==== TABLE OF CONTENTS ====
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//analytic-inputs.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//bibliography.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//conclusion.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//finite-inputs.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//introduction.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//main.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//moving-centers.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//preamble.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//reduction.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//signed.tex
Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//two-slot.tex
===========================

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//analytic-inputs.tex ====
\section{The analytic inputs and the lifting problem}\label{sec:analytic-inputs}

The geometry will produce a divisor representing $K_Y$ with coefficients
of both signs. We shall prove that its full support is of logarithmic
general type. The difficulty is to find the first section of an auxiliary
nef adjoint before invoking a semiampleness criterion. The two analytic
results below, proved in the complete companion \cite{OpenAIMetrics2026},
address exactly that difficulty. They concern actual rational line bundles.

A semipositive singular Hermitian metric has \emph{minimal singularities}
if its local weight is, up to a bounded additive error, no more singular
than the weight of any other semipositive metric on the same bundle.
For a rational line bundle this definition is made after taking one
Cartier multiple and dividing the weights by that multiple.

\begin{theorem}[Zero Lelong numbers for nef klt adjoints]
\label{thm:adjoint-metric}
Let $(H,\Theta)$ be a normal connected projective complex klt pair,
where $\Theta\ge0$ is rational and $D_H=K_H+\Theta$ is nef and
$\Q$-Cartier. Let $\pi:V\to H$ be any projective log resolution.
The rational line bundle $\pi^*D_H$ admits a semipositive metric
with minimal singularities, and every such metric has zero Lelong
number at every point of $V$. Its multiplier ideal is trivial for
every positive exponent. Neither $K_H$ nor $\Theta$ is required
to be separately $\Q$-Cartier.
\end{theorem}

This is \cite[Theorem~1.1]{OpenAIMetrics2026}. The adjoint hypothesis is material:
a general nef line bundle need not have this metric regularity
\cite[Example~1.7]{DPS1994}.

\begin{theorem}[Ordinary interior injectivity]
\label{thm:interior-injectivity}
Let $V$ be a smooth projective complex variety and $L$ an integral
divisor. Let $0\le C_0\le C_2$ be effective rational divisors whose
combined support has simple normal crossings. Suppose the actual
rational line bundles $L-C_0$ and $L-C_2$ have semipositive singular
Hermitian metrics with zero Lelong numbers at every point. For a
rational number $0<\lambda<1$ put $C_1=(1-\lambda)C_0+\lambda C_2$.
The natural inclusion of sheaves induces an injection
\[
 H^1(V,\cO_V(K_V+L-\lfloor C_1\rfloor))
 \longrightarrow H^1(V,\cO_V(K_V+L-\lfloor C_0\rfloor)).
\]
\end{theorem}

The complete proof, including its passage to ordinary coherent
cohomology, is in \cite[Theorem~1.2]{OpenAIMetrics2026}. The analytic methods have antecedents
in Fujino's singular-metric treatment of harmonic forms and Matsumura's
uniform control of primitives \cite[Theorem~1.2 and Lemmas~3.1--3.2]
{FujinoInjectivity2012} and \cite[Theorems~1.3 and~5.9]{Matsumura2018}.
Those results are not being identified with the interior comparison.

Here is the lifting problem to which we will apply the theorem.
For a nef rational adjoint $N$ on a normal projective variety $H$
and a reduced closed subscheme $S\subset H$, choose $m>0$ with
$mN$ Cartier. The restriction sequence is
\[
 0\longrightarrow\cO_H(mN)\otimes\cI_S
 \longrightarrow\cO_H(mN)
 \longrightarrow\cO_S(mN)\longrightarrow0.
\]
The obstruction to lifting a section on $S$ is its image under the
connecting homomorphism, whose image is the kernel of
\[
 H^1(H,\cO_H(mN)\otimes\cI_S)
 \longrightarrow H^1(H,\cO_H(mN)).
\]
Thus injectivity of this ordinary cohomology map suffices to lift
every section on the whole scheme $S$. In the next section, $S$ is
the reduced non-klt floor of an auxiliary lc boundary. Two nearby
nef klt adjoints supply the endpoint metrics, and multiplier-ideal
local vanishing identifies the theorem's map with this inclusion.
Threefold abundance supplies a section on the entire floor, with
the conductor identifications already imposed.
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//analytic-inputs.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//bibliography.tex ====
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==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//bibliography.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//conclusion.tex ====
\section{A section on the original normal variety}\label{sec:conclusion}

The geometric argument produces a pluricanonical section on a smooth
fourfold. Hashizume's reduction gives an effective real representative
of an lc adjoint on the original variety. The following elementary
lemma turns that representative into a rational one. It concerns
finite linear algebra and uses no nonvanishing or abundance theorem.

\begin{lemma}[Finite rational feasibility]\label[lemma]{lem:rational-feasibility}
Let $M=(m_{ij})\in\operatorname{Mat}_{w\times v}(\Q)$ and
$d\in\Q^w$. If the system
\begin{equation}\label{eq:rational-feasibility}
 d_i-y_i=\sum_{j=1}^{v}m_{ij}b_j\quad(1\le i\le w),
 \qquad y_i\ge0
\end{equation}
has a real solution, it has a rational solution. Moreover, one may
require precisely the same $y_i$ to be zero as in the given solution.
Consequently, if $D$ is a rational Weil divisor on a normal integral
variety $X$ over a field $k$ and $D\sim_{\R}G$ for an effective real Weil divisor $G$,
then $D\sim_{\Q}G_{\Q}$ for an effective rational divisor with
$\Supp G_{\Q}=\Supp G$. If $rD$ is Cartier for a prescribed
positive integer $r$, then $H^0(X,\cO_X(mrD))\ne0$ for some $m>0$.
\end{lemma}

\begin{proof}
Fix the equations $y_i=0$ at all coordinates vanishing in the given
real solution. Together with the displayed equations they define a
nonempty affine space over $\Q$. Gaussian elimination over $\Q$
gives a rational particular solution and a basis over $\Q$ for its
direction space. Thus its rational points are dense in its real
points. Approximate the given point closely enough to keep all of
the finitely many remaining coordinates $y_i$ strictly positive.
This proves the first assertion, including the case where all
$y_i$ vanish or the affine space has dimension zero.

For the divisor assertion, choose a finite expression
\[
 D-G=\sum_{j=1}^{v} a_j\Div(f_j),\qquad
 a_j\in\R,\quad f_j\in k(X)^*.
\]
List the prime divisors in the supports of
$D,G$, and these finitely many principal divisors as
$P_1,\ldots,P_w$. Write $D=\sum d_iP_i$ and let $m_{ij}\in\Z$
be the coefficient of $P_i$ in $\Div(f_j)$. The coefficients of
$G$ and the $a_j$ give a real solution of
\eqref{eq:rational-feasibility}. A rational solution gives
$G_{\Q}=\sum y_iP_i\ge0$ with the same support and
$D-G_{\Q}=\sum b_j\Div(f_j)$.

Choose a positive integer $N$ divisible by $r$ and by the
denominators of all $y_i,b_j$. Then
\[
 ND-NG_{\Q}=\Div\!\left(\prod_j f_j^{Nb_j}\right).
\]
The effective integral divisor $NG_{\Q}$ is Cartier, since $ND$
is Cartier and their difference is principal. It defines a nonzero
section of $\cO_X(ND)$. Write $N=mr$. No $\Q$-factoriality is
needed: all equalities were equalities of Weil divisors on $X$.
\end{proof}

\begin{proof}[Proof of \Cref{cor:lc-nonvanishing}]
Hashizume's reduction \cite[Theorem~1.4]{Hashizume2018} takes smooth
canonical nonvanishing in dimension four, supplied by
\Cref{thm:nonvanishing}, and gives
nonvanishing for projective lc pairs with effective real boundaries
and pseudo-effective real Cartier adjoints in dimensions at most four.
Applied to the given rational lc pair, it produces $D\sim_{\R}G\ge0$
on the original normal variety; nefness implies pseudo-effectivity.
\Cref{lem:rational-feasibility} gives a nonzero section of $mrD$
for some positive integer $m$, with the prescribed index $r$.
\end{proof}
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//conclusion.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//finite-inputs.tex ====
\section{Fixed finite data and the final contradiction}\label{sec:frobenius}

The scalar bound and the two-slot jets will now be tested against one
independent theorem. We state all its hypotheses before constructing
the remaining data. Its independent proof in
\cite{OpenAIAbundance2026} uses neither abundance nor logarithmic
subadditivity.

\begin{theorem}[Finite-data Frobenius incompatibility]
\label{thm:finite-data}
Let $W$ be a smooth connected projective complex variety of dimension
$n\ge2$. Fix rational Cartier divisors $L,H$, with $H$ ample, an
integral Cartier divisor $S$, an integer $q>0$, and real numbers
$r>1$, $\epsilon>0$. Suppose
\begin{gather*}
 H^n\le(2\epsilon)^n,\qquad
 K_WH^{n-1}\le 2H^n/r,\qquad
 (2L+qS)H^{n-1}\le4H^n,\\
 (14\epsilon)^n<1/4,\qquad80\epsilon<3/4.
\end{gather*}
The following three kinds of fixed data cannot all exist.
\begin{enumerate}[label=\textup{(\roman*)}]
\item A smooth integral complete-intersection flag ending in a curve
$C$, of successive divisor classes $l_iH$ for positive integers $l_i$,
with $\mu_{\min}(\Omega_W^1|_C)\ge0$.
\item On $Z_W=\PP(\cO(S)_1\oplus\cO(S)_2)\to W\times W$,
in the symmetric-power convention, put $M=L_1+L_2+q\xi$.
For some integer $k_0>0$ with $k_0L$ Cartier, finitely many sections of
$k_0M$ surject onto
$\cO_{Z_W,z_*}/\mathfrak m_{z_*}^{(2n+2)k_0+1}$ after
trivialization at a smooth torus point $z_*$.
\item A point $x\in W$, the blowup $\pi_x:\widehat W\to W$
with exceptional divisor $J_x$, and a curve class
$\gamma=f_*(A_1\cdots A_{n-1})$, where
$f:V\to\widehat W$ is a fixed birational morphism from a smooth
integral projective variety and the $A_i$ are ample Cartier divisors,
such that $J_x\gamma>0$ and
\[
 \pi_x^*(L+K_W/2+\lambda S)\gamma
 \le40\epsilon J_x\gamma\qquad(0\le\lambda\le q).
\]
It suffices to check the two endpoint inequalities.
\end{enumerate}
No nefness of $L,S,K_W$ is assumed, and $z_*$ need not lie over $x$.
All divisors, the flag, the point, the curve representative, and the
finite jet sections are fixed over $\C$ before the residual
characteristic varies.
\end{theorem}

The proof is \cite[Theorem~9.1]{OpenAIAbundance2026}.
We shall apply it with $n=4$, $S=S_0=\iota K$, and $r=r_t$.
The remaining work is to obtain its ample perturbation, flag, and
actual movable curve witness from the fourfold geometry.

\subsection{Fixing the polarization, flag, and movable test}

Choose a positive rational number $\epsilon$ sufficiently small that
\begin{equation}\label{eq:epsilon-choice}
 (14\epsilon)^n<\frac14,\qquad 80\epsilon<\frac34.
\end{equation}
Choose $q$ as in \Cref{prop:two-slot-jets}, then a sufficiently small
rational $t>0$ so that both jet estimates hold,
$r_t>q\iota+1$, and $L^n<(2\epsilon)^n$. Fix $t$ and $r_t$. Let
$W\to Y$ be a projective resolution, and continue to denote
pullbacks by $K,A,L$.  Terminality gives
\[
 K_W=K+E,\qquad E\ge0\text{ exceptional},\qquad S_0=\iota K\text{ Cartier}.
\]
Since $L$ is pulled back from an ample divisor on $Y$,
\[
 K_WL^{n-1}=KL^{n-1}\le L^n/r_t.
\]
Perturb $L$ by a sufficiently small ample rational class on $W$.
By continuity, the resulting ample rational divisor $H$ satisfies
\begin{equation}\label{eq:fixed-H}
 H^n\le(2\epsilon)^n,\qquad
 K_WH^{n-1}\le\frac{2H^n}{r_t},\qquad
 (2L+qS_0)H^{n-1}\le4H^n.
\end{equation}
For the last inequality, at $H=L$ use
$S_0L^{n-1}\le \iota L^n/r_t$ and $q\iota<r_t$.
We now fix $H$; all further geometric
data will be chosen once over $\C$ before reduction.

The divisor $K_W$ is pseudo-effective.  Generic semipositivity, in
the smooth empty-boundary case of
\cite[Theorem~2.1]{CampanaPaun2015}, gives nonnegative
$H^{n-1}$-slope to every positive-rank torsion-free quotient of
$\Omega_W^1$.  Apply the Mehta--Ramanathan restriction theorem to the
Harder--Narasimhan filtration and its factors
\cite[Theorem~6.1 and Remark~6.2]{MehtaRamanathan1982}.  We obtain a
fixed flag of smooth integral general very ample sections, of types
\[
 h_i=l_iH\quad(1\le i\le n-1),
\]
ending in a smooth curve $C$, such that
\begin{equation}\label{eq:fixed-curve-slope}
 \mu_{\min}(\Omega_W^1|_C)\ge0.
\end{equation}
The $l_i$ are sufficiently large divisible integers, chosen
successively.  The final general curve avoids the codimension-two
loci where the filtration factors fail to be locally free.  Set
$\Lambda=\prod_{i=1}^{n-1}l_i$.  Then
\begin{equation}\label{eq:flag-degrees}
 [C]=\Lambda H^{n-1},\qquad h_i\cdot C=l_i\Lambda H^n.
\end{equation}

Next choose a very general point $x\in W$ above the isomorphism locus
of $W\to Y$, and let
$\pi_x:\widehat W\to W$ be its blowup, with exceptional divisor $J_x$.
Put
\[
 D^+=2r_t(K+2tA)+2E.
\]
By \Cref{prop:scalar-orders}, every nonzero section of a sufficiently
divisible multiple $kD^+$ satisfies
\begin{equation}\label{eq:Dplus-order}
 \ord_x(s)\le32k\epsilon.
\end{equation}
Indeed, pushing down removes only the effective exceptional part,
and is an isomorphism near $x$.

\begin{lemma}\label[lemma]{lem:fixed-movable-test}
There is a strongly movable curve class $\gamma$ on $\widehat W$,
represented as the pushforward of an ample complete intersection on
a fixed smooth projective birational model, such that
\begin{equation}\label{eq:movable-test}
 (\pi_x^*D^+)\cdot\gamma
 <40\epsilon\,J_x\cdot\gamma,\qquad J_x\cdot\gamma>0.
\end{equation}
\end{lemma}

\begin{proof}
If $\pi_x^*D^+-40\epsilon J_x$ were pseudo-effective, then for any
rational $c$ with $32\epsilon<c<40\epsilon$ the divisor
$\pi_x^*D^+-cJ_x$ would be big: it is a positive convex combination
of the preceding pseudo-effective divisor and the big divisor
$\pi_x^*D^+$.  A sufficiently divisible effective multiple would
contradict \Cref{eq:Dplus-order}.  Thus
$\pi_x^*D^+-40\epsilon J_x$ is not pseudo-effective.

Movable-curve duality \cite[Theorem~2.2]{BDPP2013} gives a strongly
movable class with negative pairing against that divisor.  Since the
inequality is strict, we may take an actual pushforward of an ample
complete intersection on one smooth birational projective model,
approximating the ample classes rationally and clearing their
denominators if needed.  The big divisor $\pi_x^*D^+$ has positive
pairing with this nonzero class: write it as an ample class plus an
effective class and pull back to the chosen model.  The strict
inequality therefore implies $J_x\cdot\gamma>0$.
\end{proof}

We omit $\pi_x^*$ in subsequent intersection formulas involving
$\gamma$.  The concrete complete-intersection representative will
continue to test effective divisors after reduction.

\subsection{Endpoint comparison}

For $0\le\lambda\le q$, put $Q_\lambda=L+K_W/2+\lambda S_0$.
The actual divisor classes on the fixed resolution satisfy
\begin{equation}\label{eq:slot-domination}
 D^+-Q_\lambda
 =(r_t-1/2-\lambda\iota)K+3r_ttA+\tfrac32E.
\end{equation}
The first coefficient is positive because $r_t>q\iota+1$.
The pullbacks of $K,A$ are nef, and $E$ is effective. A strongly
movable curve pairs nonnegatively with each of these classes.
Consequently \Cref{lem:fixed-movable-test} gives
\[
 Q_\lambda\gamma\le D^+\gamma
 <40\epsilon J_x\gamma\qquad(0\le\lambda\le q).
\]
This verifies the curve inequalities in \Cref{thm:finite-data}.
It does not require a pseudo-effective cone to specialize to
positive characteristic: the curve has the fixed ample
complete-intersection representative required by that theorem.

Take $z_*,k_0$, and the finite sections supplied by
\Cref{cor:fixed-two-slot-jets} on this resolution. For clarity about
the finite choices in the comparison, choose also an integral ample
divisor $T_0$ so that each $T_0+aS_0$, for
$0\le a\le(k_0-1)q$, has fixed sections nonvanishing at the two
base coordinates of $z_*$. Serre generation permits these finitely
many choices. They supply the residual degrees when the common
proof puts $B_p=k_0\lfloor p/k_0\rfloor L+T_0$. The differences
$B_p-pL$ belong to a fixed finite list.

We have now verified all the hypotheses of \Cref{thm:finite-data}.
The choices were made in the order $n=4,\epsilon$; then the gap
$\delta$ and integer $q$; then $t,r_t$; then the resolution,
$H$, flag, point, and curve; and finally the finite jets and fillers.
Only after these choices does its proof let $p$ tend to infinity.

\subsection{The two orders of the determinant}

We recall the comparison in the proof of
\cite[Theorem~9.1]{OpenAIAbundance2026}, identifying the determinant
bundles to which our movable test applies. Spread the fixed data as
in that proof and work on a smooth reduction in sufficiently large
characteristic $p$. The same notation denotes the reduced divisors
and witnesses. For relative Frobenius $F:W\to W'$, let $S_0'$ be
the base-field twist of $S_0$ and put
\[
 \mathcal E=F_*\cO_W(B_p),\qquad
 s=\rk\mathcal E=p^n,\qquad R=s^2=p^8,\qquad
 U_a=H^0(W,\cO_W(B_p+paS_0)).
\]
The projection formula gives an evaluation map
\begin{equation}\label{eq:finite-evaluation}
 \Phi_p:\bigoplus_{\substack{a+b=q\\a,b\ge0}}
 U_a\otimes U_b\otimes\cO_{W'\times W'}(-aS'_{0,1}-bS'_{0,2})
 \longrightarrow\mathcal E\boxtimes\mathcal E.
\end{equation}
Products of the fixed jet sections, multiplied by the fixed fillers,
generate the local quotient by the $p$-th powers of the parameters
at $z_*$. Let $A_*$ be the tensor product of the two base
Frobenius-fiber algebras. If the torus coordinate $z$ has value
$z_0\ne0$ at $z_*$, this quotient is
$A_*[z]/(z^p-z_0^p)$. Project to the coefficient of $1$ in its
$A_*$-basis $1,z,\ldots,z^{p-1}$. Exactly the weights divisible
by $p$ survive, showing that the values of
\eqref{eq:finite-evaluation} span $A_*$. Its dimension is $R$,
so $\Phi_p$ has generic rank $R$.

On the diagonal, the columns instead restrict to global sections of
one line bundle on the Frobenius fiber product:
\[
 \Delta_F=W\times_{W'}W,\qquad
 \mathcal L_p=
 \cO_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_{0,1}).
\]
Here $pS_{0,1}$ and $pS_{0,2}$ are isomorphic line bundles on
$\Delta_F$, so all the weight pairs $a+b=q$ give this same bundle.
The rank at every diagonal point is at most
$h^0(\Delta_F,\mathcal L_p)$. The diagonal-ideal filtration has
graded bundles
\[
 \mathcal G_j=\cO_W(2B_p+pqS_0)\otimes A_j(\Omega_W^1),
 \qquad 0\le j\le n(p-1),
\]
where $A_j$ is the degree-$j$ part of the symmetric algebra modulo
the $p$-th powers of its local generators. If $e_j=\rk A_j$, then
$\sum_j e_j=p^n$. The slope estimate and our one fixed flag give
the common bound proved in the companion:
\[
 h^0(W,\mathcal G_j)
 \le e_jp^n\bigl(7^nH^n+O(p^{-1})\bigr).
\]
The error is independent of $j$. Summing therefore bounds every
diagonal rank by
$R(7^nH^n+O(p^{-1}))<R/4$, since
$7^nH^n\le(14\epsilon)^n<1/4$.

Choose $R$ generically independent columns of $\Phi_p$, and let
$m_i$ be the sum of their weights in slot $i$. Their nonzero
determinant is a section
\[
 0\ne\sigma\in H^0(W'\times W',\mathcal Q_1\boxtimes\mathcal Q_2),
 \qquad
 \mathcal Q_i=(\det\mathcal E)^{\otimes s}
                 \otimes\cO_{W'}(m_iS_0').
\]
These are actual line bundles. Under the numerical identification
with the base-field twist, their first Chern classes satisfy
\begin{equation}\label{eq:finite-determinant-classes}
 \frac{c_1(\mathcal Q_i)}R
 =\frac{B_p}p+\frac{p-1}{2p}K_W+\lambda_iS_0
 =Q_{\lambda_i}+\frac{B_p-pL-K_W/2}{p},
 \qquad \lambda_i=\frac{m_i}{R}\in[0,q].
\end{equation}
The numerator of the last term belongs to a fixed finite list.
Thus the endpoint comparison bounds the intersection of each
normalized class with $\gamma$ by
$(40\epsilon+O(p^{-1}))J_x\gamma$, uniformly in the chosen columns.

This intersection bound controls sections on the reduction itself.
Put $x'=F(x)$. For any nonzero section of $\mathcal Q_i$, the strict
transform of its zero divisor on the blowup at $x'$ is effective and
pairs nonnegatively with the
twist of $\gamma$: that curve is still the pushforward of the fixed
ample complete intersection. Since $J_x\gamma>0$, every such section
has order at $x'$ at most $R(40\epsilon+O(p^{-1}))$.
Bases adapted to vanishing order in the two factors show that a
nonzero section of their external product has order at most the sum
of these bounds. Within each bidegree the leading tensors are linearly
independent, and different bidegrees cannot cancel. Consequently
\[
 \ord_{(x',x')}\sigma\le R(80\epsilon+O(p^{-1})).
\]
But the rank of $\Phi_p$ at $(x',x')$ is less than $R/4$.
Constant row operations make more than $3R/4$ rows of the selected
matrix vanish at that point, forcing
$\ord_{(x',x')}\sigma\ge3R/4$. The two orders contradict
$80\epsilon<3/4$ for large $p$. The generic-rank calculation used
$z_*$, whereas this diagonal comparison uses the independently
chosen point $x$.

Thus the terminal counterexample in \Cref{prop:reduction} cannot
exist. Canonical section spaces are preserved by its birational
construction, so this proves \Cref{thm:nonvanishing}.
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//finite-inputs.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//introduction.tex ====
\section{Introduction}
\label{sec:introduction}

For a smooth projective variety $X$, canonical nonvanishing asks whether
pseudo-effectivity of $K_X$ guarantees a pluricanonical form: a nonzero
section of $mK_X$ for some integer $m>0$. Pseudo-effectivity says that the
numerical class is a limit of effective divisor classes. The question
concerns the canonical line bundle itself, rather than an approximation
to its numerical class. Over $\C$, pseudo-effectivity of $K_X$ is
equivalent to $X$ not being covered by rational curves
\cite[Corollary~0.3]{BDPP2013}. Nonvanishing thus asks for a
pluricanonical form on every smooth projective variety outside the
uniruled class.

Nonvanishing is weaker than abundance. For a nef canonical divisor,
abundance asks for \emph{semiampleness}: some positive multiple is
generated by its global sections and therefore defines a morphism.
Obtaining the first section is a separate difficulty. We prove the
following nonvanishing theorem in dimension four.

\begin{theorem}\label{thm:nonvanishing}
Let $X$ be a smooth connected projective complex fourfold. If $K_X$ is
pseudo-effective, then $H^0(X,mK_X)\ne0$ for some positive integer $m$.
\end{theorem}

The theorem imposes no condition on numerical dimension, irregularity,
or holomorphic Euler characteristic. Its conclusion is qualitative;
it gives no uniform bound on the integer $m$.

The threefold results explain both the history of the problem and the
lower-dimensional inputs used here. Miyaoka proved nonvanishing for
minimal threefolds \cite{Miyaoka1988Nonvanishing} and then abundance
in numerical dimension one \cite[pp.~203--204]{Miyaoka1988NuOne}.
Kawamata's work on pluricanonical systems and his abundance theorem for
minimal threefolds established the broader canonical case
\cite{Kawamata1985,Kawamata1992}. Keel--Matsuki--McKernan proved log
abundance for threefolds, with the correction included here
\cite{KMM1994,KMM2004}. Boundary arguments also require sections to
agree where components meet: Fujino's semi-log-canonical threefold
abundance theorem supplies generation on the whole reduced floor,
including its conductor identifications
\cite[Corollary~4.10]{Fujino2000}. These are established
lower-dimensional theorems used before the new fourfold argument.

Several important fourfold cases were already known. Fujino proved
semiampleness for canonical fourfolds with nef canonical divisor and
positive irregularity. He distinguished nonvanishing for non-uniruled
fourfolds of irregularity zero from the remaining abundance question
in Kodaira dimension zero
\cite[Corollary~4.7 and Problems~4.10--4.11]{Fujino2010}.
Lazi\'c--Peternell proved nonvanishing for $\Q$-factorial terminal projective minimal
varieties with canonical numerical dimension one and nonzero
holomorphic Euler characteristic
\cite[Theorem~6.7]{LazicPeternell2018}.
Ambro's canonical bundle formula gives a complementary reduction:
the lower-dimensional minimal model program and abundance settle a
nef klt adjoint whose nef reduction has lower-dimensional image
\cite[Theorem~4.3]{Ambro2005}. The nef reduction contracts the curves
of degree zero through very general points; the dimension of its image
is the \emph{nef dimension}. This curve-theoretic dimension is distinct
from numerical dimension, which is defined by nonzero intersection
powers of the nef divisor.

The smooth theorem also has a log canonical consequence. Hashizume
proved that smooth canonical nonvanishing in dimension $n$ implies
nonvanishing for log canonical pairs with real boundaries, as well as
log minimal models, in dimensions at most $n$
\cite[Theorem~1.4]{Hashizume2018}. For rational data we replace the
resulting effective real representative by a rational one, preserving
its support and working on the original variety.

\begin{corollary}\label[corollary]{cor:lc-nonvanishing}
Let $(X,\Delta)$ be a connected normal projective log canonical complex
pair of dimension at most four. Suppose that $\Delta\geq0$ is rational
and $D=K_X+\Delta$ is $\mathbb Q$-Cartier and nef. For every positive
integer $r$ such that $rD$ is Cartier, there is a positive integer $m$
with $H^0(X,\mathcal O_X(mrD))\ne0$.
\end{corollary}

The corollary does not require $X$ to be $\mathbb Q$-factorial, and
$m$ may depend on the pair and the prescribed index. Log abundance
asks for semiampleness of the nef adjoint $K_X+\Delta$. The passage
from the first section to that conclusion uses the supported-boundary
lifting and abundance-after-nonvanishing theorems in the separate
companion \emph{Lifting sections from the reduced support of an adjoint}
\cite[Theorems~1.1--1.2]{OpenAIBoundary2026}.
Together with the corollary, they give fourfold log abundance; that
consequence does not enter the proof below.

A recent comparison is Liu--Xu's preprint on good minimal models for
log canonical pairs of dimension at most five with nonnegative
invariant Iitaka dimension and numerical dimension at most one
\cite[Theorem~5.1]{LiuXu2025}. Its hypotheses already include
nonvanishing and restrict numerical dimension. The present smooth
theorem obtains the first section without either restriction.

\subsection*{Geometry of a possible counterexample}

Suppose nonvanishing fails. Existence of minimal models in dimension
four allows us to fix a terminal minimal model $Y$, retaining the
pluricanonical section spaces. The termination input used here is the
ordinary-pair case of Chen--Tsakanikas
\cite[Theorem 1.1]{ChenTsakanikas2023}; it supplies a minimal model,
without asserting semiampleness. Write $K=K_Y$. The divisor $K$ is nef
and not big. The preceding reductions give $q(Y)=h^1(Y,\mathcal O_Y)=0$
and full nef dimension. On a fixed very general locus, every curve has
positive $K$-degree and every proper positive-dimensional subvariety is
of general type. A fixed Cartier index makes the positive curve degrees
uniformly bounded away from zero.

Choose ample polarizations approaching the nef ray of $K$, scaled so
that their volumes remain small while their degrees on curves through
that locus tend to infinity. A section with unusually high vanishing
at a moving point would force a persistent component in the base locus
of the corresponding jet systems. Differentiation with respect to the
moving point controls the multiplicity along that component. This uses
the parameter-differentiation method of Ein--K\"uchle--Lazarsfeld
\cite[Proposition 2.3]{EinKuchleLazarsfeld1995}. We apply it to the
whole space of sections satisfying the jet conditions, so the estimate
controls ordinary powers of the ideal of the component. Intersection
theory bounds its degree and canonical intersection. General type then
produces a curve of bounded degree, contradicting the chosen scaling.
The resulting upper bound applies to every section in every allowed
Cartier degree.
The independent moving-center estimates and the finite-cover lemma
used in this argument and its two-factor extension are
\cite[Lemmas~7.1--7.5]{OpenAIAbundance2026}. We apply them to the
fourfold geometry and prove the required curve and correspondence
obstructions here.

A second use of this argument concerns sections over two copies of $Y$.
Fix an integer $\iota>0$ making $S_0=\iota K$ Cartier and consider
\[
 Z=\mathbb P\bigl(\mathcal O_{Y\times Y}(S_{0,1})
             \oplus\mathcal O_{Y\times Y}(S_{0,2})\bigr)
       \longrightarrow Y\times Y.
\]
A subscript indicates pullback from the corresponding factor. The two
summands give two distinguished sections of this projective bundle.
The symmetric-power formula records the distributions of a fixed
tensor exponent between the two factors: its summands are
$\mathcal O(aS_{0,1}+bS_{0,2})$, for nonnegative integers $a,b$
with $a+b$ fixed.
A polarization on $Z$ combines the two base polarizations and a multiple
of its tautological bundle. We show that it generates many jets at a
very general point away from the two distinguished sections. Its
Seshadri constant, the infimum of degree divided by multiplicity over
curves through the point, measures this local positivity.

Moving base components in $Z$ have several possible images and fibers.
One case requires an additional argument. Restrict to a slice where one
base coordinate is fixed. A multisection different from the two
distinguished sections meets them in divisors whose difference represents
a multiple of $K$. This is a \emph{signed representative}: its coefficients
may be positive or negative. We must prove that the logarithmic adjoint
on a resolution of its full support is big. This is where minimal metrics
and ordinary cohomological injectivity enter the geometry.

The companion \emph{Minimal metrics and interior injectivity
for nef adjoints}~\cite{OpenAIMetrics2026} supplies two analytic inputs. Every minimal semipositive metric on
the pullback of a nef projective klt adjoint has zero Lelong numbers.
In addition, an injectivity theorem on ordinary $H^1$ compares an
interior rational boundary with an endpoint boundary when the two endpoint
bundles carry such metrics. The exact hypotheses are recorded in
Section~\ref{sec:analytic-inputs}. These statements concern actual rational
line bundles; numerical equivalence would not suffice.

The cohomological method draws on the harmonic-form approach of Enoki,
as extended by Fujino to singular metrics, and on Matsumura's treatment
of metrics with transcendental singularities
\cite[Theorem~1.2, Lemmas~3.1--3.2, and Section~3, Claim~1]{FujinoInjectivity2012}
\cite[Theorems~1.3 and~5.9, Sections~5.2--5.3]{Matsumura2018}.
Fujino's complete metrics on an analytic complement and his comparison
with coherent cohomology with multiplier ideals give a model for the
setup. Matsumura supplies uniform estimates for primitives of exact forms.

Applied to two nearby nef klt adjoints, the cohomological injection
makes restriction to a whole reduced non-klt boundary surjective.
Lower-dimensional abundance and gluing across the conductor supply a
section on that boundary. Once this section has been lifted,
Gongyo--Matsumura's fourfold criterion converts nonvanishing and the
zero-Lelong metric into semiampleness
\cite[Corollary 5.3]{GongyoMatsumura2017}. This order matters: the
criterion is applied after the first section has been obtained.
The signed-representative argument then excludes the offending
multisection. The remaining analysis also excludes dominant finite
correspondences, using a fixed branch complement and the positive
canonical degree of curves.

\subsection*{From jets to a rank contradiction}

We now have opposing estimates. Sections on one copy of $Y$ can vanish
only to a small order, while the projective bundle over $Y\times Y$
has enough jets to give a large evaluation rank. We fix the polarization,
a smooth resolution and a flag of general hyperplane sections ending in
a curve, together with one finite system of jets. The scalar estimate
also supplies a fixed movable curve on the blowup of a point. Its
intersection with effective divisors will bound orders of sections
after reduction. We spread all these finite data and reduce modulo
sufficiently large primes.

Ordinary jets produce an evaluation matrix whose generic rank is large.
On the diagonal, the canonical Frobenius filtration expresses its
possible values through truncated powers of the cotangent bundle.
A single flag estimate bounds the sum of the dimensions of these spaces,
forcing a much smaller rank there. A nonzero maximal determinant must
therefore vanish to high order at a suitable diagonal point. Its two actual
determinant line bundles, one from each factor, have classes controlled
by the fixed movable curve. This bounds the order of every section of
either factor on the reduction itself. The resulting small determinant
order is incompatible with the order forced by the rank drop.

The filtration has a substantial history. For curves, the diagonal
filtration appears in Raynaud's work
\cite[Remarques~4.1.2(2)]{Raynaud1982}; Joshi--Ramanan--Xia--Yu
formulate it using the Cartier connection
\cite[Section 5.3]{JoshiRamananXiaYu2006}. Sun and Kitadai--Sumihiro
give the higher-dimensional filtration and its local description by
truncated monomials \cite[Theorem 3.7]{Sun2008}
\cite[Corollary 3.5]{KitadaiSumihiro2008}.
Musta\c{t}\u{a}--Schwede's comparison of ordinary and Frobenius powers
provides the local jet conversion \cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede2014},
and Langer's estimate controls the Frobenius instability error
\cite[Corollary 2.5]{Langer2004}. The argument combines these tools
with the two-factor geometry and the common flag estimate.
Section~\ref{sec:frobenius} specifies the numerical separation between
the resulting ranks and orders and verifies the inputs of the common theorem.

The complete finite-data Frobenius comparison is proved independently in
\cite[Theorem~9.1]{OpenAIAbundance2026}. It
is also applicable to the all-dimensional abundance argument. Its hypotheses are the stated
geometric data and numerical inequalities. No all-dimensional abundance
conclusion or logarithmic Iitaka subadditivity is used in the fourfold
preparation of these data.

\paragraph{Reading order.}
Section~\ref{sec:reduction} fixes the terminal model and its curve-degree
obstruction. Section~\ref{sec:moving-centers} develops the moving-center
tools and proves the scalar order bound. Section~\ref{sec:analytic-inputs}
states the analytic inputs, and Section~\ref{sec:signed-representatives}
proves logarithmic general type for signed supports.
Section~\ref{sec:two-slot-jets} excludes the product's moving centers
and obtains the two-slot jets.
Section~\ref{sec:frobenius} verifies the finite data for the common
Frobenius theorem, completing smooth nonvanishing.
Section~\ref{sec:conclusion} gives the original-variety and
prescribed-index consequence.

\paragraph{Conventions.}
Varieties are over $\C$ except when the finite-data comparison is
reduced to positive characteristic. A subvariety is of general type
when a smooth projective resolution is of general type. Rational
line bundles and their sections are interpreted after clearing
specified denominators. An inequality in nef order means that its
difference is nef. A very general locus is the complement of a
countable union of proper closed subsets. The symbols $r$ for a
prescribed Cartier index, $r_t$ for a scaling integer, $q(Y)$ for
irregularity, and $q$ for a projective-bundle weight have distinct uses.
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//introduction.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//main.tex ====
\pdfinfoomitdate=1
\pdftrailerid{}
\pdfsuppressptexinfo=15
\newcommand{\DoNotLoadEpstopdf}{}
\documentclass[11pt]{article}
\input{preamble}
\hypersetup{pdftitle={Fourfold nonvanishing by minimal metrics and moving jets},pdfauthor={OpenAI},bookmarksdepth=2}
\title{Fourfold nonvanishing by minimal metrics\\and moving jets}
\author{OpenAI}
\date{September 27, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove canonical nonvanishing for smooth connected projective complex
fourfolds: if \(K_X\) is pseudo-effective, then
\(H^0(X,mK_X)\ne0\) for some positive integer \(m\).
\end{abstract}
\setcounter{tocdepth}{1}
\begingroup\small\tableofcontents\endgroup
\clearpage
\input{sections/introduction}
\input{sections/reduction}
\input{sections/moving-centers}
\input{sections/analytic-inputs}
\input{sections/signed}
\input{sections/two-slot}
\input{sections/finite-inputs}
\input{sections/conclusion}
\input{bibliography}
\end{document}
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//main.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//moving-centers.tex ====
\section{Moving centers and scalar vanishing}
\label{sec:moving-centers}

We work on the terminal fourfold $Y$ from \Cref{sec:reduction}, write
$K=K_Y$, and put $n=4$. Write $Y^{\mathrm{vg}}$ for a fixed very
general locus on which both \Cref{prop:reduction,prop:subvarieties}
apply. Thus every curve through this locus has $K$-degree at least
$1/\iota$, and every proper positive-dimensional subvariety through
it is of general type. We will bound the order of every section of
every Cartier multiple of a suitable ample rational divisor at a
very general point. Fix a very ample Cartier divisor $A$ on $Y$.
For a fixed positive rational $\epsilon$ and positive rational $t$
tending to zero, define
\begin{equation}
\label{eq:small-volume-polarization}
 \mu_t=\bigl((K+tA)^n\bigr)^{1/n},\qquad
 r_t\in\Z_{>0},\quad r_t\mu_t\longrightarrow\epsilon,
 \qquad L=L_t=r_t(K+tA).
\end{equation}
Such integers $r_t$ exist, for example by rounding $\epsilon/\mu_t$.
Since $K$ is nef and not big, $\mu_t>0$ tends to zero. Hence
\begin{equation}
\label{eq:small-volume-curve-lower-bound}
 r_t\longrightarrow\infty,\qquad
 L_t^n\longrightarrow\epsilon^n,
 \qquad L_t\cdot C\geq r_t/\iota
\end{equation}
for every integral curve $C$ through that very general locus.
The integers $r_t$ are unrelated to the original Cartier index $r$
in \Cref{cor:lc-nonvanishing}.

A section with excessive vanishing will yield a curve through
$Y^{\mathrm{vg}}$ whose $L_t$-degree is bounded independently of $t$,
contradicting \Cref{eq:small-volume-curve-lower-bound}. To construct
that curve, we follow the base loci of the full jet kernels as their
marked point moves. A component that persists at two successive
jet levels has high multiplicity; this bounds both its degree and
its canonical intersection. General type then produces a curve
of bounded degree. We first give the three moving-center estimates
from \cite[Lemmas~7.1--7.4]{OpenAIAbundance2026}, including the
differentiation argument responsible for ordinary ideal powers.
These inputs are independent of abundance and subadditivity. We
then prove the curve obstruction and apply it to scalar vanishing.

\subsection{Numerical subadjunction at a generic lc center}

The numerical statement below only requires log canonicity near
the generic point of the center. This is the form needed for
boundaries obtained from a linear system.

The subadjunction method relates an lc center to the canonical bundle
formula. Kawamata's minimal-center theorem and the later unperturbed
form of Fujino--Gongyo are important antecedents
\cite{KawamataSubadjunction1998,FujinoGongyoSubadjunction2012}.
The numerical form below permits a center that is only generically lc
and need not be minimal. Its proof in the companion uses a dlt stratum,
the generic rank-one condition, b-nef moduli, and finite-base ramification;
it is not an application of those antecedents with their hypotheses
omitted.

\begin{lemma}[Numerical generic subadjunction]
\label[lemma]{lem:numerical-subadjunction}
Let $Z$ be a projective $\Q$-factorial klt variety over $\C$, let
$\Theta\geq0$ be a rational divisor, and let $V\subset Z$ be an
integral positive-dimensional subvariety. Suppose that $(Z,\Theta)$
is lc at the generic point of $V$ and that a divisor of log
discrepancy zero has center $V$. Let $f=\dim V$, let $P$ be a nef
rational Cartier divisor on $Z$, and let $\rho:V^*\to V$ be a smooth
projective resolution, factoring through the normalization. Then
\begin{equation}
\label{eq:numerical-subadjunction}
 K_{V^*}\cdot(\rho^*P)^{f-1}
 \leq (K_Z+\Theta)|_V\cdot(P|_V)^{f-1}.
\end{equation}
The right side is the intersection of the restricted rational
Cartier class with the cycle $[V]$. No $\Q$-Gorenstein hypothesis
on the normalization of $V$ is required.
\end{lemma}

\noindent The proof is given in the independent moving-center part of
\cite[Lemma~7.1]{OpenAIAbundance2026}.\par


\subsection{Degree and canonical bounds for a base component}

The following companion lemma applies numerical subadjunction to a boundary built from an
actual linear system. Its local multiplicity controls both the
degree of the center and the coefficient of that boundary.

\begin{lemma}[A base component of high multiplicity]
\label[lemma]{lem:base-component}
Let $Z$ be a projective $\Q$-factorial klt variety of dimension $d$,
let $P$ be an ample rational Cartier divisor, and let $k>0$ be an
integer for which $kP$ is Cartier. Let
$0\ne\mathcal H\subset H^0(Z,\cO_Z(kP))$ be a linear subspace
with proper base locus and base ideal $\mathfrak b$.
Suppose that $V$ is an integral base-locus component of dimension
$f<d$, isolated at its generic point $\eta_V$, and that
$\eta_V\in Z_{\rm sm}$. Write $a=d-f$ and
$\mathfrak m=\mathfrak m_{\cO_{Z,\eta_V}}$. If
\[
 \mathfrak b\cO_{Z,\eta_V}\subset\mathfrak m^h
 \qquad(h\in\Z_{>0}),
\]
then
\begin{equation}
\label{eq:base-component-degree}
 P^f\cdot V\leq (k/h)^aP^d.
\end{equation}
If $f>0$, there is a rational number $c$ with
$0<c\leq ak/h$ such that, on a smooth resolution $V^*$,
\begin{equation}
\label{eq:base-component-canonical}
 K_{V^*}\cdot P^{f-1}
 \leq (K_Z+cP)|_V\cdot P^{f-1}.
\end{equation}
The coefficient $c$ may be chosen with an effective rational divisor
$\Theta\sim_{\Q}cP$ such that $(Z,\Theta)$ is lc at the generic
point of $V$ and has an lc place with center $V$. Thus
\Cref{lem:numerical-subadjunction} can also be applied with a different
nef testing class. The degree estimate includes $f=0$; the canonical estimate is
asserted only for positive-dimensional $V$.
\end{lemma}

\noindent The proof is given in the independent moving-center part of
\cite[Lemma~7.2]{OpenAIAbundance2026}.\par


\subsection{Differentiating the full moving jet kernel}

We use the parameter-differentiation method developed in
\cite[Proposition~2.3]{EinKuchleLazarsfeld1995}. The form used here
retains ordinary ideal powers and isolated base components. Fixing a
high order at a varying point produces a family of
linear subspaces of one fixed section space. A component common
to two successive base loci acquires high multiplicity, because
parameter derivatives of the higher kernel still belong to the
lower kernel. We state both the parameter-family conclusion and
the local slicing property that will be used later.

\begin{lemma}[Moving multiplicity]
\label[lemma]{lem:moving-multiplicity}
Let $Z$ be an integral projective variety of dimension $d$, and
let $P$ be an ample rational Cartier divisor. Fix an integer $k$
with $kP$ Cartier and positive real numbers
\[
 \tau_0<\tau_1<\cdots<\tau_d,
 \qquad\tau_{i+1}-\tau_i=\delta>0.
\]
For a smooth point $x$, let $\mathcal H_i(x)$ be the entire
subspace of sections of $kP$ vanishing at $x$ to order at least
$\ceil{k\tau_i}$. Suppose that for very general $x$ all these
subspaces are nonzero and the base locus of $\mathcal H_0(x)$
has a positive-dimensional component through $x$.
If $k\delta\geq4$, then, after an algebraic parameter extension
and restriction to nonempty opens, there are an irreducible
parameter space $T$, a fixed adjacent pair of levels, and a
family of integral subvarieties $V_t$ common to the two base
loci such that the incidence
\[
 \Gamma=\{(t,z):z\in V_t\}\subset T\times Z
\]
dominates $Z$. Each general $V_t$ has dimension in $[1,d-1]$,
is an isolated component of the higher base locus at its generic
point, and its entire higher-series base ideal is contained
there in
\begin{equation}
\label{eq:moving-multiplicity-order}
 \cI_{V_t}^{h},\qquad h=\ceil{k\delta/2},
 \qquad k/h\leq2/\delta.
\end{equation}
These are ordinary local ideal powers in the smooth ambient open.

More precisely, the incidence base ideal lies in $\cI_\Gamma^h$
on a dense open of $\Gamma$. If a further morphism $Z\to B$ is
given, one may choose a general incidence point where the base
support is only $V_t$ and $V_t$ is smooth over the smooth open of
its image in $B$. For the fiber component $F$ through that point,
assume that $F$ is proper in the integral ambient fiber component
under consideration. The restricted series is then nonzero and
has base ideal in the ordinary
$\cI_F^h$ and has $F$ as an isolated base component there.
This last assertion concerns the scheme fiber locally at that
chosen point, and does not assert transversality of every fiber.
Smoothness near the generic point of $F$, and the projectivity and klt
hypotheses needed for \Cref{lem:base-component} on the ambient fiber,
must be checked in each application.
\end{lemma}

\begin{proof}
We recall the argument of \cite[Lemmas~7.3--7.4]{OpenAIAbundance2026}
to explain why the estimate concerns the entire kernel and ordinary
ideal powers. On the open where the jet maps have constant rank,
their kernels are vector bundles with fibers $\mathcal H_i(x)$.
Their base loci increase with $i$. Starting with a positive-dimensional
component through the mark, follow a containing component at each
level. The $d+1$ components all have dimensions in $[1,d-1]$, so
two successive ones coincide. After a finite parameter extension
and shrinking, their geometric generic components and inclusions
spread to one family $\Gamma\subset T\times Z$. Its evaluation
dominates $Z$, since each member contains its moving mark $x(t)$.

Take a local frame of the higher kernel and trivialize $kP$ near
$z$ with a frame independent of $t$. In a fixed basis of the global
section space, each frame section has the form
\[
 s(t,z)=\sum_\alpha c_\alpha(t)s_\alpha(z).
\]
Differentiation in the parameter with $z$ fixed therefore gives
another global section of $kP$. A derivative of order $r$ lowers
the vanishing order at $x(t)$ by at most $r$. It belongs to the
entire lower kernel whenever
\[
 r\leq\ceil{k\tau_{i+1}}-\ceil{k\tau_i}.
\]
The right side is at least $k\delta-1$, so every derivative of
order less than $h=\ceil{k\delta/2}$ belongs to that kernel and
vanishes on $\Gamma$. This is where using the full kernels matters:
an arbitrarily chosen subseries need not contain these derivatives.

To turn these vanishings into ordinary ideal membership, work at a
general smooth incidence point. Since $\Gamma\to Z$ is submersive,
parameter directions span its normal space in $T\times Z$.
Choose parameter coordinates $t=(t',t'')$ so that the implicit
function theorem writes $\Gamma$ as $t'=F(t'',z)$. With $t''$ and
$z$ fixed, derivatives in $t'$ are exactly derivatives in the
normal coordinates $u=t'-F(t'',z)$. All normal Taylor coefficients
of degree below $h$ therefore vanish, proving
$s\in\cI_\Gamma^h$. Faithful flatness of analytic local rings over
algebraic local rings gives the same membership algebraically.
There are finitely many frame sections, so this containment holds
for the entire incidence base ideal on a dense algebraic open.

Generic smoothness of $\Gamma\to T$ identifies its scheme fiber
there with the reduced member $V_t$. Restriction gives
$\cI_\Gamma^h\cO_{\{t\}\times Z}=\cI_{V_t}^h$, and the frame
specializes to a basis of the entire higher kernel. Isolation
holds because $V_t$ is a base-locus component. Finally remove all
other base components and choose the incidence point where
$V_t$ is smooth over its actual image in $B$. Its scheme fiber is
then reduced locally, so restriction to the ambient fiber sends
$\cI_{V_t}^h$ to $\cI_F^h$ and the base support to $F$. If $F$
is proper in the chosen integral ambient fiber component, that
support is proper there; hence the restricted series is nonzero
and has $F$ as an isolated base component at its generic point.
\end{proof}


For a fixed gap $\delta$, combining
\Cref{lem:moving-multiplicity,lem:base-component} gives degree and
canonical-intersection bounds for a repeated moving component,
with coefficients independent of the large divisible integer $k$.
The next lemma converts such intersection bounds into a curve of
bounded degree. Its general-type hypothesis supplies a birational
linear system in a degree depending only on the dimension. The
scalar application uses \Cref{prop:subvarieties} and has no boundary.
We include a reduced boundary in the statement because the
two-slot argument will also need its logarithmic form.

\subsection{A uniform obstruction from logarithmic general type}

\begin{lemma}[Bounded-degree curve obstruction]
\label[lemma]{lem:degree-obstruction}
Keep $Y$, $L_t$, and $\iota$ as in
\Cref{eq:small-volume-polarization,eq:small-volume-curve-lower-bound}.
Fix $f\geq1$ and positive constants $C_0,C_1$. Suppose, for
arbitrarily small $t$, that there are a smooth projective variety
$V_t^*$ of dimension $f$, a reduced simple normal crossings
divisor $G_t$ on it, a nef rational divisor $P_t$, and a morphism
$e_t:V_t^*\to Y$ satisfying the following conditions:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $K_{V_t^*}+G_t$ is big;
\item $0<P_t^f\leq C_0$ and
$(K_{V_t^*}+G_t)\cdot P_t^{f-1}\leq C_1P_t^f$;
\item $e_t$ is generically finite onto its positive-dimensional
image, and that image meets the fixed very general locus of $Y$;
\item $P_t-e_t^*L_t$ is nef.
\end{enumerate}
Then these conditions are impossible for all sufficiently small
$t$. The case $G_t=0$ includes ordinary general type.
\end{lemma}

\begin{proof}
Uniform effective birationality for projective lc pairs of
dimension $f$ with big adjoint and coefficients in the fixed DCC
set $\{0,1\}$ supplies an integer $b=b(f)$ such that
$|b(K_{V_t^*}+G_t)|$ is birational
\cite[Theorem~4.0.1]{HMX2018}. Resolve its base ideal and denote
the free moving divisor by $H_b$. We keep the same notation for
the pullbacks of $P_t$ and $L_t$. The fixed divisor is effective,
so
\[
 H_b\cdot P_t^{f-1}
 \leq b(K_{V_t^*}+G_t)\cdot P_t^{f-1}
 \leq bC_1P_t^f.
\]
Both $H_b$ and $P_t$ are nef and big; $P_t$ is big because
$P_t^f>0$. The Khovanskii--Teissier log-concavity inequalities for
their mixed intersections give, when $f>1$,
\begin{equation}
\label{eq:degree-obstruction-mixed}
 P_t\cdot H_b^{f-1}
 \leq
 \frac{(P_t^{f-1}\cdot H_b)^{f-1}}{(P_t^f)^{f-2}}
 \leq(bC_1)^{f-1}P_t^f
 \leq(bC_1)^{f-1}C_0.
\end{equation}
Indeed the successive ratios of the positive numbers
$P_t^{f-i}H_b^i$ decrease, which gives the first inequality.

Choose a point where the birational morphism defined by $H_b$
is an isomorphism onto an open of its image, where $e_t$ is
quasi-finite, and whose image lies in the very general locus of
$Y$. These conditions are compatible. To spell out the last
one, write the excluded subset of $Y$ as a countable union of
proper closed subsets. The image of $e_t$ is not contained in
any of them because it meets their complement. Their inverse
images are therefore proper closed subsets of $V_t^*$. They
can be avoided together with the finitely many dense-open
conditions. For a sweeping family one first chooses a very
general member not contained in any excluded subset, and then
such a point on that member.

For $f>1$, cut by $f-1$ general members of $|H_b|$ passing through
the chosen point. These give a proper effective curve cycle:
on the birational image they are general hyperplanes through a
point of the isomorphism open, and their pullbacks have no
positive-dimensional base component. A component $C$ through
the selected point maps nonconstantly to $Y$, since $e_t$ is
quasi-finite there. Nefness and
\Cref{eq:degree-obstruction-mixed} give
\[
 L_t\cdot e_t(C)
 \leq e_t^*L_t\cdot C
 \leq P_t\cdot C
 \leq P_t\cdot H_b^{f-1}
 \leq(bC_1)^{f-1}C_0.
\]
In the first inequality the positive degree of the finite map
of the normalization of $C$ onto its image has merely been
discarded. If $f=1$, use $V_t^*$ itself as $C$; the same argument
gives $L_t\cdot e_t(C)\leq P_t\cdot C\leq C_0$.

The image is an integral curve through the very general locus,
so \Cref{eq:small-volume-curve-lower-bound} bounds its degree
below by $r_t/\iota$. This tends to infinity, whereas the upper
bound depends only on $f,C_0,C_1$. The contradiction proves the
lemma. If several dimensions $1\leq f\leq n$ occur, take the
maximum of the finitely many resulting constants.
\end{proof}

\subsection{The scalar order bound}

Define
\begin{equation}\label{eq:scalar-polarization}
 P_{\mathrm s}=2r_t(K+2tA),\qquad
 \rho_t=(P_{\mathrm s}^{\,n})^{1/n}.
\end{equation}
The inequalities
\[
 2L\le P_{\mathrm s}\le4L
\]
are inequalities in nef order. After decreasing \(t\), they imply
\begin{equation}\label{eq:scalar-volume-bounds}
 \epsilon\le\rho_t\le8\epsilon .
\end{equation}
Only the existence of a positive lower bound and the displayed upper
bound will be used.

\begin{proposition}[Scalar orders]\label[proposition]{prop:scalar-orders}
For all sufficiently small positive rational \(t\), a very general
point \(x\in Y_{\mathrm{sm}}\) has the following property. For every
positive integer \(k\) for which \(kP_{\mathrm s}\) is Cartier and every
nonzero section
\(s\in H^0(Y,\cO_Y(kP_{\mathrm s}))\), one has
\begin{equation}\label{eq:scalar-orders}
 \ord_x(s)\le4k\rho_t\le32k\epsilon .
\end{equation}
\end{proposition}

\begin{proof}
Fix \(t\) for the moment. For each Cartier multiple \(kP_{\mathrm s}\)
and each integer \(a\), the condition
\[
 H^0(Y,\cO_Y(kP_{\mathrm s})\otimes\cI_x^a)\ne0
\]
is a rank condition on the jet evaluation map over \(Y_{\mathrm{sm}}\).
We choose \(x\) outside all proper rank-jumping loci of these
countably many maps. Thus if the order bound fails at such a point,
the relevant high-order kernel is nonzero for general marks as well.
Taking powers of the offending sections gives the same strict
inequality in arbitrarily large divisible degrees. This permits all
subsequent choices of a sufficiently large divisible \(k\).

Choose \(n+1\) levels equally spaced from \(2\rho_t\) to \(3\rho_t\);
their gap is
\[
 \delta_{\mathrm s}=\rho_t/n .
\]
If \(\ord_x(s)>4k\rho_t\) in a sufficiently large degree, every one of
these jet kernels is nonzero. The base locus of the lowest kernel
cannot have \(x\) as an isolated component. Indeed its base ideal at
\(x\) is contained in
\(\mathfrak m_x^{\ceil{2k\rho_t}}\), so the zero-dimensional case of
\Cref{lem:base-component} would give
\[
 1\le
 \left(\frac{k}{\ceil{2k\rho_t}}\right)^nP_{\mathrm s}^{\,n}
 \le2^{-n},
\]
a contradiction.

Apply \Cref{lem:moving-multiplicity}. It produces a dominant family
of proper positive-dimensional base components \(V\), of some
dimension \(1\le f<n\), such that the entire higher jet kernel's generic
base ideal along \(V\) is primary and contained in
\(\cI_V^h\), where
\[
 h=\ceil{k\delta_{\mathrm s}/2},
 \qquad k/h\le2/\delta_{\mathrm s}.
\]
For a general member, \Cref{lem:base-component} gives
\begin{align}
 P_{\mathrm s}^{\,f}\cdot V
 &\le(2/\delta_{\mathrm s})^{n-f}P_{\mathrm s}^{\,n},
 \label{eq:scalar-center-volume}\\
 K_{V^*}P_{\mathrm s}^{\,f-1}
 &\le
 \left(\frac1{2r_t}+\frac{2(n-f)}{\delta_{\mathrm s}}\right)
 P_{\mathrm s}^{\,f}\cdot V ,
 \label{eq:scalar-center-canonical}
\end{align}
where \(V^*\) is a smooth projective resolution. The second inequality
uses \(K\le P_{\mathrm s}/(2r_t)\).

The constants on the right are bounded independently of small \(t\),
of \(k\), and of the member \(V\), by
\Cref{eq:scalar-volume-bounds}. We choose a very general incidence
point before choosing the member \(V\). Its image belongs to
\(Y^{\mathrm{vg}}\), so \(V^*\) is of general type by
\Cref{prop:subvarieties}. Moreover \(P_{\mathrm s}\) dominates \(L\)
in nef order. The inclusion of \(V\) into \(Y\), composed with its
resolution, satisfies all the hypotheses of
\Cref{lem:degree-obstruction}. \Crefrange{eq:scalar-center-volume}{eq:scalar-center-canonical}
would therefore give a curve through \(Y^{\mathrm{vg}}\) of bounded
\(L\)-degree, contradicting
\Cref{eq:small-volume-curve-lower-bound} as \(t\) becomes small.

The bounds used in this contradiction do not depend on the initial
degree in which excessive vanishing occurred. Consequently, after
one restriction to sufficiently small \(t\), all Cartier multiples
satisfy \Cref{eq:scalar-orders} at a very general point.
\end{proof}
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//moving-centers.tex ====

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==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//reduction.tex ====
\section{A fixed minimal model of full nef dimension}\label{sec:reduction}

Assume that smooth canonical nonvanishing fails in dimension four.
We first choose a terminal minimal model on which every curve through
a very general point has a fixed positive lower bound for its
canonical degree. We then show that every proper
positive-dimensional subvariety through such a point is of general
type. These two properties will control the moving centers of the
later jet argument. The model remains fixed while its ample
perturbations vary.

\begin{proposition}\label[proposition]{prop:reduction}
If \Cref{thm:nonvanishing} fails, there is a projective
$\Q$-factorial terminal fourfold $Y$, with $K=K_Y$, such that
\begin{enumerate}[label=\textup{(\roman*)}]
\item $K$ is nef, $\kappa(Y,K)=-\infty$, and $K^4=0$;
\item $q(Y)=0$ and the nef dimension of $K$ is four;
\item for some integer $\iota>0$, $\iota K$ is Cartier and
\begin{equation}\label{eq:curve-lower-bound}
 K\cdot C\ge \frac1\iota
\end{equation}
for every integral curve $C$ through a point of a fixed very general
locus in $Y$.
\end{enumerate}
\end{proposition}

\begin{proof}
Start with a smooth counterexample $H$.  Run the canonical minimal
model program.  Existence of the necessary flips follows from
\cite[Corollary~1.4.1]{BCHM2010}.  Termination in the pseudo-effective fourfold case is
\cite[Theorem~1.1]{ChenTsakanikas2023}, applied with zero nef data to
the ordinary pair with zero boundary.  Divisorial contractions lower
the Picard number, so together these statements give termination of
the program.  Pseudo-effectivity is preserved and excludes a Mori
fiber space as its endpoint.  We obtain a projective $\Q$-factorial
terminal minimal model $Y$.

Canonical section spaces in sufficiently divisible degrees are
birationally invariant here.  On a common resolution the differences
from the respective pulled-back canonical divisors are effective
exceptional divisors, and pushing their sheaves forward does not add
sections.  Thus $\kappa(Y,K)=-\infty$.  In particular $K$ is not big;
as it is nef, this says $K^4=0$.

Terminal singularities are rational.  Hence $q(Y)$ agrees with the
irregularity of a resolution.  If it were positive,
\cite[Corollary~4.7]{Fujino2010} would make $K$ semiample, a
contradiction.  Likewise, if the nef dimension were at most three,
\cite[Theorem~4.3]{Ambro2005}, with the established log minimal model
program and log abundance in those dimensions, would make $K$
semiample.  This includes nef dimension zero.  Thus the nef dimension
is four.

The curve property of nef reduction \cite[Theorem~2.1]{BauerEtAl2002} says in this
case that $K\cdot C>0$ for every integral curve through a very general
point.  Choose $\iota$ with $\iota K$ Cartier.  Its degree on an
integral complete curve is an integer, so positivity gives
\Cref{eq:curve-lower-bound}.  This uses only the curve property of nef
reduction, not an Iitaka fibration on $Y$.
\end{proof}

\begin{proposition}\label[proposition]{prop:subvarieties}
For $Y$ as in \Cref{prop:reduction}, every proper positive-dimensional
subvariety through a point of a suitable fixed very general locus is
of general type.
\end{proposition}

\begin{proof}
We first work with one family of subvarieties, and at the end make the
very general locus independent of the family.  Hilbert schemes give
countably many finite-type parameter spaces for integral subvarieties
of $Y$.  Stratification, resolution of the universal families, and
generic smoothness give countably many smooth families
\[
 b:\mathcal V\longrightarrow B,\qquad e:\mathcal V\longrightarrow Y,
\]
whose fibers are smooth projective resolutions of their images and
whose fiber maps are birational onto those images.  To obtain this
description, resolve the total universal family on an integral
parameter stratum, shrink until its fibers are smooth and the fiber
maps birational, and repeat on the complement by noetherian
induction.  Non-dominant evaluation families can be discarded by
excluding their image closures in $Y$.

Consider a dominant family with fibers of dimension $d$, where
$0<d<4$.  We may slice the parameter space to dimension $4-d$ so
that $e$ becomes generically finite and remains dominant.  Indeed, the
parameters incident to a general point of $Y$ have a component of
dimension $\dim B-(4-d)$ on the locus where the fiber maps are
generically embeddings.  General ample cuts in a compactification of
$B$ meet this locus in finitely many points.  We take the cuts very
generally, so a very general point of the slice has the generic value
of every plurigenus of the original family.  There are only countably
many such conditions.

On the resulting smooth total space, ramification and terminality
give
\[
 K_{\mathcal V}\sim_\Q e^*K+E_{\mathcal V},\qquad E_{\mathcal V}\ge0.
\]
For completeness, the rational lift of $e$ to a resolution of $Y$ is
defined at codimension-one points by properness.  The usual
ramification formula there, followed by the effective discrepancies
of the terminal target, proves this divisor inequality.  Its
restriction to a very general fiber is
\begin{equation}\label{eq:fiber-canonical-comparison}
 K_V\sim_\Q e_V^*K+E_V,\qquad E_V\ge0.
\end{equation}
In particular $K_V$ is pseudo-effective.

The dimension of $V$ is at most three.  The minimal model program and
abundance in these dimensions give a good terminal minimal model
$V_{\min}$.  Suppose that $V$ is not of general type.  The semiample
canonical divisor of $V_{\min}$ defines a morphism with connected
positive-dimensional general fibers, on each of which a multiple of
$K_{V_{\min}}$ is trivial.  On a common resolution
$\widetilde V$ of $V$ and $V_{\min}$, there is a complete curve through
a very general point with $K_{\widetilde V}$-degree zero.  Here is the
avoidance needed for this assertion.  The singular locus of the
terminal model and the centers of the exceptional divisors have
codimension at least two.  A general fiber either misses a given
vertical center or meets a dominant center in codimension at least
two.  General sufficiently ample complete intersections in that
fiber through a prescribed very general point give a curve
avoiding all such centers; for a one-dimensional fiber the
fiber itself does so.  Its strict transform has canonical degree
zero by the exceptional discrepancy formula.

Pulling back \Cref{eq:fiber-canonical-comparison} to $\widetilde V$
and adding the effective discrepancies of
$\widetilde V\to V$ gives
\[
 K_{\widetilde V}\sim_\Q \widetilde e^{\,*}K+\widetilde E,
 \qquad \widetilde E\ge0.
\]
Choose the preceding curve through a point outside
$\Supp\widetilde E$ and outside the exceptional locus of the
birational map onto the image subvariety.  It is not contained in
$\widetilde E$, and its image in $Y$ is a curve.  Thus its pulled-back
$K$-degree is at most zero.  Dominance of the total evaluation lets
us choose the point in the very general locus of
\Cref{eq:curve-lower-bound}, which gives a contradiction.

We have proved that the very general fiber of the sliced family is
of general type.  By the choice of slice its plurigenera are the
generic plurigenera of the original smooth family.  Upper
semicontinuity shows that all fibers of that family have at least
these plurigenera, hence are of general type as well.  Finally
exclude the image closures of all non-dominant families.  Countability
of the parameter strata gives the asserted very general locus.
\end{proof}

For the rest of the contradiction argument, fix $Y$, $K$, $\iota$,
and a very general locus on which both propositions apply. We next
choose ample perturbations of $K$ with bounded volume and growing
degree on every curve through this locus. The two propositions will
then turn a section of excessive vanishing order into a contradiction.
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//reduction.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//signed.tex ====
\section{Signed representatives and logarithmic general type}
\label{sec:signed-representatives}

Fix the terminal canonical counterexample $Y$ of
\Cref{prop:reduction}, with the very general locus supplied by
\Cref{prop:subvarieties}. The particular need for a signed divisor
comes from the slice bundle
$\mathbb P(\mathcal O_Y\oplus\mathcal O_Y(\iota K_Y))$.
Its two summands determine disjoint sections. Intersecting an integral
multisection of degree $e>0$, distinct from those sections, with them
and pushing to $Y$ gives effective divisors $D_0,D_\infty$ with
$D_0-D_\infty\sim e\iota K_Y$, as verified in
\Cref{eq:axis-linear-equivalence}. Thus the representative of $K_Y$
has signed coefficients, whereas the boundary is the reduced union
of the two supports, including any components that cancel in their
difference. We prove the logarithmic bigness needed to apply
\Cref{lem:degree-obstruction} to this boundary once its intersection
degree has been bounded.

The proposition allows any reduced boundary containing the signed
support. Its proof constructs an auxiliary nef adjoint with a
nonempty reduced floor. The analytic results in
\Cref{sec:analytic-inputs} lift a section from that whole floor;
only after this establishes nonnegative Kodaira dimension do we
apply the required semiampleness theorem.

\begin{proposition}[Signed representative alternative]
\label[proposition]{prop:signed-alternative}
Let $Y$ be a projective $\Q$-factorial terminal fourfold such that $K_Y$ is
nef and $\kappa(Y,K_Y)=-\infty$. Suppose that there is a very general locus
of $Y$ through whose points every proper positive-dimensional subvariety
is of general type on resolution. Let $J$ be a rational Weil divisor,
with coefficients of either sign, satisfying $J\sim_{\Q}K_Y$.
Let $p:W\to Y$ be a projective log resolution, and let $D_W$ be a reduced
simple normal crossings divisor containing the strict support of $J$
and every $p$-exceptional divisor. Then $K_W+D_W$ is big.
\end{proposition}

\begin{proof}
Put $P_W=K_W+D_W$. Terminality gives
\[
 K_W=p^*K_Y+E_W,\qquad E_W\geq0
\]
with $E_W$ exceptional. In particular, both $K_W$ and $P_W$ are
pseudo-effective. We assume that $P_W$ is not big and derive a
contradiction. We first reduce its Kodaira dimension to at most zero,
then construct a nef boundary with a nonempty reduced floor. We will
lift a nonzero section from that entire floor before using any
semiampleness theorem that requires nonnegative Kodaira dimension.

\subsection*{Positive Kodaira dimension would already imply bigness}

Suppose first that $\kappa(P_W)>0$. Resolve the rational map defined by
a moving subsystem of a sufficiently divisible multiple $bP_W$:
\[
 \pi:\widetilde W\longrightarrow W,\qquad
 f:\widetilde W\longrightarrow U.
\]
Here $\widetilde W$ is smooth, $U$ is the projective image, and, for a
very ample line bundle $\cO_U(1)$,
\begin{equation}
\label{eq:signed-moving-system}
 b\pi^*P_W\sim f^*\cO_U(1)+F,\qquad F\geq0.
\end{equation}
If $f$ is generically finite, then $P_W$ is big, contrary to the
assumption. Otherwise $0<\dim U<4$. A smooth fiber component through a
very general point of $\widetilde W$ is birational to a proper
positive-dimensional subvariety of $Y$ meeting the very general locus.
It is therefore of general type. We work over a smooth open of $U$
where generic smoothness applies. On its smooth fibers the restriction
of $K_{\widetilde W}$ is the canonical bundle of the fiber, up to the
constant one-dimensional factor coming from the base.

Fix an ample Cartier divisor $H_0$ on $\widetilde W$. For some integer
$j>0$, the restriction of $jK_{\widetilde W}-H_0$ has a nonzero section
on such a fiber component. Distinct components of a smooth fiber are
disjoint, so the section can be taken to be zero on the other
components. Choose the fiber over a point where the fiber dimension of
the space of sections is the generic one for every integer $j$; this
requires only countably many exclusions. Consequently
\[
 f_*\cO_{\widetilde W}(jK_{\widetilde W}-H_0)
\]
is generically nonzero for one $j$. After tensoring by a sufficiently
large power $\cO_U(c)$, this coherent sheaf has a nonzero global
section. Thus
\[
 jK_{\widetilde W}+c f^*\cO_U(1)-H_0\sim E_0,
 \qquad E_0\geq0.
\]
The divisor on the left before subtracting $H_0$ is big. By
\Cref{eq:signed-moving-system}, adding an effective divisor gives
bigness of $jK_{\widetilde W}+cb\pi^*P_W$. Birational pushforward gives
bigness of $jK_W+cbP_W$. For completeness, the ample divisor $H_0$
has big pushforward: for a fixed ample divisor $H_W$ on $W$ and small
positive rational $\eta$, $H_0-\eta\pi^*H_W$ is ample, and pushing an
effective rational representative proves this assertion. Finally
\[
 (j+cb)P_W=jK_W+cbP_W+jD_W
\]
is big. This contradiction proves
\begin{equation}
\label{eq:signed-kodaira-small}
 \kappa(P_W)\leq0.
\end{equation}

\subsection*{A minimal model and a crepant klt perturbation}

Run a log minimal model program for the rational dlt pair $(W,D_W)$.
We use flip existence for klt pairs, the usual dlt contraction
properties, and termination of flips for pseudo-effective lc
fourfolds, applied with zero nef part
\cite{BCHM2010,ChenTsakanikas2023}. The dlt flips needed here are
obtained by lowering the finitely many boundary coefficients slightly
while retaining negativity of the given extremal ray. The resulting
pair is klt. On the relative Picard-number-one contraction, the
original and perturbed negative adjoints are relatively proportional
with positive rational ratio; the contraction theorem descends a
Cartier multiple of their numerically trivial difference. The klt
flip therefore has the required positivity for the original adjoint
as well. This supplies the flips used in this dlt program.

There are only finitely many divisorial contractions. Pseudo-effectivity
persists under each pushforward and strict transform and excludes a
negative fiber space, by intersection with covering curves in a
general fiber. These pushforwards on numerical divisor classes are
well-defined on the $\Q$-factorial models: on a common resolution they
are pullback followed by pushforward, and exceptional divisor classes
give no ambiguity. We obtain a projective $\Q$-factorial dlt minimal
model $(Z,D_Z)$ without extracting divisors. Write
\[
 M=K_Z+D_Z.
\]
The divisor $D_Z$ is reduced, $M$ is nef, and the standard exceptional
comparison on a common resolution preserves section spaces in
sufficiently divisible degrees. Hence
\[
 \kappa(M)=\kappa(P_W)\leq0.
\]
Moreover $K_Z$ is pseudo-effective, since $K_W$ is pseudo-effective.

The signed representative supplies an additional linear relation.
Indeed
\[
 P_W\sim_{\Q}p^*J+E_W+D_W,
\]
and the divisor on the right is supported on $D_W$. Its strict
pushforward to $Z$ is a signed rational divisor $G_Z$ supported on
$D_Z$, with
\begin{equation}
\label{eq:signed-supported-representative}
 M\sim_{\Q}G_Z.
\end{equation}

We next make a small klt perturbation while keeping a fixed Cartier
multiple of $M$. Choose an integer $\ell>0$ for which $\ell M$ is
Cartier, and fix a rational number
\begin{equation}
\label{eq:signed-delta-choice}
 0<\delta<\min\left\{\frac12,\frac{1}{16\ell+2}\right\}.
\end{equation}
The klt adjoint
\[
 M-\delta D_Z=K_Z+(1-\delta)D_Z
\]
is pseudo-effective because $K_Z$ is pseudo-effective and
$(1-\delta)D_Z$ is effective. Run its minimal model program. We show
inductively that every step is crepant for $M$, preserves nefness of
$M$, and preserves the Cartier property of $\ell M$.

At a given step let $R$ be a negative extremal ray for the driving
adjoint. Since $M$ is nef and $\delta<1/2$, the ray is also negative
for the klt adjoint $M-D_Z/2$. The length bound gives a rational curve
$C$ spanning $R$ such that
\[
 0<-(M-D_Z/2)\cdot C\leq 2\dim Z=8.
\]
Set $a_C=M\cdot C$ and $b_C=D_Z\cdot C$. Then
\[
 a_C\geq0,\qquad b_C\leq2a_C+16,
 \qquad a_C<\delta b_C.
\]
It follows that
\[
 0\leq a_C<\frac{16\delta}{1-2\delta}<\frac1\ell.
\]
Because $\ell M$ is Cartier, $a_C\in\ell^{-1}\Z$, and therefore
$a_C=0$.

The line-bundle clause of the contraction theorem now descends
$\cO_Z(\ell M)$ to an actual line bundle on the contraction base
\cite[Theorem~1.1(4)(iii)]{Fujino2011}; the same theorem's part~(5)
gives the length bound just used. For a divisorial contraction its
pullback is the original bundle. For a flip it pulls back on both
sides. Consequently $M$ remains nef, the same multiple $\ell M$ remains Cartier, and its pullbacks to a common resolution agree. This is
crepancy for the full adjoint, rather than numerical triviality
alone. More explicitly, choose a rational section of the descended
line bundle. The divisor $\ell M$ differs from its pullback divisor
by a principal divisor; transform the same rational function across
the flip. The two exact divisor pullbacks then agree on a common
resolution, with compatible canonical divisors. The driving pair
stays klt. The boundary with coefficient
$1/2$ is smaller than the driving boundary, so it is again klt at
the next step. The preceding argument therefore repeats with the
same $\ell$ and $\delta$.

Let $Z'$ be the resulting model, and denote strict transforms by
$D'$, $M'$, and $G'$. Put $K'=K_{Z'}$. We have
\begin{equation}
\label{eq:signed-nef-interval}
 \begin{gathered}
 (Z',D')\text{ is lc},\qquad K'\text{ is pseudo-effective},
 \qquad M'\sim_{\Q}G',\qquad \kappa(M')\leq0,\\
 M'-uD'\text{ is nef and }(Z',(1-u)D')\text{ is klt}
 \quad(0<u\leq\delta).
 \end{gathered}
\end{equation}
Here the last nefness assertion follows by interpolation between
$M'$ and $M'-\delta D'$. The klt assertion follows by interpolation
between the klt driving pair and the lc full-boundary pair. The
underlying variety $Z'$ is klt.

We have reduced the problem to a fixed nef interval of actual klt
adjoints and a signed representative supported on its boundary.
We next use that representative to locate a nonempty boundary floor.

\subsection*{A nonempty floor and the reduced non-klt ideal}

Some coefficient of $G'$ is positive. Suppose otherwise. Intersect
$K'+D'\sim_{\Q}G'\leq0$ with the third power of an ample divisor.
Pseudo-effectivity of $K'$ and effectivity of $D'$ show that all
inequalities are equalities; positivity for nonzero effective
divisors then gives $D'=0$ and $G'=0$.

To see why this is impossible, take a common smooth resolution $V$
of $W$ and $Z'$. The pullback to $V$ of the signed divisor $J$ is
supported over $D_W$. Since the programs extract no divisors and
$D'=0$, this pullback is exceptional over $Z'$. It is rationally
linearly equivalent to the pullback of $K_Y$, and hence is nef.
The negativity lemma makes this exceptional divisor nonpositive.
Its intersection with an ample divisor to the third power is
nonnegative by nefness and nonpositive by its sign; it must vanish.
Thus the pullback of $K_Y$ is rationally linearly trivial, contrary
to $\kappa(Y,K_Y)=-\infty$.

Write $D'=\sum_j D'_j$ and $G'=\sum_j g_jD'_j$, allowing $g_j=0$.
Let $a=\max_j g_j>0$. Choose a sufficiently small positive rational
$s$ and define
\begin{equation}
\label{eq:signed-boundary-wall}
 B=(1-s)D'+\frac{s}{a}G',\qquad
 \alpha=1+\frac{s}{a},\qquad N=K'+B.
\end{equation}
Choose $s$ so that every coefficient $1-s+(s/a)g_j$ is nonnegative
and $s/\alpha\in(0,\delta)$. All these coefficients are at most one,
with equality precisely when $g_j=a$. Therefore
\[
 0\leq B\leq D',\qquad S:=\floor{B}\ne0,
 \qquad N\sim_{\Q}\alpha M'-sD'.
\]
The divisor $N$ is nef by \Cref{eq:signed-nef-interval}. The pair
$(Z',B)$ is lc and klt outside $S$. Indeed, outside $S$ its finitely
many boundary coefficients are bounded above by a number less than
one; interpolate the lc full-boundary pair with the underlying klt
variety. The same reasoning shows that $(Z',bB)$ is klt whenever
$0<b<1$.

The multiplier ideals are actual ideals
\begin{equation}
\label{eq:signed-multiplier-ideals}
 \cJ(Z',B)=\cI_S,\qquad
 \cJ(Z',bB)=\cO_{Z'}\quad(0<b<1).
\end{equation}
Here $S$ carries its reduced induced structure. To verify the first
identity, use the discrepancy description on a log resolution.
Since the pair is lc, a regular function satisfies every positive
log-discrepancy condition automatically. At a log-discrepancy-zero
divisor it must vanish to positive order. All such centers lie in
$S$, and each component of $S$ itself gives such a condition. A
function vanishing on reduced $S$ has positive order at every
valuation centered in $S$; conversely the component valuations
force vanishing on $S$. This proves the identity. These are
multiplier ideals for an lc pair, not a substitution of a non-lc
ideal for the reduced ideal.

\subsection*{Lifting sections from the floor}

We claim that, for every sufficiently large and divisible integer $m$,
the following restriction map to the whole reduced floor $S$ is surjective:
\begin{equation}
\label{eq:signed-section-surjection}
 H^0(Z',\cO_{Z'}(mN))\longrightarrow H^0(S,\cO_S(mN)).
\end{equation}
The interior injectivity theorem will compare the boundary $B$
with $b_0B$ for $b_0<1$, where the pair is klt. It also requires
an endpoint $b_2B$ beyond $B$. The corresponding residual bundle
is $(m-1)N-(b_2-1)B$; its positivity is supplied by the interval
\Cref{eq:signed-nef-interval}. We now verify this for both endpoints.

Fix rational numbers $0<b_0<1=b_1<b_2$ and a projective
log resolution $h:R\to Z'$ of $(Z',D')$, hence also of $(Z',B)$.
Choose a fixed
effective integral $h$-exceptional divisor $A_R$, with coefficients
large enough that
\[
 C_i=A_R+h^*(K'+b_iB)-K_R\geq0\quad(i=0,1,2).
\]
Their supports lie in one simple normal crossings divisor, and
$C_0\leq C_1\leq C_2$. Moreover $C_1$ is the strict interior convex
combination of $C_0$ and $C_2$ with parameter
$(1-b_0)/(b_2-b_0)$. For $mN$ Cartier put
\[
 L_R=A_R+h^*(mN)-K_R.
\]
This is an integral divisor. Its endpoint residuals are
\begin{align}
 L_R-C_i
 &=h^*((m-1)N+(1-b_i)B) \notag\\
 &\sim_{\Q}h^*\bigl(t_i(m)M'-v_i(m)D'\bigr),
 \label{eq:signed-endpoint-residuals}\\
 t_i(m)&=(m-1)\alpha+(1-b_i)s/a,\qquad
 v_i(m)=(m-1)s-(1-b_i)(1-s). \notag
\end{align}
For $i=0,2$, $t_i(m)>0$ once $m$ is large, and
\[
 u_i(m):=\frac{v_i(m)}{t_i(m)}\longrightarrow\frac{s}{\alpha}
 \in(0,\delta).
\]
Consequently each endpoint residual is rationally linearly
equivalent to a positive rational multiple of
$h^*(K'+(1-u_i(m))D')$, a nef klt adjoint pulled back to its log
resolution. By \Cref{thm:adjoint-metric}, it admits a semipositive
singular metric with zero Lelong numbers at every point. Taking
positive rational powers and transporting by the actual rational
line-bundle isomorphisms preserves this property. The endpoint
hypotheses of \Cref{thm:interior-injectivity} are therefore satisfied.

That theorem gives injectivity from the first ordinary cohomology
group of $K_R+L_R-\floor{C_1}$ to that of
$K_R+L_R-\floor{C_0}$. The precise floor calculation is
\begin{equation}
\label{eq:signed-floor-calculation}
 K_R+L_R-\floor{C_i}
 =h^*(mN)+\ceil{K_R-h^*(K'+b_iB)}.
\end{equation}
Multiplier-ideal local vanishing and the projection formula
\cite[Theorem~3.3]{Fujino2011} identify these groups with
\[
 H^1\bigl(Z',\cO_{Z'}(mN)\otimes\cJ(Z',b_iB)\bigr).
\]
The comparison map is the map induced by inclusion of the
multiplier ideals. By \Cref{eq:signed-multiplier-ideals}, we have
proved injectivity of
\[
 H^1(Z',\cO_{Z'}(mN)\otimes\cI_S)
 \longrightarrow H^1(Z',\cO_{Z'}(mN)).
\]
The exact sequence of $S$ proves
\Cref{eq:signed-section-surjection}.

The restriction map is now surjective. To obtain a nonzero global
section, we must produce one on the whole reduced scheme $S$.
We do this on a dlt floor using adjunction and threefold abundance.
The next step therefore includes both gluing on that floor and
actual line-bundle descent to $S$.

\subsection*{Whole-floor abundance and descent}

Take a projective $\Q$-factorial dlt blowup
\[
 g:(Q,B_Q)\longrightarrow(Z',B),\qquad
 K_Q+B_Q=g^*(K'+B)=g^*N,
\]
extracting only lc places; this is the lc case of the dlt blowup
theorem \cite[Theorem~2.10]{FujinoHashizume2023}. Set
$T=\floor{B_Q}$. It is nonempty. Whole-floor dlt adjunction equips
$T$ with an effective rational different $\operatorname{Diff}_T$
such that
\begin{equation}
\label{eq:signed-whole-floor-adjunction}
 K_T+\operatorname{Diff}_T
 =(K_Q+B_Q)|_T=g^*N|_T
\end{equation}
as the actual rational adjoint line bundle, including the conductor
and residue identifications on the normalization. The pair on $T$
is semi-dlt; see \cite[Remark~1.2(3)]{Fujino2000} and
\cite[Remark~2.7]{FujinoGongyo2014}.

We record the depth condition needed for this whole-floor statement.
The underlying $Q$ is klt and $\Q$-factorial. Both $\cO_Q$ and
$\cO_Q(-T)$ are Cohen--Macaulay. For the latter assertion, locally
trivialize a Cartier multiple of $T$ and take the normal cyclic index
cover. It is finite and quasi-\'etale, hence klt and Cohen--Macaulay.
Its finite pushforward decomposes into the corresponding rank-one
reflexive sheaves, including $\cO_Q(-T)$; in characteristic zero
these are direct summands. They are therefore Cohen--Macaulay.
The sequence
\[
 0\longrightarrow\cO_Q(-T)\longrightarrow\cO_Q
 \longrightarrow\cO_T\longrightarrow0
\]
then gives $S_2$ for $T$ (in fact the required Cohen--Macaulay depth
along its support). Its codimension-one crossings and the conductor
description come from the dlt structure. Thus the semi-log-canonical
adjunction in \Cref{eq:signed-whole-floor-adjunction} is on the
whole projective threefold, not on a disjoint collection of its
components.

The adjoint in \Cref{eq:signed-whole-floor-adjunction} is nef.
Projective semi-dlt threefold abundance makes it semiample
\cite[Corollary~4.10]{Fujino2000}. Hence
\[
 H^0(T,\cO_T(mg^*N))\ne0
\]
for all sufficiently divisible positive $m$.

These sections descend to $S$. Indeed
$\cJ(Q,B_Q)=\cO_Q(-T)$, and crepancy together with multiplier-ideal
local vanishing gives
\begin{equation}
\label{eq:signed-floor-pushforward}
 g_*\cO_Q(-T)=\cI_S,\qquad
 R^i g_*\cO_Q(-T)=0\quad(i>0).
\end{equation}
One can compute both assertions on a common log resolution of
$(Q,B_Q)$ and $(Z',B)$. The equality of their pulled-back adjoints
identifies the discrepancy round-up sheaves, and local vanishing
for each log resolution followed by Leray gives
\Cref{eq:signed-floor-pushforward}. Since $Z'$ is normal,
$g_*\cO_Q=\cO_{Z'}$. Pushing forward the divisor sequence therefore
gives
\[
 g_*\cO_T=\cO_S.
\]
The projection formula identifies the nonzero section spaces above
with $H^0(S,\cO_S(mN))$. Choose $m$ both sufficiently divisible
for this semiampleness and sufficiently large for
\Cref{eq:signed-section-surjection}. A nonzero section on $T$
then descends to $S$ and lifts to $Z'$. We have proved
$\kappa(N)\geq0$.

\subsection*{The final klt adjoint}

Because $B\leq D'$, section inclusion in divisible degrees gives
$\kappa(N)\leq\kappa(M')\leq0$. Thus $\kappa(N)=0$.
Now consider the klt pair
\[
 \left(Z',\left(1-\frac{s}{\alpha}\right)D'\right).
\]
Its nef adjoint is rationally linearly equivalent to $N/\alpha$,
so its Kodaira dimension is zero. By
\Cref{thm:adjoint-metric}, on a smooth projective log resolution
a positive Cartier multiple of its pullback has a semipositive
singular metric with zero point Lelong numbers. All hypotheses of
the fourfold semiampleness result of Gongyo--Matsumura are now
satisfied: the pair is projective klt with rational boundary, the
adjoint has nonnegative Kodaira dimension, and the specified
pullback metric exists
\cite[Corollary~5.3]{GongyoMatsumura2017}.

Semiampleness and Kodaira dimension zero make this adjoint
numerically trivial. On the other hand, $K'$ is pseudo-effective,
$D'\ne0$ because $G'$ has a positive coefficient, and
$1-s/\alpha>0$. Intersecting
$K'+(1-s/\alpha)D'$ with the third power of an ample divisor gives
a strictly positive number. This contradiction proves the
proposition.
\end{proof}
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//signed.tex ====

==== BEGIN Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//two-slot.tex ====
\section{Moving jets on a two-slot bundle}
\label{sec:two-slot-jets}

We continue with the terminal fourfold \(Y\), its nef non-big canonical
divisor \(K\), and the polarizations of
\Cref{eq:small-volume-polarization}. Thus \(n=4\), the integer
\(\iota>0\) makes \(\iota K\) Cartier, and
\[
 L=L_t=r_t(K+tA),\qquad L^n\longrightarrow\epsilon^n,\qquad
 r_t\longrightarrow\infty .
\]
The positive number \(\epsilon\) is fixed throughout this section.
Retain the fixed locus \(Y^{\mathrm{vg}}\) from
\Cref{sec:moving-centers}, on which \Cref{prop:subvarieties} holds
and every integral curve has \(K\)-degree at least \(1/\iota\).
In particular,
\begin{equation}\label{eq:two-slot-curve-lower-bound}
 L\cdot C\ge r_t/\iota
 \quad\text{if \(C\) passes through a point of \(Y^{\mathrm{vg}}\).}
\end{equation}
A very general locus here means the complement of a countable union
of proper closed subsets. We may enlarge that union whenever finitely
or countably many additional conditions are imposed.

The scalar estimate of \Cref{prop:scalar-orders} limits vanishing
on one copy of \(Y\). We now seek the complementary estimate: a
polarization on a projective bundle over \(Y\times Y\) generates
many jets at a very general point. Its Seshadri constant measures
this local positivity \cite[Chapter~5]{Lazarsfeld2004I}.

Failure of the estimate would produce a moving base component.
Positive-dimensional fibers over either copy of \(Y\) are excluded
on a bundle slice, using general type and, for a multisection,
\Cref{prop:signed-alternative}. If both projections are generically
finite, the required contradiction has a different source:
ramification bounds reduce the family to finitely many fixed
covers, and their birational automorphisms cannot sweep the product.
We isolate that argument before applying the moving-center method.

\subsection{The bundle and its volumes}

Put \(S_0=\iota K\), an actual Cartier divisor on \(Y\). On
\(Y\times Y\), a subscript denotes pullback from the corresponding
factor. Let
\begin{equation}\label{eq:two-slot-bundle}
 \pi:Z=\PP\bigl(\cO(S_{0,1})\oplus\cO(S_{0,2})\bigr)
       \longrightarrow Y\times Y,\qquad
 \xi=c_1(\cO_Z(1)).
\end{equation}
We use the convention
\[
 \pi_*\cO_Z(a\xi)
 =\operatorname{Sym}^a\bigl(\cO(S_{0,1})\oplus\cO(S_{0,2})\bigr)
 \quad(a\ge0).
\]
We suppress pullback symbols for divisors on the base. For a positive
integer \(q\), define
\begin{equation}\label{eq:two-slot-polarization}
 P=L_1+L_2+q\xi,\qquad d=\dim Z=2n+1=9.
\end{equation}
This \(q\) is a bundle weight; it is unrelated to the irregularity
\(q(Y)=0\).

The two summands determine two disjoint sections, called the axes.
The complement of the axes is the torus open of \(Z\). Fixing either
base coordinate at a point and trivializing the constant line gives
a slice
\begin{equation}\label{eq:two-slot-slice}
 \pi_{\mathrm{sl}}:Z_{\mathrm{sl}}
 =\PP\bigl(\cO_Y\oplus\cO_Y(S_0)\bigr)\longrightarrow Y,
 \qquad P_{\mathrm{sl}}=L+q\xi_{\mathrm{sl}}.
\end{equation}
The restriction of \(P\) is identified with \(P_{\mathrm{sl}}\);
the remaining constant factor is trivialized when restricting
sections.

\begin{lemma}\label[lemma]{lem:two-slot-ambient}
The varieties \(Y\times Y\), \(Z\), and \(Z_{\mathrm{sl}}\) are
projective, \(\Q\)-factorial, and klt. The tautological classes of
the two bundles are nef, and \(P\) and \(P_{\mathrm{sl}}\) are
ample rational divisors.
Writing \(K_\Sigma=K_1+K_2\) on the product and \(K_\Sigma=K\) on a
slice, one has
\begin{equation}\label{eq:two-slot-canonical}
 K_{\mathrm{bundle}}
 =-2\xi+(\iota+1)K_\Sigma
 \le\frac{\iota+1}{r_t}\,P_{\mathrm{bundle}}
\end{equation}
in nef order.
\end{lemma}

\begin{proof}
Let \(p:\widetilde Y\to Y\) be a smooth projective resolution.
Rational singularities and the irregularity conclusion in
\Cref{prop:reduction} give
\(q(\widetilde Y)=0\). The product description of the Picard group,
or the seesaw principle and the vanishing of
\(\Hom(\Alb(\widetilde Y),\Pic^0(\widetilde Y))\), gives
\[
 \Pic(\widetilde Y\times\widetilde Y)
 =\operatorname{pr}_1^*\Pic(\widetilde Y)
  \oplus\operatorname{pr}_2^*\Pic(\widetilde Y).
\]
For a prime Weil divisor on \(Y\times Y\), take its strict transform
on this smooth product. Its Cartier class is a sum of pullbacks from
the two factors. Pushing down the corresponding linear equivalence
expresses the original Weil divisor class as a sum of pullbacks of
Weil divisor classes on \(Y\). These are \(\Q\)-Cartier because
\(Y\) is \(\Q\)-factorial. Thus \(Y\times Y\) is \(\Q\)-factorial.
Exceptional components have images of codimension at least two and
do not affect this divisor-class calculation.

The bundle over \(\widetilde Y\times\widetilde Y\) is a smooth
resolution of \(Z\). Its Picard group is the Picard group of the base
plus the integral tautological class. Pushing down as above proves
\(\Q\)-factoriality of \(Z\). The identical argument over
\(\widetilde Y\) proves it for the slice. A product of canonical
varieties is canonical, as follows directly from the product of
resolutions and the canonical discrepancy formula. Projective
bundles are smooth over their bases, so the bundles are klt as well.

The direct sum of nef line bundles is nef; hence its tautological
class is nef. That class is also relatively ample. Adding a positive
pullback of an ample class therefore makes it ample: for example,
write the sum as a positive multiple of a relatively ample class
made ample by a sufficiently large base twist, plus nef classes.
This proves the assertions about \(P\) and \(P_{\mathrm{sl}}\).
Finally, the canonical bundle formula for a rank-two projective
bundle gives the equality in \Cref{eq:two-slot-canonical}.
Since \(\xi\) is nef and \(K_\Sigma\le L_\Sigma/r_t\), the stated
nef inequality follows.
\end{proof}

\begin{lemma}\label[lemma]{lem:two-slot-volumes}
Assume \(q\iota/r_t\le1\). For sufficiently small \(t\), there are
positive constants \(c_n,C_n\), depending only on \(n\), such that
\begin{equation}\label{eq:two-slot-total-volume}
 c_n q\epsilon^{2n}\le P^d\le C_n q\epsilon^{2n}.
\end{equation}
On either slice,
\begin{equation}\label{eq:two-slot-slice-volume}
 0<P_{\mathrm{sl}}^{\,n+1}
 \le2^{n+2}q\epsilon^n .
\end{equation}
\end{lemma}

\begin{proof}
For a rank-two split bundle with first Chern classes \(E_1,E_2\),
the projective-bundle formula is
\[
 \pi_*(\xi^i)=\sum_{a+b=i-1}E_1^aE_2^b\quad(i\ge1),
 \qquad \pi_*(1)=0 .
\]
All classes in the expansion of \(P^d\) are nef. Put
\(L_\Sigma=L_1+L_2\). Each \(E_j=S_{0,j}\) is bounded above in nef
order by \((\iota/r_t)L_\Sigma\). Consequently
\begin{align*}
 d qL_\Sigma^{2n}
 &\le P^d\\
 &\le
 qL_\Sigma^{2n}
 \sum_{i=1}^d i\binom di(q\iota/r_t)^{i-1}
 \le d2^{d-1}qL_\Sigma^{2n}.
\end{align*}
Here \(L_\Sigma^{2n}=\binom{2n}{n}(L^n)^2\), and
\(L^n\to\epsilon^n>0\). This proves
\Cref{eq:two-slot-total-volume}.

On a slice one summand is trivial, so
\[
 P_{\mathrm{sl}}^{\,n+1}
 =\sum_{i=1}^{n+1}\binom{n+1}{i}
 q^i L^{n+1-i}S_0^{i-1}
 \le qL^n\sum_{i=1}^{n+1}\binom{n+1}{i}.
\]
For small \(t\), we may use \(L^n\le2\epsilon^n\), giving
\Cref{eq:two-slot-slice-volume}. Positivity follows either from
ampleness or from the \(i=1\) term.
\end{proof}

We now choose the parameters in the order required by the proof.
After fixing \(\epsilon\), choose \(\delta>0\) such that
\begin{equation}\label{eq:two-slot-gap-choice}
 (2/\delta)^n2^{n+2}\epsilon^n<1 .
\end{equation}
Next choose an integer \(q>0\) so large that the lower bound in
\Cref{eq:two-slot-total-volume} gives
\begin{equation}\label{eq:two-slot-jet-volume}
 P^d>(2n+3+d\delta)^d
\end{equation}
for all sufficiently small \(t\). This is possible because that
lower bound is linear in \(q\). The integer \(q\) is now fixed.
Finally decrease \(t\), imposing \(q\iota/r_t\le1\) and all the
preceding estimates. Further decreases of \(t\) do not change
\(\epsilon,\delta,\) or \(q\).

\subsection{The large Seshadri estimate}

For an ample rational divisor \(P\) and a smooth point \(z\), write
\[
 \varepsilon(P;z)
 =\inf_{C\ni z}\frac{P\cdot C}{\mult_z C},
\]
where the infimum is over integral projective curves through \(z\).

\begin{proposition}[Two-slot jets]\label[proposition]{prop:two-slot-jets}
Let \(Z\) and \(P\) be defined by
\Crefrange{eq:two-slot-bundle}{eq:two-slot-polarization}.
Fix \(\epsilon>0\), choose \(\delta\) satisfying
\Cref{eq:two-slot-gap-choice}, and then fix \(q\) large enough
for \Cref{eq:two-slot-jet-volume}. For all sufficiently small
positive rational \(t\), one has
\begin{equation}\label{eq:two-slot-seshadri}
 \varepsilon(P;z)>2n+2=10
\end{equation}
at a very general smooth point of the torus open of \(Z\).
\end{proposition}

\subsection{Dominant finite correspondences}

The final case of the jet argument will be a family of
\(n\)-dimensional subvarieties whose two projections to \(Y\) are
generically finite. The following proposition excludes such a family
using only bounds for its degree and canonical intersection. The
family itself may change with \(t\).

\begin{proposition}[Bounded correspondences cannot sweep the bundle]
\label[proposition]{prop:two-slot-correspondences}
Fix \(\epsilon>0\), an integer \(q>0\), and positive constants
\(B_0,B_1\). Keep \(Y,K,A,L_t\) and \(P=L_1+L_2+q\xi\) as above.
For all sufficiently small positive rational \(t\), there is no
irreducible finite-type parameter space \(T\) and integral closed
subvariety \(\Gamma\subset T\times Z\), dominant over \(T\), with
the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The evaluation \(\Gamma\to Z\) is dominant, and a general
fiber \(V\) over \(T\) is an integral subvariety of dimension \(n\).
\item On a smooth projective resolution \(V^*\to V\), both
projections \(g_i:V^*\to Y\), \(i=1,2\), are dominant and
generically finite.
\item Writing \(P\) also for its pullback to \(V^*\), one has
\[
 0<P^n\cdot V\le B_0,\qquad
 K_{V^*}P^{n-1}\le B_1(P^n\cdot V).
\]
\end{enumerate}
The restriction on \(t\) depends only on the fixed geometry and
\(\epsilon,q,B_0,B_1\), not on the parameter space or family.
\end{proposition}

\begin{proof}
We first rule out branch divisors that sweep \(Y\), using the
intersection bounds uniformly in \(t\). We then fix \(t\) and the
family: a common branch complement gives finitely many covers, whose
birational graph families lead to an abelian variety dominating \(Y\).

Since \(P-g_i^*L\) is nef, the projection formula gives
\begin{equation}\label{eq:two-slot-cover-degrees}
 \deg(g_i)L^n\le P^n\cdot V\le B_0.
\end{equation}
The degrees are uniformly bounded because \(L^n\to\epsilon^n>0\).
This degree bound alone does not give a finite list of covers; we
must first control their branch loci.

\paragraph{Numerical bounds for branch divisors.}
Resolve the general members of the incidence family in family.
After shrinking the irreducible parameter space, there is a smooth
projective family \(\mathcal V^*\to T\) with integral fibers
birational to the corresponding \(V\), together with the two
generically finite morphisms \(g_i:V^*\to Y\) on each fiber.
For either projection, let \(T_i\to Y\) denote the finite normal
Stein factor. Its function field is \(\C(V)\).

If \(J\) is a branch prime of \(T_i\to Y\), a ramification prime on
\(T_i\) has a strict transform \(R\) on \(V^*\). A proper birational
morphism to a normal variety is an isomorphism at the generic point
of each target divisor, so this transform is present. Over the
smooth locus of \(Y\), its ramification coefficient is at least one.
There is an effective canonical comparison
\begin{equation}\label{eq:two-slot-ramification}
 K_{V^*}\sim_{\Q}g_i^*K+E_i,\qquad E_i\ge R .
\end{equation}
For completeness, resolve the rational lift to a resolution of the
terminal target. Ramification between smooth varieties and the
effective terminal discrepancies give this formula upstairs;
pushing down to \(V^*\) gives \Cref{eq:two-slot-ramification}.

Set \(d_J=L^{n-1}\cdot J\). Projection and nef order imply
\begin{equation}\label{eq:branch-degree}
 0<d_J
 \le P^{n-1}\cdot R
 \le K_{V^*}P^{n-1}
 \le C_3 ,
\end{equation}
where \(C_3=B_1B_0\) is independent of \(t\) and of the family.
The same estimate applies to every prime
divisorial image of ramification that meets \(Y_{\mathrm{sm}}\).

We also need a canonical bound on \(J\), which need not be normal.
On \(Y_{\mathrm{sm}}\), the divisor \(J\) is a Gorenstein
hypersurface. If \(\nu:J^\nu\to J\) is its normalization, finite
duality gives, in codimension one,
\[
 K_{J^\nu}+D_{\mathrm{cond}}
 \sim_{\Q}\nu^*((K+J)|_J),\qquad D_{\mathrm{cond}}\ge0 .
\]
The conductor therefore has the sign that decreases the
normalized canonical intersection. Terminal singularities are
smooth in codimension two, so \(J\cap Y_{\mathrm{sing}}\) has
dimension at most \(n-3\). Its inverse image on the normalization
has codimension at least two. It introduces no omitted
codimension-one term in this comparison. On a resolution \(J^*\),
the exceptional canonical cycles push to zero against \(n-2\)
pullbacks of \(L\). Consequently
\begin{align}
 K_{J^*}L^{n-2}
 &\le (K+J)J L^{n-2} \notag\\
 &\le d_J/r_t+d_J^2/L^n
 \le C_4d_J .
 \label{eq:branch-canonical}
\end{align}
Here \(K\le L/r_t\). The square term is the Hodge-index inequality
\[
 J^2L^{n-2}\le\frac{(JL^{n-1})^2}{L^n}.
\]
It applies to the \(\Q\)-Cartier divisor \(J\); one can pull back
to a resolution and approximate the nef class \(L\) by ample
classes. The bound in \Cref{eq:branch-degree} and the positive lower
bound for \(L^n\) make \(C_4\) uniform.

If a family of these prime images dominates \(Y\), choose a very
general member and then a very general point on its resolution
mapping to \(Y^{\mathrm{vg}}\). Such a choice is legitimate despite
the countable exceptional set. For each excluded proper closed
subset of \(Y\), dominance says that the generic image divisor is
not contained in it; omit the corresponding proper parameter
locus. On the remaining very general member, omit the inverse
images of the countably many excluded subsets, together with the
finite birational-system and quasi-finiteness exceptions.
The chosen image divisor is of general type by
\Cref{prop:subvarieties}. Its top \(L\)-degree is \(d_J\), and
\Cref{eq:branch-canonical} is the required canonical bound.
\Cref{lem:degree-obstruction} excludes this sweeping family for
sufficiently small \(t\).

\paragraph{A fixed branch complement and finitely many covers.}
The preceding exclusion holds uniformly over all families in the
statement. Now fix one such sufficiently small \(t\) and its incidence
family. No further decrease of \(t\) will be made in this proof.
The smooth projective family
\(\mathcal V^*\to T\) has integral fibers, and both projections
to the normal variety \(Y\) are dominant and generically finite.
Their degrees are bounded by \Cref{eq:two-slot-cover-degrees}.
After a finite parameter extension and shrinking, follow the
finitely many geometric generic components of the relative
Jacobian over \(Y_{\mathrm{sm}}\). Every followed component whose
fiber image is a divisor has proper total image closure: the
preceding argument excludes a dominant such family using
\Cref{eq:branch-degree,eq:branch-canonical}.

These are exactly the hypotheses of the simultaneous finite-cover
lemma \cite[Lemma~7.5]{OpenAIAbundance2026}. It gives one smooth
nonempty open \(Y^\circ\subset Y\) over which every general finite
Stein cover in either slot is finite \'etale, and finite lists of
the resulting normal covers of \(Y\). The lemma accounts for every
branch prime by its ramification strict transform, then applies
purity and finiteness of bounded-degree \'etale covers of the fixed
open. In particular, this is a finite-type family argument; a
numerical branch bound alone would not suffice. The open and lists
may depend on the fixed \(t\) and incidence family. No uniformity
in those choices is needed.

\paragraph{Countably many birational graph families.}
Each \(V\) gives a birational map between one cover from each list:
both finite Stein factors have function field \(\C(V)\). Its
image in \(Y\times Y\) is the finite image of the closure of that
birational graph. Fix one pair of covers that are birational.
A smooth projective resolution of their common birational class
has pseudo-effective canonical class, by the effective canonical
comparison with the nef \(K\) on \(Y\). It is non-uniruled by
\cite{BDPP2013}.

The birational-group structure theorem of Hanamura
\cite{Hanamura1988}, in the form stated in
\cite[Proposition 3.7 and Theorem 3.8]{Blanc2017}, permits a smooth
projective model \(U\) of this class for which the reduced
flat-graph birational scheme is a group scheme locally of finite
type and
\[
 A_U=\Aut^0(U)
\]
is its identity component, an abelian variety. The flat graphs are
represented in graph Hilbert schemes. There are countably many
Hilbert polynomials and finitely many components in each
finite-type Hilbert scheme, so the birational maps occur in
countably many translates \(A_U b\).

For such a translate, its graphs are parametrized on a dense open
by the rational evaluation map
\begin{equation}\label{eq:abelian-graph-evaluation}
 U\times A_U\dashrightarrow U\times U,\qquad
 (x,a)\longmapsto(x,a\cdot b(x)).
\end{equation}
If this map is not dominant, its image closure is a proper closed
subset of \(U\times U\). Fix birational identifications of \(U\)
with the two covers. Every graph meets the product of their fixed
birational opens densely, because it dominates both factors.
Transporting the image closure through those opens, then taking
its closure in the product of covers, preserves its dimension.
The subsequent finite map to \(Y\times Y\) also preserves
dimension. Thus all correspondences belonging to this translate
are contained in a proper closed subset of \(Y\times Y\).
The possible boundary of an individual birational map introduces
no new component: the generic graph has already been included.

There are finitely many cover pairs and countably many translates
for each pair. If every map in \Cref{eq:abelian-graph-evaluation}
were nondominant, the image of the original incidence would be
contained in a countable union of proper closed subsets of
\(Y\times Y\). But \(\Gamma\to Z\) is dominant, so its composite
to \(Y\times Y\) is dominant and its image contains a nonempty
open subset. Over the uncountable field \(\C\), such an open
cannot be covered by countably many proper closed subsets.
Therefore the map in \Cref{eq:abelian-graph-evaluation} is dominant for at
least one translate.

\paragraph{The abelian curve contradiction.}
For a general \(x\) in this last evaluation, \(A_U\cdot b(x)\)
is a dense orbit in \(U\). A stabilizer element fixes every point
of that orbit by commutativity; it fixes \(U\) by density.
The action of \(\Aut^0(U)\) is faithful, so the stabilizer is
zero-dimensional. Hence the orbit map from \(A_U\) is generically
finite dominant. Composing with the rational map through either
fixed cover gives a generically finite dominant rational map
\[
 A_U\dashrightarrow Y .
\]
In particular \(\dim A_U=\dim Y=n\).

Resolve this map as
\(\pi_A:T_A\to A_U\) and \(g:T_A\to Y\), with \(T_A\) smooth and
projective. Since the canonical bundle of \(A_U\) is trivial,
and since \(Y\) is terminal, the canonical comparisons give
\begin{equation}\label{eq:abelian-canonical-comparisons}
 K_{T_A}\sim_{\Q}E_{\pi_A},\qquad
 K_{T_A}\sim_{\Q}g^*K+R,\qquad
 E_{\pi_A},R\ge0 ,
\end{equation}
where \(E_{\pi_A}\) is exceptional over \(A_U\).
The images of the exceptional divisors of \(\pi_A\) have
codimension at least two on the smooth abelian variety.

Choose a point in the isomorphism open of \(\pi_A\) where \(g\)
is quasi-finite, outside \(\Supp(R)\), and mapping to
\(Y^{\mathrm{vg}}\). Dominance and the countable-exclusion
description ensure that such a point exists. A general sufficiently
ample complete-intersection curve in \(A_U\) through its image
avoids all the exceptional centers: they have codimension at
least two and do not contain the prescribed point. Its strict
transform \(C\) is disjoint from \(E_{\pi_A}\) and is not contained
in \(R\). The map \(g\) is nonconstant on \(C\), since it is
quasi-finite at the chosen point. Intersecting
\Cref{eq:abelian-canonical-comparisons} with \(C\) gives
\[
 0=K_{T_A}\cdot C=g^*K\cdot C+R\cdot C>0 .
\]
The final strict inequality uses
\(K\cdot g(C)\ge1/\iota\), the defining curve property of
\(Y^{\mathrm{vg}}\), multiplied by the positive degree of
\(C\to g(C)\), together with \(R\cdot C\ge0\).
This contradiction excludes the remaining correspondence family.
\end{proof}

\subsection{Excluding moving base components}

\begin{proof}[Proof of \Cref{prop:two-slot-jets}]
All constants used to exclude moving centers below are independent
of the section degree and of small \(t\), once
\(\epsilon,\delta,q\) have been fixed. We first produce such a center
from failure of \Cref{eq:two-slot-seshadri}; we then exclude each
possible dimension of its fibers over the two factors.

\paragraph{A moving component from failure of the estimate.}
Suppose that \Cref{eq:two-slot-seshadri} fails at very general
torus points. At such a point there is a curve whose ratio is
strictly less than \(2n+3\). This conclusion does not require the
infimum defining the Seshadri constant to be attained.

Take \(d+1\) levels
\[
 \tau_j=2n+3+j\delta,\qquad 0\le j\le d .
\]
By \Cref{eq:two-slot-jet-volume}, ample Hilbert asymptotics and the
dimension of the jet space at a smooth point show that, for large
divisible \(k\), every kernel of jets of order
\(\ceil{k\tau_j}\) is nonzero. Explicitly, the leading coefficients
of the two dimensions are \(P^d/d!\) and \(\tau_j^d/d!\).
If a section in the lowest kernel does not contain the curve just
chosen, its local intersection with that curve at the marked point
is at least
\(\ceil{k\tau_0}\mult_z C\). This exceeds \(kP\cdot C\).
Thus every section in that kernel contains the curve, and the
lowest base locus has a positive-dimensional component through the
mark.

These conditions produce a dominant marked-point family in the
sense of \Cref{lem:moving-multiplicity}. For generality, choose the
marked point outside the proper
rank-jumping and base-dimension loci for all divisible degrees; there
are only countably many such conditions. For the degree \(k\) now
chosen, restrict to the open where the finitely many jet kernels have
constant rank. The dimension of the base
scheme at its marked point is upper semicontinuous. The existence
of the witnessing curve at very general marks therefore forces the
positive-dimensional-base condition on a dominant locus; no family
of the witnessing curves is assumed. The finitely many generic
base components and their inclusions can then be followed after a
finite parameter extension and shrinking.

We may increase the divisible integer \(k\) to arrange, in addition,
that all divisors
\begin{equation}\label{eq:two-slot-fillers}
 kL+(2kq-j)S_0,\qquad 0\le j\le kq ,
\end{equation}
are Cartier and globally generated. Here is one simultaneous
verification. For \(1\le a\le n\), subtract \(aA\) and then \(K\).
The resulting class is
\[
 \bigl(kr_t+(2kq-j)\iota-1\bigr)K+(kr_tt-a)A,
\]
which is ample once \(kr_tt>n\) and \(kr_t\ge1\).
Klt Kawamata--Viehweg vanishing followed by regularity with respect
to \(A\) proves global generation for every \(j\) in
\Cref{eq:two-slot-fillers}.

The moving-multiplicity lemma supplies an irreducible parameter
space \(T\), a dominant incidence
\(\Gamma\subset T\times Z\), and integral fiber components
\(V=V_u\) of a fixed dimension \(1\le f<d\). The high-kernel
subseries has \(V\) as an isolated base component generically, with
ordinary generic order at least
\[
 h=\ceil{k\delta/2},\qquad k/h\le2/\delta .
\]
For a smooth resolution \(V^*\), the base-component estimates and
\Cref{eq:two-slot-canonical} give
\begin{align}
 0<P^f\cdot V
 &\le(2/\delta)^{d-f}P^d,
 \label{eq:two-slot-center-volume}\\
 K_{V^*}P^{f-1}
 &\le
 \left(\frac{\iota+1}{r_t}+\frac{2(d-f)}{\delta}\right)P^f\cdot V .
 \label{eq:two-slot-center-canonical}
\end{align}
The right sides are bounded uniformly in the specified sense.

We first exclude positive-dimensional slot fibers. The chosen incidence
point will ensure that all image subvarieties below meet the prescribed
very general locus of \(Y\). For an intermediate fiber \(F\) in a slice, let \(W\subseteq Y\)
be its image and write \(s=\dim F\), \(w=\dim W\). The three
possibilities are summarized below; the subsequent paragraphs prove
all the indicated exclusions.
\begin{center}
\small
\begin{tabular}{@{}p{0.18\textwidth}p{0.31\textwidth}p{0.43\textwidth}@{}}
\toprule
Dimensions & Geometry & Contradiction \\
\midrule
\(s=w<n\) & Generically finite over a proper image & General type and the degree obstruction on \(F\). \\[3pt]
\(s=w+1\), \(w<n\) & Entire bundle over \(W\) & Direct slice-volume bound if \(w=0\); fillers and subadjunction on \(W\) if \(w>0\). \\[3pt]
\(s=w=n\) & Non-axis multisection over \(Y\) & Its two axis intersections give a signed representative; the logarithmic degree obstruction applies. \\
\bottomrule
\end{tabular}
\end{center}
A full slot fiber is excluded by exchanging slots. This leaves
generically finite projections. Proper images are again excluded by
general type, and dominant images are excluded by
\Cref{prop:two-slot-correspondences}.

\paragraph{Restriction to a slot fiber.}
Consider \(V\) over either factor of \(Y\times Y\). Choose a very
general incidence point first. We require that its two base
coordinates belong to \(Y^{\mathrm{vg}}\), that it lies in the torus,
and that the ordinary ideal-power and isolated-component conclusions
above hold near it. We also impose the smoothness conditions for
the incidence over \(T\) and for \(V\) over the smooth open of its
actual slot image. These are available on dense opens.

Now take the slot value of this point, and let \(F\) be the reduced
fiber component through it. Write \(s=\dim F\), the general fiber
dimension. Suppose first that
\[
 0<s<n+1.
\]
The restricted high-kernel subseries on the opposite slice is
nonzero and has \(F\) as an isolated base component generically.
Indeed, near the selected point the original base support is just
\(V\). No other base component can contain the fiber component
through that point. The scheme fiber over the ambient slot agrees
with the scheme fiber over the actual image, and smoothness over
that image makes it reduced at the selected generic component.
Thus the image of \(\cI_V^h\) in the slice is contained in
\(\cI_F^h\), as an ordinary ideal power. A restriction that vanished
identically on the whole slice would contradict this local proper
base support.

Applying \Cref{lem:base-component} in the slice gives
\begin{align}
 0<P_{\mathrm{sl}}^{\,s}\cdot F
 &\le(2/\delta)^{n+1-s}P_{\mathrm{sl}}^{\,n+1},
 \label{eq:two-slot-fiber-volume}\\
 K_{F^*}P_{\mathrm{sl}}^{\,s-1}
 &\le
 \left(\frac{\iota+1}{r_t}+\frac{2(n+1-s)}{\delta}\right)
 P_{\mathrm{sl}}^{\,s}\cdot F .
 \label{eq:two-slot-fiber-canonical}
\end{align}
Let \(W\subseteq Y\) be the image of \(F\) in the other base, and put
\(w=\dim W\). It meets the prescribed very general locus. Since the
bundle fiber has dimension one, either \(w=s\) or \(w=s-1\).
We treat every possibility.

\paragraph{A generically finite proper image.}
If \(w=s<n\), the resolution of \(W\) is of general type by
\Cref{prop:subvarieties}. The same holds for \(F^*\): after resolving
the generically finite map to a resolution of \(W\), ramification
adds an effective divisor to the pulled-back canonical divisor.
The polarization \(P_{\mathrm{sl}}\) dominates the pullback of \(L\).
\Cref{eq:two-slot-fiber-volume,eq:two-slot-fiber-canonical} therefore contradict
\Cref{lem:degree-obstruction} for sufficiently small \(t\).

\paragraph{The whole bundle over a proper image.}
Suppose \(s=w+1\), where necessarily \(w<n\). A positive-dimensional
closed subset of the generic \(\PP^1\) fiber is the entire fiber.
Consequently
\[
 F=\pi_{\mathrm{sl}}^{-1}(W)
\]
as reduced integral subvarieties. Nef intersection and the
projective-bundle formula give
\begin{equation}\label{eq:whole-bundle-base-volume}
 qL^w\cdot W\le P_{\mathrm{sl}}^{\,w+1}\cdot F .
\end{equation}
For \(w=0\), the left side is \(q\). The right side is at most
\[
 (2/\delta)^nP_{\mathrm{sl}}^{\,n+1}
 \le(2/\delta)^n2^{n+2}q\epsilon^n<q
\]
by \Cref{eq:two-slot-gap-choice}, a contradiction.

Assume \(w>0\). We will apply the degree obstruction to \(W\), rather
than to the ruled variety \(F\). Expand the restricted high sections
using the two homogeneous fiber coordinates. Their coefficients of
weight \(j\), for \(0\le j\le kq\), are actual sections of
\[
 \cO_Y(kL+jS_0).
\]
Let \(\mathfrak a\) be the ideal generated by all these coefficients
in the regular local ring
\[
 R=\cO_{Y,\eta_W},\qquad \mathfrak m=\mathfrak m_R.
\]
This ring is regular because the generic point of \(W\) lies in the
smooth locus of \(Y\).

We need both the order and the support of this coefficient ideal.
After choosing frames, the local ring of the slice at the generic
point of \(F\) is
\[
 R[u]_{\mathfrak m R[u]} .
\]
The residue of \(u\) is transcendental over the residue field of
\(R\). If a coefficient had order less than \(h\), the least
\(\mathfrak m\)-adic homogeneous part of its polynomial would remain
nonzero after adjoining that residue variable. Therefore membership
of every high section in
\(\mathfrak m^hR[u]_{\mathfrak m R[u]}\) implies
\(\mathfrak a\subseteq\mathfrak m^h\).
Moreover, if a prime \(\mathfrak p\subsetneq\mathfrak m\) contained
\(\mathfrak a\), its extension to this local polynomial ring would
be a proper prime strictly below the generic ideal of \(F\)
containing the whole high base ideal. This contradicts the isolated
support of \(F\). Hence
\begin{equation}\label{eq:two-slot-coefficient-ideal}
 \mathfrak a\subseteq\mathfrak m^h,\qquad
 \sqrt{\mathfrak a}=\mathfrak m .
\end{equation}

Multiply every weight-\(j\) coefficient by the globally generated
space in \Cref{eq:two-slot-fillers}. The products all belong to one
linear subseries of
\[
 H^0\bigl(Y,\cO_Y(2k(L+qS_0))\bigr).
\]
At \(\eta_W\), each filler system generates a unit after choosing a
frame. Thus this new subseries has precisely the joint ideal
\(\mathfrak a\), not merely an ideal with the same support. Its
threshold construction in \Cref{lem:base-component} gives an
effective rational divisor
\[
 \Theta\sim_{\Q}c\,2(L+qS_0),\qquad
 0<c\le(n-w)k/h,
\]
such that the pair is lc at the generic point of \(W\) and has
an lc place centered there. Apply
\Cref{lem:numerical-subadjunction} with the testing class \(L\).
Since \(qS_0\le L\) in nef order, we obtain
\begin{align}
 K_{W^*}L^{w-1}
 &\le\bigl(K+2c(L+qS_0)\bigr)L^{w-1}\cdot W \notag\\
 &\le\left(\frac1{r_t}+\frac{8(n-w)}{\delta}\right)L^w\cdot W .
 \label{eq:whole-bundle-canonical}
\end{align}
The top degree \(L^w\cdot W\) is bounded by
\Cref{eq:whole-bundle-base-volume} and
\Cref{eq:two-slot-fiber-volume}. The variety \(W^*\) is of general
type. \Cref{eq:whole-bundle-canonical} and the top-degree
bound now contradict \Cref{lem:degree-obstruction} on \(W^*\).

\paragraph{A non-axis multisection over all of \(Y\).}
The only remaining intermediate-fiber case is \(s=w=n\).
Here \(F\) is a multisection of positive degree \(e\) over \(Y\).
It is not an axis, since it contains the selected torus point.
Let \(\Sigma_0,\Sigma_\infty\) be the axes on the slice, with
classes
\[
 \Sigma_0\sim\xi_{\mathrm{sl}},\qquad
 \Sigma_\infty\sim\xi_{\mathrm{sl}}-S_0.
\]
The intersections with \(F\) are effective integral cycles. Their
pushforwards define effective integral Weil divisors \(D_0,D_\infty\)
on \(Y\), allowing a divisor to be zero. Projection and rational
equivalence of cycles give
\begin{equation}\label{eq:axis-linear-equivalence}
 D_0-D_\infty\sim eS_0=e\iota K .
\end{equation}
This is linear equivalence of Weil divisors: codimension-one
rational equivalence on the normal variety \(Y\) is precisely
linear equivalence. In particular,
\[
 J=\frac{D_0-D_\infty}{e\iota}\sim_{\Q}K
\]
is an actual signed rational representative.

For either axis, \(P_{\mathrm{sl}}-q\Sigma\) is nef: it is \(L\)
or \(L+qS_0\). As \(F\cap\Sigma\) is effective and
\(P_{\mathrm{sl}}\ge\pi_{\mathrm{sl}}^*L\) in nef order,
\begin{equation}\label{eq:axis-degree-bound}
 qL^{n-1}\cdot D_i
 \le qP_{\mathrm{sl}}^{\,n-1}\cdot(F\cap\Sigma_i)
 \le P_{\mathrm{sl}}^{\,n}\cdot F,\qquad i=0,\infty .
\end{equation}
Take a log resolution \(p:Y^\dagger\to Y\) of \(D_0+D_\infty\),
and let \(G\) be the reduced snc divisor consisting of their strict
supports and all exceptional divisors. By
\Cref{prop:signed-alternative},
\(K_{Y^\dagger}+G\) is big. The exceptional terms push to zero
against \(L^{n-1}\), while the reduced strict support is bounded
by the integral divisor \(D_0+D_\infty\). Hence
\begin{equation}\label{eq:multisection-log-bound}
 (K_{Y^\dagger}+G)L^{n-1}
 \le \frac{L^n}{r_t}
       +L^{n-1}\cdot(D_0+D_\infty).
\end{equation}
The second term is uniformly bounded by
\Cref{eq:axis-degree-bound} and
\Cref{eq:two-slot-fiber-volume}. Also \(L^n\) is bounded above
and bounded away from zero. Thus
\Cref{eq:multisection-log-bound} is the canonical-to-volume bound
in the log version of \Cref{lem:degree-obstruction}, with
polarization \(p^*L\) and map \(p\). That lemma gives the
contradiction in this final intermediate-fiber case.

\paragraph{Reduction to finite correspondences.}
We have excluded all slot-fiber dimensions \(1,\ldots,n\).
A fiber dimension \(n+1\) over one slot image \(W\) would force
\(V\) to be the entire bundle over \(W\times Y\).
Since \(V\ne Z\), one has \(\dim W<n\); its fiber over the other
slot then has dimension \(\dim W+1\), which has just been excluded.
It follows that both projections of \(V\) to \(Y\) are generically
finite onto their images.

If one image were proper, it would have dimension \(f<n\) and be of
general type. Its generically finite cover \(V^*\) would also be
of general type, and the total bounds
\Crefrange{eq:two-slot-center-volume}{eq:two-slot-center-canonical}
would contradict \Cref{lem:degree-obstruction}. Therefore
\[
 f=n,\qquad g_i:V^*\longrightarrow Y\quad(i=1,2)
\]
are generically finite dominant morphisms. The bounds
\Crefrange{eq:two-slot-center-volume}{eq:two-slot-center-canonical}
supply constants \(B_0,B_1\) independent of small \(t\), of the
section degree \(k\), and of the family. We are therefore in the
situation excluded by \Cref{prop:two-slot-correspondences}.

Every possibility for a moving component has now been excluded.
The slice degree obstructions and
\Cref{prop:two-slot-correspondences} impose restrictions on \(t\)
independent of \(k\) and of the moving family.
The assumed failure of \Cref{eq:two-slot-seshadri} is therefore
impossible for all sufficiently small \(t\), as required.
\end{proof}

\subsection{One finite system of jets on a smooth model}

We now convert the strict Seshadri bound into a single surjective
jet-evaluation map on a smooth model. Choose \(t\) sufficiently small
for both \Cref{prop:scalar-orders,prop:two-slot-jets}, and keep it
fixed. The next corollary then supplies a point, a Cartier multiple,
and finitely many sections whose jets span the required target.
These are the finite data used in reduction to positive
characteristic; no family of jet systems over varying \(t\) is needed.

\begin{corollary}\label[corollary]{cor:fixed-two-slot-jets}
Fix sufficiently small \(t\) as in
\Cref{prop:scalar-orders,prop:two-slot-jets}, and let
\(p:W\to Y\) be any fixed smooth projective resolution. Continue
to denote pullbacks by \(L\) and \(S_0\). On
\[
 Z_W=\PP\bigl(\cO_W(S_0)_1\oplus\cO_W(S_0)_2\bigr)
       \longrightarrow W\times W,\qquad
 M=L_1+L_2+q\xi ,
\]
there are a smooth torus point \(z_*\), over the isomorphism opens
of \(p\) in both slots, and a sufficiently divisible integer
\(k_0>0\) such that \(k_0L\) is Cartier and
\begin{equation}\label{eq:fixed-two-slot-jet-map}
 H^0(Z_W,\cO_{Z_W}(k_0M))
 \longrightarrow
 \cO_{Z_W}(k_0M)\otimes
 \bigl(\cO_{Z_W,z_*}/\mathfrak m_{z_*}^{\,10k_0+1}\bigr)
\end{equation}
is surjective. In particular, finitely many sections can be fixed
whose jets span its target.
\end{corollary}

\begin{proof}
Choose a very general smooth torus point \(z\) of \(Z\) above the
isomorphism open of \(p\) in both slots and satisfying
\Cref{eq:two-slot-seshadri}, so that \(\varepsilon(P;z)>10\). If
\(b:\widehat Z\to Z\) is its blowup and \(E_z\) is the exceptional
divisor, the Seshadri criterion shows that the following rational divisor
is ample:
\[
 b^*P-10E_z.
\]
Indeed one may choose a rational
number strictly between \(10\) and \(\varepsilon(P;z)\), and then
use the usual ample interval below the Seshadri threshold.
Choose \(a>0\) so that \(aP\) and
\(a(b^*P-10E_z)\) are Cartier. Serre vanishing, applied to the
fixed sheaf \(\cO_{\widehat Z}(-E_z)\), gives
\[
 H^1\!\left(\widehat Z,
  \cO_{\widehat Z}(kb^*P-(10k+1)E_z)\right)=0
\]
for all sufficiently large multiples \(k\) of \(a\).
The standard local calculation for the blowup of a smooth point
identifies the pushforward of
\(\cO_{\widehat Z}(-(10k+1)E_z)\) with
\(\mathfrak m_z^{10k+1}\), with vanishing higher direct images.
The restriction sequence therefore gives surjectivity onto jets
through order \(10k\) at \(z\).

The natural morphism \(Z_W\to Z\) is an isomorphism near the lift
\(z_*\) of \(z\). Pullback preserves the sections in this jet
surjection and identifies their jets. Increase the divisible
integer \(k\) if necessary so that \(kL\) is Cartier, and call it
\(k_0\). Choosing finitely many inverse images of a basis of the
finite-dimensional jet target proves the assertion.
\end{proof}
==== END Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets//two-slot.tex ====