers are Campana's
+constructions and results \cite{Campana2004,CampanaOrbifolds}.
+
+\subsection{The reduced-boundary refinement}
+
+For a rational boundary, the root form above spans one character
+subspace of the cyclic cover's highest Hodge space; the whole
+highest space can be larger. For projective reduced-boundary pairs
+of logarithmic Kodaira dimension zero, Section~\ref{cyc:section} proves more:
+the minimal root cover has a one-dimensional entire logarithmic
+top-form space. Mixed-Hodge extension and semipositivity in the
+forms of Fujino--Fujisawa control the coefficient-one boundary
+\cite{FujinoFujisawa,FujinoFujisawaVMHS}.
+We identify the resulting line with the moduli divisor in the
+projective subpair setting of Bakker--Filipazzi--Mauri--Tsimerman
+\cite[Theorem~6.28]{BFMT}. This identification does not use their
+separate semiampleness theorem, whose hypotheses include
+effectiveness over the generic point \cite[Theorem~1.5]{BFMT}.
+Section~\ref{cyc:section} also gives the exact divisorial normalization.
+Appendix~\ref{p1logcbfalt:section} collects complementary projective
+calculations by valuations and toroidal models, together with the
+comparison on higher models.
+
+The logarithmic lower bound, adjoint positivity, and this
+reduced-boundary refinement supply the stated inputs to the
+whole-fiber-variation companion \cite{IitakaVariationCompanion}. The reverse-inequality companion
+\cite{IitakaAdditivityCompanion} uses the lower bound only for additivity; its upper theorem has
+additional smoothness hypotheses on all boundary strata.
+Neither companion is used in the proof of Theorem~\ref{main:orbifold}.
+The separate article on projective Hodge lines
+\cite{IitakaProjectiveCompanion} gives a complete
+ordinary proof through rationally polarized integral variations,
+together with the distinct integral and analytic boundary arguments.

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/models.tex

@@ -0,0 +1,364 @@
+\section{The relative Iitaka reduction and its Hodge line}
+\label{setup:section}
+
+The first task is to replace the fibers by their Iitaka images while
+retaining every sufficiently divisible pluricanonical system. We first
+fix the birational models on which divisorial order tests suffice. The
+root form of a Kodaira-zero fiber will then provide a Hodge line and an
+exact comparison of sections on the intermediate base.
+
+We use additive notation for rational line bundles; equality of them
+means equality in $\operatorname{Pic}\otimes\Q$. A rational line is
+effective if a positive integral multiple has a nonzero section.
+All degree assertions below are made after clearing denominators.
+For a fibration $g:(Z,\Gamma)\to T$ and a very general smooth
+fiber $F$, adjunction gives
+\begin{equation}\label{setup:fiber-adjunction}
+ (K_Z+\Gamma)|_F=K_F+\Gamma|_F.
+\end{equation}
+
+\subsection{The fixed smooth models}
+
+A smooth-base fibration $g:(Z,\Gamma)\to T$ is \emph{neat} if there is a
+proper bimeromorphic orbifold morphism
+$w:(Z,\Gamma)\to(Z_0,\Gamma_0)$ to a smooth pair, with
+$w_*\Gamma=\Gamma_0$, such that every prime divisor sent by $g$ into
+codimension at least two is also sent by $w$ into codimension at least
+two. Campana's construction gives suitable neat models, model
+independence, and the formula
+\begin{equation}\label{setup:neat-formula}
+ \kappa(g\mid\Gamma)=\kappa\bigl(T,K_T+B(g,\Gamma)\bigr)
+ \quad\text{on a neat model}
+\end{equation}
+\cite[Definition~3.8, Proposition~3.10, Corollary~4.11, and
+Remark~6.2]{CampanaOrbifolds}.
+For the proof we only need the defining inequality, valid on every
+allowed smooth model:
+\begin{equation}\label{setup:base-comparison}
+ \kappa\bigl(T,K_T+B(g,\Gamma)\bigr)
+ \ \geq\ \kappa(g\mid\Gamma).
+\end{equation}
+
+\subsection{Reduced exceptional boundaries}
+
+\begin{lemma}[Logarithmic modifications]\label{setup:log-modification}
+Let $p:Z'\to Z$ be a proper bimeromorphic holomorphic map between smooth
+compact manifolds. Suppose that $\Gamma$ has simple normal crossing
+support and coefficients in $\Q\cap[0,1]$, and set
+\[
+ \Gamma'=p_*^{-1}\Gamma+\operatorname{Exc}(p)_{\mathrm{red}}.
+\]
+Assume, after further resolution if necessary, that this boundary has
+simple normal crossing support. There is an effective $p$-exceptional
+rational divisor $A$ such that
+\begin{equation}\label{setup:log-discrepancy}
+ K_{Z'}+\Gamma'=p^*(K_Z+\Gamma)+A.
+\end{equation}
+For every sufficiently divisible positive integer $m$, the natural
+identification of meromorphic pluricanonical forms induces
+\begin{equation}\label{setup:section-invariance}
+ p_*\mathcal O_{Z'}\bigl(m(K_{Z'}+\Gamma')\bigr)
+ =\mathcal O_Z\bigl(m(K_Z+\Gamma)\bigr).
+\end{equation}
+These identifications respect multiplication, so the divisible section
+rings and their Iitaka dimensions agree. Moreover, $p$ satisfies
+\eqref{setup:orbifold-morphism}.
+\end{lemma}
+
+\begin{proof}
+The difference in \eqref{setup:log-discrepancy} is exceptional, since
+$p$ is an isomorphism at the generic point of every target prime.
+To compute its coefficient at an exceptional prime $E$, choose
+compatible local canonical forms.
+Near a general point of its center on $Z$, let $z_1,\ldots,z_n$ be
+coordinates in which the boundary components containing that center are
+$z_i=0$ for $1\leq i\leq r$, with coefficients $\gamma_i$.
+Put $t_i=\ord_E(p^*z_i)$ and let $k_E$ be the order of the Jacobian of
+$p$ along $E$. The pullback of
+\[
+ \frac{dz_1}{z_1}\wedge\cdots\wedge\frac{dz_r}{z_r}
+ \wedge dz_{r+1}\wedge\cdots\wedge dz_n
+\]
+has at most a simple pole along $E$. Indeed, at a general smooth point of
+$E$, each logarithmic factor has the form $t_i\,du/u$ plus a holomorphic
+one-form, and a nonzero wedge product contains at most one factor
+$du/u$. Consequently $k_E+1\geq\sum_i t_i$. The coefficient in question
+is therefore
+\[
+ k_E+1-\sum_i\gamma_i t_i
+ \ \geq\ \sum_i(1-\gamma_i)t_i\ \geq\ 0.
+\]
+For an effective integral exceptional divisor $mA$, normality of $Z$
+gives $p_*\mathcal O_{Z'}(mA)=\mathcal O_Z$: a meromorphic function
+with poles only on $mA$ descends outside the codimension-two exceptional
+image and extends by Hartogs' theorem. The projection formula proves
+\eqref{setup:section-invariance}, compatibly with products and ratios.
+
+Finally, an exceptional prime has infinite boundary multiplicity.
+A nonexceptional prime is a strict transform, with pullback order one
+and unchanged coefficient. These two cases prove
+\eqref{setup:orbifold-morphism}.
+\end{proof}
+
+\begin{corollary}[Descent across exceptional centers]
+\label{setup:exceptional-descent}
+In the situation of \Cref{setup:log-modification}, a meromorphic
+$m$-pluricanonical form on $Z'$ satisfying the boundary bounds of
+$m\Gamma'$ at every nonexceptional prime descends to a section of
+$m(K_Z+\Gamma)$ whenever $m\Gamma$ and $m\Gamma'$ are integral.
+The assertion also holds after tensoring by the pullback of a line
+bundle on $Z$.
+\end{corollary}
+
+\begin{proof}
+On the complement of the exceptional image the form is a holomorphic
+section of the prescribed line bundle on $Z$: divisorial regularity
+suffices on a smooth space. A local trivialization reduces extension
+across the remaining codimension-two analytic subset to Hartogs'
+extension theorem. The tensor-twisted assertion has the same proof.
+\end{proof}
+
+\begin{corollary}[Fibers, composition, and a fixed base]
+\label{setup:modification-compatibility}
+Let $p:(Z',\Gamma')\to(Z,\Gamma)$ be a modification with the boundary
+rule of \Cref{setup:log-modification}. Then:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $p$ is a modification over a fixed base, it induces a
+logarithmic modification on a very general smooth fiber. The log
+Kodaira dimensions of the corresponding fibers agree.
+\item A finite composition of such modifications again has boundary
+equal to the strict transform of the initial boundary plus the reduced
+exceptional divisor of the composite.
+\item If $g:(Z,\Gamma)\to T$ has smooth base and $g'=g\circ p$, then
+\begin{equation}\label{setup:fixed-base-boundary}
+ B(g',\Gamma')=B(g,\Gamma).
+\end{equation}
+\end{enumerate}
+\end{corollary}
+
+\begin{proof}
+For (i), choose the parameter so that the relevant smoothness and
+dimension statements hold for the finitely many exceptional components
+and their images. A component dominating the base restricts to a
+reduced divisor, and its exceptional image restricts to codimension at
+least two. Components not dominating the base do not meet this fiber.
+The restricted map has the same boundary rule, so
+\Cref{setup:log-modification} applies on the fiber. For (ii), every
+prime exceptional for the composite is either newly exceptional or the
+transform of an earlier exceptional prime; in both cases its coefficient
+is one.
+
+For (iii), fix a prime $D\subset T$. Every old prime dominating $D$
+has a strict transform, with the same pullback order and boundary
+multiplicity. Every new prime in the minimum defining
+$m(g',\Gamma';D)$ is exceptional for $p$, so its boundary multiplicity
+is infinite. Adding these entries leaves the old minimum unchanged,
+including when that minimum was already infinite.
+\end{proof}
+
+For choice independence in the singular-base definition, take a common
+resolution of two chosen diagrams. Their induced source boundaries
+coincide: old primes retain their coefficients and all primes exceptional
+over the initial $X$ have coefficient one. The comparison maps satisfy
+\Cref{setup:log-modification}, so the smooth fibrations are equivalent
+and their total-space and very general fiber log Kodaira dimensions agree.
+
+\subsection{Flattening relative to a fixed reference pair}
+
+To compare forms over generic points of base divisors, we need control of
+source divisors over higher codimension. Proper analytic flattening gives
+a base modification whose strict transform is flat
+\cite{HironakaFlattening,FujikiDouady}. Here the strict transform is the
+closure of the original family over the isomorphism locus, with base
+torsion removed. Further base change preserves flatness. We use only
+the resulting equidimensionality before resolving this intermediate
+source.
+
+\begin{lemma}[Flattening and the exceptional reference]
+\label{setup:flattening}
+Let $g_0:(Z_0,\Gamma_0)\to T_0$ be a fibration from a smooth compact
+pair with rational SNC boundary in $[0,1]$ to a smooth compact manifold.
+There are a smooth-base modification
+$q:T\to T_0$, a smooth-source modification $p:Z\to Z_0$, and a
+fibration $g:Z\to T$ with $q\circ g=g_0\circ p$ such that, for
+\[
+ \Gamma=p_*^{-1}\Gamma_0+\operatorname{Exc}(p)_{\mathrm{red}},
+\]
+the boundary has simple normal crossing support and every prime divisor
+$E\subset Z$ satisfying $\operatorname{codim}_Tg(E)\geq2$ is
+$p$-exceptional. In particular every such prime has coefficient one in
+$\Gamma$, and the resulting fibration is neat with reference pair
+$(Z_0,\Gamma_0)$.
+
+One may resolve prescribed proper analytic bad loci on the base during
+this construction. After any finite sequence of further base
+modifications, one can choose dominating smooth source models for which
+the same exceptional assertion holds with respect to the original
+reference $Z_0$.
+\end{lemma}
+
+\begin{proof}
+Flatten first, before resolving the source. After resolving the new base
+by further base change, denote the equidimensional strict transform by
+$\bar g:\bar Z\to T$, and its modification to $Z_0$ by $\bar p$.
+Resolve $\bar Z$ and all boundary data, and write
+\[
+ Z\xrightarrow{r}\bar Z\xrightarrow{\bar p}Z_0,
+ \qquad p=\bar p\circ r,
+ \qquad g=\bar g\circ r.
+\]
+Let $N=\dim Z_0$, $b=\dim T$, and $d=N-b$. If a prime $E$ on $Z$
+has image contained in an analytic subset $C\subset T$ of codimension
+at least two, equidimensionality gives
+\[
+ \dim\bar g^{-1}(C)\leq\dim C+d\leq N-2.
+\]
+It follows that $r(E)$, and therefore also $p(E)$, has dimension at most
+$N-2$. Thus $E$ is exceptional over $Z_0$. The boundary and
+orbifold-morphism assertions follow from \Cref{setup:log-modification}.
+The new map has connected general fiber.
+Its Stein factorization has a finite bimeromorphic map to the normal
+base $T$, which is an isomorphism, so all its fibers are connected.
+
+For any further base modification $T'\to T$, pull back the flat
+intermediate family and take a common resolution dominating the previous
+smooth source. The same dimension estimate proves exceptionality over
+$Z_0$. This includes principalizations of prescribed bad loci and their
+SNC resolutions on the base. Iterating proves the persistence assertion;
+\Cref{setup:modification-compatibility}\textup{(ii)} identifies the
+stepwise boundary with that computed directly over $Z_0$.
+\end{proof}
+
+The resolution need not be flat: the conclusion we retain is
+exceptionality over $Z_0$, which permits
+\Cref{setup:exceptional-descent}.
+
+All constructions remain available in Fujiki class $\mathcal C$.
+A smooth compact member admits a compact K\"ahler modification
+\cite[Theorems~0.7 and~3.4]{DemaillyPaun2004}. Starting with one for
+$Z_0$, graph resolutions and principalizations by smooth-center blowups
+give a K\"ahler dominating source. K\"ahlerness is preserved by these
+projective modifications, and proper meromorphic images remain in
+class $\mathcal C$ \cite[Lemmas~4.4 and~4.6\textup{(3)}]{FujikiDouady}.
+Thus the bases also admit K\"ahler models. The preceding dimension
+argument still refers to the original $Z_0$.
+
+\subsection{Keeping the original base fixed}
+
+\begin{corollary}[Simultaneous model control]\label{setup:simultaneous}
+Suppose a smooth pair $(X_1,\Delta_1)$ has a fibration $f_1:X_1\to Y$
+to a fixed smooth base, and every prime divisor contracted by $f_1$ into
+codimension at least two has coefficient one in $\Delta_1$. Put
+$C=B(f_1,\Delta_1)$. Let a factorization through another smooth compact
+space be given, after an allowed source modification if necessary, by
+\[
+ X_1\xrightarrow{g_1}W_1\xrightarrow{h_1}Y.
+\]
+Apply \Cref{setup:flattening} to $g_1$ and make any finite number of
+subsequent modifications of $W_1$, always taking dominating smooth
+source models with reduced added exceptional boundaries. The resulting
+diagram $X\xrightarrow{g}W\xrightarrow{h}Y$, with source boundary
+$\Delta$, can be chosen to have all the following properties:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every prime contracted by $g$ into codimension at least two is
+exceptional over the chosen smooth reference source for $g_1$.
+\item Every prime contracted by $f=h\circ g$ into codimension at least
+two in $Y$ has coefficient one in $\Delta$.
+\item $B(f,\Delta)=C$. In particular, for each prime $D\subset Y$ and
+each prime $E\subset X$ dominating $D$,
+\begin{equation}\label{setup:retained-multiplicity}
+ \ord_E(f^*D)\,m_\Delta(E)\geq m_C(D).
+\end{equation}
+\item The log Kodaira dimensions of the total source and of its very
+general fibers over $Y$ are unchanged by these source modifications.
+\end{enumerate}
+The hypothesis on contracted divisors follows from
+\Cref{setup:flattening}, applied to the original map to its resolved
+base before fixing $Y$.
+\end{corollary}
+
+\begin{proof}
+Assertion (i) is \Cref{setup:flattening}. For (ii), a source prime
+is either newly exceptional, with coefficient one, or the strict
+transform of an old prime with unchanged image in $Y$. The hypothesis
+handles the latter case. Assertions (iii) and (iv) are
+\Cref{setup:modification-compatibility};
+\eqref{setup:retained-multiplicity} is the defining minimum for $C$.
+Flattening the original map supplies the initial coefficient-one
+condition by making its contracted primes exceptional.
+\end{proof}
+
+In applications we include the critical values and images of boundary
+strata among the base bad loci, so that $C$ has SNC support. We then
+fix $Y$ and use \Cref{setup:simultaneous} for changes of intermediate
+bases.
+
+\subsection{The relative Iitaka map}
+
+\begin{lemma}[Relative Iitaka construction]
+\label{application:relative-iitaka}
+Let \(f:(X,\Delta)\to Y\) be a fibration between smooth compact
+manifolds in Fujiki class \(\mathcal C\), with \(\Delta\) a rational
+SNC boundary in \([0,1]\). Suppose that
+\[
+ k=\kappa(F,K_F+\Delta_F)\geq 0
+\]
+on very general smooth fibers.  After modifications of the source, adding
+the reduced exceptional divisor to the strict transform of the boundary,
+there is a factorization
+\[
+ X\xrightarrow{g}W\xrightarrow{h}Y
+\]
+with smooth compact intermediate space in class \(\mathcal C\), both
+maps proper with connected fibers, and
+\[
+ \dim W_y=k,\qquad
+ \kappa(G,K_G+\Delta_G)=0
+\]
+for very general fibers of \(h\) and \(g\), respectively.  The factorization
+restricts to the log Iitaka fibration on very general \(f\)-fibers.
+These properties persist on higher models obtained by the same rule for
+the source boundary.
+\end{lemma}
+
+\begin{proof}
+Consider all degrees clearing the denominators of \(\Delta\).
+Their relative adjoint direct images are coherent. First shrink to
+the dense analytic open where the map and boundary strata are smooth.
+There the family and its adjoint line bundles are flat over the base;
+generic constancy of the fiber section dimension permits cohomology
+and base change \cite[\S7, nos.~4--6, S\"atze~3--5]{GrauertDirectImages},
+with its corrigendum \cite{GrauertDirectImagesCorr}.
+Together with the evaluation ranks this gives a dense analytic open
+for each degree;
+outside the resulting countable collection of exceptional sets, the
+maximum image dimension is \(k\), attained in one fixed degree.
+
+Choose a sufficiently divisible degree giving the stabilized maps,
+resolve the relative meromorphic map to the projective bundle of its
+direct image, and take Stein factorization onto the image. The Iitaka
+fibration theorem for line bundles on compact complex spaces identifies
+the resulting map on very general fibers
+\cite[Chapter~II, \S5; Chapter~III, \S7]{UenoClassification}.
+No finite generation is required. The image has relative dimension \(k\); its
+Stein factor is in class \(\mathcal C\), being a proper image of a space
+in that class.  Resolve it and the resulting source map.  Connectedness
+of the fibers of \(h\) follows from that of \(f\), since their images
+under \(g\) are precisely the fibers of \(h\).
+
+The assertion that the restriction has Kodaira dimension zero concerns
+the original line bundle. Its standard proof applies here: coherent
+direct images on the projective Iitaka image acquire sections after an
+ample twist. If the
+restriction had positive Iitaka dimension, multiplying those sections
+by powers of the defining system would enlarge the Iitaka image.
+A nonzero defining section restricts nontrivially to a very general
+Iitaka fiber, so the restriction has dimension exactly zero.
+
+Apply this to the resolved log adjoint on a very general \(f\)-fiber.
+Its divisible section ring is unchanged by
+\Cref{setup:log-modification}, and adjunction identifies the restriction
+with \(K_G+\Delta_G\). Further source modifications preserve the same
+section ring, hence the relative Iitaka factorization and its asserted
+fiber dimensions.
+\end{proof}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/normalization.tex

@@ -0,0 +1,428 @@
+\section{Supplementary projective calculations}
+\label{p1logcbfalt:section}
+
+The reduced-boundary construction of Section~\ref{cyc:section} admits
+complementary projective proofs and calculations. We first give
+divisorial proofs of the section comparisons and recover the
+fixed-source descent by extracting valuations on the base. The fixed
+logarithmic zero divisor then gives another discrepancy-sheaf test,
+and a toroidal calculation proves logarithmic extension of the root
+form. The final subsection tracks the auxiliary subpair when the
+minimum-rule divisor is recomputed on a higher base model. This last
+operation is distinct from the prescribed boundary modification of
+Lemma~\ref{lem:normalized-models}: its discriminant is recomputed,
+rather than defined as the strict boundary plus the reduced exceptional
+divisor. Throughout these calculations, modifications of the original
+source carry strict boundary plus reduced exceptional boundary.
+
+\subsection{Divisorial sections and relative Iitaka systems}
+
+The following elementary comparisons fix the complete systems and the
+original models used in the normalization. All varieties in this
+subsection are projective over $\C$, all rational divisor degrees are
+sufficiently divisible, and the very general fibers are taken integral.
+The logarithmic modification identity, also with a pulled-back rational
+twist, is \Cref{setup:log-modification}.
+
+\begin{lemma}[Divisorial section test]\label{p1logcbfalt:section-test}
+Let \(V\) be normal and \(D\) an integral Weil divisor. A rational section
+belongs to \(H^0(V,D)\) if and only if it satisfies the corresponding order
+inequality at every prime divisor of \(V\). If \(\pi:V'\to V\) is finite
+surjective with \(V'\) normal and \(D\) is Cartier, this test on \(V'\) needs
+only primes lying over prime divisors of \(V\). If \(\pi\) is Galois, then
+\[
+H^0(V',\pi^*D)^{\operatorname{Gal}(V'/V)}=H^0(V,D).
+\]
+\end{lemma}
+
+\begin{proof}
+The first assertion is the description of a rank-one reflexive sheaf as
+the intersection of its lattices at codimension-one points. For a finite
+morphism between normal varieties, a prime divisor upstairs lies over a
+prime divisor downstairs. The last assertion follows from
+\((\pi_*\cO_{V'})^{\operatorname{Gal}(V'/V)}=\cO_V\) and the projection
+formula.
+\end{proof}
+
+\begin{lemma}[Finite pullback]\label{p1logcbfalt:finite}
+If \(\pi:V'\to V\) is a dominant generically finite morphism of normal
+projective varieties and \(D\) is a rational Cartier divisor on \(V\),
+then
+\[
+\kappa(V',\pi^*D)=\kappa(V,D).
+\]
+If \(\kappa(V,D)=0\), every nonzero space \(H^0(V,mD)\) has dimension
+one.
+\end{lemma}
+
+\begin{proof}
+Stein factorization reduces the first assertion to a finite morphism,
+since a proper birational morphism onto a normal variety has direct image
+\(\cO_V\). After replacing \(D\) by a multiple, use the natural inclusion
+of graded section rings
+\[
+R(V,D)\ \subseteq\ R(V',\pi^*D).
+\]
+Each homogeneous section upstairs is integral over the ring downstairs.
+Indeed its monic characteristic polynomial in the finite function-field
+extension has coefficients which are rational sections of the appropriate
+multiples of \(D\); they are regular in the required lattices at every
+prime divisor, and hence are global sections by
+Lemma~\ref{p1logcbfalt:section-test}. Thus the two nonzero graded rings have the
+same transcendence degree, which gives equality of Iitaka dimensions.
+If the ring downstairs has no positive-degree section, the same norm
+argument excludes one upstairs. Finally, two independent sections in one
+degree have a nonconstant ratio and define a positive-dimensional
+rational image. This proves the last assertion.
+\end{proof}
+
+\begin{lemma}[Logarithmic pullback]\label{p1logcbfalt:log-pullback}
+Let \(\rho:V\to Y\) be a dominant generically finite morphism of smooth projective
+varieties. Let \(D_Y\) be a reduced snc divisor, and let \(B\) be a
+boundary on \(V\) satisfying \(B\ge(\rho^*D_Y)_{\red}\). Assume that the
+boundary supports on the chosen models are snc. Then pullback of rational
+pluricanonical forms gives
+\[
+H^0(Y,m(K_Y+D_Y))\ \longrightarrow\
+H^0(V,m(K_V+B))
+\]
+injectively, for divisible \(m\).
+\end{lemma}
+
+\begin{proof}
+A logarithmic differential on a smooth snc pair pulls back to a
+logarithmic differential along the reduced inverse image. In local
+coordinates this follows from
+\(d(\rho^*z)/\rho^*z\), whose divisorial poles are simple. Taking top
+exterior powers and then tensor powers proves the order inequalities.
+Equivalently, at a prime of ramification index \(e\) over a boundary
+component, the Jacobian contributes \(e-1\), and the coefficient-one
+boundary supplies the remaining one. At exceptional-image primes the
+same logarithmic differential calculation applies. Injectivity follows
+from dominance and separability.
+\end{proof}
+
+\begin{lemma}[Relative Iitaka fibration]\label{p1logcbfalt:iitaka}
+Let \(p:V\to B\) be a dominant projective morphism, with \(V\) smooth
+projective, and let \(D\) be a rational divisor whose restriction to the
+geometric generic fiber has Iitaka dimension \(k\ge0\). After birational
+modifications there is a relative Iitaka fibration
+\[
+V'\xrightarrow{q}U\longrightarrow B
+\]
+in which \(q\) has geometrically connected generic fiber, such that
+\[
+\dim U-\dim B=k,\qquad
+\kappa(G,D'|_G)=0
+\]
+for a very general fiber \(G\) of \(q\). Here \(D'=\mu^*D\), where
+\(\mu:V'\to V\) is the modification. When \(D=K_V+B_V\), one may
+instead take the full log divisor from Lemma~\ref{setup:log-modification}.
+
+If \(D'=K_{V'}+B_{V'}\) and the models are log smooth, then
+\(D'|_G\sim_{\Q}K_G+B_G\).
+\end{lemma}
+
+\begin{proof}
+Apply the Iitaka fibration theorem over the function field of \(B\)
+\cite[Section~6, Remark~1]{IitakaOriginal},
+take the relative algebraic closure of its section field in \(\C(V)\),
+and spread out. Resolve the resulting rational maps and take their
+Stein factorizations. The dimension of the relative section field is
+\(k\); the restriction of the full divisor to the Iitaka fiber has
+Iitaka dimension zero by the Iitaka fibration theorem. For the logarithmic
+replacement, the proof of Lemma~\ref{setup:log-modification} gives the sheaf identity
+\[
+ \mu_*\cO_{V'}\bigl(m(K_{V'}+B_{V'})\bigr)
+ =\cO_V\bigl(m(K_V+B_V)\bigr)
+\]
+in every sufficiently divisible degree. Pushing to the original base
+identifies the relative section systems, even if there are no global
+sections. Apply the same full-divisor Iitaka theorem to the replacement
+divisor; its relative section field identifies its fibration with the
+one already constructed and gives the asserted Iitaka dimension zero
+on the fiber.
+Generic smoothness and adjunction give the last assertion.
+The dimensions at very general complex fibers agree with the geometric
+generic values: for each divisible degree use cohomology and base
+change, and remove the union of the exceptional closed sets over the
+countably many degrees.
+\end{proof}
+
+\begin{lemma}[Easy addition]\label{p1logcbfalt:easy-addition}
+Let \(p:V\to B\) be dominant and projective, with \(V\) smooth projective
+and integral very general fiber \(G\), and let \(D\) be a rational
+divisor on \(V\). Then
+\[
+\kappa(V,D)\le \dim B+\kappa(G,D|_G).
+\]
+If \(D\) has a nonzero global section, its restriction to a general
+fiber is nonzero.
+\end{lemma}
+
+\begin{proof}
+A nonzero section cannot vanish identically on all fibers over a dense
+open subset. For each divisible \(m\) with sections, let \(\phi_m\)
+be its rational map. The image of \((p,\phi_m)\) has dimension at most
+\(\dim B+\kappa(G,D|_G)\), and dominates the image of \(\phi_m\).
+Taking the maximum over \(m\) proves the inequality. If the fiber term
+is \(-\infty\), the last assertion implies that the total term is
+\(-\infty\) as well.
+\end{proof}
+
+\subsection{Restricted valuations and the fixed original source}
+
+Let $p:H\to I$ be a dominant morphism of smooth projective varieties,
+and fix the model $H$. A divisorial valuation $v$ of $\C(H)$ whose
+restriction to $\C(I)$ is nontrivial restricts to a positive integral
+multiple of a divisorial valuation of $\C(I)$. To see this, put
+$n=\dim H$, $m=\dim I$, and denote the restricted valuation by $w$.
+The value groups have rational rank one with finite-index inclusion.
+The relative residue-transcendence-degree inequality gives
+\[
+ \operatorname{trdeg}_{\kappa(w)}\kappa(v)\le n-m.
+\]
+As $\operatorname{trdeg}_{\C}\kappa(v)=n-1$, this implies
+$\operatorname{trdeg}_{\C}\kappa(w)\ge m-1$. The Abhyankar
+inequality gives the reverse inequality. Thus $w$ is divisorial.
+Finitely many such restrictions can be extracted on one smooth
+projective birational model of $I$.
+
+Only finitely many prime divisors of the fixed $H$ have image of
+codimension at least two in $I$: they lie in the complement of a
+locus where $p$ is smooth and are components of its divisorial part.
+Extract their restrictions as above. Original horizontal primes
+remain horizontal, while original vertical primes then have
+divisorial centers on the prepared base. These centers remain
+divisorial on higher base models. Hence on any subsequent commuting
+birational diagram, a source prime whose base image still has
+codimension at least two is exceptional over the fixed $H$.
+
+For the rank-one section comparison, fix the original smooth pair
+$(X,D_X)$, and let $X'\to X$ carry strict plus reduced exceptional
+boundary $D'$. Let $x:X'\to W$ be the relative Iitaka fibration
+on smooth projective models; its geometric generic fiber $(J,D_J)$
+has logarithmic Kodaira dimension zero. Let $s$ be a nonzero
+relative rational $p$-canonical form generating the degree-$p$
+generic section space. For a source prime $E$, put
+$d_E=\operatorname{coeff}_E D'$, and for a base prime $P$ define
+\[
+ t_P=\min_{E\mapsto P}
+ \frac{p^{-1}\operatorname{ord}^{pK_{X'/W}}_E(s)+d_E}
+ {\operatorname{ord}_E(x^*P)},\qquad
+ T_W=\sum_Pt_PP.
+\]
+Here $T_W$ is the entire minimum-rule base term, before its division
+into discriminant and moduli parts. In divisible degrees, division by
+$s^{m/p}$ gives
+\begin{equation}\label{p1logcbfalt:source-descent}
+ H^0\bigl(X',m(K_{X'}+D')\bigr)
+ \lhook\joinrel\longrightarrow
+ H^0\bigl(W,m(K_W+T_W)\bigr).
+\end{equation}
+Indeed, on the geometric generic fiber the section is a constant
+multiple of $s^{m/p}$; connectedness identifies that coefficient
+with a rational $m$-canonical form $\sigma$ on $W$. At a component
+$E$ dominating $P$, write $\sigma$ in a nonvanishing local frame of
+$mK_W$, and put $a_E=\operatorname{ord}_E(x^*P)$. Its logarithmic
+regularity upstairs is equivalent to
+\[
+ a_E\operatorname{ord}_P(\sigma)
+ +(m/p)\operatorname{ord}^{pK_{X'/W}}_E(s)+md_E\ge0.
+\]
+The inequalities for all components above $P$ are exactly
+$\operatorname{ord}_P(\sigma)+mt_P\ge0$. Horizontal primes impose
+no further condition because $s$ is an allowed generic logarithmic
+pluriform. The same argument works over the generic point of the
+original base $Y$. If the relative Iitaka dimension is $k$, then
+$\dim(W/Y)=k$, so preservation of section ratios makes
+$K_W+T_W$ big over $Y$.
+
+For the converse, extract on $W$ the restrictions of all source
+divisors with exceptional image that survive as divisors on the
+original $X$. Resolve the induced diagram and recompute the minimum
+rule. Every divisor of the original $X$ is now tested either by the
+generic horizontal condition or by a base prime. A section on the
+right of \eqref{p1logcbfalt:source-descent}, multiplied by $s^{m/p}$,
+therefore satisfies the logarithmic order inequality at every
+original source prime. Any remaining untested pole lies on a
+divisor exceptional over $X$. The codimension-one criterion on the
+normal original source gives a section of $m(K_X+D_X)$; later
+models preserve this criterion. Conversely the logarithmic
+modification identity recovers its section on $X'$. The
+identifications respect powers and multiplication. Under these
+identifications, corresponding section ratios agree after pullback by
+$x:X'\to W$. Their section fields and rational-map image dimensions
+therefore agree.
+
+\subsection{The fixed zero divisor and the discrepancy-sheaf test}
+
+In the notation of Lemma~\ref{lem:reduced-cyclic-top}, the rational
+divisor $Z=E/p$ is independent of the chosen nonzero logarithmic
+pluriform and its degree. Indeed, if $\omega_m$ is a nonzero section
+in another degree $m$, the degree-$mp$ sections $\omega_m^p$ and
+$\omega^m$ are proportional, since that section space is
+one-dimensional. Consequently
+\[
+ p\bigl(\operatorname{div}(\omega_m)+mD\bigr)=mE.
+\]
+The least positive integer trivializing $K_J+D-Z$ is also $p$.
+Writing $\omega=b\eta^{\otimes p}$ in a rational canonical frame
+gives $p(K_J+D-Z)=-\operatorname{div}(b)$. Conversely, a
+trivialization of $r(K_J+D-Z)$ gives a rational $r$-canonical form
+whose logarithmic zero divisor is $rZ\ge0$, hence a nonzero
+logarithmic pluriform of degree $r$. Minimality gives $r\ge p$.
+
+Here is a second way to check the generic discrepancy-sheaf rank
+condition in Corollary~\ref{cor:reduced-integral-moduli}. Take a log
+resolution $J'\to J$, let $D_{J'}$ be the strict boundary plus the
+reduced exceptional divisor, and put
+\[
+ E_{J'}=\operatorname{div}(\omega|_{J'})+pD_{J'}\ge0.
+\]
+The crepant transform of the auxiliary generic subboundary is
+$D_{J'}-E_{J'}/p$. Its coefficients are at most one. If $A^*$ is
+the discrepancy b-divisor with coefficient $-1$ replaced by zero,
+the rounded trace is effective, and a positive coefficient can
+occur only on $\operatorname{Supp}E_{J'}$. Coefficientwise,
+\[
+ 0\le \lceil A^*\rceil_{J'}
+   \le\lceil E_{J'}/p\rceil\le E_{J'}.
+\]
+The first inequality uses the omission of discrepancy $-1$.
+Logarithmic birational invariance gives $\kappa(E_{J'})=0$.
+The associated space of rational functions therefore has dimension
+at most one and contains the constants. This proves rank one on
+every sufficiently high resolution, including when the auxiliary
+subboundary has negative coefficients.
+
+\subsection{A toroidal proof of logarithmic extension}
+
+The extension part of Lemma~\ref{lem:reduced-cyclic-top} also has a
+direct coordinate proof on a toroidal resolution of the SNC
+cyclic-cover construction. First resolve the supports on $J$,
+retaining strict plus reduced exceptional logarithmic boundary.
+The minimal index and the fixed-zero assertions are unchanged.
+In a local SNC chart write
+\[
+ \omega=\varepsilon\prod_i z_i^{a_i-pd_i}
+                (dz_1\wedge\cdots\wedge dz_r)^{\otimes p},
+ \qquad a_i\ge0,\quad d_i\in\{0,1\},
+\]
+where $\varepsilon$ is a unit. Include all coordinates; outside
+$D+\operatorname{Supp}E$ put $d_i=a_i=0$. After a local unramified
+change the unit has a root and contributes no divisorial order.
+Let $v$ be a toroidal prime on the resolved root cover, and put
+$n_i=v(z_i)\ge0$. The nonzero ray defining this prime has at least
+one positive $n_i$. Rewriting with logarithmic differentials gives
+\begin{equation}\label{p1logcbfalt:toroidal-root-order}
+ \operatorname{ord}_v(\Omega)
+   =\sum_i n_i\left(1-d_i+\frac{a_i}{p}\right)-1.
+\end{equation}
+The logarithmic volume form has order $-1$ along a toroidal prime,
+and the change of torus lattice multiplies it by a nonzero constant.
+Every coefficient in the sum is nonnegative. It vanishes exactly
+when $d_i=1$ and $a_i=0$, namely along the coefficient-one pole
+boundary of the auxiliary subpair.
+
+If $v$ is outside the inverse image of this pole boundary, then
+$n_i=0$ for each such zero coefficient. Some remaining $n_i$ is
+positive, so the sum in \eqref{p1logcbfalt:toroidal-root-order} is
+strictly positive. The order of the rational top form $\Omega$ is
+integral, so that positive sum is at least one, giving regularity.
+On the inverse pole boundary the nonnegative sum gives order at
+least $-1$. Primes in the smooth unramified locus have the
+regularity already checked on the finite normalization. Thus
+$\Omega$ extends as a top form with logarithmic poles allowed only
+along the inverse pole boundary. This provides the coordinate
+alternative to the discrepancy-scaling proof.
+
+\subsection{The normalized discriminant on higher models}
+
+Let $x:X'\to W$ and $D'$ be the projective reduced-boundary
+data of Lemma~\ref{lem:rankone-normalization}. Write $s$ for its
+relative rational $p$-canonical generator, and set
+\[
+ t_P=\min_{E\mapsto P}
+ \frac{p^{-1}\operatorname{ord}^{pK_{X'/W}}_E(s)+d_E}
+ {\operatorname{ord}_E(x^*P)},\qquad
+ T_W=\sum_Pt_PP,
+\]
+\[
+ \Delta_0=-p^{-1}\operatorname{div}_{K_{X'/W}}(s),
+ \qquad \Delta=\Delta_0+x^*T_W.
+\]
+In this subsection $T_W$ denotes the entire minimum-rule base
+term, before its decomposition into discriminant and moduli parts;
+the boundary denoted $T$ in Section~\ref{rankone:section} is the
+discriminant part. We have
+$K_{X'}+\Delta\sim_{\Q}x^*(K_W+T_W)$.
+The auxiliary divisor is required to be sub-lc over the generic
+point of $W$. Its coefficients at source divisors with image of
+codimension at least two are not asserted to be at most one.
+
+For a prime $P\subset W$, let
+\[
+ c_P=\operatorname{lct}_{\eta_P}(X',\Delta;x^*P),
+ \qquad b_P=1-c_P,
+ \qquad B_W=\sum_Pb_PP.
+\]
+Over the generic point of $P$ the minimum rule gives
+$\Delta\le D'$, including the horizontal divisors. Log canonicity
+therefore gives $c_P\ge0$. A component $E$ attaining the minimum
+has coefficient exactly $d_E\in\{0,1\}$ in $\Delta$, and hence
+\[
+ c_P\le\frac{1-d_E}{\operatorname{ord}_E(x^*P)}\le1.
+\]
+It follows that $0\le B_W\le1$. If $P$ lies in a required inverse
+boundary from a base $Y$, all components over $P$ have $d_E=1$,
+so $c_P=0$ and $b_P=1$.
+
+For the exact model comparison, let $\beta:W_1\to W_0$ be a
+higher smooth base model and let $\pi:X_1\to X_0$ resolve the
+induced diagram, with $x_i:X_i\to W_i$. Choose compatible
+rational top forms for the canonical divisors. First form the
+crepant transform
+\[
+ K_{X_1}+\Delta_{\mathrm{crepant}}
+    =\pi^*(K_{X_0}+\Delta_{\mathrm{old}}).
+\]
+Using the new logarithmic boundary and the same relative
+trivialization $s$, recompute the minimum-rule divisor $T_{W_1}$
+and the normalized auxiliary divisor $\Delta_{\mathrm{new}}$.
+Comparison of their definitions gives
+\begin{equation}\label{p1logcbfalt:model-twist}
+ \begin{split}
+ Q&=K_{W_1}+T_{W_1}-\beta^*(K_{W_0}+T_{W_0}),\\
+ \Delta_{\mathrm{new}}
+   &=\Delta_{\mathrm{crepant}}+x_1^*Q.
+ \end{split}
+\end{equation}
+Thus the twist includes the canonical discrepancy of the base.
+Adding $x^*Q$ lowers the threshold at $P$ by
+$\operatorname{coeff}_P Q$, adding $Q$ both to the discriminant
+and to the total base term. It leaves the moduli part unchanged.
+Birational pullback of the lc-trivial fibration also preserves its
+moduli b-divisor; these compatibilities agree with the Hodge
+extension description in \cite[Remarks~6.22 and~6.25]{BFMT}.
+
+Resolve the finite supports on $W$ and resolve the induced source
+diagram. At an unchanged codimension-one point, logarithmic forms
+satisfying the old tests remain logarithmic on the source
+resolution, while the strict old minimizing components persist.
+Thus the minimum and discriminant trace there are unchanged.
+New support is contained in the base-exceptional divisor. Its union
+with the old strict support is SNC on the chosen resolution;
+recomputing and applying the preceding coefficient argument yields
+an effective SNC discriminant with the same coefficient-one inverse
+boundary and the same moduli b-divisor.
+
+Finally, if $m=ap$ and $s_m=c s^a$ on the generic fiber, with
+$c\in\C(W)^*$, the minimum construction gives
+\[
+ T_W(s_m)=T_W(s)+m^{-1}\operatorname{div}(c).
+\]
+The two occurrences of $\operatorname{div}(c)$ cancel in the
+definition of the normalized auxiliary subpair. It is unchanged.
+The minimal-index cover is therefore compatible with all common
+divisible degrees used for section descent.

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/period-positivity.tex

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+\section{Adjoint positivity of an extreme Hodge line}
+\label{period:section}
+
+The relative canonical bundle calculation gives a line in the highest
+nonzero Hodge filtration step of one complex summand.  Its curvature
+is semipositive, but this alone does not compare it with a canonical
+bundle.  We construct a projective quotient on which the line descends
+and prove that adding a positive multiple of it makes the canonical
+class big.
+
+Here an \emph{extreme Hodge line} means the rank-one highest nonzero
+filtration step of a complex Hodge direct summand.  An integral
+structure is a lattice in the ambient real local system.  The chosen
+summand need not be rational, and the ambient real polarization need
+not be rational either.
+
+The \emph{parabolic extension} is obtained by local power substitutions
+that make boundary monodromy unipotent, followed by Schmid extension
+and descent with rational weights.  The monodromy theorem gives
+rational weights in the integral case.  When the local monodromy has
+a finite part, these weights are essential.  We first resolve any
+compactification whose boundary is not simple normal crossing.
+Unipotent extension commutes with pullback, so the same is true of
+the parabolic rational line after the power substitutions; see
+\cite{Schmid,CKS} and \cite[Lemma~5.6]{BFMT}.  We use version~2
+of \cite{BFMT} throughout.
+
+
+We prove the adjoint-positivity theorem stated as
+\Cref{period:adjoint}. The line in that statement is allowed to be a
+complex character line inside an integral ambient variation.
+
+There are two maps in the proof.  In a flat trivialization the
+\emph{line map} records only the chosen highest line; the \emph{period
+map} records all Hodge filtration steps.  Their ranks can differ.
+Curvature identifies the numerical dimension of the line with the
+rank of the first map.  We use the full periods of suitable rational
+adjoint factors to construct the projective quotient.  On that
+quotient, logarithmic general type gives a big canonical class with
+boundary.  The remaining argument proves that the line loses numerical
+rank on each marked boundary component, so a section count removes
+the boundary after adding a multiple of the line.
+
+\subsection{Curvature, numerical dimension, and the boundary}
+
+We first relate the curvature on the open set to intersection numbers
+on the compactification.  For a nef rational line bundle $A$ on a compact
+Kähler $n$-fold, we use
+\[
+ \nu(A)=\max\{k:c_1(A)^k\cdot[\omega]^{n-k}>0\},
+\]
+where $\omega$ is any Kähler form.  On a projective manifold this is
+the usual numerical dimension.
+
+\begin{lemma}[Curvature and numerical dimension]
+\label{period:numerics}
+Let $T$ be a smooth compact Kähler manifold, let $E\subset T$ be a
+simple normal crossing divisor, and let $A$ be the parabolic extension
+of the rank-one highest step of a polarizable pure complex variation on
+$T\setminus E$ with quasi-unipotent local monodromy.  Then $A$ is nef.
+Its numerical dimension equals the generic rank of the map that
+remembers this line in a flat trivialization.
+
+The corresponding assertion for the Griffiths line, the tensor product
+of the determinants of the Hodge filtration steps, uses the full period
+map.  In particular, its rational extension is big if the full period
+map is generically immersive.
+\end{lemma}
+
+The curvature-and-volume argument is closely related to the proofs of
+\cite[Lemmas~6.15 and~6.17]{BakkerBrunebarbeTsimerman}; here we need
+all mixed intersection numbers, since the chosen line need not be big.
+
+\begin{proof}
+We may check the assertions after a finite cover and resolution on
+which local monodromy is unipotent.  This does not require an integral
+lattice.  Indeed, $\pi_1(T\setminus E)$ is finitely generated.  Put
+the entries of finitely many monodromy generators and their inverses
+in a finitely generated $\Z$-algebra $R\subset\C$.  Adjoin the
+finite group of roots of unity generated by the eigenvalues of the
+finitely many boundary monodromies, and invert every difference
+$\zeta-1$ for a nonidentity element of this group.  Reduction at a
+maximal ideal with finite residue field gives a finite-index normal
+congruence subgroup.  Every element of a local boundary monodromy
+group that lies in this subgroup is unipotent: its eigenvalues
+belong to that finite group and reduce to one, so the inverted
+differences force them to equal one.  This also applies to products
+of commuting boundary monodromies at crossings.  The corresponding finite cover extends
+over the compactification, and a Kähler resolution gives the desired
+unipotent model; see \cite[Lemma~2.21]{VilladsenHodgeKahler}.
+These covers preserve nefness, numerical dimension, and bigness of
+the rational line.  Let $h$ be its Hodge metric and $\Theta$ its curvature
+on $T\setminus E$, normalized to represent $c_1(A)$.  Griffiths'
+curvature formula gives $\Theta\geq0$; its rank is the rank of the
+line map, since its value on a tangent vector is the squared norm of
+the second fundamental form of the highest step.  The same formula
+for the Griffiths line detects the differential of the full period
+map.
+
+The Hodge norm estimates in extending frames give, in boundary
+coordinates $t_1,\ldots,t_k$, weights whose absolute values are bounded
+by
+\begin{equation}
+\label{period:loglog}
+              C\left(1+\sum_{j=1}^k
+                   \log(-\log|t_j|)\right).
+\end{equation}
+The semipositive metric therefore extends as a positive current in
+$c_1(A)$ with zero Lelong numbers.  Regularization of positive
+currents then gives nefness; see
+\cite[Corollary~6.4]{DemaillyRegularization}.
+This extends the nefness argument of \cite[Lemma~2.16(1)]{BFMT}
+from a proper log smooth algebraic space to the present compact
+Kähler setting.
+
+To identify numerical dimension, we must also show that no
+intersection mass is lost at $E$.  Write $\alpha=c_1(A)$ and fix
+$\varepsilon>0$.  Since $\alpha$ is nef, $\alpha+\varepsilon[\omega]$
+has a smooth Kähler representative
+$\theta_\varepsilon$.  Write
+\[
+ \Theta+\varepsilon\omega=\theta_\varepsilon+\ddc u,
+ \qquad \ddc=\frac{\sqrt{-1}}{2\pi}\partial\bar\partial.
+\]
+The global potential $u$ satisfies the local bounds
+\eqref{period:loglog}. Put $n=\dim T$. If $E$ is empty, the potential
+is smooth and there is no lost mass. Otherwise write
+$E=E_1+\cdots+E_N$ and choose smooth divisor metrics, with defining
+sections $s_j$, such that
+\[
+ u_j=-\log|s_j|^2,\qquad
+ \ddc(-u_j)=[E_j]-\eta_j,
+\]
+where the $\eta_j$ are smooth curvature forms. Choose $c>0$ with
+$c\eta_j\leq\theta_\varepsilon$ for all $j$, and rescale the metrics
+so that $cu_j\geq1$. Then $\phi_j=-cu_j$ is
+$\theta_\varepsilon$-plurisubharmonic and $\phi_j\leq-1$.
+For any fixed $0<b<1$, the attenuation theorem
+\cite[Example~2.14]{GuedjZeriahiEnergy} gives
+\[
+ \psi_j=-(-\phi_j)^b=-c^bu_j^b
+\]
+with full Monge--Amp\`ere mass in the K\"ahler class
+$[\theta_\varepsilon]$. The full-mass class is convex and contains
+the zero potential \cite[Proposition~1.6]{GuedjZeriahiEnergy}.
+Thus, for $0<\delta\leq c^b/N$, the potential
+\begin{equation}\label{period:barrier}
+ v=-\delta\sum_{j=1}^N u_j^b
+   =\sum_{j=1}^N\frac{\delta}{c^b}\psi_j
+      +\left(1-\frac{N\delta}{c^b}\right)0
+\end{equation}
+is $\theta_\varepsilon$-plurisubharmonic and has full mass. A positive power grows
+faster than a logarithm, so \eqref{period:loglog} implies
+$v-C\leq u$ globally for some constant $C$.
+Full mass is preserved on passing to less singular potentials
+\cite[Propositions~2.11(i) and~2.14(i)]{BEGZ}. Hence $u$ has full mass.
+The non-pluripolar measure gives no mass to the analytic set $E$ and
+agrees with the smooth wedge product on its complement. Consequently
+\[
+ \int_{T\setminus E}(\Theta+\varepsilon\omega)^n
+                  =(\alpha+\varepsilon[\omega])^n.
+\]
+Expand both sides as polynomials in $\varepsilon$.  Their coefficients
+are finite, since all summands on the left are nonnegative.  Equality
+for all $\varepsilon>0$ gives
+\begin{equation}
+\label{period:masses}
+ \int_{T\setminus E}\Theta^k\wedge\omega^{n-k}
+                         =\alpha^k\cdot[\omega]^{n-k}
+                         \qquad(0\leq k\leq n).
+\end{equation}
+If the generic curvature rank is $r$, the left side is positive for
+$k\leq r$ and zero for $k>r$.  This proves the numerical assertion.
+For an immersive Griffiths line the top integral is positive, and
+the volume criterion for a positive current on a compact Kähler
+manifold gives bigness; see \cite{BoucksomVolume}.
+\end{proof}
+
+\begin{lemma}[Restriction to a boundary component]
+\label{period:boundary}
+In the unipotent case of \Cref{period:numerics}, let $E_i$ be a smooth
+boundary component and $N_i$ its monodromy logarithm.  On the open
+stratum of $E_i$, the limiting highest line occurs in a unique weight
+grade of the limiting mixed variation.  The restriction $A|_{E_i}$
+is the Schmid extension, across the remaining boundary of $E_i$, of
+that graded Hodge line.  Its numerical dimension is therefore the
+generic rank of the graded line map.
+\end{lemma}
+
+\begin{proof}
+The extended filtration and $W(N_i)$ give the limiting mixed
+variation along the open stratum.  Since the highest step has rank
+one, exactly one weight grade contains it.  Changes of the normal
+parameter act by $\exp(cN_i)$ and act trivially on the associated
+graded.  At a further crossing the relative monodromy filtration
+identifies the extension of this graded variation with the graded
+of the original extending filtration.  Thus the identification holds
+on the entire component.
+For log smooth algebraic spaces it is stated in
+\cite[\S2.5.3 and Lemma~2.16(3)]{BFMT}; its local proof uses the
+multivariable nilpotent orbit theorem \cite{CKS} and applies here
+as well.  The filtrations and strictness also restrict to a complex
+summand.  Apply \Cref{period:numerics} on $E_i$.
+\end{proof}
+
+\subsection{Rational adjoint periods and descent of the line}
+
+The period-quotient construction requires discrete monodromy.  A
+projection to one complex factor need not retain it.  We therefore
+construct a rationally polarized adjoint variation from the ambient
+integral local system and keep whole rational factors.  A tensor
+construction then recovers a positive power of the original line,
+including its scalar monodromy.
+
+\begin{lemma}[Rational adjoint reduction]
+\label{period:adjoint-reduction}
+In the setting of \Cref{period:adjoint}, let $G$ be the identity
+component of the rational Zariski closure of monodromy.  Then $G$ is
+semisimple.  Its adjoint local system is a rationally polarized
+integral variation of weight zero.
+
+There is a monodromy-invariant collection of rational simple adjoint
+factors, each containing a complex factor acting nontrivially on the
+highest-line orbit, with the following properties.  The resulting
+adjoint variation $\mathbb A$ has discrete integral monodromy.  After a
+positive power and up to a constant one-dimensional complex Hodge
+shift, the original line is the highest line of a complex
+subvariation of a tensor construction on $\mathbb A$.  On a finite
+cover preserving the factors, that line splits into the tensor
+product of the highest lines belonging to the individual complex
+factors.  All these highest steps have rank one.
+\end{lemma}
+
+\begin{proof}
+Semisimplicity and the theorem of the fixed part hold for polarizable
+real and complex variations on a Zariski open subset of a compact
+Kähler manifold.  We may use a Kähler modification of $B$ to apply
+them; see \cite[Lemma~2.3 and its proof]{PopaSchnellKahler} and
+\cite[Theorem~3 and Remark~4]{BakkerMatsushita}.  Thus the algebraic
+monodromy group is reductive.  Pass temporarily to a finite cover
+on which it is connected.  The irreducible complex constituents of
+the local system carry polarizable complex variations.  They can be
+defined over a number field: the representation is rational and its
+algebraic monodromy is reductive.  The determinant of each constituent
+is unitary, as are all its algebraic conjugates, which are again
+constituents of the rational representation.  Its monodromy values
+are algebraic units by the integral lattice.  Kronecker's theorem
+makes these values roots of unity in a fixed number field, hence
+makes the determinant character finite.  It is consequently trivial
+on $G$.  This is Deligne's finite-determinant argument; its classical
+algebraic formulation is \cite[Corollary~4.2.8(iii)(b)]{DeligneHodgeII}.
+The connected center of $G$ acts by scalar characters on
+the irreducible constituents; their determinants detect these
+characters.  Since the original representation is faithful, this
+connected center is trivial.  Therefore $G$ is semisimple.
+
+We next equip the monodromy Lie algebra with a rational polarization.
+In every tensor construction,
+the subspace of $G$-invariant tensors is the fixed part and has a
+constant Hodge structure.  In a flat trivialization the Hodge circles
+$h_x$ and $h_{x_0}$ therefore induce exactly the same linear operator
+on each such subspace at each parameter $z\in S^1$.  Reductivity
+identifies $G$ with the pointwise stabilizer of all its invariant
+tensors.  The circles preserve these subspaces, so normalize $G$,
+and the equality of their actions gives
+\[
+                    h_x(z)h_{x_0}(z)^{-1}\in G(\R).
+\]
+Here is a direct local conjugacy argument.  Along a smooth path in
+the base, write $a_t=\operatorname{Ad}\circ h_t:S^1\to G^{\rm ad}(\R)$
+and $u_t(z)=(\partial_t a_t(z))a_t(z)^{-1}$.  Differentiating the
+homomorphism identity gives
+\[
+ u_t(zz')=u_t(z)+\operatorname{Ad}(a_t(z))u_t(z').
+\]
+Normalized Haar averaging, with $X_t=\int_{S^1}u_t(z')\,dz'$, yields
+$u_t(z)=X_t-\operatorname{Ad}(a_t(z))X_t$.  The solution of
+$g_t'g_t^{-1}=X_t$, $g_0=1$, therefore gives
+$a_t=g_ta_0g_t^{-1}$.  Lift this path through the central isogeny
+$G(\R)^+\to G^{\rm ad}(\R)^+$.  The ratio of $h_t$ and
+$g_th_0g_t^{-1}$ lies in $G(\R)$ by the invariant-tensor equality,
+and centralizes $G$ by the equality of their adjoint actions.
+It lies in the finite center of $G$; continuity in the circle
+parameter makes it the identity.  Thus $h_t=g_th_0g_t^{-1}$,
+proving that the lifted period map lies in a single
+$G(\R)^+$-orbit.
+
+It follows that $\mathfrak g_{\Q}=\operatorname{Lie}(G)$ is a Hodge
+sub-local system of $\End(\mathbb V_{\Q})$, with its weight-zero
+grading.  The Weil involution $\theta$ preserves $\mathfrak g_{\R}$
+and is the restriction of the Cartan involution of the real
+polarization isometry group.  To check the restriction, use the
+positive Hodge metric: on its Lie algebra $\theta(X)=-X^*$.
+If $\mathfrak g_{\R}=\mathfrak k\oplus\mathfrak p$ is its
+eigenspace decomposition, then $\mathfrak k\oplus i\mathfrak p$
+is a compact real form of $\mathfrak g_{\C}$, represented by
+skew-Hermitian endomorphisms.  Thus
+$-B_{\mathfrak g}(X,\theta X)$ is positive definite, proving that
+$\theta$ is Cartan on $\mathfrak g_{\R}$.  The appropriately
+signed Killing form therefore polarizes this weight-zero variation.
+The form is rational.  Moreover
+\[
+          \mathfrak g_{\Q}\cap\End(\mathbb V_{\Z})
+\]
+is a monodromy-invariant lattice.  This gives the rationally polarized
+integral adjoint variation.
+
+Let $\ell$ be the highest line at a reference point of the chosen
+complex summand.  The non-lowering parabolic of the Hodge grading
+preserves $\ell$, so a Borel subgroup preserves it.  Its closed orbit
+\[
+                       J=G_{\C}\ell
+\]
+is a flag variety, a product of flag varieties for the complex simple
+factors.  The span $W$ of this orbit is irreducible.  Indeed, in a
+semisimple decomposition of the representation, the nonzero
+components of a Borel-eigenline all have the same highest weight;
+their span is a single diagonal copy of that irreducible
+representation.  The line is its unique highest Hodge step.  Indeed, at every point
+the Hodge circle normalizes $G$ and preserves the highest line there;
+it consequently preserves the span of its $G_{\C}$-orbit.  Thus $W$
+is a Hodge subvariation.  The lifted line map takes values in $J$,
+and the original line is the pullback of the tautological line there,
+equivariantly for monodromy.
+
+Call a complex simple factor \emph{visible} if it acts nontrivially
+on $J$.  Retain every rational simple adjoint factor containing a
+visible factor.
+This collection is invariant under the possibly disconnected
+monodromy group.  On its Lie algebra we obtain the variation
+$\mathbb A$; its monodromy is contained in the automorphisms of a
+lattice and is discrete.  Notice that an entire rational factor is
+kept.  Projection to only one real factor would not ensure
+discreteness.
+
+We now recover the line from this adjoint system.  Choose a positive
+integer $N$ for which the highest weight $N\lambda$ of $W$ belongs
+to the root lattice.  The Cartan component
+$W_N\subset W^{\otimes N}$ is generated by the orbit of
+$\ell^{\otimes N}$.  It factors through the connected adjoint group,
+and its highest Hodge line is $\ell^{\otimes N}$.  The same
+orbit-span argument just used shows that $W_N$ is a Hodge
+subvariation.
+
+We must also account for disconnected monodromy before applying
+tensor generation.  Deck transformations take the lifted line to
+another point of the same connected orbit, so they preserve $W$ and
+its Cartan components.  The kernel of the action on the retained
+adjoint factors centralizes the $G$-action on $W$, and hence acts
+scalarly by irreducibility.  Its connected part is semisimple and
+has no nontrivial scalar character.  Its component group is finite.
+Increasing $N$ therefore kills the entire kernel action on $W_N$.
+Consequently $W_N$ is a representation of the full projected
+algebraic monodromy group, not just of its identity component.
+This step rules out a discarded nontorsion scalar local system.
+
+The adjoint representation of that projected group is faithful and
+self-dual.  By tensor generation and complete reducibility,
+$W_N$ embeds as a flat direct summand in a tensor construction
+$\mathbb T(\mathbb A)_{\C}$; see
+\cite[Theorem~4.14]{MilneAlgebraicGroups}.  To make the embedding
+compatible with Hodge structures, apply the theorem of the fixed part
+to
+\[
+ H^0\bigl(B^\circ,
+       \mathcal Hom(W_N,\mathbb T(\mathbb A)_{\C})\bigr).
+\]
+This nonzero space of flat maps has a constant complex Hodge
+structure.  A nonzero Hodge-homogeneous component of any nonzero
+flat map is still flat.  Its kernel is $G$-invariant, so the map is
+injective because $W_N$ is irreducible.  A constant one-dimensional
+complex Hodge shift makes it a morphism of variations; that shift
+changes neither the underlying holomorphic line nor its boundary
+extension.  This is the constituent and fixed-part argument of
+\cite[Theorem~3]{BakkerMatsushita}.
+
+The complex summand carries a flat Hermitian polarization.  In a
+real tensor variation of weight $w$ polarized by $Q$, it is induced
+by $i^wQ_{\C}(v,\bar u)$, up to the conventional overall sign.
+Its restriction to a Hodge subvariation is nondegenerate because
+it is definite with the prescribed sign on each Hodge component.
+There is no need for the real bilinear form itself to restrict
+nondegenerately to the chosen complex summand: it may pair that
+summand with its conjugate.
+
+Finally, on a factor-preserving finite cover an irreducible
+representation is a tensor product over the simple factors.  The
+dimension of its highest step is the product of the corresponding
+dimensions.  Since that dimension is one, each factor has a unique
+rank-one highest step.  This gives the asserted factor lines.  Their
+extensions are nef and functorial by
+\Cref{period:numerics,period:boundary}.
+\end{proof}
+
+\begin{lemma}[Generic stabilizers of adjoint periods]
+\label{period:stabilizer}
+Let $G$ be a connected semisimple adjoint group over $\Q$, and let
+$\mathbb A_{\Q}$ be a rationally polarized integral variation whose
+fiber is $\mathfrak g_{\Q}=\operatorname{Lie}(G)$ and whose connected
+algebraic monodromy is $G$, acting by its adjoint representation.
+No nonidentity rational Lie algebra automorphism of
+$\mathfrak g_{\Q}$ fixes all generic lifted periods of $\mathbb A$.
+The automorphism need not belong to the monodromy group.
+\end{lemma}
+
+\begin{proof}
+Suppose $\gamma$ fixes those periods.  By equivariance, every
+monodromy conjugate of $\gamma$ is everywhere of Hodge type $(0,0)$
+in the endomorphism variation.  Their rational span is therefore a
+rational subvariation of type $(0,0)$.  Its induced polarization is
+positive definite.  Intersection with the endomorphism lattice is
+a full monodromy-invariant lattice: clear one denominator of
+$\gamma$; integral monodromy conjugation preserves that denominator.
+A positive definite integral monodromy group is finite.  Consequently
+the connected group $G$ fixes $\gamma$, so $\gamma$ centralizes every
+inner automorphism of $\mathfrak g$.  The centralizer of the inner
+group in $\operatorname{Aut}(\mathfrak g)$ is trivial: on each simple
+ideal commutation with all adjoint operators makes an endomorphism
+scalar, and compatibility with the Lie bracket makes this scalar
+one; permutations of distinct ideals do not commute with their
+separate inner actions.  Thus $\gamma=1$.
+\end{proof}
+
+\subsection{A projective quotient for the retained periods}
+
+We now quotient by connected fibers of the retained full period map.
+The preceding reduction supplies both discrete integral monodromy
+and a tensor description of a power of the line.  The first property
+allows us to compactify the quotient; the second will descend the
+line with its exact parabolic extension.  No bigness assertion about
+$K_S+aP$ is used in this construction.
+
+\begin{proposition}[The projective period quotient]
+\label{period:quotient}
+In \Cref{period:adjoint}, after modifications there is a map
+$p:B'\to S$ with connected fibers to a smooth projective variety
+and a nef rational line $P$ with $\tau^*L=p^*P$.  On a dense open
+subset $S^\circ$, the retained adjoint variation of
+\Cref{period:adjoint-reduction} descends, has the same connected
+algebraic monodromy, and has a generically immersive period map.
+The line $P$ is its parabolically extended highest-weight line,
+up to the positive scaling already described.  If the retained
+collection is empty, $L$ is rationally trivial and $S$ may be a
+point.
+\end{proposition}
+
+The construction uses the compact-period-quotient approach of
+\cite[Theorems~7.6--7.7]{BakkerBrunebarbeTsimerman}, whose
+compactification input goes back to
+\cite[Proposition~III and Remark~III-C]{SommesePeriodMapping}.
+We use the precise class-$\mathcal C$ formulation of
+\cite[Theorem~2.19]{VilladsenHodgeKahler} below.
+
+\begin{proof}
+Choose a compact K\"ahler modification of the class-$\mathcal C$
+source. Then pass to a finite level at which the projected monodromy is
+neat and preserves the factors.  Finite covers of the open locus
+extend to finite ramified covers of compactifications and then to
+Kähler resolutions; see \cite[Lemma~2.21]{VilladsenHodgeKahler}.
+Resolve the boundary to simple normal crossings.  Extend the period
+map over all finite-local-monodromy strata.  At neat level their
+monodromy is trivial and the nilpotent orbit theorem gives this
+extension.
+
+The extended period map on this locus is proper; see
+\cite[Proposition~9.11 and the proof of Theorem~9.6]{GriffithsPeriodsIII}.
+Indeed, if a sequence tending to the boundary has period images in
+a compact subset of the quotient, translate period lifts into a
+compact set.
+The punctured-disk Schwarz estimate makes the displacement of each
+vanishing-coordinate monodromy tend to zero.  The discrete group
+acts properly, since its isotropy in the real period domain is
+compact.  The local monodromies must therefore be trivial at neat
+level, so such a sequence cannot escape the extended locus.
+
+Take the Stein factorization of this proper map.  Its first map has
+compact connected fibers and its base is finite over the period
+image.  It extends after modifications to a map of compact
+class-$\mathcal C$ spaces with Kähler resolutions by
+\cite[Theorem~2.19]{VilladsenHodgeKahler} (version~1).
+That theorem applies to proper connected-fiber maps from Zariski
+open subsets of compact class-$\mathcal C$ spaces.  In terms of
+cycles, one first flattens the proper map, sends its irreducible general fibers
+to a component of the Douady space of the compact source, and takes
+the closure of that family.  The component is compact and in class
+$\mathcal C$.  Near a general fiber the map to the cycle space is an
+open embedding, since an irreducible compact cycle of the same
+dimension in a sufficiently small neighborhood maps to a compact
+analytic subset of a small Stein neighborhood in the base, hence
+to a point, and must be the whole irreducible fiber.
+The incidence map is consequently a modification of the source.
+This produces the required compact quotient and also shows its
+uniqueness as the quotient by the connected general period fibers.
+
+On a dense open subset of this quotient the period and its integral
+variation descend.  At neat level a constant period has trivial
+stabilizer; local slices to the connected-fiber map give the descent
+data.  After shrinking to a proper smooth fibration, the map on
+fundamental groups is surjective, so the descended variation has
+the same connected algebraic monodromy.  Its period differential is
+generically injective by construction.  On a Kähler compactification
+its Griffiths line is big by \Cref{period:numerics}; equivalently,
+one can apply \cite[Corollary~1.3]{BrunebarbeCadorel}.  The quotient
+therefore has a smooth projective model.  This is also the argument
+in \cite[proof of Corollary~2.22]{VilladsenHodgeKahler}.
+
+The finite deck group acts on the connected-period-fiber quotient.
+Take its finite quotient and resolve.  This again has a smooth
+projective model.  A general fiber before this finite quotient is
+the union of connected level fibers indexed by one deck-group orbit.
+The group acts transitively on these fibers, so the quotient is the
+continuous image of any one of them and is connected.  Thus the
+induced map from the original source has connected general fibers,
+hence connected fibers after its Stein factorization.  The latter
+does not change its function field.
+All maps can be resolved using modifications of the original
+source and of this projective model.
+
+There is an actual adjoint variation on a dense Zariski open subset
+of the quotient before changing level.  Write $\Gamma'$ for the
+normal level subgroup of $\Gamma$.  The induced action of
+$\Gamma/\Gamma'$ on the level quotient is faithful.  Indeed, if
+$\delta\in\Gamma$ acts trivially there, discreteness shows locally
+that its action on lifted periods agrees with a fixed
+$\eta\in\Gamma'$.  Then $\eta^{-1}\delta$ fixes a period germ,
+hence the whole generic lifted period image by analytic
+continuation.  By \Cref{period:stabilizer} it is the identity,
+so $\delta\in\Gamma'$.  Resolve the finite group action
+equivariantly and remove its proper algebraic fixed loci and the
+branch locus.  On the resulting dense Zariski open subset the level
+quotient is free and unramified.  Its equivariant integral adjoint
+local system and Hodge filtration descend there.  In particular,
+this descent uses an open subset, not merely very general points.
+
+The highest-weight construction in
+\Cref{period:adjoint-reduction} gives a line on that open subset
+whose pullback is a fixed positive power of $L$.  Take its
+parabolic extension on a log smooth projective compactification
+$S$ and divide by that power to obtain $P$.  Nefness follows from
+\Cref{period:numerics}.  Pullback of the unipotent extensions,
+followed by rational descent, gives
+$\tau^*L=p^*P$ on the compact models, including the boundary.
+Equivalently, this can be checked at level and then descended by
+the norm: a line whose finite pullback is trivial is torsion in
+the rational Picard group.  The identity therefore includes the
+boundary and persists under further modifications.
+
+If no factor is retained, the orbit is a point and the only possible
+scalar monodromy is the finite scalar group treated in
+\Cref{period:adjoint-reduction}.  Its parabolic line is rationally
+trivial, as asserted.
+\end{proof}
+
+\subsection{Loss of numerical rank on the marked boundary}
+
+Fix the projective quotient of \Cref{period:quotient} and a log smooth
+compactification $(S,E)$ of its descended variation.  Let $D\leq E$
+be the reduced sum of components with nonidentity local monodromy
+on the retained adjoint system, including finite monodromy.
+Logarithmic general type will give bigness of $K_S+D$.  The next
+lemma is what permits us to remove $D$.
+
+\begin{lemma}[Strict loss of rank at the marked boundary]
+\label{period:rankloss}
+For every component $D_i$ of $D$,
+\begin{equation}
+\label{period:strictdrop}
+                       \nu(P|_{D_i})<\nu(P).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write $n=\dim S$ and $r=\nu(P)$, the generic rank of the line map
+$j$ by \Cref{period:numerics}. The full retained period map has rank
+$n$; the argument does not require $r=n$.
+Work first on a finite Galois cover of the
+open locus with connected algebraic monodromy and no factor
+permutations.  Its connected adjoint group is denoted by $G$.
+
+\smallskip\noindent
+\emph{The Ax--Schanuel decomposition.}
+Choose a Hodge-generic point, also general for the ranks of the
+factor period maps and their joint differentials.  There are only
+finitely many rational normal-factor decompositions of $G$, so
+these requirements can be imposed simultaneously.  Let $T$ be a
+local fiber germ of $j$ there, and $Z$ its irreducible Zariski
+closure in the source of the variation.  Thus $\dim T=n-r$.
+Move within the germ to a smooth point of $Z$ and use a smooth
+open subset of a resolution when applying Hodge theory.
+
+The restricted variation on $Z$ has the same generic Mumford--Tate
+group as the ambient variation, since $Z$ contains the chosen
+Hodge-generic point.  The monodromy normality theorem therefore
+makes its connected monodromy $H$ normal in $G$; see
+\cite[\S5, Theorem~1]{AndreMonodromy}.  In adjoint notation write
+\[
+                            G=H\times Q.
+\]
+The $Q$-period is constant on the lifted $Z$, by the fixed-part
+theorem after killing finite monodromy.  The varying monodromy
+domain on $Z$ is therefore the domain for $H$.  Write $J_H$ for
+its line-orbit projection and $d_H=\dim J_H$.
+In local flat coordinates write $j=(j_H,j_Q)$, with targets $J_H$
+and $J_Q$, the products of line orbits for the two groups.  We will
+show that the $H$-line map has full rank $d_H$ on $Z$, whereas the
+$Q$-line map and the full $Q$-period map have equal generic rank
+on the ambient base.
+
+For the first assertion, put
+$m=\dim Z$ and $d=\dim\check{\mathcal D}_H$, where
+$\check{\mathcal D}_H$ is the compact dual of the connected-monodromy
+period domain.  The projection
+$\check{\mathcal D}_H\to J_H$ is a surjective morphism of
+homogeneous projective flag varieties.  All its fibers have
+dimension $d-d_H$.  Let
+$V\subset Z\times\check{\mathcal D}_H$ be the algebraic locus
+where the projected line equals its value on $T$.  It follows that
+$\dim V=m+d-d_H$.  The period graph has dimension $m$ in an
+ambient variety of dimension $m+d$, so its expected intersection
+dimension with $V$ is $m-d_H$.
+
+By \cite[Theorem~1.1]{BakkerTsimermanAxSchanuel}, an intersection
+component of larger dimension projects into a proper algebraic weak
+Mumford--Tate subvariety of $Z$.  This is impossible for the component
+containing the lifted period graph over $T$, since $T$ is Zariski
+dense in $Z$.  Conversely, a map
+to $J_H$ has fibers of dimension at least $m-d_H$.  At the chosen
+smooth point the restricted fiber is exactly $T$, because the
+ambient local fiber is $T\subset Z$.  Equality follows, proving the
+claim.  The cited theorem applies to the rationally polarized
+integral adjoint variation on a smooth algebraic base and uses
+precisely its connected-monodromy domain.
+
+Let $q$ denote the full $Q$-period map and set
+$s_Q=\rank(dq)$. Choose the smooth point $x\in T\cap Z_{\mathrm{reg}}$
+inside the ambient constant-rank neighborhood fixed above. Since
+$T$ is Zariski dense in $Z$, such a point exists. The constant-rank
+theorem gives $\ker(dj_x)=T_xT\subset T_xZ$, and hence
+\[
+ \rank(dj_x|_{T_xZ})=\dim Z-\dim T=d_H.
+\]
+The $Q$-period is constant on $Z$, so $T_xZ\subseteq\ker(dq_x)$.
+Thus, at this same point,
+\[
+ \ker(dj)\subseteq\ker(dq),\qquad
+             dj_H(\ker(dq_x))=T_{j_H(x)}J_H.
+\]
+The first inclusion uses $T_xT\subset T_xZ$; the second is the
+just-established rank calculation on $T_xZ$.
+The differential $dj_Q$ factors through $dq$.  If $dj_Q(v)=0$,
+choose $w\in\ker(dq)$ with $dj_H(w)=dj_H(v)$.  Then
+$v-w\in\ker(dj)$, and therefore $dq(v)=0$.  We conclude
+\begin{equation}
+\label{period:rankdecomposition}
+       \rank(dj_Q)=s_Q,\qquad r=d_H+s_Q.
+\end{equation}
+These are generic identities by our choice of the point.
+
+The factor-line construction, after a common positive rescaling
+and a further finite cover, now gives nef lines with
+\begin{equation}
+\label{period:linedecomposition}
+       \pi^*P=P_H+P_Q,\qquad
+       \nu(P_H)=d_H,
+       \qquad P_Q=q^*B_Q.
+\end{equation}
+Here $q:\widetilde S\to S_Q$ is a resolved projective quotient
+for the full $Q$-period, constructed by
+\Cref{period:quotient}, and $\dim S_Q=s_Q$.  The highest-line
+rank on $S_Q$ is $s_Q$ by
+\eqref{period:rankdecomposition}.  Only the quotient construction,
+not adjoint positivity, has been used to construct $S_Q$.
+
+\smallskip\noindent
+\emph{Reduction for a boundary component.}
+For a fixed marked $D_i$, use a common finite Galois cover that
+preserves the factors and makes boundary monodromy unipotent.
+Resolve it and the map $q$ equivariantly, and let $E_i$ be the
+strict transform of a component above $D_i$ that is generically
+finite over $D_i$.  The inertia group at its generic point fixes
+$E_i$ pointwise.  Further rescaling does not change any numerical
+dimension.  Nefness gives
+\[
+                     \nu(P_H|_{E_i})\leq d_H.
+\]
+For example, every intersection involving $E_i$ is bounded by
+the corresponding intersection with a sufficiently ample
+effective divisor; the vanishing of higher powers of $P_H$
+then gives this inequality.  If $E_i$ does not dominate $S_Q$,
+\[
+                     \nu(P_Q|_{E_i})\leq s_Q-1,
+\]
+and the binomial expansion for the two nef lines in
+\eqref{period:linedecomposition} proves the strict drop.
+We may therefore suppose $E_i$ dominates $S_Q$.  At its generic
+point the $Q$-variation is pulled back from the interior of
+$S_Q$, so its local unipotent logarithm vanishes.
+
+\smallskip\noindent
+\emph{Infinite local monodromy.}
+If the original local monodromy is infinite, its logarithm at
+unipotent level is a nonzero rational element
+$N\in\mathfrak h$.  Every nonzero rational element of a rational
+simple factor has nonzero projection to every conjugate complex
+factor.  Each retained rational factor has a visible complex
+factor.  Hence $N$ acts nontrivially on $J_H$.
+
+Let $w$ be the pure weight of the representation supplying $P_H$,
+and let $F^a=\C v$ be its limiting highest line at a general
+point of $E_i$.  This line has a unique weight contribution,
+say of weight $w+k$.  It is primitive and $k\geq0$: a
+nonprimitive contribution, or a contribution below the middle
+weight, would come by monodromy Lefschetz from a nonzero
+filtration piece above $F^a$.
+Moreover
+\begin{equation}
+\label{period:nilpotenttop}
+                    N^{k+1}v=0,\qquad N^kv\ne0.
+\end{equation}
+To see that this statement holds for the actual vector rather
+than only its weight-graded image, use the Deligne splitting
+of the limiting mixed Hodge structure.  The one-dimensional
+highest step occupies a single summand, and $N$ has degree
+$(-1,-1)$ for this functorial splitting.  Primitive vanishing
+on the graded piece therefore gives the first equality in
+\eqref{period:nilpotenttop}; the Lefschetz isomorphism gives
+the second.  For a complex summand the same argument is made
+in its conjugate-paired real variation and restricted to the
+summand.
+
+In nilpotent-orbit coordinates $[v]\in J_H$.  Since this orbit
+is closed in projective space and invariant under $\exp(tN)$,
+\begin{equation}
+\label{period:fixedlocus}
+       [N^kv]=\lim_{t\to\infty}\exp(tN)[v]
+                \in J_H\cap\mathbb P(\ker N).
+\end{equation}
+This fixed locus is a proper closed subset of the irreducible
+variety $J_H$, because $N$ acts nontrivially on it.  Its dimension
+is less than $d_H$.  Restrict to a simply connected open part of the boundary stratum
+on which $k$ is constant.  In a flat trivialization, both $N$ and
+$W(N)$ are constant.  Project $N^kv$ to
+$\Gr^{W(N)}_{w-k}$ and apply the inverse of the fixed Lefschetz
+isomorphism
+\[
+ N^k:\Gr^{W(N)}_{w+k}\longrightarrow\Gr^{W(N)}_{w-k}.
+\]
+This recovers the weight-graded limiting highest line.  Every actual
+vector $N^kv$ lies in $W(N)_{w-k}$, and its image in that graded
+piece is nonzero.  Thus the graded line map factors through the
+rational map on
+\[
+ J_H\cap\mathbb P(\ker N)\cap\mathbb P(W(N)_{w-k})
+\]
+induced by this fixed linear quotient and inverse.  This locus is
+contained in the proper fixed locus in \eqref{period:fixedlocus},
+and the rational map is defined at every point under consideration.
+Its rank is less than $d_H$.
+By \Cref{period:boundary},
+\[
+                       \nu(P_H|_{E_i})<d_H.
+\]
+Together with $\nu(P_Q|_{E_i})\leq s_Q$ this proves the
+required inequality in the infinite-monodromy case.
+
+\smallskip\noindent
+\emph{Finite nonidentity local monodromy.}
+Let $\gamma$ be the original finite local monodromy.  The
+disconnected-monodromy argument in \Cref{period:adjoint-reduction}
+shows that $\gamma$ preserves the orbit span $W$.  It therefore acts
+on the highest-line orbit $J$, permuting its nontrivial simple-factor
+flag varieties. Equivariance preserves the kernel of
+$G_{\C}\to\operatorname{Aut}(J)$, which is the product of the
+invisible simple factors. Thus visible and invisible factors cannot
+be exchanged. At unipotent level its logarithm is zero, so the variation extends
+with pure limiting flags.  The inertia generator fixes the generic
+boundary point; equivariance therefore makes these flags fixed by
+$\gamma$.  Suppose that the
+boundary line retained rank $r=d_H+s_Q$.  Since $E_i$ dominates $S_Q$,
+its full $Q$-period and its $Q$-line map both have rank $s_Q$.
+Along a generic fiber of this $Q$-period, the $H$-line map
+must consequently have rank $d_H$.  Its coordinates fill an
+open subset of $J_H$ while the full $Q$-flags remain fixed.
+
+The fixed-point condition for $\gamma$ rules out permutations
+involving any visible $H$-factor, including a permutation
+with a factor whose coordinate is held fixed.  On each visible
+$H$-factor of the highest-line orbit it forces the identity action.
+The action of a simple adjoint group on a nontrivial flag variety is faithful;
+equivariance then makes the induced automorphism of that
+simple group the identity as well.  Rationality forces the
+same conclusion on all conjugate factors.  Thus $\gamma$
+acts identically on $H$ and preserves $Q$.
+
+Its $Q$-action fixes every generic $Q$-period, by the domination
+of $S_Q$ and analytic continuation from the resulting open set of
+lifted periods. Apply \Cref{period:stabilizer} to the descended
+$Q$-variation.  This action is the identity.  Hence $\gamma$
+is the identity on the whole retained adjoint system,
+contradicting the marking.
+
+In both cases nef binomial expansion and
+\eqref{period:rankdecomposition} give
+$\nu(\pi^*P|_{E_i})<r$.  Numerical dimension of a nef line is
+unchanged by a generically finite pullback, so this is
+\eqref{period:strictdrop}.  If $r=0$, the local line fiber is the whole source germ, so its
+Zariski closure is the whole source and $H=G$, $Q=1$.  The
+Ax--Schanuel calculation gives $d_H=0$.  Every retained rational
+factor contains a factor acting nontrivially on the line orbit;
+hence there can be no retained factor.  Generic immersivity of the
+full retained period map then forces $S$ to be a point, with no
+marked component.
+\end{proof}
+
+\subsection{Removing a boundary by counting sections}
+
+The period argument must pass from a big logarithmic canonical
+divisor to a big adjoint divisor without its boundary. The following
+lemma isolates the last step: sections lost on the boundary grow
+more slowly in the nef direction than sections on the whole space.
+For a nef rational divisor $J$ on a smooth projective $n$-fold, set
+\[
+ \nu(J)=\max\{q\in\{0,\ldots,n\}:J^q\cdot H^{n-q}>0\},
+\]
+where $H$ is ample. This is independent of $H$ and equals zero
+on a point. The proof below fixes the nef coefficient before
+taking the asymptotic section limit.
+
+\begin{lemma}\label{period:subtract}
+Let $Z$ be a smooth projective variety, $J$ a nef rational divisor,
+$A$ a rational divisor, and
+$D=\sum_{i=1}^s d_iE_i$ an effective rational divisor with smooth
+prime components. Suppose $A+D$ is big and
+\[
+ \nu(J|_{E_i})<r:=\nu(J)
+ \quad\text{for every component of }D.
+\]
+Then $A+tJ$ is big for every sufficiently large rational $t$.
+\end{lemma}
+
+\begin{proof}
+If $r=0$, the condition forces $D=0$, so $A$ is already big;
+adding a nef divisor preserves bigness. This also covers dimension
+zero with the usual convention. Assume $r>0$ and put $n=\dim Z$.
+Choose an ample rational divisor $H_1$ with
+$A+D-H_1$ rationally effective. For fixed rational $t\geq0$
+and sufficiently divisible $m$,
+\[
+ h^0(Z,m(A+D+tJ))\geq h^0(Z,m(H_1+tJ)).
+\]
+Since $H_1+tJ$ is ample, asymptotic Riemann--Roch makes the
+leading coefficient on the right
+$(H_1+tJ)^n/n!$. As a polynomial in $t$, this intersection has
+degree $r$ and positive leading coefficient.
+
+Remove $mD$ one component at a time by divisor exact sequences.
+Every lost quotient is a line bundle on an $E_i$ of the form
+\[
+ \mathcal O_{E_i}\left(m(A+D+tJ)-\sum_h k_hE_h\right),
+ \qquad 0\leq k_h\leq md_h.
+\]
+Choose an ample integral divisor $B_0$ so large that a fixed
+denominator-clearing multiple of $B_0-A-D$ is globally generated.
+For each $h$, choose an ample integral divisor $B_h$ such that
+both $B_h$ and $B_h+E_h$ are globally generated. Put
+$H_2=B_0+\sum_h d_hB_h$. The difference between
+$m(H_2+tJ)$ and the displayed quotient divisor is
+\[
+ m(B_0-A-D)+
+ \sum_h\bigl((md_h-k_h)B_h+k_h(B_h+E_h)\bigr).
+\]
+It is globally generated in these divisible degrees. A section
+not vanishing identically on $E_i$ gives an injection into
+$\mathcal O_{E_i}(m(H_2+tJ)|_{E_i})$. There are exactly
+$md_i$ quotient terms on $E_i$. Hence
+\[
+ h^0(Z,m(A+tJ))
+ \geq h^0(Z,m(H_1+tJ))
+       -m\sum_i d_i h^0(E_i,m(H_2+tJ)|_{E_i}).
+\]
+For each fixed $t$, divide by $m^n/n!$ and let $m$ tend to
+infinity through the specified multiples. All the comparison
+divisors are ample, so the resulting lower bound is
+\[
+ (H_1+tJ)^n
+     -n\sum_i d_i(H_2+tJ)^{n-1}\cdot E_i.
+\]
+Each restriction polynomial has degree at most
+$\nu(J|_{E_i})\leq r-1$, because the restricted divisor is nef.
+The positive degree-$r$ term of the first polynomial therefore
+dominates the loss for every sufficiently large $t$. The section
+growth of $A+tJ$ is then positive in dimension $n$, which is
+exactly bigness.
+\end{proof}
+
+Taking $A=K_Z$ and $D$ reduced gives the required removal of the
+period boundary once its restriction ranks have been established.
+The weights also allow simultaneous removal of other effective
+divisors satisfying the same rank inequality, for example a
+divisorial ramification term on a generically finite cover.
+
+
+\begin{proof}[Proof of \Cref{period:adjoint}]
+Apply \Cref{period:quotient}.  If its target is a point there
+is nothing further to prove.  Otherwise mark the divisor $D$
+as above.  The adjoint variation extends across every unmarked
+boundary component, including their intersections away from
+$D$, because its local monodromies there are trivial.  Its
+period differential is generically injective.  Consequently
+\cite[Theorem~1.1]{BrunebarbeCadorel} gives
+\[
+                              K_S+D\text{ big}.
+\]
+If an initial choice omitted codimension-two loci, apply the
+theorem on a log resolution and push the sections down; the
+resulting logarithmic canonical divisor pushes forward to
+$K_S+D$.  By \Cref{period:rankloss}, each marked component
+has strictly smaller numerical rank.  Apply \Cref{period:subtract} with $A=K_S$ and the marked reduced
+divisor $D$. It gives $K_S+aP$ big.  Together with the
+rational line identity in \Cref{period:quotient}, this proves
+\eqref{period:conclusion}.  Since $P$ is nef, the same bigness
+holds for every larger rational coefficient $a$.
+\end{proof}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/reduced-root.tex

@@ -0,0 +1,391 @@
+\section{Minimal root covers for reduced boundaries}\label{cyc:section}
+
+For a smooth projective pair of logarithmic Kodaira dimension zero
+with reduced SNC boundary, the minimal cyclic cover has a
+one-dimensional space of logarithmic pluriforms in every positive
+degree. This strengthens
+the character-line statement of Proposition~\ref{rankone:comparison}.
+We prove the refinement together with its exact Hodge and divisorial
+normalization. It gives a pure integral realization of the entire
+highest line and fixes the normalization used in further
+relative-Iitaka constructions.
+The normalization lemmas below compare the minimum over actual fiber components
+with the threshold over all divisorial valuations. They show that a
+zero coefficient of the normalized base boundary yields a
+multiplicity-one component, and that later permitted modifications
+preserve the complete section spaces. These are the additional
+normalization statements used by the variation companion.
+
+\begin{lemma}[The cyclic cover of a logarithmic Kodaira-zero pair]
+\label{lem:reduced-cyclic-top}
+Let $(J,D)$ be a smooth geometrically integral projective pair over a
+field $K$ of characteristic zero. Assume that $D$ is reduced and has
+simple normal crossings and that
+$\kappa(J_{\overline K},K_{J_{\overline K}}+D_{\overline K})=0$.
+Choose the least positive integer $p$ for which
+$H^0(J,p(K_J+D))\ne0$, a nonzero section $\omega$ in that space,
+and put
+\[
+ E=\operatorname{div}(\omega)+pD,
+ \qquad \Delta=D-\frac1pE.
+\]
+Write $\omega=b\eta^{\otimes p}$ in a rational canonical frame,
+and let $\tau:N\to J$ be the normalization in
+$K(J)[t]/(t^p-b)$. Let $\rho:\widetilde N\to N$ be a log
+resolution, and set
+\[
+ P=\operatorname{Supp}(\Delta^{=1}),\qquad
+ D_{\widetilde N}=\bigl((\tau\circ\rho)^{-1}P\bigr)_{\mathrm{red}}.
+\]
+Then $N$ is geometrically integral, the rational top form
+$\Omega=t\tau^*\eta$ extends logarithmically to
+$(\widetilde N,D_{\widetilde N})$, and
+\[
+ H^0\bigl(\widetilde N,m(K_{\widetilde N}+D_{\widetilde N})\bigr)
+       =K\,\Omega^m\qquad(m\ge1).
+\]
+\end{lemma}
+
+\begin{proof}
+Every nonzero plurilogarithmic section space on $J$ is
+one-dimensional: two independent sections of one degree would
+define a nonconstant rational map. Consequently $\kappa(E)=0$,
+and every effective integral divisor supported on $E$ also has
+exactly one section, since it is bounded above by a multiple of $E$.
+These assertions and the minimal index $p$ are unchanged by field
+extension, by flat base change for global sections.
+
+Over $\overline K(J)$, reducibility of $T^p-b$ would make $b$
+an $\ell$th power for some prime $\ell\mid p$. The corresponding
+root of $\omega$ would be a nonzero logarithmic pluriform of
+degree $p/\ell$, since its logarithmic zero divisor is $E/\ell$.
+This contradicts minimality. Thus $N$ is geometrically integral.
+
+At a prime $Q$ of $N$ above a prime $R$ of $J$, write
+\[
+ a=\operatorname{coeff}_R E,\qquad
+ d=\operatorname{coeff}_R D\in\{0,1\},\qquad
+ e=\operatorname{ord}_Q(\tau^*R).
+\]
+The ramification formula for top differentials gives
+\begin{equation}\label{eq:cyclic-root-order}
+ \operatorname{ord}_Q\Omega=\frac{ea}{p}-ed+e-1.
+\end{equation}
+If $a=0$, the cover is unramified at $R$, and this order is
+$-d$. If $a>0$, then $q=ea/p$ is a positive integer: the
+integrality follows from $t^p=b$ and $d\in\mathbf Z$.
+The order is $q-1\ge0$ when $d=1$, and $q+e-1\ge0$ when
+$d=0$. Thus the divisorial poles are exactly the reduced inverse
+image $D_N=(\tau^{-1}P)_{\mathrm{red}}$.
+
+We also check extension across the exceptional divisors of $\rho$.
+The subpair $(J,\Delta)$ is sub-log-canonical and is sub-klt
+outside $P$. Indeed, its positive part is an SNC boundary bounded
+by $D$, with all coefficients strictly below one outside $P$;
+subtracting an effective divisor cannot worsen discrepancies.
+Define the crepant finite pullback by
+$K_N+\Delta_N=\tau^*(K_J+\Delta)$. With the compatible
+canonical trivialization, $\Delta_N=-\operatorname{div}(\Omega)$.
+For a prime divisor $V$ over $N$, let its restricted valuation
+on $K(J)$ be $e_Vv$. The same ramification formula gives
+\[
+ 1+\operatorname{ord}_V\Omega
+       =e_V A_{J,\Delta}(v),
+\]
+where $A$ denotes log discrepancy. The right-hand side is
+nonnegative, and is positive if the center lies outside $P$.
+Since the left-hand side is integral, $\Omega$ has at most a
+simple pole above $P$ and is regular elsewhere. This proves the
+asserted logarithmic extension.
+
+Finally let $H_N=\operatorname{div}_N(\Omega)+D_N$. Equation
+\eqref{eq:cyclic-root-order} gives the sharper bound
+\[
+             0\le H_N\le\tau^*E.
+\]
+For completeness, when $a>0$ and $d=1$ its coefficient is
+$q-1\le ea$; when $d=0$, it is
+$q+e-1\le q+p-1\le pq=ea$, because $e\le p$ and $q\ge1$.
+At $a=0$ its coefficient is zero.
+Writing a logarithmic $m$-canonical form as $u\Omega^m$ and
+testing its orders at the primes of the finite normalization gives
+\[
+ \operatorname{div}_N(u)+mH_N\ge0,
+ \qquad
+ u\in H^0(N,\mathcal O_N(m\tau^*E)).
+\]
+Finite pullback preserves Iitaka dimension, so
+$\kappa(N,\tau^*E)=\kappa(J,E)=0$. Since $\tau^*E$ is
+effective and $N$ is geometrically integral, the last space is
+$K$. Conversely $\Omega^m$ is a logarithmic pluriform for every
+$m\ge1$, proving the equality.
+\end{proof}
+
+\begin{corollary}[An integral pure realization of the moduli line]
+\label{cor:reduced-integral-moduli}
+Let $x:X\to W$ be a projective fibration between smooth complex
+projective varieties, with geometrically integral generic fiber
+$J$. Let $D_X$ be a reduced SNC boundary whose restriction $D_J$
+satisfies the hypotheses of Lemma~\ref{lem:reduced-cyclic-top}.
+Let $M_W$ be the rational Hodge divisor of the rank-one reduction,
+with its rational frame and parabolic orders as in
+Equation~\eqref{rankone:order}. On sufficiently high smooth models,
+there is a polarizable integral pure variation
+of Hodge structures on a dense smooth open $W^\circ\subset W$
+whose top nonzero Hodge filtration piece is a line, and
+\[
+ M_W\sim_{\mathbf Q}
+ \text{the rational canonical extension of that line}.
+\]
+In the canonical-bundle-formula normalization, this is the moduli
+divisor of the auxiliary generic subpair
+$\Delta_J=D_J-E_J/p$, which may have negative coefficients.
+\end{corollary}
+
+\begin{proof}
+Let $s$ be the minimal-degree logarithmic generator on the generic
+fiber, and choose a rational relative canonical frame $\eta$ on
+$X$. The rational function $s/\eta^{\otimes p}$ defines one
+cyclic cover over a dense open of $W$, not merely a collection
+of local character covers. Spread its resolution and pole boundary
+over a smaller dense open $W^\circ$ so that they form a smooth
+projective family of SNC pairs. Write $r=\dim J$, and let
+$u:\widetilde X^\circ\setminus D_{\widetilde X}^\circ\to W^\circ$
+be the complementary open family. Its geometric variation
+\[
+ \mathbb V_{\mathbf Z}=R^ru_*\mathbf Z/\mathrm{torsion}
+\]
+is admissible and graded-polarizable. Logarithmic Hodge theory
+identifies
+\[
+ F^{r+1}\mathbb V=0,\qquad
+ F^r\mathbb V_{\mathcal O}
+   =\widetilde x_*\omega_{\widetilde X^\circ/W^\circ}
+                           (D_{\widetilde X}^\circ).
+\]
+Lemma~\ref{lem:reduced-cyclic-top} makes this entire top piece a
+line. Strictness for the weight filtration therefore singles out
+one weight $w$ with
+\[
+ F^rW_{w-1}\mathbb V=0,
+ \qquad F^rW_w\mathbb V=F^r\mathbb V,
+ \qquad
+ F^r\operatorname{Gr}^W_w\mathbb V\simeq F^r\mathbb V.
+\]
+The rational variation $\operatorname{Gr}^W_w\mathbb V$ is
+polarizable. Intersecting its rational weight filtration with
+$\mathbb V_{\mathbf Z}$ and removing torsion supplies an integral
+lattice; after rescaling, its rational polarization is integral.
+Thus this pure variation has precisely the required top line.
+
+After a finite cover making local monodromies unipotent, the
+canonical extensions have locally free double gradeds for the
+Hodge and weight filtrations. The displayed identification of top
+lines consequently extends across the boundary; see
+\cite[Proposition~3.12]{FujinoFujisawa}. Descent gives the same
+identification for their rational canonical extensions.
+
+The global rational frame of this entire top line is
+$s^{1/p}$. Equation~\eqref{rankone:order} computes its parabolic
+order at every prime divisor $P$ as $\beta_P$, the order of the
+same rational frame in $M_W$. Equality of these divisorial
+orders identifies the rational extensions. In particular, it
+identifies $M_W$ itself, with no additional factor of $p$.
+We check the rank condition for the canonical-bundle-formula
+comparison. On a log resolution $\mu:J'\to J$, let
+$R'=\lceil A^*(J,\Delta_J)_{J'}\rceil$, where $A^*$ omits the
+log-discrepancy-zero components of the discrepancy divisor.
+The subpair is sub-log-canonical, so $R'\geq0$. At the strict
+transform of each original prime its coefficient is zero if
+$0\leq\Delta_J\leq1$, and is $\lceil-\Delta_J\rceil$ otherwise.
+Thus $0\leq\mu_*R'\leq E_J$. Testing original primes gives
+$H^0(J',\cO_{J'}(R'))\subseteq H^0(J,\cO_J(E_J))=\C(W)$,
+and constants give equality. This is the required rank-one condition.
+Extending the generic subpair by
+$-p^{-1}\operatorname{div}_{K_{X/W}}(s)$ makes its log canonical
+divisor rationally equivalent to $x^*K_W$. The generic sub-lc,
+relative triviality, and rank conditions of
+\cite[Definition~6.18]{BFMT} therefore hold. Adding a base pullback
+to normalize vertical orders changes the total base term and the
+discriminant equally, leaving the moduli divisor unchanged.
+
+The least positive index trivializing $K_J+\Delta_J$ is $p$.
+Indeed, a smaller such index $q$ would produce a rational
+$q$-canonical form with logarithmic zero divisor $qE_J/p\geq0$,
+contradicting the minimality of $p$. Thus the cyclic field extension
+in Construction-Definition~6.23 of \cite{BFMT} is the one used above.
+
+For its boundary comparison, put
+$P_J=\operatorname{Supp}(\Delta_J^{=1})$ and first make a common
+crepant log resolution of the auxiliary subpair and its root cover. The
+coefficient-one support on the crepant model is contained in the
+total inverse image of $P_J$: outside $P_J$ the auxiliary subpair is
+sub-klt. Every additional component allowed by our pole boundary
+has positive log discrepancy, so the ramification discrepancy
+formula in Lemma~\ref{lem:reduced-cyclic-top} makes $\Omega$
+regular there. The top-form space for the BFMT pole boundary
+therefore injects into the top-form space for our boundary and
+contains $\Omega$; both are the line generated by $\Omega$.
+After spreading over a common smooth base open, inclusion of the
+open complements induces a morphism of their admissible geometric
+mixed variations. This morphism is strict for the Hodge and weight
+filtrations, including their limiting filtrations after a finite
+cover making boundary monodromy unipotent. It therefore identifies
+the top lines in the same pure weight grade. Functoriality
+of canonical extension gives the same identification across the
+boundary and hence for the parabolic rational lines. Theorem~6.28 of
+\cite{BFMT} now identifies this Hodge divisor with the moduli
+divisor using one common positive multiple on both sides. This
+comparison permits negative coefficients in the generic subpair;
+it does not use the separate semiampleness theorem requiring
+generic effectiveness.
+\end{proof}
+
+\begin{remark}
+The hypothesis that $D$ is reduced is essential to
+Lemma~\ref{lem:reduced-cyclic-top}. For five distinct points on
+$\mathbf P^1$, the rational boundary
+$D=\frac25(P_1+\cdots+P_5)$ satisfies $K_{\mathbf P^1}+D\sim_{\mathbf Q}0$.
+Its index-five root cover has genus six and empty pole boundary.
+Thus its entire top-form space has dimension six. The refinement therefore applies to reduced boundaries; the
+character-line construction of Section~\ref{rankone:section} is used
+for general rational orbifold boundaries.
+\end{remark}
+
+\subsection{The original relative orders}
+
+
+\begin{lemma}[Normalization of the rank-one boundary]
+\label{lem:rankone-normalization}
+Let $x:X'\to W$ be a projective fibration on smooth models, let $D'$ be
+an effective reduced SNC divisor, and suppose that its geometric generic
+fiber $(J,D_J)$ has logarithmic Kodaira dimension zero. Fix a nonzero
+relative rational $p$-canonical form $s$ generating
+$H^0(J,p(K_J+D_J))$. At a prime $P$ of $W$ set
+\[
+ a_E=\operatorname{ord}_E(x^*P),\qquad
+ r_E=p^{-1}\operatorname{ord}^{pK_{X'/W}}_E(s),\qquad
+ d_E=\operatorname{coeff}_E(D'),\qquad
+ t_P=\min_{E\mapsto P}\frac{r_E+d_E}{a_E}.
+\]
+Here the minimum is over actual components on the chosen source model.
+Choose that model so that the generator divisor, boundary and inverse
+image of $P$ have SNC support over the generic point of $P$; later
+source modifications carry strict boundary plus reduced exceptional
+boundary. The coefficients in the rank-one comparison satisfy
+\[
+ \alpha_P=t_P,\qquad
+ \beta_P=\inf_{E\mapsto P}\frac{1+r_E}{a_E}-1,\qquad
+ T_P=1-c_P,
+ \quad
+ c_P:=\inf_{E\mapsto P}\frac{1+r_E}{a_E}-t_P.
+\]
+The infima allow divisorial valuations on higher smooth source models.
+Equivalently, $c_P$ is the log canonical threshold of $x^*P$ for the
+normalized auxiliary subpair
+\[
+ \Delta^{\mathrm{aux}}
+ =-p^{-1}\operatorname{div}_{K_{X'/W}}(s)+x^*\sum_Qt_QQ.
+\]
+If $T_P=0$, an actual component attaining $t_P$ has
+$a_E=1$, $d_E=0$, and attains the threshold infimum as well.
+\end{lemma}
+
+\begin{proof}
+Choose a local frame $\eta$ of $K_W$ that does not vanish at the generic
+point of $P$. The canonical line-bundle identification
+\[
+ \mathcal O_{X'}(pK_{X'/W})\otimes x^*\mathcal O_W(pK_W)
+       \simeq\mathcal O_{X'}(pK_{X'})
+\]
+identifies the order of $s$ with the order of
+$s\wedge x^*\eta^{\otimes p}$. Thus the quantity $l_E$ in the
+rank-one order formula is exactly $r_E$. This uses the actual relative
+canonical line bundle: the Jacobian contribution is already present in
+$r_E$, so no ramification term is added or subtracted here.
+Consequently its two order formulas give
+\[
+ \alpha_P=t_P,
+ \qquad
+ \beta_P=\min_E\frac{r_E+1-a_E}{a_E}.
+\]
+Put
+\[
+ \Delta_0=-p^{-1}\operatorname{div}_{K_{X'/W}}(s),
+ \qquad \lambda_P=\min_{E\mapsto P}\frac{1+r_E}{a_E},
+\]
+where the minimum is over actual vertical components. The indicated
+SNC preparation and the generic logarithmic bound give $\Delta_0$ SNC
+support and horizontal coefficients at most one. In
+$\Delta_0+\lambda_Px^*P$, each vertical coefficient is
+$-r_E+\lambda_Pa_E\leq1$. Thus this subpair is sub-log-canonical
+over the generic point of $P$, and every higher divisorial valuation
+satisfies
+\[
+                     1+r_E-\lambda_Pa_E\geq0.
+\]
+An actual component attaining $\lambda_P$ gives equality. Hence the
+infimum over all such valuations equals $\lambda_P$, and
+$\beta_P=\lambda_P-1$.
+
+For completeness, the first minimum also remains unchanged on higher
+source models. A coefficient realizing $t_P$ makes
+$t_Pa_E-r_E$ no greater than the original boundary coefficient at every
+actual component over $P$. Equivalently, after clearing denominators,
+the normalized generating form satisfies every logarithmic order test
+over the generic point of $P$. Its pullback remains logarithmic with
+strict plus reduced exceptional boundary. No new valuation can lower
+the first minimum, while an old minimizing valuation persists.
+
+At any divisorial valuation above the generic point of $P$, the
+crepant coefficient of $\Delta^{\mathrm{aux}}$ is $-r_E+a_Et_P$.
+The threshold formula therefore gives
+\[
+ \operatorname{lct}_{\eta_P}(X',\Delta^{\mathrm{aux}};x^*P)
+ =\inf_E\frac{1+r_E-a_Et_P}{a_E}=c_P.
+\]
+Since $T_P=\alpha_P-\beta_P$, we obtain $T_P=1-c_P$.
+Finally, if $T_P=0$ and $E$ attains the first minimum, then
+\[
+ 1=c_P\le\frac{1+r_E}{a_E}-t_P
+       =\frac{1-d_E}{a_E}\le1.
+\]
+All terms are equal. Since $a_E$ is a positive integer and
+$d_E\in\{0,1\}$, this forces $a_E=1$ and $d_E=0$, as claimed.
+\end{proof}
+
+\begin{lemma}[Permitted models after normalization]
+\label{lem:normalized-models}
+Suppose that $(W,T)$ is smooth with effective SNC rational boundary and
+that the moduli divisor $M$ is a rational Cartier divisor on $W$.
+For a smooth modification $\mu:W'\to W$, put
+\[
+ T'=\mu_*^{-1}T+\operatorname{Exc}(\mu)_{\mathrm{red}},
+ \qquad M'=\mu^*M,
+\]
+and assume the resulting boundary is SNC. Then pullback identifies the
+section spaces of every sufficiently divisible multiple of
+$K_W+T+M$ and $K_{W'}+T'+M'$. At unchanged codimension-one points the
+normalization in Lemma~\ref{lem:rankone-normalization} is preserved.
+Every new exceptional prime has coefficient one in $T'$.
+\end{lemma}
+
+\begin{proof}
+Log canonicity of $(W,T)$ gives
+\[
+ K_{W'}+T'+M'=\mu^*(K_W+T+M)+E,
+\]
+where $E$ is effective and $\mu$-exceptional. For divisible $m$,
+normality gives
+\[
+ \mu_*\mathcal O_{W'}(mE)=\mathcal O_W.
+\]
+The projection formula therefore identifies the section spaces.
+The modification is an isomorphism at each old generic divisorial
+point, so the old local order calculations are unchanged. The final
+assertion is the definition of $T'$.
+\end{proof}
+
+Appendix~\ref{p1logcbfalt:section} gives complementary projective
+proofs of the section comparisons and further calculations with the
+auxiliary subpair on higher models.

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/build/sections/root-comparison.tex

@@ -0,0 +1,675 @@
+\subsection{The exact comparison for Kodaira-zero fibers}
+\label{rankone:section}
+
+For fibers of logarithmic Kodaira dimension zero, the relative
+pluricanonical forms determine a single Hodge line on the base.  We
+construct that line, compute its boundary orders, and use them to
+identify the adjoint section spaces.  Coefficient-one boundary
+components introduce mixed Hodge structures, but the line itself
+lies in one pure weight grade.
+
+A pure variation of Hodge structures carries a local system and a
+varying Hodge filtration. The \emph{highest step} of a summand is its
+highest nonzero filtration piece. The \emph{parabolic extension} of
+such a line is obtained by local power substitutions making boundary
+monodromy unipotent, extension of the unipotent Hodge filtration, and
+descent with the resulting rational weights. We will compute those
+weights using the actual relative pluricanonical form.
+
+\begin{proposition}[The rank-one comparison]\label{rankone:comparison}
+Let $g:(X,\Delta)\to W$ be a fibration with connected fibers from a
+smooth compact K\"ahler manifold to a smooth compact manifold in
+class $\mathcal C$.  Suppose that $\Delta$ is an SNC boundary with
+rational coefficients in $[0,1]$ and that
+$\kappa(G,K_G+\Delta_G)=0$ on a very general smooth fiber $G$.
+After the modifications of
+\Cref{setup:log-modification,setup:flattening}, fix a smooth compact
+reference pair $(X_0,\Delta_0)$ and a modification
+\[
+ \pi:X\longrightarrow X_0,\qquad
+ \Delta=\pi_*^{-1}\Delta_0+\operatorname{Exc}(\pi)_{\mathrm{red}},
+\]
+such that every prime divisor mapped by $g$ into codimension at least two
+is $\pi$-exceptional.  Then there are a rational line bundle $M$
+and a rational divisor $T$ on $W$ with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item A positive integral multiple of $M$ is the parabolic highest Hodge
+line of a complex direct summand of a pure, real-polarizable variation
+with an integral lattice on a dense open subset of $W$.  Its parabolic extension is the line to which the adjoint-positivity
+theorem, \Cref{period:adjoint}, will apply.
+\item The divisor $T$ is supported on an SNC divisor and satisfies
+\begin{equation}\label{rankone:boundary}
+ B(g,\Delta)\leq T\leq 1.
+\end{equation}
+In particular, $T$ is effective.  If every prime over a prime $D$ of $W$
+has $\Delta$-coefficient one, then $T_D=1$.
+\item In sufficiently divisible degrees $q$, multiplication by the
+relative generating forms gives linear isomorphisms
+\begin{equation}\label{rankone:sections}
+ H^0\bigl(W,q(K_W+T+M)\bigr)
+ \simeq H^0\bigl(X,q(K_X+\Delta)\bigr)
+ \simeq H^0\bigl(X_0,q(K_{X_0}+\Delta_0)\bigr).
+\end{equation}
+In a common divisible grading they preserve products and ratios of
+sections, and hence the dimensions
+of the images of their linear systems.
+\item The same comparison holds relatively.  If $h:W\to Y$ and
+$f_0:X_0\to Y$ are holomorphic maps with $f_0\pi=hg$, then
+\begin{equation}\label{rankone:relative-sections}
+ h_*\mathcal O_W\bigl(q(K_W+T+M)\bigr)
+ \simeq (f_0)_*\mathcal O_{X_0}\bigl(q(K_{X_0}+\Delta_0)\bigr)
+\end{equation}
+for sufficiently divisible $q$.  The comparison of ratios also applies
+to systems over $Y$ and to their restrictions wherever generic base
+change holds. The relative identity remains valid after tensoring by a
+line bundle on $Y$, or a rational line bundle in degrees clearing its
+denominator.
+\end{enumerate}
+\end{proposition}
+
+The Hodge construction and its divisorial calculation are independent
+of the reference pair.  The condition on $\pi$ is used in the final
+section comparison, to descend forms across exceptional centers.
+
+\subsubsection{The root form and its eigenspace}
+
+The cyclic-root construction and its distinguished eigensheaf occur in
+\cite[Remark~2.6]{FujinoMori},
+\cite[Lemma~5.2]{AmbroBoundary}, and
+\cite[Remark~3.9]{FujinoGongyo}. Here the order calculations also
+include coefficient-one boundary and keep the character line distinct
+from the entire top Hodge space.
+
+Choose a positive integer $m$ clearing the boundary denominators such
+that the generic rank of
+\[
+ g_*\mathcal O_X\bigl(m(K_{X/W}+\Delta)\bigr)
+\]
+is one.  The same holds in every positive multiple of this degree:
+powers of a nonzero section exist, whereas two independent sections
+would contradict Kodaira dimension zero.  Its reflexive hull
+$\mathcal E_m$ is a line bundle since $W$ is smooth.  A local frame $s$ of
+$\mathcal E_m$ determines a meromorphic relative $m$-canonical form on
+$X$ over the corresponding base open set.  Two such frames differ by
+an invertible base function.
+The direct image records $m$-pluriforms. Positivity will come from the
+Hodge extension of their rational root; comparing the two extension
+orders will supply $T$.
+
+Resolve the horizontal zeros, poles, and boundary, then choose a dense
+open set $W^\circ$ on which $g$ is smooth, the resulting divisors are
+relatively SNC, and all their strata are smooth over the base.  By
+\Cref{setup:log-modification}, new exceptional components have
+coefficient one, and the logarithmic Kodaira dimension of the general
+fiber stays zero.  On each base chart take the normalized cyclic cover
+of $m$-th roots of $s$ as a relative pluriform and resolve it
+functorially.  Write $\tau$ for its tautological relative top form and
+$d=\dim X-\dim W$.
+
+At a horizontal prime of boundary coefficient $\delta$, let $l$ be
+the order of $s$ divided by $m$.
+The inequality $l\geq-\delta\geq-1$ shows that $\tau$ has at most
+logarithmic poles.  Indeed, at a covering prime of ramification index
+$j$, its order is
+\begin{equation}\label{rankone:root-order}
+ j(1+l)-1.
+\end{equation}
+This number is integral.  If $l>-1$ it is nonnegative; if $l=-1$ it
+equals $-1$.  On the resolved cover let $H$ be the reduced inverse
+image of the horizontal primes with $l=-1$.  It includes the
+exceptional components above those primes that are needed in the log
+resolution.  Logarithmic pullback preserves the bound, and no
+additional horizontal poles occur.  We use the cohomology of the
+complement of $H$.
+
+\begin{lemma}\label{rankone:eigenspace}
+The tautological form spans the root-character part of
+$F^dH^d(\widetilde G\setminus H,\C)$ on a general fiber of the cyclic
+cover.  In particular, that part has dimension one.
+\end{lemma}
+
+\begin{proof}
+For a compact K\"ahler SNC pair, logarithmic Hodge--de Rham degeneration
+identifies this top Hodge space with its global logarithmic top
+forms; it is also the smooth-fiber case of
+\cite[Theorem~1.1]{FujinoFujisawaVMHS}, by duality.  Let $\eta$ be
+another form of the tautological character.  Then $\eta/\tau$ is
+deck-invariant and descends to a meromorphic function $v$ on $G$.
+
+Consider a prime where the logarithmic section $s$ has no zero.
+There $l=-\delta$.  For $l>-1$, the integer in
+\eqref{rankone:root-order} lies between $0$ and $j-1$.  A pole of
+$v$ downstairs would lower the covering order by at least $j$, which
+is impossible.  For $l=-1$, the logarithmic pole allowance is already
+exhausted, so the same conclusion holds.  Thus the poles of $v$ can
+occur only on the positive zero divisor of $s$ as a section of
+$m(K_G+\Delta_G)$.  A sufficiently high power $s^N$ absorbs those
+poles.  Both $s^N$ and $v s^N$ are then sections of
+$Nm(K_G+\Delta_G)$.  The assumption $\kappa=0$ forces them to be
+dependent, and hence forces $v$ to be constant.
+
+The argument also applies to a disconnected cover: we retain the whole
+cover with its $\mu_m$-action, whose invariant meromorphic functions
+descend to $G$.
+\end{proof}
+
+The ranks of the Hodge and weight graded bundles are locally constant
+on $W^\circ$.  The conclusion of \Cref{rankone:eigenspace}, proved on
+a very general fiber, therefore holds throughout this smooth log
+locus.  Notice that neither the argument nor
+\eqref{rankone:root-order} requires a boundary coefficient to have
+the form $1-1/a$ with $a$ an integer.
+
+\subsubsection{Pure weights and descent of the local systems}
+
+We next place a power of the local Hodge line in a global pure
+variation.  This requires compatible polarizations and descent,
+because the cyclic covers need not form a global family.
+
+First make the local normalized-cover constructions on charts of the
+compact base $W$, using the meromorphic forms supplied by the frames
+of $\mathcal E_m$.  They extend across the missing base divisor as
+finite covers; subsequent resolutions can be projective and
+functorial in the analytic category
+\cite[Theorem~1.1]{BierstoneMilmanFunctorial}.
+The resulting maps to the K\"ahler source are projective, since the
+finite covers are projective. Thus their total spaces are K\"ahler
+over relatively compact source neighborhoods
+\cite[Definition~4.1 and Lemma~4.4]{FujikiDouady}. Properness allows
+such a neighborhood to contain the inverse image of a sufficiently
+small disk transverse to a base divisor.
+The divisor of the tensor is unchanged under multiplication
+by a base unit.  Thus the constructions on overlaps are identified
+after taking a local root of that unit, and the resolutions and their
+exceptional divisor classes are identified as well.
+
+Fix a K\"ahler class $\xi$ on $X$.  On each resolved-cover chart
+$q_i:Z_i\to X|_{U_i}$, choose the relatively ample line bundle $A_i$
+as follows.  Functoriality matches the blowup center ideals and their
+tautological line bundles under the root-change identifications.
+Insert identity stages where a center is empty; the corresponding
+tautological bundle is trivial there.  A sufficiently weighted tensor
+product gives compatible $A_i$, equivariant under deck transformations,
+by descent of the centers, blowups, and tautological bundles.  The classes
+\[
+ \theta_i=Nq_i^*\xi+c_1(A_i)
+\]
+are K\"ahler on the smooth compact fibers for $N$ sufficiently large;
+their restrictions to the compact boundary strata are K\"ahler too.
+At the intermediate finite covering stage one uses the pullback
+K\"ahler \emph{class}; the literal pulled-back form can degenerate
+at ramification.  The relatively ample class supplies positivity
+along the projective resolution.  One may choose a single $N$:
+carry out the construction over charts $U_i^+$ containing the closures
+of finitely many relatively compact charts $U_i$ covering $W$.
+Properness makes the inverse images of these closures compact, so
+each admits a bound $N_i$ in this class construction.  Taking
+$N\geq\max_iN_i$ works on all charts, since increasing $N$ adds
+the semipositive class $q_i^*\xi$.  This is why the construction was
+extended over the compact base before shrinking to $W^\circ$.
+For the invariant K\"ahler-class construction on an equivariant
+resolved cyclic cover, see also \cite[Lemma~8.1]{ChenLimits}.
+
+The classes $\theta_i$ come from total-space cohomology, so their
+restrictions to the smooth fibers and strata are flat for the
+Gauss--Manin connection.  They agree under the overlap
+identifications.  Lefschetz decomposition and cup product consequently
+give compatible flat real polarizations on the compact-stratum
+cohomology systems.  The weight spectral sequence, with its Tate
+twists, gives the pure weight grades of the open-fiber cohomology as
+subquotients of these systems.  They are real-polarizable: a real Hodge
+subvariation of a polarized pure variation is nondegenerate for the
+polarization, and a quotient can be identified with an orthogonal
+complement.  These operations are compatible with the overlap maps.
+The integral structures come from cohomology modulo torsion and the
+saturated intersections with the rational weight filtration.  We do
+not assert that the real polarizations are rational.
+
+There is a unique weight $w$ for which
+\[
+ F^d\Gr^W_w H^d(\widetilde G\setminus H,\C)_\chi\ne0.
+\]
+Here $\chi$ is the tautological character.  Uniqueness follows from
+the rank-one assertion and strictness of the weight filtration.  Let
+$V_i$ be the pure integral variation given by this weight grade on
+the $i$th chart; the chosen line is $F^d(V_i)_\chi$.
+
+\begin{lemma}\label{rankone:descent}
+A positive tensor power of these local Hodge lines is the highest
+Hodge line of a complex direct summand of a global real-polarizable
+integral pure variation on $W^\circ$.
+\end{lemma}
+
+\begin{proof}
+Refine an overlap so that a root of its transition unit can be
+chosen.  It gives an isomorphism between the cyclic-cover families,
+and therefore an isomorphism of their integral cohomology systems.
+Another choice differs by a deck transformation.  On a triple
+overlap the cocycle defect is also a deck transformation.  All these
+maps commute with the distinguished $\mu_m$-action.
+
+Put
+\[
+ \mathbb U_i=
+ \bigl((V_{i,\Z}/\text{torsion})^{\otimes m}\bigr)^{\mu_m},
+\]
+using the diagonal deck action.  The ambiguity and the cocycle defect
+act trivially on $\mathbb U_i$.  Hence the $\mathbb U_i$ descend to
+an integral local system $\mathbb U$; saturation only changes the
+lattice by an isogeny.  The compatible real polarizations, tensorized
+and restricted to invariants, polarize this pure variation.
+
+Over $\C$ the character projector $e_\chi$ is a flat Hodge
+endomorphism on each chart.  The projector
+$e_\chi^{\otimes m}$ restricts to the invariant tensor space,
+because $\chi^m=1$.  It commutes with all transition maps and
+therefore defines a global complex direct summand with local fibers
+$((V_i)_\chi)^{\otimes m}$.  Its highest step is
+$(F^d(V_i)_\chi)^{\otimes m}$, a line.  In local rooted notation
+this line is generated by $\tau^{\otimes m}$, which is identified
+meromorphically with $s$.  Thus no global root of $\mathcal E_m$ and
+no global family of cyclic covers is needed.
+\end{proof}
+
+\subsubsection{The two extension orders}
+
+Let $M$ denote one $m$th of the parabolic extension of the line just
+constructed.  We use $s^{1/m}$ for its local rational frame; the
+notation becomes literal after the powers and finite boundary covers
+above. We now compare this parabolic extension with the pole bounds
+of the original pluriform.
+
+Fix a prime divisor $D$ on $W$ and use a local ordinary base volume
+$\omega$ that does not vanish at its generic point. Work on a source
+model on which the divisor of the generator, the boundary, and the
+inverse image of $D$ have SNC support over that generic point,
+using strict transforms plus reduced exceptional boundary as before.
+All primes of this prepared source dominating $D$, including the
+exceptional primes introduced by its resolution, enter the following
+order tests. Put
+\[
+ l_E=\frac1m\ord_E(s\wedge g^*\omega^m),\qquad
+ a_E=\ord_E(g^*D),\qquad \delta_E=\Delta_E,
+\]
+and
+\[
+ \alpha_D=\min_{g(E)=D}\frac{l_E+\delta_E}{a_E},
+ \qquad \beta_D=\ord_D^M(s^{1/m}).
+\]
+The notation $s\wedge g^*\omega^m$ means the relative-times-base
+identification in the $m$th tensor power of the canonical bundle.
+For $q$ divisible by $m$ and clearing the rational data, and a
+meromorphic base function $\varphi$, the total pluriform
+$\varphi\,\omega^q s^{q/m}$ has boundary-adjusted order
+\[
+ a_E\ord_D(\varphi)+q(l_E+\delta_E)
+\]
+at $E$. It therefore satisfies the source boundary bounds at all
+these primes if and only if
+\begin{equation}\label{rankone:base-bound}
+ \ord_D(\varphi)+q\alpha_D\geq0.
+\end{equation}
+The Hodge frame contributes $q\beta_D$ to the order on the base,
+so the candidate boundary correction is $T_D=\alpha_D-\beta_D$.
+Computing $\beta_D$ will reduce the required bounds on $T_D$ to
+a comparison of two minima. We first justify computing that Hodge
+order by logarithmic orders on a semistable cover, without changing
+the chosen pure Hodge line. Global consistency of $T$ and descent
+across the untested source primes will follow afterward.
+
+\subsubsection{The extension across a transverse disk}
+
+We use the analytic canonical-extension theorem
+\cite[Theorem~1.1]{FujinoFujisawaVMHS}.  It applies to a proper
+surjective morphism of an analytic SNC pair $(Z,H)$ to a smooth base,
+with $H$ reduced, every stratum K\"ahler and dominant, and all strata
+smooth off a normal crossing discriminant.  It supplies a
+graded-polarizable real variation on compact-support cohomology,
+locally free double Hodge/weight grades on the extension, and the
+dual direct-image description by $\omega_{Z/B}(H)$.  Admissibility
+is established in \cite[Remark~4.8]{FujinoFujisawaVMHS}.
+These conclusions apply to proper K\"ahler families; projectivity
+is not required.
+
+Take a transverse disk at a general point of a base divisor. Resolve
+the cyclic-cover pair together with the central fiber, leaving its
+smooth logarithmic locus over the punctured disk unchanged. Near a
+central point the resulting map has the form
+\[
+ t=\varepsilon x_1^{a_1}\cdots x_r^{a_r},
+ \qquad H\subset\{x_{r+1}\cdots x_l=0\},
+\]
+where $\varepsilon$ is a unit and $H$ is the horizontal boundary. Absorbing it
+into a vertical coordinate gives a strict toroidal morphism. Compactness
+of the central fiber and properness allow us to shrink the disk until
+finitely many such charts cover its inverse image. The associated cone
+complex is therefore finite. Apply the combinatorial semistable
+subdivision theorem
+\cite[Theorem~2.7]{AdiprasitoLiuTemkinSemistable}. Its base is one ray,
+so the base lattice alteration is realized by $t=u^N$. Normalize this
+base change, keeping all components, and realize the projective source
+subdivision analytically
+\cite[Lemmas~2.14--2.15, 2.18, and~2.31(4)]
+{EnokizonoHashizumeSemistable}. Zero-image faces, which record the
+horizontal boundary, are allowed in the combinatorial theorem.
+
+The semistable local form is
+\[
+ u=y_1\cdots y_q,
+ \qquad H\subset\{y_{q+1}\cdots y_k=0\};
+\]
+see \cite[Lemma~2.21(4)]{EnokizonoHashizumeSemistable}.
+Here $H$ includes every horizontal exceptional component in the
+transformed boundary. The total space is smooth, the central fiber is
+reduced, and its union with $H$ is SNC. On every horizontal
+intersection stratum the displayed monomial is nonconstant;
+properness makes its image the whole disk. The composite to the
+initial K\"ahler total space is projective, so the new total space and
+its smooth strata are K\"ahler after shrinking the disk
+\cite[Lemma~4.4]{FujikiDouady}.
+
+These toroidal modifications preserve the open pair on the punctured
+fibers. Its cohomology is therefore the pullback of the original
+open-fiber cohomology, including the full direct sum if the cover
+splits. Transport the original $\mu_m$-action through this
+identification. The deck projector then acts on the variation and,
+by functoriality, on its canonical extension; no action on a selected
+component or on the chosen semistable model is needed. We apply the
+extension theorem componentwise and take the direct sum. We use this
+semistable pair itself: an arbitrary further resolution need not
+preserve reducedness.
+
+For application of the cited theorem, $H$ consists of the horizontal
+logarithmic boundary.  The central fiber is not included in $H$:
+it appears in the relative logarithmic complex, with $dt/t$ in the
+base quotient.  Write $f:Z\to B$ for this degeneration over the disk,
+and $Z_0=f^{-1}(0)$ for its central fiber.  The top relative
+logarithmic sheaf is
+\begin{equation}\label{rankone:log-volume}
+ \omega_{Z/B}\bigl(H+(Z_0)_{\mathrm{red}}-f^*[0]\bigr).
+\end{equation}
+On the semistable model the central fiber is reduced, so this is
+$\omega_{Z/B}(H)$.  For a smooth open fiber of dimension $d$,
+Poincar\'e duality identifies
+\[
+ H^d_c(Z_t\setminus H_t)^*\simeq
+ H^d(Z_t\setminus H_t)(d).
+\]
+Accordingly the dual degree-zero Hodge quotient in
+\cite[Theorem~1.1(iv)]{FujinoFujisawaVMHS} is precisely the top
+$F^d$ step of ordinary $H^d$.  The upper and lower canonical
+extensions coincide after unipotentization.  This proves that the
+canonical extension of the top step is the direct image of
+\eqref{rankone:log-volume}.  The K\"ahler form of the
+one-parameter limiting theorem is also explained in
+\cite[Remarks~2.16, 4.4, and~4.8]{FujinoFujisawaVMHS}.
+
+\begin{lemma}\label{rankone:extension}
+The projection of the rank-one top eigenspace to its nonzero pure
+weight grade is an isomorphism on canonical extensions after
+unipotentization.  Consequently it is an isomorphism of parabolic
+rational lines before that base change.
+\end{lemma}
+
+\begin{proof}
+Let $\overline{\mathcal V}_\chi$ be the extended mixed eigensystem
+over the disk.  The finite deck projector acts on the canonical
+extension and its filtrations.  By the cited extension theorem,
+each
+\[
+ \Gr_F^p\Gr^W_j\overline{\mathcal V}_\chi
+\]
+is locally free and has its rank on the punctured disk.  The
+filtrations on the weight quotients are induced filtrations.
+Thus the exact sequences
+\[
+ 0\longrightarrow F^dW_{j-1}\overline{\mathcal V}_\chi
+ \longrightarrow F^dW_j\overline{\mathcal V}_\chi
+ \longrightarrow F^d\Gr^W_j\overline{\mathcal V}_\chi
+ \longrightarrow0
+\]
+have locally free quotients.  There is no step above $F^d$, also
+on the extension, by these constant-rank assertions.  Exactly one
+of the displayed quotients has rank one, namely that for $j=w$;
+all the others have rank zero.  Induction on the weight filtration
+therefore gives
+\[
+ F^dW_{w-1}\overline{\mathcal V}_\chi=0,\qquad
+ F^dW_w\overline{\mathcal V}_\chi
+   =F^d\overline{\mathcal V}_\chi,
+\]
+and identifies the latter line with
+$F^d\Gr^W_w\overline{\mathcal V}_\chi$ across the origin, without
+an additional zero or pole.  Descent gives the parabolic assertion.
+\end{proof}
+
+\subsubsection{The divisorial order of the Hodge line}
+
+With the notation and prepared source above, we claim that the
+parabolic order of the frame $s^{1/m}$ in $M$ is
+\begin{equation}\label{rankone:order}
+ \beta_D=\ord_D^M(s^{1/m})
+     =\min_{g(E)=D}\frac{l_E+1-a_E}{a_E}.
+\end{equation}
+Equivalently, $\beta_D$ is the largest rational number $b$ such that
+$t^{-b}s^{1/m}$ extends at $D=(t=0)$ in the parabolic sense.
+
+It suffices to work over a transverse disk, suppressing the other
+base coordinates.  Write $\tau=s^{1/m}$ and
+$\Omega=\tau\wedge g^*dt$.  The multivalued total form
+$\Omega/t$ has order $l_E-a_E$ at $E$.  A logarithmic extension
+allows order $-1$, so twisting by $t^{-b}$ imposes
+\begin{equation}\label{rankone:inequality}
+ l_E+1-a_E-a_Eb\geq0.
+\end{equation}
+To verify this directly against the canonical extension, take a
+finite base change $t=u^N$ and a semistable model of the root
+cover.  At a central prime over the strict transform of $E$, let
+$e$ be the total ramification index of the composite map to $X$,
+including the cyclic-cover ramification.  Reducedness gives
+$ea_E=\ord_{E'}(t)=N\ord_{E'}(u)=N$, where $E'$ is this new prime.
+Orders of a total volume form shifted by one multiply by $e$.
+Dividing its pullback by
+$dt/du=Nu^{N-1}$ therefore gives
+\[
+ \ord(\tau\wedge du)
+ =e(l_E+1)-N
+ =e(l_E+1-a_E).
+\]
+The base twist subtracts $Nb=ea_Eb$.  Regularity in
+\eqref{rankone:log-volume} is exactly
+\eqref{rankone:inequality}.  By \Cref{rankone:extension}, testing
+the logarithmic top form tests the chosen pure Hodge line as well.
+The form being tested is the pullback of the original tautological
+form on the full cover. The identification of open-fiber cohomology
+above and the direct-image description of its canonical extension
+identify it with that same Hodge section. Keeping all normalized
+components ensures that a prime above every original $E$ is present;
+the proper toroidal modifications retain its strict transform.
+
+To prove sufficiency, continue to write $X$ for the smooth source
+over the transverse disk and put $n=d+1$.  In SNC coordinates write
+\[
+ \Omega=h\prod_i x_i^{l_i}\,dx_1\wedge\cdots\wedge dx_n,
+ \qquad t=h'\prod_i x_i^{a_i},
+\]
+where $h,h'$ are units and $a_i=0$ for horizontal components.
+Put $c_i=l_i+1-(1+b)a_i$.  The vertical inequalities
+\eqref{rankone:inequality} and the horizontal logarithmic bounds
+give $c_i\geq0$.  For any additional divisorial valuation $v$,
+write $A_X(v)$ for its log discrepancy over this smooth model,
+that is, one plus the order of the Jacobian.  Logarithmic pullback
+of the SNC coordinate divisor gives
+$A_X(v)\geq\sum_i v(x_i)$, exactly as in
+\Cref{setup:log-modification}.  Thus the shifted order of
+$t^{-(1+b)}\Omega$ is
+\[
+ A_X(v)+\sum_i\bigl(l_i-(1+b)a_i\bigr)v(x_i)
+ \ \geq\ \sum_i c_i v(x_i)\ \geq\ 0.
+\]
+These inequalities persist under finite covers, since shifted orders
+multiply by ramification indices.  Further exceptional primes therefore
+impose no stronger condition.
+Horizontally, shifted orders are strictly positive outside the
+designated log boundary; on the root cover these are positive
+integers, so there are no poles there.  Conversely, covering primes
+above strict transforms detect every
+inequality in \eqref{rankone:inequality}.  Taking their minimum
+proves \eqref{rankone:order}.
+
+\begin{remark}\label{rankone:collision}
+The use of all primes on the resolved model in
+\eqref{rankone:order} matters even for a simple logarithmic
+family.  On $\mathbf P^1_x$ over a disk, remove the horizontal
+divisor $x^2=t$ and use $\tau=dx/(x^2-t)$.  After $t=u^2$ and
+$x=uz$, one has $\tau=u^{-1}dz/(z^2-1)$, so the parabolic order
+is $-1/2$.  Resolving the tangency of the horizontal divisor with
+$t=0$ introduces a prime with $a_E=2$ and $l_E=0$, which gives
+exactly $-1/2$ in \eqref{rankone:order}.  The original central
+prime alone would not detect it.
+\end{remark}
+
+\subsubsection{The boundary and the comparison of sections}
+
+Define
+\begin{equation}\label{rankone:alpha-T}
+ T_D=\alpha_D-\beta_D.
+\end{equation}
+Changing $s$ to $v s$, for a meromorphic base function $v$, adds
+$a_E\ord_D(v)/m$ to $l_E$.  Hence it adds
+$\ord_D(v)/m$ to both $\alpha_D$ and $\beta_D$ and leaves
+$T_D$ unchanged.  This also checks the transformation rule for
+the rational Hodge frame in \eqref{rankone:order}.  Passing to
+a higher divisible generating degree has the same effect: on the
+generic fiber its generator is a base multiple of a power of $s$.
+Thus the construction is independent of these choices.
+
+For the boundary inequality, write
+\[
+ x_E=\frac{l_E+\delta_E}{a_E},\qquad
+ c_E=1-\frac{1-\delta_E}{a_E}.
+\]
+Then $0\leq c_E\leq1$, and
+\[
+ T_D=\min_E x_E-\min_E(x_E-c_E).
+\]
+For any finite collections of real numbers $x_E,c_E$, this
+difference lies between $\min_Ec_E$ and $\max_Ec_E$.
+Moreover the inf-multiplicity convention gives
+\[
+ \min_Ec_E
+ =1-\max_{g(E)=D}\frac{1-\delta_E}{a_E}
+ =B(g,\Delta)_D.
+\]
+This proves \eqref{rankone:boundary}.  If all $\delta_E=1$,
+all $c_E=1$, so $T_D=1$.
+
+Choose the discriminant to include the vertical boundary and all
+loci where the smooth log construction fails.  Off it the family
+and the generator line are smooth log-relative data, and there is
+no vertical boundary.  There $\alpha_D=\beta_D$; in a
+nonvanishing generator frame both are zero.  Thus $T$ has finite
+support in the chosen discriminant, which may be resolved to be
+SNC.  On subsequent modifications one pulls back the variation and
+its parabolic line and recomputes $\alpha_D$ and
+\eqref{rankone:alpha-T}; the order computation is unchanged on the
+resulting models.
+
+\begin{proof}[Completion of the proof of \Cref{rankone:comparison}]
+Take $q$ divisible by $m$ and by the denominators of all rational
+data.  In a local base frame, a meromorphic section of
+$q(K_W+T+M)$ is represented
+by
+\[
+ \varphi\,\omega^q s^{q/m}.
+\]
+The order of the frame in $T+M$ is
+$q(T_D+\beta_D)=q\alpha_D$.  Consequently this section is
+regular at $D$ exactly when \eqref{rankone:base-bound} holds.
+The source-side order test already proved identifies this with
+the boundary bounds at every source prime dominating $D$.
+Along horizontal primes the required bound is automatic because
+$s^{q/m}$ is a logarithmic
+pluricanonical section on the general fiber.  We have proved the
+exact section comparison in codimension one on the base, and at
+all source primes except those contracted by $g$ into codimension
+at least two.
+
+Conversely, a section of $q(K_X+\Delta)$ restricts on a general
+$g$-fiber to a multiple of $s^{q/m}$.  Their ratio is a
+meromorphic function constant on the connected general fibers,
+and therefore descends meromorphically to $W$.  The same
+divisorial inequalities make it a section of $q(K_W+T+M)$
+outside codimension two; smoothness of $W$ extends it uniquely
+across that set.  This proves the converse comparison in all
+divisible degrees under consideration.
+
+For the forward direction, a section on $W$ gives a meromorphic
+adjoint form on $X$ whose only possible remaining poles lie on
+$g$-contracted prime divisors.  By hypothesis these divisors are
+$\pi$-exceptional.  Push the form to $X_0$.  At every prime of
+$X_0$ its transform was among the tested primes of $X$, so the
+pushed form satisfies the $\Delta_0$ pole bound in codimension
+one.  Since $X_0$ is smooth and the adjoint multiple is a line
+bundle, it extends across codimension two.  Finally
+\Cref{setup:log-modification} pulls it back to a section of
+$q(K_X+\Delta)$, proving \eqref{rankone:sections}.
+
+If the diagram is over $Y$, apply this argument over an arbitrary
+open subset $U\subset Y$.  Its inverse image in $X$ is
+$\pi^{-1}(f_0^{-1}U)=g^{-1}(h^{-1}U)$, and $\pi$ is still
+proper over the smooth open space $f_0^{-1}U$.  The exceptional
+images still have codimension at least two.  The construction and
+the extension of sections commute with restriction to $U$,
+giving \eqref{rankone:relative-sections}.  Relative canonical
+versions, when the auxiliary base is smooth, follow by the
+projection formula.  All comparisons multiply by the same relative
+generator, with its powers in the common divisible grading, so they
+preserve products and ratios of sections. The projection formula also
+gives the identity after any pulled-back base twist, in degrees clearing
+the denominator for a rational twist. This completes
+\Cref{rankone:comparison}.
+\end{proof}
+
+\subsection{Retaining the original base multiplicities}
+
+\begin{lemma}[Composition of inf-multiplicity bounds]
+\label{application:composition}
+Suppose that \(X\xrightarrow{g}W\xrightarrow{h}Y\) is a diagram of
+surjective maps between smooth spaces, and \(\Delta,T,C\) are rational
+boundaries on these spaces.  If
+\[
+ T\geq B(g,\Delta),\qquad C\leq B(h\circ g,\Delta),
+\]
+then \(B(h,T)\geq C\).  If, in addition, every prime divisor of \(X\)
+mapped into codimension at least two in \(Y\) has coefficient one in
+\(\Delta\), and \(T\leq1\), then every prime divisor of \(W\) mapped
+into codimension at least two in \(Y\) has coefficient one in \(T\).
+\end{lemma}
+
+This is the inf-multiplicity composition comparison of
+\cite[Proposition~3.13(1)]{CampanaOrbifolds}; we include its order
+calculation to track coefficient-one exceptional divisors.
+
+\begin{proof}
+Let \(D\subset Y\) and \(V\subset W\) be prime divisors with
+\(h(V)=D\).  Write \(b_V=\ord_V(h^*D)\).  If \(E\subset X\) is a
+prime divisor dominating \(V\), put \(a_E=\ord_E(g^*V)\).  At their
+generic points orders multiply, so
+\[
+ \ord_E((h\circ g)^*D)=b_Va_E.
+\]
+The hypothesis on \(C\) implies
+\(b_Va_E m_\Delta(E)\geq m_C(D)\) for every such \(E\).  Taking the
+minimum over them and then using the hypothesis on \(T\) gives
+\[
+ b_Vm_T(V)\ \geq\ b_Vm(g,\Delta;V)\ \geq\ m_C(D).
+\]
+Taking the minimum over \(V\) proves the first assertion.  All these
+inequalities are valid with the prescribed convention for infinity.
+
+For the second assertion, let \(V\) be mapped into codimension at least
+two in \(Y\).  Every prime \(E\) dominating it has
+\(\Delta_E=1\), so \(m(g,\Delta;V)=\infty\).  It follows that
+\(B(g,\Delta)_V=1\), and hence \(T_V=1\).
+\end{proof}

preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b143366cc99615e19783fa075caafd8c9565e7bf5a554a612e086657c2aeeb79
+size 677817

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/README.md

@@ -0,0 +1,18 @@
+# [Ordinary two-point correlations of multiplicative functions](final.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026,
+  author = {{OpenAI}},
+  title = {{Ordinary two-point correlations of multiplicative functions}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf}{OAI:Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026}},
+  year = {2026}
+}
+```
+

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/introduction.tex

@@ -0,0 +1,278 @@
+\section{Introduction}\label{sec:introduction}
+
+For \(n\ge1\), let \(\Omega(n)\) be the number of prime factors of \(n\),
+counted with multiplicity, and let \(\lambda(n)=(-1)^{\Omega(n)}\).
+We study ordinary two-point averages: each integer up to the given
+cutoff has weight one. Thus the correlation measures whether the two prime-factor counts
+have the same parity more often than different parities on the full
+initial interval.
+The distinction from a logarithmic average is the distinction between
+\[
+ \frac1X\sum_{n\le X}a(n)
+ \qquad\text{and}\qquad
+ \frac1{\log X}\sum_{n\le X}\frac{a(n)}n.
+\]
+Logarithmic weighting combines information from many multiplicative
+scales; an ordinary estimate must control the terminal scale itself.
+Our quantitative conclusion concerns every fixed pair of
+nonproportional affine forms.
+
+\begin{theorem}[A logarithmic saving for affine Liouville correlations]
+\label{thm:q-affine}
+There is an absolute constant \(c>0\) such that, for every fixed choice
+of integers \(a_1,a_2\ge1\) and \(b_1,b_2\ge0\) with
+\(a_1b_2-a_2b_1\ne0\), there is a constant
+\(C_{a_1,a_2,b_1,b_2}\) for which
+\[
+ \left|\sum_{1\le n\le X}
+       \lambda(a_1n+b_1)\lambda(a_2n+b_2)\right|
+ \le C_{a_1,a_2,b_1,b_2}\frac{X}{(\log X)^c}
+ \qquad(X\ge3).
+\]
+The cutoff \(X\) may be any real number. No coprimality condition is
+imposed on the coefficients.
+\end{theorem}
+
+For $a_1=a_2=1$ and $b_1=0$, $b_2=h>0$, this resolves the
+ordinary two-point Chowla conjecture, with a logarithmic saving.
+The exponent is independent of the forms. The implied constants need
+not be effective, and no uniformity for coefficients growing with \(X\)
+is asserted. The proof first obtains the same saving for
+\(\lambda(n)\lambda(n+h)\) in each fixed residue class modulo each fixed
+positive integer, including nonunit and zero classes. Complete
+multiplicativity then gives the affine conclusion.
+
+We also prove a qualitative statement for general multiplicative
+functions. Let \(\mathbb D=\{z\in\C:|z|\le1\}\). A function
+\(f:\N\to\mathbb D\) is multiplicative when
+\(f(mn)=f(m)f(n)\) for coprime positive integers \(m,n\).
+For a Dirichlet character \(\chi\), a real number \(t\), and \(X\ge2\),
+define the nonnegative distance \(D\) by
+\begin{equation}\label{eq:distance}
+ D(f,\chi n^{it};X)^2
+ =\sum_{p\le X}\frac{1-\operatorname{Re}
+       (f(p)\overline{\chi(p)p^{it}})}p.
+\end{equation}
+All sums indexed by \(p\) are over primes. We call \(f\)
+\emph{uniformly nonpretentious} if, for every fixed Dirichlet character,
+\begin{equation}\label{eq:unp}
+ \inf_{|t|\le N}D(f,\chi n^{it};N)\longrightarrow\infty
+ \qquad(N\longrightarrow\infty).
+\end{equation}
+The prime cutoff and the height bound are the same \(N\).
+
+\begin{theorem}[Binary corrected Elliott]\label{thm:elliott}
+Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, and suppose that
+at least one satisfies \eqref{eq:unp}. For every fixed pair of distinct
+nonnegative integers \(h_1,h_2\),
+\[
+ \frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0
+ \qquad(N\longrightarrow\infty,\ N\in\N).
+\]
+\end{theorem}
+
+This proves the binary corrected Elliott conjecture for ordinary
+averages. The functions may be complex, need not have modulus one,
+and need not be completely multiplicative. There is no conjugation
+in the displayed product; the conjugated version follows because
+\eqref{eq:unp} is preserved by conjugation. We make no quantitative
+rate claim for this class of functions.
+
+\begin{corollary}[Affine correlations]\label{cor:affine}
+Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, with at least one
+satisfying \eqref{eq:unp}. For all fixed integers \(a_1,a_2\ge1\) and
+\(b_1,b_2\ge0\) with \(a_1b_2-a_2b_1\ne0\),
+\[
+ \frac1N\sum_{n=1}^N f_1(a_1n+b_1)f_2(a_2n+b_2)
+ \longrightarrow0.
+\]
+\end{corollary}
+
+Every fixed progression restriction is also permitted in
+\Cref{thm:elliott}; \Cref{prop:affine-progression} gives the precise
+statement through all real cutoffs and for every residue class.
+These conclusions use the hypothesis on an original factor \(f_j\).
+The finite local changes needed for dilations and residue restrictions
+preserve that hypothesis. In particular, Liouville satisfies the exact
+condition \eqref{eq:unp}; a quantitative distance bound is proved in
+\Cref{lem:affine-liouville-unp}.
+
+\subsection{Historical context and prior work}
+
+Chowla's conjecture predicts vanishing Liouville correlations at every
+fixed collection of distinct shifts \cite{Chowla65,MRT15}. Its binary
+case asks whether the parities of the numbers of prime factors of
+\(n\) and \(n+h\) become uncorrelated for each fixed \(h\ge1\).
+Partial summation transfers ordinary cancellation to logarithmic
+cancellation. An ordinary estimate at every cutoff requires more
+than this implication.
+
+Elliott extended the correlation problem to bounded multiplicative
+functions \cite{Elliott92}. The obstruction presented by the twists
+\(n^{it}\) is already central to Hal\'asz's theory of one-point mean
+values \cite{Halasz71}; see also the treatment of Granville, Harper,
+and Soundararajan \cite{GHS19}. The distance in \eqref{eq:distance}
+belongs to the pretentious approach developed by Granville and
+Soundararajan \cite[Section~3]{GS07}. Matom\"aki, Radziwi\l\l, and Tao
+showed why Elliott's original condition needs correction for complex
+functions: a function can imitate different twists at successive
+scales without imitating any one fixed twist globally. Their
+corrected condition is equivalent to \eqref{eq:unp}; see
+\cite[Section~1.1 and Appendix~B]{MRT15}.
+
+Matom\"aki and Radziwi\l\l\ proved that most short means of a real
+bounded multiplicative function approximate its long mean
+\cite[Theorem~1]{MR16}. They also obtained a nontrivial bound for each
+fixed-shift Liouville correlation: the normalized absolute value is
+at most \(1-\delta(h)\), for some \(\delta(h)>0\), at all sufficiently
+large cutoffs \cite[Corollary~2]{MR16}. Matom\"aki, Radziwi\l\l, and
+Tao then established cancellation averaged over arbitrarily slowly
+growing windows of shifts. Their exponential-sum extension to
+complex multiplicative functions supplies the short-interval
+estimates used here \cite[Theorems~1.1, 1.3, and~1.7]{MRT15}.
+
+Tao proved the logarithmically averaged binary corrected Elliott
+theorem for nonproportional affine forms
+\cite[Theorem~1.3 and Corollary~1.5]{Tao16}.
+For the logarithmically weighted Liouville sum, Helfgott and
+Radziwi\l\l\ obtained
+\[
+ \left|\sum_{n\le x}\frac{\lambda(n)\lambda(n+1)}n\right|
+ \ll \frac{\log x}{\sqrt{\log\log x}}
+\]
+by divisibility-graph expansion \cite[Corollary~1.5]{HR21}.
+Pilatte improved this bound to \((\log x)^{1-c_0}\) for an absolute
+\(c_0>0\) \cite[Theorem~1.1]{Pilatte26}.
+These estimates retain the logarithmic weight.
+
+The graph method of Helfgott and Radziwi\l\l\ uses high traces,
+recurrence constraints, and cancellation at primes occurring only
+once \cite[Sections~2--7]{HR21}. They also proposed composite steps
+and nonbacktracking operators \cite[Sections~9.1--9.2]{HR21}.
+Pilatte develops these directions through products of centered prime
+factors, prohibited sequences, triangular arithmetic constraints,
+and a rank-truncated intersection sieve
+\cite[Sections~2--13 and Appendix~B]{Pilatte26}.
+Our two graph arguments adapt these constructions to different
+weights and orders of limits. Their trace estimates, local
+expansions, and transfers to ordinary averages are proved below.
+
+For ordinary averages,
+Tao and Ter\"av\"ainen obtained binary cancellation outside
+a set of scales of logarithmic density zero, under the same
+condition \eqref{eq:unp} on one factor
+\cite[Corollary~1.13]{TT19}. Klurman, Mangerel, and Ter\"av\"ainen
+relaxed the hypothesis to nonpretentiousness against each fixed
+\(\chi(n)n^{it}\),
+obtaining cancellation along a set of scales of full upper logarithmic
+density \cite[Theorem~1.2]{KMT23}. The later quantitative progression
+estimates of Tao and Ter\"av\"ainen
+give logarithmic savings for affine Liouville correlations,
+with coefficients allowed to grow slowly, outside a quantitatively
+controlled exceptional set of scales
+\cite[Theorem~3.1 and Remarks~3.2]{TT26}. The results here concern
+every cutoff, with the affine forms fixed.
+
+The quantitative route also uses bounded independence for Boolean
+circuits. This subject developed from the work of Linial and Nisan
+\cite{LinialNisan90}, through Bazzi's depth-two theorem and
+Razborov's simpler proof \cite{Bazzi09,Razborov08}, to Braverman's
+theorem for every fixed depth \cite{Braverman10}. In applying that theorem, the
+encoded residue law is first corrected to exact bounded independence.
+The local Fourier calculation is a form of the almost-to-exact
+independence construction of Alon, Goldreich, and Mansour
+\cite[Theorem~2.1 and Section~3.1]{AGM03}.
+
+\subsection{Main ideas}
+
+Both arguments begin with multiplicative dilations. For a fixed
+shift $h\ge1$, comparison of the correlations at $n,n+h$ and at
+$dn,d(n+h)$ leads to a graph on the integers with edge displacement
+$hd$. Short-interval Fourier estimates control the change in the
+correlation when selected divisibility indicators are replaced by
+signed expressions. Estimates for closed walks then bound the resulting
+graph operator. The two proofs use different vertex weights, and their
+cancellation mechanisms must be matched to those weights.
+
+In the quantitative argument, a step contains one prime from each of
+$J$ disjoint supplies, each with reciprocal mass comparable to a large
+fixed constant $W$. These primes contribute the factors
+$\1_{p\mid n}-1/p$, which have mean zero under uniform residues.
+An additional squarefree padding divisor uses a disjoint set of primes.
+The centered supplies provide reciprocal mass at least $W^J$, while
+their contribution to the operator bound is only $(C\sqrt W)^J$, with
+an absolute $C$. Padding controls the concentration
+of divisor weight in short multiplicative intervals. Writing $L$ for
+a small fixed power of $\log X$, the edge normalization gives a factor
+$1/L$ in the estimate for each interval, compensating for the factor
+$L$ in the number of intervals. The remaining losses are exponential
+in $J$ with absolute bases. Choosing $W$ large enough therefore gives
+exponential decay in $J$; since $J$ grows proportionally to
+$\log\log X$, this is a power-of-logarithm saving.
+
+Two issues underlie this comparison. First, the prime supplies grow
+with $X$, so averaging over their full joint period would not give
+a useful finite-scale estimate. We instead compare the residue tests
+used in the proof with independent residues, correcting their encoded
+bit law to exact bounded independence before applying Braverman's
+theorem \cite{Braverman10}. Second, repeated primes constrain the
+closed walks. Many independent arithmetic relations give a direct
+saving. When few remain, a forest encodes the possible patterns of
+repetition. Its pattern count is uniform over all padding values,
+allowing the padding weights to be summed afterward in one common
+residue environment. These steps preserve the absolute constants
+needed for the choice of $W$.
+
+For general multiplicative functions, the short-interval input
+contains the nonpretentiousness distance, whose divergence has no
+prescribed rate. We therefore fix the graph before taking the long
+average. At an auxiliary scale $B$, squarefree divisors are formed from
+two bands of primes at most $e^B$. Primes in the higher band receive a
+fixed weight greater than one, and those in the lower band receive
+weight one; the weight of a divisor is the product of its prime weights.
+After dividing the ordinary divisor average by the total reciprocal
+weight of these divisors, a biased correlation subsequence produces a
+raw average of size at least a constant times $B^{-1}$ in one short
+multiplicative interval. The analytic centering error is $o(B^{-1})$,
+and the graph estimate gives $O(B^{-1-c_* /2})$ for a fixed absolute
+$c_*>0$. Taking the long limit with $B$ fixed, and then increasing $B$,
+contradicts the bias.
+
+The higher prime band, called the core band, supplies decay through
+restrictions on divisor size and on the number of core primes at a
+vertex. The lower, center band provides the additional power saving
+needed to beat the $B^{-1}$ scale. Its signed factor is
+$\1_{p\mid n}-\theta/p$, with
+$\theta$ chosen to match the vertex normalization; it is not itself
+mean zero under uniform residues. For suitable primes appearing on
+only one edge, the normalization factors at the two endpoints in the
+closed-walk product make the leading divisibility contribution match
+the subtracted constant term.
+To retain this cancellation,
+we organize a walk by the tree of its first visits and expand the
+vertex-deletion conditions while leaving these center-prime residues
+unconditioned. Their signed averages are taken before absolute values.
+This yields the qualitative theorem without a Liouville-specific rate.
+
+\subsection{Organization}
+
+Part~I proves the quantitative progression estimate and then
+\Cref{thm:q-affine}.
+\Cref{q:setup,q:finite-law-section} construct its prime supplies and
+establish the finite residue comparison; \Cref{q:analytic} bounds
+centering and deletion errors. \Cref{q:paths-trace} introduces the
+graph moment, and \Cref{q:rank-witness-section,q:forest-section} prove
+it through arithmetic constraints and pattern counting.
+\Cref{q:sec-spectral} returns to ordinary correlations, while
+\Cref{q:affine-transfer} gives the affine deduction.
+
+Part~II proves \Cref{thm:elliott} with new parameters and prime sets.
+\Cref{sec:graph-setup,sec:analytic-transfer,sec:analytic-centering}
+state the graph estimate and prove the analytic reduction to it.
+\Cref{sec:paths,sec:thinning} establish the geometry of retained walks;
+\Cref{sec:cylinder-sieve,sec:trace-count} control their conditional
+averages and sum the trace. \Cref{sec:localization} transfers that
+estimate to long integer intervals and completes the theorem.
+Part~III uses finite local expansions to prove the progression and
+affine consequences for general functions, and verifies their
+Liouville and M\"obius specializations.

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/main.tex

@@ -0,0 +1,90 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\pdftrailerid{}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.08in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{microtype,enumitem}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[hidelinks]{hyperref}
+\usepackage[capitalise,noabbrev]{cleveref}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\AddToHook{env/theorem/begin}{\crefalias{theorem}{theorem}}
+\AddToHook{env/proposition/begin}{\crefalias{theorem}{proposition}}
+\AddToHook{env/lemma/begin}{\crefalias{theorem}{lemma}}
+\AddToHook{env/corollary/begin}{\crefalias{theorem}{corollary}}
+\AddToHook{env/definition/begin}{\crefalias{theorem}{definition}}
+\AddToHook{env/remark/begin}{\crefalias{theorem}{remark}}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\T}{\mathbb R/\mathbb Z}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Pbb}{\mathbb P}
+\newcommand{\1}{\mathbf 1}
+\newcommand{\eps}{\varepsilon}
+\DeclareMathOperator{\RePart}{Re}
+\setlist{itemsep=3pt,topsep=5pt}
+\setlength{\emergencystretch}{1.5em}
+\title{Ordinary two-point correlations of multiplicative functions}
+\author{OpenAI}
+\hypersetup{
+  pdftitle={Ordinary two-point correlations of multiplicative functions},
+  pdfauthor={OpenAI}
+}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the ordinary two-point Chowla conjecture. For every fixed pair
+of nonproportional affine forms, the Liouville correlation has a
+power-of-logarithm saving at every cutoff, with an absolute exponent.
+We also prove the binary corrected Elliott conjecture for ordinary
+averages of complex multiplicative functions of modulus at most one,
+under uniform nonpretentiousness of at least one original factor.
+This qualitative conclusion holds in fixed residue classes and for
+fixed nonproportional affine forms.
+\end{abstract}
+\tableofcontents
+\input{introduction}
+\part{A logarithmic saving for Liouville correlations}
+\input{quantitative/01-setup}
+\input{quantitative/02-finite-law}
+\input{quantitative/03-analytic}
+\input{quantitative/04-paths-trace}
+\input{quantitative/05-rank-witness}
+\input{quantitative/06-forest}
+\input{quantitative/07-spectral}
+\input{quantitative/08-affine}
+\part{Qualitative correlations of general multiplicative functions}
+\noindent
+We now prove \Cref{thm:elliott}. The graph in this part uses a different
+normalization and a different order of limits. Its parameters and prime
+sets are defined afresh; none of the choices of \(A,W,L,J,\eta\) in
+Part~I is in force here. The functions and their nonpretentiousness
+condition remain those of the introduction.
+\input{qualitative/graph-setup}
+\input{qualitative/analytic-transfer}
+\input{qualitative/paths}
+\input{qualitative/thinning}
+\input{qualitative/cylinder-sieve}
+\input{qualitative/trace-count}
+\input{qualitative/localization}
+\part{Progressions and affine forms}
+\input{qualitative/affine-corollaries}
+\begingroup\small
+\bibliographystyle{alpha}
+\bibliography{refs}
+\endgroup
+\end{document}

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/affine-corollaries.tex

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+\section{Local factors and affine correlations}
+\label{sec:affine-corollaries}
+
+We now deduce correlations on arbitrary fixed progressions and affine
+forms from \Cref{thm:elliott}. The main point is to handle the primes
+dividing a progression modulus or a dilation without assuming complete
+multiplicativity. A finite expansion at these primes will suffice.
+
+\subsection{Stability and finite local expansions}
+
+We first extend \Cref{lem:finite-prime-stability} to character twists
+and complex conjugation.
+
+\begin{lemma}[Character twists and conjugation]
+\label{lem:affine-stability}
+Let $f,G\colon\N\to\mathbb D$ be multiplicative, let $\psi$ be a
+fixed Dirichlet character, and let $S$ be a finite set of primes.
+If $G(p)=f(p)\psi(p)$ for every $p\notin S$, then for every Dirichlet
+character $\chi$, every real $t$, and every $X\ge2$,
+\begin{equation}
+ \left|D(G,\chi n^{it};X)^2
+       -D(f,(\chi\overline\psi)n^{it};X)^2\right|
+ \le 2\sum_{p\in S}\frac1p.
+ \label{eq:affine-stability}
+\end{equation}
+Thus uniform nonpretentiousness of $f$ implies that of $G$.
+Complex conjugation also preserves uniform nonpretentiousness.
+\end{lemma}
+
+\begin{proof}
+The product $\chi\overline\psi$ is a Dirichlet character modulo the
+least common multiple of the two moduli. At every prime outside $S$,
+\[
+ G(p)\overline{\chi(p)p^{it}}
+ =f(p)\overline{(\chi\overline\psi)(p)p^{it}}.
+\]
+This identity holds also at primes where a character vanishes. Each
+prime in $S$ contributes at most $2/p$ to the difference of squared
+distances, proving \eqref{eq:affine-stability}. Take the infimum over
+exactly $|t|\le X$ on both sides of the resulting lower bound. For each
+fixed $\chi$, the character $\chi\overline\psi$ is fixed, and the
+bounded error cannot prevent divergence to infinity.
+For conjugation the exact identity is
+\[
+ D(\overline f,\chi n^{it};X)^2
+ =D(f,\overline\chi n^{-it};X)^2.
+\]
+The interval $|t|\le X$ is unchanged by $t\mapsto-t$.
+\end{proof}
+
+A multiplicative function is either identically zero or has value one
+at $1$: apply multiplicativity to $(1,n)$. Zero factors give zero
+correlations, so in the following local construction we may work with
+normalized functions. Write $v_p(m)$ for the exponent of $p$ in $m$.
+
+\begin{lemma}[A finite expansion for dilation]
+\label{lem:affine-dilation}
+Let $a\ge1$ be an integer and let $f\colon\N\to\mathbb D$ be
+multiplicative with $f(1)=1$. Define
+\begin{equation}
+ F_{a,f}(m)=
+ \begin{cases}
+ f(m/a),&a\mid m,\\
+ 0,&a\nmid m.
+ \end{cases}
+ \label{eq:affine-dilation-function}
+\end{equation}
+Then $F_{a,f}$ is a signed sum of $2^{\omega(a)}$ normalized,
+$1$-bounded multiplicative functions, each agreeing with $f$ at every
+prime power whose prime does not divide $a$. If $f$ is uniformly
+nonpretentious, every function in this sum is uniformly nonpretentious.
+\end{lemma}
+
+\begin{proof}
+Let $P(a)$ be the set of prime divisors of $a$. For $p\in P(a)$ set
+$\alpha_p=v_p(a)$ and define
+\begin{equation}
+ A_p(k)=
+ \begin{cases}
+ 0,&0\le k<\alpha_p,\\
+ f(p^{k-\alpha_p}),&k\ge\alpha_p.
+ \end{cases}
+ \label{eq:affine-local-sequence}
+\end{equation}
+Factorization into pairwise coprime prime powers gives the exact identity
+\begin{equation}
+ F_{a,f}(m)=
+ \prod_{p\in P(a)}A_p(v_p(m))
+ \prod_{\substack{p\mid m\\p\notin P(a)}}f(p^{v_p(m)}).
+ \label{eq:affine-local-product}
+\end{equation}
+No identity between $f(p^k)$ and $f(p)^k$ is used.
+
+The obstruction to multiplicativity in this product is that
+$A_p(0)=0$. Resolve it by setting
+\begin{equation}
+ B_p(0)=C_p(0)=1,\qquad
+ B_p(k)=A_p(k),\quad C_p(k)=0\quad(k\ge1).
+ \label{eq:affine-b-minus-c}
+\end{equation}
+Then $A_p=B_p-C_p$ at every nonnegative exponent. For
+$E\subseteq P(a)$ put
+\[
+ f_{a,E}(m)=
+ \prod_{p\in E}C_p(v_p(m))
+ \prod_{p\in P(a)\setminus E}B_p(v_p(m))
+ \prod_{\substack{p\mid m\\p\notin P(a)}}f(p^{v_p(m)}).
+\]
+All local factors have modulus at most one and value one at exponent
+zero. If $u$ and $v$ are coprime, at most one of $v_p(u),v_p(v)$ is
+positive at each prime; hence $f_{a,E}(uv)=f_{a,E}(u)f_{a,E}(v)$.
+Expanding the finite product in \eqref{eq:affine-local-product} yields
+\begin{equation}
+ F_{a,f}(m)=
+ \sum_{E\subseteq P(a)}(-1)^{|E|}f_{a,E}(m).
+ \label{eq:affine-dilation-expansion}
+\end{equation}
+Each summand agrees with $f$ at prime powers outside $P(a)$, so
+\Cref{lem:finite-prime-stability} proves the last assertion.
+\end{proof}
+
+For $a=1$ the expansion has just one term, namely $f$. For $a>1$
+and $m=1$, its signed sum is
+$\sum_{E\subseteq P(a)}(-1)^{|E|}=0$, as required. Thus the expansion
+also handles the value at $1$, despite normalizing every summand there.
+
+\begin{lemma}[Every residue class]
+\label{lem:affine-residue}
+Let $l\ge1$ and $0\le b<l$ be integers, and set
+\[
+ d=\gcd(b,l),\qquad q=l/d,\qquad c=b/d.
+\]
+For every positive integer $m$,
+\begin{equation}
+ \1_{m\equiv b\pmod l}
+ =\frac1{\varphi(q)}\sum_{\chi\bmod q}
+       \overline{\chi(c)}F_{d,\chi}(m).
+ \label{eq:affine-character-expansion}
+\end{equation}
+Each $F_{d,\chi}$ is a signed sum of $2^{\omega(d)}$ normalized,
+$1$-bounded multiplicative functions agreeing with $\chi$ at all prime
+powers outside $d$. For $q=1$, the character in the formula is the
+constant function one, including at the residue $c=0$, and
+$\varphi(1)=1$.
+\end{lemma}
+
+\begin{proof}
+If $d\nmid m$, both sides vanish. If $d\mid m$, the congruence on
+the left is equivalent to $m/d\equiv c\pmod q$. When $q>1$, the
+residue $c$ is a unit modulo $q$, so character orthogonality on
+$(\Z/q\Z)^\times$ gives
+\[
+ \1_{m/d\equiv c\pmod q}
+ =\frac1{\varphi(q)}\sum_{\chi\bmod q}
+       \overline{\chi(c)}\chi(m/d).
+\]
+For a nonunit $m/d$ every character on the right vanishes. The case
+$q=1$ is the trivial identity with the stated convention. The finite
+expansion now follows from \Cref{lem:affine-dilation}.
+\end{proof}
+
+In particular, if $p\mid d$ and $\alpha=v_p(d)$, the local sequence
+in this application is
+\[
+ A_p(k)=0\quad(k<\alpha),\qquad
+ A_p(k)=\chi(p)^{k-\alpha}\quad(k\ge\alpha),
+ \qquad 0^0=1.
+\]
+If $p$ also divides $q$, then $\chi(p)=0$ and this sequence is one
+at $k=\alpha$ and zero elsewhere. No character value is divided out.
+When $b=0$ we have $d=l$ and $q=1$, so the same formula expands
+$\1_{l\mid m}$; it includes $l=1$ and $m=1$.
+
+\subsection{Progressions and the affine deduction}
+
+The two expansions just proved reduce the desired averages to finitely
+many applications of \Cref{thm:elliott}. We first incorporate a
+progression restriction, placing its weight at the argument of a
+nonpretentious factor.
+
+\begin{proposition}[Correlations on fixed progressions]
+\label{prop:affine-progression}
+Let $f_1,f_2\colon\N\to\mathbb D$ be multiplicative, with at least
+one uniformly nonpretentious. For fixed distinct nonnegative integers
+$h_1,h_2$, an integer $l\ge1$, and any integer $b$,
+\begin{equation}
+ \frac1X\sum_{1\le n\le X}
+ f_1(n+h_1)f_2(n+h_2)\1_{n\equiv b\pmod l}
+ \longrightarrow0
+ \qquad(X\longrightarrow\infty).
+ \label{eq:affine-progression-limit}
+\end{equation}
+The limit holds through all real $X$, including nonunit residue classes.
+\end{proposition}
+
+\begin{proof}
+We may assume both functions are nonzero. Choose $j\in\{1,2\}$ for
+which $f_j$ is uniformly nonpretentious. Since
+\[
+ \1_{n\equiv b\pmod l}
+ =\1_{n+h_j\equiv b+h_j\pmod l},
+\]
+\Cref{lem:affine-residue}, using the least nonnegative representative
+of $b+h_j$ modulo $l$, expands the indicator as a function of
+$m=n+h_j$ into finitely many bounded multiplicative functions, each
+agreeing with a fixed Dirichlet character outside a fixed finite prime
+set. Their products with $f_j$ are uniformly nonpretentious by
+\Cref{lem:affine-stability}. Apply \Cref{thm:elliott} to each product
+at shift $h_j$ and the unchanged other factor at its distinct shift,
+and sum the finitely many conclusions. This proves
+\eqref{eq:affine-progression-limit} at integer $X$.
+For real $X$, replacing the normalization $X$ by $\lfloor X\rfloor$
+changes the average by at most $1/X$, proving the assertion.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{cor:affine}]
+Zero factors again give an immediate conclusion. Otherwise set
+\begin{equation}
+ \Delta=a_1b_2-a_2b_1,\qquad
+ l=a_1a_2,\qquad
+ c_0=\min(a_2b_1,a_1b_2),\qquad h=|\Delta|\ge1.
+ \label{eq:affine-parameters}
+\end{equation}
+For $\Delta>0$ put
+$U=F_{a_2,f_1}$ and $V=F_{a_1,f_2}$; for $\Delta<0$ put
+$U=F_{a_1,f_2}$ and $V=F_{a_2,f_1}$. In either case,
+\begin{equation}
+ U(ln+c_0)V(ln+c_0+h)
+ =f_1(a_1n+b_1)f_2(a_2n+b_2).
+ \label{eq:affine-exact-identity}
+\end{equation}
+Indeed, when $\Delta>0$ the two arguments on the left are
+$a_2(a_1n+b_1)$ and $a_1(a_2n+b_2)$; when $\Delta<0$ their order
+is reversed.
+
+Expand $U$ and $V$ by \Cref{lem:affine-dilation}. Each pair of
+multiplicative components has a uniformly nonpretentious factor,
+inherited from the corresponding original function. Applying
+\Cref{prop:affine-progression} termwise with shifts $0,h$ gives
+\begin{equation}
+ \frac1X\sum_{1\le m\le X}
+ U(m)V(m+h)\1_{m\equiv c_0\pmod l}\longrightarrow0.
+ \label{eq:affine-expanded-limit}
+\end{equation}
+Put $T(m)=U(m)V(m+h)\1_{m\equiv c_0\pmod l}$ and $X=lN+c_0$.
+The integers in this class satisfying $c_0<m\le X$ are precisely
+$ln+c_0$, $1\le n\le N$. Therefore
+\begin{align}
+ \frac1N\sum_{n=1}^N f_1(a_1n+b_1)f_2(a_2n+b_2)
+ &=\frac XN\left(\frac1X\sum_{1\le m\le X}T(m)\right)
+   -\frac1N\sum_{1\le m\le c_0}T(m).
+ \label{eq:affine-endpoints}
+\end{align}
+The first term tends to zero because $X/N\to l$ and
+\eqref{eq:affine-expanded-limit} holds. The second has modulus at most
+$c_0/N$ and is empty when $c_0=0$. This proves the corollary, including
+zero intercepts and coefficients with common factors.
+\end{proof}
+
+\subsection{Liouville and M\"obius correlations}
+
+To apply the preceding conclusions to the Liouville function, we verify
+the nonpretentiousness hypothesis over its full growing height interval.
+The required uniform estimate is supplied by
+Matom\"aki, Radziwi{\l}{\l}, and Tao's version of an argument of
+Granville and Soundararajan \cite[Appendix~C]{MRT15}.
+
+\begin{lemma}[Uniform nonpretentiousness of Liouville]
+\label{lem:affine-liouville-unp}
+For every fixed Dirichlet character $\chi$,
+\begin{equation}
+ \inf_{|t|\le X}D(\lambda,\chi n^{it};X)
+ \ge \frac14\sqrt{\log\log X}-O_\chi(1)
+ \qquad(X\ge100).
+ \label{eq:affine-liouville-unp}
+\end{equation}
+In particular, $\lambda$ is uniformly nonpretentious.
+\end{lemma}
+
+\begin{proof}
+For a real $1$-bounded multiplicative function $f$,
+\cite[Lemma~C.1]{MRT15} gives
+\begin{equation}
+ D(f,\chi n^{it};X)
+ \ge\frac14\sqrt{\log\log X}-O_\chi(1)
+ \quad(1\le|t|\le X).
+ \label{eq:affine-real-large-height}
+\end{equation}
+When $\chi^2$ is nonprincipal the same bound holds throughout
+$|t|\le X$. Thus taking $f=\lambda$ proves the assertion for nonreal
+$\chi$ and handles $1\le|t|\le X$ for all $\chi$.
+For real $\chi$, the other part of the cited lemma gives
+\begin{equation}
+ D(\lambda,\chi n^{it};X)
+ \ge\frac13D(\lambda,\chi;X)-O(1)
+ \quad(|t|\le1).
+ \label{eq:affine-real-small-height}
+\end{equation}
+It remains to estimate the distance at $t=0$ for these real characters.
+
+If $\chi$ is nonprincipal, the prime number theorem in arithmetic
+progressions for its fixed modulus yields, for some $c_\chi>0$,
+\[
+ A_\chi(u):=\sum_{p\le u}\chi(p)\log p
+ =O_\chi\bigl(u\exp(-c_\chi\sqrt{\log u})\bigr).
+\]
+Partial summation gives
+\[
+ \sum_{p\le X}\frac{\chi(p)}p
+ =\frac{A_\chi(X)}{X\log X}
+   +\int_2^X A_\chi(u)
+     \frac{\log u+1}{u^2(\log u)^2}\,du
+ =O_\chi(1),
+\]
+since the integral converges absolutely as $X\to\infty$.
+As $\lambda(p)=-1$, Mertens' estimate now gives
+\[
+ D(\lambda,\chi;X)^2
+ =\sum_{p\le X}\frac{1+\chi(p)}p
+ =\log\log X+O_\chi(1).
+\]
+If instead $\chi$ is principal modulo $q$, then
+\[
+ D(\lambda,\chi;X)^2
+ =2\sum_{p\le X}\frac1p
+  -\sum_{\substack{p\mid q\\p\le X}}\frac1p
+ =2\log\log X+O_q(1).
+\]
+In both cases \eqref{eq:affine-real-small-height} is at least
+$\frac13\sqrt{\log\log X}-O_\chi(1)$. Combining this with
+\eqref{eq:affine-real-large-height} proves
+\eqref{eq:affine-liouville-unp} on the entire interval $|t|\le X$.
+\end{proof}
+
+Thus $\lambda$ satisfies the exact hypothesis of the qualitative theorem.
+Applying \Cref{prop:affine-progression,cor:affine} with
+$f_1=f_2=\lambda$ independently recovers ordinary cancellation in
+every fixed residue class and along every fixed nonproportional affine
+pair. \Cref{q:progression,thm:q-affine} establish these conclusions
+with an absolute power-of-logarithm saving.
+
+The qualitative theorem also applies to the M\"obius function, defined by
+\[
+ \mu(n)=
+ \begin{cases}
+ \lambda(n),&n\text{ is squarefree},\\
+ 0,&n\text{ is not squarefree}.
+ \end{cases}
+\]
+This function is multiplicative and has $\mu(p)=\lambda(p)=-1$ at
+every prime. Since the distance uses only prime values,
+\[
+ D(\mu,\chi n^{it};X)=D(\lambda,\chi n^{it};X)
+\]
+for every Dirichlet character $\chi$, real $t$, and $X\ge2$.
+The preceding lemma therefore proves uniform nonpretentiousness of
+$\mu$ with the same growing twist range. The conclusions of
+\Cref{prop:affine-progression,cor:affine} hold for any choice
+$f_1,f_2\in\{\lambda,\mu\}$. This proves ordinary cancellation for two-point M\"obius
+and mixed M\"obius--Liouville correlations in every fixed
+residue class and along every fixed nonproportional affine pair.
+For products containing a M\"obius factor, this deduction gives
+convergence without a quantitative rate.

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/analytic-transfer.tex

@@ -0,0 +1,671 @@
+\section{Reduction to the graph estimate}\label{sec:analytic-transfer}
+
+We now explain how \Cref{prop:graph-estimate} rules out a nonzero
+multiplicative correlation. The first step produces a large sum with
+ordinary divisibility indicators. We then state the estimate that permits
+centering those indicators, and display the resulting contradiction.
+The proof of the centering estimate occupies
+\Cref{sec:analytic-centering}; the graph estimate itself will be proved
+in \Cref{sec:localization}.
+
+\subsection{Absolute-value defects and a biased sequence}
+
+A multiplicative function satisfies $f(1)=f(1)^2$. If $f(1)=0$, then
+$f(n)=f(n)f(1)=0$ for every $n$. We may therefore suppose that both
+functions in \Cref{thm:elliott} take the value $1$ at $1$.
+There is another case in which the conclusion follows without a graph.
+
+\begin{lemma}\label{lem:analytic-absolute-defect}
+Let $f\colon\N\to\C$ be multiplicative and $|f|\le1$. If
+\[
+ \sum_p\frac{1-|f(p)|}{p}=\infty,
+\]
+then $N^{-1}\sum_{n\le N}|f(n)|\to0$.
+\end{lemma}
+\begin{proof}
+Write $v_p(n)$ for the exponent of $p$ in $n$. For a finite set
+$\mathcal P$ of primes, multiplicativity gives
+\[
+ |f(n)|\le
+ \prod_{\substack{p\in\mathcal P\\v_p(n)=1}}|f(p)|.
+\]
+The majorant is periodic modulo $\prod_{p\in\mathcal P}p^2$.
+Its ordinary mean is
+\[
+ \prod_{p\in\mathcal P}
+ \left(1-(1-|f(p)|)\left(\frac1p-\frac1{p^2}\right)\right).
+\]
+These products tend to zero as $\mathcal P$ increases through the primes:
+the sum of the subtracted quantities diverges, whereas
+$\sum_p p^{-2}<\infty$. First take the long average with $\mathcal P$
+fixed, and then increase $\mathcal P$.
+\end{proof}
+
+A fixed translation affects only finitely many terms of a bounded
+average. Thus, after translating by the smaller shift and ordering the
+functions accordingly, it suffices to prove cancellation of
+$f_1(m)f_2(m+h)$ for a fixed $h\ge1$.
+By \Cref{lem:analytic-absolute-defect}, we may assume
+\begin{equation}\label{eq:analytic-finite-defects}
+ \sum_p\frac{1-|f_i(p)|}{p}<\infty\qquad(i=1,2).
+\end{equation}
+If the desired cancellation fails, there are $0<\gamma\le1$ and positive
+integers $N_j\to\infty$ such that
+\begin{equation}\label{eq:analytic-bias}
+ \left|S(N_j)\right|\ge\gamma N_j,
+ \qquad S(N)=\sum_{m\le N}f_1(m)f_2(m+h).
+\end{equation}
+We retain this same sequence throughout the argument.
+The numbers $S(N_j)$ may have varying complex arguments.
+
+Fix $1<\tau<2$ sufficiently close to $1$ in terms of $\gamma$, and then
+fix $T$ sufficiently large in terms of $\gamma$. Choose
+$\phi\in C_c^\infty(\R)$ with $0\le\phi\le1$, supported in $[-T,T]$
+and equal to $1$ on $[-T/2,T/2]$. For every admissible divisor $d$, use
+\begin{equation}\label{eq:analytic-cutoffs}
+ w_d(x)=\phi\left(\frac{k_d(x)-v_C}{\sqrt{v_C}}\right).
+\end{equation}
+These cutoffs satisfy the support and residue-dependence requirements
+of \Cref{prop:graph-estimate}.
+
+\subsection{Capturing the bias in one divisor interval}
+
+Recall that $\mathcal S=\mathcal C\cup\mathcal Z$ is the finite prime set
+at scale $B$, and that a squarefree product $d$ of these primes has
+weight $A^{\omega_C(d)}$. The normalizing factor $L_0$ has the exact
+expansion
+\begin{equation}\label{eq:raw-divisor-law}
+ L_0=\sum_{\substack{d\ \mathrm{squarefree}\\p\mid d\Rightarrow p\in\mathcal S}}
+       \frac{A^{\omega_C(d)}}d.
+\end{equation}
+The term $d=1$ is included here.
+
+\begin{lemma}[A divisor interval carrying positive mass]
+\label{lem:raw-bin}
+Let $f_1,f_2\colon\N\to\C$ be $1$-bounded multiplicative functions
+satisfying \eqref{eq:analytic-finite-defects} and $f_1(1)=f_2(1)=1$.
+Fix $1<\tau<2$. There are constants $C_0\ge1$ and
+$c>0$ such that, for every sufficiently large $B$, one can choose
+$1\le H\le\exp(C_0B)$ and a family $\mathcal D$ of squarefree
+$\mathcal S$-products satisfying
+\begin{gather}
+ H<d\le\tau H,\qquad d>1,\qquad
+ \omega(d)\le J=\lceil C_0\log B\rceil,
+ \qquad |f_1(d)f_2(d)|\ge\tfrac12,
+ \label{eq:raw-selected-divisors}\\
+ \sum_{d\in\mathcal D}\frac{A^{\omega_C(d)}}d
+ \ge c\frac{L_0}{B}.
+ \label{eq:raw-bin-mass}
+\end{gather}
+The constants are independent of $B$ and of the long averaging variable.
+\end{lemma}
+\begin{proof}
+Normalize the summands of \eqref{eq:raw-divisor-law} to a probability
+law on divisors, and denote its expectation and probability by
+$\E_{\mathrm{div}}$ and $\Pbb_{\mathrm{div}}$. Each prime $p$ in band $a$
+is included independently with
+probability $A_a/(p+A_a)$. Prime harmonic estimates give
+\[
+ \E_{\mathrm{div}}\omega(d)\ll_A\log B,
+ \qquad
+ \E_{\mathrm{div}}\log d\le A\sum_{p\le e^B}\frac{\log p}{p}\ll_A B.
+\]
+Choose $C_0$ large enough that the probabilities of
+$\omega(d)>\lceil C_0\log B\rceil$ and $\log d>C_0B$ are each at most
+$1/8$, by Markov's inequality.
+
+For $r_p=|f_1(p)f_2(p)|$, the inequality
+$1-\prod_j r_j\le\sum_j(1-r_j)$, valid when $0\le r_j\le1$, gives
+\begin{align*}
+ \E_{\mathrm{div}}\bigl(1-|f_1(d)f_2(d)|\bigr)
+ &\le A\sum_{p>P_0}\frac{1-|f_1(p)f_2(p)|}{p}\\
+ &\le A\sum_{p>P_0}\frac{(1-|f_1(p)|)+(1-|f_2(p)|)}p=o(1).
+\end{align*}
+Here squarefreeness permits the use of multiplicativity.
+Another application of Markov's inequality shows that the probability
+of $|f_1(d)f_2(d)|<1/2$ tends to zero. Also
+$\Pbb_{\mathrm{div}}(d=1)=L_0^{-1}\to0$.
+Thus the remaining divisors carry at least $L_0/2$ of the unnormalized
+mass for large $B$.
+
+There are at most $1+C_0B/\log\tau$ intervals
+$(\tau^r,\tau^{r+1}]$, $r\ge0$, meeting $(1,e^{C_0B}]$.
+One of them carries at least $cL_0/B$ of that mass, for a fixed $c>0$.
+Take its lower endpoint as $H$ and retain the divisors already satisfying
+the preceding conditions.
+\end{proof}
+
+For any such divisor family, any coefficients $|a_d|\le1$, and the
+cutoffs \eqref{eq:analytic-cutoffs}, define the raw and centered sums by
+\begin{align}
+ R_B(X)&=\sum_{x\le X}\sum_{d\in\mathcal D}
+ a_d A^{\omega_C(d)}f_1(x)f_2(x+hd)
+ w_d(x)w_d(x+hd)\1_{d\mid x},
+ \label{eq:raw-sum}\\
+ C_B(X)&=\sum_{x\le X}\sum_{d\in\mathcal D}
+ a_df_1(x)f_2(x+hd)g_d(x,x+hd).
+ \label{eq:analytic-centered-sum}
+\end{align}
+The first sum retains ordinary divisibility by all primes of $d$;
+the second is the sum bounded by \Cref{prop:graph-estimate}.
+
+\begin{lemma}[Raw lower bound]\label{lem:raw-lower-bound}
+Let $f_1,f_2\colon\N\to\C$ be $1$-bounded multiplicative functions,
+with $f_1(1)=f_2(1)=1$, satisfying \eqref{eq:analytic-bias} and
+\eqref{eq:analytic-finite-defects}. Choose $1<\tau<2$ and $T>0$ with
+$\tau-1\le\gamma/64$ and $32/T^2\le\gamma/64$, and use a cutoff
+$\phi$ as in \eqref{eq:analytic-cutoffs}, equal to $1$ on
+$[-T/2,T/2]$. With $\mathcal D,H$ supplied by \Cref{lem:raw-bin}, set
+\[
+ a_d=\overline{f_1(d)f_2(d)},\qquad X_j=HN_j.
+\]
+There is $c_\gamma>0$, independent of $B,j$, such that, for every
+sufficiently large fixed $B$,
+\begin{equation}\label{eq:raw-lower-bound}
+ \liminf_{j\to\infty}\frac{|R_B(X_j)|}{X_j}
+ \ge c_\gamma\frac{L_0}{B}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+Put
+\[
+ V_{\mathcal D}=\sum_{d\in\mathcal D}A^{\omega_C(d)},\qquad
+ K_{\mathcal D}=\sum_{d\in\mathcal D}
+   |f_1(d)f_2(d)|^2A^{\omega_C(d)}\ge\tfrac14 V_{\mathcal D}.
+\]
+For a fixed $d$, write $x=dm$. Unless
+$\gcd(d,m)>1$ or $\gcd(d,m+h)>1$, ordinary multiplicativity gives
+\[
+ a_df_1(dm)f_2(d(m+h))
+   =|f_1(d)f_2(d)|^2f_1(m)f_2(m+h).
+\]
+The exceptional set has ordinary density at most
+$2\sum_{p\mid d}p^{-1}\le2J/P_0$. At fixed $B$ there are only finitely
+many $d$, so all associated residue-counting errors vanish as
+$j\to\infty$.
+
+For $p\in\mathcal C$ not dividing $d$, divisibility of $dm$ by $p$ is
+equivalent to divisibility of $m$. Hence the mean and variance of
+$k_d(dm)$ under uniform residues are
+\[
+ \mu_d=v_C-\sum_{\substack{p\mid d\\p\in\mathcal C}}\frac1p,
+ \qquad
+ \operatorname{Var}(k_d(dm))
+ =\sum_{\substack{p\in\mathcal C\\p\nmid d}}
+       \frac1p\left(1-\frac1p\right)\le v_C.
+\]
+The same formulas hold with $m+h$ in place of $m$. For large $B$,
+$|\mu_d-v_C|\le J/P_0\le(T/4)\sqrt{v_C}$. Chebyshev's inequality and
+a union bound show that the proportion on which either cutoff is not
+$1$ is at most $32/T^2$. Independence of the two endpoints is not needed.
+
+The length $X_j/d$ lies between $N_j/\tau$ and $N_j$. Comparing each
+shorter sum with the sum up to $N_j$ therefore gives
+\begin{equation}\label{eq:raw-comparison}
+ \begin{split}
+ |R_B(X_j)-K_{\mathcal D}S(N_j)|
+ \le N_jV_{\mathcal D}\left(
+ \tau-1+\frac{32}{T^2}+\frac{4J}{P_0}+o_{j\to\infty}(1)\right).
+ \end{split}
+\end{equation}
+The factor $4J/P_0$ allows a difference of size at most $2$ on each
+coprimality failure. The endpoint rounding errors are included in $o(1)$.
+
+Choose $\tau-1\le\gamma/64$ and $32/T^2\le\gamma/64$, and then take
+$B$ and $j$ large enough for the other two errors to be at most
+$\gamma/64$ each. The reverse triangle inequality, valid for the
+complex number $S(N_j)$, yields
+\[
+ |R_B(X_j)|\ge\frac\gamma8N_jV_{\mathcal D}.
+\]
+Finally,
+$V_{\mathcal D}\ge H\sum_{d\in\mathcal D}A^{\omega_C(d)}/d
+\ge cHL_0/B$. This proves the result with $c_\gamma=\gamma c/8$.
+\end{proof}
+
+\subsection{The centering estimate and the contradiction}
+
+The remaining analytic task is to show that centering changes the raw
+sum by less than its lower bound. Its statement does not require
+\eqref{eq:analytic-finite-defects}.
+
+\begin{proposition}[Analytic centering estimate]
+\label{prop:analytic-centering}
+Let $f_1,f_2\colon\N\to\C$ be multiplicative, with $|f_i|\le1$, and
+suppose at least one is uniformly nonpretentious. Fix
+$h\ge1$, $1<\tau<2$, $C_0\ge1$, $T>0$, and a smooth function
+$0\le\phi\le1$ supported in $[-T,T]$.
+For each sufficiently large $B$, let $\mathcal D,H,J$ satisfy
+\eqref{eq:divisor-family}, let $|a_d|\le1$, and use
+\eqref{eq:analytic-cutoffs}. Define $R_B,C_B$ by
+\eqref{eq:raw-sum} and \eqref{eq:analytic-centered-sum}. Then
+\begin{equation}\label{eq:analytic-centering-bound}
+ \limsup_{X\to\infty}\frac{|R_B(X)-C_B(X)|}{X}
+ \ll L_0\left(B^{-1-\eps}+\frac{J}{P_0}\right).
+\end{equation}
+The implied constant may depend on the fixed parameters and $\phi$,
+but is independent of the divisor family and coefficients.
+The long-variable limit is taken with $B$ fixed.
+\end{proposition}
+
+\begin{proof}[Reduction of \Cref{thm:elliott}]
+We deduce \Cref{thm:elliott} from
+\Cref{prop:graph-estimate,prop:analytic-centering}.
+The zero-function and divergent-defect cases were settled above.
+Otherwise suppose \eqref{eq:analytic-bias} holds. Choose
+$\tau,T,\phi,C_0$ and, for each sufficiently large fixed $B$, the divisor
+family of \Cref{lem:raw-bin}. Combining \Cref{lem:raw-lower-bound} with
+the two propositions gives
+\[
+ c_\gamma\frac{L_0}{B}
+ \le C L_0\left(B^{-1-c_*/2}+B^{-1-\eps}+\frac{J}{P_0}\right).
+\]
+After dividing by $L_0/B$, this reads
+\begin{equation}\label{eq:analytic-contradiction}
+ c_\gamma\le C\left(B^{-c_*/2}+B^{-\eps}
+             +BJ\exp(-B^{1-\eps})\right),
+\end{equation}
+which is impossible for sufficiently large $B$.
+
+In this comparison the functions, shift, and bias $\gamma$ are fixed
+first; then $\tau,T,\phi,C_0$ are fixed. For each fixed $B$ the limit
+is taken along the original sequence $N_j$, with $X_j=HN_j$.
+Only after these inequalities hold does $B$ increase.
+Thus the contradiction excludes every alleged biased sequence.
+\end{proof}
+
+\section{The analytic centering estimate}\label{sec:analytic-centering}
+
+We prove \Cref{prop:analytic-centering}. Expanding the centered factors
+leaves sums over integers with large prime factors. We shall bound the
+Fourier multiplier of those integers and use short-interval cancellation
+for whichever multiplicative function is nonpretentious. Combining short
+exponential sums with a fourth-moment bound follows the centering
+strategy in \cite[Appendix~C]{Pilatte26}; we prove the rough-number
+estimates needed for the present weights. Throughout this section,
+$\mathrm e(t)=\exp(2\pi it)$.
+
+\subsection{Expansion and finite-prime twists}
+
+A term other than the raw term in the expansion of the center-band
+factors has $d=uw$, where $w>1$ is the product of the center primes
+whose constant terms were selected. Thus $u,w$ are coprime squarefree
+products, every core factor of $d$ belongs to $u$, and the coefficient
+of the divisibility indicator $\1_{u\mid x}$ is
+\[
+ A^{\omega_C(u)}a_{uw}\frac{(-\theta)^{\omega(w)}}w.
+\]
+For each fixed $u$, put $c_w^{(u)}=a_{uw}(-\theta)^{\omega(w)}$ on
+these admissible factorizations and set $c_w^{(u)}=0$ otherwise.
+Then $|c_w^{(u)}|\le1$, and its support is independent of $x$.
+Set $M=H/u$ and write $x=uz$. Whenever this support is nonempty,
+its values of $w$ satisfy
+\begin{equation}\label{eq:analytic-rough-range}
+ M<w\le\tau M,\qquad
+ \frac{P_0}{\tau}\le M\le e^{C_0B}.
+\end{equation}
+The lower bound follows because a nonempty term has $w>P_0$.
+Every such $w$ has no prime factor below $P_0$; we call integers with
+this property \emph{$P_0$-rough}.
+
+Since $u$ and $d$ have exactly the same core factors, the first
+cutoff becomes
+\[
+ w_{uw}(uz)=\phi\left(
+ \frac{\displaystyle\sum_{\substack{p\in\mathcal C\\p\nmid u}}
+              \1_{p\mid z}-v_C}{\sqrt{v_C}}\right).
+\]
+The second cutoff has the same expression with $z+hw$ in place of $z$.
+Apart from
+$\gcd(u,z(z+hw))>1$, multiplicativity gives
+\[
+ f_1(uz)f_2(u(z+hw))=f_1(u)f_2(u)f_1(z)f_2(z+hw).
+\]
+For a fixed $u,w$, the exceptional set has ordinary density at most
+$2J/P_0$. Since
+\begin{equation}\label{eq:analytic-sum-u}
+ \sum_{M<w\le\tau M}\frac1w\ll_\tau1,
+ \qquad
+ \sum_{\substack{u\ \mathrm{squarefree}\\p\mid u\Rightarrow p\in\mathcal S}}
+       \frac{A^{\omega_C(u)}}u=L_0,
+\end{equation}
+the total error, divided by $X$, has limsup $O(L_0J/P_0)$.
+The case of an identically zero factor is immediate, so in this
+factorization we may assume $f_1(1)=f_2(1)=1$.
+
+For $\widehat\phi(t)=\int_{\R}\phi(s)\mathrm e(-ts)\,ds$,
+Fourier inversion expresses each cutoff as an integral of constant
+phases times twists
+\begin{equation}\label{eq:analytic-twists}
+ b_i(n)=f_i(n)\mathrm e\left(
+ \frac{t_i}{\sqrt{v_C}}
+ \sum_{\substack{p\in\mathcal C\\p\nmid u}}\1_{p\mid n}\right),
+ \qquad t_i\in\R.
+\end{equation}
+For coprime integers the count in this exponent is additive. Hence
+$b_i$ is multiplicative and $|b_i|\le1$. It need not be completely
+multiplicative: the added factor has the same value at $p$ and $p^2$.
+Its prime values agree with those of $f_i$ outside $\mathcal C$.
+The two Fourier integrals have total absolute weight
+$\|\widehat\phi\|_1^2$, a fixed finite constant.
+
+It is therefore enough to show, uniformly in the twists
+\eqref{eq:analytic-twists} and in $|c_w|\le1$ supported on
+$P_0$-rough integers in $(M,\tau M]$, that
+\begin{equation}\label{eq:analytic-rough-target}
+ \limsup_{Y\to\infty}\frac1Y
+ \left|\sum_{z\le Y}b_1(z)
+       \sum_{M<w\le\tau M}\frac{c_w}{w}b_2(z+hw)\right|
+ \ll B^{-1-\eps}.
+\end{equation}
+Here $Y=X/u$, and all graph parameters are fixed before $Y$ increases.
+
+\subsection{The short-interval input}
+
+We first justify the uniformity in the Fourier parameters. Finite
+changes to prime values have only a bounded effect on the squared
+distance defining nonpretentiousness.
+
+\begin{lemma}[Stability under finitely many prime changes]
+\label{lem:finite-prime-stability}
+Let $f,b\colon\N\to\C$ be multiplicative and $1$-bounded. Suppose their
+prime values agree outside a finite set $\mathcal P$. For every
+Dirichlet character $\chi$, every real $t$, and $X\ge2$,
+\begin{equation}\label{eq:analytic-distance-stability}
+ \left|D(b,\chi n^{it};X)^2-D(f,\chi n^{it};X)^2\right|
+ \le2\sum_{p\in\mathcal P}\frac1p.
+\end{equation}
+Consequently, if $f$ is uniformly nonpretentious, the same divergence
+holds uniformly over all such $b$ and over every fixed finite family
+of Dirichlet characters.
+\end{lemma}
+\begin{proof}
+Outside $\mathcal P$ the summands in the squared distances coincide.
+At a prime in $\mathcal P$ their difference has absolute value at most
+$|b(p)-f(p)|/p\le2/p$. Sum this bound, and then take the infimum over
+$|t|\le X$ and the minimum over the finite character family.
+\end{proof}
+
+For a $1$-bounded multiplicative function $b$, define
+\[
+ \mathcal M(b;X,Q)=
+ \inf_{\substack{|t|\le X\\q\le Q,\ \chi\ ({\rm mod}\ q)}}
+     D(b,\chi n^{it};X)^2.
+\]
+The following is the general exponential-sum theorem of
+Matom\"aki, Radziwi\l\l, and Tao, in its corrected version
+\cite[Theorem~1.7]{MRT15}.
+
+\begin{theorem}[Averaged short exponential sums]\label{thm:mrt}
+Let $X\ge D\ge10$ and let $b\colon\N\to\C$ be a $1$-bounded
+multiplicative function. With
+$Q=\min((\log X)^{1/125},(\log D)^5)$, one has
+\begin{equation}\label{eq:mrt-input}
+ \begin{split}
+ \sup_{\alpha\in\T}\int_0^X
+ \left|\sum_{y<m\le y+D}b(m)\mathrm e(\alpha m)\right|\,dy
+ \ll DX\left(
+ e^{-\mathcal M(b;X,Q)/20}
+ +\frac{\log\log D}{\log D}
+ +\frac1{(\log X)^{1/700}}\right).
+ \end{split}
+\end{equation}
+The implied constant is absolute.
+\end{theorem}
+
+The interval endpoint convention does not affect the integral.
+For fixed $B$ and $D$, the characters of moduli at most $(\log D)^5$
+form a finite family. If $f_i$ is uniformly nonpretentious,
+\Cref{lem:finite-prime-stability} with $\mathcal P=\mathcal C$ gives
+$\mathcal M(b_i;Y,Q)\to\infty$ uniformly in the twists
+\eqref{eq:analytic-twists}. It follows that
+\begin{equation}\label{eq:mrt-fixed-length}
+ \limsup_{Y\to\infty}\sup_{\alpha,t_i}
+ \frac1{DY}\int_0^Y
+ \left|\sum_{y<m\le y+D}b_i(m)\mathrm e(\alpha m)\right|\,dy
+ \ll\frac{\log\log D}{\log D}.
+\end{equation}
+Both the prime cutoff and the allowed height in the distance are $Y$,
+as in the hypothesis of \Cref{thm:elliott}. For real $Y$, apply
+\Cref{thm:mrt} with $X=\lceil Y\rceil$ and enlarge the integral to
+$[0,\lceil Y\rceil]$; the normalization changes by a factor tending
+to $1$. Thus the hypothesis along integer scales is sufficient.
+The frequency supremum in
+\eqref{eq:mrt-fixed-length} is outside the integral. No bound with that
+supremum inside the integral is used below.
+
+\subsection{A rough-number Fourier multiplier}
+
+To use \eqref{eq:mrt-fixed-length} for
+\eqref{eq:analytic-rough-target}, define
+\begin{equation}\label{eq:rough-multiplier}
+ Q_M(\alpha)=\sum_{M<w\le\tau M}\frac{c_w}{w}\mathrm e(\alpha hw).
+\end{equation}
+We need its maximum and its fourth moment. The coefficients are
+arbitrary; only their rough support will be used.
+
+\begin{lemma}[Rough-number bounds]\label{lem:rough-multiplier}
+Fix $C_0\ge1$, $1<\tau<2$, and an integer $h\ge1$. Let
+$P_0=\exp(B^{1-\eps})$, where $\eps=10^{-4}$, and suppose
+$P_0/\tau\le M\le\exp(C_0B)$. If $|c_w|\le1$ and $c_w=0$ unless
+$w\in(M,\tau M]$ is $P_0$-rough, then, for all sufficiently large $B$,
+\begin{align}
+ \|Q_M\|_\infty&\ll B^{-1+\eps}\log B,
+ \label{eq:rough-sup}\\
+ \int_{\T}|Q_M(\alpha)|^4\,d\alpha
+ &\ll M^{-1}B^{-4+4\eps}(\log B)^6.
+ \label{eq:rough-fourth}
+\end{align}
+The constants are uniform in $M$ and the coefficients.
+\end{lemma}
+\begin{proof}
+We give the sieve bounds including their counting errors. Put
+\[
+ s=2\lceil100\log B\rceil,\qquad z_0=P_0^{1/(100s)}.
+\]
+For every prime $p<z_0$, forbid $\nu_p$ residue classes, where
+$0\le\nu_p\le2$. If $r(n)$ is the number of these prime conditions
+satisfied by $n$, even inclusion-exclusion gives
+\[
+ \1_{r(n)=0}\le\sum_{j=0}^s(-1)^j\binom{r(n)}j.
+\]
+Indeed, for $r>0$ the sum on the right is $\binom{r-1}s\ge0$, and for
+$r=0$ it is $1$.
+Write $E_j((a_p))=\sum_{|\mathcal I|=j}\prod_{p\in\mathcal I}a_p$
+for the elementary symmetric sum over subsets of the primes $p<z_0$.
+The Chinese remainder theorem, applied on any interval $I$ of real
+length $V$, now yields
+\begin{equation}\label{eq:rough-brun}
+ \begin{split}
+ \#\{n\in I:n\text{ avoids the forbidden classes}\}
+ \le V\sum_{j=0}^s(-1)^jE_j((\nu_p/p))
+       +O\left(\sum_{j=0}^sE_j((\nu_p))\right).
+ \end{split}
+\end{equation}
+The residue-counting error is independent of the position of $I$.
+It is at most
+\[
+ (s+1)(2z_0)^s=(s+1)2^sP_0^{1/100}\le P_0^{1/5}
+\]
+for large $B$.
+
+Let $\Lambda_0=\sum_{p<z_0}\nu_p/p$. Mertens' estimate gives
+$\Lambda_0\le2\log\log z_0+O(1)\le3\log B$ for large $B$.
+Since $E_j((\nu_p/p))\le\Lambda_0^j/j!$, the difference between the
+truncated density in \eqref{eq:rough-brun} and its full Euler product
+has absolute value at most
+\[
+ \sum_{j>s}\frac{\Lambda_0^j}{j!}
+ \le\frac{(e\Lambda_0/(s+1))^{s+1}}{1-\Lambda_0/(s+2)}
+ \le B^{-10}.
+\]
+Consequently the count in \eqref{eq:rough-brun} is at most
+\begin{equation}\label{eq:rough-sieve-bound}
+ V\left(\prod_{p<z_0}(1-\nu_p/p)+B^{-10}\right)+O(P_0^{1/5}).
+\end{equation}
+
+For one rough integer, take $\nu_p=1$. The product is
+$O(1/\log z_0)=O(B^{-1+\eps}\log B)$. Thus the number $R$ of
+$P_0$-rough integers in $(M,\tau M]$ satisfies
+\begin{equation}\label{eq:rough-one-point}
+ R\ll M B^{-1+\eps}\log B.
+\end{equation}
+Both error terms in \eqref{eq:rough-sieve-bound} are absorbed, since
+$M\ge P_0/\tau$.
+
+The one-point bound proves \eqref{eq:rough-sup}, because $w>M$.
+To estimate the fourth moment, extend $a_w=c_w/w$ by zero outside its
+support. Fourier orthogonality gives
+\[
+ \int_{\T}|Q_M(\alpha)|^4\,d\alpha
+ =\sum_k\left|\sum_w a_{w+k}\overline{a_w}\right|^2.
+\]
+There is no factor depending on $h$: multiplication by the nonzero
+integer $h$ preserves Haar measure on $\T$.
+Let $R_k$ count pairs $w,w+k\in(M,\tau M]$ that are both
+$P_0$-rough. Each inner absolute value is at most $R_k/M^2$.
+We next bound $R_k$ for $k\ne0$; only $|k|\le2M$ can contribute.
+The permitted interval for $w$ is an intersection of two intervals of
+length less than $M$. Sieve out the residues $0,-k$ modulo $p<z_0$.
+If $k$ is odd there are no pairs for large $B$, because both rough
+integers must be odd. If $k$ is even, the density product is
+\begin{align*}
+ \frac12\prod_{3\le p<z_0}(1-2/p)
+ \prod_{\substack{3\le p<z_0\\p\mid k}}
+       \frac{1-1/p}{1-2/p}
+ &\ll\frac1{(\log z_0)^2}
+       \exp\left(\sum_{p\mid k}\frac1p+O(1)\right).
+\end{align*}
+Here $(1-2/p)/(1-1/p)^2=1-1/(p-1)^2\le1$ and
+$\log((1-1/p)/(1-2/p))=1/p+O(p^{-2})$ for $p\ge3$.
+The singular factor is uniformly $O_{C_0}(\log B)$: indeed
+\[
+ \sum_{p\mid k}\frac1p
+ \le\sum_{p\le B}\frac1p+\frac{\log|k|}{B\log B}
+ \le\log\log B+O_{C_0}(1),
+\]
+since $\log|k|\le C_0B+\log2$. Applying
+\eqref{eq:rough-sieve-bound} proves
+\begin{equation}\label{eq:rough-two-point}
+ R_k\ll M B^{-2+2\eps}(\log B)^3\qquad(k\ne0).
+\end{equation}
+
+There are $O(M)$ nonzero differences, so their total contribution is
+$O(M^{-1}B^{-4+4\eps}(\log B)^6)$ by
+\eqref{eq:rough-two-point}. The term $k=0$ is at most
+$(R/M^2)^2\ll M^{-2}$, which is smaller than the same bound because
+$M\ge P_0/\tau$. This proves \eqref{eq:rough-fourth}.
+\end{proof}
+
+The fourth moment localizes the frequencies at which $Q_M$ is large.
+On the remaining frequencies its maximum is already small enough.
+We next put the target correlation into a form where these two facts
+can be combined with \eqref{eq:mrt-fixed-length}.
+
+\subsection{Two overlapping intervals and the frequency split}
+
+Let
+\[
+ D_1=\lceil M\rceil,\qquad D_2=(1+2h)D_1,
+\]
+and define
+\[
+ A_y(\alpha)=\sum_{y<n\le y+D_1}b_1(n)\mathrm e(\alpha n),
+ \qquad
+ B_y(-\alpha)=\sum_{y<m\le y+D_2}b_2(m)\mathrm e(-\alpha m).
+\]
+For each term of $Q_M$, integration in $\alpha$ imposes $m=n+hw$.
+Moreover $hw<2hD_1=D_2-D_1$, so
+\[
+ [n-D_1,n)\subseteq[n+hw-D_2,n+hw).
+\]
+Thus, if $L_Y(n)=|[0,Y]\cap[n-D_1,n)|$, orthogonality gives the exact
+identity
+\[
+ \int_0^Y\int_{\T}Q_M(\alpha)A_y(\alpha)B_y(-\alpha)
+                 \,d\alpha\,dy
+ =\sum_w\frac{c_w}{w}\sum_{n\ge1}b_1(n)b_2(n+hw)L_Y(n).
+\]
+For $D_1\le n\le Y$ one has $L_Y(n)=D_1$. The difference from
+$D_1\1_{1\le n\le Y}$ is supported on $O(D_1)$ integers at the two
+ends and has total absolute mass $O(D_1^2)$. Since
+$\sum_{M<w\le\tau M}1/w\ll1$, we obtain
+\begin{equation}\label{eq:analytic-overlap}
+ \begin{split}
+ &\frac1{D_1Y}\int_0^Y\int_{\T}
+        Q_M(\alpha)A_y(\alpha)B_y(-\alpha)\,d\alpha\,dy\\
+ &\hspace{10mm}=
+ \frac1Y\sum_{n\le Y}b_1(n)
+       \sum_{M<w\le\tau M}\frac{c_w}{w}b_2(n+hw)
+       +O(D_1/Y).
+ \end{split}
+\end{equation}
+The endpoint error tends to zero because $D_1$ is fixed before $Y$.
+
+Set $t_B=B^{-1-\eps}$ and
+$E_B=\{\alpha\in\T:|Q_M(\alpha)|>t_B\}$.
+By \Cref{lem:rough-multiplier},
+\begin{equation}\label{eq:rough-large-frequencies}
+ |E_B|\ll M^{-1}B^{8\eps}(\log B)^6,
+ \qquad \|Q_M\|_\infty\ll B^{-1+\eps}\log B.
+\end{equation}
+On $\T\setminus E_B$, Parseval and Cauchy--Schwarz give, for each $y$,
+\[
+ \int_{\T\setminus E_B}|Q_M A_y B_y|\,d\alpha
+ \le t_B\left(\int_{\T}|A_y|^2\right)^{1/2}
+             \left(\int_{\T}|B_y|^2\right)^{1/2}
+ \le t_B\sqrt{D_1D_2}.
+\]
+After integration in $y$ and division by $D_1Y$, the contribution is
+$O_h(B^{-1-\eps})$.
+
+On $E_B$, suppose first that $f_1$ is uniformly nonpretentious.
+Use $|B_y|\le D_2$, integrate first in $y$, and apply
+\eqref{eq:mrt-fixed-length} to $A_y$ at each fixed frequency. The
+normalized limsup is at most
+\begin{equation}\label{eq:analytic-large-bound}
+ O\left(\|Q_M\|_\infty\,|E_B|\,D_2
+                    \frac{\log\log D_1}{\log D_1}\right).
+\end{equation}
+If instead $f_2$ is the nonpretentious factor, use $|A_y|\le D_1$
+and apply the same estimate to $B_y$; this replaces $D_1$ by $D_2$
+inside the logarithms in \eqref{eq:analytic-large-bound}.
+In both cases Fubini's theorem is followed by the bound for the
+already integrated exponential sum, with a supremum over its fixed
+frequency. The frequency supremum has not been moved inside a
+short-interval integral.
+
+The range \eqref{eq:analytic-rough-range} implies, for $i=1,2$,
+\[
+ \frac{D_i}{M}\ll_h1,
+ \qquad
+ \frac{\log\log D_i}{\log D_i}
+ \ll_{C_0,h,\tau}B^{-1+\eps}\log B.
+\]
+Using \eqref{eq:rough-large-frequencies} in
+\eqref{eq:analytic-large-bound}, the large-frequency contribution is
+\[
+ O\left(B^{-2+10\eps}(\log B)^8\right)
+   =o(B^{-1-\eps}),
+\]
+since $11\eps<1$. Together with the small-frequency bound and
+\eqref{eq:analytic-overlap}, this proves
+\eqref{eq:analytic-rough-target}. The argument is uniform in the
+Fourier twists, since \eqref{eq:mrt-fixed-length} is uniform in them.
+
+\begin{proof}[Completion of \Cref{prop:analytic-centering}]
+Apply \eqref{eq:analytic-rough-target} with $c_w=c_w^{(u)}$ to each fixed
+$u$ in the expansion at the start of this section. Dividing its contribution by
+$X$ contributes the factor $A^{\omega_C(u)}/u$; the extracted factors
+$f_1(u)f_2(u)$ have modulus at most $1$.
+The two Fourier inversions cost at most $\|\widehat\phi\|_1^2$.
+At fixed $B$ all sums over $u,w$ are finite, and the uniform estimate
+therefore gives, by \eqref{eq:analytic-sum-u}, a total limsup
+$O(L_0B^{-1-\eps})$. Adding the earlier coprimality error
+$O(L_0J/P_0)$ proves \eqref{eq:analytic-centering-bound}.
+\end{proof}
+
+The analytic comparison required for
+\eqref{eq:analytic-contradiction} is now proved. It remains to establish
+the graph estimate; the following sections develop its closed-line
+bound and then pass from that bound to the long-average estimate.

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/cylinder-sieve.tex

@@ -0,0 +1,402 @@
+\section{A conditional cylinder sieve}\label{sec:cylinder-sieve}
+
+The restrictions $n+b_j\in Y$ still involve the residue coordinates
+outside the fixed set $\mathcal F$.  We replace their product by a sum
+of conditions from short lists of primitive specifications.  In each
+term the remaining coordinates are averaged before taking an absolute
+value; this preserves the cancellation needed in the trace estimate.
+The error will contain many specifications with distinct private prime
+coordinates, which supply enough arithmetic constraints to apply
+\Cref{lem:triangular-saving}.
+
+The intersection rank and the extraction of constraints from its
+witnesses adapt Pilatte's construction
+\cite[Definition~13.3 and Lemmas~13.4 and~13.6]{Pilatte26}.
+The associated truncation belongs to the composite-modulus sieve
+developed in \cite[Section~3]{HR21} and
+\cite[Appendix~B]{Pilatte26}.  We prove the finite-cylinder statement
+and its application here, including the effect of conditioning.
+
+\subsection{Truncating an intersection poset}
+
+Let $\Omega=\prod_{p\in P}\Omega_p$ be a finite product, with
+$|\Omega_p|\ge2$ for every $p$.  An \emph{exact cylinder} specifies one
+value at each coordinate in a subset of $P$ and imposes no condition at
+the other coordinates.  Its \emph{support} is the set of specified
+coordinates, and its \emph{width} is the size of that support.
+
+Let $\mathscr E$ be a finite family of proper, nonempty exact cylinders.
+Identify duplicate cylinders.  Let $\mathscr P$ consist of the nonempty
+intersections of subfamilies of $\mathscr E$, including the empty-family
+intersection $\widehat0=\Omega$.  Order $\mathscr P$ by reverse inclusion:
+$I\le J$ means $I\supseteq J$.  For $I\in\mathscr P$, define
+\begin{equation}\label{eq:cylinder-rank}
+ \operatorname{rk}(I)=
+ \max\left\{|\mathscr A|:\begin{array}{l}
+   \mathscr A\subseteq\{E\in\mathscr E:I\subseteq E\},\\
+   \text{each }E\in\mathscr A\text{ has a support coordinate}\\
+   \text{absent from every other member of }\mathscr A
+ \end{array}\right\}.
+\end{equation}
+The empty family is allowed in this maximum.  A coordinate as in
+\eqref{eq:cylinder-rank} is called \emph{private} to its member of
+$\mathscr A$.  All events in such a family have compatible prescribed
+values because they contain the nonempty intersection $I$.
+
+Write $\mu(\widehat0,I)$ for the M\"obius coefficients of this finite
+poset, characterized by
+\[
+ \sum_{I\le J}\mu(\widehat0,I)=\1_{J=\widehat0}.
+\]
+Their role is inclusion--exclusion with equal intersections collected
+into one term.  This is M\"obius inversion in the incidence algebra of a
+finite poset; see \cite[Section~3]{Rota1964}.  The chain expansion used
+below is the one in \cite[Section~3, Proposition~6]{Rota1964}.
+
+\begin{lemma}[Finite-cylinder truncation]\label{lem:cylinder-truncation}
+Suppose that the cylinders in $\mathscr E$ have width at most $w\ge1$.
+For an integer $t\ge1$, put $M=tw$, and let
+\[
+ \mathscr H_t=\{J\in\mathscr P:\operatorname{rk}(J)\ge t,
+       \ J\text{ is generated by at most }t\text{ members of }\mathscr E\}.
+\]
+Then the rank is nondecreasing on $\mathscr P$, every $I\in\mathscr P$
+is generated by at most $\operatorname{rk}(I)$ events, and
+\begin{equation}\label{eq:cylinder-width}
+ |\operatorname{supp}(I)|\le w\operatorname{rk}(I).
+\end{equation}
+For every $x\in\Omega$,
+\begin{equation}\label{eq:cylinder-truncation}
+ \left|\1_{x\notin\bigcup_{E\in\mathscr E}E}
+       -\sum_{\operatorname{rk}(I)<t}\mu(\widehat0,I)\1_I(x)\right|
+ \le 2^M\max(1,M^{M+1})\sum_{J\in\mathscr H_t}\1_J(x).
+\end{equation}
+Every coefficient retained in the sum on the left has absolute value
+at most $\max(1,M^{M+1})$.  Every intersection on the right has width
+at most $M$, even if its rank is greater than $t$.
+\end{lemma}
+
+\begin{proof}
+If $I\le J$, every event containing $I$ also contains $J$.  Thus every
+family admitted for the maximum defining $\operatorname{rk}(I)$ is
+admitted for $\operatorname{rk}(J)$, proving monotonicity.
+Choose an inclusion-minimal family generating $I$.  Each member must
+have a private coordinate: otherwise all its prescribed values already
+follow from the other members, since the prescriptions are compatible,
+and it can be removed.  Its size is therefore at most
+$\operatorname{rk}(I)$, proving \eqref{eq:cylinder-width} as well.
+
+Fix $J\in\mathscr P$ and let $m=|\operatorname{supp}(J)|$.  An element
+of the interval $[\widehat0,J]$ is determined by its support: its values
+must be the corresponding restrictions of those prescribed by $J$.
+Consequently this interval has at most $2^m$ elements.  Along a strict
+chain from $\widehat0$ to $J$, supports grow strictly, so there are at
+most $m$ steps.  A chain of $k$ steps is specified by recording, for
+each coordinate of $\operatorname{supp}(J)$, the step at which it
+enters.  There are at most $k^m$ such chains.  Expanding the inverse of
+the poset's upper triangular incidence matrix, or applying the M\"obius
+recursion repeatedly, gives the alternating sum over these chains.
+Thus, for $m\ge1$,
+\begin{equation}\label{eq:cylinder-mobius}
+ |\mu(\widehat0,J)|\le\sum_{k=1}^m k^m\le m^{m+1};
+\end{equation}
+the coefficient at $\widehat0$ is $1$.  This proves the asserted
+coefficient bound for intersections of rank below $t$.
+
+For $x\in\Omega$, let $J_x$ be the intersection of all events satisfied
+at $x$, with $J_x=\widehat0$ if none are satisfied.  An intersection
+$I\in\mathscr P$ contains $x$ if and only if $I\le J_x$.  Indeed any
+family generating $I$ then consists of events satisfied at $x$.
+It follows that
+\begin{equation}\label{eq:cylinder-full-sum}
+ \sum_{I\in\mathscr P}\mu(\widehat0,I)\1_I(x)
+    =\sum_{I\le J_x}\mu(\widehat0,I)
+    =\1_{J_x=\widehat0}.
+\end{equation}
+Since no forbidden event is the whole space, the last expression is
+precisely the avoidance indicator in \eqref{eq:cylinder-truncation}.
+
+If $\operatorname{rk}(J_x)<t$, monotonicity shows that the truncated
+sum contains the entire interval $[\widehat0,J_x]$, so it is exact.
+Suppose instead that $\operatorname{rk}(J_x)\ge t$.  Given any
+$I\le J_x$ of rank below $t$, successively intersect it with events
+satisfied at $x$ until reaching $J_x$.  Let $K$ be the first intersection
+whose rank is at least $t$, and let $K^-$ be its immediate predecessor.
+Replace the generating family for $K^-$ by an inclusion-minimal one.
+It has at most $t-1$ members.  Adjoining the event that produces $K$
+shows that $K$ is generated by at most $t$ events.  Thus
+\[
+       I\le K\le J_x,\qquad K\in\mathscr H_t,
+       \qquad |\operatorname{supp}(K)|\le M.
+\]
+This argument allows the rank to jump by more than one.  The
+re-minimization of the predecessor, rather than the length of the
+successive list, bounds the number of generators.
+
+Every satisfied low-rank $I$ is therefore below a satisfied member of
+$\mathscr H_t$.  For any one such member there are at most $2^M$
+possible predecessors $I$, by the interval bound already proved.
+The exact avoidance indicator is zero in the present case.  Bounding
+each retained coefficient by \eqref{eq:cylinder-mobius} now gives
+\eqref{eq:cylinder-truncation}.
+\end{proof}
+
+\subsection{Applying the truncation after conditioning}
+
+Return to a fixed numerical line $\boldsymbol d$.  Its fixed prime set
+$\mathcal F$ consists of its core labels and its repeated labels, as in
+\Cref{sec:thinning}.  In particular, core primes absent from the line
+are not added to $\mathcal F$.  Write
+$R_{\boldsymbol d}(n_{\mathcal F})$ for the indicator of the conditions
+retained in \Cref{prop:retained-lines}.
+
+Condition on $n_{\mathcal F}$ with $R_{\boldsymbol d}=1$.  At every
+$n+b_j$, take the primitive forbidden specifications compatible with
+these fixed residues.  Omit their fixed-coordinate tests, leaving
+exact cylinders on the coordinates $\mathcal S\setminus\mathcal F$.
+Each has width at most $LJ+1$.  There is no whole-space event: such an
+event would be a qualifying primitive specification using only fixed
+primes, excluded by the retained conditions.  Avoiding these residual
+events is exactly the condition $n+b_j\in Y$ for every $j$.
+
+For a list $\mathfrak L$ of primitive specifications attached at line
+indices $0,\ldots,\ell$, let $I_{\mathfrak L}(n)$ be the indicator that
+all its specifications qualify, including their fixed-coordinate
+tests.  The empty list has indicator $1$.  Let
+$\mathscr L_{<t_0}(\boldsymbol d)$ denote all ordered such lists of length
+less than $t_0$.  These are finite families at the fixed graph
+parameters.  We will use the bound
+\begin{equation}\label{eq:cylinder-budget}
+ M_0=t_0(LJ+1)=O(B^{1-\rho+4\eps}\log B),\qquad
+ M_0\log B=o(B).
+\end{equation}
+In particular, the coefficient and error factors in
+\Cref{lem:cylinder-truncation} are $\exp(O(M_0\log B))$.
+
+A low-rank intersection is generated by fewer than $t_0$ residual
+events.  Choose one realizing primitive specification for each event.
+The conjunction of their full qualification indicators equals the
+intersection indicator at the conditioned residues.  Different
+intersections cannot have the same chosen generating list, since that
+list determines its intersection.  We may therefore bound their sum
+by the sum over $\mathscr L_{<t_0}(\boldsymbol d)$ without a multiplicity
+loss.
+
+For an intersection in $\mathscr H_{t_0}$, choose at most $t_0$
+generating events and exactly $t_0$ events witnessing its rank.  The
+latter may be selected from a larger irredundant family; deleting other
+members preserves their private coordinates.  The witnessing events
+contain the intersection.  Consequently adjoining them to the
+generating list leaves its indicator unchanged.  After choosing
+realizing primitive specifications, each witness has a prime outside
+$\mathcal F$ that occurs in none of the other witness specifications.
+These choices may depend on $n_{\mathcal F}$, which is harmless:
+the preceding assertions and the uniform coefficient bounds apply
+pointwise at each fixed-coordinate assignment.
+
+We have thus reduced the truncation error to jointly qualifying lists
+of at most $2t_0$ specifications, including $t_0$ witnesses with private
+primes outside $\mathcal F$.  The next lemma shows that these error
+lists impose many triangular constraints on their numerical prime
+labels.
+
+\begin{lemma}[Constraints from private primes]\label{lem:cylinder-witnesses}
+Let $\boldsymbol d$ be a numerical closed line, and let $\mathcal F$ contain
+its core prime labels and all labels occurring at least twice on the
+line.  Suppose $t_0$ primitive specifications attached at line vertices
+qualify simultaneously.  Suppose that each has a prime outside
+$\mathcal F$ absent from every other specification in this family.
+For sufficiently large $B$, their numerical labels satisfy a triangular
+system on at least $B^{2\eps}$ distinct selected prime variables of the
+kind in \Cref{lem:triangular-saving}.  Each constraint uses one intrinsic
+suffix displacement, or the difference of two witness prefixes and a
+line segment.  Its nonzero expressions have logarithmic size
+$O(B\log B)$ even when numerical label-size restrictions are dropped.
+\end{lemma}
+
+\begin{proof}
+Order the witnesses by nondecreasing attachment indices
+$k_1,\ldots,k_{t_0}$; break ties arbitrarily.  For witness $j$, choose a
+private prime $q_j\notin\mathcal F$, write $p_j$ for its extra prime,
+and choose a tail prime $r_j$ as in
+\Cref{lem:primitive-properties}.  Thus $r_j$ occurs only in its final
+constant block.  That lemma supplies two coefficient facts:
+\begin{enumerate}[label=(\roman*)]
+\item the intrinsic suffix displacement divisible by $p_j$ is linear
+in a tail-only prime, with coefficient nonzero modulo $p_j$;
+\item if a prime occurs before the constant tail, its coefficient in
+the prefix displacement immediately preceding that tail is nonzero
+modulo $r_j$.
+\end{enumerate}
+All private primes $q_j$ are distinct.  Each occurs at most once as a
+label on the original line, because it is outside $\mathcal F$.
+We divide into three cases to control every dependence on another
+selected variable.
+
+\paragraph{Private primes in intrinsic suffix constraints.}
+Suppose at least $t_0/10$ witnesses have $q_j=p_j$ or have $q_j$
+occurring only in the constant tail.  In the first situation $q_j$
+divides the intrinsic suffix displacement, which is nonzero by
+simplicity and does not involve $q_j$.  In the second situation use
+that suffix constraint as a linear congruence in $q_j$ modulo $p_j$;
+its coefficient is nonzero by (i).  The private-prime property excludes
+every other selected $q$ from the witness, including from its modulus.
+These constraints have no dependencies on the other selected
+variables and hence form a triangular system.
+
+\paragraph{Tail primes first or last appearing among the witnesses.}
+Suppose at least $t_0/10$ witnesses have $r_j$ absent from every earlier
+witness.  Select these tail primes in increasing witness order.  They
+are distinct: equality between two would make the latter prime occur
+in an earlier witness.  Use the intrinsic suffix congruence for each
+selected $r_j$.  No later selected tail prime occurs in this witness,
+by its defining absence property.  This also excludes later selected
+variables from the modulus $p_j$.  Thus each constraint uses only its
+own variable, earlier selected variables, and nonselected variables,
+and its coefficient is nonzero modulo $p_j$ by (i).
+If instead at least $t_0/10$ witnesses have $r_j$ absent from every later
+witness, the same argument in decreasing witness order applies.
+
+\paragraph{Comparing two activities of a shared tail prime.}
+If none of these cases applies, fewer than $3t_0/10$ witnesses have
+been excluded by their three respective conditions.  Retain at least
+$t_0/2$ witnesses for which $q_j$ occurs before the tail and $r_j$
+appears in both an earlier and a later witness.  Since $q_j$ occurs
+at most once on the original line, it is absent either from all steps
+$1,\ldots,k_j$ or from all steps $k_j+1,\ldots,\ell$.  At least
+$t_0/4$ of the retained witnesses satisfy the same one of these two
+absence conditions.
+
+First consider the family absent from steps $1,\ldots,k_j$, ordered
+increasingly by attachment index.  For its witness $j$, choose an
+earlier witness $i=i(j)$ using $r_j$.  In witness $i$, choose a vertex
+at which $r_j$ is active: its starting vertex if $r_j$ is its extra
+prime, or the origin of an edge carrying $r_j$ otherwise.  Let $T_i$
+be this vertex's offset relative to that witness's start.  Let $P_j$
+be the prefix displacement of witness $j$ immediately preceding its
+constant tail.  Joint qualification implies
+\begin{equation}\label{eq:cylinder-comparison}
+ b_{k_j}-b_{k_i}+P_j-T_i\equiv0\pmod{r_j}.
+\end{equation}
+The coefficient of $q_j$ in this congruence is exactly its coefficient
+in $P_j$: privateness excludes it from witness $i$, and the selected
+absence condition excludes it from the line segment between
+$k_i$ and $k_j$.  This coefficient is nonzero modulo $r_j$ by (ii).
+
+Now take a later selected private variable $q_s$.  It occurs in
+neither witness $i$ nor witness $j$, by privateness.  Moreover
+$k_s\ge k_j$, and its own absence condition excludes it from every
+line step through $k_s$, hence from the segment in
+\eqref{eq:cylinder-comparison}.  Thus no later selected variable occurs
+in that congruence.  The modulus $r_j$ is not any selected private
+prime: it differs from $q_j$ because $q_j$ occurs before the tail,
+and it differs from all other $q_s$ by privateness.  This proves
+triangularity in increasing order.  Equal attachment indices merely
+make the corresponding line segment empty.
+
+For the family absent from steps after $k_j$, order the witnesses
+decreasingly and compare witness $j$ to a later witness containing
+$r_j$.  Equation \eqref{eq:cylinder-comparison} has the same form, now
+with its line segment after $k_j$.  A later selected variable in this
+decreasing order has attachment $k_s\le k_j$ and is absent from all
+steps after $k_s$, so it is absent from that segment.  Privateness
+excludes it from the two witness prefixes.  Fact (ii) again supplies
+the nonzero coefficient.  This gives a triangular system in decreasing
+order.  The attained comparison displacement may equal zero; that
+causes no difficulty, since we use an invertible linear congruence,
+not a divisibility constraint with the selected prime as modulus.
+
+Every case provides at least $t_0/20$ selected variables for sufficiently
+large $B$, allowing for integer parts.  This is greater than
+$B^{2\eps}$.  Each label is a squarefree product of at most $J$ primes
+at most $e^B$, so its logarithm is at most $BJ=O(B\log B)$ even without
+the divisor-bin restrictions.  A suffix or a comparison above uses
+$O(\ell+L)$ such terms.  Every nonzero expression consequently has
+logarithmic size $O(B\log B)$.  The constraint choices are specified
+by witness and line positions and prime slots, within the enumeration
+allowed in \Cref{lem:triangular-saving}.  All its hypotheses are now
+verified.
+\end{proof}
+
+\begin{proposition}[Reduction to short qualification lists]
+\label{prop:cylinder-reduction}
+For the retained line and fixed-coordinate data of
+\Cref{prop:retained-lines}, there is a factor
+$\mathfrak C_B=\exp(O(M_0\log B))$ such that
+\begin{align}
+ &\sum_{\boldsymbol d}\E_{\mathcal F}
+   \left[R_{\boldsymbol d}
+     \left|\E_{\mathcal S\setminus\mathcal F}
+          K_{\boldsymbol d}(n)\prod_{j=0}^{\ell}\1_Y(n+b_j)\right|\right]
+       \notag\\
+ &\quad\le \mathfrak C_B\sum_{\boldsymbol d}\E_{\mathcal F}
+   \left[R_{\boldsymbol d}
+     \sum_{\mathfrak L\in\mathscr L_{<t_0}(\boldsymbol d)}
+       \left|\E_{\mathcal S\setminus\mathcal F}
+                      K_{\boldsymbol d}(n)I_{\mathfrak L}(n)\right|\right]
+       +O\!\left(e^{-B^{1+\eps/2}}\right).
+       \label{eq:cylinder-reduction}
+\end{align}
+Here each sum runs over numerical closed lines, $\mathcal F$ is the
+fixed prime set of that line, and every list indicator imposes full
+qualification of its specifications.
+\end{proposition}
+
+\begin{proof}
+At fixed $\boldsymbol d,n_{\mathcal F}$ with $R_{\boldsymbol d}=1$, apply
+\Cref{lem:cylinder-truncation} to the residual cylinders with
+$t=t_0$ and $w=LJ+1$.  Multiply the identity with its pointwise error
+bound by $K_{\boldsymbol d}$ and integrate the residual coordinates.
+For the low-rank sum, take the absolute value after each such integral
+and use the generating lists already constructed.  The coefficient
+bound gives the main term on the right of
+\eqref{eq:cylinder-reduction}.
+
+For the error, take the absolute value of $K_{\boldsymbol d}$ inside the
+integral and use the generating and witnessing lists described above.
+After integrating the fixed coordinates, their indicators require
+joint qualification of at most $2t_0$ specifications, including the
+$t_0$ witnesses with private primes outside $\mathcal F$.
+We may drop the retained-data restriction in this positive bound.
+\Cref{lem:cylinder-witnesses} supplies at least $B^{2\eps}$ triangular
+constraints for each nonzero contribution.
+
+Refine the error count by the lit or unlit status of every line
+occurrence.  Joint qualification fixes one residue for each distinct
+prime used by the specifications, or is inconsistent and contributes
+zero.  For each refinement, \Cref{lem:crude-count} gives
+$A^{\ell J}\prod_p p^{-1}$, with the product over distinct prime labels,
+rather than charging a shared residue repeatedly.  The line and lists
+have at most
+\[
+       \ell J+2t_0(LJ+1)=O(B\log B)
+\]
+prime slots.  Their equality patterns, occurrence statuses and all
+choices of witness constraints have cost $\exp(O(B\log^2 B))$.
+These are exactly the
+positive majorant and description budget required by
+\Cref{lem:triangular-saving}.  Its reverse elimination applies to the
+system just supplied by \Cref{lem:cylinder-witnesses} and gives
+$O(e^{-B^{1+\eps/2}})$.
+The additional factor $\mathfrak C_B$ is absorbed by the same estimate, since
+\eqref{eq:cylinder-budget} gives $\log \mathfrak C_B=o(B)$ whereas the saving
+before this final simplification is $\exp(-\Omega(B^{1+\eps}))$.
+This proves the proposition.
+\end{proof}
+
+The reduction has preserved a signed average at the unfixed coordinates.
+For its use in the next section, it is useful to distinguish
+$\mathcal C'=\mathcal C\setminus\mathcal F$ and
+$\mathcal Z'=\mathcal Z\setminus\mathcal F$.  For each list,
+\begin{equation}\label{eq:cylinder-core-order}
+ \left|\E_{\mathcal S\setminus\mathcal F}
+              K_{\boldsymbol d}I_{\mathfrak L}\right|
+ \le \E_{\mathcal C'}
+       \left|\E_{\mathcal Z'}K_{\boldsymbol d}I_{\mathfrak L}\right|.
+\end{equation}
+This is only Fubini's theorem and the triangle inequality.  In the
+inner average all core coordinates are held fixed, so the cutoffs are
+fixed there; the free center coordinates still retain their signs.
+The core coordinates can subsequently be integrated against positive
+bounds.  In particular, \eqref{eq:cylinder-core-order} does not enlarge
+$\mathcal F$ or change the private-coordinate rank used in the sieve.

preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/build/qualitative/graph-setup.tex

@@ -0,0 +1,120 @@
+\section{The weighted divisor graph}\label{sec:graph-setup}
+
+We begin the qualitative argument with the finite-scale graph estimate
+that will rule out a correlation bias. Its test functions are arbitrary bounded sequences;
+no multiplicativity enters its statement or proof. The arithmetic
+application follows in \Cref{sec:analytic-transfer}.
+
+Fix an integer $h\ge1$. Throughout the graph construction use
+\begin{equation}\label{eq:graph-constants}
+ \eps=10^{-4},\quad \eta=\eps/100,\quad \rho=1/20,\quad
+ \kappa=400/\eta,\quad A=\exp(2\kappa),\quad c_*=\eps/100.
+\end{equation}
+The scale $B$ will be sufficiently large. Define two finite prime sets by
+\begin{equation}\label{eq:prime-bands}
+ \begin{split}
+ \mathcal C&=\{p:B^{1-\eta}<\log p\le B\},\\
+ \mathcal Z&=\{p:B^{1-\eps}<\log p\le B^{1-\eta}\},\qquad
+ \mathcal S=\mathcal C\cup\mathcal Z,\quad P_0=\exp(B^{1-\eps}).
+ \end{split}
+\end{equation}
+We refer to $\mathcal C$ as the core band and $\mathcal Z$ as the center
+band. For $a\in\{C,Z\}$, set $\mathcal P_C=\mathcal C$ and
+$\mathcal P_Z=\mathcal Z$, and put
+\[
+ v_a=\sum_{p\in\mathcal P_a}\frac1p,\qquad
+ A_C=A,\quad A_Z=1,\quad \beta_a=(1+A_a)^{-1},\qquad
+ \theta=\beta_Z^2=\tfrac14.
+\]
+Mertens' prime harmonic estimate gives
+\[
+ v_C=(\eta+o(1))\log B,\qquad
+ v_Z=(\eps-\eta+o(1))\log B.
+\]
+
+Fix also $\tau\in(1,2)$, $C_0\ge1$, and $T>0$. An admissible divisor
+family $\mathcal D$ is any collection of squarefree products of primes
+in $\mathcal S$ such that
+\begin{equation}\label{eq:divisor-family}
+ \begin{gathered}
+ d>1,\qquad H<d\le\tau H,\qquad 1\le H\le\exp(C_0B),\\
+ \omega(d)\le J:=\lceil C_0\log B\rceil.
+ \end{gathered}
+\end{equation}
+Here $\omega$ counts distinct prime factors. For any integer $x$, define
+$\omega_a(x)=\#\{p\in\mathcal P_a:p\mid x\}$, and write
+$d_C=\prod_{p\mid d,\ p\in\mathcal C}p$.
+For each $d\in\mathcal D$, let $w_d:\Z\to[0,1]$ be any function of
+all the residues $x\bmod p$, $p\in\mathcal C$, with support restricted by
+\begin{equation}\label{eq:core-cutoff}
+ w_d(x)=0\quad\hbox{unless}\quad
+ k_d(x):=\#\{p\in\mathcal C:p\mid x,\ p\nmid d\}
+       \ge v_C-T\sqrt{v_C}.
+\end{equation}
+The cutoff need not factor over primes.
+
+For $y=x\pm hd$, define the real edge weight
+\begin{equation}\label{eq:graph-kernel}
+ g_d(x,y)=w_d(x)w_d(y)A^{\omega_C(d)}\1_{d_C\mid x}
+       \prod_{\substack{p\mid d\\p\in\mathcal Z}}
+             \left(\1_{p\mid x}-\frac\theta p\right).
+\end{equation}
+For each $p\mid d$, the residues of $x$ and $y$ agree modulo $p$.
+Consequently $g_d(x,y)=g_d(y,x)$. The uncentered core factors enforce
+$d_C\mid x$, whereas the center factors can have either sign.
+
+The vertex weight and its mean are
+\begin{equation}\label{eq:graph-weights}
+ W(x)=\prod_{a\in\{C,Z\}}\beta_a^{-\omega_a(x)},\qquad
+ L_0=\prod_{a\in\{C,Z\}}\prod_{p\in\mathcal P_a}(1+A_a/p).
+\end{equation}
+Let $Q_B=\prod_{p\in\mathcal S}p$. We use the uniform probability space
+$\Z/Q_B\Z$, identified by the Chinese remainder theorem with the product
+of the uniform spaces $\Z/p\Z$. Its expectation is denoted by $\E$;
+$\E_U$ averages only coordinates indexed by $U\subseteq\mathcal S$.
+For a uniform residue $n$,
+\begin{equation}\label{eq:weight-moments}
+ \E W(n)=L_0,\qquad
+ \E W(n)^2
+ =\prod_a\prod_{p\in\mathcal P_a}
+        \left(1+\frac{\beta_a^{-2}-1}{p}\right)
+ \le \exp\bigl((A^2+2A)v_C+3v_Z\bigr)=B^{O_A(1)}.
+\end{equation}
+Both identities follow by independence of the prime coordinates.
+The exponent in the last bound is fixed, although large.
+
+\begin{proposition}[Divisor graph estimate]\label{prop:graph-estimate}
+Fix $h\ge1$, $1<\tau<2$, $C_0\ge1$, and $T>0$, and use
+\eqref{eq:graph-constants}--\eqref{eq:graph-weights}. For every sufficiently
+large $B$, every admissible family $\mathcal D$, all permitted cutoffs
+$w_d$, all coefficients $|a_d|\le1$, and all functions
+$F,G:\N\to\mathbb D$,
+\begin{equation}\label{eq:graph-estimate}
+ \limsup_{X\to\infty}\frac1X
+ \left|\sum_{1\le x\le X}\sum_{d\in\mathcal D}
+ a_dF(x)G(x+hd)g_d(x,x+hd)\right|
+ \ll L_0 B^{-1-c_*/2}.
+\end{equation}
+The implied constant and the threshold for $B$ may depend on
+$h,\tau,C_0,T$, and the fixed constants in \eqref{eq:graph-constants},
+but not on $H,\mathcal D,w_d,a_d,F,G$. All graph parameters are fixed
+when $X$ tends to infinity.
+\end{proposition}
+
+The proof occupies \Cref{sec:paths,sec:thinning,sec:cylinder-sieve,sec:trace-count,sec:localization}.
+The two bands have different roles in that proof. The core weights and
+cutoffs give savings from the short divisor interval and from the
+lower bound on $k_d(x)$ in \eqref{eq:core-cutoff}.
+The center factors provide further cancellation in closed-walk products,
+where each edge weight is divided by $W$ at its starting vertex.
+The relation $\theta=\beta_Z^2$ matches the
+centering constant to that normalization; its precise use is established
+in \Cref{sec:trace-count}, once the relevant walk configurations have
+been defined.
+
+The comparison scale is $L_0/B$. In the next section, one of $O(B)$
+short multiplicative intervals captures enough divisor weight to turn
+any persistent correlation bias into a sum of this size with ordinary
+divisibility indicators. The extra factor $B^{-c_*/2}$ in
+\eqref{eq:graph-estimate} will contradict that
+bias once the analytic cost of centering has been bounded.