Sync from openai/math@faee2cf485a8 (part 31)

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/README.md

@@ -1,20 +1,20 @@
-# [Incompressible Box Transport and Finite Computation](manuscript.pdf)
-
-**Author:** OpenAI  
-**Date:** October 6, 2026
-
-## Citation
-
-```bibtex
-@misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026,
-  author = {{OpenAI}},
-  title = {{Incompressible Box Transport and Finite Computation}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026}},
-  year = {2026}
-}
-```
-
-## Version note
-
-This version supplies effective torus-projection and common-clock decay estimates, including the projected-force and pressure bounds; see the [previous version](../Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf).
+# [Incompressible Box Transport and Finite Computation](manuscript.pdf)
+
+**Author:** OpenAI  
+**Date:** October 6, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026,
+  author = {{OpenAI}},
+  title = {{Incompressible Box Transport and Finite Computation}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf}{OAI:Incompressible-Box-Transport-and-Finite-Computation-October-6-2026}},
+  year = {2026}
+}
+```
+
+## Version note
+
+This version supplies effective torus-projection and common-clock decay estimates, including the projected-force and pressure bounds; see the [previous version](../Incompressible-Box-Transport-and-Finite-Computation-September-27-2026/manuscript.pdf).

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/analysis.tex

@@ -1,282 +1,282 @@
-\section{Analytic classes and changes of physical coordinates}\label{sec:model}
-The geometric constructions prescribe a smooth velocity first. This section
-records exactly which other solutions are excluded by the energy argument.
-Pressure assumptions are kept separate: continuity in a pressure norm, a
-norm at each individual time, and an integrable pressure gradient are
-different conditions.
-
-Write $\mathcal F_\nu[U]=U_t+(U\cdot\nabla)U-\nu\Delta U$.
-All noncompact results below use $\mathbb R^3$; their velocities and forces
-have one fixed compact spatial support unless the individual statement
-says otherwise. This property concerns the constructed solution, not a
-competing solution. On $[0,T]$, write $CH^k=C([0,T];H^k(\mathbb R^3))$
-and $C^1L^2=C^1([0,T];L^2(\mathbb R^3))$. Spatial constants in pressure
-may depend on time.
-
-\begin{lemma}[The pressure term without a pressure norm]\label{lem:b-gradient}
-If $w,g\in L^2(\mathbb R^3;\mathbb R^3)$ satisfy
-$\operatorname{div}w=0$ and $\operatorname{curl}g=0$ in distributions, then
-$\langle w,g\rangle_{L^2}=0$. In particular, any distributional gradient
-$g=\nabla p\in L^2$ has zero pairing with $w$, without an assumption that
-$p\in L^2$.
-\end{lemma}
-\begin{proof}
-An $L^2$ field is a tempered distribution. The differential identities,
-initially tested on compactly supported smooth functions, extend to
-Schwartz tests by multiplying by cutoffs tending to one and using $L^2$
-convergence of the tests and their first derivatives. Fourier transformation
-gives $\xi\cdot\widehat w=0$ and
-$\xi_j\widehat g_k-\xi_k\widehat g_j=0$. These are identities of locally
-integrable functions, so they hold almost everywhere. Away from $\xi=0$,
-$\widehat g$ is parallel to $\xi$ and $\widehat w$ is perpendicular to it.
-Their Hermitian product is zero. The exceptional point has measure zero;
-Plancherel proves the assertion.
-\end{proof}
-
-\begin{lemma}[Residual realization and separate comparison classes]
-\label{lem:b-comparison}\label{lem:p2-realization}
-Let $\nu>0$ and let $U$ be a prescribed smooth divergence-free velocity
-on $[0,\infty)\times\mathbb R^3$, with $U(0)=0$. On each finite interval,
-assume $U\in CH^2\cap C^1L^2$ and $U,\nabla U$ are bounded. Put
-$f=\mathcal F_\nu[U]$. Then $(U,0)$ is a solution. Its velocity is unique
-in each of the following separately stated comparison classes, with the
-same force and initial data:
-\begin{enumerate}[label=(\roman*),itemsep=2pt]
-\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v$, and a
-pressure representative $p\in CH^1$.
-\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$,
-and an $H^1$ pressure representative at each time, with no required time
-continuity in that pressure norm.
-\item Strong solutions with $v\in CH^4\cap C^1H^2$ and distributional
-$\nabla p\in CL^2$.
-\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$,
-and a pressure representative $p\in CL^2$.
-\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$,
-without an additional integrability or normalization condition on pressure.
-\end{enumerate}
-The pressure gradient is unique. An $L^2$ pressure representative, when
-specified, is zero; otherwise pressure is determined up to a function of
-time. The reference material flow is a smooth diffeomorphism at every
-finite time and exists for every finite forward time.
-
-On a flat torus $\T_L^3$, the same conclusion holds in the classical class
-where $v,v_t$, spatial derivatives through order two, and $p,\nabla p$ are
-continuous on finite closed cylinders, with periodic mean-zero pressure.
-No Sobolev conditions at infinity are involved in that assertion.
-\end{lemma}
-\begin{proof}
-Substitution proves existence of the prescribed solution. Set $w=v-U$.
-The difference equation is
-\[
- w_t+(v\cdot\nabla)w+(w\cdot\nabla)U
-    =\nu\Delta w-\nabla p,\qquad \operatorname{div}w=0.
-\]
-The stated velocity regularity implies $w\in C^1L^2\cap CH^2$, so
-$\frac d{dt}\|w\|_2^2=2\langle w,w_t\rangle$ at each time. In case
-(iii), the stronger assumptions imply these inclusions. They also imply
-bounded velocity and spatial derivatives through order two: for $j\le2$,
-Cauchy--Schwarz in Fourier space uses the finite integral
-$\int_{\mathbb R^3}|\xi|^{2j}(1+|\xi|^2)^{-4}\,d\xi$.
-
-For the transport term, insert a cutoff $\chi_R$ that equals one on the
-ball of radius $R$, is supported in the ball of radius $2R$, and satisfies
-$|\nabla\chi_R|\le C/R$. Its boundary error is at most
-$CR^{-1}\|v\|_\infty\|w\|_2^2$, which tends to zero. Diffusion is
-integrated by parts in $H^2$; alternatively its cutoff error is bounded by
-$CR^{-1}\|w\|_2\|\nabla w\|_2$. The deformation term is bounded by
-$\|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2$.
-
-We justify the pressure pairing according to the stated alternatives.
-For (i) and (ii), at each fixed time approximate the chosen $H^1$ pressure
-by compactly supported smooth functions; distributional incompressibility
-and $w\in L^2$ give $\langle\nabla p,w\rangle=0$. Thus no continuity
-of $p$ in time is used in (ii). For (iv), integration by parts on the
-cutoff gives a pressure boundary term bounded by
-$CR^{-1}\|p\|_2\|w\|_2$, also tending to zero. For (iii), apply
-Lemma~\ref{lem:b-gradient} directly. Finally, in (v) the equation itself
-gives at each time
-\[
- \nabla p=f-v_t-(v\cdot\nabla)v+\nu\Delta v\in L^2,
- \qquad \|(v\cdot\nabla)v\|_2\le\|v\|_\infty\|\nabla v\|_2.
-\]
-Here $f\in L^2$ follows from the identical estimate for $U$ and its stated
-time and space regularity. The smooth pressure gradient is curl free, so
-Lemma~\ref{lem:b-gradient} again cancels the term. This does not choose an
-$L^2$ representative of the pressure.
-
-In every case the resulting pointwise energy inequality is
-\[
- \frac12\frac d{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2
- \le \|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2.
-\]
-The coefficient is bounded on each finite interval and $w(0)=0$, so an
-integrating factor gives $w=0$. The pressure has already disappeared before
-time integration; the proof imposes no unlisted temporal pressure norm.
-Substitution then gives $\nabla p=0$. A spatially constant $L^2$ function
-on $\mathbb R^3$ is zero, which proves the normalization assertion.
-
-On the torus every integration by parts is periodic and has no boundary
-term. The specified classical regularity justifies norm differentiation,
-diffusion, and pressure cancellation. The same energy inequality gives
-uniqueness and the mean-zero normalization fixes pressure. In either domain,
-the smooth reference velocity and its bounded spatial derivative give
-unique material trajectories; bounded speed prevents finite-time escape.
-Solving backwards on a finite interval gives the inverse flow, and smooth
-ODE dependence gives its smoothness.
-\end{proof}
-
-Cases (i)--(v) identify the precise hypotheses used by the applications;
-their presence in a common lemma does not replace a theorem's declared
-comparison class by a different one. In particular, a pressure norm is not
-silently imposed in a gradient-only application. The proof is the classical
-energy comparison for prescribed smooth solutions \cite[Section~18]{Leray1934},
-with its noncompact pressure pairings made explicit.
-
-For the flow $\Phi_t$, differentiation of its trajectory equation gives
-\[
- U(t)=(\partial_t\Phi_t)\circ\Phi_t^{-1},\qquad
- \partial_{tt}\Phi_t(a)=
- (f-\nabla p+\nu\Delta U)(t,\Phi_t(a)).
-\]
-These identities describe the material form of the same constructed solution;
-they do not add an independent existence claim.
-
-\begin{lemma}[Physical scaling]\label{lem:p2-chart}
-For $a,b>0$, $x=x_0+ay$, and $s=bt$, let
-$U(t,x)=abV(bt,(x-x_0)/a)$. Then trajectories are transformed by this change
-of variables, incompressibility is preserved, and at the prescribed physical
-viscosity $\nu$,
-\[
- \mathcal F_\nu[U](t,x)=ab^2\left[
- V_s+(V\cdot\nabla_y)V-\frac{\nu}{a^2b}\Delta_yV
- \right](bt,(x-x_0)/a).
-\]
-\end{lemma}
-\begin{proof}
-The trajectory equation gives $\dot x=abV$; differentiating in space gives
-$\operatorname{div}_xU=b\operatorname{div}_yV$. The three residual terms
-scale by $ab^2,ab^2$, and $b/a$, respectively, proving the formula.
-\end{proof}
-Thus a side-$10$ torus is kept as a physical side-$10$ torus. If a chart
-is rescaled, the residual is recomputed with the fixed $\nu$; the equation
-is not identified with a differently normalized viscosity.
-
-\begin{proposition}[Projection on a flat torus]\label{prop:projection-l2}
-Let $g$ be a smooth force on $[0,\infty)\times\T_L^3$ with zero
-spatial mean at each time. There is a unique smooth mean-zero potential
-$\phi$ satisfying $\Delta\phi=\operatorname{div}g$. Put
-$Qg=\nabla\phi$ and $Pg=g-Qg$. Then $Pg$ is solenoidal and mean zero,
-and replacing $(g,p)$ by $(Pg,p-\phi)$ preserves the velocity and the
-pressure normalization.
-
-Use normalized spatial measure, frequencies $\xi_k=2\pi k/L$, and norms
-\[
- \|G\|_{H^s}^2=\sum_{k\in\Z^3}(1+|\xi_k|^2)^s|\widehat G(k)|^2,
- \qquad
- \|G\|_{C^r}=\max_{|\beta|\le r}\|\partial_x^\beta G\|_\infty.
-\]
-For all integers $h,r\ge0$, at each time,
-\begin{equation}\label{eq:projection-estimate}
- \begin{aligned}
- \max\{\|\partial_t^h Pg\|_{C^r},\|\partial_t^h Qg\|_{C^r}\}
-     &\le S_L\|\partial_t^h g\|_{H^{r+2}},\\
- \|\partial_t^h\phi\|_{C^{r+1}}
-     &\le S_Ld_L\|\partial_t^h g\|_{H^{r+2}},
- \end{aligned}
-\end{equation}
-where $S_L=(1+52(L/(2\pi))^4)^{1/2}$ and
-$d_L=(1+(L/(2\pi))^2)^{1/2}$.
-
-Uniform bounds, temporal $L^2$ bounds on spatial suprema, and qualitative
-uniform decay of all mixed derivatives pass from $g$ to $Pg,Qg,\phi$.
-A quantitative envelope for time order $h$ and output spatial order $r$
-passes to $Pg,Qg$ through order $r$ and to $\phi$ through order $r+1$
-when the input derivatives through spatial order $r+2$ satisfy that
-envelope. The constants may depend on the derivative order.
-Periodicity and stationarity, including either property on a time tail,
-are preserved on their respective intervals.
-
-For computable $L>0$, effective mixed-derivative evaluation and effective
-derivative bounds for $g$ on finite time slabs give effective evaluation
-and such bounds for $Pg,Qg,\phi$. Corresponding supplied global input
-bounds give effective global bounds in each assertion above.
-\end{proposition}
-\begin{proof}
-For $k\ne0$ define
-\[
- \widehat\phi(t,k)=-\frac{i\xi_k\cdot\widehat g(t,k)}{|\xi_k|^2},
- \qquad \widehat\phi(t,0)=0.
-\]
-The nonzero-frequency symbols of $Q$ and $P$ are complementary orthogonal
-projections. Their $H^s$ operator norms are at most one, while the
-multiplier for $\phi$ has norm at most $d_L$ from $H^s$ to $H^{s+1}$.
-For $|\beta|\le r$, Cauchy--Schwarz bounds the absolute sum of the
-Fourier series for $\partial_x^\beta G$ by
-\[
- \left(\sum_{k\in\Z^3}(1+|\xi_k|^2)^{-2}\right)^{1/2}
- \|G\|_{H^{r+2}}.
-\]
-The shell $|k|_\infty=n\ge1$ contains $24n^2+2\le26n^2$ points.
-Since $\sum_{n\ge1}n^{-2}\le2$, the factor is at most $S_L$.
-For the tail $|k|_\infty>N\ge1$, it is at most
-\begin{equation}\label{eq:projection-tail}
- \left(26(L/(2\pi))^4/N\right)^{1/2}.
-\end{equation}
-Applying these estimates to the multipliers proves
-\eqref{eq:projection-estimate}, including the estimate for $\phi$ with
-one additional spatial derivative. On compact time slabs these tails
-converge uniformly for every mixed derivative. Thus the series define
-smooth fields and may be differentiated term by term. The Fourier
-identities give the Poisson equation, uniqueness of its mean-zero
-solution, and $\operatorname{div}Pg=0$. Also
-$-\nabla(p-\phi)+Pg=-\nabla p+g$.
-
-For every integer $n\ge0$, the multinomial expansion and Parseval give
-\begin{equation}\label{eq:projection-derivative-bounds}
- \|G\|_{H^n}^2
- \le\sum_{|\beta|\le n}
-       \frac{n!}{(n-|\beta|)!\,\beta!}
-       \|\partial_x^\beta G\|_\infty^2.
-\end{equation}
-Use $n=r+2$ and $G=\partial_t^h g$. This finite sum proves the uniform
-bounds and qualitative decay assertions. If its input suprema are
-bounded by $C_{h,\beta}a_h(t)$, it gives the same envelope $a_h$ for
-the output, with the constants specified by
-\eqref{eq:projection-estimate}--\eqref{eq:projection-derivative-bounds}.
-Integrating the squared estimates proves the temporal $L^2$ assertion
-from the corresponding input norms. The operators act at each time,
-so they preserve the stated temporal symmetries.
-
-For effective evaluation, the supplied derivative bounds and
-\eqref{eq:projection-tail} give a computable Fourier truncation on each
-finite time slab. The finitely many coefficient integrals are computed
-by Riemann sums. After scaling the torus to $[0,1]^3$, the integrand
-for $\widehat G(k)$ has Lipschitz bound
-\[
- L\sqrt3\max_j\|\partial_{x_j}G\|_\infty
-       +2\pi|k|\|G\|_\infty.
-\]
-This gives an effective quadrature error and hence evaluation of every
-mixed derivative of all three fields. The displayed finite sums also
-compute output bounds from the supplied input bounds.
-\end{proof}
-The estimates apply to the pressure shift $\phi$; in the constructions
-with original pressure zero, the projected pressure is $-\phi$.
-Projection generally changes pressure. It supplies no compact-support
-conclusion on Euclidean space and is distinct from the direct zero-pressure
-solenoidal shear construction.
-
-\paragraph{Effective flat profiles.}
-For later cutoff formulas put
-\begin{equation}\label{eq:p2-step}
- \rho(s)=\begin{cases}e^{-1/s},&s>0,\\0,&s\le0,\end{cases}
- \qquad \sigma(s)=\frac{\rho(s)}{\rho(s)+\rho(1-s)}.
-\end{equation}
-Positive-side derivatives of $\rho$ are polynomials in $1/s$ times
-$e^{-1/s}$ and extend flatly at zero. The estimate
-$v^me^{-v}\le(m+k)!v^{-k}$ gives effective error bounds near the seam,
-and $\rho(s)+\rho(1-s)\ge e^{-2}$. Products, translations, rescalings,
-and derivatives therefore yield effective cutoffs with all requested
-derivative bounds. Periodization uses zero collars; at approximate real
-arguments a finite superset of possible translates is evaluated. No real
-equality test at a seam or execution of the encoded computation is used.
+\section{Analytic classes and changes of physical coordinates}\label{sec:model}
+The geometric constructions prescribe a smooth velocity first. This section
+records exactly which other solutions are excluded by the energy argument.
+Pressure assumptions are kept separate: continuity in a pressure norm, a
+norm at each individual time, and an integrable pressure gradient are
+different conditions.
+
+Write $\mathcal F_\nu[U]=U_t+(U\cdot\nabla)U-\nu\Delta U$.
+All noncompact results below use $\mathbb R^3$; their velocities and forces
+have one fixed compact spatial support unless the individual statement
+says otherwise. This property concerns the constructed solution, not a
+competing solution. On $[0,T]$, write $CH^k=C([0,T];H^k(\mathbb R^3))$
+and $C^1L^2=C^1([0,T];L^2(\mathbb R^3))$. Spatial constants in pressure
+may depend on time.
+
+\begin{lemma}[The pressure term without a pressure norm]\label{lem:b-gradient}
+If $w,g\in L^2(\mathbb R^3;\mathbb R^3)$ satisfy
+$\operatorname{div}w=0$ and $\operatorname{curl}g=0$ in distributions, then
+$\langle w,g\rangle_{L^2}=0$. In particular, any distributional gradient
+$g=\nabla p\in L^2$ has zero pairing with $w$, without an assumption that
+$p\in L^2$.
+\end{lemma}
+\begin{proof}
+An $L^2$ field is a tempered distribution. The differential identities,
+initially tested on compactly supported smooth functions, extend to
+Schwartz tests by multiplying by cutoffs tending to one and using $L^2$
+convergence of the tests and their first derivatives. Fourier transformation
+gives $\xi\cdot\widehat w=0$ and
+$\xi_j\widehat g_k-\xi_k\widehat g_j=0$. These are identities of locally
+integrable functions, so they hold almost everywhere. Away from $\xi=0$,
+$\widehat g$ is parallel to $\xi$ and $\widehat w$ is perpendicular to it.
+Their Hermitian product is zero. The exceptional point has measure zero;
+Plancherel proves the assertion.
+\end{proof}
+
+\begin{lemma}[Residual realization and separate comparison classes]
+\label{lem:b-comparison}\label{lem:p2-realization}
+Let $\nu>0$ and let $U$ be a prescribed smooth divergence-free velocity
+on $[0,\infty)\times\mathbb R^3$, with $U(0)=0$. On each finite interval,
+assume $U\in CH^2\cap C^1L^2$ and $U,\nabla U$ are bounded. Put
+$f=\mathcal F_\nu[U]$. Then $(U,0)$ is a solution. Its velocity is unique
+in each of the following separately stated comparison classes, with the
+same force and initial data:
+\begin{enumerate}[label=(\roman*),itemsep=2pt]
+\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v$, and a
+pressure representative $p\in CH^1$.
+\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$,
+and an $H^1$ pressure representative at each time, with no required time
+continuity in that pressure norm.
+\item Strong solutions with $v\in CH^4\cap C^1H^2$ and distributional
+$\nabla p\in CL^2$.
+\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$,
+and a pressure representative $p\in CL^2$.
+\item Smooth classical $v\in CH^2\cap C^1L^2$, with bounded $v,\nabla v$,
+without an additional integrability or normalization condition on pressure.
+\end{enumerate}
+The pressure gradient is unique. An $L^2$ pressure representative, when
+specified, is zero; otherwise pressure is determined up to a function of
+time. The reference material flow is a smooth diffeomorphism at every
+finite time and exists for every finite forward time.
+
+On a flat torus $\T_L^3$, the same conclusion holds in the classical class
+where $v,v_t$, spatial derivatives through order two, and $p,\nabla p$ are
+continuous on finite closed cylinders, with periodic mean-zero pressure.
+No Sobolev conditions at infinity are involved in that assertion.
+\end{lemma}
+\begin{proof}
+Substitution proves existence of the prescribed solution. Set $w=v-U$.
+The difference equation is
+\[
+ w_t+(v\cdot\nabla)w+(w\cdot\nabla)U
+    =\nu\Delta w-\nabla p,\qquad \operatorname{div}w=0.
+\]
+The stated velocity regularity implies $w\in C^1L^2\cap CH^2$, so
+$\frac d{dt}\|w\|_2^2=2\langle w,w_t\rangle$ at each time. In case
+(iii), the stronger assumptions imply these inclusions. They also imply
+bounded velocity and spatial derivatives through order two: for $j\le2$,
+Cauchy--Schwarz in Fourier space uses the finite integral
+$\int_{\mathbb R^3}|\xi|^{2j}(1+|\xi|^2)^{-4}\,d\xi$.
+
+For the transport term, insert a cutoff $\chi_R$ that equals one on the
+ball of radius $R$, is supported in the ball of radius $2R$, and satisfies
+$|\nabla\chi_R|\le C/R$. Its boundary error is at most
+$CR^{-1}\|v\|_\infty\|w\|_2^2$, which tends to zero. Diffusion is
+integrated by parts in $H^2$; alternatively its cutoff error is bounded by
+$CR^{-1}\|w\|_2\|\nabla w\|_2$. The deformation term is bounded by
+$\|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2$.
+
+We justify the pressure pairing according to the stated alternatives.
+For (i) and (ii), at each fixed time approximate the chosen $H^1$ pressure
+by compactly supported smooth functions; distributional incompressibility
+and $w\in L^2$ give $\langle\nabla p,w\rangle=0$. Thus no continuity
+of $p$ in time is used in (ii). For (iv), integration by parts on the
+cutoff gives a pressure boundary term bounded by
+$CR^{-1}\|p\|_2\|w\|_2$, also tending to zero. For (iii), apply
+Lemma~\ref{lem:b-gradient} directly. Finally, in (v) the equation itself
+gives at each time
+\[
+ \nabla p=f-v_t-(v\cdot\nabla)v+\nu\Delta v\in L^2,
+ \qquad \|(v\cdot\nabla)v\|_2\le\|v\|_\infty\|\nabla v\|_2.
+\]
+Here $f\in L^2$ follows from the identical estimate for $U$ and its stated
+time and space regularity. The smooth pressure gradient is curl free, so
+Lemma~\ref{lem:b-gradient} again cancels the term. This does not choose an
+$L^2$ representative of the pressure.
+
+In every case the resulting pointwise energy inequality is
+\[
+ \frac12\frac d{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2
+ \le \|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2.
+\]
+The coefficient is bounded on each finite interval and $w(0)=0$, so an
+integrating factor gives $w=0$. The pressure has already disappeared before
+time integration; the proof imposes no unlisted temporal pressure norm.
+Substitution then gives $\nabla p=0$. A spatially constant $L^2$ function
+on $\mathbb R^3$ is zero, which proves the normalization assertion.
+
+On the torus every integration by parts is periodic and has no boundary
+term. The specified classical regularity justifies norm differentiation,
+diffusion, and pressure cancellation. The same energy inequality gives
+uniqueness and the mean-zero normalization fixes pressure. In either domain,
+the smooth reference velocity and its bounded spatial derivative give
+unique material trajectories; bounded speed prevents finite-time escape.
+Solving backwards on a finite interval gives the inverse flow, and smooth
+ODE dependence gives its smoothness.
+\end{proof}
+
+Cases (i)--(v) identify the precise hypotheses used by the applications;
+their presence in a common lemma does not replace a theorem's declared
+comparison class by a different one. In particular, a pressure norm is not
+silently imposed in a gradient-only application. The proof is the classical
+energy comparison for prescribed smooth solutions \cite[Section~18]{Leray1934},
+with its noncompact pressure pairings made explicit.
+
+For the flow $\Phi_t$, differentiation of its trajectory equation gives
+\[
+ U(t)=(\partial_t\Phi_t)\circ\Phi_t^{-1},\qquad
+ \partial_{tt}\Phi_t(a)=
+ (f-\nabla p+\nu\Delta U)(t,\Phi_t(a)).
+\]
+These identities describe the material form of the same constructed solution;
+they do not add an independent existence claim.
+
+\begin{lemma}[Physical scaling]\label{lem:p2-chart}
+For $a,b>0$, $x=x_0+ay$, and $s=bt$, let
+$U(t,x)=abV(bt,(x-x_0)/a)$. Then trajectories are transformed by this change
+of variables, incompressibility is preserved, and at the prescribed physical
+viscosity $\nu$,
+\[
+ \mathcal F_\nu[U](t,x)=ab^2\left[
+ V_s+(V\cdot\nabla_y)V-\frac{\nu}{a^2b}\Delta_yV
+ \right](bt,(x-x_0)/a).
+\]
+\end{lemma}
+\begin{proof}
+The trajectory equation gives $\dot x=abV$; differentiating in space gives
+$\operatorname{div}_xU=b\operatorname{div}_yV$. The three residual terms
+scale by $ab^2,ab^2$, and $b/a$, respectively, proving the formula.
+\end{proof}
+Thus a side-$10$ torus is kept as a physical side-$10$ torus. If a chart
+is rescaled, the residual is recomputed with the fixed $\nu$; the equation
+is not identified with a differently normalized viscosity.
+
+\begin{proposition}[Projection on a flat torus]\label{prop:projection-l2}
+Let $g$ be a smooth force on $[0,\infty)\times\T_L^3$ with zero
+spatial mean at each time. There is a unique smooth mean-zero potential
+$\phi$ satisfying $\Delta\phi=\operatorname{div}g$. Put
+$Qg=\nabla\phi$ and $Pg=g-Qg$. Then $Pg$ is solenoidal and mean zero,
+and replacing $(g,p)$ by $(Pg,p-\phi)$ preserves the velocity and the
+pressure normalization.
+
+Use normalized spatial measure, frequencies $\xi_k=2\pi k/L$, and norms
+\[
+ \|G\|_{H^s}^2=\sum_{k\in\Z^3}(1+|\xi_k|^2)^s|\widehat G(k)|^2,
+ \qquad
+ \|G\|_{C^r}=\max_{|\beta|\le r}\|\partial_x^\beta G\|_\infty.
+\]
+For all integers $h,r\ge0$, at each time,
+\begin{equation}\label{eq:projection-estimate}
+ \begin{aligned}
+ \max\{\|\partial_t^h Pg\|_{C^r},\|\partial_t^h Qg\|_{C^r}\}
+     &\le S_L\|\partial_t^h g\|_{H^{r+2}},\\
+ \|\partial_t^h\phi\|_{C^{r+1}}
+     &\le S_Ld_L\|\partial_t^h g\|_{H^{r+2}},
+ \end{aligned}
+\end{equation}
+where $S_L=(1+52(L/(2\pi))^4)^{1/2}$ and
+$d_L=(1+(L/(2\pi))^2)^{1/2}$.
+
+Uniform bounds, temporal $L^2$ bounds on spatial suprema, and qualitative
+uniform decay of all mixed derivatives pass from $g$ to $Pg,Qg,\phi$.
+A quantitative envelope for time order $h$ and output spatial order $r$
+passes to $Pg,Qg$ through order $r$ and to $\phi$ through order $r+1$
+when the input derivatives through spatial order $r+2$ satisfy that
+envelope. The constants may depend on the derivative order.
+Periodicity and stationarity, including either property on a time tail,
+are preserved on their respective intervals.
+
+For computable $L>0$, effective mixed-derivative evaluation and effective
+derivative bounds for $g$ on finite time slabs give effective evaluation
+and such bounds for $Pg,Qg,\phi$. Corresponding supplied global input
+bounds give effective global bounds in each assertion above.
+\end{proposition}
+\begin{proof}
+For $k\ne0$ define
+\[
+ \widehat\phi(t,k)=-\frac{i\xi_k\cdot\widehat g(t,k)}{|\xi_k|^2},
+ \qquad \widehat\phi(t,0)=0.
+\]
+The nonzero-frequency symbols of $Q$ and $P$ are complementary orthogonal
+projections. Their $H^s$ operator norms are at most one, while the
+multiplier for $\phi$ has norm at most $d_L$ from $H^s$ to $H^{s+1}$.
+For $|\beta|\le r$, Cauchy--Schwarz bounds the absolute sum of the
+Fourier series for $\partial_x^\beta G$ by
+\[
+ \left(\sum_{k\in\Z^3}(1+|\xi_k|^2)^{-2}\right)^{1/2}
+ \|G\|_{H^{r+2}}.
+\]
+The shell $|k|_\infty=n\ge1$ contains $24n^2+2\le26n^2$ points.
+Since $\sum_{n\ge1}n^{-2}\le2$, the factor is at most $S_L$.
+For the tail $|k|_\infty>N\ge1$, it is at most
+\begin{equation}\label{eq:projection-tail}
+ \left(26(L/(2\pi))^4/N\right)^{1/2}.
+\end{equation}
+Applying these estimates to the multipliers proves
+\eqref{eq:projection-estimate}, including the estimate for $\phi$ with
+one additional spatial derivative. On compact time slabs these tails
+converge uniformly for every mixed derivative. Thus the series define
+smooth fields and may be differentiated term by term. The Fourier
+identities give the Poisson equation, uniqueness of its mean-zero
+solution, and $\operatorname{div}Pg=0$. Also
+$-\nabla(p-\phi)+Pg=-\nabla p+g$.
+
+For every integer $n\ge0$, the multinomial expansion and Parseval give
+\begin{equation}\label{eq:projection-derivative-bounds}
+ \|G\|_{H^n}^2
+ \le\sum_{|\beta|\le n}
+       \frac{n!}{(n-|\beta|)!\,\beta!}
+       \|\partial_x^\beta G\|_\infty^2.
+\end{equation}
+Use $n=r+2$ and $G=\partial_t^h g$. This finite sum proves the uniform
+bounds and qualitative decay assertions. If its input suprema are
+bounded by $C_{h,\beta}a_h(t)$, it gives the same envelope $a_h$ for
+the output, with the constants specified by
+\eqref{eq:projection-estimate}--\eqref{eq:projection-derivative-bounds}.
+Integrating the squared estimates proves the temporal $L^2$ assertion
+from the corresponding input norms. The operators act at each time,
+so they preserve the stated temporal symmetries.
+
+For effective evaluation, the supplied derivative bounds and
+\eqref{eq:projection-tail} give a computable Fourier truncation on each
+finite time slab. The finitely many coefficient integrals are computed
+by Riemann sums. After scaling the torus to $[0,1]^3$, the integrand
+for $\widehat G(k)$ has Lipschitz bound
+\[
+ L\sqrt3\max_j\|\partial_{x_j}G\|_\infty
+       +2\pi|k|\|G\|_\infty.
+\]
+This gives an effective quadrature error and hence evaluation of every
+mixed derivative of all three fields. The displayed finite sums also
+compute output bounds from the supplied input bounds.
+\end{proof}
+The estimates apply to the pressure shift $\phi$; in the constructions
+with original pressure zero, the projected pressure is $-\phi$.
+Projection generally changes pressure. It supplies no compact-support
+conclusion on Euclidean space and is distinct from the direct zero-pressure
+solenoidal shear construction.
+
+\paragraph{Effective flat profiles.}
+For later cutoff formulas put
+\begin{equation}\label{eq:p2-step}
+ \rho(s)=\begin{cases}e^{-1/s},&s>0,\\0,&s\le0,\end{cases}
+ \qquad \sigma(s)=\frac{\rho(s)}{\rho(s)+\rho(1-s)}.
+\end{equation}
+Positive-side derivatives of $\rho$ are polynomials in $1/s$ times
+$e^{-1/s}$ and extend flatly at zero. The estimate
+$v^me^{-v}\le(m+k)!v^{-k}$ gives effective error bounds near the seam,
+and $\rho(s)+\rho(1-s)\ge e^{-2}$. Products, translations, rescalings,
+and derivatives therefore yield effective cutoffs with all requested
+derivative bounds. Periodization uses zero collars; at approximate real
+arguments a finite superset of possible translates is evaluated. No real
+equality test at a seam or execution of the encoded computation is used.

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/balanced.tex

@@ -1,166 +1,166 @@
-\section{A complete balanced three-stack realization}\label{sec:bb-balanced-three-stack}
-
-In a positional code with base $B$, replacing a prefix of length $r$
-by one of length $s$ gives an affine map with slope $B^{r-s}$.
-Thus a three-stack instruction has determinant one when it preserves
-the sum of the three prefix lengths.  A third stack supplies reversible
-history under this constraint: we first move a fresh blank from beyond
-a work-tape boundary onto the history stack, then replace that blank
-by the record.
-
-\begin{theorem}[Balanced three-stack realization]
-\label{thm:bb-balanced-three-stack}
-For every machine and finite input, there are effective coefficients
-$f_0,f_1$ such that $f=f_0+\nu f_1$ on $\R^3$ has an explicit
-zero-data solution $(U,0)$.  Force and velocity have one compact
-spatial support and bounded mixed derivatives of every order, and
-are one-periodic after $t=1$.  At the fixed label $x_*=(4,0,0)$,
-\[
- \exists t\ge0:\quad X(t;x_*)\in(-1,2)^3
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-For a fixed machine its repeated field is independent of the input.
-The comparison class is $u\in C H^2\cap C^1L^2$, bounded $u,\nabla u$
-on finite intervals, and pressure modulo constants in $C H^1$.
-The formula is valid for every fixed real $\nu>0$, with effective
-evaluation when $\nu$ is computable and relative evaluation otherwise.
-\end{theorem}
-\begin{proof}
-Normalize the finite machine to one halting state $h$, directing missing
-nonhalting instructions to it by unchanged-letter stay moves. This preserves
-initial halting as well. Let $\Gamma$ be the work alphabet and
-$b\in\Gamma$ its blank. Use three infinite top-first stacks $L,R,J$.  A rule replaces three
-finite prefixes, leaving their tails untouched.  Introduce boundary
-$\star$, marked work letters $\bar a$, and controls $A_q,S_q,T_q,D_q$;
-only $A_h$ is terminal.  For each $q\ne h$, the preparatory rules are
-\begin{align}
- (A_q;\varepsilon,a,\varepsilon)&\to(S_q;\bar a,\varepsilon,\varepsilon),
- \label{eq:bb-balanced-entry}\\
- (S_q;\varepsilon,c,\varepsilon)&\to(S_q;c,\varepsilon,\varepsilon),
- \label{eq:bb-balanced-out-sweep}\\
- (S_q;\varepsilon,\star b,\varepsilon)&\to(T_q;\varepsilon,\star,b),
- \label{eq:bb-balanced-blank}\\
- (T_q;c,\varepsilon,\varepsilon)&\to(T_q;\varepsilon,c,\varepsilon),
- \label{eq:bb-balanced-return-sweep}\\
- (T_q;\bar a,\varepsilon,\varepsilon)&\to(D_q;\varepsilon,a,\varepsilon).
- \label{eq:bb-balanced-restore}
-\end{align}
-Here $a,c\in\Gamma$.  For $\delta(q,a)=(q',d,\sigma)$ add
-\begin{equation}\label{eq:bb-balanced-step-table}
-\begin{array}{c|c|c}
-\sigma&\text{source}&\text{target}\\\hline
-0&(D_q;\varepsilon,a,b)&(A_{q'};\varepsilon,d,\kappa)\\
--1&(D_q;c,a,b)&(A_{q'};\varepsilon,cd,\kappa)\quad(c\in\Gamma)\\
-1&(D_q;\varepsilon,ac,b)&(A_{q'};d,c,\kappa)\quad(c\in\Gamma)\\
-1&(D_q;\varepsilon,a\star b,b)&(A_{q'};d,b\star,\kappa).
-\end{array}
-\end{equation}
-Every individual row occurrence, including its choice of $c$, receives
-a distinct record symbol $\kappa$.
-
-\emph{Full-domain separation and initialized behavior.}
-\label{lem:bb-balanced-processor}\label{proof:bb-balanced-processor}
-Every rule preserves total prefix length: it is one in the transfer
-rows, two in the blank-supply row, and respectively two, three, three,
-four in the final rows.  Source branches at $A_q$ test the right top;
-at $S_q$ the right top separates work letters from the boundary;
-at $T_q$ the left top separates work and marked letters.  At $D_q$
-the right top fixes the work instruction, with the next right symbol
-or left top separating its remaining branches.  Target branches at
-$S_q$ have distinct marked or ordinary left tops; at $T_q$ distinct
-boundary or ordinary right tops; at $D_q$ distinct restored $a$;
-and at an $A$ control distinct history tops $\kappa$.
-This proves separation for arbitrary independent tails.
-
-Initialize
-\[
- (A_{q_0};b^\infty,v\star b^\infty,b^\infty),\qquad
- v=\begin{cases}\omega,&\omega\ne\varepsilon,\\b,&\omega=\varepsilon.
- \end{cases}
-\]
-At a checkpoint, $L$ lists cells strictly left of the head and
-$R=a_1\cdots a_m\star b^\infty$, $m\ge1$, lists the current cell
-and a finite right block; beyond it the tape is blank.  $J$ contains
-completed records, newest first.  The first rule marks $a_1$ while
-moving it left; $m-1$ forward transfers expose $\star$.  The blank
-row removes one filler blank beyond that boundary and pushes it on
-$J$, without changing the infinite blank filler.  The reverse transfers
-and restoration of $a_1$ restore exactly the original two work stacks.
-The final rule replaces the fresh history blank by its record and
-performs the work instruction.  A left move pops the next cell from
-$L$; a right move with $m\ge2$ pops the current cell from $R$;
-when $m=1$ the last row exposes a blank and retains a nonempty right
-block.  Thus the invariant holds and one work step costs
-$1+(m-1)+1+(m-1)+1+1=2m+2$ rules.  An initially halted input needs
-none.  The records recover the head displacements as well.
-
-\emph{From balanced words to solid boxes.}
-Give the enlarged alphabet $\Sigma$ the odd digits
-$1,3,\ldots,2|\Sigma|-1$ in base $B=2|\Sigma|+1$.
-For a finite word $v$ put $e(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}$;
-for an infinite stack $\xi$ put
-$e(\xi)=\sum_{j\ge1}d(\xi_j)B^{-j}$.
-The prefix interval is $I(v)=e(v)+[0,B^{-|v|}]$, with
-$I(\varepsilon)=[0,1]$.  Every infinite tail satisfies
-\[
- \frac1{B-1}\le e(\xi)\le\frac{2|\Sigma|-1}{B-1}<1,
-\]
-so its code lies strictly inside each applicable prefix interval.
-Distinct incompatible prefixes have positive gaps, since digits differ
-by at least two while the remaining interval has width at most one unit
-in the differing place. Place control $s$ at
-$o_s=(4i_s,0,0)$, with $i_{A_h}=0$ and distinct positive integers
-for the others.  The code of $(s;L,R,J)$ is
-$o_s+(e(L),e(R),e(J))$.
-A rule with input prefixes $\alpha_1,\alpha_2,\alpha_3$
-and output prefixes $\alpha'_1,\alpha'_2,\alpha'_3$ gives boxes
-\[
- Q_i=o_s+\prod_jI(\alpha_j),\qquad
- Q_i'=o_{s'}+\prod_jI(\alpha'_j).
-\]
-Both families are separately disjoint by the preceding full-prefix
-separation.  In control-relative coordinates their exact maps are
-\begin{equation}\label{eq:bb-balanced-affine-rule}
- y_j=e(\alpha'_j)+B^{|\alpha_j|-|\alpha'_j|}
-                    (x_j-e(\alpha_j)).
-\end{equation}
-The identity $e(\alpha\xi)=e(\alpha)+B^{-|\alpha|}e(\xi)$
-proves their action on every code.  Initialized stacks are eventually
-blank, with rational codes
-$e(vb^\infty)=e(v)+B^{-|v|}d(b)/(B-1)$.
-The affine determinant equals
-$B^{\sum|\alpha_j|-\sum|\alpha'_j|}=1$, exactly by balance.
-Apply Theorem~\ref{lem:bb-balanced-box-routing} to these full boxes.
-
-\emph{Every intermediate time.}
-All half-widths are at most $1/2$, so that lemma uses $R=1$.
-Sources have first coordinate at least four, since no rule leaves
-$A_h$.  Targets with nonhalt control have the same property.
-For such a branch the initial expansion and final contraction preserve
-first coordinate at least four.  During deformation its center has
-first coordinate at least $4\Lambda$.  Every scaled halt target
-center has first coordinate at most $\Lambda$, whereas the endpoints
-of each nonhalt parking path have first coordinate at least
-$4\Lambda$.  Since $\Lambda+4<4\Lambda$, the original segment
-cannot meet an obstacle cube around a halt target.  Any encountered
-obstacle has first coordinate at least $4\Lambda$, so its boundary
-detour remains at least $4\Lambda-4$.  Subtracting a half-width at
-most one still leaves first coordinate greater than two.  Hence
-\begin{equation}\label{eq:bb-balanced-no-premature-event}
- \Phi_\tau(Q_i)\subset\{x_1>2\}\quad(0\le\tau\le1)
- \quad\text{for every branch with nonhalt target}.
-\end{equation}
-This includes delivery after halt targets have already been filled.
-
-Finally load $x_*=(4,0,0)$ along the straight segment to the rational
-initial code using the translation part of Lemma~\ref{lem:bb-curl}.
-It avoids $(-1,2)^3$ for a nonhalt initial control.  Repeat the box
-field from time one.  Induction at integer times applies the exact
-full-box maps; the finite compiler cycles reach a code in $(0,1)^3$
-precisely upon halting.  Equation~\eqref{eq:bb-balanced-no-premature-event}
-excludes every intermediate false visit.  The residual proposition
-gives the force and uniqueness; compact support gives all finite-slab
-Sobolev conditions and uniformly bounded kinetic energy.  Every choice
-was finite and effective, including the obstacle detours.  The input
-enters only the first loading path, completing the proof.
-\end{proof}
+\section{A complete balanced three-stack realization}\label{sec:bb-balanced-three-stack}
+
+In a positional code with base $B$, replacing a prefix of length $r$
+by one of length $s$ gives an affine map with slope $B^{r-s}$.
+Thus a three-stack instruction has determinant one when it preserves
+the sum of the three prefix lengths.  A third stack supplies reversible
+history under this constraint: we first move a fresh blank from beyond
+a work-tape boundary onto the history stack, then replace that blank
+by the record.
+
+\begin{theorem}[Balanced three-stack realization]
+\label{thm:bb-balanced-three-stack}
+For every machine and finite input, there are effective coefficients
+$f_0,f_1$ such that $f=f_0+\nu f_1$ on $\R^3$ has an explicit
+zero-data solution $(U,0)$.  Force and velocity have one compact
+spatial support and bounded mixed derivatives of every order, and
+are one-periodic after $t=1$.  At the fixed label $x_*=(4,0,0)$,
+\[
+ \exists t\ge0:\quad X(t;x_*)\in(-1,2)^3
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+For a fixed machine its repeated field is independent of the input.
+The comparison class is $u\in C H^2\cap C^1L^2$, bounded $u,\nabla u$
+on finite intervals, and pressure modulo constants in $C H^1$.
+The formula is valid for every fixed real $\nu>0$, with effective
+evaluation when $\nu$ is computable and relative evaluation otherwise.
+\end{theorem}
+\begin{proof}
+Normalize the finite machine to one halting state $h$, directing missing
+nonhalting instructions to it by unchanged-letter stay moves. This preserves
+initial halting as well. Let $\Gamma$ be the work alphabet and
+$b\in\Gamma$ its blank. Use three infinite top-first stacks $L,R,J$.  A rule replaces three
+finite prefixes, leaving their tails untouched.  Introduce boundary
+$\star$, marked work letters $\bar a$, and controls $A_q,S_q,T_q,D_q$;
+only $A_h$ is terminal.  For each $q\ne h$, the preparatory rules are
+\begin{align}
+ (A_q;\varepsilon,a,\varepsilon)&\to(S_q;\bar a,\varepsilon,\varepsilon),
+ \label{eq:bb-balanced-entry}\\
+ (S_q;\varepsilon,c,\varepsilon)&\to(S_q;c,\varepsilon,\varepsilon),
+ \label{eq:bb-balanced-out-sweep}\\
+ (S_q;\varepsilon,\star b,\varepsilon)&\to(T_q;\varepsilon,\star,b),
+ \label{eq:bb-balanced-blank}\\
+ (T_q;c,\varepsilon,\varepsilon)&\to(T_q;\varepsilon,c,\varepsilon),
+ \label{eq:bb-balanced-return-sweep}\\
+ (T_q;\bar a,\varepsilon,\varepsilon)&\to(D_q;\varepsilon,a,\varepsilon).
+ \label{eq:bb-balanced-restore}
+\end{align}
+Here $a,c\in\Gamma$.  For $\delta(q,a)=(q',d,\sigma)$ add
+\begin{equation}\label{eq:bb-balanced-step-table}
+\begin{array}{c|c|c}
+\sigma&\text{source}&\text{target}\\\hline
+0&(D_q;\varepsilon,a,b)&(A_{q'};\varepsilon,d,\kappa)\\
+-1&(D_q;c,a,b)&(A_{q'};\varepsilon,cd,\kappa)\quad(c\in\Gamma)\\
+1&(D_q;\varepsilon,ac,b)&(A_{q'};d,c,\kappa)\quad(c\in\Gamma)\\
+1&(D_q;\varepsilon,a\star b,b)&(A_{q'};d,b\star,\kappa).
+\end{array}
+\end{equation}
+Every individual row occurrence, including its choice of $c$, receives
+a distinct record symbol $\kappa$.
+
+\emph{Full-domain separation and initialized behavior.}
+\label{lem:bb-balanced-processor}\label{proof:bb-balanced-processor}
+Every rule preserves total prefix length: it is one in the transfer
+rows, two in the blank-supply row, and respectively two, three, three,
+four in the final rows.  Source branches at $A_q$ test the right top;
+at $S_q$ the right top separates work letters from the boundary;
+at $T_q$ the left top separates work and marked letters.  At $D_q$
+the right top fixes the work instruction, with the next right symbol
+or left top separating its remaining branches.  Target branches at
+$S_q$ have distinct marked or ordinary left tops; at $T_q$ distinct
+boundary or ordinary right tops; at $D_q$ distinct restored $a$;
+and at an $A$ control distinct history tops $\kappa$.
+This proves separation for arbitrary independent tails.
+
+Initialize
+\[
+ (A_{q_0};b^\infty,v\star b^\infty,b^\infty),\qquad
+ v=\begin{cases}\omega,&\omega\ne\varepsilon,\\b,&\omega=\varepsilon.
+ \end{cases}
+\]
+At a checkpoint, $L$ lists cells strictly left of the head and
+$R=a_1\cdots a_m\star b^\infty$, $m\ge1$, lists the current cell
+and a finite right block; beyond it the tape is blank.  $J$ contains
+completed records, newest first.  The first rule marks $a_1$ while
+moving it left; $m-1$ forward transfers expose $\star$.  The blank
+row removes one filler blank beyond that boundary and pushes it on
+$J$, without changing the infinite blank filler.  The reverse transfers
+and restoration of $a_1$ restore exactly the original two work stacks.
+The final rule replaces the fresh history blank by its record and
+performs the work instruction.  A left move pops the next cell from
+$L$; a right move with $m\ge2$ pops the current cell from $R$;
+when $m=1$ the last row exposes a blank and retains a nonempty right
+block.  Thus the invariant holds and one work step costs
+$1+(m-1)+1+(m-1)+1+1=2m+2$ rules.  An initially halted input needs
+none.  The records recover the head displacements as well.
+
+\emph{From balanced words to solid boxes.}
+Give the enlarged alphabet $\Sigma$ the odd digits
+$1,3,\ldots,2|\Sigma|-1$ in base $B=2|\Sigma|+1$.
+For a finite word $v$ put $e(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}$;
+for an infinite stack $\xi$ put
+$e(\xi)=\sum_{j\ge1}d(\xi_j)B^{-j}$.
+The prefix interval is $I(v)=e(v)+[0,B^{-|v|}]$, with
+$I(\varepsilon)=[0,1]$.  Every infinite tail satisfies
+\[
+ \frac1{B-1}\le e(\xi)\le\frac{2|\Sigma|-1}{B-1}<1,
+\]
+so its code lies strictly inside each applicable prefix interval.
+Distinct incompatible prefixes have positive gaps, since digits differ
+by at least two while the remaining interval has width at most one unit
+in the differing place. Place control $s$ at
+$o_s=(4i_s,0,0)$, with $i_{A_h}=0$ and distinct positive integers
+for the others.  The code of $(s;L,R,J)$ is
+$o_s+(e(L),e(R),e(J))$.
+A rule with input prefixes $\alpha_1,\alpha_2,\alpha_3$
+and output prefixes $\alpha'_1,\alpha'_2,\alpha'_3$ gives boxes
+\[
+ Q_i=o_s+\prod_jI(\alpha_j),\qquad
+ Q_i'=o_{s'}+\prod_jI(\alpha'_j).
+\]
+Both families are separately disjoint by the preceding full-prefix
+separation.  In control-relative coordinates their exact maps are
+\begin{equation}\label{eq:bb-balanced-affine-rule}
+ y_j=e(\alpha'_j)+B^{|\alpha_j|-|\alpha'_j|}
+                    (x_j-e(\alpha_j)).
+\end{equation}
+The identity $e(\alpha\xi)=e(\alpha)+B^{-|\alpha|}e(\xi)$
+proves their action on every code.  Initialized stacks are eventually
+blank, with rational codes
+$e(vb^\infty)=e(v)+B^{-|v|}d(b)/(B-1)$.
+The affine determinant equals
+$B^{\sum|\alpha_j|-\sum|\alpha'_j|}=1$, exactly by balance.
+Apply Theorem~\ref{lem:bb-balanced-box-routing} to these full boxes.
+
+\emph{Every intermediate time.}
+All half-widths are at most $1/2$, so that lemma uses $R=1$.
+Sources have first coordinate at least four, since no rule leaves
+$A_h$.  Targets with nonhalt control have the same property.
+For such a branch the initial expansion and final contraction preserve
+first coordinate at least four.  During deformation its center has
+first coordinate at least $4\Lambda$.  Every scaled halt target
+center has first coordinate at most $\Lambda$, whereas the endpoints
+of each nonhalt parking path have first coordinate at least
+$4\Lambda$.  Since $\Lambda+4<4\Lambda$, the original segment
+cannot meet an obstacle cube around a halt target.  Any encountered
+obstacle has first coordinate at least $4\Lambda$, so its boundary
+detour remains at least $4\Lambda-4$.  Subtracting a half-width at
+most one still leaves first coordinate greater than two.  Hence
+\begin{equation}\label{eq:bb-balanced-no-premature-event}
+ \Phi_\tau(Q_i)\subset\{x_1>2\}\quad(0\le\tau\le1)
+ \quad\text{for every branch with nonhalt target}.
+\end{equation}
+This includes delivery after halt targets have already been filled.
+
+Finally load $x_*=(4,0,0)$ along the straight segment to the rational
+initial code using the translation part of Lemma~\ref{lem:bb-curl}.
+It avoids $(-1,2)^3$ for a nonhalt initial control.  Repeat the box
+field from time one.  Induction at integer times applies the exact
+full-box maps; the finite compiler cycles reach a code in $(0,1)^3$
+precisely upon halting.  Equation~\eqref{eq:bb-balanced-no-premature-event}
+excludes every intermediate false visit.  The residual proposition
+gives the force and uniqueness; compact support gives all finite-slab
+Sobolev conditions and uniformly bounded kinetic energy.  Every choice
+was finite and effective, including the obstacle detours.  The input
+enters only the first loading path, completing the proof.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/clocks.tex

@@ -1,140 +1,140 @@
-\section{Clock profiles for selected processors}\label{sec:clocks}
-
-Lemma~\ref{ba:log-clock} gives the common logarithmic change of time.
-We now apply it to two Euclidean processors with different pressure
-classes, and then use a companion's autonomous spatial clock to obtain
-stationary or decaying forcing on the torus. In every case the clock
-traverses every finite computational time; periodic, stationary and
-decaying profiles remain separate choices.
-
-\subsection{Preserving the declared comparison classes}
-\label{sec:bb-late-logclock}
-
-The two templates below use different whole-space pressure assumptions.
-Slowing their trajectories preserves those declared comparison classes
-because the new reference velocity retains the required regularity on
-every finite physical-time interval.
-
-\begin{proposition}\label{prop:bb-late-logclock}
-The template of Theorem~\ref{thm:bb-direction-shuttle} admits a force
-in $L^2([0,\infty)\times\R^3)$ with the same material label and
-event.  The template of Theorem~\ref{thm:bb-guarded-history} admits
-a force in $L^2([0,\infty);H^j(\R^3))$ for every integer $j\ge0$.
-Both retain uniform compact support, zero initial velocity, bounded
-mixed derivatives, their stated pressure and velocity comparison
-classes, and finite effective prescriptions.
-\end{proposition}
-\begin{proof}\label{proof:bb-late-logclock}
-Apply Lemma~\ref{ba:log-clock} to each template, including its loading
-interval if present. Both are initially zero, have one compact support,
-and have uniformly bounded mixed derivatives. The lemma gives the exact
-residual and all mixed bounds; fixed support also gives
-$\|f(t)\|_{H^j}\le C_j(1+t)^{-1}$ for every fixed $j$.
-
-On each finite interval the slowed reference field retains every Sobolev
-regularity and bounded derivative required by its original comparison
-class. Case (i) of Lemma~\ref{lem:b-comparison} therefore applies to the
-direction shuttle's continuous $H^1$ pressure representative, and case
-(iii) applies to the guarded-history processor's continuous $L^2$
-pressure gradient. Both constructed pressures are zero.
-
-The new path is $X(\log(1+t);a_0)$. The clock lemma preserves the event
-and the effective prescription. In particular, loaded samples $1+k$
-occur at physical times $e^{1+k}-1$, whereas the aligned construction's
-samples $k$ occur at $e^k-1$. These times have no finite accumulation.
-The new forces have the stated temporal integrability; no periodicity
-in physical time is asserted for them.
-\end{proof}
-
-\subsection{The input-offset alternative}\label{bc:input-offset-section}
-A fixed material label can also be retained by shifting the state charts
-by the initial tape code. The complete planar routing and spatial-clock
-construction for this choice belongs to
-\cite[Theorem~4.1 and Section~5]{OpenAI379StationaryA2}.
-The following application fixes its match with the present conventions.
-
-\begin{corollary}[Two profiles of the input-offset processor]
-\label{bc:input-offset-event}
-On $\T^3$, the fixed label $P=(1/4,1/4,0)$ and strip
-$O=\{1/32<Y<1/8\}$ realize halting from zero velocity by two effective
-mean-zero general forces. One is stationary after time one. The other
-tends to zero uniformly and has every mixed-derivative spatial supremum
-norm in $L^2(0,\infty)$. Both have bounded mixed derivatives, zero
-prescribed pressure, uniformly bounded kinetic energy, and uniqueness in
-the torus comparison class.
-\end{corollary}
-\begin{proof}\label{bc:input-offset-event-proof}
-Add the companion's fresh nonhalting dummy start and use its seven-row
-compiler. The permutation of (work, mark, history) into our track order
-identifies its table with Lemma~\ref{lem:bb-movefirst}, including the full-domain
-inverse. Let $K$ be the full compiler alphabet size, $m$ the number of
-controls, and $N$ the number of full branches. With odd digits in base
-$B=2K+2$ and initial rational stack coordinates $x_0,y_0$, its state-square
-parameters are
-\begin{equation}\label{bc:input-offset-state-squares}
- \lambda=\frac1{100(m+N+1)},\qquad
- a_*=1/4-\lambda x_0,\qquad b_{\rm start}=1/4-\lambda y_0.
-\end{equation}
-The halt control has lower height $1/16$; enumerate the controls other
-than start and halt by $\ell$ and give them lower heights
-$1/3+\ell/[12(m+1)]$, as in that theorem. Thus the
-initial code is $(1/4,1/4)$, and the common full branch formulas of
-Lemma~\ref{lem:bb-onecell} are precisely its inputs. The cited theorem gives
-the full closed-rectangle routing, positive cutoff neighborhoods, and the
-all-time guard in $J_*=[1/8,7/8]\times[1/32,7/8]$. Write $V(s,X,Y)$ for
-its planar velocity, one-periodic in $s$. Choose $\chi_*$ smooth,
-compactly supported in the open unit-square chart and equal to one near
-$J_*$. The autonomous field
-from the companion's Section~5 is
-\begin{equation}\label{bc:input-offset-W}
- W=(V_1(Z,X,Y),V_2(Z,X,Y),\partial_X(X\chi_*(X,Y))),
-\end{equation}
-with the compactly supported products extended periodically. It is
-solenoidal and mean zero; its phase
-speed is one along the tracked path. Its trajectory from $P$ enters $O$
-exactly upon halting. These are the specific geometric imports, rather
-than an identification based only on the headline conclusion.
-
-Take the stationary and slow profiles from
-\cite[Section~5 and Proposition~6.1]{OpenAI379StationaryA2}, respectively:
-\begin{equation}\label{bc:input-offset-profiles}
- \alpha_0=\sigma(2t-1),\qquad \alpha_1=\sigma(2t-1)/(1+t).
-\end{equation}
-Their exact force formula at the fixed physical viscosity is
-\begin{equation}\label{bc:input-offset-force}
- U_\alpha=\alpha W,\qquad
- f_\alpha=\alpha'W+\alpha^2(W\cdot\nabla)W-\nu\alpha\Delta W.
-\end{equation}
-Lemma~\ref{lem:p2-realization} supplies the stated comparison conclusion.
-For $s_\alpha(t)=\int_0^t\alpha(r)\,dr$, symmetry of $\sigma$ gives
-$s_{\alpha_0}(1)=1/4$ and $s_{\alpha_0}(t)=t-3/4$ for $t\ge1$.
-The other clock satisfies
-$s_{\alpha_1}(t)=s_{\alpha_1}(1)+\log((1+t)/2)$ there. Both are
-continuous, finite at finite time and onto $[0,\infty)$, so both preserve
-the complete trajectory event, including original initial halting through
-the dummy start. The first force is stationary after one. In the second,
-the coefficient orders are respectively $(1+t)^{-2},(1+t)^{-2}$ and
-$(1+t)^{-1}$; each time derivative gives an additional inverse power.
-The spatial factors in \eqref{bc:input-offset-force} have effective
-bounds at every order. Thus for every $h,r\ge0$,
-\begin{equation}\label{bc:input-offset-common-rate}
- \|\partial_t^h U_{\alpha_1}(t)\|_{C^r}
- +\|\partial_t^h f_{\alpha_1}(t)\|_{C^r}
- \le C_{h,r}(1+t)^{-1-h}.
-\end{equation}
-The constant is effective; increasing it over $[0,1]$ makes the estimate
-valid for all $t\ge0$. The exponent is common to all spatial orders,
-and the square of the envelope has integral $(1+2h)^{-1}$.
-This proves the asserted bounded and temporal $L^2$ norms.
-
-For the stationary profile, Proposition~\ref{prop:projection-l2}
-preserves stationarity after time one and every mixed-derivative bound.
-For the slow profile, \eqref{eq:projection-estimate} and
-\eqref{eq:projection-derivative-bounds} apply to
-\eqref{bc:input-offset-common-rate} through spatial order $r+2$.
-They retain the rate $(1+t)^{-1-h}$ for the projected force through
-order $r$ and for its pressure through order $r+1$, with effective
-constants and temporal $L^2$ bounds. In either projection choice the
-velocity and event are unchanged and the mean-zero pressure is
-$-\phi$. The stationary and slow profiles remain separate choices.
-\end{proof}
+\section{Clock profiles for selected processors}\label{sec:clocks}
+
+Lemma~\ref{ba:log-clock} gives the common logarithmic change of time.
+We now apply it to two Euclidean processors with different pressure
+classes, and then use a companion's autonomous spatial clock to obtain
+stationary or decaying forcing on the torus. In every case the clock
+traverses every finite computational time; periodic, stationary and
+decaying profiles remain separate choices.
+
+\subsection{Preserving the declared comparison classes}
+\label{sec:bb-late-logclock}
+
+The two templates below use different whole-space pressure assumptions.
+Slowing their trajectories preserves those declared comparison classes
+because the new reference velocity retains the required regularity on
+every finite physical-time interval.
+
+\begin{proposition}\label{prop:bb-late-logclock}
+The template of Theorem~\ref{thm:bb-direction-shuttle} admits a force
+in $L^2([0,\infty)\times\R^3)$ with the same material label and
+event.  The template of Theorem~\ref{thm:bb-guarded-history} admits
+a force in $L^2([0,\infty);H^j(\R^3))$ for every integer $j\ge0$.
+Both retain uniform compact support, zero initial velocity, bounded
+mixed derivatives, their stated pressure and velocity comparison
+classes, and finite effective prescriptions.
+\end{proposition}
+\begin{proof}\label{proof:bb-late-logclock}
+Apply Lemma~\ref{ba:log-clock} to each template, including its loading
+interval if present. Both are initially zero, have one compact support,
+and have uniformly bounded mixed derivatives. The lemma gives the exact
+residual and all mixed bounds; fixed support also gives
+$\|f(t)\|_{H^j}\le C_j(1+t)^{-1}$ for every fixed $j$.
+
+On each finite interval the slowed reference field retains every Sobolev
+regularity and bounded derivative required by its original comparison
+class. Case (i) of Lemma~\ref{lem:b-comparison} therefore applies to the
+direction shuttle's continuous $H^1$ pressure representative, and case
+(iii) applies to the guarded-history processor's continuous $L^2$
+pressure gradient. Both constructed pressures are zero.
+
+The new path is $X(\log(1+t);a_0)$. The clock lemma preserves the event
+and the effective prescription. In particular, loaded samples $1+k$
+occur at physical times $e^{1+k}-1$, whereas the aligned construction's
+samples $k$ occur at $e^k-1$. These times have no finite accumulation.
+The new forces have the stated temporal integrability; no periodicity
+in physical time is asserted for them.
+\end{proof}
+
+\subsection{The input-offset alternative}\label{bc:input-offset-section}
+A fixed material label can also be retained by shifting the state charts
+by the initial tape code. The complete planar routing and spatial-clock
+construction for this choice belongs to
+\cite[Theorem~4.1 and Section~5]{OpenAI379StationaryA2}.
+The following application fixes its match with the present conventions.
+
+\begin{corollary}[Two profiles of the input-offset processor]
+\label{bc:input-offset-event}
+On $\T^3$, the fixed label $P=(1/4,1/4,0)$ and strip
+$O=\{1/32<Y<1/8\}$ realize halting from zero velocity by two effective
+mean-zero general forces. One is stationary after time one. The other
+tends to zero uniformly and has every mixed-derivative spatial supremum
+norm in $L^2(0,\infty)$. Both have bounded mixed derivatives, zero
+prescribed pressure, uniformly bounded kinetic energy, and uniqueness in
+the torus comparison class.
+\end{corollary}
+\begin{proof}\label{bc:input-offset-event-proof}
+Add the companion's fresh nonhalting dummy start and use its seven-row
+compiler. The permutation of (work, mark, history) into our track order
+identifies its table with Lemma~\ref{lem:bb-movefirst}, including the full-domain
+inverse. Let $K$ be the full compiler alphabet size, $m$ the number of
+controls, and $N$ the number of full branches. With odd digits in base
+$B=2K+2$ and initial rational stack coordinates $x_0,y_0$, its state-square
+parameters are
+\begin{equation}\label{bc:input-offset-state-squares}
+ \lambda=\frac1{100(m+N+1)},\qquad
+ a_*=1/4-\lambda x_0,\qquad b_{\rm start}=1/4-\lambda y_0.
+\end{equation}
+The halt control has lower height $1/16$; enumerate the controls other
+than start and halt by $\ell$ and give them lower heights
+$1/3+\ell/[12(m+1)]$, as in that theorem. Thus the
+initial code is $(1/4,1/4)$, and the common full branch formulas of
+Lemma~\ref{lem:bb-onecell} are precisely its inputs. The cited theorem gives
+the full closed-rectangle routing, positive cutoff neighborhoods, and the
+all-time guard in $J_*=[1/8,7/8]\times[1/32,7/8]$. Write $V(s,X,Y)$ for
+its planar velocity, one-periodic in $s$. Choose $\chi_*$ smooth,
+compactly supported in the open unit-square chart and equal to one near
+$J_*$. The autonomous field
+from the companion's Section~5 is
+\begin{equation}\label{bc:input-offset-W}
+ W=(V_1(Z,X,Y),V_2(Z,X,Y),\partial_X(X\chi_*(X,Y))),
+\end{equation}
+with the compactly supported products extended periodically. It is
+solenoidal and mean zero; its phase
+speed is one along the tracked path. Its trajectory from $P$ enters $O$
+exactly upon halting. These are the specific geometric imports, rather
+than an identification based only on the headline conclusion.
+
+Take the stationary and slow profiles from
+\cite[Section~5 and Proposition~6.1]{OpenAI379StationaryA2}, respectively:
+\begin{equation}\label{bc:input-offset-profiles}
+ \alpha_0=\sigma(2t-1),\qquad \alpha_1=\sigma(2t-1)/(1+t).
+\end{equation}
+Their exact force formula at the fixed physical viscosity is
+\begin{equation}\label{bc:input-offset-force}
+ U_\alpha=\alpha W,\qquad
+ f_\alpha=\alpha'W+\alpha^2(W\cdot\nabla)W-\nu\alpha\Delta W.
+\end{equation}
+Lemma~\ref{lem:p2-realization} supplies the stated comparison conclusion.
+For $s_\alpha(t)=\int_0^t\alpha(r)\,dr$, symmetry of $\sigma$ gives
+$s_{\alpha_0}(1)=1/4$ and $s_{\alpha_0}(t)=t-3/4$ for $t\ge1$.
+The other clock satisfies
+$s_{\alpha_1}(t)=s_{\alpha_1}(1)+\log((1+t)/2)$ there. Both are
+continuous, finite at finite time and onto $[0,\infty)$, so both preserve
+the complete trajectory event, including original initial halting through
+the dummy start. The first force is stationary after one. In the second,
+the coefficient orders are respectively $(1+t)^{-2},(1+t)^{-2}$ and
+$(1+t)^{-1}$; each time derivative gives an additional inverse power.
+The spatial factors in \eqref{bc:input-offset-force} have effective
+bounds at every order. Thus for every $h,r\ge0$,
+\begin{equation}\label{bc:input-offset-common-rate}
+ \|\partial_t^h U_{\alpha_1}(t)\|_{C^r}
+ +\|\partial_t^h f_{\alpha_1}(t)\|_{C^r}
+ \le C_{h,r}(1+t)^{-1-h}.
+\end{equation}
+The constant is effective; increasing it over $[0,1]$ makes the estimate
+valid for all $t\ge0$. The exponent is common to all spatial orders,
+and the square of the envelope has integral $(1+2h)^{-1}$.
+This proves the asserted bounded and temporal $L^2$ norms.
+
+For the stationary profile, Proposition~\ref{prop:projection-l2}
+preserves stationarity after time one and every mixed-derivative bound.
+For the slow profile, \eqref{eq:projection-estimate} and
+\eqref{eq:projection-derivative-bounds} apply to
+\eqref{bc:input-offset-common-rate} through spatial order $r+2$.
+They retain the rate $(1+t)^{-1-h}$ for the projected force through
+order $r$ and for its pressure through order $r+1$, with effective
+constants and temporal $L^2$ bounds. In either projection choice the
+velocity and event are unchanged and the mean-zero pressure is
+$-\phi$. The stationary and slow profiles remain separate choices.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/coding-conventions.tex

@@ -1,211 +1,211 @@
-\section{History conventions and branch subdivision}\label{ba:codes}
-
-The general box motions now let us choose a recorder and an observer
-separately.  We begin with the two frontier recorders in
-Section~\ref{sec:bb-frontiers}, which retain history in the
-sense of Bennett~\cite{Bennett1973}.  Later we retain the distinct
-neighboring-cell guards and persistent-mark tables needed by other
-realizations.  A full-domain inverse, not merely agreement along an
-initialized run, is the criterion for sharing a compiler.
-
-For these applications let $A$ be the work alphabet with distinguished
-blank, $Q$ the states, $q_0$ the initial state, and $H\subseteq Q$
-the halting states.  Write $h_n$ for the work head after $n$ steps and
-\[
- \delta(q,a)=(q^+(r),b(r),d(r)),\qquad
- r=(q,a)\in\mathcal R=(Q\setminus H)\times A,
- \quad d(r)\in\{-1,0,1\}.
-\]
-The initial head is zero and the tape has finitely specified nonblank
-cells.  Replace missing instructions by unchanged-letter stay moves to
-a halt.  If one halt square is needed, merge halting states and redirect
-their incoming instructions; no outgoing rule is changed.  This preserves
-initial halting as well.  An unreachable halt state can be added when
-needed.  These are the same normalizations as in
-Section~\ref{sec:bb-frontiers}.
-
-\subsection{Frontier recorders and their notation}
-
-We use the move-first recorder of Lemma~\ref{lem:bb-movefirst}
-with its original notation and complete partial domain.
-The delayed-head recorder of Lemma~\ref{lem:bb-direction} records the
-work displacement until its final return step.  Its table can also be
-initialized with the frontier one cell farther to the right.
-
-\begin{lemma}[Six-rule history recorder]\label{ba:six-compiler}
-For $r_0=1$ or $r_0=2$, there is a partial table with the two incoming
-properties of Lemma~\ref{lem:bb-onecell}, frontier $r_0+n$ at checkpoint
-$n$, and checkpoint control recording both the work state and the
-preceding work displacement.  Its next checkpoint takes
-$2(r_0+n-h_n)+3$ instructions.  The work head moves only on the last
-instruction.
-\end{lemma}
-\begin{proof}\label{ba:six-proof}
-Specialize Lemma~\ref{lem:bb-direction} to a singleton passive track
-$O$ and delete that coordinate.  Rename its controls
-$(M(q,e),G(\tau),A(q,d),B(q,d))$ as
-$(M_{q,e},R_\tau,F_{q,d},L_{q,d})$, and its frontier $F$ as $P$.
-The full alphabet is
-$A\times(\{\bot,P\}\sqcup\mathcal T)\times\{0,1\}$,
-where $\mathcal T=(Q\setminus H)\times\{-1,0,1\}\times A$.
-Every row and guard is the same under this bijection, so both incoming
-properties hold on the entire partial domain, including unused controls.
-Only $M_{q,e}$ with $q\in H$ are halting controls.
-
-Initialize $M_{q_0,0}$ with all marks zero, frontier at $r_0$, and
-empty history elsewhere.  The case $r_0=1$ is the initialization already
-proved in the shared lemma.  For either allowed $r_0$, checkpoint $n$
-has frontier $g=r_0+n>h_n$.  Mark the old head, scan right to $g$,
-write the record and install the new frontier at $g+1$, then return to
-the mark.  All intervening history is nonfrontier, the new frontier
-cell is empty, and there is only one mark.  Hence the same rows apply
-when $r_0=2$.  There are $g-h_n-1$ continuing right moves,
-$g-h_n$ continuing left moves, and four other instructions, giving
-$2(g-h_n)+3$.  The last row clears the mark and makes the work
-movement; the invariant is restored with frontier $g+1$.  This proves
-finite completion, indefinite legality for a nonhalting run, and the
-stated halt equivalence, including initial halting.
-\end{proof}
-
-\subsection{From symbolic cylinders to closed rectangles}
-
-Radix encodings turn a push or pop of a tape letter into an affine map;
-this is the generalized-shift viewpoint of
-Moore~\cite{Moore1990,Moore1991}.  The following calculation adds the
-separation of full closed rectangles needed for smooth localization.
-
-\begin{lemma}[Separated rectangle codes]\label{ba:rectangles}
-Let a deterministic partial tape table on an alphabet $\Gamma$ of size
-$m$ have the two incoming properties of Lemma~\ref{lem:bb-onecell}.
-Assign distinct odd digits $e(a)\in\{1,3,\ldots,2m-1\}$ and choose
-an integer $B\ge2m+1$.  Its local rules admit effective positive diagonal
-affine maps of determinant one between finite families of closed planar
-rectangles.  Sources are pairwise separated, as are targets.  Both the
-minimal subdivision and the subdivision by both adjacent letters given
-below realize every tape in their full symbolic cylinders.
-\end{lemma}
-\begin{proof}\label{ba:rectangles-proof}
-Use Lemma~\ref{lem:bb-onecell} with digits $d(a)=e(a)$.  Its digit
-hypotheses hold for every $B\ge2m+1$.  In the notation
-\[
- C(a_1a_2\cdots)=\sum_{j\ge1}e(a_j)B^{-j},\quad
- h_a(v)=\frac{e(a)+v}{B},\quad
- I_{a_1\cdots a_l}=h_{a_1}\cdots h_{a_l}([0,1]),
-\]
-the left stream starts immediately left of the head and the right stream
-starts at the head.  Place state $s$ in $k_s+a[0,1]^2$ with a common
-rational $a>0$.  The minimal branch maps are exactly
-\eqref{eq:bb-one-cell-branches}; scaling both charts by $a$ leaves
-$\operatorname{diag}(B^{-d},B^d)$ unchanged.
-
-For the additional full subdivision, restrict the right- and stay-move
-sources to $x\in I_\gamma$, for every full letter $\gamma$.
-Their targets become $I_{\beta\gamma}\times[0,1]$ and
-$I_\gamma\times I_\beta$, respectively, for written letter $\beta$.
-The already split left-move targets remain
-$[0,1]\times I_{\gamma\beta}$.  At a fixed target state the
-incoming direction is fixed, and the pair $(\beta,\gamma)$
-distinguishes the branches in every case.  Thus the added subdivision
-preserves separate target separation, while source separation follows
-by restriction of the shared source rectangles.  All these statements
-concern the filled closed rectangles and arbitrary symbolic tails.
-
-For completeness the quantitative gaps remain valid for the entire
-range $B\ge2m+1$: distinct first-digit intervals have gap at least
-$B^{-1}$ and distinct second-digit intervals at least $B^{-2}$.
-The source and target families therefore have within-state gaps at
-least $aB^{-2}$; different state squares have their own chosen gaps.
-Tail values lie between $1/(B-1)$ and $(2m-1)/(B-1)<1$.
-An initial constant tail after $l$ digits contributes
-$B^{-l}e(a_{\rm tail})/(B-1)$, so initialized codes are rational.
-At checkpoints the history records, or the known absolute origin of
-the record block, also recover the absolute work-head position.
-\end{proof}
-
-
-\subsection{Conventions for the remaining geometric realizations}\label{bc:section}
-\label{bc:frontier-section}
-Unless a different table is specified, subsequent routes use the move-first recorder of
-Lemma~\ref{lem:bb-movefirst}, with initial frontier two. The controls
-$(C_q,J_r,S_r,P_q,T_q)$ there are written $(W_q,P_r,B_r,C_q,D_q)$
-here, and its empty/frontier/record symbols $(e,F,r)$ are written
-$(E,F,R_r)$. Permuting work, history and mark coordinates in every row
-preserves every guard. This identifies the full partial table, its inverse
-and the exact checkpoint count $2(2+n-h')+4$, including arbitrary allowed
-tapes. A nonterminal branch means one with both endpoint controls
-nonterminal; a final branch into a halt can enter the observer.
-
-The full branches are those of Lemma~\ref{lem:bb-onecell}. For an alphabet
-of size $m$ we use one of the following explicitly indicated conventions:
-\begin{equation}\label{bc:digits}
-(B,\{d_a\})=(2m+2,\{1,3,\ldots,2m-1\}),\quad
-(2m+1,\{1,3,\ldots,2m-1\}),\quad
-(2m,\{0,2,\ldots,2m-2\}).
-\end{equation}
-All satisfy its digit bound and separation hypotheses. In the last
-convention the full blank has digit zero; boundary codes remain in the
-closed-rectangle and neighborhood conclusions. Write
-$C(a_1a_2\cdots)=\sum_{j\ge1}d_{a_j}B^{-j}$ and
-$I_v=\sum_{j=1}^{|v|}d_{v_j}B^{-j}+[0,B^{-|v|}]$.
-A constant tail after $k$ positions contributes
-$d_{\rm tail}B^{-k}/(B-1)$, so the initial codes are rational.
-Guarded three-cell rules use Lemma~\ref{lem:bb-window-rectangles} with
-$(a,b)=(1,2)$ after their own full-domain inverse has been proved.
-
-\subsection{Analytic conventions for the geometric variants}\label{bc:analytic-section}
-The whole-space baseline in these variants means smooth
-$u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$ on finite intervals,
-and a pressure representative in $CH^1$. When the competitor-gradient
-condition is omitted explicitly, the other conditions remain unchanged.
-Both cases fall under Lemma~\ref{lem:b-comparison}(i); separately stated
-pressure classes are retained. The torus baseline is its classical torus
-clause. Unless a projection is specified, the force is
-\begin{equation}\label{bc:residual}
-f_U=U_t+(U\cdot\nabla)U-\nu\Delta U,\qquad (u,p)=(U,0).
-\end{equation}
-Torus projection is the separate choice in
-Proposition~\ref{prop:projection-l2}, with changed pressure.
-Localized affine motion is supplied by Lemma~\ref{lem:bb-curl} on a
-compact time interval, using effective derivative bounds, positive lower
-scale bounds and the positive cutoff margins specified in each route.
-The potential is \eqref{eq:bb-potential}; the resulting flow fixes every
-region disjoint from its support. In particular exponential diagonal
-scales and linear reciprocal scales have respectively generators
-$\theta'\log D$ and $\diag(l'/l,-l'/l,0)$.
-
-\begin{lemma}[Finite forcing mechanisms]\label{bc:analytic}
-A finite family of the preceding pulses, joined with flat time collars,
-can be used once for initialization and repeated with period one thereafter.
-It gives a smooth effective velocity and residual force with all mixed
-derivatives uniformly bounded. In $\R^3$ both have one fixed compact
-spatial support, and the velocity has uniformly bounded kinetic energy.
-In a torus chart,
-periodized curl or planar Hamiltonian pulses have zero spatial mean, as
-does their residual force. Initial velocity and pressure are zero.
-Uniqueness holds in the comparison classes of Section~\ref{sec:model};
-in the whole-space class the competitor's gradient bound can be omitted
-if the remaining conditions are retained.
-\end{lemma}
-\begin{proof}\label{bc:analytic-proof}
-Each pulse is a finite expression in rational data, effective smooth
-switches, their derivatives, positive diagonal scales and logarithms.
-Finite positive separation margins bound every cutoff derivative. A finite
-maximum bounds a period and the loading interval, and repetitions preserve
-that bound. To evaluate near a join, a coarse enclosure of time gives a
-finite superset of possibly active pulses. Summing them avoids an exact
-comparison of a computable time with an integer. The flat collars give
-smoothness. Compact support gives every required spatial Sobolev bound.
-Lemma~\ref{lem:p2-realization} gives the asserted solution and comparison;
-on a torus the integral of each curl or Hamiltonian derivative vanishes,
-and the convection term is a divergence. Its flow satisfies
-$\det D\Phi_t=1$, because differentiating this determinant gives
-$(\divg U)(t,\Phi_t)\det D\Phi_t=0$ and its initial value is one.
-The velocity and acceleration identities in that lemma consequently apply
-to these volume-preserving diffeomorphisms.
-\end{proof}
-
-For later use, spatial chart changes are always made at the velocity level.
-If $x=o+a y$, set $\widetilde U(t,x)=aU(t,(x-o)/a)$ and then compute
-\eqref{bc:residual} at the prescribed physical $\nu$. The time and convection
-terms scale by $a$, while the Laplacian term scales by $a^{-1}$.
-Thus chart rescaling does not silently change the fixed physical viscosity.
+\section{History conventions and branch subdivision}\label{ba:codes}
+
+The general box motions now let us choose a recorder and an observer
+separately.  We begin with the two frontier recorders in
+Section~\ref{sec:bb-frontiers}, which retain history in the
+sense of Bennett~\cite{Bennett1973}.  Later we retain the distinct
+neighboring-cell guards and persistent-mark tables needed by other
+realizations.  A full-domain inverse, not merely agreement along an
+initialized run, is the criterion for sharing a compiler.
+
+For these applications let $A$ be the work alphabet with distinguished
+blank, $Q$ the states, $q_0$ the initial state, and $H\subseteq Q$
+the halting states.  Write $h_n$ for the work head after $n$ steps and
+\[
+ \delta(q,a)=(q^+(r),b(r),d(r)),\qquad
+ r=(q,a)\in\mathcal R=(Q\setminus H)\times A,
+ \quad d(r)\in\{-1,0,1\}.
+\]
+The initial head is zero and the tape has finitely specified nonblank
+cells.  Replace missing instructions by unchanged-letter stay moves to
+a halt.  If one halt square is needed, merge halting states and redirect
+their incoming instructions; no outgoing rule is changed.  This preserves
+initial halting as well.  An unreachable halt state can be added when
+needed.  These are the same normalizations as in
+Section~\ref{sec:bb-frontiers}.
+
+\subsection{Frontier recorders and their notation}
+
+We use the move-first recorder of Lemma~\ref{lem:bb-movefirst}
+with its original notation and complete partial domain.
+The delayed-head recorder of Lemma~\ref{lem:bb-direction} records the
+work displacement until its final return step.  Its table can also be
+initialized with the frontier one cell farther to the right.
+
+\begin{lemma}[Six-rule history recorder]\label{ba:six-compiler}
+For $r_0=1$ or $r_0=2$, there is a partial table with the two incoming
+properties of Lemma~\ref{lem:bb-onecell}, frontier $r_0+n$ at checkpoint
+$n$, and checkpoint control recording both the work state and the
+preceding work displacement.  Its next checkpoint takes
+$2(r_0+n-h_n)+3$ instructions.  The work head moves only on the last
+instruction.
+\end{lemma}
+\begin{proof}\label{ba:six-proof}
+Specialize Lemma~\ref{lem:bb-direction} to a singleton passive track
+$O$ and delete that coordinate.  Rename its controls
+$(M(q,e),G(\tau),A(q,d),B(q,d))$ as
+$(M_{q,e},R_\tau,F_{q,d},L_{q,d})$, and its frontier $F$ as $P$.
+The full alphabet is
+$A\times(\{\bot,P\}\sqcup\mathcal T)\times\{0,1\}$,
+where $\mathcal T=(Q\setminus H)\times\{-1,0,1\}\times A$.
+Every row and guard is the same under this bijection, so both incoming
+properties hold on the entire partial domain, including unused controls.
+Only $M_{q,e}$ with $q\in H$ are halting controls.
+
+Initialize $M_{q_0,0}$ with all marks zero, frontier at $r_0$, and
+empty history elsewhere.  The case $r_0=1$ is the initialization already
+proved in the shared lemma.  For either allowed $r_0$, checkpoint $n$
+has frontier $g=r_0+n>h_n$.  Mark the old head, scan right to $g$,
+write the record and install the new frontier at $g+1$, then return to
+the mark.  All intervening history is nonfrontier, the new frontier
+cell is empty, and there is only one mark.  Hence the same rows apply
+when $r_0=2$.  There are $g-h_n-1$ continuing right moves,
+$g-h_n$ continuing left moves, and four other instructions, giving
+$2(g-h_n)+3$.  The last row clears the mark and makes the work
+movement; the invariant is restored with frontier $g+1$.  This proves
+finite completion, indefinite legality for a nonhalting run, and the
+stated halt equivalence, including initial halting.
+\end{proof}
+
+\subsection{From symbolic cylinders to closed rectangles}
+
+Radix encodings turn a push or pop of a tape letter into an affine map;
+this is the generalized-shift viewpoint of
+Moore~\cite{Moore1990,Moore1991}.  The following calculation adds the
+separation of full closed rectangles needed for smooth localization.
+
+\begin{lemma}[Separated rectangle codes]\label{ba:rectangles}
+Let a deterministic partial tape table on an alphabet $\Gamma$ of size
+$m$ have the two incoming properties of Lemma~\ref{lem:bb-onecell}.
+Assign distinct odd digits $e(a)\in\{1,3,\ldots,2m-1\}$ and choose
+an integer $B\ge2m+1$.  Its local rules admit effective positive diagonal
+affine maps of determinant one between finite families of closed planar
+rectangles.  Sources are pairwise separated, as are targets.  Both the
+minimal subdivision and the subdivision by both adjacent letters given
+below realize every tape in their full symbolic cylinders.
+\end{lemma}
+\begin{proof}\label{ba:rectangles-proof}
+Use Lemma~\ref{lem:bb-onecell} with digits $d(a)=e(a)$.  Its digit
+hypotheses hold for every $B\ge2m+1$.  In the notation
+\[
+ C(a_1a_2\cdots)=\sum_{j\ge1}e(a_j)B^{-j},\quad
+ h_a(v)=\frac{e(a)+v}{B},\quad
+ I_{a_1\cdots a_l}=h_{a_1}\cdots h_{a_l}([0,1]),
+\]
+the left stream starts immediately left of the head and the right stream
+starts at the head.  Place state $s$ in $k_s+a[0,1]^2$ with a common
+rational $a>0$.  The minimal branch maps are exactly
+\eqref{eq:bb-one-cell-branches}; scaling both charts by $a$ leaves
+$\operatorname{diag}(B^{-d},B^d)$ unchanged.
+
+For the additional full subdivision, restrict the right- and stay-move
+sources to $x\in I_\gamma$, for every full letter $\gamma$.
+Their targets become $I_{\beta\gamma}\times[0,1]$ and
+$I_\gamma\times I_\beta$, respectively, for written letter $\beta$.
+The already split left-move targets remain
+$[0,1]\times I_{\gamma\beta}$.  At a fixed target state the
+incoming direction is fixed, and the pair $(\beta,\gamma)$
+distinguishes the branches in every case.  Thus the added subdivision
+preserves separate target separation, while source separation follows
+by restriction of the shared source rectangles.  All these statements
+concern the filled closed rectangles and arbitrary symbolic tails.
+
+For completeness the quantitative gaps remain valid for the entire
+range $B\ge2m+1$: distinct first-digit intervals have gap at least
+$B^{-1}$ and distinct second-digit intervals at least $B^{-2}$.
+The source and target families therefore have within-state gaps at
+least $aB^{-2}$; different state squares have their own chosen gaps.
+Tail values lie between $1/(B-1)$ and $(2m-1)/(B-1)<1$.
+An initial constant tail after $l$ digits contributes
+$B^{-l}e(a_{\rm tail})/(B-1)$, so initialized codes are rational.
+At checkpoints the history records, or the known absolute origin of
+the record block, also recover the absolute work-head position.
+\end{proof}
+
+
+\subsection{Conventions for the remaining geometric realizations}\label{bc:section}
+\label{bc:frontier-section}
+Unless a different table is specified, subsequent routes use the move-first recorder of
+Lemma~\ref{lem:bb-movefirst}, with initial frontier two. The controls
+$(C_q,J_r,S_r,P_q,T_q)$ there are written $(W_q,P_r,B_r,C_q,D_q)$
+here, and its empty/frontier/record symbols $(e,F,r)$ are written
+$(E,F,R_r)$. Permuting work, history and mark coordinates in every row
+preserves every guard. This identifies the full partial table, its inverse
+and the exact checkpoint count $2(2+n-h')+4$, including arbitrary allowed
+tapes. A nonterminal branch means one with both endpoint controls
+nonterminal; a final branch into a halt can enter the observer.
+
+The full branches are those of Lemma~\ref{lem:bb-onecell}. For an alphabet
+of size $m$ we use one of the following explicitly indicated conventions:
+\begin{equation}\label{bc:digits}
+(B,\{d_a\})=(2m+2,\{1,3,\ldots,2m-1\}),\quad
+(2m+1,\{1,3,\ldots,2m-1\}),\quad
+(2m,\{0,2,\ldots,2m-2\}).
+\end{equation}
+All satisfy its digit bound and separation hypotheses. In the last
+convention the full blank has digit zero; boundary codes remain in the
+closed-rectangle and neighborhood conclusions. Write
+$C(a_1a_2\cdots)=\sum_{j\ge1}d_{a_j}B^{-j}$ and
+$I_v=\sum_{j=1}^{|v|}d_{v_j}B^{-j}+[0,B^{-|v|}]$.
+A constant tail after $k$ positions contributes
+$d_{\rm tail}B^{-k}/(B-1)$, so the initial codes are rational.
+Guarded three-cell rules use Lemma~\ref{lem:bb-window-rectangles} with
+$(a,b)=(1,2)$ after their own full-domain inverse has been proved.
+
+\subsection{Analytic conventions for the geometric variants}\label{bc:analytic-section}
+The whole-space baseline in these variants means smooth
+$u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$ on finite intervals,
+and a pressure representative in $CH^1$. When the competitor-gradient
+condition is omitted explicitly, the other conditions remain unchanged.
+Both cases fall under Lemma~\ref{lem:b-comparison}(i); separately stated
+pressure classes are retained. The torus baseline is its classical torus
+clause. Unless a projection is specified, the force is
+\begin{equation}\label{bc:residual}
+f_U=U_t+(U\cdot\nabla)U-\nu\Delta U,\qquad (u,p)=(U,0).
+\end{equation}
+Torus projection is the separate choice in
+Proposition~\ref{prop:projection-l2}, with changed pressure.
+Localized affine motion is supplied by Lemma~\ref{lem:bb-curl} on a
+compact time interval, using effective derivative bounds, positive lower
+scale bounds and the positive cutoff margins specified in each route.
+The potential is \eqref{eq:bb-potential}; the resulting flow fixes every
+region disjoint from its support. In particular exponential diagonal
+scales and linear reciprocal scales have respectively generators
+$\theta'\log D$ and $\diag(l'/l,-l'/l,0)$.
+
+\begin{lemma}[Finite forcing mechanisms]\label{bc:analytic}
+A finite family of the preceding pulses, joined with flat time collars,
+can be used once for initialization and repeated with period one thereafter.
+It gives a smooth effective velocity and residual force with all mixed
+derivatives uniformly bounded. In $\R^3$ both have one fixed compact
+spatial support, and the velocity has uniformly bounded kinetic energy.
+In a torus chart,
+periodized curl or planar Hamiltonian pulses have zero spatial mean, as
+does their residual force. Initial velocity and pressure are zero.
+Uniqueness holds in the comparison classes of Section~\ref{sec:model};
+in the whole-space class the competitor's gradient bound can be omitted
+if the remaining conditions are retained.
+\end{lemma}
+\begin{proof}\label{bc:analytic-proof}
+Each pulse is a finite expression in rational data, effective smooth
+switches, their derivatives, positive diagonal scales and logarithms.
+Finite positive separation margins bound every cutoff derivative. A finite
+maximum bounds a period and the loading interval, and repetitions preserve
+that bound. To evaluate near a join, a coarse enclosure of time gives a
+finite superset of possibly active pulses. Summing them avoids an exact
+comparison of a computable time with an integer. The flat collars give
+smoothness. Compact support gives every required spatial Sobolev bound.
+Lemma~\ref{lem:p2-realization} gives the asserted solution and comparison;
+on a torus the integral of each curl or Hamiltonian derivative vanishes,
+and the convection term is a divergence. Its flow satisfies
+$\det D\Phi_t=1$, because differentiating this determinant gives
+$(\divg U)(t,\Phi_t)\det D\Phi_t=0$ and its initial value is one.
+The velocity and acceleration identities in that lemma consequently apply
+to these volume-preserving diffeomorphisms.
+\end{proof}
+
+For later use, spatial chart changes are always made at the velocity level.
+If $x=o+a y$, set $\widetilde U(t,x)=aU(t,(x-o)/a)$ and then compute
+\eqref{bc:residual} at the prescribed physical $\nu$. The time and convection
+terms scale by $a$, while the Laplacian term scales by $a^{-1}$.
+Thus chart rescaling does not silently change the fixed physical viscosity.

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compact-processors.tex

@@ -1,189 +1,189 @@
-\section{Compact support and onto slow clocks}\label{ba:euclidean}
-
-A whole-space processor can use arbitrarily separated finite lanes.
-We shall construct a periodic cycle with fixed compact support and also
-give a separate decaying force with the same material event. We first
-establish the time change used for the latter choice. Its clock must tend
-to infinity: traversing only a finite amount of machine time would not
-preserve the halting event.
-
-\subsection{The logarithmic clock at every derivative order}
-
-\begin{lemma}[Logarithmic clock with full derivative bounds]
-\label{ba:log-clock}
-Let $W(s,x)$ be a smooth solenoidal field for $s\ge0$, initially zero,
-with every mixed derivative bounded and support in one compact set $K$.
-For fixed computable $\nu>0$, set
-\[
- s(t)=\log(1+t),\quad \rho(t)=(1+t)^{-1},\quad
- U(t,x)=\rho(t)W(s(t),x),\quad
- F=U_t+(U\cdot\nabla)U-\nu\Delta U.
-\]
-Every derivative $\partial_t^k\partial_x^\alpha U$ and
-$\partial_t^k\partial_x^\alpha F$ is uniformly
-$O((1+t)^{-1-k})$ and belongs to space-time $L^2$.
-Both supports lie in $K$.  If $\Xi$ is the material flow of $W$, that
-of $U$ is $\Xi(s(t),a)$, so every fixed spatial reachability event is
-unchanged.  Effective finite formulas for $W$ give effective formulas
-for $U,F$ and all derivatives.
-\end{lemma}
-\begin{proof}\label{ba:log-clock-proof}
-Since $s'=\rho$ and $\rho'=-\rho^2$,
-\begin{equation}\label{ba:clock-rule}
- \partial_t\{\rho^jY(s(t),x)\}
-       =\rho^{j+1}(\partial_s-j)Y(s(t),x).
-\end{equation}
-For $P_{k,j}(D)=\prod_{r=0}^{k-1}(D-j-r)$, with $P_{0,j}=1$,
-induction yields
-\begin{equation}\label{ba:clock-induction}
- \partial_t^k\{\rho^jY(s(t),x)\}
-       =\rho^{j+k}P_{k,j}(\partial_s)Y(s(t),x).
-\end{equation}
-Spatial differentiation commutes with this identity.  Taking $j=1$
-proves the velocity bound at every order.  The residual is exactly
-\begin{equation}\label{ba:clock-residual}
- F=\rho^2\{W_s-W+(W\cdot\nabla)W\}(s(t),x)
-       -\nu\rho\Delta W(s(t),x).
-\end{equation}
-Every mixed derivative of the braces is bounded by the finite product
-rule and the hypotheses.  Apply \eqref{ba:clock-induction} with $j=2$
-and $j=1$ to obtain orders $\rho^{2+k}$ and $\rho^{1+k}$,
-respectively.  This gives the force bound, including every spatial
-multi-index.  In particular no temporal derivative of the convection
-term has been omitted.
-
-Supports do not increase under differentiation or multiplication.
-For either field $Z$, the squared integral of an indicated derivative
-is at most
-\[
- |K|C_{k,\alpha}^2\int_0^\infty(1+t)^{-2-2k}\,dt
-       =\frac{|K|C_{k,\alpha}^2}{1+2k}.
-\]
-The zero datum remains zero, so residual realization applies.
-The chain rule shows that $\Xi(s(t),a)$ solves the $U$ trajectory
-equation with initial label $a$; uniqueness identifies it with the
-actual trajectory.  The clock is increasing onto $[0,\infty)$ with
-inverse $e^s-1$, so it loses no finite computational time.  Finally
-logarithm on positive arguments, the rational powers of $1+t$, and
-the finite-index evaluation of the template and its derivatives are
-effective.  This proves all assertions.
-\end{proof}
-
-
-\subsection{Loaded Euclidean processors}
-
-\begin{theorem}[Two full-box processors on Euclidean space]
-\label{ba:euclidean-boxes}
-For every finite machine and input, and every fixed computable $\nu>0$,
-either the move-first recorder of Lemma~\ref{lem:bb-movefirst} or the
-six-rule recorder of Lemma~\ref{ba:six-compiler} yields effective smooth forcing on $\mathbb R^3$ with one
-compact spatial support, all mixed derivatives bounded, and period one
-for $t\ge1$, such that
-\[
- \exists t\ge0:\ (\Phi_t(0))_1<-1
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-The zero-data solution is unique in Lemma~\ref{ba:realization}'s class,
-has pressure zero and uniformly bounded energy.  Separately, the six-rule
-processor admits a force square-integrable on all space-time with the
-same support and event.
-\end{theorem}
-\begin{proof}\label{ba:euclidean-boxes-proof}
-Write $\Gamma$ for the full alphabet of the recorder being used.
-For the six-rule table use initial frontier 2 and base $B=2|\Gamma|+2$.
-Give nonhalting controls lanes $3i+[0,1]$ and halting controls lanes
-$-3i+[0,1]$, enumerating each group from 1; the second coordinate is
-$[0,1]$.  The minimal subdivision has gap $B^{-2}$.  Use full thickness
-$[-1,1]$, heights $10i$, and the transported-cutoff construction in
-Section~\ref{sec:ba-transported-localization}, with initial padding $1/(10B^3)$.
-Target horizontal padding is at most $1/(10B^2)$, below half the gap;
-vertical middle intervals are also separated.
-
-For the move-first table use a single halt lane $-3+[0,1]$, other lanes
-$2+2j+[0,1]$, base $B=2|\Gamma|+1$, the minimal subdivision, thickness
-$[-1,1]$, and heights $4i$.  Use current-box padding $1/(10B^2)$.
-The middle-phase separation is
-$2(1+1/(10B^2))<4$; the planar gaps handle the other phases.
-This localization has a fixed current padding, as distinct from the
-transported initial padding in the six-rule version.
-
-The rational initial code $P_*$ is reached from zero along
-$\theta(t)P_*$ by a localized translation during $[0,1]$.
-Repeating the chosen cycle produces a template $W$ with one compact
-support and bounded mixed derivatives.  The code identity at times
-$1+n$ follows by induction through the local rules; the respective
-recorder invariant supplies every next checkpoint in finite positive
-time.  A halt lies below $-1$ in the first coordinate.  In a nonhalting
-run all branch endpoints have positive first coordinate, and
-\eqref{eq:bb-convex-coordinate} keeps it positive throughout every cycle.
-Loading is nonnegative.  Lemma~\ref{ba:realization} completes the PDE
-argument.  Uniform support and boundedness give a uniform energy bound.
-The separate square-integrable choice follows from
-Lemma~\ref{ba:log-clock}, without claiming periodicity in physical
-time for that choice.
-\end{proof}
-
-
-\subsection{Two realizations of a left frontier}
-
-\begin{proposition}[Separate columns or solid-box parking]
-\label{prop:bb-mid-stream}\label{prop:bb-mid-parking}
-Both left-frontier tables have compact whole-space realizations,
-periodic after $t=1$, with fixed label zero and event $X_1(t;0)<-1/2$.
-The stopping version can use distinct-height stream functions and
-has uniqueness with pointwise-in-time $H^1$ comparison pressure.
-The continuing version transports full solid boxes by evacuation,
-reshaping and delivery, and represents every local step, including
-post-halt idle computation.  It has in particular uniqueness with
-comparison pressure in $C([0,T];H^1)$.
-\end{proposition}
-\begin{proof}\label{proof:bb-mid-stream}\label{proof:bb-mid-parking}
-Let $\Sigma$ be the full work--history--mark alphabet of the chosen
-left-frontier table, including its record symbols and any halting idle
-records in the continuing version. Use odd digits in base $B=2|\Sigma|+2$, offset $-2$ for $C_h$ and
-$2,4,6,\ldots$ for other controls.  The two head-relative fractions
-and height zero give a rational initial code $Z_*$.  The rightmost
-nonempty history cell is the original cell $-2$, an absolute landmark.
-
-For the stopping table, apply the stream-function construction
-\eqref{eq:bb-mid-stream-path} at heights $z_i=3i$.
-Choose source and target column paddings no greater than $1/2$ and
-one third of their respective family gaps.  A middle layer padding
-$1/2$ keeps distinct layers disjoint.  The explicit thickening bound
-following that construction proves the neighborhood action.
-Write $Z_*=(x_*,y_*,0)$. With a flat switch $\theta$ from zero to one,
-load the origin along $\theta(t)Z_*$ using the velocity
-$\nabla\times(0,0,H)$, where
-$H(t,x,y,z)=\theta'(t)\chi(x,y,z)(x_*y-y_*x)$.
-Choose $\chi$ compactly supported in $\R^3$ and equal to one near the
-entire segment. The gate $\theta'$ vanishes on the time collars.
-All nonhalt source and target lower-left first coordinates are at
-least two.  Inequality~\eqref{eq:bb-mid-stream-safe} therefore
-excludes the observer during each period; loading has nonnegative
-first coordinate.  Halt codes lie in $(-2,-1)$.
-
-For the continuing table retain its unrestricted first-row history
-symbol and its idle work rules.  Thicken every rectangle by $[-1,1]$.
-Choose $L$ ten larger than the maximum of zero and every source or
-target right endpoint.  Reserve parking projections
-$(L+4i,0)+[-1,1]^2$ and let $\delta$ be the minimum of one and the
-gaps in each union of one rectangle family with these reservations.
-Use padding $\delta/10$, travel height five, and $7N$ slots: evacuate
-all sources by three translations each, reshape all parked boxes by
-$\operatorname{diag}(1+\theta(\lambda_i-1),[1+\theta(\lambda_i-1)]^{-1},1)$,
-and deliver by three translations each.  Horizontal widths stay at
-most one, vertical half-width is one, and the reservations miss both
-original families.  Lemma~\ref{lem:bb-lift-parking} proves exact action
-on the whole boxes and their neighborhoods even with source--target
-overlap.  A first localized translation loads $Z_*$ from the origin.
-The sampling identity is now valid for every $k\ge0$, not just until
-halting: $\Phi_{1+k}(0)=Z(C_k)$ for the continued table.
-
-For a nonhalting run all used endpoint centers have first coordinate
-at least two, parking centers are farther right, and half-widths are
-at most $1/2$.  Translation, parking strain, and waiting therefore
-keep first coordinate positive.  A halt checkpoint lies in $(-2,-1)$.
-Proposition~\ref{prop:bb-realization} proves the analytic assertions
-for both constructions.  Their compact support also gives uniformly
-bounded kinetic energy.
-\end{proof}
+\section{Compact support and onto slow clocks}\label{ba:euclidean}
+
+A whole-space processor can use arbitrarily separated finite lanes.
+We shall construct a periodic cycle with fixed compact support and also
+give a separate decaying force with the same material event. We first
+establish the time change used for the latter choice. Its clock must tend
+to infinity: traversing only a finite amount of machine time would not
+preserve the halting event.
+
+\subsection{The logarithmic clock at every derivative order}
+
+\begin{lemma}[Logarithmic clock with full derivative bounds]
+\label{ba:log-clock}
+Let $W(s,x)$ be a smooth solenoidal field for $s\ge0$, initially zero,
+with every mixed derivative bounded and support in one compact set $K$.
+For fixed computable $\nu>0$, set
+\[
+ s(t)=\log(1+t),\quad \rho(t)=(1+t)^{-1},\quad
+ U(t,x)=\rho(t)W(s(t),x),\quad
+ F=U_t+(U\cdot\nabla)U-\nu\Delta U.
+\]
+Every derivative $\partial_t^k\partial_x^\alpha U$ and
+$\partial_t^k\partial_x^\alpha F$ is uniformly
+$O((1+t)^{-1-k})$ and belongs to space-time $L^2$.
+Both supports lie in $K$.  If $\Xi$ is the material flow of $W$, that
+of $U$ is $\Xi(s(t),a)$, so every fixed spatial reachability event is
+unchanged.  Effective finite formulas for $W$ give effective formulas
+for $U,F$ and all derivatives.
+\end{lemma}
+\begin{proof}\label{ba:log-clock-proof}
+Since $s'=\rho$ and $\rho'=-\rho^2$,
+\begin{equation}\label{ba:clock-rule}
+ \partial_t\{\rho^jY(s(t),x)\}
+       =\rho^{j+1}(\partial_s-j)Y(s(t),x).
+\end{equation}
+For $P_{k,j}(D)=\prod_{r=0}^{k-1}(D-j-r)$, with $P_{0,j}=1$,
+induction yields
+\begin{equation}\label{ba:clock-induction}
+ \partial_t^k\{\rho^jY(s(t),x)\}
+       =\rho^{j+k}P_{k,j}(\partial_s)Y(s(t),x).
+\end{equation}
+Spatial differentiation commutes with this identity.  Taking $j=1$
+proves the velocity bound at every order.  The residual is exactly
+\begin{equation}\label{ba:clock-residual}
+ F=\rho^2\{W_s-W+(W\cdot\nabla)W\}(s(t),x)
+       -\nu\rho\Delta W(s(t),x).
+\end{equation}
+Every mixed derivative of the braces is bounded by the finite product
+rule and the hypotheses.  Apply \eqref{ba:clock-induction} with $j=2$
+and $j=1$ to obtain orders $\rho^{2+k}$ and $\rho^{1+k}$,
+respectively.  This gives the force bound, including every spatial
+multi-index.  In particular no temporal derivative of the convection
+term has been omitted.
+
+Supports do not increase under differentiation or multiplication.
+For either field $Z$, the squared integral of an indicated derivative
+is at most
+\[
+ |K|C_{k,\alpha}^2\int_0^\infty(1+t)^{-2-2k}\,dt
+       =\frac{|K|C_{k,\alpha}^2}{1+2k}.
+\]
+The zero datum remains zero, so residual realization applies.
+The chain rule shows that $\Xi(s(t),a)$ solves the $U$ trajectory
+equation with initial label $a$; uniqueness identifies it with the
+actual trajectory.  The clock is increasing onto $[0,\infty)$ with
+inverse $e^s-1$, so it loses no finite computational time.  Finally
+logarithm on positive arguments, the rational powers of $1+t$, and
+the finite-index evaluation of the template and its derivatives are
+effective.  This proves all assertions.
+\end{proof}
+
+
+\subsection{Loaded Euclidean processors}
+
+\begin{theorem}[Two full-box processors on Euclidean space]
+\label{ba:euclidean-boxes}
+For every finite machine and input, and every fixed computable $\nu>0$,
+either the move-first recorder of Lemma~\ref{lem:bb-movefirst} or the
+six-rule recorder of Lemma~\ref{ba:six-compiler} yields effective smooth forcing on $\mathbb R^3$ with one
+compact spatial support, all mixed derivatives bounded, and period one
+for $t\ge1$, such that
+\[
+ \exists t\ge0:\ (\Phi_t(0))_1<-1
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+The zero-data solution is unique in Lemma~\ref{ba:realization}'s class,
+has pressure zero and uniformly bounded energy.  Separately, the six-rule
+processor admits a force square-integrable on all space-time with the
+same support and event.
+\end{theorem}
+\begin{proof}\label{ba:euclidean-boxes-proof}
+Write $\Gamma$ for the full alphabet of the recorder being used.
+For the six-rule table use initial frontier 2 and base $B=2|\Gamma|+2$.
+Give nonhalting controls lanes $3i+[0,1]$ and halting controls lanes
+$-3i+[0,1]$, enumerating each group from 1; the second coordinate is
+$[0,1]$.  The minimal subdivision has gap $B^{-2}$.  Use full thickness
+$[-1,1]$, heights $10i$, and the transported-cutoff construction in
+Section~\ref{sec:ba-transported-localization}, with initial padding $1/(10B^3)$.
+Target horizontal padding is at most $1/(10B^2)$, below half the gap;
+vertical middle intervals are also separated.
+
+For the move-first table use a single halt lane $-3+[0,1]$, other lanes
+$2+2j+[0,1]$, base $B=2|\Gamma|+1$, the minimal subdivision, thickness
+$[-1,1]$, and heights $4i$.  Use current-box padding $1/(10B^2)$.
+The middle-phase separation is
+$2(1+1/(10B^2))<4$; the planar gaps handle the other phases.
+This localization has a fixed current padding, as distinct from the
+transported initial padding in the six-rule version.
+
+The rational initial code $P_*$ is reached from zero along
+$\theta(t)P_*$ by a localized translation during $[0,1]$.
+Repeating the chosen cycle produces a template $W$ with one compact
+support and bounded mixed derivatives.  The code identity at times
+$1+n$ follows by induction through the local rules; the respective
+recorder invariant supplies every next checkpoint in finite positive
+time.  A halt lies below $-1$ in the first coordinate.  In a nonhalting
+run all branch endpoints have positive first coordinate, and
+\eqref{eq:bb-convex-coordinate} keeps it positive throughout every cycle.
+Loading is nonnegative.  Lemma~\ref{ba:realization} completes the PDE
+argument.  Uniform support and boundedness give a uniform energy bound.
+The separate square-integrable choice follows from
+Lemma~\ref{ba:log-clock}, without claiming periodicity in physical
+time for that choice.
+\end{proof}
+
+
+\subsection{Two realizations of a left frontier}
+
+\begin{proposition}[Separate columns or solid-box parking]
+\label{prop:bb-mid-stream}\label{prop:bb-mid-parking}
+Both left-frontier tables have compact whole-space realizations,
+periodic after $t=1$, with fixed label zero and event $X_1(t;0)<-1/2$.
+The stopping version can use distinct-height stream functions and
+has uniqueness with pointwise-in-time $H^1$ comparison pressure.
+The continuing version transports full solid boxes by evacuation,
+reshaping and delivery, and represents every local step, including
+post-halt idle computation.  It has in particular uniqueness with
+comparison pressure in $C([0,T];H^1)$.
+\end{proposition}
+\begin{proof}\label{proof:bb-mid-stream}\label{proof:bb-mid-parking}
+Let $\Sigma$ be the full work--history--mark alphabet of the chosen
+left-frontier table, including its record symbols and any halting idle
+records in the continuing version. Use odd digits in base $B=2|\Sigma|+2$, offset $-2$ for $C_h$ and
+$2,4,6,\ldots$ for other controls.  The two head-relative fractions
+and height zero give a rational initial code $Z_*$.  The rightmost
+nonempty history cell is the original cell $-2$, an absolute landmark.
+
+For the stopping table, apply the stream-function construction
+\eqref{eq:bb-mid-stream-path} at heights $z_i=3i$.
+Choose source and target column paddings no greater than $1/2$ and
+one third of their respective family gaps.  A middle layer padding
+$1/2$ keeps distinct layers disjoint.  The explicit thickening bound
+following that construction proves the neighborhood action.
+Write $Z_*=(x_*,y_*,0)$. With a flat switch $\theta$ from zero to one,
+load the origin along $\theta(t)Z_*$ using the velocity
+$\nabla\times(0,0,H)$, where
+$H(t,x,y,z)=\theta'(t)\chi(x,y,z)(x_*y-y_*x)$.
+Choose $\chi$ compactly supported in $\R^3$ and equal to one near the
+entire segment. The gate $\theta'$ vanishes on the time collars.
+All nonhalt source and target lower-left first coordinates are at
+least two.  Inequality~\eqref{eq:bb-mid-stream-safe} therefore
+excludes the observer during each period; loading has nonnegative
+first coordinate.  Halt codes lie in $(-2,-1)$.
+
+For the continuing table retain its unrestricted first-row history
+symbol and its idle work rules.  Thicken every rectangle by $[-1,1]$.
+Choose $L$ ten larger than the maximum of zero and every source or
+target right endpoint.  Reserve parking projections
+$(L+4i,0)+[-1,1]^2$ and let $\delta$ be the minimum of one and the
+gaps in each union of one rectangle family with these reservations.
+Use padding $\delta/10$, travel height five, and $7N$ slots: evacuate
+all sources by three translations each, reshape all parked boxes by
+$\operatorname{diag}(1+\theta(\lambda_i-1),[1+\theta(\lambda_i-1)]^{-1},1)$,
+and deliver by three translations each.  Horizontal widths stay at
+most one, vertical half-width is one, and the reservations miss both
+original families.  Lemma~\ref{lem:bb-lift-parking} proves exact action
+on the whole boxes and their neighborhoods even with source--target
+overlap.  A first localized translation loads $Z_*$ from the origin.
+The sampling identity is now valid for every $k\ge0$, not just until
+halting: $\Phi_{1+k}(0)=Z(C_k)$ for the continued table.
+
+For a nonhalting run all used endpoint centers have first coordinate
+at least two, parking centers are farther right, and half-widths are
+at most $1/2$.  Translation, parking strain, and waiting therefore
+keep first coordinate positive.  A halt checkpoint lies in $(-2,-1)$.
+Proposition~\ref{prop:bb-realization} proves the analytic assertions
+for both constructions.  Their compact support also gives uniformly
+bounded kinetic energy.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/compilers.tex

@@ -1,307 +1,307 @@
-\section{History records and complete symbolic domains}
-\label{sec:bb-frontiers}
-
-The balanced-stack theorem is complete.  The remaining realizations
-use alternative recorders with tape data in two coordinates.
-A motion of boxes is invertible.  To use it for an ordinary machine,
-we must retain whatever information an instruction erases.  We use
-finite history records in the sense of reversible computation, but
-prove the local inverses below: reversibility of the initialized
-execution would not by itself separate the full boxes.
-
-A work instruction is $r=(q,a)\mapsto(p_r,b_r,d_r)$, with
-$d_r\in\{-1,0,1\}$.  The work tape is indexed by $\Z$, starts with
-head zero, and has only finitely many prescribed nonblank cells.
-Halting states have no outgoing work instruction.  Missing instructions
-can be replaced by write-preserving stationary steps to a fresh halt
-state.  Unless a construction explicitly adds a dummy initial step,
-an initially halting state is already a halt checkpoint. When a construction
-uses a single halting state $h$, merge the original halting states and
-redirect their incoming transitions; this preserves initial halting. In
-the recorder tables $Q$ is the work-state set and $\Gamma$ the work
-alphabet. Radix bases are chosen from the full cell alphabet, including
-all history and mark tracks.
-
-\subsection{A one-cell code and its exact inverse test}
-
-A partial one-cell table sends a control and a scanned letter to a
-new control, a written letter, and a displacement.  Its two incoming
-conditions are: the new control determines the displacement, and the
-new control together with the written letter determines the entire
-old control and read letter.
-
-\begin{lemma}[From one-cell tables to closed rectangles]
-\label{lem:bb-onecell}\label{lem:bb-mid-prefix}
-Give an alphabet of size $m$ digits separated by at least two, between
-$0$ and $B-2$.  Put
-\[
-E(A)=\sum_{j\ge1}d(A_j)B^{-j},\qquad
-I(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}+[0,B^{-|v|}].
-\]
-For a head-relative tape $(a_j)_{j\in\Z}$ use the coordinates
-\[
- (x,y)=\big(E(a_{-1}a_{-2}\cdots),E(a_0a_1\cdots)\big).
-\]
-After writing at cell zero, a displacement $e$ makes the old cell $e$
-the new cell zero.
-A deterministic partial one-cell table satisfying the incoming
-conditions acts, after separate state translations, by a finite list
-of reciprocal affine maps on full closed rectangles.  Each source
-family and each target family is separately disjoint.  For a rule
-reading $a$, writing $b$, and moving by $e$, the branches are
-\begin{equation}\label{eq:bb-one-cell-branches}
-\begin{array}{c|c|c|c}
-e&\text{source}&\text{target}&(x',y')\\\hline
-1 &[0,1]\times I(a)&I(b)\times[0,1]
- &((d(b)+x)/B,By-d(a))\\
-0 &[0,1]\times I(a)&[0,1]\times I(b)
- &(x,y+(d(b)-d(a))/B)\\
--1&I(c)\times I(a)&[0,1]\times(d(c)+I(b))/B
- &(Bx-d(c),[d(c)+(d(b)+By-d(a))/B]/B).
-\end{array}
-\end{equation}
-The last row is split over every letter $c$.  Each linear part is
-$\operatorname{diag}(B^{-e},B^e)$.  With odd digits
-$1,3,\ldots,2m-1$ and $B\in\{2m+1,2m+2\}$, every code lies in
-the interior of its applicable rectangle, and unscaled within-family
-gaps are at least $B^{-2}$.
-\end{lemma}
-\begin{proof}
-The identity $E(aA)=(d(a)+E(A))/B$ gives all three formulas and their
-action on the independent tails.  At the first differing digit,
-incompatible prefix intervals have a positive gap: the digit difference
-is at least two while the child interval has length one in that scale.
-Thus the source state, read letter, and any left split separate sources.
-At one target state the displacement is fixed.  For a right or stationary
-move the written letter separates the relevant target coordinate;
-for a left move the pair $(c,b)$ separates the two-letter prefix.
-The incoming inverse condition rules out duplicates.  These arguments
-concern the complete intervals, including their endpoints.
-For odd digits the infinite tail lies between $1/(B-1)$ and
-$(2m-1)/(B-1)<1$, proving interiority.  The longest target prefix has
-length two, giving the asserted gap.  Uniformly scaling state squares
-by $s$ scales gaps by $s$ and leaves the linear parts unchanged.
-\end{proof}
-
-\subsection{Moving the work head before recording}
-
-We first place a frontier to the right of every possible work head.
-A temporary mark lets the history excursion return to the correct
-cell even though its length grows with the computation.
-
-\begin{lemma}[Move-first recorder]
-\label{lem:bb-movefirst}\label{lem:bb-mid-right-compiler}
-There is an effective one-cell table satisfying the incoming conditions
-whose checkpoint configurations simulate the work machine.  After
-$n$ work steps its frontier is at $n+2$, its records occupy
-$2,\ldots,n+1$, and its temporary marks are all zero.  Each nonhalting
-checkpoint has a finite positive continuation to the next one, with
-no intervening checkpoint.
-\end{lemma}
-\begin{proof}
-Use letters $(c,\ell,j)$ with work letter $c$, history
-$\ell\in\{e,F\}\sqcup\{r\}$ and mark $j\in\{0,1\}$.
-The controls are $C_q,J_r,S_r,P_p,T_p$.  The complete partial table is
-\[
-\begin{array}{lll}
- C_q:(a,\ell,0)&\mapsto J_r:(b_r,\ell,0),d_r&\ell\ne F,\\
- J_r:(c,\ell,0)&\mapsto S_r:(c,\ell,1),1&\ell\ne F,\\
- S_r:(c,\ell,0)&\mapsto S_r:(c,\ell,0),1&\ell\ne F,\\
- S_r:(c,F,0)&\mapsto P_{p_r}:(c,r,0),1,&\\
- P_p:(c,e,0)&\mapsto T_p:(c,F,0),-1,&\\
- T_p:(c,\ell,0)&\mapsto T_p:(c,\ell,0),-1&\ell\ne F,\\
- T_p:(c,\ell,1)&\mapsto C_p:(c,\ell,0),0&\ell\ne F.
-\end{array}
-\]
-Initialize the frontier at two, all other history empty, and all marks
-zero.  At checkpoint $n$ the head satisfies $|h_n|\le n$.
-The first row updates the work tape and moves to $h'\le n+1$, strictly
-left of the frontier $f=n+2$.  The second row marks $h'$.  The forward
-scan writes $r$ at $f$, puts a new frontier at $f+1$, and the backward
-scan stops exactly at the mark, which it clears.  Its length is
-$2(f-h')+4$, so it is finite and positive.  This proves the invariant
-inductively, including the exact work update and halt equivalence.
-
-For the full-domain inverse, $r$ recovers the old work pair into $J_r$;
-mark one distinguishes entry into $S_r$ from its mark-zero loop;
-a written record determines entry into $P_p$; a written $F$
-distinguishes entry into $T_p$ from its loop; and entry into $C_p$
-uniquely restores mark one.  Every destination has the single incoming
-displacement displayed in the table.  No invariant on remote cells
-was used.
-\end{proof}
-
-\subsection{Recording before the work-head displacement}
-
-The next table leaves its mark at the old work head.  Its final step
-performs the work displacement; storing that displacement in the
-checkpoint control keeps incoming moves unambiguous.
-
-\begin{lemma}[Direction-recording scan]
-\label{lem:bb-direction}\label{lem:bb-mid-mark-compiler}
-Let $D=\{-1,0,1\}$ and let $O$ be either a singleton or $\{0,1\}$.
-An effective one-cell table on
-$\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}\times O$,
-where $\mathcal R=\{(q,e,a):q\text{ nonhalting},e\in D,a\in\Gamma\}$,
-satisfies the incoming conditions.  It preserves the $O$ track in
-its physical cells.  Starting with frontier at one, its checkpoint
-at work step $n$ has frontier $n+1$, records at $1,\ldots,n$, zero
-marks, and the correct work configuration.  A step from head $i$
-takes $5+2(n-i)$ local rules.
-\end{lemma}
-\begin{proof}\label{proof:bb-mid-mark-compiler}
-Use main controls $M(q,e)$, forward controls $G(r)$, and $A(p,d),B(p,d)$
-for every work state $p$ and $d\in D$, including pairs not produced by
-a work instruction. Copy the $O$ component in each row. In the first
-three rows use every $r=(q,e,a)\in\mathcal R$ with
-$\delta(q,a)=(p,c,d)$. The final three rows apply independently to
-every $(p,d)\in Q\times D$, including pairs that are not outputs of
-any work instruction. The variables $v\in\Gamma$ and
-$h\in\{\bot,F\}\sqcup\mathcal R$ range over all letters subject to
-the displayed restrictions. The whole table is
-\[
-\begin{array}{lll}
- M(q,e):(a,h,0)&\mapsto G(r):(c,h,1),1&h\ne F,\\
- G(r):(v,h,0)&\mapsto G(r):(v,h,0),1&h\ne F,\\
- G(r):(v,F,0)&\mapsto A(p,d):(v,r,0),1,&\\
- A(p,d):(v,\bot,0)&\mapsto B(p,d):(v,F,0),-1,&\\
- B(p,d):(v,h,0)&\mapsto B(p,d):(v,h,0),-1&h\ne F,\\
- B(p,d):(v,h,1)&\mapsto M(p,d):(v,h,0),d&h\ne F.
-\end{array}
-\]
-Starting at $M(q_0,0)$, the first row writes and marks the old head.
-The scan reaches frontier $j=n+1$, stores its record, creates the
-next frontier, returns to the unique mark and makes the work move.
-The count is $2(j-i)+3=5+2(n-i)$.  Since $i\le n<j$, every test is
-satisfied and both scans are finite.
-
-Into $G(r)$ the written mark distinguishes entry and loop; $r$
-recovers the old state and work symbol.  Into $A(p,d)$ the record
-recovers the old $G(r)$.  Into $B(p,d)$ a written frontier distinguishes
-arrival and loop.  Into $M(p,d)$ only clearing a mark is possible.
-All remaining components were copied.  This proves the inverse on
-arbitrary tapes and the fixed incoming displacement.  Deleting the
-passive $O$ coordinate is an exact specialization of every rule,
-not merely an equivalence of initialized runs.  When $O=\{0,1\}$,
-a unique initial one at physical cell zero records the absolute origin.
-\end{proof}
-
-\subsection{A frontier on the left and post-halt continuation}
-
-\begin{lemma}[Two left-frontier tables]\label{lem:bb-mid-left-compiler}
-There are two effective incoming-separated tables: a stopping table,
-and a table which continues with idle work steps after halting.
-At checkpoint $n$ their frontier is $f_n=-2-n$ and their marks vanish.
-If the next work move ends at $i'$, the next checkpoint is reached
-after $4+2(i'-f_n)$ local rules.
-\end{lemma}
-\begin{proof}
-For the stopping version let $J=(Q\setminus\{h\})\times\Gamma$.
-For the continuing version take $J=Q\times\Gamma$ and add
-$\delta(h,a)=(h,a,0)$.  Use letters
-$(c,\eta,j)\in\Gamma\times(\{\bot,F\}\sqcup J)\times\{0,1\}$,
-controls $C_q,D_q,T_q,P_r,S_r$, and precisely
-\begin{equation}\label{eq:bb-mid-left-table}
-\begin{aligned}
- C_q:(a,\eta,0)&\mapsto P_r:(b_r,\eta,0),d_r,\\
- P_r:(c,\eta,0)&\mapsto S_r:(c,\eta,1),-1&&\eta\ne F,\\
- S_r:(c,\eta,0)&\mapsto S_r:(c,\eta,0),-1&&\eta\ne F,\\
- S_r:(c,F,0)&\mapsto D_{p_r}:(c,r,0),-1,\\
- D_q:(c,\bot,0)&\mapsto T_q:(c,F,0),1,\\
- T_q:(c,\eta,0)&\mapsto T_q:(c,\eta,0),1&&\eta\ne F,\\
- T_q:(c,\eta,1)&\mapsto C_q:(c,\eta,0),0&&\eta\ne F.
-\end{aligned}
-\end{equation}
-In the first row only, the stopping version requires $\eta\ne F$;
-the continuing version allows every $\eta$.  These domains remain
-distinct even though initialized runs never use that difference.
-
-Initialize $F$ at $-2$, empty history elsewhere and zero marks.
-At checkpoint $n$, $i\ge-n=f_n+2$.  After the work update,
-$i'\ge f_n+1$.  Marking $i'$ and scanning left therefore reaches
-$f_n$, records $r$, puts the next frontier at $f_n-1$, and returns
-right to the mark without crossing the new frontier.  Clearing it
-completes the next work checkpoint.  The two finite traversals and
-four endpoint actions give the stated count.  The continuing version
-has a next rule even after every halt checkpoint.
-
-Into $P_r$ the record index recovers the work pair.  Into $S_r$
-mark one distinguishes entry from loop; into $D_q$ the written record
-recovers the scan; into $T_q$ a written $F$ distinguishes entry from
-loop; into $C_q$ there is only stationary unmarking.  This proves the
-incoming conditions for both full partial domains.  It also gives
-the configuration inverse: reverse the displacement, inspect the
-written cell, and invert its uniquely determined rule.
-\end{proof}
-
-\subsection{A guarded update on two neighboring history cells}
-
-One may advance the frontier and write its record in a single local
-instruction.  The resulting return loop needs a neighboring-cell guard.
-Without that guard, its image would overlap the logging image.
-
-\begin{lemma}[Three-cell guarded window]\label{lem:bb-window-compiler}
-There is a finite partial injective map on relative tapes with controls
-$\mathrm{Ready}(q,e)$, $\mathrm{Go}(r)$ and $\mathrm{Back}(q,d)$.
-Here $r=(q,e,a)$ records an instruction, with
-$\delta(q,a)=(p(r),b(r),d(r))$.
-Use work, history and mark tracks $A,H,J$.  At cursor position $i$,
-the Ready row has $a=A_i$ and $r=(q,e,a)$.  The entire table is
-\[
-\begin{array}{c|l|l}
-\text{control}&\text{test}&\text{action}\\\hline
-\mathrm{Ready}(q,e)&q\text{ nonhalting},J_i=0
- &A_i\gets b(r),\ J_i\gets1,\ \mathrm{Go}(r),+1\\
-\mathrm{Go}(r)&J_i=0,H_i\ne P&\mathrm{Go}(r),+1\\
-\mathrm{Go}(r)&J_i=0,H_i=P,H_{i+1}=E
- &H_i\gets r,H_{i+1}\gets P,\ \mathrm{Back}(p(r),d(r)),-1\\
-\mathrm{Back}(q,d)&J_i=0,H_{i+1}\ne P&\mathrm{Back}(q,d),-1\\
-\mathrm{Back}(q,d)&J_i=1&J_i\gets0,\ \mathrm{Ready}(q,d),d.
-\end{array}
-\]
-Unmentioned cells and tracks are copied.  Initialized with frontier
-at $r_0\in\{1,2\}$, other history $E$, zero marks and
-$\mathrm{Ready}(q_0,0)$, the checkpoint at step $n$ has frontier
-$p_n=n+r_0$ and correct head
-$h_n$.  Its next work step uses exactly
-$2(p_n-h_n)+1\le4n+2r_0+1$ local rules.
-\end{lemma}
-\begin{proof}
-The first rule writes and marks $h_n$.  Since $p_n>h_n$,
-the forward scan reaches the frontier, moves it one cell right, and
-the return scan reaches the mark.  Its neighboring-cell guard holds
-because the new frontier is $p_n+1$.  The last rule clears the mark
-and performs the work displacement.  Counting the two traversals and
-endpoint actions gives $2(p_n-h_n)+1$; induction gives $|h_n|\le n$.
-
-For injectivity on all relative tapes, into Go the old cursor is
-output offset $-1$: a mark there distinguishes entry from the loop,
-and $r$ restores the overwritten work letter and old Ready control.
-Into Back the old cursor is offset $+1$.  An output pointer at $+2$
-identifies a log insertion, with record at $+1$ recovering $r$ and
-old pair $(P,E)$.  A nonpointer at $+2$ identifies the guarded loop.
-Into Ready the corresponding Back control uniquely restores the
-cleared mark.  These are all possibilities, proving the full inverse.
-\end{proof}
-
-\begin{lemma}[Closed rectangles for a finite guarded window]
-\label{lem:bb-window-rectangles}
-Suppose a finite partial injective local tape map reads and writes only
-indices $[-a,b-1]$, then shifts the cursor by $e$, where
-$-a\le e\le b$.  Split its domain by every complete word in that
-window.  With gapped radix coding the input prefix lengths are
-$(a,b)$ and output lengths $(a+e,b-e)$; the affine branch has linear
-part $\operatorname{diag}(B^{-e},B^e)$.  Both full rectangle families
-are separately disjoint.
-\end{lemma}
-\begin{proof}
-After the writes the new left prefix consists of old indices
-$e-1,e-2,\ldots,-a$, and the right prefix of $e,e+1,\ldots,b-1$.
-The two remaining tails are free and unchanged, so the image is the
-whole output cylinder.  Distinct source cylinders have disjoint
-images by injectivity.  If both pairs of output prefixes were
-compatible, a common extension would belong to both images; hence
-at least one coordinate has incompatible prefixes.  The gapped
-interval argument then separates the complete closed rectangles.
-The prefix lengths give the two reciprocal scales directly.
-\end{proof}
+\section{History records and complete symbolic domains}
+\label{sec:bb-frontiers}
+
+The balanced-stack theorem is complete.  The remaining realizations
+use alternative recorders with tape data in two coordinates.
+A motion of boxes is invertible.  To use it for an ordinary machine,
+we must retain whatever information an instruction erases.  We use
+finite history records in the sense of reversible computation, but
+prove the local inverses below: reversibility of the initialized
+execution would not by itself separate the full boxes.
+
+A work instruction is $r=(q,a)\mapsto(p_r,b_r,d_r)$, with
+$d_r\in\{-1,0,1\}$.  The work tape is indexed by $\Z$, starts with
+head zero, and has only finitely many prescribed nonblank cells.
+Halting states have no outgoing work instruction.  Missing instructions
+can be replaced by write-preserving stationary steps to a fresh halt
+state.  Unless a construction explicitly adds a dummy initial step,
+an initially halting state is already a halt checkpoint. When a construction
+uses a single halting state $h$, merge the original halting states and
+redirect their incoming transitions; this preserves initial halting. In
+the recorder tables $Q$ is the work-state set and $\Gamma$ the work
+alphabet. Radix bases are chosen from the full cell alphabet, including
+all history and mark tracks.
+
+\subsection{A one-cell code and its exact inverse test}
+
+A partial one-cell table sends a control and a scanned letter to a
+new control, a written letter, and a displacement.  Its two incoming
+conditions are: the new control determines the displacement, and the
+new control together with the written letter determines the entire
+old control and read letter.
+
+\begin{lemma}[From one-cell tables to closed rectangles]
+\label{lem:bb-onecell}\label{lem:bb-mid-prefix}
+Give an alphabet of size $m$ digits separated by at least two, between
+$0$ and $B-2$.  Put
+\[
+E(A)=\sum_{j\ge1}d(A_j)B^{-j},\qquad
+I(v)=\sum_{j=1}^{|v|}d(v_j)B^{-j}+[0,B^{-|v|}].
+\]
+For a head-relative tape $(a_j)_{j\in\Z}$ use the coordinates
+\[
+ (x,y)=\big(E(a_{-1}a_{-2}\cdots),E(a_0a_1\cdots)\big).
+\]
+After writing at cell zero, a displacement $e$ makes the old cell $e$
+the new cell zero.
+A deterministic partial one-cell table satisfying the incoming
+conditions acts, after separate state translations, by a finite list
+of reciprocal affine maps on full closed rectangles.  Each source
+family and each target family is separately disjoint.  For a rule
+reading $a$, writing $b$, and moving by $e$, the branches are
+\begin{equation}\label{eq:bb-one-cell-branches}
+\begin{array}{c|c|c|c}
+e&\text{source}&\text{target}&(x',y')\\\hline
+1 &[0,1]\times I(a)&I(b)\times[0,1]
+ &((d(b)+x)/B,By-d(a))\\
+0 &[0,1]\times I(a)&[0,1]\times I(b)
+ &(x,y+(d(b)-d(a))/B)\\
+-1&I(c)\times I(a)&[0,1]\times(d(c)+I(b))/B
+ &(Bx-d(c),[d(c)+(d(b)+By-d(a))/B]/B).
+\end{array}
+\end{equation}
+The last row is split over every letter $c$.  Each linear part is
+$\operatorname{diag}(B^{-e},B^e)$.  With odd digits
+$1,3,\ldots,2m-1$ and $B\in\{2m+1,2m+2\}$, every code lies in
+the interior of its applicable rectangle, and unscaled within-family
+gaps are at least $B^{-2}$.
+\end{lemma}
+\begin{proof}
+The identity $E(aA)=(d(a)+E(A))/B$ gives all three formulas and their
+action on the independent tails.  At the first differing digit,
+incompatible prefix intervals have a positive gap: the digit difference
+is at least two while the child interval has length one in that scale.
+Thus the source state, read letter, and any left split separate sources.
+At one target state the displacement is fixed.  For a right or stationary
+move the written letter separates the relevant target coordinate;
+for a left move the pair $(c,b)$ separates the two-letter prefix.
+The incoming inverse condition rules out duplicates.  These arguments
+concern the complete intervals, including their endpoints.
+For odd digits the infinite tail lies between $1/(B-1)$ and
+$(2m-1)/(B-1)<1$, proving interiority.  The longest target prefix has
+length two, giving the asserted gap.  Uniformly scaling state squares
+by $s$ scales gaps by $s$ and leaves the linear parts unchanged.
+\end{proof}
+
+\subsection{Moving the work head before recording}
+
+We first place a frontier to the right of every possible work head.
+A temporary mark lets the history excursion return to the correct
+cell even though its length grows with the computation.
+
+\begin{lemma}[Move-first recorder]
+\label{lem:bb-movefirst}\label{lem:bb-mid-right-compiler}
+There is an effective one-cell table satisfying the incoming conditions
+whose checkpoint configurations simulate the work machine.  After
+$n$ work steps its frontier is at $n+2$, its records occupy
+$2,\ldots,n+1$, and its temporary marks are all zero.  Each nonhalting
+checkpoint has a finite positive continuation to the next one, with
+no intervening checkpoint.
+\end{lemma}
+\begin{proof}
+Use letters $(c,\ell,j)$ with work letter $c$, history
+$\ell\in\{e,F\}\sqcup\{r\}$ and mark $j\in\{0,1\}$.
+The controls are $C_q,J_r,S_r,P_p,T_p$.  The complete partial table is
+\[
+\begin{array}{lll}
+ C_q:(a,\ell,0)&\mapsto J_r:(b_r,\ell,0),d_r&\ell\ne F,\\
+ J_r:(c,\ell,0)&\mapsto S_r:(c,\ell,1),1&\ell\ne F,\\
+ S_r:(c,\ell,0)&\mapsto S_r:(c,\ell,0),1&\ell\ne F,\\
+ S_r:(c,F,0)&\mapsto P_{p_r}:(c,r,0),1,&\\
+ P_p:(c,e,0)&\mapsto T_p:(c,F,0),-1,&\\
+ T_p:(c,\ell,0)&\mapsto T_p:(c,\ell,0),-1&\ell\ne F,\\
+ T_p:(c,\ell,1)&\mapsto C_p:(c,\ell,0),0&\ell\ne F.
+\end{array}
+\]
+Initialize the frontier at two, all other history empty, and all marks
+zero.  At checkpoint $n$ the head satisfies $|h_n|\le n$.
+The first row updates the work tape and moves to $h'\le n+1$, strictly
+left of the frontier $f=n+2$.  The second row marks $h'$.  The forward
+scan writes $r$ at $f$, puts a new frontier at $f+1$, and the backward
+scan stops exactly at the mark, which it clears.  Its length is
+$2(f-h')+4$, so it is finite and positive.  This proves the invariant
+inductively, including the exact work update and halt equivalence.
+
+For the full-domain inverse, $r$ recovers the old work pair into $J_r$;
+mark one distinguishes entry into $S_r$ from its mark-zero loop;
+a written record determines entry into $P_p$; a written $F$
+distinguishes entry into $T_p$ from its loop; and entry into $C_p$
+uniquely restores mark one.  Every destination has the single incoming
+displacement displayed in the table.  No invariant on remote cells
+was used.
+\end{proof}
+
+\subsection{Recording before the work-head displacement}
+
+The next table leaves its mark at the old work head.  Its final step
+performs the work displacement; storing that displacement in the
+checkpoint control keeps incoming moves unambiguous.
+
+\begin{lemma}[Direction-recording scan]
+\label{lem:bb-direction}\label{lem:bb-mid-mark-compiler}
+Let $D=\{-1,0,1\}$ and let $O$ be either a singleton or $\{0,1\}$.
+An effective one-cell table on
+$\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}\times O$,
+where $\mathcal R=\{(q,e,a):q\text{ nonhalting},e\in D,a\in\Gamma\}$,
+satisfies the incoming conditions.  It preserves the $O$ track in
+its physical cells.  Starting with frontier at one, its checkpoint
+at work step $n$ has frontier $n+1$, records at $1,\ldots,n$, zero
+marks, and the correct work configuration.  A step from head $i$
+takes $5+2(n-i)$ local rules.
+\end{lemma}
+\begin{proof}\label{proof:bb-mid-mark-compiler}
+Use main controls $M(q,e)$, forward controls $G(r)$, and $A(p,d),B(p,d)$
+for every work state $p$ and $d\in D$, including pairs not produced by
+a work instruction. Copy the $O$ component in each row. In the first
+three rows use every $r=(q,e,a)\in\mathcal R$ with
+$\delta(q,a)=(p,c,d)$. The final three rows apply independently to
+every $(p,d)\in Q\times D$, including pairs that are not outputs of
+any work instruction. The variables $v\in\Gamma$ and
+$h\in\{\bot,F\}\sqcup\mathcal R$ range over all letters subject to
+the displayed restrictions. The whole table is
+\[
+\begin{array}{lll}
+ M(q,e):(a,h,0)&\mapsto G(r):(c,h,1),1&h\ne F,\\
+ G(r):(v,h,0)&\mapsto G(r):(v,h,0),1&h\ne F,\\
+ G(r):(v,F,0)&\mapsto A(p,d):(v,r,0),1,&\\
+ A(p,d):(v,\bot,0)&\mapsto B(p,d):(v,F,0),-1,&\\
+ B(p,d):(v,h,0)&\mapsto B(p,d):(v,h,0),-1&h\ne F,\\
+ B(p,d):(v,h,1)&\mapsto M(p,d):(v,h,0),d&h\ne F.
+\end{array}
+\]
+Starting at $M(q_0,0)$, the first row writes and marks the old head.
+The scan reaches frontier $j=n+1$, stores its record, creates the
+next frontier, returns to the unique mark and makes the work move.
+The count is $2(j-i)+3=5+2(n-i)$.  Since $i\le n<j$, every test is
+satisfied and both scans are finite.
+
+Into $G(r)$ the written mark distinguishes entry and loop; $r$
+recovers the old state and work symbol.  Into $A(p,d)$ the record
+recovers the old $G(r)$.  Into $B(p,d)$ a written frontier distinguishes
+arrival and loop.  Into $M(p,d)$ only clearing a mark is possible.
+All remaining components were copied.  This proves the inverse on
+arbitrary tapes and the fixed incoming displacement.  Deleting the
+passive $O$ coordinate is an exact specialization of every rule,
+not merely an equivalence of initialized runs.  When $O=\{0,1\}$,
+a unique initial one at physical cell zero records the absolute origin.
+\end{proof}
+
+\subsection{A frontier on the left and post-halt continuation}
+
+\begin{lemma}[Two left-frontier tables]\label{lem:bb-mid-left-compiler}
+There are two effective incoming-separated tables: a stopping table,
+and a table which continues with idle work steps after halting.
+At checkpoint $n$ their frontier is $f_n=-2-n$ and their marks vanish.
+If the next work move ends at $i'$, the next checkpoint is reached
+after $4+2(i'-f_n)$ local rules.
+\end{lemma}
+\begin{proof}
+For the stopping version let $J=(Q\setminus\{h\})\times\Gamma$.
+For the continuing version take $J=Q\times\Gamma$ and add
+$\delta(h,a)=(h,a,0)$.  Use letters
+$(c,\eta,j)\in\Gamma\times(\{\bot,F\}\sqcup J)\times\{0,1\}$,
+controls $C_q,D_q,T_q,P_r,S_r$, and precisely
+\begin{equation}\label{eq:bb-mid-left-table}
+\begin{aligned}
+ C_q:(a,\eta,0)&\mapsto P_r:(b_r,\eta,0),d_r,\\
+ P_r:(c,\eta,0)&\mapsto S_r:(c,\eta,1),-1&&\eta\ne F,\\
+ S_r:(c,\eta,0)&\mapsto S_r:(c,\eta,0),-1&&\eta\ne F,\\
+ S_r:(c,F,0)&\mapsto D_{p_r}:(c,r,0),-1,\\
+ D_q:(c,\bot,0)&\mapsto T_q:(c,F,0),1,\\
+ T_q:(c,\eta,0)&\mapsto T_q:(c,\eta,0),1&&\eta\ne F,\\
+ T_q:(c,\eta,1)&\mapsto C_q:(c,\eta,0),0&&\eta\ne F.
+\end{aligned}
+\end{equation}
+In the first row only, the stopping version requires $\eta\ne F$;
+the continuing version allows every $\eta$.  These domains remain
+distinct even though initialized runs never use that difference.
+
+Initialize $F$ at $-2$, empty history elsewhere and zero marks.
+At checkpoint $n$, $i\ge-n=f_n+2$.  After the work update,
+$i'\ge f_n+1$.  Marking $i'$ and scanning left therefore reaches
+$f_n$, records $r$, puts the next frontier at $f_n-1$, and returns
+right to the mark without crossing the new frontier.  Clearing it
+completes the next work checkpoint.  The two finite traversals and
+four endpoint actions give the stated count.  The continuing version
+has a next rule even after every halt checkpoint.
+
+Into $P_r$ the record index recovers the work pair.  Into $S_r$
+mark one distinguishes entry from loop; into $D_q$ the written record
+recovers the scan; into $T_q$ a written $F$ distinguishes entry from
+loop; into $C_q$ there is only stationary unmarking.  This proves the
+incoming conditions for both full partial domains.  It also gives
+the configuration inverse: reverse the displacement, inspect the
+written cell, and invert its uniquely determined rule.
+\end{proof}
+
+\subsection{A guarded update on two neighboring history cells}
+
+One may advance the frontier and write its record in a single local
+instruction.  The resulting return loop needs a neighboring-cell guard.
+Without that guard, its image would overlap the logging image.
+
+\begin{lemma}[Three-cell guarded window]\label{lem:bb-window-compiler}
+There is a finite partial injective map on relative tapes with controls
+$\mathrm{Ready}(q,e)$, $\mathrm{Go}(r)$ and $\mathrm{Back}(q,d)$.
+Here $r=(q,e,a)$ records an instruction, with
+$\delta(q,a)=(p(r),b(r),d(r))$.
+Use work, history and mark tracks $A,H,J$.  At cursor position $i$,
+the Ready row has $a=A_i$ and $r=(q,e,a)$.  The entire table is
+\[
+\begin{array}{c|l|l}
+\text{control}&\text{test}&\text{action}\\\hline
+\mathrm{Ready}(q,e)&q\text{ nonhalting},J_i=0
+ &A_i\gets b(r),\ J_i\gets1,\ \mathrm{Go}(r),+1\\
+\mathrm{Go}(r)&J_i=0,H_i\ne P&\mathrm{Go}(r),+1\\
+\mathrm{Go}(r)&J_i=0,H_i=P,H_{i+1}=E
+ &H_i\gets r,H_{i+1}\gets P,\ \mathrm{Back}(p(r),d(r)),-1\\
+\mathrm{Back}(q,d)&J_i=0,H_{i+1}\ne P&\mathrm{Back}(q,d),-1\\
+\mathrm{Back}(q,d)&J_i=1&J_i\gets0,\ \mathrm{Ready}(q,d),d.
+\end{array}
+\]
+Unmentioned cells and tracks are copied.  Initialized with frontier
+at $r_0\in\{1,2\}$, other history $E$, zero marks and
+$\mathrm{Ready}(q_0,0)$, the checkpoint at step $n$ has frontier
+$p_n=n+r_0$ and correct head
+$h_n$.  Its next work step uses exactly
+$2(p_n-h_n)+1\le4n+2r_0+1$ local rules.
+\end{lemma}
+\begin{proof}
+The first rule writes and marks $h_n$.  Since $p_n>h_n$,
+the forward scan reaches the frontier, moves it one cell right, and
+the return scan reaches the mark.  Its neighboring-cell guard holds
+because the new frontier is $p_n+1$.  The last rule clears the mark
+and performs the work displacement.  Counting the two traversals and
+endpoint actions gives $2(p_n-h_n)+1$; induction gives $|h_n|\le n$.
+
+For injectivity on all relative tapes, into Go the old cursor is
+output offset $-1$: a mark there distinguishes entry from the loop,
+and $r$ restores the overwritten work letter and old Ready control.
+Into Back the old cursor is offset $+1$.  An output pointer at $+2$
+identifies a log insertion, with record at $+1$ recovering $r$ and
+old pair $(P,E)$.  A nonpointer at $+2$ identifies the guarded loop.
+Into Ready the corresponding Back control uniquely restores the
+cleared mark.  These are all possibilities, proving the full inverse.
+\end{proof}
+
+\begin{lemma}[Closed rectangles for a finite guarded window]
+\label{lem:bb-window-rectangles}
+Suppose a finite partial injective local tape map reads and writes only
+indices $[-a,b-1]$, then shifts the cursor by $e$, where
+$-a\le e\le b$.  Split its domain by every complete word in that
+window.  With gapped radix coding the input prefix lengths are
+$(a,b)$ and output lengths $(a+e,b-e)$; the affine branch has linear
+part $\operatorname{diag}(B^{-e},B^e)$.  Both full rectangle families
+are separately disjoint.
+\end{lemma}
+\begin{proof}
+After the writes the new left prefix consists of old indices
+$e-1,e-2,\ldots,-a$, and the right prefix of $e,e+1,\ldots,b-1$.
+The two remaining tails are free and unchanged, so the image is the
+whole output cylinder.  Distinct source cylinders have disjoint
+images by injectivity.  If both pairs of output prefixes were
+compatible, a common extension would belong to both images; hence
+at least one coordinate has incompatible prefixes.  The gapped
+interval argument then separates the complete closed rectangles.
+The prefix lengths give the two reciprocal scales directly.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/geometry.tex

@@ -1,324 +1,324 @@
-\section{Transporting complete boxes}
-\label{sec:bb-boxes}
-
-We allow an instruction to change all three side lengths while
-preserving their product. The geometric question is concrete:
-given finitely many disjoint closed boxes and prescribed determinant-one
-affine maps onto another disjoint family, can one carry out every map by
-a smooth incompressible motion?  The two families may overlap each
-other.  Consequently a destination cannot be occupied until every source
-that obstructs it has been removed.
-
-We first extend an explicitly separated motion to a velocity field.
-We then give two ways to produce that motion.  Separate horizontal
-projections permit a simple lift and transfer.  For general solid boxes
-we use parking and detours around the other occupied boxes.
-Figure~\ref{fig:bb-routing} contrasts the height separation with the
-evacuation order that makes overlapping source and target families harmless.
-
-\subsection{An affine motion and its curl extension}
-
-Throughout, a smooth switch is obtained by translating and rescaling
-$\sigma$ in \eqref{eq:p2-step}, with its transition strictly inside
-its allotted interval.  Thus successive motions are stationary on
-neighborhoods of their joining times.
-
-\begin{lemma}[A velocity on a neighborhood of each moving box]
-\label{lem:bb-curl}
-Let $K_i(t)=c_i(t)+D_i(t)K_i^0$, $1\le i\le N$, where $K_i^0$ is
-an axis-parallel box, possibly with zero side lengths, and $D_i(t)$ is
-positive diagonal with determinant one.  Suppose the centers and scales
-are smooth and constant near the ends of a compact time interval.
-For the effective conclusion, assume computable endpoints for the
-initial boxes $K_i^0$, effective evaluation of the centers' and scales'
-derivatives, bounds for those derivatives, and positive lower bounds
-for every scale on that interval.
-If the coordinatewise $2\eta$ enlargements of the moving boxes are
-pairwise disjoint and bounded, for a known $\eta>0$, a
-compactly supported smooth divergence-free field realizes these motions
-on open neighborhoods of the initial boxes.  It vanishes near the time
-endpoints.  When all enlarged boxes lie in an interior torus chart,
-the field extends to a mean-zero field on that torus.
-The field is effective under the effective-data assumptions above;
-smooth existence itself does not require computable data.
-\end{lemma}
-\begin{proof}
-Choose a smooth cutoff $\chi_i$ which is one on the $\eta$
-enlargement of $K_i(t)$ and supported in its $2\eta$ enlargement,
-using products of the effective switches.  Put $y=x-c_i$ and
-$L_i=\dot D_iD_i^{-1}$; then $\operatorname{tr}L_i=0$.
-Take the curl of
-\begin{equation}\label{eq:bb-potential}
- \chi_i\{\tfrac12\dot c_i\times y+\tfrac13(L_i y)\times y\}.
-\end{equation}
-The identity
-$\nabla\times(G\times y)=2G-y\operatorname{div}G+(y\cdot\nabla)G$
-shows that this curl equals $\dot c_i+L_i y$ on the plateau.
-Disjoint supports permit summation over $i$.  The path
-$c_i(t)+D_i(t)D_i(t_0)^{-1}(x_0-c_i(t_0))$ has exactly that
-derivative and remains in its prescribed box.  ODE uniqueness proves
-the formula.  On the compact time interval the normalized matrix norm
-has a finite bound $M$, computable under the effective-data assumptions;
-an initial perturbation less than
-$\eta/(2M)$ remains on the plateau.  This proves the neighborhood
-claim, also for zero-width initial boxes.
-The potential is zero near time endpoints because the centers and
-scales are constant there.  Its support is bounded.  Within a torus
-chart extend the potentials by zero; periodic curls have zero integral.
-\end{proof}
-
-\begin{proposition}[Fluid realization and a pressure refinement]
-\label{prop:bb-realization}
-Fix $\nu>0$.  A finite collection of the preceding motions, repeated
-with period one, or preceded by one finite loading interval, yields a
-smooth prescribed velocity $U$ with $U(0)=0$.  Suppose its spatial
-support is contained in one compact set in $\R^3$, or its potentials
-are contained in interior torus charts.  Then
-\begin{equation}\label{eq:bb-residual}
- f=U_t+(U\cdot\nabla)U-\nu\Delta U
-\end{equation}
-has the same support and time periodicity and bounded mixed derivatives
-of every order.  The solution is $(U,0)$, with the uniqueness and
-material-flow conclusions of Lemma~\ref{lem:p2-realization}.
-In $\R^3$ its uniqueness statement also holds when the competing
-pressure has an $H^1$ representative at each time, without requiring
-continuity into $H^1$, while the other velocity hypotheses of
-Section~\ref{sec:model} are retained.  On the torus $U$ and $f$ have
-zero mean. Under the effective-data assumptions of Lemma~\ref{lem:bb-curl},
-the formula is $f_0+\nu f_1$ with effective coefficients;
-evaluation is effective for computable $\nu$, and relative to $\nu$
-otherwise.
-\end{proposition}
-\begin{proof}
-The residual identity and Lemma~\ref{lem:b-comparison} give existence,
-uniqueness and the material flow.  In the whole-space pressure refinement
-use case (ii): the reference field has compact support and smooth bounded
-mixed derivatives, hence belongs to $CH^2\cap C^1L^2$ with bounded
-velocity and gradient; the competing velocity and pointwise $H^1$
-pressure are exactly that case's hypotheses.  No pressure continuity
-in time is added.  Integrating the residual on a torus uses
-$\int U=0$, $\int\Delta U=0$ and
-$\int(U\cdot\nabla)U=\int\operatorname{div}(U\otimes U)=0$.
-
-Each derivative of $e^{-1/s}$ for $s>0$ is a polynomial in $1/s$
-times $e^{-1/s}$.  The estimate
-$v^m e^{-v}\le(m+k)!v^{-k}$ supplies effective endpoint error bounds,
-and the denominator defining $\sigma$ is at least $e^{-2}$.
-Under those effective-data assumptions all positive scales have effective
-positive lower bounds. Finite
-compositions, differentiation, and the logarithms of positive computable
-scales are therefore effective.  At an approximate time, a coarse bound
-gives a finite superset of possible translates of the zero-extended
-unit template.  This evaluates the periodic field without deciding
-whether time is an integer.  None of these operations iterates the
-machine execution: the program contains the entire finite instruction
-list.
-\end{proof}
-
-\subsection{Three geometric schedules}
-
-For later use we describe the exact hypotheses of the schedules.
-First suppose $P_i,Q_i\subset\R^2$ are finite families of closed
-rectangles, possibly with zero side lengths, separately
-pairwise disjoint, with affine maps
-\[
- F_i(x)=q_i+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(x-p_i),
- \qquad \lambda_i>0,
-\]
-where $p_i,q_i$ are their centers, and require $F_i(P_i)=Q_i$.
-Their side lengths have a common finite bound. Source and target rectangles
-belonging to different families need not be disjoint. The smooth existence
-statements allow arbitrary real rectangle coordinates and scales. For the
-effective conclusions, assume rational rectangle endpoints and rational
-scales, together with computable chosen heights and vertical half-widths;
-then a rational side-length bound and positive gap margins can be computed.
-
-\begin{lemma}[Separated projections and parking]
-\label{lem:bb-lift-parking}
-These reciprocal maps have the following smooth compactly supported
-unit-time realizations, stationary on time collars. They are effective
-under the rational-data assumptions just stated.
-\begin{enumerate}
-\item For the products $P_i\times[z_0-h,z_0+h]$, with common $h\ge0$
-and $z_0\in\R$, lift all boxes with their source projections fixed,
-transfer them at sufficiently separated private heights, and lower
-them with their target projections fixed.  The middle motion may use
-$g_i(s)=\lambda_i^s$ or $g_i(s)=1+s(\lambda_i-1)$ and matrix
-$\operatorname{diag}(g_i,g_i^{-1},1)$, with the center interpolated
-linearly in $s$.  Each horizontal width stays between its endpoint
-widths.  In the linear case every point satisfies
-\begin{equation}\label{eq:bb-convex-coordinate}
- X_1(s)=(1-s)X_1(0)+sX_1(1).
-\end{equation}
-\item Alternatively, choose disjoint parking rectangles away from both
-families.  Move every source to its parking site before delivering any
-target, using an upward, horizontal, and downward translation.  Reshape
-at the parking sites, then deliver by the same three translations.
-This uses $7N$ motions. The same schedule applies to assigned onto positive
-diagonal determinant-one maps between rational solid boxes whose centers
-lie in a common horizontal plane and whose source and target horizontal
-projections are separately disjoint. Require both horizontal widths during
-reshaping to stay inside their reserved parking rectangles and the travel
-height above the common plane to exceed twice the largest vertical
-half-width.
-\end{enumerate}
-Both constructions act on neighborhoods of the whole source boxes.
-\end{lemma}
-\begin{proof}
-In the first construction choose padding smaller than a quarter of
-the within-family horizontal gaps and choose height differences larger
-than twice the padded vertical half-width.  During lift and descent,
-horizontal separation suffices.  During transfer, vertical separation
-suffices.  The two choices of $g_i$ are positive and monotone between
-$1$ and $\lambda_i$; their reciprocals are monotone as well.  Thus the
-width claim holds.  The affine path is
-$c_i(s)+\operatorname{diag}(g_i,g_i^{-1},1)(X_0-c_i(0))$.
-Its first coordinate gives \eqref{eq:bb-convex-coordinate} exactly for
-the linear choice.  Lemma~\ref{lem:bb-curl} now applies.
-
-For the second construction choose padding smaller than the gaps in
-each union of source projections with reserved parking rectangles, and
-of target projections with those rectangles.  In the evacuation pass,
-vertical motion is separated from uncollected sources and parked boxes.
-During delivery it is separated from already filled targets and boxes
-still parked.  At travel height, the moving box is above every resting
-box.  A shape change stays within its own reserved projection.
-These are the occupied sets at each actual time; source--target
-intersection causes no interference because all sources have left
-before delivery starts.  Lemma~\ref{lem:bb-curl} applies to each motion
-with every other box fixed.  Finite concatenation gives the required
-neighborhood map.
-\end{proof}
-
-\begin{figure}[ht]
-\centering
-\begin{tikzpicture}[x=.75cm,y=.65cm,>=stealth,font=\small]
- \draw[gray] (0,0)--(5,0);
- \draw[thick,blue!65!black,->] (.5,0)--(.5,2.4)--(4.2,2.4)--(4.2,.1);
- \draw[thick,orange!80!black,->] (1.6,0)--(1.6,1.3)--(3.1,1.3)--(3.1,.1);
- \node[below] at (1,0) {sources};
- \node[below] at (3.7,0) {targets};
- \node[above] at (2.5,2.6) {Separate height layers};
- \begin{scope}[xshift=5.1cm]
-  \node[draw,rounded corners,minimum width=2.6cm] (s) at (1.8,2.8) {all source boxes};
-  \node[draw,rounded corners,minimum width=2.6cm] (p) at (1.8,1.1) {all boxes parked};
-  \node[draw,rounded corners,minimum width=2.6cm] (t) at (1.8,-.6) {all target boxes};
-  \draw[->,thick] (s)--(p) node[midway,right] {evacuate};
-  \draw[->,thick] (p)--(t) node[midway,right] {deliver};
- \end{scope}
-\end{tikzpicture}
-\caption{The two separation mechanisms in
-Lemma~\ref{lem:bb-lift-parking}.  Left: horizontal projections separate
-the lift and descent, and distinct heights separate the transfer;
-the second horizontal coordinate is suppressed.  Right: a temporal
-schedule, not a spatial drawing.  Every source is removed before any
-target is occupied, so overlap between the two families is harmless.}
-\label{fig:bb-routing}
-\end{figure}
-
-There is also a useful stream-function implementation of the first
-schedule.  Suppose $a_i,a_i'$ are the lower left corners of $P_i,Q_i$.
-At height $z_i$, put $d_i=a_i'-a_i$, $b_i=\log\lambda_i$ and
-$g_i(s)=a_i+sd_i$.  On a neighborhood of the swept rectangle use
-\begin{equation}\label{eq:bb-mid-stream-path}
- \xi_i(s)=g_i(s)+\operatorname{diag}(\lambda_i^s,\lambda_i^{-s})(\xi-a_i).
-\end{equation}
-The Hamiltonian
-\[
- H_i=d_{i1}y-d_{i2}x+b_i(x-g_{i1})(y-g_{i2})
-\]
-generates this motion. Put $\widehat H_i=\chi_iH_i$, where $\chi_i$
-is one near the whole sweep and supported in its reserved layer. Use
-\[
- \dot s(\partial_y\widehat H_i,-\partial_x\widehat H_i,0).
-\]
-Disjoint height layers separate these cutoffs.  Vertical phases use
-$\nabla\times(0,x\chi_i,0)$, equal to $(0,0,1)$ on their column
-plateaus.  If $M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and the column
-plateau margins are $\delta,\delta'$, an initial thickening smaller than
-$\min(\delta/(4M),\delta'/(4M),1/(8M))$ stays on every plateau.
-Thus the affine formula also holds on open neighborhoods. For points of
-the original rectangle it has the additional bound
-\begin{equation}\label{eq:bb-mid-stream-safe}
- X_1(s)\ge\min(a_{i1},a'_{i1}).
-\end{equation}
-The bound follows because the first relative coordinate in
-\eqref{eq:bb-mid-stream-path} is nonnegative.
-
-\subsection{Solid boxes with overlapping projections}
-
-The preceding lift requires separate horizontal projections.  We now
-remove that restriction; this is needed when all three coordinates
-encode independent stacks.
-
-\begin{theorem}[Expansion, obstacle detours, and delivery]
-\label{lem:bb-balanced-box-routing}
-Let $Q_i=a_i+\prod_{j=1}^3[-h_{ij},h_{ij}]$ and
-$Q_i'=b_i+\prod_{j=1}^3[-k_{ij},k_{ij}]$ be rational solid boxes,
-with positive half-widths, each family pairwise disjoint.  Suppose
-$s_{ij}=k_{ij}/h_{ij}$ and $\prod_js_{ij}=1$.  An effective compactly
-supported smooth solenoidal field, zero outside the time interval
-$[1/4,3/4]$, has time-one map
-\[
- a_i+y\longmapsto b_i+\operatorname{diag}(s_{i1},s_{i2},s_{i3})y
-\]
-on a neighborhood of each $Q_i$.
-\end{theorem}
-\begin{proof}\label{proof:bb-balanced-box-routing}
-For an empty family take the zero field.  Otherwise let
-$R=\max(1,h_{ij},k_{ij})$.  Choose a rational $\Lambda>10$ so that
-centers in each of the lists $(\Lambda a_i)$ and $(\Lambda b_i)$
-are separated by more than $20R$ in maximum norm.  This is possible
-because distinct disjoint boxes have distinct centers.  Let
-\[
- K=\max(4\Lambda,|\Lambda a_{i1}|,|\Lambda b_{i1}|),
- \qquad d_i=(K+30Ri,0,0).
-\]
-The $d_i$ are parking centers, separated from each other and both scaled
-families by at least $30R$ in their first coordinate.
-
-First multiply all source centers by a common factor increasing from
-$1$ to $\Lambda$, while keeping shapes fixed.  Next change all shapes
-at those separated centers by factors $s_{ij}^{\theta}$,
-$0\le\theta\le1$.  Each half-width is the geometric interpolation
-$h_{ij}^{1-\theta}k_{ij}^{\theta}\le R$.
-Move the boxes one at a time from $\Lambda a_i$ to $d_i$, then,
-after all have been parked, from $d_i$ to $\Lambda b_i$.
-Finally shrink all target centers by a common factor from $\Lambda$
-to $1$, with target shapes fixed.
-
-Here is a finite construction of every single-box path.  Around each
-stationary center place the closed cube of maximum-norm radius $4R$.
-These cubes are disjoint since stationary centers are separated by
-more than $20R$, and neither endpoint is in a cube.  Start with the
-straight segment between the endpoints.  Its intersections with the
-cubes are rational intervals, found by linear inequalities.  Replace
-each portion inside a cube by a polygonal path on its boundary: join
-the entry point to a vertex of a containing face, use cube edges to a
-face containing the exit point, and join to that exit point.  The cube
-is convex face by face, so these segments remain on its boundary;
-no other cube meets them.  A tangency needs no replacement.  The
-resulting rational polygonal path has distance at least $4R$ from
-every stationary center.  Traverse its segments with flat switches,
-stopping to all orders at vertices.  Its geometric image is unchanged,
-so smoothing the time parameter does not reduce clearance.
-
-For a uniform localization margin, let $g_a$ be the minimum over
-$i<j$ of
-\[
- \max_\ell(|a_{i\ell}-a_{j\ell}|-h_{i\ell}-h_{j\ell}),
-\]
-and define $g_b$ similarly.  These are positive; for a singleton list
-set both to $R$.  During outward expansion a separating coordinate
-gap can only increase; during final contraction the target gap is its
-minimum.  During deformation centers are more than $20R$ apart.
-During a single-box path the center distance is at least $4R$, leaving
-a coordinate gap at least $2R$.  Hence padding smaller than
-$\tfrac18\min(R,g_a,g_b)$ satisfies Lemma~\ref{lem:bb-curl}
-throughout.  Put all stages, and all straight segments, in consecutive
-subintervals of $[1/4,3/4]$.  Their flat switches give a smooth finite
-motion, with positive diagonal determinant-one linear parts.  That
-lemma provides the velocity and the neighborhood assertion.  All
-choices use rational inequalities and effective elementary functions,
-which also supply bounds for every derivative.
-\end{proof}
+\section{Transporting complete boxes}
+\label{sec:bb-boxes}
+
+We allow an instruction to change all three side lengths while
+preserving their product. The geometric question is concrete:
+given finitely many disjoint closed boxes and prescribed determinant-one
+affine maps onto another disjoint family, can one carry out every map by
+a smooth incompressible motion?  The two families may overlap each
+other.  Consequently a destination cannot be occupied until every source
+that obstructs it has been removed.
+
+We first extend an explicitly separated motion to a velocity field.
+We then give two ways to produce that motion.  Separate horizontal
+projections permit a simple lift and transfer.  For general solid boxes
+we use parking and detours around the other occupied boxes.
+Figure~\ref{fig:bb-routing} contrasts the height separation with the
+evacuation order that makes overlapping source and target families harmless.
+
+\subsection{An affine motion and its curl extension}
+
+Throughout, a smooth switch is obtained by translating and rescaling
+$\sigma$ in \eqref{eq:p2-step}, with its transition strictly inside
+its allotted interval.  Thus successive motions are stationary on
+neighborhoods of their joining times.
+
+\begin{lemma}[A velocity on a neighborhood of each moving box]
+\label{lem:bb-curl}
+Let $K_i(t)=c_i(t)+D_i(t)K_i^0$, $1\le i\le N$, where $K_i^0$ is
+an axis-parallel box, possibly with zero side lengths, and $D_i(t)$ is
+positive diagonal with determinant one.  Suppose the centers and scales
+are smooth and constant near the ends of a compact time interval.
+For the effective conclusion, assume computable endpoints for the
+initial boxes $K_i^0$, effective evaluation of the centers' and scales'
+derivatives, bounds for those derivatives, and positive lower bounds
+for every scale on that interval.
+If the coordinatewise $2\eta$ enlargements of the moving boxes are
+pairwise disjoint and bounded, for a known $\eta>0$, a
+compactly supported smooth divergence-free field realizes these motions
+on open neighborhoods of the initial boxes.  It vanishes near the time
+endpoints.  When all enlarged boxes lie in an interior torus chart,
+the field extends to a mean-zero field on that torus.
+The field is effective under the effective-data assumptions above;
+smooth existence itself does not require computable data.
+\end{lemma}
+\begin{proof}
+Choose a smooth cutoff $\chi_i$ which is one on the $\eta$
+enlargement of $K_i(t)$ and supported in its $2\eta$ enlargement,
+using products of the effective switches.  Put $y=x-c_i$ and
+$L_i=\dot D_iD_i^{-1}$; then $\operatorname{tr}L_i=0$.
+Take the curl of
+\begin{equation}\label{eq:bb-potential}
+ \chi_i\{\tfrac12\dot c_i\times y+\tfrac13(L_i y)\times y\}.
+\end{equation}
+The identity
+$\nabla\times(G\times y)=2G-y\operatorname{div}G+(y\cdot\nabla)G$
+shows that this curl equals $\dot c_i+L_i y$ on the plateau.
+Disjoint supports permit summation over $i$.  The path
+$c_i(t)+D_i(t)D_i(t_0)^{-1}(x_0-c_i(t_0))$ has exactly that
+derivative and remains in its prescribed box.  ODE uniqueness proves
+the formula.  On the compact time interval the normalized matrix norm
+has a finite bound $M$, computable under the effective-data assumptions;
+an initial perturbation less than
+$\eta/(2M)$ remains on the plateau.  This proves the neighborhood
+claim, also for zero-width initial boxes.
+The potential is zero near time endpoints because the centers and
+scales are constant there.  Its support is bounded.  Within a torus
+chart extend the potentials by zero; periodic curls have zero integral.
+\end{proof}
+
+\begin{proposition}[Fluid realization and a pressure refinement]
+\label{prop:bb-realization}
+Fix $\nu>0$.  A finite collection of the preceding motions, repeated
+with period one, or preceded by one finite loading interval, yields a
+smooth prescribed velocity $U$ with $U(0)=0$.  Suppose its spatial
+support is contained in one compact set in $\R^3$, or its potentials
+are contained in interior torus charts.  Then
+\begin{equation}\label{eq:bb-residual}
+ f=U_t+(U\cdot\nabla)U-\nu\Delta U
+\end{equation}
+has the same support and time periodicity and bounded mixed derivatives
+of every order.  The solution is $(U,0)$, with the uniqueness and
+material-flow conclusions of Lemma~\ref{lem:p2-realization}.
+In $\R^3$ its uniqueness statement also holds when the competing
+pressure has an $H^1$ representative at each time, without requiring
+continuity into $H^1$, while the other velocity hypotheses of
+Section~\ref{sec:model} are retained.  On the torus $U$ and $f$ have
+zero mean. Under the effective-data assumptions of Lemma~\ref{lem:bb-curl},
+the formula is $f_0+\nu f_1$ with effective coefficients;
+evaluation is effective for computable $\nu$, and relative to $\nu$
+otherwise.
+\end{proposition}
+\begin{proof}
+The residual identity and Lemma~\ref{lem:b-comparison} give existence,
+uniqueness and the material flow.  In the whole-space pressure refinement
+use case (ii): the reference field has compact support and smooth bounded
+mixed derivatives, hence belongs to $CH^2\cap C^1L^2$ with bounded
+velocity and gradient; the competing velocity and pointwise $H^1$
+pressure are exactly that case's hypotheses.  No pressure continuity
+in time is added.  Integrating the residual on a torus uses
+$\int U=0$, $\int\Delta U=0$ and
+$\int(U\cdot\nabla)U=\int\operatorname{div}(U\otimes U)=0$.
+
+Each derivative of $e^{-1/s}$ for $s>0$ is a polynomial in $1/s$
+times $e^{-1/s}$.  The estimate
+$v^m e^{-v}\le(m+k)!v^{-k}$ supplies effective endpoint error bounds,
+and the denominator defining $\sigma$ is at least $e^{-2}$.
+Under those effective-data assumptions all positive scales have effective
+positive lower bounds. Finite
+compositions, differentiation, and the logarithms of positive computable
+scales are therefore effective.  At an approximate time, a coarse bound
+gives a finite superset of possible translates of the zero-extended
+unit template.  This evaluates the periodic field without deciding
+whether time is an integer.  None of these operations iterates the
+machine execution: the program contains the entire finite instruction
+list.
+\end{proof}
+
+\subsection{Three geometric schedules}
+
+For later use we describe the exact hypotheses of the schedules.
+First suppose $P_i,Q_i\subset\R^2$ are finite families of closed
+rectangles, possibly with zero side lengths, separately
+pairwise disjoint, with affine maps
+\[
+ F_i(x)=q_i+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(x-p_i),
+ \qquad \lambda_i>0,
+\]
+where $p_i,q_i$ are their centers, and require $F_i(P_i)=Q_i$.
+Their side lengths have a common finite bound. Source and target rectangles
+belonging to different families need not be disjoint. The smooth existence
+statements allow arbitrary real rectangle coordinates and scales. For the
+effective conclusions, assume rational rectangle endpoints and rational
+scales, together with computable chosen heights and vertical half-widths;
+then a rational side-length bound and positive gap margins can be computed.
+
+\begin{lemma}[Separated projections and parking]
+\label{lem:bb-lift-parking}
+These reciprocal maps have the following smooth compactly supported
+unit-time realizations, stationary on time collars. They are effective
+under the rational-data assumptions just stated.
+\begin{enumerate}
+\item For the products $P_i\times[z_0-h,z_0+h]$, with common $h\ge0$
+and $z_0\in\R$, lift all boxes with their source projections fixed,
+transfer them at sufficiently separated private heights, and lower
+them with their target projections fixed.  The middle motion may use
+$g_i(s)=\lambda_i^s$ or $g_i(s)=1+s(\lambda_i-1)$ and matrix
+$\operatorname{diag}(g_i,g_i^{-1},1)$, with the center interpolated
+linearly in $s$.  Each horizontal width stays between its endpoint
+widths.  In the linear case every point satisfies
+\begin{equation}\label{eq:bb-convex-coordinate}
+ X_1(s)=(1-s)X_1(0)+sX_1(1).
+\end{equation}
+\item Alternatively, choose disjoint parking rectangles away from both
+families.  Move every source to its parking site before delivering any
+target, using an upward, horizontal, and downward translation.  Reshape
+at the parking sites, then deliver by the same three translations.
+This uses $7N$ motions. The same schedule applies to assigned onto positive
+diagonal determinant-one maps between rational solid boxes whose centers
+lie in a common horizontal plane and whose source and target horizontal
+projections are separately disjoint. Require both horizontal widths during
+reshaping to stay inside their reserved parking rectangles and the travel
+height above the common plane to exceed twice the largest vertical
+half-width.
+\end{enumerate}
+Both constructions act on neighborhoods of the whole source boxes.
+\end{lemma}
+\begin{proof}
+In the first construction choose padding smaller than a quarter of
+the within-family horizontal gaps and choose height differences larger
+than twice the padded vertical half-width.  During lift and descent,
+horizontal separation suffices.  During transfer, vertical separation
+suffices.  The two choices of $g_i$ are positive and monotone between
+$1$ and $\lambda_i$; their reciprocals are monotone as well.  Thus the
+width claim holds.  The affine path is
+$c_i(s)+\operatorname{diag}(g_i,g_i^{-1},1)(X_0-c_i(0))$.
+Its first coordinate gives \eqref{eq:bb-convex-coordinate} exactly for
+the linear choice.  Lemma~\ref{lem:bb-curl} now applies.
+
+For the second construction choose padding smaller than the gaps in
+each union of source projections with reserved parking rectangles, and
+of target projections with those rectangles.  In the evacuation pass,
+vertical motion is separated from uncollected sources and parked boxes.
+During delivery it is separated from already filled targets and boxes
+still parked.  At travel height, the moving box is above every resting
+box.  A shape change stays within its own reserved projection.
+These are the occupied sets at each actual time; source--target
+intersection causes no interference because all sources have left
+before delivery starts.  Lemma~\ref{lem:bb-curl} applies to each motion
+with every other box fixed.  Finite concatenation gives the required
+neighborhood map.
+\end{proof}
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[x=.75cm,y=.65cm,>=stealth,font=\small]
+ \draw[gray] (0,0)--(5,0);
+ \draw[thick,blue!65!black,->] (.5,0)--(.5,2.4)--(4.2,2.4)--(4.2,.1);
+ \draw[thick,orange!80!black,->] (1.6,0)--(1.6,1.3)--(3.1,1.3)--(3.1,.1);
+ \node[below] at (1,0) {sources};
+ \node[below] at (3.7,0) {targets};
+ \node[above] at (2.5,2.6) {Separate height layers};
+ \begin{scope}[xshift=5.1cm]
+  \node[draw,rounded corners,minimum width=2.6cm] (s) at (1.8,2.8) {all source boxes};
+  \node[draw,rounded corners,minimum width=2.6cm] (p) at (1.8,1.1) {all boxes parked};
+  \node[draw,rounded corners,minimum width=2.6cm] (t) at (1.8,-.6) {all target boxes};
+  \draw[->,thick] (s)--(p) node[midway,right] {evacuate};
+  \draw[->,thick] (p)--(t) node[midway,right] {deliver};
+ \end{scope}
+\end{tikzpicture}
+\caption{The two separation mechanisms in
+Lemma~\ref{lem:bb-lift-parking}.  Left: horizontal projections separate
+the lift and descent, and distinct heights separate the transfer;
+the second horizontal coordinate is suppressed.  Right: a temporal
+schedule, not a spatial drawing.  Every source is removed before any
+target is occupied, so overlap between the two families is harmless.}
+\label{fig:bb-routing}
+\end{figure}
+
+There is also a useful stream-function implementation of the first
+schedule.  Suppose $a_i,a_i'$ are the lower left corners of $P_i,Q_i$.
+At height $z_i$, put $d_i=a_i'-a_i$, $b_i=\log\lambda_i$ and
+$g_i(s)=a_i+sd_i$.  On a neighborhood of the swept rectangle use
+\begin{equation}\label{eq:bb-mid-stream-path}
+ \xi_i(s)=g_i(s)+\operatorname{diag}(\lambda_i^s,\lambda_i^{-s})(\xi-a_i).
+\end{equation}
+The Hamiltonian
+\[
+ H_i=d_{i1}y-d_{i2}x+b_i(x-g_{i1})(y-g_{i2})
+\]
+generates this motion. Put $\widehat H_i=\chi_iH_i$, where $\chi_i$
+is one near the whole sweep and supported in its reserved layer. Use
+\[
+ \dot s(\partial_y\widehat H_i,-\partial_x\widehat H_i,0).
+\]
+Disjoint height layers separate these cutoffs.  Vertical phases use
+$\nabla\times(0,x\chi_i,0)$, equal to $(0,0,1)$ on their column
+plateaus.  If $M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and the column
+plateau margins are $\delta,\delta'$, an initial thickening smaller than
+$\min(\delta/(4M),\delta'/(4M),1/(8M))$ stays on every plateau.
+Thus the affine formula also holds on open neighborhoods. For points of
+the original rectangle it has the additional bound
+\begin{equation}\label{eq:bb-mid-stream-safe}
+ X_1(s)\ge\min(a_{i1},a'_{i1}).
+\end{equation}
+The bound follows because the first relative coordinate in
+\eqref{eq:bb-mid-stream-path} is nonnegative.
+
+\subsection{Solid boxes with overlapping projections}
+
+The preceding lift requires separate horizontal projections.  We now
+remove that restriction; this is needed when all three coordinates
+encode independent stacks.
+
+\begin{theorem}[Expansion, obstacle detours, and delivery]
+\label{lem:bb-balanced-box-routing}
+Let $Q_i=a_i+\prod_{j=1}^3[-h_{ij},h_{ij}]$ and
+$Q_i'=b_i+\prod_{j=1}^3[-k_{ij},k_{ij}]$ be rational solid boxes,
+with positive half-widths, each family pairwise disjoint.  Suppose
+$s_{ij}=k_{ij}/h_{ij}$ and $\prod_js_{ij}=1$.  An effective compactly
+supported smooth solenoidal field, zero outside the time interval
+$[1/4,3/4]$, has time-one map
+\[
+ a_i+y\longmapsto b_i+\operatorname{diag}(s_{i1},s_{i2},s_{i3})y
+\]
+on a neighborhood of each $Q_i$.
+\end{theorem}
+\begin{proof}\label{proof:bb-balanced-box-routing}
+For an empty family take the zero field.  Otherwise let
+$R=\max(1,h_{ij},k_{ij})$.  Choose a rational $\Lambda>10$ so that
+centers in each of the lists $(\Lambda a_i)$ and $(\Lambda b_i)$
+are separated by more than $20R$ in maximum norm.  This is possible
+because distinct disjoint boxes have distinct centers.  Let
+\[
+ K=\max(4\Lambda,|\Lambda a_{i1}|,|\Lambda b_{i1}|),
+ \qquad d_i=(K+30Ri,0,0).
+\]
+The $d_i$ are parking centers, separated from each other and both scaled
+families by at least $30R$ in their first coordinate.
+
+First multiply all source centers by a common factor increasing from
+$1$ to $\Lambda$, while keeping shapes fixed.  Next change all shapes
+at those separated centers by factors $s_{ij}^{\theta}$,
+$0\le\theta\le1$.  Each half-width is the geometric interpolation
+$h_{ij}^{1-\theta}k_{ij}^{\theta}\le R$.
+Move the boxes one at a time from $\Lambda a_i$ to $d_i$, then,
+after all have been parked, from $d_i$ to $\Lambda b_i$.
+Finally shrink all target centers by a common factor from $\Lambda$
+to $1$, with target shapes fixed.
+
+Here is a finite construction of every single-box path.  Around each
+stationary center place the closed cube of maximum-norm radius $4R$.
+These cubes are disjoint since stationary centers are separated by
+more than $20R$, and neither endpoint is in a cube.  Start with the
+straight segment between the endpoints.  Its intersections with the
+cubes are rational intervals, found by linear inequalities.  Replace
+each portion inside a cube by a polygonal path on its boundary: join
+the entry point to a vertex of a containing face, use cube edges to a
+face containing the exit point, and join to that exit point.  The cube
+is convex face by face, so these segments remain on its boundary;
+no other cube meets them.  A tangency needs no replacement.  The
+resulting rational polygonal path has distance at least $4R$ from
+every stationary center.  Traverse its segments with flat switches,
+stopping to all orders at vertices.  Its geometric image is unchanged,
+so smoothing the time parameter does not reduce clearance.
+
+For a uniform localization margin, let $g_a$ be the minimum over
+$i<j$ of
+\[
+ \max_\ell(|a_{i\ell}-a_{j\ell}|-h_{i\ell}-h_{j\ell}),
+\]
+and define $g_b$ similarly.  These are positive; for a singleton list
+set both to $R$.  During outward expansion a separating coordinate
+gap can only increase; during final contraction the target gap is its
+minimum.  During deformation centers are more than $20R$ apart.
+During a single-box path the center distance is at least $4R$, leaving
+a coordinate gap at least $2R$.  Hence padding smaller than
+$\tfrac18\min(R,g_a,g_b)$ satisfies Lemma~\ref{lem:bb-curl}
+throughout.  Put all stages, and all straight segments, in consecutive
+subintervals of $[1/4,3/4]$.  Their flat switches give a smooth finite
+motion, with positive diagonal determinant-one linear parts.  That
+lemma provides the velocity and the neighborhood assertion.  All
+choices use rational inequalities and effective elementary functions,
+which also supply bounds for every derivative.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/guarded-history.tex

@@ -1,660 +1,660 @@
-\section{Wider guards and alternative comparison classes}
-\label{sec:bb-guarded}
-
-The transport lemmas also accept tables which differ on tapes never
-visited by the initialized computation.  Those differences matter:
-they change the complete source and image cylinders on which the
-smooth extension is prescribed.  We give the remaining tables with
-their actual guards and then isolate the extra pressure arguments.
-
-\subsection{A shuttle aligned at the origin}
-\label{sec:bb-shuttle-periodic}
-
-\begin{theorem}\label{thm:bb-shuttle-periodic}
-On the unit torus there is an effective smooth mean-zero force,
-periodic from time zero with period one and with bounded mixed
-derivatives, whose unique zero-data classical solution has
-\[
- \exists t\ge0:\quad X_1(t;0)\pmod1\in(1/16,1/4)
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-The constructed pressure is zero.  Alternatively the force may be
-projected to a solenoidal force with the same velocity and event,
-and with the correspondingly changed mean-zero pressure.
-\end{theorem}
-\begin{proof}\label{proof:bb-shuttle-periodic}
-Add a fresh nonhalting initial state $*$, without incoming transitions,
-which writes back the scanned symbol and stays while entering the
-original initial state.  This handles an originally halted input.
-Use work, flag and history tracks
-$\Sigma=\Gamma\times\{0,1\}\times(\{\perp,E\}\sqcup\mathcal R)$,
-and controls $R_q,A_r,B_r,C_q,D_q$.  Its complete table is
-\begin{equation}\label{eq:bb-shuttle-table}
-\begin{array}{lll}
- R_q:(a,0,g)&\mapsto A_r:(b_r,0,g),d_r,\\
- A_r:(a,0,g)&\mapsto B_r:(a,1,g),1,\\
- B_r:(a,0,g)&\mapsto B_r:(a,0,g),1&g\ne E,\\
- B_r:(a,0,E)&\mapsto C_{p_r}:(a,0,r),1,\\
- C_q:(a,0,\perp)&\mapsto D_q:(a,0,E),-1,\\
- D_q:(a,0,g)&\mapsto D_q:(a,0,g),-1&g\ne E,\\
- D_q:(a,1,g)&\mapsto R_q:(a,0,g),0&g\ne E.
-\end{array}
-\end{equation}
-In particular its first two rows do not exclude the frontier symbol;
-this is a wider domain than the move-first table.  Into $A_r$ the
-record identifies the work instruction; into $B_r$ the output flag
-distinguishes entry and loop; into $C_q$ the record determines the
-incoming scan; into $D_q$ a written $E$ distinguishes entry and loop;
-into $R_q$ only clearing a flag is possible.  The destination fixes
-the incoming displacement.  These checks include every copied track
-and every permitted frontier case.
-
-Initialize at $R_*$, with all flags zero and only $E$ at cell two.
-At ready step $n$ the frontier is $e_n=n+2$, the records are at
-$2,\ldots,n+1$, and the head is $h_n$, $|h_n|\le n$.
-The work move reaches $h'=h_{n+1}<e_n$; the scan marks that cell,
-records at $e_n$, creates the next frontier, returns to the mark and
-clears it.  Its count is $2(e_n-h')+4$ and all its tests hold.
-Thus it has exactly the required initialized behavior despite its
-wider full domain.
-
-Use even digits in base $b=2|\Sigma|$ and initial fractions
-$\ell_*,r_*$.  Let $S$ be the control set and $m$ count the branches in
-Lemma~\ref{lem:bb-onecell}, and put $L=100(1+|S|+m)$.
-Give $R_*$ offset $-\ell_*$, other nonhalt states offsets $-4i$,
-and ready halt states offsets $L+4i$.  The code is
-$(o_s+\ell,r-r_*,0)$, so the initial point is zero.
-The incoming conditions give separately disjoint full rectangle
-families.  Thicken each by $[-1,1]$, lift to height $10i$, and use
-the linear reciprocal middle scale.  Choose padding at most $1/10$
-small enough for both family gaps.  Vertical phases can use fixed
-column potentials $(0,\dot Z x\chi,0)$, with columns covering
-$[-3,10i+3]$; middle phases use horizontal Hamiltonians in disjoint
-height bands.  All paths and supports are contained in
-$(-2L,2L)^3$: their coordinate bounds are
-$[-4|S|-3,L+4|S|+4]$, $[-4,4]$, and $[-3,10m+3]$.
-This verifies the hypotheses of the lift schedule on neighborhoods
-of the whole boxes.
-
-Scale by $\gamma=(8L)^{-1}$ into the torus patch
-$(-1/2,1/2)^3$: $U(t,x)=\gamma W(t,x/\gamma)$.
-The support lies inside $(-1/4,1/4)^3$, the potentials scale by
-$\gamma^2$, and the residual is recomputed at viscosity $\nu$ by
-Lemma~\ref{lem:p2-chart}.  Periodicity starts at zero
-because the input is already encoded there and the pulse has idle
-endpoint collars.  At integer times the particle is the correct code.
-Halt first coordinates before scaling lie between $L+4$ and $2L$.
-For a nonhalt branch its endpoint first coordinates are at most one;
-the linear-scale identity keeps them at most one, and the weaker
-bound $3/2$ also follows from center and width estimates.  After
-scaling every nonhalt path lies in $(-1/4,1/16)$, disjoint modulo
-one from the event.  The initialized dummy step makes the implication
-valid even for an originally halted machine.
-
-For the optional projection solve $\Delta q=\operatorname{div}g$,
-$\int q=0$, where $g=\mathcal F_\nu[U]$.
-Proposition~\ref{prop:projection-l2} gives the effective force
-$f=g-\nabla q$ and pressure $p=-q$, with all mixed derivative bounds
-and periodicity retained.  The pressure change is part of this
-alternative; the direct residual had pressure zero.
-\end{proof}
-
-\subsection{A persistent marker and cursor shifts of two cells}
-\label{sec:bb-marked-window}
-
-\begin{theorem}\label{thm:bb-marked-window}
-There is a unit-torus realization, periodic for $t\ge1$, with fixed
-label $a_*=(1/4,1/4,1/4)$.  Its event
-\[ \exists t\ge0:\quad X_1(t;a_*)\pmod1\in(2/3,5/6) \]
-occurs exactly when the machine halts.
-It uses an injective marked-head table whose cursor shifts may equal
-two.  Its force is effective, smooth and mean zero, with bounded
-mixed derivatives and the zero-data uniqueness class of
-Proposition~\ref{prop:bb-realization}.
-\end{theorem}
-\begin{proof}\label{proof:bb-marked-window}
-Normalize to one halt state and use work, permanent work-head marker,
-and history tracks $(a,m,g)\in\mathcal A:=\Gamma\times\{0,1\}\times
-(\{\bot,\#\}\sqcup\mathcal R)$.
-Let $S$ be the set of controls $M_q,L_q,R_r$.  With the cursor at zero, make
-exactly the following tests and writes, then reindex by
-$T'_i=\bar T_{i+e}$ for the displayed shift $e$.
-\begin{enumerate}
-\item In $M_q$, $q$ nonhalting, require $m_0=1$ and, if $d_r\ne0$,
-$m_{d_r}=0$.  Write $b_r$ at zero, move the marker to $d_r$ if
-necessary, and enter $R_r$ with shift $e=d_r+1$.
-\item In $R_r$, a cell with $m_0=0,g_0\ne\#$ gives shift $1$
-in the same control.  If $m_0=0,g_0=\#,g_1=\bot$, write
-$g_0=r,g_1=\#$ and enter $L_{p_r}$ with shift zero.
-\item In $L_q$, if $m_0=0,g_0\ne\#$, shift $-1$ in that control.
-If $m_0=1$, enter $M_q$ with shift zero.
-\end{enumerate}
-All unmentioned entries are retained.  Into $R_r$, a main entry has
-output $m_{-1}=1$, whereas a scan has $m_{-1}=0$.
-The record recovers the overwritten work letter, displacement and
-marker changes.  Into $L_q$, a log write has output $g_1=\#$
-and record $r$ at zero, whereas a scan has $g_1\ne\#$.
-Its tests recover the overwritten pair.  Into $M_q$ only the
-matching control change is possible.  This is a full-domain inverse;
-uniqueness of a marker was not assumed on arbitrary tapes.
-
-Initially the sole marker is at zero, the sole pointer at two, and
-all other history is blank.  At work step $n$, head and marker are
-at $h$, pointer at $n+2$, and records at $2,\ldots,n+1$.
-The work update puts the marker at $h'=h+d_r$ and the cursor at
-$h'+1\le n+2$.  The forward scan reaches the pointer, logs, and
-returns to $h'$.  All tests hold since $h'<n+2$.  The cycle length is
-\[
- 1+[n+2-(h'+1)]+1+[(n+2)-h']+1.
-\]
-This proves finite continuation, exact work simulation and halting.
-
-Use even digits, blank zero, in base $B=2|\mathcal A|+1$.
-Enumerate every full word on $[-2,2]$ with an applicable rule, giving
-$J$ branches.
-Lemma~\ref{lem:bb-window-rectangles} gives prefix lengths $(2,3)$
-and $(2+e,3-e)$, with $e\in\{-1,0,1,2\}$.
-Each output length is at most four, so gaps after chart scaling
-by $l=[100(|S|+1)]^{-1}$ are at least $lB^{-4}$.
-Place the halt square at $(3/4,1/4)+l[0,1]^2$ and other squares
-at $(1/4+2il,1/4)+l[0,1]^2$; the latter lie below $x_1=1/3$.
-At base height $1/4$, lift branch $j$ to
-$1/2+j/[4(J+1)]$, use exponential scales $B^{-e_j\theta},B^{e_j\theta}$,
-and lower.  Padding
-$\min(1/200,1/[32(J+1)],lB^{-4}/3)$ separates supports.
-The middle Hamiltonian is
-\[
- \dot c_x(y-c_y)-\dot c_y(x-c_x)
-      -e_j\dot\theta\log B\,(x-c_x)(y-c_y).
-\]
-It gives the intended derivative on the full moving rectangle.
-The plateau margins and finite scale bounds extend this to an open
-neighborhood, as in Lemma~\ref{lem:bb-curl}.
-
-Load the fixed label along its segment to the rational initial code
-in time one, then repeat the unit field.  Integer samples give the
-full local execution.  In a nonhalt period the center is below $1/3$
-and the width at most $l$, so the whole path remains below $1/2$;
-loading also avoids the event.  Halt codes lie in $[3/4,3/4+l]$.
-Thus every real time is accounted for.  The pointer site $n+2$,
-record count $n$, and relative pointer offset recover the absolute
-head position at main checkpoints.  The residual construction
-proves the remaining assertions.
-\end{proof}
-
-\subsection{A split direction shuttle in the whole space}
-\label{sec:bb-direction-shuttle}
-
-\begin{theorem}\label{thm:bb-direction-shuttle}
-On $\R^3$ an effective compactly supported force with bounded mixed
-derivatives and period one after $t=1$ realizes the halting event
-$X_1(t;(1,0,0))<0$.  Uniqueness holds with bounded $u,\nabla u$,
-$u\in C H^2\cap C^1L^2$ and pressure modulo constants in $C H^1$
-on each finite interval.  An alternative force is square-integrable
-in space-time, with the same label, event, support and derivative
-bounds.
-\end{theorem}
-\begin{proof}\label{proof:bb-direction-shuttle}
-Use exactly the direction table of Lemma~\ref{lem:bb-direction}
-without its passive track, renaming
-the controls by $M\mapsto S$, $G\mapsto R$, $A\mapsto I$,
-and $B\mapsto J$, and the frontier by $F\mapsto\#$.  This is a bijection
-of its entire partial table, including the nonfrontier tests.
-Write $\Sigma$ for the resulting three-track alphabet.
-Initialize frontier at one and zero flags.  At work step $j$ the
-frontier is $j+1$ and the cycle length is
-\[2(j-h_j)+5.\]
-
-Use odd digits in base $B=2|\Sigma|+1$, offsets $3,6,\ldots$
-for nonhalt controls and $-3,-6,\ldots$ for halt main controls.
-In this realization split \emph{every} one-cell rule by every left
-symbol $c$, including the stationary and right-moving rules.
-Its source is $I(c)\times I(a)$, and the three target rectangles,
-for moves $1,-1,0$, are respectively
-\[
- \varphi_b\varphi_c([0,1])\times[0,1],\quad
- [0,1]\times\varphi_c\varphi_b([0,1]),\quad I(c)\times I(b),
- \qquad \varphi_a(t)=(d(a)+t)/B.
-\]
-The maps themselves are the restrictions of
-\eqref{eq:bb-one-cell-branches}.  Source separation follows from
-$(c,a)$, target separation from the fixed incoming move and $(c,b)$.
-Their gaps are at least $B^{-2}$, and their linear parts are
-$\operatorname{diag}(B^{-d},B^d)$.
-
-Thicken by $[-1,1]$, lift branch $i$ to $4i$, use exponential
-middle scale and center interpolation, then descend.  Padding smaller
-than $(4B^2)^{-1}$ is valid, by source gaps, height gaps, and target
-gaps in the three phases.  A first localized translation takes
-$(1,0,0)$ to the rational initial code.  Exact full-box transport
-then gives the local execution at every subsequent integer time.
-Nonhalt centers have first coordinate at least three and horizontal
-width at most one, so every intermediate point stays positive;
-the nonhalt loading segment does too.  A halt sample has negative
-first coordinate, including initial halting after loading.
-Proposition~\ref{prop:bb-realization} gives the force, pressure zero,
-and uniqueness.  The decaying alternative follows from
-Proposition~\ref{prop:bb-late-logclock} below, applied to this same
-loaded template.
-\end{proof}
-
-\subsection{Two-cell history and a gradient-only pressure condition}
-\label{sec:bb-guarded-history}
-
-\begin{theorem}\label{thm:bb-guarded-history}
-An effective smooth force on $\R^3$, periodic from time zero,
-with one compact spatial support and bounded mixed derivatives,
-realizes the fixed event
-\[
- \exists t\ge0:\quad X(t;(2,0,0))\in(-2,-1)\times(0,1)\times(-1,1)
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-Velocity is unique in the class
-$u\in C([0,T];H^4)\cap C^1([0,T];H^2)$,
-$\nabla p\in C([0,T];L^2)$ for every finite $T$, allowing a
-distributional pressure gradient.  Alternatively the force belongs
-to $L^2([0,\infty);H^j)$ for every integer $j\ge0$, with all the
-same support, event and comparison-class conclusions.
-\end{theorem}
-\begin{proof}\label{proof:bb-guarded-history}
-Add a fresh initial state which writes back and stays, and replace
-all originally terminal state-symbol pairs by stay transitions into
-one fresh terminal state $q_h$.  Normalize the other missing
-instructions similarly.  Let $r=(q,a)\mapsto(p_r,b_r,d_r)$ denote
-the resulting nonterminal instructions, and let $\mathcal A$ be the
-work alphabet. The full cell alphabet is
-$\Gamma=\mathcal A\times\{0,1\}\times(\{e,p\}\sqcup\mathcal R)$.
-Use its work, flag and log tracks $W,F,L$,
-and controls $N_q,A_r,R_r,B_q$.  The complete rules are:
-\begin{enumerate}
-\item At $N_q$, $q\ne q_h$, write $W_0=b_r$, enter $A_r$,
-and shift $d_r$, with no test on the other tracks.
-\item At $A_r$, require $F_0=0$, set it to one, enter $R_r$
-and shift one.
-\item At $R_r$, if $F_0=0,L_0\ne p$, shift one in that control.
-If $F_0=0,L_0=p,L_1=e$, write $L_0=r,L_1=p$, enter $B_{p_r}$
-and shift minus one.
-\item At $B_q$, if $F_0=0,L_1\ne p$, shift minus one in that
-control.  If $F_0=1$, clear it and enter $N_q$ without moving.
-\end{enumerate}
-The first output control records the overwritten work pair and shift.
-Into $R_r$, output flag one at $-1$ distinguishes entry from its
-loop.  Into $B_q$, a log insertion has pointer at output index two
-and record at one, whereas a return loop has a nonpointer at two;
-the record and tests recover both overwritten log entries.  Into
-$N_q$ only unmarking is possible.  Thus this map is injective on
-its whole domain, including the unrestricted first-row tracks.
-
-Initialize zero flags and a sole pointer at cell three, all other
-log cells empty.  At work step $n$, records occupy $3,\ldots,n+2$
-and pointer $g=n+3$.  The work move reaches $h'$ with $g-h'\ge2$.
-The finite scan flags $h'$, logs at $g$, advances the pointer, and
-returns to the flag.  The return guard holds because the new pointer
-is at $g+1$.  The cycle has $2(g-h'-1)+4$ steps.  This verifies
-the work invariant and eventual terminal entry.
-
-Use odd digits in radix $D=2|\Gamma|+1$ and write
-$\phi(\xi)=\sum_{j\ge1}d(\xi_j)D^{-j}$ for the radix sum.
-Let $L_*,R_*$ be the initialized left and right streams, ordered as in
-Lemma~\ref{lem:bb-onecell}.  Give the terminal state
-chart offset $(-2,0)$, the initial chart offset
-$(2-\phi(L_*),-\phi(R_*))$, and all other charts $(4+2j,0)$.
-The initial code is $(2,0,0)$, every nonterminal interval lies in
-$x_1>1$, and terminal codes lie strictly in $(-2,-1)\times(0,1)$.
-The infinite tails are in $[\delta,1-\delta]$,
-$\delta=1/(D-1)$, so finite-prefix decoding of length $b$ is possible
-from error less than $(\delta/2)D^{-b}$ without a boundary ambiguity.
-
-Split full triples $[-1,1]$.  By
-Lemma~\ref{lem:bb-window-rectangles}, source prefix lengths $(1,2)$
-become $(1+d,2-d)$, target cylinders are complete, and both rectangle
-families have gaps at least $D^{-3}$.  Thicken by $[-1/4,1/4]$,
-lift to height $4i$, use linear middle scale
-$\mu_i=1+(D^{-d_i}-1)\theta$, and descend.  Padding smaller than
-$D^{-3}/4$ separates all phases.  Repeating the pulse, zero on
-$[0,1/12]$ and $[11/12,1]$, gives periodicity from zero without
-loading.  At a nonhalting branch's lift and descent the first
-coordinate is greater than one; during the middle transfer the
-particle's third coordinate is exactly $4i$, outside $(-1,1)$.
-Thus even a horizontal route passing over the observation box cannot
-cause a false visit.  The integer samples and the compiler invariant
-prove halting equivalence.  The record count and pointer site $n+3$
-recover absolute indexing at checkpoints.
-
-For uniqueness apply Lemma~\ref{lem:b-comparison}(iii).
-The explicit compactly supported smooth field lies in $CH^4\cap C^1H^2$
-on every finite interval and has pressure zero.  The competing class
-in the theorem is precisely case (iii), including its distributional
-$CL^2$ gradient.  Lemma~\ref{lem:b-gradient} cancels that gradient
-against the solenoidal velocity difference without introducing a pressure
-$L^2$ norm.  Thus the velocity and pressure gradient are unique, while
-pressure itself remains free up to a time-dependent spatial constant.
-Proposition~\ref{prop:bb-late-logclock} gives the stated
-decaying alternative.
-\end{proof}
-
-\subsection{A flagged head and two-pass parking}
-\label{sec:bb-two-pass-parking}
-
-\begin{theorem}\label{thm:bb-two-pass-parking}
-For any machine and finitely specified initial tape, an effective
-smooth force on $\R^3$, periodic from time zero, with one compact
-spatial support and bounded mixed derivatives, realizes the event
-\[
- \exists t\ge0:\quad X(t;0)\in(-6,-4)\times(-1,1)\times(-1,1)
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-The zero-data velocity is unique among classical smooth solutions
-with bounded $u,\nabla u$, $u\in C H^2\cap C^1L^2$ and $p\in C L^2$
-on every finite interval.  The prescribed solution has uniformly
-bounded kinetic energy.
-\end{theorem}
-\begin{proof}\label{proof:bb-two-pass-parking}
-Merge terminal states into $q_\dagger$, send missing instructions
-there by a stay step, and add a fresh initial state $q_*$, with no
-incoming instruction, which writes back and stays while entering
-the old initial state.  For each normalized nonterminal rule
-$r=(q,a)\mapsto(p_r,b_r,d_r)$ use cell letters
-\[
- (w,h,j)\in\Gamma:=\mathcal A\times
- (\{\circ,\star\}\sqcup\{[r]:r\in\mathcal R\})\times\{0,1\},
-\]
-where $\mathcal A$ is the work alphabet and $\mathcal R$ is the
-finite set of normalized instruction labels. Thus $\Gamma$ includes
-all three tracks.
-The complete table, with $h\ne\star$ wherever $h$ occurs, is
-\[
-\begin{array}{lll}
- M_q:(a,h,1)&\mapsto A_r:(b_r,h,0),d_r,\\
- A_r:(w,h,0)&\mapsto R_r:(w,h,1),1,\\
- R_r:(w,h,0)&\mapsto R_r:(w,h,0),1,\\
- R_r:(w,\star,0)&\mapsto B_{p_r}:(w,[r],0),1,\\
- B_q:(w,\circ,0)&\mapsto L_q:(w,\star,0),-1,\\
- L_q:(w,h,0)&\mapsto L_q:(w,h,0),-1,\\
- L_q:(w,h,1)&\mapsto M_q:(w,h,1),0.
-\end{array}
-\]
-The destination fixes the incoming displacement.  The record index
-recovers the first row's old work pair; a flag distinguishes the
-entries into $R_r$; a record distinguishes the entries into $B_q$;
-a written frontier distinguishes entry and loop into $L_q$;
-and entry into $M_q$ is the unique cell-preserving last row.
-This proves the full one-cell inverse conditions.
-
-Initialize the sole flag at zero and the sole frontier at three,
-with empty history elsewhere.  At ready step $k$, only the work
-head $i$ is flagged, records occupy $3,\ldots,k+2$, and the frontier
-is $e_k=k+3$.  The work rule clears the flag and moves to
-$j=i+d_r\le e_k-2$, which is unflagged even for $d_r=0$.
-It is flagged anew; the scan logs at $e_k$, advances the pointer,
-returns, and preserves the ready flag.  The two loop lengths are
-$e_k-j-1$ and $e_k-j$, with five other rules, for a total
-$2(e_k-j)+4$.  Thus all scans terminate and ready halting is exact.
-
-Use odd digits in base $K=2|\Gamma|+2$, initial fractions $x_*,y_*$,
-and state offsets $a_{M_{q_*}}=0$, $a_{M_{q_\dagger}}=-5$, and
-$4,8,12,\ldots$ for the others.  The code
-$(a_s-x_*+x,y-y_*,0)$ starts at zero.  Lemma~\ref{lem:bb-onecell}
-gives $N$ full rectangle maps with scales $\lambda_i\in\{K^{-1},1,K\}$;
-thicken by $[-1/4,1/4]$.  The entire terminal block is inside the
-observation box because $0<x_*,y_*<1$.  Every nonterminal block is
-in $x_1>-1$.  Eventually constant tails are rational even for an
-arbitrary finitely specified initial tape, and the pointer and record
-count recover its absolute head position at ready checkpoints.
-
-For the required two-pass geometry, let $g$ be the minimum of one
-and all positive separating coordinate gaps in either planar family,
-and set $\epsilon=g/20$.  Choose
-\[
- C=20+\max(\{0\}\cup\{\text{right endpoints of source and target boxes}\}),
- \qquad S_i=(C+4i,0,0).
-\]
-In a first pass give each source four consecutive slots: lift by ten,
-translate above $S_i$, descend, and reshape there by the linear
-reciprocal scale.  In a second pass give each parked box three slots:
-lift by ten, translate above its target, and descend.  The total is
-$7N$ slots, and every source is gone before a target is filled.
-Widths remain at most one.  Source gaps protect the first lift,
-parking spacing protects strain and parking moves, target gaps protect
-delivery, and travel height separates every horizontal transfer from
-resting boxes.  These inequalities remain strict under $2\epsilon$
-padding.  Applying Lemma~\ref{lem:bb-curl} in chronological order
-fixes every stationary box and gives the exact full-box map to the
-moving one.  Thus overlapping source and target families cause no
-interference.
-
-Repeat the flat-ended template from zero and take its residual.
-The integer sampling induction gives the table execution.  In a
-nonhalting run every endpoint box lies in $x_1>-1$; translations
-interpolate corresponding endpoint points, strain occurs far to the
-right, and waiting leaves the point fixed.  The entire two-pass path
-therefore remains in $x_1>-1$.  A terminal sample lies inside the
-observer.  This proves the event for all times.
-
-For the exact pressure class use Lemma~\ref{lem:b-comparison}(iv).
-The prescribed compactly supported smooth velocity satisfies the
-reference $CH^2\cap C^1L^2$ and bounded-gradient conditions.  The
-competing velocity has precisely these bounds and a $CL^2$ pressure
-representative, as that case requires.  Its cutoff argument uses the
-pressure itself, with error $CR^{-1}\|p\|_2\|u-U\|_2$, and assumes
-no extra integrability of its derivative.  The lemma gives velocity
-uniqueness; the $L^2$ pressure representative is then zero.
-Compact support and
-bounded derivatives of the prescribed velocity give its uniform
-energy bound and its global material flow.
-\end{proof}
-
-
-\subsection{A two-cell append and its guarded inverse}\label{bc:two-cell-section}
-Use Lemma~\ref{lem:bb-window-compiler} with initial frontier $r_0=1$,
-renaming Ready, Go and Back as $M,R,L$ and $(E,P)$ as
-$(\perp,\star)$. The track order is work, mark, history.
-The Ready history coordinate and the marked Back row remain unrestricted;
-the unmarked Back row tests precisely the right-neighbor history.
-Thus the full table, not only its initialized execution, is identical.
-At checkpoint $n$, its frontier is $p_n=1+n>h_n$, where $h_n$ is
-the work-head position, and the exact cycle length is $2(1+n-h_n)+1$.
-Lemma~\ref{lem:bb-window-rectangles} applies with
-$(a,b)=(1,2)$: the complete allowed triple fixes both history writes,
-and its full image has prefix lengths $(1+e,2-e)$.
-
-\begin{theorem}[Guarded cylinders in a torus chart]\label{bc:guarded-torus}
-On the unit torus, the fixed label $(1/8,0,0)$ and the strip
-$\{x:x_1\pmod1\in(-1/4,0)\}$ realize halting by a general mean-zero
-force with zero initial velocity and pressure. It is periodic after time
-one and all its mixed derivatives are bounded. The torus comparison
-class gives uniqueness.
-\end{theorem}
-\begin{proof}\label{bc:guarded-torus-proof}
-Let $a$ be the augmented alphabet size and $k$ the control count in
-Lemma~\ref{lem:bb-window-compiler}. Use zero-blank base $B=2a$, and enumerate
-nonterminal and terminal states with positive integer indices $j\ge1$
-and first-coordinate offsets respectively
-$L_0+3j$ and $-L_0-3j$, where $L_0=4k+ka^3+10$.
-Refinement by triples gives at most $N=ka^3$ branches. Their source
-prefix lengths are $(1,2)$ and image lengths $(1+e,2-e)$, so the
-smallest possible grid gap is $B^{-3}$. If $P_i,Q_i$ are lower corners,
-the branch map is
-$Q_i+\diag(B^{-e_i},B^{e_i})(y-P_i)$.
-
-Choose three flat ramps $\alpha,\beta,\gamma$, active respectively in
-$(1/8,1/4)$, $(3/8,5/8)$ and $(3/4,7/8)$. Set
-\[
- c_i(t)=((1-\beta)P_i+\beta Q_i,\ i(\alpha-\gamma)),
- \quad l_i=1-\beta+\beta B^{-e_i}.
-\]
-The moving rectangle has side lengths $l_i B^{-1}$ and
-$l_i^{-1}B^{-2}$. Use a physical padding
-$\varepsilon=B^{-3}/4$ around each moving rectangle in three dimensions.
-During ascent and descent the planar endpoint gaps separate these
-neighborhoods; during the affine phase the unit height gaps separate them.
-Use Lemma~\ref{lem:bb-curl} for the paths
-$c_i+\diag(l_i,l_i^{-1},1)(y-(P_i,0))$ and sum the disjoint fields.
-The first coordinate is an endpoint convex combination even though the
-paths are based at lower corners rather than centers. All supports lie
-in $[-2L_0,2L_0]^3$, by the definitions of $L_0$ and $N$.
-
-Scale by $\rho=1/(32L_0)$, so the repeated mechanism is supported in
-$[-1/16,1/16]^3$. Load the fixed label $(1/8,0,0)$ to the scaled
-rational input code by a moving translation with cutoff radius $1/32$;
-its support stays in $(-1/4,1/4)^3$. For a nonterminal initial code,
-the entire first coordinate during loading is positive, and subsequent
-nonterminal branches have first coordinate greater than $1/32$.
-Terminal codes have first coordinate less than $-1/32$. Hence the stated
-strip detects exactly terminal entry, including initial halting, without
-any wraparound artifact. Periodize the compact curl fields and recompute
-the force at physical $\nu$. Mean zero, bounded derivatives and comparison
-follow from Lemma~\ref{bc:analytic}.
-\end{proof}
-
-\subsection{A filled half-line and mean-corrected vertical motion}
-\label{bc:halfline-section}
-The next recorder starts with an infinite constant history tail rather
-than a single frontier symbol. Its guard and its mean-zero vertical
-realization both have independent roles.
-
-\begin{lemma}[History at a filled half-line]\label{bc:halfline-table}
-A finite guarded table has a full-domain inverse and simulates the original
-machine from a tape with a separately constant tail in each direction.
-At checkpoint $n$ its nonempty history cells are exactly the absolute
-indices below $2+n$.
-\end{lemma}
-\begin{proof}\label{bc:halfline-table-proof}
-Normalize to one halt state $h$. For $i=(q,a)\in I=(Q\setminus\{h\})\times A$,
-write $\delta(q,a)=(q_i,b_i,e_i)$. History letters are
-$\mathcal H=\{\perp,\#\}\sqcup I$, with
-$\mathcal H_+=\{\#\}\sqcup I$. The tracks are work, mark and history;
-controls are $\mathsf R_q,\mathsf B_q,\mathsf T_i,\mathsf S_i$.
-Write $\ell_j$ for the history at relative cell $j$.
-Here is the complete table, with unmentioned cells unchanged:
-\begin{equation}\label{bc:halfline-rules}
-\begin{array}{c|c|c|l}
-\text{input}&\text{output}&\text{move}&\text{condition}\\\hline
-\mathsf R_q(a,0,k)&\mathsf T_i(b_i,0,k)&e_i&i=(q,a),\ q\ne h\\
-\mathsf T_i(c,0,k)&\mathsf S_i(c,1,k)&1&k\in\mathcal H_+\\
-\mathsf S_i(c,0,k)&\mathsf S_i(c,0,k)&1&k\in\mathcal H_+\\
-\mathsf S_i(c,0,\perp)&\mathsf B_{q_i}(c,0,i)&0&\ell_{+1}=\perp\\
-\mathsf B_q(c,0,k)&\mathsf B_q(c,0,k)&-1&k\in\mathcal H_+\\
-\mathsf B_q(c,1,k)&\mathsf R_q(c,0,k)&0&k\in\mathcal H_+.
-\end{array}
-\end{equation}
-Initialize $\mathsf R_{q_0}$ at head zero with marks zero, history $\#$
-at every index below two and $\perp$ elsewhere. At checkpoint $n$, the
-work head $p_n$ has $|p_n|\le n$ and the boundary is $r_n=2+n$.
-After the work move, $p'=p_n+e_i\le n+1<r_n$. Thus the mark is placed
-on nonempty history, and the right scan reaches the first empty cell.
-Its right neighbor is also empty, so the append is allowed. A finite
-left scan returns to and erases the unique mark. The new boundary is
-$r_n+1$; no other work symbol changes. The filler boundary at absolute
-index two is permanent, so it also retains the tape's absolute alignment.
-
-Into $\mathsf T_i$ the tag gives the old work instruction and shift.
-Into $\mathsf S_i$, both shifts are rightward, and the mark at the old
-cell distinguishes entry from a loop. Into $\mathsf B_q$, an output
-with empty history at offset $+1$ must be an append: its current record
-identifies $i$. An output of a leftward loop has nonempty history at
-that offset, since it is the old current cell. This is exactly why the
-append guard is necessary. Into $\mathsf R_q$ only mark erasure applies.
-Each case gives a unique predecessor on the full domain.  In particular,
-entry into $\mathsf B_q$ may have displacement zero or minus one;
-the neighboring history, rather than the target control alone, recovers
-that displacement.  Apply Lemma~\ref{lem:bb-window-rectangles} with
-$(a,b)=(1,2)$, refining by the full cells at $-1,0,1$.
-Its full-domain injectivity hypothesis has just been proved, and every
-write and displacement lies in the allowed window.  It therefore gives
-full image cylinders and separately disjoint closed rectangle families.
-\end{proof}
-
-Enumerate the resulting branches by $i=1,\ldots,N$, with displacements
-$e_i$.  Number the augmented alphabet by $0,\ldots,k-1$, use base $b=3k+1$
-and digits $1+3\ell$. Two different prefixes have gaps at least
-$2b^{-r}$ at their first differing place $r$. Codes lie in the interior
-of their intervals. If there are $K$ controls, let $\sigma_0=1/[8(K+1)]$.
-Put the terminal state's lower corner at $(3/4,1/4)$ and the other lower
-corners at $(1/8+2\ell\sigma_0,1/4)$ for $0\le\ell<K-1$, giving every state square side
-$\sigma_0$. Their branch rectangles have sides
-$\sigma_0b^{-1},\sigma_0b^{-2}$ and image sides
-$\sigma_0b^{-(1+e)},\sigma_0b^{-(2-e)}$. The families are separately
-disjoint by the full inverse. Every endpoint rectangle is contained in
-\begin{equation}\label{bc:halfline-code}
- F=[1/8,7/8]\times[1/4,3/8],\qquad z_*=1/4.
-\end{equation}
-All nonterminal state squares have first coordinate at most $3/8$.
-
-\begin{proposition}[Mean-corrected vertical transport]\label{bc:halfline-transport}
-The preceding full rectangles at height $z_*$ have an effective mean-zero
-solenoidal realization on the unit torus. Every tracked first coordinate
-is between its endpoint values throughout the transition.
-\end{proposition}
-\begin{proof}\label{bc:halfline-transport-proof}
-For a source rectangle $D$, choose a planar cutoff $\theta_D$ with
-padding $\varepsilon=\sigma_0b^{-3}/4$, equal to one near $D$. Define
-\[
- \kappa_D(x,y)=\theta_D(x,y)-\theta_D(x,y-1/2)
-\]
-periodically. It has integral zero. On its own source it equals one,
-and on all other sources it equals zero; the shifted copy lies away from
-$F$. Use the same construction for target rectangles $E$. Assign branch
-$i$ the height $z_i=1/2+i/[4(N+1)]$ and choose disjoint height cutoffs
-$\rho_i$ equal to one near $z_i$, with padding $1/[16(N+1)]$. The
-vertical fields with speeds
-\[
- Z_D(x,y)=\sum_i(z_i-z_*)\kappa_{D_i}(x,y),\qquad
- Z_E(x,y)=\sum_i(z_i-z_*)\kappa_{E_i}(x,y)
-\]
-are divergence free and mean zero, because they are independent of $z$.
-
-Let $\chi$ equal one near $F$, with planar padding $1/32$. For branch
-$i$, let $p_i,q_i$ be the source and target centers and put $d_i=q_i-p_i$, $c_i(\mu)=p_i+\mu d_i$,
-$l_i=1-\mu+\mu b^{-e_i}$ and
-\[
- H_i=\chi\left[d_{i,1}y-d_{i,2}x+
- \frac{b^{-e_i}-1}{l_i}(x-c_{i,1})(y-c_{i,2})\right].
-\]
-The middle field $W(\mu)=\sum_i\rho_i(z)(\partial_yH_i,-\partial_xH_i,0)$
-is divergence free and mean zero. The path
-\begin{equation}\label{bc:halfline-path}
- c_i(\mu)+\diag(l_i,l_i^{-1})(\zeta-p_i)
-\end{equation}
-stays in $F$: the first side is interpolated linearly and the reciprocal
-inequality \eqref{eq:ba-reciprocal-envelope} bounds the second side
-by the interpolated endpoint sides. Thus $\chi=1$ along the entire
-path, and differentiation gives its exact velocity.
-For three ordered flat ramps $\alpha_0,\alpha_1,\alpha_2$ on a unit
-interval, the complete field is
-\begin{equation}\label{bc:halfline-period}
- V=\alpha_0'(0,0,Z_D)+\alpha_1'W(\alpha_1)
-                   -\alpha_2'(0,0,Z_E).
-\end{equation}
-It lifts each source, performs its affine map in its own height layer,
-and lowers it at its target. The first coordinate is unchanged during
-vertical motion and is an endpoint convex combination in the middle.
-All statements hold on the full rectangles by the cutoff plateaus and
-ODE uniqueness. The zero-mean correction acts away from the computational
-rectangles and therefore changes none of these paths.
-\end{proof}
-
-\begin{theorem}[A fixed strip with half-line history]\label{bc:halfline-event}
-On $\T^3$, the fixed label $(1/8,1/4,1/4)$ and strip
-$2/3<x_1<15/16$ realize halting by a general mean-zero force, with zero
-initial velocity and pressure, bounded mixed derivatives and period one
-after time one. For a fixed machine only the loading field depends on the
-input. The material maps are volume-preserving diffeomorphisms, and the
-velocity and material-acceleration identities of
-Lemma~\ref{lem:p2-realization} hold.
-\end{theorem}
-\begin{proof}\label{bc:halfline-event-proof}
-The two initial tape tails are separately constant, so their radix sums
-are computable rationals, even though the left history tail is filled.
-Load the fixed label to that code at height $z_*$ by the planar Hamiltonian
-$\chi(d_x y-d_y x)$ with a flat scalar schedule. The segment stays in $F$.
-A nonterminal load and every nonterminal phase have first coordinate at
-most $3/8$; a terminal code lies in the stated strip. For an initially
-terminal machine the loading endpoint suffices. Repeat
-\eqref{bc:halfline-period} after time one. Checkpoint simulation and the
-intermediate-time guard prove the event equivalence. The common analytic
-lemma proves force effectivity, uniqueness, boundedness and the asserted
-flow identities. In particular $U=(\partial_t\Phi_t)\circ\Phi_t^{-1}$ and
-$\partial_{tt}\Phi_t=(f-\nabla p+\nu\Delta U)\circ\Phi_t$; these are
-identities for the constructed flow, not additional unknown equations.
-\end{proof}
+\section{Wider guards and alternative comparison classes}
+\label{sec:bb-guarded}
+
+The transport lemmas also accept tables which differ on tapes never
+visited by the initialized computation.  Those differences matter:
+they change the complete source and image cylinders on which the
+smooth extension is prescribed.  We give the remaining tables with
+their actual guards and then isolate the extra pressure arguments.
+
+\subsection{A shuttle aligned at the origin}
+\label{sec:bb-shuttle-periodic}
+
+\begin{theorem}\label{thm:bb-shuttle-periodic}
+On the unit torus there is an effective smooth mean-zero force,
+periodic from time zero with period one and with bounded mixed
+derivatives, whose unique zero-data classical solution has
+\[
+ \exists t\ge0:\quad X_1(t;0)\pmod1\in(1/16,1/4)
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+The constructed pressure is zero.  Alternatively the force may be
+projected to a solenoidal force with the same velocity and event,
+and with the correspondingly changed mean-zero pressure.
+\end{theorem}
+\begin{proof}\label{proof:bb-shuttle-periodic}
+Add a fresh nonhalting initial state $*$, without incoming transitions,
+which writes back the scanned symbol and stays while entering the
+original initial state.  This handles an originally halted input.
+Use work, flag and history tracks
+$\Sigma=\Gamma\times\{0,1\}\times(\{\perp,E\}\sqcup\mathcal R)$,
+and controls $R_q,A_r,B_r,C_q,D_q$.  Its complete table is
+\begin{equation}\label{eq:bb-shuttle-table}
+\begin{array}{lll}
+ R_q:(a,0,g)&\mapsto A_r:(b_r,0,g),d_r,\\
+ A_r:(a,0,g)&\mapsto B_r:(a,1,g),1,\\
+ B_r:(a,0,g)&\mapsto B_r:(a,0,g),1&g\ne E,\\
+ B_r:(a,0,E)&\mapsto C_{p_r}:(a,0,r),1,\\
+ C_q:(a,0,\perp)&\mapsto D_q:(a,0,E),-1,\\
+ D_q:(a,0,g)&\mapsto D_q:(a,0,g),-1&g\ne E,\\
+ D_q:(a,1,g)&\mapsto R_q:(a,0,g),0&g\ne E.
+\end{array}
+\end{equation}
+In particular its first two rows do not exclude the frontier symbol;
+this is a wider domain than the move-first table.  Into $A_r$ the
+record identifies the work instruction; into $B_r$ the output flag
+distinguishes entry and loop; into $C_q$ the record determines the
+incoming scan; into $D_q$ a written $E$ distinguishes entry and loop;
+into $R_q$ only clearing a flag is possible.  The destination fixes
+the incoming displacement.  These checks include every copied track
+and every permitted frontier case.
+
+Initialize at $R_*$, with all flags zero and only $E$ at cell two.
+At ready step $n$ the frontier is $e_n=n+2$, the records are at
+$2,\ldots,n+1$, and the head is $h_n$, $|h_n|\le n$.
+The work move reaches $h'=h_{n+1}<e_n$; the scan marks that cell,
+records at $e_n$, creates the next frontier, returns to the mark and
+clears it.  Its count is $2(e_n-h')+4$ and all its tests hold.
+Thus it has exactly the required initialized behavior despite its
+wider full domain.
+
+Use even digits in base $b=2|\Sigma|$ and initial fractions
+$\ell_*,r_*$.  Let $S$ be the control set and $m$ count the branches in
+Lemma~\ref{lem:bb-onecell}, and put $L=100(1+|S|+m)$.
+Give $R_*$ offset $-\ell_*$, other nonhalt states offsets $-4i$,
+and ready halt states offsets $L+4i$.  The code is
+$(o_s+\ell,r-r_*,0)$, so the initial point is zero.
+The incoming conditions give separately disjoint full rectangle
+families.  Thicken each by $[-1,1]$, lift to height $10i$, and use
+the linear reciprocal middle scale.  Choose padding at most $1/10$
+small enough for both family gaps.  Vertical phases can use fixed
+column potentials $(0,\dot Z x\chi,0)$, with columns covering
+$[-3,10i+3]$; middle phases use horizontal Hamiltonians in disjoint
+height bands.  All paths and supports are contained in
+$(-2L,2L)^3$: their coordinate bounds are
+$[-4|S|-3,L+4|S|+4]$, $[-4,4]$, and $[-3,10m+3]$.
+This verifies the hypotheses of the lift schedule on neighborhoods
+of the whole boxes.
+
+Scale by $\gamma=(8L)^{-1}$ into the torus patch
+$(-1/2,1/2)^3$: $U(t,x)=\gamma W(t,x/\gamma)$.
+The support lies inside $(-1/4,1/4)^3$, the potentials scale by
+$\gamma^2$, and the residual is recomputed at viscosity $\nu$ by
+Lemma~\ref{lem:p2-chart}.  Periodicity starts at zero
+because the input is already encoded there and the pulse has idle
+endpoint collars.  At integer times the particle is the correct code.
+Halt first coordinates before scaling lie between $L+4$ and $2L$.
+For a nonhalt branch its endpoint first coordinates are at most one;
+the linear-scale identity keeps them at most one, and the weaker
+bound $3/2$ also follows from center and width estimates.  After
+scaling every nonhalt path lies in $(-1/4,1/16)$, disjoint modulo
+one from the event.  The initialized dummy step makes the implication
+valid even for an originally halted machine.
+
+For the optional projection solve $\Delta q=\operatorname{div}g$,
+$\int q=0$, where $g=\mathcal F_\nu[U]$.
+Proposition~\ref{prop:projection-l2} gives the effective force
+$f=g-\nabla q$ and pressure $p=-q$, with all mixed derivative bounds
+and periodicity retained.  The pressure change is part of this
+alternative; the direct residual had pressure zero.
+\end{proof}
+
+\subsection{A persistent marker and cursor shifts of two cells}
+\label{sec:bb-marked-window}
+
+\begin{theorem}\label{thm:bb-marked-window}
+There is a unit-torus realization, periodic for $t\ge1$, with fixed
+label $a_*=(1/4,1/4,1/4)$.  Its event
+\[ \exists t\ge0:\quad X_1(t;a_*)\pmod1\in(2/3,5/6) \]
+occurs exactly when the machine halts.
+It uses an injective marked-head table whose cursor shifts may equal
+two.  Its force is effective, smooth and mean zero, with bounded
+mixed derivatives and the zero-data uniqueness class of
+Proposition~\ref{prop:bb-realization}.
+\end{theorem}
+\begin{proof}\label{proof:bb-marked-window}
+Normalize to one halt state and use work, permanent work-head marker,
+and history tracks $(a,m,g)\in\mathcal A:=\Gamma\times\{0,1\}\times
+(\{\bot,\#\}\sqcup\mathcal R)$.
+Let $S$ be the set of controls $M_q,L_q,R_r$.  With the cursor at zero, make
+exactly the following tests and writes, then reindex by
+$T'_i=\bar T_{i+e}$ for the displayed shift $e$.
+\begin{enumerate}
+\item In $M_q$, $q$ nonhalting, require $m_0=1$ and, if $d_r\ne0$,
+$m_{d_r}=0$.  Write $b_r$ at zero, move the marker to $d_r$ if
+necessary, and enter $R_r$ with shift $e=d_r+1$.
+\item In $R_r$, a cell with $m_0=0,g_0\ne\#$ gives shift $1$
+in the same control.  If $m_0=0,g_0=\#,g_1=\bot$, write
+$g_0=r,g_1=\#$ and enter $L_{p_r}$ with shift zero.
+\item In $L_q$, if $m_0=0,g_0\ne\#$, shift $-1$ in that control.
+If $m_0=1$, enter $M_q$ with shift zero.
+\end{enumerate}
+All unmentioned entries are retained.  Into $R_r$, a main entry has
+output $m_{-1}=1$, whereas a scan has $m_{-1}=0$.
+The record recovers the overwritten work letter, displacement and
+marker changes.  Into $L_q$, a log write has output $g_1=\#$
+and record $r$ at zero, whereas a scan has $g_1\ne\#$.
+Its tests recover the overwritten pair.  Into $M_q$ only the
+matching control change is possible.  This is a full-domain inverse;
+uniqueness of a marker was not assumed on arbitrary tapes.
+
+Initially the sole marker is at zero, the sole pointer at two, and
+all other history is blank.  At work step $n$, head and marker are
+at $h$, pointer at $n+2$, and records at $2,\ldots,n+1$.
+The work update puts the marker at $h'=h+d_r$ and the cursor at
+$h'+1\le n+2$.  The forward scan reaches the pointer, logs, and
+returns to $h'$.  All tests hold since $h'<n+2$.  The cycle length is
+\[
+ 1+[n+2-(h'+1)]+1+[(n+2)-h']+1.
+\]
+This proves finite continuation, exact work simulation and halting.
+
+Use even digits, blank zero, in base $B=2|\mathcal A|+1$.
+Enumerate every full word on $[-2,2]$ with an applicable rule, giving
+$J$ branches.
+Lemma~\ref{lem:bb-window-rectangles} gives prefix lengths $(2,3)$
+and $(2+e,3-e)$, with $e\in\{-1,0,1,2\}$.
+Each output length is at most four, so gaps after chart scaling
+by $l=[100(|S|+1)]^{-1}$ are at least $lB^{-4}$.
+Place the halt square at $(3/4,1/4)+l[0,1]^2$ and other squares
+at $(1/4+2il,1/4)+l[0,1]^2$; the latter lie below $x_1=1/3$.
+At base height $1/4$, lift branch $j$ to
+$1/2+j/[4(J+1)]$, use exponential scales $B^{-e_j\theta},B^{e_j\theta}$,
+and lower.  Padding
+$\min(1/200,1/[32(J+1)],lB^{-4}/3)$ separates supports.
+The middle Hamiltonian is
+\[
+ \dot c_x(y-c_y)-\dot c_y(x-c_x)
+      -e_j\dot\theta\log B\,(x-c_x)(y-c_y).
+\]
+It gives the intended derivative on the full moving rectangle.
+The plateau margins and finite scale bounds extend this to an open
+neighborhood, as in Lemma~\ref{lem:bb-curl}.
+
+Load the fixed label along its segment to the rational initial code
+in time one, then repeat the unit field.  Integer samples give the
+full local execution.  In a nonhalt period the center is below $1/3$
+and the width at most $l$, so the whole path remains below $1/2$;
+loading also avoids the event.  Halt codes lie in $[3/4,3/4+l]$.
+Thus every real time is accounted for.  The pointer site $n+2$,
+record count $n$, and relative pointer offset recover the absolute
+head position at main checkpoints.  The residual construction
+proves the remaining assertions.
+\end{proof}
+
+\subsection{A split direction shuttle in the whole space}
+\label{sec:bb-direction-shuttle}
+
+\begin{theorem}\label{thm:bb-direction-shuttle}
+On $\R^3$ an effective compactly supported force with bounded mixed
+derivatives and period one after $t=1$ realizes the halting event
+$X_1(t;(1,0,0))<0$.  Uniqueness holds with bounded $u,\nabla u$,
+$u\in C H^2\cap C^1L^2$ and pressure modulo constants in $C H^1$
+on each finite interval.  An alternative force is square-integrable
+in space-time, with the same label, event, support and derivative
+bounds.
+\end{theorem}
+\begin{proof}\label{proof:bb-direction-shuttle}
+Use exactly the direction table of Lemma~\ref{lem:bb-direction}
+without its passive track, renaming
+the controls by $M\mapsto S$, $G\mapsto R$, $A\mapsto I$,
+and $B\mapsto J$, and the frontier by $F\mapsto\#$.  This is a bijection
+of its entire partial table, including the nonfrontier tests.
+Write $\Sigma$ for the resulting three-track alphabet.
+Initialize frontier at one and zero flags.  At work step $j$ the
+frontier is $j+1$ and the cycle length is
+\[2(j-h_j)+5.\]
+
+Use odd digits in base $B=2|\Sigma|+1$, offsets $3,6,\ldots$
+for nonhalt controls and $-3,-6,\ldots$ for halt main controls.
+In this realization split \emph{every} one-cell rule by every left
+symbol $c$, including the stationary and right-moving rules.
+Its source is $I(c)\times I(a)$, and the three target rectangles,
+for moves $1,-1,0$, are respectively
+\[
+ \varphi_b\varphi_c([0,1])\times[0,1],\quad
+ [0,1]\times\varphi_c\varphi_b([0,1]),\quad I(c)\times I(b),
+ \qquad \varphi_a(t)=(d(a)+t)/B.
+\]
+The maps themselves are the restrictions of
+\eqref{eq:bb-one-cell-branches}.  Source separation follows from
+$(c,a)$, target separation from the fixed incoming move and $(c,b)$.
+Their gaps are at least $B^{-2}$, and their linear parts are
+$\operatorname{diag}(B^{-d},B^d)$.
+
+Thicken by $[-1,1]$, lift branch $i$ to $4i$, use exponential
+middle scale and center interpolation, then descend.  Padding smaller
+than $(4B^2)^{-1}$ is valid, by source gaps, height gaps, and target
+gaps in the three phases.  A first localized translation takes
+$(1,0,0)$ to the rational initial code.  Exact full-box transport
+then gives the local execution at every subsequent integer time.
+Nonhalt centers have first coordinate at least three and horizontal
+width at most one, so every intermediate point stays positive;
+the nonhalt loading segment does too.  A halt sample has negative
+first coordinate, including initial halting after loading.
+Proposition~\ref{prop:bb-realization} gives the force, pressure zero,
+and uniqueness.  The decaying alternative follows from
+Proposition~\ref{prop:bb-late-logclock} below, applied to this same
+loaded template.
+\end{proof}
+
+\subsection{Two-cell history and a gradient-only pressure condition}
+\label{sec:bb-guarded-history}
+
+\begin{theorem}\label{thm:bb-guarded-history}
+An effective smooth force on $\R^3$, periodic from time zero,
+with one compact spatial support and bounded mixed derivatives,
+realizes the fixed event
+\[
+ \exists t\ge0:\quad X(t;(2,0,0))\in(-2,-1)\times(0,1)\times(-1,1)
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+Velocity is unique in the class
+$u\in C([0,T];H^4)\cap C^1([0,T];H^2)$,
+$\nabla p\in C([0,T];L^2)$ for every finite $T$, allowing a
+distributional pressure gradient.  Alternatively the force belongs
+to $L^2([0,\infty);H^j)$ for every integer $j\ge0$, with all the
+same support, event and comparison-class conclusions.
+\end{theorem}
+\begin{proof}\label{proof:bb-guarded-history}
+Add a fresh initial state which writes back and stays, and replace
+all originally terminal state-symbol pairs by stay transitions into
+one fresh terminal state $q_h$.  Normalize the other missing
+instructions similarly.  Let $r=(q,a)\mapsto(p_r,b_r,d_r)$ denote
+the resulting nonterminal instructions, and let $\mathcal A$ be the
+work alphabet. The full cell alphabet is
+$\Gamma=\mathcal A\times\{0,1\}\times(\{e,p\}\sqcup\mathcal R)$.
+Use its work, flag and log tracks $W,F,L$,
+and controls $N_q,A_r,R_r,B_q$.  The complete rules are:
+\begin{enumerate}
+\item At $N_q$, $q\ne q_h$, write $W_0=b_r$, enter $A_r$,
+and shift $d_r$, with no test on the other tracks.
+\item At $A_r$, require $F_0=0$, set it to one, enter $R_r$
+and shift one.
+\item At $R_r$, if $F_0=0,L_0\ne p$, shift one in that control.
+If $F_0=0,L_0=p,L_1=e$, write $L_0=r,L_1=p$, enter $B_{p_r}$
+and shift minus one.
+\item At $B_q$, if $F_0=0,L_1\ne p$, shift minus one in that
+control.  If $F_0=1$, clear it and enter $N_q$ without moving.
+\end{enumerate}
+The first output control records the overwritten work pair and shift.
+Into $R_r$, output flag one at $-1$ distinguishes entry from its
+loop.  Into $B_q$, a log insertion has pointer at output index two
+and record at one, whereas a return loop has a nonpointer at two;
+the record and tests recover both overwritten log entries.  Into
+$N_q$ only unmarking is possible.  Thus this map is injective on
+its whole domain, including the unrestricted first-row tracks.
+
+Initialize zero flags and a sole pointer at cell three, all other
+log cells empty.  At work step $n$, records occupy $3,\ldots,n+2$
+and pointer $g=n+3$.  The work move reaches $h'$ with $g-h'\ge2$.
+The finite scan flags $h'$, logs at $g$, advances the pointer, and
+returns to the flag.  The return guard holds because the new pointer
+is at $g+1$.  The cycle has $2(g-h'-1)+4$ steps.  This verifies
+the work invariant and eventual terminal entry.
+
+Use odd digits in radix $D=2|\Gamma|+1$ and write
+$\phi(\xi)=\sum_{j\ge1}d(\xi_j)D^{-j}$ for the radix sum.
+Let $L_*,R_*$ be the initialized left and right streams, ordered as in
+Lemma~\ref{lem:bb-onecell}.  Give the terminal state
+chart offset $(-2,0)$, the initial chart offset
+$(2-\phi(L_*),-\phi(R_*))$, and all other charts $(4+2j,0)$.
+The initial code is $(2,0,0)$, every nonterminal interval lies in
+$x_1>1$, and terminal codes lie strictly in $(-2,-1)\times(0,1)$.
+The infinite tails are in $[\delta,1-\delta]$,
+$\delta=1/(D-1)$, so finite-prefix decoding of length $b$ is possible
+from error less than $(\delta/2)D^{-b}$ without a boundary ambiguity.
+
+Split full triples $[-1,1]$.  By
+Lemma~\ref{lem:bb-window-rectangles}, source prefix lengths $(1,2)$
+become $(1+d,2-d)$, target cylinders are complete, and both rectangle
+families have gaps at least $D^{-3}$.  Thicken by $[-1/4,1/4]$,
+lift to height $4i$, use linear middle scale
+$\mu_i=1+(D^{-d_i}-1)\theta$, and descend.  Padding smaller than
+$D^{-3}/4$ separates all phases.  Repeating the pulse, zero on
+$[0,1/12]$ and $[11/12,1]$, gives periodicity from zero without
+loading.  At a nonhalting branch's lift and descent the first
+coordinate is greater than one; during the middle transfer the
+particle's third coordinate is exactly $4i$, outside $(-1,1)$.
+Thus even a horizontal route passing over the observation box cannot
+cause a false visit.  The integer samples and the compiler invariant
+prove halting equivalence.  The record count and pointer site $n+3$
+recover absolute indexing at checkpoints.
+
+For uniqueness apply Lemma~\ref{lem:b-comparison}(iii).
+The explicit compactly supported smooth field lies in $CH^4\cap C^1H^2$
+on every finite interval and has pressure zero.  The competing class
+in the theorem is precisely case (iii), including its distributional
+$CL^2$ gradient.  Lemma~\ref{lem:b-gradient} cancels that gradient
+against the solenoidal velocity difference without introducing a pressure
+$L^2$ norm.  Thus the velocity and pressure gradient are unique, while
+pressure itself remains free up to a time-dependent spatial constant.
+Proposition~\ref{prop:bb-late-logclock} gives the stated
+decaying alternative.
+\end{proof}
+
+\subsection{A flagged head and two-pass parking}
+\label{sec:bb-two-pass-parking}
+
+\begin{theorem}\label{thm:bb-two-pass-parking}
+For any machine and finitely specified initial tape, an effective
+smooth force on $\R^3$, periodic from time zero, with one compact
+spatial support and bounded mixed derivatives, realizes the event
+\[
+ \exists t\ge0:\quad X(t;0)\in(-6,-4)\times(-1,1)\times(-1,1)
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+The zero-data velocity is unique among classical smooth solutions
+with bounded $u,\nabla u$, $u\in C H^2\cap C^1L^2$ and $p\in C L^2$
+on every finite interval.  The prescribed solution has uniformly
+bounded kinetic energy.
+\end{theorem}
+\begin{proof}\label{proof:bb-two-pass-parking}
+Merge terminal states into $q_\dagger$, send missing instructions
+there by a stay step, and add a fresh initial state $q_*$, with no
+incoming instruction, which writes back and stays while entering
+the old initial state.  For each normalized nonterminal rule
+$r=(q,a)\mapsto(p_r,b_r,d_r)$ use cell letters
+\[
+ (w,h,j)\in\Gamma:=\mathcal A\times
+ (\{\circ,\star\}\sqcup\{[r]:r\in\mathcal R\})\times\{0,1\},
+\]
+where $\mathcal A$ is the work alphabet and $\mathcal R$ is the
+finite set of normalized instruction labels. Thus $\Gamma$ includes
+all three tracks.
+The complete table, with $h\ne\star$ wherever $h$ occurs, is
+\[
+\begin{array}{lll}
+ M_q:(a,h,1)&\mapsto A_r:(b_r,h,0),d_r,\\
+ A_r:(w,h,0)&\mapsto R_r:(w,h,1),1,\\
+ R_r:(w,h,0)&\mapsto R_r:(w,h,0),1,\\
+ R_r:(w,\star,0)&\mapsto B_{p_r}:(w,[r],0),1,\\
+ B_q:(w,\circ,0)&\mapsto L_q:(w,\star,0),-1,\\
+ L_q:(w,h,0)&\mapsto L_q:(w,h,0),-1,\\
+ L_q:(w,h,1)&\mapsto M_q:(w,h,1),0.
+\end{array}
+\]
+The destination fixes the incoming displacement.  The record index
+recovers the first row's old work pair; a flag distinguishes the
+entries into $R_r$; a record distinguishes the entries into $B_q$;
+a written frontier distinguishes entry and loop into $L_q$;
+and entry into $M_q$ is the unique cell-preserving last row.
+This proves the full one-cell inverse conditions.
+
+Initialize the sole flag at zero and the sole frontier at three,
+with empty history elsewhere.  At ready step $k$, only the work
+head $i$ is flagged, records occupy $3,\ldots,k+2$, and the frontier
+is $e_k=k+3$.  The work rule clears the flag and moves to
+$j=i+d_r\le e_k-2$, which is unflagged even for $d_r=0$.
+It is flagged anew; the scan logs at $e_k$, advances the pointer,
+returns, and preserves the ready flag.  The two loop lengths are
+$e_k-j-1$ and $e_k-j$, with five other rules, for a total
+$2(e_k-j)+4$.  Thus all scans terminate and ready halting is exact.
+
+Use odd digits in base $K=2|\Gamma|+2$, initial fractions $x_*,y_*$,
+and state offsets $a_{M_{q_*}}=0$, $a_{M_{q_\dagger}}=-5$, and
+$4,8,12,\ldots$ for the others.  The code
+$(a_s-x_*+x,y-y_*,0)$ starts at zero.  Lemma~\ref{lem:bb-onecell}
+gives $N$ full rectangle maps with scales $\lambda_i\in\{K^{-1},1,K\}$;
+thicken by $[-1/4,1/4]$.  The entire terminal block is inside the
+observation box because $0<x_*,y_*<1$.  Every nonterminal block is
+in $x_1>-1$.  Eventually constant tails are rational even for an
+arbitrary finitely specified initial tape, and the pointer and record
+count recover its absolute head position at ready checkpoints.
+
+For the required two-pass geometry, let $g$ be the minimum of one
+and all positive separating coordinate gaps in either planar family,
+and set $\epsilon=g/20$.  Choose
+\[
+ C=20+\max(\{0\}\cup\{\text{right endpoints of source and target boxes}\}),
+ \qquad S_i=(C+4i,0,0).
+\]
+In a first pass give each source four consecutive slots: lift by ten,
+translate above $S_i$, descend, and reshape there by the linear
+reciprocal scale.  In a second pass give each parked box three slots:
+lift by ten, translate above its target, and descend.  The total is
+$7N$ slots, and every source is gone before a target is filled.
+Widths remain at most one.  Source gaps protect the first lift,
+parking spacing protects strain and parking moves, target gaps protect
+delivery, and travel height separates every horizontal transfer from
+resting boxes.  These inequalities remain strict under $2\epsilon$
+padding.  Applying Lemma~\ref{lem:bb-curl} in chronological order
+fixes every stationary box and gives the exact full-box map to the
+moving one.  Thus overlapping source and target families cause no
+interference.
+
+Repeat the flat-ended template from zero and take its residual.
+The integer sampling induction gives the table execution.  In a
+nonhalting run every endpoint box lies in $x_1>-1$; translations
+interpolate corresponding endpoint points, strain occurs far to the
+right, and waiting leaves the point fixed.  The entire two-pass path
+therefore remains in $x_1>-1$.  A terminal sample lies inside the
+observer.  This proves the event for all times.
+
+For the exact pressure class use Lemma~\ref{lem:b-comparison}(iv).
+The prescribed compactly supported smooth velocity satisfies the
+reference $CH^2\cap C^1L^2$ and bounded-gradient conditions.  The
+competing velocity has precisely these bounds and a $CL^2$ pressure
+representative, as that case requires.  Its cutoff argument uses the
+pressure itself, with error $CR^{-1}\|p\|_2\|u-U\|_2$, and assumes
+no extra integrability of its derivative.  The lemma gives velocity
+uniqueness; the $L^2$ pressure representative is then zero.
+Compact support and
+bounded derivatives of the prescribed velocity give its uniform
+energy bound and its global material flow.
+\end{proof}
+
+
+\subsection{A two-cell append and its guarded inverse}\label{bc:two-cell-section}
+Use Lemma~\ref{lem:bb-window-compiler} with initial frontier $r_0=1$,
+renaming Ready, Go and Back as $M,R,L$ and $(E,P)$ as
+$(\perp,\star)$. The track order is work, mark, history.
+The Ready history coordinate and the marked Back row remain unrestricted;
+the unmarked Back row tests precisely the right-neighbor history.
+Thus the full table, not only its initialized execution, is identical.
+At checkpoint $n$, its frontier is $p_n=1+n>h_n$, where $h_n$ is
+the work-head position, and the exact cycle length is $2(1+n-h_n)+1$.
+Lemma~\ref{lem:bb-window-rectangles} applies with
+$(a,b)=(1,2)$: the complete allowed triple fixes both history writes,
+and its full image has prefix lengths $(1+e,2-e)$.
+
+\begin{theorem}[Guarded cylinders in a torus chart]\label{bc:guarded-torus}
+On the unit torus, the fixed label $(1/8,0,0)$ and the strip
+$\{x:x_1\pmod1\in(-1/4,0)\}$ realize halting by a general mean-zero
+force with zero initial velocity and pressure. It is periodic after time
+one and all its mixed derivatives are bounded. The torus comparison
+class gives uniqueness.
+\end{theorem}
+\begin{proof}\label{bc:guarded-torus-proof}
+Let $a$ be the augmented alphabet size and $k$ the control count in
+Lemma~\ref{lem:bb-window-compiler}. Use zero-blank base $B=2a$, and enumerate
+nonterminal and terminal states with positive integer indices $j\ge1$
+and first-coordinate offsets respectively
+$L_0+3j$ and $-L_0-3j$, where $L_0=4k+ka^3+10$.
+Refinement by triples gives at most $N=ka^3$ branches. Their source
+prefix lengths are $(1,2)$ and image lengths $(1+e,2-e)$, so the
+smallest possible grid gap is $B^{-3}$. If $P_i,Q_i$ are lower corners,
+the branch map is
+$Q_i+\diag(B^{-e_i},B^{e_i})(y-P_i)$.
+
+Choose three flat ramps $\alpha,\beta,\gamma$, active respectively in
+$(1/8,1/4)$, $(3/8,5/8)$ and $(3/4,7/8)$. Set
+\[
+ c_i(t)=((1-\beta)P_i+\beta Q_i,\ i(\alpha-\gamma)),
+ \quad l_i=1-\beta+\beta B^{-e_i}.
+\]
+The moving rectangle has side lengths $l_i B^{-1}$ and
+$l_i^{-1}B^{-2}$. Use a physical padding
+$\varepsilon=B^{-3}/4$ around each moving rectangle in three dimensions.
+During ascent and descent the planar endpoint gaps separate these
+neighborhoods; during the affine phase the unit height gaps separate them.
+Use Lemma~\ref{lem:bb-curl} for the paths
+$c_i+\diag(l_i,l_i^{-1},1)(y-(P_i,0))$ and sum the disjoint fields.
+The first coordinate is an endpoint convex combination even though the
+paths are based at lower corners rather than centers. All supports lie
+in $[-2L_0,2L_0]^3$, by the definitions of $L_0$ and $N$.
+
+Scale by $\rho=1/(32L_0)$, so the repeated mechanism is supported in
+$[-1/16,1/16]^3$. Load the fixed label $(1/8,0,0)$ to the scaled
+rational input code by a moving translation with cutoff radius $1/32$;
+its support stays in $(-1/4,1/4)^3$. For a nonterminal initial code,
+the entire first coordinate during loading is positive, and subsequent
+nonterminal branches have first coordinate greater than $1/32$.
+Terminal codes have first coordinate less than $-1/32$. Hence the stated
+strip detects exactly terminal entry, including initial halting, without
+any wraparound artifact. Periodize the compact curl fields and recompute
+the force at physical $\nu$. Mean zero, bounded derivatives and comparison
+follow from Lemma~\ref{bc:analytic}.
+\end{proof}
+
+\subsection{A filled half-line and mean-corrected vertical motion}
+\label{bc:halfline-section}
+The next recorder starts with an infinite constant history tail rather
+than a single frontier symbol. Its guard and its mean-zero vertical
+realization both have independent roles.
+
+\begin{lemma}[History at a filled half-line]\label{bc:halfline-table}
+A finite guarded table has a full-domain inverse and simulates the original
+machine from a tape with a separately constant tail in each direction.
+At checkpoint $n$ its nonempty history cells are exactly the absolute
+indices below $2+n$.
+\end{lemma}
+\begin{proof}\label{bc:halfline-table-proof}
+Normalize to one halt state $h$. For $i=(q,a)\in I=(Q\setminus\{h\})\times A$,
+write $\delta(q,a)=(q_i,b_i,e_i)$. History letters are
+$\mathcal H=\{\perp,\#\}\sqcup I$, with
+$\mathcal H_+=\{\#\}\sqcup I$. The tracks are work, mark and history;
+controls are $\mathsf R_q,\mathsf B_q,\mathsf T_i,\mathsf S_i$.
+Write $\ell_j$ for the history at relative cell $j$.
+Here is the complete table, with unmentioned cells unchanged:
+\begin{equation}\label{bc:halfline-rules}
+\begin{array}{c|c|c|l}
+\text{input}&\text{output}&\text{move}&\text{condition}\\\hline
+\mathsf R_q(a,0,k)&\mathsf T_i(b_i,0,k)&e_i&i=(q,a),\ q\ne h\\
+\mathsf T_i(c,0,k)&\mathsf S_i(c,1,k)&1&k\in\mathcal H_+\\
+\mathsf S_i(c,0,k)&\mathsf S_i(c,0,k)&1&k\in\mathcal H_+\\
+\mathsf S_i(c,0,\perp)&\mathsf B_{q_i}(c,0,i)&0&\ell_{+1}=\perp\\
+\mathsf B_q(c,0,k)&\mathsf B_q(c,0,k)&-1&k\in\mathcal H_+\\
+\mathsf B_q(c,1,k)&\mathsf R_q(c,0,k)&0&k\in\mathcal H_+.
+\end{array}
+\end{equation}
+Initialize $\mathsf R_{q_0}$ at head zero with marks zero, history $\#$
+at every index below two and $\perp$ elsewhere. At checkpoint $n$, the
+work head $p_n$ has $|p_n|\le n$ and the boundary is $r_n=2+n$.
+After the work move, $p'=p_n+e_i\le n+1<r_n$. Thus the mark is placed
+on nonempty history, and the right scan reaches the first empty cell.
+Its right neighbor is also empty, so the append is allowed. A finite
+left scan returns to and erases the unique mark. The new boundary is
+$r_n+1$; no other work symbol changes. The filler boundary at absolute
+index two is permanent, so it also retains the tape's absolute alignment.
+
+Into $\mathsf T_i$ the tag gives the old work instruction and shift.
+Into $\mathsf S_i$, both shifts are rightward, and the mark at the old
+cell distinguishes entry from a loop. Into $\mathsf B_q$, an output
+with empty history at offset $+1$ must be an append: its current record
+identifies $i$. An output of a leftward loop has nonempty history at
+that offset, since it is the old current cell. This is exactly why the
+append guard is necessary. Into $\mathsf R_q$ only mark erasure applies.
+Each case gives a unique predecessor on the full domain.  In particular,
+entry into $\mathsf B_q$ may have displacement zero or minus one;
+the neighboring history, rather than the target control alone, recovers
+that displacement.  Apply Lemma~\ref{lem:bb-window-rectangles} with
+$(a,b)=(1,2)$, refining by the full cells at $-1,0,1$.
+Its full-domain injectivity hypothesis has just been proved, and every
+write and displacement lies in the allowed window.  It therefore gives
+full image cylinders and separately disjoint closed rectangle families.
+\end{proof}
+
+Enumerate the resulting branches by $i=1,\ldots,N$, with displacements
+$e_i$.  Number the augmented alphabet by $0,\ldots,k-1$, use base $b=3k+1$
+and digits $1+3\ell$. Two different prefixes have gaps at least
+$2b^{-r}$ at their first differing place $r$. Codes lie in the interior
+of their intervals. If there are $K$ controls, let $\sigma_0=1/[8(K+1)]$.
+Put the terminal state's lower corner at $(3/4,1/4)$ and the other lower
+corners at $(1/8+2\ell\sigma_0,1/4)$ for $0\le\ell<K-1$, giving every state square side
+$\sigma_0$. Their branch rectangles have sides
+$\sigma_0b^{-1},\sigma_0b^{-2}$ and image sides
+$\sigma_0b^{-(1+e)},\sigma_0b^{-(2-e)}$. The families are separately
+disjoint by the full inverse. Every endpoint rectangle is contained in
+\begin{equation}\label{bc:halfline-code}
+ F=[1/8,7/8]\times[1/4,3/8],\qquad z_*=1/4.
+\end{equation}
+All nonterminal state squares have first coordinate at most $3/8$.
+
+\begin{proposition}[Mean-corrected vertical transport]\label{bc:halfline-transport}
+The preceding full rectangles at height $z_*$ have an effective mean-zero
+solenoidal realization on the unit torus. Every tracked first coordinate
+is between its endpoint values throughout the transition.
+\end{proposition}
+\begin{proof}\label{bc:halfline-transport-proof}
+For a source rectangle $D$, choose a planar cutoff $\theta_D$ with
+padding $\varepsilon=\sigma_0b^{-3}/4$, equal to one near $D$. Define
+\[
+ \kappa_D(x,y)=\theta_D(x,y)-\theta_D(x,y-1/2)
+\]
+periodically. It has integral zero. On its own source it equals one,
+and on all other sources it equals zero; the shifted copy lies away from
+$F$. Use the same construction for target rectangles $E$. Assign branch
+$i$ the height $z_i=1/2+i/[4(N+1)]$ and choose disjoint height cutoffs
+$\rho_i$ equal to one near $z_i$, with padding $1/[16(N+1)]$. The
+vertical fields with speeds
+\[
+ Z_D(x,y)=\sum_i(z_i-z_*)\kappa_{D_i}(x,y),\qquad
+ Z_E(x,y)=\sum_i(z_i-z_*)\kappa_{E_i}(x,y)
+\]
+are divergence free and mean zero, because they are independent of $z$.
+
+Let $\chi$ equal one near $F$, with planar padding $1/32$. For branch
+$i$, let $p_i,q_i$ be the source and target centers and put $d_i=q_i-p_i$, $c_i(\mu)=p_i+\mu d_i$,
+$l_i=1-\mu+\mu b^{-e_i}$ and
+\[
+ H_i=\chi\left[d_{i,1}y-d_{i,2}x+
+ \frac{b^{-e_i}-1}{l_i}(x-c_{i,1})(y-c_{i,2})\right].
+\]
+The middle field $W(\mu)=\sum_i\rho_i(z)(\partial_yH_i,-\partial_xH_i,0)$
+is divergence free and mean zero. The path
+\begin{equation}\label{bc:halfline-path}
+ c_i(\mu)+\diag(l_i,l_i^{-1})(\zeta-p_i)
+\end{equation}
+stays in $F$: the first side is interpolated linearly and the reciprocal
+inequality \eqref{eq:ba-reciprocal-envelope} bounds the second side
+by the interpolated endpoint sides. Thus $\chi=1$ along the entire
+path, and differentiation gives its exact velocity.
+For three ordered flat ramps $\alpha_0,\alpha_1,\alpha_2$ on a unit
+interval, the complete field is
+\begin{equation}\label{bc:halfline-period}
+ V=\alpha_0'(0,0,Z_D)+\alpha_1'W(\alpha_1)
+                   -\alpha_2'(0,0,Z_E).
+\end{equation}
+It lifts each source, performs its affine map in its own height layer,
+and lowers it at its target. The first coordinate is unchanged during
+vertical motion and is an endpoint convex combination in the middle.
+All statements hold on the full rectangles by the cutoff plateaus and
+ODE uniqueness. The zero-mean correction acts away from the computational
+rectangles and therefore changes none of these paths.
+\end{proof}
+
+\begin{theorem}[A fixed strip with half-line history]\label{bc:halfline-event}
+On $\T^3$, the fixed label $(1/8,1/4,1/4)$ and strip
+$2/3<x_1<15/16$ realize halting by a general mean-zero force, with zero
+initial velocity and pressure, bounded mixed derivatives and period one
+after time one. For a fixed machine only the loading field depends on the
+input. The material maps are volume-preserving diffeomorphisms, and the
+velocity and material-acceleration identities of
+Lemma~\ref{lem:p2-realization} hold.
+\end{theorem}
+\begin{proof}\label{bc:halfline-event-proof}
+The two initial tape tails are separately constant, so their radix sums
+are computable rationals, even though the left history tail is filled.
+Load the fixed label to that code at height $z_*$ by the planar Hamiltonian
+$\chi(d_x y-d_y x)$ with a flat scalar schedule. The segment stays in $F$.
+A nonterminal load and every nonterminal phase have first coordinate at
+most $3/8$; a terminal code lies in the stated strip. For an initially
+terminal machine the loading endpoint suffices. Repeat
+\eqref{bc:halfline-period} after time one. Checkpoint simulation and the
+intermediate-time guard prove the event equivalence. The common analytic
+lemma proves force effectivity, uniqueness, boundedness and the asserted
+flow identities. In particular $U=(\partial_t\Phi_t)\circ\Phi_t^{-1}$ and
+$\partial_{tt}\Phi_t=(f-\nabla p+\nu\Delta U)\circ\Phi_t$; these are
+identities for the constructed flow, not additional unknown equations.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/initialization.tex

@@ -1,574 +1,574 @@
-\section{Choosing initialization and the observer}
-\label{sec:bb-initialization}
-
-The preceding recorders supply complete instruction tables. We now
-specify how their boxes are positioned and how the fixed material label
-becomes the initial code. We compare one-time loading, a loading branch
-repeated in every cycle, and chart translations that place the initial
-code at the label itself. The latter two choices can preserve periodicity
-from time zero. Each construction also needs a bound between integer
-rule times; the endpoint coding alone does not give a material event.
-
-Unless stated otherwise, the force in this section is the residual
-\eqref{eq:bb-residual}, with pressure zero.  Its smoothness, all-order
-mixed derivative bounds, finite effective prescription and uniqueness
-are as in Proposition~\ref{prop:bb-realization}.  Whole-space forces
-have one compact spatial support; torus forces have zero spatial mean.
-All statements use a fixed positive computable viscosity, with the
-relative interpretation of that proposition for arbitrary fixed
-positive viscosity.
-
-\subsection{Torus loading and direct chart placement}
-
-\begin{proposition}[A move-first material test]\label{thm:bb-torus33}
-The move-first recorder has a unit-torus realization, periodic after
-$t=1$, whose fixed label $(1/4,1/2,1/4)$ enters
-$\{x_1\in(1/2,1)\pmod1\}$ exactly when the work machine halts.
-For a fixed machine only the loading interval depends on the input.
-\end{proposition}
-\begin{proof}\label{proof:bb-torus33}
-Use Lemma~\ref{lem:bb-movefirst}, let $\Sigma$ be its full cell alphabet,
-and enumerate its $N$ controls by $i=1,\ldots,N$.
-Set $s=1/[16(N+1)]$ and give the $i$th control a square
-of side $s$ centered at
-\[
- m_i=(o_i+i/[4(N+1)],1/2),\qquad
- o_i=\begin{cases}5/8,&C_q\text{ with }q\text{ halting},\\1/8,&\text{otherwise}.
- \end{cases}
-\]
-Use odd digits and $B=2|\Sigma|+2$.  The code at height $z_0=1/4$
-is $(m_i+s(E(L)-1/2,E(R)-1/2),z_0)$.
-It is rational initially.  Lemma~\ref{lem:bb-onecell} gives $J$
-branches with separately separated rectangles, all side lengths at
-most $s$ and endpoints on the grid of mesh
-$[32(N+1)B^2]^{-1}$.  For branch $j$ choose height
-$z_j=1/2+j/[4(J+1)]$.  Use the three-phase lift of
-Lemma~\ref{lem:bb-lift-parking}, with exponential middle scale
-$\lambda_j^\theta$ and center interpolation.  A padding parameter
-\[
- \epsilon=\min\{1/128,[320(N+1)B^2]^{-1},[40(J+1)]^{-1}\}
-\]
-separates the enlarged sheets in the source, height, and target phases
-and keeps them inside the unit cube.  Their thickness may then be
-chosen positive by Lemma~\ref{lem:bb-curl}.  For $J=0$ the repeated
-field is zero.
-
-During $[0,1]$ a localized curl translation takes the fixed label
-along the segment to the initial code.  Both segment and a small
-padding lie in the chart.  Repeat the branch pulse thereafter.
-At each subsequent integer time the particle is the next table code.
-If both endpoint controls are nonhalting, the interpolated center and
-half-width bounds give $x_1<3/8+s<1/2$ throughout the motion.
-The loader also stays below $1/2$ in a nonhalting run.  A halting
-checkpoint lies in the upper half, reached in finitely many positive
-local steps; an initial halt is reached at the end of loading.
-This proves the event for every real time.
-\end{proof}
-
-\begin{proposition}[A guarded-window material test]\label{thm:bb-window37}
-The table of Lemma~\ref{lem:bb-window-compiler} has a unit-torus
-realization, periodic after $t=1$, with fixed label $(1/4,1/4,1/4)$
-and event $x_1\in(1/2,7/8)\pmod1$, occurring exactly when the
-work machine halts.  At work step $n$ its next
-checkpoint occurs after exactly $2(p_n-h_n)+1\le4n+5$ unit periods.
-\end{proposition}
-\begin{proof}\label{proof:bb-window37}
-Choose $r_0=2$ in Lemma~\ref{lem:bb-window-compiler}: initially the
-frontier is at cell two, all other history cells are empty, and all
-marks are zero. Thus $p_n=n+2$, with records in cells $2,\ldots,n+1$.
-Split the full three-cell windows $[-1,1]$ and apply
-Lemma~\ref{lem:bb-window-rectangles} with $(a,b)=(1,2)$.
-Let $\mathcal A$ be the full cell alphabet of this window table and use
-odd digits in base $B=2|\mathcal A|+1$.
-For $S$ controls let $\epsilon=1/[16(S+1)]$ and place their squares
-at $o_s+\epsilon[0,1]^2$, where
-\[
- o_s=(\xi_s,1/4+2\kappa(s)\epsilon),\quad
- \kappa(s)\in\{1,\ldots,S\},\quad
- \xi_s=\begin{cases}5/8,&s\text{ halting Ready},\\1/4,&\text{otherwise}.
- \end{cases}
-\]
-Write $J$ for the number of branches and $e_j$ for branch $j$'s cursor
-displacement. The full source and target rectangle gaps are at least
-$\epsilon B^{-3}$.  Put their common height at $1/4$, assign branch
-$j$ height $1/2+j/[4(J+1)]$, and use the exponential middle scale
-$\operatorname{diag}(B^{-e_j\theta},B^{e_j\theta},1)$.
-Take one tenth of
-$\min(1/32,\epsilon B^{-3},1/[4(J+1)])$ as padding.
-Horizontal source and target gaps and the private heights separate
-the three phases, while all widths remain at most $\epsilon$.
-The strain can equivalently be generated by
-$(0,0,-e_j\dot\theta\log B\,(x_1-c_{j1})(x_2-c_{j2}))$,
-retaining the explicit diagonal-strain implementation.
-
-A first unit interval loads the label along the segment to the rational
-initial code.  Thereafter the exact rectangle action and the compiler
-lemma give checkpoint times $t_0=1$ and
-$t_{n+1}=t_n+2(p_n-h_n)+1$.  A nonhalting center stays in
-$[1/4,1/4+\epsilon]$, with half-width at most $\epsilon/2$;
-thus every intermediate first coordinate remains below $1/2$.
-The loader has the same exclusion.  Halt codes are in
-$[5/8,5/8+\epsilon]$, inside the observer, including an initial halt.
-The history block starts at the known absolute cell two, so the
-head-relative code also recovers absolute indexing at checkpoints.
-\end{proof}
-
-\begin{proposition}[Two direction-recording placements]
-\label{thm:bb-aligned38}\label{thm:bb-direction39}
-The direction recorder gives both of the following realizations.
-\begin{enumerate}
-\item On $\R^3$, with its passive origin bit, there is a force periodic
-from time zero and a fixed-origin event $X_1(t;0)<-5$.  Input dependence
-can be confined to two translations of the state charts.  On every
-implemented full box the first coordinate is a convex combination
-of its endpoint first coordinates.
-\item On the unit torus, without the passive track, a force periodic
-after $t=1$ has fixed label $(1/8,1/8,1/4)$ and event
-$x_1\in(1/2,7/8)\pmod1$.  For a fixed machine its repeated part is
-independent of the input.
-\end{enumerate}
-Both events are equivalent to halting.
-\end{proposition}
-\begin{proof}\label{proof:bb-aligned38}\label{proof:bb-direction39}
-For the first case introduce a fresh terminal state $\dagger$ and
-replace every missing instruction and every originally terminal
-state--symbol pair by a write-preserving stationary transition into it.
-The original initial control is therefore nonterminal in this normalized
-table, even when the original machine is initially halted. Apply
-Lemma~\ref{lem:bb-direction} with its passive origin track, and let
-$\mathcal A$ be the resulting full cell alphabet. Use odd digits in
-base $B=2|\mathcal A|+1$
-and initial fractions $x_*,y_*$.  Index the initial main state by zero,
-the main $\dagger$ states by distinct negative integers, and the rest
-by distinct positive integers.  State $s$ has placement
-\[
- (x,y)\longmapsto(10k(s)+x-x_*,y-y_*).
-\]
-The initialized point is zero; nonhalt codes have first coordinate
-$>-1$, and halt codes $<-5$.  Thicken the rectangles of
-Lemma~\ref{lem:bb-onecell} by $[-1,1]$, give branch $i$ height $10i$,
-and use the linear scale $g_i=1+\theta(\lambda_i-1)$ in the lift
-schedule.  A padding smaller than one tenth of the minimum of one
-and all within-family planar gaps suffices.  The middle strain has
-potential $(0,0,(\dot g_i/g_i)(x_1-c_{i1})(x_2-c_{i2}))$.
-Equation~\eqref{eq:bb-convex-coordinate} excludes every nonhalting
-visit to $x_1<-5$.  The table and routing are fixed for the machine;
-only $x_*,y_*$ change with the input.  The passive origin bit remains
-at physical cell zero, so no absolute tape information is discarded.
-
-For the second case first merge all halting work states into a single
-state $q_h$, redirecting incoming transitions and retaining initial halting.
-Delete the passive track from the direction recorder, and now let
-$\mathcal A$ denote that full alphabet. Use odd digits with
-$B=2|\mathcal A|+2$ and
-$\lambda=1/[16(J+1)]$, where $J$ counts the controls.  Number them
-from zero and set
-\[
- S_s(x,y)=(A_s+\lambda x,1/4+\lambda y),\qquad
- A_s=2\lambda\operatorname{index}(s)+
- \begin{cases}5/8,&s=M(q_h,e),\\1/4,&\text{otherwise}.
- \end{cases}
-\]
-The nonhalt squares lie in $[1/4,3/8]\times[1/4,1/4+\lambda]$;
-halt squares lie in $[5/8,3/4]\times[1/4,1/4+\lambda]$.
-Write $m$ for the number of branches. Use coding height $1/4$,
-branch heights $1/2+i/[4(m+1)]$, and
-the same linear first scale.  One hundredth of the minimum of
-$1/16$, $1/[4(m+1)]$, and the source and target planar gaps is
-valid padding; absent pair gaps are omitted.  All supports lie inside
-the cube, and all horizontal widths are at most $\lambda$.
-A localized translation with padding $1/1000$ loads the rational
-initial point from the fixed label in one unit interval.
-
-In a nonhalting period the center has first coordinate at most $3/8$
-and half-width at most $\lambda/2$, hence is below $1/2$; loading
-has the same bound.  Halting codes lie in the observer.  The exact
-sampling induction gives one table step per period, with checkpoint
-cycle $2(k+1-h_k)+3$.  The record block beginning at cell one
-recovers absolute indexing at checkpoints.  An initial halt is
-observed at time one.
-\end{proof}
-
-
-\subsection{A one-time torus loader}
-
-The next placement uses a separate first interval to load the code. Only
-the instruction cycle is then repeated, so periodicity begins at time one.
-
-\begin{theorem}[A separate one-unit loader on the torus]
-\label{ba:loaded-torus}
-For every machine and finite input and every fixed computable $\nu>0$,
-there is an effective smooth
-mean-zero force, one-periodic for $t\ge1$ with bounded mixed derivatives,
-whose unique smooth zero-data solution with zero normalized pressure
-satisfies
-\[
- \exists t\ge0:\ \Phi_t(1/2,1/2,1/4)
-       \in(1/16,1/4)\times\mathbb T^2
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-Uniqueness holds in the classical class of Section~\ref{sec:model}.
-\end{theorem}
-\begin{proof}\label{ba:loaded-torus-proof}
-Use the single-halt move-first recorder of
-Lemma~\ref{lem:bb-movefirst}, without an entry state.  Let
-$\Gamma$ be its full alphabet and $N$ its number of controls.  Put
-$B=2|\Gamma|+1$, $a=1/(16N)$, the halt square at
-$(1/8,1/4)+a[0,1]^2$, and the other squares at
-$(1/2+2ja,1/4)+a[0,1]^2$, $0\le j\le N-2$.
-Use the minimal subdivision, so right and stay moves are not split by
-the left letter.  Its gaps are at least $a/B^2$.
-For $M$ branches choose
-\[
- h=\frac1{64(M+1)},\quad Z_i=\frac12+\frac{i}{4(M+1)},\quad
- \varepsilon=\min\{1/128,h/4,a/(4B^2)\}.
-\]
-Apply linear-first-scale full-box routing from height $z_0=1/4$.
-Middle heights differ by $16h$; source and target gaps handle the
-vertical phases.  Centers lie in
-$[1/8,5/8]\times[1/4,5/16]$, widths are at most $a$, and all padded
-boxes lie in $(1/16,15/16)^3$.  This is a construction on the specified
-unit torus, with no change of physical viscosity.
-
-In the first unit interval translate a box of side $1/64$ from the
-fixed label to the rational initial code, using Lemma~\ref{lem:bb-curl}.
-The segment and a small padding lie inside the same chart.  Repeat
-only the instruction cycle afterward.  Code induction proves the
-checkpoint identities.  A halting code has first coordinate in
-$[1/8,3/16]$, including an initial halt reached by loading.  For a
-nonhalting run both centers of every used branch have first coordinate
-at least $1/2$, with half-width at most $a/2$; loading has the same
-lower center bound.  Hence the particle always stays to the right of
-$1/4$.  Residual realization proves every analytic assertion.  The
-separate loading field accounts for the qualification $t\ge1$ in the
-periodicity statement.
-\end{proof}
-
-
-\subsection{Repeated torus loading branches}\label{ba:torus}
-
-A loader can instead be one branch of every period. To keep its target
-separate from all computational targets, place the initial code in a state
-square with no incoming instruction. This condition concerns the entire
-target family, not only the distinguished execution. We implement the
-same startup principle first with full boxes, then with fixed columns.
-
-\begin{theorem}[A loader repeated as part of the processor]
-\label{ba:periodic-torus}
-For every finite machine and input, and every fixed computable $\nu>0$,
-there is an effective smooth mean-zero force on the unit flat torus,
-one-periodic from time zero, with all mixed derivatives bounded.  Its
-zero-data solution has pressure zero, is unique in the classical class
-of Section~\ref{sec:model}, and satisfies
-\[
- \exists t\ge0:\ \Phi_t(1/8,1/8,1/4)
-             \in\{5/8<X_1<7/8\}
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-The one repeated cycle implements all instruction branches and its
-loading branch.
-\end{theorem}
-\begin{proof}\label{ba:periodic-torus-proof}
-Normalize to one halt state and prepend a new nonhalting state $q_I$
-whose rule preserves the scanned letter, stays, and enters the original
-initial state.  No machine rule targets $q_I$.  Apply the move-first
-recorder of Lemma~\ref{lem:bb-movefirst} and omit the unused controls
-$P_{q_I},T_{q_I}$. Then the entry control $C_{q_I}$ has no incoming local
-instruction.  Write $\Gamma$
-for this recorder's full alphabet.  Put $B=2|\Gamma|+1$, let $N$
-be the number of controls, and set
-$a=1/(16(N+1))$.  Use the halt square
-$(3/4,1/4)+a[0,1]^2$ and nonhalting squares
-$(1/4+2al,1/4)+a[0,1]^2$, $0\le l\le N-2$.
-The full subdivision of Lemma~\ref{ba:rectangles} gives source
-half-widths $a/(2B)$ in both coordinates.
-
-Add a square with those half-widths centered at $(1/8,1/8)$ and a
-translation to the square centered at the initial entry code.  Each
-initial tail is at distance at least $1/(B-1)>1/(2B)$ from the endpoints
-of $[0,1]$, so the new target lies strictly inside the entry square.
-There is no other target in that square.  Both enlarged branch lists
-therefore remain separately separated, with gaps at least $a/B^2$.
-The extra source is separated from all state squares.
-
-Number the branches, including loading, by $1\le i\le M$.  Set
-\[
- z_0=\tfrac14,\qquad h=\frac1{16(M+1)},\qquad
- Z_i=\tfrac12+\frac{i}{4(M+1)},\qquad
- \varepsilon=\frac1{10}\min(h,a/B^2).
-\]
-Thicken each planar source by $[z_0-h,z_0+h]$.  Lift the resulting
-full box to center height $Z_i$, move it horizontally, and lower it
-to $z_0$.  For an instruction with displacement $d_i$, use
-$D_i=\operatorname{diag}(B^{-d_iw_2},B^{d_iw_2},1)$;
-loading has $d_i=0$.  Its center is the planar endpoint interpolation
-plus $(Z_i-z_0)(w_1-w_3)e_3$.  Adjacent middle heights are $4h$ apart,
-while the padded half-thickness is $h+\varepsilon<2h$.
-Source and target gaps handle the other phases.  Horizontal half-widths
-never exceed $a/2$, horizontal centers lie between $1/8$ and $3/4+a$,
-and vertical centers lie between $1/4$ and $3/4$.  The padded motions
-are thus strictly inside the unit cube.  Lemma~\ref{lem:bb-curl} gives a
-chart field, extended periodically in space and time.  Its zero time
-collars give zero initial velocity.
-
-The first cycle loads the fixed particle.  Each subsequent cycle maps
-its exact code to the next local code until the first halt, or forever
-in a nonhalting run.  Lemma~\ref{lem:bb-movefirst} identifies the finite
-checkpoint times with machine steps.  The added entry state handles an
-original initial halt.  A halt code is in the target slab.  For a
-nonhalting execution every used source and target center, including the
-loader, has first coordinate at most $3/8$.  Exponential interpolation
-keeps the center below that bound and the half-width below $a/2$;
-the particle cannot reach $5/8$ at any intermediate time.  The final
-transition into a halt is not subject to this nonhalting-endpoint bound.
-Lemma~\ref{ba:realization} gives the force, its mean, its effective
-derivatives, and uniqueness.  Repeating the loading field is harmless
-to the initialized trajectory and makes the force periodic from zero.
-\end{proof}
-
-\paragraph{A fixed-column implementation.}\label{bc:unit-periodic-section}
-The same untargeted-entry condition also permits a repeated initializer
-using fixed source and target columns. The next construction retains its
-explicit column potentials and its fixed label; its middle motion uses
-linear interpolation of the first planar scale.
-
-\begin{theorem}[A periodic initializer on the unit torus]
-\label{bc:unit-periodic-event}
-There is an effective halting realization on $\T^3$ with fixed label
-$a_*=(3/8,1/4,1/2)$ and fixed strip $5/8<x_1<7/8$, using a smooth
-mean-zero general force of period one from time zero. Its initial velocity
-and prescribed pressure are zero, every mixed derivative is bounded, and
-its kinetic energy is uniformly bounded. The solution is unique in the
-torus comparison class.
-\end{theorem}
-\begin{proof}\label{bc:unit-periodic-event-proof}
-Normalize to a single halt and add a new nonhalting start $q_s$, with no
-incoming original transition, which performs a dummy unchanged stay move
-to the old initial state. This preserves even initial halting. Use the
-move-first recorder of Lemma~\ref{lem:bb-movefirst}, with the control names
-of Section~\ref{bc:section}, and retain its frontier-writing and return
-controls $C_q,D_q$
-only for $q$ in the set $Q^+$ of targets of the normalized work table,
-including the new dummy transition. In particular its destination, the
-old initial state, belongs to $Q^+$, whereas $q_s\notin Q^+$. No recorder
-rule therefore targets the ready control $W_{q_s}$. Its
-initial frontier is still two and the simulation proof is unchanged.
-
-Use odd digits in base $B=2K+2$, where $K$ is the recorder alphabet size.
-If there are $n$ nonterminal controls, put $\tau=1/[16(n+1)]$. Give
-nonterminal states squares of side $\tau$ at lower corners
-$(1/8+2\tau j,1/4)$, $0\le j<n$, and put the terminal square at
-$(3/4,1/4)$. All nonterminal first coordinates are below $1/4$.
-Refine every rule, including right and stationary rules, by the left
-letter $\ell$. The full source is $I_\ell\times I_\gamma$, and the
-three image rectangles are respectively
-\begin{equation}\label{bc:unit-rectangles}
- I_{\beta\ell}\times[0,1],\qquad
- I_\ell\times I_\beta,\qquad [0,1]\times I_{\ell\beta}.
-\end{equation}
-They follow from the same affine formulas as Lemma~\ref{lem:bb-onecell}.
-The refined sources are separately disjoint; the incoming displacement
-and written symbol, together with the extra left prefix where applicable,
-separate the images. Thus full geometric separation still holds.
-
-Let $w_*=(3/8,1/4)$ and let $w_{\rm in}$ be the rational input code in
-the untargeted start square. Add the translation from
-$w_*+[-\delta,\delta]^2$ to $w_{\rm in}+[-\delta,\delta]^2$, where
-$\delta=\tau/(10B^2)$. The source lies away from every state square.
-The target lies strictly inside the start square: odd-digit codes have
-distance at least $1/(B-1)$ from zero and at least $2/(B-1)$ from one,
-both larger than $1/(10B^2)$. It meets no computational image because
-none targets that state. This extended list still has separately disjoint
-source and target families, even though a target can overlap a source.
-
-We give the torus transport explicitly to keep the startup claim separate
-from the whole-space column construction. All rectangles initially lie in
-$z_0=1/2$. For $m$ branches assign heights
-$z_j=1/2+j/[4(m+1)]$. Choose planar source and target cutoffs equal to
-one on neighborhoods of their rectangles, with padding less than one tenth
-of every relevant pairwise gap and chart-face margin; a missing pairwise
-minimum is omitted. Let $\eta(z)$ equal one near $[1/2,3/4]$ and have
-padding $1/16$. The source column field
-\[
- Z_j=\nabla\times(0,x\chi_j(x,y)\eta(z),0)
-\]
-equals $e_3$ along its source column; the target analogue $Z'_j$ does
-so along the target column. These fields have compact support in the
-open unit cube and zero spatial mean after periodization.
-
-For the middle phase use the center path
-$c_j=(1-\mu)p_j+\mu q_j$ and scale
-$l_j=1-\mu+\mu a_j$, where $a_j$ is the branch's horizontal scale.
-Choose $\xi_j$ equal to one near the planar coordinate bounding rectangle
-of the endpoints, and disjoint height cutoffs $\rho_j$ at $z_j$ with
-padding $1/[16(m+1)]$. For three successive flat ramps $\theta_1,
-\theta_2,\theta_3$, put $c_j=c_j(\theta_2)$, $l_j=l_j(\theta_2)$ and
-\[
- \Psi_j=\xi_j\rho_j\left[
- \dot c_{j,1}(y-c_{j,2})-\dot c_{j,2}(x-c_{j,1})
-       +\frac{\dot l_j}{l_j}(x-c_{j,1})(y-c_{j,2})\right].
-\]
-The full velocity is
-\begin{equation}\label{bc:unit-V}
- V=\sum_j(z_j-z_0)\theta_1'Z_j
-       +\sum_j(\partial_y\Psi_j,-\partial_x\Psi_j,0)
-       +\sum_j(z_0-z_j)\theta_3'Z'_j.
-\end{equation}
-It lifts all full rectangles, performs the reciprocal affine motions in
-separate layers, and lowers them. In the middle phase the exact planar path
-is $c_j+\diag(l_j,l_j^{-1})(\zeta-p_j)$. The swept-rectangle bound in
-Lemma~\ref{ba:two-interpolations} keeps it inside the coordinate bounding
-rectangle of the endpoints. The first coordinate is an endpoint convex combination.
-The cutoffs are therefore one along every claimed path. Differentiation
-and uniqueness prove the assigned full maps. The entire construction is
-supported in the open unit cube and has zero collars in phase.
-
-Extend it with period one in phase and periodically in space. It is zero
-at time zero, so its residual has the required period from time zero.
-During the first period the distinguished particle follows the initializer;
-its first coordinate stays below $1/2$. Thereafter it follows the recorder.
-Every nonterminal transition remains below $1/4$, whereas a terminal code
-lies inside $5/8<x_1<7/8$. It cannot return to the initializer source,
-whose square lies at $x_1=3/8$. The dummy instruction handles initial
-halting without changing the observation rule. Lemma~\ref{bc:analytic}
-now proves all analytic assertions, including uniform energy on the compact
-torus. The force formula includes every branch, the initializer included,
-in each period; it uses no precomputed execution.
-\end{proof}
-
-
-\subsection{A repeated loading instruction}
-
-The next construction incorporates loading in a different symbolic table:
-its instructions replace finite prefixes of two stacks. A replacement can
-change planar area, so the third coordinate compensates for the product
-of the two planar scales. Evacuation and delivery then realize the
-resulting volume-preserving boxes in every period.
-
-\begin{theorem}[Periodic prefix-box realization]\label{thm:bb-prefix29}
-For every deterministic machine and finite input, a compactly supported
-smooth force on $\R^3$, periodic with period one from time zero,
-realizes the fixed-origin halting event
-\[
- \exists t\ge0:\quad X(t;0)\in(-4,-1)\times(-1,2)\times(-1,1)
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-The force has bounded mixed derivatives and the unique zero-data
-solution in the class of Proposition~\ref{prop:bb-realization}.
-Its finite program uses full volume-preserving boxes; no solenoidal
-condition is imposed on the force itself.
-\end{theorem}
-\begin{proof}\label{proof:bb-prefix29}
-Normalize to one halting work state $q_h$, redirecting incoming
-transitions to it and replacing missing nonhalting rules by unchanged-letter
-stay moves into it. This preserves initial halting.
-Write $\Gamma$ for the work alphabet, $b$ for its blank, $q_0$ for the
-initial control, and $\omega$ for the finite input word.
-For top-first stacks $L,R$, a rule
-$(s,\alpha,\beta)\rightsquigarrow(r,\gamma,\eta)$ replaces the
-two displayed prefixes and leaves their independent infinite tails
-unchanged.  Introduce fresh state $s_0$, boundary letter $S$ and mark
-letter $M$.  The preliminary rules are
-\[
- (s_0,\varepsilon,\varepsilon)\rightsquigarrow(q_0,S,\omega)
-\]
-and, for each work transition $(q,a)\mapsto(r,c,d)$,
-\[
-\begin{array}{c|l}
- d&\text{preliminary rules}\\\hline
-1&(q,\varepsilon,a)\rightsquigarrow(r,c,\varepsilon)\\
-0&(q,\varepsilon,a)\rightsquigarrow(r,\varepsilon,c)\\
--1&(q,v,a)\rightsquigarrow(r,\varepsilon,vc)\quad(v\in\Gamma),\\
-  &(q,S,a)\rightsquigarrow(r,S,bc).
-\end{array}
-\]
-The final row exposes the implicit blank beyond the left boundary.
-Index each preliminary occurrence by $j$, with target $r(j)$,
-and introduce a private control $g_j$ and record letter $\ell_j$.
-Replace that occurrence by
-$(s,\alpha,\beta)\to(g_j,\gamma,M\eta)$ and add
-\begin{equation}\label{eq:bb-prefix-table}
-\begin{aligned}
- (g_j,v,\varepsilon)&\to(g_j,\varepsilon,v)&&v\in\Gamma,\\
- (g_j,S,\varepsilon)&\to(k_{r(j)},S\ell_j,\varepsilon),\\
- (k_q,\varepsilon,v)&\to(k_q,v,\varepsilon)&&v\in\Gamma,\\
- (k_q,\varepsilon,M)&\to(q,\varepsilon,\varepsilon).
-\end{aligned}
-\end{equation}
-Sources are separated by control and tested top symbols.  Into $g_j$,
-entry has right top $M$, whereas a loop has a work letter there.
-Into $k_q$, entry has left prefix $S\ell_j$, whereas a loop has a
-work letter at left top; distinct entries have distinct $\ell_j$.
-Into a work state only the last rule enters.  Thus both source and
-target cylinders are disjoint for arbitrary tails.
-
-After a preliminary replacement the stacks have form $wSB,MY$,
-with $w\in\Gamma^*$.  The forward loop transfers $w$, leaving
-$SB,w^{\rm rev}MY$.  The boundary rule inserts $\ell_j$ behind
-$S$, the reverse loop restores $w$, and the final rule removes $M$.
-After $2|w|+2$ further instructions the stacks are
-$wS\ell_jB,Y$ at control $r(j)$.  At a work control, $w$ lists
-represented cells left of the head, nearest first; $R$ lists the
-current cell and right tail, and cells beyond $S$ on the left are
-implicit blanks.  The preliminary rules perform exactly the work
-write and move.  Starting at $(s_0,b^\infty,b^\infty)$, the first
-work visit is at rule count three, and consecutive work visits differ
-by $3+2|w_{\rm new}|$.  This proves finite continuation and the
-halting equivalence, including an initially halted input.  The
-retained records also recover all head displacements.
-
-Enumerate the enlarged alphabet, blank first, by even digits
-$0,2,\ldots,2m-2$ in base $K=2m$.  Use prefix intervals $I(v)$
-and codes $E$ as in Lemma~\ref{lem:bb-onecell}.  Set offsets
-$o_{q_h}=-3$, $o_{s_0}=0$, and $3,6,9,\ldots$ for the other controls.
-Encode $(s,L,R)$ by $(o_s+E(L),E(R),0)$; its initial value is zero.
-For a rule $(s,\alpha,\beta)\to(s',\gamma,\eta)$ put
-\[
- a_i=K^{|\alpha|-|\gamma|},\quad
- b_i=K^{|\beta|-|\eta|},\quad c_i=(a_ib_i)^{-1}.
-\]
-The full source box is
-$(o_s+I(\alpha))\times I(\beta)\times[-1,1]$;
-the target is
-$(o_{s'}+I(\gamma))\times I(\eta)\times[-c_i,c_i]$.
-The center-to-center map has matrix $\operatorname{diag}(a_i,b_i,c_i)$
-and determinant one.  It implements the prefixes on all codes.
-Disjoint cylinders give separated full planar projections in each
-family, including blank-tail endpoints.
-
-Park box $i$ at $(4i,4,0)$, reserving the horizontal rectangle
-$[4i-1/2,4i+1/2]\times[7/2,9/2]$.
-Let $R=\max(1,c_1,\ldots,c_N)$ and travel at height $Z=2R+3$.
-Evacuate all sources to their parking sites, reshape there by
-$\operatorname{diag}(a_i^s,b_i^s,c_i^s)$, then deliver all targets.
-The two horizontal widths remain at most one, and vertical half-widths
-at most $R$.  Let $g$ be the minimum of one and the within-family
-gaps in the source-plus-parking and target-plus-parking projections.
-Padding smaller than $g/8$ works during vertical motion and strain;
-at travel height the moving bottom is at least $Z-R>R+2$.
-Lemma~\ref{lem:bb-lift-parking} gives the full $7N$-motion schedule.
-
-Repeat this unit pulse from zero.  Nonhalting endpoint boxes and all
-parking boxes have nonnegative first coordinate.  Translations preserve
-that bound, and strain takes place in positive parking envelopes.
-Therefore a nonhalting path never enters the negative observation box.
-A halting code lies in $[-3,-2]\times[0,1]\times\{0\}$ inside it.
-The residual proposition finishes the proof.  Loading was a repeated
-instruction, so no exceptional first-time force is required.
-\end{proof}
-
-
-\begin{remark}[A different observation regime]
-For comparison, the distinct velocity-field detector of
-\cite[Theorem~1.1(2) and Section~5]{OpenAI379EulerianC} uses
-$\R^2\times\T$ and the event
-$\int_{\{X_2>0\}\times\T}u_3>1/2$. Its force has globally bounded
-mixed derivatives and horizontal support compact on each finite time
-interval. The nonnegative scalar velocity has diffusion tails; after
-loading its total mass is one. Its cutoff estimate gives loss
-$\delta<1/24$, nonhalting mass at most $\delta$ in the observed half-space,
-and halting mass above $3/4$. Those conclusions follow from the cited
-advection--diffusion proof and have no uniform spatial support assertion.
-They are a separate force and observation regime from the material test
-proved here.
-\end{remark}
+\section{Choosing initialization and the observer}
+\label{sec:bb-initialization}
+
+The preceding recorders supply complete instruction tables. We now
+specify how their boxes are positioned and how the fixed material label
+becomes the initial code. We compare one-time loading, a loading branch
+repeated in every cycle, and chart translations that place the initial
+code at the label itself. The latter two choices can preserve periodicity
+from time zero. Each construction also needs a bound between integer
+rule times; the endpoint coding alone does not give a material event.
+
+Unless stated otherwise, the force in this section is the residual
+\eqref{eq:bb-residual}, with pressure zero.  Its smoothness, all-order
+mixed derivative bounds, finite effective prescription and uniqueness
+are as in Proposition~\ref{prop:bb-realization}.  Whole-space forces
+have one compact spatial support; torus forces have zero spatial mean.
+All statements use a fixed positive computable viscosity, with the
+relative interpretation of that proposition for arbitrary fixed
+positive viscosity.
+
+\subsection{Torus loading and direct chart placement}
+
+\begin{proposition}[A move-first material test]\label{thm:bb-torus33}
+The move-first recorder has a unit-torus realization, periodic after
+$t=1$, whose fixed label $(1/4,1/2,1/4)$ enters
+$\{x_1\in(1/2,1)\pmod1\}$ exactly when the work machine halts.
+For a fixed machine only the loading interval depends on the input.
+\end{proposition}
+\begin{proof}\label{proof:bb-torus33}
+Use Lemma~\ref{lem:bb-movefirst}, let $\Sigma$ be its full cell alphabet,
+and enumerate its $N$ controls by $i=1,\ldots,N$.
+Set $s=1/[16(N+1)]$ and give the $i$th control a square
+of side $s$ centered at
+\[
+ m_i=(o_i+i/[4(N+1)],1/2),\qquad
+ o_i=\begin{cases}5/8,&C_q\text{ with }q\text{ halting},\\1/8,&\text{otherwise}.
+ \end{cases}
+\]
+Use odd digits and $B=2|\Sigma|+2$.  The code at height $z_0=1/4$
+is $(m_i+s(E(L)-1/2,E(R)-1/2),z_0)$.
+It is rational initially.  Lemma~\ref{lem:bb-onecell} gives $J$
+branches with separately separated rectangles, all side lengths at
+most $s$ and endpoints on the grid of mesh
+$[32(N+1)B^2]^{-1}$.  For branch $j$ choose height
+$z_j=1/2+j/[4(J+1)]$.  Use the three-phase lift of
+Lemma~\ref{lem:bb-lift-parking}, with exponential middle scale
+$\lambda_j^\theta$ and center interpolation.  A padding parameter
+\[
+ \epsilon=\min\{1/128,[320(N+1)B^2]^{-1},[40(J+1)]^{-1}\}
+\]
+separates the enlarged sheets in the source, height, and target phases
+and keeps them inside the unit cube.  Their thickness may then be
+chosen positive by Lemma~\ref{lem:bb-curl}.  For $J=0$ the repeated
+field is zero.
+
+During $[0,1]$ a localized curl translation takes the fixed label
+along the segment to the initial code.  Both segment and a small
+padding lie in the chart.  Repeat the branch pulse thereafter.
+At each subsequent integer time the particle is the next table code.
+If both endpoint controls are nonhalting, the interpolated center and
+half-width bounds give $x_1<3/8+s<1/2$ throughout the motion.
+The loader also stays below $1/2$ in a nonhalting run.  A halting
+checkpoint lies in the upper half, reached in finitely many positive
+local steps; an initial halt is reached at the end of loading.
+This proves the event for every real time.
+\end{proof}
+
+\begin{proposition}[A guarded-window material test]\label{thm:bb-window37}
+The table of Lemma~\ref{lem:bb-window-compiler} has a unit-torus
+realization, periodic after $t=1$, with fixed label $(1/4,1/4,1/4)$
+and event $x_1\in(1/2,7/8)\pmod1$, occurring exactly when the
+work machine halts.  At work step $n$ its next
+checkpoint occurs after exactly $2(p_n-h_n)+1\le4n+5$ unit periods.
+\end{proposition}
+\begin{proof}\label{proof:bb-window37}
+Choose $r_0=2$ in Lemma~\ref{lem:bb-window-compiler}: initially the
+frontier is at cell two, all other history cells are empty, and all
+marks are zero. Thus $p_n=n+2$, with records in cells $2,\ldots,n+1$.
+Split the full three-cell windows $[-1,1]$ and apply
+Lemma~\ref{lem:bb-window-rectangles} with $(a,b)=(1,2)$.
+Let $\mathcal A$ be the full cell alphabet of this window table and use
+odd digits in base $B=2|\mathcal A|+1$.
+For $S$ controls let $\epsilon=1/[16(S+1)]$ and place their squares
+at $o_s+\epsilon[0,1]^2$, where
+\[
+ o_s=(\xi_s,1/4+2\kappa(s)\epsilon),\quad
+ \kappa(s)\in\{1,\ldots,S\},\quad
+ \xi_s=\begin{cases}5/8,&s\text{ halting Ready},\\1/4,&\text{otherwise}.
+ \end{cases}
+\]
+Write $J$ for the number of branches and $e_j$ for branch $j$'s cursor
+displacement. The full source and target rectangle gaps are at least
+$\epsilon B^{-3}$.  Put their common height at $1/4$, assign branch
+$j$ height $1/2+j/[4(J+1)]$, and use the exponential middle scale
+$\operatorname{diag}(B^{-e_j\theta},B^{e_j\theta},1)$.
+Take one tenth of
+$\min(1/32,\epsilon B^{-3},1/[4(J+1)])$ as padding.
+Horizontal source and target gaps and the private heights separate
+the three phases, while all widths remain at most $\epsilon$.
+The strain can equivalently be generated by
+$(0,0,-e_j\dot\theta\log B\,(x_1-c_{j1})(x_2-c_{j2}))$,
+retaining the explicit diagonal-strain implementation.
+
+A first unit interval loads the label along the segment to the rational
+initial code.  Thereafter the exact rectangle action and the compiler
+lemma give checkpoint times $t_0=1$ and
+$t_{n+1}=t_n+2(p_n-h_n)+1$.  A nonhalting center stays in
+$[1/4,1/4+\epsilon]$, with half-width at most $\epsilon/2$;
+thus every intermediate first coordinate remains below $1/2$.
+The loader has the same exclusion.  Halt codes are in
+$[5/8,5/8+\epsilon]$, inside the observer, including an initial halt.
+The history block starts at the known absolute cell two, so the
+head-relative code also recovers absolute indexing at checkpoints.
+\end{proof}
+
+\begin{proposition}[Two direction-recording placements]
+\label{thm:bb-aligned38}\label{thm:bb-direction39}
+The direction recorder gives both of the following realizations.
+\begin{enumerate}
+\item On $\R^3$, with its passive origin bit, there is a force periodic
+from time zero and a fixed-origin event $X_1(t;0)<-5$.  Input dependence
+can be confined to two translations of the state charts.  On every
+implemented full box the first coordinate is a convex combination
+of its endpoint first coordinates.
+\item On the unit torus, without the passive track, a force periodic
+after $t=1$ has fixed label $(1/8,1/8,1/4)$ and event
+$x_1\in(1/2,7/8)\pmod1$.  For a fixed machine its repeated part is
+independent of the input.
+\end{enumerate}
+Both events are equivalent to halting.
+\end{proposition}
+\begin{proof}\label{proof:bb-aligned38}\label{proof:bb-direction39}
+For the first case introduce a fresh terminal state $\dagger$ and
+replace every missing instruction and every originally terminal
+state--symbol pair by a write-preserving stationary transition into it.
+The original initial control is therefore nonterminal in this normalized
+table, even when the original machine is initially halted. Apply
+Lemma~\ref{lem:bb-direction} with its passive origin track, and let
+$\mathcal A$ be the resulting full cell alphabet. Use odd digits in
+base $B=2|\mathcal A|+1$
+and initial fractions $x_*,y_*$.  Index the initial main state by zero,
+the main $\dagger$ states by distinct negative integers, and the rest
+by distinct positive integers.  State $s$ has placement
+\[
+ (x,y)\longmapsto(10k(s)+x-x_*,y-y_*).
+\]
+The initialized point is zero; nonhalt codes have first coordinate
+$>-1$, and halt codes $<-5$.  Thicken the rectangles of
+Lemma~\ref{lem:bb-onecell} by $[-1,1]$, give branch $i$ height $10i$,
+and use the linear scale $g_i=1+\theta(\lambda_i-1)$ in the lift
+schedule.  A padding smaller than one tenth of the minimum of one
+and all within-family planar gaps suffices.  The middle strain has
+potential $(0,0,(\dot g_i/g_i)(x_1-c_{i1})(x_2-c_{i2}))$.
+Equation~\eqref{eq:bb-convex-coordinate} excludes every nonhalting
+visit to $x_1<-5$.  The table and routing are fixed for the machine;
+only $x_*,y_*$ change with the input.  The passive origin bit remains
+at physical cell zero, so no absolute tape information is discarded.
+
+For the second case first merge all halting work states into a single
+state $q_h$, redirecting incoming transitions and retaining initial halting.
+Delete the passive track from the direction recorder, and now let
+$\mathcal A$ denote that full alphabet. Use odd digits with
+$B=2|\mathcal A|+2$ and
+$\lambda=1/[16(J+1)]$, where $J$ counts the controls.  Number them
+from zero and set
+\[
+ S_s(x,y)=(A_s+\lambda x,1/4+\lambda y),\qquad
+ A_s=2\lambda\operatorname{index}(s)+
+ \begin{cases}5/8,&s=M(q_h,e),\\1/4,&\text{otherwise}.
+ \end{cases}
+\]
+The nonhalt squares lie in $[1/4,3/8]\times[1/4,1/4+\lambda]$;
+halt squares lie in $[5/8,3/4]\times[1/4,1/4+\lambda]$.
+Write $m$ for the number of branches. Use coding height $1/4$,
+branch heights $1/2+i/[4(m+1)]$, and
+the same linear first scale.  One hundredth of the minimum of
+$1/16$, $1/[4(m+1)]$, and the source and target planar gaps is
+valid padding; absent pair gaps are omitted.  All supports lie inside
+the cube, and all horizontal widths are at most $\lambda$.
+A localized translation with padding $1/1000$ loads the rational
+initial point from the fixed label in one unit interval.
+
+In a nonhalting period the center has first coordinate at most $3/8$
+and half-width at most $\lambda/2$, hence is below $1/2$; loading
+has the same bound.  Halting codes lie in the observer.  The exact
+sampling induction gives one table step per period, with checkpoint
+cycle $2(k+1-h_k)+3$.  The record block beginning at cell one
+recovers absolute indexing at checkpoints.  An initial halt is
+observed at time one.
+\end{proof}
+
+
+\subsection{A one-time torus loader}
+
+The next placement uses a separate first interval to load the code. Only
+the instruction cycle is then repeated, so periodicity begins at time one.
+
+\begin{theorem}[A separate one-unit loader on the torus]
+\label{ba:loaded-torus}
+For every machine and finite input and every fixed computable $\nu>0$,
+there is an effective smooth
+mean-zero force, one-periodic for $t\ge1$ with bounded mixed derivatives,
+whose unique smooth zero-data solution with zero normalized pressure
+satisfies
+\[
+ \exists t\ge0:\ \Phi_t(1/2,1/2,1/4)
+       \in(1/16,1/4)\times\mathbb T^2
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+Uniqueness holds in the classical class of Section~\ref{sec:model}.
+\end{theorem}
+\begin{proof}\label{ba:loaded-torus-proof}
+Use the single-halt move-first recorder of
+Lemma~\ref{lem:bb-movefirst}, without an entry state.  Let
+$\Gamma$ be its full alphabet and $N$ its number of controls.  Put
+$B=2|\Gamma|+1$, $a=1/(16N)$, the halt square at
+$(1/8,1/4)+a[0,1]^2$, and the other squares at
+$(1/2+2ja,1/4)+a[0,1]^2$, $0\le j\le N-2$.
+Use the minimal subdivision, so right and stay moves are not split by
+the left letter.  Its gaps are at least $a/B^2$.
+For $M$ branches choose
+\[
+ h=\frac1{64(M+1)},\quad Z_i=\frac12+\frac{i}{4(M+1)},\quad
+ \varepsilon=\min\{1/128,h/4,a/(4B^2)\}.
+\]
+Apply linear-first-scale full-box routing from height $z_0=1/4$.
+Middle heights differ by $16h$; source and target gaps handle the
+vertical phases.  Centers lie in
+$[1/8,5/8]\times[1/4,5/16]$, widths are at most $a$, and all padded
+boxes lie in $(1/16,15/16)^3$.  This is a construction on the specified
+unit torus, with no change of physical viscosity.
+
+In the first unit interval translate a box of side $1/64$ from the
+fixed label to the rational initial code, using Lemma~\ref{lem:bb-curl}.
+The segment and a small padding lie inside the same chart.  Repeat
+only the instruction cycle afterward.  Code induction proves the
+checkpoint identities.  A halting code has first coordinate in
+$[1/8,3/16]$, including an initial halt reached by loading.  For a
+nonhalting run both centers of every used branch have first coordinate
+at least $1/2$, with half-width at most $a/2$; loading has the same
+lower center bound.  Hence the particle always stays to the right of
+$1/4$.  Residual realization proves every analytic assertion.  The
+separate loading field accounts for the qualification $t\ge1$ in the
+periodicity statement.
+\end{proof}
+
+
+\subsection{Repeated torus loading branches}\label{ba:torus}
+
+A loader can instead be one branch of every period. To keep its target
+separate from all computational targets, place the initial code in a state
+square with no incoming instruction. This condition concerns the entire
+target family, not only the distinguished execution. We implement the
+same startup principle first with full boxes, then with fixed columns.
+
+\begin{theorem}[A loader repeated as part of the processor]
+\label{ba:periodic-torus}
+For every finite machine and input, and every fixed computable $\nu>0$,
+there is an effective smooth mean-zero force on the unit flat torus,
+one-periodic from time zero, with all mixed derivatives bounded.  Its
+zero-data solution has pressure zero, is unique in the classical class
+of Section~\ref{sec:model}, and satisfies
+\[
+ \exists t\ge0:\ \Phi_t(1/8,1/8,1/4)
+             \in\{5/8<X_1<7/8\}
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+The one repeated cycle implements all instruction branches and its
+loading branch.
+\end{theorem}
+\begin{proof}\label{ba:periodic-torus-proof}
+Normalize to one halt state and prepend a new nonhalting state $q_I$
+whose rule preserves the scanned letter, stays, and enters the original
+initial state.  No machine rule targets $q_I$.  Apply the move-first
+recorder of Lemma~\ref{lem:bb-movefirst} and omit the unused controls
+$P_{q_I},T_{q_I}$. Then the entry control $C_{q_I}$ has no incoming local
+instruction.  Write $\Gamma$
+for this recorder's full alphabet.  Put $B=2|\Gamma|+1$, let $N$
+be the number of controls, and set
+$a=1/(16(N+1))$.  Use the halt square
+$(3/4,1/4)+a[0,1]^2$ and nonhalting squares
+$(1/4+2al,1/4)+a[0,1]^2$, $0\le l\le N-2$.
+The full subdivision of Lemma~\ref{ba:rectangles} gives source
+half-widths $a/(2B)$ in both coordinates.
+
+Add a square with those half-widths centered at $(1/8,1/8)$ and a
+translation to the square centered at the initial entry code.  Each
+initial tail is at distance at least $1/(B-1)>1/(2B)$ from the endpoints
+of $[0,1]$, so the new target lies strictly inside the entry square.
+There is no other target in that square.  Both enlarged branch lists
+therefore remain separately separated, with gaps at least $a/B^2$.
+The extra source is separated from all state squares.
+
+Number the branches, including loading, by $1\le i\le M$.  Set
+\[
+ z_0=\tfrac14,\qquad h=\frac1{16(M+1)},\qquad
+ Z_i=\tfrac12+\frac{i}{4(M+1)},\qquad
+ \varepsilon=\frac1{10}\min(h,a/B^2).
+\]
+Thicken each planar source by $[z_0-h,z_0+h]$.  Lift the resulting
+full box to center height $Z_i$, move it horizontally, and lower it
+to $z_0$.  For an instruction with displacement $d_i$, use
+$D_i=\operatorname{diag}(B^{-d_iw_2},B^{d_iw_2},1)$;
+loading has $d_i=0$.  Its center is the planar endpoint interpolation
+plus $(Z_i-z_0)(w_1-w_3)e_3$.  Adjacent middle heights are $4h$ apart,
+while the padded half-thickness is $h+\varepsilon<2h$.
+Source and target gaps handle the other phases.  Horizontal half-widths
+never exceed $a/2$, horizontal centers lie between $1/8$ and $3/4+a$,
+and vertical centers lie between $1/4$ and $3/4$.  The padded motions
+are thus strictly inside the unit cube.  Lemma~\ref{lem:bb-curl} gives a
+chart field, extended periodically in space and time.  Its zero time
+collars give zero initial velocity.
+
+The first cycle loads the fixed particle.  Each subsequent cycle maps
+its exact code to the next local code until the first halt, or forever
+in a nonhalting run.  Lemma~\ref{lem:bb-movefirst} identifies the finite
+checkpoint times with machine steps.  The added entry state handles an
+original initial halt.  A halt code is in the target slab.  For a
+nonhalting execution every used source and target center, including the
+loader, has first coordinate at most $3/8$.  Exponential interpolation
+keeps the center below that bound and the half-width below $a/2$;
+the particle cannot reach $5/8$ at any intermediate time.  The final
+transition into a halt is not subject to this nonhalting-endpoint bound.
+Lemma~\ref{ba:realization} gives the force, its mean, its effective
+derivatives, and uniqueness.  Repeating the loading field is harmless
+to the initialized trajectory and makes the force periodic from zero.
+\end{proof}
+
+\paragraph{A fixed-column implementation.}\label{bc:unit-periodic-section}
+The same untargeted-entry condition also permits a repeated initializer
+using fixed source and target columns. The next construction retains its
+explicit column potentials and its fixed label; its middle motion uses
+linear interpolation of the first planar scale.
+
+\begin{theorem}[A periodic initializer on the unit torus]
+\label{bc:unit-periodic-event}
+There is an effective halting realization on $\T^3$ with fixed label
+$a_*=(3/8,1/4,1/2)$ and fixed strip $5/8<x_1<7/8$, using a smooth
+mean-zero general force of period one from time zero. Its initial velocity
+and prescribed pressure are zero, every mixed derivative is bounded, and
+its kinetic energy is uniformly bounded. The solution is unique in the
+torus comparison class.
+\end{theorem}
+\begin{proof}\label{bc:unit-periodic-event-proof}
+Normalize to a single halt and add a new nonhalting start $q_s$, with no
+incoming original transition, which performs a dummy unchanged stay move
+to the old initial state. This preserves even initial halting. Use the
+move-first recorder of Lemma~\ref{lem:bb-movefirst}, with the control names
+of Section~\ref{bc:section}, and retain its frontier-writing and return
+controls $C_q,D_q$
+only for $q$ in the set $Q^+$ of targets of the normalized work table,
+including the new dummy transition. In particular its destination, the
+old initial state, belongs to $Q^+$, whereas $q_s\notin Q^+$. No recorder
+rule therefore targets the ready control $W_{q_s}$. Its
+initial frontier is still two and the simulation proof is unchanged.
+
+Use odd digits in base $B=2K+2$, where $K$ is the recorder alphabet size.
+If there are $n$ nonterminal controls, put $\tau=1/[16(n+1)]$. Give
+nonterminal states squares of side $\tau$ at lower corners
+$(1/8+2\tau j,1/4)$, $0\le j<n$, and put the terminal square at
+$(3/4,1/4)$. All nonterminal first coordinates are below $1/4$.
+Refine every rule, including right and stationary rules, by the left
+letter $\ell$. The full source is $I_\ell\times I_\gamma$, and the
+three image rectangles are respectively
+\begin{equation}\label{bc:unit-rectangles}
+ I_{\beta\ell}\times[0,1],\qquad
+ I_\ell\times I_\beta,\qquad [0,1]\times I_{\ell\beta}.
+\end{equation}
+They follow from the same affine formulas as Lemma~\ref{lem:bb-onecell}.
+The refined sources are separately disjoint; the incoming displacement
+and written symbol, together with the extra left prefix where applicable,
+separate the images. Thus full geometric separation still holds.
+
+Let $w_*=(3/8,1/4)$ and let $w_{\rm in}$ be the rational input code in
+the untargeted start square. Add the translation from
+$w_*+[-\delta,\delta]^2$ to $w_{\rm in}+[-\delta,\delta]^2$, where
+$\delta=\tau/(10B^2)$. The source lies away from every state square.
+The target lies strictly inside the start square: odd-digit codes have
+distance at least $1/(B-1)$ from zero and at least $2/(B-1)$ from one,
+both larger than $1/(10B^2)$. It meets no computational image because
+none targets that state. This extended list still has separately disjoint
+source and target families, even though a target can overlap a source.
+
+We give the torus transport explicitly to keep the startup claim separate
+from the whole-space column construction. All rectangles initially lie in
+$z_0=1/2$. For $m$ branches assign heights
+$z_j=1/2+j/[4(m+1)]$. Choose planar source and target cutoffs equal to
+one on neighborhoods of their rectangles, with padding less than one tenth
+of every relevant pairwise gap and chart-face margin; a missing pairwise
+minimum is omitted. Let $\eta(z)$ equal one near $[1/2,3/4]$ and have
+padding $1/16$. The source column field
+\[
+ Z_j=\nabla\times(0,x\chi_j(x,y)\eta(z),0)
+\]
+equals $e_3$ along its source column; the target analogue $Z'_j$ does
+so along the target column. These fields have compact support in the
+open unit cube and zero spatial mean after periodization.
+
+For the middle phase use the center path
+$c_j=(1-\mu)p_j+\mu q_j$ and scale
+$l_j=1-\mu+\mu a_j$, where $a_j$ is the branch's horizontal scale.
+Choose $\xi_j$ equal to one near the planar coordinate bounding rectangle
+of the endpoints, and disjoint height cutoffs $\rho_j$ at $z_j$ with
+padding $1/[16(m+1)]$. For three successive flat ramps $\theta_1,
+\theta_2,\theta_3$, put $c_j=c_j(\theta_2)$, $l_j=l_j(\theta_2)$ and
+\[
+ \Psi_j=\xi_j\rho_j\left[
+ \dot c_{j,1}(y-c_{j,2})-\dot c_{j,2}(x-c_{j,1})
+       +\frac{\dot l_j}{l_j}(x-c_{j,1})(y-c_{j,2})\right].
+\]
+The full velocity is
+\begin{equation}\label{bc:unit-V}
+ V=\sum_j(z_j-z_0)\theta_1'Z_j
+       +\sum_j(\partial_y\Psi_j,-\partial_x\Psi_j,0)
+       +\sum_j(z_0-z_j)\theta_3'Z'_j.
+\end{equation}
+It lifts all full rectangles, performs the reciprocal affine motions in
+separate layers, and lowers them. In the middle phase the exact planar path
+is $c_j+\diag(l_j,l_j^{-1})(\zeta-p_j)$. The swept-rectangle bound in
+Lemma~\ref{ba:two-interpolations} keeps it inside the coordinate bounding
+rectangle of the endpoints. The first coordinate is an endpoint convex combination.
+The cutoffs are therefore one along every claimed path. Differentiation
+and uniqueness prove the assigned full maps. The entire construction is
+supported in the open unit cube and has zero collars in phase.
+
+Extend it with period one in phase and periodically in space. It is zero
+at time zero, so its residual has the required period from time zero.
+During the first period the distinguished particle follows the initializer;
+its first coordinate stays below $1/2$. Thereafter it follows the recorder.
+Every nonterminal transition remains below $1/4$, whereas a terminal code
+lies inside $5/8<x_1<7/8$. It cannot return to the initializer source,
+whose square lies at $x_1=3/8$. The dummy instruction handles initial
+halting without changing the observation rule. Lemma~\ref{bc:analytic}
+now proves all analytic assertions, including uniform energy on the compact
+torus. The force formula includes every branch, the initializer included,
+in each period; it uses no precomputed execution.
+\end{proof}
+
+
+\subsection{A repeated loading instruction}
+
+The next construction incorporates loading in a different symbolic table:
+its instructions replace finite prefixes of two stacks. A replacement can
+change planar area, so the third coordinate compensates for the product
+of the two planar scales. Evacuation and delivery then realize the
+resulting volume-preserving boxes in every period.
+
+\begin{theorem}[Periodic prefix-box realization]\label{thm:bb-prefix29}
+For every deterministic machine and finite input, a compactly supported
+smooth force on $\R^3$, periodic with period one from time zero,
+realizes the fixed-origin halting event
+\[
+ \exists t\ge0:\quad X(t;0)\in(-4,-1)\times(-1,2)\times(-1,1)
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+The force has bounded mixed derivatives and the unique zero-data
+solution in the class of Proposition~\ref{prop:bb-realization}.
+Its finite program uses full volume-preserving boxes; no solenoidal
+condition is imposed on the force itself.
+\end{theorem}
+\begin{proof}\label{proof:bb-prefix29}
+Normalize to one halting work state $q_h$, redirecting incoming
+transitions to it and replacing missing nonhalting rules by unchanged-letter
+stay moves into it. This preserves initial halting.
+Write $\Gamma$ for the work alphabet, $b$ for its blank, $q_0$ for the
+initial control, and $\omega$ for the finite input word.
+For top-first stacks $L,R$, a rule
+$(s,\alpha,\beta)\rightsquigarrow(r,\gamma,\eta)$ replaces the
+two displayed prefixes and leaves their independent infinite tails
+unchanged.  Introduce fresh state $s_0$, boundary letter $S$ and mark
+letter $M$.  The preliminary rules are
+\[
+ (s_0,\varepsilon,\varepsilon)\rightsquigarrow(q_0,S,\omega)
+\]
+and, for each work transition $(q,a)\mapsto(r,c,d)$,
+\[
+\begin{array}{c|l}
+ d&\text{preliminary rules}\\\hline
+1&(q,\varepsilon,a)\rightsquigarrow(r,c,\varepsilon)\\
+0&(q,\varepsilon,a)\rightsquigarrow(r,\varepsilon,c)\\
+-1&(q,v,a)\rightsquigarrow(r,\varepsilon,vc)\quad(v\in\Gamma),\\
+  &(q,S,a)\rightsquigarrow(r,S,bc).
+\end{array}
+\]
+The final row exposes the implicit blank beyond the left boundary.
+Index each preliminary occurrence by $j$, with target $r(j)$,
+and introduce a private control $g_j$ and record letter $\ell_j$.
+Replace that occurrence by
+$(s,\alpha,\beta)\to(g_j,\gamma,M\eta)$ and add
+\begin{equation}\label{eq:bb-prefix-table}
+\begin{aligned}
+ (g_j,v,\varepsilon)&\to(g_j,\varepsilon,v)&&v\in\Gamma,\\
+ (g_j,S,\varepsilon)&\to(k_{r(j)},S\ell_j,\varepsilon),\\
+ (k_q,\varepsilon,v)&\to(k_q,v,\varepsilon)&&v\in\Gamma,\\
+ (k_q,\varepsilon,M)&\to(q,\varepsilon,\varepsilon).
+\end{aligned}
+\end{equation}
+Sources are separated by control and tested top symbols.  Into $g_j$,
+entry has right top $M$, whereas a loop has a work letter there.
+Into $k_q$, entry has left prefix $S\ell_j$, whereas a loop has a
+work letter at left top; distinct entries have distinct $\ell_j$.
+Into a work state only the last rule enters.  Thus both source and
+target cylinders are disjoint for arbitrary tails.
+
+After a preliminary replacement the stacks have form $wSB,MY$,
+with $w\in\Gamma^*$.  The forward loop transfers $w$, leaving
+$SB,w^{\rm rev}MY$.  The boundary rule inserts $\ell_j$ behind
+$S$, the reverse loop restores $w$, and the final rule removes $M$.
+After $2|w|+2$ further instructions the stacks are
+$wS\ell_jB,Y$ at control $r(j)$.  At a work control, $w$ lists
+represented cells left of the head, nearest first; $R$ lists the
+current cell and right tail, and cells beyond $S$ on the left are
+implicit blanks.  The preliminary rules perform exactly the work
+write and move.  Starting at $(s_0,b^\infty,b^\infty)$, the first
+work visit is at rule count three, and consecutive work visits differ
+by $3+2|w_{\rm new}|$.  This proves finite continuation and the
+halting equivalence, including an initially halted input.  The
+retained records also recover all head displacements.
+
+Enumerate the enlarged alphabet, blank first, by even digits
+$0,2,\ldots,2m-2$ in base $K=2m$.  Use prefix intervals $I(v)$
+and codes $E$ as in Lemma~\ref{lem:bb-onecell}.  Set offsets
+$o_{q_h}=-3$, $o_{s_0}=0$, and $3,6,9,\ldots$ for the other controls.
+Encode $(s,L,R)$ by $(o_s+E(L),E(R),0)$; its initial value is zero.
+For a rule $(s,\alpha,\beta)\to(s',\gamma,\eta)$ put
+\[
+ a_i=K^{|\alpha|-|\gamma|},\quad
+ b_i=K^{|\beta|-|\eta|},\quad c_i=(a_ib_i)^{-1}.
+\]
+The full source box is
+$(o_s+I(\alpha))\times I(\beta)\times[-1,1]$;
+the target is
+$(o_{s'}+I(\gamma))\times I(\eta)\times[-c_i,c_i]$.
+The center-to-center map has matrix $\operatorname{diag}(a_i,b_i,c_i)$
+and determinant one.  It implements the prefixes on all codes.
+Disjoint cylinders give separated full planar projections in each
+family, including blank-tail endpoints.
+
+Park box $i$ at $(4i,4,0)$, reserving the horizontal rectangle
+$[4i-1/2,4i+1/2]\times[7/2,9/2]$.
+Let $R=\max(1,c_1,\ldots,c_N)$ and travel at height $Z=2R+3$.
+Evacuate all sources to their parking sites, reshape there by
+$\operatorname{diag}(a_i^s,b_i^s,c_i^s)$, then deliver all targets.
+The two horizontal widths remain at most one, and vertical half-widths
+at most $R$.  Let $g$ be the minimum of one and the within-family
+gaps in the source-plus-parking and target-plus-parking projections.
+Padding smaller than $g/8$ works during vertical motion and strain;
+at travel height the moving bottom is at least $Z-R>R+2$.
+Lemma~\ref{lem:bb-lift-parking} gives the full $7N$-motion schedule.
+
+Repeat this unit pulse from zero.  Nonhalting endpoint boxes and all
+parking boxes have nonnegative first coordinate.  Translations preserve
+that bound, and strain takes place in positive parking envelopes.
+Therefore a nonhalting path never enters the negative observation box.
+A halting code lies in $[-3,-2]\times[0,1]\times\{0\}$ inside it.
+The residual proposition finishes the proof.  Loading was a repeated
+instruction, so no exceptional first-time force is required.
+\end{proof}
+
+
+\begin{remark}[A different observation regime]
+For comparison, the distinct velocity-field detector of
+\cite[Theorem~1.1(2) and Section~5]{OpenAI379EulerianC} uses
+$\R^2\times\T$ and the event
+$\int_{\{X_2>0\}\times\T}u_3>1/2$. Its force has globally bounded
+mixed derivatives and horizontal support compact on each finite time
+interval. The nonnegative scalar velocity has diffusion tails; after
+loading its total mass is one. Its cutoff estimate gives loss
+$\delta<1/24$, nonhalting mass at most $\delta$ in the observed half-space,
+and halting mass above $3/4$. Those conclusions follow from the cited
+advection--diffusion proof and have no uniform spatial support assertion.
+They are a separate force and observation regime from the material test
+proved here.
+\end{remark}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/introduction.tex

@@ -1,173 +1,173 @@
-\section{Introduction}\label{sec:introduction}
-A finite instruction can prescribe a volume-preserving affine map on a
-box. A fluid realization must do more: it must move every point of that
-box, preserve incompressibility outside it, and leave the other instructions
-available. Source boxes and destination boxes can overlap. Moreover, a
-particle used to report the answer must avoid the observation region
-throughout every nonhalting transition. We study these requirements together.
-
-Our geometric result is Theorem~\ref{lem:bb-balanced-box-routing}.
-It extends any finite collection of positive diagonal determinant-one maps
-between separately disjoint rational solid boxes to the time-one map of
-an effective smooth compactly supported incompressible velocity. The maps
-hold on neighborhoods of the entire closed boxes. The proof separates
-centers, changes shapes, evacuates the sources to storage, and delivers
-boxes along paths avoiding everything still occupied. Positive margins make
-the construction effective and permit additional observation guards.
-
-The first computational application, Theorem~\ref{thm:bb-balanced-three-stack},
-uses all three coordinates for symbolic data. Three stacks retain a tape
-and its execution history. Every local instruction preserves the total
-length of the prefixes it replaces, so its affine map has determinant one.
-A fixed particle, initially in a fluid at rest, enters the fixed cube
-$(-1,2)^3$ exactly when the encoded machine halts. The force is smooth,
-compactly supported in space and periodic after a one-unit loading interval.
-Its prescription uses the finite rule table, not the executed computation.
-This application completes the main construction before we vary
-initialization, history guards and observers.
-
-The passage from a symbolic instruction to a fluid observation begins
-with retained history, which makes the instruction injective on its
-entire domain of tapes, not just along the chosen computation. A rule
-specifies finite prefixes and leaves their infinite tails free; gapped
-radix expansions place the codes in separately disjoint closed source
-and target boxes. Prefix replacement extends to an affine map of each
-whole box, which smooth incompressible transport then realizes.
-An additional geometric bound must exclude unintended visits to
-the observer between successive rule times. Finally, the prescribed
-velocity determines a body force, and an energy comparison identifies
-the resulting solution in the declared class. Each later variant changes
-one or more of these steps while retaining this separation of tasks.
-
-The remaining constructions answer more specific questions. Two tape
-coordinates can be thickened into solid boxes, or their planar scale changes
-can be compensated in the normal direction. A temporary mark, a persistent
-mark, an explicit frontier and a boundary between occupied and unused
-history give different partial instruction maps. We prove their inverses
-on their full domains because this is what separates the target boxes.
-A bounded observation box permits horizontal motion above it; a slab needs
-a coordinate bound throughout that motion. Storage and obstacle detours
-supply further alternatives. Loading once gives eventual periodicity,
-whereas placing the initial code at the fixed label or repeating a loading
-branch gives periodicity from time zero. Onto slow clocks give separate
-decaying forces with derivative norms square-integrable in time.
-
-\subsection{Prior work and the role of forcing}
-The computational starting point is Turing's finite machine model
-\cite{Turing1936}. Landauer's discussion of retained intermediate information
-\cite[Section~3]{Landauer1961} and Bennett's explicit history recording
-\cite[Eqs.~(9)--(11)]{Bennett1973} explain how an irreversible computation
-can be embedded in an invertible mechanism. We use that principle, but
-prove the finite guarded tables and their complete inverses here. No
-complexity bound or history-erasure procedure from those works is asserted
-for these tables.
-
-Moore's generalized shifts turn symbolic updates into affine operations
-on positional expansions \cite{Moore1990,Moore1991}. Our gapped expansions
-serve the same purpose. Moore also realizes invertible generalized shifts
-by smooth planar diffeomorphisms and three-dimensional flows
-\cite[Section~6, Theorem~12 and its corollary]{Moore1991}.
-Cardona, Miranda, Peralta-Salas and Presas extend the affine maps on
-neighborhoods of separated planar blocks to an area-preserving disk
-diffeomorphism, using contraction, transport, expansion and Moser's area
-correction \cite[Proposition~5.1]{CardonaMirandaPeraltaSalasPresas2021}.
-This is a geometric predecessor of the routing problem here. Our local
-curl formulas implement positive diagonal determinant-one motions in
-three dimensions directly; the proof supplies effective support and
-neighborhood bounds and controls observation at intermediate times.
-
-Computational universality has been established for stationary Euler
-flows with adapted geometry \cite{CardonaMirandaPeraltaSalasPresas2021}.
-Cardona, Miranda and Peralta-Salas also construct universal Euclidean
-Beltrami flows, with infinite energy, and robust simulations of
-tape-bounded machines by unforced Navier--Stokes flows on a flat torus
-\cite[arXiv version~3, Theorem~1 and Remark~29]{CardonaMirandaPeraltaSalas2023Beltrami}.
-Dyhr, Gonz\'alez-Prieto, Miranda and Peralta-Salas obtain stationary
-unforced Navier--Stokes universality using an adapted metric and the
-Hodge Laplacian
-\cite[Theorem~A]{DyhrGonzalezPrietoMirandaPeraltaSalas2026}.
-
-Here the ambient metric is fixed and flat, the viscosity is
-prescribed, and the initial velocity is zero. The machine is placed in an
-external body force. Given an explicit solenoidal velocity $U$, the identity
-$f=U_t+(U\cdot\nabla)U-\nu\Delta U$ supplies that force. The mathematical
-work is the effective construction of $U$ and its all-time observation,
-followed by uniqueness in the stated comparison class. We do not assert a
-regularity theorem for arbitrary fluid data.
-
-The companion \cite{OpenAI379Shear} gives an entry construction using
-solenoidal shears, and \cite{OpenAI379Sheets} treats prefix instructions
-and other planar or normal-compensated realizations. They explain adjacent
-geometric choices; the proofs of the present full-box mechanisms are given
-here. The final input-offset application explicitly imports the planar
-processor and spatial clock of \cite{OpenAI379StationaryA2}, with the exact
-parameters and conclusions stated in Section~\ref{bc:input-offset-section}.
-The companion \cite{OpenAI379EulerianC} detects computation from an
-integral of the velocity field on $\R^2\times\T$ via an
-advection--diffusion estimate. The end of
-Section~\ref{sec:bb-initialization} compares that event with the path of
-one material particle studied here.
-
-\subsection{Equation, effectiveness and reading path}
-Throughout $\nu>0$ is fixed, $\T=\R/\Z$, and the fluid equation on
-$\R^3$ or the flat unit torus is
-\begin{equation}\label{eq:box-NS}
- u_t+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f,
- \qquad \divg u=0,\qquad u(0)=0.
-\end{equation}
-A material label $a$ has path $\dot\Phi_t(a)=u(t,\Phi_t(a))$,
-$\Phi_0(a)=a$. We also write $X(t;a)=\Phi_t(a)$. The smooth constructed
-velocities have global material flows. On the torus pressure is normalized
-to have mean zero. Whole-space pressure assumptions are specified in
-Section~\ref{sec:model} and in the individual statements. Unless explicitly
-projected, the force is a general body force and the constructed pressure
-is zero.
-
-An effective field is a finite program that evaluates every requested
-mixed derivative at computably supplied arguments to any prescribed
-rational error and computes the bounds used in the proof. The finite
-machine and its finite input determine the coefficients, cutoffs and
-routing schedule. Evaluating the field cannot first compute the selected
-machine run. All algorithmic statements concern computable $\nu$;
-the same formulas are valid for arbitrary positive $\nu$, with evaluation
-relative to that parameter. The encoded trajectories are exact real
-trajectories. No uniform precision tolerance for an unbounded computation
-is asserted.
-
-Section~\ref{sec:model} states the analytic comparison tools.
-Section~\ref{sec:bb-boxes} proves the box motions, and
-Section~\ref{sec:bb-balanced-three-stack} gives the balanced-stack application.
-Sections~\ref{sec:bb-frontiers} and~\ref{ba:codes} establish reusable
-recorders and full closed-rectangle coding; Section~\ref{ba:geometry}
-supplies the common interpolation formulas. The subsequent constructions
-can be read according to the requirement being changed:
-\begin{center}
-\begin{tabular}{@{}p{.24\textwidth}p{.71\textwidth}@{}}
-\toprule
-Choice & Where it is developed \\
-\midrule
-Initialization and time periodicity &
-Section~\ref{sec:bb-initialization} compares a separate loader, direct
-placement of the initial code, and a loading branch repeated in every
-period. An untargeted start square permits the last choice. \\[4pt]
-History domains &
-Sections~\ref{ba:marked-routes} and~\ref{sec:bb-guarded} distinguish
-persistent marks, fresh history and neighboring-cell guards. Their
-inverses on arbitrary allowed tapes determine the separated image boxes.
-\\[4pt]
-Observer geometry &
-Section~\ref{ba:ordered-routes} treats bounded boxes, slabs and
-half-spaces. Sections~\ref{sec:solid-routing} and~\ref{sec:torus-routing}
-develop storage, detours and fixed torus charts with the corresponding
-intermediate-time bounds. \\[4pt]
-Decay and stationarity &
-The logarithmic clock in Section~\ref{ba:euclidean} produces separate
-decaying forces. Section~\ref{sec:clocks} treats further profile choices,
-including the imported input-offset processor. \\
-\bottomrule
-\end{tabular}
-\end{center}
-The five whole-space comparison classes are stated in
-Lemma~\ref{lem:b-comparison}. Each alternative fixes its own label,
-observer, support, pressure class and time dependence; these are separate
-constructions rather than simultaneous guarantees for one force.
+\section{Introduction}\label{sec:introduction}
+A finite instruction can prescribe a volume-preserving affine map on a
+box. A fluid realization must do more: it must move every point of that
+box, preserve incompressibility outside it, and leave the other instructions
+available. Source boxes and destination boxes can overlap. Moreover, a
+particle used to report the answer must avoid the observation region
+throughout every nonhalting transition. We study these requirements together.
+
+Our geometric result is Theorem~\ref{lem:bb-balanced-box-routing}.
+It extends any finite collection of positive diagonal determinant-one maps
+between separately disjoint rational solid boxes to the time-one map of
+an effective smooth compactly supported incompressible velocity. The maps
+hold on neighborhoods of the entire closed boxes. The proof separates
+centers, changes shapes, evacuates the sources to storage, and delivers
+boxes along paths avoiding everything still occupied. Positive margins make
+the construction effective and permit additional observation guards.
+
+The first computational application, Theorem~\ref{thm:bb-balanced-three-stack},
+uses all three coordinates for symbolic data. Three stacks retain a tape
+and its execution history. Every local instruction preserves the total
+length of the prefixes it replaces, so its affine map has determinant one.
+A fixed particle, initially in a fluid at rest, enters the fixed cube
+$(-1,2)^3$ exactly when the encoded machine halts. The force is smooth,
+compactly supported in space and periodic after a one-unit loading interval.
+Its prescription uses the finite rule table, not the executed computation.
+This application completes the main construction before we vary
+initialization, history guards and observers.
+
+The passage from a symbolic instruction to a fluid observation begins
+with retained history, which makes the instruction injective on its
+entire domain of tapes, not just along the chosen computation. A rule
+specifies finite prefixes and leaves their infinite tails free; gapped
+radix expansions place the codes in separately disjoint closed source
+and target boxes. Prefix replacement extends to an affine map of each
+whole box, which smooth incompressible transport then realizes.
+An additional geometric bound must exclude unintended visits to
+the observer between successive rule times. Finally, the prescribed
+velocity determines a body force, and an energy comparison identifies
+the resulting solution in the declared class. Each later variant changes
+one or more of these steps while retaining this separation of tasks.
+
+The remaining constructions answer more specific questions. Two tape
+coordinates can be thickened into solid boxes, or their planar scale changes
+can be compensated in the normal direction. A temporary mark, a persistent
+mark, an explicit frontier and a boundary between occupied and unused
+history give different partial instruction maps. We prove their inverses
+on their full domains because this is what separates the target boxes.
+A bounded observation box permits horizontal motion above it; a slab needs
+a coordinate bound throughout that motion. Storage and obstacle detours
+supply further alternatives. Loading once gives eventual periodicity,
+whereas placing the initial code at the fixed label or repeating a loading
+branch gives periodicity from time zero. Onto slow clocks give separate
+decaying forces with derivative norms square-integrable in time.
+
+\subsection{Prior work and the role of forcing}
+The computational starting point is Turing's finite machine model
+\cite{Turing1936}. Landauer's discussion of retained intermediate information
+\cite[Section~3]{Landauer1961} and Bennett's explicit history recording
+\cite[Eqs.~(9)--(11)]{Bennett1973} explain how an irreversible computation
+can be embedded in an invertible mechanism. We use that principle, but
+prove the finite guarded tables and their complete inverses here. No
+complexity bound or history-erasure procedure from those works is asserted
+for these tables.
+
+Moore's generalized shifts turn symbolic updates into affine operations
+on positional expansions \cite{Moore1990,Moore1991}. Our gapped expansions
+serve the same purpose. Moore also realizes invertible generalized shifts
+by smooth planar diffeomorphisms and three-dimensional flows
+\cite[Section~6, Theorem~12 and its corollary]{Moore1991}.
+Cardona, Miranda, Peralta-Salas and Presas extend the affine maps on
+neighborhoods of separated planar blocks to an area-preserving disk
+diffeomorphism, using contraction, transport, expansion and Moser's area
+correction \cite[Proposition~5.1]{CardonaMirandaPeraltaSalasPresas2021}.
+This is a geometric predecessor of the routing problem here. Our local
+curl formulas implement positive diagonal determinant-one motions in
+three dimensions directly; the proof supplies effective support and
+neighborhood bounds and controls observation at intermediate times.
+
+Computational universality has been established for stationary Euler
+flows with adapted geometry \cite{CardonaMirandaPeraltaSalasPresas2021}.
+Cardona, Miranda and Peralta-Salas also construct universal Euclidean
+Beltrami flows, with infinite energy, and robust simulations of
+tape-bounded machines by unforced Navier--Stokes flows on a flat torus
+\cite[arXiv version~3, Theorem~1 and Remark~29]{CardonaMirandaPeraltaSalas2023Beltrami}.
+Dyhr, Gonz\'alez-Prieto, Miranda and Peralta-Salas obtain stationary
+unforced Navier--Stokes universality using an adapted metric and the
+Hodge Laplacian
+\cite[Theorem~A]{DyhrGonzalezPrietoMirandaPeraltaSalas2026}.
+
+Here the ambient metric is fixed and flat, the viscosity is
+prescribed, and the initial velocity is zero. The machine is placed in an
+external body force. Given an explicit solenoidal velocity $U$, the identity
+$f=U_t+(U\cdot\nabla)U-\nu\Delta U$ supplies that force. The mathematical
+work is the effective construction of $U$ and its all-time observation,
+followed by uniqueness in the stated comparison class. We do not assert a
+regularity theorem for arbitrary fluid data.
+
+The companion \cite{OpenAI379Shear} gives an entry construction using
+solenoidal shears, and \cite{OpenAI379Sheets} treats prefix instructions
+and other planar or normal-compensated realizations. They explain adjacent
+geometric choices; the proofs of the present full-box mechanisms are given
+here. The final input-offset application explicitly imports the planar
+processor and spatial clock of \cite{OpenAI379StationaryA2}, with the exact
+parameters and conclusions stated in Section~\ref{bc:input-offset-section}.
+The companion \cite{OpenAI379EulerianC} detects computation from an
+integral of the velocity field on $\R^2\times\T$ via an
+advection--diffusion estimate. The end of
+Section~\ref{sec:bb-initialization} compares that event with the path of
+one material particle studied here.
+
+\subsection{Equation, effectiveness and reading path}
+Throughout $\nu>0$ is fixed, $\T=\R/\Z$, and the fluid equation on
+$\R^3$ or the flat unit torus is
+\begin{equation}\label{eq:box-NS}
+ u_t+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f,
+ \qquad \divg u=0,\qquad u(0)=0.
+\end{equation}
+A material label $a$ has path $\dot\Phi_t(a)=u(t,\Phi_t(a))$,
+$\Phi_0(a)=a$. We also write $X(t;a)=\Phi_t(a)$. The smooth constructed
+velocities have global material flows. On the torus pressure is normalized
+to have mean zero. Whole-space pressure assumptions are specified in
+Section~\ref{sec:model} and in the individual statements. Unless explicitly
+projected, the force is a general body force and the constructed pressure
+is zero.
+
+An effective field is a finite program that evaluates every requested
+mixed derivative at computably supplied arguments to any prescribed
+rational error and computes the bounds used in the proof. The finite
+machine and its finite input determine the coefficients, cutoffs and
+routing schedule. Evaluating the field cannot first compute the selected
+machine run. All algorithmic statements concern computable $\nu$;
+the same formulas are valid for arbitrary positive $\nu$, with evaluation
+relative to that parameter. The encoded trajectories are exact real
+trajectories. No uniform precision tolerance for an unbounded computation
+is asserted.
+
+Section~\ref{sec:model} states the analytic comparison tools.
+Section~\ref{sec:bb-boxes} proves the box motions, and
+Section~\ref{sec:bb-balanced-three-stack} gives the balanced-stack application.
+Sections~\ref{sec:bb-frontiers} and~\ref{ba:codes} establish reusable
+recorders and full closed-rectangle coding; Section~\ref{ba:geometry}
+supplies the common interpolation formulas. The subsequent constructions
+can be read according to the requirement being changed:
+\begin{center}
+\begin{tabular}{@{}p{.24\textwidth}p{.71\textwidth}@{}}
+\toprule
+Choice & Where it is developed \\
+\midrule
+Initialization and time periodicity &
+Section~\ref{sec:bb-initialization} compares a separate loader, direct
+placement of the initial code, and a loading branch repeated in every
+period. An untargeted start square permits the last choice. \\[4pt]
+History domains &
+Sections~\ref{ba:marked-routes} and~\ref{sec:bb-guarded} distinguish
+persistent marks, fresh history and neighboring-cell guards. Their
+inverses on arbitrary allowed tapes determine the separated image boxes.
+\\[4pt]
+Observer geometry &
+Section~\ref{ba:ordered-routes} treats bounded boxes, slabs and
+half-spaces. Sections~\ref{sec:solid-routing} and~\ref{sec:torus-routing}
+develop storage, detours and fixed torus charts with the corresponding
+intermediate-time bounds. \\[4pt]
+Decay and stationarity &
+The logarithmic clock in Section~\ref{ba:euclidean} produces separate
+decaying forces. Section~\ref{sec:clocks} treats further profile choices,
+including the imported input-offset processor. \\
+\bottomrule
+\end{tabular}
+\end{center}
+The five whole-space comparison classes are stated in
+Lemma~\ref{lem:b-comparison}. Each alternative fixes its own label,
+observer, support, pressure class and time dependence; these are separate
+constructions rather than simultaneous guarantees for one force.

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/main.tex

@@ -1,56 +1,56 @@
-\documentclass[11pt]{article}
-\ifdefined\pdftrailerid\pdftrailerid{}\fi
-\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
-\usepackage[T1]{fontenc}
-\usepackage[utf8]{inputenc}
-\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,enumitem,microtype,longtable,array,tikz}
-\usepackage[margin=1in]{geometry}
-\usetikzlibrary{arrows.meta,positioning}
-\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}
-\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
-\newcommand{\R}{\mathbb R}\newcommand{\T}{\mathbb T}\newcommand{\Z}{\mathbb Z}\newcommand{\N}{\mathbb N}\newcommand{\Q}{\mathbb Q}
-\DeclareMathOperator{\supp}{supp}\DeclareMathOperator{\divg}{div}\DeclareMathOperator{\diag}{diag}\DeclareMathOperator{\id}{id}
-\hypersetup{pdftitle={Incompressible Box Transport and Finite Computation},pdfauthor={OpenAI}}
-\title{Incompressible Box Transport and Finite Computation}
-\author{OpenAI}
-\date{October 6, 2026}
-\begin{document}\maketitle
-\begin{abstract}
-We realize finite positive diagonal affine maps of determinant one by
-effective smooth incompressible flows on neighborhoods of entire closed
-rational solid boxes. The source and target families are each disjoint, but may
-overlap each other. The construction uses localized curls, evacuation to
-storage and obstacle detours. A balanced three-stack recorder then assigns
-every machine and finite input a smooth Navier--Stokes force with one
-compact spatial support, periodic after a loading interval, at any fixed
-positive computable viscosity. The fluid starts at rest, and one fixed
-particle enters one fixed open cube exactly when the machine halts.
-Further constructions give fixed torus charts, periodicity from time zero,
-alternative history guards and bounded or slab observers. Onto slow clocks
-yield separate decaying forces. Each construction includes an all-time
-observation proof, effective derivative bounds and a stated pressure
-comparison class.
-\end{abstract}
-\input{introduction}
-\input{analysis}
-\input{geometry}
-\input{balanced}
-\input{compilers}
-\input{coding-conventions}
-\input{routing-conventions}
-\input{initialization}
-\input{compact-processors}
-\input{persistent-history}
-\input{guarded-history}
-\input{observer-routing}
-\input{solid-routing}
-\input{torus-routing}
-\input{clocks}
-\bibliographystyle{plain}
-\bibliography{references}
-\end{document}
+\documentclass[11pt]{article}
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,enumitem,microtype,longtable,array,tikz}
+\usepackage[margin=1in]{geometry}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}\newcommand{\T}{\mathbb T}\newcommand{\Z}{\mathbb Z}\newcommand{\N}{\mathbb N}\newcommand{\Q}{\mathbb Q}
+\DeclareMathOperator{\supp}{supp}\DeclareMathOperator{\divg}{div}\DeclareMathOperator{\diag}{diag}\DeclareMathOperator{\id}{id}
+\hypersetup{pdftitle={Incompressible Box Transport and Finite Computation},pdfauthor={OpenAI}}
+\title{Incompressible Box Transport and Finite Computation}
+\author{OpenAI}
+\date{October 6, 2026}
+\begin{document}\maketitle
+\begin{abstract}
+We realize finite positive diagonal affine maps of determinant one by
+effective smooth incompressible flows on neighborhoods of entire closed
+rational solid boxes. The source and target families are each disjoint, but may
+overlap each other. The construction uses localized curls, evacuation to
+storage and obstacle detours. A balanced three-stack recorder then assigns
+every machine and finite input a smooth Navier--Stokes force with one
+compact spatial support, periodic after a loading interval, at any fixed
+positive computable viscosity. The fluid starts at rest, and one fixed
+particle enters one fixed open cube exactly when the machine halts.
+Further constructions give fixed torus charts, periodicity from time zero,
+alternative history guards and bounded or slab observers. Onto slow clocks
+yield separate decaying forces. Each construction includes an all-time
+observation proof, effective derivative bounds and a stated pressure
+comparison class.
+\end{abstract}
+\input{introduction}
+\input{analysis}
+\input{geometry}
+\input{balanced}
+\input{compilers}
+\input{coding-conventions}
+\input{routing-conventions}
+\input{initialization}
+\input{compact-processors}
+\input{persistent-history}
+\input{guarded-history}
+\input{observer-routing}
+\input{solid-routing}
+\input{torus-routing}
+\input{clocks}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/observer-routing.tex

@@ -1,982 +1,982 @@
-\section{Routing choices dictated by the observer}
-\label{ba:ordered-routes}
-
-The preceding processors move every branch by a prescribed affine map.
-We now ask which parts of that motion must be controlled between the
-integer times.  A bounded observation box allows horizontal work above
-the observer.  A slab observer requires a bound on the horizontal
-coordinate throughout the motion.  Finally, a half-space observer can be
-protected by moving all sources to storage before delivering any target.
-These three requirements lead to the constructions below.
-
-\subsection{Four motions above a bounded observer}
-\label{ba:four-stage-section}
-
-We first separate ascent, scaling, translation and descent.  The two
-planar motions occur at a fixed height, so their exclusion from the
-bounded observer follows without a horizontal coordinate estimate.
-
-Let $D_i,E_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational
-axis-aligned rectangles with positive side lengths.  Assume the $D_i$
-are pairwise disjoint and the $E_i$ are pairwise disjoint, and let
-\[
- A_i(b)=c_i'+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(b-c_i),
- \qquad \lambda_i\in\mathbb Q_{>0},\qquad A_i(D_i)=E_i,
-\]
-where $c_i,c_i'$ are their centers.  No separation between a source and
-a target is assumed.  All distances used to choose coordinate padding
-below are distances for the maximum norm.
-
-Let $\vartheta=\sigma$ be the effective flat switch in
-\eqref{eq:p2-step}.  For a closed interval, possibly reduced to one
-point, define
-\[
- \chi_{[a,b],\epsilon}(v)=
- \vartheta\!\left(\frac{2(v-a+\epsilon)}\epsilon\right)
- \vartheta\!\left(\frac{2(b+\epsilon-v)}\epsilon\right).
-\]
-For a box $K$, let $\chi_{K,\epsilon}$ be the product over its three
-coordinates.  It is one on the $\epsilon/2$ enlargement of $K$ and
-zero outside its $\epsilon$ enlargement.  The effective profile estimates
-following \eqref{eq:p2-step} apply to these products and rescalings.
-
-\begin{lemma}[Four-stage realization]\label{ba:four-stage}
-The rectangle data determine effectively a smooth divergence-free field
-$W(s,x)$ supported in a compact subset of $(0,1)\times\mathbb R^3$.
-Its time-one map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of
-each $D_i\times\{0\}$.  For a point of that plate, every planar
-motion occurs at height $4i$, while ascent and descent preserve its
-source and target planar coordinates, respectively.  The scaling can
-use either linear interpolation of its positive first factor or linear
-interpolation of the logarithm of that factor.
-\end{lemma}
-
-\begin{proof}\label{ba:four-stage-proof}
-The empty family is realized by zero.  Otherwise put $h_i=4i$,
-$w_i=c_i'-c_i$, and
-\[
- \Theta_k(s)=\vartheta(10s-(2k-1)),\qquad 1\le k\le4.
-\]
-Only the $k$th motion is active on the interval
-$[(2k-1)/10,2k/10]$; the four intervals are disjoint.  If
-$a_{ij},a'_{ij}$ are the source and target half-widths, respectively,
-take the four swept boxes
-\[
- \begin{aligned}
- K_{i1}&=D_i\times[0,h_i],\\
- K_{i2}&=\{c_i+(u_1,u_2):|u_j|\le\max(a_{ij},a'_{ij})\}
-                                      \times\{h_i\},\\
- K_{i3}&=\operatorname{box}(E_i-w_i,E_i)\times\{h_i\},\\
- K_{i4}&=E_i\times[0,h_i].
- \end{aligned}
-\]
-Here $\operatorname{box}$ denotes the smallest axis-aligned rectangle
-containing the two displayed rectangles.  Within each of the four
-families these boxes are disjoint: source separation handles the first,
-target separation the fourth, and distinct heights the other two.
-Choose a positive rational $\epsilon<1$ smaller than one quarter of
-every pairwise distance within these families.  A family with fewer than
-two members contributes no restriction.  Its $\epsilon$ enlargements
-are then still disjoint.
-
-Choose either
-\[
- \mu_i=1+(\lambda_i-1)\Theta_2
- \quad\hbox{or}\quad \mu_i=\lambda_i^{\Theta_2},
- \qquad \gamma_i=\dot\mu_i/\mu_i.
-\]
-The factors $\mu_i$ and $\mu_i^{-1}$ stay between their respective
-endpoint values.  Use the coordinate potentials
-\[
- \begin{aligned}
- B_{i1}&=(0,h_i x_1,0),&
- B_{i2}&=(0,0,(x_1-c_{i1})(x_2-c_{i2})),\\
- B_{i3}&=(0,0,w_{i1}x_2-w_{i2}x_1),&
- B_{i4}&=(0,-h_i x_1,0).
- \end{aligned}
-\]
-For the logarithmic-scale construction we also retain the alternative
-translation potentials, with $e_3=(0,0,1)$,
-\[
- B_{i1}=\tfrac12(h_ie_3)\times x,\qquad
- B_{i3}=\tfrac12(w_{i1},w_{i2},0)\times x,\qquad
- B_{i4}=\tfrac12(-h_ie_3)\times x.
-\]
-Their curls equal the same constant translation vectors.  Their localized
-fields need not agree away from the plates.  Either choice gives
-\[
- W=\sum_i\left[
- \dot\Theta_1\nabla\times(\chi_{K_{i1},\epsilon}B_{i1})
- +\gamma_i\nabla\times(\chi_{K_{i2},\epsilon}B_{i2})
- +\dot\Theta_3\nabla\times(\chi_{K_{i3},\epsilon}B_{i3})
- +\dot\Theta_4\nabla\times(\chi_{K_{i4},\epsilon}B_{i4})\right].
-\]
-This is a finite sum of spatial curls.  Its spatial support is compact,
-its time support is contained in $[1/10,8/10]$, and the lower bound
-$\mu_i\ge\min(1,\lambda_i)>0$ proves smoothness and effectivity.
-
-For $b\in D_i$, the successive paths are
-\[
- \begin{array}{c|c}
- \text{motion}&\text{position}\\\hline
- \text{ascent}&(b,h_i\Theta_1)\\
- \text{scaling}&(c_i+\operatorname{diag}(\mu_i,\mu_i^{-1})(b-c_i),h_i)\\
- \text{translation}&(A_i(b)-w_i+\Theta_3w_i,h_i)\\
- \text{descent}&(A_i(b),h_i(1-\Theta_4)).
- \end{array}
-\]
-Each path stays in its swept box.  On a neighborhood of that box the
-uncut curls are, in order,
-\[
- h_ie_3,\qquad (x_1-c_{i1},-(x_2-c_{i2}),0),\qquad
- (w_{i1},w_{i2},0),\qquad -h_ie_3.
-\]
-After multiplication by the time coefficients, these are the prescribed
-affine velocities.  The motion matrices are positive diagonal of determinant
-one, the plateau margin is $\epsilon/2$, and the support padding is
-$\epsilon$.  The disjoint swept boxes therefore permit the plateau
-argument of Lemma~\ref{lem:bb-curl}, with $\eta=\epsilon/2$.
-It identifies these particular localized fields with the displayed
-paths on the plateaux.  The normalized diagonal factors are bounded by
-$L_i=\max(1,\lambda_i,\lambda_i^{-1})$, so that lemma gives the
-full initial neighborhood of radius $\epsilon/(4L_i)$.
-\end{proof}
-
-We use the following precise Euclidean comparison class in the applications:
-on each $[0,T]$,
-\begin{equation}\label{ba:euclidean-comparison-class}
- \begin{gathered}
- u\in C([0,T];H^2(\mathbb R^3))\cap C^1([0,T];L^2(\mathbb R^3)),\\
- u,\nabla u\in L^\infty([0,T]\times\mathbb R^3),\qquad
- p\in C([0,T];H^1(\mathbb R^3)).
- \end{gathered}
-\end{equation}
-The pressure representative is part of this condition; it is zero for
-our constructed solution.  These comparison velocities and pressures
-satisfy case (i) of Lemma~\ref{lem:b-comparison}.  We retain the stated
-bounded-gradient requirement here, although that case does not need it.
-
-That lemma also gives a smooth material diffeomorphism on every finite
-interval.  Differentiating its trajectory equation gives
-$\dot J=(D_xu)(t,\Phi_t(a))J$, $J(0)=I$, for $J=D_a\Phi_t$.
-Consequently
-\begin{equation}\label{ba:material-volume}
- \frac d{dt}\det J=(\nabla\cdot u)(t,\Phi_t(a))\det J=0,
- \qquad \det J=1.
-\end{equation}
-The same proof applies to periodic lifts on the torus.  The residual
-formulas below are affine in viscosity, so they also define the asserted
-solutions for any real $\nu>0$, with effectivity relative to $\nu$.
-
-\begin{theorem}[Two bounded-box material tests]\label{ba:bounded-box-tests}
-Fix a positive computable viscosity $\nu$.  A deterministic one-tape
-machine and finite input effectively determine each of the following
-general body forces on $[0,\infty)\times\mathbb R^3$.  Both have zero
-initial velocity, a unique global smooth velocity in
-\eqref{ba:euclidean-comparison-class}, and constructed pressure zero.
-\begin{enumerate}
-\item The force and velocity have one fixed compact spatial support, all
-mixed derivatives bounded, and period one for $t\ge1$.  The particle
-from $0$ enters $(-5,-2)\times(-1,2)\times(-1,1)$ exactly when the
-machine halts.
-\item The force and velocity have one fixed compact spatial support and,
-for all $k\ge0$ and spatial multi-indices $\alpha$, satisfy
-\begin{equation}\label{ba:eighteen-decay}
- \|\partial_t^k\partial_x^\alpha u(t)\|_\infty+
- \|\partial_t^k\partial_x^\alpha f(t)\|_\infty
- \le C_{k,\alpha}(1+t)^{-1-k}.
-\end{equation}
-The particle from $(2,0,0)$ enters
-$(-1/2,3/2)\times(-1/2,3/2)\times(-1/2,1/2)$ exactly when the
-machine halts.  In particular $f\in L^2([0,\infty);H^m(\mathbb R^3))$
-for every integer $m\ge0$.
-\end{enumerate}
-The supports and constants may depend on the machine and input.  The
-labels and observers do not.  All force derivatives admit evaluation to
-any prescribed rational error by a finite program.
-\end{theorem}
-
-\begin{proof}\label{ba:bounded-box-tests-proof}
-Normalize to a single halt state $q_H$ and a complete nonhalting table,
-as in Section~\ref{ba:codes}.  Apply the move-first recorder of
-Lemma~\ref{lem:bb-movefirst}.  Its history frontier starts at cell $2$;
-all other history cells are empty and all mark bits are zero.  At
-checkpoint $n$, the frontier is $g_n=n+2$, the work head satisfies
-$|h_n|\le n$, and the next work instruction ending at $h'$ is completed
-in exactly $4+2(g_n-h')$ local steps.  Thus every required checkpoint
-is reached after finitely many steps.  The incoming-direction and
-written-symbol inverse properties hold on the whole partial table.
-The records recover the successive work instructions and hence the
-absolute head position, even though the geometric code is head-relative.
-
-Let $m$ be the full alphabet size.  Use odd digits $\beta(a)$ in
-$\{1,3,\ldots,2m-1\}$, with base $B=2m+1$ in the first construction
-and $B=2m+2$ in the second.  Set
-\[
- C(a_0a_1\cdots)=\sum_{r\ge0}\beta(a_r)B^{-r-1},\quad
- P_a(v)=\frac{\beta(a)+v}{B},\quad D_a(v)=Bv-\beta(a),\quad
- I_a=P_a([0,1]).
-\]
-Let $\xi$ encode the left string, nearest cell first, and $\eta$
-the string starting at the head.  The complete right, stay and left
-branch maps are \eqref{eq:bb-one-cell-branches}, with its digit
-$d(a)$ equal to $\beta(a)$ and its move $e$ equal to $d$ here.
-The left move is split over every full symbol $c$.  The shared lemma
-proves the full-rectangle endpoint formulas, arbitrary-tail action and
-linear part $\operatorname{diag}(B^{-d},B^d)$.
-
-Translate each state square horizontally.  In the first construction
-give the halt state offset $-4$ and the other states offsets $2+2r$,
-$r\ge0$; in the second use halt offset $0$ and other offsets $3r$,
-$r\ge1$.  Thus the code is $(o_s+\xi,\eta,0)$.
-Lemma~\ref{lem:bb-onecell} applies with exactly these odd digits and
-either displayed base: its incoming conditions are those supplied by
-the move-first recorder, and \eqref{eq:bb-one-cell-branches} is its minimal
-left-split branch list.  Thus both complete closed rectangle families
-are separately disjoint, with within-state gaps at least $B^{-2}$.
-This proves the hypotheses of Lemma~\ref{ba:four-stage}; no separation
-between the source and target families is required.
-
-Use its linear scale and coordinate potentials for the first construction;
-use its logarithmic scale and symmetric translation potentials for the
-second.  In the latter case the scaling coefficient is exactly
-$(\log\lambda_i)\dot\Theta_2$ and the scale matrix is
-$\operatorname{diag}(\lambda_i^{\Theta_2},\lambda_i^{-\Theta_2})$.
-Denote the resulting step field by $W_{\rm step}$.
-
-The initial code $P_{\rm in}$ is rational.  A constant full-symbol
-tail of digit $\beta_0$ after $M$ positions contributes
-$B^{-M}\beta_0/(B-1)$; all other contributions are finite.  For the
-appropriate fixed label $a_*$, put $v_0=P_{\rm in}-a_*$ and let
-$J_0$ be the smallest box containing $[a_*,P_{\rm in}]$.  The loader
-\[
- W_{\rm in}(s,x)=\frac d{ds}\vartheta(2s-1/2)\,
- \nabla\times\left(\chi_{J_0,1}(x)\frac{v_0\times x}{2}\right)
-\]
-has the exact marked path $a_*+\vartheta(2s-1/2)v_0$, since the
-uncut curl is $v_0$ and the entire segment lies in the plateau.  Form
-\[
- W(s,x)=W_{\rm in}(s,x)+\sum_{n\ge1}W_{\rm step}(s-n,x).
-\]
-The zero time collars make this smooth, initially zero, and periodic for
-$s\ge1$, with common compact spatial support and bounded mixed derivatives.
-
-At time $s=1+k$ the marked particle is exactly the code after $k$ local
-instructions, through the first halt or indefinitely in a nonhalting run.
-This follows by induction from the full-rectangle maps.  Each halt code
-lies in its stated observation box, including one reached by loading an
-initially halted machine.  In a nonhalting run, the first loader has
-$x_1\ge0$ and later nonhalt codes have $x_1\ge2$; for the second these
-bounds are $x_1\ge2$ and $x_1\ge3$.  Ascent and descent leave those
-planar coordinates unchanged.  Every planar motion has height $4i\ge4$,
-outside both observers.  The intervening stationary intervals add no new
-positions.  This proves exclusion at every real time.
-
-For the first case take $u=W$ and its residual force.  For the second set
-\[
- s(t)=\log(1+t),\quad \theta(t)=(1+t)^{-1},\quad
- u(t,x)=\theta(t)W(s(t),x),
-\]
-whose force is
-\[
- f=\theta^2\{-W+\partial_sW+(W\cdot\nabla)W\}(s(t),x)
-                         -\nu\theta\Delta W(s(t),x).
-\]
-Lemma~\ref{ba:log-clock} applies to this fixed-support, bounded-derivative
-template.  Its identity
-$\partial_t(\theta^jY(s(t)))=\theta^{j+1}(\partial_s-j)Y(s(t))$
-proves \eqref{ba:eighteen-decay} for every $k,\alpha$.  Fixed support
-then gives the stated $L^2_tH^m_x$ conclusion.  Its clock is onto, with
-inverse $t=e^s-1$, so the material path is the old path composed with
-$s(t)$ and the event is unchanged.
-
-In both cases smoothness, fixed support and bounded derivatives on
-finite time intervals put the reference velocity in $CH^2\cap C^1L^2$
-with bounded velocity and gradient.  It is solenoidal and initially zero.
-Lemma~\ref{lem:b-comparison}(i) therefore gives the exact solution
-$(u,0)$ and uniqueness in \eqref{ba:euclidean-comparison-class}.
-The effective profiles following \eqref{eq:p2-step}, applied to the
-finite table, rational initial code, finitely many boxes and repeated
-template, give all derivative evaluations.  A coarse time approximation
-encloses finitely many possible active translates, avoiding equality
-tests at seams.  The prescription never computes the machine's run.
-\end{proof}
-
-\subsection{Small plates and a slab observer on the unit torus}
-\label{ba:torus-plates-section}
-
-A slab observer has no height restriction, so elevation alone cannot
-protect it.  We now keep each nonhalting branch inside a narrow range of
-first coordinates throughout its motion.  The delayed-head recorder also
-allows arbitrary finitely supported work data on both sides of the head.
-
-\begin{theorem}[Torus plates and square-integrable forcing]
-\label{ba:torus-plates}
-Fix a positive computable $\nu$.  Every deterministic one-tape machine
-with finitely specified tape, blank outside a finite set, effectively
-determines a smooth mean-zero general force on
-$\mathbb T^3=\mathbb R^3/\mathbb Z^3$, with bounded mixed derivatives
-and period one for $t\ge1$.  Its zero-data Navier--Stokes solution is
-globally smooth and unique among smooth solutions, with pressure normalized
-to mean zero and constructed pressure zero.  For
-$P_*=(1/4,1/2,1/4)$ and $X(t)=\Phi_t(P_*)$, the event
-\begin{equation}\label{ba:torus-plate-event}
- \exists t\ge0:\quad X_1(t)\pmod1\in(1/2,7/8)
-\end{equation}
-is equivalent to halting.
-
-A separate effective force has the same zero data, event, uniqueness and
-mean-zero property, is supported in one compact sub-box of $(0,1)^3$,
-has all mixed derivatives bounded, and satisfies
-\begin{equation}\label{ba:cube-rate}
- \|\widehat f(t,\cdot)\|_\infty=O((1+t)^{-2/3}).
-\end{equation}
-In particular $\widehat f\in L^2([0,\infty)\times\mathbb T^3)$.
-\end{theorem}
-
-\begin{proof}\label{ba:torus-plates-proof}
-Use Lemma~\ref{ba:six-compiler} with the frontier initially at cell $1$.
-Its main control $M_{q,e}$ records the last work displacement $e$, initially
-zero.  In one work step it writes and marks the old head, records
-$(q,e,a)$ at the frontier, returns to the mark, and only then makes the
-work displacement.  All other initial history cells are blank and every
-mark is zero.  At checkpoint $n$ the frontier is $n+1$, $|h_n|\le n$,
-and the next checkpoint takes exactly $2(n+1-h_n)+3$ local steps.
-These scans test history and marks, so the proof is unaffected by work
-symbols on either side of the head.  Main controls with halting $q$ have
-no outgoing rules.  The full-domain incoming properties are those of
-the cited lemma.
-
-Let $S$ be the local state set, $N=|S|$, let $S_h$ be its halting main
-states, and enumerate $S$ by $n(s)=0,\ldots,N-1$.  Put
-\[
- \ell=\frac1{16(N+1)},\quad z_0=\frac14,\quad
- x_s=\frac18+2\ell n(s)+\frac12\mathbf1_{s\in S_h},\quad
- T_s(\xi,\eta)=(x_s+\ell\xi,1/4+\ell\eta).
-\]
-Nonhalting squares lie in $1/8\le x_1<1/4$ and halting squares in
-$5/8\le x_1<3/4$; distinct squares have gap at least $\ell$.
-Give the full alphabet, of size $m$, odd digits in base $\beta=2m+1$.
-Conjugating \eqref{eq:bb-one-cell-branches} by the source and target charts
-gives separately disjoint rectangle families $R_j^0,R_j^1$ with positive
-diagonal factors $\alpha_j,\alpha_j^{-1}$,
-$\alpha_j\in\{\beta^{-1},1,\beta\}$.  Every chart has the same
-scale, so this remains the physical linear part.
-Lemma~\ref{lem:bb-onecell}, with this uniform chart scaling, gives
-within-family coordinate gaps at least $\ell/\beta^2$, including the
-two-letter target intervals for left moves.
-
-For $1\le j\le L$, write the centers as $p_j^0,p_j^1$, the source
-half-widths as $a_j,b_j$, and the target half-widths as
-$\alpha_ja_j,\alpha_j^{-1}b_j$.  All endpoint half-widths are at
-most $\ell/2$.  Choose
-\[
- h_j=\frac12+\frac{j}{4(L+1)},\qquad
- \epsilon=\frac1{100}\min\left(1,\frac\ell{\beta^2},
-                                      \frac1{4(L+1)}\right).
-\]
-With $\theta(s)=\vartheta(2s-1/2)$ and
-$\Theta_k(\tau)=\theta(3\tau-k+1)$ for $k=1,2,3$, define
-\[
- \begin{aligned}
- C_j(\tau)&=\big((1-\Theta_2)p_j^0+\Theta_2p_j^1,
-                      z_0+(\Theta_1-\Theta_3)(h_j-z_0)\big),\\
- \sigma_j&=1+(\alpha_j-1)\Theta_2,\qquad
- \lambda_j=\dot\sigma_j/\sigma_j,\\
- (w_{j1},w_{j2},w_{j3})&=(a_j\sigma_j,b_j/\sigma_j,0).
- \end{aligned}
-\]
-These centers and half-widths describe moving plates.  Their
-$\epsilon$ enlargements are disjoint: the first third uses source
-projections, the middle third distinct heights, and the last third
-target projections.  The padding is at most one hundredth of every
-relevant gap.  All plates and padding lie in a compact sub-box of
-$(0,1)^3$, since their centers interpolate between the stated chart
-centers, widths are at most $\ell$, heights lie between $1/4$ and
-$3/4$, and $\ell\le1/32$, $\epsilon\le1/100$.
-
-For $w\ge0$ put
-\[
- \kappa_\epsilon(w,v)=
- \vartheta\!\left(\frac{2(w+\epsilon-v)}\epsilon\right)
- \vartheta\!\left(\frac{2(w+\epsilon+v)}\epsilon\right),\qquad
- \chi_j=\prod_{a=1}^3\kappa_\epsilon(w_{ja},x_a-C_{ja}).
-\]
-Writing $y=x-C_j$, use
-\[
- A_j=\tfrac12\dot C_j\times y+(0,0,\lambda_jy_1y_2),
- \qquad V(\tau,x)=\sum_j\nabla\times(\chi_jA_j).
-\]
-Extend the potentials by zero outside the chart.  The support remains
-away from its faces, so this gives a smooth torus field, zero near the
-time endpoints.  On the whole plateau of plate $j$ its velocity is
-$\dot C_j+(\lambda_jy_1,-\lambda_jy_2,0)$; other summands vanish
-there.  Hence the path from $(p_j^0+(r,s),z_0)$ is exactly
-\begin{equation}\label{ba:plate-path}
- C_j(\tau)+(\sigma_j(\tau)r,\sigma_j(\tau)^{-1}s,0),
- \qquad |r|\le a_j,\quad |s|\le b_j.
-\end{equation}
-It ends at the prescribed affine image.  The plateau argument of
-Lemma~\ref{lem:bb-curl} applies with
-$D_j=\operatorname{diag}(\sigma_j,\sigma_j^{-1},1)$,
-$\eta=\epsilon/2$, and diagonal bound
-$\max(1,\alpha_j,\alpha_j^{-1})$: the required enlarged-box separation
-was just proved.  It gives the same motion on an initial neighborhood
-of radius $\epsilon/[4\max(1,\alpha_j,\alpha_j^{-1})]$, with constant
-normal displacement, and therefore includes boundary codes.
-
-The initial code $P_{\rm in}$ is rational, since relative to constant
-blank tails only finitely many work and frontier digits differ.  Load it
-from $P_*$ along $C_*=(1-\theta)P_*+\theta P_{\rm in}$ using
-\[
- V_0(t,x)=\nabla\times\left[
- \prod_{a=1}^3\kappa_{1/64}(0,x_a-C_{*,a}(t))
-                \frac{\dot C_*(t)\times(x-C_*(t))}{2}\right].
-\]
-The segment and its collar stay within the chart: its first coordinate
-stays between $1/8$ and $3/4$, its second between $1/4$ and
-$1/2+\ell$, and its third is $1/4$.  Its plateau curl is $\dot C_*$.
-Take $U=V_0$ on $[0,1]$ and $U(t)=V(t-n)$ on $[n,n+1]$ for $n\ge1$.
-Time collars make this smooth and initially zero.  It has fixed compact
-chart support, bounded mixed derivatives, and period one after loading.
-Define $f=U_t+(U\cdot\nabla)U-\nu\Delta U$ using the stated physical
-viscosity.  Both $U$ and $f$ have mean zero: $U$ is a curl, the
-Laplacian integrates to zero, and
-$(U\cdot\nabla)U=\operatorname{div}(U\otimes U)$.
-The torus clause of Lemma~\ref{lem:b-comparison} applies to this
-smooth, initially zero, solenoidal reference field and gives the unique
-smooth solution $(U,0)$ with mean-zero pressure.
-
-Equations~\eqref{eq:bb-one-cell-branches} and \eqref{ba:plate-path} identify
-the particle at time $1+k$ with the $k$th local configuration.  The
-recorder invariant ensures that a nonhalting computation continues
-indefinitely.  A halt, including an initial halt reached by loading,
-puts the particle in $5/8\le X_1<3/4$, proving the event.  For a
-nonhalting computation the loader has $X_1\le1/4$ and every later
-branch has both endpoint centers below $1/4$.  Its center is their
-convex interpolation and its first half-width is at most $\ell/2$.
-Thus $0<X_1<1/4+\ell/2<1/2$ at every intermediate time.  The path
-stays inside the chart, so reduction modulo one cannot create a false visit.
-
-For the separate decaying force choose the onto clock
-\[
- s(t)=(1+t)^{1/3}-1,\qquad \widehat U(t,x)=s'(t)U(s(t),x).
-\]
-Its residual is
-\[
- \widehat f=s''U+(s')^2\{\partial_sU+(U\cdot\nabla)U\}
-                                         -\nu s'\Delta U,
-\]
-where the fields on the right are evaluated at $(s(t),x)$.  Since
-$s'=(1/3)(1+t)^{-2/3}$ and $s''=-(2/9)(1+t)^{-5/3}$, the bounded
-template derivatives prove \eqref{ba:cube-rate}.  Every positive-order
-derivative of $s$ is bounded; repeated product and chain rules therefore
-bound every mixed derivative of $\widehat U$ and $\widehat f$.  The
-square of $(1+t)^{-2/3}$ is integrable.  Support and mean zero persist,
-and $U(0)=0$ retains the zero datum.  The torus clause of
-Lemma~\ref{lem:b-comparison} therefore applies to $\widehat U$ too.
-Finally the material path is the old one at $s(t)$,
-and $s$ has inverse $t=(1+s)^3-1$, so the event is unchanged.
-
-All choices are effective finite data followed by explicit smooth
-formulas.  In particular $\sigma_j\ge\min(1,\alpha_j)>0$; the cutoff
-estimates and zero collars allow derivative evaluation at approximate real
-arguments without tests for exact boundary equality.  Only positive
-arguments enter the cube root.  The program repeats the entire local
-table and never uses a computed execution.
-\end{proof}
-
-\subsection{Storage and periodic forcing without a loader}
-\label{ba:parking-section}
-
-The next construction absorbs initialization into the positions of the
-state rectangles.  A zero digit for the full blank symbol places the
-initial left stack at zero; an input-dependent second-coordinate translation then
-places the entire initial code at one fixed particle.  To protect a
-half-space observer during every period, we first evacuate all sources
-to separate storage rectangles and only afterwards fill the targets.
-
-\begin{lemma}[Zero-blank coding]\label{ba:zero-blank-code}
-Let an alphabet of size $m>1$ have digits $0,\ldots,m-1$, with full
-blank digit zero.  Set $B=m+1$, $\kappa=(m-1)/m$, and
-\[
- \operatorname{val}(a_0a_1\cdots)=\sum_{r\ge0}a_rB^{-r-1},\qquad
- I_a=\frac{a+[0,\kappa]}B,\qquad I_{a,b}=\frac{a+I_b}B.
-\]
-The code is injective, takes values in $[0,\kappa]$, and its distinct
-one-letter and two-letter cylinders have positive rational gaps.  For a
-deterministic partial table with target-state-determined incoming direction
-and incoming rule determined by target state and written symbol, separated
-state translates of $[0,\kappa]^2$ yield separately disjoint full source
-and target rectangles by the following maps in state-relative coordinates:
-\begin{equation}\label{ba:zero-digit-maps}
- \begin{array}{c|c|c|c}
- d&\text{source}&(\xi',\eta')&\text{target}\\\hline
- 1&[0,\kappa]\times I_a&((b+\xi)/B,B\eta-a)&I_b\times[0,\kappa]\\
- 0&[0,\kappa]\times I_a&(\xi,\eta+(b-a)/B)&[0,\kappa]\times I_b\\
- -1&I_c\times I_a&(B\xi-c,c/B+(b+B\eta-a)/B^2)&
-                                      [0,\kappa]\times I_{c,b}.
- \end{array}
-\end{equation}
-The last row is split over all $c$.  Each map has positive diagonal
-linear part of determinant one and acts exactly on the full rectangle.
-\end{lemma}
-
-\begin{proof}\label{ba:zero-blank-code-proof}
-The maximum tail is $(m-1)/(B-1)=\kappa$, and
-$(m-1+\kappa)/B=\kappa$.  Consecutive first-digit intervals have
-gap $(1-\kappa)/B>0$.  Within one first-digit interval the second-digit
-gaps are this quantity divided by $B$; different first digits already
-give a gap.  The interval containing a code recovers its first digit,
-and $v\mapsto Bv-a$ then exposes the tail.  Repetition proves
-injectivity, including endpoint codes.
-
-Writing $a$ to $b$ changes the first right digit.  A right move pushes
-$b$ onto the left string and removes $a$ from the right; a left move
-removes $c$ from the left and prepends $c,b$ to the old right tail.
-These identities give \eqref{ba:zero-digit-maps}, including its images
-and reciprocal slopes.  Determinism and digit gaps separate sources by
-state, read digit, and $c$ when needed.  At one target state the common
-incoming direction determines where the written digit is visible: in
-$I_b$ on the left or right, or in $I_{c,b}$.  The incoming-rule
-hypothesis and digit gaps then separate the closed target rectangles.
-\end{proof}
-
-\begin{lemma}[Evacuation, storage and delivery]\label{ba:parking}
-Suppose $A_i:D_i\to E_i$, $1\le i\le N$, are the rational reciprocal
-rectangle maps of Lemma~\ref{ba:four-stage}, with $N\ge1$ and every
-$D_i\subset\{x_1<0\}$.  There is an effective smooth divergence-free
-field on $\mathbb R\times\mathbb R^3$, one-periodic in time, zero near
-each integer time, and uniformly compactly supported in space, whose
-one-period map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of every
-$D_i\times\{0\}$.  If $E_i\subset\{x_1<0\}$, the entire
-one-period path of each point in $D_i\times\{0\}$ remains in that
-half-space.
-\end{lemma}
-
-\begin{proof}\label{ba:parking-proof}
-Let $d_i,b_i$ be the source and target centers, and let $a_{ij},a'_{ij}$
-be their half-widths.  Put $a^*_{ij}=\max(a_{ij},a'_{ij})$.
-Choose $\epsilon$ to be one quarter of the minimum of $1$ and all
-pairwise maximum-norm distances within the source and target families;
-omit pair entries when $N=1$.  Define
-\[
- \begin{gathered}
- x_{\min}=\min\{x_1:(x_1,x_2)\in D_i\cup E_i\text{ for some }i\},\\
- R=\max(1,\max_i a^*_{i1}),\qquad J=4(R+1),\qquad
- s_i=(x_{\min}-iJ,0).
- \end{gathered}
-\]
-The storage rectangle at $s_i$ with half-widths $a^*_{ij}$ lies in
-$x_1<0$.  Its first-coordinate projection is separated by more than one
-from every source and target and from the other storage projections:
-the relevant gaps are at least $J-R>1$ and $J-2R>1$.
-
-We need only a localized translation and a localized reciprocal scaling.
-On a time segment $[t_0,t_0+\delta]$, use
-$g(t)=\vartheta(3(t-t_0)/\delta-1)$, whose derivative is supported
-in the middle third.  Using the cutoff $\kappa_\epsilon$ defined in
-the proof of Theorem~\ref{ba:torus-plates}, for half-widths $a,b$ write
-\[
- \chi_{a,b}(y)=\kappa_\epsilon(a,y_1)
-               \kappa_\epsilon(b,y_2)\kappa_\epsilon(0,y_3).
-\]
-A plate translation from center $P$ to $Q$ is produced by
-\[
- C(t)=(1-g)P+gQ,\qquad
- \nabla\times\left[
- \tfrac12\chi_{a,b}(x-C(t))\dot C(t)\times(x-C(t))\right].
-\]
-On its plateau the velocity is $\dot C$ and the exact path is
-$C(t)+(r,s,0)$.  At storage center $P=(s_i,0)$, take
-$q(t)=1+(\lambda_i-1)g(t)$ and the field
-\[
- \nabla\times\left(0,0,
- \frac{\dot q}{q}(x_1-P_1)(x_2-P_2)
-                           \chi_{a^*_{i1},a^*_{i2}}(x-P)\right).
-\]
-It has the exact path $P+(qr,q^{-1}s,0)$.  The denominator has the
-positive lower bound $\min(1,\lambda_i)$, and the whole path lies in
-the storage rectangle, because each half-width stays between its source
-and target values.
-
-Partition a period into $7N$ equal segments and take $Z=2$.  For each
-$i$ in order, use four segments for
-\[
- (d_i,0)\longrightarrow(d_i,Z)\longrightarrow(s_i,Z)
-                \longrightarrow(s_i,0),\qquad\text{then scale at }s_i.
-\]
-Only after all these evacuations, use three segments for each $i$ to
-deliver its target shape along
-\[
- (s_i,0)\longrightarrow(s_i,Z)\longrightarrow(b_i,Z)
-                                      \longrightarrow(b_i,0).
-\]
-Sum these segment fields and extend periodically.  The middle-third
-time switches give zero neighborhoods of every segment endpoint and
-every integer time.  The result is a smooth finite sum of curls with
-one compact spatial support.
-
-This is the $7N$ schedule of Lemma~\ref{lem:bb-lift-parking}(ii).
-Its separation hypotheses have explicit margins here.  Unpadded
-source-source and target-target gaps are at least $4\epsilon$, so their
-$\epsilon$ enlargements leave gap at least $2\epsilon$.  Storage
-gaps exceed one, leaving more than $1-2\epsilon>0$ after padding.
-At horizontal travel height the enlarged moving plate and every
-enlarged stationary plate have vertical gap $2-2\epsilon>0$.
-Every scaling stays in its reserved storage rectangle.  The lemma
-therefore gives simultaneous noninterference for evacuation and delivery,
-including when a source meets a target.
-
-The localized potentials above have exactly these affine generators on
-their plateaux, so the plateau argument of Lemma~\ref{lem:bb-curl}
-gives endpoint
-$(d_i+(r,s),0)\mapsto(b_i+(\lambda_i r,\lambda_i^{-1}s),0)$,
-with normal displacement unchanged.  Put
-$M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and
-$d_*=\min(2\epsilon,1-2\epsilon,2-2\epsilon)>0$.
-A neighborhood radius smaller than both $\epsilon/(8M)$ and
-$d_*/(4M)$ retains the plateaux and all exclusion margins throughout
-the concatenation.  This applies the shared proofs to the displayed
-potentials and does not identify their fields away from the plateaux.
-
-If $E_i$ is negative, each translation of an individual point joins two
-negative first coordinates, and its linear interpolation remains negative.
-Scaling occurs inside the negative storage rectangle.  The point is
-stationary during all other segments.  This proves the asserted
-whole-period exclusion.
-\end{proof}
-
-\begin{theorem}[A periodic force with no loading interval]
-\label{ba:periodic-parking}
-Fix a positive computable $\nu$.  A deterministic one-tape machine and
-finite input word in cells $0,\ldots,L-1$, with other cells blank and
-initial head zero, effectively determine a general body force on
-$\mathbb R^3$ extending smoothly and one-periodically to all real times.
-It has uniformly compact spatial support and bounded mixed derivatives.
-Its zero-data solution has a unique global smooth velocity in
-\eqref{ba:euclidean-comparison-class}, with constructed pressure zero, and
-\[
- \exists t\ge0:\quad (\Phi_t((-2,0,0)))_1>0
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-The fixed label is already the initial code; periodicity begins at zero.
-\end{theorem}
-
-\begin{proof}\label{ba:periodic-parking-proof}
-Insert a fresh nonhalting start state that preserves the read letter,
-stays put and enters the original start state.  Replace missing
-nonhalting instructions by letter-preserving stay instructions to a halt
-state.  This finite normalization preserves the halting question,
-including an initially halted machine, and guarantees a nonhalting
-initial control and at least one rule.
-
-Apply the six-row delayed-head recorder of Lemma~\ref{ba:six-compiler}
-with initial frontier $1$.  Initially every other history cell is blank
-and every mark is zero.  The main control records $(q,e)$, where $e$
-is the preceding displacement, and the history record is $(q,e,a)$.
-The old head is marked before the right scan and cleared on return,
-immediately before displacement into the next main state.  At checkpoint
-$k$ the frontier is $k+1$ and the head $i$ satisfies $|i|\le k$;
-the next checkpoint uses $2(k+1-i)+3\le4k+5$ local steps.  The
-incoming-direction and written-symbol properties hold on the full
-domain, as required by Lemma~\ref{ba:zero-blank-code}.
-
-Give the full blank symbol digit zero and the other full symbols digits
-$1,\ldots,m-1$.  Order nonhalting local states with the initial state
-first and assign offsets $-2h$, $h=1,2,\ldots$; assign halting main
-states offsets $2h$, $h=1,2,\ldots$.  Let $\eta_{\rm in}$ be the
-initial right-stack value and put $y_*=-\eta_{\rm in}$.  Only finitely
-many full symbols are initially nonblank, so this is a finite rational
-sum.  Every cell left of the initial head is fully blank.  The code
-\[
- (x_\sigma+\xi,y_*+\eta,0)
-\]
-therefore places the initial configuration exactly at $(-2,0,0)$.
-The input changes the common second-coordinate translation, while the
-observed label and zero initial velocity remain fixed.
-
-Every finitely reached tape still has finite full-symbol support, so its
-code is rational and can be decoded by rational digit extraction until
-both tails are zero.  At a checkpoint the leftmost nonempty history site
-has absolute index $1$; its relative position also recovers the head's
-absolute index.  Thus this coding retains the initialized computation.
-
-Translate the rectangles of \eqref{ba:zero-digit-maps} by the indicated
-state offsets and $y_*$.  Lemma~\ref{ba:zero-blank-code} gives their
-full affine action and separate source and target gaps.  Since
-$\kappa<1$, every nonhalting state block is strictly negative in the
-first coordinate and every halting block strictly positive.  Halting
-states have no outgoing rules, so all sources are negative.  Apply
-Lemma~\ref{ba:parking} to obtain its periodic field $v$.  Zero tails
-can place a code on a rectangle boundary, which is included in that
-lemma's full-neighborhood conclusion.
-
-Set
-\[
- f=f_0+\nu f_1,\qquad
- f_0=\partial_tv+(v\cdot\nabla)v,\qquad f_1=-\Delta v.
-\]
-The field and force are smooth, periodic, uniformly compactly supported,
-and have all mixed derivatives bounded.  Since $v$ vanishes near integer
-times, its initial value is zero.  Fixed support and bounded derivatives
-give $v\in CH^2\cap C^1L^2$ and bounded $v,\nabla v$ on every finite
-interval.  Lemma~\ref{lem:b-comparison}(i) therefore supplies the exact
-solution $(v,0)$, uniqueness in
-\eqref{ba:euclidean-comparison-class}, and its global material flow.
-
-At time zero the marked particle is the initial code.  If it is a
-nonhalting code at integer $n$, its digits select a source rectangle and
-the next period applies exactly that local instruction.  Induction
-identifies the integer-time codes through the first halt or forever.
-Finite completion of every recorder scan identifies the original
-machine at the checkpoint integers.  A halt therefore reaches a positive
-code.  If the machine never halts, each used branch has a negative target
-rectangle.  Lemma~\ref{ba:parking} keeps its marked path negative for
-that whole period, ruling out an intermediate false visit.
-
-Finally the force program uses finite normalization, the finite table,
-the rational input shift, a finite rectangle list, rational storage and
-separation data, $7N$ time segments and explicit cutoffs.  All scaling
-denominators have known positive lower bounds.  The zero-collar and flat
-profile estimates following \eqref{eq:p2-step} permit evaluation of the
-repeated formula and every derivative from approximate real arguments
-without exact seam tests.
-Every period executes all branch maps, independently of the marked
-particle's current state; the program never computes the input run.
-\end{proof}
-
-\subsection{Direction-split marking and a bounded observer}
-\label{ba:bounded-box-native}
-
-A stay instruction need not clear and reinstall a persistent mark.
-We separate that case in the next table.  Its geometric realization
-uses a bounded observation box: during horizontal transfers the routing
-height itself prevents a false visit.
-
-\begin{lemma}[Direction-split persistent marking]
-\label{ba:split-mark-compiler}
-In the notation of Lemma~\ref{ba:persistent-compiler}, there is an
-effective partial finite table with a persistent checkpoint mark,
-pointer at $p_n=n+2$, and both full-domain incoming properties of
-Lemma~\ref{ba:rectangles}.  A work step ending at $w'$ takes
-$2(p_n-w')+4+\mathbf1_{\{d_\tau\ne0\}}$ local steps.
-\end{lemma}
-
-\begin{proof}\label{ba:split-mark-compiler-proof}
-Use full symbols $[g,e,l]$, with work letter $g$, mark $e\in\{0,1\}$,
-and $l\in\{E,P\}\sqcup\{J_\tau\}$.  Controls are $A_q$,
-$B_\tau$ for moving instructions, $C_\tau,S_\tau$, and $D_q,R_q$.
-For $\tau=(q,a)$ writing $b_\tau$, the complete table is
-\[
-\begin{array}{c|c|c|c|r}
-\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline
-A_q&[a,1,l],\ d_\tau=0&C_\tau&[b_\tau,1,l]&0\\
-A_q&[a,1,l],\ d_\tau\ne0&B_\tau&[b_\tau,0,l]&d_\tau\\
-B_\tau&[g,0,l]&C_\tau&[g,1,l]&0\\
-C_\tau&[g,1,l],\ l\ne P&S_\tau&[g,1,l]&1\\
-S_\tau&[g,0,l],\ l\ne P&S_\tau&[g,0,l]&1\\
-S_\tau&[g,0,P]&D_{q^+_\tau}&[g,0,J_\tau]&1\\
-D_q&[g,0,E]&R_q&[g,0,P]&-1\\
-R_q&[g,0,l],\ l\ne P&R_q&[g,0,l]&-1\\
-R_q&[g,1,l],\ l\ne P&A_q&[g,1,l]&0.
-\end{array}
-\]
-The first two rows use nonhalting $q$.  In those rows and the
-$B_\tau$ row, $l$ is unrestricted, including $P$; no additional
-history guard is imposed.  Initialize in $A_{q_0}$ at zero, with the
-input work tape, a unique mark at zero, pointer at two, and empty
-history elsewhere.
-
-At checkpoint $n$, the work move ends at
-$w'=w_n+d_\tau\le n+1<p_n$.  A stay instruction reaches $C_\tau$
-in one step with the mark retained.  A moving instruction reaches it
-in two steps after transferring the mark.  From $C_\tau$, move right,
-scan the $p_n-w'-1$ intervening nonpointer cells, record at $p_n$,
-and extend the pointer to $p_n+1$.  The return scans $p_n-w'$
-unmarked cells and stops at the persistent mark, entering the correct
-$A_{q^+_\tau}$.  There are five other steps for a stay instruction
-and six for a moving instruction.  This yields the stated count and
-restores the invariant.  Every required guard holds, so the execution
-continues exactly until halting, including possible initial halting.
-
-For an arbitrary allowed tape, each target determines its incoming move.
-It is zero into $A_q$ or $C_\tau$, $d_\tau$ into $B_\tau$, one into
-$S_\tau$ or $D_q$, and minus one into $R_q$.  The record $\tau$ recovers the
-source of the work update.  A fixed $C_\tau$ receives only an $A_q$
-entry if $d_\tau=0$, and only $B_\tau$ entries otherwise; the written
-symbol recovers the old full symbol in either case.  At $S_\tau$,
-the written mark separates entry from scan.  At $D_q$, $J_\tau$
-identifies the recording instruction and its former pointer.  At $R_q$,
-the written pointer separates entry from the loop.  At $A_q$, the
-unique source is $R_q$ and the marked symbol is unchanged.  Undoing
-the indicated shift therefore gives at most one predecessor in every
-case, on the full stated domain.
-\end{proof}
-
-\begin{theorem}[A bounded-box event with a vertical routing guard]
-\label{ba:bounded-box-euclidean}
-For every machine and finite input, and fixed positive computable
-viscosity, there is an effective smooth force on $\mathbb R^3$ with
-uniformly compact spatial support, bounded mixed derivatives and period
-one for $t\ge1$, whose zero-data solution satisfies
-\[
- \exists t\ge0:\ X(t;(2,0,0))\in(0,1)\times(0,1)\times(-1,1)
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-The constructed solution is smooth with pressure zero.  It is unique
-among smooth solutions with $u\in C_tH^2\cap C_t^1L^2$, bounded
-$u$ on each finite interval, and $p\in C_tH^1$.  This comparison class
-requires no bound on the competing velocity's spatial gradient.
-\end{theorem}
-
-\begin{proof}\label{ba:bounded-box-euclidean-proof}
-Put $\rho(s)=\sigma(2s-1/2)$, using the effective flat switch
-from~\eqref{eq:p2-step}.
-Use the single-halt normalization and
-Lemma~\ref{ba:split-mark-compiler}.  For its full alphabet $\Sigma$,
-choose distinct odd digits $e(a)\in\{1,3,\ldots,2|\Sigma|-1\}$ and
-base $B=2|\Sigma|+1$.  For an infinite word
-$\omega=\omega_0\omega_1\cdots$, define
-\[
- c(\omega)=\sum_{j\ge0}e(\omega_j)B^{-j-1}.
-\]
-Place $A_h$ at offset zero and the
-other control squares at first-coordinate offsets $3,6,9,\ldots$.
-The code at height zero is
-\[
- (o_s+c(\text{left stream}),\ c(\text{right stream}),0).
-\]
-Every tail value lies in
-$[1/(B-1),(B-2)/(B-1)]\subset(0,1)$, so a halt code lies strictly
-inside the observed box.  The two incoming properties give the
-separated filled rectangles of Lemma~\ref{ba:rectangles}, with
-minimal left subdivision and first factors $1,B^{-1},B$ for stay,
-right and left moves.
-
-Apply the linear version of Lemma~\ref{ba:two-interpolations} to
-the full cuboids with third interval $[-1,1]$, heights $6i$, and
-switches $\rho(3s),\rho(3s-1),\rho(3s-2)$.  Source projections
-separate lifts, heights separate horizontal motion, and target
-projections separate descents.  Equation~\eqref{ba:affine-box-field}
-gives the exact affine flow on every swept cuboid and a neighborhood.
-
-The initial code $x^{\rm in}$ is rational.  For example, if
-$b=\max(3,\text{input length})$, the first $b$ right-stream symbols
-include the entire input and the pointer at cell two, after which the
-stream is constant; the left stream is constant.  On the loading
-interval use
-\[
- C_*(t)=(2,0,0)+\rho(t)(x^{\rm in}-(2,0,0))
-\]
-and
-\[
- V_*(t,x)=\nabla\times\left\{
- \chi(C_*(t),(1,1,1),1;x)
- \tfrac12 C_*'(t)\times(x-C_*(t))\right\}.
-\]
-The cutoff is the centered one defined in
-Lemma~\ref{ba:two-interpolations}.  Its plateau velocity is $C_*'$, so
-the fixed particle follows the displayed path.  This field vanishes
-outside $[1/4,3/4]$.  Extend the step field $V$ by zero and set
-$U=V_*+\sum_{k\ge1}V(t-k)$, with its residual force at the prescribed
-$\nu$.  The finite union of swept cuboids and the loading tube is
-bounded; all time joins have zero collars.  Consequently $U$ and its
-force are smooth, uniformly compactly supported, and have bounded
-mixed derivatives, with period one after loading.  Effective evaluation
-uses the finite formulas and a finite superset of potentially active
-integer shifts at any approximate time, as in
-Lemma~\ref{ba:realization}.
-
-The precise uniqueness input is Lemma~\ref{lem:b-comparison}(i).
-On each finite interval the constructed $U$ is smooth, has one compact
-spatial support, and has bounded velocity and gradient.  Consequently
-$U\in CH^2\cap C^1L^2$, $U(0)=0$, and the force is exactly
-$\mathcal F_\nu[U]$.  The theorem's competitors are smooth classical
-$v\in CH^2\cap C^1L^2$ with bounded $v$ and a representative
-$p\in CH^1$, precisely case (i).  No bound on $\nabla v$ is added.
-That case gives $v=U$ and fixes the $H^1$ pressure representative to
-zero.  The lemma also supplies the unique global material trajectories
-of the reference field.
-
-Loading and the exact full-box maps put the particle at the code after
-$k$ local steps at time $1+k$, through the first halt or forever in a
-nonhalting execution.  A halt code, including an initially halted code
-reached by loading, is inside the observation box.  In a nonhalting
-run, loading has first coordinate at least two.  During a later lift,
-the particle keeps its nonhalting source planar code, whose first
-coordinate exceeds three.  During descent it keeps the nonhalting
-target planar code with the same property.  Throughout the middle
-third its height is $6i>1$.  These three exclusions cover every real
-time and prove the event equivalence independently of horizontal
-excursions in the middle stage.  The repeated processor installs every
-local branch from the finite table; it does not depend on the executed
-run or its outcome.
-\end{proof}
-
-
-\subsection{Separating vertical and planar potentials}
-
-\begin{theorem}[A decaying processor with phasewise planar potentials]
-\label{ba:phasewise-flow}
-At fixed computable $\nu>0$ there is an effective force on $\mathbb R^3$
-with one compact spatial support and all mixed derivatives of time order
-$k$ bounded by $C_{k,\alpha}(1+t)^{-1-k}$, whose unique smooth
-zero-data solution satisfies
-\[
- \exists t\ge0:\ (\Phi_t(0))_1<-1/2
- \quad\Longleftrightarrow\quad\text{halting}.
-\]
-It uses the six-rule recorder with initial frontier 1, full subdivision
-of the rectangles, and separate vertical-translation and planar
-Hamiltonian potentials.  The constructed pressure is zero and uniqueness
-is in Lemma~\ref{ba:realization}'s whole-space class.
-\end{theorem}
-\begin{proof}\label{ba:phasewise-flow-proof}
-Let $\Gamma$ be the full alphabet of the six-rule table with $r_0=1$.
-Use odd digits in base $B=2|\Gamma|+1$ and its fully subdivided maps.  Put halt lanes at $-2,-4,\ldots$
-and the other lanes at $2,4,\ldots$, each of width one.  The second
-coordinate lies in $[0,1]$.  For the resulting rectangles $P_i,Q_i$,
-centers $p_i,q_i$, and first scale $\lambda_i$, choose rational
-$0<\varepsilon\le1/4$ smaller than one quarter of all within-source
-and within-target maximum-norm distances.  Empty pair lists impose no
-condition.  Thicken by $[-1,1]$ and put $H_i=4i$.
-
-On each third of the cycle use a flat progress function $\vartheta$
-from zero to one.  Lifting uses $z_i=H_i\vartheta$ and potential
-$A_i=(0,z_i'X_1,0)$, whose curl is $(0,0,z_i')$.
-At height $H_i$, set $c_i=(1-\vartheta)p_i+\vartheta q_i$ and
-$\mu_i=1-\vartheta+\vartheta\lambda_i$.  Use the potential
-$A_i=(0,0,\psi_i)$, where
-\[
- \psi_i=c_{i1}'(X_2-c_{i2})-c_{i2}'(X_1-c_{i1})
-       +\frac{\mu_i'}{\mu_i}(X_1-c_{i1})(X_2-c_{i2}).
-\]
-Its curl is
-$(c_{i1}'+(\mu_i'/\mu_i)(X_1-c_{i1}),
- c_{i2}'-(\mu_i'/\mu_i)(X_2-c_{i2}),0)$.
-Thus it realizes reciprocal scaling about the interpolated center.
-Descending uses the first potential with $z_i=H_i(1-\vartheta)$.
-Multiply each potential by its current-box $\varepsilon$ cutoff
-before taking the curl.  Source gaps, distinct middle heights, and
-target gaps separate the three stages.  On the whole boxes and a
-neighborhood the uncut formulas are exact.  The first coordinate is
-constant during vertical motion and is the convex interpolation of its
-endpoints during the planar stage.
-
-For loading put $\theta_0(s)=\sigma(2s-1/2)$, which is zero near zero
-and one near one. Let $P_*$ be the rational initial code. Load it in the
-first unit interval by the planar path
-$m(s)=\theta_0(s)((P_*)_1,(P_*)_2)$, localized around
-$(m(s),0)+[-1,1]^3$, using potential
-$(0,0,m_1'(X_2-m_2)-m_2'(X_1-m_1))$.
-Repeat the three-phase instruction field afterward to obtain $W$.
-Code induction and the delayed-head invariant prove the correspondence
-at times $1+n$.  Nonhalting codes have first coordinate at least 2,
-so loading and the convex-path bound exclude $X_1<-1/2$ for every real
-time.  A halt code has first coordinate at most $-1$, including one
-reached by loading an initially halted machine.  Apply
-Lemma~\ref{ba:log-clock}; physical code times are $e^{1+n}-1$.
-Its support, derivative, event, and uniqueness conclusions give precisely
-the claimed separate force choice.
-\end{proof}
+\section{Routing choices dictated by the observer}
+\label{ba:ordered-routes}
+
+The preceding processors move every branch by a prescribed affine map.
+We now ask which parts of that motion must be controlled between the
+integer times.  A bounded observation box allows horizontal work above
+the observer.  A slab observer requires a bound on the horizontal
+coordinate throughout the motion.  Finally, a half-space observer can be
+protected by moving all sources to storage before delivering any target.
+These three requirements lead to the constructions below.
+
+\subsection{Four motions above a bounded observer}
+\label{ba:four-stage-section}
+
+We first separate ascent, scaling, translation and descent.  The two
+planar motions occur at a fixed height, so their exclusion from the
+bounded observer follows without a horizontal coordinate estimate.
+
+Let $D_i,E_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational
+axis-aligned rectangles with positive side lengths.  Assume the $D_i$
+are pairwise disjoint and the $E_i$ are pairwise disjoint, and let
+\[
+ A_i(b)=c_i'+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(b-c_i),
+ \qquad \lambda_i\in\mathbb Q_{>0},\qquad A_i(D_i)=E_i,
+\]
+where $c_i,c_i'$ are their centers.  No separation between a source and
+a target is assumed.  All distances used to choose coordinate padding
+below are distances for the maximum norm.
+
+Let $\vartheta=\sigma$ be the effective flat switch in
+\eqref{eq:p2-step}.  For a closed interval, possibly reduced to one
+point, define
+\[
+ \chi_{[a,b],\epsilon}(v)=
+ \vartheta\!\left(\frac{2(v-a+\epsilon)}\epsilon\right)
+ \vartheta\!\left(\frac{2(b+\epsilon-v)}\epsilon\right).
+\]
+For a box $K$, let $\chi_{K,\epsilon}$ be the product over its three
+coordinates.  It is one on the $\epsilon/2$ enlargement of $K$ and
+zero outside its $\epsilon$ enlargement.  The effective profile estimates
+following \eqref{eq:p2-step} apply to these products and rescalings.
+
+\begin{lemma}[Four-stage realization]\label{ba:four-stage}
+The rectangle data determine effectively a smooth divergence-free field
+$W(s,x)$ supported in a compact subset of $(0,1)\times\mathbb R^3$.
+Its time-one map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of
+each $D_i\times\{0\}$.  For a point of that plate, every planar
+motion occurs at height $4i$, while ascent and descent preserve its
+source and target planar coordinates, respectively.  The scaling can
+use either linear interpolation of its positive first factor or linear
+interpolation of the logarithm of that factor.
+\end{lemma}
+
+\begin{proof}\label{ba:four-stage-proof}
+The empty family is realized by zero.  Otherwise put $h_i=4i$,
+$w_i=c_i'-c_i$, and
+\[
+ \Theta_k(s)=\vartheta(10s-(2k-1)),\qquad 1\le k\le4.
+\]
+Only the $k$th motion is active on the interval
+$[(2k-1)/10,2k/10]$; the four intervals are disjoint.  If
+$a_{ij},a'_{ij}$ are the source and target half-widths, respectively,
+take the four swept boxes
+\[
+ \begin{aligned}
+ K_{i1}&=D_i\times[0,h_i],\\
+ K_{i2}&=\{c_i+(u_1,u_2):|u_j|\le\max(a_{ij},a'_{ij})\}
+                                      \times\{h_i\},\\
+ K_{i3}&=\operatorname{box}(E_i-w_i,E_i)\times\{h_i\},\\
+ K_{i4}&=E_i\times[0,h_i].
+ \end{aligned}
+\]
+Here $\operatorname{box}$ denotes the smallest axis-aligned rectangle
+containing the two displayed rectangles.  Within each of the four
+families these boxes are disjoint: source separation handles the first,
+target separation the fourth, and distinct heights the other two.
+Choose a positive rational $\epsilon<1$ smaller than one quarter of
+every pairwise distance within these families.  A family with fewer than
+two members contributes no restriction.  Its $\epsilon$ enlargements
+are then still disjoint.
+
+Choose either
+\[
+ \mu_i=1+(\lambda_i-1)\Theta_2
+ \quad\hbox{or}\quad \mu_i=\lambda_i^{\Theta_2},
+ \qquad \gamma_i=\dot\mu_i/\mu_i.
+\]
+The factors $\mu_i$ and $\mu_i^{-1}$ stay between their respective
+endpoint values.  Use the coordinate potentials
+\[
+ \begin{aligned}
+ B_{i1}&=(0,h_i x_1,0),&
+ B_{i2}&=(0,0,(x_1-c_{i1})(x_2-c_{i2})),\\
+ B_{i3}&=(0,0,w_{i1}x_2-w_{i2}x_1),&
+ B_{i4}&=(0,-h_i x_1,0).
+ \end{aligned}
+\]
+For the logarithmic-scale construction we also retain the alternative
+translation potentials, with $e_3=(0,0,1)$,
+\[
+ B_{i1}=\tfrac12(h_ie_3)\times x,\qquad
+ B_{i3}=\tfrac12(w_{i1},w_{i2},0)\times x,\qquad
+ B_{i4}=\tfrac12(-h_ie_3)\times x.
+\]
+Their curls equal the same constant translation vectors.  Their localized
+fields need not agree away from the plates.  Either choice gives
+\[
+ W=\sum_i\left[
+ \dot\Theta_1\nabla\times(\chi_{K_{i1},\epsilon}B_{i1})
+ +\gamma_i\nabla\times(\chi_{K_{i2},\epsilon}B_{i2})
+ +\dot\Theta_3\nabla\times(\chi_{K_{i3},\epsilon}B_{i3})
+ +\dot\Theta_4\nabla\times(\chi_{K_{i4},\epsilon}B_{i4})\right].
+\]
+This is a finite sum of spatial curls.  Its spatial support is compact,
+its time support is contained in $[1/10,8/10]$, and the lower bound
+$\mu_i\ge\min(1,\lambda_i)>0$ proves smoothness and effectivity.
+
+For $b\in D_i$, the successive paths are
+\[
+ \begin{array}{c|c}
+ \text{motion}&\text{position}\\\hline
+ \text{ascent}&(b,h_i\Theta_1)\\
+ \text{scaling}&(c_i+\operatorname{diag}(\mu_i,\mu_i^{-1})(b-c_i),h_i)\\
+ \text{translation}&(A_i(b)-w_i+\Theta_3w_i,h_i)\\
+ \text{descent}&(A_i(b),h_i(1-\Theta_4)).
+ \end{array}
+\]
+Each path stays in its swept box.  On a neighborhood of that box the
+uncut curls are, in order,
+\[
+ h_ie_3,\qquad (x_1-c_{i1},-(x_2-c_{i2}),0),\qquad
+ (w_{i1},w_{i2},0),\qquad -h_ie_3.
+\]
+After multiplication by the time coefficients, these are the prescribed
+affine velocities.  The motion matrices are positive diagonal of determinant
+one, the plateau margin is $\epsilon/2$, and the support padding is
+$\epsilon$.  The disjoint swept boxes therefore permit the plateau
+argument of Lemma~\ref{lem:bb-curl}, with $\eta=\epsilon/2$.
+It identifies these particular localized fields with the displayed
+paths on the plateaux.  The normalized diagonal factors are bounded by
+$L_i=\max(1,\lambda_i,\lambda_i^{-1})$, so that lemma gives the
+full initial neighborhood of radius $\epsilon/(4L_i)$.
+\end{proof}
+
+We use the following precise Euclidean comparison class in the applications:
+on each $[0,T]$,
+\begin{equation}\label{ba:euclidean-comparison-class}
+ \begin{gathered}
+ u\in C([0,T];H^2(\mathbb R^3))\cap C^1([0,T];L^2(\mathbb R^3)),\\
+ u,\nabla u\in L^\infty([0,T]\times\mathbb R^3),\qquad
+ p\in C([0,T];H^1(\mathbb R^3)).
+ \end{gathered}
+\end{equation}
+The pressure representative is part of this condition; it is zero for
+our constructed solution.  These comparison velocities and pressures
+satisfy case (i) of Lemma~\ref{lem:b-comparison}.  We retain the stated
+bounded-gradient requirement here, although that case does not need it.
+
+That lemma also gives a smooth material diffeomorphism on every finite
+interval.  Differentiating its trajectory equation gives
+$\dot J=(D_xu)(t,\Phi_t(a))J$, $J(0)=I$, for $J=D_a\Phi_t$.
+Consequently
+\begin{equation}\label{ba:material-volume}
+ \frac d{dt}\det J=(\nabla\cdot u)(t,\Phi_t(a))\det J=0,
+ \qquad \det J=1.
+\end{equation}
+The same proof applies to periodic lifts on the torus.  The residual
+formulas below are affine in viscosity, so they also define the asserted
+solutions for any real $\nu>0$, with effectivity relative to $\nu$.
+
+\begin{theorem}[Two bounded-box material tests]\label{ba:bounded-box-tests}
+Fix a positive computable viscosity $\nu$.  A deterministic one-tape
+machine and finite input effectively determine each of the following
+general body forces on $[0,\infty)\times\mathbb R^3$.  Both have zero
+initial velocity, a unique global smooth velocity in
+\eqref{ba:euclidean-comparison-class}, and constructed pressure zero.
+\begin{enumerate}
+\item The force and velocity have one fixed compact spatial support, all
+mixed derivatives bounded, and period one for $t\ge1$.  The particle
+from $0$ enters $(-5,-2)\times(-1,2)\times(-1,1)$ exactly when the
+machine halts.
+\item The force and velocity have one fixed compact spatial support and,
+for all $k\ge0$ and spatial multi-indices $\alpha$, satisfy
+\begin{equation}\label{ba:eighteen-decay}
+ \|\partial_t^k\partial_x^\alpha u(t)\|_\infty+
+ \|\partial_t^k\partial_x^\alpha f(t)\|_\infty
+ \le C_{k,\alpha}(1+t)^{-1-k}.
+\end{equation}
+The particle from $(2,0,0)$ enters
+$(-1/2,3/2)\times(-1/2,3/2)\times(-1/2,1/2)$ exactly when the
+machine halts.  In particular $f\in L^2([0,\infty);H^m(\mathbb R^3))$
+for every integer $m\ge0$.
+\end{enumerate}
+The supports and constants may depend on the machine and input.  The
+labels and observers do not.  All force derivatives admit evaluation to
+any prescribed rational error by a finite program.
+\end{theorem}
+
+\begin{proof}\label{ba:bounded-box-tests-proof}
+Normalize to a single halt state $q_H$ and a complete nonhalting table,
+as in Section~\ref{ba:codes}.  Apply the move-first recorder of
+Lemma~\ref{lem:bb-movefirst}.  Its history frontier starts at cell $2$;
+all other history cells are empty and all mark bits are zero.  At
+checkpoint $n$, the frontier is $g_n=n+2$, the work head satisfies
+$|h_n|\le n$, and the next work instruction ending at $h'$ is completed
+in exactly $4+2(g_n-h')$ local steps.  Thus every required checkpoint
+is reached after finitely many steps.  The incoming-direction and
+written-symbol inverse properties hold on the whole partial table.
+The records recover the successive work instructions and hence the
+absolute head position, even though the geometric code is head-relative.
+
+Let $m$ be the full alphabet size.  Use odd digits $\beta(a)$ in
+$\{1,3,\ldots,2m-1\}$, with base $B=2m+1$ in the first construction
+and $B=2m+2$ in the second.  Set
+\[
+ C(a_0a_1\cdots)=\sum_{r\ge0}\beta(a_r)B^{-r-1},\quad
+ P_a(v)=\frac{\beta(a)+v}{B},\quad D_a(v)=Bv-\beta(a),\quad
+ I_a=P_a([0,1]).
+\]
+Let $\xi$ encode the left string, nearest cell first, and $\eta$
+the string starting at the head.  The complete right, stay and left
+branch maps are \eqref{eq:bb-one-cell-branches}, with its digit
+$d(a)$ equal to $\beta(a)$ and its move $e$ equal to $d$ here.
+The left move is split over every full symbol $c$.  The shared lemma
+proves the full-rectangle endpoint formulas, arbitrary-tail action and
+linear part $\operatorname{diag}(B^{-d},B^d)$.
+
+Translate each state square horizontally.  In the first construction
+give the halt state offset $-4$ and the other states offsets $2+2r$,
+$r\ge0$; in the second use halt offset $0$ and other offsets $3r$,
+$r\ge1$.  Thus the code is $(o_s+\xi,\eta,0)$.
+Lemma~\ref{lem:bb-onecell} applies with exactly these odd digits and
+either displayed base: its incoming conditions are those supplied by
+the move-first recorder, and \eqref{eq:bb-one-cell-branches} is its minimal
+left-split branch list.  Thus both complete closed rectangle families
+are separately disjoint, with within-state gaps at least $B^{-2}$.
+This proves the hypotheses of Lemma~\ref{ba:four-stage}; no separation
+between the source and target families is required.
+
+Use its linear scale and coordinate potentials for the first construction;
+use its logarithmic scale and symmetric translation potentials for the
+second.  In the latter case the scaling coefficient is exactly
+$(\log\lambda_i)\dot\Theta_2$ and the scale matrix is
+$\operatorname{diag}(\lambda_i^{\Theta_2},\lambda_i^{-\Theta_2})$.
+Denote the resulting step field by $W_{\rm step}$.
+
+The initial code $P_{\rm in}$ is rational.  A constant full-symbol
+tail of digit $\beta_0$ after $M$ positions contributes
+$B^{-M}\beta_0/(B-1)$; all other contributions are finite.  For the
+appropriate fixed label $a_*$, put $v_0=P_{\rm in}-a_*$ and let
+$J_0$ be the smallest box containing $[a_*,P_{\rm in}]$.  The loader
+\[
+ W_{\rm in}(s,x)=\frac d{ds}\vartheta(2s-1/2)\,
+ \nabla\times\left(\chi_{J_0,1}(x)\frac{v_0\times x}{2}\right)
+\]
+has the exact marked path $a_*+\vartheta(2s-1/2)v_0$, since the
+uncut curl is $v_0$ and the entire segment lies in the plateau.  Form
+\[
+ W(s,x)=W_{\rm in}(s,x)+\sum_{n\ge1}W_{\rm step}(s-n,x).
+\]
+The zero time collars make this smooth, initially zero, and periodic for
+$s\ge1$, with common compact spatial support and bounded mixed derivatives.
+
+At time $s=1+k$ the marked particle is exactly the code after $k$ local
+instructions, through the first halt or indefinitely in a nonhalting run.
+This follows by induction from the full-rectangle maps.  Each halt code
+lies in its stated observation box, including one reached by loading an
+initially halted machine.  In a nonhalting run, the first loader has
+$x_1\ge0$ and later nonhalt codes have $x_1\ge2$; for the second these
+bounds are $x_1\ge2$ and $x_1\ge3$.  Ascent and descent leave those
+planar coordinates unchanged.  Every planar motion has height $4i\ge4$,
+outside both observers.  The intervening stationary intervals add no new
+positions.  This proves exclusion at every real time.
+
+For the first case take $u=W$ and its residual force.  For the second set
+\[
+ s(t)=\log(1+t),\quad \theta(t)=(1+t)^{-1},\quad
+ u(t,x)=\theta(t)W(s(t),x),
+\]
+whose force is
+\[
+ f=\theta^2\{-W+\partial_sW+(W\cdot\nabla)W\}(s(t),x)
+                         -\nu\theta\Delta W(s(t),x).
+\]
+Lemma~\ref{ba:log-clock} applies to this fixed-support, bounded-derivative
+template.  Its identity
+$\partial_t(\theta^jY(s(t)))=\theta^{j+1}(\partial_s-j)Y(s(t))$
+proves \eqref{ba:eighteen-decay} for every $k,\alpha$.  Fixed support
+then gives the stated $L^2_tH^m_x$ conclusion.  Its clock is onto, with
+inverse $t=e^s-1$, so the material path is the old path composed with
+$s(t)$ and the event is unchanged.
+
+In both cases smoothness, fixed support and bounded derivatives on
+finite time intervals put the reference velocity in $CH^2\cap C^1L^2$
+with bounded velocity and gradient.  It is solenoidal and initially zero.
+Lemma~\ref{lem:b-comparison}(i) therefore gives the exact solution
+$(u,0)$ and uniqueness in \eqref{ba:euclidean-comparison-class}.
+The effective profiles following \eqref{eq:p2-step}, applied to the
+finite table, rational initial code, finitely many boxes and repeated
+template, give all derivative evaluations.  A coarse time approximation
+encloses finitely many possible active translates, avoiding equality
+tests at seams.  The prescription never computes the machine's run.
+\end{proof}
+
+\subsection{Small plates and a slab observer on the unit torus}
+\label{ba:torus-plates-section}
+
+A slab observer has no height restriction, so elevation alone cannot
+protect it.  We now keep each nonhalting branch inside a narrow range of
+first coordinates throughout its motion.  The delayed-head recorder also
+allows arbitrary finitely supported work data on both sides of the head.
+
+\begin{theorem}[Torus plates and square-integrable forcing]
+\label{ba:torus-plates}
+Fix a positive computable $\nu$.  Every deterministic one-tape machine
+with finitely specified tape, blank outside a finite set, effectively
+determines a smooth mean-zero general force on
+$\mathbb T^3=\mathbb R^3/\mathbb Z^3$, with bounded mixed derivatives
+and period one for $t\ge1$.  Its zero-data Navier--Stokes solution is
+globally smooth and unique among smooth solutions, with pressure normalized
+to mean zero and constructed pressure zero.  For
+$P_*=(1/4,1/2,1/4)$ and $X(t)=\Phi_t(P_*)$, the event
+\begin{equation}\label{ba:torus-plate-event}
+ \exists t\ge0:\quad X_1(t)\pmod1\in(1/2,7/8)
+\end{equation}
+is equivalent to halting.
+
+A separate effective force has the same zero data, event, uniqueness and
+mean-zero property, is supported in one compact sub-box of $(0,1)^3$,
+has all mixed derivatives bounded, and satisfies
+\begin{equation}\label{ba:cube-rate}
+ \|\widehat f(t,\cdot)\|_\infty=O((1+t)^{-2/3}).
+\end{equation}
+In particular $\widehat f\in L^2([0,\infty)\times\mathbb T^3)$.
+\end{theorem}
+
+\begin{proof}\label{ba:torus-plates-proof}
+Use Lemma~\ref{ba:six-compiler} with the frontier initially at cell $1$.
+Its main control $M_{q,e}$ records the last work displacement $e$, initially
+zero.  In one work step it writes and marks the old head, records
+$(q,e,a)$ at the frontier, returns to the mark, and only then makes the
+work displacement.  All other initial history cells are blank and every
+mark is zero.  At checkpoint $n$ the frontier is $n+1$, $|h_n|\le n$,
+and the next checkpoint takes exactly $2(n+1-h_n)+3$ local steps.
+These scans test history and marks, so the proof is unaffected by work
+symbols on either side of the head.  Main controls with halting $q$ have
+no outgoing rules.  The full-domain incoming properties are those of
+the cited lemma.
+
+Let $S$ be the local state set, $N=|S|$, let $S_h$ be its halting main
+states, and enumerate $S$ by $n(s)=0,\ldots,N-1$.  Put
+\[
+ \ell=\frac1{16(N+1)},\quad z_0=\frac14,\quad
+ x_s=\frac18+2\ell n(s)+\frac12\mathbf1_{s\in S_h},\quad
+ T_s(\xi,\eta)=(x_s+\ell\xi,1/4+\ell\eta).
+\]
+Nonhalting squares lie in $1/8\le x_1<1/4$ and halting squares in
+$5/8\le x_1<3/4$; distinct squares have gap at least $\ell$.
+Give the full alphabet, of size $m$, odd digits in base $\beta=2m+1$.
+Conjugating \eqref{eq:bb-one-cell-branches} by the source and target charts
+gives separately disjoint rectangle families $R_j^0,R_j^1$ with positive
+diagonal factors $\alpha_j,\alpha_j^{-1}$,
+$\alpha_j\in\{\beta^{-1},1,\beta\}$.  Every chart has the same
+scale, so this remains the physical linear part.
+Lemma~\ref{lem:bb-onecell}, with this uniform chart scaling, gives
+within-family coordinate gaps at least $\ell/\beta^2$, including the
+two-letter target intervals for left moves.
+
+For $1\le j\le L$, write the centers as $p_j^0,p_j^1$, the source
+half-widths as $a_j,b_j$, and the target half-widths as
+$\alpha_ja_j,\alpha_j^{-1}b_j$.  All endpoint half-widths are at
+most $\ell/2$.  Choose
+\[
+ h_j=\frac12+\frac{j}{4(L+1)},\qquad
+ \epsilon=\frac1{100}\min\left(1,\frac\ell{\beta^2},
+                                      \frac1{4(L+1)}\right).
+\]
+With $\theta(s)=\vartheta(2s-1/2)$ and
+$\Theta_k(\tau)=\theta(3\tau-k+1)$ for $k=1,2,3$, define
+\[
+ \begin{aligned}
+ C_j(\tau)&=\big((1-\Theta_2)p_j^0+\Theta_2p_j^1,
+                      z_0+(\Theta_1-\Theta_3)(h_j-z_0)\big),\\
+ \sigma_j&=1+(\alpha_j-1)\Theta_2,\qquad
+ \lambda_j=\dot\sigma_j/\sigma_j,\\
+ (w_{j1},w_{j2},w_{j3})&=(a_j\sigma_j,b_j/\sigma_j,0).
+ \end{aligned}
+\]
+These centers and half-widths describe moving plates.  Their
+$\epsilon$ enlargements are disjoint: the first third uses source
+projections, the middle third distinct heights, and the last third
+target projections.  The padding is at most one hundredth of every
+relevant gap.  All plates and padding lie in a compact sub-box of
+$(0,1)^3$, since their centers interpolate between the stated chart
+centers, widths are at most $\ell$, heights lie between $1/4$ and
+$3/4$, and $\ell\le1/32$, $\epsilon\le1/100$.
+
+For $w\ge0$ put
+\[
+ \kappa_\epsilon(w,v)=
+ \vartheta\!\left(\frac{2(w+\epsilon-v)}\epsilon\right)
+ \vartheta\!\left(\frac{2(w+\epsilon+v)}\epsilon\right),\qquad
+ \chi_j=\prod_{a=1}^3\kappa_\epsilon(w_{ja},x_a-C_{ja}).
+\]
+Writing $y=x-C_j$, use
+\[
+ A_j=\tfrac12\dot C_j\times y+(0,0,\lambda_jy_1y_2),
+ \qquad V(\tau,x)=\sum_j\nabla\times(\chi_jA_j).
+\]
+Extend the potentials by zero outside the chart.  The support remains
+away from its faces, so this gives a smooth torus field, zero near the
+time endpoints.  On the whole plateau of plate $j$ its velocity is
+$\dot C_j+(\lambda_jy_1,-\lambda_jy_2,0)$; other summands vanish
+there.  Hence the path from $(p_j^0+(r,s),z_0)$ is exactly
+\begin{equation}\label{ba:plate-path}
+ C_j(\tau)+(\sigma_j(\tau)r,\sigma_j(\tau)^{-1}s,0),
+ \qquad |r|\le a_j,\quad |s|\le b_j.
+\end{equation}
+It ends at the prescribed affine image.  The plateau argument of
+Lemma~\ref{lem:bb-curl} applies with
+$D_j=\operatorname{diag}(\sigma_j,\sigma_j^{-1},1)$,
+$\eta=\epsilon/2$, and diagonal bound
+$\max(1,\alpha_j,\alpha_j^{-1})$: the required enlarged-box separation
+was just proved.  It gives the same motion on an initial neighborhood
+of radius $\epsilon/[4\max(1,\alpha_j,\alpha_j^{-1})]$, with constant
+normal displacement, and therefore includes boundary codes.
+
+The initial code $P_{\rm in}$ is rational, since relative to constant
+blank tails only finitely many work and frontier digits differ.  Load it
+from $P_*$ along $C_*=(1-\theta)P_*+\theta P_{\rm in}$ using
+\[
+ V_0(t,x)=\nabla\times\left[
+ \prod_{a=1}^3\kappa_{1/64}(0,x_a-C_{*,a}(t))
+                \frac{\dot C_*(t)\times(x-C_*(t))}{2}\right].
+\]
+The segment and its collar stay within the chart: its first coordinate
+stays between $1/8$ and $3/4$, its second between $1/4$ and
+$1/2+\ell$, and its third is $1/4$.  Its plateau curl is $\dot C_*$.
+Take $U=V_0$ on $[0,1]$ and $U(t)=V(t-n)$ on $[n,n+1]$ for $n\ge1$.
+Time collars make this smooth and initially zero.  It has fixed compact
+chart support, bounded mixed derivatives, and period one after loading.
+Define $f=U_t+(U\cdot\nabla)U-\nu\Delta U$ using the stated physical
+viscosity.  Both $U$ and $f$ have mean zero: $U$ is a curl, the
+Laplacian integrates to zero, and
+$(U\cdot\nabla)U=\operatorname{div}(U\otimes U)$.
+The torus clause of Lemma~\ref{lem:b-comparison} applies to this
+smooth, initially zero, solenoidal reference field and gives the unique
+smooth solution $(U,0)$ with mean-zero pressure.
+
+Equations~\eqref{eq:bb-one-cell-branches} and \eqref{ba:plate-path} identify
+the particle at time $1+k$ with the $k$th local configuration.  The
+recorder invariant ensures that a nonhalting computation continues
+indefinitely.  A halt, including an initial halt reached by loading,
+puts the particle in $5/8\le X_1<3/4$, proving the event.  For a
+nonhalting computation the loader has $X_1\le1/4$ and every later
+branch has both endpoint centers below $1/4$.  Its center is their
+convex interpolation and its first half-width is at most $\ell/2$.
+Thus $0<X_1<1/4+\ell/2<1/2$ at every intermediate time.  The path
+stays inside the chart, so reduction modulo one cannot create a false visit.
+
+For the separate decaying force choose the onto clock
+\[
+ s(t)=(1+t)^{1/3}-1,\qquad \widehat U(t,x)=s'(t)U(s(t),x).
+\]
+Its residual is
+\[
+ \widehat f=s''U+(s')^2\{\partial_sU+(U\cdot\nabla)U\}
+                                         -\nu s'\Delta U,
+\]
+where the fields on the right are evaluated at $(s(t),x)$.  Since
+$s'=(1/3)(1+t)^{-2/3}$ and $s''=-(2/9)(1+t)^{-5/3}$, the bounded
+template derivatives prove \eqref{ba:cube-rate}.  Every positive-order
+derivative of $s$ is bounded; repeated product and chain rules therefore
+bound every mixed derivative of $\widehat U$ and $\widehat f$.  The
+square of $(1+t)^{-2/3}$ is integrable.  Support and mean zero persist,
+and $U(0)=0$ retains the zero datum.  The torus clause of
+Lemma~\ref{lem:b-comparison} therefore applies to $\widehat U$ too.
+Finally the material path is the old one at $s(t)$,
+and $s$ has inverse $t=(1+s)^3-1$, so the event is unchanged.
+
+All choices are effective finite data followed by explicit smooth
+formulas.  In particular $\sigma_j\ge\min(1,\alpha_j)>0$; the cutoff
+estimates and zero collars allow derivative evaluation at approximate real
+arguments without tests for exact boundary equality.  Only positive
+arguments enter the cube root.  The program repeats the entire local
+table and never uses a computed execution.
+\end{proof}
+
+\subsection{Storage and periodic forcing without a loader}
+\label{ba:parking-section}
+
+The next construction absorbs initialization into the positions of the
+state rectangles.  A zero digit for the full blank symbol places the
+initial left stack at zero; an input-dependent second-coordinate translation then
+places the entire initial code at one fixed particle.  To protect a
+half-space observer during every period, we first evacuate all sources
+to separate storage rectangles and only afterwards fill the targets.
+
+\begin{lemma}[Zero-blank coding]\label{ba:zero-blank-code}
+Let an alphabet of size $m>1$ have digits $0,\ldots,m-1$, with full
+blank digit zero.  Set $B=m+1$, $\kappa=(m-1)/m$, and
+\[
+ \operatorname{val}(a_0a_1\cdots)=\sum_{r\ge0}a_rB^{-r-1},\qquad
+ I_a=\frac{a+[0,\kappa]}B,\qquad I_{a,b}=\frac{a+I_b}B.
+\]
+The code is injective, takes values in $[0,\kappa]$, and its distinct
+one-letter and two-letter cylinders have positive rational gaps.  For a
+deterministic partial table with target-state-determined incoming direction
+and incoming rule determined by target state and written symbol, separated
+state translates of $[0,\kappa]^2$ yield separately disjoint full source
+and target rectangles by the following maps in state-relative coordinates:
+\begin{equation}\label{ba:zero-digit-maps}
+ \begin{array}{c|c|c|c}
+ d&\text{source}&(\xi',\eta')&\text{target}\\\hline
+ 1&[0,\kappa]\times I_a&((b+\xi)/B,B\eta-a)&I_b\times[0,\kappa]\\
+ 0&[0,\kappa]\times I_a&(\xi,\eta+(b-a)/B)&[0,\kappa]\times I_b\\
+ -1&I_c\times I_a&(B\xi-c,c/B+(b+B\eta-a)/B^2)&
+                                      [0,\kappa]\times I_{c,b}.
+ \end{array}
+\end{equation}
+The last row is split over all $c$.  Each map has positive diagonal
+linear part of determinant one and acts exactly on the full rectangle.
+\end{lemma}
+
+\begin{proof}\label{ba:zero-blank-code-proof}
+The maximum tail is $(m-1)/(B-1)=\kappa$, and
+$(m-1+\kappa)/B=\kappa$.  Consecutive first-digit intervals have
+gap $(1-\kappa)/B>0$.  Within one first-digit interval the second-digit
+gaps are this quantity divided by $B$; different first digits already
+give a gap.  The interval containing a code recovers its first digit,
+and $v\mapsto Bv-a$ then exposes the tail.  Repetition proves
+injectivity, including endpoint codes.
+
+Writing $a$ to $b$ changes the first right digit.  A right move pushes
+$b$ onto the left string and removes $a$ from the right; a left move
+removes $c$ from the left and prepends $c,b$ to the old right tail.
+These identities give \eqref{ba:zero-digit-maps}, including its images
+and reciprocal slopes.  Determinism and digit gaps separate sources by
+state, read digit, and $c$ when needed.  At one target state the common
+incoming direction determines where the written digit is visible: in
+$I_b$ on the left or right, or in $I_{c,b}$.  The incoming-rule
+hypothesis and digit gaps then separate the closed target rectangles.
+\end{proof}
+
+\begin{lemma}[Evacuation, storage and delivery]\label{ba:parking}
+Suppose $A_i:D_i\to E_i$, $1\le i\le N$, are the rational reciprocal
+rectangle maps of Lemma~\ref{ba:four-stage}, with $N\ge1$ and every
+$D_i\subset\{x_1<0\}$.  There is an effective smooth divergence-free
+field on $\mathbb R\times\mathbb R^3$, one-periodic in time, zero near
+each integer time, and uniformly compactly supported in space, whose
+one-period map is $(b,z)\mapsto(A_i(b),z)$ on a neighborhood of every
+$D_i\times\{0\}$.  If $E_i\subset\{x_1<0\}$, the entire
+one-period path of each point in $D_i\times\{0\}$ remains in that
+half-space.
+\end{lemma}
+
+\begin{proof}\label{ba:parking-proof}
+Let $d_i,b_i$ be the source and target centers, and let $a_{ij},a'_{ij}$
+be their half-widths.  Put $a^*_{ij}=\max(a_{ij},a'_{ij})$.
+Choose $\epsilon$ to be one quarter of the minimum of $1$ and all
+pairwise maximum-norm distances within the source and target families;
+omit pair entries when $N=1$.  Define
+\[
+ \begin{gathered}
+ x_{\min}=\min\{x_1:(x_1,x_2)\in D_i\cup E_i\text{ for some }i\},\\
+ R=\max(1,\max_i a^*_{i1}),\qquad J=4(R+1),\qquad
+ s_i=(x_{\min}-iJ,0).
+ \end{gathered}
+\]
+The storage rectangle at $s_i$ with half-widths $a^*_{ij}$ lies in
+$x_1<0$.  Its first-coordinate projection is separated by more than one
+from every source and target and from the other storage projections:
+the relevant gaps are at least $J-R>1$ and $J-2R>1$.
+
+We need only a localized translation and a localized reciprocal scaling.
+On a time segment $[t_0,t_0+\delta]$, use
+$g(t)=\vartheta(3(t-t_0)/\delta-1)$, whose derivative is supported
+in the middle third.  Using the cutoff $\kappa_\epsilon$ defined in
+the proof of Theorem~\ref{ba:torus-plates}, for half-widths $a,b$ write
+\[
+ \chi_{a,b}(y)=\kappa_\epsilon(a,y_1)
+               \kappa_\epsilon(b,y_2)\kappa_\epsilon(0,y_3).
+\]
+A plate translation from center $P$ to $Q$ is produced by
+\[
+ C(t)=(1-g)P+gQ,\qquad
+ \nabla\times\left[
+ \tfrac12\chi_{a,b}(x-C(t))\dot C(t)\times(x-C(t))\right].
+\]
+On its plateau the velocity is $\dot C$ and the exact path is
+$C(t)+(r,s,0)$.  At storage center $P=(s_i,0)$, take
+$q(t)=1+(\lambda_i-1)g(t)$ and the field
+\[
+ \nabla\times\left(0,0,
+ \frac{\dot q}{q}(x_1-P_1)(x_2-P_2)
+                           \chi_{a^*_{i1},a^*_{i2}}(x-P)\right).
+\]
+It has the exact path $P+(qr,q^{-1}s,0)$.  The denominator has the
+positive lower bound $\min(1,\lambda_i)$, and the whole path lies in
+the storage rectangle, because each half-width stays between its source
+and target values.
+
+Partition a period into $7N$ equal segments and take $Z=2$.  For each
+$i$ in order, use four segments for
+\[
+ (d_i,0)\longrightarrow(d_i,Z)\longrightarrow(s_i,Z)
+                \longrightarrow(s_i,0),\qquad\text{then scale at }s_i.
+\]
+Only after all these evacuations, use three segments for each $i$ to
+deliver its target shape along
+\[
+ (s_i,0)\longrightarrow(s_i,Z)\longrightarrow(b_i,Z)
+                                      \longrightarrow(b_i,0).
+\]
+Sum these segment fields and extend periodically.  The middle-third
+time switches give zero neighborhoods of every segment endpoint and
+every integer time.  The result is a smooth finite sum of curls with
+one compact spatial support.
+
+This is the $7N$ schedule of Lemma~\ref{lem:bb-lift-parking}(ii).
+Its separation hypotheses have explicit margins here.  Unpadded
+source-source and target-target gaps are at least $4\epsilon$, so their
+$\epsilon$ enlargements leave gap at least $2\epsilon$.  Storage
+gaps exceed one, leaving more than $1-2\epsilon>0$ after padding.
+At horizontal travel height the enlarged moving plate and every
+enlarged stationary plate have vertical gap $2-2\epsilon>0$.
+Every scaling stays in its reserved storage rectangle.  The lemma
+therefore gives simultaneous noninterference for evacuation and delivery,
+including when a source meets a target.
+
+The localized potentials above have exactly these affine generators on
+their plateaux, so the plateau argument of Lemma~\ref{lem:bb-curl}
+gives endpoint
+$(d_i+(r,s),0)\mapsto(b_i+(\lambda_i r,\lambda_i^{-1}s),0)$,
+with normal displacement unchanged.  Put
+$M=\max_i(1,\lambda_i,\lambda_i^{-1})$ and
+$d_*=\min(2\epsilon,1-2\epsilon,2-2\epsilon)>0$.
+A neighborhood radius smaller than both $\epsilon/(8M)$ and
+$d_*/(4M)$ retains the plateaux and all exclusion margins throughout
+the concatenation.  This applies the shared proofs to the displayed
+potentials and does not identify their fields away from the plateaux.
+
+If $E_i$ is negative, each translation of an individual point joins two
+negative first coordinates, and its linear interpolation remains negative.
+Scaling occurs inside the negative storage rectangle.  The point is
+stationary during all other segments.  This proves the asserted
+whole-period exclusion.
+\end{proof}
+
+\begin{theorem}[A periodic force with no loading interval]
+\label{ba:periodic-parking}
+Fix a positive computable $\nu$.  A deterministic one-tape machine and
+finite input word in cells $0,\ldots,L-1$, with other cells blank and
+initial head zero, effectively determine a general body force on
+$\mathbb R^3$ extending smoothly and one-periodically to all real times.
+It has uniformly compact spatial support and bounded mixed derivatives.
+Its zero-data solution has a unique global smooth velocity in
+\eqref{ba:euclidean-comparison-class}, with constructed pressure zero, and
+\[
+ \exists t\ge0:\quad (\Phi_t((-2,0,0)))_1>0
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+The fixed label is already the initial code; periodicity begins at zero.
+\end{theorem}
+
+\begin{proof}\label{ba:periodic-parking-proof}
+Insert a fresh nonhalting start state that preserves the read letter,
+stays put and enters the original start state.  Replace missing
+nonhalting instructions by letter-preserving stay instructions to a halt
+state.  This finite normalization preserves the halting question,
+including an initially halted machine, and guarantees a nonhalting
+initial control and at least one rule.
+
+Apply the six-row delayed-head recorder of Lemma~\ref{ba:six-compiler}
+with initial frontier $1$.  Initially every other history cell is blank
+and every mark is zero.  The main control records $(q,e)$, where $e$
+is the preceding displacement, and the history record is $(q,e,a)$.
+The old head is marked before the right scan and cleared on return,
+immediately before displacement into the next main state.  At checkpoint
+$k$ the frontier is $k+1$ and the head $i$ satisfies $|i|\le k$;
+the next checkpoint uses $2(k+1-i)+3\le4k+5$ local steps.  The
+incoming-direction and written-symbol properties hold on the full
+domain, as required by Lemma~\ref{ba:zero-blank-code}.
+
+Give the full blank symbol digit zero and the other full symbols digits
+$1,\ldots,m-1$.  Order nonhalting local states with the initial state
+first and assign offsets $-2h$, $h=1,2,\ldots$; assign halting main
+states offsets $2h$, $h=1,2,\ldots$.  Let $\eta_{\rm in}$ be the
+initial right-stack value and put $y_*=-\eta_{\rm in}$.  Only finitely
+many full symbols are initially nonblank, so this is a finite rational
+sum.  Every cell left of the initial head is fully blank.  The code
+\[
+ (x_\sigma+\xi,y_*+\eta,0)
+\]
+therefore places the initial configuration exactly at $(-2,0,0)$.
+The input changes the common second-coordinate translation, while the
+observed label and zero initial velocity remain fixed.
+
+Every finitely reached tape still has finite full-symbol support, so its
+code is rational and can be decoded by rational digit extraction until
+both tails are zero.  At a checkpoint the leftmost nonempty history site
+has absolute index $1$; its relative position also recovers the head's
+absolute index.  Thus this coding retains the initialized computation.
+
+Translate the rectangles of \eqref{ba:zero-digit-maps} by the indicated
+state offsets and $y_*$.  Lemma~\ref{ba:zero-blank-code} gives their
+full affine action and separate source and target gaps.  Since
+$\kappa<1$, every nonhalting state block is strictly negative in the
+first coordinate and every halting block strictly positive.  Halting
+states have no outgoing rules, so all sources are negative.  Apply
+Lemma~\ref{ba:parking} to obtain its periodic field $v$.  Zero tails
+can place a code on a rectangle boundary, which is included in that
+lemma's full-neighborhood conclusion.
+
+Set
+\[
+ f=f_0+\nu f_1,\qquad
+ f_0=\partial_tv+(v\cdot\nabla)v,\qquad f_1=-\Delta v.
+\]
+The field and force are smooth, periodic, uniformly compactly supported,
+and have all mixed derivatives bounded.  Since $v$ vanishes near integer
+times, its initial value is zero.  Fixed support and bounded derivatives
+give $v\in CH^2\cap C^1L^2$ and bounded $v,\nabla v$ on every finite
+interval.  Lemma~\ref{lem:b-comparison}(i) therefore supplies the exact
+solution $(v,0)$, uniqueness in
+\eqref{ba:euclidean-comparison-class}, and its global material flow.
+
+At time zero the marked particle is the initial code.  If it is a
+nonhalting code at integer $n$, its digits select a source rectangle and
+the next period applies exactly that local instruction.  Induction
+identifies the integer-time codes through the first halt or forever.
+Finite completion of every recorder scan identifies the original
+machine at the checkpoint integers.  A halt therefore reaches a positive
+code.  If the machine never halts, each used branch has a negative target
+rectangle.  Lemma~\ref{ba:parking} keeps its marked path negative for
+that whole period, ruling out an intermediate false visit.
+
+Finally the force program uses finite normalization, the finite table,
+the rational input shift, a finite rectangle list, rational storage and
+separation data, $7N$ time segments and explicit cutoffs.  All scaling
+denominators have known positive lower bounds.  The zero-collar and flat
+profile estimates following \eqref{eq:p2-step} permit evaluation of the
+repeated formula and every derivative from approximate real arguments
+without exact seam tests.
+Every period executes all branch maps, independently of the marked
+particle's current state; the program never computes the input run.
+\end{proof}
+
+\subsection{Direction-split marking and a bounded observer}
+\label{ba:bounded-box-native}
+
+A stay instruction need not clear and reinstall a persistent mark.
+We separate that case in the next table.  Its geometric realization
+uses a bounded observation box: during horizontal transfers the routing
+height itself prevents a false visit.
+
+\begin{lemma}[Direction-split persistent marking]
+\label{ba:split-mark-compiler}
+In the notation of Lemma~\ref{ba:persistent-compiler}, there is an
+effective partial finite table with a persistent checkpoint mark,
+pointer at $p_n=n+2$, and both full-domain incoming properties of
+Lemma~\ref{ba:rectangles}.  A work step ending at $w'$ takes
+$2(p_n-w')+4+\mathbf1_{\{d_\tau\ne0\}}$ local steps.
+\end{lemma}
+
+\begin{proof}\label{ba:split-mark-compiler-proof}
+Use full symbols $[g,e,l]$, with work letter $g$, mark $e\in\{0,1\}$,
+and $l\in\{E,P\}\sqcup\{J_\tau\}$.  Controls are $A_q$,
+$B_\tau$ for moving instructions, $C_\tau,S_\tau$, and $D_q,R_q$.
+For $\tau=(q,a)$ writing $b_\tau$, the complete table is
+\[
+\begin{array}{c|c|c|c|r}
+\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline
+A_q&[a,1,l],\ d_\tau=0&C_\tau&[b_\tau,1,l]&0\\
+A_q&[a,1,l],\ d_\tau\ne0&B_\tau&[b_\tau,0,l]&d_\tau\\
+B_\tau&[g,0,l]&C_\tau&[g,1,l]&0\\
+C_\tau&[g,1,l],\ l\ne P&S_\tau&[g,1,l]&1\\
+S_\tau&[g,0,l],\ l\ne P&S_\tau&[g,0,l]&1\\
+S_\tau&[g,0,P]&D_{q^+_\tau}&[g,0,J_\tau]&1\\
+D_q&[g,0,E]&R_q&[g,0,P]&-1\\
+R_q&[g,0,l],\ l\ne P&R_q&[g,0,l]&-1\\
+R_q&[g,1,l],\ l\ne P&A_q&[g,1,l]&0.
+\end{array}
+\]
+The first two rows use nonhalting $q$.  In those rows and the
+$B_\tau$ row, $l$ is unrestricted, including $P$; no additional
+history guard is imposed.  Initialize in $A_{q_0}$ at zero, with the
+input work tape, a unique mark at zero, pointer at two, and empty
+history elsewhere.
+
+At checkpoint $n$, the work move ends at
+$w'=w_n+d_\tau\le n+1<p_n$.  A stay instruction reaches $C_\tau$
+in one step with the mark retained.  A moving instruction reaches it
+in two steps after transferring the mark.  From $C_\tau$, move right,
+scan the $p_n-w'-1$ intervening nonpointer cells, record at $p_n$,
+and extend the pointer to $p_n+1$.  The return scans $p_n-w'$
+unmarked cells and stops at the persistent mark, entering the correct
+$A_{q^+_\tau}$.  There are five other steps for a stay instruction
+and six for a moving instruction.  This yields the stated count and
+restores the invariant.  Every required guard holds, so the execution
+continues exactly until halting, including possible initial halting.
+
+For an arbitrary allowed tape, each target determines its incoming move.
+It is zero into $A_q$ or $C_\tau$, $d_\tau$ into $B_\tau$, one into
+$S_\tau$ or $D_q$, and minus one into $R_q$.  The record $\tau$ recovers the
+source of the work update.  A fixed $C_\tau$ receives only an $A_q$
+entry if $d_\tau=0$, and only $B_\tau$ entries otherwise; the written
+symbol recovers the old full symbol in either case.  At $S_\tau$,
+the written mark separates entry from scan.  At $D_q$, $J_\tau$
+identifies the recording instruction and its former pointer.  At $R_q$,
+the written pointer separates entry from the loop.  At $A_q$, the
+unique source is $R_q$ and the marked symbol is unchanged.  Undoing
+the indicated shift therefore gives at most one predecessor in every
+case, on the full stated domain.
+\end{proof}
+
+\begin{theorem}[A bounded-box event with a vertical routing guard]
+\label{ba:bounded-box-euclidean}
+For every machine and finite input, and fixed positive computable
+viscosity, there is an effective smooth force on $\mathbb R^3$ with
+uniformly compact spatial support, bounded mixed derivatives and period
+one for $t\ge1$, whose zero-data solution satisfies
+\[
+ \exists t\ge0:\ X(t;(2,0,0))\in(0,1)\times(0,1)\times(-1,1)
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+The constructed solution is smooth with pressure zero.  It is unique
+among smooth solutions with $u\in C_tH^2\cap C_t^1L^2$, bounded
+$u$ on each finite interval, and $p\in C_tH^1$.  This comparison class
+requires no bound on the competing velocity's spatial gradient.
+\end{theorem}
+
+\begin{proof}\label{ba:bounded-box-euclidean-proof}
+Put $\rho(s)=\sigma(2s-1/2)$, using the effective flat switch
+from~\eqref{eq:p2-step}.
+Use the single-halt normalization and
+Lemma~\ref{ba:split-mark-compiler}.  For its full alphabet $\Sigma$,
+choose distinct odd digits $e(a)\in\{1,3,\ldots,2|\Sigma|-1\}$ and
+base $B=2|\Sigma|+1$.  For an infinite word
+$\omega=\omega_0\omega_1\cdots$, define
+\[
+ c(\omega)=\sum_{j\ge0}e(\omega_j)B^{-j-1}.
+\]
+Place $A_h$ at offset zero and the
+other control squares at first-coordinate offsets $3,6,9,\ldots$.
+The code at height zero is
+\[
+ (o_s+c(\text{left stream}),\ c(\text{right stream}),0).
+\]
+Every tail value lies in
+$[1/(B-1),(B-2)/(B-1)]\subset(0,1)$, so a halt code lies strictly
+inside the observed box.  The two incoming properties give the
+separated filled rectangles of Lemma~\ref{ba:rectangles}, with
+minimal left subdivision and first factors $1,B^{-1},B$ for stay,
+right and left moves.
+
+Apply the linear version of Lemma~\ref{ba:two-interpolations} to
+the full cuboids with third interval $[-1,1]$, heights $6i$, and
+switches $\rho(3s),\rho(3s-1),\rho(3s-2)$.  Source projections
+separate lifts, heights separate horizontal motion, and target
+projections separate descents.  Equation~\eqref{ba:affine-box-field}
+gives the exact affine flow on every swept cuboid and a neighborhood.
+
+The initial code $x^{\rm in}$ is rational.  For example, if
+$b=\max(3,\text{input length})$, the first $b$ right-stream symbols
+include the entire input and the pointer at cell two, after which the
+stream is constant; the left stream is constant.  On the loading
+interval use
+\[
+ C_*(t)=(2,0,0)+\rho(t)(x^{\rm in}-(2,0,0))
+\]
+and
+\[
+ V_*(t,x)=\nabla\times\left\{
+ \chi(C_*(t),(1,1,1),1;x)
+ \tfrac12 C_*'(t)\times(x-C_*(t))\right\}.
+\]
+The cutoff is the centered one defined in
+Lemma~\ref{ba:two-interpolations}.  Its plateau velocity is $C_*'$, so
+the fixed particle follows the displayed path.  This field vanishes
+outside $[1/4,3/4]$.  Extend the step field $V$ by zero and set
+$U=V_*+\sum_{k\ge1}V(t-k)$, with its residual force at the prescribed
+$\nu$.  The finite union of swept cuboids and the loading tube is
+bounded; all time joins have zero collars.  Consequently $U$ and its
+force are smooth, uniformly compactly supported, and have bounded
+mixed derivatives, with period one after loading.  Effective evaluation
+uses the finite formulas and a finite superset of potentially active
+integer shifts at any approximate time, as in
+Lemma~\ref{ba:realization}.
+
+The precise uniqueness input is Lemma~\ref{lem:b-comparison}(i).
+On each finite interval the constructed $U$ is smooth, has one compact
+spatial support, and has bounded velocity and gradient.  Consequently
+$U\in CH^2\cap C^1L^2$, $U(0)=0$, and the force is exactly
+$\mathcal F_\nu[U]$.  The theorem's competitors are smooth classical
+$v\in CH^2\cap C^1L^2$ with bounded $v$ and a representative
+$p\in CH^1$, precisely case (i).  No bound on $\nabla v$ is added.
+That case gives $v=U$ and fixes the $H^1$ pressure representative to
+zero.  The lemma also supplies the unique global material trajectories
+of the reference field.
+
+Loading and the exact full-box maps put the particle at the code after
+$k$ local steps at time $1+k$, through the first halt or forever in a
+nonhalting execution.  A halt code, including an initially halted code
+reached by loading, is inside the observation box.  In a nonhalting
+run, loading has first coordinate at least two.  During a later lift,
+the particle keeps its nonhalting source planar code, whose first
+coordinate exceeds three.  During descent it keeps the nonhalting
+target planar code with the same property.  Throughout the middle
+third its height is $6i>1$.  These three exclusions cover every real
+time and prove the event equivalence independently of horizontal
+excursions in the middle stage.  The repeated processor installs every
+local branch from the finite table; it does not depend on the executed
+run or its outcome.
+\end{proof}
+
+
+\subsection{Separating vertical and planar potentials}
+
+\begin{theorem}[A decaying processor with phasewise planar potentials]
+\label{ba:phasewise-flow}
+At fixed computable $\nu>0$ there is an effective force on $\mathbb R^3$
+with one compact spatial support and all mixed derivatives of time order
+$k$ bounded by $C_{k,\alpha}(1+t)^{-1-k}$, whose unique smooth
+zero-data solution satisfies
+\[
+ \exists t\ge0:\ (\Phi_t(0))_1<-1/2
+ \quad\Longleftrightarrow\quad\text{halting}.
+\]
+It uses the six-rule recorder with initial frontier 1, full subdivision
+of the rectangles, and separate vertical-translation and planar
+Hamiltonian potentials.  The constructed pressure is zero and uniqueness
+is in Lemma~\ref{ba:realization}'s whole-space class.
+\end{theorem}
+\begin{proof}\label{ba:phasewise-flow-proof}
+Let $\Gamma$ be the full alphabet of the six-rule table with $r_0=1$.
+Use odd digits in base $B=2|\Gamma|+1$ and its fully subdivided maps.  Put halt lanes at $-2,-4,\ldots$
+and the other lanes at $2,4,\ldots$, each of width one.  The second
+coordinate lies in $[0,1]$.  For the resulting rectangles $P_i,Q_i$,
+centers $p_i,q_i$, and first scale $\lambda_i$, choose rational
+$0<\varepsilon\le1/4$ smaller than one quarter of all within-source
+and within-target maximum-norm distances.  Empty pair lists impose no
+condition.  Thicken by $[-1,1]$ and put $H_i=4i$.
+
+On each third of the cycle use a flat progress function $\vartheta$
+from zero to one.  Lifting uses $z_i=H_i\vartheta$ and potential
+$A_i=(0,z_i'X_1,0)$, whose curl is $(0,0,z_i')$.
+At height $H_i$, set $c_i=(1-\vartheta)p_i+\vartheta q_i$ and
+$\mu_i=1-\vartheta+\vartheta\lambda_i$.  Use the potential
+$A_i=(0,0,\psi_i)$, where
+\[
+ \psi_i=c_{i1}'(X_2-c_{i2})-c_{i2}'(X_1-c_{i1})
+       +\frac{\mu_i'}{\mu_i}(X_1-c_{i1})(X_2-c_{i2}).
+\]
+Its curl is
+$(c_{i1}'+(\mu_i'/\mu_i)(X_1-c_{i1}),
+ c_{i2}'-(\mu_i'/\mu_i)(X_2-c_{i2}),0)$.
+Thus it realizes reciprocal scaling about the interpolated center.
+Descending uses the first potential with $z_i=H_i(1-\vartheta)$.
+Multiply each potential by its current-box $\varepsilon$ cutoff
+before taking the curl.  Source gaps, distinct middle heights, and
+target gaps separate the three stages.  On the whole boxes and a
+neighborhood the uncut formulas are exact.  The first coordinate is
+constant during vertical motion and is the convex interpolation of its
+endpoints during the planar stage.
+
+For loading put $\theta_0(s)=\sigma(2s-1/2)$, which is zero near zero
+and one near one. Let $P_*$ be the rational initial code. Load it in the
+first unit interval by the planar path
+$m(s)=\theta_0(s)((P_*)_1,(P_*)_2)$, localized around
+$(m(s),0)+[-1,1]^3$, using potential
+$(0,0,m_1'(X_2-m_2)-m_2'(X_1-m_1))$.
+Repeat the three-phase instruction field afterward to obtain $W$.
+Code induction and the delayed-head invariant prove the correspondence
+at times $1+n$.  Nonhalting codes have first coordinate at least 2,
+so loading and the convex-path bound exclude $X_1<-1/2$ for every real
+time.  A halt code has first coordinate at most $-1$, including one
+reached by loading an initially halted machine.  Apply
+Lemma~\ref{ba:log-clock}; physical code times are $e^{1+n}-1$.
+Its support, derivative, event, and uniqueness conclusions give precisely
+the claimed separate force choice.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/persistent-history.tex

@@ -1,512 +1,512 @@
-\section{Persistent marks and guarded history routes}
-\label{ba:marked-routes}
-
-The preceding recorders clear the work-head mark at each checkpoint.
-We now retain that mark, or replace the moving history pointer by a
-guarded boundary between occupied and unused cells.  These changes give
-different partial maps on arbitrary tapes, even though their initialized
-executions simulate the same machine.  We prove each inverse on its
-declared domain before using it to separate the geometric branch images.
-The use of retained history follows the reversible-computation idea of
-Bennett~\cite{Bennett1973}; the local guards and geometric interfaces
-are established here.
-
-In this section an input word occupies cells $0,1,\ldots$, the other
-work cells are blank, and the initial head is at zero.  All constructions
-include initial halting.  When one halt state $h$ is used, identify the
-original halting states with $h$ and redirect incoming instructions;
-there are no outgoing halting instructions to preserve.  If necessary
-adjoin an unreachable halt state.
-We write $X(t;a)=\Phi_t(a)$ for the trajectory from $a$.
-
-\subsection{A persistent mark and input-dependent placement}
-\label{ba:persistent-periodic}
-
-A force can be periodic from time zero if no separate loading interval
-is needed.  We achieve this by translating the initial control's coding
-square so that the prescribed particle already represents the input.
-The translation depends effectively on the finite input.  First we give
-the recorder that this periodic processor implements.
-
-\begin{lemma}[A recorder with a persistent work-head mark]
-\label{ba:persistent-compiler}
-Let a machine have work alphabet $\Gamma$, halt state $h$, and
-instructions
-$\gamma=(q,a)\mapsto(q^+_\gamma,a^+_\gamma,d_\gamma)$,
-where $q\ne h$ and $d_\gamma\in\{-1,0,1\}$.
-There is an effective finite partial one-head table with these properties.
-At the checkpoint after $n$ work steps, it has the correct work tape,
-state and head, a unique mark at that head, and a history pointer at
-$p_n=n+2$.  Each target control determines the incoming displacement;
-the target control and written full cell symbol determine the source
-control and read full symbol on the entire partial domain.  If the next
-work-head position is $w'=w_n+d_\gamma$, the next checkpoint is reached
-after exactly $2(p_n-w')+4$ local steps.
-\end{lemma}
-
-\begin{proof}\label{ba:persistent-compiler-proof}
-A full symbol is $(a,e,s)\in\Gamma\times\{0,1\}\times\mathcal H$,
-where $\mathcal H=\{E,P\}\sqcup\{r_\gamma\}$ records empty history,
-a pointer, or an instruction.  All displayed control families are
-disjoint.  The following is the complete table; free cell indices range
-over their finite alphabets:
-\[
-\begin{array}{c|c|c|c|r}
-\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline
-A_q&(a,1,s),\ s\ne P&B_\gamma&(a^+_\gamma,0,s)&d_\gamma\\
-B_\gamma&(c,0,s),\ s\ne P&R_\gamma&(c,1,s)&1\\
-R_\gamma&(c,0,s),\ s\ne P&R_\gamma&(c,0,s)&1\\
-R_\gamma&(c,0,P)&C_{q^+_\gamma}&(c,0,r_\gamma)&1\\
-C_q&(c,0,E)&L_q&(c,0,P)&-1\\
-L_q&(c,0,s),\ s\ne P&L_q&(c,0,s)&-1\\
-L_q&(c,1,s),\ s\ne P&A_q&(c,1,s)&0
-\end{array}
-\]
-The first row has $q\ne h$ and $\gamma=(q,a)$; there is no rule from
-$A_h$.  Initially the actual head and the unique mark are at zero,
-the work tape is the input, and history is $E$ except for $P$ at cell 2.
-
-At checkpoint $n$, $|w_n|\le n$, so after the first row the head is at
-$w'\le n+1<p_n$.  That row clears the old mark, and the second installs
-the new one at $w'$.  The right scan reaches the pointer without meeting
-another mark, writes $r_\gamma$, and visits the empty cell $p_n+1$.
-The next row installs the new pointer there.  The left scan returns
-through the newly written record to the unique mark and enters
-$A_{q^+_\gamma}$ without clearing it.  Hence every invoked rule is
-defined, the work instruction is correct, and the checkpoint invariant
-is restored.  The two continuing scans take $p_n-w'-1$ and $p_n-w'$
-steps; the other five steps give the stated total.  This also handles
-$d_\gamma=0$.  Induction gives a defined execution until and including
-the first halting checkpoint, and indefinitely in the nonhalting case.
-
-We check the inverse independently of this initialization.  Targets
-$B_\gamma,R_\gamma,C_q,L_q,A_q$ have incoming displacements
-$d_\gamma,1,1,-1,0$, respectively.  Undo that displacement to locate
-the written cell.  At $B_\gamma$, the index $\gamma$ recovers the old
-control and overwritten work symbol, and the old mark was one.  At
-$R_\gamma$, the written mark is one for entry and zero for the scan,
-so it identifies the source and the old mark.  At $C_q$, the written
-record identifies $\gamma$ and its former value $P$.  At $L_q$, a
-written pointer identifies entry, whereas every loop has history
-different from $P$.  At $A_q$, the unique source control is $L_q$ and
-the full symbol is unchanged.  All unspecified tracks are retained.
-Thus every possible output has at most one preimage on the full domain.
-\end{proof}
-
-\begin{theorem}[An input-dependent entry chart]
-\label{ba:periodic-entry}
-For each machine and finite input, and each fixed positive computable
-viscosity $\nu$, there is an effective smooth force on $\mathbb R^3$
-with one compact spatial support, bounded mixed derivatives, and period
-one from time zero, whose zero-data solution satisfies
-\[
- \exists t\ge0:\ X_1(t;(2,0,0))<0
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-The solution is smooth with pressure zero and is unique among smooth
-solutions for which, on every finite interval,
-$u\in C_tH^2\cap C_t^1L^2$, $u$ and $\nabla u$ are bounded, and
-$p\in C_tH^1$.
-As a separate force choice, the same event is realized with every
-mixed derivative of velocity and force in space-time $L^2$ and bounded
-by a derivative-dependent multiple of $(1+t)^{-1}$.
-\end{theorem}
-
-\begin{proof}\label{ba:periodic-entry-proof}
-Prepend a nonhalting state $q_{\rm in}$ which, on every work symbol,
-preserves it, stays put, and enters the original initial state.  This
-includes original initial halting while keeping the new initial state
-nonhalting.  Apply Lemma~\ref{ba:persistent-compiler}.  For its full
-alphabet of size $m$ use odd digits $e(\sigma)\in\{1,3,\ldots,2m-1\}$
-in base $B=2m+2$, and write
-\[
- c(\omega)=\sum_{j\ge0}e(\omega_j)B^{-j-1},\qquad
- (\xi,\eta)=
- \big(c(\sigma_{-1}\sigma_{-2}\cdots),
-       c(\sigma_0\sigma_1\cdots)\big).
-\]
-The initial values $\xi_0,\eta_0$ are rational, because the initialized
-streams have finite prefixes and known constant tails.  Set
-\[
- O(A_{q_{\rm in}})=(2-\xi_0,-\eta_0),\qquad
- O(A_h)=(-2,0),\qquad O(P)=(4+2j,0)
-\]
-for the remaining controls in a finite ordering.  All squares
-$O(P)+[0,1]^2$ are separated.  Every nonhalting square lies strictly
-to the right of $x_1=1$, and the halting square is
-$[-2,-1]\times[0,1]$.  The initial code at height zero is exactly
-$(2,0,0)$.
-
-Apply Lemma~\ref{lem:bb-onecell} with digits $d(\sigma)=e(\sigma)$,
-base $B=2m+2$, and the state translations just specified.  Its incoming
-hypotheses are exactly those of Lemma~\ref{ba:persistent-compiler}.
-With $I_\sigma=(e(\sigma)+[0,1])/B$, use the minimal left-split
-branches of \eqref{eq:bb-one-cell-branches}, including every left
-letter for a left move.  The state translations conjugate these maps
-without changing their reciprocal diagonal linear parts.  The shared
-lemma therefore gives the exact updates and separate source and target
-separation on full closed rectangles.  If an absolute head position is wanted at a checkpoint,
-the number of contiguous records gives $n$, and the pointer's relative
-position locates the head relative to its known absolute position $n+2$.
-The gapped digits permit each finite prefix needed for this recovery
-to be read with increasing precision.
-
-Apply the exponential interpolation of
-Lemma~\ref{ba:two-interpolations}, with heights $10i$ and the following
-separated phase intervals: lifting on $[1/8,2/8]$, horizontal motion
-on $[3/8,5/8]$, and descent on $[6/8,7/8]$.  On each interval use a
-rescaled flat step $\sigma$ from \eqref{eq:p2-step}.
-Let $\delta$ be one quarter of the minimum of 1
-and the positive planar source and target gaps.  Use plateau padding
-$\delta/3$ and support padding $2\delta/3$.  The same gap and height
-estimates in the lemma make these collars disjoint.  Here the local
-potential can be written
-\[
- \tfrac12 C_i'\times(x-C_i)
- +(0,0,\eta'\log\lambda_i\,(x_1-C_{i1})(x_2-C_{i2})).
-\]
-Its curl is the required translation plus
-$\eta'\log\lambda_i\,(x_1-C_{i1},-(x_2-C_{i2}),0)$.
-It realizes the whole cuboid and a neighborhood by the plateau argument.
-
-Repeat this unit-time field as $U$, and put
-$f=U_t+(U\cdot\nabla)U-\nu\Delta U$ at the prescribed physical
-viscosity.  The zero collars give $U(0)=0$ and smooth period-one gluing.
-Lemma~\ref{ba:realization} gives the solution, uniqueness, support and
-effective mixed-derivative bounds.  Its hypotheses hold because this is
-a finite curl formula on a compact spatial set and one compact time
-period.  The formula includes every branch and never requires the run
-on the selected input.
-
-At integer times, induction gives the exact local configuration through
-the first halt, or at every integer time for a nonhalting run.  A halt
-therefore reaches the negative square.  During a nonhalting run, both
-endpoint squares of every used branch are nonhalting.  Every moving
-center has first coordinate greater than one and every first half-width
-is at most $1/2$, because the endpoint widths are at most one and the
-exponential scale is monotone between them.  The actual particle stays
-positive during all phases.  This proves the equivalence for all real
-times, including the added entry step.
-
-For the separate decaying choice apply Lemma~\ref{ba:log-clock}:
-$\widehat U(t,x)=(1+t)^{-1}U(\log(1+t),x)$, with its own residual at
-the same $\nu$.  That lemma gives the stronger temporal-order-$k$
-bound $O((1+t)^{-1-k})$ for every spatial derivative, hence the stated
-bound and space-time $L^2$ conclusion on the common compact support.
-The clock is onto $[0,\infty)$, and the chain rule identifies the new
-trajectory with $X(\log(1+t);(2,0,0))$.  The fixed event is unchanged.
-\end{proof}
-
-
-\subsection{Using a fresh half-line instead of a frontier symbol}
-
-The next rule reads a neighboring history cell.  Its inverse therefore
-uses a full three-cell cylinder, rather than only the written cell.
-
-\begin{lemma}[Three-cell fresh-tail recorder]\label{ba:fresh-recorder}
-There is an effective partial rule on cells at offsets $-1,0,1$ that
-simulates the machine, is injective on its entire domain of relative
-tapes, and has incoming displacement determined by the target state.
-Its allowed cylinders admit separately separated closed-rectangle maps
-with linear parts $\operatorname{diag}(B^{-d},B^d)$.
-\end{lemma}
-\begin{proof}\label{ba:fresh-recorder-proof}
-Use symbols $c_i=(a_i,\ell_i,\mu_i)$ in
-$C=A\times(\{\star,\perp\}\sqcup\mathcal R)\times\{0,1\}$
-and controls $N_q,B_q,J_r,S_r$.  Only $N_q$ with $q\in H$ halt.
-After writing, a move $d$ reindexes a relative tape by
-$c'_j=\bar c_{j+d}$.  The following rules give the complete domain:
-\begin{enumerate}
-\item At $N_q$, $q\notin H$, require $\mu_0=0$ and
-$\ell_0\ne\perp$.  With $r=(q,a_0)$ write $b(r)$, move $d(r)$,
-and enter $J_r$.
-\item At $J_r$, require $\mu_0=0$ and $\ell_0\ne\perp$.
-Mark cell zero, move right, and enter $S_r$.
-\item At $S_r$, require $\mu_0=0$.  If $\ell_0\ne\perp$, scan
-right.  If $\ell_0=\ell_1=\perp$, write record $r$ at zero,
-move left, and enter $B_{q^+(r)}$.
-\item At $B_q$, require $\ell_0,\ell_1\ne\perp$.
-Scan left if $\mu_0=0$; if $\mu_0=1$, clear the mark, stay, and
-enter $N_q$.
-\end{enumerate}
-Initialize history as $\star$ below absolute cell 2 and $\perp$ from
-2 onward, all marks zero, the prescribed work tape, and control $N_{q_0}$.
-At checkpoint $n$ the first fresh cell is $F=n+2$, records occupy
-$2,\ldots,F-1$, and the work head satisfies $h\le F-2$.
-The work move reaches $k\le F-1$.  Mark $k$, scan to $F$, write the
-record, and return from $F-1$ to $k$.  The return's two-history guard
-holds even at its first cell because cell $F$ has just been filled.
-Clearing the mark restores the invariant with frontier $F+1$.
-There are $F-k-1$ instructions in each continuing scan and four others,
-so the duration is $2(F-k)+2$.  This proves finite completion and the
-checkpoint halt equivalence.
-
-For the full-domain inverse, targets $J_r,S_r,B_q,N_q$ determine moves
-$d(r),1,-1,0$, respectively.  Undo the indicated shift.  At $J_r$,
-$r$ recovers the overwritten letter and old control.  At $S_r$, the
-written mark distinguishes entry from the zero-mark scan.  At $B_q$,
-target offset $+2$ is the former offset $+1$; its history is fresh
-for a recording entry and nonfresh for a return scan.  In the entry
-case target offset $+1$ is the former scanned cell; its record identifies $r$.
-At $N_q$, restore the mark to one.  All other tracks are preserved.
-The neighbor guard is what makes this inverse valid on arbitrary allowed
-tapes.
-
-Use odd digits in base $B=2|C|+1$ and state offsets $e_s$ whose
-unit intervals have gaps at least two.  A permitted tuple
-$(s,c_{-1},c_0,c_1)$ has source
-$(e_s+I_{c_{-1}})\times I_{c_0c_1}$.  The full inverse just proved
-checks the injectivity hypothesis of Lemma~\ref{lem:bb-window-rectangles}
-with $(a,b)=(1,2)$ and $d\in\{-1,0,1\}$.
-Only cell zero is written, which is within that window.  The lemma
-therefore gives full target cylinders with prefix lengths
-$(1+d,2-d)$, exact affine factors $(B^{-d},B^d)$, and separate
-closed-rectangle images.  For one target state the incoming $d$ is
-fixed, so its prefix lengths agree across incoming branches.  An
-incompatible digit among at most three positions yields the additional
-quantitative target gap $B^{-3}$.  This is separation of the filled
-rectangles, including their boundaries.
-\end{proof}
-
-\begin{theorem}[Compact realization of fresh-tail cylinders]
-\label{ba:fresh-flow}
-The fresh-tail recorder has an effective compactly supported smooth
-realization on $\mathbb R^3$, with zero initial velocity and force
-periodic after one unit of loading, whose event
-$\exists t\ge0:(\Phi_t(0))_1<-1$ is equivalent to halting.
-A separate choice, with the same event and one compact support, satisfies
-for every $k\ge0$ and spatial multi-index $\alpha$
-\begin{equation}\label{ba:fresh-decay}
- \sup_x\bigl(|\partial_t^k\partial_x^\alpha u|+
-             |\partial_t^k\partial_x^\alpha F|\bigr)
- \le C_{k,\alpha}(1+t)^{-1-k}.
-\end{equation}
-Every displayed derivative is space-time square-integrable.  Both forces
-have the unique zero-pressure solution of Lemma~\ref{ba:realization}
-at the fixed computable viscosity.
-\end{theorem}
-\begin{proof}\label{ba:fresh-flow-proof}
-Use the full alphabet $C$ of Lemma~\ref{ba:fresh-recorder} and
-$B=2|C|+1$.  Place halting controls at $e_s=-3j_s$ and all others
-at $3j_s$, with distinct positive integer labels.  Use the three-cell rectangles as
-plates at height zero.  Branch $i$ has middle height $4i$ and padding
-$B^{-3}/10$.  Interpolate both planar factors independently and apply
-Lemma~\ref{ba:plate-routing}, including its reciprocal normal strain.
-Planar gaps separate the vertical phases; middle heights separate the
-padded plates, whose collar half-thickness is less than one.  Thus the
-specified motion holds on full neighborhoods.  It retains independent
-planar interpolation throughout the cycle.
-
-The initial code $P_*$ is rational.  A point-centered curl translation,
-with center $\theta(t)P_*$ and padding one, loads zero in one unit.
-Repeat the plate cycle thereafter.  The finite-step invariant and code
-maps identify the trajectory at all times $1+n$ until halt and for all
-$n$ otherwise.  A halt code has first coordinate below $-1$.  For a
-nonhalting run, loading is nonnegative and each later pair of coded
-endpoints is positive; the two-coordinate convex interpolation of
-Lemma~\ref{ba:plate-routing} proves the all-time exclusion.
-Use residual realization for the periodic choice and
-Lemma~\ref{ba:log-clock} for the separate decay choice.
-\end{proof}
-
-
-\subsection{Guarded history cylinders in a torus chart}
-\label{ba:guarded-history}
-
-An occupied prefix can replace the explicit pointer.  The resulting
-inverse must distinguish arrival from the recording cell from an
-ordinary return-scan step.  Testing the next history cell supplies that
-distinction.  Unlike the fresh-tail recorder, the marked return below
-does not test its right neighbor; we retain this larger partial domain.
-
-\begin{lemma}[An occupied-prefix history recorder]
-\label{ba:occupied-compiler}
-Let a machine have state set $Q$, halting set $H$, alphabet $\Gamma$,
-and instructions
-$\tau=(q,a)\mapsto(q^+_\tau,w_\tau,d_\tau)$.
-There is an effective partial injective map on head-relative tapes
-whose domain depends only on the control and full cells at offsets
-$-1,0,1$.  With the initialization below, occupied history at checkpoint
-$n$ consists exactly of the absolute indices less than $F_n=n+2$.
-A work step ending at head position $j$ takes exactly
-$2(F_n-j)+2$ local steps.  Halting at a checkpoint, including the initial
-one, is equivalent to halting of the original machine.
-\end{lemma}
-
-\begin{proof}\label{ba:occupied-compiler-proof}
-Full symbols are $(a,h,m)$, with work letter $a$, mark $m\in\{0,1\}$,
-and history $h\in\{\#,E\}\sqcup\mathcal R$, where
-$\mathcal R=(Q\setminus H)\times\Gamma$.  History is called occupied
-when $h\ne E$.  A move $d$ reindexes the post-write relative tape as
-$\xi^{\rm new}_k=\widetilde\xi_{k+d}$.  The complete table is
-\[
-\begin{array}{c|l|l|c|r}
-\text{control}&\text{guard}&\text{write at }0&\text{target}&d\\\hline
-\mathsf{Run}_q&m_0=0,\ h_0\ne E&a_0\leftarrow w_\tau&
- \mathsf{Mark}_\tau&d_\tau\\
-\mathsf{Mark}_\tau&m_0=0,\ h_0\ne E&m_0\leftarrow1&
- \mathsf{Seek}_\tau&1\\
-\mathsf{Seek}_\tau&m_0=0,\ h_0\ne E&\text{unchanged}&
- \mathsf{Seek}_\tau&1\\
-\mathsf{Seek}_\tau&m_0=0,\ h_0=h_1=E&h_0\leftarrow\tau&
- \mathsf{Back}_{q^+_\tau}&-1\\
-\mathsf{Back}_q&m_0=0,\ h_0\ne E,\ h_1\ne E&\text{unchanged}&
- \mathsf{Back}_q&-1\\
-\mathsf{Back}_q&m_0=1,\ h_0\ne E&m_0\leftarrow0&
- \mathsf{Run}_q&0.
-\end{array}
-\]
-The first row requires $q\notin H$ and $\tau=(q,a_0)$; all unmentioned
-tracks are unchanged.  Initialize the original work tape with head zero,
-all marks zero, history $\#$ at every index below 2, and $E$ elsewhere.
-
-At checkpoint $n$ the original head has $|j_n|\le n$.  The work step
-ends at $j\le n+1<F_n$, on occupied history.  Mark that cell and scan
-right across occupied unmarked cells to $F_n$.  Both $F_n$ and $F_n+1$
-are unused, so the recording row applies and moves left.  Every unmarked
-return-loop cell has occupied history at itself and its right neighbor,
-including the new record at $F_n$.  The return reaches $j$, clears the
-mark, and enters the new work state.  The continuing scans each take
-$F_n-1-j$ steps, with four other steps.  This proves finite completion,
-the duration, and the new occupied boundary $F_n+1$.  Induction gives
-the stated computation and halt equivalence.
-
-For the full-domain inverse, target controls determine displacement.
-The moves are $d_\tau$ into $\mathsf{Mark}_\tau$, one into
-$\mathsf{Seek}_\tau$, minus one into $\mathsf{Back}_q$, and zero
-into $\mathsf{Run}_q$.  At $\mathsf{Mark}_\tau$, undoing the
-move and restoring the read work symbol encoded by $\tau$ recovers the
-predecessor.  At $\mathsf{Seek}_\tau$, target offset $-1$ is the
-written cell; its mark distinguishes the marking entry from the scan.
-At $\mathsf{Back}_q$, target offset 2 is the former offset 1.  Its
-history is $E$ for a recording entry and is occupied for a return loop.
-In the recording case, target offset 1 contains $\tau$, identifying
-the old control and restoring its history to $E$.  Finally, the only
-entry into $\mathsf{Run}_q$ clears a mark without moving, so the mark
-is restored to one.  These inverses use only the table's guards and
-retained symbols, not the special shape of initialized history.
-\end{proof}
-
-\begin{lemma}[The full cylinders of the guarded recorder]
-\label{ba:occupied-cylinders}
-The recorder of Lemma~\ref{ba:occupied-compiler} admits a finite
-effective list of positive-area closed source and target rectangles,
-separately pairwise disjoint.  If $\mathcal A$ is its full alphabet and
-$B=2|\mathcal A|+1$, a branch of displacement $d$ has an exact affine
-map with linear part $\operatorname{diag}(B^{-d},B^d)$.
-\end{lemma}
-
-\begin{proof}\label{ba:occupied-cylinders-proof}
-Apply Lemma~\ref{lem:bb-window-rectangles} with window parameters
-$(a,b)=(1,2)$ to the complete table of
-Lemma~\ref{ba:occupied-compiler}.  That table reads only offsets
-$-1,0,1$, writes only offset zero, and has displacements
-$d\in\{-1,0,1\}\subset[-1,2]$; its full-domain injectivity was
-proved above.  Thus every hypothesis of the window lemma holds.
-
-To fix the actual rectangles, choose odd digits
-$\gamma(a)\in\{1,3,\ldots,2|\mathcal A|-1\}$ and
-$B=2|\mathcal A|+1$.  Write
-\[
- e(P_1\cdots P_k)=\sum_{j=1}^k\gamma(P_j)B^{-j},\qquad
- I(P)=e(P)+[0,B^{-k}],\qquad I(\varnothing)=[0,1],
-\]
-using the infinite series for an infinite word.  Choose separated
-state intervals $o_s+[0,1]$ and use the code
-\[
- r(s,\xi)=
- (o_s+e(\xi_{-1}\xi_{-2}\cdots),\ e(\xi_0\xi_1\cdots),0).
-\]
-For each permitted triple $(l,a,b)$ at offsets $-1,0,1$, let $a'$
-be its written full symbol.  The source prefix pair is $((l),(a,b))$,
-and the target pair is
-\[
- \begin{cases}
- ((a',l),(b)),&d=1,\\
- ((l),(a',b)),&d=0,\\
- (\varnothing,(l,a',b)),&d=-1.
- \end{cases}
-\]
-Take the corresponding products of prefix intervals with the source
-and target state offsets.  Both arbitrary tails are unchanged, so the
-window lemma gives the exact center-to-center affine map with factors
-$B^{-d},B^d$, onto the whole specified target cylinder and filled
-rectangle.  Its separation conclusion applies to these complete closed
-rectangles, including their boundaries.  All endpoints and positive
-within-family gaps are effectively rational.  The odd digits also put
-all infinite-word codes strictly in $(0,1)$ in each normalized coordinate.
-\end{proof}
-
-\begin{theorem}[A guarded processor in a fixed torus chart]
-\label{ba:occupied-torus}
-For each machine and finite input, at any fixed positive computable
-viscosity there is an effective smooth mean-zero force on the unit flat
-three-torus, with bounded mixed derivatives and period one for $t\ge1$,
-whose smooth zero-data solution has pressure zero and satisfies
-\[
- \exists t\ge0:\ X(t;[(-1/16,0,0)])\in
- \{[x]:x_1\bmod1\in(0,1/2)\}
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-The pressure is normalized to mean zero, and the solution is unique in
-the classical torus comparison class of Section~\ref{sec:model}.
-\end{theorem}
-
-\begin{proof}\label{ba:occupied-torus-proof}
-Use the occupied-prefix recorder.  Assign offsets $2j$, $j\ge1$, to
-halting run controls and $-2j$ to all other controls, with distinct
-indices within each group.  The initial code $r_{\rm in}$ is rational:
-the left stream is constant and the right stream is eventually constant.
-If $N$ is the number of cylinders, set
-\[
- O=1+\max_s|o_s|,\qquad K=O+6\max(1,N)+4,\qquad
- \sigma=(8K)^{-1},\qquad b_*=(-K/2,0,0).
-\]
-Use the linear version of Lemma~\ref{ba:two-interpolations} for the
-branch boxes of third interval $[-1,1]$, obtaining $V_1$.
-Use the same lemma with scale one on the single pair of cubes of
-half-width one centered at $b_*$ and $r_{\rm in}$, obtaining a loader
-$V_0$.  Extend both fields by zero outside their unit time intervals.
-Their supports lie in $(-K,K)^3$.  For $V_1$, first-coordinate centers
-have magnitude at most $O$, second-coordinate centers lie in $[0,1]$,
-horizontal half-widths are at most $1/2$, and heights are at most $6N$.
-For $V_0$, the horizontal center is bounded by $\max(K/2,O)$,
-half-widths are one, and height is at most six.  The padding is at most
-$1/4$ in either application.  Each bound is strictly less than $K$.
-
-Periodize the rescaled fields in space and the step field in time:
-\[
- U(t,[x])=\sigma\sum_{k\in\mathbb Z^3}
- \left[V_0(t,(x-k)/\sigma)+
-       \sum_{n\ge1}V_1(t-n,(x-k)/\sigma)\right].
-\]
-Spatial support lies in chart boxes of radius $1/8$, so distinct spatial
-copies are disjoint.  The time collars make the gluing smooth, initially
-zero and periodic after time one.  Every mixed derivative is bounded.
-The rescaled blocks remain curls of compactly supported potentials,
-so $U$ has zero spatial mean.  Define the force at the physical viscosity
-by $f=U_t+(U\cdot\nabla)U-\nu\Delta U$.  It has zero mean because
-the convection term is $\operatorname{div}(U\otimes U)$ and the
-Laplacian integrates to zero.  Lemma~\ref{ba:realization} now gives
-the unique solution $(U,0)$.  Recomputing the residual after the chart
-scaling is essential: it does not identify two different viscosities.
-
-The prescription is effective without deciding a space or time seam.
-Given approximations $\widehat x,\widehat t$ with error at most $1/4$,
-all possibly active spatial indices satisfy
-$\|\widehat x-k\|_\infty\le1$, and repeated time indices satisfy
-$|\widehat t-n|\le2$.  Evaluate this finite superset using the flat
-cutoffs and their derivative bounds.  The finite table, not an executed
-machine history, determines every coefficient.
-
-Since $\sigma b_*=(-1/16,0,0)$, loading sends the fixed particle to
-$[\sigma r_{\rm in}]$.  At time $1+j$ it is the code after $j$ local
-steps, through the first halt or forever in the nonhalting case, by the
-exact whole-box maps.  Halting, including initial halting, gives a
-positive first-coordinate representative below $1/8$, inside the target
-half-circle.  For a nonhalting input both loading endpoints and both
-endpoints of every subsequent used branch have negative first coordinate.
-The linear-scale identity keeps the first coordinate negative at every
-intermediate time.  All representatives stay in $(-1/8,0)$, disjoint
-from the target modulo one.  This proves the all-time equivalence.
-\end{proof}
+\section{Persistent marks and guarded history routes}
+\label{ba:marked-routes}
+
+The preceding recorders clear the work-head mark at each checkpoint.
+We now retain that mark, or replace the moving history pointer by a
+guarded boundary between occupied and unused cells.  These changes give
+different partial maps on arbitrary tapes, even though their initialized
+executions simulate the same machine.  We prove each inverse on its
+declared domain before using it to separate the geometric branch images.
+The use of retained history follows the reversible-computation idea of
+Bennett~\cite{Bennett1973}; the local guards and geometric interfaces
+are established here.
+
+In this section an input word occupies cells $0,1,\ldots$, the other
+work cells are blank, and the initial head is at zero.  All constructions
+include initial halting.  When one halt state $h$ is used, identify the
+original halting states with $h$ and redirect incoming instructions;
+there are no outgoing halting instructions to preserve.  If necessary
+adjoin an unreachable halt state.
+We write $X(t;a)=\Phi_t(a)$ for the trajectory from $a$.
+
+\subsection{A persistent mark and input-dependent placement}
+\label{ba:persistent-periodic}
+
+A force can be periodic from time zero if no separate loading interval
+is needed.  We achieve this by translating the initial control's coding
+square so that the prescribed particle already represents the input.
+The translation depends effectively on the finite input.  First we give
+the recorder that this periodic processor implements.
+
+\begin{lemma}[A recorder with a persistent work-head mark]
+\label{ba:persistent-compiler}
+Let a machine have work alphabet $\Gamma$, halt state $h$, and
+instructions
+$\gamma=(q,a)\mapsto(q^+_\gamma,a^+_\gamma,d_\gamma)$,
+where $q\ne h$ and $d_\gamma\in\{-1,0,1\}$.
+There is an effective finite partial one-head table with these properties.
+At the checkpoint after $n$ work steps, it has the correct work tape,
+state and head, a unique mark at that head, and a history pointer at
+$p_n=n+2$.  Each target control determines the incoming displacement;
+the target control and written full cell symbol determine the source
+control and read full symbol on the entire partial domain.  If the next
+work-head position is $w'=w_n+d_\gamma$, the next checkpoint is reached
+after exactly $2(p_n-w')+4$ local steps.
+\end{lemma}
+
+\begin{proof}\label{ba:persistent-compiler-proof}
+A full symbol is $(a,e,s)\in\Gamma\times\{0,1\}\times\mathcal H$,
+where $\mathcal H=\{E,P\}\sqcup\{r_\gamma\}$ records empty history,
+a pointer, or an instruction.  All displayed control families are
+disjoint.  The following is the complete table; free cell indices range
+over their finite alphabets:
+\[
+\begin{array}{c|c|c|c|r}
+\text{control}&\text{read and guard}&\text{target}&\text{write}&d\\\hline
+A_q&(a,1,s),\ s\ne P&B_\gamma&(a^+_\gamma,0,s)&d_\gamma\\
+B_\gamma&(c,0,s),\ s\ne P&R_\gamma&(c,1,s)&1\\
+R_\gamma&(c,0,s),\ s\ne P&R_\gamma&(c,0,s)&1\\
+R_\gamma&(c,0,P)&C_{q^+_\gamma}&(c,0,r_\gamma)&1\\
+C_q&(c,0,E)&L_q&(c,0,P)&-1\\
+L_q&(c,0,s),\ s\ne P&L_q&(c,0,s)&-1\\
+L_q&(c,1,s),\ s\ne P&A_q&(c,1,s)&0
+\end{array}
+\]
+The first row has $q\ne h$ and $\gamma=(q,a)$; there is no rule from
+$A_h$.  Initially the actual head and the unique mark are at zero,
+the work tape is the input, and history is $E$ except for $P$ at cell 2.
+
+At checkpoint $n$, $|w_n|\le n$, so after the first row the head is at
+$w'\le n+1<p_n$.  That row clears the old mark, and the second installs
+the new one at $w'$.  The right scan reaches the pointer without meeting
+another mark, writes $r_\gamma$, and visits the empty cell $p_n+1$.
+The next row installs the new pointer there.  The left scan returns
+through the newly written record to the unique mark and enters
+$A_{q^+_\gamma}$ without clearing it.  Hence every invoked rule is
+defined, the work instruction is correct, and the checkpoint invariant
+is restored.  The two continuing scans take $p_n-w'-1$ and $p_n-w'$
+steps; the other five steps give the stated total.  This also handles
+$d_\gamma=0$.  Induction gives a defined execution until and including
+the first halting checkpoint, and indefinitely in the nonhalting case.
+
+We check the inverse independently of this initialization.  Targets
+$B_\gamma,R_\gamma,C_q,L_q,A_q$ have incoming displacements
+$d_\gamma,1,1,-1,0$, respectively.  Undo that displacement to locate
+the written cell.  At $B_\gamma$, the index $\gamma$ recovers the old
+control and overwritten work symbol, and the old mark was one.  At
+$R_\gamma$, the written mark is one for entry and zero for the scan,
+so it identifies the source and the old mark.  At $C_q$, the written
+record identifies $\gamma$ and its former value $P$.  At $L_q$, a
+written pointer identifies entry, whereas every loop has history
+different from $P$.  At $A_q$, the unique source control is $L_q$ and
+the full symbol is unchanged.  All unspecified tracks are retained.
+Thus every possible output has at most one preimage on the full domain.
+\end{proof}
+
+\begin{theorem}[An input-dependent entry chart]
+\label{ba:periodic-entry}
+For each machine and finite input, and each fixed positive computable
+viscosity $\nu$, there is an effective smooth force on $\mathbb R^3$
+with one compact spatial support, bounded mixed derivatives, and period
+one from time zero, whose zero-data solution satisfies
+\[
+ \exists t\ge0:\ X_1(t;(2,0,0))<0
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+The solution is smooth with pressure zero and is unique among smooth
+solutions for which, on every finite interval,
+$u\in C_tH^2\cap C_t^1L^2$, $u$ and $\nabla u$ are bounded, and
+$p\in C_tH^1$.
+As a separate force choice, the same event is realized with every
+mixed derivative of velocity and force in space-time $L^2$ and bounded
+by a derivative-dependent multiple of $(1+t)^{-1}$.
+\end{theorem}
+
+\begin{proof}\label{ba:periodic-entry-proof}
+Prepend a nonhalting state $q_{\rm in}$ which, on every work symbol,
+preserves it, stays put, and enters the original initial state.  This
+includes original initial halting while keeping the new initial state
+nonhalting.  Apply Lemma~\ref{ba:persistent-compiler}.  For its full
+alphabet of size $m$ use odd digits $e(\sigma)\in\{1,3,\ldots,2m-1\}$
+in base $B=2m+2$, and write
+\[
+ c(\omega)=\sum_{j\ge0}e(\omega_j)B^{-j-1},\qquad
+ (\xi,\eta)=
+ \big(c(\sigma_{-1}\sigma_{-2}\cdots),
+       c(\sigma_0\sigma_1\cdots)\big).
+\]
+The initial values $\xi_0,\eta_0$ are rational, because the initialized
+streams have finite prefixes and known constant tails.  Set
+\[
+ O(A_{q_{\rm in}})=(2-\xi_0,-\eta_0),\qquad
+ O(A_h)=(-2,0),\qquad O(P)=(4+2j,0)
+\]
+for the remaining controls in a finite ordering.  All squares
+$O(P)+[0,1]^2$ are separated.  Every nonhalting square lies strictly
+to the right of $x_1=1$, and the halting square is
+$[-2,-1]\times[0,1]$.  The initial code at height zero is exactly
+$(2,0,0)$.
+
+Apply Lemma~\ref{lem:bb-onecell} with digits $d(\sigma)=e(\sigma)$,
+base $B=2m+2$, and the state translations just specified.  Its incoming
+hypotheses are exactly those of Lemma~\ref{ba:persistent-compiler}.
+With $I_\sigma=(e(\sigma)+[0,1])/B$, use the minimal left-split
+branches of \eqref{eq:bb-one-cell-branches}, including every left
+letter for a left move.  The state translations conjugate these maps
+without changing their reciprocal diagonal linear parts.  The shared
+lemma therefore gives the exact updates and separate source and target
+separation on full closed rectangles.  If an absolute head position is wanted at a checkpoint,
+the number of contiguous records gives $n$, and the pointer's relative
+position locates the head relative to its known absolute position $n+2$.
+The gapped digits permit each finite prefix needed for this recovery
+to be read with increasing precision.
+
+Apply the exponential interpolation of
+Lemma~\ref{ba:two-interpolations}, with heights $10i$ and the following
+separated phase intervals: lifting on $[1/8,2/8]$, horizontal motion
+on $[3/8,5/8]$, and descent on $[6/8,7/8]$.  On each interval use a
+rescaled flat step $\sigma$ from \eqref{eq:p2-step}.
+Let $\delta$ be one quarter of the minimum of 1
+and the positive planar source and target gaps.  Use plateau padding
+$\delta/3$ and support padding $2\delta/3$.  The same gap and height
+estimates in the lemma make these collars disjoint.  Here the local
+potential can be written
+\[
+ \tfrac12 C_i'\times(x-C_i)
+ +(0,0,\eta'\log\lambda_i\,(x_1-C_{i1})(x_2-C_{i2})).
+\]
+Its curl is the required translation plus
+$\eta'\log\lambda_i\,(x_1-C_{i1},-(x_2-C_{i2}),0)$.
+It realizes the whole cuboid and a neighborhood by the plateau argument.
+
+Repeat this unit-time field as $U$, and put
+$f=U_t+(U\cdot\nabla)U-\nu\Delta U$ at the prescribed physical
+viscosity.  The zero collars give $U(0)=0$ and smooth period-one gluing.
+Lemma~\ref{ba:realization} gives the solution, uniqueness, support and
+effective mixed-derivative bounds.  Its hypotheses hold because this is
+a finite curl formula on a compact spatial set and one compact time
+period.  The formula includes every branch and never requires the run
+on the selected input.
+
+At integer times, induction gives the exact local configuration through
+the first halt, or at every integer time for a nonhalting run.  A halt
+therefore reaches the negative square.  During a nonhalting run, both
+endpoint squares of every used branch are nonhalting.  Every moving
+center has first coordinate greater than one and every first half-width
+is at most $1/2$, because the endpoint widths are at most one and the
+exponential scale is monotone between them.  The actual particle stays
+positive during all phases.  This proves the equivalence for all real
+times, including the added entry step.
+
+For the separate decaying choice apply Lemma~\ref{ba:log-clock}:
+$\widehat U(t,x)=(1+t)^{-1}U(\log(1+t),x)$, with its own residual at
+the same $\nu$.  That lemma gives the stronger temporal-order-$k$
+bound $O((1+t)^{-1-k})$ for every spatial derivative, hence the stated
+bound and space-time $L^2$ conclusion on the common compact support.
+The clock is onto $[0,\infty)$, and the chain rule identifies the new
+trajectory with $X(\log(1+t);(2,0,0))$.  The fixed event is unchanged.
+\end{proof}
+
+
+\subsection{Using a fresh half-line instead of a frontier symbol}
+
+The next rule reads a neighboring history cell.  Its inverse therefore
+uses a full three-cell cylinder, rather than only the written cell.
+
+\begin{lemma}[Three-cell fresh-tail recorder]\label{ba:fresh-recorder}
+There is an effective partial rule on cells at offsets $-1,0,1$ that
+simulates the machine, is injective on its entire domain of relative
+tapes, and has incoming displacement determined by the target state.
+Its allowed cylinders admit separately separated closed-rectangle maps
+with linear parts $\operatorname{diag}(B^{-d},B^d)$.
+\end{lemma}
+\begin{proof}\label{ba:fresh-recorder-proof}
+Use symbols $c_i=(a_i,\ell_i,\mu_i)$ in
+$C=A\times(\{\star,\perp\}\sqcup\mathcal R)\times\{0,1\}$
+and controls $N_q,B_q,J_r,S_r$.  Only $N_q$ with $q\in H$ halt.
+After writing, a move $d$ reindexes a relative tape by
+$c'_j=\bar c_{j+d}$.  The following rules give the complete domain:
+\begin{enumerate}
+\item At $N_q$, $q\notin H$, require $\mu_0=0$ and
+$\ell_0\ne\perp$.  With $r=(q,a_0)$ write $b(r)$, move $d(r)$,
+and enter $J_r$.
+\item At $J_r$, require $\mu_0=0$ and $\ell_0\ne\perp$.
+Mark cell zero, move right, and enter $S_r$.
+\item At $S_r$, require $\mu_0=0$.  If $\ell_0\ne\perp$, scan
+right.  If $\ell_0=\ell_1=\perp$, write record $r$ at zero,
+move left, and enter $B_{q^+(r)}$.
+\item At $B_q$, require $\ell_0,\ell_1\ne\perp$.
+Scan left if $\mu_0=0$; if $\mu_0=1$, clear the mark, stay, and
+enter $N_q$.
+\end{enumerate}
+Initialize history as $\star$ below absolute cell 2 and $\perp$ from
+2 onward, all marks zero, the prescribed work tape, and control $N_{q_0}$.
+At checkpoint $n$ the first fresh cell is $F=n+2$, records occupy
+$2,\ldots,F-1$, and the work head satisfies $h\le F-2$.
+The work move reaches $k\le F-1$.  Mark $k$, scan to $F$, write the
+record, and return from $F-1$ to $k$.  The return's two-history guard
+holds even at its first cell because cell $F$ has just been filled.
+Clearing the mark restores the invariant with frontier $F+1$.
+There are $F-k-1$ instructions in each continuing scan and four others,
+so the duration is $2(F-k)+2$.  This proves finite completion and the
+checkpoint halt equivalence.
+
+For the full-domain inverse, targets $J_r,S_r,B_q,N_q$ determine moves
+$d(r),1,-1,0$, respectively.  Undo the indicated shift.  At $J_r$,
+$r$ recovers the overwritten letter and old control.  At $S_r$, the
+written mark distinguishes entry from the zero-mark scan.  At $B_q$,
+target offset $+2$ is the former offset $+1$; its history is fresh
+for a recording entry and nonfresh for a return scan.  In the entry
+case target offset $+1$ is the former scanned cell; its record identifies $r$.
+At $N_q$, restore the mark to one.  All other tracks are preserved.
+The neighbor guard is what makes this inverse valid on arbitrary allowed
+tapes.
+
+Use odd digits in base $B=2|C|+1$ and state offsets $e_s$ whose
+unit intervals have gaps at least two.  A permitted tuple
+$(s,c_{-1},c_0,c_1)$ has source
+$(e_s+I_{c_{-1}})\times I_{c_0c_1}$.  The full inverse just proved
+checks the injectivity hypothesis of Lemma~\ref{lem:bb-window-rectangles}
+with $(a,b)=(1,2)$ and $d\in\{-1,0,1\}$.
+Only cell zero is written, which is within that window.  The lemma
+therefore gives full target cylinders with prefix lengths
+$(1+d,2-d)$, exact affine factors $(B^{-d},B^d)$, and separate
+closed-rectangle images.  For one target state the incoming $d$ is
+fixed, so its prefix lengths agree across incoming branches.  An
+incompatible digit among at most three positions yields the additional
+quantitative target gap $B^{-3}$.  This is separation of the filled
+rectangles, including their boundaries.
+\end{proof}
+
+\begin{theorem}[Compact realization of fresh-tail cylinders]
+\label{ba:fresh-flow}
+The fresh-tail recorder has an effective compactly supported smooth
+realization on $\mathbb R^3$, with zero initial velocity and force
+periodic after one unit of loading, whose event
+$\exists t\ge0:(\Phi_t(0))_1<-1$ is equivalent to halting.
+A separate choice, with the same event and one compact support, satisfies
+for every $k\ge0$ and spatial multi-index $\alpha$
+\begin{equation}\label{ba:fresh-decay}
+ \sup_x\bigl(|\partial_t^k\partial_x^\alpha u|+
+             |\partial_t^k\partial_x^\alpha F|\bigr)
+ \le C_{k,\alpha}(1+t)^{-1-k}.
+\end{equation}
+Every displayed derivative is space-time square-integrable.  Both forces
+have the unique zero-pressure solution of Lemma~\ref{ba:realization}
+at the fixed computable viscosity.
+\end{theorem}
+\begin{proof}\label{ba:fresh-flow-proof}
+Use the full alphabet $C$ of Lemma~\ref{ba:fresh-recorder} and
+$B=2|C|+1$.  Place halting controls at $e_s=-3j_s$ and all others
+at $3j_s$, with distinct positive integer labels.  Use the three-cell rectangles as
+plates at height zero.  Branch $i$ has middle height $4i$ and padding
+$B^{-3}/10$.  Interpolate both planar factors independently and apply
+Lemma~\ref{ba:plate-routing}, including its reciprocal normal strain.
+Planar gaps separate the vertical phases; middle heights separate the
+padded plates, whose collar half-thickness is less than one.  Thus the
+specified motion holds on full neighborhoods.  It retains independent
+planar interpolation throughout the cycle.
+
+The initial code $P_*$ is rational.  A point-centered curl translation,
+with center $\theta(t)P_*$ and padding one, loads zero in one unit.
+Repeat the plate cycle thereafter.  The finite-step invariant and code
+maps identify the trajectory at all times $1+n$ until halt and for all
+$n$ otherwise.  A halt code has first coordinate below $-1$.  For a
+nonhalting run, loading is nonnegative and each later pair of coded
+endpoints is positive; the two-coordinate convex interpolation of
+Lemma~\ref{ba:plate-routing} proves the all-time exclusion.
+Use residual realization for the periodic choice and
+Lemma~\ref{ba:log-clock} for the separate decay choice.
+\end{proof}
+
+
+\subsection{Guarded history cylinders in a torus chart}
+\label{ba:guarded-history}
+
+An occupied prefix can replace the explicit pointer.  The resulting
+inverse must distinguish arrival from the recording cell from an
+ordinary return-scan step.  Testing the next history cell supplies that
+distinction.  Unlike the fresh-tail recorder, the marked return below
+does not test its right neighbor; we retain this larger partial domain.
+
+\begin{lemma}[An occupied-prefix history recorder]
+\label{ba:occupied-compiler}
+Let a machine have state set $Q$, halting set $H$, alphabet $\Gamma$,
+and instructions
+$\tau=(q,a)\mapsto(q^+_\tau,w_\tau,d_\tau)$.
+There is an effective partial injective map on head-relative tapes
+whose domain depends only on the control and full cells at offsets
+$-1,0,1$.  With the initialization below, occupied history at checkpoint
+$n$ consists exactly of the absolute indices less than $F_n=n+2$.
+A work step ending at head position $j$ takes exactly
+$2(F_n-j)+2$ local steps.  Halting at a checkpoint, including the initial
+one, is equivalent to halting of the original machine.
+\end{lemma}
+
+\begin{proof}\label{ba:occupied-compiler-proof}
+Full symbols are $(a,h,m)$, with work letter $a$, mark $m\in\{0,1\}$,
+and history $h\in\{\#,E\}\sqcup\mathcal R$, where
+$\mathcal R=(Q\setminus H)\times\Gamma$.  History is called occupied
+when $h\ne E$.  A move $d$ reindexes the post-write relative tape as
+$\xi^{\rm new}_k=\widetilde\xi_{k+d}$.  The complete table is
+\[
+\begin{array}{c|l|l|c|r}
+\text{control}&\text{guard}&\text{write at }0&\text{target}&d\\\hline
+\mathsf{Run}_q&m_0=0,\ h_0\ne E&a_0\leftarrow w_\tau&
+ \mathsf{Mark}_\tau&d_\tau\\
+\mathsf{Mark}_\tau&m_0=0,\ h_0\ne E&m_0\leftarrow1&
+ \mathsf{Seek}_\tau&1\\
+\mathsf{Seek}_\tau&m_0=0,\ h_0\ne E&\text{unchanged}&
+ \mathsf{Seek}_\tau&1\\
+\mathsf{Seek}_\tau&m_0=0,\ h_0=h_1=E&h_0\leftarrow\tau&
+ \mathsf{Back}_{q^+_\tau}&-1\\
+\mathsf{Back}_q&m_0=0,\ h_0\ne E,\ h_1\ne E&\text{unchanged}&
+ \mathsf{Back}_q&-1\\
+\mathsf{Back}_q&m_0=1,\ h_0\ne E&m_0\leftarrow0&
+ \mathsf{Run}_q&0.
+\end{array}
+\]
+The first row requires $q\notin H$ and $\tau=(q,a_0)$; all unmentioned
+tracks are unchanged.  Initialize the original work tape with head zero,
+all marks zero, history $\#$ at every index below 2, and $E$ elsewhere.
+
+At checkpoint $n$ the original head has $|j_n|\le n$.  The work step
+ends at $j\le n+1<F_n$, on occupied history.  Mark that cell and scan
+right across occupied unmarked cells to $F_n$.  Both $F_n$ and $F_n+1$
+are unused, so the recording row applies and moves left.  Every unmarked
+return-loop cell has occupied history at itself and its right neighbor,
+including the new record at $F_n$.  The return reaches $j$, clears the
+mark, and enters the new work state.  The continuing scans each take
+$F_n-1-j$ steps, with four other steps.  This proves finite completion,
+the duration, and the new occupied boundary $F_n+1$.  Induction gives
+the stated computation and halt equivalence.
+
+For the full-domain inverse, target controls determine displacement.
+The moves are $d_\tau$ into $\mathsf{Mark}_\tau$, one into
+$\mathsf{Seek}_\tau$, minus one into $\mathsf{Back}_q$, and zero
+into $\mathsf{Run}_q$.  At $\mathsf{Mark}_\tau$, undoing the
+move and restoring the read work symbol encoded by $\tau$ recovers the
+predecessor.  At $\mathsf{Seek}_\tau$, target offset $-1$ is the
+written cell; its mark distinguishes the marking entry from the scan.
+At $\mathsf{Back}_q$, target offset 2 is the former offset 1.  Its
+history is $E$ for a recording entry and is occupied for a return loop.
+In the recording case, target offset 1 contains $\tau$, identifying
+the old control and restoring its history to $E$.  Finally, the only
+entry into $\mathsf{Run}_q$ clears a mark without moving, so the mark
+is restored to one.  These inverses use only the table's guards and
+retained symbols, not the special shape of initialized history.
+\end{proof}
+
+\begin{lemma}[The full cylinders of the guarded recorder]
+\label{ba:occupied-cylinders}
+The recorder of Lemma~\ref{ba:occupied-compiler} admits a finite
+effective list of positive-area closed source and target rectangles,
+separately pairwise disjoint.  If $\mathcal A$ is its full alphabet and
+$B=2|\mathcal A|+1$, a branch of displacement $d$ has an exact affine
+map with linear part $\operatorname{diag}(B^{-d},B^d)$.
+\end{lemma}
+
+\begin{proof}\label{ba:occupied-cylinders-proof}
+Apply Lemma~\ref{lem:bb-window-rectangles} with window parameters
+$(a,b)=(1,2)$ to the complete table of
+Lemma~\ref{ba:occupied-compiler}.  That table reads only offsets
+$-1,0,1$, writes only offset zero, and has displacements
+$d\in\{-1,0,1\}\subset[-1,2]$; its full-domain injectivity was
+proved above.  Thus every hypothesis of the window lemma holds.
+
+To fix the actual rectangles, choose odd digits
+$\gamma(a)\in\{1,3,\ldots,2|\mathcal A|-1\}$ and
+$B=2|\mathcal A|+1$.  Write
+\[
+ e(P_1\cdots P_k)=\sum_{j=1}^k\gamma(P_j)B^{-j},\qquad
+ I(P)=e(P)+[0,B^{-k}],\qquad I(\varnothing)=[0,1],
+\]
+using the infinite series for an infinite word.  Choose separated
+state intervals $o_s+[0,1]$ and use the code
+\[
+ r(s,\xi)=
+ (o_s+e(\xi_{-1}\xi_{-2}\cdots),\ e(\xi_0\xi_1\cdots),0).
+\]
+For each permitted triple $(l,a,b)$ at offsets $-1,0,1$, let $a'$
+be its written full symbol.  The source prefix pair is $((l),(a,b))$,
+and the target pair is
+\[
+ \begin{cases}
+ ((a',l),(b)),&d=1,\\
+ ((l),(a',b)),&d=0,\\
+ (\varnothing,(l,a',b)),&d=-1.
+ \end{cases}
+\]
+Take the corresponding products of prefix intervals with the source
+and target state offsets.  Both arbitrary tails are unchanged, so the
+window lemma gives the exact center-to-center affine map with factors
+$B^{-d},B^d$, onto the whole specified target cylinder and filled
+rectangle.  Its separation conclusion applies to these complete closed
+rectangles, including their boundaries.  All endpoints and positive
+within-family gaps are effectively rational.  The odd digits also put
+all infinite-word codes strictly in $(0,1)$ in each normalized coordinate.
+\end{proof}
+
+\begin{theorem}[A guarded processor in a fixed torus chart]
+\label{ba:occupied-torus}
+For each machine and finite input, at any fixed positive computable
+viscosity there is an effective smooth mean-zero force on the unit flat
+three-torus, with bounded mixed derivatives and period one for $t\ge1$,
+whose smooth zero-data solution has pressure zero and satisfies
+\[
+ \exists t\ge0:\ X(t;[(-1/16,0,0)])\in
+ \{[x]:x_1\bmod1\in(0,1/2)\}
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+The pressure is normalized to mean zero, and the solution is unique in
+the classical torus comparison class of Section~\ref{sec:model}.
+\end{theorem}
+
+\begin{proof}\label{ba:occupied-torus-proof}
+Use the occupied-prefix recorder.  Assign offsets $2j$, $j\ge1$, to
+halting run controls and $-2j$ to all other controls, with distinct
+indices within each group.  The initial code $r_{\rm in}$ is rational:
+the left stream is constant and the right stream is eventually constant.
+If $N$ is the number of cylinders, set
+\[
+ O=1+\max_s|o_s|,\qquad K=O+6\max(1,N)+4,\qquad
+ \sigma=(8K)^{-1},\qquad b_*=(-K/2,0,0).
+\]
+Use the linear version of Lemma~\ref{ba:two-interpolations} for the
+branch boxes of third interval $[-1,1]$, obtaining $V_1$.
+Use the same lemma with scale one on the single pair of cubes of
+half-width one centered at $b_*$ and $r_{\rm in}$, obtaining a loader
+$V_0$.  Extend both fields by zero outside their unit time intervals.
+Their supports lie in $(-K,K)^3$.  For $V_1$, first-coordinate centers
+have magnitude at most $O$, second-coordinate centers lie in $[0,1]$,
+horizontal half-widths are at most $1/2$, and heights are at most $6N$.
+For $V_0$, the horizontal center is bounded by $\max(K/2,O)$,
+half-widths are one, and height is at most six.  The padding is at most
+$1/4$ in either application.  Each bound is strictly less than $K$.
+
+Periodize the rescaled fields in space and the step field in time:
+\[
+ U(t,[x])=\sigma\sum_{k\in\mathbb Z^3}
+ \left[V_0(t,(x-k)/\sigma)+
+       \sum_{n\ge1}V_1(t-n,(x-k)/\sigma)\right].
+\]
+Spatial support lies in chart boxes of radius $1/8$, so distinct spatial
+copies are disjoint.  The time collars make the gluing smooth, initially
+zero and periodic after time one.  Every mixed derivative is bounded.
+The rescaled blocks remain curls of compactly supported potentials,
+so $U$ has zero spatial mean.  Define the force at the physical viscosity
+by $f=U_t+(U\cdot\nabla)U-\nu\Delta U$.  It has zero mean because
+the convection term is $\operatorname{div}(U\otimes U)$ and the
+Laplacian integrates to zero.  Lemma~\ref{ba:realization} now gives
+the unique solution $(U,0)$.  Recomputing the residual after the chart
+scaling is essential: it does not identify two different viscosities.
+
+The prescription is effective without deciding a space or time seam.
+Given approximations $\widehat x,\widehat t$ with error at most $1/4$,
+all possibly active spatial indices satisfy
+$\|\widehat x-k\|_\infty\le1$, and repeated time indices satisfy
+$|\widehat t-n|\le2$.  Evaluate this finite superset using the flat
+cutoffs and their derivative bounds.  The finite table, not an executed
+machine history, determines every coefficient.
+
+Since $\sigma b_*=(-1/16,0,0)$, loading sends the fixed particle to
+$[\sigma r_{\rm in}]$.  At time $1+j$ it is the code after $j$ local
+steps, through the first halt or forever in the nonhalting case, by the
+exact whole-box maps.  Halting, including initial halting, gives a
+positive first-coordinate representative below $1/8$, inside the target
+half-circle.  For a nonhalting input both loading endpoints and both
+endpoints of every subsequent used branch have negative first coordinate.
+The linear-scale identity keeps the first coordinate negative at every
+intermediate time.  All representatives stay in $(-1/8,0)$, disjoint
+from the target modulo one.  This proves the all-time equivalence.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/references.bib

@@ -1,206 +1,206 @@
-@article{Turing1936,
-  author = {Turing, A. M.},
-  title = {On computable numbers, with an application to the {Entscheidungsproblem}},
-  journal = {Proceedings of the London Mathematical Society},
-  series = {2}, volume = {42}, pages = {230--265},
-  year = {1936--1937}, doi = {10.1112/plms/s2-42.1.230}
-}
-
-@article{Landauer1961,
-  author = {Landauer, Rolf},
-  title = {Irreversibility and heat generation in the computing process},
-  journal = {IBM Journal of Research and Development},
-  volume = {5}, number = {3}, pages = {183--191}, year = {1961},
-  doi = {10.1147/rd.53.0183}
-}
-
-@article{Bennett1973,
-  author = {Bennett, Charles H.},
-  title = {Logical reversibility of computation},
-  journal = {IBM Journal of Research and Development},
-  volume = {17}, number = {6}, pages = {525--532}, year = {1973},
-  doi = {10.1147/rd.176.0525}
-}
-
-@article{Moore1990,
-  author = {Moore, Cristopher},
-  title = {Unpredictability and undecidability in dynamical systems},
-  journal = {Physical Review Letters},
-  volume = {64}, number = {20}, pages = {2354--2357}, year = {1990},
-  doi = {10.1103/PhysRevLett.64.2354}
-}
-
-@article{Moore1991,
-  author = {Moore, Cristopher},
-  title = {Generalized shifts: unpredictability and undecidability in dynamical systems},
-  journal = {Nonlinearity},
-  volume = {4}, number = {2}, pages = {199--230}, year = {1991},
-  doi = {10.1088/0951-7715/4/2/002}
-}
-
-@article{Leray1934,
-  author = {Leray, Jean},
-  title = {Sur le mouvement d'un liquide visqueux emplissant l'espace},
-  journal = {Acta Mathematica}, volume = {63}, pages = {193--248},
-  year = {1934}, doi = {10.1007/BF02547354}
-}
-
-@article{Aronson1968,
-  author = {Aronson, Donald G.},
-  title = {Non-negative solutions of linear parabolic equations},
-  journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
-  series = {3}, volume = {22}, number = {4}, pages = {607--694}, year = {1968},
-  url = {https://www.numdam.org/item/ASNSP_1968_3_22_4_607_0/}
-}
-
-@article{CardonaMirandaPeraltaSalas2022,
-  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel},
-  title = {Turing universality of the incompressible {Euler} equations and a conjecture of {Moore}},
-  journal = {International Mathematics Research Notices},
-  volume = {2022}, number = {22}, pages = {18092--18109}, year = {2022},
-  doi = {10.1093/imrn/rnab233}
-}
-
-@article{Tao2016,
-  author = {Tao, Terence},
-  title = {Finite time blowup for an averaged three-dimensional {Navier--Stokes} equation},
-  journal = {Journal of the American Mathematical Society},
-  volume = {29}, number = {3}, pages = {601--674}, year = {2016},
-  doi = {10.1090/jams/838}
-}
-
-@article{Nash1958,
-  author = {Nash, John},
-  title = {Continuity of solutions of parabolic and elliptic equations},
-  journal = {American Journal of Mathematics},
-  volume = {80}, number = {4}, pages = {931--954}, year = {1958},
-  doi = {10.2307/2372841}
-}
-
-@article{Moser1965,
-  author = {Moser, J{\"u}rgen},
-  title = {On the volume elements on a manifold},
-  journal = {Transactions of the American Mathematical Society},
-  volume = {120}, pages = {286--294}, year = {1965},
-  doi = {10.1090/S0002-9947-1965-0182927-5}
-}
-
-@article{Cardona2025Advice,
-  author = {Cardona, Robert},
-  title = {Hydrodynamic and symbolic models of computation with advice},
-  journal = {Revista Matem{\'a}tica Iberoamericana},
-  volume = {41}, number = {1}, pages = {313--338}, year = {2025},
-  doi = {10.4171/RMI/1506}
-}
-
-@article{EtnyreGhrist2000,
-  author = {Etnyre, John and Ghrist, Robert},
-  title = {Contact topology and hydrodynamics: {I}. {Beltrami} fields and the {Seifert} conjecture},
-  journal = {Nonlinearity},
-  volume = {13},
-  number = {2},
-  pages = {441--458},
-  year = {2000},
-  doi = {10.1088/0951-7715/13/2/306},
-  eprint = {dg-ga/9708011},
-  archivePrefix = {arXiv}
-}
-
-@article{CardonaMirandaPeraltaSalasPresas2021,
-  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel and Presas, Francisco},
-  title = {Constructing {Turing} complete {Euler} flows in dimension 3},
-  journal = {Proceedings of the National Academy of Sciences},
-  volume = {118},
-  number = {19},
-  pages = {e2026818118},
-  year = {2021},
-  doi = {10.1073/pnas.2026818118},
-  eprint = {2012.12828},
-  archivePrefix = {arXiv}
-}
-
-@article{CardonaMirandaPeraltaSalasPresas2023,
-  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel and Presas, Francisco},
-  title = {Universality of {Euler} flows and flexibility of {Reeb} embeddings},
-  journal = {Advances in Mathematics},
-  volume = {428},
-  pages = {109142},
-  year = {2023},
-  doi = {10.1016/j.aim.2023.109142},
-  eprint = {1911.01963},
-  archivePrefix = {arXiv}
-}
-
-@article{DyhrGonzalezPrietoMirandaPeraltaSalas2026,
-  author = {Dyhr, S{\o}ren and Gonz{\'a}lez-Prieto, {\'A}ngel and Miranda, Eva and Peralta-Salas, Daniel},
-  title = {{Turing} complete {Navier--Stokes} steady states via cosymplectic geometry},
-  journal = {PNAS Nexus},
-  volume = {5},
-  number = {5},
-  pages = {pgag131},
-  year = {2026},
-  doi = {10.1093/pnasnexus/pgag131},
-  eprint = {2507.07696},
-  archivePrefix = {arXiv}
-}
-
-@article{CardonaMirandaPeraltaSalas2023Beltrami,
-  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel},
-  title = {Computability and {Beltrami} fields in {Euclidean} space},
-  journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
-  volume = {169},
-  year = {2023},
-  pages = {50--81},
-  doi = {10.1016/j.matpur.2022.11.007},
-  eprint = {2111.03559},
-  archivePrefix = {arXiv},
-  primaryClass = {math.AP},
-  url = {https://arxiv.org/abs/2111.03559v3}
-}
-
-@article{CardonaMirandaPeraltaSalas2025FluidComputer,
-  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel},
-  title = {Towards a Fluid Computer},
-  journal = {Foundations of Computational Mathematics},
-  volume = {25},
-  year = {2025},
-  pages = {1921--1937},
-  doi = {10.1007/s10208-025-09699-6},
-  eprint = {2405.20999},
-  archivePrefix = {arXiv},
-  primaryClass = {math.DS},
-  url = {https://link.springer.com/article/10.1007/s10208-025-09699-6}
-}
-
-@misc{OpenAI379Shear,
-  author = {{OpenAI}},
-  title = {{Finite Instructions and Solenoidal Shear Flows}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf}{OAI:Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026}},
-  year = {2026}
-}
-
-@misc{OpenAI379Sheets,
-  author = {{OpenAI}},
-  title = {{Prefix Instructions and Incompressible Flows}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf}{OAI:Prefix-Instructions-and-Incompressible-Flows-September-27-2026}},
-  year = {2026}
-}
-
-@misc{OpenAI379StationaryA2,
-  author = {{OpenAI}},
-  title = {{Universal Computation with Eventually Stationary Navier--Stokes Forcing}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf}{OAI:Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026}},
-  year = {2026}
-}
-
-@misc{OpenAI379EulerianC,
-  author = {{OpenAI}},
-  title = {{Velocity-Field Detection of Computation in Forced Navier--Stokes Flows}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/manuscript.pdf}{OAI:Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026}},
-  year = {2026}
-}
-
+@article{Turing1936,
+  author = {Turing, A. M.},
+  title = {On computable numbers, with an application to the {Entscheidungsproblem}},
+  journal = {Proceedings of the London Mathematical Society},
+  series = {2}, volume = {42}, pages = {230--265},
+  year = {1936--1937}, doi = {10.1112/plms/s2-42.1.230}
+}
+
+@article{Landauer1961,
+  author = {Landauer, Rolf},
+  title = {Irreversibility and heat generation in the computing process},
+  journal = {IBM Journal of Research and Development},
+  volume = {5}, number = {3}, pages = {183--191}, year = {1961},
+  doi = {10.1147/rd.53.0183}
+}
+
+@article{Bennett1973,
+  author = {Bennett, Charles H.},
+  title = {Logical reversibility of computation},
+  journal = {IBM Journal of Research and Development},
+  volume = {17}, number = {6}, pages = {525--532}, year = {1973},
+  doi = {10.1147/rd.176.0525}
+}
+
+@article{Moore1990,
+  author = {Moore, Cristopher},
+  title = {Unpredictability and undecidability in dynamical systems},
+  journal = {Physical Review Letters},
+  volume = {64}, number = {20}, pages = {2354--2357}, year = {1990},
+  doi = {10.1103/PhysRevLett.64.2354}
+}
+
+@article{Moore1991,
+  author = {Moore, Cristopher},
+  title = {Generalized shifts: unpredictability and undecidability in dynamical systems},
+  journal = {Nonlinearity},
+  volume = {4}, number = {2}, pages = {199--230}, year = {1991},
+  doi = {10.1088/0951-7715/4/2/002}
+}
+
+@article{Leray1934,
+  author = {Leray, Jean},
+  title = {Sur le mouvement d'un liquide visqueux emplissant l'espace},
+  journal = {Acta Mathematica}, volume = {63}, pages = {193--248},
+  year = {1934}, doi = {10.1007/BF02547354}
+}
+
+@article{Aronson1968,
+  author = {Aronson, Donald G.},
+  title = {Non-negative solutions of linear parabolic equations},
+  journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
+  series = {3}, volume = {22}, number = {4}, pages = {607--694}, year = {1968},
+  url = {https://www.numdam.org/item/ASNSP_1968_3_22_4_607_0/}
+}
+
+@article{CardonaMirandaPeraltaSalas2022,
+  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel},
+  title = {Turing universality of the incompressible {Euler} equations and a conjecture of {Moore}},
+  journal = {International Mathematics Research Notices},
+  volume = {2022}, number = {22}, pages = {18092--18109}, year = {2022},
+  doi = {10.1093/imrn/rnab233}
+}
+
+@article{Tao2016,
+  author = {Tao, Terence},
+  title = {Finite time blowup for an averaged three-dimensional {Navier--Stokes} equation},
+  journal = {Journal of the American Mathematical Society},
+  volume = {29}, number = {3}, pages = {601--674}, year = {2016},
+  doi = {10.1090/jams/838}
+}
+
+@article{Nash1958,
+  author = {Nash, John},
+  title = {Continuity of solutions of parabolic and elliptic equations},
+  journal = {American Journal of Mathematics},
+  volume = {80}, number = {4}, pages = {931--954}, year = {1958},
+  doi = {10.2307/2372841}
+}
+
+@article{Moser1965,
+  author = {Moser, J{\"u}rgen},
+  title = {On the volume elements on a manifold},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {120}, pages = {286--294}, year = {1965},
+  doi = {10.1090/S0002-9947-1965-0182927-5}
+}
+
+@article{Cardona2025Advice,
+  author = {Cardona, Robert},
+  title = {Hydrodynamic and symbolic models of computation with advice},
+  journal = {Revista Matem{\'a}tica Iberoamericana},
+  volume = {41}, number = {1}, pages = {313--338}, year = {2025},
+  doi = {10.4171/RMI/1506}
+}
+
+@article{EtnyreGhrist2000,
+  author = {Etnyre, John and Ghrist, Robert},
+  title = {Contact topology and hydrodynamics: {I}. {Beltrami} fields and the {Seifert} conjecture},
+  journal = {Nonlinearity},
+  volume = {13},
+  number = {2},
+  pages = {441--458},
+  year = {2000},
+  doi = {10.1088/0951-7715/13/2/306},
+  eprint = {dg-ga/9708011},
+  archivePrefix = {arXiv}
+}
+
+@article{CardonaMirandaPeraltaSalasPresas2021,
+  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel and Presas, Francisco},
+  title = {Constructing {Turing} complete {Euler} flows in dimension 3},
+  journal = {Proceedings of the National Academy of Sciences},
+  volume = {118},
+  number = {19},
+  pages = {e2026818118},
+  year = {2021},
+  doi = {10.1073/pnas.2026818118},
+  eprint = {2012.12828},
+  archivePrefix = {arXiv}
+}
+
+@article{CardonaMirandaPeraltaSalasPresas2023,
+  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel and Presas, Francisco},
+  title = {Universality of {Euler} flows and flexibility of {Reeb} embeddings},
+  journal = {Advances in Mathematics},
+  volume = {428},
+  pages = {109142},
+  year = {2023},
+  doi = {10.1016/j.aim.2023.109142},
+  eprint = {1911.01963},
+  archivePrefix = {arXiv}
+}
+
+@article{DyhrGonzalezPrietoMirandaPeraltaSalas2026,
+  author = {Dyhr, S{\o}ren and Gonz{\'a}lez-Prieto, {\'A}ngel and Miranda, Eva and Peralta-Salas, Daniel},
+  title = {{Turing} complete {Navier--Stokes} steady states via cosymplectic geometry},
+  journal = {PNAS Nexus},
+  volume = {5},
+  number = {5},
+  pages = {pgag131},
+  year = {2026},
+  doi = {10.1093/pnasnexus/pgag131},
+  eprint = {2507.07696},
+  archivePrefix = {arXiv}
+}
+
+@article{CardonaMirandaPeraltaSalas2023Beltrami,
+  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel},
+  title = {Computability and {Beltrami} fields in {Euclidean} space},
+  journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
+  volume = {169},
+  year = {2023},
+  pages = {50--81},
+  doi = {10.1016/j.matpur.2022.11.007},
+  eprint = {2111.03559},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AP},
+  url = {https://arxiv.org/abs/2111.03559v3}
+}
+
+@article{CardonaMirandaPeraltaSalas2025FluidComputer,
+  author = {Cardona, Robert and Miranda, Eva and Peralta-Salas, Daniel},
+  title = {Towards a Fluid Computer},
+  journal = {Foundations of Computational Mathematics},
+  volume = {25},
+  year = {2025},
+  pages = {1921--1937},
+  doi = {10.1007/s10208-025-09699-6},
+  eprint = {2405.20999},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DS},
+  url = {https://link.springer.com/article/10.1007/s10208-025-09699-6}
+}
+
+@misc{OpenAI379Shear,
+  author = {{OpenAI}},
+  title = {{Finite Instructions and Solenoidal Shear Flows}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf}{OAI:Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026}},
+  year = {2026}
+}
+
+@misc{OpenAI379Sheets,
+  author = {{OpenAI}},
+  title = {{Prefix Instructions and Incompressible Flows}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf}{OAI:Prefix-Instructions-and-Incompressible-Flows-September-27-2026}},
+  year = {2026}
+}
+
+@misc{OpenAI379StationaryA2,
+  author = {{OpenAI}},
+  title = {{Universal Computation with Eventually Stationary Navier--Stokes Forcing}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf}{OAI:Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026}},
+  year = {2026}
+}
+
+@misc{OpenAI379EulerianC,
+  author = {{OpenAI}},
+  title = {{Velocity-Field Detection of Computation in Forced Navier--Stokes Flows}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/manuscript.pdf}{OAI:Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026}},
+  year = {2026}
+}
+

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/routing-conventions.tex

@@ -1,255 +1,255 @@
-\section{Affine motion on neighborhoods of boxes}\label{ba:geometry}
-
-Rectangle separation controls the beginning and end of a transfer; distinct
-heights separate its middle. We now specify the paths and cutoff margins
-used by the applications. Interpolating one planar scale and its reciprocal
-preserves thickness and, for a linear first scale, gives a convex formula
-for each first coordinate. Interpolating both planar scales independently
-gives that formula in both coordinates, at the cost of a compensating
-normal strain during the motion.
-
-\subsection{Reciprocal paths and their localization}
-\label{sec:ba-localization}
-
-Use the flat step $\sigma$ of \eqref{eq:p2-step} and put
-\[
- \theta(s)=\sigma(2s-1/2),\qquad
- w_j(s)=\theta(3s-j+1)\quad(j=1,2,3).
-\]
-The switches $w_1,w_2,w_3$ perform the lift, horizontal transfer and
-descent in separate thirds of the unit interval. All are constant near
-their phase endpoints. The following version of
-Lemma~\ref{lem:bb-lift-parking}(i) records a common vertical half-width
-of one and explicit neighborhoods; the same formulas allow other heights
-and thicknesses as described immediately afterward.
-
-\begin{lemma}[Two explicit interpolations of separated boxes]
-\label{ba:two-interpolations}
-Let $S_i,T_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational
-rectangles of positive side lengths.  Suppose each list is pairwise
-disjoint and
-\[
- F_i(y)=c_i^T+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(y-c_i^S),
- \qquad\lambda_i\in\mathbb Q_{>0},
-\]
-maps $S_i$ onto $T_i$, with $c_i^S,c_i^T$ their centers.  There is a
-smooth nondecreasing switch $\eta:[0,1]\to[0,1]$, zero near 0 and one
-near 1, for which either scale
-path $r_i=\lambda_i^\eta$ or $r_i=1-\eta+\eta\lambda_i$ yields an
-effective smooth compactly supported solenoidal unit-time field, zero
-on time collars, realizing $(y,z)\mapsto(F_i(y),z)$ on the whole box
-$S_i\times[-1,1]$ and a neighborhood of it.  The paths are affine;
-each horizontal half-width remains between its endpoint values, and the
-horizontal sweep of each original box lies in the coordinate bounding
-rectangle of $S_i$ and $T_i$. With the linear choice, a particle's first
-coordinate is the convex
-interpolation of its endpoint first coordinates.
-\end{lemma}
-
-\begin{proof}\label{ba:two-interpolations-proof}
-For $N=0$ take the zero field.  For $N\ge1$ apply
-Lemma~\ref{lem:bb-lift-parking}(i) with vertical half-width one.
-Use the shared phase switches with $\eta=w_2$ and set
-\[
- C_i(s)=\big((1-\eta)c_i^S+\eta c_i^T,\,6i(w_1-w_3)\big),
- \qquad D_i(s)=\operatorname{diag}(r_i,r_i^{-1},1).
-\]
-The affine path is
-$G_i(s,y)=C_i(s)+D_i(s)(y-(c_i^S,0))$.
-Choose $\epsilon$ to be one quarter of the minimum of 1 and all
-pairwise max-norm distances in the two planar lists, omitting empty
-pair lists.  The $\epsilon$-padded moving boxes are disjoint: source
-and target gaps are at least $4\epsilon$ during the vertical phases,
-and the middle-phase height spacing is $6>2(1+\epsilon)$.
-
-For later loading formulas, record the centered cutoff explicitly:
-\[
- \chi(C,L,\epsilon;x)=
- \prod_{j=1}^3\sigma\!\left(\frac{L_j+\epsilon+x_j-C_j}{\epsilon/2}\right)
- \sigma\!\left(\frac{L_j+\epsilon-x_j+C_j}{\epsilon/2}\right).
-\]
-Here $L$ is the vector of half-widths; the plateau has padding
-$\epsilon/2$ and the support has padding $\epsilon$.
-If $L_i(s)$ are the moving half-widths, $Y=x-C_i(s)$ and
-$A_i=D_i'D_i^{-1}$, the field supplied by
-\eqref{eq:bb-potential} is
-\begin{equation}\label{ba:affine-box-field}
- V(s,x)=\sum_i\nabla\times\left\{
- \chi(C_i,L_i,\epsilon;x)
- \left[\tfrac12 C_i'\times Y+\tfrac13(A_iY)\times Y\right]
- \right\}.
-\end{equation}
-Indeed $\det D_i=1$, and the preceding separation verifies
-Lemma~\ref{lem:bb-curl} with localization parameter $\epsilon/2$.
-Its neighborhood conclusion applies with
-$L_* =\max_i\max(1,\lambda_i,\lambda_i^{-1})$ because
-$\|D_i(s)\|_\infty\le L_*$ and $D_i(0)=I$.
-It therefore gives the stated affine motion on the full boxes and
-on their initial max-norm neighborhoods of radius $\epsilon/(4L_*)$.
-The paths are stationary for $s\le1/12$ and $s\ge11/12$.
-Their finite center ranges and endpoint width bounds give common
-compact support.  The effective positive lower bound
-$r_i\ge\min(1,\lambda_i)$ verifies the scale requirement.
-For the linear scale, the first-coordinate identity is
-\[
- (G_i(s,y))_1=(1-\eta)y_1+\eta(F_i(y_1,y_2))_1.
-\]
-The second coordinate need not interpolate its own endpoint values.
-Instead, if $b_{i2}$ is the source's second half-width, convexity of the
-reciprocal function gives
-\begin{equation}\label{eq:ba-reciprocal-envelope}
- \frac{b_{i2}}{(1-\eta)+\eta\lambda_i}
- \le (1-\eta)b_{i2}+\eta\frac{b_{i2}}{\lambda_i}.
-\end{equation}
-The first half-width is exactly its endpoint interpolation. For the
-exponential scale, convexity of the exponential gives the corresponding
-upper bounds in both coordinates, since
-$\lambda_i^{\pm\eta}\le(1-\eta)+\eta\lambda_i^{\pm1}$.
-In either case the center is interpolated linearly and each half-width
-is at most the interpolation of its endpoint half-widths. Therefore each
-lower edge is at least the interpolation of the endpoint lower edges,
-and each upper edge is at most the corresponding interpolation of upper
-edges. This proves the claimed bounding rectangle for the entire sweep.
-The individual endpoint-width bounds also follow directly from monotonicity
-of the positive scales and their reciprocals.
-\end{proof}
-
-The same construction applies to $P_i\times[-h,h]$ for any common
-$h\ge0$, including plates and rectangles with zero side lengths, under
-the hypotheses of Lemma~\ref{lem:bb-lift-parking}(i). Replace $6i$ by
-heights $H_i$ whose pairwise gaps exceed $2h+2\varepsilon$, and choose
-$\varepsilon$ below one third of every within-family planar gap. The
-formula for $C_i$ uses the source and target centers in place of
-$c_i^S,c_i^T$; its support padding is $\varepsilon$ and plateau padding
-is $\varepsilon/2$. Source projections, distinct heights and target
-projections separate the three phases in that order. Empty pair lists
-impose no gap restriction, and the empty family gives the zero field.
-Arbitrary real data give smooth existence; the effective-data assumptions
-of the shared lemma give computable scales, derivative bounds and margins.
-The same perturbation argument gives an open neighborhood even when a
-side length is zero.
-
-\subsection{Transporting an initial cutoff}
-\label{sec:ba-transported-localization}
-
-The transported-initial-cutoff convention is also useful.  Write the
-affine path as $G_i(t)x=J_i(t)x+g_i(t)$ and localize by
-$\psi_i(G_i(t)^{-1}x)$, where $\psi_i$ has support padding
-$\varepsilon_0$ around the initial box and equals one on a smaller
-positive padding.
-If $\lambda_i\in[B^{-1},B]$, its initial padding
-$\varepsilon_0$ expands horizontally by at most $B$.  Choose
-$2B\varepsilon_0$ below every target gap,
-$2\varepsilon_0$ below every source gap, and
-$\varepsilon_0\le\varepsilon$.  These transported supports stay
-separated in the same three phases.  The affine generator is
-$v_i+A_ix$, with $A_i=\dot J_iJ_i^{-1}$ and
-$v_i=\dot g_i-A_ig_i$.  Its potential is
-\[
- \tfrac12v_i\times x+\tfrac13(A_ix)\times x.
-\]
-The identity proved in Lemma~\ref{lem:bb-curl} gives the exact
-generator on the transported plateau.  Its neighborhood argument then
-proves the same full-box endpoint map.  The factor $B$ in the margin
-choice is essential: transported padding does not stay fixed under
-stretching.
-
-\subsection{Independent planar interpolation}
-
-For an observer that needs both planar coordinates to follow their own
-endpoint interpolations, the reciprocal path is insufficient. The next
-construction interpolates the two factors independently and compensates
-the temporary planar area change in the third direction. Here the named
-targets may be disjoint containers of the actual affine images; only the
-image centers enter the motion.
-
-\begin{lemma}[Independent planar scales and normal compensation]
-\label{ba:plate-routing}
-Let $P_i,Q_i\subset\R^2$ be finite families of closed rectangles,
-separately pairwise disjoint, and let assigned positive diagonal affine
-maps $F_i$ satisfy $F_i(P_i)\subseteq Q_i$. Write their diagonal factors as
-$\lambda_{i1}^*,\lambda_{i2}^*>0$ with product one.
-There is a smooth compactly supported solenoidal unit-time field, zero on
-time collars, whose endpoint map is $(y,z)\mapsto(F_i(y),z)$ on a
-neighborhood of each entire plate $P_i\times\{0\}$. During the horizontal
-phase both planar factors interpolate linearly, while the normal factor
-compensates their product. Each planar coordinate of a point on the plate
-is its convex endpoint interpolation.
-\end{lemma}
-\begin{proof}\label{ba:plate-routing-proof}
-Use the height phases in Section~\ref{sec:ba-localization}. For source
-center $p_i$, end its center path at its actual image $F_i(p_i)$; in the
-onto case this is the named target center. Choose separated heights $H_i$
-as above and set
-\[
- c_i=\bigl((1-w_2)p_i+w_2F_i(p_i),\,H_i(w_1-w_3)\bigr),
-\]
-\[
- \lambda_{ij}=1-w_2+w_2\lambda_{ij}^*,\qquad
- D_i=\operatorname{diag}\bigl(
- \lambda_{i1},\lambda_{i2},(\lambda_{i1}\lambda_{i2})^{-1}\bigr).
-\]
-The volume-preserving path has logarithmic derivative
-\[
- T_i=\dot D_iD_i^{-1}=\operatorname{diag}\left(
- \frac{\dot\lambda_{i1}}{\lambda_{i1}},
- \frac{\dot\lambda_{i2}}{\lambda_{i2}},
- -\frac{\dot\lambda_{i1}}{\lambda_{i1}}
- -\frac{\dot\lambda_{i2}}{\lambda_{i2}}\right).
-\]
-This matrix has trace zero, so \eqref{eq:bb-potential} localized
-around the moving plate gives the desired field.  Its planar motion is
-$c_i+(\lambda_{i1}(y_1-p_{i1}),\lambda_{i2}(y_2-p_{i2}),0)$.
-Normal displacement changes by the factor
-$(\lambda_{i1}\lambda_{i2})^{-1}$. The factors have positive
-lower bounds; choose the initial thickness less than their product's
-minimum times the normal plateau width.  A small enlargement of the
-planar rectangle is allowed by the positive collars as well.  Thus a
-whole thin neighborhood follows the volume-preserving affine motion.
-The normal factor returns to one at the endpoint. The final plate image
-is contained in $Q_i\times\{0\}$. Source and target-container
-gaps protect the vertical phases and distinct heights protect the
-middle phase.  Linear interpolation of each planar factor gives both
-convex-coordinate identities on the plate.
-This proves smooth existence for arbitrary real data. Under the effective
-input assumptions of Lemma~\ref{lem:bb-curl}, the positive scale bounds and
-finite gap margins can be computed and the construction is effective.
-\end{proof}
-
-\subsection{Forcing and effective evaluation}
-
-\begin{lemma}[Residual forcing for the finite processors]\label{ba:realization}
-Fix computable $\nu>0$.  Let $W$ be smooth, solenoidal, initially zero,
-with support in one compact subset of $\mathbb R^3$ and every mixed
-derivative bounded.  Then
-\begin{equation}\label{ba:residual}
- F_W=W_t+(W\cdot\nabla)W-\nu\Delta W
-\end{equation}
-has the same support and derivative properties.  The zero-data solution
-$(W,0)$ is unique among smooth solutions with
-$u\in C_tH^2\cap C_t^1L^2$, bounded $u$ on finite intervals, and
-$p\in C_tH^1$.  The corresponding torus assertion holds in the
-classical comparison class of Section~\ref{sec:model}.  A sum of curls
-in a torus chart gives both $W$ and $F_W$ mean zero.  Finite formulas
-from the effective-data instances of the preceding constructions, finite
-loading, and repetition of one cycle give effective evaluation of every
-force derivative.
-\end{lemma}
-\begin{proof}\label{ba:realization-proof}
-Fixed compact support and bounded mixed derivatives imply all reference
-velocity hypotheses of Lemma~\ref{lem:b-comparison}.  Case (i) gives
-exactly the stated whole-space comparison class, without a competing
-gradient bound; its torus clause gives the stated classical comparison
-class.  The same lemma provides the pressure normalization and global
-material flow.  The product rule applied to \eqref{ba:residual}
-proves the force's support and derivative bounds.  A chart curl has
-zero mean, and integration of the residual uses
-$\int\Delta W=0$ and
-$\int(W\cdot\nabla)W=\int\operatorname{div}(W\otimes W)=0$.
-For these effective-data instances, gate derivatives and seam evaluation are
-those of \eqref{eq:p2-step} and Proposition~\ref{prop:bb-realization}.
-The explicit affine scales and every reciprocal denominator have positive
-computable bounds.  Thus differentiating the finite formulas, using a
-finite superset of possible period indices, evaluates every derivative
-without a machine execution or a real equality test at a seam.
-\end{proof}
+\section{Affine motion on neighborhoods of boxes}\label{ba:geometry}
+
+Rectangle separation controls the beginning and end of a transfer; distinct
+heights separate its middle. We now specify the paths and cutoff margins
+used by the applications. Interpolating one planar scale and its reciprocal
+preserves thickness and, for a linear first scale, gives a convex formula
+for each first coordinate. Interpolating both planar scales independently
+gives that formula in both coordinates, at the cost of a compensating
+normal strain during the motion.
+
+\subsection{Reciprocal paths and their localization}
+\label{sec:ba-localization}
+
+Use the flat step $\sigma$ of \eqref{eq:p2-step} and put
+\[
+ \theta(s)=\sigma(2s-1/2),\qquad
+ w_j(s)=\theta(3s-j+1)\quad(j=1,2,3).
+\]
+The switches $w_1,w_2,w_3$ perform the lift, horizontal transfer and
+descent in separate thirds of the unit interval. All are constant near
+their phase endpoints. The following version of
+Lemma~\ref{lem:bb-lift-parking}(i) records a common vertical half-width
+of one and explicit neighborhoods; the same formulas allow other heights
+and thicknesses as described immediately afterward.
+
+\begin{lemma}[Two explicit interpolations of separated boxes]
+\label{ba:two-interpolations}
+Let $S_i,T_i\subset\mathbb R^2$, $1\le i\le N$, be closed rational
+rectangles of positive side lengths.  Suppose each list is pairwise
+disjoint and
+\[
+ F_i(y)=c_i^T+\operatorname{diag}(\lambda_i,\lambda_i^{-1})(y-c_i^S),
+ \qquad\lambda_i\in\mathbb Q_{>0},
+\]
+maps $S_i$ onto $T_i$, with $c_i^S,c_i^T$ their centers.  There is a
+smooth nondecreasing switch $\eta:[0,1]\to[0,1]$, zero near 0 and one
+near 1, for which either scale
+path $r_i=\lambda_i^\eta$ or $r_i=1-\eta+\eta\lambda_i$ yields an
+effective smooth compactly supported solenoidal unit-time field, zero
+on time collars, realizing $(y,z)\mapsto(F_i(y),z)$ on the whole box
+$S_i\times[-1,1]$ and a neighborhood of it.  The paths are affine;
+each horizontal half-width remains between its endpoint values, and the
+horizontal sweep of each original box lies in the coordinate bounding
+rectangle of $S_i$ and $T_i$. With the linear choice, a particle's first
+coordinate is the convex
+interpolation of its endpoint first coordinates.
+\end{lemma}
+
+\begin{proof}\label{ba:two-interpolations-proof}
+For $N=0$ take the zero field.  For $N\ge1$ apply
+Lemma~\ref{lem:bb-lift-parking}(i) with vertical half-width one.
+Use the shared phase switches with $\eta=w_2$ and set
+\[
+ C_i(s)=\big((1-\eta)c_i^S+\eta c_i^T,\,6i(w_1-w_3)\big),
+ \qquad D_i(s)=\operatorname{diag}(r_i,r_i^{-1},1).
+\]
+The affine path is
+$G_i(s,y)=C_i(s)+D_i(s)(y-(c_i^S,0))$.
+Choose $\epsilon$ to be one quarter of the minimum of 1 and all
+pairwise max-norm distances in the two planar lists, omitting empty
+pair lists.  The $\epsilon$-padded moving boxes are disjoint: source
+and target gaps are at least $4\epsilon$ during the vertical phases,
+and the middle-phase height spacing is $6>2(1+\epsilon)$.
+
+For later loading formulas, record the centered cutoff explicitly:
+\[
+ \chi(C,L,\epsilon;x)=
+ \prod_{j=1}^3\sigma\!\left(\frac{L_j+\epsilon+x_j-C_j}{\epsilon/2}\right)
+ \sigma\!\left(\frac{L_j+\epsilon-x_j+C_j}{\epsilon/2}\right).
+\]
+Here $L$ is the vector of half-widths; the plateau has padding
+$\epsilon/2$ and the support has padding $\epsilon$.
+If $L_i(s)$ are the moving half-widths, $Y=x-C_i(s)$ and
+$A_i=D_i'D_i^{-1}$, the field supplied by
+\eqref{eq:bb-potential} is
+\begin{equation}\label{ba:affine-box-field}
+ V(s,x)=\sum_i\nabla\times\left\{
+ \chi(C_i,L_i,\epsilon;x)
+ \left[\tfrac12 C_i'\times Y+\tfrac13(A_iY)\times Y\right]
+ \right\}.
+\end{equation}
+Indeed $\det D_i=1$, and the preceding separation verifies
+Lemma~\ref{lem:bb-curl} with localization parameter $\epsilon/2$.
+Its neighborhood conclusion applies with
+$L_* =\max_i\max(1,\lambda_i,\lambda_i^{-1})$ because
+$\|D_i(s)\|_\infty\le L_*$ and $D_i(0)=I$.
+It therefore gives the stated affine motion on the full boxes and
+on their initial max-norm neighborhoods of radius $\epsilon/(4L_*)$.
+The paths are stationary for $s\le1/12$ and $s\ge11/12$.
+Their finite center ranges and endpoint width bounds give common
+compact support.  The effective positive lower bound
+$r_i\ge\min(1,\lambda_i)$ verifies the scale requirement.
+For the linear scale, the first-coordinate identity is
+\[
+ (G_i(s,y))_1=(1-\eta)y_1+\eta(F_i(y_1,y_2))_1.
+\]
+The second coordinate need not interpolate its own endpoint values.
+Instead, if $b_{i2}$ is the source's second half-width, convexity of the
+reciprocal function gives
+\begin{equation}\label{eq:ba-reciprocal-envelope}
+ \frac{b_{i2}}{(1-\eta)+\eta\lambda_i}
+ \le (1-\eta)b_{i2}+\eta\frac{b_{i2}}{\lambda_i}.
+\end{equation}
+The first half-width is exactly its endpoint interpolation. For the
+exponential scale, convexity of the exponential gives the corresponding
+upper bounds in both coordinates, since
+$\lambda_i^{\pm\eta}\le(1-\eta)+\eta\lambda_i^{\pm1}$.
+In either case the center is interpolated linearly and each half-width
+is at most the interpolation of its endpoint half-widths. Therefore each
+lower edge is at least the interpolation of the endpoint lower edges,
+and each upper edge is at most the corresponding interpolation of upper
+edges. This proves the claimed bounding rectangle for the entire sweep.
+The individual endpoint-width bounds also follow directly from monotonicity
+of the positive scales and their reciprocals.
+\end{proof}
+
+The same construction applies to $P_i\times[-h,h]$ for any common
+$h\ge0$, including plates and rectangles with zero side lengths, under
+the hypotheses of Lemma~\ref{lem:bb-lift-parking}(i). Replace $6i$ by
+heights $H_i$ whose pairwise gaps exceed $2h+2\varepsilon$, and choose
+$\varepsilon$ below one third of every within-family planar gap. The
+formula for $C_i$ uses the source and target centers in place of
+$c_i^S,c_i^T$; its support padding is $\varepsilon$ and plateau padding
+is $\varepsilon/2$. Source projections, distinct heights and target
+projections separate the three phases in that order. Empty pair lists
+impose no gap restriction, and the empty family gives the zero field.
+Arbitrary real data give smooth existence; the effective-data assumptions
+of the shared lemma give computable scales, derivative bounds and margins.
+The same perturbation argument gives an open neighborhood even when a
+side length is zero.
+
+\subsection{Transporting an initial cutoff}
+\label{sec:ba-transported-localization}
+
+The transported-initial-cutoff convention is also useful.  Write the
+affine path as $G_i(t)x=J_i(t)x+g_i(t)$ and localize by
+$\psi_i(G_i(t)^{-1}x)$, where $\psi_i$ has support padding
+$\varepsilon_0$ around the initial box and equals one on a smaller
+positive padding.
+If $\lambda_i\in[B^{-1},B]$, its initial padding
+$\varepsilon_0$ expands horizontally by at most $B$.  Choose
+$2B\varepsilon_0$ below every target gap,
+$2\varepsilon_0$ below every source gap, and
+$\varepsilon_0\le\varepsilon$.  These transported supports stay
+separated in the same three phases.  The affine generator is
+$v_i+A_ix$, with $A_i=\dot J_iJ_i^{-1}$ and
+$v_i=\dot g_i-A_ig_i$.  Its potential is
+\[
+ \tfrac12v_i\times x+\tfrac13(A_ix)\times x.
+\]
+The identity proved in Lemma~\ref{lem:bb-curl} gives the exact
+generator on the transported plateau.  Its neighborhood argument then
+proves the same full-box endpoint map.  The factor $B$ in the margin
+choice is essential: transported padding does not stay fixed under
+stretching.
+
+\subsection{Independent planar interpolation}
+
+For an observer that needs both planar coordinates to follow their own
+endpoint interpolations, the reciprocal path is insufficient. The next
+construction interpolates the two factors independently and compensates
+the temporary planar area change in the third direction. Here the named
+targets may be disjoint containers of the actual affine images; only the
+image centers enter the motion.
+
+\begin{lemma}[Independent planar scales and normal compensation]
+\label{ba:plate-routing}
+Let $P_i,Q_i\subset\R^2$ be finite families of closed rectangles,
+separately pairwise disjoint, and let assigned positive diagonal affine
+maps $F_i$ satisfy $F_i(P_i)\subseteq Q_i$. Write their diagonal factors as
+$\lambda_{i1}^*,\lambda_{i2}^*>0$ with product one.
+There is a smooth compactly supported solenoidal unit-time field, zero on
+time collars, whose endpoint map is $(y,z)\mapsto(F_i(y),z)$ on a
+neighborhood of each entire plate $P_i\times\{0\}$. During the horizontal
+phase both planar factors interpolate linearly, while the normal factor
+compensates their product. Each planar coordinate of a point on the plate
+is its convex endpoint interpolation.
+\end{lemma}
+\begin{proof}\label{ba:plate-routing-proof}
+Use the height phases in Section~\ref{sec:ba-localization}. For source
+center $p_i$, end its center path at its actual image $F_i(p_i)$; in the
+onto case this is the named target center. Choose separated heights $H_i$
+as above and set
+\[
+ c_i=\bigl((1-w_2)p_i+w_2F_i(p_i),\,H_i(w_1-w_3)\bigr),
+\]
+\[
+ \lambda_{ij}=1-w_2+w_2\lambda_{ij}^*,\qquad
+ D_i=\operatorname{diag}\bigl(
+ \lambda_{i1},\lambda_{i2},(\lambda_{i1}\lambda_{i2})^{-1}\bigr).
+\]
+The volume-preserving path has logarithmic derivative
+\[
+ T_i=\dot D_iD_i^{-1}=\operatorname{diag}\left(
+ \frac{\dot\lambda_{i1}}{\lambda_{i1}},
+ \frac{\dot\lambda_{i2}}{\lambda_{i2}},
+ -\frac{\dot\lambda_{i1}}{\lambda_{i1}}
+ -\frac{\dot\lambda_{i2}}{\lambda_{i2}}\right).
+\]
+This matrix has trace zero, so \eqref{eq:bb-potential} localized
+around the moving plate gives the desired field.  Its planar motion is
+$c_i+(\lambda_{i1}(y_1-p_{i1}),\lambda_{i2}(y_2-p_{i2}),0)$.
+Normal displacement changes by the factor
+$(\lambda_{i1}\lambda_{i2})^{-1}$. The factors have positive
+lower bounds; choose the initial thickness less than their product's
+minimum times the normal plateau width.  A small enlargement of the
+planar rectangle is allowed by the positive collars as well.  Thus a
+whole thin neighborhood follows the volume-preserving affine motion.
+The normal factor returns to one at the endpoint. The final plate image
+is contained in $Q_i\times\{0\}$. Source and target-container
+gaps protect the vertical phases and distinct heights protect the
+middle phase.  Linear interpolation of each planar factor gives both
+convex-coordinate identities on the plate.
+This proves smooth existence for arbitrary real data. Under the effective
+input assumptions of Lemma~\ref{lem:bb-curl}, the positive scale bounds and
+finite gap margins can be computed and the construction is effective.
+\end{proof}
+
+\subsection{Forcing and effective evaluation}
+
+\begin{lemma}[Residual forcing for the finite processors]\label{ba:realization}
+Fix computable $\nu>0$.  Let $W$ be smooth, solenoidal, initially zero,
+with support in one compact subset of $\mathbb R^3$ and every mixed
+derivative bounded.  Then
+\begin{equation}\label{ba:residual}
+ F_W=W_t+(W\cdot\nabla)W-\nu\Delta W
+\end{equation}
+has the same support and derivative properties.  The zero-data solution
+$(W,0)$ is unique among smooth solutions with
+$u\in C_tH^2\cap C_t^1L^2$, bounded $u$ on finite intervals, and
+$p\in C_tH^1$.  The corresponding torus assertion holds in the
+classical comparison class of Section~\ref{sec:model}.  A sum of curls
+in a torus chart gives both $W$ and $F_W$ mean zero.  Finite formulas
+from the effective-data instances of the preceding constructions, finite
+loading, and repetition of one cycle give effective evaluation of every
+force derivative.
+\end{lemma}
+\begin{proof}\label{ba:realization-proof}
+Fixed compact support and bounded mixed derivatives imply all reference
+velocity hypotheses of Lemma~\ref{lem:b-comparison}.  Case (i) gives
+exactly the stated whole-space comparison class, without a competing
+gradient bound; its torus clause gives the stated classical comparison
+class.  The same lemma provides the pressure normalization and global
+material flow.  The product rule applied to \eqref{ba:residual}
+proves the force's support and derivative bounds.  A chart curl has
+zero mean, and integration of the residual uses
+$\int\Delta W=0$ and
+$\int(W\cdot\nabla)W=\int\operatorname{div}(W\otimes W)=0$.
+For these effective-data instances, gate derivatives and seam evaluation are
+those of \eqref{eq:p2-step} and Proposition~\ref{prop:bb-realization}.
+The explicit affine scales and every reciprocal denominator have positive
+computable bounds.  Thus differentiating the finite formulas, using a
+finite superset of possible period indices, evaluates every derivative
+without a machine execution or a real equality test at a seam.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/solid-routing.tex

@@ -1,668 +1,668 @@
-\section{Evacuation, columns and obstacle detours}\label{sec:solid-routing}
-The following refinements choose the occupied regions and the storage order
-explicitly. Ordered low extraction controls a height observer; simultaneous
-lifting controls a first-coordinate observer; expanded centers permit
-three-dimensional detours. Fixed columns and recorded pointers give further
-ways to keep a whole transition outside a detector. Each proof specifies
-which parts of the geometry act on solid neighborhoods and which bounds
-are needed only for the coded central plane.
-
-Unless an individual theorem specifies another class, whole-space
-comparison means smooth $u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$
-on finite intervals, and a pressure representative in $CH^1$, as in
-Section~\ref{bc:analytic-section}. The torus comparison class is the
-classical class of Lemma~\ref{lem:b-comparison}. The distinct pressure
-conditions stated below remain part of their respective results.
-
-\subsection{Ordered extraction below the computation}\label{bc:ordered-section}
-\begin{proposition}[Ordered full-box transport]\label{bc:ordered}
-Let $D_i,S_i$ be finite families of closed rational axis-aligned boxes in $\R^3$
-with positive side lengths, pairwise disjoint within each family.
-Suppose the assigned center-to-center affine map $F_i$ satisfies
-$F_i(D_i)=S_i$ and has positive diagonal linear part $A_i$ of
-determinant one. There is an effective compactly supported solenoidal
-unit-time field, zero near the time endpoints, whose time-one map has
-this action on every full $D_i$. If both centers of a pair have negative
-third coordinate, every point of $D_i$ whose third coordinate equals that
-of its center follows a path with negative third coordinate.
-\end{proposition}
-\begin{proof}\label{bc:ordered-proof}
-Choose an integer $L>1$ larger than the absolute values of all endpoint
-coordinates. The storage columns have first coordinates $U_i=4(i+1)L$
-and common height $Z=-4L$. Write $[\ell_i,r_i]$ for the first-coordinate
-interval of $D_i$. Extract sources in decreasing order of $r_i$.
-Translate each horizontally right to its column and then vertically to
-$Z$, retaining its second coordinate. If an unextracted box meets the
-active box in both the second and third projections, disjointness forces
-their first intervals to be separated. That box cannot lie to the right:
-its right endpoint would then exceed $r_i$, contrary to the extraction
-order. It lies strictly to the left of $\ell_i$ and misses the entire
-rightward sweep. Stored boxes lie below the original boxes; vertical
-moves occur in distinct columns outside them.
-
-At storage height, adjust the second center coordinate to its target
-value and apply $A_i^{\theta}$. Each intermediate half-width lies between
-its two endpoint values, hence is less than $L$. The columns remain
-disjoint throughout these operations. Deliver boxes in increasing order
-of the right endpoint of their target first interval: raise a box to its
-target height and translate it left into its target. The raising column
-is empty. As above, a previously delivered box whose last two intervals
-meet the active ones must be separated in the first coordinate. Its
-smaller right endpoint puts it wholly to the left of the active target
-interval, so the leftward sweep stops before reaching it.
-Undelivered boxes remain at storage height. These observations give
-positive rational clearances for all swept bounding boxes. Choose cutoff
-padding below each clearance and use Lemma~\ref{lem:bb-curl} in successive
-flat slots. Induction over the slots proves the full-box assertion.
-A point initially in the central third-coordinate plane retains zero
-third-coordinate offset during each diagonal deformation. Its height is
-interpolated only between the two endpoint heights and $Z$. If those
-endpoint heights are negative, every intermediate height is negative.
-\end{proof}
-
-The onto hypothesis in the preceding proposition is a useful exact-box
-specialization. Its target-container version, including the effective
-input conditions, is given in Section~\ref{bc:contained-targets}.
-
-\begin{theorem}[A positive-height observation]\label{bc:positive-height}
-On $\R^3$, the fixed label $0$ and observation $\{x_3>0\}$ realize
-halting effectively by a compactly supported general force with zero
-initial velocity and pressure, bounded mixed derivatives, and period one
-after time one. The force has a separate alternative in
-$L^2(0,\infty;H^j)$ for every $j$, tending to zero in every spatial
-supremum norm at rate $O((1+t)^{-1})$.
-\end{theorem}
-\begin{proof}\label{bc:positive-height-proof}
-Use Lemma~\ref{lem:bb-movefirst} and odd digits in base $2m+2$. Give every
-simulator state a distinct positive integer index $i_s$ and put its
-third-coordinate center at $z_s=i_s$ for ready halting states and
-$z_s=-i_s$ otherwise. Thicken the full rectangles of
-Lemma~\ref{lem:bb-onecell} by $[z_s-1/4,z_s+1/4]$. The assigned third
-scale is one, so the maps satisfy Proposition~\ref{bc:ordered}.
-The rational initial code $Y_0$ is loaded from the origin along the
-straight segment, by a localized translation during $[0,1]$. Repeat the
-unit-time mechanism after time one. At time $1+k$ the particle is exactly
-the code after $k$ defined simulator transitions. If the initial machine
-halts, the loading endpoint already has positive height. If it never
-halts, loading has nonpositive height and all later transitions have
-negative endpoint centers; the proposition excludes a positive height
-at every intervening time. Conversely, a halting checkpoint has positive
-height. Lemma~\ref{bc:analytic} gives the force and uniqueness, and
-Lemma~\ref{ba:log-clock} gives the separate temporal alternative.
-\end{proof}
-
-\subsection{Simultaneous lifting with a convex-coordinate guard}
-\label{bc:simultaneous-section}
-\begin{proposition}[Simultaneous full-box transport]\label{bc:simultaneous}
-Suppose finite planar source and target rectangle families $S_i,T_i$
-have positive side lengths and endpoints on the grid $B^{-2}\Z^2$,
-with integer $B\ge2$, and are separately
-disjoint. Require their assigned center-to-center maps to satisfy
-$F_i(S_i)=T_i$ and to have linear part $\diag(\lambda_i,\lambda_i^{-1})$ with
-$\lambda_i\in\{B^{-1},1,B\}$. There is an effective solenoidal motion
-of the full boxes obtained by taking the product with $[-1,1]$.
-For every transported point its first coordinate is a convex combination
-of its initial and final first coordinates at every time.
-\end{proposition}
-\begin{proof}\label{bc:simultaneous-proof}
-Let $p_i^-,p_i^+$ be the centers in the plane $z=0$, set $h_i=5i$, and
-choose a smooth $\Theta$ equal to zero on $(-\infty,1/4]$ and one on
-$[3/4,\infty)$. For $0\le t\le1$, put
-\[
- \theta=\Theta(3t-1),\quad l_i=1-\theta+\theta\lambda_i,
- \quad c_i=(1-\theta)p_i^-+\theta p_i^+
-       +h_i\{\Theta(3t)-\Theta(3t-2)\}e_3.
-\]
-The desired path is $g_i(t,y)=c_i(t)+\diag(l_i,l_i^{-1},1)(y-p_i^-)$.
-Use a source padding $\varepsilon=1/(10B^3)$, transported by this same
-affine map, for the cutoff neighborhood. During lifting the planar
-source gap is at least $B^{-2}$. During lowering the enlarged target
-padding is at most $B\varepsilon<B^{-2}/2$. During the middle phase
-the height difference is at least five, whereas the sum of third
-half-widths is $2+2\varepsilon<5$. Thus these moving cutoff supports
-can be chosen disjoint. Sum the localized curls of
-Lemma~\ref{lem:bb-curl}. Differentiation and uniqueness give the displayed
-paths for the full boxes. Finally
-\[
- g_{i,1}(t,y)=(1-\theta)y_1+
- \theta\{p_{i,1}^++\lambda_i(y_1-p_{i,1}^-)\},
-\]
-which proves the asserted guard. Flat time collars hold automatically.
-\end{proof}
-
-\begin{theorem}[A zero-blank half-space test]\label{bc:zero-blank}
-The fixed label $0$ and event $\{x_1<-1\}$ in $\R^3$ realize halting
-with the force and comparison conclusions of Theorem~\ref{bc:positive-height}.
-This construction uses a zero joint-blank digit, so every initial tape
-coordinate is a finite radix sum. Its temporal alternative belongs to
-$L^2(0,\infty;H^j)$ for every $j$.
-\end{theorem}
-\begin{proof}\label{bc:zero-blank-proof}
-Use the frontier recorder, zero-blank base $B=2m$, and state first-coordinate
-offsets $3j$ for nonterminal controls and $-3j$ for terminal controls,
-with positive distinct indices. Encode a state by $(o_s+L,R,0)$.
-The full branches are on the $B^{-2}$ grid and satisfy the preceding
-proposition. Load the rational initial code from zero along a straight
-segment. Nonhalting codes have nonnegative first coordinate; the whole
-segment and every nonhalting transition therefore avoid $x_1<-1$.
-Terminal codes have first coordinate at most $-2$, including an initially
-terminal loading endpoint. Exact integer-time simulation gives both
-implications. The force conclusions follow from the common analytic and
-logarithmic-clock lemmas, with the compact support needed for all $H^j$
-norms. No claim of periodicity is made for the slowed alternative.
-\end{proof}
-
-\subsection{Expanded centers and obstacle detours}\label{bc:expanded-height-section}
-\begin{proposition}[Full solid boxes with a height guard]
-\label{bc:expanded-height-routing}
-Let the source and target families be finite collections of full rational
-axis-aligned solid boxes, each family pairwise disjoint. Write their centers
-as $a_i,b_i$ and their positive half-widths as $r_{i,j},w_{i,j}$.
-Require each prescribed affine map to send its entire source box onto
-its specified target box:
-\[
- F_i(a_i+h)=b_i+D_i h,\qquad
- D_i=\diag(w_{i,1}/r_{i,1},w_{i,2}/r_{i,2},w_{i,3}/r_{i,3}),
- \qquad \det D_i=1.
-\]
-Suppose every source center has height at most $-1$, and every target
-center has height at most $-1$ or exactly $1$. They admit an effective
-compactly supported solenoidal unit-time realization. If a target center
-has negative height, every point $a_i+h$ of its source box with $h_3=0$
-remains at negative height throughout. A rational bound for the
-entire support is computable from the boxes.
-\end{proposition}
-\begin{proof}\label{bc:expanded-height-routing-proof}
-The empty family is realized by zero. Otherwise let $R=2+$ the largest
-endpoint half-width, and choose an integer $H>20R$ so large that the
-expanded source centers $A_i^*=Ha_i$ and target centers $B_i^*=Hb_i$
-are separately more than $20R$ apart in the sup norm. Set
-\[
- Z_0=1+100R+\max_i\{\|A_i^*\|_\infty,\|B_i^*\|_\infty\},
- \qquad P_i=(0,0,-Z_0-100Ri).
-\]
-Each of the unions $\{A_i^*\}\cup\{P_i\}$ and
-$\{B_i^*\}\cup\{P_i\}$ has separation greater than $20R$.
-The construction has five phases: expand source centers by a factor
-from one to $H$ while keeping shapes fixed; move them successively to
-$P_i$; change each parked shape by the prescribed positive diagonal
-power; move them successively to $B_i^*$; contract target centers to
-$b_i$ while keeping the final shapes fixed.
-
-The simultaneous expansion is collision free. For each pair of source
-boxes some coordinate has positive gap
-$|a_{i,j}-a_{k,j}|-r_{i,j}-r_{k,j}$; multiplication of the center
-difference by a factor at least one preserves that gap. The same
-argument applies to contraction toward the disjoint targets. Choose
-padding less than one quarter of the minimum such gap, and less than
-one. During a parked deformation the $j$th half-width is
-$r_{i,j}^{1-\theta}w_{i,j}^{\theta}\le\max(r_{i,j},w_{i,j})<R$.
-Thus parked deformations have all half-widths less than $R$ and
-supports of half-width at most $2R$; the storage separation suffices.
-
-For a successive transfer, surround each stationary center by its closed
-sup-norm cube of radius $4R$. These cubes are pairwise disjoint. Start
-with the straight segment between the moving centers. Each nontrivial
-intersection with such a cube can be replaced by a polygonal path on
-its boundary: join the entrance to a vertex of its face, follow cube
-edges to an exit-face vertex, and join to the exit. Fixed index orders
-choose the vertices and edges. All coordinates and intersection parameters
-are rational. Tangencies need no detour. Since different obstacle cubes
-are disjoint, the resulting finite polygon stays at distance at least
-$4R$ from every stationary center. Move along its segments in flat slots,
-using a cutoff support of half-width at most $2R$. Stationary boxes have
-half-width less than $R$, so no such support touches them. Lemma~\ref{lem:bb-curl}
-realizes each translation and deformation.
-
-For the height assertion, a transfer with negative endpoints has endpoints
-of height at most $-H$. Its straight segment is negative. It cannot meet
-an obstacle centered at positive height $H$, whose bottom is $H-4R>0$.
-Every boundary used around a negative-height obstacle lies below
-$-H+4R<0$. Storage heights are negative as well. Expansion, contraction,
-and positive diagonal deformation preserve the central-plane condition.
-Thus all these paths are negative. Finitely many rational vertices,
-endpoint widths, and cutoff paddings give a rational support bound.
-\end{proof}
-
-\begin{theorem}[A height test inside one torus chart]\label{bc:expanded-height-event}
-On the unit torus, the fixed label $o=(1/2,1/2,1/2)$ and event
-\begin{equation}\label{bc:expanded-height-fixed-event}
- \{x:x_3\pmod1\in(1/2,1)\}
-\end{equation}
-realize halting by a smooth general mean-zero force, periodic after time
-one, with zero initial velocity and pressure and bounded mixed derivatives.
-The repeated mechanism depends on the machine alone; only initialization
-depends on its finite input. The solution is unique in the torus class.
-\end{theorem}
-\begin{proof}\label{bc:expanded-height-event-proof}
-Use the frontier recorder, odd base $B=2m+1$, distinct negative integer
-heights for nonterminal states and height one for the terminal ready state.
-Thicken every branch by a third half-width $1/4$. The reciprocal tape
-map is onto its target rectangle; adjoining the unchanged third interval
-gives an onto box map with linear part $\diag(B^{-e},B^e,1)$ and
-determinant one. All source centers are
-negative because terminal controls have no outgoing rules. Apply the
-preceding proposition. The logical initial code $Y_0$ is rational and has
-a uniform machine-dependent bound as the input varies: its two stack
-coordinates lie in $[0,1]$ and its height is fixed by the initial control.
-A moving translation loads the logical origin to $Y_0$, with a fixed
-source half-width one and padding one. This gives a machine-dependent
-rational bound $K$ for the supports of all such loaders and the repeated
-mechanism; for example take $K=1+2R+M$ with $M$ an upper bound for all
-routing vertices and loading coordinates and $R$ enlarged to cover their
-half-widths. Choose $\lambda=1/(4K)$ and use the chart $x=o+\lambda Y$.
-Everything remains strictly inside $(1/4,3/4)^3$.
-
-Periodize the resulting compactly supported curl fields, loading during
-the first unit interval and repeating thereafter. Compute the residual
-at physical viscosity $\nu$, after the chart change. The fields and force
-have zero mean and all the bounds in Lemma~\ref{bc:analytic}.
-For a nonhalting run, loading has nonpositive logical height and each
-later segment remains negative by the proposition. A terminal code has
-physical height $1/2+\lambda$, which lies in the event. There is no
-chart-boundary crossing that could create a spurious observation.
-An initially terminal input is detected at the loading endpoint.
-The initialized correspondence is exact; at checkpoints the number of
-history records gives $n$, and the frontier's relative position together
-with its absolute index $2+n$ also recovers the absolute work-head position.
-\end{proof}
-
-\subsection{Fixed columns and separate layers}\label{bc:columns-section}
-\begin{proposition}[Column and layer transport]\label{bc:columns}
-Let $S_i,T_i$ be finite rational planar rectangle families of positive side
-lengths, each separately disjoint, with endpoints on the $B^{-2}$ grid,
-where $B\ge2$ is an integer.
-Let their assigned center-to-center positive reciprocal diagonal affine
-maps satisfy $F_i(S_i)=T_i$. A finite effective
-solenoidal field in $\R^3$ realizes all these maps on the rectangles in
-$z=0$. Along each tracked path the first coordinate stays between its
-initial and final values. The field has compact support and flat time
-collars.
-\end{proposition}
-\begin{proof}\label{bc:columns-proof}
-Give rectangle $i$ height $h_i=4i$. A planar cutoff equal to one near
-its source and supported within padding $1/(4B^2)$ has support disjoint
-from the other source cutoffs. Multiply it by a third-coordinate cutoff
-which is one near the entire interval $[0,h_i]$. The curl of the potential
-$(0,h_i x,0)$ times these cutoffs equals $(0,0,h_i)$ along that source
-column. A common flat schedule lifts all rectangles. The corresponding
-target cutoffs give a later descent by $-h_i$.
-
-In the middle phase, let $p_i,q_i$ be the centers and $a_i>0$ the
-first-coordinate scale. With phase parameter $\mu\in[0,1]$, set
-$c_i=p_i+\mu(q_i-p_i)$, $l_i=1+\mu(a_i-1)$ and
-\[
- H_i(\mu,x,y)=d_{i,1}y-d_{i,2}x+
-       \frac{a_i-1}{l_i}(x-c_{i,1})(y-c_{i,2}),
- \qquad d_i=q_i-p_i.
-\]
-On a neighborhood of the desired path, the planar Hamiltonian field
-$(\partial_yH_i,-\partial_xH_i)$ generates
-$c_i+\diag(l_i,l_i^{-1})(\zeta-p_i)$.
-The first coordinate is the convex interpolation of its endpoint values.
-The second half-width obeys
-\[
- \frac{r_{i,2}}{(1-\mu)+\mu a_i}
- \le (1-\mu)r_{i,2}+\mu r_{i,2}/a_i,
-\]
-by convexity of the reciprocal function. Consequently the swept rectangle
-lies in the coordinate bounding rectangle of the two endpoints. Localize
-$H_i$ around that rectangle in the layer $z=h_i$, with third padding less
-than one, and take the curl of $(0,0,H_i)$ times that cutoff. For a
-flat middle ramp $\mu=\eta_2(t)$, multiply the resulting phase field by
-$\eta_2'(t)$. Layers are disjoint, so the sum has the required action. Lift and descent preserve
-planar coordinates. The exact paths and ODE uniqueness establish the
-full-rectangle and intermediate-coordinate assertions.
-\end{proof}
-
-\subsection{Target containers and prescribed affine images}
-\label{bc:contained-targets}
-The named targets in Propositions~\ref{bc:ordered}, \ref{bc:simultaneous}
-and~\ref{bc:columns} can also be rational \emph{containers} for the actual
-images. More precisely, replace the onto condition by
-\[
- F_i(D_i)\subseteq S_i\quad\text{in Proposition~\ref{bc:ordered}},
- \qquad
- F_i(S_i)\subseteq T_i\quad\text{in the other two propositions}.
-\]
-Keep the other geometric hypotheses. For ordered transport retain the
-center-to-center condition. For simultaneous and column transport the
-actual endpoint center is $F_i(p_i)$, where $p_i$ is the source center;
-it need not be the center of the container. The maps have the same positive
-diagonal determinant-one linear parts as before, including the specified
-three scale choices in Proposition~\ref{bc:simultaneous}.
-All transport and path conclusions remain valid. For arbitrary real map
-coefficients this is a smooth existence statement. The construction is
-effective when those coefficients are computable, and otherwise its
-evaluation is relative to them. An empty list again uses the zero field.
-
-Here are the necessary modifications of the proofs. In ordered transport,
-write $r_{i,j}$ for a source half-width, $w_{i,j}$ for the corresponding
-container half-width, and $a_{i,j}>0$ for the diagonal factor. Containment
-gives $a_{i,j}r_{i,j}\le w_{i,j}$, hence
-\[
- r_{i,j}a_{i,j}^{\theta}
- \le\max(r_{i,j},a_{i,j}r_{i,j})
- \le\max(r_{i,j},w_{i,j})<L,\qquad 0\le\theta\le1.
-\]
-The rational source and container widths therefore bound every parked
-deformation. After deformation, use translates of the rational target
-containers as envelopes for the actual images. Deliver in increasing
-order of the containers' right endpoints. If an earlier delivered image
-meets the moving image in the last two projections, their containers meet
-in those projections. Disjointness and the endpoint order force the earlier
-container wholly to the left of the active target container. Its image
-therefore misses the entire leftward sweep, which stops in that active
-container. The original extraction and storage arguments are unchanged.
-The same rational envelopes give positive cutoff margins. They do not
-require the actual image endpoints to be rational. The central-cross-section
-height guard is unchanged because its third offset remains zero.
-
-For simultaneous transport, interpolate toward $F_i(p_i)$. The transported
-source padding at descent has width at most $B\varepsilon$ outside its
-actual image, so it lies inside the $B\varepsilon$ enlargement of the
-target container. The original grid-gap and private-height estimates
-still separate all supports. The first-coordinate convex identity uses
-the actual value $F_i(y)$ and remains exact.
-
-For column transport use the same actual endpoint center. The reciprocal
-convexity estimate bounds each moving second half-width by the linear
-interpolation of the source and actual-image half-widths. Thus the entire
-middle sweep lies in the coordinate bounding rectangle of the source and
-its rational target container. The target-column cutoff equals one on
-the actual image, so the descent is unchanged. This also proves the stated
-first-coordinate bound for each point. Finally, a positive computable
-scale admits computable positive lower and finite upper bounds; its
-logarithm, powers and the displayed reciprocal interpolations are
-computable. The containment and determinant identities are hypotheses,
-not tests performed by the construction.
-
-\begin{theorem}[A half-space test by fixed columns]\label{bc:column-event}
-On $\R^3$, the frontier recorder with odd digits in base $2m+1$ realizes
-halting from the fixed label $0$ by the event $x_1<-1$. It gives a compactly
-supported general force, periodic after time one, with zero initial
-velocity and pressure and all mixed derivatives bounded.
-\end{theorem}
-\begin{proof}\label{bc:column-event-proof}
-Put nonterminal state offsets at $3,6,\ldots$ and terminal offsets at
-$-3,-6,\ldots$ in the first coordinate. Lemma~\ref{lem:bb-onecell} and
-Proposition~\ref{bc:columns} give a whole-rectangle phase map. A localized
-planar Hamiltonian translation loads the origin to the rational initial
-code in $[0,1]$ time. A nonterminal loading segment has nonnegative first
-coordinate, and every later nonterminal segment has first coordinate at
-least three. Terminal codes have first coordinate below $-1$. This proves
-both directions, including initial halting. Repeat the phase map and use
-Lemma~\ref{bc:analytic}. Although this route only requires planar coding
-rectangles, its cutoff plateaus also give a positive neighborhood on which
-the same affine formulas hold.
-\end{proof}
-
-\subsection{Delaying the original head movement}\label{bc:delayed-column-section}
-Use Lemma~\ref{lem:bb-direction} with singleton passive track removed.
-Rename $(M(q,s),G(r),A(q,d),B(q,d))$ as
-$(R_{q,s},F_r,C_{q,d},G_{q,d})$ and $(\bot,F)$ as $(\perp,*)$.
-These bijections identify every row and nonfrontier guard on the full
-domain, including all $q,d$ controls. The frontier starts at one, and the
-work movement occurs only on return. At checkpoint $n$, write $i$ for
-the work-head position and $j=n+1$ for the frontier position. The exact count is
-$2(j-i)+3$: the right and left continuing loops have respectively
-$j-i-1$ and $j-i$ instructions, with four other instructions.
-
-\begin{theorem}[The delayed-head column realization]\label{bc:delayed-column-event}
-The table of Lemma~\ref{lem:bb-direction} has a realization on $\R^3$
-with fixed label $0$ and detector $x_1<-1$, compact spatial support,
-period one after time one, bounded mixed derivatives and uniformly bounded
-kinetic energy. The prescribed pressure is zero. Uniqueness holds in the
-whole-space comparison class, in particular when the competitor and its
-first spatial derivatives are bounded on every finite time interval.
-\end{theorem}
-\begin{proof}\label{bc:delayed-column-event-proof}
-Assign odd digits $1,3,\ldots,2m-1$ in base $B=2m+2$, with state
-offsets positive multiples of three except for negative terminal offsets.
-The tail of the blank symbol contributes
-$d_{\rm blank}B^{-k}/(B-1)$ after a finite prefix, so initialization is
-rational even though the blank digit is nonzero. Full affine branches
-follow from the lemma and Lemma~\ref{lem:bb-onecell}. Use the fixed-column
-proof with heights $4i$ and the smaller padding $1/(10B^2)$; schedule a
-slot $[a,b]$ by $\sigma(2(t-a)/(b-a)-1/2)$, which is flat on collars
-of both endpoints. These choices preserve the same separation proof.
-A planar Hamiltonian loading segment starts at zero. The convex first-
-coordinate guard excludes the detector throughout every nonhalting
-transition and throughout a nonhalting load; a terminal code lies below
-$-1$. Finite checkpoint simulation gives the equivalence. The field has
-one compact support and uniform pointwise bounds, hence uniformly bounded
-kinetic energy. Lemma~\ref{bc:analytic} supplies all remaining assertions.
-\end{proof}
-
-\subsection{Recorded pointers and seven-slot storage}\label{bc:recorded-pointer-section}
-\begin{theorem}[A fixed box test with recorded pointers]
-\label{bc:recorded-pointer-event}
-On $\R^3$, the fixed label $(2,0,0)$ and open box
-$(0,1)^2\times(-1/2,1/2)$ realize halting by an effective general force
-with compact spatial support, zero initial velocity and pressure, bounded
-mixed derivatives, and period one after time one. The construction moves
-whole solid boxes. Uniqueness holds among smooth competitors with
-\[
- u\in C^1([0,T];L^2),\quad \nabla u,\Delta u\in C([0,T];L^2),
- \quad p,\nabla p\in C([0,T];L^2),\quad
- \sup_{[0,T]\times\R^3}(|u|+|\nabla u|)<\infty
-\]
-for every finite $T$.
-\end{theorem}
-\begin{proof}\label{bc:recorded-pointer-event-proof}
-Normalize to one halt $h$. For $c=(q,b)\in\mathcal D=(Q\setminus\{h\})\times\Gamma$
-write the instruction as $(q^+(c),b^+(c),m(c))$.
-Use histories
-\[
- \Lambda_0=\{\varnothing\}\sqcup\{H_e:e\in\mathcal D\sqcup\{\star\}\},
- \qquad \Lambda=\Lambda_0\sqcup\{P_e:e\in\mathcal D\sqcup\{\star\}\}.
-\]
-The full alphabet is $\mathcal A=\Gamma\times\Lambda\times\{0,1\}$,
-whose tracks are data, history and return mark. Controls are
-$S_q,L_q,M_c,R_c,F_c$, and the complete table is
-\[
-\begin{array}{c|c|c|l}
-\text{input}&\text{output}&\text{move}&\text{restriction}\\\hline
-S_q(b,l,0)&M_c(b^+(c),l,0)&m(c)&c=(q,b),\ l\in\Lambda_0\\
-M_c(b,l,0)&R_c(b,l,1)&1&l\in\Lambda_0\\
-R_c(b,l,0)&R_c(b,l,0)&1&l\in\Lambda_0\\
-R_c(b,P_e,0)&F_c(b,H_e,0)&1&e\in\mathcal D\sqcup\{\star\}\\
-F_c(b,\varnothing,0)&L_{q^+(c)}(b,P_c,0)&-1&\\
-L_q(b,l,0)&L_q(b,l,0)&-1&l\in\Lambda_0\\
-L_q(b,l,1)&S_q(b,l,0)&0&l\in\Lambda_0.
-\end{array}
-\]
-Initially put $P_\star$ at absolute cell three, empty history elsewhere,
-marks zero, and control $S_{q_0}$. At ready checkpoint $k$, the pointer
-is at $r=3+k$, all cells to its right are empty, and the work configuration
-is the original one. After the first row the work head satisfies
-$j'\le k+1\le r-2$. The scan marks it, reaches the pointer, changes
-$P_e$ into $H_e$, writes $P_c$ in the next empty cell, and returns to
-erase the mark. Both scans are finite and positive. This proves the
-initialized simulation. The initial pointer tag is retained as $H_\star$
-after the first cycle, so the absolute cell-three reference is permanent.
-
-The inverse works on all allowed tapes. Into $M_c$ the tag fixes the old
-instruction and move; into $R_c$ mark one distinguishes entry from a loop;
-into $F_c$ the written $H_e$ identifies the old pointer $P_e$; into $L_q$
-a pointer output identifies entry, including the old $F_c$, whereas a loop
-writes a nonpointer; into $S_q$ only mark erasure applies. Incoming shifts
-are fixed in every target control. The remaining written components retain
-all unchanged symbols, proving the two properties of
-Lemma~\ref{lem:bb-onecell}.
-
-Let $m_Q$ be the number of controls, and number them bijectively by
-$\iota(s)\in\{0,\ldots,m_Q-1\}$ with $\iota(S_h)=0$.
-Use odd digits in base $B=2|\mathcal A|+1$ and state offsets
-$o_s=3\iota(s)$. The physical
-code is $(o_s+L,R,0)$. Terminal codes lie strictly in $(0,1)^2\times\{0\}$;
-nonterminal first coordinates are at least three. Form the full branch
-boxes by taking the product with $[-1,1]$. Their planar sides have length
-at most one. Choose $H_x=3m_Q+1$ and
-parking centers $\pi_i=(H_x+10+4i,0)$, all at height zero.
-
-Here is a storage construction distinct from ordered low extraction.
-Evacuate each source in three successive operations: lift its center to
-height ten, translate horizontally there to $\pi_i$, and lower it to
-zero. Once every box is parked, process each in four operations: deform
-its planar shape at the parking center, lift it to height ten, translate
-it there to its target center, and lower it into the target. This uses
-$7N$ slots for $N$ branches. During deformation use the positive scale
-$l=1+\theta(\lambda_i-1)$ and its reciprocal, not an exponential scale.
-Both side lengths remain bounded by the maximum of their endpoint lengths,
-which is at most one. The third half-width remains one.
-
-Let $\delta$ be the minimum of one and all positive source-source and
-target-target planar distances, omitting empty sets of pairs. Use cutoff
-plateaus with padding $\varepsilon=\delta/8$ and support padding
-$2\varepsilon$. During a lift or descent the planar projection separates
-the moving box from the current sources or delivered targets. Parked
-boxes lie outside their entire first-coordinate range, and parking centers
-have spacing four. During a horizontal translation the box lies in
-$9\le z\le11$, whereas all stationary boxes lie in $-1\le z\le1$.
-These positive gaps also separate the padded supports. Lemma~\ref{lem:bb-curl}
-therefore realizes every operation without disturbing a stationary box.
-The full assigned affine map follows by composition.
-
-For a branch whose target control is nonterminal, every full point has
-first coordinate at least three, including the storage and transfer paths.
-Load the fixed label
-$(2,0,0)$ to the rational initial code by a localized straight translation.
-If the code is nonterminal, the loading segment has first coordinate at
-least two; if it is terminal, the endpoint is in the observation box.
-At later integer samples the exact recorder code gives terminal entry,
-and the preceding bounds exclude the observation at all other times of a
-nonhalting run. Repeat the finite transport field and apply
-Lemma~\ref{bc:analytic}. Finally the displayed comparison class implies
-$u\in C H^2$, since $\|u\|_{H^2}\le C(\|u\|_2+\|\Delta u\|_2)$
-by the Fourier multiplier identity; its pressure assumptions give $p\in C H^1$.
-Thus the stated uniqueness follows from the common whole-space comparison,
-without changing this route's original hypotheses.
-\end{proof}
-
-\subsection{Planar detours behind a vertical observation guard}
-\label{bc:guarded-planar-section}
-
-
-\begin{theorem}[A guarded planar observation with unrestricted pressure]
-\label{bc:guarded-planar-event}
-On $\R^3$, the fixed label $(-4,0,0)$ and fixed box
-$(-1,2)^2\times(-1,1)$ realize halting by a compactly supported general
-force with zero initial velocity and prescribed pressure and all mixed
-derivatives bounded. One choice is periodic after time one. A separate
-choice satisfies $\|f(t)\|_\infty=O((1+t)^{-2/3})$ and
-$f\in L^2([0,\infty)\times\R^3)$.
-Uniqueness holds among smooth competitors with
-$u\in C([0,T];H^2)\cap C^1([0,T];L^2)$ and bounded $u,\nabla u$
-on each finite interval, without any spatial integrability condition on
-pressure. Pressure is unique modulo a function of time.
-\end{theorem}
-\begin{proof}\label{bc:guarded-planar-event-proof}
-Use the frontier recorder after normalization to one halting state, and
-write $\mathcal A$ for its full cell alphabet and $m_Q$ for its number
-of controls. Number the controls bijectively by
-$\iota(s)\in\{0,\ldots,m_Q-1\}$, giving the terminal control index zero.
-Use odd base $B=2|\mathcal A|+1$ and first-coordinate offsets
-$o_s=4\iota(s)$. Code the
-configuration by $(o_s+L,R)$. The origin of the original tape can be
-recovered from the number of history records and the frontier position,
-so this head-relative coding loses no initialized information.
-The full planar branches have reciprocal positive diagonal maps.
-
-We first realize finite separately disjoint source and target rectangle
-families $S_i,T_i$ with rational endpoints and positive side lengths,
-equipped with assigned positive
-reciprocal diagonal affine maps $F_i$ satisfying $F_i(S_i)=T_i$, by
-compactly supported planar Hamiltonian motion. Write $c_i,q_i$ for the
-source and target centers. With no
-branches use zero. With one branch, interpolate its center directly and
-use a positive linear first-coordinate scale and its reciprocal; there
-are no stationary obstacles. For $N\ge2$, let $R=1+$ the largest endpoint
-half-width and let $m_0>0$ be the minimum sup-norm separation within the
-two center families. Put $K=1+32R/m_0$. Expand all source centers by
-factors from one to $K$, leaving their shapes unchanged. Let
-$M=\max_i\{|(Kc_i)_1|,|(Kq_i)_1|\}$ and choose parking centers
-$P_i=(M+32Ri,0)$. Each union of expanded endpoint centers and parking
-centers has separation at least $32R$.
-
-Move expanded sources successively to their parking centers, deform them
-there using the linear reciprocal scale, move them successively to the
-expanded target centers, then contract those centers to their targets.
-For each successive transfer put a square of sup-radius $4R$ around
-every stationary center. Replace each nontrivial segment intersection
-with an obstacle square by a counterclockwise path along its boundary.
-The intersection points and corners are rational; the squares are disjoint,
-so the resulting finite polygon stays at distance at least $4R$ from
-all stationary centers. Flat schedules on its segments give smooth center
-motion. The simultaneous expansion and contraction preserve a positive
-coordinate gap for every pair, exactly as in the solid-box argument.
-If $g_0,g_1$ are their minimum positive gaps, choose padding
-$\varepsilon=\min(g_0,g_1,2R)/4$ for these phases. During individual
-transfers use padding below $R$; the stationary half-widths are below $R$.
-Thus all required cutoff supports avoid stationary rectangles.
-
-For a moving center $c$ and horizontal scale $l$, the potential
-\[
- H=\chi\left[\dot c_x(y-c_y)-\dot c_y(x-c_x)
-                    +\frac{\dot l}{l}(x-c_x)(y-c_y)\right]
-\]
-gives the required motion on the cutoff plateau. The other current
-rectangles are fixed. This proves every full planar branch, including
-its boundary, by induction through the operations. A one-square loading
-translation with half-width $1/4$ takes $(-4,0)$ to the initial rational
-code. All these supports lie in a square $[-L,L]^2$, where one can choose
-$L=10+S_0+4m_Q$ with $S_0$ a finite bound for routing vertices and cutoff
-widths. The same $L$ covers every input of a fixed machine, since initial
-codes are uniformly bounded.
-
-The planar detours need not avoid the horizontal observation box.
-We instead make them at height three. Choose a switch $\eta$ with
-collars and set
-$h(s)=3\{\eta(3s)-\eta(3s-2)\}$ on a unit interval. It lifts in the
-first third, stays at three in the middle, and lowers in the last third.
-Choose $\Gamma$ equal to one near $[-L,L]^2\times[-1,4]$, with compact
-support, and $\beta(z)$ compactly supported and equal to one near three.
-For any one of the planar fields $w$, zero outside its phase interval,
-put
-\[
- Z(s,x)=h'(s)\nabla\times(0,x_1\Gamma(x),0),\qquad
- V_w(s,x)=Z(s,x)+3\beta(x_3)(w(3s-1,x_1,x_2),0).
-\]
-Along each tracked path the first term is exact vertical translation.
-The second term acts only in the middle third, at height three, and runs
-the full planar phase. It is divergence free because $w$ is planar
-solenoidal and $\beta$ is independent of the planar coordinates.
-Thus the trajectory lifts at its exact source, performs every detour
-outside the vertical detector interval, and lowers at its exact target.
-Use this construction for the loader and for each repeated phase. While
-its height can lie in $(-1,1)$, a nonterminal computational particle has
-first coordinate at least four; the initial loading source has first
-coordinate $-4$. A terminal code lies in $(0,1)^2\times\{0\}$.
-These facts prove the all-real-time observation equivalence, including
-an initially terminal input.
-
-For the comparison assertion, the constructed field has compact support
-and bounded mixed derivatives, so it satisfies the reference hypotheses
-of Lemma~\ref{lem:b-comparison}. The displayed competitor class is exactly
-case (v) of that lemma. Its proof obtains $\nabla p\in L^2$ from the
-equation and cancels the pressure term by Lemma~\ref{lem:b-gradient}; it
-does not require an integrable pressure representative. Hence the velocity
-is unique, and pressure is determined up to a function of time.
-
-Finally let $U(s)$ denote the eventually periodic field just constructed
-and set $\rho(t)=(1+t)^{1/3}-1$, $\widehat U=\rho' U(\rho(t))$.
-The new residual is
-\[
- \widehat f=\rho''U+(\rho')^2\{U_s+(U\cdot\nabla)U\}
-                          -\nu\rho'\Delta U,
- \quad \rho'=\tfrac13(1+t)^{-2/3},\quad
- \rho''=-\tfrac29(1+t)^{-5/3}.
-\]
-Uniform derivative bounds and the fixed compact support give the stated
-decay and space-time $L^2$ estimate. Chain differentiation also gives all
-mixed bounds. The clock is effective on $[0,\infty)$, starts at zero,
-is finite at finite time and is onto; hence it preserves precisely the
-same material event. The initial zero collar is preserved as well.
-This slower force is an alternative to, not a periodic version of, the
-first force.
-\end{proof}
+\section{Evacuation, columns and obstacle detours}\label{sec:solid-routing}
+The following refinements choose the occupied regions and the storage order
+explicitly. Ordered low extraction controls a height observer; simultaneous
+lifting controls a first-coordinate observer; expanded centers permit
+three-dimensional detours. Fixed columns and recorded pointers give further
+ways to keep a whole transition outside a detector. Each proof specifies
+which parts of the geometry act on solid neighborhoods and which bounds
+are needed only for the coded central plane.
+
+Unless an individual theorem specifies another class, whole-space
+comparison means smooth $u\in CH^2\cap C^1L^2$, bounded $u,\nabla u$
+on finite intervals, and a pressure representative in $CH^1$, as in
+Section~\ref{bc:analytic-section}. The torus comparison class is the
+classical class of Lemma~\ref{lem:b-comparison}. The distinct pressure
+conditions stated below remain part of their respective results.
+
+\subsection{Ordered extraction below the computation}\label{bc:ordered-section}
+\begin{proposition}[Ordered full-box transport]\label{bc:ordered}
+Let $D_i,S_i$ be finite families of closed rational axis-aligned boxes in $\R^3$
+with positive side lengths, pairwise disjoint within each family.
+Suppose the assigned center-to-center affine map $F_i$ satisfies
+$F_i(D_i)=S_i$ and has positive diagonal linear part $A_i$ of
+determinant one. There is an effective compactly supported solenoidal
+unit-time field, zero near the time endpoints, whose time-one map has
+this action on every full $D_i$. If both centers of a pair have negative
+third coordinate, every point of $D_i$ whose third coordinate equals that
+of its center follows a path with negative third coordinate.
+\end{proposition}
+\begin{proof}\label{bc:ordered-proof}
+Choose an integer $L>1$ larger than the absolute values of all endpoint
+coordinates. The storage columns have first coordinates $U_i=4(i+1)L$
+and common height $Z=-4L$. Write $[\ell_i,r_i]$ for the first-coordinate
+interval of $D_i$. Extract sources in decreasing order of $r_i$.
+Translate each horizontally right to its column and then vertically to
+$Z$, retaining its second coordinate. If an unextracted box meets the
+active box in both the second and third projections, disjointness forces
+their first intervals to be separated. That box cannot lie to the right:
+its right endpoint would then exceed $r_i$, contrary to the extraction
+order. It lies strictly to the left of $\ell_i$ and misses the entire
+rightward sweep. Stored boxes lie below the original boxes; vertical
+moves occur in distinct columns outside them.
+
+At storage height, adjust the second center coordinate to its target
+value and apply $A_i^{\theta}$. Each intermediate half-width lies between
+its two endpoint values, hence is less than $L$. The columns remain
+disjoint throughout these operations. Deliver boxes in increasing order
+of the right endpoint of their target first interval: raise a box to its
+target height and translate it left into its target. The raising column
+is empty. As above, a previously delivered box whose last two intervals
+meet the active ones must be separated in the first coordinate. Its
+smaller right endpoint puts it wholly to the left of the active target
+interval, so the leftward sweep stops before reaching it.
+Undelivered boxes remain at storage height. These observations give
+positive rational clearances for all swept bounding boxes. Choose cutoff
+padding below each clearance and use Lemma~\ref{lem:bb-curl} in successive
+flat slots. Induction over the slots proves the full-box assertion.
+A point initially in the central third-coordinate plane retains zero
+third-coordinate offset during each diagonal deformation. Its height is
+interpolated only between the two endpoint heights and $Z$. If those
+endpoint heights are negative, every intermediate height is negative.
+\end{proof}
+
+The onto hypothesis in the preceding proposition is a useful exact-box
+specialization. Its target-container version, including the effective
+input conditions, is given in Section~\ref{bc:contained-targets}.
+
+\begin{theorem}[A positive-height observation]\label{bc:positive-height}
+On $\R^3$, the fixed label $0$ and observation $\{x_3>0\}$ realize
+halting effectively by a compactly supported general force with zero
+initial velocity and pressure, bounded mixed derivatives, and period one
+after time one. The force has a separate alternative in
+$L^2(0,\infty;H^j)$ for every $j$, tending to zero in every spatial
+supremum norm at rate $O((1+t)^{-1})$.
+\end{theorem}
+\begin{proof}\label{bc:positive-height-proof}
+Use Lemma~\ref{lem:bb-movefirst} and odd digits in base $2m+2$. Give every
+simulator state a distinct positive integer index $i_s$ and put its
+third-coordinate center at $z_s=i_s$ for ready halting states and
+$z_s=-i_s$ otherwise. Thicken the full rectangles of
+Lemma~\ref{lem:bb-onecell} by $[z_s-1/4,z_s+1/4]$. The assigned third
+scale is one, so the maps satisfy Proposition~\ref{bc:ordered}.
+The rational initial code $Y_0$ is loaded from the origin along the
+straight segment, by a localized translation during $[0,1]$. Repeat the
+unit-time mechanism after time one. At time $1+k$ the particle is exactly
+the code after $k$ defined simulator transitions. If the initial machine
+halts, the loading endpoint already has positive height. If it never
+halts, loading has nonpositive height and all later transitions have
+negative endpoint centers; the proposition excludes a positive height
+at every intervening time. Conversely, a halting checkpoint has positive
+height. Lemma~\ref{bc:analytic} gives the force and uniqueness, and
+Lemma~\ref{ba:log-clock} gives the separate temporal alternative.
+\end{proof}
+
+\subsection{Simultaneous lifting with a convex-coordinate guard}
+\label{bc:simultaneous-section}
+\begin{proposition}[Simultaneous full-box transport]\label{bc:simultaneous}
+Suppose finite planar source and target rectangle families $S_i,T_i$
+have positive side lengths and endpoints on the grid $B^{-2}\Z^2$,
+with integer $B\ge2$, and are separately
+disjoint. Require their assigned center-to-center maps to satisfy
+$F_i(S_i)=T_i$ and to have linear part $\diag(\lambda_i,\lambda_i^{-1})$ with
+$\lambda_i\in\{B^{-1},1,B\}$. There is an effective solenoidal motion
+of the full boxes obtained by taking the product with $[-1,1]$.
+For every transported point its first coordinate is a convex combination
+of its initial and final first coordinates at every time.
+\end{proposition}
+\begin{proof}\label{bc:simultaneous-proof}
+Let $p_i^-,p_i^+$ be the centers in the plane $z=0$, set $h_i=5i$, and
+choose a smooth $\Theta$ equal to zero on $(-\infty,1/4]$ and one on
+$[3/4,\infty)$. For $0\le t\le1$, put
+\[
+ \theta=\Theta(3t-1),\quad l_i=1-\theta+\theta\lambda_i,
+ \quad c_i=(1-\theta)p_i^-+\theta p_i^+
+       +h_i\{\Theta(3t)-\Theta(3t-2)\}e_3.
+\]
+The desired path is $g_i(t,y)=c_i(t)+\diag(l_i,l_i^{-1},1)(y-p_i^-)$.
+Use a source padding $\varepsilon=1/(10B^3)$, transported by this same
+affine map, for the cutoff neighborhood. During lifting the planar
+source gap is at least $B^{-2}$. During lowering the enlarged target
+padding is at most $B\varepsilon<B^{-2}/2$. During the middle phase
+the height difference is at least five, whereas the sum of third
+half-widths is $2+2\varepsilon<5$. Thus these moving cutoff supports
+can be chosen disjoint. Sum the localized curls of
+Lemma~\ref{lem:bb-curl}. Differentiation and uniqueness give the displayed
+paths for the full boxes. Finally
+\[
+ g_{i,1}(t,y)=(1-\theta)y_1+
+ \theta\{p_{i,1}^++\lambda_i(y_1-p_{i,1}^-)\},
+\]
+which proves the asserted guard. Flat time collars hold automatically.
+\end{proof}
+
+\begin{theorem}[A zero-blank half-space test]\label{bc:zero-blank}
+The fixed label $0$ and event $\{x_1<-1\}$ in $\R^3$ realize halting
+with the force and comparison conclusions of Theorem~\ref{bc:positive-height}.
+This construction uses a zero joint-blank digit, so every initial tape
+coordinate is a finite radix sum. Its temporal alternative belongs to
+$L^2(0,\infty;H^j)$ for every $j$.
+\end{theorem}
+\begin{proof}\label{bc:zero-blank-proof}
+Use the frontier recorder, zero-blank base $B=2m$, and state first-coordinate
+offsets $3j$ for nonterminal controls and $-3j$ for terminal controls,
+with positive distinct indices. Encode a state by $(o_s+L,R,0)$.
+The full branches are on the $B^{-2}$ grid and satisfy the preceding
+proposition. Load the rational initial code from zero along a straight
+segment. Nonhalting codes have nonnegative first coordinate; the whole
+segment and every nonhalting transition therefore avoid $x_1<-1$.
+Terminal codes have first coordinate at most $-2$, including an initially
+terminal loading endpoint. Exact integer-time simulation gives both
+implications. The force conclusions follow from the common analytic and
+logarithmic-clock lemmas, with the compact support needed for all $H^j$
+norms. No claim of periodicity is made for the slowed alternative.
+\end{proof}
+
+\subsection{Expanded centers and obstacle detours}\label{bc:expanded-height-section}
+\begin{proposition}[Full solid boxes with a height guard]
+\label{bc:expanded-height-routing}
+Let the source and target families be finite collections of full rational
+axis-aligned solid boxes, each family pairwise disjoint. Write their centers
+as $a_i,b_i$ and their positive half-widths as $r_{i,j},w_{i,j}$.
+Require each prescribed affine map to send its entire source box onto
+its specified target box:
+\[
+ F_i(a_i+h)=b_i+D_i h,\qquad
+ D_i=\diag(w_{i,1}/r_{i,1},w_{i,2}/r_{i,2},w_{i,3}/r_{i,3}),
+ \qquad \det D_i=1.
+\]
+Suppose every source center has height at most $-1$, and every target
+center has height at most $-1$ or exactly $1$. They admit an effective
+compactly supported solenoidal unit-time realization. If a target center
+has negative height, every point $a_i+h$ of its source box with $h_3=0$
+remains at negative height throughout. A rational bound for the
+entire support is computable from the boxes.
+\end{proposition}
+\begin{proof}\label{bc:expanded-height-routing-proof}
+The empty family is realized by zero. Otherwise let $R=2+$ the largest
+endpoint half-width, and choose an integer $H>20R$ so large that the
+expanded source centers $A_i^*=Ha_i$ and target centers $B_i^*=Hb_i$
+are separately more than $20R$ apart in the sup norm. Set
+\[
+ Z_0=1+100R+\max_i\{\|A_i^*\|_\infty,\|B_i^*\|_\infty\},
+ \qquad P_i=(0,0,-Z_0-100Ri).
+\]
+Each of the unions $\{A_i^*\}\cup\{P_i\}$ and
+$\{B_i^*\}\cup\{P_i\}$ has separation greater than $20R$.
+The construction has five phases: expand source centers by a factor
+from one to $H$ while keeping shapes fixed; move them successively to
+$P_i$; change each parked shape by the prescribed positive diagonal
+power; move them successively to $B_i^*$; contract target centers to
+$b_i$ while keeping the final shapes fixed.
+
+The simultaneous expansion is collision free. For each pair of source
+boxes some coordinate has positive gap
+$|a_{i,j}-a_{k,j}|-r_{i,j}-r_{k,j}$; multiplication of the center
+difference by a factor at least one preserves that gap. The same
+argument applies to contraction toward the disjoint targets. Choose
+padding less than one quarter of the minimum such gap, and less than
+one. During a parked deformation the $j$th half-width is
+$r_{i,j}^{1-\theta}w_{i,j}^{\theta}\le\max(r_{i,j},w_{i,j})<R$.
+Thus parked deformations have all half-widths less than $R$ and
+supports of half-width at most $2R$; the storage separation suffices.
+
+For a successive transfer, surround each stationary center by its closed
+sup-norm cube of radius $4R$. These cubes are pairwise disjoint. Start
+with the straight segment between the moving centers. Each nontrivial
+intersection with such a cube can be replaced by a polygonal path on
+its boundary: join the entrance to a vertex of its face, follow cube
+edges to an exit-face vertex, and join to the exit. Fixed index orders
+choose the vertices and edges. All coordinates and intersection parameters
+are rational. Tangencies need no detour. Since different obstacle cubes
+are disjoint, the resulting finite polygon stays at distance at least
+$4R$ from every stationary center. Move along its segments in flat slots,
+using a cutoff support of half-width at most $2R$. Stationary boxes have
+half-width less than $R$, so no such support touches them. Lemma~\ref{lem:bb-curl}
+realizes each translation and deformation.
+
+For the height assertion, a transfer with negative endpoints has endpoints
+of height at most $-H$. Its straight segment is negative. It cannot meet
+an obstacle centered at positive height $H$, whose bottom is $H-4R>0$.
+Every boundary used around a negative-height obstacle lies below
+$-H+4R<0$. Storage heights are negative as well. Expansion, contraction,
+and positive diagonal deformation preserve the central-plane condition.
+Thus all these paths are negative. Finitely many rational vertices,
+endpoint widths, and cutoff paddings give a rational support bound.
+\end{proof}
+
+\begin{theorem}[A height test inside one torus chart]\label{bc:expanded-height-event}
+On the unit torus, the fixed label $o=(1/2,1/2,1/2)$ and event
+\begin{equation}\label{bc:expanded-height-fixed-event}
+ \{x:x_3\pmod1\in(1/2,1)\}
+\end{equation}
+realize halting by a smooth general mean-zero force, periodic after time
+one, with zero initial velocity and pressure and bounded mixed derivatives.
+The repeated mechanism depends on the machine alone; only initialization
+depends on its finite input. The solution is unique in the torus class.
+\end{theorem}
+\begin{proof}\label{bc:expanded-height-event-proof}
+Use the frontier recorder, odd base $B=2m+1$, distinct negative integer
+heights for nonterminal states and height one for the terminal ready state.
+Thicken every branch by a third half-width $1/4$. The reciprocal tape
+map is onto its target rectangle; adjoining the unchanged third interval
+gives an onto box map with linear part $\diag(B^{-e},B^e,1)$ and
+determinant one. All source centers are
+negative because terminal controls have no outgoing rules. Apply the
+preceding proposition. The logical initial code $Y_0$ is rational and has
+a uniform machine-dependent bound as the input varies: its two stack
+coordinates lie in $[0,1]$ and its height is fixed by the initial control.
+A moving translation loads the logical origin to $Y_0$, with a fixed
+source half-width one and padding one. This gives a machine-dependent
+rational bound $K$ for the supports of all such loaders and the repeated
+mechanism; for example take $K=1+2R+M$ with $M$ an upper bound for all
+routing vertices and loading coordinates and $R$ enlarged to cover their
+half-widths. Choose $\lambda=1/(4K)$ and use the chart $x=o+\lambda Y$.
+Everything remains strictly inside $(1/4,3/4)^3$.
+
+Periodize the resulting compactly supported curl fields, loading during
+the first unit interval and repeating thereafter. Compute the residual
+at physical viscosity $\nu$, after the chart change. The fields and force
+have zero mean and all the bounds in Lemma~\ref{bc:analytic}.
+For a nonhalting run, loading has nonpositive logical height and each
+later segment remains negative by the proposition. A terminal code has
+physical height $1/2+\lambda$, which lies in the event. There is no
+chart-boundary crossing that could create a spurious observation.
+An initially terminal input is detected at the loading endpoint.
+The initialized correspondence is exact; at checkpoints the number of
+history records gives $n$, and the frontier's relative position together
+with its absolute index $2+n$ also recovers the absolute work-head position.
+\end{proof}
+
+\subsection{Fixed columns and separate layers}\label{bc:columns-section}
+\begin{proposition}[Column and layer transport]\label{bc:columns}
+Let $S_i,T_i$ be finite rational planar rectangle families of positive side
+lengths, each separately disjoint, with endpoints on the $B^{-2}$ grid,
+where $B\ge2$ is an integer.
+Let their assigned center-to-center positive reciprocal diagonal affine
+maps satisfy $F_i(S_i)=T_i$. A finite effective
+solenoidal field in $\R^3$ realizes all these maps on the rectangles in
+$z=0$. Along each tracked path the first coordinate stays between its
+initial and final values. The field has compact support and flat time
+collars.
+\end{proposition}
+\begin{proof}\label{bc:columns-proof}
+Give rectangle $i$ height $h_i=4i$. A planar cutoff equal to one near
+its source and supported within padding $1/(4B^2)$ has support disjoint
+from the other source cutoffs. Multiply it by a third-coordinate cutoff
+which is one near the entire interval $[0,h_i]$. The curl of the potential
+$(0,h_i x,0)$ times these cutoffs equals $(0,0,h_i)$ along that source
+column. A common flat schedule lifts all rectangles. The corresponding
+target cutoffs give a later descent by $-h_i$.
+
+In the middle phase, let $p_i,q_i$ be the centers and $a_i>0$ the
+first-coordinate scale. With phase parameter $\mu\in[0,1]$, set
+$c_i=p_i+\mu(q_i-p_i)$, $l_i=1+\mu(a_i-1)$ and
+\[
+ H_i(\mu,x,y)=d_{i,1}y-d_{i,2}x+
+       \frac{a_i-1}{l_i}(x-c_{i,1})(y-c_{i,2}),
+ \qquad d_i=q_i-p_i.
+\]
+On a neighborhood of the desired path, the planar Hamiltonian field
+$(\partial_yH_i,-\partial_xH_i)$ generates
+$c_i+\diag(l_i,l_i^{-1})(\zeta-p_i)$.
+The first coordinate is the convex interpolation of its endpoint values.
+The second half-width obeys
+\[
+ \frac{r_{i,2}}{(1-\mu)+\mu a_i}
+ \le (1-\mu)r_{i,2}+\mu r_{i,2}/a_i,
+\]
+by convexity of the reciprocal function. Consequently the swept rectangle
+lies in the coordinate bounding rectangle of the two endpoints. Localize
+$H_i$ around that rectangle in the layer $z=h_i$, with third padding less
+than one, and take the curl of $(0,0,H_i)$ times that cutoff. For a
+flat middle ramp $\mu=\eta_2(t)$, multiply the resulting phase field by
+$\eta_2'(t)$. Layers are disjoint, so the sum has the required action. Lift and descent preserve
+planar coordinates. The exact paths and ODE uniqueness establish the
+full-rectangle and intermediate-coordinate assertions.
+\end{proof}
+
+\subsection{Target containers and prescribed affine images}
+\label{bc:contained-targets}
+The named targets in Propositions~\ref{bc:ordered}, \ref{bc:simultaneous}
+and~\ref{bc:columns} can also be rational \emph{containers} for the actual
+images. More precisely, replace the onto condition by
+\[
+ F_i(D_i)\subseteq S_i\quad\text{in Proposition~\ref{bc:ordered}},
+ \qquad
+ F_i(S_i)\subseteq T_i\quad\text{in the other two propositions}.
+\]
+Keep the other geometric hypotheses. For ordered transport retain the
+center-to-center condition. For simultaneous and column transport the
+actual endpoint center is $F_i(p_i)$, where $p_i$ is the source center;
+it need not be the center of the container. The maps have the same positive
+diagonal determinant-one linear parts as before, including the specified
+three scale choices in Proposition~\ref{bc:simultaneous}.
+All transport and path conclusions remain valid. For arbitrary real map
+coefficients this is a smooth existence statement. The construction is
+effective when those coefficients are computable, and otherwise its
+evaluation is relative to them. An empty list again uses the zero field.
+
+Here are the necessary modifications of the proofs. In ordered transport,
+write $r_{i,j}$ for a source half-width, $w_{i,j}$ for the corresponding
+container half-width, and $a_{i,j}>0$ for the diagonal factor. Containment
+gives $a_{i,j}r_{i,j}\le w_{i,j}$, hence
+\[
+ r_{i,j}a_{i,j}^{\theta}
+ \le\max(r_{i,j},a_{i,j}r_{i,j})
+ \le\max(r_{i,j},w_{i,j})<L,\qquad 0\le\theta\le1.
+\]
+The rational source and container widths therefore bound every parked
+deformation. After deformation, use translates of the rational target
+containers as envelopes for the actual images. Deliver in increasing
+order of the containers' right endpoints. If an earlier delivered image
+meets the moving image in the last two projections, their containers meet
+in those projections. Disjointness and the endpoint order force the earlier
+container wholly to the left of the active target container. Its image
+therefore misses the entire leftward sweep, which stops in that active
+container. The original extraction and storage arguments are unchanged.
+The same rational envelopes give positive cutoff margins. They do not
+require the actual image endpoints to be rational. The central-cross-section
+height guard is unchanged because its third offset remains zero.
+
+For simultaneous transport, interpolate toward $F_i(p_i)$. The transported
+source padding at descent has width at most $B\varepsilon$ outside its
+actual image, so it lies inside the $B\varepsilon$ enlargement of the
+target container. The original grid-gap and private-height estimates
+still separate all supports. The first-coordinate convex identity uses
+the actual value $F_i(y)$ and remains exact.
+
+For column transport use the same actual endpoint center. The reciprocal
+convexity estimate bounds each moving second half-width by the linear
+interpolation of the source and actual-image half-widths. Thus the entire
+middle sweep lies in the coordinate bounding rectangle of the source and
+its rational target container. The target-column cutoff equals one on
+the actual image, so the descent is unchanged. This also proves the stated
+first-coordinate bound for each point. Finally, a positive computable
+scale admits computable positive lower and finite upper bounds; its
+logarithm, powers and the displayed reciprocal interpolations are
+computable. The containment and determinant identities are hypotheses,
+not tests performed by the construction.
+
+\begin{theorem}[A half-space test by fixed columns]\label{bc:column-event}
+On $\R^3$, the frontier recorder with odd digits in base $2m+1$ realizes
+halting from the fixed label $0$ by the event $x_1<-1$. It gives a compactly
+supported general force, periodic after time one, with zero initial
+velocity and pressure and all mixed derivatives bounded.
+\end{theorem}
+\begin{proof}\label{bc:column-event-proof}
+Put nonterminal state offsets at $3,6,\ldots$ and terminal offsets at
+$-3,-6,\ldots$ in the first coordinate. Lemma~\ref{lem:bb-onecell} and
+Proposition~\ref{bc:columns} give a whole-rectangle phase map. A localized
+planar Hamiltonian translation loads the origin to the rational initial
+code in $[0,1]$ time. A nonterminal loading segment has nonnegative first
+coordinate, and every later nonterminal segment has first coordinate at
+least three. Terminal codes have first coordinate below $-1$. This proves
+both directions, including initial halting. Repeat the phase map and use
+Lemma~\ref{bc:analytic}. Although this route only requires planar coding
+rectangles, its cutoff plateaus also give a positive neighborhood on which
+the same affine formulas hold.
+\end{proof}
+
+\subsection{Delaying the original head movement}\label{bc:delayed-column-section}
+Use Lemma~\ref{lem:bb-direction} with singleton passive track removed.
+Rename $(M(q,s),G(r),A(q,d),B(q,d))$ as
+$(R_{q,s},F_r,C_{q,d},G_{q,d})$ and $(\bot,F)$ as $(\perp,*)$.
+These bijections identify every row and nonfrontier guard on the full
+domain, including all $q,d$ controls. The frontier starts at one, and the
+work movement occurs only on return. At checkpoint $n$, write $i$ for
+the work-head position and $j=n+1$ for the frontier position. The exact count is
+$2(j-i)+3$: the right and left continuing loops have respectively
+$j-i-1$ and $j-i$ instructions, with four other instructions.
+
+\begin{theorem}[The delayed-head column realization]\label{bc:delayed-column-event}
+The table of Lemma~\ref{lem:bb-direction} has a realization on $\R^3$
+with fixed label $0$ and detector $x_1<-1$, compact spatial support,
+period one after time one, bounded mixed derivatives and uniformly bounded
+kinetic energy. The prescribed pressure is zero. Uniqueness holds in the
+whole-space comparison class, in particular when the competitor and its
+first spatial derivatives are bounded on every finite time interval.
+\end{theorem}
+\begin{proof}\label{bc:delayed-column-event-proof}
+Assign odd digits $1,3,\ldots,2m-1$ in base $B=2m+2$, with state
+offsets positive multiples of three except for negative terminal offsets.
+The tail of the blank symbol contributes
+$d_{\rm blank}B^{-k}/(B-1)$ after a finite prefix, so initialization is
+rational even though the blank digit is nonzero. Full affine branches
+follow from the lemma and Lemma~\ref{lem:bb-onecell}. Use the fixed-column
+proof with heights $4i$ and the smaller padding $1/(10B^2)$; schedule a
+slot $[a,b]$ by $\sigma(2(t-a)/(b-a)-1/2)$, which is flat on collars
+of both endpoints. These choices preserve the same separation proof.
+A planar Hamiltonian loading segment starts at zero. The convex first-
+coordinate guard excludes the detector throughout every nonhalting
+transition and throughout a nonhalting load; a terminal code lies below
+$-1$. Finite checkpoint simulation gives the equivalence. The field has
+one compact support and uniform pointwise bounds, hence uniformly bounded
+kinetic energy. Lemma~\ref{bc:analytic} supplies all remaining assertions.
+\end{proof}
+
+\subsection{Recorded pointers and seven-slot storage}\label{bc:recorded-pointer-section}
+\begin{theorem}[A fixed box test with recorded pointers]
+\label{bc:recorded-pointer-event}
+On $\R^3$, the fixed label $(2,0,0)$ and open box
+$(0,1)^2\times(-1/2,1/2)$ realize halting by an effective general force
+with compact spatial support, zero initial velocity and pressure, bounded
+mixed derivatives, and period one after time one. The construction moves
+whole solid boxes. Uniqueness holds among smooth competitors with
+\[
+ u\in C^1([0,T];L^2),\quad \nabla u,\Delta u\in C([0,T];L^2),
+ \quad p,\nabla p\in C([0,T];L^2),\quad
+ \sup_{[0,T]\times\R^3}(|u|+|\nabla u|)<\infty
+\]
+for every finite $T$.
+\end{theorem}
+\begin{proof}\label{bc:recorded-pointer-event-proof}
+Normalize to one halt $h$. For $c=(q,b)\in\mathcal D=(Q\setminus\{h\})\times\Gamma$
+write the instruction as $(q^+(c),b^+(c),m(c))$.
+Use histories
+\[
+ \Lambda_0=\{\varnothing\}\sqcup\{H_e:e\in\mathcal D\sqcup\{\star\}\},
+ \qquad \Lambda=\Lambda_0\sqcup\{P_e:e\in\mathcal D\sqcup\{\star\}\}.
+\]
+The full alphabet is $\mathcal A=\Gamma\times\Lambda\times\{0,1\}$,
+whose tracks are data, history and return mark. Controls are
+$S_q,L_q,M_c,R_c,F_c$, and the complete table is
+\[
+\begin{array}{c|c|c|l}
+\text{input}&\text{output}&\text{move}&\text{restriction}\\\hline
+S_q(b,l,0)&M_c(b^+(c),l,0)&m(c)&c=(q,b),\ l\in\Lambda_0\\
+M_c(b,l,0)&R_c(b,l,1)&1&l\in\Lambda_0\\
+R_c(b,l,0)&R_c(b,l,0)&1&l\in\Lambda_0\\
+R_c(b,P_e,0)&F_c(b,H_e,0)&1&e\in\mathcal D\sqcup\{\star\}\\
+F_c(b,\varnothing,0)&L_{q^+(c)}(b,P_c,0)&-1&\\
+L_q(b,l,0)&L_q(b,l,0)&-1&l\in\Lambda_0\\
+L_q(b,l,1)&S_q(b,l,0)&0&l\in\Lambda_0.
+\end{array}
+\]
+Initially put $P_\star$ at absolute cell three, empty history elsewhere,
+marks zero, and control $S_{q_0}$. At ready checkpoint $k$, the pointer
+is at $r=3+k$, all cells to its right are empty, and the work configuration
+is the original one. After the first row the work head satisfies
+$j'\le k+1\le r-2$. The scan marks it, reaches the pointer, changes
+$P_e$ into $H_e$, writes $P_c$ in the next empty cell, and returns to
+erase the mark. Both scans are finite and positive. This proves the
+initialized simulation. The initial pointer tag is retained as $H_\star$
+after the first cycle, so the absolute cell-three reference is permanent.
+
+The inverse works on all allowed tapes. Into $M_c$ the tag fixes the old
+instruction and move; into $R_c$ mark one distinguishes entry from a loop;
+into $F_c$ the written $H_e$ identifies the old pointer $P_e$; into $L_q$
+a pointer output identifies entry, including the old $F_c$, whereas a loop
+writes a nonpointer; into $S_q$ only mark erasure applies. Incoming shifts
+are fixed in every target control. The remaining written components retain
+all unchanged symbols, proving the two properties of
+Lemma~\ref{lem:bb-onecell}.
+
+Let $m_Q$ be the number of controls, and number them bijectively by
+$\iota(s)\in\{0,\ldots,m_Q-1\}$ with $\iota(S_h)=0$.
+Use odd digits in base $B=2|\mathcal A|+1$ and state offsets
+$o_s=3\iota(s)$. The physical
+code is $(o_s+L,R,0)$. Terminal codes lie strictly in $(0,1)^2\times\{0\}$;
+nonterminal first coordinates are at least three. Form the full branch
+boxes by taking the product with $[-1,1]$. Their planar sides have length
+at most one. Choose $H_x=3m_Q+1$ and
+parking centers $\pi_i=(H_x+10+4i,0)$, all at height zero.
+
+Here is a storage construction distinct from ordered low extraction.
+Evacuate each source in three successive operations: lift its center to
+height ten, translate horizontally there to $\pi_i$, and lower it to
+zero. Once every box is parked, process each in four operations: deform
+its planar shape at the parking center, lift it to height ten, translate
+it there to its target center, and lower it into the target. This uses
+$7N$ slots for $N$ branches. During deformation use the positive scale
+$l=1+\theta(\lambda_i-1)$ and its reciprocal, not an exponential scale.
+Both side lengths remain bounded by the maximum of their endpoint lengths,
+which is at most one. The third half-width remains one.
+
+Let $\delta$ be the minimum of one and all positive source-source and
+target-target planar distances, omitting empty sets of pairs. Use cutoff
+plateaus with padding $\varepsilon=\delta/8$ and support padding
+$2\varepsilon$. During a lift or descent the planar projection separates
+the moving box from the current sources or delivered targets. Parked
+boxes lie outside their entire first-coordinate range, and parking centers
+have spacing four. During a horizontal translation the box lies in
+$9\le z\le11$, whereas all stationary boxes lie in $-1\le z\le1$.
+These positive gaps also separate the padded supports. Lemma~\ref{lem:bb-curl}
+therefore realizes every operation without disturbing a stationary box.
+The full assigned affine map follows by composition.
+
+For a branch whose target control is nonterminal, every full point has
+first coordinate at least three, including the storage and transfer paths.
+Load the fixed label
+$(2,0,0)$ to the rational initial code by a localized straight translation.
+If the code is nonterminal, the loading segment has first coordinate at
+least two; if it is terminal, the endpoint is in the observation box.
+At later integer samples the exact recorder code gives terminal entry,
+and the preceding bounds exclude the observation at all other times of a
+nonhalting run. Repeat the finite transport field and apply
+Lemma~\ref{bc:analytic}. Finally the displayed comparison class implies
+$u\in C H^2$, since $\|u\|_{H^2}\le C(\|u\|_2+\|\Delta u\|_2)$
+by the Fourier multiplier identity; its pressure assumptions give $p\in C H^1$.
+Thus the stated uniqueness follows from the common whole-space comparison,
+without changing this route's original hypotheses.
+\end{proof}
+
+\subsection{Planar detours behind a vertical observation guard}
+\label{bc:guarded-planar-section}
+
+
+\begin{theorem}[A guarded planar observation with unrestricted pressure]
+\label{bc:guarded-planar-event}
+On $\R^3$, the fixed label $(-4,0,0)$ and fixed box
+$(-1,2)^2\times(-1,1)$ realize halting by a compactly supported general
+force with zero initial velocity and prescribed pressure and all mixed
+derivatives bounded. One choice is periodic after time one. A separate
+choice satisfies $\|f(t)\|_\infty=O((1+t)^{-2/3})$ and
+$f\in L^2([0,\infty)\times\R^3)$.
+Uniqueness holds among smooth competitors with
+$u\in C([0,T];H^2)\cap C^1([0,T];L^2)$ and bounded $u,\nabla u$
+on each finite interval, without any spatial integrability condition on
+pressure. Pressure is unique modulo a function of time.
+\end{theorem}
+\begin{proof}\label{bc:guarded-planar-event-proof}
+Use the frontier recorder after normalization to one halting state, and
+write $\mathcal A$ for its full cell alphabet and $m_Q$ for its number
+of controls. Number the controls bijectively by
+$\iota(s)\in\{0,\ldots,m_Q-1\}$, giving the terminal control index zero.
+Use odd base $B=2|\mathcal A|+1$ and first-coordinate offsets
+$o_s=4\iota(s)$. Code the
+configuration by $(o_s+L,R)$. The origin of the original tape can be
+recovered from the number of history records and the frontier position,
+so this head-relative coding loses no initialized information.
+The full planar branches have reciprocal positive diagonal maps.
+
+We first realize finite separately disjoint source and target rectangle
+families $S_i,T_i$ with rational endpoints and positive side lengths,
+equipped with assigned positive
+reciprocal diagonal affine maps $F_i$ satisfying $F_i(S_i)=T_i$, by
+compactly supported planar Hamiltonian motion. Write $c_i,q_i$ for the
+source and target centers. With no
+branches use zero. With one branch, interpolate its center directly and
+use a positive linear first-coordinate scale and its reciprocal; there
+are no stationary obstacles. For $N\ge2$, let $R=1+$ the largest endpoint
+half-width and let $m_0>0$ be the minimum sup-norm separation within the
+two center families. Put $K=1+32R/m_0$. Expand all source centers by
+factors from one to $K$, leaving their shapes unchanged. Let
+$M=\max_i\{|(Kc_i)_1|,|(Kq_i)_1|\}$ and choose parking centers
+$P_i=(M+32Ri,0)$. Each union of expanded endpoint centers and parking
+centers has separation at least $32R$.
+
+Move expanded sources successively to their parking centers, deform them
+there using the linear reciprocal scale, move them successively to the
+expanded target centers, then contract those centers to their targets.
+For each successive transfer put a square of sup-radius $4R$ around
+every stationary center. Replace each nontrivial segment intersection
+with an obstacle square by a counterclockwise path along its boundary.
+The intersection points and corners are rational; the squares are disjoint,
+so the resulting finite polygon stays at distance at least $4R$ from
+all stationary centers. Flat schedules on its segments give smooth center
+motion. The simultaneous expansion and contraction preserve a positive
+coordinate gap for every pair, exactly as in the solid-box argument.
+If $g_0,g_1$ are their minimum positive gaps, choose padding
+$\varepsilon=\min(g_0,g_1,2R)/4$ for these phases. During individual
+transfers use padding below $R$; the stationary half-widths are below $R$.
+Thus all required cutoff supports avoid stationary rectangles.
+
+For a moving center $c$ and horizontal scale $l$, the potential
+\[
+ H=\chi\left[\dot c_x(y-c_y)-\dot c_y(x-c_x)
+                    +\frac{\dot l}{l}(x-c_x)(y-c_y)\right]
+\]
+gives the required motion on the cutoff plateau. The other current
+rectangles are fixed. This proves every full planar branch, including
+its boundary, by induction through the operations. A one-square loading
+translation with half-width $1/4$ takes $(-4,0)$ to the initial rational
+code. All these supports lie in a square $[-L,L]^2$, where one can choose
+$L=10+S_0+4m_Q$ with $S_0$ a finite bound for routing vertices and cutoff
+widths. The same $L$ covers every input of a fixed machine, since initial
+codes are uniformly bounded.
+
+The planar detours need not avoid the horizontal observation box.
+We instead make them at height three. Choose a switch $\eta$ with
+collars and set
+$h(s)=3\{\eta(3s)-\eta(3s-2)\}$ on a unit interval. It lifts in the
+first third, stays at three in the middle, and lowers in the last third.
+Choose $\Gamma$ equal to one near $[-L,L]^2\times[-1,4]$, with compact
+support, and $\beta(z)$ compactly supported and equal to one near three.
+For any one of the planar fields $w$, zero outside its phase interval,
+put
+\[
+ Z(s,x)=h'(s)\nabla\times(0,x_1\Gamma(x),0),\qquad
+ V_w(s,x)=Z(s,x)+3\beta(x_3)(w(3s-1,x_1,x_2),0).
+\]
+Along each tracked path the first term is exact vertical translation.
+The second term acts only in the middle third, at height three, and runs
+the full planar phase. It is divergence free because $w$ is planar
+solenoidal and $\beta$ is independent of the planar coordinates.
+Thus the trajectory lifts at its exact source, performs every detour
+outside the vertical detector interval, and lowers at its exact target.
+Use this construction for the loader and for each repeated phase. While
+its height can lie in $(-1,1)$, a nonterminal computational particle has
+first coordinate at least four; the initial loading source has first
+coordinate $-4$. A terminal code lies in $(0,1)^2\times\{0\}$.
+These facts prove the all-real-time observation equivalence, including
+an initially terminal input.
+
+For the comparison assertion, the constructed field has compact support
+and bounded mixed derivatives, so it satisfies the reference hypotheses
+of Lemma~\ref{lem:b-comparison}. The displayed competitor class is exactly
+case (v) of that lemma. Its proof obtains $\nabla p\in L^2$ from the
+equation and cancels the pressure term by Lemma~\ref{lem:b-gradient}; it
+does not require an integrable pressure representative. Hence the velocity
+is unique, and pressure is determined up to a function of time.
+
+Finally let $U(s)$ denote the eventually periodic field just constructed
+and set $\rho(t)=(1+t)^{1/3}-1$, $\widehat U=\rho' U(\rho(t))$.
+The new residual is
+\[
+ \widehat f=\rho''U+(\rho')^2\{U_s+(U\cdot\nabla)U\}
+                          -\nu\rho'\Delta U,
+ \quad \rho'=\tfrac13(1+t)^{-2/3},\quad
+ \rho''=-\tfrac29(1+t)^{-5/3}.
+\]
+Uniform derivative bounds and the fixed compact support give the stated
+decay and space-time $L^2$ estimate. Chain differentiation also gives all
+mixed bounds. The clock is effective on $[0,\infty)$, starts at zero,
+is finite at finite time and is onto; hence it preserves precisely the
+same material event. The initial zero collar is preserved as well.
+This slower force is an alternative to, not a periodic version of, the
+first force.
+\end{proof}

preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/build/torus-routing.tex

@@ -1,365 +1,365 @@
-\section{Torus charts and phasewise localization}\label{sec:torus-routing}
-
-The next constructions keep the whole motion inside a specified torus
-chart. A scaled linear interpolation preserves a convex first coordinate;
-an exponential interpolation in a fixed chart instead uses endpoint-width
-bounds. A phasewise construction then exhibits the vertical and planar
-potentials separately, and the final variant retains a permanent origin
-track. These choices give different path estimates for their observers.
-In every case the force is recomputed at the prescribed physical viscosity
-after any spatial rescaling.
-
-\subsection{Physical chart scaling and thickened torus boxes}
-
-\begin{proposition}[Linear scaling into the torus]
-\label{prop:bb-mid-scaled-torus}
-The direction recorder without its passive track has a unit-torus
-realization periodic after $t=1$, with fixed label $(1/8,0,0)$ and
-halting event $x_1\in(-1/4,0)\pmod1$.  Each period acts on full
-solid boxes; the pressure is zero and the force has zero mean.
-\end{proposition}
-\begin{proof}\label{proof:bb-mid-scaled-torus}
-Use the controls of Lemma~\ref{lem:bb-direction}. Let
-$\Sigma=\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}$
-be its full cell alphabet after deleting the singleton passive track.
-Let $S$ be the finite set of recorder controls and choose a bijection
-$j:S\to\{1,\ldots,|S|\}$.
-Use odd digits with $B=2|\Sigma|+1$, and offsets $o_s=-2j(s)$ for halting main
-controls and $o_s=2j(s)$ for the others. Number the resulting reciprocal branches
-$i=1,\ldots,N$, and write $k_i$ for branch $i$'s first-coordinate scale.
-Thicken each reciprocal branch by
-$[-1,1]$.  Its horizontal family gaps are at least $B^{-2}$.
-Lift branch $i$ to $4i$, use the linear first scale
-$\kappa_i=1+\theta(k_i-1)$, and descend.  Padding smaller than
-$(4B^2)^{-1}$ separates all phases: in the middle the half-height
-is one and height gaps are four; in the other phases horizontal
-gaps suffice.  The two horizontal half-widths are at most $1/2$.
-Use the true-plateau cutoff of Lemma~\ref{lem:bb-curl}; it supplies
-at least the full-box guarantee, together with a neighborhood extension.
-The exact first coordinate obeys
-\begin{equation}\label{eq:bb-mid-scaled-convex}
- X_1(\tau)=(1-\theta)X_1(0)+\theta X_1(1).
-\end{equation}
-
-Let $V$ be the period-one extension of the unit-time field just constructed.
-All codes, paths and padded supports lie in $(-L,L)^3$ for
-$L=10+2|S|+4N$.  Put $a=1/(16L)$ and define in the torus patch
-$(-1/2,1/2)^3$ the repeated field $U(t,x)=aV(t,x/a)$.
-Its support lies in $(-1/16,1/16)^3$ and its curl potential is the
-scaled potential $a^2A(t,x/a)$.  In particular its mean is zero.
-The residual must be recomputed at the physical viscosity:
-\begin{equation}\label{eq:bb-mid-viscosity-scale}
- \mathcal F_\nu[U](t,x)=a\{V_t+(V\cdot\nabla)V-(\nu/a^2)\Delta V\}(t,x/a).
-\end{equation}
-This keeps the specified viscosity rather than silently rescaling it.
-During the first unit interval load from $(1/8,0,0)$ to the scaled
-initial code using a curl translation with a box of half-width $1/32$
-and padding $1/32$.  Its support lies inside $(-1/4,1/4)^3$.
-
-For a nonhalting run both loading endpoints have positive first
-coordinate, and every subsequent pair of coded endpoints is positive.
-Equation~\eqref{eq:bb-mid-scaled-convex} preserves that sign throughout.
-A halting checkpoint has negative first coordinate in $(-1/16,0)$.
-All paths stay in the stated patch, so passage modulo one introduces
-no extra visit.  Checkpoint times are
-$1+\sum_{l<n}(5+2(l-h_l))$.  This proves the all-time equivalence,
-including initial halting; Proposition~\ref{prop:bb-realization}
-finishes the proof.
-\end{proof}
-
-\begin{proposition}[Exponential deformation in a fixed chart]
-\label{prop:bb-mid-direct-torus}
-The move-first recorder has a unit-torus realization periodic after
-one loading interval, with fixed label $(1/4,1/4,1/4)$ and halting
-event $x_1\in(2/3,5/6)\pmod1$.  The repeated motion acts on thickened
-closed rectangles and their neighborhoods.  Its mean-zero residual
-has the form $f_0+\nu f_1$ with coefficients independent of $\nu$.
-\end{proposition}
-\begin{proof}\label{proof:bb-mid-direct-torus}
-First normalize to one halting work state and use the move-first recorder
-of Lemma~\ref{lem:bb-movefirst}; let $\Sigma$ be its full three-track
-cell alphabet. Initial halting is preserved by this normalization.
-For $m$ controls take $\ell=1/[64(m+1)]$.  Nonhalt squares have
-lower left corners $(1/4+2j\ell,1/4)$, $0\le j\le m-2$;
-the halt square has lower left corner $(3/4,1/4)$.
-Use odd digits and $b=2|\Sigma|+1$, with coding height $1/4$.
-There are $N$ reciprocal branches with gaps at least $\ell b^{-2}$.
-Thicken them by half-height $h_0=1/[64(N+1)]$ and choose heights
-$z_i=1/2+i/[4(N+1)]$.  Use exponential scale $k_i^\theta$,
-center interpolation, and
-$\epsilon=\min(h_0/2,\ell/(8b^2))$.
-The first and last phases have horizontal gaps exceeding
-$2\epsilon$; middle height gaps equal $16h_0$ and exceed
-$2(h_0+\epsilon)$.  Widths never exceed their endpoint values.
-A concrete plateau cutoff for a box centered at $C_i$, with
-half-widths $w_{ij}$, is
-\[
- \chi_i=\prod_{j=1}^3\sigma\!\left(
- \frac{(w_{ij}+\epsilon)^2-(x_j-C_{ij})^2}
- {(w_{ij}+\epsilon)^2-(w_{ij}+\epsilon/2)^2}\right).
-\]
-Use its product with
-$\tfrac12\dot C_i\times(x-C_i)+(0,0,(\dot a_i/a_i)(x_1-C_{i1})(x_2-C_{i2}))$,
-where $a_i=k_i^\theta$.  Its curl gives the exact affine motion on
-an open neighborhood of each box.  All supports remain inside the
-unit cube.
-
-Load the initial rational code along its segment from the fixed
-label, using a radial cutoff of radius $1/32$ and a flat switch
-$\sigma(4t-1)$.  Then repeat the unit motion, with phase switches
-$\sigma(8\tau-1),\sigma(8\tau-3),\sigma(8\tau-5)$.
-These choices give spatial zero collars of width $1/8$ and temporal
-zero collars of width $1/16$ at nonnegative integer joins, including
-loading.  The force follows from
-$f_0=U_t+(U\cdot\nabla)U$, $f_1=-\Delta U$.
-
-Nonhalt centers have first coordinate at most $1/4+1/32$ and
-half-width at most $\ell/2$, hence their entire trajectories satisfy
-$x_1\le1/4+1/32+\ell/2<2/3$.  This is a width bound, not a convexity
-claim for exponential scaling.  Loading also avoids the slab for a
-nonhalting input.  Halt codes enter it at a finite integer sample,
-including time one for an initial halt.  The record block length and
-the known frontier site $n+2$ recover absolute head position at each
-work checkpoint.  All force and uniqueness assertions now follow
-from Proposition~\ref{prop:bb-realization}.
-\end{proof}
-
-\subsection{Thin-sheet curls and a slab observer}
-\label{ba:thin-sheet}
-
-The next realization stays directly inside the unit cube.  It uses a
-vertical translation potential in the first and last phases and a planar
-Hamiltonian potential in the middle.  Because its observer ignores
-height, the proof must keep the first coordinate safe throughout all
-three phases.
-
-\begin{theorem}[A direct unit-cube implementation]
-\label{ba:thin-sheet-torus}
-At each fixed positive computable viscosity, every machine and finite
-input has an effective smooth mean-zero force on the unit flat
-three-torus, with bounded mixed derivatives and period one for $t\ge1$,
-whose smooth zero-data solution satisfies
-\[
- \exists t\ge0:\ X(t;(1/4,1/4,1/4))\in
- (2/3,5/6)\times\mathbb T^2
- \quad\Longleftrightarrow\quad\text{the machine halts}.
-\]
-The normalized pressure is zero and the solution is unique in the
-classical torus comparison class of Section~\ref{sec:model}.
-\end{theorem}
-
-\begin{proof}\label{ba:thin-sheet-torus-proof}
-For the cutoffs in this proof, recall
-\[
- E(r)=\begin{cases}e^{-1/r},&r>0,\\0,&r\le0,\end{cases}
- \qquad \Theta(r)=\frac{E(r)}{E(r)+E(1-r)}.
-\]
-Use the single-halt normalization and the seven-row recorder in
-Lemma~\ref{lem:bb-movefirst}, with frontier initially at two and all marks
-initially zero.  In the present notation its controls are
-$S_q,U_r,W_r,C_q,D_q$ (ready, work-move completed, right scan,
-frontier extension, left scan), and its full symbols are $(a,g,m)$
-with history $g\in\{E,F\}\sqcup\{\widehat r\}$.
-The first row writes and moves, the second marks and starts the scan,
-and the last clears the mark and stays.  These are exactly the seven
-rows already proved, after renaming.  Thus the new work head $j'$ is
-less than $n+2$, a work step takes $2(n+2-j')+4$ local steps, and
-both full-domain incoming properties hold.
-
-Let $M$ count the partial controls, let $H=S_h$, and put
-\[
- \ell=\frac1{100(M+1)},\qquad o_H=(3/4,1/2),\qquad
- o_{s_i}=(1/4+2i\ell,1/2)\quad(0\le i<M-1),\qquad z_*=1/4.
-\]
-In the state frames $o_s+\ell[0,1]^2$, use odd digits in base
-$B=2|\Sigma|+1$ for the full alphabet $\Sigma$.  The minimal
-left-split subdivision in Lemma~\ref{ba:rectangles} gives separately
-separated rectangles $E_i^-,E_i^+$, centers $c_i^-,c_i^+$, and maps
-\[
- y\longmapsto c_i^+
- +\operatorname{diag}(a_i,a_i^{-1})(y-c_i^-),
- \qquad a_i\in\{B^{-1},1,B\}.
-\]
-The common frame scale leaves these linear parts unchanged; all
-rectangle side lengths are at most $\ell$.
-
-For $N$ branches set $z_i=1/2+i/(4(N+1))$, $1\le i\le N$.
-Let $\delta$ be one tenth of the minimum of $1/100$, all pairwise
-source gaps, all pairwise target gaps and all $|z_i-z_j|$; omit empty
-pair lists.  For a box $D=\prod_j[l_j,r_j]$, allowing zero widths,
-define
-\[
- \chi_D^\delta(x)=\prod_{j=1}^3
- \Theta\!\left(2(x_j-l_j+\delta)/\delta\right)
- \Theta\!\left(2(r_j+\delta-x_j)/\delta\right).
-\]
-It equals one on the $\delta/2$ padding and is supported in the
-$\delta$ padding.  Put $\theta(s)=\Theta(3s-1)$.
-On the first third of a unit step use
-\[
- z_i(s)=z_*+(z_i-z_*)\theta(3s),\qquad
- D_i(s)=E_i^-\times\{z_i(s)\},\qquad
- P_i=\chi_{D_i(s)}^\delta(0,\dot z_i(s)x_1,0).
-\]
-Its curl on the plateau is $(0,0,\dot z_i)$; source gaps separate
-these supports.  Hence every point of the whole source plate lifts
-with planar coordinates unchanged.
-
-On the middle third set
-\[
- \eta=\theta(3s-1),\quad c_i=(1-\eta)c_i^-+\eta c_i^+,
- \quad\mu_i=1-\eta+\eta a_i,
-\]
-\[
- D_i(s)=
- \big[c_i+\operatorname{diag}(\mu_i,\mu_i^{-1})(E_i^--c_i^-)\big]
- \times\{z_i\}.
-\]
-With $\gamma_i=\dot\mu_i/\mu_i$, use
-\[
- P_i=\chi_{D_i(s)}^\delta(0,0,\Psi_i),\qquad
- \Psi_i=\dot c_{i1}(x_2-c_{i2})-\dot c_{i2}(x_1-c_{i1})
-          +\gamma_i(x_1-c_{i1})(x_2-c_{i2}).
-\]
-Its plateau curl is
-$(\dot c_{i1}+\gamma_i(x_1-c_{i1}),
-  \dot c_{i2}-\gamma_i(x_2-c_{i2}),0)$,
-which is precisely the reciprocal-scale affine velocity.  The distinct
-heights separate the supports during this phase.  In the last third use
-the vertical potential again, with $E_i^+$ and
-$z_i(s)=z_i+(z_*-z_i)\theta(3s-2)$; target gaps separate the descents.
-Summing the curls gives the required field in each phase, with zero
-collars at every join.
-
-All supports lie strictly inside the unit cube: horizontal centers lie
-in $[1/4,3/4+\ell]\times[1/2,1/2+\ell]$, half-widths are at most
-$\ell/2$, heights lie between $1/4$ and $3/4$, and
-$\delta\le10^{-3}$.  Thus the field extends smoothly and with zero
-mean to the torus.  The construction acts on neighborhoods, not only
-plates.  An initial perturbation of max norm less than $\delta/(4B)$
-stays within $\delta/4$ of its affine plate path: vertical phases
-preserve the perturbation, and both horizontal scale factors are at
-most $B$.  It remains on the plateau, so uniqueness of the smooth ODE
-gives that same affine continuation throughout.
-
-Write the rational initial code as $(b,z_*)$.  During one unit of time
-load the fixed particle using
-\[
- c_0(t)=(1-\theta(t))(1/4,1/4)+\theta(t)b
-\]
-and the curl of
-\[
- \chi_{\{(c_0(t),z_*)\}}^{1/100}
- (0,0,\dot c_{01}(x_2-c_{02})-\dot c_{02}(x_1-c_{01})).
-\]
-This has plateau velocity $(\dot c_0,0)$, support inside the cube,
-and zero endpoint collars.  Repeat the step field thereafter and define
-its residual force at $\nu$.  The finite compact phase formulas give
-effective mixed-derivative bounds; the only scale denominators satisfy
-$\mu_i\ge\min(1,a_i)>0$.  Curls have zero mean, and incompressibility
-gives zero mean for the residual.  Lemma~\ref{ba:realization} supplies
-the unique zero-data PDE solution and its global trajectories.
-
-At times $1+k$, induction through the full branch maps gives the exact
-local code through the first halt, or for all $k$ in the nonhalting
-case.  A halt code has first coordinate in
-$[3/4,3/4+\ell]\subset(2/3,5/6)$, including a code reached solely by
-loading an initially halted machine.  For a nonhalting input the loader
-stays below $1/2$ in first coordinate.  Lift and descent retain the
-nonhalting planar endpoint codes.  In the middle phase the centers are
-convex combinations of nonhalting centers and the first half-width is
-at most $\ell/2$, so
-\[
- X_1\le1/4+2M\ell+\ell/2<1/2.
-\]
-This excludes the height-independent observer at every intermediate
-time.  Only the loader depends on the input; the repeated field depends
-on the finite machine table.
-\end{proof}
-
-
-\subsection{A permanent origin marker and moving cutoff neighborhoods}
-\label{bc:height-separated-section}
-\begin{theorem}[A height-separated mechanism with an origin track]
-\label{bc:height-separated-event}
-On the unit torus, the fixed label $(1/8,3/8,1/4)$ and strip
-$2/3<x_1<7/8$ realize halting by a smooth mean-zero general force,
-periodic after time one, with zero initial velocity and pressure and bounded
-mixed derivatives. A permanent tape track identifies the original absolute
-origin. For a fixed machine the repeated field is independent of the input.
-\end{theorem}
-\begin{proof}\label{bc:height-separated-event-proof}
-Normalize the work machine to one halting state as in
-Section~\ref{sec:bb-frontiers}, preserving initial halting.
-Use four tracks: work, head mark, history, and an origin bit. The history
-is empty except for a frontier at cell two; the origin bit is one only
-at absolute cell zero. In a ready control $S_q$, the head mark is one
-at the work head. At intermediate controls use the seven-row recorder
-unchanged, with the origin track copied in every row. The first row now
-requires ready mark one and writes mark zero at the old head before the
-work move. The final return row leaves mark one when entering $S_q$.
-These are the only two changes to the seven rows.
-
-For precision, this table is the conjugate of the frontier table on its
-full configuration domain by the following involution: toggle the mark
-at the current head exactly when the control is ready, and leave it
-unchanged at every other control. Also take the direct product with the
-passive origin track. This is a bijection of all configurations, not merely
-of initialized ones. It proves the full-domain inverse. The incoming-rule
-properties also persist: into the work-tag control the first-row output
-mark is zero; into the right scan the marking row and its loop still write
-one and zero; into the return scan frontier and nonfrontier outputs remain
-distinct; into the ready control only the last row applies. The initial
-configuration is the conjugate of the frontier initialization. Consequently
-its cycle length is $2(2+n-h')+4$, its absolute frontier is $2+n$, and
-its origin bit remains fixed at the absolute origin. This verifies the
-distinct initialization without another appeal to an existential compiler.
-
-Denote the resulting full four-track alphabet by $\mathcal A$, and let
-$K=2|\mathcal A|+1$ be its odd-digit radix.
-For $m$ controls take state-square side $r=1/[16(m+1)]$, with nonterminal
-lower corners $(1/8+2(i-1)r,1/8)$ and terminal lower corner $(3/4,1/8)$.
-Their common coding height is $z_0=1/4$. Lemma~\ref{lem:bb-onecell} supplies
-separated full planar rectangles; let $N$ be the number of branches.
-Give branch $i$ height
-$h_i=1/2+i/[4(N+1)]$. During a first phase lift its center from $z_0$
-to $h_i$, keeping its planar shape; during a middle phase interpolate
-its planar center, with width
-$w_i=(1-\mu)w_i^-+\mu w_i^+$ and height
-$v_i=w_i^-v_i^-/w_i$; during a final phase lower it to $z_0$.
-The planar first coordinate is again an endpoint convex combination.
-Choose a fixed positive padding less than one tenth of all source and
-target gaps, all chart-face margins and all height separations. The
-moving padded rectangles are disjoint at every phase. During the middle
-phase this follows from the different heights; during the first and last
-phases it follows from endpoint planar separation.
-
-Here is a direct potential for these moving neighborhoods. With moving
-center $c=(c_x,c_y,c_z)$ and $\lambda=\dot w/w=-\dot v/v$, set
-\[
- A=(0,\dot c_z(x-c_x),
-       \dot c_x(y-c_y)-\dot c_y(x-c_x)
-                         +\lambda(x-c_x)(y-c_y)).
-\]
-Its curl on the plateau is
-$(\dot c_x+\lambda(x-c_x),\dot c_y-\lambda(y-c_y),\dot c_z)$.
-Multiply it by a cutoff supported in the moving padded neighborhood and
-sum the curls. All supports remain in the open cube. The displayed
-center-and-shape paths solve this field on the full planar rectangles;
-uniqueness proves their exact endpoint maps. The positive padding and
-bounded planar scales also give a uniform initial neighborhood that stays
-on every plateau, by the perturbation argument of Lemma~\ref{lem:bb-curl}.
-Each component of the total
-field has zero mean after periodization, since the entire field is a curl.
-
-For initialization, choose equal small squares about the fixed label and
-the rational input code at height $z_0$, with half-side less than one
-quarter of their minimum chart-face margin. Move one to the other by the
-same localized potential for translation during $[0,1]$. A nonterminal
-loading first coordinate lies between $1/8$ and $1/4$, and every subsequent
-nonterminal phase lies below $1/4$. A terminal endpoint lies in the detector.
-The initialized symbolic equivalence, including initial halting, therefore
-holds at all real times. Repeating the finite middle mechanism and applying
-Lemma~\ref{bc:analytic} gives the asserted force and comparison properties.
-Only the loading translation and its input code depend on the finite word.
-\end{proof}
+\section{Torus charts and phasewise localization}\label{sec:torus-routing}
+
+The next constructions keep the whole motion inside a specified torus
+chart. A scaled linear interpolation preserves a convex first coordinate;
+an exponential interpolation in a fixed chart instead uses endpoint-width
+bounds. A phasewise construction then exhibits the vertical and planar
+potentials separately, and the final variant retains a permanent origin
+track. These choices give different path estimates for their observers.
+In every case the force is recomputed at the prescribed physical viscosity
+after any spatial rescaling.
+
+\subsection{Physical chart scaling and thickened torus boxes}
+
+\begin{proposition}[Linear scaling into the torus]
+\label{prop:bb-mid-scaled-torus}
+The direction recorder without its passive track has a unit-torus
+realization periodic after $t=1$, with fixed label $(1/8,0,0)$ and
+halting event $x_1\in(-1/4,0)\pmod1$.  Each period acts on full
+solid boxes; the pressure is zero and the force has zero mean.
+\end{proposition}
+\begin{proof}\label{proof:bb-mid-scaled-torus}
+Use the controls of Lemma~\ref{lem:bb-direction}. Let
+$\Sigma=\Gamma\times(\{\bot,F\}\sqcup\mathcal R)\times\{0,1\}$
+be its full cell alphabet after deleting the singleton passive track.
+Let $S$ be the finite set of recorder controls and choose a bijection
+$j:S\to\{1,\ldots,|S|\}$.
+Use odd digits with $B=2|\Sigma|+1$, and offsets $o_s=-2j(s)$ for halting main
+controls and $o_s=2j(s)$ for the others. Number the resulting reciprocal branches
+$i=1,\ldots,N$, and write $k_i$ for branch $i$'s first-coordinate scale.
+Thicken each reciprocal branch by
+$[-1,1]$.  Its horizontal family gaps are at least $B^{-2}$.
+Lift branch $i$ to $4i$, use the linear first scale
+$\kappa_i=1+\theta(k_i-1)$, and descend.  Padding smaller than
+$(4B^2)^{-1}$ separates all phases: in the middle the half-height
+is one and height gaps are four; in the other phases horizontal
+gaps suffice.  The two horizontal half-widths are at most $1/2$.
+Use the true-plateau cutoff of Lemma~\ref{lem:bb-curl}; it supplies
+at least the full-box guarantee, together with a neighborhood extension.
+The exact first coordinate obeys
+\begin{equation}\label{eq:bb-mid-scaled-convex}
+ X_1(\tau)=(1-\theta)X_1(0)+\theta X_1(1).
+\end{equation}
+
+Let $V$ be the period-one extension of the unit-time field just constructed.
+All codes, paths and padded supports lie in $(-L,L)^3$ for
+$L=10+2|S|+4N$.  Put $a=1/(16L)$ and define in the torus patch
+$(-1/2,1/2)^3$ the repeated field $U(t,x)=aV(t,x/a)$.
+Its support lies in $(-1/16,1/16)^3$ and its curl potential is the
+scaled potential $a^2A(t,x/a)$.  In particular its mean is zero.
+The residual must be recomputed at the physical viscosity:
+\begin{equation}\label{eq:bb-mid-viscosity-scale}
+ \mathcal F_\nu[U](t,x)=a\{V_t+(V\cdot\nabla)V-(\nu/a^2)\Delta V\}(t,x/a).
+\end{equation}
+This keeps the specified viscosity rather than silently rescaling it.
+During the first unit interval load from $(1/8,0,0)$ to the scaled
+initial code using a curl translation with a box of half-width $1/32$
+and padding $1/32$.  Its support lies inside $(-1/4,1/4)^3$.
+
+For a nonhalting run both loading endpoints have positive first
+coordinate, and every subsequent pair of coded endpoints is positive.
+Equation~\eqref{eq:bb-mid-scaled-convex} preserves that sign throughout.
+A halting checkpoint has negative first coordinate in $(-1/16,0)$.
+All paths stay in the stated patch, so passage modulo one introduces
+no extra visit.  Checkpoint times are
+$1+\sum_{l<n}(5+2(l-h_l))$.  This proves the all-time equivalence,
+including initial halting; Proposition~\ref{prop:bb-realization}
+finishes the proof.
+\end{proof}
+
+\begin{proposition}[Exponential deformation in a fixed chart]
+\label{prop:bb-mid-direct-torus}
+The move-first recorder has a unit-torus realization periodic after
+one loading interval, with fixed label $(1/4,1/4,1/4)$ and halting
+event $x_1\in(2/3,5/6)\pmod1$.  The repeated motion acts on thickened
+closed rectangles and their neighborhoods.  Its mean-zero residual
+has the form $f_0+\nu f_1$ with coefficients independent of $\nu$.
+\end{proposition}
+\begin{proof}\label{proof:bb-mid-direct-torus}
+First normalize to one halting work state and use the move-first recorder
+of Lemma~\ref{lem:bb-movefirst}; let $\Sigma$ be its full three-track
+cell alphabet. Initial halting is preserved by this normalization.
+For $m$ controls take $\ell=1/[64(m+1)]$.  Nonhalt squares have
+lower left corners $(1/4+2j\ell,1/4)$, $0\le j\le m-2$;
+the halt square has lower left corner $(3/4,1/4)$.
+Use odd digits and $b=2|\Sigma|+1$, with coding height $1/4$.
+There are $N$ reciprocal branches with gaps at least $\ell b^{-2}$.
+Thicken them by half-height $h_0=1/[64(N+1)]$ and choose heights
+$z_i=1/2+i/[4(N+1)]$.  Use exponential scale $k_i^\theta$,
+center interpolation, and
+$\epsilon=\min(h_0/2,\ell/(8b^2))$.
+The first and last phases have horizontal gaps exceeding
+$2\epsilon$; middle height gaps equal $16h_0$ and exceed
+$2(h_0+\epsilon)$.  Widths never exceed their endpoint values.
+A concrete plateau cutoff for a box centered at $C_i$, with
+half-widths $w_{ij}$, is
+\[
+ \chi_i=\prod_{j=1}^3\sigma\!\left(
+ \frac{(w_{ij}+\epsilon)^2-(x_j-C_{ij})^2}
+ {(w_{ij}+\epsilon)^2-(w_{ij}+\epsilon/2)^2}\right).
+\]
+Use its product with
+$\tfrac12\dot C_i\times(x-C_i)+(0,0,(\dot a_i/a_i)(x_1-C_{i1})(x_2-C_{i2}))$,
+where $a_i=k_i^\theta$.  Its curl gives the exact affine motion on
+an open neighborhood of each box.  All supports remain inside the
+unit cube.
+
+Load the initial rational code along its segment from the fixed
+label, using a radial cutoff of radius $1/32$ and a flat switch
+$\sigma(4t-1)$.  Then repeat the unit motion, with phase switches
+$\sigma(8\tau-1),\sigma(8\tau-3),\sigma(8\tau-5)$.
+These choices give spatial zero collars of width $1/8$ and temporal
+zero collars of width $1/16$ at nonnegative integer joins, including
+loading.  The force follows from
+$f_0=U_t+(U\cdot\nabla)U$, $f_1=-\Delta U$.
+
+Nonhalt centers have first coordinate at most $1/4+1/32$ and
+half-width at most $\ell/2$, hence their entire trajectories satisfy
+$x_1\le1/4+1/32+\ell/2<2/3$.  This is a width bound, not a convexity
+claim for exponential scaling.  Loading also avoids the slab for a
+nonhalting input.  Halt codes enter it at a finite integer sample,
+including time one for an initial halt.  The record block length and
+the known frontier site $n+2$ recover absolute head position at each
+work checkpoint.  All force and uniqueness assertions now follow
+from Proposition~\ref{prop:bb-realization}.
+\end{proof}
+
+\subsection{Thin-sheet curls and a slab observer}
+\label{ba:thin-sheet}
+
+The next realization stays directly inside the unit cube.  It uses a
+vertical translation potential in the first and last phases and a planar
+Hamiltonian potential in the middle.  Because its observer ignores
+height, the proof must keep the first coordinate safe throughout all
+three phases.
+
+\begin{theorem}[A direct unit-cube implementation]
+\label{ba:thin-sheet-torus}
+At each fixed positive computable viscosity, every machine and finite
+input has an effective smooth mean-zero force on the unit flat
+three-torus, with bounded mixed derivatives and period one for $t\ge1$,
+whose smooth zero-data solution satisfies
+\[
+ \exists t\ge0:\ X(t;(1/4,1/4,1/4))\in
+ (2/3,5/6)\times\mathbb T^2
+ \quad\Longleftrightarrow\quad\text{the machine halts}.
+\]
+The normalized pressure is zero and the solution is unique in the
+classical torus comparison class of Section~\ref{sec:model}.
+\end{theorem}
+
+\begin{proof}\label{ba:thin-sheet-torus-proof}
+For the cutoffs in this proof, recall
+\[
+ E(r)=\begin{cases}e^{-1/r},&r>0,\\0,&r\le0,\end{cases}
+ \qquad \Theta(r)=\frac{E(r)}{E(r)+E(1-r)}.
+\]
+Use the single-halt normalization and the seven-row recorder in
+Lemma~\ref{lem:bb-movefirst}, with frontier initially at two and all marks
+initially zero.  In the present notation its controls are
+$S_q,U_r,W_r,C_q,D_q$ (ready, work-move completed, right scan,
+frontier extension, left scan), and its full symbols are $(a,g,m)$
+with history $g\in\{E,F\}\sqcup\{\widehat r\}$.
+The first row writes and moves, the second marks and starts the scan,
+and the last clears the mark and stays.  These are exactly the seven
+rows already proved, after renaming.  Thus the new work head $j'$ is
+less than $n+2$, a work step takes $2(n+2-j')+4$ local steps, and
+both full-domain incoming properties hold.
+
+Let $M$ count the partial controls, let $H=S_h$, and put
+\[
+ \ell=\frac1{100(M+1)},\qquad o_H=(3/4,1/2),\qquad
+ o_{s_i}=(1/4+2i\ell,1/2)\quad(0\le i<M-1),\qquad z_*=1/4.
+\]
+In the state frames $o_s+\ell[0,1]^2$, use odd digits in base
+$B=2|\Sigma|+1$ for the full alphabet $\Sigma$.  The minimal
+left-split subdivision in Lemma~\ref{ba:rectangles} gives separately
+separated rectangles $E_i^-,E_i^+$, centers $c_i^-,c_i^+$, and maps
+\[
+ y\longmapsto c_i^+
+ +\operatorname{diag}(a_i,a_i^{-1})(y-c_i^-),
+ \qquad a_i\in\{B^{-1},1,B\}.
+\]
+The common frame scale leaves these linear parts unchanged; all
+rectangle side lengths are at most $\ell$.
+
+For $N$ branches set $z_i=1/2+i/(4(N+1))$, $1\le i\le N$.
+Let $\delta$ be one tenth of the minimum of $1/100$, all pairwise
+source gaps, all pairwise target gaps and all $|z_i-z_j|$; omit empty
+pair lists.  For a box $D=\prod_j[l_j,r_j]$, allowing zero widths,
+define
+\[
+ \chi_D^\delta(x)=\prod_{j=1}^3
+ \Theta\!\left(2(x_j-l_j+\delta)/\delta\right)
+ \Theta\!\left(2(r_j+\delta-x_j)/\delta\right).
+\]
+It equals one on the $\delta/2$ padding and is supported in the
+$\delta$ padding.  Put $\theta(s)=\Theta(3s-1)$.
+On the first third of a unit step use
+\[
+ z_i(s)=z_*+(z_i-z_*)\theta(3s),\qquad
+ D_i(s)=E_i^-\times\{z_i(s)\},\qquad
+ P_i=\chi_{D_i(s)}^\delta(0,\dot z_i(s)x_1,0).
+\]
+Its curl on the plateau is $(0,0,\dot z_i)$; source gaps separate
+these supports.  Hence every point of the whole source plate lifts
+with planar coordinates unchanged.
+
+On the middle third set
+\[
+ \eta=\theta(3s-1),\quad c_i=(1-\eta)c_i^-+\eta c_i^+,
+ \quad\mu_i=1-\eta+\eta a_i,
+\]
+\[
+ D_i(s)=
+ \big[c_i+\operatorname{diag}(\mu_i,\mu_i^{-1})(E_i^--c_i^-)\big]
+ \times\{z_i\}.
+\]
+With $\gamma_i=\dot\mu_i/\mu_i$, use
+\[
+ P_i=\chi_{D_i(s)}^\delta(0,0,\Psi_i),\qquad
+ \Psi_i=\dot c_{i1}(x_2-c_{i2})-\dot c_{i2}(x_1-c_{i1})
+          +\gamma_i(x_1-c_{i1})(x_2-c_{i2}).
+\]
+Its plateau curl is
+$(\dot c_{i1}+\gamma_i(x_1-c_{i1}),
+  \dot c_{i2}-\gamma_i(x_2-c_{i2}),0)$,
+which is precisely the reciprocal-scale affine velocity.  The distinct
+heights separate the supports during this phase.  In the last third use
+the vertical potential again, with $E_i^+$ and
+$z_i(s)=z_i+(z_*-z_i)\theta(3s-2)$; target gaps separate the descents.
+Summing the curls gives the required field in each phase, with zero
+collars at every join.
+
+All supports lie strictly inside the unit cube: horizontal centers lie
+in $[1/4,3/4+\ell]\times[1/2,1/2+\ell]$, half-widths are at most
+$\ell/2$, heights lie between $1/4$ and $3/4$, and
+$\delta\le10^{-3}$.  Thus the field extends smoothly and with zero
+mean to the torus.  The construction acts on neighborhoods, not only
+plates.  An initial perturbation of max norm less than $\delta/(4B)$
+stays within $\delta/4$ of its affine plate path: vertical phases
+preserve the perturbation, and both horizontal scale factors are at
+most $B$.  It remains on the plateau, so uniqueness of the smooth ODE
+gives that same affine continuation throughout.
+
+Write the rational initial code as $(b,z_*)$.  During one unit of time
+load the fixed particle using
+\[
+ c_0(t)=(1-\theta(t))(1/4,1/4)+\theta(t)b
+\]
+and the curl of
+\[
+ \chi_{\{(c_0(t),z_*)\}}^{1/100}
+ (0,0,\dot c_{01}(x_2-c_{02})-\dot c_{02}(x_1-c_{01})).
+\]
+This has plateau velocity $(\dot c_0,0)$, support inside the cube,
+and zero endpoint collars.  Repeat the step field thereafter and define
+its residual force at $\nu$.  The finite compact phase formulas give
+effective mixed-derivative bounds; the only scale denominators satisfy
+$\mu_i\ge\min(1,a_i)>0$.  Curls have zero mean, and incompressibility
+gives zero mean for the residual.  Lemma~\ref{ba:realization} supplies
+the unique zero-data PDE solution and its global trajectories.
+
+At times $1+k$, induction through the full branch maps gives the exact
+local code through the first halt, or for all $k$ in the nonhalting
+case.  A halt code has first coordinate in
+$[3/4,3/4+\ell]\subset(2/3,5/6)$, including a code reached solely by
+loading an initially halted machine.  For a nonhalting input the loader
+stays below $1/2$ in first coordinate.  Lift and descent retain the
+nonhalting planar endpoint codes.  In the middle phase the centers are
+convex combinations of nonhalting centers and the first half-width is
+at most $\ell/2$, so
+\[
+ X_1\le1/4+2M\ell+\ell/2<1/2.
+\]
+This excludes the height-independent observer at every intermediate
+time.  Only the loader depends on the input; the repeated field depends
+on the finite machine table.
+\end{proof}
+
+
+\subsection{A permanent origin marker and moving cutoff neighborhoods}
+\label{bc:height-separated-section}
+\begin{theorem}[A height-separated mechanism with an origin track]
+\label{bc:height-separated-event}
+On the unit torus, the fixed label $(1/8,3/8,1/4)$ and strip
+$2/3<x_1<7/8$ realize halting by a smooth mean-zero general force,
+periodic after time one, with zero initial velocity and pressure and bounded
+mixed derivatives. A permanent tape track identifies the original absolute
+origin. For a fixed machine the repeated field is independent of the input.
+\end{theorem}
+\begin{proof}\label{bc:height-separated-event-proof}
+Normalize the work machine to one halting state as in
+Section~\ref{sec:bb-frontiers}, preserving initial halting.
+Use four tracks: work, head mark, history, and an origin bit. The history
+is empty except for a frontier at cell two; the origin bit is one only
+at absolute cell zero. In a ready control $S_q$, the head mark is one
+at the work head. At intermediate controls use the seven-row recorder
+unchanged, with the origin track copied in every row. The first row now
+requires ready mark one and writes mark zero at the old head before the
+work move. The final return row leaves mark one when entering $S_q$.
+These are the only two changes to the seven rows.
+
+For precision, this table is the conjugate of the frontier table on its
+full configuration domain by the following involution: toggle the mark
+at the current head exactly when the control is ready, and leave it
+unchanged at every other control. Also take the direct product with the
+passive origin track. This is a bijection of all configurations, not merely
+of initialized ones. It proves the full-domain inverse. The incoming-rule
+properties also persist: into the work-tag control the first-row output
+mark is zero; into the right scan the marking row and its loop still write
+one and zero; into the return scan frontier and nonfrontier outputs remain
+distinct; into the ready control only the last row applies. The initial
+configuration is the conjugate of the frontier initialization. Consequently
+its cycle length is $2(2+n-h')+4$, its absolute frontier is $2+n$, and
+its origin bit remains fixed at the absolute origin. This verifies the
+distinct initialization without another appeal to an existential compiler.
+
+Denote the resulting full four-track alphabet by $\mathcal A$, and let
+$K=2|\mathcal A|+1$ be its odd-digit radix.
+For $m$ controls take state-square side $r=1/[16(m+1)]$, with nonterminal
+lower corners $(1/8+2(i-1)r,1/8)$ and terminal lower corner $(3/4,1/8)$.
+Their common coding height is $z_0=1/4$. Lemma~\ref{lem:bb-onecell} supplies
+separated full planar rectangles; let $N$ be the number of branches.
+Give branch $i$ height
+$h_i=1/2+i/[4(N+1)]$. During a first phase lift its center from $z_0$
+to $h_i$, keeping its planar shape; during a middle phase interpolate
+its planar center, with width
+$w_i=(1-\mu)w_i^-+\mu w_i^+$ and height
+$v_i=w_i^-v_i^-/w_i$; during a final phase lower it to $z_0$.
+The planar first coordinate is again an endpoint convex combination.
+Choose a fixed positive padding less than one tenth of all source and
+target gaps, all chart-face margins and all height separations. The
+moving padded rectangles are disjoint at every phase. During the middle
+phase this follows from the different heights; during the first and last
+phases it follows from endpoint planar separation.
+
+Here is a direct potential for these moving neighborhoods. With moving
+center $c=(c_x,c_y,c_z)$ and $\lambda=\dot w/w=-\dot v/v$, set
+\[
+ A=(0,\dot c_z(x-c_x),
+       \dot c_x(y-c_y)-\dot c_y(x-c_x)
+                         +\lambda(x-c_x)(y-c_y)).
+\]
+Its curl on the plateau is
+$(\dot c_x+\lambda(x-c_x),\dot c_y-\lambda(y-c_y),\dot c_z)$.
+Multiply it by a cutoff supported in the moving padded neighborhood and
+sum the curls. All supports remain in the open cube. The displayed
+center-and-shape paths solve this field on the full planar rectangles;
+uniqueness proves their exact endpoint maps. The positive padding and
+bounded planar scales also give a uniform initial neighborhood that stays
+on every plateau, by the perturbation argument of Lemma~\ref{lem:bb-curl}.
+Each component of the total
+field has zero mean after periodization, since the entire field is a curl.
+
+For initialization, choose equal small squares about the fixed label and
+the rational input code at height $z_0$, with half-side less than one
+quarter of their minimum chart-face margin. Move one to the other by the
+same localized potential for translation during $[0,1]$. A nonterminal
+loading first coordinate lies between $1/8$ and $1/4$, and every subsequent
+nonterminal phase lies below $1/4$. A terminal endpoint lies in the detector.
+The initialized symbolic equivalence, including initial halting, therefore
+holds at all real times. Repeating the finite middle mechanism and applying
+Lemma~\ref{bc:analytic} gives the asserted force and comparison properties.
+Only the loading translation and its input code depend on the finite word.
+\end{proof}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/README.md

@@ -1,21 +1,21 @@
-# [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](main.pdf)
-
-**Author:** OpenAI
-
-**Date:** October 6, 2026
-
-## Citation
-
-```bibtex
-@misc{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026,
-  author = {{OpenAI}},
-  title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}},
-  year = {2026}
-}
-```
-
-## Version note
-
-This version supplies base-presentation and multiplier-ideal arguments for the induction, including the adjoint pullback and contraction-descent steps; see the [previous version](../Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026/main.pdf).
+# [Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity](main.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 6, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026,
+  author = {{OpenAI}},
+  title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}},
+  year = {2026}
+}
+```
+
+## Version note
+
+This version supplies base-presentation and multiplier-ideal arguments for the induction, including the adjoint pullback and contraction-descent steps; see the [previous version](../Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026/main.pdf).

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/figures/two-diagonals.tex

@@ -1,33 +1,33 @@
-\begin{minipage}[c]{0.48\linewidth}
-\centering
-\small
-\begin{tikzcd}[column sep=large,row sep=large]
- V \arrow[r,hook] \arrow[d,"\Bl_{\Delta_X\times\PP^1}"']
-   & B \arrow[d,"\Bl_{\Delta_Z}"] \\
- \mathcal V_0 \arrow[r,hook] & Z\times Z
-\end{tikzcd}
-\[
- \begin{gathered}
- V_z=\Bl_xX,\\
- \begin{aligned}
- (V\cap E)_z&=\PP(T_z^*D_\lambda)\\
-             &\subset E_z=\PP(T_z^*Z).
- \end{aligned}
- \end{gathered}
-\]
-\end{minipage}
-\hfill
-\begin{minipage}[c]{0.48\linewidth}
-\centering
-\small
-\begin{tikzcd}[column sep=large,row sep=large]
- E^+ \arrow[r] \arrow[d] & E=\PP_Z(\Omega_Z) \arrow[d] \\
- Z^+ \arrow[r,"p_+=\Bl_{U_0}"] & Z
-\end{tikzcd}
-\[
- \begin{gathered}
- J_+=p_+^{-1}(U_0),\qquad U_0=X\times\PP^1,\\
- \mathcal J=(\mathcal I_V\cdot\OO_E)\cdot\OO_{E^+}.
- \end{gathered}
-\]
-\end{minipage}
+\begin{minipage}[c]{0.48\linewidth}
+\centering
+\small
+\begin{tikzcd}[column sep=large,row sep=large]
+ V \arrow[r,hook] \arrow[d,"\Bl_{\Delta_X\times\PP^1}"']
+   & B \arrow[d,"\Bl_{\Delta_Z}"] \\
+ \mathcal V_0 \arrow[r,hook] & Z\times Z
+\end{tikzcd}
+\[
+ \begin{gathered}
+ V_z=\Bl_xX,\\
+ \begin{aligned}
+ (V\cap E)_z&=\PP(T_z^*D_\lambda)\\
+             &\subset E_z=\PP(T_z^*Z).
+ \end{aligned}
+ \end{gathered}
+\]
+\end{minipage}
+\hfill
+\begin{minipage}[c]{0.48\linewidth}
+\centering
+\small
+\begin{tikzcd}[column sep=large,row sep=large]
+ E^+ \arrow[r] \arrow[d] & E=\PP_Z(\Omega_Z) \arrow[d] \\
+ Z^+ \arrow[r,"p_+=\Bl_{U_0}"] & Z
+\end{tikzcd}
+\[
+ \begin{gathered}
+ J_+=p_+^{-1}(U_0),\qquad U_0=X\times\PP^1,\\
+ \mathcal J=(\mathcal I_V\cdot\OO_E)\cdot\OO_{E^+}.
+ \end{gathered}
+\]
+\end{minipage}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/main.tex

@@ -1,33 +1,33 @@
-\documentclass[11pt,letterpaper]{article}
-\input{preamble}
-
-\title{Log abundance for compact Kähler spaces\\
-under logarithmic Iitaka subadditivity}
-\author{OpenAI}
-\date{October 6, 2026}
-
-\begin{document}
-\maketitle
-
-\begin{abstract}
-Assume logarithmic Iitaka subadditivity for surjective morphisms with
-connected fibers between smooth projective complex varieties with compatible
-reduced simple normal crossing boundaries. We prove log abundance for normal
-irreducible compact Kähler spaces in every dimension: for a log canonical
-pair \((X,\Delta)\) with effective rational boundary and \(K_X+\Delta\)
-\(\Q\)-Cartier, analytic nefness of \(K_X+\Delta\) implies semiampleness.
-\end{abstract}
-
-\tableofcontents
-
-\input{sections/01-introduction}
-\input{sections/02-inductive-reduction}
-\input{sections/03-programs}
-\input{sections/04-boundary}
-\input{sections/05-rank-one}
-\input{sections/06-simple-nonvanishing}
-\input{sections/07-signed-rigidity}
-
-\bibliographystyle{plain}
-\bibliography{references}
-\end{document}
+\documentclass[11pt,letterpaper]{article}
+\input{preamble}
+
+\title{Log abundance for compact Kähler spaces\\
+under logarithmic Iitaka subadditivity}
+\author{OpenAI}
+\date{October 6, 2026}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+Assume logarithmic Iitaka subadditivity for surjective morphisms with
+connected fibers between smooth projective complex varieties with compatible
+reduced simple normal crossing boundaries. We prove log abundance for normal
+irreducible compact Kähler spaces in every dimension: for a log canonical
+pair \((X,\Delta)\) with effective rational boundary and \(K_X+\Delta\)
+\(\Q\)-Cartier, analytic nefness of \(K_X+\Delta\) implies semiampleness.
+\end{abstract}
+
+\tableofcontents
+
+\input{sections/01-introduction}
+\input{sections/02-inductive-reduction}
+\input{sections/03-programs}
+\input{sections/04-boundary}
+\input{sections/05-rank-one}
+\input{sections/06-simple-nonvanishing}
+\input{sections/07-signed-rigidity}
+
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/preamble.tex

@@ -1,67 +1,67 @@
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+  pdfauthor={OpenAI},
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+\newtheorem{remark}[theorem]{Remark}
+
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+\DeclareMathOperator{\codim}{codim}
+
+\renewcommand{\labelenumi}{(\roman{enumi})}
+\allowdisplaybreaks[1]
+\emergencystretch=1.5em

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/references.bib

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-  howpublished = {OpenAI Math Release preprint
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+}
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+}
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+
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+  author = {Hacon, Christopher and Xie, Lingyao},
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+  year = {2026},
+  eprint = {2607.24986},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
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+  doi = {10.48550/arXiv.2607.24986},
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+}
+
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+  author = {Hacon, Christopher and Li, Yi and Xie, Lingyao},
+  title = {{Fujiki} class {$\mathcal C$} varieties and a {K{\"a}hler} criterion},
+  year = {2026},
+  eprint = {2608.20588},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
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+
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+  author = {Das, Omprokash and Hacon, Christopher and P{\u a}un, Mihai},
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+  journal = {Advances in Mathematics},
+  volume = {443},
+  year = {2024},
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+  archivePrefix = {arXiv},
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+}
+
+@misc{FujinoAnalyticBCHM,
+  author = {Fujino, Osamu},
+  title = {Minimal model program for projective morphisms between complex analytic spaces},
+  year = {2022},
+  eprint = {2201.11315},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  note = {arXiv:2201.11315v1},
+  doi = {10.48550/arXiv.2201.11315},
+  url = {https://arxiv.org/abs/2201.11315v1}
+}
+
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+  author = {Fujino, Osamu},
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+  number = {3},
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+  archivePrefix = {arXiv},
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+
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+  journal = {Mathematische Zeitschrift},
+  volume = {307},
+  year = {2024},
+  note = {Article 20; arXiv:2301.09186v3},
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+  eprint = {2301.09186},
+  archivePrefix = {arXiv},
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+}
+
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+  archivePrefix = {arXiv},
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+}
+
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+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
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+}
+
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+
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+  primaryClass = {math.AG},
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+}
+
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+}
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+}
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+
+@misc{LI,
+  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}
+}
+
+@misc{CI,
+  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}
+}
+
+@misc{CB,
+  author = {{OpenAI}},
+  title = {{Conditional good minimal models for compact K{\"a}hler fourfolds}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf}{OAI:Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026}},
+  year = {2026}
+}
+
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+  author = {{OpenAI}},
+  title = {{Abundance after nonvanishing for compact K\"ahler fourfolds}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf}{OAI:Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026}},
+  year = {2026}
+}
+
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+  author = {{OpenAI}},
+  title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}},
+  howpublished = {OpenAI Math Release preprint
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+  year = {2026}
+}
+
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+  url = {https://arxiv.org/html/1504.01095v3}
+}
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+}
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+  title = {Projectivity criteria for {K{\"a}hler} morphisms},
+  year = {2024},
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+  archivePrefix = {arXiv},
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+}
+
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+}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/01-introduction.tex

@@ -1,238 +1,238 @@
-\section{Introduction}\label{sec:introduction}
-
-The abundance problem asks when the numerical positivity of a log canonical
-bundle produces a holomorphic map. More precisely, a nef adjoint should have
-a positive multiple generated by global sections. On a compact Kähler space,
-nefness is an analytic condition on a cohomology class, whereas generation
-is a statement about an actual holomorphic line bundle. The distinction is
-essential in the nonprojective setting.
-
-We prove log abundance for compact Kähler spaces under logarithmic Iitaka
-subadditivity for smooth projective varieties with reduced boundary. The
-argument passes through a stronger birational statement: every
-pseudo-effective smooth log adjoint has a semiample positive part and an
-effective rational fixed divisor equal to its analytic divisorial negative
-part. This form both supports the induction and retains the line-bundle
-information needed for generation on the original space.
-
-\subsection{The assumption and the theorem}
-
-A compact Kähler space is a compact complex analytic space with a Kähler
-form in the sense of local strictly plurisubharmonic potentials on local
-embeddings. For a rational line bundle \(L\), analytic nefness means that
-\(c_1(L)\) belongs to the closure of the Kähler cone. Equivalently, fix a
-Kähler form \(\omega\) and an integer \(r>0\) for which \(rL\) is a line
-bundle. For every \(\varepsilon>0\), that line has a smooth Hermitian metric
-\(h_\varepsilon\) satisfying
-\[
- \frac{\sqrt{-1}}{2\pi r}\Theta_{h_\varepsilon}
- \geq-\varepsilon\omega.
-\]
-Smooth metrics and forms on a singular space are understood through local
-embeddings. We call \(L\) semiample if an actual positive integral multiple
-is a holomorphic line bundle generated by its global sections.
-
-\begin{assumption}[Logarithmic Iitaka subadditivity]
-\label{asm:log-iitaka}
-Let \(f:X\to Y\) be a surjective morphism with connected fibers between
-smooth connected projective complex varieties. Let \(D_X\) and \(D_Y\) be
-reduced effective simple normal crossing divisors, allowing zero divisors,
-such that
-\[
- \Supp(f^*D_Y)\subseteq\Supp(D_X).
-\]
-For a very general smooth fiber \(F\), put \(D_F=D_X|_F\). Then
-\[
- \kappa(X,K_X+D_X)\geq
- \kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y).
-\]
-Here \(D_F\) is reduced SNC and
-\(K_F+D_F\sim (K_X+D_X)|_F\). The Iitaka dimension is \(-\infty\) when
-all positive integral systems are empty, with
-\((-\infty)+b=-\infty\) also for \(b=-\infty\); the zero divisor on a
-point has Iitaka dimension zero.
-\end{assumption}
-
-\begin{theorem}[Log abundance for compact Kähler spaces]\label{thm:main}
-Assume Assumption~\ref{asm:log-iitaka}. Let \(X\) be a normal irreducible
-compact Kähler complex analytic space, and let \(\Delta\geq0\) be a rational
-divisor such that \((X,\Delta)\) is log canonical and \(K_X+\Delta\) is
-rational Cartier. If \(K_X+\Delta\) is analytically nef, then it is
-semiample. Explicitly, there is an integer \(m>0\) such that
-\(m(K_X+\Delta)\) is Cartier and
-\[
- H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)\otimes_{\C}\OO_X
- \longrightarrow \OO_X(m(K_X+\Delta))
-\]
-is surjective at every point of \(X\).
-\end{theorem}
-
-The theorem includes every finite dimension and the zero-boundary case.
-Assumption~\ref{asm:log-iitaka} is used through the projective good-model
-theorem stated in Proposition~\ref{proj:good-models}. The cited proof of
-that projective theorem uses only its zero-boundary case, on a resolved
-projective Albanese fibration.
-
-\subsection{Context and relation to earlier work}
-
-In the minimal model program, abundance complements the construction of
-minimal models: the nef adjoint on a minimal model should determine a
-canonical fibration. For minimal projective threefolds, Miyaoka proved the
-case of numerical dimension one and Kawamata completed canonical abundance
-\cite{Miyaoka1988,Kawamata1992}. Keel, Matsuki, and McKernan established
-log abundance for threefolds
-\cite{KMM1994,KMM2004}. In arbitrary dimension, Birkar, Cascini, Hacon, and
-McKernan established log terminal models for projective klt pairs with big
-boundary and pseudo-effective adjoint, and finite generation for big
-adjoints \cite{BCHM2010}.
-
-The compact Kähler history already points to the importance of simple
-spaces. Peternell's threefold theorem isolated the possible case of a simple
-space not bimeromorphic to a finite quotient of a torus
-\cite{Peternell2001}; Demailly and Peternell subsequently proved canonical
-nonvanishing for nef terminal Kähler threefolds, including that case
-\cite{DemaillyPeternell2003}. Höring and Peternell constructed minimal models
-for normal \(\Q\)-factorial terminal compact Kähler threefolds with pseudo-effective canonical class
-\cite{HoeringPeternell2016}. Campana, Höring, and Peternell treated canonical abundance
-\cite{CHP2016,CHP2023}, and Das and Ou
-established log abundance for compact Kähler log canonical threefolds
-\cite[Corollary~1.3]{DasOu2026}.
-
-With \(D_X=D_Y=0\), Assumption~\ref{asm:log-iitaka} is Iitaka's classical
-\(C_{n,m}\) inequality
-\(\kappa(X)\geq\kappa(F)+\kappa(Y)\) for algebraic fiber spaces
-\cite[Equation~(1.0.1)]{CaoPaun2017}. Cao and Păun proved the analogous
-logarithmic inequality over an abelian base when the pair on the total space
-has an effective rational boundary and is klt
-\cite[Theorem~1.1]{CaoPaun2017}. Fujino derives
-logarithmic subadditivity from the conjectured equality
-\(\kappa_\sigma(X,K_X+D)=\kappa(X,K_X+D)\) for smooth projective
-\(X\) with reduced SNC \(D\)
-\cite{Fujino2017Subadditivity,Fujino2020Corrigendum}. Hashizume proves the
-reduced-SNC subadditivity statement when the log canonical divisor of the
-very general fiber is abundant \cite[Theorem~1.2]{Hashizume2020}. Here the
-implication is used in the other direction: the projective companion
-\cite{LI} derives good models for pseudo-effective projective lc adjoints
-from Assumption~\ref{asm:log-iitaka}, and those good models anchor the
-Kähler induction.
-
-Hacon and Xie's cone theorem, adjunction, relative positivity, and
-modified-big preparation \cite{HX} provide analytic inputs for our
-program constructions. Section~\ref{sec:programs} constructs effective
-generalized base data on a higher carrier, with an exact adjoint
-pullback identity and a multiplier-ideal inclusion. Fibrewise
-$L^2$ positivity and regularization give this presentation; its
-Stein-factor consequence supplies the non-klt induction.
-The section then proves the restricted semiampleness, proper relative
-good-model, and finite-model statements needed here, following the
-dimension-induction strategy of \cite{HX}.
-The nef data remain globally nef on a fixed carrier, and the total
-generalized boundary is globally modified big. Actual rational lines
-make the selected ray contractions projective; ordinary projective
-analytic results \cite{FujinoAnalyticBCHM} then construct their flips
-and the relative models. This produces a Kähler pullback description of
-the relevant Bott--Chern class. Generation of the prescribed holomorphic
-line requires the additional divisorial decomposition and boundary
-arguments. The fibration comparison and period positivity come from
-\cite{CI}; the boundary and lifting constructions adapt inputs in
-\cite{CB,BL}. The precise imported results are stated at their points
-of use.
-
-\subsection{The proof and its main constructions}
-
-For a pseudo-effective \((1,1)\)-class \(\alpha\), \(N(\alpha)\) records
-the least generic vanishing forced along prime divisors. Given a smooth
-compact Kähler manifold \(X\) and a rational SNC boundary \(B\) with
-\(J=K_X+B\) pseudo-effective, the induction constructs a modification
-\(\mu:Y\to X\) with \(Y\) smooth and
-\[
- \mu^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad
- R=N(c_1(\mu^*J)),
-\]
-as an identity of rational holomorphic line bundles, with \(R\) an effective
-rational divisor. Section~\ref{sec:inductive-reduction} uses transfer rules,
-algebraic reduction, and generically finite covers to reduce the induction
-to projective manifolds, nontrivial fibrations, and simple spaces of
-algebraic dimension zero. Here simple means that no positive-dimensional
-proper compact subvariety passes through a very general point.
-
-Section~\ref{sec:programs} supplies two program results. For the fibration
-case it contracts a known negative part while preserving the prescribed nef
-line. For the simple case it proves special termination for a chosen scaling
-and reaches a nef model when the pseudo-effective adjoint is rationally
-equivalent to a divisor supported on the reduced boundary, allowing negative
-coefficients. The termination argument uses lower-dimensional induction to
-construct one model on which an interval of perturbed adjoints is nef. On a
-common resolution their negative multiplicities are affine, whereas every
-nontrivial wall of the restricted program changes a slope. Hence only
-finitely many such walls occur.
-
-Section~\ref{sec:boundary} turns generation on separate strata into generation
-on the entire reduced dlt boundary, adapting Fujino's method of admissible
-sections \cite[Section~4]{Fujino2000}. When a comparison is needed, a Mori
-contraction realizes Kollár's residue comparison at the two coefficient-one
-points of a general \(\PP^1\) fiber
-\cite[Definition~13 and Proposition~14]{Kollar2013Sources}. On a common
-resolution, the pulled-back restrictions of the ambient Kähler class differ
-by a real linear combination of Chern classes of line bundles. On a stratum
-whose boundary dominates the image of the map
-defined by its semiample adjoint, restriction determines sections. For the
-remaining strata, we prove finite image for self-comparisons on sufficiently
-divisible pluricanonical systems using an invariant integral, period and
-lattice arguments, and a uniform cohomological bound for cyclic covers.
-Products over these finite images give compatible generating sections, used
-in both closing arguments.
-
-Section~\ref{sec:rank-one} first reduces to fibrations whose very general
-fiber has log Kodaira dimension zero. On a prepared fibration \(g:X\to W\),
-a relative generating form gives
-\[
- K_X+B\sim_{\Q}g^*H+A_*,\qquad H=K_W+T+M,
-\]
-where \(T\) is a rational SNC boundary and \(M\) is pulled back from a nef
-rational line on a projective quotient of \(W\). Fiber induction allows us
-to subtract the part of \(A_*\) dominating \(W\) from a positive current for
-\(K_X+B\). The resulting metric descends along connected smooth fibers; a
-zero among the nonnegative coefficients at primes dominating each base prime
-allows extension, proving \(H\) pseudo-effective. If \(a(W)=0\), the
-projective quotient is a point, so \(M\sim_{\Q}0\) and lower-dimensional
-induction makes \(H\) rationally equivalent to its negative divisor. For
-projective \(W\), a reduction via a chosen generalized program derived from
-Assumption~\ref{asm:log-iitaka} either lowers the positive base dimension,
-handled by secondary induction, or reaches a nef line that is big or torsion.
-At a stopping case, an intersection argument identifies the full negative
-divisor upstairs. A torsion positive part completes the decomposition
-directly. In the big nef case, Section~\ref{sec:programs} contracts that
-divisor while preserving the nef line; boundary generation, extension, and
-a new log canonical place at a hypothetical base locus prove semiampleness.
-
-Section~\ref{sec:nonvanishing} proves, independently of
-Assumption~\ref{asm:log-iitaka}, that a positive canonical power has a nonzero
-meromorphic section on every smooth connected simple compact Kähler manifold
-of algebraic dimension zero. Ou's uniruledness and foliation results supply
-the cotangent slope control \cite{Ou2025}. A point-threshold bound for big
-classes of volume one is contradicted using an auxiliary projective bundle
-over \(X\times X\). The diagonal in its square produces a subsheaf occupying
-a fixed positive fraction of a symmetric cotangent power of that bundle.
-A second construction over the diagonal of \(X\times X\) forces determinant
-vanishing which, after restriction to a blowup of \(X\), violates the point
-bound.
-
-Finally, Section~\ref{sec:signed} uses the meromorphic section to obtain,
-after resolving and enlarging to a reduced SNC boundary, a pseudo-effective
-adjoint with a signed representative supported there.
-Section~\ref{sec:programs} gives a nef dlt model, and
-Section~\ref{sec:boundary} gives generation on its reduced boundary.
-Pseudo-effectivity of the canonical pullback and an intersection argument
-isolate positive-dimensional boundary fibers away from components whose
-signed coefficients are nonpositive. A local root construction adapted from
-\cite[Proposition~3.3]{LI} separates the positive and negative divisor
-supports. An SNC Hodge-module calculation extends the lifting method of
-\cite[Sections~10--12]{BL} to residual poles and kills every finite-order
-obstruction to lifting such a fiber. Douady space and Artin approximation
-give compact deformations leaving the boundary; compactness of the cycle-space
-components and relative compactness of bounded-volume cycles allow a Baire
-argument to produce a covering family. Simplicity
-forces the actual nef line to be torsion. The program comparison and
-Lemma~\ref{bir:unique-current} then permit subtraction of the added boundary,
-yielding the inductive decomposition.
+\section{Introduction}\label{sec:introduction}
+
+The abundance problem asks when the numerical positivity of a log canonical
+bundle produces a holomorphic map. More precisely, a nef adjoint should have
+a positive multiple generated by global sections. On a compact Kähler space,
+nefness is an analytic condition on a cohomology class, whereas generation
+is a statement about an actual holomorphic line bundle. The distinction is
+essential in the nonprojective setting.
+
+We prove log abundance for compact Kähler spaces under logarithmic Iitaka
+subadditivity for smooth projective varieties with reduced boundary. The
+argument passes through a stronger birational statement: every
+pseudo-effective smooth log adjoint has a semiample positive part and an
+effective rational fixed divisor equal to its analytic divisorial negative
+part. This form both supports the induction and retains the line-bundle
+information needed for generation on the original space.
+
+\subsection{The assumption and the theorem}
+
+A compact Kähler space is a compact complex analytic space with a Kähler
+form in the sense of local strictly plurisubharmonic potentials on local
+embeddings. For a rational line bundle \(L\), analytic nefness means that
+\(c_1(L)\) belongs to the closure of the Kähler cone. Equivalently, fix a
+Kähler form \(\omega\) and an integer \(r>0\) for which \(rL\) is a line
+bundle. For every \(\varepsilon>0\), that line has a smooth Hermitian metric
+\(h_\varepsilon\) satisfying
+\[
+ \frac{\sqrt{-1}}{2\pi r}\Theta_{h_\varepsilon}
+ \geq-\varepsilon\omega.
+\]
+Smooth metrics and forms on a singular space are understood through local
+embeddings. We call \(L\) semiample if an actual positive integral multiple
+is a holomorphic line bundle generated by its global sections.
+
+\begin{assumption}[Logarithmic Iitaka subadditivity]
+\label{asm:log-iitaka}
+Let \(f:X\to Y\) be a surjective morphism with connected fibers between
+smooth connected projective complex varieties. Let \(D_X\) and \(D_Y\) be
+reduced effective simple normal crossing divisors, allowing zero divisors,
+such that
+\[
+ \Supp(f^*D_Y)\subseteq\Supp(D_X).
+\]
+For a very general smooth fiber \(F\), put \(D_F=D_X|_F\). Then
+\[
+ \kappa(X,K_X+D_X)\geq
+ \kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y).
+\]
+Here \(D_F\) is reduced SNC and
+\(K_F+D_F\sim (K_X+D_X)|_F\). The Iitaka dimension is \(-\infty\) when
+all positive integral systems are empty, with
+\((-\infty)+b=-\infty\) also for \(b=-\infty\); the zero divisor on a
+point has Iitaka dimension zero.
+\end{assumption}
+
+\begin{theorem}[Log abundance for compact Kähler spaces]\label{thm:main}
+Assume Assumption~\ref{asm:log-iitaka}. Let \(X\) be a normal irreducible
+compact Kähler complex analytic space, and let \(\Delta\geq0\) be a rational
+divisor such that \((X,\Delta)\) is log canonical and \(K_X+\Delta\) is
+rational Cartier. If \(K_X+\Delta\) is analytically nef, then it is
+semiample. Explicitly, there is an integer \(m>0\) such that
+\(m(K_X+\Delta)\) is Cartier and
+\[
+ H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)\otimes_{\C}\OO_X
+ \longrightarrow \OO_X(m(K_X+\Delta))
+\]
+is surjective at every point of \(X\).
+\end{theorem}
+
+The theorem includes every finite dimension and the zero-boundary case.
+Assumption~\ref{asm:log-iitaka} is used through the projective good-model
+theorem stated in Proposition~\ref{proj:good-models}. The cited proof of
+that projective theorem uses only its zero-boundary case, on a resolved
+projective Albanese fibration.
+
+\subsection{Context and relation to earlier work}
+
+In the minimal model program, abundance complements the construction of
+minimal models: the nef adjoint on a minimal model should determine a
+canonical fibration. For minimal projective threefolds, Miyaoka proved the
+case of numerical dimension one and Kawamata completed canonical abundance
+\cite{Miyaoka1988,Kawamata1992}. Keel, Matsuki, and McKernan established
+log abundance for threefolds
+\cite{KMM1994,KMM2004}. In arbitrary dimension, Birkar, Cascini, Hacon, and
+McKernan established log terminal models for projective klt pairs with big
+boundary and pseudo-effective adjoint, and finite generation for big
+adjoints \cite{BCHM2010}.
+
+The compact Kähler history already points to the importance of simple
+spaces. Peternell's threefold theorem isolated the possible case of a simple
+space not bimeromorphic to a finite quotient of a torus
+\cite{Peternell2001}; Demailly and Peternell subsequently proved canonical
+nonvanishing for nef terminal Kähler threefolds, including that case
+\cite{DemaillyPeternell2003}. Höring and Peternell constructed minimal models
+for normal \(\Q\)-factorial terminal compact Kähler threefolds with pseudo-effective canonical class
+\cite{HoeringPeternell2016}. Campana, Höring, and Peternell treated canonical abundance
+\cite{CHP2016,CHP2023}, and Das and Ou
+established log abundance for compact Kähler log canonical threefolds
+\cite[Corollary~1.3]{DasOu2026}.
+
+With \(D_X=D_Y=0\), Assumption~\ref{asm:log-iitaka} is Iitaka's classical
+\(C_{n,m}\) inequality
+\(\kappa(X)\geq\kappa(F)+\kappa(Y)\) for algebraic fiber spaces
+\cite[Equation~(1.0.1)]{CaoPaun2017}. Cao and Păun proved the analogous
+logarithmic inequality over an abelian base when the pair on the total space
+has an effective rational boundary and is klt
+\cite[Theorem~1.1]{CaoPaun2017}. Fujino derives
+logarithmic subadditivity from the conjectured equality
+\(\kappa_\sigma(X,K_X+D)=\kappa(X,K_X+D)\) for smooth projective
+\(X\) with reduced SNC \(D\)
+\cite{Fujino2017Subadditivity,Fujino2020Corrigendum}. Hashizume proves the
+reduced-SNC subadditivity statement when the log canonical divisor of the
+very general fiber is abundant \cite[Theorem~1.2]{Hashizume2020}. Here the
+implication is used in the other direction: the projective companion
+\cite{LI} derives good models for pseudo-effective projective lc adjoints
+from Assumption~\ref{asm:log-iitaka}, and those good models anchor the
+Kähler induction.
+
+Hacon and Xie's cone theorem, adjunction, relative positivity, and
+modified-big preparation \cite{HX} provide analytic inputs for our
+program constructions. Section~\ref{sec:programs} constructs effective
+generalized base data on a higher carrier, with an exact adjoint
+pullback identity and a multiplier-ideal inclusion. Fibrewise
+$L^2$ positivity and regularization give this presentation; its
+Stein-factor consequence supplies the non-klt induction.
+The section then proves the restricted semiampleness, proper relative
+good-model, and finite-model statements needed here, following the
+dimension-induction strategy of \cite{HX}.
+The nef data remain globally nef on a fixed carrier, and the total
+generalized boundary is globally modified big. Actual rational lines
+make the selected ray contractions projective; ordinary projective
+analytic results \cite{FujinoAnalyticBCHM} then construct their flips
+and the relative models. This produces a Kähler pullback description of
+the relevant Bott--Chern class. Generation of the prescribed holomorphic
+line requires the additional divisorial decomposition and boundary
+arguments. The fibration comparison and period positivity come from
+\cite{CI}; the boundary and lifting constructions adapt inputs in
+\cite{CB,BL}. The precise imported results are stated at their points
+of use.
+
+\subsection{The proof and its main constructions}
+
+For a pseudo-effective \((1,1)\)-class \(\alpha\), \(N(\alpha)\) records
+the least generic vanishing forced along prime divisors. Given a smooth
+compact Kähler manifold \(X\) and a rational SNC boundary \(B\) with
+\(J=K_X+B\) pseudo-effective, the induction constructs a modification
+\(\mu:Y\to X\) with \(Y\) smooth and
+\[
+ \mu^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad
+ R=N(c_1(\mu^*J)),
+\]
+as an identity of rational holomorphic line bundles, with \(R\) an effective
+rational divisor. Section~\ref{sec:inductive-reduction} uses transfer rules,
+algebraic reduction, and generically finite covers to reduce the induction
+to projective manifolds, nontrivial fibrations, and simple spaces of
+algebraic dimension zero. Here simple means that no positive-dimensional
+proper compact subvariety passes through a very general point.
+
+Section~\ref{sec:programs} supplies two program results. For the fibration
+case it contracts a known negative part while preserving the prescribed nef
+line. For the simple case it proves special termination for a chosen scaling
+and reaches a nef model when the pseudo-effective adjoint is rationally
+equivalent to a divisor supported on the reduced boundary, allowing negative
+coefficients. The termination argument uses lower-dimensional induction to
+construct one model on which an interval of perturbed adjoints is nef. On a
+common resolution their negative multiplicities are affine, whereas every
+nontrivial wall of the restricted program changes a slope. Hence only
+finitely many such walls occur.
+
+Section~\ref{sec:boundary} turns generation on separate strata into generation
+on the entire reduced dlt boundary, adapting Fujino's method of admissible
+sections \cite[Section~4]{Fujino2000}. When a comparison is needed, a Mori
+contraction realizes Kollár's residue comparison at the two coefficient-one
+points of a general \(\PP^1\) fiber
+\cite[Definition~13 and Proposition~14]{Kollar2013Sources}. On a common
+resolution, the pulled-back restrictions of the ambient Kähler class differ
+by a real linear combination of Chern classes of line bundles. On a stratum
+whose boundary dominates the image of the map
+defined by its semiample adjoint, restriction determines sections. For the
+remaining strata, we prove finite image for self-comparisons on sufficiently
+divisible pluricanonical systems using an invariant integral, period and
+lattice arguments, and a uniform cohomological bound for cyclic covers.
+Products over these finite images give compatible generating sections, used
+in both closing arguments.
+
+Section~\ref{sec:rank-one} first reduces to fibrations whose very general
+fiber has log Kodaira dimension zero. On a prepared fibration \(g:X\to W\),
+a relative generating form gives
+\[
+ K_X+B\sim_{\Q}g^*H+A_*,\qquad H=K_W+T+M,
+\]
+where \(T\) is a rational SNC boundary and \(M\) is pulled back from a nef
+rational line on a projective quotient of \(W\). Fiber induction allows us
+to subtract the part of \(A_*\) dominating \(W\) from a positive current for
+\(K_X+B\). The resulting metric descends along connected smooth fibers; a
+zero among the nonnegative coefficients at primes dominating each base prime
+allows extension, proving \(H\) pseudo-effective. If \(a(W)=0\), the
+projective quotient is a point, so \(M\sim_{\Q}0\) and lower-dimensional
+induction makes \(H\) rationally equivalent to its negative divisor. For
+projective \(W\), a reduction via a chosen generalized program derived from
+Assumption~\ref{asm:log-iitaka} either lowers the positive base dimension,
+handled by secondary induction, or reaches a nef line that is big or torsion.
+At a stopping case, an intersection argument identifies the full negative
+divisor upstairs. A torsion positive part completes the decomposition
+directly. In the big nef case, Section~\ref{sec:programs} contracts that
+divisor while preserving the nef line; boundary generation, extension, and
+a new log canonical place at a hypothetical base locus prove semiampleness.
+
+Section~\ref{sec:nonvanishing} proves, independently of
+Assumption~\ref{asm:log-iitaka}, that a positive canonical power has a nonzero
+meromorphic section on every smooth connected simple compact Kähler manifold
+of algebraic dimension zero. Ou's uniruledness and foliation results supply
+the cotangent slope control \cite{Ou2025}. A point-threshold bound for big
+classes of volume one is contradicted using an auxiliary projective bundle
+over \(X\times X\). The diagonal in its square produces a subsheaf occupying
+a fixed positive fraction of a symmetric cotangent power of that bundle.
+A second construction over the diagonal of \(X\times X\) forces determinant
+vanishing which, after restriction to a blowup of \(X\), violates the point
+bound.
+
+Finally, Section~\ref{sec:signed} uses the meromorphic section to obtain,
+after resolving and enlarging to a reduced SNC boundary, a pseudo-effective
+adjoint with a signed representative supported there.
+Section~\ref{sec:programs} gives a nef dlt model, and
+Section~\ref{sec:boundary} gives generation on its reduced boundary.
+Pseudo-effectivity of the canonical pullback and an intersection argument
+isolate positive-dimensional boundary fibers away from components whose
+signed coefficients are nonpositive. A local root construction adapted from
+\cite[Proposition~3.3]{LI} separates the positive and negative divisor
+supports. An SNC Hodge-module calculation extends the lifting method of
+\cite[Sections~10--12]{BL} to residual poles and kills every finite-order
+obstruction to lifting such a fiber. Douady space and Artin approximation
+give compact deformations leaving the boundary; compactness of the cycle-space
+components and relative compactness of bounded-volume cycles allow a Baire
+argument to produce a covering family. Simplicity
+forces the actual nef line to be torsion. The program comparison and
+Lemma~\ref{bir:unique-current} then permit subtraction of the added boundary,
+yielding the inductive decomposition.

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/02-inductive-reduction.tex

@@ -1,586 +1,586 @@
-\section{Birational decompositions and the geometric reduction}
-\label{sec:inductive-reduction}
-
-The proof will produce a semiample line bundle on a smooth model, together
-with its entire fixed divisorial part. This section makes that statement
-precise and reduces the induction to two cases: spaces admitting a
-nontrivial fibration, and simple spaces of algebraic dimension zero.
-
-\subsection{The inductive decomposition and its negative part}
-
-We use additive notation for rational holomorphic line bundles. Thus
-\(L\sim_{\Q}L'\) means that \(mL\) and \(mL'\) are holomorphically
-isomorphic for some positive integer \(m\). If a rational divisor occurs in such
-an identity, it denotes its associated rational line bundle. In contrast,
-\(c_1(L)\) and \(\{D\}=c_1(\OO_X(D))\) denote real Bott--Chern classes.
-Canonical comparisons are made with the usual local meromorphic canonical
-identifications. In particular, an identity of lines below contains more
-information than equality of their Chern classes.
-
-Let \(X\) be a smooth compact Kähler manifold and let \(\alpha\) be a
-pseudo-effective real \((1,1)\)-class. Fix a Kähler form \(\omega\). For a
-prime divisor \(D\), its minimal multiplicity is
-\[
- \nu_D(\alpha)=\lim_{\varepsilon\downarrow0}
- \inf\bigl\{\nu_D(T):T\in\alpha,\ T\geq-\varepsilon\omega\bigr\}.
-\]
-Here \(T\) ranges over closed currents, and \(\nu_D(T)\) is its generic
-Lelong number; one may equivalently use currents with analytic singularities.
-The limit is independent of \(\omega\). Boucksom's divisorial decomposition
-is
-\[
- N(\alpha)=\sum_D\nu_D(\alpha)D,
- \qquad Z(\alpha)=\alpha-\{N(\alpha)\}.
-\]
-The sum is a finite effective real divisor, and \(Z(\alpha)\) is modified
-nef: its minimal multiplicity at every prime is zero. We write \(N(L)\) and
-\(\nu_D(L)\) for \(N(c_1(L))\) and \(\nu_D(c_1(L))\). These are the
-analytic notions, with small Kähler
-perturbations, throughout the paper. We use the foundational results in
-\cite[Sections~2--3 and~5]{Boucksom2004}.
-
-The induction asks for a decomposition that retains both the actual line
-bundle and this analytic negative divisor.
-
-\begin{definition}\label{def:good-decomposition}
-For \(n\geq0\), let \(\Ggood_n\) be the following assertion. If \(X\)
-is a connected smooth compact Kähler manifold of dimension \(n\), \(B\)
-is a rational simple normal crossing boundary with coefficients in
-\([0,1]\), and \(J=K_X+B\) is pseudo-effective, then there is a smooth
-compact Kähler modification \(\mu:Y\to X\) and an actual rational line
-identity
-\begin{equation}\label{bir:good-decomposition}
- \mu^*J\sim_{\Q}P+R,
- \qquad P\text{ semiample},\qquad R=N(\mu^*J)\geq0,
-\end{equation}
-where \(R\) is a rational divisor.
-\end{definition}
-
-The next lemmas make this decomposition stable under further resolutions
-and allow exceptional resolution errors to be removed. They also provide
-the fixed-section statement needed to descend the case in which the
-positive part is torsion.
-
-\begin{lemma}[Negative-part calculus]\label{bir:calculus}
-Let \(\alpha\) be pseudo-effective on a smooth compact Kähler manifold.
-\begin{enumerate}
-\item Every positive current in \(\alpha\) contains the divisorial current
-\([N(\alpha)]\). If \(0\leq F\leq N(\alpha)\), then
-\[
- \alpha-\{F\}\text{ is pseudo-effective},\qquad
- N(\alpha-\{F\})=N(\alpha)-F.
-\]
-\item Minimal multiplicities are homogeneous and subadditive. In
-particular, if \(\beta\) is nef, then
-\(N(\alpha+\beta)\leq N(\alpha)\).
-\item If \(\mu:Y\to X\) is a smooth modification and \(D'\) is the strict
-transform of a prime \(D\), then
-\[
- \nu_{D'}(\mu^*\alpha)=\nu_D(\alpha).
-\]
-\item If \(\gamma\) is modified nef and \(j:\widetilde D\to X\) is a
-resolution of a prime divisor, then \(j^*\gamma\) is pseudo-effective.
-\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-The first assertion about currents follows from the definition of minimal
-multiplicity and Siu decomposition. Homogeneity and subadditivity follow by
-scaling and adding testing currents. The subtraction identity is the
-corresponding property in the big cone, followed by a small Kähler
-perturbation. More explicitly, for \(\alpha_\varepsilon=\alpha+
-\varepsilon\{\omega\}\), subtract
-\(F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\}\) coefficientwise.
-In the big cone all positive currents contain this divisor, so subtracting
-it translates each minimal multiplicity by its coefficient. As
-\(\varepsilon\downarrow0\), \(F_\varepsilon\to F\). The class of
-\(F-F_\varepsilon\) can be absorbed in a Kähler perturbation tending to
-zero (there are only finitely many components). This gives the upper bound
-for the asserted identity; subadditivity applied after adding \(F\) gives
-the reverse bound. Weak compactness of positive currents gives
-pseudo-effectivity of the limit. This is also the subtraction property of
-the divisorial decomposition in \cite[Section~3]{Boucksom2004}.
-
-For the strict-transform formula, pull almost-positive testing currents
-to \(Y\). Near the generic point of \(D'\), the map is an isomorphism,
-so the generic order is unchanged. This proves one inequality. Conversely,
-push a testing current on \(Y\) to \(X\). To control its negative error,
-write the error as a small multiple of the fixed positive current
-\(\mu_*\omega_Y\), and add a fixed smooth Kähler representative of
-\(C\omega_X-\{\mu_*\omega_Y\}\) for \(C\) sufficiently large. The
-result is a positive test in a Kähler perturbation tending to zero. The
-pushforward of \(\omega_Y\) has no divisorial order at the generic point
-of \(D\), where \(\mu\) is an isomorphism. Thus this test has the same
-generic order as the original one at \(D'\). Regularization, if needed,
-returns to tests with analytic singularities. This proves the other
-inequality and the formula. It is the analytic form of the strict-transform
-comparison in \cite[Lemma~5.8]{Boucksom2004}.
-
-Finally, choose analytic-singularity tests for a modified-nef class whose
-generic order at the chosen prime tends to zero. Subtract that generic
-divisorial order before restricting to a resolution of the prime. The
-remaining singular set does not contain its generic point, so restriction
-gives an almost-positive current there. The subtracted multiple and the
-Kähler error tend to zero. Taking the limit proves the last assertion; see
-also \cite[Propositions~2.4 and~3.8]{Boucksom2004}.
-\end{proof}
-
-For a rational line \(J\) on a normal compact Kähler space \(X\) whose pullback
-to a smooth resolution is pseudo-effective, we also use minimal
-multiplicities for divisorial places. If \(Q\) appears on a smooth
-resolution \(p:Y\to X\), set
-\[
- \nu_Q(J):=\nu_Q(p^*J).
-\]
-The strict-transform formula on a common smooth refinement makes this
-independent of the chosen resolution. When a real class is already on a
-fixed smooth model, \(\nu_Q\) continues to denote its coefficient in the
-negative part on that model.
-
-\begin{lemma}[Exceptional translation]\label{bir:exceptional}
-Let \(p:Y\to X\) be a proper bimeromorphic morphism from a smooth compact
-Kähler manifold to a normal compact Kähler space. Let \(M\) be a real
-\((1,1)\)-class on \(X\) represented by smooth local potentials, and let
-\(E\geq0\) be a real \(p\)-exceptional divisor. If
-\(\alpha=p^*M+\{E\}\) is pseudo-effective, then \(p^*M\) is
-pseudo-effective and
-\[
- N(\alpha)=N(p^*M)+E.
-\]
-\end{lemma}
-
-\begin{proof}
-It suffices by Lemma~\ref{bir:calculus} to prove \(E\leq N(\alpha)\).
-Write
-\[
- E-N(\alpha)=U-V,
- \qquad U,V\geq0,
-\]
-with no common component, and suppose \(U\neq0\). Its components are
-\(p\)-exceptional. Put \(n=\dim Y\) and
-\(\ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}\). Normality
-gives \(\ell\leq n-2\). Let \(\eta\) be the pullback of a Kähler form
-on \(X\), let \(\omega\) be Kähler on \(Y\), and set
-\[
- q(\gamma,\delta)=
- \int_Y\gamma\,\delta\,\eta^{\ell}\omega^{n-\ell-2}.
-\]
-The restriction property in Lemma~\ref{bir:calculus} gives
-\(q(Z(\alpha),U)\geq0\). The term \(q(p^*M,U)\) is zero: on each
-component of \(U\), it contains \(\ell+1\) factors pulled back from an
-image of dimension at most \(\ell\). Distinct effective divisors have
-nonnegative intersection against the semipositive and Kähler factors in
-\(q\). Hence
-\[
- 0\leq q(Z(\alpha),U)=q(U,U)-q(V,U)\leq q(U,U).
-\]
-On the other hand, \(q(U,\eta)=0\) by the same dimension count, whereas
-\(q(\eta,\eta)>0\). The mixed Hodge index theorem, applied first with
-Kähler factors and then by a semipositive limit, says that \(q\) is
-negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\)
-and that \(U\) is in the radical of \(q\): the radical assertion follows
-from negative semidefiniteness on \(\eta^\perp\), and then from
-\(q(U,\eta)=0\). But
-\[
- q(U,\omega)=\sum_i u_i\int_{U_i}
-       \eta^{\ell}\omega^{n-\ell-1}>0.
-\]
-Indeed, a component with image dimension \(\ell\) has strictly positive
-integrand on a dense open. This is a contradiction. The mixed Hodge index
-statement used here is the mixed Hodge--Riemann relation
-\cite[Theorems~A and~C]{DinhNguyen}; the limit preserves the assertion that
-there is at most one positive direction.
-\end{proof}
-
-\begin{lemma}[Pulling up a known decomposition]\label{bir:pullup}
-Let \(\alpha\) be a pseudo-effective real \((1,1)\)-class on a smooth compact
-Kähler manifold, and suppose
-\[
- \alpha=\beta+\{R\},\qquad \beta\text{ nef},\qquad R=N(\alpha)\geq0.
-\]
-For every smooth modification \(\mu:Y\to X\),
-\[
- N(\mu^*\alpha)=\mu^*R.
-\]
-In particular, this applies to the Chern classes of an actual rational line
-decomposition with a nef rational positive part.
-\end{lemma}
-
-\begin{proof}
-Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the
-strict-transform formula, the difference
-\(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover
-\(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply
-Lemma~\ref{bir:exceptional} to this class. It gives
-\(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion.
-\end{proof}
-
-\begin{lemma}[Fixed sections]\label{bir:sections}
-Let \(L\) be a pseudo-effective rational line on a smooth compact Kähler
-manifold. Every section \(s\) of an integral multiple \(mL\) satisfies
-\[
- \ord_D(s)\geq m\nu_D(L)
-\]
-at every prime \(D\). If \(L\sim_{\Q}P+R\), with \(P\) semiample and
-\(R=N(L)\) rational, then in every sufficiently divisible degree
-multiplication by the canonical section of \(mR\) identifies
-\(H^0(X,mP)\) with \(H^0(X,mL)\).
-\end{lemma}
-
-\begin{proof}
-The divisor current \([\operatorname{div}(s)]/m\) is a positive current
-in \(c_1(L)\), so Lemma~\ref{bir:calculus} gives the order bound. In the
-stated decomposition, every section therefore divides holomorphically by
-the canonical section of \(mR\). Conversely, multiplication gives a
-section of \(mL\). The actual rational line identity makes these operations
-inverse after clearing denominators.
-\end{proof}
-
-\begin{lemma}[Uniqueness when the positive part is torsion]
-\label{bir:unique-current}
-Suppose \(L\sim_{\Q}R\), where \(R=N(L)\) is an effective rational
-divisor on a smooth compact Kähler manifold. Then the only positive current
-in \(c_1(L)\) is \([R]\). If \(F\geq0\) is a rational divisor and
-\(L-F\) is pseudo-effective, then
-\[
- F\leq R,\qquad L-F\sim_{\Q}R-F=N(L-F).
-\]
-The same conclusions hold if \(L\sim_{\Q}P+R\) with \(P\) torsion.
-\end{lemma}
-
-\begin{proof}
-Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\),
-so assume \(\dim X>0\).
-Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The
-residual \(T-[R]\) is positive with zero cohomology class, so its mass
-against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes.
-For the second assertion, add \([F]\) to any positive current in
-\(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in
-Lemma~\ref{bir:calculus} gives the asserted negative part. A torsion line
-has zero Chern class and becomes trivial after taking a positive multiple,
-so the last statement is the same argument.
-\end{proof}
-
-\subsection{Birational transfer of the decomposition}
-
-All modifications resolving spaces, maps, or ideals can be chosen
-projective. A projective modification of a compact Kähler space is Kähler;
-graphs between compact Kähler models can likewise be resolved by smooth
-compact Kähler manifolds. We use these standard analytic resolution and
-flattening results without further mention.
-
-For log discrepancies our convention is
-\[
- a(E;X,\Delta)=1+\ord_E\bigl(K_Y-p^*(K_X+\Delta)\bigr)
-\]
-on a smooth model \(p:Y\to X\). In particular, on a log resolution of an
-lc pair, giving each new exceptional divisor coefficient one produces an
-effective exceptional error. The next proposition explains why this
-convention is harmless.
-
-\begin{proposition}[Resolution and descent]\label{bir:resolution-transfer}
-Let \((X,\Delta)\) be a normal compact Kähler lc pair with rational
-boundary and rational Cartier adjoint \(J\). Let \(p:Y\to X\) be a log
-resolution, and put
-\[
- B_Y=p_*^{-1}\Delta+\sum_{E\text{ exceptional}}E.
-\]
-If \(p^*J\) is pseudo-effective and the conclusion of
-Definition~\ref{def:good-decomposition} holds for \((Y,B_Y)\), then there
-is a smooth compact Kähler modification \(r:V\to X\) with an actual
-rational line identity
-\[
- r^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad
- R=N(r^*J)\geq0,
-\]
-where \(R\) is a rational divisor.
-If \(J\) is analytically nef, it is semiample on \(X\).
-\end{proposition}
-
-\begin{proof}
-With compatible canonical choices there is an actual rational divisor
-identity
-\[
- K_Y+B_Y=p^*J+E_p,\qquad
- E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0.
-\]
-Suppose \(q:V\to Y\) realizes the decomposition for the left side.
-Lemma~\ref{bir:exceptional}, applied over \(X\), gives
-\[
- N\bigl(q^*(K_Y+B_Y)\bigr)
- =N(q^*p^*J)+q^*E_p.
-\]
-Cancelling \(q^*E_p\) in the actual rational line identities proves the
-first assertion. This argument also shows that further resolutions using
-the reduced-exceptional convention do not change the assertion
-\(\Ggood_n\).
-
-If \(J\) is nef, its pullback is nef, so its negative part is zero. The
-semiample line in the decomposition is therefore \(q^*p^*J\) itself, up
-to rational line isomorphism. Choose an actual Cartier multiple generated
-on \(V\). Normality gives \((p q)_*\OO_V=\OO_X\), and projection formula
-identifies its sections with the sections downstairs. A base point
-downstairs would make every pullback section vanish on its nonempty fiber.
-There is no such point, so that multiple of \(J\) is globally generated.
-\end{proof}
-
-The second transfer concerns a torsion positive part. Its proof uses the
-following local construction for finite maps, which will also be used for
-meromorphic sections and for covers of spaces of algebraic dimension zero.
-
-\begin{lemma}[Finite analytic norms and characteristic polynomials]
-\label{bir:finite-norm}
-Let \(\nu:Z\to X\) be a finite surjective morphism of normal irreducible
-complex spaces, of degree \(d>0\), and let \(L\) be a holomorphic line
-bundle on \(X\). A nonzero meromorphic section \(s\) of \(\nu^*L\)
-has a nonzero meromorphic norm \(\operatorname{Nm}_\nu(s)\) in
-\(L^{\otimes d}\), with
-\[
- \operatorname{div}(\operatorname{Nm}_\nu(s))
-   =\nu_*\operatorname{div}(s).
-\]
-If \(s\) is holomorphic, its norm is holomorphic. If, in addition,
-\(s\) is nonzero at every point of \(\nu^{-1}(x)\), then its norm is
-nonzero at \(x\).
-
-For a global meromorphic function \(f\) on \(Z\), there is a monic
-polynomial \(\chi_f(T)\) of degree \(d\) with global meromorphic
-coefficient functions on \(X\) such that \(\chi_f(f)=0\) after pulling
-the coefficients to \(Z\).
-\end{lemma}
-
-\begin{proof}
-At \(x\in X\), choose a holomorphic frame of \(L\) and put
-\[
- R=\OO_{X,x},\qquad K=\operatorname{Frac}(R),\qquad
- A=(\nu_*\OO_Z)_x.
-\]
-The generic algebra \(A\otimes_RK\) is a product of finite field
-extensions of \(K\), of total dimension \(d\). The local meromorphic
-coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is
-irreducible and \(s\) is not identically zero, its restriction is nonzero
-on every generic local component. Thus \(a\) is nonzero in every field
-factor, and the determinant of multiplication by \(a\) is nonzero.
-Under a change of frame by a unit \(g\), this determinant changes by
-\(g^{-d}\). The determinants therefore glue to the asserted
-meromorphic section. The valuation formula for a norm over a discrete
-valuation ring gives the displayed divisor identity at every prime of
-the normal space \(X\).
-
-If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\).
-Its norm is integral over \(R\) and belongs to \(K\), hence belongs to
-\(R\) by normality. This does not require \(A\) to be locally free.
-If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit
-in the finite semilocal algebra \(A\); applying the same argument to
-\(a^{-1}\) shows that its norm is a unit in \(R\).
-
-For a meromorphic function \(f\), use instead the determinant of
-\(T\) minus multiplication by its local coefficient. These monic
-polynomials agree on overlaps, and Cayley--Hamilton shows that they
-annihilate \(f\). All these constructions use fractions of local analytic
-germs, so they remain available when the global meromorphic functions on
-\(X\) are constant.
-\end{proof}
-
-We now descend a decomposition whose positive part is torsion, the form
-needed for a covering family on a space of algebraic dimension zero.
-
-\begin{proposition}[Descent of a purely negative decomposition]
-\label{bir:finite-descent}
-Let \(e:Y\to X\) be a proper generically finite surjective morphism
-between connected smooth compact Kähler manifolds. Let \(L\) be a
-pseudo-effective rational line on \(X\), and let \(F\geq0\) be a
-rational divisor on \(Y\). Suppose that on a smooth model over \(Y\),
-the line \(e^*L+F\) is rationally linearly equivalent to its rational
-negative part. Then there is an effective rational divisor \(D\) on
-\(X\) such that
-\[
- L\sim_{\Q}D=N(L).
-\]
-\end{proposition}
-
-\begin{proof}
-Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back.
-Write \(e^*L+F\sim_{\Q}R=N(e^*L+F)\). Pull back a positive current in
-\(c_1(L)\) and add \([F]\). Lemma~\ref{bir:unique-current} gives
-\(F\leq R\). Put \(D_Y=R-F\). Subtraction in
-Lemma~\ref{bir:calculus} gives
-\[
- e^*L\sim_{\Q}D_Y=N(e^*L)\geq0.
-\]
-In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\).
-
-Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is
-the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\)
-through its normal Stein space:
-\[
- Y\xrightarrow{h}Z\xrightarrow{\nu}X,
- \qquad \nu\text{ finite of degree }d=\deg e.
-\]
-The connected birational map \(h\) descends \(s\) to a holomorphic
-section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula.
-Lemma~\ref{bir:finite-norm} gives a holomorphic section of \(mdL\) whose
-divisor is
-\[
- \operatorname{div}(\operatorname{Nm}_\nu(s_Z))
-   =\nu_*\operatorname{div}(s_Z)=m e_*D_Y.
-\]
-Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\Q}D\).
-
-Both \(e^*D\) and \(D_Y\) are positive divisor currents in
-\(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know
-\(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\),
-choose analytic-singularity tests for \(L\) whose generic orders at \(A\)
-tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\)
-dominating \(A\), the generic orders are multiplied by
-\(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be
-enlarged to Kähler errors upstairs tending to zero. Consequently
-\[
- \nu_{A'}(e^*L)
- \leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L)
- <\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D),
-\]
-contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\).
-\end{proof}
-
-We will also use Proposition~\ref{bir:finite-descent} when \(e\) is a
-modification and \(F=0\). Once the positive part of a decomposition is
-known to be torsion, it gives \(L\sim_{\Q}N(L)\) on the original smooth
-space; Lemma~\ref{bir:pullup} then identifies the negative divisor upstairs
-with the pullback of \(N(L)\).
-
-\subsection{The projective anchor and the two geometric cases}
-
-\begin{proposition}[Projective good models]\label{proj:good-models}
-Under Assumption~\ref{asm:log-iitaka}, every projective lc pair over
-\(\C\) with rational boundary and pseudo-effective rational Cartier
-adjoint has a good log minimal model. In particular, \(\Ggood_n\) holds
-for projective manifolds in every dimension, and every nef rational lc
-adjoint on a projective variety is semiample as an actual rational line.
-\end{proposition}
-
-\begin{proof}
-The good-model assertion is the rational-boundary consequence of the full
-argument in \cite[Proposition~2.5, Theorem~9.6, and Section~10]{LI}. That
-argument establishes the additional inductive and nonvanishing premises of
-its Proposition~2.5 before applying it, and proves a stronger real-boundary
-induction. Its only use of Assumption~\ref{asm:log-iitaka} is in its
-Lemma~6.1, on a resolved projective Albanese fibration with both boundaries
-zero. Thus its premise is precisely available here.
-
-For clarity, its good-model conclusion gives on a common smooth projective
-resolution
-\[
- u^*(K_X+B)=v^*(K_{X_m}+B_m)+F,
- \qquad F\geq0\text{ and }v\text{-exceptional},
-\]
-as an actual rational divisor identity; the line on \(X_m\) is semiample.
-This is the comparison in \cite[Lemma~2.2]{LI}. Algebraic nefness of a
-rational line on a projective variety implies analytic nefness, by adding
-arbitrarily small ample rational classes. Lemma~\ref{bir:exceptional}
-therefore identifies \(F\) with the analytic negative part of the left
-side. On a smooth projective variety, analytic pseudo-effectivity of a
-divisor class agrees with algebraic pseudo-effectivity
-\cite[Proposition~1.2]{BDPP}; hence the good-model assertion applies to
-every projective instance of \(\Ggood_n\). This proves the stated form of
-\(\Ggood_n\). If the original adjoint
-is nef, the same comparison and normal descent give its semiampleness.
-\end{proof}
-
-The remainder of the paper proves the following two propositions. They
-are stated here so that the global induction can be completed before its
-technical components are developed. A compact space is \emph{simple} if
-no positive-dimensional proper compact analytic subvariety passes through
-a very general point. We write \(a(X)\) for its algebraic dimension.
-
-\begin{proposition}[The fibration case]\label{rank:fibration-case}
-Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume
-\(\Ggood_j\) for \(j<n\). Let \(X\) be a connected
-smooth compact Kähler manifold of dimension \(n\), let \(B\) be a rational
-SNC boundary with coefficients in \([0,1]\), and suppose \(K_X+B\) is pseudo-effective. If \(X\) admits
-a dominant meromorphic map to an irreducible compact space \(Y\) in Fujiki
-class \(\mathcal C\), with \(0<\dim Y<n\), then the conclusion of
-\(\Ggood_n\) holds for \((X,B)\).
-\end{proposition}
-
-\begin{proposition}[The simple case]\label{simple:case}
-Fix \(n>0\) and assume \(\Ggood_j\) for \(j<n\). Let \(X\) be a connected
-smooth simple compact Kähler manifold of dimension \(n\) with \(a(X)=0\).
-For a rational SNC boundary \(B\) with coefficients in \([0,1]\) and
-\(J=K_X+B\) pseudo-effective,
-there is a smooth compact Kähler modification \(\mu:Y\to X\) such that
-\[
- \mu^*J\sim_{\Q}N(\mu^*J),
-\]
-and this negative part is a rational divisor. In particular, the conclusion
-of \(\Ggood_n\) holds with torsion positive part.
-\end{proposition}
-
-Proposition~\ref{rank:fibration-case} is proved in
-Section~\ref{sec:rank-one}; it reduces the fibration to relative log Kodaira
-dimension zero and then descends the adjoint to its base.
-Proposition~\ref{simple:case} is proved in Section~\ref{sec:signed}, using
-the meromorphic nonvanishing theorem of Section~\ref{sec:nonvanishing}.
-Both propositions use \(\Ggood\) only in dimensions below \(n\).
-
-\begin{theorem}[Good divisorial decomposition]
-\label{thm:good-decomposition}
-Under Assumption~\ref{asm:log-iitaka}, the assertion \(\Ggood_n\) holds
-for every finite \(n\).
-\end{theorem}
-
-\begin{proof}
-We use Propositions~\ref{rank:fibration-case} and~\ref{simple:case}, whose
-proofs occupy the rest of the paper. The assertion is immediate in dimension
-zero. Assume it in smaller dimensions and let \((X,B)\) be as in
-Definition~\ref{def:good-decomposition} in dimension \(n>0\).
-
-If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and
-Proposition~\ref{proj:good-models} applies. If \(0<a(X)<n\), the algebraic
-reduction of \(X\) is a dominant meromorphic map to a projective model of
-dimension \(a(X)\) \cite[Theorem and Definition~3.1]{CampanaPeternell1999}.
-Proposition~\ref{rank:fibration-case} applies.
-
-It remains to consider \(a(X)=0\). If \(X\) is simple, use
-Proposition~\ref{simple:case}. Otherwise Campana's maximal covering-family
-theorem supplies a generically finite evaluation map from an incidence
-space which itself has a nontrivial fibration; see
-\cite[Lemma~17 and Corollary~18]{Campana2026}. After resolving the incidence,
-the graph, and the base, we obtain a smooth compact Kähler source \(X'\) and maps
-\[
- e:X'\longrightarrow X,\qquad f:X'\longrightarrow Y',
- \qquad 0<\dim Y'<n,
-\]
-where \(e\) is proper generically finite and \(Y'\) is in class
-\(\mathcal C\) \cite[(1.1.1), (1.5)(3)--(4), (2.2)(1),(3), and (2.3)]{Fujiki1982}.
-The algebraic dimension of \(X'\) is zero. Indeed, a
-meromorphic function on \(X'\) descends through the birational part of the
-normal Stein factorization of \(e\). Lemma~\ref{bir:finite-norm} gives its
-characteristic polynomial over the finite part, with meromorphic
-coefficients on \(X\). These coefficients are constant because \(a(X)=0\),
-so the function is constant on the irreducible space \(X'\). Bimeromorphic
-modifications do not change this conclusion.
-
-Choose a reduced SNC divisor \(B'\) containing the inverse image of
-\(\Supp B\), after a further resolution. Pullback of logarithmic
-differentials gives the actual rational identity
-\[
- K_{X'}+B'=e^*(K_X+B)+F,\qquad F\geq0.
-\]
-For example, this follows locally by pulling back logarithmic top forms
-for the reduced support of \(B\); decreasing its coefficients to those of
-\(B\) only increases the error. Thus the adjoint upstairs is
-pseudo-effective. Proposition~\ref{rank:fibration-case} applies to
-\((X',B')\). Its semiample positive part is torsion, since a nonconstant
-semiample map would give a nonconstant meromorphic function on \(X'\).
-Proposition~\ref{bir:finite-descent} now gives the required decomposition
-for \(K_X+B\). This exhausts the cases and proves the induction.
-\end{proof}
-
-\begin{proof}[Proof of Theorem~\ref{thm:main}]
-Apply Theorem~\ref{thm:good-decomposition} on a log resolution of the
-given lc pair, with the reduced-exceptional boundary. The adjoint is
-pseudo-effective because the original one is nef.
-Proposition~\ref{bir:resolution-transfer} then gives generation of an
-actual Cartier multiple at every point of the original normal space.
-\end{proof}
+\section{Birational decompositions and the geometric reduction}
+\label{sec:inductive-reduction}
+
+The proof will produce a semiample line bundle on a smooth model, together
+with its entire fixed divisorial part. This section makes that statement
+precise and reduces the induction to two cases: spaces admitting a
+nontrivial fibration, and simple spaces of algebraic dimension zero.
+
+\subsection{The inductive decomposition and its negative part}
+
+We use additive notation for rational holomorphic line bundles. Thus
+\(L\sim_{\Q}L'\) means that \(mL\) and \(mL'\) are holomorphically
+isomorphic for some positive integer \(m\). If a rational divisor occurs in such
+an identity, it denotes its associated rational line bundle. In contrast,
+\(c_1(L)\) and \(\{D\}=c_1(\OO_X(D))\) denote real Bott--Chern classes.
+Canonical comparisons are made with the usual local meromorphic canonical
+identifications. In particular, an identity of lines below contains more
+information than equality of their Chern classes.
+
+Let \(X\) be a smooth compact Kähler manifold and let \(\alpha\) be a
+pseudo-effective real \((1,1)\)-class. Fix a Kähler form \(\omega\). For a
+prime divisor \(D\), its minimal multiplicity is
+\[
+ \nu_D(\alpha)=\lim_{\varepsilon\downarrow0}
+ \inf\bigl\{\nu_D(T):T\in\alpha,\ T\geq-\varepsilon\omega\bigr\}.
+\]
+Here \(T\) ranges over closed currents, and \(\nu_D(T)\) is its generic
+Lelong number; one may equivalently use currents with analytic singularities.
+The limit is independent of \(\omega\). Boucksom's divisorial decomposition
+is
+\[
+ N(\alpha)=\sum_D\nu_D(\alpha)D,
+ \qquad Z(\alpha)=\alpha-\{N(\alpha)\}.
+\]
+The sum is a finite effective real divisor, and \(Z(\alpha)\) is modified
+nef: its minimal multiplicity at every prime is zero. We write \(N(L)\) and
+\(\nu_D(L)\) for \(N(c_1(L))\) and \(\nu_D(c_1(L))\). These are the
+analytic notions, with small Kähler
+perturbations, throughout the paper. We use the foundational results in
+\cite[Sections~2--3 and~5]{Boucksom2004}.
+
+The induction asks for a decomposition that retains both the actual line
+bundle and this analytic negative divisor.
+
+\begin{definition}\label{def:good-decomposition}
+For \(n\geq0\), let \(\Ggood_n\) be the following assertion. If \(X\)
+is a connected smooth compact Kähler manifold of dimension \(n\), \(B\)
+is a rational simple normal crossing boundary with coefficients in
+\([0,1]\), and \(J=K_X+B\) is pseudo-effective, then there is a smooth
+compact Kähler modification \(\mu:Y\to X\) and an actual rational line
+identity
+\begin{equation}\label{bir:good-decomposition}
+ \mu^*J\sim_{\Q}P+R,
+ \qquad P\text{ semiample},\qquad R=N(\mu^*J)\geq0,
+\end{equation}
+where \(R\) is a rational divisor.
+\end{definition}
+
+The next lemmas make this decomposition stable under further resolutions
+and allow exceptional resolution errors to be removed. They also provide
+the fixed-section statement needed to descend the case in which the
+positive part is torsion.
+
+\begin{lemma}[Negative-part calculus]\label{bir:calculus}
+Let \(\alpha\) be pseudo-effective on a smooth compact Kähler manifold.
+\begin{enumerate}
+\item Every positive current in \(\alpha\) contains the divisorial current
+\([N(\alpha)]\). If \(0\leq F\leq N(\alpha)\), then
+\[
+ \alpha-\{F\}\text{ is pseudo-effective},\qquad
+ N(\alpha-\{F\})=N(\alpha)-F.
+\]
+\item Minimal multiplicities are homogeneous and subadditive. In
+particular, if \(\beta\) is nef, then
+\(N(\alpha+\beta)\leq N(\alpha)\).
+\item If \(\mu:Y\to X\) is a smooth modification and \(D'\) is the strict
+transform of a prime \(D\), then
+\[
+ \nu_{D'}(\mu^*\alpha)=\nu_D(\alpha).
+\]
+\item If \(\gamma\) is modified nef and \(j:\widetilde D\to X\) is a
+resolution of a prime divisor, then \(j^*\gamma\) is pseudo-effective.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The first assertion about currents follows from the definition of minimal
+multiplicity and Siu decomposition. Homogeneity and subadditivity follow by
+scaling and adding testing currents. The subtraction identity is the
+corresponding property in the big cone, followed by a small Kähler
+perturbation. More explicitly, for \(\alpha_\varepsilon=\alpha+
+\varepsilon\{\omega\}\), subtract
+\(F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\}\) coefficientwise.
+In the big cone all positive currents contain this divisor, so subtracting
+it translates each minimal multiplicity by its coefficient. As
+\(\varepsilon\downarrow0\), \(F_\varepsilon\to F\). The class of
+\(F-F_\varepsilon\) can be absorbed in a Kähler perturbation tending to
+zero (there are only finitely many components). This gives the upper bound
+for the asserted identity; subadditivity applied after adding \(F\) gives
+the reverse bound. Weak compactness of positive currents gives
+pseudo-effectivity of the limit. This is also the subtraction property of
+the divisorial decomposition in \cite[Section~3]{Boucksom2004}.
+
+For the strict-transform formula, pull almost-positive testing currents
+to \(Y\). Near the generic point of \(D'\), the map is an isomorphism,
+so the generic order is unchanged. This proves one inequality. Conversely,
+push a testing current on \(Y\) to \(X\). To control its negative error,
+write the error as a small multiple of the fixed positive current
+\(\mu_*\omega_Y\), and add a fixed smooth Kähler representative of
+\(C\omega_X-\{\mu_*\omega_Y\}\) for \(C\) sufficiently large. The
+result is a positive test in a Kähler perturbation tending to zero. The
+pushforward of \(\omega_Y\) has no divisorial order at the generic point
+of \(D\), where \(\mu\) is an isomorphism. Thus this test has the same
+generic order as the original one at \(D'\). Regularization, if needed,
+returns to tests with analytic singularities. This proves the other
+inequality and the formula. It is the analytic form of the strict-transform
+comparison in \cite[Lemma~5.8]{Boucksom2004}.
+
+Finally, choose analytic-singularity tests for a modified-nef class whose
+generic order at the chosen prime tends to zero. Subtract that generic
+divisorial order before restricting to a resolution of the prime. The
+remaining singular set does not contain its generic point, so restriction
+gives an almost-positive current there. The subtracted multiple and the
+Kähler error tend to zero. Taking the limit proves the last assertion; see
+also \cite[Propositions~2.4 and~3.8]{Boucksom2004}.
+\end{proof}
+
+For a rational line \(J\) on a normal compact Kähler space \(X\) whose pullback
+to a smooth resolution is pseudo-effective, we also use minimal
+multiplicities for divisorial places. If \(Q\) appears on a smooth
+resolution \(p:Y\to X\), set
+\[
+ \nu_Q(J):=\nu_Q(p^*J).
+\]
+The strict-transform formula on a common smooth refinement makes this
+independent of the chosen resolution. When a real class is already on a
+fixed smooth model, \(\nu_Q\) continues to denote its coefficient in the
+negative part on that model.
+
+\begin{lemma}[Exceptional translation]\label{bir:exceptional}
+Let \(p:Y\to X\) be a proper bimeromorphic morphism from a smooth compact
+Kähler manifold to a normal compact Kähler space. Let \(M\) be a real
+\((1,1)\)-class on \(X\) represented by smooth local potentials, and let
+\(E\geq0\) be a real \(p\)-exceptional divisor. If
+\(\alpha=p^*M+\{E\}\) is pseudo-effective, then \(p^*M\) is
+pseudo-effective and
+\[
+ N(\alpha)=N(p^*M)+E.
+\]
+\end{lemma}
+
+\begin{proof}
+It suffices by Lemma~\ref{bir:calculus} to prove \(E\leq N(\alpha)\).
+Write
+\[
+ E-N(\alpha)=U-V,
+ \qquad U,V\geq0,
+\]
+with no common component, and suppose \(U\neq0\). Its components are
+\(p\)-exceptional. Put \(n=\dim Y\) and
+\(\ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}\). Normality
+gives \(\ell\leq n-2\). Let \(\eta\) be the pullback of a Kähler form
+on \(X\), let \(\omega\) be Kähler on \(Y\), and set
+\[
+ q(\gamma,\delta)=
+ \int_Y\gamma\,\delta\,\eta^{\ell}\omega^{n-\ell-2}.
+\]
+The restriction property in Lemma~\ref{bir:calculus} gives
+\(q(Z(\alpha),U)\geq0\). The term \(q(p^*M,U)\) is zero: on each
+component of \(U\), it contains \(\ell+1\) factors pulled back from an
+image of dimension at most \(\ell\). Distinct effective divisors have
+nonnegative intersection against the semipositive and Kähler factors in
+\(q\). Hence
+\[
+ 0\leq q(Z(\alpha),U)=q(U,U)-q(V,U)\leq q(U,U).
+\]
+On the other hand, \(q(U,\eta)=0\) by the same dimension count, whereas
+\(q(\eta,\eta)>0\). The mixed Hodge index theorem, applied first with
+Kähler factors and then by a semipositive limit, says that \(q\) is
+negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\)
+and that \(U\) is in the radical of \(q\): the radical assertion follows
+from negative semidefiniteness on \(\eta^\perp\), and then from
+\(q(U,\eta)=0\). But
+\[
+ q(U,\omega)=\sum_i u_i\int_{U_i}
+       \eta^{\ell}\omega^{n-\ell-1}>0.
+\]
+Indeed, a component with image dimension \(\ell\) has strictly positive
+integrand on a dense open. This is a contradiction. The mixed Hodge index
+statement used here is the mixed Hodge--Riemann relation
+\cite[Theorems~A and~C]{DinhNguyen}; the limit preserves the assertion that
+there is at most one positive direction.
+\end{proof}
+
+\begin{lemma}[Pulling up a known decomposition]\label{bir:pullup}
+Let \(\alpha\) be a pseudo-effective real \((1,1)\)-class on a smooth compact
+Kähler manifold, and suppose
+\[
+ \alpha=\beta+\{R\},\qquad \beta\text{ nef},\qquad R=N(\alpha)\geq0.
+\]
+For every smooth modification \(\mu:Y\to X\),
+\[
+ N(\mu^*\alpha)=\mu^*R.
+\]
+In particular, this applies to the Chern classes of an actual rational line
+decomposition with a nef rational positive part.
+\end{lemma}
+
+\begin{proof}
+Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the
+strict-transform formula, the difference
+\(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover
+\(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply
+Lemma~\ref{bir:exceptional} to this class. It gives
+\(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion.
+\end{proof}
+
+\begin{lemma}[Fixed sections]\label{bir:sections}
+Let \(L\) be a pseudo-effective rational line on a smooth compact Kähler
+manifold. Every section \(s\) of an integral multiple \(mL\) satisfies
+\[
+ \ord_D(s)\geq m\nu_D(L)
+\]
+at every prime \(D\). If \(L\sim_{\Q}P+R\), with \(P\) semiample and
+\(R=N(L)\) rational, then in every sufficiently divisible degree
+multiplication by the canonical section of \(mR\) identifies
+\(H^0(X,mP)\) with \(H^0(X,mL)\).
+\end{lemma}
+
+\begin{proof}
+The divisor current \([\operatorname{div}(s)]/m\) is a positive current
+in \(c_1(L)\), so Lemma~\ref{bir:calculus} gives the order bound. In the
+stated decomposition, every section therefore divides holomorphically by
+the canonical section of \(mR\). Conversely, multiplication gives a
+section of \(mL\). The actual rational line identity makes these operations
+inverse after clearing denominators.
+\end{proof}
+
+\begin{lemma}[Uniqueness when the positive part is torsion]
+\label{bir:unique-current}
+Suppose \(L\sim_{\Q}R\), where \(R=N(L)\) is an effective rational
+divisor on a smooth compact Kähler manifold. Then the only positive current
+in \(c_1(L)\) is \([R]\). If \(F\geq0\) is a rational divisor and
+\(L-F\) is pseudo-effective, then
+\[
+ F\leq R,\qquad L-F\sim_{\Q}R-F=N(L-F).
+\]
+The same conclusions hold if \(L\sim_{\Q}P+R\) with \(P\) torsion.
+\end{lemma}
+
+\begin{proof}
+Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\),
+so assume \(\dim X>0\).
+Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The
+residual \(T-[R]\) is positive with zero cohomology class, so its mass
+against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes.
+For the second assertion, add \([F]\) to any positive current in
+\(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in
+Lemma~\ref{bir:calculus} gives the asserted negative part. A torsion line
+has zero Chern class and becomes trivial after taking a positive multiple,
+so the last statement is the same argument.
+\end{proof}
+
+\subsection{Birational transfer of the decomposition}
+
+All modifications resolving spaces, maps, or ideals can be chosen
+projective. A projective modification of a compact Kähler space is Kähler;
+graphs between compact Kähler models can likewise be resolved by smooth
+compact Kähler manifolds. We use these standard analytic resolution and
+flattening results without further mention.
+
+For log discrepancies our convention is
+\[
+ a(E;X,\Delta)=1+\ord_E\bigl(K_Y-p^*(K_X+\Delta)\bigr)
+\]
+on a smooth model \(p:Y\to X\). In particular, on a log resolution of an
+lc pair, giving each new exceptional divisor coefficient one produces an
+effective exceptional error. The next proposition explains why this
+convention is harmless.
+
+\begin{proposition}[Resolution and descent]\label{bir:resolution-transfer}
+Let \((X,\Delta)\) be a normal compact Kähler lc pair with rational
+boundary and rational Cartier adjoint \(J\). Let \(p:Y\to X\) be a log
+resolution, and put
+\[
+ B_Y=p_*^{-1}\Delta+\sum_{E\text{ exceptional}}E.
+\]
+If \(p^*J\) is pseudo-effective and the conclusion of
+Definition~\ref{def:good-decomposition} holds for \((Y,B_Y)\), then there
+is a smooth compact Kähler modification \(r:V\to X\) with an actual
+rational line identity
+\[
+ r^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad
+ R=N(r^*J)\geq0,
+\]
+where \(R\) is a rational divisor.
+If \(J\) is analytically nef, it is semiample on \(X\).
+\end{proposition}
+
+\begin{proof}
+With compatible canonical choices there is an actual rational divisor
+identity
+\[
+ K_Y+B_Y=p^*J+E_p,\qquad
+ E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0.
+\]
+Suppose \(q:V\to Y\) realizes the decomposition for the left side.
+Lemma~\ref{bir:exceptional}, applied over \(X\), gives
+\[
+ N\bigl(q^*(K_Y+B_Y)\bigr)
+ =N(q^*p^*J)+q^*E_p.
+\]
+Cancelling \(q^*E_p\) in the actual rational line identities proves the
+first assertion. This argument also shows that further resolutions using
+the reduced-exceptional convention do not change the assertion
+\(\Ggood_n\).
+
+If \(J\) is nef, its pullback is nef, so its negative part is zero. The
+semiample line in the decomposition is therefore \(q^*p^*J\) itself, up
+to rational line isomorphism. Choose an actual Cartier multiple generated
+on \(V\). Normality gives \((p q)_*\OO_V=\OO_X\), and projection formula
+identifies its sections with the sections downstairs. A base point
+downstairs would make every pullback section vanish on its nonempty fiber.
+There is no such point, so that multiple of \(J\) is globally generated.
+\end{proof}
+
+The second transfer concerns a torsion positive part. Its proof uses the
+following local construction for finite maps, which will also be used for
+meromorphic sections and for covers of spaces of algebraic dimension zero.
+
+\begin{lemma}[Finite analytic norms and characteristic polynomials]
+\label{bir:finite-norm}
+Let \(\nu:Z\to X\) be a finite surjective morphism of normal irreducible
+complex spaces, of degree \(d>0\), and let \(L\) be a holomorphic line
+bundle on \(X\). A nonzero meromorphic section \(s\) of \(\nu^*L\)
+has a nonzero meromorphic norm \(\operatorname{Nm}_\nu(s)\) in
+\(L^{\otimes d}\), with
+\[
+ \operatorname{div}(\operatorname{Nm}_\nu(s))
+   =\nu_*\operatorname{div}(s).
+\]
+If \(s\) is holomorphic, its norm is holomorphic. If, in addition,
+\(s\) is nonzero at every point of \(\nu^{-1}(x)\), then its norm is
+nonzero at \(x\).
+
+For a global meromorphic function \(f\) on \(Z\), there is a monic
+polynomial \(\chi_f(T)\) of degree \(d\) with global meromorphic
+coefficient functions on \(X\) such that \(\chi_f(f)=0\) after pulling
+the coefficients to \(Z\).
+\end{lemma}
+
+\begin{proof}
+At \(x\in X\), choose a holomorphic frame of \(L\) and put
+\[
+ R=\OO_{X,x},\qquad K=\operatorname{Frac}(R),\qquad
+ A=(\nu_*\OO_Z)_x.
+\]
+The generic algebra \(A\otimes_RK\) is a product of finite field
+extensions of \(K\), of total dimension \(d\). The local meromorphic
+coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is
+irreducible and \(s\) is not identically zero, its restriction is nonzero
+on every generic local component. Thus \(a\) is nonzero in every field
+factor, and the determinant of multiplication by \(a\) is nonzero.
+Under a change of frame by a unit \(g\), this determinant changes by
+\(g^{-d}\). The determinants therefore glue to the asserted
+meromorphic section. The valuation formula for a norm over a discrete
+valuation ring gives the displayed divisor identity at every prime of
+the normal space \(X\).
+
+If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\).
+Its norm is integral over \(R\) and belongs to \(K\), hence belongs to
+\(R\) by normality. This does not require \(A\) to be locally free.
+If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit
+in the finite semilocal algebra \(A\); applying the same argument to
+\(a^{-1}\) shows that its norm is a unit in \(R\).
+
+For a meromorphic function \(f\), use instead the determinant of
+\(T\) minus multiplication by its local coefficient. These monic
+polynomials agree on overlaps, and Cayley--Hamilton shows that they
+annihilate \(f\). All these constructions use fractions of local analytic
+germs, so they remain available when the global meromorphic functions on
+\(X\) are constant.
+\end{proof}
+
+We now descend a decomposition whose positive part is torsion, the form
+needed for a covering family on a space of algebraic dimension zero.
+
+\begin{proposition}[Descent of a purely negative decomposition]
+\label{bir:finite-descent}
+Let \(e:Y\to X\) be a proper generically finite surjective morphism
+between connected smooth compact Kähler manifolds. Let \(L\) be a
+pseudo-effective rational line on \(X\), and let \(F\geq0\) be a
+rational divisor on \(Y\). Suppose that on a smooth model over \(Y\),
+the line \(e^*L+F\) is rationally linearly equivalent to its rational
+negative part. Then there is an effective rational divisor \(D\) on
+\(X\) such that
+\[
+ L\sim_{\Q}D=N(L).
+\]
+\end{proposition}
+
+\begin{proof}
+Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back.
+Write \(e^*L+F\sim_{\Q}R=N(e^*L+F)\). Pull back a positive current in
+\(c_1(L)\) and add \([F]\). Lemma~\ref{bir:unique-current} gives
+\(F\leq R\). Put \(D_Y=R-F\). Subtraction in
+Lemma~\ref{bir:calculus} gives
+\[
+ e^*L\sim_{\Q}D_Y=N(e^*L)\geq0.
+\]
+In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\).
+
+Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is
+the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\)
+through its normal Stein space:
+\[
+ Y\xrightarrow{h}Z\xrightarrow{\nu}X,
+ \qquad \nu\text{ finite of degree }d=\deg e.
+\]
+The connected birational map \(h\) descends \(s\) to a holomorphic
+section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula.
+Lemma~\ref{bir:finite-norm} gives a holomorphic section of \(mdL\) whose
+divisor is
+\[
+ \operatorname{div}(\operatorname{Nm}_\nu(s_Z))
+   =\nu_*\operatorname{div}(s_Z)=m e_*D_Y.
+\]
+Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\Q}D\).
+
+Both \(e^*D\) and \(D_Y\) are positive divisor currents in
+\(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know
+\(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\),
+choose analytic-singularity tests for \(L\) whose generic orders at \(A\)
+tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\)
+dominating \(A\), the generic orders are multiplied by
+\(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be
+enlarged to Kähler errors upstairs tending to zero. Consequently
+\[
+ \nu_{A'}(e^*L)
+ \leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L)
+ <\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D),
+\]
+contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\).
+\end{proof}
+
+We will also use Proposition~\ref{bir:finite-descent} when \(e\) is a
+modification and \(F=0\). Once the positive part of a decomposition is
+known to be torsion, it gives \(L\sim_{\Q}N(L)\) on the original smooth
+space; Lemma~\ref{bir:pullup} then identifies the negative divisor upstairs
+with the pullback of \(N(L)\).
+
+\subsection{The projective anchor and the two geometric cases}
+
+\begin{proposition}[Projective good models]\label{proj:good-models}
+Under Assumption~\ref{asm:log-iitaka}, every projective lc pair over
+\(\C\) with rational boundary and pseudo-effective rational Cartier
+adjoint has a good log minimal model. In particular, \(\Ggood_n\) holds
+for projective manifolds in every dimension, and every nef rational lc
+adjoint on a projective variety is semiample as an actual rational line.
+\end{proposition}
+
+\begin{proof}
+The good-model assertion is the rational-boundary consequence of the full
+argument in \cite[Proposition~2.5, Theorem~9.6, and Section~10]{LI}. That
+argument establishes the additional inductive and nonvanishing premises of
+its Proposition~2.5 before applying it, and proves a stronger real-boundary
+induction. Its only use of Assumption~\ref{asm:log-iitaka} is in its
+Lemma~6.1, on a resolved projective Albanese fibration with both boundaries
+zero. Thus its premise is precisely available here.
+
+For clarity, its good-model conclusion gives on a common smooth projective
+resolution
+\[
+ u^*(K_X+B)=v^*(K_{X_m}+B_m)+F,
+ \qquad F\geq0\text{ and }v\text{-exceptional},
+\]
+as an actual rational divisor identity; the line on \(X_m\) is semiample.
+This is the comparison in \cite[Lemma~2.2]{LI}. Algebraic nefness of a
+rational line on a projective variety implies analytic nefness, by adding
+arbitrarily small ample rational classes. Lemma~\ref{bir:exceptional}
+therefore identifies \(F\) with the analytic negative part of the left
+side. On a smooth projective variety, analytic pseudo-effectivity of a
+divisor class agrees with algebraic pseudo-effectivity
+\cite[Proposition~1.2]{BDPP}; hence the good-model assertion applies to
+every projective instance of \(\Ggood_n\). This proves the stated form of
+\(\Ggood_n\). If the original adjoint
+is nef, the same comparison and normal descent give its semiampleness.
+\end{proof}
+
+The remainder of the paper proves the following two propositions. They
+are stated here so that the global induction can be completed before its
+technical components are developed. A compact space is \emph{simple} if
+no positive-dimensional proper compact analytic subvariety passes through
+a very general point. We write \(a(X)\) for its algebraic dimension.
+
+\begin{proposition}[The fibration case]\label{rank:fibration-case}
+Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume
+\(\Ggood_j\) for \(j<n\). Let \(X\) be a connected
+smooth compact Kähler manifold of dimension \(n\), let \(B\) be a rational
+SNC boundary with coefficients in \([0,1]\), and suppose \(K_X+B\) is pseudo-effective. If \(X\) admits
+a dominant meromorphic map to an irreducible compact space \(Y\) in Fujiki
+class \(\mathcal C\), with \(0<\dim Y<n\), then the conclusion of
+\(\Ggood_n\) holds for \((X,B)\).
+\end{proposition}
+
+\begin{proposition}[The simple case]\label{simple:case}
+Fix \(n>0\) and assume \(\Ggood_j\) for \(j<n\). Let \(X\) be a connected
+smooth simple compact Kähler manifold of dimension \(n\) with \(a(X)=0\).
+For a rational SNC boundary \(B\) with coefficients in \([0,1]\) and
+\(J=K_X+B\) pseudo-effective,
+there is a smooth compact Kähler modification \(\mu:Y\to X\) such that
+\[
+ \mu^*J\sim_{\Q}N(\mu^*J),
+\]
+and this negative part is a rational divisor. In particular, the conclusion
+of \(\Ggood_n\) holds with torsion positive part.
+\end{proposition}
+
+Proposition~\ref{rank:fibration-case} is proved in
+Section~\ref{sec:rank-one}; it reduces the fibration to relative log Kodaira
+dimension zero and then descends the adjoint to its base.
+Proposition~\ref{simple:case} is proved in Section~\ref{sec:signed}, using
+the meromorphic nonvanishing theorem of Section~\ref{sec:nonvanishing}.
+Both propositions use \(\Ggood\) only in dimensions below \(n\).
+
+\begin{theorem}[Good divisorial decomposition]
+\label{thm:good-decomposition}
+Under Assumption~\ref{asm:log-iitaka}, the assertion \(\Ggood_n\) holds
+for every finite \(n\).
+\end{theorem}
+
+\begin{proof}
+We use Propositions~\ref{rank:fibration-case} and~\ref{simple:case}, whose
+proofs occupy the rest of the paper. The assertion is immediate in dimension
+zero. Assume it in smaller dimensions and let \((X,B)\) be as in
+Definition~\ref{def:good-decomposition} in dimension \(n>0\).
+
+If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and
+Proposition~\ref{proj:good-models} applies. If \(0<a(X)<n\), the algebraic
+reduction of \(X\) is a dominant meromorphic map to a projective model of
+dimension \(a(X)\) \cite[Theorem and Definition~3.1]{CampanaPeternell1999}.
+Proposition~\ref{rank:fibration-case} applies.
+
+It remains to consider \(a(X)=0\). If \(X\) is simple, use
+Proposition~\ref{simple:case}. Otherwise Campana's maximal covering-family
+theorem supplies a generically finite evaluation map from an incidence
+space which itself has a nontrivial fibration; see
+\cite[Lemma~17 and Corollary~18]{Campana2026}. After resolving the incidence,
+the graph, and the base, we obtain a smooth compact Kähler source \(X'\) and maps
+\[
+ e:X'\longrightarrow X,\qquad f:X'\longrightarrow Y',
+ \qquad 0<\dim Y'<n,
+\]
+where \(e\) is proper generically finite and \(Y'\) is in class
+\(\mathcal C\) \cite[(1.1.1), (1.5)(3)--(4), (2.2)(1),(3), and (2.3)]{Fujiki1982}.
+The algebraic dimension of \(X'\) is zero. Indeed, a
+meromorphic function on \(X'\) descends through the birational part of the
+normal Stein factorization of \(e\). Lemma~\ref{bir:finite-norm} gives its
+characteristic polynomial over the finite part, with meromorphic
+coefficients on \(X\). These coefficients are constant because \(a(X)=0\),
+so the function is constant on the irreducible space \(X'\). Bimeromorphic
+modifications do not change this conclusion.
+
+Choose a reduced SNC divisor \(B'\) containing the inverse image of
+\(\Supp B\), after a further resolution. Pullback of logarithmic
+differentials gives the actual rational identity
+\[
+ K_{X'}+B'=e^*(K_X+B)+F,\qquad F\geq0.
+\]
+For example, this follows locally by pulling back logarithmic top forms
+for the reduced support of \(B\); decreasing its coefficients to those of
+\(B\) only increases the error. Thus the adjoint upstairs is
+pseudo-effective. Proposition~\ref{rank:fibration-case} applies to
+\((X',B')\). Its semiample positive part is torsion, since a nonconstant
+semiample map would give a nonconstant meromorphic function on \(X'\).
+Proposition~\ref{bir:finite-descent} now gives the required decomposition
+for \(K_X+B\). This exhausts the cases and proves the induction.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Apply Theorem~\ref{thm:good-decomposition} on a log resolution of the
+given lc pair, with the reduced-exceptional boundary. The adjoint is
+pseudo-effective because the original one is nef.
+Proposition~\ref{bir:resolution-transfer} then gives generation of an
+actual Cartier multiple at every point of the original normal space.
+\end{proof}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03a-program-foundations.tex

@@ -1,693 +1,693 @@
-\subsection{Polarizations, ordinary replacements, and descent}
-\label{prog:foundations}
-
-The program constructions below use the analytic cone
-\(\overline{\mathrm{NA}}(T)\), dual to the nef cone in
-\(H^{1,1}_{\BC}(T,\R)\) for spaces with rational singularities
-\cite[Proposition~2.4]{DHP}. Degrees of global line bundles form a
-possibly smaller numerical space. We pass from the analytic cone to
-projective geometry by producing an actual relatively ample line.
-The ensuing descent statements apply on singular working models as
-well as on smooth initial spaces.
-
-\begin{lemma}[Degree-zero Bott--Chern descent]
-\label{prog:bc-descent}
-Let \(f:T\to Z\) be a proper surjective morphism with connected fibers
-between normal compact complex spaces with rational singularities.
-Suppose either
-\begin{enumerate}
-\item \(f\) is bimeromorphic and both spaces belong to Fujiki's class
-\(\mathcal C\); or
-\item \(f\) is projective and an effective rational boundary
-\(\Delta\) makes \((T,\Delta)\) klt with
-\(-(K_T+\Delta)\) relatively nef and big.
-\end{enumerate}
-Pullback is injective on both \(H^2(-,\R)\) and
-\(H^{1,1}_{\BC}(-,\R)\). In either group its image consists exactly
-of the classes with degree zero on every curve contracted by \(f\).
-\end{lemma}
-
-\begin{proof}
-These are, respectively, the two cases of
-\cite[Lemma~2.6(1),(2)]{DH2024}. The first case is birational descent
-between spaces with rational singularities and does not require a
-projectivity criterion. The second case uses relative vanishing for
-the ordinary rational klt pair and the projective relative curve
-space. In particular, it can be applied to a projective log-Fano
-contraction before constructing generalized data on its base.
-\end{proof}
-
-In applications to an ordinary birational negative contraction,
-the rationality hypotheses are automatic. The source is locally klt.
-For a dlt pair, first decrease its rational Cartier floor slightly;
-relative antiampleness persists. Relative vanishing gives
-\(R^if_*\OO_T=0\) for \(i>0\)
-\cite[Theorem~5.2]{FujinoAnalyticBCHM}. Composing a resolution of \(T\)
-with \(f\), the Leray spectral sequence proves that \(Z\) has rational
-singularities. The same reasoning works when the klt boundary is
-chosen separately on finitely many Stein base neighborhoods. For
-fiber-type contractions in case~(ii), rationality of the base follows
-from \cite[Lemma~8.8(i)]{DHP}; thus that application also precedes
-the construction of generalized data on the base.
-
-\begin{lemma}[Projectivity from a rational line detecting the ray]
-\label{prog:detected-projectivity}
-Let $f:T\to Z$ be a proper contraction between normal compact
-K\"ahler spaces, with $T$ having rational singularities. Suppose
-$\gamma$ is K\"ahler on $Z$ and
-\[
- \alpha=f^*\gamma,\qquad
- \overline{\mathrm{NA}}(T)\cap\alpha^\perp=R
-\]
-is a nonzero ray. If a global rational line bundle $D$ has
-$D\cdot R<0$, then $-D$ is relatively ample and $f$ is projective.
-\end{lemma}
-\begin{proof}
-Fix a K\"ahler class $\omega$ on $T$ and normalize $\overline{\mathrm{NA}}(T)$ by
-$\omega\cdot z=1$. The resulting slice $S$ is compact. Its unique
-point on $R$ has negative $D$-degree, so $d=c_1(D)$ is negative on
-a neighborhood of that point in $S$. On the complementary compact
-set, $\alpha$ has a strictly positive minimum, while $d$ is bounded.
-For all sufficiently small $s>0$, the class $\alpha-sd$ is therefore
-strictly positive on $S$ and is K\"ahler by cone duality.
-
-Choose an integer $m>0$ making $mD$ integral. Then
-\[
- c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd)
-\]
-is K\"ahler. On a neighborhood in the base with a potential for
-$\gamma$, absorb that potential in a smooth metric on the same global
-line bundle $-mD$. Its curvature is positive on the fibres. The analytic
-positive-line-bundle criterion and the fibrewise relative-ampleness
-criterion make $-mD$ relatively ample. The line bundle was global
-throughout, so no gluing of unrelated local polarizations is required.
-\end{proof}
-
-
-An analytic contraction supplied by a supporting class therefore becomes
-projective as soon as one rational line has negative degree on its ray.
-For a small contraction, that same line will define the flip globally.
-The local ordinary adjoints used to prove finite generation need not
-glue; their section algebras will be Veronese subalgebras of one algebra.
-
-\begin{lemma}[Divisor representatives over a Stein base]
-\label{prog:stein-representatives}
-Let $\pi:Y\to U$ be a projective surjective morphism with connected fibres
-between normal irreducible complex spaces, where $U$ is Stein, and let
-$\mathcal L$ be a holomorphic line bundle on $Y$.
-If $\pi$ is bimeromorphic, $\mathcal L$ has a holomorphic section
-which is not identically zero, and hence an effective Cartier divisor
-representative. More generally, after shrinking $U$ around any specified
-point, $\mathcal L$ has a nonzero meromorphic section.
-\end{lemma}
-
-\begin{proof}
-Suppose first that $\pi$ is bimeromorphic. Grauert's proper direct-image
-theorem makes $\mathcal F=\pi_*\mathcal L$ coherent. On the nonempty
-open subset $U^\circ$ where $\pi$ is an isomorphism, it is a line bundle.
-Choose $u_*\in U^\circ$. Cartan's theorem~A says that global sections
-generate $\mathcal F_{u_*}$, so one global section has nonzero image in
-$\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}$.
-Under
-\[
- H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L)
-\]
-it gives a section $s$ nonzero at the unique point above $u_*$.
-Its zero divisor is effective Cartier and represents $\mathcal L$.
-The section may vanish elsewhere; only its meromorphic inverse is
-being used to obtain a meromorphic trivialization.
-
-For the general case, fix $u\in U$ and $y\in\pi^{-1}(u)$, and choose
-a $\pi$-ample line bundle $\mathcal H$. Relative Serre generation,
-after shrinking to a Stein neighbourhood of $u$, gives an integer
-$n$ for which both $\mathcal L\otimes\mathcal H^n$ and
-$\mathcal H^n$ are relatively generated. Their proper direct images
-are coherent. Cartan's theorem~A and the evaluation maps supply
-sections $s$ and $t$ of these two bundles which are both nonzero at
-$y$. The quotient $s/t$ is the required nonzero meromorphic section
-of $\mathcal L$. At no point is $Y$ assumed Stein.
-\end{proof}
-
-\begin{proposition}[A global algebra for a detected flip]
-\label{prog:detected-flip}
-Let $(T,B+\mathbf M)$ be a gklt pair on a normal compact K\"ahler
-space which is globally strongly $\Q$-factorial, and let $A\in H^{1,1}_{\BC}(T,\R)$
-be its adjoint class. Let $f:T\to Z$ be a projective small contraction
-onto a normal compact K\"ahler space. Suppose that the classes of all
-$f$-vertical curves lie on one ray $R\subset\overline{\mathrm{NA}}(T)$, that $A\cdot R<0$,
-and that a global line bundle $L$ satisfies $L\cdot R<0$.
-
-Then $\bigoplus_{m\geq0}f_*L^m$ is locally finitely generated. The
-relative Proj of a sufficiently divisible Veronese is a projective
-small contraction $f^+:T^+\to Z$, where $T^+$ is compact K\"ahler and
-globally strongly $\Q$-factorial. The reflexive transform $L^+$ is a
-rational line bundle, with a positive power equal to the relatively
-ample tautological bundle. The transformed boundary and the same
-b-nef datum define a gklt pair on $T^+$, with relatively K\"ahler
-adjoint $A^+$. This is the generalized flip, with the usual
-strict discrepancy increase at centres in either exceptional locus.
-
-Moreover, if $t=(A\cdot C)/(L\cdot C)>0$ for an $f$-vertical curve $C$,
-then there is a unique $\eta\in H^{1,1}_{\BC}(Z,\R)$ such that
-\[
- A=t\,c_1(L)+f^*\eta,\qquad
- A^+=t\,c_1(L^+)+(f^+)^*\eta.
-\]
-Every class on $T$ has the corresponding forward transport with an
-actual exceptional-divisor correction. No assertion of surjectivity
-on Bott--Chern groups is required.
-\end{proposition}
-
-\begin{proof}
-Since $f$ has a global relatively ample bundle and all its vertical
-curves lie on $R$, numerical invariance of ampleness on the projective
-fibres shows that $-L$ is $f$-ample.
-
-\smallskip\noindent
-\emph{A local rational ordinary adjoint.}
-Fix $z\in Z$ and a relatively compact Stein neighbourhood $U$.
-Take a projective log resolution $p:V\to T_U$ carrying $\mathbf M$,
-and put $\rho=f p$. Write its actual structure boundary as $B_V$, so
-\[
- [K_V+B_V]+[\mathbf M_V]=p^*A.
-\]
-Every coefficient of $B_V$ is less than one. Apply
-Lemma~\ref{prog:stein-representatives} to obtain a Cartier
-representative $D_L$ of $L$ on $T_U$. The same birational direct-image
-argument applies to the coherent rank-one reflexive canonical sheaf:
-it supplies a canonical form nonzero at a chosen point of the common
-smooth isomorphism locus. Use this meromorphic form on $T_U$ and its
-pullback to $V$ to choose compatible canonical representatives, and set
-\[
- N=t p^*D_L-K_V-B_V.
-\]
-For every $\rho$-vertical curve, its degree equals that of $\mathbf M_V$.
-Thus $N$ is $\rho$-nef.
-
-The relative bigness of $N$ uses the additional fact that
-$\rho=f\circ p$ is \emph{bimeromorphic}: both the small contraction
-$f$ and the resolution $p$ are bimeromorphic. It does not follow from
-relative nefness alone. In fact every real Cartier divisor is relatively
-big for this projective bimeromorphic map over the Stein neighbourhood.
-Here is the required ample-plus-effective decomposition explicitly.
-Write $N=\sum_j r_jN_j$, where $r_j>0$, $\sum_jr_j=1$, and the $N_j$
-are rational Cartier divisors in its finite Cartier span. Choose a
-$\rho$-ample line bundle and a Cartier representative
-$P$ for it using Lemma~\ref{prog:stein-representatives}.
-If $q_jN_j$ is integral, apply the birational case of that lemma to
-$\mathcal O_V(q_jN_j-P)$. Its nonzero holomorphic section has an
-effective Cartier zero divisor $E_j$, giving the actual relation
-$q_jN_j\sim P+E_j$. This use depends on $\rho$ being bimeromorphic;
-the section is allowed to vanish along exceptional fibres. Therefore
-\[
- N\sim_{\R}
- \underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H}
- +\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}.
-\]
-Here $H$ is $\rho$-ample and $E\geq0$, proving $\rho$-bigness.
-For a map with positive-dimensional general fibres the direct image
-used above can be zero; no such bigness claim is made for that setting.
-Make the finitely many section choices on a slightly larger Stein
-neighbourhood, and now shrink $U$ so its closure is compact in that
-neighbourhood. Properness makes the inverse image of this closure
-compact. A log resolution of the finitely many resulting divisors
-therefore gives a uniform positive bound for the following choice.
-Choose $\epsilon>0$ small enough that $(V,B_V+\epsilon E)$ is sub-klt.
-The divisor $(1-\epsilon)N+\epsilon H$ is relatively ample. Express it
-as a positive combination of rational ample divisors and take general
-members of sufficiently high multiples. Relative Bertini
-\cite[Theorem~2.20]{DHP}, applied on a log resolution of $B_V+E$,
-gives an effective $\Theta\sim_{\R}N$ such that $(V,B_V+\Theta)$ is
-sub-klt. The high multiples make the added coefficients arbitrarily
-small. All choices are made near the compact fibre; shrink $U$ once
-to retain them.
-
-Put $\Delta=p_*(B_V+\Theta)\geq0$. Write the actual linear equivalence
-as $\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)$.
-The bimeromorphic map $p$ identifies meromorphic function fields, so
-$g_j=p^*h_j$. Pushing down gives
-$K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)$;
-in particular this divisor is real Cartier. Pulling back this same
-identity gives
-\[
- K_{T_U}+\Delta\sim_{\R}tD_L,
- \qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta.
-\]
-Thus $(T_U,\Delta)$ is klt. Any finitely many base line bundles in a
-relative linear equivalence can first be trivialized by shrinking $U$.
-Record the resulting equality as
-\begin{equation}\label{prog:detector-rationalization}
- K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j),
- \qquad d_i>0,
-\end{equation}
-using the positive support of $\Delta$. After a further relatively
-compact shrinking, the displayed divisors have finitely many prime
-components: their locally finite supports meet a compact inverse image.
-Equality of coefficients in \eqref{prog:detector-rationalization} is a
-finite rational affine system in $(d_i,t,b_j)$. The klt condition is
-open within this system, as seen on one log resolution; the pullbacks
-of its adjoints vary linearly by the displayed identity. A nearby
-rational solution therefore gives a rational effective klt boundary
-$\Delta_q$ and $t_q\in\Q_{>0}$ with
-\begin{equation}\label{prog:detector-local-adjoint}
- K_{T_U}+\Delta_q\sim_{\Q}t_qD_L.
-\end{equation}
-The individual local primes $D_i$ need not be $\Q$-Cartier: the identity
-ensures that the total adjoint is $\Q$-Cartier.
-
-\smallskip\noindent
-\emph{One global flip algebra.}
-The adjoint in \eqref{prog:detector-local-adjoint} is $f_U$-antiample.
-Its ordinary flip and local finite generation follow from
-\cite[Theorems~1.14 and~1.18]{FujinoAnalyticBCHM}. Clearing the displayed
-linear equivalence identifies a Veronese of its canonical algebra
-with a Veronese of $\mathcal R:=\bigoplus_{m\geq0}f_*L^m$ over $U$.
-Hence $\mathcal R$ is locally finitely generated by
-\cite[Lemma~2.26]{FujinoAnalyticBCHM}. The graded pieces are coherent. Over a connected normal base open
-the inverse image is irreducible; products of nonzero sections of line
-bundles are nonzero there. Thus its graded section algebra is an
-integral domain, without choosing any global meromorphic frame of $L$.
-Compactness supplies one sufficiently divisible Veronese
-$\mathcal R^{(r)}$ generated in degree one. Its relative Proj $T^+$
-restricts to the ordinary flip over every such $U$. It is therefore
-normal, locally klt, and small over $Z$, and has the global relatively
-ample bundle
-$\mathcal O_{T^+}(1)$. On the common big open this bundle is $L^r$.
-Reflexivity gives
-\[
- (L^+)^{[r]}\simeq\mathcal O_{T^+}(1).
-\]
-This proves that $L^+$ is a rational line bundle before any assertion
-of factoriality. Relative positivity over the compact K\"ahler base
-makes $T^+$ compact K\"ahler.
-
-For any global rank-one reflexive sheaf $\mathcal F^+$ on $T^+$,
-take its coherent reflexive transform $\mathcal F$ on $T$ using a
-common resolution. Some $M=\mathcal F^{[m]}$ is a line bundle.
-Choose $c\in\Q$ so that $(M+cL)\cdot C=0$, and clear its denominator
-to obtain an integral bundle $J=n(M+cL)$ numerically trivial over $f$.
-The local rational klt pairs \eqref{prog:detector-local-adjoint} have
-antiample adjoint, so Lemma~\ref{prog:integral-descent} yields
-one global line bundle $J_Z$ with $J=f^*J_Z$.
-On $T^+$, pull back $J_Z$, undo the rational twist $ncL^+$, and compare
-on the common big open. After clearing the already established index
-of $L^+$, reflexivity gives an invertible positive reflexive power of
-$\mathcal F^+$. This proves global strong factoriality.
-
-\smallskip\noindent
-\emph{The original generalized pair.}
-The local log-Fano pairs above give $R^if_*\mathcal O_T=0$ for $i>0$
-by \cite[Theorem~5.2]{FujinoAnalyticBCHM}. They make $T$ locally klt, hence rational.
-Leray for a projective resolution then shows that $Z$ is rational.
-Lemma~\ref{prog:bc-descent}(i) therefore gives
-$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$, with injective pullback. Thus $A=t c_1(L)+f^*\eta$ for a unique $\eta$.
-Define $A^+=t c_1(L^+)+(f^+)^*\eta$; it is relatively K\"ahler.
-
-To construct its actual discrepancy divisor, let $\mathcal I$ be the
-ideal defined by
-\[
- \operatorname{im}(f^*f_*L^r\longrightarrow L^r)
-   =\mathcal I\otimes L^r.
-\]
-Principalize $\mathcal I$ on a common resolution $p:W\to T$,
-$q:W\to T^+$ carrying $\mathbf M$. If
-$\mathcal I\mathcal O_W=\mathcal O_W(-F_L)$, the tautological quotient
-gives the actual bundle identity
-\[
- q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L).
-\]
-Hence $E_L=F_L/r$ is effective and exceptional over both sides, and
-$[E_L]=p^*c_1(L)-q^*c_1(L^+)$. If $B_W$ is the original structure
-boundary, put $B_W^+=B_W-tE_L$. Then
-\[
- [K_W+B_W^+]+[\mathbf M_W]=q^*A^+.
-\]
-Its pushforward is $B^+$, and its discrepancies do not decrease.
-The local ordinary comparison in \eqref{prog:detector-local-adjoint}
-has discrepancy divisor $t_qE_L$, so its strictness proves the stated
-strictness for $tE_L$ as well. Thus the original pair remains gklt.
-
-Finally, write any $\theta\in H^{1,1}_{\BC}(T,\R)$ uniquely as
-$\theta=f^*\eta_\theta+s c_1(L)$ and set
-$\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)$. Its correction is the actual
-divisor $sE_L$. For a rational line $M$, the number $s$ is rational. Applying
-Lemma~\ref{prog:integral-descent} to a Cartier multiple of $M-sL$
-gives $M=sL+f^*M_Z$ as actual rational lines. Reflexive extension
-on the common big open gives $M^+=sL^++(f^+)^*M_Z$.
-Consequently the cohomological transport agrees with the reflexive
-transform on Chern classes.
-\end{proof}
-
-
-\subsubsection{Real boundaries and their linear transport}
-
-The rational ordinary step constructed below also determines the step
-for every sufficiently close real boundary. The relevant comparison is
-an identity of actual rational lines over the contraction base.
-
-\begin{proposition}[Ordinary real boundaries]
-\label{prog:ordinary-replacement}
-Let $T$ be normal compact K\"ahler and globally Weil $\Q$-factorial,
-with canonical sheaf a rational line bundle. Let $B\geq0$ be a real
-boundary with $(T,B)$ klt. Assume the rational ordinary-step result
-of Proposition~\ref{prog:ordinary-step} for $T$ and its subsequent models. Then every
-$(K_T+B)$-negative extremal analytic ray has an ordinary step with
-projective contractions, compact K\"ahler models, and the expected
-opposite relative ample signs. Global Weil $\Q$-factoriality is preserved;
-global strong $\Q$-factoriality is preserved when imposed initially.
-\end{proposition}
-\begin{proof}
-Put $D=K_T+B$ and fix a $D$-negative extremal ray $R$. On the finite
-positive support of $B$, effectiveness, the klt condition and negativity
-on $R$ persist under small coefficient changes. Klt openness is checked
-on one log resolution of this support. Choose nearby effective rational
-klt boundaries $B_0,\ldots,B_m$ and positive real numbers $u_j$ with
-\[
- B=\sum_j u_jB_j,\qquad \sum_j u_j=1,
- \qquad D_j\cdot R<0,\quad D_j=K_T+B_j.
-\]
-For $B=0$ take just $B_0=0$. Every $D_j$ is a global rational line
-bundle. The rational-step theorem for $(T,B_0)$ provides a projective
-contraction $f:T\to Z$ onto a compact K\"ahler base, contracting
-exactly $R$, with $-D_0$ relatively ample. The base is rational by
-\cite[Lemma~8.8]{DHP}, so Lemma~\ref{prog:bc-descent} gives
-$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$; use case~(ii) when $f$ is of
-fiber type.
-Let $C$ be a rational curve spanning $R$, and set
-$a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}$. On every projective
-fibre, $-D_j$ is numerically equivalent to the positive multiple
-$-a_jD_0$. Numerical invariance of ampleness on that fibre, followed
-by the relative-ampleness criterion, makes $-D_j$ relatively ample.
-The positive combination $-D$ is relatively ample as a real line
-bundle. This also handles a fibre-type contraction.
-
-Suppose henceforth that $f$ is birational. A Cartier multiple
-$M_j=n_j(D_j-a_jD_0)$ is numerically trivial over $Z$ and
-$M_j-D_0$ is relatively ample. Lemma~\ref{prog:integral-descent},
-applied to the rational klt pair $(T,B_0)$ gives a rational line bundle
-$A_j$ on $Z$ with the actual identity
-\begin{equation}\label{prog:real-line-descent}
- D_j=a_jD_0+f^*A_j.
-\end{equation}
-Here and below an identity of rational line bundles means an isomorphism
-after a common integral multiple. The descent lemma is used only for
-birational $f$.
-
-For a small $f$, let $f^+:T^+\to Z$ be its rational $D_0$-flip.
-The rational theorem supplies a global relatively ample adjoint
-$D_0^+$ and preserves the stated factoriality and canonical-sheaf
-conditions. The other transformed adjoints $D_j^+$ are therefore
-rational line bundles. Transform \eqref{prog:real-line-descent} on the common
-big open and extend by reflexivity to obtain
-\[
- D_j^+=a_jD_0^++(f^+)^*A_j.
-\]
-All $D_j^+$ are relatively ample, as is
-$D^+=\sum_j u_jD_j^+$. Thus this same small map is the required
-ordinary real-boundary flip. The real ordinary discrepancy comparison
-proves that $(T^+,B^+)$ is klt and gives the strict increases at
-exceptional centres.
-
-For a divisorial contraction, write
-$D_0=f^*D_0'+e_0E$ with $e_0>0$. Transforming
-\eqref{prog:real-line-descent} on the target and summing yields
-\[
- D-f^*D'=\left(\sum_j u_ja_j\right)e_0E.
-\]
-This coefficient is positive. The same discrepancy comparison proves
-the required assertions. The ambient category was already preserved
-by the rational step in both cases.
-\end{proof}
-
-
-For later use, let \(D=K_T+B\) drive one of the birational steps just
-constructed, and write \(f':T'\to Z\) for its positive morphism
-(or the identity for a divisorial contraction). Lemma~\ref{prog:bc-descent}
-gives the unique decomposition and forward transform
-\begin{equation}
- \theta=f^*\eta+s\{D\},\qquad
- \theta'=(f')^*\eta+s\{D'\}.
- \label{prog:ordinary-forward-transport}
-\end{equation}
-On a common resolution \(p:W\to T\), \(q:W\to T'\), the ordinary
-comparison \(p^*D-q^*D'=F\) is an actual effective real divisor,
-exceptional over \(T'\) and over both sides for a flip. Hence
-\(p^*\theta-q^*\theta'=s\{F\}\). On Chern classes this is the
-reflexive transform, by the line identities in the proof above and
-Lemma~\ref{prog:integral-descent}. This construction gives forward
-transport without assuming surjectivity onto the next Bott--Chern group.
-
-\subsubsection{Actual real lines and relative numerical spaces}
-
-A second polarization argument will be used for maps whose fibers are
-Moishezon. It applies to a real combination of global line bundles,
-provided that its degrees on the fibers agree with a relatively Kähler
-class.
-
-\begin{lemma}[Polarization by an actual real line bundle]
-\label{prog:actual-line-polarization}
-Let $f:T\to Z$ be a proper Moishezon map, where $T$ is a compact
-K\"ahler space. Suppose that $L\in\operatorname{Pic}(T)\otimes\R$
-and a relatively K\"ahler class $\omega$ satisfy
-\[
- L\cdot C=\omega\cdot C
- \quad\text{for every compact curve $C$ contracted by $f$.}
-\]
-Then $L$ is relatively ample and $f$ is projective.
-\end{lemma}
-
-\begin{proof}
-Let $V$ be an irreducible positive-dimensional subspace of a reduced
-fibre. It is Moishezon. Normalize $V$ and take a smooth projective
-modification $\pi:\widetilde V\to V^{\mathrm n}$, chosen so that
-an effective exceptional divisor $E$ has $-E$ relatively ample.
-The pullback of the K\"ahler class from $V$ to its normalization is
-K\"ahler. Consequently
-$\pi^*\omega-\delta[E]$ is K\"ahler for sufficiently small
-$\delta>0$. On the projective manifold $\widetilde V$ the actual
-real line bundle $\pi^*L-\delta E$ has the same curve degrees as
-that K\"ahler class, and is therefore ample. For example, subtract
-a sufficiently small positive multiple of a fixed ample class from
-the K\"ahler class; the corresponding real line bundle is nef by
-the projective numerical criterion. Also $\pi^*L$ is nef, and the
-decomposition
-\[
- \pi^*L=(\pi^*L-\delta E)+\delta E
-\]
-shows that it is big. Thus
-\[
- L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0.
-\]
-The real Nakai--Moishezon criterion for proper algebraic spaces
-\cite[Theorem~1.6]{FujinoMiyamoto2020}, applied to the algebraizations
-of the Moishezon fibres, proves that $L$ is ample on every reduced
-fibre. The criterion is insensitive to nilpotents and reducible
-components. In the coefficient space of the finitely many global
-line bundles occurring in $L$, choose, for each fibre, a rational
-simplex containing $L$ in its interior whose vertices restrict to
-ample rational lines on that fibre. Openness of fibrewise ampleness
-for each of these finitely many rational lines gives a base
-neighbourhood on which the entire simplex is relatively ample.
-Choose finitely many such neighborhoods covering the base and
-intersect their coefficient neighborhoods of $L$. Every rational
-point of the resulting neighborhood is relatively ample globally.
-A rational simplex in this intersection also expresses $L$ as a
-positive combination of relatively ample rational lines. Clearing
-the denominator of one rational point proves projectivity.
-\end{proof}
-
-\begin{remark}
-The proof uses only positivity of the top self-intersections of
-$L$. Equality of curve degrees with $\omega$ does not assert
-equality of their higher intersection numbers or of their
-Bott--Chern classes.
-\end{remark}
-
-
-
-
-The projective morphism needed for the relative program can be chosen
-on a smooth model of the rational quotient. Here the Hodge-theoretic
-polarization has an elementary construction, independent of any
-birational projectivity-descent assertion.
-
-\begin{lemma}[Smooth Hodge-theoretic polarization]
-\label{prog:smooth-polarization}
-Let \(g:W\to S\) be a surjective morphism with connected fibers
-between smooth compact Kähler manifolds. If
-\(g^*:H^0(S,\Omega_S^2)\to H^0(W,\Omega_W^2)\) is an isomorphism,
-then \(g\) is projective. This applies when its general fiber is
-rationally connected.
-\end{lemma}
-
-\begin{proof}
-We use the smooth argument of \cite[Theorem~3.1, Step~1]{CH}.
-Let \(d=\dim W-\dim S\). Choose a rational class \(u\in H^2(W,\Q)\)
-close to a Kähler class. Write its Hodge decomposition as
-\(u=b+g^*v\), where \(b\) is Kähler and \(v\) has types \((2,0)\)
-and \((0,2)\). Hodge type and the projection formula give
-\[
- c=g_*(u^d)=g_*(b^d)>0,\qquad
- g_*(u^{d+1})=g_*(b^{d+1})+(d+1)c\,v.
-\]
-Since pushforward is rational, the class
-\[
- \ell=u-g^*\frac{g_*(u^{d+1})}{(d+1)c}
-      =b-g^*\frac{g_*(b^{d+1})}{(d+1)c}
-\]
-is rational and of type \((1,1)\). Lefschetz's theorem makes it the
-Chern class of a rational line. The displayed equality supplies a
-metric with positive curvature locally over \(S\), by adding a
-potential for the base class. Thus this line is relatively ample.
-For rationally connected general fibers, holomorphic two-forms are
-pulled back from the smooth base, so the hypothesis holds.
-\end{proof}
-
-\begin{lemma}[A rational Hodge correction over a smooth base]
-\label{prog:relative-hodge}
-Let $f:T\to S$ be an already projective surjective morphism of normal
-compact K\"ahler spaces. Assume that $S$ is smooth, $T$ has globally
-strongly $\Q$-factorial klt singularities, and, for a projective resolution
-$r:W\to T$, the morphism $g=f\circ r$ satisfies
-\[
- g^*:H^0(S,\Omega_S^2)\xrightarrow{\ \simeq\ }
-                         H^0(W,\Omega_W^2).
-\]
-Then every $\alpha\in H^{1,1}_{\BC}(T,\R)$ can be written
-\[
-                         \alpha=c_1(L)+f^*\gamma,
-\]
-where $L\in\operatorname{Pic}(T)\otimes_{\mathbb Z}\mathbb R$ is a
-finite real combination of global line bundles and
-$\gamma\in H^{1,1}(S,\mathbb R)$.
-\end{lemma}
-
-\begin{proof}
-The composition $g$ is projective, since all the spaces are compact.
-Choose a $g$-ample line bundle on $W$, with integral Chern class $H$.
-Set $d=\dim W-\dim S$ and
-\[
-                            c=g_*(H^d)>0.
-\]
-Thus $c$ is the positive degree of $H^d$ on a general fibre. The
-cohomological pushforward here is the usual pushforward between the
-smooth compact manifolds $W$ and $S$, defined over $\mathbb Q$ and of
-Hodge bidegree $(-d,-d)$.
-
-Define a rational linear operator on degree-two cohomology by
-\[
- \Pi(\eta)=\eta-g^*\!\left(\frac{g_*(\eta\smile H^d)}{c}\right).
-\]
-Every class of type $(2,0)$ on $W$ is pulled back from $S$, and the
-same holds for type $(0,2)$. The projection formula therefore shows
-that $\Pi$ kills both types. It preserves type $(1,1)$. Consequently
-\[
-        \Pi(H^2(W,\mathbb Q))\subset
-                         H^2(W,\mathbb Q)\cap H^{1,1}(W).
-\]
-By the Lefschetz $(1,1)$ theorem the image consists of rational Chern
-classes of global line bundles. Applying $\Pi$ to $r^*\alpha$ gives
-\[
- r^*\alpha=c_1(L_W)+g^*\gamma,
- \qquad L_W\in\operatorname{Pic}(W)\otimes\mathbb R,
- \qquad
- \gamma=\frac{g_*(r^*\alpha\smile H^d)}{c}\in H^{1,1}(S,\mathbb R).
-\]
-For example, express $r^*\alpha$ in a rational basis of $H^2(W,\mathbb Q)$
-and apply $\Pi$ to each basis vector. This gives the required finite
-real combination $L_W$.
-
-Write $L_W=\sum_j a_jL_j$ with actual line bundles $L_j$ on $W$.
-For each $j$, let
-\[
-                   \mathcal F_j=(r_*L_j)^{**}.
-\]
-It is a rank-one reflexive coherent sheaf. Strong
-$\mathbb Q$-factoriality gives an integer $m_j>0$ for which
-$M_j=\mathcal F_j^{[m_j]}$ is a line bundle on $T$.
-The canonical identification over the isomorphism locus of $r$ gives
-\[
-             L_j^{\otimes m_j}\simeq r^*M_j\otimes\mathcal O_W(E_j)
-\]
-for an integral $r$-exceptional divisor $E_j$.
-One may obtain this comparison directly from the evaluation map for
-$r_*L_j$: the two coherent rank-one sheaves agree away from the
-exceptional locus, so their invertible transforms differ by an
-exceptional divisor. No global meromorphic frame of $L_j$ or of the
-canonical sheaf is required.
-
-Set $L=\sum_j(a_j/m_j)M_j$ and
-$E=\sum_j(a_j/m_j)E_j$. The preceding identities yield
-\[
-                r^*(\alpha-c_1(L)-f^*\gamma)=[E].
-\]
-The divisor $E$ is $r$-exceptional and is numerically trivial on every
-$r$-contracted curve. The exceptional negativity lemma applied in both
-signs gives $E=0$. Injectivity of pullback by a resolution, or pushforward
-of the equality of currents, now gives
-$\alpha=c_1(L)+f^*\gamma$.
-\end{proof}
-
-
-The hypothesis of Lemma~\ref{prog:relative-hodge} persists on each
-projective globally strongly \(\Q\)-factorial klt model over this fixed
-smooth base: on a common smooth resolution, holomorphic two-forms are
-invariant under modifications. Thus the lemma applies separately on
-every working model. It supplies real combinations of global lines;
-Lemma~\ref{prog:stein-representatives} supplies their Cartier-divisor
-representatives on sufficiently small Stein base neighborhoods.
-
-\begin{lemma}[The projective relative cone in Bott--Chern cohomology]
-\label{prog:projective-cone}
-Let \(f:T\to S\) be projective between normal compact Kähler spaces
-with rational singularities, and assume
-\[
- H^{1,1}_{\BC}(T,\R)
-   =c_1\bigl(\Pic(T)\otimes\R\bigr)
-          +f^*H^{1,1}_{\BC}(S,\R).
-\]
-Let \(N_1^{\mathrm{line}}(T/S)\) denote the space of real combinations
-of \(f\)-contracted curves modulo degrees of global line bundles,
-and let \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) be their
-closed effective cone. Their analytic classes give an isomorphism
-onto the span of the face
-\[
- \mathcal F_f=
- \overline{\mathrm{NA}}(T)\cap(f^*\omega_S)^\perp,
-\]
-where \(\omega_S\) is any Kähler class on \(S\); this isomorphism
-identifies \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) with
-\(\mathcal F_f\). In particular, relative extremal rays in the line
-space are extremal rays of the full analytic cone.
-\end{lemma}
-
-\begin{proof}
-The assumed decomposition makes analytic equivalence and line-degree
-equivalence identical on combinations of vertical curves. Their map
-into the analytic numerical space is therefore well-defined and
-injective. We must show that its closed effective cone is the entire
-face, including its classes represented by positive currents.
-
-Fix \(z\in\mathcal F_f\). For every \(\gamma\in
-H^{1,1}_{\BC}(S,\R)\), both \(a\omega_S+\gamma\) and
-\(a\omega_S-\gamma\) are Kähler for large \(a\). Positivity then
-shows that \(z\) annihilates \(f^*\gamma\). Fix a global relatively
-ample line \(H\). If a real line \(D\) has nonnegative degree on
-every vertical curve, the projective relative numerical criterion
-makes \(D+\epsilon H\) relatively ample for every \(\epsilon>0\).
-Its Chern class becomes Kähler after adding a sufficiently large
-multiple of \(f^*\omega_S\). Therefore
-\[
-                 (c_1(D)+\epsilon c_1(H))\cdot z\geq0,
- \qquad c_1(D)\cdot z\geq0.
-\]
-Applying this to both signs of a relatively numerically trivial line
-shows that pairing with \(z\) factors through the finite-dimensional
-space of relative line degrees. It is nonnegative on its nef cone.
-By duality for this projective relative space, it is represented by
-an element of \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\).
-The decomposition in the statement makes its analytic image equal
-to \(z\). Conversely every vertical effective curve lies in
-\(\mathcal F_f\), and finite-dimensional injectivity preserves
-closedness. This proves the cone equality and the extremality claim.
-\end{proof}
+\subsection{Polarizations, ordinary replacements, and descent}
+\label{prog:foundations}
+
+The program constructions below use the analytic cone
+\(\overline{\mathrm{NA}}(T)\), dual to the nef cone in
+\(H^{1,1}_{\BC}(T,\R)\) for spaces with rational singularities
+\cite[Proposition~2.4]{DHP}. Degrees of global line bundles form a
+possibly smaller numerical space. We pass from the analytic cone to
+projective geometry by producing an actual relatively ample line.
+The ensuing descent statements apply on singular working models as
+well as on smooth initial spaces.
+
+\begin{lemma}[Degree-zero Bott--Chern descent]
+\label{prog:bc-descent}
+Let \(f:T\to Z\) be a proper surjective morphism with connected fibers
+between normal compact complex spaces with rational singularities.
+Suppose either
+\begin{enumerate}
+\item \(f\) is bimeromorphic and both spaces belong to Fujiki's class
+\(\mathcal C\); or
+\item \(f\) is projective and an effective rational boundary
+\(\Delta\) makes \((T,\Delta)\) klt with
+\(-(K_T+\Delta)\) relatively nef and big.
+\end{enumerate}
+Pullback is injective on both \(H^2(-,\R)\) and
+\(H^{1,1}_{\BC}(-,\R)\). In either group its image consists exactly
+of the classes with degree zero on every curve contracted by \(f\).
+\end{lemma}
+
+\begin{proof}
+These are, respectively, the two cases of
+\cite[Lemma~2.6(1),(2)]{DH2024}. The first case is birational descent
+between spaces with rational singularities and does not require a
+projectivity criterion. The second case uses relative vanishing for
+the ordinary rational klt pair and the projective relative curve
+space. In particular, it can be applied to a projective log-Fano
+contraction before constructing generalized data on its base.
+\end{proof}
+
+In applications to an ordinary birational negative contraction,
+the rationality hypotheses are automatic. The source is locally klt.
+For a dlt pair, first decrease its rational Cartier floor slightly;
+relative antiampleness persists. Relative vanishing gives
+\(R^if_*\OO_T=0\) for \(i>0\)
+\cite[Theorem~5.2]{FujinoAnalyticBCHM}. Composing a resolution of \(T\)
+with \(f\), the Leray spectral sequence proves that \(Z\) has rational
+singularities. The same reasoning works when the klt boundary is
+chosen separately on finitely many Stein base neighborhoods. For
+fiber-type contractions in case~(ii), rationality of the base follows
+from \cite[Lemma~8.8(i)]{DHP}; thus that application also precedes
+the construction of generalized data on the base.
+
+\begin{lemma}[Projectivity from a rational line detecting the ray]
+\label{prog:detected-projectivity}
+Let $f:T\to Z$ be a proper contraction between normal compact
+K\"ahler spaces, with $T$ having rational singularities. Suppose
+$\gamma$ is K\"ahler on $Z$ and
+\[
+ \alpha=f^*\gamma,\qquad
+ \overline{\mathrm{NA}}(T)\cap\alpha^\perp=R
+\]
+is a nonzero ray. If a global rational line bundle $D$ has
+$D\cdot R<0$, then $-D$ is relatively ample and $f$ is projective.
+\end{lemma}
+\begin{proof}
+Fix a K\"ahler class $\omega$ on $T$ and normalize $\overline{\mathrm{NA}}(T)$ by
+$\omega\cdot z=1$. The resulting slice $S$ is compact. Its unique
+point on $R$ has negative $D$-degree, so $d=c_1(D)$ is negative on
+a neighborhood of that point in $S$. On the complementary compact
+set, $\alpha$ has a strictly positive minimum, while $d$ is bounded.
+For all sufficiently small $s>0$, the class $\alpha-sd$ is therefore
+strictly positive on $S$ and is K\"ahler by cone duality.
+
+Choose an integer $m>0$ making $mD$ integral. Then
+\[
+ c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd)
+\]
+is K\"ahler. On a neighborhood in the base with a potential for
+$\gamma$, absorb that potential in a smooth metric on the same global
+line bundle $-mD$. Its curvature is positive on the fibres. The analytic
+positive-line-bundle criterion and the fibrewise relative-ampleness
+criterion make $-mD$ relatively ample. The line bundle was global
+throughout, so no gluing of unrelated local polarizations is required.
+\end{proof}
+
+
+An analytic contraction supplied by a supporting class therefore becomes
+projective as soon as one rational line has negative degree on its ray.
+For a small contraction, that same line will define the flip globally.
+The local ordinary adjoints used to prove finite generation need not
+glue; their section algebras will be Veronese subalgebras of one algebra.
+
+\begin{lemma}[Divisor representatives over a Stein base]
+\label{prog:stein-representatives}
+Let $\pi:Y\to U$ be a projective surjective morphism with connected fibres
+between normal irreducible complex spaces, where $U$ is Stein, and let
+$\mathcal L$ be a holomorphic line bundle on $Y$.
+If $\pi$ is bimeromorphic, $\mathcal L$ has a holomorphic section
+which is not identically zero, and hence an effective Cartier divisor
+representative. More generally, after shrinking $U$ around any specified
+point, $\mathcal L$ has a nonzero meromorphic section.
+\end{lemma}
+
+\begin{proof}
+Suppose first that $\pi$ is bimeromorphic. Grauert's proper direct-image
+theorem makes $\mathcal F=\pi_*\mathcal L$ coherent. On the nonempty
+open subset $U^\circ$ where $\pi$ is an isomorphism, it is a line bundle.
+Choose $u_*\in U^\circ$. Cartan's theorem~A says that global sections
+generate $\mathcal F_{u_*}$, so one global section has nonzero image in
+$\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}$.
+Under
+\[
+ H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L)
+\]
+it gives a section $s$ nonzero at the unique point above $u_*$.
+Its zero divisor is effective Cartier and represents $\mathcal L$.
+The section may vanish elsewhere; only its meromorphic inverse is
+being used to obtain a meromorphic trivialization.
+
+For the general case, fix $u\in U$ and $y\in\pi^{-1}(u)$, and choose
+a $\pi$-ample line bundle $\mathcal H$. Relative Serre generation,
+after shrinking to a Stein neighbourhood of $u$, gives an integer
+$n$ for which both $\mathcal L\otimes\mathcal H^n$ and
+$\mathcal H^n$ are relatively generated. Their proper direct images
+are coherent. Cartan's theorem~A and the evaluation maps supply
+sections $s$ and $t$ of these two bundles which are both nonzero at
+$y$. The quotient $s/t$ is the required nonzero meromorphic section
+of $\mathcal L$. At no point is $Y$ assumed Stein.
+\end{proof}
+
+\begin{proposition}[A global algebra for a detected flip]
+\label{prog:detected-flip}
+Let $(T,B+\mathbf M)$ be a gklt pair on a normal compact K\"ahler
+space which is globally strongly $\Q$-factorial, and let $A\in H^{1,1}_{\BC}(T,\R)$
+be its adjoint class. Let $f:T\to Z$ be a projective small contraction
+onto a normal compact K\"ahler space. Suppose that the classes of all
+$f$-vertical curves lie on one ray $R\subset\overline{\mathrm{NA}}(T)$, that $A\cdot R<0$,
+and that a global line bundle $L$ satisfies $L\cdot R<0$.
+
+Then $\bigoplus_{m\geq0}f_*L^m$ is locally finitely generated. The
+relative Proj of a sufficiently divisible Veronese is a projective
+small contraction $f^+:T^+\to Z$, where $T^+$ is compact K\"ahler and
+globally strongly $\Q$-factorial. The reflexive transform $L^+$ is a
+rational line bundle, with a positive power equal to the relatively
+ample tautological bundle. The transformed boundary and the same
+b-nef datum define a gklt pair on $T^+$, with relatively K\"ahler
+adjoint $A^+$. This is the generalized flip, with the usual
+strict discrepancy increase at centres in either exceptional locus.
+
+Moreover, if $t=(A\cdot C)/(L\cdot C)>0$ for an $f$-vertical curve $C$,
+then there is a unique $\eta\in H^{1,1}_{\BC}(Z,\R)$ such that
+\[
+ A=t\,c_1(L)+f^*\eta,\qquad
+ A^+=t\,c_1(L^+)+(f^+)^*\eta.
+\]
+Every class on $T$ has the corresponding forward transport with an
+actual exceptional-divisor correction. No assertion of surjectivity
+on Bott--Chern groups is required.
+\end{proposition}
+
+\begin{proof}
+Since $f$ has a global relatively ample bundle and all its vertical
+curves lie on $R$, numerical invariance of ampleness on the projective
+fibres shows that $-L$ is $f$-ample.
+
+\smallskip\noindent
+\emph{A local rational ordinary adjoint.}
+Fix $z\in Z$ and a relatively compact Stein neighbourhood $U$.
+Take a projective log resolution $p:V\to T_U$ carrying $\mathbf M$,
+and put $\rho=f p$. Write its actual structure boundary as $B_V$, so
+\[
+ [K_V+B_V]+[\mathbf M_V]=p^*A.
+\]
+Every coefficient of $B_V$ is less than one. Apply
+Lemma~\ref{prog:stein-representatives} to obtain a Cartier
+representative $D_L$ of $L$ on $T_U$. The same birational direct-image
+argument applies to the coherent rank-one reflexive canonical sheaf:
+it supplies a canonical form nonzero at a chosen point of the common
+smooth isomorphism locus. Use this meromorphic form on $T_U$ and its
+pullback to $V$ to choose compatible canonical representatives, and set
+\[
+ N=t p^*D_L-K_V-B_V.
+\]
+For every $\rho$-vertical curve, its degree equals that of $\mathbf M_V$.
+Thus $N$ is $\rho$-nef.
+
+The relative bigness of $N$ uses the additional fact that
+$\rho=f\circ p$ is \emph{bimeromorphic}: both the small contraction
+$f$ and the resolution $p$ are bimeromorphic. It does not follow from
+relative nefness alone. In fact every real Cartier divisor is relatively
+big for this projective bimeromorphic map over the Stein neighbourhood.
+Here is the required ample-plus-effective decomposition explicitly.
+Write $N=\sum_j r_jN_j$, where $r_j>0$, $\sum_jr_j=1$, and the $N_j$
+are rational Cartier divisors in its finite Cartier span. Choose a
+$\rho$-ample line bundle and a Cartier representative
+$P$ for it using Lemma~\ref{prog:stein-representatives}.
+If $q_jN_j$ is integral, apply the birational case of that lemma to
+$\mathcal O_V(q_jN_j-P)$. Its nonzero holomorphic section has an
+effective Cartier zero divisor $E_j$, giving the actual relation
+$q_jN_j\sim P+E_j$. This use depends on $\rho$ being bimeromorphic;
+the section is allowed to vanish along exceptional fibres. Therefore
+\[
+ N\sim_{\R}
+ \underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H}
+ +\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}.
+\]
+Here $H$ is $\rho$-ample and $E\geq0$, proving $\rho$-bigness.
+For a map with positive-dimensional general fibres the direct image
+used above can be zero; no such bigness claim is made for that setting.
+Make the finitely many section choices on a slightly larger Stein
+neighbourhood, and now shrink $U$ so its closure is compact in that
+neighbourhood. Properness makes the inverse image of this closure
+compact. A log resolution of the finitely many resulting divisors
+therefore gives a uniform positive bound for the following choice.
+Choose $\epsilon>0$ small enough that $(V,B_V+\epsilon E)$ is sub-klt.
+The divisor $(1-\epsilon)N+\epsilon H$ is relatively ample. Express it
+as a positive combination of rational ample divisors and take general
+members of sufficiently high multiples. Relative Bertini
+\cite[Theorem~2.20]{DHP}, applied on a log resolution of $B_V+E$,
+gives an effective $\Theta\sim_{\R}N$ such that $(V,B_V+\Theta)$ is
+sub-klt. The high multiples make the added coefficients arbitrarily
+small. All choices are made near the compact fibre; shrink $U$ once
+to retain them.
+
+Put $\Delta=p_*(B_V+\Theta)\geq0$. Write the actual linear equivalence
+as $\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)$.
+The bimeromorphic map $p$ identifies meromorphic function fields, so
+$g_j=p^*h_j$. Pushing down gives
+$K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)$;
+in particular this divisor is real Cartier. Pulling back this same
+identity gives
+\[
+ K_{T_U}+\Delta\sim_{\R}tD_L,
+ \qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta.
+\]
+Thus $(T_U,\Delta)$ is klt. Any finitely many base line bundles in a
+relative linear equivalence can first be trivialized by shrinking $U$.
+Record the resulting equality as
+\begin{equation}\label{prog:detector-rationalization}
+ K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j),
+ \qquad d_i>0,
+\end{equation}
+using the positive support of $\Delta$. After a further relatively
+compact shrinking, the displayed divisors have finitely many prime
+components: their locally finite supports meet a compact inverse image.
+Equality of coefficients in \eqref{prog:detector-rationalization} is a
+finite rational affine system in $(d_i,t,b_j)$. The klt condition is
+open within this system, as seen on one log resolution; the pullbacks
+of its adjoints vary linearly by the displayed identity. A nearby
+rational solution therefore gives a rational effective klt boundary
+$\Delta_q$ and $t_q\in\Q_{>0}$ with
+\begin{equation}\label{prog:detector-local-adjoint}
+ K_{T_U}+\Delta_q\sim_{\Q}t_qD_L.
+\end{equation}
+The individual local primes $D_i$ need not be $\Q$-Cartier: the identity
+ensures that the total adjoint is $\Q$-Cartier.
+
+\smallskip\noindent
+\emph{One global flip algebra.}
+The adjoint in \eqref{prog:detector-local-adjoint} is $f_U$-antiample.
+Its ordinary flip and local finite generation follow from
+\cite[Theorems~1.14 and~1.18]{FujinoAnalyticBCHM}. Clearing the displayed
+linear equivalence identifies a Veronese of its canonical algebra
+with a Veronese of $\mathcal R:=\bigoplus_{m\geq0}f_*L^m$ over $U$.
+Hence $\mathcal R$ is locally finitely generated by
+\cite[Lemma~2.26]{FujinoAnalyticBCHM}. The graded pieces are coherent. Over a connected normal base open
+the inverse image is irreducible; products of nonzero sections of line
+bundles are nonzero there. Thus its graded section algebra is an
+integral domain, without choosing any global meromorphic frame of $L$.
+Compactness supplies one sufficiently divisible Veronese
+$\mathcal R^{(r)}$ generated in degree one. Its relative Proj $T^+$
+restricts to the ordinary flip over every such $U$. It is therefore
+normal, locally klt, and small over $Z$, and has the global relatively
+ample bundle
+$\mathcal O_{T^+}(1)$. On the common big open this bundle is $L^r$.
+Reflexivity gives
+\[
+ (L^+)^{[r]}\simeq\mathcal O_{T^+}(1).
+\]
+This proves that $L^+$ is a rational line bundle before any assertion
+of factoriality. Relative positivity over the compact K\"ahler base
+makes $T^+$ compact K\"ahler.
+
+For any global rank-one reflexive sheaf $\mathcal F^+$ on $T^+$,
+take its coherent reflexive transform $\mathcal F$ on $T$ using a
+common resolution. Some $M=\mathcal F^{[m]}$ is a line bundle.
+Choose $c\in\Q$ so that $(M+cL)\cdot C=0$, and clear its denominator
+to obtain an integral bundle $J=n(M+cL)$ numerically trivial over $f$.
+The local rational klt pairs \eqref{prog:detector-local-adjoint} have
+antiample adjoint, so Lemma~\ref{prog:integral-descent} yields
+one global line bundle $J_Z$ with $J=f^*J_Z$.
+On $T^+$, pull back $J_Z$, undo the rational twist $ncL^+$, and compare
+on the common big open. After clearing the already established index
+of $L^+$, reflexivity gives an invertible positive reflexive power of
+$\mathcal F^+$. This proves global strong factoriality.
+
+\smallskip\noindent
+\emph{The original generalized pair.}
+The local log-Fano pairs above give $R^if_*\mathcal O_T=0$ for $i>0$
+by \cite[Theorem~5.2]{FujinoAnalyticBCHM}. They make $T$ locally klt, hence rational.
+Leray for a projective resolution then shows that $Z$ is rational.
+Lemma~\ref{prog:bc-descent}(i) therefore gives
+$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$, with injective pullback. Thus $A=t c_1(L)+f^*\eta$ for a unique $\eta$.
+Define $A^+=t c_1(L^+)+(f^+)^*\eta$; it is relatively K\"ahler.
+
+To construct its actual discrepancy divisor, let $\mathcal I$ be the
+ideal defined by
+\[
+ \operatorname{im}(f^*f_*L^r\longrightarrow L^r)
+   =\mathcal I\otimes L^r.
+\]
+Principalize $\mathcal I$ on a common resolution $p:W\to T$,
+$q:W\to T^+$ carrying $\mathbf M$. If
+$\mathcal I\mathcal O_W=\mathcal O_W(-F_L)$, the tautological quotient
+gives the actual bundle identity
+\[
+ q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L).
+\]
+Hence $E_L=F_L/r$ is effective and exceptional over both sides, and
+$[E_L]=p^*c_1(L)-q^*c_1(L^+)$. If $B_W$ is the original structure
+boundary, put $B_W^+=B_W-tE_L$. Then
+\[
+ [K_W+B_W^+]+[\mathbf M_W]=q^*A^+.
+\]
+Its pushforward is $B^+$, and its discrepancies do not decrease.
+The local ordinary comparison in \eqref{prog:detector-local-adjoint}
+has discrepancy divisor $t_qE_L$, so its strictness proves the stated
+strictness for $tE_L$ as well. Thus the original pair remains gklt.
+
+Finally, write any $\theta\in H^{1,1}_{\BC}(T,\R)$ uniquely as
+$\theta=f^*\eta_\theta+s c_1(L)$ and set
+$\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)$. Its correction is the actual
+divisor $sE_L$. For a rational line $M$, the number $s$ is rational. Applying
+Lemma~\ref{prog:integral-descent} to a Cartier multiple of $M-sL$
+gives $M=sL+f^*M_Z$ as actual rational lines. Reflexive extension
+on the common big open gives $M^+=sL^++(f^+)^*M_Z$.
+Consequently the cohomological transport agrees with the reflexive
+transform on Chern classes.
+\end{proof}
+
+
+\subsubsection{Real boundaries and their linear transport}
+
+The rational ordinary step constructed below also determines the step
+for every sufficiently close real boundary. The relevant comparison is
+an identity of actual rational lines over the contraction base.
+
+\begin{proposition}[Ordinary real boundaries]
+\label{prog:ordinary-replacement}
+Let $T$ be normal compact K\"ahler and globally Weil $\Q$-factorial,
+with canonical sheaf a rational line bundle. Let $B\geq0$ be a real
+boundary with $(T,B)$ klt. Assume the rational ordinary-step result
+of Proposition~\ref{prog:ordinary-step} for $T$ and its subsequent models. Then every
+$(K_T+B)$-negative extremal analytic ray has an ordinary step with
+projective contractions, compact K\"ahler models, and the expected
+opposite relative ample signs. Global Weil $\Q$-factoriality is preserved;
+global strong $\Q$-factoriality is preserved when imposed initially.
+\end{proposition}
+\begin{proof}
+Put $D=K_T+B$ and fix a $D$-negative extremal ray $R$. On the finite
+positive support of $B$, effectiveness, the klt condition and negativity
+on $R$ persist under small coefficient changes. Klt openness is checked
+on one log resolution of this support. Choose nearby effective rational
+klt boundaries $B_0,\ldots,B_m$ and positive real numbers $u_j$ with
+\[
+ B=\sum_j u_jB_j,\qquad \sum_j u_j=1,
+ \qquad D_j\cdot R<0,\quad D_j=K_T+B_j.
+\]
+For $B=0$ take just $B_0=0$. Every $D_j$ is a global rational line
+bundle. The rational-step theorem for $(T,B_0)$ provides a projective
+contraction $f:T\to Z$ onto a compact K\"ahler base, contracting
+exactly $R$, with $-D_0$ relatively ample. The base is rational by
+\cite[Lemma~8.8]{DHP}, so Lemma~\ref{prog:bc-descent} gives
+$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$; use case~(ii) when $f$ is of
+fiber type.
+Let $C$ be a rational curve spanning $R$, and set
+$a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}$. On every projective
+fibre, $-D_j$ is numerically equivalent to the positive multiple
+$-a_jD_0$. Numerical invariance of ampleness on that fibre, followed
+by the relative-ampleness criterion, makes $-D_j$ relatively ample.
+The positive combination $-D$ is relatively ample as a real line
+bundle. This also handles a fibre-type contraction.
+
+Suppose henceforth that $f$ is birational. A Cartier multiple
+$M_j=n_j(D_j-a_jD_0)$ is numerically trivial over $Z$ and
+$M_j-D_0$ is relatively ample. Lemma~\ref{prog:integral-descent},
+applied to the rational klt pair $(T,B_0)$ gives a rational line bundle
+$A_j$ on $Z$ with the actual identity
+\begin{equation}\label{prog:real-line-descent}
+ D_j=a_jD_0+f^*A_j.
+\end{equation}
+Here and below an identity of rational line bundles means an isomorphism
+after a common integral multiple. The descent lemma is used only for
+birational $f$.
+
+For a small $f$, let $f^+:T^+\to Z$ be its rational $D_0$-flip.
+The rational theorem supplies a global relatively ample adjoint
+$D_0^+$ and preserves the stated factoriality and canonical-sheaf
+conditions. The other transformed adjoints $D_j^+$ are therefore
+rational line bundles. Transform \eqref{prog:real-line-descent} on the common
+big open and extend by reflexivity to obtain
+\[
+ D_j^+=a_jD_0^++(f^+)^*A_j.
+\]
+All $D_j^+$ are relatively ample, as is
+$D^+=\sum_j u_jD_j^+$. Thus this same small map is the required
+ordinary real-boundary flip. The real ordinary discrepancy comparison
+proves that $(T^+,B^+)$ is klt and gives the strict increases at
+exceptional centres.
+
+For a divisorial contraction, write
+$D_0=f^*D_0'+e_0E$ with $e_0>0$. Transforming
+\eqref{prog:real-line-descent} on the target and summing yields
+\[
+ D-f^*D'=\left(\sum_j u_ja_j\right)e_0E.
+\]
+This coefficient is positive. The same discrepancy comparison proves
+the required assertions. The ambient category was already preserved
+by the rational step in both cases.
+\end{proof}
+
+
+For later use, let \(D=K_T+B\) drive one of the birational steps just
+constructed, and write \(f':T'\to Z\) for its positive morphism
+(or the identity for a divisorial contraction). Lemma~\ref{prog:bc-descent}
+gives the unique decomposition and forward transform
+\begin{equation}
+ \theta=f^*\eta+s\{D\},\qquad
+ \theta'=(f')^*\eta+s\{D'\}.
+ \label{prog:ordinary-forward-transport}
+\end{equation}
+On a common resolution \(p:W\to T\), \(q:W\to T'\), the ordinary
+comparison \(p^*D-q^*D'=F\) is an actual effective real divisor,
+exceptional over \(T'\) and over both sides for a flip. Hence
+\(p^*\theta-q^*\theta'=s\{F\}\). On Chern classes this is the
+reflexive transform, by the line identities in the proof above and
+Lemma~\ref{prog:integral-descent}. This construction gives forward
+transport without assuming surjectivity onto the next Bott--Chern group.
+
+\subsubsection{Actual real lines and relative numerical spaces}
+
+A second polarization argument will be used for maps whose fibers are
+Moishezon. It applies to a real combination of global line bundles,
+provided that its degrees on the fibers agree with a relatively Kähler
+class.
+
+\begin{lemma}[Polarization by an actual real line bundle]
+\label{prog:actual-line-polarization}
+Let $f:T\to Z$ be a proper Moishezon map, where $T$ is a compact
+K\"ahler space. Suppose that $L\in\operatorname{Pic}(T)\otimes\R$
+and a relatively K\"ahler class $\omega$ satisfy
+\[
+ L\cdot C=\omega\cdot C
+ \quad\text{for every compact curve $C$ contracted by $f$.}
+\]
+Then $L$ is relatively ample and $f$ is projective.
+\end{lemma}
+
+\begin{proof}
+Let $V$ be an irreducible positive-dimensional subspace of a reduced
+fibre. It is Moishezon. Normalize $V$ and take a smooth projective
+modification $\pi:\widetilde V\to V^{\mathrm n}$, chosen so that
+an effective exceptional divisor $E$ has $-E$ relatively ample.
+The pullback of the K\"ahler class from $V$ to its normalization is
+K\"ahler. Consequently
+$\pi^*\omega-\delta[E]$ is K\"ahler for sufficiently small
+$\delta>0$. On the projective manifold $\widetilde V$ the actual
+real line bundle $\pi^*L-\delta E$ has the same curve degrees as
+that K\"ahler class, and is therefore ample. For example, subtract
+a sufficiently small positive multiple of a fixed ample class from
+the K\"ahler class; the corresponding real line bundle is nef by
+the projective numerical criterion. Also $\pi^*L$ is nef, and the
+decomposition
+\[
+ \pi^*L=(\pi^*L-\delta E)+\delta E
+\]
+shows that it is big. Thus
+\[
+ L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0.
+\]
+The real Nakai--Moishezon criterion for proper algebraic spaces
+\cite[Theorem~1.6]{FujinoMiyamoto2020}, applied to the algebraizations
+of the Moishezon fibres, proves that $L$ is ample on every reduced
+fibre. The criterion is insensitive to nilpotents and reducible
+components. In the coefficient space of the finitely many global
+line bundles occurring in $L$, choose, for each fibre, a rational
+simplex containing $L$ in its interior whose vertices restrict to
+ample rational lines on that fibre. Openness of fibrewise ampleness
+for each of these finitely many rational lines gives a base
+neighbourhood on which the entire simplex is relatively ample.
+Choose finitely many such neighborhoods covering the base and
+intersect their coefficient neighborhoods of $L$. Every rational
+point of the resulting neighborhood is relatively ample globally.
+A rational simplex in this intersection also expresses $L$ as a
+positive combination of relatively ample rational lines. Clearing
+the denominator of one rational point proves projectivity.
+\end{proof}
+
+\begin{remark}
+The proof uses only positivity of the top self-intersections of
+$L$. Equality of curve degrees with $\omega$ does not assert
+equality of their higher intersection numbers or of their
+Bott--Chern classes.
+\end{remark}
+
+
+
+
+The projective morphism needed for the relative program can be chosen
+on a smooth model of the rational quotient. Here the Hodge-theoretic
+polarization has an elementary construction, independent of any
+birational projectivity-descent assertion.
+
+\begin{lemma}[Smooth Hodge-theoretic polarization]
+\label{prog:smooth-polarization}
+Let \(g:W\to S\) be a surjective morphism with connected fibers
+between smooth compact Kähler manifolds. If
+\(g^*:H^0(S,\Omega_S^2)\to H^0(W,\Omega_W^2)\) is an isomorphism,
+then \(g\) is projective. This applies when its general fiber is
+rationally connected.
+\end{lemma}
+
+\begin{proof}
+We use the smooth argument of \cite[Theorem~3.1, Step~1]{CH}.
+Let \(d=\dim W-\dim S\). Choose a rational class \(u\in H^2(W,\Q)\)
+close to a Kähler class. Write its Hodge decomposition as
+\(u=b+g^*v\), where \(b\) is Kähler and \(v\) has types \((2,0)\)
+and \((0,2)\). Hodge type and the projection formula give
+\[
+ c=g_*(u^d)=g_*(b^d)>0,\qquad
+ g_*(u^{d+1})=g_*(b^{d+1})+(d+1)c\,v.
+\]
+Since pushforward is rational, the class
+\[
+ \ell=u-g^*\frac{g_*(u^{d+1})}{(d+1)c}
+      =b-g^*\frac{g_*(b^{d+1})}{(d+1)c}
+\]
+is rational and of type \((1,1)\). Lefschetz's theorem makes it the
+Chern class of a rational line. The displayed equality supplies a
+metric with positive curvature locally over \(S\), by adding a
+potential for the base class. Thus this line is relatively ample.
+For rationally connected general fibers, holomorphic two-forms are
+pulled back from the smooth base, so the hypothesis holds.
+\end{proof}
+
+\begin{lemma}[A rational Hodge correction over a smooth base]
+\label{prog:relative-hodge}
+Let $f:T\to S$ be an already projective surjective morphism of normal
+compact K\"ahler spaces. Assume that $S$ is smooth, $T$ has globally
+strongly $\Q$-factorial klt singularities, and, for a projective resolution
+$r:W\to T$, the morphism $g=f\circ r$ satisfies
+\[
+ g^*:H^0(S,\Omega_S^2)\xrightarrow{\ \simeq\ }
+                         H^0(W,\Omega_W^2).
+\]
+Then every $\alpha\in H^{1,1}_{\BC}(T,\R)$ can be written
+\[
+                         \alpha=c_1(L)+f^*\gamma,
+\]
+where $L\in\operatorname{Pic}(T)\otimes_{\mathbb Z}\mathbb R$ is a
+finite real combination of global line bundles and
+$\gamma\in H^{1,1}(S,\mathbb R)$.
+\end{lemma}
+
+\begin{proof}
+The composition $g$ is projective, since all the spaces are compact.
+Choose a $g$-ample line bundle on $W$, with integral Chern class $H$.
+Set $d=\dim W-\dim S$ and
+\[
+                            c=g_*(H^d)>0.
+\]
+Thus $c$ is the positive degree of $H^d$ on a general fibre. The
+cohomological pushforward here is the usual pushforward between the
+smooth compact manifolds $W$ and $S$, defined over $\mathbb Q$ and of
+Hodge bidegree $(-d,-d)$.
+
+Define a rational linear operator on degree-two cohomology by
+\[
+ \Pi(\eta)=\eta-g^*\!\left(\frac{g_*(\eta\smile H^d)}{c}\right).
+\]
+Every class of type $(2,0)$ on $W$ is pulled back from $S$, and the
+same holds for type $(0,2)$. The projection formula therefore shows
+that $\Pi$ kills both types. It preserves type $(1,1)$. Consequently
+\[
+        \Pi(H^2(W,\mathbb Q))\subset
+                         H^2(W,\mathbb Q)\cap H^{1,1}(W).
+\]
+By the Lefschetz $(1,1)$ theorem the image consists of rational Chern
+classes of global line bundles. Applying $\Pi$ to $r^*\alpha$ gives
+\[
+ r^*\alpha=c_1(L_W)+g^*\gamma,
+ \qquad L_W\in\operatorname{Pic}(W)\otimes\mathbb R,
+ \qquad
+ \gamma=\frac{g_*(r^*\alpha\smile H^d)}{c}\in H^{1,1}(S,\mathbb R).
+\]
+For example, express $r^*\alpha$ in a rational basis of $H^2(W,\mathbb Q)$
+and apply $\Pi$ to each basis vector. This gives the required finite
+real combination $L_W$.
+
+Write $L_W=\sum_j a_jL_j$ with actual line bundles $L_j$ on $W$.
+For each $j$, let
+\[
+                   \mathcal F_j=(r_*L_j)^{**}.
+\]
+It is a rank-one reflexive coherent sheaf. Strong
+$\mathbb Q$-factoriality gives an integer $m_j>0$ for which
+$M_j=\mathcal F_j^{[m_j]}$ is a line bundle on $T$.
+The canonical identification over the isomorphism locus of $r$ gives
+\[
+             L_j^{\otimes m_j}\simeq r^*M_j\otimes\mathcal O_W(E_j)
+\]
+for an integral $r$-exceptional divisor $E_j$.
+One may obtain this comparison directly from the evaluation map for
+$r_*L_j$: the two coherent rank-one sheaves agree away from the
+exceptional locus, so their invertible transforms differ by an
+exceptional divisor. No global meromorphic frame of $L_j$ or of the
+canonical sheaf is required.
+
+Set $L=\sum_j(a_j/m_j)M_j$ and
+$E=\sum_j(a_j/m_j)E_j$. The preceding identities yield
+\[
+                r^*(\alpha-c_1(L)-f^*\gamma)=[E].
+\]
+The divisor $E$ is $r$-exceptional and is numerically trivial on every
+$r$-contracted curve. The exceptional negativity lemma applied in both
+signs gives $E=0$. Injectivity of pullback by a resolution, or pushforward
+of the equality of currents, now gives
+$\alpha=c_1(L)+f^*\gamma$.
+\end{proof}
+
+
+The hypothesis of Lemma~\ref{prog:relative-hodge} persists on each
+projective globally strongly \(\Q\)-factorial klt model over this fixed
+smooth base: on a common smooth resolution, holomorphic two-forms are
+invariant under modifications. Thus the lemma applies separately on
+every working model. It supplies real combinations of global lines;
+Lemma~\ref{prog:stein-representatives} supplies their Cartier-divisor
+representatives on sufficiently small Stein base neighborhoods.
+
+\begin{lemma}[The projective relative cone in Bott--Chern cohomology]
+\label{prog:projective-cone}
+Let \(f:T\to S\) be projective between normal compact Kähler spaces
+with rational singularities, and assume
+\[
+ H^{1,1}_{\BC}(T,\R)
+   =c_1\bigl(\Pic(T)\otimes\R\bigr)
+          +f^*H^{1,1}_{\BC}(S,\R).
+\]
+Let \(N_1^{\mathrm{line}}(T/S)\) denote the space of real combinations
+of \(f\)-contracted curves modulo degrees of global line bundles,
+and let \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) be their
+closed effective cone. Their analytic classes give an isomorphism
+onto the span of the face
+\[
+ \mathcal F_f=
+ \overline{\mathrm{NA}}(T)\cap(f^*\omega_S)^\perp,
+\]
+where \(\omega_S\) is any Kähler class on \(S\); this isomorphism
+identifies \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) with
+\(\mathcal F_f\). In particular, relative extremal rays in the line
+space are extremal rays of the full analytic cone.
+\end{lemma}
+
+\begin{proof}
+The assumed decomposition makes analytic equivalence and line-degree
+equivalence identical on combinations of vertical curves. Their map
+into the analytic numerical space is therefore well-defined and
+injective. We must show that its closed effective cone is the entire
+face, including its classes represented by positive currents.
+
+Fix \(z\in\mathcal F_f\). For every \(\gamma\in
+H^{1,1}_{\BC}(S,\R)\), both \(a\omega_S+\gamma\) and
+\(a\omega_S-\gamma\) are Kähler for large \(a\). Positivity then
+shows that \(z\) annihilates \(f^*\gamma\). Fix a global relatively
+ample line \(H\). If a real line \(D\) has nonnegative degree on
+every vertical curve, the projective relative numerical criterion
+makes \(D+\epsilon H\) relatively ample for every \(\epsilon>0\).
+Its Chern class becomes Kähler after adding a sufficiently large
+multiple of \(f^*\omega_S\). Therefore
+\[
+                 (c_1(D)+\epsilon c_1(H))\cdot z\geq0,
+ \qquad c_1(D)\cdot z\geq0.
+\]
+Applying this to both signs of a relatively numerically trivial line
+shows that pairing with \(z\) factors through the finite-dimensional
+space of relative line degrees. It is nonnegative on its nef cone.
+By duality for this projective relative space, it is represented by
+an element of \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\).
+The decomposition in the statement makes its analytic image equal
+to \(z\). Conversely every vertical effective curve lies in
+\(\mathcal F_f\), and finite-dimensional injectivity preserves
+closedness. This proves the cone equality and the extremality claim.
+\end{proof}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03b-projective-good-model.tex

@@ -1,439 +1,439 @@
-\subsection{Good models for already projective relative adjoints}
-
-\begin{proposition}[Projective relative completion]
-\label{prog:projective-good-model}
-Let \(f:X\to S\) be an already projective surjective morphism of
-normal compact Kähler spaces, with \(X\) smooth. Let
-\((X,B+\mathbf M)\) be an effective generalized klt pair with adjoint
-\(A\), carried by a projective resolution \(p:W\to X\) with smooth
-\(W\), so that
-\[
- p^*A=c_1(K_W)+[B_W]+\mathbf M_W.
-\]
-Suppose that there are an actual real line bundle
-\(J\in\Pic(X)\otimes\R\) and
-\(\beta\in H^{1,1}_{\BC}(S)\) such that
-\[
- A=c_1(J)+f^*\beta,
- \qquad
- N:=p^*J-K_W-B_W
- \text{ is nef and big over }fp.
-\]
-Assume that \(J\) is pseudoeffective over \(S\).
-Then there is a finite chosen \(A\)-negative projective program over
-\(S\), with globally strongly \(\Q\)-factorial compact Kähler
-working spaces, and a nonextracting endpoint \(X_m\) such that
-\[
- J_m\sim_{\R}g^*H,
- \qquad
- A_m=g^*\bigl(c_1(H)+\rho^*\beta\bigr).
-\]
-Here \(J_m\) is the actual real-line transform of \(J\),
-\(g:X_m\to Z\) is a projective connected-fibre morphism,
-\(\rho:Z\to S\) is projective, and \(H\) is an actual real line
-bundle ample over \(S\). Thus the class on \(Z\) in this formula is
-relatively Kähler over \(S\). The generalized pair on \(X_m\) is
-gklt. On a common resolution the comparison with \(A\) is effective
-and exceptional over \(X_m\), with strictly positive coefficient
-at the strict transform of every prime contracted from \(X\).
-\end{proposition}
-
-\begin{proof}
-We reduce to fixed ordinary pairs, and construct one global program.
-Choose finitely many Stein open sets \(U_\ell\subset S\) and
-semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose
-interiors cover \(S\). Such compacts are obtained by intersections
-with closed polydiscs in local embeddings of \(S\). In particular
-they satisfy condition~(P) of \cite{FujinoAnalyticBCHM}: the source
-is normal, the base is Stein, the compact is Stein, and its
-intersection with any analytic subset defined near it has finitely
-many connected components. This remains true on every normal
-projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves
-the finite cover.
-
-\smallskip\noindent
-\emph{Actual ordinary replacements.}
-On each chart represent the finitely many line bundles in the data
-by Cartier divisors. For a projective map to a Stein space this can
-be done by twisting a line bundle and its prospective meromorphic
-frame by sufficiently positive relative lines and using nonzero
-sections. The generic-rank-one direct-image argument is an
-alternative only when the map is bimeromorphic.
-Relative bigness gives
-\(N\sim_{\R}P+E\), with \(P\) relatively ample and \(E\geq0\).
-For sufficiently small \(\epsilon>0\),
-\((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image.
-The line \((1-\epsilon)N+\epsilon P\) is relatively ample.
-General divided relative sections give an effective representative
-\(T\) such that
-\[
- \Theta:=\epsilon E+T\sim_{\R}N,
- \qquad (W,B_W+\Theta)\text{ is sub-klt}.
-\]
-All supports and coefficient margins are fixed near that compact.
-With compatible canonical representatives, push the finite
-principal-divisor expression for \(\Theta-N\) down and pull it
-back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives
-\[
- \Delta_\ell\geq0,
- \qquad K_X+\Delta_\ell\sim_{\R}J|_{X_{U_\ell}},
- \qquad
- p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta.
-\]
-In the last equality the chosen principal correction is included
-in the representative of \(K_X+\Delta_\ell\).
-Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary
-is relatively big. Lemma~11.15 of \cite{FujinoAnalyticBCHM}, with
-its big part equal to \(\Delta_\ell\) and its remaining part zero,
-replaces it, in its actual real linear equivalence class, by a klt
-boundary with an effective general relatively ample rational
-summand. Its condition on non-klt centres is vacuous.
-Choose the reserved ample summand to be \(\Q\)-linearly equivalent
-to a small rational multiple of a fixed global \(f\)-ample line.
-Its later strict transform is consequently \(\Q\)-Cartier by the
-transport of that global line, without a local factoriality claim.
-
-We make these choices for a finite rational polytope of actual
-global line data. Start with a finite rational span containing
-\(J\), and add relatively ample rational lines spanning the global
-relative degree space. The above construction is valid in a
-neighbourhood of \(J\): retain a small part of the ample summand
-and use openness of ampleness and the strict klt inequalities.
-It is also valid along a segment from \(J\) to a sufficiently
-positive adjoint, by adding divided general ample divisors.
-Use finitely many such representatives and one log resolution on
-each chart. More explicitly, represent a finite simplex inside the
-open ample neighbourhood used above; varying its positive
-coefficients represents all nearby global line parameters on the
-same finite support. Form the joint finite affine system consisting
-of the divisor identities on all charts, with common global line
-coordinates and separate boundary and principal-divisor
-coordinates. This system is rational. Allow all boundary
-coefficients to vary, including those originally contributed by
-the real boundary \(B\); fix only the reserved rational ample part
-and the zero-support constraints. Its projection to the global
-line coordinates contains the just constructed neighbourhood, so
-it has a rational affine section. Choose that section sufficiently
-close to the constructed real affine lift, by rational approximation
-in its affine space of sections, so that the strict klt and
-effectiveness margins are retained along the compact segment.
-In the joint solution space the
-strict klt and effectiveness margins permit a rational polytope
-containing the scaling segment. This supplies on every chart
-a rational polytope of ordinary klt boundaries with a fixed
-effective relatively ample rational summand, representing the
-restrictions of the same global line parameters. These boundaries,
-including the scaling representatives, are fixed before the
-program starts. Their divisors need not agree on different charts.
-
-\smallskip\noindent
-\emph{One global scaling construction.}
-Apply the global degree-space construction in the proof of
-Proposition~\ref{prog:relative-klt} to \(J\) and a general ample
-scaling direction in the chosen span. Here are the modifications
-that allow real line data and a nonbirational map to \(S\).
-The global relatively ample line still gives a compact normalized
-slice of the curve cone. The fixed ordinary chart pairs give finite
-negative-ray truncations. A general direction avoids equal wall
-parameters for independent ray-degree functionals, exactly as in
-that proof. A positive wall therefore selects a single ray in the
-global degree space. A rational nef support is represented by a
-global rational line; a large multiple minus the chartwise ordinary
-adjoint is relatively ample. Ordinary relative base point freeness
-contracts the ray using the evaluation of this one global line.
-
-If the contraction is small, choose a global rational line having
-negative ray degree and construct its flip by its one global
-graded algebra. The local ordinary proof of
-Proposition~\ref{prog:detected-flip} applies to this algebra:
-all curves of the contraction have proportional degrees for global
-lines, so the detector and the driving line are positively
-proportional over the contraction; ordinary rational replacement
-gives local finite generation. This part of that proof needs only
-the already projective contraction and these line degrees.
-It does not require a prior assertion about the full Bott--Chern
-cone. The relative Proj is the next global model. In a divisorial
-step the exceptional prime and
-Lemma~\ref{prog:integral-descent} give the same continuation as in
-Proposition~\ref{prog:relative-klt}. Killing the degree of any global
-line by a rational multiple of the detector, then applying that
-lemma, preserves global strong \(\Q\)-factoriality in both cases.
-The global lines continue to span the relative degree spaces.
-The rational support-simplex argument in that proposition makes
-the wall a descended relatively ample real line on its contraction
-base and gives a nonempty interval of ampleness immediately after
-each step. This argument uses relative line data and projectivity,
-not birationality of the map to \(S\).
-
-We check explicitly the persistence of the ordinary pairs. Expand
-the line parameters in finitely many global rational lines. Their
-reflexive transforms are actual rational lines by the preceding
-strong property. Pushing forward the fixed principal-divisor
-identities gives
-\[
- K_{X_i}+\Delta_{\ell,i}\sim_{\R}J_i|_{(X_i)_{U_\ell}}.
-\]
-Hence these ordinary adjoints are \(\R\)-Cartier; no local
-factoriality of individual prime divisors is being assumed.
-On a common resolution their comparison minus the appropriate
-positive multiple of the detector comparison is exceptional and
-relatively numerically trivial. Negativity in both signs makes
-this difference zero. Every chosen step is consequently negative
-for the fixed ordinary pairs. They remain effective and klt, and
-their boundaries remain relatively big. The same equality of
-comparison divisors, after adding the fixed base pullback,
-preserves the generalized pair and its discrepancy inequalities.
-
-For termination, every already constructed working model is a weak
-log canonical model on each chart of a boundary in the fixed
-polytope: choose a parameter in its nonempty ample scaling interval.
-Indeed, actual descent at a preceding wall \(\lambda_j\) makes
-the comparison for \(J+tH\) equal to
-\((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter
-in the current interval all preceding \(\lambda_j\) are at least
-\(t\), so these comparisons are effective.
-Theorem~E of \cite{FujinoAnalyticBCHM} gives finitely many marked
-models on each chart. That theorem does not require locally
-\(\Q\)-factorial targets. There are therefore finitely many possible
-global marked working models. Indeed, isomorphisms between the
-restrictions of two existing marked models agree on their common
-dense open, and hence on overlaps. A repeated marked working model
-would have the same boundary transform and discrepancies, contrary
-to strict discrepancy increase at an intervening nontrivial step.
-The global program is finite. Relative pseudoeffectivity excludes
-a Mori fibre endpoint, so \(J_m\) is nef over \(S\).
-This is a finite-cover argument for a single constructed program;
-it does not glue independent minimal models.
-
-\smallskip\noindent
-\emph{Semiampleness and the global polarization.}
-On the endpoint the fixed ordinary pair is klt, its effective
-boundary is relatively big, and its adjoint is nef over
-\(W_\ell\). Lemma~11.16 of \cite{FujinoAnalyticBCHM} applies to
-this already existing weak model. More explicitly, Lemma~11.15
-replaces the endpoint boundary by \(A_\ell+B_\ell\), where
-\(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the
-pair is klt. Theorem~8.3 then gives semiampleness near \(W_\ell\):
-its log canonical, klt-existence, real Cartier, ample-summand and
-nef-over-compact hypotheses have all been checked.
-
-The actual equivalences identify these local semiample adjoints
-with restrictions of \(J_m\). Their ample models agree on overlaps
-by \cite[Lemma~11.3]{FujinoAnalyticBCHM}. To retain a global
-polarization, shrink the initial parameter polytope around \(J\)
-after the finite program: all its finitely many strict step signs
-persist, so the transformed ordinary boundaries remain klt and big.
-On each chart the pushforward of its initial common ample rational
-summand is effective and relatively big. Apply
-\cite[Lemma~11.13]{FujinoAnalyticBCHM} to this common big rational
-part and the small endpoint boundary polytope. All pairs are klt,
-so the condition on non-klt centres is vacuous. The lemma gives a
-common general relatively ample rational summand by one
-\(\Q\)-linearly trivial translation. In particular the rational
-parametrization is preserved; separate applications of
-Lemma~11.15 to individual parameters would not suffice here.
-In this finite rational span Theorem~11.17 of \cite{FujinoAnalyticBCHM}
-makes the endpoint nef region a rational polytope. Intersect the
-finitely many inverse images of these regions and express \(J_m\)
-as a positive combination of rational points in its minimal face.
-The corresponding global rational lines are semiample on every
-chart by the same ordinary base point free argument. After clearing
-denominators and taking common sufficiently divisible powers over
-the finite cover, each is relatively generated globally: generation on
-the charts is generation of its global coherent relative
-evaluation map. The product of these finitely many relative
-morphisms, followed by Stein factorization, gives a projective
-\(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended
-global lines whose positive combination \(H\) is relatively ample.
-It realizes the glued ample model and gives
-\(J_m\sim_{\R}g^*H\). Adding back \(f_m^*\beta\) proves the
-asserted Bott--Chern identity. Composing the effective step
-comparisons proves the final discrepancy statement.
-\end{proof}
-
-\begin{corollary}[Semiampleness on an existing projective model]
-\label{prog:projective-semiample}
-Let \(f:T\to S\) be a projective morphism of normal compact Kähler
-spaces with \(T\) globally strongly \(\Q\)-factorial, and let
-\(J\in\Pic(T)\otimes\R\) be nef over \(S\). Suppose that on a
-finite Stein-compact cover satisfying condition~(P), there are
-effective ordinary klt boundaries \(\Delta_\ell\), big over the
-base, such that
-\(K_T+\Delta_\ell\sim_{\R}J|_{T_{U_\ell}}\).
-Then \(J\) has the projective relative semiample contraction and
-actual descended ample real line of
-Proposition~\ref{prog:projective-good-model}.
-In particular this applies to any already existing weak model of
-the fixed ordinary chart pairs in that proof; one need not run a
-new program for the adjoint in question.
-\end{corollary}
-
-\begin{proof}
-Lemma~11.15 and Theorem~8.3 of \cite{FujinoAnalyticBCHM} give
-local semiampleness. For an existing weak model this is also
-Lemma~11.16. To globalize the polarization, take a finite rational
-span of the global lines defining \(J\). On each chart the ample
-part of the normalized boundary allows small perturbations in
-this span. The joint rational affine construction in the preceding
-proof gives a rational family with a common rational ample part;
-all boundary coefficients are allowed to vary. Theorem~11.17 then
-gives a rational nef region on each chart. The finite-intersection,
-rational-vertex and global-evaluation argument in that proof gives
-one projective contraction and its actual descended ample real
-line. Only its local ample models are identified by uniqueness.
-\end{proof}
-
-\begin{remark}[The two uses of projective completion]
-\label{prog:projective-good-model-uses}
-For a selective extraction over an already constructed weak model,
-the carrier map is projective bimeromorphic. The inherited
-exceptional splitting expresses the nef datum, modulo this base,
-by actual real lines. Relative nefness follows from the carrier
-datum, while relative bigness follows from birationality, using the
-ample-plus-effective construction in the ordinary replacement.
-The prepared adjoint is a base pullback plus an effective
-exceptional divisor supported on exactly the unwanted primes.
-Proposition~\ref{prog:projective-good-model} applies. On its endpoint
-the remaining error is effective, exceptional over the fixed base,
-and relatively nef; negativity makes it zero. Strict discrepancy
-improvement prevents contraction of a requested zero-error prime.
-The chosen extraction boundary must be effective and klt; in
-particular its prescribed primes have the usual admissible
-discrepancy range.
-
-For the graph construction, suppose the already projective graph
-resolution \(h:V\to Y_m\) has smooth source and
-\[
- A_V=h^*A_{Y_m}+[G],\qquad G\geq0.
-\]
-Then the global relative real line is \(J=\OO_V(G)\), the base
-class is \(A_{Y_m}\), and its carrier line is
-\(N=G-K_V-B_V\). Its relative degrees equal those of the prepared
-nef datum. That datum is nef and big on the prepared carrier, for
-example a pullback of its prepared Kähler class; thus \(N\) is nef
-and big over \(Y_m\). This bigness is a separate hypothesis check,
-since \(h\) can have positive-dimensional general fibres.
-Effectivity of \(G\) gives relative pseudoeffectivity. Apply the
-proposition over \(Y_m\). The subsequent negativity argument must
-still remove \(G_m\); only then is the endpoint adjoint the pullback
-of \(A_{Y_m}\) and good over the original base.
-
-For transport of all classes, verify separately that all
-Bott--Chern classes on the graph carrier are real-line classes
-modulo \(Y_m\), using the smooth rationally connected fibration
-and the inherited exceptional splitting on the lower-dimensional
-base model. Lemma~\ref{prog:projective-cone} then identifies
-the relative degree rays with full Bott--Chern rays, so the
-detected-step transport applies. The driving identity involving
-\(G\) alone does not give this additional property.
-\end{remark}
-
-\begin{corollary}[Ordinary projective Mori output]
-\label{prog:projective-mori}
-Let \(f:X\to S\) be an already projective surjective morphism of
-compact Kähler spaces, with \(X\) smooth. Let \((X,B)\) be an
-effective ordinary real klt pair, and let \(D=K_X+B\). If \(D\)
-is not pseudoeffective over \(S\), then its projective program over
-\(S\), with scaling of a relatively ample actual real line \(N\),
-terminates with a Mori fibre contraction. The initial scaling
-parameter is chosen so that \(D+TN\) is nef over \(S\).
-The working spaces are globally strongly \(\Q\)-factorial and
-compact Kähler, and the actual line comparisons and integral
-descent of Proposition~\ref{prog:projective-good-model} apply.
-\end{corollary}
-
-\begin{proof}
-The line \(N\) is fixed; it need not be a general scaling direction.
-In particular several extremal rays may have the same wall
-parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is
-still not pseudoeffective over \(S\). On the finite Stein-compact
-cover choose divided general effective representatives of \(N\)
-so that all the resulting ordinary pairs representing \(D+tN\),
-\(0\leq t\leq T\), are klt. For
-\(\epsilon\leq t\leq T\) reserve a common effective relatively
-ample rational summand. The joint rational-family construction in
-Proposition~\ref{prog:projective-good-model} puts these fixed pairs
-in the polytopes required by Theorem~E of
-\cite{FujinoAnalyticBCHM}.
-
-Suppose a finite prefix has reached \(X_i\), and put
-\(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed
-\(D_i+\mu_iN_i\) is nef. Define
-\[
- \lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i
-                         \text{ is nef over }S\}.
-\]
-The feasible set is a nonempty closed interval. Inductively
-\(\lambda_i>2\epsilon\): all previous steps have nonpositive
-comparison for \(D+2\epsilon N\), so its transform is still not
-pseudoeffective. A first threshold at most \(2\epsilon\) would
-make this transform nef by convexity between the current and
-preceding thresholds, a contradiction.
-
-There is a \(D_i\)-negative extremal ray \(R_i\) with
-\((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a
-generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\).
-A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree,
-because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is
-also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed
-ordinary klt big-boundary representatives on the charts, since
-\(\lambda_i/2>\epsilon\). Normalize each of these boundaries by
-\cite[Lemma~11.15]{FujinoAnalyticBCHM} as \(A_\ell+B'_\ell\),
-with \(A_\ell\geq0\) relatively ample and the new pair klt.
-The \(A_\ell\)-truncation of the cone theorem for
-\(K+B'_\ell\) gives finitely many rays negative for the entire
-adjoint in question. The finite cover therefore gives finitely
-many global negative rays. One of these rays attains the wall,
-as required.
-
-Choose one such ray, even if the wall vanishes on other rays.
-The rational-support argument for an individual negative ray in
-Proposition~\ref{prog:relative-klt} gives a global rational nef
-support annihilating exactly \(R_i\), and ordinary relative base
-point freeness contracts it. For the real ordinary boundary one
-may first choose a nearby rational klt boundary that is still
-negative on \(R_i\). The contraction therefore has a rational
-ordinary antiample adjoint. Its flip, when small, is constructed
-from one global detector algebra as above.
-
-If the contraction is of fibre type, stop. Otherwise the wall
-\(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\).
-In the finite rational span of its actual line factors, this
-zero-degree subspace is rational: its defining functional has
-rational values on rational lines. Thus \(w_i\) is a real
-combination of rational lines of zero ray degree.
-Lemma~\ref{prog:integral-descent} descends those lines to the
-contraction base. Their combination is nef over \(S\), by lifting
-curves through the projective contraction. Pulling it to the
-positive side shows that the transformed wall stays nef. The next
-threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\);
-equality is permitted. No interval of ampleness between successive
-thresholds is required.
-
-If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common
-resolution, this actual wall descent gives, for every real \(t\),
-\[
- p_i^*(D_i+tN_i)
- \sim_{\R}q_i^*(D_{i+1}+tN_{i+1})
-                 +(1-t/\lambda_i)E_i.
-\]
-For \(t\leq\lambda_i\) the error is effective. This proves all
-the nonpositivity and fixed-pair persistence assertions used in
-the induction, including the lower bound on thresholds.
-
-At every working model use the \emph{wall parameter} \(t=\lambda_i\),
-rather than an interior ample parameter. Since
-\(\lambda_i\leq\lambda_j\) for every earlier step, the displayed
-comparisons show that \(X_i\) is a weak log canonical model of
-the initial fixed chart pairs representing \(D+\lambda_iN\).
-Its trace is nef by definition. Theorem~E counts all these marked
-weak log canonical models, including the ones at repeated wall
-parameters. The finite chart cover gives a finite global marked
-list. Repetition would contradict the strict ordinary
-\(D\)-discrepancy increase at an intervening step. Thus this
-program with the specified \(N\) is finite. Non-pseudoeffectivity
-excludes a nef endpoint for \(D\), so the last contraction is a
-Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is
-crepant: such a step supplies no strict discrepancy improvement
-for \(D+N\), while it is still strictly negative for \(D\).
-\end{proof}
+\subsection{Good models for already projective relative adjoints}
+
+\begin{proposition}[Projective relative completion]
+\label{prog:projective-good-model}
+Let \(f:X\to S\) be an already projective surjective morphism of
+normal compact Kähler spaces, with \(X\) smooth. Let
+\((X,B+\mathbf M)\) be an effective generalized klt pair with adjoint
+\(A\), carried by a projective resolution \(p:W\to X\) with smooth
+\(W\), so that
+\[
+ p^*A=c_1(K_W)+[B_W]+\mathbf M_W.
+\]
+Suppose that there are an actual real line bundle
+\(J\in\Pic(X)\otimes\R\) and
+\(\beta\in H^{1,1}_{\BC}(S)\) such that
+\[
+ A=c_1(J)+f^*\beta,
+ \qquad
+ N:=p^*J-K_W-B_W
+ \text{ is nef and big over }fp.
+\]
+Assume that \(J\) is pseudoeffective over \(S\).
+Then there is a finite chosen \(A\)-negative projective program over
+\(S\), with globally strongly \(\Q\)-factorial compact Kähler
+working spaces, and a nonextracting endpoint \(X_m\) such that
+\[
+ J_m\sim_{\R}g^*H,
+ \qquad
+ A_m=g^*\bigl(c_1(H)+\rho^*\beta\bigr).
+\]
+Here \(J_m\) is the actual real-line transform of \(J\),
+\(g:X_m\to Z\) is a projective connected-fibre morphism,
+\(\rho:Z\to S\) is projective, and \(H\) is an actual real line
+bundle ample over \(S\). Thus the class on \(Z\) in this formula is
+relatively Kähler over \(S\). The generalized pair on \(X_m\) is
+gklt. On a common resolution the comparison with \(A\) is effective
+and exceptional over \(X_m\), with strictly positive coefficient
+at the strict transform of every prime contracted from \(X\).
+\end{proposition}
+
+\begin{proof}
+We reduce to fixed ordinary pairs, and construct one global program.
+Choose finitely many Stein open sets \(U_\ell\subset S\) and
+semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose
+interiors cover \(S\). Such compacts are obtained by intersections
+with closed polydiscs in local embeddings of \(S\). In particular
+they satisfy condition~(P) of \cite{FujinoAnalyticBCHM}: the source
+is normal, the base is Stein, the compact is Stein, and its
+intersection with any analytic subset defined near it has finitely
+many connected components. This remains true on every normal
+projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves
+the finite cover.
+
+\smallskip\noindent
+\emph{Actual ordinary replacements.}
+On each chart represent the finitely many line bundles in the data
+by Cartier divisors. For a projective map to a Stein space this can
+be done by twisting a line bundle and its prospective meromorphic
+frame by sufficiently positive relative lines and using nonzero
+sections. The generic-rank-one direct-image argument is an
+alternative only when the map is bimeromorphic.
+Relative bigness gives
+\(N\sim_{\R}P+E\), with \(P\) relatively ample and \(E\geq0\).
+For sufficiently small \(\epsilon>0\),
+\((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image.
+The line \((1-\epsilon)N+\epsilon P\) is relatively ample.
+General divided relative sections give an effective representative
+\(T\) such that
+\[
+ \Theta:=\epsilon E+T\sim_{\R}N,
+ \qquad (W,B_W+\Theta)\text{ is sub-klt}.
+\]
+All supports and coefficient margins are fixed near that compact.
+With compatible canonical representatives, push the finite
+principal-divisor expression for \(\Theta-N\) down and pull it
+back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives
+\[
+ \Delta_\ell\geq0,
+ \qquad K_X+\Delta_\ell\sim_{\R}J|_{X_{U_\ell}},
+ \qquad
+ p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta.
+\]
+In the last equality the chosen principal correction is included
+in the representative of \(K_X+\Delta_\ell\).
+Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary
+is relatively big. Lemma~11.15 of \cite{FujinoAnalyticBCHM}, with
+its big part equal to \(\Delta_\ell\) and its remaining part zero,
+replaces it, in its actual real linear equivalence class, by a klt
+boundary with an effective general relatively ample rational
+summand. Its condition on non-klt centres is vacuous.
+Choose the reserved ample summand to be \(\Q\)-linearly equivalent
+to a small rational multiple of a fixed global \(f\)-ample line.
+Its later strict transform is consequently \(\Q\)-Cartier by the
+transport of that global line, without a local factoriality claim.
+
+We make these choices for a finite rational polytope of actual
+global line data. Start with a finite rational span containing
+\(J\), and add relatively ample rational lines spanning the global
+relative degree space. The above construction is valid in a
+neighbourhood of \(J\): retain a small part of the ample summand
+and use openness of ampleness and the strict klt inequalities.
+It is also valid along a segment from \(J\) to a sufficiently
+positive adjoint, by adding divided general ample divisors.
+Use finitely many such representatives and one log resolution on
+each chart. More explicitly, represent a finite simplex inside the
+open ample neighbourhood used above; varying its positive
+coefficients represents all nearby global line parameters on the
+same finite support. Form the joint finite affine system consisting
+of the divisor identities on all charts, with common global line
+coordinates and separate boundary and principal-divisor
+coordinates. This system is rational. Allow all boundary
+coefficients to vary, including those originally contributed by
+the real boundary \(B\); fix only the reserved rational ample part
+and the zero-support constraints. Its projection to the global
+line coordinates contains the just constructed neighbourhood, so
+it has a rational affine section. Choose that section sufficiently
+close to the constructed real affine lift, by rational approximation
+in its affine space of sections, so that the strict klt and
+effectiveness margins are retained along the compact segment.
+In the joint solution space the
+strict klt and effectiveness margins permit a rational polytope
+containing the scaling segment. This supplies on every chart
+a rational polytope of ordinary klt boundaries with a fixed
+effective relatively ample rational summand, representing the
+restrictions of the same global line parameters. These boundaries,
+including the scaling representatives, are fixed before the
+program starts. Their divisors need not agree on different charts.
+
+\smallskip\noindent
+\emph{One global scaling construction.}
+Apply the global degree-space construction in the proof of
+Proposition~\ref{prog:relative-klt} to \(J\) and a general ample
+scaling direction in the chosen span. Here are the modifications
+that allow real line data and a nonbirational map to \(S\).
+The global relatively ample line still gives a compact normalized
+slice of the curve cone. The fixed ordinary chart pairs give finite
+negative-ray truncations. A general direction avoids equal wall
+parameters for independent ray-degree functionals, exactly as in
+that proof. A positive wall therefore selects a single ray in the
+global degree space. A rational nef support is represented by a
+global rational line; a large multiple minus the chartwise ordinary
+adjoint is relatively ample. Ordinary relative base point freeness
+contracts the ray using the evaluation of this one global line.
+
+If the contraction is small, choose a global rational line having
+negative ray degree and construct its flip by its one global
+graded algebra. The local ordinary proof of
+Proposition~\ref{prog:detected-flip} applies to this algebra:
+all curves of the contraction have proportional degrees for global
+lines, so the detector and the driving line are positively
+proportional over the contraction; ordinary rational replacement
+gives local finite generation. This part of that proof needs only
+the already projective contraction and these line degrees.
+It does not require a prior assertion about the full Bott--Chern
+cone. The relative Proj is the next global model. In a divisorial
+step the exceptional prime and
+Lemma~\ref{prog:integral-descent} give the same continuation as in
+Proposition~\ref{prog:relative-klt}. Killing the degree of any global
+line by a rational multiple of the detector, then applying that
+lemma, preserves global strong \(\Q\)-factoriality in both cases.
+The global lines continue to span the relative degree spaces.
+The rational support-simplex argument in that proposition makes
+the wall a descended relatively ample real line on its contraction
+base and gives a nonempty interval of ampleness immediately after
+each step. This argument uses relative line data and projectivity,
+not birationality of the map to \(S\).
+
+We check explicitly the persistence of the ordinary pairs. Expand
+the line parameters in finitely many global rational lines. Their
+reflexive transforms are actual rational lines by the preceding
+strong property. Pushing forward the fixed principal-divisor
+identities gives
+\[
+ K_{X_i}+\Delta_{\ell,i}\sim_{\R}J_i|_{(X_i)_{U_\ell}}.
+\]
+Hence these ordinary adjoints are \(\R\)-Cartier; no local
+factoriality of individual prime divisors is being assumed.
+On a common resolution their comparison minus the appropriate
+positive multiple of the detector comparison is exceptional and
+relatively numerically trivial. Negativity in both signs makes
+this difference zero. Every chosen step is consequently negative
+for the fixed ordinary pairs. They remain effective and klt, and
+their boundaries remain relatively big. The same equality of
+comparison divisors, after adding the fixed base pullback,
+preserves the generalized pair and its discrepancy inequalities.
+
+For termination, every already constructed working model is a weak
+log canonical model on each chart of a boundary in the fixed
+polytope: choose a parameter in its nonempty ample scaling interval.
+Indeed, actual descent at a preceding wall \(\lambda_j\) makes
+the comparison for \(J+tH\) equal to
+\((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter
+in the current interval all preceding \(\lambda_j\) are at least
+\(t\), so these comparisons are effective.
+Theorem~E of \cite{FujinoAnalyticBCHM} gives finitely many marked
+models on each chart. That theorem does not require locally
+\(\Q\)-factorial targets. There are therefore finitely many possible
+global marked working models. Indeed, isomorphisms between the
+restrictions of two existing marked models agree on their common
+dense open, and hence on overlaps. A repeated marked working model
+would have the same boundary transform and discrepancies, contrary
+to strict discrepancy increase at an intervening nontrivial step.
+The global program is finite. Relative pseudoeffectivity excludes
+a Mori fibre endpoint, so \(J_m\) is nef over \(S\).
+This is a finite-cover argument for a single constructed program;
+it does not glue independent minimal models.
+
+\smallskip\noindent
+\emph{Semiampleness and the global polarization.}
+On the endpoint the fixed ordinary pair is klt, its effective
+boundary is relatively big, and its adjoint is nef over
+\(W_\ell\). Lemma~11.16 of \cite{FujinoAnalyticBCHM} applies to
+this already existing weak model. More explicitly, Lemma~11.15
+replaces the endpoint boundary by \(A_\ell+B_\ell\), where
+\(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the
+pair is klt. Theorem~8.3 then gives semiampleness near \(W_\ell\):
+its log canonical, klt-existence, real Cartier, ample-summand and
+nef-over-compact hypotheses have all been checked.
+
+The actual equivalences identify these local semiample adjoints
+with restrictions of \(J_m\). Their ample models agree on overlaps
+by \cite[Lemma~11.3]{FujinoAnalyticBCHM}. To retain a global
+polarization, shrink the initial parameter polytope around \(J\)
+after the finite program: all its finitely many strict step signs
+persist, so the transformed ordinary boundaries remain klt and big.
+On each chart the pushforward of its initial common ample rational
+summand is effective and relatively big. Apply
+\cite[Lemma~11.13]{FujinoAnalyticBCHM} to this common big rational
+part and the small endpoint boundary polytope. All pairs are klt,
+so the condition on non-klt centres is vacuous. The lemma gives a
+common general relatively ample rational summand by one
+\(\Q\)-linearly trivial translation. In particular the rational
+parametrization is preserved; separate applications of
+Lemma~11.15 to individual parameters would not suffice here.
+In this finite rational span Theorem~11.17 of \cite{FujinoAnalyticBCHM}
+makes the endpoint nef region a rational polytope. Intersect the
+finitely many inverse images of these regions and express \(J_m\)
+as a positive combination of rational points in its minimal face.
+The corresponding global rational lines are semiample on every
+chart by the same ordinary base point free argument. After clearing
+denominators and taking common sufficiently divisible powers over
+the finite cover, each is relatively generated globally: generation on
+the charts is generation of its global coherent relative
+evaluation map. The product of these finitely many relative
+morphisms, followed by Stein factorization, gives a projective
+\(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended
+global lines whose positive combination \(H\) is relatively ample.
+It realizes the glued ample model and gives
+\(J_m\sim_{\R}g^*H\). Adding back \(f_m^*\beta\) proves the
+asserted Bott--Chern identity. Composing the effective step
+comparisons proves the final discrepancy statement.
+\end{proof}
+
+\begin{corollary}[Semiampleness on an existing projective model]
+\label{prog:projective-semiample}
+Let \(f:T\to S\) be a projective morphism of normal compact Kähler
+spaces with \(T\) globally strongly \(\Q\)-factorial, and let
+\(J\in\Pic(T)\otimes\R\) be nef over \(S\). Suppose that on a
+finite Stein-compact cover satisfying condition~(P), there are
+effective ordinary klt boundaries \(\Delta_\ell\), big over the
+base, such that
+\(K_T+\Delta_\ell\sim_{\R}J|_{T_{U_\ell}}\).
+Then \(J\) has the projective relative semiample contraction and
+actual descended ample real line of
+Proposition~\ref{prog:projective-good-model}.
+In particular this applies to any already existing weak model of
+the fixed ordinary chart pairs in that proof; one need not run a
+new program for the adjoint in question.
+\end{corollary}
+
+\begin{proof}
+Lemma~11.15 and Theorem~8.3 of \cite{FujinoAnalyticBCHM} give
+local semiampleness. For an existing weak model this is also
+Lemma~11.16. To globalize the polarization, take a finite rational
+span of the global lines defining \(J\). On each chart the ample
+part of the normalized boundary allows small perturbations in
+this span. The joint rational affine construction in the preceding
+proof gives a rational family with a common rational ample part;
+all boundary coefficients are allowed to vary. Theorem~11.17 then
+gives a rational nef region on each chart. The finite-intersection,
+rational-vertex and global-evaluation argument in that proof gives
+one projective contraction and its actual descended ample real
+line. Only its local ample models are identified by uniqueness.
+\end{proof}
+
+\begin{remark}[The two uses of projective completion]
+\label{prog:projective-good-model-uses}
+For a selective extraction over an already constructed weak model,
+the carrier map is projective bimeromorphic. The inherited
+exceptional splitting expresses the nef datum, modulo this base,
+by actual real lines. Relative nefness follows from the carrier
+datum, while relative bigness follows from birationality, using the
+ample-plus-effective construction in the ordinary replacement.
+The prepared adjoint is a base pullback plus an effective
+exceptional divisor supported on exactly the unwanted primes.
+Proposition~\ref{prog:projective-good-model} applies. On its endpoint
+the remaining error is effective, exceptional over the fixed base,
+and relatively nef; negativity makes it zero. Strict discrepancy
+improvement prevents contraction of a requested zero-error prime.
+The chosen extraction boundary must be effective and klt; in
+particular its prescribed primes have the usual admissible
+discrepancy range.
+
+For the graph construction, suppose the already projective graph
+resolution \(h:V\to Y_m\) has smooth source and
+\[
+ A_V=h^*A_{Y_m}+[G],\qquad G\geq0.
+\]
+Then the global relative real line is \(J=\OO_V(G)\), the base
+class is \(A_{Y_m}\), and its carrier line is
+\(N=G-K_V-B_V\). Its relative degrees equal those of the prepared
+nef datum. That datum is nef and big on the prepared carrier, for
+example a pullback of its prepared Kähler class; thus \(N\) is nef
+and big over \(Y_m\). This bigness is a separate hypothesis check,
+since \(h\) can have positive-dimensional general fibres.
+Effectivity of \(G\) gives relative pseudoeffectivity. Apply the
+proposition over \(Y_m\). The subsequent negativity argument must
+still remove \(G_m\); only then is the endpoint adjoint the pullback
+of \(A_{Y_m}\) and good over the original base.
+
+For transport of all classes, verify separately that all
+Bott--Chern classes on the graph carrier are real-line classes
+modulo \(Y_m\), using the smooth rationally connected fibration
+and the inherited exceptional splitting on the lower-dimensional
+base model. Lemma~\ref{prog:projective-cone} then identifies
+the relative degree rays with full Bott--Chern rays, so the
+detected-step transport applies. The driving identity involving
+\(G\) alone does not give this additional property.
+\end{remark}
+
+\begin{corollary}[Ordinary projective Mori output]
+\label{prog:projective-mori}
+Let \(f:X\to S\) be an already projective surjective morphism of
+compact Kähler spaces, with \(X\) smooth. Let \((X,B)\) be an
+effective ordinary real klt pair, and let \(D=K_X+B\). If \(D\)
+is not pseudoeffective over \(S\), then its projective program over
+\(S\), with scaling of a relatively ample actual real line \(N\),
+terminates with a Mori fibre contraction. The initial scaling
+parameter is chosen so that \(D+TN\) is nef over \(S\).
+The working spaces are globally strongly \(\Q\)-factorial and
+compact Kähler, and the actual line comparisons and integral
+descent of Proposition~\ref{prog:projective-good-model} apply.
+\end{corollary}
+
+\begin{proof}
+The line \(N\) is fixed; it need not be a general scaling direction.
+In particular several extremal rays may have the same wall
+parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is
+still not pseudoeffective over \(S\). On the finite Stein-compact
+cover choose divided general effective representatives of \(N\)
+so that all the resulting ordinary pairs representing \(D+tN\),
+\(0\leq t\leq T\), are klt. For
+\(\epsilon\leq t\leq T\) reserve a common effective relatively
+ample rational summand. The joint rational-family construction in
+Proposition~\ref{prog:projective-good-model} puts these fixed pairs
+in the polytopes required by Theorem~E of
+\cite{FujinoAnalyticBCHM}.
+
+Suppose a finite prefix has reached \(X_i\), and put
+\(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed
+\(D_i+\mu_iN_i\) is nef. Define
+\[
+ \lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i
+                         \text{ is nef over }S\}.
+\]
+The feasible set is a nonempty closed interval. Inductively
+\(\lambda_i>2\epsilon\): all previous steps have nonpositive
+comparison for \(D+2\epsilon N\), so its transform is still not
+pseudoeffective. A first threshold at most \(2\epsilon\) would
+make this transform nef by convexity between the current and
+preceding thresholds, a contradiction.
+
+There is a \(D_i\)-negative extremal ray \(R_i\) with
+\((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a
+generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\).
+A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree,
+because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is
+also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed
+ordinary klt big-boundary representatives on the charts, since
+\(\lambda_i/2>\epsilon\). Normalize each of these boundaries by
+\cite[Lemma~11.15]{FujinoAnalyticBCHM} as \(A_\ell+B'_\ell\),
+with \(A_\ell\geq0\) relatively ample and the new pair klt.
+The \(A_\ell\)-truncation of the cone theorem for
+\(K+B'_\ell\) gives finitely many rays negative for the entire
+adjoint in question. The finite cover therefore gives finitely
+many global negative rays. One of these rays attains the wall,
+as required.
+
+Choose one such ray, even if the wall vanishes on other rays.
+The rational-support argument for an individual negative ray in
+Proposition~\ref{prog:relative-klt} gives a global rational nef
+support annihilating exactly \(R_i\), and ordinary relative base
+point freeness contracts it. For the real ordinary boundary one
+may first choose a nearby rational klt boundary that is still
+negative on \(R_i\). The contraction therefore has a rational
+ordinary antiample adjoint. Its flip, when small, is constructed
+from one global detector algebra as above.
+
+If the contraction is of fibre type, stop. Otherwise the wall
+\(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\).
+In the finite rational span of its actual line factors, this
+zero-degree subspace is rational: its defining functional has
+rational values on rational lines. Thus \(w_i\) is a real
+combination of rational lines of zero ray degree.
+Lemma~\ref{prog:integral-descent} descends those lines to the
+contraction base. Their combination is nef over \(S\), by lifting
+curves through the projective contraction. Pulling it to the
+positive side shows that the transformed wall stays nef. The next
+threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\);
+equality is permitted. No interval of ampleness between successive
+thresholds is required.
+
+If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common
+resolution, this actual wall descent gives, for every real \(t\),
+\[
+ p_i^*(D_i+tN_i)
+ \sim_{\R}q_i^*(D_{i+1}+tN_{i+1})
+                 +(1-t/\lambda_i)E_i.
+\]
+For \(t\leq\lambda_i\) the error is effective. This proves all
+the nonpositivity and fixed-pair persistence assertions used in
+the induction, including the lower bound on thresholds.
+
+At every working model use the \emph{wall parameter} \(t=\lambda_i\),
+rather than an interior ample parameter. Since
+\(\lambda_i\leq\lambda_j\) for every earlier step, the displayed
+comparisons show that \(X_i\) is a weak log canonical model of
+the initial fixed chart pairs representing \(D+\lambda_iN\).
+Its trace is nef by definition. Theorem~E counts all these marked
+weak log canonical models, including the ones at repeated wall
+parameters. The finite chart cover gives a finite global marked
+list. Repetition would contradict the strict ordinary
+\(D\)-discrepancy increase at an intervening step. Thus this
+program with the specified \(N\) is finite. Non-pseudoeffectivity
+excludes a nef endpoint for \(D\), so the last contraction is a
+Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is
+crepant: such a step supplies no strict discrepancy improvement
+for \(D+N\), while it is still strictly negative for \(D\).
+\end{proof}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/03c-base-presentations.tex

@@ -1,644 +1,644 @@
-\subsubsection{Base presentations and multiplier ideals}
-
-The following constructions take place in the category of generalized
-data specified above. In particular, the whole adjoint is a
-Bott--Chern class, whereas the boundary and nef traces on a singular
-space are interpreted as currents. On a smooth carrier the structure
-boundary is a real divisor; its negative components may be exceptional
-over the space of the pair. The nef part is specified on that carrier
-and pulled back to higher models.
-
-\begin{lemma}[Adjoint-preserving preparation]
-\label{prog:base-presentation-preparation}
-Let $(X,B+\mathbf M)$ be a generalized pair on a normal compact
-K\"ahler space, with globally modified-big total boundary
-$B+\mathbf M_X$. Write
-\[
- A=[K_X+B+\mathbf M_X],\qquad
- \mathcal J_o=\mathcal J(X,B+\mathbf M).
-\]
-There are generalized data on $X$ with the same adjoint class, a
-projective carrier $p:U\to X$ with $U$ smooth and compact K\"ahler, an SNC real
-divisor $C$ whose negative components are $p$-exceptional, and a
-K\"ahler class $\beta$ such that
-\begin{equation}\label{prog:base-presentation-prepared}
- p^*A=[K_U+C]+\beta,\qquad
- \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor)
-                 \supseteq\mathcal J_o.
-\end{equation}
-The carrier may be required to dominate any prescribed projective
-modification of $X$, with the structure and exceptional divisors
-having simultaneous SNC support. The prepared total boundary is
-globally modified big. In particular, the preparation preserves the
-gklt condition.
-\end{lemma}
-
-\begin{proof}
-Choose a common projective log resolution $p_0:U_0\to X$ of the
-original nef carrier and the prescribed modification, and write
-\[
- p_0^*A=[K_{U_0}+C^0]+\mathbf M_{U_0}.
-\]
-The class $\mathbf M_{U_0}$ is nef, and the negative part $(C^0)^-$
-is exceptional over $X$. The modified-big preparation
-\cite[Lemmas~2.10 and~2.23]{HX} can be made with the prescribed
-resolution. We give its divisor and ideal details. Choose an effective divisor $E_0$ with sufficiently large coefficients
-and full $p_0$-exceptional support so that the class
-\[
- [(C^0)^+]+\mathbf M_{U_0}+[E_0]
-\]
-is big. Resolve a Kodaira decomposition of this class. On a further
-projective modification $a:U\to U_0$ with $U$ smooth one obtains
-\[
- a^*\bigl([(C^0)^+]+\mathbf M_{U_0}+[E_0]\bigr)
-       =\kappa+[D],\qquad D\geq0,
-\]
-where $\kappa$ is K\"ahler. The support can be made SNC together
-with all the other divisors. On any additional blowup, subtract a
-sufficiently small effective exceptional class from the pulled-back
-K\"ahler class and add it to the divisor; an exceptional divisor
-whose negative is relatively ample makes the resulting class
-K\"ahler again.
-
-Put $p=p_0a$ and
-\[
- C^{\mathrm{alt}}=D-a^*(C^0)^--a^*E_0-K_{U/U_0}.
-\]
-Then
-\[
- p^*A=[K_U+C^{\mathrm{alt}}]+\kappa,
-\]
-and every negative component of $C^{\mathrm{alt}}$ is
-$p$-exceptional. Let $C^{\mathrm{old}}$ be the crepant structure
-boundary of the original data on this same model. For $0<t\ll1$,
-set
-\[
- C=(1-t)C^{\mathrm{old}}+tC^{\mathrm{alt}},\qquad
- \beta=(1-t)a^*\mathbf M_{U_0}+t\kappa.
-\]
-The class $\beta$ is K\"ahler, the adjoint identity is unchanged,
-and negative coefficients still occur only on exceptional divisors.
-There are only finitely many coefficients. Thus $t$ can be chosen so
-that
-\[
- \lfloor C\rfloor\leq\lfloor C^{\mathrm{old}}\rfloor.
-\]
-Pushing forward the corresponding inclusion of invertible sheaves
-gives~\eqref{prog:base-presentation-prepared}. The K\"ahler class
-$\beta$ plus the strict transform of the effective trace boundary is
-big; its pushforward is the prepared total boundary. If the original pair
-is gklt, its multiplier ideal is the unit ideal, and hence so is
-$\mathcal J_1$.
-\end{proof}
-
-\begin{proposition}[Base presentations for projective contractions]
-\label{prog:base-presentation}
-Let $f:X\to Z$ be a projective contraction between normal irreducible
-compact K\"ahler spaces. Suppose that $(X,B+\mathbf M)$ has globally
-modified-big total boundary and that
-\[
- A=[K_X+B+\mathbf M_X]=f^*\gamma,
- \qquad \gamma\in H^{1,1}_{\mathrm{BC}}(Z).
-\]
-There is an adjoint-preserving preparation as in
-Lemma~\ref{prog:base-presentation-preparation}, with multiplier ideal
-$\mathcal J_1\supseteq\mathcal J_o$, for which
-\begin{equation}\label{prog:base-presentation-Stein}
- R^if_*\mathcal J_1=0\quad(i>0),\qquad
- f_*\mathcal O_{N_1}=\mathcal O_Z/(f_*\mathcal J_1),
- \qquad N_1=V(\mathcal J_1).
-\end{equation}
-If $N_1$ dominates $Z$, then $f_*\mathcal J_1=0$.
-Otherwise there are generalized data $(Z,B_Z+\mathbf N)$, a
-projective carrier $\tau:T\to Z$ with $T$ smooth and compact K\"ahler, a real
-SNC structure divisor $C_T$, and a K\"ahler class $\beta_T$ such that
-\begin{equation}\label{prog:base-presentation-identities}
- \begin{gathered}
- [K_Z+B_Z+\mathbf N_Z]=\gamma,\qquad
- \tau^*\gamma=[K_T+C_T]+\beta_T,\qquad
- \mathbf N_T=\beta_T,\\
- B_Z=\tau_*C_T\geq0,\qquad
- B_Z+\mathbf N_Z\ \text{is globally modified big}.
- \end{gathered}
-\end{equation}
-The negative components of $C_T$ are $\tau$-exceptional, and
-\begin{equation}\label{prog:base-presentation-ideal}
- \mathcal J_Z:=\mathcal J(Z,B_Z+\mathbf N)
-       =\tau_*\mathcal O_T(-\lfloor C_T\rfloor)
-       \supseteq f_*\mathcal J_1.
-\end{equation}
-In particular the base data are gklt wherever $f_*\mathcal J_1$ is
-the unit ideal. If the source data are gklt, the base data are gklt.
-All adjoint identities are preserved by pullback to higher models.
-\end{proposition}
-
-\begin{proof}
-Choose a projective resolution $\mu:W\to Z$, with $W$ smooth
-compact K\"ahler and reduced exceptional divisor $F$ having SNC
-support. Apply Lemma~\ref{prog:base-presentation-preparation} on a
-carrier dominating the main component of the base change. We obtain
-projective maps $p:U\to X$ and $q:U\to W$ with $fp=\mu q$ and
-\begin{equation}\label{prog:base-presentation-source}
- [K_U+C]+\beta=p^*A=q^*\mu^*\gamma,
- \qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor).
-\end{equation}
-Here $U$ is smooth compact K\"ahler, $\beta$ is K\"ahler, and all
-negative components of $C$ are $p$-exceptional. The maps are
-projective by composition and factorization over the separated base
-$W$. The map $q$ has connected fibres: this holds over the open set
-where $\mu$ is an isomorphism, and its Stein factorization is finite
-and bimeromorphic over the normal space $W$.
-
-The integral divisor $-\lfloor C\rfloor$ has class
-\[
- [-\lfloor C\rfloor]
-   =[K_U+\{C\}]+\beta-(fp)^*\gamma.
-\]
-The pair $(U,\{C\})$ is klt with SNC boundary. Relative
-Kawamata--Viehweg vanishing, first for $p$ and then for $fp$, applies
-because the positive class modulo pullbacks is K\"ahler
-\cite[Theorem~2.2]{HX}. Equivalently, the real line bundle
-$-K_U-C$ is relatively ample,
-so the vanishing follows from \cite[Theorem~1.2]{FujinoVanishing2025}
-with the identity as its first morphism. Leray gives $R^if_*\mathcal J_1=0$ for
-$i>0$. Applying $f_*$ to the sequence defining $N_1$, and using
-$f_*\mathcal O_X=\mathcal O_Z$, proves
-\eqref{prog:base-presentation-Stein}. If $N_1$ dominates $Z$, the
-ideal $f_*\mathcal J_1\subseteq\mathcal O_Z$ vanishes generically
-and therefore vanishes. This proves that alternative.
-
-Suppose from now on that $N_1$ does not dominate $Z$. Every
-$q$-horizontal coefficient of $C$ is then less than one. Indeed,
-a component with coefficient at least one forces the multiplier
-ideal to be proper along its image in $X$, and its image would make
-$N_1$ dominate $Z$. If $Z$ is a point, the asserted base presentation
-is the dimension-zero presentation. We therefore assume
-$b=\dim Z>0$.
-
-\paragraph{The fibre metric.}
-Choose a smooth representative $\rho$ of
-$\mu^*\gamma-c_1(K_W)$, a K\"ahler form $\omega_W$, and a
-K\"ahler form representing $\beta$, still denoted by $\beta$.
-For some $a>0$,
-\[
- \beta\geq a q^*\omega_W.
-\]
-Consider the holomorphic line bundle
-\[
- L=\mathcal O_U(-K_{U/W}-\lfloor C\rfloor).
-\]
-Equation~\eqref{prog:base-presentation-source} gives a singular
-Hermitian metric $h$ on $L$ whose curvature current is
-\begin{equation}\label{prog:base-presentation-curvature}
- \Theta_h(L)=\beta+[\{C\}]-q^*\rho.
-\end{equation}
-Its weights are smooth functions plus the divisorial logarithms
-with coefficients in $\{C\}$. We normalize $dd^c$ so that
-$dd^c\log|z|^2=[z=0]$ and write squared-norm weights as $e^{-\varphi}$.
-Let $e$ be the canonical meromorphic section of
-\[
- K_{U/W}\otimes L=\mathcal O_U(-\lfloor C\rfloor),
-\]
-so that $\operatorname{div}(e)=-\lfloor C\rfloor$.
-
-Choose an analytic Zariski open set $W^\circ\subseteq W$ on which
-$q$ is a submersion, the horizontal boundary is relatively SNC,
-there are no vertical boundary components, $\mu$ is an isomorphism
-onto an open subset of $Z$, and coherent base change holds in degree
-zero. On $q^{-1}(W^\circ)$, the divisor $-\lfloor C\rfloor$ is
-effective and exceptional over $X$. Normality and the contraction
-property therefore identify
-\[
- q_*(K_{U/W}\otimes L)|_{W^\circ}
-       =\mathcal O_{W^\circ}e.
-\]
-Here $K_{U/W}$ is viewed as the relative canonical line bundle. The
-fibre norm
-\[
- I(w)=\int_{U_w}|e_w|_h^2
-\]
-is positive and finite. The fractional SNC weights are integrable;
-any zeros of $e_w$ only improve integrability. Relative SNC
-coordinates and a partition of unity show that $I$ is smooth:
-parameter derivatives act on smooth factors multiplying fixed
-integrable monomial weights.
-
-Set $\psi=-\log I$ on $W^\circ$. Then
-\begin{equation}\label{prog:base-presentation-open-positivity}
- \rho+dd^c\psi\geq a\omega_W.
-\end{equation}
-Here are the details for the singular weights. Locally choose
-$\ell$ with $dd^c\ell=\rho-a\omega_W$, and put
-$h'=he^{-q^*\ell}$. Its weights are psh, and its fibre norm satisfies
-$I'=e^{-\ell}I$. Restrict the family to a small coordinate line disk
-$\Delta_R$, with centre $0$. The sharp $L^2$ extension theorem gives
-a twisted top form extending $e_0$ and satisfying
-\begin{equation}\label{prog:base-presentation-sharp-extension}
- \int_{q^{-1}(\Delta_R)}|E|_{h'}^2
-        \leq\pi R^2 I'(0).
-\end{equation}
-Top-form norms here are normalized to use Euclidean area in the
-base coordinate. We use the Stein hypersurface form of the optimal
-extension theorem \cite[Theorem~2.2 and \S3.5, (3.1)]{GuanZhouOptimal2015},
-with defining function the disk coordinate divided by $R$.
-
-To check applicability, embed the restricted projective family into
-projective space over the disk and remove a relative hyperplane
-section that does not contain the central fibre. The complement is
-Stein. A holomorphic section of $L$ on this Stein space can be chosen
-nonzero at a point of the central fibre; deleting its zero divisor
-trivializes $L$ and leaves a Stein space. The central fibre is smooth
-and connected, hence irreducible, so both deletions leave a dense
-open subset of it. For a zero-dimensional fibre choose the deletions
-to avoid its point. In the resulting trivialization the weight is
-psh, and the sharp extension theorem applies to canonical forms.
-The extension extends across the deleted hypersurfaces in reverse
-order by local $L^2$ removability: in every original frame the metric
-is bounded below by a smooth positive metric. The extension retains
-its norm bound and its prescribed restriction to the central fibre.
-
-After dividing $E$ by the base differential, the rank-one direct
-image identifies it with $b(t)e$, where $b$ is holomorphic and
-$b(0)=1$. Thus the disk average of $|b(t)|^2I'(t)$ is at most $I'(0)$.
-Jensen's inequality and the submean inequality for $\log|b|^2$ give
-\[
- \log I'(0)
- \geq\operatorname{avg}_{\Delta_R}
-             \bigl(\log|b(t)|^2+\log I'(t)\bigr)
- \geq\operatorname{avg}_{\Delta_R}\log I'(t).
-\]
-The same argument applies to every sufficiently small disk at every
-centre and in every complex line direction. Hence $-\log I'=\psi+\ell$ is psh, which proves
-\eqref{prog:base-presentation-open-positivity}.
-
-\paragraph{Extension over the base.}
-Let $D$ be a prime divisor in $W\setminus W^\circ$. Choose a component
-$Q$ of $q^*D$ dominating $D$. When $D$ is not $\mu$-exceptional, choose
-$Q$ not $p$-exceptional: take a component over the corresponding prime
-on $Z$ already on $X$ and then its strict transform. Put
-\[
- u=\operatorname{ord}_Q(q^*D),\qquad c=\operatorname{coeff}_Q(C).
-\]
-At a general point of $Q$, avoiding all other relevant divisors,
-there are local coordinates with
-\[
- t_1=z_1^u,\qquad t_j=z_j\quad(2\leq j\leq b).
-\]
-Indeed the map $Q\to D$ is submersive at a general smooth point,
-and a local unit in the first equation is absorbed by a holomorphic
-$u$th root. Such points can be chosen above every point of a dense
-analytic open subset of $D$ by generic smoothness and properness.
-Integration over a fixed small polydisk in the remaining fibre
-coordinates gives
-\begin{equation}\label{prog:base-presentation-divisor-bound}
- I(t)\geq c_*|t_1|^{-2(c+u-1)/u}
-\end{equation}
-for some $c_*>0$ near each such base point. The section and metric
-contribute the exponent $c$, while division by the base differential
-in the relative canonical form contributes $u-1$.
-
-For a nonexceptional $D$ the chosen $Q$ has $c\geq0$, so
-\eqref{prog:base-presentation-divisor-bound} bounds $\psi$ from above
-near its general points. For the finitely many exceptional divisors,
-the same estimate supplies a nonnegative integer $m$ such that, if
-$k$ is a smooth metric on $\mathcal O_W(F)$ and $s_F$ its divisor
-section,
-\[
- \varphi=\psi+m\log|s_F|_k^2
-\]
-is locally bounded above near general points of every divisor.
-Adding local potentials of $\rho+m c_1(k)-a\omega_W$ to
-$\varphi$ gives psh weights by
-\eqref{prog:base-presentation-open-positivity}. These weights
-extend across the divisors and then across the remaining analytic
-set of codimension at least two, giving
-\begin{equation}\label{prog:base-presentation-positive-current}
- S=\rho+m c_1(k)+dd^c\varphi\geq a\omega_W
-\end{equation}
-on all of $W$.
-
-\paragraph{A K\"ahler carrier.}
-Regularize the current in
-\eqref{prog:base-presentation-positive-current} in its class, retaining
-a lower bound for the potentials. More precisely, choose an
-approximation with analytic singularities such that
-\begin{equation}\label{prog:base-presentation-regularization}
- S'=\rho+m c_1(k)+dd^c\varphi',\qquad
- S'\geq\frac a2\omega_W,\qquad
- \varphi'\geq\varphi-O(1).
-\end{equation}
-These simultaneous properties are the Bergman regularization
-properties of \cite{DemaillyRegularization1992}; see also
-\cite[pp.~59--60]{PopoviciRegularization2004}.
-Resolve the analytic singularities and the support of $F$ by a
-sequence of blowups with smooth centres $r:T\to W$, with simultaneous
-SNC support, and write
-\[
- r^*S'=[H]+\zeta,\qquad H\geq0.
-\]
-The residual current $\zeta$ has locally bounded potentials. Removing
-the divisorial part from the positive current
-$r^*S'-(a/2)r^*\omega_W$ shows that
-\[
- \zeta\geq\frac a2r^*\omega_W.
-\]
-The difference has vanishing Lelong numbers, so its class is nef by
-regularization. Choose an effective $r$-exceptional divisor $G$ with
-$-G$ relatively ample. For $0<\eta\ll1$, the class
-$(a/2)r^*\omega_W-\eta[G]$ is K\"ahler. Therefore
-\begin{equation}\label{prog:base-presentation-carrier-data}
- \begin{split}
- \tau&=\mu r,\qquad
- \beta_T=[\zeta]-\eta[G],\\
- C_T&=H+\eta G-mr^*F-K_{T/W}
- \end{split}
-\end{equation}
-has K\"ahler $\beta_T$. If $r$ is an isomorphism, omit $G$ and its
-correction. The choice of $\eta$ will be decreased once more below.
-
-The exact class calculation is
-\[
- \begin{split}
- [K_T+C_T]+\beta_T
- &=r^*c_1(K_W)+[H]+[\zeta]-m[r^*F]\\
- &=r^*(c_1(K_W)+[\rho])=\tau^*\gamma.
- \end{split}
-\]
-All possibly negative terms in $C_T$ are $\tau$-exceptional.
-Consequently $B_Z=\tau_*C_T$ is effective. Specify the nef b-data by
-$\mathbf N_T=\beta_T$; the displayed identity defines generalized
-data on $Z$ with adjoint class $\gamma$. At the level of currents,
-pushing the identity gives the required trace identity on $Z$:
-exceptional divisor currents push to zero, and a $dd^c$ difference
-pushes to a $dd^c$ difference. Canonical traces can be computed using
-local meromorphic canonical frames; a change of frame changes the
-corresponding Chern potentials by the usual logarithmic transition
-term. Thus only the whole adjoint, rather
-than its individual traces, needs to be a Bott--Chern class. The
-class $\beta_T$ plus the strict transform of $B_Z$ is big on $T$,
-so its pushforward $B_Z+\mathbf N_Z$ is globally modified big.
-These are~\eqref{prog:base-presentation-identities}.
-
-\paragraph{The multiplier ideal.}
-Take a local holomorphic function $v\in f_*\mathcal J_1$. By the
-pushforward description of $\mathcal J_1$,
-\begin{equation}\label{prog:base-presentation-source-orders}
- \operatorname{div}(q^*\mu^*v)\geq\lfloor C\rfloor.
-\end{equation}
-Thus $q^*\mu^*v\,e$, tensored with a local base canonical frame,
-is a holomorphic $L$-valued top form on $U$. Its squared norm is
-locally integrable by the fractional SNC weights in
-\eqref{prog:base-presentation-curvature}. Over the submersion locus,
-fibre integration followed by base integration is precisely this
-top-form integral. Properness and a finite coordinate covering
-therefore give
-\begin{equation}\label{prog:base-presentation-base-integral}
- \int |\mu^*v|^2e^{-\psi}<\infty
-\end{equation}
-locally on $W$; the integral is taken where the original fibre
-metric is defined.
-
-The lower bound on $\varphi'$ in
-\eqref{prog:base-presentation-regularization} preserves this
-integrability after replacing $\psi$ by
-$\varphi'-m\log|s_F|_k^2$. Change variables through $r$. Along a
-prime divisor of $T$, let $h,j,k_D,d$ be respectively the coefficients
-of $H$, $\operatorname{div}(\tau^*v)$, $K_{T/W}$, and $r^*F$.
-The local integrand has exponent
-$2(j-h+md+k_D)$, up to a bounded positive factor. Its integrability
-implies $j-h+md+k_D>-1$, or, since $j$ is integral,
-\[
- j\geq\lfloor h-md-k_D\rfloor.
-\]
-Therefore
-\[
- \operatorname{div}(\tau^*v)
-       \geq\lfloor H-mr^*F-K_{T/W}\rfloor.
-\]
-Choose $\eta>0$ sufficiently small that adding $\eta G$ does not
-change any of these finitely many round-downs. The resulting
-inequality is exactly
-$\operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor$, proving
-\eqref{prog:base-presentation-ideal} as an inclusion of ideal sheaves.
-In particular, if $f_*\mathcal J_1$ is the unit ideal on an open set,
-then so is $\mathcal J_Z$, and the base data are gklt there.
-
-Finally, the nef data descend to $T$, and the structure boundaries
-on higher models are defined by crepant pullback of
-\eqref{prog:base-presentation-identities}. On every common refinement
-of the source and base carriers, both adjoints are the pullback of
-$\gamma$. Thus the identities remain exact on higher models.
-\end{proof}
-
-\begin{lemma}[Contractions of closed subspaces]
-\label{prog:base-emc}
-Let $D$ be a compact complex space, possibly nonreduced, and let
-$\delta\in H^2(D,\R)$ have nonnegative degree on compact curves.
-Use the definitions of a Moishezon morphism and of a contraction
-defined by a class in \cite[Definitions~2.32 and~2.34]{HX}.
-Then the following hold.
-\begin{enumerate}
-\item If $\delta$ is endowed with a Moishezon contraction, so is
-its restriction to every closed analytic subspace of $D$.
-\item Let $g:D\to V$ be a projective contraction of compact
-complex spaces, so $g_*\OO_D=\OO_V$, and suppose
-$\delta=g^*\epsilon$ for a class $\epsilon\in H^2(V,\R)$
-having nonnegative degree on compact curves. Then $\delta$ is
-endowed with a Moishezon contraction if and only if $\epsilon$
-is endowed with one.
-\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-Write $h:D\to S$ for a Moishezon contraction defined by
-$\delta$. For a closed subspace $D_0\subset D$, take the Stein
-factorization
-\[
-                 D_0\longrightarrow S_0\longrightarrow S
-\]
-of its restriction. The first map is a contraction, and the second
-is finite. Thus a curve in $D_0$ is contracted by the first map
-exactly when its $\delta$-degree is zero. Choose a projective
-surjection $P\to D$ witnessing that $h$ is Moishezon, with
-$P\to S$ projective. The closed subspace
-$P_0=P\times_D D_0$ is projective over $S$ and projective
-and surjective over $D_0$. It is also projective over $S_0$:
-a relatively ample line for $P_0\to S$ remains ample on the
-fibres of $P_0\to S_0$. This proves the first assertion,
-including its scheme structure.
-
-For the descent direction of the second assertion, every curve
-in a fibre of $g$ has $\delta$-degree zero. A connected
-projective complex space is connected by chains of compact curves
-after passing to its reduction. Therefore $h$ is constant on
-the underlying set of each fibre of $g$. It factors as
-$h=k\circ g$ for a holomorphic map $k:V\to S$.
-To see that this factorization holds also for nonreduced spaces,
-choose a coordinate neighbourhood in $S$ of the image of a fibre.
-Properness permits restriction to a neighbourhood of its point
-in $V$ whose inverse image maps into that coordinate
-neighbourhood. The coordinate functions of $h$ then descend
-uniquely through $g_*\OO_D=\OO_V$; their analytic relations
-descend through the same equality. These local factorizations
-glue. Furthermore,
-\[
-                 k_*\OO_V=h_*\OO_D=\OO_S.
-\]
-Every irreducible compact curve $C\subset V$ is dominated by
-a compact curve $C'\subset D$. Indeed, after base change to
-the normalization of $C$, take a component dominating that curve
-and a projective multisection. The projection formula gives
-\[
-       \delta\cdot C'
-          =\deg(C'\to C)\,\epsilon\cdot C.
-\]
-Consequently $k$ contracts exactly the curves of
-$\epsilon$-degree zero. The composite $P\to D\to V$
-is a projective surjection, and $P\to S$ is projective, so
-it witnesses that $k$ is Moishezon.
-
-Conversely, suppose $k:V\to S$ is a Moishezon contraction
-defined by $\epsilon$. Then $k\circ g$ is a contraction,
-and the projection formula verifies its curve criterion, treating
-separately curves contracted by $g$. If $Q\to V$ is a
-projective surjection witnessing that $k$ is Moishezon, then
-$D\times_V Q\to D$ is projective and surjective, while
-$D\times_V Q\to Q\to S$ is projective. This supplies the
-required witness.
-\end{proof}
-
-\begin{lemma}[Crepant transport of the multiplier closed subspace]
-\label{prog:multiplier-transport}
-Let $g:X\to Y$ be a projective bimeromorphic morphism of
-normal compact K\"ahler spaces. Suppose effective generalized
-pair data on $X$ and $Y$ have the same globally nef b-part
-and are crepant over $Y$. In particular their adjoint classes
-$A_X$ and $A_Y$ satisfy $A_X=g^*A_Y$ in Bott--Chern
-cohomology. Write $J_X,J_Y$ for their multiplier ideals and
-$N_X=V(J_X)$, $N_Y=V(J_Y)$ for their non-klt closed
-subspaces. Then
-\[
-       g_*J_X=J_Y,\qquad R^ig_*J_X=0\quad(i>0),
-       \qquad g_*\OO_{N_X}=\OO_{N_Y}.
-\]
-The restriction $N_X\to N_Y$ is a projective contraction.
-Hence, whenever $A_Y$ is nef, the restriction of $A_X$ to
-$N_X$ is endowed with a Moishezon contraction if and only if
-the restriction of $A_Y$ to $N_Y$ is endowed with one.
-\end{lemma}
-
-\begin{proof}
-Take a projective common log resolution $r:U\to X$ carrying
-the nef datum, and put $s=g\circ r$. By crepancy there is
-one real SNC divisor $C$ such that
-\[
-            K_U+[C]+\beta=s^*A_Y=r^*A_X,
-\]
-where $\beta$ is globally nef. The multiplier ideals are
-\[
- J_X=r_*\OO_U(-\lfloor C\rfloor),\qquad
- J_Y=s_*\OO_U(-\lfloor C\rfloor).
-\]
-Their direct-image equality follows at once. For vanishing, set
-$L=\OO_U(-\lfloor C\rfloor)$ and
-$\Delta=C-\lfloor C\rfloor$. The pair $(U,\Delta)$
-is klt, and the actual real line
-\[
-                  L-(K_U+\Delta)=-K_U-C
-\]
-has class $\beta-s^*A_Y$. It is relatively nef over $Y$
-and relatively big because $s$ is bimeromorphic. The same
-statements hold over $X$. Relative Kawamata--Viehweg vanishing,
-in the form of \cite[Theorem~1.2]{FujinoVanishing2025} with
-the first morphism the identity and the real line
-$\mathcal H=-K_U-C$, therefore gives
-\[
- R^is_*L=R^ir_*L=0\qquad(i>0).
-\]
-The real-line identity in that theorem is the displayed definition
-of $\mathcal H$; its relative positivity is checked by the class
-identity above. The klt condition means that there are no proper log
-canonical strata to test for relative log bigness. Leray gives
-$R^ig_*J_X=0$ for $i>0$. Pushing forward
-$0\to J_X\to\OO_X\to\OO_{N_X}\to0$ and using
-$g_*\OO_X=\OO_Y$ now proves the asserted structure-sheaf
-equality. It also identifies the scheme-theoretic image with
-$N_Y$, giving the projective contraction between these closed
-subspaces. Apply Lemma~\ref{prog:base-emc} to conclude.
-\end{proof}
-
-\begin{remark}
-The preceding lemma also transports the contraction hypothesis
-through a crepant projective flip: apply it to the two contractions
-to their common base, equipped with the descended adjoint and
-the common nef b-data. On that base the boundary is the effective
-pushforward of the source boundary, and the nef data remain on
-the same carrier; the descended Bott--Chern adjoint and its
-identity on a common resolution define the crepant base data
-in the whole-adjoint and current-trace category used here.
-All assertions concern the fixed adjoint
-and its multiplier ideal. No conclusion about the multiplier
-ideal of a different driving adjoint is used.
-\end{remark}
-
-\begin{corollary}[The contraction step over a lower-dimensional base]
-\label{prog:base-emc-induction}
-Use the hypotheses and notation of
-Proposition~\ref{prog:base-presentation}. Suppose that
-$A=f^*\gamma$ is nef and that its restriction to the original
-non-klt closed subspace $N_o=V(J_o)$ is endowed with a
-Moishezon contraction. Put
-\[
-                I=f_*J_1,\qquad N_*=V(I)\subset Z.
-\]
-Then $\gamma|_{N_*}$ is endowed with a Moishezon contraction.
-If the prepared non-klt closed subspace $N_1=V(J_1)$ dominates
-$Z$, then $\gamma$ and $A$ are endowed with Moishezon
-contractions. Otherwise the generalized pair supplied on $Z$
-has a non-klt closed subspace endowed with the contraction
-defined by the restricted class of $\gamma$. In this latter
-case, if $\gamma$ is nef and NQC and the assertion
-$\mathsf C_{\dim Z}$ is available, then $\gamma$ and $A$
-are endowed with Moishezon contractions.
-\end{corollary}
-
-\begin{proof}
-The inclusion $J_o\subset J_1$ gives $N_1\subset N_o$
-as closed analytic subspaces. By
-Lemma~\ref{prog:base-emc}, the given contraction restricts to
-$N_1$. Proposition~\ref{prog:base-presentation} identifies
-\[
-              f_*\OO_{N_1}=\OO_Z/I=\OO_{N_*}.
-\]
-Thus the restriction $N_1\to N_*$ is a projective
-contraction. The restricted adjoint is the pullback of
-$\gamma|_{N_*}$, so Lemma~\ref{prog:base-emc} descends
-the contraction to $N_*$. Nonnegative degrees on curves of
-$N_*$ follow from projective multisections and nefness of
-$A$.
-
-If $N_1$ dominates the normal irreducible space $Z$, then
-$I=0$: a local function in $I$ pulls back to zero on the
-dominating closed subspace and hence vanishes generically on
-$Z$. Normality, and in particular reducedness, gives its
-vanishing. Therefore $N_*=Z$, which proves the assertion in
-this case.
-
-If $N_1$ does not dominate $Z$, the constructed base ideal
-satisfies $I\subset J_Z$. Hence
-\[
-                 V(J_Z)\subset V(I)=N_*
-\]
-as closed analytic subspaces. Restriction by
-Lemma~\ref{prog:base-emc} provides the required contraction
-on $V(J_Z)$, with its complete scheme structure. The
-generalized pair on $Z$ has globally nef carrier data and
-globally modified-big total boundary by
-Proposition~\ref{prog:base-presentation}. When the stated
-nef and NQC conditions hold, these facts are precisely the
-hypotheses of $\mathsf C_{\dim Z}$, which gives the
-contraction on $Z$. Finally Lemma~\ref{prog:base-emc}
-pulls it back through the projective contraction $f$ to $X$.
-The case of an empty non-klt closed subspace has the same
-interpretation, with its contraction condition vacuous.
-\end{proof}
+\subsubsection{Base presentations and multiplier ideals}
+
+The following constructions take place in the category of generalized
+data specified above. In particular, the whole adjoint is a
+Bott--Chern class, whereas the boundary and nef traces on a singular
+space are interpreted as currents. On a smooth carrier the structure
+boundary is a real divisor; its negative components may be exceptional
+over the space of the pair. The nef part is specified on that carrier
+and pulled back to higher models.
+
+\begin{lemma}[Adjoint-preserving preparation]
+\label{prog:base-presentation-preparation}
+Let $(X,B+\mathbf M)$ be a generalized pair on a normal compact
+K\"ahler space, with globally modified-big total boundary
+$B+\mathbf M_X$. Write
+\[
+ A=[K_X+B+\mathbf M_X],\qquad
+ \mathcal J_o=\mathcal J(X,B+\mathbf M).
+\]
+There are generalized data on $X$ with the same adjoint class, a
+projective carrier $p:U\to X$ with $U$ smooth and compact K\"ahler, an SNC real
+divisor $C$ whose negative components are $p$-exceptional, and a
+K\"ahler class $\beta$ such that
+\begin{equation}\label{prog:base-presentation-prepared}
+ p^*A=[K_U+C]+\beta,\qquad
+ \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor)
+                 \supseteq\mathcal J_o.
+\end{equation}
+The carrier may be required to dominate any prescribed projective
+modification of $X$, with the structure and exceptional divisors
+having simultaneous SNC support. The prepared total boundary is
+globally modified big. In particular, the preparation preserves the
+gklt condition.
+\end{lemma}
+
+\begin{proof}
+Choose a common projective log resolution $p_0:U_0\to X$ of the
+original nef carrier and the prescribed modification, and write
+\[
+ p_0^*A=[K_{U_0}+C^0]+\mathbf M_{U_0}.
+\]
+The class $\mathbf M_{U_0}$ is nef, and the negative part $(C^0)^-$
+is exceptional over $X$. The modified-big preparation
+\cite[Lemmas~2.10 and~2.23]{HX} can be made with the prescribed
+resolution. We give its divisor and ideal details. Choose an effective divisor $E_0$ with sufficiently large coefficients
+and full $p_0$-exceptional support so that the class
+\[
+ [(C^0)^+]+\mathbf M_{U_0}+[E_0]
+\]
+is big. Resolve a Kodaira decomposition of this class. On a further
+projective modification $a:U\to U_0$ with $U$ smooth one obtains
+\[
+ a^*\bigl([(C^0)^+]+\mathbf M_{U_0}+[E_0]\bigr)
+       =\kappa+[D],\qquad D\geq0,
+\]
+where $\kappa$ is K\"ahler. The support can be made SNC together
+with all the other divisors. On any additional blowup, subtract a
+sufficiently small effective exceptional class from the pulled-back
+K\"ahler class and add it to the divisor; an exceptional divisor
+whose negative is relatively ample makes the resulting class
+K\"ahler again.
+
+Put $p=p_0a$ and
+\[
+ C^{\mathrm{alt}}=D-a^*(C^0)^--a^*E_0-K_{U/U_0}.
+\]
+Then
+\[
+ p^*A=[K_U+C^{\mathrm{alt}}]+\kappa,
+\]
+and every negative component of $C^{\mathrm{alt}}$ is
+$p$-exceptional. Let $C^{\mathrm{old}}$ be the crepant structure
+boundary of the original data on this same model. For $0<t\ll1$,
+set
+\[
+ C=(1-t)C^{\mathrm{old}}+tC^{\mathrm{alt}},\qquad
+ \beta=(1-t)a^*\mathbf M_{U_0}+t\kappa.
+\]
+The class $\beta$ is K\"ahler, the adjoint identity is unchanged,
+and negative coefficients still occur only on exceptional divisors.
+There are only finitely many coefficients. Thus $t$ can be chosen so
+that
+\[
+ \lfloor C\rfloor\leq\lfloor C^{\mathrm{old}}\rfloor.
+\]
+Pushing forward the corresponding inclusion of invertible sheaves
+gives~\eqref{prog:base-presentation-prepared}. The K\"ahler class
+$\beta$ plus the strict transform of the effective trace boundary is
+big; its pushforward is the prepared total boundary. If the original pair
+is gklt, its multiplier ideal is the unit ideal, and hence so is
+$\mathcal J_1$.
+\end{proof}
+
+\begin{proposition}[Base presentations for projective contractions]
+\label{prog:base-presentation}
+Let $f:X\to Z$ be a projective contraction between normal irreducible
+compact K\"ahler spaces. Suppose that $(X,B+\mathbf M)$ has globally
+modified-big total boundary and that
+\[
+ A=[K_X+B+\mathbf M_X]=f^*\gamma,
+ \qquad \gamma\in H^{1,1}_{\mathrm{BC}}(Z).
+\]
+There is an adjoint-preserving preparation as in
+Lemma~\ref{prog:base-presentation-preparation}, with multiplier ideal
+$\mathcal J_1\supseteq\mathcal J_o$, for which
+\begin{equation}\label{prog:base-presentation-Stein}
+ R^if_*\mathcal J_1=0\quad(i>0),\qquad
+ f_*\mathcal O_{N_1}=\mathcal O_Z/(f_*\mathcal J_1),
+ \qquad N_1=V(\mathcal J_1).
+\end{equation}
+If $N_1$ dominates $Z$, then $f_*\mathcal J_1=0$.
+Otherwise there are generalized data $(Z,B_Z+\mathbf N)$, a
+projective carrier $\tau:T\to Z$ with $T$ smooth and compact K\"ahler, a real
+SNC structure divisor $C_T$, and a K\"ahler class $\beta_T$ such that
+\begin{equation}\label{prog:base-presentation-identities}
+ \begin{gathered}
+ [K_Z+B_Z+\mathbf N_Z]=\gamma,\qquad
+ \tau^*\gamma=[K_T+C_T]+\beta_T,\qquad
+ \mathbf N_T=\beta_T,\\
+ B_Z=\tau_*C_T\geq0,\qquad
+ B_Z+\mathbf N_Z\ \text{is globally modified big}.
+ \end{gathered}
+\end{equation}
+The negative components of $C_T$ are $\tau$-exceptional, and
+\begin{equation}\label{prog:base-presentation-ideal}
+ \mathcal J_Z:=\mathcal J(Z,B_Z+\mathbf N)
+       =\tau_*\mathcal O_T(-\lfloor C_T\rfloor)
+       \supseteq f_*\mathcal J_1.
+\end{equation}
+In particular the base data are gklt wherever $f_*\mathcal J_1$ is
+the unit ideal. If the source data are gklt, the base data are gklt.
+All adjoint identities are preserved by pullback to higher models.
+\end{proposition}
+
+\begin{proof}
+Choose a projective resolution $\mu:W\to Z$, with $W$ smooth
+compact K\"ahler and reduced exceptional divisor $F$ having SNC
+support. Apply Lemma~\ref{prog:base-presentation-preparation} on a
+carrier dominating the main component of the base change. We obtain
+projective maps $p:U\to X$ and $q:U\to W$ with $fp=\mu q$ and
+\begin{equation}\label{prog:base-presentation-source}
+ [K_U+C]+\beta=p^*A=q^*\mu^*\gamma,
+ \qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor).
+\end{equation}
+Here $U$ is smooth compact K\"ahler, $\beta$ is K\"ahler, and all
+negative components of $C$ are $p$-exceptional. The maps are
+projective by composition and factorization over the separated base
+$W$. The map $q$ has connected fibres: this holds over the open set
+where $\mu$ is an isomorphism, and its Stein factorization is finite
+and bimeromorphic over the normal space $W$.
+
+The integral divisor $-\lfloor C\rfloor$ has class
+\[
+ [-\lfloor C\rfloor]
+   =[K_U+\{C\}]+\beta-(fp)^*\gamma.
+\]
+The pair $(U,\{C\})$ is klt with SNC boundary. Relative
+Kawamata--Viehweg vanishing, first for $p$ and then for $fp$, applies
+because the positive class modulo pullbacks is K\"ahler
+\cite[Theorem~2.2]{HX}. Equivalently, the real line bundle
+$-K_U-C$ is relatively ample,
+so the vanishing follows from \cite[Theorem~1.2]{FujinoVanishing2025}
+with the identity as its first morphism. Leray gives $R^if_*\mathcal J_1=0$ for
+$i>0$. Applying $f_*$ to the sequence defining $N_1$, and using
+$f_*\mathcal O_X=\mathcal O_Z$, proves
+\eqref{prog:base-presentation-Stein}. If $N_1$ dominates $Z$, the
+ideal $f_*\mathcal J_1\subseteq\mathcal O_Z$ vanishes generically
+and therefore vanishes. This proves that alternative.
+
+Suppose from now on that $N_1$ does not dominate $Z$. Every
+$q$-horizontal coefficient of $C$ is then less than one. Indeed,
+a component with coefficient at least one forces the multiplier
+ideal to be proper along its image in $X$, and its image would make
+$N_1$ dominate $Z$. If $Z$ is a point, the asserted base presentation
+is the dimension-zero presentation. We therefore assume
+$b=\dim Z>0$.
+
+\paragraph{The fibre metric.}
+Choose a smooth representative $\rho$ of
+$\mu^*\gamma-c_1(K_W)$, a K\"ahler form $\omega_W$, and a
+K\"ahler form representing $\beta$, still denoted by $\beta$.
+For some $a>0$,
+\[
+ \beta\geq a q^*\omega_W.
+\]
+Consider the holomorphic line bundle
+\[
+ L=\mathcal O_U(-K_{U/W}-\lfloor C\rfloor).
+\]
+Equation~\eqref{prog:base-presentation-source} gives a singular
+Hermitian metric $h$ on $L$ whose curvature current is
+\begin{equation}\label{prog:base-presentation-curvature}
+ \Theta_h(L)=\beta+[\{C\}]-q^*\rho.
+\end{equation}
+Its weights are smooth functions plus the divisorial logarithms
+with coefficients in $\{C\}$. We normalize $dd^c$ so that
+$dd^c\log|z|^2=[z=0]$ and write squared-norm weights as $e^{-\varphi}$.
+Let $e$ be the canonical meromorphic section of
+\[
+ K_{U/W}\otimes L=\mathcal O_U(-\lfloor C\rfloor),
+\]
+so that $\operatorname{div}(e)=-\lfloor C\rfloor$.
+
+Choose an analytic Zariski open set $W^\circ\subseteq W$ on which
+$q$ is a submersion, the horizontal boundary is relatively SNC,
+there are no vertical boundary components, $\mu$ is an isomorphism
+onto an open subset of $Z$, and coherent base change holds in degree
+zero. On $q^{-1}(W^\circ)$, the divisor $-\lfloor C\rfloor$ is
+effective and exceptional over $X$. Normality and the contraction
+property therefore identify
+\[
+ q_*(K_{U/W}\otimes L)|_{W^\circ}
+       =\mathcal O_{W^\circ}e.
+\]
+Here $K_{U/W}$ is viewed as the relative canonical line bundle. The
+fibre norm
+\[
+ I(w)=\int_{U_w}|e_w|_h^2
+\]
+is positive and finite. The fractional SNC weights are integrable;
+any zeros of $e_w$ only improve integrability. Relative SNC
+coordinates and a partition of unity show that $I$ is smooth:
+parameter derivatives act on smooth factors multiplying fixed
+integrable monomial weights.
+
+Set $\psi=-\log I$ on $W^\circ$. Then
+\begin{equation}\label{prog:base-presentation-open-positivity}
+ \rho+dd^c\psi\geq a\omega_W.
+\end{equation}
+Here are the details for the singular weights. Locally choose
+$\ell$ with $dd^c\ell=\rho-a\omega_W$, and put
+$h'=he^{-q^*\ell}$. Its weights are psh, and its fibre norm satisfies
+$I'=e^{-\ell}I$. Restrict the family to a small coordinate line disk
+$\Delta_R$, with centre $0$. The sharp $L^2$ extension theorem gives
+a twisted top form extending $e_0$ and satisfying
+\begin{equation}\label{prog:base-presentation-sharp-extension}
+ \int_{q^{-1}(\Delta_R)}|E|_{h'}^2
+        \leq\pi R^2 I'(0).
+\end{equation}
+Top-form norms here are normalized to use Euclidean area in the
+base coordinate. We use the Stein hypersurface form of the optimal
+extension theorem \cite[Theorem~2.2 and \S3.5, (3.1)]{GuanZhouOptimal2015},
+with defining function the disk coordinate divided by $R$.
+
+To check applicability, embed the restricted projective family into
+projective space over the disk and remove a relative hyperplane
+section that does not contain the central fibre. The complement is
+Stein. A holomorphic section of $L$ on this Stein space can be chosen
+nonzero at a point of the central fibre; deleting its zero divisor
+trivializes $L$ and leaves a Stein space. The central fibre is smooth
+and connected, hence irreducible, so both deletions leave a dense
+open subset of it. For a zero-dimensional fibre choose the deletions
+to avoid its point. In the resulting trivialization the weight is
+psh, and the sharp extension theorem applies to canonical forms.
+The extension extends across the deleted hypersurfaces in reverse
+order by local $L^2$ removability: in every original frame the metric
+is bounded below by a smooth positive metric. The extension retains
+its norm bound and its prescribed restriction to the central fibre.
+
+After dividing $E$ by the base differential, the rank-one direct
+image identifies it with $b(t)e$, where $b$ is holomorphic and
+$b(0)=1$. Thus the disk average of $|b(t)|^2I'(t)$ is at most $I'(0)$.
+Jensen's inequality and the submean inequality for $\log|b|^2$ give
+\[
+ \log I'(0)
+ \geq\operatorname{avg}_{\Delta_R}
+             \bigl(\log|b(t)|^2+\log I'(t)\bigr)
+ \geq\operatorname{avg}_{\Delta_R}\log I'(t).
+\]
+The same argument applies to every sufficiently small disk at every
+centre and in every complex line direction. Hence $-\log I'=\psi+\ell$ is psh, which proves
+\eqref{prog:base-presentation-open-positivity}.
+
+\paragraph{Extension over the base.}
+Let $D$ be a prime divisor in $W\setminus W^\circ$. Choose a component
+$Q$ of $q^*D$ dominating $D$. When $D$ is not $\mu$-exceptional, choose
+$Q$ not $p$-exceptional: take a component over the corresponding prime
+on $Z$ already on $X$ and then its strict transform. Put
+\[
+ u=\operatorname{ord}_Q(q^*D),\qquad c=\operatorname{coeff}_Q(C).
+\]
+At a general point of $Q$, avoiding all other relevant divisors,
+there are local coordinates with
+\[
+ t_1=z_1^u,\qquad t_j=z_j\quad(2\leq j\leq b).
+\]
+Indeed the map $Q\to D$ is submersive at a general smooth point,
+and a local unit in the first equation is absorbed by a holomorphic
+$u$th root. Such points can be chosen above every point of a dense
+analytic open subset of $D$ by generic smoothness and properness.
+Integration over a fixed small polydisk in the remaining fibre
+coordinates gives
+\begin{equation}\label{prog:base-presentation-divisor-bound}
+ I(t)\geq c_*|t_1|^{-2(c+u-1)/u}
+\end{equation}
+for some $c_*>0$ near each such base point. The section and metric
+contribute the exponent $c$, while division by the base differential
+in the relative canonical form contributes $u-1$.
+
+For a nonexceptional $D$ the chosen $Q$ has $c\geq0$, so
+\eqref{prog:base-presentation-divisor-bound} bounds $\psi$ from above
+near its general points. For the finitely many exceptional divisors,
+the same estimate supplies a nonnegative integer $m$ such that, if
+$k$ is a smooth metric on $\mathcal O_W(F)$ and $s_F$ its divisor
+section,
+\[
+ \varphi=\psi+m\log|s_F|_k^2
+\]
+is locally bounded above near general points of every divisor.
+Adding local potentials of $\rho+m c_1(k)-a\omega_W$ to
+$\varphi$ gives psh weights by
+\eqref{prog:base-presentation-open-positivity}. These weights
+extend across the divisors and then across the remaining analytic
+set of codimension at least two, giving
+\begin{equation}\label{prog:base-presentation-positive-current}
+ S=\rho+m c_1(k)+dd^c\varphi\geq a\omega_W
+\end{equation}
+on all of $W$.
+
+\paragraph{A K\"ahler carrier.}
+Regularize the current in
+\eqref{prog:base-presentation-positive-current} in its class, retaining
+a lower bound for the potentials. More precisely, choose an
+approximation with analytic singularities such that
+\begin{equation}\label{prog:base-presentation-regularization}
+ S'=\rho+m c_1(k)+dd^c\varphi',\qquad
+ S'\geq\frac a2\omega_W,\qquad
+ \varphi'\geq\varphi-O(1).
+\end{equation}
+These simultaneous properties are the Bergman regularization
+properties of \cite{DemaillyRegularization1992}; see also
+\cite[pp.~59--60]{PopoviciRegularization2004}.
+Resolve the analytic singularities and the support of $F$ by a
+sequence of blowups with smooth centres $r:T\to W$, with simultaneous
+SNC support, and write
+\[
+ r^*S'=[H]+\zeta,\qquad H\geq0.
+\]
+The residual current $\zeta$ has locally bounded potentials. Removing
+the divisorial part from the positive current
+$r^*S'-(a/2)r^*\omega_W$ shows that
+\[
+ \zeta\geq\frac a2r^*\omega_W.
+\]
+The difference has vanishing Lelong numbers, so its class is nef by
+regularization. Choose an effective $r$-exceptional divisor $G$ with
+$-G$ relatively ample. For $0<\eta\ll1$, the class
+$(a/2)r^*\omega_W-\eta[G]$ is K\"ahler. Therefore
+\begin{equation}\label{prog:base-presentation-carrier-data}
+ \begin{split}
+ \tau&=\mu r,\qquad
+ \beta_T=[\zeta]-\eta[G],\\
+ C_T&=H+\eta G-mr^*F-K_{T/W}
+ \end{split}
+\end{equation}
+has K\"ahler $\beta_T$. If $r$ is an isomorphism, omit $G$ and its
+correction. The choice of $\eta$ will be decreased once more below.
+
+The exact class calculation is
+\[
+ \begin{split}
+ [K_T+C_T]+\beta_T
+ &=r^*c_1(K_W)+[H]+[\zeta]-m[r^*F]\\
+ &=r^*(c_1(K_W)+[\rho])=\tau^*\gamma.
+ \end{split}
+\]
+All possibly negative terms in $C_T$ are $\tau$-exceptional.
+Consequently $B_Z=\tau_*C_T$ is effective. Specify the nef b-data by
+$\mathbf N_T=\beta_T$; the displayed identity defines generalized
+data on $Z$ with adjoint class $\gamma$. At the level of currents,
+pushing the identity gives the required trace identity on $Z$:
+exceptional divisor currents push to zero, and a $dd^c$ difference
+pushes to a $dd^c$ difference. Canonical traces can be computed using
+local meromorphic canonical frames; a change of frame changes the
+corresponding Chern potentials by the usual logarithmic transition
+term. Thus only the whole adjoint, rather
+than its individual traces, needs to be a Bott--Chern class. The
+class $\beta_T$ plus the strict transform of $B_Z$ is big on $T$,
+so its pushforward $B_Z+\mathbf N_Z$ is globally modified big.
+These are~\eqref{prog:base-presentation-identities}.
+
+\paragraph{The multiplier ideal.}
+Take a local holomorphic function $v\in f_*\mathcal J_1$. By the
+pushforward description of $\mathcal J_1$,
+\begin{equation}\label{prog:base-presentation-source-orders}
+ \operatorname{div}(q^*\mu^*v)\geq\lfloor C\rfloor.
+\end{equation}
+Thus $q^*\mu^*v\,e$, tensored with a local base canonical frame,
+is a holomorphic $L$-valued top form on $U$. Its squared norm is
+locally integrable by the fractional SNC weights in
+\eqref{prog:base-presentation-curvature}. Over the submersion locus,
+fibre integration followed by base integration is precisely this
+top-form integral. Properness and a finite coordinate covering
+therefore give
+\begin{equation}\label{prog:base-presentation-base-integral}
+ \int |\mu^*v|^2e^{-\psi}<\infty
+\end{equation}
+locally on $W$; the integral is taken where the original fibre
+metric is defined.
+
+The lower bound on $\varphi'$ in
+\eqref{prog:base-presentation-regularization} preserves this
+integrability after replacing $\psi$ by
+$\varphi'-m\log|s_F|_k^2$. Change variables through $r$. Along a
+prime divisor of $T$, let $h,j,k_D,d$ be respectively the coefficients
+of $H$, $\operatorname{div}(\tau^*v)$, $K_{T/W}$, and $r^*F$.
+The local integrand has exponent
+$2(j-h+md+k_D)$, up to a bounded positive factor. Its integrability
+implies $j-h+md+k_D>-1$, or, since $j$ is integral,
+\[
+ j\geq\lfloor h-md-k_D\rfloor.
+\]
+Therefore
+\[
+ \operatorname{div}(\tau^*v)
+       \geq\lfloor H-mr^*F-K_{T/W}\rfloor.
+\]
+Choose $\eta>0$ sufficiently small that adding $\eta G$ does not
+change any of these finitely many round-downs. The resulting
+inequality is exactly
+$\operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor$, proving
+\eqref{prog:base-presentation-ideal} as an inclusion of ideal sheaves.
+In particular, if $f_*\mathcal J_1$ is the unit ideal on an open set,
+then so is $\mathcal J_Z$, and the base data are gklt there.
+
+Finally, the nef data descend to $T$, and the structure boundaries
+on higher models are defined by crepant pullback of
+\eqref{prog:base-presentation-identities}. On every common refinement
+of the source and base carriers, both adjoints are the pullback of
+$\gamma$. Thus the identities remain exact on higher models.
+\end{proof}
+
+\begin{lemma}[Contractions of closed subspaces]
+\label{prog:base-emc}
+Let $D$ be a compact complex space, possibly nonreduced, and let
+$\delta\in H^2(D,\R)$ have nonnegative degree on compact curves.
+Use the definitions of a Moishezon morphism and of a contraction
+defined by a class in \cite[Definitions~2.32 and~2.34]{HX}.
+Then the following hold.
+\begin{enumerate}
+\item If $\delta$ is endowed with a Moishezon contraction, so is
+its restriction to every closed analytic subspace of $D$.
+\item Let $g:D\to V$ be a projective contraction of compact
+complex spaces, so $g_*\OO_D=\OO_V$, and suppose
+$\delta=g^*\epsilon$ for a class $\epsilon\in H^2(V,\R)$
+having nonnegative degree on compact curves. Then $\delta$ is
+endowed with a Moishezon contraction if and only if $\epsilon$
+is endowed with one.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Write $h:D\to S$ for a Moishezon contraction defined by
+$\delta$. For a closed subspace $D_0\subset D$, take the Stein
+factorization
+\[
+                 D_0\longrightarrow S_0\longrightarrow S
+\]
+of its restriction. The first map is a contraction, and the second
+is finite. Thus a curve in $D_0$ is contracted by the first map
+exactly when its $\delta$-degree is zero. Choose a projective
+surjection $P\to D$ witnessing that $h$ is Moishezon, with
+$P\to S$ projective. The closed subspace
+$P_0=P\times_D D_0$ is projective over $S$ and projective
+and surjective over $D_0$. It is also projective over $S_0$:
+a relatively ample line for $P_0\to S$ remains ample on the
+fibres of $P_0\to S_0$. This proves the first assertion,
+including its scheme structure.
+
+For the descent direction of the second assertion, every curve
+in a fibre of $g$ has $\delta$-degree zero. A connected
+projective complex space is connected by chains of compact curves
+after passing to its reduction. Therefore $h$ is constant on
+the underlying set of each fibre of $g$. It factors as
+$h=k\circ g$ for a holomorphic map $k:V\to S$.
+To see that this factorization holds also for nonreduced spaces,
+choose a coordinate neighbourhood in $S$ of the image of a fibre.
+Properness permits restriction to a neighbourhood of its point
+in $V$ whose inverse image maps into that coordinate
+neighbourhood. The coordinate functions of $h$ then descend
+uniquely through $g_*\OO_D=\OO_V$; their analytic relations
+descend through the same equality. These local factorizations
+glue. Furthermore,
+\[
+                 k_*\OO_V=h_*\OO_D=\OO_S.
+\]
+Every irreducible compact curve $C\subset V$ is dominated by
+a compact curve $C'\subset D$. Indeed, after base change to
+the normalization of $C$, take a component dominating that curve
+and a projective multisection. The projection formula gives
+\[
+       \delta\cdot C'
+          =\deg(C'\to C)\,\epsilon\cdot C.
+\]
+Consequently $k$ contracts exactly the curves of
+$\epsilon$-degree zero. The composite $P\to D\to V$
+is a projective surjection, and $P\to S$ is projective, so
+it witnesses that $k$ is Moishezon.
+
+Conversely, suppose $k:V\to S$ is a Moishezon contraction
+defined by $\epsilon$. Then $k\circ g$ is a contraction,
+and the projection formula verifies its curve criterion, treating
+separately curves contracted by $g$. If $Q\to V$ is a
+projective surjection witnessing that $k$ is Moishezon, then
+$D\times_V Q\to D$ is projective and surjective, while
+$D\times_V Q\to Q\to S$ is projective. This supplies the
+required witness.
+\end{proof}
+
+\begin{lemma}[Crepant transport of the multiplier closed subspace]
+\label{prog:multiplier-transport}
+Let $g:X\to Y$ be a projective bimeromorphic morphism of
+normal compact K\"ahler spaces. Suppose effective generalized
+pair data on $X$ and $Y$ have the same globally nef b-part
+and are crepant over $Y$. In particular their adjoint classes
+$A_X$ and $A_Y$ satisfy $A_X=g^*A_Y$ in Bott--Chern
+cohomology. Write $J_X,J_Y$ for their multiplier ideals and
+$N_X=V(J_X)$, $N_Y=V(J_Y)$ for their non-klt closed
+subspaces. Then
+\[
+       g_*J_X=J_Y,\qquad R^ig_*J_X=0\quad(i>0),
+       \qquad g_*\OO_{N_X}=\OO_{N_Y}.
+\]
+The restriction $N_X\to N_Y$ is a projective contraction.
+Hence, whenever $A_Y$ is nef, the restriction of $A_X$ to
+$N_X$ is endowed with a Moishezon contraction if and only if
+the restriction of $A_Y$ to $N_Y$ is endowed with one.
+\end{lemma}
+
+\begin{proof}
+Take a projective common log resolution $r:U\to X$ carrying
+the nef datum, and put $s=g\circ r$. By crepancy there is
+one real SNC divisor $C$ such that
+\[
+            K_U+[C]+\beta=s^*A_Y=r^*A_X,
+\]
+where $\beta$ is globally nef. The multiplier ideals are
+\[
+ J_X=r_*\OO_U(-\lfloor C\rfloor),\qquad
+ J_Y=s_*\OO_U(-\lfloor C\rfloor).
+\]
+Their direct-image equality follows at once. For vanishing, set
+$L=\OO_U(-\lfloor C\rfloor)$ and
+$\Delta=C-\lfloor C\rfloor$. The pair $(U,\Delta)$
+is klt, and the actual real line
+\[
+                  L-(K_U+\Delta)=-K_U-C
+\]
+has class $\beta-s^*A_Y$. It is relatively nef over $Y$
+and relatively big because $s$ is bimeromorphic. The same
+statements hold over $X$. Relative Kawamata--Viehweg vanishing,
+in the form of \cite[Theorem~1.2]{FujinoVanishing2025} with
+the first morphism the identity and the real line
+$\mathcal H=-K_U-C$, therefore gives
+\[
+ R^is_*L=R^ir_*L=0\qquad(i>0).
+\]
+The real-line identity in that theorem is the displayed definition
+of $\mathcal H$; its relative positivity is checked by the class
+identity above. The klt condition means that there are no proper log
+canonical strata to test for relative log bigness. Leray gives
+$R^ig_*J_X=0$ for $i>0$. Pushing forward
+$0\to J_X\to\OO_X\to\OO_{N_X}\to0$ and using
+$g_*\OO_X=\OO_Y$ now proves the asserted structure-sheaf
+equality. It also identifies the scheme-theoretic image with
+$N_Y$, giving the projective contraction between these closed
+subspaces. Apply Lemma~\ref{prog:base-emc} to conclude.
+\end{proof}
+
+\begin{remark}
+The preceding lemma also transports the contraction hypothesis
+through a crepant projective flip: apply it to the two contractions
+to their common base, equipped with the descended adjoint and
+the common nef b-data. On that base the boundary is the effective
+pushforward of the source boundary, and the nef data remain on
+the same carrier; the descended Bott--Chern adjoint and its
+identity on a common resolution define the crepant base data
+in the whole-adjoint and current-trace category used here.
+All assertions concern the fixed adjoint
+and its multiplier ideal. No conclusion about the multiplier
+ideal of a different driving adjoint is used.
+\end{remark}
+
+\begin{corollary}[The contraction step over a lower-dimensional base]
+\label{prog:base-emc-induction}
+Use the hypotheses and notation of
+Proposition~\ref{prog:base-presentation}. Suppose that
+$A=f^*\gamma$ is nef and that its restriction to the original
+non-klt closed subspace $N_o=V(J_o)$ is endowed with a
+Moishezon contraction. Put
+\[
+                I=f_*J_1,\qquad N_*=V(I)\subset Z.
+\]
+Then $\gamma|_{N_*}$ is endowed with a Moishezon contraction.
+If the prepared non-klt closed subspace $N_1=V(J_1)$ dominates
+$Z$, then $\gamma$ and $A$ are endowed with Moishezon
+contractions. Otherwise the generalized pair supplied on $Z$
+has a non-klt closed subspace endowed with the contraction
+defined by the restricted class of $\gamma$. In this latter
+case, if $\gamma$ is nef and NQC and the assertion
+$\mathsf C_{\dim Z}$ is available, then $\gamma$ and $A$
+are endowed with Moishezon contractions.
+\end{corollary}
+
+\begin{proof}
+The inclusion $J_o\subset J_1$ gives $N_1\subset N_o$
+as closed analytic subspaces. By
+Lemma~\ref{prog:base-emc}, the given contraction restricts to
+$N_1$. Proposition~\ref{prog:base-presentation} identifies
+\[
+              f_*\OO_{N_1}=\OO_Z/I=\OO_{N_*}.
+\]
+Thus the restriction $N_1\to N_*$ is a projective
+contraction. The restricted adjoint is the pullback of
+$\gamma|_{N_*}$, so Lemma~\ref{prog:base-emc} descends
+the contraction to $N_*$. Nonnegative degrees on curves of
+$N_*$ follow from projective multisections and nefness of
+$A$.
+
+If $N_1$ dominates the normal irreducible space $Z$, then
+$I=0$: a local function in $I$ pulls back to zero on the
+dominating closed subspace and hence vanishes generically on
+$Z$. Normality, and in particular reducedness, gives its
+vanishing. Therefore $N_*=Z$, which proves the assertion in
+this case.
+
+If $N_1$ does not dominate $Z$, the constructed base ideal
+satisfies $I\subset J_Z$. Hence
+\[
+                 V(J_Z)\subset V(I)=N_*
+\]
+as closed analytic subspaces. Restriction by
+Lemma~\ref{prog:base-emc} provides the required contraction
+on $V(J_Z)$, with its complete scheme structure. The
+generalized pair on $Z$ has globally nef carrier data and
+globally modified-big total boundary by
+Proposition~\ref{prog:base-presentation}. When the stated
+nef and NQC conditions hold, these facts are precisely the
+hypotheses of $\mathsf C_{\dim Z}$, which gives the
+contraction on $Z$. Finally Lemma~\ref{prog:base-emc}
+pulls it back through the projective contraction $f$ to $X$.
+The case of an empty non-klt closed subspace has the same
+interpretation, with its contraction condition vacuous.
+\end{proof}

preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/build/sections/05-projective-program-input.tex

@@ -1,206 +1,206 @@
-\subsection{Projective programs under the assumption}
-\label{rank:projective-program-input}
-
-The argument for a projective base uses a terminating program to reach a nef
-adjoint and, after decreasing the nef data, a program ending in a Mori fiber
-space. The conditional input is termination in the pseudo-effective case.
-We first state the exact program result and then derive that case from the
-projective good models in Proposition~\ref{proj:good-models}.
-
-\begin{proposition}[A conditional generalized program]
-\label{proj:generalized-mmp}
-Assume Assumption~\ref{asm:log-iitaka}. Let
-\((V,B+\boldsymbol M)\) be a projective generalized dlt rational pair,
-where \(V\) is \(\Q\)-factorial, \(B\geq0\), and the nef b-divisor
-\(\boldsymbol M\) is determined by a nef rational Cartier divisor on a
-projective birational model. Put \(D=K_V+B+M_V\).
-
-There is a choice of \(D\)-MMP with ample scaling which terminates. If
-\(D\) is pseudo-effective, its endpoint \(V_m\) has nef adjoint \(D_m\).
-If \(D\) is not pseudo-effective, its endpoint has a Mori fiber contraction
-to a projective variety of smaller dimension. The birational steps and their
-endpoints stay \(\Q\)-factorial generalized dlt, and the forward birational
-map extracts no divisors.
-In the pseudo-effective case, on a common resolution one has an actual
-rational divisor identity
-\[
- u^*D=v^*D_m+F,\qquad F\geq0\text{ and }v\text{-exceptional}.
-\]
-\end{proposition}
-
-For the pseudo-effective alternative, we first derive the relative weak
-Zariski decompositions used by the minimal-model existence theorem.
-
-For a projective morphism, first replace its image by its normal Stein
-factor. We use the relative convention of \cite[Section~2]{TsakanikasXie}:
-a divisor is pseudo-effective over the base when its restriction to a very
-general fiber of this surjective morphism is pseudo-effective. An NQC weak Zariski decomposition of \(D\) over the
-base is a birational equality \(h^*D\equiv_Z P+N\), where \(P\) is a
-nonnegative real combination of relatively nef rational Cartier divisors
-and \(N\geq0\). The decomposition below has rational Cartier parts.
-
-\begin{lemma}[Relative canonical decomposition]\label{proj:relative-wzd}
-Assume Assumption~\ref{asm:log-iitaka}. Let \(V\) be a smooth
-quasi-projective complex variety and let \(f:V\to Z\) be projective, with
-\(Z\) normal and quasi-projective. If \(K_V\) is pseudo-effective over
-\(Z\), then on a normal variety \(Y\) with a projective birational morphism \(h:Y\to V\) there
-are rational Cartier divisors \(P,N\) such that
-\[
- h^*K_V\equiv_Z P+N,\qquad P\text{ nef over }Z,\qquad N\geq0.
-\]
-\end{lemma}
-
-\begin{proof}
-We use the construction in the proof of
-\cite[Proposition~3.3]{External037}. Its absolute input is a nef-plus-
-effective rational decomposition for the canonical divisor of a smooth
-projective variety with pseudo-effective canonical class. This is supplied
-here by Proposition~\ref{proj:good-models} and its common-resolution
-comparison.
-
-Replace the image of \(f\) by its normal Stein factor. This does not change
-relative numerical classes or contracted curves, and now \(f\) is
-surjective with connected fibers. Its very general fiber is smooth
-projective with pseudo-effective canonical divisor, hence is non-uniruled
-by \cite[Theorem~0.2 and Corollary~0.3]{BDPP}. Compactify the projective
-morphism and resolve away from \(V\). We obtain
-\(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and
-\(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\),
-\(V\) is already projective and the absolute input proves the lemma.
-
-Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth
-projective resolution. Choose a sufficiently positive very ample divisor
-\(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems
-on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes
-\(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double
-covers
-\[
- \pi_Z:Z^\sharp\to\widetilde Z,
- \qquad \pi:V^\sharp\to\bar V
-\]
-are smooth integral projective varieties. The canonical formulas are
-\[
- K_{Z^\sharp}\sim_{\Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z),
- \qquad
- K_{V^\sharp}\sim_{\Q}\pi^*(K_{\bar V}+\bar f^*A_Z).
-\]
-The first line is big when \(A_Z\) is sufficiently positive. The rational
-map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers
-as \(f\). A covering family of rational curves on \(V^\sharp\) would
-either dominate a covering family of rational curves on \(Z^\sharp\), or
-cover its very general fibers by vertical rational curves. Both are
-impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is
-pseudo-effective by BDPP.
-
-Apply the absolute decomposition to \(V^\sharp\). Moving the rational
-principal difference in the canonical formula into the nef part gives an
-actual rational divisor decomposition for a pullback of
-\(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of
-that model and its conjugate under the covering involution. On this smooth
-projective resolution \(g:T\to V^\sharp\), average the two decompositions.
-We obtain
-\[
- g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N,
- \qquad \bar P\text{ nef},\quad \bar N\geq0,
-\]
-where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors.
-
-Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant
-composite \(\pi g\) induces a projective birational morphism
-\(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant
-rational divisor descends by dividing its coefficient at an upstairs prime
-by the ramification index. The descended divisor is rational Cartier when
-the original divisor is: after clearing denominators, choose one Cartier
-equation on a semilocal neighborhood of the finite orbit above a point
-(a Cartier divisor is principal on a semilocal ring), and multiply all its
-translates. The product is invariant and has divisor
-equal to the group order times the original divisor. It descends to a local
-equation for a multiple of the downstairs divisor. This argument includes
-stabilizers and ramification primes.
-
-Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier
-divisors on \(\bar Y\). Lifting curves through the finite map shows that
-\(P\) is nef, and coefficientwise descent shows that \(N\) is effective.
-Finite pullback is injective on rational divisors, so the displayed identity
-descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled
-back from \(A_Z\) is numerically trivial over \(Z\), giving
-\(h^*K_V\equiv_ZP+N\), as required.
-\end{proof}
-
-\begin{proof}[Proof of Proposition~\ref{proj:generalized-mmp}]
-Lemma~\ref{proj:relative-wzd} and
-\cite[Theorem~5.2]{TsakanikasXie}, applied to the generalized pair
-\((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties
-by dimension induction. The hypothesis of that theorem is relative smooth
-minimal-model existence one dimension lower; the lemma supplies its other
-hypothesis, an NQC weak Zariski decomposition. The induction begins in
-dimension zero. Then \cite[Theorem~5.4]{TsakanikasXie} supplies minimal
-models for every pseudo-effective NQC generalized lc pair in the relevant
-dimension. Their Theorem~2.7 gives existence in the Birkar--Shokurov sense.
-
-The nef data in the statement are NQC, because a nef rational Cartier
-divisor is a positive multiple of a nef Cartier divisor. In the
-pseudo-effective case the preceding paragraph therefore gives the
-Birkar--Shokurov model required by
-\cite[Theorem~4.2]{TsakanikasXie}. In the other case, the non-pseudo-effective
-alternative of that theorem applies directly. To meet its scaling
-hypothesis, take an effective sufficiently positive ample rational divisor
-\(A\) whose general components have sufficiently small coefficients. On
-a fixed log resolution carrying \(\boldsymbol M\), these components are
-transverse to the boundary and preserve generalized log canonicity, while
-their ample class can be chosen large enough that \(D+A\) is nef. The
-theorem supplies a terminating MMP starting on \(V\).
-
-We verify the step types inductively.  Suppose \(V_i\) is
-\(\Q\)-factorial generalized dlt.  By
-\cite[Remark~2.15]{TsakanikasXie}, each birational step has the form
-\[
- V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1},
-\]
-where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective.
-Suppose \(g_i\) is divisorial, and write \(R\) for its ray.  An exceptional
-prime \(E\) has \(E\cdot R<0\): otherwise it would be effective,
-exceptional, and \(g_i\)-nef, contrary to negativity.  There is only one
-exceptional prime.  Indeed, a combination of two distinct exceptional
-primes can be chosen to have degree zero on \(R\); negativity applied to
-both signs of that combination would make it zero.
-
-We show directly that \(Z_i\) is \(\Q\)-factorial.  For a prime divisor
-\(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a
-curve \(C\) spanning \(R\), and put
-\[
-                 a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\Q.
-\]
-Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is
-Cartier.  Its line has degree zero on \(R\), so
-\cite[Theorem~1.5]{Xie2022} gives an actual Cartier line \(\mathcal L_T\)
-on \(Z_i\) and an isomorphism
-\[
- \OO_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T.
-\]
-On an open set trivializing \(\mathcal L_T\), the Cartier divisor
-\(m(\widetilde T+aE)\) is principal upstairs.  Pushing this principal
-divisor identity down through the birational \(g_i\) gives a principal
-divisor \(mT\) on that open set.
-Thus \(T\) is rational Cartier.  It follows that \(h_i\) is an isomorphism:
-the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier
-on \(Z_i\), and its pullback is the original divisor because \(h_i\)
-is small.  It has degree zero on every contracted curve, so relative
-ampleness rules out positive-dimensional fibers; a finite birational
-map to the normal \(Z_i\) is an isomorphism.  If \(g_i\) is small,
-the relatively ample canonical model \(h_i\) is the flip of
-\cite[Definition~3.2]{HanLi}.  Thus the chosen birational steps are
-divisorial contractions and flips.  The underlying variety of a
-generalized dlt pair is klt, and these steps preserve the
-\(\Q\)-factorial generalized dlt category
-\cite[Remark~2.3 and Lemma~3.7]{HanLi}.  They introduce no prime
-divisors on the new models. For completeness, on a common resolution
-of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\),
-taking the difference of the two generalized discrepancy formulas cancels
-their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new
-model and anti-nef over it. The negativity lemma makes this difference
-effective. Composing these actual rational divisor comparisons gives
-\(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint.
-The endpoints in the two cases are the nef and Mori fiber endpoints supplied
-by the terminating program theorem.
-\end{proof}
+\subsection{Projective programs under the assumption}
+\label{rank:projective-program-input}
+
+The argument for a projective base uses a terminating program to reach a nef
+adjoint and, after decreasing the nef data, a program ending in a Mori fiber
+space. The conditional input is termination in the pseudo-effective case.
+We first state the exact program result and then derive that case from the
+projective good models in Proposition~\ref{proj:good-models}.
+
+\begin{proposition}[A conditional generalized program]
+\label{proj:generalized-mmp}
+Assume Assumption~\ref{asm:log-iitaka}. Let
+\((V,B+\boldsymbol M)\) be a projective generalized dlt rational pair,
+where \(V\) is \(\Q\)-factorial, \(B\geq0\), and the nef b-divisor
+\(\boldsymbol M\) is determined by a nef rational Cartier divisor on a
+projective birational model. Put \(D=K_V+B+M_V\).
+
+There is a choice of \(D\)-MMP with ample scaling which terminates. If
+\(D\) is pseudo-effective, its endpoint \(V_m\) has nef adjoint \(D_m\).
+If \(D\) is not pseudo-effective, its endpoint has a Mori fiber contraction
+to a projective variety of smaller dimension. The birational steps and their
+endpoints stay \(\Q\)-factorial generalized dlt, and the forward birational
+map extracts no divisors.
+In the pseudo-effective case, on a common resolution one has an actual
+rational divisor identity
+\[
+ u^*D=v^*D_m+F,\qquad F\geq0\text{ and }v\text{-exceptional}.
+\]
+\end{proposition}
+
+For the pseudo-effective alternative, we first derive the relative weak
+Zariski decompositions used by the minimal-model existence theorem.
+
+For a projective morphism, first replace its image by its normal Stein
+factor. We use the relative convention of \cite[Section~2]{TsakanikasXie}:
+a divisor is pseudo-effective over the base when its restriction to a very
+general fiber of this surjective morphism is pseudo-effective. An NQC weak Zariski decomposition of \(D\) over the
+base is a birational equality \(h^*D\equiv_Z P+N\), where \(P\) is a
+nonnegative real combination of relatively nef rational Cartier divisors
+and \(N\geq0\). The decomposition below has rational Cartier parts.
+
+\begin{lemma}[Relative canonical decomposition]\label{proj:relative-wzd}
+Assume Assumption~\ref{asm:log-iitaka}. Let \(V\) be a smooth
+quasi-projective complex variety and let \(f:V\to Z\) be projective, with
+\(Z\) normal and quasi-projective. If \(K_V\) is pseudo-effective over
+\(Z\), then on a normal variety \(Y\) with a projective birational morphism \(h:Y\to V\) there
+are rational Cartier divisors \(P,N\) such that
+\[
+ h^*K_V\equiv_Z P+N,\qquad P\text{ nef over }Z,\qquad N\geq0.
+\]
+\end{lemma}
+
+\begin{proof}
+We use the construction in the proof of
+\cite[Proposition~3.3]{External037}. Its absolute input is a nef-plus-
+effective rational decomposition for the canonical divisor of a smooth
+projective variety with pseudo-effective canonical class. This is supplied
+here by Proposition~\ref{proj:good-models} and its common-resolution
+comparison.
+
+Replace the image of \(f\) by its normal Stein factor. This does not change
+relative numerical classes or contracted curves, and now \(f\) is
+surjective with connected fibers. Its very general fiber is smooth
+projective with pseudo-effective canonical divisor, hence is non-uniruled
+by \cite[Theorem~0.2 and Corollary~0.3]{BDPP}. Compactify the projective
+morphism and resolve away from \(V\). We obtain
+\(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and
+\(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\),
+\(V\) is already projective and the absolute input proves the lemma.
+
+Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth
+projective resolution. Choose a sufficiently positive very ample divisor
+\(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems
+on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes
+\(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double
+covers
+\[
+ \pi_Z:Z^\sharp\to\widetilde Z,
+ \qquad \pi:V^\sharp\to\bar V
+\]
+are smooth integral projective varieties. The canonical formulas are
+\[
+ K_{Z^\sharp}\sim_{\Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z),
+ \qquad
+ K_{V^\sharp}\sim_{\Q}\pi^*(K_{\bar V}+\bar f^*A_Z).
+\]
+The first line is big when \(A_Z\) is sufficiently positive. The rational
+map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers
+as \(f\). A covering family of rational curves on \(V^\sharp\) would
+either dominate a covering family of rational curves on \(Z^\sharp\), or
+cover its very general fibers by vertical rational curves. Both are
+impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is
+pseudo-effective by BDPP.
+
+Apply the absolute decomposition to \(V^\sharp\). Moving the rational
+principal difference in the canonical formula into the nef part gives an
+actual rational divisor decomposition for a pullback of
+\(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of
+that model and its conjugate under the covering involution. On this smooth
+projective resolution \(g:T\to V^\sharp\), average the two decompositions.
+We obtain
+\[
+ g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N,
+ \qquad \bar P\text{ nef},\quad \bar N\geq0,
+\]
+where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors.
+
+Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant
+composite \(\pi g\) induces a projective birational morphism
+\(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant
+rational divisor descends by dividing its coefficient at an upstairs prime
+by the ramification index. The descended divisor is rational Cartier when
+the original divisor is: after clearing denominators, choose one Cartier
+equation on a semilocal neighborhood of the finite orbit above a point
+(a Cartier divisor is principal on a semilocal ring), and multiply all its
+translates. The product is invariant and has divisor
+equal to the group order times the original divisor. It descends to a local
+equation for a multiple of the downstairs divisor. This argument includes
+stabilizers and ramification primes.
+
+Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier
+divisors on \(\bar Y\). Lifting curves through the finite map shows that
+\(P\) is nef, and coefficientwise descent shows that \(N\) is effective.
+Finite pullback is injective on rational divisors, so the displayed identity
+descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled
+back from \(A_Z\) is numerically trivial over \(Z\), giving
+\(h^*K_V\equiv_ZP+N\), as required.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{proj:generalized-mmp}]
+Lemma~\ref{proj:relative-wzd} and
+\cite[Theorem~5.2]{TsakanikasXie}, applied to the generalized pair
+\((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties
+by dimension induction. The hypothesis of that theorem is relative smooth
+minimal-model existence one dimension lower; the lemma supplies its other
+hypothesis, an NQC weak Zariski decomposition. The induction begins in
+dimension zero. Then \cite[Theorem~5.4]{TsakanikasXie} supplies minimal
+models for every pseudo-effective NQC generalized lc pair in the relevant
+dimension. Their Theorem~2.7 gives existence in the Birkar--Shokurov sense.
+
+The nef data in the statement are NQC, because a nef rational Cartier
+divisor is a positive multiple of a nef Cartier divisor. In the
+pseudo-effective case the preceding paragraph therefore gives the
+Birkar--Shokurov model required by
+\cite[Theorem~4.2]{TsakanikasXie}. In the other case, the non-pseudo-effective
+alternative of that theorem applies directly. To meet its scaling
+hypothesis, take an effective sufficiently positive ample rational divisor
+\(A\) whose general components have sufficiently small coefficients. On
+a fixed log resolution carrying \(\boldsymbol M\), these components are
+transverse to the boundary and preserve generalized log canonicity, while
+their ample class can be chosen large enough that \(D+A\) is nef. The
+theorem supplies a terminating MMP starting on \(V\).
+
+We verify the step types inductively.  Suppose \(V_i\) is
+\(\Q\)-factorial generalized dlt.  By
+\cite[Remark~2.15]{TsakanikasXie}, each birational step has the form
+\[
+ V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1},
+\]
+where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective.
+Suppose \(g_i\) is divisorial, and write \(R\) for its ray.  An exceptional
+prime \(E\) has \(E\cdot R<0\): otherwise it would be effective,
+exceptional, and \(g_i\)-nef, contrary to negativity.  There is only one
+exceptional prime.  Indeed, a combination of two distinct exceptional
+primes can be chosen to have degree zero on \(R\); negativity applied to
+both signs of that combination would make it zero.
+
+We show directly that \(Z_i\) is \(\Q\)-factorial.  For a prime divisor
+\(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a
+curve \(C\) spanning \(R\), and put
+\[
+                 a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\Q.
+\]
+Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is
+Cartier.  Its line has degree zero on \(R\), so
+\cite[Theorem~1.5]{Xie2022} gives an actual Cartier line \(\mathcal L_T\)
+on \(Z_i\) and an isomorphism
+\[
+ \OO_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T.
+\]
+On an open set trivializing \(\mathcal L_T\), the Cartier divisor
+\(m(\widetilde T+aE)\) is principal upstairs.  Pushing this principal
+divisor identity down through the birational \(g_i\) gives a principal
+divisor \(mT\) on that open set.
+Thus \(T\) is rational Cartier.  It follows that \(h_i\) is an isomorphism:
+the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier
+on \(Z_i\), and its pullback is the original divisor because \(h_i\)
+is small.  It has degree zero on every contracted curve, so relative
+ampleness rules out positive-dimensional fibers; a finite birational
+map to the normal \(Z_i\) is an isomorphism.  If \(g_i\) is small,
+the relatively ample canonical model \(h_i\) is the flip of
+\cite[Definition~3.2]{HanLi}.  Thus the chosen birational steps are
+divisorial contractions and flips.  The underlying variety of a
+generalized dlt pair is klt, and these steps preserve the
+\(\Q\)-factorial generalized dlt category
+\cite[Remark~2.3 and Lemma~3.7]{HanLi}.  They introduce no prime
+divisors on the new models. For completeness, on a common resolution
+of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\),
+taking the difference of the two generalized discrepancy formulas cancels
+their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new
+model and anti-nef over it. The negativity lemma makes this difference
+effective. Composing these actual rational divisor comparisons gives
+\(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint.
+The endpoints in the two cases are the nef and Mori fiber endpoints supplied
+by the terminating program theorem.
+\end{proof}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/README.md

@@ -1,21 +1,21 @@
-# [Milne's rationality conjecture for abelian varieties](paper.pdf)
-
-**Author:** OpenAI
-
-**Date:** October 7, 2026
-
-## Citation
-
-```bibtex
-@misc{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026,
-  author = {{OpenAI}},
-  title = {{Milne's rationality conjecture for abelian varieties}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf}{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026}},
-  year = {2026}
-}
-```
-
-## Version note
-
-This version updates a citation to a revised manuscript in this collection; see the [previous version](../Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf).
+# [Milne's rationality conjecture for abelian varieties](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 7, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026,
+  author = {{OpenAI}},
+  title = {{Milne's rationality conjecture for abelian varieties}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf}{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026}},
+  year = {2026}
+}
+```
+
+## Version note
+
+This version updates a citation to a revised manuscript in this collection; see the [previous version](../Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf).

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/figures/two-transports.tex

@@ -1,28 +1,28 @@
-\begin{tikzpicture}[
- >=Latex, font=\small,
- variety/.style={draw=MidnightBlue!60,rounded corners=2pt,
-                 fill=MidnightBlue!3,align=center,minimum width=43mm,
-                 minimum height=11mm},
- every edge/.style={draw,->,thick},
- note/.style={font=\footnotesize,align=center}]
-\node[font=\bfseries] at (3,1.0) {CM transport};
-\node[variety] (A) at (0,0) {$A^n/\Qbar$};
-\node[variety] (C) at (6,0) {$C/\Qbar$\quad(CM)};
-\node[variety] (A0) at (0,-1.7) {$A_0^n$};
-\node[variety] (C0) at (6,-1.7) {$C_0$};
-\path (A) edge node[left,note] {reduction} (A0);
-\path (C) edge node[right,note] {reduction} (C0);
-\path (A0) edge node[above] {$a_{\mathrm{CM}}$} (C0);
-\node[note] at (3,-2.65) {$\gamma_0=(a_{\mathrm{CM}}j)^*\kappa_0$};
-\node[font=\bfseries] at (3,-3.3) {Prescribed-filtration transport};
-\node[variety] (Z) at (0,-4.4) {$Z^m/\Qbar$\quad(balanced Weil)};
-\node[variety] (B) at (6,-4.4) {$B/K$, $K/K_0$ finite};
-\node[variety] (Z0) at (0,-6.1) {$Z_0^m$};
-\node[variety] (B0) at (6,-6.1) {$B_0$};
-\path (Z) edge node[left,note] {reduction} (Z0);
-\path (B) edge node[right,note] {reduction} (B0);
-\path (Z0) edge node[above] {$a_{\mathrm{fl}}$} (B0);
-\node[note] at (3,-7.05)
- {$a_{\mathrm{fl}}^*\Fil^1H^1_{\mathrm{dR}}(B)
-       =(\mathcal F_Z^1)^{\oplus m}$};
-\end{tikzpicture}
+\begin{tikzpicture}[
+ >=Latex, font=\small,
+ variety/.style={draw=MidnightBlue!60,rounded corners=2pt,
+                 fill=MidnightBlue!3,align=center,minimum width=43mm,
+                 minimum height=11mm},
+ every edge/.style={draw,->,thick},
+ note/.style={font=\footnotesize,align=center}]
+\node[font=\bfseries] at (3,1.0) {CM transport};
+\node[variety] (A) at (0,0) {$A^n/\Qbar$};
+\node[variety] (C) at (6,0) {$C/\Qbar$\quad(CM)};
+\node[variety] (A0) at (0,-1.7) {$A_0^n$};
+\node[variety] (C0) at (6,-1.7) {$C_0$};
+\path (A) edge node[left,note] {reduction} (A0);
+\path (C) edge node[right,note] {reduction} (C0);
+\path (A0) edge node[above] {$a_{\mathrm{CM}}$} (C0);
+\node[note] at (3,-2.65) {$\gamma_0=(a_{\mathrm{CM}}j)^*\kappa_0$};
+\node[font=\bfseries] at (3,-3.3) {Prescribed-filtration transport};
+\node[variety] (Z) at (0,-4.4) {$Z^m/\Qbar$\quad(balanced Weil)};
+\node[variety] (B) at (6,-4.4) {$B/K$, $K/K_0$ finite};
+\node[variety] (Z0) at (0,-6.1) {$Z_0^m$};
+\node[variety] (B0) at (6,-6.1) {$B_0$};
+\path (Z) edge node[left,note] {reduction} (Z0);
+\path (B) edge node[right,note] {reduction} (B0);
+\path (Z0) edge node[above] {$a_{\mathrm{fl}}$} (B0);
+\node[note] at (3,-7.05)
+ {$a_{\mathrm{fl}}^*\Fil^1H^1_{\mathrm{dR}}(B)
+       =(\mathcal F_Z^1)^{\oplus m}$};
+\end{tikzpicture}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/main.tex

@@ -1,73 +1,73 @@
-\documentclass[11pt]{article}
-\usepackage[a4paper,margin=28mm]{geometry}
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{amsmath,amssymb,amsthm,mathtools}
-\usepackage{microtype}
-\usepackage{enumitem}
-\usepackage{booktabs}
-\usepackage[dvipsnames]{xcolor}
-\usepackage{tikz}
-\usetikzlibrary{arrows.meta,positioning}
-\usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue,breaklinks=true]{hyperref}
-\hypersetup{pdftitle={Milne's rationality conjecture for abelian varieties},
- pdfsubject={Rational pairings of specialized Hodge classes with Lefschetz classes},
- pdfauthor={OpenAI}}
-\setlength{\emergencystretch}{2em}
-\setlist[enumerate]{itemsep=3pt,topsep=5pt}
-\setlist[itemize]{itemsep=3pt,topsep=5pt}
-\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}
-\newcommand{\Q}{\mathbb Q}
-\newcommand{\Z}{\mathbb Z}
-\newcommand{\R}{\mathbb R}
-\newcommand{\C}{\mathbb C}
-\newcommand{\F}{\mathbb F}
-\newcommand{\Qbar}{\overline{\mathbb Q}}
-\newcommand{\Kzero}{K_0}
-\newcommand{\Cw}{C_w}
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-\DeclareMathOperator{\Lef}{Lef}
-\DeclareMathOperator{\NS}{NS}
-\DeclareMathOperator{\Gal}{Gal}
-\DeclareMathOperator{\id}{id}
-\DeclareMathOperator{\Spec}{Spec}
-\title{Milne's rationality conjecture\protect\\ for abelian varieties}
-\author{OpenAI}
-\date{October 7, 2026}
-\begin{document}
-\maketitle
-\vspace{-1em}
-\begin{abstract}
-We prove Milne's rationality conjecture for abelian varieties, including
-residue characteristic \(2\). After good reduction, the pairing of a
-rational Hodge class with any complementary product of divisor classes
-on the reduction is the same rational number in every prime-to-\(p\)
-realization and in crystalline cohomology. Using the Hodge theorem for CM
-abelian varieties, we also show that every such specialized Hodge class
-is represented by a single rational algebraic cycle in all these realizations.
-\end{abstract}
-\input{sections/00_intro}
-\input{sections/01_preliminaries}
-\input{sections/02_transport}
-\input{sections/03_filtration}
-\input{sections/04_weil}
-\input{sections/05_reduction}
-\input{references}
-\end{document}
+\documentclass[11pt]{article}
+\usepackage[a4paper,margin=28mm]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs}
+\usepackage[dvipsnames]{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,urlcolor=MidnightBlue,breaklinks=true]{hyperref}
+\hypersetup{pdftitle={Milne's rationality conjecture for abelian varieties},
+ pdfsubject={Rational pairings of specialized Hodge classes with Lefschetz classes},
+ pdfauthor={OpenAI}}
+\setlength{\emergencystretch}{2em}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\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}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\F}{\mathbb F}
+\newcommand{\Qbar}{\overline{\mathbb Q}}
+\newcommand{\Kzero}{K_0}
+\newcommand{\Cw}{C_w}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\Fil}{Fil}
+\DeclareMathOperator{\Res}{Res}
+\DeclareMathOperator{\GSp}{GSp}
+\DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\SL}{SL}
+\DeclareMathOperator{\SU}{SU}
+\DeclareMathOperator{\MT}{MT}
+\DeclareMathOperator{\Lef}{Lef}
+\DeclareMathOperator{\NS}{NS}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\Spec}{Spec}
+\title{Milne's rationality conjecture\protect\\ for abelian varieties}
+\author{OpenAI}
+\date{October 7, 2026}
+\begin{document}
+\maketitle
+\vspace{-1em}
+\begin{abstract}
+We prove Milne's rationality conjecture for abelian varieties, including
+residue characteristic \(2\). After good reduction, the pairing of a
+rational Hodge class with any complementary product of divisor classes
+on the reduction is the same rational number in every prime-to-\(p\)
+realization and in crystalline cohomology. Using the Hodge theorem for CM
+abelian varieties, we also show that every such specialized Hodge class
+is represented by a single rational algebraic cycle in all these realizations.
+\end{abstract}
+\input{sections/00_intro}
+\input{sections/01_preliminaries}
+\input{sections/02_transport}
+\input{sections/03_filtration}
+\input{sections/04_weil}
+\input{sections/05_reduction}
+\input{references}
+\end{document}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/references.tex

@@ -1,274 +1,274 @@
-\begin{thebibliography}{Stacks}
-
-\bibitem[And92]{Andre1992}
-Y.~Andr\'e,
-\emph{Une remarque \`a propos des cycles de Hodge de type CM},
-in S.~David (ed.), \emph{S\'eminaire de Th\'eorie des Nombres,
-Paris, 1989--90}, Progress in Math. \textbf{102},
-Birkh\"auser Boston, Boston, MA, 1992, 1--7.
-
-\bibitem[And96]{Andre1996}
-Y.~Andr\'e,
-\emph{Pour une th\'eorie inconditionnelle des motifs},
-Publ. Math. Inst. Hautes \'{E}tudes Sci. \textbf{83} (1996), 5--49.
-\href{https://doi.org/10.1007/BF02698643}{doi:10.1007/BF02698643}.
-\href{https://www.numdam.org/item/10.1007/BF02698643.pdf}{Original article}.
-
-\bibitem[Bla94]{Blasius1994}
-D.~Blasius,
-\emph{A $p$-adic property of Hodge classes on abelian varieties},
-in \emph{Motives} (Seattle, WA, 1991), Part~2,
-U.~Jannsen, S.~Kleiman, and J.-P.~Serre (eds.),
-Proc. Sympos. Pure Math. \textbf{55}, Part~2,
-American Mathematical Society, Providence, RI, 1994, 293--308.
-\href{https://doi.org/10.1090/pspum/055.2/1265557}{doi:10.1090/pspum/055.2/1265557}.
-\href{https://archive.mpim-bonn.mpg.de/id/eprint/3859/1/preprint_1992_59.pdf}{Author's preprint}.
-
-\bibitem[CF00]{ColmezFontaine2000}
-P.~Colmez and J.-M.~Fontaine,
-\emph{Construction des repr\'esentations $p$-adiques semi-stables},
-Invent. Math. \textbf{140} (2000), no.~1, 1--43.
-\href{https://doi.org/10.1007/s002220000042}{doi:10.1007/s002220000042}.
-\href{https://www.imo.universite-paris-saclay.fr/~fontaine/CF.pdf}{Author's version}.
-
-\bibitem[Con03]{Conrad2003}
-B.~Conrad,
-\emph{Minimal models for elliptic curves}, notes, 20 October 2003.
-\href{https://math.stanford.edu/~conrad/vigregroup/vigre03/minimalmodel.pdf}{Author's notes}.
-
-\bibitem[Del82]{Deligne1982}
-P.~Deligne,
-\emph{Hodge cycles on abelian varieties} (notes by J.~S.~Milne),
-in P.~Deligne, J.~S.~Milne, A.~Ogus, and K.-y.~Shih,
-\emph{Hodge Cycles, Motives, and Shimura Varieties},
-Lecture Notes in Math. \textbf{900}, Springer-Verlag,
-Berlin--New York, 1982, 9--100.
-\href{https://www.jmilne.org/math/Documents/Deligne82.pdf}{Corrected online version}.
-
-\bibitem[FR05]{FontaineRapoport2005}
-J.-M.~Fontaine and M.~Rapoport,
-\emph{Existence de filtrations admissibles sur des isocristaux},
-Bull. Soc. Math. France \textbf{133} (2005), no.~1, 73--86.
-\href{https://doi.org/10.24033/bsmf.2479}{doi:10.24033/bsmf.2479}.
-\href{https://www.numdam.org/article/BSMF_2005__133_1_73_0.pdf}{Published version}.
-
-\bibitem[GLX26]{GleasonLimXu2026}
-I.~Gleason, D.~G.~Lim, and Y.~Xu,
-\emph{The connected components of affine Deligne--Lusztig varieties},
-Invent. Math. \textbf{243} (2026), 805--861.
-\href{https://doi.org/10.1007/s00222-025-01386-1}{doi:10.1007/s00222-025-01386-1}.
-\href{https://arxiv.org/abs/2208.07195v3}{arXiv:2208.07195v3}
-(10 November 2025); preprint locators refer to
-\href{https://arxiv.org/pdf/2208.07195v3}{this version}.
-
-\bibitem[Gro71]{Grothendieck1970}
-A.~Grothendieck,
-\emph{Groupes de Barsotti-Tate et cristaux},
-in \emph{Actes du Congr\`es International des Math\'ematiciens}
-(Nice, 1970), Tome~1, Gauthier-Villars, Paris, 1971, 431--436.
-\href{https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GroupesB-T.pdf}{Original article}.
-
-\bibitem[Har08]{Hartl2008}
-U.~Hartl,
-\emph{On period spaces for $p$-divisible groups},
-C. R. Math. Acad. Sci. Paris \textbf{346} (2008), nos.~21--22, 1123--1128.
-\href{https://doi.org/10.1016/j.crma.2008.07.003}{doi:10.1016/j.crma.2008.07.003}.
-\href{https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2008.07.003.pdf}{Published version}.
-
-\bibitem[Ill15]{IllusiePisa}
-L.~Illusie,
-\emph{Grothendieck at Pisa: crystals and Barsotti--Tate groups},
-in U.~Zannier (ed.), \emph{Colloquium De Giorgi 2013 and 2014},
-Publications of the Scuola Normale Superiore \textbf{5},
-Edizioni della Normale, Pisa, 2015, 79--107.
-\href{https://doi.org/10.1007/978-88-7642-515-8_5}{doi:10.1007/978-88-7642-515-8\_5}.
-\href{https://www.imo.universite-paris-saclay.fr/~luc.illusie/Illusie-Pisa5.pdf}{Author's version}.
-
-\bibitem[Kim12]{Kim2012}
-W.~Kim,
-\emph{The classification of $p$-divisible groups over $2$-adic discrete
-valuation rings},
-Math. Res. Lett. \textbf{19} (2012), no.~1, 121--141.
-\href{https://doi.org/10.4310/MRL.2012.v19.n1.a10}{doi:10.4310/MRL.2012.v19.n1.a10}.
-\href{https://arxiv.org/abs/1007.1904v3}{arXiv:1007.1904v3}
-(30 December 2011); preprint locators refer to
-\href{https://arxiv.org/pdf/1007.1904v3}{this version}.
-
-\bibitem[Kis06]{Kisin2006}
-M.~Kisin,
-\emph{Crystalline representations and $F$-crystals},
-in V.~Ginzburg (ed.), \emph{Algebraic Geometry and Number Theory:
-In Honor of Vladimir Drinfeld's 50th Birthday},
-Progress in Math. \textbf{253}, Birkh\"auser Boston, Boston, MA,
-2006, 459--496.
-\href{https://doi.org/10.1007/978-0-8176-4532-8_7}{doi:10.1007/978-0-8176-4532-8\_7}.
-
-\bibitem[Kis17]{Kisin2017}
-M.~Kisin,
-\emph{Mod $p$ points on Shimura varieties of abelian type},
-J. Amer. Math. Soc. \textbf{30} (2017), no.~3, 819--914.
-\href{https://people.math.harvard.edu/~kisin/dvifiles/lr.pdf}{Author's version used for all locators}.
-
-\bibitem[KZ25a]{KisinZhouEarlier}
-M.~Kisin and R.~Zhou,
-\emph{Independence of $\ell$ for Frobenius conjugacy classes attached to
-abelian varieties},
-Ann. of Math. (2) \textbf{202} (2025), no.~3, 1077--1156.
-\href{https://doi.org/10.4007/annals.2025.202.3.3}{doi:10.4007/annals.2025.202.3.3}.
-\href{https://arxiv.org/abs/2103.09945v2}{arXiv:2103.09945v2} (7 October 2024).
-All locators refer to
-\href{https://arxiv.org/pdf/2103.09945v2}{arXiv:2103.09945v2}.
-
-\bibitem[KZ25b]{KisinZhou2025}
-M.~Kisin and R.~Zhou,
-\emph{Strongly compatible systems associated to semistable abelian varieties},
-preprint, 2025,
-\href{https://arxiv.org/abs/2505.02165v1}{arXiv:2505.02165v1}
-(deposited 4 May 2025).
-\href{https://arxiv.org/pdf/2505.02165v1}{PDF version used for all locators},
-dated 10 June 2025.
-
-\bibitem[Mar26]{Markman2026}
-E.~Markman,
-\emph{Secant sheaves and Weil classes on abelian varieties},
-preprint, contribution to the ICM 2026,
-\href{https://arxiv.org/abs/2509.23403v2}{arXiv:2509.23403v2}
-(11 February 2026).
-\href{https://arxiv.org/pdf/2509.23403v2}{Version used for all locators},
-dated 13 February 2026.
-
-\bibitem[Mil86]{MilneAV}
-J.~S.~Milne,
-\emph{Abelian varieties},
-in G.~Cornell and J.~H.~Silverman (eds.), \emph{Arithmetic Geometry}
-(Storrs, CT, 1984), Springer-Verlag, New York, 1986, 103--150.
-\href{https://doi.org/10.1007/978-1-4613-8655-1_5}{doi:10.1007/978-1-4613-8655-1\_5}.
-\href{https://www.jmilne.org/math/xnotes/AVs.pdf}{Corrected author's version},
-2 January 2022; original theorem numbering retained.
-
-\bibitem[Mil99]{Milne1999}
-J.~S.~Milne,
-\emph{Lefschetz classes on abelian varieties},
-Duke Math. J. \textbf{96} (1999), no.~3, 639--675.
-\href{https://www.jmilne.org/math/articles/1999aP.pdf}{Author's version};
-\href{https://www.jmilne.org/math/articles/1999a.html}{author's correction to
-the proof of Theorem~5.9}.
-
-\bibitem[Mil00]{Milne2000}
-J.~S.~Milne,
-\emph{Towards a proof of the conjecture of Langlands and Rapoport},
-notes for a talk at the Institut Henri Poincar\'e, 28 April 2000,
-version dated 30 April 2000.
-\href{https://www.jmilne.org/math/articles/2000.pdf}{Author's notes}.
-
-\bibitem[Mil05]{MilneShimura}
-J.~S.~Milne,
-\emph{Introduction to Shimura varieties},
-in \emph{Harmonic Analysis, the Trace Formula, and Shimura Varieties},
-Clay Math. Proc. \textbf{4}, American Mathematical Society,
-Providence, RI, 2005, 265--378.
-\href{https://www.jmilne.org/math/xnotes/svi.pdf}{Author's revision},
-16 September 2017; locators refer to this revision.
-
-\bibitem[Mil08a]{MilneAIM2007}
-J.~S.~Milne,
-\emph{The Tate conjecture over finite fields (AIM talk)},
-notes for the American Institute of Mathematics workshop, July 2007,
-version~2.2, 7 May 2008.
-\href{https://www.jmilne.org/math/articles/2007e.pdf}{Version used for all locators}.
-
-\bibitem[Mil08b]{MilneAVNotes}
-J.~S.~Milne,
-\emph{Abelian Varieties}, course notes, version~2.00, 2008, vi+166~pp.
-\href{https://www.jmilne.org/math/CourseNotes/AVc.pdf}{Author's course notes}.
-
-\bibitem[Mil09]{Milne2009}
-J.~S.~Milne,
-\emph{Rational Tate classes},
-Mosc. Math. J. \textbf{9} (2009), no.~1, 111--141.
-\href{https://www.jmilne.org/math/articles/2009b.pdf}{Author's version};
-page numbers refer to this version.
-
-\bibitem[Mil13]{MilneLEC}
-J.~S.~Milne,
-\emph{Lectures on \'{E}tale Cohomology},
-course notes, version~2.21, 22 March 2013.
-\href{https://www.jmilne.org/math/CourseNotes/LEC.pdf}{Author's notes}.
-
-\bibitem[MilANT]{MilneANT}
-J.~S.~Milne,
-\emph{Algebraic Number Theory}, course notes, version~3.08, 19 July 2020.
-\href{https://www.jmilne.org/math/CourseNotes/ANTc.pdf}{Author's course notes}.
-Locators refer to this version.
-
-\bibitem[Mil20]{Milne2020}
-J.~S.~Milne,
-\emph{Hodge classes on abelian varieties},
-preprint, 2020,
-\href{https://arxiv.org/abs/2010.08857v2}{arXiv:2010.08857v2}
-(11 November 2020).
-\href{https://arxiv.org/pdf/2010.08857v2}{Version used for all locators}.
-
-\bibitem[Mil22]{MilneAG2022}
-J.~S.~Milne,
-\emph{Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field},
-Cambridge Studies in Advanced Mathematics \textbf{170},
-Cambridge University Press, Cambridge, 2017; corrected reprint, 2022.
-\href{https://doi.org/10.1017/9781316711736}{doi:10.1017/9781316711736}.
-\href{https://www.jmilne.org/math/Books/iAG2022.pdf}{Corrected 2022 version used for all locators}.
-
-\bibitem[Mil25]{Milne2025}
-J.~S.~Milne,
-\emph{Abelian motives and Shimura varieties in nonzero characteristic},
-preprint, version~2.2, 2 November 2025.
-\href{https://www.jmilne.org/math/articles/SVp.pdf}{Author's version used for all locators}.
-
-\bibitem[Moo98]{Moonen1998}
-B.~Moonen,
-\emph{Models of Shimura varieties in mixed characteristics},
-in A.~J.~Scholl and R.~L.~Taylor (eds.),
-\emph{Galois Representations in Arithmetic Algebraic Geometry}
-(Durham, 1996), London Math. Soc. Lecture Note Ser. \textbf{254},
-Cambridge University Press, Cambridge, 1998, 267--350.
-\href{https://doi.org/10.1017/CBO9780511662010.008}{doi:10.1017/CBO9780511662010.008}.
-\href{https://www.math.ru.nl/personal/bmoonen/Papers/SMCfinal.pdf}{Author's version}.
-
-\bibitem[OAI26]{OpenAICMHodge2026}
-OpenAI,
-\emph{The rational Hodge conjecture for CM abelian varieties},\newline
-OpenAI Math Release preprint
-\href{https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf}{OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026},
-2026.
-
-\bibitem[PR24]{PappasRapoport2024}
-G.~Pappas and M.~Rapoport,
-\emph{$p$-adic shtukas and the theory of global and local Shimura varieties},
-Camb. J. Math. \textbf{12} (2024), no.~1, 1--164.
-\href{https://doi.org/10.4310/CJM.2024.v12.n1.a1}{doi:10.4310/CJM.2024.v12.n1.a1}.
-\href{https://arxiv.org/abs/2106.08270v6}{arXiv:2106.08270v6}
-(26 October 2023); preprint page numbers refer to
-\href{https://arxiv.org/pdf/2106.08270v6}{this version}.
-
-\bibitem[Stacks]{StacksProject}
-The Stacks Project Authors,
-\emph{The Stacks Project},
-\href{https://stacks.math.columbia.edu/tag/0A42}{Lemma~30.28.3, Tag~0A42},
-accessed 23 September 2026.
-
-\bibitem[Tat67]{Tate1967}
-J.~T.~Tate,
-\emph{$p$-Divisible Groups},
-in T.~A.~Springer (ed.), \emph{Proceedings of a Conference on Local Fields}
-(Driebergen, 1966), Springer-Verlag, Berlin--Heidelberg, 1967, 158--183.
-\href{https://doi.org/10.1007/978-3-642-87942-5_12}{doi:10.1007/978-3-642-87942-5\_12}.
-\href{https://www.math.purdue.edu/~tongliu/teaching/598/p-divisible.pdf}{Original article}.
-
-\bibitem[Win97]{Wintenberger1997}
-J.-P.~Wintenberger,
-\emph{Propri\'et\'es du groupe tannakien des structures de Hodge $p$-adiques
-et torseur entre cohomologies cristalline et \'etale},
-Ann. Inst. Fourier (Grenoble) \textbf{47} (1997), no.~5, 1289--1334.
-\href{https://doi.org/10.5802/aif.1600}{doi:10.5802/aif.1600}.
-\href{https://www.numdam.org/item/10.5802/aif.1600.pdf}{Original article}.
-
-\end{thebibliography}
+\begin{thebibliography}{Stacks}
+
+\bibitem[And92]{Andre1992}
+Y.~Andr\'e,
+\emph{Une remarque \`a propos des cycles de Hodge de type CM},
+in S.~David (ed.), \emph{S\'eminaire de Th\'eorie des Nombres,
+Paris, 1989--90}, Progress in Math. \textbf{102},
+Birkh\"auser Boston, Boston, MA, 1992, 1--7.
+
+\bibitem[And96]{Andre1996}
+Y.~Andr\'e,
+\emph{Pour une th\'eorie inconditionnelle des motifs},
+Publ. Math. Inst. Hautes \'{E}tudes Sci. \textbf{83} (1996), 5--49.
+\href{https://doi.org/10.1007/BF02698643}{doi:10.1007/BF02698643}.
+\href{https://www.numdam.org/item/10.1007/BF02698643.pdf}{Original article}.
+
+\bibitem[Bla94]{Blasius1994}
+D.~Blasius,
+\emph{A $p$-adic property of Hodge classes on abelian varieties},
+in \emph{Motives} (Seattle, WA, 1991), Part~2,
+U.~Jannsen, S.~Kleiman, and J.-P.~Serre (eds.),
+Proc. Sympos. Pure Math. \textbf{55}, Part~2,
+American Mathematical Society, Providence, RI, 1994, 293--308.
+\href{https://doi.org/10.1090/pspum/055.2/1265557}{doi:10.1090/pspum/055.2/1265557}.
+\href{https://archive.mpim-bonn.mpg.de/id/eprint/3859/1/preprint_1992_59.pdf}{Author's preprint}.
+
+\bibitem[CF00]{ColmezFontaine2000}
+P.~Colmez and J.-M.~Fontaine,
+\emph{Construction des repr\'esentations $p$-adiques semi-stables},
+Invent. Math. \textbf{140} (2000), no.~1, 1--43.
+\href{https://doi.org/10.1007/s002220000042}{doi:10.1007/s002220000042}.
+\href{https://www.imo.universite-paris-saclay.fr/~fontaine/CF.pdf}{Author's version}.
+
+\bibitem[Con03]{Conrad2003}
+B.~Conrad,
+\emph{Minimal models for elliptic curves}, notes, 20 October 2003.
+\href{https://math.stanford.edu/~conrad/vigregroup/vigre03/minimalmodel.pdf}{Author's notes}.
+
+\bibitem[Del82]{Deligne1982}
+P.~Deligne,
+\emph{Hodge cycles on abelian varieties} (notes by J.~S.~Milne),
+in P.~Deligne, J.~S.~Milne, A.~Ogus, and K.-y.~Shih,
+\emph{Hodge Cycles, Motives, and Shimura Varieties},
+Lecture Notes in Math. \textbf{900}, Springer-Verlag,
+Berlin--New York, 1982, 9--100.
+\href{https://www.jmilne.org/math/Documents/Deligne82.pdf}{Corrected online version}.
+
+\bibitem[FR05]{FontaineRapoport2005}
+J.-M.~Fontaine and M.~Rapoport,
+\emph{Existence de filtrations admissibles sur des isocristaux},
+Bull. Soc. Math. France \textbf{133} (2005), no.~1, 73--86.
+\href{https://doi.org/10.24033/bsmf.2479}{doi:10.24033/bsmf.2479}.
+\href{https://www.numdam.org/article/BSMF_2005__133_1_73_0.pdf}{Published version}.
+
+\bibitem[GLX26]{GleasonLimXu2026}
+I.~Gleason, D.~G.~Lim, and Y.~Xu,
+\emph{The connected components of affine Deligne--Lusztig varieties},
+Invent. Math. \textbf{243} (2026), 805--861.
+\href{https://doi.org/10.1007/s00222-025-01386-1}{doi:10.1007/s00222-025-01386-1}.
+\href{https://arxiv.org/abs/2208.07195v3}{arXiv:2208.07195v3}
+(10 November 2025); preprint locators refer to
+\href{https://arxiv.org/pdf/2208.07195v3}{this version}.
+
+\bibitem[Gro71]{Grothendieck1970}
+A.~Grothendieck,
+\emph{Groupes de Barsotti-Tate et cristaux},
+in \emph{Actes du Congr\`es International des Math\'ematiciens}
+(Nice, 1970), Tome~1, Gauthier-Villars, Paris, 1971, 431--436.
+\href{https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GroupesB-T.pdf}{Original article}.
+
+\bibitem[Har08]{Hartl2008}
+U.~Hartl,
+\emph{On period spaces for $p$-divisible groups},
+C. R. Math. Acad. Sci. Paris \textbf{346} (2008), nos.~21--22, 1123--1128.
+\href{https://doi.org/10.1016/j.crma.2008.07.003}{doi:10.1016/j.crma.2008.07.003}.
+\href{https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2008.07.003.pdf}{Published version}.
+
+\bibitem[Ill15]{IllusiePisa}
+L.~Illusie,
+\emph{Grothendieck at Pisa: crystals and Barsotti--Tate groups},
+in U.~Zannier (ed.), \emph{Colloquium De Giorgi 2013 and 2014},
+Publications of the Scuola Normale Superiore \textbf{5},
+Edizioni della Normale, Pisa, 2015, 79--107.
+\href{https://doi.org/10.1007/978-88-7642-515-8_5}{doi:10.1007/978-88-7642-515-8\_5}.
+\href{https://www.imo.universite-paris-saclay.fr/~luc.illusie/Illusie-Pisa5.pdf}{Author's version}.
+
+\bibitem[Kim12]{Kim2012}
+W.~Kim,
+\emph{The classification of $p$-divisible groups over $2$-adic discrete
+valuation rings},
+Math. Res. Lett. \textbf{19} (2012), no.~1, 121--141.
+\href{https://doi.org/10.4310/MRL.2012.v19.n1.a10}{doi:10.4310/MRL.2012.v19.n1.a10}.
+\href{https://arxiv.org/abs/1007.1904v3}{arXiv:1007.1904v3}
+(30 December 2011); preprint locators refer to
+\href{https://arxiv.org/pdf/1007.1904v3}{this version}.
+
+\bibitem[Kis06]{Kisin2006}
+M.~Kisin,
+\emph{Crystalline representations and $F$-crystals},
+in V.~Ginzburg (ed.), \emph{Algebraic Geometry and Number Theory:
+In Honor of Vladimir Drinfeld's 50th Birthday},
+Progress in Math. \textbf{253}, Birkh\"auser Boston, Boston, MA,
+2006, 459--496.
+\href{https://doi.org/10.1007/978-0-8176-4532-8_7}{doi:10.1007/978-0-8176-4532-8\_7}.
+
+\bibitem[Kis17]{Kisin2017}
+M.~Kisin,
+\emph{Mod $p$ points on Shimura varieties of abelian type},
+J. Amer. Math. Soc. \textbf{30} (2017), no.~3, 819--914.
+\href{https://people.math.harvard.edu/~kisin/dvifiles/lr.pdf}{Author's version used for all locators}.
+
+\bibitem[KZ25a]{KisinZhouEarlier}
+M.~Kisin and R.~Zhou,
+\emph{Independence of $\ell$ for Frobenius conjugacy classes attached to
+abelian varieties},
+Ann. of Math. (2) \textbf{202} (2025), no.~3, 1077--1156.
+\href{https://doi.org/10.4007/annals.2025.202.3.3}{doi:10.4007/annals.2025.202.3.3}.
+\href{https://arxiv.org/abs/2103.09945v2}{arXiv:2103.09945v2} (7 October 2024).
+All locators refer to
+\href{https://arxiv.org/pdf/2103.09945v2}{arXiv:2103.09945v2}.
+
+\bibitem[KZ25b]{KisinZhou2025}
+M.~Kisin and R.~Zhou,
+\emph{Strongly compatible systems associated to semistable abelian varieties},
+preprint, 2025,
+\href{https://arxiv.org/abs/2505.02165v1}{arXiv:2505.02165v1}
+(deposited 4 May 2025).
+\href{https://arxiv.org/pdf/2505.02165v1}{PDF version used for all locators},
+dated 10 June 2025.
+
+\bibitem[Mar26]{Markman2026}
+E.~Markman,
+\emph{Secant sheaves and Weil classes on abelian varieties},
+preprint, contribution to the ICM 2026,
+\href{https://arxiv.org/abs/2509.23403v2}{arXiv:2509.23403v2}
+(11 February 2026).
+\href{https://arxiv.org/pdf/2509.23403v2}{Version used for all locators},
+dated 13 February 2026.
+
+\bibitem[Mil86]{MilneAV}
+J.~S.~Milne,
+\emph{Abelian varieties},
+in G.~Cornell and J.~H.~Silverman (eds.), \emph{Arithmetic Geometry}
+(Storrs, CT, 1984), Springer-Verlag, New York, 1986, 103--150.
+\href{https://doi.org/10.1007/978-1-4613-8655-1_5}{doi:10.1007/978-1-4613-8655-1\_5}.
+\href{https://www.jmilne.org/math/xnotes/AVs.pdf}{Corrected author's version},
+2 January 2022; original theorem numbering retained.
+
+\bibitem[Mil99]{Milne1999}
+J.~S.~Milne,
+\emph{Lefschetz classes on abelian varieties},
+Duke Math. J. \textbf{96} (1999), no.~3, 639--675.
+\href{https://www.jmilne.org/math/articles/1999aP.pdf}{Author's version};
+\href{https://www.jmilne.org/math/articles/1999a.html}{author's correction to
+the proof of Theorem~5.9}.
+
+\bibitem[Mil00]{Milne2000}
+J.~S.~Milne,
+\emph{Towards a proof of the conjecture of Langlands and Rapoport},
+notes for a talk at the Institut Henri Poincar\'e, 28 April 2000,
+version dated 30 April 2000.
+\href{https://www.jmilne.org/math/articles/2000.pdf}{Author's notes}.
+
+\bibitem[Mil05]{MilneShimura}
+J.~S.~Milne,
+\emph{Introduction to Shimura varieties},
+in \emph{Harmonic Analysis, the Trace Formula, and Shimura Varieties},
+Clay Math. Proc. \textbf{4}, American Mathematical Society,
+Providence, RI, 2005, 265--378.
+\href{https://www.jmilne.org/math/xnotes/svi.pdf}{Author's revision},
+16 September 2017; locators refer to this revision.
+
+\bibitem[Mil08a]{MilneAIM2007}
+J.~S.~Milne,
+\emph{The Tate conjecture over finite fields (AIM talk)},
+notes for the American Institute of Mathematics workshop, July 2007,
+version~2.2, 7 May 2008.
+\href{https://www.jmilne.org/math/articles/2007e.pdf}{Version used for all locators}.
+
+\bibitem[Mil08b]{MilneAVNotes}
+J.~S.~Milne,
+\emph{Abelian Varieties}, course notes, version~2.00, 2008, vi+166~pp.
+\href{https://www.jmilne.org/math/CourseNotes/AVc.pdf}{Author's course notes}.
+
+\bibitem[Mil09]{Milne2009}
+J.~S.~Milne,
+\emph{Rational Tate classes},
+Mosc. Math. J. \textbf{9} (2009), no.~1, 111--141.
+\href{https://www.jmilne.org/math/articles/2009b.pdf}{Author's version};
+page numbers refer to this version.
+
+\bibitem[Mil13]{MilneLEC}
+J.~S.~Milne,
+\emph{Lectures on \'{E}tale Cohomology},
+course notes, version~2.21, 22 March 2013.
+\href{https://www.jmilne.org/math/CourseNotes/LEC.pdf}{Author's notes}.
+
+\bibitem[MilANT]{MilneANT}
+J.~S.~Milne,
+\emph{Algebraic Number Theory}, course notes, version~3.08, 19 July 2020.
+\href{https://www.jmilne.org/math/CourseNotes/ANTc.pdf}{Author's course notes}.
+Locators refer to this version.
+
+\bibitem[Mil20]{Milne2020}
+J.~S.~Milne,
+\emph{Hodge classes on abelian varieties},
+preprint, 2020,
+\href{https://arxiv.org/abs/2010.08857v2}{arXiv:2010.08857v2}
+(11 November 2020).
+\href{https://arxiv.org/pdf/2010.08857v2}{Version used for all locators}.
+
+\bibitem[Mil22]{MilneAG2022}
+J.~S.~Milne,
+\emph{Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field},
+Cambridge Studies in Advanced Mathematics \textbf{170},
+Cambridge University Press, Cambridge, 2017; corrected reprint, 2022.
+\href{https://doi.org/10.1017/9781316711736}{doi:10.1017/9781316711736}.
+\href{https://www.jmilne.org/math/Books/iAG2022.pdf}{Corrected 2022 version used for all locators}.
+
+\bibitem[Mil25]{Milne2025}
+J.~S.~Milne,
+\emph{Abelian motives and Shimura varieties in nonzero characteristic},
+preprint, version~2.2, 2 November 2025.
+\href{https://www.jmilne.org/math/articles/SVp.pdf}{Author's version used for all locators}.
+
+\bibitem[Moo98]{Moonen1998}
+B.~Moonen,
+\emph{Models of Shimura varieties in mixed characteristics},
+in A.~J.~Scholl and R.~L.~Taylor (eds.),
+\emph{Galois Representations in Arithmetic Algebraic Geometry}
+(Durham, 1996), London Math. Soc. Lecture Note Ser. \textbf{254},
+Cambridge University Press, Cambridge, 1998, 267--350.
+\href{https://doi.org/10.1017/CBO9780511662010.008}{doi:10.1017/CBO9780511662010.008}.
+\href{https://www.math.ru.nl/personal/bmoonen/Papers/SMCfinal.pdf}{Author's version}.
+
+\bibitem[OAI26]{OpenAICMHodge2026}
+OpenAI,
+\emph{The rational Hodge conjecture for CM abelian varieties},\newline
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf}{OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026},
+2026.
+
+\bibitem[PR24]{PappasRapoport2024}
+G.~Pappas and M.~Rapoport,
+\emph{$p$-adic shtukas and the theory of global and local Shimura varieties},
+Camb. J. Math. \textbf{12} (2024), no.~1, 1--164.
+\href{https://doi.org/10.4310/CJM.2024.v12.n1.a1}{doi:10.4310/CJM.2024.v12.n1.a1}.
+\href{https://arxiv.org/abs/2106.08270v6}{arXiv:2106.08270v6}
+(26 October 2023); preprint page numbers refer to
+\href{https://arxiv.org/pdf/2106.08270v6}{this version}.
+
+\bibitem[Stacks]{StacksProject}
+The Stacks Project Authors,
+\emph{The Stacks Project},
+\href{https://stacks.math.columbia.edu/tag/0A42}{Lemma~30.28.3, Tag~0A42},
+accessed 23 September 2026.
+
+\bibitem[Tat67]{Tate1967}
+J.~T.~Tate,
+\emph{$p$-Divisible Groups},
+in T.~A.~Springer (ed.), \emph{Proceedings of a Conference on Local Fields}
+(Driebergen, 1966), Springer-Verlag, Berlin--Heidelberg, 1967, 158--183.
+\href{https://doi.org/10.1007/978-3-642-87942-5_12}{doi:10.1007/978-3-642-87942-5\_12}.
+\href{https://www.math.purdue.edu/~tongliu/teaching/598/p-divisible.pdf}{Original article}.
+
+\bibitem[Win97]{Wintenberger1997}
+J.-P.~Wintenberger,
+\emph{Propri\'et\'es du groupe tannakien des structures de Hodge $p$-adiques
+et torseur entre cohomologies cristalline et \'etale},
+Ann. Inst. Fourier (Grenoble) \textbf{47} (1997), no.~5, 1289--1334.
+\href{https://doi.org/10.5802/aif.1600}{doi:10.5802/aif.1600}.
+\href{https://www.numdam.org/item/10.5802/aif.1600.pdf}{Original article}.
+
+\end{thebibliography}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/00_intro.tex

@@ -1,280 +1,280 @@
-\section{Introduction}
-
-Suppose an abelian variety has good reduction at a prime. A rational
-algebraic cycle on the variety specializes to a cycle on the reduction.
-Its intersection with any complementary product of divisors has a rational
-degree, and every cycle-class realization computes that same number.
-Milne's rationality conjecture asks for this conclusion when the original
-class is only known to be a rational Hodge class. We prove it for every
-residue characteristic.
-
-The distinction matters because good reduction can acquire divisors that
-do not lift to the original variety. Deligne's absolute-Hodge theorem gives
-compatible realizations of the Hodge class, but does not by itself identify
-its pairing with these new divisors as the degree of a rational cycle.
-The conjecture isolates this rationality question from the stronger problem
-of representing the Hodge class by an algebraic cycle.
-
-Fix \(\Qbar\subset\C\), a prime \(p\), and a \(p\)-adic place \(w\)
-of \(\Qbar\). Write
-\[
- \F=\overline{\mathbb F}_p,\qquad
- \Kzero=\operatorname{Frac}W(\F),\qquad
- \Cw=\widehat{\Qbar}_w.
-\]
-We regard \(\Kzero\) and an algebraic closure of it as subfields of
-\(\Cw\). Good reduction means good reduction after descent to a number
-field and finite extension at the chosen place.
-
-\begin{theorem}\label{thm:main}
-Let \(A/\Qbar\) be an abelian variety of dimension \(d\) with good
-reduction \(A_0/\F\) at \(w\). Let \(0\leq r\leq d\), and let
-\[
- \gamma\in H^{2r}(A(\C),\Q(r))
-\]
-be a rational Hodge class. For every list of Cartier divisors
-\(D_1,\ldots,D_{d-r}\) on \(A_0\), there is a number \(q\in\Q\)
-such that
-\begin{align*}
- \Tr_\ell\bigl(\gamma_{0,\ell}
-       c_{1,\ell}(D_1)\cdots c_{1,\ell}(D_{d-r})\bigr)&=q
-       &&\text{in }\Q_\ell\quad(\ell\ne p),\\
- \Tr_p\bigl(\gamma_{0,p}
-       c_{1,\mathrm{cris}}(D_1)\cdots
-       c_{1,\mathrm{cris}}(D_{d-r})\bigr)&=q
-       &&\text{in }\Cw.
-\end{align*}
-Here \(\gamma_{0,\ell}\) is the smooth proper specialization of the
-\(\ell\)-adic realization, and \(\gamma_{0,p}\) is the
-de Rham--crystalline specialization in
-\[
- H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
-       \otimes_{\Kzero}\Cw.
-\]
-The traces use the top Tate twist and send the class of a zero-cycle to
-its degree. If \(r=d\), the divisor product is \(1\).
-\end{theorem}
-
-By linearity the same conclusion holds for every rational Lefschetz class
-of complementary codimension, that is, every rational linear combination
-of products of divisors. The number \(q\) is common to all realizations
-for the fixed data; no comparison between different reduction places is
-required.
-
-\subsection{Origins and consequences}
-
-Milne introduced a weak-rationality formulation while studying the
-reduction of Shimura varieties and the Langlands--Rapoport conjecture
-\cite[Conjecture~6.1(A)]{Milne2000}. It asks that a specialized CM
-Hodge class whose components lie in the local spans of divisor products
-in every realization arise from one rational Lefschetz class.
-The pairing formulation appears in his
-AIM notes and his study of rational Tate classes
-\cite[Section~10]{MilneAIM2007}, \cite[Conjecture~4.1]{Milne2009}.
-A previous positive case is a CM abelian variety with simple ordinary
-reduction, together with all its powers: in that case every divisor
-class on the reduction lifts \cite[Example~4.2]{Milne2009}.
-
-The broader purpose is to obtain rational cohomological correspondences
-for abelian motives in characteristic~\(p\), compatible with reduction
-from characteristic zero \cite[Sections~5--6]{Milne2000}; see also
-\cite[Introduction and Section~2]{Milne2025}. For \(\ell\ne p\),
-Tate classes are twisted \(\ell\)-adic classes fixed by a power of
-Frobenius; there is a corresponding \(\Q_p\)-algebra of crystalline
-Tate classes. A \emph{good theory of rational Tate classes} chooses
-one graded \(\Q\)-algebra on each variety whose scalar extensions
-give these Tate algebras. The theory is stable under pullback and
-pushforward, contains divisor classes, and contains specializations of
-CM Hodge classes \cite[Definitions~2.1 and~3.1]{Milne2009}.
-These are cohomological correspondences; their definition does not
-assert that they are algebraic.
-
-Milne proved that the rationality assertion for all CM abelian varieties
-is equivalent to the existence of such a good theory on varieties over
-\(\F\) whose connected components are products of abelian varieties
-and projective spaces \cite[Theorem~4.5]{Milne2009}.
-Our proof also expresses each specialized Hodge class as the pullback,
-along a rational homomorphism of special fibers, of a specialized Hodge
-class from a CM abelian variety. This extends the theory's
-specialization property beyond the CM case.
-In Corollary~\ref{cor:all-hodge-specialization}, every rational
-Hodge class on a good-reduction abelian variety specializes to a unique
-class in one fixed good theory. Its crystalline component gives the
-stated specialization after extension to \(\Cw\).
-
-A distinct additional input gives algebraicity on the special fiber.
-Combining the same transport with the Hodge theorem for CM abelian
-varieties \cite[Theorem~1.1]{OpenAICMHodge2026},
-Corollary~\ref{cor:algebraic-specialization} represents each specialized
-Hodge class by one rational algebraic cycle in all the stated
-realizations. Its pairing with any complementary algebraic cycle is
-therefore a rational intersection number. This consequence uses the CM
-Hodge theorem; the proof of Theorem~\ref{thm:main} is independent of it.
-
-\subsection{The method and its antecedents}
-
-Three foundational results make this question accessible. Deligne proved
-that every Hodge class on an abelian variety is absolute Hodge
-\cite[Main Theorem~2.11]{Deligne1982}. Blasius established its compatibility
-with \(p\)-adic comparison \cite[Theorems~0.3 and~5.3]{Blasius1994};
-Moonen presents the Blasius--Wintenberger comparison theorem, with
-Ogus's simplification, in the perfect-residue-field form needed for
-the local auxiliary lifts below \cite[Theorem~5.6.3]{Moonen1998}. Milne's theory of
-Lefschetz classes supplies a common rational divisor algebra with perfect
-complementary pairing, independent of the Weil cohomology
-\cite[Section~5]{Milne1999}.
-
-The determinant classes used in the proof belong to the theory of Weil
-classes developed by Weil, Deligne and Andr\'e; see
-\cite[Section~4 and Endnotes~M.10--M.12]{Deligne1982}.
-Andr\'e's refinement of the CM reduction expresses Hodge classes by
-pullbacks from varieties of split Weil type \cite{Andre1992}; a detailed
-account is \cite[Theorem~1]{Milne2020}. We prove directly the balanced
-form of that reduction needed here in Lemma~\ref{lem:cm-reduction}.
-Balanced signatures assert equal Hodge multiplicities at conjugate
-embeddings; split Weil type also requires the associated Hermitian
-form to have a half-dimensional isotropic subspace.
-Andr\'e's later theory of motivated cycles establishes a broader
-unconditional motivic setting for abelian Hodge classes
-\cite[Theorem~0.6.2]{Andre1996}, without identifying them with algebraic
-cycles.
-
-For an imaginary quadratic field, Markman has proved algebraicity of
-Weil classes on abelian fourfolds of Weil type and on abelian sixfolds
-of split Weil type \cite[Theorem~1.2]{Markman2026}. These are
-characteristic-zero algebraicity results. The pairing argument here
-applies in arbitrary dimension and begins only with the Hodge class
-and its good reduction.
-
-There is also a direct antecedent for the lifting strategy. Milne asked
-whether a specialized CM Hodge class and a divisor could be lifted
-together, up to isogeny, to a CM variety, and explained the resulting
-rationality implication \cite[Question~10.6 and Proposition~10.7]{MilneAIM2007}.
-His current account gives a conditional criterion through weak lifting
-of Lefschetz classes on CM Weil triples and describes the reduction
-from general Hodge classes through CM varieties to Weil classes
-\cite[Introduction, Question~2.17, and Theorem~2.18]{Milne2025}.
-Our proof follows this general route, using two auxiliary varieties
-with different purposes. The
-first is CM; the second realizes a prescribed admissible filtration
-over a local field and need not be CM. The necessary realization
-machinery comes from Grothendieck--Serre--Tate deformation theory,
-Tate's extension theorem, and the integral-model and local-Shimura
-results of Kisin--Zhou and Pappas--Rapoport
-\cite{Grothendieck1970,Tate1967,KisinZhou2025,PappasRapoport2024}.
-For the prescribed filtration, we choose a framed local Shimura
-point over its admissible period and use integral uniformization;
-the required uniformization condition is supplied by
-\cite[Corollary~6.3]{GleasonLimXu2026}. We state the precise inputs
-where they enter and prove the connecting arguments for the
-prescribed flag.
-
-The contribution along this route is to construct a filtration that
-preserves the extra divisor and then obtain a rational divisor identity
-on its lift. A separate normalization by rational intersection numbers
-makes the tensor transports simultaneous in all realizations.
-Kisin--Zhou's compatibility theorem concerns Weil--Deligne
-representations across coefficient fields
-\cite[Theorem~1.2]{KisinZhou2025}; the additional issue here is the
-rational value of a Hodge--divisor intersection.
-
-The key step is a dichotomy for balanced Weil classes: either all the
-complementary Lefschetz pairings vanish, or the specialized class is
-itself a rational polynomial in divisors, simultaneously in every
-realization.
-
-\subsection{Proof overview}
-
-The proof has four stages. The order below describes the deduction;
-Sections~\ref{sec:transport}--\ref{sec:weil} establish its reusable
-constructions before Section~\ref{sec:reduction} assembles them.
-
-\paragraph{1. Transport to a CM variety.}
-Corollary~\ref{cor:cm-transport} constructs a CM
-abelian variety \(C/\Qbar\), a quasi-isogeny
-\(a_{\mathrm{CM}}:A_0^n\to C_0\), and a rational Hodge class
-\(\kappa\) such that
-\[
- \gamma_0=(a_{\mathrm{CM}}j)^*\kappa_0,
-\]
-where \(j\) includes the first factor. This is one rational algebraic homomorphism on the
-special fibers and one equality in every realization. Kisin--Zhou's
-special-point theorem supplies the CM lift
-\cite[Theorem~2.2.7]{KisinZhou2025}. The local tensors initially determine
-similitude scalars; a ratio of rational divisor intersections shows
-that all those scalars are the same rational number. This normalization
-is what transports the original rational class, rather than a separate
-local multiple in each cohomology theory.
-
-\paragraph{2. Reduce CM classes to balanced Weil classes.}
-Lemma~\ref{lem:cm-reduction} constructs rational-idempotent factors of
-actual powers of \(C\). Their homology has rank \(2s\) over a CM field
-\(E\), with signatures \((s,s)\) at every embedding, and their
-polarizations induce conjugation on \(E\). Their Weil classes are the
-rational forms whose components, after splitting \(E\), are top
-alternating forms on individual \(E\)-eigenspaces. Pullbacks of these
-Hodge classes span the given CM Hodge class over \(\Q\). Good reduction
-is retained. Having one rational value for each complementary
-Lefschetz pairing is preserved by rational pullback
-(Lemma~\ref{lem:pairing-functorial}). It therefore remains to prove
-that assertion for balanced Weil classes.
-
-\paragraph{3. Find a divisor and a compatible filtration.}
-For a balanced Weil variety \(Z\), let \(\dagger\) denote Rosati on its
-reduction and consider
-\[
- \mathcal U=\{u\in\End^0(Z_0):u^\dagger=u,\quad
-                         ue=\bar e u\text{ for all }e\in E\}.
-\]
-Proposition~\ref{prop:weil} proves a dichotomy. If \(\mathcal U\) has no
-invertible element, every complementary Lefschetz pairing of a
-specialized Weil class is zero. If it has an invertible element, the
-specialized Weil class is itself a simultaneous rational Lefschetz
-class: one rational polynomial in divisors gives its class in every
-realization. For the second branch, such a \(u\) supplies a divisor
-\(D^u\) whose alternating form is nondegenerate on each \(E\)-eigenspace.
-Theorem~\ref{thm:filtration} constructs a weakly admissible balanced
-filtration on the fixed crystalline module that preserves \(E\), \(u\),
-and the polarization. Generic transversality has a classical role in
-admissible-filtration constructions
-\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}.
-Here bounded-degree avoidance makes all the constrained choices over
-one finite extension, and a maximality argument reduces the
-weak-admissibility test to subobjects stable under the endomorphisms.
-
-\paragraph{4. Lift the divisor and extract a rational identity.}
-The second case of Proposition~\ref{prop:transport} realizes this flag
-on a local auxiliary abelian variety \(B\), with a marked quasi-isogeny
-\(a_{\mathrm{fl}}:Z_0^m\to B_0\). The prescribed flag makes the
-first-summand projector, its algebraic \(E\)-action, and the transported
-divisor filtered. These specified algebraic maps lift to \(B\).
-After choosing an abstract complex realization of \(B\), polynomial
-coefficient extraction expresses the relevant rational Betti Weil class
-as a rational linear combination of divisor products
-(Lemma~\ref{lem:weil-divisor-span}). The same rational identity specializes
-in every realization. Pullback returns it to \(Z_0\), completing the
-case of an invertible element in the dichotomy and, through the first two stages, the
-main theorem.
-
-Figure~\ref{fig:two-transports} separates the two lifts and their
-special-fiber markings. Section~\ref{sec:preliminaries} fixes comparisons,
-traces, and rational pullback. Section~\ref{sec:transport} proves both
-transport alternatives; Section~\ref{sec:filtration} constructs the
-prescribed flag; Section~\ref{sec:weil} proves the balanced-Weil dichotomy;
-and Section~\ref{sec:reduction} proves the CM reduction and finishes the
-argument.
-
-\begin{figure}[tbp]
-\centering
-\input{figures/two-transports}
-\caption{The two uses of transport. Each lower horizontal arrow is one
-actual quasi-isogeny preserving the specified tensors in all
-realizations; the vertical arrows denote good reduction. There is no
-claimed map between the generic fibers. The upper construction starts
-from the original variety. The lower construction is applied separately
-to a balanced Weil variety \(Z\) arising in the CM reduction; its lift
-\(B\) realizes the chosen filtration \(\mathcal F_Z^{\oplus m}\) under
-\(a_{\mathrm{fl}}^*\). The CM and prescribed-filtration auxiliaries need
-not coincide.}
-\label{fig:two-transports}
-\end{figure}
+\section{Introduction}
+
+Suppose an abelian variety has good reduction at a prime. A rational
+algebraic cycle on the variety specializes to a cycle on the reduction.
+Its intersection with any complementary product of divisors has a rational
+degree, and every cycle-class realization computes that same number.
+Milne's rationality conjecture asks for this conclusion when the original
+class is only known to be a rational Hodge class. We prove it for every
+residue characteristic.
+
+The distinction matters because good reduction can acquire divisors that
+do not lift to the original variety. Deligne's absolute-Hodge theorem gives
+compatible realizations of the Hodge class, but does not by itself identify
+its pairing with these new divisors as the degree of a rational cycle.
+The conjecture isolates this rationality question from the stronger problem
+of representing the Hodge class by an algebraic cycle.
+
+Fix \(\Qbar\subset\C\), a prime \(p\), and a \(p\)-adic place \(w\)
+of \(\Qbar\). Write
+\[
+ \F=\overline{\mathbb F}_p,\qquad
+ \Kzero=\operatorname{Frac}W(\F),\qquad
+ \Cw=\widehat{\Qbar}_w.
+\]
+We regard \(\Kzero\) and an algebraic closure of it as subfields of
+\(\Cw\). Good reduction means good reduction after descent to a number
+field and finite extension at the chosen place.
+
+\begin{theorem}\label{thm:main}
+Let \(A/\Qbar\) be an abelian variety of dimension \(d\) with good
+reduction \(A_0/\F\) at \(w\). Let \(0\leq r\leq d\), and let
+\[
+ \gamma\in H^{2r}(A(\C),\Q(r))
+\]
+be a rational Hodge class. For every list of Cartier divisors
+\(D_1,\ldots,D_{d-r}\) on \(A_0\), there is a number \(q\in\Q\)
+such that
+\begin{align*}
+ \Tr_\ell\bigl(\gamma_{0,\ell}
+       c_{1,\ell}(D_1)\cdots c_{1,\ell}(D_{d-r})\bigr)&=q
+       &&\text{in }\Q_\ell\quad(\ell\ne p),\\
+ \Tr_p\bigl(\gamma_{0,p}
+       c_{1,\mathrm{cris}}(D_1)\cdots
+       c_{1,\mathrm{cris}}(D_{d-r})\bigr)&=q
+       &&\text{in }\Cw.
+\end{align*}
+Here \(\gamma_{0,\ell}\) is the smooth proper specialization of the
+\(\ell\)-adic realization, and \(\gamma_{0,p}\) is the
+de Rham--crystalline specialization in
+\[
+ H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
+       \otimes_{\Kzero}\Cw.
+\]
+The traces use the top Tate twist and send the class of a zero-cycle to
+its degree. If \(r=d\), the divisor product is \(1\).
+\end{theorem}
+
+By linearity the same conclusion holds for every rational Lefschetz class
+of complementary codimension, that is, every rational linear combination
+of products of divisors. The number \(q\) is common to all realizations
+for the fixed data; no comparison between different reduction places is
+required.
+
+\subsection{Origins and consequences}
+
+Milne introduced a weak-rationality formulation while studying the
+reduction of Shimura varieties and the Langlands--Rapoport conjecture
+\cite[Conjecture~6.1(A)]{Milne2000}. It asks that a specialized CM
+Hodge class whose components lie in the local spans of divisor products
+in every realization arise from one rational Lefschetz class.
+The pairing formulation appears in his
+AIM notes and his study of rational Tate classes
+\cite[Section~10]{MilneAIM2007}, \cite[Conjecture~4.1]{Milne2009}.
+A previous positive case is a CM abelian variety with simple ordinary
+reduction, together with all its powers: in that case every divisor
+class on the reduction lifts \cite[Example~4.2]{Milne2009}.
+
+The broader purpose is to obtain rational cohomological correspondences
+for abelian motives in characteristic~\(p\), compatible with reduction
+from characteristic zero \cite[Sections~5--6]{Milne2000}; see also
+\cite[Introduction and Section~2]{Milne2025}. For \(\ell\ne p\),
+Tate classes are twisted \(\ell\)-adic classes fixed by a power of
+Frobenius; there is a corresponding \(\Q_p\)-algebra of crystalline
+Tate classes. A \emph{good theory of rational Tate classes} chooses
+one graded \(\Q\)-algebra on each variety whose scalar extensions
+give these Tate algebras. The theory is stable under pullback and
+pushforward, contains divisor classes, and contains specializations of
+CM Hodge classes \cite[Definitions~2.1 and~3.1]{Milne2009}.
+These are cohomological correspondences; their definition does not
+assert that they are algebraic.
+
+Milne proved that the rationality assertion for all CM abelian varieties
+is equivalent to the existence of such a good theory on varieties over
+\(\F\) whose connected components are products of abelian varieties
+and projective spaces \cite[Theorem~4.5]{Milne2009}.
+Our proof also expresses each specialized Hodge class as the pullback,
+along a rational homomorphism of special fibers, of a specialized Hodge
+class from a CM abelian variety. This extends the theory's
+specialization property beyond the CM case.
+In Corollary~\ref{cor:all-hodge-specialization}, every rational
+Hodge class on a good-reduction abelian variety specializes to a unique
+class in one fixed good theory. Its crystalline component gives the
+stated specialization after extension to \(\Cw\).
+
+A distinct additional input gives algebraicity on the special fiber.
+Combining the same transport with the Hodge theorem for CM abelian
+varieties \cite[Theorem~1.1]{OpenAICMHodge2026},
+Corollary~\ref{cor:algebraic-specialization} represents each specialized
+Hodge class by one rational algebraic cycle in all the stated
+realizations. Its pairing with any complementary algebraic cycle is
+therefore a rational intersection number. This consequence uses the CM
+Hodge theorem; the proof of Theorem~\ref{thm:main} is independent of it.
+
+\subsection{The method and its antecedents}
+
+Three foundational results make this question accessible. Deligne proved
+that every Hodge class on an abelian variety is absolute Hodge
+\cite[Main Theorem~2.11]{Deligne1982}. Blasius established its compatibility
+with \(p\)-adic comparison \cite[Theorems~0.3 and~5.3]{Blasius1994};
+Moonen presents the Blasius--Wintenberger comparison theorem, with
+Ogus's simplification, in the perfect-residue-field form needed for
+the local auxiliary lifts below \cite[Theorem~5.6.3]{Moonen1998}. Milne's theory of
+Lefschetz classes supplies a common rational divisor algebra with perfect
+complementary pairing, independent of the Weil cohomology
+\cite[Section~5]{Milne1999}.
+
+The determinant classes used in the proof belong to the theory of Weil
+classes developed by Weil, Deligne and Andr\'e; see
+\cite[Section~4 and Endnotes~M.10--M.12]{Deligne1982}.
+Andr\'e's refinement of the CM reduction expresses Hodge classes by
+pullbacks from varieties of split Weil type \cite{Andre1992}; a detailed
+account is \cite[Theorem~1]{Milne2020}. We prove directly the balanced
+form of that reduction needed here in Lemma~\ref{lem:cm-reduction}.
+Balanced signatures assert equal Hodge multiplicities at conjugate
+embeddings; split Weil type also requires the associated Hermitian
+form to have a half-dimensional isotropic subspace.
+Andr\'e's later theory of motivated cycles establishes a broader
+unconditional motivic setting for abelian Hodge classes
+\cite[Theorem~0.6.2]{Andre1996}, without identifying them with algebraic
+cycles.
+
+For an imaginary quadratic field, Markman has proved algebraicity of
+Weil classes on abelian fourfolds of Weil type and on abelian sixfolds
+of split Weil type \cite[Theorem~1.2]{Markman2026}. These are
+characteristic-zero algebraicity results. The pairing argument here
+applies in arbitrary dimension and begins only with the Hodge class
+and its good reduction.
+
+There is also a direct antecedent for the lifting strategy. Milne asked
+whether a specialized CM Hodge class and a divisor could be lifted
+together, up to isogeny, to a CM variety, and explained the resulting
+rationality implication \cite[Question~10.6 and Proposition~10.7]{MilneAIM2007}.
+His current account gives a conditional criterion through weak lifting
+of Lefschetz classes on CM Weil triples and describes the reduction
+from general Hodge classes through CM varieties to Weil classes
+\cite[Introduction, Question~2.17, and Theorem~2.18]{Milne2025}.
+Our proof follows this general route, using two auxiliary varieties
+with different purposes. The
+first is CM; the second realizes a prescribed admissible filtration
+over a local field and need not be CM. The necessary realization
+machinery comes from Grothendieck--Serre--Tate deformation theory,
+Tate's extension theorem, and the integral-model and local-Shimura
+results of Kisin--Zhou and Pappas--Rapoport
+\cite{Grothendieck1970,Tate1967,KisinZhou2025,PappasRapoport2024}.
+For the prescribed filtration, we choose a framed local Shimura
+point over its admissible period and use integral uniformization;
+the required uniformization condition is supplied by
+\cite[Corollary~6.3]{GleasonLimXu2026}. We state the precise inputs
+where they enter and prove the connecting arguments for the
+prescribed flag.
+
+The contribution along this route is to construct a filtration that
+preserves the extra divisor and then obtain a rational divisor identity
+on its lift. A separate normalization by rational intersection numbers
+makes the tensor transports simultaneous in all realizations.
+Kisin--Zhou's compatibility theorem concerns Weil--Deligne
+representations across coefficient fields
+\cite[Theorem~1.2]{KisinZhou2025}; the additional issue here is the
+rational value of a Hodge--divisor intersection.
+
+The key step is a dichotomy for balanced Weil classes: either all the
+complementary Lefschetz pairings vanish, or the specialized class is
+itself a rational polynomial in divisors, simultaneously in every
+realization.
+
+\subsection{Proof overview}
+
+The proof has four stages. The order below describes the deduction;
+Sections~\ref{sec:transport}--\ref{sec:weil} establish its reusable
+constructions before Section~\ref{sec:reduction} assembles them.
+
+\paragraph{1. Transport to a CM variety.}
+Corollary~\ref{cor:cm-transport} constructs a CM
+abelian variety \(C/\Qbar\), a quasi-isogeny
+\(a_{\mathrm{CM}}:A_0^n\to C_0\), and a rational Hodge class
+\(\kappa\) such that
+\[
+ \gamma_0=(a_{\mathrm{CM}}j)^*\kappa_0,
+\]
+where \(j\) includes the first factor. This is one rational algebraic homomorphism on the
+special fibers and one equality in every realization. Kisin--Zhou's
+special-point theorem supplies the CM lift
+\cite[Theorem~2.2.7]{KisinZhou2025}. The local tensors initially determine
+similitude scalars; a ratio of rational divisor intersections shows
+that all those scalars are the same rational number. This normalization
+is what transports the original rational class, rather than a separate
+local multiple in each cohomology theory.
+
+\paragraph{2. Reduce CM classes to balanced Weil classes.}
+Lemma~\ref{lem:cm-reduction} constructs rational-idempotent factors of
+actual powers of \(C\). Their homology has rank \(2s\) over a CM field
+\(E\), with signatures \((s,s)\) at every embedding, and their
+polarizations induce conjugation on \(E\). Their Weil classes are the
+rational forms whose components, after splitting \(E\), are top
+alternating forms on individual \(E\)-eigenspaces. Pullbacks of these
+Hodge classes span the given CM Hodge class over \(\Q\). Good reduction
+is retained. Having one rational value for each complementary
+Lefschetz pairing is preserved by rational pullback
+(Lemma~\ref{lem:pairing-functorial}). It therefore remains to prove
+that assertion for balanced Weil classes.
+
+\paragraph{3. Find a divisor and a compatible filtration.}
+For a balanced Weil variety \(Z\), let \(\dagger\) denote Rosati on its
+reduction and consider
+\[
+ \mathcal U=\{u\in\End^0(Z_0):u^\dagger=u,\quad
+                         ue=\bar e u\text{ for all }e\in E\}.
+\]
+Proposition~\ref{prop:weil} proves a dichotomy. If \(\mathcal U\) has no
+invertible element, every complementary Lefschetz pairing of a
+specialized Weil class is zero. If it has an invertible element, the
+specialized Weil class is itself a simultaneous rational Lefschetz
+class: one rational polynomial in divisors gives its class in every
+realization. For the second branch, such a \(u\) supplies a divisor
+\(D^u\) whose alternating form is nondegenerate on each \(E\)-eigenspace.
+Theorem~\ref{thm:filtration} constructs a weakly admissible balanced
+filtration on the fixed crystalline module that preserves \(E\), \(u\),
+and the polarization. Generic transversality has a classical role in
+admissible-filtration constructions
+\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}.
+Here bounded-degree avoidance makes all the constrained choices over
+one finite extension, and a maximality argument reduces the
+weak-admissibility test to subobjects stable under the endomorphisms.
+
+\paragraph{4. Lift the divisor and extract a rational identity.}
+The second case of Proposition~\ref{prop:transport} realizes this flag
+on a local auxiliary abelian variety \(B\), with a marked quasi-isogeny
+\(a_{\mathrm{fl}}:Z_0^m\to B_0\). The prescribed flag makes the
+first-summand projector, its algebraic \(E\)-action, and the transported
+divisor filtered. These specified algebraic maps lift to \(B\).
+After choosing an abstract complex realization of \(B\), polynomial
+coefficient extraction expresses the relevant rational Betti Weil class
+as a rational linear combination of divisor products
+(Lemma~\ref{lem:weil-divisor-span}). The same rational identity specializes
+in every realization. Pullback returns it to \(Z_0\), completing the
+case of an invertible element in the dichotomy and, through the first two stages, the
+main theorem.
+
+Figure~\ref{fig:two-transports} separates the two lifts and their
+special-fiber markings. Section~\ref{sec:preliminaries} fixes comparisons,
+traces, and rational pullback. Section~\ref{sec:transport} proves both
+transport alternatives; Section~\ref{sec:filtration} constructs the
+prescribed flag; Section~\ref{sec:weil} proves the balanced-Weil dichotomy;
+and Section~\ref{sec:reduction} proves the CM reduction and finishes the
+argument.
+
+\begin{figure}[tbp]
+\centering
+\input{figures/two-transports}
+\caption{The two uses of transport. Each lower horizontal arrow is one
+actual quasi-isogeny preserving the specified tensors in all
+realizations; the vertical arrows denote good reduction. There is no
+claimed map between the generic fibers. The upper construction starts
+from the original variety. The lower construction is applied separately
+to a balanced Weil variety \(Z\) arising in the CM reduction; its lift
+\(B\) realizes the chosen filtration \(\mathcal F_Z^{\oplus m}\) under
+\(a_{\mathrm{fl}}^*\). The CM and prescribed-filtration auxiliaries need
+not coincide.}
+\label{fig:two-transports}
+\end{figure}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/01_preliminaries.tex

@@ -1,227 +1,227 @@
-\section{Comparisons and rational Lefschetz classes}
-\label{sec:preliminaries}
-
-We first put the realizations in Theorem~\ref{thm:main} in a common
-notation and formulate its conclusion as a property of cohomology classes.
-The essential functorial fact is that this property is preserved by rational
-pullback. We also record the criterion that will lift an additional divisor
-once a suitable Hodge filtration has been realized.
-
-We work in the category of abelian varieties up to isogeny. Thus
-\(\Hom^0(X,Y)=\Hom(X,Y)\otimes_\Z\Q\), and divisors may have rational
-coefficients. Pullback by a rational homomorphism is defined on \(H^1\)
-and extended to its exterior algebra. If \(Nf\) is an actual homomorphism,
-then on \(H^{2r}\)
-\begin{equation}\label{eq:rational-pullback}
- f^*=N^{-2r}(Nf)^*.
-\end{equation}
-This convention is the same in all realizations.
-
-For an abelian variety \(X/\F\), set
-\[
- k_\ell=\Q_\ell\quad(\ell\ne p),\qquad k_p=\Cw,
-\]
-and write \(H_v^{2r}(X)(r)\) for its \(\ell\)-adic cohomology if
-\(v=\ell\ne p\), and for crystalline cohomology extended to \(\Cw\)
-if \(v=p\). Cup products will usually be written without a cup symbol.
-All trace maps and divisor classes have their cycle-class normalizations.
-The Betti Tate line \(\Q(r)\) supplies the twists in the other
-realizations through comparison. Thus \(\Tr_v\) is the trace on top
-cohomology with its top Tate twist and sends the class of a zero-cycle to
-its degree. For a good-reduction lift, \(\gamma_{0,\ell}\) denotes
-smooth proper specialization of its absolute-Hodge realization, and
-\(\gamma_{0,p}\) denotes de Rham--crystalline specialization extended
-from \(\Kzero\) to \(\Cw\). We keep first crystalline cohomology
-over \(\Kzero\) when constructing filtrations, and specify each
-extension of the filtration field separately.
-
-We use the exterior-algebra description of abelian cohomology and
-faithfulness of rational homomorphisms on \(H^1\); see
-\cite[Theorems~12.5 and~15.1]{MilneAV} for the prime-to-characteristic
-realizations. The uniform Weil-cohomology formalism, including
-crystalline cohomology, exterior powers, cycle-normalized traces and
-the projection formula, is set out in
-\cite[Section~1 and Appendix, Proposition~(A.2)]{Milne1999}. Smooth
-proper specialization away from \(p\) is
-\cite[Theorem~20.4]{MilneLEC}.
-
-\subsection{Comparison of Hodge tensors}
-
-\begin{lemma}\label{lem:comparison}
-For an abelian variety with good reduction, the realizations of an
-absolute Hodge class are compatible with the de Rham and crystalline
-comparison isomorphisms. In particular, the \(p\)-adic crystalline tensor
-obtained from its \(p\)-adic \'etale realization agrees, after extension of
-scalars, with the de Rham--crystalline realization in
-Theorem~\ref{thm:main}. These compatibilities respect tensor operations,
-duals, Tate twists, divisor classes, and rational homomorphisms.
-
-The same assertion applies to auxiliary abelian varieties with good
-reduction over a finite extension of \(\Kzero\), using a complex
-embedding of an algebraic closure of their field of definition.
-\end{lemma}
-
-\begin{proof}
-For abelian varieties over \(\Qbar\), Deligne's absolute-Hodge theorem
-\cite[Main Theorem~2.11]{Deligne1982} and Blasius's comparison theorem
-\cite[Theorems~0.3 and~5.3]{Blasius1994} give the assertion. The
-compatibility of the crystalline comparison with the de Rham comparison
-is part of \cite[\S5.1]{Blasius1994}. The formulation for a complete
-mixed-characteristic discrete valuation ring with perfect residue field
-is given in \cite[Theorem~5.6.3]{Moonen1998}; the paragraph following
-that theorem explains removal of the number-field hypothesis. A finite
-extension of \(\Kzero\) is within that scope. The stated functorialities
-are those of the comparison isomorphisms.
-\end{proof}
-
-An abstract complex embedding used here need not be continuous. To see
-that it can be chosen compatibly with \(\Qbar\subset\C\), take a
-transcendence basis of \(\overline{\Kzero}/\Qbar\), embed it in a
-transcendence basis of \(\C/\Qbar\), and extend algebraically. The
-cardinalities permit this construction. No descent of a general
-auxiliary lift to \(\Qbar\) is inferred.
-
-\subsection{The common rational Lefschetz space}
-
-Let \(\Lef^r(X)\) be the space of rational polynomial expressions in
-divisors of codimension \(r\), modulo numerical equivalence. We use the
-following established properties:
-\begin{enumerate}[label=(\roman*)]
- \item its cycle map is injective, and
- \(\Lef^r(X)\otimes_\Q k_v\) is the divisor-generated subspace of
- \(H_v^{2r}(X)(r)\);
- \item intersection and degree give a perfect pairing
- \[
-   \Lef^r(X)\times\Lef^{\dim X-r}(X)\longrightarrow\Q;
- \]
- \item pushforward by a homomorphism of abelian varieties preserves the
- divisor-generated subspaces in each realization;
- \item hard Lefschetz and its inverse restrict to these subspaces.
-\end{enumerate}
-These are \cite[Proposition~5.2, Corollaries~5.3 and~5.5,
-Theorem~5.9]{Milne1999}, with the author's correction to the proof of
-Theorem~5.9. Crystalline cohomology is included. Extension of its
-coefficient field to \(\Cw\) preserves these statements. We call the
-tuple of realizations of a fixed element of \(\Lef^r(X)\) a
-\emph{simultaneous rational Lefschetz class}.
-
-\begin{definition}
-A tuple \(\alpha=(\alpha_v)_v\), with
-\(\alpha_v\in H_v^{2r}(X)(r)\), has the \emph{Lefschetz pairing
-property} if, for every \(\delta\in\Lef^{\dim X-r}(X)\), the numbers
-\(\Tr_v(\alpha_v\delta_v)\) are the images of one number in \(\Q\).
-\end{definition}
-
-Simultaneous rational algebraic classes have this property, by the
-intersection formula. The property is closed under rational linear
-combinations. We shall need the following additional closure property.
-
-\begin{lemma}\label{lem:pairing-functorial}
-If \(\alpha\) has the Lefschetz pairing property on \(Y\) and
-\(f\in\Hom^0(X,Y)\), then \(f^*\alpha\) has that property on \(X\).
-\end{lemma}
-
-\begin{proof}
-If \(r>\dim Y\), the input class is zero; if \(r>\dim X\),
-its pullback is zero. We may therefore suppose that \(f\) is an actual
-homomorphism and \(r\leq\min(\dim X,\dim Y)\).
-For \(\delta\in\Lef^{\dim X-r}(X)\), the class \(f_*\delta_v\)
-lies in the Lefschetz subspace of codimension \(\dim Y-r\) in every
-realization. Choose rational bases \(a_i\) of
-\(\Lef^{\dim Y-r}(Y)\) and \(b_j\) of \(\Lef^r(Y)\). The matrix
-\(M_{ij}=\deg(a_i b_j)\) is an invertible rational matrix. If
-\(f_*\delta_v=\sum_i c_{i,v}(a_i)_v\), the projection formula gives
-\[
- \sum_i c_{i,v}M_{ij}=\deg(\delta f^*b_j).
-\]
-The right side is a rational number independent of \(v\). Solving this
-one rational system shows that \(c_{i,v}=c_i\in\Q\) for every \(v\).
-Thus \(f_*\delta\) is simultaneous rational Lefschetz. Applying the
-projection formula once more proves the assertion. For rational \(f\),
-use \eqref{eq:rational-pullback}.
-\end{proof}
-
-\subsection{Lifting a prescribed rational homomorphism}
-
-The second transport construction will produce an abelian lift with a
-chosen Hodge filtration. The following criterion then lifts the algebraic
-projector, field action, and divisor already specified on its special fiber.
-Its input is a rational algebraic homomorphism, rather than an arbitrary
-map between cohomology spaces.
-
-In the next lemma \(R\) is a complete mixed-characteristic discrete
-valuation ring finite over \(W(\F)\), with fraction field \(K\).
-For an abelian scheme \(\mathcal X/R\), the Hodge filtration is regarded
-as a filtration of
-\(H^1_{\mathrm{cris}}(X_0/W(\F))[1/p]\otimes_{\Kzero}K\).
-
-\begin{lemma}\label{lem:lift-hom}
-Let \(\mathcal X,\mathcal Y\) be abelian schemes over \(R\), with
-special fibers \(X_0,Y_0\), and let
-\(f_0\in\Hom^0(X_0,Y_0)\). If its contravariant crystalline map
-preserves the Hodge filtrations, then \(f_0\) lifts uniquely to a
-rational homomorphism \(\mathcal X\to\mathcal Y\).
-
-In particular, a symmetric rational homomorphism \(X_0\to X_0^\vee\)
-with this property lifts to a rational divisor class on the generic
-fiber. Algebraic identities among finitely many such lifted maps remain
-valid.
-\end{lemma}
-
-\begin{proof}
-The crystalline map already commutes with Frobenius. Full faithfulness
-for rational filtered Dieudonn\'e modules gives a rational morphism of
-the associated \(p\)-divisible groups; see
-\cite{Grothendieck1970} and the perfect-residue-field formulation in
-\cite[\S4.2(v)]{IllusiePisa}. Equivalently, crystalline comparison first
-gives the rational map on Tate modules. Multiply by an integer \(N\)
-that clears both its Tate-lattice denominator and the denominator of
-\(f_0\). Tate's extension theorem
-\cite[Theorem~4]{Tate1967} extends the resulting integral generic-fiber
-map of \(p\)-divisible groups over \(R\). Faithfulness of the
-special-fiber Dieudonn\'e functor identifies its reduction with the
-map induced by \(Nf_0\). Here equality after inverting \(p\) is
-already equality of integral maps, because the integral Dieudonn\'e
-modules are torsion free. The integer \(N\) is fixed for the remainder
-of the construction.
-
-The specified-map form of Serre--Tate deformation theory now gives a
-lift over each \(R/\mathfrak m_R^n\)
-\cite[p.~432]{Grothendieck1970}. One can also deduce it from the
-fixed-object formulation \cite[Theorem~2.3.3]{IllusiePisa} as follows.
-On \(X_0\times Y_0\) form the shear
-\(s_0(x,y)=(x,y+Nf_0(x))\). The lifted \(p\)-divisible-group map gives
-the same shear on the product \(p\)-divisible group. View the existing
-product deformation with two special-fiber markings differing by
-\(s_0\). Full faithfulness lifts the shear to an algebraic isomorphism
-between these marked deformations; its second component restricted to
-\(X\) lifts \(Nf_0\). Uniqueness makes these maps compatible at every
-finite order.
-
-Formal existence algebraizes the resulting formal map
-\cite[Tag~0A42]{StacksProject}: its source is proper over the complete
-noetherian ring \(R\), and its target is separated and of finite type.
-Dividing by the fixed \(N\) gives the desired rational lift. Rigidity
-gives uniqueness and injectivity of specialization on rational
-homomorphisms \cite[Proposition~20.1 and Corollary~20.2]{MilneAV}.
-
-If \(f_0\) is symmetric, injectivity shows that its lift is symmetric.
-The usual identification
-\[
- \NS(X_K)\otimes\Q
-   \simeq\Hom(X_K,X_K^\vee)^{\mathrm{sym}}\otimes\Q
-\]
-gives the rational divisor class, after a finite extension if
-necessary \cite[Proposition~17.2]{MilneAV}. Finally an identity between the maps specializes to the
-given identity, so injectivity proves it upstairs as well.
-\end{proof}
-
-Only prescribed rational algebraic maps on the special fiber are being
-lifted in Lemma~\ref{lem:lift-hom}. An arbitrary \(\Q_p\)-linear map of
-\(p\)-divisible groups is not asserted to be a rational algebraic map.
-We also use the extension of generic-fiber homomorphisms between abelian
-schemes over a discrete valuation ring: their N\'eron mapping property
-extends such a homomorphism uniquely over the base
-\cite[Definition~5.1 and Example~5.3]{Conrad2003}. Clearing one
-rational denominator therefore makes specialization of these maps
-well defined.
+\section{Comparisons and rational Lefschetz classes}
+\label{sec:preliminaries}
+
+We first put the realizations in Theorem~\ref{thm:main} in a common
+notation and formulate its conclusion as a property of cohomology classes.
+The essential functorial fact is that this property is preserved by rational
+pullback. We also record the criterion that will lift an additional divisor
+once a suitable Hodge filtration has been realized.
+
+We work in the category of abelian varieties up to isogeny. Thus
+\(\Hom^0(X,Y)=\Hom(X,Y)\otimes_\Z\Q\), and divisors may have rational
+coefficients. Pullback by a rational homomorphism is defined on \(H^1\)
+and extended to its exterior algebra. If \(Nf\) is an actual homomorphism,
+then on \(H^{2r}\)
+\begin{equation}\label{eq:rational-pullback}
+ f^*=N^{-2r}(Nf)^*.
+\end{equation}
+This convention is the same in all realizations.
+
+For an abelian variety \(X/\F\), set
+\[
+ k_\ell=\Q_\ell\quad(\ell\ne p),\qquad k_p=\Cw,
+\]
+and write \(H_v^{2r}(X)(r)\) for its \(\ell\)-adic cohomology if
+\(v=\ell\ne p\), and for crystalline cohomology extended to \(\Cw\)
+if \(v=p\). Cup products will usually be written without a cup symbol.
+All trace maps and divisor classes have their cycle-class normalizations.
+The Betti Tate line \(\Q(r)\) supplies the twists in the other
+realizations through comparison. Thus \(\Tr_v\) is the trace on top
+cohomology with its top Tate twist and sends the class of a zero-cycle to
+its degree. For a good-reduction lift, \(\gamma_{0,\ell}\) denotes
+smooth proper specialization of its absolute-Hodge realization, and
+\(\gamma_{0,p}\) denotes de Rham--crystalline specialization extended
+from \(\Kzero\) to \(\Cw\). We keep first crystalline cohomology
+over \(\Kzero\) when constructing filtrations, and specify each
+extension of the filtration field separately.
+
+We use the exterior-algebra description of abelian cohomology and
+faithfulness of rational homomorphisms on \(H^1\); see
+\cite[Theorems~12.5 and~15.1]{MilneAV} for the prime-to-characteristic
+realizations. The uniform Weil-cohomology formalism, including
+crystalline cohomology, exterior powers, cycle-normalized traces and
+the projection formula, is set out in
+\cite[Section~1 and Appendix, Proposition~(A.2)]{Milne1999}. Smooth
+proper specialization away from \(p\) is
+\cite[Theorem~20.4]{MilneLEC}.
+
+\subsection{Comparison of Hodge tensors}
+
+\begin{lemma}\label{lem:comparison}
+For an abelian variety with good reduction, the realizations of an
+absolute Hodge class are compatible with the de Rham and crystalline
+comparison isomorphisms. In particular, the \(p\)-adic crystalline tensor
+obtained from its \(p\)-adic \'etale realization agrees, after extension of
+scalars, with the de Rham--crystalline realization in
+Theorem~\ref{thm:main}. These compatibilities respect tensor operations,
+duals, Tate twists, divisor classes, and rational homomorphisms.
+
+The same assertion applies to auxiliary abelian varieties with good
+reduction over a finite extension of \(\Kzero\), using a complex
+embedding of an algebraic closure of their field of definition.
+\end{lemma}
+
+\begin{proof}
+For abelian varieties over \(\Qbar\), Deligne's absolute-Hodge theorem
+\cite[Main Theorem~2.11]{Deligne1982} and Blasius's comparison theorem
+\cite[Theorems~0.3 and~5.3]{Blasius1994} give the assertion. The
+compatibility of the crystalline comparison with the de Rham comparison
+is part of \cite[\S5.1]{Blasius1994}. The formulation for a complete
+mixed-characteristic discrete valuation ring with perfect residue field
+is given in \cite[Theorem~5.6.3]{Moonen1998}; the paragraph following
+that theorem explains removal of the number-field hypothesis. A finite
+extension of \(\Kzero\) is within that scope. The stated functorialities
+are those of the comparison isomorphisms.
+\end{proof}
+
+An abstract complex embedding used here need not be continuous. To see
+that it can be chosen compatibly with \(\Qbar\subset\C\), take a
+transcendence basis of \(\overline{\Kzero}/\Qbar\), embed it in a
+transcendence basis of \(\C/\Qbar\), and extend algebraically. The
+cardinalities permit this construction. No descent of a general
+auxiliary lift to \(\Qbar\) is inferred.
+
+\subsection{The common rational Lefschetz space}
+
+Let \(\Lef^r(X)\) be the space of rational polynomial expressions in
+divisors of codimension \(r\), modulo numerical equivalence. We use the
+following established properties:
+\begin{enumerate}[label=(\roman*)]
+ \item its cycle map is injective, and
+ \(\Lef^r(X)\otimes_\Q k_v\) is the divisor-generated subspace of
+ \(H_v^{2r}(X)(r)\);
+ \item intersection and degree give a perfect pairing
+ \[
+   \Lef^r(X)\times\Lef^{\dim X-r}(X)\longrightarrow\Q;
+ \]
+ \item pushforward by a homomorphism of abelian varieties preserves the
+ divisor-generated subspaces in each realization;
+ \item hard Lefschetz and its inverse restrict to these subspaces.
+\end{enumerate}
+These are \cite[Proposition~5.2, Corollaries~5.3 and~5.5,
+Theorem~5.9]{Milne1999}, with the author's correction to the proof of
+Theorem~5.9. Crystalline cohomology is included. Extension of its
+coefficient field to \(\Cw\) preserves these statements. We call the
+tuple of realizations of a fixed element of \(\Lef^r(X)\) a
+\emph{simultaneous rational Lefschetz class}.
+
+\begin{definition}
+A tuple \(\alpha=(\alpha_v)_v\), with
+\(\alpha_v\in H_v^{2r}(X)(r)\), has the \emph{Lefschetz pairing
+property} if, for every \(\delta\in\Lef^{\dim X-r}(X)\), the numbers
+\(\Tr_v(\alpha_v\delta_v)\) are the images of one number in \(\Q\).
+\end{definition}
+
+Simultaneous rational algebraic classes have this property, by the
+intersection formula. The property is closed under rational linear
+combinations. We shall need the following additional closure property.
+
+\begin{lemma}\label{lem:pairing-functorial}
+If \(\alpha\) has the Lefschetz pairing property on \(Y\) and
+\(f\in\Hom^0(X,Y)\), then \(f^*\alpha\) has that property on \(X\).
+\end{lemma}
+
+\begin{proof}
+If \(r>\dim Y\), the input class is zero; if \(r>\dim X\),
+its pullback is zero. We may therefore suppose that \(f\) is an actual
+homomorphism and \(r\leq\min(\dim X,\dim Y)\).
+For \(\delta\in\Lef^{\dim X-r}(X)\), the class \(f_*\delta_v\)
+lies in the Lefschetz subspace of codimension \(\dim Y-r\) in every
+realization. Choose rational bases \(a_i\) of
+\(\Lef^{\dim Y-r}(Y)\) and \(b_j\) of \(\Lef^r(Y)\). The matrix
+\(M_{ij}=\deg(a_i b_j)\) is an invertible rational matrix. If
+\(f_*\delta_v=\sum_i c_{i,v}(a_i)_v\), the projection formula gives
+\[
+ \sum_i c_{i,v}M_{ij}=\deg(\delta f^*b_j).
+\]
+The right side is a rational number independent of \(v\). Solving this
+one rational system shows that \(c_{i,v}=c_i\in\Q\) for every \(v\).
+Thus \(f_*\delta\) is simultaneous rational Lefschetz. Applying the
+projection formula once more proves the assertion. For rational \(f\),
+use \eqref{eq:rational-pullback}.
+\end{proof}
+
+\subsection{Lifting a prescribed rational homomorphism}
+
+The second transport construction will produce an abelian lift with a
+chosen Hodge filtration. The following criterion then lifts the algebraic
+projector, field action, and divisor already specified on its special fiber.
+Its input is a rational algebraic homomorphism, rather than an arbitrary
+map between cohomology spaces.
+
+In the next lemma \(R\) is a complete mixed-characteristic discrete
+valuation ring finite over \(W(\F)\), with fraction field \(K\).
+For an abelian scheme \(\mathcal X/R\), the Hodge filtration is regarded
+as a filtration of
+\(H^1_{\mathrm{cris}}(X_0/W(\F))[1/p]\otimes_{\Kzero}K\).
+
+\begin{lemma}\label{lem:lift-hom}
+Let \(\mathcal X,\mathcal Y\) be abelian schemes over \(R\), with
+special fibers \(X_0,Y_0\), and let
+\(f_0\in\Hom^0(X_0,Y_0)\). If its contravariant crystalline map
+preserves the Hodge filtrations, then \(f_0\) lifts uniquely to a
+rational homomorphism \(\mathcal X\to\mathcal Y\).
+
+In particular, a symmetric rational homomorphism \(X_0\to X_0^\vee\)
+with this property lifts to a rational divisor class on the generic
+fiber. Algebraic identities among finitely many such lifted maps remain
+valid.
+\end{lemma}
+
+\begin{proof}
+The crystalline map already commutes with Frobenius. Full faithfulness
+for rational filtered Dieudonn\'e modules gives a rational morphism of
+the associated \(p\)-divisible groups; see
+\cite{Grothendieck1970} and the perfect-residue-field formulation in
+\cite[\S4.2(v)]{IllusiePisa}. Equivalently, crystalline comparison first
+gives the rational map on Tate modules. Multiply by an integer \(N\)
+that clears both its Tate-lattice denominator and the denominator of
+\(f_0\). Tate's extension theorem
+\cite[Theorem~4]{Tate1967} extends the resulting integral generic-fiber
+map of \(p\)-divisible groups over \(R\). Faithfulness of the
+special-fiber Dieudonn\'e functor identifies its reduction with the
+map induced by \(Nf_0\). Here equality after inverting \(p\) is
+already equality of integral maps, because the integral Dieudonn\'e
+modules are torsion free. The integer \(N\) is fixed for the remainder
+of the construction.
+
+The specified-map form of Serre--Tate deformation theory now gives a
+lift over each \(R/\mathfrak m_R^n\)
+\cite[p.~432]{Grothendieck1970}. One can also deduce it from the
+fixed-object formulation \cite[Theorem~2.3.3]{IllusiePisa} as follows.
+On \(X_0\times Y_0\) form the shear
+\(s_0(x,y)=(x,y+Nf_0(x))\). The lifted \(p\)-divisible-group map gives
+the same shear on the product \(p\)-divisible group. View the existing
+product deformation with two special-fiber markings differing by
+\(s_0\). Full faithfulness lifts the shear to an algebraic isomorphism
+between these marked deformations; its second component restricted to
+\(X\) lifts \(Nf_0\). Uniqueness makes these maps compatible at every
+finite order.
+
+Formal existence algebraizes the resulting formal map
+\cite[Tag~0A42]{StacksProject}: its source is proper over the complete
+noetherian ring \(R\), and its target is separated and of finite type.
+Dividing by the fixed \(N\) gives the desired rational lift. Rigidity
+gives uniqueness and injectivity of specialization on rational
+homomorphisms \cite[Proposition~20.1 and Corollary~20.2]{MilneAV}.
+
+If \(f_0\) is symmetric, injectivity shows that its lift is symmetric.
+The usual identification
+\[
+ \NS(X_K)\otimes\Q
+   \simeq\Hom(X_K,X_K^\vee)^{\mathrm{sym}}\otimes\Q
+\]
+gives the rational divisor class, after a finite extension if
+necessary \cite[Proposition~17.2]{MilneAV}. Finally an identity between the maps specializes to the
+given identity, so injectivity proves it upstairs as well.
+\end{proof}
+
+Only prescribed rational algebraic maps on the special fiber are being
+lifted in Lemma~\ref{lem:lift-hom}. An arbitrary \(\Q_p\)-linear map of
+\(p\)-divisible groups is not asserted to be a rational algebraic map.
+We also use the extension of generic-fiber homomorphisms between abelian
+schemes over a discrete valuation ring: their N\'eron mapping property
+extends such a homomorphism uniquely over the base
+\cite[Definition~5.1 and Example~5.3]{Conrad2003}. Clearing one
+rational denominator therefore makes specialization of these maps
+well defined.

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/02_transport.tex

@@ -1,814 +1,814 @@
-\section{Transport through Hodge-type integral models}
-\label{sec:transport}
-
-After taking a power of the original abelian variety, we transport a
-rational Hodge class to an auxiliary variety by one quasi-isogeny on
-special fibers, with the same equality in every
-realization. The auxiliary variety can be chosen either of CM type or
-with a prescribed admissible Hodge filtration. These choices serve
-different parts of the proof, but share the tensor construction and the
-rational normalization in Proposition~\ref{prop:transport} below.
-
-Throughout this section, a polarization on a rational homology space is
-written as a rational alternating form, using the fixed Betti
-trivialization of the Tate line.  If its similitude character is $c$,
-an alternating form $\xi:V^{2r}\to\Q$ of multiplier $c^r$ means that
-\begin{equation}
- \xi(gv_1,\ldots,gv_{2r})=c(g)^r\xi(v_1,\ldots,v_{2r}).
- \label{eq:transport-multiplier}
-\end{equation}
-Such a form is regarded as a tensor with its codimension-$r$ Tate
-twist.  All homomorphisms and polarizations may be rational: an
-isomorphism in the isogeny category induces the usual isomorphisms on
-cohomology, including its tensor constructions.
-
-We say that filtered isocrystal data have \emph{finite local-field
-descent} if they are obtained by scalar extension from an isocrystal
-with its tensors over the maximal unramified subfield \(H_0\) of a
-finite extension \(H/\Q_p\), together with a filtration over \(H\).
-The scalar extension is to \(\Kzero H\). This condition will let us
-apply weak admissibility over a finite local field.
-For a group in a faithful representation, a tensor-preserving
-identification means one respecting a finite family of rational tensors
-whose stabilizer is that group, together with their comparison realizations.
-
-\begin{proposition}[Transport]
-\label{prop:transport}
-Let $A/\Qbar$ be an abelian variety of positive dimension with good
-reduction at $w$, and put $V=H_1(A(\C),\Q)$. Suppose
-\[
- G\subset\GSp(V,\psi)
-\]
-is connected reductive, its Hodge homomorphism $h$ gives the
-Hodge structure of $A$, and its conjugacy class $X$ makes this
-inclusion a Hodge embedding. Let $\gamma$ be a rational Hodge
-class of codimension $r$, represented on $V$ by an alternating
-form satisfying \eqref{eq:transport-multiplier}.
-
-Then there exist a positive integer $n$, an auxiliary
-good-reduction abelian variety $B$, a quasi-isogeny
-$a:A_0^n\to B_0$, and a rational Hodge class $\kappa$ of
-codimension $r$ on $B$ such that, for the first-factor inclusion
-$j:A_0\to A_0^n$,
-\begin{equation}
- (aj)^*\kappa_0=\gamma_0.
- \label{eq:transport-class}
-\end{equation}
-The equality holds simultaneously in every $\ell$-adic
-realization with $\ell\ne p$ and in the crystalline realization
-after extension to $\Cw$. Either of the following two further
-specifications can be imposed.
-\begin{enumerate}
-\item The variety $B$ is of CM type and is defined over $\Qbar$,
-with its indicated good reduction at $w$.
-\item Put $D_A=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$. Suppose
-$\mathcal F_A$ is a filtration in degrees $0,1$ on $D_A$, over a
-finite extension of $\Kzero$, which is weakly admissible and has
-finite local-field descent together with the specialized tensors
-specifying the $G$-structure. Suppose
-also that, after a geometric tensor-preserving identification, it
-lies in the $G$-orbit of the original Hodge filtration. Then $B$
-can be defined over a finite extension $K/\Kzero$ inside $\Cw$,
-and $a$ can be chosen so that
-\begin{equation}
- a^*\Fil^1H^1_{\mathrm{dR}}(B/K)
-       =(\mathcal F_A^1)^{\oplus n}.
- \label{eq:transport-prescribed-filtration}
-\end{equation}
-Here the equality is made after a common finite extension and
-crystalline--de Rham identification. To discuss Hodge classes on
-this $B$, choose a complex embedding of an algebraic closure of
-$K$ extending the fixed embedding of $\Qbar$.
-\end{enumerate}
-In the second case the realizations of $\kappa$ are supplied by
-the canonical tensor sections of the Hodge-type model and the
-polarization. In the first case these agree with the usual
-realizations and specialization of its absolute Hodge class.
-\end{proposition}
-
-We prepare two ingredients for the proof. First, we enlarge the given
-Hodge datum so that integral-model theorems apply at every prime while
-its rational representation remains a sum of copies of \(V\).
-Second, we realize a prescribed admissible filtration by choosing a
-framed local Shimura point above its crystalline period and applying
-integral uniformization.  The resulting abelian lift retains that
-point's integral tensors and special-fiber marking.  After these
-preparations, the proof constructs the transported tensor and normalizes
-it using rational divisor intersections.
-
-\subsection{An auxiliary strongly admissible datum}
-
-We recall the hypotheses on integral models that we will use.
-For a reductive group \(H_p/\Q_p\), a full Bruhat--Tits stabilizer
-is the smooth group scheme fixing a point of its extended building.
-Its identity component is the associated parahoric group scheme.
-Now let \((H,X_H)\) be a Shimura datum over \(\Q\). In the terminology
-of \cite[Definition~2.1.6]{KisinZhou2025}, the triple
-\((H,X_H,\mathcal H)\) is \emph{strongly admissible} if \((H,X_H)\)
-is of Hodge type, \(\mathcal H\) is a full stabilizer of \(H_{\Q_p}\)
-with connected special fiber, the centralizer of a maximal
-\(\Kzero\)-split torus of \(H_{\Kzero}\) is \(R\)-smooth, and the
-relative root system over \(\Kzero\) is reduced when \(p=2\).  Here $R$-smoothness is the condition on tori used in
-\cite[\S2.4]{KisinZhouEarlier}.  We will use it only for products and
-extensions of restrictions of scalars of split tori, for which it
-holds by \cite[Proposition~2.4.6]{KisinZhouEarlier}.
-
-\begin{lemma}[Auxiliary Hodge datum]
-\label{lem:auxiliary-datum}
-Let
-\[
- (G,X)\lhook\joinrel\longrightarrow
- (\GSp(V,\psi),X_{\mathrm{Siegel}})
-\]
-be a Hodge embedding, with $G$ connected and similitude character $c$.
-For every prime $p$ there exist a totally real field $P$, a connected
-reductive group $J$, and a Hodge embedding
-\[
- (J,X_J)\lhook\joinrel\longrightarrow
- (\GSp(W,\psi_W),X'_{\mathrm{Siegel}})
-\]
-with the following properties.
-\begin{enumerate}
-\item There are inclusions of Shimura data induced by
-\[
- G\longrightarrow J\longrightarrow\Res_{P/\Q}G_P,
-\]
-where the first map is diagonal and $J$ is the subgroup on which the
-similitude multipliers in the restriction of scalars agree.
-\item The group $J_{\Q_p}$ is quasi-split and has a full connected
-stabilizer $\mathcal J$ making $(J,X_J,\mathcal J)$ strongly admissible.
-\item As a rational $J$-representation,
-\begin{equation}
- W\simeq (V\otimes_{\Q}P)^{\oplus m}
- \label{eq:transport-copies}
-\end{equation}
-for some positive integer $m$.  The form $\psi_W$ has multiplier the
-common character $c$, and $\mathcal J\to\GL(W_{\Q_p})$ extends to a
-closed immersion into $\GL(\Lambda_p)$ for a self-dual $\Z_p$-lattice
-$\Lambda_p$.
-\end{enumerate}
-\end{lemma}
-
-\begin{proof}
-Choose a finite extension of $\Q_p$ splitting $G_{\Q_p}$.  Polynomial
-approximation, together with Krasner's lemma
-\cite[Propositions~7.60--7.61 and Corollary~7.62]{MilneANT}, gives a
-totally real number field $P$ whose completions at $p$ split $G_{\Q_p}$.  For
-example, approximate the minimal polynomial of a primitive element
-of a splitting extension at $p$, and a polynomial with distinct real
-roots at the real place.  The resulting number field can have just
-one place above $p$; this additional condition is not needed.
-
-Define the multiplier fiber product
-\begin{equation}
- J=(\Res_{P/\Q}G_P)
-       \mathbin{\times}_{\Res_{P/\Q}\mathbf G_m}\mathbf G_m,
- \label{eq:equal-multiplier-group}
-\end{equation}
-where the last map is diagonal.  The Hodge cocharacter, or its
-inverse according to convention, pairs to $1$ with $c$.  Since $c$
-factors through $G^{\mathrm{ab}}$, its character on this torus is
-primitive.  Thus $\ker(G^{\mathrm{ab}}\to\mathbf G_m)$ is a torus,
-and $\ker(c:G\to\mathbf G_m)$ is connected reductive.  The exact
-sequence
-\[
- 1\longrightarrow\Res_{P/\Q}(\ker c)_P
- \longrightarrow J\xrightarrow{c}\mathbf G_m\longrightarrow1
-\]
-proves connectedness and reductivity of $J$.  Moreover,
-\begin{equation}
- J^{\mathrm{der}}=\Res_{P/\Q}G_P^{\mathrm{der}}.
- \label{eq:auxiliary-derived}
-\end{equation}
-
-The action on $U=V\otimes_\Q P$, with alternating form
-$\Tr_{P/\Q}\psi_P$, gives a symplectic embedding of $J$ with
-similitude character $c$.  The diagonal Hodge homomorphism gives
-$X_J$.  The adjoint Hodge-type and Cartan-involution axioms follow
-componentwise from those for $X$.  We also check that extending the
-totally real field introduces no rational adjoint factor on which
-the Hodge homomorphism is trivial.  Write a rational simple adjoint
-factor as $\Res_{L/\Q}H$, with $L$ totally real.  Every field factor
-$L'$ of $P\otimes_\Q L$ is totally real, and every real embedding of
-$L$ extends to a real embedding of $L'$.  Hence the existence of a
-nontrivial Hodge component for $H$ persists in each resulting
-rational factor.  Finally the diagonal weight is rational and the
-trace form is a polarization.  This proves that $(J,X_J)$ is a
-Shimura datum of Hodge type.
-
-Here are the local conditions explicitly.  Put $P\otimes\Q_p=
-\prod_{v\mid p}P_v$ and choose a split maximal torus $T_v$ of
-$G_{P_v}$.  The character $c$ on $T_v$ has a cocharacter section:
-a conjugate of the Hodge cocharacter pairs to $1$ with it, and all
-cocharacters of a split torus are defined over $P_v$.  The same
-argument applies to $G_{P_v}^{\mathrm{ab}}$.  Consequently the
-maximal torus of $J_{\Q_p}$ obtained from the $T_v$, and the
-abelianization of $J_{\Q_p}$, have the forms
-\begin{align}
- T_J&\simeq\mathbf G_m\times
-       \prod_{v\mid p}\Res_{P_v/\Q_p}\mathbf G_m^{\operatorname{rk}G-1},
-       \label{eq:auxiliary-torus}\\
- J_{\Q_p}^{\mathrm{ab}}&\simeq\mathbf G_m\times
-       \prod_{v\mid p}\Res_{P_v/\Q_p}
-                    \mathbf G_m^{\dim G^{\mathrm{ab}}-1}.
-       \label{eq:auxiliary-abelianization}
-\end{align}
-The split Borels over the $P_v$ give a Borel of $J_{\Q_p}$, so this
-group is quasi-split.  Its derived group is a product of restrictions
-of split groups by \eqref{eq:auxiliary-derived}.  The vertices from
-the hyperspecial vertices in those split groups give a very special
-parahoric of $J_{\Q_p}$.  The inertia coinvariants of the cocharacter
-module in \eqref{eq:auxiliary-abelianization} are torsion-free, since
-that module is a direct sum of permutation modules.  The full
-stabilizer is therefore connected by
-\cite[Lemma~4.2.4]{KisinZhouEarlier}.
-
-The centralizer of a maximal $\Kzero$-split torus is obtained from
-\eqref{eq:auxiliary-torus} by unramified base extension; its torus
-factors are still induced split tori and $\mathbf G_m$.
-It is $R$-smooth by
-\cite[Proposition~2.4.6]{KisinZhouEarlier}.  Finally restrictions of
-split root systems give reduced relative root systems, also at
-$p=2$.  These are precisely the strong-admissibility checks in the
-construction of \cite[Proposition~5.3.8]{KisinZhou2025}.  They use
-the stipulated Hodge embedding, not the assertion that $G$ is the
-full Mumford--Tate group of a particular point.
-
-Apply \cite[Proposition~2.1.9]{KisinZhou2025}, starting with the
-representation $U$ just constructed.  Its proof first tensors $U$
-with an auxiliary totally real $\Q$-vector space, then takes direct
-sums, and finally applies the symplectic self-duality construction.
-The group acts trivially on that extra vector-space factor.  In the
-last construction the dual representation twisted by $c$ is
-identified with $U$ by its symplectic form.  Thus every resulting
-rational summand is a copy of $U$, with the same multiplier.  This
-gives \eqref{eq:transport-copies}, together with $\psi_W$ and
-$\Lambda_p$.  The assertion of the cited proposition includes
-$p=2$; its proof uses the reduced-relative-root condition through
-\cite[Lemma~2.1.8]{KisinZhou2025}.
-\end{proof}
-
-Choose a sufficiently small neat prime-to-$p$ level $K^p$ and put
-$K_p=\mathcal J(\Z_p)$.  The adapted embedding gives the integral
-model $\mathcal S_K$ and its universal abelian scheme.  A rational
-$\Z_{(p)}$-lattice can be chosen with completion $\Lambda_p$.
-We use a finite family of tensors defining its stabilizer
-$\mathcal J$, as in \cite[\S2.1.4]{KisinZhou2025}.
-Any additional rational $J$-invariant tensors may be included in
-this family after clearing denominators.  Indeed their integral
-stabilizer condition already holds on the generic fiber; flatness
-of $\mathcal J$ makes it hold on $\mathcal J$ itself.  Adding them
-therefore changes neither the group scheme nor its tensor
-trivializations.
-
-The added tensors have compatible sections in every realization, as
-follows.  Use the homological degree-one realizations in this paragraph,
-that is, the duals of the corresponding $H^1$, so that their reference
-representation is $W$.  For such a realization $H_\nu$ over its
-coefficient field $k_\nu$, let
-\[
- \mathcal P_\nu=\operatorname{Isom}_{\{s_\alpha\}}
-                    (W\otimes k_\nu,H_\nu)
-\]
-be its torsor of frames preserving the original defining tensors.
-For a rational $J$-invariant tensor $u$ and a frame $e$, set
-$u_\nu=e(u)$.  Two frames differ by an element of $J$, so this
-definition is independent of the frame and descends from a
-trivialization of the torsor.  It also works integrally after clearing
-the fixed denominator of $u$.
-
-The Betti and \'etale frame torsors are supplied by the canonical
-tensor sections of \cite[\S2.1.4]{KisinZhou2025}.  The construction
-of the universal Breuil--Kisin--Fargues module and its structure
-tensors gives the crystalline torsor and its compatibility with
-those sections
-\cite[\S\S2.1.13--2.1.14]{KisinZhou2025}.
-Indeed, a comparison map carrying every $s_\alpha$ to its specified
-realization carries a frame $e$ to the comparison of $e$; it therefore
-carries $e(u)$ to the same tensor evaluated in the new frame.
-Thus it preserves $u_\nu$ for every added invariant $u$, without
-requiring $u$ to be an expression in the chosen defining tensors.
-At a lift over $\Qbar$, these are the realizations of its rational
-Hodge tensor, by Lemma~\ref{lem:comparison}.  Their crystalline
-specializations are independent of the lift because they are
-evaluations in the fixed crystalline frame torsor.
-
-The same observation applies to isogenies.  One quasi-isogeny
-preserving the defining tensors identifies their frame torsors in
-every realization in which those tensors are preserved.  It
-therefore preserves all the added invariant tensors as well.
-This proves the tensor comparison and simultaneous preservation
-that will be used below; flatness alone is used only to keep the
-integral stabilizer unchanged.
-
-Specification~\textup{(1)} of Proposition~\ref{prop:transport}
-uses the special-point theorem
-\cite[Theorem~2.2.7]{KisinZhou2025} directly.  The next lemma is
-needed only for specification~\textup{(2)}, where the filtration
-has been prescribed in advance.
-
-\subsection{Realizing a prescribed admissible filtration}
-
-For a point $x\in\mathcal S_K(\F)$ let $C_x$ be the corresponding
-abelian variety and put
-\[
- \mathbb D_x=H^1_{\mathrm{cris}}(C_x/W(\F)),\qquad
- D_x=\mathbb D_x[1/p].
-\]
-The crystalline structure tensors are denoted $s_{\alpha,0,x}$;
-the tensors away from $p$ are denoted $s_{\alpha,\ell,x}$.
-They are the specializations of the canonical tensor sections and
-are independent of the chosen lift of $x$.
-
-The \emph{tensor isogeny class} $\mathcal I_x$ consists of the points
-$y\in\mathcal S_K(\F)$ for which a single quasi-isogeny
-$C_x\to C_y$ preserves $s_{\alpha,0}$ and every
-$s_{\alpha,\ell}$ for $\ell\ne p$.  We emphasize that this
-definition concerns actual quasi-isogenies of abelian varieties.
-It is the class defined in \cite[\S2.2.4]{KisinZhou2025}.
-
-\begin{lemma}[Realization of an admissible filtration]
-\label{lem:admissible-realization}
-Let $(J,X_J,\mathcal J)$ be strongly admissible, with $J_{\Q_p}$
-quasi-split, and choose the adapted embedding and neat level above.
-Let $x\in\mathcal S_K(\F)$ have a lift to characteristic zero.
-Suppose a filtration $\mathcal F$ on $D_x$, after a finite extension
-of $\Kzero$, has the following properties:
-\begin{enumerate}
-\item the resulting filtered isocrystal is admissible;
-\item geometrically its filtration is induced by a $J$-valued
-cocharacter in the conjugacy class prescribed by $X_J$.
-\end{enumerate}
-Then, after extending that field finitely, there are $y\in\mathcal I_x$,
-a tensor-preserving quasi-isogeny $b:C_x\to C_y$, and a lift
-$\widetilde y$ over the extended valuation ring such that
-\[
- b^*\Fil^1 H^1_{\mathrm{dR}}(C_{\widetilde y})=\mathcal F^1
-\]
-under the crystalline--de Rham identification.
-
-In particular, condition \textup{(1)} follows if the filtered
-isocrystal and its tensor data have finite local-field descent and
-$\mathcal F$ is weakly admissible.
-\end{lemma}
-
-\begin{proof}
-Write $k=\F$ and let $\breve E$ be the completed local reflex field.
-We work over a common finite extension $K$ of $\Kzero$ containing
-$\breve E$ and a field of definition of $\mathcal F$.  Every further
-field extension below is finite.  Put $K_p=\mathcal J(\Z_p)$.
-
-We first fix the relation between the group and cohomological
-conventions.  Let $\rho:J_{\Q_p}\to\GL(W_{\Q_p})$ be the adapted
-homological representation and let $\rho^\vee$ be its dual.  Choose
-the tensor-preserving identification
-$\mathbb D_x\simeq\Lambda_p^\vee\otimes_{\Z_p}W(k)$ of
-\cite[\S2.1.14]{KisinZhou2025}.  The local shtuka parameter is
-$b_x\in J(\Kzero)$, whereas crystalline Frobenius on this
-cohomological Dieudonn\'e module is
-\[
- F_x=\rho^\vee(\sigma(b_x))\circ\sigma.
-\]
-The additional $\sigma$ on $b_x$ comes from the comparison with
-the vector shtuka: its special-fiber module is
-$(\sigma^{-1})^*(\mathbb D_x^\vee)$
-\cite[Example~2.3.4 and Remark~2.3.10]{PappasRapoport2024}.
-Write $\{\mu\}$ for the class prescribed by $X_J$; then
-$[b_x]\in B(J,\mu^{-1})$.
-
-For precision, identify the general linear crystalline period
-flag with the appropriate filtration variety on $D_x$ by its
-marked Rapoport--Zink interpretation.  If a deformation
-$\mathcal H$ has special-fiber marking
-$q:\mathcal H_k\dashrightarrow C_x[p^\infty]$, this identification
-sends its period to
-$(q^*)^{-1}\Fil^1H^1_{\mathrm{dR}}(\mathcal H)$.
-Pushout through the faithful Hodge representation gives the
-commutative period square
-\[
-\begin{array}{ccc}
-M_{\mathcal J,b_x,\mu,K_p}&\longrightarrow&M_{\GL,\rho(b_x),\rho(\mu)}\\
-\pi_{\mathrm{GM}}\downarrow&&\downarrow\pi_{\mathrm{GM}}\\
-\mathcal F_{J,\mu}&\longrightarrow&\operatorname{Flag}(D_x).
-\end{array}
-\]
-The lower map is a closed immersion: the parabolic stabilizing
-the faithful filtration intersects $J$ in the parabolic for
-$\mu$.  Define $f_{\mathcal F}$ to be the unique $J$-flag with
-image $\mathcal F$.  It exists by condition~\textup{(2)}.  The
-square uses crystalline periods and the dual Frobenius descent
-above, so no extension of $\sigma$ to the ramified filtration
-field is chosen.  When $q=b^{-1}$ its lower right period is
-exactly $b^*\Fil^1$.
-
-\medskip\noindent
-\emph{A framed point over the prescribed period.}
-Admissibility of the faithful filtered isocrystal implies that
-the image of $f_{\mathcal F}$ in the general linear period flag
-variety is in its admissible locus.  To see this in the
-cohomological coordinates of the square, let $U$ be a crystalline
-representation realizing $(D_x,\mathcal F)$.  Its crystalline
-comparison identifies the associated vector bundle away from
-the untilt with $U\otimes\mathcal O$.  The filtered de Rham
-comparison identifies the modification at the untilt with the
-standard lattice $U\otimes B_{\mathrm{dR}}^+$.  They glue to the
-trivial bundle $U\otimes\mathcal O$ on the Fargues--Fontaine
-curve, which is the general linear admissibility condition.
-Since $\rho$ is faithful,
-$\rho^{-1}(Z_{\GL(W)})\subset Z_J$: an element acting as a scalar
-commutes with the faithful image of $J$.  The admissible-locus
-identity in \cite[Proposition~3.1.1(i), (3.1.9)]{PappasRapoport2024}
-therefore puts $f_{\mathcal F}$ in
-$\mathcal F_{J,\mu}^{\mathrm{adm}}$.
-
-This locus is the image of the crystalline period map
-\[
- \pi_{\mathrm{GM}}:
- M_{\mathcal J,b_x,\mu,K_p}
-       \longrightarrow\mathcal F_{J,\mu}^{\mathrm{adm}}.
-\]
-The map is \'{e}tale
-\cite[\S3.1.1, (3.1.4)]{PappasRapoport2024}.
-Its fiber at the classical point $f_{\mathcal F}$ is nonempty;
-an \'{e}tale rigid space over $K$ has a point over a finite
-extension of $K$.  After making that extension, choose
-\begin{equation}
- z\in M_{\mathcal J,b_x,\mu,K_p}(K),
- \qquad \pi_{\mathrm{GM}}(z)=f_{\mathcal F}.
- \label{eq:transport-framed-point}
-\end{equation}
-In particular $z$ supplies the integral level as well as the
-rational isocrystal marking.  If $C$ is the completed algebraic
-closure of $K$, its moduli description gives an actual
-$\mathcal J$-torsor $\mathcal P_z$, a Frobenius isomorphism
-bounded by $\mu$, and, for sufficiently large $a$, an isomorphism
-\[
- i_a:\mathcal J_{Y_{[a,\infty)}(C^\flat)}
-      \xrightarrow{\ \sim\ }
-      \mathcal P_z|_{Y_{[a,\infty)}(C^\flat)},
- \qquad
- \varphi_{\mathcal P_z}\circ\varphi^*i_a
-       =i_a\circ\varphi_{b_x}.
-\]
-These are precisely the torsor and annular framing in
-\cite[\S3.1.1]{PappasRapoport2024}.  Its associated
-pro-\'{e}tale $\mathcal J(\Z_p)$-torsor gives integral tensor
-frames.  Thus the integral structure and Frobenius-compatible
-framing are part of the chosen point, with all their tensor
-compatibilities.
-
-\medskip\noindent
-\emph{Integral realization and its special fiber.}
-Our adapted datum is of global Hodge type: the group scheme is
-a full connected Bruhat--Tits stabilizer, and the chosen Hodge
-embedding is integral on $\Lambda_p$.
-The center also satisfies the split-rank hypothesis of
-\cite[\S4.1]{PappasRapoport2024}.  Indeed its real subgroup of
-multiplier one preserves the positive definite form
-$\psi_W(v,h(i)w)$ and is compact.  Its real split rank is
-therefore at most one, while the rational weight torus supplies
-one split central direction over $\Q$; both split ranks are one.
-The model here is the one in
-\cite[\S2.1.11 and Theorem~2.1.12]{KisinZhou2025}, obtained
-from this adapted Hodge embedding and satisfying the
-Pappas--Rapoport characterization.  Condition $(U_x)$ holds
-by \cite[Corollary~6.3]{GleasonLimXu2026}; see also
-\cite[Remark~4.10.4]{PappasRapoport2024}.  Consequently
-\cite[Theorem~4.10.6]{PappasRapoport2024} represents the integral
-local Shimura variety by a normal formal scheme
-$\mathfrak M_{\mathcal J,b_x,\mu}$ and gives the morphism
-\[
- \Theta_x:\mathfrak M_{\mathcal J,b_x,\mu}
-          \longrightarrow\widehat{\mathcal S_K}.
-\]
-More precisely, \cite[\S4.10.2, (4.10.2)--(4.10.6)]{PappasRapoport2024}
-identifies this formal scheme with the tensor-compatible open and
-closed part of the pullback of the Siegel Rapoport--Zink space.
-This construction identifies the pullback of the universal
-$\mathcal J$-shtuka with its tautological framed shtuka and the
-faithful vector realization with the universal $p$-divisible group.
-
-Its rigid generic fiber is the local Shimura variety
-\cite[Proposition~3.5.1(i)]{PappasRapoport2024}.
-The classical point $z$ of this rigid generic fiber gives
-$\operatorname{Spf}\mathcal O_K\to\mathfrak M_{\mathcal J,b_x,\mu}$.
-Indeed it lies in the generic fiber of a formal affine chart;
-for a special formal chart one may take an affinoid member of
-its standard generic-fiber covering containing $z$.
-Evaluation of the formal chart's ring is integral, and its
-ideal of definition maps to topologically nilpotent elements of
-$\mathcal O_K$.  Let $z_0$ be the special point and put
-$y=\Theta_x(z_0)$ and $\widetilde y=\Theta_x(z)$.
-The formal map $\widetilde y$ gives a point of the integral model
-over $\mathcal O_K$ by restricting to an affine neighborhood of
-$y$.  Pullback of the universal abelian scheme gives
-$C_{\widetilde y}$ and its $p$-divisible group $\mathcal H_z$.
-
-We record explicitly the integral comparison at this point.
-For a uniformizer $\pi$ of $K$, put
-$\mathfrak S=W(k)[[u]]$, with $u\mapsto\pi$, and let
-$E_\pi(u)$ generate its kernel.  The crystalline representation
-and integral level of $z$ give the $\mathcal J$-Breuil--Kisin
-torsor of \cite[\S3.5]{PappasRapoport2024}, with
-\[
- \Phi:\varphi^*\mathcal P_{\mathrm{BK}}[1/E_\pi]
-             \xrightarrow{\ \sim\ }
-             \mathcal P_{\mathrm{BK}}[1/E_\pi].
-\]
-Its pushout to $\GL(\Lambda_p)$ is the Breuil--Kisin module of
-$\mathcal H_z$.  Reduction at $u=0$, followed by the dual
-Frobenius descent fixed above, therefore gives the actual
-Dieudonn\'e lattice of $\mathcal H_{z,k}$.
-Proposition~3.5.1 of \cite{PappasRapoport2024} compares the two
-specializations of this already constructed framed point.  In
-the coordinates of \cite[Theorem~2.2.5]{KisinZhou2025} it gives
-\[
- g\in X_{\mathcal J}(\mu^{-1},b_x),\qquad
- \mathbb D(\mathcal H_{z,k})
-       =\rho^\vee(\sigma(g))\mathbb D_x\subset D_x.
-\]
-Indeed the Frobenius matrix on this lattice, written in its
-induced basis, is
-\[
- \rho^\vee\!\left(
-       \sigma(g^{-1}b_x\sigma(g))\right)\circ\sigma,
-\]
-which explains the same Frobenius descent in the
-Deligne--Lusztig condition and in the lattice formula.
-
-\medskip\noindent
-\emph{The abelian marking and the filtration.}
-Keep the identity prime-to-$p$ coordinate in the uniformization.
-The description of its special fiber in
-\cite[\S4.10.3]{PappasRapoport2024}, or equivalently the
-isogeny-class map of \cite[Theorem~2.2.5]{KisinZhou2025}, gives
-$y=i_x(g)$ and a single abelian quasi-isogeny
-$b:C_x\to C_y$ preserving the crystalline tensors and every
-prime-to-$p$ tensor.  Its crystalline marking is the one just
-used for $\mathcal H_z$, because the uniformization is induced
-by the marked Siegel Rapoport--Zink space.
-
-The crystalline period map of that space is the Hodge filtration
-of its universal deformation pulled back by its marking.
-Compatibility of $\Theta_x$ with the tautological shtuka and
-\eqref{eq:transport-framed-point} consequently give
-\[
- b^*\Fil^1H^1_{\mathrm{dR}}(C_{\widetilde y}/K)
-       =\mathcal F^1.
-\]
-The deformation criterion
-\cite[Proposition~2.2.3]{KisinZhou2025} is satisfied concretely:
-the filtration is of type $\mu$, and the crystalline structure
-tensors compare with the integral \'{e}tale tensors in the
-$\mathcal J(\Z_p)$-frames of $z$.  The bounded integral shtuka
-and its annular framing were obtained before using this
-criterion or comparing specializations.  All cited global
-Hodge-type and strongly admissible results apply at $p=2$ as
-well.
-
-Finally, under finite local-field descent, the weak-admissibility
-inequalities restrict to the descended Frobenius subobjects.
-Colmez--Fontaine's theorem
-\cite[Theorem~A]{ColmezFontaine2000} makes those descended data
-admissible, and scalar extension supplies condition~\textup{(1)}.
-\end{proof}
-
-\subsection{Proof of the transport proposition}
-
-\begin{proof}[Proof of Proposition~\ref{prop:transport}]
-\medskip\noindent
-\emph{The original point and its reduction.}
-Apply Lemma~\ref{lem:auxiliary-datum}.  Choose a rational
-$J$-equivariant identification
-$W=(V\otimes_\Q P)^{\oplus m}$ and set $n=m[P:\Q]$.
-Choose a basis of $P$ whose first vector is $1$.  The resulting
-identification of the Hodge structure at the diagonal point $h$
-with $V^{\oplus n}$ makes the first-factor map equal to
-\begin{equation}
- v\longmapsto(v\otimes1,0,\ldots,0).
- \label{eq:transport-first-factor}
-\end{equation}
-The rational homology/isogeny correspondence
-\cite[Corollary~6.9]{MilneShimura} therefore makes the abelian variety
-\(C\) at that point isogenous to \(A^n\). Fix a rational isogeny
-\(f:A^n\to C\) realizing this identification.
-
-We explain the field and reduction assertions for this starting
-point.  The Hodge embedding maps it to the Siegel point of the
-polarized variety $C$ with finite level.  The abelian variety $C$
-is defined over $\Qbar$: clearing a denominator in $f$ realizes
-it as a quotient of $A^n$ by a finite subgroup of its torsion, and
-every such subgroup is defined over $\Qbar$.  Its rational
-polarization and its finite level also descend to $\Qbar$, since
-homomorphisms between abelian varieties over $\Qbar$ do not
-enlarge after extending algebraically closed fields
-\cite[Corollary~20.4]{MilneAV}.  The Hodge
-embedding induces a finite morphism onto its
-image at sufficiently small finite level
-\cite[Theorem~5.16]{MilneShimura}. Its fiber over a
-$\Qbar$-point therefore consists of $\Qbar$-points.  This places
-the diagonal point on the canonical model over $\Qbar$.
-
-After a finite number-field extension the variety $C$ has good
-reduction, since it is isogenous to $A^n$
-\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}.  Its polarization and
-prime-to-$p$ level extend after a further finite extension.
-The integral Hodge-type model is the normalization of the closure
-of the generic fiber in the adapted Siegel model.  The valuative
-criterion for this finite normalization therefore extends our
-point to its integral model.  Denote the reduction by $x$.
-The rational isogeny $f$ specializes to a quasi-isogeny
-$f_0:A_0^n\to C_x$.
-
-\medskip\noindent
-\emph{The trace tensor and its descent.}
-Next extend the alternating form defining $\gamma$
-$P$-multilinearly and take its trace.  Composing with projection
-onto the first copy in \eqref{eq:transport-copies} gives
-\begin{equation}
- \eta=\Tr_{P/\Q}(\gamma_P)\circ
-              (\operatorname{pr}_1)^{\otimes 2r}
-       \in\bigwedge^{2r}W^*.
- \label{eq:transport-trace-tensor}
-\end{equation}
-For $r=0$ this is interpreted as the scalar trace.
-The equal-multiplier condition implies that $\eta$ has multiplier
-$c^r$ under $J$.  On \eqref{eq:transport-first-factor}, its
-restriction is
-\begin{equation}
- j^*f^*\eta_C=[P:\Q]\gamma.
- \label{eq:transport-trace-restriction}
-\end{equation}
-The class $\eta_C$ is understood with its Tate twist.
-
-We turn this semi-invariant into the untwisted tensors to which
-the integral-model theorems directly apply.  Write
-$\psi_W^{-1}\in\bigwedge^2W$ for the inverse symplectic
-bivector, and include
-\begin{equation}
- u_\psi=\psi_W\otimes\psi_W^{-1},\qquad
- u_\eta=\eta\otimes(\psi_W^{-1})^{\otimes r}
- \label{eq:transport-untwisted-tensors}
-\end{equation}
-among the structure tensors.  Both are $J$-invariant tensors in
-ordinary tensor constructions of $W$ and $W^*$; no separately
-trivialized Tate object is needed.  They can be made integral and
-added to the defining family as explained above.  Their
-crystalline and prime-to-$p$ realizations at $x$ are the
-specializations from the original point, with the comparison
-conventions fixed above.
-
-\phantomsection
-\label{par:transport-tensor-descent}
-The entire finite family of structure tensors, including
-$u_\psi$ and $u_\eta$, has finite local-field descent.  To see
-this, choose a finite extension $L/\Q_p$ over whose valuation
-ring the original number-field point is defined, and let $L_0$
-be its maximal unramified subfield.  The canonical \'etale
-tensor sections of \cite[\S2.1.4]{KisinZhou2025}, pulled back
-to that point, are $\Gal(\overline L/L)$-invariant.  Applying
-the tensor functor $D_{\mathrm{cris},L}$ to these morphisms
-from the unit object places them in the crystalline tensor
-spaces over $L_0$.  Their extension to $\Kzero$ is exactly
-the family $s_{\alpha,0,x}$, by good-reduction comparison.
-The map $f_0$ also descends after a finite residue-field
-extension, so this descent is retained under its marking.
-Taking a compositum with the finite field of definition of a
-prescribed filtration therefore gives one finite local field
-for all the data.  No tensor-preserving trivialization over
-$L_0$ is being asserted or needed.
-
-\medskip\noindent
-\emph{The two choices of lift.}
-For specification~(1), apply
-\cite[Theorem~2.2.7]{KisinZhou2025} to $x$.  It gives a point
-$y\in\mathcal I_x$ with a special-point lift and a
-tensor-preserving quasi-isogeny $b:C_x\to C_y$.
-The special-point lift supplies a CM abelian variety $B$.  Special
-points on the canonical Shimura variety are algebraic over its
-reflex field by the canonical-model theory
-\cite[Definition~12.8 and Proposition~14.14]{MilneShimura}, so $B$
-and its level data can be taken over $\Qbar$.  The given local lift specifies their embedding at $w$
-and their good reduction $B_0=C_y$.
-
-For specification~(2), identify $D_x$ with $D_A^{\oplus n}$ using
-$f_0^*$ and prescribe the direct-sum filtration
-$\mathcal F_A^{\oplus n}$.  It is weakly admissible, with the
-required finite local-field descent.  It is also in the required
-$J$-cocharacter orbit.  Indeed a geometric element of the
-transported $G$ taking the original filtration on $D_A$ to
-$\mathcal F_A$ acts diagonally on all the copies of $V$ in $W$;
-its image in $J$ takes the original filtration on $D_x$ to the
-prescribed one.  The tensor-preserving identifications used here
-come from the initial point and comparison, so they include the
-$J$-structure tensors as well.  Geometric conjugacy can be
-realized over a finite extension, since the corresponding
-isomorphism scheme is of finite type.
-Lemma~\ref{lem:admissible-realization} now gives $y$, its lift
-$B$, and a tensor-preserving quasi-isogeny $b:C_x\to B_0$ with
-the prescribed marked filtration.  This proves
-\eqref{eq:transport-prescribed-filtration} for
-$a=bf_0$.
-
-\medskip\noindent
-\emph{A common rational normalization.}
-It remains, in both specifications, to recover the twisted class
-and normalize it by a single rational scalar.  Choose a complex
-description of the lifted Shimura point.  Its rational marking by
-$W$ supplies a rational Hodge class $\eta_B$ corresponding to
-$\eta$, and a rational divisor class $\theta_B$
-corresponding to $\psi_W$.  Write $\theta_C$ for the analogous
-class on $C$ and put
-\[
- \theta_*=f_0^*\theta_{C,0}\in\NS(A_0^n)\otimes_\Z\Q.
-\]
-The class $\theta_*$ is rational ample.  Tensor preservation of
-$u_\psi$ says that, in each realization,
-\begin{equation}
- a^*\theta_{B,0}=\lambda_v\theta_*,
- \label{eq:transport-local-similitudes}
-\end{equation}
-with a nonzero local scalar $\lambda_v$.  Let $D=n\dim A$.
-Pairing with $\theta_*^{D-1}$ gives
-\begin{equation}
- \lambda_v=
- \frac{\deg\bigl(a^*\theta_{B,0}\cdot\theta_*^{D-1}\bigr)}
-      {\deg(\theta_*^D)}=:\lambda\in\Q^\times.
- \label{eq:transport-rational-similitude}
-\end{equation}
-Both intersection numbers refer to rational divisor classes on
-the same abelian variety; their cycle-class traces are identical
-in every realization.  The denominator is positive because
-$\theta_*$ is ample, and the numerator is nonzero by
-\eqref{eq:transport-local-similitudes}.  Thus all the scalars in
-\eqref{eq:transport-local-similitudes}, including the crystalline
-one, are the single rational number $\lambda$.
-
-Preservation of $u_\eta$, together with
-\eqref{eq:transport-rational-similitude}, then yields
-\begin{equation}
- a^*\eta_{B,0}=\lambda^r f_0^*\eta_{C,0}
- \label{eq:transport-eta-scaling}
-\end{equation}
-simultaneously.  In fact the inverse bivector scales by
-$\lambda^{-1}$, so that the equality for the untwisted tensor
-in \eqref{eq:transport-untwisted-tensors} gives exactly
-\eqref{eq:transport-eta-scaling}.  Set
-\[
- \kappa=\frac{\lambda^{-r}}{[P:\Q]}\eta_B.
-\]
-Specializing \eqref{eq:transport-trace-restriction} and using
-\eqref{eq:transport-eta-scaling} proves
-\eqref{eq:transport-class}.
-
-For the arbitrary local lift, the crystalline realization just
-used is unambiguous: the canonical model supplies the untwisted
-tensor $u_\eta$, and the algebraic polarization supplies its Tate
-line and inverse bivector.  The comparison isomorphism and these
-operations define $\eta_B$ in the crystalline realization and
-agree with its canonical \'{e}tale tensors.  Consequently any
-identity expressing this class by algebraic divisor operations
-specializes with the same rational coefficients in all
-realizations.  For the $\Qbar$-valued CM lift, these are also the
-usual absolute-Hodge and de Rham realizations.  This completes
-the two constructions.
-\end{proof}
-
-\begin{corollary}[CM transport of a Hodge class]
-\label{cor:cm-transport}
-Let \(A/\Qbar\) be an abelian variety of positive dimension with
-good reduction at \(w\), and let
-\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class,
-where \(0\leq r\leq\dim A\). There are a positive integer \(n\),
-a CM abelian variety \(B/\Qbar\) with good reduction at \(w\),
-a rational Hodge class \(\kappa\) of codimension \(r\) on \(B\),
-and a quasi-isogeny \(a:A_0^n\to B_0\) such that
-\[
- \gamma_0=(aj)^*\kappa_0,
-\]
-where \(j:A_0\to A_0^n\) is the first-factor inclusion. This is an
-equality of the usual absolute-Hodge specializations in every
-prime-to-\(p\) realization and in crystalline cohomology extended
-to \(\Cw\).
-\end{corollary}
-
-\begin{proof}
-Choose a polarization \(\psi\) on \(V=H_1(A(\C),\Q)\), and let
-\(G=\MT(V)\). The Mumford--Tate group is connected reductive and
-is characterized by its Hodge tensors
-\cite[Section~3, Propositions~3.4 and~3.6]{Deligne1982}.
-The Hodge homomorphism and its \(G(\R)\)-conjugacy class define
-the Mumford--Tate Shimura datum of Hodge type, with faithful
-symplectic representation
-\[
- G\hookrightarrow\GSp(V,\psi).
-\]
-The polarization supplies the rational weight and the Cartan
-involution condition. The definition of the Mumford--Tate group
-excludes an adjoint factor on which the Hodge homomorphism is
-trivial. The class \(\gamma\), with its Tate twist, is fixed by
-\(G\). Equivalently, its alternating form on \(V\) has multiplier
-\(c^r\), where \(c\) is the similitude character. Thus the first
-alternative of Proposition~\ref{prop:transport} applies and gives
-the asserted data and simultaneous equality.
-\end{proof}
+\section{Transport through Hodge-type integral models}
+\label{sec:transport}
+
+After taking a power of the original abelian variety, we transport a
+rational Hodge class to an auxiliary variety by one quasi-isogeny on
+special fibers, with the same equality in every
+realization. The auxiliary variety can be chosen either of CM type or
+with a prescribed admissible Hodge filtration. These choices serve
+different parts of the proof, but share the tensor construction and the
+rational normalization in Proposition~\ref{prop:transport} below.
+
+Throughout this section, a polarization on a rational homology space is
+written as a rational alternating form, using the fixed Betti
+trivialization of the Tate line.  If its similitude character is $c$,
+an alternating form $\xi:V^{2r}\to\Q$ of multiplier $c^r$ means that
+\begin{equation}
+ \xi(gv_1,\ldots,gv_{2r})=c(g)^r\xi(v_1,\ldots,v_{2r}).
+ \label{eq:transport-multiplier}
+\end{equation}
+Such a form is regarded as a tensor with its codimension-$r$ Tate
+twist.  All homomorphisms and polarizations may be rational: an
+isomorphism in the isogeny category induces the usual isomorphisms on
+cohomology, including its tensor constructions.
+
+We say that filtered isocrystal data have \emph{finite local-field
+descent} if they are obtained by scalar extension from an isocrystal
+with its tensors over the maximal unramified subfield \(H_0\) of a
+finite extension \(H/\Q_p\), together with a filtration over \(H\).
+The scalar extension is to \(\Kzero H\). This condition will let us
+apply weak admissibility over a finite local field.
+For a group in a faithful representation, a tensor-preserving
+identification means one respecting a finite family of rational tensors
+whose stabilizer is that group, together with their comparison realizations.
+
+\begin{proposition}[Transport]
+\label{prop:transport}
+Let $A/\Qbar$ be an abelian variety of positive dimension with good
+reduction at $w$, and put $V=H_1(A(\C),\Q)$. Suppose
+\[
+ G\subset\GSp(V,\psi)
+\]
+is connected reductive, its Hodge homomorphism $h$ gives the
+Hodge structure of $A$, and its conjugacy class $X$ makes this
+inclusion a Hodge embedding. Let $\gamma$ be a rational Hodge
+class of codimension $r$, represented on $V$ by an alternating
+form satisfying \eqref{eq:transport-multiplier}.
+
+Then there exist a positive integer $n$, an auxiliary
+good-reduction abelian variety $B$, a quasi-isogeny
+$a:A_0^n\to B_0$, and a rational Hodge class $\kappa$ of
+codimension $r$ on $B$ such that, for the first-factor inclusion
+$j:A_0\to A_0^n$,
+\begin{equation}
+ (aj)^*\kappa_0=\gamma_0.
+ \label{eq:transport-class}
+\end{equation}
+The equality holds simultaneously in every $\ell$-adic
+realization with $\ell\ne p$ and in the crystalline realization
+after extension to $\Cw$. Either of the following two further
+specifications can be imposed.
+\begin{enumerate}
+\item The variety $B$ is of CM type and is defined over $\Qbar$,
+with its indicated good reduction at $w$.
+\item Put $D_A=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$. Suppose
+$\mathcal F_A$ is a filtration in degrees $0,1$ on $D_A$, over a
+finite extension of $\Kzero$, which is weakly admissible and has
+finite local-field descent together with the specialized tensors
+specifying the $G$-structure. Suppose
+also that, after a geometric tensor-preserving identification, it
+lies in the $G$-orbit of the original Hodge filtration. Then $B$
+can be defined over a finite extension $K/\Kzero$ inside $\Cw$,
+and $a$ can be chosen so that
+\begin{equation}
+ a^*\Fil^1H^1_{\mathrm{dR}}(B/K)
+       =(\mathcal F_A^1)^{\oplus n}.
+ \label{eq:transport-prescribed-filtration}
+\end{equation}
+Here the equality is made after a common finite extension and
+crystalline--de Rham identification. To discuss Hodge classes on
+this $B$, choose a complex embedding of an algebraic closure of
+$K$ extending the fixed embedding of $\Qbar$.
+\end{enumerate}
+In the second case the realizations of $\kappa$ are supplied by
+the canonical tensor sections of the Hodge-type model and the
+polarization. In the first case these agree with the usual
+realizations and specialization of its absolute Hodge class.
+\end{proposition}
+
+We prepare two ingredients for the proof. First, we enlarge the given
+Hodge datum so that integral-model theorems apply at every prime while
+its rational representation remains a sum of copies of \(V\).
+Second, we realize a prescribed admissible filtration by choosing a
+framed local Shimura point above its crystalline period and applying
+integral uniformization.  The resulting abelian lift retains that
+point's integral tensors and special-fiber marking.  After these
+preparations, the proof constructs the transported tensor and normalizes
+it using rational divisor intersections.
+
+\subsection{An auxiliary strongly admissible datum}
+
+We recall the hypotheses on integral models that we will use.
+For a reductive group \(H_p/\Q_p\), a full Bruhat--Tits stabilizer
+is the smooth group scheme fixing a point of its extended building.
+Its identity component is the associated parahoric group scheme.
+Now let \((H,X_H)\) be a Shimura datum over \(\Q\). In the terminology
+of \cite[Definition~2.1.6]{KisinZhou2025}, the triple
+\((H,X_H,\mathcal H)\) is \emph{strongly admissible} if \((H,X_H)\)
+is of Hodge type, \(\mathcal H\) is a full stabilizer of \(H_{\Q_p}\)
+with connected special fiber, the centralizer of a maximal
+\(\Kzero\)-split torus of \(H_{\Kzero}\) is \(R\)-smooth, and the
+relative root system over \(\Kzero\) is reduced when \(p=2\).  Here $R$-smoothness is the condition on tori used in
+\cite[\S2.4]{KisinZhouEarlier}.  We will use it only for products and
+extensions of restrictions of scalars of split tori, for which it
+holds by \cite[Proposition~2.4.6]{KisinZhouEarlier}.
+
+\begin{lemma}[Auxiliary Hodge datum]
+\label{lem:auxiliary-datum}
+Let
+\[
+ (G,X)\lhook\joinrel\longrightarrow
+ (\GSp(V,\psi),X_{\mathrm{Siegel}})
+\]
+be a Hodge embedding, with $G$ connected and similitude character $c$.
+For every prime $p$ there exist a totally real field $P$, a connected
+reductive group $J$, and a Hodge embedding
+\[
+ (J,X_J)\lhook\joinrel\longrightarrow
+ (\GSp(W,\psi_W),X'_{\mathrm{Siegel}})
+\]
+with the following properties.
+\begin{enumerate}
+\item There are inclusions of Shimura data induced by
+\[
+ G\longrightarrow J\longrightarrow\Res_{P/\Q}G_P,
+\]
+where the first map is diagonal and $J$ is the subgroup on which the
+similitude multipliers in the restriction of scalars agree.
+\item The group $J_{\Q_p}$ is quasi-split and has a full connected
+stabilizer $\mathcal J$ making $(J,X_J,\mathcal J)$ strongly admissible.
+\item As a rational $J$-representation,
+\begin{equation}
+ W\simeq (V\otimes_{\Q}P)^{\oplus m}
+ \label{eq:transport-copies}
+\end{equation}
+for some positive integer $m$.  The form $\psi_W$ has multiplier the
+common character $c$, and $\mathcal J\to\GL(W_{\Q_p})$ extends to a
+closed immersion into $\GL(\Lambda_p)$ for a self-dual $\Z_p$-lattice
+$\Lambda_p$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Choose a finite extension of $\Q_p$ splitting $G_{\Q_p}$.  Polynomial
+approximation, together with Krasner's lemma
+\cite[Propositions~7.60--7.61 and Corollary~7.62]{MilneANT}, gives a
+totally real number field $P$ whose completions at $p$ split $G_{\Q_p}$.  For
+example, approximate the minimal polynomial of a primitive element
+of a splitting extension at $p$, and a polynomial with distinct real
+roots at the real place.  The resulting number field can have just
+one place above $p$; this additional condition is not needed.
+
+Define the multiplier fiber product
+\begin{equation}
+ J=(\Res_{P/\Q}G_P)
+       \mathbin{\times}_{\Res_{P/\Q}\mathbf G_m}\mathbf G_m,
+ \label{eq:equal-multiplier-group}
+\end{equation}
+where the last map is diagonal.  The Hodge cocharacter, or its
+inverse according to convention, pairs to $1$ with $c$.  Since $c$
+factors through $G^{\mathrm{ab}}$, its character on this torus is
+primitive.  Thus $\ker(G^{\mathrm{ab}}\to\mathbf G_m)$ is a torus,
+and $\ker(c:G\to\mathbf G_m)$ is connected reductive.  The exact
+sequence
+\[
+ 1\longrightarrow\Res_{P/\Q}(\ker c)_P
+ \longrightarrow J\xrightarrow{c}\mathbf G_m\longrightarrow1
+\]
+proves connectedness and reductivity of $J$.  Moreover,
+\begin{equation}
+ J^{\mathrm{der}}=\Res_{P/\Q}G_P^{\mathrm{der}}.
+ \label{eq:auxiliary-derived}
+\end{equation}
+
+The action on $U=V\otimes_\Q P$, with alternating form
+$\Tr_{P/\Q}\psi_P$, gives a symplectic embedding of $J$ with
+similitude character $c$.  The diagonal Hodge homomorphism gives
+$X_J$.  The adjoint Hodge-type and Cartan-involution axioms follow
+componentwise from those for $X$.  We also check that extending the
+totally real field introduces no rational adjoint factor on which
+the Hodge homomorphism is trivial.  Write a rational simple adjoint
+factor as $\Res_{L/\Q}H$, with $L$ totally real.  Every field factor
+$L'$ of $P\otimes_\Q L$ is totally real, and every real embedding of
+$L$ extends to a real embedding of $L'$.  Hence the existence of a
+nontrivial Hodge component for $H$ persists in each resulting
+rational factor.  Finally the diagonal weight is rational and the
+trace form is a polarization.  This proves that $(J,X_J)$ is a
+Shimura datum of Hodge type.
+
+Here are the local conditions explicitly.  Put $P\otimes\Q_p=
+\prod_{v\mid p}P_v$ and choose a split maximal torus $T_v$ of
+$G_{P_v}$.  The character $c$ on $T_v$ has a cocharacter section:
+a conjugate of the Hodge cocharacter pairs to $1$ with it, and all
+cocharacters of a split torus are defined over $P_v$.  The same
+argument applies to $G_{P_v}^{\mathrm{ab}}$.  Consequently the
+maximal torus of $J_{\Q_p}$ obtained from the $T_v$, and the
+abelianization of $J_{\Q_p}$, have the forms
+\begin{align}
+ T_J&\simeq\mathbf G_m\times
+       \prod_{v\mid p}\Res_{P_v/\Q_p}\mathbf G_m^{\operatorname{rk}G-1},
+       \label{eq:auxiliary-torus}\\
+ J_{\Q_p}^{\mathrm{ab}}&\simeq\mathbf G_m\times
+       \prod_{v\mid p}\Res_{P_v/\Q_p}
+                    \mathbf G_m^{\dim G^{\mathrm{ab}}-1}.
+       \label{eq:auxiliary-abelianization}
+\end{align}
+The split Borels over the $P_v$ give a Borel of $J_{\Q_p}$, so this
+group is quasi-split.  Its derived group is a product of restrictions
+of split groups by \eqref{eq:auxiliary-derived}.  The vertices from
+the hyperspecial vertices in those split groups give a very special
+parahoric of $J_{\Q_p}$.  The inertia coinvariants of the cocharacter
+module in \eqref{eq:auxiliary-abelianization} are torsion-free, since
+that module is a direct sum of permutation modules.  The full
+stabilizer is therefore connected by
+\cite[Lemma~4.2.4]{KisinZhouEarlier}.
+
+The centralizer of a maximal $\Kzero$-split torus is obtained from
+\eqref{eq:auxiliary-torus} by unramified base extension; its torus
+factors are still induced split tori and $\mathbf G_m$.
+It is $R$-smooth by
+\cite[Proposition~2.4.6]{KisinZhouEarlier}.  Finally restrictions of
+split root systems give reduced relative root systems, also at
+$p=2$.  These are precisely the strong-admissibility checks in the
+construction of \cite[Proposition~5.3.8]{KisinZhou2025}.  They use
+the stipulated Hodge embedding, not the assertion that $G$ is the
+full Mumford--Tate group of a particular point.
+
+Apply \cite[Proposition~2.1.9]{KisinZhou2025}, starting with the
+representation $U$ just constructed.  Its proof first tensors $U$
+with an auxiliary totally real $\Q$-vector space, then takes direct
+sums, and finally applies the symplectic self-duality construction.
+The group acts trivially on that extra vector-space factor.  In the
+last construction the dual representation twisted by $c$ is
+identified with $U$ by its symplectic form.  Thus every resulting
+rational summand is a copy of $U$, with the same multiplier.  This
+gives \eqref{eq:transport-copies}, together with $\psi_W$ and
+$\Lambda_p$.  The assertion of the cited proposition includes
+$p=2$; its proof uses the reduced-relative-root condition through
+\cite[Lemma~2.1.8]{KisinZhou2025}.
+\end{proof}
+
+Choose a sufficiently small neat prime-to-$p$ level $K^p$ and put
+$K_p=\mathcal J(\Z_p)$.  The adapted embedding gives the integral
+model $\mathcal S_K$ and its universal abelian scheme.  A rational
+$\Z_{(p)}$-lattice can be chosen with completion $\Lambda_p$.
+We use a finite family of tensors defining its stabilizer
+$\mathcal J$, as in \cite[\S2.1.4]{KisinZhou2025}.
+Any additional rational $J$-invariant tensors may be included in
+this family after clearing denominators.  Indeed their integral
+stabilizer condition already holds on the generic fiber; flatness
+of $\mathcal J$ makes it hold on $\mathcal J$ itself.  Adding them
+therefore changes neither the group scheme nor its tensor
+trivializations.
+
+The added tensors have compatible sections in every realization, as
+follows.  Use the homological degree-one realizations in this paragraph,
+that is, the duals of the corresponding $H^1$, so that their reference
+representation is $W$.  For such a realization $H_\nu$ over its
+coefficient field $k_\nu$, let
+\[
+ \mathcal P_\nu=\operatorname{Isom}_{\{s_\alpha\}}
+                    (W\otimes k_\nu,H_\nu)
+\]
+be its torsor of frames preserving the original defining tensors.
+For a rational $J$-invariant tensor $u$ and a frame $e$, set
+$u_\nu=e(u)$.  Two frames differ by an element of $J$, so this
+definition is independent of the frame and descends from a
+trivialization of the torsor.  It also works integrally after clearing
+the fixed denominator of $u$.
+
+The Betti and \'etale frame torsors are supplied by the canonical
+tensor sections of \cite[\S2.1.4]{KisinZhou2025}.  The construction
+of the universal Breuil--Kisin--Fargues module and its structure
+tensors gives the crystalline torsor and its compatibility with
+those sections
+\cite[\S\S2.1.13--2.1.14]{KisinZhou2025}.
+Indeed, a comparison map carrying every $s_\alpha$ to its specified
+realization carries a frame $e$ to the comparison of $e$; it therefore
+carries $e(u)$ to the same tensor evaluated in the new frame.
+Thus it preserves $u_\nu$ for every added invariant $u$, without
+requiring $u$ to be an expression in the chosen defining tensors.
+At a lift over $\Qbar$, these are the realizations of its rational
+Hodge tensor, by Lemma~\ref{lem:comparison}.  Their crystalline
+specializations are independent of the lift because they are
+evaluations in the fixed crystalline frame torsor.
+
+The same observation applies to isogenies.  One quasi-isogeny
+preserving the defining tensors identifies their frame torsors in
+every realization in which those tensors are preserved.  It
+therefore preserves all the added invariant tensors as well.
+This proves the tensor comparison and simultaneous preservation
+that will be used below; flatness alone is used only to keep the
+integral stabilizer unchanged.
+
+Specification~\textup{(1)} of Proposition~\ref{prop:transport}
+uses the special-point theorem
+\cite[Theorem~2.2.7]{KisinZhou2025} directly.  The next lemma is
+needed only for specification~\textup{(2)}, where the filtration
+has been prescribed in advance.
+
+\subsection{Realizing a prescribed admissible filtration}
+
+For a point $x\in\mathcal S_K(\F)$ let $C_x$ be the corresponding
+abelian variety and put
+\[
+ \mathbb D_x=H^1_{\mathrm{cris}}(C_x/W(\F)),\qquad
+ D_x=\mathbb D_x[1/p].
+\]
+The crystalline structure tensors are denoted $s_{\alpha,0,x}$;
+the tensors away from $p$ are denoted $s_{\alpha,\ell,x}$.
+They are the specializations of the canonical tensor sections and
+are independent of the chosen lift of $x$.
+
+The \emph{tensor isogeny class} $\mathcal I_x$ consists of the points
+$y\in\mathcal S_K(\F)$ for which a single quasi-isogeny
+$C_x\to C_y$ preserves $s_{\alpha,0}$ and every
+$s_{\alpha,\ell}$ for $\ell\ne p$.  We emphasize that this
+definition concerns actual quasi-isogenies of abelian varieties.
+It is the class defined in \cite[\S2.2.4]{KisinZhou2025}.
+
+\begin{lemma}[Realization of an admissible filtration]
+\label{lem:admissible-realization}
+Let $(J,X_J,\mathcal J)$ be strongly admissible, with $J_{\Q_p}$
+quasi-split, and choose the adapted embedding and neat level above.
+Let $x\in\mathcal S_K(\F)$ have a lift to characteristic zero.
+Suppose a filtration $\mathcal F$ on $D_x$, after a finite extension
+of $\Kzero$, has the following properties:
+\begin{enumerate}
+\item the resulting filtered isocrystal is admissible;
+\item geometrically its filtration is induced by a $J$-valued
+cocharacter in the conjugacy class prescribed by $X_J$.
+\end{enumerate}
+Then, after extending that field finitely, there are $y\in\mathcal I_x$,
+a tensor-preserving quasi-isogeny $b:C_x\to C_y$, and a lift
+$\widetilde y$ over the extended valuation ring such that
+\[
+ b^*\Fil^1 H^1_{\mathrm{dR}}(C_{\widetilde y})=\mathcal F^1
+\]
+under the crystalline--de Rham identification.
+
+In particular, condition \textup{(1)} follows if the filtered
+isocrystal and its tensor data have finite local-field descent and
+$\mathcal F$ is weakly admissible.
+\end{lemma}
+
+\begin{proof}
+Write $k=\F$ and let $\breve E$ be the completed local reflex field.
+We work over a common finite extension $K$ of $\Kzero$ containing
+$\breve E$ and a field of definition of $\mathcal F$.  Every further
+field extension below is finite.  Put $K_p=\mathcal J(\Z_p)$.
+
+We first fix the relation between the group and cohomological
+conventions.  Let $\rho:J_{\Q_p}\to\GL(W_{\Q_p})$ be the adapted
+homological representation and let $\rho^\vee$ be its dual.  Choose
+the tensor-preserving identification
+$\mathbb D_x\simeq\Lambda_p^\vee\otimes_{\Z_p}W(k)$ of
+\cite[\S2.1.14]{KisinZhou2025}.  The local shtuka parameter is
+$b_x\in J(\Kzero)$, whereas crystalline Frobenius on this
+cohomological Dieudonn\'e module is
+\[
+ F_x=\rho^\vee(\sigma(b_x))\circ\sigma.
+\]
+The additional $\sigma$ on $b_x$ comes from the comparison with
+the vector shtuka: its special-fiber module is
+$(\sigma^{-1})^*(\mathbb D_x^\vee)$
+\cite[Example~2.3.4 and Remark~2.3.10]{PappasRapoport2024}.
+Write $\{\mu\}$ for the class prescribed by $X_J$; then
+$[b_x]\in B(J,\mu^{-1})$.
+
+For precision, identify the general linear crystalline period
+flag with the appropriate filtration variety on $D_x$ by its
+marked Rapoport--Zink interpretation.  If a deformation
+$\mathcal H$ has special-fiber marking
+$q:\mathcal H_k\dashrightarrow C_x[p^\infty]$, this identification
+sends its period to
+$(q^*)^{-1}\Fil^1H^1_{\mathrm{dR}}(\mathcal H)$.
+Pushout through the faithful Hodge representation gives the
+commutative period square
+\[
+\begin{array}{ccc}
+M_{\mathcal J,b_x,\mu,K_p}&\longrightarrow&M_{\GL,\rho(b_x),\rho(\mu)}\\
+\pi_{\mathrm{GM}}\downarrow&&\downarrow\pi_{\mathrm{GM}}\\
+\mathcal F_{J,\mu}&\longrightarrow&\operatorname{Flag}(D_x).
+\end{array}
+\]
+The lower map is a closed immersion: the parabolic stabilizing
+the faithful filtration intersects $J$ in the parabolic for
+$\mu$.  Define $f_{\mathcal F}$ to be the unique $J$-flag with
+image $\mathcal F$.  It exists by condition~\textup{(2)}.  The
+square uses crystalline periods and the dual Frobenius descent
+above, so no extension of $\sigma$ to the ramified filtration
+field is chosen.  When $q=b^{-1}$ its lower right period is
+exactly $b^*\Fil^1$.
+
+\medskip\noindent
+\emph{A framed point over the prescribed period.}
+Admissibility of the faithful filtered isocrystal implies that
+the image of $f_{\mathcal F}$ in the general linear period flag
+variety is in its admissible locus.  To see this in the
+cohomological coordinates of the square, let $U$ be a crystalline
+representation realizing $(D_x,\mathcal F)$.  Its crystalline
+comparison identifies the associated vector bundle away from
+the untilt with $U\otimes\mathcal O$.  The filtered de Rham
+comparison identifies the modification at the untilt with the
+standard lattice $U\otimes B_{\mathrm{dR}}^+$.  They glue to the
+trivial bundle $U\otimes\mathcal O$ on the Fargues--Fontaine
+curve, which is the general linear admissibility condition.
+Since $\rho$ is faithful,
+$\rho^{-1}(Z_{\GL(W)})\subset Z_J$: an element acting as a scalar
+commutes with the faithful image of $J$.  The admissible-locus
+identity in \cite[Proposition~3.1.1(i), (3.1.9)]{PappasRapoport2024}
+therefore puts $f_{\mathcal F}$ in
+$\mathcal F_{J,\mu}^{\mathrm{adm}}$.
+
+This locus is the image of the crystalline period map
+\[
+ \pi_{\mathrm{GM}}:
+ M_{\mathcal J,b_x,\mu,K_p}
+       \longrightarrow\mathcal F_{J,\mu}^{\mathrm{adm}}.
+\]
+The map is \'{e}tale
+\cite[\S3.1.1, (3.1.4)]{PappasRapoport2024}.
+Its fiber at the classical point $f_{\mathcal F}$ is nonempty;
+an \'{e}tale rigid space over $K$ has a point over a finite
+extension of $K$.  After making that extension, choose
+\begin{equation}
+ z\in M_{\mathcal J,b_x,\mu,K_p}(K),
+ \qquad \pi_{\mathrm{GM}}(z)=f_{\mathcal F}.
+ \label{eq:transport-framed-point}
+\end{equation}
+In particular $z$ supplies the integral level as well as the
+rational isocrystal marking.  If $C$ is the completed algebraic
+closure of $K$, its moduli description gives an actual
+$\mathcal J$-torsor $\mathcal P_z$, a Frobenius isomorphism
+bounded by $\mu$, and, for sufficiently large $a$, an isomorphism
+\[
+ i_a:\mathcal J_{Y_{[a,\infty)}(C^\flat)}
+      \xrightarrow{\ \sim\ }
+      \mathcal P_z|_{Y_{[a,\infty)}(C^\flat)},
+ \qquad
+ \varphi_{\mathcal P_z}\circ\varphi^*i_a
+       =i_a\circ\varphi_{b_x}.
+\]
+These are precisely the torsor and annular framing in
+\cite[\S3.1.1]{PappasRapoport2024}.  Its associated
+pro-\'{e}tale $\mathcal J(\Z_p)$-torsor gives integral tensor
+frames.  Thus the integral structure and Frobenius-compatible
+framing are part of the chosen point, with all their tensor
+compatibilities.
+
+\medskip\noindent
+\emph{Integral realization and its special fiber.}
+Our adapted datum is of global Hodge type: the group scheme is
+a full connected Bruhat--Tits stabilizer, and the chosen Hodge
+embedding is integral on $\Lambda_p$.
+The center also satisfies the split-rank hypothesis of
+\cite[\S4.1]{PappasRapoport2024}.  Indeed its real subgroup of
+multiplier one preserves the positive definite form
+$\psi_W(v,h(i)w)$ and is compact.  Its real split rank is
+therefore at most one, while the rational weight torus supplies
+one split central direction over $\Q$; both split ranks are one.
+The model here is the one in
+\cite[\S2.1.11 and Theorem~2.1.12]{KisinZhou2025}, obtained
+from this adapted Hodge embedding and satisfying the
+Pappas--Rapoport characterization.  Condition $(U_x)$ holds
+by \cite[Corollary~6.3]{GleasonLimXu2026}; see also
+\cite[Remark~4.10.4]{PappasRapoport2024}.  Consequently
+\cite[Theorem~4.10.6]{PappasRapoport2024} represents the integral
+local Shimura variety by a normal formal scheme
+$\mathfrak M_{\mathcal J,b_x,\mu}$ and gives the morphism
+\[
+ \Theta_x:\mathfrak M_{\mathcal J,b_x,\mu}
+          \longrightarrow\widehat{\mathcal S_K}.
+\]
+More precisely, \cite[\S4.10.2, (4.10.2)--(4.10.6)]{PappasRapoport2024}
+identifies this formal scheme with the tensor-compatible open and
+closed part of the pullback of the Siegel Rapoport--Zink space.
+This construction identifies the pullback of the universal
+$\mathcal J$-shtuka with its tautological framed shtuka and the
+faithful vector realization with the universal $p$-divisible group.
+
+Its rigid generic fiber is the local Shimura variety
+\cite[Proposition~3.5.1(i)]{PappasRapoport2024}.
+The classical point $z$ of this rigid generic fiber gives
+$\operatorname{Spf}\mathcal O_K\to\mathfrak M_{\mathcal J,b_x,\mu}$.
+Indeed it lies in the generic fiber of a formal affine chart;
+for a special formal chart one may take an affinoid member of
+its standard generic-fiber covering containing $z$.
+Evaluation of the formal chart's ring is integral, and its
+ideal of definition maps to topologically nilpotent elements of
+$\mathcal O_K$.  Let $z_0$ be the special point and put
+$y=\Theta_x(z_0)$ and $\widetilde y=\Theta_x(z)$.
+The formal map $\widetilde y$ gives a point of the integral model
+over $\mathcal O_K$ by restricting to an affine neighborhood of
+$y$.  Pullback of the universal abelian scheme gives
+$C_{\widetilde y}$ and its $p$-divisible group $\mathcal H_z$.
+
+We record explicitly the integral comparison at this point.
+For a uniformizer $\pi$ of $K$, put
+$\mathfrak S=W(k)[[u]]$, with $u\mapsto\pi$, and let
+$E_\pi(u)$ generate its kernel.  The crystalline representation
+and integral level of $z$ give the $\mathcal J$-Breuil--Kisin
+torsor of \cite[\S3.5]{PappasRapoport2024}, with
+\[
+ \Phi:\varphi^*\mathcal P_{\mathrm{BK}}[1/E_\pi]
+             \xrightarrow{\ \sim\ }
+             \mathcal P_{\mathrm{BK}}[1/E_\pi].
+\]
+Its pushout to $\GL(\Lambda_p)$ is the Breuil--Kisin module of
+$\mathcal H_z$.  Reduction at $u=0$, followed by the dual
+Frobenius descent fixed above, therefore gives the actual
+Dieudonn\'e lattice of $\mathcal H_{z,k}$.
+Proposition~3.5.1 of \cite{PappasRapoport2024} compares the two
+specializations of this already constructed framed point.  In
+the coordinates of \cite[Theorem~2.2.5]{KisinZhou2025} it gives
+\[
+ g\in X_{\mathcal J}(\mu^{-1},b_x),\qquad
+ \mathbb D(\mathcal H_{z,k})
+       =\rho^\vee(\sigma(g))\mathbb D_x\subset D_x.
+\]
+Indeed the Frobenius matrix on this lattice, written in its
+induced basis, is
+\[
+ \rho^\vee\!\left(
+       \sigma(g^{-1}b_x\sigma(g))\right)\circ\sigma,
+\]
+which explains the same Frobenius descent in the
+Deligne--Lusztig condition and in the lattice formula.
+
+\medskip\noindent
+\emph{The abelian marking and the filtration.}
+Keep the identity prime-to-$p$ coordinate in the uniformization.
+The description of its special fiber in
+\cite[\S4.10.3]{PappasRapoport2024}, or equivalently the
+isogeny-class map of \cite[Theorem~2.2.5]{KisinZhou2025}, gives
+$y=i_x(g)$ and a single abelian quasi-isogeny
+$b:C_x\to C_y$ preserving the crystalline tensors and every
+prime-to-$p$ tensor.  Its crystalline marking is the one just
+used for $\mathcal H_z$, because the uniformization is induced
+by the marked Siegel Rapoport--Zink space.
+
+The crystalline period map of that space is the Hodge filtration
+of its universal deformation pulled back by its marking.
+Compatibility of $\Theta_x$ with the tautological shtuka and
+\eqref{eq:transport-framed-point} consequently give
+\[
+ b^*\Fil^1H^1_{\mathrm{dR}}(C_{\widetilde y}/K)
+       =\mathcal F^1.
+\]
+The deformation criterion
+\cite[Proposition~2.2.3]{KisinZhou2025} is satisfied concretely:
+the filtration is of type $\mu$, and the crystalline structure
+tensors compare with the integral \'{e}tale tensors in the
+$\mathcal J(\Z_p)$-frames of $z$.  The bounded integral shtuka
+and its annular framing were obtained before using this
+criterion or comparing specializations.  All cited global
+Hodge-type and strongly admissible results apply at $p=2$ as
+well.
+
+Finally, under finite local-field descent, the weak-admissibility
+inequalities restrict to the descended Frobenius subobjects.
+Colmez--Fontaine's theorem
+\cite[Theorem~A]{ColmezFontaine2000} makes those descended data
+admissible, and scalar extension supplies condition~\textup{(1)}.
+\end{proof}
+
+\subsection{Proof of the transport proposition}
+
+\begin{proof}[Proof of Proposition~\ref{prop:transport}]
+\medskip\noindent
+\emph{The original point and its reduction.}
+Apply Lemma~\ref{lem:auxiliary-datum}.  Choose a rational
+$J$-equivariant identification
+$W=(V\otimes_\Q P)^{\oplus m}$ and set $n=m[P:\Q]$.
+Choose a basis of $P$ whose first vector is $1$.  The resulting
+identification of the Hodge structure at the diagonal point $h$
+with $V^{\oplus n}$ makes the first-factor map equal to
+\begin{equation}
+ v\longmapsto(v\otimes1,0,\ldots,0).
+ \label{eq:transport-first-factor}
+\end{equation}
+The rational homology/isogeny correspondence
+\cite[Corollary~6.9]{MilneShimura} therefore makes the abelian variety
+\(C\) at that point isogenous to \(A^n\). Fix a rational isogeny
+\(f:A^n\to C\) realizing this identification.
+
+We explain the field and reduction assertions for this starting
+point.  The Hodge embedding maps it to the Siegel point of the
+polarized variety $C$ with finite level.  The abelian variety $C$
+is defined over $\Qbar$: clearing a denominator in $f$ realizes
+it as a quotient of $A^n$ by a finite subgroup of its torsion, and
+every such subgroup is defined over $\Qbar$.  Its rational
+polarization and its finite level also descend to $\Qbar$, since
+homomorphisms between abelian varieties over $\Qbar$ do not
+enlarge after extending algebraically closed fields
+\cite[Corollary~20.4]{MilneAV}.  The Hodge
+embedding induces a finite morphism onto its
+image at sufficiently small finite level
+\cite[Theorem~5.16]{MilneShimura}. Its fiber over a
+$\Qbar$-point therefore consists of $\Qbar$-points.  This places
+the diagonal point on the canonical model over $\Qbar$.
+
+After a finite number-field extension the variety $C$ has good
+reduction, since it is isogenous to $A^n$
+\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}.  Its polarization and
+prime-to-$p$ level extend after a further finite extension.
+The integral Hodge-type model is the normalization of the closure
+of the generic fiber in the adapted Siegel model.  The valuative
+criterion for this finite normalization therefore extends our
+point to its integral model.  Denote the reduction by $x$.
+The rational isogeny $f$ specializes to a quasi-isogeny
+$f_0:A_0^n\to C_x$.
+
+\medskip\noindent
+\emph{The trace tensor and its descent.}
+Next extend the alternating form defining $\gamma$
+$P$-multilinearly and take its trace.  Composing with projection
+onto the first copy in \eqref{eq:transport-copies} gives
+\begin{equation}
+ \eta=\Tr_{P/\Q}(\gamma_P)\circ
+              (\operatorname{pr}_1)^{\otimes 2r}
+       \in\bigwedge^{2r}W^*.
+ \label{eq:transport-trace-tensor}
+\end{equation}
+For $r=0$ this is interpreted as the scalar trace.
+The equal-multiplier condition implies that $\eta$ has multiplier
+$c^r$ under $J$.  On \eqref{eq:transport-first-factor}, its
+restriction is
+\begin{equation}
+ j^*f^*\eta_C=[P:\Q]\gamma.
+ \label{eq:transport-trace-restriction}
+\end{equation}
+The class $\eta_C$ is understood with its Tate twist.
+
+We turn this semi-invariant into the untwisted tensors to which
+the integral-model theorems directly apply.  Write
+$\psi_W^{-1}\in\bigwedge^2W$ for the inverse symplectic
+bivector, and include
+\begin{equation}
+ u_\psi=\psi_W\otimes\psi_W^{-1},\qquad
+ u_\eta=\eta\otimes(\psi_W^{-1})^{\otimes r}
+ \label{eq:transport-untwisted-tensors}
+\end{equation}
+among the structure tensors.  Both are $J$-invariant tensors in
+ordinary tensor constructions of $W$ and $W^*$; no separately
+trivialized Tate object is needed.  They can be made integral and
+added to the defining family as explained above.  Their
+crystalline and prime-to-$p$ realizations at $x$ are the
+specializations from the original point, with the comparison
+conventions fixed above.
+
+\phantomsection
+\label{par:transport-tensor-descent}
+The entire finite family of structure tensors, including
+$u_\psi$ and $u_\eta$, has finite local-field descent.  To see
+this, choose a finite extension $L/\Q_p$ over whose valuation
+ring the original number-field point is defined, and let $L_0$
+be its maximal unramified subfield.  The canonical \'etale
+tensor sections of \cite[\S2.1.4]{KisinZhou2025}, pulled back
+to that point, are $\Gal(\overline L/L)$-invariant.  Applying
+the tensor functor $D_{\mathrm{cris},L}$ to these morphisms
+from the unit object places them in the crystalline tensor
+spaces over $L_0$.  Their extension to $\Kzero$ is exactly
+the family $s_{\alpha,0,x}$, by good-reduction comparison.
+The map $f_0$ also descends after a finite residue-field
+extension, so this descent is retained under its marking.
+Taking a compositum with the finite field of definition of a
+prescribed filtration therefore gives one finite local field
+for all the data.  No tensor-preserving trivialization over
+$L_0$ is being asserted or needed.
+
+\medskip\noindent
+\emph{The two choices of lift.}
+For specification~(1), apply
+\cite[Theorem~2.2.7]{KisinZhou2025} to $x$.  It gives a point
+$y\in\mathcal I_x$ with a special-point lift and a
+tensor-preserving quasi-isogeny $b:C_x\to C_y$.
+The special-point lift supplies a CM abelian variety $B$.  Special
+points on the canonical Shimura variety are algebraic over its
+reflex field by the canonical-model theory
+\cite[Definition~12.8 and Proposition~14.14]{MilneShimura}, so $B$
+and its level data can be taken over $\Qbar$.  The given local lift specifies their embedding at $w$
+and their good reduction $B_0=C_y$.
+
+For specification~(2), identify $D_x$ with $D_A^{\oplus n}$ using
+$f_0^*$ and prescribe the direct-sum filtration
+$\mathcal F_A^{\oplus n}$.  It is weakly admissible, with the
+required finite local-field descent.  It is also in the required
+$J$-cocharacter orbit.  Indeed a geometric element of the
+transported $G$ taking the original filtration on $D_A$ to
+$\mathcal F_A$ acts diagonally on all the copies of $V$ in $W$;
+its image in $J$ takes the original filtration on $D_x$ to the
+prescribed one.  The tensor-preserving identifications used here
+come from the initial point and comparison, so they include the
+$J$-structure tensors as well.  Geometric conjugacy can be
+realized over a finite extension, since the corresponding
+isomorphism scheme is of finite type.
+Lemma~\ref{lem:admissible-realization} now gives $y$, its lift
+$B$, and a tensor-preserving quasi-isogeny $b:C_x\to B_0$ with
+the prescribed marked filtration.  This proves
+\eqref{eq:transport-prescribed-filtration} for
+$a=bf_0$.
+
+\medskip\noindent
+\emph{A common rational normalization.}
+It remains, in both specifications, to recover the twisted class
+and normalize it by a single rational scalar.  Choose a complex
+description of the lifted Shimura point.  Its rational marking by
+$W$ supplies a rational Hodge class $\eta_B$ corresponding to
+$\eta$, and a rational divisor class $\theta_B$
+corresponding to $\psi_W$.  Write $\theta_C$ for the analogous
+class on $C$ and put
+\[
+ \theta_*=f_0^*\theta_{C,0}\in\NS(A_0^n)\otimes_\Z\Q.
+\]
+The class $\theta_*$ is rational ample.  Tensor preservation of
+$u_\psi$ says that, in each realization,
+\begin{equation}
+ a^*\theta_{B,0}=\lambda_v\theta_*,
+ \label{eq:transport-local-similitudes}
+\end{equation}
+with a nonzero local scalar $\lambda_v$.  Let $D=n\dim A$.
+Pairing with $\theta_*^{D-1}$ gives
+\begin{equation}
+ \lambda_v=
+ \frac{\deg\bigl(a^*\theta_{B,0}\cdot\theta_*^{D-1}\bigr)}
+      {\deg(\theta_*^D)}=:\lambda\in\Q^\times.
+ \label{eq:transport-rational-similitude}
+\end{equation}
+Both intersection numbers refer to rational divisor classes on
+the same abelian variety; their cycle-class traces are identical
+in every realization.  The denominator is positive because
+$\theta_*$ is ample, and the numerator is nonzero by
+\eqref{eq:transport-local-similitudes}.  Thus all the scalars in
+\eqref{eq:transport-local-similitudes}, including the crystalline
+one, are the single rational number $\lambda$.
+
+Preservation of $u_\eta$, together with
+\eqref{eq:transport-rational-similitude}, then yields
+\begin{equation}
+ a^*\eta_{B,0}=\lambda^r f_0^*\eta_{C,0}
+ \label{eq:transport-eta-scaling}
+\end{equation}
+simultaneously.  In fact the inverse bivector scales by
+$\lambda^{-1}$, so that the equality for the untwisted tensor
+in \eqref{eq:transport-untwisted-tensors} gives exactly
+\eqref{eq:transport-eta-scaling}.  Set
+\[
+ \kappa=\frac{\lambda^{-r}}{[P:\Q]}\eta_B.
+\]
+Specializing \eqref{eq:transport-trace-restriction} and using
+\eqref{eq:transport-eta-scaling} proves
+\eqref{eq:transport-class}.
+
+For the arbitrary local lift, the crystalline realization just
+used is unambiguous: the canonical model supplies the untwisted
+tensor $u_\eta$, and the algebraic polarization supplies its Tate
+line and inverse bivector.  The comparison isomorphism and these
+operations define $\eta_B$ in the crystalline realization and
+agree with its canonical \'{e}tale tensors.  Consequently any
+identity expressing this class by algebraic divisor operations
+specializes with the same rational coefficients in all
+realizations.  For the $\Qbar$-valued CM lift, these are also the
+usual absolute-Hodge and de Rham realizations.  This completes
+the two constructions.
+\end{proof}
+
+\begin{corollary}[CM transport of a Hodge class]
+\label{cor:cm-transport}
+Let \(A/\Qbar\) be an abelian variety of positive dimension with
+good reduction at \(w\), and let
+\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class,
+where \(0\leq r\leq\dim A\). There are a positive integer \(n\),
+a CM abelian variety \(B/\Qbar\) with good reduction at \(w\),
+a rational Hodge class \(\kappa\) of codimension \(r\) on \(B\),
+and a quasi-isogeny \(a:A_0^n\to B_0\) such that
+\[
+ \gamma_0=(aj)^*\kappa_0,
+\]
+where \(j:A_0\to A_0^n\) is the first-factor inclusion. This is an
+equality of the usual absolute-Hodge specializations in every
+prime-to-\(p\) realization and in crystalline cohomology extended
+to \(\Cw\).
+\end{corollary}
+
+\begin{proof}
+Choose a polarization \(\psi\) on \(V=H_1(A(\C),\Q)\), and let
+\(G=\MT(V)\). The Mumford--Tate group is connected reductive and
+is characterized by its Hodge tensors
+\cite[Section~3, Propositions~3.4 and~3.6]{Deligne1982}.
+The Hodge homomorphism and its \(G(\R)\)-conjugacy class define
+the Mumford--Tate Shimura datum of Hodge type, with faithful
+symplectic representation
+\[
+ G\hookrightarrow\GSp(V,\psi).
+\]
+The polarization supplies the rational weight and the Cartan
+involution condition. The definition of the Mumford--Tate group
+excludes an adjoint factor on which the Hodge homomorphism is
+trivial. The class \(\gamma\), with its Tate twist, is fixed by
+\(G\). Equivalently, its alternating form on \(V\) has multiplier
+\(c^r\), where \(c\) is the similitude character. Thus the first
+alternative of Proposition~\ref{prop:transport} applies and gives
+the asserted data and simultaneous equality.
+\end{proof}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/03_filtration.tex

@@ -1,482 +1,482 @@
-\section{An endomorphism-stable weakly admissible filtration}
-\label{sec:filtration}
-
-We construct a filtration on first crystalline cohomology that preserves an
-additional divisor.  The construction takes place entirely on the given
-isocrystal.  Its realization by an abelian lift will be a separate step.
-Generic transversality is a basic ingredient of the admissible-filtration
-existence theorems of Fontaine--Rapoport
-\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}.
-We impose the additional endomorphism and polarization constraints
-explicitly. The bounded-degree argument below chooses the required
-coordinates over a finite algebraic extension, while testing every
-subisocrystal over the fixed coefficient field.
-
-Throughout this section, $\Kzero=\operatorname{Frac}W(\overline{\F}_p)$,
-and valuations are normalized by $v(p)=1$.  If $D$ is an isocrystal over
-$\Kzero$ and $K/\Kzero$ is finite, a subspace $L\subset D_K$ defines the
-one-step filtration
-\[
- \Fil^iD_K=
- \begin{cases}
- D_K,&i\leq 0,\\
- L,&i=1,\\
- 0,&i\geq 2.
- \end{cases}
-\]
-For an isocrystal subobject $P\subset D$, its Hodge number for this
-filtration is $t_H(P)=\dim_K(P_K\cap L)$.  Its Newton number $t_N(P)$ is
-the sum of its slopes with multiplicities.  Weak admissibility means that
-$t_H(D)=t_N(D)$ and $t_H(P)\leq t_N(P)$ for every isocrystal subobject
-$P$ over $\Kzero$.
-
-Finite local-field descent is understood as in Section~\ref{sec:transport}:
-the isocrystal and its indicated tensors descend to the maximal unramified
-subfield of a finite extension of \(\Q_p\), and the filtration descends
-to that finite extension.
-
-\begin{theorem}[The filtration construction]
-\label{thm:filtration}
-Let $A_0/\overline{\F}_p$ be an abelian variety equipped with a rational
-action of a CM field $E$ and a polarization whose Rosati involution
-restricts to conjugation on $E$.  Suppose that
-\[
- D=H^1_{\mathrm{cris}}(A_0/W(\overline{\F}_p))[1/p]
-\]
-has rank $2s$ over $E\otimes_\Q\Kzero$, where $s\geq 1$.  Suppose also
-that there is an invertible $u\in\End^0(A_0)$ such that
-\[
- u^\dagger=u,\qquad ue=\bar e u\quad(e\in E).
-\]
-Let $\beta$ be the dual polarization pairing on $D$, with values in
-the isocrystal line of slope $1$.
-
-Then there are a finite extension $K/\Kzero$ and a subspace $L\subset
-D_K$ with the following properties:
-\begin{enumerate}
-\item The one-step filtration with $\Fil^1D_K=L$ is weakly admissible.
-\item $L$ is stable under the cohomological actions of $E$ and $u$,
-and is Lagrangian for $\beta$.
-\item After extending to an algebraically closed field $\Omega/K$,
-one has $\dim_\Omega L_\tau=s$ for every embedding
-$\tau:E\hookrightarrow\Omega$.
-\item The isocrystal, its indicated tensors, and this filtration have
-finite local field descent.
-\end{enumerate}
-Moreover, geometrically $L$ belongs to the balanced polarized
-cocharacter orbit.  More explicitly, define
-\[
- G_D=\left\{g\in\GSp(D,\beta)\cap
- \GL_{E\otimes_\Q\Kzero}(D):
- \det_{E\otimes_\Q\Kzero}(g)=c(g)^s\right\},
-\]
-where the scalar on the right is viewed in
-$(E\otimes_\Q\Kzero)^\times$.  The balanced $E$-stable
-$\beta$-Lagrangians form a single $G_D(\Omega)$-orbit.  Each is the
-weight-$1$ space of a cocharacter of $G_{D,\Omega}$ with weights $0,1$.
-In particular $L$ belongs to the orbit of any original filtration of
-this type under a tensor-preserving comparison.
-\end{theorem}
-
-The proof will construct \(L\) from Lagrangians in finitely many
-symplectic spaces. The two lemmas below supply the required simultaneous
-choice: minimum intersection with a subspace is ensured by a nonzero
-minor of bounded degree, and all such minors over a fixed coefficient
-field can be made nonzero over one finite extension.
-
-\begin{lemma}[Bounded-degree avoidance over a finite extension]
-\label{lem:avoidance}
-Let $K_1/\Kzero$ be a finite extension, and let $r,b\geq 1$ be integers.
-There exist an element $\theta$ algebraic over $\Q_p$ and a point
-\[
- (x_0,\ldots,x_{r-1})\in K_1(\theta)^r
-\]
-at which every nonzero polynomial in $K_1[X_0,\ldots,X_{r-1}]$ of total
-degree at most $b$ is nonzero.  The coordinates $x_i$ can all be chosen
-in $\Q_p(\theta)$.
-\end{lemma}
-
-\begin{proof}
-Choose an integer $N>b$ and put
-\[
- B=b\sum_{i=0}^{r-1}N^i.
-\]
-Let $e$ be the ramification index of $K_1/\Kzero$, so that
-$v(K_1^\times)=\frac1e\Z$.  Choose a prime $q>\max(B,e)$, and choose
-$\theta$ with $\theta^q=p$ in an algebraic closure of $K_1$.  The value
-$v(\theta)=1/q$ has order $q$ modulo $\frac1e\Z$.  Consequently the
-ramification index, and hence the degree, of $K_1(\theta)/K_1$ is at
-least $q$.  The polynomial $X^q-p$ gives the reverse inequality, so
-\[
- [K_1(\theta):K_1]=q>B.
-\]
-
-Set $x_i=\theta^{N^i}$.  Under the substitution $X_i=T^{N^i}$,
-distinct monomials of total degree at most $b$ have distinct exponents:
-each exponent of an individual variable is smaller than $N$, and
-uniqueness follows from base-$N$ expansion.  Thus a nonzero polynomial
-of total degree at most $b$ becomes a nonzero polynomial in $T$ of
-degree at most $B$.  It cannot vanish at $\theta$, whose degree over
-$K_1$ exceeds $B$.
-\end{proof}
-
-\begin{remark}
-The coefficient field in Lemma~\ref{lem:avoidance} is the fixed field
-$K_1$.  The conclusion concerns all bounded-degree polynomials over that
-field, with no restriction on their cardinality.  It would be false with
-$K_1(\theta)$ in place of $K_1$ as the coefficient field.
-\end{remark}
-
-\begin{lemma}[Intersections with Lagrangians]
-\label{lem:lagrangian-intersection}
-Let $M$ be a symplectic space of dimension $2m$ over a field $k$ of
-characteristic zero, and let $W\subset M$ have dimension $a$.  A nonempty
-open subset of the Lagrangian Grassmannian consists of subspaces $L$
-such that
-\[
- \dim(L\cap W)=\max(0,a-m).
-\]
-On any fixed symplectic affine big cell, at least one nonzero polynomial
-of degree at most $m$ has nonvanishing locus contained in this open
-subset.  This degree bound is independent of $W$.
-\end{lemma}
-
-\begin{proof}
-We first work over an algebraic closure.  The Lagrangian Grassmannian
-is geometrically integral of dimension $m(m+1)/2$; its affine big cells
-are parametrized by symmetric $m\times m$ matrices.  If $1\leq a\leq m$,
-consider the projective incidence variety of pairs $(\ell,L)$ with
-$\ell\subset W$ a line and $\ell\subset L$ a Lagrangian.  The fiber over
-$\ell$ is the Lagrangian Grassmannian of the symplectic space
-$\ell^\perp/\ell$, and therefore the incidence variety has dimension
-\[
- a-1+\frac{m(m-1)}2<\frac{m(m+1)}2.
-\]
-Its image is a proper closed subset.  Its complement consists precisely
-of the Lagrangians missing $W$.  The case $a=0$ is immediate.  For
-$a>m$, use
-\[
- \dim(L\cap W)=a-m+\dim(L\cap W^\perp)
-\]
-and apply the preceding argument to $W^\perp$.
-
-For the assertion on equations, choose coordinates on the fixed big
-cell in which $L$ is the column space of
-\[
- \begin{pmatrix}I_m\\ T\end{pmatrix},\qquad T=T^{\mathsf t}.
-\]
-The minimum-intersection condition says that the composite of this
-matrix with the quotient map $M\longrightarrow M/W$ has rank
-$\min(m,2m-a)$.  Its entries are affine linear polynomials in the
-independent entries of $T$.  The nonempty open subset already proved
-meets this big cell, so some minor of the required size is a nonzero
-polynomial over $k$.  Its degree is at most $m$, and its nonvanishing
-implies the required rank.  If the required rank is zero, use the
-constant polynomial $1$.
-\end{proof}
-
-
-The two lemmas reduce simultaneous transversality to a uniform bound on
-the degrees of the relevant minors. We now identify the symplectic
-spaces on which to apply them and the inequalities they must enforce.
-
-\begin{proof}[Proof of Theorem~\ref{thm:filtration}]
-We write $e$ and $u$ also for their cohomological actions on $D$.
-These actions commute with Frobenius and satisfy
-\begin{equation}
-\label{eq:filtration-adjoints}
- \beta(ux,y)=\beta(x,uy),\qquad
- \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u.
-\end{equation}
-There is no change to the last relation from contravariance.  Indeed,
-pulling back $ue=\bar e u$ gives $e^*u^*=u^*\bar e^*$; replacing $e$
-by $\bar e$ gives $u^*e^*=\bar e^*u^*$.  Likewise, if $J$ is the
-homological polarization matrix, $U^{\mathsf t}J=JU$ implies
-$UJ^{-1}=J^{-1}U^{\mathsf t}$, which is the self-adjointness of the
-cohomological action $U^{\mathsf t}$ for the dual pairing.  The
-Rosati-conjugate adjoint identity for $E$ follows in the same way.
-
-The pullback $U^{\mathsf t}$ of an individual endomorphism is
-distinct from the contragredient group action $g\mapsto g^{-\mathsf t}$.
-If a homological similitude $g$ has multiplier $c_V$ and satisfies
-$\det_E(g)=c_V^s$, its dual action has multiplier $c_D=c_V^{-1}$
-and satisfies $\det_E(g^{-\mathsf t})=c_D^s$.
-Thus $G_D$ is the dual form of the homological group defined by this
-determinant condition, with its cohomological multiplier.
-
-\medskip\noindent
-\emph{Reduction to the stable subobjects.}
-First consider any $E,u$-stable candidate $L\subset D_K$, where
-$K/\Kzero$ is finite. We show that the weak-admissibility inequalities
-need only be tested on $E,u$-stable subobjects. This resembles the use of canonical
-destabilizing objects in \cite[Section~2]{FontaineRapoport2005},
-but we use total defect, not defect divided by dimension, and give
-the required elementary argument.
-For any isocrystal subobject $P\subset D$ over $\Kzero$, set
-\[
- f(P)=t_H(P)-t_N(P).
-\]
-These values form a finite set: the Hodge number is a bounded integer,
-and the Newton number is determined by bounded integer multiplicities
-among the finitely many slopes of $D$.  Let $a$ be their maximum.
-Newton numbers are additive in short exact sequences, while
-\[
- (P_K\cap L)+(Q_K\cap L)\subset (P+Q)_K\cap L.
-\]
-Consequently
-\begin{equation}
-\label{eq:filtration-supermodularity}
- f(P)+f(Q)\leq f(P\cap Q)+f(P+Q).
-\end{equation}
-If $P$ and $Q$ both attain $a$, the right-hand terms are each at
-most $a$, so both attain $a$.  Thus maximizers are closed under
-intersection and sum.  Choose a maximizer $P_{\max}$ of largest
-dimension.  Adding any other maximizer to it shows that the other
-one is contained in it.  Hence $P_{\max}$ is the unique largest
-maximizer.
-
-The operators $u$ and $e\in E^\times$ are filtration-preserving
-isocrystal automorphisms.  They preserve $f$ and hence fix
-$P_{\max}$, by its uniqueness.  Thus $P_{\max}$ is $E,u$-stable.
-Consequently, if $t_H(D')\leq t_N(D')$ for every $E,u$-stable
-subobject $D'$, then $a=f(P_{\max})\leq0$. All the
-weak-admissibility inequalities follow. We must therefore construct
-an $E,u$-stable Lagrangian with these restricted inequalities and
-$t_H(D)=t_N(D)$.
-
-\medskip\noindent
-\emph{Semisimplicity and symplectic blocks.}
-The Rosati trace form
-\[
- \langle a,b\rangle=\Tr(ab^\dagger)
- \quad\text{on }\End^0(A_0)\otimes_\Q\R
-\]
-is positive definite \cite[Theorem~17.3]{MilneAV}.  Left multiplication
-by $u$ is self-adjoint for this form, since trace cyclicity gives
-\[
- \langle ua,b\rangle=\Tr(uab^\dagger)
- =\Tr(ab^\dagger u)=\langle a,ub\rangle.
-\]
-It is therefore diagonalizable.  Its minimal polynomial equals the
-minimal polynomial of $u$ in the endomorphism algebra, as is seen by
-applying a polynomial in the multiplication operator to the identity.
-Thus $u$, and hence $u^2$, acts semisimply on $D$.  The relation in
-\eqref{eq:filtration-adjoints} also shows that $u^2$ commutes with $E$.
-
-Put
-\[
- B(x,y)=\beta(x,uy).
-\]
-This is a nondegenerate alternating pairing.  Indeed, invertibility is
-immediate, and
-$B(y,x)=-\beta(ux,y)=-B(x,y)$.  In addition,
-\[
- B(ex,y)=\beta(x,\bar euy)=\beta(x,uey)=B(x,ey),
- \qquad B(u^2x,y)=B(x,u^2y).
-\]
-After a finite splitting extension $K_1/\Kzero$, write
-\[
- D_{K_1}=\bigoplus_{\tau,z}M_{\tau,z},
- \qquad
- M_{\tau,z}=\{x:ex=\tau(e)x\ (e\in E),\ u^2x=zx\}.
-\]
-We omit zero summands.  The self-adjoint identities for $B$ show that
-distinct summands are orthogonal.  Its restriction to each summand is
-therefore nondegenerate.  Write $\dim M_{\tau,z}=2m_{\tau,z}$.
-
-For $\bar\tau=\tau\circ\overline{(\cdot)}$, the operator $u$ maps
-$M_{\tau,z}$ isomorphically to $M_{\bar\tau,z}$.  On these spaces,
-\begin{equation}
-\label{eq:filtration-u-similitude}
- B(ux,uy)=zB(x,y).
-\end{equation}
-Here $z\neq 0$, and $\tau\neq\bar\tau$.  Choose a $B$-Lagrangian
-$L_{\tau,z}$ in one member of each opposite pair, and prescribe
-\begin{equation}
-\label{eq:filtration-opposite-choice}
- L_{\bar\tau,z}=uL_{\tau,z}.
-\end{equation}
-Equation~\eqref{eq:filtration-u-similitude} shows that the latter is
-Lagrangian as well.  Their direct sum $L$ is $E$-stable and $u$-stable;
-on applying $u$ twice, one obtains $zL_{\tau,z}=L_{\tau,z}$.  Since
-\[
- \beta(x,y)=B(x,u^{-1}y),
-\]
-$L$ is also $\beta$-Lagrangian.  Its dimension at each $E$-embedding
-is $s$.  It remains to make the independent choices appropriately.
-
-\medskip\noindent
-\emph{The inherited Newton grading.}
-Use the canonical slope decomposition
-\[
- D=\bigoplus_\lambda D_\lambda.
-\]
-The isocrystal category over $\overline{\F}_p$ is semisimple, and
-this decomposition is functorial
-\cite[\S3.2, Proposition~3.3]{ColmezFontaine2000}.  The slopes of $D$
-belong to $[0,1]$: its integral Dieudonn\'e operators \(F,V\)
-satisfy \(FV=p\), so both the slopes of \(F\) and those of \(V\)
-are nonnegative.  All the endomorphisms above preserve every
-$D_\lambda$, and the projectors onto the simultaneous eigenspaces
-preserve its scalar extension.  Thus each block has a grading
-\[
- M_{\tau,z}=\bigoplus_\lambda M_{\tau,z,\lambda},\qquad
- M_{\tau,z,\lambda}=M_{\tau,z}\cap(D_\lambda)_{K_1}.
-\]
-We use this grading even if Frobenius permutes the labels $(\tau,z)$.
-No extension of Frobenius to $K_1$, and no isocrystal structure on an
-individual block, is required.
-
-Before scalar extension, $B$ is a pairing of isocrystals into the
-slope-$1$ line.  Hence $B(D_\lambda,D_\mu)=0$ unless
-$\lambda+\mu=1$.  Nondegeneracy on an individual block $M$ then gives
-perfect pairings between $M_\lambda$ and $M_{1-\lambda}$.  In
-particular, if $\dim M=2m$,
-\begin{equation}
-\label{eq:filtration-block-newton}
- \sum_\lambda\lambda\dim M_\lambda=m.
-\end{equation}
-
-Let $D'\subset D$ be an isocrystal subobject over $\Kzero$ stable
-under $E$ and $u$.  Its intersections $M'=D'_{K_1}\cap M$ decompose
-$D'_{K_1}$ and inherit the same slope grading.  Put $a=\dim M'$.
-The slope sum of the graded quotient $M/M'$ is at most $2m-a$,
-because all its slopes belong to $[0,1]$.  By
-\eqref{eq:filtration-block-newton},
-\begin{equation}
-\label{eq:filtration-block-bound}
- \sum_\lambda\lambda\dim M'_\lambda
- \geq \max(0,a-m).
-\end{equation}
-The additional bound by zero follows from the nonnegative slopes.
-The right side is also the smallest possible dimension of the
-intersection of a Lagrangian in $M$ with $M'$.
-Thus attaining this minimum in every block will give the
-weak-admissibility inequalities for all the stable subobjects.
-
-\medskip\noindent
-\emph{Simultaneous choices and finite local field descent.}
-We may arrange the splitting field and coordinates used above as
-follows.  Choose a finite field $k_0\subset\overline{\F}_p$ over
-which $A_0$, its polarization, $u$, and a rational basis of $E$ are
-defined.  The underlying isocrystal and the indicated tensors then
-descend to $\operatorname{Frac}W(k_0)$.  Choose a finite local field
-$H/\Q_p$ containing this field and splitting the actions of $E$ and
-$u^2$.  The eigenspaces and their symplectic bases can be constructed
-over $H$ by linear algebra.  We take $K_1=\Kzero H$.  In particular,
-the independent Lagrangian Grassmannians, fixed symplectic big cells
-in them, and the maps induced by $u$ all have coordinates over $H$.
-
-Take the product of these big cells, one for each independent choice
-in \eqref{eq:filtration-opposite-choice}, and let $r$ be its number
-of affine coordinates.  For every $E,u$-stable isocrystal subobject
-$D'$ over $\Kzero$ and each independent block $M$, the space
-$M'=D'_{K_1}\cap M$ is defined over the \emph{fixed} field $K_1$.
-Lemma~\ref{lem:lagrangian-intersection} supplies a nonzero polynomial
-over $K_1$, of degree at most $m$, whose nonvanishing ensures
-\begin{equation}
-\label{eq:filtration-minimal-intersection}
- \dim(L_{\tau,z}\cap M')=\max(0,\dim M'-m).
-\end{equation}
-We regard this as a polynomial on the whole product by ignoring the
-other coordinates.  A single bound $b=\max m_{\tau,z}$ works for all
-these polynomials, independently of $D'$.  On the opposite block,
-the same intersection equality follows from $u$-stability of $D'$
-and \eqref{eq:filtration-opposite-choice}.
-
-Apply Lemma~\ref{lem:avoidance} to $K_1,r,b$.  Its one point satisfies
-\eqref{eq:filtration-minimal-intersection} for every one of the
-subobjects just specified, simultaneously.  This use of the lemma
-does not assume that the collection of subobjects is countable or
-that its members descend to $H$.  Their definition over $K_1$ and
-the uniform polynomial degree bound are the only requirements.
-There are nonzero blocks with $m\geq1$, so $r,b\geq1$ as required.
-
-Put $K=K_1(\theta)=\Kzero H(\theta)$.  The resulting Lagrangian is
-defined over $H(\theta)$ in the chosen coordinates, and $H(\theta)$
-is finite over $\Q_p$.  Extending the original isocrystal to the
-maximal unramified subfield of $H(\theta)$ therefore gives finite
-local field descent of the complete filtered data.  In particular,
-the construction has not obtained its generic point by introducing
-transcendental parameters.
-
-Summing \eqref{eq:filtration-minimal-intersection} and
-\eqref{eq:filtration-block-bound} over all blocks gives
-\begin{equation}
-\label{eq:filtration-stable-test}
- t_H(D')\leq t_N(D')
- \quad\text{for every $E,u$-stable isocrystal subobject $D'\subset D$.}
-\end{equation}
-For $D$ itself, both numbers equal $\frac12\dim D$ by
-\eqref{eq:filtration-block-newton}. The reduction to stable subobjects
-proved at the start now gives all the weak-admissibility inequalities
-for subobjects of $D$ over $\Kzero$. Each subobject over the smaller unramified
-field of descent extends to one of these subobjects. Hence the
-descended filtered isocrystal is weakly admissible as well.
-
-\medskip\noindent
-\emph{The geometric cocharacter orbit.}
-Over $\Omega$, the polarization pairs $D_\tau$ only with
-$D_{\bar\tau}$.  A balanced $E$-stable Lagrangian is determined by an
-$s$-dimensional subspace of $D_\tau$ for one member of each pair;
-the subspace in the opposite member is its annihilator.  The
-subgroup of $G_D$ with multiplier $1$ contains the product of
-$\SL(D_\tau)$ acting on opposite spaces by the inverse dual
-representations.  These special linear groups are transitive on
-the $s$-dimensional subspaces.  Thus all the indicated Lagrangians
-form one $G_D(\Omega)$-orbit.
-
-Finally choose an $E$-stable $\beta$-Lagrangian complement to one
-such $L$.  Such a complement is obtained separately on each pair
-of opposite embedding spaces.  Acting by $t$ on $L$ and by $1$
-on the complement gives a cocharacter with multiplier $t$ and
-$E$-determinant $t^s$ at each embedding.  It lies in $G_D$ and has
-the stated filtration.  This proves the final assertions.
-\end{proof}
-
-\begin{corollary}
-\label{cor:filtration-admissible}
-The filtration of Theorem~\ref{thm:filtration}, with its finite local
-field descent, is admissible: the descended filtered isocrystal is
-the filtered crystalline module of a crystalline representation.
-\end{corollary}
-
-\begin{proof}
-Apply the weakly-admissible-implies-admissible theorem of
-Colmez--Fontaine with monodromy operator zero
-\cite[Theorem~A]{ColmezFontaine2000}.
-\end{proof}
-
-\begin{corollary}[The divisor filtration condition]
-\label{cor:filtration-divisor}
-Let \(D,L\) be as in Theorem~\ref{thm:filtration}, and let \(\psi\)
-and \(U\) denote the homological polarization and the homological
-action of \(u\) on \(D_K^\vee\). The untwisted divisor form
-\(\psi(\cdot,U\cdot)\) belongs to
-\[
- \Fil^1\bigl(\bigwedge^2D_K\bigr)=L\wedge D_K,
-\]
-and hence to filtration degree zero after its codimension-one Tate
-twist. If a marked power is realized by an abelian lift, the
-corresponding pulled-back divisor satisfies the filtration condition
-for its symmetric homomorphism in Lemma~\ref{lem:lift-hom}.
-\end{corollary}
-
-\begin{proof}
-Let \(H\subset D_K^\vee\) be the annihilator of \(L\). It is
-Lagrangian for \(\psi\) and stable under \(U\): for \(a\in L\)
-and \(h\in H\),
-\[
- a(Uh)=(u^*a)(h)=0.
-\]
-Therefore \(\psi(h,Uh')=0\) for \(h,h'\in H\). The divisor form
-thus lies in
-\[
- \ker\bigl(\bigwedge^2D_K\longrightarrow\bigwedge^2H^\vee\bigr)
-       =L\wedge D_K.
-\]
-This proves the assertion on the fixed filtered vector space. After
-realization of a marked power, pullback respects this filtration
-condition, which is precisely the condition on the associated
-symmetric homomorphism in Lemma~\ref{lem:lift-hom}.
-\end{proof}
+\section{An endomorphism-stable weakly admissible filtration}
+\label{sec:filtration}
+
+We construct a filtration on first crystalline cohomology that preserves an
+additional divisor.  The construction takes place entirely on the given
+isocrystal.  Its realization by an abelian lift will be a separate step.
+Generic transversality is a basic ingredient of the admissible-filtration
+existence theorems of Fontaine--Rapoport
+\cite[Theorem~1 and Lemmas~1.1--1.2]{FontaineRapoport2005}.
+We impose the additional endomorphism and polarization constraints
+explicitly. The bounded-degree argument below chooses the required
+coordinates over a finite algebraic extension, while testing every
+subisocrystal over the fixed coefficient field.
+
+Throughout this section, $\Kzero=\operatorname{Frac}W(\overline{\F}_p)$,
+and valuations are normalized by $v(p)=1$.  If $D$ is an isocrystal over
+$\Kzero$ and $K/\Kzero$ is finite, a subspace $L\subset D_K$ defines the
+one-step filtration
+\[
+ \Fil^iD_K=
+ \begin{cases}
+ D_K,&i\leq 0,\\
+ L,&i=1,\\
+ 0,&i\geq 2.
+ \end{cases}
+\]
+For an isocrystal subobject $P\subset D$, its Hodge number for this
+filtration is $t_H(P)=\dim_K(P_K\cap L)$.  Its Newton number $t_N(P)$ is
+the sum of its slopes with multiplicities.  Weak admissibility means that
+$t_H(D)=t_N(D)$ and $t_H(P)\leq t_N(P)$ for every isocrystal subobject
+$P$ over $\Kzero$.
+
+Finite local-field descent is understood as in Section~\ref{sec:transport}:
+the isocrystal and its indicated tensors descend to the maximal unramified
+subfield of a finite extension of \(\Q_p\), and the filtration descends
+to that finite extension.
+
+\begin{theorem}[The filtration construction]
+\label{thm:filtration}
+Let $A_0/\overline{\F}_p$ be an abelian variety equipped with a rational
+action of a CM field $E$ and a polarization whose Rosati involution
+restricts to conjugation on $E$.  Suppose that
+\[
+ D=H^1_{\mathrm{cris}}(A_0/W(\overline{\F}_p))[1/p]
+\]
+has rank $2s$ over $E\otimes_\Q\Kzero$, where $s\geq 1$.  Suppose also
+that there is an invertible $u\in\End^0(A_0)$ such that
+\[
+ u^\dagger=u,\qquad ue=\bar e u\quad(e\in E).
+\]
+Let $\beta$ be the dual polarization pairing on $D$, with values in
+the isocrystal line of slope $1$.
+
+Then there are a finite extension $K/\Kzero$ and a subspace $L\subset
+D_K$ with the following properties:
+\begin{enumerate}
+\item The one-step filtration with $\Fil^1D_K=L$ is weakly admissible.
+\item $L$ is stable under the cohomological actions of $E$ and $u$,
+and is Lagrangian for $\beta$.
+\item After extending to an algebraically closed field $\Omega/K$,
+one has $\dim_\Omega L_\tau=s$ for every embedding
+$\tau:E\hookrightarrow\Omega$.
+\item The isocrystal, its indicated tensors, and this filtration have
+finite local field descent.
+\end{enumerate}
+Moreover, geometrically $L$ belongs to the balanced polarized
+cocharacter orbit.  More explicitly, define
+\[
+ G_D=\left\{g\in\GSp(D,\beta)\cap
+ \GL_{E\otimes_\Q\Kzero}(D):
+ \det_{E\otimes_\Q\Kzero}(g)=c(g)^s\right\},
+\]
+where the scalar on the right is viewed in
+$(E\otimes_\Q\Kzero)^\times$.  The balanced $E$-stable
+$\beta$-Lagrangians form a single $G_D(\Omega)$-orbit.  Each is the
+weight-$1$ space of a cocharacter of $G_{D,\Omega}$ with weights $0,1$.
+In particular $L$ belongs to the orbit of any original filtration of
+this type under a tensor-preserving comparison.
+\end{theorem}
+
+The proof will construct \(L\) from Lagrangians in finitely many
+symplectic spaces. The two lemmas below supply the required simultaneous
+choice: minimum intersection with a subspace is ensured by a nonzero
+minor of bounded degree, and all such minors over a fixed coefficient
+field can be made nonzero over one finite extension.
+
+\begin{lemma}[Bounded-degree avoidance over a finite extension]
+\label{lem:avoidance}
+Let $K_1/\Kzero$ be a finite extension, and let $r,b\geq 1$ be integers.
+There exist an element $\theta$ algebraic over $\Q_p$ and a point
+\[
+ (x_0,\ldots,x_{r-1})\in K_1(\theta)^r
+\]
+at which every nonzero polynomial in $K_1[X_0,\ldots,X_{r-1}]$ of total
+degree at most $b$ is nonzero.  The coordinates $x_i$ can all be chosen
+in $\Q_p(\theta)$.
+\end{lemma}
+
+\begin{proof}
+Choose an integer $N>b$ and put
+\[
+ B=b\sum_{i=0}^{r-1}N^i.
+\]
+Let $e$ be the ramification index of $K_1/\Kzero$, so that
+$v(K_1^\times)=\frac1e\Z$.  Choose a prime $q>\max(B,e)$, and choose
+$\theta$ with $\theta^q=p$ in an algebraic closure of $K_1$.  The value
+$v(\theta)=1/q$ has order $q$ modulo $\frac1e\Z$.  Consequently the
+ramification index, and hence the degree, of $K_1(\theta)/K_1$ is at
+least $q$.  The polynomial $X^q-p$ gives the reverse inequality, so
+\[
+ [K_1(\theta):K_1]=q>B.
+\]
+
+Set $x_i=\theta^{N^i}$.  Under the substitution $X_i=T^{N^i}$,
+distinct monomials of total degree at most $b$ have distinct exponents:
+each exponent of an individual variable is smaller than $N$, and
+uniqueness follows from base-$N$ expansion.  Thus a nonzero polynomial
+of total degree at most $b$ becomes a nonzero polynomial in $T$ of
+degree at most $B$.  It cannot vanish at $\theta$, whose degree over
+$K_1$ exceeds $B$.
+\end{proof}
+
+\begin{remark}
+The coefficient field in Lemma~\ref{lem:avoidance} is the fixed field
+$K_1$.  The conclusion concerns all bounded-degree polynomials over that
+field, with no restriction on their cardinality.  It would be false with
+$K_1(\theta)$ in place of $K_1$ as the coefficient field.
+\end{remark}
+
+\begin{lemma}[Intersections with Lagrangians]
+\label{lem:lagrangian-intersection}
+Let $M$ be a symplectic space of dimension $2m$ over a field $k$ of
+characteristic zero, and let $W\subset M$ have dimension $a$.  A nonempty
+open subset of the Lagrangian Grassmannian consists of subspaces $L$
+such that
+\[
+ \dim(L\cap W)=\max(0,a-m).
+\]
+On any fixed symplectic affine big cell, at least one nonzero polynomial
+of degree at most $m$ has nonvanishing locus contained in this open
+subset.  This degree bound is independent of $W$.
+\end{lemma}
+
+\begin{proof}
+We first work over an algebraic closure.  The Lagrangian Grassmannian
+is geometrically integral of dimension $m(m+1)/2$; its affine big cells
+are parametrized by symmetric $m\times m$ matrices.  If $1\leq a\leq m$,
+consider the projective incidence variety of pairs $(\ell,L)$ with
+$\ell\subset W$ a line and $\ell\subset L$ a Lagrangian.  The fiber over
+$\ell$ is the Lagrangian Grassmannian of the symplectic space
+$\ell^\perp/\ell$, and therefore the incidence variety has dimension
+\[
+ a-1+\frac{m(m-1)}2<\frac{m(m+1)}2.
+\]
+Its image is a proper closed subset.  Its complement consists precisely
+of the Lagrangians missing $W$.  The case $a=0$ is immediate.  For
+$a>m$, use
+\[
+ \dim(L\cap W)=a-m+\dim(L\cap W^\perp)
+\]
+and apply the preceding argument to $W^\perp$.
+
+For the assertion on equations, choose coordinates on the fixed big
+cell in which $L$ is the column space of
+\[
+ \begin{pmatrix}I_m\\ T\end{pmatrix},\qquad T=T^{\mathsf t}.
+\]
+The minimum-intersection condition says that the composite of this
+matrix with the quotient map $M\longrightarrow M/W$ has rank
+$\min(m,2m-a)$.  Its entries are affine linear polynomials in the
+independent entries of $T$.  The nonempty open subset already proved
+meets this big cell, so some minor of the required size is a nonzero
+polynomial over $k$.  Its degree is at most $m$, and its nonvanishing
+implies the required rank.  If the required rank is zero, use the
+constant polynomial $1$.
+\end{proof}
+
+
+The two lemmas reduce simultaneous transversality to a uniform bound on
+the degrees of the relevant minors. We now identify the symplectic
+spaces on which to apply them and the inequalities they must enforce.
+
+\begin{proof}[Proof of Theorem~\ref{thm:filtration}]
+We write $e$ and $u$ also for their cohomological actions on $D$.
+These actions commute with Frobenius and satisfy
+\begin{equation}
+\label{eq:filtration-adjoints}
+ \beta(ux,y)=\beta(x,uy),\qquad
+ \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u.
+\end{equation}
+There is no change to the last relation from contravariance.  Indeed,
+pulling back $ue=\bar e u$ gives $e^*u^*=u^*\bar e^*$; replacing $e$
+by $\bar e$ gives $u^*e^*=\bar e^*u^*$.  Likewise, if $J$ is the
+homological polarization matrix, $U^{\mathsf t}J=JU$ implies
+$UJ^{-1}=J^{-1}U^{\mathsf t}$, which is the self-adjointness of the
+cohomological action $U^{\mathsf t}$ for the dual pairing.  The
+Rosati-conjugate adjoint identity for $E$ follows in the same way.
+
+The pullback $U^{\mathsf t}$ of an individual endomorphism is
+distinct from the contragredient group action $g\mapsto g^{-\mathsf t}$.
+If a homological similitude $g$ has multiplier $c_V$ and satisfies
+$\det_E(g)=c_V^s$, its dual action has multiplier $c_D=c_V^{-1}$
+and satisfies $\det_E(g^{-\mathsf t})=c_D^s$.
+Thus $G_D$ is the dual form of the homological group defined by this
+determinant condition, with its cohomological multiplier.
+
+\medskip\noindent
+\emph{Reduction to the stable subobjects.}
+First consider any $E,u$-stable candidate $L\subset D_K$, where
+$K/\Kzero$ is finite. We show that the weak-admissibility inequalities
+need only be tested on $E,u$-stable subobjects. This resembles the use of canonical
+destabilizing objects in \cite[Section~2]{FontaineRapoport2005},
+but we use total defect, not defect divided by dimension, and give
+the required elementary argument.
+For any isocrystal subobject $P\subset D$ over $\Kzero$, set
+\[
+ f(P)=t_H(P)-t_N(P).
+\]
+These values form a finite set: the Hodge number is a bounded integer,
+and the Newton number is determined by bounded integer multiplicities
+among the finitely many slopes of $D$.  Let $a$ be their maximum.
+Newton numbers are additive in short exact sequences, while
+\[
+ (P_K\cap L)+(Q_K\cap L)\subset (P+Q)_K\cap L.
+\]
+Consequently
+\begin{equation}
+\label{eq:filtration-supermodularity}
+ f(P)+f(Q)\leq f(P\cap Q)+f(P+Q).
+\end{equation}
+If $P$ and $Q$ both attain $a$, the right-hand terms are each at
+most $a$, so both attain $a$.  Thus maximizers are closed under
+intersection and sum.  Choose a maximizer $P_{\max}$ of largest
+dimension.  Adding any other maximizer to it shows that the other
+one is contained in it.  Hence $P_{\max}$ is the unique largest
+maximizer.
+
+The operators $u$ and $e\in E^\times$ are filtration-preserving
+isocrystal automorphisms.  They preserve $f$ and hence fix
+$P_{\max}$, by its uniqueness.  Thus $P_{\max}$ is $E,u$-stable.
+Consequently, if $t_H(D')\leq t_N(D')$ for every $E,u$-stable
+subobject $D'$, then $a=f(P_{\max})\leq0$. All the
+weak-admissibility inequalities follow. We must therefore construct
+an $E,u$-stable Lagrangian with these restricted inequalities and
+$t_H(D)=t_N(D)$.
+
+\medskip\noindent
+\emph{Semisimplicity and symplectic blocks.}
+The Rosati trace form
+\[
+ \langle a,b\rangle=\Tr(ab^\dagger)
+ \quad\text{on }\End^0(A_0)\otimes_\Q\R
+\]
+is positive definite \cite[Theorem~17.3]{MilneAV}.  Left multiplication
+by $u$ is self-adjoint for this form, since trace cyclicity gives
+\[
+ \langle ua,b\rangle=\Tr(uab^\dagger)
+ =\Tr(ab^\dagger u)=\langle a,ub\rangle.
+\]
+It is therefore diagonalizable.  Its minimal polynomial equals the
+minimal polynomial of $u$ in the endomorphism algebra, as is seen by
+applying a polynomial in the multiplication operator to the identity.
+Thus $u$, and hence $u^2$, acts semisimply on $D$.  The relation in
+\eqref{eq:filtration-adjoints} also shows that $u^2$ commutes with $E$.
+
+Put
+\[
+ B(x,y)=\beta(x,uy).
+\]
+This is a nondegenerate alternating pairing.  Indeed, invertibility is
+immediate, and
+$B(y,x)=-\beta(ux,y)=-B(x,y)$.  In addition,
+\[
+ B(ex,y)=\beta(x,\bar euy)=\beta(x,uey)=B(x,ey),
+ \qquad B(u^2x,y)=B(x,u^2y).
+\]
+After a finite splitting extension $K_1/\Kzero$, write
+\[
+ D_{K_1}=\bigoplus_{\tau,z}M_{\tau,z},
+ \qquad
+ M_{\tau,z}=\{x:ex=\tau(e)x\ (e\in E),\ u^2x=zx\}.
+\]
+We omit zero summands.  The self-adjoint identities for $B$ show that
+distinct summands are orthogonal.  Its restriction to each summand is
+therefore nondegenerate.  Write $\dim M_{\tau,z}=2m_{\tau,z}$.
+
+For $\bar\tau=\tau\circ\overline{(\cdot)}$, the operator $u$ maps
+$M_{\tau,z}$ isomorphically to $M_{\bar\tau,z}$.  On these spaces,
+\begin{equation}
+\label{eq:filtration-u-similitude}
+ B(ux,uy)=zB(x,y).
+\end{equation}
+Here $z\neq 0$, and $\tau\neq\bar\tau$.  Choose a $B$-Lagrangian
+$L_{\tau,z}$ in one member of each opposite pair, and prescribe
+\begin{equation}
+\label{eq:filtration-opposite-choice}
+ L_{\bar\tau,z}=uL_{\tau,z}.
+\end{equation}
+Equation~\eqref{eq:filtration-u-similitude} shows that the latter is
+Lagrangian as well.  Their direct sum $L$ is $E$-stable and $u$-stable;
+on applying $u$ twice, one obtains $zL_{\tau,z}=L_{\tau,z}$.  Since
+\[
+ \beta(x,y)=B(x,u^{-1}y),
+\]
+$L$ is also $\beta$-Lagrangian.  Its dimension at each $E$-embedding
+is $s$.  It remains to make the independent choices appropriately.
+
+\medskip\noindent
+\emph{The inherited Newton grading.}
+Use the canonical slope decomposition
+\[
+ D=\bigoplus_\lambda D_\lambda.
+\]
+The isocrystal category over $\overline{\F}_p$ is semisimple, and
+this decomposition is functorial
+\cite[\S3.2, Proposition~3.3]{ColmezFontaine2000}.  The slopes of $D$
+belong to $[0,1]$: its integral Dieudonn\'e operators \(F,V\)
+satisfy \(FV=p\), so both the slopes of \(F\) and those of \(V\)
+are nonnegative.  All the endomorphisms above preserve every
+$D_\lambda$, and the projectors onto the simultaneous eigenspaces
+preserve its scalar extension.  Thus each block has a grading
+\[
+ M_{\tau,z}=\bigoplus_\lambda M_{\tau,z,\lambda},\qquad
+ M_{\tau,z,\lambda}=M_{\tau,z}\cap(D_\lambda)_{K_1}.
+\]
+We use this grading even if Frobenius permutes the labels $(\tau,z)$.
+No extension of Frobenius to $K_1$, and no isocrystal structure on an
+individual block, is required.
+
+Before scalar extension, $B$ is a pairing of isocrystals into the
+slope-$1$ line.  Hence $B(D_\lambda,D_\mu)=0$ unless
+$\lambda+\mu=1$.  Nondegeneracy on an individual block $M$ then gives
+perfect pairings between $M_\lambda$ and $M_{1-\lambda}$.  In
+particular, if $\dim M=2m$,
+\begin{equation}
+\label{eq:filtration-block-newton}
+ \sum_\lambda\lambda\dim M_\lambda=m.
+\end{equation}
+
+Let $D'\subset D$ be an isocrystal subobject over $\Kzero$ stable
+under $E$ and $u$.  Its intersections $M'=D'_{K_1}\cap M$ decompose
+$D'_{K_1}$ and inherit the same slope grading.  Put $a=\dim M'$.
+The slope sum of the graded quotient $M/M'$ is at most $2m-a$,
+because all its slopes belong to $[0,1]$.  By
+\eqref{eq:filtration-block-newton},
+\begin{equation}
+\label{eq:filtration-block-bound}
+ \sum_\lambda\lambda\dim M'_\lambda
+ \geq \max(0,a-m).
+\end{equation}
+The additional bound by zero follows from the nonnegative slopes.
+The right side is also the smallest possible dimension of the
+intersection of a Lagrangian in $M$ with $M'$.
+Thus attaining this minimum in every block will give the
+weak-admissibility inequalities for all the stable subobjects.
+
+\medskip\noindent
+\emph{Simultaneous choices and finite local field descent.}
+We may arrange the splitting field and coordinates used above as
+follows.  Choose a finite field $k_0\subset\overline{\F}_p$ over
+which $A_0$, its polarization, $u$, and a rational basis of $E$ are
+defined.  The underlying isocrystal and the indicated tensors then
+descend to $\operatorname{Frac}W(k_0)$.  Choose a finite local field
+$H/\Q_p$ containing this field and splitting the actions of $E$ and
+$u^2$.  The eigenspaces and their symplectic bases can be constructed
+over $H$ by linear algebra.  We take $K_1=\Kzero H$.  In particular,
+the independent Lagrangian Grassmannians, fixed symplectic big cells
+in them, and the maps induced by $u$ all have coordinates over $H$.
+
+Take the product of these big cells, one for each independent choice
+in \eqref{eq:filtration-opposite-choice}, and let $r$ be its number
+of affine coordinates.  For every $E,u$-stable isocrystal subobject
+$D'$ over $\Kzero$ and each independent block $M$, the space
+$M'=D'_{K_1}\cap M$ is defined over the \emph{fixed} field $K_1$.
+Lemma~\ref{lem:lagrangian-intersection} supplies a nonzero polynomial
+over $K_1$, of degree at most $m$, whose nonvanishing ensures
+\begin{equation}
+\label{eq:filtration-minimal-intersection}
+ \dim(L_{\tau,z}\cap M')=\max(0,\dim M'-m).
+\end{equation}
+We regard this as a polynomial on the whole product by ignoring the
+other coordinates.  A single bound $b=\max m_{\tau,z}$ works for all
+these polynomials, independently of $D'$.  On the opposite block,
+the same intersection equality follows from $u$-stability of $D'$
+and \eqref{eq:filtration-opposite-choice}.
+
+Apply Lemma~\ref{lem:avoidance} to $K_1,r,b$.  Its one point satisfies
+\eqref{eq:filtration-minimal-intersection} for every one of the
+subobjects just specified, simultaneously.  This use of the lemma
+does not assume that the collection of subobjects is countable or
+that its members descend to $H$.  Their definition over $K_1$ and
+the uniform polynomial degree bound are the only requirements.
+There are nonzero blocks with $m\geq1$, so $r,b\geq1$ as required.
+
+Put $K=K_1(\theta)=\Kzero H(\theta)$.  The resulting Lagrangian is
+defined over $H(\theta)$ in the chosen coordinates, and $H(\theta)$
+is finite over $\Q_p$.  Extending the original isocrystal to the
+maximal unramified subfield of $H(\theta)$ therefore gives finite
+local field descent of the complete filtered data.  In particular,
+the construction has not obtained its generic point by introducing
+transcendental parameters.
+
+Summing \eqref{eq:filtration-minimal-intersection} and
+\eqref{eq:filtration-block-bound} over all blocks gives
+\begin{equation}
+\label{eq:filtration-stable-test}
+ t_H(D')\leq t_N(D')
+ \quad\text{for every $E,u$-stable isocrystal subobject $D'\subset D$.}
+\end{equation}
+For $D$ itself, both numbers equal $\frac12\dim D$ by
+\eqref{eq:filtration-block-newton}. The reduction to stable subobjects
+proved at the start now gives all the weak-admissibility inequalities
+for subobjects of $D$ over $\Kzero$. Each subobject over the smaller unramified
+field of descent extends to one of these subobjects. Hence the
+descended filtered isocrystal is weakly admissible as well.
+
+\medskip\noindent
+\emph{The geometric cocharacter orbit.}
+Over $\Omega$, the polarization pairs $D_\tau$ only with
+$D_{\bar\tau}$.  A balanced $E$-stable Lagrangian is determined by an
+$s$-dimensional subspace of $D_\tau$ for one member of each pair;
+the subspace in the opposite member is its annihilator.  The
+subgroup of $G_D$ with multiplier $1$ contains the product of
+$\SL(D_\tau)$ acting on opposite spaces by the inverse dual
+representations.  These special linear groups are transitive on
+the $s$-dimensional subspaces.  Thus all the indicated Lagrangians
+form one $G_D(\Omega)$-orbit.
+
+Finally choose an $E$-stable $\beta$-Lagrangian complement to one
+such $L$.  Such a complement is obtained separately on each pair
+of opposite embedding spaces.  Acting by $t$ on $L$ and by $1$
+on the complement gives a cocharacter with multiplier $t$ and
+$E$-determinant $t^s$ at each embedding.  It lies in $G_D$ and has
+the stated filtration.  This proves the final assertions.
+\end{proof}
+
+\begin{corollary}
+\label{cor:filtration-admissible}
+The filtration of Theorem~\ref{thm:filtration}, with its finite local
+field descent, is admissible: the descended filtered isocrystal is
+the filtered crystalline module of a crystalline representation.
+\end{corollary}
+
+\begin{proof}
+Apply the weakly-admissible-implies-admissible theorem of
+Colmez--Fontaine with monodromy operator zero
+\cite[Theorem~A]{ColmezFontaine2000}.
+\end{proof}
+
+\begin{corollary}[The divisor filtration condition]
+\label{cor:filtration-divisor}
+Let \(D,L\) be as in Theorem~\ref{thm:filtration}, and let \(\psi\)
+and \(U\) denote the homological polarization and the homological
+action of \(u\) on \(D_K^\vee\). The untwisted divisor form
+\(\psi(\cdot,U\cdot)\) belongs to
+\[
+ \Fil^1\bigl(\bigwedge^2D_K\bigr)=L\wedge D_K,
+\]
+and hence to filtration degree zero after its codimension-one Tate
+twist. If a marked power is realized by an abelian lift, the
+corresponding pulled-back divisor satisfies the filtration condition
+for its symmetric homomorphism in Lemma~\ref{lem:lift-hom}.
+\end{corollary}
+
+\begin{proof}
+Let \(H\subset D_K^\vee\) be the annihilator of \(L\). It is
+Lagrangian for \(\psi\) and stable under \(U\): for \(a\in L\)
+and \(h\in H\),
+\[
+ a(Uh)=(u^*a)(h)=0.
+\]
+Therefore \(\psi(h,Uh')=0\) for \(h,h'\in H\). The divisor form
+thus lies in
+\[
+ \ker\bigl(\bigwedge^2D_K\longrightarrow\bigwedge^2H^\vee\bigr)
+       =L\wedge D_K.
+\]
+This proves the assertion on the fixed filtered vector space. After
+realization of a marked power, pullback respects this filtration
+condition, which is precisely the condition on the associated
+symmetric homomorphism in Lemma~\ref{lem:lift-hom}.
+\end{proof}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/04_weil.tex

@@ -1,439 +1,439 @@
-\section{Balanced Weil classes}\label{sec:weil}
-
-Let $A/\Qbar$ have good reduction at the fixed place $w$, and let a CM
-field $E$ act on $A$ up to isogeny.  We write $e\mapsto\bar e$ for its
-complex conjugation.  Suppose that a polarization $\lambda$ induces this
-involution on $E$.  Put
-\[
- V=H_1(A(\C),\Q),\qquad \dim_E V=2s,\qquad
- g=\dim A=s[E:\Q],
-\]
-where $s\geq 1$.  We assume that the Hodge multiplicities at every
-embedding of $E$ are $(s,s)$.  The polarization is an alternating form
-$\psi:V\times V\longrightarrow\Q(1)$ satisfying
-\[
- \psi(ex,y)=\psi(x,\bar e y).
-\]
-
-\begin{definition}\label{def:weil-space}
-The rational space of \emph{Weil classes} associated with $E$ is the
-subspace
-\[
- W_E^s(A)\subset
- \bigwedge\nolimits_{\Q}^{2s} V^\vee\otimes\Q(s)
-       =H^{2s}(A(\C),\Q(s))
-\]
-whose scalar extension to a splitting field $\Omega$ of $E$ is
-\begin{equation}\label{eq:weil-pure-space}
- W_E^s(A)\otimes_\Q\Omega
- =\bigoplus_{\tau:E\hookrightarrow\Omega}
-       \bigwedge\nolimits_\Omega^{2s}V_\tau^\vee
-          \otimes\Q(s)_\Omega.
-\end{equation}
-Here each summand is extended by zero on the other eigenspaces.  The
-right side is Galois stable, so it defines the indicated rational
-subspace.
-\end{definition}
-
-The balanced multiplicities imply that every element of $W_E^s(A)$ is
-a Hodge class: each determinant in \eqref{eq:weil-pure-space} has
-untwisted type $(s,s)$.  We use its compatible realizations as an
-absolute Hodge class; see \cite{Deligne1982}.  The same eigenspace
-description of its support holds after specialization, since the
-$E$-action specializes compatibly.
-
-Our goal is the Lefschetz pairing property for these specialized
-classes.  In fact, the reduction satisfies a dichotomy determined by
-its rational endomorphisms.  Let $\dagger$ be Rosati for the reduced
-polarization and put
-\begin{equation}\label{eq:weil-U}
- \mathcal U=
- \{u\in\End^0(A_0):u^\dagger=u,\quad
-                      ue=\bar e u\ \text{for every }e\in E\}.
-\end{equation}
-
-\begin{proposition}\label{prop:weil}
-Every $\gamma\in W_E^s(A)$ satisfies the Lefschetz pairing property
-at $w$.  More precisely, if $\mathcal U$ in \eqref{eq:weil-U} has no
-invertible element, all complementary pairings are zero.  If it has
-an invertible element, $\gamma_0$ is a simultaneous rational
-Lefschetz class.
-\end{proposition}
-
-We first prove the vanishing alternative.  For the other alternative,
-an invertible $u$ supplies a divisor whose form is nondegenerate on
-each $E$-eigenspace separately.  We will lift this divisor and use it
-to obtain a rational identity for the determinant classes.
-
-\subsection{The orthogonality alternative}
-
-\begin{lemma}\label{lem:weil-orthogonality}
-If $\mathcal U$ contains no invertible element, every specialization of
-a class in $W_E^s(A)$ is orthogonal to all complementary Lefschetz
-classes, in each cohomological realization.
-\end{lemma}
-
-\begin{proof}
-We prove the contrapositive.  A nonzero pairing will produce a divisor
-form nondegenerate on one $E$-eigenspace.  We will then show that a
-rational endomorphism in $\mathcal U$ with this property is invertible.
-
-Fix one realization and extend its coefficient field to an
-algebraically closed field $\Omega$.  At the crystalline component we
-also include the coefficient extension used for the specialized
-class.  In this proof, $V_\tau$ also denotes the corresponding
-eigenspace in the dual of first cohomology of $A_0$, identified by
-specialization and comparison.  Write $L^j_\Omega(A_0)$ for the $\Omega$-span of products of
-$j$ divisor classes.  Since $[E:\Q]\geq2$, we have $g-2s\geq0$.
-Hard Lefschetz on the Lefschetz algebra gives
-\begin{equation}\label{eq:weil-hard-lefschetz}
- L^{g-s}_\Omega(A_0)
-        =\psi_0^{g-2s}L^s_\Omega(A_0).
-\end{equation}
-The restriction of hard Lefschetz to this algebra, including for
-crystalline cohomology, follows from the Lefschetz correspondences
-of \cite[Theorem~5.9]{Milne1999}.
-
-Suppose a Weil class pairs nontrivially with
-$D_1\cdots D_s\psi_0^{g-2s}$, and choose a pure term, supported on
-$V_\tau$, whose contribution is nonzero.  This term already has
-degree $2s$ on $V_\tau$, so no further factor in a nonzero top-degree
-product can use that space.  A factor from $\psi_0$ which uses
-$V_{\bar\tau}$ also uses $V_\tau$; hence the polarization factors
-contribute to neither space in this pair.  All $2s$ degrees on
-$V_{\bar\tau}$ must therefore come from the $s$ divisor factors.
-Each has degree two, so each must contribute its restriction to
-$V_{\bar\tau}$, and those restrictions have nonzero product.
-The coefficient of $t_1\cdots t_s$ in
-\[
- (t_1D_1+\cdots+t_sD_s)^s\big|_{V_{\bar\tau}}
-\]
-is $s!$ times that product.  Some $\Omega$-linear combination of the
-divisors therefore restricts to a nondegenerate alternating form on
-$V_{\bar\tau}$.
-
-We identify which endomorphisms can give these restrictions.  Let
-$p_\xi\in E\otimes_\Q\Omega$ be the eigenspace projectors.  Rosati
-satisfies $p_\xi^\dagger=p_{\bar\xi}$.  A divisor corresponds under
-$\psi_0$ to a Rosati-symmetric endomorphism $t$.  Define
-\begin{equation}\label{eq:weil-block-projection}
- \Pi(t)=\sum_\xi p_{\bar\xi}t p_\xi.
-\end{equation}
-This operation takes place inside
-$\End^0(A_0)\otimes_\Q\Omega$.  Moreover,
-\[
- (p_{\bar\xi}t p_\xi)^\dagger
-       =p_{\bar\xi}t^\dagger p_\xi,
- \qquad
- \Pi(t)e=\bar e\Pi(t).
-\]
-Thus $\Pi(t)\in\mathcal U\otimes_\Q\Omega$.  Conversely, these are
-exactly the blocks allowed by the second equation in
-\eqref{eq:weil-U}.  For $x,y\in V_\xi$, the form
-$\psi_0(x,ty)$ depends only on $p_{\bar\xi}t p_\xi$.
-It follows that projecting by \eqref{eq:weil-block-projection}
-preserves all the same-embedding restrictions in question.
-
-Nondegeneracy on $V_{\bar\tau}$ is a polynomial open condition on
-$\mathcal U\otimes_\Q\Omega$, and we have shown this open is
-nonempty.  The rational points of the rational vector space
-$\mathcal U$ are Zariski dense after extension to $\Omega$.  Hence
-there is a rational $u\in\mathcal U$ for which
-$\psi_0(\cdot,u\cdot)$ is nondegenerate on $V_{\bar\tau}$.
-
-To prove invertibility, we use the fact that a nonzero $E$-stable
-abelian subvariety has a nonzero homology space at every embedding
-of $E$.  Suppose that $u$ is not invertible, and choose an integer
-$N>0$ such that $Nu$ is an endomorphism.  The reduced identity
-component $C=((\ker(Nu))^0)_{\mathrm{red}}$ is a
-positive-dimensional abelian subvariety.  The relation $ue=\bar e u$ makes $C$ stable under
-the rational action of $E$.  This action is unital and therefore
-faithful.  Choose a primitive element $e$ of $E$ and an integer
-$M>0$ such that $Me$ acts by an integral endomorphism of $C$.
-The exterior algebra description of cohomology and normalized
-top-degree pullback give, in every realization
-\cite[Appendix, Proposition~(A.2)(a),(b)]{Milne1999},
-\[
- \det\bigl(n-e\mid H^1(C)\bigr)
-    =M^{-2\dim C}\deg\bigl([Mn]-Me\bigr)
-\]
-for every integer $n$ for which the map on the right is an
-isogeny.  All but finitely many integers have this property.
-The right side is rational and independent of the realization.
-Polynomial interpolation therefore gives one characteristic
-polynomial in $\Q[T]$, identical also in crystalline cohomology.
-Its roots lie among
-the conjugates of $e$, and rationality forces their multiplicities to
-be equal.  Since $C\ne0$, every embedding of $E$ occurs with positive
-multiplicity.  The same statement holds on first homology, in the
-chosen realization. Rational Poincar\'e reducibility
-\cite[Proposition~12.1]{MilneAV} makes the inclusion
-\(C\hookrightarrow A_0\) injective on first homology in every
-realization. This inclusion is \(E\)-equivariant, and its
-image is killed by \(u\). Thus the nonzero \(\bar\tau\)-eigenspace
-of \(H_1(C)\) shows that the kernel of \(u\) meets
-\(V_{\bar\tau}\) nontrivially.  Such a vector lies in the radical of
-$\psi_0(\cdot,u\cdot)|_{V_{\bar\tau}}$, a contradiction.
-
-A nonzero pairing would therefore give an invertible element of
-$\mathcal U$.  Taking the contrapositive proves the lemma.  The
-argument applies separately to every realization, so all the
-pairings in this case have the common rational value zero.
-\end{proof}
-
-\subsection{A divisor identity on a rational summand}
-
-For the rest of the section, suppose that $u\in\mathcal U$ is
-invertible.  It gives a rational divisor $D^u$ on $A_0$ with
-alternating form
-\begin{equation}\label{eq:weil-extra-divisor}
- D^u(x,y)=\psi_0(x,uy).
-\end{equation}
-This form is nondegenerate and pairs each embedding of $E$ only with
-itself: $u$ exchanges opposite eigenspaces, whereas $\psi_0$ pairs
-opposite eigenspaces.
-
-Prescribed-filtration transport will replace $A_0$ by a power and
-then by an isogenous special fiber $B_0$.  We will retain the first
-factor through a rational projector on $B_0$, and lift that projector,
-the $E$-action on its image, and the transported divisor to $B$.
-The next lemma explains the resulting divisor identity: a divisor
-nondegenerate separately on those eigenspaces generates their
-determinant classes.  A rational projector here means an idempotent
-in the rational endomorphism algebra.
-
-\begin{lemma}\label{lem:weil-divisor-span}
-Let $B$ be a complex abelian variety, let $R\in\End^0(B)$ be a
-rational projector, and suppose that there is an algebra action
-$E\longrightarrow R\End^0(B)R$ whose unit maps to $R$.
-Let $T=\operatorname{im}(R)$ in rational first homology, with
-$\dim_E T=2s$; the indicated endomorphisms act by zero on
-$\ker(R)$.  Suppose there is a rational divisor class $D$ on
-$B$ such that $R^*D=D$ and, after splitting $E$, its alternating form
-is
-\[
- D=\sum_\tau D_\tau,
- \qquad D_\tau\in\bigwedge\nolimits^2 T_\tau^\vee\otimes\Q(1),
-\]
-with each $D_\tau$ nondegenerate.  Then the rational subspace whose
-scalar extension is
-\[
- \bigoplus_\tau\bigwedge\nolimits^{2s}T_\tau^\vee\otimes\Q(s),
-\]
-extended by zero on the complementary summand, consists of rational
-Lefschetz classes on $B$.
-\end{lemma}
-
-\begin{proof}
-For $b\in E$, let $\widetilde b\in\End^0(B)$ act by $b$ on $T$ and
-by zero on the complement.  Its pullback of a rational divisor is a
-rational divisor: clear a denominator in $\widetilde b$, and use
-that multiplication by $N$ acts as $N^2$ on $H^2$.  Over a splitting
-field $\Omega$, choosing Tate-line bases, we have
-\[
- (\widetilde b^*D)^s
-       =\left(\sum_\tau\tau(b)^2D_\tau\right)^s.
-\]
-The coordinates $x_\tau=\tau(b)$ are independent coordinates on the
-scalar extension of the rational vector group underlying $E$.
-In the corresponding vector-valued polynomial
-\begin{equation}\label{eq:weil-coefficient-polynomial}
- P((x_\tau)_\tau)=\left(\sum_\tau x_\tau^2D_\tau\right)^s,
-\end{equation}
-the coefficient of $x_\tau^{2s}$ is $D_\tau^s$.  Nondegeneracy makes
-this coefficient nonzero, so it spans
-$\bigwedge^{2s}T_\tau^\vee\otimes\Q(s)_\Omega$.
-
-Let $M$ be the rational span of the Lefschetz classes
-$(\widetilde b^*D)^s$ for $b\in E$.  The set $E$ of rational points
-is Zariski dense in its vector group, also after extension to
-$\Omega$.  Any linear functional annihilating $M\otimes_\Q\Omega$
-annihilates all evaluations of \eqref{eq:weil-coefficient-polynomial}
-on those points.  It therefore annihilates the polynomial, and hence
-each coefficient.  Every $D_\tau^s$ belongs to
-$M\otimes_\Q\Omega$.
-
-This proves the required inclusion after scalar extension.  Both
-subspaces are rational; applying faithful scalar extension to their
-quotient proves the inclusion over $\Q$.  In particular it gives
-rational coefficients for every rational class in the asserted
-subspace, without requiring an individual embedding line to be
-defined over $\Q$.
-\end{proof}
-
-\subsection{A lift preserving the divisor}
-
-We now construct the auxiliary variety to which
-Lemma~\ref{lem:weil-divisor-span} applies.  To transport the Weil class,
-we use the $E$-linear polarization similitudes whose determinant has
-the same transformation law as a codimension-$s$ Tate twist.
-Choose a rational basis
-for the Betti Tate line, let $c$ be the multiplier of
-$\GSp(V,\psi)$, and set
-\begin{equation}\label{eq:weil-group}
- G=\{a\in\GSp(V,\psi)\cap\GL_E(V):
-                         \det_E(a)=c(a)^s\}.
-\end{equation}
-On points over a $\Q$-algebra $R$, the determinant equality is in
-$(E\otimes_\Q R)^\times$, with the right side embedded diagonally.
-
-\begin{lemma}\label{lem:weil-group}
-The Hodge homomorphism of $A$ factors through $G$, and its conjugacy
-class defines a Shimura datum of Hodge type through the displayed
-embedding.  Over an algebraically closed coefficient field, the orbit
-of the corresponding filtration on $H^1$ consists exactly of the
-$E$-stable Lagrangians for the dual polarization pairing whose
-dimension at each embedding of $E$ is $s$.  Using a basis of its Tate
-line, every alternating form in \eqref{eq:weil-pure-space} transforms
-by $c^s$.
-\end{lemma}
-
-\begin{proof}
-The polarization pairs $V_\tau$ with $V_{\bar\tau}$, and no other pair
-of eigenspaces.  Over an algebraically closed field, choose one
-embedding in each conjugate pair.  An element of $G$ is specified by
-$c$ and matrices $a_\tau\in\GL_{2s}$ with
-$\det(a_\tau)=c^s$; the matrices on the opposite spaces are then
-determined by the polarization.  In particular,
-\[
- \ker(c:G\longrightarrow\mathbb G_m)
-       \simeq\prod_{\{\tau,\bar\tau\}}\SL_{2s}.
-\]
-The multiplier is surjective geometrically.  Thus $G$ is connected
-reductive, with the same adjoint group as the group of $E$-linear
-polarization similitudes.
-
-The two Hodge characters occur $s$ times at every embedding.  Their
-product is the polarization multiplier, so the Hodge homomorphism
-satisfies the determinant condition in \eqref{eq:weil-group}.  Its
-weight is rational and central.  On the adjoint Lie algebra the Hodge
-types are among $(-1,1),(0,0),(1,-1)$, as follows from the weight-one
-representation.  Conjugation by the image of $i$ is a Cartan
-involution, by the polarization.  At every real embedding of the
-maximal totally real subfield of $E$, the adjoint group has unitary
-signature $(s,s)$.  Its Hodge homomorphism is nontrivial there, so no
-rational simple adjoint factor has trivial Hodge homomorphism.
-These are the Shimura-datum conditions for the given Hodge embedding.
-
-On $H^1$ the polarization still pairs opposite eigenspaces.  An
-$E$-stable Lagrangian is specified by an $s$-dimensional subspace at
-one embedding of each pair; the subspace at the opposite embedding
-is its annihilator.  The factor $\SL_{2s}$ acts transitively on these
-subspaces, as in the cohomological orbit calculation of
-Theorem~\ref{thm:filtration}.  Finally, an alternating determinant form
-evaluated on $a_\tau v_1,\ldots,a_\tau v_{2s}$ is multiplied by
-$\det(a_\tau)=c^s$.  This proves the asserted transformation law.
-\end{proof}
-
-\begin{proof}[Proof of Proposition~\ref{prop:weil}]
-The vanishing alternative is Lemma~\ref{lem:weil-orthogonality}.
-Suppose that $u$ is invertible, and use the divisor $D^u$ from
-\eqref{eq:weil-extra-divisor}.
-
-\medskip\noindent
-\emph{A filtration preserving the divisor.}
-
-Let $\mathcal D=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$, with its
-contravariant convention, and let $\beta$ be the dual polarization
-pairing, taking values in the isocrystal line of slope one.  The
-actions on $\mathcal D$ satisfy
-\[
- \beta(ux,y)=\beta(x,uy),\qquad
- \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u.
-\]
-All these endomorphisms and the polarization have finite field
-descent.  Theorem~\ref{thm:filtration} supplies, over a finite
-extension with finite local field descent, a weakly admissible
-filtration $\Fil^1\mathcal D=L$ which is $E$- and $u$-stable,
-is Lagrangian for $\beta$, and has dimension $s$ at every embedding
-of $E$.  By Lemma~\ref{lem:weil-group} it lies geometrically in the
-required $G$-orbit.  The
-\hyperref[par:transport-tensor-descent]{finite-descent argument} in
-Proposition~\ref{prop:transport} shows that its full finite family of
-Hodge-type structure tensors descends from the original number-field
-point by crystalline comparison.  Taking a compositum with the local
-field of definition of $L$ gives the common finite local-field descent
-required by its prescribed-filtration clause.
-
-\medskip\noindent
-\emph{Transport and the first rational summand.}
-Apply Proposition~\ref{prop:transport} to $G$ and $\gamma$ with this
-filtration.  It gives a lift $B$, a
-quasi-isogeny $a:A_0^n\longrightarrow B_0$, and a rational Hodge
-class $\kappa$ on $B$, with
-\[
- f_0=aj:A_0\longrightarrow B_0,\qquad
- f_0^*\kappa_0=\gamma_0,
-\]
-simultaneously in the realizations under consideration.  The
-filtration on $B$, marked by $a$, is the direct sum of $n$ copies of
-the chosen filtration.  Here $j$ includes the first factor.  The map
-$f_0$ identifies $A_0$ with a rational direct summand of $B_0$.
-Its retraction $q_0$ and the corresponding projector $R_0$ are
-\begin{equation}\label{eq:weil-projector-identity}
- q_0=\operatorname{pr}_1a^{-1},\qquad R_0=f_0q_0.
-\end{equation}
-Thus $q_0f_0=\id$ and $R_0^2=R_0$.  Pulling back
-$f_0^*\kappa_0=\gamma_0$ by $q_0$ gives
-\[
- R_0^*\kappa_0=q_0^*\gamma_0.
-\]
-Act by $E$ on $\operatorname{im}(R_0)$ using $e\mapsto f_0e q_0$,
-and by zero on the complement.  Also put $D_0=q_0^*D^u$.
-
-\medskip\noindent
-\emph{Lifting the projector, action, and divisor.}
-The projector and this $E$-action preserve the repeated filtration.
-By Corollary~\ref{cor:filtration-divisor}, $D^u$ satisfies the divisor
-lifting condition for the chosen filtration on the first summand.
-The crystalline map induced by $q_0=\operatorname{pr}_1a^{-1}$
-respects that filtration and the marked repeated filtration on $B_0$.
-Hence $D_0=q_0^*D^u$ satisfies the same condition on $B$.
-Apply Lemma~\ref{lem:lift-hom} to \(R_0\), to a finite
-\(\Q\)-basis of the action of \(E\) on \(\operatorname{im}(R_0)\), and to the
-symmetric homomorphism \(h_{D_0}:B_0\to B_0^\vee\) defined by \(D_0\).
-Denote their lifts by \(R\), the corresponding basis of maps, and
-\(h_D\). Injectivity of specialization preserves \(R^2=R\), the
-rational multiplication table of that basis, and its unit \(R\).
-It therefore gives an algebra action \(E\to R\End^0(B)R\).
-The symmetric map \(h_D\) defines a rational divisor class \(D\).
-Moreover, \(q_0R_0=q_0\), so
-\[
- R_0^\vee h_{D_0}R_0=h_{D_0}.
-\]
-Injectivity lifts this identity to \(R^\vee h_DR=h_D\), which is
-exactly \(R^*D=D\). Thus the required projector, action, and divisor
-all exist on the one auxiliary variety $B$.
-
-\medskip\noindent
-\emph{The rational identity and its specialization.}
-Consider now rational Betti cohomology at a complex realization of
-$B$.  The summand $T=\operatorname{im}(R)$ has dimension $2s$ at
-each embedding of $E$.  The form of $D$ is supported on this summand
-and pairs each embedding with itself nondegenerately.  Also
-$R^*\kappa$ is supported on the pure determinant spaces of this
-summand.  These assertions can be checked after extension to any
-one prime-to-$p$ realization: for the divisor they follow from its
-specified specialization, and for the Hodge class they follow from
-the equality $R_0^*\kappa_0=q_0^*\gamma_0$ and the
-pure support of $\gamma_0$.  Compatibility of the transported
-realizations then gives these support and nondegeneracy assertions
-in Betti cohomology.
-
-Lemma~\ref{lem:weil-divisor-span} now expresses the rational Betti
-class $R^*\kappa$ as a finite rational linear combination of products
-of rational divisor classes on $B$.  The realization and
-specialization compatibilities in
-Proposition~\ref{prop:transport}, together with those for divisors
-and homomorphisms, specialize this identity with the same rational
-coefficients in every component.  Hence $R_0^*\kappa_0$ is a
-simultaneous rational Lefschetz class.  Finally,
-\[
- \gamma_0=f_0^*R_0^*\kappa_0,
-\]
-because $R_0f_0=f_0$.  Pullback by a rational homomorphism preserves
-simultaneous rational Lefschetz classes, so the same is true of
-$\gamma_0$.  Its complementary traces are therefore one common
-rational intersection number, proving the second alternative and
-the proposition.
-\end{proof}
+\section{Balanced Weil classes}\label{sec:weil}
+
+Let $A/\Qbar$ have good reduction at the fixed place $w$, and let a CM
+field $E$ act on $A$ up to isogeny.  We write $e\mapsto\bar e$ for its
+complex conjugation.  Suppose that a polarization $\lambda$ induces this
+involution on $E$.  Put
+\[
+ V=H_1(A(\C),\Q),\qquad \dim_E V=2s,\qquad
+ g=\dim A=s[E:\Q],
+\]
+where $s\geq 1$.  We assume that the Hodge multiplicities at every
+embedding of $E$ are $(s,s)$.  The polarization is an alternating form
+$\psi:V\times V\longrightarrow\Q(1)$ satisfying
+\[
+ \psi(ex,y)=\psi(x,\bar e y).
+\]
+
+\begin{definition}\label{def:weil-space}
+The rational space of \emph{Weil classes} associated with $E$ is the
+subspace
+\[
+ W_E^s(A)\subset
+ \bigwedge\nolimits_{\Q}^{2s} V^\vee\otimes\Q(s)
+       =H^{2s}(A(\C),\Q(s))
+\]
+whose scalar extension to a splitting field $\Omega$ of $E$ is
+\begin{equation}\label{eq:weil-pure-space}
+ W_E^s(A)\otimes_\Q\Omega
+ =\bigoplus_{\tau:E\hookrightarrow\Omega}
+       \bigwedge\nolimits_\Omega^{2s}V_\tau^\vee
+          \otimes\Q(s)_\Omega.
+\end{equation}
+Here each summand is extended by zero on the other eigenspaces.  The
+right side is Galois stable, so it defines the indicated rational
+subspace.
+\end{definition}
+
+The balanced multiplicities imply that every element of $W_E^s(A)$ is
+a Hodge class: each determinant in \eqref{eq:weil-pure-space} has
+untwisted type $(s,s)$.  We use its compatible realizations as an
+absolute Hodge class; see \cite{Deligne1982}.  The same eigenspace
+description of its support holds after specialization, since the
+$E$-action specializes compatibly.
+
+Our goal is the Lefschetz pairing property for these specialized
+classes.  In fact, the reduction satisfies a dichotomy determined by
+its rational endomorphisms.  Let $\dagger$ be Rosati for the reduced
+polarization and put
+\begin{equation}\label{eq:weil-U}
+ \mathcal U=
+ \{u\in\End^0(A_0):u^\dagger=u,\quad
+                      ue=\bar e u\ \text{for every }e\in E\}.
+\end{equation}
+
+\begin{proposition}\label{prop:weil}
+Every $\gamma\in W_E^s(A)$ satisfies the Lefschetz pairing property
+at $w$.  More precisely, if $\mathcal U$ in \eqref{eq:weil-U} has no
+invertible element, all complementary pairings are zero.  If it has
+an invertible element, $\gamma_0$ is a simultaneous rational
+Lefschetz class.
+\end{proposition}
+
+We first prove the vanishing alternative.  For the other alternative,
+an invertible $u$ supplies a divisor whose form is nondegenerate on
+each $E$-eigenspace separately.  We will lift this divisor and use it
+to obtain a rational identity for the determinant classes.
+
+\subsection{The orthogonality alternative}
+
+\begin{lemma}\label{lem:weil-orthogonality}
+If $\mathcal U$ contains no invertible element, every specialization of
+a class in $W_E^s(A)$ is orthogonal to all complementary Lefschetz
+classes, in each cohomological realization.
+\end{lemma}
+
+\begin{proof}
+We prove the contrapositive.  A nonzero pairing will produce a divisor
+form nondegenerate on one $E$-eigenspace.  We will then show that a
+rational endomorphism in $\mathcal U$ with this property is invertible.
+
+Fix one realization and extend its coefficient field to an
+algebraically closed field $\Omega$.  At the crystalline component we
+also include the coefficient extension used for the specialized
+class.  In this proof, $V_\tau$ also denotes the corresponding
+eigenspace in the dual of first cohomology of $A_0$, identified by
+specialization and comparison.  Write $L^j_\Omega(A_0)$ for the $\Omega$-span of products of
+$j$ divisor classes.  Since $[E:\Q]\geq2$, we have $g-2s\geq0$.
+Hard Lefschetz on the Lefschetz algebra gives
+\begin{equation}\label{eq:weil-hard-lefschetz}
+ L^{g-s}_\Omega(A_0)
+        =\psi_0^{g-2s}L^s_\Omega(A_0).
+\end{equation}
+The restriction of hard Lefschetz to this algebra, including for
+crystalline cohomology, follows from the Lefschetz correspondences
+of \cite[Theorem~5.9]{Milne1999}.
+
+Suppose a Weil class pairs nontrivially with
+$D_1\cdots D_s\psi_0^{g-2s}$, and choose a pure term, supported on
+$V_\tau$, whose contribution is nonzero.  This term already has
+degree $2s$ on $V_\tau$, so no further factor in a nonzero top-degree
+product can use that space.  A factor from $\psi_0$ which uses
+$V_{\bar\tau}$ also uses $V_\tau$; hence the polarization factors
+contribute to neither space in this pair.  All $2s$ degrees on
+$V_{\bar\tau}$ must therefore come from the $s$ divisor factors.
+Each has degree two, so each must contribute its restriction to
+$V_{\bar\tau}$, and those restrictions have nonzero product.
+The coefficient of $t_1\cdots t_s$ in
+\[
+ (t_1D_1+\cdots+t_sD_s)^s\big|_{V_{\bar\tau}}
+\]
+is $s!$ times that product.  Some $\Omega$-linear combination of the
+divisors therefore restricts to a nondegenerate alternating form on
+$V_{\bar\tau}$.
+
+We identify which endomorphisms can give these restrictions.  Let
+$p_\xi\in E\otimes_\Q\Omega$ be the eigenspace projectors.  Rosati
+satisfies $p_\xi^\dagger=p_{\bar\xi}$.  A divisor corresponds under
+$\psi_0$ to a Rosati-symmetric endomorphism $t$.  Define
+\begin{equation}\label{eq:weil-block-projection}
+ \Pi(t)=\sum_\xi p_{\bar\xi}t p_\xi.
+\end{equation}
+This operation takes place inside
+$\End^0(A_0)\otimes_\Q\Omega$.  Moreover,
+\[
+ (p_{\bar\xi}t p_\xi)^\dagger
+       =p_{\bar\xi}t^\dagger p_\xi,
+ \qquad
+ \Pi(t)e=\bar e\Pi(t).
+\]
+Thus $\Pi(t)\in\mathcal U\otimes_\Q\Omega$.  Conversely, these are
+exactly the blocks allowed by the second equation in
+\eqref{eq:weil-U}.  For $x,y\in V_\xi$, the form
+$\psi_0(x,ty)$ depends only on $p_{\bar\xi}t p_\xi$.
+It follows that projecting by \eqref{eq:weil-block-projection}
+preserves all the same-embedding restrictions in question.
+
+Nondegeneracy on $V_{\bar\tau}$ is a polynomial open condition on
+$\mathcal U\otimes_\Q\Omega$, and we have shown this open is
+nonempty.  The rational points of the rational vector space
+$\mathcal U$ are Zariski dense after extension to $\Omega$.  Hence
+there is a rational $u\in\mathcal U$ for which
+$\psi_0(\cdot,u\cdot)$ is nondegenerate on $V_{\bar\tau}$.
+
+To prove invertibility, we use the fact that a nonzero $E$-stable
+abelian subvariety has a nonzero homology space at every embedding
+of $E$.  Suppose that $u$ is not invertible, and choose an integer
+$N>0$ such that $Nu$ is an endomorphism.  The reduced identity
+component $C=((\ker(Nu))^0)_{\mathrm{red}}$ is a
+positive-dimensional abelian subvariety.  The relation $ue=\bar e u$ makes $C$ stable under
+the rational action of $E$.  This action is unital and therefore
+faithful.  Choose a primitive element $e$ of $E$ and an integer
+$M>0$ such that $Me$ acts by an integral endomorphism of $C$.
+The exterior algebra description of cohomology and normalized
+top-degree pullback give, in every realization
+\cite[Appendix, Proposition~(A.2)(a),(b)]{Milne1999},
+\[
+ \det\bigl(n-e\mid H^1(C)\bigr)
+    =M^{-2\dim C}\deg\bigl([Mn]-Me\bigr)
+\]
+for every integer $n$ for which the map on the right is an
+isogeny.  All but finitely many integers have this property.
+The right side is rational and independent of the realization.
+Polynomial interpolation therefore gives one characteristic
+polynomial in $\Q[T]$, identical also in crystalline cohomology.
+Its roots lie among
+the conjugates of $e$, and rationality forces their multiplicities to
+be equal.  Since $C\ne0$, every embedding of $E$ occurs with positive
+multiplicity.  The same statement holds on first homology, in the
+chosen realization. Rational Poincar\'e reducibility
+\cite[Proposition~12.1]{MilneAV} makes the inclusion
+\(C\hookrightarrow A_0\) injective on first homology in every
+realization. This inclusion is \(E\)-equivariant, and its
+image is killed by \(u\). Thus the nonzero \(\bar\tau\)-eigenspace
+of \(H_1(C)\) shows that the kernel of \(u\) meets
+\(V_{\bar\tau}\) nontrivially.  Such a vector lies in the radical of
+$\psi_0(\cdot,u\cdot)|_{V_{\bar\tau}}$, a contradiction.
+
+A nonzero pairing would therefore give an invertible element of
+$\mathcal U$.  Taking the contrapositive proves the lemma.  The
+argument applies separately to every realization, so all the
+pairings in this case have the common rational value zero.
+\end{proof}
+
+\subsection{A divisor identity on a rational summand}
+
+For the rest of the section, suppose that $u\in\mathcal U$ is
+invertible.  It gives a rational divisor $D^u$ on $A_0$ with
+alternating form
+\begin{equation}\label{eq:weil-extra-divisor}
+ D^u(x,y)=\psi_0(x,uy).
+\end{equation}
+This form is nondegenerate and pairs each embedding of $E$ only with
+itself: $u$ exchanges opposite eigenspaces, whereas $\psi_0$ pairs
+opposite eigenspaces.
+
+Prescribed-filtration transport will replace $A_0$ by a power and
+then by an isogenous special fiber $B_0$.  We will retain the first
+factor through a rational projector on $B_0$, and lift that projector,
+the $E$-action on its image, and the transported divisor to $B$.
+The next lemma explains the resulting divisor identity: a divisor
+nondegenerate separately on those eigenspaces generates their
+determinant classes.  A rational projector here means an idempotent
+in the rational endomorphism algebra.
+
+\begin{lemma}\label{lem:weil-divisor-span}
+Let $B$ be a complex abelian variety, let $R\in\End^0(B)$ be a
+rational projector, and suppose that there is an algebra action
+$E\longrightarrow R\End^0(B)R$ whose unit maps to $R$.
+Let $T=\operatorname{im}(R)$ in rational first homology, with
+$\dim_E T=2s$; the indicated endomorphisms act by zero on
+$\ker(R)$.  Suppose there is a rational divisor class $D$ on
+$B$ such that $R^*D=D$ and, after splitting $E$, its alternating form
+is
+\[
+ D=\sum_\tau D_\tau,
+ \qquad D_\tau\in\bigwedge\nolimits^2 T_\tau^\vee\otimes\Q(1),
+\]
+with each $D_\tau$ nondegenerate.  Then the rational subspace whose
+scalar extension is
+\[
+ \bigoplus_\tau\bigwedge\nolimits^{2s}T_\tau^\vee\otimes\Q(s),
+\]
+extended by zero on the complementary summand, consists of rational
+Lefschetz classes on $B$.
+\end{lemma}
+
+\begin{proof}
+For $b\in E$, let $\widetilde b\in\End^0(B)$ act by $b$ on $T$ and
+by zero on the complement.  Its pullback of a rational divisor is a
+rational divisor: clear a denominator in $\widetilde b$, and use
+that multiplication by $N$ acts as $N^2$ on $H^2$.  Over a splitting
+field $\Omega$, choosing Tate-line bases, we have
+\[
+ (\widetilde b^*D)^s
+       =\left(\sum_\tau\tau(b)^2D_\tau\right)^s.
+\]
+The coordinates $x_\tau=\tau(b)$ are independent coordinates on the
+scalar extension of the rational vector group underlying $E$.
+In the corresponding vector-valued polynomial
+\begin{equation}\label{eq:weil-coefficient-polynomial}
+ P((x_\tau)_\tau)=\left(\sum_\tau x_\tau^2D_\tau\right)^s,
+\end{equation}
+the coefficient of $x_\tau^{2s}$ is $D_\tau^s$.  Nondegeneracy makes
+this coefficient nonzero, so it spans
+$\bigwedge^{2s}T_\tau^\vee\otimes\Q(s)_\Omega$.
+
+Let $M$ be the rational span of the Lefschetz classes
+$(\widetilde b^*D)^s$ for $b\in E$.  The set $E$ of rational points
+is Zariski dense in its vector group, also after extension to
+$\Omega$.  Any linear functional annihilating $M\otimes_\Q\Omega$
+annihilates all evaluations of \eqref{eq:weil-coefficient-polynomial}
+on those points.  It therefore annihilates the polynomial, and hence
+each coefficient.  Every $D_\tau^s$ belongs to
+$M\otimes_\Q\Omega$.
+
+This proves the required inclusion after scalar extension.  Both
+subspaces are rational; applying faithful scalar extension to their
+quotient proves the inclusion over $\Q$.  In particular it gives
+rational coefficients for every rational class in the asserted
+subspace, without requiring an individual embedding line to be
+defined over $\Q$.
+\end{proof}
+
+\subsection{A lift preserving the divisor}
+
+We now construct the auxiliary variety to which
+Lemma~\ref{lem:weil-divisor-span} applies.  To transport the Weil class,
+we use the $E$-linear polarization similitudes whose determinant has
+the same transformation law as a codimension-$s$ Tate twist.
+Choose a rational basis
+for the Betti Tate line, let $c$ be the multiplier of
+$\GSp(V,\psi)$, and set
+\begin{equation}\label{eq:weil-group}
+ G=\{a\in\GSp(V,\psi)\cap\GL_E(V):
+                         \det_E(a)=c(a)^s\}.
+\end{equation}
+On points over a $\Q$-algebra $R$, the determinant equality is in
+$(E\otimes_\Q R)^\times$, with the right side embedded diagonally.
+
+\begin{lemma}\label{lem:weil-group}
+The Hodge homomorphism of $A$ factors through $G$, and its conjugacy
+class defines a Shimura datum of Hodge type through the displayed
+embedding.  Over an algebraically closed coefficient field, the orbit
+of the corresponding filtration on $H^1$ consists exactly of the
+$E$-stable Lagrangians for the dual polarization pairing whose
+dimension at each embedding of $E$ is $s$.  Using a basis of its Tate
+line, every alternating form in \eqref{eq:weil-pure-space} transforms
+by $c^s$.
+\end{lemma}
+
+\begin{proof}
+The polarization pairs $V_\tau$ with $V_{\bar\tau}$, and no other pair
+of eigenspaces.  Over an algebraically closed field, choose one
+embedding in each conjugate pair.  An element of $G$ is specified by
+$c$ and matrices $a_\tau\in\GL_{2s}$ with
+$\det(a_\tau)=c^s$; the matrices on the opposite spaces are then
+determined by the polarization.  In particular,
+\[
+ \ker(c:G\longrightarrow\mathbb G_m)
+       \simeq\prod_{\{\tau,\bar\tau\}}\SL_{2s}.
+\]
+The multiplier is surjective geometrically.  Thus $G$ is connected
+reductive, with the same adjoint group as the group of $E$-linear
+polarization similitudes.
+
+The two Hodge characters occur $s$ times at every embedding.  Their
+product is the polarization multiplier, so the Hodge homomorphism
+satisfies the determinant condition in \eqref{eq:weil-group}.  Its
+weight is rational and central.  On the adjoint Lie algebra the Hodge
+types are among $(-1,1),(0,0),(1,-1)$, as follows from the weight-one
+representation.  Conjugation by the image of $i$ is a Cartan
+involution, by the polarization.  At every real embedding of the
+maximal totally real subfield of $E$, the adjoint group has unitary
+signature $(s,s)$.  Its Hodge homomorphism is nontrivial there, so no
+rational simple adjoint factor has trivial Hodge homomorphism.
+These are the Shimura-datum conditions for the given Hodge embedding.
+
+On $H^1$ the polarization still pairs opposite eigenspaces.  An
+$E$-stable Lagrangian is specified by an $s$-dimensional subspace at
+one embedding of each pair; the subspace at the opposite embedding
+is its annihilator.  The factor $\SL_{2s}$ acts transitively on these
+subspaces, as in the cohomological orbit calculation of
+Theorem~\ref{thm:filtration}.  Finally, an alternating determinant form
+evaluated on $a_\tau v_1,\ldots,a_\tau v_{2s}$ is multiplied by
+$\det(a_\tau)=c^s$.  This proves the asserted transformation law.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:weil}]
+The vanishing alternative is Lemma~\ref{lem:weil-orthogonality}.
+Suppose that $u$ is invertible, and use the divisor $D^u$ from
+\eqref{eq:weil-extra-divisor}.
+
+\medskip\noindent
+\emph{A filtration preserving the divisor.}
+
+Let $\mathcal D=H^1_{\mathrm{cris}}(A_0/W(\F))[1/p]$, with its
+contravariant convention, and let $\beta$ be the dual polarization
+pairing, taking values in the isocrystal line of slope one.  The
+actions on $\mathcal D$ satisfy
+\[
+ \beta(ux,y)=\beta(x,uy),\qquad
+ \beta(ex,y)=\beta(x,\bar e y),\qquad ue=\bar e u.
+\]
+All these endomorphisms and the polarization have finite field
+descent.  Theorem~\ref{thm:filtration} supplies, over a finite
+extension with finite local field descent, a weakly admissible
+filtration $\Fil^1\mathcal D=L$ which is $E$- and $u$-stable,
+is Lagrangian for $\beta$, and has dimension $s$ at every embedding
+of $E$.  By Lemma~\ref{lem:weil-group} it lies geometrically in the
+required $G$-orbit.  The
+\hyperref[par:transport-tensor-descent]{finite-descent argument} in
+Proposition~\ref{prop:transport} shows that its full finite family of
+Hodge-type structure tensors descends from the original number-field
+point by crystalline comparison.  Taking a compositum with the local
+field of definition of $L$ gives the common finite local-field descent
+required by its prescribed-filtration clause.
+
+\medskip\noindent
+\emph{Transport and the first rational summand.}
+Apply Proposition~\ref{prop:transport} to $G$ and $\gamma$ with this
+filtration.  It gives a lift $B$, a
+quasi-isogeny $a:A_0^n\longrightarrow B_0$, and a rational Hodge
+class $\kappa$ on $B$, with
+\[
+ f_0=aj:A_0\longrightarrow B_0,\qquad
+ f_0^*\kappa_0=\gamma_0,
+\]
+simultaneously in the realizations under consideration.  The
+filtration on $B$, marked by $a$, is the direct sum of $n$ copies of
+the chosen filtration.  Here $j$ includes the first factor.  The map
+$f_0$ identifies $A_0$ with a rational direct summand of $B_0$.
+Its retraction $q_0$ and the corresponding projector $R_0$ are
+\begin{equation}\label{eq:weil-projector-identity}
+ q_0=\operatorname{pr}_1a^{-1},\qquad R_0=f_0q_0.
+\end{equation}
+Thus $q_0f_0=\id$ and $R_0^2=R_0$.  Pulling back
+$f_0^*\kappa_0=\gamma_0$ by $q_0$ gives
+\[
+ R_0^*\kappa_0=q_0^*\gamma_0.
+\]
+Act by $E$ on $\operatorname{im}(R_0)$ using $e\mapsto f_0e q_0$,
+and by zero on the complement.  Also put $D_0=q_0^*D^u$.
+
+\medskip\noindent
+\emph{Lifting the projector, action, and divisor.}
+The projector and this $E$-action preserve the repeated filtration.
+By Corollary~\ref{cor:filtration-divisor}, $D^u$ satisfies the divisor
+lifting condition for the chosen filtration on the first summand.
+The crystalline map induced by $q_0=\operatorname{pr}_1a^{-1}$
+respects that filtration and the marked repeated filtration on $B_0$.
+Hence $D_0=q_0^*D^u$ satisfies the same condition on $B$.
+Apply Lemma~\ref{lem:lift-hom} to \(R_0\), to a finite
+\(\Q\)-basis of the action of \(E\) on \(\operatorname{im}(R_0)\), and to the
+symmetric homomorphism \(h_{D_0}:B_0\to B_0^\vee\) defined by \(D_0\).
+Denote their lifts by \(R\), the corresponding basis of maps, and
+\(h_D\). Injectivity of specialization preserves \(R^2=R\), the
+rational multiplication table of that basis, and its unit \(R\).
+It therefore gives an algebra action \(E\to R\End^0(B)R\).
+The symmetric map \(h_D\) defines a rational divisor class \(D\).
+Moreover, \(q_0R_0=q_0\), so
+\[
+ R_0^\vee h_{D_0}R_0=h_{D_0}.
+\]
+Injectivity lifts this identity to \(R^\vee h_DR=h_D\), which is
+exactly \(R^*D=D\). Thus the required projector, action, and divisor
+all exist on the one auxiliary variety $B$.
+
+\medskip\noindent
+\emph{The rational identity and its specialization.}
+Consider now rational Betti cohomology at a complex realization of
+$B$.  The summand $T=\operatorname{im}(R)$ has dimension $2s$ at
+each embedding of $E$.  The form of $D$ is supported on this summand
+and pairs each embedding with itself nondegenerately.  Also
+$R^*\kappa$ is supported on the pure determinant spaces of this
+summand.  These assertions can be checked after extension to any
+one prime-to-$p$ realization: for the divisor they follow from its
+specified specialization, and for the Hodge class they follow from
+the equality $R_0^*\kappa_0=q_0^*\gamma_0$ and the
+pure support of $\gamma_0$.  Compatibility of the transported
+realizations then gives these support and nondegeneracy assertions
+in Betti cohomology.
+
+Lemma~\ref{lem:weil-divisor-span} now expresses the rational Betti
+class $R^*\kappa$ as a finite rational linear combination of products
+of rational divisor classes on $B$.  The realization and
+specialization compatibilities in
+Proposition~\ref{prop:transport}, together with those for divisors
+and homomorphisms, specialize this identity with the same rational
+coefficients in every component.  Hence $R_0^*\kappa_0$ is a
+simultaneous rational Lefschetz class.  Finally,
+\[
+ \gamma_0=f_0^*R_0^*\kappa_0,
+\]
+because $R_0f_0=f_0$.  Pullback by a rational homomorphism preserves
+simultaneous rational Lefschetz classes, so the same is true of
+$\gamma_0$.  Its complementary traces are therefore one common
+rational intersection number, proving the second alternative and
+the proposition.
+\end{proof}

preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/build/sections/05_reduction.tex

@@ -1,448 +1,448 @@
-\section{CM reduction and consequences}
-\label{sec:reduction}
-
-The remaining step in the main proof is to express CM Hodge classes
-as pullbacks of balanced Weil classes.  Together with CM transport
-and Proposition~\ref{prop:weil}, this will prove
-Theorem~\ref{thm:main}.  We then derive two specialization consequences.
-
-\subsection{CM classes as pullbacks of Weil classes}
-
-We give the CM reduction in the form needed here.  The stronger
-reduction to classes on abelian varieties of split Weil type is due to
-Deligne and Andr\'e \cite{Andre1992}; see the reconstruction in
-\cite[Theorem~1 and Section~4]{Milne2020}. Only balanced signatures
-and a compatible polarization are needed for our application, and
-we prove this form directly using rational idempotents on powers.
-
-\begin{lemma}[CM reduction]
-\label{lem:cm-reduction}
-Let $A/\Qbar$ be an abelian variety of CM type with good reduction at $w$,
-and let $r>0$.  For every rational Hodge class
-$\gamma\in H^{2r}(A(\C),\Q(r))$, there are finitely many abelian varieties
-$B_i/\Qbar$ with good reduction at $w$, rational homomorphisms
-$f_i\colon A\to B_i$, and rational Weil classes
-$\eta_i\in H^{2r}(B_i(\C),\Q(r))$ such that
-\begin{equation}
-\label{eq:cm-pullback-decomposition}
-                 \gamma=\sum_i f_i^*\eta_i.
-\end{equation}
-Each $B_i$ has an action of a CM field $K$ for which $H_1(B_i(\C),\Q)$
-has rank $2r$ over $K$, the Hodge signatures are $(r,r)$ at every
-embedding of $K$, and a polarization induces complex conjugation on $K$.
-The equality specializes simultaneously in all the realizations used in
-Theorem~\ref{thm:main}.
-\end{lemma}
-
-\begin{proof}
-We first identify the eigenspace components that can support a rational
-Hodge class.  For each such component we construct a balanced abelian
-factor of a power of $A$, whose Weil classes pull back onto that
-component.  Taking the sum of these rational pullback spaces will then
-give the decomposition over $\Q$.
-
-\medskip\noindent
-\emph{Balanced eigenspace components.}
-Set $V=H_1(A(\C),\Q)$.  We work throughout up to isogeny.  There is a
-CM algebra $E'$, a product of CM fields, acting on $A$ such that $V$ is
-free of rank one over $E'$.  To obtain it, decompose $A$ up to isogeny
-into simple CM factors and take the product of their CM fields, with a
-separate label for each repeated factor.  Transporting this action back
-along an isogeny gives the asserted action on $A$ itself.
-
-Choose a finite Galois CM field $K\subset\Qbar$ containing the images of
-all the factors of $E'$, and put $S=\Hom_{\Q\text{-alg}}(E',K)$.  Then
-\[
-                V_K:=V\otimes_{\Q}K=\bigoplus_{\rho\in S}V_\rho,
-                \qquad \dim_K V_\rho=1.
-\]
-The Hodge decomposition is diagonal for this decomposition after
-extension to $\C$.  We suppress the common Tate twist in the following
-linear-algebra calculations.  Alternating forms of degree $2r$ decompose
-as
-\[
- \bigwedge^{2r}V_K^*
-     =\bigoplus_{\substack{\Delta\subset S\\ |\Delta|=2r}}L_\Delta,
- \qquad
- L_\Delta=
- \bigwedge^{2r}\left(\bigoplus_{\rho\in\Delta}V_\rho\right)^*,
-\]
-where each form on the right is extended by zero on the other summands.
-Every $L_\Delta$ is one-dimensional and has a definite Hodge type.
-
-Suppose the component of $\gamma$ in $L_\Delta$ is nonzero.  Since
-$\gamma$ is rational, its component in $L_{\sigma\Delta}$ is the
-semilinear conjugate of that component and is nonzero for every
-$\sigma\in\Gal(K/\Q)$.  The Hodge condition therefore implies that
-every $\sigma\Delta$ contains $r$ lines of each of the two Hodge types.
-We call such a subset balanced under all conjugates.
-
-\medskip\noindent
-\emph{An abelian factor with balanced signatures.}
-Fix one of these subsets $\Delta$.  Form the abelian variety
-$A\otimes_{\Q}K$ in the category up to isogeny: it is a power of $A$
-whose index space is the underlying rational vector space of $K$.
-The regular action of $K$ on this index space is an action by rational
-endomorphisms of the power.  Its rational homology is
-\[
- T=V\otimes_{\Q}K,\qquad
- \dim_\Q T=[K:\Q]\dim_\Q V.
-\]
-In contrast with the coefficient space $V_K$, we regard $T$ here as a
-$\Q$-vector space, with $K$ acting on its second factor.  The abelian
-variety remains over $\Qbar$.  The actions on the two factors give
-an action of the rational algebra
-\[
- E'\otimes_{\Q}K\simeq\prod_{\rho\in S}K,
- \qquad e\otimes k\longmapsto\bigl(\rho(e)k\bigr)_{\rho\in S}.
-\]
-Let $e_\Delta$ be the idempotent with coordinates $1$ on $\Delta$ and
-$0$ elsewhere.  It is a rational endomorphism of $A\otimes_{\Q}K$,
-so its image is an abelian variety $B_\Delta$ up to isogeny.  In
-particular no Galois invariance of $\Delta$ is required for this
-construction.  Write
-\[
-                   U=H_1(B_\Delta(\C),\Q)=e_\Delta T.
-\]
-The index-field action of $K$ preserves $U$.
-
-Now extend the coefficients of $U$ to $K$, and use the fixed inclusion
-$K\subset\C$ to identify the embeddings of $K$ with $\Gal(K/\Q)$.
-Evaluation of the index factor at $\tau$ identifies its eigenspace in
-$T\otimes_\Q K$ with $V_K$.  In that eigenspace the idempotent retains
-exactly the summands indexed by $\tau\Delta$.  Thus
-\begin{equation}
-\label{eq:cm-embedding-summand}
-                  U_\tau=\bigoplus_{\rho\in\Delta}V_{\tau\rho},
-\end{equation}
-and the same description holds after extension to $\C$.  Explicitly,
-evaluation of the idempotent at the pair of embeddings $(\xi,\tau)$ is $1$ exactly
-when $\xi=\tau\rho$ for some $\rho\in\Delta$.  Thus
-$\dim_K U_\tau=2r$ for every $\tau$.  Since $K$ is Galois and
-$\Delta$ is balanced under all conjugates, each $U_\tau$ has $r$ lines
-of each Hodge type.  Consequently $U$ has rank $2r$ over $K$,
-$\dim B_\Delta=r[K:\Q]$, and the required Hodge signatures are
-$(r,r)$.
-
-\medskip\noindent
-\emph{A compatible polarization.}
-Let $\psi$ be a
-rational polarization form on $V$, and equip the index space $K$ with
-the positive definite rational symmetric form
-\[
-                       t(k,l)=\Tr_{K/\Q}(k\bar l).
-\]
-The form $\Psi=\psi\otimes t$ is a rational polarization form on the
-power $A\otimes_{\Q}K$: its alternating and Hodge properties follow
-from those of $\psi$, and its positivity follows by tensoring the
-positive symmetric form associated to $\psi$ with $t$.  A positive
-integral multiple gives an actual polarization.  For $a\in K$,
-\[
-                       t(ak,l)=t(k,\bar a l),
-\]
-so the Rosati involution of $\Psi$ induces conjugation on the
-index-field action.  Restrict $\Psi$ to $B_\Delta$.  This is again a
-polarization, and because $U$ is $K$-stable its adjoint identity still
-induces conjugation on $K$.  This restriction does not require
-$e_\Delta$ to be self-adjoint.
-
-\medskip\noindent
-\emph{Pullback and descent of the rational span.}
-The factor $B_\Delta$ now has the field action, balanced signatures,
-and polarization needed to define its Weil classes.  The rational
-homomorphism $f_\Delta\colon A\to B_\Delta$ defined on
-homology by
-\[
-                      v\longmapsto e_\Delta(v\otimes 1)
-\]
-has the needed effect on Weil classes.  After splitting coefficients,
-its component at the identity embedding of $K$ is the projection
-\[
-                 \bigoplus_{\rho\in S}V_\rho
-                    \longrightarrow
-                 \bigoplus_{\rho\in\Delta}V_\rho.
-\]
-Thus pullback maps $\bigwedge^{2r}U_{\id}^*$ isomorphically onto
-$L_\Delta$.  More generally the component indexed by $\tau$ pulls back
-onto $L_{\tau\Delta}$.  The rational Weil-class space $W_K^r(B_\Delta)$
-is characterized, with the twist restored, by
-\[
- W_K^r(B_\Delta)\otimes_{\Q}K
-       =\bigoplus_{\tau:K\hookrightarrow K}
-                         \bigwedge^{2r}U_\tau^*(r).
-\]
-In particular $f_\Delta^*W_K^r(B_\Delta)$ is a rational subspace whose
-extension to $K$ contains $L_\Delta(r)$.
-
-Take the sum of these rational pullback spaces over the finitely many
-subsets supporting $\gamma$.  Its scalar extension contains every
-component of $\gamma$, and therefore contains $\gamma$ itself.  Since
-extension of scalars is injective on the quotient by a rational
-subspace, the sum already contains $\gamma$ over $\Q$.  This proves
-\eqref{eq:cm-pullback-decomposition} with rational Weil classes.  Notice
-that an individual embedding component need not descend to $\Q$;
-only the entire Weil-class space and its pullback are used in this
-descent argument.
-
-\medskip\noindent
-\emph{Good reduction and specialization.}
-All the constructions use finitely many rational endomorphisms of
-powers of $A$ and their images.  They therefore exist over $\Qbar$ and
-descend, together with the maps, to a common number field after a finite
-extension.  Choose that extension so that $A$ has good reduction at the
-place specified by $w$.  For any $\ell\ne p$, the rational Tate module
-of $B_\Delta$ is a Galois-stable direct summand of the rational Tate
-module of the power of $A$.  It is unramified at this place.  The
-N\'eron--Ogg--Shafarevich criterion gives good reduction of
-$B_\Delta$ there; it also shows that the isogenies used in the
-construction do not affect this conclusion
-\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}.  Rational homomorphisms between
-the resulting good-reduction varieties specialize after clearing
-denominators.  The equality of rational Hodge classes
-\eqref{eq:cm-pullback-decomposition} is respected by their compatible
-realizations and by specialization, with the same rational coefficients
-in every component.  This proves the last assertion.
-\end{proof}
-
-\begin{corollary}
-\label{cor:cm-pairing}
-The Lefschetz pairing property holds for the specialization of every
-rational Hodge class on a CM abelian variety over $\Qbar$ with good
-reduction at $w$.
-\end{corollary}
-
-\begin{proof}
-In codimension zero the class is a rational multiple of the unit, so
-the assertion follows from the compatibility of divisor intersection
-numbers with the normalized traces.  In positive codimension use
-Lemma~\ref{lem:cm-reduction}.  Proposition~\ref{prop:weil} gives the
-Lefschetz pairing property for the specialization of each $\eta_i$.
-Lemma~\ref{lem:pairing-functorial} preserves this property under
-rational pullback and rational linear combinations.  Applying it to
-\eqref{eq:cm-pullback-decomposition} proves the assertion.
-\end{proof}
-
-\subsection{Proof of the main theorem}
-
-\begin{proof}[Proof of Theorem~\ref{thm:main}]
-The case \(r=0\) follows from divisor intersection numbers as in the
-preceding corollary. This includes \(\dim A=0\), where the abelian
-variety is a point. If \(r=\dim A\), a rational top class is a
-rational multiple of the point class; its normalized trace is
-preserved by specialization, and the complementary product is \(1\).
-We may therefore assume \(0<r<\dim A\).
-
-Corollary~\ref{cor:cm-transport} gives a CM abelian variety $B/\Qbar$ with good
-reduction at $w$, a rational Hodge class $\kappa$ on $B$, a
-quasi-isogeny $a\colon A_0^n\to B_0$, and the first-factor inclusion
-$j\colon A_0\to A_0^n$, such that
-\[
-                            \gamma_0=(aj)^*\kappa_0
-\]
-simultaneously in all the realizations under consideration.
-Corollary~\ref{cor:cm-pairing} gives the Lefschetz pairing property for
-$\kappa_0$, and Lemma~\ref{lem:pairing-functorial} gives it for
-$\gamma_0$.  Hence, for each prescribed product of complementary
-divisor classes on $A_0$, all the normalized traces in
-Theorem~\ref{thm:main} equal one rational number.  This proves the
-theorem.
-\end{proof}
-
-\subsection{Specialization in a fixed good theory}
-
-We now use CM transport to place the specialization of every rational
-Hodge class in one fixed theory of rational Tate classes.
-
-Continue to fix \(\Qbar\subset\C\), \(p\), and \(w\), with
-\(\F\), \(\Kzero\), and \(\Cw\) as in the introduction. Let
-\(\mathcal S_0\) be Milne's class of \(\F\)-varieties whose connected
-components are products of abelian varieties and projective spaces.
-The CM case of Theorem~\ref{thm:main}, by linearity for complementary
-Lefschetz classes, satisfies the hypothesis of
-\cite[Theorem~4.5]{Milne2009}. Fix one good theory \(\mathcal R\) of
-rational Tate classes on \(\mathcal S_0\) supplied by that theorem,
-before varying any abelian variety or Hodge class.
-
-For \(\rho\in\mathcal R^r(X)\), write \((\rho)_\ell\) for its
-\(\ell\)-adic component when \(\ell\ne p\), and \((\rho)_p\) for
-its native crystalline Tate component in
-\[
- H^{2r}_{\mathrm{cris}}(X/W(\F))[1/p](r).
-\]
-We use three properties of the theory.  Its spaces are rational vector
-spaces preserved by pullback, its realization maps are injective, and
-specializations of CM Hodge classes belong to it.  These are the
-properties in axioms \textup{(R1)}, \textup{(R4)}, and the goodness
-axiom \textup{(R3)}, respectively
-\cite[Definitions~2.1 and~3.1]{Milne2009}.  The following consequence
-extends the last property to all rational Hodge classes, while keeping
-this one theory fixed.
-
-\begin{corollary}[Specialization of all rational Hodge classes]
-\label{cor:all-hodge-specialization}
-For every abelian variety \(A/\Qbar\) of dimension \(d\) with good
-reduction \(A_0/\F\) at \(w\), every \(0\leq r\leq d\), and every
-rational Hodge class
-\[
- \gamma\in H^{2r}(A(\C),\Q(r)),
-\]
-there is a unique class \(\rho_\gamma\in\mathcal R^r(A_0)\) such that
-\[
- (\rho_\gamma)_\ell=\gamma_{0,\ell}\quad(\ell\ne p),\qquad
- (\rho_\gamma)_p\otimes 1=\gamma_{0,p}.
-\]
-The prime-to-\(p\) equalities are in
-\(H^{2r}_{\acute{e}t}(A_0,\Q_\ell(r))\); the crystalline equality is in
-\[
- H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
-       \otimes_{\Kzero}\Cw.
-\]
-The specialization components are those of Theorem~\ref{thm:main}.
-The assertion holds for every prime \(p\),
-including \(p=2\), with the prime, place, and theory fixed before
-\(A\) and \(\gamma\) vary.
-\end{corollary}
-
-\begin{proof}
-In codimension zero, take the corresponding rational multiple of the
-unit, which belongs to the theory by pullback from the point.  This
-also covers $d=0$.  In top codimension $r=d>0$, take $q$ times the
-point class, where $q\in\Q$ is the normalized trace of $\gamma$.
-The point class belongs to $\mathcal R^d(A_0)$
-\cite[\S2.3]{Milne2009}, and its normalized trace is one.  Compatibility
-of traces with specialization gives the required components in these
-cases.
-
-Suppose now that $0<r<d$.  Choose the CM variety $B$, Hodge class
-$\kappa$, and rational homomorphism
-\[
- b=aj\in\Hom^0(A_0,B_0),\qquad
- \gamma_0=b^*\kappa_0
-\]
-from Corollary~\ref{cor:cm-transport}.  The equality holds in all
-the stated realizations, with the usual absolute-Hodge specializations
-of $\kappa$.  Axiom \textup{(R3)} gives one class
-\(\rho_\kappa\in\mathcal R^r(B_0)\) with
-\[
- (\rho_\kappa)_\ell=\kappa_{0,\ell}\quad(\ell\ne p),\qquad
- (\rho_\kappa)_p\otimes1=\kappa_{0,p}.
-\]
-
-Choose an integer \(N>0\) such that \(h=Nb\) is an actual
-homomorphism \(A_0\to B_0\). Pullback by \(h\) preserves the theory
-by axiom \textup{(R1)} of \cite[Definition~2.1]{Milne2009}. Since
-\(\mathcal R^r(A_0)\) is a \(\Q\)-vector space, we may set
-\[
- \rho_\gamma=N^{-2r}h^*\rho_\kappa\in\mathcal R^r(A_0).
-\]
-Functoriality of the component maps and
-\eqref{eq:rational-pullback} give
-\begin{align*}
- (\rho_\gamma)_\ell
-   &=N^{-2r}h^*\kappa_{0,\ell}
-     =b^*\kappa_{0,\ell}=\gamma_{0,\ell}
-       &&(\ell\ne p),\\
- (\rho_\gamma)_p\otimes1
-   &=N^{-2r}h^*\kappa_{0,p}
-     =b^*\kappa_{0,p}=\gamma_{0,p}.
-\end{align*}
-The second line is taken after extension from $\Kzero$ to $\Cw$.
-Pullback in \textup{(R1)} applies to any homomorphism $h$, and rational
-scalar multiplication allows any denominator $N$.  Thus the argument
-requires neither an isogeny nor separability, and remains valid when
-$p\mid N$, including at $p=2$.
-
-Finally, two classes with the required components have the same
-\(\ell\)-adic component for any one \(\ell\ne p\). Injectivity of
-that component map by \textup{(R4)} makes them equal. Thus the class
-is unique in the fixed theory, independently of the auxiliary choices.
-\end{proof}
-
-This cohomological membership statement alone gives no algebraic cycle
-and proves neither the Hodge nor the Tate conjecture.  The construction
-takes place at the fixed
-reduction place: its CM auxiliary may have dimension $n\dim A$, and
-the map $b$ is between special fibers.  It supplies no same-dimensional
-CM lift or map between the generic fibers.
-
-\subsection{Algebraicity after good reduction}
-
-The CM Hodge theorem \cite[Theorem~1.1]{OpenAICMHodge2026}, combined
-with our CM transport, strengthens the preceding specialization
-statement. It is an additional input to the following consequence,
-not to the proof of Theorem~\ref{thm:main}.
-
-\begin{corollary}[Algebraic specialization of rational Hodge classes]
-\label{cor:algebraic-specialization}
-Fix the prime and place as in Theorem~\ref{thm:main}. Let
-\(A/\Qbar\) be an abelian variety of dimension \(d\) with good
-reduction \(A_0/\F\) at \(w\), and let
-\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class,
-where \(0\leq r\leq d\). There is one rational algebraic cycle
-\(z_\gamma\in\operatorname{CH}^r(A_0)_{\Q}\) such that
-\[
- \operatorname{cl}_\ell(z_\gamma)=\gamma_{0,\ell}
-       \quad(\ell\ne p),\qquad
- \operatorname{cl}_{\mathrm{cris}}(z_\gamma)\otimes1=\gamma_{0,p}.
-\]
-The crystalline equality is in
-\[
- H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
-       \otimes_{\Kzero}\Cw.
-\]
-This holds for every prime \(p\), including \(p=2\).
-Consequently, for every
-\(y\in\operatorname{CH}^{d-r}(A_0)_{\Q}\), the traces of
-\(\gamma_{0,\ell}\operatorname{cl}_\ell(y)\) for \(\ell\ne p\)
-and of \(\gamma_{0,p}\operatorname{cl}_{\mathrm{cris}}(y)\)
-are all the same rational number \(\deg(z_\gamma\cdot y)\).
-\end{corollary}
-
-\begin{proof}
-In codimension zero or top codimension, use the corresponding rational
-multiple of the unit or the point class, as in the preceding proof.
-This includes dimension zero.  For $0<r<d$, use the CM variety $B$,
-rational Hodge class $\kappa$, and map
-\[
- b=aj\in\Hom^0(A_0,B_0),\qquad \gamma_0=b^*\kappa_0
-\]
-given by Corollary~\ref{cor:cm-transport}, with the equality in every
-stated realization.
-
-By \cite[Theorem~1.1]{OpenAICMHodge2026}, \(\kappa\) is the
-Betti class of a rational algebraic cycle on \(B_{\C}\). A
-representative can be chosen over \(\Qbar\): spread each of its
-finitely many components in the Hilbert scheme of the fixed projective
-variety \(B\), and choose a \(\Qbar\)-point on the same
-geometrically connected component. The universal flat family preserves
-the Betti cycle class. Thus there is
-\(Z\in\operatorname{CH}^r(B)_{\Q}\) with class \(\kappa\).
-Descend \(B\) and \(Z\) to a number field and extend the chosen
-place so that \(B\) has an abelian scheme model. Specialization gives
-\(Z_0\in\operatorname{CH}^r(B_0)_{\Q}\). Compatibility of
-cycle classes with smooth proper specialization and the
-de Rham--crystalline comparison identifies its classes with
-\(\kappa_{0,\ell}\) and \(\kappa_{0,p}\), respectively.
-In particular the same rational cycle \(Z_0\) gives every realization.
-
-Choose \(N>0\) with \(h=Nb\) an actual homomorphism
-\(A_0\to B_0\), and put
-\[
- z_\gamma=N^{-2r}h^*Z_0\in\operatorname{CH}^r(A_0)_{\Q}.
-\]
-Here \(h^*\) is the Chow pullback between smooth varieties, defined
-by the graph and refined intersection; it requires neither flatness
-nor separability of \(h\). Functoriality of cycle classes and
-\eqref{eq:rational-pullback} give the required equalities. Rational
-coefficients permit this denominator even if \(p\mid N\); the
-crystalline realization has already inverted \(p\). Finally,
-compatibility of cycle classes with products and normalized traces
-gives the asserted intersection number for every \(y\).
-\end{proof}
-
-This conclusion concerns only the special fiber at the fixed place.
-It neither asserts the Hodge conjecture for arbitrary abelian varieties
-in characteristic zero nor lifts \(z_\gamma\) to the original
-generic fiber. The representing cycle is not asserted to be unique
-modulo rational equivalence, and no comparison between different
-reduction places is made.
+\section{CM reduction and consequences}
+\label{sec:reduction}
+
+The remaining step in the main proof is to express CM Hodge classes
+as pullbacks of balanced Weil classes.  Together with CM transport
+and Proposition~\ref{prop:weil}, this will prove
+Theorem~\ref{thm:main}.  We then derive two specialization consequences.
+
+\subsection{CM classes as pullbacks of Weil classes}
+
+We give the CM reduction in the form needed here.  The stronger
+reduction to classes on abelian varieties of split Weil type is due to
+Deligne and Andr\'e \cite{Andre1992}; see the reconstruction in
+\cite[Theorem~1 and Section~4]{Milne2020}. Only balanced signatures
+and a compatible polarization are needed for our application, and
+we prove this form directly using rational idempotents on powers.
+
+\begin{lemma}[CM reduction]
+\label{lem:cm-reduction}
+Let $A/\Qbar$ be an abelian variety of CM type with good reduction at $w$,
+and let $r>0$.  For every rational Hodge class
+$\gamma\in H^{2r}(A(\C),\Q(r))$, there are finitely many abelian varieties
+$B_i/\Qbar$ with good reduction at $w$, rational homomorphisms
+$f_i\colon A\to B_i$, and rational Weil classes
+$\eta_i\in H^{2r}(B_i(\C),\Q(r))$ such that
+\begin{equation}
+\label{eq:cm-pullback-decomposition}
+                 \gamma=\sum_i f_i^*\eta_i.
+\end{equation}
+Each $B_i$ has an action of a CM field $K$ for which $H_1(B_i(\C),\Q)$
+has rank $2r$ over $K$, the Hodge signatures are $(r,r)$ at every
+embedding of $K$, and a polarization induces complex conjugation on $K$.
+The equality specializes simultaneously in all the realizations used in
+Theorem~\ref{thm:main}.
+\end{lemma}
+
+\begin{proof}
+We first identify the eigenspace components that can support a rational
+Hodge class.  For each such component we construct a balanced abelian
+factor of a power of $A$, whose Weil classes pull back onto that
+component.  Taking the sum of these rational pullback spaces will then
+give the decomposition over $\Q$.
+
+\medskip\noindent
+\emph{Balanced eigenspace components.}
+Set $V=H_1(A(\C),\Q)$.  We work throughout up to isogeny.  There is a
+CM algebra $E'$, a product of CM fields, acting on $A$ such that $V$ is
+free of rank one over $E'$.  To obtain it, decompose $A$ up to isogeny
+into simple CM factors and take the product of their CM fields, with a
+separate label for each repeated factor.  Transporting this action back
+along an isogeny gives the asserted action on $A$ itself.
+
+Choose a finite Galois CM field $K\subset\Qbar$ containing the images of
+all the factors of $E'$, and put $S=\Hom_{\Q\text{-alg}}(E',K)$.  Then
+\[
+                V_K:=V\otimes_{\Q}K=\bigoplus_{\rho\in S}V_\rho,
+                \qquad \dim_K V_\rho=1.
+\]
+The Hodge decomposition is diagonal for this decomposition after
+extension to $\C$.  We suppress the common Tate twist in the following
+linear-algebra calculations.  Alternating forms of degree $2r$ decompose
+as
+\[
+ \bigwedge^{2r}V_K^*
+     =\bigoplus_{\substack{\Delta\subset S\\ |\Delta|=2r}}L_\Delta,
+ \qquad
+ L_\Delta=
+ \bigwedge^{2r}\left(\bigoplus_{\rho\in\Delta}V_\rho\right)^*,
+\]
+where each form on the right is extended by zero on the other summands.
+Every $L_\Delta$ is one-dimensional and has a definite Hodge type.
+
+Suppose the component of $\gamma$ in $L_\Delta$ is nonzero.  Since
+$\gamma$ is rational, its component in $L_{\sigma\Delta}$ is the
+semilinear conjugate of that component and is nonzero for every
+$\sigma\in\Gal(K/\Q)$.  The Hodge condition therefore implies that
+every $\sigma\Delta$ contains $r$ lines of each of the two Hodge types.
+We call such a subset balanced under all conjugates.
+
+\medskip\noindent
+\emph{An abelian factor with balanced signatures.}
+Fix one of these subsets $\Delta$.  Form the abelian variety
+$A\otimes_{\Q}K$ in the category up to isogeny: it is a power of $A$
+whose index space is the underlying rational vector space of $K$.
+The regular action of $K$ on this index space is an action by rational
+endomorphisms of the power.  Its rational homology is
+\[
+ T=V\otimes_{\Q}K,\qquad
+ \dim_\Q T=[K:\Q]\dim_\Q V.
+\]
+In contrast with the coefficient space $V_K$, we regard $T$ here as a
+$\Q$-vector space, with $K$ acting on its second factor.  The abelian
+variety remains over $\Qbar$.  The actions on the two factors give
+an action of the rational algebra
+\[
+ E'\otimes_{\Q}K\simeq\prod_{\rho\in S}K,
+ \qquad e\otimes k\longmapsto\bigl(\rho(e)k\bigr)_{\rho\in S}.
+\]
+Let $e_\Delta$ be the idempotent with coordinates $1$ on $\Delta$ and
+$0$ elsewhere.  It is a rational endomorphism of $A\otimes_{\Q}K$,
+so its image is an abelian variety $B_\Delta$ up to isogeny.  In
+particular no Galois invariance of $\Delta$ is required for this
+construction.  Write
+\[
+                   U=H_1(B_\Delta(\C),\Q)=e_\Delta T.
+\]
+The index-field action of $K$ preserves $U$.
+
+Now extend the coefficients of $U$ to $K$, and use the fixed inclusion
+$K\subset\C$ to identify the embeddings of $K$ with $\Gal(K/\Q)$.
+Evaluation of the index factor at $\tau$ identifies its eigenspace in
+$T\otimes_\Q K$ with $V_K$.  In that eigenspace the idempotent retains
+exactly the summands indexed by $\tau\Delta$.  Thus
+\begin{equation}
+\label{eq:cm-embedding-summand}
+                  U_\tau=\bigoplus_{\rho\in\Delta}V_{\tau\rho},
+\end{equation}
+and the same description holds after extension to $\C$.  Explicitly,
+evaluation of the idempotent at the pair of embeddings $(\xi,\tau)$ is $1$ exactly
+when $\xi=\tau\rho$ for some $\rho\in\Delta$.  Thus
+$\dim_K U_\tau=2r$ for every $\tau$.  Since $K$ is Galois and
+$\Delta$ is balanced under all conjugates, each $U_\tau$ has $r$ lines
+of each Hodge type.  Consequently $U$ has rank $2r$ over $K$,
+$\dim B_\Delta=r[K:\Q]$, and the required Hodge signatures are
+$(r,r)$.
+
+\medskip\noindent
+\emph{A compatible polarization.}
+Let $\psi$ be a
+rational polarization form on $V$, and equip the index space $K$ with
+the positive definite rational symmetric form
+\[
+                       t(k,l)=\Tr_{K/\Q}(k\bar l).
+\]
+The form $\Psi=\psi\otimes t$ is a rational polarization form on the
+power $A\otimes_{\Q}K$: its alternating and Hodge properties follow
+from those of $\psi$, and its positivity follows by tensoring the
+positive symmetric form associated to $\psi$ with $t$.  A positive
+integral multiple gives an actual polarization.  For $a\in K$,
+\[
+                       t(ak,l)=t(k,\bar a l),
+\]
+so the Rosati involution of $\Psi$ induces conjugation on the
+index-field action.  Restrict $\Psi$ to $B_\Delta$.  This is again a
+polarization, and because $U$ is $K$-stable its adjoint identity still
+induces conjugation on $K$.  This restriction does not require
+$e_\Delta$ to be self-adjoint.
+
+\medskip\noindent
+\emph{Pullback and descent of the rational span.}
+The factor $B_\Delta$ now has the field action, balanced signatures,
+and polarization needed to define its Weil classes.  The rational
+homomorphism $f_\Delta\colon A\to B_\Delta$ defined on
+homology by
+\[
+                      v\longmapsto e_\Delta(v\otimes 1)
+\]
+has the needed effect on Weil classes.  After splitting coefficients,
+its component at the identity embedding of $K$ is the projection
+\[
+                 \bigoplus_{\rho\in S}V_\rho
+                    \longrightarrow
+                 \bigoplus_{\rho\in\Delta}V_\rho.
+\]
+Thus pullback maps $\bigwedge^{2r}U_{\id}^*$ isomorphically onto
+$L_\Delta$.  More generally the component indexed by $\tau$ pulls back
+onto $L_{\tau\Delta}$.  The rational Weil-class space $W_K^r(B_\Delta)$
+is characterized, with the twist restored, by
+\[
+ W_K^r(B_\Delta)\otimes_{\Q}K
+       =\bigoplus_{\tau:K\hookrightarrow K}
+                         \bigwedge^{2r}U_\tau^*(r).
+\]
+In particular $f_\Delta^*W_K^r(B_\Delta)$ is a rational subspace whose
+extension to $K$ contains $L_\Delta(r)$.
+
+Take the sum of these rational pullback spaces over the finitely many
+subsets supporting $\gamma$.  Its scalar extension contains every
+component of $\gamma$, and therefore contains $\gamma$ itself.  Since
+extension of scalars is injective on the quotient by a rational
+subspace, the sum already contains $\gamma$ over $\Q$.  This proves
+\eqref{eq:cm-pullback-decomposition} with rational Weil classes.  Notice
+that an individual embedding component need not descend to $\Q$;
+only the entire Weil-class space and its pullback are used in this
+descent argument.
+
+\medskip\noindent
+\emph{Good reduction and specialization.}
+All the constructions use finitely many rational endomorphisms of
+powers of $A$ and their images.  They therefore exist over $\Qbar$ and
+descend, together with the maps, to a common number field after a finite
+extension.  Choose that extension so that $A$ has good reduction at the
+place specified by $w$.  For any $\ell\ne p$, the rational Tate module
+of $B_\Delta$ is a Galois-stable direct summand of the rational Tate
+module of the power of $A$.  It is unramified at this place.  The
+N\'eron--Ogg--Shafarevich criterion gives good reduction of
+$B_\Delta$ there; it also shows that the isogenies used in the
+construction do not affect this conclusion
+\cite[Chapter~IV, Theorem~3.5]{MilneAVNotes}.  Rational homomorphisms between
+the resulting good-reduction varieties specialize after clearing
+denominators.  The equality of rational Hodge classes
+\eqref{eq:cm-pullback-decomposition} is respected by their compatible
+realizations and by specialization, with the same rational coefficients
+in every component.  This proves the last assertion.
+\end{proof}
+
+\begin{corollary}
+\label{cor:cm-pairing}
+The Lefschetz pairing property holds for the specialization of every
+rational Hodge class on a CM abelian variety over $\Qbar$ with good
+reduction at $w$.
+\end{corollary}
+
+\begin{proof}
+In codimension zero the class is a rational multiple of the unit, so
+the assertion follows from the compatibility of divisor intersection
+numbers with the normalized traces.  In positive codimension use
+Lemma~\ref{lem:cm-reduction}.  Proposition~\ref{prop:weil} gives the
+Lefschetz pairing property for the specialization of each $\eta_i$.
+Lemma~\ref{lem:pairing-functorial} preserves this property under
+rational pullback and rational linear combinations.  Applying it to
+\eqref{eq:cm-pullback-decomposition} proves the assertion.
+\end{proof}
+
+\subsection{Proof of the main theorem}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+The case \(r=0\) follows from divisor intersection numbers as in the
+preceding corollary. This includes \(\dim A=0\), where the abelian
+variety is a point. If \(r=\dim A\), a rational top class is a
+rational multiple of the point class; its normalized trace is
+preserved by specialization, and the complementary product is \(1\).
+We may therefore assume \(0<r<\dim A\).
+
+Corollary~\ref{cor:cm-transport} gives a CM abelian variety $B/\Qbar$ with good
+reduction at $w$, a rational Hodge class $\kappa$ on $B$, a
+quasi-isogeny $a\colon A_0^n\to B_0$, and the first-factor inclusion
+$j\colon A_0\to A_0^n$, such that
+\[
+                            \gamma_0=(aj)^*\kappa_0
+\]
+simultaneously in all the realizations under consideration.
+Corollary~\ref{cor:cm-pairing} gives the Lefschetz pairing property for
+$\kappa_0$, and Lemma~\ref{lem:pairing-functorial} gives it for
+$\gamma_0$.  Hence, for each prescribed product of complementary
+divisor classes on $A_0$, all the normalized traces in
+Theorem~\ref{thm:main} equal one rational number.  This proves the
+theorem.
+\end{proof}
+
+\subsection{Specialization in a fixed good theory}
+
+We now use CM transport to place the specialization of every rational
+Hodge class in one fixed theory of rational Tate classes.
+
+Continue to fix \(\Qbar\subset\C\), \(p\), and \(w\), with
+\(\F\), \(\Kzero\), and \(\Cw\) as in the introduction. Let
+\(\mathcal S_0\) be Milne's class of \(\F\)-varieties whose connected
+components are products of abelian varieties and projective spaces.
+The CM case of Theorem~\ref{thm:main}, by linearity for complementary
+Lefschetz classes, satisfies the hypothesis of
+\cite[Theorem~4.5]{Milne2009}. Fix one good theory \(\mathcal R\) of
+rational Tate classes on \(\mathcal S_0\) supplied by that theorem,
+before varying any abelian variety or Hodge class.
+
+For \(\rho\in\mathcal R^r(X)\), write \((\rho)_\ell\) for its
+\(\ell\)-adic component when \(\ell\ne p\), and \((\rho)_p\) for
+its native crystalline Tate component in
+\[
+ H^{2r}_{\mathrm{cris}}(X/W(\F))[1/p](r).
+\]
+We use three properties of the theory.  Its spaces are rational vector
+spaces preserved by pullback, its realization maps are injective, and
+specializations of CM Hodge classes belong to it.  These are the
+properties in axioms \textup{(R1)}, \textup{(R4)}, and the goodness
+axiom \textup{(R3)}, respectively
+\cite[Definitions~2.1 and~3.1]{Milne2009}.  The following consequence
+extends the last property to all rational Hodge classes, while keeping
+this one theory fixed.
+
+\begin{corollary}[Specialization of all rational Hodge classes]
+\label{cor:all-hodge-specialization}
+For every abelian variety \(A/\Qbar\) of dimension \(d\) with good
+reduction \(A_0/\F\) at \(w\), every \(0\leq r\leq d\), and every
+rational Hodge class
+\[
+ \gamma\in H^{2r}(A(\C),\Q(r)),
+\]
+there is a unique class \(\rho_\gamma\in\mathcal R^r(A_0)\) such that
+\[
+ (\rho_\gamma)_\ell=\gamma_{0,\ell}\quad(\ell\ne p),\qquad
+ (\rho_\gamma)_p\otimes 1=\gamma_{0,p}.
+\]
+The prime-to-\(p\) equalities are in
+\(H^{2r}_{\acute{e}t}(A_0,\Q_\ell(r))\); the crystalline equality is in
+\[
+ H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
+       \otimes_{\Kzero}\Cw.
+\]
+The specialization components are those of Theorem~\ref{thm:main}.
+The assertion holds for every prime \(p\),
+including \(p=2\), with the prime, place, and theory fixed before
+\(A\) and \(\gamma\) vary.
+\end{corollary}
+
+\begin{proof}
+In codimension zero, take the corresponding rational multiple of the
+unit, which belongs to the theory by pullback from the point.  This
+also covers $d=0$.  In top codimension $r=d>0$, take $q$ times the
+point class, where $q\in\Q$ is the normalized trace of $\gamma$.
+The point class belongs to $\mathcal R^d(A_0)$
+\cite[\S2.3]{Milne2009}, and its normalized trace is one.  Compatibility
+of traces with specialization gives the required components in these
+cases.
+
+Suppose now that $0<r<d$.  Choose the CM variety $B$, Hodge class
+$\kappa$, and rational homomorphism
+\[
+ b=aj\in\Hom^0(A_0,B_0),\qquad
+ \gamma_0=b^*\kappa_0
+\]
+from Corollary~\ref{cor:cm-transport}.  The equality holds in all
+the stated realizations, with the usual absolute-Hodge specializations
+of $\kappa$.  Axiom \textup{(R3)} gives one class
+\(\rho_\kappa\in\mathcal R^r(B_0)\) with
+\[
+ (\rho_\kappa)_\ell=\kappa_{0,\ell}\quad(\ell\ne p),\qquad
+ (\rho_\kappa)_p\otimes1=\kappa_{0,p}.
+\]
+
+Choose an integer \(N>0\) such that \(h=Nb\) is an actual
+homomorphism \(A_0\to B_0\). Pullback by \(h\) preserves the theory
+by axiom \textup{(R1)} of \cite[Definition~2.1]{Milne2009}. Since
+\(\mathcal R^r(A_0)\) is a \(\Q\)-vector space, we may set
+\[
+ \rho_\gamma=N^{-2r}h^*\rho_\kappa\in\mathcal R^r(A_0).
+\]
+Functoriality of the component maps and
+\eqref{eq:rational-pullback} give
+\begin{align*}
+ (\rho_\gamma)_\ell
+   &=N^{-2r}h^*\kappa_{0,\ell}
+     =b^*\kappa_{0,\ell}=\gamma_{0,\ell}
+       &&(\ell\ne p),\\
+ (\rho_\gamma)_p\otimes1
+   &=N^{-2r}h^*\kappa_{0,p}
+     =b^*\kappa_{0,p}=\gamma_{0,p}.
+\end{align*}
+The second line is taken after extension from $\Kzero$ to $\Cw$.
+Pullback in \textup{(R1)} applies to any homomorphism $h$, and rational
+scalar multiplication allows any denominator $N$.  Thus the argument
+requires neither an isogeny nor separability, and remains valid when
+$p\mid N$, including at $p=2$.
+
+Finally, two classes with the required components have the same
+\(\ell\)-adic component for any one \(\ell\ne p\). Injectivity of
+that component map by \textup{(R4)} makes them equal. Thus the class
+is unique in the fixed theory, independently of the auxiliary choices.
+\end{proof}
+
+This cohomological membership statement alone gives no algebraic cycle
+and proves neither the Hodge nor the Tate conjecture.  The construction
+takes place at the fixed
+reduction place: its CM auxiliary may have dimension $n\dim A$, and
+the map $b$ is between special fibers.  It supplies no same-dimensional
+CM lift or map between the generic fibers.
+
+\subsection{Algebraicity after good reduction}
+
+The CM Hodge theorem \cite[Theorem~1.1]{OpenAICMHodge2026}, combined
+with our CM transport, strengthens the preceding specialization
+statement. It is an additional input to the following consequence,
+not to the proof of Theorem~\ref{thm:main}.
+
+\begin{corollary}[Algebraic specialization of rational Hodge classes]
+\label{cor:algebraic-specialization}
+Fix the prime and place as in Theorem~\ref{thm:main}. Let
+\(A/\Qbar\) be an abelian variety of dimension \(d\) with good
+reduction \(A_0/\F\) at \(w\), and let
+\(\gamma\in H^{2r}(A(\C),\Q(r))\) be a rational Hodge class,
+where \(0\leq r\leq d\). There is one rational algebraic cycle
+\(z_\gamma\in\operatorname{CH}^r(A_0)_{\Q}\) such that
+\[
+ \operatorname{cl}_\ell(z_\gamma)=\gamma_{0,\ell}
+       \quad(\ell\ne p),\qquad
+ \operatorname{cl}_{\mathrm{cris}}(z_\gamma)\otimes1=\gamma_{0,p}.
+\]
+The crystalline equality is in
+\[
+ H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
+       \otimes_{\Kzero}\Cw.
+\]
+This holds for every prime \(p\), including \(p=2\).
+Consequently, for every
+\(y\in\operatorname{CH}^{d-r}(A_0)_{\Q}\), the traces of
+\(\gamma_{0,\ell}\operatorname{cl}_\ell(y)\) for \(\ell\ne p\)
+and of \(\gamma_{0,p}\operatorname{cl}_{\mathrm{cris}}(y)\)
+are all the same rational number \(\deg(z_\gamma\cdot y)\).
+\end{corollary}
+
+\begin{proof}
+In codimension zero or top codimension, use the corresponding rational
+multiple of the unit or the point class, as in the preceding proof.
+This includes dimension zero.  For $0<r<d$, use the CM variety $B$,
+rational Hodge class $\kappa$, and map
+\[
+ b=aj\in\Hom^0(A_0,B_0),\qquad \gamma_0=b^*\kappa_0
+\]
+given by Corollary~\ref{cor:cm-transport}, with the equality in every
+stated realization.
+
+By \cite[Theorem~1.1]{OpenAICMHodge2026}, \(\kappa\) is the
+Betti class of a rational algebraic cycle on \(B_{\C}\). A
+representative can be chosen over \(\Qbar\): spread each of its
+finitely many components in the Hilbert scheme of the fixed projective
+variety \(B\), and choose a \(\Qbar\)-point on the same
+geometrically connected component. The universal flat family preserves
+the Betti cycle class. Thus there is
+\(Z\in\operatorname{CH}^r(B)_{\Q}\) with class \(\kappa\).
+Descend \(B\) and \(Z\) to a number field and extend the chosen
+place so that \(B\) has an abelian scheme model. Specialization gives
+\(Z_0\in\operatorname{CH}^r(B_0)_{\Q}\). Compatibility of
+cycle classes with smooth proper specialization and the
+de Rham--crystalline comparison identifies its classes with
+\(\kappa_{0,\ell}\) and \(\kappa_{0,p}\), respectively.
+In particular the same rational cycle \(Z_0\) gives every realization.
+
+Choose \(N>0\) with \(h=Nb\) an actual homomorphism
+\(A_0\to B_0\), and put
+\[
+ z_\gamma=N^{-2r}h^*Z_0\in\operatorname{CH}^r(A_0)_{\Q}.
+\]
+Here \(h^*\) is the Chow pullback between smooth varieties, defined
+by the graph and refined intersection; it requires neither flatness
+nor separability of \(h\). Functoriality of cycle classes and
+\eqref{eq:rational-pullback} give the required equalities. Rational
+coefficients permit this denominator even if \(p\mid N\); the
+crystalline realization has already inverted \(p\). Finally,
+compatibility of cycle classes with products and normalized traces
+gives the asserted intersection number for every \(y\).
+\end{proof}
+
+This conclusion concerns only the special fiber at the fixed place.
+It neither asserts the Hodge conjecture for arbitrary abelian varieties
+in characteristic zero nor lifts \(z_\gamma\) to the original
+generic fiber. The representing cycle is not asserted to be unique
+modulo rational equivalence, and no comparison between different
+reduction places is made.

preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/README.md

@@ -1,21 +1,21 @@
-# [Numerical semiampleness of nef adjoint classes on compact Kähler manifolds](numerical-generalized-abundance.pdf)
-
-**Author:** OpenAI
-
-**Date:** October 6, 2026
-
-## Citation
-
-```bibtex
-@misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026,
-  author = {{OpenAI}},
-  title = {{Numerical semiampleness of nef adjoint classes on compact K{\"a}hler manifolds}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf}{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026}},
-  year = {2026}
-}
-```
-
-## Version note
-
-This version states the Iitaka, model, and contraction inputs used in the divisorial-decomposition argument explicitly; see the [previous version](../Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf).
+# [Numerical semiampleness of nef adjoint classes on compact Kähler manifolds](numerical-generalized-abundance.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 6, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026,
+  author = {{OpenAI}},
+  title = {{Numerical semiampleness of nef adjoint classes on compact K{\"a}hler manifolds}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf}{OAI:Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026}},
+  year = {2026}
+}
+```
+
+## Version note
+
+This version states the Iitaka, model, and contraction inputs used in the divisorial-decomposition argument explicitly; see the [previous version](../Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf).

preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/companions.bib

@@ -1,41 +1,41 @@
-@misc{K,
-  author = {{OpenAI}},
-  title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}},
-  year = {2026},
-  note = {Theorem 2.13, Lemma 3.6, Proposition 3.8, Proposition 3.30, and Theorem 3.55}
-}
-
-@misc{LI,
-  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},
-  note = {Corollary 6.2; Lemmas 4.1 and 4.3 and Proposition 4.5}
-}
-
-@misc{P,
-  author = {{OpenAI}},
-  title = {{Numerical Semiampleness of Nef Adjoint Divisors}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf}{OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026}},
-  year = {2026}
-}
-
-@misc{MM,
-  author = {{OpenAI}},
-  title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}},
-  howpublished = {OpenAI Math Release preprint
-                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}},
-  year = {2026}
-}
-
-@misc{LA,
-  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}
-}
+@misc{K,
+  author = {{OpenAI}},
+  title = {{Log abundance for compact K{\"a}hler spaces under logarithmic Iitaka subadditivity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf}{OAI:Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026}},
+  year = {2026},
+  note = {Theorem 2.13, Lemma 3.6, Proposition 3.8, Proposition 3.30, and Theorem 3.55}
+}
+
+@misc{LI,
+  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},
+  note = {Corollary 6.2; Lemmas 4.1 and 4.3 and Proposition 4.5}
+}
+
+@misc{P,
+  author = {{OpenAI}},
+  title = {{Numerical Semiampleness of Nef Adjoint Divisors}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf}{OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026}},
+  year = {2026}
+}
+
+@misc{MM,
+  author = {{OpenAI}},
+  title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}},
+  year = {2026}
+}
+
+@misc{LA,
+  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}
+}

preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/figures/zero-two-diagonals.tex

@@ -1,42 +1,42 @@
-\begin{minipage}[c]{0.47\linewidth}
-\centering
-\small
-\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)]
-\node (v) at (0,1.5) {$V$};
-\node (b) at (2.7,1.5) {$B$};
-\node (v0) at (0,0) {$\mathcal V_0$};
-\node (zz) at (2.7,0) {$Z\times Z$};
-\draw[->] (v)--node[above] {$\subset$} (b);
-\draw[->] (v0)--node[above] {$\subset$} (zz);
-\draw[->] (v)--node[left] {$b|_V$} (v0);
-\draw[->] (b)--node[right] {$b$} (zz);
-\end{tikzpicture}
-\[
- \begin{gathered}
- V_z=\operatorname{Bl}_xX,\\
- (V\cap E)_z=\mathbb P(T_z^*D_\lambda)
-       \subset E_z=\mathbb P(T_z^*Z).
- \end{gathered}
-\]
-\end{minipage}
-\hfill
-\begin{minipage}[c]{0.50\linewidth}
-\centering
-\small
-\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)]
-\node (ep) at (0,1.5) {$E^+$};
-\node (e) at (3,1.5) {$E=\mathbb P_Z(\Omega_Z)$};
-\node (zp) at (0,0) {$Z^+$};
-\node (z) at (3,0) {$Z$};
-\draw[->] (ep)--(e);
-\draw[->] (ep)--(zp);
-\draw[->] (e)--(z);
-\draw[->] (zp)--node[below] {$p_+=\operatorname{Bl}_{U_0}$} (z);
-\end{tikzpicture}
-\[
- \begin{gathered}
- J_+=p_+^{-1}(U_0),\qquad U_0=X\times\mathbb P^1,\\
- \mathcal J=(\mathcal I_V\cdot\cO_E)\cdot\cO_{E^+}.
- \end{gathered}
-\]
-\end{minipage}
+\begin{minipage}[c]{0.47\linewidth}
+\centering
+\small
+\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)]
+\node (v) at (0,1.5) {$V$};
+\node (b) at (2.7,1.5) {$B$};
+\node (v0) at (0,0) {$\mathcal V_0$};
+\node (zz) at (2.7,0) {$Z\times Z$};
+\draw[->] (v)--node[above] {$\subset$} (b);
+\draw[->] (v0)--node[above] {$\subset$} (zz);
+\draw[->] (v)--node[left] {$b|_V$} (v0);
+\draw[->] (b)--node[right] {$b$} (zz);
+\end{tikzpicture}
+\[
+ \begin{gathered}
+ V_z=\operatorname{Bl}_xX,\\
+ (V\cap E)_z=\mathbb P(T_z^*D_\lambda)
+       \subset E_z=\mathbb P(T_z^*Z).
+ \end{gathered}
+\]
+\end{minipage}
+\hfill
+\begin{minipage}[c]{0.50\linewidth}
+\centering
+\small
+\begin{tikzpicture}[>=stealth, baseline=(current bounding box.center)]
+\node (ep) at (0,1.5) {$E^+$};
+\node (e) at (3,1.5) {$E=\mathbb P_Z(\Omega_Z)$};
+\node (zp) at (0,0) {$Z^+$};
+\node (z) at (3,0) {$Z$};
+\draw[->] (ep)--(e);
+\draw[->] (ep)--(zp);
+\draw[->] (e)--(z);
+\draw[->] (zp)--node[below] {$p_+=\operatorname{Bl}_{U_0}$} (z);
+\end{tikzpicture}
+\[
+ \begin{gathered}
+ J_+=p_+^{-1}(U_0),\qquad U_0=X\times\mathbb P^1,\\
+ \mathcal J=(\mathcal I_V\cdot\cO_E)\cdot\cO_{E^+}.
+ \end{gathered}
+\]
+\end{minipage}

preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/literature.bib

@@ -1,536 +1,536 @@
-
-@article{Boucksom2004,
-  author = {Boucksom, S{\'e}bastien},
-  title = {Divisorial {Zariski} decompositions on compact complex manifolds},
-  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
-  series = {4},
-  volume = {37},
-  number = {1},
-  year = {2004},
-  pages = {45--76},
-  doi = {10.1016/j.ansens.2003.04.002},
-  url = {https://numdam.org/articles/10.1016/j.ansens.2003.04.002/}
-}
-
-@article{DemaillyPaun2004,
-  author = {Demailly, Jean-Pierre and P{\u a}un, Mihai},
-  title = {Numerical characterization of the {K{\"a}hler} cone of a compact {K{\"a}hler} manifold},
-  journal = {Annals of Mathematics},
-  series = {2},
-  volume = {159},
-  number = {3},
-  year = {2004},
-  pages = {1247--1274},
-  doi = {10.4007/annals.2004.159.1247},
-  url = {https://annals.math.princeton.edu/2004/159-3/p05}
-}
-
-@misc{Ou,
-  author = {Ou, Wenhao},
-  title = {A characterization of uniruled compact {K{\"a}hler} manifolds},
-  year = {2025},
-  eprint = {2501.18088},
-  archivePrefix = {arXiv},
-  primaryClass = {math.AG},
-  note = {arXiv:2501.18088v1},
-  url = {https://arxiv.org/abs/2501.18088v1}
-}
-
-@misc{Vu2023,
-  author = {Vu, Duc-Viet},
-  title = {Derivative of volumes of big cohomology classes},
-  year = {2023},
-  eprint = {2307.15909},
-  archivePrefix = {arXiv},
-  primaryClass = {math.AG},
-  note = {arXiv:2307.15909v1},
-  url = {https://arxiv.org/abs/2307.15909v1}
-}
-
-@article{CollinsTosatti2018,
-  author = {Collins, Tristan C. and Tosatti, Valentino},
-  title = {Restricted volumes on {K{\"a}hler} manifolds},
-  journal = {Annales de la Facult{\'e} des sciences de Toulouse: Math{\'e}matiques},
-  series = {6},
-  volume = {31},
-  number = {3},
-  year = {2022},
-  pages = {907--947},
-  doi = {10.5802/afst.1708},
-  eprint = {1608.07089},
-  archivePrefix = {arXiv},
-  url = {https://www.numdam.org/articles/10.5802/afst.1708/}
-}
-
-@article{LevyGRR,
-  author = {Levy, Roni N.},
-  title = {The {Riemann--Roch} theorem for complex spaces},
-  journal = {Acta Mathematica},
-  volume = {158},
-  year = {1987},
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+}
+
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+  author = {Fujiki, Akira},
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+}
+
+@misc{HX,
+  author = {Hacon, Christopher and Xie, Lingyao},
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+  year = {2026},
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+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  note = {arXiv:2607.24986v1},
+  doi = {10.48550/arXiv.2607.24986},
+  url = {https://arxiv.org/abs/2607.24986v1}
+}
+
+@article{DinhNguyen,
+  author = {Dinh, Tien-Cuong and Nguyen, Viet-Anh},
+  title = {The mixed {Hodge--Riemann} bilinear relations for compact {K{\"a}hler} manifolds},
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+  number = {4},
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+  pages = {838--849},
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+  archivePrefix = {arXiv},
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+}
+
+@article{Tosatti2016,
+  author = {Tosatti, Valentino},
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+}
+
+@article{BEGZ2010,
+  author = {Boucksom, S{\'e}bastien and Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed},
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+  journal = {Acta Mathematica},
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+  year = {2010},
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+}
+
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+  author = {Demailly, Jean-Pierre},
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+  pages = {361--409},
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+}
+
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+  author = {Lieberman, David I.},
+  title = {Compactness of the {Chow} scheme: applications to automorphisms and deformations of {K{\"a}hler} manifolds},
+  booktitle = {Fonctions de plusieurs variables complexes {III}},
+  editor = {Norguet, Fran{\c c}ois},
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+  volume = {670},
+  publisher = {Springer},
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+}
+
+@article{KebekusSchnell2021,
+  author = {Kebekus, Stefan and Schnell, Christian},
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+  journal = {Journal of the American Mathematical Society},
+  volume = {34},
+  number = {2},
+  year = {2021},
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+  eprint = {1811.03644},
+  archivePrefix = {arXiv},
+  note = {Numbered citations refer to arXiv:1811.03644v4},
+  url = {https://arxiv.org/abs/1811.03644v4}
+}
+
+@article{Miyaoka1988,
+  author = {Miyaoka, Yoichi},
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+}
+
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+}
+
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+}
+
+@article{KMM2004,
+  author = {Keel, Sean and Matsuki, Kenji and McKernan, James},
+  title = {Corrections to: \textquotedblleft Log abundance theorem for threefolds\textquotedblright},
+  journal = {Duke Mathematical Journal},
+  volume = {122},
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+  year = {2004},
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+  url = {https://www.math.purdue.edu/people/profile/matsuki.html}
+}
+
+@article{HoeringPeternell2016,
+  author = {H{\"o}ring, Andreas and Peternell, Thomas},
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+}
+
+@article{CHP2016,
+  author = {Campana, Fr{\'e}d{\'e}ric and H{\"o}ring, Andreas and Peternell, Thomas},
+  title = {Abundance for {K{\"a}hler} threefolds},
+  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
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+}
+
+@misc{CHP2023,
+  author = {Campana, Fr{\'e}d{\'e}ric and H{\"o}ring, Andreas and Peternell, Thomas},
+  title = {Erratum and addendum to the paper: {Abundance for K{\"a}hler threefolds}},
+  year = {2023},
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+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  note = {arXiv:2304.10161v1},
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+}
+
+@article{DasOu2026,
+  author = {Das, Omprokash and Ou, Wenhao},
+  title = {On the log abundance for compact {K{\"a}hler} threefolds {II}},
+  journal = {Proceedings of the London Mathematical Society},
+  volume = {132},
+  number = {3},
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+  pages = {e70141},
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+  archivePrefix = {arXiv},
+  note = {Numbered citations refer to arXiv:2306.00671v4},
+  url = {https://arxiv.org/abs/2306.00671v4}
+}
+
+@article{LP1,
+  author = {Lazi{\'c}, Vladimir and Peternell, Thomas},
+  title = {On Generalised Abundance, {I}},
+  journal = {Publications of the Research Institute for Mathematical Sciences},
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+  eprint = {1808.00438}, archivePrefix = {arXiv},
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+}
+
+@article{LP2,
+  author = {Lazi{\'c}, Vladimir and Peternell, Thomas},
+  title = {On Generalised Abundance, {II}},
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+  volume = {3}, number = {1}, year = {2020}, pages = {1--46},
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+  eprint = {1809.02500}, archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1809.02500v4}
+}
+
+@article{DasOu2024,
+  author = {Das, Omprokash and Ou, Wenhao},
+  title = {On the log abundance for compact {K{\"a}hler} threefolds},
+  journal = {Manuscripta Mathematica},
+  volume = {173}, number = {1--2}, year = {2024}, pages = {341--404},
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+  eprint = {2201.01202}, archivePrefix = {arXiv},
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+}
+
+@misc{DHY,
+  author = {Das, Omprokash and Hacon, Christopher and Y{\'a}{\~n}ez, Jos{\'e} Ignacio},
+  title = {{MMP} for Generalized Pairs on {K{\"a}hler} $3$-folds},
+  year = {2026},
+  note = {arXiv:2305.00524v3; to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
+  doi = {10.2422/2036-2145.202506_004},
+  eprint = {2305.00524}, archivePrefix = {arXiv},
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+}
+
+@article{Ambro,
+  author = {Ambro, Florin},
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+}
+
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+  author = {Cattani, Eduardo and Kaplan, Aroldo and Schmid, Wilfried},
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+}
+
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+  title = {Bimeromorphic smoothing of a complex-analytic space},
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+}
+
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+  author = {Bierstone, Edward and Milman, Pierre D.},
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+  volume = {128}, number = {2}, year = {1997}, pages = {207--302},
+  doi = {10.1007/s002220050141},
+  eprint = {alg-geom/9508005}, archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/alg-geom/9508005}
+}
+
+@misc{BBYv2,
+  author = {Matsumura, Shin-ichi and Wang, Juanyong and Wu, Xiaojun and Zhang, Qimin},
+  title = {{Beauville--Bogomolov--Yau} decomposition for {K{\"a}hler} generalized pairs},
+  year = {2026},
+  note = {arXiv:2506.23218v2, September 9, 2026},
+  eprint = {2506.23218}, archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/2506.23218v2}
+}
+
+@article{COP,
+  author = {Campana, Fr{\'e}d{\'e}ric and Oguiso, Keiji and Peternell, Thomas},
+  title = {Non-algebraic hyperk{\"a}hler manifolds},
+  journal = {Journal of Differential Geometry},
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+  doi = {10.4310/jdg/1292940689},
+  eprint = {0804.1682}, archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/0804.1682}
+}
+
+@book{SGA1,
+  author = {Grothendieck, Alexander and others},
+  title = {Rev{\^e}tements {\'e}tales et groupe fondamental ({SGA} 1)},
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+  eprint = {math/0206203}, archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math/0206203v2}
+}
+
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+  author = {Ueno, Kenji},
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+  doi = {10.1007/BFb0070570}
+}
+
+@misc{DemaillyCADG,
+  author = {Demailly, Jean-Pierre},
+  title = {Complex Analytic and Differential Geometry},
+  year = {2012}, note = {Version of June 21, 2012},
+  url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf}
+}
+
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+  author = {Pareschi, Giuseppe and Popa, Mihnea and Schnell, Christian},
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+  journal = {Geometry {\&} Topology},
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+  doi = {10.2140/gt.2017.21.2419},
+  url = {https://msp.org/gt/2017/21-4/p12.xhtml}
+}
+
+
+@incollection{Barlet,
+ author={Barlet, Daniel},
+ title={Espace analytique r\'eduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie},
+ booktitle={Fonctions de plusieurs variables complexes, II},
+ editor={Norguet, Fran\c{c}ois},
+ series={Lecture Notes in Mathematics}, volume={482},
+ publisher={Springer}, address={Berlin}, year={1975}, pages={1--158},
+ doi={10.1007/BFb0080243}
+}
+
+@incollection{BarletSurvey,
+ author={Barlet, Daniel},
+ title={How to use the cycle space in complex geometry},
+ booktitle={Several Complex Variables},
+ series={Mathematical Sciences Research Institute Publications}, volume={37},
+ publisher={Cambridge University Press}, address={Cambridge},
+ year={1999}, pages={25--42},
+ url={https://library.slmath.org/books/Book37/files/barlet.pdf}
+}
+
+@article{Grauert,
+ author={Grauert, Hans},
+ title={Ein {Theorem} der analytischen {Garbentheorie} und die {Modulr\"aume} komplexer {Strukturen}},
+ journal={Publ. Math. Inst. Hautes \'Etudes Sci.},
+ volume={5}, year={1960}, pages={5--64},
+ doi={10.1007/BF02684746},
+ note={Correction: ibid. 16 (1963), 35--36, doi:10.1007/BF02684291}
+}
+
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+  author = {Hironaka, Heisuke},
+  title = {Flattening Theorem in Complex-Analytic Geometry},
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+}
+
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+  author = {Bierstone, Edward and Milman, Pierre D.},
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+  journal = {Publications of the Research Institute for Mathematical Sciences},
+  volume = {44},
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+  year = {2008},
+  doi = {10.2977/PRIMS/1210167338},
+  eprint = {math/0702375},
+  archivePrefix = {arXiv}
+}
+
+@misc{FujinoAnalyticBCHM,
+  author = {Fujino, Osamu},
+  title = {Minimal model program for projective morphisms between complex analytic spaces},
+  year = {2022},
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+  archivePrefix = {arXiv},
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+  note = {arXiv:2201.11315v1},
+  doi = {10.48550/arXiv.2201.11315},
+  url = {https://arxiv.org/abs/2201.11315v1}
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+
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+  doi = {10.1090/S0894-0347-09-00649-3},
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+}
+
+@article{GrauertCorrection,
+  author = {Grauert, Hans},
+  title = {Berichtigung: {Ein Theorem der analytischen Garbentheorie und die Modulr{\"a}ume komplexer Strukturen}},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume = {16}, year = {1963}, pages = {35--36},
+  doi = {10.1007/BF02684291},
+  url = {https://www.numdam.org/item/PMIHES_1963__16__35_0/}
+}
+
+@article{Campana1981,
+  author = {Campana, Fr{\'e}d{\'e}ric},
+  title = {Cor{\'e}duction alg{\'e}brique d'un espace analytique faiblement {K{\"a}hl{\'e}rien} compact},
+  journal = {Inventiones Mathematicae},
+  volume = {63}, number = {2}, year = {1981}, pages = {187--223},
+  doi = {10.1007/BF01393876}
+}

preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/build/main.tex

@@ -1,30 +1,30 @@
-\documentclass[11pt]{article}
-\input{preamble}
-\title{Numerical semiampleness of nef adjoint classes\\on compact K\"ahler manifolds}
-\author{OpenAI}
-\date{October 6, 2026}
-\begin{document}
-\maketitle
-\begin{abstract}
-Let \(X\) be a smooth connected compact K\"ahler manifold, let \(B\) be an
-effective rational simple normal crossing divisor with coefficients less
-than one, and let \(M\) be a nef rational holomorphic line bundle on \(X\).
-If \(K_X+B\) is pseudo-effective and \(K_X+B+M\) is nef, we prove that its
-first Chern class in real Bott--Chern cohomology is represented by a semiample
-rational line bundle. This numerical statement allows a flat change of line
-bundle; it does not assert semiampleness of the original adjoint.
-\end{abstract}
-\tableofcontents
-\input{sections/01-introduction}
-\input{sections/02-preliminaries}
-\input{sections/03-descent}
-\input{sections/04-adjoint}
-\input{sections/05-zero}
-\input{sections/06-cycles}
-\input{sections/07-completion}
-\begingroup
-\raggedright
-\bibliographystyle{plain}
-\bibliography{literature,companions}
-\endgroup
-\end{document}
+\documentclass[11pt]{article}
+\input{preamble}
+\title{Numerical semiampleness of nef adjoint classes\\on compact K\"ahler manifolds}
+\author{OpenAI}
+\date{October 6, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+Let \(X\) be a smooth connected compact K\"ahler manifold, let \(B\) be an
+effective rational simple normal crossing divisor with coefficients less
+than one, and let \(M\) be a nef rational holomorphic line bundle on \(X\).
+If \(K_X+B\) is pseudo-effective and \(K_X+B+M\) is nef, we prove that its
+first Chern class in real Bott--Chern cohomology is represented by a semiample
+rational line bundle. This numerical statement allows a flat change of line
+bundle; it does not assert semiampleness of the original adjoint.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-preliminaries}
+\input{sections/03-descent}
+\input{sections/04-adjoint}
+\input{sections/05-zero}
+\input{sections/06-cycles}
+\input{sections/07-completion}
+\begingroup
+\raggedright
+\bibliographystyle{plain}
+\bibliography{literature,companions}
+\endgroup
+\end{document}