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\title{Prescribed large-volume charges on threefolds with trivial canonical bundle}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $X$ be a smooth projective complex threefold with trivial canonical
bundle. We construct a numerical Bridgeland stability condition at every
sufficiently large volume whose central charge is exactly the
large-volume charge: either the ordinary one, or its square-root-Todd
correction. One threshold works on an open set of real twists and real
ample directions. The charge has the support property on the full
numerical Grothendieck group, and point sheaves are stable. Separately,
on every smooth projective complex threefold we prove a strong tilt
inequality: with a fixed rational correction at every volume, and with
no correction at all above a volume threshold that is uniform in the
object and in the twist along a fixed polarization.
\end{abstract}
\tableofcontents

\section{Introduction}
\label{sec:intro}

\subsection{A stability condition in one minute}
\label{sec:intro-bridgeland}

Let $X$ be a smooth projective complex variety and $\Db(X)$ its bounded
derived category of coherent sheaves. Bridgeland \cite{Bridgeland2007}
introduced the following.

A \emph{numerical central charge} is a homomorphism
$Z : \Knum(X) \to \CC$ whose image is not dense in a half-plane; the
group $\Knum(X)$, formalized in \S\ref{sec:setup}, records the
intersection-theoretic data of a sheaf and is the natural home for
charges on a fixed numerical type.

A \emph{slicing} assigns to each real $\phi$ a full subcategory
$\mathcal P(\phi) \subset \Db(X)$ such that
\begin{enumerate}[label=(\roman*)]
\item $\mathcal P(\phi+1) = \mathcal P(\phi)[1]$ (shift);
\item $\Hom(E,F) = 0$ whenever $E \in \mathcal P(\phi)$ and
      $F \in \mathcal P(\psi)$ with $\phi > \psi$ (phase ordering);
\item every nonzero object of $\Db(X)$ admits a finite
      Harder--Narasimhan filtration by the $\mathcal P(\phi)$
      (finiteness).
\end{enumerate}
Local finiteness adds that, for any $a<b$ with $b-a<1$, the category
generated by semistable objects with phases in $(a,b]$ is of finite
length.

A \emph{stability condition} is a pair $(Z,\mathcal P)$ where $Z$ is a
numerical central charge, $\mathcal P$ is a slicing, and for every
nonzero $E \in \mathcal P(\phi)$ one has $Z(E) = m(E)e^{i\pi\phi}$ with
$m(E)>0$.

Informally, $Z$ assigns to each object a positive real weight $m(E)$
and a phase angle $\pi\phi$; the slicing says which objects have which
phase. An object is semistable of phase $\phi$ when it belongs to
$\mathcal P(\phi)$. Every object is built from semistable factors with
strictly decreasing phases, the categorical analogue of a
Jordan--H\"older filtration.

Two refinements matter throughout. A stability condition is
\emph{geometric} when every point sheaf $\OO_x$ is stable of phase $1$.
It has the \emph{support property} of Kontsevich--Soibelman
\cite[Sections 1.2 and 2.1]{KontsevichSoibelman2008} when, for some norm
on the real vector space $\Knum(X)_\RR$, there is $C>0$ with
$\|[E]\| \le C|Z(E)|$ for every semistable $E$. Full numerical support
(in the sense of \cite[Definition 1.1]{BayerDeformation2019}) is
equivalent to: the quadratic form $C^2|Z(v)|^2 - \|v\|^2$ is negative
definite on the kernel of $Z$. The support property is what makes
$\Stab(X)$ a complex manifold and forces Harder--Narasimhan
filtrations to behave well under deformation.

\subsection{The large-volume charge}
\label{sec:intro-LV}

A \emph{threefold} is a smooth projective complex variety of dimension
three. Fix a real numerical twist $B \in N^1(X)_\RR$, a real ample
class $H \in \operatorname{Amp}(X)$, and $t > 0$. Put
$\ch^B(E) = e^{-B}\ch(E)$. The \emph{ordinary physical charge} is
\begin{equation}\label{eq:intro-physical}
 Z_{B,tH}(E)
 = -\ch_3^B(E) + \tfrac{t^2}{2} H^2 \ch_1^B(E)
 + i\Bigl( t H \ch_2^B(E) - \tfrac{t^3}{6} H^3 \ch_0(E) \Bigr).
\end{equation}
The \emph{square-root-Todd charge} is
\begin{equation}\label{eq:intro-Todd}
 Z^{\mathrm{LV}}_{B,tH}(E)
 = -\int_X e^{-B - itH} \ch(E) \sqrt{\td(X)},\qquad
 \sqrt{\td(X)} = 1 + \tfrac{c_2(X)}{24}.
\end{equation}

Both charges are exponential in the twisted Chern character, and
in both cases the modulus grows like $t^3$ as $t\to\infty$, which is
why the region $t\gg 1$ is the \emph{large-volume limit}: the charge is
dominated by the top Chern character $H^3 \ch_0(E)$, i.e. by the
product of volume and rank. Heuristically, at large volume a sheaf
looks like a point on the ``rank'' axis, and its first, second, and
third Chern characters organize themselves along transverse axes
rescaled by $t$, $t^2$, and $t^3$ respectively.

The two charges differ only in the Todd factor. On a threefold with
trivial canonical bundle, $\td(X) = 1 + c_2(X)/12$ in the relevant
degrees, so $\sqrt{\td(X)} = 1 + c_2(X)/24$. The correction is a fixed
real multiple of $\ch_1^B$ plus a fixed real multiple of $\ch_0$; it
does not scale with $t$.

Both charges are natural objects of mirror symmetry: $Z_{B,tH}(E)$ is
the central charge of the D-brane of class $[E]$ on the $B$-model side,
and the Todd factor is what Grothendieck--Riemann--Roch naturally
produces when $X$ is embedded in a product of projective spaces and
later restricted back. The prescription problem, in this language, is
to build a stability condition whose central charge is exactly the
\emph{specified} charge at a finite volume, not just asymptotically as
$t \to \infty$.

\subsection{Theorem A: exact charges on threefolds with trivial $K_X$}
\label{sec:intro-A}

Before stating Theorem A we need the heart $\mathcal A_{B,tH}$ that
appears in it. Full details are in \S\ref{sec:setup}, but a summary
will fix terminology.

For a real twist $B$, a real ample class $H$ and a nonzero coherent
sheaf $F$ of positive rank, define the slope
\[
 \mu_{H,B}(F) = \frac{H^2 \ch_1^B(F)}{H^3 \ch_0(F)},
 \qquad
 \text{nonzero torsion } F: \mu_{H,B}(F) = +\infty.
\]
Write $\mu^+_{H,B}(F)$ and $\mu^-_{H,B}(F)$ for the largest and
smallest slopes in the slope Harder--Narasimhan filtration. The
\emph{slope torsion pair} and its tilt (the \emph{first tilt}) are
\[
 \mathcal T_{H,B} = \{F : \mu^-_{H,B}(F) > 0\}, \qquad
 \mathcal F_{H,B} = \{F : \mu^+_{H,B}(F) \le 0\},
\]
\[
 \mathcal B_{H,B} = \langle \mathcal F_{H,B}[1],\ \mathcal T_{H,B} \rangle.
\]
Here the zero sheaf belongs to both $\mathcal T$ and $\mathcal F$,
$[1]$ is the cohomological shift in $\Db(X)$, and
$\langle -, - \rangle$ denotes extension closure. This is a heart of
a bounded $t$-structure on $\Db(X)$. The boundary convention is
strict positive on the left and nonpositive on the right.

On $\mathcal B_{H,B}$ the \emph{second slope} is
\begin{equation}\label{eq:intro-second-slope}
 \nu_{B,tH}(E) = \frac{t H \ch_2^B(E) - t^3 H^3 \ch_0(E)/6}
                     {t^2 H^2 \ch_1^B(E)},
\end{equation}
with value $+\infty$ at zero denominator. Semistability and stability
for $\nu_{B,tH}$ use subobject-versus-quotient comparison in
$\mathcal B_{H,B}$, with strict inequality required for stability.
By \cite[Section 2]{BMS2016}, Harder--Narasimhan filtrations exist
for every finite-slope object of $\mathcal B_{H,B}$ at every real
ample class $tH$ and real twist $B$. Let $\mathcal T'_{B,tH}$ be the
subcategory whose smallest Harder--Narasimhan slope is strictly
positive, and $\mathcal F'_{B,tH}$ the subcategory whose largest
Harder--Narasimhan slope is nonpositive. The \emph{standard double
tilt} is
\[
 \mathcal A_{B,tH} = \langle \mathcal F'_{B,tH}[1],\ \mathcal T'_{B,tH} \rangle.
\]
The same strict-positive/nonpositive boundary convention is used at
both tilts.

\begin{theorem}[Exact large-volume charges on a real sector]
\label{thm:geometric-sector}
Let $X$ be a smooth projective complex threefold with $K_X \simeq
\OO_X$. For every $B_* \in N^1(X)_\RR$ and every $H_* \in
\operatorname{Amp}(X)$, there exist open neighborhoods $V \ni B_*$,
$W \ni H_*$, and a single threshold $T > 0$ such that, for every
$B \in V$, $H \in W$, and $t \ge T$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $(Z_{B,tH}, \mathcal A_{B,tH})$ is a geometric numerical
stability condition with support on $\Knum(X)_\RR$, where
$Z_{B,tH}$ is the ordinary physical charge \eqref{eq:intro-physical}
and $\mathcal A_{B,tH}$ is the standard double tilt defined above.
\item There is a geometric numerical stability condition with support
on the same full numerical group whose central charge is exactly the
square-root-Todd charge $Z^{\mathrm{LV}}_{B,tH}$ of
\eqref{eq:intro-Todd}, and whose heart is the one constructed in
\S\ref{sec:ambient}.
\end{enumerate}
\end{theorem}

\noindent\emph{What this means.} On a threefold with trivial canonical
bundle, the double tilt really is a Bridgeland stability condition,
at all sufficiently large volumes, and its central charge is the
ordinary large-volume charge. A second, independently constructed,
stability condition has the square-root-Todd charge on the same
sector. Both are geometric, both have full numerical support, and
both have stable point sheaves. The quantifiers are the content: a
single $T$ works on the whole open sector, not point-by-point in
$(B,H,t)$.

Two remarks on the hypotheses. Only $K_X \simeq \OO_X$ is assumed;
the stronger Calabi--Yau condition $H^1(X,\OO_X) = H^2(X,\OO_X) = 0$
is not needed. And the two hearts differ: the ordinary charge uses
the standard double tilt defined from slopes, while the
Todd-corrected one uses the geometric heart from \S\ref{sec:ambient};
they agree on point sheaves but not in general.

\subsection{Theorem B: a uniform strong tilt inequality}
\label{sec:intro-B}

The second theorem is independent of the first. It applies on every
smooth projective complex threefold, Calabi--Yau or not, and its
statement is purely numerical: no Bridgeland stability condition need
exist.

Fix an ample integral divisor class $H$ and a rational class $B_0 \in
N^1(X)_\QQ$. For $a>0$ and $b\in\RR$, put $B = B_0+bH$ and define
\begin{equation}\label{eq:intro-tilt-a-b}
 \nu_{a,b}(E) =
 \frac{H \ch_2^B(E) - \tfrac{a^2}{2} H^3 \ch_0(E)}
      {H^2 \ch_1^B(E)},
\end{equation}
with value $+\infty$ at zero denominator. This is exactly
\eqref{eq:intro-second-slope} after $t = \sqrt3 a$, up to an overall
factor $1/(\sqrt3 a)$ that does not affect stable or semistable
objects. Semistability and stability for $\nu_{a,b}$ are defined by
subobject-versus-quotient comparison in $\mathcal B_{H,B}$, exactly
as for $\nu_{B,tH}$.

\begin{theorem}[Uniform strong inequality]
\label{thm:uniform-inequality}
Let $X$ be a smooth projective complex threefold, $H$ an ample
integral divisor, and $B_0 \in N^1(X)_\QQ$. There exist a rational
$k \ge 0$ and an $R_* > 0$, depending only on $(X,H,B_0)$, such that
every finite-slope $\nu_{a,b}$-semistable object
$E \in \mathcal B_{H,B_0+bH}$ with $\nu_{a,b}(E) = 0$ satisfies
\begin{equation}\label{eq:intro-corrected}
 \ch_3^{B_0+bH}(E)
 \le \Bigl( \frac{a^2}{6} + k \Bigr) H^2 \ch_1^{B_0+bH}(E)
 \qquad\text{for all } a>0,\ b\in\RR.
\end{equation}
For the same fixed data, the stronger bound
\begin{equation}\label{eq:intro-tail}
 \ch_3^{B_0+bH}(E)
 \le \frac{a^2}{6} H^2 \ch_1^{B_0+bH}(E)
\end{equation}
holds for every $b \in \RR$ whenever $a > R_*$.
\end{theorem}

\noindent\emph{What this means.} The classical Bogomolov--Gieseker
inequality controls the discriminant
$\ch_1^2 - 2 \ch_0 \ch_2$ on slope-semistable sheaves. There is no
analogue for the third Chern character, and any bound obtained purely
from earlier characters is called a \emph{strong} (second-tilt)
inequality. Schmidt \cite[Theorem 3.1]{Schmidt2017} and
Martinez--Schmidt \cite[Theorem 1.1]{MartinezSchmidt2019} exhibit
examples in which the uncorrected inequality
$\ch_3 \le (a^2/6) H^2 \ch_1$ fails, at small volume. The theorem
says: (i) a fixed rational correction $k$ rescues the inequality at
every volume; (ii) above a threshold $a > R_*$ the correction can be
removed entirely and the inequality survives uncorrected. Both $k$ and
$R_*$ are uniform in the object $E$, in its rank and Chern character,
and in the twist $b$. No hypothesis on $K_X$ is used.

The correction is by the curve class $\Gamma = kH^2 \in
N_1(X)_\QQ$, which contributes
$\Gamma \cdot \ch_1^{B_0+bH}(E)$ and satisfies
$\Gamma\cdot H = kH^3 \ge 0$. This is precisely the shape of
correction asked for by Bernardara--Macr\`i--Schmidt--Zhao in
\cite[Question 2.4]{BMSZ2017} and Martinez--Schmidt in
\cite[Question 2.6]{MartinezSchmidt2019}. The theorem answers both
questions affirmatively and adds the uncorrected tail above $R_*$.
The known counterexamples are all in the bounded region $a < 1/2$
(see \S\ref{sec:counterexamples}), so the restriction $a > R_*$ on
the tail is not an artifact of the proof. The specific value of
$R_*$ produced by the proof may be larger than $1/2$.

Two further remarks. First, the argument uses only the tilt
semistability machinery of \S\ref{sec:tilt} and never invokes the
existence of a Bridgeland stability condition. Second, the
coefficient $a^2/6$ is nine times smaller than the coefficient
$3a^2/2$ that the categorical argument of \S\ref{sec:heart} yields
from positivity of the ordinary physical charge. The sharper
coefficient $a^2/6$ is what the numerical argument of
\S\ref{sec:analysis}--\ref{sec:wall} establishes.

\subsection{A quadratic corollary of the tail}
\label{sec:intro-C}

After a change of coordinates, the uncorrected tail takes the form
traditionally used for generalized Bogomolov--Gieseker inequalities.
For $v = (v_0,v_1,v_2,v_3) \in \RR^4$, put
\begin{equation}\label{eq:intro-quadratic}
 Q_{\alpha,\beta}(v) = \alpha(v_1^2 - 2v_0v_2) + \beta(3v_0v_3 - v_1v_2)
 + 2v_2^2 - 3v_1v_3.
\end{equation}

\begin{corollary}[A parabolic region with trivial correction]
\label{cor:quadratic-tail}
For every polarized smooth projective threefold $(X,H)$ there is
$R_* > 0$ such that the following holds. If
$\alpha > f(\beta) := (\beta^2 + R_*^2)/2$, then
$Q_{\alpha,\beta}(v_H(E)) \ge 0$ for every nonzero semistable
object $E \in \mathcal B_{H,\beta H}$ for the slope
\[
 \hat\nu_{\alpha,\beta}(E) =
 \frac{v_2 - \beta v_1 + (\beta^2-\alpha) v_0}{v_1 - \beta v_0},
 \qquad
 v_H(E) = (H^3 \ch_0,\, H^2 \ch_1,\, H \ch_2,\, \ch_3)(E),
\]
interpreted as $+\infty$ at zero denominator.
\end{corollary}

\noindent\emph{What this means.} The function $f(\beta)$ is a
parabola in the $(\beta,\alpha)$-plane. Above the parabola, the
standard quadratic inequality $Q_{\alpha,\beta}(v_H(E)) \ge 0$ holds
with \emph{trivial correction}: $v_H$ uses the ordinary Chern
character, with no Todd terms and no extra curve class. The slope in
the corollary is exactly $\nu_{a,b}$ after $a = \sqrt{2\alpha -
\beta^2}$ and $b = \beta$; the condition $\alpha > f(\beta)$ becomes
$a > R_*$, and the conclusion follows from \eqref{eq:intro-tail}.

\subsection{Two independent proofs}
\label{sec:intro-two-proofs}

Theorems \ref{thm:geometric-sector} and \ref{thm:uniform-inequality}
are proved by separate arguments. The distinction is mathematical: on
a threefold of Picard rank $>1$, four intersection degrees along one
polarization do not control the full numerical subgroup of $\Knum$,
so the numerical and the charge-existence problems are genuinely
different. The two arguments meet only in the tilt conventions of
\S\ref{sec:tilt}.

\paragraph{Proof of Theorem A (Sections \ref{sec:ambient}--\ref{sec:heart}).}
Choose very ample line bundles $L_1,\ldots,L_q$ whose positive span
contains the given ample directions, so that the induced
$i : X \hookrightarrow Y = \prod_h \mathbb P^{d_h}$ has $i_*$
injective on $\Knum$ (Lemma \ref{lem:full-numerical-embedding}).
On $Y$ one builds, using product and elliptic-factor stability and
the deformation theorem, a stability condition whose charge is the
plain exponential $\ch$ at large volume. A weighted support bound,
with the Chern character rescaled by powers of the volume, is
independent of the volume; hence the ambient charge deforms
uniformly at large volume. A fixed polynomial correction $\gamma$,
chosen from the Todd factor of the embedding, makes the ambient
charge restrict exactly to $Z_{B,tH}$ and to $Z^{\mathrm{LV}}_{B,tH}$.
Restriction back to $X$ through a truncated finite-map criterion
gives both stability conditions on the full numerical group, with
stable point sheaves and full numerical support
(Section \ref{sec:ambient}).

The heart of the ordinary charge requires more. Restriction to
smooth surfaces $S_h \in |L_h|$ gives a known surface stability
condition whose heart is the classical geometric tilt; the positive
weights $\ell_h$ in $H = \sum_h \ell_h L_h$ combine the surface
phase bounds into a bound on the two outer cohomology sheaves of
each object of the threefold heart (\S\ref{sec:heart}). A real-axis
argument, with positive weights, fixes the zero-slope boundary of
the second tilt. Together these identify the heart as the standard
double tilt $\mathcal A_{B,tH}$.

\paragraph{Proof of Theorem B (Sections \ref{sec:tilt}--\ref{sec:wall}).}
The inequality is first proved at one fixed volume $A$
(Sections \ref{sec:analysis} and \ref{sec:apex}). At that volume,
each finite-slope tilt object of slope zero has an ordinary slope
filtration; truncation leaves a \emph{narrow sheaf tail} plus a
\emph{residual complex} whose rank, first character, and $H$-degree
of the second character are bounded by the discriminant error
$e = d_B(E) - A|r(E)|$. Successive restriction to smooth surfaces
and curves reduces the tail to rank-uniform sheaf bounds;
Hermitian--Einstein theory and the Bochner--Kodaira--Nakano formula
produce curvature bounds, and a matrix-valued
Cwikel--Lieb--Rozenblum estimate converts them into dimension bounds
on the residual cohomology (\S\ref{sec:analysis}). The narrow tail is
instead estimated from its narrow range of slopes, and the two
successive restrictions produce the coefficient $A^2/6$. For each
fixed object, spread to positive characteristic; Frobenius pullback
and Riemann--Roch turn the cohomology bounds into the inequality
$\ch_3^B(E) \le \frac{A^2}{6} d_B(E) + Ce$ in characteristic zero
(\S\ref{sec:apex}).

The all-volume inequality follows by wall transport
(\S\ref{sec:wall}). Along the zero-slope hyperbola the differential
inequality preserves a strictly positive violation of the corrected
bound as the volume parameter moves toward $A$. When the object
changes stability, one continues with a violating stable factor,
chosen by weighted Cauchy--Schwarz. A discrete discriminant strictly
decreases at every wall splitting, so the process is finite and the
violation cannot reach $A$. Uniformity in $b$ gives the corrected
inequality at every volume, and the uncorrected tail above $R_*$.
Corollary \ref{cor:quadratic-tail} then follows from
Theorem \ref{thm:uniform-inequality} by the algebraic identity
proved in \S\ref{sec:wall}.

\subsection{Prior work}
\label{sec:intro-history}

Nonemptiness of $\Stab(X)$ on projective varieties was already known:
Li \cite{Li2026} constructed stability conditions on smooth
projective varieties; Li--Liu--Liu--Macr\`i--Perry--Stellari--Zhao
\cite{LiEtAl2026} developed the family-and-deformation framework
with a {\it mass-Hom\/} estimate; and Cheng \cite{Cheng2026}
established full numerical support and general nonemptiness.
Nonemptiness in the setting of Theorem A is therefore already
known; Theorem A's content is the \emph{prescription} of the two
exact large-volume charges on a uniform open sector, together with
the standard double-tilt heart for the ordinary charge.

Bayer's polynomial stability conditions \cite{BayerPolynomial2009}
already construct large-volume asymptotic phases. What Theorem A
adds is a Bridgeland stability condition at each \emph{finite}
volume in the sector, which requires controlling the kernel of the
charge map on the full numerical group. On a threefold of Picard
rank $\ge 2$, the four intersection degrees along one fixed
polarization do not distinguish all classes of $\Knum$.

The product-embedding route to full numerical support in
\S\ref{sec:ambient} adapts the construction of
Giovenzana--Robotis--Rota--Zuliani
\cite{GiovenzanaRobotisRotaZuliani2026}. What we use from those
works is the product-and-elliptic-factor input to quantitative
stability on the ambient $Y$; the prescribed-charge and
real-sector refinements are new here.

Cheng--Feyzbakhsh \cite{ChengFeyzbakhsh2026} identify a large-volume
double-tilt heart for the character
$\td(N_{X/\mathbb P^n})^{-1}\ch$, where $N_{X/\mathbb P^n}$ is the
normal bundle of a fixed projective embedding. A
reparameterization expresses that heart as the ordinary double tilt
but leaves a curve-class correction in the charge. Theorem B removes
the correction at every volume (with a fixed rational $k$) and
removes it entirely above $R_*$.

The corrected strong tilt inequality \eqref{eq:intro-corrected} was
proved by Bernardara--Macr\`i--Schmidt--Zhao
\cite[Theorem 1.1]{BMSZ2017} for Fano threefolds with
$H \propto -K_X$, and Martinez--Schmidt
\cite[Question 2.6]{MartinezSchmidt2019} stated the version with a
fixed rational transverse twist. The counterexamples of
Schmidt \cite[Theorem 3.1]{Schmidt2017} and Martinez--Schmidt
\cite[Theorem 1.1]{MartinezSchmidt2019} show that the uncorrected
inequality fails at some $a < 1/2$, even on Calabi--Yau threefolds.
Theorem B answers both corrected-inequality questions and adds the
independent uncorrected tail above $R_*$.

Restriction to lower-dimensional varieties has also led to corrected
tilt inequalities: Feyzbakhsh--Koseki--Liu--Rekuski
\cite[Theorems 1.1, 1.3, 1.4]{FeyzbakhshEtAl2025} derive one from a
Brill--Noether bound on curves, and obtain a correction on families
of Calabi--Yau threefolds. Our numerical argument also uses
successive restrictions, but the input is a rank-uniform
cohomology estimate on surfaces.

\subsection{Plan of the paper}
\label{sec:intro-plan}

Section \ref{sec:setup} fixes notation: the numerical Grothendieck
group, the two tilt hearts, discriminants, and the norm used for
support. Section \ref{sec:ambient} proves Theorem A up to the heart
computation: the injection $\Knum(X) \to \Knum(Y)$, the weighted
support bound on products of projective spaces, the uniform
deformation, the Todd charge correction, and the restriction back
to $X$. Section \ref{sec:heart} identifies the heart of the
ordinary charge with the standard double tilt.
Sections \ref{sec:tilt}--\ref{sec:wall} prove Theorem B: tilt
machinery and the reduction to a fixed apex, the surface
cohomology estimate, the apex proposition at one volume, and the
wall transport. Section \ref{sec:counterexamples} computes the
known counterexamples, shows they all lie in the bounded region
$a<1/2$, and proves the uniform bound on the reduced exceptional
surface for sheaves of arbitrary rank.

\begin{figure}[htbp]
\centering
\begin{tabular}{@{}p{0.44\linewidth}@{\qquad}p{0.44\linewidth}@{}}
\textbf{Prescribed charges (Theorem A)}
  & \textbf{Strong inequality (Theorem B)} \\
\hline
Embed $X \hookrightarrow Y = \prod_h \mathbb P^{d_h}$ with $i_*$
injective on $\Knum$
  & Work at fixed apex $a = A$ \\
Weighted support bound on $Y$
  & Discriminant error $e = d_B - A|r|$ \\
Uniform ambient deformation
  & Surface/curve cohomology, HE + BKN + CLR \\
Correct charge by $\gamma = \td(Y)(1-\delta/12)$
  & Narrow tail gives $A^2/6$, residual gives $Ce$ \\
Restrict via finite-map criterion
  & Frobenius: char.\ 0 inequality at $A$ \\
Surface sections $\Rightarrow$ heart
  & Wall transport $\Rightarrow$ all $a$, tail above $R_*$ \\
\end{tabular}
\caption{The two independent arguments, side by side. The left
   column builds the charges of Theorem \ref{thm:geometric-sector};
   the right column proves the inequality of
   Theorem \ref{thm:uniform-inequality}. They share only the tilt
   conventions of \S\ref{sec:tilt}.}
\label{fig:proof}
\end{figure}